electronic structure of noble metals and polariton-mediated light scattering
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Point Defects in Metals I: Introduction to the Theory
Electronic Structure of Noble Metals, and Polariton-Mediated Light Scattering
Electronic Structure of Noble Metals and
Polariton-Mediated Light Scattering
Contributions by B.Bendow B.Lengeler
With 42 Figures
Springer-Verlag Berlin Heidelberg NewYork 1978
Dr. Bernard B e n d o w
Rome Air Development Center, Deputy for Electronic Technology, Hanscom AFB, MA 01731, USA
Dr. Bruno Lenge le r
Institut fer Festk6rperforschung der Kernforschungsanlage J(Jlich Postfach 1913, D-5170 Jelich (Present address: Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA)
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ISBN 3-540-08814-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08814-8 Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data. Bendow, Bernard, 1942 --. Electronic structure of noble metals and polariton-mediated light scattering. (Springer tracts in modern physics; v. 82) Bibliography: p. includes index. 1. Polaritons. 2. Precious metals. 3. Electronic structure, h Lengeler, B., 1939--. joint author. I1. Title. II1. Series. QCl.STg7 vol. 82 [(~C176.8.P6] 539'.08s [530.4'1] ISBN 0-387-08814-8 78-18848
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Contents
de Haas-van Alphen Studies of the Electronic Structure of the Noble Metals and Their Dilute Alloys
By B. Lengelero With 26 Figures
1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. The de Haas-van Alphen (dHvA) E f fec t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 L i f sh i t z -Kosev i ch Expression f o r the dHvA E f fec t . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Conduction Elect rons in a Homogeneous Magnetic F ie ld . . . . . . . . . 5
2.1.2 Densi ty o f States o f the Electrons in the Magnetic F ie ld . . . . . 9
2 .1 .3 Or ig in o f the dHvA O s c i l l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 .1 .4 Frequency of the dHvA O s c i l l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . I0
2 .1 .5 Ampl i tude o f the dHvA O s c i l l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Damping of the dHvA O s c i l l a t i o n s by F i n i t e Temperature . . . . . . . 12
Damping of the dHvA O s c i l l a t i o n s by Elect ron Sca t te r ing . . . . . . 12
In f luence of the Elect ron Spin on the dHvA E f fec t . . . . . . . . . . . . 13
2 .1 .6 L i f s h i t z - K o s e v i c h Expression f o r the dHvA E f f ec t . . . . . . . . . . . . . 13
2.2 In f luence o f Electron-Phonon I n t e r a c t i o n on the dHvA E f fec t . . . . . . . . . 14
2.3 In fo rmat ion Der ivab le from the dHvA E f fec t . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Geometry of the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 .3 .2 Cyclotron Masses and Fermi V e l o c i t i e s . . . . . . . . . . . . . . . . . . . . . . . . 18
2 .3 .3 Dingle Temperatures and Sca t te r ing Rates o f the Conduction
Elect rons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.4 g-Factor o f the Conduction Electrons . . . . . . . . . . . . . . . . . . . . . . . . . 19
3. Experimental Setup f o r dHvA Measurements in Cu, Ag, and Au . . . . . . . . . . . . . . . 20
3.1 F ie ld Modulat ion Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Magnet and Cryosta t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Sample Holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 S ing le Crys ta ls of the Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 P i t f a l l s in dHvA Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.1 Skin E f f ec t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.2 Harmonic dHvA Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 .5 .3 Magnet ic I n t e r a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 . 5 . 4 Phase Smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4. The Fermi Surface o f the Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5. Cyc lo t ron Masses and Fermi V e l o c i t i e s o f the Noble Metals . . . . . . . . . . . . . . . . 30
5.1 Cyc lo t ron Masses of Cu, Ag, and Au . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 De te rmina t ion o f Energy Surfaces Ad jacent to the Fermi Surface . . . . . . 33
5.3 Angular Dependence o f the Cyc lo t ron Masses in Cu, Ag, and Au . . . . . . . . 34
5.4 Fermi V e l o c i t i e s in the Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5 Elect ron-Phonon Coupl ing Constant ~(k) in Cu . . . . . . . . . . . . . . . . . . . . . . . . 42
5.6 C o e f f i c i e n t o f E l e c t r o n i c S p e c i f i c Heat f o r Cu, Ag, and Au . . . . . . . . . . 43
6, D ing le Temperatures and Sca t t e r i ng Rates o f Conduct ion E lec t rons in the
Noble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1 D ing le Temperatures and the L i f e t i m e of E lec t ron States . . . . . . . . . . . . . 45
6.2 An i so t ropy o f the Sca t t e r i ng Rates in the Noble Metals . . . . . . . . . . . . . . 47
6.3 Phase S h i f t Ana lys i s o f the S c a t t e r i n g o f Conduct ion E lec t rons a t
Defects in the Nnble Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 .3 .1 S u b s t i t u t i o n a l Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 .3 .2 Defects on Octahedral I n t e r s t i c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 .3 .3 S c a t t e r i n g of the Conduct ion E lec t rons by Hydrogen in Cu
Occupying Octahedral I n t e r s t i c e s and L a t t i c e S i tes . . . . . . . . . . . 60
6.4 Phase S h i f t Ana lys is o f Defec t - lnduced Fermi Surface Changes . . . . . . . . 60
L i s t o f Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Polariton Theory of Resonance Raman Scattering in Solids
By B. Bendow, With 16 Figures
1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
1.1 Purpose und Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
1.2 Review of P e r t u r b a t i o n Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2, P o l a r i t o n s and The i r S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.1 Fundamentals o f P o l a r i t o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2 Formalism o f Po la r i t on -Med ia ted S c a t t e r i n g . . . . . . . . . . . . . . . . . . . . . . . . . 81
3. P o l a r i t o n Theory o f the Resonance Raman E f fec t . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1 General P rope r t i es o f the S c a t t e r i n g Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2 Ca l cu l a t i ons f o r Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3 Spa t i a l D ispers ion and F i n i t e - C r y s t a l E f fec ts . . . . . . . . . . . . . . . . . . . . . . i01
3.4 S c a t t e r i n g by P o l a r i t o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
VI
de Haas-van Alphen Studies of the Electronic Structure of the Noble Metals and Their Dilute Alloys
Bruno Lengeler
1. Introduction
The de-Haas-van Alphen (dHvA) e f fec t is one of the quantum o s c i l l a t i o n phenomena
which are characterized by the red i s t r i bu t i on of conduction electron states on
Landau cyl inders in a magnetic f i e l d . The Landau cyl inders expand with increasing
f i e l d and leave the Fermi surface one by one. As a resu l t , the e lec t ron ic density
of states at the Fermi level changes per iod ica l l y with the f i e l d . Thus quantum os-
c i l l a t i o n s occur in a l l physical quant i t ies which contain the density of states.
Among these are the magnetoresistance, the Hall e f fec t , the thermoelectr ic e f fec t ,
the contact potent ia l between two metals, the e lect ron ic spec i f ic heat, and the u l -
t rasonic absorption in metals. The dHvA ef fec t is the quantum o s c i l l a t i o n of the
magnetization of the conduction electrons. The e f fec t has evolved from a cu r i os i t y ,
f i r s t observed in bismuth, to one of the most powerful methods for the invest iga-
t ion of the e lec t ron ic st ructure of pure metals, i n te rmeta l l i c compounds, and d i l u te
a l loys. The geometry of the Fermi surface can be deduced from the frequencies of the
dHvA osc i l l a t i ons , and the Fermi surfaces of nearly a l l pure metals and of many
ordered al loys have been determined in th is manner. These measurements have had a
great inf luence on our understanding of the e lec t ron ic st ructure of metals. In more
recent years, the in teres t has shi f ted towards the information contained in the
dHvA amplitudes. Cyclotron masses and Fermi ve loc i t i es can be derived from the tem-
perature dependence of these amplitudes, whereas the i r f i e l d dependence determines
the Dingle temperatures and the scatter ing of the conduction electrons at defects.
Two charac ter is t i c features of the dHvA e f fec t , and of quantum osc i l l a t i ons in
general, should be emphasized. F i r s t , only the e lec t ron ic states at the Fermi level
and in i t s immediate v i c i n i t y can be invest igated by the dHvA e f fec t because only
those states are affected by the deplet ion of the Landau levels when a cy l inder
leaves the Fermi surface. Electronic states which are fa r ther away from the Fermi
level than kBT must be invest igated by other methods, fo r instance, by opt ical spec-
troscopy. Nonetheless, the propert ies of the e lec t ron ic system at the Fermi level
can be measured by dHvA e f fec t wi th great accuracy. The l inear dimensions of the
Fermi surface for cer ta in metals are known with an accuracy of I part in 10 4 or
better. The immediate v i c i n i t y of the Fermi surface is also accessible to the dHvA
ef fect . By thermal exc i ta t ion of the states, a range of width kBT around E F can be
scanned. Thus the temperature dependence of the dHvA amplitudes contains the gra-
dients on the Fermi surface, i . e . , the cyclotron masses and the Fermi ve loc i t i es .
A second charac ter is t i c feature of the dHvA e f fec t is that a l l quant i t ies derived
from the dHvA e f fec t are averages over the extremal cross section on the Fermi sur-
face for the f i e l d d i rec t ion under consideration. Thus the dHvA frequency is an or-
b i ta l average of the rad i i of the extremal cross section. The cyclotron masses are
orb i ta l averages of the Fermi ve loc i t i es and the Dingle temperatures are o rb i ta l
averages of the electron scat ter ing rates. Since only a r e l a t i v e l y small number of
states at the Fermi level are involved in the e f fec t at one time, local values of
the r a d i i , Fermi ve loc i t i es , and scatter ing rates can be obtained by measuring the
or ien ta t ion dependence of the o rb i ta l averages and by deconvoluting them. The app l i -
c a b i l i t y of th is procedure is one of the great advantages of the dHvA ef fect .
The present paper is concerned with the invest igat ion of the e lec t ron ic st ructure
of the noble metals - copper, s i l v e r , and gold - and the i r d i l u te a l loys by means
of the dHvA ef fect . The paper is organized as fo l lows. In Section 2, the dHvA e f fec t
is explained in a semiclassical way and the L i fsh i tz-Kosevich expression for the os-
c i l l a t o r y magnetization is given. The various appl icat ions of the dHvA ef fec t are
described, and the inf luence of the electron-phonon in terac t ion on the dHvA ef fec t
is treated at some length. In Section 3, de ta i l s are given of the f i e l d modulation
technique by which most of the frequency and amplitude measurements have been made.
In Section 4, a detai led descr ipt ion of the geometry of the Fermi surface of the
noble metals is given. The anisotropy of the Fermi surface is explained w i th in a
band structure ca lcu la t ion by the hybr id iza t ion of the s- , p-, and d-bands. Sec-
t ion 5 gives a descr ipt ion of detai led cyclotron mass measurements in Cu, Ag, and
Au. From these data are derived values of the Fermi ve loc i t i es and of the coe f f i -
c ient y* of the spec i f i c heat. For Cu, the anisotropy of the electron-phonon coup-
l i ng constant is obtained by comparing the Fermi ve loc i t i es derived from cyclotron
masses with those obtained from a band st ructure ca lcu la t ion . F i na l l y , in Section 6
measurements of Dingle temperatures for some d i l u te a l loys of the noble metals are
discussed. Only a l loys in which the electron scat ter ing is spin independent are con-
sidered. The inf luence of the scat ter ing strength of the defect, of i t s posi t ion in
the l a t t i c e , and of the wave character of the conduction electrons on the observed
scat ter ing rates is explained in deta i l by means of a generalized phase s h i f t ana-
l ys is .
2 The de Haas-van Alphen (dHvA) Effect
In 1930, DE HAAS and VAN ALPHEN observed that the suscep t i b i l i t y of s ingle crysta l
bismuth varied at low temperature in an osc i l l a t o r y way with the magnetic f i e l d
/2 .1 / . The amplitude of the osc i l l a t i ons decreased with increasing temperature and
the e f fec t disappeared at about 35 K. PEIERLS correlated the osc i l l a t i ons with the
quant izat ion of the orb i ts of the free conduction electrons in the magnetic f i e l d
/2 .2 / . The f i r s t e x p l i c i t expression for the var ia t ion of the magnetization with
the f i e l d was given by LANDAU for e l l i pso ida l energy surfaces /2 .3 / . ONSAGER showed
that the frequency of the dHvA osc i l l a t i ons fo r a rb i t ra ry energy surfaces is pro-
port ional to the extremal cross section of the Fermi surface for a given f i e l d d i -
rect ion /2 .4 / . LIFSHITZ and KOSEVlCH have extended LANDAU's expression fo r the f i e l d
dependence of the magnetization for a rb i t ra ry energy surfaces /2 .5 / . Pioneering work
in the determination of Fermi surfaces of metals has been done by SHOENBERG /2 .6 / .
To date, the Fermi surfaces of nearly a l l pure metals and of many ordered compounds
have been determined /2 .7 / . In more recent years, the dHvA e f fec t has been used to
determine cyclotron masses, Fermi ve loc i t i es , and scat ter ing rates of conduction
electrons at defects.
Typical dHvA osc i l l a t i ons in gold and copper are shown schematical ly in Fig. 2.1.
When the magnetic f i e l d is para l le l to a <I00> crysta l lographic d i rec t ion , the mag-
ne t iza t ion contains two per iodic contr ibut ions (Fig. 2.1a). dHvA osc i l l a t i ons can
also be observed i f the crysta l is rotated in a constant magnetic f i e l d . Fig. 2.1b
shows the osc i l l a t i ons observed i f a Cu crystal is turned around an axis <II0>.
2.1 L i fsh i tz-Kosevich Expression for the dHvA Effect
The o s c i l l a t o r y var ia t ion of the magnetization of the conduction electrons is des-
cribed quan t i t a t i ve l y be the L i fsh i tz-Kosevich theory of the dHvA ef fec t . In th is
theory, the free energy of the conduction electrons is calculated for a rb i t ra ry ener-
gy surfaces as a funct ion of the magnetic f i e l d H. The magnetization of the conduc-
t ion electrons in a s ingle crysta l contains an osc i l l a t o r y part M which can be de-
termined from the o s c i l l a t o r y part of the free energy G according to
: -BGI~H . (2.1)
The period of the osci l lat ions is.correlated with the extremal cross sections of the
Fermi surface. The temperature and f ie ld dependence of the amplitudes of the osci l la-
tions is determined by the cyclotron masses and by the electron l i fetimes, There
exist a number of review art icles in which the Lifshitz-Kosevich expression of the
dHvA effect is presented. An excellent review has been given by GOLD /2.8/. In this
paper, we confine ourselves to a representation in which the major physical aspects
of the dHvA effect are derived in a semiclassical way.
AU # <1oo>
I H (G) I 55906 ,~ 56141
Ca)
Cu ~1101 669gOG 1.225 K
11i , (11o]
(b) [111] I00'~
Fig. 2.1a and b. dHvA osc i l l a t i ons in gold. (a) Field dependence of the osc i l l a - t ions at T = 1.179 K. The magnetic f i e l d is para l le l to a crysta l lographic d i rec- t ion <I00>. The magnetization contains two contr ibut ions (B <I00> and R <I00>). (b) Angular dependence of the osc i l l a t i ons . The crystal is rotated in a constant f i e l d through i00 ~ around an axis <110>
2.1.1 Conduction Electrons in a Homogeneous Magnetic Field
In the absence of a magnetic f i e l d , the conduction electron states in an ideal me-
t a l l i c s ingle crysta l are characterized by the wave vectors k and the spin quantum
numbers s. The k-vectors specifying the d i f f e ren t e lec t ron ic states are confined to
the f i r s t B r i l l o u i n zone. At T=O, a l l states with energies up to the Fermi energy E F
are occupied. The Fermi surface separates the occupied and unoccupied states. Each
state (k,s) can be occupied only once.
A magnetic f i e l d H red is t r ibu tes the possible e lec t ron ic states and a l ters the i r
degeneracy. The Lorentz force describes how a state k changes in time with the mag-
net ic f i e l d
h~ = -(eo/C)V x B ~ -(eo/C)V x H (2.2)
where e o is the charge of the proton, c is the ve loc i t y of l i g h t , and h is Planck's
constant, v(k) is the ve loc i t y of an electron in the state k. The f i e l d experienced
by the electrons is the magnetic induct ion B = H + 4~(M o+M), where ~o is the non-
osc i l l a t o r y part of the magnetization. Generally, M ~ + M ~H in nonferromagnetic
mater ials. Therefore B is replaced by H in the fo l lowing discussion (see also Sec-
t ion 3.5.3). In a s ta t ionary magnetic f i e l d , an electron moves on a path of con-
stant energy, the cyclotron o rb i t . In tegrat ion of (2.2) gives
: - (eo/~C)rxH . (2.3)
y (~) k ky
kx Fi~. 2.2. Cyclotron orb i ts in r - and in k-space. Only the-pro ject ion of the motion on a plane perpendicular to the f i e l d is shown
The cyclotron o rb i t in k-space is obtained from that in real space by ro ta t ion
through 7/2 and by scal ing with eoH/~c (F ig.2.2) . The electrons move on the cyclo-
tron o rb i t wi th the cyclotron frequency m given by C
m c = eoH/mcC . (2.4)
The cyclotron mass m c is obtained in the fo l lowing manner. According to (2.2) and
(2.4) the cyclotron period T c is
T c = 2~1% = 2~Cmc/eoH
= f d t = (hc leoH) fdk / v~ �9
Hence
m c = ( ~ / 2 ~ ) f d k / v •
where ~vz = aE/~k• Therefore
= ( ~ 2 / 2 ~ ) ~ d k ~k /~E = (~2/2~)~A/~E m c o
Y
(2.5a)
(2.5b)
(2.6)
(2.7)
dk
H ( E ) ~ J S - k • -(( ~ ~ 6 A = ~ d k 5kz
Fig. 2.3. The cyclotron mass is pro- portional to the energy der ivat ive of the area enclosed by a cyclotron orb i t
Fig. 2.3 i l l us t ra tes the signi f icance of the cyclotron mass which is proportional to
the energy der ivat ive of the area enclosed by the cyclotron orb i t . For free electrons,
m c reduces to the free electron mass m o. Using Fig. 2.3, m c can be wr i t ten
m c = ( ~ / 2 ~ ) / d ~ k2/(v.kx) . (2.8)
The cyclotron orb i ts are not only l ines of constant energy in k-space. In addi-
t ion, the area enclosed by them must be quantized. The Bohr-Onsager quantization for
the project ion of the orb i t in real space on a plane perpendicular to the magnetic
f i e l d H = rot A is wr i t ten
( h k - e o u [ = 2x~(n+1/2) . (2.9)
I f the areas enclosed by the cyclotron orb i t in real space and in k-space are denoted
by S n and A n , respect ively, (2.9) can be wr i t ten
S n = %(n + I /2) /H (2.10)
and
A n = 4~2H(n + 1 /2) /% . (2.11)
The f l ux quantum @o = 2~C/eo has the value 4.1356-10 -7 G cm 2. Eq. (2.11) is of p r i -
mary importance for the dHvA ef fect . I t indicates that the cyclotron orb i ts in k-space
expand with the f i e l d so that the area enclosed by them increases l i n e a r l y with H.
The energy of the states on the n-th o rb i t is (disregarding the spin of the electrons
and the contr ibut ions of the motion para l le l to the f i e l d ) ,
E n = ~ c ( n + 1 / 2 ) . (2.12)
The red i s t r i bu t i on of the allowed states by the magnetic f i e l d is shown in Fig. 2.4
for free electrons. The quant izat ion para l le l to the magnetic f i e l d is not al tered
by the magnetic f i e l d . Hence, the e lec t ron ic states are arranged on a system of cy l -
inders, the Landau cy l inders. For free electrons, these are concentric cy l inders
with c i r cu la r cross sections and axes para l le l to the magnetic f i e l d . For a rb i t ra ry
energy surfaces, the cy l inder axis must not coincide with the magnetic f i e l d as shown
in Fig. 2.5. For any f i e l d d i rec t ion there exists an extremal cross section on the
e l l i p so id which is perpendicular to the f i e l d .
The degeneracy d of the e lec t ron ic states is also changed by the magnetic f i e l d .
In the absence of a f i e l d , a state k can be occupied twice. With a f i e l d (H para l le l
kz), the degeneracy of the state (n,kz) is
d = LxLyH/Oo where LxLyH is the f l ux through the crystal cross section LxLy perpendicular to the
f i e l d , d increases l i n e a r l y with the f i e l d H. This is a consequence of the l o c a l i -
zat ion of the charge in the f i e l d . Whereas a conduction electron in state k is spread
out through the whole c rys ta l , i t is local ized on a cy l inder with radius v/m c in the
state (n,kz). Thus a number of electrons increasing l i n e a r l y in H can be accommodated
in the same state in the same crystal cross section wi thout v io la t i ng Pauli exclu-
sion.
At T = O, a l l those states on the Landau cyl inders with energies not exceeding
the Fermi energy E F are occupied. I f a large number of Landau cyl inders are cut by
the Fermi surface, the Fermi energy is somewhat f i e l d independent. At increasing
f i e l d , the ef fects of decreasing number of Landau cyl inders and increasing degene-
racy compensate, so that the same number of electrons can be accommodated inside
the Fermi surface as at H = O. I f , on the other hand, the f i e l d has been increased
to such a degree that merely one Landau cy l inder is cut by the Fermi surface, E F
must inCrease with H. This quantum l i m i t is not real ized with conventional magnets
in the noble metals and w i l l thus be not considered here. In the fo l low ing , the
Fermi energy and the Fermi surface of these metals are treated as f i e l d independent.
H=O
ky
iii!iiiiii!iiiiiiiiiiiiiiiii!i!iiiiiiiiiiiiiiiiiiiiiiiiiiiii!ii ii{~4i{ii!i{ii{iiiiiiiiii{ii!iiiiiiii!iiiiiiiiiiiiiiii!ii{ili iiiiiiiii~i~iii!iiiiiiiiiiiiiiliiiiiiiiiiiiiiiii~iiiiii~ii~ E,-- i~F~ !!~!~i!!!!ii!!l~ii!!~ii~!~!i!i~F~ii!!i r ii!iiiiii?iiiiiiiiii~li~iiiiiiiiiiiiiii!iiiil
~x
ky
QH_ 4rl;2H/~ o
EF kx
~ Wc (n +II 2)
Fi 9. 2.4. Redistr ibut ion of the elec- t ronic states in a magnetic f i e l d , drawn for free electrons. With magnetic f i e l d the states form c i rc les in the planes perpendicular to H. The area enclosed by neighboring c i rc les increases l i n - early with the f ie ld . The degeneracy of the states increases also l i near ly with the f ie ld . Thus the Fermi energy and the Fermi surface are p rac t i ca l l y f i e l d independent (except for the quantum l im i t )
I H H I
E = consL ~
Fig. 2.5. Landau cyl inders for e l l i ps - oidal energy surfaces. The cyl inder
/ axis coincides with the f i e l d direc-
4 t ion when a pr incipal axis of the e l - l ipsoid coincides with the f i e l d d i rec- t ion
2.1.2 Density of States of the Electrons in the Magnetic Field
The red i s t r i bu t i on of the e lec t ron ic states in k-space af fects the density of states
D(E) in a drast ic manner. I t is obvious from Fig. 2.5 that D(E) has s i ngu la r i t i e s
each time a Landau cy l inder is tangent to the energy she l l . Fig. 2.6 shows the en-
ergy spectrum of free electrons with and wi thout f i e l d . The density of those states
D(E,kz=0)
I I I I I
(a) 0 1 2 3 4 E / ~ c
DIE)
H=0
/ / i I n 0 1 2 (b) 3 4 E/hto c
Fig. 2.6a and b. Density of states of free conduction electrons wi thout and with f i e l d (b). The density of the states in the extremal cross section is a sum of delta funct ions separated by ~ ^ . Scattering of the conduction electrons at defects causes a Loren- tz ian broadening of the levels (a)
which are n the extremal cross section is a per iodic funct ion of the energy with
period hw c. For sharp Landau cy l inders , i t is a sum of delta funct ions as indicated
in Fig. 2.6a. The cont r ibu t ion of the states above and below the extremal cross sec-
t ion to the to ta l density leads to a smooth decrease in D(E) on the upper sides of
the s i ngu la r i t i e s . The density of states at the Fermi level changes pe r iod ica l l y
wi th increasing f i e l d , and drops abrupt ly each time a Landau cy l inder leaves the
Fermi surface. This per iodic var ia t ion of D(EF) with H is the cause of the quantum
o s c i l l a t i o n s , and in pa r t i cu la r , of the dHvA e f fec t .
2.1.3 Origin of the dHvA Osc i l la t ions
Figure 2.7 i l l u s t r a t e s the o r ig in of the dHvA osc i l l a t i ons . In part (a) i t is as-
sumed that at a f i e l d strength HI, the n-th Landau cy l inder is tangent to the Fermi
9
\
J n
I E F
Fig. 2.7. Origin of the dHvA osc i l l a t i ons . Each time a Landau cy l inder leaves the Fermi surface with increasing f i e l d , the free energy of the electrons drops abrupt- l y . This causes the per iodic var ia t ions of the magnetization with the f i e l d . I f the Fermi surface is smeared by f i n i t e temperature or i f the Landau cyl inders are smeared by the scat ter ing of the conduction electrons, the free energy varies less abrupt ly . Hence the amplitudes of the osc i l l a t i ons are reduced. This argument is va l id for ar- b i t ra ry Fermi surfaces and not only for spherical surfaces as shown here for reasons of s imp l i c i t y
surface. At T = O, the Fermi surface is sharp. I f the electrons are not scattered
at defects, the Landau cyl inders are sharp as wel l . The free energy G of the con-
duction electrons has a maximum at H 1 because the states on the equator at the Fer-
mi surface are occupied and have the highest energy of a l l occupied states. I f the
f i e l d is increased from H I to H 2, the n-th Landau Cylinder leaves the Fermi surface.
The states on the equator l i ne of the Fermi surface are depleted and the correspon-
ding electrons are red is t r ibu ted , mostly on lower energy states. Thus the free en-
ergy decreases to a minimum in a small f i e l d i n te rva l . With fu r ther increase of the
f i e l d , the Landau cyl inders fu r ther expand, the free energy increases again and
reaches another maximum when the (n-1) th cy l inder is tangent to the Fermi surface.
This completes a cycle of an o s c i l l a t i o n of the free energy and of the magnetiza-
t ion of the conduction electrons.
2.1.4 Frequency of the dHvA Osc i l la t ions
The frequency of the dHvA osc i l l a t i ons may be deduced immediately from Fig. 2.7.
When the n-th Landau cy l inder is tangent to the Fermi surface the energy of the
states on the contact l i ne is given by
10
~ c ( n + 1/2) = (h2/2Xmc)Aex . (2.14)
Here Aex is the area of the extremal cross section of the Fermi surface for the
given f i e l d d i rec t ion . With (2.4) , (2.14) can be wr i t ten
F/H : n+1/2 (2.15)
where
F = r 2 (2.16)
is cal led the dHvA frequency which is proport ional to the extremal cross section
Aex at the Fermi surface. The phase of the osc i l l a t i ons is
2~n = 2~(F/H-1/2) . (2.17)
From Fig. 2.7, i t is obvious that only the electrons in the extremal cross section
of the Fermi surface contr ibute to the dHvA e f fec t . The density of states in the
extremal cross section is a periodic funct ion in energy with period ~w c (Fig. 2.6a).
I t can therefore be expanded in a Fourier series containing terms cos(2xn) and the
corresponding higher harmonics. This Fourier series enters the magnetization (2.1)
through the free energy G as a sum of harmonics with fundamental s in [2~(F/H- 1 /2) ] .
Thus the magnetization varies s inusoidaly wi th the dHvA period F and i t s higher
harmonics. Since th is var ia t ion is sinusoidal in I /H, and not in H, the period AH
of the osc i l l a t i ons is f i e l d dependent
AH = H2/F . (2.18)
= 6.1022 cm-3), For a free electron metal with the electron density of gold (nel
the values of k F and F are
k F = (3~2nel) I /3 ~ 1.2 ~-1 (2.19)
F ~ 4 .8 -108 G . (2.20)
A't 105 G, the dHvA osc i l l a t i ons have a period of AH ~ 20 G. For a sample volume of
i cm 3, there are 2k F• I cm/2~ m 4• states on a diameter of the Fermi surface at
H = O. In a f i e l d of 105 G, on the other hand, there are only n ~ F/H = 4800 Landau
cyl inders which are cut by the Fermi surface. The strong diminut ion of the number
of Landau levels and the corresponding increase of the degeneracy of the states in
the f i e l d imply that a r e l a t i v e l y large number of electrons are involved in the de-
population of the Landau cy l inders. I t is the combination of these two ef fects which
makes the dHvA e f fec t observable experimental ly. In the case mentioned above, about
2.10 -6 of a l l conduction electrons are involved in the red i s t r i bu t i on when a Landau
cy l inder leaves the Fermi surface.
11
2.1.5 Amplitude of the dHvA Osc i l la t ions
Only those electrons in the extremal cross section of the Fermi surface contr ibute
to the dHvA ef fec t . The thickness of th is s l i ce is inversely proport ional to the
curvature C = ~2Aex/~k ~ ~ of the area in the d i rec t ion of the f i e l d . The magnetization
of the electrons in the extremal cross-sectional s l i ce is
D(N) = -(2eo~/~Cmc)F~o3/2(N/~)i/2 (2.21)
I t is larger for larger cross sections Aex. D(H) is the maximum value of the ampli-
tude which can only be real ized at T = 0 and for sharp Landau cy l inders.
Damping of the dHvA Osc i l la t ions by F in i te Temperature
At f i n i t e temperature, the Fermi surface is smeared out according to the Fermi d is-
t r i bu t i on . This smearing implies a less abrupt decrease of the free energy when a
Landau cy l inder leaves the Fermi surface (Fig. 2.7). Thus, the magnetization M =
-~G/~H is reduced in amplitude. The damping depends on the ra t i o of the Fermi sur-
face smearing, kBT, and the energy d i f ference, ~mc ' of neighboring Landau cy l i n -
ders
2~2kBT/~mc = bmcT/mo H . (2.22)
The constant b has the value
= 2~2kBCmo/~eo = 146.925 kG/K . (2.23) b
The temperature damping factor I I has the form
11 = (bmcT/moH)[sinh(bmcT/moH)]-I At 105 G and for m c = m o , i t is I I ( 0 K) = i , 11(1 K) = 0.71, and 11(4.2 K) = 0.026.
These numbers show that the temperature damping can be appreciable at 4.2 K. This
is the reason why the dHvA e f fec t is observable only at low temperatures.
Damping of the dHvA Osc i l la t ions by Electron Scattering
Due to the scat ter ing at defects, a conduction electron exists only for a mean time
T(k) in a state k before i t is scattered in to another state. According to the un-
cer ta in ty re la t i on , the scatter ing broadens the Landau levels . In a phenomenological
approach DINGLE has described th is broadening by a Lorentzian of width 2xkBX /2 .9 / .
X is cal led Dingle temperature. Since the broadening of the Landau levels reduces
the dHvA amplitudes in a way s im i la r to the smearing of the Fermi surface at f i n i t e
temperature (Fig. 2.7) , i t is natural to associate i t with a temperature, the Dingle
12
temperature. The Dingle damping fac torK 1o f the dHvA amplitude depends on the ra t i o
kBX/~m c which is s im i la r to the ra t i o (2.22). The density of the Landau levels in
the extremal cross section which are broadened according to a Lorentzian (Fig. 2.6a)
can again be expanded in a Fourier series. Since the Fourier transform of a Loren-
tz ian is an exponential funct ion, the Dingle damping fac tor K I becomes
K 1 = exp(-bmcX/moH ) . (2.25)
MANN has shown how DINGLE'S assumption can be derived from f i r s t pr inc ip les /2 .10/ .
The Dingle temperature is an average of the scatter ing rates I /T(k) of the electrons
over the extremal cross sect ion.
X = (~/2~kB) <I/T(k)> . (2.26)
Inf luence of the Electron Spin on the dHvA Effect
The magnetic moment ~ associated with the electron spin s can occupy two states in
a magnetic f i e l d (O,O,H) wi th the energies
• c = • c . (2.27)
The spin sp l i t s the Landau cyl inders in two systems of Landau cy l inders sh i f ted in
energy by the amounts given in (2.27). An electron in a state (n,k z=O,s) has the
energy
E = ~ c [n+1 /2 • 1/2(gcmc/2mo)] . (2.28)
The sp in -o rb i t coupling in a metal can cause deviat ions of the s p i n - s p l i t t i n g factor
gc from the value 2 for free electrons. The spin s p l i t t i n g reduces the dHvA ampli-
tude according to the factor
S I = cos(~gcmc/2mo) . (2.29)
2.1.6 L i fsh i tz-Kosevich Expression for the dHvA Effect
The main cont r ibut ions to the magnetization have now been introduced. The o s c i l l a t o -
ry part of the magnetization para l le l to the f i e l d is wr i t ten (neglect ing higher
dHvA harmonics)
M = D(H) I IK IS ls in [2~(F/H-1 /2) • ~/4] . (2.30)
The magnetization varies s inuso ida l l y in I/H with the dHvA frequency F and the am-
p l i tude
A(T,H,X) = D(H)IIKlS I (2.31)
13
which depends on the temperature, f i e l d , Dingle temperature, and or ientat ion of the
crystal in the f i e l d . I t is assumed that the Fermi surface has only one extremal
cross section for a given or ientat ion. Otherwise, the contr ibut ions from the d i f -
ferent cross sections must be added�9
2.2 Influence of the Electron-Phonon Interact ion on the dHvA Effect
The electrons in a metal can in teract with phonons by electron-phonon interact ions
and with other electrons by Coulomb repulsion and Pauli exclusion. Fig. 2.8 shows
how these interact ions af fect the probab i l i t y f(E) for an electron to occupy a state
fiE)
1
E F E electron- electron
interact ion
fiE) ,kB~, __f_-"
E F electron-phonon
interaction
=E
Fi 9. 2.8. Influence of electron-electron and electron-phonon interact ion on the probabi l i ty f(E) for an electron to occupy a state of energy E
with energy E /2.11/ . The interact ions smear out the step at E F in the Fermi d is-
t r ibu t ion (at T = 0). States above E F are par t ly occupied and states below E F are
par t ly depleted. The degree of smearing depends on the strength of the interact ions.
In the noble metals the average distance of the conduction electrons is 3a o (ao:
Bohr radius). Thus, the i r mean Coulomb repulsion is e~/3a ~ ~ 9 eV. This is more
than the Fermi energy. The electron-electron interact ion therefore affects even the
electrons at the bottom of the conduction band. The electron-phonon in teract ion, on
the other hand, can only a f fec t the electrons in a range of the order kBe D (eD: Debye
temperature) around E F. The electron-phonon interact ion reduces the step at E F from
i to i / ( I + ~ ) where the electron-phonon coupling constant ~ for energies near E F is
given by mmax
~(k) = 2 ~ I d~ ~ (k ,~ )F (~)/w �9 (2.32)
~ J
14
9 The Eliashberg function ~(k,m)F (m) describes the coupling of an electron state
with a l l other states by phonons of frequency m and polar izat ion ~. ~ can range from
0.04 to 1.5 depending on the strength of the electron-phonon interact ion.
Although the probab i l i t y f(E) can be affected strongly by the two interact ions,
the geometry of the Fermi surface is not affected by them. The magnetic f i e l d H en-
ters the Hamiltonian through the k inet ic energy [p + (eo/c)A]. The quantization of
the cyclotron orb i ts fol lows from th is Hamiltonian, and therefore the period of the
dHvA osc i l l a t ions is independent of the Coulomb and electron-phonon interact ion
/2.12/. On the other hand, those quant i t ies which contain gradients on the Fermi
surface l i ke the Fermi ve loc i t ies or cyclotron masses are affected by electron-elec-
tron and electron-phonon interact ions. Real is t ic band structure ca lcu lat ions take
into account the electron-electron interact ion whereas they normally do not consider
the electron-phonon interact ion. A comparison of cyclotron masses measured by dHvA
ef fect with those determined from a band structure calculat ion, therefore, gives the
p o s s i b i l i t y of estimating the electron-phonon coupling constant Z(k). Since the elec-
tron-phonon interact ion changes the dispersion re la t ion E(k) of the conduction elec-
trons in a range of width kB0 D at the Fermi leve l , the Fermi ve loc i t y is reduced by
the factor I / ( i + ~ ) and the density of states is increased by ( I + ~ ) . By ~ is de-
noted the average of X(k) over the Fermi surface (Fig. 2.9). I t should be emphasized
again that the value of the Fermi energy and the geometry of the Fermi surface are
unaffected, whereas the gradients ( l i ke the Fermi ve loc i ty or the density of states
of the Fermi leve l ) are. Quantit ies which are renormalized by electron-phonon in te r -
action w i l l be denoted in the fol lowing by an aster isk. The influence of the elec-
tron-phonon interact ion on the amplitude of the dHvA osc i l la t ions is shown in Fig.
2.10. Near E F i t is /2.11/
E ~ = E F + (E-EF) / ( I+<~>) (2.33)
where <x> is the average of ~(k) around the corresponding cross section
<~> = ( f d k ~ ( k ) / v • -1 (2.34)
Since the slope of E ~ versus E is i / ( I + < ~ > ) the distance between neighboring Landau
leve ls , the level broadening and spin s p l i t t i n g are reduced by the factor I / ( I + < ~ > )
by electron-phonon interact ion
hm c ~ h~ c = h~c/(Z + <x>) (2.35)
X ~ X ~ = X/( l+<~>) (2.36)
gc ~ gc : gc/(1 + <x>) " (2.37)
15
E E"
F . . . . . . . . .
k F __k
E ,
E*
E F
/
DIE)" D*(E)
Influence of electron-phonon interact ion on the dispersion re la t ion and on the density of states of the electrons
E* I E~EF + E - E.F 2..rl;kBX 1 +<~k> J
L / 1.<X> , J
X:'k I-
H gcl'J'B H ~"
T
J
h 00 c
Z I I I I I I I I I I
/2T[,kBX
I \
d , E
E F
Fig. 2.10. The distance, broadening, and spin sp l i t t i ng of Landau levels are re- duced by i / (1+<~>) due to electron-phonon interact ion. The level broadening has only been drawn for the unspl i t levels
16
The temperature damping of the dHvA amplitudes depends on the ra t io of kBT and the
distance between neighboring Landau levels. This ra t io now becomes
2~2kBT/hm~ = bm~T/moH (2.38)
The temperature damping factor 11 (2.23) therefore contains the cyclotron mass
m ~ = mc(Z + <~>) (2.39) C
which is enhanced by electron-phonon interact ion. In Azbel-Kaner cyclotron resonance,
electronic t rans i t ions between neighboring Landau levels are induced by an e lec t r i c
r f f i e l d . I t is therefore again the renormalized mass m c which is determined by th is
technique.
The Dingle damping factor K 1 (2.25) depends on the ra t io kBX/~ c. Since these
factors are both reduced by the factor I / ( i + < ~ > ) the i r ra t io is independent of the
electron-phonon interact ion. This can be wr i t ten in the form
mcX = m~X ~ . (2.40) C
The same argument holds for the sp in -sp l i t t i ng factor S 1
gcmc = gcmc �9 (2.41)
I t should be noted that the renormalization of the electronic energies discussed
here involves only t rans i t ions by Vir tual phonons. The influence of real phonons
on the amplitude of the dHvA ef fec t is not yet completely understood /2.13,14/. In
the noble metals, real phonon effects can be neglected at temperatures below 4 K.
2.3 Information Derivable from the dHvA Effect
Including the electron-phonon in teract ion, the osc i l l a to ry part of the magnetization
for one extremal cross section and for negl ig ib le higher dHvA harmonics is given by
M : A(T,H,X ~) sin[2~(F/H - 1/2) • x/4]
A(T,H,X *) = D(H)IIKIS I
D(H) = -(2eo~F/~cm~@o3/2)(H/~) 1/2
11 : (bm~T/moH) [sinh(bm~T/moH)]
K 1 = exp(-bm~X~/moH )
S I : cos(~g~m~/2mo)
- I
(2.42)
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
17
F = r 2 . (2.48)
A l l the quant i t ies which can be derived from the dHvA e f fec t are averages of local
values around an extremal cross section on the Fermi surface. The thickness of
these cross sections depends on the curvature C of the Fermi surface. In the noble
metals, th is thickness is usual ly of the order of one degree.
2.3.1 Geometry of the Fermi Surface
According to (2.48), the area Aex of an extremal cross section can be derived d i -
rec t l y from the corresponding dHvA frequency F. To obtain the l i nea r dimensions of
the Fermi surface, i t is necessary to measure the angular dependence of the dHvA
frequencies. From Fig. 2.3 i t can be seen that
Aex = 1 / 2 / d ~ k~ . (2.49)
To obtain the geometry of the Fermi surface from (2.49), i t is very helpful to have
some idea of the shape of the Fermi surface. This can be obtained, for instance,
from a band st ructure ca lcu la t ion . In these ca lcu la t ions , cer ta in parameters ( in
the electron po ten t ia l ) are f i t t e d in such a way that the calculated cross sections
agree opt imal ly with the measured cross sections. In recent years, the Fermi sur-
faces of most pure metals and of many ordered al loys have been determined by the
dHvA e f fec t /2 .7 / . In some cases, the Fermi surface is known to 1 part in 105 . This
accuracy is achieved at the present time by no other experimental technique. Only
for disordered al loys and fo r those metals for which the preparation of s ingle crys-
ta ls is d i f f i c u l t is i t necessary to use other techniques.
2.3.2 Cyclotron Masses and Fermi Veloc i t ies
The cyclotron mass m ~ for a cer ta in extremal cross section can be obtained from the c temperature dependence of the dHvA amplitude A(T) at f ixed f i e l d H. For su i tab le
experimental condi t ions, the hyperbol ic sine in (2.45) can be replaced by an expo-
nent ia l . A p lo t of the dHvA amplitude versus the temperature according to
Zn(A/T) = ZnA o - (bm~/moH)T (2.50)
gives a s t ra igh t l i ne with a slope proport ional to m~/m o. The cyclotron mass is an
average of the reciprocal Fermi ve loc i t y v~(k) over an extremal cross section
f m c = (h12~) d~ [k~/v*(k)(! 'k•
where ~ is a unit vector in the direction of the gradient on the Fermi surface. The
weighting factor k~/(~.k• depends only on the geometry of the Fermi surface. I f
18
th is is known, and i f the masses have been measured for a su f f i c i en t number of cross
sections, then a deconvolution of (2.51) gives the local values of the Fermi velo-
c i t i es v*(k) . The measured values m~ and v*(k) are renormalized by electron-phonon
in teract ion. I f the band ve loc i t ies v(k) (not renormalized by electron-phonon in te r -
action) are known from a band structure calculat ion, local values of the electron-
-phonon coupling constant ~(k) can be obtained from
I + ~(k) = v (k ) /v* (k ) . (2.52)
2.3.3 Dingle Temperatures and Scattering Rates of the Conduction Electrons
From measurements of the f i e l d dependence of the dHvA amplitudes A(H) at f ixed tem-
peratures T, the Dingle temperature X* can be obtained. A plot
Zn [AH I /2 sinh(bm~T/moH)] = InA o - bm~X*/moH (2.53)
versus I/H y ie lds a s t ra igh t l ine with slope -bm*X*/mA. I f m* is known for the c u c given extremal cross section, X* is also known. X* is the average of the local scat-
tering rates i /~ * (k ) of the conduction electrons around the extremal cross section
X* = (~/2~kB) <l/z*> (2.54)
= (h /2~)2 (Z /kBm~) fdk / (v~ *) (2.55)
or
m'X* = (h/2~)2(Z/kB)Q~dm [k~/ (v* .k• . (2.56) c Y
The weighting factor of the local scattering rate depends on the geometry of the
Fermi surface and on the Fermi ve loc i t ies . The determination of the scattering an-
isotropy therefore requires a detai led knowledge of the electronic structure of the
host l a t t i c e in which the scatter ing defects are d is t r ibuted. I t is not yet complete-
ly clear whether the scatter ing of the electrons by real phonons contributes to the
product m'X* in (2.56) /2.13,14/. However, in the noble metals, only impuri t ies or c structural defects contribute to the scattering rates at temperatures below 4.2 K.
2.3.4 g-Factor of the Conduction Electrons
I f the absolute values of the dHvA amplitudes are known, the product * * gcmc can be determined from the sp in -sp l i t t i ng factor S 1. This allows the invest igat ion of the
k-dependence of the sp in-orb i t coupling of the conduction electrons at the Fermi
leve l .
19
3. Experimental Setup for dHvA Measurements in Cu, Ag, and Au
3.1 Field Modulation Technique
Four experimental techniques have been applied for the measurement of the dHvA ef fect
in the noble metals. These are inductive magnetometers in a pulsed f i e l d /3 .1 / , v i -
brational magnetometers /3 .2 / , torque magnetometers /3 .3 / , and inductive magneto-
meters with f i e l d modulation /3 .4 / . Since the a v a i l a b i l i t y of highly homogeneous
and strong superconducting solenoids the f i e l d modulation technique has been used
in most invest igat ions on noble metals /3 .5 / . Since th is technique has been used in
the present invest igat ion the method w i l l be b r i e f l y described.
A small a l ternat ing f i e l d h sin mt of frequency m is superposed on the large
f i e l d H o which is produced by a superconducting solenoid
H(t) = H ~ + h s i n~ t (3.1)
This f i e l d creates in a single c rys ta l l i ne meta l l i c sample a magnetization which
according to (2.42) is
M(t) = A(T,Ho,X* ) sin[2~(F/H o - 1/2) • 7/4 - 2~(Fh/H~)sin~t] (3.2)
The high dHvA frequency in the noble metals allows the time dependence of the f i e l d
in the dHvA amplitude to be neglected re la t i ve to that of the phase of the sine.
dM/dt induces in a pick-up coi l which surrounds the sample a voltage which is pro-
portional to
A ~ wJv(2~Fh/H~) sin(umt+~/2) sin[2~(F/H o - I/2) • 7/4 + ~ - ] . (3.3)
v=1 Since the magnetization depends in a nonlinear way on the magnetic f ield, the volt-
age contains harmonic contributions of the modulation frequency ~ which are weighted
by Bessel functions Jv of integer index v. The harmonic contributions can be f i l -
tered out by means of suitable electronic equipment (like, for example, lock-in
amplifiers). In most cases, as in the present experiment, the second harmonic is
chosen because i t gives an optimal signal to noise ratio. The corresponding ampli-
tude is proportional to
A(T,Ho,X~)2~J2(2~Fh/H~) sin[2~(F/H ~ - 1/2) • 7/4] . (3.4)
When the magnetic f i e l d H is swept, the dHvA osc i l la t ions can be recorded on a o XY recorder and Fourier analyzed in subsequent electronic equipment. The amplitude
h of the modulation f i e l d is chosen in such a way that J2 is at i t s f i r s t maximum. 2 h must be control led by the sweep generator to keep 2~Fh/H ~ at the value of the
f i r s t maximum of J2" To regis ter the dHvA osc i l la t ions at a constant rate in time
20
I I
~ S
qu
are
r ,
t
Plo
tter
~
20o
< ~
~r~
H-[
_~
10
0~
.j-t
-<-t
~
I "~
1 ~1
1~1
p.uLJ
T M-o
d.
"~1
I I
I
Sam
ple
Fig
. 3.
1.
Ele
ctro
nic
setu
p fo
r ou
r dH
vA e
xper
imen
t (f
ield
m
odul
atio
n te
chni
que)
i t is necessary to sweep the f i e l d at a rate proportional to 1 / t . The pick-up coi l
used was a system of balanced coi ls one of which contained the sample. A diagram
of the electronic setup of the present dHvA experiment is shown in Fig. 3.1.
3.2 Magnet and Cryostat
The superconducting magnet used in this invest igat ion had a maximum f i e l d of 76 kG.
Since the noble metals have dHvA frequencies of about 5.108 G, the period of the
osc i l la t ions at 50 kG is about 5 G. This imposes s t r i c t requirements on both the
crystal qua l i ty and the homogeneity of the magnetic f i e l d . The homogeneity of the
magnetic f i e l d has been measured by means of an NMR probe (27AI) /3 .6 / . In the re-
gion covered by the sample the f i e l d was homogeneous to a few parts in 106 over the
dimensions of the sample. This is small enough to avoid perceptible reductions of
the dHvA amplitudes. The cryostats for the magnet and for the sample holder were
conventional bath cryostats. In the inner cryostat which contains the sample, any
temperature between 1.1 and 4.2 K can be generated by pumping 4He and kept for hours
wi th in a few mK.
3.3 Sample Holder
To measure the angular dependence of the dHvA frequencies, the cyclotron masses and
the Dingle temperatures the crystal must be oriented in the f i e l d . This is achieved
by means of the gear system shown in Fig. 3.2. The sample can be turned around a
horizontal axis through 360 ~ by means of a worm gear. A second gear system allows
t i l t i n g by • o about an axis perpendicular to the f i e l d H. Part (b) Of Fig. 3.2
(which is viewed at 90 o re la t i ve to part (a)) shows the pick-up and the modulation
coi ls . Orientation of a given crystal lographic axis along the f i e l d is achieved by
using the dHvA ef fec t i t s e l f . The magnetization is measured as the crystal is turned
in the f i e l d . Such a measurement is shown in Fig. 2.1. A copper crystal was turned
by I00 ~ around an axis <110>. When the symmetry direct ions [001], [111], and [ i i 0 ]
are paral le l to the f i e l d , the magnetization goes through a stat ionary value. These
marks allow any crystal lographic d i rect ion to be aligned paral le l to the f i e l d to
wi th in one minute of arc. Fig. 3.3 is a detai led view of the angular dependence of
the magnetization near <100>. Notice the jerk- f ree rotat ion of the crystal in the
f i e l d at l iqu id helium temperature.
3.4 Single Crystals of the Noble Metals
The high phase of the dHvA osc i l la t ions in the noble metals and the strong angular
dependence of the dHvA frequencies necessitate high qua l i ty single crysta ls . Indeed,
i f a crystal is composed of two grains which are t i l t e d against one another by only
92
ion
I l
1 cm
J M
-C-R
(bl
Fig. 3.2a and b. Sample holder used to o r ien t the crysta ls in the magnetic f i e l d ,
The sample (S) can be rotated through 360 o around the axis R and t i l t e d by •
around the axis T. P and M stand for pick-up and modulation co i l s
23
Cu ~Io}
1.1~ K
Fig. 3.3. Angular dependence of the magnetization near [100]. The or3entation of the crystal in the f i e l d can be done w i th in a few minutes of arc
a few minutes of arc, then the dHvA amplitude is markedly reduced for a nonstat ion-
ary cross section. For that reason the most accurate amplitude measurements can be
made fo r s ta t ionary orb i ts l i ke the high symmetry o rb i ts . High qua l i t y s ingle crys-
ta ls wi thout subgrain boundaries have been grown by the Czochralski technique by
UELHOFF and co-workers /3 .7 / and were grown as wires i mm in diameter and were cut
in pieces 5 mm long by an acid layer saw. The width of the rocking curves of these
crysta ls was only 10-20 seconds of arc. The d is locat ion densi ty was as low as
103 cm -2. The pure Cu, Ag, and Au crysta ls had residual r e s i s t i v i t i e s of 0.5, 0.5,
and 0.4 nohmcm. Their Dingle temperatures were t y p i c a l l y 0.03 K for a l l o rb i ts .
3.5 P i t f a l l s in dHvA Measurements
3.5.1 Skin Ef fect
Accurate amplitude measurements are possible only i f the modulation f i e l d penetrates
homogeneously in to the samples. For reasons of signal to noise ra t i o i t was not
possible to use modulation frequencies appreciably below 30 Hz in the experimental
setup used in th is inves t iga t ion . Although fo r samples with a r e s i s t i v i t y of
0.5 nohmcm the skin depth at 30 Hz is only 0.2 mm, the magnetoresistance in the
noble metals increases th is skin depth to such a degree that~a homogeneous penetra-
t ion is guaranteed for a l l o rb i ts except fo r the be l l y <100> and the dogsbone <110>
orb i ts . Thus in the pure noble metals, o rb i t s 0.5 o away from these high symmetry
d i rec t ions have been invest igated. Here the magnetoresistance is already large
enough. In most doped samples the skin e f fec t creates no problems.
3.5.2 Harmonic dHvA Components
The L i fsh i tz -Kosev ich expression of the dHvA e f fec t contains higher harmonics in
the dHvA frequency F in addi t ion to the fundamental term given in (2.42). These
24
components are separated from the fundamental in a Fourier analysis of the dHvA
s ignal . Because they are more s t rongly damped than the fundamental they can be ne-
glected under su i tab le experimental condi t ions. To keep the second harmonic smaller
than 1% of the fundamental, the condi t ion m~(T_ +X)/m o > 1.4 must be f u l f i l l e d at
50 kG. This can be eas i ly achieved.
3,5.3 Magnetic In teract ion
I t was f i r s t emphasized by SHOENBERG /3 .1 / that the e f fec t i ve f i e l d acting on the
electrons is not H but H+4~( I -D)M. Thus the magnetization (2,42) is given by the
i m p l i c i t equation
M : A s in [2~F/ IH+4~(1-D)MI • ~ / 4 - 7 ] . (3.5)
D is the demagnetization factor which depends on the geometry of the sample and
which is a scalar for e l l i p s o i d a l l y shaped samples. When 4~( I -D)M is comparable
to the dHvA period H2/F, then appreciable self-modulat ion occurs which can d i s t o r t
the s igna l , in an extreme case, from a sine to a sawtooth-shaped curve (magnetic
in te rac t ion) /3 .8 / . Deviations of the signal from a pure sine can be most eas i ly
monitored by d i f f e r e n t i a t i n g the s ignal . Because the magnetic in te rac t ion depends
on the ra t i o of the magnetization to the period of the o s c i l l a t i o n s , i t is better
to reduce the magnetization by increasing the temperature rather than by decreasing
the f i e l d . The l a t t e r would reduce at the same time the period. The e f fec t is most
c r i t i c a l fo r the be l l y orb i ts in the noble metals because they have the highest
frequencies and thus the smallest periods.
3.5.4 Phase Smearing
The inf luence of f i e l d and crystal inhomogeneities on the dHvA amplitudes has been
pointed out in the Sections 3.2 and 3.4.
4. The Fermi Surface of the Noble Metals
In Section 2.3.1 i t was shown how the geometry of the Fermi surface can be deduced
from the angular dependence of the dHvA frequencies. The f i r s t dHvA measurements
of a noble metal were made on Cu by SHOENBERG /4 .1 / . They showed that the Fermi
surface has protrusions in the d i rect ions <111> which contact the B r i l l o u i n zone.
These protrusions form the necks of the Fermi surface of copper. These resul ts con-
firmed the model of the Fermi surface of Cu proposed by PIPPARD on the basis of
anomalous skin e f fec t measurements /4 .2 / . A graph of the model is shown in Fig.
4.1. In the per iodic zone scheme, the Fermi surfaces of the d i f f e ren t B r i l l o u i n
zones are connected at the necks. This leads to both electron orb i ts (be l l y B and
25
Fig. 4 . ! . Model of the Fermi surface of the noble metals. The Fermi surface con- tacts the B r i l l o u i n zone in the d i rect ions < i i i > . This gives r ise to the necks. Three cyclotron orb i ts are shown (the neck o rb i t N<111> and the two be l l y o rb i ts B<IO0> and B<111>)
i [ooi}
[1101
(a) (b)
i [oio]
{tOOl
Fig. 4.2a and b. The dogsbone D<110> and the rosette R<IO0> orb i ts are observable when the f i e l d is para l le l to the d i rect ions <110> and <100>, respect ive ly
neck N orb i ts ) and to hole orb i ts with t ra jec tor ies going through d i f f e ren t B r i l -
lou in zones. In Fig. 4.2 are shown the dogsbone o rb i t D<110> when the f i e l d is
para l le l to <110> and the four cornered rosette R<IO0>. When the magnetic f i e l d
is para l le l to <i00>, the Fermi surface has two extremal cross sections, the B<IO0>
and the R<IO0>. The superposit ion of the corresponding osc i l l a t i ons in Au is shown
in Fig, 2.1. Meanwhile the Fermi surface of the noble metals is among the best in -
vestigated of a l l metals. A detai led reference l i s t up to 1971 is given in the
book by CRACKNELL /2 .7 / . Later very precise measurements on gold were published by
BOSACCHI et a l . / 4 .4 / . COLERIDGE and TEMPLETON /4 .5 / increased the accuracy of
the frequency determination in Cu, Ag, and Au to 1 part in 106 using an NMR probe
for the f i e l d measurements. An accurate and detai led determination of the Fermi
surface of the noble metals which is used extensively in the fo l lowing sections is
the work done by HALSE /4 .6 / . In his paper an analy t ica l expression for the form
26
of the Fermi surface is given which is based on a symmetrized Fourier series. Since
the Fermi surface is a periodic funct ion in k-space, i t can be expanded in a Fou-
r i e r ser ies. In add i t ion, the Fourier series must be invar ian t against the opera-
t ion of the t rans la t iona l and point groups of the f . c . c , l a t t i c e of the noble met-
als. I t turned out that an ansatz with f i ve coef f i c ien ts of the form (4.1) for the
Fermi surface can describe the rad i i to I part in 103 .
o = F ( k )
= - COO 0 + (3 - Z cos s a 2 x cos 7ky)
+ C200(3 - Z cos akx) a a (4.1)
+ C211(3 - ~ cos ak x cos ~ky cos ~kz)
+ C220(3 - ~ cos ak x cos aky)
+ c310(6 -Zcos x cos -Z cos ky) 2 Y 2 x cos .
The sums in (4.1) denote cyc l ic interchange of x, y, and z. The values of the l a t -
t i ce parameters a and of the coef f ic ients C m n used by HALSE are given in Table 4.1.
Table 4.1. Lat t ice parameters a and coef f i c ien ts CLm n
t ion (4.1) according to HALSE /4 .6 /
Cu
Ag
Au
a at 0 K
3.6030 • 0.0004
4.0692 • 0.0008
4.0652 • 0.0004
CO00 C200 C211 C220 C310
Cu5
Ag5
Au5
1.69167 0.00693 -0.42501 -0.01679 -0.03772
-0.89789 -0.12030 -0.90187 -0.14086 -0.09483
-2.26213 -0.16635 -1.25516 -0.09914 -0.12704
of the Fermi surface descrip-
This representation of the Fermi surfaces of Cu, Ag, and Au has been used below in
the determination of Fermi ve loc i t i es from cyclotron masses and of scat ter ing rates
from Dingle temperatures. Fig. 4.3 shows the cross sections of the Fermi surfaces
in the planes { i00} and {110} drawn from the data published by HALSE. S i l ve r shows
the weakest and gold the strongest anisotropy of the noble metals.
Besides the experimental invest iga t ions, band structure calculat ions have in -
creased the understanding of the Fermi surface of the noble metals. There ex i s t a
great number of ca lculat ions fo r the noble metals. Copper is the model substance
27
Cu
[lOO]., / ~ x
Ag
[100] X
Au
[lOO], X
001]
'X U
K
IX 011 U
K OOl]
U
F K
, [110]
[110]
, [11o]
Fi 9. 4.3. Anisotropy of the Fermi surface of Cu, Ag, and Au in the planes {100} and { I I 0 } according to HALSE /4 .6 /
of a d-band metal for which many band structure techniques have been tested. The
most usual ones are the augmented plane wave method (APW) /4.7/ and the Green's
function method (KKR) /4 .8 / . The main problem in ab i n i t i o calculat ions is the con-
struct ion of an electron potential which takes into account the Coulomb and the ex-
change interact ions /4.9-13/ . Even i f these calculat ions are not able to give the
Fermi surface with the same accuracy with which they can be determined experimen-
t a l l y , they show in a simple way the or ig in of the anisotropy sketched in Fig. 4.3.
As described in SEGALL's paper for Cu /4 .10/ , the hybr id izat ion of the s- , p-, and
d-bands is responsible for the deviations of the Fermi surface from a sphere. In
the direct ions <100> and < I i i> the s- and p-bands hybridize. In the center r of the
Br i l l ou in zone the wave functions have pure s-character. At the boundary of the
Br i l l ou in zone (points X and L in Fig. 4.3), the wave functions have pure p-charac-
ter . On the other hand the wave functions have s-character a11 along the d i rect ion
<110> from r to the boundary point K. The s- , p-bands hybr idize, in addit ion, with
the d-bands which are completely occupied in the noble metals but which l i e only a
few eV below the Fermi level . The hybr id izat ion of the sp- and d-bands increases
28
the energy of the states near the Fermi level (antibonding hybr ids). Because the
d -o rb i ta l s do not have lobes along the d i rec t ion < I i i > there is no sp-d hybr id iza-
t ion along these d i rec t ions in contrast to a l l other d i rec t ions . This implies that
the states along <111> are not energe t ica l l y enhanced. Consequently, they w i l l be
occupied up to higher k-values compared to the other d i rec t ions. That is the reason
for the protrusions along the d i rec t ions <111> which, due to the contact wi th the
B r i l l o u i n zone, give r ise to the necks. The sp-d hybr id iza t ion is not equal ly strong
in the d i f f e r e n t b e l l y regions of the Fermi surface. Again fo r symmetry reasons the
energy of the states <110> on the Fermi surface is enhanced more than the energy
at the points <100>. For th is reason the Fermi surface of the noble metals is bulged
inwards at the points <i i0> and outwards at the points <I00> (Fig. 4.3). The argu-
ment given here shows that general symmetry arguments can explain essent ial features
of the Fermi surface of the noble metals.
Equation (4.1) is a phenomenological ansatz fo r the Fermi surface. The coe f f i -
c ients C~m n have no physical meaning. A t heo re t i ca l l y more sa t i s fac to ry parameteri-
zat ion of the Fermi surface can be given by means of a KKR-band structure calcula-
t ion . For that purpose, cer ta in parameters l i ke the Fermi energy and the phase h sh i f t s n~(EF) of the potent ia l are chosen in such a way that the band structure re-
produces the measured Fermi surface in an optimal way. Such a set of parameters fo r
the noble metals has been calculated by LEE et a l . /4.14/ and is given in Table 4.2.
Table 4.2. Fermi energy and phase sh i f t s from a n o n r e l a t i v i s t i c band st ructure
ca lcu la t ion for the noble metals according to LEE et a l . /4 .14/ . The phase sh i f t s
for ~ > 3 are neglected
Cu
Ag
Au
h h h EF[Ry] n o n I n 2
0.55 0.0755 0.1298 -0.1186
0.41 0.2097 0.1188 -0.1019
0.53 0.2496 0.0632 -0.2426
These data w i l l be used in Section 6 fo r the analysis of Dingle temperatures. Since
the Fermi energy is not determined by the dHvA e f fec t , i t must be considered as a
free parameter.
29
5. C y c l o t r o n M a s s e s and Fermi V e l o c i t i e s of the N o b l e M e t a l s
5.1 Cyclotron Masses of Cu, Ag, and Au
According to Section 2.3.2 the cyclotron masses m ~ can be determined from the tem- c perature dependence of the dHvA amplitudes. In an extensive invest igat ion we have
measured a great number of cyclotron masses in Cu, Ag, and Au /5 .1 / . Fig. 5.1 shows
the temperature dependence of the amplitudes for one orb i t in the three metals. The
7.0
6.0
5.[
4.0
3.0 1.0
\ Au B<111> ~,~ mc=(O.s ~ Ca N~11.1> _ . ~,m~: (,.066+- 0.002)~o',~
I ! I I I I
1.5 2.0 25 3.0 T[K] 35 4.0
Fi 9. 5.1. Cyclotron masses for three extremal cross sections in Cu, Ag, and Au determined from the temperature dependence of the dHvA amplitudes. The slope of the l ines is -bmm/moHc
temperature T has been determined from the vapor pressure of pumped 4He /5.2/ by
means of a capacitance manometer. The accuracy of the temperature reading is i mK
below and a few mK above the ~-point T~ = 2.172 K. Special at tent ion has been given
to the influence of systematic errors in the amplitude measurements. The greatest
problems arose from the skin e f fec t of the modulation f i e l d in the high pur i ty sam-
ples. The magneteresistance increases the skin depth to such a degree that a homo-
geneous penetration is guaranteed for a l l o rb i ts , except for the orbi ts B<IO0> and
D<110>, where i t has a minimum /5 .3 / . Here the skin e f fec t is s t i l l disturbing at
30 Hz. For noise reasons i t was not possible to choose modulation frequencies ap-
30
Table 5.1. Cyclotron masses in copper. The experimental data have been determined
from the temperature dependence of the dHvA amplitudes. The masses characterized by §
Ccm n have been calculated from the energy surfaces adjacent to the Fermi surface
and d i f f e r i ng in energy by 56.10 -4 E F from E F
Cu
{110}
B<IO0>
BO.5
BIO
BTP
B50
B<111>
B65
R<IO0>
RO,5
N<III>
N65
m~/m o
Experiment C • %mn
1.341
1 343 • 0.004 1.341
1 315 • 0.002 1.317
Cu
m Jmo Experiment C •
~mn
{110}
N75 0.648 • 0.004 0.645
D85 1.288 • 0.004 1.287
D89.5 1.260 • 0.002 1.262
1 310 • 0.003
1 388 • 0.003
1 378 • 0.003
1 431 • 0.004
1.307 • 0.002
0.444 • 0.001
0.478 • 0.002
1.309
1.391
1.375
1,433
1,306
i . 306
0.444
0.480
D<110> 1.262
{100}
B7 1,326 • 0,003 1.328
BSP 1.320 • 0,003 1.319
BI8 1.327 • 0.004 1.325
B29.2 1.468 • 0.005 1.468
D40 1.309 • 0,004 1.309
Table 5.2. Cyclotron masses in s i l ve r , See Table 5,1
Ag l x iment C mo {110}
B<IO0> 0.938 • 0,003 0.936
B7 0.923 • 0,002 0.924
B8 0.919 • 0,002 0.921
BIO 0.916 • 0,003 0.915
B12 0,911 • 0.002 0.909
BTP 0,904 • 0.003 0.903
B24 0.934 • 0.002 0.934
B50 0.928 • 0.002 0.929
B<111> 0.923 • 0.003 0,920
B60 0.927 • 0.003 0.929
Im~/m o
Ag IExperiment C~m n
{ I I 0 }
B65 0.954 • 0.003 0.953
R<IO0> 1,044 • 0.004 1.044
N<111> 0.365 • 0.001 0.366
N60 0.375 • 0.001 0.374
D85 1.030 • 0.002 1.031
D<IIO> 1,001 • 0.002 1.000
{ i00}
B4 0,933 • 0.002 0.932
B7 0.923 • 0.002 0.925
BSP 0.912 • 0.002 0.912
31
Table 5.3~ Cyclotron masses in gold. See Table 5.1
m~/m o
Au Experiment C~m n
{110}
B<IO0> 1.142
B1 1.140 • 0.004 1.141
B5 1.121 • 0.003 1.122
BIO 1.084 • 0.002 1.083
BI5 1.051 • 0.003 1.050
B23 1.052 • 0.002 1.053
B50 1.074 • 0.005 1.074
B<I l I> 1.066 • 0.002 1.065
B60 1,071 • 0.002 1,072
R<IO0> 1.014
Au
R1
N<111>
N60
D85
D89.5
D<110>
{100}
B8
BSP
B25
D40
m*/m C 0
Experiment C • ~mn
1.014 • 0.004 1.014
0.280 • 0.001 0.281
0.286 • 0.001 0.286
1.003 _+ 0.003 1.003
0.983 • 0.002 0.983
0.983
1.107 • 0.002 1.106
1.067 • 0.002 1.067
1.073 • 0.003 1.072
1.018 m 0.003 1.018
preciably smaller than 30 Hz. Thus the masses have been measured for those orbits
which are 0.50 away from <100> and <110> instead of the orbits B<IO0> and D<110>
themselves. There the magnetoresistance is already large enough so that the skin
effect can be neglected. In s i lver , a sample containing about 80 ppm of vacancies
has been investigated. The vacancy induced res is t i v i t y made measurements also pos-
sible for the orbits B<IO0> and D<110>. Details of the temperature and amplitude
measurements are described in /5.1/.
The Tables 5.1, 5.2, and 5.3 contain the values of the cyclotron masses for Cu,
Ag, and Au measured in the planes {100} and {110}. B, R, N, and D denote bel ly,
four-cornered rosette, neck, and dogsbone orbits, respectively. The numbers without
brackets after the symbol give the angle in degrees by which the magnetic f ie ld is
t i l t ed against the crystallographic axis [001]. The positions of the planes {100}
and {110} and of the angles @ and r are given in Fig. 5.2 to i l lus t ra te the orien-
[001]
[100]
[010]
{001]
11 .H__ (I)0~ [100]
[010] L
Fig. 5.2. The orientation of
the magnetic f ie ld H relative
to the crystal axes kx, ky,
and k z in the planes {100}
and {110} is given by the an-
gles @ and @
32
tat• of the d i f f e ren t extremal cross sections. BTP and BSP are two stat ionary bel-
ly orb i ts with the or ientat ions given in Table 5.4.
Table 5.4. Posit ion of the be l l y turning point and be l ly saddle point orb i ts in the
planes { I i 0 } and { i00}
Cu Ag Au
BTP {110} C) = 16.2 • 0.2 18.4 • 0.1 21.5 • 0.2
BSP {100} ~ = 11.8 • 0. I 13.6 • 0. I 16.3 • 0. I
The accuracy of the measured cyclotron masses is • 0.3%. A comparison with previous:
mass data published in the l i t e r a t u r e is given in Section 5.3.
5.2 Determination of Energy Surfaces Adjacent to the Fermi Surface
According to (2.51) the cyclotron masses are o rb i ta l averages of the reciprocal Fer-
mi ve loc i t i es v*(k). In p r inc ip le , another symmetrized Fourier series can be chosen
which parameterizes the Fermi ve loc i t i es , s imi la r to that used by HALSE to parame-
ter ize the Fermi surface. But an ansatz with f i ve coef f ic ients for v*(k) produces
poor cor re la t ion with the'data although the same ansatz describes the Fermi surface
quite wel l . The reason for th is is the stronger anisotropy of the Fermi ve loc i t i es .
To keep the number of f i t parameters as small as possible, the parametrization scheme
proposed by HALSE has been adopted here. I t consists of constructing energy surfa-
ces adjacent to the Fermi surface which d i f f e r in energy from the Fermi energy E F
by ~E/E F = • 6.10 -4 (Fig. 5.3). These two surfaces are described again by a symme-
6k ~ lA~x - Aex I
\ \ \ -
"EF EF -OE
Fi 9. 5.3. The distance ak of the two sur-
faces E F • aE in k is inversely proport ion-
al to the Fermi ve loc i ty v*(k)
+ A + - t r ized Fourier series with coef f ic ients C~m n. I f ex 'Aex 'and Aex are the areas of
the extremal cross sections for a given f i e l d di rect ions on the surfaces E F + aE,
E F - ~E,and E F then
33
mc/mo : IAex- AexJEF/~k~ 6E . (5.1)
+
The coef f ic ients C~m n are determined in such a way that the A• f i t the measured
masses according to (5.1). This has been done by means of a nonlinear least squares
f i t t i n g program (VAO5AD Harwell Subroutine L ibrary) . Al l masses from the Tables 5.1,
5.2 and 5.3 have been used in the f i t . The values of the f i t t e d parameters C -+ 91fin are given in Table 5.5. The masses, which have been calculated from the coef f ic ients
+
Table 5.5. Coefficients C~m n of the Fourier series which describe the energy sur-
faces adjacent to the Fermi surface. The energy difference between the surfaces
E F • aE and E F is • 6.10 -4 E F. The coefficients Cm~ n for the Fermi surface are those
by HALSE /4.6/
Cu5
Cu5+
Cu5-
Ag5
Ag5+
Ag5-
Au5
Au5+
Au5-
CO00 C200 C211 C220 C310
1.69167 0.00693 -0.42501 -0.01679 -0.03772
1.705925 0 ~ -0.422119 -0.016660 -0.037677
1.680147 0.006285 -0.427345 -0.016834 -0.037728
-0.89789 -0.12030 -0.90187 -0.14086 -0.09483
-0.872823 -0.118934 -0.896995 -0.140175 -0.094590
-0.911864 -0.121070 -0.904573 -0.141142 -0.094881
-2.26213 -0.16635 -1.25516 -0.09914 -0.12704
-2.385641 -0.172520 -1.280091 -0.102875 -0.129762
-2.141185 -0.160308 -1.230744 -0.095484 -0.124378
C+zmn and C~m n - ,are compared to the experimental data in Tables 5.1 , 5 . 2 , a n d 5.3.
The f i t t e d masses agree wi th in the accuracy of the data with the measured ones.
5.3 Angular Dependence of the Cyclotron Masses in Cu, Ag, and Au
Once the coef f ic ients C • Lmn are known, the cyclotron masses can be determined for
any or ientat ion of the crystal in the f i e l d using (5.1). In Table 5.6 a number of
masses are quoted which have been calculated in th is way. Using these data a plot
of the cyclotron masses in the planes {100} and {110} has been drawn in Fig. 5.4
for Cu, Ag, and Au. For comparison, some data from the l i t e ra tu re are shown as wel l .
These are the cyclotron resonance data by KOCH et a l . /5 .4/ for Cu, by HOWARD /5.5 /
for Ag, and by LANGENBERG et a l . /5 .6/ for Au; and dHvA data by COLERIDGE et al /5 .7/
34
Table 5.6. Cyclotron masses m~/m ~ for Cu, Ag, and Au calculated from the energy
surfaces E F • 6E. The angles 0 and @ are those of Fig. 5.2
Cu Ag Au Cu Ag Au
BO = DO =
B~=
0 1.341 0.936 1,142
5 1,333 0.930 1.122
10 1.317 0.915 1.083
15 1,308 0.903 1.050
20 1.325 0.907 1,038
25 1.481 0,949 1.096
26 1.714 0.970 1.207
27.5 - 1,035 1.426
45 1.664 0.977 1.165
47 1.441 0.949 1.098
50 1.391 0.929 1.074
54,74 1.375 0.920 1.065
60 1.388 0.929 1.072
65 1.433 0.953 1.095
70 1.558 1.004 1.155
73 2.013 1.061 1.286
5 1.333 0.930 1.124
i0 1.322 0.918 1.094
15 1.319 0,911 1,071
20 1.333 0.917 1.061
25 1.377 0.941 1.072
30 1,500 0.998 1.130
32 1.653 1.041 1.199
33 1.956 1,072 1.282
D~ =
R@=
NO=
90 1.262 1.000 0.983
88 1.266 1.005 0.986
85 1.287 1.031 1.003
83 1.313 1.066 1.025
80 1.377 1,178 1.078
78 1.446 - 1.137
43 1.269 1.009 0.988
40 1.309 1.065 1.018
37.5 1.381 1.218 1.075
35 1,541 - 1.217
0 1.306 1,044 1.014
5 1.417 1.218 1.102
30 0.952 0.849 0.576
32 0.774 0.662 0.478
35 0.638 0.534 0.398
40 0.529 0.438 0.333
45 0.476 0.393 0.300
50 0.451 0.372 0.285
54,74 0.444 0,366 0,281
60 0,453 0.374 0.286
65 0.380 0.397 0.303
70 0.534 0.444 0.336
75 0.645 0.546 0.404
77 0.722 0.624 0.450
for Cu, by JOSEPH et a l . /5 .8 / for Ag, and by BOSACCHI et a l . /4 .4/ fo r Au. In Sec-
t ion 2.2 i t has been pointed out that the same enhanced mass is measured in the
dHvA e f fec t and in cyclotron resonance. Mass measurements by the dHvA e f fec t have
two advantages compared to those by cyclotron resonance. F i rs t , masses deduced from
the Azbel-Kaner theory of the cyclotron resonance are only correct when the magne-
t i c f i e l d is para l le l to the crystal surface. Even small t i l t s such as those pro-
duced by surface roughness can ser iously a f fec t the data /5 .4 / . This problem does
not arise in the dHvA ef fect . Secondly, the dHvA e f fec t i t s e l f can be used to or ien t
the crystals in the f i e l d . Due to very careful temperature and amplitude measurements
the masses quoted in Tables 5.1, 5.2, and 5.3 are probably the most r e l i ab l e set
of data for the noble metals ava i lab le at the present time.
35
1,=.
m
mo
t4
Cu
v COLERIDGE et oL o KOCH et al.
' 4 ~oo} ~io}
i I 1.0~ D
m~__l
Ag
i i i
o B
(a)
9O
(b)
0.6 o HOWARD JOSEPH et ol.
0..:
I
O./,i
0.3
{lOO; ' 4
3u 4J5 60 {110}
90
Fig. 5 . 4 a - c . Angular depen- dence of the cyc lo t ron masses in Cu, Ag, and Au. The pre- sent resu l t s are ind icated by c i r c l e s . For comparison some cyc lo t ron resonance data (squares) and some dHvA data ( t r i a n g l e s ) are shown as wel l
36
1,2
m c T o
11
1.0
a LANGENBERG e ta l .
0.5 ,~ BOSACCH] et at.
O.Z
0.:
l p r i
/ I (c)
~ ~ 3'0 & ~ 1~ {100}
30 & & e 7'5 {110}
13
9o Fig. 5.4c
5.4 Fermi Ve loc i t i e s in the Noble Metals
I f ak is the distance of the two surfaces E F+~E and E F -aE at the po in t k of the
Fermi surface (Fig. 5 .3 ) , the Fermi v e l o c i t y in k is given by
v * ( k ) / v s = 6.10 -4 ks/ak (5.2)
wi th
v s : ~ks/m ~ (5.3)
k s = (12~2)1/31/a . (5.4)
k s and v s are the radius and the v e l o c i t y fo r the f ree e lec t ron sphere of energy
E F. For an energy d i f f e rence aE/E F = 6.10 -4, the grad ien t ~v* = [~E/~k[ can be re-
placed in (5.2) by the d i f f e rence quo t ien t aE/6k. The Fermi v e l o c i t i e s ca lcu la ted
in th is way are given in the Tables 5 . 7 , 5 . 8 , and 5.9 fo r Cu, Ag, and Au. ~ and e
are the usual po lar coordinates wi th the pole a t [001]. ~ = ~/2 and e = 0 are the
coordinates o f the po in t [ I 00 ] . Since the necks are c i r c u l a r to I par t in 10 -4 , the
Fermi v e l o c i t y at the neck per iphery is
* = /A /~k2~l/2/m /m* VN/Vs ~ exN" s j ~ o" cN ) " (5.5)
37
Table 5.7. Fermi ve loc i t i es v * (k ) / v s in Cu deduced from the cyclotron masses in
Table 5.1 (v s = 1.5779-108 cm s - I )
0 = 0
5
i0
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
= 0 5 i0 15 20 25 30 35 40 45
0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684 0.684
0.727 0.727 0.727 0.727 0.727 0.727 0.727 0.727 0.727 0.727
0.790 0.790 0.790 0.790 0.790 0.790 0.790 0.790 0.790 0.790
0.813 0.813 0.813 0.813 0.814 0.814 0.814 0.815 0.815 0.815
0.800 0.800 0.801 0.803 0.804 0.806 0.808 0.809 0.810 0.810
0.772 0.773 0.775 0.778 0.782 0.785 0.788 0.791 0.792 0.793
0.743 0.744 0.747 0.752 0.757 0.763 0.767 0.770 0.771 0.772
0.718 0.720 0.724 0.730 0.736 0.742 0.744 0.744 0,741 0.740
0.702 0.704 0.708 0.715 0.720 0.721 0.714 0.696 0.674 0.664
0.696 0.698 0.703 0.709 0.711 0.700 0.664 0.585 0.477 0.428
0.702 0.704 0.708 0.713 0.711 0.685 0.600 - -
0.718 0.720 0,724 0.727 0.722 0,689 0.581 - -
0.743 0.744 0.747 0.748 0.741 0,714 0.634 0.439
0.772 0.773 0.773 0.771 0.763 0.742 0.701 0.625 0.522 0.464
0.800 0.800 0,798 0.792 0.781 0.763 0.739 0.709 0,680 0.667
0.813 0.814 0.813 0.807 0.793 0.773 0.752 0,731 0.715 0,708
0.790 0.799 0.813 0.814 0.798 0.775 0.750 0.728 0.713 0.708
0.727 0.756 0.799 0.814 0,800 0.773 0.745 0.721 0.705 0.700
0.684 0.727 0.790 0.813 0.800 0,772 0.743 0.718 0.702 0.696
The value of VN/V s is (0,425 • 0.001) for Cu, (0.371 • 0.001) for Ag, and (0.638 •
0.002) for Au. The accuracy of the ve loc i t i es is better than i%, espec ia l ly at the
neck where i t is w i th in 0,3%. These data are s i g n i f i c a n t l y more accurate than the
data published by HALSE which were deduced from older mass data /4 .6 / . They had an
accuracy of 3% for Cu and Ag and of 10% for Au. The anisotropy of the Fermi ve loc i -
t ies v * (k ) / v s along some symmetry d i rect ions is shown in Fig. 5.5. The qua l i t a t i ve
behavior of the ve loc i t i es is the same in the three metals.
Fermi ve loc i t i es can also be deduced from Landau surface states /5 .9 / . I f the
Fermi surface is known, the ve loc i t i es can be determined from the posi t ion of the
38
Table 5.8. Fermi ve loc i t i es v*(k)/Vs in Ag deduced from the cyclotron masses in
Table 5.2 (v s = 1.3971.108 cm s - I )
0 = 0
5
i0
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
@ = 0 5 i0 15 20 25 30 35 40 45
0.927 0.927 0.927 0.927 0.927 0.927 0.927 0.927 0.927 0.927
0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976
1.067 1.067 1.067 1.067 1.068 1.068 1.068 1.069 1.069 1.069
1.129 1.129 1.130 1.131 1.133 1.134 1.136 1.137 1.138 1.138
1.145 1.146 1.148 1.151 1.155 1.159 1.162 1.165 1.167 1.168
1.132 1.133 1.136 1.141 1.147 1.153 1.159 1.163 1.156 1.167
1.106 I . I 08 1.112 1.118 1.125 1.131 1.136 1.139 1.141 1.141
1.081 1.082 1.087 1.092 1.097 1.100 1.098 1.092 1.087 1.084
1.063 1.064 1.068 1.071 1.071 1.061 1.039 1.007 0.975 0,962
1.056 1.058 1.061 1.061 1.051 1.021 0.957 0.855 0.740 0.683
1.063 1.064 1.067 1.064 1.046 0.992 0.872 0.638
1.081 1.082 1.085 1.083 1.061 0.997 0.850 0.529
1.106 1.108 1.112 i . I I i 1.092 1.037 0.919 0.699
1.132 1.135 1.140 1.141 1.127 1.088 1.016 0.907 0.785 0.725
1.145 1.150 1.159 1.164 1.154 1.127 1.084 1.031 0.982 0.962
1.129 1.138 1.158 1.170 1.165 1o144 1.113 1.080 1.053 1.043
1.067 1.087 1.129 1.158 1.161 1.143 1.116 1.089 1.068 1.061
0.976 1.014 1.087 1.138 1.151 1.136 1.110 !.084 1.065 1.059
0.927 0.976 1.067 1.129 1.145 1.132 1.106 1.081 1.063 1.056
resonance in the surface impedance. Whereas in the dHvA e f fec t the measured masses
are averages of the reciprocal Fermi ve loc i t y averaged over the f u l l o rb i t of an ex-
tremal cross sect ion, the ve loc i t i es are averaged only over a s t r i p of 5 - i 0 ~ in the
Landau surface states. This is an advantage for the determination of the ve loc i t i es .
On the other hand the s t r ingen t requirements concerning the pu r i t y and surface con-
d i t i ons of the sample l i m i t th is method to a small number of metals. Some Fermi ve-
l o c i t i e s have been measured for Cu and Ag /5 .9 ,10 / . The measured values are shown
in Fig. 5.5. For Cu, the surface preparation is well establ ished. Here the agreement
with our data is sa t i s fac to ry . In Ag the surface preparation creates problems /5.10/
39
Table 5.9. Fermi ve loc i t i es v * (k ) / v s in Au deduced from the cyclotron masses in
Table 5.3 (v s = 1.3985.108 cm s - I )
0 = 0
5
10
15
= 0
0.736
0.891
1.014
1.028
20 0.997
25 O.952
30 0.905
35 0.866
40 0.839
45 0.830
50 O.839
55 0.866
60 0.905
65 0.952
70 0.997
75 1.028
80 1.014
85 0.891
90 O.736
5 10 15 20 25 30 35 40 45
0.736 0.736 0.736 0.736 0,736 0.736 0.736 0.736 0.736
0.891 0.891 0.891 0.891 0.891 0.891 0.891 0 ,891 0,891
1.014 1.015 1,015 1.017 1.018 1.019 1.020 1.021 1.021
1.029 1.031 1.035 1.040 1.045 1.050 1,054 1.057 1.058
0.999 1.004 1.013 1.023 1.034 1.045 1.054 1.061 1.063
0.955 0.963 0.976 0.993 1.012 1.030 1.046 1.056 1,059
0.909 0.920 0.938 0.961 0.987 1.012 1.034 1.048 1.053
0.870 0.884 0.906 0.934 0.964 0.993 1.016 1.030 1.035
0.845 0,860 0.885 0.915 0.946 0.970 0.978 0.974 0.969
0.836 0.853 0.879 0.910 0.934 0.931 0.872 0,757 0.690
0.846 0,865 0.893 0.922 0,934 0.878 0.640
0.873 0.894 0.924 0.953 0.956 0.868
0,913 0.936 0.967 0.994 0.998 0.935 0.708
0.960 0.983 1.012 1.033 1.035 1.002 0.923 0.806 0.737
1.006 1.026 1.047 1,055 1.045 1.017 0.979 0.942 0.925
1.036 1.054 1.062 1.053 1.026 0.990 0.954 0,927 0,916
1.031 1.055 1.055 1.031 0.993 0.950 0.911 0.884 0.874
0.961 1.031 1.037 1.007 0.963 0.917 0.878 0.851 0.842
0.891 1.014 1.028 0.997 0.952 0.905 0.866 0.839 0.830
and discrepancies of up to 13% occur wi th our data. This value is much greater than
the errors quoted. In Au i t has not yet been possible to observe magnetic surface
states.
40
1.0 v * / v s
0,9
0.81
0.7
0.6
0.5
0.4
(a ) 03 , [101]
1.3
1.2
1.1
1,0 v*/v s
0.9
0.8
0.7
Neck 0.6
0.5
0.4 I
[101]
v s: 1.57785• cm/s
LENGELER et el, . . . . LEE (bend velocities} . . . . DOEZEMA et uL,
J i r I [ I I J
[O01]
' r I
/ t
i//I
LENGELER et ol. --,-- DEIMEL et o[.,
I I r I I [ I I
[001]
1.3
12
1.1 v " /v s
1.0
0.9
0.8
0.7
( c ) 0.8 [101]
A u Vs= 139845•
i i I I L I I I i
[001]
Neck
i I
[111]
I I
[111]
I
i I
[101]
Neck
/
/ I I
[101]
(b)
I I I I I
[111] [lOl]
F i g . 5 . 5 a - c . Fermi v e l o c i t i e s i n Cu, Ag, and Au a long some h igh symmetry d i r e c - t i o n s . Our d a t a a re g i v e n by the s o l i d l i n e s . Data deduced f rom m a g n e t i c s u r f a c e s t a t e s a re d o t - d a s h e d . Band v e l o c i t i e s by LEE / 5 . 1 4 / a re dashed
41
5.5 Electron-Phonon Coupling Constant k(k) in Cu
Cyclotron masses and Fermi ve loc i t i es which have been determined from band struc-
ture calculat ions are not renormalized by electron-phonon in te rac t ion . A comparison
with dHvA data can therefore give values of k(k) and <k>.
i + k(k) : v (k ) Iv * (k ) (5.6)
1 + <k> = m~/m c . (5.7)
As explained in Section 4, there are no potent ia ls avai lable which al low one to
calculate the Fermi surface and the Fermi ve loc i t i es with the same accuracy with
which they can be measured. Calculat ions of the ve loc i t i es have been carr ied out
by LEWIS et a l . /5 .11 / , O'SULLIVAN et a l . /5 .12 / , and JANAK et a l . /5 .13/ . The most
accurate and detai led calculat ions have been made for Cu by LEE /5.14/ . Table 5.10
gives some of the calculated masses and the corresponding constants l+<k> obtained
Table 5.10. Measured and calculated /5.14/ cyclotron masses in Cu and corresponding
coupling constants l+<k> fo r some orb i t s in the plane {110}
* 1 + <k> Cu mc/m ~ mc/m o /5.14/ Table 5.6
B<IO0> 1.238 1.341 1.083
B8 1.229 1.323 1.076
B16 1.223 1.308 1.070
B24 1.296 1.420 1.096
B<111> 1.277 1.375 1.077
E<lO0> 1.183 1.306 1.104
N30 0.827 0,952 1.151
N40 0.448 0.529 1.181
N50 0.380 0.451 1.187
N<III> 0.374 0.444 1.187
N60 0,383 0,453 1.183
N70 0.452 0.534 1.181
D<110> 1.126 1.262 1.121
by comparison with the present dHvA cyclotron masses. The band ve loc i t i es calculated
by LEE and the electron-phonon coupling constants l + k ( k ) deduced by comparison with
the dHvA data are given in the Figs. 5.5 and 5.6. k(k) is s t rongly anisotropic and
has maxima at the necks and at the be l l y points <i00>. These are the regions where
42
1.1
1+ Mk)
1.2
1.o [lOl
I I I I I I I I I
[ool]
1,3
L I
[111] i i i
[lOl]
Fig. 5.6. Local values of the electron-phonon coupling constants l + ~ ( k ) in Cu calculated from the Fermi ve loc i t i es v*(k) and the band ve loc i t i es v(k) of Fig. 5.5
the Fermi surface deviates the most from the free electron sphere. To our knowledge
there are no experimental values of the k-dependence of X, but there are mean values
of X averaged over the whole Fermi surface as shown in Table 5.11. The mean value
obtained from the present x(k) is 0.11, which is somewhat lower than the values from
the table. In view of the l im i ted accuracy of the band ve loc i t i es on which the va l -
ues x(k) are based, th is discrepancy is not s i gn i f i can t .
Table 5.11. Values of the electron-phonon coupling ~ averaged over the Fermi sur-
face of Cu
Experiment ~ Theory
0.16 /5.15/
0.13 • 0.03 /5.16/
0.11 th is work
0.15 /5.17/
0.15 _+ 0.02 /5.18/
0.12 +_ 0.02 /5.19/
O. 14 /5.20/
5.6 Coef f i c ien t of the Electronic Speci f ic Heat for Cu, Ag, and Au
At low temperatures the e lec t ron ic cont r ibut ion Cel to the spec i f i c heat increases
l i n e a r l y wi th T. The p ropor t i ona l i t y constant y* is given by
�9 i ~ (5.8) = -~ ~2k D*(EF) .
43
The density of states D*(EF) at the Fermi level is enhanced by electron-phonon in-
teract ion. I f 6V k is the volume between the two surfaces E F • aE characterized by +
the coef f ic ients C~m n
D*(EF) = (27) -3 ~Vk/aE (5.9)
and thus
Y*/Ys = 104 aVk/18Vs (5.10)
with
Ys = k~moks/3~2 (5.11)
k s = (12~2) 1/3 I /a (5.12)
V s = 4~k~/3 . (5.13)
k s 5.12 are given the coef f ic ients y /Ys calculated from the present dHvA data. They
agree very well with the values obtained by MARTIN from speci f ic heat measurements
/5.21/. The close agreement supports the r e l i a b i l i t y of the present cyclotron mass
measurements. Ear l ie r calculat ions of u by HALSE /4 .6 / and by BOSACCHI et a l .
/4.4, 5.22/ are given for comparison in Table 5.12.
and V s are the radius and the volume of the free electron sphere for E F, In Table
Table 5.12. Coeff ic ients y /Ys
units Ys for Cu, Ag, and Au. Ys
Ag, and Au
of the electronic speci f ic heat in free electron
is 0.49954, 0.63718, and 0.63592 mJ/mole. K 2 for Cu,
Y*/Ys
Cu 1.382 • 0.010
1,383 • 0,002
1.397 • 0.016
1.400
Ag 1.008 • 0.007
1.004 • 0.002
1.021
Au 1,074 • 0.007
1.083 • 0.002
1.077
1.093
th is work
MARTIN /5.21/
BOSACCHI et a l . /5.22/
HALSE /4.6/
th is work
MARTIN /5.21/
HALSE /4 .6 /
th is work
MARTIN /5.21/
BOSACCHI et a l . /4 .4 /
HALSE /4.6/
44
6. Dingle Temperatures and Scattering Rates of Conduction Electrons in the Noble Metals
6.1 Dingle Temperatures and the Li fet ime of Electron States
The conduction electrons are scattered by impuri t ies and i n t r i n s i c defects (vacan-
cies, i n t e r s t i t i a l s , d is locat ions, e tc . ) . The resul t ing reduced l i f e t ime of the
electrons in a given state causes a broadening of the Landau leve ls , which is des-
cribed by a Dingle temperature X*. The product m*X* can be determined from the f i e l d c dependence of the dHvA amplitudes. I f m* is taken from the temperature dependence c of the dHvA amplitudes, then X* can be deduced from
+X* X (6.1) m c = m c ,
since th is product is independent of the electron-phonon interact ion. The Dingle
temperature X * is an average of the local scattering rates of the conduction elec-
trons for a given extremal cross section
X* = (~12~kB) <l lT*(k)> . (6.2)
The individual elements dk of the extremal cross section are weighted in (6.2) by
the time spent in them by the k-vector /6 .1/ (Fig. 2.3).
X* = (~ /4~2kB) f d t I / ~ * [ k ( t ) ] (6.3)
o r
X m c = (h2/4~2kB
= (h2/4~2kB
] dk/v*~* (k ) (6 .4)
f d~ k~[(v* 'k• -1 (6.5)
The weighting factor k~/(v*.k• of the scattering rate depends on the geometry of
the extremal cross section and on the associated Fermi ve loc i t ies . Just l i ke X*m + c the product v + + " ]n (6.5) is independent of the electron-phonon interact ion
T'v* = ~v . (6.6)
The enhancement of the electron l i f e t ime T*(k) by electron-phonon interact ion
z+(k) : T(k) [ l + ~ ( k ) ] (6.7)
is a consequence of the scattering of the conduction electrons by v i r tua l phonons
which increase the iner t ia of the electrons and thereby reduce the scattering pro-
bab i l i t y .
45
According to Bohr-Onsager quantizat ion, the interference of the electron wave
with i t s e l f on the cyclotron o rb i t brings about the red is t r ibu t ion of the e lectronic
states on Landau cyl inders in a magnetic f i e l d . Every scatter ing event which de-
stroys the phase coherence of the wave produces a reduction of the dHvA amplitude.
This reduction is par t i cu la r l y large for scatter ing angles e
e = ~/n = ~H/F (6.8)
where the phase of the wave is shi f ted by ~. Since H/F is t yp ica l l y 10 -3 to 10 -4 in
the noble metals, the phase coherence is destroyed for scatter ing angles even smal-
l e r than 0. I ~ For that reason, T(k) can be considered as the real l i fe t ime of an
electron in state k. Here i t should be emphasized again that the scattering of the
electrons at real phonons seems not to contr ibute to the reduction of T(k) (see
Sec. 2.2). I f P(k,k ' ) is the t rans i t ion rate from a state k to a state k' then # 4
Z/~(k) = ( 2 ~ ) - 3 J d 3 k ' P(k,k ' ) . (6.9)
According to Fermi's golden rule /6 .2/ the t rans i t ion rate is
P(k,k ' ) = (2~Cd/~) ITkk, I 2 6 [ E ( k ) - E ( k ' ) ] (6.10)
where c d is the concentration of scattering centers in the l a t t i ce in the d i lu te
l im i t . Thus
Z/~(k) = (2~)-3(2~Cd/~) f dSk, ITkk,12/hv(k ') .
FS
(6.11)
The surface integral extends over the Fermi surface. Using the optical theorem,
which expresses the conservation of par t ic les ,
( 2 ~ ) - 3 f d S k , JTkk, 12/hv(k ') = -Im{Tkk/~} (6.12)
(6.11) can be wr i t ten
1/T(k) = (2Cd/~) Im{Tkk} . (6.13)
The scatter ing of the conduction electrons by magnetic impurit ies w i l l not be con-
sidered in th is a r t i c le . This has been described in some detai l by SHIBA /6 .3 / . The
scattering of the electrons by nonmagnetic impurit ies in the noble metals is f i e l d
independent. This is again a consequence of the high phase 2~F/H of the dHvA osc i l -
lat ions. In this case, the radius of a cyclotron orb i t is appreciably larger than
the l inear dimensions of the scatter ing potent ia ls. Hence the curvature of the o rb i t
over the range of the potential can be neglected. In the quantum l im i t , which is
not achieved here, the cyclotron orb i t has atomic dimensions. Then this assumption
does not hold any longer and ~(k) w i l l be f i e l d dependent.
4G
The l i f e t ime T(k) which enters the dHvA ef fect d i f fe rs generally from the trans-
port relaxat ion time Ttr(k ) which enters in transport coef f ic ients such as the elec-
t r i ca l r e s i s t i v i t y . Small angle scattering processes do not contribute e f f ec t i ve l y
to the resistance, because they scatter the electrons only s l i g h t l y from the d r i f t
d i rect ion. This is taken into account by a weighting factor which can be often ap-
proximated by ( l - c o s 0) where 0 is the scattering angle. In th is case, the l i f e t ime
becomes
= (2~ ) -31d3 ~' P (k ,k ' ) (1 -cos e) . (6.14) 1/Ttr(~)
Electrons are scattered p re fe ren t ia l l y at small angles by the long-range stra in
f i e l d of dis locat ions. Hence, isolated dislocat ions are hardly v i s ib le in the elec-
t r i ca l r e s i s t i v i t y . On the other hand, they destroy the phase coherence of the elec-
trons on a cyclotron o rb i t /6 .4 ,5 / . The dHvA ef fect is therefore a sensi t ive probe
for dislocat ions in single crystals (Sec.3.4). There is another essential di f ference
between the electron l i f e t ime T(k) and the transport relaxat ion time ~tr(k ). ~tr is
introduced as an approximate solution of the Boltzmann equation. Only for the t r i v i -
al case of isotropic scattering and a spherical Fermi surface is Ttr an exact solu-
t ion of the Boltzmann equation. In general, the solution of th is equation produces
great d i f f i c u l t i e s . In contrast to th is , the l i f e t ime T(k) which enters the dHvA
ef fect is much more easi ly accessible from a theoret ical standpoint. Furthermore,
the transport coef f ic ients contain averages of ~tr(k ) over the whole Fermi surface,
whereas in the dHvA ef fec t the average is only over an extrema] cross section.
6.2 Anisotropy of the Scattering Rates in the Noble Metals
Dingle temperature determinations for d i lu te noble metal al loys and for noble metals
containing i n t r i ns i c defects (d is locat ions, vacancies) have been reported by a num-
ber of authors. These include the invest igat ions of POULSEN et a l . on C_uuAu, CuGe,
and C__~uNi /6 .6 / ; TEMPLETON et a l . on C__uuAl and C uNi /6 .7 / ; WAMPLER et al . on Cull
/6 .8 / ; BROWN et a l . on AgAu, AgCd, AgGe, and AgSn /6 .9 / ; LOWNDES et a l . on AuAg,
AuCu, AuZn /6 .1 / ; and CHUNG et a l . on AuCo /6.10/ . Dislocations in Cu have been
studied by COLERIDGE et al . /6 .4 / and CHANG et a l . /6 .5 / . Vacancies in Au have been
investigated by LENGELER /6.11/ and CHANG et a l . /6.12/. An example of the determi-
nation of Dingle temperatures from the f i e l d dependence of the dHvA amplitude is
shown in Fig. 6.1. Although the error in the determination of the slope is t yp i ca l l y
0.5%, the Dingle temperatures per at % defects are not nearly so accurately known.
This is par t ly due to inaccuracies in the determination of the concentration of de-
fects. The main contr ibut ion to the errors is due to the scattering of the electrons
at defects (mainly dis locat ions) which are also present in the sample, often in un-
known concentrations. A p o s s i b i l i t y to separate the contributions of the d i f fe ren t
defects is to measure the dependence of the Dingle temperatures on the defect con-
47
0.03
CI ,,,:(
0.04 I
0.05 i
Au Vacancies
N<111> c v: 255 ppm
~ D X% (1.114• O.O03)K <110>
~ O051K
I I I
001 0.02 I/H[kG -I] 0.03
Fig. 6.1. Dingle plots for four extremal cross sections in gold containing 255 ppm
of la t t ice vacancies. The slope of the lines is -bm~X*/m o. The lower abscissa holds
for the bellies and dogsbone and the upper for the neck orbit
centration. In our LuH and Au vacancy measurements /6 .8 ,11/ , we have measured the
Dingle temperatures for I I d i f fe ren t hydrogen and vacancy concentrations. Fig. 6.2
shows some Dingle temperatures for vacancies in gold as a function of the vacancy
concentration. The vacancies present in thermal equi l ibr ium at high temperatures
(600- i000 ~ are quenched into the samples by quick cooling /6.13/. In the quench-
ing process, dislocat ions are created which give r ise to addit ional scattering and
which manifest themselves as an intercept in Fig. 6.2 at c v = O. Di lute al loys can
show such intercepts as we l l , as shown by BROWN et a l . in A__ggAu and A gSn /6 .9 / . Here
also the intercept is mainly due to dislocat ions created during the crystal pul l ing
process. Since in many invest igat ions the Dingle temperatures have been measured
only for one concentration of scattering centers, the Dingle temperatures per at %
are rather uncertain. Table 6.1 summarizes the Dingle temperatures for typical or-
48
1.5
1.0
0~5
o 6
i I I i I
Au Vacancies o X~[K]
[] N < 1 1 ~ ~ ~ /
I I I I 50 100 150 200 Cv[ppm] 250
Fig. 6.2. Concentration dependence of the Dingle temperatures in the system Au vacancy. The l i n e a r l y increasing part of X * is due to the scat ter ing of the c~duc- t ion electrons by the vacancies and the l a t t i c e d i s to r t i on surrounding them. The in tercept at c v = 0 is due to the scat ter ing at d is locat ions which are created in the Au crysta l during the quenching process
Table 6.1. Dingle temperatures in K/at% for some nonmagnetic scatter ing centers in
Cu, Ag, and Au. The Dingle temperatures are quoted only for the s ta t ionary cross
sections in the plane { I i 0 } . The residual r e s i s t i v i t y p, the chemical valence d i f -
ference AZ, and the charge AZ B to be screened are quoted as well
X* C uAu C uNi C uGe C_uuH AgAu AgGe A_ggSn AuAg A uCu A uZn A_uu vacancy
K/at% /6 .6 / /6 .6 / /6 .6 / /6 .8 / /6 .9 / /6 .9/ /6 .9/ /6.1/ /6 .1 / /6 .1 / /6 .11/
9.8 9.2 35.6 B<IO0>
BTP
B<111>
D<110>
R<IO0>
N<111>
P [ijQcm/at%]
AZ
AZ B
13.2 26.6 109 37.9 8.7 300 212 38.3
13.9 28.8 119 48.4 8.7 273 210 38.0
15.1 30.9 114 65 8.7 224 193 9.3 10.1 38.7 38.7
12.4 26.6 161 77.3 7.0 332 272 7.6 8.6 36,3 37.0
10.2 25.0 119 89.8 7.0 203 212 9.0 9.8 39.8 38.6
7.9 14.9 188 112.6 3.6 351 318 2.8 4.5 24,5 33.8
0.55 1.11 3.79 1.50 0.38 5.5 4.3 0.36 0.45 0.95 1.69
0 - I 3 I 0 3 3 0 0 1 - I
-0.32 -0.94 2.81 0.89 0 3 2.76 0 0.21 1.1 -0.6
49
bi ts in a few Cu, Ag, and Au systems. A l l the defects are subs t i tu t iona l except
fo r hydrogen which occupies octahedral in te rs t i ces /6 .8 / . The d i f f e ren t defect s i tes
in an f . c . c , l a t t i c e are shown in Fig. 6.3. The scat ter ing strength is nearly pro-
_ _ %-----~
substitutionol
defect
A i
�9 o D
J. ; . . . . - , - . . . . T 7
defect on octahedrol interM ice
Fi 9. 6.3. Defects on a l a t t i c e s i te and on an octahedral i n te r s t i ce in an f . c . c . lat t ice
port ional to the residual resistance p of the defects. On the other hand, the an-
isotropy of the scat ter ing depends essent ia l l y on the s i te occupied by the defects,
Subst i tu t ional defects wi th a small valence di f ference compared to the host l a t t i c e ,
IAZI = O. l , scat ter more strongly the be l l y electrons than the neck electrons. Hy-
drogen occupying octahedral i n te rs t i ces scatters the neck electrons more strongly.
Local values of the scat ter ing rates I / z * (k ) can be obtained by deconvolution of
Dingle temperatures according to (6,5). A symmetrized Fourier series expansion with
f i t t i n g coef f i c ien ts Tzm n has been used to parameterize the scat ter ing rates
1/T*(k) = TO00 + T l l 0 Z cos ~k2 x cos ~k2 Y
+ T200 ~ cos ak x + T21 t ~ cos ak x cos ~-ky cos 2kz + . . .
(6.15)
Local scat ter ing rates for three character is t icsystems from Table 6.1 are shown
in Fig. 6.4 for some high symmetry d i rec t ions. Ag in Au /6 .1 / is a weak scatter ing
center. The chemical valence di f ference is zero and the l a t t i c e d i s to r t i on produced
by the defect is neg l ig ib le . Ag in Au is therefore a short-range scatter ing center
on a subs t i tu t iona l l a t t i c e s i te . I t scatters p re fe ren t i a l l y the be l l y electrons,
The absolute values of the scatter ing rates are rather small. Hydrogen in copper
/6 .8 / const i tu tes a short-range potent ia l on an octahedral i n te rs t i ce . I t scatters
p re fe ren t i a l l y the neck electrons. The overal l scat ter ing rates are larger than for
a subs t i tu t iona l defect with I~Z I = I ( l i ke Zn in Au or Ni in Cu). F i na l l y , vacan-
cies in gold are characterized by an especia l ly strong d i s to r t i on f i e l d . The atoms
surrounding the vacancy relax in to the empty l a t t i c e s i te so that the e f fec t i ve
volume of the vacancy is not i but only 0.6 atomic volumes /6.11,14/ . For vacancies
50
9
8
7
5
5
Z~
3
40 20 [101]
i l i I I i I I i i , I
~_k) [1013s_ I at~ Cu H Neck
I I
I I I I I I I T I I
0 20 40 60 80 [001] [111] [110]
Fig. 6.4. Local scat ter ing rates of the conduction electrons in A uAg, C uH, and Au vacancy. Ag is a we l l - l oca l i zed defect on a l a t t i c e s i te in Au and scatters the neck electrons only weakly. H is a we l l - l oca l i zed defect on an octahedral i n te r - s t ice in Cu and scatters most the neck electrons. Vacancies in Au create a strong d i s to r t i on around the defect which causes the scatter ing anisotropy to be only weak
in gold, the scat ter ing anisotropy is only weak. A__uuAg and C uH are two extreme cases
of pronounced scat ter ing anisotropy. The anisotropy of a l l other systems in the
Table 6.1 is between these two extremes. Obviously the posi t ion and the strength
of the defect inf luence the absolute values and the anisotropy of the scat ter ing.
These re la t ions can be described quan t i t a t i ve l y by a phase s h i f t analysis of the
Bloch waves of the host l a t t i c e at the defects.
6.3 Phase Sh i f t Analysis of the Scattering of Conduction Electrons at Defects
in the Noble Metals
MORGAN has given a formulat ion of the scat ter ing of Bloch waves at a l a t t i c e defect
/6 .15/ . In a series of papers, COLERIDGE, HOLZWARTH, and LEE have extended th is
theory to the descr ipt ion of scat ter ing rates and Dingle temperatures in d i l u te
noble metal a l loys /6.16-22/ . The resul ts of Section 6.2 can be explained quant i -
t a t i v e l y w i th in the framework of th is theory. In the fo l low ing , i t w i l l be assumed
that the host l a t t i c e and the defects can be treated in the mu f f i n - t i n approxima-
t ion (spher ica l ly symmetric potent ia l inside a mu f f i n - t i n sphere of radius R s and
constant potent ia l outside).
51
At f i r s t , the scatter ing of a plane wave at a single muf f in - t in potential in
vacuum wi l l betreated. For a spher ical ly symmetric potent ia l , the solut ion of the
Schr~dinger equation can be separated into par t ia l waves and each part ia l wave fac-
tored into a radial and spherical harmonic contr ibut ion. The asymptotic solut ion of
the Schr~dinger equation is wr i t ten
~ i ~ a L ( k ) [ j ~ ( K r ) + i sin n~ exp(in~)h~(<r)] YL(~) (6.16) L
where L ~ ~,m, E = ~2K2/2mo is the energy of the wave measured from the muf f in - t in
zero, j~ and h~ are spherical Bessel and Hankel funct ions, and YL(~) are spherical
harmonics. Due to the conservation of angular momentum in a scatter ing process at
a spher ical ly symmetric potent ia l , the waves with d i f fe ren t ~ do not mix. The scat-
ter ing potential V(r) scatters the incoming wave J~YL" An outgoing spherical wave
hJL or iginates from the scatter ing center with amplitude i sin n~ exp(in~). The
phase sh i f ts n~ depend on the potential through an element of the t-matr ix
t~ m sin nc exp(in~)
Rs (6.17)
= - (2moK/~2) f j g (Kr )V( r )R~(~ , r ) r 2 dr
o
where R~(K,r) is the solut ion of the radial Schr~dinger equation inside the muff in-
t in sphere.
In an ideal crystal l a t t i c e , the potent ials of the l a t t i ce atoms are again approx-
imated by muf f in - t in potent ia ls. In close-packed la t t i ces with high electron concen-
t rat ions l i ke the noble metals, th is assumption is rather well j us t i f i ed . The ampli-
tudes a L and b L of the wave aLJ~(<l ~-RN I) entering the N-th cel l at ~N and of the
wave bLh~(Kl ~-RN I) leaving i t are related again by
h exp(" h i t~ (6.18) bL/a L i sin q~ = l q ~ ) =
where ~ and m are no longer good quantum numbers. The nonspherical symmetry of the
crystal potent ial removes pa r t i a l l y the m degeneracy. In a cubic crys ta l , the wave
functions are c lass i f ied according to the angular momenta ~ and the i r reducib le re-
presentations r of the cubic point group O h belonging to the d i f f e ren t ~. The f i r s t
four are ?1 for ~ = O, r15 for ~ = 1 and 712 and r25, for ~ = 2. In the fo l lowing,
L ~ c, ? denotes this new set of quantum numbers. Whereas for the scattering at an
isolated potential the amplitude a L is a free parameter, i t is bu i l t up in the l a t -
t ice by superposition of the scattered waves which emerge from al l other l a t t i c e
s i tes.
a L : i ~ B L L , b L , . (6.19) L'
52
The structure~ factors BLL, depend on k, <, and on the l a t t i c e but not on the phase
sh i f t s n~. I t should be noted that waves with d i f f e ren t L mix because the l a t t i c e
is not spher ica l l y symmetric as seen from a mu f f i n - t i n ce l l . Combining (6.18) and
(6.19) gives the KKR-band st ructure equations
~ ( B L L , ( k , < ) t ~, + 6LL,)a L, = 0 . (6.20) L'
Nont r iv ia l solut ions ex is t only i f
h + i = 0 . (6.21) det IBLL,t~, ~LL'
The structure matrices BLL, are known for the noble metals. The Fermi energy E F and h the phase sh i f t s n~(EF) are determined in such a way that the rad i i k(E~)h f i t op t i -
the dHvA measurements. In th is way, the set of parameters (EF,n~)'in~ Table 4.2 mally
has been determined. The amplitudes aL(k) of the L-th par t ia l wave can now be cal -
culated from (6.20). laL(k) i 2 describes the anisotropy of the wave character of the
conduction electrons in a l a t t i c e ce l l .
In a t h i r d step a defect is imbedded in an otherwise ideal l a t t i c e . The defect
potent ia l is also described in the mu f f i n - t i n approximation. In the fo l lowing sub-
s t i t u t i o n a l and i n t e r s t i t i a l defects are treated separately.
6.3.1 Subst i tu t ional Defect
Consider a s i tua t ion where a defect replaces a l a t t i c e atom in the N'- th ce l l .
According to MORGAN /6 .15/ , the wave j ~ ( K i r - R N , i ) coming into th is ce l l no longer
has the amplitude a L. Because the wave bLh~(Ki~-RN, i) emerging from N' is no longer
a Bloch state, i t creates by scatter ing at the surrounding l a t t i c e an addi t ional
cont r ibu t ion to the incoming wave. This e f fec t is cal led backscattering and con-
s t i t u tes the essential d i f ference between the scat ter ing of plane waves at a defect
in vacuum and the scat ter ing of Bloch waves at a defect in a l a t t i c e . I f the crys-
tal can be treated n o n r e l a t i v i s t i c a l l y and i f only phase sh i f t s wi th ~ ~ 2 have to
be considered, the backscattering can be taken into account by renormalizing the
amplitude a L by a complex factor A L. There is a close re la t ionsh ip between A L and
the st ructure matrices BLL,. Both describe how the emerging spherical waves super-
pose to construct an incoming wave, BLL, in the ideal l a t t i c e and A L for the defect
ce l l . A L depends on the defect and on the host l a t t i c e
h " (s in h i - . (6.22) AL(E F) = sin 2 n~ exp(-IAn~) n~ sin n~ - • sin An~) 1
Here
= h 6.23) An~ q~ - n~
53
is the d i f ference of the phase shi f ts for the defect and for the host atoms at the
Fermi energy E F. The Br i l l ou in zone integral
Xk = (2~)-3VEz I d 3 k (BLL + Z/t~) - I (6.24)
BZ
expresses the close re la t ionship between BLL, and A L. In the free electron gas,
where there is no backscattering (BLL, = 0 , A L = i ) , th is integral is
h . h xLFE = t~h = sin nz exp(Inz) . (6.25)
The complex integrals • depend only on the propert ies of the host l a t t i ce and the
posit ion of the defect, and not on the phase sh i f ts n~ of the defect. The values of
• have been calculated for Cu, Ag, and Au by LEE, HOLZWARTH and COLERIDGE /6.21/ .
An equivalent representation of A L is
AL(E F) = sin 2 n~ e x p ( - i ~ n ~ ) [ ( ~ - X L) sin Anal -1
(6.26) = IALI exp(io L)
with
IAL(EF)I = sin 2 n~ (sin A n ~ ) - I [ ( ~ - R e { x L } ) 2 + (Im{XL})2] -1/2 (6.27)
e L = arctg[Im{XL}/(r z - Re{XL})] -Anc (6.28)
: ( c o t g - cotg ( 6 . 2 9 )
The factor A L renormalizes the amplitude and the phase of the L-th wave entering the
defect ce l l . Thus the total phase sh i f t ~L produced by the defect contains two con-
t r ibu t ions . There is f i r s t the usual phase sh i f t An~ between the outgoing and the
incoming waves. In addi t ion, the incoming wave is already phase shi f ted by an amount
e L compared to a wave entering an ideal l a t t i ce ce l l . The total phase sh i f t eL is
cal led Friedel phase sh i f t
~L = Anz + o L (6.30)
= arctg[Im{XL}/(r z - Re{XL} )] .
The conduction electrons create by scatter ing a charge t ransfer F which screens the
defect-induced charge AZ B and sa t is f ies the Friedel sum rule
= ~2nL@L/~ = AZ B (6.31) L
where n L is the degeneracy of the representation L. [n(O,r l ) = I , n (1 , r l5 ) = 3 ,
54
n(2,r12 ) = 2 , and n(2,r25, ) = 3]. The scattering rate i / z (k ) is related to the diagonal elements Tkk of the t-ma-
t r i x by (6.13). The analogue expression to (6.17) is /6.15/
Tkk(EF) = -(h2/2moK)~IaL(k,EF)I2AL(EF ) sin An~ exp(iAn~) . (6.32) - - L
L Each part ial wave contribution in (6.32) is the product of an anisotropy factor t k
and a scattering parameter S L. t~ depends only on the host la t t i ce and can be cal-
culated from the solution of the KKR equation (6.20),
k (~2/2moK) laL(k)[2 sin 2 h t k = ~ . (6.33)
The scattering parameter S L
S L = A L sin An~ exp(iAn~)(sin n~) -2 = (~ - XL )-I (6.34)
is k-independent, describes the properties of the defect, and depends on the host
la t t i ce through n~ and XL. According to (6.13) the scattering rate is
1/T(k) = ( 2 C d / ~ ) ~ t ~ Im{S L} (6.35) L
Im{S L} = Im{XL} [ ( ~ - R e { • 2 + (Im(XL})2] - I . (6.36)
Equations (6.33,35,36) are the essential result of the phase shift analysis for the
scattering of Bloch waves at a defect. The scattering anisotropy of the individual
partial waves L depends on the host lattice and is given by the factor t~. Im{S L}
is the weight factor with which the rates of the different L are superposed to form
the total scattering rate. I t depends on the defect (n~) and on the host latt ice h (n~,• Using (6.4) and (6.35), the Dingle temperatures can be written
Xmc/m ~ = (eo /kB)(ao /a)2Cd~WL Im{S L} . (6.37) k
Here E ~ is 1Rydberg, a o is the Bohr radius, and a the la t t i ce parameter.
W L = (a/~)2fdk t~/~v• (6.38)
is the average of the anisotropy factors t~ over the appropriate extremal cross
section. The coefficients W L have been calculated for stationary extremal cross
sections and for substitutional defects in Cu, Ag, and Au /6.17,20/. They are listed
in Table 6.2 for some orbits in Cu. I t can be seen that the neck electrons have main-
ly p-character at a latt ice site with no s-character and a l i t t l e d-admixture. The
B<IO0> electrons have mainly p- and d-character.
In the present description, the lattice and the defect have been treated nonrela-
t i v is t i ca l l y . For elements with a high Z as host (e.g., Au) or as defect, the spin-
55
Table 6.2. B r i l l ou in zone integrals x L, E L and averages W L, WL of the anisotropy
factors t , t for some extremal cross sections in Cu for subst i tut ional defects (S)
and for defects on octahedral in ters t ices (0). t~ and ~Ltk give the contr ibut ion of
the wave character L (st I , P r 1 5 , d r 1 2 , d r 2 5 , ) in the Bloch wave with wave vector k
at the posit ion of the defect
Re{x L }
Re{# L }
Im{x L }
Im{# L }
WE' WL
B<IO0>
B<111>
D<110>
R<IO0>
N<III>
s i te OF 1 1F15 2F12 2r25,
S 0,07607 0.12412 -0.13233 -0.12862
0 -0.51768 -0.92621 -5.55897 -5.61006
S 0.00318 0.01392 0.01502 0.01564
0 0.74896 0.86023 0.45958 1.36575
S 0.01317 0.12156 0.09939 0.14539
0 1.42530 12.59223 5.00423 3,61711
S 0.01144 0.10998 0.10695 0.18667
0 2.39290 6.63995 1.90415 11.51961
S 0.00817 0.14234 0.08301 0.11562
0 2.45904 7.84367 2.47674 14.68939
S 0.00799 0.13371 0.09156 0.14747
0 2.98703 4.80336 1.17912 17.43271
S 0 0.07396 0.02020 0.01872
0 1.59115 0.66813 0.09496 12.10754
-o rb i t coupling must be taken into account. The r e l a t i v i s t i c formulation of the
phase s h i f t analysis has been given by HOLZWARTH and LEE /6.20/. Since i t is s imi la r
to the non re la t i v i s t i c one, i t w i l l not be discussed fur ther here.
n~,~ • and W L have been determined for the noble The host l a t t i ce parameters
metals from nonre la t i v i s t i c and from r e l a t i v i s t i c band structure calculat ions. Ex-
perimental Dingle temperatures can be parametrized according to (6.36-38). By means
of nonlinear least squares f i t t i n g procedures ( for instance the program VAO5AD from
the Harwell Subroutine L ibrary) , the defect phase sh i f ts n~ are chosen in such a
way that the calculated Dingle temperatures agree with the measured ones. The Frie-
del phases @L' the backscattering phase sh i f ts o L, and the Friedel sum F can be
56
Table 6.3. Phase s h i f t analysis for three typical d i lu te al loys in Table 6.1. Ni
in Cu scatters most strongly the d-waves, Ag in Au the s-wave, and Ge in Cu the
p-wave. Backscattering creates a substantial contr ibut ion in the Friedel phases.
The Friedel sum F and the charge AZ B agree quite well for CuNi and for AuAg; not
so for C uGe where the l a t t i c e d is to r t ion is rather high
C_uuNi
Au__._Ag
CuGe
AZ L ~L n~ Anz e L IALI F AZ B
-1 Or I 0.00 0.076 0 0 1 -0.96 -0.94
1F15 -0.061 0.058 -0.072 0.01 1.02
2F12 -0.26 -0.31 -0.19 -0.07 1.24
2r25, -0.27 -0.32 -0.20 -0.07 1.18
0 Or I -0.26 -0.28 -0.53 0.27 0.75 -0.13 0
IF15 -0.067 -0.06 -0.12 0.06 1.03
2r12 0.04 -0.20 0.048 -0.01 0.94
2F25, 0.06 -0.19 0.05 0.01 0.94
3 Or I 0.09 0.23 0.16 -0.07 1.03 2.14 2.81
IF15 0.73 1.11 0.98 -0.25 0.97
2r12 0.21 0.13 0.25 -0.04 0.80
2r25, 0.22 0.12 0.24 -0.02 0.84
calculated from (6.29-31). The experimental data of Table 6.1 have been analyzed in
th is way. Table 6.3 shows the resul ts of f i t s for three typical subst i tut ional a l loys
(CuNi ,AuAg, and CuGe). The f i r s t column AZ gives the chemical valence dif ference
between the impurity and the host l a t t i ce . The Friedel phase sh i f t s indicate that
Ni in Cu is mainly a d-scat terer , Ag in Au is mainly an s-scat terer , and Ge in Cu
is a pa r t i cu la r l y strong p-scatterer. Backscattering creates s ign i f i can t differences
between the ~L and An~. In the system AuAg, for instance, the s-wave is shi f ted by
backscattering by 0.27 rad and reduced in amplitude by 25%. The charge t ransfer due
to the scattering is given by the Friedel sum F. I t must screen the defect-induced
charge AZ B. I f the defect has no l a t t i c e d is to r t ion , AZ B is equal to AZ. But in gen-
eral the defect d is tor ts the l a t t i ce . ESHELBY /6.23/ has shown that a center of d i -
la ta t ion in a f i n i t e an~sotropic continuum with a st ress- f ree surface creates a l a t -
t ice d i l a ta t ion AVfi n which contains a contr ibut ion AVIo c confined to the impurity
s i te and a long-range contr ibut ion from the relaxat ion of the stress-free surface.
57
The re la t i ve volume change (AV/V)f in is given by the re la t i ve l a t t i c e parameter
change 3Aa/a and the local volume change is given in terms of Aa/a by
(AV/V)Ioc = 3Aa/~a . (6.39)
The constant ~ has the values 1.45, 1 .42,and 1.23 for Cu, Ag, and Au, respect ive ly
/6.24/ . BLATT has shown that the charge to be screened by the conduction electrons
is approximately /6.25/
AZ B = AZ - (AV/V)Ioc (6.40)
When the l a t t i c e d i s to r t i on is not too large, the Friedel sum agrees qui te well
wi th AZ B. For strong l a t t i c e d i s t o r t i on , the phase s h i f t analysis in the present
form is no longer adequate for the descr ipt ion of the scat ter ing. Up to now, i t has
been assumed that the defect is confined to a mu f f i n - t i n sphere in an ideal l a t t i c e .
I f th is is no longer appropriate for reasons of strong l a t t i c e d i s t o r t i on , the de-
fec t potent ia l and the backscatte~ing are changed. These ef fects are not included
in the ex is t ing theory. An example of a defect with strong l a t t i c e d i s to r t i on is
the vacancy in Au /6 .11/ . The local volume change (AV/V)Ioc = -0.40, i . e . , by l a t -
t i ce re laxat ion the volume of a vacancy is reduced to 0.60 atomic volume. The phase
s h i f t analysis no longer works for th is system. Model ca lcu lat ions for a vacancy in
Au have shown that the neck scatter ing rates for a "vacancy" wi thout d i s to r t i on are
only about ha l f as large as the be l l y scat ter ing rates /6 .11/ . The l a t t i c e relaxa-
t ion pa r t i cu l a r l y enhances the neck scat ter ing rates so that the measured scatter ing
rates show only very l i t t l e anisotropy (Fig. 6.4). A p o s s i b i l i t y to separate the
ef fects of the vacancy and of the surrounding d i s to r t i on f i e l d on the scat ter ing
rates is given in Section 6.4.
6.3.2 Defects on 0ctahedral In ters t ices
i A defect inf luences the scat ter ing rates not only by the phase sh i f t s nc but also
by i t s posi t ion in the l a t t i c e . The cont r ibu t ion of the d i f f e ren t par t ia l waves L
to the scat ter ing rates 1/T(k) (6.35) is the product of two factors, t~ gives the
cont r ibu t ion of the L-th wave in the Bloch wave vector k at the posi t ion of the de-
fect . Since the Bloch wave at a l a t t i c e s i te is d i f f e ren t from that at an i n t e r s t i t i a l
pos i t ion, t~ depends on the posi t ion of the defect. The same holds for the scat ter -
ing parameter S L. The superposit ion of the scattered waves which are ref lected from
the surroundings has to be considered at the defect posi t ion. Hence, the B r i l l o u i n
zone in tegra ls • and S L depend on the defect posi t ion. HOLZWARTH and LEE /6.22/
have formulated the phase s h i f t analysis for the case of a defect occupying octahe-
dral in te rs t i ces in Cu. The re la t ions for the renormalizat ion factors AL' for the
scatter ing rates 1/~(k), and for the Dingle temperatures X are now
58
AL = [s in n~ exp(in~)(~z - ~L)] -1 = I~LI exp(i~L) (6.41)
with
]~LI = [sin n~{(~ - Re{xL})2 + (Im{~L})2}I/2] -1 (6.42)
_ i ~ = arctg[Im{XL}/(~L - Re{xL})] - ni~ : ~L n~ (6.43)
l l= (k ) : ( 2 C d l h ) ~ ~ Im{S L} (6.44) L
Xmc/m o = (~o/kB)(ao/a)2Cd~W L Im{S L} (6.45) L
~ = cotg n~ �9 (6.46)
The quanti t ies marked by a t i lde depend on the posit ion of the defect in the la t t i ce .
The coeff ic ients WL and • calculated by HOLZWARTH and LEE for octahedral in terst ices
in Cu are l is ted in Table 6.2. I t should be emphasized that W L and WL have d i f ferent
magnitudes and d i f ferent anisotropy. The neck electrons, for instance, have mainly
p-character on subst i tut ional sites and mainly s- and dF25,-character on octahedral
in terst ices. The analysis of the C_uuH results is given in Table 6.4. The hydrogen on
Table 6.4. Phase sh i f t analysis for hydrogen in Cu occupying octahedral interst ices
OF 1
lr15
2r22
2F25,
i eL ~L
1.641 1.032 -0.609 1.149
0.152 0.115 -0.037 0.880
0 0 0 1
0 0 0 I
an octahedral in terst ice is a defect of short range creating mainly s-scatter ing.
That is the reason for the stronger neck and the weaker B<IO0> scattering (Fig. 6.4).
The Friedel phases create a charge transfer of F = 0.88, in good agreement with the
charge to be screened AZ B = 0.89 /6 .8/ . The backscattering changes the amplitudes of the incoming s- and p-waves by +15% and -12%. The phase shi f ts ~L are appreciable.
i The phase sh i f ts n~ alone would give a Friedel sum of 1.34. This d i f fers from the
real Friedel sum by 0.46 and corresponds to the charge transfer by backscattering.
5 9
6.3.3 Scattering of the Conduction Electrons by Hydrogen in Cu Occupying Octa-
hedral Interstices and Lattice Sites
The influence of the position of the defects on the Dingle temperatures can be
i l lus t ra ted by hydrogen occupying octahedral interst ices and la t t i ce sites in Cu.
Loading copper with hydrogen at temperatures around 650~ leads to the occupation
of octahedral interst ices /6.26/. But there are indications that loading near the
melting point of copper can lead to the occupation of la t t i ce sites by the hydrogen.
I t is known that hydrogen can be trapped by impurities at low temperatures (-100~
/6.26/. On the other hand, i t is possible to quench hydrogen and vacancies into
copper by appropriate loading conditions /6.27/. I f the hydrogen could be trapped
by the vacancies (thereby becoming a substitutional defect), this would have sub-
stantial consequences for the scattering of the conduction electrons. Using the
n~ of Table 6.4 for both la t t i ce si tes, the Dingle temperatures l is ted phase shi f ts
in Table 6.5 are obtained. They d i f fe r appreciably for the two positions especially
Table 6.5. Dingle temperatures in CuH for hydrogen occupying octahedral interst ices
and la t t i ce si tes, calculated with the phase shi f ts of Table 6.4. The neck scattering
rates are especially strongly affected by the position of the defect
X* Octahedral Lattice
K/at.% interst ice si te
B<IO0> 40.3 56.8
BTP 48.4 54.3
B<111> 61.2 49.9
D<110> 68.7 38.7
R<IO0> 78.5 37.6
N<111> i20.7 2.8
for the neck orbits. I t is obvious that measurements of the Dingle temperatures give
the poss ib i l i t y to determine whether hydrogen can be trapped by a vacancy.
6.4 Phase Shif t Analysis of Defect-lnduced Fermi Surface Changes
Besides reducing the l i fe t ime of conduction electrons, defects also a l ter the geome-
t ry of the Fermi surface /2.10,6.19,28/. When each impurity is screened separately
by the conduction electrons (di lute l im i t ) and when the defects create no la t t ice
distort ion, the Fermi energy is not changed by the defects. Indeed, in this case
60
the l a t t i c e is undisturbed between the defects and, since the Fermi energy must be
constant throughout the l a t t i c e , i t s value remains that of the ideal l a t t i c e . On
the other hand, the geometry of the Fermi surface is changed by the defects. I f k
is the wave vector in the perfect c rys ta l , the wave vector q in the defect l a t t i c e
is given by /6.19,29/
9 : ~ - Cd dk/dE Re{Tkk} (6.47)
or using (6.32-34)
9 = ~ + Cd dk/dE ~] t~ Re{S L} (6.48) [
where
Re{S L} = (~g - Re{XL}) [(~g - Re{xL}) 2 + (Im{XL})2] -1 (6.49)
The s i m i l a r i t y of (6.48,49) with (6.35,36) for the scattering rates is obvious. The
real part of Tkk is correlated with the defect-induced Fermi surface changes and
the imaginary part with the l i f e t ime of the electrons. The defect-induced changes
of the dHvA frequencies are obtained from (6.48) by averaging over an extremal cross
section
AF = (hc/2~eo)(~/a)2 c d ~W L Re{S L} .
L
(6.50)
The coef f ic ients W L are given by (6.38). Thus the Dingle temperature X and the Fermi i surface changes AF can both be described by the phase sh i f ts nL and ~L"
i Two differences in the determination< of qL and ~L from X and AF should be empha-
sized, F i r s t , the phase sh i f ts n# can be determined from AF including the i r sign,
in contrast to those determined from Dingle temperatures. The reason is that Re{S L}
contains the ~ l i nea r l y while Im{S L} contains them quadrat ical ly . In the analysis i of the Dingle temperatures, the sign of n~ must be determined from other c r i t e r i a ,
such as the Priedel sum rule or the r e s i s t i v i t y .
The second dif ference in the information contained in the Dingle temperatures
and in the Fermi surface changes concerns the influence of the defect-induced l a t -
t ice d is tor t ion . In the simplest approach, the l a t t i c e d is to r t ion created by defects
is treated in an e las t i c continuum model. Within this model the l a t t i c e expansion
AVfi n in a f i n i t e crystal with stress-free surface can be separated into a d i l a ta -
t ion AVIo c confined to the defect s i te and a long-range expansion AV I due to the re-
laxat ion of the stress- f ree surface /6.23,24/
AVfi n = AVIo c + AV I . (6.51)
AVfi n is related to the l a t t i ce parameter change
61
(AV/V)fin = 3 Aa/a . (6.52)
There exists experimental evidence /6.19/ that the local l a t t i c e d is tor t ion around
the defects contributes to the dHvA frequency changes AF only through i t s contribu-
t ion to the homogeneous l a t t i c e expansion expressed by the l a t t i ce parameter change.
Since Aa/a is known experimentally for many al loys the defect-induced l a t t i ce dis-
tor t ion can be taken into account rather eas i ly in the determination of phase sh i f ts
from the frequency changes AF. In contrast to th is , the local l a t t i ce d is to r t ion
affects the scatter ing rates in a more subtle way, by changing the scattering poten-
t i a l and the backscattering from the atoms surrounding the defect. I t has not yet
been possible to t reat these changes in a sat is factory way. Thus the Friedel phase
sh i f ts derived from Dingle temperatures d i f f e r in general from those derived from
Fermi surface changes /6.19/. I f th is concept is confirmed by fur ther experiments,
a comparison of the phase sh i f ts may help to c l a r i f y the influence of l a t t i ce dis-
tor t ions on the scattering rates.
Acknowledgements
I would l i ke to thank the fol lowing persons who have contributed to th is a r t i c l e :
Dr. W.R. Wampler, Prof. R.R. Bourassa, Dr. P.H. Dederichs and Dr. R.O. Jones for
many helpful discussions,
Dr. W. Uelhoff, M. Abdel-Fattah and G. Hanke for the preparation of the high qual i -
ty single crystals used in the experiments,
Dr. K. Mika and K. Wingerath for help in the numerical evaluation of the data.
Dr. J. Mundy, Dr. J.B. Roberto and Dr. E. Seitz for care fu l l y reading the manuscript
and Mrs. G. Hahn and Mrs. M. Klein for typing i t .
F ina l ly I would l i ke to express my special grat i tude to Prof. W. Schi l l ing who has
i n i t i a t ed and continuously supported our de Haas-van Alphen work.
62
List of Symbols
A
A, A n
Aex, A~x A(T,H,X*)
AL' AL a
a o
aL(~) B
BLL, b
bL(~) Cs C~mn
c
c d D(E)
D(H)
d
E, E n
E F
e o F ?
G
gc ~, H 1, H 2 h
I 1
K 1
k F , k s
k B
L %
M
m o
vector potent ial
area of cyclotron o rb i t in k-space
area of extremal cross section of the Fermi surface
dHvA amplitude (2.43)
backscattering renormalization factor (6.22,41)
l a t t i ce parameter
Bohr radius
amplitude of the L-th par t ia l wave J~YL (6.16)
magnetic induction
structure matrix of KKR-band structure (6.19)
2~2kBCmo/~e ~ = 146.925 kG/K (2.23)
amplitude of the L-th par t ia l wave h~Y L (6.161
coef f ic ients of Fourier series representation of the Fermi
surface (4.1)
ve loc i ty of l i gh t
concentration of defects
density of states
magnetization of electrons in extremal cross section (2.44)
degeneracy of Landau levels
electron energy
Fermi energy
charge of proton
dHvA frequency (2.16)
Friedel sum (6.31)
osc i l l a to ry part of electron free energy
electron g- factor
magnetic f ie lds
modulation f i e l d amplitude
Planck's constant
temperature damping factor of dHvA amplitudes (2.45)
Dingle damping factor (2.46)
wave vectors
radius of free electron Fermi surface
Boltzmann's constant
~m or ~ , r
angular momentum quantum number
osc i l l a to ry part of magnetization
free electron mass
63
m c
n
nel P(k,k')
9 r
S 1
SL' ~k S n S
I
T c
Tkk, t
h t~, t~ tk
V S
WE' ~k •
r
~ ' YS
AF
AH
AVfi n, AVlo c , AV I AZ
AZ B h i
o
o D
eL' ~ e
8
x(~)
cyclotron mass (2.7)
integer
electron density scattering transition rate (6.10)
wave vector in defect la t t ice
radius vector
spin-spl i t t ing factor in dHvA amplitude (2.47)
scattering parameter (6.34)
area enclosed by cyclotron orbit in r-space
spin quantum number
temperature cyclotron period (2.5)
element of t-matrix (6.10)
time
t-matrix (6.17,18)
anisotropy factor of Tkk (6.33) electron velocity
free electron Fermi velocity
orbital averages of t~ and ~ (6.38)
Dingle temperature
angle characterizing a point on the cyclotron orbit
representations of the cubic group
coefficients of electronic specific heat (5.8)
defect-induced changes of dHvA frequency
dHvA period (2.18)
defect-induced volume changes in a lat t ice
valence difference between defect and host
charge to be screened by conduction electrons
scattering phase shifts
angle characterizing a cyclotron orbit in the {110} plane
(Fig. 5.2)
Debye temperature
backscattering phase shifts (6.28,43)
polar angle
scattering angle (2moE/~2)Z/2
curvature of extremal cross section along f ie ld direction electron-phonon coupling constant (2.32)
electron-phonon coupling constant averaged over the Fermi surface
64
~, ~
P ~(~)
Ttr(~) @
@L' ~L @o
x L , x L CO
~C
< >
integer
(6.29) and (6.46)
residual res i s t i v i t y produced by defects
electron l i fet ime (6.11)
transport relaxation time (6.14)
angle characterizing a cyclotron orb i t in the plane {I00}
(Fig. 5.2)
Friedel phase shi f ts
f lux quantum
polar angle
Br i l lou in zone integrals (6.24,25)
modulation frequency
cyclotron frequency (2.4)
quantity renormalized by electron-phonon interaction
orbital average of a quantity defined on the Fermi surface
(2.34)
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66
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67
Polariton Theory of Resonance Raman Scattering in Solids
Bernard Bendow
1. Introduction
1.1 Purpose and Scope
Polaritons are the composite quasipart ic les formed by the coupling of l i gh t with
c rys ta l l i ne exci tat ions (see, for example BURSTEIN and DE~ARTINI, 1974). Since po-
la r i tons are derived from an exact solut ion of an interact ion Hamiltonian, they
provide the most physical ly sat is fy ing basis for the descript ion of a var ie ty of
optical e f fects , among them absorption (HOPFIELD, 1958), luminescence (TAIT and
WEIHER, 1969; BENOIT A LA GUILLAUME, 1970), nonlinear processes (OVAIIDER, 1965),
and l i gh t scattering (OVANDER, 1962). A good deal of l i t e ra tu re has been directed
at the problem of resonant l i gh t scatter ing mediated by excitons, and th is topic
w i l l const i tute the main subject of this review as wel l .
The in terest in polar i ton approaches to resonant Raman scatter ing (RRS) can be
understood from a var ie ty of standpoints. As intimated above, i t is very appealing
physical ly to pursue polar i ton descript ions, since polari tons are the fundamental
exci tat ions of l i gh t interact ing with a crys ta l . One expects polar i ton formulations
to be more rigorous than those u t i l i z i n g noninteracting quasipart ic les. I t is rather
i n t u i t i v e , for example, that in the resonance regime where photon and exciton dis-
persions cross, a perturbation approach which treats these exci tat ions as uncoupled
must break down. Polariton approaches, on the other hand, o f fer the prospect of an
exact treatment of the dispersion induced by the (b i l i near ) photon-ex~iton interac-
t ion. Other advantages of a polar i ton formulation include the correct and consistent
determination of k-dependent factors in scattering rates, and the p o s s i b i l i t y of
accounting for multimode exc i ta t ion effects at boundaries (stemming from a number
of polar i ton branches being degenerate at a par t icu lar frequency). A f ina l aspect
that is not spec i f i ca l l y associated with resonance condit ions, but is nonetheless
of s ign i f i can t in teres t , is the incorporation of f i na l - s ta te polar i ton ef fects . We
shall touch on th is subject only b r i e f l y here, since our main concern is with the
mediation of scatter ing by polar i tons, and because i t has been covered very ade-
quately in a number of recent reviews (see, for example, BARKER and LOUDON, 1972;
MILLS and BURSTEIN, 1974; and CLAUS et a l . , 1975).
69
The principal purpose of the present review is to organize and sort the existing
l i te ra ture on the subject of polariton-mediated l igh t scattering, and to c r i t i c a l l y
evaluate the implications and the u t i l i t y of the polariton approach. We shall espe-
c i a l l y inquire whether or not polariton predictions d i f fe r from those of perturbation
theory, and whether or not any predicted polariton effects can in fact be d is t in-
guished experimentally. Hopefully, the present review can provide insights that w i l l
aid experimentalists as well as theorists in answering questionssuch as these.
While the prospect of a polariton formulation of RS might seem very at t ract ive
indeed, an assortment of d i f f i c u l t i e s arises in connection with i ts implementation.
These d i f f i c u l t i e s , which wi l l be explored in some detail in this review, are con-
fined mostly to frequencies in the resonance regime (as one might expect, polariton
results simply reduce to those of perturbation theory far from resonance). Stated
b r ie f l y , these d i f f i cu l t i e s are associated with properly relat ing the scattering
inside of crystals by polaritons to measurements of photons outside of the crystal .
Moreover, depending on the formulation u t i l i zed , i t is necessary to consider the
scattering of damped quasiparticles, for which single well-defined quantum states
do not exist , and which are characterized by unphysical group veloci t ies.
The nature and role of these d i f f i cu l t i e s w i l l be made more exp l i c i t further on
in the development. We simply note here that we w i l l be describing two characteris-
t i c a l l y d i f ferent approaches to the formulation of polariton-mediated RS. In the
f i r s t , the eff iciency is assumed to factorize into appropriate transmission factors
(which relate photons outside the crystal to polaritons inside), and into polariton
scattering eff ic iencies within the crystal . The second approach, on the other hand,
attempts to treat the entire scattering process as a single unified event, u t i l i z i ng
a combination of quantum and semiclassical techniques. We shall find certain s ign i f -
icant differences implied by the two approaches.
Because the physically important properties of the various possible polaritons
are closely s imi lar, we find i t convenient, for definiteness and s impl ic i ty , to re-
s t r i c t the present discussion to scattering mediated by photon-exciton polaritons,
with the possible production of f ina l -s ta te photon-phonon polaritons. Extensions to
scattering by other polaritons follows d i rect ly from information readily available
in the l i te ra ture (see, for example, MILLS and BURSTEIN, 1974).
The reader who lacks a general f am i l i a r i t y with RRS in solids is directed to a
variety of previous reviews on the subject, such as those by LOUDON, 1963, 64; MARTIN
and FALICOV, 1975; and RICHTER and ZEYHER, 1975, for example. More detailed accounts
of original research in the f ie ld over the last decade are available in the pro-
ceedings of the Light Scattering Conferences (WRIGHT, 1967; BALKANSKI, 1971; BAL-
KANSKI et ai.,1976), and the US-USSR Symposium on Light Scattering (BEHDOW et a l . ,
1976).
The plan of the paper is as follows: In Section 1.2 we very b r ie f l y review certain
aspects of the perturbation theory approach to resonant RS. Section 2.1 reviews the
70
fundamentals of polari tons and Section 2.2 introduces the two pr incipal approaches
to {he formulation of the polar i ton theory of RRS. The predict ions and impl icat ions
of polar i ton theory are examined in some deta i l in Section 3, while Section 4 con-,
s is ts of some concluding remarks on the subject.
1.2 Review of Perturbation Theory
The perturbation approach to RRS and i t s implications have been described in deta i l
in a wide var ie ty of papers (see, for example, LOUDON, 1963; GANGULY and BIRMAN,
1967; PLATZMAN and TZOAR, 1969; MARTIN, 1971; BARKER and LOUDON, 1972). We shall
here d i rect our at tent ion to jus t certain sa l ient features which are useful for com-
parison purposes when discussing the polar i ton approach to RRS.
In th is section and throughout the remainder of th is paper we shall employ a sim-
p l i f i e d model Hamiltonian which w i l l allow us to deduce the pr incipal features of
RRS with a minimum of mathematical and notational complexity. Extensions to more
general cases w i l l be straightforward from the development.
Let us take for the Hamiltonian H (see, for example, GANGULY and BIRMAN, 1967):
H = H o + V; V = V I + V 2
V I : Vl(a) + Vl(b)
R = H o + V I
k ky ks
Vl(a) = E (+) i l g3 ' I (kc ) - l /2 at c~ - + k
_+ ,.kY (b) _ k_ I C t_
Vl : E (+)mP 2 a_ k a+-k _+,k
(1.1)
t b s + h.c v 2 : f y ( k , . k ' ) c y,Ck _ ,k-k'_ _ kk 'yy 's
- 2 m 4~ N e2/m e gy = gy Ev~y , mp
where (at,a) are creation-annihilation operators for photons, (cf,c) for excitons, and (bt,b) for phonons; Ey(k) and Qs(k) are the exciton and phonon energies, re-
71
spect ively (~ = 1), with k the wave vector and ~ and s branch indices (we note that
y can be e i ther a discrete or continuous index for the exci tons); mp is the exciton
plasma frequency and N the electron density. H ~ is the noninteracting Hamiltonian
three f ie lds ; v1(a) stems from the p.A in teract ion between the crystal elec- for the
trons and the electromagnetic f i e l d ; Vl(b) comes from the A 2 interact ion term; and
V 2 is the exciton-phonon in teract ion. We have l imited the Hamiltonian to jus t a
single polar izat ion of l i gh t , although the polar izat ions must of course be reinstated
to obtain the tensorial character of the scatter ing amplitude. For s impl ic i ty , we
have omitted b i l inear phonon-exciton in teract ions; the i r omission does not qual i ta-
t i ve l y change the essential resonance propert ies of the predicted cross section
(see Eqs. 29-31 of GANGULY and BIRMAN, 1967, for example). The R defined above, which
incorporates al l b i l inear in teract ions, is the polar i ton Hamitonian to be treated
in Section 2.
The perturbation picture of a f i r s t order RS event is indicated in Fig. I and
consists of:
a) transmission of the photon m into the crystal
b) annih i la t ion of the photon with creation of an exciton (via V1)
c) scatter ing of the exciton accompanied by creation (Stokes) or annih i la t ion (ant i -
Stokes) of a phonon (via V2)
d) annih i la t ion of the scattered exciton and creation of the photon m' (via VI)
e) transmission of the photon m' out of the crysta l .
z/,~,
�9 POLARITON PICTUREI"~__
Fi~. 1. Perturbation theory and polar i ton pictures of single-pho- non Stokes RS. Single wavy l ines: photons; double wavy l ines: po- la r i tons ; s t ra ight sol id l ines: excitons; dashed l ines: phonons
72
We l im i t ourselves here to Stokes RS at zero temperature, which is su f f i c i en t to
demonstrate a l l the resonance propert ies we need be concerned with. The probabi l i ty
per uni t time for RS is taken to be simply the t rans i t ion rate from the i n i t i a l state
~i containing one photon m and no excitons, and the f ina l state @f containing one
photonm~ one phonon Q, and no excitons. According to the usual scatter ing theory
(MESSIAH, 1962) this rate is given by (~=i)
dPif ]2[ a )I = 2 z ~ I T i f ( ~ ) ~-m' - ~s(k
f A
T i f = <~fITl~i >
^ where T is the s ca t t e r i ng -ope ra to r ,
(1.2)
T(m) = V + VGV = V + VGoV + VGoVGoV + . . .
G(m) = (m-H + i~) - I , Go(u ) = (m-H ~ + i~) - I .
(1.3)
�9 The lowest order contribution to RS from the above expansion must contain two Vl(a)
interaction terms (corresponding to incident and scattered photons), and the inter- .(a) . . . . Ca) action V 2 (to create or annih i la te phonons). We thus require the term "I UoV2~oVl
in T, which leads to
Ti f + ~ <ilVzl~>l~_r<yiv21y,s> I <Y,slVzlfs> (1.4) ~y' ~
where an abbreviated, schematic notation has been employed. Assuming f(k,k')=f(k',k),
restricting ourselves to energy-conserving transitions, and taking the two time or-
derings for creation and annihilation of m and m' into account, we obtain (41", c = 1)
I I ~ . IgT ] lgy ' l f yy ' ( k ' k ' ) (m-Ey) (m'-E ,) +(m+Ex) (m'+E T i f = ~ yy,
I f we now consider a single phonon n o and neglect i ts dispersion, then
, ] . (1.5)
dPif k,2 2 k' 2 ~ ]Ti f ] ~(m-m'-~o)~ IT- [R(k,k ' ) ]
R ( k , k ' ) : ~ f~"Ig~llgx'l (m~'§ TY' (m2-E~ 2) (m'2-Ey '2)
(1.6)
where m' = m-n o . In the present perturbat ion analysis k' and k are the wave vectors
of the noninteract ing photons, so that k ' / k = m'/m z I , since m o << m,m'. Thus, when
?3
the couplings f and g are independent of , or depend only weakly on wave vector, then
the resonance_ behavior is nearly en t i re ly determined by the factor :r(m2-E#2)(m '2-
Ey,2)] -2 above. When y and ~' are d iscrete, then a second-order pole arises whenever
e i ther ~ ~ Ey or m' - E , ( inc ident and scattered resonances); note also that the
resonance becomes "doubly strong" i f the l a t t e r conditions are sa t i s f i e ld simulta-
neously for some set of variables (see, for example, YU et a l . , 1973) I f the levels
form a continuum, then the strong resonance associated with discrete states w i l l
be suppressed, and replaced instead by an extended regime of enhancement for values
of m and ~' f a l l i n g into the frequency range spanned by the Ey (LOUDON, 1963; BENDDW
et a l . , 1970; BENDOW, 1970). For the polar i ton problem, one is natura l ly concerned
pr imar i ly with discrete state resonances.
Several remarks are in order. I f the b i l inear phonon-exciton term is retained in
V, then six terms rather than two contr ibute to T i f (see GANGULY and BIRMAN, 1967,
for example), although the resonance behavior is not s ign i f i can t l y af fected. An-
other point is that the f~y , ' s in (1.6) can have d i f fe ren t signs for d i f fe ren t com-
binations of ( y ,# ' ) , and thus introduce antiresonances into dP/dt (RALSTON et a l . ,
1970). In fact , the contr ibut ions from the discrete and continuous portions of a
single exciton band may in ter fe re construct ively and destruct ive ly in d i f f e ren t f re -
quency regimes (BENDOW, 1970). Another consideration of importance is the necessity
to include l i fe t ime ef fects in the energy denominators (m2-Ey2) -2, via the prescrip-
t ion E + E + & (m) + iTy (m) (BENDOW and BIRMAN, 1971). In th is way the obvious-
ly unphysical s ingu la r i t ies in dP/dt for discrete y are el iminated. The detai led
shape of dP/dt vs m at resonance is determined in pr inc ip le by the frequency depen-
dence of T(m), although th is dependence may often be neglected in pract ice. Note
that for a continuum of levels the inclusion of an imaginary part in m2-E 2 is es- Y
pec ia l ly c r i t i c a l , since this leads to terms describing creation of real (as opposed
to v i r tua l ) excitons (see, for example, MARTIN, 1974). RSS within a continuum of
levels actual ly represents a mixture of v i r tua l -exc i ton (Raman-type) processes and
absorptions followed by frequency-shif ted reemission (resonance f luorescence), a l l
of these being contained in dP/dt when the imaginary part referred to above are re-
tained. Again, these considerations are not par t i cu la r l y germane as far as polar i ton
ef fects are concerned, although continuum ef fects do inf luence polar i ton dispersion,
for example.
The present resul ts , although s impl i f ied, should provide an adequate backdrop for
discussions of the polar i ton formulation of RRS.
74
2. Polaritons and Their S c a t t e r i n g
2.1 Fundamentals of Polaritons
The concept of mixed electromagnetic and c rys ta l l ine modes appears to have been f i r s t
explored c lass ica l l y by HUANG, 1951a, b and POULET, 1955, and quantum-mechanically by
FANO, 1956. The extension to coupling of l i gh t with excitons was carr ied out by HOP-
FIELD, 1958 and AGRANOVlCH, 1960, both u t i l i z i n g a second quantized representat ion.
The development given here fol lows closely that of the l a t t e r papers, in a form pre-
sented previously by BENDOW, 1970.
Transformation Coefficients and Polariton Dispersion. The Hamiltonian R for a single polar izat ion of l i gh t in teract ing with excitons in an isot rop ic crystal has
been given above in ( I . i ) . Since we do not include the b i l inear phonon-exciton in te r -
action in H, the phonon port ion of H o may be ignored in the considerations that f o l -
low. I t should be remarked that there is absolutely no d i f f i c u l t y formally with
including these terms; they are omitted for reasons of convenience alone. R defined
in th is manner const i tutes the polar i ton port ion of the Hamiltonian which is central
to the present paper, t
We search for the normal modes of R by def ining polar i ton operators (Ak , Akv )
which are l inear combinations of the a's and c 's , i . e . ,
A*_k,a = Xo(_k)atk + ~X-.y(k)c~, 4 _ _ Y #po(_k)a_k_ +~.@y(_k)c__k, Y
and determine the coeff ic. ients (X,@) which sat is fy the equations of motion
(2.1)
, : _+ m A- (2.2) i
This leads, in matrix form, to
2 2mp 2 k_i/2 2mp zm.-1/2
<c + ~ ~y kc YX~
zm~-l/2 ~k -1/2 Ey yyK
2m 2 2m 2 kc p -gyk -1/2 -(kc + - - ~ - ) - ~mk-1/2 I E i ]
Xo
~o
Cy
=CJ
•
~o I
- gyk - I /2 0 - gyk -1/2
(2.3)
where the summation convention is to be used for the products gy • and gy #X' but
not otherwise. The determinant of the above matrix minus Im yie lds the dispersion
re lat ion determining the polar i ton modes, namely
75
4 2 /,._\(~)2 = 1 + ~z_~ ~ g Y -= c(k,~)_ (2.4)
EN-~
where the sum rule ~p2 = ~lgyi2 has been u t i l i zed ; e is the contr ibut ion to the d i -
e lec t r i c function of the crystal from interact ion of l i gh t with excitons. To account
for a background d ie lec t r i c constant eo' the replacement c§ o and 9-*9/~ o should
be made above. The manipulation of (2.3) can be shown to lead to Maxwell's equations
for the photon amplitude X o, which represents the admixture of (a k + a_k ) present
in the polar i ton (see ZEYHER e t a ] . , 1974).
We denote the solutions for m of the dispersion re lat ion (kc/~) 2 = e(k~), i . e . ,
the polar i ton energies, by E(ks). Then apart from an i r re levant constant, the polar-
i ton Hamiltonian becomes
: ~ E(ks)A* A , (2.5) ks ks ks
i . e . , H is comprised of a co l lec t ion of noninteracting (harmonic) modes of energy
E(k~). The polari tons are related to the noninteracting f ie lds through the coe f f i -
cients (X,~), which fol low as
xy = (~-Ey) -1 gy A -1 /2 , ~y : _(~+Ey)-1 ~y A ~-:/2
Xo = i /2 (kc+m) (kc~)- l /2A, ~o = 1/2 (kc-m) (kcm)-l/2A (2.6)
A -2= 1 + ~ 4E~21g~] 2 ( 2 _ E 2)-2. Y
Physical ly, the (X,@) measure the f ract ion of photon or exciton contained in a given
polar i ton mode, as w i l l be seen more e x p l i c i t l y below. One may s imi lar ly obtain the
inverse relat ions expressing a's and c's in terms of A's, but we do not do th is here.
As examples of the appl icat ion of the above formalism, le t us consider the polar-
i ton dispersion and transformation coef f ic ients for various s impl i f ied cases. For a
hydrogenic exciton band consist ing of discrete levels plus a continuum, one obtains
the resul ts for E vs k indicated in Fig. 2. The "repulsion" of the polari tons from
the bare quasipart ic les in the crossover regions is c lear ly evident; far from the
crossover, on the other hand, the polar i ton takes on the dispersion character is t ic
of the bare quasipart ic les. The requency regime in which dispersion ef fects are sig-
n i f i can t is given by Im2-Ey21#4g2/m o. We note that in addit ion to al l the discrete
modes i l l us t ra ted , a l l energies in the continuum are eigenvalues of the polar i ton
dispersion re lat ion as wel l . Turning to the transformation coef f ic ients (• which
measure the "amount of photon" in the polar i ton, i t is clear from (2.6) that these
are large everywhere except when Im2-E 2I#4g2/e ; correspondingly, the exciton co- 2 2 ~Y 2 o
e f f i c ien ts are large only for ~ - E l<4g /~o"
76
i 2.62
!
2.60
"~ 2.56 Z3-
2 Ez-
,.o, 2.5e
s ...., ~. 2,54
2.52
2.50
I I P
ii li
i l
I , f I i / il ,I
i ]
, , ! l , I 4 6 iO
POLARITON WAVE VECTOR "tick (eV)
[ i i
I i t2 14
Calculated polariton frequency vs wave vector for CdS A-exciton parameters. Solid l ine: fu l l resul t ; dashed l ine: omitting continuum; dash-dot l ine: photon l ine (from BENDOW, 1970)
)- o z 14J
E2(0) h Z .111
I1:
. J o
WAVE VECTOR k
~ Po la r i t on d ispers ion fo r two nondispersive leve ls ( so l i d l i nes ) and parabo l ic (dashed l i nes ) (from BENDOW and BIRMAN, 1970)
77
The modifications of the polar i ton dispersion for the case of dispersive exci-
tons (E=E(k)) is indicated in Fig. 3. The var ia t ion of the coef f ic ient I•
which measures the "amount of exciton" in the polar i ton, is i l l us t ra ted in Fig. 4 for
the lower polar i ton branch. This coef f ic ient is small for m<<E(O), but becomes large
and constant for m~E(O), where the polari ton becomes exc i ton- l ike in character. Note
that in the dispersive exciton case a m u l t i p l i c i t y of denegerate polar i ton modes
may coexist for a given frequency m. For photons incident at such frequencies the
appropriate admixture of each of the polaritons excited must be determined in order
to calculate the cross section (PEKAR, 1958). The poss ib i l i t y of creating mul t ip le
propagating modes, even when just a single dispersive exciton is present, is pecul iar
to the polar i ton view of the scattering event. Within a bare exciton approach the
photon is assumed to excite a l l possible exciton states as intermediate states, and
the matter of mult ip le mode exci tat ion (and, as we shall see la te r , the correspon-
ding necessity for addit ional boundary conditions) need not be taken into consider-
at ion.
0.01
0.00
"~-0.01
"6 -0.02
-0.03
/ E(O}
t 2.55 2.57 2. 9
Frequency (u(eV)
Fi~. 4. Logan of exciton polar i ton transformation coef f ic ient vs frequency for CdS parameters ~from BENDOW and BIRMAN, 1970)
Strength Functions. Although calculated scattering amplitudes w i l l , in general,
involve quadratic expressions in the (X,@)'s with arbitrary coefficients, i t is never-
theless useful for many purposes to consider simpler functions constructed from the
(X,@)'s. Among these are the "strength functions" (MILLS and BURSTEIN, 1969), which
measure the fraction of interacting quasiparticle contained in the polariton. The
definit ion of strength functions is not unique (see, e.g., BENDOW, 1971b); we adopt
the following ones because of their relations to certain important matrix elements.
78
For a single dispersionless exciton E 1 the exciton and photon strengths are given by
i<n +iIc~+ c kln >i 2 4g2Elm - = I A2 Se(~) = n+T - (E12-~2)2 ~o
OJ : ~ ( IX112 I~i 12) (2 .7 )
l<n +l lat+a In,>l 2
SR(~m) = n +1 : A 2 : IXo 12 - I@o 12
where n is the occupancy of the polar i ton corresponding to m. Note that EISe+mSR=m,
which is equivalent to the normalization condit ion ~( l• 12 l@i 12) : I . A more de- I
ta i led exposit ion of the propert ies of the S's and the i r in terpretat ion may be found
in MILLS and BURSTEIN, 1974. For our purposes the important point is simply that S e
is maximal at frequencies close to E I , while S R displays the opposite trend.
I t is perhaps appropriate at th is point to note that much of the present develop-
ment could have been equally well deduced from a Green's funct ion analysis, as per-
formed by MAVROYANNIS, 1967, for example. The advantage of such a formulation is that
one may be able to re late scatter ing e f f i c ienc ies d i rec t l y to appropriate Green's funct ions, or the i r anc i l la ry funct ions, without separately calculat ing transforma-
t ion coef f ic ients and the l i ke . Nevertheless, we believe that the combination of the
equation-of-motion approach presented above with scatter ing theory for the polar i tons
provides the most physical ly transparent approach. On the other hand, Green's func-
t ions may prove especial ly useful with respect to exciton damping ef fects in polar-
i ton scat ter ing, although these have yet to be worked out e x p l i c i t l y wi th in any method.
Po~in#. I f the exciton-phonon interact ion V 2 were added to H, then the polari tons
would become damped quasipart ic les, since the i r exciton port ion would acquire a f i n i t e
l i fe t ime. Formally, the e f fec t of the in teract ion is to modify the polar i ton eigen-
energies through a complex proper self-energy of the form
~(k ,~ ) : Z(k,m) + i r ( k , ~ ) . (2.8)
Z, the real energy sh i f t , is usually absorbed into ff to produce, at least conceptu-
a l l y , the "observed" polar i ton energy. The ef fects on F, on the other hand, may be
much more s ign i f i can t . For example, the polar i ton dispersion may become substant ia l ly
a l tered; sharp- l ine spectra w i l l acquire f i n i t e widths, and possibly detai led struc-
ture as wel l . Unfortunately, the propert ies of F(k,~) have not been extensively in-
vestigated, and at present e x p l i c i t resul ts are avai lable only for highly s impl i f ied
models. Nevertheless, i t is possible to obtain insight into some damping-induced
phenomena from jus t certain qua l i ta t i ve considerations, which is the course we shall
fo l low here. For example, i t is clear that ?(k~) is smallest when the polar i ton is
79
l i g h t l i k e , since only the exciton portion of the polar i ton is subject to decay. Cor-
respondingly, ~ is maximal when the polar i ton is exc i ton- l i ke ; in fac t , we expect
that r is roughly proportional to the exciton strength S e, and that F§ e, where r e
is the bare exciton damping constant, for Se§
The semiclassical approach may be u t i l i zed to incorporate the ef fects of damping
on polar i ton dispersion. I f we add a f r i c t i o n term to the c lassical equations of
motion then the dispersion re la t ion becomes
Y Ex2(k)_m2_imru (2.9)
where r is the f r i c t i ona l damping constant for exciton X. A typical dispersion curve Y
for a single level E I and constant F is i l l u s t ra ted in Fig. 5. One notes that the
re f lec t ion gap present when F=O now disappears, and that the polar i ton dispersion
"bends back" at some f i n i t e value of k, in contrast to the undamped case. Detailed
curves for various cases are given by AGRANOVICH and GINSBURG, 1966. The strength
functions for the present case may be obtained by resort ing to a de f in i t i on of these
quant i t ies in terms of c rys ta l l i ne and electromagnetic energy. The resu l t for a single
level (MILLS and BURSTEIN, 1974) is obtained by replacing ~o§ and (E2-~2) 2 §
(E2-m2) 2 + m2r2 in A, in (2.6). Increasing r , i t turns out, decreases S e, with S e
remaining less than uni ty even for o~+E. I t must be remembered, however, that these
properties are qua l i ta t i ve in nature. The quantum theory must u l t imate ly be employed
to obtain the rigorous (k,m) dependence of ?, and i t would not be surprising i f con-
clusions based on constant F approximations turn out to be inaccurate in certain
instances. I t should also be pointed out that scatter ing e f f ic ienc ies do not simply
involve strength functions alone, but more general corre lators, or response functions,
possessing a more complicated structure.
P o l ~ i t o n Veloeit ies. Polariton scattering e f f ic ienc ies are related to the rate
at which polari tons are transported through the c rys ta l . Some of the ve loc i t ies
associated with the polar i ton are the phase ve loc i ty Vp:E(k)/k; group ve loc i ty Vg :
@E(k)/@k; and energy ve loc i ty VE, which is defined as the (time-averaged) Poynting
vector divided by the energy density. Clear ly , the phase ve loc i ty is a useful de-
scr ipt ion of wave propagation only when m depends l i nea r l y on k (when the polar i ton
is l i gh t l i ke )o Indeed, for the l a t t e r case a l l three ve loc i t ies are nearly equal
to Vp = c/n, where the re f rac t ive index n is very slowly varying as a function of
frequency. In the photon-exciton crossover region, however, t he i r behavior can
d i f f e r widely. For dispersionless excitons, for example, al~ three ve loc i t ies tend
to zero as ~r*Ey, but at d i f fe ren t rates. Obviously, none of the ve loc i t ies ex is t
in the re f lec t ion gap for the dispersionless case. However, for a dispersive exciton,
the gap is f i l led and Vg § @Ey(k)/~k along the exciton-like portion of the lower polariton branch.
80
When damping is included, the polar i ton dispersion may conceivably bend around
for mSEy (see Fig. 5). Thus, Vg w i l l take on unphysical i n f i n i t e and negative values.
The only ve loc i ty of the three retaining i t s physical s igni f icance is v E, which can
be shown to always remain less than c (BRILLOUIN, 1960). For constant damping (n+n+iK)
300
(JJTO
u
200
c
o
100
10 z' 5'103 0 5"103 104 Imaginary part of the propagation vector Real part of the propagation vector
Fi B . 5. Frequency vs real and imaginary parts of the wave vector q, for a polar i ton formed by the TO phonon mode of a zincblende crystal from M. BALKANSKI, in "Optical Properties of Sol ids", F. Abeles, ed. (N. Holland, Amsterdam, 1972)
VE = c [n(w)+2mK(m)/s -1
which at ta ins i t s maximum for m~E . A comparison of v E and Vg
value of r is indicated in Fig. 6.
2.2 Formalism of Polariton-Mediated Scatterin9
(2.10)
for a f a i r l y large
In th is subsection we explore two formulations of polariton-mediated RS, on which
the resul ts in Section 3 w i l l be based. A RS event may be pictured in the fashion
indicated in Fig. ib, as opposed to the perturbation picture in Fig. la. A central
issue in obtaining a polar i ton formulation is the re la t ion between polar i ton scat-
ter ing probab! l i t ies and experimentally observed e f f i c ienc ies , and we therefore
focus some at tent ion on th is question here. The effects of f i na l - s t a te polari tons (creation of IR-active phonons) does not require fundamental revisions of the usual
scattering theory, and therefore we defer th is topic en t i re l y unt i l Section 3. In
th is section we l i m i t the discussion to polar i ton ef fects associated with states
mediating the scatter ing. 81
1 . 5 0 - - I i I
1.25
1.00 >. h-
~ 0 . 7 5 . J I,LI
0 .50
0.25
0.00 0.80
- GROUP VELOCITY-----.
t _ , ~,
I l , ',
l i
VELOCITY _ _ I l i I t
0.85 0.90 0.95 1.00 1.05 1.10 FREQUENCY o~
~ q . 6, Cgmpariso~ of group and energy ve loc i t ies for a material with e(m) = eo g (E ~ _ mz + iF)- for mo = I0, g2 = 0.2, E = 1.0, and g = 0.1
Gener~ Considerations. Why, in general, should one expect scatter ing calculated
u t i l i z i n g polari tons to d i f f e r from that obtained by perturbation theory? A pr incipal
reason is that the quasimomenta (wave vectors) of the scatter ing quasipart ic les are
no longer constrained to the corresponding external photon momenta, nor do the quasi-
momenta of the phonons produced in the scatter ing need to sum to the di f ference in
the external photon momenta. (Of course, quasimomentum must s t i l l be conserved within
the c rys ta l . ) Polaritons for frequencies near resonance are characterized by large
values, as opposed to the small values constraining the excitons wi th in the per-
turbat ive approach. Variations in ~olar i ton momentum may be es~ecial ly s ign i f i cant
in the case of spatial dispersion, and in cases where the coupling functions (photon-
exciton, exciton-phonon) are strongly k ~ (k,@,#) dependent.
The de f in i t ion of the quantum scatter ing rate for polari tons inside of an i n f i n i t e
crystal is straightforward and unambiguous. Namely, i f the i n i t i a l and f inal states
are (~,~ ' ) and the i r energies (E,E' ) , then the ~robabi l i ty per uni t time and e~ergy
for scatter ing from ~b-~' via the interact ion V is given in lowest order by the Golden
rule [ re tent ion of jus t the V term in T in (1 .3) ] :
d2p = 2~l<elv [ ~'>12a(d-E ')
dP / d2p 7T~= dE' ~ .
(z.11)
82
Typ ica l l y , we take 4 to be a single polar i ton and 4' a po lar i ton plus a phonon. How-
ever, i t is now necessary to re late th is p robab i l i t y for in ternal po lar i ton scat ter ing
to the observed scat ter ing of photons external to the c rys ta l . Not su rp r i s ing ly , i t
turns out that any reasonable d e f i n i t i o n of the observed scat ter ing reduces to the
same expression, equivalent to that obtained by perturbat ion theory, when a l l polar-
i tons in the scat ter ing are l i g h t l i k e in nature. As is always the pattern in po lar i ton approaches, dif ferences ar ise only near to resonance (m~Ey), where the po lar i ton
dispersion is subs tan t ia l l y al tered from that of the bare quasipar t ic les.
One way to appreciate the ef fects of po lar i ton dispersion on observed scat ter ing
e f f i c ienc ies is to consider the p robab i l i t y P of scat ter ing during traversal of a
photon through a crystal of thickness L. Ignoring at tentuat ion ef fects ,
dP L dP (2.12) P : ~ t ~ : ~ dt
where At is the po lar i ton traversal time, and v the po lar i ton ve loc i t y . We note that
P involves the ve loc i ty v (v E in absorbing regions, Vg otherwise); i f one could some-
how measure the po lar i ton scatter ing p robab i l i t y dP/dt d i r e c t l y , then the dependence
of P on v would be sidestepped. In any case, the above considerations indicate how
po lar i ton dispersion may inf luence observed RS cross sections.
Formulation~ o f Polc~iton RS. Basica l ly , two approaches have been taken in the
l i t e r a t u r e to define RS cross sections w i th in the po lar i ton framework. The g is t of
these approaches is the fo l lowing:
a) App. I . The RS process is viewed as i n i t i a t e d by an external photon m which
is e i ther re f lec ted with r e f l e c t i v i t y R or transmitted with t r a n s m i t t i v i t y T (R+T=I)
upon crossing the c r ys ta l l i ne boundary . I t i t is transmitted, then a unique po la r -
i ton state ~ is created which, in general, is a l i near superposit ion of a l l of the
n polar i tons degenerate wi th frequency w [= El(k) = E2(k2). . . = En(kn)]. In the
absence of at tenuat ion, the p robab i l i t y of experimental ly obsering a scat ter ing
event is proport ional to the product of the t r a n s m i t t i v i t y of the i n i t i a l photon
across the entrance boundary of the c rys ta l , the p robab i l i t y of 4 scatter ing to 4 ' ,
and the t r a n s m i t t i v i t y of po lar i ton 4' across the ex i t boundary of the c rys ta l . More
general ly , one must correct fo r attenuat ion by reducing the e f fec t ive propagation
length of the crystal fo r both inc ident and scattered po lar i tons; so l id angle ef fects
due to the discontinuous change in momentum across the boundary need also be con-
sidered (see, e .g . , LAX and NELSON, 1974, 1975). Nevertheless, the essential point
i s that in App. I , RS is viewed as a sequence of quantum events: Transmission of
the inc ident photon, which creates a s ingle unique po lar i ton state or none at a l l ,
fol lowed by scat ter ing to a wel l -def ined f i na l state, and f i n a l l y , transmission lead-
ing to detect ion of the scattered photon.
b) App. I I . The RS of photons in a crysta l is viewed as a s ingle un i f ied process
invo lv ing re f rac t ion , absorption, and scat ter ing, rather than one consist ing of a
succession of quantum steps. The asymptotic states are pure photons, which become
83
polaritons in the region of space defined by the crystal. The two interactions af-
fecting the asymptotic states are the photon-exciton interaction which produces the
polariton, and the exciton-phonon interaction, which scatters them.
I t is immediately obvious that App. I I is simply an exact description of the scat-
tering event, and that i f such a formulation could be successfully implemented, then App. I would be unnecessary. Such an attempt has been carried out in recent years
by one group of workers (BRENIG et a l . , 1972; ZEYHER et a l . , 1972, 1974). Historical-
ly, all of the rest of a rather extensive body of l i terature on polariton RS has been
pursued within the framework of App. I (for example, OVANDER, 1962; MILLS and BUR-
STEIN, 1969; HOPFIELD, 1969; BENDOW and BIRMAN, 1970). This situation is probably
due in part to the relative ease of implementing App. I. Our emphasis in this paper
wi l l largely reflect the preponderance of this approach in the l i terature.
In what follows we discuss the form of the cross section within the two approaches.
For simplicity, we restr ic t attention to single-phonon Stokes scattering at T=O K,
for the case where there is just one polariton from a single branch which corresponds
to the incident and scattered photons; moreover, we consider the creation of only
IR-inactive phonons. This allows us to focus on the physics of the problem rather
than on the many complicating details which often tend to cloud the significant
properties of the scattering event.
Approach I . Within th is approach one uses the arguments stated above to deduce
the probabi l i ty P1 that a single photon incident on a crystal is scattered from
to m'. The incident photon is transmitted with t ransmi t t i v i t y T, and produces a
polar i ton k of energy m = E(k), which is subsequently scattered via the interact ion
V, producing the state Ik ' , k-k '> which contains a polar i ton of energy E(k') and
a phonon of energy Q(k-k ' ) . F ina l ly , the polar i ton E(k') is transmitted out of the
crystal with t ransmi t t i v i t y T' to produce the photon m'. Thus
dP1 ~ dP ~ kllk I IE )] = TT' = TT' 2~ ~ <_ V_'> 2 ~ (k)-E(k')+~(k-_k'
K '
d2P1 = [ @[E(l~)-f;(k-~)]. - t TT' 2~T k'21<klVl_k'>l 2
~=k'
(2.13)
where k' is determined by the energy conservation a-functions; in general, several
such k' may exist and a sum over them need be taken. Also, in general, dE is not the
observed solid angle; one must use methods such as those described by LAX and NELSON,
1974, to rewrite (2.13) in terms of the observed solid angle d~outside of the crys-
ta l . I f we define f= = dE/~then, formally,
d2P1/dt~= f~ d2P1/dtdE (2.14)
84
d2P1/dt~ is now the probab i l i t y per uni t time that a scattered photon is de- where
tected within an element of observed sol id angle d~. This probabi l i ty is independent
of time only as long as the photon ( i . e . , the polar i ton) is within the crys ta l .
Therefore one is forced to consider the tota l p robab i l i t y dP1/d~-= that a scatter ing event occurs during the photon's traversal of the c rys ta l . Neglecting a l l at tentu-
ation e f fec ts , then for a crystal of length L,
L d2Pl (2.15) dP I
d~ ~ v dtd~ =
where v is the ve loc i ty of the incident polar i ton beam in the c rys ta l . Since there
is no f i r s t pr inc ip les prescr ipt ion which specif ies v uniquely, i t must be chosen
on physical grounds instead, a matter which we address fur ther on in the develop-
ment. The d i f f e ren t i a l Raman e f f i c iency , which is defined as the number of scattered
photons in sol id angle d~ ~, re la t i ve to the number incident, is c lear ly also given
by dPl/d~. Moreover, under steady-state condit ions, dP1/d~may also be interpreted
as the number of photons detected per uni t time, re la t i ve to the number incident
per uni t time,
dPl NdPl/Atd~
d~ N__ (2.16) At
I t should be noted that dPl/d-~ times the number of photons incident per uni t time
is not equivalent to the p robab i l i t y per uni t time of polar i ton scatter ing.
I f primes denote the f ina l state quant i t ies , then for Q(k) = Qo one obtains d i -
rec t l y from (2.13 and 15),
dP 1 2~f-TT'k'2L - - = = v' l<klVlk'>12" (2.17) d~ v g - -
HOPFIELD, 1969, obtained an equivalent expression with V=Vg s tar t ing from the Born
approximation for polar i ton scat ter ing, i . e . , Vg was chosen as the physical ly appro-
pr iate choice for v in (2.15). When attenuation is present, the e f fec t ive thickness
L of the sample is al tered. Such corrections have been described by LOUDON, 1964,
1965; for backward scatter ing from normal incidence, for example, L is replaced by [~(~) + K(~ ' ) ] -1 ( for K-l<< L).
Can dP/dt be measured d i rec t ly? There is one highly ideal ized case in which i t
would be possible in pr inc ip le , namely, backscattering from a semi - in f in i te crystal
in the absence of attenuation. The polar i ton would be assumed to propagate un t i l
a scatter ing event occurs, in which case i t is detected as a backscattered photon.
In th is instance, since the crystal is i n f i n i t e l y long, the experimentally observed
85
scattering probabi l i ty I per unit time and sol id angle, d2P1/dtd~, is simply equal
to the corresponding photon scattering probabi l i ty. Thus In(k) = n o]
d2p I d2p I 2~f, TT'k '2 - 2
=~l~-d-~-~ = ~ , I<kIVIK'> I (2.181 Vg
Although a veloci ty vG appears above as a consequence of the density of f inal states,
the observed scattering rate does not depend on the incident polariton veloci ty v
(associated with polariton kinematics) which appears in (2.15 and 17). This resul t
is an a r t i fac t of the in f i n i te propagation length available to the polari ton, and
does not correspond to conditions which would normally be real izable in practice.
For example~ although very long optical f ibers could provide the necessary path
length, the f i n i t e attenuation which is always present and, in fact , maximal in
the resonance regime, reduces the effect ive path length to a small f ract ion of the
nominal length in even the most favorable of cases.
Approach I I . We consider the application of App. I I to RS developed in various
papers authored by ZEYHER, BIRMAN, BRENIG and TING (BRENIG et a l . , 1972; ZEYHER et
a l . , 1972, 1974; ZEYHER and BIRMAN, 1974). We shall attempt as much as is possible
to omit mathematical detai ls , concentrating instead on just the essence of the de-
velopment. One begins by wri t ing the d i f ferent ia l scattering eff ic iency for single-
phonon Stokes scattering as
d2p k '2 2
9
T(ko, k~) : <Olako b T(m) a~; I0> (2.191
T(x) = V + V(x-H+iE)-Iv
V = V 1 + V 2
I Note that one could in pr inciple define an eff ic iency for this case as the number
of scattered photons per unit time divided by the incident photon f lux ,
d2Pl I d2P1 dP/d~ = N ~-d~s'/Nc = ~
which, however, possesses dimensions of inverse length. Since the crystal is con- sidered to be i n f i n i t e l y long, dP/d~ is physically a less appealing quantity to work with than is d2Pl/dtdE.
86
where I0> is the noninteracting vacuum, and k ' s represent the bare photon momenta; -o is the scatter ing operator. The scatter ing is then recast in terms of the in and
out eigenstates ~ which are exact in the photon-exciton in teract ion V I. These cor-
respond to incoming and outgoing waves which reduce asymptotical ly to free photons
far from the crys ta l , but are polar i tons inside of the crysta l . Defining creat ion-
annihi. lation operators A ~ corresponding to ~ , and denoting the interact ing vacuum
as I0>, then one f inds
T(ko,ko) = <~<IbqVI~>> = <01A<(_ko) bq VA>t(_ko)lO>. (2.20)
Thus the e f f ic iency is proportional to the square of the matrix element of V between
A t l0>, which is an incident photon at t = -= and a polar i ton within the crys ta l , f t and the f ina l state bqA<[0>, containing a polar i ton and phonon within the crystal
and a pure photon out~ide the crystal at t = +~. The v i ta l information regarding
propagation across thetcrystal boundaries and within the crystal must now be con-
tained in the states A~I0>. Clearly, the crucial element in the present formulation
w i l l be the procedure u t i l i zed to obtain these states. A
I t is useful to cast T e x p l i c i t l y in terms of the coef f ic ients ~i of the exciton
amplitude operators ~i = E i l /2 (c i + c i ) ; the ~'s are proportional to • of Section
2.1 for polar i tons within an i n f i n i t e crysta l . Although a set of equations analogous
to (2.3) obtains for the polar i ton coef f ic ients in th is case, the boundary condi-
t ions and labe l l ing of the modes are al tered because the polari tons are res t r ic ted
to a f i n i t e crystal region, and must connect asymptotical ly to pure photons in the
ex ter io r of the crysta l . To express the scatter ing in terms of the ~ i ' s , one must
specify the form of V 2, which we take to be simi lar to that of ( i . I ) , namely
V 2 : ~. f i j (q)c~c~b~ + h.c. (2.21) - ~ J H
i j q
Then
>
T(ko,k~) : ~. mi(k~) m mj(~o ) (EiEj)-3/2
i j (2.22)
x [ f i j (9 ) (E i+m' ) (E j+m)+f j i (q ) (m-E j ) (m ' -E i ) ] �9
Note that as before the square of the m's should measure the amount of exciton pre-
sent at m and m'. However, since the m's now correspond to exci tat ions in a spat ia l -
ly inhomogeneous medium (crystal plus vacuum), they cannot be obtained d i rec t l y in
k-space as was previously the case; the transformation to I-space y ie lds a set of
87
coupled Maxwell's equations for the amplitudes a ~ and the photon amplitude corre-
sponding to the vector potent ia l , which may be solved with f u l l account of the bound-
ary conditions for the problem.lf we make the ansatz that the quantum amplitudes a i
may be replaced by the i r c lassical analogues in ~-space, then the scatter ing may
be calculated from the ~-space version of (2.3) whence
T(ko- '-ok') = 2 ~ fyy,(q)(EyEy,+ozo')_ (~Ey,)-3/2 Syy,(q)_ yy' (2.23)
/d i 9 " ~ ; ( ~ ) ~ , ( ~ ) s~y,(q) : c e
I t is also useful to note that a is related to the photon amplitude A(~) by
E 2
~y(~) ~ Y gY A([). (2.24) E 2 _ ( ~ i ~ ) 2
Perhaps the most s t r ik ing feature of (2.23 and 24) and one which c lear ly empha-
sizes the differences between Apps. I and I I , is the behaviour predicted for m+E . T
Since A is normalized to a constant outside of the c rys ta l , the a 's , and conse-
quently T(ko,k~), blow up as e i ther m or~§ This feature, which is a consequence of the semiclassical determination of the amplitudes in App. I I , contrasts strongly
with the maximum f i n i t e strengths of the exciton amplitudes, and the matrix elements
occurring in App. I . While a classical osc i l l a to r driven at resonance absorbs energy
without l i m i t , in the quantum scatter ing formulation of App. I a single photon can
at most create a single polari ton of f ixed energy and, therefore, f i n i t e amplitude.
Within App. I resonance may be considered to be a consequence of the decreased
ve loc i ty , and therefore longer time, spent by polaritons in the crystal when m~E
[see (2.15) and (2.17) for example]. Y
~raetioal Considerations. Before concluding th is section, i t is worth pointing
out some of the pract ical d i f f i c u l t i e s encountered when applying the polari ton for -
mulation of RS. One important problem is how to include f i n i t e polari ton damping
which, as indicated previously, may s u b s t a n t i a l l y a l t e r the polari ton dispersion
for m~Ey. More s ign i f i can t l y , however, damping may compete with (undamped) polari ton
dispersive ef fects in determining the frequency dependence of the scattering near
to resonance, and unless both can be handled simultaneously, a detai led interpre-
tat ion of the observed frequency dependence at resonance w i l l not be possible. In-
clusion of damping cannot be accomplished by straightforward modif ication of the
matrix elements, since the eigenstates Ik> and Ik'> are no longer well defined in
the presence of damping. This d i f f i c u l t y could, in pr inc ip le , be overcome through
a Green's function approach (MAVROYANNIS, 1967; MILLS and BURSTEIN, 1969), by cal-
culat ing the photon proper self-energy in the presence of polari ton damping. How-
ever, e x p l i c i t calculat ions of th is sort have not been reported in the l i t e ra tu re .
88
I t thus appears that although actual inclusion of damping effects is r e l a t i v e l y
straightforward within perturbation approaches (BENDOW and BIRMAN, 1971b; FERRARI
et a l . , 1974), i t becomes more complicated within the polar i ton formalism. The rea-
son is that despite the presence of damping the scatter ing states are pure photons
in iperturbation theory; in polar i ton theory the scattering states are themselves
damped to begin with. A qua l i ta t i ve picture of damping effects may be deduced from
the modif ication in the strength functions in th is instance, but such predictions
are not quant i ta t i ve ly accurate. I t is easier to incorporate damping within App. I I ,
i f one assumes that the pr incipal e f fect of damping is the modif ication of the semi-
c lassical amplitudes (as before, one may append appropriate f r i c t i ona l terms to
the equations of motion, say). However, i t is uncertain whether such a procedure
provides a rigorous quantum account of the scattering in the presence of damping.
This concludes our necessari ly b r ie f sketch of the formal aspects of RS mediated
by polari tons which w i l l serve as a framework for the calculat ions described in the
fol lowing section. Readers interested in fur ther elaborations of the subject are d i -
rected, for example, to HOPFIELD, 1969; MILLS and BURSTEIN, 1970; and BENDOW, 1971a
for App. I ; and BRENIG et a l . , 1972 and ZEYHER et a l . , 1972, 1974, for App. I I .
3 Polariton Theory of the Resonance Raman Effect
This section examines the consequences and predictions of the polar i ton approaches
to RRS delineated in Section 2. Comparisons between polari ton predict ions, and those
of other theories and experiment w i l l be indicated. Sections 3.1 - 3.2 w i l l be re-
s t r ic ted to a discussion of polariton-mediated RRS, assuming the quasipart ic les
which scatter l i gh t to be IR inact ive. A b r ie f description of the modifications
ar is ing when the scattering is caused by (rather than mediated by) polari tons is
given in Section 4.
3.1 General Properties of the Scatterinq Rate
Various features of the frequency dependence of the RS ef f ic iency calculated using
App. ~ (see Section 2.2) are well revealed by inspection of (2.17), which we wri te
as
dP k '2 2 ~ - ~ JVkk, l , Vkk, ~<klVlk'>. (3.1)
Note that in contrast to the corresponding photon matrix element ar is ing in th i rd
order perturbation theory (see Section 1.2), as long as i t s dependence on k and k'
is weak, then Vkk, remains bounded for a l l ~,~ ' . Vkk, reaches a maximum when both
89
k,k ' correspond to exc i ton- l i ke polar i tons ( i . e . , near to resonance) and drops o f f
quickly outside the resonance regime. Very strong resonant enhancement is associated l not with var iat ions in Vkk,, but rather with the smallness of v and Vg, and the large-
ness of k ' , when m andg§ While the consequences are ul t imately s imi lar to those
obtained from a perturbation approach, the physical or ig in of the ef fects and the i r
in terpreta t ion are c lear ly d i f fe ren t .
Since Vkk, is the matrix element of an in teract ion consist ing of products of
exciton operators taken between polar i ton states k and k ' , one would expect that
the RS e f f i c iency might be expressibl~ as an appropriately weighted sum of products
of exciton strength functions for k and k ' . This is essent ia l ly the case for typical
choices of V, as demonstrated in detai l by MILLS and BURSTEIN, 1969. In fac t , i f the exciton-phonon coupling is only weakly k-dependent, then the pr incipal frequency
dependence of Vkk, is
Vkk, ~ Se(km)Se(k 'm' ) ( 3 . 2 )
where jus t single excitons k and k' have been assumed to correspond to the polari tons
k and k ' . The propert ies of S e have been discussed previously in Section 2; essen-
t i a l l y , one obtains a near-Lorentzian l ine shape near resonance,
S ~ 2 - E 1 2 ) 2 + 4E12,gl,2 I -I ~ Co - I (3.3)
for a single level E I . Thus IVkk, 12 is a product of Lorentzians of width 2E I Ig I I~oI/~
which are displaced by the phonon frequency ~o ( for single-phonon scat ter ing) . I f
the coupling coef f i c ien t f in V 2 is k-dependent, than i t becomes an addit ional source
of frequency dependence which must be accounted for . In a l l instances, strong en-
hancement of dP/dEwi l l ar ise from the polar i ton ve loc i t ies . For example, when ex- 2)-3/2 citon damping is ignored V=VE=Vg, so v -I diverges as (2 -E I as ~-*E I , for a
single dispersionless level .
To examine some ef fects of exciton damping, le t us f i r s t assume that al l exciton-
phonon couplings are much weaker than the exciton-photon in teract ion, so that to a
f i r s t approximation we may neglect the ef fects of damping on the strength functions.
Thus, i f the interact ion V 2 does not involve e x p l i c i t l y k-dependent couplings, the
ef fects of damping on V.., may be neglected. For example, for typical values of g and F, replacement ofS~(E2-w2) 2 by (E2r-~2) 2 + F 2 in ( 3 . 3 ) w i l l have l i t t l e e f fec t .
For k-dependent coupling, the frequency dependence w i l l be al tered due to changes
in k:k(m) induced by damping. However, in a l l cases, the pr incipal inf luence of
exciton damping is expected to be through the dynamical factor k'2/vv~. For example,
for a dispersionless leve l , k' w i l l no longer vanish at the longitudinal exciton
frequency or increase without l im i t as m'~Eu The ve loc i ty v must be chosen as v E
when damping is present (see Section 2.1); for small F, v becomes small but not
90
zero as m~Ey (for a dispersionless level). However, the final state velocity v~
is not a physically proper velocity when damping is present; this di f f iculty arises
because damping was not incorporated consistently from the start in calculating
dP/dE. Explicit calculations within App. I which remedy this state of affairs are
not available in the literature. Actually, as evident from (2.13), e.g., in the i presence of phonon dispersion the factor Vg in (3.1) will be replaced by @E(k')/@k'+
this removes the singularity from (v~) -I for m'~E I, but does not ~Pb(k-k' )/~k' ; remedy the unphysical values of @E/Bk which s t i l l arise when damping is included.
All that can be reliably concluded in this connection within App. I is that damping
weakens the photon resonances as m,m'~E from below. Although i t is reasonable to
suppose that similar effects occur throughout the resonance regime, the explicit
form of the modifications has yet to be established.
When phonon damping r L is included but exciton damping neglected, one obtains
d2p ~ ~ / 2 rL 2 dk'k' [%2_(E(k)_E(k,))212+rL 2 IVkk, I , (3.4)
i . e . , the 6-function in (2.13) is replaced, as usual, by the imaginary part of the
interact ing phonon Green's function. The double displaced Lorentzians composing 12 IVkk, are now d is t r ibuted over a width r L centered about ~o' thereby suppressing
I the-frequency dependence of dP/dE. S imi la r l y , Vg is now replaced by a more smoothly
varying factor as a function of frequency. Deductions of general app l i cab i l i t y are
not eas i ly made; the speci f ic form of the frequency dependence must be worked out
separately for each par t icu lar case.
In a spa t ia l l y dispersive medium Vg approaches the exciton ve loc i ty ~E(k)/~k in
the region where the polar i ton becomes exc i ton- l ike (see Fig. 3). Thus the l im i t i ng
factor at resonance in the absence of damping is the smallness of the exciton veloc-
i t y . However, i f exciton damping is included, the group ve loc i ty once again takes
on unphysical character is t ics , and cannot be u t i l i zed d i rec t l y in the formula for
RS. This state of a f f a i r s is not surpr is ing, since once again the damping has not
been accounted for consistent ly to begin with.
To summarize then, within polar i ton App. I RSS is due to the maximization of the
exciton portion of the polar i ton states near resonance, combined with the smallness
of the polar i ton ve loc i ty and/or density of states. The sharpness and strength of
the resonance are determined by the size of the exciton-photon coupling, the exciton
and phonon dampings, and the exciton and phonon dispersion. Al l of the l a t t e r ef -
fects compete to determine the overal l frequency dependence of the RS e f f i c iency .
At present, e x p l i c i t resul ts are avai lable only for specialized cases, although
qua l i ta t i ve considerations can suggest the possible behavior under more general
conditions.
91
In App. I I the Raman ef f ic iency is eventual ly expressed in terms of semiclassical
exciton amplitudes ~ [see (2.23)]. In the absence of spat ial dispersion and damping,
App. I I predicts resonance behavior equivalent to that of perturbation theory, in
contrast to App. I . Damping is easi ly taken into account phenomenologically via
modif ication of the suscep t ib i l i t y , and hence the mi 's. However, App. I I does not
t e l l us how to calculate the damping; any prescript ion for the l a t t e r must be for -
mulated separately. The results for the ef f ic iency when temporal damping is included
are equivalent, in the simplest instances, to those obtained when the bare exciton
approach has been modified to include exciton damping (BENDOW and BIRMAN, 1971b).
The pr incipal manifestation of polari ton ef fects in the l i m i t of small damping and weakly dispersive excitons is through the k-dependence of the interact ions and the
phonon dispersion. The k's involved are no longer determined by conservation of
photon momentum, but conservation of the polari ton quasimomentum, i . e . , #(+_kk i ) =
k -k ' , where ~i are the phonon momenta. 1
3.2 Calculations for Model S%stem~
In th is subsection we invest igate the frequency dependence of resonance RS predicted
by polari ton approaches for various s impl i f ied cases, and compare the resul ts to
the corresponding predictions of perturbation theory.
Consider scattering mediated by a series of dispersionless, undamped excitons.
In the present discussion we shall be concerned with the "uncorrected" e f f ic iency
dP/dE which we define via
dP1 dP T E = T T ' f ~ - - ; (3.5) a dE
the exciton-phonon interact ion w i l l be taken to have the form in (1.1). Then the
Stokes scatter ing at T=O K follows as
dP k '2 = 2 ~
VgV v
where
I~ fyy,(k,k') Uyy,(k,k' YY
2 1 (3.6)
, �9 (k). Uyy,(k,k') ~ Xy (E)X~,(~) + ~y (~) ~X, (3.7)
We have above assumed that just single polari tons correspond to (have the same ener-
gy as) mand m', and have neglected phonon dispersion as wel l . An especial ly simple
resul t which reveals many general properties of (3.6) fol lows when one takes f , = YY ~yy, f y , whence
92
2 dP k' = 2~ "E'- ' ' " ' " " " Ir~"E'r"J 1
R(k,k') -= ~] f((k,k')Ig~(12(com'+Eu 2) (3.8)
_ _ u (~2-E.(2) (~'2-E 2)
R is id.entical, to within a constant, to the Raman amplitude calculated by pertur-
bation theory (see Section 1.2; the equivalence holds fqr f y,# 6u ,fu as wel l ) .
Denoting perturbation results by "B" and polariton results by "P" then
P k' : ~-- (3.9)
B
Essential ly, this factor enhances the scattered resonance and suppresses the inc i -
dent one, as compared to the perturbation theory results.
The factor k ' /k is displayed as a function of frequency in Fig. 7 for CdS para- meters. The scattering for a single level predicted by (3.8) varies as (m2-E12)-3/2 as m+E I, and as (m'2-E12)-5/2 as m'-~E I, as opposed to the (m2-E12)-2 and (m;2-E12)-2
behaviour predicted by perturbation theory. These same results can be anticipated
d i rect ly from (3.1); for example, since Vkk, varies slowly compared to Vg, dP/dE ~ -I -3/2 - - Vg ~ (m2-E12) as m+E I.
1.4
'.• 1.2
'.'- 1.0
:m
o.e
I
1.6
0.6
0.4
I ! 2,51 2.53 2.55
Y .... y
II 2.E ;'.59 1' 2.61
t E I E 2 E3 El+W 0 E2+~ o E3+u o
INCIDENT PHOTON FREQUENCY oJ (eVl
I 2.63 2.65
Fiq. 7. Dynamical factor k ' /k vs photon frequency for CdS A-exciton parameters (from BENDOW, 1970)
93
Note that the eff ic iency wi l l be influenced by exactly the same factors involved
in calculating R(k,k') within perturbation theory including, e.g., multiple-band
effects, discrete-continuum interferences, etc. (see Fig. 8). The frequency depen-
dence of R wi l l be substantially influenced by polariton effects only i f R is strong-
ly wave vector dependent. For example, for forbidden phonon scattering where the
leading term in f ( k , k ' ) = I~ -~ ' I , we may expect substantial additional enhancement for m or ~'§ . Moreover, the scattering kinematics Decome those of polaritons rather
than photons.YThis situation contrasts with the perturbation approach, where just
the difference in (bare) photon momenta, ko-k'o ~0 is involved. For the intraband
Frohlich scattering experiment reported by MARTIN and DAMEN, 1971, for example, the
coupling function f peaks at values of wave vector substantially greater than the
values of I~o-~'ol characterist ic of the experiment. The la t te r authors obtained a
resonable f i t to three widely Spaced data points without including polariton cor-
rections. Nevertheless, i t is possible for polariton effects to become signi f icant
in various instances; a more detailed extension of the measurements closer to re-
sonance may have been desirable to establish thei r role in the la t te r case.
Another source of k-dependence is the photon-exciton coupling gy(k). In this
instance, both the polariton group velocity and the function R(k,k') wi l l be modi-
f ied. For the case of quadrupole excitons, for example, gu ~ k is the leading
dependence of gy. One finds (BENDOW, 1971a) that the rat io of quadrupole (Q) to
dipole (D) exciton-mediated eff ic iencies varies as
dP (d-~)Q k4k,4
~D (3.10)
in the regime where the relation g#~k is val id. Far from resonance the above rat io
reduces to k2k '2, which is the perturbation theory result . The frequency dependence
of the Q to D rat io is exhibited for a single level and typical parameters in Fig. 9.
Existing experiments on forbidden exciton scattering (BALKANSKI et a l . , 1972; COM-
PAANS and CUMMINS, 1973) do not appear to display resonances substantial ly sharper
than those predicted by perturbation theory. However, i t should be noted that (3.10)
is only valid over a restr icted region in k(w) and, moreover, does not include damp- ing effects which could be crucial in actual observed cases.
When exciton dispersion is present, then the calculated dP/dE no longer diverges
as ~,~'+Eyas in the dispersionless case. Rather than Vg-H], Vg+Vex c ~ ~Eik)/~k near resonance; thus dP/dZ ~ k -I for ~+Ey(O) and ~k'2/v~~k ' for~'+ET(O), apart from any dependence stemming from Vkk,. For a single parabolic exciton, for example, k ~
2m*[~-E1(O)]1/2 for ~>EI(O~ i . e . , in the region where the polariton is exciton-l ike.
94
4.5 I ~ i [ i I
4.0 (a) iN II
\ PI! N \ \ I t \
_ 5.5 \\\ !
)o ', t ',,- 3.0 , ; /
# / / U . . / t I
2 . 5 " / / / I(
2.o- . - " ,j / " ii
~ I I I I I& III 2.46 2.48 2.50 2,52 2.54 2.561
' IP I '1 f+ i I st iI
N / ' , I' ~" ; I II I " ; t II '-'J , ] ~P II | ; / ~. I i /
| i i ]
" I ~ I I \ , , ^ , I
\ ',, I/!\ 1
2.58 2.60 I 2,62 2.64
E 2 E 5 El+@ 0 E2+~oE3+w 0
INCIDENT PHOTON FREQUENCY ~(eV)
-,m
o ,=,-, o
( b )
2.53
El
I 2.55
/ /
2.57 T 2.59
E 2 E 3 El+Ce 0 Eg
IY 2.6t 2.63
E2+~ 0 E3+~O
INCIDENT PHOTON FREQUENCY wIeV)
Fi 9. 8~ (a) Contribution to Raman tensor from discrete (broken line) and continuum (solid l ine) states for A-exciton in CdS. "P" and "N" indicate regions where the contributions are positive and negative, respectively. (b) The polariton RS cross section vs incident frequency ~, calculated from (a)
95
" 7 0 I -
m , -
=, -6 I--
u.I
~ 5
o o = - 4
A
' / ~ ' ,
J
I 2.40 2.50
INCIDENT PHOTON FREQUENCY ~ (eV)
S 2.60 2.70
Fiq. 9. Logln of quadrupole to dipole in tens i ty ra t io F vs incident photon frequency m. Here the ~honon frequency is taken as ~ = 0.08, the remaining parameters are typical of CdS. "A" indicates the exciton ~from BENDOW, 1971)
The resonance enhancement for w,~'<El(O) w i l l be s im i la r to that in the nondispersive
case, except that Vg at ta ins i t s maximum near to E I rather than increasing without
l im i t . For zero damping, the in and out resonances occur at the somewhat higher
energies E~ and E~ + ~o where E~ is the longitudinal exciton, rather than at E I
and E I + n o (see BENDOW and BIRMAN, 1970). The var ia t ion of Vg with frequency for
a parabolic exciton is indicated in Fig. I0. Actual ly , in the case of dispersive
excitons, a m u l t i p l i c i t y of degenerate polari tons may ar ise, a circumstance which w i l l be noted in Section 3.3.
- 2 .0
>=
E - 3 .0
"6
o ~ .J
-Z,.O
2.55 2.57 2.59 Frequency ~ ( e V )
Logio of group velocity vs frequency for A-exciton in CdS; E(O) is the exclton energy for k=O
96
Al l of the above resul ts have been obtained for cases where exciton and phonon
dampings are negl ig ib le . Unfortunately, damping may very l i k e l y have a major e f fect
on the resonance properties of the e f f i c iency . I t is therefore disappointing that
e x p l i c i t calculat ions have not been carr ied out within App. I for scatter ing med-
iated by damped excitons. In view of the l a t t e r , we must necessari ly r e s t r i c t the
discussion of damping effects to an extension of the qua l i ta t i ve analysis outl ined
in Section 3.1. As stated here, unless e i ther the damping is unusually large or Vkk,
is strongly _k-dependent, we expect the influence of damping on Vkk, to be r e l a t i v e l y ' In the presence of damping v must minor compared to that on the ve loc i t ies v and Vg.
be chosen as v E, so that
d__PP =VE-1 = n+2wK/s (3.11) dE c
as m~E. The e x p l i c i t frequency dependence of v E is somewhat complicated, even for
a single dispersionless leve l . Nevertheless, the general behaviour is rather simple
to deduce (see Fig. 6 . ) : far from resonance v E is s imi lar to the Vg calculated for
zero-damping; however, as w~Ey, v E does not tend to zero in the same manner as
Vg(~O), but bottoms out to some f i n i t e value near to Ey. The smaller the damping,
the lower the minimum value of v E, and hence the greater the resonant enhancement.
The trends implied by the l a t t e r are s im i la r to those predicted by including damping
within a perturbation approach (BENDOW and'BIRMAN, 1971b), namely,
dP) _[(2_ )2 r2] -1 (TE B E12 +
(3.12)
for m~E1; however, the detai led form of the frequency dependence is c lear ly d i f f e -
ent. The damping may also be included phonomenologically in the polar i ton cross
section by changing the various k's in the dynamical factors to the real part of
k(~), and replacing E~ + Ey+iPT (BENDOW, 1971b). Results for the calculated e f f i -
ciency are displayed in Fig. 11.
Although paral le l arguments to the above cannot be u t i l i zed for the scattered
photon resonances, i t is apparent that damping w i l l el iminate the s ingu lar i t y in I the f ina l state polari ton density of states (which equals Vg for p= 0), and that
k '2 w i l l now be bounded as m'~Ey. The smoothing out of v~(m') is analogous to the
broadening of harmonic densit ies of state induced by anharmonic interact ions. VERLAN and OVANDER, 1967, in considering scattering for the case where exciton damping is
so large that Vg is everywhere less c, u t i l i zed Vg in place of v E to calculate dP/dE.
However, not only are these conditions not commonly met in pract ice, but even i f
they were, i t is far from certain that the choice of V=Vg is j u s t i f i e d for the large
damping case. In cases where Vkk, is strongly -k-dependent (such as for intraband
Frohlich coupling), i t is evident that damping may substant ia l ly suppress the f re -
quency var ia t ion exhibited by Vkk,. By u t i l i z i n g the real part of k and k' in place
97
~- 5
4
,g r ~2
( I J I
14
12
]I
I 0
. . . . 9
0 - - 8 2.51 2.53 2.55 2.57 2.59 2.61 2.63 2,65
INCIDENT PHOTON FREQUENCY ~o (eV)
~ Polariton Raman cross section with the inclusion of damping, for CdS A-ex- ee Fig. 8). The exciton is represented as a three- level system as indicated
by arrows in the f igure (a l l levels for N>3 are collapsed into one). The broken l ine is a calculat ion for the forbidden exciton case, for which the N=~ level does not couple to l i gh t
of the zero-damping values, one can obtain a qua l i ta t i ve indicat ion of the modif i -
cation which w i l l be induced by damping. Expressions obtained by the l a t t e r proce-
dures, however, do not necessari ly y ie ld quant i ta t i ve ly accurate predictions of the
RS ef f ic iency.
As indicated previously, in the absence of spatial dispersion the ef f ic iency pre-
dicted by App. I I is the same as that obtained in perturbation theory (assuming
appropriate transmission and attenuation corrections have been made) for the case
where Vkk, depends weakly on k. For the k-dependent case, i t is merely necessary
to replace a l l bare wave vectors in perturbation theory by the i r polar i ton counter-
parts. Other modif icat ions, some of which w i l l be described la te r , occur for large
values of damping and for scattering in crystals of f i n i t e size in special circum-
stances (ZEYHER et a l . , 1974). Within App. I I the modifications of dP/d~ which are
induced by exciton damping are usually accounted for via modif ication of the ~ i ' s , which stem from the par t icu lar form of the suscep t ib i l i t y chosen for the c rys ta l .
In the presence of damping
X ~ Z g i 2 ( J Ei21-1 ~ gi 2 [ 2 Ei2 i r i ( ~ ) ] - I - § . - - ( 3 . 1 3 )
1 1
98
with the eventual resu l t that the factors (Ex2-m 2) in (3.8) are replaced by (Ex2-m2+
i t ) . This resu l t is essent ia l ly the same as one would obtain from perturbation theory
when damping has been included. Spatial dispersion ef fects within App. I I w i l l be
discussed in Section 3.3.
In Table I we contrast the predict ions of polar i ton approaches with those of per-
turbat ion theory, for the frequency dependence of dP/d~ as m§ I for a single exciton
leve l , under a var iety of assumed condit ions. I t is to be noted that the assertions
by various authors such as BARKER and LOUDON, 1972, that the predict ions of pertur-
bation theory and polar i ton theory for resonant R~ are equivalent, are not s t r i c t l y
correct . The formulae used by Barker and Loudon are semiclassical expressions which
appear designed to y ie ld the polar i ton resul ts , and which cannot be obtained from
f i r s t - p r i n c i p l e s quantum treatments. Moreover, even granting the va l i d i t y of the i r
s tar t ing expressions, agreement w i l l not be obtained between polar i ton and perturba-
t ion predict ions for any other conditions except dipo~e-exciton mediated, a11owed-
phonon scat ter ing; predict ions for the quadrupole-exciton mediated case, for example,
w i l l d i f f e r non t r i v ia l l y [see (3.10)] . And although the di f ferences obtained here between polar i ton and perturbation approaches may be small from a pract ical stand-
point, they nevertheless represent s ign i f i can t di f ferences in pr inc ip le , and are
th~s of in terest from a fundamental standpoint.
Table I . Incident photon resonance behaviour [~§ Raman ef f ic iency for an isolated exciton level a
D ~ 2 _ E12; table displays resul ts for undampe~ dipole excitons and a]lowed-phonon
scatter ing except where otherwise stated.
Case Perturbation Polariton Polari ton theory Approach I Approach I I
Allowed D-2 D-3/2 phonon
Dispersionless b Forbidden D-2 D-5/2 excitons phonon
b Quadrupole D-2 D-7/2 exciton
Dispersive exciton D -2 BE(k)-1
( 2_E23+F2)-2 -I Fin i te damping VE
a Not including re f lec t ion or attenuation correct ions. b For regime where coupling is l inear in wave vector.
c k is the polar i ton wave vector determined from E(k) =w.
D-2
D-3
D-4
c [ J - E 1 2 ( k ) ] -2
(J-E12+r2) "2 i
99
An interest ing question within App. I to polar i ton RS is the l inewidth of the
scattered radiat ion induced by temporal damping. Since within perturbation theory
the only real c rys ta l l i ne quasipart ic les involved in the scattering are phonons,
the scattered l inewidth w i l l simply be that of phonons (LOUDON, 1963). In the polar-
iton picture, however, both a phonon and a polar i ton const i tute the real f ina l state
(at least within the c rys ta l ) , and one might expect the damping to be a sum of phonon
(L) and polar i ton (P) dampings, F = FL+F P. The polar i ton damping is large only near
Im 2 - El21 ~ 2glE1/~o I /2) where i t achieves a value close to the resonance ( i . e . , for
exciton damping F E. Combined with the fact that F E and F L are generally of the same
order of magnitude, i t may be d i f f i c u l t to test the l ine shape dependence carefu l ly
enough for a wide var ie ty of cases to dist inguish between the two predict ions. There
are, of course, many other sources of experimental l ine shape broadening, be they
instrumental or inherent (spat ial damping, for example, contributes a l inewidth, as
w i l l be pointed out in Section 3.3). Exist ing experiments have not indicated any
marked increase in scattered l inewidth for o~+E . For example, the l inewidth remains
nearly constant for the Is forbidden exciton resonance in Cu20 (COMPAANS, 1973), and
appears to increase s l i g h t l y in CuCl (OKA et a l . , 1973). Recent data of SCHMIDT,
1975, for the TO phonon in ZnSe, where F L << ?p, suggest that the exciton width does
not a f fec t the scattered l ine shape. I t is l i k e l y that a more careful theory, which
incorporates a l l damping effects in a rigorous fashion, w i l l be required to y ie ld
predictions consistent with these observations.
I t is evident from the present discussion that most of the pr incipal features
predicted within perturbation and polari ton approaches are very s im i la r , such as
the nature of the in (m) and out (m') resonances, the interferences between dis-
crete and continuum contr ibut ions, and the effects of wave vector dependent couplings
on the Raman spectrum. Quanti tat ive differences arise because the polar i ton scat ter-
ing is governed by the polar i ton wave vectors k and k ' , which become large compared
to the photon momenta in the v i c i n i t y of resonance. The calculat ions of BENDOW et
a l . , 1970, indicate that in the case of allowed scatter ing such differences become
pronounced only for large values of exciton-photon coupling gu For wave vector
dependent couplings, however, the differences could be more pronounced even for
smaller values of gy. In a l l cases, the major ef fects would occur very close to re-
sonance, where there does not appear to be any ex is t ing experiments with su f f i c ien t
deta i l and/or accuracy to decide on the differences in exponents n in the energy
denominators (E2-J) -n predicted'wi th in the d i f fe ren t approaches. Moreover, the
broadening effects of damping and spat ial dispersion require a very accurate and
detai led determination of frequency dependence in order to dist inguish between the various theories. As w i l l be pointed out in Section 3.3, spat ial dispersion and
f i n i t e crystal ef fects may o f fe r the poss ib i l i t y of detecting the polar i ton nature
of the scat ter ing, although to date only l imi ted data is avai lable. In summary, i t
seems f a i r to say that both polar i ton and perturbation predictions of resonance RS
share the same successes as well as fa i lu res in the i r in terpretat ion of experimental
100
results. I t is as yet uncertain to what extent the polariton character of the scat-
tering leads to s igni f icant , or even unambiguously detectable effects in observed
scattering ef f ic iencies.
3.3 Spatial Dispersion and Finite Crystal Effects
As pointed out previously, a feature peculiar to polariton formulations of RS is
the possib i l i ty of multichannel scattering in the case of spatial dispersion, where
a number of degenerate polaritons exist in the incident and/or scattered channels.
For example, the four possible channels for scattering via a single dispersive ex-
citon are indicated in Fig. 12. Aspects of single branch scattering in the presence
of spatial dispersion were discussed in Section 3.2; in this section we explore
multichannel scattering effects. Brief consideration wi l l also be given to f i n i t e
crystal size and spatial damping effects in polariton RS.
/
E[O: q ~z
WAVE VECTOR k
Fi 9. 12. Four scattering channels for the A-exciton in CdS (from BENDOW and BIRMAN, 1970)
We begin with a description of multichannel scattering within App. I . In the
absence of damping, an incident photon m wi l l create an admixture of N degenerate
polaritons c i (k i ) = m, i = I, 2 . . . . N. The i n i t i a l polariton state ~ becomes a
l inear combination of degenerate polariton states ~ i ' i . e . , ~ = Sidi~i with the
coeff icients d i determined by appropriate boundary conditions (bc). I t is apparent
that additional boundary conditions (abc) besides the usual Maxwell or Fresnel re-
lations wi l l be required to uniquely specif iy the d i 's (see, e.g. PEKAR, 1958; AGRA-
NOVICH and GINSBURG, 1966). The nature of the abc has been the subject of some con-
101
troversy over the years, with r~gorous solutions avai lable only in model cases (ZEY-
HER et a l . , 1971). Recently, bc's which seem independent of the detai led properties
of the surface potential have been proposed for the center-of-mass motion of the
exciton (SEIN, 1969, 1970; BIRMAN and SEIN, 1972; AGRAWAL et a l . , 1971; MARADUDIN
and MILLS, 1973). The derivat ion of these abc actual ly assume that exciton re f lec-
t ion at the boundary can be neglected (ZEYHER et a l . , 1972). From the experimental
standpoint, most attempts to discern ef fects of mult ip le wave exci tat ion in optical
spectra appear to be inconclusive. For example, damping and spat ial dispersion both
exert s im i la r influences on re f lec t ion spectra (see SKETTRUP and BALSEV, 1970), so
i t is d i f f i c u l t to d i f fe ren t ia te between them. Observations of interference between
mult ip le polar i ton waves has been claimed by various invest igators (BRODIN and PEKAR,
1960; GORBAN and TIMOFEEV, 1962), with the recent work of KISELEV et a l . , 1974, be-
ing perhaps the most convincing. Rather then delve more deeply into the theoret ical
and experimental aspects of the abc and multimode exci tat ion problems, we refer the interested reader to the extensive l i t e ra tu re on the subject, some of which has been
referenced above. We shall here rather be concerned with the general implications of
multimode exci tat ion for polariton-mediated RS.
While we expect that multichannel effects w i l l be manifested in pr inc ip le in the
RS e f f i c iency , two factors w i l l strongly influence the i r role in practice. These
the re la t i ve exci tat ion e f f ic ienc ies .[di 12 and the re la t i ve values of the mode are
damping functions ~ . The contr ibution of a given channel w i l l be substant ia l ly
suppressed for large r i and small Idi 12. For example, exc i ton- l ike polaritons w i l l
be damped much more strongly than photon-like polari tons ( t yp i ca l l y by a few orders of magnitude).
The RS ef f ic iency formulated within App. I is not straight forwardly generalizable
to the case of multibranch scatter ing. Note that the only quantity describing the
scattering which remains well-defined is the quantum probabi l i ty for t ransi t ions to
a f ina l state ~ ' , which is given as usual by the Golden Rule,
dP ~ f d 2 (3.14) = 2 J E , ]<2di~i[Vl~,> I 6(~-E~,).
i
In order to deduce the observed scattering e f f i c iency , one must specify the rate
at which energy is transported in the polar i ton state ~ = z id i~i . Since the d i f -
ferent portions of the wave propagate with d i f fe ren t ve loc i t i es , i t is not obvious
whether, or how, a single physical ly meaningful ve loc i ty characterizing the wave can be defined. Moreover, i t is unclear how to properly account for attenuation
corrections, since each polari ton is characterized by a d i f fe ren t damping function.
These questions can be simply resolved only i f interference between channels can
be neglected. In th is instance the ef f ic iency reduces to just a sum of individual channel e f f i c ienc ies ,
102
dP 2~(I-R) Z T j ' I d i 12 kj 2 i<~i I V ,>j2 dPij T_= = i j ~ l~j z ~. d~ (3.15)
i j
where R is the incident photon r e f l e c t i v i t y and Tj the transmission coef f i c ien t of
polar i ton j . To obtain R, T j , and d i above, one must f i r s t solve the multimode
boundary problem, as discussed above. The only exist ing multichannel calculat ions
within App. I are those of BENDOW and BIRMAN, 1970, for CdS. Consider the ind iv idual -
channel, unweighted ef f ic iences defined by
2 dPij _ kj 2 dE Vg(i)Vg(j) l<~i lVl~j >] ; (3.16)
the calculat ion of the dP/dE's for the four channels in Fig. I are i l l us t ra ted
in Fig. 13a; the f ina l e f f ic iency incorporating the d's and T's is indicated in
Fig. 13b. Below E L only S I (purely lower branch) scatter ing can contr ibute to the
o
== o b
w o
o ~
2.54 2.56 2.58 2.60 2.62 Frequency o~ (eV)
Fig. 13a. Log1^ of unweighted e f f i - ciencies dPiJ~E , vs frequency for the CdS A-ex~iton, for the four channels indicated in Fig. 10
Stokes ef f ic iency. We note that for EL<m<E~+~ o, S 2 dominates, while for ~>E~+~ o,
S 3 (purely upper branch) dominates. Typical ly ( inter ference ef fects aside), one
expects ident ical trends in the presence of small, but f i n i t e damping, ~ Ie88 those
channels which were dominant for ? = 0 turn out to be much more strongly damped
103
8.00
(b) 730
6.60
5.90 o u g E
i1:
~ 5 .20
.J
&50
3 80 I E ( O ) ~ t , Frequency ~) (eV)
E(O]+(~ ,I
Fi 9. 13b. Loglo of the f inal Raman e f f i - ciency vs frequency calculated from the dPi j /dE's (from BENDOW and BIRMAN, 1970)
for s # 0 (dominated by exc i ton- l i ke polar i tons). The l a t t e r is usually not the case
in typical instances.
We note that the multichannel calculat ions of BENDOW and BIRMAN, 1970, predict
e f f i c ienc ies which are very s imi lar to those obtained by single channel theory ( i . e . ,
ignoring spatial dispersion), and thus perturbation theory, for typical parameters,
such as in CdS. Greater di f ferences between various calculat ions are shown to occur
for larger photon-exciton couplings. I t is doubtful that any of the commonly in-
vestigated semiconductors are characterized by su f f i c i en t l y large couplings to easi-
ly d i f fe ren t ia te between the various predict ions.
App. I I is especial ly well suited to tackle multimode scatter ing problems, since
the pr incipal d i f f i c u l t i e s surrounding attentuat ion and propagation of the polar i ton
waves which trouble App. I are, in fact , absent from App. I I . In the v i c i n i t y of a
single level the e f f ic iency for backscattering from a semi- in f in i te crystal takes
the fol lowing approximate form (ZEYHER et a l . , 1974)
2 Im(k i + kj + 2k') dP o~ ~, A.A. d.~ 1 j Re(ki_kj)2 im(ki+kj+2k,)2 i j= l +
2 ozo' + E I
Di(m) = Ig I I 2 [E12(ki) - j ] [ E 1 2 ( k ' ) - ~jj I 2 ]
Di(~)D;(~) (3.17)
104
where k' denotes the f ina l state polar i ton, and q = 0 has been assumed. The f i e l d
amplitudes A i are chosen to provide the appropriate admixture of polar i ton modes
obeying the bc's. We have above also incorporated the effects of spat ial damping
stemming from S(q) in (2.23). In general, the interference terms ( i # j ) are small,
since usually Re(kl k2 ) >> Im(k I + k2). The diagonal terms ( i = j ) correspond to
in t ra - or interbranch scatter ing (depending on k ' ) ; t he i r re la t i ve magnitude depends
on the abc u t i l i zed to determine the A i ' s . ZEYHER et a l . , 1974, found that in f re -
quency regions where both polari tons can propagate, then for equal skin depth the
two diagonal terms contribute equally to dP/dE. Nevertheless, the skin depth is
usually much greater for the phonon-like branch, so i t s contr ibut ion is generally
favored (except, perhaps, in very thin samples with L-Imk<<1). Note that transmission and attenuation factors do not separate out from dP/d~ as they do in s impl i f ied
cases in the absence of spat ial dispersion. Computations of dP/dZvs w in the v i -
c i n i t y of E I for CdS parameters are indicated in Fig. 14. Note that the sharp reso-
nance peak obtained in the absence of spat ial dispersion is strongly suppressed in
i t s presence. I t is found that for the l a t t e r case the extra-channel contr ibut ions
have a negl ig ib le e f fect on dP/dE. Differences in dP/dE with and without spat ial
dispersion are pr imar i ly a resul t of differences in the polar i ton dynamics (disper-
sion) in the two cases. The resul ts in Fig. 14 are not eas i ly compared to the App. I
resul ts which were displayed in Fig. 11, for example, since attenuation ef fects have
~0-2 t /~
_ / / I
10 -4
-40
L t
-30 -2Q -10 0 10 I L -
20 310 Z,O
~ - 0)15 [cm -1 ]
Sum of both polar i ton branch contr ibut ions to the cross section for f i r s t - lowed scatter ing. Solid l ine : calculated with spat ial dispersion; dashed
l ine : calculated without spat ial dispersion. Corrections for absorption and re f lec- t ion are included (from ZEYHER et a l . , 1974)
105
not been included in the l a t t e r . Nevertheless, both approaches appear to agree that
multichannel effects t yp i ca l l y do not have a s ign i f i can t ef fect on the frequency
dependence of the RS e f f i c iency .
Multimode scattering calculat ions have been pursued by BRENIG et a l . , 1972 for
the case of Br i l l ou in scatter ing in resonance with a diScrete dispersive exciton
leve l . The pr incipal conclusion is that an octet of l ines, rather than the usual
1o-S
10-6
10-7
10 .8 ,
I0_9' I -30
LToto.I efficiency 22
J //"
/ \
~L \"-~\\\
\
13)
(2)
I I l I I I ~-~is -20 -10 0 10 20 30 [cm-i~
Lower branch Br i l l ou in scattering e f f ic iency vs frequency calculated for three d i f fe ren t abc (from BRENIG et a l . , 1972). The three abc's take the form
(1) (P1-Xo E 1) + (P2-Xo E 2) = 0
(2) nl(Pl-Xo El) + nz(P2-Xo E2) = 0
PI P2 (3) + = 0
(nl-ne)(n12-1) (n2-ne)(n22-1)
where ~ is the background suscep t ib i l i t y ; n., n^ and n are the re f rac t ive indices corresponding to the polari tons and the noni~ter~cting ~xciton; and the Pi 's are the p o l a r i z a b i l i t i e s of the polari tons
106
doublet, should be observed. Determination of the width and sh i f t of the l ines per-
mits, in pr inc ip le , a complete determination of the dispersions of the various po-
la r i ton branches. An interest ing feature of the calculat ion is that the re la t i ve
in tens i t ies calculated for the Br i l l ou in l ines d i f f e r quant i ta t i ve ly depending on
the abc's u t i l i z e d , so that resonance Br i l l ou in scattering measurements may aid in
resolvi.ng the abc controversy. Computed resul ts for the resonance behaviour are in-
dicated in Fig. 15. Recent experiments in GaAs (ULBRICH and WEISBUCH, 1977) and in
CdS (WINTERLING and KOTELES, 1977) have corroborated the predict ion of a mu l t ip le t
of l ines at frequencies consistent with scatter ing from mul t ip le channels. In neither
instance were the measurements s u f f i c i e n t l y detai led to allow quant i ta t ive comparison
with theory, and to thus dist inguish between the various abc's. And although the
GaAs data appear to conform well to the overal l predict ion of BRENIG et a l . , 1972,
the l ines appearing for m>E~ in the CdS study do not appear to be consistent with
these predict ions.
App. I I suggests that in s i tuat ions characterized by spat ial inhomogeneity, in-
cluding scattering in thin samples, i t may be necessary to take e x p l i c i t account of
boundary ef fects in calculat ing e f f i c ienc ies . Within App. I I th is is accomplished
by solving the (inhomQgeneous) Maxwell equations for the polari tons subject to appro-
pr iate bc. As noted previously, the fac tor izat ion of the e f f ic iency into an uncor-
rected term times transmission and attenuation factors does not obtain in general. To th is time, the detai led ef fects of spatial inhomogeneity and bc on RS have been
worked out only in a l imi ted number of specialized cases, from which i t is d i f f i c u l t
to draw conclusions of a general app l i cab i l i t y . One ef fect of a general nature,
however, is the spread in phonon momentum q induced by absorption (BRENIG et a l . ,
1972, DRESSELHAUS and PINE, 1975; DERVISCH and LOUDON, 1976). I f the absorption
is not exceedingly large [Im(k)<<Re(k)] then the pr incipal influence of the dis-
t r ibu t ion in q-values w i l l be to contribute a l ine shape to the scattered radiat ion.
The nature of the d is t r ibu t ion may be inferred from an inspection of S(q) in (2.23). 1
In the presence of absorption A(r) ~ exp l i ( k I + i k2 ) . r I for both incident and scat-
tered waves; employing (2.24), one eventual ly f inds that S leads to a factor pro-
portional to
Im (k'2+ k) (3 18) Re(k' k - q) + Im(k' + k) 2
in dP/dz. This resul t must be modified for very strong absorption, such as in metals,.
for example, and may also depend on phonon boundary conditions (see MILLS et a l . ,
1968; DRESSELHAUS and PINE, 1975; DERVISCH and LOUDON, 1976). Although detai led
invest igat ions are lacking, i t appears that the existence of a momentum l ine shape
w i l l not, under usual condit ions, s i gn i f i can t l y a f fect the resonance properties of
dP/d~. Cases involving strongly k-dependent couplings may, however, provide excep-
t ions.
107
In summary, whi le po lar i ton approaches predict the existence of a var ie ty of novel
features in scat ter ing from spa t i a l l y inhomogeneous media, only a few isolated cases
have been i den t i f i es in which the ef fects appear c lea r l y capable of detect ion. The
one case where the experiments have corroborated the predicted ef fects is with re-
spect to resonant B r i l l o u i n scat ter ing. Exist ing data are far from adequate to cor-
roborate the detai led nature of the frequency dependence of the RS e f f i c iency . The
predict ions of the ex is t ing theories and the i r companion experiments w i l l have to
be pushed considerably fu r ther before the existence and/or nature of various pheno-
mena discussed here can be appropr iately ascertained.
3.4 Scattering by Polar i tons
When the quas ipar t ic le responsible fo r scat ter ing couples to l i g h t d i r ec t l y (e .g . ,
TO phonons), then a po lar i ton formulat ion of the scattered spectrum becomes appro-
pr ia te. The l i ne shape w i l l no longer be charac ter is t i c of the bare quas ipar t ic le
but w i l l take on a po lar i ton character instead. Moreover, the scat ter ing may con-
s i s t of contr ibut ions stemming from both the bare quas ipar t ic le f i e l d as well as
from the electromagnetic f i e l d associated with the po lar i ton. For scat ter ing by TO
phonons, fo r example, we may wr i te the change in the e lect ron ic suscep t i b i l i t y in
the schematic form
AX = au + bE (3.19)
where E is the e lec t r i c f i e l d ; here a is the usual atomic displacement coe f f i c i en t ,
whi le b is the e lect roopt ic coe f f i c i en t (see BARKER and LOUDON, 1977 for example).
The appearance of the two terms in A X creates a p o s s i b i l i t y for resonant cancel la-
t ions in the scattered spectrum. In general, a and b w i l l depend on both m and ~ ' ,
which w i l l resu l t in resonance propert ies s im i la r to those described heretofore.
We l i m i t the present discussion p r i n c i p a l l y to f i n a l - s t a t e ef fects in the nonre-
sonant regime, although some b r ie f comments regarding changes associated with reso-
nance condit ions w i l l be made. The present section has been included mainly as a
matter of in te res t , but also for reasons of completeness and perspective. An exten-
sive l i t e r a t u r e exists on the subject ( fo r example, BENSON and MILLS, 1970; GIALLO-
RENZI, 1971; BARKER and LOUDON, 1972), to which the interested reader is directed
for fu r the r deta i ls . Our b r ie f development here w i l l fo l low c losely that of Benson
and M i l l s (BM).
Consider, for s imp l i c i t y , zero damping and a case where s ingle a and b coe f f i -
c ients describe the scat ter ing. The E- f ie ld may then be related to the displacement
u in the standard fashion (see BORN and HUANG, 1956, Ch. I I , for example); then one
obtains fo r the change in suscep t i b i l i t y A•
108
gX = Zp, Z : a + ~ b(m~0 - Am 2) (3.20)
where Am is the frequency sh i f t , p is the reduced mass, e* the e f fec t ive transverse
charge, and a and b have been defined above. Under nonresonant conditions dP/dE
depends on jus t the sh i f ts Am and Ak, and is proportional to the frequency trans-
form of the suscep t ib i l i t y cor re la tor , i . e . ,
dEdP =<Ax(t ) AX(0)>Am = iZl2u(t)u(o)>A~ = i z 12SL(Am) (3.21)
where S L is the phonon strength function (see Section 2.1). For large angle scat ter-
ing Am=~T0, S L= I , and the scatter ing becomes simi lar to that in the absence of
polar i ton ef fects . For smaller angles dP/dEwi l l display construct ive or destruct ive
inter ference, with the nature and extent depending on the sign and size of the ra t io
ae*/b (construct ive for posi t ive sign, destruct ive for negative). Interference ef-
fects of th is type appear to have been f i r s t reported by FAUST and HENRY, ~966, for
experiments on GaP. The propert ies of the Raman spectrum for GaP have been analyzed
in some detai l by BARKER and LOUDON, 1972, (see the i r Fig. 7, for example). The
analogous interference e f fec t , as observed in ZnSe is i l l us t ra ted in Fig. 16.
When phonon damping is present, (3.20) and (21) need to be generalized by f i r s t
wr i t ing dP/dE as a l inear combination of the four correlators <u(t)u(0)>Am,
<E(t)u(0)>Am, <u(t)E(0)>Amand <E(t)E(0)>Am. These functions are proportional to
the imaginary part of the corresponding Green's funct ions. For the present case
where a b i l inear in teract ion describes the polar i ton, the equations of motion for
the set of coupled Green's functions can be solved exact ly, as detai led in MB, for
example. We here merely state the resul ts corresponding to the general izat ion of
(3.21) to nonzero damping:
dP la + 4~e*Nm2 bl 2 C(Ak,A~ ~ k2c2_Som2 - (3.22)
C(Ak, Am) = / d s dt e i (~ ' [ -mt ) <u(r , t )u(0,0)>
where C is the displacement corre la tor of the l a t t i ce in (k,m) space. For a TO phonon
mode one obtains (E I and E 2 are here phonon-photon polar i tons)
~TOF(m ) ( 2 _ k2c2 0-I )
2 2 - k2c2 -I o
Between them (3.22) and (23) embody both polar i ton ef fects ar is lng through C(km)
and those associated with interference between atomic displacement (a) and e lec t ro-
opt ic (b) induced scatter ing mechanisms. The typical var iat ion in scattered l ine-
width with angle is highly parameter dependent. Results for ZnSe are indicated in
109
0!.1.
(OL6L '$391I~ pue NOSN38 moa~) po6ueyo St e/q ~0 U61S a~; uaff~ ;Lnsa~ sa~e~Lpu[ A -asu Z UL auL[ u~ 8 aq~ ;o mnmLxe~ >[ey ;e'4;p~ LLn;'OLSULJ~U[ (q) "(A) aSuZ o~ $~eLadoJdde
s~a~eweaed ao; 'a[6u~ 6utaa~eos uo ~LSUe~UL ue~e~ ~eod ;o aouapuadao (e) "9i ~6~3
S'Z 0"~ S'| O'I I r I -- W
__w
~-....
S'O 0 f "~0 '9
t (q)
0"5
$$ ~,
0'9 i
S'9
0's
""-- o@ S'I 0'1
§ "----... \
S'O -- ;0'0
I'O
0'I
Fig. 16b, and display a substantial increase in l inewidth for angular values corres-
ponding to cancel lat ion, as would be expected. For one par t icu lar configuration in
Zn0, on the other hand, BM found a r e l a t i v e l y slow var ia t ion for the width. MB also
found, in te res t ing ly enough, that even for f a i r l y large values of damping the peak
posit ion in the scattered spectrum is determined very nearly by the undamped polar-
i ton dispersion re la t ion . For large angles one is outside the polar i ton regime (Am2<<c2Ak2~o 1 ) " and dp/dE reduces to
dP § a 2 r (3.24) (m2,~0)2 + 4~T0 r 2 2
which is formal ly the same as for scattering by nonpolar phonons. Thus polar i ton
ef fects are manifested p r inc ipa l l y in small angle (nearly forward) scatter ing.
Although the assumption that dP/dE depends only on Am and Ak breaks down far
from resonance, one may nevertheless ant ic ipate some of the ef fects ar is ing from
resonance conditions for the case of scatter ing by polar i tons. These would be ex-
pected to occur p r inc ipa l l y from the modif ication of Ak from the bare phonon momentum
transfer to that of polar i tons. Since Ak may then become very large near to reso-
nance, the scatter ing may transform from polar type to nonpolar as m or m' is tuned
through resonance. Of course, when the attenuation of the modes k and k' are s ign i -
f i can t , then the spread in wave vectors induced by attenuation must also be accounted
fo r , as discussed previously in Section 3.3. Detailed experiments and calculat ions
of th is type for resonant scatter ing by polari tons have yet to be performed.
4. Concluding Remarks
In the preceding we have reviewed two polar i ton approaches to the theory of the
f i r s t order RRS in crysta ls . For most common cases of in teres t , polari ton theory
is found to predict resonant scatter ing properties which are qua l i t a t i ve l y , although
not necessari ly quan t i ta t i ve ly , s imi la r to the perturbation theory results (see
Section 3.2 and Table I ) . Polariton theory thus serves to confirm and val idate cer-
ta in aspects of perturbation theory predict ions. The pr incipal differences occur
for strongly k-dependent couplings and weak exciton damping, for which polar i ton
theory predicts markedly enhanced scatter ing near resonance. I t is doubtful, how-
ever, whether purely pol~ri ton ef fects can be re l i ab l y distinguished within ex is t ing
experimental data, although i t is cer ta in ly possible in pr inc ip le to perform experi-
ments where th is can be done. The optimal experiments to be performed for th is purpose would be those u t i l i z i n g resonance with well-separated ( for example, Frenkel-
type) exci tat ions which also possess large optical osc i l l a to r strengths.
Although a polar i ton formulation is obviously required in pr inc ip le to account
for interference effects associated with multichannel scat ter ing, i t has not yet
been established whether any such effects have in fact been manifested in ex is t ing
111
spectra; they may be d i f f i c u l t to observe experimentally in any case. The appearance
of new scattering channels seems to have been observed in the case of Br i l l ou in
scattering from a dispersive exciton,,where a mul t ip le t of l ines is predicted. Never-
theless, to th is point in time multichannel and spat ia l-dispersion effects in RRS
s t i l l remain r e l a t i v e l y unexplored topics for experimental invest igat ion.
Another area where polariton-mediated effects are involved is scattering by TO
phonon polar i tons. When ~or ~' are tuned through resonance, the transferred momentum
Ak can become large, thus transforming the nature of the scattering from polar to
nonpolar (see Section 3.4). Again, these effects have yet to be investigated in
detai l on an experimental basis.
Turning b r i e f l y to the theoret ical side, a var ie ty of s ign i f i can t problems de-
serve fur ther at tent ion, especial ly with respect to the or igins of the differences
between, and d i f f i c u l t i e s with, App. I and I I . For example, the question of the
effects of damping on the scattering e f f ic iency, and the el iminat ion of the singular
behavior stemming from the polar i ton ve loc i ty for undamped, dispersionless excitons,
are matters which merit fur ther attent ion within App. I . For App. I I , the appearance of s ingu la r i t ies in the scattering amplitude for the undamped, dispersionless ex-
citon case should be addressed fur ther. One more area worthy of pursuit is the po-
la r i ton formulation of higher-phonon scattering rates. Here certain economies in
complexity may be achieved from a reduction of the order of perturbation theory
needed to describe the processes under consideration (from th i rd to f i r s t , fourth
to second, e tc . ) . And although many of the experimental and theoret ical considera-
tions outl ined in this paper may appear somewhat subtle in nature, they nevertheless
provide a var ie ty of in terest ing, and cer ta in ly challenging, opportunit ies for fur -
ther invest igat ion.
Acknowledgements
The author thanks Drs. J.L. Birman, R.K. Chang, A.K. Ganguly, D.L. M i l l s , and R.
Zeyher for enlightening discussions.
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