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.