Polarizability in Quantum DotsPolarizability in Quantum Dots

via via

Correlated Quantum Monte CarloCorrelated Quantum Monte Carlo

Leonardo CollettiIstituto Nazionale di Fisica Nucleare, Sezione di Padova, Gruppo di

Trento, ItalyFree 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

SDE CDE

quantum dots’ collective excitations 1) Raman scattering exp.

incident light beam: polarized

scattered light beam...

...polarization

is lost ...conserving the polarization

ω Schüller et al. PRL 80, 2673 (1998)

2) The aim of this work is to carefully analyze the

role of Coulomb interactionrole of Coulomb interaction

in the excitation of such collective modes.

3) Devising a Quantum Monte Carlo

algorithm foralgorithm for correlated quantitiescorrelated quantities.

Indeed, QMC great for ground state, not for excited states...

Backbone of the approach

1

1ω

m

m Sum rulesω ?

analytic

2α/1 m

polarizability numerical

excitation

correlated quantity…no QMC

still a correlated quantity…but feasible

QMC

Model independent

Coulomb interaction

Using sum rules to get ωRatios 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

the energy weighted sum rule m1:

20

01 0ωωω)(ω

nn nDdSm

the inverse-energy weighted sum rule m :

n n

nDd

Sm

0

2

0

1 ω

|0|ω

ω

)(ω

-1

Polarizability?

Electric field in the dipole approximation( λ ~ 50μm >> 100nm)

Polarizability

λλλint0 )( EHH

N charged particles under the effect of a small constant electric field:

||λ λ intint0 EqDHHHH

dipole operatordipole operator

N

iixD

1

iE ˆ|| E

unperturbed Hamiltonian

polarizabilitypolarizability :λ

00limα λλ

0λ

DD

THEN

Here we assume that for =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 energyn0:

n n

nD

0

2

ω

|0|2α

But recall that

n n

nDm

0

2

1 ω

|0|

2α/1 m

Computingis therefore equivalent to compute m-1, without determining all the eigenstates |n and eigenvalues n0.

then

QMC unfeasibleHow to QMC ?

How to simulate Polarizability?

“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”

E

P

Induced dipole moment

External electric field

λlimα λλ

0λ

D

Polarizability in a Quantum Dot

Harmonic for N < 30

Electrons (conducting band)

or

Holes (valence band)

20 - 100 nm

2 - 10 nm

• Low density

• Shell structure

E

the picture

Polarizability in a Quantum Dot

The QD Hamiltonian is assumed to be:

ji ji

N

ii

i erm

m

pH

rr

1

εω

2

1

2

2

1

20

*2

0

in the effective mass/dielectric constant approximation (for GaAs m*=0.067, =12 .3). The parameter 0 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.

r/a*0

ra0*-

3

the formalism

Polarizability in QDs

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

0

ji ji

N

iii

i

ji ji

N

iii

i

e

mym

mxm

m

p

exrm

m

p

HH

rr

rr

1

εω

λω

2

1

ω

λω

2

1

2

1

ελω

2

1

2

2

14

02*

22

0*

2

0*0

*2

2

1

20

*2

int0

x

=0

Electric field

Constant shift in E

Polarizability in QDsThe 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 interactiontranslational invariance of the Coulomb interaction prevents it to influence such expectations. These considerations would yield:

0* ω

αm

N

The same result can be rigorously obtained by applying to the Hamiltonian a unitary transformation and solving for at first order in .

20

*0020

*λλ

0λ

020

*0λ0λ

λ λλ1

20ω*

λ

ω,

ωλlimα

ω

λ λin order 1st at

E~

~

m

NDP

m

iD

Pm

i

HUHUHeU

x

x

xPm

i

Note: still speaking about charge density polarizability

1m

Estimate of CDE excitations

The energy-weighted sum rule can be computed analytically for the QD. Note that m1 is model independentmodel independent!

*000

201

2,,

2

1|0|ω

m

NDHDDnm

nn

The estimate for the CDE average energy is therefore

01

1 ωα*

ω m

N

m

m

In agreement with the KohnKohn Theorem Theorem !

= frequency of confinement

Recall:

seeking 1

1ω

m

m

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

ii

iz xD

1

σ σ

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

i

zi

xip

meU 120

σ σω*

λ

does not commute with the does not commute with the Coulomb interactionCoulomb 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

0*σ ω

ααm

N i.e. ω = ω SDE

CDE

Role of the e-e interactionThe 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.

SDECDE

We use the scheme devised by C.J. Umrigar and C. Filippi (PRB 61,

R16291 (2000)) for forces, indeed:

Correlated sampling VMC

λlimα λλ

0λ

D It’s a It’s a correlated correlated quantityquantity

get V(d)± δV get V’(d’) ± δV’

d d’F = - (V-V’)/(d-d’)

Computationally Computationally expensive: need several expensive: need several dd and and δδVV << (V-V’)<< (V-V’)

Sample only a “primary” geometry; and “link” secondary geometries to this one

λ λ’

get D(λ)± δD get D(λ’)± δD

etc…

α = (D-D’)/(λ-λ’)

Correlated sampling VMC

In the linear regime and 0 are very close. The idea is to compute the matrix elements of D for different fields using only the configurations sampled* from the unperturbed ground state. In Variational Monte Carlo this procedure is defined as follows:

i

N

i

N

jj

zj

conf

wxN

Dconf

1 1λ

σλ σ

1σσ

Where Nconf is the number of configurations sampled, and the wi is a weight of the configuration defined by:

NiN

j jj

iiconf

iconf

Nw rrR

RR

RR1

1

2

0λ

2

0λ

)(/)(

)(/)(

σ

σ

Displaces each electron wrt spin,

in each configuration sampled*

Note: sampled from || 2

0

Correlated sampling Correlated sampling VVMCMC

01

20

σ0

σω*

λ

λ σω*

λ1

20

σ

σ

N

i

zi

xi

pm

i

pm

ie

N

i

zi

xi

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 translationrigid translation of the coordinates:

20

σ

ω*

λσ

mzii

si rr

SIGN DEPENDS ON SPIN

Evaluate <D> on each secondary

geometry

|| 0 || 2s

o

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 using an importance sampled approximate Green’s Function of the Schrödinger equation:

where

Δτ,,'expΔτ2

Δτ)('exp

Δτ)2(

1Δτ),,'(

2

23RR

RVRRRR SG

N

Δτ)'(

)'(

)(

)(

2

1Δτ,,'(

)()(

0

0

0

0

00

R

R

R

RRR

RRV

HHE)S T

Drift - Diffusion processMultiplicity of the walker (“branching” process)

for the primary geometryfor the primary geometry

Correlated sampling DMC

The secondary walkssecondary walks, used to project out the | states, are generated from the primary walk applying the translationtranslation previously defined.

Averages are obtained by reweighting with the ratio of the primary and secondary reweighting with the ratio of the primary and secondary wavefunctionswavefunctions, as in the VMC case.

We must, however, take into account the different multiplicitydifferent multiplicity of the primary and secondary walkers due to the different G(R’,R,) that should be effectively used for propagation. This is obtained redefining the weights as

projN

ssss S

Sww

Δτ),,(exp

)Δτ,,(exp

RR

RR

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 CORRELATEDWALKERS REMAIN EFFECTIVELY CORRELATED.

Effective time step: takes into account modifications to the width of the proposed move due to the coordinates transformation:

22ττ RRss

Wave Functions

The Correlated Walkers scheme illustrated is efficient if the branching is small

• 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

WE NEED VERY OPTIMIZED WAVEFUNCTIONS

Results

We performed simulations for closed shell QD

with N = 6, 12, and 20 electrons

and for different values of the external confinement

0 = 0.21, 0.28, and 0.35 H*

To compute the polarizability and check the linear regime the expectation value D| was computed for four different values of , 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.

0(H*) N (VMC) (DMC) 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)

HUGE EFFECT of INTERACTION!

Results

)2( 85.1 )1( 165.1 r

The ratio is equal to the ratio (d/), and gives us

information about the

splitting between the charge and spin collective modes.

We get

r

Results

4.2r

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.

Exact diagonalization for a QD with N=6 electrons indicate

5.3r

ResultsExperimental data obtained on quantum dots with N200 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!