Maximally localized Wannier functions: Theory and applications

may be expressed as
where c = (µ 0 ϵ 0 ) −1/2 is the speed of light.
∇ × ( ϵr
−1 (r)∇ × H(r) ) = ω2
c 2 H(r), (142) of the vectors A and B, with Cartesian components {A i } and
{B i }, respectively.
ê ν qs = ∑ e iq·R′ δ RR ′δ ss ′e ν For a perfect periodic dielectric structure, ϵ r (r) =
qs ′. (137)
ϵ r (r + R), where R is a lattice vector. Application of
s ′ R ′ Bloch’s theorem leads to solutions that are indexed by
Eq. (137) stands in direct correspondence to the electronic
analogue given by the inverse of Eq. (10), from
which we conclude that the LWFs do indeed correspond
to individual atomic displacements δ RR ′δ ss ′ and, furthermore,
wavevector k, which may be chosen to lie in the first
Brillouin zone, and a band index n. For example
H nk
(r) = e ik·r u nk
(r),
that the required unitary transformation is the ma-
where u nk
(r) = u nk
(r + R) is the periodic part of
trix of eigenvectors [e q ] µν
. As discussed in Sec. VI.D, the magnetic field Bloch function. The electromagnetic
Giustino et al. (2007a) exploit this property for the efficient
interpolation of dynamical matrices and calculation
of electron-phonon couplings.
wave equations can be solved, and hence the Bloch functions
obtained, by a number of methods including finitedifference
time domain (Taflove and Hagness, 2005; Yee,
1966), transfer matrix (Pendry, 1996; Pendry and Mackinnon,
1992), empirical tight-binding methods (Lidorikis
B. Photonic crystals
et al., 1998; Yariv et al., 1999), and Galerkin techniques
in which the field is expanded in a set of orthogonal basis
functions (Mogilevtsev et al., 1999). Within the latter
Photonic crystals are periodic arrangements of dielectric
materials that are designed and fabricated in order to
class, use of a plane-wave basis set is particularly common
(Ho et al., 1990; Johnson and Joannopoulos, 2001).
control the flow of light (John, 1987; Yablonovitch, 1987).
They are very much to light what semiconductors are to
The operators ∇ × ∇ and ∇ × ϵ −1
r (r)∇ are self-adjoint
electrons and, like semiconductors that exhibit an electronic
band gap in which an electron may not propagate,
and, therefore, the fields satisfy orthogonality relations
given by 21
photonic crystals can be engineered to exhibit photonic
∫
band gaps: ranges of frequencies in which light is forbidden
to propagate in the crystal. In the electronic case,
nn ′δ(k − k′ ), (143)
dr H ∗ nk(r) · H n′ k ′(r)
∫
a band gap results from scattering from the periodic potential
due to the ions in the crystal; in the photonic
dr ϵ r (r) E ∗ nk(r) · E n′ k ′(r) = δ nn ′δ(k − k′ ). (144)
case, it arises from scattering from the periodic dielectric Leung (1993) first suggested that transforming to a
interfaces of the crystal. Again by analogy with electronic
materials, localized defect states can arise in the be advantageous for computational efficiency, especially
basis of Wannier functions localized in real space would
gap by the deliberate introduction of defects into a perfect
photonic crystal structure. The ability to control the ing conventional methods, require very large supercells
when dealing with defects in photonic crystals which, us-
nature of these states promises to lead to entirely lightbased
integrated circuits, which would have a number of justifying the existence of a suitable localized basis, and
for convergence. Although of great formal importance for
advantages over their electronic counterparts, including hence the tight-binding approach, the non-uniqueness of
greater speeds of propagation, greater bandwidth, and the transformation between Bloch states and Wannier
smaller energy losses (Joannopoulos et al., 1997). functions caused difficulties. As a result, early work was
In SI units, Maxwell’s equations in source-free regions limited to the case of single, isolated bands (Konotop,
of space are
1997; Leung, 1993) or composite bands in which the matrix
∇ · E = 0, ∇ · B = 0, (138)
elements U mn (k) were treated as parameters to fit the
∇ × E = − ∂B
∂t , ∇ × H = ∂D
tight-binding band structure to the plane-wave result.
∂t , (139) The formalism for obtaining maximally-localized Wannier
functions, however, removed this obstacle and several
applications of MLWFs to calculating the proper-
where the constitutive relations between the fields are
D = ϵ r ϵ 0 E, B = µ r µ 0 H. (140) ties of photonic crystals have been reported since, in
both two-dimensional (Garcia-Martin et al., 2003; Jiao
Considering non-magnetic materials (µ r = 1) with a position
dependent dielectric constant ϵ r (r), and fields that dimensional (Takeda et al., 2006) photonic crystal struc-
et al., 2006; Whittaker and Croucher, 2003) and three-
vary with a sinusoidal dependence e −iωt , it is straightforward
to derive electromagnetic wave equations in terms et al., 2008) (see Busch et al. (2003) for an early review).
tures, as well as for the case of entangled bands (Hermann
of either the electric field E or the magnetic field H,
∇ × (∇ × E(r)) = ω2
c 2 ϵ r(r)E(r), (141)
The notation A·B = ∑ 3
i=1 A iB i , and denotes the scalar product
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