Communications in Applied and Industrial Mathematics, DOI: 10.1685/2010CAIM482ISSN 2038-0909, 1, 1, (2010) 110–127

Modeling thermal effects in submicron

semiconductor devices

Vincenza Di Stefano

Dipartimento di Matematica e Informatica, Facolta di Scienze MM.FF.NN.

Universita degli Studi di Catania, Italy

vdistefano@dmi.unict.it

Communicated by Roberto Natalini

Abstract

In this paper an Extended Hydrodynamic Model for the coupled system formed by

electrons and phonons has been formulated, in which the problem of the closure for the

high-order moments is solved by means of the Maximum Entropy Principle of extended

thermodynamics. Simulation results for a 1D n+− n− n

+ silicon diode are presented.

Keywords: Semiconductor devices, hydrodynamic models, maximum entropy

principle, self-heating.

AMS Subject Classification: 65C05, 82D37

1. Introduction.

The technological revolution that started with the introduction of thetransistor just over half a century ago is without parallel in the way ithas shaped our economy and our daily lives. The current trend towardnanoscale electronics is expected to have a similar impact into the thirdmillennium. Commercial integrated circuits are currently available withtransistors whose smallest lateral feature size is less than 100 nm and thethinnest material films are below 2 nm, or only a few atomic layers thick.Such miniaturization has led to tremendous integration levels, with a hun-dred million transistors assembled together on a chip area no larger thana few square centimeters. Due to this nanometric scale, large electric fieldsand field gradients are produced inside the material, which generate a verylarge quantity of heat as well as hot or energetic electrons, that will preventthe reliable operation of the integrated circuits. Overheating is one of themost common causes of device failure; the efficient removal of heat fromsuch submicron devices is a daunting task.The heat conduction in crystalline materials is related to the lattice vibra-tions. According to quantum mechanics, these oscillations are quantized

Received 2010 01 20, in final form 2010 06 12

Published 2010 06 21

Licensed under the Creative Commons Attribution Noncommercial No Derivatives

DOI: 10.1685/2010CAIM482

and the associated particle is called phonon. Phonons travel through spaceand engage anharmonic interactions with one another, and with electrons,impurities and geometric boundaries. These interactions are called scatte-

rings.The transport mechanism starts by applying an external bias voltage. Theelectrons are accelerated by the self consistent and external electric field,thanks to which they gain energy. This energy is lost by inelastical scatte-ring with geometric boundaries and phonons, heating up the lattice throughthe mechanism known as Joule heating. If the characteristic size of a solidmaterial is much larger than the mean free path of the resident phonons,which is approximately 300 ÷ 400 nm in bulk silicon at room tempera-ture, then the number of scattering events is large. Moreover, if there aresmall temperature gradients, the local thermodynamic equilibrium is re-stored. This regime (diffusion-like) can be described accurately using thenon-isothermal drift-diffusion model, in which the classical drift-diffusionequations are coupled with the Fourier law of heat conduction, via a heatgeneration rate term [1]:

(1) H = J · E+ (R−G)(EG + 3kBT ) .

In the previous equation, the dot product of the electric field E and thecurrent density J represents the Joule heating and the second right-hand-side term represents the heating rate due to the generation-recombinationprocesses. With continued scaling into the nanometric scale, this non-isothermal drift diffusion model is no more valid because, as said, highelectric field and high field-gradient conditions in the active region of thedevice (whose length is of the order of 10÷ 100 nm), drive the system outof local thermal equilibrium. Moreover, in spite of the small electron meanfree path (approximately 5 ÷ 10 nm in bulk silicon at room temperature),a small amount of hot electrons, which belong to the tail of the electrondistribution function, can travel in the device without losing appreciableenergy (in a ballistic regime). In this picture, degradation effects, substratecurrent, injection into the gate oxide regions, impact ionization and latticeheating are produced. Consequently, the commercial TCAD tools, basedon the drift-diffusion equations, are not able to describe accurately theseregimes; for this reason other transport models are needed.The most complete description of a system formed by electrons andphonons, is based on the Bloch-Boltzmann-Peierls (BBP), coupled withthe Poisson equation. To solve the BBP is not an easy task also from thenumerical point of view. The Monte Carlo solution of the BBP has beenrecently achieved [2], at expenses of extensive computational times. For thisreason, it is important to have alternative models which are sufficiently ac-

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V. Di Stefano

curate, and with a lower computational cost.The hydrodynamic models represent a good tool to simulate submicrondevices. The main problems to tackle are the closure for the high-ordermoments as well as the closure of the production terms (i.e. the momentson the collisional operator). A simple coupled thermal and electrical modelhas been developed [3], consisting of the hydrodynamic equations for theelectron transport and energy conservation equations for different phononmodes. The hydrodynamic model for the electrons was closed by using theFourier law for the electron heat flow. It is well known that the Fourier lawis not the best way to get a closed form [4], because it greatly overestimatesthe heat current near the channel-drain junction of the device.An alternative is to formulate a hydrodynamic model by means of the Maxi-mum Entropy Principle (hereafter MEP) of extended thermodynamics [5].In Sec. 2 we introduce the kinetic equations for electrons and phonons. InSec. 3 we write the moment equations and in Sec. 4 we deal with the MEPand the closure relations for the fluxes. An Extended Hydrodynamic Model(EHM) is formulated in Sec. 5. Finally in Sec. 6 this model is used to si-mulate a 1D n+ − n − n+ silicon diode and conclusions are drawn in Sec.7.

2. Kinetic equations

The non-equilibrium behaviour of the system formed by the electronsand phonons is described by a system of coupled Boltzmann equations,called Bloch-Boltzmann-Peierls (BBP) equations. In principle the interac-tions in such a system are phonon-phonon (p-p), phonon-electron (p-e) andelectron-electron (e-e). In the sequel, we shall consider the electrons as ararefied gas in a sea of phonons, which means that the e-e interactions canbe neglected. In e-p interactions only the electrons number and energy areconserved. In p-p interactions only energy is conserved. Let f(t, x, k) be theprobability density to find an electron at time t, position x = (x1, x2, x3),with wave vector k=(k1, k2, k3) and energy ε(k), Ng = Ng(t, x, q) the pro-bability density to find a phonon at time t, position x with wave vectorq=(q1, q2, q3) and energy ~ωg(q) of type g (i.e. the branch g of the phononspectrum). The BBP equations read [6]

∂f

∂t+ v(k) · ∇xf −

e

~E · ∇kf =

∑

g

Qepg [f,Ng](2)

∂Ng

∂t+ ug(q) · ∇xNg = Qpe

g [f,Ng] +Qppg [Ng] , g = {g1, g2, g3, ...}(3)

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where e is the absolute value of the electron charge, ~ the Planck con-stant divided by 2π, v=(v1, v2, v3) and ug=(u1, u2, u3) are the electron andphonon group velocities

v(k) =1

~∇kε , ug(q) = ∇qωg .(4)

Scattering processes due to the electron-phonon interaction are modeledby the collisional operators Qep

g [f,Ng], Qpeg [f,Ng], whereas the phonon-

phonon collision operator isQppg [Ng]. The electric field E(t, x) ≡ (E1, E2, E3)

satisfies the Poisson equation

∆(ǫφ) = e [n(t, x)−ND(x) +NA(x)]

E = −∇xφ(5)

where n(t, x) is the electron density, φ(t, x) is the electric potential, ND andNA are the donor and acceptor densities respectively, ǫ the permittivity.We treat the electrons in the parabolic band approximation, in which thedispersion relation is given by

(6) ε(k) =~2k2

2m⋆,

where m⋆ denotes the effective electron mass, which is 0.32 me in silicon.Then we apply the Debey approximation with the sound velocity vs to theacoustic phonon system and use the Einstein dispersion relation for theoptical phonons, as given in the following formulas:

(7) ωac(q) = vs|q|

(8) ωop(q) = const.

The collisional operator e-p, in the low density approximation (not degene-rate case), writes:

Qepg [f,Ng] =

V

8π3

∫

dk′{

w(k′, k)f(k′)− w(k, k′)f(k)}

(9)

where V is the crystal volume, and w(k′, k) is the scattering rate e-p which,according to the Fermi rule, writes

w(k, k′) =∑

q

s(q)

[

Ng(q) +1

2∓

1

2

]

δk′,k±qδ[

ε(k′)− ε(k) ∓ ~ω(q)]

.(10)

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V. Di Stefano

s(q) is the scattering function, δ is the Dirac’s Delta, the upper and lowersigns refer to phonon absorption or emission processes respectively. Thescattering function for acoustic phonons is given by

(11) sac(q) =πD2

A|q|

ρvsV

where DA is the electron acoustic-phonon deformation potential (9 eV), ρthe silicon density (2.33 g/cm3). For optical phonons is:

(12) sop(q) =πD2

ηZη

ρωηV

where the coupling constants are given in [7]. The collisional operator p-eis

Qpeg [f,Ng] = 2

∑

k,k′

s(q)f(k){

[Ng(q) + 1] δk′,k−qδ[

ε(k′)− ε(k) + ~ω(q)]

−Ng(q)δk′,k+qδ[

ε(k′)− ε(k) − ~ω(q)}

.(13)

The physics of the phonon-phonon scattering is very complicated. Opticalphonons do not readily migrate to the surface, as their non-dispersive na-ture near the zone center gives them a very low group velocity, and theydo not carry very much heat [8]. According to the scattering theory, opticalphonons decay mostly into two acoustic phonons. Of this process is knownthe experimental relaxation time τop (4 ÷ 20 ps) which is known fromRaman spectroscopy [9]; consequently this process will be modelled as arelaxation term in the optical energy density equation. Hence the transferof heat to the package is mostly governed by acoustic phonons, with highgroup velocity. In general, their scattering mechanisms can be distinguishedinto Normal and Resistive processes (which do not conserve momentum).Resistive processes are the combination of defect scattering, boundary scat-tering, and Umklapp processes. In silicon, at room temperature and above,Resistive processes are expected to be dominant. Due to the already men-tioned difficulties, we shall approximate again this scattering with a relaxa-tion. Regarding to the acoustic phonon relaxation time τac, some analyticalformula have been published, but we prefer to use an experimental valueτac = 50÷ 80 ps, obtained for silicon films at room temperature.

3. Moment equations

Starting from the transport equations (2),(3) one can obtain balanceequations for macroscopic quantities associated to the flow. By multiplying

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(2) by a weight function ψA(k) and (3) by φB(q), and integrating over R3,

one finds :

(14)∂MA

∂t+

∫

R3ψA(k)v

i(k)∂f

∂xidk−

e

~Ei

∫

R3ψA(k)

∂f

∂kidk

=∑

g

∫

R3ψA(k)Q

epg [f,Ng]dk

(15)∂MB

∂t+

∫

R3φB(q)u

ig(q)

∂Ng

∂xidq =

∫

R3φB(q)

{

Qpeg [f,Ng] +Qpp

g [Ng]}

dq

where

MA =

∫

R3ψA(k)fdk , MB =

∫

R3φB(q)Ngdq(16)

are the moments relative to the weight functions ψA(k) and φB(q). Since∫

D

ψA(k)∂f

∂kidk =

∫

∂D

ψA(k)fnidσ −

∫

D

f∂ψ(k)

∂kidk

with ni the outward unit normal field on the boundary ∂D of the domainD and dσ as the surface element of ∂D, supposing that f tends to zerosufficiently fast as k → ∞, the integral over ∂D vanishes and eq.(3) becomes

∂MA

∂t+

∂

∂xi

∫

R3ψA(k)v

i(k)fdk+e

~Ei

∫

R3f∂ψA(k)

∂kidk(17)

=∑

g

∫

R3ψA(k)Q

epg [f,Ng]dk .

If we choose 8-moments model with

ψA = (1, ~k, ε, εv)

one obtains from eq.(18) the following balance equations

∂n

∂t+∂(nV i)

∂xi= 0(18)

∂(nP i)

∂t+∂(nU ij)

∂xj+ neEi = nCi

P(19)

∂(nW )

∂t+∂(nSi)

∂xi+ neViE

i = nCW(20)

∂(nSi)

∂t+∂(nF ij)

∂xj+ neEjG

ij = nCiW(21)

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V. Di Stefano

where n is the electron density, V i the electron velocity, W the electronenergy, Si the electron energy flux, P i the electron crystal momentum, U ij

the flux of crystal momentum, F ij the flux of electron energy flux, CiP the

production of electron crystal momentum, CW the production of electronenergy, Ci

W the production of electron energy flux.To describe the phonon transport we have chosen a 4-moments model with

φB = (~ωg, ~ωgug) ,

which in our case writes

(22) φB = (~vs|q|, ~v2sq

i, ~ωop) .

Then equations (15) yield:

∂Wac

∂t+∂Qi

∂xi= C(23)

∂Qi

∂t+∂Mij

∂xj= Ci(24)

∂Wop

∂t= Cop(25)

where Wac is the acoustic phonon energy density, Qi the acoustic phononheat flux, Mij the acoustic phonon flux of heat flux, C the productionof acoustic phonon energy density, Ci the production of acoustic phononheat flux, Wop the optical phonon energy density and Cop the production ofoptical phonon energy density.

4. Maximum Entropy Principle and closure relations.

The moment equations (18)-(21),(23)-(25) are not closed. If we assumeas fundamental variables n, V i, W , Si, Wac, Q

i, Wop, which have a directphysical interpretation, the closure problem consists in expressing the high-order fluxes U ij , Gij , F ij ,Mij and the moments of the collisional termsCiP , CW , C

iW , C, C

i, Cop as function of the fundamental variables. A good,physically motivated way to obtain constitutive equations is by means ofthe Maximum Entropy Principle (MEP) [5,10]. It gives a form of the distri-bution function that makes the best use of the knowledge of a finite numberof moments. By using this ansatz, the closure equations for the fluxes ofthe electron transport write [11]

U ij =2

3Wδij , m⋆F ij =

10

9W 2δij , m⋆Gij =

5

3Wδij(26)

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and those for the acoustic phonons [12]

Mij =1

3v2sWacδij +

1

2v2sWac(3χ− 1)

(

QiQj

Q2−

1

3δij

)

(27)

χ =5

3−

4

3

√

1−3

4

(

Q

vsWac

)2

, Q =√

QiQi.(28)

Moreover, following a very crude approximation used in most hydrody-namic models [3,13], we shall assume the energy production terms to be ofrelaxation type. Since the electrons lose their energy to optical phonons, weshall assume

(29) nCW = H = −nW −W ⋆

op

τW, W ⋆

op =3

2~ωop

where W ⋆op is the optical phonon energy (59.7 meV), and τW the electron

energy relaxation time. Then, as said, optical phonons decay into acousticphonons, consequently the production term C is given by

C =Wop −Wac

τop(30)

and, since the energy of any system can only change because of theinteractions with some external agent, i.e. the total energy production(Rc = nCW + C + Cop ) must be zero, we obtain

Cop = nW −W ⋆

op

τW−Wop −Wac

τop(31)

where τop is the optical phonon relaxation time. The remaining productionterms are modeled as

nCiP = −m⋆nV

i

τP, nCi

W = −nSi

τS, Ci = m⋆v2s

nV i

τP−Qi

τac.

The relaxation times for the energy and momentum are those obtainedin [14], i.e.

τP = m⋆µ0e

TLTe

, τW =m⋆

2

µ0e

TLTe

+3

2

µ0kBev2s

TeTLTe + TL

(32)

where Te is the electron temperature which is related to the electron energyby

W =3

2kBTe +

1

2m⋆V 2,

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V. Di Stefano

TL is the lattice temperature and µ0(TL, ND) is the Caughey-Thomas elec-tron mobility. In order to have an estimate of the relaxation time for theenergy flux τS , we have introduced the ratio r=τS/τP . MC simulations havebeen performed in intrinsic (bulk) silicon [15], showing a weak dependenceon the lattice temperature. In the sequel we shall take r= 0.8.

5. The Extended Hydrodynamic model

By using the results of the previous sections, substituting eq.(19) witha linear combination of equations (19) and (21), we have the followingExtended Hydrodynamic Model (hereafter EHM):

∂n

∂t+∂(nV i)

∂xi= 0(33)

∂(nm⋆V i)

∂t+

∂

∂xj[

n(U ij − 2αm⋆F ij)]

+ neEj(1− 2αm⋆Gij)(34)

= −nP i

τP+ 2αm⋆nS

i

τS

∂(nW )

∂t+∂(nSi)

∂xi+ neViE

i = −nW −W ⋆

op

τW(35)

∂(nSi)

∂t+∂(nF ij)

∂xj+ neEjG

ij = −nSi

τS(36)

∂Wac

∂t+∂Qi

∂xi=Wop −Wac

τop(37)

∂Qi

∂t+∂Mij

∂xj= v2s

nP i

τP−Qi

τac(38)

∂Wop

∂t= n

W −W ⋆op

τW−Wop −Wac

τop(39)

together with the closure relations for the fluxes (26), (27), (28).The mathematical properties of the above system have been studied in [16],showing that the system is of hyperbolic type.

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Moreover, from eq.(35), it is possible to observe that the right-hand-siderepresents the heat generation rate

H = −nW −W ⋆

op

τW,(40)

which is different from the heat generation rate given in (1).In the one-dimensional and stationary case the model writes as follows:

d(nV )

dx= 0(41)

2

3

d(nW )

dx− ne

dφ

dx= −

nm⋆V

τP(42)

d(nS)

dx− neV

dφ

dx= −n

W −W ⋆op

τW(43)

10

9

d(nW 2)

dx−

5

3neW

dφ

dx= −

nm⋆S

τS(44)

dQ

dx=Wop −Wac

τop(45)

dM

dx= v2s

nm⋆V

τP−

Q

τac(46)

nW −W ⋆

op

τW−Wop −Wac

τop= 0 .(47)

From eq. (47) we obtain

Wop =Wac + nτopτW

(W −W ⋆op)

and the system, in the unknowns (n, V,W, S,Wac, Q, φ), yields

fn =d(nV )

dx= 0(48)

fV =2

3

d(nW )

dx− ne

dφ

dx+nm⋆V

τP= 0(49)

fW =d(nS)

dx− neV

dφ

dx+ n

W −W ⋆op

τW= 0(50)

fS =10

9

d(nW 2)

dx−

5

3neW

dφ

dx+nm⋆S

τS= 0(51)

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V. Di Stefano

fWac=dQ

dx− n

W −W ⋆op

τW= 0(52)

fQ =dM

dx− v2s

nm⋆V

τP+

Q

τac= 0(53)

fφ = ǫd2φ

dx2+ e(ND − n) = 0(54)

together with the closure relations (27),(28). The acoustic phonon energydensity can be related to the acoustic phonon temperature, via the specificheat at constant volume cV [17], i.e.

dWac = cV (Tac)dTac(55)

and in a neighborhood of the room temperature T0 we have

Tac = T0 +1

cV (T0)(Wac −W 0

ac)

where W 0ac is the acoustic phonon energy density at equilibrium.

For boundary conditions we take charge neutral contacts in thermal equi-librium with the ambient temperature, i.e.

n(0) = n(L) = ND , Tac(0) = T0 = 300

W (0) =W (L) =3

2kBT0 , S(0) = S(L) = 0 , Wac(0) =W 0

ac

φ(0) =kBT0e

logn

ni, φ(L) =

kBT0e

logn

ni+ eVb

where L is the device length, ni is the intrinsic carrier concentration insilicon (1.45 · 1010cm−3), and Vb is the bias potential.The main problem is how to fix the phonon heat flux boundary condition,because in the Extended Hydrodynamic Model (EHM) this quantity repre-sents an independent variable.Physically, one expects some heat flow at the boundaries, which in principleis not known. If the following Dirichlet boundary conditions are fixed

Q(0) = Q(L) = 0 ,

Wac(L) =W 0ac ,

then, the results obtained with the EHM model show oscillations, whichare likely due to insufficient modeling of the boundary.

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One possibility is to fix Robin boundary conditions, which interpolate bet-ween Dirichlet and Neumann boundary conditions. These conditions are:

dWac

dx+ α1(Wac −W 0

ac) = 0 at x = L

dQ

dx+ α2Q = 0 at x = 0, L

dTacdx

+ α3(Tac − T0) = 0 at x = L

where α1, α2, α3 are suitable parameters.By using this ansatz, the oscillations in the EHM model disappear.In order to find a numerical solution, this system has been discretized byusing central differences and the box integration method, following the nu-merical scheme used in [18,19]. The variables n, φ, W , S, Tac are defined atthe grid points i = 0, 1, ..., N −1, N . The boundary conditions specify thesevariables at i = 0 and i = N . The velocity V is defined at the midpointsof the elements li (i = 1, ..., N) connecting the grid points i− 1, i. Then allequations are enforced at the interior grid points i = 1, 2, ....N − 1, whileeq.(49) is enforced at the midpoints of the elements li, i = 1, ..., N .The linearization of the system is done via Newton’s method

Jδ~x = −~f

where J is the Jacobian , δ~x is the correction vector, and ~f is the residualvector

δ~x = (δn, δV, δW, δS, δφ, δTac)T , ~f = (fn, fV , fW , fS , fφ, fTac

)T .

In the Jacobian, after discretization, each block is diagonal, bi-diagonal ortri-diagonal. As an initial guess for the solution we have taken

n = ND , V = 0 , W =3

2kBT0 , S = 0 ,

φ =kBT0e

logn

ni, Tac = T0 = 300K .

Starting from this initial guess, the new solution is given by

~xnew = ~xold + t δ~x

where t ∈ (0,1) is a damping factor, chosen to ensure that the norm ofthe residual vector ~f decreases monotonically. However, because the initialguess is very far from the real solution, we have used the continuationmethod: at the beginning the applied external potential is set to a low value,then it is increased, step by step, to reach the given potential difference.

121

V. Di Stefano

6. Simulations and results

The simplest benchmark used in the literature is given by the one-dimensional n+ − n − n+ silicon diode. This unipolar device consists oftwo highly doped regions n+, called cathode and anode, connected by aless doped region n, called channel. In our simulations, the n+ regions are100 nm-long doped to a density ND = 1019cm−3, while the channel is 100nm-long doped to a density ND = 1016cm−3, and the applied bias Vb = 1.2V.Such device has been simulated with EHM model. Moreover, in the samegraphics there are the data obtained by using an isothermal Monte Carlosimulator [20], just to have a more physical comparison.In the figure 1 we plot the electron velocity along the device. It is evidentthat the behaviour of the EHM is similar to the behaviour of the MonteCarlo simulation, with a difference in correspondence of the second junc-tion, in which EHM produces a little spike.The figure 2 shows that the electron temperature for EHM is below theisothermal MC data, for which the maximum is about 0.33 eV. This is dueto the fact that, since the lattice is heated up, the atoms oscillate more fre-quently from their equilibrium position: consequently there are more scat-terings with the electrons and more energy dissipation.In figure 3 we plot the potential. It is evident that there are not relevantdifferences between the EHM model and the Monte Carlo simulator.

The figure 4 shows the heat generation rate H produced in the device,

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

x (nm)

velo

city

(1

07 c

m/s

ec)

EHMMC isoth.

Fig. 1. The mean electron velocity versus the position, evaluated with EHM and an

isothermal MC simulator.

given by eq. 40 and eq.(1) (with R = G = 0). This figure reveals a smallnegative heat generation at the beginning of the channel, with a minimum

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DOI: 10.1685/2010CAIM482

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x (nm)

ele

ctro

n te

mp

(e

V)

EHMMC isoth.

Fig. 2. The electron temperature versus the position, evaluated with EHM and an

isothermal MC simulator.

0 50 100 150 200 250 300−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x (nm)

po

ten

tial (

eV

)

EHMMC isoth.

Fig. 3. The electric potential versus the position, evaluated with EHM and an isothermal

MC simulator.

of -0.5 · 1011 W/cm3 for the formula given in eq.(1) and about -0.3 · 1011

W/cm3 for the EHM model. This is a thermoelectric (Peltier) effect dueto electron injection across the potential barrier from the cathode to thechannel. This is the so called built-in potential, shown in figure 3 at x =100 nm. Electrons diffusing against the energy barrier extract the energyrequired to move up the conduction band slope from the lattice, throughnet phonon absorption. Moreover, the EHM model predicts that most ofthe Joule heat is dissipated into the anode region, and not in the channel,as shown by the eq.(1) and confirmed by MC studies [21].The Fig. 5 shows the acoustic phonon temperature.In figure 6 we plot the heat flux produced in the device by using the EHM

123

V. Di Stefano

0 50 100 150 200 250 300−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

x (nm)

He

at G

en

era

tion

ra

te (1

01

1 W

/cm

3)

EHM J x E

Fig. 4. The heat generation rate H versus the position, evaluated with EHM by using

eq. 40 and an isothermal MC simulator.

0 50 100 150 200 250 300299.8

300

300.2

300.4

300.6

300.8

301

301.2

301.4

301.6

301.8

x (nm)

Aco

ust

ic p

ho

no

n te

mp

(K

)

EHM

Fig. 5. The acoustic phonon temperature Tac versus the position obtained by using the

EHM model.

model: in this case the heat flux is an independent variable of the model.The sign of the heat flux is consistent with the corresponding acousticphonon temperature, as in Fig. 5.

7. Conclusions

We have introduced an Extended Hydrodynamic model to study theheat conduction in thin silicon semiconductor devices.In this model the closure of the high-order moments has been achieved withthe Maximum Entropy Principle, whereas the closure of the productionterms with relaxation terms, consistently to the first principle of Thermo-

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0 50 100 150 200 250 300−0.89

−0.88

−0.87

−0.86

−0.85

−0.84

x (nm)

Aco

ustic

hea

t flu

x (1

05 W/c

m2 )

EHM

Fig. 6. The acoustic heat flux density versus the position evaluated with EHM.

dynamics.The most important differences between the EHMmodel and the isothermalMC simulator are in the electron velocity and in the electron temperature.In fact, in the first case, as seen, EHM produces a bigger spike with respectto MC, near the second junction, while the electron temperature is smaller.Moreover, the EHM model predicts that most of the Joule heat is dissipatedinto the anode region and not in the channel.These behaviours should be studied also in the 2D case for more realisticdevices. Work along this line is in progress and will be presented in the nextfuture.

Acknowledgements.

This work has been supported by ”Progetti di ricerca di Ateneo” Uni-versita degli Studi di Catania, and ”Progetto Giovani Ricercatori GNFM2009”.

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