Ann. Physik 4 (1995) 191-201 Annalen der Physik @ Johann Ambrosius Barth 1995 Conductivity of a small metallic particle F.C Schaefer and R. v. Baltz Institut fur Theorie der Kondensierten Materie, Universitlt Karlsruhe, D-76128Karlsruhe. Germany Received 10 November 1994, accepted 20 January 1995 Abstract. The size-dependent uc-conductivity of a small metallic particle is studied for electrons enclosed in a box with random impurities. The exact wave functions and energies are calculated numerically up to 5 0 0 electrons, then the Kubo-formula is waluated, and an ensemble average is performed. A strong size- and frequency dependence of the conductivity is found which is in good agreement with random matrix results. The size dependence of the dc-conductivity agrees qualitatively with microwave absorption measurements on submicron particles. Keywords: Small particles; Conductivity; Numerical simulation. 1 Introduction Systems which are smaller than the phase coherence length of the electrons cannot be described by classical transport theory, and striking new phenomena such as universal conductivity fluctuations were discovered [ 11. In addition, the discreteness of the energy levels changes the electric properties drastically compared to the bulk material in which energy bands determine the electronic behavior. Still, the size- and frequency-dependence of the conductivity ( = mean conductance) or polarizability is not fully understood [2-61. Within a Drude-description =, a(w) = -, 1 +iwr a,=- ne2r 9 (1) m0 it is plausible that the scattering rate y = 1/r is increased by surface-processes [4] 1 - 1 rbulk + -1, Tb R rb=a- (2) “F thus reducing the dc-conductivity of a particle with radius R. a = 0.29. . .0.39 [9]. n, mo,vF, rbulk respectively, denote the density, mass, Fermi-velocity, and scattering time of the electrons in the bulk. 56 is the “ballistic” (or transit) time of the electrons inside the particle. 192 Ann. Physik 4 (1 995) Random matrix theories [7 - 101, on the other hand, lead to essentially different results. For w > 6 / h , where 6 denotes the mean level-spacing, Frahm et al. [9] obtained for the ballistic and diffusive limits 72 R4 R e a d ( w )= a O - ~ =w 2 , 175 V F T , ] 7,1 14R (4) and I = ~~7~~are the elastic scattering time and mean free path, on = D / R 2 and D = v ~ r e 1 / 3denote the Thouless frequency and the diffusion coefficient. Eqs. (4) holds only, if additionally o < min(mThrl / ~ , , ) . Our aim is to present a quantum mechanical investigation of the conductance of a small particle and to compare it with the theoretical [7- 101 and experimental [ l l , 13, 141 results on submicron particles. The model consists of independent electrons enclosed in a box with random impurities. The exact wave-functions and energies are calculated numerically up to 500 electrons, then the Kubo-formula is evaluated and the ensemble average is performed. This model has been used by Genzel et al. [4]and by Wood and Ashcroft (151 some time ago for an analytical study of the same problem. But within their approach, impurity scattering was treated only phenomenologically, whereas in our calculation, surface-effects as well as electron-impurity scattering is incorporated from the very beginning. Inelastic scattering, however, will be considered phenomenologically, too. 2 Electrons in a cubic box with impurities We consider N independent electrons confined to a cubic box of linear dimension L , including Ni randomly positioned impurities with individual potential VJr), where, for simplicity, delta-function potentials will be used, V,(r) = V56(r).Without impurities, the wave functions and energies are given by w,(r) = (+) 312 sin t ~ (y) ) (r> sin sin n nz , O s x , y , z s L , j = ( l , m , n ) , and l , m , n = 1,2,3 ... To obtain the wave functions and energies of the electron impurity system, we set up a matrix representation of the Hamiltonian in terms of the base states (6) 193 F. C. Schaefer, R. v. Baltz, Conductivity of a small metallic particle The Hamiltonian matrix (8) will be truncated to M x M , with A4= 2 N so that there are sufficiently many excited states to saturate thef-sum rule to better than 1Yo (see (16)). Then the truncated matrix is diagonalized with standard routines yielding the energies E, and eigenvectors Cy for a selected realization of impurity configurations. Finally, an ensemble average is performed (about 25 realizations). H', has the properties of a real random matrix so that all degeneracies of (7) are lifted (Figs. 1, 2). Apparently, however, the spacing distribution (Fig. l), does neither coincide with the Wigner surmise [ 161 1 .o n a 0.5 0.0 0.0 1 .o 2.0 3.0 S Fig. 1 Distribution of adjacent energy levels. Full line: numerical results (100 electrons), dashed line: Wigner-distribution, dotted line: exponential distribution. 1.5 z M 'W 2. 1.0 N A 3 0.5 0 0.0 0.0 0.5 1 .o 1.5 2.0 E/EF Fig. 2 Normalized density of states. Full and dotted lines: numerical results for 200 and 500 electrons, dashed line: free electron parabola. 194 Ann. Physik 4 (1995) nor with the exponential distribution for uncorrelated levels, which are supposed to hold for an ensemble of Gaussian orthogonal matrices or integrable systems, respectively. S = 6 E / 6 E is the separation of adjacent levels normalized malized to their mean value. This can be understood by the estimated values of the ballistic time and elastic scattering time, r e I = 2 q , , so that even the largest systems studied are more ballistic than diffusive (see next section). Therefore, the level statistics is expected to be strongly influenced by the properties of the clean system which, due to the cubic geometry, is integrable. 3 Conductivity To calculate the size-dependent ac-conductivity ( = mean conductance) we start from the Kubo-formula [ 171, where, as usual, the influence of field inhomogeneities inside the specimen are neglected [7 - 10, 151. I a ) , a = 1 , 2 . . . M denote the exact eigenstates in terms of the eigenvectors of (8), M f, = f ( E a - p ) is the Fermi-function, Ea, = Ea- E,, and 4'0'. In the next step, the dipole matrix elements are calculated for a selected impurity configuration, Then, the Kubo formula is evaluated, and, finally, an ensemble average is performec (about 25 realizations). To compare with random matrix theories, e.g. [7 - lo], we alsc give the magnitude of the dipole matrix elements, as a function of transition energie: (Fig. 3). The dashed line shows the analytical approximation used by Frahm et al. [9] F.C. Schaefer, R. v. Baltz, Conductivity of a small metallic particle 7 1--,_ 195 7 . X \ \ Fig. 3 Dipole matrix elements. Full line: numerical results (100 electrons), dashed line: analytical approximation (IS). 1o-& \ I \ L I L 3 R = -L = 0.62L. a=0.35 from (2). AS q+O+, the real part of the conductivity (11) of a single particle would consist of a series of &functions. This structure is smeared-out when performing an ensemble average on infinitely many samples whereas our study merely allowed for 25 samples at each frequency. In addition, the so far neglected inelastic scattering processes would smoothen the singular structure, too. On a phenomenological level, this effect can be obtained by identifying q with the inelastic scattering rate yin = l/rin. In general, such a simplified treatment of a relaxation phenomenon would spoil the f-sum rule and would violate number-conservation. Nevertheless, the conductivity as given by (1 1) turns out to be number-conserving even for finite values of q. We note that this description is equivalent to Mermin’s result [18] for the dielectric-function of the electron gas with damping in terms of the Lindhard-function and using this result for a finite system, too. At zero wavevector, we have E~(o)= ( :) 1 + 1+- [~(u+iy)-l] With E = 1 +ia/o~,,,(17) leads to (11) with the infinitesimal q replaced by a finite y (9,151. 196 Ann. Physik 4 (1995) 4.0 b" 3.0 13: w b 2.0 a, IY 1 .o 0.0 0.0 0.2 0.4 0.6 0.8 1 .o 0.6 0.8 1 .o Q/WP 3.0 2 .o n b 1 1.0 3W - b -E 0.0 -1 .o -2.0 0.0 0.2 0.4 W/WP Fig. 4 Normalized conductivity. a, = ne*T,/rn, = 3.87 x 105/Clm. a) Real, b) imaginary part. Full lines: numerical results (100 electrons), dashed lines: random matrix theory [9]. The frequency-dependent conductivity and dielectric functions are displayed in Figs.4, 5 . If possible, parameters suitable for bulk indium have been chosen [19]: electron density n = N I L 3 = 1.15 x Id3cm-3, plasma frequency up= 1.9 x lot6s-'. Fermi-energy, velocity and wave-vector EF.= 8.6 eV, uF = 1.74 x lo6 m/s, kF = 1.5 x 10'om-t. From the &-conductivity 60= 1.25 X 107/Rm (at room temperature) we have for the bulk scattering time wp?bulk = 73 and mean free path Ibulk = 6.7 nm. rin=100 reI. For our numerical studies an impurity concentration of N i / N = 0.1 is chosen. Then the scattering potential V, may be estimated within the Born-approximation [20] 1 N, --1 --?bulk 4n L 3 2 F. C. Schaefer, R. v. Baltz. Conductivity of a small metallic particle 40 1 197 I I I - \ Q) [r , 0 -10 - -20 0.0 0.2 0.4 0.6 0.8 1 .o 0.6 0.8 1 W/QP 40 30 A 3 W 20 E 10 0.0 0.2 0.4 .o W/WP Fig. 5 Dielectric function. a) Real, b) imaginary part. Full lines: numerical results (100 electrons), dashed lines: random matrix theory [9]. to Vs/EF= 0.8 A [3]. But then, (1 1) would give a much too small dc-conductivity accompanied by unacceptable large statistical fluctuations! Therefore, merely for computational reasons, we took a much larger value for the scattering potential at the price of an estimated bulk dc-conductivity of about one tenth of the experimental value. For 100 electrons in a box of size L = 0.96 nm the remaining parameters are: level spacing 6 = 0.11 eV, estimates of characteristic times: ~ ~ ~ ~ 5 wpre,=5, 2 . 3 , and o T h = 0.02 up. In the high frequency limit, the response is Drude-like. For w-+O, however, a(o) and E ( O ) are drastically reduced due to the coherence of wave functions over the size of the specimen. In order to analyse this phenomenon in more detail, we use the memory-function M ( w ) defined by a(o)= n e2/mo M ( o )- io Ann. Physik 4 ( 1 995) 198 Eq. (19) generalizes the Drude-function (1) to frequency-dependent scattering rates. Causality requires that M ( o ) , like a(&), are analytic functions in the upper-half of the complex a-plane. In the ballistic and diffusive limits, the random matrix theory results (3, 4) yield At low frequencies (but still h a > & , the magnitude of R e M ( o ) as well as the strong increase of I m M ( o ) , Figs. 6a, b, agree well with (20). Notice the qualitatively different 0.5 0.4 3" A 0.3 3 W I 0.2 Q) t Y 0.1 0.0 ' 0.0 I I I I 0.5 1 .o 1.5 2.0 W/WP 1 .o I-I I I I 5 I I 1 I 0.8 3" 0.6 3 W 2 0.4 E - 0.2 r= 0.0 I 1 0.0 0.5 n I I 1 1 .o 1.5 2.0 W/WP Fig. 6 Memory-function. a) Real, b) imaginary part. Full lines: numerical results (100 electrons), dashed lines: random matrix theory [9]. Inset: low frequency behaviour. F. C. Schaefer, R. v. Baltz, Conductivity of a small metallic particle 199 n C L C W r= Fig. 7 dc-Conductivity. Squares: numerical results (10. . .SO0 electrons). Dashed lines: extrapolated region of experimental data [ I I]. structures of (20, 21) and (2). From our numerical results we estimate ReM/oP=0.15, which is about one third of the analytical result, whereas there is perfect agreement for the imaginary part at low frequencies. An approximate result for the memory function is worked out in the appendix. In electrotechnical terms, M ( o ) describes an impedance shunted in series to the (-inductive) free-electron part - io.(Notice the different sign-convention with respect to i when compared with electrotechnics). The i/o-term in (20, 21) describes a “capacitive” behaviour, which originates from the surface-layer where the carriers are immobile and from the tendency of the impurity potential to localize the electron states inside the particle. For small particles we expect an increase of the inelastic scattering time with decreasing particle size (freeze-out of transition-channels). 2 qn(L)= rin+-X E , DL where D is the diffusion coefficient and xc denotes the coherence length. At roomtemperature xe= lo-*. ..lO-’m [14]. For very small frequencies (and finite, size-dependent 4, (L))our numerical results lead to a dc-conductivity which is in qualitative agreement with experimental results (11 - 141 (Fig 7). 4 Discussion A nearly quantitative agreement between our numerical data and random matrix theory results is found (in the ballistic regime). This holds because the analytical treatment is essentially based on sum rules and deviations mainly originate from differences between the matrix elements (Fig. 3). The frequency dependence of the conductivity, however, is much stronger than for the random matrix theory results. Some of the peak structure in E (0)and R e M ( o ) around the plasma frequency is also resolved in Wood et al. [15] analytical results and, in some respect, it resembles the bulk and surface plasma modes in thin metallic films [21]. Yet E ( o ) # O , 00 at these frequencies, hence, 200 Ann. Physik 4 (1995) it is not a collective phenomenon but originates from the structure in the dipole matrix elements (Fig. 3). Remarkably, the microwave absorption measurements on submicron indium particles [9- 1 I , 13, 141 are described qualitatively correct when extrapolated to smaller size. Our numerical study, however, is limited to rather small particles. The conductivity obtained above describes the response to a local electric field which, for a metallic particle, is much weaker than the “bare” field applied far from it. This screening of the applied field greatly reduces the absorption and must be taken into accdunt within macroscopic electrodynamics. However, the actual fields inside a metallic particle are very inhomogeneous, at least at low frequencies, so that spatial dispersion as well as local field effects are neglected (cfm. [IS]). The mean field approach poses some limitations, too [22]. In addition, conductance measurements using leads or contact-free methods (e.g. microwave or IR investigations) may not give the same result [6, 11 - 14). Deviations between experimental and theoretical studies may be due to these shortcomings. It is also not quite clear if the (size-dependent) dc-conductivity of a small particle is nonzero or not. It had been stressed by Kawabata and Kubo [24] a long time ago that the surface is not really a scatterer but does determine the eigenstates of the system which are bound to a finite volume rather than free states scattered by the surface. This aspect, however, is incorporated within our approach from the very beginning and a finite dc-conductivity is essentially due to inelastic scattering processes. Appendix An approximate evaluation of the Kubo-formula may be obtained by replacing the transition energies Eaa with the mean level separation, hwo=EF/N [7, 91, and then using the sum-rule (16). This leads to Within this approximation, the scattering rate is enhanced at low frequencies with respect to yin. For large radii, however, M ( o )= yin. Obviously an elastic contribution to the scattering rate is missing so that this result can be reasonable only for w > o o . We thank G. Nimtz and P. Marquardt for helpful discussions. Part of this work was supported b j the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 195. References [ l ] G. W. Bryant, Comments Cond. Mat. Phys. 14, 367 (1989) [2] J.A.A.J. Perenboom, P. Wyder, F. Meier, Phys. Rep. 78, 174 (1981) [3] W.P. Halperin, Reviews of Modern Physics, 58, 533 (1986) (4) L. Genzel, T.P. Martin, U. Kreibig, 2. Physik B 21, 339 (1975) (51 U. Kreibig, L. Genzel, Surf. Sci. 156, 678 (1985) [6] D.B.Tanner, Phys. Rev. B30, 1042 (1984) [7] L.P. Gorkov, G.M. Eliashberg, Sov. Phys. JETP 21, 940 (1965) [8] U. Sivan, Y. Imry, Phys. Rev. B35, 6074 (1987) [9] K. Frahm, B. Miihlschlegel. R. Nemeth. 2. Physik B78 (1990) F. 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