Hot Electron Modelling of HEMTs

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 287-293Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Hot Electron Modelling of HEMTsERIC A. B. COLEa’*, CHRISTOPHER M. SNOWDENb’* and SHAHZAD HUSSAINb’*

aDepartment of Applied Mathematics, bSchool of Electronic and Electrical Engineering,University of Leeds, Leeds LS2 9JT, UK

The hot-electron two-dimensional HEMT with recessed gate is modelled by solvingthe Poisson, current continuity and energy transport equations consistently with theSchr6dinger equation using a finite difference scheme. New expressions are used for theenergy densities inside and outside the quantum wells. A method is described forpinning the conduction band at the contact edge to produce an extremely stablenumerical solution. Results are presented for an eight layer GaAs-A1GaAs-InGaAsdevice.

Keywords: HEMT; Schr6dinger; Poisson; Current-continuity; Hot electron

1. INTRODUCTION

Considerable problems are encountered in themathematical and numerical modelling of recessedgate High Electron Mobility Transistors (HEMTs)in which hot electrons are considered. Even whenthe most exacting initial conditions are used, it isoften difficult to get the quantum well structureestablished at the outset for the simulation tocontinue in a robust fashion. Even without a

recess, there is difficulty in pinning the conductionband high enough on the free surfaces of the edgecontaining the contacts. The problem is made even

more difficult with the recess in place because thewell structure is compressed into the device. Manyattempts have been made to obtain rapid andaccurate solutions for these and similar devices

[1-14]. A multigrid method has been developedfor the solution of the coupled one-dimensionalPoisson-Schr6dinger equations [1], and applied tothe isothermal HEMT model [2]. The currentcontinuity and energy transport equations havebeen discretised in a consistent manner using theC-function method [3], and it has been shown thatthe interpolation of the electron temperaturebetween grid points must be done using a new

logarithmic mean rather than the arithmeticmean [4].

In this paper we give details of a simulationinvolving the solution of the coupled Poisson,Schr6dinger, current continuity and energy trans-

port equations for hot electrons and show how thesteady state system may be solved numerically in a

robust fashion using a modified Newton iteration

*Corresponding author. Tel.: 0113 233 5117, Fax: 0113 233 5090, e-mail: [email protected]." 0113 233 2001, Fax: 0113 233 2032, e-mail: [email protected].: 0113 233 5117, Fax: 0113 242 9925, e-mail: [email protected]

287

288 E.A.B. COLE et al.

scheme. The purpose of this paper is to describehow a stable smooth solution may be obtained,not necessarily the most rapid, although it isshown how the runtime may be reduced byintroducing techniques which do not compromisethe stability. The ideas presented here could thenbe applied to faster methods, for example to themultigrid method. Results are obtained for thedevice shown in Figure 1.The band offset structure illustrated in Figure 2

will depend on the proportion u of A1 and v of Inin the layers of A1,Gal_, As and InvGal_v Asrespectively. The dependent variables for which asolution is found are the electrostatic potential ,

GaAs substrate

A1GaAs buffer

A1GaAs delta-dopedA GaAs spacer

InGaAs channel

A1GaAs spacerA1GaAs delta-dop_edA1GaAs undoped

GaAsgate

source drain x

FIGURE Cross-section of an eight-layer GaAs-A1GaAs-InGaAs HEMT showing the positions of the recess, contactsand layer structure.

II EhEc2

the electron temperature T, and the quasi Fermilevel EF----q where q is the magnitude of theelectron charge. The equations to be solved are:

(i) The Poisson equation

V. (oV) -q(ND -n- T-) (1.1)

for the solution of the electrostatic potential. Here, n is the total electron density and e0 isthe permittivity of free space. The conductionband edge is then Ec Eh-q.

(ii) The current continuity equation

On-V. J, J -q#nV + #knVT (1.2)Ot q

for the current density J, where k isBoltzmann’s constant. Generally, the mobil-ity # will depend on the electric fieldr,= -v.

(iii) The energy transport equation

( 0)OCot qv. E- v. Vc -- V. (nvkBT)

(1.3)

(iv)

where is the average electron energy density,0 is the average electron energy evaluated atthe lattice temperature To, and -e is theelectron energy relaxation time depending on. It should be noted that the use of the energybalance model and its underlying assumptionsmay in some circumstances overestimate thevelocity overshoot. Specifically, the energybalance model described here neglects con-vective effects (v. V)v and kinetic energy.The Schr6dinger equation

Ev2Evl

Layer Layer 2

FIGURE 2 The band offset structure between two layersshowing the conduction band Ec and step Eh.

-- (Vxc -- Eh qff))i ii

(1.4)

where i and Ai (i-0, 1,2...) are the energyeigenfunctions (normalised) and eigenvalues

HEMT MODELLING 289

respectively. The forms of the exchangecorrelation energy Vxc, effective mass m,relative permittivity er, conduction band dis-continuity Eh, N and T- are taken fromAdachi [5]. The form of the kinetic energyoperator is taken for its hermitian qualities[15,16]. Since we are dealing with an

essentially two-dimensional problem, theSchr6dinger equation must be solved in twodimensions. Because of the slope of E downtowards the drain end of the device (seeFig. 3a). many eigensolutions must be ob-tained in order to reach energy eigenvalueswhich intersect with the quantum well towardsthe source end of the device. Obtaining therequired number of eigensolutions is very time

consuming. In order to obtain a numericalsolution in a realistic time, we assume that thepotential is much more slowly varying inthe x-direction than the y-direction since theoverall device length in the x-direction is of anorder of magnitude greater. This. enables us tosolve the Schr6dinger equation in one-dimen-sional slices perpendicular to the layer struc-ture (that is, in the y-direction). Thus for a

given slice, the lowest eigenvalue is obtainedimmediately rather than having to be built upfrom many eigensolutions of the full two-dimensional model. However, further workmust be done to show how this approachcan be justified in a more rigorous fashion.We therefore solve the one dimensional

1.5

0.5

--0.5

(a)

FIGURE 3 Results of the simulation with Vgs 0.5 V and Vas 3.0 V viewed in the direction of the arrow in Figure 1, with alldistances in Angstroms, showing (a) the conduction band Ec, (b) the electron density n, and (c) the electron temperature T.


-1

(b)

FIGURE 3 (Continued).

equation

h2d( ldi )2 dy --d-fy + (Vx + Eh q)i )kiWi

(1.5)

by imposing the boundaryi(0) i(Y) 0 (i- 0, 1,2,...).

conditions

The electron density n inside the quantum wellswill be given in terms of the eigensolutions of thisequation. The total electron density is given by n--

n2+n3 where n2 is the contribution from thesub-bands given by the solution of the Schr6dingerequation and n3 is the bulk electron density. Wefirst choose a maximum number L of eigensolu-tions of the Schr6dinger equation with which towork [1]. Outside the potential well defined by

/L-1 < Ec we have n--n2-k-n3 with

n2 O, n3 Nc3F(1/2) (- (EF Ec)). (1.6)

Inside the well we must avoid the doublecounting of contributions, and thus we have n--

n2+ n3 with

L-1

rt2 Nc2Z [i(y)[2 In (l + e(1/:r)(e-’))i=O

(8m)3/2 (E-Ec) 1/2

n3---Tr -- L-, +exp((1/kT)(E-EF))dE

(1.7)

In the above expressions, we have Nc2 =_ (47rmkT/h2) and No3--2(2rmkT/h2)a’5. The expressions for

HEMT MODELLING 291

5000

4000

3000

2000

1000

0

-1000

0

(c) 18000.......

y(A)

FIGURE 3 (Continued).

the total energy density W=n will take similarforms. By performing integrals involving thedensity of states, it can be verified that outsidethe wells,

(EF-- Ec)) (1 8)W2 O, W3 Wc3F(3/2) -while inside the wells,

W2=Wc2li(y)’2(Fl(-(EF-/i))i=0

(Ec-Ai)ln(1 +e(1/Ic)(FF-’3))kT

(8m)3/2 (E-Ec)3/2

W3--’7"f -- +exp((1/kT)(E-EF))(1.9)

where We2-(47rmk2T2/h:) and Wc3-3(27rmkT/h2)l.SkT.

2. SOLUTION OF THE EQUATIONS

The integrals in Eqs. (1.7) and (1.9) are notstraightforward Fermi integrals since they havenon-zero lower limits, and it is necessary to obtainfast and accurate methods of evaluating theseintegrals. They may be written in terms of thegeneral integral

f zrdzIr(a, b) =_

r(r + 1) + exp(z- (a + b))(2.1)


Inside the well for example,

n =n2+Nc3I(1/2)(---(EF-AL),T (AL Ec)).(2.2)

It may be shown [6] that Ir(a, b) may be written

It(a, b) Fr(a) + c(a, b) ln(1 + ea)

where the function Cr(a, b) may be approximatedby

c(a, b) bpr (0[, + a)

for some values Pr, or and/3 which are found bya least mean square search over some suitablydefined physical ranges of a and b. In the iso-thermal case previously considered [2]. only Cl/2was needed, and it was found there that al/2=0.782 and /31/2 -0.023 when it was assumedthat Pl/2 --0.5. A more accurate calculation showsthat we must take al/2 0.514. /31/2=-0.011and pl/2--0.7. For this new hot electron simula-tion we must take a3/2--1.420. /33/2--0.028 and

P3/2-- 1.293.A rectangular uniform mesh was used on which

to discretise the equations. Expressions for J and sat the half points were found in terms of the C-functions introduced earlier [3]; this method hasbeen shown [4] to be a continuous upwindingmethod in which the electron temperature at thehalf point (i+ 1/2, j) has to be written as thelogarithmic mean

Ti+l,j Ti,jTi+(1/2),j In Ti+l,j In Ti,j

At each grid point (i,j) the dependent variableswere written as a vector vi,j (, , T)i,j and aNewton iteration was used on this vector. Theeigenvalues of the Schr6dinger equation werefound using a QL algorithm with implicit shiftsand then ordered. The first L eigenfunctions werethen found by back substitution using the Newtonmethod.

The following boundary conditions wereapplied: T was set equal to the lattice temperatureTo on the three contacts, with zero derivatives nor-mal to the free surfaces. On the source and drain,

was put equal to the applied voltages Vs andVa respectively, while on the gate it was put equalto Vg+b where )b is the built-in potential. Itsnormal derivative along the free surfaces along thecontact edge was given an artificial slope 0Ca inorder to pin E up to an acceptable level for thequantum wells to form. Its normal derivatives onthe remaining free surfaces were taken as zero.Given the value of n on these surfaces, thecorresponding boundary values for were foundby inverting Eq. (1.6) since these surfaces lieoutside the quantum wells.

3. SIMULATION RESULTS

Steady state results are produced for the deviceshown in Figure at lattice temperatureTo 300 K. The thicknesses of the layers, startingfrom the contact edge, are 350, 320 ]k, 20 A.,45]k, 125]k, 50]k, 50 and 1000]k. The propor-tion u of A1 in layers 2, 3, 4, 6, 7 and 8 is 0.24, andthe proportion v of In in layer 5 is 0.22. Dopingdensities are 4.0 1024 m- 3, 1.0 102o m- 3, 2.01025 rn- 3 1.0 1020 m- 3 1.0 1020 m- 3 1.0 x102o rn- 3 1.8 x 1024 rn- 3 and 1.0 1020 rn- 3

The recess depth is 400/1 and the overalllength in the x-direction is 18000. The built-involtage Cb is taken as -0.7V. Data from MonteCarlo simulations performed on the steady statewere used to provide the dependence of m* andr on { [3]. Results shown in Figure 3 relate tothe case Vgs -0.5 V and Vd, 3.0 V. The itera-tions were terminated when the residuals reachedprescribed minimum values. It was found thattaking the artificial slope value of &b=-5.010VVm produced a rapidly-forming quantumwell profile. The factor a in the Newton upgrade

k+l .k. O6/k,j increased automati-step i,j t,j + wascally as the iterations progressed. It was started ata very low value and increased to a value not

HEMT MODELLING 293

greater than 0.2- a larger value produced anunstable solution. The calculation of the eigen-solutions of the Schr6dinger equation is computa-tionally very expensive, and it was found that thesimulation time could be reduced by iterating toconvergence with only three eigensolutions (L 2)and then performing extra iterations using teneigensolutions (L 9).

References

[1] Cole, E. A. B., Snowden, C. M. and Boettcher, T. (1997).Int. J. Num. Modelling: Electronic Networks, Devices andFields, 10, 121.

[2] Cole, E. A. B., Boettcher, T. and Snowden, C. M. (1998).VLSI Design, 8, 29.

[3] Cole, E. A. B. and Snowden, C. M. (1995). Int. J. Num.Modelling: Electronic Networks, Devices and Fields, 8, 13.

[4] Cole, E. A. B. and Snowden, C. M. (2000). Math. andComputer Modelling, 31, 15.

[5] Adachi, S. (1985). J. Appl. Phys., 58, R1.[6] Cole, E. A. B., Boettcher, T. M. and Snowden, C. M.

(1997). Semicond. Sci. Technol., 12, 100.[7] Cole, E. A. B. (1993). In: Compound Semiconductor De-

vice Modelling, Snowden, C. M. and Miles, R. E. Eds.,Springer Verlag, New York, p. 1.

[8] Tang, T.-W. (1984). IEEE Trans. Electronic Devices,ED-31, 1912.

[9] Snowden, C. M. and Loret, D. (1987). IEEE Trans.Electronic Devices, ED-34, 212.

[10] Feng, Y.-K. and Hintz, A. (1988). IEEE Trans. ElectronicDevices, ED-35, 1419.

[11] McAndrew, C. C., Heasell, E. L. and Singhal, K. (1988).Semicond. Sci. Technol., 3, 758.

[12] Zhou, J.-R. and Ferry, D. K. (1995). VLSI Design, 3, 159.[13] Ando, Y. and Itoh, T. (1988). IEEE Trans. Electron

Devices, 35, 2295.[14] Ng, S.-H. and Khoie, R. (1991). IEEE Trans. Electron

Devices, 38, 852.[15] Stern, F. (1984). Phys. Rev. B, 30, 840.[16] von Roos, O. (1983). Phys. Rev. B, 27, 7547.

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