Surface resistivity induced by isolated atoms and atomic clusters on metallic surfaces

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

~Received 17 May 1996!

We investigate the surface resistivity induced by isolated atoms and atomic clusters adsorbed on semi-infinite flat jellium substrates by a first-principles density-functional calculation. For this purpose, the originalexpression of the surface resistivity derived in a recent work@Phys. Rev. B52, 10 819~1995!# is recast in aform suitable for surface Green function calculations based on the Dyson equation. For single atoms anddiatomic molecules, we study the dependence of the adsorbate-induced resistivity on the adatom-substratebond length and the adatom-adatom bond length. In addition, we perform extensive resistivity calculations foratomic clusters consisting ofn atoms in various atomic configurations (n<5) in order to clarify the depen-dence of the adsorbate-induced resistivity on the size and shape of atomic clusters.@S0163-1829~96!00139-7#

I. INTRODUCTION

Adsorbates on metal surfaces scatter conduction electronsimpinging on the surface from the interior of the metal andchange their momentum in the plane. A result of this scat-tering process is the increase in electrical resistivity. At lowtemperatures (;10 K!, the resistivity of a thin metallic filmwith thicknessl f51022103 Å is increased by;10% byadsorbates at less than monolayer coverages.1 If each adsor-bate works as an independent scattering center, theadsorbate-induced resistivityrs may change in inverse ratioto the volume per adsorbate as in the case of the impurity-induced residual resistivity. This implies thatrs increaseslinearly at low coverages and thatrs at a fixed coveragescales as 1/l f . These behaviors both have been confirmedexperimentally.1 Persson2 derived a simple relationship be-tween the adsorbate-induced resistivity and the electron-holepair contribution to the lifetime of the parallel vibrationalmotion of adsorbates,t, which is one of the key parametersgoverning adsorbate diffusion and reaction processes. Hisrelationship is

1

t5ne2e2

Mna@ l frs#, ~1!

wherene stands for the electron density in the metal,na isthe two-dimensional number density of adsorbates, andM isthe adsorbate mass. By analyzing experimental data, heshowed that~1! is useful for estimatingt from the measuredresistivity change.2 It was also shown that the adsorbate-induced resistivity is related with the ‘‘electron-wind’’ forceexerted on adsorbates located in current densities by a simpleequation.3

As the diffusive motion of adsorbates is limited at lowtemperatures, the distance between neighboring adsorbates ata given coverage may distribute in a certain range. Thus, inorder to be able to utilize resistivity measurements as a quan-titative method of surface spectroscopy, it is necessary toclarify fundamental questions such as to what extent theadsorbate-induced resistivityrs varies as a function of thenearest-neighbor distance between adsorbates and how muchrs would differ from the sum ofrs of a single atom if several

atoms form a cluster on the surface. These questions aredifficult to answer experimentally, since the resistivity mea-surement outputs only an average over all the adsorbates.

The purpose of the present paper is to address these ques-tions based on a first-principles density-functional calcula-tion. We evaluate the adsorbate-induced resistivityrs of at-oms and atomic clusters adsorbed on semi-infinite metalsubstrates and investigate the dependence ofrs on theadatom-substrate bond length, the adatom-adatom distance,and the size and shape of atomic clusters. For simplicity, thesubstrate is modeled by a semi-infinite flat jellium surface.4

The ground-state electronic structure is calculated self-consistently with the local density approximation5 where thepotential perturbation induced by the adsorbate is treatedwith the use of a surface Green function technique based onthe Dyson equation.6 In the present work, we treat the zero-temperature limit and do not consider inelastic scattering ofconduction electrons via adsorbate vibrational excitations.As in the case of the impurity resistivity, the adsorbate-induced resistivity in this limit is determined by asymptoticbehaviors of one-electron wave functions at the Fermi en-ergy eF alone.

The plan of the present paper is as follows: recently, wederived a microscopic expression of the surface resistivity bylinear response theory.7 In Sec. II, we first rewrite this ex-pression in a form suitable for surface Green function calcu-lations. Next, we explain the numerical method to calculatethe electronic structure of isolated atoms and atomic clusterson a semi-infinite jellium surface. Section III is the main partof the present paper and contains results and discussion ofthe numerical calculations. Finally, a short summary is givenin Sec. IV.

II. THEORY

A. Surface resistivity

First, we give a short summary of our recent work7 wherea microscopic expression of the surface resistivity was de-rived. Let us consider a semi-infinite metal surface. Thezaxis is chosen as the surface normal pointing toward thevacuum. For simplicity, the interior of the metal is treated asjellium. We apply a uniform electric fieldE5(Ex ,Ey) with

PHYSICAL REVIEW B 15 OCTOBER 1996-IVOLUME 54, NUMBER 15

540163-1829/96/54~15!/10905~7!/$10.00 10 905 © 1996 The American Physical Society

frequencyv parallel to the surface. By introducing the sur-face resistivity in tensor form, the energy dissipation in thesystem per unit time is written as

P5@Slf #(a,b

rsab~v!Ja* Jb , ~2!

whereS is the area of the surface andJa5 inee2Ea /(mv) is

thea (x or y) component of the macroscopic current densityin the bulk. It was shown thatrs

ab is given by

@Slf #rsab~v!5

2v

ne2e2

Imx0~ pa ,pb ,v!

52pv

ne2e2(i , j d~e j2e i2\v!~ f i2 f j !

3^c i u pauc j&^c j u pbuc i&, ~3!

where the bare polarization function is defined by

x0~A,B,v!52(i , j

f i2 f je i2e j1\v1 id

^c i uAuc j&^c j uBuc i&.

~4!

Here,c i denotes the wave function of one-electron stateiwith energye i , and f i5u(eF2e i) is the occupation at zerotemperature. The prefactor 2 accounts for spin degeneracy.By substituting appropriate wave functions for semi-infinitegeometry in~3!, one obtains in the dc limit~hereafter weomit the indexv for the dc limit!,

@Slf #rsab5

\

2pne2e2(k,k8

@DkaDkb#p~eF ,k,k8!, ~5!

where p(e,k,k8) denotes the probability that a conductionelectron incident from the interior of the metal with energye and the parallel wave vectork is elastically scattered at thesurface and changes its parallel wave vector tok8, andDka5ka82ka . The current conservation implies(k8p(e,k,k8)51. It was shown that~5! coincides with thesemiclassical expression of the surface resistivity8 if theprobability of off-specular scattering (kÞk8) was indepen-dent of the direction ofk8. We note that the surface resistiv-ity for a flat jellium surface vanishes becausep(e,k,k8)5dk,k8 .

Next, we rewrite~5! in a form which is more suitable forevaluating the resistivity induced by an isolated adsorbate ona flat jellium surface. For this purpose, we replace the matrixelement c i upauc j& in ~3! by

^c i u@H,pa#uc j&e i2e j

5i\^c i u]Veff /]xauc j&

e i2e j, ~6!

where Veff is the self-consistent effective potential, andH52\2D/(2m)1Veff is the one-electron Hamiltonian, i.e.,Hc i5e ic i . Inserting~6! into ~3!, one obtains in the dc limit,

@Slf #rsab5

2p\

ne2e2(i , j d~e i2eF!d~e j2eF!K c iU ]Veff

]xaUc j L

3K c jU ]Veff

]xbUc i L . ~7!

For an isolated adsorbate on flat jellium substrates,]Veff /]xa is localized in the vicinity of the adsorbate. Thus,rs

ab can be evaluated from the local electronic structure nearthe adsorbate without the knowledge of asymptotic behaviorsof scattered electron wave functions.

Here we discuss the relation between the surface resistiv-ity and the frustrated translation of an atomic cluster on jel-lium. We consider a parallel vibrational mode where all theconstituent atoms in the cluster are assumed to oscillate inthe a direction without changing their relative coordinates.Then, one can show that the electron-hole pair contributionto the lifetime of this parallel vibration is given by9

1

ta52p\

M (i , j

d~e i2eF!d~e j2eF!U K c iU(i51

Na ]Veff

]Rai Uc j L U2,

~8!

whereRai denotes thea coordinate of thei th constituent

atom of the adsorbate (1< i<Na). Using ~7!, ~8!, and therelation

]Veff

]xa52(

i51

Na ]Veff

]Rai , ~9!

which holds for aflat jellium substrate, one immediately ob-tains the relationship proposed by Persson,2 i.e.,1/ta5ne

2e2@ l frsaa#/(Mna), wherena51/S.

B. Surface Green function formulation

We calculate the elecronic sructure of isolated atoms andatomic clusters on semi-infinite jellium with the local densityapproximation in density-functional theory.5 The effectiveone-electron potential is written asVeff5V01DVeff , whereV0 is the potential for the unperturbed jellium surface. Thepotential changeDVeff upon adsorption is localized in theneighborhood of the adsorbate, and the Green functionG(l)51/(l2H) in this region is expanded by a nonor-thogonal basis set$fn% as

G~r ,r 8,l!5 (n,n8

Gnn8~l!fn~r !fn8* ~r 8!, ~10!

wherer5(x,z). According to the formulation of Wachutkaet al.,6 Gnn8 satisfies the Dyson equation in matrix form,

Gnn85Gnn80

1Gnm0 DVmm8Gm8n8,

Gnn85Snm21^fmuGufm8&Sm8n8

21 ,

Gnn80

5Snm21^fmuG0ufm8&Sm8n8

21 ,

DVnn85^fnuDVeffufn8&, ~11!

wherel is omitted for simplicity, and summation is impliedfor repeated indices. In~11!, G0 denotes the Green functionof the unperturbed substrate, andS21 stands for the inverseof the overlap matrixSnn85^fnufn8&. The ground-stateelectron density is calculated from the Green function as

n0~r !521

p E eFde ImG~r ,r ,e1 id!, ~12!

10 906 54H. ISHIDA

whered is an infinitesimal positive number, and the effectiveone-electron potential is given by

Veff~r !5Vext~r !1E d r 8e2n0~r 8!

ur2r 8u1Vxc@n0~r !#, ~13!

whereVxc(n) denotes the exchange-correlation potential of auniform electron gas with densityn ~Ref. 10! andVext is theexternal potential due to the adsorbate ion cores and thebackground charge of jellium.~11!–~13! constitute a set ofself-consistent equations to determinen0 andVeff .

We choose the basis set such that the complex conjugateof each basis funcionfn is included in the same basis set.Then, with the definitionGnn8

I5@Gnn82(Gn n8)* #/(2i ),

wheref n5fn* one has

(i

d~e i2e!c i~r !c i* ~r 8!521

pImG~r ,r 8,e1 id!

521

p (n,n8

Gnn8I

~e1 id!

3fn~r !fn8* ~r 8!. ~14!

By substituting~14! in ~7!, we obtain

@Slf #rsab5

2\

pne2e2

Tr@GI~eF1 id!DVaGI~eF1 id!DVb#,

~15!

where (DVa)nn85^fnu](DVeff)/]xaufn8&. ~15! is used inthe next section for the numerical evaluation of theadsorbate-induced resistivity.

In the present work, the adsorbate ion cores are repre-sented by norm-conserving pseudopotentials in theKleinman-Bylander form.11 As for its nonlocal partVnl , thepotential derivative with respect to the electron coodinatexa appearing in~15! should be replaced byi @ pa ,Vnl#/\.

C. Basis functions

In the present work, we employ a plane-wave-like basisset,

fg,n~r !5A2

Vexp~ ig•x!sin~pnz!, ~16!

where g5(2pnx /Lx ,2pny /Ly) (nx ,ny50,61, . . . ),pn5pn/Lz (n>1), andV5LxLyLz . Correspondingly, thecharge density is expanded as

n0~r !5 (g,n~>0!

n0~g,n!exp~ ig•x!cos~pnz!. ~17!

A similar basis set was used previously in the electronicstructure calculation of atomic overlayers3 based on the em-bedding method of Inglesfield.12 We emphasize, however,that in the present work we treat a truly isolated adsorbate ona semi-infinite substrate and do not utilize super-cell tech-niques in any direction. The basis set~16! could express onlyfunctions that are periodic in the plane and vanish at thelower and upper edges (z50, Lz) if it was used in the boxwith volumeV. In order to be able to represent the Green

function whose boundary condition varies with energy, thebasis functions are actually used in a smaller box:uxu<Lx8/2, uyu<Ly8/2 and b1<z<b2, where Lx8,Lx ,Ly8,Ly and 0,b1,b2,Lz . Lx8, Ly8, andb22b1 should belarge enough so thatDVeff(r ) may be negligibly small out-side this smaller volumeV85Lx8Ly8(b22b1). Needless tosay, the change density~17! is physically meaningful in thesame smaller box.

The merit of the present basis set is that one can utilizenumerical techniques developed for standard plane-waveband structure calculations. For example,n0(r ) defined by~17! is periodic in the box with volume 2V5LxLy(2Lz).Thus, using the fast Fourier transform~FFT!, one can con-vertn0(g,n) to the charge density on real-space mesh points.Then, evaluating the exchange-correlation potential on eachmesh point and performing an inverse FFT, one obtains eas-ily Vxc(r ) expanded in the same form as~17!.

III. RESULTS AND DISCUSSION

In this section, we use the Hartree atomic units withm5e5\51. We present numerical results for isolated atomsand atomic clusters on semi-infinite jellium withr s52.07a.u., which corresponds to the valence electron density of Al.

For adsorption of single atoms, the parameters are chosenas b151.5, b2512.5, Lz514, Lx5Ly512, andLx85Ly8510a.u. For adsorption of the largest cluster treated in the presentwork, Al 5, they areb151.5, b2516.5,Lz518, Lx5Ly518,andLx85Ly8516 a.u. The cutoff energy for the basis set~16!is taken as 6.25 Ry. With this value, the number of basisfunctions amounts to;1500 for Al5. The iteration proce-dure toward self-consistency is continued until the meansquare difference between the input and output charge den-sities becomes less than 1026 a.u.

A. Single atom

Figure 1 shows the calculated adsorbate-induced resistiv-ity rs of isolated Mg, Al, and Si atoms on jellium as a func-tion of the adatom-substrate distancedas5Rz2zj . Here,zjdenotes thez coordinate at which the positive backgroundcharge of the jellium substrate drops fromne to 0. For thepresent case, the resistivity tensor~15! is diagonal and iso-

FIG. 1. Surface resistivity induced by single Mg, Al, and Siatoms on jellium withr s52.07 a.u. as a function of the adatom-substrate bond lengthdas5Rz2zj .

54 10 907SURFACE RESISTIVITY INDUCED BY ISOLATED . . .

tropic. As stated in the Introduction, the adsorbate-inducedresistivity scales in inverse ratio to the volume per adsorbate@Slf #. If we adopt the conventional units for impurity resis-tivity where @Slf # is taken for convenience as 100 times theatomic volume of Al in bulk fcc Al, 100 a.u. in Fig. 1 is readas 0.19mV cm/at. %. The calculated resistivity in Fig. 1 isof the same order as measured values, for example, 0.660.2mV cm/at. % for monolayer Cu on Cu~111! ~Ref. 1!and 1.160.1mV cm/at. % for monolayer Au on Au~111!.13

Persson2 defined the effective cross section for diffusiveelectron scattering,S, by

S516nee

2

3mvF@Slf #rs , ~18!

wherevF is the Fermi velocity. In the present system, 100a.u. in Fig. 1 corresponds toS54.33 Å2. For example, thecross section for Si at the equilibrium bond lengthdas52.3a.u is 14.8 Å2.

Toward the interior of the metal, the calculatedrs gradu-ally approaches the resistivity of a single impurity atom em-bedded in a uniform electron gas with the samer s value. ForSi, the 3s resonance shifts downward with decreasingdasandbecomes a discrete level below the bottom of the jelliumbands atdas;0. Apparently, this change in the electronicstructure has no significant effects on the scattering ampli-tude of one-electron wave functions ateF . The calculatedrs of Si exhibits no singularity around thisdas value.

Toward the vacuum, the adsorbate-induced resistivitytends to zero. It decreases more slowly than the ground-statecharge density profile. For Mg, the small decay rate may beattributed to a large orbital size of Mg 3s. On the other hand,it is a little puzzling that there is an appreciable differencebetween Al and Si. The valence orbitals of Al are slightlymore extended than those of Si. Nevertheless, the adsorbate-induced resistivity of Al in Fig. 1 decays more rapidly thanthat of Si. Based on a Newns-Anderson-type model Hamil-tonian, Persson2 derived a simple expression of theadsorbate-induced resistivity,

@Slf #rs52meF\ne

2e2^sin2u&Gra~eF!, ~19!

whereG is the width of the adsorbate resonance,ra(e) is thedensity of states projected on the adatom orbital, and an av-erage^sin2u& defined in Ref. 2 is a moderate function ofdas. Figure 2 shows the adsorbate-induced density of statesof single Al and Si atoms for threedasvalues. Here the originof the energy is taken as the bottom of the jellium bands, andthe Fermi energy is 0.43 a.u. The density of states was cal-culated using the generalized phase shift14 as

ra~e!522

p

d

deIm$ lndet@12G0~e1 id!DVeff#%. ~20!

With increasingdas, the Si 3p resonance becomes sharperand shifts upwards. As a result, the density of states of Si atthe Fermi energy rises steeply asdas is increased from 1–3a.u. ~19! indicates that the increase inra(eF) partly cancelsthe exponential decay ofG with increasingdas. For Al, thevariation of ra(eF) in the samedas range is much smaller.

This explains why the adsorbate-induced resistivity for Aldecreases toward the vacuum more rapidly than that for Si.

It should be noted that the surface resistivity induced bysingle atoms shown in Fig. 1 is much larger than the resis-tivity per atom of ordered atomic overlayers at high cover-ages. Recently, we evaluated the overlayer resistivity of one-monolayer Al~001! on jellium with r s52.07 a.u.~Ref. 7!.Fordas51.91 a.u., we obtainedl frs51.9 a.u. per unit surfacearea. By multiplying the area of the unit cell of Al~001!, 29.3a.u., we obtain 55.7 a.u. as the surface resistivity per atom.This value is 3.8 times smaller than the resistivity of a singleAl atom with the samedas, 213 a.u. shown in Fig. 1.

B. Atomic clusters

Next, we study the dependence of the adsorbate-inducedresistivity on the adatom-adatom distancedaa. For this pur-pose, we consider a diatomic molecule Al2 adsorbed on aflat semi-infinite jellium surface with varying Al-Al bondlengths. In order to be able to focus on the dependence ofrs on daa, the adatom-jellium distancedas is chosen asdas51.91 a.u. in this subsection, which corresponds to halfthe layer spacing between adjacing Al~001! layers. Figure 3shows contour maps of the ground-state chargen0(r ) ofAl 2 on a vertical cut-plane containing Al nuclei for threevalues ofdaa. The solid circles indicate the positions of Alnuclei and the origin of thez coordinate is chosen as thejellium edge. Atdaa510.5 a.u., the charge density distributesitself almost independently around each atom, whereas, atdaa54.5 a.u., there appears a bonding charge with a largeamplitude between the two atoms due to the strong Al-Alchemical interaction.

The solid lines in Fig. 4 show the calculated adsorbate-induced resistivity of Al2 as a function ofdaa. For thepresent case, the diagonal elements ofrs

ab are nondegener-ate. The solid circles and squares arers

xx and rsyy , respec-

tively, where thex axis is chosen as the molecular bond axis.The cross on the vertical axis indicates two timesrs of a

FIG. 2. Adsorbate-induced density of states for single~a! Al and~b! Si atoms on jellium withr s52.07 a.u. Solid, dashed, and dottedlines correspond todas53, 2, and 1 a.u., respectively.

10 908 54H. ISHIDA

single Al atom on jellium with the samedas. The calculatedresistivity of Al2 reaches this value atdaa;10 a.u. Withdecreasingdaa, the calculatedrs starts to deviate from theasymptotic value because of three reasons:~1! the multiplescattering of conduction electrons involving the two atomicsites, ~2! the interference of scattered electron waves fromthe two sites, and~3! the change in the scattering potentialDVeff due to the strong Al-Al chemical ineraction, whichmay be significant only at shortdaa.

Here, we demonstrate the importance of the multiple scat-tering of conduction electrons in adsorbate-induced resistiv-ity. The one-electron wave function of the unperturbed flatjellium surface is written as

c0~e,k,r !51

ASexp~ ik•x!f@kz~e,k!,z#, ~21!

where kz(e,k)5A2me/\22uku2, and f(kz ,z) takes anasymptotic form 2sin@kzz2g(kz)#/kz in the interior of themetal. With this normalization of the wave function, the scat-tering probabilityp(e,k,k8) in ~5! is calculated as

p~e,k,k8!5Rekz~e,k!Rekz~e,k8!ur ~e,k,k8!u2, ~22!

r ~e,k,k8!5idk,k8kz~e,k!

1m

\2E dr dr 8c0* ~e,k8,r 8!t~e,r 8,r !c0~e,k,r !,

~23!

where thet matrix is defined by

t~e,r ,r 8!5DVeff~r !d~r2r 8!

1DVeff~r !G~e1 id,r ,r 8!DVeff~r 8!. ~24!

In the Born approximation, the second term of~24! which isresponsible for all the intra- and interatomic multiple-scattering processes is omitted. The expression of theadsorbate-induced resistivity in this approximation is ob-tained simply by replacingGI in ~15! by a correspondingquantity for the jellium Green functionG0

I .The dashed lines in Fig. 4 show the calculatedrs of

Al 2 in the Born approximation as a function of the Al-Albond lengthdaa. At daa;10 a.u. where the intra-atomic mul-tiple scattering is dominant, the Born approximation repro-duces;50% of the resistivity in the full Green functioncalculation. With decreasingdaa, rs

xx in the Born approxima-tion decreases monotonously, while the correspondingrs

yy

increases slightly. For relatively largedaa, the scattering po-tentialDVeff for Al 2 may be approximated by the superpo-sition ofDVeff of two independent atoms. Then, without hav-ing multiple-scattering terms, the variation of the resistivitywith daa in the Born approximation arises solely from theinterference of scattered electron waves from the two atomicsites, i.e., the scattering probabilityp(e,k,k8) forAl 2 is given byu11exp@i(kx82kx)daa#u2 times that of a singleatom. This phase factor together with (kx82kx)

2 in ~5! leadsto the larger variation ofrs

xx as a function ofdaaas comparedwith rs

yy . Similar phase factors containingkx and kx8 alsoappear in the multiple-scattering terms and may be respon-sible for the larger variation ofrs

xx thanrsyy in the full Green

function calculation. At shortdaa, the discrepancy betweenthe Born approximation and the full calculation is especiallylarge for thexx component. This means that the adsorbate-induced resistivity can be largely enhanced by the inter-atomic multiple scattering when the current flows in the di-rection of the molecular axis.

FIG. 3. Contour maps of the charge density of Al2 on jelliumwith r s52.07 a.u. on a vertical cut-plane containing Al nuclei. Solidcircles indicate the positions of Al nuclei and the origin of thez axisis the edge of the positive charge of jellium. The contour spacing is0.005 a.u.~a! daa510.5, ~b! daa57.5, and~c! daa54.5 a.u.

FIG. 4. Surface resistivity induced by diatomic molecule Al2 onjellium with r s52.07 a.u. as a function of the Al-Al bond lengthdaa ~solid lines!. Solid circles and squares correspond to thexx andyy components, respectively. Dashed lines indicate the resistivity ofAl 2 calculated with the Born approximation.

54 10 909SURFACE RESISTIVITY INDUCED BY ISOLATED . . .

One may define the effective scattering radius of a singleadatom byr a5AS/p. From Fig. 1, the radiusr a for Al atdas51.91 a.u. is estimated as 3.26 a.u. Naively thinking, onewould expect that the resistivity of Al2 would be constant fordaa>2r a and decrease when the scattering cross sections ofthe two atoms start to overlap (daa<2r a). Figure 4 indicatesthat such a simplified picture actually does not hold. Espe-cially, the calculatedrs of Al 2 increases at shortdaa.

Finally, we study the dependence of the adsorbate-induced resistivity on the size of atomic clusters. We con-sider isolated atomic clusters Aln adsorbed on semi-infinitejellium, wheren ranges from 2 to 5. Figure 5 shows chargecontour maps of Aln (2<n<4) on the horizontal plane con-taining Al nuclei. Here, the adatom-substrate distancedas is1.91 a.u. as in the case of Al2, and the Al-Al bond length ischosen as the nearest neighbor bond distance in the bulk Al,5.41 a.u.~structural optimization would be necessary if quan-titative comparison with experiments were intended!. ForAl 3, we consider both linear and triangular clusters, andAl 4 is assumed to be square. In addition to the two-dimensional clusters shown in Fig. 5, we consider three-dimensional Al4 and Al5 in pyramidal configurations. Theyare made from triangular Al3 and square Al4 by adding oneAl atom on top of the hollow site formed by the other Alatoms in the first layer.

Figure 6 shows the calculated adsorbate-induced resistiv-ity of Al n as a function of the cluster sizen. The resistivitytensor is diagonal and isotropic, except for the linear Al2 andAl 3 clusters. Bothrs

xx andrsyy are plotted for these clusters.

Accidentally, for the particular Al-Al bond length chosen,the two components are nearly degenerate for linear Al3. Anadditional calculation was performed for linear Al3 with ashorter Al-Al bond lengthdaa54.5 a.u. In that case, therewas a;10% difference between thexx andyy components.Roughly speaking, the adsorbate-induced resistivity in-creases monotonously with increasingn as far as the clusteris planar. Yet, depending on the shape of the cluster, there isa considerable deviation from the independent-atom limit,n timesrs of a single atom, which is indicated by a dashedline in Fig. 6. For example, the resistivity induced by trian-

gular Al3 and square Al4 is significantly larger than the sumof the resitivity of isolated Al atoms.

As for three-dimensional clusters, one sees that the Aladatom in the second layer has very little positive contribu-tions to the adsorbate-induced resistivity. The calculatedrsof pyramidal Al5 is very close to that of square Al4, namely,the bottom layer of Al5. As for tetrahedral Al4, its resistivityis even smaller than that of triangular Al3. It may be con-ceivable that the top-layer Al atom in tetrahedral Al4 par-tially weakens the chemical bond between the substrate andAl 3 in the bottom plane, thus reducing the probability ofoff-specular scattering of conduction electrons.

IV. SUMMARY

Measurements of the surface resistivity provide useful in-formation on the electronic structure and vibrational proper-ties of adsorbates on solid surfaces. In the present work, wehave studied the adsorbate-induced resistivity of isolated at-oms and atomic clusters on flat semi-infinite jellium sub-strates at zero temperature by a first-principles density-functional calculation. For this purpose, we derived amicroscopic expression of the surface resistivity which canbe evaluated from the local electronic structure in the vicin-ity of the adsorbate alone without knowing asymptotic be-haviors of one-electron wave functions. The electronic struc-ture of isolated adsorbates on jellium was calculated self-consistently using a surface Green function technique basedon the Dyson equation. We investigated the dependence ofthe adsorbate-induced resistivity on the adatom-substrate dis-tance, the adatom-adatom bond length, and the size andshape of atomic clusters.

The measured adsorbate-induced resistivity in many ex-periments exhibited a parabolic dependence with increasingcoverage. Thus, one may have the impression that the resis-

FIG. 5. Contour maps of the charge density of planar Aln clus-ters (2<n<4) on jellium with r s52.07 a.u. on a horizontal cut-plane containing Al nuclei. The contour spacing is 0.005 a.u.

FIG. 6. Surface resistivity induced by atomic clusters Aln as afunction of the cluster sizen. The Al-Al bond lengthdaa55.41 a.u.,and the distance between the substrate and the first layer Al atoms,das, is 1.91 a.u. for all the clusters. For linear Al2 ~‘‘2 L ’’ ! andAl 3 ~‘‘3 L ’’ ! clusters, both thexx andyy components are plotted.The symbols ‘‘3T,’’ ‘‘4 S,’’ ‘‘4 T,’’ and ‘‘5 P’’ designate triangularAl 3, square Al4, tetrahedral Al4, and pyramidal Al5, respectively.The dashed line indicatesn times the resistivity of an isolated Alatom.

10 910 54H. ISHIDA

tivity per atom should be the largest for a single adatom. Thepresent calculation demonstrated that the resistivity inducedby atomic clusters can exceed the limit of independent at-oms. It was found that the interatomic multiple scattering ofconduction electrons is mainly responsible for the enhance-ment of the resistivity in atomic clusters.

ACKNOWLEDGMENTS

This work was supported by a Grant-in-Aid from theMinistry of Education, Science, and Culture of Japan. Part ofthe numerical calculation was performed at the Institute ofMolecular Science.

1For review, see D. Schumacher,Surface Scattering Experimentswith Conduction Electrons, Springer Tracts in Modern Physics,Vol. 128 ~Springer, Berlin, 1993!.

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54 10 911SURFACE RESISTIVITY INDUCED BY ISOLATED . . .