Sergei Matinyan1,2
1Department of Physics, North Carolina Central University,
Durham, NC, USA2 Alikhanian National Laboratory ( YerPhI ) ,Yerevan, Armenia
Quantum Mechanics of Electron
Transfer between Double Concentric
Quantum Rings in Magnetic Field
September 28, 2011Tbilisi, Georgia
(In collaboration with Igor Filikhin and Branislav Vlahovic)
September 28, 2011Tbilisi, Georgia
Content
1 . QR as meso and nano structures of non simply connected topology
2 . Aharonov - Bohm effect
3 . Progress in fabrication of meso and nano rings
4 . Meso nano: coherence , few (even single ) electron problem
5 . Magnetic field. Level crossing and anti - crossing .Tunneling .Double concentric QR (DCQR )
6 . Electron transition between QR. Anti-crossing as a way of transfer electron between rings
7 .Electron trapping in the system of QR and Central QD
8 .Conclusions
QR as meso and nano - structures of non simply
connected topology
3
•Aharonov - Bohm (AB) effect : vector potential �phase of w/function �
non trivial interference effects in B 0.
•Progress in fabrication of meso and nano single and double concentric QR
( A . Lorke at al, 1997,1998 ,2000 ) ( T. Mano ,T . Kuroda et al. 2005; M . Abbachi et al. 2010 )
•Today are produced more “exotic triple” and even quintuple concentric QR (method of droplet epitaxy (S.Sanguinetti, 2009).
First self-assembled InAs/GaAs QRs were fabricated by solid source MBE in Stranski-Krastanow grows mode
GaAs/Al0.70Ga0.30As Double Concentric Quantum Ring
≠
(Aharonov, Y. and D. Bohm, «Significance of electromagnetic potentials in quantum theory», Phys. Rev. 115, 485—491 (1959))
QR as meso and nano - structures of non simply
connected topology
PERSPECTIVES : in optics, optoelectronics, quantum cryptography and computing etc
PERSISTENT CURRENT (M. Buttiker, Y. Imry, and R. Landauer, 1983 )
Meso nano is essential : (ring radius ); few or even single electron problem
Atomic analogy; electron in ring Bloch periodic structure
Transverse magnetic field in DCQR
Connectedness : QD ( e.g. circular (2D) spherical ( 3D ))
Degeneracy in radial and orbital quantum numbers ( ), ( )
QR : only degeneracy in
Rl >
,...2,1±±=l,...2,1,0=nln
l
Transverse magnetic field in DCQR
5
0BTransverse B : possibility of level crossing at some connection between and .
Possibility transfer (jump ) of electron with gain in energy
Crossing only for levels of different symmetry (Wigner, von Neumann, 1929)
QD : no crossing, QR crossing possible
Nano-ring has richer crossings.
.effR0B
B
B
B
DCQR : new phenomenon: transfer of
electron between ringsFor DCQR 3D treatment important and interesting: aperiodic and
saturated AB oscillations of magnetization anddependence of geometry of rings and shape
( O. Voskoboinikov et al., 2002 ; V.Tkachenko, 2004; I. Filikhin et al., 2004)
Numerous papers in theory of related subjects :B. Szafran, F. Peeters, 2005,M .Bayer et al., 2003; J.L. Zhu et al., 2005;.B. Szhafran,.2008 ;S . Sanguinetti et al., 2008 ;V . Arsoski, M. Tadich, and F. Peeters, 2010 .
Remark :Separate electron and hole motion : H << R. Kinetic energy quantization is stronger than Coulomb interaction ( R << Bohr radius -about several hundred of nm) effective masses of e and hole very different :Motion in the different circles independently. Small H - small mixing between electron (e) and holes (lh, hh) ( Arsoski et al, 2010 ).
e flexible, hole (hh) is not, localized in the rings ( A.B. Kalameitsev et al. 1998; A. Govorov et al., 2002).
Formalism:
• In the single subband approach the problem becomes to Schrodinger equation :
•
• With the vector potential: , one has in cylindrical coordinates
•• Solved using FEM utilizing Ben-Daniel-Duke boundary
conditions (BDD).
( ) ( ) Ψ=Ψ+∂
Ψ∂−Ψ+∂
Ψ∂+
∂Ψ∂+
∂Ψ∂
∂∂− ,
28
2
1 1
2 2
2
*
222**
*2
2
2**
2
EzVzm
mqBm
m
qBi
mm c ρρϕϕρρ
ρρρ
hhh
( )( ) ( ) ( )rrr Ψ=Ψ+ ˆ EVH ckp
ϕρ ˆ2
1B=A
=substrate inside meV 262
rings inside 0cV
=substrate inside 0.093m
rings inside 0.067m*
0
0m
a) term Includes first derivation dividing on ρ
b) term Includes the centrifugal potential
c) Magnetic field terms
Competition between a) b) and c) gives the effect of the electron transfer between inner
and outer rings
For GaAs/Al0.70Ga0.30As QR
Solved by Finite Elements Method (I. Filikhin, V. Suslov, and B. Vlahovic (2005) )
Formalism
Single electron energies inDCQR and electron rmsradius R(n,l) for the stateswith radial quantum numbersn=1,2,…6 andl=0
(T. Mano, et al. 2005 Nano Letters 5 425)
Cross section profile of the DCQR
Electron localizations in the experimentally
fabricated DCQR
Weakly and strongly coupled electron levels in DCQR.
n=1
n=2
n=3
n=4
n=5
n=6
Wave function ofelectron in DCQR
Single electron energies and rms of DCQR in magnetic field B
anti-crossing of levels
Profiles of the normalized square wave function of electron
Initial state
Final state
Strongly coupling state
Geometry factor
Anti-crossing ofSingle electronlevels
Cross section of the DCQR
n=2,3
n=1,2
n=1,2
Different radial quantum numbers
RMS radius of electron in DCQR as a function of magnetic field
( )∫ Φ= dzdzR lnN
ln |,| 32,,
2 ρρρ
Like a jump !
Geometry factor
nm 3=S
nm 17D 21 == D
nm 5=H
Geometry is taken fromV. Arsoski, M. Tadic and F.M. Peeters, Acta Physica Polonica, Vol. 117 No. 5, 733 (2010).
rdrdzzrzrVzr innerlms
pinnerouterp
outerlns ),(),(),( ),,(
,
),,( ΨΨ∑ ∫=nmE∆ ~
(see G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, New York: Halsted Press, 1988)
Energy gap in anti-crossing:
Single electron energies of DCQR as function of magnetic field for 6,5,4 −−−=l
a) b)n=1,2 n=3,4
Energy gap of anti-crossing for n=1,2 (squares) and n=3,4 (circles) states with different orbital momenta
Energy gap as a function of ring separation (S)
AB effect in DCQR and ideal ring approximation
Rectangular shaped quantum ring composed of InGaAs in a GaAs substrate
AB effect and the localization of the single electron:
Why the ideal ring approximation is working?
Averaged radius does not depend on the magnetic field B !
����. � �����.���
����.���� 9%, when B< 8 T
Single electron energies of the QD and QR complex
Final stateInitial state
Cross section with Contour Plot
Initial state
Final state
Wave function of the anti-crossed states:
Conclusions:
1. Inter-ring electron transfer occurs to avoid energy level crossing evident in level anti-crossing
2. Tunneling is the mechanism for the electron transfer.
3. The levels anti-crossed have different radial quantum numbers and the same orbital momentum
4. The energy gap is strong influenced by the radial quantum number (n) and is weakly dependent on the orbital quantum number (l).
5. Electron transition in the DCQR (and QD+QR complex) can be use for qubit.
This work is supported by the NSF (HRD-0833184) and NASA (NNX09AV07A).
