Polarizability in Quantum Dots via Correlated Quantum Monte Carlo

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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
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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

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quantum dots collective excitations
1) Raman scattering exp.
incident light beam polarized
scattered light beam...
...polarization is lost
...conserving the polarization
SDE
CDE
Schüller et al. PRL 80, 2673 (1998)
?
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• 2) The aim of this work is to carefully analyze
  the
• role of Coulomb interaction
• in the excitation of such collective modes.
• Devising a Quantum Monte Carlo
• algorithm for correlated quantities.
• Indeed, QMC great for ground state, not for
  excited states...

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Backbone of the approach
Sum rules
? ?
excitation
correlated quantity no QMC
analytic
Model independent
still a correlated quantity but feasible QMC
polarizability
numerical
Coulomb interaction
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Using sum rules to get ?
Ratios of sum rules can be used to estimate the
mean excitation energy of collective modes. If
S(?) is the dynamic structure factor of the
system, then we define the energy weighted sum
rule m1
Electric field in the dipole approximation ( ?
50µm gtgt 100nm)
the inverse-energy weighted sum rule m
-1
Polarizability?
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Polarizability
N charged particles under the effect of a small
constant electric field
dipole operator
unperturbed Hamiltonian
THEN
polarizability
Here we assume that for l0 the system is in its
ground state, and ?0D0? 0 for parity.
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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
But recall that
How to QMC ?
then
QMC unfeasible
Computing a is therefore equivalent to compute
m-1, without determining all the eigenstates n?
and eigenvalues wn0.
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How to simulate Polarizability?
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

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Polarizability in a Quantum Dot
the picture
E
20 - 100 nm
Electrons (conducting band) or Holes (valence
band)
2 - 10 nm
Harmonic for N lt 30

• Low density

• Shell structure

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Polarizability in a Quantum Dot
the formalism
The QD Hamiltonian is assumed to be
ra0-3
r
r/a0
in the effective mass/dielectric constant
approximation (for GaAs m0.067, e12 .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 (a097.9Å ). The parabolic
confinement is a realistic choice only for
small dots (N lt 30 electrons). For larger dots
some more appropriate form must be chosen.
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Polarizability in QDs
Electric field
Constant shift in E
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!
x
l?0
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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
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 ?? at first order
in ?.
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Estimate of CDE excitations
Recall seeking
The energy-weighted sum rule can be computed
analytically for the QD. Note that m1 is model
independent!
The estimate for the CDE average energy is
therefore
frequency of confinement
Kohn PR 123, 1242 (1961) Maksym, Chakraborty PRL
65, 108 (1990)
In agreement with the Kohn Theorem !
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Is it the same for spin-density polarizability?
The spin dipole operator is defined as follows
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!
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Spin polarizability computationally
The spin polarizability cannot be computed
analytically. The reason is that the unitary
operator that would define the transformed
Hamiltonian
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 interaction we would have
? ?
i.e.
SDE
CDE
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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
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Correlated sampling VMC
Its a correlated quantity
We use the scheme devised by C.J. Umrigar and C.
Filippi (PRB 61, R16291 (2000)) for forces,
indeed
?
?
etc
d
d
F - (V-V)/(d-d)
a (D-D)/(?-?)
get V(d) dV
get V(d) dV
get D(?) dD
get D(?) dD
Sample only a primary geometry and link
secondary geometries to this one
Computationally expensive need several d and dV
ltlt (V-V)
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Correlated sampling VMC
In the linear regime ?? 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
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
Note sampled from
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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 ?? 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
That defines a rigid translation of the
coordinates
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Evaluate ltDgt on each secondary geometry
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Correlated sampling DMC
In Diffusion Monte Carlo the primary walk that
projects the unperturbed ground state of the dot
is generated according to the standard procedure,
i.e. a population of walkers is evolved for an
imaginary time Dt using an importance sampled
approximate Greens Function of the Schrödinger
equation
for the primary geometry
where
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Correlated sampling DMC
The secondary walks, used to project out the Yl?
states, are generated from the primary walk
applying the translation previously
defined. Averages are obtained by reweighting
with the ratio of the primary and secondary
wavefunctions, as in the VMC case. We must,
however, take into account the different
multiplicity of the primary and secondary walkers
due to the different G(R,R,Dt) that should be
effectively used for propagation. This is
obtained redefining the weights as
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.
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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

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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 ?YlDYl?
was computed for four different values of l,
namely 10-2,10-3,10-4,10-5.
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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)
0.28 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)
0.35 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)
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Results
The ratio is equal to the
ratio (wd/ws), and gives us information about
the splitting between the charge and spin
collective modes. We get
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Results
Exact diagonalization for a QD with N6 electrons
indicate
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 QDs, finding a ratio between
polarizabilities of about 3.
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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
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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!