SAP Project:
Mesoscopic Correlations in Many-Electron Quantum Dot Systems
Jean-Pierre Leburton
Beckman Institute for Advanced Studies and Technology
University of Illinois at Urbana-Champaign
Research Objectives
SCIENTIFIC GOALS
Since the pioneering work by Tarucha et al. [1] demonstrating the possibility
of tuning the electron number in quantum dots (QD) all the way through to zero,
there has been a growing interest in understanding the behavior of electron system
in zero-dimensional systems. This interest is primarily fueled by the idea of
realizing a scalable solid state quantum computer where operations are performed
with qubits, each of which is represented by an electron spin confined in QDs
[2]. The realization of the quantum computer relies on the successful measurement
and control of the exchange energy (energy difference between two electron singlet
and triplet states) in the double quantum dot system, which so far has been eluding
the direct observation, though recent indirect estimates gave a value of about
20 meV [3].
The interplay among electron-electron interaction, spin, and coupling to a Fermi sea a in the leads makes investigation of single quantum dots (quantum dot atoms) and coupled quantum dot systems (quantum dot molecules) in the few-electron regime a subtle problem in many-body physics. Frequently, this problem is tackled by various approximate methods assuming simplistic electron confinement potentials in the QDs ( a simple parabola or their superposition is usually used). However, real QDs (especially double and triple QD systems) are not just parabolas; they represent an excellent example of the novel quantum dot device where interaction between the QD and the surrounding regions (leads and substrtate0 plays an important and sometimes definitive factor in the properties of the electron system [4].
Therefore, the development of accurate theoretical and computational methods capable of unraveling the many-electron problem by fully accounting for the correlation effects among electrons confined in the quantum-dot device is of the great fundamental and practical importance.
COMPUTATIONAL GOALS AND METHODS
Our approach is based on the numerically exact
diagonalization of the many-electron Hamiltonian which in the case
of two electrons confined in the QD takes the following form:
(1)
where here m* and 
stand
for the electron effective mass and dielectric constant respectively.
A = (1/2) (Bx, -By, 0) is the vector potential in
the symmetric gage for the magnetic field B oriented along
the z-direction (Zeeman splitting is neglected at present).
The single particle confinement potential Vcon f (r) can
be obtained from the multiscale density functional theory modeling
taking into account full electron and dopant distribution outside the
active QD region.
As we are interested in the energy separation between triplet an singlet states of the two-electron system (exchange energy), we diagonalize the above Hamiltonian (1) using a two-particle spin-free wave function with the definite value of the total spin S:
(2)
which is symmetric for a singlet state (S = 0) and antisymmetric for a triplet state (S = 1).
The expansion coefficients cij are
determined from the minimization of the total energy 
for
a given spin state which leads to the generalized eigenvalue problem
with dense Hermitian matrices (filling factor is about 1/2) which in
the current realization of the method is diagonalized using algorithm zhegvd in
LAPACK.
At present, the basis set of single-particle wave functions 
is
that of a three-dimensional anisotropic harmonic oscillator with frequencies
being adjustable parameters. We found that the frequencies smaller
than the confinement strengths work best due to the fact that the Coulomb
interaction tends to flatten out the effective potential. To simulate
the extension of the wave function in the z-direction, a single Gaussian
was used, whose frequency was also adjusted to get the minimal value
of the total energy ES. In case
of a 2D circular confinement, the Coulomb matrix elements can be evaluated
analytically yielding four-fold series. In the case of the anisotropic
2D potential (and/or to account for the 3D confinement), the matrix
elements are also expressed through the four (six)-fold series but
the auxiliary one-dimensional exponential integral evaluated numerically
by means of the Gauss-Kronrod qauadrature has to be used to compute
their final values.
Current single-processor memory limitations effectively preclude utilization of more than 9 x 9 harmonic oscillator states (producing the two-electron matrix of about 3000 x 3000) to form a two-particle wave function expansion (2). However, clearly, as the system becomes more complicated (single QD vs. double QD vs. triple WD, more basis states becomes necessary for accurate predictions of the exchange energy values.
Several possible approaches can be outlined to overcome the bottlenecks:
- Keeping LAPACK as a diagonalization routine, we use either a distributed memory (on tungsten and mercury clusters at NCSA) or shared memory approach and compare there respective performance. This should allow diagonalization and storage of larger matrices without substantial modifications of the existing software.
- Test other diagonalization packages for both sparse and dense matrix calculation and compare their respective performance.
- The major factor limiting performance of the method is the calculations of the Coulomb interaction matrix elements:
(3)
The usage of the harmonic oscillator basis set in this case is justified only by the fact that it is possible to calculate these elements analytically.
However, in the extended multiple QD systems such as double dots, the two basis sets centered in each of the dots could potentially be a better approach matching the physics of the system, and eventually giving rise to smaller matrices. The drawback here is the inability to perform analytical calculations for Vijkl, while the brute-force numerical calculation of the six-dimensional integral is cumbersome. This problem can be circumvented, in part, by solving numerically Poisson equation with the right-hand site given by the product:
(4)
by standard preconditioned conjugate-gradient methods with suitable boundary conditions, and then evaluating the remaining three-dimensional integral on the same grid:
(5)
Note that by using the same grid and the finite elements method, it may be easier to couple the exact diagonalization method with the already existing multi-scale device simulation code (which now relies on the density functional theory in the description of the QD electrons).
POTENTIAL BENEFITS
Accurate and fast evaluation of the exchange
energy in the two electron system confined tin the 3E double quantum
dot system without usual a priori assumptions on the shape and strength
of the confinement potential. Investigate evolution of this quantity
in magnetic fields, in order to better understand its properties,
gage the validity of the approximate models (Heitler-London, extended
Hubbard, and so on), and optimize device geometry to maximize its
value.
Investigate evolution of the many-electron system in the single and multiple coupled quantum dots in the Quantum Hall regime. Study formation of the so-called maximum density droplet (electron system is fully spin-polarized) and its subsequent breakdown into the lower density droplet with the increase of the applied magnetic field [5].
ACCOMPLISHMENTS AND SIGNIFICANCE
Fast and efficient
tool for the direct diagonalization of the many-electron Schrodinger
equation within the full device environment.
Performance comparison of various eigenvalue solvers using the dense Hermitian matrices obtained in this approach.
Comparison of effects that various approaches to the large-scale memory usage (distributed vs. shared memory) exert on the efficiency of the method.
Investigation of the role that various basis sets play in the overall performance of the method.
Development of the visualization tools to comprehensively assess evolution of the system properties.
PUBLICATIONS
1. S. Tarucha, D.G. Austing, T. Honda, R.J.
van der Hage, and L.P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996).
2. D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
3. T. Hatano, M. Stopa, and S. Tarucha, Science 309, 268 (2005).
4. D.V. Melnikov, P. Matagne, J.-P. Leburton, D.G. Austing, G. Yu, S. Tarucha, J.Fettig, and N. Sobh, Phys, Rev. B 72, 085331 (2005).
5. T.H. Oosterkmp, J.W. Jansen, L.P. Kouwenhoven, D.G. Austing, T. Honda, S. Tarucha, Phys. Rev. Lett. 82, 2931 (1999).
6. J. Kim, D.V. Melnikov, J.-P. Leburton, D.G. Austing, and S. Tarucha, Phys. Rev. B, to be submitted.
Status Report
October 12.2006
A general study of the singlet-triplet energy separation (exchange energy)
in the two-electron
system confined in a realistic double quantum dot system is performed
using a hybrid multiscale
approach where the many-body Schr¨odinger equation is solved exactly
within the full quantum dot
device environment. The exchange energy is computed as a function of the
gate confinement and
magnetic field. It is found, in particular, that at zero magnetic field
the exchange energy varies
from meV to sub-µeV value as the confinement gate biases (tunneling
barrier) are changed and the
system is driven from a single quantum dot to two coupled quantum dots.
The small values of the
exchange coupling in this structure are attributed to the large inter-electron
separation arising when
the Coulomb repulsion dominates tunneling. [more (PDF)]
