Nonlinear Exact Closure for the Hydrodynamical Model of Semiconductors Based on the Maximum Entropy Principle

March 20, 2007 16:30 WSPC - Proceedings Trim Size: 9in x 6in SIMAI06˙mascali˙romano

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NONLINEAR EXACT CLOSURE FOR THEHYDRODYNAMICAL MODEL OF SEMICONDUCTORSBASED ON THE MAXIMUM ENTROPY PRINCIPLE

G. MASCALI

Department of Mathematics, University of Calabria and INFN-Gruppo c.Cosenza,Cosenza, 87036, Italy

∗E-mail: [email protected]

V. ROMANO

Department of Mathematics and Informatics, University of CataniaCatania, 95125, Italy

E-mail: [email protected]

An exact closure is obtained for the 8-moment model of semiconductors basedon the maximum entropy principle.

Keywords: Semiconductors; hydrodynamical models; maximum entropy prin-ciple.

1. Kinetic model

Semiconductors are characterized by a sizable energy gap between the va-lence and the conduction bands. The energy band structure of crystalscan be obtained at the cost of intensive numerical calculations (and alsosemi-phenomenologically) by means of the quantum theory of solids.1 Theelectrons, which mainly contribute to the charge transport, are those withenergy near the lowest conduction band minima, each neighborhood beingcalled valley. In silicon, which is the material we will deal with in this paper,there are six equivalent ellipsoidal valleys along the main crystallographicdirections ∆ at about 85 % from the center of the first Brillouin zone, nearthe X points, which, for this reason, are termed as X-valleys.

In the derivation of macroscopic models, usually, the energy in eachvalley is represented by analytical approximations. Among these, the mostcommon one is the Kane dispersion relation, which describes the energy EA


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of the A-valley, measured from the bottom of the valley EA, as

EA(kA) [1 + αAEA(kA)] =~2k2

A

2m∗A

, kA ∈ R3. (1)

kA is the electron wave vector in the A-valley and kA its modulus, m∗A is the

effective electron mass in the A-valley and ~ the reduced Planck constant.αA is the non parabolicity parameter. In the sequel, in order to simplifythe notation, the valley index is omitted.

The electron velocity v (k) depends on the energy E by the quantumrelation

v (k) =1~∇kE .

Explicitly, we get in the Kane approximation of the dispersion relation

vi =~ki

m∗ [1 + 2αE(k)]. (2)

In the semiclassical kinetic approach the charge transport in semicon-ductors is described by the Boltzmann equationa

∂f

∂t+ vi(k)

∂f

∂xi− eEi

~∂f

∂ki= C[f ], (3)

where f(x,k, t) is the one electron distribution function and e the absolutevalue of the electron charge. In a multivalley description one has to considera transport equation for each valley.

The electric field E is calculated by solving the Poisson equation for theelectric potential φ

Ei = − ∂φ

∂xi, (4)

∇ · (ε∇φ) = −e(N+ −N− − n), (5)

N+ and N− being the donor and acceptor density respectively (which de-pend only on the position) and n the electron number density

n =∫

fdk.

C[f ] represents the effects due to scattering of electrons with phonons,impurities and with other electrons. After a collision the electron can remainin the same valley (intravalley scattering) or be drawn into another valley(intervalley scattering).

aHereafter summation over repeated indices is understood


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Under the assumption that the electron gas is dilute, the collision oper-ator can be assumed in the linear form

C[f ] =∫

[P (k′,k)f(k′)− P (k,k′)f(k)] dk. (6)

For the sake of brevity, we will consider only electron-phonon scatteringswhich can be summarized as follows:

• scattering with intravalley acoustic phonons (approximately elas-tic);

• electron-phonon intervalley inelastic scatterings, for which there aresix contributions: the three g1, g2, g3 and the three f1, f2, f3 opticaland acoustical intervalley scatterings2

me electron rest mass 9.109510−28 gm∗ effective electron mass 0.32 me

TL lattice temperature 300 Kρ density 2330 g/cm3

vs longitudinal sound speed 9.18 105 cm/secΞd acoustic-phonon deformation potential 9 eVα non parabolicity factor 0.5 eV−1

εr relative dielectric constant 11.7ε0 vacuum dieletric constant 1.24 × 10−22 C/( eV cm)

α Z f ~ω(meV ) (Dt K) (108 eV/cm)g1 1 12 0.5g2 1 18.5 0.8g3 1 61.2 11f1 4 19 0.3f2 4 47.4 2f3 4 59 2

In the elastic case

P (ac)(k,k′) = Kac δ(E − E ′), (7)

while for the inelastic scatterings

Pα(k,k′) = Kα

[(N (α)

B + 1)δ(E ′ − E + ~ωα) + N(α)B δ(E ′ − E − ~ωα)

], (8)


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where α = g1, g2, g3, f1, f2, f3, N(α)B is the phonon equilibrium distribution

obeying the Bose-Einstein statistics and ~ωα the phonon energy.The parameters that appear in the scattering rates can be expressed in

terms of physical quantities characteristic of the considered material

Kac =kBTLΞ2

d

4π2~ρv2s

, Kα =Zf α(DtK)2α

8π2ρωα,

where kB is the Boltzmann constant, TL the lattice temperature, Ξd thedeformation potential of acoustic phonons, ρ the mass density of the semi-conductor, vs the sound velocity of the longitudinal acoustic mode, (DtK)α

the deformation potential relative to the interaction with the α intervalleyphonon and Zf α the number of final equivalent valleys for the consideredintervalley scattering.

2. 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 itat a suitable order N. Such a truncation introduces two main problems dueto the fact that the number of unknowns exceeds that of the equations:

i) the closure for higher order fluxes;ii) the closure for the production terms.As in gasdynamics, multiplying eq. (3) by a sufficiently regular func-

tion ψ(k) and integrating with respect to k, one gets the generic momentequation

∂Mψ

∂t+

∫ψ(k)vi(k)

∂f

∂xidk +

e

~Ej

∫f

∂ψ(k)∂kj

dk =∫

ψ(k)C[f ]dk, (9)

with

Mψ =∫

ψ(k)fdk,

the moment relative to the weight function ψ.Various models employ different expressions of ψ(k) and numbers of

moments.

3. The maximum entropy principle

The maximum entropy principle (hereafter MEP) leads to a systematicway for obtaining constitutive relations on the basis of information theory(see3–8 for a review).


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According to the MEP if a given number of moments MA, A = 1, . . . , N ,are known, the distribution function which can be used to evaluate theunknown moments of f , corresponds to the extremal, fME , of the entropyfunctional under the constraints that it exactly yields the known momentsMA ∫

ψAfMEdk = MA, A = 1, . . . , N. (10)

fME is the least biased distribution, which can be used to estimate f , whenonly a finite number of moments of this latter are known.

Since the electrons interact with the phonons which describe the thermalvibrations of the ions placed at the points of the crystal lattice, in princi-ple we should deal with a two component system (electrons and phonons).However, if one considers the phonon gas as a thermal bath at constanttemperature TL, only the electron component of the entropy must be max-imized. Moreover, by considering the electron gas as sufficiently dilute, onecan take the expression of the entropy obtained as limiting case of thatarising in the Fermi statistics, that is

s = −kB

∫(f log f − f) dk. (11)

If we introduce the Lagrangian multipliers ΛA, the problem of maximiz-ing s under the constraints (10) is equivalent to maximizing

s = ΛA

(MA −

∫ψAfdk

)− s,

the Legendre transform of s, without constraints, so that the equation

δs = 0

has to be solved. This gives[log f +

ΛAψA

kB

]δf = 0.

Since the latter relation must hold for arbitrary δf , it follows that

fME = exp[− 1

kBΛAψA

]. (12)

In order to get the dependence of the ΛA’s on the MA’s, one has to invertthe constraints (10). Then by taking the moments of fME and C[fME ], onefinds the closure relations for the fluxes and the production terms appearingin the balance equations. On account of the analytical difficulties this, ingeneral, can be achieved only with a numerical procedure. However, apart


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from the computational problems, the balance equations are now a closedset of partial differential equations and with standard considerations inextended thermodynamics4 it is easy to show that they form a quasilinearhyperbolic system.

When the Kane dispersion relation is used, the solvability of the maxi-mum entropy problem has been proved in.9

4. The 8-moments model

Let us consider the balance equations for the density, the velocity, theenergy and the energy flux, which correspond to the kinetic variables1,v, E , Ev

∂n

∂t+

∂(nV i)∂xi

= 0, (13)

∂(nV i)∂t

+∂(nU ij)

∂xj+ neEj Hij = nCV i , (14)

∂(nW )∂t

+∂(nSj)

∂xj+ neVkEk = nCW , (15)

∂(nSi)∂t

+∂(nF ij)

∂xj+ neEjG

ij = nCSi . (16)

The macroscopic quantities involved in the balance equations are related tothe one particle distribution function of electrons f(x,k, t) by the followingdefinitions

n =∫

R3fdk is the electron density,

V i =1n

∫

R3fvidk is the average electron velocity,

W =1n

∫

R3E(k)fdk is the average electron energy,

Si =1n

∫

R3fviE(k)dk is the energy flux,

U ij =1n

∫

R3fvivjdk is the velocity flux,

Hij =1n

∫

R3

1~f

∂

∂kj(vi)dk,

F ij =1n

∫

R3fvivjE(k)dk is the flux of the energy flux,

Gij =1n

∫

R3

1~f

∂

∂kj(Evi)dk,


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CV i =1n

∫

R3C[f ]~vidk is the velocity production,

CW =1n

∫

R3C[f ]E(k)dk is the energy production,

CSi =1n

∫

R3C[f ]viE(k)dk is the the energy flux production.

These moment equations do not constitute a set of closed relations be-cause of the fluxes and production terms. Therefore constitutive assump-tions must be prescribed.

If we assume as fundamental variables n, V i, W and Si, which havea direct physical interpretation, the closure problem consists of expressingU ij , Hij , F ij , Gij and the moments of the collision term CV i , CW and CSi

as functions of n, V i, W and Si.If we use the MEP to get the closure relations, we have to face with the

problem of inverting the constraints (10) with ψA = 1,v, E , Ev.This problem has been overcame in10,11 upon the ansatz of small

anisotropy for fME since Monte Carlo simulations for electron transportin Si show that the anisotropy of f is small even far from equilibrium.

Here we will show that it is possible to invert the constraints (10) in anexact way assuming what follows

BASIC ASSUMPTION:

V and S are collinear. (17)

REMARK. The previous assumption is valid in the one dimensional case.In general it is not true. However, apart from the specific interest in semi-conductor mathematical modeling, getting exact closure relations is itselfof great interest in thermodynamical theories of non equilibrium and inparticular gives relevant insights into the influence of the non linear terms.

5. Closure relations

The constraints (10) in the case under consideration explicitly read

n =∫

R3fMEdk, V i =

1n

∫

R3fMEvidk, (18)

W =1n

∫

R3E(k)fMEdk, Si =

1n

∫

R3fMEviE(k)dk, (19)

where

fME = exp[−

(λ

kB+ λWE + λV · v(k) + λS · v(k) E)

)], (20)


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λ, λW , λV and λS being the Lagrangian multipliers relative to the density,energy, velocity and energy-flux respectively.

Thanks to the assumption (17),

λV · v(k) + λS · v(k) E =(|λV |+ |λS | E

)|v| cosϑ,

ϑ being the angle between V and v.By expressing the elementary volume dk as g(E) dΩ, where g(E) =√

2(m∗)3/2 ~−3√E(1 + αE)(1 + 2αE) is related to the density of states and

dΩ is the element of solid angle, the constraints, after some algebra, become

n =2π(2m∗)3/2

~3e− λ

kB d0, (21)

V =1d0

√2

m∗

∫ ∞

0

E(1 + αE)e−λW E[sinhA(E)

A(E)2− cosh A(E)

A(E)

]dE , (22)

W =1d0

∫ ∞

0

Ee−λW E sinhA(E)A(E)

√E(1 + αE)(1 + 2αE) dE , (23)

S =1d0

√2

m∗

∫ ∞

0

E2(1 + αE)e−λW E[sinhA(E)

A(E)2− cosh A(E)

A(E)

]dE ,(24)

with

d0 =∫ ∞

0

e−λW E sinh A(E)A(E)

√E(1 + αE)(1 + 2αE) dE ,

where A(E) = (|λV |+ |λS | E)|v| and V and S are the relevant component ofV and S in the chosen frame (of course here time and position are frozen).

The previous relations define the fields n, V , W and S in terms of theLagrangian multipliers apart from an integration with respect to E whichcan be efficiently performed with Gauss-Laguerre quadrature formulas.

Inserting fME in the definition of fluxes one has

m∗U =2d0

∫ ∞

0

e−λW E [E(1 + αE)]3/2

1 + 2αE B(E)d E , (25)

m∗F =2d0

∫ ∞

0

E e−λW E [E(1 + αE)]3/2

1 + 2αE B(E)d E , (26)

m∗H =1d0

∫ ∞

0

e−λW E[√

E(1 + αE)sinhA(E)

A(E)− 4α

[E(1 + αE)]3/2

(1 + 2αE)2B(E)

]d E , (27)

m∗G =12α

(1−m∗H) . (28)

with

B(E) =[sinhA(E)

A(E)− 2

A(E)2

(coshA(E)− sinhA(E)

A(E)

)].


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Here U , F , G and H are the relevant components (that is those in thedirection of |V |) of the tensors Uij , Fij , Gij and Hij , respectively, .

Similarly inserting fME in the definition of the production terms, onehas for the significant components in the case of elastic phonon scattering

CV = −8πKacm∗

~3 d0

∫ ∞

0

e−λW E [E(1 + αE)]3/2(1 + 2αE) B1(E) d E , (29)

CW = 0, (30)

CS = −8πKacm∗

~3 d0

∫ ∞

0

E e−λW E [E(1 + αE)]3/2(1 + 2αE)B1(E) d E ,(31)

with

B1(E) =(

sinhA(E)A(E)2

− cosh A(E)A(E)

),

and in the case of inelastic phonon scattering

CV = −8πKαm∗

~3d0

∫ ∞

0

[e−λW E E(1 + αE)N (α)

B N+(E)B1(E)

+(N (α)B + 1)E+(1 + αE+)e−λW E+N (E)B1(E+)

]d E , (32)

CW =8πKα(m∗)3/2

√2~3d0

~ωα

∫ ∞

0

e−λW EN+(E)N (E)[N

(α)B

sinhA(E)A(E)

− (N (α)B + 1)

sinhA(E+)A(E+)

]d E , (33)

CS = −8πKαm∗

~3d0

∫ ∞

0

[e−λW E E2(1 + αE)N (α)

B N+(E)B1(E)

+ (N (α)B + 1)E2

+(1 + αE+)e−λW E+N (E)B1(E+)]

d E , (34)

with

N (E) =√E(1 + αE)(1 + 2αE), E+ = E + ~ωα, N+(E) = N (E+).

6. Comparison of linear and nonlinear closure in the onedimensional bulk case

In the one dimensional homogenous problem the density equals the constantdoping, while the balance equations of velocity, energy and energy-flux lead


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to the following system of ODEs

d

dtV = −eEH + CV , (35)

d

dtW = −eEV + CW , (36)

d

dtS = −eEG + CS , (37)

where the electric field E enters as a parameter. Once all the variableshave been expressed in terms of the Lagrangian multipliers, the balanceequations (35) -(37) can be rewritten as

d

dt

λV

λW

λS

= J−1

0

−eEH + CV

−eEV + CW

−eEG + CS

, (38)

with J0 Jacobian matrix

∂(V, W, S)∂(λV , λW , λS)

.

As initial conditions we consider the equilibrium state

V (0) = 0, W (0) = W0 =32kBTL, S(0) = 0.

We recall that TL is considered as constant.In terms of Lagrangian multipliers the previous conditions read

λV = 0, λW =1

kBTL, λS = 0.

For the evaluation of the integrals the Gauss-Laguerre formulas withweights e−x and

√x e−x have been adopted. A Runge-Kutta method has

been used for the numerical integration of the evolution equations.In fig. 1 we compare the drift velocity and average energy in bulk silicon

versus the electric field, obtained by using respectively the approximatedclosure based on the small anisotropy ansatz (AM) and the exact closurepresented in this paper (EM). As can be seen the results are remarkablydifferent at high fields, moreover if they are compared with the MonteCarlo ones shown in K. Tomizawa12(p. 100, figure 3.11) together with theexperimental data, it is possible to conclude that the results with the exactclosure are considerably better. This shows that the anisotropy effects andthe nonlinearity play an important role.


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0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Electric Field(V/µ m)

Vel

ocity

(10

7 cm

/s)

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Electric Field(V/µ m)

Ene

rgy

Flu

x (e

V 1

07 cm

/s)

Fig. 1. Velocity and Energy vs Electric Field (bulk silicon), continuous line: theAM model, crosses: the EM model.


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References

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Sounders CollegePublishing International Edition, Philadelphia, 1976).

2. C. Canali, C. Jacoboni, F. Nava, G. Ottaviani, A. Alberigi-Quaranta, Phys.Rev B 12 2265 (1975).

3. E. T. Jaynes, Phys. Rev B 106 620 (1957).4. I. Muller and T. Ruggeri, Rational Extended Thermodynamics, (Springer-

Verlag, Berlin, 1998).5. D. Jou, J. Casas-Vazquez and G. Lebon, Extended irreversible thermodynam-

ics, (Springer-Verlag, Berlin, 1993).6. C. D. Levermore, J. Stat. Phys 83 331 (1996).7. N. Wu, The maximum entropy method, (Springer-Verlag, Berlin, 1997).8. A.M. Anile, G. Mascali and V. Romano, in Mathematical Problems in Semi-

conductor Physics, Lecture Notes in Mathematics 1832, (Springer, Berlin,2003).

9. M. Junk and V. Romano, Cont. Mech. Thermodyn. 17 247 (2005).10. A. M. Anile and V. Romano , Cont. Mech. Thermodyn. 11 307 (1999).11. V. Romano, Cont. Mech. Thermodyn. 12 31 (2000).12. K. Tomizawa, Numerical simulation of sub micron semiconductor devices,

(Artech House, Boston 1993).

Documents

Nonlinear Exact Closure for the Hydrodynamical Model of Semiconductors Based on the Maximum Entropy Principle