Nonlinear Electrical Transport in Semiconductors a Nonequilibrium Thermodynamics Approach

the coupled electron-phonon system in the external DCfield E , the distribution function is determined from the Boltzmann-Peierls equation [10] ∂ ∂ eE ∂ + v g (k) · − · ∂t ∂r ¯ ∂k h
α) v( g (k) =
1 ∂ Eα (k) h ∂k ¯
(3)
where Eα (k) is the band structure of the material under consideration. For a single parabolic band with constant spherical energy surface, E (k) = ¯ 2 k2 h , 2m∗ e (4)
The electrical transport problems in semiconductors are nonequilibrium phenomena and most conveniently described by the hydrodynamic equations (or balance equations) that involve the directly measurable, macroscopic variables of the system [1–8]. Such nonequilibrium state-variables are the carrier density, drift velocity, and temperature in the conventional hydrodynamic description [1–4] and also momentum and heat fluxes are included in the generalized hydrodynamic equations approach [5–8]. Due to the many-particle nature, the observables are defined microscopically as the statistical average of the relevant operators over the nonequilibrium occupancy, the single-particle distribution function. In the linear response regime (or near equilibrium) where the external perturbation is weak, there exists the firm foundation between the desired macroscopic quantities and the underlying microscopic physics. For instance, the Ohmic conductivity can be calculated by either solution to the Boltzmann equation or the Kubo formula, equivalently [9]. However, in the high-field regime there remain the fundamental difficulties in understanding the nonlinear transport properties. First of all, it is not a priori obvious what are the relevant quantities to be specified, since physics depends on space and time scales under consideration. Also, it is not known how to comply with the macroscopic law of thermodynamics under general nonequilibrium situation, in particular the second law of thermodynamics. On the other hand, the modern miniaturization trend of semiconductor devices demands a better understanding of this long-standing problems ever since. In this paper, we attempt to tackle this nonlinear transport problem in semiconductors while giving much effort to constructing thermodynamic consistencies in the theory. In solids, the electrical current density j is most mi∗
0 Re [fe , fph ] = V
dq A(q ) (2π )3
0 1 + fph (q ) fe (r , k + q , t) [1 − fe (r , k, t)]
0 − fph (q )fe (r , k, t) [1 − fe (r , k + q , t)] δ [E (k + q ) − E (k) − ¯ hωq ] dq 0 A(q ) fph (q )fe (r , k − q , t) [1 − fe (r , k, t)] +V (2π )3 0 − 1 + fph (q ) fe (r , k, t) [1 − fe (r , k − q , t)] δ [E (k − q ) − E (k) + ¯ hωq ]
E-mail: cskim@boltzmann.chonnam.ac.kr
-S310-
The Influence of a Gaussian Beam on the Read-out Signal in · · · – Seong Jong Park and Chang Sub Chung
-S311-
Deug Yong Kim and Chang Sub Kim∗
Department of Physics, Chonnam National University, Kwangju 500-757 We present a transport theory of semiconductors in the high-field regime where the usual linearresponse theory breaks down. We start from the Boltzmann-Peierls equation for the coupled electron-phonon system to generate the macroscopic transport equations. We use the idea of the extended irreversible hydrodynamics in selecting the relevant nonequilibrium state variables and also incorporate the nonequilibrium ensemble method that introduces the phenomenological ansatz imposing the second law of thermodynamics by hands. In order to manifest the utility of the theory constructed, we sketch how to calculate the nonlinear drift velocity.
α
dk fe (Eα (k)) (2π )3
(2)
where the summation is over all filled bands and the integration is to be carried out in the full first Brillouin zone. The group velocity is given by the Hellmann-Feynman theorem as
Journal of the Korean Physical Society, Vol. 34, No. , June 1999, pp. S310∼S314
Nonlinear Electrical Transport in Semiconductors : a Nonequilibrium Thermodynamics Approach
0 = Re [fe , fph ],
and
0 Rph [fe , fph ] = 0,
(8)
fe (r , k, t) (7)
where the phonon equation is simply given by the condition of vanishing the collision operator because phonons 0 . The collision opare assumed to be in equilibrium, fph erator in the right side of Eq. (7) is given as [10]
α
dk α) (−e)v ( g (k)fe (Eα (k)) = n(−e)u(1) (2π )3
where u is the drift velocity and the carrier density n is defined through the normalization condition for fe as n=2