Theoretical Materials Science

1
Theoretical Materials Science

• 2nd lecture
• The basic concept of the density functional
  theory II

Associate Professor Koichi Kusakabe
Graduate School of Engineering Science, Osaka
University
2
The local density approximation IV.

• LDA is given by expressiones for the exchange and
  correlation energy density, ex(n(r)), ec(n(r)).
  They are derived from information on the electron
  gas.
• The Wigner interpolation formula (1934) The
  low-density regime (a BCC lattice).
• Nozières and Pines (1958) Splitting the Coulomb
  potential into a short-range (2nd order
  perturbation)and a long-range part (plasmon
  theory).
• Hedin and Lundqvist (1971) Inclusion of the
  local field correction of Singwi et al.
• Vosko, Wilk and Nusair (1980) von Barth-Hedins
  interpolation with Padé approx. and incorporation
  of the Monte Carlo result by Ceperley and Alder.
• Perdew and Zunger (1981) Parametrization of the
  Monte-Carlo results by Ceperley and Alder.

3
Correlation energy density for LDA
Vosko, Wilk Nusair (1980)
Hedin Lundqvist (1971)
Perdew Zunger (1981)
Nozières Pines (1958)
Wigner (1934)
4
Band structure of cubic diamond
s bands (anti-bonding bands)
s bands (bonding bands)
G
G
An energy gap appears and the system is a
wide-gap semiconductor.
LDA by PW91. Plane-wave expansion with ultra-soft
PP.
5
Bonding charge in hex-diamond

• In a covalent crystal, we can see charge density
  of electrons at each bond connection.
• Yellow object represents charge density and white
  spheres are carbons.

6
Band structure of graphite
s bands (anti-bonding bands)
p bands (anti-bonding)
p bands (bonding bands)
s bands (bonding bands)
G
The p-band is half-filled and there are
small Fermi pockets both for electrons and holes.
(Semimetal)
7
Bonding charge in graphite

• Bonding charge comes from s-electrons.
• This system is a semimetal where the Fermi
  surface is made of p-bands.

8
Landaus theory of the phase transition

• Order parameter F (r)
• Liquid-Gas F (r)rliq(r)-rgas(r)
• Solid-Liquid F (r)rsol(r)-rliq(r)
• Magnetic MQ(r) or
• Superconducting D (r)
• Free energy

Gorkovs derivation of the Ginzburg-Landau
equation is valid, when

• The superconducting order parameter and the
  vector potential are both small.
• The integral kernels in the expressions are
  short ranged and they are damped in a distance
  shorter than the coherence length and the
  penetration depth.

9
The density functional theory
Y?n

• N-representability by Harriman
• Variational principle w.r.t. the density
• Kohn-Sham scheme as a theory of the
  phasetransition

Y?n
Phase space of Y
F ? nF(r), DF(r) ? Fn, Excn ? Free energy
Order parameters
10
The Harriman construction
11
2p
12
The spin-density functional theory
To introduce spin density as a basic variable in
DFT, we have to modify the theory.

• Current-DFT formulation
• Extension of Levys functional to relativistic
  version (Rajagopal-Callaway, and
  other works)

• DFT with arbitral basic variables
• Extension of Levys constrained search
  (Higuchi-Higuchi)

Excn,m is obtained by e.g. fitting the
numerical data of QMC for a spin polarized
electron gas. ? LSDA, spin-GGA Solvers for DFT
are immediately applied for the spin-DFT
calculations.
13
Transition metal elements

• Characteristics of 3d transition metals
• Spins in an atom or in an ion align by Hunds
  coupling
• The Hund rule tells that if there is degeneracy
  w.r.t. L S,
• The maximum S appears.
• The maximum allowed L for given S appears.
• d-orbitals have characters below.
• They are rather localized around the nucleus.
• They form narrow bands.
• To explain ferromagnetism in 3d transition
  metals, we have to consider at least by an
  itinerant electron picture, since

14
DFT-GGA calculation of Fe
NM BCC
AF BCC
AF HCP
NM FCC
NM HCP
FM BCC
T. Asada K. Terakura, PRB 46 (1992) 13599.
15
GGA calculations of tetragonal manganites
Fang, Solovyev Terakura, PRL 84 (2000) 3169.
GGA may reproduce the orbital ordered (OO)
magnetic phases. Metal-insulator transition with
OO may also be found.
16
Exchange-Correlation hole for atoms

• The exchange-correlation energy functional may be
  written as,
• with
• In LDA, we approximate nxc by that of the
  homogeneous electron gas as,

Fluctuation!
17
Exchange-correlation hole in H
Gunnarsson, Jonson Lundqvist, PRB 20 (1979)
3136.
If one looks at spherical average of the XC hole,
the LDA result is close to the exact one.
Excn is proportional to an integral of the XC
hole. Thus the total energy and its parameter
derivatives, i.e. atomic forces, internal stress.
r
r
Atomic center
18
Exchange hole for a neon atom
Gunnarsson, Jonson Lundqvist, PRB 20 (1979)
3136.
19
GEA and GGA

• Gradient expansion approximation GEA
• Generalized gradient approximation GGA

20
GEA exchange hole
21
Difficulty in the gradient expansion approximation
Cf. K. Burke, J.P. Perdew and Y. Wang In
electronic Density Functional Theory edt. Dobson
et al. (1998).

• The gradient expansion approximation (GEA) fails
  due to
• Impossibe to full fill next equality and an
  inequality.

negativity of exchange hole
sum rule for exchange hole
sum rule for correlation hole
Coupling constant is proportional to l keeping
r(r).
22
GGA given by cutoff procedure to GEA

• In GGA, starting from GEA, cuttoff procedure is
  introduced to keep

negativity of exchange hole
sum rule for exchange hole
sum rule for correlation hole
For the case of exchange hole,
For the case of correlation hole, similar
expressions with a cutoff in a reduced separation
on the Thomas-Fermi length scale, when it is
integrated.
23
Difficulty in GGA

• Impossible to reproduce cohesion of layered
  materials, graphite, hex-BN and CF.
• Due to two dimensionality.
• Due to Van-der Waals nature (even worse for
  one-dimensional materials including metallic
  nanotubes)
• Impossible to reproduce magnetism of weak
  ferromagnets including ZrZn2, meta-magnetic
  paramagnets including YCo2 in the Laves phase.
• Due to strong tendency to stabilize
  ferromagnetism.
• Interestingly, L(S)DA reproduces qualitative
  features of these problematic materials.
• Much accurate calculations (DMC for 2DEG, DMC
  with backflow effects etc for EG) are required as
  references.

24
Various methods to overcome difficulty of
DFT-LDA, DFT-GGA

• Excitation spectrum is not properly described by
  DFT-LDA, DFT-GGA.
• This is partly because DFT is only for the ground
  state. However, sometimes, DFT-LDA DFT-GGA
  incorrectly conclude a metal rather than gapped
  excitations (Motts insulator).
• For excitations GW, GWT, EXXRPA (perturbative
  methods.)
• For Motts insulator LDAU, LDA (a model
  description introduced in DFT.)