The second Born approximation and the Friedel sum rule

Volume 137, number 3

PHYSICSLETTERSA

8 May 1989

THE SECOND BORN APPROXIMATION AND THE FRIEDEL SUM RULE J.D. PATTERSON Physics~SpaceSciences Department, Florida Institute of Technology, Melbourne, FL 32901, USA and S.L. LEHOCZKY ES75, Space Science Laboratory, Marshall Space Flight Center, AL 35812, USA Received 16 January 1989;acceptedfor publication 6 March 1989 Communicatedby A.A. Maradudin

We improve the results of Agarwal and Singh for the screening length of an impurity potential. FollowingAgarwaland Singh we use the Friedel sum rule and evaluate the phase shift through secondorder in the Born approximation. The results still distinguish between donor and acceptorscatteringcenters, but with a different sign than givenby Agarwaland Singh.

oo

The Friedel sum rule for conduction electrons described by spherical energy surfaces in a semiconductor with an ionized impurity can be written as

f rfor 0

[1,2] 2 dSt -n~(21+I)~f(E)dE=Z,

oo

X U(r' )jl(kr' )rZr 'e •

(I)

tan 81= - k A l ( 1 - B I / A I ) - l ,

(2)

where k is the magnitude of the wave vector, oo

AI = f j 2 ( k r ) U ( r ) r 2 d r , 0

and

(3)

(4)

In eqs. ( 3 ) and (4) the jr are the customary spherical Bessel functions, Gt(r, r ' ) = k j t ( k r < )qz(kr> ) ,

where ~ is the phase for t h e / t h partial wave, f ( E ) is the customary Fermi-Dirac distribution function, E is the energy of the electron which is scattered and Z measures the charge on the impurity (donors which donate one electron have Z = 1 ). From ref. [3], the phase shifts 8z can be determined by

,

0

(5)

where rh are the spherical Neumann functions and r< and r> represent the lesser and greater of r and r' respectively. Also, U ( r ) represents essentially the screened Coulomb potential, U(r) -=aZ[exp( - r / L ) /r] ,

(6a)

me 2 a =- - 2n~h2,

(6b)

where m is the effective electronic mass, e is the electronic charge, h is Planck's constant divided by 2n, e is the static dielectric constant, and L is the screening length. Eq. (2) is derived by a variational principle and is identical to the Born series through second order. Also correct to second order we have tan Jt-~St. Eq. (2) to first order gives S t = - k A l , the standard first

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Volume 137, number 3

PHYSICSLETTERSA

order Born approximation result. We follow Agarwal and Singh in only using l= 0 (at low temperatures) for the corrections to the first order terms. So we write approximately

6l~- -kA/(1-Bo/Ao) -~

(7)

Using Et(2/+ 1 )j2(kr) = 1, it is then elementary to show from the Friedel sum rule that

2~ dk aZL z Z= - ~ j dE f(E) dE 1-Bo/ao "

(8)

8 May 1989

where

V= ¼aZL< ~).

(9b)

Since a < 0 we see from eq. (9) that L 0 and L>LtI) for Z < 0 . Thus through the second order Born approximation, donors are predicted to scatter less effectively than acceptors in so far as screening is concerned. This disagrees with the result of Agarwal and Singh. Their result appears to arise from an inaccurate expression for the phase shift in the second Born approximation.

0

In evaluating Bo/Ao, we again follow Agarwal and Singh and assume only kr << 1 gives important contributions. Noting in eq. (8) that in the first Born approximation (with B o = 0 ) L=_L¢~), we find after an elementary calculation from eq. (8) that

L=L(,) (x/1 + V 2 "q- V )

138

,

(9a)

References [ 1 ] F. Stem, Phys. Rev. 158 (1967) 697. [2] D. Chattopadhyay and H.J. Queisser, Rev. Mod. Phys. 53 (1981) 745. [3] C.J. Joachain, Quantum collision theory (North-Holland, Amsterdam, 1975) p. 236. [4 ] B.K. Agarwal and N. Singh, Phys. Len. A 95 ( 1983 ) 319.