Electron transport through quantum dots

Transcription

1 Electron transport through quantum dots S. Rotter, B. Weingartner, F. Libisch, and J. Burgdörfer Inst. for Theoretical Physics/E136 December 3, 2002 Closed billiards have long served as prototype systems in the field of classical and quantum dynamics [1]. Studying the motion of a particle on a twodimensional confined surface contributed enormously to an understanding of dynamical phenomena. Since the behaviour of the billiard system strongly depends on the geometry of the limiting boundary, regular and chaotic motion can be investigated by shaping this boundary accordingly. One of the systems that has been studied extensively in the literature is the so-called Bunimovich stadium [2]. In spite of its simple shape this structure can be proven to be a strongly chaotic billiard [3]. The contrast to its regular counterpart (i.e. the circle) has received considerable attention in the past. In addition to a thorough investigation of numerous classical features, billiards were subject to a quantum mechanical study. Many interesting results on the statistics of energy levels [4], and the density distribution of the wavefunction ( scars ) [5, 6] have been obtained. Only a few years ago, the advances in the fabrication of semiconductor structures allowed the experimental realization of phase coherent scattering devices [7]. Of particular interest are billiards formed in high-mobility GaAs/Al x Ga 1 x As heterostructures of high purity. At low temperatures their elastic and inelastic mean free paths can be made larger than the system dimensions. The electronic motion through these devices is thus ballistic (phase-coherent) rather than diffusive. Since the two-dimensional electron gas in these devices has a very low density, one may neglect many-particle effects. A very strong influence on the electron is the particular shape of the confining potential. To make such structures accessible to measurements, one needs to connect

2 e - Figure 1: The stadium billiard with perpendicular leads. The arrows indicate the directions of electron transport through the microcavity. The quantum wave gets injected from the left and can either be reflected or transmitted through the attached waveguides. them to the outside world. This is done by attaching to the billiard domain different electro-chemical potentials by means of two semi-infinite waveguides (leads). By doing so, one can study the transport of electrons through these systems. The aim of investigating such devices is to find out how the transport properties are determined by the system boundary. Experiments [8, 9] as well as numerical investigations [10, 11] showed strong conductance fluctuations as a function of the electron wavelength or a perpendicular magnetic field. One of the standard geometries investigated is shown in Fig. (1). In that case two leads which are attached to a Bunimovich stadium are orthogonal to each other. A common approach for solving the quantum transport problem through mesoscopic scattering devices is by means of the Green s function. One way of calculating this constant energy propagator is the so-called Recursive Green s Function Method (RGM). This method proceeds by a discretization of configuration space. But even in a discrete regime it is a non-trivial problem to calculate the Green s function for an arbitrarily shaped scattering structure. We notice however that the geometry depicted in Fig. (1) is made up of substructures that are separable in either cartesian or polar coordinates. For these subsystems, i.e. the semi-infinite lead, the semi-circle and the rectangle, the Green s function is either known analytically or can be obtained numerically. The Green s function of the whole scattering geometry can then be calculated recursively out of the various sub-pieces. The corresponding recursion is set up as a matrix equation in which the discrete Green s functions enter as matrices. The known variables of the equation are the Green s functions of the sub-pieces and the unknowns are

3 those of the total geometry. The solution of the recursion involves the inversion of the Green s function matrices. In Fig. (2) we give a graphical illustration of the connection of the lead (cartesian grid) with the semi-circle (polar grid). For this particular scenario the Green s functions entering the recursion for the connection of A and B would be 4 4 matrices, corresponding to the 4 grid points involved.fig. (3) A B Figure 2: The discrete transition from the lead to a circular cavity. The Dirichlet boundary conditions are indicated by empty discs. For calculating transport coefficients through such a junction, the Green s functions at A and B are written in a matrix scheme. illustrates the connection of the semi-circle to the rectangle. The number of grid points must be very high for a good approximation of the continuum solution and for high magnetic field. The involved matrices reach a size of about which have to be evaluated for every gridpoint (1500) simultaneousely. Their inversion is the computational bottleneck that sets a time limit for going to smaller wave-lengths or to higher magnetic fields in our investigations. It is therefore crucial to use fast matrix inversion routines in the framework of our fortran codes. It is most important to note that in our given physical context these routines need several GigaBytes of memory. This can only be done at facilities that dispose of large and fast memory devices. The success of our project is thus strongly dependent on the availability of computers with large-scale linear algebra capabilities as provided by the ZID. The machines used for performing the above computations are FP98 and

4 Figure 3: The semi-circle can be connected to a rectangle by means of an additional link module [12]. For a good approximation of the physical transport the mesh size must be chosen in accordance with the electron wavelength. HAL. We mainly employ the fortran facilities installed on these computers. The duration of the above project will be several years. References [1] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer Verlag, New York, [2] L. A. Bunimovich, Funct. Anal. Appl. 8, 254 (1974). [3] G. Benettin and J.-M. Strelcyn, Phys. Rev. A 17, 773 (1978). [4] O. Bohigas, M. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984). [5] S. McDonald and A. Kaufman, Phys. Rev. Lett. 42, 1189 (1979). [6] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984). [7] See, e.g., Quantum Transport in Ultrasmall Devices, edited by D. Ferry, H. Grubin, C. Jacobini and A.-P. Jauho, NATO ASI Series B, Vol. 342, Plenum, NY, 1995.

5 [8] C. M. Marcus, A.J. Rimberg, R.M. Westervelt, P.F. Hopkins, and A.C. Gossard, Phys. Rev. Lett. 69, 506 (1992). [9] M. W. Keller, O. Millo, A. Mittal, D.E. Prober, and R.N. Sacks, Surf. Sci. 305, 501 (1994). [10] H. U. Baranger, R. A. Jalabert, and A. D. Stone, Chaos 3, 665 (1993). [11] H. Ishio and J. Burgdörfer, Phys. Rev. B 51, 2013 (1995). [12] S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgdo rfer, Phys. Rev. B 62, 1950 (2000).