VLSI (Very Large Scale Integrated) chipsincorporate hundreds of millions ofsemiconductor devices (transistors,diodes, optical devices, etc.). To predictthe performance of the VLSI circuits, thecurrent-voltage (I-V) characteristics ofthe semiconductor device are required.Semiconductor device simulation codesprovide a way of predicting I-V curvesas device parameters are varied, withouthaving to fabricate the device first.(These parameters include semiconduc-tor material, size, doping and geometry.)Thus, many different designs for devicesand circuits can be explored efficientlyusing computer simulations. Promisingdesigns then can be selected for actualfabrication and testing.
Many devices including MOSFETs(metal oxide semiconductor field effecttransistors) and MESFETs (metal semi-conductor field effect transis-tors) can be modeled using asemiclassical approach. Thatis, the charge transport ofelectrons and/or holes isdescribed by the classicalBoltzmann equation, or aclassical hydrodynamicmodel with effective massesfor electrons and holes inputfrom quantum theory. Thecharge transport equations are then cou-pled to Poisson's equation for the elec-trostatic potential. A Monte-Carlo (parti-cle based) approach to solving theBoltzmann equation is presented byGoodnick, Saraniti, Vasileska and Aboudstarting on page 12. Quantum semicon-ductor devices like resonant tunnelingdiodes and transistors, HEMTs (highelectron mobility transistors), and super-lattice devices are increasingly beingused in VLSI chips. These devices relyon quantum tunneling of charge carriersthrough potential barriers for their oper-ation. Advanced microelectronic applica-tions include multiple-state logic andmemory devices, and high frequencyoscillators and sensors. In addition, theincreasing miniaturization enhances“unwanted” quantum effects in standardMESFETs and MOSFETs (for instance,current leakage in the MOSFETs due toquantum tunneling through the oxideinsulator between the gate and the chan-nel). Both types of quantum effects mustbe simulated in order to design robustultra-small semiconductor devices.
A fundamental approach to model-ing quantum transport of electrons andholes in semiconductor devices is the
Wigner-Boltzmann equation, the quan-tum generalization of the Boltzmannequation. The Wigner-Boltzmann equa-tion differs from its classical counterpartprincipally in that particle transportcouples to the potential energy in anon-local way; i.e., the values of theelectrostatic potential energy integratedover a finite region in space determinethe transport at a point in space.
Simulating the kinetic equations (theclassical Boltzmann equation or thequantum Wigner-Boltzmann equation)is computationally expensive. This isbecause the distribution function for theelectrons or holes is a function of six
variables (three spaceand three momentum)plus time for a three-dimensional device.Thus, a hydrodynamicapproximation to thekinetic equations,where the density,velocity and tempera-ture of a charge carrier are functionsonly of three spatial dimensions plustime, offers enormous computationalspeedups in simulating devices. Thisapproach has worked well in modelingsemiclassical devices using the classicalhydrodynamic model, and quantumdevices using the quantum hydrody-namic model.
The classical hydrodynamic modelhas become a standard industrial simu-lation tool that incorporates important“hot electron” phenomena in submicronsemiconductor devices. Hot electroneffects are missing in the simpler drift-diffusion model, which assumes theelectron gas is always at ambient tem-
perature. The hydrodynamic modelconsists of nonlinear hyperbolic conser-vation laws for particle number,momentum, and energy (with a heatconduction term), coupled to Poisson'sequation for the electrostatic potential.In the momentum and energy conserva-tion equations, charge carrier scatteringby phonons is modeled by relaxationtime approximations. The electric fieldappears as terms on the right-handsides (see equations athttp://math.asu.edu/~gardner/IMA.pdf).
The nonlinear hyperbolic modessupport shock waves or “velocity over-shoot” in the parlance of semiconductordevice physicists. The hydrodynamicmodel can be extended to includequantum-tunneling effects by addingquantum corrections.
To accurately compute solutionsincluding the sharp reso-lution of waves, a con-servative hyperbolicmethod from gasdynamics can beemployed. Hyperbolicmethods like the essen-tially non-oscillatory(ENO), piecewise para-bolic, and Tadmor cen-tral methods are well
suited for simulating the transientclassical and quantum hydrody-namic models. Steady-state solu-tions may be obtained as theasymptotic large time limit; i.e.,the time-dependent equationsare simulated to a large value ofthe time at which the solutionremains steady.
The classicalhydrodynamic model
The hydrodynamic modeltreats the propagation of electronsand/or holes in a semiconductordevice as the flow of a charged com-pressible fluid. The model exhibits hotcarrier effects missing in the standarddrift-diffusion model. The hydrody-namic description should be valid fordevices with active regions greaterthan 0.05 microns.
The hydrodynamic model is equiva-lent to the equations of electro-gasdynamics. The electron gas has a soundspeed and the electron flow may beeither subsonic or supersonic. In general,a shock wave develops at the transitionfrom supersonic flow to subsonic flow.
extensively used to study the n+/n/n+diode that models the channel of a fieldeffect transistor. The diode begins witha heavily doped n+ source region, fol-lowed by a lightly doped n channelregion, and ends with an n+ drainregion.
A hydrodynamic model simulation ofa steady-state electron shock wave,shown here in a plot of the electronvelocity, in an n+/n/n+ diode is pre-
sented in Fig. 1. With a 1 volt biasacross the diode, the shock wave isfully developed in Si (silicon) at 77 Kfor a 1.0 micron channel, and at 300 Kfor a 0.1 micron channel (not shown).The hydrodynamic simulation is com-pared with a Monte Carlo simulation ofthe Boltzmann equation by Steven Laux(IBM Thomas J. Watson ResearchCenter) using the DAMOCLES program.
The hydrodynamic model may beused to simulate Si and GaAs (galliumarsenide) MESFETs (see Fig. 2). ForGaAs devices, the hydrodynamic modelincorporates conservation laws forupper and lower valley electrons. Thevarious charge carriers are coupledthrough the electric field and throughsource terms in the conservation laws.
In submicron gate Si and GaAsMESFETs, the hydrodynamic model
does a better jobof predictingenhancements inthe drain currentvs. drain voltagethan the drift-dif-fusion model. Theh yd r od yn am i cmodel results pre-dict more accu-rately the experi-mentally mea-sured fasterswitching timesfor GaAs devicesthan the drift-dif-fusion model.
Two -d imen -sional hydrody-namic simulationsof electron flowin a submicrongate MOSFEThave been car-ried out by anumber ofresearchers. Thehyd rodynam i csimulations cor-rectly show thatelectron concen-trations areenhanced aroundthe MOSFETchannel by a fac-tor of 10 or morewith respect tod r i f t - d i f f u s i onsimulations. Thesimulations alsoreveal the largetemperature vari-
ations that are known to occur in theMOSFET, with a sharp peak betweengate and drain.
The quantum hydrodynamic model
A new version of the quantumhydrodynamic (QHD) model, called thesmooth QHD model, was recentlydeveloped by Christian Ringhofer and
the author. The goal is to rigorouslyhandle discontinuities in the potentialenergy which occur at heterojunctionbarriers in quantum semiconductordevices; these discontinuities causemathematical problems for hydrody-namic equations. We constructed a“quantum Maxwellian” solution to theBloch equation, which governs thermalequilibrium in quantum statisticalmechanics. Dynamic evolution of thedistribution function for charge carriers(including departures from thermalequilibrium) is governed by the Wigner-Boltzmann equation. The smooth QHDmodel is derived from a momentexpansion of the Wigner-Boltzmannequation using the quantum Maxwelliansolution to close the moments. (Amoment expansion involves takingaverages of the kinetic equation toobtain transport equations for chargecarrier density, momentum density andenergy density. It is the standardmethod for deriving fluid dynamicsfrom the Boltzmann equation.)
Feynman introduced into quantumstatistical mechanics a smoothing of theclassical potential as a way of partlyaccounting for the long range effects ofquantum mechanics. However, as henoted, his smoothed potential “fails inits present form when the (classical)potential has a very large derivative asin the case of hard-sphere interatomicpotential” or heterojunction potentialbarriers in semiconductor devices.
The smooth QHD model involves asmoothing of the classical potentialover both space and inverse tempera-ture. The double integration providessufficient smoothing so that the leadingsingularity in the smooth potential can-cels the leading singularity in the classi-cal potential at a barrier (see Fig. 3).This cancellation leaves a residualsmooth effective potential with a lowerpotential height in the barrier region.The lower barrier height, along with thesmoothing, makes the barriers partiallytransparent to the particle flow and pro-vides the mechanism for particle tunnel-ing in the QHD model. Note that theeffective barrier height approaches zeroas the ambient temperature T goes tozero. This effect explains why particletunneling is enhanced at low tempera-tures. As T goes to infinity, the effectivepotential approaches the classical doublebarrier potential and quantum effects inthe QHD model are suppressed.
The smooth QHD equations havethe same form as classical electro-gas
18 IEEE POTENTIALS
Fig. 1 Hydrodynamic (blue) and DAMOCLES (red) simulations ofan electron shock wave (velocity overshoot) in a Si diode.Velocity is in 107 cm/s.
Fig. 2 Hydrodynamic simulation of electron density in 1018 cm-3in a Si MESFET at 300 K by Anne Gelb (Arizona State University)and the author. x and y are in microns. S, G, and D label thesource, gate, and drain, respectively.
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ocity
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dynamics (see http://math.asu.edu/~gardner/IMA.pdf for the smooth QHDequations, as well as the Wigner-Boltzmann equations). Scattering ismodeled by the standard relaxationtime approximation, with momentumand energy relaxation times. Explicitfactors of Planck's constant � appearonly in the fourth and higher momentequations. In the first three moments, �2
appears in the expressions for the stresstensor, energy density and heat flux.The heat conduction term consists of aclassical Fourier law plus a new quan-tum contribution.
Simulations of a GaAs resonant tun-neling diode with AlGaAs (aluminumgallium arsenide) double barriers at 300K are presented in Fig. 4. The barrierheight is equal to 280 meV. The diodeconsists of n+ source (at the left) anddrain (at the right) regions, and an nchannel. The channel is 200 Angstromslong, the barriers are 25 Angstromswide, and the quantum well betweenthe barriers is 50 Angstroms wide.There are 50 Angstrom spacers betweenthe barriers and the contacts.
Figure 4 displays the experimentalsignal of quantum resonance—nega-tive differential resistance: a region ofthe current-voltage curve where thecurrent decreases as the applied volt-age is increased. The resonant peak ofthe current-voltage curve occurs asthe electrons tunneling through thefirst barrier come into resonance withthe energy levels of the quantum well.The resonant peak in Fig. 4 occurs at150 millivolts. As the voltage biasincreases above 150 millivolts, the res-onance effect rapidly decreasesbecause the right barrier height is pro-gressively reduced. As a result, elec-trons tunnel out of the well throughthe thin portion of the effective para-bolic well (see Fig. 5).
ConclusionThere is a technologically important
range of parameters (device size, ambi-ent temperature, potential barrierheight, applied voltage, semiconductormaterial, etc.) in which either the clas-sical hydrodynamic model or the quan-tum hydrodynamic model gives solu-tions and current-voltage curves thatare very close to those given by the(Wigner-) Boltzmann-Poisson system.In this parameter regime, semiclassicaland quantum semiconductor devicescan be efficiently simulated using thehydrodynamic approximation. This
approximation isorders of magni-tude faster thansimulating thekinetic equa-tions. (A MonteCarlo simulationof the Boltz-mann equationfor a 2D FETtakes about aweek of CPUtime on thefastest worksta-tions, while thehydrodynamicmodel simula-tion runs inabout 15 min-utes.) This speed allows the deviceengineer to rapidly explore many differ-ent device designs.However, when theh y d r o d y n a m i capproach fails to bevalidæfor ultra-smalllength scales or forvery small numbers ofthe majority chargecarrieræthe more fun-damental and morec o m p u t a t i o n a l l yexpensive Boltzmannor Wigner-Boltzmannequation should besolved.
Read more about it
• P. A. Markowich, C. Ringhofer,and C. Schmeiser. SemiconductorEquations . Springer Verlag,Wien, 1990.
• C. L. Gardner and C.Ringhofer, “Smooth quantumpotential for the hydrodynamicmodel,” Physical Review, vol. E53, pp. 157-167, 1996.
• <http://math.asu.edu/~gard-ner/IMA.pdf>
About the authorCarl L. Gardner is a Professor
of Mathematics at Arizona StateUniversity. He obtained his B.A.in Mathematics from DukeUniversity (1973) and his Ph.D.in Physics from MIT (1981). His currentresearch interests involve the modelingand simulation of classical and quan-tum semiconductor devices, ion trans-port in biological channels, and super-sonic astrophysical jets.
1
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Fig. 3 Smooth effective potential forelectrons in GaAs for 50 Angstrom wideunit potential double barriers and 50Angstrom wide well.
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100
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Fig. 4 Current density in kiloamps/cm2vs. voltage using the smooth QHDmodel for the resonant tunneling diode.
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Fig. 5 Smooth effective potential U in eVfor applied voltages between 0 and 400millivolts for the resonant tunnelingdiode. x is in 5 Angstrom.
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