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Impurity Scattering in Semiconductors

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76

Impurity Scattering in SemiconductorsBY R. MANSFIELDBedford College, London

M S. recaved 11th August 1955Abstract . The theory of impurity scattering in semiconductors has beendeveloped, and the combination of impurity and lattice scattering has been con-sidered for the general case of any degree of degeneracy of the charge carriers.A comparison is made between theory and experimental results obtained withgermanium and indium antimonide.

Q 1. INTRODUCTIONH E theory of impurity scattering in non-ionic semiconductors has beendiscussed by Debye and Conwell (1954) and a formula has been given, whichT as derived by Herring, for the case of scattering by impurity centres havinga screened Coulomb field, when the charge carriers are not degenerate. Mott (1936)

has considered the scattering of electrons by a screened Coulomb field in thecase of metals when the electron gas is degenerate. Similarly, the combinationof impurity and lattice scattering has been considered for the limiting cases ofdegeneracy. Frequently, however, impurity scattering has to be considered when itcannot be assumed that the charge carriers are either degenerate or "degenerate.T h e object of this paper is to consider the general case of arbitrary degeneracy a n dthe combination of lattice and impurity scattering under these conditions.

$2 . THEORY2.1. Impuri ty Scattering

T h e expression for the conductivity U is as follows :

whereelectrons.given by

and 7" are the reduced energy and reduced chemical potential of theFollowing the theory derived by Mott (1936), the relaxation time is=Nz* (1- os 0) [ ( B ) 27~in 0dO.7

N i s he concentration of impurity centres.potential given by For scattering by a screened Coulomb

where E is the dielectric constant.. . (2 )

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Impurity Scattering in Semiconductors 77hence

where1= N u 2 ~ { ~ ~ } ~ f2 (x)7

Xf (x )= log ( l+x ) - -+ x16n2m*2cv2q2h2=

2/(2m*)E2(77k~)312nNe4f(x)=and

Substitution of this value of T in equation (1) gives3 2 ~ ~ m * ( k T ) ~ q3exp(q-q*)dq

3Ne2h3 {exp(v-q*)+l}2f(x) *'31= . . . . 3)To evaluate the integral it is necessary to assume that, sincef(x) is a slowlyvarying function of q, t may be taken outside the integral and calculated from the

value of q which makes the integral a maximum. Equation (3) then becomesafter integrating by parts 3 2 ~ ~ m * ( k T ) ~ q 2 d q . . . (4)

. . . . . . (5 )uI= N e"3f(x) , exp (7 - *) + 1

where .I, he value of q to be used to calculate f (x) is given by the equation(7- 3)exp ( { -q* )= t +3.

This equation may be solved graphically.Equation (4) is similar to the expression derived by Johnson and Lark-Horovitz(1947), who have considered scattering by a Coulomb field for any degeneracy.In order to calculate the potential field near an ionized impurity centre andhence to determine the screening constant q in equation (2) it is necessary to solvePoisson's equation. For the conduction band, let no be the concentration ofelectrons in the unperturbed lattice and n the concentration of electrons in theperturbed lattice, then, since no s also the density of positive charge, Poisson'sequation becomes

4veV2(+) = -- no- ).If F (0 ) no and F(O++)= 2 assume that F (@++)= F (0)k+ F' (0)wheree@=maximum energy of the electrons= q* kT and4n(2m*k T)3f2F ( @ )= h3 h3 filZ (?*I

a l s o 4 ~ ( 2 m * k T ) ~ ' ~ eF' (@)= h3 f i ' z ' (7") k Tthen 4~ev"+) =- F ( 0 ) q 2 + .Th e solution of Poisson's equation then becomeswhere

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7 8 R . MansfieldLimiting case 7 a0.

and 16n2m*e2 3n0 113E2h2 (7)p2=

This is the value obtained by Mott (1936) except for the introduction of thedielectric constant. If q* >6 and no= N then

The equation for the conductivity becomes3h3e2n0u - ...... 6 )

I- 167i2e2m"~(x)with x=(a)2s(,) 3n, 1'3 .

This equation shows that for complete degeneracy, the mobility becomesdependent on the concentration of scattering centres only through the term f(s).This arises because although the number of collisions between charge carriers andimpurity centres is increased, the energy of the charge carriers also increases,resulting in a greater amount of small angle scattering.Limiting case q*< 0.

(2nm*k T)31*2 7") = 2 exp q* =and equation (5) givesHence the mobility

= 3 .U 27124 T)Wz L = -n,e A77i32m*12f (3 )Gnz*e(kT)'where x =

This is the expression derived by Herring (Debye and Conwell 1954).Genera1 case.written a s

V % 7 2 , @ 2 .

The general equation for the conductivity due to impurity scattering can be...... 7)32e2n2*(k T)3f , (q* )A'e2h3f(x )1 =

where xf ( x )= og (1 +x) -7$xnz".c )2

l + X.I(T)l 6 h. y*=-( qh ) e2(2712*)12f'1,,(7*)

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Impurity Scattering in Semiconductors2.2. Combination of Impurity and Lattice Scattering

The expression for th e total conductivity can be written as follows :

7 9

where uL s the conductivity which would arise if lattice scattering only occurredat the appropriate value of v*, andf 4L 3h3Ne2log (1+ exp 7") 32e2m*(kT)3=

.....( )Assuming thatf x) can be treated as a constant, c( was calculated from equation

( 9 ) , and the integral in equation (8) determined, for different values of 7" andU ~ , U ~ . igure 1 illustrates the correcting factor F=(pL+pI) /p which has tobe applied to th e sum of the resistivities pL due to lattice scattering and pI due toimpurity scattering, to obtain the total resistivity p. When 7" +O the totalresistivity is given by the sum of the resistivities and when 7*< 0 the theory ofJones (1951)and Johnson and Lark-Horovitz (1951) can be used.

Figure 1. Combmation of lattice and impurity scattering.In order to apply these results i t must be remembered that u r should bec&ulated from equation (7) using the value of 7=.Ihich makes the mtegrand of

equation ( 5 )a inaxiinum. T he appropriate value of 5 9 given by the solution ofthe equation . .. . (10)'i'he solution of this equation for the various values of 7" and u,,/'uI which have beenconsidered is given in the table and is illustrated in figure 2.

Th e error introduced by assuming that {(x) can be treated as a constant in theIntegral was jnvestlgated by determining the value of the integral for a few valueso f ? * and vJu1, choosing the special case of s= 107. The difference between the

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80 R. Mansfieldtrue values and the approximate values was less than 3 %. The error, however,becomes appreciable if the value of { determined from equation (5) instead ofequation (10) is used.

Values of .Isatisfying equation (10)4 5F2*0394210.50.250.1110

99.6439.6009.5589.5029.4339.3649.3069.2639.218

66.9286.8786.8046.7266.6306.5336.4526.3886.320

34.5724.4644.3664.2454-0993.9503.8203.7083.575

1 53.7253.5753 e4463.2923 - 1 1 22.9272.7602.6072.391

03.2443 o 4 42.8852.7032.4942.2812.0821 8881 4 3

77 *Figure 2. Solution of equation (10).

5 3. C OMP A R ISONITH EXPERIMENTALESULTSFor a comparison with experimental results it is useful to consider sample..in which the impurities are all ionized over a considerable range of temperatures,

then the concentration of scattering centres is the same as the concentration O!charge carriers and scattering by neutral impurities can be neglected. If theresidual conductivity is known for the limiting case 7" $0 then it is possibleto calculiite the effective mass of the charge carriers using equation (6), providedthe dielectric constant and concentration of charge carriers is known. Thtlatter quantity can usually be calculated from the Hall constant and the formerdetermined from the refractive index, if a direct measurement of the dielectrlrconstant has not been made.calculate 7" using the expressionHaving determined a value of na*, it is possible

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Impurity Scattering in Senticonductors 81j 2 (v* ) nd f1,2'(7*) may then be determined from tables of these functions (Johnsonand Shipley 1953, McDougall and Stoner 1938). Hence the conductivityarising from impurity scattering for any temperature may be calculated.Over the temperature range in which impurity and lattice scattering arecomparable it is necessary to calculate uL using the expression

2 ( 2 i~ m* k T) ~ /~h3 eulog(1 + exp ~]* )L=where U is the lattice mobility at temperature T when the charge carriers arenot degenerate. The ratio uL/uImay then be determined, and hence he appru-priate value of obtained and a new value of u1 calculated. It is generallyunnecessary to redetermine a value of using the new value of q. Th e correctingfactor F can now be determined from figure 1 and the value of the total resistivityobtained.This procedure was adopted in the case of germanium and indium anti-monide. Debye and Conwell (1954) have reported measurements on an impuren-type sample of germanium (specimen 58) in which the conductivity is constantover a considerable range of temperatures. Agreement is obtained betweenthecry and experiment if it is assumed that E =16.1 and m*jm=0.29.In the case of indium antimonide a direct determination of the dielectricconstant has not been made, and although the refractive index has been measured,the values obtained by different authors differ appreciably (Breckenridge et al.1954, Avery e t al. 1954, Oswald and Schade 1954). For a comparison withexperiment, values of the dielectric constant of 12.5 and 16 were chosen. Barrieand Edmond (1955)have reported values of the Hall mobilities at room temperatureof n-type specimens having a range of impurity content. Figure 3 illustratesthe theoretical variation of the mobility with concentration and gives theexperimentally determined values of the Hall mobilities, assuming a latticemobility of 80 000cm2v-1 sec-1 at room temperature for a non-degenerate sampleand that all the impurities are ionized. A comparison in this case is complicated

01 I I , I I I 110'6 IO" 10'8 IOUConcentration o f Electrons (cm-'}

Figure 3 . Variation of mobility with concentration of electrons in InSb.The experimental points are those of Barrie and Edmond (1955).by the uncertainty of the relation between the Hall and conductivity mobilitiesand also in the actual value of the concentration of charge carriers. For the purerSpecimens the concentration of scattering centres is less than the concentration

PRoC. PHYS. SOC LXIX, I-B F

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82 R.Mansfieldof charge carriers, since at room temperature the contribution of intrinsic electronswill become important for n less than 101' ~ m - ~ . n this case the calculated valueof the mobility assuming n o = & will be too small. When the values of E a dm l m * used in figure 3 are applied to the n-type specimens studied by Madelungand Weiss (1954) reasonable agreement is obtained. I t is seen that the con-ductivity depends niainly on the product c m j m * and consequently it is notpossible to decide on the correct value of m fm * from these measurements.

REFERENCESAWRY,D. G., GOODWIN, . W , LAWSON,W. D., and Moss, T. S., 1954, Proc. PhpBARRIE,. , and EDMOND,. T , 1955, J Electronacs, 1, 161.BRECKENRIDGE,. G ., BLUNT, . F., HOSLER, . R . , FREDERIKSE,. P. R . , BECKER, H,and OSKINSKY,., 1954, Phys. Rev., 96, 571.DEBYE,. P., and CONWELL,. M., 1954, Phys. Rev., 93, 693.JOHNSON, V. A ., and LARK-HOROVITZ,. , 1947, Phys. Rev., 71, 374; 1951, i b i d . , 82, 977JOHNSON, V. A., and SHIPLEY, M., 1953, Phys Rev., 90, 523JONES, H., 1951, Phys. Rev., 81, 149.MCDOUGALL,. , and STONER,. C., 1938, Phil. Trans. Roy. Soc . A, 237, 67MADELUNG,. , and WEISS,H., 1954, 2. N a t u r f . , 9a,527Mom, N F , 1936, Proc. C a m b . Phil. Soc., 32, 281OSWALD,. , and SCHADE,., 1954, 2. N a t u r f . , ?a, 611.

SOCB, 67, 761.