VOLUME 2, NUMBER 9, SEPTEMBER 2002 © Copyright 2002 by the American Chemical Society
Spectrum of Electrons Confined to Rotationally Symmetric Nanoparticles P. Malits,†,‡ A. Kaplunovsky,*,†,§ I. D. Vagner,†,§,| and P. Wyder† Grenoble High Magnetic Field Laboratory, Max-Planck-Institut fu¨ r Festko¨ rperforschung and Centre Nationale de la Recherche Scientifique, BP 166, 38042 Grenoble Cedex 09, France, PERI at Ruppin Academic Center, Emek Hefer, 40250, Israel, RCQCE at Department of Communication Engineering, Holon Academic Institute of Technology, 52 Golomb Str., Holon 58102, Israel, and Department of Physics, Clarkson UniVersity, Potsdam, New York, 13699 Received May 16, 2002; Revised Manuscript Received July 6, 2002
ABSTRACT The energy spectrum of an electron confined to an arbitrary surface of revolution in an external magnetic field, parallel to the symmetry axis, is studied analytically and numerically. Via conformation mapping of the cross section of the circle, the problem is reduced to one on the surface of the sphere. The case of a spheroid is considered in details, and the dependence on parameters is discussed. In the high magnetic field limit a regular structure in the energy spectrum, resembling the Landau levels, is obtained.
Recent technological progress in fabrication of semiconducting and metallic nanostructures opened a vast field of research of their electron properties.1,2 Traditionally, application of high magnetic fields is an extremely powerful method for experimental studies of electronic properties in solids. Detailed theoretical study of the electron spectrum of the nanostructures under strong magnetic fields is therefore of primary importance for the future progress in this field. Initially the solutions for an electron in confined plane geometries, such as a disk, ring, cylinder, or oval shaped stadium,3-5 were proposed. As it was shown in ref 5, these models are relevant to the notion of chaos in the level statistics and related thermodynamics of such systems. * Corresponding author. † Max-Planck-Institut fu ¨ r Festko¨rperforschung and Centre Nationale de la Recherche Scientifique. ‡ PERI at Ruppin Academic Center. § Holon Academic Institute of Technology. | Clarkson University. 10.1021/nl020214+ CCC: $22.00 Published on Web 08/07/2002
© 2002 American Chemical Society
Among already studied three-dimensional systems are electrons on a simple surface such as a sphere.6,7 A challenging problem is an adequate quantum mechanical description of noninteracting electrons on a nanoparticle of an arbitrary shape. Here we consider a single electron confined to the surface of revolution placed in an axial uniform magnetic field. Our goal is to treat the general case of the arbitrary shaped surface of revolution r ) f (z) ((r, φ, z) are cylindrical coordinates) and to investigate the influence of its geometrical characteristics upon a quantum mechanical spectrum. Further, we suppose the surface to be smooth, closed, and crossing z-axes only in two points. The uniform magnetic field B is defined to point in the z-direction. The problem is described by the Hamiltonian H)
1 [ih∇ - A]2 + V 2m
(1)
where, for simplicity, we ignore spin dependent terms. A ) B(-y,x,0)/2 is the symmetric gauge, and (x,y) are Cartesian coordinates. This is leading to the Schro¨dinger equation on the surface r ) f(z) ∂ - B21 r2 - V1 ψ ) -E1ψ ∆ + 2iB1 ∂φ
(
)
(2)
where E1 ) 2mE/h2, B1 ) eB/2ch, V1 ) 2mV/h2. We introduce new orthogonal coordinates by z + ir ) F(u + iV), where the function F(u + iV) maps conformally the domain of the (u,V) plane containing the unit circle onto the domain of the (r,z) plane containing the closed curve r ) (f (z). This curve is the image of the circle u2 + V2 ) 1 with the arc 0 e θ e π (u + iV ) R exp iθ) corresponding to r g 0. Since conformal mapping conserves a normal to the surface, it enables us to write eq 2 on the surface R ) 1 neglecting derivatives in R. Thus, the three-dimensional Schro¨dinger operator has been reduced to a two-dimensional operator in (θ,φ) variables. Due to conservation of the z component of the angular momentum, the cyclic coordinate φ can be separated in the Fourier series development +∞
ψ(θ,φ) )
∑
m ) -∞
ψm(θ) exp (imφ)
(3)
Further simplification x ) cos θ results in the ordinary differential equation of the second order
where |x| e 1, |ψm((1)| < ∞; F0 ) b, ξ ) a2b-2 - 1. This problem has an infinite discrete spectrum λlm. Its eigenfunctions ψlm(x) have l zeroes in the interval (-1, 1). One can see that if l is an even (odd) integer, then these functions are even (odd). It can be shown that all eigenvalues λlm are positive. They are large if one of the following conditions is fulfilled: (1) l . 1, (2) m . 1, (3) B ˜ . 1. Below we point out leading terms of the corresponding asymptotics. As l . 1, the spectrum can be obtained by the method described in ref 8. Particularly, the leading term is given by E ˜ lm )
(4)
F0 ) max Im F(exp(iθ)), G0(x) ) Φ(x)[λ -
2 2 - m2F-2(x)]F-2 0 , Φ(x) ) |F′(x + ix1-x )| , λ ) 2 2 E ˜ - 2B ˜ m, E ˜ ) (E1 - V1)F0, B ˜ ) B1 F0. A low field (B ˜ , 1) asymptotics of the spectrum and eigenfunctions may be found in the traditional way by the perturbation method. It is much more difficult to suggest some general approach to indicate a high field (B ˜ . 1) asymptotics. This is governed by coefficients of eq 4 or, in other words, by the surface shape. Below we consider closely a spheroidal surface whose equation is
B ˜ 2F2(x)
2
2
r z + )1 a2 b2
(5)
w j a-b + (a + b) , w j )R 2w j 2 exp(iθ), is a one-to-one mapping of the unit circle R ) 1 onto this ellipse of the (r,z) plane. Equation 4 can be written in the form Conformal mapping: z + ir )
916
(x )
+ 2Bm + O
ξ 1+ξ
2
(1l )
(7)
where E(x) is a complete elliptic integral of the second kind. As m . 1, the asymptotic expansion may be found with ˜ lm - 2B ˜ m ) m2 + (2l + 1) a stretched variable λlm ) E
xξ |m| + O(1). Eigenfunctions are expressed by Hermite 1+ξ -2 polynomials ψlm(x) ) exp(-x2xλlm)H(xλ1/4 lm ) + O(m ), x∈(-, ). In the high field limit (B ˜ . 1) the spectrum is given by an asymptotic formula:
()
N ) l + cos2
Here G1(x) ) x - (1 - x2)F′(x)F-1(x), F0F(x) ) Im F(x + i
x1-x2),
π2 (2l+2|m|+1)2 16(1 + ξ)E
|x| e 1 |ψm((1)| < ∞
]
1 1 1 E ˜ lm ) 2NB ˜ - N(N - 2m)(ξ + 1) + (ξ - 1) + O 2 2 B ˜
d2ψm
dψm - G1(x) + G0(x)ψm ) 0 (1 - x ) 2 dx dx 2
[
dψm d m2 (1 - x2) + λ-B ˜ 2(1 - x2) [ξ(1 - x2) + 2 dx dx 1-x 1]ψm ) 0 (6)
(8)
πl + |m| + m 2
It is readily seen that the expression for the energy given in eq. 8 describes, in the case ξ ) 0, the spectrum of an electron on a sphere, presented in ref 7. The corresponding asymptotic expansion of the eigenfunctions is expressed by Laguere polynomials ψlm(x) )
()
x l (1 - x2)|m|/2 × |x|
(
)
()
1 1 2 exp - (1 - x2)B ˜ L|m| ˜) + O n ((1 - x )B 2 B ˜ n)
πl 1 l - sin2 2 2
(
)
x ∉ (-, ) Hence it appears that bunches of the energy levels resembling the Landau levels are formed in the high field limit. Every bunch consists of the parallel equidistant levels with the same number N. Their leading term is irrespective of a spheroidal geometry and coincide with the spectrum for the plane. The energy level corresponds to two quasi-degenerated bound states labeled (2k, m) and (2k + 1, m). A disk of radius F0 is a limiting case of a strongly flattened spheroidal shell (ξ ) - 1). In this limit, the values of the Nano Lett., Vol. 2, No. 9, 2002
eigenfunctions on both sides of the disk (x > 0 and x < 0) are added, and according to the above formula the antisymmetric eigenfunctions are canceled out. Since F20(1 - x2) ) r2, we obtain E1 - V1 ) 2B1(2n + |m| + m + 1) +
(
)
( ) ()
1 1 +O 2 F0 B ˜ F20
(9)
1 1 2 ψnm(r) ) r|m| exp - r2B1 L|m| n (r B1) + O 2 B ˜ r < F0 -
These relationships turn into the well-known Landau solution as F0 ) ∞. In the high magnetic field, the disk (circle billiard) spectrum coalesces into the straight lines nearly the same that in the classical Landau problem. This confirms results of the numerical calculations by K. Nakamura and H. Thomas.5 To calculate the spectrum, we represent ψlm(x) by the iβ ∞ cnPn+|m||m| (x) exp x2 ], β ) expression ψlm(x) ) Re[∑s)0 2 B ˜ xξ. Here indices n ) 2s + sin2 (π/2)l are either even or odd integers corresponding to the to symmetric and antisymmetric solutions, respectively. Substituting this expansion into the eq 6 yields two recurrence formulas (separately for even and odd integers n)
( )
Ascn-2 + Jscn + Dscn+2 ) 0
(10)
where As ) Js )
n(n - 1) 4(n+|m|-1)2 -1
[χ - iβ(2n + 2|m| -1)]
χ[2n(2|m| + n + 1) + 2|m| - 1] (2n + 2|m| + 1) - 4 (n + |m|)(n + |m| + 1) + λ1
Ds )
(n + 2|m| + 2)(n + 2|m| + 1) 4(n + |m| + 2)2 - 1
×
[χ + iβ(2|m| + 2n + 3)] λ1 ) (λ - B ˜ 2)(1 + ξ) - m2ξ, χ ) B ˜ 2(1 + ξ) - λξ The spectrum is determined by equating the infinite determinants of these equations to zero and is given, therefore, by the roots of the following continued fractions: 0 ) J0 (A1D0)/(J1-) (A2D1)/(J2-...). These continued fractions are real because AsDs-1 are real values. Results of the calculations for (ξ ) 3, ξ ) 0, and ξ ) 0.7) are presented in Figure 1. The spectrum for ξ ) 0 (sphere) is the same as that presented on Figure 1 of ref 6. We observe the behavior of the spectrum that was derived above analytically. As ξ or N increases, the distance between lines of the same bunch (the splitted Landau level) enlarges. Irregular crossings intensify as well. These crossings arise from dropping and intermingling lines of the different bunches. Examples of the wave functions for the various parameters are shown in Figure 2. Nano Lett., Vol. 2, No. 9, 2002
Figure 1. Spectra for |m| ) 0,..., 4; l ) 0,..., 7 in the following cases: (a) ξ ) - 0.7; (b) ξ ) 0; (c) ξ ) 3.0. Intensifying of irregular crossings is observed.
To conclude, it is shown that the electron motion on an arbitrary shaped convex surface exhibits spectrum behavior similar to the spectrum behavior of the electron on the spheroid. In the high field region, the spectrum behavior is 917
constitute bunches of straight lines that coalesce into the Landau levels as the surface is flattened (ξ f - 1). References
Figure 2. Wave functions for B ˜ ) 3; m ) 3; l ) 0; ξ ) - 0.7; 0; 1; 3.
predetermined by the surface flatness in the vicinity of the poles (where the electron is trapped), and energy levels
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(1) Perenboom, J. A. A. J.; Wyder, P.; Meier, F. Phys. Rep. 1981, 78, 173. (2) van Ruitenbeek, J. M.; van Leeuwen, D. A. Phys. ReV. Lett. 1991, 67, 640. (3) Bu¨ttiker, M.; Imry, Y.; Landauer, R. Phys. Lett. 1983, 96A, 365. (4) Wohlleben, D.; Esser, M.; Freche, P.; Zipper, E.; Szopa, M. Phys. ReV. Lett. 1991, 66, 3191. (5) Nakamura, K.; Thomas, H. Phys. ReV. Lett. 1988, 61, 247. (6) Kim, Ju H.; Vagner, I. D.; Sundaram, B. Phys. ReV. B 1992, 46, 9501. (7) Aoki, H.; Suezava, H. Phys. ReV. A 1992, 46, R1163. (8) Fedorjuk, M. V. Diff. Equations 1982, 18, 2166-2173. Fedorjuk, M. V. Diff. Equations 1983, 19, 278-286.
NL020214+
Nano Lett., Vol. 2, No. 9, 2002
