On a One-Dimensional Hydrodynamic Model for Semiconductors with Field-Dependent Mobility

: We consider a one-dimensional, isentropic, hydrodynamical model for a unipolar semiconductor, with the mobility depending on the electric ﬁeld. The mobility is related to the momentum relaxation time, and ﬁeld-dependent mobility models are commonly used to describe the occurrence of saturation velocity, that is, a limit value for the electron mean velocity as the electric ﬁeld increases. For the steady state system, we prove the existence of smooth solutions in the subsonic case, with a suitable assumption on the mobility function. Furthermore, we prove uniqueness of subsonic solutions for sufﬁciently small currents.


Introduction
The hydrodynamic model for semiconductors was first introduced by Bløtekjaer in 1970 [1]. It describes the dynamics of charged fluid particles, such as electrons and holes in semiconductor devices [2,3], and positively and negatively charged ions in plasmas [4]. The ruling equations are the Euler-Poisson equations, which in their scaled one-dimensional form, for a unipolar semiconductor, are: where n(x, t) is the electron number density, j(x, t) is the electron flux density, φ(x, t) is the electric potential, p(n) is the pressure and µ is mobility. The mobility is related to the momentum relaxation time, and in (1) they coincide because of the scaling. The device domain is the x-interval (0, ν), where ν is the device length measured in terms of the Debye length, and N = N(x) is the number density of the background ions (doping profile), with N(x) > 0. The steady-state version of this one-dimensional model has been extensively studied, starting from the existence result, and uniqueness for small currents, in the subsonic case [5]. The existence of transonic solutions, with subsonic constant doping profile, was addressed in [6], with an approach based on the construction of the orbits of the system in the electron density-electric field phase plane. The same approach was later used in [7] for a more general case. A different approach was used in [8] for the construction on transonic solutions with transonic shocks, by using the vanishing viscosity limit method. The same method was used in [9], with prescribed current and periodic boundary conditions. We also refer to [10] for a non-isentropic case. More recently, the existence of supersonic solutions was proven in [11], and the existence of solutions for sonic boundary conditions was studied in a series of papers [12][13][14].
The hydrodynamic system for semiconductors has been studied also in higher dimension and in the time-dependent case. We can refer to [15] for potential flow in three dimensions with a fully subsonic doping profile, and to [15][16][17][18][19][20][21] for more general analysis of the subsonic case. For supersonic flows, the existence and uniqueness of supersonic solutions in two and three dimensions was studied in [22,23]. The transonic solutions have also been studied in relation to the formation of shock waves [24][25][26][27][28].
In all the above results, the mobility, or momentum relaxation time, is either assumed constant or a function of n, j. In this paper, we consider the effect of an electric-field dependent mobility, µ = µ(E), with E = φ x . A preliminary discussion of this problem has been given in [29], with a phase-plane approach. Here, we prove the existence of subsonic solutions to the semiconductor hydrodynamic model with field-dependent mobility, for sufficiently small currents. Moreover, we prove the uniqueness of solutions for small currents. Both proofs are inspired by the classical results in [5], with some substantial changes due to the particular nature of the source term considered in our case. In particular, at variance with the standard case, it is not obvious how to relate the current with the electric potential at the boundaries, so some additional introductory work is needed.
The plan of the paper is the following. After Section 2, where the steady-state semiconductor hydrodynamical model is introduced and the problem is stated, Section 3 is devoted to proving the existence of subsonic solutions of the system. Section 4 shows a uniqueness result for regular solutions of the problem, while Section 5 concludes the paper with a short summary of the obtained findings.

Statement of the Problem
We consider the steady-state, one-dimensional semiconductor hydrodynamical model: with the following boundary conditions: The pressure depends on the electron density by the constitutive relation p = p(n), which satisfies the condition n 2 p (n) is strictly monotonically increasing from R + onto R + .
A common choice for the pressure relation is p(n) = kn γ , with γ ≥ 1, k > 0. The mobility is of the form with α, q 0 , c q positive constants. The above condition holds for the most used high-field mobility models, like the Caughey-Thomas [30] or the Canali model [31], which can be written as: with µ 0 low-field mobility, E c critical field value, related to the saturation velocity, and β ≥ 1 a real exponent. After scaling, for this specific model we get which satisfies (5). The doping profile N(x) satisfies the condition: The first equation in (2) gives j = constant. We are going to see whether we can prescribe j instead of φ 1 , as in the case where the mobility-that is, the momentum relaxation time-is independent of the electric field. For regular solutions, we can expand the derivative on the left-hand side of the second equation in (2), divide by n and obtain with Integrating (8) over (0, ν) and using the boundary conditions (3), we obtain On the other hand, using the third equation in (2), we can represent φ and φ x as integral functionals of n − N, depending on the boundary data φ 1 . The result is where the functional ρ(x) → I ρ (x) is defined by For a later use, it is simple to see that (12), the relation (9) can be written in the form We can use the implicit function theorem to show that (14) defines implicitly φ 1 as a function of j. First of all, we observe that which represents the so-called "built-in potential". Then, we can compute It follows that (14) defines implicitly a function g : I → J, with I ⊂ R neighborhood of 0 and J ⊂ R neighborhood of φ bi , such that G(j, g(j)) = 0 for all j ∈ I. Moreover, we have which implies that is, the function φ 1 = g(j) is locally increasing around j = 0.
If we know an a priori bound of the type 0 < n ≤ n(x) ≤ n for x ∈ (0, ν), using assumption (5), for positive j we find which implies that the implicit function φ 1 = g(j) can be extended to all positive values of j such that j n ≤ 1 αc q .
The constant 1 αc q represents the saturation velocity, which is encoded in the fielddependent mobility model.
The above discussion implies that (14) can be used to replace the boundary conditions for φ with the assignment of the current j, when j is small enough. Then, we can differentiate (8) and use the third equation in (2) (Poisson's equation) to obtain with φ x given by (11), and φ 1 related to j by (14). Equation (18) is supplemented with boundary conditions n(0) = n 0 , n(ν) = n 1 .
In conclusion, the original problems (2) and (3) can be replaced by the integraldifferential Equation (18) with boundary conditions (19), where the current j is assigned. The resulting solution will still depend on the boundary value φ 1 , which will be adjusted so that (14) is satisfied.

Existence of Subsonic Solutions
In this section we prove an existence result for subsonic solutions of system (2). Before stating the main theorem, we clarify the notion of subsonic solution. For regular solutions, the higher order term in (18) that is, n 2 p (n) > j 2 . Condition (4) implies that there exists a unique minimal value n min , depending on j, such that condition (20) holds for all n > n min . As a function of j, the minimal value n min is strictly decreasing, and for j = 0 we have n min = 0. Thus, we can conclude that Equation (18) is uniformly elliptic for n ≥ n * > n min , for some constant n * to be determined. This bound on the electron density can be interpreted by saying that the solution is subsonic, since it amounts to the condition |j/n| < (p (n)) 1 2 , with j/n mean electron velocity and (p (n)) 1 2 speed of sound. As a last remark, we note that we can limit ourselves to study the case j > 0, because the case j < 0 can be recovered from the same system (2) after a transformation x → ν − x. Theorem 1. Let us assume (4), (5) and (7), and let p (n) and q (E) be continuous functions of their arguments. Let us denote N(x), n = min{n 0 , n 1 , N}, n = max{n 0 , n 1 , N}.
For j > 0, under the assumption and for any φ 1 , problems (18) and (19) have a solution which satisfies n ≤ n(x) ≤ N.
Proof. The proof is based on the construction of a fixed point map T which, to a function n with n ≤n(x) ≤ n for all x ∈ (0, ν), associates the solution n of the problem withÊ(x) = 1 ν φ 1 + In −N (x). We notice that (13) ensures thatÊ is bounded, and so is q(Ê) and q (Ê). Thanks to the first assumption in (21), Equation (24) is a linear, uniformly elliptic equation, which admits a unique solution n = T(n) in H 1 (0, ν). Moreover, thanks to the second assumption in (21), we have so the maximum principle implies that n satisfies the bounds (22). Recalling the definition (12), and (23), it is not difficult to use (24) and obtain H 1 -bounds for n independent ofn. By the compact imbedding of H 1 (0, ν) into C 0 ([0, ν]), and by a standard continuity argument we can use Schauder's theorem to conclude the existence of a fixed point of the map T.
All the steps of this proof can be followed if, in the definition of the map T, we choose φ 1 as the minimum value greater than φ bi such that This concludes the proof.

Uniqueness of Solutions
Theorem 2. Under the same assumptions of Theorem 1, there exists a constant j 0 > 0 such that for any j < j 0 there exists at most one regular solution n of problems (18), (19), (9) and (11), which satisfies the bound n(x) ≥ n > n min for all x ∈ (0, ν).
Proof. Let n 1 and n 2 be two solutions of problems (18), (19), (9) and (11), such that n i (x) ≥ n > n min for all x ∈ (0, ν), i = 1, 2. Thus, we have with Subtracting the Equation (27) and introducing the variable with Moreover, from (28) we find Multiplying (30) by a(x)u and integrating over (0, ν), we find We can estimate On the other hand, from (29) and (13), we find and using (32) we obtain Recalling the second assumption in (21), we get which, together with (34) yields The constant c φ (j) tends to 0 as j tends to zero. Finally, we can estimate with The constant c * φ (j) tends to a strictly positive value as j tends to zero. Moreover, we can write with a = min x∈(0,ν) a(x) > 0. Thus, using (37) and (38) in (33), we find It is possible to find a constant j 0 > 0 such that For instance, if j ≤ n 2αc q , we have c * φ (j) ≤ 2c * φ (0), so we can choose It follows that, for j ≤ j 0 , we have which implies that u(x) ≡ 0 and thus the thesis.

Conclusions
The hydrodynamic model for semiconductors describing the dynamics of charged fluid particles, such as electron and holes in semiconductor devices and ions in plasmas, has attracted the attention of many scholars. In such studies the mobility, or momentum relaxation time, is either assumed constant or a function of the electron number density and the electron flux density. Here, we have considered the effect of an electric field dependent mobility. This kind of mobility model was introduced in the literature after empirical studies, to take into account the occurrence of a saturation velocity, that is, a maximum electron mean velocity which is reached as the electric fields grows and tends to infinity in modulus. The second condition in (21), which plays a key role in both the existence and the uniqueness proof, expresses precisely the upper bound of the electron velocity given by the saturation velocity. In our case, the saturation velocity v s can be recovered as the high-field limit of the velocity j n = µ(φ x )φ x , which follows by equating to zero the right-hand side of (1) 2 . We obtain This conclusion is accurate if we use the explicit form (6), while in general the condition (5) only yields a lower bound 1 αc q for v s , unless we add specific hypothesis on the behaviour of q (E) at infinity.
In this paper, we have examined a one-dimensional, isentropic, hydrodynamical model for a unipolar semiconductor, with the relaxation time depending on the electric field. In this framework, for the steady-state version of the problem, we have proved the existence and uniqueness of smooth solutions in the subsonic case for sufficiently small currents. This promising result can be considered as the basis for a further exploration of the implication of a field-dependent mobility on the solutions of the hydrodynamic model for semiconductors, not only in the subsonic case but also in the transonic and supersonic case, addressing the stability of the steady-state solution and the asymptotic behaviour of general solutions for long times. These will be the topics for subsequent papers.