Physzca 117B & 118B (1983) 290-292 290 North-Holland Pubhshin 8 Company

THE ORIGIN OF THE URBACH TAIL

- A MONTE-CARLO STUDY OF THE EXCITON ABSORPTION LINESHAPE -

Michael Schreiber* and Yutaka Toyozawa

Institute for Solid State Physics, University of Tokyo 7-22-I Roppongi, Minato-Ku, Tokyo 106, Jcqoan

A standardized formulation of the lineshape problem, i.e. the effect of lattice vibra- tions on the exciton absorption, is used in a Monte-Carlo-investlgation of the direct and indirect absorption lines for different dlmensionalities and temperatures. With respect to the low-energy tails the empirical Urbach rule is established. The analysis of the average-oscillator-strength-per-state (AOSPS) spectra allows us to unambiguous- ly ascribe this behaviour to momentarily localized states. The introduction of a steep- ness index which can be deEermined from the calculated Urbach tails enables us to pre- dict whether exciton self-trapplng takes place. The predictions are in excellent agree- m~nt with various experimental observations.

I. Introduction

Usually, the low-energy tail of the exciton ab- sorption spectra decreases exponentially in ener- gy with a decay constant inversely proportional to (high) temperatures T:

F(E) = A . exp (- ~ ) (i) z J

This behaviour, known as the Urbach I rule, has been observed for many direct as well as indirect absorption edges 2 .

By the present calculation we demonstrate that the Urbach tall can be explained as an effect of the phonon field on the translational motion of the exciton. Then, a posteri- ori, the origin of the Urbach rule is found in the momen- tary localization of the ex- cltonlc states at energies below the renormallzed band edge. Finally, we correlate experimental data on the Ur- bach tall and on the self- trapping of excitons.

If. Calculation of the Llneshape

The standardized formulation of the llneshape problem has been discussed elsewhere 5. It is characterized by (i)har- monic lattice vibrations, (2) linear and (3) on-site exclton-phonon interaction, (4) neglection of intersite correlation, and (5)adiaba- tic or Franck-Condon-approx- imatlon for the lattice vi- brations. In slte-represen- tation the Hamiltonian can

*Permanent address: Institut ~r Physik, Universit~t Dortmund, Postfach 500 500, D-4600 Dortmund 50, Germany

A

=

=

= =

o

=

<

=

-14 - 1 0 - 0 8 eV

Energy Fig. l" Absorption (Eq.4), direct edge

be written as

H = ~n'~m [n>V<m[ - ~n [n>Oqn<n] (2)

with nearest neighbour exciton transfer V. The configurational coordinates qn are independent random variables governed by thermal distribution:

P(q n) = exp ( -qn 2 / 2k~ ) (3)

We have solved the corresponding secular equation for finite samples of 30, 156, and 720 atoms in a linear, square, and simple cubic lattice,re- spectively. From the eigenvalues E i and respective elgenvectors ~i= (ezl , .... eiN) the normalized lineshape function

F(E) = N -1 ~ ~i O ( E - E i ) l[nezn [2~ (4)

zs calculated, where ~ .... * denotes the average over all possible configurations {ql,q2,...} for which the Monte-Carlo method is employ- ed. It is easily seen from eqs. 2, 3, and 4 that the llneshape is uniquely deter- mined by the parameter Tc2/V 2 or, equivalently, gT/B where g =c2/2B is the exciton- phonon coupling constant and B=~]V] Is half the width of the rigid lattice band with

nearest neighbours. We can therefore arbitrarily choose g E 1 and 2B ~leV.

Typical examples for the di- rect and the indirect edge (which can be realized by taking V<O and V)O, respect.) are shown in Figs.l and 2 on a semilogarlthmio scale. The straight lines in these plots establish the energy- dependence of the Urbach rule over several decades in absorptlon intensity for var- ious temperatures. The exten- sions of the straight lines

0 378-4363/83/0000-0000/$03 00 © 1983 North-Holland

M. $chrelber, Y Toyozawa / The origin of the Urbach tail 2 9 1

converge in one point at energy E~ near the band edge at vanishing temperature (Eo =-B =-0. SeV), also in agreement with experiments 2 and eq.l.

For g El, Fig.3 shows the steepness coefficient ~ versus tem- perature T calculated from the straight lines of Figs.l and 2 by the Urbach formula. In two or three dimensions it is independent of temperature as expected whereas the Tll3-be - haviour in one dimension meaning a T-213-exponent of the Ur- bach rule awaits experimental checking.

The high energy side of the asymmetric absorption peaks in Fig.l was found to be approximately of Lorentzian shape as anticipated by various exper- iments 4 . This llneshape typ- ical for life- time broaden- ing suggests that indirect transitions are responsible for the high energy side.

-14 - t 0 -06 eV E n e r g y

Fig. 2 : Absorption (Eq. 4), indirect edge

lIT. The AOSPS

To pursue this reasoning further, we define the average oscillator strength per state (AOSPS)

f(E) = F(E) / p (E) (5)

where p is the normalized density of states. The investi- gation of the A0gPS-spectra for the direct edge (see e.g. Fig.4) shows that they depend only on temperature above

2 0 ¸

=

ffl

o o

z i i i

Y Z / / /

. . . . . . . . ~ . . . . . - ; . . . . - ; - - - ; ~ . . . . . . .

/ . / / ~(

..x" z

;,t.. d= 1 x (8.T) 1/3 2 v

3 z

-150

' | 2 4

.OB5 "OBO

~'o 2;0 ' 8;0 Coup l ing C o n s t a n t . T e m p a r a t u r

Fig. 3: Steepness of grbach tails (Eq. 1) for direct (upper symbols, broken lines) and indirect (lower sym- bols, dotted lines) edges

the absorption peak, and only on dimensionality below. The behaviour above the peak agrees well with second-order perturbation theory which yields the AOSPS

f(E) =T/(E-E~) 2 for the indirect transitions.

For the region below the peak we consider discrete localized states momentarily trapped at each instant t by the spatial fluctuations of the potential w=- Cqn(t) in eq.2. The os- cillator strength of a state ~ with energy E were given by

f e E ) = I In <nl e > 12 C6) Considering one state bound by the perturbing potential w at site 0 and using the resolvent Q°(E) E (E-H°) -I of the unper- turbed Hamiltonian H ° we get

<n G ° O> w<O ~> = <n G ° O> (7) <nl~> = [ T n l < n Go O>w<O ~ > l Z ] , , ~ [ [ n l <n G° O>lZ) " ~

With <olco • colo> = - a/a~ <oloo(~)lo> = -c/¢~o-E in one,

c. lnlE=-EI in two, and A + C.~-E in three dimensions for energies Just below the edge Ee we determine the oscillator strength zn a d-dlmenslonal lattice

f ( E ) = ( % _ E ) 2 - d / 2 (8) (Ek= 0 - E )z

which the state ~ borrows from the k=0-exciton to which the whole oscillator strength were concentrated in a rigid lat- tice.

10

5

0

A O S P S A

d = l

t -1 -05 av

E n e r g y

Fig. 4: AOSPS (Eq.5), dim~ct edge

292 M Schrelber, Y Toyozawa / The o r~n o f the Urbach tazl

02_

01-

0 - 1 5

~ , A O S P S - 2

d - 1

,%

I - t I - OL5

E n e r g y

~Fig. 5:

(AOSPS) -2/d_ spectra for the analysis of the direct edge tails (see Eq.9)

Fig. 6:

(AOSPS)~4-d~ spectra for

the analysis of indirect edge tails

,v (see Eq. lO)

A O S P S 2 " ~

- t 4 -1 0 -0 6 eV E n e r g y

For the dlrect edge with Eo=Ek= 0 we get

y-2/d = ~_ E (9)

Thls behavlour was Indeed found in all dimensions (see, e.g., Fig.5). For the indirect edge we ap- proximate Ek= O-E = Ek= O-Eo = 2B and obtain

--~/(4-d) = ~_ E (lO)

which is also obeyed by the calculated AOSPS- spectra (see, e.g., Fig.6).

In conclusion, we have seen that the notion of (momentarily) localized states is appropriate for the explanation of the Urbach tails of the direct and indirect absorption edges.

IV. The Steepness Index

In earlier investigations 5 based on the model of momentarily localized states it was argued that the steepness coefficient of the Urbach tail should be inversely proportional to the exclton- phonon coupling constant

a = 8/g (n)

and the "steepness index" 8 should be of the order of unity.

A reexamination of Fig.3 for variable g estab- lishes this relation and determines indices of 1.50 (I. 24) and 0.85 (0.80) for the direct and indirect edges in three (two) dlmens~ons, respec- tively. We can therefore determine the coupling constant from experimental data on the Urbaeh tall and compare it with the critical coupling constant go=l-(29) -I to predict whether self- trapping of excltons takes place (g>gc) or not.

The above indices explain the recently observed self-trapplng in GeS 6 (~ = 7.45, g = 1.0S > g~--0.92) and older data in PbI27 (o = 1.48) for the first time. Recent experiments on e- and 8-type pery- lenes 8(o =0.9S and 1. $8) show that excltons in 8-perylene self-trap more shallowly than in

~-perylene in accordance with the determined cou- pling constant which is much closer to the crit- ical value in the former. In CdS 2 no self-trap- ping occurs In agreement with ~ = 2.2 leading to a value of g= 0.68 < gc. In all other cases where

is known our predictions about self-trapplng agree with experiment. This agreement is espe- cially distinct in the indirect absorption of the mixed crystals AgClxBrl_x 9, where a discontinous transition from free to self-trapped states is observed at x c=0.45, thecorrespondingac=0.88 yields gc = 0.97 which is to be compared with the theoretical gc = 1-(29)-I = 0.96.

V. Concluslon

Without resortlng to the ad-hoc picture 5 of mo- mentary localization, but simply by solving the standard mode~ of exciton-phonon interaction for the absorption llneshape numerically, we derived the empirically established Urbach rule 12 for lattzces with direct and indirect edges. A pos- teriori, the investigation of the AOSPS furnish- ed evidence that the Urbach behavlour originates from momentarily localized states. In addition we demonstrated the usefulness of the steepness index in deriving exciton-phonon coupling con- stants from the measured steepness coefficients.

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H., J. Phys. Soc. Jpn. 35 (1973) 495. 5 Toyozawa, Y., Techn. Rep. ISSP All9 (1964) 1 ;

Sumi,H.,Toyozawa,Y.,J.Phys. Soc. Jpn.31(1971)342 6 Wiley, D., Thomas, D., Schonherr, E., Breit-

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