a four-electron artificial atom in the hyperspherical function method new york city college of...

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

R.Ya. Kezerashvili Bonn, 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.


Quantum DotProgress 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.


• 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 electronsFaddeev 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.


zBzcji

iji

iBeff

i BSgLrVrmm

pH

2

1

2

1

2

34

1

222

4

1

34

1

220

2

2

1

2

1i

izBji

iji

iieff

sBgrVrmAc

ep

mH

Zeeman 0,,2

1

2

1iii xyBrBA

Parabolic trap

c is the cyclotron frequency,

Let us consider a system of four electrons with effective mass meff, moving in the xy-plane subject to parabolic confinement with frequency

0 in the presence of an external perpendicular magnetic field. The

Hamiltonian is

4/220

2cB


2

16/11 ji rr

2

16/12 lk rr

)(2

13/23 jilk rrrr

)(2

143213/2rrrrR

c.m IH H H

22 20

1

2 2c.m.

c.m. c.m.i

PH M R

M

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

zBcCi

ieff

i BSgLVmm

pH

2

1

2

1

2

3

1

220

2 Hamiltonian of relative motion of four electrons


Theoretical Formalism

),()(),,( ][

][

][

i

SfKLM

fKLM

SfKLM

β,α 2/,0

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

cos ;sinsin|| ;sincos|| ; 3212

32

22

12

Step 1:


),(][ i

SfKLM

)(),(1

),( ][][

][

][ fSi

fKLM

f

iSf

KLM

h

)(),()( 12321

312321

12321][][

illllLMKK

llll

fKLi

fKLM KllllC

)()( 321321

3321

12321

3 31231212122111 immmlll

KKmmm

illllLMKK LMmmllmlmmll

)()7()( 321321

3

321321

3 immmlll

KKimmmlll

KK KKM

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.

),( 12321][ KllllC f

KL


EBSgLbmK

B 0

23/22

3/1

2

2

2

2

2

1

2

24/1)2(

)ρ(χ 321 lll

KL

''''

)(''

))(('''

21

)()(MllK

lLK

llMMLLKKW

33

3

333333332

342 K~)l~(K

~)l~(

'MM'LL'KKjKL

ijKL

K~)l~(

iK)l(K)l('MM'LL'KK

/K)l(K)l('MM'LL'KK JK)'l(|K

~)'l~(K)l(|K

~)l~(J

m)(W

-are four-body unitary coefficients of Reynal-Revai

...JK)'l(|K~~

)'l~~

(K)l(|K~~

)l~~

( 33

3

K~~

)'l~~

(K~~

)l~~

(MM'LL'KK

jKL33

ijKL3

K~~

)l~~

(

3i

jKL

i KlKl 33 )(|~

)~

(),( 12321][ KllllC f

KL =


EBSgLbmK

B 0

23/22

3/1

2

2

2

2

2

1

2

24/1)2(

)(321 lll

KL 0

We expand hyperradial function in terms of functions

N

KlKN

KlKN

KlKL a )()( 333 )()()(

)(321 lllKN

)(321 lllKL

)( 2/1

2/)52(4/)3(

)1(

N

G KK

)2/)(exp( 22/1 G 22/12 )()3(

1

GL

KNKN

This equation has the analytical solution

Step 2:

Step 3:

33KlKNa

0

2133

N

KlKNa

the coefficients obey the normalization condition

.


0)1(IEEdet NNKK)'l)(l(KKKK);'l)(l(

MM;LL;KKNNKK)'l)(l(KKNK

o 33

33

33

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:


0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

5

10

15

20

25

30

35

40

45 (0,0) 2,0) (0,1) (1,1) (2,2)

Ene

rgy

spec

trum

, mev

mev

the evolution of the lowest-energy states for different L and S


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.


B(Tes la)

E (mev )

1 2 3 4 5

50

10 0

15 0

20 0

25 0

30 0

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


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


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.

a four-electron artificial atom in the hyperspherical function method new york city college of...

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