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

Anh Ngo, Eugene Kim and Sergio Ulloa, Submitted (arXiv:1103.3265).

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)