CDN THE ROLE OF TWO-PHONON PROCESSES IN
THE ENERGY RELAXATICDN OF A HEATED ELECTRON
DESTREBUTION

Thesis for Hm Degree of Ph. D.
MKHIGAN STATE UNIVERSITY
Geraitfi P. Aildma’ge
191615

  
 
   

LIBRARY

Michigan State
University

all“

This is to certify that the

thesis entitled

On the Role of Two-Phonon Processes in the Energy

Relaxation of a Heated Electron Distribution

presented by

Gerald P. Alldredge

has been accepted towards fulfillment
of the requirements for

—P)_h.'_.ll'._ degree in- Phys ic S

I a \

) (‘Ll{

- Major professor

 

Date 2/28/66

0-169

F” THE POLE ”F TWO-PHUTQH PROCESSES 1? TH; TVF"GY
PTLRXRTI“H ?F F hCFTTU TLECTCOC DISTQITUTI‘I

ld P. filldredne

(I)

Y‘

(D

by E

An investigation is rade of the role of multinle acoustical
nhonon scatterinc of carriers as a mechanism contributing to
increased efficiency of energy loss in "warm“ electron ohencnera.
The method adopted comnares the single- and two-phonon contrihutions
to power lost to a cold lattice by a warm Taxwellian distriLution
of carriers. The model of electron-nhonon interaction develooed is
a kind of rinid ion model, which for one—ohonon processes coincides
with the usual standard single valley model for semiconductors.
Two-shonon processes arising from first order and from second order
perturbation theory are considered, and account is taken of their
statistical interference. Expressions for the contribution to sewer
loss from one-ononon processes and from the various kinds of two-
Chonon orcczsses are develooed and reduced to cuadratures. Tae
nuadratures are estimated numerically by random samnling (Monte
Carlo) and by Gauss-Leoendre aonroximation. The estimates sugnest
that the two-ononon orocesses involvino an intermediate state
(second order oerturbation theory) dive a cover loss comnaratle to
that of one-ohonon processes. The contributions to oo~er loss fror

the two-phonon processes of first order perturbation theory and fror

?erald P. [lldredte

the statistical interference between the two tvnes of two-nhonon

nrocesses are insignificant.

7:1; "“LF :gF T"O-Pi" ‘-
Ft‘L;'.x.”:.T1“=si

F”°CFSCL" I“ TEL fN?”CV
1F fl anT"m FerTooq

ISTQIEUTIOS
bv

J

” {lldredoe

. V
Gerald P.”

n
l

TPESI?

Sutmitted to

iichigan State University
in nartial fulfillment of tke resuirements

for the denree of

DOCT07 CF DHlLaST7FY

l

f.

KC)

\JV

DE”artment of Physics and fistronony

{CKJQVLELGEMT‘TS

I wish to thank Professor F. J. ilatt for suhnestino this
research rroblem and for his encouraoewent and stimulatino discus-
sion durino the course of the work. This work was suoaorted in
oart by the U.S. Air Force Office of Scientific Research and in
part by a HSF Cooperative Graduate Fellowshio. I also wish to
acknowledoe the assistance of the Vichioan State University

Comruter Center.

ii

TELLE 5F C77TEETC

CHAPTEP
I. IETR33UCTICT . . . . . . . . . . . . . .
II. PCELI I WPV iEflARKS “T T"C-PH ECU
fiVOCESSES . . . . . . . . . . . . . . . .
III. Ti; STRTICTICfiL VIE FCI T n“; TV? C“ITE“ICH
F”R IVPlfTA CE CF TU“-PV”N“N TCAVVITIOVS.
IV. DEViLfidegT OF THE “UE- 5V9 TVT-?L Hit
IVTCFACTlfld “VFVATODS . . . . . . . . . .
V. TCTAL CL CTPQNIC TRLFSITI?T ”CTES DUE To
wC-“HTTST TPfihSITIO S . . . . . . . . .
VI. ThE T U-PM nor P“ EC LCSSES--“E$ULTS END

CSNCLUSIQJ . . . . . . . . . . . . . . .

IPCEIDIX
3. )4 THE CUESTIfi CF VIRTUAL PQSCFSSES IV
n-PHCgJH OPEQXTQES . . . . . . . . . . .

C. TCF”SFOQ¥ATIOJ TC PPOLFTE SDHECOIUfiL

C. WUTERICWL ESTligTIOH CF TFE FUADCATUPES . .
Listino of rtroorarr SSPOUEU . . . . . . .

Listing of Prooram GLDIFF4 . . . . . . .

ll

l7

58

(O
6‘)

J

FICURE

LIST OF FIGU?ES

Final state surfaces for the electronic transi-

tion % + - é via one-phonon processes . . . .

k
’L
Schematic representation of tVe natrix elements

(I) (2)
Ufi and Ufi

amplitudes Kfi derived from them . . . . . . .

and the second order transition

Rectangular and spherical coordinates for
description of the two-phonon geometry . . . .

Lines of resonance, V and 9+, for the contrac-

1
tion over phonon degrees of freedom in tne

. . c
tranSition rate w . . . . . . . . . . . . . .

iv

67

7l

l . IriTECDUCTI if

Conduction electrons in bulk semiconductors mav, by the use cf
high electric fields, be driven into statistical distributions

that differ appreciably from the ecuilibrium distribution. This is
1,2,3

the inference drawn from the various "hot electron' phenomena

which have been investigated since the initial vork of Ryder and

4’5’6 The term "hot electron" indicates that in such

Shockley.
situations the conduction electrons aceuire an average energy--
temperature, in a sense-—which is significantlv greater than that
obtaining when in the absence of the external field the carriers

are in thermodynamic eouilibrium with the lattice.

Among the hot electron phenomena are broadening of cyclotron
resonance lines,7 enhanced electronic noise,8 and hot electron
thermoelectric power,9 but foremost of the hot electron effects
are the deviations from 0hm's law even at moderate fields. Ryder/l"6
found for fairly pure n-type germanium that the characteristic
curve of current versus applied electric field began deviating
markedly from linearity at a field of about 500 V/cm, and there-
after to a field of about 3000 V/cm the curve could be approximated
by a mobility propertional to the reciprocal of the snuare root of
the field; at the upper field, the current saturated. The term
”hot electron" is often restricted to refer to the range of applied

fields in which occur such major deviations from Chm's law as

observed in Vyder's initial experiments. Then the term ”warm
l

‘ u . ' " ' A»! "V‘ ' -- -:. l '.' '1‘ " ',
rlcctron 15 use» t r {CF to The lever ranwe s” f el.s en -hrc
~- v'- (TIT; A .\ {a :n- Am ~'0,' r ‘ ,4,~.1 "
Sfdll, bua a-: - t "J.u+ '2C ~.c s‘ ,, . a xa.,.h f
1r
the lgtt r, 'unr ‘ invised a sensitive exc.rir it to neasure small

I

deviaci'ns fr:n rm's law at or fields; to fiu d that the mo'ilit”

u “f n-‘e cculi '3 clos’lw fifte' h" the cuadritic drviaticr f rnpli

‘ l

) (I)

from C to a‘out l7 V/cn. Tn Ec. (l), ”o is the zero field mo'ilitv,
F is the applied field, and 8 is a constant fnupd to ie -2.?
-r 7 -9
x l0 ‘ cm“V “ for lattice scattering at lattice temperature
TL = 90°K.

In the regime of very shall electric fielfs, the main interect
is in the morentum exchange tetween carriers ard lattice, and the
cuestion of energy exchange is not as important--eneroy eouilitriur
can he assumed between carriers and lattice. But as the electric
field increases, carriers of (lattice) thermal energies may ac uire
from the field much higher energies before they can mate a colli-
sion Hith the lattice. The steadv state statistical distribution
of the carriers t'en begins to differ from thermal ecuilibrium
With the lattice, and one mav expect that the macroscopic response
to the external field will be essentially different from the small
field resnonse. The onset of such field dependent phenomena
should occur when the mechanisms of energy exchange can no lcnger
maintain a thermal energy eouilibrium between carriers and lattice.
For this reason, theoretical approaches to the hot electron

problem have been concerned esneciallv wit” rechanisns by which

carriers exchange energv with the lattice.

3
This r“aner is concerned Jith estinotinn the importance, as
energy loss Mechanisms for heated electrons, of transitions

ll .. ,
Te swall concen-

involving two acoustical ohonons at a tire.
trate on the regime of warm electrons in a cold lattice, since it
is here that energy loss mechanisms comnarable to transitions
involving single acoustical ohonons will be significant. it
higher electric fields and higher lattice temperatures there is
sufficient evidence that the optical nhonons--in one-ononon transi-
tions--are dominant.]2’]3
The remainder of this oaoer is divided into five maior sections
as follows: Chapter II contains a orelirinary discussion of the
nuestion wzy tvo-phonon transitions may be imeortant in energy
loss. In Chapter III we discuss the viewnoint adopted toward the
orohlem of the statistical distritution of electrons, and we state
our criterion for judnino the imoortance of two-rhonon transitions
as energy loss mechanisms. Chapter IV is devoted to developing
a model for one- and two-phonon interaction terms in the hamil-
tonian, which is then anelied in Chapter V to the oroblem of
calculating one- and two-phonon contributions to the electronic
transition rate. In Chapter VI expressions for the electronic
energy loss rates Ly one- and two-ohonon nrocesses are obtained,
and numerical estimates of heir values over a range of electronic

temoeratures are oresented.

II. 9RFLI“I‘“PY “*”APKS of T““-Eh‘““” “PiffSSES

Shockley's initial theory4’5, utilizing only one-phonon colli-
sions to interpret Pyder's data4’6, contained a hint that a correct
explanation of high field phenomena may reouire more careful atten-
tion to the energy loss aspect of the physical processes of con—
ductivity theory. Shoctley's result for the high field region--
based on predominant optical phonon collisions-—gave good agree-
ment with the saturated-current part of Pyder's curves. However,
his results for the intermediate-field region ioining the more or
less ohmic part at low fields to the high-field saturated current
“ere not in ouantitative agreement with experiment. thably, the
intermediate-field theory could be fitted to the experimental curves
only by assuming the velocity of sound in the crystal to be 3.2
times greater than the actual velocity of sound. Since the velocity
of sound 5 enters the theory only in the energy, hwh = hos, of the
acoustical phonons that scatter electrons, the discrepancy
suggests that the energy loss of electrons in this region is
due to mechanisms that are three times more efficient in energy
exchange than the usual single acoustical phonon collisions.

This inadequacy of the one-phonon theory is further suggested

l0

by the small deviation from Fhm's law found by Gunn in the low

field region [[o. (l)]:

no = no (1+ o2)

Cunn's neasuroments fitted an equation of this form over a field
range of O to lC ”/cm. at 90°K for n-tyoc Ce in whic' lattice

scattering overshadowed ionized imnurity scattering; hU“QVCF,

-5. 9 _’3
the fit of data required P = —2.? x l? ” cmL V L

‘ :I (:19?
“EXT-t ’ ‘z'her\. xi}

I.
b
l

:is theory based on single acoustical phonon collisions nrtoicted

r) —

-4 s d . .. . ,.
-2.8 x l0 cm V . That 15, the theory preoicted a dev12ti“n

i}:
II

‘\I D

from anm 5 law that was l0 times greater than was found by experi-
ment.

Stratton12

added to the theory the energy loss due to excita-
tion of single optical phonons hy the hot electrons. He found
that although in the low-field region the number of electrons
having energies high enough to excite optical p“onons is ouite
small with respect to the total numter,-nevertheless these electrors
could lose such large fractions of their energy that they contribu—
ted greatly to the total energy loss. Stratton thus removed most
of the discrepancy between theory and experiment. However, shall
discrepancies remained12, and it should be useful to examine
further energy loss mechanisms.

The energy soectrum of the longitudinal acoustical branch
of Ge contains ohonons of energies as high 300°K. However, the
selection rules for one-phonon collisions prevent the more energetic
acoustical nhonons from scattering electrons. In the simple
effective mass model for the electrons and bebye model for the

-. .—

phonons that ye shall adopt here, the transition, £¥==é;==£; with

emission (a = +l) or absorption (a = -l) of an acoustical phonon

of wave vector o and energy hon = KS n, is governed by the follow-

6

ino selection rules of conservation ef energy and wave vector:

1. ._. la _ c.
a; (a, 5) T “Q, (._)
2 ,2
M 2 h [I 2 I
aa— m'tt! Ht“ (3)

These may be combined to give tge e=urgion cf the final state

surface in t-snace:

where
u = cosine of angle betyeen h and .

k
S

m*s/h (

01
V

wave vector of electron movino at Steed of sound.
The ouantity k, is very small comoared to most electron wave
3

5 l

vectors (for n-tvae Ge with s = 5.4 x l0 cm -sec' and

m* = O.l2 me, kS corres ends to a tenterature of O.ll5 deg K).
In most studies of mCEility, the scattering of the carriers by
acoustical rhonons is assumed to be elastic, “hich amounts te
neglecting the term invdlvinn Rs in Ed. (4).

Eouation (4) defines two surfaces in k-seace on which final
electron states lie (Fig. 1). For phonon enission (a = +l),
the final electron state lies on the surface M, inside the constant
energy S"here I of the initial state, whereas for absorption
(a = -l) the final state lies on the surface A, outside. For
erission of an acoustical ohonon, the greatest ”ave vector transfer,

5, has magnitude K = 2(k - ks)’ corresoonding to an energy loss of

th = 2hs (k - ks) . (5)

 

 

Fig. l. Final state surfaces for the electronic transition
h + k - g yja_one-nhonon processes. Surface I is the initial
electron energy surface, M is the final stite surface for oh n;:
emission, and A is the final state surface for phonon absor“ti n.
The shaded volume (of radius k = ks) corresponds to “cold electron"

initial states.

If the electrons have a temperature of lOO°K, the average wave
, - 6 '1 . .- . ' .. u . . L
vector 18 k m l0 Ch , lhlch 13 only a-cut one per cent of toe
radius of the “rillcuin zone; thus the maximum energy loss (on a
temperature scale) suffered by a tynical electron at this temcer-
ature is only aLout 4 deg K, i.e., only 4% of its initial energy!
0n the other hand, if the electron is scattered by two

acoustical ohonons at once, the selection rules read:

t = (t ' 5) + O‘+r€3+ + a_g_ (7)
n2 2 "AZ . 2 , a
W )1 '-' 2-HT; (")3 " ()6) + ”5(O.+Q+ + 0-5)“) (V)

w ere a: = l signifies emission of the C: :nonon and ai = -l
signifies absorption. The additional phonon adds extra degrees
of freedom so that the final electron states are no longer con-
strained to lie on the surfaces A and V of Fig. l. By simple
manipulations of the rules of Tos. (7) - (8) one may show that
for two-oh+-on absornticn the volume of h-snace outside the sur-
face f is available to final states. For one ohonon absorbed and
one emitted the final states are confined to the volure tet een
surfaces f and H.

For the case that orimarily interests us, that of two-chanon
emission, most of the volume interior to the surface i is available
to final states, so that acoustical phonons having energies greater
than those permitted in one-oaonon transitions may particioate in

scattering of electrons. This may be seen by corbinino the relaticrs

Wl til
$2
- §E;-K(K — 2ku) = 58(g+ + C_)
to give the inevuality
Kifiw-kg, W)
where eguality holds when lo + o | = g + 0 .
m+ m- + _

From Ed. (9) and the natural bound on the cosine u there follow

the ineoualities

l_>_ p _>_ kS/ff. (10)

which imnly that there can be no thonon emission for k < kg.
(The resemblance of this emission threshold condition to the
“Cold neutron" scattering threshold has been oointed out by
Seitz;M it also reselees the threshold condition for Cerenkov
radiation, the oheton emission by fast electrons passing through

condensed hatter.) Furthermore, from EC. (9) we see that the

electron may lose all its momentum and energy (§_= 5) only then

klm; NH

5
Since at the electron temneratures that interest us, most electrons
have wave vectors considerably larger than ts, we then conclude
that in tug-nhonon emission the selection rules oermit electronic
transitions having enormous energy loss as comnared to energy loss
in one-phonon transition.

This license for large energy loss must he bought at a high
orice; we should excect that the rate of occurrence of tro-ahdnen

transitions will be much less than that of one-ohonon transitions

l0

for the follbwin reasons. Eve: one—phonon transitions core about
only through : slibht failure of electrons in the conduction band
to follow adiabatically the heavy ions in their therral notions;
this failure gives rise to an electron—nhonon interaction energy
that is linear in lattice distortion. “no source of tuo-nhonon
transitions is an electron-lattice vibration interaction bilinear
in lattice distortion; if this were comnarable to the linear inter-
action, it would indicate an unexpectedly severe failure of the
adiabatic orinciple. The other source of two-ohonon transitions
aooears in second order of oerturbation theory; this should not be
comoarable to the first order one-thonon transitions if berturbation
theory works, as it seems to for low field mobility. Finally, the
two kinds of two-rhonon transitions are coherent. It will be
shown that this leads to destructive interference between the two
kinds of two-phonon orocesses, although in our motel this inter-
ference is not so strong as to give the almOSt comolete cancellation
sneculated by herring.1S

The task is then to see how the balance is struck between the
two ooposed ororensities of two-phonon orocesses: low nrobability
of occurence of such collisions versus possibility of larne energy
loss in each such collision. is irolied above, ye assure a Sim/l:
oarabolic band centered at k = 0. le describe the acoustical
modes of lattice vibration in the framework of the simele Debye
nodel. “e concentrate on the regime in which two-phonon orocesses
should be most important--that of a cold lattice and only warm

electrons; conseouently, we ignore ontical nodes.

111. TV? STPTISTICAL VIEHFlI T [ks TL? CFITEFIPI FUR

IflF“PT/NCE FF TUO-FHC C” TFF;SITIOVS

Before taking up the develeoment of the interaction energy
ooerators (see Charter IV), we discuss the statistical hnsatz which
disooses of the r-roblem of obtaining the distribution function for
the carriers. This chatter concludes with the statement of our
criterion for judging the imoortance in energy loss via multirhcnon
transitions.

The most comolete, most satisfying method of attack on a
transrort problem, of course, is to obtain the statistical distribu—
tion by solving a kinetic eouation that includes all significant
transition mechanisms as well as the driving fields. The resulting
distribution function can then be used to calculate the macroscotic
resoonse to the driving fields. The best work to date utilizing
this approach to study the hot electron oroblem is that of Refer-
ences l3 and l6. In these works the Boltzmann eouation, including
acoustical and octical ohonon scattering (and intervalley scatter-
ing where anorooriate), is solved. But only one-nhonon transitions
are considered, and carrier-carrier scattering is neglected. In
general, the well-known difficulties in solving the Boltzmann
eouation are now aggravated since in the hot electron oroblem
noneruilibrium must apoear explicitly. Thus, in our attemot to
Study mechanisms with transition rates as corplicated as those of

ll

l2
the two-phonon processes, an aeoroach via solution of a kinetic

eouation annears intractable.
Instead, we assume, from the start, a simole distribution

function as a working hypothesis. We assume that the part of the

distribution function that is srherically svnmetric, fs, is a heated

maxwellian distribution:

fs(k) a. exo (-h2k2/2m*kBT) (12)

where the electron temperature T, a function (unsnecified in this
oaoer) of the arelied electric field, is generally greater than the

temrerature of the lattice TL, and kg is Boltzmann's constant.
This hot electron assumotion has been argued by Frohlich and
3) on

. . l . . . l6 -
Parangane for rather moderate carrier denSities (m l0 cm

the basis that electron-electron collisions occur so much more
freouently than electron-phonon collisions that the energy a rartic-
ular electron gains from the electric field is thoroughly distrib-

uted throughout the electron distribution before the electron can

lose a significant fraction of that energy by a ohonon process. If

1:he efficiency in energy loss through the two-ohonon nrocesses

cxynsidered here were of the same order of magnitude as that due to

(Brie-phonon orocesses, the carrier density criterio of Frohlich

arid Paranjaoe should remain valid on inclusion of two-nhonon colli-

S‘ions.

This justification of the hot maxwellian distribution nas

b€3EH1 criticized, mainly on the grounds that it ignores the distor-

tTKDn of the high energy tail of the distribution, where the elec-

tr‘Ons have energies exceeding the threshold for excitation of an

l3
ontical ohonon. When an electron can lose a very large fraction
of its energy by the excitation of a single ontical nhonon, one
can no longer argue that the energy an electron gains from the
field will be randorized before it can te lost to the lattice.
Cn the other hand, hot maxwellian distributions have been obtained
by solving the Boltzmann eouation with electron-electron colli-

16 Furthermore, the carrier distribution

sions comgletely ignored.
function may be obtained as a hot maxwellian on the basis of a

maximum entropy argument subject to the macroscooic constraint of

p.

stationary average energy. The maxwellian distribution seems to
be remarkably independent of the details of a oarticular situation
in which it is obtained, and for this reason the hot maxwellian
has been chosen as a simole working hyoothesis.

'hen the electron distribution function is known, the never

balance can he found as follows. The average energy of the distri-

bution is

E = ne<F(ff)> = ne( 1. L(k)f(k) (l3)

vvhere the last ecualitv follows on assuming that the energy func-
‘tion E(k) is soberically symmetric. here the distribution func-
‘tion f(k) is normalized to one narticle, and n8 is the number
(iensity of carriers. The angular brackets will always be taken in

'ttiis never to mean the average over the (known) electron distribu-

tlom

<...>

(gfo)r“) . no

l4

In a steady-state, the energy [Ei. (l3)] is constant in time:

d '7 - l/ ? 8f(k‘l — —~ 7:
ne ‘t‘ <L> ‘ i“’e J Pia; E(k) 7371' - (M)
53‘1”?
_: 2 (at) . (2:)
I 3" field c 1' (is)
and
(if z _ E‘ 8f
at field W (17)
fl— = dwvuuwwa> «ow+wn (r)
at CF], mt ‘ ” a m ‘ ' 4 E e - .,

Here w(¥+k') is the total transition rate for electronic transi-
tion from state a to state k'- 3y means of an integration by

narts, the field term of the rower eeuation [Eo. (l5)] may ;e

written in the form

- 3f
.- l _ = n .
"e I 9a, E(k) (at): e 1:. Xd (19)
where the drift velocity v, is the average of the electron grouo

velocitv:

m
/\
8<
7s‘
\/
m
Q)
Q.)
\_
\/

(2’?)

ta

The collision term of the power ecuation Cin be rewritten as

‘Follows:
' ' 3' .31: = n ' ' l’ la: + ' t: ' —'-
ne j 9,, A...) (3 )CO1] lg, J g, f( l[ (k k ) (k)
-u(k+!;')f(k)] = -ne < ,9,» e(l:,k')w(k+k')> (21)

15

where the electron energy loss in the transition k+£ is
e(k,k') = E(k) - E(k') . (22)
The power balance thus reads
”e fit-.1 = “e <( $32; E(k’k') "'"1att(r'i‘*’é')> ° (23)

That is, the energy fed into the carrier distribution “v the field
E is in turn fed by the carriers into the lattice as a net energy
loss in collisions with lattice vibrations. Tnly he transition
rate due to lattice vibration anpears in E0. (23), tecause in
electron-electron scattering no energy is lost from the carrier
distribution.

He shall take as our criterion of the imoortance of two-ehonon
processes the contribution that they mate to the right hand side
of Eo. (23). The standard chosen will be the one-ehonon contribu-
tion to EC. (23). In Chapter V we obtain a one-ohonon transition

(1) (2)

and a total two-phonon transition rate w . The two-

(2) : wa + WC

rate w
phonon transition rate is the sum of three terms, w
+ wb, rhich are, resoectively, the contributions from two-jhonon
transitions arising solely in first order eerturbation theory,
from two-ohonon transitions arising solely from second order
oerturbation theory, and from the interference arising from the
coherence of the two kinds of transitions.

The :0 er loss due to the one-phonon transitions is tien

Pm = n. <( 9,; e(!<.,:¢'>w(”<g+k')> . (24)

’b 'b
-rhe other nower loss contritutions are similarly defined. Ye then

ledqe the imoortance of the various two-Phonon contributions by

l6

their bower loss in coroarison to the one-ohonon oower loss. For

(C)/¢(l)

and conclude that

(c)/,(1) << 1,

examole, we shall calculate the ratio P
c-tyne two-nhonon nrocesses are (i) insignificant if 9

mph)»

(ii) overwhelning if F l, or (iii) noderately inrortant

if P(C)/9(1)m l.

Another irnortant statistical assunotion made in this eerer
is the "cold lattice" assumption. Since we are nrinarilv interested
in the low field, low (lattice) tennerature discrenancy between

10’13 we shall assune that ini-

exneriment and one-ohoron theories,
tial phonon occupation numbers are maintained effectively at

zero. That is, we assume that rhonons ehitted by one carrier

are absorbed by the heat bath before they can interact with
another carrier. This re-interaction that we ignore seems cer-
tain to be significant in semiconductors such as CdS which have
biezoelectric coupling between carriers and :honons. In such
materials, when the field is so high that the carrier drift
velocity exceeds the sneed of sound, the re-interaction of rhonons
with electrons arrears reshonsible for a current instability.19
This latter effect has not been found in atomic seniconductors

such as germanium, either experimentally or theoretically,19 and in

any case the drift velocity will not exceed the soeed of sound

in the moderate field region.

IV. CEVELC? T_T ’F TH‘ ”FE- ATU T”’- DH’V'" I’l’:’7'-‘”l"l”I’, “CinT

to

This chaater is devoted to develonirg a model f;r one- and
twa-nhonon terms to the interaction Hariltanian. Ve first
establish our nrtation. Te c ncentrate on acoustical vibraticr;
by treating the structure as a r-‘ravais lattice described by
direct lattice vectors % = 21%] + 2232 + e3é3 = (21,22,23). The
2i(i=l,2,3) are integers and the Q1 are the rrimitive translatian
vectors defining a r‘rimitive unit cell of volume Qa = a] x tz'é3°
The reciorocal lattice is defined by the vectors g = 61b] + GZRZ
+ G3b3 = (C],CZ,G3) where Qi = 2na5 x ak/Qa and the Si are integers.

He shall have occasion to do Fourier analysis in a large
volune Q (Q>>na) using the cyclic Sorn-von Varnan boundary condi-

tions. The volume and sign conventions are fixed by the relations

(2U)

where the fine cloud of soints in wave vector snace is indicated

Q“:
by the vectors f = (QC/Q)(f]gi + fzbz + f3b3) = _:.( f

52 19 29f3)9

2, ...). This volume convention is chosen in

I+

(f1.=o, :1,

(erer that as g + m the f-sum goes over to an integral
’b

l7

Lastly we note that the crystal delta function
N .
-i - .,
Melsrl'gelf‘ ,N=0."‘2, , (737)
m a
t

is zero unless i is ecual to one of the recinrocal lattice

vectors g in which case it is ecual to unity. A convenient retre-

sentation of a(f) is the sunerrosition of ordinary (three diwen-

), centered on the recinrocal

(J

sional) Kronecker delta functions 6(f,f

lattice:

ME) =

Cur central interest is in the nuestign of whether acoustical
two—ohonon transitions, for which the selection rules are much
less restrictive in energy loss comeared to acoustical one-nhonon
transitions, are nechanisms of aenreciable energy transfer conoared
to acoustical one-nhonon transitions. Consecuently, we shall
ignore many comnlications that arise in real -emiconductors,
such as ontical nhonon interactions, intervallev scattering,
anisotronic effective nass, traobing, and imourity scattering.

“e assume a nrotgtyne material with a conduction band of standard
form, whose effective carrier nass and sound velocity are anorexi-
fnately eoual to those for germanium. The carriers are fircvided

tN’ doning, but we shall inagine that the ionized donors are

l9

sreared out into a uniform pcsitive bactoround. lhese carriers
experience a self-consistent notential, arising vainly from their
interaction with the neutral "ions" in t'e unit cells; we shall
assume that this rotential can he written for a narticular carrier
as a suneroosition of I‘ionic" notentials in the unit cells. There
is a residual interaction between the carriers, but in good ideal-
gas fashion it is never written down and makes its anoearance
only in arguments about the nassage to eouilibrium in the carrier
distribution function f(k). By the foregoing assumption, we have
now reached a one—electron gicture in Which we shall continue the
develogment.

Let us now consider the nroblem of an electron in a deforned

lattice. e write the Hamiltonian

n2
:: L.— ,_D ’7
H HL + 2m + E V(}; mg) (1.9)

where the first term, H is the lattice energy in the absence

L’
of the electron, the second term is the kinetic energy of the
electron and the last term is the potential energy of the electron
in the field of the lattice. It should be noted that the last term
can be wore general than the rigid-ion model of .‘lordheim,20 for the
electron-ion ootential v(r-P£) may be treated as a functional of
lattice deformation in addition to the exnlicit denendence on fig
that is disolayed. However, we will not make use of this possible
generalization.

The next sten is to seoarate out of the electron-lattice

notential that hart having the translational symmetry of the oerfect

20
crystalline lattice and thus giving rise to Bloch states. It
is a standard result in solid state tVeorv that a function f({)
having the translational symmetry of the lattice may be written

as a Fourier series in which only the wave vectors of the recié—

rocal lattice, g, annear:
, -l iG-r
at.) =2 f e . r» . <30)
m . G
.-,=0 g,
’b

Conseouentlv, the periodic eart of the electrcn lattice inter-
action may be obtained by oicking out from the Fourier series
representation of the interaction all terms involving recinrocel
lattice wave vectors.

Ve write the electron-ion potential in its Fourier series

refreswatetion:

Then t'e electron-lattice r‘otential is

-1 ‘f.
; v(r-§Q) = :2 vf e‘ r of , (32)
2 ’b m
tiere
p 5 Z exo(-imQ-:) (33)
’y 2

‘is the density fluctuation ooerator for the arrangenent of ions,
ithich, considering the ions as noint particles, has a number

<iensitv p(;) = 2 6(;-;£). (If the electron-ion interaction is
t

considered as a functional of lattice displacerent--thus goino
beyond the rigid ion arnroximation--the Fourier coefficient v;
is also a functional of lattice displacement with the result that
not all of the lattice coordinate denendence is in the density

7‘

fluctuation oeerator pf.) The ion coordinate $2 is written as
’b

an eouilibrium lattice vector % nlus u the displacement from

Q,

the ecuilibrium nosition: P 2 + u . The electron-lattice
mg m m2

potential then becomes

2 v(r-i£) = “'0 + U (34)
t
where
| _ ‘1 iG'Y‘
H n :2 9 £5 0,. (35)
' £20 ’ ’b

is the oeriodic oart which together with the electron kinetic
2
Fl . . . . . ,
energy, fifi, is the Hamiltonian whose eigenfunctions are tlocn waves.
-l if-r 92 -l i(f+G)-r
U a :2 vi e pF =:E:§Z§2 vf+n e pf+G (36)
{(762) m G'O: m 't m 9i.

‘is the non-neriodic oart, which induces transitions between the

f3loch states. here the density fluctuation onerators

Eire functionals of ion disrlacement.

’ l
\a lo

“ote tyat tae a ove exnre sion i'r for tTe neriodic
hart of tie electron notential differs from thjt f iugt the

array of ionic rotentials on tie lattice:

X v(g- ) = z 9-1 ve e1h°r o/zia . (3L)

81

Tnat is, h'” reduces to to. (38) only in the limit QAEngj] = 9/9,,
‘.' ‘1, u

and generally some oart of ionic disolacement from the lattice is

included in H' . If we consider that the lattice, as a massive

system, drives the electron with relatively no reaction, then

" will give rise to (lattice) temnerature denehcent carrier

L l
i

levels. Uithin the conduction band w ere the levels are a nuasi-
continuum, this makes little difference, tut there may be temrera-
ture shifts of the band edges thus yielding ternerature deeendent
gaps.2]
”e will assume that the lattice Hamiltonian HL of En. (29)
is adeouately reeresented in the harmonic aperoximation. Tius
"the lattice states can be described in terms of hhonons of wave
riumbers g and nolarizations e and he ion disnlacement ooerators
iri terms of :honon annihilation and creation ooerators aqp and

+— . 02
a pm, resoectivelyz‘

lattice states: In): ln1,n¢,...,nnq,..;> (39)
1
. p . 5 'oot + -'.-£ .
u”: =2 (ii/L’Daiwan) {334th e1 + a m e W- ] . (4o)
n,o ‘ “ ' “

lieere nrm is the number of nhonons of the (g,r)-mode nresent
1Y1 the state In), D is the mass density of the crystal, he?“ is

tfmg energy cf a (g,n)-nhonon, and gap is the eolarization vector

.23
of the (g,r)-mede. The following Debye annroximetien of the

lattice vibrational srectrum is also assumed:

longitudinal branch: org = cs
._ . w. -.-. - (41)
s — average sneec 9: sound
(longitudinal);
transverse branches: wpt = csf, t = 2,3
(42)
s < s .

t

Use of the Detve snectrum 'nvolves ignoring two maior
characteristics of real acoustical spectra in crystals. First,
there is the anisotropy that prevents decomeosition into longi—
tudinal and transverse branches exceot for certain symmetry
directions and which is further revealed in the variation with
direction of the slooe near o=o of the various branches. Second,
there is the dispersion of the w(0) vs. 0 curves as o annroaches
the edge of the Vrillouin zcne. 9n the first noint we must
resign ourselves to the limitations of the model, else the nroblem
becomes intractable. 5n the second point, we are fortunate, for
in the next chapter we show that in two-Dhenon orocesses the maxi-
nmm energy of a nhonon that can narticinate is one half the initial
electron energy. Thus if we limit ourselves to average electron
energies of m lOO deg K, we can draw comfort from the vibrational

. . 23 .
spectra of germanium ehtained by Brockhcuse and Ivennar which
are cuite flat out to "longitudinal" nhonon energies of m l50 deg h
before they start to tend over noticeably. Fven the "transverse"
r‘honon branches are nuite flat out to ohonon energies of m 50 deg K.

iJe conclude from such curves that the velocity of long wavelength

f‘.
I

acoustical vibrations must he used for s rather than some averaged

ouantity such as one might derive from a snecific hea Cebve
temoerature.
The electronic hamiltenian

2

He : 2m + H o (‘3)

is all of that hart of the total Hamiltonian of Fe. (29) that
is invariant under onerations of the lattice translation grouo
aonlied to the electron coordinates. Py Bloch's theorem, the

eigenstates of He may thus be represented in the form

-1,
<E|t> = yk({) = o 291k'r xk(() (44)

’b

where xk(r) is invariant under the translation grouo. The
eigeneneroies will be denoted by E(h). In this work we are inter-
ested only in orocesses involving a nondenenerate band--conse-
ouently the band indices will be suppressed. 7f course, the wave

vectors are then limited to the first Brillouin zone (82):
1 a l— 1 o l
- - - _L. - _:._'
lg- (a,m>(k1,k2.k3>. -, g,a)3< k, _<_ 4,3)3 . <45)

lle further assume that E(k) has a minimum at k = o and that in
‘this neighborhood it can be apnroximated in an isotropic effective

niass form:

E(y) = h2k2/2m* . (46)

The stationary states of the electron-lattice system, in

‘the absence of the interaction U of Ed. (36), are described by

25

the product of the state vectors (39) and (44):

|§,Q> e |k>ln> . (47)

Resitivitv will arise due to the transitions between states of
this basis set induced by the interaction U.

Despite the fact that of generally does have a dianonal tart
in the basis (47) as is shown in Annendix A, U is strictly
nondiagonalas nay be shown by the following. Denote the diagonal
part of lattice onerators by the brackets <...>(Oz [ccording
to results due to Glauber24 discussed in Ponendix 5, the diaoonal

oart of the lattice fluctuation onerator is

(o%>(0) : Z E-TT-t <e-Tug.f>(o) (4g)
2

with
(e-iui°f>(o) = exn - 3(:(u -f)2>(0) = indeoendent of Q (49)
u 2 mg m I m.

The factor <exr(-if-u2)>(0) may thus be taken out of the sur over

2 in En. (48) to give
’L

"(Wu
<pf>(0) = (o/a ) a(f) e 2 t

a m

[But when f cannot be one of the recinrocal lattice vectors--a
icestriction contained in U--then the diagonal cart of the lattice
(density fluctuation of vanishes and with it, the diagonal rart
(3f U. It is a comfort to see that the interaction as so defined
y/ill not give annarent transitions in the case when all ion dis-

??lacements are identical; this result can be obtained without

l\)

ll

8C

recourse to ‘lauber's :nalvsis, for one need only set g2

factur

C.)

= (independent of g) in too eXprCSSion for pf(#c) to get
cf a(f) which must vanish for f f G.
’b 'b ’L

The transition inducitg property of the interaction d
' .. _. ° 1 ‘ __ if‘Y‘
is mad fare transoarent by ex,YESSlnC the erectron o erator e

. . 25 .
in terms of construction operators for carrier states:

U =ZE: 2Z:: ch <t'tleif.rlt) Vf pf (51)

k, K f (#G)

+, for the rurposes of the

where the construction operators c,c
one-electron, One-band picture used here, are defined sufficiently

bv the following properties:

ck|§d> = (O), (the state having no electron), & = 5'
= 0 . 5 f 5' ;
_ . (52)
CkIO> - 05
i" ..
C k-Klo) ‘ lt't>
The carrier matrix element is
' ' -l 3
Q’filelf VHS) 3; J2 d Y‘ X V_V(Y‘)x(,( ) 1(K+f)
s
53)
_ i(K+f) r (
- a(§+{) (g-gle lfi)
where
(K+f) _ -l 3 (K+f) r
(§-§|e l5) : Qa Qd rx . , Xk e (54)

31ince Uikla on srccesses--correspontino G¢O in tle A(K+f) in

Lo. (53)--are not appreciable in semiconductors having narrow

27

vallevs in h-snace, we henceforth consider only the case g=0;
that is, A(§+:) = 6(é,—£). Then Eq. (Sl) for U may be rewritten:

._.-.
s

: " .I'W ...
U [*2 (5251:) ”K O_KC k-ch; (55)

where the ccuoling function 9, is defined as

l

_ _ -- /

n : 3.3 V ',
l‘\

1,
Ifi—

. V
‘mm

) . (56)

 

'l
\v
N

In calculating the transition orobatilities for scatterino
of the carrier, we shall need the matrix elements of the inter-
action U between latticn states that differ by one, two, or were

nhonons. That is, be {rene’ese U into e "um

(57)

U z Um + mm +

in which U(]) has non-zero matrix elements only between lattice
states differing by one phonon, U(2) has non-zero matrix elements
(only between states differino by two phonons, and so on.

it first glance, it mioht seem that the ooerators U(]),
LJ(2), ..., could be obtained by takinn for, say U(n), the n-th
(jeqree tern of th; power series for the exoonential exo(-i:°£fl) in
C‘f(#G)° This will nct do, for on examinino the terms of degrees
ri+2, n+4, n+6, ..., one can see that thew also have non-zero
rhatrix elements between states differinn by n ohonons; thus rarts
(if the n+2, n+4, ... denree terms of the exponehtial nower series
PHJst be grouoed with the n degree tern to obtain U(n). Tiny of

tfiiese higher denree terms will be of order 013-1) with resnect to

ttne nrimarv term, but not all cf them will be.

28
The problem of sorting out fro m a function of Rose f eld the
part that refers to a change in excitation of the field by a given
number of euanta is a problem that also occurs in such contexts as
Dion theory of nuclear interaction and theory of neutron diffraction

24 method of attacking this

from many-particle systems. Glauber' 5
problem is discussed in Appendix i. It is shown there that the eart
of the operator exp(-i£°g£) that will have non-zero matrix elements

only between states differing by n phonCis car be written as

<e-if'u£>(n (4?: Lawn <-if° u2>(o) (58)

r.

He must exphasize that the right hand side of EC. (38) is an
operational representation for the left side to be used only when
we are taking matrix elements between states differing by n r‘honens;
it is obvious that the right side will also have matrix elements
between states differing by (n-Zj) chonons (j=l, 2,...[n/2]),
whereas the left side (by definition) has zero matrix elements for

such states. The diagonal orerator en the right side of EC. (58) is
1
— O -- g 2 (O) "
(e f u,>(o) = e 2((f up) = e“ (f) (59)

Ch averaging over the thermal ecuilibrium state of the lattice
the nuantity in Ee. (59) becomes the Debye-“aller factor familiar
in x-ray and neutron diffraction work.

The main effect of the higher degree corrections to the simple
{Bower series expansion form, (if-u2 )n /n!, fthe n-phonon o:_ eratc r
tlius is just to multiply that term by the Detye-“aller exnonential.

f:

Fk)r a Debye model2

 

 

 

x q 2 o/T
l(f) — 3 ”le - TL L {l + 1 } xdx ’60)
' ZMGk- 0 r 2 ' eX-l )

. . V . . . O
wiere 0 l8 the Desye tam erature of tne crystal (9 2 400 K),

t is the mass of an ion, and T is the lattice temperature.

L

The limit of ” fcr lcw lattice tehperature is2L

3 hzrz
”(f) -> 2-1- 0 m as TL *0, (61)

which will tend to reduce matrix elements for large wave nunber
change f. The effect of the Debye- aller exponentials may be
quite ccnsideratle at high temperatures and/or in metals where,
because of degeneracy, Wave vector transfers i=5 may be large.:7
In the present situatio , they are not significant except as a
matter of comoleteness, for the following reason: The wave vector

transfers =K are of t”e same order as the initial rave Vector g

m m g ¢
for whicn en the average, §E%E" g 100 deg K. Tue mass ratio
_5 B ]21.2
is m*/M 2 l0 ; therefore, §fi7—-<< O = 90 def K.
a,
L)

Arplying Glauber's analysis to the orerator U, we find (details
in incendix A) trat the one- and two-phonon operators can be

written as follows:

1
U(]) = i2:: gKK e'H(K) (h/ZDiwe)2 cf“ Kc,[a+K + a K] . (62)
k’K f\ \- l< -
U(2) = -% QYKZ e' (K) (h/CDQ) cf, KcK
k h ‘ " r
’ (33)

N

-"[3

.h ‘ n'n' ,,
COS<K,q:>C°5(}<,-_~ )<;_ a . .§(g+g‘.-:) +

.00
"U

(mommy

In Eq. (62) phonon polarization is limited to the longitudinal
mode, and lattice ouantities for which the oolarization index 2
is omitted refer to this mode. In En. (63) the arguments of
the cosines are the anoles between the electron (crystal-)
momentum loss é and the polarization vectors of the respective
nhonons. Note that on the right side of to. (C3) the guestion

of noncommutativity of a7 and a+C does not arise because we are

)

to take the two-hhonon oart (...)(Z .
The question now arises: ”hat "ionic" potential v(£) to

use? Thus far, we have not said much about v(£) aside from

assuming that it and its Fourier transform vf exist and that it

will give us a narrow valley conduction band as a solution of the

namiltonian he[Ee. (43)]. In valence crystals such as germanium

the unit cells are electrically neutral (except for the occasional

impurity which we shall neglect). Thus, the potential v(£)

is not really ionic but is rather the aotential of a positive core

screened by an equal negative charge in the form of the filled

valence states. The screening distance can he considered to be

(of the order of the lattice constant. Such a short screening

ciistance reguires that the Fourier transform vf vary little with

1: until f becomes of the order of magnitude of the first reciprocal

leattice vectors g(#3) and larger. This is a common feature of

FOurier traasforms (compare the uncertainty principle in wave

rAechanics), and it can be illustrated by a Thomas-Fermi screened

C<>int Ciarge notential:

_ e -r/r
{V({)}TF "- ‘7" e 0 (64)
3 Z -” -l
{v } — -4nZeC (f + r a )
f TF 0
= ~4nZe2r2 , for f<<r'1 . (cs)
0 0

he do not attempt to calculate the approximately constant vf
by the use of some definite model of valence electron distribution.

Instead, a semiemoirical approach is adopted: we assume

_ x -l
(It, - JLa V_K ()é'fi

I\

l.
’9

. of interest and

7

 

g) is constant for all

t
choose its value by a corresrondence between the one-phonon transi-

tion probability developed in the next chapter and the one-phonon
transition probability used in the usual low-field motility
theory. A constant oK is consistent with a constant V-K for the

small g of interest tecause of the standard result for intra-band

Bloch overlaps:28

, for small K. (at)

 

ilere K g 2k, and since the k's themselves are drawn from a small

 

:ieiehborhood of the conduction band minimum, (&-§ h) is nearly

constant.

V. TIT L FLECTWCHIC TifVSIllpfl “FTES WEE T' IV7-33‘“ H TVAFSITIC S

In this chapter he use tfle interaction operators dove oped
in Chatter IV tc build up microscopic transition rates. From
these two-nhOzon transition rates we form the total transition
rate for a oiven electronic transition by summino over all phonon:
contribution to teat electronic transition.

To beqin, we shall assume that we may describe the effect of
the electron-lattice interactions by a transition probability per

29,30 T

' for convenience). .he

unit time (called ”transition rate'
transition rate for the trarsition from the initial state Ii) of
total eneroij to the final state If} of eneroyf;f is to be given

by
‘her=%l fl “SFéQ ’ “7)

where hfi is the scuare of tte absolute magnitude of the transition
amplitude Tfi’ The transition amplitudes are matrix elements of
the transition operator T, which is oiven as an implicit function

of the perturbing energy Operator U accordino to the inteoral

eouation
T u ; Tfjujl (6“)
f1 'f1 J é:-é:+i_y . U
l J

Iri the present context, the peak function, ath-éz), is taken as

tiie Dirac delta function. Tle danpino rate v, which oenerally is

32

33
deoendent on the statesli) and |j>, will be taken as a constant
determined by the total transition rate from an initial state,

averaged over initial states:
Y = <; Wi+f>C’-‘.v(over i) (69)

He shall also work under tte hypothesis that low order pertur-
bation theory is valid here. This will be taken to mean first
that Eo. (68) can be solved to an adeouate aoproximation by the

first iteration:

Tfi z Ufi T Kfi (70)

where K . is the effective second order transition amplitude
f1

in E0. (7l) we have introduced as a matter of notational con-
venience the damoino energy F e hy/Z.

The second implication of our assumption of valid low order
perturbation theory is that we shall ignore all self-energy effects
mediated by the electron-phonon interaction; these can presumably
be adeouately accounted for by using empirical effective mass.
There is thus no residual level shift from this source, and neither
is there a shift from the diagonal part of our strictly nondiagonal
interaction operators.

The perturbation operator that enters in this work is the
ceerator U [Eo. (55)], and our particular interest lies in the

2)

operators U(]) and U( [Eos. (62) and (63), respectively].

34

Lon-vanishing matrix elements of C(1) and U(2) are schematically

n) )

represented by Figs. (2a, b). Use of U t ice in EC.

(7l
- . .. ...- . (2)
produces the e.fect.ve two-phonon tranSitlon amplitude Kf, ,
.0
represented in Fig. (2c). Figures (2d, e, f, 0) represent the
rest of the second-order transition amplitudes following from the

use of u(]) and u(2)

in varying combinations in Ed. (7l). These
are neglected either for the reason that they are self-energy
effects that can be accounted for by using the empirical (renorma-
lized) effective mass [Figs. (2d, e)] or as a conseouence of our
hypothesis that we need keep only the lowest order transitions
that relax the stringent one-phonon condition on available phase
space for final electron states [Figs. (2f, 9)].

Vith the preceding restrictions, we thus find for the one-

phonon transition rate
l 2 l 2 .
IT$1)| = IU§1)| - (74)

And for the two-phonon transition rate, we find

 

lighz = high-2 + IKE) 9 + 2 (eating?)
3 ”ii + ”gi + Vii (73)
Here
(I) (l)
Kg?) hi??? :3}? . (74)
In E0. (73) the term $21 a |U$§)|2 is the contribution to the

'tyno-phonon transition rate arising from what He shall call "direct"

‘tVMD-phonon processes, considered independently of the other kinds

 

I1) (2) .(23
I;

a: an’
Kg? "fi'
F 6
Km K“)

J? :6

(l)

fi

r~ *
._
I

':n. 2. {Clematic re ras,rtation of the ratrix elerents U
2 . , . . . . _

arid Ufi) one toe second order tranSition au*litu-es K51 deriVed

fioom them. The straight lines represent electron romanti; ”av;

l‘ines renrc.- t phonzn fiOiENt7.

36

of two-phonon processes; this will also be called the a-type
contritution to the two-phonon transition rate.

Similarly, the c-tyne contribution cores from the term
,c (2) 2 L. , . n.“ . . "
nfi a [Kfj I , WHlCfl represents the intermediate-state two—
phonon processes--so called because of the oassane (for very short
periods of time) through the intermediate states, |j>, [Ec. (74)].

Lastly, the direct and intermediate-state processes are not
statistically independent. The term V21 5 26E2,UJ$:)*K$§))
represents the interference between these two processes; it will
be called the b-type contribution. The b-type contribution will
often in the following be referred to as arising from "interference"
two-phonon processes--as if it were truly the Square medulus of a
transition amplitude, in spite of thc facc that for many physical
transitions, 2:1 is negative. This terminology is adooted only

for brevity, and no real confusion should result.

The non-vanishing matrix elements of the operator U(1) are,
by EC. (62)
1 { l+n1
(y-' nf|U(])|k pl) = i e‘“ K(h/2“ow )5 ‘»’T’K (7n)
6’m m’& gK ””WK {'nv ’ ”

venere tne final lattice state (Q > differs from the initial
i . . . .
In ) by only one phonon in the o = K, longitudinal mooe. The
“a" ' 'b’b
LJpner storey of the brackets {...} in En. (75) refers to phonon
eemission; the lover, to phonon absorption. The one-phonon transi-

tLion rate may thus be written

 

(l) , :21: K. 4,42,,
W (ktt't) “h 2025 (qu ) '4 , l, ’

"4

37
2n the other hand, the result of the usual one-enonon theory

. 3l
is

 

x 5(F(k)-f.(lq<J-rlg) ; th). (77)

In Eq. (77) C1 is a constant coupling energy32 that is determined
empirically by fitting the expression for mobility (derived from

this transition rate) to low-field mobility data. For n-type Ge,

33

C 1_l3 eV.

l
he want the present theory to duplicate the usual mobility

theory in the regime of low applied fields, where only momentum
scattering is important. Ve thus ecuate the transition rates

Eos. (76) and (77) which results in the corresoondence
-p ll

C1 = ngle = 'GK e . (78)
The negative sign has been introduced in Kg. (78) because we wish
to keep Cl positive, and the Fourier coefficient v_K in gK is nega-
tive--coming as it does from the potential of a positive core
screened by a negative cloud.

If the one-phonon transitions dominate the total number of

ilransitions per unit time as we here assume, then the damping energy

\Jill be, accordion to Egs. (69), (77), (78), and (l4)

 

 

m2 k T 2 '>
1 l l B h kL 3
I. E _ m = . (l-k no ) . (79)
2 6" DS(h2/2m*)2 <2m*kBT S "

Tine use of a maxvellian distribution for tte average <2'2>k yields

 

 

 

 

 

 

, ere
2
[(31
I‘d : '7) (3])
4nDS(flb/2F*)h
and
l m 3/2 3 ”2kg
: "-X 7-, o : S : T T 0-
Ir (3777T (x dx e x (. dxs7x) ,(xs - Cm*kDT 15/.). (o2)
s s
The ouantity IP can be expanded as a series of incomplete catta
functions:
1
= - 2 7
IF [f(5/2.x5) 3 xS r(2.x5) + 3xS r(3/«.xs)
, o/.‘ . \ ,, .
-..», weal/«57-) , (82)
Selected values fellow:
= T =
xS Ts/' O.l 0.0l 0.00l
(3 )
IF = 0.46 .7: .9o
Since IF is so insensitive to the temperature, in sutseruent
nurerical work we shall give it a constant value of unity.
The discussion now turns to the tic-phonon transition rite
(2) _ (2) i f _ a c .,
wif _ w (Q ’§ + Q ’t'é) ' W1; + wff + wjf (it)
vehere
a _ 2n ,(2) 2 r
wif ' lin I 6ngi) (8d)

and

To get the total rate for a given electronic transition K + 5-5,
it is necessary to perform a contraction over the phonon degrees
of freedom inplicitly contained in Eg. (84). This contraction
generally will take the fonn of (l) a thennal average over initial
states of the lattice (in the atsence of mutually strong electron-
phonon coupling) and (2) a sum over the r‘ossible final states of

the lattice. Thit is,

w(2)(¥ + 5—g) = E Z. 0(Q1) W(2)( jg£ + f: ¥'§)9 (89)
n

in which p(pi) is the thermal weight factor for the initial lattice
states.

Invoking the cold lattice assumption, we nenlect all factors
p(pi) except that for the lattice ground state IQ) for which we

'take p(Q) = l. Eouation (83) then becomes

171 which the two ohonens in the final state are of the (g+,p+)-
iirid (g_,p )- modes. Ye remark that becaUSe of the large number
C>f’ modes ( «o/Qa+m), the coincidences (3+,o+) = (g_,:_) in En. (90)

”“33! be safely neglected.

40

There are thus eight degrees of freedom iivolved in the Coh-
traction indicated in Fe. (93): three comn nents of the wave
vector and the polarization for eact phonon. Hot all of these
degrees of freedom are inderendent. Some constraints are provided
by the basic selection rules of energy conservation as enforced
by the peak function 6(émi) and of momentum conservation (neglect-
ing umklapp processes) as enforced by a Kronecker delta,
6(g,k-g+g++g_), in the transition amplitudes Téfi). lfiere is also
the particular feature of the rigid ion interaction with carriers
from a parabolic band that only longitudinal phonons contribute

..(2)

to the operator E(1). Because the transition amplitude “f1 is

built up from the operator U(]), it then follows in our model
that the b- and c-tyne contributions to the two-phonon transition
rate involve only longitudinal phonons. Transverse phonons will

contribute to the a-type rate, Ha. The transverse sound velocity

5 is typically within fifty per cent of the longitudinal scund

t
velocity 5. Hence the use of s in place of st ought not change

the transition rate wa and its conseouences by more than an order

of magnitude. he shall find that setting st = s in evaluating w“
is a luxury we can well afford, for the a-type contribution proves
b c

‘to be many orders of magnitude below those of either w or w .
In perforning the sum over pnonons [Eg. (90)], it is convenient
txo describe the electronic transition k’+ K - K in terms of the

iriitial i, and the losses of momentum, E, and of energy, e, where

on - egg-p -- 2:2: (on-K2) (9i)

 

aificj u = cos(h,§). Energy and momentum conservation rules then read

e = hs(c++e_) (92)
ES: m+ + g- (93)

It will nrove c nVEDTELt to rewrite Ers. (92) n3 (93) in dihen—

sionless fore:

go = 3+ + ;_ (94)
5 : P+ + Q- (95)

where
50 2 e/th , it: gt/K , g = g/K. (96)

The paraneter 5 can be th;uoht of as a reasure of the energy

lcss efficiency of a t'o-hhonon erissicn nrocess in comoarison
to one-:honon emission, since the enerny loss in one-nhonon

emission is e = th. Note that the r nentun less K may be written

as
K = zen-50kg). (97>

ivhich is the neneralization for two-rhonon erission of the one-
ishonon condition [E0. (4)] and from which Eo. (9) follows.
Equations (94), (95) establish the neonetry for the sur
(2ver narticioatinq ohonons. The ends of the unit vector E are the
flaci of the hrolate spheroid K [see Fin. 4, (on. C] which is the
1Ocus of the canton ooint of the (reduced) wave vectors i+, §_.
Theeccentricity of the ellintical crtss section in any plane

. . - . -l . . . . . .
COntaininn 5 15 e = go , and toe semimagtr aXis lS a = -g . {e

42
stall Parametrize the sum over narticioatinn nhcnons in tenis of
:vrolate snheroidal coordinates g, n, ;, fcr which we ntte the

follo“ino bronerties [mere details are c:llected in finnendix F]:
t = n + n . (l:t<w)

; = azimuthal anole of (n ,o ) olane about the K axis
' m+ m- ' m

(O:§<2n)

«138, = <-§-)3(g2-n2wgdndc .

i hen the sum over anontns is carried out by intenratind over 2+,
ssubiect to conditions Ens. (94) and (95), we find that this is

eacuivalent to inteorating over 5, n. and C-

fnp' -;

The wave vectors a, have ‘ litur;lly hon-n . i,, .-z t.;

[Sr‘illouin zone (82). If this additional ccneraint rust be te en

ee)<nlicitly into account, much of the simnlicitw of the intenra-
‘tfion over orolate snheroidal coordinates nculd be lost. He
FH3\' show that initial electron enernies orovide a bound on the
rneznrfitudes of £1. so that when the initial electron distribution
fEIT ls off raridly (as in the maxwellian) we nay rely on this dis-
‘tr~il3ution to provide a cutoff for the nhonon snectrun. Referring
ti) Ens. (94), (95) with the factor K restored, we see that the

"h3><innm*negnitude taken by one nhonen wave vectcr, say 5+, is

: (0+) 5 %K(l+go) (99)

0
max

1+

'38 kriow that K g 2k and e g “2k2/2m*. so that EP- (94) gives “5

43

the bound on ohonon energy:

“30+ ”S M2K2

k < k i ' 77??"
B “ B ” 3

7<

(lOO)

 

 

For electron eneroies above the cold-electron threshold, the first
term on the rioht of En. (lOO) is less than the second tern, and
the latter is tvnicallv g 50 deo K in the regions of oresent
interest. That is, for the averane initial electron energy, re

may write (since kBT=h2k2/2m*)

hso+

l
(-
kBe m 2

 

®|—1

(lOl)

where e is the lattice Debye temoerature. This last condition
shows that there is no need of an exnlicit cutoff of ot at the
Pl so long as T < e m 400 deg K.

(2)

In the terms of w , the sums over two-ohonon degrees of

freedom take two Prototype form

.6 )A(K-n -0 )a(e-ns(n++c_))

m+ m- m m+ @-

(l02)

 

:_)6(e"S(Q++C_)):

in which nolarization indices have been suoorzssed for the moment.
In the scaled coordinates [Eo. (96)], the rioht side of EC. (l02)

becomes

 

m M'

3
K 3- - - -- - - -
v I Q)3 d n+ . Z F(Kg+,Kg_)A(K-c+-n )9(€0'(Q++c-))’
n

44

and the change to coordinates (g,n,;) yields

 

 

m+ m- 0
(2" (103)
- 2 l 211
. e K d d 2
- Q 3 his - eeO-n ( 2"] 5%‘(EE-n may, )1,
(4n) -] O

in E0. (l03), the vectors Q: are consicered as functions of

(g,n,g) as given in Appendix B, 9(50-1) is the usual step function

6(X) = 4'
L = 0 , X < 0

and [...] denotes that the enclosed cuantity is evaluated at
o
The function 6(gO-l) in EC. (lO3) provides the cold

5 = to.

electron threshold.

- “1'

Hakino the replacement C1 = -oKe ‘ [Eo. (78)] in the operator
U(2), we may write the matrix elements
11(2) - “€le (K )5 (K )
fi — —EE§— COS m’g+ CUS m’g-
u; (104)
X (w+w_) A(K-g+-0_) .

The direct two-phonon transition rate then follows:

hC K
2 l 2 -l 2 2
wif = “1.. (ZEEE) (9+9-) COS (K,e+)cos (§,g_)
(lOS)
x A(g-g+-g_)6(s-h5(q++q_)).

where we have not distinguished between the differ;nt sound
velocities for different acoustical branches. This neglect of

the difference in sound velocities has decoupled the cosine

45
polarization factors from the rest of the expression, allowing
the sum over polarizations of the ohoncns to be done separately.
The deneneracy of tie transverse nodes in the isotropic model
of the lattice spectra allows us to further simplify the nclari—
zations sum by choosing one transverse polarization for each wave
vector to be normal to the (K,g+) nlane. The remaining transverse

oolarizations, is then in tVis plane, and the polarization

gtt’
cosine is simply related to the sine of the associated lonni-

tudinal polarization, e :

cos(K,eti) = tsin(K,e£i) = tsin ¢1° (lOo)

With this convention, tae sum over polarization is just

cos2¢+c052¢_ + cosZ¢+sin2¢_ + sin2¢+cos¢_ + sin2¢+sin2¢ = l. (lO7)

-- a ,_, . . ,. ., ..
thus for w , one orototyoe function r(g+,;_) introduced in

Eo. (l02) becomes

 

C)
CO
V

HKC
2n l 2 l l
[Fak ,0 l] = T ( > ° — ' -——-¢ ' 9(E -l)- (1
m+ m- £0 2“) Q2 £;_n2 0

Fe then find the a-tyne contribution to the electronic transition:

 

C2U4
a _ l l l“
W (AS-*lejé) - .— 0 3- ' 7'3 ° 9(50-1). (109)

3 s
211 US

This last exoression, when substituted into the average power
loss formula [Fo. (23)], will yield Pa, the a-tyoe contribution
to two-ohonon power loss.

The second order transition amolitude K(2) [Eo. (74)] is now

fi
. . . l a 2
reou1red. Being made up from the one-ononon operator U( ), kgi)

46

;k-K) has only two terms in
'N’b

for the transition (Q;§) + (lr+,lo

the sum over interoediate states:

m = lln+’oo+3t‘8+> and |j> = l0p+,lc it?)

 

We thus find
(2) _ (2) .
Kn - K (event-Wt)
hf? l 1 (110)
- _ _;__.. 2 - -
2059 (q+q_) [aE++ir AE_+ir]A(5 3+ 8-)
in which
aEt 25(k30) -€(,§-gi;ni)
2 (lll)
= E——-[2tc cos(k o )-02-°k 0 ]
2m* “"1: m’q'jt 't L 5 it °
Use of Eq. (llO) and Eq. (lO4) gives
5 _ ,. (2) (2)
'qki:=ilgftg ‘ Ccfi2(uif Kfi )
C?h2K2 at,
= - ————————-. cos(K,o )cos(K,o ) ) ' (ll?)
(2bso)2 m m+ m m- i (aEi)2+r2

x HIS-revs-)

Cnly longitudinal phonons are involved. The sum, 2, is eouivalent
2
ti

to twice one of the terms because the lateling of ie thonons is

arbitrary. The b-type transition rate is then

2C3h2K2 AE

wgf = - Z%—. —-l——-§-cos(K,o )cos(K,o
(ZDSo) ”

(ll3)

X A(§‘g+'g_)6xe‘$9(9++q_))o

47

The integrated electronic transition rate of the b-tyne is

C3
b -l l 4
iv (K+k-K) = -———-- ~———— - K ° 9(5 -l)
m m m 24"Q D283 0 (114)

X
I
._a _1
0.

2n q A5
a] e-u-agn- ———————2 2
0 (AE+) +r g

. . . . . .b
Jote the expliCit negative Sign of the guantity Wfi [Eq. (ll2)],

O

arising from the sign convention of Eo. (78); this acts to reduce
the effect on the total two-nhonon rate due to the positive
definite parts “ii and “$1“ It is worth noting, however, that this
interference is not always destructive. Either one of tne polari-
zation factors or the factor AE+ can become negative, making the
interference constructive. Finally, there seems to Le no adenuate
way of approximating the ouadraturcs in E0. (ll4), and we resign
ourselves to the ultimate necessity of using numerical methods.

The derivation of the c-type transition rate follows the by-

now-familiar pattern of the a- and b-tyne. The microsc09ic rate is

 

 

 

 

 

n2c4
wif=g%w]2'(n+c)°[ 12 2+ 12 2
(ZDso) (AE+) +r (aE ) +r
2 (ll5)
AEiAE'+F ] ( ) < n < ))
+ 2 A K-g -n_ 5 8- s o +g_ .
[(AE+)2+F2][(AE_)2+F2] “ ”i “ *
And the integrated electronic rate is
4
C l 2n
C l l 4 d 2 2 2
w (k+k-K) = ' ' K ‘8(5 ‘1)[ do] ’£'(€ '9 )
m m m 27"Q D253 0 O 0 2n 0
r 2 HO
[ 1 1 2 AE+A»_+F ( )
x —_——__-+ ——————-+
Azi+r2 AE§+r2 (AE§+ 2)( r§+r2)

:3
3

 

where AEE E (Aft)2. The term
AL+AE +r2 (
2 ‘ ii7)
(atf+r2)(atf+r2)

reoresents the interference between the two ways of passage through
the intermediate state; we may expect this interference to make
only a small contributioa if the width of the resonance arising
from AEi = O is so small that the AE+ = O resonance does not
apprecially overlap the AE_ = 0 resonance.

The integrated electronic transition rates [Egs. (lO9),
(ll4), (ll7)] will be used in the next chapter to derive expres-
sions for the average energy loss rates as functions of electron

temperature T.

VI. ThE T”9-PHOICJ PTUEP LOSSES-~“ESULTS AND CCkCLUSIWN

”e may now, using the electronic transition probabilities
developed in Chapter V and the (4axwellian) statistical distribu-
tion of initial states, write expressions for the average rate of
energy loss due to the various components of the electronic transi-

tion rate. First, for comparison, the average one-phonon power

 

 

loss is
( l "l
P 1) = n Udt.,£,2e'Ek/kal]
e :\ L
%_ TTCZ
_ l;
x JdEkfik e Ek/“BT JQK E§%-K . (Ek-Ek_K)a(Ek-Ek_K-th)
. 2 c.
n3/ZD(h2/2m*)5/2 B s
where QK = V 3 d3K.
(2")
Here we have introduced the dimensionless variables x = h2k2/2m*kBT,
‘2 2 *. T “2kg 0
y- K/2m..3., XS :TS/sz (]]J)

l/

in terms of waich the dorain of the pggcos(k,K) and n integra-

tions are

lipiudei/XSH,
(l20)

O :_y :_y E 4X(”-ud) ;

the function I

49

s (lZl)
% i
= r(3,XS)/r(3) - (3n /2) XS r(5/2,XS)/r(5/2)
l
+ 3xsr(2.XS)/r(2) - n 2 xs3/2 r(3/2.XS)/r(3/2)
+ x 2 r(l,X mm
S S

where r(n), r(n,x) are tie usual complete and incomolcte maria
functions which are tabulated, for example, in fleference 34.

The a-tyne comnonent of co“er loss, arising from direct tio-
ohonon transitions considered indeoendently of the intermediate-

state transitions, may also be written in terms of tabulated func-

 

tions:
on _1_ E /k .. -l °° % E /' --
Pa = n J dE,E,2 e‘ k 'B' I dE F e"; ‘r'
e [(3 {,2, [k k k
'd
2
x [av ——El-——-V4 (E - E ) (122)
m 25113253 k k-K
2
4n C ,
= e 1 9/4 a
3n7/20253(h2/2m *)7/2 (kBT) I (Xd) ’
wnere
m l -
— 5 ,2 5'.
Ia(T) = %T'J dx c X {x (l - (XS/X; )
Xd
l g l
+ lOXE x (l - (XS/x)2)9}
(i"3l

F(G.Xd)/r(7) - 9X,r(5.X;)/r(5)
l 3

4 (1‘én2/1)x52r(9/2,xd)/r(9/2) - (xv/:)x;r(.,x,‘ ri'

Sl
%

I: _ \
’2 [I ‘. (I
+ (\ v n f . (I A

U1 NU‘I

r(7/:.Xd)/r(7/Z) - (35/2)x§r(3,xd)/r(3)
“ (123)

[\HH

7
2 a . . h
+ (9“ /::)xS r(5/z,xd>/r(s a) - (2)/d)x:r(2.xd)

l
+ (NZ/3)X

m NIKD

F(3/2,xd)/r(3/3) - (3/4e)x§r(1,xd)

In the second eouality of Eq. (l23), the threshold has been raised

to k = 2kg, or xd = 4x5, to take into account the condition that

d

not all states with initial wave vector k in the recion

kS i k _<_ 2kS (l2?)

can be taken to the final state k = O bv two-phonon transitions.
By so doing, we are neglecting these initial states in the region
of En. (l24) that can lose all tteir energy in a two-ehonon transi-
tion, thus making an underestimate of the bower loss. This, how—
ever,is a small point because our interest is in situations in
which the average wave vector lies an aporeciable distance outside
this region.

The ratio of this direct two-ehonon nover loss to the one-

phonon loss is then
9(a)/P(1) = (k7T/Ea)3 1(a)(T)/I(1)(T) (125)
where the comnariscn rarameter (having dimensions of energy) is
[a E (3"2h3Ds3/4m*)1/3 ‘ (‘25)
*r “;r n Ears 1 *r *rewgr i* n-tfi”: germ uiu ,

M J
E = 2.55 eV = 2.05 x 15’ deg Vofih . (127)

r‘.
5.

L.

The ratio 1(a)(T)/I(])(T) is of the order of unity for T > T ;

 

5
selected values are given in the following short table:
TS/T = 0.023 0.0l 0.0025 0.00l
(lZ‘d)
1(a)/1(]) = .90 1.02 l.06 1.05

 

 

 

 

We thus conclude from Egs. (l25), (l27), and (l28) that a-tyée
two-phonon power loss would not become comearable to the one-
phonon eower loss for any reasonable case.

He turn next to the b- and c-tyee two-phonon contributions
to the power less. They cannot be reduced to tabulated functions,
and no adeeuate analytical approximations have been found for
them. He can do no more than reduce them to euadratures which
are then estimated numerically. Thus, for the b-type (inter-

ference) two-ehonon power loss we find, using Ee. (lld),

1 -l
b E -E k T
P = ne UdEkEk e k/ B]

 

 

 

1
°0 ' I, C 4
x I dEkEvz e-Ek/kBT [Q5 (Ek‘Ek-K) {' “2"12“§" E T
1 2n D
d i 2 2 +
x BRO-‘14 “3‘! %% (1'50?) )[fl]
.1 0 D++pO J50
2n,%c2(k T)”2 c (k i)2
_ _ ti 1 B . 1 B . Ib(T)
n3/Zn(n2/2m*)5/2 E:
where
b 3 m -x 2 1 yu 3 2 1 d 2 2
I = _§-J dx e x J du I dy y 2xS g I —%—(l-go n )
2 Xd ud O -
x 2n g5. 0+ (13‘)
Zn 2 2 ’ ’
o D++rO

53

and where we have introduced the following dimensionless forns of

the energy changes at, and line width r:

U
m

H-

AE+/kBT
" (l3l)

“"1
ll

1 .
F/hBT

(where we have set Ir=l, according to the discussion following

En. (82)).
(The constants are grouped so as to facilitate the comparison
with the power losses P(a) and P(1).)

The multiple cuadrature ID was estimated by Monte Carlo
technioue (see Appendix C) over a wide range of the ratio T/TS.

For the following nominal values of the parameters, Ea = 2.55 eV,

C = l0 eV, r0 = 2 x l0'4, the results follow:

1
15/1 = J- 0.1 _J 0.01 J 0.001

-P(b)/P(1) = I 510’7 ‘ 2.8(:.3) x 10‘7( l.60(¢.03) x 10‘

(132)
7

(humbers in parentheses are the probable errors, obtained in the
usual way from the empirical standard deviation.)
Lastly, the average power loss contributed by intermediate-

state transitions is, from the transition rate of E0. (ll6),

 

.1. a -1 o. - , ._
PC = ne [ (dEkE,2 e-Ek/kf‘] J dEkEk2 e'Ek/LB'
’ o (133)
4
C 4 l 211 r
l K do 2 2 2
x dK-(EfiE/ )°{ ° ' I d0! — (E '0 )

B

f q ' t ' Eo _ (133)

= ZhefiC](aBT) . -
o

n3/20(/f/2m*)5/2 L3

 

 

c - ..
I 15 the multiple Quadrature

1

m - l
dx e-X x2 I
d Ud

Here

c _ 3
I “ '11'(
X

y
du J udy V
2 o

A = D + r. . (135)

The ratio of c-type two-phonon power loss t: one-phonon power loss

is then

PC/P(1) = (C]/Ea)2(kBT/E3) - IC/I(]) . (136)

-.A

Preliminary Monte-Carlo estimates of the multiple cuadrature
IC indicated that the ratio P(C)/P(]) was within an order of nagni-
tude of unity for nominal values of the parameters C], m*, p, ard
s. The integrand has a great deal of variation, and even with a
high sreed cam:uter such as the Control Data 3600 it proved
impractical to carry the computations t0 the point where the
uncertainties of the estimations were less than about 40%. Such
uncertainty, ho ever, may still hermit one to determine whether
‘the (c-tyoe) two-phonon transitions are significant in the energy

r“elaxation of a heated electron gas. The details of the numerical

I”

35

estimation methods used are presented in fl:pendix C. lhen the

nominal parameters C1 = l0 eV, [a = 2.55 eV, to = 2 x l0—4, and
TS = 0.ll5 deg K are used, the best estimations for a range of

electron temperatures T follow:

 

 

(‘37)
T(deg K) = 5.3 11.5 25. 53. 11C
PC/P(1) = 1.94 2.68 3.25 1.02 2.30
(% interference) = 0.3 0.2 0.6 0.6 l.0

 

 

 

 

 

I37
The line of Eq. (+98) labelled "% Interference" represents
that fraction of the total intermediate-state two-phonon power
loss arising from the interference between the two possible inter-
”) ' ’Q o . '2‘. + I + n + ' KI
med1ate states. (l) 5] + & q+ + hf &+ g_ and (2) 51 + &
+ o + h + o + c . That is, the dominant contribution comes
f «5+ g.-

N,-
1 + o:], of Eq. (l34), whereas the inter-

from the two terms, A;
ference term, 2(D+D_+F02)/(A+A_), contributes only one per cent

or less. The two intermediate states are thus almost statistically
independent.

(own),

The uncertainty of about 40 per cent in the ratio P
\31
of course, means that the three figures in E0. (+38) are not all
significant. In particular, the dip at 53 deg K is likely spurious.
Nevertheless, the conclusion seems justified that the intermediate-
state two-phonon transition contribution to power loss is at least
(:omoarable to--and perhaps greater than-~the one-phonon power loss.
he thus conclude that accordino to the model developed in

this paper, direct two-ononon transition contribute very little to

ECNNer loss. The interference tetween the direct and intermediate-

56

state two-phonon process has tye proper sion for cancellation,
but it lacks the magnitude to be significant. ind the intermediate-
state two—phonon transitions contribute a power loss comparable to
that of one-phonon transitions. It would seem that at low tenpera-
tures and moderate fields, where optical phonons cannot be active
in energy loss, theories aiming for cuantitative agreement with
experiment ought to take into account the energy loss due to two-
phonon transition.

fit first glance, the conclusion that c-type two—phonon proc—
esses are comparable (in energy loss) to one-phonon processes
might appear to depend fairly strongly on the particular value
of the interaction energy C], and hat the nominal value, C1 = 105V,
used in the numerical estipates would not apply euite so well to,
say germanium where the value derived from pobility exoerinents

33

is C1 = l3.6 eV. Fumerical estimates, however, indicated that

the product, C? IC, has much less variation than either of its

factors. Recall that to a 0?, and denote the multiple guadrature
IC as a functional: IC = IC[FO]. The ratio P(C)/P(1) is then
proportional to rolcfro], and numerical estimates made for

T = ll.5 deg K are summarized as follows:

 

 

r0 = 1 x 10"4 2 x 10'4 3 x 10'4
1C[ro] = 9;3 409 255 (l38)
FOIC[FO] = 0.0913 0.0513 0.0755

 

 

 

The result that the intermediate-state two-phonon power

lczss is of the order of magnitude of the power loss be one- gonzo

57

processes also lends suiport to our original assumption that
perturbation theorr ”orhs. For, even after weighting the
interwediate-state t o-phopon transition pro ability with energy
losses that greatly exceed those of one-phonon transitions and
after summing over a domain of final states that also greatly
exceeds that of one-phonon transitions, we still obtain an averape
two-phonon power loss comparable to one-phonon power loss. 'e
may thus infer that our moiel provides that individual two-phonon
transitions are much less pro‘able than one-phonon transitions.

For contrast with our conclusicn on the relative importance
of two-phonon processes in energy exchange, we may mention the
work of Franzah and Tailyn35 in which the relative importance
of two-phonon processes in romentum exchange in tie alkali metals

is small. These workers found that the a-t re c1: 4,..1 '1" it

. A ‘ " ' ' “ “ ‘ ‘ ’ " l- \ ’ - I" i r" :5 T v ' ‘ 4 I H ‘ * ' .
15 t9? e. in t.« 'res.n. - e1) F t.e .gu5_e .pn « -r..,1oi‘

~ 3! . n : 4' ‘ -' ‘V - >- - v ' v . . . ‘ a I ‘I . q' a. - . f, :
T.L;, x.t. no .. r c s e 1nclu-e , contriiu-eo - c --0- n
L. .5. .' .— ‘:' \ A, . ' 4- ,. .- . . E. .' 0/ 1,. ”' - . ,.
ego (l: . .,u, Cg_aUCt’#lL§ af al~ a~eu_ as . 19$ , even at

FPPEEDIX A

Oh THE QULSTION OF VIVTUSL PVCCESSES IL V-PHfiNTK “DERKTCT

The interaction operator in this borer has the form

U =ZELSE-o c+ C
k K9 «K k-K k 0-K(¢e) (r l)
where
' -K ‘ oK ;
Q-K =; e12 e1u2 = p+K . (n.2)

We are interested in the ratrix elements of U between lattice
states differing by n ohnnons; for convenience, in this hrpendix

only, we use the tenn n-bhonon processes" to designate the non-
vanishing matrix elements between such states (this temoorary
terminology is not to be confused with nhvsical n-bhonon transi-

tions).

Consider the operator:

1
1 l h E .
/‘ 51Kou -1249 (3‘ U K'E (TO-9,_ ”110211.. l
m m? Z'w m mto "
Q.“ :1 ”
_, (A.3)
_ ‘ 2 * ,+
_ lhiQ (AC?a‘“ + A PC 0,)
where
l
x = (n/zna )2 K-e e“2 = indenendent of (;.4)
on ” ' op m map ' ”

volune o.

It is readilv annarent that "n-nhonen nrocesses" will be

SJenerated by tie n-th degree term of the power series represente-
58'

59

tion of the onerator, exp(fi). However, the terns of degree n+2,
n+4, n+6, ... will also generate "n-phonon processes" because cf

what we call virtual processes. Virtual processes correspond to

nn+2

the presence in, say, the (n+2)-degree operator, . /(n+2)!, of

terms in which both the destruction operator an and the creation

Operator a'O of the same mode anpear as factors and there is LO

..f.

other such mode pairing between a and a in that term. This situa-

tion corresponds to the creation of a phonon cf the paired mode
and the immediate destruction of a phonon of this mode, while the
remaining n unpaired Operators in that term generate an "n-ehonon

process." Similarly, in the (n+4)-degree term there will be terms

involving two virtual processes that will contribute n-phonon

processes,‘ and so on.

Glauber developed an analysis of multiple boson processes
that is useful in sorting out "n-boson processes" from fairly
general functions of the Rose field.24 There it is shown that

1
*

.a+,) the

. . '2
for an operator haVing the form A = 12 n (Xiai + A 1
l

. m .
effective part of tne power, A , for a "n-boson process" 15

’.m (n) _ m n m-n (0) _ l |n dn m (0) ,
<.«> — (n) I. <A > “Fifi {Ii-AHA (.—.5)
where (E) is the binominal coefficient--the probability of select-

ing n things from m things without regard to the order of selection,
and <...>(0) indicates the diagonal part of the enclosed onerator.
The argument leading to this formula goes as follows: ”e select
‘from Am the n factors of h which are allowed to generate the

"n-boson process," he remaining (m-n) factors being allowed to

o0
carry out virtual processes only. There are (E) ways of carrying
out this selection, and if re set aside for the morent guestions
of commutahility, the n factors of t can be collected as L” thus
giving Eo. ( .5).
It must be noted that the right hand sides of To. (fl.5)
are rerresentations of (Rm>(n) that are to be used only when one

is taking matrix elements Letueen states differing by n otonons;

 

. . .. . . - . n
tne pos51ble ambigUity can oe renoved by replaCing P by the
' " '- ". An (n)
somewiat redundant notation (m > :
m (n) l n (n) n m (0) . .
(A ’> = —— </, > < A > . (A.5 )
n! d.n

The argument leading up to Eg. (3.5) does not take account
of the noncomoutation of creation and annihilation operators
belonging to the same mode but coming from different factors
of A in the power Am. Those corrections arising from the commu-

tation relations, [aA, ax.+] = 6 are not significant, for

AA"
these correction terns are of order o/oa with resoect to the
right hand side of En. (A.5). That is, of the m factors of F
in Am, let us consider two given factors of h: one which will
go into the operator (An>(n) and which will contribute to the
change in phonon states, and the other factor A which is in the

operator (Am-n>(o)

and which will give only virtual changes in
Dhonon states. According to Fe. (A.3), each A contains a large
(N m Q/oa) number of phonon operator terms, and the product of
tflie two A's we are considering contains “2 terms. Corrections

to) E0. (h.5) arise only when a creation (annihilation) operator

in one A belongs to tie same node as an annihilation (creation)
operator in the other A; there are only N such coincidences in
the total of K2 erms so that corrections to Ed. (l.5) are only
of order l/N with respect to the right hand side. Since it is
necessary to use all the degrees of freedom of an n-phonon orocess
to obtain a non-zero result in the linit, N m a + m, then the
correction terms, being of order l/T, will vanish in this limit.
One may say, therefore, that in the limit Q + m, the right hand
side of En. (l.5) and the correct expression are equal except on
a set of “measure zero" (0(l/ ) + O).

The argument of the proceeding eiragraph can be simply
illustrated by considering the one-phonon operator of the operator

A3. Applying Ed. (5.5), one finds

.3 (l)

<_.’ >

-'_\
- -3/2 *_+ * + *,+ \(il
-1<%:E Q (Aiai + Ajai)(xjaj + Ajaj)(xkak + ikak), /

. ,*+ -l '...
-13 g o (Aiai T xiai) 2 9 (ii) (2“. + l) .

ll
NIH

L4

The sum over j in the diagonal part will remain finite in the
limit o + w because the measure associated with each point in
— ° 3 A r 9 i . 0 - '
g~space lS o/(Zn) . kn the other hand if one JUSt multiplies

out the first line of Eo. (A.6) and considers the phonon annihi-

lation terms one finds

. -3/2 * t * 1+ * + (l)
-13 <E;%(Aixjxkaiajak + Aixjxkaiajak + Aiijkaiajak)>
‘ 1 (A.7)
Zni + l) - o'llxilz}

62
The last term in the brackets is the correction from the commuta-
tors, and it is readily evident t'at this va isnes as Q + m.
Once Ec. (A.;) is established, it can be extended immediately

to analytic functions of the Operator A:

 

(0)

The diagonal operator <exp w> can be evaluated by applying
Es. (fl.9) to the exoression d/dx <exp XF>(O) = <fl exn xfi>(0)

to obtain the differential eauatien in the parameter x:

g1'<exf>(0) = A<P2>(O)<eNA>(O) (5-10)

with the boundary condition: <exn x >(O) = l at A = C.
The solution of E0. (A.lO) is

(ef>(0) = exp[% <A2>(O)] . (8.11)

Thus the operator 0; of E0. (A.2) does have a diagonal part when

E is a vector of the reciprocal lattice, a result which follows

(0)

is inde-

)2>(0).

9 S

itmediately once it is established tflat (exp if-ug>
nendent of the cell index 2. Consider the coerator %<(fou

it may be written

W0) = e7)

NIH

1 n T009,
5 <(£°%9 (fi/zaaaoc) f°gcc(e a

CT?

-'M

L)

(A.12)

63

m - -)2
: g :97] W01?” (2n

 

L3u~ on + 1) (5-12)
on or.“ .
3 m - >2
2 l 2 I d c g gap (n + l)
2 C (2n)3 amt? C? 2

which is independent of the cell index &. If we perform a
thermal average for the lattice and invoke a Sebye model, this

last expression becomes the neoative of the exhonent of the

 

 

Debye-Valler factor, exp[-!J(f)];26
2 2
2 2 T 6/;
(f) : % nzf ('61:)J L { 1 + %} ZdZ (13.13)
.4 Z
l kB(} 0 e -1
which in the cold lattice limit (TL + 0) 15
2 2 2 2 *
ll(f) §fl_f_= .. é: . Tl— . EL. (Ami?)
8 MkBB Li Li” {\pe I4

Gne can see from Ec. (3.l4) that in the present case the Debye-
Waller factor is essentially unity, because i = 5 is of the sate
order of magnitude as the initial electron wave vector &; we

g < 1150K
l) ’b

'k
are interested in electron energies such that h2k2/2m k

*
and where m /V << 1. Then
.2 2 *
(j . .1;— «i , (p.15)
2m kBe ‘

3

2':

 

.c'luu

(In metals where the Fenni energy is much greater than kBe,
")
the Debve-Valler factor is significantL7.)

Lastly, the one- and two-phonon parts of tie interaction

operator U [Eq. (55)] must be established. First, the one-nhonon

operator

0' - o . 0k _"
(O-K>(]) 5 X 12 h <eiu K>(l) : 1K. E t» e12 A e
t 1 Q “ (/ l7)
= 1 '2E: e / ‘)2 2"l 5(” “ l + + *(‘ /)} e-)
n: rm m 'l 2,0qu , $7 171 m’ m o”, “ ’b”\;
l
2
r \ T -
= —— \ d / + F1
(:%I)]Y (ZDQng) [‘- ,t ,,] 9
One thus obtains the formula
m 2: l
= - , -” o. 2 + +
U l ”K K e (K/_UwK) Ck-K CV (a_K + aK) ( .l8)

k,K

wiere the longitudinal polarization index is suppressed. Next,

the two-nhonon operator

U(2) EEEi(Qa/Q) 9K CE-K CK (O-K>(2) ' (7°19)
Here
<048(2) = _% §<(§'%2)2 (2) eig.K e-“
= -éo;:;;' (o/oa) ”x Kze')(%/CEQ) (£.zo)

 

f)
'5(R+Crt"r|$) + 2 aoca:'r'é(%'-Q’Kz)>( )

‘0

2)

where the <...>( notation is preserved in the last ecuation

65

 

as a reminder that the diagonal oart (a aT + a+ a ) has teen
cp on op op
taken out and rut in e'”. Tyen one finds
' -L«
”(2) = -§:E:(a§;) 9K K2 9 I Cl—K Cr
k,K L “ ‘ (A.2l)
E cos(K,cn)cos(K,q_'p') n. ,
x an C'F' Ev <aFTaO'E'6(%+& ’-&)
VF,' .
(”an wa'n')
+ + \(2)

i . u , .
C¢a0.D.6(g+g ’h) + 2 aC.D.acn6(g -g.§)/

TWAXSFOPHflTICJ T? PSTLRTE SPHEVOIDKL COCTCILATES

In this appendix are collected the details of the transfor-
mation to nrclate soheroidal (PS) coordinates to be used in the
sum over two phonon wave vectors for fixed electron transition:
h + 5 - §- Fe choose axes as shown in Fig. 3, with the polar z-
axis along the wave vector transfer g and with the (5,§)-olane

as the reference for azimuthal angles. Then the appropriate

rectangular representations of the vectors of interest are

h K (0,0,l); E = 5 (-sine, 0, cose)

0+ (iSlneiCOSQ, tsineising, cosoi)

0
q, t ..

2 . 1 r ‘
where c aZimuthal angle of Q+. cecause K, %+’ and g_ are
conlanar, (n+c) = azimuth of g_.

He now go to the PS coordinates g,n,; according to the follow-

ing transformation:36

m
H
A
..O
+
+
f)

l
v
\

7?
v
....o
A
m
A
8

.3
II
A

D
+
I
O
v
\
7:
a
I
-—J
A
J
A
+
_a
A
CD
0
N
v

(as above).

\f‘l
ll

azimuth of o
m+

Elementary trigonometry then yields the following expression

for the more important functions of the angles:

66

 

 

 

 

Fin. 3. Pectangular and spherical coordinates for descriztion

of the two-phonon geometry.

67

 

 

 

1
2 2 2
_lttn. .- _[( -l)(l- )]
cos ¢¢ ~ gin , Slfl at - 5 gin —fl~ (8.3)
1
n 2 i’
, _ ltgn - [(gF-1)(i-n )1
cos(k,oi) — cos a tin t Sln 9 cos g gin

The inverse to the transformation Eo. (8.2) is

l
q,(;) = . K [(gZ-i)<i-n2)12 (33:) ( )
C2.,(2) = £— (ltgn) (8.4)
13.1 =-§ (gin)

By direct calculation of the Jacotian, one finds for the volume

element

0

d30+ = (3)3 (EZ'nL) dcdndc (2.5)

he PS coordinates [Eo. (8.2)] refer to a system of confocal
ellipsoids and hynerboloids of revolution about the major axis
of the ellipses. (In this case, the foci are the ends of 5
and the axis of revolution is along 5.) A fixed value of 5
defines a orolate ellipsoid of eccentricity e=g-]. As one
might guess from the domain of n. this variable is related to
the cosine of an anole that varies from n to 0; it is most closely
related to the cosine of the angle between the polar axis and the
vector (0 -%§):

m &+ i“) = n[1+(n/g)2 - (l/t)2] (8.5)

NIH

FlXEd n defines a hypertoloid, the sfeet of which is determined

by the sign of n.

9PPELDIX C

[ULERICKL ESTI'7TICH OF THE CURJEKTCRFS

In Chapter VI we reduced the formula for the interference
and intermediate-state power loss to five-fold ouadratures,

which, for the latter, is of tke form

00 1 V 1 2‘”
C = I dx J du I Udy J dh J gZ-G (x,u,y,h,z)
xd ud o o 0 L"

where the variaLles and limits are the following:

: 2 ’) *l ' 1,2 _ 2 l, 2 —_.
X (“I /LmI<B~[)I\,xd-‘rk/I\S 4X5
1
u = cos(k,K) , ud = k /k = (XS/X)2
y = (h2/2m*krT)K2, yu = 4 x (u-ud)2
q:n ’ch

 

 

0 O A+ _ [34-A—
l l l
where 50 = (ZXZU'Y2)/X52
l l l
2 2 2
D: = 2(X COS(%agi)-XS ) a: - at
212 a (HZ/2m*k3T) 01h = %'(€otn)2 1
E
1.50%. [(goz-lHl-hzfl
Cosmo aotsficosm "” ' €01h

70
)2 + 2 («3.4)

He concentrate the discussion on the internediate-state power
loss, because it is of the order of maonitude of the one-rnonon
power loss. The interference :ower loss oroves to be many orders
of magnitude smaller. (The method used to estimate the inter-
ference two—phonon power loss was a variant of the Vonte Carlo
method described below.)

Because of the resonance-like behavior of the ouantities
(Ai)-], it is quite out of the Question to estimate the five-fold
Quadrature Eo. (C.l) bv t”a use of numerical inteoration metwods
such as Causs's method-~simoly too many evaluations of the inte-
orand would be require'. The two attacks we made on the orotlem
of estimation were (l) to use a random sampling ( onte Carlo)
metnod to estimate tne five-fold form [Eo. (C.l)] and (2) to
reduce this form by a heuristic "mean-value" argument to the

four-fold quadrature

l
dh 5 (C.5)

which is then estimated by four-fold Gaussian cuadrature.
The mean-value argument may be visualized with the aid
F :ig. (4). here is reoresented an electron transition by two-
phonon processes. The initial energy surface is I, and F is
the final-states surface forced on one-phonon orocesses by energy
conservation. To reduce clutter in the diadram, the wave vector

transfer 5 has been moved so that its tail coincides with the

 

 

Fig. 4. L nes of resonance, 2+ and R+, for the contrsction

. . ‘ . . . . c -
over o.onon deora;s of free ~r in tie tranSitiun rate w . Sur.a:s
is electron initial enerff surface, M (and fi') is the final at to

surface for one-niooéi EPlSSTCH, and N is the t --‘honon ellirsoid.

7l

72
head of 5 and the appropriate energy conservation surface n
drawn. The two-phonon ellipsoid is reoresented by U.
The dominant contribution to the ouadrature 0 comes about

when, in the integration over h,z, first a and then g nave

+
in the narrow strip on the ellipsoid that surrounds the line
of intersection of the ellipsoid V and the energy conservation

surface M'. This line will he termed the "line of resonance"

of o and it is labelled R

m+ +; the line R_, traced out by 2+ when

is called the "line of resonance" of o

g_ 15 moved along P+, . ¢_.

It seems reasonable to expect that we would get arproximately

the same over-all contribution if we replaced G with the integrand
G = G(Z=n/2) which has its lines of resonance at 3+, P_; this

is eouivalent to forming G from G by reolacino the functions

Di by their averages over the azimuth:

a = e[ii]
_ 2“ dz (C.6)
With U E J T D+ .
0 " ‘

The validity of this mean-value approximation will depend
primarily uoon two criteria. One criterion is that the strips
of ellipsoid surface wnich provide the dominant contributions
are approximately eoual in area along the lines ”1 and 51. If
the areas of these strips are prooortional to the lengths of
the lines of resonance, we see that the areas will be approxi-
inately eoual, because generally the actual lines Rt tend to oass

around the ellipsoid the long way but do only halfway around

because of the condition, 0 < h < 1, whereas the mean lines it

73
go all the way around the ellipsoid but along a shorter bath.

The other criterion for the validity of the mean-value
aoproximation is tlat the h-dependence of the integrand--excluding
the resonant factor--is such that the values taken by the inte-
grand do not differ significantly along the actual lines

.5 l. ’ v .. 2 L‘ ‘L 2 L. 2 2
and along toe mean lines a+. THE factor (go ‘u )

approximately independent of h over the integration because

i

will be

lhl :_l and go is usually appreciably greater than unity. The

only other h-dependence of possible consequence is in the term

2(D+D_+F02)/A+A_ which represents the lack of statistical inde-

pendence in the two ways to the intermediate state. When one

phonon, say 0+, is near resonance, this term becomes approximately

2/A_, and if the o_ resonance is not nearby, then this term will

be overwhelred ty the q+ resonance: 2/A_ << ro'z. is a matter

of fact, in the honte Carlo estimates, the contribution of the

2
)

term 2(D+D_+1"O /A+A_ was always less than one per cent of the

contribution of the term (l/A+ + l/A_).
3f course, the above argument is only suggestive--not con-
clusive. It was checked by comparing values of 0 obtained by

\

Gaussian ouadrature with values of Q obtained by Fonte Carlo.

J

The best Wonte Carlo results for P were consistent with the

\

Gaussian results for Q within the statistics of the sampling.

fit the same time E was estimated by the same sampling to give

another indicator of the reliatility of the Monte Carlo results.
The random samplinn method used is known as donte Carlo with

importance sampling. Ve will briefly outline the method and

refer to the monograph by Hammersley and Handscomb37 for further

74
details. ”a deal With n-fold ouadratures that, after agnronriate
changes of variables, may be written as inteorals over the unit

cube in n dimensions:

). (o7)

There are many ways that this quadrature can be written as the
mean of a function 0(x) over a cumulative distribution function

(c.d.f.) F(x):

1
I = <n>F = [a any «up (2.8)
where F(x) is a non-negative, semi-monotonically increasinq
function such that
l
J dF(x) = l . (C.9)
0

If we chose F(x) such that its first derivative exists, we can
just as well work with the freouency distribution function

(f.d.f.), f(x) 2 %;-, so that C(x) = f(x)o(x). The random

A

variable n has a variance with respect to the c.d.f. F(x) qiven

by

(C.l0)

The method of random samplind is built on the followino
result: If x],...,xN is a seouence of random nunbers drawn
independently from the distribution F(x) (i.e., we have a random

7._ 1

sample of size of the variable x), then the random variable

,>g-) . (CM)

in the limit V + w, approaches <g>F with probability unity; the

variance of the random variable 6F is

0 7 .1 n
O'EaF] = OF /I‘. . (v.12)

As samplino is carried out, it is a simple matter to calculate

the guantities

and

/H ; (C.li)

these are actually random variables on the same footing as gF,
but one can expect that they will give at least a rough estimate
, . . . 2 2 . .
or the true statistical parameters CF and OEa ] . Tae quantity
“F

§[6 ] will be called the "empirical" standard deviation to
“F

contrast it with the theoretical standard deviation o[§ ]'
F
Tne decompoSition G(x)dx = g(x)dF(x) 15 of course largely

arbitrary, and in practice one makes the decomposition for con-
venience in calculation. So called "crude" honte Carlo corres-
ponds to taking dF(x) = dx (that is, uniform sampling). Impor-
tance sampling consists in choosing the c.d.f. F(x) so that (l)
it is relatively easy in terms of computing time to generate
random numbers x distributed according to it and (2) at the sane
time F(x) concentrates x in regions that contribute most to the

ouadrature. The importance sampling actually done for our two-

76

phonon power loss expressions was of a very rudiicntary kind.
It would have been est to sample the variables h,z in the
sum over the two-phonon ellipsoid from a distribution peaked
around the lines of resonance R1[Fig. (5)], but to locate these
lines involves the solution of ouartic ecuations with variable
coefficients. Ve settled for uniform sampling of the variables
u, h, z. The sampling of x was according to the exponential
F] = exp(xd - x), and that of y according to F3 = (yZ/yuz)n
with n = l, 4, 8 being tried to see which was best. As it turned
out, the case n = 4 proved best.

The integral Q [Eq. (C.l)] was reduced to an integral
over the five-dimensional unit cube accordino to the following

transformations of variables:

v1 = exp(-x + xd)
v2 = (”lud)<(1-Ud)
2 -
v3 = (y /y.2)". n = l. 4. 3 ((3.15)
u
v. = h
L?
V5 = Z/Zn .

Then

1 1 l l 1
Q = J dv] J dv, J dv J dv4 I dv5 exp(-xd)°(l-Ud)

 

Q 1 01‘- 0 3 o 0
5 n i l-fi (C 16)
(yu ) (y l
x n O C(X,U,yghsz)

where the extra factors in tke integrand are components of the

Jacotian of the transformation. The variables v], vq, v4, v5

77

are obtained from the standard nifcrm random hunber function of

the computer, and then the variatles x, u, n, 2 calculated from

them according to the inverses of tie transr ions [Ec. (C.l3)].
Rather than sample according to F3 = (y /vu )n by taking
the n-th root of a uniform random number v3 (root-taking is a
relatively lengthy process for the comcuter), the program did

this sampling by a rejection technique. 5ne obtains a distritu-
tion according to the c.d.f. F(x) = xn by selecting the largest

of a secuence of n independently generated uniform random

numbers.

At the end of this flppendix is a listing of a program,
MSPUNEP, that estimates by random sampling not only the quadrature
Eq. (C.l6) for intermediate-state two—phonon power loss but also
the analogous quadrature for the four-fold 5- ft the same time,
the statistics are accumulated in the form of fractional empirical
standard deviations. The computation proceeds bv breaking the
total number of points to be sampled into subsamples and outputting
at the end of each subsample not only the estinates for that sub-
sample alone but also the cunulated estim: as for all samelino
up to that point. The input data are the number of subsamples
to be taken (waD), the nurber of points per subsample (frfDSB),
the cold-electron-threshold temperature (T57), the electron
temperature (TE), thelvalue of PO (Gf”f), and the exponent n

a -
of the c.d.f. F = (y /yu2)n ( F3).

3
It is convenient, in sumnarizing the cuantities computed
by the orooram, to introduce the functional L for the intermediate-

state two-phonon quadrature:

 

 

+-o
L [A+ + X; + 2 a+A_ ] (C.l7)
o _ w l V l 2n 1 l
_ (29n7/‘) ‘ J dx I du J ”cw I an I 92 e'x x2 v 2x 2
xd u. 0 O 0 d” S
2
D D +r
2 2 2 l l + - O
x a (a -h ) [ - + —- + 2 l
0 O A+ a_ A+A_ J
so that the cuantity IC cf Ec (l35) 15
2
D g +F 7/2
IC=L[J-+l+2+'0].3 (C.lt?)
+ A_ A+A_ 4

flso we recall from the earlier paragraphs t'pt the substituticn

D+ + D+ 3 (0+)z- /2 produces the "mean-value" four-fold quadrature.
- - -n

Then the cuantities that are output along witi their emoirical

 

fractional standard deviation ("FSD" = s[;]/F) are these:
for the subsaople:
D D +r ‘
ycgggg = L [l +.l + 2.;:;;_Jl_]
A+ A_ A+A_
z 1 1
‘I.o B L [A+ + a_]
D+D_+1‘O2
”I TSB = L [2 ]
A+A_
_ _ 2 (C.l9)
W W +r
ucsaa = L [% + l- + 2 +_'- 0 ]
A+ 5_ «+A_
-l l 1
z I 4. p ;.1
T A A“-
“'El’r,
1 : l r 1 ‘ L,"
‘ A A

for to: cumul‘teo :

filial-{... ull|-‘. 1%! lllu .Illll. .{

~ 2
L D +t
mic-Tn = L [1 +1 + 2 + - a]
A+ A- A+A
NTND=L[l +1
A+ A
..JD-l-I‘d
WIFT = L [2 _i;:__9_J
A+A' (c.20)
-— ’)
DD+f“
l-IC4 =L[J__ +1_+2 +_-_o]
+ A- A+A_
wo4 -L[l +17]
A+ -
fi+5 +f02
HT4 = L [2 _'_ ]
A+A_

The method of Gauss-Legendre numerical quadrature approxi—

mates a one-dimensional integral of the standard form

l
1=[ CV on) ((3.21)
-l

by the m-th order anproximant
m
I = 2:: f. E(v.) (C.23)
m ._ l l

i—l
where the abscissas vi are the m real roots of the m-th degree

. . . . . 38
Legendre polynomial and the 91 are the assoc1ated weights.

Tables of abscissas and weights for orders up to m = 96 are avail-

able.38’39

The extension to n dimensions is straightforward in
principle, although the computing time recuired raoidly becomes

unreasonable. Thus for the n-dimensional ouadrature

80

”e may apply tLe one-dimensional Gauss-Legendre formula of orderrfi
to the integration over v], the formula of order m2 to the v2
integration, etc. he then denote the collective order of the
n-dimensional guadrature by the ordered secuence m = (m1, m2,

., mn), and the m-th approximant to I is

. ) (C.22')

After introducing a large, but finite, upper limit xu (= lOCO,
nominally) in the x-integration of 5 [Eq. (C.l6)] we reduce the
four-fold QUadrature to the standard form for the Gauss-Legendre
method by the following transformation of variables:

exp(-x) - exp(-xu)
vl - exp(ixd) - exp(Lxu) ' ]

 

[\3

v2 = 2 (u-ud)/(l-ud) - l

(C.23)
v3 = 2(y/yU) - l
V4 = 21" '1
Then
- - 2
D D +F
L[-1— +1— +2 25-0]
4- A- A+13.
., x l y l 1 g 1
: (29117/‘1-1 I udx J u I udy J dh x e-x y5/‘ ”XS y2
x u 0 0
d d p w
F} E +P 2 (u.(.t)
9 w '_
x50(502-“‘)2 1— +!— +2 +__0]
A4. A_ A+A_
9 7/2 -1 .1? 4 l l
(2 N ) 2xS 2 [exp(-xd) — exo(-x )] I 1dv1(l-ud)J1dv2 v x

8l
,.-., .- 2

l l U +1“
3 - 2 2 2 l l + - O
X dV 3' g I (JV (5 "h ) [1: + =- + 2 —-—.—-..—~—-—-] ((3.24)
)_1 3 o _] 4 o A+ A_ A+¢

The prograr "LDIFFfl (listing at end of this Appendix)
approximates the functionals cf EC. (6.26) and outouts the

aporoxinants as follows:

- - 2
C +r
wCTi = L [l- + l— + 2 +_-_ 0 ]
A+ A_ A+A_
H = L [l- + l- (6.25)
13+ A-
— - 2
D (l +I‘
Fl = L [2 —ie;er£L— .
A+A_

The integrand is most variatle for the v4 integration, and it

will become orogressively smoother with succeeding integrations.
Conseouently, low order should suffice for the v1 integration and
very hioh order will be reeuired for the v4 integration. Pro-
vision was Wade for orders up to (24, 36, 96, 96), but the acproxi-
mants aopeared to stabilize with increasing order at (l2, 24,

64, 96) and no higher orders were taken.

The question of reliability of the nurerical estimates has
two parts. First, how good is the mean-value reduction to a four-
fold ouadrature? Second, was the order of approximation high
enough that approximation schenes adequatelv sensed the sharply
peaked nature of the integrand? The first ouestion, on a purely
analytical approximation, arises in this discussion of numerical
estimates because we chose to use the Gauss-Legendre nethod to
approximate the four-fold ouadrature for the intermediate state

two-phonon power loss at different temperatures.

82

4 study of the reliability was nade by comparing Gauss-
Legendre and “onte Carlo results at a prctcty:e terrerature,
T = l00 TS (z ll.5 deg K for n-type Ge). First, the :uestion
of whether the order of the Causs-Leqendre method was adecuate
was attacked by increasing the erer in stages. Because the
sharply peaxcd integrand is Positive definite, one exnects that
the arproximants Vlll increase with increasing order as the roints
at which the integrand is evaluated are crowded closer together;
then as the order increases still further, the abnroximants will
tend to stabilize. This behavior was indeed observed; the aonroxi-
mants increased with increasing order until the order (l2, 24,
64, 96) at which the approximant actually decreased about 20%.
The last three approximants to 0 (in arbitrary units) with their

orders follow:
(3.26)
(l2,24,24,96) (l2,24,36,96) (l2,24,64,96)

order

 

0 = 7.4l 9.93 8.24

 

 

 

“ext, Monte Carlo estimates of Q and 0 Here made for saroles
of uo to l05 ooints in subsamples gf 4000 points for various
methods of sampling the variable v5 (still for T = 100 Ts).

From the related criteria (l) of least fractional standard devia-
tion at the end of the sampling and (2) of least fluctuation of
the estimator as samplieg oroceeded (aoplied to the estimator

for thelfive-fold euadrature Q), we concluded that the samoling

2 . . ..
F3 = (y /yu )4 was best. The results of thlS estimation (same

units as Eq. (C.26)) follow:

83

I.)

= 5.71 (i 361)
(C.27)

II

7.35 (t 44%)

The errors in parenthesis are the probable fractional error
(p.f.e.) obtained from the empirical fractional standard devia-
tion (f.s.d.) by the usual formula (p.f.e.) = 0.C74Ex(f.s.d.).
Thus we see from Eqs. (C.26) and (C.27) both that the Gauss-
Legendre and Monte Carlo estimates of 0 are consistent and that

0 is probably a fair approximation to Q.

ul'lnl'll.‘l]llllvill"lulll‘!lil

 

wean thZ Oh OMIZRFZOU QMPOOmE

muz..nh «m co

o.cnnm.>

QQOZ.«anCC

OHmFDZ fl OoOuUPZHW & CoCuUCZHu

c.cuquesz»c.ouvogz~u

C c CHUUe/wc . imp. >9: Sn); s>

c.cneuu>eo.ouvoe>mc.auvuc>
.\.*mHZ?nmm .uHZZHQu .AI_#.N~.* CF fl
cum~<0v>uru .L_2:uwu .nom xvuaxwuum mag m.omocou*.x_.\.mom_m.*uqzcm
U.*.xu.c.aaw.#n<:qc.#.xm.m.mm.*uozwh CJOImwaIP.k.xm.m.®u.*uazMFt.xN
~.\.*uccamlm 2h mhzhoa*.xm.oqw. * 00¢ ouaqu wkZO$*.x~vk<£cOu
mLZ.UZOU.<E<0.0Zb.wh.kOhZ.u FZmdQ

N¢$<z<0ucu<§qc

mou<>*o7ouuq<um3 a mN~.o*ZOUuUZOU

Am**wu*ahuvukacmto.cc.\0o~uéou

mu2u\«04>#2XkoomanU<>

me ”Nexu.uaxm*m.cu_u<>
HQ*C.NH~QOE_

«*acxumcx u 2x*c.uu:x

ZX\C0HHZ¥C F .vaukacuuZX W wF\DZFNOX

GMCOCZ*CUCZuFOFZ

Anuz.uk<OJuumuzu

,m_.m.m~m.(.u—u.c.a~u.m~_.N~H.eqsaou
FuZ.<§<0.wH.C2_.meuCZ.CUCZ.~OC~ Cdma

.u.>20~¢2u2~c

Acugmmdcuwvaomuuuvchc

Gflzo¢2§4quOUQ>h

22.Q244mflufl>k

ouzcams S<Uocou

AZ<GFHCK COG“ UOUV
mmCJ ERECQ ZCZCIQIOEF uFCFMILFaHChhflhFZH QCk meDF<QC<DG DJOu
lm>~u C24 IQDOu uIF OZ~F<EHme QCK E<OOOQQ OJQCU mFZOE uO 02~P¢~J

H

ace“

.1).
00

h.u\_<0 FXLZ CF CLArL~FZCQ ,Ohr..._ODUP
aa*flzc*ocvw~X*mu2L>*muz:>*"6*XHFZFL
Au>au+zxv*>QCquc*run¢0C
Au>20+2xvfi>zclzuc*fiuuqzc
A>OCIQZXIQX*CEV*u>oCuQC
.>:0l¢2qux*sz*u>éou§C
au\AU<hcluuu "a? e £w\anuu+Euu.nZE
“Nvuuoc*aAm**:IO.~v*am**llc.~.*hoofilflu*wxvvuhaomuu<uu
>*xuec e ou+3uuLc w mulbuguc
>00*uo.cuu>ac & >2c*um.cuu>§u
CQ$>N>QG m ZOx>u>§0

2*I*_xueu a IIHXHEC w I+~quc
A>*U.CID*X.*ZXCu~x

~003F*au.>uN

AQV>NI

.«uznmv**>nmuzc> e mu2**:>umu23>
amv>*:>n>

ACDIDV*X*Comu:>

am.>*fio+03u:

C2I0.~u~c a X\Zx"c:
2x*c.¢uq2x a x*c.cncx
Ah.~.>.uOOJlmcxvueccmux
amca02.~n§ K CC
Cooudu>wcocucb>mCocqu>
CCCNU>9C0CHF>$C¢CHC>
c.0uqez~mmc.cu¢c2_m

o.cunmuec2 e c.0uez~w e c.cuoz_u
A\.A~F.r_u.xa..*u uoz Ecc2<o uqhefiz~*.x~ce<zaou
Am.~u~.A—v>v.mh2~0a
m32_kzcu
A_nvu2<eu.m+wv> a .Huvu2<au.~c>
No mum "CN CO

wDZ~FZOU

o2me>uamv>

NN..N ..m»>.ee.o:we>vu_ a «anymZaauazue>
veo<o m no m wean awzoom:

fl

_OK

«a
"a

8C

uuca exuz Ck CUDZHFZCU Buaouuz
FE*U§:VHLU<W $ FE*UFZ~WNCF<% m F3$UC2~UHUC<U
mak3*¢§3muqu<u w mme3*¢h2~much<w e aur3*¢CZ_mudC<m
E¢F3$$Dwuu<m w auk3*FZ~wu#<m m flmFE#CZ~muC<m
QUFZ~m+GUQZ~mHQUEDm

¢k2~m+¢02_mn¢23u

(CZ _ U.+UFZ H muUEDU,

F2~m+02~mu230
QL>+vLU>uqcc>mcp>+¢ce>uque>eco>+cuc>neuc>
u>+uu>uuu>me>+ue>uue>ec>+uo>uoc>
ceZHm+<cFZHmu¢UkZHmeqC2Hm+vuczHmuduczhm
F2nm+0h2~mu0kzfim m CZ_m+UOZHmuUCZ~m
mth\oouuF3 e mmc&02\co~nmmk3
mmca02+meu2umkuz

mDZHPZCu

wzzahzou
“givu2<auam+wv> m A_lvuZ<ana_v>

u.~u_ ~c¢ OC

w32_kzou

azwk>naww>

¢¢._a hamv>okconzuh>vum w “alvu2<auazuh>
mu2.~u_ ¢¢ 0C

C.cuflmv>
«*xfi¢FF+QCFV+vb>nqu>wm**fiee+okv+c>nu>
o**qek+¢k>uQ»>$N**¢CF+¢O>H¢C>
a**kk+b>ue>mu**ce+o>uc>
€kk+¢k2~mudkznwm¢CF+vCZ~mMQCZHu
FF+F2_mukZHumCF+CZ~mnCZHm
canC*Q024C*ACaczaw+vfir*cacv*o.m*FZ~UHQHe
AQQQJC+VQEJCV*FZ~UH¢CF
CCJQ¢C:;C*fiCU<S<c+EQ*QDV*o.n*kzuuukh
.deuo+mzqov*e2~uuce
acu<§<c+m**vacv\c.wunaQJC$ACu<5<c+a**czcv\c.~uqach
Acu<iae+d**acv\o.~uach
Acuqf<u+n$*zcv\0.~u024c

Uw0<0 m kc r w0<c GHBOQUE

"cc

vq
_q

d» Mp..- Swr.r5l".
..

87

mw<fl FXUZ OF CEDZ—FZCU OMBCQm>

.XU~.\.G.E~&.* n QUE*.XF—.¢oU_uo* n cmw03*.XvaF<EQCU (0
ebkhu.¢kkk.¢k3.¢EUh3—
oQUCku.QCNk.QOSo<0¢DE.QUQNE.¢CNko¢b3odflmUBoCOquflfl
a\\.moW~Uo* fl Cmu*o>F~omoU_mo* u Owu*.xmwo\.moM~um
.* k2~3*.xmu.m.m~m.* u mue2~3*.xn~.\.m.m—m.# u cmu*.xrhq
.a.u~m.* u Cmu*.xm~.\.mouam.* n CZHB*.Xu_.q.K~m.* u dmcz_?*.xn—m
.\.m.u_w.* u Cmu*.XE~.a.U~u.* u Cmu*.xu~.\.q.u~m.* u zou3*.xmfiu
.aomumo* u mmZCUB*.XN~.\.*m><I UR .mkfl *.O-.* no JakCFfi

CZ< men *.©~.* no uJazqmm:m Ikl*.®~.* no 62m Fq*.xuvk<zocu m
UFNu.FNuoquiommFZ~3.UONuoOth
.c2~3.dmc2~3.uuwu.cNu.zou3.mmzcu3.meaz.mncecz.4.@ez~co
qe2+ac3uqus e aqumeequeqmuaes e Jqum3*euo<mu¢03
QEMF3+¢EmOEHQEmU3 w chm3*dkdmufiflmk3 & J<bW3$¢O<mu¢mmC3
FZ~B+CZ~3u20v3 u JaUfiE$Uhcmnk2~3 fi J<U¢3*UC<muCZ—2
CMPZ~3+Q¢CZ~EHEWZCCE & J<Um3*kqwuflmkzuz w Jqum3*0cmumm02~?
A¢UU<m\¢UUNVumm<u¢UUNu
AQUFGM\¢UFN.umE<u¢UFNu a “QUC<m\¢UCNVumm<nQUCNu
AF3*.N**¢UU<mlk3*¢UU>VvuhaOWHGUUN
ak3*an**QUk<mlh3*¢UF>v.uhcomudUFN
nk3*nm**cuc<mse2*dco>v.uhaomn6UCN
A¢U<m\¢UNvumm<u¢UNu
.¢F<M\quvhac<uveNu m .chm\qCN.ume<u¢CNu
acmh3*am**QUqUIamF3$cU>v.ueccmnqum
“Gmk3*an**qequtmukE*ve>vvueaomuvem
.CWFP*AN**QG<mIFMF3*QC>VquQCMNGCN
AQL<M\QUNvumO<uUUNu
Auk<m\UFthUfl<uckNu & AUC<m\tCthqunUCNu
“h3*an**uu<mle3*uu>.quUOmHUUN
AF3*no**uk<mlk3*ue>..hkocmuLeN w ak3$»n**LC<mlhE*LC>vvuhacmuucm
acqm\UNvumm<uUNu
Ah<m\FNVuMQ<flFNh 6 AC<m\CNVumU<HCNu
Amme2*.m**o<mnqmee*u>vcmeecmucw
nGUF3*AO**F<mIUWF3*F>VVHFQCWHFN w AmWF3*nQ**C<m‘EWF3*C>vquUCmNCN
es*qcs:muaooqm e ez*qeezhmnaoeqm a e3*cuoz~muqoo<m
mmo<o m no a mean awzoomz

If
(\

Q . QCCIuCoR Wow" mwfio OCCQ

DuECQUE ch thC <F<C COQL 02C uIF uC J<U~Q>F m~ GZ~ECJJOH mIF

02m
mDZHFZCU F
~\\.¢.m~m.* u cmu*.XE_q
.m.m_w.* n Cmu*.xm~.\.a.u~u.* : ce3*.xrw.uom—m.* u cdmk3*.xr~m
.\.a.m_m.* n Cmu*.xrw.m.mwm.* n cwu*.xm—.\.mom~w.* n «03*.XVHN
.Q.K*m.* u uddcz*.xm~.\.m.u~w.* u cmu*.xr~.mom_u.* u Cmu*_
ouc<c U uC m m0<a DmEOumz U

ucca exmz or cuDZHFZOU quuHCJC
IXZ.X<I.<CQCXZ.~Ccua
AIxzouu~ofiwvxxvomf<uO
IXZ.XXI.CUCXZ.~C<&Q
“aw".muucvveazacu m
Ar~.m<m.ruvh<:ucu H
mdmcoc*oku~u<>
A\.<.H~m.uczcc Im.xm.m.c~m.~
HQDX Im.xr.m.0~u.ncx IQ.XF.Q.-U.H<S<G I®.XF.moC~u.uazuk I®vk<§mcu «H
LZCU.QZX.CX.<Eq0.Q§wH.H~ FZHUQ
a.*mam
newscaqo no mu34<> c2_;0440u mIe eou*.x_.\.*mwoe ameoo ZOZOIQTOBF g
no S~CIQ Z~ mUZF<QC<3C uaczmoqumv3<wllcuuHCJw §<u00&0*.~I~VF<200u «00¢
mocq k2_oa "CCQ
um_.o*§OUuUZCU €-
am**~u*.wuvukfiom* Coqcv\ oonuscu
(0(CQFOPCFGGUflummow—Q—omuHO mpfi
q2q®*q2<0ncu<§qw
% moum~.cu<s<e aw“
«FIACXIvuoxmume
.QDleuaxwuuk
006*CXNCX
ACXVKFUCUHIFD
OEtF\C<FhIFuCX
Coocwuabx Us"
u.~.ouc<FMIe my”
mofinucsue HH~
“NVI<I..Nva1.nm.><I.Amv>x1.amvzcl..«VEXI.“QVXQ1.ANVXXI7o—m2uz~c
“(OV1<.AQGVIxonOOV><.nomv>x.awmv3<.ADVVDX.Agavxc..vQVXXZC~m2u§~C

quu~cqo $<eccuo

.Zqflkaou Ccofi UCUV
uucu au3co ZCZCIoucse ue<eunuechcuzcue2_ ecu ue:edocq:c
coounasou ale oz_e<z~emu ecu scoeoea ueczweeunuuzqw ac oz~em_4

ECQQ exuz Ck cuDZMFZOb dun—C46
Am—.a<u.m~vquacu ~O

MDZHFZOU «CCU
anvlcuaIIZ+~vI< e A_VIXIu.IIz+_va
Ilzonuudooooo

wDZHFZOU mace
._v><unI>Z+~v>< w ._V>XIu.I>Z+~.>x
I>Z._u_mhoooc

m32~FZOU «CCU
AHVDQHAIDZ+~VD< & A~VZXIN~IDZ+~V3X

IDZ.~H~RCOOOC
MDZHFZOU deco
.~.X<u‘1x2+_VX< e .~.XXIu.Ixz+~.xx
.Ix2.—u-ccooc
wtzghzcu a
m.m.<c&CIz.cw.c&CIZ.u_
.IIZ.«u_.._~I<v.No<ma
9 IIZ.I<I.<OQOIZ._c<wa
.IIZ.~n~.a_vav.mo<mo
IIZ.IXI.OQCIZ.—C<m0 h
m.baqcoo>z.cw.oao>zvu~
.I>z.~n_..~.><v.mo<wa
I)Z.><I.<C&O>Z.~Cqma
AI>z.~u~.a~.>x..«c<we
I>Z.>XI.QUO>Z.~C4&U @
m.o.<oao:z.cw.oaoazqu
AIDZ.~u~..~.D<V.mo<mo
Izz.:<1.<oa032.~04wo
AIDZ.~n~..~v:xv.mo<wo
IDZ.DXI.GQODZ._Q<wU
oo~_ceoo
A*muawd F200 ml< CZq m|x to oOleqde Cdmtvh<i00u c
aezuao m
m.m.<cacx2.cm.caoxz.u~
.Ixz.~u~.a_.x<v.mo<ma
mmo<o b no N ww<a qua—GJQ

LC

.*.r..*.*.r_.*v.

uec<o b no r uo<o

ucqu exuz OF CUEZHFZOU cuu~CJu

3*eaxuuee
5......- Oo~v*C3+._.2u3
.A—v:x+.~v*m.cue:
caC32.~u~oobOC
Coouunm
OocuDu
FHX\IkauG:
C.0*IFQ*FQX*~U<>u~u
.xvueacuukox
“WF*FX+~FVu¢CJIHX
AAWVXX+o_.*rooukx
CQszonuUOOQCC
Cocn~xm

A<CUCIZ._n_.A~cI<v.mo
IIZ.I<I.<C&CIZ._0
ACOOIZ.—n~.a_.Ix..mc
IIZ.IXI.CQCIZ._O
.qcuo>z.~u_..~.>av.mo
I>z.><I.<cec>z.~o
Aceo>z.—u...~.>xv.no
I>2.>XI.COC>Z.~O
A<cooaz.~u~.a_v:<..mo
IDZ.DqI.<CaODZ.H0
aca032._u~..~.:xv.«o
132.:x1.cac:z.wo
.qcaox2.~u_..~vx<v.mc
Ixz.XqI.<anxz._o
.C00x2.~u~.5_vxx..mo
Ixz.xx1.caox2 .wo

..*.*.m~.*.*.r~.*w
oucco ocu*.x~.\.*CILu «ecc uezeqoc<ac*.x_.eqsacu
cac:2.coc>2.cec:z.caox2.cccc e2_oe

,oCNXU

F2~a0
FZHQC
FZ~Q0
k2~au
FZHUO
F7300
ezaeo
hZHOQ
ezpau
F2~a0
...Z—Qfl
FZ—OQ
FZ~UG
k2~au
PZ~GQ
FZHQQ

a.c_.m,uc.x~..eqzocu

vuu~c40

U

«00¢
rccq
no

u

been exwz Ce cu22~ezou quu~04u u
mu*.y.>dnyq<
u22_k2Cu CCU
QCPF*02hk*nCm<5cu+$4uc*04ucv*c.n*4<<+—Im"~Im
AGQFF+a5hk.*J<<+ImuIm
ree*.JvI<nJ«<
acu<s<o+cw04mcv\ocwuaube
ACU<5<U+CmSJmC.\o.~uUEhe
uuwc*04moucm04wc
24m0*24mcuom24uc
Aaee+mee*auev*uuelche+«henn4mc
Adhh+fikh*2mkv*IUFICPFIKFFHEJMC
I*mmeeucee
u**aaue*smk.umeh
I+OHXuame
IIQ~Xu2me
A.J.Ix+.~v*u.cu1
QQOIZ.~choucc
o.cu~Im
oocuIm
ka>*IkUu¢hh
>* mmooumkk
c~x*AHhuNNPF
ha>*~eeumee
m**>*o_x*muumu
IFQ\AFU>* W.Cl~kk.flC—X
A>Vukacmuha>
a3>*k>u>
nayv>x+.—.*Kocuk>
COC>2.~uyccmoc
oocu_>m
oocn>m
du<>*~uu«u
o:>*nc:: oouvu0U<>
«**ACDIDV*X* o.¢u0:>
quda b to v wwqa QuuueJG U

93

02m
EDZHFZCU 00"“
A\.m.unu.*u~xm*.xm.a.m~m.*uxm*.x~.\.m.m—m.*u_3*.m
xwomomfiw.*n3 C2<*.x_.\.m.m—u.*uay wuc *.r.mu.*.zou2 ..*.mH.*.*.m~.~
*.*.r~#.*.m~.*vuawcao no uQDk<QCq20 quwmaqw oqculc acu*.xwvh<saou Kocw
. ~Xm.xm.«3.3.ZCUE.QEKF.OOCIZ.CCO>Z.DUODZ.C&CXZ.moo~FZ~30
~E+BHZOU3
~XW*UZCUM~3
XW*UZOUn3
sznkZOU Ocm
~3m*afivxq+~xmumxm
Dm* A ..J x<+xmuxm
wZZHFZOU CCh
~>M*A~VD<+HDWN~Dm
>m*.~v3<+:mu3m
MSZ~FZOU CCO
"Im*y<<+~>mu~>m
Im*1<<+>mu>m
wueao e ac m ueqo auuficqe

94

\
II
memmerCUPOo

\.

PQPVCCGflrCo
GxdmmGCLOFuo

L
-

NMLCmUumommo+mwFF~GCwPRU.+NOmF
a.xcucumfro+CJROMUc

FCFmGGCW uoc.
[FNCQVMPCNCQ
mmmm MCNmCo

«mmommw m.<c.
mecapoacqec.

cqfihrrflcficc.
KOF¢FGRUC~Oo
omec00_cnmr.
NCQQGCPNrMUo

mochcquao

CGOnmpcmofic.
C—OuwupfFOCo

:Cm m:moo.
GG.U rOCCGW.

CrmceUPFPGC.

uncemcmupco.

A(0.G€.Gn.o.:

CUMFHG®MOmfio
UVUWLV(PFQGO
meKPCCQGWGCo
QFGGFUCPCCUo

¢CC.mcrrmmco

mmomeeemacc.
.wmhficemmmc.
emwupmcommc.

NwL—m WFQNGCO
"PLIpCGLFQCo

QQVUQerrmO.
comGtuwroga.
rpcmcwmgo_bo
pFUGwFOGOQGo
GrerrmGhnm.

ruaarth 00C.
0~CFOQQQFCuo

FCUUUCQFGFGo
CrwaanKGUo

CCWNFCVOCO—o

remuorpnGoco

mmm.+u

OOCGCHQPUCC
UGHCNCCCULC.
CCPwU
FUMGOthCmF

\lx,.\.l. .

-.. ...
{CCCCC

temmpm¢~_Pc.
FmNCbNmmeoo
(mmvmmrcmmc.

mmGNFmQWMGCo
C(UGCCCCFGOo

CFF\Qr .OPPFCO
UHCCeoccucm o
mcmpmwrauuvo
UWF_CGIWGGGo
«GORGGGGO®~0

ocarcbrmGGoo
Ecuwwucuuwmo

CCOUUGFGGPC.
F(FCKFOFKKG.

GUCOFQFcowmo

C—GF€RCCO(F¢

UtCQC QCU kaC

Q0+P~.t<COnCU
”GOO—WFOQ+u~H:m~

C G
MGGOGKKGCKo+CI.r
. ”H

04"»;

necoomech.
6:F— RFCCrLC
~thv.cv~rc.
—:QNOC\L{FO
.,r_(.

nwuaar.

mfixFCO

Ccrcomwcmwo.
.hcmcremwmo.
GbmmmeCFmOo
«figmmooraqc.
mNmIPGGemGCo

mcowFQCOHQCo
U.(O&CmmCGmo
rOFFGU(CUG(o
l0FL~CFC00€o
CKO_QR¢G.N~0

WOGQUGCQCUOo
reth0h(_m~.

oorcuupewcao
CFCFWNGOW_mo

obvmdh6_eomo

GQGUOwaFGU

(FCC QUU—CJU

fimmm_GHC&OUo
FFCFFCFWGfiOo
FGFVCOLCGHQ.
fiFFQQOPCQPFO
ethrmrowcmo

fUMCfiF—WWQO.
RmeGfiuFFNGo
WkfiwamfiFmio
Who—mflFwflwFo
WWQHGOflmQCKo+

G.+~GmwochumGo+mmCG©FOFmOmo+
GN_GN0+CGO~N
.cco+¢~vmeoFoaGoo+um¢thhm®~o.+

~fi00~N-+

0G mammmuvflq JC 00

eromurob_cco
mvcmcwm.m#00
WQNFOCM mccoo
mem—Cflm CMOQ
MOFN (HGn UGO.
GGFKGLFLOGOo
Kr

GFHGCUCflCOGo
(mehmfifiqoqo.

N~U ~mU1CN—ao
Ffiturt upnwflo
(POL—COOFUFO
OF~Q~W00NFOO

Wk10~m3

emrrcec_aoc.
aeu~h_m0b~c.

hermeqrmcmo.
mfiwohmwoomo.
muwoqumeqo.
Horwooocch.
4w «0
remaficamcoo.
namefiobmomo.
aommmmeommb.
omwvemcm_bm.
oofihmmmm~mm.
memomcmmcmc.

me w<m¢~Um¢< JG Gm

~G~GGfGWmFOo
UPCUCFFGUQu.

mmuo—chcmco
mmm®~mm0FN~o

«w mhoums 46 cm
range "ccou a. mu_o_cmc~cb.
rahw¢m-_o~. emmomomoqoc.

«H mammhummq 40 «m
auwrmmoaemm. "mucoeqfioqm.

c mere—us 40 a"

OOCCG~flmF¢mo

amGOGFWNUmgo

€ u<um~umflq JG a"

UIF W—

CZ~ECJJou wIF

PLQOPQOFCCC.
OtCFGQOCPCC-
PmCmmmmwvrCo
flOCFOCGOGHCo
CCUONMWGWGC.
th00n—GPNC0
oqmfiooomovc.
mmmmccmcmmc.

Ohcwommawcc.
rcwbmmm_mcc.
rbwqococcmcc
G—HWCCFWC~C0
9Pw<QGCNGOCo
NGFCCCNOFQC.
nfimnr~w1cno.
GPGORQCGQFO.

NGHMFcpfimcco
rnpfiqu—CCCo
NCHNFCDOU—Co
mfirOFQUccac.
mmmw¢c00qmc.
«eugenCQQuo.
cumrrcwcfloo.

PmoonGFOPCo

GMGmmGGChOC.
cmcrrcw.cwc.
oomqworomwc.
cacrnmrprnc.
wcemccrmmmc.
m.q~mcocmmc.
CCDOG¢~F~FO¢
fibrwfinqqoro.

GPNCNGHCUCO.
¢~m0~0q_a_c.
@mOPGMCQCFOo
«cchwmopmo.
<rcqmm~m¢mo.
wccuowwqomo.
rrovmoam_no.
~rm_nwmumno.
an

mhIonm?

CKUGUEUC¢CC0
F(Gccfiwumwc.

mrNcmmahh~oo
cmocorrrmmo.

flfifwmenmCNCo
GfiGGFCCmOmCo
CQCUFQRC—FC.
o<q~cowmnmc.
40 CO

REFERTTCTS

1For a review of hct e1ectron work rrior to 1957 see J. B. Gunn

in Progress in Semiconductors edited by f. F. Gitson, R. E. Eur-

 

gess, and P. Aigrain (Heywood, London, 1957), V01. 2, no. 211-297.

2A survey of 1ater hot e1ectron work, through 1962, in the narticu-

1ar case of germanium is given Ly E. G. S. Daige, The E1ectrica1

 

Conductivity of Gernanium which anneared as V01. 8 of Progress

 

in Semiconductors (Heywood, London, 1964).

 

3 .
For reference to even ear11er work on the sonewhat ana1ogous
prob1em in oaseous e1ectronics, see S. Channan and T. G. Cow1ing,

The Mathewatica1 Theory of Non-Uniform Gases (Cambridge Universitv

 

Press, Canbridoe, 1952), Section 18.7.

4E. J. Ryder and w. Shock1ey, Phys, Rev. $1, 139 (1951).

5w. Shock1ev, Je11 System Tech. J. go, 990 (1951).

6E. J. Ryder, Phys. Rev. go, 766 (1953).

7H. Kawamura, M. Fukai, Y. Hayashii, and T. Hashimura, J. Phys.

Soc. Japan 16, 1646 (1961).
’b’b

35. Er1bach and J. L. Gunn, Phys, Rev. Letters a, zoo (1962).

9J. Zucker, J. Aee1. Phys. 35, 61; (1964).

10Reference 1, on. 230-231.

97

11 . . , - . .
F. J. B1att, Proceed1ngs 9v 1ntornatitna1 Conference on Ser1con-

r‘ A

ductors, Prague, 1JLU (Czechos1ovakian fcajeny of Sciences,
Prague, 1962), o. 101.

12?. Stratton, in So1id State Physics in E1ectronics and Te1e-

 

communications edited by E. Désirant and J. L. kichie1s (Acaderic

 

fress, new York, 1990). V01. 1, PP- 343‘355-
‘31. J. Morgan and C. E. ke1iy. Dhys. 9ev. 137, P1573 (1965).
mm

r. Seitz, Phys. Rev. 7%, 539 (1948).

C. Herring, Proceedings of Internationa1 Conference on Senicon-

 

ductors, Prague, 1960 (Czechos1ovakian Pcalemy of Sciences,

 

Prague, 1962). pp. 62-63.

16H. G. Peik, H. Risken, and 0. Finger, Phys. Pev. Letters 6,

423 (1960); H. G. Peik and H. Risken, Phys. Pev. 121, 777 (1961);
’V'Vb
H. G. Peik and H. Risken, ibid. 126, 1737 (1962).
—-—-— ’b’b’b

17H. Froh1ich and B. Paranjape, Proc. Phys. Soc. (London) 069,

21, 866 (1956).

18;. Takinoto, Prog- Theoret- Phys- (KYOtO) Eg' 433 (1962)'

19J. Yamashita and K. Hakamura, Prog. Theoret. Phys. (Kyoto) mg,

1022 (1965). Contains references to ear1ier experimenta1 and

theoretica1 work.

20L. Nordheim, Ann. Physik [5] 9, 607 (1931).

Z‘COmoare H. Y. Fan, Phys. Pev. 6g, 900 (1951), in which tennera-

ture denendent band gaps are found usinn one-nhonon orocesses in

second order se1f-energy processes.

98

22 , . , . . . . . ,
J. P. Z1man, .1eCtrors and Pnorons (“xfnrd Univer51tv Press,

 

Txford, 1960).
a. H. rrockhouse and P. K. Ivengar, Piys. Pev. ill, 747 (195;).
R. J. G1auber, Phys. Pev. ké’ 395 (1951); ibid. 90, 1692 (1955).

J. de Boer in Studies in Statistica1 ‘echanics, Vo1. III edited

 

by J. de Boer and G. Uh1enbeck (John Vi1ev, iew York, 1965),
a. 213, advocates the term "construction onerator" for annihi1aticn
and creation 0 erators, co11ective1v.

26 n

J. . Ziman, Princip1es of the Theory of So1ids (Cambridge

 

University Press, Cambridge, 1964), Section 2.9.

27 . - . , .
L. J. Sham and J. M. Ziman, "The E1ectron-Puoncn Interaction”

in So1id State Physics edited by F. Seitz and D. Turnbu11 (Tcademic

 

Press, New York, 1963), V01. 15, Section 2.

28C. Fntoncik and P. T. Landsberg, Proc. Phys. Soc. (London)

f‘

803

337 (1963).

2 . , ., . . . , -
9M. L. Go1dberger and k. M. atson, Co111s1on Theory (Joan ‘11ey

 

and Sons, flew York, 1964), Chaeter 8.

30, . . . . . . . .
L. He1t1er, quantum Theory of Radiation (0xforo Un1vers1ty

 

Press, Oxford, 1954), 3rd. ed., Section 16.

3:Reference 14. Note that the constant C actua11v used by Seitz

in Ref. 14 differs by a facter of g-from the 01 used in the nresent

naner [E0. (77)].

‘ 0
LD

0
”For the connection between t’e interbretations of C] in the

rigid ion, deformab1e ion, and deformation notentia1 mode1s,
see C. E. Pikus, Zh. Techn. Fiz. 25, 612 (1958) (Eng1ish trans1.:
Soviet Phys.--Tech. Phys. 3, 2194 (1958)).

33H. Crooks, in Advances in E1ectronics and E1ectron Physics edited

 

by L. Parton (Academic Press, Lew York, 1955), V01. 7, on. 85-182.

34handbook of Mathematica1 Functions, Apn1ied Tathematics Series 55

 

edited by M. Abramouitz and I. A. Stequn (Tationa1 Pureau of
Standards, Nashinnton, D. C., 1964), Chan. 26, Tab1e 26.7.

35E. Franzak, Ph.D. thesis, Korthwestern University, 1960 (unnuL-

1ished); E. Franzax and M. Cai1yn, ths. Pev. 126, 2033 (1962).

3C. .. . ., . . . .
H. margenau and G. M. Murphy, Matnematics of Phy51cs and Cmenistrz

 

(D. van Eostrand, Princeton, 1956), Section 5.7, on. 180-182.

37J. H. Pamvers1ey and D. C. Handscomb, Honte Car1o Pethods

 

(fiethuen, London, 1964).

(3

UV. 1. Kry1ov, prroxieate Ca1cu1ation of Inteqra1s (Vacfii11an,

 

new York, 1963), Appendix A.

39Peference 34, Chaoter 25, TatTe 25.4.

 

111111H11|||1|1|ll|1|||111||111|||H1||1|1|1||11|
3 1293 03037 9501