Hot-electron numerical modelling of short gate length pHEMTs applied to novel field plate structures

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. 2003; 16:15–28 (DOI: 10.1002/jnm.462)

Hot-electron numerical modelling of short gate length pHEMTsapplied to novel field plate structures

Shahzad Hussain1,2,3,*,y, Eric A. B. Cole1,z and Christopher M. Snowden2,3,}

1Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, U.K.2School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, U.K.

3Filtronic plc., U.K.

SUMMARY

Hot-electron numerical simulations were carried out in order to simulate the DC parameters ofpseudomorphic high electron mobility transistors (pHEMTs). The hot-electron effects were studied bysimulating several HEMT device structures. Hot-carrier injection in the substrate and the formation of thepeak of electric field in the channel were studied in detail. The inclusion of a field-plate contact in a multiplerecessed pHEMT structure lowered the peak value of the electric field by 24% compared with theconventional pHEMT. These devices were modelled by solving the two-dimensional Poisson, currentcontinuity and energy transport equations consistently with the time-independent Schr .oodinger waveequation. Appropriate Ohmic boundaries are discussed here and implemented in the simulations ofpHEMT structures. A new integral approximation is used to calculate electron densities and electronenergy densities for degenerate approximations. Copyright # 2002 John Wiley & Sons, Ltd.

KEY WORDS: hot-electron; HEMT; field-plate; numerical modelling; semiclassical; Schrodinger

1. INTRODUCTION

Digital, and many analogue, circuits require non-linear models capable of representing sub-micron gate-length devices operating at microwave frequencies. A typical AlGaAs/InGaAs/GaAs HEMT is illustrated in Figure 1. Many of these devices experience high electric field in theactive channel. Drift-diffusion and quasi-two-dimensional models do not provide sufficientlyaccurate information about carrier dynamics and transportation throughout the devicerespectively. Hot-electron modelling is required to provide valuable insight into the operationof small-scale semiconductor devices and to assist the development of improved devicestructures. The comprehensive numerical HEMT model described here is capable of simulatingmultiple recessed sub-micron gate contact pseudomorphic high electron mobility transistors(pHEMT) structures. This model is based on a quasi-Fermi level approach, where thetwo-dimensional Poisson, current continuity and energy transport equations are solvedself-consistently with the Schr .oodinger equation which is solved in one-dimensional slices

Published Online: 24 July 2002Copyright # 2002 John Wiley & Sons, Ltd. Received 1 January 2002

*Correspondence to: Dr S. Hussain, Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, U.K.yE-mail: s hussain @hotmail.comzE-mail: [email protected]}E-mail: [email protected]

perpendicular to the layer structures [1]. The hot-electron numerical simulation studies werecarried out for various device structures in order to study short channel effects.

Several pHEMT device structures were simulated and the results revealed the existence ofhighly energetic electrons at the drain side of the gate contact. The ‘hot spot’ at this region of thedevice has already been confirmed by optical measurements at higher bias potentials as well asby some hydrodynamic models for the simplest structures [2,3]. Our comprehensive modelindicates the region of largest energy peaks as well as the effects on the electron dynamics, andmaps the hot-carrier injection into the substrate. The hot-carrier injection process was studiedby examining different thicknesses of buffer layer and by varying the level of band discontinuity.The variations in the peaks of electric field were studied here with multiple recesses for the gatecontact and also by introducing a ‘field-plate’ contact on the free surface.

Figure 1. Schematic of a typical pseudomorphic HEMT structure, quantizationand the co-ordinate system used.

Copyright # 2002 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2003; 16:15–28

S. HUSSAIN, E. A. B. COLE AND C. M. SNOWDEN16

The highest electron energy peaks were found in the reverse biased region of the HEMTstructures. Hence, some of the highly energetic carriers can penetrate into the AlGaAs bufferlayer. Once the carriers manage to cross the barrier (interface between channel layer andundoped buffer layer) they are to more likely to flow toward the substrate and accumulate intothe second potential well, which is formed between the buffer and substrate layers. Hence, theuse of a thicker buffer did not show any significant reduction in the substrate injection. However,the introduction of a larger band discontinuity showed a significant reduction in the substrateinjection. It showed that the band discontinuity and carrier energies have to be comparable inorder to control the substrate injection. It is also demonstrated that the drain current neversaturates completely in the saturation region of the IDS � VDS profiles; this is due to substratecurrent, which increases continuously with the increases of drain–source potential VDS:

A recently introduced high power pHEMT structure using a novel field plate (FP) [4,5] is alsostudied here. High power pHEMT structures, with large breakdown voltages, are required forthird-generation wireless-communications base stations. High power GaAs-based FETamplifiers can improve the overall system performance in terms of efficiencies and the runningcosts of these communication systems. In order to develop high power field effect transistorswith improved breakdown voltages, field-modulating contacts have recently been introduced forMESFETs and heterostructure devices (Figure 4). These contacts are formed on the drain sideof the gate contact, where the gate-to-drain potential is modified and allows the devices tooperate at higher drain biases. The simulated electric field profile of the field-plated pHEMTindicates the potential of fabricating GaAs-based high power field effect transistors for the nextgeneration wireless-communication systems. The simulated parameters of three differentpHEMT structures are presented here.

2. HOT-ELECTRON MATHEMATICAL FORMULATION

The classical transport model is based on a fundamental assumption that the carrier energydistribution stays close to its equilibrium values. However, modern sub-micron devices haveregions of very high electric field and high current densities which cause substantial electronheating. The electrons experience transient transport in these regions exceeding the steady-statevalues. There are many numerical formulations and solution techniques that have beendeveloped in order to model FET devices accurately [6–10].

In this present work the hot-electron model formulation is devised by combining ahydrodynamic formulation, with Monte-Carlo calculations in order to model the non-equilibrium carrier transportation. The Schr .oodinger wave equation is used to calculate thequantized electron density and energy densities for the field effect heterojunction devices.Expressions for degenerate electron concentration and the electron energy density are also usedin this formulation.

2.1. Poisson’s equation

r � ðercÞ ¼ �qðNd � nÞ ð1Þ

where c is the electrostatic potential, q is the electron charge, Nd and n are the doping andconduction electron densities, respectively. Since the investigated devices were unipolar andbased on majority carriers, the minority carriers (holes) were ignored in simulation.


HOT-ELECTRON NUMERICAL MODELLING OF SHORT GATE LENGTH pHEMTs 17

2.2. The current continuity equation

@n@t

¼1

qr �

%J ð2Þ

where%J is current density. The expression of the current density is used as

%J ¼ �qmnrfþ kBmnrTe ð3Þ

The quantity m is the energy-dependent mobility, kB is the Boltzman’s constant, f is the Fermipotential and Te is the electron temperature.

Here the current density expression includes the drift-diffusion and electron temperaturegradients. But, in heterostructure devices, potential barriers are formed across theheterointerfaces which could considerably affect the carrier transport in some devices.Therefore, normally an extra potential term is required to be included in the current densityexpression in order to model inhomogeneous device structures. However, the effects of thesebarriers can be ignored in planar devices (FETs), where the carrier transport is mainly along thechannel. In our simulation, the deep Ohmic source and drain contacts were used, and hencethere is no potential barrier formed down to the channel layer. But the energy barrierdiscontinuity term is necessary if the current transport is only across epitaxial growth of thedevices, such as transport in heterojunction bipolar transistors.

2.3. Energy transport equation

@W@t

¼%J �

%E �r �

%S �

ðW � W0ÞtðxÞ

ð4Þ

where W is the non-equilibrium energy density, W0 is a constant energy density,%S is the energy

flux @W =@t is a rate of change of energy with time, r �%S is energy flux across body,

%J �

%E is Joule

heating and ððW � W0Þ=tðxÞÞ is the energy lost to the fixed energy lattice, and x is the averagedelectron energy. The expression of the energy flux is given as

%S ¼ mWrc�

kBqr � ðmWTeÞ ð5Þ

2.4. The Schr .oodinger equation

�_2

2

d

dy1

mn

dxidy

� �þ ðVxc þ Eh � qcÞxi ¼ lixi ð6Þ

where xi and li ði ¼ 0; 1; 2 . . .Þ are the energy eigenfunctions and eigenvalues, respectively, mn isthe effective mass of electrons, Eh is the equilibrium band discontinuity. The exchangecorrelation energy Vxc; is given by

VxcðxÞ ¼ �½1þ 0:7734b logeð1þ b�1Þ�2

pars

� �Ryn ð7Þ



where

a ¼4

9p

� �1=3

; bðxÞ ¼rsðxÞ21

rsðxÞ ¼ ð43pa*3nðxÞÞ�1=3; an ¼

4pe0er_2

mnðxÞq2ð8Þ

and Ryn is the effective Rydberg constant, where Ryn ¼ e2=8pe0eran; which is approximately5 meV for GaAs.

3. THE ELECTRON DENSITIES

The total electron density is n ¼ n2 þ n3; whereIn non-quantized region:

n2 ¼ 0

n3 ¼ 22pmnkBTe

h2

� �1:5

F1=21

kBTeðEF � EcÞ

� �ð9Þ

where Ec ¼ Eh � c is the conduction band edge and EF ¼ �f; is the quasi-Fermi level.In quantized region:

n2 ¼4pmnkBTe

h2XL�1

i¼0

jxiðyÞj2 lnð1þ e1=kBTeðEF�liÞÞ ð10Þ

n3 ¼ 12p

8mn

h2

� �3=2 Z 1

lL�1

ðE � EcÞ1=2

1þ expðð1=kBTeÞðE � EFÞÞdE ð11Þ

where the first L eigensolutions have been used in the calculation.

3.1. The energy densities

The total energy density is W ¼ W2 þ W3; whereIn non-quantized region:

W2 ¼ 0

W3 ¼ 32pmnkBTe

h2

� �1:5

kBTF3=21

kBTeðEF � EcÞ

� �ð12Þ

In quantized region:

W2 ¼Wc2

XL�1

i¼0

jxiðyÞj2 F1

1

kBTeðEF � liÞ

� ��

�1

kBTeðEc � liÞ lnð1þ e1=kBTeðEF�liÞÞ

�



W3 ¼ 12p

8mn

h2

� �3=2 Z 1

lL�1

ðE � EcÞ3=2

1þ expðð1=kBTeÞðE � EFÞÞdE ð13Þ

where Wc2 � ð4pmnk2T 2e Þ=h

2:The above equations comprise a full description of carrier dynamics in a HEMT. The energy-

dependent mobility mðxÞ is calculated by using the expression

mðxÞ ¼vssðxÞEssðxÞ

¼300mdT0

1þ vsE3ss=½mdE

4s ð1� 5:3� 10�4T0Þ�

1þ ðEss=EsÞ4

" #ð14Þ

where the dependence of the steady-state electric field Ess on the electron average energy x ismodelled by fitting a relationship to the corresponding Monte-Carlo data. The form of therelationship is similar in all three materials. The energy relaxation time constant tðxÞ is alsocalculated by fitting the relationship to the corresponding Monte-Carlo data for the bulkmaterials.

Table I shows the constant values of saturated velocities vs; low-field mobilities m0 (mobility inundoped bulk-materials) and the saturated electric fields Es for various materials which wereused in the above mobility expression.

The doping-dependent, low-field mobility md; is expressed as a function of doping density ND

as [6]

md ¼m0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ND=1023p ð15Þ

4. SOLUTION METHOD

It was found convenient to collect the quantities c; f and Te at each grid point ði; jÞ into a 3-component vector

%xi;j. A modified Newton method was used, in which the calculated correction

%di;j at the kth iteration was multiplied by a factor ak ð05ak51Þ in order to avoid overshoot:

%xkþ1i;j ¼

%xki;j � ak

%dki;j ð16Þ

At the start of simulation, following an arbitrary initial guess of the independent variables, theresiduals are large. Therefore, the damping factor must start with a small value, and was thenincremented linearly with the number of iteration from a ¼ 0:001 to 0.6 in the steps of 0.001.Note, a small value for a increases the convergence time, but it gives improved convergence froman arbitrary initial guess. If the initial guess is obtained from a previously saved data file, thenthe minimum a can be started from 0.1 to its maximum value with larger increments, i.e. 0.01.

Table I. Steady-state parameters.

AlGaAs InGaAs GaAs

vs ðm s�1Þ 5:0� 104 1:0� 105 8:5� 104

m0 ðm2 V�1 s�1Þ 0.2 1.3 0.8

Es ðV m�1Þ 7:0� 105 3:0� 105 4:0� 105



5. BOUNDARY CONDITIONS

The most complicated boundary is the top surface which significantly influences the operation ofthe device and consequently the numerical results. The top surface boundary was further dividedinto five different regions. These are the metal contacts and the free surfaces. The contacts arethe regions specified as source, gate, FP and drain contacts.

The sides of the device were modelled as interfaces between the bulk semiconductor and itssurrounding insulating materials, where the current flow is zero normal to these boundaries. Afew microns depth of substrate (GaAs) was included in our numerical modelling. It was assumedthat sufficient depth of substrate must be included to achieve approximately neutral conditions,i.e. the total current density must be close to zero at that depth and electron density must beclose to the dopant density. This reduced the computational time. Hence the Neumannboundary conditions were applied for the independent variables.

@c@n

¼@f@n

¼@Te@n

¼ 0 ð17Þ

where n represent a unit normal vector.In device fabrications processes, the source S and drain D contacts are normally formed by

evaporating gold alloys on a highly doped GaAs (capping) layer. Since the diffusion rate ofthese alloys into GaAs crystals is very high, the gold atoms diffuse deep down into the deviceand hence form Ohmic regions [11]. The diffusion of these alloys, into the device layer structure,was also confirmed by a manufacturer of compound semiconductor devices [12]. Therefore, theboundaries of these contacts were taken down into the device and hence modelled as uniformOhmic contacts, Figure 1. These contacts were modelled using Dirichlet boundary conditions.

The electron densities nS and nD on the source and drain regions, respectively, were kept at thedoping density of the capping layer. The electrostatic potentials and electron temperatures arealso fixed in these regions. The Fermi potentials fS and fD in these regions were then calculatedby rearranging the electron density expression. The gate contact is modelled as a Schottkycontact formed between the gate metal and semiconductor material. The FP contact, forsimplicity, modelled as a second insulated contact formed on the free surface.

cS ¼ VS; Te ¼ 300 K

cD ¼ VD; Te ¼ 300 K

cg ¼ Vg � fb; Te ¼ 300 K

where fb ¼ 0:7 V is the built-in potential.The free surface between the contacts is modelled as the interface between the vacuum and the

semiconductor surface and hence the electrostatic potential is calculated by assuming a depletedsurface due to surface states in the compound semiconductor materials. The Neumannconditions are used to calculate f and Te at the free surface.

csurface ¼ fsurface þkBTsurface

qF �11=2

nSNCd

� �þ

1

qEh ð18Þ

where NCd is the density of states and nS ¼ 1016 m�3:



Figure 2. A single recessed deep gate-contact HEMT structure, gate-length LG ¼ 0:15 mm:

Figure 3. Double recessed gated HEMT structure.



6. SIMULATION ROUTINE AND RESULTS

The hot-electron numerical formulation scheme was implemented on three different HEMTstructures, Figures 2–4. For steady-state calculations, the independent variables were computedfor given DC bias values, ðVDS ¼ 2 V VGS ¼ �0:4 VÞ: Mixed boundary conditions wereimplemented on the edges of the device, where the surface depletion and the neutral innerboundaries were considered.

In each iteration, the independent variables c; f and Te were simultaneously updated usingNewton’s modified scheme, Equation (16). The quantum eigensolutions were then calculated bysolving the Schr .oodinger equation in one-dimensional slices perpendicular to the epitaxialgrowth. The dependent variables were calculated by substituting the updated values of theindependent variables in the relevant equations. The energy-dependent variables mðxÞ and tðxÞwere calculated prior to the next iteration. The routine was repeated, until the residual errorswent lower than the specified minimum fixed values ð10�3Þ: The choice of the number of discretequantum energy levels and corresponding eigenfunctions in the HEMT structures variesaccording to the depth of the wells. However, it is found that the two eigensolutions providegood approximations for the quantized energy region. The inclusion of larger number ofquantum solutions in calculations increases the simulation time without making any significantchanges in the overall results.

The simulated electron temperature profiles reveal a large sharp peak of electron temperaturein the channel, near the drain side of the gate contact. Narrow sharp peaks indicate that the

Figure 4. Double recessed gated HEMT structure with a field-plate contact.



electrons gets hot over a short distance in the gate-to-drain reverse bias region, and then rapidlyloses energy. Hence, highly energetic carriers have a high probability of moving towards thesurface and down into the substrate, crossing interfaces. This clearly demonstrates that hot-electron modelling is necessary in order to design and model more complex heterostructure sub-micron devices.

Simulated hot-electron results indicate that the hot-carrier injection into the substrate occurseven at low bias potentials. These injected carriers then contribute to the total drain current,which strongly influences the device performance. In order to study the reduction of thesubstrate injection, the simulations were repeated using a thicker buffer layer. This did not showany significant reduction in substrate injection. The effects of larger and smaller conductionband discontinuities were studied. The larger discontinuity in conduction band, especially thelarger step formed between channel and the buffer layer, revealed a significant reduction incarrier injection.

Figure 5. X-components of e-field profiles in the pHEMT devices shown in Figures 2–4, respectively.



7. CONCLUSION

A hot-electron numerical formulation and simulation technique is described here which issuitable for modelling sub-micron gate contact HEMT structures. The hot-electron formulationsolves the hydrodynamic transport equations self-consistently with the time-independentSchr .oodinger wave equation. The electron density and electron energy densities wereapproximated with fully degenerate statistics [13]. A novel model for Ohmic contacts wasused for the source and drain boundary conditions, which improved the representation of thedevice and enhanced simulation stability. The hot-electron formulation is suitable for multiplerecessed gate-contact structures with a choice of field-plate contact. The model incorporatedsurface depletion, new Ohmic contacts boundaries, quantum calculations and a detailedtreatment of non-equilibrium transport.

The model has been applied to various pHEMT structures with various arrangements of top-surface boundaries (Figures 5–7). The simulated results showed considerable variation in the

Figure 6. Electron temperature profiles in all pHEMT structures, respectively.



electric-field profiles. The multiple recessed gate contact showed smaller peak value of the fieldthan the corresponding peak formed in the single recesses gate contact structure. The novel fieldplate technique showed further reduction in the peak of electric field by 24% than is found in thebasic multiple recess structure. Therefore, it is confirmed that the InGaAs-based channeldevices, with an appropriate field-plate contact, can operate at larger terminal potentials.However, although the device can potentially be operated at very large bias potentials, othercharacteristics of the device may be adversely effected, such as operating frequencies of thedevice [5].

Source

Drain

Gate

n

m-3

Source

Gate

Drain

n

m-3

Source

Drain

Gate

n

m-3

FP

Figure 7. Electron density profiles in all pHEMT structures, respectively.



REFERENCES

1. Cole EAB, Snowden CM, Hussain S. Hot electron modelling of HEMTs. VLSI Design 2001; 13(1–4):287–293.2. Snowden CM, Loret D. Two-dimensional hot-electron models for short-gate-length GaAs MESFET’s. IEEE

Transactions on Electron Devices 1987; ED-34(2):212.3. Woolard DL, Threw RJ, Liitlejohn MA, Hydrodynamic hot-electron transport model with Monte Carlo-generated

transport parameters. Solid State Electronics 1988; 31(3/4):571–574.4. Wakejima A, Ota K, Matsunaga K, Contrata W, Kuzuhara M. Field-modulating plate (FP) InGaAs MESFET with

high breakdown voltage and low distortion. IEEE Radio Frequency Integrated Circuits Symposium, System Devicesand Fundamental Research, NEC Corporation, 2-9-1 Seiran, OTsu, Shiga 520-0833, Japan, 2001.

5. Sakura N, Matsunaga K, Ishikura K, Takenaka I, Asano K, Iwata N, Kanamori M, Kuzuhara M. 100W L-BANDGaAs POWER FP-HFET OPERATED AT 30 V: NEC Corporation, ULSI device development laboratories, pre-published, 2001.

6. Snowden CM. Introduction to semiconductor device modelling. World Scientific Publishing Co Pte Ltd., ISBN9971-50-142-2, Leeds 1986; p. 42–59.

7. Lee SC, Tang T-w. Transport coefficients for silicon hydrodynamic model extracted from inhomogeneous MonteCarlo calculations. Solid State Electronics 1992; 35:561–569.

8. Cole EAB. In: Compound semiconductor device modelling. Snowden CM, Miles RE (eds.). Springer: New York,1993; p. 1.

9. Cole EAB, Boettcher T, Snowden CM. Two-dimensional modelling of HEMTs using multigrid with quantumcorrection. VLSI DESIGN 1998; 8(1–4):29–34.

10. Snowden CM, Howes MJ, Morgan DV. A large-signal physical MESFET operation. IEEE Transactions on ElectronDevices 1983; 30:1817–1824.

11. Goronkin H et al. Ohmic contact penetration and encroachment in GaAs=AlGas and GaAs FET’s. IEEETransactions on Electron Devices 1989; 36:281–287.

12. O’Keefe M. Private Communication, Filtronic Compound Semiconductor Ltd, UK, Filtronic Plc.13. Cole EAB. Integral evaluation in the mathematical and numerical modelling of high electron mobility transistors.

Journal of Physics: Condensed Matter 2001; 13:515–524.

AUTHORS’ BIOGRAPHIES

Shahzad Hussain received his BSc(Hons) in Physics from the Sheffield HallamUniversity in 1993, and then in 1994 he completed his MSc in process controlengineering from the University of Bradford. He received a PhD degree from theUniversity of Leeds, where he worked jointly in the Applied Mathematics and theElectronics and Electrical Engineering Departments. His PhD studies wereconcerned with the development and modeling of sub-micron pHEMTs transistors.He is currently a Research Associate in the Laser Photonics Research group in theDepartment of Physics and Astronomy, University of Manchester, England. He isworking on simulation, development and characterization of polymer compositedevices.

Eric A. B. Cole received a BSc degree in Mathematics from the University of Wales,and subsequently obtained a PhD in Mathematics after studying in the Departmentof Applied Mathematics and Theoretical Physics at the University College, Cardiff.The subject of the research leading to this degree was semiconductor noise. Thiswas followed by a year as Research Fellow at the same institution. He joined theDepartment of Mathematics at the University of Leeds as a lecturer in 1967, andwas later appointed as a senior lecturer in the Department of AppliedMathematical studies. He has been the Deputy Chairman of the school ofMathematics, and was the first Director of the centre for Nano-Device modeling atthe University of Leeds. His main research interests include the Mathematical andnumerical modeling of semiconductor devices, and multitemporal relativity.



Christopher Snowden received the BSc(Hons), MSc and PhD degrees from theUniversity of Leeds, England. He is currently Chief Executive of CompoundSemiconductors, Filtronic plc and Professor of Microwave Engineering at theUniversity of Leeds. He is a Fellow of the Royal Academy of Engineering, Fellow ofthe IEEE, and a Fellow of the IEE. He is currently a Distinguished Lecturer for theIEEE (Electron Devices Society). He was awarded the 1999 Microwave Prize of theIEEE Microwave Theory and Techniques Society. After graduating in 1977 heworked as an Applications Engineer for Mullard. His PhD studies were laterconducted in association with Racal-MESL and were concerned with the large-signal characterisation and design of MESFET microwave oscillators. He has heldthe personal Chair of Microwave Engineering at the University of Leeds since 1992.

During the period 1995–98 he was Head of the Department and subsequently Head of the School ofElectronic and Electrical Engineering. He was the first Director of the Institute of Microwaves andPhotonics located in the School. He was a Consultant to M/A-COM Inc., from 1989 to 1998. In 1998 hejoined Filtronic as Director of Technology. His main personal research interests include semiconductordevice and circuit modelling (CAD), and microwave, millimetre-wave and optoelectronic circuittechnology. His main research interests include compound semiconductor device modelling, microwave,terahertz and optical nonlinear subsystem design and advanced semiconductor devices. He has written 8books, over 250 refereed journal and conference papers, and many other articles.



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Hot-electron numerical modelling of short gate length pHEMTs applied to novel field plate structures