Download Quantum Monte Carlo Study of two dimensional electron gas with

Quantum Monte Carlo study of
two dimensional electron gas in
presence of Rashba interaction
A. Ambrosetti, F. Pederiva and E. Lipparini
The Rashba Interaction

Rashba interaction has been proved to exist in semiconductor
heterostructures, where electrons are subject to a quantum
well confinement and therefore move in a 2D space (plane).

It is a spin-orbit-like interaction, coupling momentum with
spin.

It can be tuned in strenght through gate voltage.
The Rashba Interaction


Due to the well asimmetry, electrons are subject to
an electric field perpendicular to their plane of
motion.
This causes electrons to sense an in-plane effective
magnetic field because of relativistic effects
ˆ  Px y
ˆ
B  Py x

The electron spin couples to the magnetic field
giving rise to the Rashba interaction:
VSO   ( Px y  Py x )
Switching off Coulomb

In absence of Coulomb interaction the problem is exactly
solvable
2
N
H 
i 1

What we get is two different eigenstates for each wavevector k,
consisting of different k-dependent spin states with two different
energies
 (k )1, 2

P i
  ( Px y  Py x ) i
2m
k2

k
2m
This generates two energy
bands, giving “quasi up –
quasi down” spin
polarization
Switching Coulomb on


When Coulomb interaction is introduced the
solution to this problem is not known
analytically
We need to use a numerical approach.
Diffusion Monte Carlo (DMC) is our method
of choice:



Widely used for electrons
Very accurate
We know how to treat SO interactions
HOW DOES DMC WORK?
 ( 0)

Take an initial wave function

Make it evolve in imaginary time   it

Expand over the Hamiltonian eigenstates:
 ( )   cii e 
i
i


Multiply by e 0
where  0 = ground state energy:
Let 
 ( )  e ( H  ) (0)
0
go to infinity
All excited states will be multiplied by the factor
e  ( i   0 )
Projection over the ground state is obtained!
DMC algorithm
Suppose that our Hamiltonian contains only a kinetic term

2 N 2
  ( R, )  
i  ( R, )


2m i 1

 ( R, )   G( R  R' , ) ( R' ,0)dR'
The solution is given by
where we used the free particle Green’s function
 ( R  R' ) 2 
1
G ( R  R' ) 
exp 

(4D ) N
4
D



2
D


2m
In terms of walkers, free propagation
means generating displacements
-> DIFFUSION.
DMC algorithm

Now suppose we have a kinetic term plus a central potential

2 N 2

 ( R,  )  
  i  ( R, )  V ( R) ( R, )

2 m i 1

From Trotter’s formula
e H  e  (T V )  e V e T  o( )

Take into account then the effect of the
interaction term over
 0
the “renormalized” wavefunction e  ( R, )
e [V ( R ) 0 ]  ( R, )

This can be seen as a weight,
i.e. the probability for the walker
in R to survive after a time 
Implementation of DMC
A possibile implementation of the projection
algorithm is:

Generate initial walkers distribution according to  (R,0)

Diffuse walkers due to free propagation

Kill or multiply walkers due to weight

Repeat steps until convergence is achieved
Spin-Orbit propagator
H  T  VRashba  VCoulomb
e H  e VCoulomb e VRashba e T


For small time steps

The idea is using Coulomb potential as a weight, and
applying the Rashba term right after the free propagator
e


 ( Py x  Px y )
G0 ( R , R ' , )  [ I   (i


 x  i  y )]e
y
x
We can thus rewrite this as
e
i

D

( R  R ') 2
2 D
( x y  y x )
e

( R ) 2
2 D
Which means we will need to sample displacements with
the free propagator, and then rotate spins according to the
just sampled.
R
Checking DMC with SO propagator



In absence of Coulomb
potential the problem is
analytically solvable.
The exact ground state
solution is a slater
determinant of plane waves.
Modify it multiplying by a
jastrow factor:
 ( R, s)  exp(  u (rij )) * DSlater
i j

DMC must be able to project
over ground state (red dot)
CHECK
RESULTS


Ground state
energies are
shown at
constant density
for different
values of Rashba
strenght.
The minimum is
shifted when
interaction
strenght
increases
rs  V / N
2
RESULTS
Hartree-Fock
energy is
known
analytically, like
the energy in
absence of
Coulomb
interaction. In
such cases
solutions are
made of plane
waves Slater
determinants.
CONCLUSIONS

We developed a functioning algorithm based on
previous work in nuclear physics, using spin-orbit
propagation

We have made some tests on method and trial
wavefunction

We are calculating the equation of state for the
2D electron gas in presence of Rashba interaction

We expect to use this method for further research
on other systems in presence of spin and
momentum dependent interactions