Solid State Communications Vol. 2, pp. 191-195,1964. Pergamon Press, Inc. Printed in the United States,

ELECTRICAL PROPERTIES AND RESONANCE SCATTERING

IN HEAVILY DOPED SEMICONDUCTORS

Donald Long, J. David Zook, P. W. Chapman and 0. N. Tufts

Honeywell Research Center, Hopkins, Minnesota

(Received 5 June 1964 by J. A. Krumhansl)

New resistivity vs. temperature data for heavily doped single crystal p-type InSb, Ge, GaAs, and GaSb and n-type GaAs are reported, which establish the generality among many semiconductors of the characteristic curve shapes published earlier for n- and p-type Si. The observed resistivity maxima near the degeneracy temperature are believed to be caused by resonance impurity scattering, of which the temperature dependences in the degenerate range are an additional consequence.

WE recently published’ a set of resistivity p and Hall coefficient RH vs. temperature T curves for heavily doped samples of n- and p-type sili- con between 4.2’ and -9OO’K. Heavily doped defines a sample in which the ionization energy has vanished and the carrier concentration n is constant according to the RH vs. T data; statis- tical degeneracy is not necessary. The Si curves exhibited two noteworthy and puzzling features: (1) a resistivity maximum or “hump” near or somewhat below the degeneracy tem- perature TD (where the Fermi level cF = 0) in each sample, which is most prominent in the least heavily doped samples and fades out with increasing concentration, and (2) an extension of the positive P vs. T dependence below the hump surprisingly far into the degenerate range. 2 The first purpose of this communica- tion is to report new data from a number of other semiconductors, which establish the generality of the resistivity curve shapes des- cribed above and add at least one more note- worthy feature. The second is to propose a consistant explanation for all the results based on a resonance scattering model.

Figure 1 presents the new data, two of the earlier Si curves, 1 and a curve for n-Ge first published by Koe

q g2.

listed in Table 1. -5 Sample parameter6 are

All the samples except the n-GaAs exhibit hump6 and positive P vs. T slopes below the hump6 extending well into the degenerate range, as were observed earlier in n- and p-type Si. Because the various semi- conductors have different band structures, one

191

concludes that the “hump-like” curve shapes are due to a scattering mechanism which is common to all the materials and is insensitive to band shapes. However, the n-GaAs samples have quite different curves with a third note- worthy feature, a negative o vs. T slope exten- ding far into the degenerate range; this pecu- liarity of n-GaAs appear6 not to have been recognized previously. All the curves of Fig. 1 must be governed by scattering mecha- nisms, since n is constant in each sample.

Any theoretical model must give a compat- ible description of both the hump-like curves and the n-GaAs curves. Previous model6 have purported to explain the OSitiVe P vs T slopes in the degenerate range, 9 6 but they do not s

seem capable of accounting for the humps and the negative n-GaA6 slopes. Let us now hypothesize that the humps in Fig. 1 are due to scattering resonances of the type predicted by 6tXidard collision theory for an attractive short range potential, 7 the potential being provided by the individual ionized impurity atoms of density NI and their screening charge;5 this screened Coulomb impurity potential is of the same form a6 that which gives the well known Brooks-Herring formula in the Born approxima- tion for lightly doped semiconductors. In the present situation, however, a partial wave analysis of the scattering must be Used instead of the Born approximation because we no longer have k2as >>l, where k is the carrier wave- number and as is the impurity screening radius. 5 When k2a2s is enough less than unity

ELECTRICAL PROPERTIES AND RESONANCE SCATTERING Vol. 2, No. 7 192

5. I I I I

4-

conduction)

7 /-

n-Gc(aflrr Koenig)

I +\

0 100 200 300 400 500

1EMPERATURE;K

FIG. 1 Resistivity vs. temperature in heavily doped semiconductors. Data points are not shown because they had no observ- able scatter on the scale of this figure. The positive P vs. T slopes above the degeneracy temperatures are undoubt- edly due to lattice scattering.

that only the zeroth-order phase shift 6, need be gonsidered, the relaxation time T is given by:

(1 + cot* 6J -1

(1)

Here v is the carrier velocity. A resonance occurs when bo=n/2. In order to calculate the resonance mobility c’max from (l), we re-ex- press v and k in terms of c, using an “isotropic model” of each semiconductor in which an average effective mass represents the actual anisotropic band structure, 3~ 4 and then inte- grate over c for arbitrary statistics. The result

h = 0.86 x lo=

where rn% is the conductivity mass, g m* is a mass which we choose to be the density-of- states mass in this simple isotropic model (see Table 1 for values), and .(r,) are the Fermi functions on the reduced Fhmi level n = c _ ,

The urnax values calculated from (2) are listed in the sixth column of Table 1. We see that urnax is higher than the experimental tihump in all sam

8 les except the two (N9 and n-Ge) in

which k a% << 1; presumably the higher-order phase shifts are not negligible in the other samples and contribute to the scattering cross- section. Uncertainties in average effective mass could also cause discrepancies. The near agreement Of urnax and tihump for the n-Si and n-Ge samples is a factor supporting the reson- ance hypothesis, since no other known model (e. g., the Brooks-Herring formulag) predicts nearly low enough a mobility at the hump.

Under the resonance scattering hypothesis, the positive P vs. T slope below the hump is explained as a movement away from resonance as T drops. The few tractable expressions derived to date for phase shifts in semiconduc- tors8 show that cot 6, depends primarily on a, when b. is near n/2, and secondarily on kas; these two quantities depend in turn on T and CF. Assuming that cot 6. = 0 at the hump and that - k2ai .=K 1 and using the expressions in ref. 8 for cot bo, we have been able to reproduce fairly well the temperature dependence below the hump in sample N9. (See Fig. 2.) We have chosen to let the dielectric constant K be the one adjustable parameter needed in the calculations and have found that values lower than the normal 11.7 must be used; however, a lower effective n is reasonable since a is only about 4-6 atomic spacings for n = 11.9. Even assuming a which locates the resonance at a temperature of, say, a factor of 2 lower than the observed

Thu the shape of the experimental curve is still %’ lrly well reproduced at low temperatures, as shown in Fig. 2. Thus a simple model based on the resonance hypothesis can explain the observed temperature dependxce and the ex- planation is not very sensitive to the parameters used. hump

au2 /T-*ec (2)

The near-resonance condition-below the essentially magnifies the very small shift

of CF which occurs as degeneracy becomes stronger, and thereby produces a relative- ly large p vs. T dependence.

The negative slope in n-GaAs is then ex-

TA

BL

E

1

Sample Parameters

c 0 N

Sample

p-InSb

p-Gs

p-GaAs

p-C

aSb

r&e

n-GaAs-1

nCsAs-2

N9 (Si)

Plr (Si)

Impurity

cd

Ga

Zn

cd

Sb

T0

Ts

P

B

Impurity

concentration'

NI' in cm -3

h.3 x lOl7

L.9 x 1017

3.7 x l0 18

1.05 x IO 18

1.0 x 10 18

lr.3 x Id7

8.5 x Id7

5.k x d8

1.311 x l0 19

Ctmsity-of-states

effective mass,

in units of free-

electron mass

0.26

0.37

0.71

0.26

0.27

0.07

0.07

0.32

0.51

Measured

hump mobility,

j.n -‘

/, -

. se

t

108

36

Calculated

hum

p mobilityb,

in -'/,-set .

9LO

ll,Q

270

500

7LO

_

138

155

-1

a determined frcm the b.2'KHal.l coefficient by NI - (e RH).

b Masses used in Gq. (2) to calculate these values were obtained l'ran two review articles 34

' where as is the screening radius calculated by Dingle's method 5

.

Degenerate

(k2as2)'

0.70

0.bl

.l

O.?

?

0.75'

0.15

1.59

1.09

O&75

0.28

E

194 ELECTRICAL PROPERTIES AND RESONANCE SCATTERING Vol. 2, No.7

CALCULATED CURVES

j

P

N/d ,x/c = 5.35

4- .’ *&aM

3.5 r. lc7”, 1 1 I I

20 40 60 80 100

TEMPERATURE,‘K

FIG. 2

Fit of simple resonance scattering model to P vs. T curve for sample N9 at low temperatures, using Gulyaev’s8 Eq. (3.2) for cot 6. and our Eq. (1). The open circles re- present calculated points.

plained as a movement toward resonance with decreasing temperature. A resonance cannot be reached, however, before complete degen- eracy freezes the Fermi level, because the low density of states in the [000] conduction

band minimum of Ga.As prevents ka, from ever becoming small enough. Any other appropria- tely doped semiconductor having a single low- mass band extremum should exhibit a curve like those for n-GaAs, while any high-mass, complex band structure semiconductor should exhibit a hump-like curve.

The above discussion implicitly assumes the use of standard collision theory, even though some of its basic assumptions are not stri tly satisfied in heavily doped semiconductors. ?? In particular, the deBroglie wavelength of the car- riers covers several impurity spacings, leading to possible correlation effects in the scattering. However, the result may not be greatly different from the scattering by the same number of widely separated ions, because the impurities are randomly distributed and should therefore produce mostly incoherent scattering. I0 In any case, the simple model used here is a reasonable first step toward explaining the observed elec- trical properties;

Resonance scattering in semiconductors has been predicted for certain situations by several investigators, 8j g but to our knowledge ; vs.T curves in these heavily doped semi- conductors provide the firstpossible experimen- tal evidence of such scattering. We are present- ly attempting to develop the resonance Scat-

tering model more quantitatively and hope to publish additonal results at a later date.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

References

CHAPMAN P.W., TUFTE O.N., ZOOK

: Soviet Phys. -Solid State 1, 381 (1959-1960)]. =

BLATT F. J., Solid State Physics 4, 199 (1957).

LAX M., Rev. Mod. Phys. 23, 287 (1951), gives a review of problems in multiple scattering. -

Vol. 2, No. 7 ELECTRICAL PROPERTIES AND RESONANCE SCATTERING

Eine Veriiffentlichung von neuen Werten des spesifischen Widerstandes in AbhXngigkeit von Temperatur fiir schwer dotierte Einkristalle in p-Mb, p-Ge, p-GaSb und n-GaAs, welche die Grundlage unter vielen Halbleitern fiir die frilher verijffentlichten charakteristischen Kurven von n-Si und p-Si bilden. Von dem beobachteten Widerstandsmaximum n%chst der Entartungstemperatur wird angenommen, dass es durch Resonanz Frem- datom Streuung verursacht wird, wobei die Temperaturabl%ngigkeit in dem Gebiet der Entartung eine ZusPtsliche Folge ist.

195