Boundary Layer Analysis in the Semiclassical Limit of a Quantum Drift Diffusion Model

Transcription

1 Boundary Layer Analysis in the Semiclassical Limit of a Quantum Drift Diffusion Model Shen Bian, Li Chen, and Michael Dreher March 1, 11 Abstract We study a singularly perturbed elliptic second order system in one space variable as it appears in a stationary quantum drift diffusion model of a semiconductor. We prove the existence of solutions and their uniqueness as minimizers of a certain functional and determine rigorously the principal part of an asymptotic expansion of a boundary layer of those solutions. We prove analytical estimates of the remainder terms of this asymptotic expansion, and confirm by means of numerical simulations that these remainder estimates are sharp. 1 Introduction and Known Results The distribution of charged particles in a semiconductor can be described by various systems of partial differential equations, for instance the drift diffusion equations. Typically, the relevant classes of particles are the movable electrons in the conduction band which is an energy band at a higher level) and the so-called holes which are vacant positions in the next lower energy band, with positive charge). Additionally, charged ions may exist at fixed positions in the semiconductor crystal. For smaller devices, it may be necessary to consider also quantum mechanical effects. Then, in the stationary case, the quantum drift diffusion model reads after scaling) F = V + h n n) n n, 1.1) G = V + h p p) ξ p p, 1.) div µ n n F ) = R n, p)r 1 F, G), 1.3) div µ p p G) = R n, p)r 1 F, G), 1.4) λ V = n p Cx), 1.5) Department of Mathematical Sciences, Tsinghua University, Beijing, 184, P.R. China, bs9@mails.tsinghua.edu.cn Department of Mathematical Sciences, Tsinghua University, Beijing, 184, P.R. China, lchen@math.tsinghua.edu.cn Corresponding author. Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany, michael.dreher@uni-konstanz.de 1

2 1 INTRODUCTION AND KNOWN RESULTS for the unknowns n, p, F, G, V. The functions n and p describe the densities of electrons and holes, and F, G are known as quantum quasi Fermi levels. Finally, V is the electric potential as generated from the charged particles via the Poisson equation 1.5). Here C characterizes the known density of positive ions. The functions h n, h p are called the enthalpy functions of the electrons and holes; typically they are of the form hs) = T ln s with some positive temperature constant T. The positive parameter is proportional to the Planck constant and describes quantum effects, and the positive constant ξ is related to the quotient of the effective masses of the electrons and the holes. Next, the functions R, R 1 are known expressions, which model generation recombination effects, they satisfy certain monotonicity assumptions, and an example is R n, p)r 1 F, G) = 1 expf + G) δ ), a + a 1 n + a p with δ > chosen in such a way that R R 1 in thermal equilibrium. The constants µ n, µ p are related to the mobilities of the particles, and λ is known as Debye length. This system has been studied extensively in [6], [8], [1]; and we also refer to [5], [4]. The above system can be complemented with certain boundary conditions, for instance Dirichlet boundary conditions of n, p, V, F, G) on a boundary part Γ + with n, p positive there), homogeneous Neumann boundary conditions of n, p, V, F, G) on a boundary part Γ N, and n, p, V ) =,, V extern + V equil ) on a further boundary part Γ note that the elliptic equations 1.3) and 1.4) degenerate at points where n = p = ). Several results were proved in [1] under appropriate assumptions: the full system has a solution n, p, V, F, G) L Ω) H 1 Ω) CΩ) with non-negative n and p. And if F, G L Ω) are given, then a solution n, p, V ) to 1.1), 1.), 1.5) exists which is uniquely determined by the condition that a certain functional shall be minimized. Finally, the semiclassical limit has been performed, under the assumption that the above mentioned boundary part Γ is empty. It is one of the goals of this article to remove this restriction on Γ, for a sub-class of the systems studied in [1] which we describe now. S G D Figure 1: A rough schematic sketch of a MOSFET, with fictitious boundary in the bulk of the material Our studies are related to MOSFET 1 devices, whose structure is sketched in Figure 1. At one end of the device, there are contacts called source, gate, drain, of which the gate contact is insulated 1 metal-oxide-semiconductor field-effect transistor

3 3 by means of an oxide, explaining the name of the device. Depending on an applied voltage V GS between gate and source, the density of movable charge carriers changes, and this effect decides whether a current can flow from source to drain. If such movable particles are available between source and drain in abundance, we say that an inversion layer has formed. A good knowledge of this inversion layer, be it analytical or numerical, is of high importance to applications. In an opposite situation, when no movable particle are available in a certain region, we say that this region depleted of particles. Various asymptotic expansions for such layers have been proposed in [9], [1], []; we also refer to [3] for a model involving quantum effects. The modifications of [3] to the model 1.1) 1.5) can be summarized as follows. The holes are supposed to be in thermal equilibrium, which implies G. In many situations, the parameter ξ from 1.) is small, which motivates us to neglect quantum effects for the holes, and then 1.) turns into = V + h p p), or equivalently p = expv/t p ). Moreover, generation recombination events are ignored: R R 1. Next we assume that the domain Ω of the device is a square, 1), 1) R, and we write the spatial variable as x, y), with x running from the contacts into the bulk of the crystal. Concerning the electron density n, we assume thermal equilibrium in direction of the x variable, which makes F a function of y, 1) only, and this function is supposed to be known. The quasi 1D approximation is supposed, which says that all functions depend only weakly on the variable y. Hence we will neglect the derivatives with respect to y from now on, and the system becomes F = V x) + T n ln nx) nx)) xx nx), < x 1, 1.6) λ V xx x) = nx) expv x)/t p ) Cx), x ) In the sequel, the dependence of the functions on the y variable is no longer mentioned. The boundary conditions at the fictitious boundary x = 1 in the bulk material are n1) = n B, V 1) = V B, 1.8) and the constants n B R + and V B R satisfy the compatibility condition F = V B + T n ln n B, 1.9) which expresses that quantum effects do not appear there. And the boundary conditions at the location of the gate x = ) are n) =, V x ) = βv ) V GS ), 1.1) with given constants β and V GS R. The vanishing of the particle density at x = makes two terms in 1.6) singular, and the formation of an additional layer inside the inversion layer is to be expected: the quantum layer. The objective of this paper is to study this quantum layer with the rigor of analysis. Our modelling follows [3], where an asymptotic expansion was conjectured, but the key improvement presented in

4 4 MAIN RESULTS n p Figure : Electron and hole densities, and electric potential in case of an inversion layer this article is a precise discussion of the error terms. This enables us to perform the limit rigorously, even in the presence of a zero boundary value of the electron density. We sketch the results presented in this paper. First, we prove the existence of a solution to the system 1.6), 1.7), 1.8), 1.1), which is unique as a minimizer to a certain functional. Our approach builds upon [1]; however, the additional nonlinearity in our equation 1.7) which is not present in [1] requires several changes. Second, we prove rigorously that the electric potential V converges to the corresponding solution V of the classical model, with an analytic estimate of the error V V. We also determine the profile of the quantum layer of the electron density, again with several analytical error estimates. And our third result are numerical simulations which confirm that our analytical error estimates are sharp. Main Results Our first result is about the existence of solutions n, V ), which are constructed in such a way that ϱ := n is the unique non-negative minimizer of a certain functional, see Remark 3.7. Theorem.1. Let us be given positive constants T n, T p,, λ, and real constants F, n B, V GS, and V B, and a non-negative constant β. Suppose that C = Cx) is continuous and real-valued. Then the boundary-value problem 1.6), 1.7), 1.8), 1.1) possesses a solution n, V ) C [, 1]) with nx) > for < x 1, such that ϱ := n is the unique non-negative minimizer to F from 3.8). There is a constant C, independent of, with n) x) ) dx + V x) ) dx C,.1)

5 5 for this solution n, V ). Our second result will describe the shape of n and V, in particular for small values of. The precise formulation requires some preparations. By variational methods, we will show in Proposition 3.6 that unique functions V, n C [, 1]) exist which solve λ V x) = n x) expv x)/t p ) Cx), x 1, n x) := expf V x))/t n ), x 1,.) V ) = βv ) V GS ), V 1) = V B. Let Z = Zy) be the unique function from C [, )) that solves Z y) = Zy) ln Zy), < y <, Z) =, lim Zy) = 1..3) y + This function approaches the value 1 exponentially for y +, and all its derivatives decay exponentially, as will be shown in Lemma A.1. Theorem.. Assume 1.9). Then there is a positive, such that, for <, we have n n) L,1)) + V V) W 1,1)) C, where the zero-th order approximations n ), V ) ) of n, V ) are defined as follows: n ) x) := n x)z Tn x ), V ) x) := V x). Theorem.3. The approximations n ) satisfy the uniform estimates n n) L,1)) C3/4, <, and in particular, we have nx) n) x) C 3/4 x, x 4..4) We follow the standard notation conventions. In particular, C denotes a generic positive constant that is independent of the unknown functions and may change its value from line to line. 3 Existence of Solutions In this section, we demonstrate Theorem.1, whose proof is split into several parts. First we consider the boundary value problem λ U x) = qx) px, Ux)), x 1, U ) = βu), β, U1) =, 3.1)

6 6 3 EXISTENCE OF SOLUTIONS with a given function q C[, 1]), under the following assumptions on the nonlinearity p: p C 1 [, 1] R), p U x, U) > x, U) [, 1] R, U P x, U) := px, u) du C x, U) [, 1] R. u= 3.) We start our considerations with a simple observation. Lemma 3.1. Solutions U C [, 1]) to 3.1) are unique, and they satisfy an estimate U W 1,1)) C1 + q L,1)) ), with a constant C depending only on p, ) L,1)) and λ >. Proof. The uniqueness is proved by usual monotonicity arguments. solution, then And if U C [, 1]) is a λ U x)) dx + λ βu)) = qx) px, ))Ux) dx px, Ux)) px, ))Ux) dx, and then the monotonicity of p and β imply λ U L,1)) q p, ) L,1)) U L,1)). It remains to exploit Poincare s inequality, which is possible since U1) =. To prove the existence of a solution to 3.1), we choose the variational ground space X U := { U W 1, 1)): U1) = }. Lemma 3.. For a fixed function q C[, 1]), define a functional F ) U) := λ U x)) dx + λ βu)) + qx)ux) + P x, Ux)) dx. 3.3) This functional possesses a unique minimizer U X U, and this minimizer is a classical solution to the boundary value problem 3.1). Proof. By Poincare s inequality and β, the functional F ) is coercive: F ) U) λ 4 U W 1 C, U X U,

7 7 and then the existence of a minimizer U X U follows by standard arguments, and U X U C[, 1]) is bounded. Take ϕ C [, 1]) with ϕ1) =, and consider F ) U + δϕ): F ) U + δϕ) = F ) U ) δ ) λ U + q px, U ) ϕ dx + δλ ) U ) + βu ) ϕ) + Oδ ), and therefore U solves the differential equation λ U = q px, U ) in the distributional sense, and it follows that U C [, 1]), as well as U ) + βu ) =. Next we discuss how this minimizer U of F ) depends on q. Lemma 3.3. Define a mapping Φ: C[, 1]) CB [, 1]), with C B [, 1]) as the vector space of all functions U from C [, 1]) with U ) = βu) and U1) =, via the relation Φ{q} := U, and U is defined as the unique minimizer of the functional F ) from 3.3). Then Φ is a homeomorphism. Proof. Clearly, Φ is bijective, and Φ 1 is continuous. It remains to establish the continuity of Φ. To this end, let q 1 C[, 1]) be given, and q C[, 1]) with q 1 q L 1. Define U k := Φ{q k }. Then λ U k + px, U k) = q k, and we quickly find ) λ U 1 ) U dx + λ β U 1 ) U ) + px, U 1 ) px, U ))U 1 U ) dx = q 1 q )U 1 U ) dx, which brings us, together with β, the monotonicity of p, and Poincare s inequality, that U 1 U W 1 C q 1 q L, hence also U 1 U L C q 1 q L. Next it follows that p, U 1 ) p, U ) L C q 1 q L, and then also U 1 U L C q 1 q L. For more information, we determine the Fréchet derivative of Φ. Lemma 3.4. Take q C[, 1]). Then there is a constant C such that for all q C[, 1]) with q q C[,1]) 1 we have the expansion Φ{q} = Φ{q } + W + R, with W C [, 1]) as the unique solution to λ W + p U x, Φ{q })W = q q, W ) = βw ), W 1) =, satisfying W C [,1]) C q q C[,1]), and R C [,1]) C q q C[,1]).

8 8 3 EXISTENCE OF SOLUTIONS Proof. From Lemma 3.3 we know already that Φ{q} Φ{q } C [,1]) C q q C[,1]). define R := Φ{q} Φ{q } W and have We λ R = λ Φ{q} Φ{q } W ) = q px, Φ{q})) q px, Φ{q })) + p U x, Φ{q })W q q ) = p U x, Φ{q }) Φ{q} Φ{q } W ) + O Φ{q} Φ{q } C[,1]) ) = p U x, Φ{q })R + O q q C[,1]) ). Note that p U x, Φ{q }) is positive, hence we can conclude that R C [,1]) C q q C[,1]). Lemma 3.5. Define a function K = Kx, U) := Upx, U) P x, U), and set F 1) q) := λ Φ{q}) x)) dx + λ βφ{q})) + Kx, Φ{q}x)) dx. Then Kx, U) for all x, U) [, 1] R, and the Fréchet derivative of F 1) is given via F 1) q) = F 1) q ) + Φ{q } q q ) dx + O q q C[,1]) ). Here the remainder term is positive for q q, and F 1) is strictly convex. Proof. Concerning the bound for K, we remark that P x, ) =, p U > and Kx, U) = U u= up U x, u) du. For the proof of the second claim, we put U = Φ{q } and U := Φ{q}. Then U = U + W + R with W and R as given in Lemma 3.4, and it follows that F 1) q) = λ U x)) dx + λ βu)) + Kx, Ux)) dx = F 1) q ) + λ U U U ) dx + λ βu ) U) U )) + λ U U ) dx + λ βu) U )) + F 1) q ) λ U U U ) dx + = F 1) q ) + + U q q ) dx Kx, U) Kx, U ) dx U px, U) px, U )) + Kx, U) Kx, U ) dx. Kx, U) Kx, U ) dx

9 9 Now we have, with some Ũ between U and U, U px, U) px, U )) + Kx, U) Kx, U ) = U px, U) + Upx, U) P x, U) + P x, U ) ) = P x, U) + px, U) U U) + P x, U ) = 1 P UUx, Ũ) U U), because of P UU = p U >. Proposition 3.6. Suppose that λ and T p are positive, V GS, V B are real, and n C is a continuous given real-valued function on [, 1]. Then there is exactly one solution V C [, 1]) to 1.7) with the boundary conditions V 1) = V B and V ) = βv ) V GS ). Moreover, the nonlinear boundary value problem.) possesses exactly one solution V C [, 1]). Proof. To solve 1.7) with the mentioned boundary conditions, we put V = V inh + U with V inh as the unique solution to { λ V inh x) = Cx), x 1, V inh ) = βv inh) V GS ), V inh 1) = V B. 3.4) We then have U = Φ{n}, with Φ as defined in Lemma 3.3, and px, U) := expv inh x)/t p ) expu/t p ). 3.5) The uniqueness of V is proved as in Lemma 3.1. The result concerning.) is proved likewise. For n we may write n = ϱ, and then we transform 1.6), 1.7), 1.8), 1.1) into { ϱf = ϱvinh + Φ{ϱ }) + T n ϱ ln ϱ ϱ, on [, 1], ϱ) =, ϱ1) = ϱ B := n B, 3.6) with V inh given by 3.4). Set hs) = T n ln s for s > and Hs) := s σ=1 hσ) dσ = T n s ln s s + 1). 3.7) Our intuition is to look for ϱ as a non-negative minimizer to the functional Fϱ) := ϱ + Hϱ ) + ϱ V inh ϱ F ) dx + F 1) ϱ ), 3.8) over the set X ϱ := {ϱ W 1, 1)): ϱ) =, ϱ1) = ϱ B}.

10 1 3 EXISTENCE OF SOLUTIONS Remark 3.7. For the convenience of the reader, we collect all information about how to construct F in one place: for functions ϱ from X ϱ we shall discuss the functional Fϱ) := ϱ + Hϱ ) + ϱ V inh ϱ F ) dx + λ + Φ{ϱ }) ) dx + λ βφ{ϱ })) Φ{ϱ }x) px, Φ{ϱ }x)) P x, Φ{ϱ }x)) dx, with V inh as the unique solution to 3.4), p = px, U) defined in 3.5), and P = P x, U) from 3.). Finally, U = Φ{q} is defined as the unique solution to 3.1). For the definition of F, we take q := ϱ. However, some problem occurs here. The pole of h = hs) at s = and the boundary values of ϱ at x = make the functional F irregular, and then the Euler Lagrange equations can not be derived. To overcome this difficulty, we perform a regularization step: for a parameter γ, 1), we set hγ) : s γ, h γ s) := hs) : γ s γ 1, hγ 1 ) : γ 1 s, and the we put H γ s) := s σ=1 h γσ) dσ. The functional for which we wish to find a non-negative minimizer is F γ ϱ) := ϱ + H γ ϱ ) + ϱ V inh ϱ F ) dx + F 1) ϱ ), 3.9) where we restrict γ to the interval, γ ), and γ with < γ 1 is selected by the condition H γ s) s V inh F L,1)) s for s γ 1. Lemma 3.8. For functions ϱ taking only non-negative values, the functionals F and F γ from 3.8) and 3.9) are strictly convex functionals of ϱ, in the following sense: if ϱ 1, ϱ X ϱ with ϱ 1, and ϱ 1 ϱ, and if < t < 1, then ) F tϱ t)ϱ < tfϱ 1 ) + 1 t)fϱ ), ) F γ tϱ t)ϱ < tf γ ϱ 1 ) + 1 t)f γ ϱ ). Non-negative minimizers of F or F γ are unique. Proof. First, the strict convexity of the functional ϱ ϱ dx was shown in [6]. Second, the scalar functions H and H γ are weakly convex functions of their arguments. And third, we recall the strict convexity of F 1) from Lemma 3.5.

11 11 Lemma 3.9. For < γ < γ, the functional F γ possesses a non-negative minimizer ϱ γ C [, 1]) X ϱ, and each such minimizer satisfies the uniform in γ estimate ϱ γ W 1,1)) + Φ{ϱ γ } W 1 C. 3.1),1)) Proof. We clearly have F γ ϱ) ϱ W 1,1)) C, ϱ X ϱ, with some C R + depending only on V inh F L,1)), but not on γ. Then m γ := inf{f γ ϱ): ϱ X ϱ } exists. Let ϱ 1, ϱ,... ) be a sequence in X ϱ with lim j F γ ϱ j ) = m. Then this sequence is bounded in W 1, 1)), hence we can assume strong convergence in C[, 1]), and weak convergence in W 1, 1)) of this sequence to some limit ϱ γ X ϱ. Then F 1) ϱ j ) has the limit F 1) ϱ γ ) for j, because Φ: C[, 1]) CB [, 1]) is continuous, see Lemma 3.3. By classical arguments [7], we conclude that F γ is weakly sequentially lower semi-continuous on X ϱ, and consequently F γ has a minimizer on X ϱ. If ϱ γ X ϱ is such a minimizer, then ϱ γ also belongs to X ϱ, and it has the same value of F γ. Independent of γ estimates of such non-negative minimizers ϱ γ can be found by choosing a function ϱ X ϱ that is non-negative and bounded from above. Then F γ ϱ γ ) F γ ϱ ), and the right-hand side is bounded from above independently of γ because H γ is uniformly bounded from below. This gives us the uniform estimates 3.1). Lemma 3.1. Let ϱ γ C [, 1]) X ϱ be a non-negative miminizer of F γ, and V γ := V inh + Φ{ϱ γ }. Then ϱ γ satisfies ϱ γ + h γ ϱ γ) + V γ F )ϱ γ =. 3.11) Proof. By Lemma 3.5, 3.11) is just the Euler Langrange equation of the functional F γ. Proof of Theorem.1. By the uniform bound of ϱ γ W 1 W 1,1)) and the compactness of the embedding, 1)) C[, 1]), we can assume that we have a sequence ϱ γ) γ of non-negative solutions to 3.11) that converges in C[, 1]) to a non-negative limit ϱ. Since the function s shs ) is continuous on [, ), we deduce that h γ ϱ γ) V γ + F )ϱ γ hϱ ) V + F )ϱ, V := V inh + Φ{ϱ }, with convergence in C[, 1]), for γ. Then the limit ϱ is a non-negative distributional solution to 3.6), and by elliptic regularity, ϱ C [, 1]), and ϱ γ ϱ L,1)). To demonstrate the positivity of ϱx) at all x, 1), we recall that lim s + hs) =. Assume that ϱx ) = at an interior point x, 1). By the continuity of s shs), the function x ϱx)hϱ x)) + V x) F ) takes only non-positive values near x. Therefore ϱ in an open neighbourhood Ω, 1) of x. On the other hand, ϱx) in Ω, which is only possible if ϱ in Ω. Therefore, the set of zeroes of ϱ is an open subset of, 1). On the other hand, it is a closed subset of, 1) in the

12 1 4 ASYMPTOTICS OF THE SOLUTIONS relative topology because ϱ is continuous. Hence the set of zeroes of ϱ in, 1) is either empty or all of, 1). The latter is impossible because ϱ > at x = 1. Therefore ϱ > in, 1), and from Fσ) = lim γ F γ σ), with F from 3.8) and any function σ X ϱ, we learn that ϱ is indeed a minimizer of F. It remains to show.1). Choose ϱ X ϱ, for instance ϱ x) = ϱ B x. Then Fϱ) Fϱ ), which is bounded independent of, 1]. Observe that there is a constant C R + such that Hψ ) + ψ V inh F ) dx C, F 1) ψ ) C, for all ψ C[, 1]), from which.1) quickly follows. 4 Asymptotics of the Solutions Now we begin to demonstrate Theorem., whose proof will span the whole Section 4. From now on, let n, V ) be the solution to 1.6), 1.7), 1.8), 1.1), as constructed in Theorem.1. We introduce the notation ϱ := n. Then the pair ϱ, V ) solves { ϱ = gϱ ) + ϱ V F ), λ V = ϱ pv ) Cx), 4.1) with gs) := T n s ln s and ps) := exps/t p ). Moreover, we have the boundary conditions { ϱ ) =, ϱ 1) = ϱ B := n B, V ) = βv ) V GS ), V 1) = V B. 4.1 Properties of the Solutions Lemma 4.1. There is a constant C 1, independent of, such that V C [,1]) + n C[,1]) C 1. Proof. By the embedding W 1, 1)) C[, 1]),.1), and V 1) = V B, we find V C[,1]) C. Suppose that ϱ takes a local maximum at an inner point x, 1). Then ϱ x ), hence hϱ x )) + V x ) F. We then have ϱ x ) expf V x ))/T n ), and then also n C[,1]) C. Then 1.7) gives us the remaining bound for V C [,1]). The next result gives us a first information on the graph of n, depending on.

13 4.1 Properties of the Solutions 13 Lemma 4.. There is a positive constant g such that ϱ x) g, x 1, ϱ x) g, x 4. Proof. The function ϱ solves 4.1). Define g as g = 1 )) 3 min ϱb 1, exp 1 C 1 ), T n with C 1 from Lemma 4.1 as an upper estimate of V F C[,1]). Then it follows that whenever < ϱ x) 3g, then T n ln ϱ x) + V x) F 1, and consequently ϱ x) <. Then ϱ x) is impossible because this would imply ϱ x) < for all x x, 1], contradicting ϱ 1) = ϱ B > 3g. Hence there is a uniquely determined number x 3, 1) with { < 3g : x < x 3, ϱ x) > 3g : x 3 < x 1. Moreover, on the interval [, x 3 ], ϱ is strictly increasing. Define x 1, x, x 3 ) uniquely by ϱ x 1 ) = g and ϱ x ) = g. Then ϱ x) g, x [x 1, x 3 ]. Now we make use of a simple fact: if ψ x) d on [a, b] for some ψ C [a, b]), then max [a,b] ψ min [a,b] ψ d b a) /4. Hence we conclude that x 3 x and x x 1. Since ϱ x,3 ) are independent of, and because of ϱ < on, x 3 ], we find that ϱ x) g, x [, x ]. From the definition of x 1 and ϱ ) = we then get x 1, hence x 4 as desired. The next result is a first step in showing that the quantum term ϱ x O ). is of less relevance for Lemma 4.3. Assume 1.9). Then there is a constant C, independent of, 1/8], such that ϱ T n ln ϱ + V F ) L,1)) C1/4, 4.) ϱ T n ln ϱ + V F ) L,1)) C3/4, 4.3) ϱ T n ln ϱ + V F ) L 4,1)) C1/, 4.4) ϱ expf V )/T n ) L 4,1)) C1/, 4.5) ϱ L,1)) C 3/4. 4.6)

14 14 4 ASYMPTOTICS OF THE SOLUTIONS Proof. Take a function χ C [, 1]) with χ 1, χ, { 3x : x 1/4, χx) = 1 : 1/3 x 1, and set χ x) = χx ) for x [, 1]. Then we conclude, using T n ln ϱ + V F ) x=1 =, that χ ϱ ϱ ) dx = = T n Now we estimate χ ϱ T n ln ϱ + V F ) dx χ ϱ ϱ ) dx χ ϱ V dx χ ϱ T n ln ϱ + V F ) dx. χ ϱ V ϱ χ T ) n + C Tn χ ϱ V ϱ ) χ T n ϱ ϱ ) + CV ), χ ϱ ln ϱ dx = χ ϱ ln ϱ ϱ ) dx ) = χ ϱ ln ϱ ϱ ) x= χ ϱ V dx = χ ϱ V ) x= and consequently we have χ ϱ ϱ ) + T n ϱ )) dx C. Choose a positive σ). Then Lemma 4. brings us to +σ) We clearly have ϱ ϱ ) + T n ϱ ) dx C σ). ϱ L +σ),1)) = ϱ T n ln ϱ + V F )), hence C σ 1/ ). χ ϱ ln ϱ ϱ ) dx, ϱ χ V + χ V ) dx, Interpolating the estimates ϱ W 3 C σ 1/ ) and ϱ W 1 Cσ 1/ ), we then derive ϱ W +σ),1)) C σ 1/ ). By Nirenberg Gagliardo interpolation, u L C u 1/ W u 1/ W 1 C u 3/4 u 1/4, 4.7) W L

15 4. First Remainder Estimates 15 we then have L +σ),1)) ϱ C 1/ σ 1/ ). Now the differential equation implies ϱ T n ln ϱ + V F ) L +σ),1)) C σ 1/ ) Then 4.) follows from choosing σ) = and interpolating this estimate with the inequality ϱ T n ln ϱ + V F ) W 1,1)) C 1/4 ; and this choice of σ) also gives 4.6). Finally, 4.4) is found when we take σ) =. 4. First Remainder Estimates We continue our preparations for the proof of Theorem.. Lemma 4.4. The sequence V ) converges to a limit V C [, 1]), V V W 1,1)) C1/, 4.8) and V solves.). Moreover, the sequence ϱ ) converges uniformly on compact subsets of, 1) to the limit ϱ := n in the sense of ϱ ϱ L,1)) = ϱ expf V )/T n )) L,1)) C1/4. 4.9) See.) for the definition of n. Proof. For parameters < < 1 < 1/8, we conclude that λ V 1 V ) = ϱ 1 ϱ expv 1 /T p ) + expv /T p ), λ V1 V ) ) dx + λ βv 1 ) V )) = + expv 1 /T p ) expv /T p )) V 1 V ) dx 41 ϱ 1 ϱ )V 1 V ) dx + ϱ 1 ϱ )V 1 V ) dx, 4 1 and then the representation 4.5) and monotonicity arguments bring us to λ V1 V ) ) dx C1, hence there is a limit V W 1, 1)), and 4.8) holds. Combined with the uniform bound V C [,1]) C from Lemma 4.1, then also lim V V C 1 [,1]) =. Now Lemma 4.1, Lemma 4., and 4.) give us ϱ expf V )/T n )) L,1)) C1/4, and combining this estimate with 4.8) then yields 4.9). It follows that V is a distributional solution to the differential equation.), and then V C [, 1]) by elliptic regularity.

16 16 4 ASYMPTOTICS OF THE SOLUTIONS Next we discuss the behaviour of ϱ near the boundary x =, and our result is: Proposition 4.5. On the interval [, 4], we have the uniform expansion ϱ x) = ϱ x)z Tn x ) + O 1/ x ), 4.1) ϱ Tn ϱ )Z ) C 1/, 4.11) C 1 [,4]) with Z = Zy) as in.3). And on the intermediate interval [4, 3 ], we uniformly have ϱ x) = ϱ x)z Tn x ) + O 1/4 ). 4.1) Proof. Observe that, for < x 1, ϱ x) = gϱ ) + ϱ V F ), ϱ ) ) = gϱ )ϱ + 1 ϱ ) V F ) 1 = T n ϱ ln ϱ 1 ) 4 ϱ + 1 ϱ ) V F ), ϱ x) ) ϱ ) ) = Tn ϱ x) ln ϱ x) 1 ) ϱ x) + ϱ x)v x) F ) x ϱ t)v t) dt = T n ϱ x) ln ϱ x) 1 + V ) x) F V T ξ) n with ξ, x). If x 3, then 4.8) and V C [, 1]) imply V x) = V ) + V x) V x)) + V x) V )) = V ) + O 1/ ) = F T n ln ϱ ) + O 1/ ). x ϱ t) dt, 4.13) In the following, let R be a generic function of x with R L,3 )) C3/4. For such R, we then also obtain R L 1,3 )) C. On the interval [, 3 ], we then have, by 4.6) and 4.3), ϱ x)) C 1/, ϱ x) = exp F V x) T n ln ϱ x) = F V x) + R x), T n x V ξ) ϱ t) dx C 1/. ) + R x) = ϱ ) + R x),

17 4. First Remainder Estimates 17 Therefore, for x 3, we conclude that 3 ϱ ) ) = Tn ϱ x) ϱ ) ) Tn ϱ x) dx 3 ϱ ) ϱ x) dx = ) + R x) + O 1/ ), C 1/ dx + R L 1,3 )) C, R x) dx C, and then we ultimately deduce that ϱ )) = T n ϱ ) + O1/ ) and under the assumption x [, 3 ]) that ϱ )) = T n ϱ x) + O1/ ). Now the differential equation becomes ϱ x) ) = Tn ϱ x) ln ϱ x) + 1 ϱ ) 1 ) ϱ x) ϱ x) ln ϱ ) + O 1/ ) ϱ = T n ϱ x) ϱ ) ) ϱ ln x) ) ϱ ) + 1 ϱ x) )) ϱ ) + O 1/ ), x 3. We introduce the scaling y = x, ϱ x) =: ϱ I,) y) ϱ ), and then we are led to a discussion of a differential equation ϱ I,) y) ) = Hϱ I,) y)) + O 1/ ), y 3 1/, 4.14) compare 3.7) for the definition of H. Now we restrict our interest to the shorter interval [, 4]. By Lemma 4., we have ϱ I,) y) and this brings us to g, y 4, 4.15) ϱ ) ϱ I,) y) = Hϱ I, y)) + O1/ ), y 4. Note that the radicand is bounded from below by a positive constant, by 4.15), which makes the function s Hs ) uniformly Lipschitz continuous on the relevant interval. The mentioned function Z = Zy) from.3) solves Z y) = 1 Tn HZ y)), y <, Z) =. Put W y) = ϱ I,) y) Z T n y). Then we have W y)) = W y) Hϱ I,) HZ y)) ) T n y)) + O 1/ ) C W y) + O),

18 18 4 ASYMPTOTICS OF THE SOLUTIONS hence W y) C 1/ by Gronwall s Lemma. For 4.11), we only note that W y) C 1/ on [, 4]. From 4.11) we then conclude that ϱ x) = ϱ )Z Tn x ) + O 1/ x ), which implies 4.1) via the Lipschitz continuity of ϱ. Now we come to the proof of the uniform estimate of W on [4, 3 1/ ], from which then 4.1) will follow. It is already known from 4.11) and 4.9) that W 4) C 1/ and W 3 1/ ) C 1/4. We split the interval as [4, y ] [y, 3 1/ ], with y as the smallest number such that ϱ I,) y) > on [4, y ). If y = 3 1/, then ϱ I,) y) ϱ I,) 3 1/ ) 1 + C 1/4, 4 y 3 1/. And if y < 3 1/, then ϱ I,) y ) =, and 4.14) as well as the convexity of H give ϱ I,) y ) 1 CHϱ I,) y )) C 1/. In both cases, we conclude that ϱ I,) y) 1 + C 1/4 for 4 y y. For the discussion of W on the interval [4, y ], we determine a number y with max W y) = W y ). y [4,y ] If y {4, y } which is the only interesting case), then W y ) =, and therefore Hϱ I,) y )) HZ T n y )) C 1/, which implies W y ) C 1/4, since s Hs ) has a second-order zero at s = 1. And here we have also made use of the fact that c Z T n y ) < 1 on the relevant interval, and ϱ I,) y ) 1+C 1/4. The result then is W y) C 1/4 on [4, y ]. It remains to discuss W on the interval [y, 3 1/ ]. We begin with Z T n y ) 1 = ϱi,) y ) 1 W y ) C 1/4. Now determine a number y with max W y) = W y ). y [y,3 1/ ] If y {y, 3 1/ }, then W y ) =, hence Hϱ I,) y )) HZ T n y )) C 1/, and from 1 C 1/4 Z T n y ) < 1 we then find that W y ) C 1/4, as desired.

19 4.3 Refined Remainder Estimates Refined Remainder Estimates Now we are in a position to finish the proof of Theorem.. The zero-th order approximations are ϱ ) x) := ϱ x)zαx/), α := T n, V ) x) := V x), and the remainders R,ϱ := ϱ ϱ ), R,V := V V ) have the following values at the boundaries: R,ϱ ) =, R,ϱ ) = O1/ ), R,V ) = βr,v ), R,ϱ 1) = Oe c/ ), R,ϱ 1) = O 3/4 ), R,V 1) =, compare 4.6) and 4.11). The estimates from Lemma 4.4 and Proposition 4.5 read R,ϱ L,1)) C1/4, R,V L,1)) C1/. We define a reduced remainder R,ϱ via R,ϱ x) =: ϱ x)zαx/) R,ϱ x), and from Proposition 4.5 we find that R,ϱ has the properties R L,ϱ,1)) C1/4, R L,ϱ,4)) C1/, Z R,ϱ L,4)) C 1/, R,ϱ 1) = Oe c/ ), R,ϱ 1) = O 3/4 ). Lemma 4.6. The remainders fulfill the differential equations R,ϱ = α ϱ ln ϱ ln ϱ ) ) + R,ϱ α ln Z + ϱ R,V ϱ Z αϱ Z, 4.16) λ R,V = ϱ R,ϱ R,ϱ + ϱ Z 1) pv ) + pv ) ), 4.17) with pv ) = expv/t p ). Here Z means Z αx/) = d α dx Zαx/). And the reduced remainder R,ϱ solves Z Z ϱ R,ϱ ) ) = α ϱ ) 1 + R,ϱ ) ln1 + R,ϱ ) + ϱ R,V Z Z ϱ ). 4.18) Proof. Observe that gϱ ) + ϱ V F ) =, by the very definition of ϱ := n in.), with gs) = T n s ln s. Then we calculate as follows: R,ϱ = ϱ ϱ Z αϱ Z ϱ α Z = gϱ ) + ϱ V F ) ϱ Z αϱ Z α ϱ Z ln Z = gϱ ) ϱ gϱ ) ϱ + ϱ R,V α ϱ ) ln Z ϱ Z αϱ Z = α ϱ ln ϱ ln ϱ ) ) + R,ϱ α ln Z + ϱ R,V ϱ Z αϱ Z.

20 4 ASYMPTOTICS OF THE SOLUTIONS Similarly, we have λ R,V = λ V + λ V ) = ϱ ) + R,ϱ ) ϱ pv ) + pv ) ) = ϱ R,ϱ R,ϱ + ϱ Z 1) pv ) + pv ) ), which is what we wanted to show. Concerning 4.18), we only note ϱ = ϱ ) 1 + R,ϱ ). Lemma 4.7. For sufficiently small, we have the estimate 1 1 Zϱ R,ϱ ) ) α dx + R,ϱ L,1)) + λ R,V C. + L,1)) pv ) pv ) )) V V ) ) dx + λ β R,V ) 4.19) Proof. We multiply 4.18) with R,ϱ and integrate over [, 1]: Z ϱ R,ϱ ) ) 1 ϱ R,ϱ dx = α ϱ R ),ϱ 1 + R,ϱ ) ln1 + R,ϱ ) dx + ϱ R,V R,ϱ dx Z ϱ ) ϱ R,ϱ dx. Now we have s1 + s) ln1 + s) s / for s near, and therefore Zϱ R,ϱ ) ) α dx + ϱ R ),ϱ dx ) ϱ R,V R,ϱ dx Zϱ Zϱ R,ϱ ) ) dx + Oe c/ ), which brings us to 1 1 Zϱ R,ϱ ) ) α dx + R,ϱ L,1)) Next we multiply 4.17) with R,V / and integrate over [, 1]: λ R =,V R,V + pv ) pv ) )) V V ) ) dx ϱ R,ϱ R,V R,ϱR,V + ϱ Z 1)R,V dx, from which we then deduce that λ R,V L,1)) + λ β R,V )) + ϱ R,ϱ R,V dx + R,V L,1)) ϱ R,V R,ϱ dx + C. 4.) pv ) pv ) )) V V ) ) dx ) R,ϱ L,1)) + Z 1 L 1,1)). Adding this to 4.), R,V L,1)) C1/, Z 1 L 1,1)) = O) and Young s inequality then conclude the proof.

21 4.3 Refined Remainder Estimates 1 Proof of Theorem.. It suffices to show ) Z ϱ R,ϱ dx C. However, we van write ) 4 Z ϱ R,ϱ dx = Z ϱ R,ϱ Z dx + 4 Z R,ϱ dx, and now we make use of R,ϱ x) C 1/ on [, 4] and Zαx/) const on [4, 1]. Additionally, Z const on [, 1]. This completes the proof of Theorem.. Proof of Theorem.3. We start with Z R L,ϱ C Zϱ L R,ϱ = C R,ϱ,1)),1)) L,1)) C, 4.1) R L,ϱ L R,ϱ + L R,ϱ,1)),4)) 4,1)) 1/ R,ϱ L,4)) + C Z R,ϱ L 4,1)) C, 4.) Z R,ϱ C Zϱ L R C Zϱ,ϱ R,ϱ ) L Zϱ + C L R,ϱ L,1)),1)),1)),1)) C Zϱ R,ϱ ) L + C. 4.3),1)) To the equation 4.18), we choose its left-hand side without one factor ) as a multiplier: Z ) ϱ R,ϱ ) ) dx Z = α ϱ 1 + R,ϱ ) ln1 + R,ϱ ) Z ϱ R,ϱ ) ) dx + = α = α ϱ 1 + R,ϱ )R,V Z ϱ R,ϱ ) ) 1 dx ϱ 1 + R,ϱ ) ln1 + R,ϱ )) Z ϱ R,ϱ ) dx + Oe c/ ) ϱ 1 + R,ϱ )R,V ) Z ϱ R,ϱ ) dx α Z Z ) ) Z ϱ R,ϱ dx + Oe c/ ) 1 Z ϱ 1 Z ) Z ϱ R,ϱ ) ) dx Z Z ϱ Z ) Z ϱ R,ϱ ) ) dx Z ϱ 1 + R,ϱ ) ln1 + R,ϱ ) 1 + R,ϱ ) + 1)) Zϱ R,ϱ ) dx ϱ 1 + R,ϱ )R,V ) Zϱ R,ϱ ) dx αz ϱ Z ϱ R,ϱ ) ) dx. Z Zϱ Z ϱ R,ϱ ) ) dx Z

22 4 ASYMPTOTICS OF THE SOLUTIONS Now we have s ln s s + 1 = Os 1) ) for s near 1, hence ϱ 1 + R,ϱ ) ln1 + R,ϱ ) 1 + R,ϱ ) + 1)) C R,ϱ + Cϱ R,ϱ R,ϱ, and then 4.1) and 4.3) imply Z ϱ 1 + R,ϱ ) ln1 + R,ϱ ) 1 + R,ϱ ) + 1)) L,1)) C R Z ) L,ϱ Z R,ϱ + L R,ϱ,1)) L,1)),1)) ) C 1/4 + Zϱ R,ϱ ) L.,1)) We also have Z ϱ 1 + R ) L Z,ϱ )R,V C R,V W 1,1)) + C L R R,ϱ,1)),V,1)) L,1)) ) C + Zϱ R,ϱ ) L,,1)) Zϱ L,1)) C, αz ϱ L,1)) C, hence we conclude that Z ϱ R,ϱ ) ) ) + α Z ϱ R,ϱ Z L,1)) L,1)) Zϱ C 1/4 + Zϱ R,ϱ ) L R,ϱ ),1))) L +,1)) Oe c/ ) Zϱ + C + Zϱ R,ϱ ) L R,ϱ ),1))) L + C Zϱ R,ϱ ) L,,1)),1)) and therefore Z ϱ R,ϱ ) ) Z L,1)) which implies R,ϱ L 4,1)) C1/. ) + Z ϱ R,ϱ L,1)) C, Interpolating this estimate with 4.) then brings us to R L,ϱ 4,1)) C3/4, hence R,ϱ L 4,1)) C3/4.

23 By 4.16) we then have R,ϱ C [4,1]) C 5/4, hence R,ϱ L 4,1)) C 1/4, by interpolation. Now we go back to 4.13), which we write as T n ϱ x) ϱ ) ) ) = ϱ x) T n ln ϱ x) + V x) F V ξ) x ϱ t) dt ϱ ) x) + R,ϱx)). We choose x [ 3/4, 3/4 ] and find out that then ϱ )) = T n ϱ x) + O 3/4 ), hence also ϱ )) = T n ϱ ) + O 3/4 ). In the same way as during the proof of Proposition 4.5, we then show.4). 3 5 Numerical Simulations To find an approximate solution to.), we apply a difference method on an equidistant grid and solve the resulting nonlinear system by Newton s method. The obtained zero order approximate solution can be used as initial values when we attack 4.1) via a difference method on a grid which is refined near the gate and via Newton s method. We choose the following parameters: V GS T n T p λ β C ϱ B Then we determine V B from 1.7) by the condition that V 1) shall vanish, and F =.978 is then given by 1.9). Results for =.3 are in Figure 3. The numerically computed errors are as follows. n n) L n n) L V V) L V V) L A Appendix Lemma A.1. There is exactly one function Z C [, )) that solves.3), and all its derivatives Z k) y) decay exponentially for y.

24 4 A APPENDIX n * n n V * V Figure 3: Electron density and electric potential Proof. Assume that Z is a solution taking only non-negative values. The system to the differential equation Z = Z ln Z possesses the Hamiltonian Z y)) Z y) ln Zy) + Z y)/ as a conserved quantity. Sending y + brings us to Z ) = 1/, and we find that Zy) takes values only in [, 1). Then Z y) = 1 Z y) ln Z y) + 1 Z y)), Z yz) = dσ, Z < 1. σ ln σ + 1 σ σ= And for σ 1, we have σ ln σ + 1 σ = O1 σ)), 1 σ and therefore we can write yz) = ln1 Z) + σ= ) σ ln σ + 1 σ 1 dσ + O1 Z) 1 σ =: ln1 Z) + C Z + O1 Z), C Z.44, which brings us to 1 Zy) = e y e C Z 1 + Oe y ) ) valid for y. Acknowledgements Li Chen is partially supported by the National Natural Science Foundation of China NSFC), grant numbers and Michael Dreher is supported by DFG 446 CHV 113/17/-) and by the Center of Evolution Equations of the University of Konstanz.

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