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Coulomb and spin-orbit interactions in nanorings
WIAS Workhop 20115 February
Andrei Manolescu
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Persistent charge and spin currents
Charge current
Spin current
1 2 0( ) sin( )cos sin( )cos( )tpulseV t Ae t t
1D ring with SOI interaction excited by an external EM pulse
M. Niţă, D.C. Marinescu, A. Manolescu, V. Guðmundsson, e-print arXiv:1012.4952
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SOI models
RR x y y xH p p
DD x x y yH p p
Rashba - Due to the vertical confinement
Dresselhaus – Due to the electric field in the crystal
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Known results on the 1D ring
3 electrons, no Coulomb interaction
Rashba
Dresselhaus
J. S. Cheng and K. Chang PRB 74, 235315 (2006)
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Charge and spin currents for R+D SOI
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The sample
Between 1-10 rings and 20-100 angular sites all connected by hopping matrix elements
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The sample Hamiltonian
2 2 2
2 2
, 1 , 1
, 1 , 1
2
2
1 1
2 2
Matrix elements for discrete polar coordinates:
| | 2
2
2
r rr r r r r
rr
R r R
H rm m r r r r
r H r t
t
Rt t t
rmR
22
R
Rt t
r
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The Coulomb interaction
1
1† † †
,2, ; ,
2* *
,
,
One particle:
N particles:
Coulomb matrix elements: ( ) ( ) ( ) ( )
Many-body states for independent electrons:
a a a
N a a a ab cd a b d c
a a b c d
ab cd a b c d
ab cd
H
H c c V c c c c
eV drdr r r r r
r r
V
,
0
: 1,0,0,... 0,1,0,... 1,1,0,... 1,1,1,...
"bit strings" with N particles
0
States for interacting electrons: and
ab cd NV H E
A
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The “exact diagonalization”
† †
,
†
1 1
We want eigenvalues/vectors of the matrix
All we need are the matrix elements and
Creation/destruction (annihilation) operators:
0,1,0,... 1,1,0,... 1,1,0,... 0,
N
N
ab cd a b c d
H E
H
V c c c c
c c
†
1 1
† †
1,0,...
1,1,0,... 0 0,1,0,... 0
| 0, 1a b c d a b c d
c c
c c c c c c c c
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Combined Coulomb and SOI effects (groundstate)
Rashba
Dresselhaus
N=3 no interaction N=3 with interaction
The spin polarization may be reduced by the Coulomb interaction ...
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Combined Coulomb and SOI effects
Rashba
N=4 no interaction N=4 with interaction
...but the spin polarization may also be amplified by the interaction ...
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Combined Rashba and Dresselhaus SOI in 1D
N=3, no interaction, inhomogeneous charge density and spin density
N=3, Coulomb interaction, screened charge density,unscreened spin density?
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N=4, no interaction, inhomogeneous spin density
N=4, Coulomb interaction, unscreened spin density?
Combined Rashba and Dresselhaus SOI
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Combined Coulomb and SOI in 2D
N=3 no interaction N=3 with interaction
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N=4 no interaction N=4 with interaction
Combined Coulomb and SOI in 2D
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Total spin and excited states
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Excited states in the presence of an external field. “Collective modes” ?
Increased energy of the “collective modes” with the SOI parameter. For plasma oscillations: D.C. Marinescu and F.Lung, PRB 82, 205322 (2010)
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Continuous 2D ring model
00
0
Lateral confinement ( )
Analytical single-particle wave functions
rrV r V
r r
Charge density
Spin density GS Spin density ES1 Spin density ES2
Tan & Inckson, Semic. Sci. Technol., 11 1635 (1996)
We include a Coulomb impurity
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(Preliminary) conclusions
• The Coulomb interaction has strong effects on the spin polarization• The spin density may propagate on longer distances than the charge density• The spin density may have a richer structure in space than the charge density• Collective modes have higher energy due to SOI ?
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Co-workers
• Csaba Daday (Reykjavik)• Marian Niţă (Bucharest)• Catalina Marinescu (Clemson)• Viðar Guðmundsson (Reykjavik)