P H Y S I C A L R E V I E W V O L U M E 1 6 6 , N U M B E R 3 I S F E B R U A R Y 1 9 6 8

Phonon Excitation by Electric Field in Semiconductors V. V. PARANJAPE

Department of Physics, Lakehead University, Port Arthur, Canada

AND

B. V. PARANJAPE

Department of Physics, University of Alberta, Edmonton, Canada (Received 28 July 1967)

Calculations of electron and hole mobilities in semiconductors are generally based on the assumption that the phonon distribution is maintained at a fixed temperature TQ even when the applied electric field is large. This assumption, however, is not always valid. One of us has shown that the phonon distribution deviates appreciably from its equilibrium value at low temperature (r<)~40K) and in strong electric fields. It was in fact shown that an increase in phonon density affects the mobility of electrons in high fields. Experiments on the mobility of w-type germanium were in reasonable agreement with the theory. We propose that a finite time is necessary for the phonons to build up to their saturation value. In the present paper, we cal­culate the phonon density and the electric current as a function of time. We show that, for a range of electric field F and for semiconductors with dominant acoustic mode scattering, the current density decreases from the initial value j(0) to its saturation value j (°°) m a time rPh of the order of the average phonon relaxa­tion time.

1. INTRODUCTION

IT is known that, for a nondegenerate semiconductor maintained at temperature To, the energy distribu­

tions of the phonons and the electrons are described, respectively, by the Planck distribution and by the Maxwell-Boltzmann distribution corresponding to the temperature To. Interaction between electrons and phonons leads to the emission and absorption of phonons by electrons. The rates of phonon emission and absorption by electrons are, however, equal, giving a time-independent phonon distribution equal to the Planck distribution, provided that the electrons and phonons are in thermal equilibrium with each other.

In sufficiently strong fields the mean electronic energy is very much higher than § KTQ and can be described in terms of an electron temperature T>T0. The phonon excitation number Nq in this case may deviate appreci­ably from that given by Planck's law. It was observed by one of the authors1 that at low temperatures and in strong electric fields the electrons interact with phonons that are weakly excited in equilibrium. Electron-phonon interaction produces a higher density of these phonons. These phonons created by the electrons react with the electrons and reduce their mobility. A steady state calculation of electron mobility, taking into account the effect of these induced phonons, was presented1 and later found to be in agreement with the experiments.2,3

We propose that if a constant electric field F is switched on at /=0 it will take a certain time T for the phonons to build up to their saturation value. Thus the electric current will not become stable until a time T. In the present paper we investigate the time dependence of the electric current in the time interval 0 to F. When

1 V. V. Paranjape, Proc. Phys. Soc. (London) 80, 971 (1962). 2 E . M. Conwell, Phys. Rev. 135, A814 (1964); Zylbersgtejn

and E. M. Conwell, Phys. Rev. Letters 11, 417 (1963). 3 E. I. Zavaritskaya, Fiz. Tverd. Tela 3, 1887 (1961) [English

transl: Soviet Phys.—Solid State 3, 1374 (1961)].

a strong field is applied the electrons will acquire a large drift velocity in a relatively short time f. In nonpolar crystals it is of the order4 f = (i£To/W2)ro, where TO is the relaxation time of thermal electrons, m is the effective mass of the electrons, and s is the speed of sound in the medium. In this short time the temperature of the electrons will have risen to a value T> T0, but distribution of the phonons will still be close to its equilibrium value. At t^T, the phonons will build up to a stationary value [see Eq. (2.16)]. The electrons will then be scattered more because of the increased interaction. Thus the current density will decrease from its instantaneous value to a stationary value in a time F. In this paper we follow the procedure presented earlier.1 An experimental confirmation of the transient electric currents calculated in the present paper will strengthen our confidence in the model presented earlier.1

2. PHONON DISTRIBUTION

We consider an ^-type semiconductor and assume a free-electron model. Let the electron wave vector be k, the corresponding momentum be hk, and the energy be tt2k2/2m} where m is the effective mass. In thermal equilibrium at a temperature To, the phonon excitation number nq{To) of the phonon wave vector q is

nq(T0) = {expthuiqyKTol-l}-1, (2.1)

where oo(q) is the phonon frequency. In the presence of a strong external electric field the distribution function of electrons may be written as5

/ ¥ \3/2 r P(k-ko)2 i /(k) = 4*W ) exp , (2.2)

XlrmrKT/ L 2mKT J 4 B. V. Paranjape, Phys. Rev. 122, 1372 (1961). 5 H. Frohlich and B. V. Paranjape, Proe. Phys. Soc. (London)

B69, 29 (1956).

166 757

758 V. V. P A R A N J A P E A N D S . V. P A R A N J A P E 166

provided the electron density ne\ is sufficiently large. Here #ko is the average electron momentum in the direction of the field, and T is the electron temperature. I t is known5 that the values of ko and T will depend on the strength of the electric field F. For F=0, obviously T— TQ and ko— 0.

Interaction of electrons with phonons leads to emission and absorption of phonons. By the principal of detailed balancing, the number of phonons does not differ from that given by Eq. (2.1) for the case of thermal equilibrium. In sufficiently strong fields, the phonon excitation number Nq may vary appreciably from nq(To). At time /, Nq(t) is determined by Boltz-mann's equation:

dNq(t) dNq(t)-\ dNq(t)

dt dt dt J P h

(2.3)

where the first and the second terms stand for the rate of change of Nq(t) due to the interaction of phonons with electrons and with other phonons or with the heat bath, respectively. Assume

dNq(t)

dt Jph

Nq(t)-nq(TQ)

^ph(g) (2.4)

where rph(q) is the phonon relaxation time. We write dNq(t)/df\e\ in the following form1:

dNq(t)

dt \-mj {(Nq(t)+l)f(k+q)-Nq(t)f(k)}

Xtt(E(k+q)-E(k)-ho>(q))k2dk sin0d6d<p, (2.5)

where k, 0, and <p are the polar coordinates with the axis along q. Here

sin( xt/h) Q(x)= (2.6)

and

B(q) = C2q

lSphsTt*

2hq2

x/h

for acoustic scattering, (2.7)

e*u(q)/l 1\ B (q) = 1 ] for polar optical scattering. (2.8)

In Eq. (2.7) C2 is the interaction constant, p is the mass density, and s the speed of sound in the medium. In Eq. (2.8) € and e^ are the static and high-frequency dielectric constants.

dNq(t)/df]Q\ is finite provided that either (a) the electron temperature T is different from the phonon temperature T0, or (b) the electrons have a drift velocity relative to the lattice, or both. In strong fields the electron temperature T can be very much larger than To. In this case the contribution due to (a) is expected to be very much stronger than that of (b).

For To^To Yamashita and Nakamura6 have considered the effect of (b) only on Nq(t). In the present paper, the effect of (a) on Nq(t) is considered. We therefore substitute (2.2) in Eq. (2.5) with &0=0 and integrate. We have

dNq(t)-\ ^Tr2B,(q)neim

dt I- q(2mwKT)m

X{(2V,(0+1) exp(-hu(q)/KT)-Nt(t)}

r h2 /q2 m2oo2

Xexp] /q1 mw moS\ 1

r \ 4 h2q2 h / J I 2mKT\^ h2q2

Substituting Eqs. (2.4) and (2.9) in (2.3) we obtain

dNq(t)

(2.9)

where dt

-PNq(t)+Q,

P= Z(T,q) 1

nq{T) Tph(q)

nq(T0)

and

Z(r ,g) = 4*<

Q = Z(T,q)-l

B(q)ne\m

Tph(q)

{-

(2.10)

(2.11)

(2.12)

q(2mwKT)^

The formal solution of Eq. (2.10) is

Nq(t)

h2 /q moSv

(2.13)

expl 1 - + — ) I 2mKT\2 hq/ J

= exp(- f PdA

x\nq(T0)+f Qexpff PdAdt' (2.14)

If P and Q are assumed to be independent of time we have

r Qi Q Nq(t) = \ nq(T0) 6T*H—. (2.15)

For £S>(1/JP) , Eq. (2.15) gives the saturation value

NM = Q/P. (2.16)

The time T required to reach a value near the satura­tion value in this case is of the order of 1/P.

3. ELECTRIC CURRENT

We consider in this section the time dependence of the current. Boltzmann's equation for the electrons is

d/(k) «F3/ (k ) d/(k)

dt 1 ,

h dkx dt Jcoii 6 J. Yamashita and K. Nakamura, Progr. Theoret. Phys.

(Kyoto) 33, 1022 (1965).

P H O N O N E X C I T A T I O N S BY E L E C T R I C F I E L D

where

2J 5(?){[/(k+q)(^ e(0+l)-/(k)^(/)]

166

where

a/flOn

dt -loll

XO(E(k+q)-£(k)-fa))+[/(k-q)^(0

- / (k) (^(0+l) ]Q(£(k-q)- -E(k)+M Xq2dqsmd'd6'd<p', (3.1)

where the factor J arises from the difference in the density of states between phonons and electrons due to spin. In Eq. (3.1), qf 0'', and <pf are the polar coor­dinates, the axis is along k and the direction of the electric field F is along x. If Nq(i) varies with time sufficiently slowly, then the electron distribution will attain a given stationary state according to the instanta­neous Nq(t) in a time of the order of r e (re is the average time between two successive electron-phonon collisions.) If Nq(t) does not vary appreciably in time Te we can neglect df/dt in Eq. (3.1). We further develop f(k) in powers of [h2k0

2/2mKT21/2 and retain only the linear terms.5 We have

/ &2k-ko\ f<*) = f(k)(l+ ) , (3.2)

V mKT I where

h2 \ 3 ' 2 f h2k2 1 /°(k) = 47T^el|

2tmrKT/ exp

ImKTA (3.3)

Multiplying (3.1) by h2k2/2m and fik separately, integrating with respect to k, and using (3.2) and (3.3), we obtain5

eFne\hk0

m

and

- 1 f

h2k2df(k)-]

2m dt Jcoii k2dk s\rti"d6"d<p" (3.4)

a/OOn

- • c < dt k2dksind"dd"d<P"> (3.5)

where k, 6", and <p" are polar coordinates with polar axis along k0. Equations (3.4) and (3.5) can be written conveniently as7,8

eFneihko

Sir3 J / «

eFnei=— /

kHk $in6"d6"d<p"f(k) / hu{q)B(q)

X{lNq{t)+\Jl{y.)-Nqm(y+))

Xq2dqsmB'd6fd<p'9 (3.6)

h2kk0 cos0" k2dk $m6"dB"d<p" f(k)-

mKT

X [B(q)hq cos0 cosd'{ZNq(t) + lJl(yJ)

-Nq(tMy+)}q2dq sinB'de'd<p', (3.7) 7 V. V. Paranjape, Phys. Rev. 150, 608 (1966). 8E. M. ConweU, Phys. Rev. 143, 657 (1966).

y ± = £ ( k ± q ) ~ £ ( k ) ^ & o ( g ) .

759

(3.8)

Integration over the angle <p' gives 2w. Following the procedure of Sec. 2 the integrations with respect to 0' can be performed.

eFne\hko m f = - / f(k)kdk

m h J

qdqhu>{q)B(q)(Nq(t) + \)

Uqdqhu{q)B{q)Nq(t)\ 1 , (3.9)

eFn, f Wkko

i = J w / / > ( A ) dk mKT

x\\J qdqB{q)(^+\q^(Nq(t)+\)^

\fqdqB(g)(^~iANt(t)\ 1, (3.10)

where the ranges of integration in brackets with subscripts A and B are

[ A - (k2-2mo)/ti)l^q^ [k+ {k2-2mo>/fiy2~], (3.11)

and

[ ( ^ 2 + 2 m ; / ^ ) 1 / 2 - ^ ] ^ g ^ [ ( ^ + 2 m c o A ) 1 / 2 + ^ ] , (3.12)

respectively. In the limit of large k, i.e., fi2k2^>mfia), the ranges of q in Eqs. (3.11) and (3.12) reduce to a simple form

mo) — ^q^2k. hk

(3.13)

Since we shall be concerned in this paper with strong electric fields, the average energy of electrons will be much greater than fiu. Hence the range of q given by (3.13) will be used. In view of (3.13) we have

eFneihko m qdqhu(q)B{q), (3.14)

eFn.

= — J f(k)kdk k J J mo)/hk

/

h2kko f r2k

f(k) dk{ / tfdqBig) mKT {

[ 1 warn

N M + ^ \ - ( 3 - i 5 )

The three unknowns Nq(t), T(t)i and ko(f) can be determined from Eqs. (2.14), (3.14), and (3.15).

760 V. V. P A R A N J A P E A N D B . V. P A R A N J A P E

Combining Eqs. (4.4) and (4.6) we obtain

16 mv(t) ms/eFr^iS) 3w\m

166

4. SPECIAL CASES

The current density J{f) = fiko{t)enB\/m can in prin­ciple be found from Eqs. (2.14), (3.14), and (3.15). This calculation is complicated. In order to obtain reasonable estimates of / ( / ) we make some simplifying approximations. Let P arid Q be slowly varying with time so that Nq(t) is given by (2.15). Further, Q is always larger than PNq{t). Thus

eF= 3 (2MTKTQ) 1/2

s/ePTaGv(t) Sny

c \ ms2 16/

ne\hz / ms2 \ 1/2

and Nq(t)~Qt+nq(T0)(l-Pt) for Pt<l, (4.1)

37r(mKTo)m\2KTo/

/ ms2 16\x/3 t X( ) — H

\eFTacv(l) 3T/ rac

(4.7)

Nq(t)^j for Pt> 1. (4.2)

Let P= ( l /rPh) be assumed to be independent of q (see discussion). We also make the assumption that P and Q are constants in time (only) to solve for Nq(t) in Eqs. (2.15) and (4.1). In the rest of the calculation, however, explicit dependence of P and Q on time is considered. We assume KT^>hco throughout this paper.

Acoustic mode scattering: Here o)(q) = qs. (mco/ftk) = (ms/M)qo^0 since (ms/hk)<£l. Then Eqs. (3.14) and (3.15) reduce to :

provided t<rVh, where the electron drift velocity v(t) at time t is related to the current density J(/) by the relationship

J(f) = nev(t). (4.8)

Polar mode scattering: In this case <a(q) — wo, a constant frequency. Equations (3.14) and (3.15) reduce to

m {2fmtKT)m\€,

eF= e2o)Q2mhkQ

€

1 1

e—log- (4.9)

eFv(t) = 8 mC2{2mKT)

9T^2 ¥ p

16 ms2/TV2

3/2 3kT(2mwKT)ll2\

/ 1 l\re'nei/1 1\

( ) — ( )

2KT hw i -e—log

4KT. he* (4.10)

and

10 ms*/

3ir rac V

eF = -8 mkQC2(2mKT)r2

TJ '

C2ne\mt

(4.3)

9w phh L.9 T^2phs(2mKT)1/2

16 mv(t) ms T

where 6 is Euler's constant. The maximum contribution to the integrals in (3.15) is from q^mw/hk for the first and the third term and from q^2k for the second term in the integration over q (in the square bracket). We therefore can replace the exponential terms in P and Q by unity and neglect nq(T0) in comparison with unity.

We rewrite Eq. (4.9) in the following form:

X

3 (2mwKTo)l/2

ne\h3

To

ms

L3w(mKTo)m (2mKT0) \ * / rac

\n Jv 1/2 m / 1 1\

( ) e . (4.11) XCoo € /

(4.4)

/ (2mwKTo)m eFv(t)

Combining Eqs. (4.10) and (4,11), a relation between v(t) and t can be written as

provided / < rph, where rac is the relaxation time due to acoustic scattering in Ohmic conductivity. eF~-

e2o)o2m2v(t)

1 (2mwKTo)mC2

rac 6w2¥phs

m / 1 1

(4.5)

The maximum contribution to the integrals in (3.15) Is from the region .q~2k in integration over q. Therefore, in obtaining (4.4) nq(To) is neglected in comparison with unity, and the exponential factor of P and Q is replaced by unity.

We may now rewrite Eq. (4.3) in the following form:

3KT0(2mwKTQ)m[(2m7rKTo)1/2 eFv(t)

e2nei/1 1\ 2KTo ( )t+

XCco € /

X

X nw^ m / I eiH (4.12)

T reFTmv(t) 3TT-]2/3

T0 L ms2 16. (4.6)

{(2mwKTo)meFv(t)

provided £<rph.

5. DISCUSSION

In Sec. 2 we have calculated the phonon excitation number Nq(t) as a function of time. The exact expression

166 P H O N O N E X C I T A T I O N S BY E L E C T R I C F I E L D 761

for Nq(i) is given formally by Eq. (2.14) while its approximate form which is considered in this paper is given by Eq. (2.15). We notice from (2.15) that the phonon excitation number Nq(t) increases with time from the thermal equilibrium value nq(To) to the saturation value Q/P in a time of the order of T = (1/P) . We refer to this time as the phonon build-up time. Since T depends on q, phonons with different wave numbers will take different times to reach their respective saturation values. We observe from (2.11) that 1/r is composed of two terms. The first term arises from the electron-phonon interaction, while the second term depends on Tph(g) the phonon relaxation time due to its interaction with other phonons and the heat bath. At sufficiently low temperatures (~4°K) the phonon-phonon interaction is very weak and hence rPh(q) is determined by the interaction of a phonon with wave number q with the heat bath. In this case rph.(q) depends on the size of the specimen, the heat absorbing proper­ties of the heat bath, and hence may not strongly depend on q.

We divide the range of q into two parts. We assume that in range I

T = rph(q), provided [rph(q)Z(T,q)~]<nq(T), (5.1)

and in range I I

V = nq(T)/Z(T,q), provided lTph(q)Z(T,q)2>nq(T). (5.2)

Using Eqs. (5.9) and (5.2) the saturation values ^V9(») in the two ranges may be written as

NQ(C0): =**(r0)[i-nq(T) Z(T,q)rvh(qy

and

1+-

nq(T0) nq{T) J

in range I (5.3)

nq{T^) nq(T) -1

nq(T) Z(T,q)rph(q) J '

in range I I . (5.4)

rph(#) is small or large according as the interaction between phonon q with the heat bath is strong or weak. rph(#) = 0 implies infinitely strong interaction, while Tph(q) = °° implies no interaction between phonons and the heat bath. If rPh(#) = 0 is substituted in (5.3), Nq(co) becomes equal to nq(To). This result is expected, since it means that interaction between phonons and the heat bath is so strong that phonons are in complete equilibrium with the heat bath, even though the electrons are at temperature T>T0. If rPh(q)=<^ is substituted in (5.4), we obtain Nq(<v) — nq(T). Here the electrons and the phonons are in complete equi­librium with each other. In normal experimental situa­tions TPh(q) is neither zero not infinite, but is finite and depends on the size of the specimen. At sufficiently low

lattice temperatures (~4°K) and for sufficiently high electric fields the factor [nq(T)/nq(To)2 is very much larger than unity. Hence the saturation values can be approximated by

and Nq(<x>) = Z(T,q)rPh(q) in range I (5.5)

Nq(co) = nq(T) in range I I . (5.6)

In view of this discussion it is now easy to follow the growth of Nq(t) from nq(T0) to the saturation value Nq(<*>). At t=0 the electrons are at a temperature T>T0 while the phonons are at a temperature T0. The hot electrons for which T^>>T0 will emit phonons, thus increasing their number. The emitted phonons in range I are absorbed by the heat bath. The phonon number increases as long as the rate of phonon emission by electrons is larger than the rate at which they are absorbed by the heat bath. In this range, the phonons attain a saturation value given by Eq. (5.5) in a time of the order of rph(#). In range I I the hot electrons emit phonons, but the interaction between the phonons and the electrons is so strong that the emitted phonons attain thermal equilibrium with the hot electrons by emission or absorption. The time required to attain equilibrium is the average of Y given by (5.2) and the saturation number Nq(<*>) is given by (5.6).

We have so far considered the case in which rPh(q) is determined by the specimen dimensions and by the interaction of a phonon q with the heat bath. This is a reasonable limitation, since most of the phonons involved in electron-phonon interaction at 4°K can have strong interaction with the heat bath. For strong electric fields, however, the electron-phonon interaction may extend to phonons having large wave numbers q. These phonons may interact more strongly with other phonons than with the heat bath. In these circumstances rph will be determined by the phonon-phonon interac­tion. Our calculations in Sec. 4 can be modified to include this possibility. There are nevertheless several difficulties. For example, the assumption of a relaxation time for phonons as described by Eq. (2.4) can not be fully justified. Furthermore, our understanding of phonon-phonon interaction is not sufficient to give with confidence the dependence of rPh(q) on q. In view of these difficulties, we have restricted our calculation to the case in which rPh(#) is independent of q.

In Sec. 4 we have assumed that the build-up time T for all phonons is given by rph(#). This means that the phonons of interest in this paper are entirely in range I, prescribed by Eq. (5.1). This requirement can be easily met in an experimental situation by choosing a specimen of sufficiently small dimensions. If this requirement is not satisfied the phonon build-up time T will depend on q making calculations more complicated.

In Sec. 2 we have shown that the phonon excitation number Nq(t) may increase under the action of an electric field from Nq(0) to A^(<*>), in a time of the

762 V. V. P A R A N J A P E A N D B . V. P A R A N J A P E 166

order of T. The change in Nq(t) with time will also affect the current density. Assuming that the current density J(t) is determined by the instantaneous value of Nq(t), one would observe a variation of current density / ( / ) from 7(0) to /(<*>) in a time of the order of T. The change in the current density J(0) — /(<*>) would be large if the changes in Nq{t) were large. A necessary condition for producing strong transient currents may be derived for acoustic scattering from Eq. (4.17). In this equation the first term in the squared bracket is time-dependent and arises from the time dependence of Nq(f). Hence for large / ( 0 ) —/(«>) we must have

1/2 r nei¥ / rns* \

UT(mKTo)m\2KTo/

X-/ ms

Wac*(

i6y/3 i (oo) 3TT/

>>*, (5.7)

where Atac=^rac/m is the electron mobility and 0(00) is the saturated drift velocity given by

»(«>)= W W

x-

•3(2wxii:ro)1/2/16\1/3

16ms \ 3 x /

37r(mKTa)m/2KTa\

1/2T

njiz \ ms2 I TphJ (5.8)

I t must be emphasized that v(oo) is given by (5.8) only if (5.7) holds. If (5.7) is not satisfied, then fl(oo) does not differ from v(0), which [from (4.17)] is given as

r (2w7r^ro) 1 / 2 3/16\ 2 / 3 f / 5

v(0)=bJ?sM ( - ) . (5.9) L m 8x3x7 J

Provided (5.7) is satisfied, variation of current density J{t) from nev(0) at / = 0 to nev(<x>) in a time of the order of T should be observed.

When the scattering is due to optical polar modes, the variation of the drift velocity v(t) with time is expressed theoretically by Eq. (4.12). Examination of Eqs. (4.9), (4.10), and (4.12) would, however, show

that the electrons can not be in a stationary state for KT>fto). This is, of course, a well-known result. In practical situations, the electron scattering may not be entirely due to optical polar modes but may be due to a combination of optical and acoustic modes. In this case electrons may have KT> hoi, and the calculation of v(f) can be accomplished by combining Eqs. (4.3), (4.4), (4.9), and (4.10). This would be a lengthy calcula­tion and will be done in a subsequent publication. For the sake of this section we shall assume that (4.10) holds and (4.9) does not. Reasons for this assumption obviously follow from the fact that if (4.9) and (4.10) are simultaneously true then electrons are unstable for KT>hco. From Eq. (4.10) it follows that time-dependent currents would be produced if

eneirph/1 1\ 2KT

mo)o

/ 1 1\ IKT

\e«> e/ noo 10)

where T is considered a parameter. Finally, it may be remarked that the transient current

proposed in this paper will have a decay time rPh which at low temperatures may be of the order of 10~6

sec. Observation of this effect is, therefore, well within the present day experimental technique.

The conclusions of this paper are based on the assumption that the phonon distribution is now strongly affected by the isotropic part of the electron distribution function (hot electrons) than by the anisotropic part (drift). Even though this assumption is reasonable in most of the nonpiezoelectric semiconductors, exceptions may exist9 in which case the conclusions of this paper will not apply.

I t is hoped that this work will stimulate interest in its experimental confirmation.

ACKNOWLEDGMENTS

We wish to thank the National Research Council of Canada for making this collaboration possible through its financial support. One of us (V. V. P.) wishes to thank the Institute of Theoretical Physics at the University of Alberta for hospitality.

9 E. M. Conwell, Phys. Letters 13, 285 (1964).