PH 7 8 I CA I RK VIE% 8 &OI UMK 16, N UMBER 12

~1Selds in electron transport: Application to electronnlration*

R. S. Sorbcllo and Basab DasguptaDepartment of Physics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

(Received 6 July 1977)

A quantum-mechanical expression is given for the local electric field in dc electron transport. The localfield is shown to provide the driving force for the migration of atoms during dc electron transport (electro-migration). %e express the local field in terms of the electron charge density, which we obtain by solvingthe Liouville equatiofi for the single-particle density matrix. The solutions are found within a self-consistentfield weak-scattering approximation scheme for an impurity in a jellium background. Electron-phononinteraction is included via a phenomenological relaxation time. It is shown that the local field arises fromboth static and dynamic screening. The static screening is associated with the screening of the external fieldnear the impurity, while the dynamic screening is associated with the "electron-wind force, " or themomentum transferred by the electrons in collisions with the impuri'ty. It is found that the "electron-windforce*' dominates when kfl &1, where k~ is the Fermi wave vector and I is the electron mean free path.Only when krl is of order unity are the static and dynamic screening contributions comparable. Landauer'sresidual-resistivity dipoles and carrier-density-modulation effect are investigated, and are found to contributeto the local field. The carrier-density modulation effect is shown to lead to deviations from Matthiessen srule by shifting the Fermi energy relative to the band bottom. Within the local-field framework, we discussthe distinction between the external field and long-range macroscopic fields in quantum-mechanicalformulations of dc electron transport.

I. INTRODUCTION

In discussing dc electron transport near inhomo-geneities it may be convenient to introduce theconcept of the local electric fieM. The local-fieldconcept turns out to be especially useful in de-scribing the migration of impurities and other de-fects during electron transport. In fact, the localfield is the driving force for the migration of atomsin the presence of the external electric field andthe accompanying electron current. This migra-tion phenomenon is known as electromigration.The local electric field is also a relevant drivingforce in thermomigration, or atomic migrationin a temperature gradient. Theories of electro-migration and thermomigration have been reviewedin the recent literature. ' '

Th'e local electric field arises from all electricalcharges in the system, including the sources ofthe external field. Part of the electrical chargedistribution may be associated with the electroncurrent flow, and corresponds to the "electronwind" or dynamic screening in electromaigrationtheories. An additional contribution may be as-sociated with static screening of the eternalfieM. The statically screened external field givesrise to the "direct-field" force in electromigrationtheories. The magnitude of the "direct-field" forcehas been controversial. For example, accordingto some theories6 ' the "direct-field" force van-ishes (complete static screening) in the vicinityof an interstitial impurity, while in other theo-ries~'6 there is no static screening.

To settle these controversies it is necessary todetermine the local electric field by solving thequantum-mechanical transport problem for theresponse of electrons in an external dc electricfield. Several powerful techniques have been in-troduced for treating the electron-transport prob-lem."" It has been found that for sufficientlylong electronic mean free paths, the electricalconductivity calculated quantum mechanically re-duces to the electrical conductivity obtained fromthe Boltzman equation. " This reduction has beenpossible only in the weak-scattering or dilute-scattering limit. When interference effects be-tween scatterers are allowed, extra contributionsto the transport equation emerge. " Additionalcorrections to the Boltzmann equation might alsoarise from current-flow inhomogeneities, "orpossibly from Landauer's residual-resistivity di-poles, "'"and his carrier-density modulation ef-fect 24, 26

Further difficulties in the Boltzmann-equationapproach arise when quantities such as chargedensity and local fields are to be determined. Theproblem is that such quantities involve off-diagonalelements of the density matrix in the Bloch repre-sentation, whereas strictly speaking the Boltz-mann equation is only relevant for the diagonalelements. " While it may be legitimate to ignorethis difficulty within a fully semiclassical problemin which the scattering potentials are slowly vary-ing in space, this semiclassical picture is highlysuspect for the case of highly localized scattererssuch as atomic defects. For this reason, charge

R. S. SORBKI LO AN D BASAB DASGUPTA

densities and local fields which are derived fromthe use of the Boltzmann equation may be of verylimited validity.

To avoid the difficulties inherent in semiclassi-cal pictures, we base our calculation on the quan-tum-mechanical Liouville equation for the densitymatrix. Our calculation is an application of themethod introduced by Kohn and Luttinger" (KL).one advantage of this method is that we shall beable to extend into the quantum-mechanical re-gime the very physical semiclassical analysis ofDas and Peierls. " This will lead to a betterphysical understanding of the various sources ofthe local electric field. In particular, we shallbe able to isolate the contributions due to the "di-rect field" and the "electron wind, " as well a,s in-vestigate the effects of the residual-resistivitydipoles and carrier-density modulation. "'

The present calculations, like those of Das andPeierls, "will be confined to the model of diluteweakly scattering impurities in jellium, with pho-non scattering introduced via a phenomenologiealrelaxation time. The results we shall obtain areconsistent with recent results obtained within theframework of linear-response theory. ""

II. LOCAL ELECTRIC FIELD

In analogy with classical electrostatics we candefine the local microscopic electric field at someposition X by introducing a point test charge ofmagnitude Q at X. If the force exerted on the testcharge Is denoted by Fo(X}, we define the localelectric field E~(X) by the expression

E,(X)= Iim F,(X)/q. (I)

The mass M of the test particle is assumed tobe very large (formally M —~) so that in calculating the force we can ignore the motion of the testparticle. %e then visualize the test particle as asharply localized wavepacket, which represents aclassical particle essentially fixed at position X.

In order to obtain Fo(X) we apply the resultsgiven elsewhere for the force on an ion." Theforce on any ion of valence 'Z at position X in thepresence of an external field E is given by~3

sity which depends on E. In Eq. (2) and henceforthwe take n(x) to be the E-dependent electron densityin the absence of the test charge.

In deriving Eq. (2) one assumes that the ions arerigid and essentially fixed in position. The coreelectrons are not treated dynamically; their polar-ization in the external field is ignored.

It is clear from the derivation" of Eq. (2) thatthis equation is valid for any external field of ar-bitrary space and time dependence, i.e. , we eantake E = E(X,f}. In this case n(x) will of coursebe time dependent due to the coupling of the elec-trons with the external field. In the present dctransport problem we take E to be constant in

space and time.Result (2) is a generalization of the Feynman-

Hellman theorem" to a nonequilibrium, dissipa-tive system. In the usual derivation of that theoremone starts with the expression F = -8(X}/SX, where(K) is the expectation value of the total systemHamiltonian X. Such an expression is not valid foropen, dissipative systems such as that consideredin electron transport. The correct starting pointfor dissipative systems is the equation of motionof the expectation value of the momentum of theion. The ion is taken to be moving infinitely slowlyas a sharply localized wavepacket whose dynamicsare governed by X. This is the approach used toderive Eq. (2}."

We now apply Eq. (2) to the particular case ofthe force on the test particle of charge Q. Theinteraction between the test charge and an electronis V,(x-x)=-ge/~X-x~. Using hqs. (I) and (2)we obtain

E,(X)=E— n(x) - - d'x~X tX-xl

where the Q -0 limit has brought in n(x), theelectron density in the absence of the test particlebut in the presence df all the ions in the system.

Since n(x) is independent of X we can recast Eq.(3) in the suggestive form

E,(X)=E "(R)BX

where C (X) is the local electrostatic potential de-fined by

F=ZeE — n x ' d'x.BX

(2) 4(X)=-e n(x) - d'x.I X-xl

Here e is the charge of the proton, n(x) is the ex-act quantum-mechanical electron density at positionx, and V,(X—x) is the bare interaction energy be-tween an electron at x and the ion in question. Theintegral is over all space. Since we are interestedonly in that part of the force F which depends onE, we need only retain in n(x) that part of the den-

It should be emphasized that despite the classicalform of Eqs. (2)—(5), these equations include theelectronic many-body effects. The latter are con-tained .within n(x).

It is interesting that Eqs. (4) and (5} imply thatone could have obtained the correct force on thetest particle by simply computing the change in

LOCAL FIELDS IN ELECTRON TRANSPORT: APPLICATION. . .

the expectation value of the interaction energy asthe particle is moved. '7hat is, one can write

Fo(X)=— (6)BX

where V' = -QX E —~, Qe~IX —x' "

Iand the sum

is over all electrons, with the position of the lthelectron denoted by x'". Equation (6) is valid onlyfor the force on the test particle to order Q. Qnecannot apply Eq. (6) to the force on an ion by simplychanging Q to Ze in the expression for V'. Theresulting expression would agree with the correctexpression (2) only to terms linear in Z in the Z-0 limit.

The connection between the force on an ion andthe local field follows immediately by comparingEqs. (2) and (3). For a point-ion potential theforce expression (2) gives F=ZeE~. In the moregeneral case of an ion core of finite size, we canwrite

F= AXE~X dx,where N(X} is the charge density of the bare ion(nucleus plus core electrons). N(X) and Vo arerelated by the Poisson equation V'V, (X}= 4weN(X).

It should be emphasized that by assuming all ionsto be rigid we are explicitly ignoring the dynamicsof the core electrons in the external field. If wewish not to invoke this rigid-ion picture, then Eqs.(3) and (7) will remain valid provided that in Eq.(3}, n(x} includes the correct core-electron den-sity in the presence of E, and in Eq. (7}, N(X} isthe nuclear charge only.

Now, for core levels well separated from theconduction band, the core electrons are expectedto polarize essentially as in a free ion. This im-plies that for an ion of valence Z, the core polari-zation cancels the effect of the field on all but Zprotons in the nucleus. The core polarization isa very slight dipolar deformation in the core-elec-txon wave functions. " This polarization producesa field near the nucleus which is comparable to theexternal field E." However, in the important re-gion outside the core this polarization field is notsignificant. There are two reasons for this: First,the dipolar sources of the core-polarization fieldare localized w1th1n the core,"and hence the f1eldis substantially weakened at distances of two orthree core radii from the nucleus. Second, thecole-polRrlzatlon field 18 screened out by the coll-duction electrons at distances greater than ascreening length from the nucleus. Thus it appearsto be R good Rpprox1IIlRt1on to use the r1gld-loll Rp-proximation and ignore the dynamics of those coreelectrons coming from core levels well below theconduction band bottom. (Of course, core elec-

trons that fall within the conduction band are to betreated dynamically and are not considered as partof the rigid ion. }

It is worth remarking that the electric field Eappearing in our equations is to be regarded Rs anexternal uniform field whose sources are not to bescreened. It would be incorrect to regard E as alongitudinal field arising from external chargesimbedded in the medium since static electronicscreening would prevent the field from penetratingvery far into the metal. The imposition of the no-screening condition on E for dc transport is typi-cally done in an ad hoe way, ' or equivalently, a'

transverse field is used. " In either case an in-finite medium (or periodic boundary conditions} isRssuIIled.

Using the no-screening condition in an infinitemedium is a device to enable one to avoid the com-plications of the physical boundary conditions in thedc transport problem. For example, we need notexplicitly consider the self-consistent chargeswhich are set up at an interface between two dis-similar metals, as at the electrode-metal inter-face at the ends of the sample. These charges areresponsible for the electric field which drives thecurrent through the sample, and arise not fromimbedded external charges, but rather from theself-consistent electron- scattering process asenvisioned by Landauer" (see Sec. IVC). In ourmodel of an infinite medium there is no polariza-tion-charge buildup at the ends -of the sample.(These charges can be obtained macroscopicallyfor a finite sample of course by solving the mac-roscopic boundary-value problem. } We emphasizethat variations in the local field due to scatterers(microscopic or macroscopic inhomogeneities)within the system will be correctly obtained fromEq. (3) provided that these scaiterers are includedin the system Hamiltonian.

A further advantage of considering the model ofan infinite medium is that if there are only ran-domly distributed microscopic scatterers in themedium the external field E is precisely the mac-roscopic electric field. This is easily seen bynoting that the macroscopic field is the spatialaverage of the local field E» and that by sym-metry the spatial average of the integral in Eq.(3) vanishes for the inifinite medium. "

A more rigorous justification for considering Eto be the macroscopic electric field and for con-sidering an infinite medium can be deduced fromKL." They considered the solution of the trans-port problem in a finite toroidal geometry, withthe electric field caused by an axial magneticfield increasing with time. They found that thesolution was consistent with that obtained for aninifinite medium (periodic boundary conditions). "

5196 R. S. SORBELLO AND BASAB DASGUPTA 16

Moreover, in the toroidal geometry it is clearfrom symmetry that no distribution n(x) can con-tribute to an average field, or net voltage drop,in a closed path around the torus. Hence, the ex-ternal field E equals the macroscopic field here.

In quantum- mechanical calculations for anyphysical system the safest course to follow is toavoid identifying E with the macroscopic field.Instead the latter is determined in terms of E atthe end of the calculation by setting the macro-scopic field equal to the spatial average of E~ inthe medium, where E~ is given by Eq. (3). Thisprocedure is valid for any system regardless ofthe boundary conditions assumed. As an example,consider the model of a single localized scattererin jellium. In this case E will equal the macro-scopic field because the integral in Eq. (3}givesa field which is not appreciable far from the im-purity (Sec. III}.

These remarks concerning the distinction inquantum-mechanical theories between the externalfield and the macroscopic field have been moti-vated by questions raised by Landauer. ~

III. DENSITY-MATRIX CALCULATION OF LOCAL FIELD

A. Liouville equation in selfwonsistent-field approximation

H~=Hp+H'+ex E, (8)

where Hp is the kinetic energy of an electron and

H is the electron-impurity interaction. x is theelectron coordinate.

We are interested in the total one-electron den-sity matrix, which we denote by p~. This quan-tity is separated into a term p, which is the equili-brium density matrix (in the absence of E, but inthe presence of H'), and a term f which is linearin the field. Corrections which are second orderor higher in E are considered negligible. Wetherefore write

Pr=P+ f. (9}

Similarly, it is convenient to separate the elec-tron-impurity interaction H into an equilibrium

We determine the electron density n(x) using asingle-particle density-matrix approach similar tothat of KL." The system we shall consider is thatof a weakly scattering impurity in a jellium back-ground. Electron-phonon scattering is includedthrough a relaxation time. Electron-electron in-teraction is treated within a self-consistent screen-ing approximation. Electron-phonon and electron-electron interactions were not considered in KL.

Following for the most part the KL notation, weexpress the total one-electron Hamiltonian H~ asfollows:

and field-dependent part as follows:

H'= V'+ V' (10)

Spr[H )

(Pr —P (12)

where we have introduced a relaxation time & todescribe the decay to equilibrium by dissipativeprocesses. These processes may be electron-phonon scattering, or in general, anybackground-scattering mechanism not included in the interac-tion H' between the electron and the impurity.

In the steady state, Spr/St=0 and Eq. (12) canbe cast into the form

if/T=[ex E, p]+[V', f]+[V',p]+[H„f].(13)

In obtaining Eq. (13) we have used Eqs. (8), (9),and (10) in Eq. (12}and have systematically dis-carded terms which are higher order in E.

It is most convenient to express Eq. (13) in theplane-wave representation since plane waves areeigenfunctions of H„ that is, H,

~

k) = g;~

k) where~k) is a plane wave state and q;= k'/2m is the en-ergy eigenvalue. In the plane-wave representationEq. (13) leads immediately to

p p[»'"'E]-J kk' (.—f, —1/7' ("—(r —z/7'k' k

1 1~km Vk-e- V~a-&ok

k ' k'e- —e- —&/r (14)

The unknown quantity V' appearing in Eq. (14}isdetermined self-consistently from the Poissonequation

V'V'=-4we'n(x), (15)

where the electron density n(x) refers to that partof the total electron density which depends li-nearly on E. The self-consistency cycle is closedby requiring that n(x) arises directly from thequantum-mechanical wave functions, or equiva-lently, from the density matrix. This condition

where V' is the screened electron-impurity potential in the absence of E, and is to be treated asa known quantity. V' is the screening part whichis linear in E and is to be determined self-con-sistently.

With the separation (10), the explicit form of pat temperature T can be expressed as

p=(exp[(H, + V'- q~)/keT]+ I] ',where q~ is the Fermi energy and k~ is Boltz-mann's constant. We use units where 8 = 1.

The density matrix satisfies the Liouville equa-tion

LOCAL FIELDS IN EI ECTRON TRANSPORT: APPLICATION ~ . .

is conveniently expressed as

II(q) g fk k-II (16)

where n(q) is the Fourier transform of n(x), thatlsd

n(x)=g n(q)exp(iq x}. (1V)

In Eqs. (16) and (1V) the sums are over all k and

q, respectively. For convenience we have chosena normalization which takes the crystal volume tobe unity. The transformation from sum to integralis accomplished via the rule Z;-(I/8)i') f {f'q,with an additional factor of 2 needed for Zk due tospin degeneracy.

It is convenient to rewrite the Poisson equation(15) in terms of Fourier-transformed quantities.Consistent with Eq. (1V), we define the transform

(14), the diagonal element f„„is O(X'}. Using theKi notationwe w~ite f-. as f an-duse Eqs. (14}and (19) to find that

(21)

Ef-.= f-,+ f-+ f-„kk' kk' kk' kk'

where

(22)

It is clear from Eqs. (14) and (20) that the off-diagonal f ., begins at O(X). Consequently to Obtain

f... to O(X), one needs to consider only those termskk'

in the second sum on the right-hand side of Eq.(14) in which the diagonal element f., or f.„, appears.Similarly, since the off-diagonal elements of p areO(A} we keep only the diagonal part of p in the lastterm on the right-hand side of Eq. (14). We thusobtain for the off-diagonal elements to O(A.), theresult

'v'(I)= I v'(x)azp{-'i x)d'x, [P~ ex 'E]rkkk' e q i/Tk'

(23)

where the integration is over all coordinate space.We can then cast Eq. (15) into the Fourier-trans-formed form

V'( q) = 4))e'n( q )/(I'.

The quantity V (q} is identical to the matrix ele-ment V~ &; irrespective of the value of k.

B. Weak-scattering approximation

Using KL as a guide we simultaneously solveEqs. (14}, (16}, and (18) in the weak-scatteringlimit. We consider H' to be weak, with the strengthof H' designated by the small parameter A.. Equa-tion (14) can now be used to show that the diagonalelement f.,„ is of order X' and the off-diagonal ele-ment f.„, is of order ~. To establish this we needto coIlsldei' tile coilllIllltatol' [p ex 'E]. KL sllow

that the commutator may be evaluated by introduc-ing the commutation rule x = i&/Sp, where p is theelectron momentum. This leads immediately tothe following expression for the diagonal elementto lowest order in X. (Refs. 1V and 35}:

8p~[p, ex E]-=-ieE.

kk Bk

whe~e O„=(exp[(&„.—ez)/ksT]+ I) ' denotes the Fer-mi-Dirac distribution.

The off-diagonal (k 4 k ) elelileilt of 'tile coII1IIlu-tator, again to lowest order, is given by""

[p ex ~ E]- =ieEH'-. =+ p' "' . (20)kk' kk'

Now it is apparent that expression (19) is O(X')and expression (20) is O(I{). Therefore, by Eq.

ns(q) =q'[I - &(q)]V'(q)/4)ie', (2V)

( ) 14)ie g Pk —Pk"q' - (--k- --I/T

k k kq

is the Lindhard dielectric function. " When ns(q)is. eliminated from Eq. (26) by means of Eqs. (2V)and (18), we find

(28)

is dependent on the electric field explicitly and thecommutator is given by expression (20). Thesecond part of f,, is given by

fW (fk- fk)&,k (24)kk' g - g —i/T

and will be shown to be associated with the "elec-tron wind. " The third part of f, is given by

yl~S (Pk ilk) ~-;k.

(25a,--e;-&/T'

which exp].icitly contains that part of the screeningpotential which is linear in the field. %e there-fore identify f ., as the self-consistent screening

kk' gresponse to the perturbations f-„, and f-.„.We can now easily determine n(q) from™Eq. (16}

and Eqs. (22)-(25). The important step is to re-place the matrix element V' .„ in f .by theright-hand side of the Poisson equatio~n (18}.

In analogy with expression (22}, we express theresult of the calculation as

n(q) =ns(q)+n~(q)+ns(q), (26}

where ns(q} =2-, ffs, and simil. a-rly for n~(q} andns(q). It is particu~iarly useful to introduce theexplicit form of ns(q). From Eq. (25), we have

5198 R ~ S ~ SORBELLO AND BASAB DASGUPTA

We can numerically determine n(q) by perform-ing th«ums n ' (q}=Z.„f -;. Transforming thesums to iotegrals and evaluating them, we obtain

n(q}=-[ieEr/2'(q)jV'(q)a(q) cosQ, (30)

where n(q) is a function we have evaluated snd (t(

is the angle between E and q. We have taken Vo(x}to be spherically symmetric.

The function a(q) can be separated into an elec-tric-field part o(s(q) and an electron-wind partn~(q) by writing

where ne(q} and ns(q) satisfy Eq. (30) when theyreplace n(q) and n( q), respectively, in that equa-tion. c.~(q) and n~(q) are related by Eq. (30) in thesame way.

Both ns(q) and o(~(q) depend on q, &, and theFermi wave vector k~, but only in combinationsof the dimensionless variables q and $, where g= q/k~ snd (=k~l, with I = k~r/m being the meanfree-path. The explicit expressions are

ae(q) = —1 ——ln + —1 —~+q' @+2 ( q' 14 g-2 2p T

&( In (I - kn)'+ I/n'8(1+—,'q)'+ I/q' t'2

Assuming that the scattering potential V' isspherically symmetric and centered at the originof X space, the angular integrations in Eq. (34}can be performed using the n(q) expression (30).The local field can then be cast into the from givenin Eq. (4), where the explicit expression for thelocal potential 4(X) is

@(x)=(,. ~ ~(s) i, (sx)&s (3&(Jg

Here X=X/~X~

and the Bessel function j, is de-fined by j,(x) =x 'sinx-x ' cosx.

The force on an impurity ion is of special inter-est. Expressing the integral in Eq. (2) in q space,we obtain

F=«E-(2 ) 'I (t(~(t()&'(e(~(s)&r

where we have taken the impurity to be at theorigin and have replaced the bare electron-ion

(36)

cx (q)

conveniently expressed in terms of n(q) ratherthan n{x}. For example, Eq. (3}becomes

E~(X)=E+ 2 Jt Pq 2 exp(tq'X)d q ~ (34)

0.8- '

0.6-

, (=&0

,' K=3

where 6(x) =1 for x~ 0 and mero for x&0. &aluesfor the tan ' functions are restricted to lie betweens-,'7(. Results (32) and (33) follow from Eqs. (20)-(31) with no approximation except for invoking theusual condition that ksT«q„. Both c.e(q) and c((q)are plotted as a function of q for $ = 3, 10, and 100in Fig. 1.

Equations (30}-(33}together with Eq. (17}givethe electron density in the weak-scattering ap-proximation. In the (» I limit, a(q}-e(2 —q} andn(x) take the form of the Bosvieux-Friedel dipole'associated with the electron-wind force. See Ref.12 for a picture of n(x) as calculated from pseudo-potential theory in this $» 1 bmit.

C. Local field and force on an impurity ion

Having determined the electron density, we nowturn to calculating the local field and the force onan impurity using Eqs. {3)and {2), respectively.For computations, the latter equations are more

0.4-, i

I

I

0.2-II

0

0.)0--

0.05--

I WW T I

1.02.0 3.0

-0.05*-

-0.10--

/ (=10/

/

ii l

i

FIQ. 1. Functions e(q) and n (q) for different valuesof the parameter $ =ATE. Variable q is q/kz.

LOCAL FIELDS IN ELECTRON TRANSPORT: APPLICATION. . . 5 I99

interaction V, by the screened interaction V' usingthe linear-screening relation V, (q) /&(q).

Taking the expression (30) for n(q) and perform-ing the angular integration in Eq. (36), we find

F=ZeE —,n(q)IV (q)lq "q (37)0

tion F~. Here E is the contribution from the sec-ond term in Eq. (37) when ns(q) is used in the in-tegrand, and similarly Ew a,rises from using u (q)in the integrand. Note that the wind force I ~ dom-inates, although at smaller $, the electric-fieldcontribution I~ begins to become appreciable.

We have computed the local potential 4(X) andthe force F using Eqs. (35) and (37), respectively.In our calculations we used the Ashcroft empty-core pseudopotential" for V'(q)." Explicitly, wetook

V'(q) = [4me-'Z/q'e(q)] cosqfI, , (38)

X

2 4 6 8 10 12 14

where A, is the core radius.A plot of 4(X) is given in Fig. 2 for X along the

direction of the electron drift velocity vD, wherevc= -eEr/m. In this calculation we used the val-ues k~=0.92'?3 a.u. , A, =1.12 a.u. , and Z=3. Thesevalues are appropriate for an aluminum impurity(interstitial) in aluminum metal. The same generalfeatures are observed for other choices of theseparameters. In the (» 1 regime and at large dis-tances from the impurity (k~» I), 4(X) varieslike (E X/X') sin2kr X. A map of 4 (X) in thenear-field region is shown in Fig. 3. .

The force on an impurity has been evaluated forkz values appropriate to aluminum (k+ =0.9273a.u. ) and potassium (k~= 0.3878 a,u. ) for variousvalues of A,. For convenience, we have taken Z=1in the impurity pseudopotential expressions (38).The results are shown in Table I for (-values of3, 10, and 100. The table gives the electric-fieldcontribution E~ and the electron-wind contribu-

IV. DISCUSSION

A. Sources, (()f the local field

The contributions to the loca, l field arising fromfs, and f-"- have been associated with the statickk'

screening in the external field and with the dy-namic screening in the "electron wind, " respec-tively. We investigate this further. ,

The association of f~-„, with the static screeningin the electric field was suggested by the presenceof E in the defining Eq. (23). It turns out that theexpression (23) is precisely what one would obtainif one were to compute in O(XE) the change in elec-tron wave functions due to an ex E perturbation onthe unperturbed plane-wave states in the absenceof current flow. Thus ns(x) is the O(XE) contribu-tion to static screening, .by which we mean screen-ing in the absence of current flow. Note that thereis no static screening in O(A'E) in our transportproblem. Terms of O(X'E) appear only in the di-agonal element f-„and not in f=„, for k. w k '. Theclaim made by Sorbello is thus correct": The pri-mary role of the electric field is in setting up thecurrent, or f-. To allow the electrons locally toscreen E directly in O(}cE) is to double count theresponse of the electrons to the electric field.

The contribution to the local field arising fromf~

, was assoc-iated with the electron current orkk'

"wind" because of the presence of the current-carrying distribution f, in the defining Eq. (24).

-0.1-

-0.2O

-0.3.

-04 ~

FIG. 2. Local potential near an interstitial aluminumion during dc transport. The potential is measured at adistance X from the ion and along a vector in the direc-tion of the electron drift velocity v D. The distance X ismeasured in units of 1/kz. The quantity plotted is thelocal potentia1 4 multiplied by (volvo}, where v z is theFermi velocity. Values are in atomic units. The dashedand full curves correspond to $ values of 3 and 100, res-pectively.

FIG. 3. Equipotential lines for the local potential nearan aluminum ion during dc transport. The plane forwhich the lines are shown is parallel to the electricfield and contains the ion at the origin. The electrontransport is from left to right. The distance betweenscale markings on the axes is 2/k~. Numerical valuesare in atomic units and represent the quantity 4 (X)x (vPvv} for the case of long mean free path ($ =100}.

5200 R-. S. SORBELLO AND BASAB DASGUPTA l6

TABLE I. Forces on an interstitial impurity in aluminum and potassium metal. The force due to the static screeningin the electric field is denoted by F; the force due to the "electron wind" is denoted by F . Forces are shown for dif-ferent values of the parameters ( and R„where $ = kp and R, is the pseudopotential core radius. R, is given in units of1/kz for aluminum metal and 1/2k+ for potassium metal. Positive values for the forces indicate forces in the directionof the electron drift velocity, negative values indicate forces in the direction of the external field. The force units aresuch that when the values shown are multiplied by the ratio of drift to Fermi velocity (vD/vz) one obtains the force inatomic units (double rydberg per Bohr radius). The valence of the impurity ion has been taken to be unity.

R Fw

Aluminum

$ =10FE F

g =100FE FW FE

Potassium)=10

FE F(= 100

0.20.40.60.81.01.21.41.61.82.0

-0.0102-0.0028-0.0008-0.0049-0.0104-0.0128-0.0130-0.0087-0.0043-0.0024

0.1216 -0.0032 0.12630.0862 -0.0004 0.09440.0527 -0.0001 0.05900.0338 -0.0029 0.03360.0325 -0.0039 0.02600.0423 -0.0093 0.03890.0586 -0.0047 0.05270.0704 -0.0028 0.07040.0756 -0.0008 0.08110.0744 +0.0005 0.0819

-0.0004-0.0001-0.0001-0.0002-0.0004-0.0006-0.0008-0.0004-0.0001+0.0004

0.13110.10030.06380.03470.02310.03050.05040.07200.08620.0881

-0.0058-0.0043-0.0026-0.0013-0.0007-0.0007-0.0011-0.0021-0.0032-0.0042

0.03690.03290.02770.02190.01680.01240.00950.00810.00820.0093

-0.0018-0.0013-0.0008-0.0003-0.0002-0.0001-0.0003-0.0006-0.0010-0.0015

0.03460.03180.02770.02270.01780.01320.00950.00700.00590.0061

-0.0003-0.0002-0.0000-0.0000-0.0000-0.0000-0.0000-0.0001-0.0001-0.0002

0.03440.03190.02810.02370.01880.01400.00990.00680.00500.0046

Moreover, the electron density n~(q) is preciselywhat one would obtain by performing a calculationof the electron density on a coordinate systemmoving with the electron drift velocity vv = -erE/mbut with no electric field explicitly present.

To see how n~( q) is associated with a calculationin the moving coordinate system, we use Eq. (24)with f. and f described as shifted Fermi-Diracdistributions. That is, we write f-„=pft- p-, , whereK=k —mvv, and similarly for f.. . [This form isequivalent to Eq. (21}for qr» keT. ) The chargedensity n~(q) according to rule (16) becomes

nw( q) Vo(q) Q PK PK-i(» v ~ '/~

}f K+mvD K-q+mvD —&f '

Vo(q) g P~- P~-a (39)e- —t- ~-i/rwhere we have changed to a summation over K inplace of k in the first sum. To terms linear in vD,the denominator in the first sum is easily expandedto give a-„- q„-

&—co —i /r, where we define u& =

—q vD. The first sum is then precisely q'[1q(q, &u}]/4we', where q(q, &u) is the frequency-de-

pendent Lindhard function and is given by the e(q}formula (28) except that i/r is replaced by &@+i/rin that formula. Expression (39} is now reducedto

n (q) = V'(q)q'[e(q, 0) —g(q, &u)]/4''. (40)

Since we are keeping only terms linear in E or vDthroughout our calculation, we can in Eq. (40) makethe replacement q(q, 0) —q(q, &u) - -&usq(q, ~)/S&u,where the derivative is evaluated at (d =0. Whenthe resulting n~(q) is used in the integral in Eq.(36) for the force, it yields precisely the "drag"

or "stopping force" on an impurity removing at velo-city -vD in a stationary electron gas." It thus be-comes clear that the force arising from nv(q} canbe thought of as a viscosity or "drag force" whichdepends on the relative velocity between the elec-tron gas and the impurity. The presence of &(q, &o)

in Eq. (40) is typical of dynamic screening re-sponse. "

We can extend our analysis to the more generalcase of an impurity moving at velocity u in anelectron gas drifting at velocity vD. The potentialof the moving impurity is of the form V (x-u t),which introduces an extra time dependence in itscourier transform, which now becomesV,(q}exp(-iQt}, where Q=q u. If we now solvethe Liouville equation (12) with the ansatz f~ e '"',we obtain a consistent solution for n(q} to O(Xu) and

O(Avv) provided that we ignore the external-fieldinteraction ex E. The solution at t =0 can bewritten n( q ) = n~( q }/& (q), where

n (q) = Vo(q)q2[~(q, O) «(q, ~+ n)]/4re (41)

plays the role of a new "electron wind" or dynamicscreening density. Since &u+ 0=-q ~ (vv —u), Eq.(41) shows that vv and -u cause equivalent effectsThis equivalence is at the basis of the Das-Peierls""equivalence theorem. "

We emphasize, however, that the equivalencebetween the effects of vD and -u is only valid forthe ri ( q} contribution to the local field or force onan impurity. The additional contribution ne(q)arising from the commutator [ex E,p] in the Liou-ville equation vitiates the "equivalence theorem. "However, since the ne(q) contribution turns out tobe generally small, the "equivalence theorem"

l6 LOCAL FIELDS IN ELECTRON TRANSPORT: APPI ICATION. . -

holds to a very good approximation in our model.In a more realistic model in which ~ depends onthe state k, the "equivalence theorem" fails dueto an additional contribution arising from 7-„, in

~Ne' K+75yg)

place of ~ in the denominator of the K sum in Eq.(39). We are thus in agreement with discussions'pointing out the importance of a constant relaxationtime for the validity of the "equivalence theorem. "

8. Numerical results

Our numerical calculations displayed in Table Ishow that for k~/» 1 the "electron-wind" forcedominates the force arising from any static screen-ing in the external field. The origin of this effectcan be seen in Fig. 1, where we have plotted n(q)and ns(q}. These quantities when used in Eqs.(35) or (37) give the total screening and static-screening contributions, respectively. Figure Iimplies that e.s(q) is much smaller than e. (q), ex-cept for q ~ 2k~ in the small-k~/ regime. This re-gime where k~/ is or order unity is not normallyachievable for pure metals. Even at the highesttemperatures where the ideal resistivity domi-nates, k~/ is typically around 50. It appears,therefore, that the static-screening contributionwill not become comparable to the wind force ex-cept for very impure metals.

Note that the force arising from ns(q) may turnout to be along E or opposite to E, depending onwhether the q &2k~ contribution is larger than the

q &2k+ contribution. The numerical results in Ta-ble I show that the q &2k~ contribution dominatesexcept at higher 8, values. The force due to staticscreening in the external field, therefore, actuallyintensifies the field slightly if A, is not too large.This behavior is contrary to what one would expectfor static screening of an external field in thermo-dynamic equilibrium. The latter behavior does notoccur here because the contribution to ns of O(A.'E}ls mlsslng.

It is also seen from Fig. 1 or from Eqs. (32) and(33) that in the kzi -~ limit, n (q) -0, and n (q)-8(2k+- q). The force obtained in this limit fromEq. (O'I) is identical to the results obtained pre-viosuly from the ballistic'" and charge-polariza-tion models'" (without static screening of the ex-ternal field). We note that the k~1-~ limit forn(q} is reached for virtually all q when k+1 = 100.However, there are always some significant de-partures from the k~l -~ limit of e.{q) in theneighboxhood of q =0 and 2k+. The departuresaround q =0 are not effective in contributing tothe integrals of Eqs. (35) or (37), however, be-cause of the phase-space factors q'dq at small q.The departures around q = 2k~ are also washedaway by the integrations. These departures would

show up more strongly if the pseudopotential V'(q)were very sharply peaked (within a 4 range equalto I '} around q=0 or 2k~. This behavior does notoccur for pseudopotentials of atomic impurities.

C. Higher-order contributions: Residual-resistivity dipoles

Within a semiclassical analysis Landauer origi-nally showed that there is a significant contributionto the local field arising from his so-called resi-dual-resistivity dipoles (RRDs)." According tolater discussions by Landauer4' and 3chaich, 4' theRRD is the electron density arising from the am-plitude of the scattered wave functions near theimpurity. That is, if we wx'ite the total wave func-tion as P(x) = P,(x)+ P,(x), where P,. is the incidentplane wave and $, is the scattered wave, then theRRD is the dipolarlike charge distribution-e

~ g,(x) ~'. The RRD is thus O(A.'), or one orderhigher than we have gone in Sec. IDB.

It is straightforward to write the form of theelectron density in O(X'). Denoting this quantity

5n(q), we can use Eq. (14}and the rule {16}to de-duce that

5n(q) = 5n~(q)+ 5n""n(q)+ 5no(q)+ 5n'(q), (42}

where all terms are electron densities of O(X') andlinear in E. 5ns(q} and 5n""n(q) arise from thefirst and second terms, respectively, on the right-hand side of Eq. (14). Both of these terms repre-sent densities which would be set up in an indepen-dent-particle picture (no self-consistent screeningpotential). 5ns(q) is easily found from the expres-sion given by KL" for [ p, ex E] to O(&'). 5n"" (q)is the RRD electron distribution. It is obtained byusing f~ from Eq. (24) in place of the off-diagonalelements of f in the second term on the right-handside of Eq. (14) and by using the O(X) correction tothe diagonal elements of f when they appear in thesecond term. [There are no such O(X) correctionsto f- if we choose VO. =O. ] 5no(q) arises fromusing f~ and f~ from~Eqs. (23) and (25) in placeof f in this second term of Eq. (14). We also in-clude in 5no( q) the contribution from the last termof Eq. (14) when the off-diagonal matrix elementsof p in O(A) are used along with the values of V'previously calculated in O(A). Finally 5n~(q)arises from using p„- and p-,, for the diagonal ele-ments p-„-, and p„-, , respectively, in the last termof Eq. (14).

In Eq. (42), 5n~( q) is the self-consistent screen-ing contribution in O(X') since it arises from thatpart of V'(q) which is O(X') in the last term of Eq.(14). Denoting this O(X') potential by 5V'(q), thePoisson equation (18) implies that 5V'(q)=4ve'5n(q}/q'. We can now eliminate 5n~(q) inEq. (42) by combining the Poisson equation and the

R. S. SORBELLO AND BASAB DASG UPTA

relation On'(q) =q'[1 —e(q)]&V'(q)/4~&'. In an»ogywith Eq. (29), we obtain

en(q)=[5nE(q)+5n""n(q)+5no(q))/e(q). (43)

gn( q ) = canaan( q }/~ (q}, with the RRD now givingthe total contribution in O(&').

D. Carrier4ensity modulation effect

All the terms on the right-hand side are knownin terms of explicit k-space integrals. Unfortu-nately, these integrals are too complicated to beevaluated analytically. Here me settle for a dis-cussion of the effect of the screening factor q(q) in

Eq. (43). Note that when 5n(q) acts on a bare po-tential V,(q}, the e(q) factor in Eq. (44} will screenVo(q). The net effect will be that 5ns, 5n""n, and5no will act on the screened potential V'(q) = V,(q)/&(q) [this can be seen directly from Eqs. (36} and

(43) (Refs. 43 and 44)]. As a result of this screen-ing, the force contributed by the RRD on Rn im-purity can be incorporated as a higher o-rder cor-rection to the pseudopotential V'(q) in the lower-order expression (37), just as suggested by3chaich. "

Now if 6n"an(x) were localized in a finite regionof space around the impurity, the screening im-plicit in the e(q) factor in Eq. (43) would preventthe creation of any long-range macroscopic polari-zation field. Instead, the RRD field mould be ef-fectively screened out mithin a distance of the or-

' der of a screening length from the region in which5n"" (x}were localized. " This would seem tocontradict Landauer's picture'4 "'"which has theRRDs centered orl eRch impurity Rnd glvlng x'lse toRn Rvel"Rge polRx'lzRtlon field in RQRlogy withclassical dielectric theory. Landauer's picture isvalid, however, because 5n"" (x) is not localizedin a finite volume, but rather has the, long-rangeasymptotic form4' E x/x'. The potential due tothis extended electron density is not effectivelyscreened out at large distances by the q(q) factorin Eq. (43).

Since 5n"" (x) is not a localized distributionclose to the impux"ity, estimates" which treat theRRD field at the origin as if it mere caused by alocalized dipolar distribution may be inaccurate.~ithin R Fermi- ThonlRs approximation, mhex'e

5V'(x) ~ 6n(x), it is legitimate to calculate theaverage Dlacl"oscoplc field due to Otal by ple-tending that 5V'(x) is caused by a very localizeddipolar charge distribution. This Fermi- Thomaspicture is consistent with Landauer's analysis. "

Finally me remark that in the k~l-~ limit of Eq.(43); the "electron-wind" related terms are domi-nant because they involve f-„, which is proportionalto r. In this limit 5n~(q) can be ignored, and5n (q) reduces to the nonlinear corrections forthe self-consistent-field response to the "elec-tron-mind" charge. Ignoring these nonlinearscreening corrections within 5n (q), we have

Ipp

[p cxaE] — gcE ~ (44)

where ~=(exp[(e~ —&z)/ksT]+ I}' is the Fermi-Qirm: distribution appropriate to a new Fermi en-ergy E~ =&~ —V'gg measured from the band bottom.This shift of the Fermi surface is precisely whatone would expect if the extra carriers introducedby the impurities were directly put into the con-duction band. It follows that to O(X) the diagonalelement f~ is given by Eq. (21) with p~ replaced by

pr in that equation. This change in fz representsan added current which leads to a conductivity in-crease (4n/n)e, exactly as predicted from theCDM effect by Landauer" to lowest order in Z.

'The CDM contribution to the local field EL isdifficult to isolate. We have just seen that CDMeffects on the conductivity are built into our equa-tions, through an effective shift of z~. In highexorders of &, further corrections to E~ will occur.These e~ corrections affect f&, and will begint:o affect fpt, in O(X') within the RRD contribution.

Landauer has pointed out" that the local field isinfluenced by carrier-density modulation (CDM).The CDM effect arises from local changes in theelectron density caused by the introduction of theimpurity. The density affects the local field bymaking it easier or harder to overcome the latticescatter'ing. According )o Landauer, "the CDMeffect would show up as a deviation from Matthies-sen's rule by increasing the number of carriersavailable for conduction. 'The increase in conduc-tivity, to lowest order in the impurity scatteringstrength, would be (& /n)no„whree hn is the num-

ber of valence electrons contributed by the irnpuri-ties and n is the number of conduction electronsoriginally present. o, is the conductivity in theabsence of the impurities. This addition to theconductivity seems to have been generally ignoredby workers in the field presumably because thereis a local pile up of &n electrons around the impu-rities and this localization would seem to disqual-ify the added carriers from the conduction pro-cess. We have found no quantum-mechanicalstudy of this question in the literature. We under-take such a study here.

We trace the CDM effect on the conductivity tothe diagonal elements of [p, ex E] in Eq. (14). Thecontribution 'to [p,cx'E]rr 1n O(X) is easily obtainedfrom the k- k' limit of expression (20)." Thisterm plus the O(X') contribution from Eq. (19) eanbe combined and expressed to O(&) as

LOCAL FIELDS IN ELECTRON TRANSPORT: APPLICATION. . .

We also suggest that some CDM-like contributionmay be contained within the term f~, of Eg. (23).This suggestion is based on the fact that the CDMcorrection to fr is precisely the diagonal elementf~-„. It is natural to expect some vestige of theCDM effect in the off-diagonal elements fg... andhence inn (j). Note also that n (j), like a CDMcontribution, depends explicitly upon the scat-tering time 7', the screening associated with thepotential 1', and the external field E. However,as we have pointed out, n (j) is more properlyinterpreted as the static screening response tothe external field. Strictly speaking, in the lat-ter interpretation ~ represents the lifetime of theelectron states and not a transport scatteringtime. However, in our model there is no distinc-tion between these two quantities since ~ entersthe i,iouville equation (12) as a universal relax-ation time valid for all perturbations.

Higher-oxder CDM contributions are still moredifficult to isolate. Furthermore, any rigorouscalculation of the CDM effect should be done out-side the relaxation-time approximation so as toallow for all interference effects between electron-phonon and electron-impurity scattering. " Weconclude that while certain CDM-like effects areidentifiable, it is generally difficult, if not impos-sible, to separate out from a quantum-mechanicalcalculation those terms associated with the semi-classical CDM effect. Fortunately, it is not nec-essary to do this.

V. CONCLUSION

We have seen that the local electric field can beexpressed exactly in terms of the electron densityaccording to Etl. (3). We have evaluated the chargedensity by solving the I.iouville equation in theweak-scattering limit. Qur calculation representsthe quantum-mechanical generalization of thesemiclassical Das-Peierls analysis, "and is thefirst correct Bosvieux-F riedel-type calcula-tion""" insofar as no ad hoe assumptions aremade concerning the static screening of the exter-nal field.

The local field, as well as the force on an im-purity, contains contributions from the externalfield, from the static screening in the presence of

the external field, and from the dynamic screeningin the presence of the electron current or "elec-tron wind. " In the k~l » 1 regime the "electron-wind" term dominates the static-screening re-sponse. There is no screening 0(Z). This is indisagreement with the Bosvieux-Friedel analysis,but it is in agreement with the semiclassical ana-lyses of Das and Peierls" and Rorschach. " Ourresult is also in agreement with the work of Kumarand Sorbello" and Sham" who used linear-re-sponse theory for an interacting electron gas. '""The contribution which we have denoted by theterm "static screening" is not present in thefinal linear-response expressions, ""but thiscontribution is negligible in the A~E» 1 regime.In general, this contribution does not necessarilyweaken the electric field at the impurity, in cer-tain situations, it may actually intensify the localfield. For this reason, it may be preferable toabandon the term "static-screening*' contributionin favox" of the somewhat less suggestive term"electrostatic" contribution.

Qur calculations have been performed in theweak-scattering limit, i. e., to lowest order inthe pseudopotential. " 'This is formally equivalentto performing the calculations to the leading orderin Z. However, it is axgued that the validity oflowest-order pseudopotential theory for impurityscattering is not limited to the Z ~1 regime. "'

We have considered contributions which arehigher order in the electron-impurity interaction.Specifically, we found that I.andauer's residual-resistivity dipoles" contribute to the local fieldnear an impurity and give rise to long-range mac-roscopic fields. These fields are not screened outbecause of the extended nature of the dipolarcharge.

Qur calculations show that I andauer's carrier-density modulation effect contributes to the localfield in higher orders of the impuxity-scatteringpseudopotential. The CDM effect contributes tothe conductivity by shifting the Fermi energy withrespect to the band bottom. The number of car-riers taking part in the conduction process is ef-fectively augmented by the number of valence elec-trons brought in by the impuxities. The resultingincrease in conductivity is in agreement with re-cent estimates by I andauer. 26

*Research sponsored by the Air Force Office ofScientific Research, Air Force Systems Command,USAF, under Grant No. AFOSR-76-3082.

~H. B.Huntington, in ASM Seminar Book on Dsffusion,edited by H. LAaronson (American Society for Metals,Metals Park, Ohio, 1974}, p. 155; Diffusion in So/ids:

Recent g)evelopments, edited by A. S. Nowick, andJ. J. Burton (Academic, New gork, 1974), p. 303.

~H. B. Huntington, Thin Solid Films 25, 265 (1975).3H. Wever, E/ectro- und ThermotransPort in Metallen

{Johann Ambrosius Barth, leipzig, 1973);J. N. Prattand R. G. R. Sellors, E/ectrotransport in Metals and

R. S. SORBKLLO AND BASAB D AS 6 U I'TA

AEEoys, Diffusion and Defect Monograph Series (Trans.Tech. SA, Biehen, Switzerland, 1973).

4B. S. Sorbello, Comments Solid State phys. 6, 117(1975).

SB. S. Sorbello, in EEectxo- and Thermo-twanspoH inMetaEs and AEEoys, edited by B. E. Hummel and H. B.Huntington {The Metallurgical Society of AINE, NewYork, 1977).

C. Bosvieux and J. Friedel, J. Phys. Chem. Solids 23,123 (1962).

~M. Gerl, Z. Naturforsch A 26, 1 (1971).,l. Turban, P. Nozieres, and M. Gerl, J. Phys. (Paris)37, 159 (1976).

SH. B. Huntington and A. B. Qrone, J. Phys. Chem.Solids 20, 76 (1961}.V. B. Fiks, Sov. Phys. -Solid State'1, 14 (1959).H. B. Huntington, Trans. AIME 248, 2571 (1969).

~2B. S. Sorbello, J. Phys. Chem. Solids 34, 937 {1973).Note the following typographical errors: In Eq. (4.3),ex e~~'R~ should read xe '~'+&. In Eq. (4.9) a fac-tor of 0+8 should multiply the term ge(2k&)2 in the de-nominator, while mk& should read mk& in that equa-tion.P. Kumar and H. S. Sorbello, Thin Solid Films 25,25 (1975).

~4L. Sham, Phys. Bev. B 12, 3142 (1975).Vf. L. Schai|(:h, Phys. Bev. 8 13, 3350 {1976);13,3360 (1976).

~SA. K. Das and B. Peierls, J. Phys. C 6, 2811 {1973).'7%. Kohn and J. M. Luttinger, Phys. Bev. 108, 590

(1957).8I,. Van Hove, Physica (Utr. ) 21, 517 {1955).H. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).

OJ. S. Langer, Phys. Bev. 120, 714 (1960).Q. Bickayzen, in I.ectuxes Notes on the Many Body~owem, edited by C. Frensdal (Benjamin, NewYork, 1961).

22Q. V. Chester, Proc. Phys. Soc. Lond. 81, 938 {1963);Bep. Prog. Phys. 26, 411 (1963).

23B. Landauer and J. %. F. %oo, Phys. Bev. B 5, 1189(1972).B. Landauer and J. %'. F. %'oo, Phys. Bev. B 10, 1266(1974).

25B. Landauer, IBM J. Bes. Dev. 1, 223 (1957);28B. Landauer, Phys. Bev. B 14, 1474 (1976).27B. P. Feynman, Phys. Bev. 56, 340 (1939);B.M. Deb,

Bev. Mod. Phys. 45, 22 (1973).The localized nature of the core polarization ean beseen from the exact solution for the ground state ofthe hydrogen atom in an electric field. It is known thatthe wave function at position I is given in atomic unitsby g g) = r '~t [1—(r 4 a r )E ~ r ] exp(- v') correct to0(E). (This is easily verified by substitution into theSchrodinger equation. ) The density

~ () (f) ~gives rise

to a classical electrostatic force -eE on the nucleus,as expected.

2 The net force on the nucleus of an isolated ion in freespace is geE. This force is produced by the sum ofthe core-polarization field and the external field. Itfollows that the field at the nucleus due to core polari-zation is (g/A —l)E, where g is the atomic number.

3~D. Pines and P. Nozibres, The 7'heqxy of quantumIiqgids (Benjamin, New York, 1966), Vol. I, Chap. 3.S. Doniach and E. H. Sondheimer, Green's Shnctionsfor SoBd State Pkysicists '(Benjamin, Heading, Mass. ,

1974).I~The integral in Eq. {3)will generally depend on the

geometry even for a very large metal sample sincethere are geometry-dependent surface charges at themetal-electrode inter&. ce at the sample boundary.This means that unless some boundary condition isprescribed, the value of the integral is ill de6ned inthe limit 0 ~, Where 9 is the sample volume. Theboundary condition we are assuming is the no-screen-ing condition discussed earlier, i.e., we are allowingno surfac@ polarization-charge at the sample boundaryas 0 ~. This is consistent with the periodic bound-ary conditions of Ref. 17 and the transverse fieldassumed in Ref. 31.An advantage of visualizing the transport problem ina finite toroidal geometry, is that the hermiticity ofoperators is preserved. It has been pointed out byBorsehach (private communication) that fax the scat-tering problem in an infinite medium, there are cor-rections to equations of motion due to the non-Her-miticity of X with respect to scattering states. {Thereare contributions from incident and scattered wave-functions at infinity. ) This difficulty does not existfor a finite system such as the toroidal model, orof course, the real physical circuit. For the latter,Xis Hermitian but generally very complicated due tosuch eternal details as the battery and connectingwires. One can argue, however, that there is no dir-ect intexaction in Xbetween the test particle at X andthe external complications. This implies that thereare no direct contributions from the latter in the equa-tion of motion from which Eq. (2) is derived. Oneagain arrives at Eq. (2) but now s ()() is to be computedin the presence of the full Hamiltonian X. One canthen argue that the external complications do not affects ()I) near the impurity and arrive at Eq. (2) withoutexplicitly considering the external complications.H. Landauer, Phys. Bev. B (to be publiShed.

35See Bef. 17, Appendix C. Note that although Hef. 17deals explicitly with p within Boltzmann rather thanFermi-Dirac statistics, the results derived in Bef.17 are also valid for the latter case. This is dis-cussed in Appendix F of Hef. 17.J. Lindhard, Kgl. Danske Mat. Fys. Medd. 28, 8(1954).

3~N. %. Ashcx"oft, Phys. Lett. 23, 48 (1966).@%e are here assuming that the interaction V is a

local pseudopotential. It was shown in Bef. 12 thatthe nonlocal effects need not be explicitly consideredin calculating the "electron-mind" force.

398ee, for example, Bef. 30, and note that the part of& {q,(d) which is linear in cg as cg 0 is Imp (q, co).

40B. McCraw and %'. Sehaich, J. Phys. Chem. Solids 38,193 (1977).

4~H. Landhuer, Thin Solid Films 26, L1 (1975).2%'. L. Schaieh, Phys. Bev. B 13, 3350 (1976}. As isshown in this reference the HBD charge must alsocontain second-order interference contributions arisingfrom (( &~ (x)g ~ (E). In our discussion and in Eq. (42) weare considering 6n~ to be the total second-order den-sity arising from both ) P, ()I)

~

t and (t &* ()f)g, (g) terms:43See also, L. Turban and M. Gexl, Phys. Bev. B 13,

939 (1976).4 The effective screening of 5m "~occurs in Sham's

work (Hef. 14) when he follows Langer's prescription

LOCAL FIELDS IN ELECTRON TRANSPORT: APPLICATION. . . 5205

(Ref. 20) and dresses all bare interaction lines withthe dielectric function.Any localized-charge distribution 4n g) wiQ give riseto a potential whicb is screened out due to the dielec-tric screening described by q(q). For example, ifd,s (%) is point charge at x= 0, then within a Fermi-Thomas approximation the potential wQl decay likex 'exp(-x/e), where a is the screening length. Bysuperyosition, similar decaying behavior is obtainedoutside any region where be g) is localized. [TheLindhard response gives Friedel osqillations ratherthan exponential decay, of course, but for localizedhag), the iong-range dipolar fields are ettectivelyscreened out. ]

48A. K. Das and B. E. Peierls, J. Phys. C 8, 3348(1975}.

47There are no additional contributions in QP) as canbe seen directly from Eq. (C7) in Bef. 17. We are alsoassuming that qz does not change appreciably upon in-troduction of an impurity. (See E. A. Stern [Phys'.Bev. 8 5, 366 (1972)] or J. Friedel [Adv. Phys. 3, 446@&M)j for a proof.)

48H. E. Borschach, Ann. Phys. (¹Y.) 98, 70 (1976).W. A. Harrison, 9olkf Salute 7'@co~ (McGraw-Hill,New York, 1970), p. 186, and Sec. 8.6.

5~See also, B. S. SorbeQo, Phys. Rev. 8 15, 3045 (1977),and references cited therein.