IOnized impurity scattering in semiconductors

J. Phys. Chem. Solids

Pergamon

Printed in Great Britain.

Press 1962. Vol. 23, pp. 1147-1151.

IONIZED

IMPURITY

SCATTERING

IN

SEMICONDUCTORS G. L. HALL Kansas State University,

Manhattan,

Kansas

(Received 8 January 1962; revised 12 March 1962) Abstract-Takimoto’s treatnient of the mobility of conduction electrons in semiconductors, of which the Conwell-Weisskopf and Brooks-Herring formulae are special cases, is extended by evaluation of a certain integral. Improved BH, CW, and Takimoto formulae are presented and the associated scattering potentials are derived. An incidental feature is that the cut-off inherent in the CW formulation is eliminated. 1. INTRODUC’I’ION

TIIE SEMI-CLASSICAL scattering theory of CONWELLWEISSKOPF(~)and the quantum-mechanical treatment of BROOKS-HERRING@)yield essentially the same mobility for conduction electrons when the density of the electrons is of the same order as the density of the ionized scattering centers, but yield different mobilities when the density of electrons is considerably less than the density of the impurities. Discussions of these facts have been presented by HERRINGand DEBYE and CONWELL.@) More recently TAKIMOTO(~)has treated the problem by the methods of NAKAJIMA’~) and BARDEEN and PINES.@) He showed the CW and BH results to be special approximations to his more general formulation which includes certain correlation effects between the electrons. From his formulation, Takimoto also derives a third intermediate approximation. The CW, BH, and Takimoto treatments, result from different approximations to two difficult integrals, Takimoto’s equations (6.11) and (7.11), which read, respectively. co

F(t) =

._L -1s dm to

x

x+E exp( - x2) log IX-6

dx,

(1.1)

I

(1.2)

The two quantities F(e) and JJd/rl) can be related to the mobility through Takimoto’s equation (7.10) so that

25&2(kT)3/2 pr = 3*12,;$qN~

* $e-7 dv

’os J,h’d

(1.3)

where y is defined by ?Tneslis

(I.41

y2 = (kT)22m&

and all symbols are defined in Appendix B. Tslrimoto, as well as CW and BH, approximates the integral in (1.3) by replacing Jr(d~) everysince the rest of the integrand where by JM3) of (1.3) is sharply peaked at q = 3. With this approximation (1.3) reduces to 27/2$(kT)W

1

‘I = &Wm~2@N~ * Jy(2/3)

(1.5)

The CW formula for the mobility is obtained by setting F(x) equal to zero and introducing a rather artificial cut-off on the lower limit of integration in equation (1.2). The BH results follow from setting F(x) equal to unity, and Takimoto’s results follow from approximating F(x) with a unit step function in the range of x between zero and one. In Section 2 of this paper, it is shown how F(x) and Jr(dv) can be evaluated to any desired degree

1147

1148

G.

L.

of accuracy. The results of Section 2 are then applied in Sections 3 and 4 to find analytical expressions for the first corrections to the BH and CW formulae, respectively. Section S is devoted to the derivation of a relatively simple expression which is more accurate than Takimoto’s formula. Even further improvements may be had if one is prepared to accept more complicated formulae or resort to numerical methods. In all these comparisons between theories the expression (1.5) is used to approximate (1.3), but at the end of Section 5 a short discussion is given on how even this approximation can be removed. 2. THE EVALUATION It is shown in the appendix written as, F(x)

1 = 2+

x

B

OF F(r) that F(x)

OOexp(- ts) dt xs- t2

s

-m

Jy(l/rl)

(1.2) by F(0)

”

= 2 J

= log (1+-J

--$. (3.1)

Equation (2.2) can now be used to show that the next higher order approximation to F(x) is F(x) 1: 1-2x2/3. Substitution of this expression into (1.2) yields the improved BH formula. Jrl

Jr(&)

x3 dx

= 2 [(1-~)xa+y2]2

can be 0

=-

1

=

(l-y)-’

exp(ts) dt

(log[l+(l-7)

;]

(3.2)

22/n

dt

29

( 1 1 -3

z s

d dx (x2+y2)2

= 1. One has immedi-

0

xs- t2

s

-co

X

in equation ately

i

m(exp(-ts)-exp(-~s))

= Aexp(-9)

HALL

(2.2)

0

The forms in equation (2.1) are convenient for finding asymptotic expansions good for large x, and (2.2) is convenient for expansions about the origin in x. Moreover, form (2.2) can be used to express F(x) in terms of the error function with complex argument or Fresnel integrals.(T) As Takimoto’s Fig. 2, obtained numerically, indicates, P(x) is a positive-valued, monotonically decreasing function of x which is equal to one at the origin and zero at infinity. Although the curve is roughly gaussian in shape, it falls to zero at infinity much slower than a gaussian, namely as 4x2. Equations (2.1) and (2.2) are much more convenient than the defining equation (1.1) for either precise numerical computations or for analytical approximations. 3. THE IMPROVED BROOKS-HERRING FORMULA The BH formula is obtained by replacing F(X)

rl+y2

A similar, but somewhat lengthier, formula is also easily derived for the even better approximation F(x) 2: 1 -2x2/3 +4x4/15. This is a very good approximation to F(x) in the region 0 < x < 3, which is all that is required for equation (1.2). For simplicity in presentation, this approximation will not be pursued here. The BH formula corresponds to the DebyeHuckel potentials*

(3.3) so it is of some interest to potential associated with mula of equation (3.1). sional Fourier inversion (6.12) [G+&F

(?)I-‘;

compare it with the new the improved BH forFrom the three-dimenof Takimoto’s equation h = &-,

(3.4)

* See Ref. 4 for a discussion of the scattering potential and definition of all the parameters appearing in the potential.

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IMPURITY

SCATTERING

The new potential is found to be proportional to

IN

1149

SEMICONDUCTORS

The first term in equation (4.4) is of the same form as (4.1) so a cut-off is effectively encompassed in the identification xx = y/2/2,

The new potential is roughly of the same form as the Debye-Huckel apart from the fact that it now has the possibility of oscillating if 4: > 6/h. For small A& the new potential is more rapidly damped than is the old potential. 4. THE MODIFIED CONWELL-WEISSKOPF

FORMULA The CW formula is obtained by setting F(x) equal to F( co) = 0, and introducing a cut-off xs on the lower limit of integration in equation (1.2). It is then found that J&1/71)

= log(?/$,

(4.1)

and Takimoto has discussed the choice for x0 which produces the usual form of the CW formula for the mobility (or conductivity). Note that the CW approximation gives a lower bound to F(x) and hence an upper bound to JY(dv). One would expect a somewhat better approximation to follow from the asymptotic form of F(x), namely *x2, which is apparently an upper bound to F(x) and hence a lower bound to J,(1/77). When this asymptotic expression is substituted into equation (1.2), it is found that,

_&A+?) = 2

, (4.2) 0

(4.5)

if the remaining terms in (4.4) can be neglected. Inspection of (4.4) immediately reveals that the additional terms are negligible if 7 9 y, a condition which is satisfied for the entire range of the curves in Takimoto’s Figs. 3 and 4. The CW formula corresponds to the bare coulomb potential which is now to be compared with the modified CW potential. By treating the potential problem as outlined in Section 3, it is found that the new potential is given by 1 - e-ar cos ar; Y

a2 = Aq&Z,

(4.6)

which is essentially the potential of equation (7) in MC~RVINE’S@) discussion of TAKIMOTO’S paper.(*) The improved BH formula of the preceding section is undoubtedly more accurate than equation (4.4), but this derivation is presented to demonstrate that the need for a cut-off is eliminated and for reasons to appear in the next section. 5. AN IMPROVED

TAKIMOTO FORMULA

The basic idea in the derivation of TAKIMOTO’S formula,(*) with results falling between those of CW and BH, is the use of the CW approximation over part of the domain and the use of the BH approximation over the remainder of the domain. Now the same idea can be carried out using the improved BH approximation in the domain 0 < x < 1, and the asymptotic approximation in the domain 1 < x < d7. The results are,

where the substitution J,(d7)

=

log [--$]+log

(l+&)1’2--&

(4.3)

(4.4)

log$=f] (5.1)

has been used. The integration is readily performed giving, -72/b2+272)

= (l-:)-2(

+

Y2

1 +-log-.

y2 + 2v2

4(~~+27~)

2

2y2

It is clear that the dividing point x = 1 could be varied for “best fit” to the F(x) curve, or other functional forms could be used to approximate F(x).

11.50

G.

L.

It was stated in Section 1 that CW, BH, and Takimoto approximate equation (1.3) with equation (1.Q which is to say that Jr(dq) is everywhere set equal to its value at 7 = 3. Now that this paper affords additional knowledge of JJdv), it is entirely feasible to numerical integrate the integral in (1.3).

HALL

Thus for the principal branch one has, log E)

--?I < er, es < 7r. Let P be a contour running Then it follows that

REFERENCES

J r

388 (1950). 2. BROO& H.,Phys. Rev. 83, 879 (1951). 3. DEBYE P. and CONWELL E. M., Phys. Rev. 93, 693 (1954). 4. TAKIMOTO N., J. phys. Sot. Japan 14, 1142 (1959). 5. NAKAJIMA S., Proceedings of the International Conference of Theoretical Physics, Kyoto and Tokyo (1954). 6. BARDEEN J. and PINES D., Phys. Rev. 99, 1140 (1955). 7. SANSONE G. and GERRETSEN J., Lectures on the Theory of Functions of a Complex Variable p. 479.

m

=

A

s s

dx

x exp( -x2) log

00

=

-03

P. Noordhoff (1960). 8. MCIRVINE E. C., J. phys. Sot. Japan 15,928 (1960). 9. B~CHNER S., Vorlesungen Uber Fouriersche Integrale p. 57. Chelsea (1948).

Derivation

(A-4)

from - co + h to to + ic.

lim r e+O

1. CONWELL E. and WEISSKOPFV. F., Phys. Rew. 77,

APPENDIX

= log ~~+i(&&);

x+l

I I

dx

x exp( -x2) log X-I f -irr

x exp(-x2)dx,

(A.5)

s -f

since no contribution arises from the branch points. The last integral on the right hand side vanishes by virtue of the oddness of its integrand. Consequently, it is found that,

of Equations (2.1) and (2.2)

Consider the defining integral for F(t), Xi!$

dx x exp( -x2) log IX-5 I

=-

1

*

%V9r s -co

dx. (A.6)

Equation

x exp(-xs)

x+s dx. log IX-5 I

(A.l) 69

It is convenient to replace the non-analytic logarithmic function in the integrand by the analytic function Iog(x + R/(x - I). The latter function possesses branch points at x equal to f 6, so a cut along the real axis joining these two points follows from the choices:

Consequently,

= Iz-ifj

=-

exp(iO2);

-m < e2 < ?r.

1 2+

(A-3)

by parts to yield,

m exp( -x2) dx .+s

52-x2

(A-7)

-

it is established that,

1 F(x) = -9 22/71.

x+S = la+61 exp(i81);

z-f

(6.6) can be integrated

.ZJexp( - ts) dt f -co

9-P

m(exp(-t2)-exp(--x”)) s -co

xs-

t2

dt ,

tA 8j

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IMPURITY

SCATTERING

Which is equation (2.1). It is clear that the last integral exists as a proper integral except with respect to the infinite limits. To further simplify F(X) to the form (2.2), it is convenient to introduce the Fourier transform of exp(--x2), namely cc

I’(p) =

k

IN

1151

SEMICONDUCTORS

theory of residues to give (n/x) sin Ifl1 X. There results,

s 00

F(x) = i X

exp( - t2) sin 2xt . dt

(A.1 1)

0

Finally, from BOCHNER’S@)evaluation of the last integral, one has, t

exp( - tz)exp( - $2) dt

F(x) = f. exp( - x2) X

s

exp(t2) dt,

(A.12)

0

(A-9)

which is identical to equation (2.2). Substitution

F(x)

of these results into equation (6.8) yields,

1 = 2&7

d/?

-co -co (A.lO)

m = i/

dflexp(-j32/4)B/

-cm The improper

APPENDIX B List of Symbols*

Q)ex&Pt) -

-ca

x2- t2

dt

.

integral over t is easily evaluated by the

T

-

k NI

-

PI

-

K

-

mn

-

11

-

absolute temperature Boltzman constant density of ionized impurities (cm-s) ionized impurity mobility (ems/volt set) dielectric constant effective mass of conduction electron (g) density of conduction electrons (cmbs)

* The notation of reference 3 is followed because there are several small errors in Takimoto’s equations.