# Optical properties of solids

S. Sharma, K. Dewhurst, and A. Sanna
The *ab-initio* calculation of optical absorption spectra of nano-structures
and solids is a formidable task. The current state-of-the-art is based on many-body
perturbation theory: one solves the Bethe-Salpeter equation (BSE) using the one-body
Green's function obtained from the *GW* approximation. Resonances, corresponding to
bound electron-hole pairs called excitons, which have energies inside the gap, can
then appear in the spectrum.
This procedure is a well-established method for yielding
macroscopic dielectric tensors which are generally in good agreement with
experiment. Unfortunately, solving the BSE involves diagonalizing a large matrix making
this method computationally very expensive.

Time-dependent density functional theory (TDDFT)^{[1]}, which
extends density functional theory into the time domain, is another method
able, to determine neutral excitations of a system.
Although formally exact, the predictions of TDDFT are only as good as the approximation of
the exchange-correlation (xc) kernel:
*f*_{xc}(**r**,**r'**,t-t') ≡
δv_{xc}(**r**,t)/δρ(**r'**,t')
,
where v_{xc} is the TD exchange-correlation potential
and ρ is the TD density.
There are several such approximate kernels in existence, the earliest of which
is the adiabatic local density approximation (ALDA)
^{[2]}.
However in the dielectric function
calculated using ALDA the physics of the bound electron-hole pair is totally missing. There have been previous
attempts to solve this problem, and there exist kernels which correctly
reproduce the peaks in the optical spectrum associated with bound excitons- The nano-quanta
kernel^{[3]}
derived from the four-point Bethe-Salpeter kernel,
is very accurate but has the drawback of being nearly as computationally demanding as
solving the BSE itself. The long-range correction (LRC) kernel
has a particularly simple form, *f*_{xc} = -α /q^{2},
which limits its
computational cost. This kernel produces the desired excitonic peak, but depends on the
choice of the parameter α
which turns out to be strongly material-dependent, thereby limiting
the predictiveness of this approximation. In our latest work
^{[4,,
5]
}
we propose a new parameter-free
approximation for *f*_{xc}, and demonstrate that this kernel is nearly as accurate as BSE with a
computational cost of ALDA.

The exact relationship between the dielectric function ε and the kernel *f*_{
xc} for a periodic solid can be written as

where v is the bare Coulomb potential,
χ is the full response function,
and χ_{0} is the response function of the non-interacting Kohn-Sham system.
All these quantities are matrices in the basis of reciprocal lattice vectors **G**.
The bootstrap kernel is a frequency-independent approximation given by:

- (2)

where ε^{-1}(**q**,ω = 0) is determined
*self-consistently* with Eq. (1).
We note that although Eq.(1) is exact, it is useful only when either
f_{xc} or
ε is given; if neither are
available then obviously it cannot be used as a generating equation
for both quantities.
With the addition of the approximation given by Eq. (2) however, both
f_{xc}
and ε can be determined from knowledge of
χ_{0} exclusively.
The advantages of this form for the kernel is that 1. it has the
correct 1/q^{2}
behavior;
2. as ε improves from cycle to cycle
so does f_{xc};
3. the computation cost is minimal and 4. most importantly, no system-dependent
external parameter is required.
Using the method the optical spectra for various extended systems have been
calculated using the ELK code^{
[6]}.

- Fig. 1: Imaginary part of the dielectric tensor (ε
_{2}) as as function of photon energy (in eV).

Presented in Fig. 1 are the results for some small (Ge ∼ 0.67 eV) to medium (diamond ∼ 5.47 eV)
bandgap semiconductors.
For comparison, experimental data as well as the RPA spectra are also plotted.
The experimental data clearly show that all these materials have weakly bound
excitons leading to a small shifting of the spectral
weight to lower energies compared to RPA.
The results from TDDFT with the new kernel exactly follow this trend and
are in excellent overall agreement with experiment.

A stringent test for any approximate xc-kernel is in its ability to treat
materials with strongly bound excitons. In these cases a new resonant peak
appears in the bandgap itself and represents the bound state of an electron-hole
pair. Perhaps the most studied test case for this phenomenon is the ionic solid LiF.
Other excitonic materials which have also attracted attention and are considered
particularly difficult to treat are the
noble gas solids. Plotted in the first column of Fig. 2 are the
results for three materials of this class: LiF, solid Ar and Ne.
What is immediately clear is that the bootstrap procedure, which gave only a
slight shift of spectral weight for
Ge, now gives rise to an entirely new bound excitonic peak inside the
gap in all three cases. The location of the peak, which corresponds to the
excitonic binding energy, is also very well reproduced for all these materials.

- Fig. 2: Imaginary part of the dielectric tensor (ε
_{2}) as function of photon energy (in eV).

The second column of Fig. 2 consists of some special cases - NiO has
an anti-ferromagnetic ground state
and provides the bootstrap technique
with a test of its validity for magnetic materials. It is clear from Fig. 2 that the bootstrap
method leads once again to the correct excitonic binding energy.
Results for the anatase phase of TiO_{2} are also presented in Fig. 2.
TiO_{2} is a useful test for the bootstrap method due to its non-cubic unit cell, which
leads to directional anisotropy in the optical spectrum. As can be seen the bootstrap method
captures this anisotropy very well. Even subtle features like the small shoulder at
∼ 4 eV in the out-of-plane dielectric function, which is missing in the
in-plane case, are well reproduced.

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## References

- [1]
- E. Runge and E. K. U. Gross.
*Phys. Rev. Lett.*, **52**, 997, (1984).
- [2]
- E. K. U. Gross, F. J. Dobson, and M. Petersilka.
Topics in Current Chemisty
**181**, 81, (1996) (and references
therein).
- [3]
- F. Sottile, V. Olevano, and L. Reining.
Phys. Rev. Lett.,
**91**, 056402, (2003).
- [4]
- S. Sharma, J. K. Dewhurst, A. Sanna, E. K. U. Gross, Phys. Rev. Lett.,
**107**, 186401 (2011).
- [5]
- S. Sharma, J. K. Dewhurst, A. Sanna, A. Rubio and E. K. U. Gross,
New J. Phys.
**14**, 053052 (2012)
- [6]
- http://elk.sourceforge.net/ (2004).

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