Localization tensor calculations onquantum dots in DFT and VMC

“Quantum Monte Carlo in the Apuan Alps”

Valico Sotto, Tuscany - 27th July 2005

Quantum Dot Arrays

Electrons Tied to specific centres byharmonic restoring force: Localized

Electrons free to move within the structure: Delocalized

Metals and Insulators

• Classical Models: Lorentz and Drude

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?

Qualitatively, can still explain the difference in terms of low-lying excitations – but only for independent electrons in crystalline systems

Insulator Metal

…no direct link to localization

E

k

E

k

Band Theory

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

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)

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

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

Wannier Functions

• Wannier Function in cell R associated with band n is

• Can define a spread functional to measure the localization of this orbital

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:

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.

“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).

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

Calculating Localization Tensors in Density Functional Theory

No suitable 2D DFT Program available, so I wrote

DOTDFT

Suggestions of better names welcome…

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

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 zN

• Overlap of two Slater Determinants is the Determinant of the individual overlaps

• Individual overlaps are zero except for adjacent k-points

Decreasing Dot Depth ->

DFT Results

Inverse Energy Gap Eg-1 (Ha*-1)

<x2 >

c (B

ohr*

2)

Behaviour of localization tensor with decreasing energy gap (approaching band insulator -> metal transition by varying QD confinement)

DFT Results

• 1D Chain of dots– simpler dispersion curve – band crossings

kx (Bohr*-1)

ε(kx)

DFT Results

• 1D Chain of dots– simpler dispersion curve – band crossings

kx (Bohr*-1)

ε(kx)

DFT Results

• 1D Chain of dots– simpler dispersion curve – band crossings

kx (Bohr*-1)

ε(kx)

DFT Results

• 1D Chain of dots– simpler dispersion curve – band crossings

kx (Bohr*-1)

ε(kx)

DFT Results

• 1D Chain of dots– simpler dispersion curve – band crossings

kx (Bohr*-1)

ε(kx)

DFT Results

<x2 >

c1/2 (B

ohr*

)

Harmonic confining potential Omega (Bohr*-1)

More Confined Dot->

Band Crossing

Approach to band crossing shows up clearly in <x2>c

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?

Some (early) QMC Results sans Jastrow

Adjacent Harmonic wells, 50x50 supercell

Errors not too bad with enough steps<

x2 >c (B

ohr*

2 )

Harmonic confining potential Omega (Bohr*-1)

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?

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!)