A Four-Electron Artificial Atom in the Hyperspherical Function Method

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A Four-Electron Artificial Atom in the
Hyperspherical Function Method
R.Ya. Kezerashvili, and Sh.M. Tsiklauri

• New York City College of Technology
• The City University of New York

Bonn, Germany, August 31 - September 5, 2009
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Objectives

• To develop the theoretical approach for
  description trapped four fermions within method
  of hyperspherical functions
• To study the dependence of the energy spectrum on
  magnetic field
• To study the dependence of the energy spectrum on
  the strength of the external potential trap.

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Quantum Dot
Progress in experimental techniques has made it
possible to construct an artificial droplet of
charge in semiconductor materials that can
contain anything from a single electron to a
collection of several thousand. These droplets of
charge are trapped in a plane and laterally
confined by an external potential. The systems of
this kind are known as "artificial atoms" or
quantum dots.
The structure contains a quantum dot a few
hundred nanometres in diameter that is 10 nm
thick and that can hold up to 100 electrons. The
dot is sandwiched between two non-conducting
barrier layers, which separate it from conducting
material above and below. By applying a negative
voltage to a metal gate around the dot, its
diameter can gradually be squeezed, reducing the
number of electrons on the dot - one by one -
until there are none left.
Kouwenhoven, Marcus, Phys. World, 1998.
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• S.M. Reimann and M. Manninen, Rev. Mod. Phys. 74,
  1283, 2002.
• C. Yannouleas and U. Landman, Rep. Prog. Phys.
  70, 2067, 2007
• 2D electrons organize themselves in electronic
  shells associated with a confining central
  potential (quantum dots in semiconductors,
  graphene) or boson quasi particles (excitons,
  magnetoexcitons, polaritons, magnetopolaritons)
  forming a Bose-Einstein condensate (graphene, QW)
• O. L. Berman, R. Ya. Kezerashvili, Yu. E.
  Lozovik, PLA, 372, 2008 PRB, 78, 035135, 2008.
• O. L. Berman, R. Ya. Kezerashvili, Yu. E. Lozovik
  PRB, 80, 2009
• Few electron quantum dot
• Three electrons
• Faddeev equations
• M. Braun, O.I. Kartavtsev, Nucl Phys A 698, 519,
  2001 PLA 331, 437, 2004.
• Hyperspherical functions method
• N.F. Johnson, L. Quiroga, PRL 74, 4277, 1995.
• W. Y Ruan and H-F. Cheung J. Phys. Condens.
  Matter 1, 435, 1999.

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Let us consider a system of four electrons with
effective mass meff, moving in the xy-plane
subject to parabolic confinement with frequency
w0 in the presence of an external perpendicular
magnetic field. The Hamiltonian is
wc is the cyclotron frequency,
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We introduce Jacobi coordinates for 2D four body
system to describe the relative motion of four
electrons and separate the CM motion.

Hamiltonian of CM motion
Hamiltonian of relative motion of four electrons
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Theoretical Formalism
Step 1
We introduce the hyperspherical coordinates as
and expand the four electron wave function in
term of the symmetrized four-body
hyperspherical functions
f and l are the Young scheme and the weight of
representation, L, M and S total orbital angular
momentum and its projection and spin
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Construction of the symmetrized four-electron
functions
The symmetrized four-particle hyperspherical
functions are introduced as follows
are four-body
Reynal-Revai symmetrization coefficients
introduced by Jibuti and Shubitidze, 1979.
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-are four-body unitary coefficients of
Reynal-Revai
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Step 2
This equation has the analytical solution
Step 3
We expand hyperradial
function in terms of functions
.
the coefficients
obey the normalization condition
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Step 4
Then the energy eigenvalues of the relative
motion are obtained from the requirement of
making the determinant of the infinite system of
linear homogeneous algebraic equations vanish
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the evolution of the lowest-energy states for
different L and S
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meV (0,0) (2,0) (0,1) (1,1) (2,2)

0.01 0.657 0.658 0.663 0.66 0.654
0.05 1.977 1.982 1.987 1.982 1.977
0.1 3.203 3.213 3.213 3.21 3.211
0.2 5.223 5.241 5.227 5.234 5.248
0.3 6.977 6.999 6.97 6.989 7.02
0.4 8.581 8.606 8.563 8.598 8.647
0.5 10.087 10.112 10.054 10.108 10.177
0.6 11.52 11.543 11.471 11.546 11.636
0.7 12.895 12.916 12.83 12.927 13.041
0.8 14.225 14.242 14.141 14.263 14.401
0.9 15.516 15.528 15.413 15.562 15.725
1 16.774 16.781 16.652 16.828 17.018
1.5 22.697 22.664 22.465 22.8 23.14
2 28.193 28.109 27.841 28.363 28.876
2.5 33.399 33.261 32.925 33.656 34.36
3 38.391 38.196 37.795 38.751 39.962
Table shows the energy spectrum of the states
(0,0), (2,0), (0,1), (1,1) and (2,2) as a
function of the confined potential with the
strength from 0.01 to 3 mev.
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The energy of a spin configurations as a function
of the magnetic field (L,S)(2,0) - orange
solid curve (L,S)(0,1) - dashed
curve (L,S)(0,1) - Bold curve
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Formation of a Wigner crystal
With increasing magnetic field we observe
formation of a Wigner state, when four electrons
are located on the corners of the square
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Conclusions

• we have demonstrated a procedure to solve the
  four-electron QD problem within the method of
  hyperspherical functions.
• ground state transitions in the absence of
  magnetic field are affected by the confinement
  strength
• we obtained the energy spectrum of the four
  electron quantum dot as a function of the
  magnetic field
• We observed the formation of a Wigner crystal by
  increasing the magnetic field.