Polarizability in Quantum Dots via Correlated Quantum Monte Carlo

1. 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

2. 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

3. 1)Raman scattering exp. quantum dots’ collective excitations incident light beam: polarized scattered light beam... ...polarization is lost ...conserving the polarization SDE CDE Schüller et al. PRL 80, 2673 (1998) ω

4. 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 forcorrelated quantities. • Indeed, QMC great for ground state, not for excited states...

5. Backbone of the approach ω? Sum rules excitation correlated quantity… no QMC analytic Model independent still a correlated quantity… but feasible QMC polarizability numerical Coulomb interaction

6. 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 the energy weighted sum rule m1: Electric field in the dipole approximation ( λ~ 50μm >> 100nm) the inverse-energy weighted sum rule m : -1 Polarizability?

7. Polarizability N charged particles under the effect of a small constant electric field: unperturbed Hamiltonian dipole operator THEN polarizability: Here we assume that for l=0 the system is in its ground state, and 0|D|0 = 0 for parity.

8. 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 energywn0: But recall that How to QMC ? then QMC unfeasible Computingais therefore equivalent to compute m-1, without determining all the eigenstates |n and eigenvalues wn0.

9. 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” Induced dipole moment External electric field

10. Polarizability in a Quantum Dot the picture E 20 - 100 nm Electrons (conducting band) or Holes (valence band) 2 - 10 nm Harmonic for N < 30 • Low density • Shell structure

11. Polarizability in a Quantum Dot the formalism The QD Hamiltonian is assumed to be: ra0*-3 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.

12. x l=0 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

13. 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 .

14. 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 KohnTheorem !

15. 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!

16. 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 polarizabilitytakes contributions from the interaction, which plays a fundamental role. Note that in absence of interaction we would have ω = ω i.e. SDE CDE

17. 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

18. Correlated sampling VMC 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 λ’ etc… d d’ 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 Sample only a “primary” geometry; and “link” secondary geometries to this one Computationally expensive: need several d and δV<< (V-V’)

19. 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 Nconfis the number of configurations sampled, and the wi is a weight of the configuration defined by: Note: sampled from

20. SIGN DEPENDS ON SPIN 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:

21. Evaluate <D> on each secondary geometry

22. Multiplicity of the walker (“branching” process) Drift - Diffusion process Correlated samplingDMC 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 Green’s Function of the Schrödinger equation: for the primary geometry where

23. Effective time step: takes into account modifications to the width of the proposed move due to the coordinates transformation: Correlated sampling DMC The secondary walks, used to project out the |Yl states, are generated from the primary walk applying thetranslationpreviously 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 Nprojis 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.

24. 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

25. 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 Yl|D|Yl was computed for four different values of l, namely 10-2,10-3,10-4,10-5.

26. 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!

27. 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

28. Results Exact diagonalization for a QD with N=6 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 QD’s, finding a ratio between polarizabilities of about 3.

29. 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

30. 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!