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Dielectrics in a time-dependent electric field:
density-polarization functional theory approach
C. Attaccalite1, D. Sangalli2, M. Grüning3
1) CNRS/CINaM Aix-Marseille Université, (France)
2) ISM, CNR, Montelibretti (Italie)
3) Queen's University Belfast, (UK)
Why DPFT?
Linear optics results
Non-linear optics
Real-time DPFT
DPFT functionals
In presence of a time-dependent macroscopic electric field the electron dynamics of dielectrics cannot be described by the time-dependent density only. We present a real-time formalism that has the density and the macroscopic polarization P as key quantities. We show that a simple local function of P already captures long-range correlation in linear and nonlinear optical response functions.
Conclusions:
References:
Time-dependent density functional theory (TD-DFT) is an extension of the ground-state formalism that allows to investigate the dynamics of many-body systems in the presence of time-dependent potential.
TDDFT is based on the Runge-Gross (RG) theorem that establishes a one-to one correspondence between time-dependent densities and time-dependent one-body potentials.
This correspondence is broken in periodic systems.
The RG theorem first establishes a one-to-one correspondence between potentials and currents. Then the continuity equation is used to relate currents to densities. This second part is not valid in periodic systems. In order to solve this problem an extension of TDDFT was presented some years ago, the Time Dependent Current Density Functional Theory(TD-CDFT)[3].
This formulation uses the direct mapping between the external potential and the current density.
In this work we will use a simplified versions of TD-CDFT, i.e. the Density-Polarization Functional Thery (DFTP). In DPFT one uses the relation between polarization and current to construct a theory that relies on density and polarization instead of current density.
This relation is valid when the transverse current can be disregarded as in the case of the optical response.
Experimental optical absorption spectra (open circles) are compared with real-time simulations at different levels of approximation: TD-LDA (continuous orange line), RPA (green dash-dotted line), IPA (blue dotted line) and RPA (green dashed line) with scissor correction.
Experimental optical absorption spectra (open circles) are compared with real-time simulations at different levels of approximation: opt-PF (blue dashed line), JGM-PF (pink continuous line), RPA (gray dotted line).
All approximations include the scissor operator correction..
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Here we present results for the linear optical response of GaAs. Results for other bulk materials as AlAs, CdTe, ZnS, ZnSe, ZnTe, Silicon, are presented in references 4 and 5.
In the left panel we compare the experiments with calculations without excitonic effects.
In the right panel we include the
electron-hole interactions within the DPFT approach.
We have implemented a real-time density functional approach suitable for infinite periodic crystals. This approach, in addition to the electron density considers also the macroscopic polarization as a main variable and extends to the time-dependent case the DPFT introduced in the nineties.
We have derived approximations for the xc-electric field exploiting the connection with long-range corrected approximations for xc kernel within the linear response theory. We have considered two approximations, the optimal polarization functional, linked to the long-range corrected xc kernel[6] and the Jellium with a gap model[7].
For systems here studied the opt-PF approximation works well, but such a good performance cannot be expected in general. However notice that in the opt-PF approximation there is a material dependent parameters a while the JGM-PF is fully ab-initio.
IPA (dotted violet), RPA (dashed green) and TD-LDA (continuous orange), all calculations are without scissor operator correction. For comparison we included the RPA spectrum of GaAs calculated by Luppi et al.(open triangles)
Opt-PF (dashed blue) and JGM-PF (continuous pink) are compared with IPA (dotted gray) and RPA for GaAs. Available experimental results are shown for GaAs (open circles).
Non-linear response of GaAs is calculated by means of real-time DPFT. Second harmonic coefficients (SHG) are extracted from the Fourier analysis of the total polarization as described in Ref. 8:
For comparison we include the results of Luppi et al. [9] and the experimental measurements of Ref. [10].
In Density Polarization Functional Thery (DPFT) the exchange correlation(xc) functional depends from both density and polarization. The part that depends only from the density can be obtained from the standard TD-DFT functionals, but we still miss the polarization dependence.
In order to find the dependence from the polarization of the xc-kernel Fxc we use the relation between
the long-range exchange correlation functionals of the TDDFT fxc(q→0) and the Fxc.[2]
Where G are reciprocal lattice vectors and q is defined in the first Brillouin zone.
In this work we tested two xc kernels:
JGM-PF
opt-PF
There is a large literacture on long-range(LR) correction to the exchange correlation functional in TD-DFT.
In particular it has been shown that these corrections are crucial to describe excitons in both bulk and molecular systems. Here we use the formulation of Botti et al.[6] that
is based on the following exact relation to define the LR-correction:
This exchange-correlation functional is derived from the electron-gas with gap model[7].
The only paramter is the gap that we calculate within GW approximation.
As it is currently done for TD-DFT we reformulated DPFT in real-time:
Where unk are the time-dependent valence bands, Hks,0 is the
independent particle Hamiltonian and:
vs(r,t) is the Kohn-Sham potential that is the sum of the Hartree, the external and the exchage-correlation one. In our approximation vs(r,t) depends from only from the density.
Es(r,t) is the Kohn-Sham electric field that is coupled with the polarization P and includes the long-range corrections described in the previous box
In our simulations we excite solids with different laser fields and then extract
the linear and non-linear response functions
from the analysis of the outgoing polarization.[4,8]