Dielectric Properties of Semiconductors by TDDFT in Real-Space and Real-Time Approach
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Dielectric Properties of Semiconductors by TDDFT in Real-Space and Real-Time Approach Yasunari Zempo and Nobuhiko Akino Tsukuba Research Laboratory Sumitomo Chemical Co., Ltd. 6 Kitahara, Tsukuba, 300-3294 JAPAN ABSTRACT The dielectric responses of semiconductors such as C, Ge, Si, and AlGaAs are studied by the time-dependent density-functional theory. In our study, the real-space grid representation of the electron wave functions is used and the real-time approach is employed for the dynamics of the system. Both the static and dynamic dielectric functions are calculated, and we yielded that the static dielectric constants ε (0) are especially in good agreement with the experimental values. The effect of Al-component in the compound semiconductor AlxGa1-xAs is also studied. The peaks of the imaginary part of the dielectric function change with the band separation as a function of the Al-component. Furthermore, the static dielectric constants show the expected change due to the band separation as a function of the Al-component. INTRODUCTION In the fabrications of electronic devices, the dielectric response is one of the most important physical properties of semiconductors. To achieve the higher transconductance, which is a measure of the current carrying capability, the material that has a higher dielectric response is required. The time-dependent density-functional theory (TDDFT) has been recognized as a useful method to study the time-dependent phenomena such as the optical properties and the dielectric properties of atoms and molecules, and solids. Due to the periodicity requirement of the system, the electromagnetic interaction in the Hamiltonian is separated into a v Coulomb field V (r ) that satisfies periodic boundary conditions in the unit cell and a vector v gauge field zˆA(t ) . The latter is uniform and has no dependence on r . Then, the electric field is given by, v v v dA E = −∇V − zˆ (1) dt With these fields, our Lagrangian in a unit cell is given by the following equation1. 1 2 ⎞ ⎛ 1 ∇φi / i − eAzˆφi − ∇V (r ) ⋅ ∇V (r ) ⎟ ⎜ ∑ 8π L = ∫ dr ⎜ 2m i ⎟ Ω ⎜ ′ + en(r )V (r ) + en(r )V (r ) + Vxc [n(r )] + Vion [ρ (r, r )]⎟⎠ (2) ⎝ Ω ⎛ dA ⎞ * ∂φ i , − ⎜ ⎟ − i ∫ ∑ φi Ω 8π ⎝ dt ⎠ ∂t i 2
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where the φi are the Bloch wave functions of the electrons and n(r ) = ∑i φ i (r )
2
is the
electron density. The Ω shows the volume of the unit cell. From the Lagrangian in eq.(2), one can obtain the dynamic equations for φ i and A, −
δV δV ⎞ ∇2 e e2 2 ∂ ⎛ A φ i + ⎜ eV + ion + xc ⎟φ i = i φ i φ i − A∇ zφ i + im 2m 2m ∂t δn δn ⎠ ⎝
(3)
Ω d2A e e2 δ − ∇ / i + AN e + A V dr = 0 , φ φ ∑ i z i 2 4π dt mi i m δA ∫Ω ion
(4)
v
v
where N e = ∫ d 3 r n(r ) is the number of electrons per unit cell. Ω
METHODS AND NUMERICAL DETAILS In our calculation, we first solve the static equations with A=0 to obtain the optimized electron density at the ground state by the uniform spatial grid representation for the electron wavefunctions2. The size of grid is related to the accuracy in energy. For example, from our prelimin
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