Hot and warm electrons — A review

J. Phys. Chem. Solids

I.2

HOT

Pergamon

AND

Press 1959. Vol. 8. pp. 227-234.

WARM

Printed in Great Britain

ELECTRONS

- A REVIEW

S. H. KOENIG IBM Watson Laboratory

at Columbia University,

New York 25, New York

Abstract-There are two categories of problems relating to the study of hot and warm carriers; i.e., the situation in which the mean carrier kinetic energy deviates measurably from its equilibrium thermal value. The first of these relates to the manner in which a steady state distribution is maintained. The pertinent work to date, both experimental and theoretical, is reviewed. The second set of problems involves the phenomena which result from a non-equilibrium distribution, but that do not play a significant role in determining it. The velocity dependence of recombination cross sections, impact ionization of neutral impurities and enhanced thermionic emission are amone these. The uresent exnerimental situation is discussed and related to what relevant theory exists for the various cases. 1. INTRODUCTION

ONE cannot adequately review all the aspects of the hot electron* (or hole) problem in a talk of this length. I have, therefore (to some extent arbitarily), chosen to restrict the discussion to hot and warm carrier phenomena in bulk material where the electric field dependent momentum distribution function for the carriers is independent of position. The current density and conductivity will then be functions only of the applied field, which will be uniform throughout the sample. Discussion of across the gap avalanching either in junctions or in the bulk is thus precluded. For metals (large Fermi energies) or semiconductors at ordinary temperatures and not too high fields, Ohm’s Law holds quite well. However, for semiconductors(l) and semimetals in the temperature range from 50”-300°K at electric fields - 100 V/cm, and for semiconductors at helium temperatures at fields(l) - 1 V/cm, significant deviations from Ohm’s Law are observed. In these cases, for the electron distribution to dissipate the energy it gains from the applied electric field, the average carrier energy must increase from its thermal equilibrium value. Hence the terms warm and hot electrons, according as one is considering small or large increases in the mean energy. A feeling for the magnitudes involved for the case of interactions with acoustic phonons may be had in the following manner. Consider a * The term electron will be used throughout in a generic sense to mean either sign of carrier except when discussing a specific conductivity type material.

carrier with initial momentum p and energy E which undergoes a collision with a phonon and either gains or loses momentum 2~. Then Ae, the energy change is given by A, = 2pc = (~~~/~wz)(Pzc”/P) = 4~ me/p where c is the sound velocity. phonons of energy 2pc is n -

[exp(2pc/kT)-11-l

The

number

N (KT/2pc).

(1) of (2)

The fraction of energy losing collisions is [(n+l)--n]/(2n+l)

N&f2 =pc/kT.

(3)

The energy loss per collision is then

= (4mc2/kT)c N T-1~ for Ge. (4) A more exact calculation alters the numerical factor 4 in equation (4) by a small amount.(s) One sees then that for acoustic scattering the rate of energy relaxation is considerably less than the rate of momentum relaxation. If the power gained by the carriers from the field is equated to that lost by scattering, then to this approximation:

epE2 s (4mc2/kT)(+);

(c/kT) z po E/2c

(5)

where 7 is the mean free time between collisions aIiTr is the low field mobility and (p/~)s E. Table 1 gives values of (c/kT) for several reasonable experimental situations. It is often convenient to express E in units of kT, and thus speak of 227

228

SESSION

I

: TRANSPORT

electron “temperature.” Only under special circumstances, however, when the distribution is essentially isotropic and definable by a single scalar parameter can the actual concept of temperature be generalized to describe the hot electron situation. For the case of mixed scattering, the problem is more involved, and will subsequently be discussed in more detail. It should be noted, though, that the collisions most effective for momentum change (those that determine mobility) need not be those most effective for energy transfer. (Consider for example, a ping-pong ball rebounding from either a billiard ball or another ping-pong ball.) Table 1. Approximate electron “temperature” to be expected on the basis of Equations l-5 for several experimental conditions. Material

T(“K)

E(V/cm)

n-Ge

300

100

n-Ge

10

10

4

1

---

c/‘kT -1

---

n-InSb

-15 N 20

The effort to date on warm and hot electrons may be divided into two main classes: (1) Studies of the manner by which a hot distribution is achieved and maintained, i.e., what the energy loss mechanisms are and what form the distribution function takes. Experimental investigations generally involve measuring the dependence of drift or Hall mobility@) on the power input to the distribution. (2) Investigation of phenomena that are observable as a result of a hot distribution, but that play little or no direct role in determining the energy distribution; e.g., the velocity dependence of lattice and impurity scattering,(s) of recombination processes, (6) impact ionization of neutral impurities,(l+s) and electron emission from bulk materials. c7) 2. ACOUSTIC

SCATTERING

2.1. Theoretical situation An analytic expression for the distribution function for arbitrary electric field may be obtained for the case of longitudinal acoustic scattering, spheri-

cal energy surfaces and Boltzmann statistics, assuming the mean energy loss per collision is small and that equipartition holds for the phonons. These assumptions limit the range of validity of the result to mc2/kT G$ TE/T < kTlmc2, where TE is the electron “temperature”, c the sound velocity, T the lattice temperature, k Boltzmann’s constant and m an effective mass. The solution was first discussed by PISAEENKo(8)* in 1938 and independently derived by YAMASHITA and WATANABEcg)in 1954. Fig. 1 shows the shape of the distribution for several values of electric field, calculated for parameters appropriate to Ge at 10°K. If one ignores what complications the multivalleyed conduction band of Ge may contribute, high purity n-type Ge in the temperature region of 8”-15°K probably affords the best experimental approximation to the theoretical model. In the limit of large electric fields, corresponding to mean electron energies 5 8 times their thermal value, but not so large as to invalidate the equipartition assumption, the Pisarenko distribution approaches that obtained by DRUYVESTEYN(~@for hard sphere scattering in gases. In this limit the drift mobility varies as E-t, as has been shown by SHOCKLEY.(~) More recently, SODHA and EASTMAN have shown that the Hall to drift mobility ratio approaches a constant in the same limit, under the implicit assumption that the magnetic field doesn’t effect the distribution. For still larger electric fields, one would on a priori grounds expect a variation between E-t and E-1. The power loss must be more rapid than for the E-r range since the collision frequency has increased and all collisions now involve spontaneous phonon emission only and are, therefore, all lossy. On the other hand, a variation faster than E-1 would correspond to drift velocity decreasing with increasing E and, therefore a negative differential resistance. The only relevant theoretical, work is the estimate of PAEANJAEE,(~~)~who, assuming a “hot” Maxwell distribution, found a mobility variation proportional to E-o.8. * It was noticed after this paper was written that this distribution was discussed by F. B. PIDDUCK, Proc. Lond. Math. Sot. 59, 89 (1915). See E. GuTHand 3. MAYERHBFER, Phys. Rew. 57, 914 (1940), footnote (11) for a more complete history. t See also R. STRATTON Proc. Roy. Sot. A 242, 355 (1957).

SESSION

I:

conductivity measurement combined with a measurement of Hall coefficient in the ohmic range. The proper Hall to drift mobility ratio to apply in this case is not certain. In addition, the 4°K data were taken on a
It should be emphasized, though, that an experimentally observed power law variation of mobility p with E in agreement with that discussed above is not by itself sufficient to indicate that acoustic scattering is the dominant energy loss mechanism. For example, YAMASHITA and wATANABEcg) have shown that for certain cases at high fields the paE-* law will hold for an arbitrary amount of mixed acoustic and optical phonon scattering.

ELECTRIC FIELD DEPENDENCE OF PlSARENitO ISVESTIA

ACAD. NWK.

229

TRANSPORT

USSR

DISTRIBUTION

PHYS. SER. p.631 (1938) Nt)d(=A(‘(E+y,Ye-cdt E= E/hT y =(3rv’l6)( /loE/c

VALUES

f

OF E FOR /&,=79x105

CM?‘VOLTSEC

c = 54 x IO’CM/SEC T: IO” K yzo.292 E=0.5

I

2

3

------DRUYVESTEYN

4

5

6

7

E

9

8

IO

DISTRIBUTION

II

I

1

1

1

12

13

I4

I5

I6

FIG. 1. The variation with electric field of the electron energy distribution appropriate to acoustic scattering and spherical bands.

2.2. Experimental situation Hot electron effects have been observed in nand p-type Ge,(lJs) Si,(lJ3) InSb,@M in InP,W InAs, and Bi.@) The early measurements on n-type Ge at 20°K by RYDER and recent work by KOENIGW and by BoK(~) are, I think, the only cases in which one may be reasonably certain that acoustic scattering controls both the momentum and energy balance. (This is discussed more fully below.) Fig. 2 shows this data in the dimensionless units appropriate to the theory. On a simple picture the two E-+ regions would be expected to superpose. There are, however, several complications. The Koenig data is Hall mobility, whereas the Ryder data is drift mobility obtained by a

direction used. The deviations from E-4 behavior of the mobility due to failure of equipartition are expected to occur at a value of E/E0 proportional to lattice temperature. This is seen to be approximately the case for the data of Fig. 2. Similar shaped curves have been obtained for p-type Ge at 77°K by Bray,@) but it is hard to see why acoustic scattering should play a dominant role for this case.

3. WAFtM ELECTRONS AND OPTICAL MODES Recent

theoretical

and experimental work by warm electrons in n-type temperature range 77”-300°K has

MORGAN(~) on slightly

Ge

in

the

230

SESSION

I:

TRANSPORT

demonstrated the importance of optical modes. The results may be summarized as follows: at 300”K, acoustic scattering contributes 80 per cent of the mobility but N 2 per cent to the energy loss. At lOO”K, acoustic scattering contributes 95 per cent of the mobility, but still only N 12 per cent of the energy loss. The remainder of the loss is caused by optical phonons and to a lesser extent by intervalley scattering. It is only at N 60°K that acoustic

IONIZED

4. ELECTRON

TEMPERATURE

Attempts have been made to obtain a more direct measurement of the electron “temperature” for a hot distribution. GIBsoN(21) has pointed out that the piezoresistance of n-type germanium should be a function of electron temperature, and has been measuring the variation of piezoresistance with electric field with some success. BoK(~)has measured the “thermoelectric power”

------

IMPURITY

AFTER

RYDER

KOENIG

IkO

T=20”K-

‘\

‘., I

I

I

I

I

lo

&E/c)

100

1000

FIG. 2. The electric field dependence of mobility in reduced units. Ideally the two E-* regions would superpose (see text). The deviation from the E-* law at high fields is most probably due to failure of the assumption of equipartition for the phonons involved in the scattering. The temperature dependence is roughly correct.

and optical scattering make equal contribution to the energy loss. These figures were obtained by Morgan using the distribution that resulted from solving the Boltzmann equation, ignoring electronelectron scattering, rather than assuming a Maxwellian distribution, which gives a loss at 100°K greater by a factor of 4. The efficiency of electronelectron scattering in tending to restore a Maxwellian distribution centered about a drift velocity has been pointed out by FR~~HLICHand PARANlAPE.(1s) Recent work by SEEGER(3*20) shows a variation with electron density in the ease with which electrons may be heated, other parameters remaining fixed, suggesting, in accordance with the preceeding ideas, the importance of electronelectron scattering.

between hot and cold electrons for extrinsic n-type Ge by using a T-shaped sample and by making contact to the cold electrons in the arm of the T, and to the hot electrons in the cross bar. This type of measurement has been extended by KOENIG and BROWN@~)to n-type Ge at 4°K in the breakdown region (Fig. 3). Though the magnitude and sign of the effect is of the order that one might expect from initial naive considerations, because in this case there is a field dependent carrier density gradient as well as energy gradient, the interpretation will not be straight forward. The observed voltage will be the sum of two terms, the difference in barrier voltage developed at the two contacts and a diffusion potential in the bulk, to some extent analogous to the Peltier and Thomson

SESSION

acceptor

contributions to the thermoelectric voltage under normal conditions. These two contributions may well be of opposite sign, and highly field dependent, which may account for the sudden drop observed at high fields (Fig. 3).

“THERMOELECTRICITY”

231

TRANSPORT

I:

densities

Nn

and NA:

AT(T>(ND-NA>+'&(~)(ND--NA)-

-~BT(T,~)NA-GBI(T,~)NA The dependence

OF HOT

of the coefficients

= dn/dt. (6) on the lattice

ELECTRONS

Tz 42°K 9

ND-N, N,-

0

- 6 x lO%M’

2 x 10’?‘CM3

EBsKDOWhl = 8-08

0

I

2

I

3

C”OLT&M,

VOLT/CM

I

I

5

E

I

6

I

7

FIG. 3. The variation of the “thermoelectric” voltage developed between hot and cold electrons for the configuration shown. For E = SV/cm, the electron “temperature” is ~100°K. 5. LOW

TEMPERATURE

BREAKDOWN

The most intensive investigations of phenomena resulting from the existence of hot electrons have been those relating to low temperature breakdown.(l.a) At N 4X, the conductivity of Ge is highly non-linear, and is characterized by a reversible breakdown region in which the current increases by orders of magnitude for small changes in applied voltage (Fig. 4). It is possible to understand the shape of the current voltage characteristics by considering the balance of the rate processes that produce and annihilate carriers as these rates vary with the electron distribution function. The rate processes that need be considered are thermal ionization AT, impact ionization of neutral donors by energetic electrons AI, and the corresponding inverse processes, single electron recombination BT, and Auger recombination BI. The rate equation becomes, for carrier density n small compared to both the donor and

temperature T and distribution function f is indicated. For small n, the term in ns may be ignored, and one obtains fl =

AT(ND--NA)/[BT--&(ND--A)/NA]NA. (7)

Breakdown occurs at an electric field such that the denominator of equation (7) approaches zero. Equation (7) predicts that the breakdown field should increase as (ND-NA)/NA (or for fixed compensation, the donor concentration) is decreased contrary to some ideas on the subject.(sa) To demonstrate such a dependence requires samples of purity sufficiently high for the mobility to be mainly lattice determined. The data of Table 2 clearly demonstrate the effect. The applicability of equation (6) to transient and steady state conditions, under the assumption that the distribution function “follows” changes in

232

SESSION

I:

Table 2. Data showing that the low temperature breakdown jield for n-type Ge may be increased by an increase in purity, in accordance with predictions of equation 7 Sample

0% --Nn)

nWLB

28-6

nWLP

33A

N*

Ewtdoan

1.5 x lOI cmw3 5 x 1012 cmm3 4.8V/cm 1 X 1012 cmm3

4 X 1012 cme3

14V/cm

applied fields in 3 10-s set, has previously been quantitatively justified and will not be discussed here.@) By creating a non steady-state value of n, and observing the decay of conductivity in a sample with the electric field sufficiently low for f to have essentially its thermal equilibrium value, it has been possible to measure & for thermal electrons as a function of T. The significance of this quantity has been discussed by Melvin Lax earlier in this conference in his talk on “giant traps.” Suffice

$

N-

nWLB Iv,-2 N,-

z

28-6

TRANSPORT

it to say that the cross-section for capture is orders of magnitude larger than one might expect if direct capture of the carrier to the ground state were the dominant process. The data and the derived cross-section are compared with theory in Fig. 5. Recently, it has been possible at Watson Laboratories(s4) to measure directly the magnitude BI, the Auger recombination term, for a range of lattice temperature of 4-lO”K, but only for an electron temperature N 100°K. The result for BI is N lo-17/cma set, which means at n N lore/ cm3 the Auger recombination rate and single electron capture rates are roughly equal. Calculations by SCLAR and h.fRsTEIN(23) of BI yield the same order of magnitude as that measured here, but the agreement is probably due to the fortuitous cancellation of several large factors in the theoretical approximations, inasmuch as the giant trap type of cascade process must be operative in Auger recombination.

I-

x IOYCM” 5 x 10’%M3

T=4*55”K e-3 LINEAR 1.0X&

08

-

S/CM)

I

E

FIG. 4. Variation of current density with electric field for a high purity n-Ge sample at low temperature. The curves, however, typify a wide range of n- and p-type Ge.

IO

SESSION

I : TRANSPORT N,

4

I

I 5

I 6

I 7

I 0

I 9

I IO

T (“K)

I

I

4

I

Illll

6

0

IO

T(“K)

FIG. 5. The temperature dependence of the recombination time for an electron and an ionized donor for thermal electrons, and the cross section calculated from the smoothed data. The tendency to saturate at the lower temperatures is associated with an overlap of the orbits on adjacent recombination centers (-1~ apart!). For the purer sample, the effect is less. 6. THERMIONIC

EMISSION

If the electron affinity of a material is fairly low, thermionic emission may be increased by heating the electron distribution. This mechanism has been suggested by GINSBURG and SHABANSKII(7) to explain the observed anomalously high electron emission of tungsten at high current densities.(T) More recently, BoK(‘) has observed such emission from bulk Si with the surface treated to lower its electron affinity. 7. ANISOTROPY In the region where Ohm’s Law applies, the symmetry of the conductivity tensor for a given material must be at least that of the point group of the lattice regardless of the complexities of the band structure. This ceases to be the case for hot electrons, so that from measurements of the anisotropy of conduction in the hot electron range

one may obtain information regarding the band structure. SASAKI, SHIBUYA and co-workers@@ have extensively studied the anisotropy in n-type Ge from 77”-300”K, and will discuss their recent work at the present conference. Pertinent calculations have also been made by GoLD,(~@ and observations on the variation of electric breakdown with crystalline orientation have been reported.(s7) Needless to say, the results are consistent with the known multivalley structure of the conduction band of Ge. Such anisotropy measurements may be one of the simplest ways to get an idea of the symmetry of the band structure of a new semiconductor. 8. CYCLOTRON

RESONANCE

A recent interesting application of hot electrons was made by ZEIGERtB8) and collaborators, who have observed the variation in d.c. conductivity due to the absorption, at the cyclotron resonance,

234

SESSION

I:

of r.f. power. The sensitivity surpasses that of the conventional methods of detecting cyclotron resonance, but the complexity of the initial published results suggests that experiments of this type will yield information on both cyclotron resonance and hot electrons. Acknowledgments-I should like to acknowledge the many interesting, and useful, conversations with P. J. PRICE, and in particular his critical perusal of what was to have been the final manuscript. The help of Messrs. BROWN, FRIEDMAN, HALL and SCHILLINGERin obtaining data has at times been invaluable. REFERENCES 1. See GUNN J. B., Progress in Semiconductors, Vol. 2, p. 246. John Wiley, New York (1957); KOENIG S. H., and GUNTHER-M• HR G. R., .J. Phys. Chem. Solids 2, 268 (1957); SCLAR N. and B&STEIN E., I. Phvs. Chem. Solids 2. 1 (1957): for references io earlier work. More recent work will be referred to below. 2. BOROVIK E. S., Dokl. Akad. Nauk. SSSR 91, 771 (1953); KOENIG S. II., HALL J. and FRIEDMANA. (unpublished). 3. SHOCI(LEY W., Bell Syst. Tech. J. 30, 990 (1951). 4. MORGANT., Report. Sig. Corp. Contr. DA-36-039SC-52670, May (1957), Univ. of 111; Bull. Amer. Phys. Sot. Ser. II, 2, 265 (1957); Ibid., 3, 13 (1958); BRAY R., Proc. Phys. Sot. B, 70, 899 (1957); GLICKSMAN M. and STEELE M. C., Phvs. Rev. 111. 1204 (19581: STEELE M. C.. Bull. Amer. Ph$s. Sot. ‘Ser. ‘iI, 3, 112 (1958); SEECER K., Bull. Amer. Phys. Sot. Ser. II, 3, 112 (1958). Also reference (1). 5. CON~ELL E. M., Phys. Rev. 90, 769 (1953). 6. KOENIG S. H., Phys. Rev. 110, 986 (1958); Phys. Rev. 110, 988 (1958). 7. BOK J., International Congress on Solid State Physics, Brussels, June (1958) to be published; GINZBURG

J. Phys.

Chem. Solids

Pergamon

LATTICE

TRANSPORT V. L. and SHABANSKIIV. P., Dokl. Akad. Nauk. SSSR 100, 3, 445 (1955); LEVEDEV S. V. and KHAIKIN S. E., Zh. eksp. teor. jiz. 12, 26, 723 (1954).

8. PISARENKON. L., Isv. Akad. Nauk. SSSR, f;z. ser. p. 631 (1938). 9. YAMASHITAJ. and WATANABEM., Prog. Theor. Phys. Osaka 12, 443 (1954). 10. DRUVVESTEYNM. J., Physica 10. 61 (1930). See also

CHAPMANS. and COWLING T., The Mithematical Theorv of Non-Uniform Gases. D. 351. Cambridge Univ&si;y Press (i939). ’_ 11. SODH.~M. S. and EASTMANP. C., Phvs. Rev. 110,

1314 (1958). 12. PARANJAPE,B. V., Proc. Phys. Sot. B, 70,628 (1957). 13. FINKE G. and LAUTZ G., Z. Naturf. 12, 223 (1957); BRAY R. and MENDELSONK., Bull. Amer. Phys.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

26. 27. 28.

Sot., Ser. II, 2, 153 (1957), BRAUNSTEINR. et al. Signal Corp. Contr. DA36-039-SC-5548, 11th and 12th quarterly report (1953-1954). BOK J. and AIGRAIN P., private communication. STEELE M. C., Bull. Amer. Phys. Sot. Ser. II, 3, 112 (1958). PRIOR A. C., J. Electronics and Control 4, 16.5 (1958). RYDER E. J., Phys. Rev. 90, 766 (1953). KOENIG S. H., unpublished results. FR~HLICH H. F. and PARANJAPEB. V., Proc. Phys. Sot. B69, 21 (1956). SEEGER K., private communication. GIBSON A. F., private communication. KOENIG S. H. and BRO~W R., unpublished results. SCLAR N. and BURSTEINE., reference 1. KOENIG S. H., to be published. SASAKI W., SHIBUYA M. and MIZUGUCHI K., J. Phys. Sot. Japan 13, 456 (1958); SASAKIW. and SHIBUYA M., J. Phys. Sot. Japan 11, 1202 (1956); SHIBUYA M., Phys. Rev. 99, 1189 (1955). GOLD I~., Phys. Rev. 104, 1580 (1956). KOENIG S. H. and GUNTHER-M• HR G. R., SCLAR N. and BURSTEIN E., reference 1. ZEIGER H. J., RAUCHC. J. and BEHRPIIDT M. E., Plqu. Rev. Letters 1, 59 (1958).

Press 1959. Vol. 8. pp. 234-239.

MOBILITY

Printed in Great Britain

OF HOT CARRIERS

E. CONWELL Research Laboratories, Sylvania Electric Products Bayside, New York

Inc.,

A THEORY of the variation of lattice mobility with electric field intensity was developed some years ago by SHOCKLEY.(~) The agreement of this theory

satisfactory in that it predicted deviations from Ohm’s law at fields about one-third as high as

with

the

experimental

data

for

germanium

was

not

actually

required.

Ohm’s

law

Also, region

it predicted drift

velocity

that

beyond

should

be