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Linear Response: The Kubo Formula

Consider a fully interacting electronic system described by the Hamiltonian H under the action of an external time-dependent field

The density operator is defined as

while the induced density is

In the linear response regime the external field is assumed to be weak, we can expand the exact time-dependent ground state at the first order in the field

with the external Hamiltonian in the interaction picture.The Kubo formula states that

where we have defined the response function

that is more extensively described in Xd .

Optical Properties

When we talk about calculating optical properties, we usually refer to a calculation of the dielectric function, ε(q,ω), as a function of the frequency, ω, and the momentum transfer, q.

In the Kubo formalism to corresponds to take as external perturbing potential the electromagnetic field. In what follows we assume to be in the longitudinal gouge where the light polarization is parallel to the electric field momentum q.

The cross section for optical absorption, σ(ω), i.e., the optical absorption spectrum, is then proportional to the imaginary part of the macroscopic dielectric function:

where

For a justification of Eq. (1) see [1]. The limit q→0 is taken because the momentum carried by a photon is vanishingly small compared to the crystal momenta of a bulk material.

Another important quantity which is often measured in experiments is the energy loss function. The loss function, Γ(q,ω), is related to the imaginary part of the inverse dielectric function:

Note that, contrary to the absorption cross section, the loss function is also defined for finite momentum transfer q. The momentum transfer can be measured in electron energy loss spectroscopy (EELS) through the deflection of the electron beam.We also note that Eq. (2) is only valid for angular resolved EELS on bulk materials and not for spatially resolved EELS on isolated nanoobjects.

Isolated systems: the polarizability

It is well known that the macroscopic dielectric function is not well defined for isolated systems, as in the numerical simulation the density tends to zero increasing the simulation volume (and, thus, the systems isolation). For these systems a more appropriate quantity is the polarizability, that we introduce with simple dimensional arguments. Let's consider for an isolated molecule contained in a simulation volume Ω the reducible polarizability ( Xd )

If now we increase Ω , as

we have that

So we define the complex polarizability α as

So, if in the case of a molecule α is a volume, it becomes an area in 1D systems, a length in 2D and, as we know, and adimensional number in 3D.

References

  1. H. Ehrenreich, The Optical Properties of Solids,Academic, New York (1965).