BRYMWCS QF LEEBEUM 1R
GERMAMUM Mil} COBQPENSATEB SILICON

Thesis ‘30:? “no Degree 0% pk. D.
WCMGW SEME BNE‘JERSETY
Eric AEexanéer Bounce

i972

 

This is to certify that the
thesis entitled
DYNAMICS 0‘5‘ LITHIUM I'\l
GER‘JEAVIUl-l A‘ID COTx’EPE‘ISATED LITHIUM
presented by
ERIC ALEXANDER DC‘IWCE

has been accepted towards fulfillment
of the requirements for

Ph. D. Jame in E. E.

fl; w/f/X/I

Major professor

Date 2/10/72

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ABSTRACT

DYNAMICS OF LITHIUM IN GERMANIUM AND

COMPENSATED SILICON

By

Eric Alexander Dounce

A study of the dynamics of lithium in germanium and
silicon is important in helping to establish practical limits
on the drift rate of lithium ions in silicons or germanium
during the fabrication of nuclear detectors. The systems
of lithium in germanium and lithium in silicon have been
studied with this goal in mind. IsotOpic diffusion studies
of lithium in germanium have been made at relatively low
diffusion temperatures, 3.3., 2OD-HOOOC. Measurements of

the ratio of 6Li to 7

Li diffusion constants provide evi—
dence for the significance of quantum effects in the dif-
fusion process at these temperatures. We have proposed a
tunneling mechanism as the probable explanation and we
compare our measurements with the results predicted by

a general diffusion theory which includes tunneling
effects. Our results, when extrapolated to detector

drift temperatures, show that the lighter 6Li isotOpe

drifts only 12% faster than 7Li.

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Eric Alexander Dounce

The detector fabrication process requires drifting
lithium ions into the silicon or germanium to form pairs
or complexes with negatively charged trap sites. The
dynamics of the lithium-acceptor complexes then becomes
important in fully understanding the compensation process.
We have found the present model describing pair dynamics
in a crystal lattice to be inadequate. A modification of
the existing model is proposed which takes into account
the actual C3V symmetry of the complex. The possibility of
interactions between different local modes at the same
physical site is also considered. Our analysis is, however,
only approximate and a more rigorous formalism is indicated
by comparison with measurements of line width and centroid
variations with temperature. These measurements have allow-
ed us to determine the coefficients of anharmonic terms in
the local-mode potential.

If a local mode of free lithium exists in silicon or
germanium it is important to locate its frequency so that
it may be excited directly. We did not find a local mode
of free lithium in silicon and expect that the free lithium
exhibits only resonant modes in silicon. A review of the
literature indicates a probable free lithium local mode in
germanium near 36H cm"1 for 7Li.

Finally, the effects of uniaxial stress on the lithium-
boron local modes in silicon have been studied. We observed
that uniaxial stress near the fracture limit did not produce

a detectable splitting of the two-fold degenerate lithium

iii

Lric Alexander Bounce

or boron local modes. Similarly, it did not produce a
measurable shift in the local-mode centroids. It is un-
likely then, that uniaxial stress will produce significant

variations in the lithium drift rate in silicon.

iv

DYNAMICS OF LITHIUM IN

GERMANIUM AND COMPENSATED SILICON

Bv

Eric Alexander Dounce

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1972

Q“;

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ACKNOWLEDGMENTS

I would like to thank: Dr. William Hartmann for his
help with the theoretical interpretations, my committee
co-chairman Dr. David Fisher for his critical reviews of
the manuscript, Dr. Gary Cloud for his help with the bi-
refringence work, and to Mrs. Marcella Williams for typing
the final draft.

A special thank you is given to: my thesis advisor
and committee co—chairman Dr. Charles R. Cruhn for his
direction, help, and enthusiasm with the experimental
phase of this project, Dr. D. J. Montgomery for his de-
tailed evaluation of the manuscript on such short notice,
and to Dr. G. E. Leroi for the donation of the FEE—112
spectrOphotometer and for the use of the FEE—225 spectro—
photometer.

Finally, I would like to thank my wife for typing the
first drafts and for helping to make this project more

meaningful.

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H.434

.11

TABLE OF CONTENTS

 

Chapter Page
1 INTRODUCTION . . . . . . . . . . . . . . . . l
2 THEORETICAL HISTORY. . . . . . . . . . . . . 5
2.1 Lithium in Germanium - Diffusion Studies . . 5
2.2 The Boron Local Mode in Silicon. . . . . . . 9
2.3 The Lithium Local Mode in Silicon. . . . . . 10
2.u Lithium-Boron Complexes. . . . . . . . . . . 11
2.5 Anharmonic Effects on a Td Symmetric Local

Mode . . . . . . . . . . . . . . . . . . . . 15
2.6 Experimental Difficulties in the Observation

of Local Modes . . . . . . . . . . . . . . . 19
2.7 Complexing Donors with Acceptor Ions . . . . 22
3 SAMPLE PREPARATION . . . . . . . . . . . . . 26
3.1 Compensation Methods . . . . . . . . . . . . 26
3.2 Material, Mechanics and Preparation. . . . . 28
3.3 The Lithium Deposition . . . . . . . . . . . 29
3.u Diffusion and Lithium Solubility . . . . . . 31
3.5 The Hot-Point Probe. . . . . . . . . . . . . 38
3.6 Precipitation Kinetics . . . . . . . . . . . 39
u INSTRUMENTATION. . . . . . . . . . . . . . . H8
H.l The Spectrophotometer. . . . . . . . . . . . U8
u.1.1 The Fore Prism . . . . . . . . . . . . . . . MB
u.1.2 The Source . . . . . . . . . . . . . . . . . Hg

vi

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TABLE OF CONTENTS (continued)
The Grating Monochromator. . . . .
The Infrared Detector. . . . . . . . .
Double Beam Modification . .
The Measurement System . . . . . .

The Lock-in Amplifier. . . . . . . . . .

LOCAL-MODE EXPERIMENTS AND EXPERIMENTAL RESULTS

The Germanium System . . . . . . . . .
The Silicon Svstem . . . . . . . . . .
The Temperature Dependence Measurements.
The Pressure Dependence Measurements . .
DISCUSSION OF LOCAL MODES. . . . . . .
Local—Mode Assignments in Silicon. . . .
Lithium—Gallium Complexes in Germanium .

Temperature Dependences of Centroids and
Line Widths in Silicon . . . . . . . .

Temperature Dependence of Integrated
Intensities. . . . . . . . . . . . . .

The Uniaxial-Stress Results. . . . .
ISOTOPIC DIFFUSION STUDIES . . . . . . .
Reasons for Measuring the Mobility Ratio
The Ratio Measurement. . . . . . .

Comparison of Experiment with Tunneling
Calculations . . . . . . . . . . . . .

SUMMARY AND CONCLUSIONS. . . . . . . . .
BIBLIOGRAPHY . . . . . . . . . . . . . .
APPENDIX A . . . . . . . . . . . . . .
APPENDIX B . . . . . . . . . . . . .
APPENDIX C . . . . . . . . . . . . . .

A.PT)EP‘IDIX D O O O O O O O O O O O O O O 0

APPENDIX E . . . . . . . . . . . . . . .

57

6O

60

64

67

67

7O

7O

87

CO
CO

96

98

99

101

107

111

11A

117

120

122

128

IBM

Table

 

II

III

IV

LIST OF TABLES

Line—Width Coefficients
Experimental Results of Isotopic Studies

Labeling of Local Modes.

Phonon Spectrum.

viii

for Two-Phonon Decay

Page

 

92

103

132

13H

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LIST OF FIGURES

F'gure Page

 

2.1 Interstitial Mass M vs. Relative Local—Mode
Frequency. . . . . . . . . . . . . . . . . . . . 12

3.1 The Evaporation System . . . . . . . . . . . . . 30
u.1 The Spectrophotometer. . . . . . . . . . . . . . 50
u.2 The Grating Equation . . . . . . . . . . . . . . 5H
H.3 Indene Calibration of Modified Spectrophotometer 58
u.u Double-Beam Modification . . . . . . . . . . . . 61
U.5 The Overall Measurement System . . . . . . . . . 63
u.6 The Phase Sensitive Amplifier. . . . . . . . . . 65
5.1 Results of I-R Absorption Measurements in

Germanium. . . . . . . . . . . . . . . . . . . . 69

5.2 The Heating Apparatus. . . . . . . . . . . . . . 71
5.3 Absorption Spectra at Different Temperatures . . 73
5.u Centroid Shifts vs. Temperature. . . . . . . . . 7H
5.5 Line Width vs. Temperature . . . . . . . . . . . 75
5.6 Time vs. Sample Temperature. . . . . . . . . . . 78
5.7 Relative Integrated Absorption Intensity vs.

Temperature. . . . . . . . . . . . . . . . . . . 80

5.8 The Pressure Applicator. . . . . . . . . . . . . 81
5.9 Calibrations of Pressure Gauge (18%) . . . . . . 83
5.10 Line Width vs. Pressure. . . . . . . . . . . . . 8H
7.1 Tunneling Calculations Compared with Experi-

mental Migh and Low Temperature Data . . . . . . 109

7.2 Ratio vs. Local-Mode Frequency . . . . . . . . . 110

ix

 

 

B.1

C.1

D.1

LIST OF FIGURES (continued)
Fraction of Paired Ions (6) vs. Temperature
for Equal Concentrations of Lithium and Boron
of lOlg/cm3 vs. Temperature. . . . . . . . . . 121

Birefringence Measurement Apparatus. . . . . . 123

The 6-Atom Model Used As A Basis for Lithium—
Boron Local-Mode Calculations. . . . . . . . . 129

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CHAPTER 1

INTRODUCTION

The work described in this thesis began with the
snpecific goal of establishing limits on the practicality
(3f increasing the diffusion rate for lithium ions in a
senniconductor matrix. Two questions were fundamental to
'this goal: can some means of selectively heating the
ilithium.ion in the semiconductor lattice be found? Or,
iJi a more direct approach, is an advantage gained if a
jlighter isotope of lithium is used?

Our interest in the motion of the lithium ion in a
senniconductor matrix was initially generated by the role
gilayed by lithium in the fabrication of nuclear detectors.
'This fabrication involves a low-temperature drift of
lgithium ions through silicon or germanium to compensate
ruegatively-charged trap sites. Low drift temperatures
Eire used so as to reduce the precipitation of lithium on
riucleation sites. This drifting process consequently
irequires considerable time and is limited by the low-
‘tenmerature diffusion constant. Therefore, any increase
iJl the diffusion rate of lithium ions may be directly
'translated into a more efficient detector fabrication.

In an effort to answer the question on selectively
lqeating the lithium ion in the semiconductor lattice, we
sset out to study the local—vibrational-modes of lithium

jjl germanium and silicon by means of infrared absorption.

1

2
The introduction of lithium into silicon or germanium
results in approximately equal concentrations of free
electrons and lithium local modes. The free electrons
have a much greater absorption coefficient than the local-
modes and, therefore, mask them. The systems actually
studied were electrically compensated with equal concen-
trations of acceptors and donor ions. The acceptors and
donors generally form pairs which tend to complicate the
local-mode spectrum. Of these systems, lithium and boron
in silicon has been the most extensively studied.l"10
However, relatively little has been done to evaluate the
effects of lattice coupling on these local modes. Lattice
coupling to the local modes is, however, fundamental to the
question of selectively heating the lithium ion in a
semiconductor lattice. The measurement of second harmonics
by Waldner10 does give an indicator of lattice coupling
effects; but, no indication of the lithium-boron pair
dynamics is evident. We have, therefore, set out to
learn about how the lithium ion reacts with its surroundings
through measurements of the centroid shifts and line widths
of the local modes as a function of temperature.

The spectrum of lithium-boron local modes in silicon
has one uncertain feature; that is, the designation of the
local mode at 522 cm_1 as a singly or doubly-degenerate
local mode of lithium in a paired configuration. It is
generally thought to be a doubly-degenerate local mode

3a,3b,9a,9b

of lithium paired with boron. However, Hayes8

 

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8 to determine the degeneracy of this mode.

suggested7’
Since we are primarily interested in local modes due to the
lithium motion, we set out to determine the degeneracy of
the 522 cm"1 mode through the application of uniaxial
stress.

Prior to our infrared absorption studies, our work
involved the more direct approach to the problem of an
increased diffusion rate; we studied the difference in
diffusion constant (or mobility) between 6Li and 7Li
isotopes. The ratio of diffusion constants of 6Li to
7Li had already been measured in silicon at 800°C by

11 . . .
His measurement showed no variation from the

Pell.
classical ratio. This result is not unexpected in view
of the high temperature of measurement. The classical
ratio of diffusion constants is only 1.080, and little
advantage would be gained by drifting detectors with 6Li
instead of 7Li. But for such light ions, the possibility
of tunneling exists which may substantially increase this
ratio at the low drift temperature involved. Evidence
supporting this possibility was available in measurements
of the low-temperature diffusion constant of 7Li in ger-
manium by Sher}2 These measurements showed about a 15%
decrease in activation energy in the range of the low-

temperature measurement; such an effect might be accounted

for by a tunneling mechanism. The parameter of greatest

 

 

 

 

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uncertainty in subsequent diffusion-ratio calculation
turned out to be the lithium local-mode frequency; this
uncertainty further motivated our infrared absorption
studies which followed.

In order to make the account of these experiments
more meaningful, the details of the necessary background
material will first be presented. The sample preparation
and instrumentation are then described, followed by a
description of the local-mode experiments and their
results. These results are then discussed in terms of
theoretical calculations. Finally, the isotopic diffusion

studies are presented and discussed.

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CHAPTER II

THEORETICAL HISTORY

2.1 Lithium and Germanium Diffusion Studies
The lithium atom enters a silicon or germanium lattice
as a singly-ionized donor impurity. Most of the lithium

13

is ionized for temperatures above 500K. The diffusion

constant for natural lithium as a function of temperature

ll’lQ’ll‘lThese studies show that the

has been well studied.
lithium ion must move among interstitial lattice sites to
account for the high mobility observed. Both silicon and
germanium have a basic diamond structure, so that inter-
stitial equilibrium sites can have only a tetrahedral (Td)
or hexagonal symmetry. Various studies, including stress
effects on electron excitations, show that the lithium must
be in a Td-symmetry site.3’ls
The potential well in which the lithium ion (Li+) is
bound in the diamond-structure crystals is the key to under-
standing the motion of lithium in silicon or germanium.
We have considered two methods for studying this potential.
The first method involves a study of tunneling effects
through the well as indicated by the diffusion constant
at low temperatures. The second method involves the
infrared absorption in a transition from the Li+ vibrational

ground state to the first excited state.

Diffusion constants have been described classically by

D(T) = Doexp(Q/kT), 2.1

 

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where k is Boltzmann's constant and T is the temperature
in °K. Q represents the classical activation energy, i.e.
the height of a sinusoidal barrier between equilibrium

sites. D0 is given by the expression16

D0 = Ad2v exp (AS/R), 2.2
where A depends on the unit—cell geometry, and equals 1/8

for silicon and germanium; d is the lattice constant; R

is the gas constant, and AS is the change in entropy per

mole of diffusing ions transferred to the activated tran—
sition state. AS is related to the elastic modulus u and

the activation energy by AS = —Q 5; du/dT; no is the elastic

o
modulus at absolute zero, i.e. "o = ll.2x10-ll

dyne/cm2 and

du/dT = —6.0x107 dyne/cm2 —deg. for germanium. These ideas

were first put forth by Wert and Zener,l7 who assumed that

the interstitial diffusant jumped from one hollow region

in the lattice to an adjacent hollow region. The parameter

of greatest uncertainty in this formulation is the vibrational
frequency of the lithium ion.2 In a localized harmonic approxi-
mation it is given by

v = (Q/2Ma2)l/2 2.

(A)

where 'a' is the distance between adjacent interstitial sites,
and M is the mass of the ion. This uncertainty in v further

motivated our subsequent infrared absorption studies. The

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Li dlfoSlOn constant in Silicon follows this claSSical

expression (2.1) for all temperatures for which it has been

measured;" it is described by D0 = 23x10-1+ cm2/sec and Q =
0.655 ev}2 In germanium, however, a somewhat smaller

activation energy is found at temperatures below 155°C
than at higher temperatures. The classical expression at
higher temperatures is given by D0 = 13x10—u cm2/sec and
Q = 0.H6 ev.ll

We have proposed that tunneling of the lithium through
the activation energy barrier explains the low-temperature
discrepancies of the Li+ diffusion constant in germanium.

If this is indeed the case, the ratio of the diffusion
constants of 6Li to 7Li at low temperatures should be larger
than the 1.080 value predicted by the classical inverse-
square—root mass ratio. We therefore set out to measure
this ratio as a function of temperature. If this effect

had turned out to be large, a time saving in detector
fabrication would result if the drift were done with 6Li
instead of 7Li.

A general diffusion theory,18 beginning with a sinu-
soidal potential and including tunneling through the barrier
at low temperatures, is applicable to this problem. This
sinusoidal potential is approximated by a harmonic potential
well plus a parabolic barrier. The ion jump frequency 1/1

is taken as prOportional to a sum over all N oscillator levels

9
{This includes temperatures down to 2730K.52

 

 

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EN of a Boltzmann distribution times a tunneling probability

T(EN),

o-ap—a

= g exp(—EN/kT)T(EN) 2.u

The resulting diffusion constant is

epreO(B—l/kT)]-exp[(2N-3)eO(B-l/kT)]
l-exp[2eo(B-1/kT)I

 

D(T) = B{eXp(-BQ)

expE-(2N+3)eO/kT]
I—exp(—2sO/kT)

I } 2.5

 

where B = (2eO/kT)DO, e = (2n2(0.35a)/h)°(2M/Q)l/2, so =
hv/2 and h is Planck's constant. Q is the full height of

the sinusoidal potential and (a) is the distance between
interstitial sites. The number of oscillator levels N is
determined from the barrier height by N = (2Q/hv)-l/2. The
first term in brackets in eq. 2.5 represents the contribution
to D due to tunneling; the second term reduces to the clas-
sical expression at high temperature. This expression for
D(T) is compared with the full range of experimental diffusion
constants in germanium; the ratio of diffusion constants for

6

Li to 7Li is then simultaneously compared with measured

ratios.

 

    

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2,2 The Boron Local Mode in Silicon

A local-vibrational-mode is a vibrational state of
an :impurity ion and its immediate surroundings. The ground
vibinational state corresponds to the zero-point motion of
the ion. Crystals with point impurities may have other
vibisational modes, called band modes, with frequencies
belLJW'wM.6 The first studies of local modes in silicon
were: theoretical, and they predicted reasonably well the
locaLL-mode frequency of the substitutional boron ion.19
The sipproximation was made that the silicon-boron force
constxants were the same as the silicon-silicon force

consteants; the results were a local-mode absorption

coefficient a(w) given by

2,2
a(w) = 2"n:e A |x<o)|2 g(w) 2.6

 

 

wherwg N is the number of impurities per unit volume, e is
the (Zliarge of an electron, A is a local field correction
Cfi:[:(r12+2)/3]2, n is the refractive index, c is the velocity
Of light in vacuum, Ix(o)l2 is the vibrational amplitude

Of t}1€2 impurity atom, and g(w) is a normalized line-shape
functZion. This work predicted that a localized-mode did

“Qt Gixist unless the impurity mass were less than 0.925
tbnegs the mass of the host atom. For very light impurities
the lxocalized impurity vibrations are much faster than those
0f tIie host lattice. One may then assume that these impurity
ions; ‘vibrate in a rigid lattice, in which case the isotOpic

Charlgkg should cause a frequency shift determined only by

‘2

’5" N“ 13""—

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10
the square root of the mass ratio. Such boron local modes
have since been observed in silicon,20 with isotopic mass

shifts within 30% of the rigid lattice approximation.

2.3 The Lithium Mode in Silicon

Bellomonte and Pryce did further theoretical work on
the free interstitial lithium ion in a silicon lattice.
Theyr began with the assumption that the lithium ion occupied

a T symmetry site. To calculate restoring forces on the

d
0+ 0 O O 0
Li they conSidered the primary energy contributions to

be frnom.over1ap repulsion due to the exclusion principle,
and ffirom.induced lattice polarization due to the charge
on‘thea lithium ion. The first of these effects stabilizes
the ikon.in the Td site, and the second effect tends to de-
stabiLLize it. The polarization energy falls off as the
fOUPtfli power of the distance from the ion as eXpected.

The cnverlap energy was found to be approximately proportional
to thee density of silicon electrons at the lithium nucleus.
Slat£!r~ wave functions were used to determine the silicon
electnhon density. Bellomonte and Pryce found that the
overliap repulsion exceeds the polarization attraction,
causing them to believe that nearest-neighbor silicon
atOng are displaced outward from the lithium by an amount
Of‘athout 0.1K. Calculations were based on a numerical
Silicflon phonon spectrum, and included lithium-silicon
inter‘actions to second nearest-neighbors. Results were
pre5“311ted in the form of a plot of the impurity mass vs.

frecquancy relative to mm. The essential features are

 

o .: u. never!- was-«n;

 

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reproduced in Fig. 2.1. The calculations for a rigid-
lattice approximation are shown for comparison. These
calculations predicted a critical mass of 12 AMUs above
which no local mode would exist. Comparison of the theory

with eXperimental data on isotopic shift in the 522 cm"1

line gave a critical mass of about 8.1 AMU. The 522 cm—1
line , however, is a Li+ mode perturbed by a paired boron
ion and consequently does not represent the free interstitial
lithium. The fact that no isolated Li+ local mode has yet
been observed, even though samples studied by Spitzer at high
temperature should have a significant number of unpaired
lithium ions, suggests that the critical mass is less than

7 AMU . Hence, the effects of lattice coupling are apparently

considerably different than these calculations predict.

2.1+ Lithium-Boron Complexes

Experimental observations of local modes in silicon have
mOSt Often been made in compensated material in which Li+ and 87
ions are present in equal numbers. The Oppositely charged
ions have a strong affinity for each other, and the Li+ is

0
captured by the boron at the radius of 33 to 39 A;21 as a

22’23 The orientation

I“BSLIZL't:, pairing or complexing occurs.
0f the Li+ - 8‘ pair axis has previously been found to be
along a [111] direction with an equilibrium separation of
2'5 - 2,7 21,11’21’23 The boron remains in a substitutional
lattice site whereas the interstitial lithium is surrounded

by three silicon atoms and the boron ion as nearest-neighbors.

The effect of the Li+ ion on the boron local mode is to perturb

12

(AMU)

 

 

 

FIGURE 2.| Interstitial mass M vs. relative local-
mode frequency in silicon, where (a) is the result of
calculations made by Bellomonte and Pryce and includes
anharmonic terms in the Hamiltonian and (b) is the re—

sult of calculations assuming a rigid-lattice i.e. no
anharmonic terms.

7396

o

028

U

‘
HF
VHO ‘v'

,eak i

R

...15 _

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.Ll

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,v. b

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13
its Td symmetry to C3v axial symmetry. The three-fold
degeneracy of the Td symmetry is therefore split into an
upper-frequency doubly degenerate band, and a lower-frequency
singly-degenerate band. It is expected that the boron ion
would have the same effect on the lithium local mode. However,
only one peak at 522 cm.1 can be attributed to the 7Li motion.
This peak is just above the maximum lattice-phonon frequency
mm, and is thought to be the two-fold-degenerate high-frequency

2” The lower—frequency peak would

perturbed lithium local mode.
then lie in the band of host lattice frequencies.6 The isotopic
frequency shift in the 522 cm.1 line is observed to be approxi-
mately one third of the inverse—square-root mass ratio predicted
in the rigid—lattice approximation. The boron peaks, on the
other hand, have isotopic frequency shifts within 30% of the
rigid lattice approximation. This may indicate a relatively
strong anharmonic coupling of the lithium to the lattice.9a
A Green's function treatment of free interstitial Li+,
substitutional B-,and Li+-B- pairs in silicon has been done
taking into account variations in force constants.7’12
Accurate prediction of the boron local-mode frequencies
required about a 10% decrease in force constants due to
local deformation of the lattice. Calculations were made
for an interstitial impurity (lithium) in a Td symmetry
coupled to its four nearest neighbors by a force constant 8;
these calculations showed that the existence of a 7Li local-
mode required B>23 kerg/cmz. Bellomonte and Pryce's calcu-

lations, however, showed coupling to second nearest-neighbors

to be larger than to nearest-neighbors, but the sum of the

 

uration

,yii. ‘-

:cnstants aI‘OU

‘- O- : 1'
$33.32. w J?
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9-9 H 7‘ ‘7
:ZLEHJOs S, a ’

the boron and
it: 1 between
:crs. in; Inc
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Mel to give

shculd be keni

7'.

once constan‘

rqnw .
......al coordii
c. tne defect

|.-'

19
interactions was in the range 20-30 kergs/cmz. Calculations
by Elliott and Pfeuty7 for boron and lithium in the paired
configuration required considerable modification of force
constants around the pair. The model used considered force
constants 8 between the lithium and its three nearest—silicon—
neighbors, a+B between the lithium and boron, and e between
the boron and silicon pair co-linear with the Li-B axis,
and y between the boron and the three silicon nearest—neigh—
bors. This model was chosen because it was the simplest
model to give good agreement (0.3%) with eXperiments. It
should be kept in mind, however, that second-nearest-neighbor
force constants are important. In this model each of the
normal coordinates of the defect space involves only one
of the defect atoms. Only anharmonicity would result in
terms in the hamiltonian which mix the atomic coordinates.
The lithium line at 522 cm-1 was fitted with a B of 23 kerg/
cm2 for which the isotopic frequency shift agreed quite
well with experiments. The boron local modes and isotopic
frequency shifts were fitted by weakening the force constant
¢ of the Si-B bond. Variations of other force constants
in the basic six atom model above did not give the prOper
combination of intensity and isotopic shift of the paired
boron local modes. Calculations of intensity were found
to agree best with experiments when the lithium charge was

replaced by an effective charge of 0.7e.

 

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15
2.5 Anharmonic Effects on a Td Symmetric Local Mode
Line width and centroid shift with temperature are
the result of anharmonic coupling of the free or paired
ions to the lattice. Theoretical results for complexes
in silicon have not as yet been presented, although
residual line widths at zero °K have been discussed by

Dawber and Elliottlga

for the free boron local mode. They
conclude that the residual line width is "probably" deter—
mined by decay of the local mode into two band modes through
anharmonic forces.

Anharmonic effects on local modes in ionic crystals
have been studied theoretically by Elliott et al.25 These
calculations are for charged impurities in a Td symmetry,
but many similarities exist with the paired lithium and
boron ions in silicon; the pertinent consequences of their
theory are presented here. Their calculations begin with
an expansion of the potential out to fourth order in atomic
displacement divided by interatomic distance. The symmetry
of the Td site is used to determine which expansion coefficients
are zero. Anharmonic thermal effects are the result of inter-
actions between local modes (Operators b, b+) and lattice
modes (Operators a(R)). Possible third-order interactions
are represented by

(b+b+)3

(b+b+)2(a(R) + a+(R))

+

(b+b+) (a(R)+a+(R))(a(R’) + a (E'>>

(a (K) + a+ <E>> (a <E’) + a+ (E’)) (a(R”)+a+(R”)) 2.7

m
h

<t+

.3 .
.0. AI.
.h e + ..
( e

s
can"

0
an

s c

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i

Q“

 

¢.
5

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reina

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2’ u

 

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16
In fourth order the combinations are
(b+b+)u
(b+b+)3 (a(R) + a+(R))
(b+b*>2 (a(R) + a+(R))(a(R’) + a+(R’))
etc., 2.8

where R is the phonon wave vector and branch index.

The resulting third- and fourth—order anharmonic Hamiltonians
applicable to displacements quadratic in the local-mode

coordinates are

:3:
I

+ 1/2 1
3 - Eja'ntém:‘( ’fi%+§’) { z (bx+b;)2i(a(k)+a+(k)) 2.9
xyz

 

+ +, 1/2
H = I C h(m(k);(k )> Z (bx+b;)2(
+ +, MM’QwM MN xyz

(a(R)+a+(R))(a(R’)+a+(R’)). 2.10

The number of unit cells is given by N. The silicon atom

has a mass M, and the impurity atom mass M’; w(R) is the
lattice-mode frequency and Q is the local~mode frequency.

The Spatial coordinates x,y and 2 refer to the local modes.
Perturbation theory is applied to these Hamiltonians to
determine thermal shifts of the local modes and line-
broadening effects due to elastic scattering of band phonons.
In the ionic crystal CaF2 the expansion coefficients c and

.B have been measured to be c = 1021 erg/cml4 andfi = 2x1012

erg/CUP and similar orders of magnitude are expected for

g

I
vyfl"
-Eai-.~

‘
.err

siere

 

1..

S

bets:

0.
n
o‘«
Yu
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an»

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V

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u

l

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l7
homopolar crystals. Elliott at aZ.found that the thermal
shift in local-mode frequency is the result of fourth-
order effects i.e. HQ. Hu was averaged over band modes,

leaving local-mode operators unchanged to give

 

.+
H = Z r fiw(k’ 2 (b b++b+b )<a(k)a+(R)+a+(R)a(R)> 2.11
H . 2 x x x x
+ MMM NQw xvz
k M ’
where terms like a(k) a+(k’) have averaged to zero. Com-

parison of eq. 2.11 with the ordinary oscillator Hamiltonian

given by %fi02(bxb;+b;bx) gave a relative frequency shift of
x

M '12

where n(k) is the phonon number operator. This expression
becomes linear in temperature at high temperature i.e. above
6d/2.

Line width may also be explained in terms of anharmonic
coupling to the band modes. Spontaneous decay of the ex-
cited local mode into several band modes requires that the
band—mode frequencies satisfy the energy—conservation re-
quirement that {n(R) = Q, where the sum includes the energies
of the several band modes created. The local-mode frequencies
of lithium and boron in silicon are between wM and sz’ so
that decay into two band modes through the third-order an-
harmonic potential is likely. This is represented by the
product (b+b+)(a(R)+a+(R)(a(R’)+a+(R’)). The resulting

Hamiltonian given by Elliott at aZ.is

l8

/2

 

, _ ’ h w(R)w(R’) +
H3 7 2+ 2MN 2(fi 2M’Q Z (bx+bx)
k,k’ “M xyz
+ ++ + ++
{a(k)+a (k)}{a(k’)+a (k’)} 2.13

Perturbation theory with this Hamiltonian gives rise to

a two-phononwdecay line width of

 

,2
%.=45 "2; u I 3; w(R)m(R’){[n(R)+l][n(R’)+l]
MM QM wM E,E, N
- n(R)n(R’)}6(w(R)+w(R’)-Q) 2.1a

where v is the local-mode oscillator level, taken as one
for decay from the lowest excited state to the ground
state. At low temperatures % becomes constant, whereas
at high temperatures it varies linearly with temperature.
In the ionic crystals the dominant line-width broadening
was found to be due to elastic scattering of band phonons
in transition Iv,n(R), n(R’)> + Iv,n(R): l, n(E’)¥ 1>
with conservation of energy requiring w(R) = n(R’). The

resulting width is

2

 

2
% = 2.(t - 23 2) (fiiliii%) I if w2(R)n(R)
M 9 NM QMwM E,E,N

[n(R’)+l] 6(w(R)-w(R’)) 2.15

C.
c.
r
.
‘71
.9
'- n
A V

t

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-
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EH nng

l

.iLS 6‘
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pa

 

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,cv‘ ~M n
bx-er-me.

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19
This elastic scattering line width goes to zero as T7, and
at high temperatures it varies as T2.
One of the best experimental means of further vali-
dating the above line width and centroid expression may
be through measurements of line width and centroid of
carbon local modes in silicon. Carbon enters silicon

substitutionally; hence, it has a Td symmetry, and all

basic conditions of the theory are satisfied.

2.6 Experimental Difficulties in the Observation of Local Modes
Observation of the free—interstitial-lithium local mode

in silicon or germanium presents several difficulties. The

most serious of these is the presence of free electrons

associated with the ionized lithium. These free electrons

can satisfy energy and momentum conservation by the absorption

or emission of a lattice phonon simultaneously with the absorp-

tion of a photon.27 The resulting experimental absorption

coefficient per unit lithium concentration in germanium varies

as v-l’g from l.lxlO-16 cm-l/cm3 at 500 cm"1 to 7.0x10"l7

cm-l/cm3 at 800 cm-1. In silicon28 the absorption coefficient

varies as v-2'0 from l.lxlO-16 cm_l/cm3 at 500 cm.1 to 7.0xlO—

cm-l/cm3 at 800 cm-1. If the free interstitial local mode

17

is assumed to have an absorption intensity similar to that
of lithium—boron-pair local modes then a value of approximately

19 cm—l/cm3 would be expected. The free-electron-

10x10-
absorption coefficient would then be about 170 times larger.
These absorption coefficients are additive in the attenuation

factor e-ax, so that making the sample thin enough to transmit

an Optical signal reduces the local-mode absorption strength

 

 

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20

to undetectable levels. There is a temperature dependence
to the free-electron absorption in germanium. At 700 cm“1
the absorption is constant with temperature, whereas at
higher wave numbers it increases with temperature; at
lower wave numbers, it decreases slightly with temperature.
Cooling would therefore be of little benefit until temperatures
below about u0°K where the lithium re-ionizes,l3 but the un—
ionized lithium represents a different problem.

We have considered the possibility of shining light
through the charge—depleted region of a reverse—biased diode
junction where lithium was used as one of the dopants. The
depletion region is so small, however, that the resulting
intensity levels would not be detectable.

The standard method for reducing the free—carrier
absorption is the introduction of an equal concentration
of an acceptor impurity. Ideally this addition would per—
fectly compensate the lattice electrically; in practice,
the compensation is usually not complete, and a residual
free-carrier-absorption background remains. These free
carriers may be either electrons or holes, depending on
the compensation processing. If the remaining free-carriers
v-2.2

are holes in silicon, a wavelength dependence29 of is

~16 1

found, which varies from 9.2x10 cm.l/cm3 at 500 cm- , to

’16 cm-l/cm3 at 1000 cm-1.

2.0xlO
A perfectly-compensated lattice will still have an
absorption background, due primarily to the lattice itself.

Silicon and germanium are covalent or homopolar crystals,

 

 

 

.3: be ‘Vj.SZJéil:
:cie induce S ‘
simultaneousl‘

an electric I?

izgt the t1»
5:23 or dIfI‘E
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5:5;
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21
and have no net dipole moment per unit cell. In these
crystals lattice absorption is a two-phonon process which
can be visualized as follows: A first lattice—vibrational
mode induces charge on the lattice atoms. A second mode
simultaneously causes the charge to vibrate, thus producing
an electric moment which couples to the radiation field.30
The structure of this absorption spectrum then varies accord—
ing to the two phonon density of states, and represents
sums or differences of two or more lattice phonon interactions.
The main absorption peak in silicon, at 610 cm—1, is the two—
phonon summation band of TA and TO phonons; similarly, the
maximum absorption band in germanium at 360 cm.1 is a summation
of TA and TO phonons. The dispersion relations obtained from
neutron Spectrometry31 show these branches to have the highest
density of states. Typical peak lattice absorption coefficients
for both silicon and germanium are about 10 cm—l. In a double-
beam spectrophotometer, the lattice absorption can be sub-
tracted out by placing an equal thickness of undoped crystal
in the reference beam.

The system of lithium in silicon differs from that of
lithium in germanium in two important respects. The Debye
temperature in silicon is 658°K, in germanium 3620K. As a
result, thermal broadening of the local-mode peaks occurs
at much lower temperatures in germanium than in silicon.

The germanium system must therefore be cooled below 2000K

in order to observe the local modes. The two systems also

differ in the diffusion constant of lithium. Hence, in

....¢ ‘

 

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9 0m.

 

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Litnnrn cc“... e.

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.4

:I‘. LIEdCCGDTdLT.

":1 ' :4 .

w‘Ls‘ I Of‘n.‘v‘£
h§ull-\lla \vb“ ‘
”:7 ‘ A

“h,“
‘-"“‘eG \AC;u»/L
‘ I
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“H ”‘38 have

u,
..f ~QMN‘Q.,:
l.‘ ‘
:‘~.}“-
M‘ '
PK
C. ‘ _
c. . *
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\
L'.‘ 1
.HQ 3*‘I
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sx‘ve ~‘
1)‘
h“ I
13“.”) _
‘ ~‘I .r‘n '_
~V.|
‘
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‘3
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n“~\-‘ ‘ r
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22
the rate of precipitation of lithium from super-saturated
germanium is faster than that from super-saturated silicon,
by approximately the inverse ratio of diffusion constants.
The precipitation of lithium in germanium continues at
temperatures down to room temperature; consequently,
lithium compensated germanium must be maintained at a low
temperature until all measurements on the sample are com-
plete. The precipitation of lithium would otherwise leave
large concentrations of uncompensated acceptors, and hence
an unacceptable free-carrier background absorption. More
will be said about the precipitation process later.

The two systems are similar in several important ways.
Lithium occurs in both silicon and germanium as a singly-
ionized donor in an interstitial lattice site. Both
lattices have the basic diamond structure with lattice
constants of 5.H3A for Si and 5.663 for Ge. Furthermore,
the outer electronic shells of silicon and germanium both
contain two 5 electrons and two p electrons, hence similar
repulsive forces on the lithium ion due to the exclusion
principle may be expected. Many aspects of the local-mode
studies on silicon can therefore be generalized to germanium

through appropriate frequency or temperature shifts.

2.7 Complexing of Donors with Acceptor Ions

The introduction of acceptor compensating ions into
the lattice gives rise to the formation of positive and
negative ion pairs or complexes. Calculations of the rela-
tive percent of complexed ions, for any fixed temperature

and substitutional acceptor concentration, have been done

' :
trancn oi

antor‘ ions,

Luv-i“ "

QRR
v- v

A

.-

2(T), a

 

 

 

T).

H n
M u(

.2
I‘A

,f"
‘m

CPS an

“‘5!
"J"‘

23
. 22 , . .
by Reiss et a1, They assumed that paired and unpaired
ions were in dynamic equilibrium hence, a law of mass action
could be written down. Thus, if P represents the concen—
tration of paired ions, NA—P, the concentration of unpaired

acceptor ions, and N -P the concentration of unpaired donor

D

ions, the mass action expression is

P -
(NA—P7 (ND—P7

 

9(T) , 2.17

where 0(T), an equilibrium constant independent of concen-

tration, is given by

2
e

n(T) = Mfr? exp (1??) dr . 2.18

E I‘

These calculations included only the effect of coulomb

interaction between ions. The fraction of paired donors
(P/ND) was then a function of the acceptor concentrations
and n(T). In the case of nearly equal concentrations of

donors and acceptors, (P/N) is given by

) . 2.19

 

Appendix B contains a somewhat modified form of this expression
evaluated as a function of temperature for equal concentrations
of lithium and boron in silicon. As would be expected, the
fraction of paired ions increases rapidly with impurity con-

centrations (N) for a given temperature. For a fixed N,

p
5
A

110 che

V
_.o- .,.

II'

‘u

wfipnsat

“ravat

0!] CC)“:
D at t'

.

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in

.19
.i

 

 

'7

Ike‘

§ 0
|
\

Cu
l~ £

2n
the fraction paired decreases with temperature. The possi—
bility of observing the unassociated Li+ or B- local mode
in a compensated system must subsequently be considered.
That is, is it possible to quench a compensated sample from
a high temperature fast enough to prevent pair formation?
Reiss et aL22 have made calculations of the average lifetime
of an unpaired ion, and denoted it as the relaxation time T

for ion pairing. They found T to be closely approximated

 

P

by

T = —Ekg(§-M) 2.20

p une N D (T)

0
where
M - JL Lip? ( e2 )dr 2 21
- u" exp EETF’ . .
a

The distance of closest approach (a) between Oppositely-
0
charged ions can be taken as 2.6A. The Debye length L is

1/2

given by L = (ekT/8ne2N) , and DO(T) is the lithium

diffusion constant. For impurity concentrations of N =

lOlglcm3

, M is small compared to N so that Tp varies as
T/NDO(T). To freeze-in a sizable fraction of unpaired
ions, i.e. 10% at ”00°K, it would be necessary to quench
the sample in a time small compared with the relaxation
time I at that temperature. At MOO°K, Tp has a value of
1.3x10-3 sec.; at room temperature Tp is 0.76 seconds.
Hence, a quench fast enough to freeze—in unpaired ions

is unlikely. At room temperature about 1% of the ions

remain unpaired. As a result, the free-boron local mode

a!

[as—-

I ~. an] 1‘.

 

 

u-

25

has been observed in compensated silicon.u’5’ga’gb’10

However, a free-lithium local mode has not been observed,

and it is thought that it must lie in the lattice continum.5’6’9a

 

.F““ “1' 4-; ran
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~JI"

CHAPTER III

SAMPLE PREPARATION

The infrared study of local modes in silicon and
germanium required a sample uniformly doped with equal
amounts of lithium and an appropriate acceptor impurity.
Concentrations were required to be equal within 1% in
order to eliminate the free—carrier absorption background.
Impurity concentration and sample thickness were then of
primary importance in determining the observed local-

mode absorption intensity.

3.1 Compensation Methods

Three basic methods for producing high-resistivity
silicon doped heavily with lithium donors have been reported.
These processes are all similar in that they introduce trap—
ping sites for the donated lithium electron. The earliest
method involved a radiation of the sample with nucleons.32
The silicon would first be diffused with lithium at high
temperatures to produce the desired concentration.* The
samples were then irradiated with nucleons to produce
lattice defects. The defect site then captures one of
the lithium donor electrons. In this manner the silicon
was compensated to give resistivities near that of pure
silicon. Spitzer5 later described a similar compensation

technique using electron irradiation in the neighborhood

of l Mev. This method is thought to leave a larger number

 

V—

::For solubility of Li in Si see E. M. Pell, J. Phys. Chem.
Solids, 3° 77 (1957)

26

 

 

 

‘ +
5?. accelera LC:

A k .-
reputed by S.

w-

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'4
l

aCllEVEd by a
El'v’e Uniform}

2339, of the S'

t: be heaViep

27

of unpaired ions than techniques based on a final thermal
diffusion of lithium to achieve compensation. This is
because the nucleons, e.g. neutrons, will leave defects
randomly distributed throughout the silicon and not neces-
sarily near a lithium ion. The irradiation method, on the
other hand, is not so convenient because of the need for
an accelerator. A second method of compensation has been

5’33 It consists of growing the silicon

reported by Spitzer.
with a substitutional donor and a substitutional acceptor

such that the donor concentration is about 10% less than

the acceptor concentration. The final compensation is then
achieved by a diffusion of lithium. This is reported to

give uniformly—compensated samples; furthermore, no local

mode of the substitutional donor will occur if it is chosen

to be heavier than the host silicon atom.7 Another early
methodau starts with silicon uniformly dOped with an acceptor
such as boron. Lithium is then diffused in excess throughout
the material. The final compensation is produced by annealing
the sample, thereby causing the excess lithium to precipitate.
More will be said about the precipitation process later.

The diffusion-precipitation technique has many problems
associated with it, but requires less apparatus than the

other techniques. It does not cause nearly as much damage

to the lattice as the irradiation methods and involves
generally fewer uncertainties. The double-doping during
crystal growth involved technologies not available to us;

as a result, the diffusion-precipitation compensation tech—

nique was chosen. These techniques apply equally as well

for the compensation of germanium.

q

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28

3.2 Material, Mechanics and Preparation

The initial material* was single—crystal silicon grown
in the [111] direction and dOped with 1019 atoms of boron
per cm3. A concentration of 1019 was chosen because it was
reported?” to produce local—mode coefficients on the order
of 10 cm"1 which could be readily observed. Higher concen-
trations would produce larger absorption coefficients, but
were not as readily available. The three symmetry axes of
the [111] direction were evident along the sides of the
crystal, and the orientation could be readily ascertained.
X-ray diffraction photographs were taken in order to con-
firm the orientation, and show that the material was single—
crystal. The sample was then sliced in the desired orientation
on a diamond saw. The resulting wafers were typically 0.7
to 1.5 mm thick. The infrared radiation is attenuated while

(1X

traveling through the silicon by e- , where a is the absorption

coefficient and x is the sample thickness. The maximum atten-

uation of the signalnu that we could easily work with and also
the maximum sample thickness that didn't require excessive
preparation time, techniques, and cost was 1.5 mm. The thinner

samples, although easier to make, gave smaller signal variations

a,- 0'0 0’.
lb 0‘ l!

between the local-mode peaks and the background.

 

“Purchased from Materials Research Corporation.

:': 2': .
Additional signal losses were caused by reflection from
the surfaces and the small cross-sectional area of the
sample compared with the spectrometer beam.

I'D Q'. J.
0 a.

"This is primarily due to thermal surface effects dis-
cussed later.

. .
r0 v" IF

I
.6“ .2- .‘

-~
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.th'!

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ed

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.1

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ustra'

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1.

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33

it w

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A
UP

Cc

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d haVQ t
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h

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h“?

““39 C111

«911
t“

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29
3.3 The Lithium Deposition

7Lithium* was deposited on the boron-doped silicon by
an evaporation process. The silicon was first lapped with
#600-grit carborundum, and then with aluminum oxide. It
was finally cleaned in an ultrasonic cleaner containing
methal alcohol just prior to the lithium deposition. The
lithium metal was placed in a tantalum boat, which was then
mounted in a bell jar. A tantalum boat was chosen because
it wouldn't react rapidly with the lithium at the vaporization
temperature (1609°K). A grounding path to the sample was
provided to prevent a possible surface charge build up due
to deposition of thermally-ionized lithium atoms. Figure
3.1 illustrates the evaporation system. The bell jar was
pumped down to two microns or below before the lithium was
heated. Vaporization of the lithium was achieved by adjust-
ing a current through the tantalum boat until the boat began
to glow a faint red. If the boat was allowed to get too hot,
or if it was placed too close to the sample, the deposited
lithium layer was likely to boil off, and the sample
would have to be reprocessed. Samples were typically kept
2.25" or more from the lithium boat. We generally tried
to get a thick lithium deposition, i.e. greater than 10
mils, during the evaporation. The heating current was then
turned off, and argon was let into the bell jar. When
atmospheric pressure was reached, the sample was quickly
removed from the vacuum chamber and placed in xylene. The

amount of time that the evaporated sample spent in the

 

hThe lithium isotopes were 99.99% pure obtained from Ortec.

 

 

A\

 

Vacuum Belljar

thhlum In Tantalum Bani

rung! rE‘LIFI
[ LE" \ ‘5‘

Sample

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     
  

(3mg) Meter

Current
Source.»

 

 

FIGURE 3.| Evaporation system.

farm-I "

 

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surface when 1

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(r “'1‘: ' ‘
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The over.
and about 3 .5
ie removed f0

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rule prot‘vlCe:

..
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_;;vf\ r“
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6
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9 (
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31
atmosphere was kept to a minimum, because the lithium

O CO

2’ 2’ 2

and especially water vapor in the air. Mineral oil was

surface layer would react quickly with the N

also tried as protection for the lithium surface, but it
was found that cracks and peeling occured in the lithium
surface when the sample was placed in the oven. If the
mineral oil was burned off slowly at about 260°C, the sur-
face remained quite good even though this temperature is

above the melting point of lithium metal.

3.H Diffusion and Lithium Solubility

The oven consisted of a quartz tube about 6 feet long
and about 3.5" inner diameter. A joint allowed one end to
be removed for insertion of samples. Gas-line connections
were provided at both ends. At the high diffusion temperatures
involved, lithium reacts readily with most substances except
inert gases, consequently an argon atmosphere was provided
during the diffusion. Three independent heating coils in
the oven provide adjacent temperature zones of one foot in
length and thermally regulated to :0.1°C. Insulation and
electrical controls make up most of the main unit. This
oven tube and the quartz sample mount were preheated to
375°C. The mounting pad was then withdrawn into the re-
movable end of the furnace tube and the tube was Opened.
An argon flow of 25 kcm3/min was maintained over the pad.
Then, the evaporated sample was removed from the xylene
and placed on the hot pad. The furnace tube was quickly

closed up again, and after allowing two minutes to vent

 

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32

the atmOSpheric gases from the furnace, the argon flow was
reduced to 10 kcm3/min. If the sample was not initially
heated from the quartz pad, it would have to be heated very
slowly: otherwise, the lithium surface has a tendency to
heat faster than the bulk silicon, and the lithium film
is likely to boil, thus leaving only small beads of lithium
on the silicon. The sample was left at 375°C for ten min-
utes to form the initial lithium-silicon pre-alloy.3H
The lithium was then firmly attached to the silicon, and
the oven temperature was then raised to its diffusion
temperature of 550°C. The argon flow rate was maintained
at 10 kcm3/min. throughout the diffusion.

Several factors had to be considered in the selection
of the diffusion temperature. Perhaps the most important
factor was the concentration of lithium desired in the sample.
Clearly, the final lithium concentration must closely approxi-
mate the boron concentration. A compensation to within 1%
of equal concentration would reduce the number of free carriers
to a manageable level of 1017/cm3. It is conceivable that
the diffusion temperature could be chosen such that the
solubility of lithium in the doped silicon was equal to

22’35 that an

the boron concentration. But, it is known
indeterminate amount of lithium will precipitate upon

cooling, leaving the sample strongly p-type. Thus, it was
decided to use the technique of diffusing in an excess of

lithium and annealing out the excess at a lower temperature

following the diffusion.

o
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. 33

It has been shown22 that an acceptor impurity in silicon
or germanium has a pronounced effect on the solubility of
lithium. This effect may be understood by considering the
analogy with impurity ions in an aqueous solution repre-
sented by the silicon. It can be seen in the following
chemical-equilibrium expression that the presence of the

+ . .
free hole "h " from the boron impurity greatly enhances the

solubility of lithium;

B 2::8 + h
+ + -
Li ext.‘—-=7—Li‘-—:_-; Li + e

J
h+e—

At the dOping levels involved, the dilute-solution condition
is valid, and mass-action relationships can be written for
each reversible reaction. The mass—action law states that
in a dilute solution, at equilibrium, the product of the
concentrations of the unassociated ions divided by the con-
centration of initial associated reactant is equal to a
constant. Such relationships can be written for the boron
acceptors, the lithium donors and the associated electron-
holes e-h+. Calculations based on these equations, including
also the charge—neutrality condition, predict quite well the
solubility of lithium in acceptor-doped silicon.22 These
results show it to be a simple matter to match lithium and
boron concentrations, but in order to reach uniform lithium

19

concentrations significantly above the 10 /cm3 boron level,

O a"! a V

;:::“C
C-..J-7'

allc'

v.

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m

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O

1' ‘

is

 

 

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Q4.

an
a temperature near the eutectic point of the lithium-silicon
system is indicated. At this temperature, lithium would
diffuse away from a high concentration silicon-lithium sur-
face-layer alloy. The temperature is high enough to allow
sufficient lithium to remain in the silicon following pre-
cipitation caused by cooling to room temperature. However,
it would not leave the very large precipitation sites and
resulting local strain of a much higher diffusion temperature.
The eutectic point is reported2H to occur at 590 : 10°C and
a lithium concentration of 58 i 5%.

The total diffusion time also had to be considered in
selecting the diffusion temperature. The diffusion time td
may be approximated by td = d2/HD(T), where "d" is the dif-
fusion depth and D(T) is the temperature—dependent diffusion
constant. D(T) has been measured at high temperatures,“l
and is found to closely approximate the temperature dependence

given by the classical model D(T) : Doe-Q/kT

, where D0 is
23x10-” cmz-sec, and the classical activation energy Q is
0.65 ev.2 Unfortunately, the diffusion does not take place

3a’3b This is partly due to non-uniform wetting

uniformly.
of the lithium to the silicon during the pre-alloying phase,
and partly due to oxygen impurities33 in the original silicon
material. Our own experiments show that 5td gives a reasonably
uniform lithium distribution in the material that we used.
During this time (Std), either a continuous and rather expen-

sive argon flow must be maintained, or else the sample must

be placed in a sealed argon atmosphere. The sealed vessel

_.ach

"an.“

8"

,
‘o-H +“i
‘~.‘ b_,...

w

+.»
a;
up;

uated.

O'rzc
v ' ab

‘7"

A

out of its

512053.78
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.1

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3‘. the

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35

approach is reasonable, but somewhat difficult. It would
be necessary to prepare a quartz vessel and a mount made
of pure silicon.* If the sample were allowed to make con—
tact with the quartz vessel, the quartz would tend to draw
lithium from the sample at the point of contact. A sample
coated with lithium and protected from the atmosphere with
xylene could be mounted in a quartz vessel which is then
evacuated. After allowing sufficient time for degasing, 1/2
atmosphere of argon would enter the tube. If the tube were
left evacuated, the diffusion would be very difficult because
of the evaporation of lithium.2u The tube would then be
sealed, and care taken not to overheat the sample or knock
it out of its mount. The whole quartz vessel could then
be placed in the diffusion oven. If required, quenching
of the sample, though difficult, could be done. Of course,
any errors in the sealing process, particularly gas leaks,
would not show up until after the diffusion. This process
was found to be troublesome, and as expensive as maintaining
a continuous argon flow. Hence, a higher diffusion temperature
and a subsequently shorter td is obviously advantageous in
terms of cost.

High diffusion temperatures, above 600°C, introduce
some new problems, however. Evaporation of the lithium21+
from the sample becomes serious, as does the increased

reaction rate of lithium with impurities in the argon. The

precipitation of lithium during cooling of the sample, is

 

x“ u

"SiC is also a good mounting material.3

 

“fix-hr 5 f-"1

‘ . 1.\.lf‘

 

 

A '5'..r-
fit a dl‘fv‘:

fl

4

A : Ir V
Tate O- 13 Act: ’/
1231's was reculf

;.-1 ' ‘
5:531:85. Dv eve.”

#7") an; d:::.
‘-:V., 4-H Ln.-

. I- ‘g
.e..per~at.1re, th:

 

36

u . .
also much more severe,2 ’3u’35 so that it is not at all cer-

tain that sufficient lithium remains dissolved in the sample.
Furthermore, the presence of a large number of precipitation
sites in the sample could cloud the interpretation of the
data.

At a diffusion temperature of 550°C and an argon flow
Pate of 10 kcm3/min., we found that of the order of six
hours was required for the surface layer of lithium to be

depleted by evaporation, reaction with impurities in the

argon, and diffusion into the sample. At this time and

temperature, the lithium could potentially diffuse through

a one-ndllimeter-thick sample. The sample was quenched by

dPOppiJig it into room—temperature mineral oil. It was then
Preparwad for another lithium deposition, by a coarse lapping

with #tSOO—grit carborundum to remove about 2-to-3 mils from

the ditffusion surface. This step was taken because the sili-

con inunediately below the evaporated lithium layer came out
Of the diffusion as p-type material for a depth of 1- to—2
mils. 'Ihis is thought to be the result of diffusion of
Vacancies into the silicon, which combine with the lithium.
Best rwasults in the final lithium distribution were obtained
when tlie first, second, and all other evaporations were made
on the same surface.
A£3 mentioned earlier, most materials in contact with

the Séunple in the diffusion oven tend to draw the lithium
from tflie sample; furthermore, many materials will diffuse

into tflie silicon. We also found, and it was reported by

 

..Ja'»l"-T’.' '7" '

_. '44---

 

 

 

~p‘

5.1 w

, . fl,
.tn‘u- \4
....A- AO‘.

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‘u

:1- were fou

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‘q"
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§ 0 1 .
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“‘Ou‘t *L
'Q
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3. of “"n
»,.‘,
":3 3:.

37

others,5 that any free surface of the sample acted as a
sink, or zero boundary condition, to the diffusion of
lithium. Out-diffusion could occur as fast as in-diffusion,
leaving unacceptably large lithium concentration gradients
throughout the sample. Use of a piece of pure or lithium-
doped silicon turned out to be the simplest solution. If
good contact was made between the sample and pure silicon,
the sample came out sufficiently uniform in lithium con-
centration. Contact surfaces flat to within about one
mil were found to be sufficient to remove the apparent zero
boundary condition at the sample surface. The somewhat
lower solubility of lithium in the pure silicon tended to
prevent considerable lithium losses to the pure silicon
mount. Another more expensive approach would have been to
simply make the sample four to six times thicker than de-
sired; then, when the diffusion process was complete, a
piece of the desired thickness would be sawed off.

The samples were quenched upon removal from the oven
by first being moved to a cool--about 350°C-—section of
the furnace tube for 15-30 seconds. They were then drOpped
immediately into room-temperature mineral oil. Precipi-
tation of lithium in silicon cooled from 550°C is not severe,
but the probability of ending up with strongly n—type material
after three to five diffusions would be considerably reduced
without the quenching. At the diffusion temperature of 550°C,
most of the lithium and boronis unassociated.22 Some thought

was given to the possibility of freezing in the unassociated

 

 

I

u v!“

0' ' ut I! ‘
oh I . t

' l-H' ‘ I

.Appnfj relax
;:‘.VAA~A .

T”the? cor:

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qui a
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38

lithium ions by rapid quenching. However, our quenching
Cprocess turned out to be too slow compared with the milli-
second relaxation times22 of the pair-formation process.
Further complications would have been introduced by the
necessity of annealing out the excess lithium. As a
result of the short pair-formation relaxation times, the
quenching can be safely assumed to have had little effect
on the ratio of paired to unpaired ions. This was later
confirmed by observation of relative absorption intensities
in the local-mode spectrum.

Much of the germanium sample preparation was exactly
the same as the silicon sample preparation just described.
Because of lithium precipitation and lack of solubility, we

were not able to compensate germanium dOped to 2x1018 Ga/cm3.

l7

Concentrations of 2.5-3x10 Ga/cm3 germanium could be com-

pensated with a lithium diffusion, but the subsequent local-
mode absorption coefficient would be less than 3 cm.1 and
difficult to detect. Samples were compensated by lithium
diffusion at u20-u60°C with total diffusion times of 1.25-
l.5 hours. The relatively short diffusion time was possible

because a signal-maximization technique (see Appendix A)

showed that sample thickness should be typically 100—600 u.

3.5 The Hot-Point Probe
A check of carrier type was necessary after the diffusion
and also during the subsequent precipitation process. This

check was performed with a simple hot-point probe. The probe

 

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39
was a small pencil-type soldering iron powered by a variac
set to 6U% of full output. The sample was placed on an
etched metal plate, and the hot probe was placed on the
surface of interest. The voltage between the probe and
the metal plate was measured with a Keithley model 155
null-detector microvoltmeter with a minimum DC input im-
pedance of 106 ohms. A positive voltage reading, relative
to the etched metal plate, indicated an n-type surface
under the probe, and a negative voltage indicated a p-type
surface. This measurement was found to be most sensitive
for lightly-doped or nearly compensated semiconductors;
it was, therefore, especially useful for our purposes.
The magnitude of the thermal voltage was found to depend
strongly on the contact pressure. As a result, a mechanical—
probe arm was constructed which allowed equal probe pressure

for each measurement.

3.6 Precipitation Kinetics

The kinetics of the precipitation process have been
reviewed for lithium in silicon, and for lithium and copper
in germanium.35 Only some of the highlights will be mentioned
here. The precipitation kinetics for lithium in silicon is
essentially the same as for lithium in germanium; the primary
differences are in the rate at which precipitation occurs at
any given temperature. Our main purpose was simply to remove
all excess free carriers associated with the lithium ions in

the silicon. It turned out, however, that one of our principal

 

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spectroscopic background problems could be attributed to
a precipitation effect at the surface of the sample.

The precipitation of lithium in silicon or germanium
is a heterogeneous process in which the nucleation sites
are lattice defects and lattice impurities. When lithium
is diffused into silicon or germanium, and then quenched
rapidly, the system is very much analogous to a super-
saturated aqueous solution. The lithium tends to diffuse
toward the nucleation sites and congregate about them.
It is obvious that such a process is diffusion-limited, i.e.
the precipitation time is primarily determined by the
length of time that it takes for the bulk lithium to diffuse
onto a nucleation site. Equally important in the precipi-
tation process is the density of nucleation sites; this
density may vary several orders of magnitude from one crystal
to another. For crystals refined by the same process, the
density of nucleation sites is approximately of the same
order of magnitude. In the case of an equal number of
nucleation sites in samples of silicon and germanium, the
precipitation rates can be scaled as the ratio of diffusion
constants. Furthermore, the rapid increase in lithium pre—
cipitation with diffusion temperature can be attributed to
the increase in the diffusion constant with temperature.
The precipitation of lithium continues until the solution
is no longer supersaturated. That is, precipitation stOps
when the density of lithium in the solution becomes equal

to the solubility of lithium in silicon at room temperature,

 

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or when the lithium concentration becomes equal to or slight-
ly less than the acceptor (boron) concentration.

The principal nucleation site in silicon is thought to
be a substitutional oxygen site.35 Oxygen is one of the
most abundant impurities in silicon, and also one of the
most difficult to remove. It typically exists in the sili-
con lattice at interstitial sites. These sites will readily
capture a lithium ion to form Li+O complexes, but these are
not the nucleation sites for the precipitation of lithium.
The substitutional oxygen (08) is formed by the reversible
reaction of an interstitial oxygen (Oi) with a lattice

vacancy (V) as follows:

[Oi] + [V] <=> [OS] . 3.1

The corresponding equilibrium mass—action relation can be

written as

[O ] N
._§____ : exp (ASb/k) exp (—AHb/kT) = K = equilibrium 3.2
[OiJEV] constant

where Ash is the change in entrOpy of the oxygen, AHb is the
binding enthalpy of the vacancy, and N is the number of
lattice sites/cm3. The concentration of vacancies has been
given by Logan37 for the condition of thermal equilibrium
to be

[V] = 1.3x102” exp(-2.0/kT) cm'3. 3.3

 

 

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Then, by assuming that the number of nucleation sites N5

is equal to [08], it is seen that
NS = 39 [O] exp(ASb/k) exp(-(Hb+2.0)/kT) cm.3 3.u

Interstitial lithium then tends to congregate in a spheroid
about the substitutional oxygen site with concentrations
near ten atomic percent of lithium.

Other shapes are possible for the precipitation center
and the rate at which precipitation occurs can be shown to
depend on the shape of these sites. For the spheroidal
precipitation particles, the kinetics of precipitation is

approximated by

C-Cf n C-Cf
E—T: exp (“t/T) n = 3/2 for m- > 0 5 ,
o f o f
o-cf
n - l for fit— < 0.5 , 3.5
o f

where CO is the high-temperature (diffusion-temperature)
concentration of lithium, Cf is the final annealing-tempera—
ture concentration of lithium, and O is the average concen-
tration of lithium at time t. In the case of precipitation
particles having the shape of long rods growing only radially,
or disk shapes only growing thicker, the precipitation kin—
etics follows a simple exponential decay with n = 1. In any
case, all precipitation kinetics follow a simple exponential

decay during the latter phase of the process. Measured

 

 

 

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precipitation rates35 and optical observations35 of these
sites in silicon and germanium confirm that most of the
lithium precipitation occurs on spherical sites. Further-
more, the rate constant 1 governing the latter phase of
precipitation has been measured to be inversely proportional
to the 2/3 power of oxygen concentration. This is consistent
with the number of precipitation sites being directly pro-
portional to the oxygen content, and independent of the
initial high-temperature lithium concentration.

An additional but relatively minor precipitation
mechanism occurs in silicon but not in germanium. The
introduction of lithium into silicon causes a high density
of stacking faults which are not found in the original
silicon. Photographic evidence exist83§.indicating the
precipitation of lithium onto these faults. These stack-
ing faults, although not important in precipitation kinetics,
may reduce considerably the stress which can be applied to
such a sample without causing fracturing.

After the lithium diffusion, our samples were usually
uniformly n-type with low resistivity; they then required
annealing to precipitate the excess lithium. The resistivity
as a function of time was used initially to monitor the
precipitation process. This resistivity measurement did
not require an accurate measurement of resistivity, since
the detection of a maximum of resistivity was sufficient.
Hence, an ohmmeter was used to indicate relative changes in

resistivity. The n-type sample was initially prepared for

Q N. .Y. r . V a ‘1 .
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the precipitation by lapping and cleaning in methanol.
It was then placed in point-contact pressure clamps and
placed in 162°C mineral oil. At this temperature the
diffusion constant for lithium in silicon is about equal
to the diffusion constant of lithium in germanium at BHOC.
Since lithium was known to take several hours to precipi-
tate in germanium at 3u°C, it was expected that resistance
variations over five-minute intervals would be small. We
observed, however, that within the first minute after im-
mersion into the hot mineral oil, the resistance increased
from 289 ohms at room temperature to 890 ohms; it then de-
creased steadily to 7M6 ohms at four minutes into the ex-
periment. The resistance then increased quite rapidly to
1,100 ohms at 6.5 minutes, where it remained relatively
constant. The resistance did not change significantly
upon cooling to room temperature. Charge-carrier type
was then measured with the hot-point probe, and it was
found to be p-type. This effect was not expected to be a
bulk effect so the surfaces of the sample were lapped to
remove approximately one mil and the carrier type was
checked again. The bulk of the sample was still n-type
without significant change in the resistivity. It is likely
that the lithium was congregating on the high surface con-
centration of nucleation sites. The effective density of
lithium near the surface would then be considerably reduced.
This results in a lithium concentration gradient, hence

the out-diffusion of lithium. The depth of this surface

 

 

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effect would then be determined by the lithium diffusion
constant at the annealing temperature. It would be inter—
esting to compare this effect in a similar sample which
had been etched previous to annealing. This should reduce
the number of defect nucleation sites on the surface which,
in turn, should reduce the rate of out-diffusion.

The resistivity of the bulk material during the pre—
cipitation phase was measured in a multistep process. The
n-type sample would first be heated for five to ten minutes
in mineral oil at 200°C. It was then cooled, cleaned and
lapped with #BOO-grit carborundum to remove surface effects.
The resistivity was then measured with a four-point probe
and recorded. Finally, the charge—carrier type was measured
with the hot-point probe. This entire process was repeated
until the resistivity became very large, or until the carrier
type began to change.

A four-point probe was now needed, in order to elimin-
ate differences in contact potential and surface effects
from one measurement to the next. A switch was provided on
the four-point probe to reverse the polarity of the current
source on the two outer probes. If the voltage reading had
the same magnitude after switching the current source polarity,
then contact potentials were not a problem. Further checks
on surface effects were provided by varying the source current.
Since the resistivity is independent of current, the measured
value should not change with current unless surface problems

occur. For the actual determination of resistivity, the

 

 

 

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approximation for a thin infinite plane was used. This is
reasonably accurate for measurement at more than 5d from
the sample edges, where d is the spacing between probes.
The resistivity p, in the case of all edges being far from
the point of measurement, is given by p = 2ndV/I. For our
thin samples, a correction factor based on the ratio of
sample thickness w to probe spacing d is needed. The
four-point probe used was a Dumas model G-25-C with a point
spacing of .025" and loaded for 25 grams of contact force.
For samples of one millimeter thickness, w/d is equal to

38 of 0.85, assuming non-

1.57, giving a correction factor
conducting surfaces.

Sample resistivities were found to increase monoton-
ically, eventually becoming too high to be measured on our
simple four—point probe system. Since we could make reason-
able measurements of 0 up to 100 ohm-cm, the compensation
for the best samples was comparable within one part in 105
or possibly better. Ohmmeter checks showed maximum resis-
tance to be in the neighborhood of one megohm. The total
precipitation time at 200°C generally was 15 to 30 minutes.
One of the best compensated samples was used in our experi-
mental determination of temperature dependences of line-
width and centroid. Unfortunately, not all samples were
this good. Sometimes a sample would begin to turn p-type
before reaching a very high value of p. This change is
thought to be the result of non-uniformities in the lithium

diffusion. If the compensation was better than one part in

100, these samples were still useful.

.1

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The primary spectral background in the thermal studies
experiment was an effect similar to the surface precipitation
phenomenon described earlier. As the temperature was raised,
lithium diffused toward the surfaces, leaving behind excess
"holes" associated with the uncompensated boron. Since
this effect was limited to a thin surface layer, the spectral
signal was not totally blocked out, but instead was modulated
with a free carrier type of absorption background. This
background could, however, be readily subtracted out.

The germanium samples were precipitated at room temp-
erature for convenience, hence several hours were required.
To prevent the precipitation process from continuing past
compensation, it was necessary to cool the sample immed-
iately to -60°C or lower. Consequently, precipitation in
the germanium was monitored by measuring the intensity of
the spectrophotometer signal. The sample was mounted on
a dewar cold finger in the spectrometer; when the intensity
stOpped increasing with time, the dewar was filled with
liquid nitrogen. Samples, when not in use, were stored at

-90°C to prevent further precipitation.

 

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CHAPTER IV

IN STRUMEI‘ITATION

H.l The SpectrOphotometer

Much of the initial exploratory work was done on a
modified Perkin Elmer model 1126 grating spectrOphotometer.
The original 1120 spectrometer consisted of three distinct
units; the source section, the fore—prism, and a double-
pass grating monochromator. It was designed primarily for
high-resolution work over a narrow wavelength region. For
the narrow spectral—scanning ranges, the optical background
variations were small, hence the single beam, without back—
ground cancellation, was sufficient. However, the local-
mode spectrum of interest covers a much wider spectral
range and, furthermore, does not require the very high
resolution provided. It was apparent, then, that some

major modifications were in order.

u.l.l The Fore-Prism

The fore-prism was intended to act as a sharply-tuned
band-pass filter with a pass band on the order of two microns.
Its function was to eliminate higher orders passed by the
grating. This was done by passing the signal back and forth
through a prism of KBr. The angle at which the signal
struck the prism could be varied by means of an external

drum; in this manner the central band-pass frequency could

H8

 

 

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be varied. To avoid a large background effect from the
fore—prism section, it would be necessary to change the
fore—prism drum setting for nearly each micron scanned.
Instead, the fore-prism section was removed and replaced
with a set of three filtersfi covering the ranges 10.2 -
18u, lu.3 - 2H.3p, and 2u.3 - Now. The filters were con-
structed by deposition of thin films of selected filter
materials;39 they were consequently degradable by prolonged
exposure to atmospheric water vapor. Most of the silicon
local-mode spectrum could be covered with the lu- to-2up
filter. The remainder of the spectrum was covered by the
10.2- to- 18u filter. The 2u-to—u0u filter allowed us to
look for similar local modes in germanium. An additional
benefit from the removal of the fore-prism section was a
two-to three-fold gain in signal level. The increase was
due to removal of the KBr prism, a decrease in the optical

path length, and to lower atmospheric absorption and scatter-

ing.

u.1.2 The Source
The source section, shown in Figure n.1, includes
simply a black-body radiation source, viz., a Globar, a

flat mirror m and a spherical mirror m2. The Globar

1’

source is heated to about 1600°K by an electrical power

of 280 watts. The resulting radiated power is given by

Planck's black-body radiation formula,”0

 

"The filters were models number 221-1790, 221-2009, and
221-2010 obtained from Perkin Elmer, Inc.

 

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_ 2WhC d)
FdA - —;5- (expThv7kT7—l) ’ ”‘1

 

where F is the power per unit area in the wavelength inter-
val A to A+dl given in watts/m2. The spectral background
between 1” and 20p is seen to decrease uniformly with wave—
length, the total variation being a factor of 3.1. A
single-beam spectrometer must, therefore, have a large
background due to source variations. The Nernst glower is
the other principal source of infrared radiation in this
wavelength region; although somewhat greater intensities
are possible, fluctuations in intensity make it less
suitable for this work. The total optical power delivered
between 17 and 17.20 from the Globar source is about F-A-dA,
where A is the area of the aperture at the source in square
meters. This source area A, however, radiates into an area

of 3300 cm2 at the distance of mirror m from the source;

1
therefore, mirror ml with an area of 15.1 cm2 collects and
sends on to the sample a power of 14.6x10-3 F-A-dl. This

value comes out to be 14x10"5 watt, and will be reduced
considerably by attenuation in the sample, monochromator
losses, spectral-resolution requirements, and scattering.
Attenuation through the sample is described by Lambert's

law, for which the attenuation factor is given by exp(-a(A)x).

Here a(k) is the absorption coefficient in cm-l, and x is

the sample thickness.

o
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52
Alignment procedures for the source and monochromator
sections were simple and are well described in the Perkin
Elmer manual for the 112G. Mirror m2 in the source section
was aligned so that the source image focused on the mono-
chromator slits. A focus on the slits was necessary for
minimum signal losses and for the most uniform slit dispersion.

Similarly, the sample should be near a focal point to reduce

scattering losses.

u.1.3 The Grating—Monochromator

The grating-monochromator section of the 1126 is a design
of the Littrow type intended as a high—resolution instrument.
The mechanical layout and optical path are similar to those
in a Perkin Elmer model—99 monochromator in which the prism
and Littrow mirror have been replaced by a grating (see Figure
u.1). Since our germanium samples were expected to have small
local-mode absorption coefficients, a maximum signal-to-noise
ratio was needed. The expected low signal also points to the
need of estimates of the monochromator efficiency and signal
losses.

The infrared beam enters the monochromator through en—
trance slits 81' These slits have fixed height and variable
width, and are controlled by an external drum; a maximum slit
width of 2mm is provided with a precision of one micron. A
similar set of slits 82 is located near the output and is
mechanically driven in parallel with the input slits. The

monochromator generates a spectrum at the exit slits, the

intensity of which depends linearly upon the signal passed

g

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sru S :I‘ Nu AM h. c ¢ A.» a e CC +» h; S n
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53
by the entrance-slit area or slit width.”1 The amount of
this spectrum passed through the exit slits depends linearly
on the exit-slit width. Hence, the power passed by the
monochromator depends on the product of input and output
slit widths. Observation shows that about l/H of the input
beam is stopped by the entrance slits set at 2 mm. The
output slits are observed to block approximately one third
of the first-order spectrum. The variation of signal in-
tensity with slit width w is then approximately W2/l2.
The assumption here is that the slits are the principal
factor determining resolution. This is true except for
very narrow (less than lOOu) slit widths. Adequate reso-
lution for the local—mode spectra was obtained with slits
of 900m or more. Calculations of resolution based on slit
setting and grating parameters can be made,”2 but the stand-
ard procedure is, instead, to run a spectrum, often water
vapor, and observe minimum peak separations.

The gratings were the echelette type in a Littrow
mounting. The echelette grating consists of many parallel
grooves or lines. Each line gives rise to a single-slit
diffraction pattern, and the whole grating represents an
interference of z-Wg such patterns, where 2 is the number
of lines/mm and wg is the grating width. In this type of
grating the intensity of higher orders does not fall off
as a single-slit pattern does. On the contrary, the
intensity of higher orders is similar to that of the first

order, with efficiencies generally greater than 60%. By

 

 

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changing the monochromator from a double—pass to a single-
pass instrument, a signal gain of more than 60% was attained.
The monochromator efficiency E, from input slits to the
detector, is approximated by E = (W2/12)(.60) where W is in
mm. It has been assumed that there are no atmospheric losses
and that the aluminum mirrors are perfectly reflecting. The

delivered to the detector, in the wavelength region
5

power Pd
l7-to-l7.2u, is less than or nearly equal to Pd = luxlO-
(W2/12)(0.6). The 0.2u resolution occurs for a slit setting
of about 1 mm, so Pd = luxlO-5(O.60/12) = 7x10.6 watts with-
out the sample in place. This will be further reduced by

a factor of u as a result of the action of the external
chopper.

The change from a double-pass to a single-pass mode
also allowed the built-in chopper to be by-passed. Since
this chopper was a large source of noise, a further gain
in signal-to-noise ratio resulted. An external variable speed
chopper, PAR model -222, was mounted at the monochromator
input. The chopping speed was still set at 13 Hz because
of the limited detector response time.

A few more details on the grating are in order to
facilitate an explanation of the calibration. The grating
equation describing wavelength vs. grating angle 6 is 2d sine
= nx, where d is the distance between adjacent grating lines
and n is an integer corresponding to the various "orders".

A plot of grating angle vs. wavelength is shown in Figure H.2

for n = 1,2,3, and u. The cut-off wavelengths of the lu.3-

‘1;
a: “
O-h

o-ZUu

O
b

:EXLSUIH

 

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8 E L c e a l V a h. a

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.3 T. e s . FD e a.” .1 +1. .1 e n“ t U h;

T w L o \A F .d P. n5 . i .G .1 C C .3
O .3 e O a R a C on 2 : . e
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56
to-QM.3u filter are shown as dotted horizontal lines. This
grating was blazed for ”Bu i.e. the first order efficiency
is a maximum at “Bu. The efficiency of the nth order is a
maximum at ”Bu/n. Thus for local-mode spectra in the range
lu-to-20u the lu-to-2Hu filter should be used with this
grating in second order. The grating should be driven through
7° from 16.8°-to—23.8°. Two gratings were selected to cover
the range 2-tO-50u, with each grating requiring the use of
three orders. The grating discussed above was blazed for
HSu and the other grating selected was blazed for 8n. The
orders were considered efficient for wavelengths within 33%

of the blaze wavelength A for the first order and within

B
23% of AB/n for higher orders. In practice, usable efficien-
cies were obtained within :70% of the blaze wavelength, although
background effects became large. The grating is driven with
a mechanical sine function so that a linear calibration of
wavelength vs. drum setting could be obtained.

Calibration was necessary to adjust the grating angle a
and to determine wavelength vs. drum reading. This was done
by placing a thin film of indene between two NaCl discs and
running the spectrum. Indene has a known spectrum in the
lH-to-ZOu range, with absorption peaks located to a hundredth
of a micron. The grating with 20 lines/mm blazed to usp
was used in first order. One of the absorption peaks occurred
at 16.95u and a drum reading of 700. This drum setting cor-

responds to a grating angle of 15°. But 16.95u corresponds

to an actual grating angle of 9.750 from the grating equation.

57

Rotating the grating mount by 5.250 would give agreement
between the measured angle and the actual angle. Seven
indene peak centroids were recorded vs. drum setting; the
results are given in Figure u.3. This plot confirms the

linearity of the drive mechanism.

u.1.u The Infrared Detector

Infrared detectors are classified in one of two categor-
ies: thermal or photoelectric.”3 Thermal detectors include
the thermocouple, bolometer, Golay cell, and the pyroelectric
detector. The thermocouple depends on a thermo-electric
effect at the junction of two different metals. The bolometer
depends on a resistance change due to infrared heating. The
Golay cell depends on the thermal expansion of a small volume
of gas. These detectors all have similar thermal response
times, i.e. they respond in tens of milliseconds or slower.
The maximum chopping speed is therefore limited, allowing
greater l/f noise in the system. Their spectral response,
on the other hand, is quite flat over the entire local-mode
spectrum range. The photoelectric detectors employ a ferro-
electric crystal which exhibits a spontaneous polarization
that is temperature dependent. No bias current is required
hence no l/f detector noise. ChOpping frequencies can be
used out to 2 kHz without signal degradation; furthermore,
they are generally more sensitive than the other thermal
detectors and can also be operated at room temperature.
The photoelectric detectors are generally characterized

by their narrow spectral region and extremely high sensi—

58

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59

tivity; they most often require cooling to liquid-nitrogen
or liquid-helium temperatures. Chopping frequencies, however,
are generally as high as 100 kHz. Their narrow spectral
response and cooling requirements make them unacceptable for
this local-mode study.

Infrared detectors are generally compared on the basis
of a figure of merit called Detectivity 0*. Detectivity is
equal to the square root of the sensitive area of the detector
divided by the noise equivalent power, i.e. 0* = Al/2/NEP.
The noise equivalent power (NEP) is defined as the square
root of the mean-square thermal noise of the detector per
unit band width, divided by the detector sensitivity, i.e.

l/2

NEP = (v-Q/Af) /sensitivity.

A thermocouple detector with a CsI window was employed
in the local-mode studies. The sensitivity was H microvolts/

microwatt with an active target area of 0.2 mm by 2 mm; the

resistance was 12 ohms. The resulting MB? is then (l.tkT(l2))]/2

10 watts/(Hz)l/2. The Detectivity is then 5.7xlO8

cm-watts/(Hz)l/2. A comparison of thermal detectorsuushows

/u

or 1.11x10'

this detector to be one of the better detectors for this
wavelength region. This detector was used in conjunction
with a matched-input-impedance preamplifier built into the
monochromator. Phase-sensitive electronics were connected
to the plate of the preamp tube through an a.c. coupling
capacitor to complete the measurement system.

There was still one serious defect in the system as it

now stood; that was the single-beam aspect. A single-beam

60
instrument has large background variations, primarily due
to source variations and grating efficiency variations.
These effects are greatly reduced or eliminated in a double-

beam instrument.

0.2 The Double-Beam Modification

It was decided then to divide the beam into upper and
lower halves. This division was done by designing a chOpper
blade with two apertures 180° out of phase and at different
radii. The chopper blade is shown in Figure u.u. Figure u.u
also shows the chOpped signal for both upper and lower beams
along with the chopper reference output. The chopper refer-
ence signal is obtained by photoelectric feedback from a
light mounted in the chopper. The chopper blade was con-
structed so that the reference light passed through only
one of the apertures per cycle. The chOpper reference sig-
nal then indicated the fundamental chopper frequency.

The lower half beam was assigned as a reference beam
and the sample was placed in the upper beam. A black wedge
was constructed which could be moved along a threaded rod
back and forth across the lower beam. This allowed for
the adjustment of an optical null at which the intensity of

the upper and lower beams were equal.

0.3 The Measurement System

The key to this system was the lock-in amplifier. A
PAR model—220 was used in conjunction with a PAR model-210
tuned amplifier. The lock-in amplifier was tuned to the
chopper reference frequency. A glance at the monochromator

detector output shown in Figure A.” will show that the only

61

 

 

 

 

 

 

 

 

 

 

fo-f. = lmm
h = 0.8 cm
upper _ _
beam I l
Intensity - I I
I I
90° [80° 270° 360°
lower L
beam
lntenslty lr 1' l l
90° I80° 270° 360°
detector 1 - 4
output I 1 1 1
l l l l
90° l80° 270° 360°
chopper
ref. 1 l I
beam
l I ,1. t
90° l80° 2 0° 360°

FIGURE 4.4 Double beam modification.

62

signal available at that frequency is generated by the
difference in intensity of the upper beam compared with
the lower beam. Since the two beams have been Optically
nulled, the largest intensity differences are due to ab-
sorption in the sample.

The overall system is sketched in Figure u.5. The
d.c. "block" is simply an R-C filter to block the bias
voltage, 15 volts, from the output of the preamp tube.

The 213 preamp shown was Operated at a gain of 1 pri-
marily because of its ground—loop-isolation properties.

The 212 amplifier was used for its variable—gain function;
it was generally run at a gain of 200 or less. The tuned
amplifier served to limit the noise bandwidth seen by the
lock-in amplifier. It was generally run at a Q value of 10
and a gain of 1.

It should be pointed out that for the double-beam
modification the necessity of decreasing the area of the
lower beam to achieve an optical null had two undesirable
effects on the system. First, it represented additional
intensity losses. Secondly, it meant that the areas of
the upper and lower beams striking the filter were different.
This difference in areas, resulted in a background due to
wavelength variations in the filter. Consequently, the
optical-null condition was modified to mean a minimization
of all background effects. This minimization was performed

with the sample in place by first selecting equal beam

 

 

Infrared
Source

 

 

 

 

 

 

 

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sample Fi r
.._. -. Monochromator preamp
* (modlfted) output
Chopper -
PAR 222 'P'°°"‘P
‘ I PAR 2H3
Chopper H. P. Function Amp
drlve ”—4 Generator
PAR 222 PAR 2|2
reference ;
“’9‘" Tuned
Amp
PAR 2|0
f Lock-In
re erencL Amp. si nal
Input pm; 220 Input
output
Strlp
Chart
Recorder
FIGURE 4.5 The overall measurement

system.

60

intensities and scanning the spectrum. The reference-beam
area was then changed slightly and the spectrum was scanned
again. After comparing the two backgrounds, the reference-
beam area would be changed accordingly. A minimum background
was soon obtained.

In addition to use of the lower beam as a simple reference,
it may also serve to cancel out any material backgrounds. For
example, if a piece of pure silicon of the same thickness as
the sample is placed in the lower beam, the effects of lattice
absorption can be canceled. This is especially useful for
looking at the free-llboron peak occuring on top of the two-
phonon lattice—absorption peak in silicon. Another possibility
is to make up two samples, one compensated with 6Li, the other
with 7Li. These samples could then be placed in the upper
and lower beams and only the differences due to the lithium

isotOpe observed.

u.3.1 The Lock-in Amplifier

The PAR model-220 lock-in amplifier is similar to the
basic homodyne amplifier;LLO a sketch of this type of ampli-
fier is shown in Figure 0.6. The reference tuned amplifier
is tuned to the fundamental of the input reference signal.
This is amplified and limited to give a bipolar square wave.
This phase is adjusted to match that of the signal. This
bipolar reference square wave can be thought of as the switch
controls for a double-pole, double-throw switch. The switch-

ing action is illustrated in Part (b) of Figure 0.6. Parts

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Slgnal
from
Output Meter
Detector Amp 8 Phase
tSensitivity Sensitive “
39'9” Detector
Ref '1: -
from
Cho er
pp Referencel (a)
Tuned If
Amp
.1. 7 (b)
V out
"J—

 

l

Wh\//\ (c) +1
Hit... i J 1[ (d)

V'for L +t
V ref in ,
phase

V'for

V ref 90° [\ \ [\ (f) ’b t
out of

phase N \ \1

FIGURE 4.6 The phase sensitive

amplifier.

 

 

 

 

 

O3
(.1

(c) and (d) of Figure ”.6 show an input signal in phase with
the reference signal. The switching then produces the signal
V’ shown in Part (e). The d.c. value of V is fed to the
output meter and is proportional to the component of the
input signal which has the same phase and the same frequency
as the reference signal. Part (f) of Figure u.6 shows what
happens to a signal that is 90° out of phase with the refer-
ence. The resultant V’ has a zero d.c. level, and no output
is produced. The principal feature of such a system is its
low-noise figure owing to the narrow effective bandwidth
allowing considerable external amplification.

The system described above was used primarily in inves—
tigations of local-modes in germanium. With some additional
modifications it was also used in some laser Raman experi-
ments. The silicon samples did not require cooling, and
hence considerably less time was involved with use of the
spectrometer. For the decreased spectrometer time required
for the silicon samples, a Perkin Elmer model-225 spectro—
photometer was available. The main feature of the 225 are
its ease in handling, its double-beam design, the degree to
which it is automated, and its wide spectral range from 200

to 5000 cm'l. It employs phase-sensitive detection with a

Golay detector; D“ for the Golay detector is nearly the same
as for the thermocouple detector. All data on local—modes

in silicon were taken with the 225 spectrograph.

CHAPTER V

LOCAL-MODE EXPERIMENTS AND EXPERIMENTAL RESULTS

5.1 The Germanium System

Our initial work began with germanium doped with gal-
lium and compensated with lithium. We were interested in
locating either a free—lithium local mode, or the local
mode of lithium in a paired configuration. The two
lithium peaks of a paired configuration would have allowed
a good estimate of the location of the free-lithium local~
mode. Once the lithium local modes were located, thermal
and stress dependencies could have been studied. Local
modes of gallium paired with lithium in germanium have been
reported by Spitzer;33 our preliminary work was intended to
produce similar local modes. The absorption coefficient
for these complexes were reported to be in the neighborhood
of 2 - u cm”1 for a lithium concentration of 3xlOl7/cm3.
Spitzer chose a double-dOping compensation technique with
Ga and P such that [PJSEGa]. Spitzer used a gallium concen-

18

tration of 2x10 atoms/cm3; the phosphorus concentration

was about 1.7x1018 atoms/cma. Our samples, with 2.5-3x10l7
Ga/cm3, were expected to give similar results when compen-
sated with lithium. A sample compensated with 7Li was

fastened to a copper plate which, in turn, connected to the

cold finger of a liquid-nitrogen dewar. The sample, thus

67

68

mounted, was placed in the upper beam of the spectrograph
and the optical background balanced to roughly 1%. A
room-temperature spectrum was then recorded; all local-
mode peaks would be thermally broadened beyond distinction,
serving as a background reference. Liquid—nitrogen was
then poured into the dewar, and enough time was allowed
for the system to reach thermal equilibrium. The spectrum
was then rerun with all settings the same except for the
amplifier gain and d.c. offset. The comparative results
are shown in Figure 5.1. There is an indication of a Ga -
7Li local-mode at 28u on top of the lattice peak. This
indication is, however, too small and too poorly defined
for any meaningful studies. A similar spectrum of the
sample compensated with 6Li showed no indication of a peak
at 26.3u. Thin samples of pure germanium doped with
lithium also showed no local-mode peak.

The local—mode absorption intensities of lithium-
gallium complexes, observed by Spitzer, were small. Conse-
quently, precipitation of lithium had to be held to a
minimum in order to preserve the local—mode signal strength.
Two possible explanations for the success of Spitzer's
double-doping technique are apparent: first, Spitzer's
germanium samples may have had a lower oxygen impurity
concentration than our samples. Secondly, the phosphorus
and gallium impurities are substitutional; hence, they do
not have a strong tendency to form pairs. As a result,

as much as 8 times more unpaired gallium may have been

Absorption Coefficient or (relative units)

69

7Ll-Ga
Local Mode?

Ge(7Li-Gq)

filter
cut off

 

 

|00°K

 

l l l l l I

 

3| 30 29 28 27 26
(microns)

FIGURE 5.| Results of I-R absorption measurements
in germanium.

70
available to pair with the lithium ions. Hence, the for-

mation of lithium—gallium pairs was greatly enhanced.

5.2 The Silicon System

Our attention then shifted to the boron—dOped lithium-
compensated silicon system for which the lithium site sym-
metry is the same as in germanium. The silicon system
is, however, easier to study. Larger concentrations of
lithium—boron complexes are attainable, hence larger local-
mode absorption coefficients result; furthermore, cooling
is not required.

Thermal studies of the local-modes in silicon were
made for the purpose of exploring anharmonic terms in the
local potential. Similarities and differences in the
lithium mode at 522 cm.1 compared with the boron modes
were also of interest. A good measure of the effects of
anharmonic terms is the change with temperature of the
local-mode centroid and line width. Other measures of
anharmonic effects already evaluated,10 are the position,
width,and intensity of the second harmonic of the local~
modes. Line widths and centroids have been measured for
temperatures below 300°C, and variations are reported to
be small. We have extended this temperature range up

near 500°K where variations are expected to be larger.

5.2.1 The Temperature-Dependence Measurements
Measurements were done on the Perkin Elmer 225 spec-
trometer with a resolution of 0.8 cm-l. The device for

heating the sample is shown in Figure 5.2. This device is

Heating
Rod

71

 

 

Variac

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1r———

“-__—-

 

 

 

 

Thermometer

FIGURE 5.2 The heating apparatus.

Q

72

a block of brass with a hole in it to pass the infrared
radiation. The sample is placed over the hole in good
contact with all edges. A heating element, powered by a
variac, was placed in another hole through the block as
shown. A third hole passing very close to the sample
contained a thermometer to record the sample temperature.
Silicon grease insured a good thermal contact.

The measurement temperature was selected by adjusting
the variac and monitoring the temperature until thermal
equilibrium was reached. All spectrometer settings were
maintained fixed for the entire temperature range of
measurements. The resulting spectra are shown on Fig. 5.3.
A scale is not indicated on absorption coefficient, but
peak values can be estimated from the height of the lattice
peak at 610 cm_l, which is known to be approximately 9.5
cm- at room temperature. Cross verification on the complex
local-mode intensities can be made from work reported by
Spitzerzu in which absorption coefficient vs. doping concen—
tration is plotted. Plots of the centroid shift and line-
width variations are shown in Figs. 5.0 and 5.5 respectively.

Error bars are shown to indicate errors from instrumen-
tal resolution, pen line width, and electronic-noise back—
ground.

There are four basic contributions, from the sample,
to the background of these measurements; the first is the
thermal generation of free carriers and the presence of

free carriers due to incomplete compensation. As seen

(RELATIVE UNITS)

ABSORPTION

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' T = 394K
“r $
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450 500 550 600 650 700 750
V (cm")

FIGURE 5.3 Absorption spectra at
different temperatures.

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76

earlier, the free-carrier absorption forms a relatively

smooth background and can be readily subtracted out.

Another background effect has been observed to occur at

the surface of the sample. As the measurement temperature

approaches 500°K, the surfaces tend to become p-type due

to a loss of lithium compensation. This surface effect is

the same effect observed in the precipitation process.

Since the effect occurs in such a thin surface layer, less

than one mil in depth, the sample remains transparent to

the infrared but has a free-carrier type background due

to the p—type surfaces. A third possible background

effect is the thermal breaking of Li—B complexes, causing

an increase in the free-boron local mode and subsequent

decreases in the local modes of the complexes. Calculations

show a maximum effect of about 22% at 500°K and doping

levels of lOlg/cm3. Calculations are given in Appendix B

for the relative number of pairs as a function of temper-

ature. This unpairing will not affect the local—mode

centroid, but the decrease in amplitude will show up as a

loss of accuracy in the line width. Finally, the background

can be shifted by infrared emittance from the heated silicon;

this effect is not expected to alter the local-mode spectrum.
Immediately after the temperature data was recorded,

the sample was cooled to 351°K and the spectrum was recorded.

The resulting background had a similar appearance to the

background at 082°K. This background was the result of the

non-reversible loss of lithium compensation at the surfaces.

77

The surfaces were later checked with the hot—point probe
and found to be p-type. Removal of about one mil of
surface by lapping restored the sample to nearly its
previous level of compensation. The line—width and
centroids of this final spectrum fit the plot of the rest
of the data well within the limits of error, indicating
that background effects had been effectively subtracted
out. The total time at “82° K, the highest temperature,
was only 16 minutes, so that the total surface effect was
not large enough to prevent a good measurement. A plot of
time vs. sample temperature is shown in Fig. 5.6. As a
further control on background effects, a spectrum of pure
silicon was run in the same temperature range. Calculations
of a background normalization factor based on the ratio
of the doped-silicon-lattice peak centroid divided by the
centroid shift of the pure—silicon-lattice peak showed no
appreciable lepe with temperature, and therefore were not
incorporated into the data.

Immediately prior to recording the temperature data,
but with the same spectrometer settings and resolution,
a spectrum of indene was run. This gave us a calibration
of the spectrometer and showed the need of adding 1 cm-1
to the wavenumber readings.

The area under the local-mode peaks was observed to
be very sensitive to background effects. For this reason
we adopted a normalization factor which is the product of

two terms. The first term is the ratio of the pure-lattice

peak area to that of the doped-lattice peak area. The

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second term is the effect of thermal breaking of the com-
plexes calculated in Appendix B. The normalized local—mode

peak areas and normalization factors are shown in Fig. 5.7.

5.2.2 The Pressure-Dependence Measurements

The pressure measurements were made on the same spec-
trometer (P 8 E 225) with a resolution of 0.8 cm-l. The
pressure applicator is sketched in Fig. 5.8. The associated
pressure meter was calibrated to an accuracy of 18%. A
slightly compressible backing, such as bakelite, was needed
to allow a more uniform distribution of the load across the
sample.*

The pressure gauge on our stress-applicator device was
calibrated with two strain gauges attached to the sides of
a small aluminum cylinder of about 1/2" in diameter and one
inch long. The two strain gauges were connected in OppOSite
arms of a four-arm bridge. The cylinder was first loaded
with 5-pound weights on the end of a lS-to—l mechanical-
advantage lever system. A plot of pounds loaded on the
aluminum cylinder vs. bridge reading was then made for
loading and unloading of the cylinder. The aluminum cyl-
inder was placed in the pressure-applicator device, and
several plots of the pressure gauge reading in psi vs. the
bridge readings were made. A sizable hysteresis was ob-

served between loading and unloading of the cylinder. This

hysteresis accounted for the 18% error in the pressure deter-

 

“Pieces of bakelite l/2" thick were necessary to prevent
their breaking under pressure.

80

X 522. cm
654 cm“
Normahmtion tact r for ll 10
Tem I’Kl N
SITE) \.0i
35\ \.0‘5
393 \.\2
444 \.52
2 2.60

b
m
R‘G‘

 

 

300

___><___-
____><_

_ _><—

V

  

 

FIGURE

81

Sampm

HI

  
 
 

Pressure
Meter(psi)

 

Hydraulic Jack

5.8 The pressure applicator.

  
 

   
 

82

mination. Both the loading and unloading plots were found
to be linear. From the above measurements a plot of actual
pounds of applied force vs. the pressure-gauge reading was
made. This is shown in Fig. 5.9. A scale of kg/mm2 is
also shown for a sample cross—sectional area of 10 mm2,
which was in the range of most of our experiments.

Our spectroscopic pressure experiments did not give
us any conclusive results on line width or centroid vari-
ations. Centroids were found to be constant with stress
perpendicular to the [111] direction or parallel to the
[112] direction. The line width variations were inconclu-
sive. A plot of line-width vs. pressure is shown in Fig.
5.10 to illustrate this. The maximum pressure that we ob-
tained in the [112] direction was 1.22x109 dynes/cm2 with
unpolarized radiation propagating in the [111] direction.
The sample was 10.5 mm in the [112] direction with a cross
section of 9.65 by 0.79 mm. Another experiment was done
with the applied pressure perpendicular to the [111] dir-
ection and a maximum value of 0.9x109 dynes/cm2 was ob—
tained. The sample dimensions were 15.0 mm in the dir-
ection of pressure with a cross section of 12.0 by 1.09 mm.
In each case the experiment concluded with the fracturing

of the sample.

83

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CHAPTER VI

DISCUSSION OF LOCAL MODES

6.1 Local-Mode Assignments in Silicon

Before discussing the results of our local-mode meas—
urements, we review the assignments of the local modes in
the silicon system. A good discussion of these assignments

9b This paper includes a study of

is in a paper by Spitzer.
isotOpic shifts in the local—mode frequency and a measure-
ment of local—mode intensities as a function of the equal
concentrations of lithium and boron. The free-substitutional-
boron local mode had already been observed in silicon by
Angress et al.;20 no other constituents had been added to

the silicon. Spitzer96 observed this free-boron peak in
reduced intensity at 620 cm-1 for 11B. This is between the
two boron modes of the lithium-boron complexes, and is

closer to the doubly-degenerate 65H cm“l peak. When the

boron isotope was changed from 11 to 10, the 56H cm—l peak
shifted to 58A cm”1 and the 65“ cm-1 peak shifted to 681

cm—1. When the lithium isotope was changed, these peaks

shifted less than 3 cm-l. This identified these peaks as
boron local modes. Their intensity was found to be directly

proportional to the concentration of lithium-boron pairs.

The degeneracy of the upper band was assigned as two-fold

85

86
degenerate, and the lower band as one-fold degenerate, on
the basis of relative integrated intensities of 5 to 3.
Axial symmetry of a paired configuration predicts an in-
tensity ratio of 2 to 1, but local polarization due to
atomic distortion may account for the difference.* The 522
cm“1 line was found by Spitzer to shift to 534 cm_1 when 6Li
was substituted for 7Li. It is insensitive to changes in
the boron isotope, and appears at 515 cm"1 for gallium-
doped 7Li compensated silicon. The intensity of this mode
is also directly prOportional to the number of lithium-boron
pairs. The isotopic shifts and intensity ratio identified
the 522 cm"1 peak as a lithium mode. By analogy with the
two boron local modes, the 522 cm_1 peak is expected to be
the two—fold degenerate peak of a paired lithium ion in a
C3v configuration. Further studies on the breaking of the
complexes with temperature have been done by Spitzer and
Waldner.9a They found that as the temperature was raised
to 150°C, the peaks assigned to lithium—boron complexes de-
creased in intensity whereas the free-boron peak increased
in intensity. This is in agreement with predictions on
the thermal breaking of complexes given by Reiss et al.22
This comparison between experimental and theoretical thermal
effects further reduced the possibility of the 522 cm"1 peak

belonging to the free-interstitial lithium since such a peak

would also have increased in intensity as the sample temper-

 

‘0

"For example, carbon impurities in silicon are substitutional
and un-ionized but they have an effective local charge of
about e.“7

87
ature was raised. The experiment also confirmed the identity
of the 65A and 56” cm_1 as boron paired to lithium. The
temperature at which the lithium is diffused into the silicon
does not significantly alter the intensity of the observed
local modes. Therefore, it is unlikely that the 522 cm"1
peak could be a lithium-vacancy complex local mode. The
small but finite shift in frequency of the 522 cm-1 peak for
different acceptor impurities confirms the designation of
the 522 cm-1 peak as a paired lithium local mode. The free
interstitial lithium must then be a resonance mode within

the band of allowed lattice frequencies. We did not observe

any such resonance mode in our samples.

6.2 The Lithium-Gallium Complexes in Germanium

Our attempted measurements of lithium-gallium complexes
in germanium were unsuccessful because of problems in sample
preparation. However, some observations on work reported
by A. E. Cosand33 will be made.

The two local modes of lithium in a paired configuration
have apparently been observed by Cosand33 in germanium at
liquid-nitrogen temperatures. The lithium is paired to

gallium, and 7Li local modes occur at 356 and 380 cm-l.“

When 6Li was used, these peaks occurred at 379 and HOS cm”l
reSpectively. Cosand suggests, on the basis of integrated

intensities, that the lower—frequency peak is the two-fold

degenerate mode. His suggestion indicates that a free-

 

.7

"The maximum single phonon lattice frequency in germanium
is 300 cm-1.

88

lithium local mode does exist in germanium and should be
between the two lithium complex local modes close to 36”
cm-l. It was not observed in Cosand's samples because the
solubility of lithium in germanium limited the amount of
lithium; in fact, sample preparation was the major problem
in this work. The complex local modes had only a 2 to H cm"l
absorption coefficient, and at the liquid-nitrogen tempera-
ture over 99.9% of the lithium was paired to acceptors.
The location of the free-lithium local mode might be confirmed
by slowly heating such a sample to break the complexes. How-
ever, it appears that thermal broadening may mask such an
effect.
6.3 Temperature Dependences of Centroids and Line Widths

in Silicon

Figures 5.5 and 5.6 show that thermal effects are most
pronounced above room temperature.* Centroids are observed
to shift to lower wave numbers with temperature, as expected.
The H82°K data point in the centroid shift of the 522 cm"1
line seems to represent a rather sudden departure from the
linear approximation. It is at about this temperature that
the centroid has been shifted to the edge of the lattice-
band modes. Therefore, the effect of the lattice phonons
on the centroid appears to account for the deviation from

linearity of the M82°K point of this line. The data on

centroid shifts can be fitted with a linear approximation

 

7.

"Low-temperature measurements of line width and centroids
have be n done with :lcm‘l resolution by Balkanski and Naz-
arewicz b

89

by lines varying in slope by 110% from the lines shown and
still fall within the limits of the error bars. Variations
in the fit to line-width data of :l5% in slope are within
the corresponding error bar limits.

The paired substitutional-boron and interstitial-lithium
ions are each in essentially a perturbed Td symmetry. The

resulting symmetry is C , but the dynamics of the system

3v
are quite complex; the theory necessary to explain the

centroid shifts and line-width variations with temperature

has not as yet been fully developed. It is expected, however,
that such a theory will take the form of the effects of an-
harmonic terms in atomic displacement described by Elliott

et al.25 for an isolated impurity, applied to the Green's
function treatment of an impurity pair by Elliott and Pfeuty.7
A preliminary review of these ideas showed that such a cor-
rectly done formalism was likely to result in third—and fourth-
order Hamiltonians having the same form as those presented

by Elliott et al. The interpretation, however, would be

quite different, i.e. the local-mode operators must be re-
labeled to apply to each local mode observed. The relabeling
should also include a distinction between the possible one

and two-fold degeneracies. Details of how such a relabeling
of the local-mode Operators occurs, and what the relabeling
means, are given in Appendix D. In this new representation,

the two-fold-degenerate boron local—mode operator is subscript-

2
3

is labeled with F

ed with F , and the singly—degenerate boron local-mode Operator

2

l . The 522 cm.1 lithium local-mode operator

90

is labeled with F31. The superscripts l and 2 refer to

lithium and boron respectively; F1 and F3 refer to the
singly and doubly degenerate infrared-active representations
of the CBV symmetry.

With this interpretation and relabeling of local-mode
operators, equation 2.12 approximates the centroid shifts.
Since our own measurements did not determine the zero-temp-
erature local-mode frequencies 90, this parameter was varied
to obtain a best fit to our data. Equation 2.12 is repro-

duced here for convenience with the indicated labeling of

the local modes.

 

AQF i rr i

_._l.. = 1 , {.1— Z’hm(k)(n(k)+1/2) 6.1
:2 l Mlm light) 2 N ->

on 'Fj M k

Evaluation of this expression requires a knowledge of
the silicon phonon spectrum g(w). This has been evaluated
numerically by E. A. Johnson (unpublished) for a mesh of

61 points on the interval Oiwib A modification giving

M'
closer agreement with experiments has been presented by

Dawber and Elliott.19b

These calculations represent one
direction in the Brillouin, zone and all other directions
are assumed to give the same density of states. The
bracketed term of eq. 6.1 then becomes

”M

y g(m)’fiw[n(w)+l/2]ng_ 6 2

(1)

o M
f g(w)dw

O

91

where the number operator n(w) is given by l/(exp(hw/kT)-l).
The factor of 6 occurs because the original sum over k in—
cluded the sum over the 3 optical and 3 accoustical lattice
dispersion branches. This expression can be easily pro-
grammed to give the full range of temperature variations.
The fortran program and further details are given in Appen-
dix E along with the values of the silicon phonon spectrum
taken from Dawber and Elliott's paper.

The expansion coefficients eri and 90 were evaluated
by requiring that the calculations agree in both magnitude
and slope with the data. The results of these calculations
are shown as dashed lines in Fig. 5.9. Calculations in the
Debye approximation, for which g(w) = 3w2/wD3 *, have also
been done. The results in all cases were very close to the
results based on the silicon phonon spectrum. Agreement
with our experimental results is very good for the 522 cm"1
lithium mode, and reasonably good for the two boron modes
at 6H5 and 565 cm_l. The resulting expansion coefficients
eri and zero-temperature local-mode frequencies 90 are
summarized in Table 1.

These calculations have not included such possible
effects as the interaction between the various local modes.

It is expected that the interaction between the two boron

local modes would be the largest of such interactions be-

.7

“The Debye frequency for silicon is ”57 cm.1 for a Debye
temperature of 6580K.

Line-Width Coefficients for Two—Phonon Decay

TABLE 1

21 * l

 

Local c (erg/emu) x 10 Q (cm_ )
F . 0

Mode 3

522 cm‘1 a l = 2.1a 525
r
3

55u.5 cm'l c 2 = 3.52 557.9
I‘
1

65a cm'l gr 2 = 5.00 557.7
3

 

 

93
cause both modes are at the same physical location. Inter-
action between the lithium mode and a boron mode would be
a smaller effect because the ions are separated by 2.5 A.
These effects would then be small for the 522 cm"1 line,
possibly accounting for the good agreement of the data with
the simplified theory. Similarly, such effects could
account for some Of the disagreement between the data and
the calculations for the boron local modes.

The line-width processes also arise from the relabeled
local-mode Operators described in Appendix D. Again, although
the interpretation is different, the expressions for line
width take the form of eq. 2.19 for the two-phonon decay
process, and eq. 2.15 for the elastic-scattering process.

To evaluate eq. 2.1M as a function of temperature, we first
converted the sum over k’ to an integral over w’ making use
of the phonon density Of states ng). The delta function
allows this integral to be easily evaluated as the value of
the integrand at w’ = eri-w. For example, ng) becomes
g(9p.i -w). The sum over k is then converted to an integral

3
over w by making use of g(w). Equation 2.1M then becomes

 

 

i 2 i
1 _ (231:3. > rm “M gummy]. -w) i
.. .. if 36 — 03(521" . -w)
T uM’Q l M2w 2 N 2 3
rj M o P
X[(n(w)+l)(n(flpjl-w)+l)-n(w)n(9rjl-w)]dm . 6.3

The phonon density functions g(w) and g(erl-w) have been

9H

6
normalized so thatjf.h(g(w)/Np)dw = 1. Since g(iji-w) is
zero for frequencigs below eri-wl’ the lower limit on the
integral in eq. 6.3 can be replaced by (eri-mm). The
program for the temperature dependence of eq. 6.3 is given
in Appendix E.

The elastic-scattering process has the form of eq. 2.15.
The sum over k and k’ can be replaced by the following
integral:

“’I

'2 ( ) 2 2
36/ (%—Ui—) w'n(w)[n(w)+l]dm 6.1+

{JP
0

The fortran program for elastic—scattering temperature-
dependent line width is also given in Appendix E. When

the parameter iji - ZCfifji)2/erji is evaluated to give
agreement with the magnitude of the experimental results,
the calculated line width is found to cross the experimental
data sharply. The elastic-scattering mechanism is therefore
considered an unlikely explanation of the observed line-
width variations.

The results of the two-phonon-decay calculations are
shown as dashed lines on PigureSJi The expansion parameter
.S’rji was fitted to give reasonable agreement with the
magnitude of the experimental results. The calculations
give approximate agreement with the measurements at tem-
peratures above 3000K. However, agreement is poor below
300°K. As mentioned earlier, there are likely to be inter-
actions between the two boron local modes. This would give

a line—width contribution to the 65M and 56h cm_l lines

proportional to

AQ[n(AQ)+l][g(AQ)/NP][n(Q’)+l] ,

where Q’is the frequency Of the boron local mode which is
not decaying and A9 is the difference between the two boron
local—mode frequencies. This expression has a T2 dependence
at high temperatures. However, the effects of the thermal
population term [n(Q’)+l] are small below about 9500K or
most of our temperature range of measurements. As a result
the variations appear nearly linear in temperature between
300 and “500K. This temperature dependence is similar

to that of the two-phonon-decay calculations and therefore
cannot account for the large low-temperature line widths.
Accurate prediction of the observed line widths will have

to wait for a formal treatment of the problem based on the
anharmonic factors of the actual symmetry of the lithium-boron

pairs.

6.” Temperature Dependence Of the Integrated Intensity

The relative integrated absorption intensity I/Io is
shown on FfiynwaSJ as a plot of (1.0 - I/IO) vs. temperature.
In this form, the dependence is linear on a log-log plot.
The integrated intensity has incorporated into it the cor-
rection factor mentioned in section (5.2.1). The relative
integrated absorption intensity was evaluated by first

plotting I vs. T on a linear scale. The zero-temperature

96

value IO was then extrapolated for each set of local-mode
data. The dependence Of (1.0 - I/IO) is approximately Tn
where n = u.2 through 5.1. This plot indicates that no
local modes exist above a temperature Of about 5300K. This
temperature dependence is, however, the least accurate of
our data because of the correction factors involved, and
because Of some uncertainty in IQ. Nevertheless, we do
believe that a non-linear decrease in integrated intensity

occurs 0

6.5 The Uniaxial-Stress Results

The results of the uniaxial-stress measurements were
inconclusive. The centroid shifts were constant for stress
applied in the [112] direction and perpendicular to the [111]
direction. A similar experiment by Hayes and MacDonald21
on H2 and D- ions in CaE2 had shown a linear variation of
singly-degenerate local-mode centroids with uniaxial stress,
and a splitting of multiply-degenerate modes. Line-width
variations were expected to be of a similar order of magni-
tude. It now appears that not enough stress can be applied
to the lattice to shift the lithium—boron local-mode frequency
by a measurable amount, i.e. more than 0.8 cm—l. The limit-
ing factor is, of course, the fracturing of the samples.
Evidence supporting this conclusion can be found in a paper
by Hayes.“8 Hayes has applied uniaxial stress to oxygen
doped silicon with a resonance mode due to oxygen at 29.3
cm-l. At stresses near the breaking point (2% - 30 kg/mm2)

he Observed a frequency shift of only one cm-l. This small

 

97
a variation in the lithium—boron local modes would remain
essentially undetected by our system. On the other hand,
if the frequency shift scales with the impurity vibrational
frequency, the results Of uniaxial stress should have been
easily detected and the fact that we Observed no shift would
be paradoxial. It appears that uniaxial-stress effects on
Li-B local modes will require a high-resolution spectrometer
(.2 cm"1 or better). The degeneracy of the local modes
is unlikely to be determined by infrared-absorption tech-
niques since the amount of splitting, near the fracture

load, is much less than the local-mode line width.

CHAPTER 7

ISOTOPIC DIFFUSION STUDIES

Our isotopic studies on the mobility of lithium in
germanium were initiated by a desire to obtain a faster
drift rate of lithium in the fabrication Of nuclear
"charged-particle" detectors. 6Li will drift faster than
7Li because mobility is inversely proportional to the
square root of the ion mass. Measurements on the ratio of
diffusion constantsfi were made primarily by means of a
diffusion type measurement, although some effort was also
put into a drift technique. Both techniques required
correction factors in the evaluation of the magnitude of
the diffusion constant. The diffusion constant of lithium
in germanium, at temperatures in the neighborhood of 50°C,
showed about 15% smaller lepe than at higher temperatures.
In the classical picture this means that the activation
energy has decreased by 15% with temperature. This suggests
the possibility of a tunneling effect's becoming important
at the low-drift temperatures. Such a tunneling effect
would further increase the mobility Of 6Li compared to that
of 7Li. The ciiffusion constant of 7Li has already been

measured in the low-temperature range11 so that only a ratio

of diffusion constants needed to be measured.

 

“Extensive use has been made of the Einstein relationship
between diffusion constant and mobility, applicable to the
"dilute semiconductor solutions'" it is given by u = eD/kT.

98

 

99

7.1 Reasons for Measuring the Mobility Ratio

Since an absolute magnitude of the diffusion constant
was not required, the measurement was simplified for several
reasons: First, a measurement of the diffusion constant
would have required a determination of the internal enhance-
ment field across the junction at the diffusion front. This
effect tends to increase the diffusion rate by a factor of
[1+ [2ni(T)/N)2+l]-l/2], where ni(T) is the density of in-
trinsic carriers.ug Thus an accurate knowledge of the donor
concentration N would be required. This, in turn, would re-
quire an accurate determination of the diffusion temperature
and some means of assuring its stability. Secondly, silicon
and germanium are difficult to Obtain in ultrapure form;

there are typically 1013 to 1012 traps/cm3 (usually oxygen)

remaining in the best crystals.* Consequently, during the
drift or diffusion process the lithium tends to form pairs,
thereby effectively reducing the diffusion rate. This
effect can be neglected if the lithium concentration is
large compared with the concentration of trapping sites,

as is the case for diffusion temperatures above 150°C.
Finally, a measurement based on a drift technique would
require an additional correction factor to account for

that component Of mobility due to diffusion. In the de-

termination Of the ratio Of mobilities of 6Li to 7Li, all

 

5.
I‘

It has recently become possible to produce germanium
crystals with less than 1010 traps/cm3; such material
does not require a lithium drift to compensate trap
sites in the production of nuclear detectors.S

:i‘J'TE-n
.. L.

100
of the effects mentioned above are equal for both isotopes
and therefore cancel. It is only necessary to insure that
both samples are drifted or diffused at the same temperature.
In an effort to assure equal drift temperatures, a
drift experiment was done with both 6Li and 7Li deposited
on the surface of the same crystal. All germanium was

13 gallium/cm3.

detector-grade, doped with typically 5x10
Half Of one surface was coated with 6Li and the other half
with 7Li, followed by a lO-minute diffusion at 900°C. The
sample was then drifted at about 25°C. The results showed
nearly equal drift depths for 6Li and 7Li regions. A
closer study showed two mechanisms which tended to equalize
the drift depths: First, the field across the junction

depended upon the drift depth. If 6Li had drifted further

than 7Li, the resulting decreased E field would reduce the
6Li drift rate. Also, the 12R heating would be larger in
the region of shortest drift depth, tending to thermally
increase the drift rate. Differences in drift depth were,
therefore, quickly reduced. Further drift measurements
were tried based on the technique of Fuller and Severins;lH
however, the CuSOu stain test used to define the junction
region did not give sufficient accuracy. All drift meas-
urements were further complicated by variations in contact

resistance. This effect was minimized by coating the

sample contacts with a liquid mixture of indium and gallium.

 

101

7.2 The Ratio Measurement

Most of our ratio measurements were done with a dif-
fusion process. Two samples Of 12 Ohm-cm gallium-doped
detector—grade germanium were lapped and cleaned. One
sample was evaporated with 6Li, then placed in mineral
oil while the next sample was evaporated with 7Li.* The
two samples were then placed side-by-side in the preheated
diffusion oven. An argon flow of 10,000-15,000 cm3/min.
was maintained throughout the diffusion. After completing
the diffusion, the samples were quenched to room temperature.
Temperature gradients in the oven were the most likely
source of error, since the two samples could consequently
be at slightly different temperatures. Efforts were there-
fore made to eliminate or determine any such temperature
gradients. For example, one diffusion was done on a
single crystal, one surface of which was half 6Li and half
7Li. Another diffusion was done with four pairs of samples,
each pair consisting Of one 6Li and one 7Li specimen. Each
sample was typically Ux12 mm of surface area and about 2mm
thick. Thermal variations were found to be small, but a
possible temperature gradient may have existed along the
direction of argon flow. Several measurements on single pairs

of samples were done at 200°C and at 900°C. Diffusion times

at 200°C were typically on the order of two hours or more.

 

"The lithium isotopes were 99.99% pure and Obtained from
ORTEC.

 

102
This gave diffusion depths on the order of 8 mils or more.
Two measurements at 200°C for six hours were tried in order
to improve the accuracy by diffusing to a greater depth, i.e.
about 1H mils. At uoooc the diffusion time was generally
20 minutes, and diffusion depths of typically 25 mils were
measured. The results of these measurements are given in
Table II. The ratio of diffusion constants was taken as
the square of the ratio of diffusion depths, i.e. the
diffusion depth x was approximated as x = 2/Df so that D is
prOportional to x2. The correction factors belonging to
this expression then cancelled in the ratio.

An accurate measurement of x required that the surface
of the sample, upon which lithium was to be evaporated, be
parallel to its opposite surface. After some practice,
we found that two surfaces could be made parallel to within
1 mil by successive grinding and checking for variations
with calipers. Better accuracy was then obtained by using
a sensitive pressure gauge.2 This gauge had a full-scale
variation of 12.0 mils; each mil was divided into 10 parts,
so 0.1 mil accuracy was possible. Variations in the sample
thickness were measured with this gauge; if the two surfaces
Were not parallel, one of them would be lapped with more
Pressure on the thickest area. The sample was then cleaned
and the uniformity of thickness checked again with the
Pressure gauge. In this manner we were able to Obtain

surfaces parallel to within 0.1 mil. More typical variations,

M

The pressure gauge was a style 1000 gauge made by the
Sheffield Corporation with serial number 2201RS.

 

103

Table II - Experimental Results of IsotOpic Studies

Diffusion Temperature 673°K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes Diffusion Depth (mils) Ratio DB/D7 Time (min)
7
6Li I 2Li
31.51 i .3 ,30.31 i .15 1.050 1 .012 23
35.00 i .15 I30.55 1 .15 1.091 1 .005 29
25.00 i .3 :20.70 i .20 1.125 1 .015 20
Average : Ratio 1.091 1 .02
Diffusion Temperature 073°K
Diff. exp 7.50 i . : 7.09 i .2 1.095 1 .09 120
7.50 i . ' 7.01 i .2 1.120 1 .09" 120
9.71 + .1 | 5.53 + .05 1.210 + .02 120
_ , _ _
5.05 i . ; 5.09 i 0.1 1.102 1 .00. 120
5.05 i . , 7.59 1 .1 1.155 1 .05" 120
10.30 i .2 113.50 1 .1 1.105 1 .05 350
2 pairs 5.51 + .2 I 5.35 + .2 1.07 + .10
diffused — — _
together 9.15 i .2 , 5.91 i .2 1.05 i .10 120
0 pairs 5.01 i .15 ' 7.50 i .15 1.11 i .05
diffused 7.59 i .15 : 7.35 i .15 1.05 i .05
simultaneously 7.37 i .15 ' 6.68 i .15 1.22 i .08 120
5.55 i .15 t 7.00 i .15 .95 i .05
average 7.16 ‘ 7.08 1.09 i .08
Average I Ratio 1.096+ .06
, _
Drift Temperature 318°K
Drift exp. 50.20 i .3 ' 1.110 1 .02 23 hrs.

 

I50.00 i .3

 

 

 

"This is a repeat of the data on the above line with a correction

factor to account for room temperature drift and diffusion.

100
however, were 0.2 or 0.3 mils (0.1 mil = 2.50 0). At this
point the thickness of the sample was measured. This meas-
urement was made with the pressure gauge and a set Of reference
blocks which varied in thickness by one—mil steps. The
pressure gauge was first adjusted to read zero on some com—
bination of reference blocks whose thickness was near that
Of the sample. The reference blocks were then removed and
the sample was inserted under the pressure gauge. The gauge
then measured the difference in thickness between the refer-
ence blocks and the sample. For samples which were not flat
within 0.2 mil, the thickness was recorded at the center and
four corners of the sample. The sample was then beveled at
one edge to assure its orientation. This procedure was
followed for both samples in the experiment. The samples
were then ready for the lithium deposition without further
surface treatment.

Following the diffusion, we generally observed a sur—
face layer Of black material one mil thick or less. Measure-
ments indicated 0.5 mil increase over the original thickness
before diffusion. This thin black layer was probably a
Ge-Li alloy with some residue from the mineral Oil burned
Off during diffusion; it could be easily rubbed off with
CH30H and a Q-tip. Measurements were taken from a zero
diffusion depth at the cleaned surface. The cleaned surface
often had small pits which were avoided in the measurement.

Mineral Oil was used to coat the lithium surface follow—

ing the evaporation. Also, the sample evaporated first was

 

105
stored in mineral Oil for the nearly one hour needed to per—
form the second evaporation. The amount Of mineral Oil on
the sample surfaces was reduced to a minimum immediately
prior to placing the samples in the oven, in order to mini-
mize the time needed for it to be burned Off. One possible
source of error may be due to the mineral oil's burning
Off one sample sooner than the other, hence allowing that
sample to become slightly hotter than the other. In retro-
spect, it appears that xylene should have been used instead
of mineral oil for this reason.

The measurement technique was similar to that used by
Pratt and Friedmann51 and consisted of measuring the sample
thickness and the conductivity of the lithium surface be-
tween successive lappings of the lithium surface. Each
lapping removed 0.3 to 0.5 mil of material; the sample was
cleaned before each measurement with trichloroethylene.

The conductivity was measured with the hot—point probe,

which was found to give readings that were strongly depen-
dent on the applied pressure. Consequently, a mechanical
support was arranged to apply equal pressure on each
measurement. The hot-point probe, as described earlier,

was connected to a voltmeter to indicate the thermally-
induced voltage. The location of the junction at the diffus—
ion front required only monitoring variations in the thermal
voltage, hence the actual conductivity was not determined.
The thermal voltage was found to be constant away from the

junction; it began to decrease at about 1 to 2 mils from

 

106
the junction. At that point, the amount of material ground
Off after each measurement was reduced to 0.1 to 0.3 mil.
Grinding and measuring were continued until the thermal vol-
tage changed polarity. In this manner we were generally
able to measure the diffusion depth to 0.1 mil. The measure-
ment process generally took one-tO—two hours. If work was
done on one sample until the measurement was complete, the
sample with the other isotope would be left standing.

During this time, lithium continued to drift through the

 

crystal due to the built-in field of the junction. Measure-
ment Of the diffusion depth in one crystal obviously does
not stOp this drift effect in the other crystal. An esti—
mate of the largest effect to be expected can be made by
assuming a step junction. The internal junction voltage is
then about V = 0.0 volt and the resulting drift depth is

given by w = quit)1/2. At room temperature the mobility u

is 3x10"10 cm2/volt-secll, and a drift depth Of 0.0 mil
results in one hour. The actual junction is graded so that
drift depths of 0.2 mil or less may be expected in one hour.
Similarly, the diffusion would continue at room temperature
and produce an additional .13 mil in the diffusion depth in
one hour. Most of our measurements, with the exception of

a few of the earlier ones, were done on the 6Li and 7Li
samples simultaneously. That is, a thin layer was ground
off the 6L1 sample, and the thickness and thermal voltage

were recorded. The same was then done for the 7Li sample,

and the measurements alternated back and forth until the

107

junction depth of both samples was recorded. The difference
in time between the location of the two junction depths was
consequently reduced to five minutes or less.

Measurements were also performed on crystals diffused
at 100°C; however, the solubility of lithium in germanium
at 100°C is only about 2x1012/cm3. This is not significantly
different from the original acceptor concentration. The hot-
point-probe measurements therefore showed large variations in

thermal voltage, and the junction could not be located within

1 mil.

7.3 Comparison of Experiment with Tunneling Calculations

The 200°C and 000°C measurements did show variations
from the classical prediction in the ratio of diffusion
constants. This led us to try and fit the tunneling theory
(eq. 2.5) to the existing data on 7L1 diffusion constant
simultaneously with our own data on the ratio. To do this
we first needed the distance (a) between adjacent lithium
interstitial sites in germanium. The value finally arrived
at was scaled from existing data on lithium localemodes in
silicon.3b This left us with the barrier height 0 and the
local-mode frequency as parameters to be scanned. The best
fit of the diffusion constant was obtained for Q in the
range 0.515 to 0.530 electron volt, and any value of local-
mode frequency in the neighborhood of 1013 Hz. The classical
activation energy (Q) given by Fuller and Sevrienslu was 0.065
ev; Q determined by the tunneling expression, (eq. 2.5) is

expected to be larger because it includes the energy between

 

108
the bottom of the potential well and the ground state of the
ion. The fit of the tunneling expression to 7Li diffusion
data is shown in Figure 7.1. Error bars indicate the range
of variation in the calculated values as Q varies from 0.515
to 0.530 ev. The local—mode frequency (v) was scanned for
a constant ratio of 06 to v7 equal to 1.08. The range of
frequencies scanned was determined from the coefficient D0
in the classical expression for the diffusion constant (see
eq. 2.2). We found that any frequency in this range would
give agreement with the experimental data on diffusion
constants but only a discrete set Of frequencies would fit
both the magnitude and ratio of diffusion constants. These
results are shown as a plot of ratio vs. local-mode frequency
(v) in Figure 7.2. The shaded area represents the error in
our measurements of the ratio. The best fit of the calcu-
lations to all the data indicates a ratio (D6/D7) = 1.12
at room temperature. This effect is considerably smaller

than hoped for, and represents a savings Of only 10 hours

in a ten-day detector drift.

109

 

       
  

 

 

 

(°K) IOOO 700 500 400 350 300 250
I T I l I I I I I
(9312) Solid curve I: tunnellng
sec calculation Q= 5231.007“
'0-6 V7 = .494 XIO 233°C-| —
.7 '—
IO r-
:0"8 — "
I0'9 - I
' Fuller and Severiens
'0 Physical Review 96, 21 _‘
m" r- 1950
X A. H. Sher
Journal Of Applied Physics
E, 2500 (1955) X‘
-ll "
IO r-
l I l l l L I I l l I 1 l
LO 2.0 3.0 “91:92 oK-l)
FIGURE 7,I Tunneling calculations compared with

experimental high and low temperature data.

110

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I

CHAPTER 8

SUMMARY AND CONCLUSIONS

The purpose of our isotOpic-diffusion studies has been
to determine whether a significant time saving could result
in nuclear—detector fabrication if 6Li is used instead Of
7Li. A review of existing data showed that the apparent
activation energy Q for 7Li in germanium increased with
increasing temperatures. Such an effect might be caused
by barrier tunneling at the lower temperatures; such a
tunneling mechanism would tend to increase the ratio of
diffusion constants D6/D7 at the lower temperatures. We
have measured this ratio and found it to be 1.091 at 000°C
and 1.096 at 200°C. These ratios are greater than the 1.080
value obtained from classical calculations and therefore
indicate the possibility of tunneling. Calculations based
on a diffusion theory which incorporated tunneling effects
were performed. Good agreement was found to exist between
the calculations and existing 7Li diffusion data with our
measured ratios. These calculations and experimental data
indicate a ratio of about 1.12 at room temperature. This
effect is smaller than had been hoped for and translates
into a saving of only 10 hours in a ten-day detector drift.
These calculations did not evaluate the 7Li local-mode
frequency or any of the effects of the lattice on the local

mode. Consequently, the infrared-absorption studies were

undertaken. 111

 

112

The study of local modes in germanium proved to be too
difficult because of sample—preparation problems; we there-
fore studied local modes in silicon. These local modes
are similar in many important respects to the local modes
in germanium. For this work it was necessary to compensate
the lithium with boron in order to reduce the free-carrier—
absorption background. We then set out to evaluate the
effects of anharmonic terms in atomic displacements on the
local modes through application Of a uniaxial stress. The
uniaxial-stress experiments, however, did not show any
splitting Of the local modes. In fact, the stress necessary
to split the two-fold degeneracies appears to be considerably
larger than the stress necessary to fracture the sample.

The thermal experiments determined the line width and
centroid shifts of the local modes as a function of temper—
ature. A linear temperature dependence in line width and
centroid shift was observed at temperatures above 300°K.
A theory describing these anharmonic effects on local modes
in a C3v symmetry has not yet been developed. We modified
the theory of Elliott et al. in an effort to account for
the actual C3V symmetry of the lithium-boron system. We
then compared calculations based on these ideas with the
Observed temperature dependences.

Data and calculations for centroid shift were found
to agree reasonably well. On the basis of the modified
theory of Elliott et al., this agreement indicated that the

centroid shifts were the result of anharmonic effects due

tO fourth-order terms in atomic displacement.

pv

113

Two processes were considered in calculating the local-
mode line widths; two—phonon decay and elastic-scattering
of band phonons. The elastic-scattering mechanism was re—
jected on the basis of its T2 dependence at high temperature.
The two—phonon-decay process was then fit to the experimental
data at 300°K and above. The agreement between calculations
and experiments at temperatures above 300°K was fair; however,
below 300°K there was no agreement. It will therefore be
necessary for a rigorous theory, based on the actual C3V sym—
metry of the local mode, to be deveIOped in order to fully

explain both the line width and centroid shift processes.

 

 

BIBLIOGRAPHY

10.

ll.
12.

13.

BIBLIOGRAPHY

R. Chrenko, R. McDonald and E. Pell, Phys. Rev. 38,
A1770 (1965)

P. M. Pfeuty, Proc. Int. Conf. on Localized Excitations
in Solids, Irvine, p. 193 (1967)

M. Balkansik and W. Nazarewicz
a. J. Phys. Chem. Solids 25, 070 (1960)
b. J. Phys. Chem. Solids 31, 571 (1955)

c. and Pfeuty, Lattice Defects in Semiconductors,
R. R. Hasiguti, ed., The Pennsylvania State
University Press, Univ. Park, Md., p.3 (1968)

V. Tsevetou, W. Allrod, and W. G. Spitzer, Proc. Int.

Conf. on Localized Excitations in Solids, Irvine p.

185 (1967)

W. G- Spitzer, NASA Report No. Cr-97309 (1968)

L. Bellomonte and M. H. L. Pryce, Pro. Phys. Soc. 8:,
pp. 967 and 973 (1966)

R. J. Elliott and P. Pfeuty, J. Phys. Chem. Solids 28,
1789 (1967)

W. Hayes, Proc. Int. Conf. on Localized Excitations in
Solids, Irvine, p. 159 (1967)

W. A. Spitzer and M. Waldner
a. Phys. Rev. Letters 1i, 223 (1965)
b. J. Appl. Phys. 36, 2050 (1965)

M. Waldner, M. A. Hiller and W. G. Spitzer, Phys. Rev.
100A, 172 (1955)

A. H. Sher, J. Appl. Phys 19, 2500 (1955)
s. M. Pell, Phys. Rev. 119, 1010 (1950)

L. J. Giacoletto, Proc. IRE 39, 921 (1961)

110

10.

15.

16.

17.

18.

19.

20.

21.

22.

23.

20.

25.

26.

27.

28.

29.
30.

31.

32.

33.

115
C. S. Fuller and J. C. Sevriens, Phys. Rev 96, 21 (1950)

R. Aggarwal, P. Fisher, V. Mourzine and A. Ramdas,
Phys. Rev. 138, A882 (1955)

B. I. Boltaks, Diffusion in Semiconductors, Academic
Press, Inc. ch. 3 71963)

 

 

C. Wert and C. Zener, Phys. Rev. 76, 1169 (1909)

. S. Yeremeyev, Phys. of Metals and Metallography
5, 139 (1968

m<

. Dawber and R. Elliott
. Proc. Roy. Soc. A273, 222 (1963)
. Proc. Phys. Soc. 81, 053 (1963)

U'DJ'U

J. Angress, A. R. Goodwin and S. D. Smith, Proc. Roy.
Soc. A287, 60 (1965)

E. M. Pell, Appl. Phys. 31, 1675 (1960)

H. Reiss, C. S. Fuller and F. J. Morin, Bell System
Tech. J. 35, no. 3 535 (1956)

F. A. Kroger, Chemistry 9: Imperfect Crystals, North
Holland Publishing Co., Amsterdam (I969I’ch. 9, sec. 2

 

 

E. M. Pell, J. Phys. Chem Solids 3, 77 (1957)

R. J. Elliott, W. Hayes, G. D. Jones, H. F. MacDonald
and C. T. Sennett, Proc. Roy. Soc. Lon. 289A, 1 (1965)

W. Hayes and H. F. MacDonald, Proc. Prog. Sec. (London)
A297, 503 (1957)

H. Y. Fan, W. Spitzer and R. J. Collins, Phys. Rev. 101,
566 (1955)

H. F. Wolf, Silicon Semiconductor Data, Pergamon Press,
(1969) pp. 1 2-110. “—

 

Ibid., p. 115

M. Lax and E. Burstein, Phys. Rev. 97, 39 (1955)

B. N. Brockhouse and P. K. Iyenger, Phys. Rev. 111,
707 (1958)

R. J. Colling and H. Y. Fan, Phys. Rev. 93, 670 (1953)

A. E. Cosand and W. G. Spitzer, Appl. Phys. Lett. 11,
279 (1967)

30.

35.

36.

37.

38.

39.

0O.

01.

02.
03.
00.
05.

06.

07.

08.

09.

50.

51.

52.

116

E. M. Pell, Symposium on Solid State Physics in
Electronics and Telecommunications, Academic Press,
New York, 6, 261 (1960)

M. Kahlweit, R. A. Swalin, and R. D. Weltzin, Progress
ip_Solid State Chemistry, (Reiss, ed.), 2, ch. 5 and'6

 

 

A. M. Sher and J. A. Coleman IEEE Trans. Nucl. Sci,
NS-l7, 125 (1970)

 

R. A. LOgan, Phys. Rev. 101, 1055 (1956)

 

L. P. Hunter, Handbook g£_Semiconductor Electronics,
second ed., McGraw—Hill ——

 

 

Donald E. McCarthy, App. Optics 2, 591 (1963)

Smith, Jones and Chasmar, The Detection and Measurement
of Infrared Radiation, , Oxford Press, ch. 2—(1963)

——

 

 

 

 

R. P. Bauman, Absorption Spectroscopy, John Wiley and
Sons Inc., (1963) ch. 2

 

Fritz Kneubuhl, Appl. Optics 3, 505 (1969)
Smith, Jones, and Chasmar gp. cit., ch. 3
Kneubuhl, 923 cit.

Wolf, gp. cit., p. 200

G. L. Cloud and J. T. Pindera, Experimental Mechanics I,
(May 1968)

R. C. Newman and J. B. Willis, J. Phys. Chem. Solids 26,
373 (1965)

W. Hayes and D. R. Bosomworth, Phys. Rev. Letters 23,
551 (1959) _‘

F. M. Smits, Proc. IRE 06, 1009 (1958)

R. N. Hall and T. J. Solyts, IEEE Trans. Nucl. Sci.
05-15, 150 (Feb. 1971)

B. Pratt and F. Friedmann, J. Appl. Phys. 37, 1893 (1966)

J. P. Maita, J. Phys. Chem Solids 0, 68 (1958)

APPENDICES

Because of the small local-mode absorption coefficients

in germanium, an Optimization technique was used to deter-
Inine the best sample thickness. This technique was carried

OLrt for the split-beam system described in section 0.2.

lie ihave considered the system in which a difference measure-

. O > 7 0 O

nmant'ls formed with a compensated L1 sample in the upper

b . . . 6 .
«seam, and either a pure Silicon sample or a compensated L1

searnple in the lower beam. The intensity of the upper beam

(5311) and (SL) of the lower beam, upon entering the monochro—

nhaftor, can be expressed as

and SQL are the

Vvliexre x is the sample thickness and SO
and aL refer

:irliqtial upper and lower beam intensities; a
Back—

‘tCD .absorption coefficients due to sample attenuation.

XZrWDLnid effects are tuned to a zero on a spectral region of

FND lIDcal-modes, i.e. we adjust SO by means of a variable

we d 88 so that

117

The absorption coefficient aFC and ab, refer to the free—
carrier-background absorption and lattice absorption respec—

tively. We then have

where
: —. .. ‘(
AA (“rcxu aFCXL+aL’Xu “5"L) '
'Thue detected signal S is then the difference between Su and
ESL.’ For a compensated sample in the upper beam and pure

ggerrwanium, of the same thickness, in the lower beam

8 = SOu e-GBXIe-aLMX-l] ,
vvlfieere aLM is the local-mode absorption coefficient and GB
$153 the background absorption, i.e. aFC +aL,. For
aLMX<<l
Vie have X
-a
s = sou e B aLMX .

TWIea Optimum signal with respect to sample thickness is then

gziayend by

Jveuiflflw

 

119

The Optimum sample thickness is X0 given by

l

onj— +
“5 “PC “L:

+ a in a reasonably compensated

The typical values of aFC L’

so that X0 = 330 u. However, the

sample are 20-00 cm-
+ i) is slowly varying near the maximum so

function (~08
These thicker

tfllat 600-7000 thicknesses are also adequate.

smamples are much easier to prepare.

APPENDIX B

In order to evaluate the thermal breaking of Li-B
complexes, eq. 2.19 is first expressed in terms Of

6 = P/N and D(T) in the form

2 3

'Tkne integral 0(a) is then defined by eq. 2.18 where

 

<1 = e2/ekTa. The form
1 ekT 2 8
log 0(a) = log {——( ) ——-———7}
”“ e2 N(l-6)
19 3
1:}1en results, where, for our samples, N = 10 /cm .

TFéaliles Of log 0(a) vs. a then allow 9 to be evaluated for
Eirisr temperature. This has been done for several temperatures
IJESqug as a distance of closest ion-to—ion approach 2.6 A.
'quea results are plotted in Figure B.1. The Observed thermal

lDlseLaking of complexes would be much more rapid for larger

Values of N .

120

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pom OLSPOLOQEOO. .m> on mcofl omswmd mo cowpomsm _.m MEDGE

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A. _ _ ___ ___ _a__3___1__

 

 

APPENDIX C

Use of bakelite contacts between the sample and the pressure

clamp, raised the question of friction at the sample contacts

avid the resulting possibility of large components of stress
Exarpendicular to the direction of application. When uniaxial-

stloess is applied to silicon, it expands in directions per-

pueridicular to the applied force. But, friction at the pressure

agbp>1icator contacts opposes this expansion. The result may

1362 local stress components considerably larger than the applied
satiress. The amount of lateral expansion is indicated by Pois-
sscxn's ratio, defined as the change in sample width perpendicular

1:c> the applied force, divided by the change in length along

For stress applied to the

‘tlie directions Of applied force.
05

[:]_10] direction of silicon, Poisson's ratio is 0.33. It

also possible that stress variations may exist in the

ixs
sseunple around regions of lithium precipitation discussed

decided to evaluate the stress

eéirlier. It was therefore

Var‘iations across the loaded sample by birefringence tech-
rliques.26
The test set-up for the birefringence work is shown in

Irigyare C.1. In this system circularly polarized and nearly

monoChromatic light is incident upon the sample such that

t22e 'two components of the light are in the plane of the

122

123

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120
sample. The silicon crystal under applied stress is Opti—
cally anisotropic, 1.3., it has different values of refrac-
tive index if different directions, or in other words, the
two components of light will prOpagate through the sample
at slightly different velocities.2 When these two com-
ponents are remixed to produce a plane polarized wave, an
interference pattern results. The pattern of a loaded
sample (one which has uniaxial stress applied) indicates the
amount of applied stress when compared to the pattern of
an unloaded sample.

The components of the birefringence system are standard,
and their function is easily explained. The incandescent
source has approximately a black—body radiation characteris-
tic. The filter is used to block most of the unwanted
visible radiation which would not pass through the silicon
sample. The first polarizer,22 set at zero degrees, is used
to establish a polarization reference; it produces plane-
polarized light which can be represented by PX. The x axis
has been arbitrarily chosen as vertical, and the z axis is
in the direction Of propagation. The first quarter—wave
platefiflfi then produces a sum of left (P_) and right (P+)
Circularly polarized light. Each circularly polarized wave
can be represented by components PX and Py, of plane polari—

zation as P+ = (l/2)l/2

(PX: i Py). The sample then retards

If the crystal is viewed as a continum of refractive index
.na the velocity of propagation is given by c/n.
"WAhren's-type polarizers made of calcite crystals are used.
LThey have typically 00% transmittance out to 2.3 microns.

"“*The necessary retardation material is available from
p<31aroid Corporation.

125

the propagation of Px with respect to PV as determined by

the corresponding anisotrOpy in refractive index. The

second quarter-wave plate combines the resultant field into

«a plane-polarized wave. The second polarizer can be rotated

tc> determine the angle of maximum intensity or to determine

(wirdations in intensity vs. polarization angle. Since

si_licon doesn't transmit in the visible, infrared radiation

hand been used throughout. The photo-converter tube is

neacessary to convert the infrared light to visible light
fc>r1the measurements.
The intensity maxima or minima are referred to as

They arise as the x and y components of P+ and P_

If

fxninges.

ear‘e recombined following passage through the sample.
‘lc3ading produces a uniform stress across the sample, the

tvliole sample surface would appear to change uniformly from

1Light to dark as the second polarizer angle is varied. A

IWCDnuniform variation in applied stress gives rise to light

aIId dark regions or fringes as the different polarization

CKJmponents are delayed by different amounts. To evaluate

tile total stress variation across a loaded sample, a fringe

1~irle is located at one edge of the sample by rotation of

'tI1e second polarizer. The polarizer angle 61 is then re-

c3C>rded. The second polarizer is then rotated slowly so

'tlieat the fringe line appears to move across the sample.

‘Vifieen the fringe line appears to reach the Opposite edge

(355‘ the sample, the polarizer angle 02 is again recorded.

“7}1ifile the polarizer is being rotated, the number of times n

126
that the original fringe line position changes from light
to dark to light again is also noted. This is usually ob-
served to be less than one cycle; hence it doesn't enter
the calculations. The total stress variation across the

loaded sample is then simply

%% = (n + 1%0) 100% .

This technique was applied to pure silicon and to boron—
doped lithium-compensated silicon samples. NO significant
differences were observed. In particular, we did not Ob-
serve any local stresses which could be attributed to
lithium precipitation sites. We found that the stress
variations were determined primarily by how uniformly the
ends of the sample made contact with the bakelite. The
sample ends were ground with aluminum oxide on a glass
plate until they appeared to be parallel to the bakelite
contacts in the pressure clamp. In order to ascertain an
upper limit on the possible stress variations, a compensated
sample was made up for the birefringence testing which had
a more non-uniform contact with the bakelite than did any
of the actual experimental samples. These results showed
that the stress variations decrease with applied pressure.
At 7.6 kg/mm2 the stress variations were :100%, while at
18 kg/mm2 the variations were 150% and at 23 kg/mm2 the
variations were :01%. When some care was taken to fit the

sample into the pressure clamp, the typical stress variation

127
at 23 kg/mm2 was :10%. The decrease in stress variations
with pressure is likely to be due to compression of the
bakelite causing more uniform contacts to the sample.
Friction between the sample and the bakelite was found to
be a relatively small effect, possibly as a result of

expansion of the bakelite.

APPENDIX D

In their analysis of lithium—boron pairs in silicon,
Elliott and Pfeuty7 have considered a basic six—atom model
described in the text (see Figure D.l). The pair symmetry
in C which has a mechanical representation for displace-

3v

ment vectors given by Srl + F? + 6F3 for the six-atom model

chosen. The ri's are irreducible representations Of the

C symmetry with F and F being one dimensional and F

3v 1 2 3
two dimensional. Both F1 and F3 are infrared active. The
symmetry coordinates given by Elliott and Pfeuty for F1 are

C1 = -%f-(X1+Yl+Zl) ,
/3
c2 = J— (X5+Y5+Z§) ,
/Ԥ' \
c3 = ——l—— (X2+Y3+Zu) ,
V3
C0 = ._.__1 (Y2+27+23+X3+X0+Y0)
/€ , .

and

is

z; (X+Y+Z) .

128

129

 

 

Si
0

B
(:flfl

I V
LI
.5 :
1”[
FIGURE D.I The 6-atom model used as

basis for Lithium-boron local—mode calcu-
lations.

130

1 . .2
: —— \ + + '
n1 /§ (21 jYl 3 Z1 ’
_ 1 . .2
_ 1 . .2
n3 - Tr;— (X2+]Y3+j Z”) ,
_ 1 . .2
nM - —;— (Y2+jz3+j X“) ,
_ 1 . .2
T15 - 7:31- (Zz+jx3+j Y”) ,
arud
_ l . .2
Mtith a second coordinate for F3 of Ci = nix, where i =

1.,2,3,0,5,6. The coordinates X,Y,Z are cartesian coordinates
C>f the atoms, where each atom is represented by a subscript,
2;.3. the subscript 1 refers to the lithium ion and 5 refers
‘tO the boron ion: j represents the complex cube root of
llnity, and 2,3,0 and 6 refer to the surrounding silicon
éitons. The two-fold-degenerate boron local mode is then
I‘epresented by one of the two dimensional r3's while the
CHie-fold degenerate boron local mode is represented by one

fo the F The two—fold-degenerate lithium local mode

l's'
ifs similarly represented by one Of the remaining five F3,S.
ITt is observed that the above symmetry coordinates do not

irlvolve a mixing of the silicon, lithium, and boron coord—

lhates. Therefore, in a first-order approximation to the

cactual dynamics of the lithium—boron system, each local>

131
mode may be considered independently of the other local
modes. An analysis of anharmonic effects similar to that
of Elliott et al.25 can be done for each local mode, taking
care to note the appropriate representations as F1 or F3.
Whereas, Elliott et al. find a Green's function which trans-
forms like the irreducible representation F5 Of Td symmetry,
we would have a Green's function which would transform like
the irreducible representation F1 or F3 of the C3v symmetry.
.After performing symmetric products on these representations
and.by use of symmetry arguments to determine which terms
«are zero, a third—and fourth—order Hamiltonian of the same
.form found by Elliott et al. is expected. However, the
local-mode operators and expansion coefficients should be

labeled according to which representation and which ion they

represent. Equations 2.9 and 2.10 would then appear as

 

 

 

..)
H=Z(f-1§é}r<~)—)£B 2( ’I‘ )[S ,+ 2
3 4 Lhm 13 2” Q 2 X- (bxr 2 + b r 2) J
k ‘ 2“r3 mu ’y 3 X 3
I1
+8 2 (~ )(b 1 2
F1 2M 0 2e. 21‘3 + sz 2)
2 r n 1
3
+8 1( 2‘ )[Z(b 1 + >21}
r3 2M 9 2 x XF3 + bxr l
1. F3 (0M ’y 3

(a(k)+a+(k))

132

 

, _ h(b(i€)b(fi'))1/2 CH2 .A .+ 2
L - 2 —~ __0_—— I (b 2 +b 2)
2 + + 00 “MW M 0 2 x y X123 xr3
k,k’ M " “2”r3 ’
Cr12 + 2
+ (b F 2 +b F 2)
1(2st 2 1 z 1
1
(,1. l
+ “3 I“) 1" 1 + 5* 1)2}( (15+ (Em (k’)+ +010)
M Q 1 x,y X 3 x? a ‘ a a a ’
l P 3
3
where M1 is the lithium mass and M2 is the boron mass. The

notation of Elliott and Pfeuty has been changed slightly
so that the boron ion is designated by a 2 instead of a 5.
The labeling Of each of the observed local modes is summar-

ized in Table 3.

Table III — Labeling of Local Modes

 

local _
mode (cm ) label
1
522 F3
2
560.5 F1
2
1
65+ F3

 

The temperature dependence of the centroid-shift and
line width then takes on the form Of Eq. 6.1 and 6.5 for
each individual local mode in this approximation.

The presence of two boron local modes at the same
coordinate location makes it seem likely that their inter—
action is comparable with the above effects. By analogy

with the development of Elliott et aZ., a line-width

133

temperature dependence is expected to be given by

.)

b(k)(n(E)+l)(n(n')+1)5(b(fi)_0 2 +0? 2) ,

I
I“
k 3 l

where 0’ is the boron local-mode frequency which is not

decaying.

APPENDIX E

The programs for centroid shifts and line widths in-

corporated a silicon phonon density of states taken from

a paper by Dawber and Elliott.19a

below for Aw = 8 1/3 cm-l.

The values are given

 

Table IV - Phonon Spectrum

i g(iAw) i gCiAw) i g<iAw>

1 0.0 21 0.20 91 1.u0

2 0.0 22 0.05 H2 1.80

3 0.10 23 0.25 H3 0.80

u 0.10 2H 0.25 an 0.90

5 0.0 25 0.u5 MS 0.35

6 0.10 26 0.30 H6 0.55

7 0.90 27 0.60 H7 1.80

8 0.00 28 0.u5 98 1.55

9 1.00 29 0.60 H9 2.30

10 1.35 30 0.70 50 2.80
11 2.90 31 0.85 51 2.85
12 3.50 32 0.95 52 3.00
13 7.80 33 1.10 53 3.05
1a 9.50 3H 1.u5 SH 3.20
15 10.60 35 2.00 55 2.30
16 12.15 36 2.55 56 1.95
17 9.55 37 2.85 57 1.u0
18 0.25 38 2.90 58 1.20
19 0.0 39 2.15 59 18.60
20 0.25 MO 2.u0 60 31.00
61 9.80

 

 

13H

135

For normalization purposes, it can be noted that

Z g(iAw) = 173.55 .
i=1

The equation to be programmed for centroid shift is

”M
1. g(w)fiw[n(w)+ l/2]dw

w
I M ng) dw

O

t)
K)

= constant.

on]

To program this equation it was first necessary to convert
the integrals to sums; then dm becomes Aw, w becomes iAw for

integer values of i, and w“ becomes 61Aw. Equation e1 then

 

becomes
61
Z 8(iAwyhiAw[n(iAw+1/2]Am
t t . i=1
CODS an 61
X g(iAw)Am
i=1

which simplifies to

 

i g(iAw)[n(iAw+1/2)] .

The term c/M1002 HwM2 determines the leading constant and
has been denoted as GAMMA in this program. It is evaluated
in terms of the local—mode frequency OMEGA on line 5 of the

program for centroid shift which follows. As a first try c

9
was taken as 10“1 ergs/cmu. GAMMA is multiplied by 6 on

136

line 8, to account for the six branches in the phonon-
dispersion relation. The loOp over temperature, in 200K
steps, begins on line 11; the sum over the lattice spectrum
begins on line 13. The term AA represents the temperature-
dependent factor. On line 17 the coefficients GAMMA and
‘hAw/Z g(iAw) are multiplied in to form AQ/QO. The program

1
was then run with lines 7 and 18 deleted. The results of
this calculation were then expressed as AQ/QO = f(T) and
compared with the experiments. The constants C and 90 were
adjusted to fit our high—temperature data in magnitude and
slope. The constant C then represented the variation of c

from the value of 1021

ergs/cmu. For the purpose of calcu-
lating Q, the zero temperature shift in AQ/QO was subtracted
out by adding QOC f(T=0) to 00. This sum is denoted OMEGO

in the following program where line 18 calculates

Q

a +9 Cf(T=0) - Q Cf(T) - Q C2f(T=0) f(T)
O O O 0

ll

0 +0 Cf(T=0) - 0 CfCT)
o o o

The second-order term -QOC2f(T=0) f(T) represents an error

of typically less than 1% in calculations of the high-
temperature slope. The scale factor C can be conveniently
multiplied into GAMMA on line 8. The program for centroid

shift of the 659 cm-1 local mode is

10

11

12

13

1”

15

16

17

18

19

2O

21

22

23

data card.

30

MO

100

200

137
DIMENSION G(62),ANS(M0),GG(62)
READ 100, (G(I),I=l,61)
CENTROID SHIFT - 65a - SILICON FNONON SPECTRUM
OMEGA = osu

GAMMA

(1.0/(10.81*1.6*28.09*1.6*((6*3.1ul6*5.18

C*6*3.1H16*OMEGA)**2)))*(10**25)

OMEGO = 663 .0
GAMMA = 6*GAMMA
GG(1) = 0.0

DP = (6.625*2.5/l.38)

00 no N=l,36

T=20*0*N

DO 30 I=l,60

AA: (l.0/(EXP(I*DP/T)-l.0))+0.5
GG(I+1)=GG(I)+I*G(I)*AA
CONTINUE
ANS<N>=CO<61)*CANNA*50*3.1u16*1.05u/(<IO**173.55)
ANS(N)=OMEGO*(1.0-ANS(N))

PRINT 200,(T,ANS(N))

CONTINUE

FORMAT (6F10.3)

FORMAT (8X,I3,10X,E10.5)

END

DATA

The data is right justified in each of six fields per

There are three decimal places to the right of

the decimal point, and each field is 10 characters wide.

138

The calculations for the 560.5 cm.1 local mode can be done
by simply replacing line u with OMEGA = 569.5 once the
appropriate factors C and 00 have been determined as above.
Calculations for the 522 cm.1 local-mode are similar,
except that GAMMA must be multiplied by 10.81/6.9u to account
for the change to a lithium atomic mass.

The equations to be programmed for the two-phonon-decay

line width is

 

 

61
1 'M . . . . .
% = CA“‘“ A“ , 36 Z g(1Aw) gCMAw-iaw)1(M—i){(n(IAw)+l)
[Z g<iAw)]‘ i=M-61
X(n(MAw-iAw)+l)-n(iAw)n((M-i)Aw)},
where
,2
GAMMA =41'3 "fiv2 u .
MM’Q M w
M
and
M is an integer such that MAw = QF i .
j

In this program the total temperature dependence is calcu—
lated to be the variable AC. The sum over the phonon
spectrum is done in the variable X(J). In line 21 the

sum over the phonon spectrum is multiplied by GAMMA and Am
in CM"1 and divided by the phonon-spectrum normalization
factor squared. The coefficient13’ was varied until a best
fit in magnitude and slope was obtained. The resulting

program is

10

11

12

13

1M

15

16

17

18

19

2O

21

22

23

25

2O

30

1+0

50

139
DIMENSION X(61),G(6l),ANS(u0)
READ 50, (C(I),I=1,61)
LINENIDTM-Q PNONON DECAY — SILICON SPECTRUM
OMECA = 65H
M = OMEGA/8.333

GAMMA=((“*3.1U16*1.05H)/(U*10.81*1.6*OMEGA*6*3.1H16*
((2.809*1.6)

C**2)*((5.18*2*3.lul6*3)**u)))*(10**9)
GAMMA = 36*GAMMA

DP=12.01

L=(M—61)

DO 30 K=1,36

T=20.0*K

DO 20 JzL,61

AA: l.0/(EXP(DP*J/T)-1.0)

AB: 1.0/EXP(DP*(M—J)/T)-1.0)

AC: (AA+1.0)*(AB+1.0)-AA(AB

X(L-l)=0.0
X(J)=X(J-1)+G(J)*G(M-J)*J*(M-J)*AC
CONTINUE
ANS(K)=(X(61)*GAMMA)/((173.55)**2)*8.33
PRINT H0,(T,ANS(K)

CONTINUE

FORMAT (8X,I3,10X,E10.3)

FORMAT (6F10.3)

END

IDATA

IUD
The equation to be programmed for the elastic—scattering

line width is

_1_ z 2" [(r, jg? )(h(?v+3)\2]3mm
T

2 2 . 2

5 n
X g4(iAw)i2n(iAw)(n(iAw)+1) .

The coefficient in square brackets is equal to (89.0/OI‘GEGA)xl0_2
for (C - ZBQ/M’02) = 1021 ergs/cmu and OMEGA in cm’l. The
sum Zg(jAw) is done with the variable 8(J) with the result
stored in 8(61). The temperature dependence is given by

the variable AA and the sum over the phonon spectrum is done
in the variable GA(J). In line 19, GA(61) is multiplied

by Am and GAMMA, and divided by (8(61))2. The program is

10

11

12

13

19

15

16

17

18

19

2O

21

22

23

2M

10

30

H0

50

60

1N1

DIMENSION S(62),G(62),GAS(N0),GA(62)

READ 50,(G(I),I=l,61)

LINEWIDTM, ELASTIC SCATTERING-SILICON PHONON

DO 10 J=1,6O
S(1)=9.0

S(J+l)=°(J)+G(J+1)

CONTINUE
OMEGA = 659
GAMMA = 2*3.1916*(89.0/(OMEGA*10**2))*36

GAMMAzGAMMA/OMEGA
DP=12.01
GA(l)=0.0

DO uO K=1,36
T=20.0*k

DO 30 J=l,60

AA= 1.0/(EXP(DP*(J+1)/T)-l.0)

GA(J+1)=GA(J)+((G(J+1))**2)*(J+1)*(J+1*AA*(AA+1.0)

CONTINUE

GAS(K)=GAMMA*GA(61)*8.3333/((S(61))**2)

PRINT 60, (T,GAS(K))
CONTINUE

FORMAT (6F10.3)

FORMAT (8X,I3,10X,E10.3)
END

DATA

SPECTRUM

.1111

m
M
M
m
NI||"|||
H

I

INN”)!

|1
2

VIHIHINIW