An Efficient Method for the Coupling of a Fully-Explicit Time-DomainSolid-State Hydrodynamic simulator with FDTD EM Solvers

Brian S. McGarvey, Manos M. Tentzeris

GEDC, School ofECE, Georgia Institute of Technology, Atlanta, Georgia, 30332, U.S.A.

B. Previous Efforts

The HDM model itself is non-trivial to discretize andachieve stable, consistent and convergent results. Previous

(6)

(5)

(4)

(1)

Wiedemann-Franz law5kBnT

K=----..;;;;....--2rn*vp {w)

Poisson's Equation

V2lp q-=-(N -n.)ax 2 BoEr D I

another, re-combination, and generation effects. The collisionterms in equations (2), (3) describe the average momentum orenergy lost or gained via interactions with the lattice or carrierinteraction [4]. The process of taking the mathematicalmoments of the BTE always leaves undefined variables thatneed to be approximated. As a result of the calculation of themoments of the BTE, two variables are left undefined: Heatflux and Electrostatic forces. The Heat flux is approximatedusing Fourier's Law (4) and Wiedemann-Franz Law (5) toexpress the internal thermal conductivity [2,3]. Poisson'sEquation (6) is used to approximate the internal electrostaticeffects of the plasma. With equations (1-6) the hydrodynamic(HD) model is fully defined and mathematically self­consistent and ready for discretization and simulation prior tointegration into an FDTD-EM solver.

Conservation of Mass:

On =-V.(v n)+(anJat d at c

Conservation of Velocity

ovd _ Vd n (_) eF----v· Pd +-

at m * m * (2)

2 ( 1 (- )2 J (a(vd )J---\7 nw--m*n V d +--3nm * 2 at c

Conservation of Energy

~ = -vd'Vw- 3~ V-[(nvd- ~ VJ(w-~m*(vJ2 )](3)

+ eft . v + (a(w)Jd at c

Heat Flux Approximation: Fourier's Law

nq=-KVT

I. INTRODUCTION

Abstract This paper examines the implications ofdecoupling the FDTD grids between and FDTD Electromagnetic(FDTD-EM) and an FDTD Hydrodynamic semiconductorsimulator. Excitation methods and responses are examined todetermine feasibility for decoupling the space and time grids forsignificant computational savings. Additionally, a new fully­explicit leap-frog discretization of the Hydrodynamic model is isbenchmarked with respect to several well known discretizationmethods.

Index Terms FDTD methods, Time-Domain,Hydrodynamics, Boltzmann equation, electromagnetic coupling,Magnetohydrodynamics.

The advent of nanostructures and the integration of exoticsolid state devices in state-of-the-art RF, microwave andoptoelectronic devices have dictated the integration ofelectromagnetic solvers with solid-state simulators. Despitethe numerous efforts in the past, there are still issuesremaining with the numerical accuracy and stability of thecoupled schemes. The dominant challenge is the fact that thetime and space gridding requirements usually differ by severalorders of magnitude for the two systems. This paper willdemonstrate an effective method for decoupling the time andspace discretization of the two systems. The hydrodynamicsimulator was chosen as it provides a wider set of submicronor nano-scale effects that can be included in the simulationgaining a more accurate result of the device or circuitcharacteristics in-situ.

A. Model Description

The hydrodynamic model (HDM) approximates thesemiconductor as a highly charged gas or plasma flowingthrough the semiconductor lattice. The model is typically usedwhen a Monte-Carlo approximation is not practical, such as intime-domain simulations. The original mathematical modelwas developed by Blotekjaer [1] for two-valley GaAssemiconductors and was extended to single-band siliconsemiconductors later [2].

The mathematical model is derived by taking the firstseveral moments of the Boltzmann Transport Equation (BTE)producing equations (I) - (3) [3]. The collision terms in eachof these equations describe the average carrier-carrier, carrier­lattice interaction. In a multi-band device the collision term inthe conservation of mass would allow for the loss or gain ofmass due to the carriers' transition from one energy band to

978-1-4244-1780-3/08/$25.00 © 2008 IEEE 49

Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on December 15, 2008 at 17:27 from IEEE Xplore. Restrictions apply.

approaches to discretize the highly coupled non-linear modelshown in (1 )-(6) were typically based on the staggering of thescalars and vector parameters at full and half nodal points orsetting all variables at the nodal points at the same time step.This approximation is based on the assumption the variablesare unable to be decoupled in time causing the model to be 2nd

order accurate in space, but only 1st order accurate in time.This discretization fails to provide a numerically stable model.To correct for this problem advanced discretization techniquesare typically used in time. As examples, Tomizawa used aCrank-Nicolson semi-implicit method [3], Aste & Vahldieckused a weighted upwind scheme [5], and EI-Ghazaly used ahybrid method that included both a standard upwind methodand a Lax-Wendroffmethod [2].

II. NEW HDM METHODOLOGY

The novel method shown here extends the basic tenets ofVee's Leapfrog method in space and time Maxwell's Curlequation discretization to the HDM [6,7]. If one accepts thatthe equations can be sufficiently decoupled to stagger thegridding in space, the staggering of these variables in time isthe next natural extension. The developed simulator staggersthe scalar variables on the full nodal points and the vectorvariables on the half nodal points of the HDM grid in time andspace as seen in Fig. 1.

1D Gridding~

1M

;-1 i ;+1Spatial DIscretization

Vector GridScalar Grid

Fig. 1. Staggered Gridding technique showing vector and scalaroffsets in time and space.

III. MODEL COMPARISON

Typically any new HD model is tested against the wellpublicized n-i-n diode case. Several of the previously

978-1-4244-1780-3/08/$25.00 © 2008 IEEE

discussed methods were implemented and compared by theauthor.

Three separate models were implemented. A reproductionof the time-domain discretization as described by Tomizawawas used for the Lax- WendrofJ. The asymmetric Upwindscheme described by Aste & Valhdieck. And the Leap Frogdiscretization described in the previous section. All modelsused the same physical constants, and included EI-Ghazaly'seffective mass approximation [2].

Evaluating a model for integration into the FDTD-EMlattice requires the model to be stable, convergent, andconsistent with physics. At a minimum the models need toproduce valid results for zero bias, DC ramp to somereasonable DC bias voltage, and for the DC bias plus a sineexcitation of reasonable amplitude.

A. Zero DC Bias Steady State

All hydrodynamic models provided reasonable results at theinput and output ports of the ballistic diode for zero DC bias.The Upwind model has excessive carrier velocities at thetransitions regions between the highly doped areas and thenon-doped transition regions; however, when a DC bias wasapplied the results improve.

B. Ramp Excitation to 1VDC Steady State

The next test excited each of the implemented discretizationmethods for the same n-i-n diode with a basic DC ramp fromOV steady state to 1V. Ramp times were varied from lOOns tolOps. Fig 2 shows the results for the fastest excitation rateeach model could support for stability up to some reasonabletime and acceptable convergence.

Ramp Excitation

,//

0.8 f //

......-... /GQ) 0.6 / /0) I /m

//o 0.4

I/ -LeapFrog> /

I / --UpWind0.2 .I / - - Lax-Wendroff/

°0 20 40 60 80Time (ps)

Fig. 2. Maximum stable ramp excitations showing the maximumramp rates the produced reasonably stable or convergent solutions.

The results of this test provide unique insight into eachmodel's behavior. The Lax- WendrofJ discretization providesstable and convergent results provided the ramp rate is eitherslow enough or the steady state time is limited. Fig 3 shows

50

Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on December 15, 2008 at 17:27 from IEEE Xplore. Restrictions apply.

Time Step(s)5.0e-13

4.08e-17

Space Step (m)0.150e-34.08e-10

FDTD-EMFDTD-HDM

C. AC Excitation Plus IV DC-Bias

The unstable Lax-Wendroffmethod results were omitted asthe results were omitted due to the fact that this technique isnot convergent. A 20GHz sine wave was used to validate andcompare the initial results the excitation was a piecewiselinear approximation similar results and the DC excitation tothe ramp in which the excitation step size was dependent uponthe HDM timestep, effectively making the excitation quitesmooth. Both models show similar and good results theresultant curves are shown in Fig 5 with the results for themethod for coupling the system.

V. RESULTS

Fig 5 shows the current density response of the HDMmodels to the excitation. Both models under investigationprovide similar results that are expected and consistent withprevious published results [2,3,5] and theory; thereby,validating the new discretization method.

The models were excited with three approximations of thepiecewise continuous signal. The best approximation was theSmooth signal that is considered the reference as the piecewise

TABLE ISPACE AND TIME STEP DISPARITY

Decoupling the time and space grids requires an effectivemethod of exchanging energy between the models at theHDM input and output ports. Picket-May [8] described astraightforward method for introducing Thevenin and Nortonequivalent circuits into the FDTD-EM lattice and SPICEmodels. The previous section demonstrated the stability of theHD models under test to a very smooth excitation with a­priori knowledge of the excitation in a Thevenin circuitequivalent. To evaluate coupling methodologies, the ACexcitation (Smooth-HDM) used in the previous section will bediscretized' in a stepwise (Step-EMS) and piecewise linear(Linear-EMS) approximation with a time step for an FDTD­EM with a maximum frequency of 100 GHz.

IV. COUPLING METHODOLOGY

EI-Ghazaly's approach for coupling the systems included aunified grid in time and space between the two models. Thetwo models CFL conditions have disparate space and timestepping requirements as shown in Table 1, with a maximumfrequency of interest in the FDTD-EM simulator of 100 GHz.Eliminating the unified grid requirement significantly reducesthe computational requirements of an integrated simulator.

Supersonicegion

Uncontrolled OscllationFrcm La-WendrottMethod

Velocity (vdlc

): Vbias = 1V

100 200 300 400 500 600Device Length (nm)

Fig. 4. Steady-state Velocity profile of n-i-n diode for a muchslower ramp excitation and much longer execution time. The Lax­Wendroff method tends to oscillate when sufficient execution timehas elapsed.

5

°0 100 200 300 400 500 600Devk;e Length (nm)

2.5

x 10$ Velocity (vCIX): Vbias:: 1V6 4'''''''W··'-···~······-·'··T·-·"········· ,,·········'..'f·········..·,···,·,,···,',..••..··'''r···.. ·•

Fig. 3. Steady-state Velocity profile of n-i-n diode modelsshowing the non-physical convergent results for the Upwind methodwhen the ramp excitation rate is too steep.

good results. However, Fig 4 shows the effects of allowing forlonger execution time. The model begins to oscillate in adivergent manner.

The Upwind method shows other unique results. Uponcloser examination of Fig 3, the results show the non-physicaleffects of using an asymmetric upwind method. The devicebecomes a perfect current sink at both the input and outputports as the carrier velocity is positive at one end and negativeat the other end. The result was convergent even for excessivesimulation time (large number of time-steps). Fig 4 shows thatincreasing the Zero DC Bias time and slowing the ramp ratevalid results can be obtained for Upwind model; however, theLax-Wendroff method's tendency to oscillate andsubsequently diverge.

978-1-4244-1780-3/08/$25.00 © 2008 IEEE 51

Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on December 15, 2008 at 17:27 from IEEE Xplore. Restrictions apply.

VI. CONCLUSION

Fig. 5. Current density response to the reference excitation, apiecewise linear approximation and a stepwise linear approximation.The piecewise linear approximation and the reference excitation aresimilar; however the stepwise approximation shows significantoverestimation in the change induced by the excitation.

REFERENCES

[1] K. BlotekJaer, "Transport Equations for Electrons in Two­Valley Semiconductors," IEEE Trans. On Electron Devices,Vol. 12, pp. 38-47, 1970.

[2] A.Taflove and S.EI-Ghazaly, "Advances in ComputationalElectro-dynamics: The finite-diffemce time domain method",Norwood, Ma: Artech House Inc, 1988.

[3] K. Tomizawa, "Numerical Simulation of Submicron Devices",Norwood Ma: Artech House Inc, 1993.

[4] G.Baccarani and M.R.Wordeman, "Investigation of steady-statevelocity overshoot in Silicon," Solid-State Electronics, Vol 28,1985, pp. 407-416.

[5] A.Aste and R.Vahldieck, "Time-domain simulation of fullhydrodynamic model," Int. 1. Numer. Model. 2003; Vol. 16,pp.l61-174, 2003.

[6] B.McGarvey and M.Tentzeris, "Coupling of Solid-State andElectromagnetic Equations for Simulation of Wireless PackagedGeometries", Proc. of the 2001 European MicrowaveSymposium, VoI.I, pp.217-220, September 2001, London,England.

[7] F. Chen, "Introduction to Plasma Physics and ControlledFusion", New York: Springer, 2nd. Ed., 2006.

[8] A.Taflove et aI., "Advances in Computational Electro-dynamics: The finite-difference time domain method",Norwood, MA: Artech House Inc, 2nd Ed., 2000.

ACKNOWLEDGEMENT

The authors wish to acknowledge the NSF CAREER ECS­9984761, the NSF ECS-0313951, and the Georgia ElectronicDesign Center (GEDC)

Coupling the energy between the models can be easilyimplemented by using Norton or Thevenin equivalent circuitsif a piecewise linear approximation is used to excite the HDMmodel embedded in the FDTD-EM lattice.

the computational savings is approximately 10 orders ofmagnitude per FDTD-EM cell per FDTD-EM time step as theHDM can be typically embedded in a single FDTD-EM cell.This reduces each of the models to their respective CFLconditions instead of imposing the smallest CFL condition onboth models.

80

LP: Leapfrog. UP: Upwind - Current Density Response

·_·'UP:Smooth.....UP:Step EMS- .' UP:Linear EMS

,,-, LP:Smooth~'lP:Step EMS

,"-,wlP:Linear EMS----

-12000 -----w-'~'-·--'30"'----~4...L.o-·-50-~oo___ro_

Time (ps)

·1(X)()

A novel method for coupling a FDTD-EM and FDTD­HDM models has been presented removing the previousrequirement for a unified spatial and temporal grid betweenthe models. Decoupling the grids leads to significantcomputational savings. For practical RF devices «200 GHz)

approximation is based on the HDM time step. The results areshown in red and both models have smooth and valid results.The next best approximation is the piecewise linearapproximation. This method, while it is slightly sensitive tothe relative quick change in slope of the excitation, hasacceptable results that are consistent with the referenceresults. Applying a single-step excitation voltage to the HDMyields unacceptable oscillations in comparison to the referencesolutions.

978-1-4244-1780-3/08/$25.00 © 2008 IEEE 52

Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on December 15, 2008 at 17:27 from IEEE Xplore. Restrictions apply.