1. A Four-Electron Artificial Atom in the Hyperspherical Function Method New York City College of Technology The City University of New York R.Ya. Kezerashvili, and Sh.M. Tsiklauri Bonn, Germany, August 31 - September 5, 2009

2. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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.

3. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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.

4. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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. R.Ya. Kezerashvili, L.L. Margolin, and Sh.M. Tsiklauri, Few-Body Systems, 44, 2008. Four electrons Wenfang Xie, Solid-State Electronics 43, 2115, 1999 M. B. Tavernier, at.el, PRB 68, 205305 2003.

5. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 Zeeman Parabolic trap  c is the cyclotron frequency, Let us consider a system of four electrons with effective mass m eff, moving in the xy-plane subject to parabolic confinement with frequency  0 in the presence of an external perpendicular magnetic field. The Hamiltonian is

6. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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

7. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 Theoretical Formalism We introduce the hyperspherical coordinates as and expand the four electron wave function in term of the symmetrized four-body hyperspherical functions : [f] and  are the Young scheme and the weight of representation, L, M and S total orbital angular momentum and its projection and spin Step 1:

8. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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.

9. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 -are four-body unitary coefficients of Reynal-Revai =

10. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 0 We expand hyperradial function in terms of functions This equation has the analytical solution Step 2: Step 3: the coefficients obey the normalization condition.

11. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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: Step 4:

12. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 the evolution of the lowest-energy states for different L and S

13. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 meV (0,0)(2,0)(0,1)(1,1)(2,2) 0.010.6570.6580.6630.660.654 0.051.9771.9821.9871.9821.977 0.13.2033.213 3.213.211 0.25.2235.2415.2275.2345.248 0.36.9776.9996.976.9897.02 0.48.5818.6068.5638.5988.647 0.510.08710.11210.05410.10810.177 0.611.5211.54311.47111.54611.636 0.712.89512.91612.8312.92713.041 0.814.22514.24214.14114.26314.401 0.915.51615.52815.41315.56215.725 116.77416.78116.65216.82817.018 1.522.69722.66422.46522.823.14 228.19328.10927.84128.36328.876 2.533.39933.26132.92533.65634.36 338.39138.19637.79538.75139.962 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.

14. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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

15. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009

16. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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

17. R.Ya. KezerashviliBonn, Germany, August 31 - September 5, 2009 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.