Maximally localized Wannier functions: Theory and applications

40
menta of the surface BZ, and in the vicinity thereof they
disperse linearly, forming Dirac cones.
First-principles calculations of the surface states for
known and candidate TI materials are obviously of
great interest for comparing with ARPES measurements.
While it is possible to carry out a direct first-principles
calculation for a thick slab in order to study the topologically
protected surface states, as done by Yazyev et al.
(2010), such an approach is computationally expensive.
Zhang et al. (2010) used a simplified Wannier-based approach
which succeeds in capturing the essential features
of the topological surface states at a greatly reduced computational
cost. Their procedure is as follows. First, an
inexpensive calculation is done for the bulk crystal without
spin-orbit interaction. Next, disentangled WFs spanning
the upper valence and low-lying conduction bands
are generated, and the corresponding TB Hamiltonian
matrix is constructed. The TB Hamiltonian is then augmented
with spin-orbit couplings λL·S, where λ is taken
from atomic data; this is possible because the WFs have
been constructed so as to have specified p-like characters.
The augmented TB parameters are then used to
construct sufficiently thick free-standing “tight-binding
slabs” by a simple truncation of the effective TB model,
and the dispersion relation is efficiently calculated by interpolation
as a function of the wavevector k ∥ in the surface
BZ.
It should be noted that this approach contains no
surface-specific information, being based exclusively on
the bulk WFs. Even if its accuracy is questionable, however,
this method is useful for illustrating the “topologically
protected” surface states that arise as a manifestation
of the bulk electronic structure (Hasan and Kane,
2010).
Instead of applying the naive truncation, it is possible
to refine the procedure so as to incorporate the changes
to the TB parameters near the surface. To do so, the
bulk calculation must now be complemented by a calculation
on a thin slab, again followed by wannierization.
Upon aligning the on-site energies in the interior of this
slab to the bulk values, the changes to the TB parameters
near the surface can be inferred. However, Zhang
et al. (2011b) found that the topological surface states
are essentially the same with and without this surface
correction.
The truncated-slab approach was applied by Zhang
et al. (2010) to the stoichiometric three-dimensional TIs
Sb 2 Te 3 , Bi 2 Se 3 , and Bi 2 Te 3 . The calculations on Bi 2 Se 3
revealed the existence of a single Dirac cone at the Γ
point as shown in Fig. 30, in agreement with ARPES
measurements (Xia et al., 2009).
An alternative strategy for calculating the surface
bands was used earlier by the same authors (Zhang et al.,
2009a). Instead of explicitly diagonalizing the Wannierbased
Hamiltonian H(k ∥ ) of a thick slab, the Green’s
function for the semi-infinite crystal as a function of
Energy (eV)
0.8
0.6
0.4
0.2
0
-0.2
-0.4
Γ
FIG. 30 Wannier-interpolated energy bands of a free-standing
(111) slab containing 25 quintuple layers of Bi 2 Se 3 plotted
along the Γ–K line in the surface Brillouin zone. A pair of
topologically-protected surface bands can be seen emerging
from the dense set of projected valence and conduction bulklike
bands and crossing at the time-reversal-invariant point Γ.
Adapted from Zhang et al. (2010).
atomic plane is obtained via iterative methods (Lopez-
Sancho et al., 1984, 1985), using the approach of Lee
et al. (2005). Here the localized Wannier representation
is used to break down the semi-infinite crystal into
a stack of “principal layers” consisting of a number of
atomic planes, such that only nearest-neighbor interactions
between principal layers exist (see Ch. VII for more
details).
Within each principal layer one forms, starting from
the fully-localized WFs, a set of hybrid WFs which are
extended (Bloch-like) along the surface but remain localized
(Wannier-like) in the surface-normal direction (see
Secs. II.H and V.B.2). This is achieved by carrying out
a partial Bloch sum over the in-plane lattice vectors,
|l, nk ∥ ⟩ = ∑ R ∥
e ik ∥·R ∥
|Rn⟩, (103)
where l labels the principal layer, k ∥ is the in-plane
wavevector, and R ∥ is the in-plane component of R. The
matrix elements of the Green’s function in this basis are
G nn′
ll ′
(k 1
∥, ϵ) = ⟨k ∥ ln|
ϵ − H |k ∥l ′ n ′ ⟩. (104)
The nearest-neighbor coupling between principal layers
means that for each k ∥ the Hamiltonian has a block tridiagonal
form (the dependence of the Hamiltonian matrix
on k ∥ is given by the usual Fourier sum expression).
This feature can be exploited to calculate the diagonal elements
of the Green’s function matrix very efficiently using
iterative schemes (Lopez-Sancho et al., 1984, 1985). 18
18 A pedagogical discussion, where a continued-fractions expansion
is used to evaluate the Green’s function of a semi-infinite linear
chain with nearest-neighbor interactions, is given by Grosso and
Parravicini (2000).
K