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Spatial correlations in chaotic nanoscale systems with spin-orbit coupling
Anh Ngo 1, Eugene Kim2 and Sergio Ulloa 1
1Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio. USA.
2 Department of Physics, University of Windsor, Windsor, Ontario, Canada N9B 3P4.
Supported by NSF-DMR WMN
MotivationsQPC
Vg1
Vg2
J.A. Folk, et al. PRL, 76, 1699 (1996) and A. M. Chang, et al. PRL, 76,1695, (1996).
~
Fluctuations and spatial correlations
of wave functions
Microscopic Model for Chaotic Nanoscale Systems
Random matrix Universal quantities
Theoretical model
Anh Ngo, Eugene Kim and Sergio Ulloa, Submitted (arXiv:1103.3265).
Random matrix ensembles
GOE
GSE
Small SOC
Stadium billiard
GOE-GSE
Large SOC
0.0 0.6 1.2 1.8 2.4 3.0 3.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
RMT for GOE- GSE
Numerical results RMT for GOE
=0
=1.5 0
=0.5 0
=0.2 0
P(
)
RMT for GSE
One point functionA is the area of the system
0.0 0.3 0.6 0.9
1.0
1.5
2.0
2.5
3.0
3.5
=0 (GOE)
=1.5 (GSE)
=0.5
=0.2
C
'(|r-
r'|)
|r-r'|(R0)
Numerical results RMT
The amplitude correlation
GOE
GSE
A is the area of the system
Two point correlation function for up and down spin
The same point in space R=0
Crossover regime
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
P
Spin-orbit coupling =0.50
P
=1, Up-up spin
=0.18 Up-down spin
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
P
P
=0.9 RMT
Spin-orbit coupling =0.10
Spin up and spin downbecome decorrelated
Spin up and spin down
SUMMARY
• We investigate the statistical properties of wave functions in two-dimensional chaotic nanostructures with spin-orbit interactions and magnetic fields.
• Weak spin-orbit coupling is expected to have consequences for spin-polarized tunneling.
• We present results for the evolution of the wave function statistics as a function
of the parameter regimes.
• Numerical calculations for a chaotic stadium billiard are compared with analytic results from random matrix theory.
THANK YOU VERY MUCH !
Spatial correlation of chaotic eigenfunctions
GSE
GOE
One point function
GOE
GUE
General one-point function is for the GOE-GUE crossover
GSE
GOE
GSE
Anh Ngo, Eugene Kim and Sergio Ulloa, Submitted.
GUE
Coral’s wall potential :
Anh Ngo, Eugene Kimand Sergio Ulloa ‘s calculations
Heller, et. al. Nature 369, 464
Boundary wall :
GUE
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
P
Microscopic calcualations RMT results
Anh Ngo, Eugene Kim and Sergio Ulloa, Submitted.
One point function
20 40 60 80 100
20
40
60
80
100
X
Y
0
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
Gaussian symplectic ensemble (GSE)