Onsager Relations and Hydrodynamic Balance Equations in 2D Quantum Wells

In this letter we clarify the role of heat flux in the hydrodynamic balance equations in 2D quantum wells, facilitating the formulation of an Onsager relation within the framework of this theory. We find that the Onsager relation is satisfied within the framework of the 2D hydrodynamic balance equation transport theory at sufficiently high density. The condition of high density is consonant with the requirement of strong electron-electron interactions for the validity of our balance equation formulation. PACS number(s): 72.10.Bg, 72.20.Ht, 05.70.Ln Typeset using REVTEX 1 The Lei-Ting balance equation transport theory [1,2] has achieved much success in hotelectron transport of homogeneous semiconductors. This theory was subsequently generalized to deal with weakly nonuniform, inhomogeneous systems [3]. The resulting hydrodynamic balance equations consist of continuity equation, momentum balance equation, and energy balance equation. A salient feature of this hydrodynamic approach is that it includes a microscopic description of scattering in the form of a frictional force function due to electron-impurity and electron-phonon scattering, as well as an electron energy loss rate function due to electron-phonon interaction. These hydrodynamic balance equations have recently been applied to device simulations [4–6] and they have been further developed to include phonon-drag effects [7] and applied to discussion of thermoelectric power under both linear [7] and nonlinear [8] transport conditions. Until recently, the important issue concerning the capability of this theory to lead to the correct form of Onsager relations [9] and/or how to determine Onsager relations within the framework of this theory has not been addressed. There is even some misunderstanding that the energy flux predicted by this theory is zero. It is well known that the Onsager relation is a manifestation of microscopic irreversibility for any statistical system near thermal equilibrium. Therefore any properly formulated statistical physics model should satisfy this relation. It is very easy to verify this relation in the framework of Kubo linear response theory. Moreover, if one can determine the distribution function from the Boltzmann equation, it is also straightforward to verify the Onsager relation by calculating the pertinent moments of the distribution function. However, for the traditional hydrodynamic model, [10–21] which is derived from Boltzmann equation, verification has been elusive. In fact, in a very recent article [22], it has been argued that the Onsager relation breaks down in this model. Recently we [23] clarified the role of heat flux in this theory, and, by introducing the fourth balance equation, ie., energy flux balance equation, we were able to show how to generate Onsager relations within the framework of this theory. Moreover, we closely checked the Onsager relation predicted by this theory for bulk semiconductors and found, that for 2 any temperature, when electron density is sufficiently high, the balance equation theory satisfies Onsager relations exactly. To our knowledge, this is the first set of hydrodynamic equations which has been shown to obey Onsager relation exactly. The purpose of the present letter is to clarify the role of heat flux in hydrodynamic balance equations in quantum wells (and other 2D systems), facilitating the formulation of an Onsager relation within the framework of this theory, and to verify the validity of this theory in regard to the Onsager relation. Following the procedure set forth in Ref. [23], the hydrodynamic balance equations which describe a weekly inhomogeneous electron system under the influence of an electron E and a small lattice temperature gradient ∇T in two dimensional (2D) quantum wells can be written as ∂n ∂t +∇ · (nv) = 0 , (1) ∂ ∂t 〈J〉+∇ · (〈J〉v) = − 1 m ∇u+ f m , (2) ∂u ∂t +∇ · 〈JH〉 = v · ∇u+ 1 2 mv∇ · (nv) + 1 2 mnv · ∇v − w − v · f , (3) which describe continuity, momentum balance and energy balance respectively. In these equations, the carrier drift velocity v, the electron temperature Te, the average relative electron energy u, the carrier density n and the chemical potential μ, together with the particle flux 〈J〉 and energy flux 〈JH〉 are all field quantities weakly dependent on the spatial coordinate, such that their spatial gradients are small and therefore are retained only to first order in Eqs. (1-3). The energy flux 〈JH〉 appearing in Eq. (3) is just the energy flux predicted by hydrodynamic balance equation theory. It relation to the other field quantities in 2D case can be derived following Ref. [23] and be written as 〈JH(R)〉 = 2u(R)v(R) + 1 2 mn(R)v(R)v(R) . (4) Substituting this relation, together with the relation of density flux 〈J(R)〉 = n(R)v(R) , (5) 3 into Eqs. (2) and (3), one arrives at the original hydrodynamic balance equations [24]. In order to obtain the Onsager relation in the framework of balance equation theory, we need first derive the energy-flux balance equation [23]. This in the case of 2D electrons in quantum wells, is given as ∂ ∂t 〈JH〉+∇ · 〈A〉 = 〈B〉+ 2 m euE + enE · vv + 1 2 envE + 1 2 vf − wv . (6) The expression of 〈B〉 is composed of two parts. One is due to collisions with impurities (〈Bi〉), and the other is due to interaction with phonons (〈Bph〉). They are given by 〈Bi〉 = 2πni ∑ kq |u(q)||F (q, z0)| (εk+q − εk) k + q/2 m δ(εk+q − εk + q · v) × [ f( εk − μ Te )− f( εk+q − μ Te ) ]