Quantum–Corrected Drift–Diffusion Models forTransport in Semiconductor Devices

aDipartimento di Matematica “F.Enriques”, Universita degli Studi di Milano,via Saldini 50, 20133 Milano, Italy

b DEI–Dipartimento di Elettronica e Informazione, Politecnico di Milano,piazza Leonardo da Vinci 32, 20133 Milano, Italy

c MOX–Modeling and Scientific Computing, Dipartimento di Matematica “F.Brioschi”,Politecnico di Milano, via Bonardi 9, 20133 Milano Italy

Carlo de Falcoa Emilio Gattib Andrea L. Lacaitab

Riccardo Saccoc

Abstract

In this article, we propose a unified framework for Quantum–Corrected Drift–Diffusion(QCDD) models in nanoscale semiconductor device simulation. QCDD models are pre-sented as a suitable generalization of the classical Drift–Diffusion (DD) system, each par-ticular model being identified by the constitutive relation for the quantum–correction tothe electric potential. We examine two special, and relevant, examples of QCDD models;the first one is the modified DD model named Schrodinger–Poisson–Drift–Diffusion, andthe second one is the Quantum–Drift–Diffusion QDD model. For the decoupled solutionof the two models, we introduce a functional iteration technique that extends the classicalGummel algorithm widely used in the iterative solution of the DD system. We discuss thefinite element discretization of the various differential subsystems, with special emphasison their stability properties, and illustrate the performance of the proposed algorithms andmodels on the numerical simulation of nanoscale devices in two spatial dimensions.

Key words: Quantum and Drift–Diffusion Models, Density–Gradient,Schrodinger–Poisson, Functional Iterations, Finite Element Method, NanoscaleSemiconductor Devices

1 Introduction and Motivations

The accurate prediction of the electrical behavior of up-to-date nanoscale semicon-ductor devices demands for the inclusion of quantum effects, such as, for example,increased equivalent oxide thickness due to strong electron confinement at silicon-silicon dioxide interface or direct tunneling through the channel potential barrier

Preprint submitted to Journal of Computational Physics 11th May 2004

[1,2,3]. As these effects cannot be appropriately dealt with by the classical Drift–Diffusion (DD) model, alternative and more sophisticated mathematical modelsmust be adopted. In this respect, two main approaches can be taken:

(i) employing a full quantum description of charge transport;(ii) adding a suitable correction to the basic DD model to include quantum electro-

statics.

The first approach typically relies on quantum models based on the non-equilibriumGreen’s function [4] or the Wigner function [5]. Another model is adopted in [6]to study the behaviour of devices in the quantum ballistic limit and requires strongassumptions on the wave functions in the metal leads. All of these models provide avery accurate and complete physical information on the quantum mechanical phe-nomena occurring in the device. However, a certain lack of numerical robustnessand the intensive computational cost make these models still unsuitable for routineindustrial semiconductor device simulation. These considerations strongly prompttowards investigating the approach (ii). This latter has the advantage to exploit allthe benefits arising from the well established mathematical and numerical experi-ence on the basic DD model, allowing at the same time to design a state-of-the-artsimulation tool.

In this work we focus on two relevant modifications of the basic DD system,namely, the Quantum–Drift–Diffusion (QDD) model proposed in [7] and the Schro-dinger–Poisson–Drift–Diffusion (SPDD) model proposed in [8].

The QDD model emanates from a self-consistent derivation of a generalized equa-tion of state for the electron gas which includes a dependence on the gradient ofelectron density. This, in turn, allows to incorporate quantum phenomena descrip-tion into the classical DD model by means of a quantum correction to the electricpotential, the so calledBohm potential[9].

The SPDD model is based instead on afully two-dimensionalconsistent solutionof the Schrodinger–Poisson subsystem within a subregion of the semiconductordevice, and incorporates quantum effects into the classical DD framework by meansof a modified density of states. The SPDD model is more accurate than previoussimulation schemes based on the coupling of the DD model with a one-dimensionalsolution of the Schrodinger equation [1,10], requiring at the same time a reducedcomputational cost compared to the solution of a full quantum model.

In this article, we show that the modification of the density of states in the SPDDmodel can indeed be regarded as asuitable quantum correction of the electricpotential in the DD current relation. This allows to interpret the SPDD modelas a variation of the QDD model, and, more generally, to view both the SPDDand the QDD models as special instances of a unified mathematical frameworkfor nanoscale semiconductor device simulation, denoted henceforth as Quantum–

2

Corrected Drift–Diffusion (QCDD) approach. This latter family of models thusrepresents a suitable generalization of the classical Drift–Diffusion (DD) system,each particular model being identified by the constitutive relation for the quantum–correction to the electric potential.

Following the derivation of a unified framework for QCDD transport models, wepropose to apply a functional iteration technique, which is customarily and success-fully used in standard DD-based semiconductor device simulation, to the decouplednumerical solution of the nonlinear boundary value problems deriving from boththe SPDD and QDD models. The iteration procedure is a generalized Gummelalgorithm [11,12,13] and, to our knowledge, represents the first fully–decoupledfunctional iteration procedure applied to the QDD model (see also [14,15] forthe use of other partially decoupled iterative maps). The advantage of adoptinga Gummel-type iteration is twofold: on the one hand, it considerably saves compu-tational effort and memory storage at each step, which is of relevant importance inmultidimensional simulations; on the other hand, it leads to successively solving el-liptic quasi-linear and linear boundary value problems for which efficient and stablediscretization methods can be employed [16,17]. Moreover, the performance of theiterative procedure can be properly improved by resorting to suitable accelerationtechniques, for example, Newton-Krylov subspace iterations [18,19].

The outline of the article is as follows. In Sect. 2 we introduce the DD, SPDD andQDD transport models, while in Sect. 3 we present a generalized framework forQCDD models, that encloses as special cases both the SPDD and QDD models.Then, in Sect. 4 we introduce a functional iteration to construct a decoupled algo-rithm for the iterative solution of the SPDD and QDD systems, and in Sect. 5 wediscuss the finite element discretization of the various differential subproblems ob-tained after decoupling. The detailed description of the numerical algorithms usedin the computations is discussed in Sect. 6. Finally, in Sect. 7 we illustrate andcompare the performance of the proposed algorithms and models on the numeri-cal simulation of nanoscale devices in two spatial dimensions. Some concludingremaks and perspectives on future work are addressed in Sect. 8.

2 Physical Models

In this section we present the classical DD transport model and introduce the no-tation that will be used for the SPDD and QDD models. For ease of presentation,we develop in detail the derivation of the models only for electron carriers, as asimilar treatment holds for hole carriers. Throughout the section, we shall assumethat the device domainΩ is an open bounded set divided into two subregions, asemiconductor region,ΩSi, and an oxide region,ΩOX , such thatΩ = ΩSi ∪ ΩOX .

3

In each of these two regions we consider the material to be homogeneous andisotropic; this implies, in particular, that the electric permittivityε is a scalar piece-wise constant quantity overΩ. The above device structure is an appropriate modelfor a MOSFET (Metal–Oxide–Semiconductor Field–Effect Transistor), one of themost widely used component in nowadays semiconductor device technology (seeSects. 2.2, 3.2 and 7 for a more detailed description).

2.1 The DD Model

The classical DD model is the zeroth order moment in the expansion of the Boltz-mann Transport Equation (see [5,13]) and, in the stationary case, it consists of thefollowing system

(1)

div Jn = qU in ΩSi

div Jp = −qU in ΩSi

− div (ε∇ϕ) = q (p− n+D) in Ω,

whereq > 0 is the electron charge,Jn andJp are the electron and hole currentdensities,D is the net doping profile in the device andU is the net recombinationrate. In (1), the first two equations represent the continuity equations for electronsand holesn andp, while the third equation is the Poisson equation for the electro-static potentialϕ. The issue of a suitable set of boundary conditions for (1) will beaddressed in Sect. 3.2.

The following constitutive relation for the electron current densityJn in terms ofnand of the electron quasi–Fermi potentialϕn can be derived through a linearizationof the distribution function in phase space around its equilibrium value (see [11],Ch. 2)

(2) Jn (x) = −q µn n (x) ∇ϕn (x) ,

wherex denotes the spatial coordinate of a point in the device domain, andµn isthe electron mobility. In (2),ϕn describes the deviation of the distribution functionfrom its equilibrium value and it reduces to the constant Fermi potentialϕn0 at equi-librium when no current flow is expected. The electron densityn can be expressedas

(3) n (x) =

+∞∫Ec

g(E,Ec (x))f(ϕn (x) , E)dE in ΩSi,

whereE denotes energy,g(E,Ec (x)) is the density of states,f(ϕn (x) , E) is theoccupation probability density andEc(x) is the energy at the bottom of the conduc-tion band.

4

Let x be a fixed point inΩSi. Then, for a non–degenerate semiconductor and as-suming a single parabolic energy band, we have

(4) g(E,Ec (x)) =1

2π2

(2m∗

n

~2

)3/2

(E − Ec (x))12 , E ≥ Ec (x) ,

and

f(ϕn (x) , E) = exp

(−E + qϕn (x)

kbT

),

wherem∗n is the (scalar) electron effective mass,~ is the reduced Planck constant,

kb is the Boltzmann constant andT is the lattice temperature. The computation ofthe integral in (3) yields

(5) n (x) = Nc exp

(−qϕn (x)− Ec (x)

kbT

),

whereNc is the effective density of states in the conduction band.

Let us introduce the intrinsic carrier concentration

nint =√NcNv exp

(−EGap

2kb T

)in ΩSi,

whereEGap is the (constant) energy band gap andNv is the effective density ofstates for holes of scalar effective massm∗

p in the valence band. Then, relation (5)can be written as

(6) n (x) = nint exp

(ϕ (x)− ϕn (x)

Vth

)in ΩSi,

whereVth =kbT

qis the thermal potential and

(7) ϕ (x) = −Ec (x) + Ev (x)

2 q+ Vth ln

(√Nc/Nv

),

Ev (x) = Ec (x)−EGap being the energy at the top of the valence band. Replacing(6) into (2) yields the familiar DD expression of the electron current density

Jn (x) = −qµn (n (x)∇ϕ (x)− Vth∇n (x)) .

Repeating the above procedure for the hole densityp, yields the following expres-sion of the hole current density

Jp (x) = −qµp (p (x)∇ϕ (x) + Vth∇ p (x))

whereµp is the hole mobility.

5

2.2 The SPDD Model

The SPDD model proposed in [8] generalizes the DD system by replacing the ex-plicit expression (4) ofg(E,Ec (x)) in terms ofE andEc (x) with a direct com-putation of the discrete spectrum of energy states for electron particles. The setof admissible energy statesEi, i ∈ N, is computed through the solution of thestationary eigenvector/eigenvalue Schrodinger problem

(8) − div

(~2

2m∗n

∇ψ (x)

)+ Ec (x)ψ (x) = Eψ (x) in ΩSchr

where, by using (7), the relation betweenEc (x) andϕ (x) can be written as

Ec (x) = −qϕ(x) +

(EGap

2+ kbT ln

√Nc

Nv

),

andΩSchr⊆ ΩSi is a computational domain to be properly defined.

To simplify the presentation, we introduce the following assumptions:

(a) ΩSchr≡ ΩSi;(b) ΩSchr⊂ R3.

We defer to the end of this section the discussion of these two assumptions and oftheir associated physical consequences. To recover a density of states from the linespectrum obtained by solving (8), we introduce the following new definition forg

(9) g(E, x) = 2∑

i

δ (E − Ei) |ψi(x)|2 .

In (9) the summation is performed over all the possible energy states andδ (E − Ei)denotes the Dirac delta function centered atEi. Notice that thismodified densityof statesdoes not depend explicitly onEc (x), as in the case of the DD model.Rather, its spatial variation is given by the wave functionsψi (x) which arenonlocal functions ofϕ (x) Also notice that the factor2 in (9) accounts for the spindegeneracy of the energy eigenstates. Inserting (9) into (3), yields the followingmodified expression of the free electron density(10)

n (x) =

+∞∫Ec

g(E, x)f(ϕn (x) , E)dE =∑

i

|ψi (x)|2 f(ϕn (x) , Ei) in ΩSi.

The choice of appropriate boundary conditions for (8) is a critical issue. A straight-forward approach, which turns out to be computationally efficient, has been adoptedin [8] and consists in setting

(11) ψ (x)|∂ΩSi≡ 0,

6

where∂ΩSi is the boundary of the semiconductor domainΩSi. Enforcing homo-geneous Dirichlet boundary conditions for the wave function has the unphysicalconsequence thatn|∂ΩSi

≡ 0. To solve this inconsistency, the classical expression(6) is forced to hold in regions of the device where quantum effects are assumedto be negligible. This, in turn, demands to define a quantum regionΩQ ⊂ ΩSi

where (10) holds, and a classical regionΩcl ⊂ ΩSi where (6) holds instead, so thatΩSi = ΩQ ∪ Ωcl. To properly account for the resultingsplittedconstitutive relationfor n, a quantum correction factoris defined as follows. We introduce aquantumelectron density

(12) nq (x) = 2∑

i

|ψi (x)|2 f(ϕn (x) , Ei) in ΩQ,

and aclassical electron densityncl (x) given by relation (6). Then, we set

(13) γn (x) =

1 in Ωcl

nq (x)

ncl (x)in ΩQ,

and, accordingly, we define

(14) n (x) = γn (x)nint exp

(ϕ (x)− ϕn (x)

Vth

)= γn (x)ncl (x) (x) in ΩSi.

By settingGn (x) = Vth ln (γn (x)) ,

relation (14) can be rewritten as

(15) n (x) = nint exp

(ϕ (x) +Gn (x)− ϕn (x)

Vth

)in ΩSi

so that, repeating the derivation carried out in Sect. 2.1, with (14) in place of (6),yields the followingmodifiedconstitutive relation for the electron current density

Jn (x) = −qµnn (x)∇ϕn (x)

= −qµn (n (x)∇ (ϕ (x) +Gn (x))− Vth∇n (x)) .(16)

Following a completely similar procedure, the hole current density reads

Jp (x) = −qµpp (x)∇ϕp (x)

= −qµp (p (x)∇ (ϕ (x) +Gp (x)) + Vth∇ p (x)) .(17)

7

Let us now come back to assumptions (a) and (b) introduced at the beginning of thesection. As for assumption (a), we notice that the computation of the eigenvaluesof the discretized Hamiltonian is a rather intensive numerical task. Therefore, it isconvenient to solve (8) only in a smaller subdomainΩSchr ⊂ Ω that needs to beproperly defined in such a way that the closed boundary condition (11) does notrelevantly affect the quantum chargenq (x) in ΩQ, which is the device subregionunder the interface between silicon and silicon dioxide where quantum effects areexpected to be important. A representation of the device subdvision into the severalphysical subdomains described in this section is shown in Fig. 1.

Figure 1. Subdomain division of a two–dimensional cross–section of a MOSFET device.

As for assumption (b), it must be noted that if the Schrodinger equation is solvedin Ω ⊂ Rd, d = 1, 2, then quantization effects are accounted for only inRd, so thata continuum approximation similar to (9) must be adopted for the energy levelsassociated with the remaining spatial directions. Therefore (9) and (10) only holdif Ω ⊂ R3, while in the caseΩ ⊂ Rd, d = 1, 2, we have

(18) g(E, x) =∑

i

H (E − Ei) |ψi(x)|2 g3−d(E),

whereH (E − Ei) denotes the Heavyside step–function centered atEi. Accord-ingly, (12) becomes(19)

nq (x) =

+∞∫Ec

g(E)f(ϕn (x) , E)dE =∑

i

|ψi (x)|2∞∫

Ei

g3−d(E)f(ϕn (x) , E)dE,

8

where

(20)

g2 =

m∗n

π~2if d = 1

g1 =

√m∗

n

2π2~2(E − Ei)

−12 if d = 2.

2.3 The QDD Model

The QDD model was introduced in [20] and originally named Density Gradientmodel because it was obtained by allowing the electron gas equation of state torelate the quasi–Fermi potentialϕn not only to the electron density but also to itsgradient. According to the single band version of the QDD model, the electrondensity can be expressed using the modified Maxwell–Boltzmann relation (15) andthe quantum correctionGn to the electric potentialϕ (usually referred to as theBohm potential) is defined by the relation

(21) Gn =1√n

div

(~2

6qm∗n

∇√n

).

For a microscopic derivation of (21) from the Wigner equation we refer to [9] and[7], while in [20] a macroscopic approach is adopted. Notice that the QDD expres-sions for the current densities are formally identical to (16) and (17). This formalanalogy will be exploited in Sect. 6 to construct numerical algorithms for the solu-tion of the SPDD and QDD transport models.

2.4 Scaling

Before proceeding, it is useful to rewrite the DD, QDD and SPDD systems in ascaled form. The scaling has the advantage to emphasize thesingularly perturbednature of the equations, which can be used as in [12] to perform an a–priori quali-tative analysis of their solutions.

Our choice for the scaling parameters is as follows [11,12]:

• the scaling factor for the carrier densities is chosen to be the maximum value ofthe doping throughout the device:n = ‖D‖∞,Ω, where‖·‖∞,Ω is the norm forthe spaceL∞ (Ω) of essentially bounded functions overΩ;

• voltages are scaled with respect to the thermal voltage, thereforeϕ = Vth;• lengths are scaled with respect to the diameter of the device:x = max

x1,x2∈Ω|x1 − x2|;

• mobilities are scaled with respect to the zero field electron mobility:µ = µn0.

9

Accordingly, we define the following nondimensional quantities(22)

n =n

n, ϕ =

ϕ

ϕ, ϕn =

ϕn + Vth ln(nint

n

)ϕ

, x =x

x, µn =

µn

µ, U = U

x2

qµ n

and the nondimensional coefficients

(23) δ2n =

~2

6qm∗nx

2ϕ, β =

Vth

ϕ= 1, λ2 =

ϕε

qx2n.

Using (22) and (23), equation (21) becomes

Gn =1√n

div(δ2n∇

√n).

Similarly, the Poisson equation (1)3 reduces to

−div(λ2∇ ϕ

)= p− n+ D,

while the continuity equation (1)1 reads

−div(µn

(n∇

(ϕ+ Gn

)− β∇ n

))= U .

As for the Schrodinger equation (8), we set

η2n =

~2

2qm∗nx

2ϕ,

and we obtain−η2

ndiv ∇ ψ + Ecψ = Eψ.

Furthermore, we define the nondimensional coefficient

(24) θ =nint

n,

which will be useful in the definition of the boundary conditions.

In Table 1 we summarize the nondimensional coefficients for electrons and holesappearing in the QDD and SPDD equations, providing their numerical values in thecase of a nanoscale device.

3 Unified Framework for Quantum–Corrected DD Models

We have shown in the preceeding sections that the QDD and SPDD models can beregarded as generalizations of the classical DD model, only differing by the choice

10

symbol meaning value

η2n ~2/

(2qm∗

nϕx2)

1.46 ∗ 10−5

η2p ~2/

(2qm∗

pϕx2)

4.25 ∗ 10−5

β Vth/ (ϕ) 1

δ2n ~2/

(6qm∗

nx2ϕ)

1.88 ∗ 10−4

δ2p ~2/

(6qm∗

px2ϕ)

5.48 ∗ 10−4

λ2 (ϕε) /(qx2n

)1.7 ∗ 10−3

θ nint/n 6.1401 ∗ 10−9

Table 1Nondimensional coefficients in QDD and SPDD equations for a device withn = 1024m−3

andx = 10−7m.

of the constitutive relation for the quantum correctionsGn andGp to the electric po-tential. In this section we will present a unified framework for Quantum–CorrectedDD models which includes both QDD and SPDD as special cases. These lattermodels can be derived by an appropriate definition of the constitutive relations forthe quantum correction potentialsGn andGp. For sake of completeness, we con-sider both electron and hole contributions to charge transport. For ease of notationwe will use henceforth for any scaled quantity the same symbol as in the unscaledcase.

3.1 The Unified Form of Quantum–Corrected DD Models

All of the three models discussed in Sect. 2 can be written in the following unifiedform

(25)

− div (λ2∇ϕ) + n− p−D = 0

− div (µn (∇n− n∇ (ϕ+Gn))) = −U

− div (µp (∇ p+ p∇ (ϕ+Gp))) = −U

n = exp ((ϕ+Gn)− ϕn)

p = exp (ϕp − (ϕ+Gp))

Gn = Gn (ϕ, ϕn, n)

Gp = Gp (ϕ, ϕp, p) .

11

In (25), the quantityU represents the scaled net recombination rate in the semicon-ductor and is a function of the carrier densitiesn, p and quasi–Fermi potentialsϕn

andϕp. A constitutive relation for this term in quantum–corrected models is notwell established. In [21] the authors propose the following formal expression

(26) U =1

a0 + a1n+ a2p(exp (ϕp − ϕn)− a3)

wherea0, a1, a2 anda3 are positive constants which should be chosen in such away thatU vanishes at equilibrium. An example of a recombination model whichfits (26) and respects the latter condition is proposed in [22] as a QDD extensionof the classical Shockley–Read–Hall theory [11]. As the focus of this article is oncomparing different choices for the quantum correction terms, in what follows wewill neglect recombination phenomena and setU = 0.

Each particular model is characterized by the constitutive relations (25)6 and (25)7for the correction termsGn andGp. The DD model corresponds to setting

(27)

GDD

n (ϕ, ϕn, n) = 0

GDDp (ϕ, ϕp, p) = 0,

while the QDD model corresponds to setting

(28)

GQDD

n (ϕ, ϕn, n) = δ2n

div (∇√n)√

n

GQDDp (ϕ, ϕp, p) = −δ2

p

div(∇√p

)√p

.

GettingGn from (25)4 andGp from (25)5, and substituting them into (27) yieldsthe following equivalent relations for computing the Bohm potentials, which aremore amenable to numerical implementation

(29)

− div (δ2n∇

√n) +

√n (ϕn − ϕ+ ln (n)) = 0

− div(δ2p∇

√p)

+√p (−ϕp + ϕ+ ln (p)) = 0

GQDDn = (ϕn − ϕ+ ln (n))

GQDDp = (ϕp − ϕ− ln (p)) .

Finally, denoting byEni , ψni, the eigenvalues and the corresponding eigen-functions of the HamiltonianHn = [−η2

n4+ Ec] for electrons, and byEpi , ψni

12

the eigenvalues and eigenfunctions ofHp =[−η2

p4− Ev

]for holes, the SPDD

model corresponds to setting

(30)

GSPDDn (ϕ, ϕn, n) =

ϕn − ϕ+ ln

(∑i

|ψni|2 f (ϕn, Eni

)

)in ΩQ

0 in Ωcl,

GSPDDp (ϕ, ϕp, p) =

ϕp − ϕ− ln

(∑i

|ψpi|2 f (ϕp, Epi

)

)in ΩQ

0 in Ωcl,

which can be cast in the following form more similar to (29)

(31)

n =

2∑

i

|ψni|2 f (ϕn, Eni

) in ΩQ

exp (ϕ− ϕn) in Ωcl,

p =

2∑

i

|ψpi|2 f (ϕp, Epi

) in ΩQ

exp (ϕp − ϕ) in Ωcl

GSPDDn = (ϕn − ϕ+ ln (n))

GSPDDp = (ϕp − ϕ− ln (p)) .

As already described in Sects. 2.2 and 2.3, both QDD and SPDD systems modifythe classical DD model by introducing a certain amount of nonlocality in the re-lation that links the carrier densities to the electrical and quasi–Fermi potentials.More precisely, in the QDD model this nonlocality effect is obtained through a de-pendance ofϕn on the gradient of the concentrations, while in the SPDD model itis obtained through a more detailed physical description of the density of states. Across–validation and mutual comparison among the three models discussed in thissection will be the object of thorough investigation in the numerical experimentsshown in Sect. 7.

3.2 Boundary Conditions

In this section we define the proper boundary conditions for the QCDD modelsintroduced in Sect. 3.1.

In the case of the QDD model, we letΩ ⊆ (0, 1)d, d = 1, 2, 3, be the scaled devicecomputational domain, withΩSi ⊂ Ω andΩOX ≡ Ω\ΩSi (see Fig. 3.2). Moreover,

13

let ΓSi ≡ ∂ΩSi be the boundary ofΩSi, ΓOX ≡ ∂ΩOX the boundary ofΩOX

andΓI ≡ ΓOX ∩ ΓSi the interface betweenΩSi andΩOX , νOX andνSi being theoutward unit normal vectors toΓOX andΓSi, respectively. These latter are dividedinto three disjoint subsets in such a way thatΓOX ≡ ΓOXD

∪ΓOXN∪ΓI andΓSi ≡

ΓSiD ∪ ΓSiN ∪ ΓI . The subsetsΓOXDandΓSiD model the ohmic contacts, where

ΓOXD= FG

ΓOXN= EF ∪GH

ΓSiD = CD ∪ IL ∪ AB

ΓSiN = BC ∪DE ∪HI ∪ LA

Figure 2. Boundary subdivision for a two–dimensional cross–section of a MOSFET device.

Dirichlet boundary conditions are given for carrier densities and potentials, whileΓOXN

andΓSiN are the remaining part of the boundary where Neumann conditionsare enforced. More precisely, the boundary conditions for the QDD model are asfollows:

• at the Ohmic contacts a Dirichlet boundary condition is given for both the carrierconcentrations and for the electric and quasi-Fermi potentials

(32)

n|

ΓSiD

= nD, p|ΓSiD

= pD

ϕ|ΓSiD

= ϕSiD , ϕn|ΓSiD

= ϕnD, ϕp|

ΓSiD

= ϕpD, ϕ|

ΓOXD

= ϕOXD.

Furthermore, we assume that thermal equilibrium conditions and charge neutral-ity hold at the contacts, and that quantum effects are negligible so that

(33)

pD nD = θ2

nD − pD − D|ΓSiD

= 0

ϕn|ΓSiD

= ϕa − ln (θ)

ϕp|ΓSiD

= ϕa + ln (θ)

ϕ|ΓSiD

= ϕa − ln (θ) + ln (nD),

whereϕa is the scaled external applied potential at the ohmic contacts;• at the artificial boundariesΓOXN

andΓSiN , the normal components of the currentdensities and of the electric potential and carrier density gradients are set equal

14

to zero

(34)

Jn · νSi|ΓSiN

= 0, Jp · νSi|ΓSiN

= 0

∇ϕ · νSi|ΓSiN

= 0, ∇ϕ · νOX |ΓOXN

= 0

∇n · νSi|ΓSiN

= 0, ∇ p · νSi|ΓSiN

= 0;

• at the boundary interface between oxide and semiconductor, imposing a properset of boundary conditions is a more involved and delicate issue. On the onehand, in a quantum model, one would expect the carrier densities to become verysmall in the vicinity of the very high potential barrier given by the gate oxide,and in the limit of an infinite barrier one should predict that both carrier densitiestend to zero. On the other hand, the condition of zero free charge carriers at theinterface is incompatible with the modified Maxwell–Boltzmann statistics forelectrons and holes, because it would requireGn andGp to tend to−∞ and+∞,respectively. To circumvent this problem, one can set the interface densities equalto a small but non–zero valuecI [23], which might be estimated by a–priori 1Dcomputations with a model including tunneling of free carriers through the oxidebarrier [22]. In the numerical computations presented in Sect. 7 we have imposeda value of10−2m−3 for the charge densities at the interface, which correspondsto a nondimensional concentrationcI = 10−27. Furthermore, at the interface weimpose the normal component of the current densities to vanish and the normalcomponent of the electric displacement vector to be continuous, i.e.

(35)

n|

ΓI= p|

ΓI= cI

Jn · νSi|ΓI

= 0, Jp · νSi|ΓI

= 0

[λ2∇ϕ · νSi]ΓI= 0,

where[f ]γ denotes the jump of a functionf across thed− 1-dimensional manifoldγ.

In the case of the SPDD model, the mathematical formulation of the boundary-value problem requires a further subdivision ofΩ as anticipated in Sec. 2.2. Withthis purpose, letΩQ ⊂ Ω be the device subdomain in which we expect (by a–priori physical considerations) the quantum effects to be relevant. LetΩcl ≡ Ω \ΩQ, and letΩSchr be a suitably chosen subdomain such thatΩ ⊇ ΩSchr ⊃ ΩQ (seeFig. 1). The choice ofΩSchr is the result of a careful trade–off. On the one hand,ΩSchr should be large enough to ensure that the closed boundary conditions (11) forthe equation (8) do not significantly affect the quantum charge distribution. On theother hand, the choice of a too wideΩSchr can greatly increase the computational costof the eigenvalue problem. Conditions (32)–(35) still hold for the SPDD model. Inaddition, homogeneous boundary conditions are enforced on the wave functions on∂ΩSchr as in (11).

15

3.3 Recovering The Classical Limit

As we have presented both the QDD and SPDD models essentially asperturbationsof the classical DD model, it is natural to briefly address the issue of how those twoformer models reduce to the latter in the classical limit.

In the case of the QDD model, we start noticing that, by formally settingδ2n =

δ2p = 0, we immediately recover the DD model. The mathematical analysis of the

limiting behaviour of the QDD model has been carried out in [21], where it wasshown that the solution of the unipolar QDD model converges to the solution of theclassical DD model asδ2

n, δ2p → 0. From a quantitative point of view, letting

LQDDcl =

√√√√ ~2

6m∗nkbT

,

we note thatδ2n 1 whenx LQDD

cl , so that, at room temperature (T = 300K ),the classical limit is attained when the characteristic device lengthx is much largerthanLQDD

cl = 1.4nm.

For the SPDD case, it is well known (see for example [24]) that for avery largecrystal and near equilibrium the summation in (9) reduces to the expression in (4)so thatγn = 1. As an example, we consider a1D case withϕ = 0, for which thefollowing exact expression of the energy eigenvalues is available

(36) Ei = Ec + η2nπ

2i2, i = 1, 2, . . .

Introducing the previous relation into (19) withd = 1 yields

g (E) = 〈g (E, x)〉 =∞∑i=1

H (E − Ei)⟨∣∣∣ψ2

i (x)∣∣∣⟩ g2 (E)(37)

=∑

i< 1πηn

√E−Ec

H (E − Ei) g2 (E) = int

(1

πηn

√E − Ec

)g2 (E) ,

where〈f(x)〉 is the integral mean value of a functionf over the device domain and∀z ∈ R, int (z) stands for the greatest integer smaller thanz. Thus, for

(38)1

πηn

√E − Ec 1,

one gets

g (E) '(

1

πηn

√E − Ec

)g2 (E) = g3 (E) ,

16

which is the same density of states function used in (4). Letting

LSPDDcl =

√√√√ π2~2

2m∗nkbT

,

we note that condition (38) is attained for energiesE > Ec if 1 1πηn

=

x/LSPDDcl , i.e. for x LSPDD

cl = 7.5nm, which is essentially the same condi-tion obtained in the case of the QDD model.

Figure 3 shows the quantum correctionGn in the channel of a1D n+–n–n+ devicefor different values of the channel length. Concerning with the top figures, on thex-axis we indicate the length of the device (inm), while on they-axis we denote thescaled position along the device. The bottom figures display the quantityGn at themiddle of the channel as a function of the device length. Notice that, as expected,the quantum correction becomes negligible asx increases.

−1

−0.5

0

0.5

1

1.5

2

2.5

x 10−4

1 2 3 4 5 6 7 8 9 10

x 10−7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Device Length [m]

QDD Bohm Potential [V]

Po

sitio

n (

x/D

evic

e L

en

gth

)

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

x 10−4

1 2 3 4 5 6 7 8 9 10

x 10−7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Device Length [m]

SPDD Bohm Potential [V]P

ositio

n (

x/D

evic

e L

en

gth

)

0 0.2 0.4 0.6 0.8 1

x 10−6

0

1

2

3

4

5

6

7

8x 10

−5 QDD

0 0.2 0.4 0.6 0.8 1

x 10−6

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−4 SPDD

Figure 3. The quantum correction potentialGn in the channel of a1D n+–n–n+ device asa function of the channel length (figures on the left are obtained by QDD simulations andthose on the right by SPDD simulations).

17

3.4 A Proper Choice for the Quantum Region

In the example discussed in Sect. 3.3 we have considered, for sake of simplicity,a spatial average of themodified density of statesg (E, x), which, for a uniformlydoped material, leads to a constant quantum charge densitynq. Actually, even in thecaseϕ = 0 andD constant, the quantum charge density isnotconstant inΩSchr, but,if the widthx of ΩSchr satisfies the conditionx LSPDD

cl , it is almost flat about thecenter of the well, where the value ofnq is close tonint, and rapidly goes to zero atthe boundaries. For this reason, as described in Sect. 3.2, one can obtain an almostconstant charge density by forcingγn = 1, i.e. nq = ncl = nint in a subregionΩSchr \ ΩQ of ΩSchr. The choice of the subregionΩQ has to be made in such a waythat the discontinuity suffered by the charge densityn across its boundary∂ΩQ is

small. In Fig. 4 the jump|nint − nq|

nint

= |1− γn| for the caseϕ = 0 is plotted as a

function of the widthx of the well. The white and red line show the width of thequantum region for which|1− γn| ≤ 0.01 and|1− γn| ≤ 0.001 respectively.

Figure 4. Relative difference|1− γn| between quantum and classic charge in an intrinsicSi sample withϕ = 0, as a function of the widthx.

Notice that, asx increases, the distance from the boundary at whichnq approachesnint decreases very rapidly. To understand this fact, we can use the exact expressionof the wave functions

|ψi (x)|2 =sin2 (iπx)

1∫0

sin2 (iπx) dx

= 2 sin2(2π

x

Λi

)i = 1, 2, . . . .

where we have introduced the scaled wavelengthsΛi = 2/i. It can be easilychecked that, by inserting (20)2 in (19) and applying the scaling procedure de-scribed in Sec. 2.4, one gets

18

Figure 5. Comparison of the productf (E) 〈g (E)〉 for DD and1D SPPD models for dif-ferent values ofx (the gray area is the difference between〈nq〉 andncl).

(39) nq = 2∑

i

ζn sin2(2π

x

Λi

)exp (−Ei − ϕn) , ζn =

m∗nπkbT

x~2.

From (39) and (36) it becomes evident that, increasing the well widthx the energyEi associated with a given wave functionψi of wavelengthΛi (with i fixed), de-creases rapidly so that its contribution to the summation in (39) becomes more rel-evant. Therefore, while for smallx low wavelength functions are strongly damped,asx becomes larger they become more important. This can be also observed inFig. 5 where the productf (E) 〈g (E)〉 for DD and1D SPPD models is comparedfor different values ofx.

4 Functional Iterations

In this section we discuss the decoupled functional iteration that will be used tosolve the unified quantum-corrected model (25). This model constitutes a highlynonlinear system of boundary-value problems. The choice of the Newton methodfor the linearization of (25), although attractive due to the quadratic convergence,is affected by several drawbacks. First, its high computational effort (in terms ofmemory storage, ill–conditioning of the Jacobian matrix and linear system solutionat each step of the iterative procedure). Second, the need of availing (or construct-ing) a good initial guess to fully exploit second-order convergence. Third, the strongrequest on computing strictly positive carrier densities, which is a–priori impossibleto ensure when dealing with a fully coupled solution approach.

To overcome these limitations, a decoupled algorithm, well-known asGummel map

19

[25], is usually employed in the numerical approximation of the Drift-Diffusionmodel. The Gummel map can be interpreted as a nonlinear block Gauss-Seidelfunctional iteration, which consists in the successive solution of the nonlinear Pois-son’s equation (25)1 coupled with (25)4-(25)5, and of the two linearized continu-ity equations (25)2 and (25)3. The method typically exhibits a rapid convergenceand a good robustness with respect to the choice of the initial guess. Moreover,stable and monotone discretization schemes can be applied to the numerical solu-tion of the linearized continuity equations, which allow to compute strictly pos-itive carrier concentrations with appropriate current conservation properties (see[26,27,28,29,30]). A detailed study of the Gummel map can be found in the refer-ences [31,32,12,33,18,13].

In the following, we propose a generalization of the Gummel decoupled algorithmto the iterative solution of the quantum-corrected model (25) With this aim, we let

K = (ϕn, ϕp) | infΓSiD

ϕnD≤ ϕn ≤ sup

ΓSiD

ϕnD, inf

ΓSiD

ϕpD≤ ϕp ≤ sup

ΓSiD

ϕpD.

Then, we construct a functional iteration

(40) (ϕn, ϕp) → T (ϕn, ϕp)

mapping a given element(ϕn, ϕp) ∈ K into T (ϕn, ϕp) ∈ K, as follows:

(Step 1) compute

(41) (ϕ,Gn, Gp) = Φ(ϕn, ϕp)

by iteratively solving the subsystem (25)1-(25)6-(25)7, where the generalizedMaxwell–Boltzmann statistics (25)4–(25)5 is used forn andp in (25)1;

(Step 2) compute

(42) ϕn = Φn(ϕ,Gn)

by solving the linearized electron continuity equation (25)2 with respect to thevariablen and then setting

ϕn = (ϕ+Gn)− ln(n);

(Step 3) compute

(43) ϕp = Φp(ϕ,Gp)

by solving the linearized hole continuity equation (25)3 with respect to thevariablep and then setting

ϕp = (ϕ+Gp) + ln(p).

20

The boundary conditions for each differential subproblem involved in (40) are thesame as described in Sect. 3.2. During the execution of Step 1, the functionsϕn

andϕp are given and kept fixed, while in Steps 2 and 3, the functionsϕ, Gn andGp resulting from Step 1 are plugged into (42) and (43) and kept fixed, as in astandard Gauss–Seidel procedure (see [13] for a thorough discussion of this latteraspect). Special attention must be paid to the treatment of the net recombinationrate in the solution of the linearized electron and hole continuity equations. As amatter of fact, the expressions (42) and (43) have rigorous validity only if, as isthe case for this article, the recombination term is neglected, otherwise one shouldalso linearizeU with respect toϕn andϕp. A possible strategy could be to resort tothe lagging procedure described in [13], although a more thorough investigation isneeded about this issue (see [22] for some numerical results in the case of the QDDmodel). The above decoupled iteration consists of anouter loop(Steps 1, 2 and3) and of aninner loop(Step 1). This latter loop can be regarded as a consistentgeneralization of the solution of the nonlinear Poisson equation (25)1, where thealgebraic Maxwell–Boltzmann relations (25)4–(25)5 for n andp are replaced bythe solution of the differential problems (29) (in the QDD case) and (30) (in theSPDD case).

In the case of the DD model we haveGn = Gp = 0, and the algorithm (40) reducesto the classical Gummel map.

In the case of the QDD model, the inner loop (41) is defined by the following fixedpoint map:

(Step 1.1) InitializeGn andGp and compute

(44) ϕ = Φϕ(ϕn, ϕp, Gn, Gp)

by solving with respect toϕ the nonlinear Poisson equation (25)1 coupled with(25)4-(25)5;

(Step 1.2) compute

(45) Gn = Gn(ϕ, ϕn)

by solving the nonlinear equation (29)1 with respect to the variable√n, and

settingGn = ϕn − ϕ+ ln (n) ;

(Step 1.3) compute

(46) Gp = Gp(ϕ, ϕp)

by solving the nonlinear equation (29)2 with respect to the variable√p, and

settingGp = ϕp − ϕ− ln (p) .

21

In the case of the SPDD model, Steps 1.2 and 1.3 are replaced by the computationof the eigenvalues and eigenvectors for the Hamiltonian operatorsHn andHp (seeSect. 3.1) and by the computation ofGn andGp through relations (30). In thesimulation ofn–type devices, such as ann–channel MOS transistor, where currenttransport is mainly due to electrons, the solution of the SPDD equations for thehole carriers can be conveniently replaced by the solution of the correspondingDD equations. This allows to significantly reduce the high computational effort ofthe two–carrier SPDD model, maintaining at the same time a reasonable physicalaccuracy of the model. This simplified SPDD model is discussed in Sects. 6.2 and7. An analogous procedure can be obviously carried out when simulatingp–typedevices, by neglecting quantum transport effects due to electron contribution.

The convergence analysis of the generalized Gummel map (40) and of its corre-sponding finite element discretization (discussed in Sect. 5) will be object of aforthcoming paper. At the present stage, the main difficulty is to deal with Steps1.2 and 1.3, that are the major (nonlinear) modification to the standard Gummelmap used in the DD model. In particular, it is crucial to establish that a maximumprinciple is satisfied by the solutions

√n and

√p of (29)1 and (29)2, proving as

a consequence thatn andp are positive quantities. This property, albeit necessarybecause computing negative carrier densities is clearly unphysical, is mandatory inorder to maintain the compactness property(ϕn, ϕp) ∈ K through the functionaliteration (40). The issue of positiveness of the carrier densities will be addressed inmore detail in Sect. 5, where a (properly) damped Newton method will be adoptedto construct iteratively a sequence of positive approximants to the solutions of thenonlinear equations (29)1 and (29)2.

Other functional block nonlinear iterations for the QDD model, related to (40),were proposed and analyzed in [14] and [15]. In the first reference, a generalizedGummel map is considered which consists of the successive solution of twonon-linear blocks, each one through afully coupledapproach. The first block amountsto carrying out Step 1 through the coupled solution of equations (25)1, (29)1 and(29)2 with respect to the variablesϕ, n and p, while the second block amountsto carrying out Steps 2 and 3 through the coupled solution of equations (25)2 and(25)3 with respect to the dependent variables. In the second reference, a unipolar1D QDD model (for electrons) is dealt with. The structure of the proposed iterativemapping is similar to that in [14], the main difference being the use of the (gener-alized) Slotboom variableun = n e−(ϕ+Gn), instead of the quasi-Fermi levelϕn, inthe solution of the (linear) electron continuity equation.

22

5 Finite Element Discretization

In this section we describe the finite element discretization of the differential sub-problems involved in the generalized Gummel map introduced in Sect. 4.

5.1 Approximation of the QDD model

Step 1.1. amounts to solving the nonlinear Poisson equation(47)− div

(λ2∇ϕ

)+ exp ((ϕ+Gn)− ϕn)− exp (ϕp − (ϕ+Gp)) = 0 in Ω

with respect to the unknownϕ, for givenϕn, ϕp, Gn, Gp, and subject to mixedDirichlet–Neumann boundary conditions as discussed in Sect. 3.2. A damped New-ton method is adopted for the linearization of (47), and the corresponding dis-cretization is carried out using piecewise linear continuous finite elements as donein [34] in the case of the DD model. The resulting linear algebraic system to besolved at each step of the damped Newton iteration is characterized by having asymmetric positive definite and diagonally dominant coefficient matrix, providedthat a suitable lumping procedure is employed to treat the zeroth order term in(47). The solution of the system is then efficiently performed by a preconditionedConjugate Gradient method.

The iterative solution and the finite element discretization of Steps 1.2 and 1.3 is acritical issue, because of the need of maintaining positive solutions for the (squareroot) of the carrier densities. With this aim, we have modified the standard New-ton procedure by introducing a relaxation parameter, to be chosen in such a waythat the linearized boundary value problem (and the corresponding finite elementapproximation) enjoys a maximum principle. This is a sufficient condition to en-sure positivity of the computed approximate carrier concentrations. We describethe novel methodology in the case of the equation (29)1 for electrons, a completelysimilar treatment being used to solve the equation (29)2 for holes.

Letw =√n; then, the standard Newton iteration applied to problem (29)1 reads:

given an initial guessw(0), at each stepk ≥ 0 solve for the unknownw(k+1) thelinear elliptic problem(48)− div

(δ2n∇w(k+1)

)+(ϕn − ϕ+ 2 ln

(w(k)

))w(k+1)+2w(k+1) = 2w(k) in ΩSi

subject to mixed Dirichlet–Neumann boundary conditions (for the variable√n in-

stead ofn) as discussed in Sect. 3.2. If the following condition is satisfied

(49)(ϕn − ϕ+ 2 ln

(w(k)

))+ 2 ≥ 0 in ΩSi

23

then (48) defines a linear continuous mapping from the function spaceH1ΓSiD

(ΩSi)

into itself, whereH1ΓSiD

(ΩSi) is the Sobolev space of order1 accounting for non

homogeneous boundary conditions associated withw(k+1) (see [35,36] for a defi-nition of this functional space and a discussion of its properties). Moreover, it canbe checked that a maximum principle holds for the weak solution of (48), whichimplies thatw(k+1) > 0 in ΩSi provided thatw(k), k ≥ 0, and the boundary data arepositive.

In the case where condition (49) is not satisfied, we can still ensure positivity ofw(k+1) by introducing adamping parametertk, 0 < tk ≤ 1 in such a way that

tk <2∣∣∣∣inf

ΩSi

(ϕn − ϕ+ 2 ln

(w(k)

))∣∣∣∣ ,and modifying (48) as follows(50)

− div(δ2n∇w(k+1)

)+(ϕn − ϕ+ 2 ln

(w(k)

))w(k+1)+

2

tkw(k+1) =

2

tkw(k) in ΩSi,

where the use oftk on the right hand side is required to maintain the consistency ofthe iterative procedure.

Problem (50) defines a modified fixed–point iteration with relaxation which, albeitnot enjoying in general the quadratic convergence of Newton method, ensures thatthe square root of the electron carrier concentration remains positive at each step.The finite element discretization of problem (50) with piecewise linear continuousfinite elements proceeds as in the case of the nonlinear Poisson equation (47). Inparticular, by lumping the mass matrix corresponding to the zeroth order term in(50), the resulting linear algebraic system is characterized by having a symmetricpositive definite and diagonally dominant coefficient matrix, with positive diago-nal entries and nonpositive off–diagonal entries, and by a positive right–hand side.These properties ensure that the coefficient matrix is an M–matrix and that the pos-itivity property ofw still holds on the discrete level.

Once Step 1 has been solved as described above, the numerical approximation ofthe remaining linear continuity equations (Steps 2 and 3) is carried out by a finiteelement scheme using piecewise linear continuous elements with exponential fit-ting as done in [30] in the case of the DD model. This approach is a consistentgeneralization of the classical Scharfetter-Gummel difference scheme to triangulardecompositions of the device domain. It has the advantage of automatically intro-ducing an upwinding treatment of the carrier densities along triangle edges, whichin turn ensures that the method satisfies a discrete maximum principle with positivenodal values of the carrier densitiesn andp (see [37] for a thorough discussion ofthis latter subject).

24

5.2 Approximation of the SPDD model

As it has been pointed out before, the algorithm used for solving the SPDD sys-tem only differs from that for the QDD system by the method used to computethe quantum correction potential. In particular, the solution of the nonlinear ellipticboundary value problem (29)1 is substituted by the computation of the eigenvaluesand eigenvectors of the HamiltonianHn followed by the summation (31)1. Thisapproach has two main effects which are relevant for the implementation of thenumerical algorithm; on the one hand, (31)1 guarantees that the quantum chargeis always positive, on the other hand the heavy cost of the eigenvalue/eigenvectorproblem makes it very convenient to neglect the quantum correction term for oneof the carriers, when it is not expected to have a high impact on the value of thecurrents. For example, in the device simulation presented in Sect. 7 we have treatedthe holes using the DD model. To discretize the operatorHn we have used piece-wise linear finite elements, and we have solved the discrete eigenvalue/eigenvectorproblem using Arnoldi’s method.

6 Numerical algorithms

In this section, we provide a detailed description of the algorithms emanating fromthe iterative procedure (40) for the approximate solution of the QDD and SPDDmodels.

6.1 Detailed Description of the Algorithm for the QDD Model

• Input:ϕ(0), n(0), ϕ(0)

n , Gn(0), p(0), ϕ(0)

p , Gp(0), toll, kmax, jmax

• setγ(0)

n = ln (Gn)(0), γ(0)p = ln

(G(0)

p

)• for k = 1, . . . , kmax ( k is the outer iteration counter)

(1) for j = 1, . . . , jmax ( j is the inner iteration counter)(a) Solve for ϕ (using a damped Newton method):

− div(λ2∇ϕ

)+γn

(k)j+1 exp

(ϕ− ϕ(k)

n

)−γp

(k)j+1 exp

(−ϕ+ ϕ(k)

p

)−D = 0

(b) set:

ϕ(k)j+1 = ϕ, n

(k)

j+12

= γn(k)j exp

(ϕ

(k)j+1 − ϕ(k)

n

), p

(k)

j+12

= γp(k)j exp

(−ϕ(k)

j+1 + ϕ(k)p

)

(c) Solve for w (using the modified Newton method):

− div(δ2n∇w

)+ w

(ϕ(k)

n − ϕ(k)j+1 + 2 ln (w)

)= 0

25

(d) Solve for v (using the modified Newton method):

− div(δ2p∇ v

)+ v

(−ϕ(k)

p + ϕ(k)j+1 + 2 ln (v)

)= 0

(e) set:

nq = w2, pq = v2

ncl = exp(ϕ

(k)j+1 − ϕ(k)

n

), pcl = exp

(ϕ(k)

p − ϕ(k)j+1

)Gn

(k)j+1 = ϕ(k)

n − ϕ(k)j+1 + 2 ln (w) , Gp

(k)j+1 = ϕ(k)

p − ϕ(k)j+1 − 2 ln (w)

γn(k)j+1 =

nq

ncl

= exp(Gn

(k)j+1

), γp

(k)j+1 =

pq

pcl

= exp(Gp

(k)j+1

)n

(k)j+1 = γn

(k)j+1 ncl, p

(k)j+1 = γp

(k)j+1 pcl

(f) if ‖ϕ(k)j+1 − ϕ

(k)j ‖∞,Ω ≤ toll set:

ϕ(k+1) = ϕ(k)j+1

n(k+12) = n

(k)j+1, p(k+

12) = p

(k)j+1

Gn(k+1) = Gn

(k)j+1, Gp

(k+1) = Gp(k)j+1

γn(k+1) = γn

(k)j+1, γp

(k+1) = γp(k)j+1

and proceed to step 2(g) else : repeat steps a) → e)

(2) Solve for n:

− div(µn∇n− µn n∇

(ϕ(k+1) +Gn

(k+1)))

= 0

(3) set:

nk+1 = n, ϕ(k+1)n = ϕ(k+1) − ln

(n(k+1)

γn(k+1)

)(4) Solve for p:

− div(µp∇ p+ µp p∇

(ϕ(k+1) +Gp

(k+1)))

= 0

(5) set:

p(k+1) = p, ϕ(k+1)p = ϕ(k+1) + ln

(p(k+1)

γp(k+1)

)(6) if ‖ϕ(k+1)−ϕ(k)‖∞,Ω ≤ toll and ‖ϕ(k+1)

n −ϕ(k)n ‖∞,ΩSi

≤ toll and‖ϕ(k+1)

p − ϕ(k)p ‖∞,ΩSi

≤ toll exit

(7) else : go back to step 1

26

6.2 Detailed Description of the Algorithm for the SPDD Model

• Input: Input:ϕ(0), n(0), ϕ(0)

n , p(0), ϕ(0)p , toll, kmax, jmax

• for k = 1, . . . , kmax ( k is the outer iteration counter)

(1) for j = 1, . . . , jmax ( j is the inner iteration counter)(a) Solve for ϕ (using a damped Newton method):

− div(λ2∇ϕ

)+γn

(k)j+1 exp

(ϕ− ϕ(k)

n

)−γp

(k)j+1 exp

(−ϕ+ ϕ(k)

p

)−D = 0

(b) set:

ϕ(k)j+1 = ϕ, n

(k)

j+12

= γn(k)j exp

(ϕ

(k)j+1 − ϕ(k)

n

), p

(k)

j+12

= γp(k)j exp

(−ϕ(k)

j+1 + ϕ(k)p

)

(c) Solve for the eigenvalues Eni and eigenvectors ψni:

− div(η2

n∇ψ)

+ Ec ψ = E ψ

(d) set:

nq =∑

i

|ψni|2∫ +∞

Eni

f (E) g2D (E) dE, ncl = exp(ϕ

(k)j+1 − ϕ(k)

n

)

γ(k)j+1 =

nq

ncl

in ΩQ

1 in Ωcl

Gn(k)j+1 = ln

(γ

(k)j+1

), n

(k)j+1 = γ

(k)j+1 ncl

(e) if ‖ϕ(k)j+1 − ϕ

(k)j ‖∞,Ω ≤ toll set:

ϕ(k+1) = ϕ(k)j+1, n(k+

12) = n

(k)j+1

p(k+12) = p

(k)j+1, Gn

(k+1) = Gn(k)j+1

γ(k+1) = γ(k)j+1

and proceed to step 2(f) else : repeat steps a) → e)

(2) Solve for n:

− div(µn∇n− µn n∇

(ϕ(k+1) +Gn

(k+1)))

= 0

(3) set:

n(k+1) = n, ϕ(k+1)n = ϕ(k+1) − ln

(n(k+1)

γ(k+1)

)

27

(4) Solve for p:

− div(µp∇ p+ µp p∇ϕ(k+1)

)= 0

(5) set:

p(k+1) = p, ϕ(k+1)p = ϕ(k+1) + ln

(p(k+1)

)(6) if ‖ϕ(k+1)−ϕ(k)‖∞,Ω ≤ toll and ‖ϕ(k+1)

n −ϕ(k)n ‖∞,ΩSi

≤ toll and‖ϕ(k+1)

p − ϕ(k)p ‖∞,ΩSi

≤ toll exit(7) else : go back to step 1

7 Numerical Results

In this section we discuss the numerical results concerning the simulation of ananoscale MOSFET device similar to that studied in [8]. In doing this we will com-pare the computational performance and the physical accuracy of the DD, QDD andSPDD models introduced in Sect. 2. The device geometry, the finite element trian-gulation and the doping profile are shown in Fig. 6. In all the reported figures theinternational system of units (SI) is used. The device has a15nm long channel witha uniformp–type doping of2 · 1025m−3; the drain and source contact regions havea uniformn–type doping of5 · 1025m−3 and reach20nm down into the bulk. Thegate oxide is1.5nm thick and is25nm long, so that it has an overlap of5nm withboth the source and drain regions. The triangulation consists of 3284 elements and1726 nodes.

Figure 6. Left: MOSFET Geometry and finite element triangulation, with highlighting ofthe different subdomains. Right: net doping profile (log–scale); positive doping isn–type,negative isp–type.

Figs. 7 shows the electric potential in the device with grounded bulk and sourcecontacts and with1.04V and0.01V voltages applied to the gate and drain contacts,respectively. Fig. 8 shows the electron density as computed by the QDD (left) andSPDD (right) models and Fig. 10 shows the same quantities along the followingcross–sections:

28

• in the oxide–bulk direction at the middle of the channel (left);• in the source–drain direction at a position corresponding to the electron density

peak (right).

Both models exhibit a stiff boundary layer in the electron concentration at theSi/SiO2 interface. No oscillations arise in the computed solution, due to the mono-tone exponentially–fitted finite element method used to discretize the continuityequations. Notice that, although the potential distributions in the bulk–to–gate di-rection are very similar, the stiffness of the boundary layer computed by the QDDmodel is much stronger than for the SPDD model. This can be explained by look-ing at Fig. 9 that shows the computed Bohm potential distributions. Note that theBohm potential computed by the SPDD model is much smoother near theSi/SiO2

interface. For the QDD model, it is also to be noted that the strongly negative valueof the correction factor (Gn ' −1.5V ⇒ Gn ' −60) forces the damping param-

etertk defined in Sec. 5 to become very small (2

tk> 60 ⇒ tk <

1

30). This has

the drawback of slowing the convergence of the algorithm, but, at the same time,it ensures the strict positivity of the electron concentration which is a mandatoryrequirement in the physical problem.

Figure 7. Electric potential. The Gate–to–Source Voltage is1.04 V .

Figure 8. Electron concentration. The Gate–to–Source Voltage is1.04 V .

A close–up on the channel of the transistor is given in Figs. 10 and 11. In particu-lar, from Fig. 10 (right) it becomes evident that the inversion layer computed by theSPDD simulation is much wider than that of the classical simulation and the peak

29

Figure 9. Electron Bohm potential. The Gate–to–Source Voltage is1.04 V .

of the electron concentration is lower and it is attained a fewnm’s away from theinterface. The QDD solution exhibits the same qualitative features but the quantita-tive prediction of the charge peak shift is much smaller. The same phenomenon isobserved by comparing the computed DD current field shown in Fig. 11 (left) withthe QDD current field shown in Fig. 11 (middle) and the SPDD current field shownin Fig. 11 (right): in the former plot the current flow is concentrated at the interfacewhile in the latter two it is moved down towards the bulk because very few carriersare available at the interface. By inspecting Fig. 10 (left) one may note that bothquantum–corrected models predict a smaller charge density in the middle of thechannel than classical simulations, but the QDD result appears to be smoother inthis direction than the SPDD one.

0 1 2 3 4 5 6 7

x 10−8

14

16

18

20

22

24

26

Section Along the Channel

Position [m]

Ele

ctro

n D

ensi

ty

DDQDDSPDD

2.5 3 3.5 4

x 10−8

12

14

16

18

20

22

24

26

28

Section Perpendicular to the Channel

Position [m]

Ele

ctro

n D

ensi

ty

DDQDDSPDD

Figure 10. Electron Density in a Section Along the Channel (right) and Through theChannel (left) of a MOSFET Device Close to the Silicon/Silicon–Dioxide Interface. TheGate–to–Source Voltage is1.04 V . Dashed line: DD model; solid line: QDD model.

Fig. 12 shows the I–V curve of the transistor. Notice that the QDD simulation re-sults display both the relevant quantum–mechanical effects observed in [8] andconsisting in a threshold voltage shift and the degraded subthreshold slope (seeFig. 14) in comparison with DD results. On the other hand, by comparing this re-sults with the SPDD curve, we may note that the QDD model underestimates botheffects.

30

3 3.5 4 4.5

x 10−8

3.5

4

4.5x 10

−8 DD − Current Density

3 3.5 4 4.5

x 10−8

3.5

4

4.5x 10

−8 QDD − Current Density

3 3.5 4 4.5

x 10−8

3.5

4

4.5x 10

−8 SPDD − Current Density

Figure 11. Electron Current Density in the Channel of the Device (left DD Model, centerQDD Model, right SPDD Model). The Gate–to–Source Voltage is1.04 V .

An explanation of these two latter phenomena can be attempted by comparing thelog–scale curve in Fig. 12 (right) with Fig. 13 where we show on the left the totalcharge in the device (qSi =

∫ΩSi

q (p− n+D) dx) against the gate voltage andon the right the capacitance (CMOS = dqSi

dVg) against the gate voltage. As noted

before, on the one hand, due to the strong confinement in the channel in the bulk–to–gate direction, the QDD and SPDD electron concentration in the channel areconsistently lower than the DD value, on the other hand the QDD and SPDD in-version layers are wider and their width increases with the increase of gate voltage,because lowering the potential barrier between source and drain regions allows car-riers to directly tunnel through the channel, as is also observed in [3] for double–gate MOSFETs. Comparing QDD and SPDD predictions seems to indicate that theQDD model underestimates the peak shift effect while it overestimates the effect ofcharge penetration under the channel barrier. This suggests that, to force the QDDmodel to produce results which are closer to the more accurate SPDD predictions,one might choose to take the quantum diffusion coefficientδ2

n (which is tipicallyused as a fitting parameter) as a rank–two tensor instead of a scalar. This couldhave a strong beneficial impact in routine device simulations because, although theSPDD model is less expensive than more advanced quantum models, the compu-tational effort required for the simulation of one bias point with SPDD is about10times more than required by QDD (approximately 5 minutes running aMatlabcomputer code on a PC with a 700 MHz PPC-G3 processor).

8 Conclusions and Future Work

In this article, we have proposed a unified framework for Quantum–CorrectedDrift–Diffusion (QCDD) models in nanoscale semiconductor device simulation.QCDD models are presented as a suitable generalization of the classical Drift–Diffusion (DD) system, each particular model being identified by the constitu-tive relation for the quantum–correction to the electric potential. We have exam-ined two special, and relevant, examples of QCDD models; the first one is theSchrodinger–Poisson–Drift–Diffusion model, and the second one is the Quantum–

31

0 0.5 1 1.50

100

200

300

400

500

600

700

800

900I−V curve (linear scale)

Vg [V]

I d [A/m

]

DDQDDSPDD

0 0.5 1 1.510

−8

10−6

10−4

10−2

100

102

104

I−V curve (log scale)

Vg [V]

I d [A/m

]

DDQDDSPDD

Figure 12. I–V Characteristics of a MOSFET device. Comparison between DD (dashedline) and bipolar QDD (solid line) models. The Drain Current is plotted versus theGate–to–Source Voltage at0.01 V Drain–to–Source Voltage. Left: linear scale; right:log–scale.

0 0.5 1 1.5−9

−8

−7

−6

−5

−4

−3

−2

−1x 10

−10 Total charge in the Device

Vg [V]

q tot [C

oul/m

]

DDQDDSPDD

0 0.5 1 1.53

4

5

6x 10

−10 d qtot

/d Vg

Vg [V]

d q to

t/d V

g [Far

ad/m

]

DDQDDSPDD

Figure 13. Left: total charge in the quantum box. Right: capacitance.

Drift–Diffusion model. Both approaches provide a more accurate description ofthe density of states than the DD model, in order to represent the effect of two-dimensional quantum confinement on the charge densities in stationary regime.However, they neglect quantum effects in charge transport modeling because theyuse a classical DD expression for the current densities. For the numerical treatmentof the two models, we have introduced a functional iteration technique that extendsthe classical Gummel decoupled algorithm widely used in the iterative solutionof the DD system. This extension represents, to our knowledge, the first fully–decoupled procedure applied to the QDD model that ensures strict positivity of thecharge densities at each step of the iteration. We have discussed the finite elementdiscretization of the various differential subsystems, with special emphasis on theirstability properties, and we have successfully validated the performance of the pro-posed algorithms and models on the numerical simulation of nanoscale devices intwo spatial dimensions.

32

0.2 0.3 0.4 0.5 0.6

100

110

120

130

140

150

160

170

180

190

200

I−V Slope

Vg [V]

I−V

slo

pe [m

V/d

ec]

DDQDDSPDD

Figure 14. Subthreshold Slopes Computed by DD and QCDD Models.

Possible extensions in further investigation of QCDD models could be:

• the use of a Schrodinger equation with open boundary and direct quantum cal-culation of the current densities to simulate quantum transport effects in deviceregions where the ballistic limit holds [6];

• the use of Fermi–Dirac statistics to account for short–range scattering phenom-ena related to the Pauli exclusion principle;

• the inclusion of temperature effects connected with lattice heating (in the station-ary case) and electron gas heating (in transient computations).

Acknowledgments

We gratefully acknowledge Dr. R. Gusmeroli, Dr. A. Pirovano and Prof. A. S.Spinelli, Dipartimento di Elettronica e Informazione, Politecnico di Milano, formany fruitful discussions on the research object of this article.

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