数理解析研究所講究録
1495 巻 2006 年 51-59

51

A Numerical Scheme for Quantum Hydrodynamics
in a Semiconductor
大阪大学サイバーメディアセンター

小田中紳二 (Shinji Odanaka)

Computer-Assisted Science Division, Cybcrmcdia Center, Osaka University
Dcdicated to Professor Tai-Ping Liu on the occasion of his 60th birthday

Abstract
We study existence and numerical approximation of solutions to the quantum
drift-diffusion equation arising in a sellliconduc ,tor at the room temperature(i.e. at
the high tcmperature). We obtain thc existence of stational solutions by means of anumerically motivated solution map, introducing the gencralized chemical potetltial
i-FerIni-levcl). This
leads to an iterative solution method of the QDD
equation coupled with the Poisson equation. An implicit scmidiscretization of thetime-dependent QDD equation is discusscd. The new discretization in space of the
QDD equation is proposed.
$\cdot$$($$\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}t1\kappa^{\backslash }\mathrm{J}1$$\mathrm{q}\mathrm{u}\mathrm{a}_{\iota}\mathrm{s}$1Introduction. at
The semiconductor transport in ultra-small structures at the room
) is modeled by the quantum hydrodynamics. The quantum driftthe highcliffusion model is viewed as one of the hicrarchy of the quantum hydrodynamic models,
which is derived from a momcnt expansion of Wigncr-Boltzmann equation including collisional effects[l]. It is known that this model is a quantum corrected version of the classicalcorrections to the stress tensor[2]. Wc study the QDD
drift-diffusion model, with
with piecowise smooth boundary, assuming
system in a bounded domain
Boltzmann statistics.
$\mathrm{t}\mathrm{e}\iota \mathrm{n}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}(\mathrm{i}.\mathrm{e}$$\mathrm{t}\mathrm{e}\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{C}^{-\backslash },$$\mathrm{O}(\hslash^{2})$$\Omega\subset R^{d}(d\geq 1)$$\epsilon\Delta\varphi=n-f$(1),$\partial_{t}n+div(n\nabla(\varphi-ln(n)+\lambda^{2}\frac{\Delta\sqrt{n}}{\sqrt{n}}))=0$.

(2)

Here is the electron density and the electrostatic potential . is the density of ionized
impurities. is the semiconductor permittivity. A is the scaled Planck constant.
The fourth-order continuity equation (2) for the electron dcnsity is split into two
second-order equations, using the generalized chemical potcntial$\mathrm{f}$$\mathrm{n}$$\varphi$$\epsilon$$(\mathrm{q}\iota \mathrm{l}\mathrm{a}\mathrm{s}\mathrm{i}- \mathrm{F}\mathrm{c}\mathrm{r}\mathrm{m}\mathrm{i}- \mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l})v=$

52$\varphi-\ln(n)+\lambda^{2}\frac{\Delta\sqrt{n}}{\sqrt{n}}[2]$.
$\epsilon\Delta\varphi=n-f$,(3)$\partial_{t}n+div(n\nabla v)=0$,$\lambda^{2}\frac{\Delta\sqrt{n}}{\sqrt{n}}-\ln(n)+\varphi=v$

(4)

.

(5)

In this paper we present the cxistence of stational solutions by means of a numericallymotivated solution map, which leads to an iterative solution method of the QDD system.
An implicit semidiscretization of the time-dependent QDD, which results in theelliptic system, is discussed. The new discretization in space of the elliptic system is
proposed.
$6\mathrm{q}\iota \mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$2Existence of a solution and an iterative solution
methodThe analysis for the steady statc of the QDD model was performed by a variational
approach in the previous. We obtain the existcnce of stational solutions bymeans of nulnerically motivated solution lll\‘app. The result provides an iterative solution
method of the QDD system. The analysis of stational solutions is })on the following
assumptions.
, or
(A.1)
is a bounded domain with piecewise smooth boimdary.
(A.2)
is a set of measure zero.
(A.3) There exists for all
and alla function $u\in W^{1,q}(\Omega)$with$\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{s}[3],[4]$$\mathrm{a}_{\iota}\mathrm{s}\mathrm{t}^{\backslash },\mathrm{c}1$$\Omega\subset R^{d},$$\mathrm{d}=3$$\mathrm{d}=1,2$$\partial\Omega\backslash (\partial\Omega_{D}\cup\partial\Omega_{N})$$a,$ $u_{D}\in W^{1,q}(\Omega)$$f\in L^{\infty}(\Omega)$$div(a\nabla u)=f,$(6)$u-u_{D}\in H_{0}^{1}(\Omega\cup\partial\Omega_{N})$$||u||_{W^{1,q}(\Omega)}\leq C(||u_{D}||_{W^{1,q}(\Omega)}+||f||_{L^{\infty(\Omega)}})$.

(7)(St) $-L^{p}(\Omega)$ andHere $q\in(2, \infty]$ withsuch that.
(A.4) The Diriehlet boundary condition data
, on
. The Neumann boundary conditions
, on
.
(A.5)
.
We cmploy the exponetial transformation of variables
to construct thesolution mapping. (5) is replaced by the equivalent form (9). In this case, if is uniformly
bounded, the positivity of the root-density $\rho=e^{u}$ is ensured.
$p,$$H^{1}$$\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$$H^{2}(\Omega)\mathrm{L}arrow W^{1,q}(\Omega)$$(\varphi_{D}, v_{D}, u_{D})\in(H^{1}(\Omega)\cap L^{\infty}(\Omega))^{3}$$\partial\Omega_{D}$$\frac{}\partial\varphi}{\partial_{l\text{ノ}}=\frac{\partial v}{\theta\nu}=\frac{\theta u}{\theta\nu}=0$$\partial\Omega_{N}$$f\in L^{\infty}(\Omega)$$\rho=\sqrt{n}=e^{\mathrm{u}}$$\mathrm{t}1$$\epsilon\Delta\varphi$$=$$e^{2u}-f$ ,$-\lambda^{2}\nabla(\rho\nabla u)+\rho u$$=$$\frac{\rho}{2}(\varphi-v)$$-div(n\nabla v)$

$=$

$0$

.

(8)
in Sl,

(9)

(10)

53
The system is supplelrlentcd with the set of boundary conditions:$\varphi=\varphi_{D},$$u=u_{D},$ $v=v_{D}$ ,on$\nabla\varphi\cdot\nu=\nabla u\cdot\nu=\nabla v\cdot\nu=0$$\partial\Omega_{D}$

on

(11)

,

$\partial’\Omega_{N}$

(12)

.

To solve iteratively, we dcfine the following solution mapping. Let
in$\Omega$$\mathrm{w}\in L^{2}(\Omega)$be given.

Algorithm 2.1 (Solution map)$(Pl)$

Solve$\epsilon\Delta\varphi=e^{2w}-f$

(13)

,

subj$ect$ to the boundary $c,ondition,s(ll),$ (12) for .
$\varphi$$(P\mathit{2})$Solve$-div(e^{2u)}\nabla v)=0$

subject to the boundary $c,onditions(ll),$ (12) for
Solve$\mathrm{t}\mathit{1}$

(14)

,

.

$(P\mathit{3})$$-2\lambda^{2}\nabla(e^{w}\nabla u)+2e^{w}u=e^{w}(\varphi-v)$

subject to the$boundan/(jond,itions(\mathit{1}\mathit{1}),(\mathit{1}\mathit{2})$

(15)

,

for .
$u$

It readily follows from Lax-Milgram theorem that there exists a unique solution toeach linear problem. Thus the mapping $T:Xarrow X,$ $T(w)=u$ , with $X=\{w\in L^{2}(\Omega)$ :, wc define the cut-off functionin }, is well defined. )$-\underline{U}\leq w\leq\overline{U}$$\Omega$$\mathrm{F}\mathrm{e}$$\mathrm{r}\overline{U},$if$\overline{U}$$Py:=\{$$L^{\infty}$$y>\overline{U}$

.

(16)

$\mathrm{i}\mathrm{f}-\underline{U}\leq y\leq\overline{U}$$y$$-\underline{U}$To obtain the(17)$\underline{U}>0$

if$y<-\underline{U}$and , we employ the following truncatcd system ofbounds for-off function.
$\varphi,$$\iota x$$\mathrm{v}$$-(19)\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{t})\mathrm{y}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{t}$$\epsilon\Delta\varphi$$=$$-\lambda^{2}\nabla(e^{\mathit{1}^{\mathrm{J}}u}\nabla u)+e^{Pu}u=$$-\nabla(e^{2Pu}\nabla v)$$=$$e^{2Pu}-f$$\frac{e^{P\mathrm{u}}}{2}(\varphi-v)$

$0$

,

(17)

,

,

(18)
(19)

subject to the boundary conditions (11) $-(12)$ . The proof of existence of solutions to thetruncated system is based on the following a priori estimates. The result is obtained byapplying Stampacchia’s lemma[5] and maximum principle type arguments.

54
Lemma 2.1 (A priori estimate) Let
the truncated system. Then there exist positive$(\varphi, v, u)\in(H^{1}(\Omega)\cap L^{\infty}(\Omega))^{3}$$-\underline{\varphi}\leq\varphi\leq\overline{\varphi},$

be a weak solution tosuch that$\overline{U}$$constr|,ants\underline{\varphi},$$-\underline{v}\leq v\leq\overline{v},$$\overline{\varphi},$$\underline{v},\overline{v},$$\underline{U},$

(20)

in S)

$-\underline{U}\leq u\leq\overline{U}$where$\overline{\varphi}$$=$$\varphi_{0}+c_{1}(\Omega, \epsilon)||f||,$$c_{1}(\Omega)>0$$\varphi_{0}+c_{2}(\zeta], \epsilon)(e^{2\overline{U}}+||f||_{\infty}),$$\underline{\varphi}=$$\overline{v}=\underline{v}=v_{0}$$\overline{U}$$=$$\underline{U}=$$\varphi_{0}$Let$\psi\geq\varphi_{0}$$c_{2}(\Omega)>0$

(22)
(23)

,

,

$\max(\frac{\overline{\varphi}+\underline{v}}{2}, U_{0})$

,

(24)

’

(25)

$\max(, U_{0})\underline{\underline{\varphi}+\overline{v}}2$$=$$v^{11}\mathrm{s}_{\Omega_{D}}\mathrm{p}|\varphi|,$

Proof.

(21),$v_{0}= \sup_{\theta\Omega_{D}}|v|,$ $U_{()}= \sup_{\prime,d\Omega_{D}}|u|$. Selecting the test function$\int_{\Omega}\epsilon|\nabla(\varphi-\psi)^{+}|^{2}dx$$(\varphi-\psi)^{+}$

.

(26)

, where $t^{+}= \max(t, 0)$ ,

(27)

$=$

$-. \int_{\Omega}(e^{2Pu}-f)(\varphi-\uparrow/))^{+}dx$$\leq$$||f||_{\infty}||(\varphi-l/))^{+}||_{1}(meas(\varphi>\psi))^{1/2}$

.

(28)

By a Poincare type inequality, we obtain$C_{1}(\Omega)||(\varphi-\uparrow/))^{+}||_{1}\leq\epsilon^{-1}||f||_{\infty}(meas(\varphi>\psi))^{1/2},$$C_{1}>()$

Let $2<r\leq 6$ . By Sobolev’s imbedding theorem, it follows for all

C2$(\Omega)^{r}||(\varphi-\uparrow/))^{+}||_{1}^{r}\geq$$\eta>\psi\geq\varphi_{0}$(30)$||(\varphi-\psi)^{+}||_{0,r}^{r}$$\geq$$\int_{\{\varphi>\eta\}}(\varphi-\psi)^{r}dx$$\geq$$\int_{\{\varphi>\eta\}}(\eta-\psi)^{r}dx$

(31)
(32)

.

(33)

,

meas
where

(29)

$\eta>\psi$$(\eta-\uparrow/))^{r}m\mathrm{e}as(\varphi>\eta)$

For.

$K=C(\Omega)^{r}\epsilon^{-r}$llfll . Applying Stampacchia’s lemma for$f\infty$$g(\varphi_{0}+\psi^{*})=0$(34)$(\varphi>\eta)\leq K(\eta-\psi)^{-r}(meas(\varphi>\mathrm{t}/)))^{\mathrm{r}/2}$with th’$=\mathrm{c}_{1}(\Omega, \epsilon)||f||_{\infty}$, where$g(\eta)=meas(\varphi>\eta)$$c_{1}(\Omega, \epsilon)=C(\Omega)\epsilon^{-1}$

.

:
(35)

55
We obtain
a lower bound for .
lower bound for .

. We need a upper bound for$\varphi\leq\varphi_{0}+\psi*=\varphi_{0}+c_{1}(\Omega, \epsilon)||f||_{\infty}\equiv\overline{\varphi}$$\mathrm{L}\mathrm{c}\mathrm{t}\overline{U}=\max(\frac{\overline{\varphi}+\underline{v}}{2}, U_{0})$$\varphi$. For$u\geq\overline{U}$to find, the maximum principle gives a$\mathrm{u}$$\mathrm{v}$$0= \int_{\Omega}-div(e^{2Pu}\nabla v)(-v-\underline{v})^{+}$

We obtain$v\geq-\underline{v}$. Selecting the test function$=$$. \int_{\Omega}e^{2Pu}|\nabla(-v-\underline{v})^{+}|^{2}dx$

(36)

$=$

$\int_{\Omega}e^{2Pu}|\nabla(-v-\underline{v})^{+}|^{2}dx$(37)$\geq$$\int_{\Omega}e^{2\overline{U}}|\nabla(-v-\underline{v})^{+}|^{2}dx$(38)$(u-\overline{U})^{+}\in H_{0}^{1}(\Omega\cup\partial\Omega_{N})$

,

.

(39)

$\int_{\Omega}\lambda^{2}e^{Pu}|\nabla(u-\overline{U})^{+}|^{2}dx+\int_{\Omega}e^{Pu}(2u-(\varphi-v))(u-\overline{U})^{+}dx=0$

Then we have the following estimates$\int_{\Omega}e^{P\mathrm{u}}(2u-(\varphi-v))(u-\overline{U})^{+}dx\geq\int_{\Omega}e^{\overline{U}}(2\overline{U}-(\overline{\varphi}+\underline{v}))(u-\overline{U})^{+}dx\geq 0$

It follows that$u\leq\overline{U}$. Using$(-\varphi-\psi)^{+}with$$\int_{\Omega}\epsilon|\nabla(-\varphi-\uparrow/))^{+}|^{2}dx$Cb$\geq\varphi_{0}$

.

(40)

,

$=$

$\int_{\Omega}(e^{2Pu}-f)(-\varphi-\psi)^{+}dx$(41)$\leq$$\int_{\Omega}(e^{2\overline{U}}+||f||_{\infty})(-\varphi-\psi)^{+}dx$(42)

We recall the Stampacchia’s lemma.

measThis implies
Similarly we obtainfunction$(-\varphi>\varphi_{0}+\psi^{*})=\mathrm{t}),$ $\psi^{*}=e^{2\overline{U}}+c_{2}(\Omega, \epsilon)||f||_{\infty}$$\varphi\geq-(\varphi 0+\uparrow/\text{ノ^{}*})\equiv-\underline{\varphi},$$v\leq\overline{v}$$\psi^{*}=e^{2U}+c_{2}(\Omega, \epsilon)||f||_{\infty}$

. Let

.

(43)

$\underline{U}=\max(^{\varphi v}\pm_{2^{-}}, U_{0})$.

from the maximum principle arguments. Applying the test
, we obtain.

$(-u-\underline{U})^{+}\in H_{0}^{1}(\Omega\cup\partial\Omega_{N})$$\square$$u\geq-\underline{U}$Theorem 2.1 (Existence of a weak solution) There exists a solution of the boundary
value problem (17)utth th, boundary conditions (11) and (12) in the regularity class$-(\mathit{1}\mathit{9})$of$H^{1}(\Omega)$A$L^{\infty}(\Omega)$$e$.

Proof. The mapping is well-defined. It can }) scen that the mapping is continuous and compac, . In fact,
is bounded. Selecting the test fimction $u-u_{D}\in$,$\mathrm{T}$$\mathrm{T}$$(^{\backslash }$$\mathrm{t}$$H_{0}^{1}(\Omega\cup\partial\Omega_{N})$$||u||_{1}$,$\int_{\Omega}\lambda^{2}e^{Pw}|\nabla(u-u_{D})|^{2}dx+\int_{\Omega}\lambda^{2}e^{Pw}\nabla u_{D}\nabla(u-u_{D})dx+/\Omega e^{Pw}u(u-u_{D})dx$$\int_{\Omega}\frac{e^{Pw}}{2}(\varphi-v)(u-u_{D})dx$$C_{1}||u-u_{D}||_{1}^{2}$

.

(44)

(45)

$\leq$$\int_{\Omega}e^{Pw}|\nabla(u-u_{D})|^{2}dx$$\leq$$C_{2}e^{\overline{U}}( \overline{U}+\frac{\overline\varphi+\overline{v}}{2}+||u_{D}||_{1})||u-u_{D}||_{1}$

.

(46)

56
bounds for and . The Scha.uder
Similarly, by (13) and (14) we conclude uniform
to (8)
fixed point theorem gives the existcnce of a fixcd point, , a solution(10).
$H^{1}$$\mathrm{v}$$\varphi$$(\varphi, u, v)$$\mathrm{i}.\mathrm{e}.$$-$$\square$Lemma 2.2 (Pinnau) The quanturn operator-norm.
to the$A(n)=- \frac{\Delta\sqrt{n}}{\sqrt{n}}$is rnonotonic utth respect$L^{p}(\Omega)$$n_{2})$

Proof. The operator A(n) is Gate\^aiix differetiable. For $h\in[0,1]$ , let $\xi:=n_{1}-h(n_{1}-$. By the mean value theorem,$\langle A(n_{1})-A(n_{2}), n_{1}-n_{2}\rangle=\langle A’(\xi), n_{1}-n_{2}\rangle$

.

(47)

The straight forward calculation gives$\langle A’(\xi), n_{1}-n_{2}\rangle=\int_{\Omega}\xi|\nabla(\frac{n_{1}-n_{2}}{\xi})|^{2}dx$

.

(48)

From [4, lemma 24], there exists a constant $C>0$ such that$\langle A’(\xi), n_{1}-n_{2}\rangle\geq C||n_{1}-n_{2}||_{\int_{\lrcorner}\mathrm{p}}^{2}$

where

$p,$

$q\in(2, \infty]$

satisfies$\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$Theorem 2.2 The mapping

Proof.

Let$w_{1},$$w_{2}\in X=$

solution of$T${$(4^{(}\mathit{3}.)$.

$\square$is a contraction with respect to the$w\in L^{2}(\Omega)$:$-\underline{U}\leq w\leq\overline{U}$$-\nabla(e^{2w}\nabla v)=0$

Taking the difference and.

$\mathrm{e}1^{-}\mathrm{n}\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{H}_{\acute{\mathrm{t}})}1\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{r}’ \mathrm{s}$in$\Omega$} and let.

$\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$$(u_{i}, \varphi_{i}),$$i=1,2$ be theinequality, we get(51). We have from the elliptic estimate$||v_{1}-v_{2}||_{H^{1}}\leq C||v_{D}||_{W^{1,q}}||w_{1}-w_{2}||_{L^{\mathrm{p}}}$Let$v_{i},$.

(50)$||v_{1}-v_{2}||_{\mathit{1}\mathit{1}1}\leq C||\nabla v_{2}||_{L^{q}}||e^{2w_{1}}-\mathrm{e}^{2w_{2}}||_{L^{\mathrm{p}}}$where$IP(\Omega)- nor\pi\iota$

.

(52)

$i=1,2$ bc the solution of$\epsilon\Delta\varphi=e^{2u}-f$

(53)

,

$- \lambda^{2}\frac{\nabla(\rho\nabla u)}{\rho}+u=\frac{1}{2}(\varphi-v),$$\rho=e^{u}$

.

(54)

57
Selccting the test fuction$n_{1}-n_{2}\in H_{0}^{1}(\Omega\cup\partial\Omega_{N})$,$0=- \lambda^{2}\int_{\mathrm{r}\iota}(\frac{\nabla^{2}\rho_{1}}{\rho_{1}}-^{\underline{\nabla}_{A\underline{2}}^{2}}\rho_{2})(n_{1}-n_{2})dx+\frac{1}{2}\int_{\Omega}(\ln(n_{1})-\ln(n_{2}))(n_{1}-n_{2})dx$$- \frac{1}{2}\int_{\Omega}(\varphi_{\perp}-\varphi_{2})(n_{1}-n_{2})dx+\frac{1}{2}.\int\Omega(v_{1}-v_{2})(n_{1}-n_{2})dx$$=I_{1}+I_{2}+I_{3}+I_{4}$

.

(55)
(56)

We estimate term wise.
$I_{1}$$=$$\langle A’(\xi), n_{1}-n_{2}\rangle\geq C||n_{1}-n_{2}||_{L^{p}}^{2}$$I_{2}$$\geq$$0$$I_{3}$$=$$\int_{\Omega}|\nabla(\varphi_{1}-\varphi_{2})|^{2}dx\geq \mathrm{t})$$I_{4}$$\leq$$||v_{1}-v_{2}||_{L^{2}}||n_{1}-n_{2}||_{L^{\mathrm{p}}}$

(57)
(58)

,

,
,

(59)

,

(60)

Combining these estimates, we have$||u_{1}-u_{2}||_{L^{\mathrm{p}}}\leq C||n_{1}-n_{2}||_{L^{\mathrm{p}}}\leq||v_{1}-v_{2}||_{H^{1}}$

.

(61)

From (52), we finally obtain$||u_{1}-u_{2}||_{L^{\mathrm{p}}}\leq C||v_{D}||_{W^{1,q}}||w_{1}-w_{2}||_{L^{\mathrm{p}}}$

The a.sscrtion follows by choosing 1

3$v_{D}||_{W^{1_{\mathrm{I}}q}}\leq V_{0}<1$

.

(62)

small enough.

$\square$Discretization in time

. For $k\in N$ ,Let $\tau_{k}=t_{k}-t_{k-1}>0$ be the time step. Set the initial condition
we discretize (3),(4) and (9), cmploying an implicit time discrctization by a backward
Euler scheme as follows.
$\rho_{0}=\sqrt{n_{0}}$$\frac{n^{k}-n^{k-1}}{\tau_{k}},+div(n^{k}\nabla v^{k})=0$

(63)

,

$-2\lambda^{2}\nabla(\rho^{k}\nabla u^{k})+2\rho^{k,}u^{k}=\rho^{k}(\varphi^{k}-v^{k})$$\epsilon\Delta\varphi^{\mathrm{k}}=n^{k}-f$

,

(64)
(65)

,

subject to the simplified boundary conditions:$\varphi^{k}=0,$ $u^{k}‘=u_{D},$ $\nabla v^{k}\cdot v=0$

on$\partial\Omega$

.

(66)

. As pointed out in the
The transient problcm is rcplaced by a sequence of elliptic
previous works[6], the following entropy dissipation property is presonted.This provides
the stability boimds of thc approximate solutions.
$\mathrm{p}\mathrm{r}\mathrm{o}\}_{)}1\mathrm{e}\mathrm{m}\mathrm{s}$58Lemma 3.1 (Discrcte entropy estimate). Let
subject to the boundary conditions
solution of (63)(free energy) estimates holds:

be a. Then the following entropy$(\varphi^{k}, u^{k}, v^{k})\in H_{0}^{1}((l)\cross H^{1}(\zeta l)\cross H^{1}(\Omega)$$-(\mathit{6}\mathit{5})$$(\theta \mathit{6})$(67)$S^{k}\leq S^{k-\perp}$

where$S^{k}= \lambda^{2}\int_{\Omega}|\nabla\rho^{k}|^{2}dx+/\Omega(n^{k}(\ln(n^{k})-1)+1)dx+\frac{\epsilon}{2}\int_{\Omega}|\nabla\varphi^{k}|^{2}dx$

Proof.

.

The straightforward calculation using the chemical potential
tion gives the entropy estimates.

4$v^{k}$

as a test$\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{I}\mathrm{l}\mathrm{C}^{\backslash },-$$\square$Discretization in space

Finally, we propose a new nonlinear scheme to the QDD equation[7] for discretizationdifference schemes. The
in space. The QDD equation is discretized by. Assuming
simulation region is partitioned into cornputational cells , i.e.,$J=e^{G}\nabla\eta$$\eta=exp(-v)$, and the fluxin (63), where
andthat the flux
$F=\rho\nabla u$ in (64), we obtain by Green’s formula over each computational cellthat$\mathrm{e}\cdot \mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$$\Omega$$\Omega=\bigcup_{i}\Omega_{i}$$\Omega_{i}$$G= \varphi+\lambda^{2}\frac{\Delta\rho}{\rho}$$\Omega_{i}$$\int_{\Omega_{j}}n_{t}dx-\int_{\partial\Omega_{i}}e^{G}\frac{\partial\eta}{\partial\nu}ds=0$$J_{\partial\Omega}.

The numerical fliixes$J_{i+1/2}$

,

(68)

\lambda^{2}\rho\frac{\partial u}{\partial\nu}ds-\int_{\Omega}.\rho udx=-\frac{1}{2}\int_{\Omega}.\rho(\varphi-v)dx$and$J_{i+1/2}$or$F_{i+1/2}$

.

(69)

yield

$F_{\iota’+1/2}= \frac{\theta_{i+1}-\theta_{t}}{\int_{\Omega_{1}}e^{-\theta}dx},$$\theta=G$

or .
$u$

(70)

lcads to a class ofSubstituting in (68) and (69) the average fluxes
andconservative schemes, following Tikhonov and Samarskii in [8]. In this case, the key)
integration of the exponetial function with respect to thc
ingredient is to find an
potentials. Assuming the piecewise-linear representation of , wc have$F_{1\pm 1/2}$$J_{1\pm 1/2}$$1\mathrm{i}(^{\backslash }.\mathrm{i}\mathrm{t}$$\mathrm{e}\mathrm{x}\iota$$\theta$$\int_{\Omega}.e^{-\theta}dx=\frac{he^{-\theta_{i}}}{B(\theta_{i+1}-\theta_{i})}$where$\mathrm{B}$(71)

is the Bernoulli function. This yields the fully discrete system as follows.

$\frac{n_{i}^{k}-n_{i}^{k-1}}{\tau_{k}}=.\frac{B(G_{i+1}^{k}-G_{i}^{k})n_{i+1}^{k}-(B(G_{i}^{k}-G_{i+1}^{k})+B(G_{i}^{k}-G_{i-1}^{k,}))n_{i}^{k}+B(G_{i-1}^{k}-G_{i}^{k})n_{i-1}^{k}}{h^{2}}.$(72)$- \frac{\rho_{i+1}^{k}B(u_{\mathrm{t}+1}^{k}-u_{i}^{k})(u_{i+1}^{k}-u_{i}^{k})-\rho_{i}^{kr}B(u_{i}^{k}-u_{i-1}^{k,})(u_{i}^{k}-u_{i-1}^{k})}{h^{2}}.+\Lambda_{i}^{k}u_{i}^{k}=\frac{\Lambda_{i}^{k}}{2}(\varphi_{i}^{k}-v_{i}^{k})$

,

(73)

. This nonlinear scheme is a consistent generalization of the Scharffeterwhere
the classical drift-diffusion equation[7].
schemetoGummel$\Lambda=\int_{\Omega}.\rho dx$,

59

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