complex permittivity of germanium at microwave … · abstract a. number of microwave measuring...
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COMPLEX PERMITTIVITY OF GERMANIUM
AT MICROWAVE FREQUENCIES
THE EFFECT OF FREQUENCY, DOPING AND TEMPERATURE ON THE . . . COMPLEX PERMITTIVITY OF N-TYPE GERMANIUM
By
RIAZ HUSSAIN SHEIKH, B.SC. ENGG. {PUNJAB,PAKIS~AN)
. D.R.c: {GLASGOW); M. ENG. {McMASTER)
A Thesis
Submitted to the Faculty of Graduate Studies
· in Partial Fulfilment of the Requirements
for the degree of
Ph. D.
McMaster University
March 1968
DOCTOR OF PHILOSOPHY (1968)
.(ELECTRICAL ENGINEERING)
McMaster University
Hamilton', Ont
TITLE: The Effect of Frequency; Doping and Temperature on the
Complex Permittivity of N-Type Germanium.
AUTHOR: Riaz Hussain Sheikh, B. Sc., (University of Punjab, Pakistan)
O.R.C. (Royal College of ·Science &
Technology, Glasgow)
M. ENG. (McMaster University, Hamilton·,
Ont.)
. SUPERVISOR: Professor t4. W. Gunn
NUMBER OF PAGES:. viii, 151
SCOPE AND CONTENTS: A number of microwave measuring techniqu~s for the
measurement of the complex permittivity of semi-conductors have been
investigated and a new method of measurement is described; Also a
study has been made of wave propagation .and higher order mode effects
in rectangular waveguides partially filled with a semi-conductor
togeth~.r with a theoretical and practical investigation of the
dependence of the complex permittjvity on temperature1frequency and·
doping.
(ii)
ACKNmiLEDGEMENTS
The author expresses his sincere gratitude to Dr. M. W. GUNN
for his continued advice and encouragement. 'Thanks are also due to the
National Research Council of Canada for a gr-ant for the equipment and
to the Government of Canada for_ the award_ of a scholarship under the
Colombo-Plan. The author is grateful to Dr. C. K. CAMPBELL for the
loan of the Delta Design chamber and to Dr. W. R. DATARS for the loan
of a dewar. Special thanks are also due to Computer Centre Staff,
especially Dr. D. J. KENWORTHY, for tiJ"Qely help in computations,
__ . ______ -~-l.'IQ -~Q.. ~he__ frtenc!..s_wh_e> ... baye_b_e_l_ped_t_hEL~u.t.hPr __ jo__a.rt.Y w~Y . in._ connection
with the project. And 1 ast, but' not least, the au'thor expresses his .. -
very· special thanks to his wife, who has held her patience during
this period.
. .
(iii)
..
ABSTRACT
A. number of microwave measuring techniques for the measurement
of the complex permittivity {£ = e er - j ~) have been investigated 0 w
and a new method based on the replacem~nt of the narrow wall of a
·rectangular wave-guide by a block of semi-conductor has been developed.
This technique is shown to be suitable for the measurement of a when
a >> we0 tr and for the measurement of a and tr for a ~ we0 er·
An investigation has been made of the propagation characteristics
of a rectangu)ar wave-guide containing a centrally placed slab of .
semi-conductor parallel to the narrow walls of the guide. A
comparison of exact solutions for the propagation constant in such
a structure with the approximate solutions normally used has shown
that the conditions for the validity of the approximate solutions
are much more stringent than has been reported previously. It is
further shown that under certain conditions the structure offers a
convenient method of measuring the conductivity of a semi-conductqr.
In addition, a theoretical and experimental investigation of the
effects of the higher order modes excited at the interfa-ce of such
a structure with an empty wave-guide has been made. The study has . . shown that under certain conditions, the effects of these modes can
be significant.
A theoretical and experimental study has also been made of the . .
effects of temperature, frequency a.nd doping on the complex permittivity
( iv)
of 1 ightly ·doped n-type gennanium. Measurements of these effects
which h.ave not been reported p.reviously have been made over a temperature
range 100°K- 500°K at frequencies.9.25 and 34.5 GHz and confirm the . .
theoretical model used.
. .
(v)
, CHAPTER I
CHAPTER II
TABLE OF CONTENTS
INTRODUCTION .
GENERAL THEORY OF HAVE PROPAGATION IN A WAVE
GUIDE CONTAINING SHU-CONDUCTOR
2.1 Introduction
2.2 Derivation of Hertzian· Potentials . 2.2.1 Magnetic Type Hertzian Potential
2.2.2 Electric Type Hertzian Potential
2.3 Special Cases
1
9
9
10
11
2.3.1 log£r =constant 12
--------~-3.1 __ €r-- =_Er(x) 14
2.3.3 £r = €r(Y) 15
~HAPTER I II l·JAVE PROPAGATION IN A RECTANGULAR WAVE-GUIDE
WITH A SEMI-CONDUCTOR WALL
3.1 ·Intr~duction
3 .. 2 Theory
3.3 Theoretical Results
· 3.4 Measuring Technique
3.5 Experimental Results & Discussion
CHAPTER IV HAVE PROPAGATION IN A RECTANGULAR WAVE-GUIDE
__ CONTAINING_A .. CENTRALLYJJ.ACEO_SEMICONDUCTOR
. 4.1 Introduction
· - -- 4.-2---Theoretical Considerations
4.2.1 Exact Solution
{vi)
17
19
22
28
36
. 39
42
42
·- 4.2.2 Approximate Solution 45
4.2.3 Discussion on Exact and Approximate Solution 48
4.2.4 Calculation of Junction Impedance 64
4.2.5 Calculation of Reflection Coefficient· 67
4.3 Neasurement. of Reflection Coefficient
4.4 Results and Discussion
CHAPTER V THE CONDUCTIVITY AND DIELECTRIC CONSTANT OF
GERMANIU~1 AT MICROWAVE FREQUENCIES
5.1 Introduction
5.2 Theoretical Considerations
.. ___ -------~-~-~~LMi_~_rq_~~y~_l1~bi }j_~Y_. a_l}_d _ _[)i_~l~c-~_r.i~-~o_nstant
·5.2.2 Ionized Impurity Scattering
· ~.2.3 Lattice Scattering
70
70
81
82
82
85
86 ...
5.2.4 Intra-valley Acoustical Scattering 88
5.2.5 Intervalley Scattering 89
~ _____ 5.2_..6 ·Intra-valley Optical Mode Scattering 90
5.2.7 Energy Bands & Lattice Vibrations in Germanium 90
5.2.8 Variation of ~t
5.3 Theoretical Results and Discussion
5.3.1 Electron Mobility in Germanium
______ ___5....3 ..2.. .Ho 1 ulob i 1 i ty_in _Germani urn
5.3.3 Microwave Conductivity and ·Permittiyity of - -- -· .. ~
Germanium
5.4 The ·~1easurement of a(w) and £rm
5.5 Experimental Results and Discussion
(vii)
94
95
96
'' --~- --102
104
108
110
CONCLUSIONS 117
APPENDIX A MICROWAVE REFLECTION BRIDGE 123
APPENDIX B COMPUTER PROGRAMMES 133 ~
i) Lossy Wall Wave-Guide 134
ii) Partially Filled Wave-Guide 136
iii) Microwave Conductivity of Germanium 143
REFERENCES 146
(viii)
CHAPTER 1
INTRODUCTION
In recent years an increasing interest has been shown in the
microwave properties of semi-conductors (1-8) • Due to rapidly
advancing technology, semi-conducting devices are finding growing
application at micrO\tave frequencies and also m1crowave measurement
techniques provide a convenient mearis of studying charge transport
phenomenon in semi-conducting materials. The electrical parameters
such as conductivity a and perm)ttivity t .. may be different at high
frequencies from those at static fields, because at sufficiently
high frequencies the inertia of the charge carriers becomes observable.
Thus at microwave frequencies a -and·£r for a one carrier semi
conductor are given by (9)
£""" = t/ = tR./t- -g2n • to o *
me ~.
q = magnitude of electronic charge
t = t(t) = relaxation time of the charge carrier as a
function of energy
n = concentration of ~he charge carriers
_ me* = conductivity effective mass Df the carriers
w • radian frequency of the waves
1
- 1.1
- 1.2
a = d.c. conductivity 0
<T> denotes the Maxwellian average of T(t)
= J: ~(t) t3/2exp-t/KT dt/ J: t3,12exp-tfkTdt
2
From observations of the changes in a and tr at high frequencies
from their d.c. values, one can study the scattering phenomena.
Further, the changes in a and tr at mm wave length (or even at em
wave lengths) may be significant. As these parameters play an
important role in the design of semi-conducting devices, accurate data
on their frequency d~pendence is desirable.
The u~e of semi-conductors at microwaves is of interest
from another point of view. The conductivity of semi-conductors
can be varied over a wide range, comparatively easily. By changing
the conductivity and hence the loss in a semi-conductor, ·one can
vary .the propagation characteristics of a given guided wave structure,
which·may be us.eful in the design of microwave devices {'lO-i2).
The objects of this thesis are three-fol~, as follows.
(i) Because of the wide range of conductivities possible in a semi
conductor,mfferent methods of measurements of its electrical properties
are required. Experiments involving the measurement of VSWR ( 5 .•.13 ~, reflection coefficient.<14) and transmission coefficient (15 ~ are
r:-eported in the literature. These methods are mostly useful when the
"' conductivity of the semi-conductor is relatively low (< 10 mho/m).
At higher conductivities, the measurement~ are effected by the unavoidable
gap present between the semi-conductor and the broad walls of the wave
guide when the completely filled wave-guide is used (16,17} (fig. 1.1 ).
For measuring high conductivities, cavity perturbation (lS,19! or
substitution methods (lO) are used. These methods are most useful for
"' a > 1000 mho/m.
3
Thus the development of a new method that could provide accurate
·results of 7 and a in the middle range of conductivities, is desirable.
Such a method, based on the replacement of one . narrow wa 11 of
a rectangular wave-guide by a semi-conductor block (fig. 1.2) has been • developed and is described in this thesis. It is found that this method
is accurate when a~w£O£r· This method is free from the gap effect
discussed in the previous paragraph and it is shown to be more accurate
at high frequencies, where wave-guide dimension~become small.· The theory
of this "lossy wall" wave-guide structure is developed and exact and
approximate methods of calculations of the propagation constants are
discussed. Finally the results of measurements which confinm the theory
at 9.25GHZ are presented.
(ii) I
.The second object of this thesis is to study the propagation _.
characteristic of the wave~guide structure shown in the fig 1.3. This
configuration has not only been used for the measurement of tr and a
(21 - 23 ) but is also useful for devices such as attenuators or
modulators (lO) In all of these reported measurements, the propagation
constants have been calculated using approximate techniques based on . perturbational or variational methods which are subject to errors.
A study of these methods and their errors has been made and
calculations of the propagation constant of such a system have been
carried out for t/a and a ranges of 0.001 ~ t/a ~0.25, 0.1 ~ a(""i'm)
4
~10\r/m, for 10, 34.5 and 70.5 GHz with £r = 12 and 16. The computations
were made using the exact equations governing wave propagation in the
structure and the results have been compared with'those obtained from
the approximate techniques reported in the literature. It is .shown
that there is considerable error in the approximate calculations, unless
t/a < < 1. It is further shown that 'in certain ranges of a and t/a the·
attenuation constant varies linearly with a and for small t/a, the
phase constant varies very slowly with a. It is proposed that these
ranges are useful for measurements of a.
In practical measurements the structure ·shown in fig. 1.3
forms a junction with an empty guide, in which the dominant mode TE10
is propagating. Higher order modes are excited at such a junction and
an investigation of these modes has been carried out. It has been
shown that unless t/a ~ 0.08 and a ~ 2.5~m, the higher order modes
are not negligible. Experiments which. confirm the theory have been
performed at 9.25GHz.
(iii) Finally, the microwave measurement techniques have been used
to determine. the conductivity and dielectric constant of lightly doped
n type germanium. Benedic,t and Shockley (l ,2} used such a technique
to determine the conductivity effective mass me* of electrons and holes·
in germanium. However, their results are not in good agreement with the
values obtained from subsequent cyclotron resonance experiments. This
dis agreement is probably due to experimental errors. Further work
on ntype germanium using microwave techniques has been reported by
Druesne (24 ). His work involved measur~ents on 'n and p type germanium
5
(20, 10 and l/2 ohm-em) at 61.3 and 92 GHz using VSWR technique. His
results are also in disagreement with the theory. This is because. the
VSWR measurement technique is probably not very ac;curate at these
frequencies and because the gap effect seems to have been neglected in
these.measurements,rather than the inadequacy of the mobility theory
as he has concluded. Thus there is a need of fresh experimental and
theoretical work.
This thesis gives the computations.of the microwave conductivity
and dielectric co~stant of n type germanium as functions of doping
(Nd ~ 1ol 61 ':)).frequency (lo9 ~ f ~ 101Z Hz) and temperature (100 ~ T Cm"l
~ 500°k). This has not been reported in literature. The theoretical model
used to compute a and £r includes the scattering by ionized im~urities and
lattice vfbrations. The latter included the.low energy acoustical
phonon and' high energy optical phonons. Inter-valley scattering by
acoustical and optical modes was considered but it was found that
neglecting them probably does not cause a serious error. At high
temperatures, the holes inn type· germanium are not negligible. Their
effect on both a and ~ has been taken into account in the computations. r .
Experiments have been performed with a reflection type microwave.
bridge on n type germanium samples (a = 10 and 4.5 mhos/m) between the
temperature range of about 100° - 500°K, at two frequencies 9.25 and
34.5 GHz. The measurements are compared with theory.
..
~~~~~~~~~~~~---~>X z~----a
..
Figure 1.1 Completed Filled Wave-Guide
·.
1
'·
..
y
J;------, j ._L_Q S. S Y WA. L L
z&-------------------~~~---+ x
Figure 1.2 Lossy Wall Wave~Guide System
..
c 0
•r-+I IU s..
' :::1 en
•r-'+-c 0
)( u (IJ
"'0 ex: ..... :::1
C) 0 I
'ft (IJ t-
> u IU :::>
Cl :3: z "0 0 (IJ --u ~
.... LL..
~ >t w -4./) -·-· --~ IU .... +I s.. '0 IU a.
M· . -(IJ s.. :::1 en >- .,..
N .....
CHAPTER II
GENERAL THEORY OF ~~AVE PROPAGATION IN A WAVE-GUIDE
CON-TAINING SENI-CONDUCTOR
2.1 INTRODUCTION
In this chapter the t.hepry of electromagnetic \'taves in uniform ..
rectangular wave guides containing an isotropic semi-conductor with relative
permittivity ~(x,y) = tr(x,y) - j a(x,y) and permeability ~ 0 is
-:developed. The-longitudinal axis -of Wto ·-the· waveguide is al~ng the
direction of propagation \'Jhich is assumed to be in z-direct-ion. The
- -------walls-of- the \'lave. guides are.assumed __ to .be_perfectly __ conducti_ng so that
electric and magnetic fields satisfy the following boundary conditions, -~ .. n X E = 0 .. -+· n • H = 0 (2.1)
n being normal to the wall7pointing outward from wall surface.
2.2 . DERIVATION OF HERTZIAN POTENTIALS
·-------The ·electromagnetic· fields and the -propagation-constants in the o.Jl.(_ McWv..eet .
system under consideration/!rom the f·1axwe11' s equations for the case of
time variation according to exp(j~t)a~ given below, -;to • ..
V X t:. = - J~o H
v x H = j c.v t 0 £r t .. .. .. V D = V.! E = V.t0~r E = 0
· v • ~ = v. ~0H' = 0 ·
9
(2.2a)
(2.2b)
(2.3a)
.(2.3b)
£0 and. 'IJo ~re respectively the permi.ttivity and permeability of free
space. £r·is the complex dielectric constant of the semi-~onductor and
will be assumed to be a function of x and y but not z.
The solution of fields may be obtained in terms of electric •
and magnetic Hertzian potentials ne and nh respectively. The special
case for Er =constant has been dealt in the reference 27. In the
follovting sections, the treatment is generalized to the case \'lhen £r
· is not a constant.
2.2·.1 Magnetic Type Hertzian Potential ·since from .(2~3a)
~ ~~ Er E may be taken as the curl of a vector say nh£r so that
~r E = -jW'IJoV X nh1r
and ~ 1 ~ E = -jw'IJ0 ~~ v x nh£r (2.4)
where tth is called the magnetic type Hertzian ~otential' f27}.
Taking the curl of the preceding equation·and substituting
the result into 2.2a, one gets
.. H = V X (~~ 1 V X nh £r)
where v log~ = £-l v '£ = vt · r r r
Novt from 2. 2b, ~ ~.
v X H = jW£0 £ E . r
(2.5)
10
11
so that
whe,re k 2 =w2ll £ and~ is ;n arbitrary function (v xv~= 0). 0 0 0
Equating this to 2.5 yields
..... So far ~and ~h are arbitrary functions. If a condition is imposed on
the·se so that
..... _____ or --~ = __ v_~_Jih t __ constant
then 2.6 reduces to
. . 2.... + k 2" ..... - ( ..... ) . v llh o £r llh - v x v~ x nh (2.7)
2.2.2 Electric Type Hertzian Potentials
A similar set of equations as that-in the previous section:
may be obtain.ed from electric type Herfzfan potentiallfe (27).
----One· no\·1 starts from 2.3b -and takes
(2.8)
so that
and (2.9)
Nov1 from 2. 2a .....
VxE = ..... .....
- j [,.)JJ H = k 2 V X ll o o e k 2 i1 + V4l o e This yields
..... E =
9 again being an arbitrary function (VxV4l a 0). Equating this to 2.9 gives
(2.9a)
I' /' A
Again v e:r 9 = cr v ~ + ~Ve: r " v A ·A
So that £ ~ = v~~- 4>Ve:r r .
Substituting this into 2.9a and taking
~
or v. IIe =
(2.10)
2.3 SPECIAL CASES
2.3.1 A
t =loge £r·= Constant
In this case Vt = 0 and the equations 2.7 and 2.10 reduce
respectively to ·
v2ii + k 2" · n • o h o £r h
v2~e + ko 2~r iie • 0
and from 2.4 and 2.5
~ . ~ H = V X V X IIh
and (2.8) and {2.9) give ~ ~
H = +j~A>t V XII o e E ="";.-lv xv x ii ""r e
(2.1la)
(2.1lb)
(2.12)
(2.13)
The solution to these equations can be divided into two basic sets
of solutions. One such mode is TE to z or simply TE or H mode and· the
12
13
other is TM to z or TM or E mode. 'l'OOse modes can respectively be .... .... .
obtained from a single component of nh or ne directed in the z-direction
and the equations 2.12 and 2.13.
The TE or H mode in the fig. 1.1 will be treated here for
reference in later applications Thus let .... .... -'k nh = a 1/J ( x ,y) e
z (2.14)
where y = propagation constant • u+ j B.
Substitution of 2.14 into (2.lla) gives
v t 2 1/J + (y 2 + ko 2 ~ r:> 1/J • 0
where V.t2.a2 + a2 a;z ay2
7 . The solution for 1/J andy for the.structure shown in the fi'g. ·1.1 are
given below (28).
1/J n ,m = cos -
ni\x a
1\ cos !J!fiL
The dominant mode in the structure is when m • 0 and n • 1. For the
dominant mode, the above equations reduce to
1/J = cos n:X. . a 2 .
y = (~/a) 2 - k 2 ~r . (2.15) 0
where the subscripts ~~~ n and m have been omitted. From the fo~
ofljl·n,m •ljlp' it is obvious that
J J ljlp lll P • dx dy • 6 PP 1
I
6 PP • 0 l f P f. PI PI + n I ,m I •
2.3.2 tr = ~r {x)
This case is app11cable to the structures shown in the figs.
1.2 and 1.3. The solutions to the wave equations can again be divided
into two basic sets, TE to X and TM to X. Here onl.Y TE to X will be
considered. The s.olution to this mode can be obtained from equations
2.4, 2.5 a'nd 2.7 with a singl_e component of nh • ax nhx· If it is
assume that · · · · -Yz
nhx = ljl(x,y)e (2.16)
then 2.7 reduces to
vt2ljl + (y2 + ko2 ~)ljl = 0 ' (2.17)
The solution for y and ljl may be obtained for a particular arrangement.
14
In a rectangular wave-guide, it is found that ~·infinite ~Ru-t. fiJf
solutions for y and ljl are possible. It will be shown here that these
solutions are orthogonal to each other.~ Let ljli and ljlj be two di~ferent
solutions with propagation constants Yi and Yj .respectively. Then
' 1\ v t 2 ljl j + ( Y j 2 + ko 2 e r )ljl j = 0
Multiplying the first by ljl. and the second by •i and subtracting one . J
obtains
2 2 ( 2 2) ljl j v t ljl i - ~ i v t ljl j = y j . - y i .. i ljl j
integrating this over the cross section of the wave guide, applying
Green's second theorem and the boundary conGition 2.1, yields ·
= f (!pi V!pj - !pj V!pi )d1 c
= 0
because of the boundary condition 2.1 GR010S
where s denotes the surface of theL -section of the wave guide and c
is the contour enclosing it. Thus if Y; 1 yj.
· II !pi 111j ds = o s
t~~'s'may be normalized so that
II 111; !pj ds = 6ij • s
r i • j
~ i 1 j (2.18)
The solution for fields and propagation constants can be
obtained for TM toy mode from·equations 2.8 through 2.10 by taking
single component of fie along y direc~ion. If
Ue·= ay !p(X,Y)e-yZ
then 2.10 reduces to
or 6_ + a2 111 _ a~ '.!! + (k 2~ + y2) 111 • 0 a 2 ay2 ay ay 0
.
multiplying .throughout by e-~ = ~;1 • gives
or
(2.19)
(2.20)
15
16'
Again in a wave-guide, there are an infinite possible values of y QNI""'
IP and again it can be shown that different 1/I'S are orthogonal to each
.other. However, the ort~ogon~l·~elationship is now
~ i=j
If e-~1/I.~J~.dS = JJd~-l 1/1·1/1· • 6 .. = s 1 J s r 1 J lJ 0 iFj
(2.21)
CHAPTER III
WAVE PROPAGATION IN RECTANGULAR WAVE-GUIDE WITH A SEMI-CONDUCTOR WALL
3. 1 H~TRODUCT ION
When a perfectly conducting wall of a wave guide is replaced by
a semi-conducting one, a tangential electric field can be supported and f
losses occur in that guide. From measurements of the changes in the
propagation constant in a guide when one of its wall is replaced by a
semi-conductor, it is possible to deduce the properties of the semi
conducting material. This principle has been used to measure highly
conducting semi-conductors from the observation of the changes in the
resonant frequency and the Q-factor of a microwave cavity (l 9). This
method has been successful' for conductivities greater than 1000 mhos/m
approximately. .
For materials of lower conductivity the method of measurement
described in this chapter has been investigated and developed. An
analysis has been made of the wave propagation of a rectangular guide
with one of its narrow wall5replaced by a thick semi-conducting slab
as shown in the fig. 3.10. Computations of the dominant mode propa~ation
constant have been made for 1 ' a ' 1000 mhos/m, 0 ~ er ~ 16 at three
dif:t"erent frequencies of 9.25, 34.5.and 70.5 GHz. Exact and ·approximate
methods of computations are discussed and exper i~nts which confirm the
theory at 9.25 GHz, have been made.
17
y
•.. Figure 3.1 Wave-Guide System Analysed Theoret.ica11Y
19
3.2 THEORY
The arrangement illustrated in the fig. 3.1 will be considered.
The region (1) is empty and the reg,ion (2) consists of a semi-conducting
material which is homogeneous and isotropic with complex permittivity
~.A- .=t.£r-j -E.£v. It will be assumed. that the thickness of the semi-W£o
conductor is much larger than the skin depth in the material so that
the fields vanish at x = d.
Because of the non-uniformity in the x-direction, in general
a hybriQ mode (a combination of E and H modes) will be propagating. One
such set is an LSE mode with no electric field perpendicular to the air
semi-conductor interface <29). The fields and the propagation constants
may be obtained from the solution of equations 2.4, 2.5 and 2.7 •. When
~ is a function of x alone and variation in z-direction as e-vz is
assumed,2.7 reduces to
v t 2ny + (ko2£r +y2) ny = - ..1.. (ny !i ) ax ax
(3.1)
vt2nz + (ko 2cr + y2) nz = -....a.. (n:z U.) ax ax .
+ For an L.S.E. only one component of n·is required. This is obtained
by taking ny = nz = 0 SO that the above equation for nx • ~(x,y)e·YZ
reduces to
~ + ~ + (k 2~- +y2) •• 0 ax ;yz o T
A {1 0 '< X < a where c (x) • ' '
r c -j_.a_a<x<d r ' c WE:o
(3.2)
. .
Elect~ic and magnetic field components are then obtained from 2.4 and
2.5 as follows
Ex = 0 , -Yz
Ey = + jwMoY"''e.; Hx =: .+1 (DEZ - e>Ey ). · JW.~ 7JY C>z
E . C}fil -Yz z = + ·JW~ -- e o?Jy
Hy ~ _L eEz ) jw~ ~x
Hz=_:]_ ~ jw~ DX (3.3)
1he boundary conditions on £y and Ez at x • 0, d, and y • O,b
require that ...., be of the form
til = A sin k1x cos mp 'f = B e-k2x cos J:!!!!Y
b
. .
where A and B are constants. k1 and k2
are wave vectors in the
transverse direct.ion (x direction) and· m is the wave number in y
(3.4)
direction. m=O,l;2,3, .... , k1 and k2 satisfy the following
equations
~2 = k12 - k 2 + (m~/b)2 . 0
(3.5)
Y is the same in both the regions of fig. 3.1 so as to satisfy the
boundary conditions for all values of z.
Now the continuity of Ez and Hy at x • a gives the following
conditions ..
20
so that (3.6)
Equations 3.5 and 3.6 give the eigenvalues Y n,m~.? doubly
infinite values. However, if m = Q, the structure can support only
Hno modes. For Hno modes, Ez and Hy vanish and the fields and.the
propagation constants are as given below
Ex = 0 Hx· = .:. , ~~1~ y 2,.y e -iz
Ey = jWA; Vny e-Yz • _i!_ ~ (3.7) 0 jwfAo oz
Ez = 0 Hz • ...::.L ~ jw~ C>x
~2 = k 2 - k 2 =-k 2 - k 2; 1 • o 2 o r (3.8)
The lowest solution of 3.6 and 3.8 gives the dominant mode in the
system and is H10•
It may be seen from 3.6 and 3.8 that a complex transcendental
equation must be solved to obtainY. This is often. a laborious task.
Therefore, an approximate e.xpression fo~ "/ 2 is ,desirable which can
adequately give the value ofY. Methods based on perturbational and
variational techniques are available for this purpose (30).
However, a simpler technique can be used in this case·under
21
certain assumptions. If it is assum~d that the conductivity is sufficiently
high so that the field distribution in the region 0 ~'X~ a is approximate1Y
the same as in an ideal empty guide of broad dimension a, then the wave
vector k1 will not be much different from its value of (n~/a) in the
ideal case. Under this condition the equation 3.6 can ·be expanded in
Taylor's series about K10 = n~/a. Thus
22
f( k1) = k1 + k2 tan k1 a = f (k1 0) + (k.1-k10) of (3.8) ~kl k1=k10
If it is further assumed that \~ 2 \ <<\k0 2 &'rl , then· the approximate
expression for k1
and ~are as follows _, k1 . (n~a) [1 ~ jt./ho2 zrJ
and 'i2 =- (n~/a) 2 {1 + · ),\ ~-} a ko ~r
3.3 THEORETICAL RESULTS
-k 2 0
(3.9)
1(3.10)'
The equations3.6 and 3.8 were numerically solved. for 0 ~ Er ~ 16
and 1 ~ a ~ 1000 mho/m \'lith the help of a 7040 IBM computer using
the Newton-Raphson technique. The computations were made for three
different frequencies, namely, 9.25, 34.5 and 70.5GH~ The results in
the form of "6'1 - '/0 , '/ being the propagation constant· in an· ideal . 0
empty wave-guide of broad dimension a, are shown in the. figs. 3.2
through 3.4. The abscissa represents the change in attenuation constant
So<.= o<.~- 0(0
= o<, (as o<_0
= 0) nep/m and the ordinate represents the change
in the phase constant o8 = ~ - 80
-lad/m. The £r and log a are taken as
parameters.
In all the three figures, it may be observed thatJ in the . . .
range of a considered) as ~he conductivity is decreased (resistivity of
the semi-conducting sample is increased), the attenuation constant
~ -....... 0 <r cc
-,. 1 • 1 1 -r··~·-,.----..---1·-.. r---....-·-r·-,~r-·~-r~·r~··~~·--r"\..-·-·T~··--,-~-,.- ···_;_:r.~-.- ·yT--" ,_ 'j
.f.:::::9.25GHZ
(;..~//
v~/ {;~/
I// .,._
16 Q
-
,, I J I I I 1 I I .. I I 1 I t t.. , 1 --l 1 I 2 3 4 :> 6 I 7 8 . _,.,.._J. - -L--a{ N E P/ M 9 "" V'
Figure 3.2 6y in the system shorm in the Figure 3.1 at f = 9.25 GHz
t _! i i
-r
f f I I
-,
J
. '-'
oCL ~
---1---r---,-r--r-~r-·r~~~·l ···r·-·-r~·-~~"'·r-r·w·-···,.~-.. ··;---·-··-"r'"~~~·-·1- ,..,J-.· .. · .. ·r·~·~''"'. r·····"···'"'~('~·· "·~'J'' ·.
f:: 3 4.5 G HZ
4
C' I I I L I 0 4 8 L I I. I I • I L- I I I t
I 2 I b 2 0 2 4 2 0 ~ I - L l, ___ ,l_._ 0( N E P/M .. 0 :0 ~ ,._,
Figure 3.3 oy in the system shown in the Figure 3.1 at f = 34.5 GHz +=
~ a--< a:
20
1 1 1 1 1~ 1 1 1 r r r -
.f,.. 70.5.GHZ
,
o• 1 r 1 1 1 . 1 1 r 1 t 1 1 t 1 r 1 , ' 0 20 4 0 6 0 60 I 00 120 14 0 160 o< N E F'i
Figure 3.4 oy in the system shown in the Figure 3.1 at 70.5Qiz
')J V)
-·~ncreases monotonically. The change in the phase constant, however,
first increases with ,o. , attains a maxfmum and theQ decreases. This
observation is true for tr ;- l. However, forE r < 1, the things
reverse and ~attains a maxima and ~ increases monotonically as the
resistivity is increased. It is fur.ther observed that maxima occur
near, though not quite at o = wE E-. the 1 ocus of this condition is o r. also plotted in the figures 3.2 to 3.4.
26
It may also be noted that at the higher end of the conductivity
range considered, the curves for various- values of~ are crowded and . r . ultimately merge into each other. This is.because as c>increases,
O'/w~0 )) E and E -7 -j 6"", independent of E"=r" · r r w~
0
The changes in the propagation constant Y were also compu~ed
-----from the expression 3.10. The results of computations are compared
to those frot:n exact equations in the fig·: 3.5 for E-r ·= 16 and ~r = 1 . .
and f = 9.25GH~. The solid and dotted lines give the values of~ and
&~ respectively as obtained from the exact equations as a function of
~- The circles and crosses give the corresponding values from equation
· _ ..... ------3-:lo:-·-It-ma.f-beooserv-ea-thaf-for high-values···o-rE r-~the-equation· ·3.-10
_____ is adequate to give 0( and b"~_within ~ 3% for the w~ole range of ($'"
considered. However, as ~r decreases the errors increase foro~ 4.0
___ m~o/_m. For o ;:- 4 mho/m, .. ~rrors are ·negligible •
. For semi-conductors, the value of dielectric constant is given -lltc-
. -~y the contributions from~lattice and the free carriers. If it is
. .
16
14
12
10
6
4
2
. t-
.
t-
\ '
\ \ \
t- \ -f'79.25GH \ \ <\ .. ::. J . o<
~ EXACT ----- ~r> \
\&f3 A PPROX I ~ ATE 0 0(
" ~~ r- \ .,. \
~ a \ \
'Y=I
\ 1- " \ o< \
\
~ '
t. 0(
t-
,_
t-
~':'""-~ ....
r-
..1
' ~
~~ '
"'~ . "' t6 _b) ..
~ .......... -,.- ---· r-- " ...... ~~
~ £ * ....... ~ "{ ,. v .... ,..
... ~ ~
' ....___
I I _1 I
.4 .8 1.2. 1.6 2.·0 2.4 L 0 G ( ·o- -v-lm)
Figure 3.5 Comparison between exact and approximate values of 6y as a function of £r and a
. •
..
assumed that there is only one type of carrier, say electrons, the
conductivity and dielectric constant at frequency w are given by the
equati~ns l.l and 1.2. If, for the moment, it is assumed that i = constant, i.e. does not change wit.h energy, and further assumed that
w is such that (wr) 2<<l then these expressions reduce.to
(3.11)
(3.12)
ttl £O = 16.0
A plot of £ with a for various values of T is given in the r fig. 3.6. The propagation constant at f = 9.25GHz for 0 ~ log a <3.0
(l.O<a< lOOO'V/m) and for values of T and tr (given by 3.12)is shown in
·the fig. 3.7.
?II
The values of k1 and k2 for~arious values of a and T = 3.0 x lp-13
(approximate value for n type Gc.at room temperature) are plotted in . . .
the fig. 3.8a and b. It may be. observed that magnitude of k1 is not
very different from that for empty guide over the whole range of a
considered and that its angle does not vary more than a few degrees.
The magnitude of the electric field in y-direction as a function
of x/a is plotted in the fig. 3.9-for 2'a~· mho/m.·
3.4 MEASURING TECHNIQUE
For the measurement of the propagation constant a section of a
wave-guide was milled as shown in the fig. j.lo. Into this milled out
section could be clamped either a polished brass block.~r the semi-
1 -.d"
16J --
Figure 3.6 Er as a function of a and T
f = 9.25 G HZ
. -
~
~ 0 ~ cr ....., .
cO.. ~
oo_ 3 6 7 o< (l"tE P/M}
Figure 3.7 6y .in the system of Figure 3.1 as a function of a and 'T
;
9
\r' 0
l40
1:36
~ ~1.32
'o ')( --~ -
1.2 B
. 1.24
i-
'""
1-
f-
,...
'-
;1·2~. I
..
f:::.9.2S GHZ -1:!> . '"(',. 3x 10 SEC
~
~ ~ ~
r--- \k_tL---
~ - -.
.~ -.
~ ' ~ "
- . .
~ ........
~ r--
I I I I I I
.4 .a 1.2 -- - -1.'6 - -- ----;to--- 2.4 2.8 L 0G(<> -v-jn\)
Figure 3.8 (a) I k1
1 and /:._l as functions of a ..
•1 ., I 4.0
-
3.2
. -
2.4 . 0 --· ~ -
I .6
- -, . 8
--~
I 0
v)
....
10 J
I 80
J. .a"' 60 'o -
X -;r :X:: -
40
2C
~~
""' -
-
-
:
-
-
I 00 .4
I
~ ,/ t::: 9.25 G Hz.
. ~3 't'.,.3)l I 0 SEC
~ ~
. -
.
~ . ~ ~
t/4 v
--
~
~ ~
I I I I I I ~ - -.a 1.2 - 1.6 2.0 2.4 2.8
LOG (~-v-I~)
Figure 3.8 (b) IK21 and ~2 as functions of a
. -~· ..
-
-- '
-
-
-
-.
-
I -- -3.2.
-j(j
c
o·
0
00 w a
"' ~ o·
0
0
VJ t'
>-w
ob .a . Va
. --- -- ·--·-· ·- ----- -----+-----
f=9.25 GHiZ -13
'T= 3x I 0 SE1 C
1.4
Figure 3.9 Distribution of Nonnalized f·lagnitude of Electric Field Intensity Ey as a Function of x/a
\1-l v
conductor block. This section of the wave guide was placed. in one ann
of a transmission bridge described by Montgomery (3l)_ The second arm
of the bridge contained a precision attenuator and a phase shifter.
The zero balance of the bridge was first carried out with ~
brass block ~~nto the milled section. A second balance was
obtained when the brass block was replaced by a sem-i-conductor block.
Both the readings were taken when the bridge balance was· independent of
34
the clamping pressure. If A and<P are the readings of the atten-Uator . 0 0
and the phase shifter in dbs and degrees respectively with brass . .
block and A1 and~l are corresponding readings with semi-conductor
block, then neglecting the internal reflections and reflect.ions at the
interfaces at z = 0 and z = ~. the propagation constant in the lossy
wall guide is given by
a = (A1
- A0)'/(8.686"2) nep/m
B = (~1 w ~0)(57.296! rad/m
where ~ = length of the sample in,meters.
The ,neglecting of the internal reflections and those at
z = 0 a.nd z. = 1 was justified because these reflections· were observed
to be small (< .03). Theoretically if z0 and z. are wave impedances
in the empty guide and the guide containg the semi-conductor, the
re·fiection coefficient at z 11 1 is given by
R = u~~~ J It is assumed here that in the empty guide section (z ~ 1) there is no
reflected wave travelling and that the higher order modes which m~ be
. t.
-:---
i ___ [ __ I
... !_ i •
-----·-·· ·--!
---~ .'
'! ... _., _______ !:_ ___ -.-.:
I , I I ··-"-! _:.__-: .. --1 __ . _____ l ' ~-·!-·-,
I
I ·-· .'
I
i ___ , __ ..
'I I· v
-N
1/l ot-)
c CJ E .,_ s...
>< • ClJ 0.. X
I.JJ
GJ ..c. ot-)
c .,_
"0 <lJ 1/l ::J
<lJ .l_ • "0 ....
::J (..!)
I . i <lJ
--. ·-· > ~ ClJ -·
~....:..:.~ ... --·.-·---: .. . z -----
•'
' ---~.
·····-'
_____ .. __
'to c 0 .....
ot-) u ClJ
V)
' ot-) ::J 0
"0 <lJ --.... :::
.
-- -·
__ j
·----:
__ :......... . '•
-- --·---- 0 ------- ---(---X
.. ·-··--~ -- --. - ... ,-------1,
I·--·-.
- .. ·-~----
,.
:· - .. ' .. -·
It
I-----
!
I ~---~-
.--·--;-' - j .J
.1
excited at the junction are negligible. Now for.TE10 wave
as ·S = s l 0
·a /s IRI = J 1 0
2+j'i/ s0
for a1
<<B . 0
36
At 9.25 GHz, for a = 2vim and tr = 16, this evaluates to 0.03 approximately
to give VSWR = 1.05. This agreed well with the experiment.-
. . 3.5 EXPERIMENTAL RESULTS AND DISCUSSION
The measurements were carried out at 9.25 GHz on 6 samples, one
of intrinsic germanium (a = 2 mho/m) and five of n type germanium having ·• ..
d.c. conductivities ranging from about 4 mho/~ to 400 mho/m. The
results of measurements are given in the table, 3.1.
Inspection of the results shows that over the resistivity range
5 ~f~ 25Acm(20 ~ a ~ 4 "'f/m) both tr and a can be measured acc-urately.
The microwave measurement of~ agreed well with the d.c. measurements
and it was found that the measurements were repeatable with the limits of
3%. For f < 5 .n.-cm (a > 20 mho/m, 1 oga > 1.3) agreement between microwave l. .
and d.c. values of i is also very good but accurate measurement\ of er
is not possible. This is bec~use as a increases·er < < _Q_w and the •• £0
contribution of er to the propagation constant becomes small as compared
to that d~e to a. From fig. 3.2 it may be observed that the
TABLE 3.1
MEASURED VALUES OF tr AND p
SM1PLE SAI1PLE ATIENUATION PHASE MEASURED MICROWAVE RESISTIVITY LENGTH CHANGE CHANGE Po(d.c.) MEASUREMENTS
(nominal) t(cms) A1-A0
(dB) t61-t6o(o) (o-cm) p(o-cm) cr
50 4.27 . 2.85 3.35 50.7 99.4 16.1
25 2.79 1.66 4.00 24.2 23.8 15.9
10 5.10 2.47 9.10 11.4 11.8 15.3
5 2.79 0.890 4.54 5.00 5.00 15.7 .
1 5.10 0.632 4.82 0.9.9 1.01 -19.0
0.25 2.79 0.176 1.29 0.258 0.244 -67.0
.:
..
. .
curves of a and s for various values of £r merge into each other for "' . . a > 50 mho/m. An extremely accurate method indeed would be required to
measure £r at high conductivities at· X-band;such a method is unknown
at the present time. However, fig. 3.3 and 3.4 suggest that at higher
frequencies it may be possible to measure Er at· higher values of a.
This assumes, of course that at high frequencies· apparatus of the same
accuracy_as at X-band is available.
38
At higher end of resistivity range (25-50 o-cm) measured values
of £r are in good agreement with the theoretical values (fig. 3.6)
but there is a large discrepancy between d.c. and microwave measurements
of f. This can be attributed to two reasons. Firstly, the measurement
accuracy is small. It is found that errors of± ~.5% and± 3% in the
measurement of (A1 - A0) -and (~1 - ~0 ) respectively caused an err9r of
"-±14% "inS andt0.7% in £r· Secondly as the resistivity goes higher
the perturbation~. in field distribution in .the system become greater
and the higher order modes excited at the junctions at z • 0 and 1
may not be negligible.
f . '
CHAPTER IV
HAVE PROPAGATION IN A RECTANGULAR WAVE-GUIDE CONTAINING A
CENTRALLY PLACED SEMI-CONDUCTOR
4.1 INTRODUCTION
The wave-guide system shown in· the fig. 4.1 has been used by a
number of workers to determine the se~i-con~uctors properties such as
a and e:r (32 ), magneto-resistance (33 ), h~t electron effect (23 ). It has ~lso
bee.n proposed f~ruse as a microwave m'odulator (l 0>-. . .
The theory of the fields present a~d the propagation constants
in such a system may be obtained fr~m the solution of the equations 2.4,
2.5 and 2.17. It is sho\·m in section 4.2 that the propagation constant is
given by the simultaneous solution of the equations ...
y2 = k 2 - k 2 = k 2 - k 2 £~ 1 0 2 0 '
(4.1)
where k1 and k2 are the transver.se wave numbers in the regions (1)
and (2) respectively
y = a + jB = prdpagation constant
a = attenuation constant B = phase constant
£ = e:r - j ~ = complex permittivity '(' W£0
The solution of these equations for the case.when.the displace
ment current is negligible (we:<< a) with t/a of the order of lo-6 and
"39
y
bl !Z ... <i < I I
3
$ I'\ ,, ' "' I X· z d dtt d
(a) Infinite Structure
SHORT
Z=l
y
~"""" X ' " " " ">\ l>' Z
(b) Junction of Empty and Partially Filled Wave-~uides
Figure 4.1 Partially Filled Rectanqular Wave-Guide with Se~i-Conductor Specimen in the Reqion (2). The regions (1) and (3) are empty. ~
~ ~
for the lossless case (a= 0) have been reported in literature33 ,34.
The solution in both of these cases is comparatively easily obtained.
However, for the case when the value of a lies in the semiconducting
range and the displacement and conduction currents are comparable, the
computations of the propagation constant from equations 4.1 become very
laborious. Because of this, in almost all of the measurements referred
41
to above, recourse has been_ made to some sort of approximation. In
particular, Nag et a132 and Lilja and Stubb35 used the following expression
for y:
. y2 = (tr/a)2- k 2- k 2 c~-- l)(t/a + l sin trt) o o ~ ,.. . a (4.2) .
where k 2 = w2~ t 0 0 0
This equation can be obtained either by variational theory using
the Rayleigh-Ritz technique with single ~ode approximation(JG,J]) or
by the first order. perturbation theory(35,38,39). ,For a • 0 it was
first obtained by Berk36 and was found to give the propagation constant
with an accuracy of a few percent for tr • 2.45. For complex permittivity
values applicable to semiconductors, however, doubts have been expressed
about the validity ·of (4.2) unless t/a is negligibly small <40 •41).
The computations.of the propagation constant in the configuration -~
shown in Figure(4.1)~discussed in thiS. chapter. Numerical solutions of
equation 4.1 are given for dielectric constant, conductivity and t/a
ranges of 12 ~ £ ~ 16, 0.1 'a~ 10 mhos/m and .001 't/a ~ 0.25 r respectively, for t~e three frequencies.lo.o; 34.5 and 70.5 GH·z. These
results are compared with the approximate values obtained from equation
(4.2) and also with those obtained u~ing the two mode approximation in
the Rayleigh-Ritz technique.
42
Further for practical measurements, the structure shown. in the
f1g. 1.3 forms a junction with an empty guide supporting H10 mode. In
all earlier measurements, the effect of the higher order modes at the
junction has been assumed to be negligible. However, it is shown that
this assumption may not be valid in view of the high values of d1electric
constant and conductivity applicable to semiconductors. An analysis of
these higher order modes has been carried out and the computations show
that unless t/a < .08 and d ~ 2.5 mho/m, the effect of the higher order
modes is not negligible.
Experiments have been performed for three values of t/a at
9.25 GHz and the results ·of the measurements verify the above observations.
4.2 THEORETICAL CONSIDERATIONS
4.2.1 Exact Solution
The solutions to the Maxwell's equations in the system shown
in the fig. 1.3 may be divided into two basic sets, TE to x and TM to x.
The dominant mode in the system for lossless case (a = 0) is given by
TE to x mode. For the lossy case it will be the sam~ and only this mode
will, therefore, be considered.
The solution for the TE to· x mod_es· may be obtained from equations
2.4, 2.5 and 2.17 by taking a single component of nh. Thus if
flh • aX~ (x,y) e·yZ
so that vt2~ + ·(k02 lr(x) + r 2), • 0
(4.3)
{4·4)
43
where er (x} = £r -j ~ (d ' x ' d + t) w£o
then in. o~der to satisfy the boundary conditions 2.1 at x = 0, a; y = 0,
band x = d, d + t must be of the form
sin k1 x 0 ' X ~ d
tP = A cos mry b
sin k1d [sin k2(x-d) - sin k2 (x-t-d~J. sin k2t sin k1(a-x) + t
where A = constant
m = 0, 1, 2, 3, .... k1 and k
2 are wave vectors in x-direction in·regions (1),
(3) and (2) respectively and.satisfy the following equations
~ = k 2 + (ffl1(/ )2 - k 2 y 1 b 0 . (4.6) ..
_ k 2 + (m/x )2 k 2 A
- 2 b - o £r
(,4.5)
y is the same in the three regions so that the b'oundary conditions are
satisfied for all z.
A further condition on ~l and k2 may be obtained from the
transverse resonance condition. Thus for symmetrical modes it is required. L..t.,CNY\
that there~ open circuit at~= af2· This condition gives the following
relation
(4.7)
-H·
ll11: rqii.-H.hm •Ll yivr~ il\{l\\\(c! l"'~·;ib\,! V·"'\ur!"\ ''{ l.. t
,_
and hence there are doubly infinite ·possible values of y. If, however,
m = 0, these reduce to singly infinite values. When m = 0; the TE to x
modes reduce to TE~0 modes. The dominant mode.propagation constant is .
given by the lowest order solution of 4.7 and is designated with n. 1.
Thus for TEn0 modes,
2 - k 2 k 2 y - 1 -n n o
- k 2 - 2n (4.8)
For given values ofw, t/a and~ the values of the propagation
constants may be obtained ,by solving 4.8 numerically. In the pres'ent
work, this was done by using the Newton-Raphson iteration technique42,
in which the (M+l) th iteration of the unknown root of a transcendental
equation f(x) = cf is given by:
The convergence of the method is good provided f' • ~ is not too small
and the initial guess of the root x0 is not ,too far off. Equation 4.8
was accordingly put into the form: . k t
f ( k1 ) = k1 cot k1 d • k2 tan + • 0
with k 2 • k 2 + k 2 (t - 1) 2 1 o r
' '.
45
and the value of k1 determined accurately, which then enabled k2 and y
to be calculated. The functions involved are periodic in nature and
hence an infinite number of roots of. the equation exist. The particular
root obtained depends on the initial guess. In the present case, the
initial value of the root was obtained from an approximate.method,.for
example, the two mode approximation in the·Rayleigh-Ritz t~ch~ique
discussed in the next section.
4.2.2 Approximate Solution
There are two approximate methods of solution that are.
commonly used, namely, variational and perturbational methods. The
former method has been used for determin~:_:_. y in a wave-guide system
containing pure dielectrics and ferrites36 and is known to give good
results.· Perturbational techniques have also been developed for such
calculations38 ,39 • It is found that the first order perturbational
technique gives the same formula for y as obtained from the variational
method using a single mode approximatio~35. For higher accuracy either
a two or higher mode approximation or a second or higher order perturbation
theory may be ~played. It should be pointed out; however, that unless
t/a is negligibly small, the high value of complex permittivity of semi
conductors alters a ·system so much that the perturbations are too large
to be consistent with the accuracy requirement of the perturbation theo~y.
In view of this, the use of two mode approximations with the Rayleigh
Ritz technique was preferred for these calculations. ,
The ap~lication of the Rayleigh-Ritz theory to inhomogeneously
filled wave-guides has been discussed by .Collin29 , and the salient steps
46
in the present analysis are as follows:
The one dimensional Helmholtz equation
2 ~2 2 u , ~ + y + k & (x) ' = 0 2 o. r
dx ·
where &r(x) = 1 0 < x ' d; d + t ' x ~ a
· = &r d ~ X ' 'd + t
(4.9)
can be obtained from equation 4.4 by taking m = 0 so that ~(x,y) = ~(x) The function 0 is to satisfy the condition that 0 = 0 at x = 0 and
x • a, to make the ·tangential component of t_he ·electric field .Ey vanish
at the wave-guide walls.
The variational integral is obtained _by multiplying 4.9 by
0 and integrating with respect to x from x • 0 to x • a. The resulting
expression is:
(4.10)
The next step consists of expanding 0 as a Fourier series and terminating
the series at a finite number of terms. Thus one may obtain for the
nth eigen-function:
r . 0n = arn f (x)
r=l,3 r ( 4.11)
where fr(x) = 1 i sin ~x (eigen-functions for the empty guide). The
s~mmetry of the system penmits only odd values of r. The function ~ may n
be nonnal ized to give: ,
fa 0 2 dx • 1 • ~ a 2 o n r•1,3 rn
' .
47
and substitution of equation 4.11 into 4.10 gives:
N I
r=l
. ~ ua df df 2 a a J (_!.-2..- k 2 &(x) frfJ
s=l ,3 rn sn ·o dx dx 0
{
0 r 1 s where ~rs =
1 is the Kronecker delta function
r = s
Now it is required that:
N N l L a a (T - y
2 6 ) = a stationary quantity r=l s=l rn sn rs n rs
= Ja·[~~- 2 where Trs 0
dx dx k0 &Jx) frfs] dx = Tsr . (4.11a)
For this equation to repr~sent a stationary value oft 2, the -
partial derivatives _!__ for i = 1 to N must vanish. When this is done ~ aain
the following set of N homogeneous equations is obtained.
N L arn (Trs - Yn2 c5rs> = 0 s = 1 to N
r=l (4.12)
RecallinQ that the modes ·being considered are synmetrical about the
point x = a/2, the indices r and s should take only odd values, that ts
r = 1, 3, 5, ••
s = 1, 3, 5, ••
. . . • • •
, N (odd)
, N (odd)
For a non-tri.vial solution of '(12), the detenninant of the
coefficients must vanish. Therefore, for the single mode approximation . (N=l) one has:
48
(4.13)
~or the two mode approximation, N = 3, a~d y12 is given by the solution
of:
(4.14)
Integration of equation (4.11 a) over the wave-guide cross
section shown in the Figure (4.1) gives
T = (r1r/a)2 - k 2 - k 2 (£1 - 1) (t/a + 1/rw sin rw t/aJ (4.15) rr o o
T = T = k 2 (£...- 1). [(-l)P sin pnt/a + (-l)q sin gwtt.aJ rt-s rs sr o -T · pw qw .
where p = r + s , q = r - s 2 2
In the present study, the dominant mode propagation constant was computed
from both equations (4.13) and (4.14) and the results compared with those
obtained from (4.8).
4.2.3 DISCUSSION ON EXACT AND APPROXIMATE SOLUTION
The dominant mode propagation constant y = a.+ ja was computed
from equations 4.8 as a function of t/a and a at three different
frequencies of 1o.o, 34.5 and 70.5 GHz for £r = 12 and 16. The variations
oft/a and a were over the ranges 0.001 ~ t/a ~ 0.25 and 0.1 ~a~ 10
mho/m, respectively and the results of the calculations are presented in
the form of curves shown in Figures 4.2 through 4.4.
It may be observed from these figures that the phase constant 8
varies very slowly with conductivity a for small values of t/a and a. In
fact the variations in the phase constant 8 are practically negligible
fort/a< 0.1 and log a • 0.50 (a • 3.16 mhos/m) at 10 GHz. This also
... ~· .
-.
f=IOGHZ. . ~~ %, ,-o~6:;
1 ~
~I () fy = 12 ------
~1 I cy = /6
"> N . ()
I <::>
~ C"',J
<::1
- o.a ~ 1~ f~ I~ I ? " . .
~ tl .;:) ~ (;) II I· . I I ~.or--.--,--- -T----y---
I I I ~ I f
'
l. 4 6 8 ~~I
Figure 4.2 (a) The exact I '!f cl. N€ P /.,..,., values~dominant mode Propagation constant y = a + ja at 10 GHz
I ~-~
I' I .. J-0 .:; 'I I I I I I I J-1· 0
/I I
' I
~ ......t\
! .
)
~ ~
~
·7
2
16 ---2"X/o:
~
<}
' f:::: lo Gllz. I II ~
f-y::/2.0.------- ~~
l-)~/f,.t) . 'JJ%=-2·0 tS" ia.v ;r/->r~' , I
1
()
. l I ....... I ~~ ") () :f) !'J •.r· ,. I II ,.,_: • .'1 (J r:c '? I
,. . ~ . - " . (! ~ ~ ~ I \; t)
•"'-1
1 --r----1 ----r- r' "-:; \> ----r ---,-- o I I '',,-,
,- - --T I '-'> I I l, I I 0 I l ,,.., ' I I I I fi I C> I • I "' " ' ,· "1 ~ l"j /,_~~-,I. I" I '\,!
' ~ : <, " I ,.; I " : "' I ,; I Q ';::. I (J I ; I , I I • ~ I \} '· I I 1 I ' ' ' 1 ' I I I / I . .:
/ /
/ /
I I
/ I I I
~-/-2. 2
/
I /
/
I
I I
I
I I .
figure 4.2 (b) The exact values~dominant mode propagation constant y = a + jB at 10 GHz
I..''\ ~
~ I ~
........ ~
~ -~
~
~
L I
.......... _ _,._.._.-~......_ ____ ._.: ___ ---~·--- ---,.-----:.-- --··----- -·--·
~ ~~--------r----~~-:~~::::-r,-----~~-:--.---~~~r-rc~ll--~~-,rc~~---r-:r-r-r-\:ll----~--r--cCT---r-------e--. < .. I ' I I 1 1 1 I ' j I I n ~ I I I '"' I ~ ~ ~'I l~ I \.')I I I I 'J ........ V) '''I L~.? l7. --o / )-. "' "-. ~ ';. ~' 1
'
1
I 1 1 n-r u let - . (;> r , , ~ ..
. ~ I I I I ~I
II
..f:::: 31·561/z
.,, \.:
...:..
~"'l
·20
/6
... 1'2.
8
-3·0
4,"()'
Ey:/2.o
q·:; 15. .o / .
6" '""" ...,,../)1"1
..
-c.$ I' II '0 ~~ ···r--
1
~) i'-. 0
•
~ v) ·~
I T--•
\., ('~
<.:, 'CI
":-~ ,.., 7:) 1.."')
.-_) --.--1
.·- /. " I ! . I I .l , I . I I ~ I , J-I
--- / "' ,.7 -1 ,... __ ...- _,.,. . .a
·- 3·o
4 ~ 8 10
(ttu. · o< Ne P/m Figure 4.3 (a) The exact values~ominant mode propagation constant y = a+ je at 34.5 GHz
.. /
I . <(~: :/,7 -r--n -;.-',)
"'
' _j_r .,. ~.'
I I I
~~
"'
/0
!?J,
{· ~ 8 ·~
<::\c
~
I T
~
- • 1 . 1 1 1 1 1 1 1 1 1 1 1 rrr 1 1 1 1 1 1 1 IIJ
•
f :::= 34.~ 5' G II~
(y == 12.- o-- -~./-f.y-:;;.. /(.0
6' l ~-~ -v-1 ~..,.. .
,
()
r \l
'o.
Jo;t . ~ v~=.-,.G
'-'> f' . ~
'
~ l') . t'"'
<:} . <:l
<:1 <.1
' I ·~
·. -_........... ...... ~ <::::,. .,.-:-= -- .....-; .... - <::: ..... .r ,...-r -- ~~_,.- _ ... .....,.. - ;roo» ........... ,.. :=? - .......... ::::o:>" - :::;>' -- ,. .,.,.
- ..- .- :.----'- ;,? • .[>
- -~:.;t-;- ~ , ' u .··~- - ......
-.- 1 1 r rrn
~ ~
\.') ~
,t vj .
() <J
l) I'
<:i
--/
~, ,._, /
" 7~-~-·0 " " /
:;;~-2. 2..
-5 I I '"' I I I I I I I I I I I I I I I I I ! I ~ 2 XI o 2 4 6 8 1 0 I 2- 4 ~ 8 1 0 ° 2. 4 6 8 I o 2.. 4 6 · 8 I c-
Figure 4.3 (b) l;f-Jto.. ~ N~ P /m
The exact values[dominant mode propagation constant y = . i
,/
a + jS at 34.5 GHz ~
t\
I' N I() -X
~ ~
.~ _. ~ "30
~
: .i 2
., '•
' '
' . 5zxto
.,.~
l = '1 0 · 5 3/lz
Ey:: 12-0 ------fy::. 16· 0 ----
~
? ~ ~
~s(tfi)=-b·tl~ I~ It ~~ /~ I~ I~ I~ F· -0.8
1 , --=LL , . -1-6. (,
-=-, , ___ , _ _,. =L I -- L ' - --1= =~=~ ='- I o -~=--L_j o I -·t. · t
I
-~ 8 Jo 2. .or II 0( NEP/m
· Figure 4.4 (a) The exact .,.,~ .
values~dominant mode propagation constant y= a + j~ at 70.5 GHz ·
\;:,"5
...,.., v
·-
i ' I . : !! ..,
" T I I I I I I
f
I I I ••• , I I
7
- \) I I I I I I
~ - C • i.: r_-;:; f/·• 0 l I ~ . - I (" - 1.2·0 ,,· I I I I • .
f;;,t. o -------- " t, I ~ ~ I
~ ...
~~~-'·~~~~·-~~~~~~~~~~~~~~~~ ([" "":., v ;., .... , .
l') ~~ 1 •• ~ . ., v i.r) ! C)
0 ('., ~ c· . (• ~ I I
12
II -3-0 ...=:::. .... - --::=::: .... -. ~---- <:::. .... - ""'= J
I I I I I I I I I I I I I I I I l I L__! 2 X I{)- 2 4 6 8 I 0 -I 2. 4 6 8 I 0 ° 2 4 ~ 8 I (1 • -
. · ;· . o( N!P/n"~ · 1 ·i=iq~~1e-:~.!4 · (~) -·:fh~ ex~ct 'val.ue'sl!ortlinant mode )propagation .constant' r = ,,.. . · •• _. ... , •.. , ,l. •. • . \... ._-:·
'; '\ ..... :~ • •4 ~ . • ·. ~ - \ ..
a + jB at )0.5 GHz
""\ ..s
55
applies for all the values of tfa and a considered at the higher frequencies
34.5, 70 GHz. At 10 GHz, the variations i~ B with a for log a ~ 0.5
and log t/a ~- 1.8 {t/a = .0158) become significant as can be seen
from Figure (4.2a).
It may be observed from Figures ( 4.3) and (4 .• 4) that for the
ranges of a and t/a chosen, the relationship between log a and log a is
a linear one. In these ranges, one may write:
log a = log M + n log a (4. 16)
where n is the slope of the curves and M is.the intercept with log a • a.
This relationship is more clearly shown in Figures (4.5a) and (4.5b), and
. examination of the curves shows that n = 1.00 which means that the
attenuation constant a varies linearly with a, i.e.
a= Ma (4.17)
At 10 GHz (see Figure 4.2a) the relationship (4.17 )also holds for t/a
< 0.05 and 0.1 <a< 10 mhos/m. For higher values oft/a, the value of ' ' ' n departs from unity.· At t/a • 0.05 its value is 1.01 and increases
as t/a ·is increased. At this frequency, equation (4.17) holds over the·
whole range of t/a considered but only up to a ~ 1.0 mhos/m. For
higher values oft/a, the slope of the log a ~·log t/a curve changes and
equation (4.17) no longer holds. For values of.t/a • 0.25 the slope is
very small and practically equals that fort/a =. 1. That is, the
attenuation is very close to that of a completely filled guide.
It is interesting to note that a similar relationship between
a and a holds for very thin films of considerably higher conductivity
than has been considered in the present work. Gunn44 has carried out
" 0.. w z 0
2 10
7
3
5
3
7
5
_,
f = 3 4.5 G HZ
E"'= I 6
' .
-3.0
12l.~O~~--~~--~--~----~--~--~----~--~--~~------~ -t-•2
LOG ( ~ -v-J-m) Figure 4.5 (a) The attenuation constant a as a function _of.t/a and 61
\
'- I I , I I ' I I
-- Li
- .(; L b G(t:/a.) -1.0 .
. .
lL/ f-f : 70.5 GHZ
E""-; I 6 -1.4
. / v/ f- 1// /v / -l.a f- Li v L L. f- J / /
~/ / v -2·2.
f-
# / v v v ..:2-6
2
2. I('
s 6
4
2
Vv ~ / / L_ // / .' L L L
I ·\ 10
1-
8 ~ /L / / / / ~
v v v -~.0 t • I·\ 6 ~ , I \
·~-· . . / L '·/ L L v v v v v ~4
0... w z
0 2
/ / / / 0
2
v v lL v 1- L / L ~/ / / v v v f-
/ / v v I I/ .•
I I I
10
8
6
4
-10 -1.0 - .6 -.2 +· 2· -r•o
'
_I_ I
· L 0 G ( 6'" -v-/m) Figure 4.5 (b) The attenuation constant a as a function of t/a,and cr
•f I ,
\
58
·computations for thin films of semi-conducting m~terials, and found that
theoretically and experimentally the·re1ation (4.17) holds for R=l/at~40.0n/
square. However in the present work it is found that equation~l7) is
(a~ 1.0 v/m, t/a ~ 0.25)
valid for R ~ 175n/square~at 9.25 GHz and to·at least 55n/square and
130n/square at 34.5 and 70 GHz respectively (a ~ 10 v/m, t/a ~ 0.25).
To the extent that the phase constant 8 is nearly independent
of a, the propagation constant as a function of conductivity a may be
written as:
y(a) = y(O) + a ay Ia= 0 a a
(4.18)
where~= M is evaluated from equations (1) and (2) with. a • 0, and is: a a
.....
jwll 2 ·0/
k10 d cosec2(k10
d) - cot (k10
d) -
-where the subscript 0 denotes that quantities are evaluated w.ith a= 0.
. . Equation (4.18) for y(a) reduces much of the computation work
as the numerical solution of (1) and (2) with a= 0 is much. easier and as
a matter of fact may be done with a desk calcula.tor. The calculation of
M is then straightforward. This linear relationship between a and a
may prove useful in the measurement of magneto-conductivity,high field
carrier mobility, the temperature dependence o~ conductivity, etc., of
the semi-conductor bulk material.
The propagation constant was computed for two values only of
dielectri.c constant (&r), namely 12, and 16. It may be observed from
Figures 4.2 to 4.3 that the variations in a and 8 with £ are significant r
only at the higher end of ranges of t/a and a considered. For. lower
values of a and t/a; the changes in £r do not cause appreciable changes
in either a or B·
Figures 4.6 and 4.7 show the percent~ge differences between
the exact values of the dominant mode propagation constant as obtained
from equation 4.8 and those obtained from the single mode and two mode
approximations given by equations A.l3 and. 4.14 respectively. The
errors shown are for a = 10 mhos/m and f = 10.0 and 70.5 GHz. For
59
lower values' of a in the range 0.1 , a, 10 mhos/m, the errors are not
very different from the ones shown, for example in the range 0.03 ~ t/a
~ 0.1, this difference is not more than 3%. The errors in y at 34.5 GHz
are not shown here as these are found to·be of the same nature and order
as those at 70.5 GHz.
It may be observed from Figure 4.6 that the single mode
approximation giv~s errors in a that are greater than 5% when t/a is as
small as 0.0025. Fort/a > 0.0025, the ·errors increase rapidly and then
decrease at t/a ~ 0.06 but are not less than 33% at t/a • 0.25. The
·two mode approximation gives better results, but the errors in u are
significant unless t/a < 0.003 when these are about 5%. The error in
a is reduced with a decrease in'the dielectric constant.
Figure 4.·7 shows that the approximate methods, both single mode
and two mode, appear to give better results for s than for «• It may be
seen that the errors are less than 5% for t/a < 0.015 •. For higher
values of t/a, the errors become large but the maximum error in s is·
about one half of that in u. As t/a is incre~sed beyond 0.1, the
~
r:::: ..... s... 0 s... s...
LJ.J
~ r:::: cv u s... 'IV 0..
80
60
40
20
f:: 10 GHZ o::. I 0 -v/tn
'Single Mode in R-R Technique Two Mode~ in .R-R Techniaue
' ' " ' ' \, '
-2.6 -2-2 -1.8 " -1.4 -1.0 -·6 --2 LOG ( t/a.)
Fiqure 4.6 (a) Percent difference between exact and approximate values of a
--~ 0
---------------------------·
f = 70.5 GHZ a'= 10 T/"fYl Single Mode in R-R Technique
----- ;_ Two Modes in R-RTe
-------:--------~---·----------·- .. -
dGO~------t-------~-------t-------4--------~------+-------~------J c ·~
tcLJ~-4--l75~~i~~~1~·~ cu u s.. cu
0.. ---- •' - "'L..' ' '
....
2CI 1 .......-:, I / / I J' r ""'' 7 7 ~ __ , I I ~ ~ ~ I I I ~
' \
(
:::::"!.D -2·6 -2-2 -1·8 -1.4 -1.0 -.6
LOG ( t/a) ,
Fiqure 4.6 (b) i .
Percent difference between exact and approximate values of a ~
~~ ~
c . ,...
40 )
) 30
20 I
~10 s.. s.. LIJ
~ c Q)
" u s.. Q)
0.. 0
-IG
I
~
-
1-
~
l
1-
~
I -15 -3.0 -2.6
I I I I I I I
Single Mode in R-R Technique f:IOGHZ Iwo _Modes_ jn~ R~R Techr ique _______ . _______ ---~-------()-;:: to-v-/..,
. ! L ~
12 Eyd6i r\ / ~-6/-r-: ....... ·., -- ' . 'I f 2 ... " " I ~ ',\ I
{ / ',~ .. '\~ I I I
/1 / I . .1
I I i I 1 L ./ I ;. . . -~ ..... ..... ...._ I I ... ........ ...... / I ....
~ I
~ \~ ·-.,
J I I I L__ -2.2. -1. 8 -1.4 -1-0 -.6
LOG ( tja) Figure 4.7 (.a) Percent difference bet\'teen exact and approximate values of s
I
.
1
I
--
-
-
-
-
r--. ~
en s::: .,.. ~ 0 ~ ~
LLJ ..., s::: C1l u ~ C1l 0.
f-
40
f-
30
r-
20
-
10
'-
0 -3.0
..
I I . I I I I ' I ' I . I I
f=705GHZ c>::: 10 -v-jm . ·single Mode in R-R Technique
- - Two Modes in R-R Technique
..
.. ~..., = 16
-~ ·/ -~ 16.,....- ........
. ,' ' ' , ,, I ___ 12;..-~-, , . /~ _.... ..... ' /// / -,', . , / . ' ~ ~ ,,
~ ,,
~ v, \ ..... ~ ,.,.... ..... .,. _ ....
~ · I I I I 1 . . . -2.6 2.2 1-8 0 6
LOG(t/a) Figure 4.7 (b) Percent difference between exact and approximate values of B
--··-------~ ..... __., ~----__..,..... - -........--~---....-~~--~--- .. - , .. ~---·--~-'--· ------~~---·-
I
-
. -
-
.
-
.
-
I
.. ···""'·.-,....; ..•.
errors reduce. This is because as t/a is increased, the condition
approaches that of a completely filled guide in which case the errors
should vanish.
4.2.4 Calculation of Junction 'Impedance
In this section the expression for the input impedance Zin
at z = 0 of the system shown in the fig. 4.1 which has a short circuit
at z = 1 will be derived.
In the analysis, it will be assumed that the input guide is
supporting _an H10 mode while in the output guide (O' z' 1) Hno
64
modes are present. If ~nand ~n are the functions-representing the Hno th
mode in the input and outp~t guides respectively and rn and Yn are
their respective propagation constants, then the electric field EY
and the magnetic field Hx are given by the following expressions(43).
For z < 0 .. Ey __ = al·(e-rlz·+ R erlz) +l + L an +nernz
n=3,5
. Hx • ~alYol(e-rlz- R erlz) +ran +n Y~n ernz
For 0 ~ z ~ 1
Ey = L bm ~m sinh '1m (1-z) m=1,3,·
Hx ;. r- bm Ym ~m cosh"tm (1-z)
where a's and b's are constants
R • reflection coefficient of the dominant mode in the input
guide
' 1 n Y0n = -2 = J.W'IJ = wave admittance for H
0 modes in the
on · o n empty guide
Yn = iL ~~=~ave admittance· for H modes in the output n JW o no
guide
Now at z =·o Ey and Hx must be.continuous and these boundary
condit.ions give
Ey = a1(1+R)41 1 + l a~ = l bm sinh(yml)pm. n=3,5 n n m=l,3
65
These two equations can be solved either simultaneously or by
the use of variational techniques. The latter method is easier and has
been discussed by Collin<43 l. Foilowing his arguments, one obtains . .
the expression for the input i~pedance Zin = (l+R)/(1-R) as follows:
a · a · 2 zin = i- If G(x;x' )Hx(x) Hx(x' )dxdx'/ riJ Hx(x)~ldJ
Ol 0 ~0 J
Now assuming that N
Hx(x) = ) an ~n(x) n=l,3,5
and substituting this into 4.12, one gets
NN" zin Zol a12 - ~ ~ ar as grs = 0
... ... grs = L 2on cSnr cSns + }: · Zm tanhym l p nl sm
n=3,5 m=1,3,5 where
where {
1 n=m 6 =
rvn 0 nlm
Now equat.ing the partial deri·vatives w.r.t. as to zero so as , . , . to render·zin a stationary quantity, one gets the sets of equation~·
N 2in 2olal - Y ar glr • 0.
y•f,J ..
66
s = 3,5,7 •••
whose determinant must be zero for non-vanishing valu~s of a's,
Therefore;
gll g13 glN
67
z. =-l-1n z
ol (4.19)
. gN] gN3 . ' . . · gNN
g33 g35 g3N
gN3 . . . gNN
4.2.5 Calculation of Reflection Coefficient
The computations of the junction impedance were done with the
help of a 7040 IBM computer, the equation 4.19 being solved for different
values of t/a, cr and N at 9.25 GHz. The results are plotted for N = 7*
in fig. 4.8 in the form of the refJection coefficient R, . . '
R =(Zin -l}/(Zin·+.l)
which is a directly measurable quantity. The length of the sample was
assumed to be one quarter wave length (=}l;/2e) at each point. The
figure indicates the extent to which the reflection coefficient may vary.
In fig. 4.9 the magnitude of the reflection coefficient is
plotted as a funct1on of cr~ for various values' of t/a and N. It. may
be observed that as cr increases, the higher modes which are excited
considerably change the reflection coefficient at.z = 0. It may
further be observed from the computational results that unless t/a
* The reason for t~e choice of N=7 in the fig. 4.8 is that in the range
considered, this value gives good agreement with the experimentalresults • •
This agreement may be better with N=9 but the numerical ·computations are then too cumbersome to handle.
l t
.... i
J
"\
•
•
w
) .; .,
'! ~ 1 -!) : 1 ; , 0 ... .J ~ ~ <{ ,, z ' c. ·~
~
~
0 .. v :l ...
" <{ ..J 0
. ~ ' ~-< ·t'
' ~
,...
-=l -j ·- ! ·-
--- ! i __ ,
__ ,
--· ' -i
., _) I
I
-I
__j
I
! _. _J
-;
"""i
' '
--1
I -~
!
--=1 j
I
~ I __J
I
~ --;
---1 --i
I .---, _, ~ --; _J
:
I
--~
'
--j
-;
_j
--.-, _j _j __ , --1
' -·
I
' ___J
270"
..;o.,
jl" I I I !
200" I 60°
i l i ,l-1
I 90" 170° 180°
170" /')00
160' . 2oo•
ISO' 210'
r. ".)
9()•
270"
-
N.l
-·-·-· N~
------ Na7
----·~ ---...:-..::-· -_________ _,
0~--~--~--._--~--~--~--~--~--~--~--~ . 0 ·4 . 6
LOG ( ($" -v-f-n~) Figure 4.9 The reflection coefficie~t at z•o {Fiq. 4.1)
as a function of a, t/a and N.
~ 0.08 and a ~ 2.5 v/m, the effect of the higher order modes is not
negligible.
4.3 MEASUREMENT OF REFLECTION COEFFICIENT
70
The reflection coefficient at the junction of an empty guide and
the semi-conductor loaded guide as shown in the fig. 4.1, was measured
by means of the reflection bridge as described in the Appendix A.
This bridge directly measures the reflection coefficient R = rRiej~ The short circuited wave-guide section containing the semi
conductor sample formed part of one of the side armsof the bridge and
placed in a Delta Design oven. The arrangement is shown in the fig. 4.'10 ·
The zero balance of the bridge was carried out at each temperature with
and without sample.
For practical convenience· a -slot of desired t and 1 was milled
in the centre of the broad wall of a wave-guide section to assist in
centrally locating the sample. At z = 1, the guide was terminated with
a solid short circuit plate. The samples used were of intrinsic
germanium whose conductivity was varied with temperature which was
varied from about 80°F. to 300°F.
4.4 RESULTS AND DISCUSSION
The experimental verification of equation 4.19was carried out
on three samples of intrinsic germanium with t/a ratios of ·.0133,
.026 and .0515, at 9.25 GHz. The conductivity.of the sample was
varied from about 2 mho/m to over 100 mho/m by the variation of the . t
temperature of the sample.
Fi gure 4.10 The photograph showing the microwave reflection bridge. The sample is in the Delta Design oven on the right hand side.
72
The theoretical value of conductivi~at ~he temperature T0 k was . . (45)
calculated from the following relationships
a(T) = n.(T)q {~ (T) + ~ (T)} 1 n p
where q =electronic charge= (1.6 x lo-19 coul} . 3j~l ~
n.(T) =intrinsic carrier density= 1.76 x lo22e-4550/T T 1~ 1
~n(T} =mobility of electrons = .38 (300/T)+l.66 m2tvolt-sec
Pp(T) =mobility of holes= 0.18 (300/T)+2.33 m2/volt-sec
In the computations, it was assumed that over the temperature
range used in these· experiments, the dielectric constant'remained
constant.
The results of the measurements are shown in the figs. 4.11a
and b. The solid curves show the quantity ~omputed from 4.19 with
.N=7 and the dotted ones with N=l. The crosses are experimental points. . . .
~ . ~t may be observed that for a> 10 mho/m (loga = l.O),'the
results agree well with N=7 curve. As the c~nductivity decreases, the
solid curves and dotted curve approach each other and remain close to
each other except in the case of t/a = .0515. In the. latter case, ·the
experimental points are close to the solid curve but lie above it. At
the lower end of the conductivity range, the experimental points do not
lie on these curves and there appears to be a small discrepancy between
the theoretical curves and the experimental points.
i <r
,.o
~
. s
.6
f.
.4
f-
• 2
f-
02
.-.___.. .. __ , I
I
~-.-----f= 9.25GH2 -t/a. =. 0 I 3 3 ,.. ----- . --l=·8651 CM .,..""
,. ,.. .
/ /
/
• ~" ---1- vv, / , , ,. ,
~' , ,.
/ ,
~ , .... ,
_"/...
'~ ., . /r
~ ' -
v ' ,' . "I '
' I ...
~ , ,
/ /
~ "",. v ' ~ Jy '' .... 'II"'- . . _ ... . . .... __
..
·~
....L f _! I I I I t -4 6 8 1.0 1.2. 1.4 1.6 1-B
L 0 G C a- -y-f""") Ffqure 4.11 a-1 The Practical and Theoretical Values of IRI in R = IRiejQ for tja = ~133
-
I I
2.b
--
---
-..1 V>
t -·-cc -1.0
.8
.6
.4
f = 9.25 GHZ
tjd :::.026-
. I :; .6688 CM
.1 .,
~ ,'
·/ /·
~ / . '/...
/ ~
~
. I I
'(_
I
, I
~' "' \ , .
J~ I A
1--- .... - -- r- - - - - --i
I , ---/ r-~
//
/
.2 I I ·,t:-_ ~ ::f/ I I - I I I I -
I I I I I ' I I I i I ' I I I I t ~-
.4 .6 .a 1.0 1·2 . 1.4 1.6 J.8 2-0 LOG ( () -v-1~) .
Figure 4.11 a-2 The Practical and Theoretical Values of IRI in R = IRieJ 9 for t/a = .026 I
,
/
~ -.t::-
I -E"
lO
1-
.e
1-
.6
t-
.4
1-:
.2
1-
-~2
l . I
.f ~ 9. 2 5 G r\7.. I
l.-ja.-::. .0 514 --· t -:.A5920"' --- - _:-------;-__ ,..
~ ~ - :X. ')( _..--- ~ . X ,, , . . v ~ . ,.
;'
"" ""
v"/ ~
v v
. ~
,.)I v "'- . )C. . / -·.,. () I .. r-. . ___.,..,. / .
I . I . .
\ /
' / . ' /
\ . // ' ,_-""
• t I I I I. I I I
·4 .6 .a 1.0 1.2 · 1.4 I.G 1.8 LOG ( 6"" -v-f-m) . g
Figure. 4.11 a-3 The Practical and Theoretical Values of IRI in R = IRiej fort/a= .0514
2.0
"'
'
-----··
·-
-
-----·
~ v"\
.... ot Q,)
. ~ teo
f = 9.25GH2 - Va.=-0133 .
f~-8 651 C M _..-
~...----_.. '
_.. ~ ... ~
140 ' .. ,.
1-/
·/ ~ . /
,' 100
, 1- . / ,
I
Go I I v
:
I !-
/' ..
~ 1-)I '( )(
')C --- ...- ..... ---
2:J
. 0
- ,--- - ...----I I I I I I I --20
.. 2. .4 .6 . a 1.0 1. 2. 1.4- 1.6
LOG co 'V"/n\) 4{,.
Figure 4.11 b-1 The Practical and Theoretical Values of Q in R~=·IRiejQ for t;a = .0133
-
-----
-- ,.........,.--------
.. ----
v X ')(
.
'
- .
_____l_____ --- --- l -
1.8 2.0
...l
"
I
l I I
I
f =925GHZ -t/o=-02 6
I
()t Q)
ISO I. :::.-B8G6 CM
- ---·---\'\~ -- __.
~
...-
0
.,;"' . ,.....,... N~7 ,.
){ / ~ .,,....
~ . ,. I •
1-
1// I v~ I
,t/ - .
;~ ~
>
1-
. . ~ . ./ I
[7/ II' ----,.... ./ .,.
·140
100
60
20
---- . -I I I ' I I ~ I
-20.2 -4. .6 -8 l6 1.0 1~2 1.4
LOG < 6-.r/..,..,..)
~igure 4.11 b-2 The Practical and Theoretical Values of Q in~= IRiejQ for t;a =
I
- - ~-
i i
.
.
I
I.S
.026
_ . .._ . ....,
i
_ _I
---
• ____ J( ____
)( .
..
I -·--
2.0 I
-1 ...._}
ot Q)
200
f-
160 -
~
120
1-
so
40
1-
~2
./
f ... 9.2 5 GHZ t;a .... o s r 4
I ____ j t~.4592CM
. . . --------------- --.-., ~ _.. ----~-----
..1 .' --,.,. 1.,. y
v-;-- I l! / )1.
/ /
/""" / / v· I
// "' . . .
/ v .
I I . I
I .
' : '
I I I . I I I I I - I
A .G .8 1.0 1.2 1.4. 1.6 1.8 2.0
Figure 4.11 b-3 The Practical and Theoretical Values of Q in R = IRiejQ fort/a= .0514
·-~
I
X
-----·---
'
I
'
'
c ... 'lo..,.
~ 0<::)
79
This discrepancy may be attributed to the effect of the slot
in the broad walls of the guide. In the' computations the effect of the
slot was not taken into account. The slot may have two-fold effect,
i} The assumed boundary condition at y = 0 and ,b is not satisfied for
d-~ x =.; ·d + t,and ii} Radiation may take place through the small
unavoidable air.gaps present between' the sample and the wave-guide wall.·
It \'las shovm that this discrepancy \'las due to the slot by applying silver
. paint ·to the top and the bottom of the sample and air gap was also .
filled with silver paint. The .resulting points are shown by ~ircles
on the fig. 4.1la and lie nearer the th~oretical curve.
Fig. 4.12 gives the computations of f. • l/a from these measure
ments, together with the theoretical curve and the measurements made
with t/a = 1.0 (i.e. completely filled guide). For a partially fill~d
guide, cr was computed by using the approximate relationship 4.2 and
assuming N=l. I~ may be observed that the results are different
from either the theoretical values or the practical measurements for
the completely filled guide. At.higher end ofT and hence a, the
discrepancy is due to two reasons, i) use of approximate expression
for Y for the deduction of a and ii) neglect of higher order modes.
At lower end of a, the disagreement may also be due to the slot for
yeasons similar to those discussed above.·
' .
60
50
40
~ u
I
~30 0.....
20
10
-
-
-
,...
1-
1-
eo
'\
-t c
../
\ d.C.R E F <4&\ . MICROWAVE (THIS WORK)
-t ~/a = t'-0
" t/a ::0 -0514 s ..,
:::::< .026 E) , =- ·0 133
'
~
~
0
0 ~ a . ... .
"" • (I
G
-4 0 ~
. .
0
-1- m
' ... . ., .,..
t-~ l( .
v , v )(
~ Cl IS
Q a 0 •
ti. tQ 0 0
t t t ' t 1 I i I . -100 140 1.80 2 20 260 -300
_To F F\qure 4,12 The Practical and Theoretical Values of P = 1/a
I
Q!) 0
CHAPTER V
· THE CONDUCTIVITY AND DIELECTRIC CONSTANT OF GERMANIUf1 AT
MICR9WAVE FREQUENCIES
5.1 INTRODUCTIOrl
A general treatment of the frequency dependence of the high
frequency transport properties of cubic crystals has been reported by __ .· _______ (46) --- -------. . ------ . --
Champlin and the high frequency dielectric constant of germanium
is given in the references· l, 2 and 7. However, some gross simplifica
tions have been made in these investigations. For example, -in the
references (l) and (2) relaxatio~ time of charge carriers has been
assumed constant and in the reference (7) only scattering by acoustic · . -- .. -· - - ·---- - ---· --~----- ··-··---- - ----· ---- -
mode has been considered. Th~se assumptions are not justified. The
·--·relax·ation time of carriers varies· with. their energy. ··Also· the . . . . . consideration of only .acoystical phonon scattering leads to the-~ c;
mobility. of ~arriers varying. with temperature as T-312 . (47) •. Tnis is
in contradiction with the experimen~ally observed variations of T-1· 66 ---- - ..
for electrons and T-2.33 for holes ·;n.germanium(48 >.
A number of theoretical models have been suggested to
-interpret this observation. These include intra-valley-optical phonon
scattering (26 ,48 )_, intervalley scattering b~ acoustical and optical
modes .(~!L~n.d_the __ v~riation_Q..Lthe. ef_fectiVELmass _wi_th _temperature ( 46 ).
In this chapter, calculations have beery made of the micro\'lave
·mobility and dielectric constant of 1·ightly doped n-type germanium
using the following scattering.mechani~ms.
81
For electrons
1 - Ionized impurity
2 - Intra-valley ac~oustical and optical phonons
3 - Inter-valley
For holes
1 - Ionized impurity
2- AC.oustical and optical phonons·
82
'
The computations have been made for 100° , T ' 500°K, 1019 ~ Nd ~ lo25/m3
and 109.: f ~ 1ol2 Hz. The effective masses.have been assumed constant
and scattering by neutral impurities has· been neglected. \
5.2 THEORETICAL CONSIDERATIONS
5.2.1 Microwave Mobility and Dielectric Constant
The expression for microwave mobility is the same as that
for d.c. mobility~ except that<~> in the latter is replaced by <. ~ 'l+jw~
for the applied electr~c field varying as. exp(jwt). Thus·
~ I * ~ac = q < > m l+jw~ . C
(5.1)
where me~= conductivity"effective mass. The conductivity a is then a·
function of frequency
a(w) = q2 n < T l+j...,~
/ m * > c
n • charge carrier density
and the relaxation time average <~> is give~ by (~O)
(5.2) .
(19) >
83
(5.3)
To evaluate <t>, one needs to know the variation of t with energy and that
depends on the scattering mechanism.' These aspects are considered later.
' If in a system both electrons and holes are present, a(w)
assumes the following form
a(w) = q2 {n < t~ > /m* + p < t~ > /m* } l+jwt en l+jwtn · cp . n '
(5.4)
where n and p are respectively electron and hole concentrations
Tn and tp ~re their respective ~elaxation times
m~n and m~p are their respective conductivity e:fective masses
Now the current density with alternating fields is given by
- a£ + J • £ · £ - + aE o r at .
+
= (jw £ 0 £r + a)t
+
= jw £0 £r E
where £r = £r - j~ ~
£r = £ 1/£0
lattice contribution to the dielectric constant·
Substituting a from 5.4, gives
cont'd
..
- j -;/- [m n * ( ;n 2 ) o en 1+1.1) ~n
+ _P_
* mcp
84
{5.5)
(5.6)
Once t~e.form of T(r.) is known, the equations 5.5 and 5.6 are readily
evaluated.
The relaxation time T(r.) may be calculated from the scattering
or transition probability for a given scattering mechanism. Thus. if •
H' is the perturbing potential and causes a charge carrier to make a
transition from state k to k', then the scattering probabi·lity per
unit time S(k,k') is proportional to the transition matrix element
M(k,k') given by{5l)
M(k,k') ·= J~k~ H' ljlk d-:; .
ljlk, and ljlk are the wave functions of the charge ca.rriers in
~he state k' and k respectively. Then
2· S(k,k') = £l! IM(k,k') I N(r.)
1\ '
·where N(r.) =density of final states
~ £1/2
I
(5.7)
{5.8)
85
If e is the angle between k and k', then. T(k) can be written, using
·Maxwellian statistics, as
T(k)-1 = J(l-cose) S(k,k')dk'
Now if it is assumed that the scattering proces$es conserve
energy, or nearly so, then the above equation'can be reduced to simpler . . fonm. Thus if k' lies on the same energy surface as k and e is the
angle between k and k', then
T(k)-1 ~ J(l-cose) S(k,{) ~..n..·
cUL= 2i\ sinede
For isotropic scattering S(k,k') is independent of e.
(5.9)
Thus the evaluation of T(k) (= T(lkl) = T(&)) red~ce~ to the
,• evaluation of M(k,k'). The expressionsfor this will be given for
various scattering processes in the following sections.
5.2.2 Ionized !~purity Scattering <52 ,53 )
When a semi-conductor is doped, the doping atoms become ionized
either by giving up an electron or taking an electron.. These charged
particles produce coulomb fields and a perturbing potential •
. V = ze/4')(. exp( -q,r) £o£r r .
where · ze = effective charge of the impurity atom
£r = dielectric constant
r· • distance from the centre of the atom , y~ = screening length
'l 2.. = L Nd (2~) (mks units) V E.,~.kT_ Nd
Nd = ionized donors and acceptors al~s~
This leads from (5.7)
S(k,k') 2.7\ =-.r;
and to imp~rity relaxation time 'I
. 2b = Sm*E-
q_~2
or
c1 and c2 being constants, independent of energy
5.2.3 Lattice Scattering(4l,S2)
Due. to thermal .energy, the atoms in a solid vibrate about
86
(5.10)
their mean position and produce a l~cal variation of potential. This
periodic potential causes the electrons (and holes) to make transitions
from one state to another state. In the process the electrons either
gain or lose energy and momentum to the crystal lattice. If the
electrons lose energy, a phonon of equival~nt energy is emitted and if
it gains energy• a phonon is absorbed. Thus if k and k' are initial and
final wave vectors of an electron and Q is the wave vector of the
phonon involved, then the following relations hold. ~ ~ ~ ~
87
K - K' = ±Q + G ( 5. }1)
where + on the R.H.S. o·+ both equa~ions is for phonon emission and.
-ve for phonon. absorption. A1w is the energy of the phonon and G is ~
any reciprocal lattice vector. If G = 0, the scattering process is
normal and if no~-zero, then the process is an Umklapp one.
·From equation 5.11, it ~ppears tha·t the energy of the
carriers is no longer conserved in the scattering process. However,
if the fields are small, (which is ~he case treated here), it may be
shown that
.flw « e:
and thus e:(k') = e:(k)
The phonon involved in the scattering process may be of
different types. For example, it may be of low energy and low
momentum - (<1t.·oustical scattering) or it may be of low momentum and
high energy (optical scattering). Further, if the semi-conductor has
different equivalent conduction band minima (as germanium has), the
electrons may be scattering from one valley to the other (intervalley
scattering). Again the. phonons involved may be acoustical or.optical,
but momentum and energy now depend on the value of Q involved which in
turn depends on the position of the energy minima. ·These processes will '
now be considered in a little more detail.
5.2.4 Intra-valley Acoustical Scattering I
This has been best treated by Bardeen and Shockley using the
concept of deformation potential. The elastic displacements ~f-r) of
the atoms would produce changes in the position of.energy band minima
and maxima. If E. = energy of minimum or maximum with lJ.W) and m . • .
£ . is the equilibrium value, then it is shown that (S2) mo
o£ = 1:, - £ = E1 l11J (Y) m mo m
88
(E1 is being a constant)may be treated effectively as the perturbing
potential of the charge carriers near the minimum or maximum. By
expanding v.;(y) in a Fourier Series, it is shown then that the matrix •
element M(k,k') is given by(SJ)
2 ~(2N +1) IM(k,k')l = Q
. 2 f E,~ = !€:'(1\/!:. ; ll = llV
.. v J = density of material·
V = .volume of the crystal
. ~ ~. ]-1 NQ = exp -1 kT .
if hwQ<<kT, then (2N +1)~2kT Q hwQ
Then assuming isotropic scattering, equation 5.9 gives
tL = lattice scatt~ring time constant -!.: = constant X£
2/T
T = temperatu're in °k
(5.12)
. (5.13)
The constant is independent of temperature and energy of the carrier . '
~ssuming constant effective masses).
5.2.5 Intervalley Scattering . It is now known that germanium has a conduction band minimum in
(111) direction of k-space. Thus there are eight equivalent minima.
These different minima are called valleys and an electron in ith valley .....
89
with wave vector ki may make a transition to another valley j with final .....
wa.ve vector kj. In the process the electron either absorbs· or emits a
phonon. The probability'of this process is given bY the matrix element
M(k,k') which is as· follows(25)
~~(k,k')i2= NQ X C,k,k') ·6 . NQ +1. WQ ~ '
t · ~ absorbed· (k) - &(k') ±. hwo
emitted
where C(k,k')~independent of Q and & or T~NQ is given by equ~tion 5.12
and .±. hwQ is the energy of the phonon absorbed (+sign) or the phonon
emitted (-ve sign). Then the relaxation time due to such scattering
is obtained from 5.9 as
-1 \ ·[(& + hw;)l/2 (Tiv> = t. c. +
i 1 1+ exp hw; kT
(&- hwi)l/2
1 - exp - hwi Kr
or zero J . '
where i varies over different phonons possible
hw; = energy of ith phonon
(5~14)
Ci = "coupling constar;ts'' of ith phonon expressing the scattering
strength
The second tenn on the R.H.·s. is zero if c < hw1 because in that case
no phonon can be ernitt.ed. ..
90
5.2.6 Intravalley Optical Mode Scattering.
The electrons or holes may be sca·ttered by optical phonons
(electrons remaining in the same valley after scattering). For
germanium, which is a nonpolar crystal, the matrix 'element is essentially
the same as for intervalley scattering as k~k' (25) and hence ~opt' is of
the same form as 5.14.
5.2.7 Energy Bands and Lattice Vibrations in Germanium
Before one can proceed to compute the mobility, a knowledge of
the conduction and valence bands and 'the lattice vibrational spectrum
is necessary for two reasons.
(i) The form of the energy bands gives information
about the effective masses and enables the computations
of carrier concentrations and the evaluation of the
relaxation time integrals.
(ii) The location of the energy bands and the lattice
vibrational spectrum give the information on the
phonons involved in the scattering and their energies.·
It is known t.hat gennanium has two valence bands degenerate . ~ ' +
at k=O. The form of these energy bands may be approximated near k • 0
as f o 11 ows (53) •
{ - - 1/2} (; • - ~2 A k2 .:!:. [s2k4 + c2 (kx2 ky2 + ky2 kz2 + kz2 kx2) J
2m0
. _
. A•13.1, B•8.3andC•l2.5 (5.1.5")
m0 • free e 1 ectron mass ·
•
The holes with+ sign in 5.15 are referred to as light~ holes and
with - sign as heavy mass holes. These masses are given respectively
by the following equations
__ ....;l ____ = .044
m;h/m0
= __ ___...;1~--- .. 0.30
A_ {B2+C2/G)l/2
The conduction band in germanium has 8 equivalent minima· in
the direction (111) located at the zone boundary. Near the minimum the
. £-k relationship has the.form <54)
{. 2 2} ·= ~2 k12 . k2 ~3
£ - -+-+-2 m~ mt . mt
m1 = 1.64m0
..
mt = 0.819m0
91
The phonons involved in the intervalley scattering are given in
the table 5.1. It may be observed that all the non-zero phonons involved
are in (100) plane and have maximum values in the (100) direction~
· The energies of the phonons are given by the lattice vibrational
spectrum. This has been determined by Brockhouse (55) from neutron
diffraction.experiments. In the lattice spectrum, there are six·
branches. three. ;~7·oustical and three· optical. Energies of. the various
phonons relative to the present work.are given in the table 5.2 • .
TABLE 5.1
VARIOUS. PHONONS INVOLVED IN THE INTERVALLEY SCATTERING
IN N-TYPE GERMANIUM
(lli)
{lTl)
. ('ill)
(lH}
(ili)
(ill}
(iii)
+ + + .. Q = K. - K·- G .
1 J
+ 2-K. = ~
1 a
+. + K. - K • . 1 J
27'-/a x
(002)
(020)
(020)
(022) .
(202)
(220)
(222)
(1,1,1)
+ .Q
(002).
(020)
(200)
(200)
(020)
(002)
(000)
92
TABLE 5.2
DIFFERENT PHONON$ AND THEIR ENERGIES FOR GERMANIUM AT ZONE EOG&
Phonon
Transverse Optical
Longi tudi na 1 .
TO[lOO]
L[lOO]
Transverse Ac·oustic TA[lOO],
Optical ',· [Q=O]
* Table II of reference (55)
Frequency • (~/h) x 1o-12Hz*
8.25 :!:. 0.3
6.9 :!:. 0.4
2.45:!:. 0.15
9.0 :!:.. 0.30
,
I I I
Although the intervalley scattering could involve phonons
corresponding to all the branches of the lattice vibrational spectrum,
it is shown by Herrig and Vogt(54) that the contribution due to two
lowest branches (Transverse acoustic) is very small and can be
neglected. This leaves five phonons (1 longitudinal 'I acoustical, 3
optical at Q ~ [001] and 1 at Q = 0. that is. the intravalley optical)
which can take part in the scattering of electrons.,
For holes there is no· intervalley scatter1ng and the only
phonons involved, are optical and longitudinal acoustical ones. -
However, for holes, interband transition can'take place but these
transitions have not been observed in Haynes-Shockley experiments and
are assumed to be very rapid.
5.2.8 Variation of t1
The latti.~e contribution £1/.t0 to the dielectric constant.
has been observed to vary with temperature by Cardona et al (56_l and
have given the following formula.
l dn - 6 7 1 o·Sj9 n dT - • X c
where
n • /£~
94
the total relaxation time constant T as follows,
_1_:.:...1 __ + , + _....;1;...._
T · (t) ph
where t h = relaxation time due to phonon scattering other than low . p
energy acoustical scattering.
[
1/2 . 1/2 ].} x (dhwi·+l) +. (e:/hwi -1) or zero hw· ·
l+exp --l l-exp -hw; kT ~
(5.15q,)
The first term on the right hand side of the above equation
represents tne contribution due to ionized impurity scattering and is
given by 5.10. The first term in the curly bracket represents intra
valley ~coustical scattering relaxation time obtained from 5.13. The
terms in the summation are contributions due to intravalley optical
and intervalley scattering .by phonon of energy ~Wi• The constaQtS ci
measure the strengths of coupling relative to acoustic phonons. T0
is the reference temperature which will be taken as 300°K. The constant
C fixes the absolute value of mobility at 300°k.
The values of ci•s are not know~but these can be left in as
adjustable parameters. These can be varied ~o that the relaxation time
obeys the same variation with temperature as has been. observed in the
experiments, ie. T-1•66 for electrons and. T-2•33 for holes. These
two cases will be treated separately in the following sections.
5.3. 1 Electrons Mobility in Germanium
It has been mentioned in art. 5.2.7 that there are five phonons
96
which can take part in the scattering and hence five adjustable constants
ci. It is qaite laborious to handle this situation. However, if it '
is observed that the three phonons (2 optical at Q • 100 and 1 optical
at Q = 0) have nearly equal energies· .and- the other two have same energy,
then these phonons could be grouped together in two groups without
making.any serious error. This leaves two constants c1 and c2 to be
adjusted which is ~omparatively an easy task. The values of c1 and.c2 were
varied and T was computed as a function of temperature (between 1oo• to 300°k)~ Each time c was adjusted to give
llJOO = • 3800 = Q<T>/m~ . The resu 1 ts of co~puta'ti on are p 1 otted in fig. 5.1 • The abscissa in
the figure represents c1 and the ordinate -n, the exponent ofT. c2 was taken as a parameter.
It may be obse~ved that there are different sets of c1 and c2 which can give the correct value of n = 1.66. No unique combination is
possible using only the d.c. mobility data.
Because of this it is convenient to neglect all the scattering
by all the intervalley phonons and assume that the scatteri.ng is only
by optical mode scattering of frequency 9 x 1012 Hz. (or that all . phonons.are grouped together having one energy). If this is done and c1
- ...
t - I
~
1.9~- c, = 0· 3
1·8 ~- I I 7.,c I 7/ I 7'-c . I ,.. . 0 I I I C 1f"' i'=~·~l ~z.
• :z.. .fov Y: 8· t:l::!> }-lz.
I 1·71 / I 7/ I 7/ I 7/ I I I I . .(_
J.41 I ! I I I I I l I I I I I _1 J , 0 •I ·2 ·3· ·4 ·5 ·
, . c2 Figure· 5.1 Variation of the Index n in T-n for n type Germanium as a function of· c1 and c2
~ -l
98
is varied over values from zero to 1, it may be seen from ~he figures
5.2a and b that the value of c1 = 0.2735jto give n = -1.66. This means
that 27.4% of the lattice scattering is due to optical and or inter
valley scattering.
The linearity of the log P ~ log T relationship in the ranges
of c1 and c2 considered, was checked and is shown i.n the fig. 5.3.
With these values of c1 and w, one is able to compute <t> and
* <t/l+jwt> and hence a(w). In the actual computations, the use of me was
avoided by using
• QPno(T) n < tn > l<tn> l+jwt n
where Pn0
(T) = d.c. mobility of electrons
= 0.3800 (T/300)·1•66 m2/volt sec.
n • electron concentration 1 m3
(5.16)
For temperatures greater than ~ut l00°k, almost all donors ~ .
are ionized·. Then for T>100°k, the carrier concentrations were obtained
as follows.
The charge neutrality condition gives
n - p = Nd
also np • n12
whence n • ~ [1+. }+4n12tNi ] (5.17)
·'
I I I ' I I I ' . I ! I I ··! ' .
1 l I
~~l----1----~---~----~---~----r---1
l j
I I 1" '~I I
1 I : 1.6J I
~ I i~ I' l:
./ v
/
v L~--~--~--~~/~--~--~---1
t6o!l I . //
. 1.5 64-;· ---1--~ .. '/-.-/-. -+----t----r------t---,
1.s.:l·----4---~/~-t----t----r----r---1---/
1.46• / .
'i;/.· I f
ill
nr I
t ·---~--~----+----r--~----~--1 \.44 ;
1.4 °~·---LJ---: .. 0~' 5:----LI--."":'1~0--_i.J...-.~~ ~s--_LJ....-. z~o~-"-'-. z~s----'-:."t3oo--__l--:-. ~3 s
Figure 5.2a Variation of the index n in r-n as a function of c1
c,~
q4
loO
Figure 5.2b n "-Cl on enlarged scale
'McMASTER UNIVERSITY LIBRARY
.I
~ ... II)
I
to-~
~
" 0 .... ~
~ ..._....,
<..9 o-·1 _j
-·3·~------+-------~-------+------~~------+-------~
. --4~------~~-------+---------+--------~----~~~--------~
Figure 5.3 in Germanium with Temperature
102
Nd [ I 2 2 ] .• p = ~ . -1 + 1+·4ni /Nd (5.18)
For germanium ni2 = 3.1 x lo44r:xp(-9100/T)/m~. · The val u.e of Nd can be computed from the known conductivity and mobilities
of. carriers at a particular temperature. Thus if a sample has 00 d.c.
conductivity at 300°k, then
ao = q (n ~n-+ p ~p)
~n = .38 m2/volt-sec.
~P = .18m2/volt-sec.
Substituting n and p from 5.17 and 5.18 into the above equation yields
a quadratic equation in Nd which is easily solved. Once Nd is known,
values of n and p as a function of temperature could be evaluated. In
the limiting'cases, 5.17 and 5.18 reduce to
( i) when ni <<· Nd
.. n = ~; P • ni2/Nd
(ii) when n; >> Nd
n = ni = p
The condition (i) ~s valid at low temperatures and (ii) at high temperatures.
5.3.2 Hole Mobility in Germanium
·In the case of holes, there is no intervalley scattering so that
there is only one phonon (i.e. optical one) which can take part in the
scattering and hence only one adjustable parameter. The computati~n~,
were carried out in the same manner as for electrons. The values ~~ 'YI ~ ••
exponent t.. ~- plotted in the fig. 5.4 for various values of c1• It 1s
found that for. n • - 2·.33 c1 • 1.89. This means that relatively large
I I
I . 2.4
1,9
~ v v .. I /
v ~
/ I v ... .. -""' .... .
/ . .
. . ..
" ~
I
v . ~-: .. :
,/ .
V·
2.3
2.2
\ 2. s::.
' 2.0
I f - . - . " - . ~ '~s· . 7 l.O 1.2 1.4 1.6 c,
Figure.5.4 Variation of n in r-n for p type Germanium as a Function of c1
- .. ~l ~ l
~ l _/"
--
. -
-·
-t.S . -2..0
I
.
-
-. ---------·----·-.
.•
- --
. ---
.. - t ! • ') ) c..·-
0 v-
amount of scattering is due to optical mode scattering. This has
.also b'een the conclusion of Conwe11(26). With values of c1•1.89
12 . and .~/2."".= 9 x 10 Hz, one can again compu~e
ap(w) = q )A (T) ·p < Tp_ )' /cT > (5.20) po l'j(l)t P
p -2.33
~po = .18 (T/200)
p :: hale concentration /m 3 ,
p can be calculated from the equation 5.18
5.3.3 Microwave Conductivity and Permittivity of Germanium
The computations of a((l))/a(o) and trm for n type germanium
were carried out with the help of.a 7040 IBM computer using equations
5.16 and 5.20 as a function of frequency, doping and temperature.
104'
The results of the computations are plotted in figures 5.5 through 5.7.
The fig. 5.5.gives a((l))/a(o) as a function of temperature and
f. and doping as p~rame_ter. The temperature _was varied between 100°k to
500°k and the frequency from 109 Hz to 1012 Hz. The computations were
made with two values of donor densities Nd so as to· give the room
temperature conductivities of 10 and 100 mhos/m
It may be seen that for all values of T and Nd c_onsidered, the
microwave conductivity is essentially equal to d.c. conductivity for
frequencies less than about 10 GHz •. The effect of increasin~ frequency
becomes evident above about 10 GHz and the.effect increases at lower ' temperatures because of the increase. in t and hence (l)t. As the
frequency goes higher. a((l))/a(o) decreases even at room temperature.
----~--~----T---rjr~--~~--~--~----r---,---~----~--~~~ I ~
t l I f
I
-I •
F
~ -< f? 0
0 0 -
If II
/" I"" ~ :L.
0 00 0 0 0 !(') tl)
...... \,;
b '<>
(..1 .
\ \
\ \ \
', \
\
\
' \
'
'
'
\ \ \ ' .
\ \ \
\ \
' ' ' ' ', '
' ' ' ~l ~·)\
'
.......... .....
' ,,
' ' ' ' ' ' ' ' ' ' ' ',
•. '+-
"0 c ro
.,._ '+-0
c 0 .... ....., u c ~
I.L.
ro VI ro ~ ....., .... > .... ....., u ~
"0 c 0 u
r .
u . "0
0 .....,
a .... > . ... ....., u ~
"0 c 0 u ~ u c Q) ~ o-Q) s..
I.L.
.s::: Cl ....
:I:
'+-0
0 .... ·~
ro ex
U') . U')
Q)' s.. ~
"' .... u..
106
The effect of the doping is seen to be more significant at lower temperatures.
The effect of frequency on the dielectric constant of germanium
is shown in ·fig. 5.6, in which the abscissa represents the log f, the
ordinate the erm' with T as ~ parameter. The doping was kept low so
that the doping effect on erm is negligible. It may be observed that
again· the effect of the frequency becomes significant for f ~1o10 Hz for
T ~ 200°k. For high temperatures, the dielectric constant begins to
change at higher frequencies than 1010 Hz. This is to be expected
because both Tn and TP decrease with temperature. It may be further
observed that at higher frequencies, the values of £~ for all values
ofT~ 200°k approach a limiting value.
This condition applies when ~T >> 1 so that
T 2 1
The changes in £r with temperature are due to two,reasons
(i) The value of <T> and hence <T/l+~~T 2) changes with
temperature
(ii) At high temperature, the carrier concentration also changes.
The effect of doping ~n £r is shown in the fig. 5.7. The
frequency in these computations was taken to be 9.25 GHz so as to minimize
the frequency effect. The effect·of doping is seen to be more pro
nounced at lower temperatures.
II
10
0
T- 300 K
IS 0
41)0 . - --. -- -.- - - --- ---- .... --12 5
5 oo . . ------ ---- --·- --:- -------roo
"' .,-"' _ _,
I I
I
I I
I
I I
I
Figure 5.6 Variation of Dielectric Constant of n type Germanium as a Function of f and T
I I
f=9.25GH:z
I '-
0
T -= 300 K j f;::;.~=====~=~~----1 200
4JO _ --------- ---....... ...... .........
Figure 5.7 Var1ation of £ of n type Germanium with Doping. . r"""
', ' ' ' ',
\ \
\ \
5
\
5.4 THE MEASUREMENT OF o(w) and erm
The measurement of the microwave conductivity and dielectric
constant was carried out on samples -of n type Ge of resistivities of
22n-cm and 10 ncm at 9.25 and 34.5 GHz. The wave-guide configuration
used for these measurements is shown in fig. 1.1. This configuration
has the advantage that the material properties can be expressed
explicitly in terms of propagation constants and no transcendental
equation has to be solved after the propagation constant has been
computed.
If y is the propagation constant in the section of the wave
guide filled with semi-conductor in fig. l.land r is the propagation
constant in the empty guide, then ~r is given by
__ y2-r2 ~ - 1 -~ k 2
0
2 2 . = x -r 2
w lloto •. 2 2 2
108
so that er.· 1 = S -80 -a (5.21a) 2
w lloto
a = (5.2lb)
a+ja = y
jB0
= r
·For a given system y was computed from the measured value of the
reflection coefficient at the interface of an empty quide and the semi- . .. conductor loaded guide at z • o with a short at z • , ,(fig. 5.8).
y
I
Ideal Arrangement
y
I·
Actual arrangement (SHoWIJ'.IG AI~ GA-P)
Figure 5.8 Completely Filled Wave-Guide Configuration used in the Experiments on n type Germanium
110
The propagation constant y in terms of reflection coefficient R at z=o
is given by the solution of the follm·ling equation
tanhy.t -- = z I js .t = - 1-
n o ·a t J 0 yR.
li-R
1- R
From this equation y can be computed numerically or graphically
using the graphs of Von Hippel (El). In the present work, ho\'rever,
numerical solution \·las preferred to obtain a higher accuracy.
The reflection coefficient R \'/as measured by means of the . .
reflection bridge described in the appendix A. For optimum accuracy,
the sample length R. was chosen to be a quarter wave length approximately.
The measurement of the reflection coefficient at different
temperatures \•Jas carried out by placing the. section of the \'lave-guide
containing the semi-conductor saMple in a "Delta Design· Chamber" (fig. 4.10) '
in which the temperature could be caried from -300°F to +600°F by the . .
application of liquid nitrogen or electric heaters. The measurements
were carried out bet\-1een the tempe·ratures of about lOQOkto 500°k
and the temperature was measured with a copper constantan thenno-cou.pl e ~ ~ .
and an H.P. meter, type 425A, giving an accuracy of about +2°K~
5.5 EXPERIMENTAL RESULTS ~NO DISCUSSION
The results of measurement of mi.crO\'Jave. conductivity and
dielectric constant at 9.25 and 34.5 GHz are presented in the figures
5.9 through 5.12. The solid lines represent the theoretical curves of
a and £nn as computed from equations given 'in previous sections!' The
crosses or the circle.~ represent the experimental points. The probable
* ~he behaviour of a \'tith temperature_ as shown in figures 5.9 and 5.10
may f>e explained as followed. As the temperature is decreased from the/co NTD
'! -~ ~ i
! )
l ...,L .::. j
L I
',•
f=9.2 5 G HZ
2.6
Figure 5.9 The Theoretical (Solid Curves) and Practical (Circles .and Crosses)
Values .of o(w) as a Function of Temperature
?..7
,, L- I
.f =- 3 4.5 G H Z
I()~ .. "'
Figure 5.10 The Theoretical (Solid Curves) and Practical {Circles and Crosses)
Values of a(w) as a F~nctton'of Temperature
J J
1 .....
\ll
·- - -- --·- - ~~--- ~ .. ~,.--· .
·------ ------------------------------ -·---------------·· - --- ------ ---· - -------------- - ----
- i~ 9.2 5 G HZ
·)(
12
I 6
~ ·----t--- -.;~ -
6' (5o o0~ ~ ;;r=--- ~ -r ---------
1- e:' 4". 6 s ~ _.,..,..,...... ~ . .
~,----··-- /
fl .
0""(30c'l0 J<)
- v = 9 -v-;..,"'1'\ . .r-
f/ ------a
1-.,_; ;:
.. .
- ' . f.";
4 --·-····
... . "
. 1-.
, .
'~ .
. ' I I . -%'5 160 .120 --200
I 240
T°K
I ..... - -280
I I :: I t -- - -_.,, __ ..J.
360 320 400 4-h) 45(
Figure 5.11 The Theoretical (Solid Curves) and Practical (Circles and Crosses) Values of Eras a Function of Temperature
·-· ....... / \J'J
-- -~ . .-............._.._ ,. ..... ,..
------- __________ l_ _____________ ----- ------- -----
t- +: 3 4.5 6HZ
!.,-
~ (' -~
~ 6":?-Do ~---f.65 .
~~ _/ - <>(3oo°K)=9 -v-; ........ LI· v
1 6
r2 ~ 1- 1
6
1/ . ·---.; f
:'-
4-
. 1-
' . • ' () 100 140 ISO .220
-
---y
~
~ 0
~ .
.
.
'
' f".l
~
--· ..
' ___l ' I I
26<!> 300 340 380 420
TCIK Figure 5.12 The Theoretical (Solid Curves) and Practical (Circles and Crosses} Values of
er as a Function of Temperature
,/
----.
~ -
-- ------
'
--
_t __ t __
460 5J~
4:"
115
errors were computed from the estimated errors in the reflection coefficient
and the uncertainty in the b djmeris ion of the samp 1 e.
To interpret the data, the gap effect was taken into account.
This was necessary because the sample dimensions were not exactly
equal to the waveguide dimensions. The gap bebteen the sa1nple and the
broad \'Ja 1l s has been found to have a considerab-1 e effect on the measure-.
ment of e: . r (16, 17} In the present \-Jork; the following two equations.
were considered,
£r~~) =. (€r-1) b'/b (5.22)
€ (m) = € /[ .. 1+(€ -1) b-b'] r r r · b ' (5.23)
where £r is the actual permittivity, €r(m) the measured value of . permittivity and b' is the actual narrow dimension of the sample •
.. ~Jhile applying the correction for the gap effect,it was found
that for low values of conductivities, equation (5.22) gives adequate
results. Tbe equation (5.23) hm·tever, was found to be valid only at
higher conductivities, say greater than about 25 mhos/m. At lower
values of conductivities, (5.23) \'tas found to over~orrect the measured . .
values. For example, at room temperat~re, and for bbb' = l/160 and
measured values of e:r = 15.7 and f = 10.4 ohms-em, the equat·ion (5~22)
, and (5.23) gave the values of e:r = 15~8, f= 10.3 and e:r = 14.4 . .
~= 8.6 Qhm-cm respectively. The former values are closer to' the
theoretiGal values.
room temperature, the mobility of tne carriers increases while the number of carriers in a doped.material remains practically constant - about 100°K. Thus the conductivity increases with the decrease in temperature. The rise ·
' . . in the conductivity above room temperature is due to the rapid increase in the numbers of intrinsic carriers.
116
It may be seen from figures 5.9 through 5.12 that the aqreement
between the measured values and theoretical values of a is quite good.
There seems to be some discrepancy between the measured values' of a and
the theoretical curve at 9.25 GHz at higher end of the temperatures.
This is possibly due to the gap effect. The equation 5.23 which is
obtained from the first order perturbation theory {l.6) probably fails
at high values of w~ that occur at higher-temperatures. 0
There is seen to be a fair agreement of £r between theory and
experiment. The discrepancy at loNer end of temperature is. again
attributed to the gap effect. At higher end of. temperature, it is
found that it is not possible to get accurate values of £r even
after applying the correction due to the gap effect. The reason for
this is that as T and hence a ?oes high, the contribution of £r to
the propagation constant becomes negligible; &r+ -j _Q_ and a+B. W£0
When this happens, it may be seen from the equation (5.2la) that even
a small measuring error of a or B would cause a large error in the
measurement of £r· A very high accuracy, indeed, wouid be required
to measure £r in this situation. .
..
CHAPTER VI
CONCLUSIONS .
6.1 GENERAL
A variety of microwave measuring techniques are required to measure
the complex permittivity(£= £o£r- j:) of semi-conductors because o~.
large possible variation in the conductivity. The accuracy of measurement
depends on the choice of a particular technique for a given .conductivity
vaiue. A number of methods of measurement have been investigated to
determine the range of conductivities to which they are best suited.
Particular attention has been given to systems which involve the
measurement of the reflection coefficient.A+·ully filled wave-guide
configuration has been used to investigate the dependence of the complex
permittivity of n type germanium on temperature, frequency and doping.
An investigation of the exact and the approximate expressions used for
computations of the dominant mode propagation constant in a partially filled . wave-guide configuration has been carried out together with the effects of .
the higher order modes which a~e excited at,the junction of such a guide and
an empty one. During the investigations a new method of measurement
involving the replacement of one narrow wall of a rectangular wave-guide
was peveloped. This· method has been· termed the "lossy wall measuring
technique".
117
118
6.2 LOSSY WALL MEASURING TECHNIQUE
The theory of the wave propagation in such a system and the exact
and the approximate expressions for the propa~ation constants of Hno
modes are developed in the Chapter III. Numerical computations of • the·dominant mode propagation constant are presented for 1 ~ a~
1000 mhos/m and 0 ~ £r ~ 16. The calculations were done for three
frequencies 9.25, 34.5 and 70.5 GHz. Experiments to confirm the theory
were performed at 9.25 GHz with germanium samples of conductivity in
the range 2 .; a .; 400 mhos/m and the measurements were made with a·
microwave transmission bridge.
The computations show that the approximate expressions for the
propagation constant give adequate results at th~ higher end of the
range of £r considered and over the whole range of conductivity
considered. The computations further show that this method of measure
ment is most useful when a~ w£o£r and becomes . more accurate with the
increasing frequency. " .
The experimental results a~ 9.25 GHz give accurate results fo~
both £rand a in the conductivity range 4 ~a ~-20,mhos/m. However,
for lo_wer values of a, the conductivity measurement is not accurat~
but the dielectric constant term is in good agreement with the expected
value. This disagreement is attributed to errors in the measurement of
the small phase change produced in such a system and the inadequacy of
the theor~tical model which assumes only the TE10 mode propagating in
the system. Also, the experimental r~sults show that for a ~ 20
mhos/m the measurement of & is not accurate. This is attributed to r
119
the fact that in this range of a, small errors in the measurement of the
transmission coefficient .can cause large errors in the calculated values
of e: . ·r
6.~ PARTIALLY FILLED WAVE-GUIDE .
6.3.1 Exact and Approximate Solution of the Dominant Mode ·Propagation
Constant
A wave-guide configuration commonly used in the measurement of
various electrical properties of semi-conductors is the one shown in
the fig. 1.3. In·the measurements.previously reported in literature,
the propagation constant has been computed from approximate expressions,
the accuracy ·o_f which has been doubtful. Computations of the dOfllinant
mode propagation constant for 0.1 $a~ 10 mhos/m, .001 ~ t/a ~ 0.25
and e:r = 16, 12 have been carried out at three different frequencies
10, 34.5 and 70.5 .. GHz using the exact and approximate expressions. The
computations show that for'all values of a and t/a considered at 34.5 . at lO Gliz.
and 70.5 GHz and t/a < .05 and a ~ 10 mhos/rnA the attenuation constant , varies linearly with a so·that a= constqnt x a· The variations of y
with e:r are found to be significant only.at higher end of the ranges of
t/a and a.
The comparison of the exact and the approximate values of the
propagation constant shows that unless t/a is very small (<.0025), the
approximate expressions do not give adequate : · i· · re$ults for the
ranges of t/a, a and e:r considered. ...
These calculations, which have not been previously reported, show
120
that approximate methods of solution for the propagation constant of a .
wave-guide partially fil1ed with a semi-conductor should be used with
due caution.
6.3.2 Higher Order Modes Effect
In a practical experimental arrangement, this partially filled
structure fo~s a junction with an empty guide. At such a junction
higher order modes are excited •. The effect of Hno higher order modes has
been studied and the expression for'the input impedance at the junction
(z=o) is obtained for such a structure terminated with a short circuit.
calculationsof the reflection coefficient at the junction z = o are
obtained for 1 ~ a ~ 10 mhos/m, 0.01 ~ t/a ~ 0.25 and £r = 16, at 9.25 GHz.
The mode number n was varied from 1 to 7 (1, 3, 5, 7). The calculations ~ ~ show that unless t/a <DB and a < 2.5 mhos/m, the effect of the higher .
order modes becomes significant.
Experiments were also performed to measure the reflection
coefficient at such a junction at 9.25 GHz for t/a values of .0133~ .026
and .0515 and 2 ~ a ~ 100 mhos/m. The measurement results were found
to confirm the above observations about the higher order modes. The
best fit between theory and experiment is obtained with N=7 (the order
the higher mode) for higher values. of a and t/a.
This wave-guide system with semi-conductor has possible application
to microwave modulating devices and this study was undertaken to determine
characteristics of such a structure which have not been previously reported.
6.4 EFFECT OF FREQUENCY, DOPING AND TEMPERATURE ON THE COMPLEX
PERMITTIVITY OF N-TYPE GERMANIUM
121
Theoretical computations have been made of microwave dielectric
constant and conductivity of n type germanium as a function of temperature
(100° ~ T ~ 500°~), frequency (109 ~ f ~1012 Hz) and doping (1019 ~.Nd ~ lo22;m3). The theoretical model used to evaluate the average relaxation
time constant includes the scattering by ionized impurities, acoustical
and optical mode scattering of charge carriers. . . These calculations show that the effect of the frequency for,
f ~ 1010 Hz on both a and Er is negligible. At. higher frequencies
·the conductivity decreases and the dielectric constant increases from its
low frequency vaJue •. The effect is more pronounced at lower temperatures I .
due to the increase in <T> and hence <w2
T2>., The effect of the doping is
also found·to be significant at high impurity contentratio~ and lo~
temperature.
The effect of temperature is such that at a given frequency, the
dielectric constant decreases and conductivity increases with a decrease
in temperature •. This is due to the increase in the value of <T>.· Above
room temperature and with light doping, the thermally excited carriers
have a considerable effect on the dielectric co~stant.
The associated experiments for measuring.the dielectric constant
and the conductivity of n type germanium samples of a ~ 9 and 4.7 mhos/~
at 9.25 and 34.5 GHz were carried out with the microwave reflection bridge.
The completely filled wave-guide configuration was used and the temperature
range was 100° ~ T ~ 500°K. Such measurements for n type germanium
122
have not been reported previously.
It was found necessary to apply a correction factor for the
effect of the unavoidable air gap between the sample and the wave-guide
walls. The investigations showed that the first order perturbation
theory was valid for a/w£0 ~ 4. With such corrections good agreement
between theory and experiment was found between 100° ~ T ~ 400°Kat
9.25 GHz and 100° ~ T ~ 450°~at 34.5 GHz and the vaHdi:ty of the
theoretical model was confirmed.
.•
....
APPENDIX A
MICROWAVE REFLECTION BRIDGE
Al BRIDGE DESCRIPTION
The schematic diagram of the reflection bridge used for the . experiments described in the chapters 4 and 5 is given in the fig. Al.
The microwave power. source feeds the H-arm of a magic tee. Arm I of
the tee has coupled to it a reflection coefficient reference which
consists of a precision type variable calibrated ·attenuator followed·
by a precision type variable short circuit. The second arm is coupled
to the network under test and a matcned detector is connected to
the E-arm.
The measurement of the reflection coefficient under test is
carried out in two steps, as follows.
(a) First a fixed short is coupled to the arm II of the tee and the
.attenuator and the short in arm·I are varied to obtain
minimum power output in the detector in the E~arm. Let Ao(nep)
and lo(meters) be the readings of the attenuator and the short
respectively~- for this condition.
(b) Now the test assembly with a reflection coefficient R is placed
in arm II, at the same plane as the short in the st·ep· I.
Again the attenuator. and short in the ann I are varied to
obtain minimum output in the E-ann. Let these readings be ..
123
1 - Power System
2 - Tuned Tee
3 - Test Arm
4 - Reference Arm
----I --
Reference Planes
Tuners ..
./
" ';: :; -tr'
4
figure Al
.
I
I I
I I I
I I 2 . :;
I I I I I
I
' .
~ .... , E-Arm
Armli_L Arm I
.. - -H-Arm
Details of 2
Schematic Di_agra.m of the Microwave Reflection Bridge Used in the Experiments.
I-' -C
125
A1 neps and r.1 meters resp.ectively. Then·, as shown later, Ris given by
R = - exp - 2(A+j$) (Al)
where A = A1-A0 nepers
$ = s0 (t1-l0
) radians
so = phase constant in the variable short
A2 SCATTERING MATRIX OF TEE
Equation A1 may be proved by considering the scattering matrix
of a hybrid tee. This is· as follows for the tee shown in th~ fig. A2.
sn 512 513 514
S= 521 522 523 524
(A2) 531 532 533 534 5 ,41 542 543 544
If the system is lossless and isotropic, then the reciprocity theorem
gives
5· · = 5· · ilj lJ J 1 . Now if it is assumed that the tee has been tuned·-:. so that
533 = 0 = 544
534 = 0 = 543
then the scattering matrix reduces to
511 512 513 ~14 512 522 523 524
S• 513 523 o· 0 (A3)
514 524 0 0
'·
SIOE·ARM
E-ARM. ( 3")'
ill . nr · · (
Figure A2 Three Dimensional View of a Hybrid Tee
S JOE ARM 4t-
(~)
' .
127
The scattering matrix is a unitary· one{3l)and the following theorem
applies. . "The sum of the squares of the absolute magnit~des of elements
in any row ·(or column) is unity". This theorem gives from equation (A3)
l511 12
+ l512l2
+ IS13l2
+ 151412
a 1 2 2 2 2
IS121 + IS221 + IS231 + 15241 • 1 2 2
IS131 + IS231 = 1 2 2
IS141 + 15241 = 1
Adding (A4a) and (A4b) and using (A4c) a_nd (A4d) gives
1Sl11 2 + 1522 12 + 215121 2 • 0 .,
.: 511 = 522 = 512 = 0
Thus (A3) reduces to
-0 0 513 514
S= 0 0 523 524 .. 513 5
23 ° 0
514 s24 o 0
(A4a)
. (A4b)
(A4c)
(A4d)
(~5)
Now if ai is the incident waves at the port i and b1 is the reflected
wave at that port, . then
bl = .a3 S13 + a4 S14
b2 = a3 S23 + a4 S24
b3 = a1 S13 ~ a2 523
b4 • a1 514 + -a2 524 :..
128
Assuming that the detector, which is coupled to arm 3 is matched
so that a~ = 0, the above equations ~ive
bl 514 . -=
If~ 1 and R2 are the reflection·. coefficients at the arms· I and
II respectively so that
·a1· = R1 b
1 and a2 = R 2 b2 ,
then b3 c R 1 513 b1 + ~ 2 523 b2
Now if
II Rl sl3 514 b2 + ~2 S.23 b2 524
• b R' sl3 514 + s R 2 1
524 23 · 2
(A6)
(a) R2 = -1 (short circuit plate), and R1 •. ~lO so that b3
• 0, ..
(balanced condition), then
513 514 • +'s 10 23
524
and (b) R 2 = R (under test) and R1
• R11 so that again
b3 = 0, then
The above two equations give .. R II !u
R1o r-
(A7)
129
A3 REFLECTION COEFFICIENT OF THE REFERENCE ARM
The reference arm consists of a precision variable oJX&-~ . . and· a precision variable short connected in ·cascade as shown in fig. A3.
The scattering matrix of the attenuator is for an isotropic system
. s • 1511 512
~12 $22
and the incident and reflected waves at the ports are given by . .
where
. . .
• . .
or
or
[::} rsll ~12
When a calibrated sliding sh~rt is connected at the port 2. then
a2 = - [exp ( -2j8ac )]b2
~ =phase constant:in the sliding short system ro . . . x = distance of the short from the reference plane
[
S 2 e -2j s0x J $11 - __ 1_2 __ _
l+S -2jsox 22
If s1l and s22 are smal_l as compared to unifiy, the above equation' .
I I I I I I I
~l. ->PRECISION
-:::.I~ 1-\ -~ .\ ~· ..,- ': '' l' •\ T· Q '"\ ... ':" l I '-·· :' ~ 0
I
i I I
:~REFERENCE I
l PLANES~
I 2
I . •'wo
I l
·ATTENUATOR ' VAR IA13LE -,_ SHORT I
I
2
Figure A3 The Details of the Reference Arm
•e
.
reduces to 131
• ..
Now if the wave suffers an attenuation of y nepers while travelling in
the attenuator, then
. . . sl2 = e-Y
~ E - e-2(y+j Box)
al D 0
• ~10 = _ e -2(Al-Ao)-2jBo{l·r-R.o) • • R 11 . .
= -e{l\+j~)
A4 TUNING OF A HYBRID TEE
(AS)
In the above analysis it is assumed that 533 = 544
• 534 = 0
and in a commercial grade tee it is sel.dom so. In order to meet this
condition.it is necessary to place lossless ·tuners in theE and H
arms and make the VSWR looking into these arms unity with arms I and II .
tenni nated with matched 1 oads. In practice it is very difficult to get a
VSWR less than 1.02 to 1.03. After this adjustment 534 is checked by Q
measuring the output in the E-arm while feeding the H-arm with te~inations
placed on arms I and II. If 534 is not zero, it can be reduced to a
mi.nimum by placing a tuner in one of the side arms. This adjustment will
usually disturb the tuning of E- and H-arms so these must again be
tuned and the process repeated till the effe~t the side arm tuner on
132
E and H arms tuning becomes negligible. If the ~ee is of good quality
s34 will normally be very small and 'need only very small amount of
tuning to make ~34 negligible.
l
. . . . . .. . .
APPE~DIX B
C~1PUTER PROGRAMMES
133
..
. .
FLoW C HAfZ 1'
FoR
Lossy WALl Gu\l>E.
C::.'TAR T
COMPUTE ,... .:y ~oL
"t ~\w()o')(~~
CALL
l\At>A~
BADA~ To
C:oMPUT£
i€.~.
ENP
SI8JOS S I 2FTC ','if.~,LLOS
C LOSSY ~ALL ~AVE GUIDE .:: .S2:...V:rc~>l OF DO>iH:ANT r·~ODE PROPAGATION CONSTANT AS A FUNCTION OF .:':: v c-""'"'l-:;-
-:::.:;:<;;L::x-c,-cp-,--co--~y, Dfy, u, AKCt DY, GAMEt GAMA C. I ; : ::: i' ~ S 1 0 ~~ V ;:: R ( 3 l , V S I 0 ( 3 l
~EAD C5dl f.\,F 1 F J:\>'1.:\ T ( 2 F 1 0 • 5 l
'\ E A) ( 5 ~ 2 l ( '/ E R ( I l ' I = 1 ' 3 l ' ( V S I 0 C I ) t I = 1 ' 3·) ·---2--F:}·i=:\'/~T-13-r ~ :__..-;-sJ
C:~ = \IE::\Cl) S I = VS I 0 ( 1 l \/ = 2.993C:l0 PI = 3el41593 .A:<U = 4.·H·PP·l.E-9
------~~PSv----t.~~~~h~,----------------------------------------------------------
F = F~q. • E9 A = A-:~-2.54 OHEGA = 2.·::-PI*F OMEGS = O~EGA*OMEGA BETAO = SQRT<OMEGS/V/V - PI*PI/A/A)
--------~x~~e6s~·~----------------------------------------------~---------------
3 CONTINUE· ':JRITi: (6,8) ER
8 FO~MAT (4H ER=,Fl0.2/) 16 CO!·:T I 0!UE
CP = CMPLXCER ·, -SI/OMEGA/EPSO) --------cD· · :::--~w.<-1\* .
C = CSQ~TC-1./CD) .G/1; ·.,\ == CSQ::ZJ ( P P}P I I C A 'I:- A* C 1. ~2 = AIMAG(GAMA) - BETAO
+ Cl*Cl.+ C})-XXl
·.·:::.I TE ( 6, 4 l (:/..,~•1.4, QB ..,. rO:~/.\T (30X, 3El5 • .5l
C~LL BADAR( A,XXt GAMA, BETAOt CP) CJ = hiMAG(GAMA> - B~TAO ~RI1~C6t51 sr, ROt GAMA, DB
5 Fc::,:o:AT (IX, 2El2e3t ·5X,3El5e5l lZ = ALOGlO(Sil
. '
-----z z--=--z-z.-+--vs-!·ftf~,-------------------------,--------------------------------s : - 1 0 • oH~ Z.Z IF(SI .LE.VSI0(3)) GO TO 16 .SI = VSIO(l) ;:;::::; = ER- VEFH2) IFCC:R.GE.VER(3)) GO TO 3.
------·---·.·:f:11t..
1'? GO TO
21 STO?
·------ ---- -·-· ~~; __ :[ ~~CCTTt~-t:U7\-DAR (A,
::o:<PLEX Gfl.t</'., GAtv'E' . "' " = \_,
2 CC;'-!T I ~~~UE
xx,-c;AMAt 8ETAOt AKit AKJt FK' DFKt CPt Yt yy, DK
IF (t!.GT. lC) GO TO 3
G;\i<~:: = GA:<;\ 1-\K I = cso:-n ( GMiE·:<GM1E + XX l Y =-A~I*AKI - XX*ICP- lel /1KJ = CSQRT ( Y l y = ;-\:<I 1:-A
---·yy ;, ·c:s-fH ('(l7C"(osTYT-- -------------·---------------
FK = AKI + AKJ*YY OFK = 1• + AKJ*A*(l.+YY*YY) D:< = -Ff-:./DFK <\:(I= ,t.\KI+ Df( G .V: ,'\ = C S C ;.; T ( M~ I -o~ i'\ K I -X X )
i" AKI*YY/AKJ
--- ---- --- P = ;\ 3 S ( R E/\ L ( G .. \i-1 E -- Gf'\iv1A) r ----· -----------------------------
.122
0 = ~SSIAI~AGIGA~E- GAMAl) IF IP.LT.1.E-6.AND.O.LT.l·E-6l GO to 1 N = N + 1
GO TO 2 3 ~v R I T E ( 6 ' 4 l 4 FOR:<.-'\ I ( 26H i'-.;0 CC)NVE-R-GI:NCE IN .THE SUB) 1 D3 = AIMAGIGAMAl - BETAO
RETURN END
--16--~- ----,;-.---~n-:--.-----------------------------
.01
.122 ..., ::; . .01 3IBSYS
• 2 70.5 1. .2
1 • 10.2
·----· -------------·---------------------------------------
____________________________________________ .___ _______________ _
------------------------------------------------
..... .. -----------. ------·---------------------------------
S.r.AR T.
(Glob.'\
(~h)
FLoW CHARI FoR
PA~'"fiALL'( fiLLEO
WAVE G-VlbE
~-~=L~ (..,JG-)
WRITE
C AL.L
G E ')( AC:.i
C()fir'\f>\)TES '(,, Y~ .. "ls, "'(''7
(a.~$')(~~ v~)
C.Ot-'\PtJTE S. 'Y 1 , Y~, ..-~, '"(7
(w~v~)
C.ON T f:!/
CALL
G ::;o RT
~~------------
Pt)TS '(.., Y~, '(~, 'h trJ PR,.oPE~ ORDE.R
G-O TO
YEs
c;..oTO CoMP'-'TE
z • .;.. """"~ R ..- w~ 1Rt,&
')(, = ea ,. ("')-+1,
'It,, lo"'
" cs-,: \o
$IBFTC, C SOLUTION FOR CENTRALLY LOADED WAVEGUIDE
CO~PLEX T, cp, TI, TJ, GAS, GA ' GAP' GAMt AKit AKJ COMPLEX FK' DFK, u, VtTAA' COt RF
0// oss OTT
---- CO\iPLcx--_n,A<·6-I-,-----yY19-r-------- ----- ________ __;_____:.__ -- o uu- ---D!f!:ENSION AKII9h AKJ(9l 'GAC9) ,T(9,9) DI~E~SION VFI5), VXC5l, VER(5lt VSI(5) DI~E~SION AKIA319l, AKJAB(9), AKIAR(9), AKJAR(9)
10 RE;\)I·Sdl [R, ERl' A, CODE' AL 1 FOR~AT 15Fl0.51
--------1.-.'F: I T E · -<·o--;, 2 )-·E R 'E ;:<1,-A-,-A-t 2 FCR~AT 14~ ER=,2Fl0.3,Fl0•2' F12.5/)
ovv 0 t-!1:1 oxx OYY ozz 0/t··---
::\EAD <·5,621 IV~IIIt I=l,3),CVX((), I=lt3h CVSICI)t I=lt3) 055 62 FORMAT 13F20.101 044
F = VFCl> 044 SI = VSI(l) 055
----x---=-y)\rrr---------:~--------------------o71 ___ _
65 A = A·::-2.54 077 PI = 3.1415928 088 S = F?l/A 099
68 WRITE (6,200) F OSS XX = .043929*F*F 055
-~---------------------------~-zz--
cP = CHPLX( ER, -1800.*51/F) 066 205 ',/RITE (6,206) X 4077
69 Y = P H<-X T S = A~<-X 0 = lA- TSJ/2.
055 066 077
·---;;ET o--~-s-oKfi-x·x-=--~-:S--t-----'----------------------A~---Do 3 N=l'7'2 JO 3 i•~=1,7t2 A~~= FLOAT(N) IF IN.NE.Ml GO TO 4 T(i-.;,r.~l= Af~·::·Af~·*S*S- XX*ERl- XX*CCP-ERl)*CX+SIN·CAN*Y)/(AN*PI))
099 OTT 044 066 033
------GO- To-s--------------------------------.f)-c'.t-4----+ L = I N + ~-~ l I 2 0 7 7
< = IN - ~l/2 066 ALL= FLOATIL) 077 AK = FLOATIKl 0== TI1•;,NJ= XX~HCP -ERll/PI*CSINCY*ALLl/ALL*«-l•l**L> + 0==
------1--- S-IN<-Y*A-K--+-1-Ai<.-*f--h) ** ( K+3)) OU\J---3 CONTINUE 044
A A ( 5 l = 0~ P L X ( 1 • ' 0 • 0 l · 0 7 7 AA(4l = -Til,l)- Tl3t3)- T(5,5l- TC7t7l 044 AA<3l = Tll,ll*ITC3,3l + T(5,5l + TC7t7ll + TC3t3l*( 044
1 T(5,5l +'T(7,7ll + TC5,5l*TC7t7l- TC3tlHE-T(l,3l- TClt5l*TC5tll 077 - -----~----T+t·-,---7+*-1-<--?-, 1 l -- T C;, ' 5 l x T C 5 ' 3 ) - T ( 3 ' 7 ) lET C 7 '3 ) -T ( 5 '7 ) l< T ( 7' 5) Q--71--'77-----
AA 12) = -TC1,l)·Y..(TC3,3l{~(T(5,5) "+ TC7t7) l + TC5t5)-¥-·TC7t7)) 077 1 -T(3,3l*TC5t5l*T(7,7) Qt t
2 + T ( 1 ' 1 l :< ( T ( 3 ' 5 l .J: T ( 5 ' 3 ) + T C 3 ' 7 l * T ( 7 , 3 ) + T ( 5 t 7 i * T ( 7 , 5 ) ) 0 ' ' 3 + T C 3 ' 3 l -:~ ( T I 1., 5 l ~:- T C 5 ' 1 l + T ( 1 '7 l * T ( 7 ' 1 ) · + T C 5 '7 l * T ( 7 '5 l ) ·0 VV 4 + T(S,5l*(TC3t7l*TC7t3) + TClt3)*TC3tl) + TClt7l*T(7,lll· 055
---;,- ----t-Tf=t--y-?--}*~-i-+i-Y-5--r*-i"-< 5' 1) I T ( 3' 5 l * T ( 5 t 3 l -+ T ( 1 t 3) ~H ( 3 t 1) ) --{)5-5-----6-TClt3l*T(3,5l*T(5,1) - TClt3l*TC3t7l*TC7tll-TClt5l*TC5t3l*TC3tll 077 7-Til,5l*TIS,7l*T(7,ll-TC1t7l*TC7,5)*TC5,1l-TC1t7l*TC7t3l*T(3•1) 077 2-T(3,7l*TC7,5l*TI5,3l - TC5t7l*TC7,3l*TC3t5l 055
AA<ll = TCltll*CTC3,3l*TC5t5)*TC7t7) + T(3,5l*TC5,7l*T(7t3) + 077
1 T (·3, 7 l -1:- T (7, 5 l -:~ T ( 5, 3) ) + 2 T(3,3l*(i(l,5l*T(5,7l*T(7,1l + TC1t7l*TC7t5l*T<5•ll l 3 + Tl5,5l*(T(l,3l*T(3,7)*T(7•1l + T<1•7>*TC7t3l*T<3•1 l I
4 + T I 7 '7 l -::-IT ( 1' 3 l .;:. T ( 3' 5 l -:c· I ( 5' 1 l +. T ( 1 '5 L* T C 5 '3 l * T C 3 '1 l l +
137
077 0 I I
0 I I
0'.•":! S Tll,5l*T(5,1l*T(7,3l*T(3,7l + TC1,7l*TC7tll*TC3t5)*TC5t3l 077
·~-----61Tn-,1T>rtT\·'3·-,-Tl1Fil5-,rn~ T < 7' 5 l + T ( 5 '5 l"* T ( 7 '3T*IG,-I·~...-+----- 0·6 ~- .. 7 i I 7, 7 H:- T ( :·h 5 I -r, T I 5 f3 l J + 0 66 S T ( 3' 3 H· ( T C S '5 )-l:- T I 7 d I* T I 1 '7 J + T C 7 '7 H~ T C 1' 5 l * T ( 5' 1) r 0 77 9+ Tl5•5l*T<l•3l*T<3,ll*TC7,7l+TC1t3l*TC3t5l*TC5t7l*T(7,1l l 077
AAill = AA(l) - CTC1,3l*TC3,7l*TC7,5l*TC5tll + TC1t5l•T·C5,7l* 077 1 T!7,3)*T(3,ll + T(1,7l*T(7,5l*TC5t3l*TC3t1l + TC3t5l*TC5tll* 066 2 -r-q ~ Tl-::,-T ( 7, 3 l -+ T-( 3mf/'h:-T·{·7·,-l-t"-!(·T C 1 '5-T-*-r-t-5 t 3 ) ) 07?---3 + T(l,3l*TIJ,ll*T(5,7l*T17,5l 0==
T I = T' ( l , 1 l ~~- T ( 3 , 3 l - T ( 3 , 1 l ·:: T I 1 ' 3 ) 0 = = TJ = (i{l,1l +T(3,3l)/2. OXX YYill = TJ - CSORTCTJ*TJ -Til 077 YYI3l = TJ + CSQRT(TJ*TJ -Til 077 CA [ L- "SPUT_l'-41-.rrA-,-y 7 7 COi·'~PLEX zo, z, Gt p, TNt NF' cs, cc 077 DI~ENSION Z0(9), ZC9lt G(9,9lt P(9,9)t DELC9~9ltTN(9) 077 DO 5 N=1,7•2 '077 Al~IL\!l= CSORTIYYINl*YYCNl + XX*ERl) 877 AKJ(Nl = CSQRTCYY<Nl*YYCNl + XX*CP ) ''
o--s r< = 1 , f., _ o·-t-t---IF CN.NE.Ml GO TO 8 ·OYY DEL(N,Nl= 1. 088 GG TO 5 099
8 DELINtMI= 0.0 0== 5 CCi\:T H,lUE . OH-
-----c o;..::;·..;oi,~+! E-1-A*i-r-A* J.r~· -----t:rD-r, -t-T~s~,_,x,.;r.,x,..,,~c;fpl;,~G~AIIr---9-• -Et::-RR-l-1-----...,...------{0}-t-• L' --
CALL EXACTG 0== CALL GSORT<GAI IFICODE.GT.O.Ol GO TO 202 0'' DO 207M = 1t7t2 . 088 AKIAB(Ml = CABSCAKICMll 099
-----<';-K~f.\S+M+-=--€-A£-S--AK-di'--t-~.., f+----------------------41Zl;F---AKIAR<Ml = ATAN2(AIMAGCAKICMllt REAL(AKICM)ll*57e296 099 A i<.J A::.; Ul: > . = AT AN 2 ( A HM G ( A K J C M l l ' REAL C A K J ( M ) ) l *57 • 2 9 6 0 9 9
207 ~·.'::\ITE (6,208 l AKIABCM>, AKIARC~n, AKJAB(M), AKJAR(t-1), GA(M) 0== GC TO 201 O••
202 UC = 4.*PI*l·E-9 0' 1
·----{};'v'!{j--·.:·2-e~I "' " ~ 0-,,..;:-r----T! = CMPLXCO.Ot1.0l*OMG*UO Ott DO 9 ~=1,7,2 099
-.. ~N = FLOAT ·(Nl 099 Z(~)= TI/GA(N) . 100 TJ = CSQRTICtv:PLX(AN1:-AN*S*S- XX, O.Ol) '1// Z:"c;(-;-~y::-T-r-; 1-Sc«S;----IF<AL.GT.O.Ol GO TO 11 lTT AL = PI/2./AIMAG(GA(lll *CODE lUU
11 U = .GA ( ~n -::·AL lUU V = L~XP(UJ lVV T,':<i<l= CV- 1./VJ/CV + le/Vl 1Wh'
-----r-;-.j l ~: r=-Trrt-:--:-y:-:.z· t7n ____________ _.._ _____________ txx-------
~0 14 >!==lt7,·2 ,~<I( ... :) = CSQRT(Gt\(:w,~)·X.GACt-1) + XX*ERl) ~~J(~J = CS~RT<GA(Ml*GACMl + XX*CP)
'" "'" "'" .. ·····---
1YY lZZ 111 111
!38 ----------------------------------------------
U = AKICM>*D V = AKJ(IJ.l*TS CO = CCOSCV) TAA = CSINCU) CS = CSINCVl ---cc--=-cccrsrUl'-_ ------------ ------------- -----·-·· · NF= 1e/CSQRTCD- TAA*CC/AKICMl + TAA*TAA/CCS*CSl*(1.-COl*
1 CTS + CS/AKJ( Mll) GAP = AKJ(Ml*AKJ(M) GAM= AKICMl*AKICMl DO 14 N=1t7,2 AN = FLOAT< N) ANT = AN*TS*S/2• PCNtM)= SQRTC2.1Al*NF*2e*SIN(AN*PI/2.l*(AN*S*SIN(ANTl*TAA
1 - AKI(Ml*CC*COSCANT ll*
155
l t~/~
177 177 lE3 1S"9 !. ss 1 r:: ~ ,;.....,./_./
l 66 ~·
177 177 :!.55
2 (GAP - GAM)/CGAP - AN*AN*S*Sl/(GAM - AN*AN*S*S) 14 CONTINUE
DO 15 I-I,7t2 DO 15 J=1t7t2
l66 177
-------- 199
GCitJl= CMPLXCOeOtOeOl DO 16 lv1=1t7t2
16 GCitJl= GCltJl + ZO(Ml*DELCMtll*DEL(M,Jl + TNCM)*P(I,Ml*P(J,~l G(I,Jl = G(l,J) - ZOCll *DELCitll*DEL(J,1l
--l"t-5~C""'0'"T1Ni
COMPLEX ZNt REC, DET, DETT, DET37 DIMENSION ZNC9lt RECCgl, RECAB(g), RECAR(9) ZNC1) = TNC1l/Z0(1) ZN ( 2 l = G ( 1 '1 l I ZO C 1 l . ZNC3l= CGClt1l*GC3t3) - GClt3l*GC3tlll/GC3t3l/ZOC1)
----TC~Att DET3 (G, DETl ZNC4) = DETIZOC1)/CGC3t3l*G<5•5l - G(3,5l *G(5,3ll CALL DET7CGt DETTt DET37l ZNC5l = DETT/ZOC1l/DET37 DO 17 N= lt5 RECCNl=. CZNCNl - 1el/CZNCNl + 1.1
---~R~E~S~H(R~E~C~(H~~~)~)~-----------------------~-
17 RECAR(Nl= ATAN2CAIMAGCREC(N)lt REALCRECCNl)l*57e296 WRiTE (6,18) <RECABCilt I= lt5), CRECAR(J), J= 1,5)
201 IFCVXC2l.LE.O.OlGO TO 71 ZZ = ALOG10CXl ZZ = ZZ + VX(2)
------x~-----+l~~z~~.---------------------------IF CX.CE.VXC3)) GO TO 205 X = VXC1l
71 IFC VSIC2l·LE.~.Ol GO To 72 ZS = ALOGlOCSil ZS = ZS + VSIC2l
199 l TT
166 , ~ ')
.!.. ---
. - -··l f.~~ 177 1 SG 177 1 ----1-... --~ : , ~ I ....,.• ... v
l 77 1'"-'~ 477 177 l 77 1 I I
1 I I ... :vv 155 1~5
--l 77 177
177 ~- 77 l I t
---~I - 10·"**·-T-5~-------------------~-------------· ______ ,.I---IF CSI.LE.VSI<3ll GO TO 203 SI =VSICll
72 IFCVFC2l.LE.O.O) GO TO 74 F = F + VF(Z) IFCF.LE.VFC3)) GO TO 68
-~7n4~C~T~~~--------~--~---------------------------·,.; -;_ : T E ( 6 ' 51.1
;1 FC~···AT (:l.H~·)
?.00 r:c;;;:,\ T < 1>:, Fl 0. 3 l
2.77 l6S lE-6
~ ==
' --------------------·----------· . _____ " _____ ------~ ------·-·-:._·':.!.: ·~OR~~,\ T 2 C 0 F C '~' ~ ~' T 2CS FJ.~:~,\T 2 C 9 != 8 :< '-', ,~. T 21C ;=c~.·<,~T
( ll x., E 1 0 • 4 l 1 = = (21Xt E10.4l lXX (3lX, 6El2.4l -177 (FlO$~' Fl6.2l 177 <F10.3l 177
----zrr?:JF; :Ar-rE .L o-.-:;---,----------------------------- rrr- ------212 r=-c·~:·,'\r <3El5.5l 177
GO io 10 177 13 STOP
c: ·'~ c $I GFTC EX!,CTG
177 1 t t
-- ·SU8~J0TI"NE~XACTG · ------~-------------- tvv--cc:·<PLEX AKI' AKJ, CP, GAt u, Vt COt TAA• FKt DFKt RFt GAS Jl>l:::,,:S·IOI-i Gt\(9), AKI(9), AKJ(9) co::.':01\J /?.i"!E/ AKI, AKJ, D• TS, XX, CPt GA , ERl DO SO ~1=1'7'2
60 GA(~)= CSQRTCAKI<Ml*AKICMl - XX*ERl)
188 199 1==
1 t I
~-6~~~~~-----------------------~~~--·r==-----
~! - " " - v
GO TO 31 32 AKJ(~l= CSORT<AKI<Ml*AKI(M) + XX*CCP -ERl)) 31 U = AKI <rviJoi-D
V = AkJ(Ml*TS/2.
1== 1 t t
' 188 199 lZZ
----rA~~!N~~-e~o~Sn(+U~l-------------------------~~---
co = CCOS<Vl/CSINCVl FK = AKJ(~l*TAA- AKICMl*CO DFK = AKI(M)/AKJ(Ml*<TAA+ AKI(Ml*TS/2•*<1• + CO*CO)l +
1 AKJCMI*D*(l. + TAA*TAAl - CO ?\F = -FIUDFK
----A.K-I-{-ivi-)-=-A-1'-2-I-<+i \---~----cR-F-----------------------------GAS = CSQRT < AK I ( f~ l -~AK I ( M j - XX*ERll PP = ABS<REAL(GAS-GACM) )) Q = ABSCAIMAG<GAS-GA(Mlll IF (~.GT.15lGO TO 33 IF<PP.LT.l.E-6.AND.Q.LT.l.E-6l ~0 TO 54
----------,.~
'' 'l •
GAC,il = CSQRT<AKI (fv.)i:·AKI CMl - XX*ERl) GO TO 32
54 GAIMI= CSQRTCAKICMl*AKICMl - XX*ERl) AKJ(Ml= CSORTCAKICMl*AKICMl + XX*CCP -ERll)
· GO TO 36 ~t!Ri-T t.'-----tQ-,~r+---------------------------------
34 rCI~>~AT ( 15H NO CONVERGENCE) 36 CCNTINU[
f~ ETU~\N
SI3FTC DCT3 ------sv"J;;vJT1llr2lJ1:4'-?J--t-f';~-f"li:~r--------------------------
cc:·iPLEX G, DC:T D I.". : ;.: .S I 0 r~ G < 9 , 9 l DET = G(l,ll*{G(3,3l*GC5t5l - GC3t5l*GC5t3l)
1 -Gil,3l*IG{3,ll*G(5,5l- G(3,5)*G(5,lll. 2 + Gll,5l*(G{3,1l*G(5,3l - G(3,3l*GC5tlll
------------~cru~: .. ~--------------------------------[f.: r)
~ Sl:.:FTC SPC~Y4
S:.JSROUT I ~!E SPOL Y 4 ( AA 'YY l (C~PLEX AA,YY, FF' OFF, RF, YX, BSt YYUt YYUN, CCtOYtTC9t9l
------.. ----·---------------------------------------------
---------- ----·-------------------------------- -------------------------------------------------------
C' D 1 1\l = 1 ' 3 ' 2
5 IFINN.GT.15l GO TO 2 YYU = CSQRTIYY(Nl l
-------·-r-::-- = --n'.PIX ro·-;--,-o-;·1 DFF = CMPLXCO.,O.l JO 3 1-'l=lt5 :·. ~ ;~-'; = {v; - 1 L L =
FF + AA(Ml*YYIN l**MM ' ~ =
-------;.,:>~- = FDJ1\T( 1'~7-~r------- ----------------------DFF ~ JFF + A~*AA(~l*YYCN l**LL
RF ...:: -FF/::FF YYINl = YYCN) + ~F YYU~= CSQ~TIYYCNll
-----D1~-~~,------------------------------------
PP = A3SIREALCDYI l QQ:: ASSCAir-iAGIDYll IF IPP. LT·1·E-6.AND.QQ.LT.1.E-6l GO TO 1 NN = ~\N + 1
GO TO 5 ---~·m1TE\oT0.---------------------------------------
6 FORi<AT I 25H NO CONVERGENCE IN SPOL Y.4) 1 CONT Ii'·lUE
CCC2l = AAC4l + YY(l) + YYC3l . CCill = /\t\13)- YYC1H~YY(3) + CYY(l) + YY(3ll*CC(2) YYI51 = -CCI21/2•- CSQRTICCC2l*CCC2l/4•- CCClll
---t'.' nr-= - CC.\7T I 2 • + C S Q R T\-t'("T("-1(~2,_..)n*r:-tC"""~C""-~-( 27---t-) ,~· 4-r.-=-• ----;;;;;----("-(-f"(-t-(--t-1--t)--t)--------------D C t, :·i = 5 , 7 , 2 y;:c··.i = CSQRTIYYU-"ll
4 YX;-;(i~l = REF.LIYYI1·1l l IF CYXRI5l·LT.YXRI7ll YYC5) IF IYXRC5).LT.YXR(7)) YYC7l
= YXC5) = YXC7)
YXC7) --------IF- { -Y-X;'{-t-;--t-.--Grl'T--=.4'f~X~-ERH(c-=:7'-')t-')1--''.Pt''rf r<( 5;;..-r) -=-¥*+~---'-------------------I,:.· (YXRI5l;GT.YXRC7ll YYC7l :::c S N= 1,3,2
9 YY\:~1= CSQRTCYYCNll
Ei:::
= YX(5)
---:s I :JFTC -:>=:T?-------------------------------------SUoROUTHlE DC:T71Gt DETT, DET37l CO~PLEX G(9,9l, DETT, DET37 DETT = G(l,ll*(G(3,3l*(G(5,5l*G(7,7l- GC5t7l*G(7,5)l-
l Gl3,5l*IG(5,3l*G(7,7l - Gl7t3l*G(5,7ll+G(3,7l*CGC5t3l*G<7•5l-2 G{5,5l*G(7,3lll -GC1,3l*CGC3,ll*CGC5,5l*GC7t7l-GC5t7l*GC7t5ll-
----:;--G { 3 ,~5~""G+-5 'l Ht·G ( 7 '7) -G ( :n;) ~G ( 7 '1) ) +G ( 3' 7) ~H G { 5 t 1 H}G { 7 '5 l -4 G(5,5l*G(7,llll + G(l,5l*CG(3,1l*(G(5t3l*GC7t7l-GC5t7l*GC7t3ll-5 G:3,3l~(G(5tll*GC7t71-G(S,7l*GC7,lll + GC3t7l*CGC5tll*GC7t3l-6 GIS,3l* Gl7,11 ll-G(l,7l*(G(3,ll*CGC5t3l*G(7,5l-G(5,5l*G(7t3)l-7 013~3l*IGI5,ll*G(7,5l-G(5,5l*GC7tlll+ GC3t5l*(GC5tll*GC7t3l-8 (: ( ::J ' 3 I -:: G ( 7 , 1 l l )
--------- ;;::_T ::n---=---G·{-n""3+-\('+6-<--s-' 5 > :~6 C 7'? l -G ( 5 '7 H~G f7 '5 l ) -G ( 3 '5 H( f G ( 5' 5) ll 1 G(J,7l-G(5,7l*G(7,3ll+ G(3•7l*CGC5t3l*GC7t5l-GC5t5l*GC7t3)l -·f.- \1 I
\...:., I ·_) '\, l
:..--I8F"T( CSC?T ..
------~ -· ~ -- ---~---------~--------~--~~ ·- ---~-------------~ -~--- - ----- ~--~ J4-'
su::~OUT I f\!E GSORT (GAl ~I~E~SION QA (9}, GAR(9), GAI(9l CC:.i?L:. EX G!, ( 9 l , G.L\X ( 9 l DC 1 :':=1'7'2 G;~,x < rvi l = G/\ < r~ l
---~u~n~T~= 1\[ALIGA'~l.~l ~---------------------------------------------------
1 G A I ( r< l = .fl. Ht,\ G ( G A (iv\ l ) G/\i/X =J\1·1..-'\Xl(Gt'\I(l), G/\!(3), GAIC5lt GA!(7))
2~ D.~ t;.; J = GA::x - G/\I U·1 J IF(DA(3l.EQ.O.OJ GO TO 3
----rrts .. :-li-s~e:::c-•r.(;)-Go-r'r'r~---------------------'-------------r=l:~<7l.Eoeo.ol. GO To 5 G~(l) ·= G.!.\X(l) GA:::: = ;:,:~It'!l <GAR(3l, Gr\R(5), GAR(7l)
6 DA(~l=· GA~M - GAR(MJ --------rr~~\liiee~'-.fCMJ-f.G~Or4T~0~7~--------------------~--------------
IF(DA(5J.EQ.O.Ol GO TO 8 G.i'd 3 l = GAX ( 3 l IF (GAR(5l.LT.GAR(7ll GA(5J = GAXC5) IF (GA.~ ( 5 l • L T • GAR ( 7 l l GA ( 7) = GAX ( 7 l IF (GAR(5l.GT.GAR(7l) GA(5) = GAXC7)
---~rp-\GAR(5l.GT.GAR<7>l G~7~l~-~G~,AHX~(H5~;-------------------------------
GO TO 9 8 G~, ( 3 I = GAX ( 5 l
IF CGAR(3leLT.GAR(7ll GAC5l = GAX(3) IF (GAR(3l .• LT.GAR(7J) GA(7J = GAX(7) IF (GAR(3l.GT.GAR(7ll GA(5J = GAX(7)
-----r-F--t Gtrttt-:3 l • Gi'.-GAAT7--H-Gf.\i/'1')--:----nG-trA*X+C "'!t3+) ------------------GO TO 9
7 Gf--.(Jl = GM~(7) IF (GAR(3l.LT.GAR(5Jl IF (GAR(3l.LT.GAR(5)J I r (GAR ( 3 l • G T. GAR ( 5 l l
----rF-TGAR ( 3). G1. GAR ( 5 l) GO TO 9
3 Cf.\(11 = G/..,:\(3) G2~~ = AMINl(GARClJ, DO lG ~··:= 1,7,2
10 DA(~l. = GR~~ -GAR(Ml
G1\ ( 5 l = GAX(3) GAC7) = GAX(5) GA(5l = GAXC5l GA ( 7) = GAX(3)
GARC5), GAR ( 7) )
r--F+C-ft+7 l • ~ e--:;;--1. vf'r.:-. r::r-t---F-cl'T--'f-fi-~T-----------------------'-----------
IF ( C;;l( 5 ) • [Q • 0. 0) GO TO 12 c; ( 3) = GA:<Cll ~ ,.... i. r (GAR(5l.LT.GARC7ll GA(5) = GAX!5) IF (GAR(5l.LT.GAR(7l) GA (7) = GAX(7l I F IGAR(5l.GT.GAR(7ll GAC5l = GAX(7l • I CGAi"{ ( 5 l .l.J T. GAR ( 7l ) GA ( I) - uAX(5) (j() TO 9
ll G;; ( 3 l = GAX(7) rr· (GARill.LT.GARI5l) GA(5l = GAXC 1"> IF IGA~(ll.LT.GAR(Sll GAI7l = GAXCS) IF IGARill.GT.GAR(Sl) GA(5l = GAX(Sl r:=-TGfi."RTJT;-GT~-G7"R ( 5 l l GA(7J = GAX ~ ~ TO 9 ~-...~~ . .., ::; . \ ( :. ) = G;\X (::;) .l.C..
l i= IGAR(ll.~T.GAR(7l l GA(5l = GAX(ll T ;: . ' ( G.;R I 1 l • LT. GAR ( 7 l l GAC7l = GAXC7l
----· ----· ----·~--
142-·------------,----~"- -·
IF ( Gi\R ( ll. G T. GAR ( 7 l l G.'\ ( S l = GAX ( 7 l IF IGARI1l.CT.GAR(7l) GA(7) = GAX(l) GO TO 9
~ G~(ll = GAXI5l G:~:.:;.; = A:\1 I f'.j1 (GAR ( 1 l ' GAR ( 3 l t GAR ( 7) )
--------D·o-rT:v·.=-1-,,;-2·--- ------------------------l3 OAL<l ::: G!-(fvi:': - GAR(,~·il
I~(8A(3l.CQ.O.Ol GO TO 14 :FIDAI7l.[J.O.Ol GO TO 15 G r-'- ( 3 l = G ;\ X ( l l I~ {G~RI3l.LT.GAR(7l l GA(5) = GAXC3)
--- 1 r \ ~"f'I~ITl~T;G·,t,;:<tTl""IGA-('7')--;;;;=;-r:G~A~X'-f(1/>;)---------------I~ IGAR(jJ.GT.GARI71l GAI5l = GAXI7) IF IGAR(3J.GT.G/I.RI7ll GAI7) = GAX(5l GO TO 9
14 GAI3l = GAX(3l l F I GAR I 1 l • LT. GAR ( 7 l l G.'\ ( 5 l = GAX ( 1) I;- (GAR< 1 J. [T;-t;11."K( 7 I l GA { /J = GAX { 7) IF IGAR(lloGT.GAR(?ll GA(5) = GAX(?)
·IF (GAR(ll.GT.GAR(?l l GA(7) = GAX(ll GO TO 9
15 GAI3l = GAX(7) IF IGARI1l.LT.GAR13ll GA(5) = GAX(ll
GAXC3l ----n:--IG"fiR\TJ.l_-r-;-G"t\JTITI,----c;-:rAI( '7'} ----:::-r::~--t-":1:-r------------------IF (GAR(lloGT.GAR(31 l GA(5) IF IGAR(1l.GT.GAR(3l l GA(?l GO TO 9
5 G!'.l1l:: GAXI71 GR~~ = AMIN~(GARI1), GAR(3),
-------oo-16 1'1= 1--,-1; · 16 D.4. { ij l = Gr~;,1f.1• - GAR ( t-1 l
IFIDAI3l.EQ.O.O) GO TO 17 IFIDAI5l.EQ.O.Ol GO TO 18 G,C,, ( 3 l = GAX ( 1)
= GAX(3) = GAX(ll
GAR(5))
IF IGAR(3l.LT.GARI5l) GA<5l = GAX(3l ---- I·f--1-G-A-R-f-3 l • LT. GA~P~, P( 5T+-l t--l --iGSi;.~\ t-'( 71+) _,_,....._,G~A~Xx-, t-():;-~ -1-) --------------,--------
1 F ( U1R ( 3 l • G T. GAf~ ( 5 l l GA ( 5 l = GAX ( 5) lF {GAR(3l.GT.GAR(5l) GA(7l = GAX(3) G:J TO 9
17 GAI3l = GAXI3l IF IGAR(ll.LT.GARI5ll GA(5l = GAX(l) IF-tu·A-R(l).tT.GAR(5ll G/d7l- GAX(5) IF (GAR(ll.GT.G,\R(5) l GA(5) = GAX(5) IF IGAR(ll.GT.GAR(S}l GA(7l = GAX(l) GO TO 9
13 G/,(3) = G;\X(5) I r ( G:\R I 1 l • LT. G.~R ( 3 l l GAX ( 1 l GA ( 5 l =
----~--n~~R(llotT.G~~(~3~)~)-f.~~~~6~AMXH(~3H)~------------------------------------GA(7) IF IGAR(1l.GT.GAR(5l) GAX(3) GA(Sl = IF IGAR(ll.GT.GAR(3l) GAX(ll GA(7l =
'i CC.'!T I ;\iUE
-- ;:-;-;T;:~y-------------------------------------------------------· , .1. • .9
.1 lel
Flotv CHJ\R.'T t:o~ MIC.PoWAVE.
COtJb\J~"TIVl,.Y C)f G-EfU,IAtJ\~~
Co MPVTE
C.o N5r AN,.S
')-'L (£.)
tT(t)
")" ( t-) _,
= ['it. (c)~ i'rf£ ]
co.....rrb /
CoNTb/
<::oM PU TE
/ '"""( > ~lfY ""'· p . <.. .l +t.~~...,.\..
ND: "'~_.~ Nb
NJ> ~ lo
ENt>
\, I., ' I
i l I ~ I I •
C , ii Ci :cY.I;\V[ ·~·Ji :tAJC. T l V ITY or:- GUd·~/\i\11 Ui~ REAL ~L' ~T, MC' ~D' MCH• MDH• Ml• M2• MISt MISHt ND DI.'<ENSION VNDI3h VTC3lt VCI(3)t VCJC3hVROC3lt VF(3l
FE(~ = 9.108E-31 :;T = .0819 ~',L = 1.64 EPSO = 8.S5L,E-12 3:< = l• 3802SE-23
----~~ _- 6-~-6-2'J7~1 t.- 34.---, -----------------------------
\..! = - H I 2 • I ? I :::vc'< =· SKIQ ;'"lj_ = • 35 :<2 = .044 ~C = 3.*FEM/C2.1~T + 1.1Mll
· , -~;:>--.::~i*lTT1:1B~ t'"ll ) ~Hi" ( l • T:r.J ~CH = 1~1**1.5 + M2**le5)/CM1**•5 + M2**e5)*FEM ''DH = CH1->H1.5 + t~2**1.5l**(2.13.)*FEt~ f·i!S = 3.111./SQRTIMLl + 2.1CSQRTCMT)))
1 -::-sc;;.;H <FE!'~ l >q S!-l== U-':1 *-;~. 5 + ~12**. 5 l *SORT ( FEM l
---------'.-Jr.::. TTlTb'Zl l MC' MD' MISt MCH t ~~DR' M ISH VI = 9.El2 CI = .2735 CIH = leS9 /,; : 0 S ( T l = • 3 8 5 -:< C 3 0 0 • I T ) ·lH~ 1 • 6 6 A~O~~CTTl = .18*(300./TTl**2•33
--- --,.c_-~;r I TTl = (I. 7 6El6 l -:;-( TT** 1. 5*EXP ( -2275 .ITT l ) /EXP ( 2275 e/TT) ·.·: = .2123El3 '. ~~ ·= .8Lt30El2 :;·s = 16.->:-EPSO
13 CG~<TI:'iUE ~~AD C5t6l NCODE
-----~;-~ l:'. r\ D-rs , 2"'.-l ---.( "V.,...F_,.t...,.I'>-,~I..-:==--1.---, .,..3') _______ .....:;_ ________________ _
F = VF(l) REA) C5,15l CVND(Ilt I= 1,3) ?EA~ (5,2) (VROCIJ, I= 1,3) IF (VND(ll.GT.O.Ol GO TO 1 ~0 = VRO(l)
---l-9- '•i~-r T c:-r-6-,"'3/.___,.r,~--------------------------------
SIO = 1./R.O t.-._':J':-J.f·. = 6 • 8LrE6 .~ :: T = 2 0 0 0 • ~~ S I 0 I Q
GA:-~:~ = 5.6~<5.6E6~~ANI (300. H~ANI (300.) - (S!O/Q)**2 X = 8ET 12.1ALPHA
----~~-[)-- ~--S::STITir-;-;.:-_,*"'2.-------.G"""A .. M,.iA....-r/-.-A-=-L""P-.-R,.,.A_,)-----,x~----------------------
:: ) = t'!D -li·l • E 6 SC TO 5
l CC~. i I I·!UE ~~D = W:D ( 1 l
S C).\ T HlUE ----- -- :'\ E'.l :::;·--r,' 2 l (VI ( I l' I= 1 '3)
T =- Vi(ll.
··.:::rrc C6,3l rm , • CC:':T!:iUE
TL = ~.L:JG:;.0:T)
CD:= NDI2•*( 1. + SQRT(1• + XXll CDh = ND/2.*(-1. + SQRT(1, + XXll
----£';1 ~ :...: M~l ( T l -;:-r. c.o Ii= Ci•iCODE.i::Q.ll CDH = 0, 1;= CNCODE.EQ.2J CDE = O. IF CRQ.GC:.SO .. l CDE ~ llr'HI IF CG.·O.GE.SC.l CDH = i\r"<I!
':. ;:; :: I::: C 6 , 7 l ;\ :1 I I , CD E: , · CD 11 , T , T L
~- ::: T /-. I = l • I t3 C: T ·"'· DN = ~.*SO~TCPili4.*8ETA**2a5 XI = VI-:<H/SKIT DII = EXPCXIl- 1. DIJ = 1. - EXPC-XI>
AA = CTI3CO.liSQRT(300,*EVBKJ CS = CXI/300.IEV3Kl**1•5 TAL1(EJ = AA*SORT(El + CI*BS*SQRTCE/XI + le>IDII TALZ{EJ = TALl(El + CI*BB*SQRTCE/XI -l.J/DIJ T~LHliEl = AA*SCRTCEJ + CIH*BB*SORTCE/XI + 1,>/DII
----.:-;- ,;-LJ-:2 ( E l =-'ff·tH1.-<--E-r-+--e-I-H*frB*S-"OR-=tT,C-TE;.,,M' )(1{-f{----1•.....--Tl••-l)-i/LfDrti=-:-Jt--------------FFC~l = EXPC-E/8ETAl*<E**1.5l A =CB.*SORTC2.l/ND*EPSIO*EPS/Ql*CMIS/QJ/Q *2e*PI*O**le5 AH=<8.*SORT(2.l/ND*EPSIQ*EPS/Ql*CMISH/Q)/0*2•*PI*O**l•5 S = (B•*MDIHSl*(EPS/HGl*CBK/Q)*(T/CDE/Ql*O
. SH= ( 8 .-:cMD/HC l -~- ( EPS/HB l * C BK/Q) * C T /CDH/Q) *0 ----i;'d-1-El = U\L-o-6+1-.o-1-8 ~~El -B. :.•E/(1• + B ME) l/A /EK1tle5
T:d:-:C~l = {!,LQGC1.+8H·:<-El- OH*E/(1, + BH*E))/AHIE**l•5 T i\ i ( E l = 1. I ( ':! -l:· TAL 1 < E) + T A I C E l ) T . ·, 2 \ C: l = 1 • I C \"! ~- T /\ L 2 C E l + T A I C E l J T /, H 1 C E l = 1 • I C\·' H -:} T A L H 1 C E l + T A I H C E ) ) L',Hz <El = 1.1C\'Ji-i-l~T/\LH2CEl + TAIH(E))
FF2(El = T~2(~l*E**1.5 t:?=i:j_CEJ = TliHl(E)-l:FFCEl ;= ::= :-; -~ < E J = T M-: 2 ( ~ l -l~ E.;~ -x- 1 • 5 T~L = CFINT14<FF1,0.0,XIJ + FLIN14(FF2,XI,BETAI))/DN -;- • I~ I I ::;: ( F H: T 1 4 { F F H 1 ' 0 • 0 ' X I ) + F L I N 14 ( F F H 2 ' X I ' BET A I ) ) I D N C'. ~ c: =--Q*-(·":D-E--!;7\MOB C T) or CDH M-AfvtOBH ( Tl )
:::.:·~;_j,'\ = 2.::+09~~PP}F :.' :~GS = c;·iEGI\*Ol'·~EG,\ T:.:-'l(Zl = 'T.\l(EJ/(1. + m1EGS*TA1(El**2J
... -- 7 .' ... 2 ( t:-r-:-r-:Q-\2) I i 1 • + 01"1 EG s-:;tri'-~~A,...2rl(~E~)Hi~d!!:'~ ..,..2-t-)-------------------;,'::. ;-:~it:l- T;'\H1CEl/Cl. + Oi'v1EGS*TAHl(E)**2l 7;:·;.2U::l = T!.H2(EJI(l• + Oi'v1EGS*TAH2(E)**2) ·: ·. . . ' : i E l = T.U C E l -::· T Mq ( E l -lf F F C E ) -·.·~ ·.~ iEl = TA2tEJ->-:·TAii.2(E){fE**1•5 7 ;. . . , ; l( C: l = T /il-11 ( c l >t· T/\1··1 H 1 C E ) * F F C E ) .,. • .... :· :2 i c i--:.:---TAH2~-:::-l-*"r;'\1\if'l.-,2-t(-1:EM)M~~-1:._~~~--------------------::-:·.•l(:J = Ti,:.il<EP·FFCEl :-";:-··.~:::l .: r·,:.:2CEP·E~H~·1·5
~- .--: · ; ~ ( ~ l = T f\;-: .-! 1 < E l ,'( .. F F C E l ::-~:· ::U:l ;.; T:,:.::-J2(EJ.J:·E~·*1•5
Tr\;.· = IFii'\T14{FFtv;1,0.0,XIl + FLIN14CFFM2 ,xi,BETAI1l/DN T A; ·. :--1 = ( F I :·: T 14 I F F 1•~ i-11 , 0 • 0 , X I l + F L I N 14 ( F F M H 2 ' X I ' BET A I> l I D N TAl<:.;= (FHlTlLdTAr,i~H•O•O'XIl + FLIN141TM1fv12•XI,BETAill/DN TAi•i:-i:-l = (FINT14(TA,~1HHl•O.O,XIl + FliN14CT,AMMH2,XI,BETAill/DN ,\ C: = T J\1.1/ L\ L .
---------·· \H-=----r-A"RR /TfiTH.
SI = O*ICDE*A~OBITl*RE + CDH*AMOBHCT)*RHl I~.S = SI/SIO DE~ = G*ICDE*AI~OB(Tl*TAMMITAL + CDH*AMOBH(Tl*TAMMH/TALH)/EPSO ~R = 16.*EXP(l.34E-04*1T-300.J) - DER R~ = TAMM/TAL/TAL
·-----,-,-:,.._r·IH-=-r-A'>1i~H1 T At:FrtT"Arf, \'if\IT2: (6,8) RE, RH; SI, RS, ER,RM,.RMH ZF = ALOGlO!Pl ZF = ZF + VF(2J F == lO • .;H~·ZF IFIF.LE.VF(3) l GO TO 4 ---··---- r---=·--vnJ:T T = T + VT!2) IF (T.LE.VT(3J l GO TO 11 T = VTI1) IF IVNDill.GT.O.Ol GO TO 10 RO = RO + VROI2)
---- ----TF { RO~TI: ~V?JJ'( '>3') ') -r::G~"'~O;---rT'7"0.....,.1-r9~--------------------'------
20 = VRO ( 1l GO TO 12
10 ZN = ALOG101ND) Zi·~ = ZN + VND(2l fW = 10.-~H<·ZN
------- IF { NDiit-E-•Vi\tV-~:A--'H'~t--------------------------2 Fo::;··.r_,r <3F10·5> 3 F 0;.; ·: t.. T · ( 1 X ' E 1 2 • 4 ) 6 F 0 ::; '; ;.. T ( I 1 l
15 FCR~AT 13El0.5l 7 F C R ::AT ( 1 X ' 5 E 1 2 • 4 l
------6--FOR·'·~A-"f-{-:t-X• 39Xr-f-E:t-z:~H---------------------------21 FC;~~·.,:,r llX, 3El2e4) 12 CO:H I NUE .
GO TO 13 14 STOP
END
-~---------------·---~--------------'----------------
·----- -~------------
REFERENCES
(1) T. S. Benedict and W. Shockley, 11Microwave Observation of
Collision Frequency of Electrons in Germanium", Phy. Rev. Vol. 89
pp. 1152-1153, 1953.
(2) T. S. Benedict, "Microwave Observation of Collision Frequency of
Holes in Germanium", Phy. Rev. Vol 91, pp. 1565-1566, 1953.
(3) J. M. Goldley and S. c. Brown, "Microwave Detenmination of
Average Masses of electrons and holes in Germanium", Phy. Rev.·
Vol. 98, pp. 1761-1763, 1955.
'(4) F. 01Altroy and H. Y Fan, "Microwave Transmission in p-type
Germanium", Phy. Rev. Vol. ~4, pp. Kf·-5", 1954.
(5) J. Lindmayer and M. Kutsko, "Reflection of Microwaves from Semi
Conductors", Solid State Electronics, Vol. 6, No.4, pp. 377-381,
1963.
{6) K. S. Champlin, "Frequency Dependence of High Frequency Transport
Properties of Cubic Crystals", Phy. Rev. Vol. 130, pp. 1374-1377,
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15· .I
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' .