BOUND STATES INELECTRON SYSTEMS INDUCED

BY THE SPIN-ORBIT INTERACTION

Magarill L.I. in collaboration with Chaplic A.V.

A shallow and narrow potential well

3D: No bound states

2D, axially symmetric well: one bound s-state, )/1exp(|~| E

1D, symmetric potential: one bound state, 2

0|~| UE

Valid with SOI neglected

=

Hamiltonian of 2D electrons with SOIin the Bychkov-Rashba form interacting with an axially symmetric potential well

-1

-0.5

0

0.5

1

00.25

0.50.75

1

0

0.2

0.4

0.6

-1

-0.5

0

0.5

1

00.25

0.50.75

1

0

0.2

0.4

0.6

Dispersion relation for 2D with SOIDispersion relation for 2D with SOI

0,0

-

+

p

0p

2

2em

loop of extrema

p0

p0=me

The lower branch of the dispersion law of 2D electrons has

a form

and corresponds to a 1D particle at least in the sense of density of states. Formally the particle has anisotropic effective mass:

radial component is me , azimuthal component = (the dispersion law is independent of the angle in the p-plane).

p-representation of the Schrodinger equation:

Cylindrical harmonics of the spinor wave functions:

0 0 0(| ' |) ( ) ( ' ) cos( )k

J J pr J p r k

p p

0( ) ( ) 2 ( ) ( )id e U r drrU r J pr prp rU

For m-th harmonic:

s-states-state

0 2 4 6 8 10 12 14 16 18 20-40

-30

-20

-10

0

1/

meR=0.01

meR=0.1

=0

ln(2

me|E

|R2 )

2

2

2exp( 4 / )

e

Em R

2 22

32 eE m

meU0R2

p-state

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0-1,0x10 -6

-8,0x10 -7

-6,0x10 -7

-4,0x10 -7

-2,0x10 -7

0,0

4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5 8,0-1,0

-0,8

-0,6

-0,4

-0,2

0,0

lower p-state

2m

eER

2

meR = 0.1

c

2m

eER

2 upper p-state

No bound states for zero SOI at c =x1

2; J0(x1)=0

j=1/2j=3/2

23

64 | |e

e

m eB

E m c

Splitting

Effect of the magnetic field

ground state (s-level)

Direct Zeemann contribution neglected (g=0),

only SOI induced effect

2 lfold degeneracy

Liquid He-4

Roton dispersion relation:

20( ) / 2E p p M 2

2D electrons with B-R SOI in 1D short-range potentialy

x

UU(x)(x)

U0 =U0

There exists pc and at |py| > pc

«+»-state becomes delocalized.

U0 =U0 =0.5U0 =U0

py =0.5 meU0 py =1.5 meU0

z

VSO=ypyxpx)pz2 z px

2-py2 pzpx py(xpy-ypx)

Dresselhaus SOI

U=-U0(z)

Narrow quantum well and 3D electrons

Localization or delocalization of an electron in z-direction depends on the orientation of longitudinal momentum p||:

[110] - lower subband: localization for all p||

upper subband: termination point 2 1/3|| 0( / 2 )c ep mU

[100] – both subbands for all values of p|| relate to the localized states

Small longitudinal momenta

Two independent equations for two components of the wave function:

2

22 ( ( )) 0

1 11 ( 1)

e

dm E U z

dz

m m

2 1em p

Asymmetric well

GaxAl1-xAs/A3B5/GayAl1-yAs

U1

U2 1

1

U0U0

a

Two identical wells

0 01 1 1e emU a mU a

For two localized states, for only one.

Conclusion Conclusion

We have shown that 2D electrons interact with impurities in a very special way if one takes into account SO coupling: because of the loop of extrema, the system behaves as a 1D one for negative energies close to the bottom of continuum. This results in the infinite number of bound states even for a short-range potential. 1D potential well in 2DEG and 3DEG for proper values of characteristic parameters form bound states for only one spin state of electrons. The ground state in a short-range 2D potential well possesses the anomalously large effective g-factor.