A study of two-dimensional quantum dot helium in a magnetic field
Golam Faruk
*and Orion Ciftja,
Department of Electrical Engineering and Department of Physics
Prairie View A&M University, Prairie View, Texas 77446
Introduction
Quantum dots, namely two-dimensionally confined
electrons in a semiconductor interface, appear to be the
building blocks on designing a quantum computer and they can be used to build switching and memory devices for
high speed applications.
In this work we study a simple quantum dot system
consisting of two interacting electrons trapped in a two-dimensional parabolic potential. This system is called quantum dot helium and its properties are fascinating especially in a magnetic field.
We applied the exact numerical diagonalization technique to study several properties of quantum dot helium subject to a perpendicular magnetic field. We obtained the ground level energy and angular momentum for various
combinations of Coulomb correlation strengths and
magnetic field. Our results show that there are multiple spin singlet-to-triplet transitions as function of the
magnetic field, suggesting a possible use of this system as a quantum bit (“qubit”) of a quantum computer.
Objective
The objective of this work is to study ground state and
excited state properties of two-dimensional quantum dot helium as a function of the magnetic field
for a variety of values of the Coulomb correlation relative to the confinement energy.
By adopting the exact numerical diagonalization
method we can determine to a high degree of accuracy the critical magnetic fields at which a spin singlet to
triplet transition happens. At the same time we want to obtain almost exact estimates for the ground and
excited state energies of this system.
Model
Quantum dot helium consists of two electrons that are confined in a two-dimensional parabolic potential and repel each other
through a Coulomb potential. When this system is subjected to a perpendicular magnetic field, the quantum Hamiltonian can be written as:
where m is the mass of the electron, –e (e>0) is
the electron’s charge,
0is the angular
frequency of the two-dimensional parabolic
potential, A(r) is the vector potential generating
a magnetic field, B in the z direction, and the
linear momentum operator, p and position, r are
given in a polar system of coordinates.
Method
The exact numerical diagonalization (ED) method is a standard technique used to solve numerically the
Schrödinger equation for a quantum system. The key idea of the method is to diagonalize the Hamiltonian matrix in a suitable chosen basis. The resulting matrix eigenvalues
correspond to the numerically exact energy eigenvalues of the quantum system.
The only uncertainty associated with ED results comes from the statistical error that can be reduced as much as we
want to. In this sense the ED method is extremely useful to obtain very accurate estimates for energies and related
quantities of a given system. Contrary to other methods, the results obtained from the ED method are unaffected by approximations or limitations, except the computer power. The ED method works very effectively for small systems of up to eight electrons, therefore is perfectly suitable to our study that includes only two electrons. Most of the
calculations do not require excessive computer power and can be performed in conventional computers.
Graph-1
Graph-2
Graph-3
Results and conclusions
By using large matrices we numerically diagonalized the
Hamiltonian of quantum dot helium in the basis functions of Fock-Darwin eigenstates.
We obtained the ground level energy and angular momentum for different values of Coulomb correlation and magnetic field.
Our results show that in absence of magnetic field, or in absence of Coulomb interaction the ground state always has zero angular momentum therefore it corresponds to a singlet (spin S=0) state.
However when both magnetic field and Coulomb interaction are present, we observed multiple spin singlet (S=0) to spin triplet (S=1) transitions as function of the magnetic field.
As the magnetic field increases, the ground state changes to
nonzero angular momentum values with alternating odd and even parity, thus suggesting a possible use of this system as a “qubit” of a quantum computer, with S=0 and S=1 being the two possible “on” and “off” states.
Selected References
• A. Harju, V.A. Sverdlov, B. Barbiellini, and R.M.
Nieminen, “Variational wave function for a two-electron quantum dot”, Physica B 255 (1998) 145-149
• L.P. Kouwenhoven and C.M. Marcus, “Quantum Dots”, Physics World vol. 11 no. 6, 35-39 (1998)
• Hans-Andreas Engel, L.P. Kouwenhoven, Daniel Loss, and C.M. Marcus, “Controlling Spin Qubits in Quantum Dots”, Condense-Matter/0409294 vol. 1 (Sept. 2004)
• F. Troiani, U. Hohenester, and Elisa Molinari, “Electron-hole localization in coupled quantum dots”,Physical
Review B, Vol. 65, 161301 (R)
• S. Tarucha, D.G. Austing, S. asaki, Y. Tokura, J.M. Elzerman, W. van der Wiel, S. de Franseschi, L.P.
Kouwenhoven, “Spin effects in semiconductor quantum dot structures” Physica E 10 (2001) 45-51
Acknowledgements
Part of this work was supported by the office of the Vice-President for Research and Development of Prairie View A&M University through a 2003-2004 Research
Enhancement Program grant. One of the authors (F. G.) would like to acknowledge the support given through the Title III Program of the U.S. Ministry of Education.
Ground state energy as a function of magnetic field,
c0 for values of from 0 (bottom) to 10 (top)
Energy spectrum, ε = /ĥo in zero magnetic field for
two values of Coulomb correlation parameter,
=e^2/ĥ0 as a function of angular momentum, |mz|
2 1
2 2
0 2 2 2
2 2
2 0 2 1 2
1 1
ˆ
ˆ
2
2
)
ˆ(
.
2
2
)ˆ
(
.
ˆ
r
r
e
r
m
m
r
A
e
p
r
m
m
r
A
e
p
H
The relative motion (=|r1-r2|) ground state wave function for
quantum dot helium in zero magnetic field for several
values of Coulomb parameter,. The parameter μ is the
parabolic oscillator’s length.
=3
=2
=1
