The Dielectric Permittivity of Crystals in the Reduced Hartree–Fock Approximation

Abstract

In a recent article (Cancès et al. in Commun Math Phys 281:129–177, 2008), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree–Fock model, the ground state electronic density matrix is decomposed as ${\gamma = \gamma^0_{\rm per} + Q_{\nu,\varepsilon_{\rm F}}}$, where ${\gamma^0_{\rm per}}$ is the ground state density matrix of the host crystal and ${Q_{\nu,\varepsilon_{\rm F}}}$ the modification of the electronic density matrix generated by a modification ν of the nuclear charge of the host crystal, the Fermi level ε _F being kept fixed. The purpose of the present article is twofold. First, we study in more detail the mathematical properties of the density matrix ${Q_{\nu,\varepsilon_{\rm F}}}$ (which is known to be a self-adjoint Hilbert–Schmidt operator on ${L^2(\mathbb{R}^3)}$). We show in particular that if ${\int_{\mathbb{R}^3}\,\nu \neq 0, Q_{\nu,\varepsilon_{\rm F}}}$ is not trace-class. Moreover, the associated density of charge is not in ${L^1(\mathbb{R}^3)}$ if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect ν, the linear and nonlinear terms of the resolvent expansion of ${Q_{\nu,\varepsilon_{\rm F}}}$. Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the Adler–Wiser formula.