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The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. (English) Zbl 1197.82113

Summary: In a recent article [the authors and A. Deleurence, Commun. Math. Phys. 281, No. 1, 129–177 (2008; Zbl 1157.82042)], 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 \({\gamma = \gamma^0_{\text{per}} + Q_{\nu,\varepsilon_{\text F}}}\), where \({\gamma^0_{\text{per}}}\) is the ground state density matrix of the host crystal and \({Q_{\nu,\varepsilon_{\text F}}}\) the modification of the electronic density matrix generated by a modification \(\nu \) of the nuclear charge of the host crystal, the Fermi level \(\varepsilon _{F}\) being kept fixed.
The purpose of the present article is twofold. First, we study in more detail the mathematical properties of the density matrix \({Q_{\nu,\varepsilon_{\text F}}}\) (which is known to be a self-adjoint Hilbert-Schmidt operator on \({L^2(\mathbb{R}^3)}\)). We show in particular that, if \(\int_{\mathbb{R}^3}\,\nu \neq 0\), \(Q_{\nu,\varepsilon_{\text F}}\) 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 \(\nu \), the linear and nonlinear terms of the resolvent expansion of \({Q_{\nu,\varepsilon_{\text F}}}\). 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.

MSC:

82D25 Statistical mechanics of crystals
82D37 Statistical mechanics of semiconductors
81V10 Electromagnetic interaction; quantum electrodynamics
35Q55 NLS equations (nonlinear Schrödinger equations)
78A30 Electro- and magnetostatics
47N50 Applications of operator theory in the physical sciences

Citations:

Zbl 1157.82042
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References:

[1] Adler S.L.: Quantum theory of the dielectric constant in real solids. Phys. Rev. 126, 413–420 (1962) · Zbl 0108.44003 · doi:10.1103/PhysRev.126.413
[2] Bach V., Lieb E.H., Solovej J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994) · Zbl 0839.60095 · doi:10.1007/BF02188656
[3] Baroni S., Resta R.: Ab initio calculation of the macroscopic dielectric constant in silicon. Phys. Rev. B 33, 7017–7021 (1986) · doi:10.1103/PhysRevB.33.7017
[4] Cancès É., Deleurence A., Lewin M.: A new approach to the modelling of local defects in crystals: the reduced Hartree–Fock case. Commun. Math. Phys. 281, 129–177 (2008) · Zbl 1157.82042 · doi:10.1007/s00220-008-0481-x
[5] Cancès É., Deleurence A., Lewin M.: Non-perturbative embedding of local defects in crystalline materials. J. Phys. Condens. Matter 20, 294213 (2008) · Zbl 1157.82042 · doi:10.1088/0953-8984/20/29/294213
[6] Catto I., Le Bris C., Lions P.-L.: On the thermodynamic limit for Hartree–Fock type models. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 687–760 (2001) · Zbl 0994.35115 · doi:10.1016/S0294-1449(00)00059-7
[7] Dreizler R., Gross E.: Density Functional Theory. Springer, Berlin (1990) · Zbl 0723.70002
[8] Engel G.E., Farid B.: Calculation of the dielectric properties of semiconductors. Phys. Rev. B 46, 15812–15827 (1992) · doi:10.1103/PhysRevB.46.15812
[9] Gajdoš M., Hummer K., Kresse G., Furthmüller J., Bechstedt F.: Linear optical properties in the projector-augmented wave methodology. Phys. Rev. B 73, 045112 (2006) · doi:10.1103/PhysRevB.73.045112
[10] Gravejat P., Lewin M., Séré É.: Ground state and charge renormalization in a nonlinear model of relativistic atoms. Commun. Math. Phys. 286, 179–215 (2009) · Zbl 1180.81155 · doi:10.1007/s00220-008-0660-9
[11] Hainzl C., Lewin M., Séré É.: Existence of a stable polarized vacuum in the Bogoliubov–Dirac–Fock approximation. Commun. Math. Phys. 257, 515–562 (2005) · Zbl 1115.81061 · doi:10.1007/s00220-005-1343-4
[12] Hainzl C., Lewin M., Séré É.: Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Ration. Mech. Anal. 192, 453–499 (2009) · Zbl 1173.81025 · doi:10.1007/s00205-008-0144-2
[13] Hainzl C., Lewin M., Séré É., Solovej J.P.: A minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics. Phys. Rev. A 76, 052104 (2007) · Zbl 1113.81126 · doi:10.1103/PhysRevA.76.052104
[14] Hybertsen M.S., Louie S.G.: Ab initio static dielectric matrices from the density-functional approach. I. Formulation and application to semiconductors and insulators. Phys. Rev. B 35, 5585–5601 (1987) · doi:10.1103/PhysRevB.35.5585
[15] Hybertsen M.S., Louie S.G.: Ab initio static dielectric matrices from the density-functional approach. II. Calculation of the screening response in diamond, Si, Ge, and LiCl. Phys. Rev. B 35, 5602–5610 (1987) · doi:10.1103/PhysRevB.35.5602
[16] Kunc K., Tosatti E.: Direct evaluation of the inverse dielectric matrix in semiconductors. Phys. Rev. B 29, 7045–7047 (1984) · doi:10.1103/PhysRevB.29.7045
[17] Lieb E.H., Simon B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977) · doi:10.1007/BF01609845
[18] Panati G.: Triviality of Bloch and Bloch–Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007) · Zbl 1375.81102 · doi:10.1007/s00023-007-0326-8
[19] Pick R.M., Cohen M.H., Martin R.M.: Microscopic theory of force constants in the adiabatic approximation. Phys. Rev. B 1, 910–920 (1970) · doi:10.1103/PhysRevB.1.910
[20] Reed M., Simon B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978) · Zbl 0401.47001
[21] Resta R., Baldereschi A.: Dielectric matrices and local fields in polar semiconductors. Phys. Rev. B 23, 6615–6624 (1981) · doi:10.1103/PhysRevB.23.6615
[22] Seiler E., Simon B.: Bounds in the Yukawa2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound. Commun. Math. Phys. 45, 99–114 (1975) · doi:10.1007/BF01629241
[23] Simon, B.: Trace Ideals and Their Applications. In: London Mathematical Society Lecture Note Series, vol. 35. Cambridge University Press, Cambridge, 1979 · Zbl 0423.47001
[24] Solovej J.P.: Proof of the ionization conjecture in a reduced Hartree–Fock model. Invent. Math. 104, 291–311 (1991) · Zbl 0732.35066 · doi:10.1007/BF01245077
[25] Thomas L.E.: Time dependent approach to scattering from impurities in a crystal. Commun. Math. Phys. 33, 335–343 (1973) · doi:10.1007/BF01646745
[26] Wiser N.: Dielectric constant with local field effects included. Phys. Rev. 129, 62–69 (1963) · Zbl 0121.44901 · doi:10.1103/PhysRev.129.62
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