ELECTRICAL RESISTIVITY of a THIN METALLIC FILM 15 783 Cally Zero; However, This Line of Reasoning Is Simplistic: the Tum Loss As Outgoing Boundary Conditions

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12-15-1999 Electrical Resistivity of a Thin etM allic Film Horacio E. Camblong University of San Francisco, [email protected]

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Recommended Citation Camblong, Horacio E., "Electrical Resistivity of a Thin eM tallic Film" (1999). Physics and Astronomy. Paper 9. http://repository.usfca.edu/phys/9

This Article is brought to you for free and open access by the College of Arts and Sciences at USF Scholarship: a digital repository @ Gleeson Library | Geschke Center. It has been accepted for inclusion in Physics and Astronomy by an authorized administrator of USF Scholarship: a digital repository @ Gleeson Library | Geschke Center. For more information, please contact [email protected]. PHYSICAL REVIEW B VOLUME 60, NUMBER 23 15 DECEMBER 1999-I

Electrical resistivity of a thin metallic ﬁlm

Horacio E. Camblong Department of Physics, University of San Francisco, San Francisco, California 94117

Peter M. Levy Department of Physics, New York University, New York, New York 10003 ͑Received 19 July 1999͒ The electrical resistivity of a pure sample of a thin metallic film is found to depend on the boundary conditions. This conclusion is supported by a free-electron model calculation and confirmed by an ab initio relativistic Korringa-Kohn-Rostoker computation. The low-temperature resistivity is found to be zero for a free-standing film ͑reflecting boundary conditions͒ but nonzero when the film is sandwiched between two semi-infinite samples of the same material ͑outgoing boundary conditions͒. In the latter case, this resistivity scales inversely with the number of monolayers and is due to the background diffusive scattering by a finite lattice. ͓S0163-1829͑99͒00147-2͔

I. INTRODUCTION conditions, one gets unequal resistivities. In other words, the It is well known that the low-temperature electrical resis- resistivity depends on the boundary conditions. tivity of an infinite pure metal is essentially zero because the The results described above raise a number of questions Bloch waves associated with the underlying periodic struc- that we address in this paper. Our goal is to explore electrical ture undergo no diffusive scattering in the absence of impu- conduction in a pure sample of a thin metallic film and re- rities and imperfections. Then, one could ask what the cor- solve the following fundamental issues. ͑i͒ Is the resistivity responding result would be for a thin metallic film. This indeed dependent on the boundary conditions? ͑ii͒ How can question, which has not been addressed in the literature, is one understand the finite resistivity of a thin, yet otherwise the central problem tackled in this paper. Our conclusion is perfect, film? ͑iii͒ How can one measure the finite resistivity that the electrical resistivity of a thin film depends upon how of a thin film? the experimental situation is set up, namely, upon the bound- As we will see in Sec. II, a proper understanding of ary conditions; in particular, with an appropriate choice of boundary conditions is a prerequisite for a thorough discus- boundary conditions, the resistivity is not zero. sion of these questions. This analysis will be followed in The problem of the electrical resistivity of a pure sample Sec. III by the application of the Kubo formula to the case of a thin metallic film is of great current interest. It can be with outgoing boundary conditions, in Sec. IV by the math- viewed as a fundamental question for which an exploratory ematical characterization of the finite periodicity of the lat- analytic calculation may provide deeper insight as well as a tice, in Sec. V by the implications of the film finiteness on limiting test of complex ab initio methods. In fact, this ques- transport properties, and in Sec. VI by concluding remarks. tion can be posed in the context of the first-principle fully relativistic layered version of the Korringa-Kohn-Rostoker II. RESISTIVITY AND BOUNDARY CONDITIONS ͑KKR͒ method, which has been recently developed,1 to per- form ab initio computations of magnetotransport in magnetic Electrical conduction in a thin metallic film can be mod- multilayers in the coherent-potential approximation. In ef- eled by representing the film in terms of a finite number M of fect, this method can be easily applied to the computation of monolayers or perfect atomic planes arranged periodically in the electrical resistivity of a thin film consisting of a finite the direction perpendicular to the boundaries of the film. number M of monolayers of a metal, by properly specifying Moreover, a pure sample is characterized by the absence of the conditions enforced at the boundaries of the system. The impurities or imperfections; in addition, at low temperatures, first result was obtained for a thin film of copper sandwiched other resistivity sources are rendered ineffective. between two semi-infinite metallic blocks of the same mate- At first sight, one might just apply the standard folklore, rial; the resistivity was computed for a film consisting of namely, that quantum-mechanical Bloch waves in a periodic between Mϭ6 and 45 monolayers and for current in the structure propagate without electrical resistance. In effect, plane of the layers ͑CIP͒ and was found to decrease approxi- electrical resistance arises from the loss of linear momentum mately as 116/M ␮⍀ cm, for MϾ20 ML—a result that information due to diffusive scattering, i.e., scattering that clearly extrapolates to zero only as M approaches infinity. randomizes the electron’s momentum, so that the outgoing This nonzero value is somewhat unusual and led us to ask electron has no knowledge of the direction of its incoming that the computation be done under different boundary con- momentum.3 In fact, in a pure metallic sample, electrons ditions. For a free-standing slab, namely, when the copper undergo Bragg scattering, which being highly directional, is film is surrounded by vacuum on both sides, it was found ineffective as a momentum-randomizing mechanism ͑as we that the resistivity is essentially zero.2 In conclusion, when will discuss in greater detail in Sec. V͒. This would seem to one performs these two computations with unequal boundary imply that the resistivity of a perfect film should be identi-

0163-1829/99/60͑23͒/15782͑8͒/$15.00PRB 60 15 782 ©1999 The American Physical Society PRB 60 ELECTRICAL RESISTIVITY OF A THIN METALLIC FILM 15 783 cally zero; however, this line of reasoning is simplistic: the tum loss as outgoing boundary conditions. standard folklore applies only to an infinite periodic sample, Transport theory with outgoing boundary conditions is for which the electrical resistivity is indeed zero, under the based on the following ideas. Transport is described via re- conditions described above. Instead, when one analyzes a tarded Green’s functions that represent the propagation of finite film, the question ‘‘What is the resistivity of a thin electrons in the environment provided by the sample, with metallic film?’’ is immediately replaced by ‘‘Is the film re- the condition that the electrons leave the film irreversibly at ally periodic?’’ Then, from a mathematical viewpoint, the the boundaries. This implies the use of Green’s functions film fails to be strictly periodic because of the boundaries; by corresponding to an infinite medium ͑bulk͒—at this level the abuse of language, one could describe the film as exhibiting calculation does not acknowledge the finiteness of the film ‘‘finite periodicity.’’ It turns out that the finiteness of the film whose conductivity is calculated. Then, as the propagators implies the existence of background diffusive scattering in themselves do not satisfy boundary conditions that keep addition to ordinary Bragg scattering ͑see Sec. V͒, and it is track of the momentum of electrons scattered within the film, this diffusive scattering that becomes the source of a finite- the calculation yields a nonzero resistivity ͑see Sec. V͒.In size resistivity if the boundary conditions are appropriately other words, all finite conductors have self-energy terms in selected. their propagators that describe their contact with leads or The simplest boundary condition is provided by the ideal reservoirs. The resistance, which is proportional to the free-standing slab when a film with perfect boundaries is imaginary part of the self-energy, reflects the fact that an inserted in a vacuum. Then, the electrons undergo specular electron in a finite conductor will eventually leak out into the reflections at the boundaries ͑with infinitely high potentials leads attached to it.6 representing the onset of a vacuum͒ and effectively ‘‘probe’’ The discussion above has dealt successfully with the first a truly periodic potential. This amounts to repeating the film two questions posed in Sec. I: the resistivity is indeed depen- periodically; periodicity is restored by the boundary condi- dent on the boundary conditions, and we have understood tions, and the standard folklore applies: the resistivity ␳ is conceptually how the finite resistivity of a perfect thin film indeed zero. In fact, the absence of electrical resistivity can arises under outgoing boundary conditions. However, the is- be traced back again to the electrons keeping their memory sue of how to measure the finite resistivity of a thin film has of linear momentum. In the old Fuchs-Sondheimer transport not yet been clarified. From the experimental viewpoint, the model,4 this reflecting boundary condition, which amounts to finite electrical resistivity of a perfect thin film still remains the absence of diffuse scattering at the boundaries ͑100% puzzling, even when the concept of finite periodicity is in- specular reflection͒, is parametrized by pϭ1, where p is the troduced. In effect, if we isolate the film from a bulk sample coefficient characterizing the specularity of scattering off the of copper and maintain it in contact with the ‘‘remainder’’ of surfaces of the film. The ensuing resistivity ␳ϭ0 has been the system ͑two perfectly conducting semi-infinite copper confirmed by the relativistic layered KKR method.2 blocks͒, then the current is shunted by these contacts. In Does this mean that the resistivity is always zero? Of other words, it would seem that it is not possible to measure course not; the same relativistic layered KKR method the resistivity of the film in this straightforward way. In fact, showed that finite-size effects are not negligible when the one concludes that, for a measurement of electrical resistiv- film is sandwiched between two semi-infinite samples of the ity, the boundary contacts should not be of the same mate- same material. The novelty lies here in the use of different rial. boundary conditions. In effect, if the boundary conditions So how does one observe the calculated resistance? As dictate that the electrons cannot keep their memory of mo- mentioned in the previous paragraph, the shunting of the mentum, then the background scattering of the finite lattice current by the contacts prohibits one from measuring the becomes the source of a nonzero electrical resistivity. resistance of a portion of a metal. However, if the current An extreme form of this loss of momentum information is probe is sufficiently narrow to contact only the finite layer, achieved when the momentum of electrons entering the finite and appropriate boundary conditions are enforced, then the sample is totally uncorrelated with that of electrons leaving current will be limited only to the finite layer. Then, one can the sample. One can conceptualize this extreme loss of mo- either measure the resistance of a free-standing film or have mentum information in three equivalent ways: in terms of it supported on an insulating substrate. If the boundaries are reservoirs, in terms of boundary scattering, and in terms of ideal so that they have no appreciable roughness, i.e., for p boundary conditions for the electron propagators ͑Green’s ϭ1, the boundaries simulate reflecting or free-standing functions͒. First, momentum loss is effectively implemented boundary conditions and lead to zero resistivity. Alterna- by having reservoirs that absorb the electrons upon leaving tively, if the boundaries are sufficiently roughened, they the sample, when the electrons probe a reservoir, they get out simulate current flow only in the finite layer subject to the of synch with respect to their ‘‘proper’’ behavior in the pϭ0 boundary condition—this amounts to outgoing bound- sample. Second, from the scattering viewpoint, this informa- ary conditions. In fact, the roughened interface could even tion loss can be modeled by perfectly diffuse scattering at the separate the finite layer from a semi-infinite sample of the boundaries of the film; this condition is precisely equivalent same material on either side, a situation that is modeled by to the choice pϭ0 in the Fuchs-Sondheimer model4 ͑0% the corresponding relativistic KKR computation.2 It is in this specular reflection, a condition that totally erases momentum way that we understand the paradox of using a film with a memory͒. The third viewpoint is needed when applying the rough surface to measure the resistance calculated for a per- Kubo formula in terms of electron propagators; in our origi- fectly flat film; it is the boundary condition for transport on nal theory of transport in metallic superlattices,5 we de- the surface of the film that happens to be the same in both scribed the required condition associated with this momen- cases. 15 784 HORACIO E. CAMBLONG AND PETER M. LEVY PRB 60

A parenthetical remark is in order. The statement that p ⑀ ϭ⑀ ϭ⑀ ϩ⑀ ͑ ͒ k kʈk kʈ k , 2 ϭ0 corresponds to outgoing boundary conditions should not z z 2 2 2 2 signify that the resistivity coming from the ‘‘bulk’’ of the where ⑀ ϭប kʈ /2m and ⑀ ϭប k /2m, with m being the kʈ kz z sample is directly related to the scattering at the boundaries; effective electron mass. Our calculation of electrical conduc- the latter are there merely to simulate the boundary condi- tivity will be performed by applying perturbation theory tions that enable the ‘‘bulk’’ scattering to produce resistance. within the framework of the Kubo formula.7,8 The required In other words, the surface should have a roughness profile potential matrix elements are given by sufficient to guarantee the boundary condition pϭ0; however, further increasing the amplitude of the roughness, 1 Ϫ Ϫ Ј 1 ϭ ͵ 3 i(k k )•r ͑ ͒ϭ ˜ ͑ Ϫ Ј͒ ͑ ͒ while increasing the resistivity due to the surface scattering, VkkЈ ⍀ d re V r ⍀V k k , 3 does not increase the resistivity coming from the bulk scattering. Therefore, for the case at hand, the actual scattering where ˜V(k) is the Fourier transform of the potential. Then, does not come from randomly situated impurities but from to second order in perturbation theory, the diagonal the potential of the positively charged background ions that momentum-space elements of the t matrix are form a finite, but otherwise perfect, lattice. A perfect periodic lattice has no resistivity, and it is easy to overlook the fact t ͑⑀͒ϭV ϩ͚ V G0 ͑⑀͒V , ͑4͒ that electrons are scattered by it, i.e., they undergo Bragg k kk kkЈ kЈ kЈk kЈ scattering. As we will show in Sec. V, a finite but otherwise perfect lattice scatters electrons for all momenta, with two where G0(⑀) is the free-particle propagator. In order to main contributions: constructive interference or Bragg scat- evaluate the Kubo formula for the electrical conductivity, the tering as well as background diffusive scattering. It is the negative imaginary part of the t matrix, ⌬ϭϪIm(t)is * ϭ latter that leads to electrical resistivity and vanishes in the needed. Then, from the condition VkkЈ VkЈk , it follows that infinite-thickness limit; unlike the case of impurity scatter- ⌬ ⑀͒ϭ⌬ ⑀͒ ing, its resistivity is inversely proportional to the thickness of ͑kʈ ,kz ; ͑k; the film. ϭϪ ͓ ͑⑀͔͒ In the remainder of this paper, we will show the details of Im tk the calculation of the electrical resistivity of a thin film using ϭ␲ J Ϫ ͒␦ ⑀Ϫ⑀ Ϫ⑀ ͒ ͑ ͒ the Kubo formula within the free-electron model. Specifi- ͚ ͑k kЈ ͑ kЈ kЈ , 5 Ј Ј ʈ z cally, we will show that outgoing boundary conditions do kʈ ,kz indeed imply the existence of a finite electrical resistivity. in which the effect of the potential on the conductivity is Remarkably, the free-electron result agrees in form and rea- now summarized by the function sonably well in magnitude with the one calculated ab initio by Blaas et al.1,2 for a perfect slab of copper embedded in ͉˜V͑q͉͒2 copper. J͑q͒ϭ , ͑6͒ ⍀2 III. KUBO FORMALISM FOR THE CONDUCTIVITY with qϭkϪkЈ and where the density of states ␦(⑀Ϫ⑀ Ј OF A THIN METALLIC SLAB kʈ Ϫ⑀ Ϫ ͓ 0 ⑀ ͔ ␲ kЈ) has been directly extracted from Im GkЈkЈ( ) / . As discussed in Sec. II, a perfectly diffuse boundary z ʈ z 7 amounts to outgoing boundary conditions, which are imple- The Kubo formula gives the zero-temperature in-plane ͑ ͒ mented with the corresponding infinite-medium retarded CIP dc conductivity in terms of the in-plane current-current Green’s functions. As the scattering ultimately leading to correlation function by means of the formal double-limit 8 resistivity is due to a potential V(r) with ‘‘finite periodic- expression ity,’’ we resolve the Hamiltonian in the form 1 ␤ ␴ϭ ␶ i␻␶͗ ͑␶͒ ͑ ͒͘ ϭ ϩ ͑ ͒ ͑ ͒ lim lim ͫ ͵ d e T␶jʈ jʈ 0 ͬ , H H0 V r , 1 ␤→ϱ ␻→ 2␻⍀ 0 • 0 i␻→␻ϩi0ϩ ͑7͒ where the unperturbed Hamiltonian H0 corresponds to free ␸ electrons characterized by the eigenfunctions k which in the independent-electron approximation reduces ϭ⍀Ϫ1/2 ik r e • ; in all quantum-mechanical computations, we straightforwardly to the familiar form will use particle-in-a-box normalization with finite volume 2 ⑀ץ ⍀. However, given a thin film of cross-sectional area A, 2 e 1 kʈk ប ⍀ϭ ␴ϭ ͫ z ͬ ␦͑⑀ Ϫ⑀ Ϫ⑀ ͒ thickness L, and volume AL, the in-plane dimensions ͚ F k k , បkʈ͒ 2⌬͑kʈ ,k ;⑀ ͒ ʈ z͑ץ ⍀ 2 ͑defining the area A) will be effectively regarded as infinite, kʈ ,kz ,s z F ͑ ͒ whereas the finiteness of the perpendicular or ‘‘longitudinal’’ 8 dimension L will become the source of finite-size effects; where s stands for the electron-spin index. Equation ͑8͒ can this longitudinal direction will be chosen to correspond to the be conveniently rewritten by evaluating the group velocity as z axis. Accordingly, whenever appropriate, a generic vector ⑀ 2 ⑀ץ V will be resolved into its in-plane component Vʈ and its k 2 k ͫ ʈ ͬ ϭ ʈ ͑ ͒ longitudinal component VЌϭzˆV ; with this notation, the en- , 9 បkʈ͒ m͑ץ z ergy of the free-particle or unperturbed state of momentum បk is and applying the in-plane continuum limit PRB 60 ELECTRICAL RESISTIVITY OF A THIN METALLIC FILM 15 785

2 the three directions need not be perpendicular to each other. 1 d kʈ ͚ →2 ͵ ϭ ͵ d⑀ gʈ͑⑀ ͒ ͑10͒ In other words, if the number of primitive cells stacked in 2 kʈ kʈ A kʈ ,s ͑2␲͒ ӷ direction j is N j , then N1 ,N2 N3, and, in practice, we will with the only restriction that the integrand be a spin- regard N1 and N2 as effectively infinite but N3 as a finite ͉ ͉ ϭ independent function of the momentum variables kʈ and kz number; notice that N N1N2N3 is the total number of ͑but not of the corresponding angular in-plane variable͒.In primitive cells in the film. The finite periodicity in the third ͑ ͒ ⑀ ͑ ͒ ϭ Eq. 10 , gʈ( kʈ) is the in-plane two-dimensional 2D den- direction can be described by counting the number M N3 ϩ Ϸ sity of states per spin degree of freedom and per unit area, 1 N3 of stacked infinite lattice planes or monolayers; no- which is given by tice that, for the sake of simplicity, we will assume Mӷ1. For instance, if the film has thickness L and consists of ex- 1 2m actly M monolayers from one boundary to the other, then L gʈ͑⑀͒ϭ , ͑11͒ ϭ Ϸ 4␲ ប2 N3d Md, with d being the distance between consecutive monolayers. provided that the in-plane dimensions be effectively infinite. The basic periodicity of the crystal structure within its ⑀ Notice that this density of states gʈ( ) is actually a constant boundaries can be described by means of a finite generaliza- gʈ , which from now on will be factored out of the corre- tion of the concept of Bravais lattice. As usual, given the sponding integrals. Then, Eqs. ͑8͒–͑11͒ imply that the con- ϭ primitive translation vectors aj , with j 1,2,3, it follows that ductivity is given by the expression ϭ the set of all translation vectors R is of the form R n1a1 ϩ ϩ 2 ⑀ Ϫ⑀ n2a2 n3a3, where the integers n j are limited to the values e 1 F kz р р Ϫ ␴ϭ ͚ , ͑12͒ 0 n j N j 1 for a finite lattice. Then, for any local func- h L k ⌬ k͑⑀ Ϫ⑀ ͒,k ;⑀ z „ F kz z F… tion of the position, such as the lattice potential, the property ⑀ ⑀ ϭ⑀ where k( ) is the positive root of the equation k , i.e., k(⑀)ϭͱ2m⑀/ប; in particular, V͑rϩR͒ϭV͑r͒ ͑17͒ k͑⑀ Ϫ⑀ ͒ϭͱk2 Ϫk2. ͑13͒ F kz F z ͑ ͒ ⌬ ⑀ In Eq. 12 , it is assumed that (kʈ ,kz ; F) is a function of remains valid within the boundaries of the film. In particular, ͉ ͉ kʈ and kz alone; thus, to simplify the notation, from now on for Fourier transforms, finite periodicity guarantees the iden- ⌬ ͉ ͉ ⑀ this quantity will be represented as ( kʈ ,kz ; F). tity Equation ͑12͒ will be the starting point for our conductivity calculation in Sec. V, where we will straightforwardly apply its counterpart in the longitudinal continuum limit, Ϫ Ϫ Ϫ ͵ d3re iq•rV͑r͒ϭe iq•R ͵ d3re iq•rV͑r͒, ͑18͒ C C 1 ϱ dk 1 ϱ TR ͑ ͒→ ͵ z ͑ ͒ϭ ͵ ⑀ ͑⑀ ͒ ͚ f kz f kz d k gЌ k Ϫϱ ␲ z z L kz 2 2 0 where an arbitrary primitive cell C can be related to any other ϫ ⑀ ͒ ϩ Ϫ ⑀ ͒ ͑ ͒ ͓ f k͑ k f k͑ k ͔, 14 „ z … „ z … primitive cell via a translation TR by a Bravais lattice vector where R. This property leads to the characteristic Bragg peaks with respect to electronic conduction, as shown below. In effect, 1/2 1 2m the Fourier transform of the potential becomes Ϫ1/2 gЌ͑⑀͒ϭ ͩ ͪ ⑀ ͑15͒ 2␲ ប2 Ϫ Ϫ Ϫ N1 1 N2 1 N3 1 is the longitudinal density of states per spin degree of free- 3 Ϫiq r ⑀ ˜V͑q͒ϭ ͵ d re • V͑r͒ϭ ͚ ͚ ͚ dom and per unit length; notice that, explicitly, k( k ) V ϭ ϭ ϭ z n1 0 n2 0 n3 0 ϭ͉ ͉ ͑ ͒ kz . Under this approximation, Eq. 12 turns into Ϫ ϩ ϩ Ϫ ϫ iq•(n1a1 n2a2 n3a3) ͵ 3 iq•r ͑ ͒ ͑ ͒ 2 ⑀ e d re V r , 19 e F 1 C ␴ϭ ͵ d⑀ ͑⑀ Ϫ⑀ ͒gЌ͑⑀ ͒ͫ k F k k 2h 0 z z z ⌬ k͑⑀ Ϫ⑀ ͒,k͑⑀ ͒;⑀ „ F kz kz F… 1 where V is the entire sample and C is a reference primitive ϩ ͑ ͒ ⌬ ⑀ Ϫ⑀ ͒ Ϫ ⑀ ͒ ⑀ ͬ. 16 cell. Then, summing the geometric progressions involved in k͑ F k , k͑ k ; F „ z z … Eq. ͑19͒ and replacing in Eq. ͑6͒, one finds

IV. FINITE PERIODICITY 3 A thin film can be regarded as built out of primitive cells J͑ ͒ϭͫ F ͑ ͒ͬJ (0)͑ ͒ ͑ ͒ q ͟ N q aj q , 20 assembled into an effectively infinite arrangement in two di- jϭ1 j • rections ͑for which we will use symbols 1 and 2͒ and a finite layering in a third direction ͑for which we will use the sym- bol 3͒. As we will consider a generic ‘‘finite Bravais lattice,’’ where 15 786 HORACIO E. CAMBLONG AND PETER M. LEVY PRB 60

3 2 However, in a ﬁnite lattice, it is legitimate to apply the d r Ϫ J (0)͑q͒ϭͯ͵ e iq•rV͑r͒ͯ ͑21͒ limit of Eq. ͑29͒ only with respect to the directions deﬁned C v by the primitive vectors a1 and a2. Then, the component of q is the corresponding quantity for just one primitive cell, v on the plane spanned by the primitive reciprocal vectors b1 ϭ ϫ and b is a1•(a2 a3) is the volume of a primitive cell, and 2 1 sin͑N ␰ /2͒ 2 ͑q a ͒ F ͑␰ ͒ϭ ͫ j j ͬ ͑ ͒ ϭ Ϫ • 3 ϭ ͑ ͒ N j 22 q q b G , 30 j 2 sin͑␰ /2͒ in 2␲ 3 n1n20 N j j stands for the interference factor associated with N j identical which is a 2D reciprocal vector, primitive cells aligned in the direction of the primitive translation vector a , with g ϭG ϭn b ϩn b , ͑31͒ j n1n2 n1n20 1 1 2 2 ␰ ϭ ͑ ͒ j q•aj . 23 so that

The interference factors are the origin of the Bragg scattering 2 ϱ ␦ peaks, which are represented by functions and amount to F ␰ ͒ϳ ␦ ͑ ͒ ͟ N ͑ j ͚ q ,g . 32 the selection of the conditions ␰ ϭq a ϭ2␲n ͑with n in- ϭ j ϭϪϱ in n1n2 j • j j j j 1 n1 ,n2 teger numbers͒, as can be seen from the vanishing of the denominator in Eq. ͑22͒. These ␦ functions can be explicitly Next we will assume that, for qϭkϪkЈ, with k and kЈ on displayed by means of the expansion in a series of simple the Fermi surface, the only allowed 2D reciprocal-lattice vector g is the zero vector. To see that this assumption is fractions n1n2 ϱ reasonable, let us consider, for example, a thin film of cop- 1 1 per, for which one can choose the set of primitive vectors ϭ ͚ ͑24͒ 2 ␰ ͒ ϭϪϱ ␰ Ϫ ␲͒2 ϭ ˆϪ ˆ ϭ ˆϩ ˆ ϭ ˆϩ ˆ sin ͑ j/2 n j ͑ j/2 n j a1 (a/2)(x y), a2 (a/2)(x y), a3 (a/2)(x z), and ϭ ␲ ˆϪ ˆϪ ˆ and from the familiar asymptotic result primitive reciprocal vectors b1 (2 /a)(x y z), b2 ϭ ␲ ˆϩ ˆϪ ˆ ϭ ␲ ˆ Ϸ (2 /a)(x y z), b3 (4 /a)z, with a 3.61 Å and kF 1 sin͑N ␰ /2͒ 2 Ϸ Ϫ1 ϭ Ϫ Ј Ј ͫ j j ͬ ϳ␲␦͑␰ ͒ ͑ ͒ 1.36 Å . Then, for q k k , with k and k on the ␰ ͒ j/2 25 N j ͑ j/2 Fermi surface, it follows that →ϱ for N j ; then, the interference factors become ͉ ͉р Ͻ͉ ͉ ͉ ͉ ͑ ͒ q 2kF b1 , b2 , 33 ϱ 1 sin͑N ␰ /2͒ 2 ϭ ϭ F ͑␰ ͒ϭ ͫ j j ͬ implying that the only acceptable choice is n1 n2 0, N j ͚ ϭ ϭ j 2 ϭϪϱ ͑␰ /2Ϫn ␲͒ namely, gn n 0. Under these conditions, qin 0 and also N j n j j j 1 2 ϭ ϭ ϭ ϱ qʈ 0, because q•a1 0 and q•a2 0 simultaneously; thus, 2␲ ϭ ϳ ␦ ␰ Ϫ ␲ ͒ ͑ ͒ q qЌ . Then, the argument of the interference factor ͚ ͑ j 2 n j , 26 ϭϪϱ F (␰ )is N j n j N3 3 where the second expression is to be understood as the ␰ ϭ ϭ ͑ ͒ 3 qЌ a3 qzd, 34 asymptotic form for N j ‘‘sufficiently large.’’ The continuum • limit implicit in Eq. ͑26͒ can be reversed by replacing the with d being the distance between consecutive monolayers. Dirac delta function by a Kronecker delta and recalling Eq. As a consequence, Eq. ͑20͒ becomes ͑23͒, whence J͑ ͒ϭJ (Ќ)͑ ͒␦ ͑ ͒ ϱ q qz qʈ ,0 , 35 F ␰ ͒ϳ ␦ ͑ ͒ N ͑ j ͚ q a ,2␲n . 27 j ϭϪϱ • j j n j where In particular, the combination of the three structure factors in J (Ќ)͑q ͒ϭF ͑q d͒J (0)͑q zˆ͒, ͑36͒ Eq. ͑20͒ amounts to z N3 z z 3 ϱ with F ␰ ͒ϳ ␦ ␦ ␦ ͟ N ͑ j ͚ q a ,2␲n q a ,2␲n q a ,2␲n . ϭ j ϭϪϱ • 1 1 • 2 2 • 3 3 j 1 n1 ,n2 ,n3 3 2 d r Ϫ ͑28͒ J (0)͑ ˆ͒ϭͯ͵ ͑ ͒ iqzzͯ ͑ ͒ qzz V r e 37 C v Equation ͑28͒ can be reinterpreted by expanding q in terms ϭ␯ ϩ␯ ϩ␯ of primitive reciprocal vectors, i.e., q 1b1 2b2 3b3, ͓cf. Eq. ͑21͔͒. Equations ͑35͒–͑37͒ give just a Bragg scatter- with a b ϭ2␲␦ , whence ␯ ϭq a /2␲. Then, Eq. ͑28͒ →ϱ j• h jh j • j ing contribution and zero resistivity as N3 , but fall short states that ␯ ϭn is an integer, so that q is indeed a j j of that singular behavior for N3 finite. Based on the preced- reciprocal-lattice vector G , i.e., J n1n2n3 ing analysis, the Bragg scattering contributions, Bragg(q) J (Ќ) 3 ϱ and Bragg(qz), are defined to be the asymptotic forms of the J J (Ќ) →ϱ F ␰ ͒ϳ ␦ ͑ ͒ functions (q) and (qz)asN3 ; thus, they are re- ͟ N ͑ j ͚ q,G . 29 ϭ j ϭϪϱ n n n ͑ ͒ j 1 n1 ,n2 ,n3 1 2 3 lated by Eq. 35 , with PRB 60 ELECTRICAL RESISTIVITY OF A THIN METALLIC FILM 15 787

ϱ ␦͑⑀Ϫ⑀ ͒ϭ ␲ (Ќ)͑⑀Ϫ⑀ ͒ 2␲ 2␲n k 2 g k J (Ќ) ͑ ͒ϭ ␦͑ Ϫ ␲ ͒J (0)ͩ 3 ˆͪ ʈ Bragg qz ͚ qzd 2 n3 z ; N ϭϪϱ d 3 n3 1 ͑38͒ ϫ ͓␦ k Ϫk͑⑀Ϫ⑀ ͒ ϩ␦ k ϩk͑⑀Ϫ⑀ ͒ ͔ 2 „ z kʈ … „ z kʈ … in particular, L ϭ (Ќ)͑⑀Ϫ⑀ ͓͒␦ ϩ␦ ͔ g kʈ k ,k(⑀Ϫ⑀ ) k ,Ϫk(⑀Ϫ⑀ ) . 2 z kʈ z kʈ (Ќ) ͓J ͑qʈϭ0,q ͔͉͒ ͉Ͻ ␲ ϭ͓J ͑q ͔͉͒ ͉Ͻ ␲ Bragg z qz 2 /d Bragg z qz 2 /d ͑42͒ 2␲ ϭ ␦ ͒J (0) ͒ ͑ ͒ ͑qzd ͑0 , 39 Then, N3 ␲ L (Ќ) ⌬͑kʈ ,k ;⑀͒ϭ gЌ͑⑀Ϫ⑀ ͓͒J k Ϫk͑⑀Ϫ⑀ ͒ an expression that will be important in the derivation of the z 2 kʈ diff „ z kʈ … electrical resistivity in the next section. Ќ ϩJ( ) k ϩk͑⑀Ϫ⑀ ͒ ͔. ͑43͒ diff „ z kʈ … V. FINITE ELECTRICAL RESISTIVITY Finally, for the calculation of the conductivity, Eqs. ͑12͒ and ͑ ͒ ͑ ͒ Equation ͑12͒ will give a nonzero electrical resistivity 16 dictate that Eq. 43 be evaluated for electrons on the Fermi surface, for which the conditions ͉k ͉ϭk(⑀ Ϫ⑀ ) and only when the t matrix develops a nonzero imaginary part, z F kʈ ϭ ⑀ Ϫ⑀ i.e., ⌬ (⑀)Þ0. This imaginary part may arise in the pro- kʈ k( F k ) apply; then, kʈkz z cess of replacing discrete sums by energy integrals ͓accord- ␲ ing to the rule deﬁned by Eq. ͑14͔͒, if the self-energy ac- L (Ќ) (Ќ) ⌬ k͑⑀ Ϫ⑀ ͒,k ;⑀ ϭ gЌ͑⑀ ͓͒J ͑0͒ϩJ ͑2k ͔͒. quires an analytic structure characterized by a branch cut that „ F kz z F… 2 kz diff diff z is effectively generated by the merging of the discrete poles ͑44͒ of the discrete self-energy. However, the scattering by a pe- Substitution of Eq. ͑44͒ into Eq. ͑16͒ leads to riodic lattice generates constructive interference in discrete directions ͑represented by ␦ functions͒; this Bragg scattering 2 ⑀ ⑀ Ϫ⑀ fails to produce an imaginary self-energy. e 2 F F kz ␴ϭ ͵ d⑀ , ͑45͒ ␲ kz (Ќ) (Ќ) Due to the directional nature of Bragg scattering, the term h L 0 J ͑0͒ϩJ ͑2k ͒ J (Ќ) Ϫ Ј ͑ ͒ diff diff z Bragg(kz kz ) given in Eq. 38 does not contribute to the J (Ќ) Ϯ ϭ͓J (Ќ) ͔ sum of Eq. ͑5͒, namely, where the properties diff ( 2kz) diff (2kz) * have been J (Ќ) applied. Finally, the term diff (2kz) can be approximated J (Ќ) ϭJ with its value diff (0) diff(0), because it is partially sup- Ϫ J ͑ Ϫ Ј͒ ͓ 0 ͑⑀͔͒ϭ ͑ ͚ Bragg k k Im GkЈkЈ 0. pressed by the numerator as kz approaches kF in fact, the Ј Ј ʈ z kʈ ,kz exponential in the corresponding Fourier integral does not complete one entire cycle even as z approaches d and kz ͒ Then, we are led to a resolution of the function J(q)of approaches kF and is approximated as taking the value one ; Eq. ͑6͒ into two parts, then,

2 ⑀2 Ќ Ќ e F J͑q͒ϭJ ͑q͒ϩJ ͑q͒ϭ͓J ( ) ͑q ͒ϩJ ( )͑q ͔͒␦ , ␴Ϸ . ͑46͒ Bragg diff Bragg z diff z qʈ ,0 h 2␲LJ ͑0͒ ͑40͒ diff

͑ ͒ In order to evaluate the final conductivity expression of corresponding to Bragg scattering, given by Eq. 38 , and the Eq. ͑46͒, the value of remainder, which we identify as background diffusive scattering. This procedure, which is based on the analysis of Sec. J ͒ϭ J ϭ ͒ϪJ ϭ ͒ ͑ ͒ diff͑0 lim ͓ ͑qʈ 0,qz Bragg͑qʈ 0,qz ͔ 47 IV, amounts to isolating the diffusive part of the scattering q →0 J ϭJ ϪJ z diff(q) (q) Bragg(q), which leads to a finite resistivity via the term is required. This can be obtained from the interference factor, ͑ ͒ ␰ Eq. 22 , which can be approximated for N3 large and 3 small via the expansion ⌬ ⑀͒ϭ␲ J Ϫ ͒␦ ⑀Ϫ⑀ Ϫ⑀ ͒ ͑kʈ ,kz ; ͚ diff͑k kЈ ͑ kЈ kЈ ʈ z 2͑ ␰ ͒ ␰ 2 kЈʈ ,kЈ 1 sin N3 3 /2 2 3 z F ͑␰ ͒ϭ ͫ1ϩ ͩ ͪ ͬϩO͑␰2͒ N3 3 2 ͑␰ ͒2 3! 2 3 N3 3/2 ϭ␲ J (Ќ)͑ Ϫ Ј͒␦͑⑀Ϫ⑀ Ϫ⑀ ͒ ͑ ͒ ͚ diff kz kz k kЈ . 41 Ј ʈ z ␲ ␰ k 2 1 N3 3 z ϳ ␦͑␰ ͒ϩ sin2ͩ ͪ ϩO͑␰2͒, ͑48͒ N 3 2 2 3 3 3N3 Equation ͑41͒ can be further simplified by either applying the longitudinal continuum approximation, Eq. ͑14͒ with respect where the denominator has been expanded in power series of ␰ ͑ ͒ to zЈ, or explicitly rewriting the ␦ function as 3/2, and Eq. 25 has been applied to provide the asymptotic 15 788 HORACIO E. CAMBLONG AND PETER M. LEVY PRB 60 form of F (␰ ). The derivation above focuses directly on Equation ͑54͒ can be evaluated numerically by introduc- N3 3 ␰ → ing a natural atomic resistivity the 3 0 limit and does not emphasize the resolution of the spectrum into infinitely many Bragg peaks, each of which h has a long tail that contributes to the background diffusive ␳ ϭ ͒Ϸ ␮⍀ ͑ ͒ 0 ͑1Å 258 cm, 56 scattering. Instead, a more illuminating approach would be to e2 display all the Bragg peaks explicitly from the start by con- ␰ → whence sidering the limit 3 0 of the simple fraction expansion of ͑ ͒ Eq. 24 , namely, 270⌫2d͓Å ͔ ␳Ϸ ␮⍀ ͑ ͒ ϱ cm. 57 1 1 2 1 2 1 M lim ͫ Ϫ ͬϭ ͚ ϭ ␨͑2͒ϭ , ␰ → 2͑␰ ͒2 ͑␰ ͒2 ␲2 nϭ1 2 ␲2 3 Equation ͑57͒ summarizes one of the main results of this 3 0 sin 3/2 3/2 n ͑49͒ paper. It displays a characteristic inverse proportionality with ͑ ͒ ͑ ͒ respect to the number M of monolayers. In addition, notice a result that is in agreement with Eqs. 47 and 48 .Inthe that: →ϱ 2 ␰ formulas above, in the limit M , the factor sin (M 3/2) ͑i͒ The period d is not a free parameter, as it is uniquely oscillates fast about its average value 1/2, which would determined from the crystal structure. therefore be effectively achieved in all physical measure- ͑ii͒ As the atomic length scale d͓Å ͔ is always of the ͑ ͒ ments; thus, one concludes that Eq. 48 is simplified to order of unity, the largest variations in the resistivity scale of ͑ ͒ ⌫ ␲ Eq. 57 will come from the dimensionless parameter . 2 1 ͑ ͒⌫ F ͑␰ ͒ϭ ␦͑␰ ͒ϩ ϩO͑␰2͒. ͑50͒ iii is indeed the only free parameter within the frame- N3 3 N 3 2 3 3 6N3 work of approximations used in this model. ͑iv͒ The value of ⌫ can be independently estimated from Notice the characteristic appearance of the Bernoulli number ϭ ͑ ͒ calculations of cohesive energy. B2 1/6, associated with either the expansion of Eq. 48 or ␨ ϭ ␲2 ␨ For example, for copper, reasonable estimates are pro- with the value (2) B2 of the Riemann function in Eq. 9 ͑ ͒ ␰ ϭ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ vided by the following values of the relevant parameters: 49 . Then, for 3 qzd small, Eqs. 35 , 36 , 39 , and 50 dϷ1.8 Å and ⌫Ϸ2/7; then, the predicted CIP resistivity is imply that approximately 40/M ␮⍀ cm, a value reasonably close to the 2 (0) ␮⍀ J͑qʈϭ0,q ͒ϭF ͑q d͒J ͑0͒ 116/M cm found by the ab initio method. z N3 z

ϭ͓J ͑qʈϭ0,q ͔͉͒ ͉Ͻ ␲ VI. CONCLUSIONS Bragg z qz 2 /d

1 It is generally recognized that a perfect infinite periodic ϩ J (0)͑ ͒ϩ ͑ 2͒ ͑ ͒ structure leads to no electrical resistance because of the con- 2 0 O qz ; 51 6N3 dition of constructive interference or Bragg scattering. In this paper, we have shown that, at sufficiently low temperatures, therefore, from Eqs. ͑37͒ and ͑47͒, and from N ϷM, the 3 for sufficiently clean samples, and for outgoing boundary diffusive part of J(0) for M monolayers is conditions, finiteness of a metallic film will prevent the po- 1 1 tential from being perfectly periodic, will produce an effec- J ͒ϭ J (0) ͒ϭ 2 ͑ ͒ diff͑0 ͑0 ͗V͘ , 52 tive background diffusive scattering, and will cause a size- 6M 2 6M 2 dependent resistivity inversely proportional to the number of with monolayers. Our free-electron estimate is in close agreement with similar results from ab initio calculations. It would be d3r interesting to see if this resistivity can be measured experi- ͗V͘ϭ ͵ V͑r͒ ͑53͒ mentally by properly simulating outgoing ͑perfectly diffu- C v sive͒ boundary conditions. being the average potential. As a coda, it is worth mentioning that the Boltzmann Finally, replacing Eqs. ͑52͒ and ͑53͒ into Eq. ͑46͒, and equation approach with the conventional relaxation time ap- recalling that LϷMd, we get a remarkably simple expres- proximation fails for transport through a finite, yet otherwise sion for the resistivity, perfect, film. The usual ansatz for the nonequilibrium distri- 10 ϭ ϩ bution function f (k) f 0(k) k•EC(k) is not applicable, ␲ h ⌫2d because the deviation from equilibrium depends on the ori- ␳Ϸ , ͑54͒ entation of the electric field relative to the crystal axes of the 3 e2 M film as well as on the angle between k and E. where ACKNOWLEDGMENTS 1 d3r V͑r͒ ⌫ϭ ͵ V͑r͒ϭͳ ʹ , ͑55͒ ͑ ͒ ⑀ C v ⑀ This work was supported in part P.M.L. by the Office of F F Naval Research together with the Defense Advanced Re- which is the average potential relative to the Fermi energy, is search Projects Agency ͑Grant No. N00014-96-1-1207͒, the a dimensionless parameter characterizing the relative National Science Foundation ͑Grant No. INT-9602192͒, and strength of the periodic potential. NATO ͑Grant No. CRG 960340͒. PRB 60 ELECTRICAL RESISTIVITY OF A THIN METALLIC FILM 15 789

1 C. Blaas, P. Weinberger, L. Szunyogh, P. M. Levy, C. B. Som- 4 E. H. Sondheimer, Adv. Phys. 1,1͑1952͒. mers, and I. Mertig, Philos. Mag. B 78, 549 ͑1998͒; P. Wein- 5 H. E. Camblong, Phys. Rev. B 51, 1855 ͑1995͒. berger, P. M. Levy, J. Banhart, L. Szunyogh, and B. Ujfalussy, 6 S. Datta, Electronic Transport in Mesoscopic Systems ͑Cam- J. Phys.: Condens. Matter 8, 7677 ͑1996͒. bridge University Press, Cambridge, England, 1995͒. 2 C. Blaas, P. Weinberger, L. Szunyogh, P. M. Levy, and C. B. 7 R. Kubo, J. Phys. Soc. Jpn. 12, 570 ͑1957͒. Sommers, Phys. Rev. B 60, 492 ͑1999͒. The resistivities quoted 8 G. D. Mahan, Many-Particle Physics, 2nd ed. ͑Plenum, New in this publication should all be increased by a factor of four, P. York, 1990͒. ͑ ͒ Weinberger private communication . Therefore the quoted re- 9 F. Seitz, The Modern Theory of Solids ͑McGraw-Hill, New York, sistivity for copper embedded in copper of 29/M ␮⍀ cm should ͒ ⑀ 1940 , pp. 369–370; the approximate values given for F and read 116/M ␮⍀ cm. ͗V͘ are, respectively, 7 and 2 eV. 3 S. Doniach and E. H. Sondheimer, Green’s Functions for Solid 10 G. D. Mahan, Many-Particle Physics ͑Ref. 8͒, pp. 602–606. State Physicists ͑Benjamin, Reading, MA, 1974͒.