1. Localization tensor calculations on quantum dots in DFT and VMC “Quantum Monte Carlo in the Apuan Alps” Valico Sotto, Tuscany - 27 th July 2005

2. Quantum Dot Arrays

3. Electrons Tied to specific centres by harmonic restoring force: Localized Electrons free to move within the structure: Delocalized Metals and Insulators Classical Models: Lorentz and Drude

4. Bloch Functions Introduce periodic boundary conditions on supercell of size L=Ma Solutions of Schrödinger Equation for independent particles are then Bloch Wavefunctions … Always delocalized Localized/Delocalized distinction between metals and insulators apparently lost?

5. Qualitatively, can still explain the difference in terms of low-lying excitations – but only for independent electrons in crystalline systems InsulatorMetal …no direct link to localization E k E k Band Theory

6. Equal except for terms which vanish exponentially with system size Kohn’s Disconnectedness Kohn (1964) revives link between localization and insulating behaviour is disconnected if it breaks down into the sum of terms which are localized in nonoverlapping regions of the 3N dimensional hyperspace defined by N electron coordinates

7. Modern Theory of Localization Recent approach to localization (1999 onwards) stems from theory of polarization (1993 onwards) Based on Berry’s Phases and proper treatment of position operator within periodic boundary conditions Resta, Sorella, PRL 82, 370 (1999); Souza, Wilkens, Martin PRB 62, 1666 (2000) Vanderbilt, King-Smith PRB 48, 4442 (1993)

8. Choosing boundary conditions for a system defines the Hilbert space for which solutions of the Sch. Eq. are defined Operator R applied to a function obeying PBC returns a function which does not obey PBC and hence does not belong to the same space R is a forbidden operator within PBC… but the associated probability distribution still has meaning Position Operator within PBC

9. Electron Localization Most natural quantity to measure localization is the quadratic spread, second cumulant moment (basically the variance). Simple definition in a finite, single particle system: Several routes to a comparable expression within PBC and for a many- particle system – most rigorous is a rather involved generating function approach Also a more intuitive formulation in terms of Maximally Localized Wannier Functions

10. Wannier Functions Wannier Function in cell R associated with band n is Can define a spread functional to measure the localization of this orbital

11. Can prove that the spread, minimized with respect to gauge transformations among the orbitals, is no smaller than the trace of this gauge invariant tensor (M&V): The Localization Tensor This expression is of the same form as the expression for the second cumulant moment of the wavefunction arrived at through the generating function approach on a general (correlated) wavefunction:

12. Many Body Phase Operator Define unitary many-body operator This is acceptable within PBC for certain values of k With ground state expectation values Need to recast in terms of periodic boundary conditions only (not twisted) and so that it can be evaluated from the expectation value of some operator.

13. “Single-Point” Formula With off-diagonal components To provide a “single-point” formula for periodic boundary conditions requires an ansatz about the form the correlation takes (assumes short range correlations).

14. So… Yes Is this a measurable quantity? Souza, Wilkens, Martin (2000) link to a conductivity integral, via Linear Response Theory This demands that for a finite gap insulator, the localization tensor is limited by the inequality

15. Calculating Localization Tensors in Density Functional Theory No suitable 2D DFT Program available, so I wrote DOTDFT Suggestions of better names welcome…

16. Calculating Localization Tensors in Density Functional Theory Represents wavefunctions and potentials using plane waves, on a real and reciprocal space grid, and using k-points on a grid in the BZ FFTs to switch between representations Calculates Hartree energy with reciprocal space sum, uses Local (Spin) Density Approx. to Exchange and Correlation Construct Kohn-Sham Hamiltonian Solve for Kohn-Sham Wavefunctions Mix with input density (Broyden Method) and repeat until converged

17. Overlap from opposite spins identical in pairs Calculating Localization Tensors in Density Functional Theory Put these Kohn-Sham wavefunctions in a Slater Determinant and evaluate z N Overlap of two Slater Determinants is the Determinant of the individual overlaps Individual overlaps are zero except for adjacent k-points

18. Decreasing Dot Depth -> DFT Results Inverse Energy Gap E g -1 (Ha* -1 ) c (Bohr*2) Behaviour of localization tensor with decreasing energy gap (approaching band insulator -> metal transition by varying QD confinement)

19. DFT Results 1D Chain of dots– simpler dispersion curve – band crossings k x (Bohr* -1 ) ε(kx)ε(kx)

20. DFT Results 1D Chain of dots– simpler dispersion curve – band crossings k x (Bohr* -1 ) ε(kx)ε(kx)

21. DFT Results 1D Chain of dots– simpler dispersion curve – band crossings k x (Bohr* -1 ) ε(kx)ε(kx)

22. DFT Results 1D Chain of dots– simpler dispersion curve – band crossings k x (Bohr* -1 ) ε(kx)ε(kx)

23. DFT Results 1D Chain of dots– simpler dispersion curve – band crossings k x (Bohr* -1 ) ε(kx)ε(kx)

24. DFT Results c 1/2 (Bohr*) Harmonic confining potential Omega (Bohr* -1 ) More Confined Dot-> Band Crossing Approach to band crossing shows up clearly in c

25. Calculating Localization Tensors with Quantum Monte Carlo QMC ideal for the evaluation of expectation values of many-body operators on many-body wavefunctions Slater-Jastrow wavefunctions can be used to include exchange and correlation What does this do to localization tensor components?

26. Some (early) QMC Results sans Jastrow Adjacent Harmonic wells, 50x50 supercell Errors not too bad with enough steps c (Bohr* 2 ) Harmonic confining potential Omega (Bohr* -1 )

27. Why Comparing DFT/HF/QMC is interesting Underestimation of bandgaps within DFT, overestimation within HF. QMC can usually calculate bandgap correctly – so might expect the localization tensor behave correctly too – but in this form it may be more dependent on the orbitals. Can we change a metal/insulator question to a lowest energies given certain occupation schemes question?

28. Further Work More VMC work – further investigation of effect of Jastrow Factors Another strand we’ve been following: Double dot systems (spin filters), collaboration with experimentalists (hard!)