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30 N.A.W. Holzwarth / Physics Procedia 57 (2014) 29 – 37 x = 2y + 3z − 5; those with the highest ionic conductivities (10−6 S/cm) have a glassy structure and a range of values of 0.2 ≤ z ≤ 0.7 representing the ideal nitrogen contribution (Dudney (2000, 2008); Mascaraque et al. (2013)). At the present time, we know of no experimental evidence that crystalline members of the LiPON family of materials can approach the conductivities of the LiPON glasses, however, a systematic study of the LiPON family of crystalline ma- terials (Du and Holzwarth (2010b)) has proven useful for developing an understanding of the fundamental structures and properties of LiPON electrolytes. Meanwhile, the structurally and chemically related Li thiophosphate family of materials have recently received attention as promising candidates for solid-state electrolytes (Mizuno et al. (2005b,a, 2006); Yamane et al. (2007); Hayashi (2007); Tatsumisago and Hayashi (2008); Minami et al. (2008); Hayashi et al. (2008, 2009); Trevey et al. (2009); Liu et al. (2013)) where increased ionic conductivities as large as 10−3 S/cm have been reported. These materials are characterized by the composition LivPSw. The comparison between the crystalline Li phosphates and corresponding thiophosphates has provided further insight into solid electrolyte development. In Section 2 we describe “first principles” computational techniques and their validation. In Section 3 we present some results from our studies of lithium phosphate and thiophosphate electrolytes. Section 4 contains a brief summary and discussion. 2. Calculational methods In the context of electronic structure calculations, the term “first-principles” techniques implies a series of well- developed approximations to the exact quantum-mechanical description of a material with Ne electrons and NN nuclei. Denoting the electron coordinates by {ri} (i = 1, 2, . . . Ne) and nuclear coordinates by {Ra} (a = 1, 2, . . . NN ), the many- particle Schro ̈dinger equation takes the form H ({ri}, {Ra}) Ψα ({ri}, {Ra}) = EαΨα ({ri}, {Ra}) , (1) where H denotes the quantum mechanical Hamiltonian, and Eα and Ψα ({ri}, {Ra}) denote the energy eigenvalue and the corresponding eigenfunction, respectively. The solution of Eq. (1) with NN Ne coupled variables, is intractable for all but the smallest systems. The analysis of Born and Oppenheimer (Born and Huang (1954)) noting that the electron mass is 10−3 times smaller than the nuclear mass, leads to an approximate separation of the nuclear and electronic motions. Operationally, the nuclei are treated as classical particles with interaction energies consistently determined by expectation values of the electronic Hamiltonian. The electronic Hamiltonian and the corresponding Schro ̈dinger equation should be solved for each set of nuclear positions {Ra}. The solution of the Born-Oppenheimer electronic Schro ̈dinger equation is further approximated with the use of density functional theory developed by Kohn, Hohenberg, and Sham (Hohenberg and Kohn (1964); Kohn and Sham (1965)), representing the effects of the Ne electrons in terms of their density and the corresponding a self-consistent mean-field. The reliability of density functional theory in the representation of real materials depends on the development of the exchange-correlation functional form. While this remains an active area of research, the local density approximation (LDA) (Perdew and Wang (1992)) and the generalized gradient approximation (GGA) (Perdew et al. (1996)) often work well, particularly for modeling the ground state properties of solid electrolytes. Within density functional theory, the electronic energy of a system of Ne electrons can be expressed as a sum of contributions: E(ρ,{Ra})=EK +Eee +Exc +EeN +ENN, (2) representing the electronic kinetic energy, the coulombic electron-electron repulsion, the exchange-correlation energy, the electron-nuclear interaction energy, and the nuclear-nuclear interaction energies respectively. The electron density ρ(r) is self-consistently determined from Kohn-Sham single particle wavefunctions for each state n: at self-consistency. Here, HKS denotes the Kohn-Sham Hamiltonian. In addition to well-controlled mathematical and physical approximations, numerical approximations are needed to solve the density functional equations. There are many successful numerical schemes, most of which grew from the frozen-core approximation (von Barth and Gelatt (1980)) and from the refinement of the pseudopotential formalism Ne ∂E(ρ,{R }) HKS ψn = εnψn where ρ(r) = |ψn(r)|2 and HKS = a (3) n=1 ∂ρ(r)PDF Image | First Principles Modeling of Electrolyte Materials in All-Solid-State Batteries
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