Mathematical Problems in Semiconductor Physics pot

The increasing demand on ultra miniturized electronic devices for ever proving performances has led to the necessity of a deep and detailed under-standing of the mathematical theory of c

J. M Morel, Cachan

F Takens, Groningen

B Teissier, Paris

Subseries:

Fondazione C.I.M.E., Firenze

Adviser: Pietro Zecca

Berlin Heidelberg New York Hong Kong London Milan Paris

Tokyo

Problems in

Semiconductor Physics Lectures given at the

C.I.M.E Summer School

held in Cetraro, Italy,

With the collaboration of

G Mascali and V Romano

Editor: A M Anile

1 3

Arizona State University

Tempe, Arizona 85287-1804, USA

e-mail: ringhofer@asu.edu

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The increasing demand on ultra miniturized electronic devices for ever proving performances has led to the necessity of a deep and detailed under-standing of the mathematical theory of charge transport in semiconductors.Because of their very short dimensions of charge transport, these devices must

im-be descriim-bed in terms of the semiclassical Boltzmann equation coupled withthe Poisson equation (or some phenomenological consequences of these equa-tions) because the standard approach, which is based on the celebrated drift-diffusion equations, leads to very inaccurate results whenever the dimensions

of the devices approach the carrier mean free path

In some cases, such as for very abrupt heterojunctions in which tunnelingoccurs it is even necessary to resort to quantum transport models (e.g theWigner-Boltzmann-Poisson system or equivalent descriptions)

These sophisticated physical models require an appropriate mathematicalframework for a proper understanding of their mathematical structure as well

as for the correct choice of the numerical algorithms employed for tional simulations

computa-The resulting mathematical problems have a broad spectrum of theoreticaland practical conceptually interesting aspects

From the theoretical point of view, it is of paramount interest to investigatewellposedness problems for the semiclassical Boltzmann equation (and also forthe quantum transport equation, although this is a much more difficult case).Another problem of fundamental interest is that of the hydrodynamical limitwhich one expects to be quite different from the Navier-Stokes-Fourier one,since the collision operator is substantially different from the one in rarefiedgas case

From the application viewpoint it is of great practical importance to studyefficient numerical algorithms for the numerical solution of the semiclassicalBoltzmann transport equation (e.g spherical harmonics expansions, MonteCarlo method, method of moments, etc.) because such investigations couldhave a great impact on the performance of industrial simulation codes for

The CIME summer course entitled MATHEMATICAL PROBLEMS

IN SEMICONDUCTOR PHYSICS dealt with this and related

ques-tions It was addressed to researchers (either PhD students, young post-docs

or mature researchers from other areas of applied mathematics) with a stronginterest in a deep involvement in the mathematical aspects of the theory ofcarrier transport in semiconductor devices

The course took place in the period 15-22 July 1998 on the premises of theGrand Hotel San Michele di Cetraro (Cosenza), located at a beach of astound-ing beauty in the Magna Graecia part of southern Italy The Hotel facilitieswere more than adequate for an optimal functioning of the course About 50

“students”, mainly from various parts of Europe, participated in the course

At the end of the course, in the period 23-24 July 1998, a related workshop ofthe European Union TMR (Training and Mobility of Researchers) on “Asymp-totic Methods in Kinetic Theory” was held in the same place and several ofthe participants stayed for both meetings Furthermore the CIME course wasconsidered by the TMR as one of the regular training schools for the youngresearchers belonging to the network

The course developed as follows:

• W Allegretto delivered 6 lectures on analytical and numerical problems

for the drift-diffusion equations and also on some recent results concerningthe electrothermal model In particular he highlighted the relationshipwith integrated sensor modeling and the relevant industrial applications,inducing a considerable interest in the audience

• F Poupaud delivered 6 lectures on the rigorous derivation of the

quan-tum transport equation in semiconductors, utilizing recent developments

on Wigner measures introduced by G´erard, in order to obtain the classical limit His lectures, in the French style of pure mathematics, werevery clear, comprehensive and of advanced formal rigour.The lectures wereparticularly helpful to the young researchers with a strong background inAnalysis because they highlighted the analytical problems arising from therigorous treatment of the semiclassical limit

semi-• C Ringhofer delivered 6 lectures which consisted of an overview of the

state of the art on the models and methods developed in order to studythe semiclassical Boltzmann equation for simulating semiconductor de-vices He started his lectures by recalling the fundamentals of semicon-ductor physics then introduced the methods of asymptotic analysis in or-der to obtain a hierarchy of models, including: drift-diffusion equations,energy transport equations, hydrodynamical models (both classical andquantum), spherical harmonics and other kinds of expansions His lec-tures provided comprehensive review of the modeling aspects of carriertransport in semiconductors

applications of the moment methods He presented in detail and depth theconcepts of exponential closures and of the principle of maximum entropy.

In his lectures he gave several physical examples of great interest arisingfrom rarefied gas dynamics, and pointed out how the method could also

be applied to the semiclassical Boltzmann equation He highlighted the lationships between the method of moments and the mathematical theory

re-of hyperbolic systems re-of conservation laws

During the course several seminars on specialized topics were given by ing specialists Of particular interest were these of P Markowich (co-director

lead-of the course) on the asymptotic limit for strong fieds, lead-of P Pietra on thenumerical solution of the quantum hydrodynamical model, of A Jungel onthe entropy formulation of the energy transport model, of O Muscato on theMonte Carlo validation of hydrodynamical models, of C Schmeiser on ex-tended moment methods, of A Arnold on the Wigner-Poisson system, and of

A Marrocco on the mixed finite element discretization of the energy transportmodel

A M Anile

CIME’s activity is supported by:

Ministero dell’Universit`a Ricerca Scientifica e Tecnologica;

Consiglio Nazionale delle Ricerche; E.U under the Training and Mobility ofResearchers Programme

Recent Developments in Hydrodynamical Modeling of

Semiconductors

A M Anile, G Mascali and V Romano 1

1 Introduction 1

2 General Transport Properties in Semiconductors 2

3 H-Theorem and the Null Space of the Collision Operator 5

4 Macroscopic Models 7

4.1 Moment Equations 7

4.2 The Maximum Entropy Principle 8

5 Application of MEP to Silicon 11

5.1 Collision Term in Silicon 11

5.2 Balance Equations and Closure Relations 13

5.3 Simulations in Bulk Silicon 15

5.4 Simulation of a n+− n − n+ Silicon Diode 21

5.5 Simulation of a Silicon MESFET 26

6 Application of MEP to GaAs 34

6.1 Collision Term in GaAs 34

6.2 Balance Equations and Closure Relations 36

6.3 Simulations in Bulk GaAs 38

6.4 Simulation a GaAs n+− n − n+ Diode 43

6.5 Gunn Oscillations 45

References 54

Drift-Diffusion Equations and Applications W Allegretto 57

1 The Classical Semiconductor Drift-Diffusion System 57

1.1 Derivation 57

1.2 Existence 58

1.3 Uniqueness and Asymptotics 63

2 Other Drift-Diffusion Equations 66

2.1 Small Devices 66

2.3 Avalanche Generation 70

3 Degenerate Systems 70

3.1 Degenerate Problems: Limit Case of the Hydrodynamic Models 70 3.2 Temperature Effects 73

3.3 Degenerate Problems: Thermistor Equations and Micromachined Structures 74

4 Related Problems 80

5 Approximations, Numerical Results and Applications 82

References 89

Kinetic and Gas – Dynamic Models for Semiconductor Transport Christian Ringhofer 97

1 Multi-Body Equations and Effective Single Electron Models 98

1.1 Effective Single Particle Models – The BBGKY Hierarchy 101

1.2 The Relation Between Classical and Quantum Mechanical Models104 2 Collisions and the Boltzmann Equation 107

3 Diffusion Approximations to Kinetic Equations 111

3.1 Diffusion Limits: The Hilbert Expansion 113

3.2 The Drift Diffusion Equations: 114

3.3 The Energy Equations: 115

3.4 The Energy Transport – or SHE Model 116

3.5 Parabolicity 118

4 Moment Methods and Hydrodynamic Models 120

References 130

A M Anile, G Mascali and V Romano

Dipartimento di Matematica e Informatica,

Universit`a di Catania

viale A Doria 6 - 95125 Catania, Italy

anile@dmi.unict.it, mascali@dmi.unict.it, romano@dmi.unict.it

Summary We present a review of recent developments in hydrodynamical

mod-eling of charge transport in semiconductors We focus our attention on the modelsfor Si and GaAs based on the maximum entropy principle which, in the framework

of extended thermodynamics, leads to the definition of closed systems of momentequations starting from the Boltzmann transport equation for semiconductors.Both the theoretical and application issues are examined

a dynamical variable

Furthermore, for many applications in optoelectronics it is necessary todescribe the transient interaction of electromagnetic radiation with carriers incomplex semiconductor materials Since the characteristic times are of order ofthe electron momentum or energy flux relaxation times, some higher moments

of the carrier distribution function must be necessarily involved These are themain reasons why more general models have been sought which incorporateenergy as a dynamical variable and whose validity, at variance with the drift-diffusion model, is not restricted to quasi-stationary situations

These models are, loosely speaking, called hydrodynamical models andthey are usually derived by suitable truncation procedures, from the infi-nite hierarchy of the moment equations of the Boltzmann transport equation.However, most of these suffer from serious theoretical drawbacks due to the

A.M Anile, W Allegretto, C Ringhofer: LNM 1821, A.M Anile (Ed.), pp 1–56, 2003 c

Springer-Verlag Berlin Heidelberg 2003

semiconductors, a moment approach has been introduced [2, 3] (see also [4] for

a complete review) in which the closure procedure is based on the maximumentropy principle, while the conduction bands in the proximity of the localminima are described by the Kane dispersion relation Later on, [5, 6], the

same approach has been employed for GaAs In this case both the Γ -valley and the four equivalent L-valleys have been considered Therefore electrons in

the conduction band have been treated as a mixture of two fluids, one

repre-senting electrons in the Γ -valley and the other electrons in the four equivalent

L-valleys.

Both in the Si and in the GaAs case, the models comprise the balanceequations of electron density, energy density, velocity and energy flux Theonly difference is that for GaAs both electron populations are taken intoaccount These equations are coupled to the Poisson equation for the electricpotential Apart from the Poisson equation, the system is hyperbolic in thephysically relevant region of the field variables

In this paper we present a general overview of the theory underlying dynamical models In particular we investigate in depth the closure problemand present various applications both to bulk materials and to electron de-vices

hydro-The considerations and the results reported in the paper are exclusivelyconcerned with silicon and gallium arsenide

2 General transport properties in semiconductors

Semiconductors are characterised by a sizable energy gap between the lence and the conduction bands Upon thermal excitation, electrons from thevalence band can jump to the conduction band leaving behind holes (in thelanguage of quasi-particles) Therefore the transport of charge is achieved boththrough negatively charged (electrons) and positively charged (holes) carri-ers The conductivity is enhanced by doping the semiconductor with donor oracceptor materials, which respectively increase the number of electrons in theconduction band or that of holes in the valence band Therefore it is clear whythe energy band structure plays a very important role in the determination ofthe electrical properties of the material The energy band structure of crys-tals can be obtained at the cost of intensive numerical calculations (and alsosemiphenomenologically) by the quantum theory of solids [7] However, formost applications, a simplified description, based on simple analytical mod-els, is adopted to describe charge transport In this paper we will be essentiallyconcerned with unipolar devices in which the current is due to electrons (semi-conductors doped with donor materials) Electrons which mainly contribute

va-to the charge transport are those with energy in the neighborhoods of thelowest conduction band minima, each neighborhood being called a valley In

graphic directions ∆ at about 85 % from the center of the first Brilloiun zone,

near the X points, which, for this reason, are termed the X-valleys In GaAs

there is an absolute minimum at the center of the Brillouin zone, the Γ -point, and local minima at the L-points along the Λ cristallographic orientations.

As mentioned above, in the simplified description employed, the energy

in each valley is represented by analytical approximations Among these, themost common are the parabolic and the Kane dispersion relation

In the isotropic parabolic band approximation, the energy E A of the

A-valley, measured from the bottom of the valley E A, has an expression similar

to that of a classical free particle

E A (k A) =
2|k A |2

2m ∗ A

where α A is the non parabolicity parameter

The electron velocity v(k) 1 in a generic band or valley depends on the

energyE by the relation

In the semiclassical kinetic approach the charge transport in

semiconduc-tors is described by the Boltzmann equation For electrons in the conduction

effects due to scattering with phonons, impurities and with other electrons.

In a multivalley description one has to consider a transport equation for eachvalley

The electric field, E, is calculated by solving the Poisson equation for the

electric potential φ

E i=− ∂φ

N+and N −denote the donor and acceptor density respectively (which depend

only on the position),  the dielectric constant and n the electron number

electron-of the collision operator on the RHS electron-of the semiclassical Boltzmann equation

In fact the collision operator for the electron-electron scattering is a highlynonlinear one, being quartic in the distribution function

After a collision the electron can remain in the same valley (intravalleyscattering) or be drawn in another valley (intervalley scattering)

The general form of the collision operatorC[f] for each type of scattering

transition probability from the state k to the state k.

Under the assumption that the electron gas is dilute, the collision operator

can be linearized with respect to f and becomes

whereE = E(k) and E =E(k ).

3 H-theorem and the null space of the collision operator

In [9, 10, 11] an H-theorem has been derived for the physical phonon operator in the homogeneous case without electric field The sameproblem has also been discussed in [12] in the parabolic case

electron-Here we review the question in the case of an arbitrary form of the energyband and in the presence of an electric field, neglecting the electron-electroninteraction and assuming the electron gas sufficiently dilute to neglect thedegeneracy effects By following [13] a physical interpretation of the results issuggested

The transition probability from the state k to the state k has the general

form [14]

P (k, k ) =G(k, k  ) [(N

B + 1)δ( E  − E +
ω q ) + N B δ( E  − E −
ω q)] (11)

where δ(x) is the Dirac distribution and G(k, k ) is the so-called overlap factor

which depends on the band structure and the particular type of interaction[14] and enjoys the properties

G(k, k ) =G(k  , k) and G(k, k )≥ 0.

N B is the phonon distribution which obeys the Bose-Einstein statistics

exp(
ω q /k B T L)− 1 , (12)

where
ω q is the phonon energy

Given an arbitary function ψ(k) for which the following integrals exist,

the chain of identities [9, 10, 11]

the phonons Ψ represents the nonequilibrium counterpart of the equilibrium

Helmholtz free energy, divided by the lattice temperature It is well known inthermostatics that for a body kept at constant temperature and mechanically

insulated, the equilibrium states are minima for Ψ

A strictly related problem is the one of determining the null space of thecollision operator It consists in finding the solutions of the equation C(f) =

0 The resulting distribution functions represent the equilibrium solutions.Physically one expects that, asymptotically in time, the solution to a giveninitial value problem will tend to such a solution

operator was tackled and solved in general in [11] where it was proved thatthe equilibrium solutions are not only the Fermi-Dirac distributions but form

an infinite sequence of functions of the kind

1 + h(k) exp E(k)/k B T L

(15)

where h( E) = h(E +
ω q) is a periodic function of period
ω q /n, n ∈ N.

This property implies a numerable set of collisional invariants and hence ofconservation laws The physical meaning is that the density of electrons whoseenergyE differs from a given value u by a multiple of
ω q is constant However

if there are several types of phonons, as in the real physical cases, and theirfrequencies are not commensurable, the kernel of the collision operator reduces

to the Fermi-Dirac distribution

4 Macroscopic models

4.1 Moment equations

Macroscopic models are obtained by taking the moments of the Boltzmanntransport equation In principle, all the hierarchy of the moment equationsshould be retained, but for practical purposes it is necessary to truncate it at

a suitable order N Such a truncation introduces two main problems due tothe fact that the number of unknowns exceeds that of the equations: these arei) the closure for higher order fluxes;

ii) the closure for the production terms

As in gasdynamics [15], multiplying eq (5) by a sufficiently regular

func-tion ψ(k) and integrating over B, the first Brillouin zone, one obtains the

generic moment equation

with n outward unit normal field on the boundary ∂ B of the domain B and

dσ surface element of ∂ B, eq (16) becomes

vanishes both when B is expanded to R3, as in the parabolic and Kane

ap-proximations, ( because in order to guarantee the integrability condition f must tend to zero sufficiently fast as k → ∞ ) and when B is compact and

ψ(k) is periodic and continuous on ∂ B This latter condition is a consequence

of the periodicity of f on B and the symmetry of B with respect to the origin.

Various models employ different expressions of ψ(k) and number of

mo-ments

4.2 The maximum entropy principle

The maximum entropy principle (hereafter MEP) leads to a systematicway of obtaining constitutive relations on the basis of information theory (see[16, 17, 18, 19] for a review)

According to MEP if a given number of moments M A , A = 1, , N , are

known, the distribution function which can be used to evaluate the unknown

moments of f , corresponds to the extremal, f M E, of the entropy functional

under the constraints that it yields exactly the known moments M A

Since the electrons interact with the phonons describing the thermal vibrations

of the ions placed at the points of the crystal lattice, in principle we shoulddeal with a two component system (electrons and phonons) However, if one

considers the phonon gas as a thermal bath at constant temperature T L, onlythe electron component of the entropy must be maximized Moreover, byconsidering the electron gas as sufficiently dilute, one can take the expression

of the entropy obtained as limiting case of that arising in the Fermi statistics

s = −k B

If we introduce the lagrangian multipliers Λ A, the problem of maximizing

s under the constraints (18) is equivalent to maximizing

prob-In order to get the dependence of the Λ A ’s on the M A’s, one has to invert

the constraints (18) Then by taking the moments of f M E and C[f M E], onefinds the closure relations for the fluxes and the production terms appearing

in the balance equations On account of the analytical difficulties this, ingeneral, can be achieved only with a numerical procedure However, apartfrom the computational problems, the balance equations are now a closed set

of partial differential equations and with standard considerations in extendedthermodynamics [16], it is easy to show that they form a quasilinear hyperbolicsystem

Let us set

η(f ) = −k B (f log f − f)

The entropy balance equation is obtained multiplying the equation (5) by

η  (f ) = ∂ f η(f ) and afterwards integrating with respect to k, one has

By taking into account the periodicity condition of f on the first Brillouin

zone, the integral

ψ A(k)C(f)dk.

It is easy to prove, by multiplying (21) by Λ A and taking the sum over

A, that the entropy balance equation is a consequence of the solution of (21)

with the MEP closure relations More in particular the field equations andthe entropy balance equations are related by the condition that

is defined in sign, one can globally invert [20] and express the moments M A

as function of the lagrangian multipliers Λ B As shown in [21] the previouscondition is equivalent to require that

∂Λ A ∂Λ B ∂t ∂Λ A Λ B ∂x i ∂Λ C

In this form it is immediate to recognize that the balance equations constitute

a symmetric quasilinear hyperbolic system [22] The main consequence of thisproperty is that according to a theorem due to Fisher and Marsden [23] theCauchy problem is well-posed for the system (24), at least in the simple casewhere the electric field is considered as an external field

5 Application of MEP to silicon

5.1 Collision term in Silicon

In Silicon the electrons which give contribution to the charge transport arethose in the six equivalent valleys around the six minima of the conductionband One assumes that the electron energy in each valley is approximated

by the Kane dispersion relation Concerning the collision term, the phonon scatterings which occur can be summarized as follows:

electron-• scattering with intravalley acoustic phonons (approximately elastic);

• electron-phonon intervalley inelastic scatterings, for which there are six

contributions: the three g1, g2, g3and the three f1, f2, f3optical and

acous-tical intervalley scatterings [24]

m e electron rest mass 9.109510−10g

m ∗ effective electron mass 0.32 m e

T L lattice temperature 300o K

v s longitudinal sound speed 9.18 105 cm/sec

Ξ d acoustic-phonon deformation potential 9 eV

α non parabolicity factor 0.5 eV−1

 r relative dielectric constant 11.7

0 vacuum dieletric constant 1.24× 10 −22C/( eV cm)

Table 1 Values of the physical parameters used for silicon

A , k 

A) = K imp[|k A − k 

A |2+ β2]2δ( E 

A − E A ), (27)

where β is the inverse Debye length.

The parameters that appear in the scatterings rates can be expressed interms of physical quantities characteristic of the considered material

density of the semiconductor, v sthe sound velocity of the longitudinal acoustic

mode, (D t K) αthe deformation potential realtive to the interaction with the

α intervalley phonon and Z f α the number of final equivalent valleys for the

considered intervalley scattering N I and Z q are respectively the impurity

concentration and charge

The deformation potentials are not known from calculations by means offirst principles because the perturbation theory employed to evaluate the tran-sition probabilities is not able to calculate them from the quantum theory ofscattering In all the simulators, even the Monte Carlo ones, these quantitiesare considered as fitting parameters Their values depend on the approxima-tion used for the energy bands, on the specific characteristics of the materialand on the energy range of interest in the applications

The values of all these quantities as well as the silicon bulk constants aregiven in [25] For the sake of completeness we summarize them in tables 1 and2

5.2 Balance equations and closure relations

As already stated, the macroscopic balance equations are deduced by takingthe moments of the Boltzmann transport equation for electrons in semicon-ductors [8] We will consider the balance equations for density, momentum,

energy and energy flux, which correspond to the weight functions 1,
k, E, Ev

R 3C[f]v i E(k)dk is the production of the energy flux,

These moment equations do not constitute a set of closed relations because

of the fluxes and production terms Therefore constitutive assumptions must

be prescribed

If we assume as fundamental variables n, V i , W and S i, which have a

direct physical interpretation, the closure problem consists of expressing P i,

U ij , F ij , G ij and the moments of the collision term C P i , C W and C S i as

functions of n, V i , W and S i

If we use MEP to get the closure relations, we face the problem of inverting

the constraints (18) with ψ A = 1, v, E, Ev.

This problem has been overcome in [2, 3] with the ansatz of small

anisotropy for f M E, since Monte Carlo simulations for electron transport in

Si show that the anisotropy of f is small even far from equilibrium.

Formally a small anisotropy parameter δ has been introduced and explicit

constitutive equations have been obtained in [2] for the higher order fluxes and

in [3] for the production terms up to second order in δ However it has been

found in [26] that the first order model is sufficiently accurate for numericalapplications and avoids some irregularities due to nonlinearities as occur inthe parabolic band case [27]

Since the closure relations are an approximation of the exact MEP ones,the hyperbolicity is not guaranteed, but must be checked As proved in [26],

2k B T L equilibrium energy.

For the numerical integration we use the scheme developed in [28, 29, 30],see appendix, which is based on the Nessyhau and Tadmor scheme [31, 32]

5.3 Simulations in bulk silicon

The physical situation is represented by a silicon semiconductor with auniform doping concentration, which we assume sufficiently low so that thescatterings with impurities can be neglected On account of the symmetry withrespect to translations, the solution does not depend on the spatial variables

The continuity equation gives n = constant and from the Poisson equation

one finds that E is also constant Therefore the remaining balance equations

reduce to the following set of ODEs for the motion along the direction of the

electric field which is chosen as x-direction

i, j = 1, 2 are production terms whose expressions can be found in [3].

As initial conditions for (33)-(35) we take

W (0) = 3

The stationary regime is reached in a few picoseconds

The solutions of (33)-(35) for several values of the applied electric field arereported in figs 1 (velocity), 2 (energy) and 3 (energy flux)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5

time (ps)

Fig 2 energy (eV) versus time (ps) for the same values of the of the electric field

as in figure 1

qual-Similar results were reported in [3], but there a different modeling of thecollision terms has been considered and, moreover, instead of taking into ac-count all the intervalley and intravalley scatterings, mean values of the cou-

pling constant Ξ and D t K have been introduced The inclusion of all the

scattering (intervalley and intravalley) mechanisms notably improves the sults

re-For the sake of completeness, the parabolic band case has been also grated, figs 4, 5, 6 The differences, especially in the energy, with respect tothe Kane case, confirm that the parabolic band is an oversimplification of thereal band structure

inte-Fig 4 velocity (cm/sec) versus time (ps) in the parabolic band case (dashed line)

and for the Kane dispersion relation

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0.1 0.2 0.3 0.4 0.5 0.6 0.7

time (ps)

Electric field 70 kV/cm

Fig 5 energy (eV) versus electric field (kV/cm) in the parabolic band case (dashed

line) and for the Kane dispersion relation

0.05 0.1 0.15 0.2 0.25

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (ps)

7 eV cm/sec)

Electric field 70 kV/cm

Fig 6 energy flux (eV cm/sec) versus electric field (kV/cm) in the parabolic band

case (dashed line) and for the Kane dispersion relation

Here we present the simulation of a ballistic n+− n − n+ silicon diode as

follos [26] (see also [34] for a comparison with MC data) The n+ regions are





Î

Fig 7 Schematic representation of a n+− n − n+diode

0.1µm long, while the various lengths of the channel are taken into account.

Moreover several doping profiles will be considered as reported in table 3

Table 3 Length of the channel, doping concentration (respectively in the n+ and

n regions) and applied voltage in the test cases for the diode

Initially the electron energy is that of the lattice in thermal equilibrium at

the temperature T L, the charges are averagely at rest and the density is equal

to the doping concentration

n(x, 0) = N+(x), W (x, 0) = 3

2k B T L , V (x, 0) = 0, S(x, 0) = 0,

where V and S are the only relevant component of velocity and

energy-flux

dent conditions on each boundary should be equal to the number of acteristics entering the domain However we impose, in analogy with similarcases [26, 27, 35, 36, 37, 38], a double number of boundary conditions Moreprecisely, we give conditions for all the variables in each boundary, located at

char-x = 0 and char-x = L, the total length of the device,

where V b is the applied bias voltage In all the numerical solutions there

is no sign of spurious oscillations near the boundary, indicating that the ditions (39)-(42) are in fact compatible with the solution of the problem.The doping profile is regularized according to the function

con-N+(x) = N++− d0

tanhx − x1

L c , with L c channel length The total length of the device is L = L c + 0.2µm.

A grid with 400 spatial nodes has been used The stationary solution isreached within a few picoseconds (about five), after a short transient withwide oscillations

As first case we consider the test problem 1 (length of the channel 0.4

micron) with V = 2 Volts

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

micron

Fig 8 numerical results of the test case 1 after 5 picoseconds in the parabolic band

case (dashed line) and for the Kane dispersion relation (continuous line)

0.05 0.1 0.15 0.2 0.25

micron

Fig 9 numerical results of the test case 2 after 5 picoseconds in the parabolic band

case (dashed line) and for the Kane dispersion relation (continuous line)

0.05 0.1 0.15 0.2 0.25 0.3

micron

0.1 0.2 0.3 0.4 0

micron

Fig 10 numerical results of the test case 3 after 5 picoseconds in the parabolic

band case (dashed line) and for the Kane dispersion relation (continuous line)

The simulation for the parabolic band approximation is also shown ( fig 8dashed line), but it is evident, like in the bulk case, that the results are ratherpoor

The other test cases have been numerically integrated with V b = 1 Volt(fig.s 9,10)

Fig 11 Schematic representation of a bidimensional MESFET

In this section we check the validity of our hydrodynamical model and theefficiency of the above-mentioned numerical method by simulating a bidimen-sional Metal Semiconductor Field Effect Transistor (MESFET), see [30] Theshape of the device is taken as rectangular and it is pictured in fig 11.The axes of the reference frame are chosen parallel to the edges of thedevice We take the dimensions of the MESFET to be such that the numericaldomain is

Ω = [0, 0.6] × [0, 0.2]

where the unit length is the micron

The regions of high doping n+ are

[0, 0.1] × [0.15, 0.2] ∪ [0.5, 0.6] × [0.15, 0.2].

The contacts at the source and drain are 0.1 µm wide while the contact at the gate is 0.2 µm wide The distance between the gate and the other two contacts

is 0.1 µm A uniform grid of 96 points in the x direction and 32 points in the y

direction is used The same doping concentration as in [39, 40, 41] is considered

N+− N −=



3× 1017cm−3 in the n+ regions

1017cm−3 in the n region

with abrupt junctions

We denote by Γ D that part of ∂Ω, the boundary of Ω, which represents

the source, gate and drain

Here∇ is the bidimensional spatial gradient operator while n and t are the

unit outward normal vector and the unit tangent vector to ∂Ω respectively.

n+ is the doping concentration in the n+ region and n g is the density at the

gate, which is considered to be a Schottky contact [42],

n g = 3.9 × 105cm−3 .

φ b is the bias voltage and φ g is the gate voltage In all the simulations we

set φ g=−0.8V while Φ b = 1V

In the standard hydrodynamical model considered in the literature (e.g

[43, 44]), the energy flux S is not a field variable and it is not necessary to

prescribe boundary conditions for it The relations (46)4 and (47)5 are not

based on the microscopic boundary conditions for the distribution function,but they may be justified [30] in a heuristic way with the same approachfollowed in [45] The numerical scheme can be found in [30]

We start the simulation with the following initial conditions:

n(x, y, 0) = N+(x, y) − N − (x, y), W = W0=32k B T L ,

The main numerical problems arise from the discontinuous doping and theboundary conditions at the Schottky barrier which give rise there to sharp

changes in the density of several orders of magnitude The use of a

shock-capturing scheme is almost mandatory for this problem The stationary

solu-tion is reached in a few picoseconds (less than five) The code takes about 9minutes and 10 seconds in a PC with 1 Ghz Pentium III microprocessor Afterthe initial behaviour, the solution becomes smooth and no signs of spuriousoscillations are present The numerical scheme seems suitably robust and it isable to capture the main features of the solution Only the Kane dispersionrelation will be considered here because the results obtained in the parabolicband approximation are rather unsatisfactory when high electric fields are

involved, as shown in the previous section for the n+− n − n+ diode.

The density is plotted in figure 12 As expected there is a depletion regionbeneath the gate Moreover one can see that the drain is less populated thanthe source

Concerning the energy (figure 13) there are steep variations near the gateedges The mean energy of the electrons reaches a maximum value of about0.35 eV in the part of the gate closest to the source

The results for the velocity are shown in figure 14 The higher values ofthe x-component are at the edges of the gate contact This happens also forthe y-component, but with a huge peak at the gate edge closest to the source.The behaviour seems to indicate that there is a loss of regularity at the edge

The results are qualitatively similar to those presented in [40, 41] for allthe variables except the y-component of the velocity in account of the hugepeak at the edge of the gate