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BOUND STATES IN
ELECTRON 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, | E |~ exp( 1 /  )
1D, symmetric potential:
2
one bound state, | E |~ U 0
Valid with SOI neglected
Hamiltonian of 2D electrons with SOI
in the Bychkov-Rashba form interacting
with an axially symmetric potential well
Dispersion relation for 2D with SOI
0.75
0.5
0.25
1
0
+
0.6
-
0.4
0.2
0
-1
0,0
me 2

2
p
-0.5
0
0.5
p0
1
p0
loop of extrema
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:
U(p)   dreiprU (r )  2  drrU (r ) J 0 ( pr )
Cylindrical harmonics of the spinor wave functions:
J 0 (| p  p ' |) 

J
k 
0
( pr ) J 0 ( p ' r ) cos(k )
For m-th harmonic:
s-state
2
ln(2me|E|R )
2 2
E
exp(4 /  )
2
me R
E
 2 2
32
me 2
0
meR=0.1
-10
-20
meR=0.01
=0
2meU0R2
-30
-40
0
2
4
6
8
10
12
14
16
18
1/
20
p-state
2meER
2
No bound states for zero SOI
at  < c =x12; J0(x1)=0
0,0
meR = 0.1
-2,0x10-7
2meER
2
lower p-state
upper p-state
0,0
-4,0x10
-7
-6,0x10
-7
-0,2
-0,4
-8,0x10
j=1/2
-0,6
j=3/2
-0,8
c
-7
-1,0
4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5 8,0

-1,0x10-6
0,0
0,5
1,0
1,5
2,0
2,5

3,0
3,5
4,0
Effect of the magnetic field
ground state (s-level)
Direct Zeemann contribution neglected (g=0),
only SOI induced effect
^
Splitting
3 me eB

64 | E | me c
2
Liquid He-4
Roton dispersion relation:
E    ( p  p ) / 2M
2 2
0
2 l
fold degeneracy
2D electrons with B-R SOI in 1D short-range potential
y
U (x)
x
0.5U00
=0.5U
=U
=U00
There exists pc and at |py| > pc
«+»-state becomes delocalized.
py =0.5 meU0
py =1.5 meU0
Narrow quantum well and 3D electrons
z
U=-U0d(z)
Dresselhaus SOI
VSO=g[(sypysxpx)pz2 sz (px2-py2) pzpx py(sxpy-sypx)]
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 p  (m U 2 / 2g )1/3
||c
e 0
[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:
d 
2
dz
2
 2m ( E  U ( z ))    0
1
1

(1  d )
m me
(   1)
d  2meg p
1
Asymmetric well
  1
U2
U1
  1
GaxAl1-xAs/A3B5/GayAl1-yAs
Two identical wells
a
U0
U0
meU0a (1  d ) < 1 < meU0a (1  d )
For 1 two localized states, for 1 only one.
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.