Lecture9: Polarization th. II

Macroscopic Polarization
from Electronic Wavefunctions
Raffaele Resta
Dipartimento di Fisica Teorica, Università di Trieste
<[email protected]>
Sanibel Symposium 1999
The dipole of a microscopic system
(atom, molecule....)
Neglecting (irrelevant) spin variables:
n(r) = N
Z
dr2dr3 . . . drN |Ψ(r, r2, · · · rN )|2;
d=
Z
dr r n(r).
Alternatively:
d = hΨ|R̂|Ψi;
R̂ ≡
N
X
i=1
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ri
(a one-body multiplicative operator).
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The polarization of a macroscopic solid
(The Feynman Lectures in Physics, Vol.2, Ch.11)
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Incorrect (and unusable) definitions
• Kittel, Introduction to Solid State Physics:
A ferroelectric crystal exhibits an electric dipole moment even in
the absence of an external electric field. In the ferroelectric
state the center of positive charge does not coincide with the
center of negative charge.
• Ashcroft & Mermin, Solid State Physics:
Crystal whose natural primitive cells have a nonvanishing
dipole moment p0 are called pyroelectric.
• Lines & Glass, Principles and Applications of Ferroelectrics
and Related Materials (1977):
If and when good electron-density maps become available for
ferroelectrics, expressing charge density ρ(r) as a function of
position vector r throughout the unit cell, more quantitative
estimates of spontaneous polarization might be envisaged as
1
rρ(r) dr.
Ps =
V V
Z
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Outline
• What makes the polarization problem difficult for theorists?
• How is macroscopic polarization measured?
• Theory of polarization (& calculations) after 1993:
• A novel view: the ultimate solution:
Very general: either periodic or disordered
either independent–electron or correlated.
• Relationship to the (discretized) Berry phase.
• Some comments about polarization within DFT.
• Latest development: wavefunction localization in insulators.
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Back to basics:
the QM position operator
• It is the simple multiplicative operator which maps:
ψ(x) → x ψ(x).
• The expectation value is:
hxi = dx x |ψ(x)|2.
R
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Fundamental issues
• What goes wrong?
• What is the correct definition of bulk
macroscopic polarization?
• How to compute the polarization from the
wavefunction?
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The Hilbert space of condensed matter
BvK boundary conditions:
ψ(x + L) = ψ(x)
(L large with respect to atomic dimensions).
The multiplicative operator x is not
a legitimate operator in this Hilbert space.
x ψ(x) does not belong to the Hilbert space
whenever ψ(x) does.
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Reconsidering some basic facts
• The “absolute” macroscopic polarization of a solid
has never been measured (as a bulk quantity).
• The experimental quantities are invariably
derivatives (piezolelectricity, pyroelectricity,
dielectric permittivity.....) or finite differences
(ferroelectricity measured through polarization
reversal).
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Basic facts cont’d
• These derivatives/differences are obtained by
measuring currents.
• The current transported by a Bloch orbital (within
BvK) is a well defined quantity.
• The current is essentially related to the phase of
the wavefunction.
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The modern theory (since 1993)
Macroscopic polarization is a Berry phase of the
Bloch orbitals (in the uncorrelated case).
In general, a Berry phase is an observable which
cannot be cast as the expectation value of any
operator.
Instead it is a gauge–invariant phase of the
wavefunction.
Basic review: Rev. Mod. Phys. 66, 899 (1994).
Nontechnical review: Europhys. News 28, 18 (1997).
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Berry phase: Actual calculations
• DFT-LDA, HF, tight-binding, Hubbard model.
• PW (norm-conserving pseudo, ultrasoft pseudo), FLAPW,
LCAO (contracted Gaussians).
• Properties: piezoelectricity, dynamical charges,
ferroelectricity.
• Permittivity ε∞ not accessible.
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A precursor: “Elettrone in brodo”
(Selloni, Carnevali, Car, Parrinello, PRL 1987)
A lone electron in a large periodic cell
(rari nantes in gurgite vasto)
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Position defined modulo L:
The wrong approach:
Any variation on this theme.
ξ(x) = x
hxi =
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Z L
0
mod
L
dx ξ(x)|ψ(x)|2
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Position defined modulo L
0
L
hxi
L
hxi =
Im log
2π
Z L
0
dx e
i 2π
Lx
|ψ(x)|2
Example:
|ψ(x)|2 =
∞
X
δ(x − x0 − mL)
m=−∞
L
i 2π
Im log e L x0 ≡ x0 (mod L)
hxi =
2π
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Q: Why the “electron–in–broth” formula is the
correct one?
A: Because it provides the macroscopic current
(for L → ∞):
d
hx(t)i = hj(t)i
dt
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N electrons in a box of size L
Naı̈f generalization: |ψ(x)|2 → n(x):
Z L
L
i 2π
Im log
dx e L x n(x)
hXi =
0
2π
Wrong:
The Fourier coefficients of the periodic density
do not carry any information about polarization.
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Many–body BvK boundary conditions
Ψ(x1, . . . , xi, . . . , xN ) = Ψ(x1, . . . , xi +L, . . . , xN ).
(periodic on each electronic variable separately).
Thermodynamic limit:
N → ∞, L → ∞,
N/L = n0 constant.
The operator X̂ ≡
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PN
i=1 xi
becomes meaningless.
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The ultimate solution
[PRL 80, 1800 (1998)]
L
i 2π
hXi =
Im log hΨ|e L X̂ |Ψi
2π
e
i 2π
Im log hΨ|e L X̂ |Ψi
Pel = lim
L→∞ 2π
A tricky limit.
hXi is defined modulo L.
A Berry phase in disguise.
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Within poor man’s first quantization:
L
hXi =
Im log zN
2π
zN =
i 2π
L X̂
e
Z L
0
PN
2π
dx1dx2 . . . dxN |Ψ(x1, x2, . . . xN )|2 ei L i=1 xi .
is a genuine many–body (multiplicative)
operator.
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Crystalline case, independent electrons
Born-von-Kàrmàn boundary conditions;
M crystal cells of size a; total length L = M a.
Allowed Bloch vectors in the reciprocal cell [0,2π/a):
2π
2π s
=
s,
qs =
a M
L
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s = 0, 1, . . . , M −1
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The one-particle orbitals may be chosen in the Bloch
form:
ψqs,m(x + τ ) = eiqsτ ψqs,m(x)
N/M occupied bands:
| Ψi = A
N/M
−1
Y MY
m=1 s=0
zN =
i 2π
hΨ|e L X̂ |Ψi
=
MY
−1
s=0
Sm,m′ (qs, qs+1) =
Z L
0
dx
ψqs,m.
det S(qs, qs+1)
−i 2π
∗
ψqs,m(x)e L xψqs+1,m′ (x)
−→ King-Smith & Vanderbilt discrete Berry phase.
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Berry phase via an expectation value:
Is it “legally accurate” or “improper”?
γ = Im log
Z L
0
dx1 . . . dxN |Ψ(x1, x2, . . . xN
i 2π
2
)| e L X̂ .
Berry phase expressed in terms of the modulus of
the wavefunction.
The modulus of the many–body wavefunction
contains interference terms due to the relative
phases of the Bloch orbitals: these are “extracted”
in a gauge–invariant manner.
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What about DFT?
The HK theorem implicitly assumes a finite system
of N electrons, with a square–integrable (L2)
wavefunction Ψ(r1, r2, · · · rN ).
The same HK proof carries over with no change
assuming instead BvK boundary conditions for the
finite system of N electrons. Then one can proceed
as in KS.
But......
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Using L2 boundary conditions the dipole of the
system is a function of the density.
Caveat: Of the whole density, surface and bulk.
Using instead BvK boundary conditions:
(i) There is no surface by construction;
(ii) Macroscopic polarization has nothing to do with
the density of the polarized solid:
polarization = phase of Ψ; density = modulus of Ψ.
Ergo......
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The dipole of a finite (possibly large) correlated
system coincides with the dipole of the corresponding
noninteracting KS system.
The macroscopic polarization of a correlated solid
does not coincide with the polarization of the
corresponding noninteracting KS system (both
evaluated within BvK).
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Hot topic these days:
“Polarization DFT”
• “Ultranonlocality” of the KS exchange–correlation potential?
• Solution of the “permittivity problem”?
• Recipes for practical calculations in periodic solids?
• Related to Current Density Functional Theory?.
Several papers continue to appear....
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Latest developments: Localization
0
L
hxi
What about hx2i − hxi2 ?
Marzari & Vanderbilt localization (alias Boys localization).
R. Resta & S. Sorella, PRL 82, 370 (1999).
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z=
Z L
0
dx e
i 2π
Lx
|ψ(x)|2
0 ≤ |z| ≤ 1
,
Complete localization:
|ψ(x)|2 =
∞
X
2π
δ(x − x0 − mL) −→ z = ei L x0
m=−∞
Complete delocalization:
1
|ψ(x)| =
L
2
hxi =
−→
z=0
L
Im log z
2π
2
L
hx2i − hxi2 = −
log |z|2
2π
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