PHYSICAL REVIEW B 15 AUGUST 1999-IVOLUME 60, NUMBER 7

Semiclassical derivation of the surface-resistivity formula

H. IshidaCollege of Humanities and Sciences, Nihon University, Sakura-josui, Tokyo 156, Japan

~Received 9 April 1999!

The contribution of the surface to the resistivity of metallic films is discussed in the limit of large filmthickness by a simple semiclassical approach. The resistivity formula obtained coincides with a rigoroussolution of the Boltzmann equation in the limit of large film thickness as well as with a more general resistivityexpression which we recently derived for semi-infinite metallic systems by a quantum-mechanical linear-response method.@S0163-1829~99!02432-7#

xth

ueip

. Iith-rei

is

thheoneniiv.d

p

ltzsug

cathx

soof-

f

e

o-t to

n of.

f thelassi-ticalnen-noticalrief

nee re-u-tedit.

es--

on-

es at

The resistivity of thin metallic films has been studied etensively as functions of temperature and thickness offilm.1 But it is rather recently that surface-science techniqin ultrahigh vacuum are utilized to explore the relationshbetween the resistivity and surface conditions of the filmhas been found that the resistivity of a thin metallic film wthicknessd51022103 Å is highly sensitive to surface defects such as adsorbates at low temperatures where thetive ratio of the temperature-dependent bulk resistivityreduced.2 An additional incentive to measuring surface restivity was recently given by Persson3 who proposed a simpleequation relating the adsorbate-induced resistivity toelectronic friction force. There is now a controversy on torigin of the friction force acting on inert atoms adsorbedmetal surfaces.4–6 Since quartz crystal microbalancmeasurements7 cannot separate the electronic and phonocontributions to the total friction force, independent resistity experiments may help determine the electronic friction

Most of the previous theories of film resistivity adoptesemiclassical models.8–10 Within the relaxation-time ap-proximation, the electron scattering in the bulk metal is reresented by a single parmetert, whereas the film surfaceenters the theory through boundary conditions of the Bomann equation. The only parameter characterizing theface isps , the probability that a conduction electron strikinthe surface is specularly reflected.ps may depend on theincident angles, but it is assumed that the electron is stered in off-specular directions completely diffusively wiprobability 12ps . With these assumptions, an analytical epression of the film resistivity can be obtained from thelution of the Boltzmann equation. Especially, in the limitlarge film thickness (d@vFt), the contribution of each surface to the film resistivityrs is given by

rs53m* vF

16nee2d

^12ps&, ~1!

which was first derived by Fuchs9 for the simplest casewhereps is constant. In Eq.~1!, ne is the number density oconduction electrons,m* is the effective mass,vF is theFermi velocity, and 12ps& is the probability of off-specularscattering averaged with a weight functiong(V)5cos2 f sin2 u cosu over all the incident electrons on thFermi surface, i.e.,

PRB 600163-1829/99/60~7!/4532~3!/$15.00

-es

t

la-s-

e

c-

-

-r-

t-

--

^12ps&5

E0

2pE0

p/2

dVg~V!~12ps!

E0

2pE0

p/2

dVg~V!

, ~2!

where (u,f) are the polar and azimuthal angles of the mmentum of an electron incident on the surface with respecthe surface normal.

Recently we derived a quantum-mechanical expressiothe surface resistivity for semi-infinite metallic surfaces11

Our formula is reduced to Eq.~1! if the probability of off-specular scattering at the surface is really independent oscattering angles as was assumed in the previous semiccal models. To our knowledge, there has been no theorework on film resistivity in which the Boltzmann equatiowas solved by taking into consideration the angular depdence of the off-specular scattering. Thus, at present, it isclear whether the semiclassical and quantum-mechantreatments agree even in such general cases. In this BReport, we show that in the limit of large film thickness, ocan derive essentially the same expression of the surfacsistivity as that in Ref. 11 from simple semiclassical argments. Therefore there is no need to perform a complicatask of solving the Boltzmann equation at least in this lim

Consider a metallic film of thicknessd ~see Fig. 1! andsuppose that thez axis is perpendicular to the surface planof the film. In the equilibrium, the electron distribution function is the Fermi-Dirac functionf 0 . We assume that the energy of a conduction electron with the crystal momentump5(pi ,pz) is given bye5upu2/(2m* ). We apply a dc electricfield Ex in the x direction. Without the film surface, theFermi surface is shifted in the momentum space by a c

FIG. 1. Metallic film with thicknessd. Thez axis is perpendicu-lar to the surface planes, and the electric field is applied in thxdirection. Electrons are scattered partly in off-specular directionthe surface.

4532 ©1999 The American Physical Society

r

e

id

n, tag-

um

g

seth

ng

acud

r

ita

df-

e

iten

bil-ing

he

allel

ear-

re-ne

ro-of

re-

tax-

re-i-the

e

PRB 60 4533BRIEF REPORTS

stant Dpx5teEx in the x direction. The resultant lineachange in the electron distribution function is

f 152] f 0

]eteExvx , ~3!

wherev is the electron velocityv5p/m* . The film surfaceinduces an additional term inf 1 , which depends on both thmomentum and space coordinates.8–10 This term is of theorder ofvFt/d when averaged in thez direction and may beignored in the subsequent discussion as far as one consthe limit of large film thickness.

In the equilibrium, the total momentum of conductioelectrons is conserved at the surface. On the other handmomentum balance is disturbed when an electric field isplied. LetDN be the number of conduction electrons impining on the top surface of the film in a short-time intervalDt.We consider only the contribution fromf 1 . Then,

DN52Dt

d (p

vzf 1 , ~4!

where the prefactor 2 accounts for electron spin and the smation is taken over conduction electrons withvz.0. At lowtemperature,2] f 0 /]e in Eq. ~3! may be approximated byd(e2eF), whereeF is the Fermi energy. Then, performinthe integration with respect topz yields

DN5teEx

p\Dt(

pi

vx , ~5!

whereupiu<m* vF . First, we consider a most simplified cawhere the electron is completely diffusively scattered atsurface with probability 12ps . ps itself may be a functionof the incident angles. Then, the crytal momentum chaassociated with thoseDN electrons is

DPx5teEx

p\Dt(

pi

vx@m* vx~12ps!#, ~6!

whereps is the specular parameter for the electron witheFand the parallel crystal momentumpi . DPx does not vanishand results in a net forceFx5DPx /Dt acting on the elec-trons. Incidentally, due to the action-reaction law, the surfatoms should experience a force in the direction of the crent density, which is known as electromigration winforce.12,13 Its magnitude should equalFx when summed oveall the surface atoms.

The work done byFx on conduction electrons per untime should be identified with the additional energy dissiption due to the surface resistivity. Thus,

W5FxDvx5@Sd#rsJx2 , ~7!

where S is the area of the surface,Dvx5teEx /m* is thedrift velocity of the center of gravity of the electrons, anJx5nete2Ex /m* is the current density in the film. The efects of the surface toJx are of the order ofv ft/d and maybe ignored in the limit of large film thickness. It is to bnoted that Eq.~7! can also be written as

ers

hisp-

-

e

e

er-

-

f x

Jx5neers , ~8!

where f x5Fx /(Sd) is the averaged atomic force per unvolume. This relationship was known for a long time in ththeory of electromigration for impurity atoms in the electrogas.12,13

From Eqs.~6! and ~7!, one has

rs5~m* !2

p\ne2e2@Sd# (pi

vx2~12ps!. ~9!

Using (pivx

2(12ps)53p\nevFS^12ps&/(16m* ), we findthat Eq.~9! coincides exactly with Eq.~1!.

Next we consider a more general case where the probaity of off-specular scattering depends on the scatterangles. We introduce a parameterp(e,pi ,pi8), the probabil-ity that a conduction electron incident from the interior of tmetal with energye and the parallel crystal momentumpi iselastically scattered at the surface and changes its parmomentum topi8 . Then, Eq.~6! is replaced by

DPx5teEx

p\Dt (

pi ,pi8vx@m* ~vx2vx8!p~eF ,pi ,pi8!#.

~10!

Following the same argument as before, we obtain

rs5~m* !2

2p\ne2e2@Sd# (

pi ,pi8~vx82vx!

2p~eF ,pi ,pi8!, ~11!

where we used the identity,

(pi8

p~eF ,pi ,pi8!5(pi

p~eF ,pi ,pi8!51. ~12!

Equation~11! coincides exactly with the expression of thsurface resistivity based on a quantum-mechanical lineresponse approach in Ref. 11. The factor (vx82vx)

2 in Eq.~11! indicates that conduction electrons contribute to thesistivity of the system if their velocities in the surface plaare changed when there are scattered at the surface.

In Ref. 11 no electron-scattering mechanism was intduced in the interior of bulk metal. Thus, the conductivitythe metal diverged in the dc limit as 1/v. In order to avoidthis inconvenience, we applied an ac electric field with fquencyv and took the limit ofv˜0 in extracting the sur-face resistivity Eq.~11!. The present derivation signifies thaEq. ~11! holds true in more general cases where the relation time in the bulk metalt is finite.

In summary, we derived an expression of the surfacesistivity in the limit of large filmthickness by a simple semclassical method. The expression obtained coincides withquantum-mechanical one which we derived previously.

This work was supported by a Grant-in-Aid from thMinistry of Education, Science, and Culture of Japan.

-

a

tss

ys

.

4534 PRB 60BRIEF REPORTS

1For experimental data, seeNumerical Data and Functional Relationships in Science and Technology, edited by O. Madelung,Landolt-Bornstein, New Series, Group III, Vol. 15, Part~Springer, Berlin, 1982!.

2For a review, see D. Schumacher,Surface Scattering Experimenwith Conduction Electrons, Springer Tracts in Modern PhysicVol. 128 ~Springer, Berlin, 1993!.

3B. N. J. Persson, Phys. Rev. B44, 3277~1991!; J. Chem. Phys.98, 1659~1993!.

4M. Cieplak, E. D. Smith, and M. O. Robbins, Phys. Rev. B54,8252 ~1996!.

5B. N. J. Persson and A. Nitzan, Surf. Sci.367, 261 ~1996!.6M. S. Tomassone, J. B. Sokoloff, A. Widom, and J. Krim, Ph

Rev. Lett.79, 4798~1997!.

.

7E. T. Watts, J. Krim, and A. Widom, Phys. Rev. B41, 3466~1990!; A. Widom and J. Krim, Phys. Rev. E49, 4154~1994!; J.Krim, D. H. Solina, and R. Chiarello, Phys. Rev. Lett.66, 181~1991!.

8E. H. Sondheimer, Adv. Phys.1, 1 ~1952!.9K. Fuchs, Proc. Cambridge Philos. Soc.34, 100 ~1938!.

10M. S. P. Lucas, J. Appl. Phys.36, 1632~1965!.11H. Ishida, Phys. Rev. B52, 10 819~1995!; 54, 10 905~1996!; 57,

4140 ~1998!.12V. B. Fiks, Fiz. Tverd. Tela~Leningrad! 1, 16 ~1959! @Sov. Phys.

Solid State1, 14 ~1959!#; H. B. Huntington and A. R. Grone, JPhys. Chem. Solids20, 76 ~1961!.

13For a review, see J. van EK and A. Lodder,Defect and DiffusionForum ~Scitec, Switzerland, 1994!, Vols. 115 and 116.