We compare the optical absorption of extended systems calculated starting from the density-density and current-current linear response formalisms within the equilibrium many-body perturbation theory(MBPT). We show how, using the latter, one can incur in errors due to quasiparticle lifetimes, electron-hole interaction or the presence of a Drude tail. We present a solution for each one of these problems.

1. Excitons, lifetime and Drude tail
within the current current response framework~
authors
Davide SANGALLI
Claudio ATTACCALITE
Arjan J. BERGER
Pina ROMANIELLO Myrta GRUNING
Introduction
RESPONSE FUNCTIONS
THE DRUDE TAIL
ELECTRON HOLE INTERACTION~
CONCLUSIONS
We compare the optical absorption of extended systems calculated starting from the
density-density and current-current linear response formalisms within the
equilibrium many-body perturbation theory(MBPT). We show how, using the latter,
one can incur in errors due to quasiparticle lifetimes, electron-hole interaction or the
presence of a Drude tail. We present a solution for each one of these problems :-)
Within Many-Body Perturbation Theory(MBPT) response functions are obtained
from the so called Bethe-Salpter Equation(BSE)2
:
Where L0
is the independent particle solution, without electron-hole interaction and
the term
Takes into account the variation of the Hartree term vH
and the self-energy . The
self-energy is usually approximated in term of a static screened Coulomb interaction.
Within this approximation the BSE has a structure similar to the response function
in Hartree-Fock. The BSE can be mapped into an effective two-particle Hamiltonian,
written in a basis of single particle excitations:
From the solution this matrix equation we obtain response functions as:
Where the sum is on the eigenvalues of the BSE matrix and aExc is the expectation
value of the density or current operators. From this expression it is possible to
calculate the dielectric constant in the two formalism:
The problem arises - how to define the velocity operator in presence of interaction. In
literature, in analogy with dipole operator, this definition is found:
In our work, we show that this definition is
wrong, and that the velocity operator must
be defined through the commutator:
In this work we discuss the equivalence between the current-current and density-
density formalisms in cases more complicated than the simple independent particle
approximation. We show how to correctly handle quasi-particle lifetimes and how to
reproduce the Drude tail in the two formalisms. Finally we provide the correct
formula to calculate the optical absorption within the current-current formalism by
means of the Bethe-Salpeter Equation.
Density and current–density variations are induced
as response to the perturbing potentials. In presence
of a macroscopic longitudinal perturbations. They
can be described by using only a scalar potential or
by using only a longitudinal vector potential. Then
we can define two response functions the density-
density and current-current response:
The dielectric constant of a solid can be expressed in terms of the above response
functions as:
and
These two definition of the dielectric constant are related by the equation:
In the limit of q that goes to zero, the two dielectric constants should give the same
description1, but how we show in this work, in practical calculations care has to be
taken to make for this to be the case.
References
QUASIPARTICLE LIFETIME
1) W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena: From
Fundamentals to Current Topics (Springer, 2002)
2) G. Strinati, Application of the Green’s functions method to the study of the optical
properties of semiconductors, Riv. Nuovo Cimento 11, 1 (1988)
3) Aryasetiawan, F., & Gunnarsson, O. The GW method. Reports on Progress in Physics,
61(3), 237. (1998)
4) N. Raimbault et al. Gauge-Invariant Calculation of Static and Dynamical Magnetic
Properties from the Current Density, Phys. Rev. Lett. 114, 066404 (2015)
In real systems single particle energies are complex quantities due to the scattering
of the electrons with other electrons3
and with phonons:
In MBPT they are described by means of a self-energy operator that contains an
imaginary part:
Inserting this complex self–energy in the bare
KS hamiltonian becomes non hermitian and the
corresponding velocity reads:
Neglecting that the velocity is non-hermitian
generates a fictitious Drude peak in the
absorption.
The characteristic behavior of a Drude metal in the frequency
domain is a divergence of the dielectric constant () for that goes
to zero. This term originates from the intra-bands transitions. The
difficult in calculating this term is that the dielectric constant at
q=0 in density-density formalism is different from its limit:
Therefore particular care has to be taken to obtain the Drude tail by expanding
occupations and energies at small momentum:
Current-current response is a more natural choice for metals, since it includes the
Drude tail both in the limit and in the dielectric function at q=0.
However care has to be taken because the
terms that contribute to the Drude tail
converge in different way with the number of
bands and a straightforward calculation will
never converge. We found that the best
strategy is to explicitly include intra-bands
transitions as in the density-density case4
.
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