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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 S. 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
W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena: From Fundamentals to Current Topics (Springer, 2002)
G. Strinati, Application of the Green’s functions method to the study of the optical properties of semiconductors, Riv. Nuovo Cimento 11, 1 (1988)
Aryasetiawan, F., & Gunnarsson, O. The GW method. Reports on Progress in Physics, 61(3), 237. (1998)
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 e(w) for w 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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