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Polarizability in Quantum Dots
via
Correlated Quantum Monte Carlo
Leonardo Colletti
Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Gruppo di Trento, Italy
Free University of Bozen-Bolzano, Italy
F. Pederiva (Trento)
E. Lipparini (Trento)
C. J. Umrigar (Cornell)
Recent Progress in Many-Body Theory, 17Jul07 Barcelona
Outline
•
Motivation: experimental data, challenge for QMC
•
Theory: Sum rules, linear response, polarizability
and collective excitations
•
Computation: a correlated sampling DMC
•
Results: comparison with literature
1)
Raman scattering exp.
quantum dots’ collective excitations
incident light beam: polarized
scattered light beam...
...polarization
...conserving the
polarization
is lost
SDE
CDE
ω
Schüller et al. PRL 80,
2673 (1998)
2) The aim of this work is to carefully analyze the
role of Coulomb interaction
in the excitation of such collective modes.
3) Devising a Quantum Monte Carlo
algorithm for correlated quantities.
Indeed, QMC great for ground state, not for excited
states...
Backbone of the approach
ω
excitation
m1
ω
m1
?
correlated
quantity…
no QMC
Sum rules
analytic
Model independent
m1   α/ 2
polarizability numerical
Coulomb interaction
still a correlated
quantity…
but feasible
QMC
Using sum rules to get ω
Ratios of sum rules can be used to estimate the mean excitation energy of
collective modes.
If S(ω) is the dynamic structure factor of the system, then we define
Electric field in the
dipole approximation
( λ ~ 50μm >> 100nm)
the energy weighted sum rule m1:

m1   S (ω) ω d ω   ω0 n 0 D n
2
n
0
the inverse-energy weighted sum rule m -1 :

S (ω)
m1  
dω
ω
n
0
| 0 D n |2
ω0 n
Polarizability
?
Polarizability
| E | Eˆi
N charged particles under the effect of a small constant electric field:
H  H 0  H int
H int  λ D
λ  q|E|
dipole operator D 
unperturbed Hamiltonian
N
x
i 1
i
THEN
( H 0  H int ) λ  Eλ λ
polarizability :
λ D λ  0 D 0
α  lim
λ 0
λ
Here we assume that for l=0 the system is in its ground state, and 0|D|0 = 0 for parity.
In the linear regime the polarizability is a sum of matrix elements
between the ground state and the excited states |n of the system
with excitation energy wn0:
α  2
n
But recall that
m1  
n
then
QMC unfeasible
| 0Dn |
2
ωn0
| 0Dn |
2
ω0 n
m1   α/ 2
How to QMC ?
Computing a is therefore equivalent to compute m-1, without
determining all the eigenstates |n and eigenvalues wn0.
How to simulate Polarizability?
α  lim
λ 0
P
a
E
λ D λ
λ
Induced dipole moment
External electric field
“The relative tendency of the electron cloud of an
atom to be distorted from its normal shape by the
presence of an external electric field”
Polarizability in a Quantum Dot
the picture
E
20 - 100 nm
Electrons (conducting band)
2 - 10 nm
or
Holes (valence band)
Harmonic for N < 30
• Low density
• Shell structure
Polarizability in a Quantum Dot
the formalism
2
 pi2 1 *

e
H 0   
 m ω0 ri 2  
2
i 1  2m
 ε
N

i j
1
ri  r j
ra0*-3
The QD Hamiltonian is assumed to be:
r
r/a*0
in the effective mass/dielectric constant approximation (for GaAs m*=0.067, e=12
.3). The parameter w0 controls the confinement of the system (typically 2-3 meV).
In the following effective atomic units will be used. Energies are given in H*
(~11.9meV for GaAs), and length in effective Bohr radii (a0*=97.9Å ).
The parabolic confinement is a “realistic” choice only for small dots (N < 30
electrons). For larger dots some more appropriate form must be chosen.
Polarizability in QDs
Electric field
H 0  H int 
p 1 *
 e
2
  
 m ω0 ri  λ xi  
2
i 1  2m
 ε
N
2
i
2
1


i  j ri  r j
Constant
shift in E
2
2
 p2 1
 e2


λ
1
λ
1
*
*
2
i



 m ω0  xi  *   m ω0 yi  *2 4  

2
m ω0  2
m ω0  ε i j ri  r j
i 1 2m



N
x
x
l=0
l0
The application of an electric field
displaces the confining potential and
the density proportionally to its
intensity. However, due to the parabolic
approximation, the shape of the
confinement does not change!
Polarizability in QDs
The polarizability can be inferred by the new position of the minimum of the
confining potential, which is related to the expectation values x and y.
Moreover the translational invariance of the Coulomb interaction prevents it to
influence such expectations. These considerations would yield:
N
α *
m ω0
i
U e
λ
Px
m* ω02
~
H  UHU 1
~
H λ  Eλ λ
 λ  0 at 1st order in λ  λ  0 
α  lim
λ 0
λ D λ
λ

m1
iλ
P 0
* 2 x
m ω0
i
N



P
,
D



0
x
0
m* ω02
m* ω02
Note: still
speaking about
charge density
polarizability
The same result can be
rigorously obtained by
applying to the Hamiltonian
a unitary transformation and
solving for l at first order
in l.
Recall:
seeking
ω
m1
m1
Estimate of CDE excitations
The energy-weighted sum rule can be computed analytically for the
QD. Note that m1 is model independent!
1
N
m1   ωn 0 | n D 0 |  0 D, H 0 , D 0  *
2
2m
n
2
The estimate for the CDE average energy is therefore
ω
m1

m1
N

 ω0
m*α
In agreement with the Kohn Theorem !
= frequency of
confinement
Kohn PR 123, 1242 (1961)
Maksym, Chakraborty PRL 65, 108 (1990)
Is it the same for
spin-density polarizability?
The spin dipole operator is defined as follows:
N
D   σ xi
σ
i 1
i
z
This operator describes the response of a field that displaces
electrons with opposite spin in opposite directions
The response to a spin dipole operator
is connected to the energy of spin
density waves!
Spin polarizability: computationally
The spin polarizability cannot be computed analytically. The
reason is that the unitary operator that would define the transformed
Hamiltonian
N
U e
λ
 σ2
pix σiz
m*ω0 i 1

does not commute with the Coulomb interaction. This fact implies
that the spin dipole polarizability takes contributions from the
interaction, which plays a fundamental role.
Note that in absence of
N
ασ  α   *
i.e.
interaction we would have
m ω0
ω SDE = ω CDE
Role of the e-e interaction
The interaction will give therefore a split between the peaks
corresponding to the CDE and the SDE. This is exactly what is
observed in Raman scattering experiments.
SDE
CDE
Correlated sampling VMC
α  lim
λ D λ
λ 0
λ
It’s a correlated
quantity
We use the scheme devised by C.J. Umrigar and C. Filippi (PRB 61,
R16291 (2000)) for forces, indeed:
λ
d
λ’
d’
etc…
F = - (V-V’)/(d-d’)
α = (D-D’)/(λ-λ’)
get V(d)± δV
get V’(d’) ± δV’
get D(λ)± δD
get D(λ’)± δD
Computationally expensive: need
several d and δV << (V-V’)
Sample only a “primary”
geometry; and “link”
secondary geometries to
this one
Correlated sampling VMC
In the linear regime l and 0 are very close. The idea is to compute the matrix
elements of Ds for different fields using only the configurations sampled* from the
unperturbed ground state. In Variational Monte Carlo this procedure is defined as
follows:
λ σ D σ λ σ 
1
N conf
 N z 
  σ j x j wi



i 1  j 1

N conf
Displaces each
electron wrt spin,
in each configuration
sampled*
Where Nconf is the number of configurations sampled, and the wi is a weight of the
configuration defined by:
wi 
N conf λ (R i ) / 0 (R i )
Note: sampled from
2
σ

N conf
j 1
λ (R j ) / 0 (R j )
σ
2
R i  r1 rN 
| 0 |2
Correlated sampling VMC
In order to increase the efficiency of the sampling it is possible to introduce a
coordinate transformation that maps the sampled configurations in a region of
space where the probability defined by the secondary wave function l is larger.
In our case the natural transformation is defined by the unitary operator used for
transforming the Hamiltonian. For the noninteracting system we have:
N
λ σ  e
λ
i σ2
pix σiz
m*ω0 i 1

λσ N x z
  0  i
p σi 0
2  i
m * ω0 i 1
That defines a rigid translation of the coordinates:
SIGN DEPENDS ON SPIN
λσ
r  ri  σ
m * ω02
s
i
z
i
Evaluate <D> on
each secondary
geometry
| 0 |
s 2
| o |
Correlated sampling DMC
Multiplicity of the
In Diffusion Monte Carlo the primary walk that projects
the unperturbed
ground
walker
(“branching”
Drift - Diffusion process
state of the dot is generated according to the standard procedure, i.e. a population
process)
of walkers is evolved for an imaginary time Dt using an importance sampled
approximate Green’s Function of the Schrödinger equation:
 R 'R  V (R ) Δτ 2 
1
G (R ' , R, Δτ) 
exp 
 exp S R ' , R, Δτ 
3N 2
(2 Δτ)
2 Δτ


where
for the primary geometry
V  0 (R ) 0 (R )

1  H0 (R ) H0 (R ' ) 
 Δτ
S (R ' , R, Δτ )   ET  

2  0 (R )
0 (R ' ) 

Correlated sampling DMC
The secondary walks, used to project out the |l states, are generated from the
primary walk applying the translation previously defined.
Effective time step: takes into
Averages are obtained by reweighting with the ratio ofaccount
the primary
and secondary
modifications
to the width
wavefunctions, as in the VMC case.
of the proposed move due to the
coordinates transformation:
We must, however, take into account the different multiplicity of the primary and
τ s beτ effectively
DRs2 DR 2 used
secondary walkers due to the different G(R’,R,Dt) that should
for propagation. This is obtained redefining the weights as
exp S (Rs , R s , Δτ s )
ws  w
exp S (R, R, Δτ) 
N proj
where Nproj is a customary number of walkers generations, long enough to project
the secondary ground state, but small enough to avoid too large fluctuations in the
weights  WALKERS REMAIN EFFECTIVELY CORRELATED.
Wave Functions
The Correlated Walkers scheme illustrated is efficient if the branching is small
WE NEED VERY OPTIMIZED WAVEFUNCTIONS
•
Jastrows are taken as in C.Filippi, C.J. Umrigar, JCP 105,213 (1996)
The single particle wave functions are taken from an LDA calculations
for a dot with the same geometry. For the secondary wavefunctions the
origins are translated according to the unitary transformation previously
defined
Results
We performed simulations for closed shell QD
with N = 6, 12, and 20 electrons
and for different values of the external confinement
w0 = 0.21, 0.28, and 0.35 H*
To compute the polarizability and check the linear regime the
expectation value l|D|l was computed for four different
values of l, namely 10-2,10-3,10-4,10-5.
Spin polarizability computed for different N and confinements in VMC
and DMC. Note the large discrepancy in the values obtained with the
two methods. The DMC results are corrected mixed estimates.
HUGE EFFECT of
INTERACTION!
w0(H*)
N
as (VMC)
as (DMC)
a
r
0.21
6
-300(50)
-306(2)
-136.1
1.497(3)
12
-830(50)
-929(18)
-272.1
1.85(2)
20
-1520(70)
-1561(8)
-453.5
1.855(5)
6
-150(20)
-179.1(3)
-76.5
1.530(1)
12
-430(30)
-424.6(7)
-153.1
1.666(2)
20
-660(20)
-609(6)
-255.1
1.543(6)
6
-93(8)
-91.64(5)
-49.0
1.3678(3)
12
-210(60)
-132.9(3)
-98.0
1.165(1)
20
-400(20)
-379.0(5)
-163.3
1.524(1)
0.28
0.35
Results
The ratio r  as a
is equal to the ratio (wd/ws), and gives us
information about the
splitting between the charge and spin collective modes.
We get
1.165 (1)  r  1.85 (2)
Results
Exact diagonalization for a QD with
N=6 electrons indicate
r  2.4
Results from TDLSDA (L. Serra, M.
Barranco, A. Emperador, and E.
Lipparini, Phys. Rev. B59 (1999), 1529)
who computed the CDE and SDE spectra
for several QD’s, finding a ratio between
polarizabilities of about 3.
r  3.5
Results
Experimental data obtained on quantum dots with N200 electrons (C. Schüller et
al. Solid State Comm. 119, 323 (2001)) give a ratio between the two modes which is
about 2.
However, we have
indications that the ratio
grows with the number of
electrons, and it is difficult to
establish from the present
calculations which is the
asymptotic value . Moreover
for such a large number of
electrons the confinement
cannot be realistically
approximated with an
harmonic potential
Conclusions
• Solving a constrained Schrödinger equation and
computing polarizabilities is a way to obtain information
about collective excited states in QD (and electron gas in
general).
• Correlated Sampling DMC is an effective way to
compute polarizabilities in QDs.
• Results are reasonably in agreement with experiments. In
order to have a more realistic comparison several steps
need to be taken (like simulating larger dots, changing the
shape of the confining potential....)
•Better (energy- rather than energy variance-) DMC
optimization!