AD-A271 663

# '" A P- nTechnical Reports Processed

k, 13 G 2 ( 1993 D Office of Naval Research-Contract N00014-84-K-0548

Task No. NR372-160

#1. E. Krotscheck, G.-X. Qian, & W. Kohn, Theory of inhomogeneous quantum systems.

I. Static properties of Bose fluids Physical Review B 31 4245 (1985).

#2. E. Krotscheck, G. -X. Qian, & W. Kohn, Theory of inhomogeneous quantum systems

I. Linear response and collective excitations in Bose systems Physical Review B 31

4258 (1985).

#3. E. Krotscheck, Theory of inhomogeneous quantum systems. III. Variational wave func.

tions for Fermi fluids, Physical Review B 31 4267 (1985).

#4 Yasutami Takada & W. Kohn, Interaction potential between a helium atom and metal

surfaces, Physical Review Letters 54 470 (1985).

#5 W. Kohn, Discontinuity of the exchange correlation potential from a density functional

view point, Physical Review B, Rapid Publications, 33 4331 (1986).

#6 E. K. U. Gross and W. Kohn, Local Density-Functional Theory of Frequency-Dependent

Linear Response, Physical Review Letters 55 2850 (1985).

#7 E. Krotscheck and W. Kohn, Nonlocal Screening in Metal Surfaces, Physical Review

Letters 57 862 (1986).

iq #8 W. Kohn, Density-functional theory for excited states in a quasi-local density approxi.Ný • ,mation, Physical Review A 34 737 (1986).

#9 L.D. Chang & W. Kohn, Semiclassical theory of Inelastic Scattering of a Particle in

the Near-Adiabatic Limit, Physical Review A 36 1618 (1987).

#10 Olle Heinonen and W. Kohn, Internal structure of a Landau Quasi-Particle Wave

0) ME Packet, Physical Review B 36 3565 (1987).

This doctument has be~en approved 18 01for pubhi releaie and sale; its 1distribution is unlimited. 9.1

# 11 D. Mtearns & W. Kohn, Frequency-Dependent v- Representability in Density Functional

Theory, Physical Review A 35 4796 (1987).

#12 D. Mearns, Inequivalence of the physical and density- functional Fermi surfaces, Phys-

ical Review B15 (1988).

#13 Y. Takada, Time-Independent Variational Approach to Inelastic Collisions of a Particle

with a Harmonic Oscillator, Phys. Rev. A, 38, 98 (1988).

#14 W. Kohn, Semiclassical Description of Inelastic Atom Scattering by Surfaces, Journal

of Statistical Physics, 52 1197 (1988).

#15 W. Kohn, J. Jensen, and P. Chang, Semiclassical Corrections for Inelastic Atom-

Surface Scatteering from Classical Trajectories, Phys. Rev. A 40 1198 (1989).

#16 Y. Takada and W. Kohn, Scattering of Evanescent Waves with Application to Atom-

Surface Interactions, Phys. Rev. B 37 826 (1988).

#17 Y. Takada and W. Kohn, Interaction of Helium with a Corrugated Surface, Phys. Rev.

Lett. 55 140 (1985).

#18 E. Krotscheck, W. Kohn and G. X.-Qian, Theory of Inhomogeneous Quantum Systems.

IV. Variational Calculations of Metal Surfaces, Phys. Rev. B 32 5693 (1985).

#19 W. Kohn, Shallow Impurity States in Semiconductors-The Early Years, Physica B &

C, 146 1 (1987).

#20 E. K. U. Gross, D. Mearns, and L. N. Oliveira, Zeros of the Frequency-Dependent

Linear Density Response, Phys. Rev. Lett., 61 1518 (1988).i-]

#21 D. Mearns and W. Kohn, The Fermi.Surface in Local Density- Functional Theory and J

in Gradient Expansions, Phys. Rev. B 39 10669 (1989).

#22 Q. Niu, Effect of an Electric Field on a Split Bloch Band, Phys. Rev. B 40 3625. ,Cles

(1989). • ~ ~~~~~i ........to

#23 J. Jensen and W. Kohn, Quantum Corrections for Inelastic Atom-Surface Scattering,

Phys. Rev. A 40 2309 (1989).

#24 S. A. Gould, K. Burke, and P. K. Hansma, Simple Theory for the Atomic-Force Micro-

scope with a Comparison of Theoretical and Ezperimental Images of Graphite, Phys.

Rev. B 40 5363 (1989).

#25 W. Kohn, Interaction of Atoms and Molecules with Solid Surfaces, Theory of Atom-Surface

Scattering, Interaction of Atoms, and Molecules with Solid Surfaces, eds. Bortolani,

March and Tosi, Hebrew University of Jerusalem, (Plenum Publishing Co.: New York

(1990)).

#26 E.K.U. Gross and W. Kohn, Time-Dependent Density-Functional Theory, Density Functional

Theory of Many- Fcrrnmon Systems, ed. S. B. Trickey, (Academic Press: Boston (1990)).

(accepted)

#27 Q. Niu, Towards a Quantum Pump of Electric Charges and Currents, Phys. Rev.

Lett., 64, 1812 (1990).

#28 J. H. Jensen and Q. Niu, Wigner Symbols, Quantum Dynamics, and the Kicked Rotator,

Phys. Rev. Lett., (1990). (submitted)

#29 J. M. MacLaren, D. P. Clougherty and R. C. Albers, Local Spin-Density Calculation

for Iron: Effect of Spin Interpolation on Ground State Properties, Phys. Rev. B, 42,

3205 (1990).

#30 K. Burke and W. Kohn, Finite Deb ye- Waller Factor for "Classical" Atom-Surface

Scattering, Phys. Rev. B, 43, 2477 (1991).

#31 Q. Niu, Quantum Coherence of a Narrow Band Particle Interacting with Phonons and

Static Disorder, Phys. Rev. B, (1990). (submitted)

#32 M. Flatte, Exact Theory of One-Phonon Amplitudes in Diffractive Atom-Surface Scat-

tering, The Structure of Surfaces III, eds. M. A. Van Hove et al., Proceedings of the

Third International Conference on the Structure of Surfaces, Springer Verlag (New

York (1991)).

#33 M. Flatte and W. Kohn, Exact Theory of Long-Wavelength One-Phonon Amplitudes

in Atom-Surface Scattering, Phys. Rev. B, 43, 7422 (1991).

#34 K. Burke, J. H. Jensen and W. Kohn Exploration of surfaces by atomic scattering in

the almost classical regime, Surface Science, 241, 211 (1991).

#35 M. E. Eberhart, D. P. Clougherty, and J. N. Louwen Geometrical Origins of Interfacial

Strength, Materials Research Society Bulletin, 16, 53 (1991).

#36 J. Dobson The Embedded Electron-gas Boundary: A New Standard Problem in Surface

Physics, Phys. Rev. B (1991). (submitted)

#37 J. Dobson Interpretation of the Fermi Hole Curvature, J. Chem. Phys. 94 (6) 4328,

(1991).

#38 J. F. Dobson, G. H. Harris and A. J. O'Connor, The Effect of a Frequency-Dependent

Ezchange and Correlation Kernel on the Multipole Surface Plasmon Frequency of .

Bare Jellium Aluminium Surface, J. Phys. Condens. Matter 2, 6461 (1990).

#39* Q. Niu Quantum Coherence of a Narrow-Band Particle Interacting with Phonona and

Static Disorder, J. Statistical Physics, 65, 317 (1991).

#40 Y. Meir Landauer Formula for the Current through an Interacting Electron Region,

Phys. Rev. Lett., 68, 2512 (1992).

#41. D. P. Clougherty and W. Kohn Quantum theory of sticking, Phys. Rev. B, 46, 4921

(1992).

#42 0. Heinonen and W. Kohn Surface Effects on Bulk Plasmons, (1992). (submitted to

Phys. Rev. Lett.)

*Indicates (1991/1992) October Reporting Year Requirements.

VOLUMIE 68. NUMBtR 16 PHYSICAL REVIEW LETTERS 20.APRIL 1992

Landauer Formula for the Curreat through an Interacting Electron Region

Yigal MeirDepartment of Physics. Massachusetts Institute of Technology. Cambridge', Massachusetts 02139

and Department of Physics. Unirersity of California. Santa Barbara. California 93106

Ned S. Wingreen(a)Department of Physics. Massachusetts Institute of Technology. Cambridge, Massachusetts 02139

(Received 21 January 1992)

A Landauer formula for the current through a region of interacting electrons is derived using thenonequilibrium Keldysh formalism. The caso of proportionate coupling to the left and right leads, wherethe formula takes an especially simple form. is studied in more detail. Two particular examples whereinteractions give rise to novel effects in the current are discussed: In the Kondo regime, an enhancedconductance is predicted, while a suppressed conductance is predicted for tunneling through a quantumdot in the fractional quantum Hall regime.

PACS numbers 72.10.3g. 72.1S.Qm. 73.40.Gk. 73.30.1k

The formulation by Landauer [I( and Buttiker [21 of teracting. The formula we derive [Eq. (6)1 expresses thethe current through a finite, possibly disordered region of current, as in the noninteracting case, in terms of the Fer-noninteracting electrons has tremendously enhanced the mi functions in the leads and local properties of the in-understanding of transport in mesoecopic systems (31. teracting region. (b) We show how the noninteractingThe Landauer formula, which expresses the current in case and the results of Langreth and of Hershfield, Davis,terms of local properties of the finite region (such as the and Wilkins follow as special cases. (c) The current willtransmission coefficient) and the distribution functions in be written in a particularly simple form for the case of aconnected reservoirs, has been used extensively and suc- constant asymmetry factor relating the coupling to theces•fully in many areas, including the scaling theory of left lead to the coupling to the right lead. This case willlocalization 14,51. universal conductance fluctuations [61, be presented and investigated in some detail. (d) Two ex-Aharonov-Bohm conductance oscillations 171, the integer ample. where the interactions lead to interesting resultsquantum Hall effect 181 and its quenching [91. the quant- for the current will he discussed: transport in the Kondoization of ballistic conductance 1101, and recently in the regime, where an emhamed conductance has been predict-field of quantum dynamics of driven systems (quantum ed [17), and tunneling through a correlated electronicchaos) [Il]. state, such as the fractional quantum Hall state, where a

While both the derivation of the Landauer formula for suppression of the conductance is expected I 181.noninteracting electrons [31 and its application are well Our starting point is the Hamiltonianestablished, an apt formulation of the current when in- H-- 7 ek.4C.C*.+Hi(1d.1;1d,)teractions between electrons are involved has been lack- A.& G L.Ring. In view of the recent technological progress in + 1 (Vk.i.c1A.+H.c.), (1)confinement of electrons into small regions, where the C.c L.nelectron-electron interactions plays a major role in thetransport [121, it is quite clear that a Landauer-type for- where CL (ca&) creates (destroys) an electron withmula for the transport through such an interacting region momentum k in channel a in either the left (L) or theis highly desirable. Several attempts have been made to right (R) lead, and W.1 and Ii.1 form a complete, ortho-deal with special case 1131. Langreth [141 was able to normal set of single-electron creation and annihilationexpress the linear conductance through a single site with operators in the interacting region. The channel index in-an on-site interaction (the Anderson model) at zero tern- dudes spin and all other quantum numbers which, in ad-perature in terms of phase shifts and thus relate the con- dition to k, are necessary to define uniquely a state in theductance to a scattering matrix. Apel and Rice [151 ap- leads. This Hamiltonian corresponds to an experimentalproximated the interaction in one dimension by the values situation (see, e.g., Refs. I101 and [121) where two metal-of the momentum transfer Sq at Sfq -0 and Sq "2kf and lic, multichannel leads are connected to thewere able to derive a Landauer-like formula. Unfor- system under study (see Fig. I). Since the leads are me-tunately, this approximation is unsuitable for electrons tallic, the interaction between electrons in the leads andconfined into a small region. More recently, Hershfield, the interaction between electrons in the intermediate re-Davis, and Wilkins [161 have been able to derive a for- gion and electrons in the leads are strongly screened andmula for the current in the Anderson model, can be neglected. However, in the presn c of barriers

In this Letter (a) we derive an exact formula for the between the leads and the interm€ediate regi--', electronscurrent through a region of interacting cictrons coupled cannot flow freely to screen the interactions in the inter-to two multichannel leads where the electrons are not in- mediate region. Accordingly, the interactions between

2512 0 1992 The American Physical Society

VOLUME 68, NUMBER 16 PHYSICAL REVIE* LETTERS 20 -kPRII ,q2

(3)P..

where G<()mi(d~d.(-)). and G'.. (,), to be used later.. m f ' Ca

is equal to -i(d.(l)d2). The Green functions with su-

FIG. I. Schematic diagram of the experimental configu- perscripts i and iare the time-ordered and the anti-time-ration for which an interacting Landauer forniula for the ordered Green functions, respectively (221, and the Greencurrent is derived. Two leads. characterized by chemical poten- functions denoted with small g are the unperturbed Greentials pt. and p. are connected to a mesoucopic region where functions (i.e.. in the uncoupled system). Using th',electrons may interact. If pl. > pi. an electron current J will equalities (221 G '(w)+G <(o)"Gr(co))+G"(a) andflow from left to right. G () -)G < (op) -G'(a) - G"(w), where G' (GO) are

the usual retarded (advanced) Green functions, and the

electrons in the intermediate region need to be treated relationsdynamically and are included in Eq. (I). This treatment &L. W(-)2xifW(w) -- cid).

is similar to the one leading to the Anderson model of a g 21 -fL) o-fe.)(4)

single magnetic impurity in a metal 1191.The logic of our approach follows that used in Ref. where a e L we find

(201 for the one-dimensional noninteracting case. The . Le ())funperturbed system (taken to exist at t - - -. ) consists ALof three uncoupled regions: a left lead and a right lead,both described by the first term in (I). and an interact-ing, intermediate region described by the second term in (5)(I). Since the leads are not coupled at t - - a. each one where p.(e) is the density of states in channel a andmaintains its own thermal equilibrium and one can asso- V.,,(e) equals Va.,, for eme. An equivalent formulaciate chemical potentials, IlL and pR, with the left and can be derived for the current between the intermediateright leads, respectively. As the coupling turns on be- region and the right lead. Since, in steady state, thetween the intermediate region and the two leads [the last current is uniform, one can symmetrize Eq. (5). and us-term in (1)). then, if PL > pr. an electron current J starts ing matrix notation for the level indices in the interactingto flow from the left lead to the right lead. After some region, we findtime the system achieves a steady state. Our aim is to re- "r dL,,o f,(P'T 'G)

late the steady state current to ML and pR, or equivalent- Jam-dL(trllfL(E)rfa(E)rI(G G)Ily. to fL(e) and fi(e), the unperturbed Fermi-Dirac dis- +tr[(rL-rR)G 1), (6)tribution functions in the leads. (We assume that thereservoirs are large enough that the bulk JL and pi are where r...m2X,•,LP.(e)V.Y.()V:.(e), with rf..not perturbed by the current J.) defined similarly. In equilibrium, fL(e) "[ft(e) --fq(e),

To this end we write the current between the inter- one has G < - -fq(G'-G') and the current vanishes.mediate region and the left lead as Equation (6) is the central result of this work. It

expresses the current through the interacting region inj-1 !e (V&.n(cIad,)- Vr.,,(dc1 ,)) terms of the distribution functions in the leads and local

"h" t.L properties of the intermediate region, such as the occupa-tion and the density of states. (The local density of states

f, '. kWWis proportional to the diagonal part of G'-GO, while G <is a product of the density of states and the occupation

[231.) Note that these are to be calculated in the pres-(2) ence of the leads.

For the noninteracting case one can write down Dyson

The first line in (2) can be easily checked by writing the equations for the Green functions in the intermediate re-continuity equation for the current 1201, while to get the gion [201, G < "ift.(e)GTLG'+ifn(e)GT"G° and G'second line we have used the definition of the Keldysh -G- -iG,(rL+rn)G', which enable us to rewriteGreen function 120,211, G.<.(t)ni(cZd1(t)). In the (6) asKeldysh formalism, since the Hamiltonian describing the J-eLf df(e) _fi(w)tri{,GerGTI (7)leads is noninteracting, one has the Dyson equations h

2513

VOLUME 68. NL MBER 16 PHYSICAL REVIEW LE 'TERS 20 -%PRIH 142

Since the transmission coefficient from left to right is at zero temperature. At finite temperature. or at finitgiven by t P.'P. VG¶,G,,, V.,,,, with a E R voltage, inelastic processes have to be taken into accountand a' E L, Eq. (7) reduces to and the usual Landauer formula (8) breaks down for ,in

J - ± fdf(f )_ (e) ] tritt +(01 (8) interacting system.h(8 In order to demonstrate the new features of Eq. (9). we

which is the usual two-terminal Landauer formula for the now study two specific examples. The first is transportnoninteracting case 1241. through a quantum dot in the Coulomb blockade regime

The conductance formula. Eq. (6). takes an especially 1121. Recently. the Anderson Hamiltonian [191,simple form (251 for the case that the couplings to theleads differ only by a constant factor. r"L(E) " C)

J" f*def,(e)-f,(e)]tr{r(G'-G@) + (VcZA.+H.c.), (II)h JA G L

"-2• fe derL(e)(-f(e)]lm~trlrG11, (9) has been employed 1261 to describe transport in this re-n,- gime. Equation (11) is a special case of (I) where the

where rIrLrR/(rL+rR). While this innocent looking different channels are the two spin directions, and the in-formula resembles the noninteracting one, it should be teracting region is a single site with an on-site Coulombemphasized that even though there is a single integral repulsion U. In this case the current takes-the formover energy in (9), this formula includes, by means of thefull Green function G', inelastic processes, spin flips, and J .- L ifd~IE f(I.(1 Ii mG',(e)even processes where several electrons are scattered. ' fde [ft (e-

In order to illustrate how the additional processes dueto interactions are reflected in the formula for the (12)current, we use the usual definition of the self-energy V, where - (I/x) ImG;.(e) is just the local density of statesnamely, we- (g') -t h (G') -u with io defined similarly, of electrons with spin a. In linear response, for tempera-

to write G'-G'-G'XGI, where Z-Z'-V. This al- tures larger than the Kondo temperature, the conduc-lows us to rewrite Eq. (9) as tance will consequently exhibit resonant tunneling peaks

only at to and eo+U, which correspond to resonances in"JM- 'f j ddf.L (e) -f (e)I tr (rG'ZGtl the density of states of the uncoupled site 1261. However,

Al below the Kondo temperature, the density of states devel.M e_. de1fL(E)-fR(e)Jtr1G~rRGTLTi}J (10) ops a peak at the Fermi energy [271, for e < #u < o+U.

h 'Consequently. Eq. (12) predicts a greatly enhanced con-where Io, the self-energy for the nonintcracting case, is ductance over this entire range, in agreement with earlierequal to -i(rL+rR). Comparing Eq. (10) to the studies 1171. Equation (12), which has been indepen-noninteracting results, Eqs. (7) and (8). we see that in dently derived by Hershfield, Davis, and Wilkins 1161 forthe presence of interactions the current cannot in general constant r. provides a framework to study the crossoverbe recast in terms of the transmission matrix, and that from high to low temperatures, and to calculate thethe additional processes included in X are directly respon- current in a Kondo system out of equilibrium. Detailedsible for the deviation of the exact formula for the in- studies will be presented elsewhere (28).teracting case from the usual Landauer formula [Eq. A second example where nontrivial effects due to in-(8)1. Note, however, that at zero temperature and in teractions appear in the conductance is the case of tunnel-linear response only single-electron, elastic processes are ing through a quantum dot in a highly correlated state,allowed by energy conservation. In this case ;(p) such as a fractional quantum Hall state. In the case-Eo(M) and Eq. (10) reduces to the usual Landauer for- where the coupling to the leads is weak, i.e., the elasticmula (8) (this is a generalization of the Langreth (141 re- broadening of the levels is smaller than the excitation en-suit for the Anderson model). Thus the Landauer formu- ergy, the correlated eigenstates arc only weakly perturbedla for the linear-response conductance holds not only for by the leads. When the temperature is larger than thethe noninteracting case, but also for the interacting case elastic broadening one can rewrite Eq. (9) for the linear-

I response conductance G in the form

G -- !--,. .,.° r...(Es -E,) (P,+Pj) -Ot (Ei -E,) (pJd.' ,d(wId.I 9,j). (13)

where the W, are eigenstates, with energies E,, of the uncoupled interacting region, and Pi is the equilibrium probabilityof state W,. Hence, the conductance consists of thermally broadened resonant tunneling peaks which occur whenever thechemical potential u suffices to add another electron to the interacting region. For noninteracting electrons, one canchoose the d's to correspond to single-particle eigenstates of the uncoupled system, and the overlap factor in each term,(W, Jd.'l ,,)(w, Id.Iwj is trivially 0 or I. On the other hand, adding an electron in a single-particle state to a correlatedN-particle eigenstate- which cannot be written as a Slater determinant of single-particle states- will generally not pro-

2514

VOLUME 68, NUMBER 16 PHYSICAL REVIEW LETTERS 20 APRIL 142

duce an (N+O)-particle eigenstate. and consequently Lett. ", 1973 (1987): M. Buttiker. Phys Rev 8 38.these overlap factors can reduce the conductance 9375 (199K).

significantly. In particular, the conductance through a 191 H. Baranger and A. D. Stone. Phys. Rev. Lctt. 63. 414

quantum dot in the fractional quantum Hall regime has (1989).been studied in detail in Ref. 1181 where it was shown 1101 B. J. van Wen et al.. Phys. Rcv. Lett. 60,848 (1988), D

A. Wharam et a.. J. Phys. C 21. L209 (1988)that the overlap factors reduce the condluctance peaks by (Il S. Fishman, M. Griniasty. and R. E. Prange. Phys. Rev.

a factor of l/N"- , for the v-l/p Laughlin state, A 39. 1628 (1989).where N is the number of electrons. Thus Eq. (13) pre- 1121 U. Meirav, M. Kastner, and S. J. Wind. Phys. Rev Lett.

dicts a strong suppression of the conductance in this re- 5, 771 (1990).

gime, an effect of considerable experimental- a,4 niflcance. 1131 It should be noted that the derivation of a conductance

Moreover, formula (13) provides a way to calculate the formula by A. Groshev. T. Ivanov. and V. Valtchinovconductance in more complicated fractional quantum [Phys. Rev. Lett. 66. 1082 (1991)1 ignored the inapplica-

Hall states (such as v- I ), where even more interesting bility of Wick's theorem to systems whose unpenurbed

effects, such as an alternating suppression of the conduc- Hamiltonian includes interactions. Consequently, their

ta1= peaks, are predicted to occur (1]. formula is incorrect and is different from the exact for-tacepeks ae reicedtoocurmula we€lerive.To conclude, we have derived a Landaucr-type formula m141 D.C. La nrh, Phys. Rev. 30. 516 (1966).di

for the current through an interacting electron region. (151 W. Apel and T. M. Rice, Phys. Rev. B 26. 7063 (1982):

This formula provides a new framework to study trans- w. Apel. J. Phys. C 16 2907 (1913).

port in mesoscopic systems where interaction effects are 1161 S. Hershfield, J. H. Davis, and J. W. Wilkins, Phys. Rev.

important. We have demonstrated the usefulness of this Lett. 67, 3720 (1991).

new tool for interacting systems by identifying novel (171 L 1. Olasman and M. E. Raikh, Pis'ma Zh. Eksp. Tear.

features in the conductance for two examples: the Kondo FiL. 47, 378 (1938) (JETP lAt. 47, 452 (1988)1; T. K.

regime and the fractional quantum Hall regime. Further Ng and P. A. Lee, Phys. Rev. LWtt. 61, 1768 (1918).

investigations are currently under way. (181 J. Kinarct, Y. Meir. N. S. Wingreen, P. A. Lee. and X.Wetha.Lee for raising our interest in this [ . Wen, Phys. Rev. B (to be publMshed).

We thank Patrick A.lu e oising or at IT was 119] P. W. Anderson, Phys. Rev. 124,41 (1961).

problem and for valuable discussions. Work at MIT was [ C. CarolL R. Combescot. P. Noziere and D. Saint-

supported under Joint Services Electronic Program Con- James, J. Phys. C 4.916 (1971).

tract No. DAAL 03-89-0001. Work at the University of (211 L V. Keldysh, Zh. Eksp. Toor. Fmz. 47, 1515 (1%5) (soa.

California at Santa Barbara was supported by NSF Phys. JETP , 1018 (1%5)].

Grant No. NSF-DMR90-0102 and ONR Grant No. [211 G. D. Mahan, May-Pwticie Physics (Plenum, New

N00014-89J-1 530. York, 1990). 2nd ed., Set. 2.9.1231 Explicitly, if one defines the local distribution matrix F by

OG < -F(G'-G'). then Eq. (6) take the form

2k i-:t f &htrlrLVfL(e) - FiOG'-GO')

42)Current addreos: NEC Research Institute, 4 Indepen- +trlrmLFL f&(e) (G,_-G,)ldence Way. Princeton, NJ 08540.

III R. Landauer, IBM J. Res. Dew. 1, 233 (1957); Philos. which show that the current can be expremed in terms of

Mag. 21. 863 (1970). a difference in distribution functions times the local densi-

(21 M. Oidttiker, Phys. Rev. Lett. 57, 1761 (1986). ty of states. Note, h vr, that in order to calculate the

131 For a review, me A. D. Stone and A. Szafer, IBM J. Res. local density of states and the local distribution matrix

Dev. 32. 384 (1938). one has to calculate the Green functions G, W', and G 0.(41 P. W. Anderson, E Abrahams, 0. S. Fisher. and D. J. (241 For a discussion of the experimental relevance of the vani-

Thouless. Phys. Rev. B 22 3519 (1980); P. A. Lee and D. am forms of the Landaser formula, we Y. imry, in

S. Fisher, Phys. Rev. Lett. 47, 832 (1981); B. Shapiro, Directions in Condensed Matter Physics. edited by G.

Phys. Rev. Lett. 43. 823 (1982). Grinstein and G. Mazenko (World Scientific, Singapore,

151 J. L. Pichard and G. Sarma, J. Phys. C 14, L127 (1981); 1980). and Ref. (31.14. L617 (1931): A. Mackinnon and B. Kramer. Phys. (251 In order toderive Eq. (9) weusethe fact that thecurrent

Rev. Lett. 47.1546 (1981). to the rtK, IA. is equal to the current from the left, JA,

(61 Y. Imry, Europhys. Lett. I. 249 (1986): P. A. Lee. H. W we may writeJ[1I/(0 +)])JL+ L/(l )]JV.

Fukuysma, and A. D. Stone, Phys. Rev. 8 35, 1039 (261 Y. Mcir. N. S. Wingen, and P. A. Lee, Phys. Rev. Lett.

(1987). 66k 3049 (1991).(71 Y. Gefen, M. Ya. Azbel. and Y. Imry, Phys. Rev. Lett. (271 A. A. Abrikosov. Physics (Loan Island City, N.Y.) 2. 5

52, 129 (19t4): M. Bittiker, M. Ya. Azbel, and Y. imry, (1%5); 2. 61 (1965); H. SOhL in Theawy of Magewisin

Phys. Rev. A 31, 1912 (19U4); 0. Entin-Wohlman. C. in Transition Metals. Cours 37, edited by W. Marshall

Hartisein, and Y. Imry, Phys. Rev. B 34.921 (1986). (Academic, New York, 1907).(81 P. Strada. J. Kucera, and A. H. MacDonald, Phys. Rev. (281 N. S. Wingmen, Y. Mair. and P. A. Lee (unpublished).

2515

Surface Effects on Bulk Plasmons

Olle Heinonen' and Walter Kohn2

1 Department of Physics, University of Central Florida, Orlando, FL 328162Department of Physics, and NSF Center for Quantized Electronic Structures

University of California, Santa Barbara, CA 93106

We discuss the effects of surfaces on bulk plasma oscillations in a metallic film ofthickness L. The leading corrections to the plasma frequencies due to the surfaces areproportional to L-'. The frequencies of low-lying long wavelength modes are givenby the dispersion relation of bulk modes, w = wp[1 + a(q2/k1)), where the allowedq-values are q(n) = (nr/T'), I =_ L - 2db, and db is a complex length characteristic ofthe surface (reminiscent of but different from Feibelman's d±). An explicit expressionfor db is derived. Resonant excitation by an external field, Eoe-'wt, is calculated.

1. . . .. . ... " _

The widespread interest in nanostructures (thin films, layer structures. quarntumwires, quantum dots etc.) as well as in atomic and molecular clusters, in which atleast one dimension is "mesos--opic" (10 A - 1000 A), has led us to re-examine theeffects of finite size on charge collective modes (plasmons). In an infinite system.these modes have frequencies given by the bulk plasmon relation [11

La(q) = wp + a L p W(19

where w, is the classical plasma frequency, w2 = 47re 2n/m, with n the electron den-sity, kF the Fermi wavevector, and a a (complex) number. The imaginary part ofa describes the damping of bulk plasmons, which in an infinite system is due to ex-citations of multiple electron-hole pairs. In a finite system, we expect the followingqualitative changes in the plasmon spectrum due to the presence of surfaces. First,only a discrete set of frequencies will be allowed. Second, there will be additionaldamping due to the decay of plasmons into (single) electron-hole pairs. In this Let-ter, we will show how both the real and imaginary parts of the allowed frequenciesare completely determined by a single (complex) length. Specifically, we consider athin slab with a positive jellium charge background, n+ = NT between z = 0 and z = Land infinite in the x and y directions, and a neutralizing electron liquid described bythe equilibrium density distribution n_(z) (See Fig. I). We shall discuss bulk-likecollective modes of the form

n(r, t) = n(z)e-i-)t, (2)

where the eigenfrequencies W(") are near the classical bulk plasma frequency associatedwith the density 7F, w -= (47re2W/m)'/ 2.

This problem has been previously addressed experimentally [2,3] and theoretically[4] for modes of the form (2) and of the more general form

n(r, t) = n(z)ei(PX -D0. (3)

In the present Letter, limited to the case p = 0, we derive the following new result:the (complex) eigenfrequencies w("}, near w,, are determined by the bulk plasmondispersion Eq. (1) for small q, and by an effective, complex, slab thickness

S=- L - 2db, (4)

where d4, precisely defined below, is an effective, complex, surface thickness. Thethickness db is reminiscent of the parameter dj. introduced by Lang and Kohn [5] forW = 0 and generalized for arbitrary w by Feibelman [6] but is not the same quantity.The allowed (complex) interior wavenumbers, q("), are given by

q (")T =nr" (n =1,2,...) (5

and the corresponding (complex) eigenfrequencies are given by Eq. (1):

2

We now derive the above-stated results. We take the equilibrium electron density,n_(z), to vanish for z < -c and z > L + c, and introduce an auxiliary thickness(71 a, large compared to the surface thickness but much smaller than L. We thendistinguish two surface regions, -c < z < a and L - a < z < L + c, and the bulkregion a < z < L - a, in which n_(z) = ff (see Fig. 1).

We consider a collective mode of frequency w, with self-consistent electric field,E(z), parallel to z. This field induces a current density,

L+e

j(z) = dz'a(z,z')E(z'), (7)

where a(z, z') = f dx'd y 'ao,,(z - - y'; z, z'); a,, is the zz-component of the non-local conductivity tensor. (For ease of notation, the time dependence, e-t, and thefunctional dependence of fields, currents etc. on w will generally not be explicitlywritten out.) The quantity a(z, z') is short-ranged in the difference variable (z - z');also, when both z and z' are in the bulk region,

(zz') = a(z - z'), a < z,z' < L- a (8)

equal to the bulk conductivity correrponding to W" and w. Of course a(z, z') vanisheswhen either z or z are outside the slab (-c, L + c), as do E(z) (due to chargeneutrality) and j(z).

Denoting the induced charge density by n(z), the continuity equation and Gauss'law are

iwen(z) + -jz = 0 (9)

and

dE(z) - 4ren(z) (10)dz

(where e has been taken as positive). Since at z = -c both j(z) and E(z) vanish, (9)and (10) imply that, for all z,

j(z) = i-E(z). (1

41r

We can eliminate j(z) from (7) and (11) which results in the integral equation

i,, [ L+ o:FE~) = dz'o.(z, z')E(z'). (2-E(z) = d'(12)

We are looking for solutions, which, in the bulk, behave as e*iqz, with q <. kF.In this limit of long wavelengths in the bulk, we can, with Eq. (8) in Eq. (12) obtainthe frequencies of these solutions by expanding E(z') about z when z and z' are inthe bulk. The result is

3

47r 2

where

ao(j) = Jdz'a(z - z'; w)= ip- (14)

a, (W) = I dz'(z' - z) 2a(z - z'; W). (15)

Equation (13) is the bulk plasmon dispersion relation of an infinite system of conduc-tivity a(z - z'). In the limit L > a, the frequencies of the lowest-lying eigenmodesof Eq. (12) are very near wp and the shape of E(z) in the surface region tends to thelimiting shape

E(z) --+ E(z) - E(z;,w,) (16)

shown schematically in Fig. 2. We arbitrarily take E(a) = 1 and write, near z = a

Cz) = - (17)a - db

where d4 is the, as yet undetermined, point where E(z) extrapolates to 0. When z isin the bulk the solution which joins (17) near z = a is

1E(z) - 1 sin q(z - db). (18)q( a -- rib)

ThuE, db plays the role of a (complex) scattering length.The length d6 can be simply expressed in terms of the limiting surface density

distribution, i(z), associated with collective modes with q -- 0 and W -4 wp. FromEq. (10) and the form of E(z) in the surface region we see that •i(z) has the generalform shown in Fig. 2. Thus, for z near a, we have

E(z) P [~ d ,z,(f)zdfi(z/)1-S"dz(z) dz -('('))- -

(-4re) -c -o [dz'd j

C dz'Z d--z7 - dz'z- ' (19)

Z 1 d' Jc dz'

The upper limits oo are to be understood as > kF? but < L. The linear form ofE(z) near a, Eq. (19), extrapolates to 3ero when z equals

"fb dz z"'W (20)f_: dz du

4

Thus, db is the center of mass of the function dýi(z)/dz.The eigenmodes are either even or odd about L/2 so that, by Eq. (18), the

allowed values of q are

q(n)T = nir, (n = 1,2,...) (21)

where "E is give by Eq. (4). The corresponding frequencies are given by Eq. (1).Using the notation d6 = d'6 + idg' etc, we have

W =W(('p•+ k2 -4a"n t (22)

I(~= (," + 4a=db)(), (23)

where T = L - 2db. We observe that the real parts Eq. (22) of the frequenciescorrespond to bulk plasmon modes with the boundary conditions E(4) = E(L-,d') =0. The imaginary parts, (23), have contributions from the bulk (oc a") and from thesurface (cc (da''/T)).

We have solved the integral equation (12) in the discrete form

N-ZEn = b F, O.n En, (4nE (24)

where b = LIN is the interval length. We have chosen L = 35.4 A, N = 248, b = 0.14A, and c = 0. For the sake of illustration, we have chosen the following values forOamn. In the "bulk", 4 _ m < 125, we have chosen

=r"n :p(w) = W-, [a2 - i*1], amom+2 = =,m-2 =,4irwb 37rkb 2_ _p 1.

m =m+ .+1,÷m = i, = 21rk•b 3 C [C2 - /•a 1 , m,..+3 = O.nm-3 = VL, (25)

and symmetrically for m > 125, with kF = 1.75 x 10s cm-', hwp = 15 eV, anda = 0.57 + iA.025 in Eq. (1) (appropriate [1) for Al in the limit of q/kF < 1). Thefunctions p(w) and v are chosen so that the average and second moment of am,, areoro(w) and a2 (wp). In the "surface", 0 < m < 3, we have chosen

aOn = Ono = 0m .O',m, = T (ip(w) + E)4

a'.=,++ 1 = O1,+i. = (iv + f)

am,.+2 = 0am+2,m = 7 (iv + C)

om,m+3 = O"m+3,m = T' (iv + ,) (26)

5

and symmetrically near the other surfaces. The small real part f =,,a"/(6kb3 )was added" to the conductivity in the surface regions as a simple model of dampingin the surfaces due to decay into electron-hole pairs. Apart from this small realpart, equation (26) is a simple interpolation between no,, = a,,o = 0 and the bulkconductivity. The results for the lowest three modes are plotted in Figs. 3 and 4, andconfirm our conclusions, Eqs. (5), (6), (22) and (23).

How do our results relate to charge density waves in classical Maxwell theory?"Classical" here means a local a(z, z')

i W2od...(Z, z') = aoot(z - z') = 4- b(z - Z'), (27)

so that (12) becomesiEw z iw_E(z)= 4- E(z). (28)

4v 4rw

The boundary conditions [8] are, from charge neutrality, E(0) = E(L) = 0. Thus,classically the general solution is w = wp and E(z) = F(z), an arbitrary function of zsatisfying the boundary conditions E(0) = E(L) = 0. In particular, the functions

N" nrFl(z)=sin(q (z) q~• -.- , n = 0,1,... (29)

iD('•) _wp, all n, (30)

are solutions. Thus, our non-local and translationally non-invariant o(z, z') leads to:(1) replacement of L by the complex TL = L - 2db; (2) a finite dispersion of w(');and, if a(z, z') is known in detail in the surface region, the detailed field and densitydistributions in the surface region.

We now turn to resonant response to a uniform external field, Eoe-i'. Eqs. (7),(9) and (10) remain unchanged, but in Eq. (11) E(z) is replaced by E(z) - Eo andEq. (12) becomes

iw jL+cj- [E(z) - Lol -- dz'a(z, z';w )E(z'). (31)

Near w = wp we can write this as

al ,'_r(z z- P ++-LZ(z) - )' E(z') +I, IE(z)- jdz, %(z') "447 47r Ez 47r

(32)

where o'(z, z') =_ a(z, z'; w,), r(z, z') - [da(z, z';w)/dw1_., and q W - Wp. On theright-hand side, r7 can be neglected. - O

Setting E0 = 0, for a moment, gives us the equation for the slightly complexnormal modes, E,(z), and eigenvalues, j7,, = 17' + iri. The kernels oa(z, z')/i, r(z, z')/iare symmetric and (slightly) complex. The En(z) satisfy the orthonormality relations

6

_C-' dzdz'E,(:) [6( - i 1 E-() (33)

We now substitute the Ansatz

E(z) = • A.En(z) (34)

into (32), giving

A.q- qj,~E,,(z) = ý'o (3-5)n4

Using the orthonormality relation Eq. (33) leads to

An p•, - 4• : ,,) - dz i-+ dz'E,,(z') 6(z - z') ('(z') (36)

Only even modes, n - 2,4... are excited. A well defined resonance occurs for q - j7,,if 41Z < It" - r'?n*2, In this case as single term in (34) is dominant, giving

E.,(z) = Lo w4 [(07- i,)l+ (37)

In order to resolve the discrete modes experimentally, the spacing between the modesmust be larger than the width. From Eqs. (22) and (23) we find that the spacingand width are given by

( I- --) a' (1)2 -4)77n• 17n•, k 2 (4n - )(38)

,7n = - n. (39)

Thus, as n increases, the width gains on the spacing. For a film of width L' = 30 A,db = db' = 1 A, and ci' = 0.57, a" = 0.025 (Al), the highest well defined resonance isn 20.

We plan to address the considerably more complex theory of modes with finitewavevector p (Eq. 3) in the near future.

We thank Professor G.C. Strinati for discussions. Support by the Office of NavalResearch, Grant No. ONR-N00014-89-J-1530 and the National Science FoundationGrant No. NSF-DMR-90-01502 and NSF Science and Technology Center for Quan-tized Electronic Structures DMR91-20007 is gratefully acknowledged.

7

REFERENCES

[1] S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982); K. Utsumi and S. Ichimaru, Phys.Rev. B23, 3291 (1981).

[21 M. Anderegg, B. Feuerbacher and B. Fitton, Phys. Rev. Lett. 27, 1565 (1971).[3] P. Ynsunkanjana et aL, Phys. Rev. B46, 7284 (1992).[4] P. Feibelman, Phys. Rev. Lett. 35, 617 (1975); Phys. Rev. B12, 4282 (1975).[5] N.D. Lang and W. Kohn, Phys. Rev. B7, 3541 (1971).[6] P. Feibelman, Prog. Surf. Sci. 12, 287 (1982).

[71 Our final result is independent of the exact choice of a and c.[8] K.L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966), incorrectly use E'(0) =

E'(L) =0.

8

FIGURES

FIG. 1. Positive background density, n+(z) (dashed line), and equilibrium electrondensity, n_(z) (dotted line).

FIG. 2. The limiting (w -- Wp) form, E(z), and mode density h(z) in the surfaceregions. The mode density i(z) approaches the constant value no in the bulk.

FIG. 3. (a) The real part of the electric field of the three lowest bulk-like eigenmodesnear the surface. In the surface, the fields all approach the limiting form k(z). Insert Thereal parts of the three lowest bulk-like eigenmodes for 0 < z < L/2. In the bulk region, thefields are sinusoidal. The vertical dashed line marks the position of a.

FIG. 4. (a) Real and (b) imaginary parts of the plasmon eigenfrequencies. The straightlines are the dispersions Eqs. (22) and (23) with L' = 34.8 A and db" = 6 x 10-2 A obtainedfrom Eq. (20).

9

\ N

+

cri--- I

I - - 0

N

- -,- - 0 o

C.)4

/ N

NN

S• 0

0

0

I -

0

Co

C\2

0

0

CiQ

d

0

NO"0

-0

0 00 0

co 0\2

d