Post on 16-Dec-2015
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Localization tensor calculations onquantum dots in DFT and VMC
“Quantum Monte Carlo in the Apuan Alps”
Valico Sotto, Tuscany - 27th July 2005
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
<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?