Linear density response function in the projector-augmented wave method: Applications to solids, surfaces, and interfaces

Abstract

We present an implementation of the linear density response function within
the projector-augmented wave (PAW) method with applications to the linear
optical and dielectric properties of both solids, surfaces, and interfaces. The
response function is represented in plane waves while the single-particle
eigenstates can be expanded on a real space grid or in atomic orbital basis for
increased efficiency. The exchange-correlation kernel is treated at the level
of the adiabatic local density approximation (ALDA) and crystal local field
effects are included. The calculated static and dynamical dielectric functions
of Si, C, SiC, AlP and GaAs compare well with previous calculations. While
optical properties of semiconductors, in particular excitonic effects, are
generally not well described by ALDA, we obtain excellent agreement with
experiments for the surface loss function of the Mg(0001) surface with plasmon
energies deviating by less than 0.2 eV. Finally, we apply the method to study
the influence of substrates on the plasmon excitations in graphene. On
SiC(0001), the long wavelength $\pi$ plasmons are significantly damped although
their energies remain almost unaltered. On Al(111) the $\pi$ plasmon is
completely quenched due to the coupling to the metal surface plasmon.

Full text

Page 1

arXiv:1104.1273v1 [cond-mat.mtrl-sci] 7 Apr 2011
Linear density response function in the projector-augmented wave method:
Applications to solids, surfaces, and interfaces
Jun Yan, Jens. J. Mortensen, Karsten W. Jacobsen, and Kristian S. Thygesen∗
Center for Atomic-scale Materials Design, Department of Physics
Technical University of Denmark, DK - 2800 Kgs. Lyngby, Denmark
(Dated: April 8, 2011)
We present an implementation of the linear density response function within the projector-
augmented wave (PAW) method with applications to the linear optical and dielectric properties
of both solids, surfaces, and interfaces. The response function is represented in plane waves while
the single-particle eigenstates can be expanded on a real space grid or in atomic orbital basis for
increased efficiency. The exchange-correlation kernel is treated at the level of the adiabatic local
density approximation (ALDA) and crystal local field effects are included. The calculated static
and dynamical dielectric functions of Si, C, SiC, AlP and GaAs compare well with previous cal-
culations. While optical properties of semiconductors, in particular excitonic effects, are generally
not well described by ALDA, we obtain excellent agreement with experiments for the surface loss
function of the Mg(0001) surface with plasmon energies deviating by less than 0.2 eV. Finally, we
apply the method to study the influence of substrates on the plasmon excitations in graphene. On
SiC(0001), the long wavelength π plasmons are significantly damped although their energies remain
almost unaltered. On Al(111) the π plasmon is completely quenched due to the coupling to the
metal surface plasmon.
PACS numbers: 73.20.Mf, 71.15.-m, 78.20.-e.
I.INTRODUCTION
Time-dependent density functional theory (TDDFT)1
has been widely used to calculate optical excitations in
molecules and clusters as well as the optical and electron
energy loss spectra of bulk semiconductors, metals and
their surfaces2.The excitation energies and oscillator
strengths of both single-particle and collective electronic
excitations are determined by the frequency-dependent
linear density response function χ(r,r′,ω) giving the den-
sity response at point r to first order in a time-dependent
perturbation of frequency ω applied at point r′,
δn(r,ω) =
?
drχ(r,r′,ω)δVext(r′,ω). (1)
For finite systems, χ can be efficiently calculated by in-
verting an effective Hamiltonian in the space of particle-
hole transitions.For the practically relevant case of
frequency-independent exchange-correlation kernels this
formulation leads to the well known Casida equation3.
For extended systems, it is more convinient to express χ
in a basis of plane waves4–6where it has the generic form
χGG′(q,ω), with G being reciprocal lattice vectors and
q being wavevectors in the first Brillouin zone (BZ).
In this paper we focus on the electronic response func-
tion of extended systems treating electron-electron in-
teractions at the level of the random phase approxi-
mation (RPA) and the adiabatic local density approx-
imation (ALDA). For many extended systems such a
description is insufficient to account for optical excita-
tions because the electron-hole attraction is not prop-
erly accounted for.However, dielectric properties, in
particular collective plasmon excitations, are generally
accurately reproduced by this approach7,8, and quanti-
tative agreement with electron energy loss experiments
have been reported for bulk metals9,10, surfaces11,12,
graphene-based systems13,14, semiconductors15,16and
even supercondutors17. Furthermore, the accurate eval-
uation of the density response function at the RPA or
ALDA level is a prerequisite for implementation of most
post-DFT schemes, such as RPA correlation energy18,
exact-exchange optimized-effective-potential methods19,
the GW approximation for quasi-particle excitations20,21,
and the Bethe-Salpeter equation21,22for optical excita-
tions.
Here we present an implementation of the density
response function within the electronic structure code
gpaw23,24which is based on the projector augmented
wave (PAW) methodology25,26and represents wave func-
tions on real space grids or in terms of linear combina-
tions of atomic orbitals (LCAO)27. Within the PAW for-
malism one works implicitly with the all-electron wave
functions and has access to the (frozen) core states. This
makes the method applicable to a very broad range of sys-
tems including materials with strongly localized d or f
electrons which can be problematic to describe with pseu-
dopotentials. An additional advantage of the PAW for-
malism, with respect to linear response theory, is that the
optical transition operator in the long wavelength limit
can be obtained directly due to the use of all-electron
wavefunctions28. The non-interacting response function,
χ0, is built from the single-particle eigenstates obtained
either on a real space grid, which is the standard repre-
sentation in the GPAW code, or in terms of a localized
atomic orbital (LCAO) basis. We have found that the
latter choice reduces the computational cost of χ0con-
siderably while still preserving the high accuracy of the
grid calculation.

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The method is used to calculate the macroscopic di-
electric constants of a number of bulk semiconductors,
showing very good agreement with previous calculations
as well as experiments. For the surface plasmons of the
Mg(0001) surface we find, in agreement with previous
studies, that the ALDA kernel lowers the plasmon en-
ergies by around 0.3 eV relative to the RPA values and
thereby reduces the deviation from experiments from 4%
to 1-2%. Very good agreement with experiments is also
found for the plasmon energies of graphene which are
shown to exhibit a linear dispersion with a value of 4.9 eV
in the long wave length limit. The deposition of graphene
on a SiC substrate is shown to have little effects on the
plasmon energies but leads to significant broadening of
the plasmon resonances. In contrast deposition on an Al
surface completely quenches the graphene plasmons due
to strong non-local electronic screening.
The rest of this paper is organized as follows. Section
II introduces the theoretical framework, where the PAW
methodology, the density response function for both fi-
nite q and q → 0, and the ALDA kernel in the PAW
method are discussed. The details of the implementation
and parallelization in gpaw and other technical details
are presented in section III. Section IV presents appli-
cations for optical properties and plasmon excitations of
bulk and surfaces, where comparison with other calcula-
tions and experiments are given. Our recent investigation
on the effect of a semiconducting and metallic substrate
on the plasmon excitations in graphene is also briefly
discussed in this section. Finally, a summary is given in
section V.
II.METHOD
A.Basics of the PAW formalism
In the PAW formalism25,26, a true all-electron Kohn-
Sham wavefunction ψnk is obtained by a linear trans-
formation from a smooth pesudo-wave-function˜ψnkvia
ψnk=ˆT˜ψnk. The transformation operator is chosen in
such a way that the all-electron wavefunction ψnkis the
sum of the pseudo one˜ψnkand an additive contribution
centered around each atom written as
ψnk(r) =˜ψnk(r)+
?
a,i
?˜ pa
i|˜ψnk?[φa
i(r−Ra)−˜φa
i(r−Ra)]
(2)
The pseudo-wave-function˜ψnk matches the all-electron
one ψnk outside the augmentation spheres centered on
each atom a at position Ra. Their differences inside the
augmentation region are expanded on atom-centered all-
electron partial waves φa
iand the smooth counterparts
˜φa
˜ pa
iis chosen as a dual basis to the pseudo-partial wave
and is called a projector function. A frequently occuring
term is the all-electron expectation value for a semilocal
i. The expansion coefficient is given by ?˜ pa
i|˜ψnk?, where
operator A written as
?ψnk|A|ψnk? = ?˜ψnk|A|˜ψnk?
+
?
a,ij
?˜ψnk|˜ pa
i??˜ pa
j|˜ψnk?[?φa
i|A|φa
j? − ?˜φa
i|A|˜φa
j?] (3)
B.Density response function and dielectric matrix
A key concept in TDDFT is the density response func-
tion χ. It is defined as χ(r,r′,ω) = δn(r,ω)/δVext(r′,ω),
where Vextis the external perturbing potential and δn is
the induced density under the perturbation. For periodic
systems, χ can be written in the form
χ(r,r′,ω) =
1
NqΩ
BZ
?
q
?
GG′
ei(q+G)·rχGG′(q,ω)e−i(q+G′)·r′,
(4)
where G,G′are reciprocal lattice vectors, q is a wave
vector restricted to the first Broullion Zone (BZ), Nq is
the number of q vectors and Ω is the volume of the real
space primitive cell.
The density response function of the interacting elec-
tron system, χ, can be obtained from the non-interacting
density response function of the Kohn-Sham system, χ0,
and a kernel, K, describing the electron-electron interac-
tions by solving a Dyson-like equation
χGG′(q,ω) = χ0
GG′(q,ω)
+
?
G1G2
χ0
GG1(q,ω)KG1G2(q)χG2G′(q,ω). (5)
The expression for the non-interacting density re-
sponse function in the Bloch representation of Adler and
Wiser4,5, is
χ0
GG′(q,ω) =
2
Ω
?
k,nn′
(fnk− fn′k+q)
×
nnk,n′k+q(G)n∗
ω + ǫnk− ǫn′k+q+ iη
nk,n′k+q(G′)
,(6)
where
nnk,n′k+q(G) ≡ ?ψnk|e−i(q+G)·r|ψn′k+q?
is defined as the charge density matrix. Its evaluation
within the PAW formalism is explained in detail in the
following subsection. ǫnk, fnk and ψnk are the Kohn-
Sham eigen-energy, occupation and wave function for
band index n and wave vector k, and η is a broadening
parameter. The summation over k runs all over the BZ
and?
factor of 2 accounts for spin (we assume a spin-degenerate
system).
The kernel in Eq. (5) consists of both a Coulomb and
an exchange-correlation(xc) part. The Coulomb kernel is
diagonal in the Bloch representation and written as
(7)
kfnk= 1 is satisfied for the occupied states. The
KC
G1G2(q) =
4π
|q + G1|2δG1G2,(8)

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while the xc kernel evaluated within ALDA is given by
Kxc−ALDA
G1G2
(q) =1
Ω
?
drfxc[n(r)]e−i(G1−G2)·r,(9)
with
fxc[n(r)] =∂2Exc[n]
∂n2
????
n0(r)
.(10)
Details on the evaluation of the xc kernel in the PAW
method can be found in a following subsection.
The Fourier transform of the microscopic dielectric ma-
trix, defined as ǫ−1(r,r′,ω) = δVtot(r,ω)/δVext(r′,ω), is
related to the density response function via
ǫ−1
GG′(q,ω) = δGG′ +
4π
|q + G|2χGG′(q,ω) (11)
where χ is obtained from χ0according to Eq. (5). The
off-diagonal elements of the χ0
response of the electrons at wave vectors different from
the external perturbing field and thus contain informa-
tion about the inhomogeneity of the microscopic response
of electrons known as the ’local field effect’6. The macro-
scopic dielectric function is defined as
GG′ matrix describes the
ǫM(q,ω) =
1
ǫ−1
00(q,ω), (12)
and is directly related to many experimental properties.
For example, the optical absorption spectrum (ABS) is
given by ImǫM(q → 0,ω). The electron energy loss spec-
trum (EELS29) is propotional to −Im(1/ǫM). Both spec-
tra reveal information about the elementary electronic
excitations of the system. EELS is especially useful in
probing the collective electronic excitations, known as
plasmons, of bulk and low-dimensional systems29.
C.Charge density matrix in the PAW method
In this subsection, we will discuss the charge density
matrix nnk,n′k+q(G), which is defined in Eq. (7) and is
a crucial quantity for the evaluation of χ0. Care must
be taken for the long wavelength limit (q → 0) since the
Coulomb kernel, 4π/|q + G|2, diverges at q → 0 and
G = 0; while the charge density matrix approaches zero
at this limit. As a result, we separate the discussion into
two parts: finite q and q → 0.
1. Finite q
Considering the transformation between the pseudo-
wavefunction and the all-electron wavefunction in Eq. (2)
and employing Eq. (3) yields
nnk,n′k+q(G) = ˜ nnk,n′k+q(G)(13)
+
?
a,ij
?˜ψnk|˜ pa
i??˜ pa
j|˜ψn′k+q?Qa
ij(q + G)
with
˜ nnk,n′k+q(G) ≡ ?˜ψnk|e−i(q+G)·r|˜ψn′k+q?
Qa
(14)
ij(K) ≡ ?φa
i|e−iK·r|φa
j? − ?˜φa
i|e−iK·r|˜φa
j?(15)
and K ≡ q + G.
The pseudo-density matrix in Eq. (14) is calculated
using a mixed space scheme. First, the cell periodic func-
tion˜ψ∗
grid; then it is Fourier transformed to get
nk(r)˜ψn′k+q(r)e−iq·ris evaluated on a real-space
˜ nnk,n′k+q(G) = F
?˜ψ∗
nk(r)˜ψn′k+q(r)e−iq·r?
(16)
The augmentation part in Eq. (15) is calculated on
fine one-dimensional radial grids centered on each atom.
Such fine grids are required to represent accurately the
oscillating nature of the all-electron partial wave in the
augmentation region. The plane wave term e−iK·ris ex-
panded using real spherical harmonics by
e−iK·r= 4π
?
lm
(−i)ljl(|K|r)Ylm(ˆ r)Ylm(ˆK), (17)
where jlis spherical Bessel function for angular momen-
tum l andˆK = K/|K|.
tions and the expression for the partial wave |φa
φa
nili(r)Ylimi(ˆ r), we can write
Combining the above equa-
i? =
Qa
ij(K) = 4πe−iK·Ra?
?
lm
(−i)lYlm(ˆK)
?
dˆ r YlmYlimiYljmj
×dr r2jl(|K|r)
?
φa
nili(r)φa
njlj(r) −˜φa
nili(r)˜φa
njlj(r)
?
(18)
2.Long wave length limit
In the long wave length limit, the G ?= 0 components
of the density matrix nnk,n′k+q(G) remain the same as
that for finite q. Only the G = 0 components need to be
modified and are written as
nnk,n′k+q(0)|q→0≡ ?ψnk|e−iq·r|ψn′k+q?q→0.
In Ref. 30, the above so called longitudinal form is de-
rived in the PAW framework by using Taylor expansion
of the eiq·rto the first order. Here we adopt an alter-
native but equivalent form which can be derived using
the second order k ·p perturbation theory31as described
below.
Expressing the wavefunction using Bloch’s theorem as
ψnk(r) = unk(r)eik·r, where unk(r) is the periodic Bloch
wave, the dipole transition element in Eq. (19) becomes
(19)
?ψnk|e−iq·r|ψn′k+q? = ?unk|un′k+q?.
For vanishing q, the wavefunction for |un′k+q? can be
obtained in terms of those for |umk? through second order
(20)

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4
perturbation theory:
|un′k+q? = |un′k? +
?
m?=n′
?ψmk|˜V |un′k?
ǫn′k− ǫmk
|umk?(21)
The perturbing potential˜V in the above equation is ob-
tained through
˜V = H(k + q) − H(k) = −iq · (∇ + ik),(22)
where
H(k) = −1
2(∇ + ik)2+ V (r)(23)
is the k ·p hamiltonian31and V (r) is the effective Kohn-
Sham potential.
Combining Eq. (20) - (22), the charge density matrix
at the long wavelength limit becomes
nnk,n′k+q(0)|q→0 =
−iq · ?nnk|∇ + ik|un′k?
ǫn′k− ǫnk
−iq · ?ψnk|∇|ψn′k?
ǫn′k− ǫnk
,
=
. (24)
The above expression for the charge density matrix in the
PAW method has an advantage over the pseudopotential
method, where the nabla operator has to be corrected by
the commutator of the non-local part of pseudopotential
with the position operator r28. In the PAW method, the
matrix element ?ψnk|∇|ψn′k? is given by
?ψnk|∇|ψn′k? = ?˜ψnk|∇|˜ψn′k?
+
?
a,ij
?˜ψnk|˜ pa
i??˜ pa
j|˜ψn′k?
?
?φa
i|∇|φa
j? − ?˜φa
i|∇|˜φa
j?
?
,
(25)
In GPAW, where the pseudo wave functions,˜ψnk, are
represented on a real space grid, the first matrix element
is calculated using a finite difference approximation for
the nabla operator. The augmentation part is evaluated
on fine one dimensional radial grids. The nabla operator
combined with partial waves φa
and φa
n2l2(r)Yl2m2(ˆ r) is written as
i(r) = φa
n1l1(r)Yl1m1(ˆ r)
j(r) = φa
?φa
= ?φa
i|∇|φa
i|∂
j?
∂r(φa
n2l2
rl2)∂r
∂rrl2Yl2m2? + ?φa
i|φa
n2l2
rl2∇(rl2Yl2m2)?.
(26)
Since real spherical harmonics are employed, we get
∂r
∂r= (x
r,yr,zr) =
?4π
3(Y1mx,Y1my,Y1mz) (27)
Substitute the above equation into Eq. (26) and split the
integration into radial and angular parts,we get for the
x-component
?φa
i|∂
?4π
∂x|φa
j?
?
=
3
dr r2φa
n1l1
∂
∂r(φa
?
n2l2
rl2)rl2
dˆ r Yl1m1r1−l2∂
?
dˆ r Yl1m1Yl2m2Y1mx
+
?
dr r2φa
n1l1
φa
n2l2
r∂x(rl2Yl2m2)
(28)
The derivation for the y- and z-component and for the
pseudo-partial-wave follows in a similar way.
D.The ALDA xc kernel in the PAW method
The ALDA xc kernel, expressed in Eq. (9), is evaluated
using the all-electron density, which takes the form
n(r) = ˜ n(r) +
?
a
[na(r − Ra) − ˜ na(r − Ra)],(29)
where
˜ n(r) =
?
nk
?
ij
?
ij
fnk|˜ψnk(r)|2+
?
a
˜ na
c(|r − Ra|), (30)
na(r) =
Da
ijφa
i(r)φa
j(r) + na
c(r),(31)
˜ na(r) =
Da
ij˜φa
i(r)˜φa
j(r) + ˜ na
c(r), (32)
with Da
all-electron core density and ˜ na
smooth continuation of na
sphere since it will be canceled out in Eq. (30).
The ALDA xc kernel can also be separated into smooth
and atom-centered contributions
Kxc−ALDA
G1G2
=˜Kxc−ALDA
G1G2
+
ij=?
nk?˜ψnk|˜ pa
i?fnk?˜ pa
j|˜ψnk?. Here na
c(r) can be chosen as any
c(r) inside the augmentation
c(r) is the
?
a
∆Ka,xc−ALDA
G1G2
.(33)
The smooth part is constructed from pseudo-density and
by utilizing a Fourier transform
1
Ω
1
ΩF {fxc[˜ n(r)]}|G1−G2
The atom-centered contribution is evaluated on 1D grids
1
Ω
×[fxc[na] − fxc[˜ na]]
˜Kxc−ALDA
G1G2
=
?
drfxc[˜ n(r)]e−i(G1−G2)·r
=
(34)
∆Ka,xc−ALDA
G1G2
=
?
r2drdˆ re−i(G1−G2)·r
(35)
III.NUMERICAL DETAILS
In this section we describe the most important numer-
ical and technical aspects of our implementation; in par-
ticular the Hilbert transform used to obtain χ0from the
dynamic form factor (spectral function) and the applied
parallelization scheme.

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5
A. Symmetry
For each wave vector q, the evaluation of χ0involves a
summation over occupied and empty states in the entire
BZ. By exploiting the crystal symmetries, however, we
need only calculate the wave functions and energies in
the irreducible BZ. This is because the wave function at
a general k-point can always be obtained from a wave
function in the irreducible part of BZ by application of
a symmetry transformation, T. In general we have the
relation
ψn,Tk(r) = ψn,k(T−1r)(36)
where k belongs to the IBZ. The above relation can be
directly verified by considering how the right hand side
transforms under lattice translations. In addition to the
crystal symmetries, time reversal symmetry applies to
any system in the absence of magnetic fields
ψ−k(r) = ψ∗
k(r)(37)
B. Hilbert transform
Rather than constructing χ0directly from Eq. (6) we
obtain it as a Hilbert transform of the (non-interacting)
dynamic form factor, S0.33,34The latter is given by
S0
GG′(q,ω) =
2
Ω
?
k,nn′
(fnk− fn′k+q)δ(ω + ǫnk− ǫn′k+q)
× nnk,n′k+q(G)n∗
nk,n′k+q(G′). (38)
In practice S0(ω) is evaluated on a uniform frequency
grid extending from 0 to around 40-60 eV with a grid
spacing in the range 0.01-0.1 eV, and the delta functions
are approximated by triangular functions following Ref.
32. The non-interacting response function is obtained as
χ0
GG′(q,ω) =
?∞
0
?
dω′S0
GG′(q,ω′)
×
1
ω − ω′+ iη−
1
ω + ω′+ iη
?
. (39)
The above Hilbert transform is performed directly on the
frequency grid setting the broadening parameter η equal
to the grid spacing.
C.LCAO vs grid calculations
It is well known that the use of localized atomic or-
bitals as basis functions can significantly reduce the com-
putational effort of groundstate electronic structure cal-
culations. For calculations of the density response func-
tion the use of localized basis functions is complicated
by the fact such basis sets are typically not closed un-
der multiplication35–37. As a consequence the size of the
FIG. 1: (Color online) The imaginary part of the dielec-
tric function (a) and energy loss function (b) of graphene at
q = 0.046˚ A−1along¯Γ −¯
M direction of its surface Broullion
zone (SBZ) calculated with 3D uniform grid (GRID, black
solid line) and localized atomic orbital (LCAO) using dzp
(red dashed line) and qztp (blue dash-dotted line) basis, re-
spectively .
product basis needed to represent the response function
grows as N2
µ, where Nµis the number of basis functions
used to represent the wave functions (we note that for
strictly localized basis functions, the effective size of the
“product basis” grows only linearly with the system size
because pair densities of non-overlapping orbitals van-
ishes, however, the prefactor is typically very large). A
further challenge is the computation of the Coulomb in-
teraction kernel, 1/|r − r′|, in the product basis leading
to six-dimensional multi center integrals. These inter-
grals must be performed either by using efficient Poisson
solvers or by resorting to analytical techniques. The lat-
ter is extensively used in quantum chemistry codes ap-
plying Gaussian basis sets.
For these reasons we have chosen to represent the
density response function in a plane wave basis. The
plane wave basis is closed under multiplication and the
Coulomb kernel is simply given by Eq. (8). However,
we still keep the advantage of using an LCAO as basis
in the calculation of the Kohn-Sham wave functions and
energies which enter the construction of χ0.27Apart from
reducing the computational effort of the groundstate cal-
culation (which must include many unoccupied bands),