Monte Carlo Simulation of Electron Transport in th

and Ternaries

Maziar Farahmand ,Member,IEEE ,Carlo Garetto,Enrico Bellotti,Kevin F.Brennan ,Senior Member,IEEE ,Michele Goano ,Member,IEEE ,Enrico Ghillino,Giovanni Ghione ,Senior Member,IEEE ,John D.Albrecht,and

P.Paul Ruden ,Senior Member,IEEE

Abstract—We present a comprehensive study of the transport dynamics of electrons in the ternary compounds,

Al

Ga 1N.Calculations are made using a nonparabolic effective mass energy band model,Monte Carlo simulation that includes all of the major scattering mechanisms.The band parameters used in the simulation are extracted from optimized pseudopotential band calculations to ensure excellent agreement with experimental information and ab initio band models.The effects of alloy scat-tering on the electron transport physics are examined.The steady-state velocity field curves and low field mobilities are calculated for representative compositions of these alloys at different temper-atures and ionized impurity concentrations.A field dependent mo-bility model is provided for both ternary compounds AlGaN and InGaN.The parameters for the low and high field mobility models for these ternary compounds are extracted and presented.The mo-bility models can be employed in simulations of devices that incor-porate the ternary III-nitrides.

Index Terms—Monte Carlo method,semiconductor materials,wide bandgap semiconductors.

I.I NTRODUCTION

T

HE WIDE bandgap semiconductor materials,particularly the III-nitrides,are of emerging importance in many semiconductor device applications.The III-nitride materials system includes the binary compounds,InN,GaN,and AlN

and their associated ternaries,

In

N,

Al N,and

In

N.The large direct energy gap of these materials Manuscript received April 1,2000;revised October 30,2000.The work at

Georgia Tech was sponsored in part by the Office of Naval Research through Contract E21-K19,through Subcontract E21-K69made to Georgia Tech from the Office of Naval Research MURI Program at UCSB,the National Science Foundation through Grant ECS-9811366,by the National Phosphor Center of Excellence through Contract E21-Z22,and by the Yamacraw Initiative.The work at the University of Minnesota was supported by the National Science Foundation through Grant ECS-9811366,by the Office of Naval Research and by the Minnesota Supercomputer Institute.The work at Politecnico di Torino was partially supported by CNR (National Research Council)through the MADESS II Project.The review of this paper was arranged by Editor U.Mishra.

M.Farahmand is with Movaz Networks,Norcross,GA USA.

E.Bellotti and K.

F.Brennan are with the School of Electrical and Com-puter Engineering,Georgia Tech,Atlanta,GA 30332-0250USA (e-mail:fm@ieee.org).

C.Garetto,M.Goano,E.Ghillino,and G.Ghione are with the Dipartimento di Elettronica,Politecnico di Torino,Torino,Italy.

J.D.Albrecht and P.P.Ruden are with the Department of Electrical and Com-puter Engineering,University of Minnesota,Minneapolis,MN 55455USA.Publisher Item Identifier S 0018-9383(01)01465-4.

makes them highly attractive for both optoelectronic and electronic devices [1]–[7].The large band gap energy of the III-nitrides insures that the breakdown electric field strength of these materials is much larger than that of either Si or GaAs [8],[9],enabling,at least in principle,much higher maximum output power delivery in power amplifiers.Additionally,it has been found that at least the binary compounds,GaN and InN,have higher electron saturation drift velocities and lower dielectric constants that can lead to higher frequency performance of devices made from these materials [10].

The fact that GaN,InN and AlN,can form Type I hetero-junctions with their related ternaries provides an additional advantageous quality for device design.Type I heterojunctions enable modulation doping techniques and their exploitation in MODFET devices.In addition,the lattice mismatch between GaN and the ternary compound AlGaN in appropriately designed structures produces strain induced polarization fields [11],[12].These strain induced polarization fields can alter the band bending and carrier concentration at the heterointerface.In this way,the free carrier concentration can be increased within the channel region of a heterojunction field effect transistor,HFET,beyond that achievable by modulation doping alone [13].The strain induced polarization fields can be further exploited to enhance the base conductivity in HBTs [14],and to alter the local impact ionization rates in a multiquantum well structure [15].

For the above stated reasons,the III-nitride materials are of great interest for power FET and optoelectronic device struc-tures.To clarify the expected performance of these materials,transport as well as device studies are critical.Though there has been recent theoretical [16]–[27]and experimental [28]–[32]work on the transport properties of the binary compounds and their related devices [33]–[35],there has been little informa-tion about the transport properties of the ternary compounds [36].It is the purpose of this paper to provide a comprehen-sive Monte-Carlo-based study of the key,low field,transport parameters of the III-nitride ternary compounds useful in device level simulation.Specifically,we present Monte-Carlo-based calculations of the electronic mobility and steady-state velocity field curves for various compositions of the InGaN and AlGaN ternary compounds.Using the parameters evaluated herein,sim-ulation of devices that include the III-nitride ternary compounds can now proceed.

0018–9383/01$10.00©2001IEEE

II.E LECTRONIC S TRUCTURE,S CATTERING M ODELS AND

O THER P ARAMETERS

Electronic transport is studied using the ensemble Monte Carlo simulation.The band structures of the materials under study are approximated with an analytical formulation using nonparabolic spherical valleys.Though usage of an analytical band structure is questionable at high-applied electric field strengths wherein impact ionization can occur,we adopt its usage here for the following reasons.First,due to the large number of compositions examined,it is too computationally expensive to utilize full band models with their concomitant numerically derived scattering mechanisms.Second,we have found that the analytical model well reflects the low field dynamics critical for assessing the carrier mobility.Since we restrict our work here only to low field phenomena,an analytical band structure is satisfactory.For each material four nonequivalent valleys have been included,as shown in Fig.1.The primary valley for all materials occurs

at

.It should be noted that their relative energy ordering varies among the materials studied.The U minima are located between the M and L symmetry points,and consist of six equivalent valleys.The K minima consist of two equivalent valleys.The electron energy dependence as a function of wave-vector in each valley is approximated using a first order nonparabolic relation.The band structure data used in these calculations have been determined from a novel adaptation of the pseudopotential method.For each constitutive atom in the compounds,the effective potentials are optimized using an iterative scheme in which the band structures are recursively calculated and selected features are compared to experimental and/or ab initio results.Using this technique,we have obtained an updated,highly accurate description of the band structures of the binary compounds.The band structures of the ternary compounds are determined from the binaries using the virtual crystal approximation.The nonparabolicity factors,effective masses and intervalley energy separations for the binary compounds,AlN,GaN,and InN wurtzite phase materials have been reported in[37].The same parameters have been reported in[38]for the ternary compounds,

Al N,

In N, and

In N.

The scattering mechanisms included within the simulation are:acoustic phonon scattering,nonpolar optical phonon (equivalent and nonequivalent intervalley)scattering,polar optical phonon scattering,ionized impurity scattering,piezo-electric scattering,and alloy scattering[39].The scattering parameters for the binaries are taken from[40]–[47].The scat-tering parameters for the ternary alloys are determined using linear interpolation.The parameters used in the calculations are reported in Table I for convenience.In the case of alloy scattering,the scattering rate is given

as

random alloy

potential;

Al

Ga

Ga

Ga N,

Al

Ga N,

Al

Ga N,and AlN materials.This set of calculations is made assuming the maximum alloy scattering rate.In this way,we can bracket the effect of alloy scattering by

FARAHMAND et al.:III-NITRIDE WURTZITE PHASE MATERIALS SYSTEM537 TABLE I

M ATERIAL P ARAMETERS U SED IN OUR M ONTE C ARLO C ALCULATION.

h!—P OLAR O PTICAL P HONON E NERGY,"

—D EFORMATION P OTENTIAL,

—S OUND V ELOCITY,P

v ve

538IEEE TRANSACTIONS ON ELECTRON DEVICES,VOL.48,NO.3,MARCH

2001

Fig.4.Calculated electron drift velocity versus applied electric field for

GaN,In

N,In N,In N,and InN.For this calculation,the random alloy potential was set equal to conduction band offsets.Lattice temperature is at 300K,and electron concentration is equal to 10cm

.

Fig.5.Calculated electron drift velocity versus applied electric field for GaN,In N,In N,In N,and InN.For this calculation,the random alloy potential was set to zero.Lattice temperature is at 300K,and electron concentration is equal to 10cm .

the fact that the peak velocity depends nonlinearly on the elec-tron mobility in the lowest valley and the energy separation be-tween this valley and the secondary valleys.By changing the In

mole fraction in

In

N from 1to 0,the mobility decreases from that of InN to that of GaN,however the energy separation between the lowest valley and secondary valleys increases.Due to the nonlinear dependence of the peak velocity on these two factors,an initial increase followed by a decrease in the peak velocity with increasing Ga composition is observed,as shown in Fig.5.

The low field electron mobility is also calculated for

the binary compounds,and the ternaries,

Al

N and

In

N

(,0.5,0.8).For

Al N and

In

N the mobility is calculated for the two bracketing TABLE II

C ALCULATE

D L OW F IELD M OBILITY FOR L ATTIC

E T EMPERATURE O

F 300K,AND A SET OF I ONIZED I MPURITY C ONCENTRATIONS OF 10cm ,10cm ,AND 10cm .U

IS THE B AND GAP O FFSET .FOR Al N,AND

In N M OBILITY H AS B BEEN C ALCULATED FOR THE T WO B RACKETING

C ONDITIONS OF U =1E

C ALCULATE

D L OW F IELD M OBILITY FOR I ONIZED I MPURITY C ONCENTRATION

OF10cm,AND A SET OF L ATTICE T EMPERATURE OF300K,450K and

600K.U IS THE

B AND GAP O FFSET.FOR Al N,AND In N M OBILITY

H AS B EEN C ALCULATED FOR THE T WO B RACKETING C ONDITIONS OF

U=1

E

,IS10

cm

TABLE V

P ARAMETERS FOR THE P ROPOSED H IGH F IELD M OBILITY M ODEL IN

(3)

In

(2),is the total doping density,

and

,are parameters that can be determined either from experiment or from Monte Carlo simu-lation.In this case,we have used the Monte Carlo simulation to determine these parameters.We have extracted these parameters by a least squares fit of(2),to mobility data obtained from our Monte Carlo calculations.The extracted parameters for the rep-resentative compositions of the ternary compounds are shown in Table IV.

No preexisting mobility model provides a satisfactory de-scription of the field dependence of the Monte Carlo calculated GaN mobility.Therefore,a new field dependent mobility model has been developed which is given

by

is the low field mobility as expressed in(2).There are five parameters in the new model,which are determined from a least squares fit to the results of our Monte Carlo simulation. These parameters

are

540IEEE TRANSACTIONS ON ELECTRON DEVICES,VOL.48,NO.3,MARCH 2001

these parameters,extracted for both ternary compounds for the two bracketing cases of alloy scattering,are shown in Table V .

IV .C ONCLUSIONS

We have presented a comprehensive study of the low field electron transport dynamics in the III-nitride ternary

compounds,

Al

N and

In N.The calculations are made using a nonparabolic effective mass band model,ensemble Monte Carlo simulation.The band parameters used in the calculation are extracted from a pseudopotential band structure calculation optimized to yield excellent agreement with experimental data and ab initio band structure models.The Monte Carlo model includes all of the important scattering mechanisms.Owing to the fact that there is no generally accepted theory of alloy scattering,we bracket its effects on the calculations by considering two extreme cases.The maximum and minimum influence of alloy scattering is examined by studying the transport without alloy scattering and with alloy scattering using the largest predicted value of the random alloy potential.Making worst case assumptions,it is found that alloy scattering can become dominant.

Calculations of the steady-state velocity field curves are presented for the binary and some representative compo-sitions of the III-nitride ternary compounds,

In

N and

Al

N.The inclusion of alloy scattering greatly influences the transport dynamics greatly changing the peak velocity and threshold electric field.We have further provided calculations of the low field mobility for the binary and ternary compounds.Using these calculations,we have derived a formula for the low field mobility that reflects the influence of temperature and ionized impurity concentration.In addition,we have also provided a field dependent model of the carrier mobility including parameters for the ternary compounds.Excellent agreement between the model and the Monte Carlo calculations has been obtained.The field dependent mobility model provides a highly useful and accurate description of the electron mobility in the III-nitride ternary compounds that can be incorporated into semiconductor device simulators.

Finally,it should be mentioned that the paucity of experi-mental data renders it presently impossible to fully calibrate our calculations.In addition,most of the experimental data for the electron mobility is determined through Hall measurements making direct comparison of the drift mobility difficult.Nev-ertheless,where possible we have provided comparison of our model to other calculations and experiments indicating good agreement.In addition,some clear trends have been identified that suggest interesting experiments.The effect of alloy scat-tering on the transport predicted by the model is obvious.The calculations predict that if alloy scattering is weak,a mono-tonic progression in the velocity field curves from one binary to the other occurs.However,if the alloy scattering is strong,the ternary compounds all exhibit peak velocities below that of the constituent binaries.Similar behavior occurs for the mobility.

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thin films on sapphire(0001),”J.Cryst.Growth,vol.171,pp.12–20, Jan.1997.Maziar Farahmand(M’95)received the B.S.and M.S.degrees in electronics from the Sharif University of Technology,Iran,in1992and1994,respectively, and the Ph.D.degree in2000from Georgia Institute of Technology,Atlanta. From1994to1996,he was a Faculty Member at Razi University,Iran,and then joined the Physical Electronics Research Group,Chalmers University, Sweden.In1996,he joined the Computational Electronics Research Group. He is currently with Movaz Networks,Norcross,GA.

Carlo Garetto received the Laurea degree in electronics engineering from Po-litecnico di Torino,Torino,Italy in2000.

From June to December,2000,he was with the Department of Information Technology,Mid-Sweden University,Sundsvall,Sweden.He is currently en-gaged in research on the electrical and transport properties of wide bandgap semiconductors.

Enrico Bellotti was born in Italy in1963.He received the“Laurea in Ingegneria Elettronica”degree from the Politecnico di Milano,in19,and the Ph.D.de-gree in1999from the Georgia Institute of Technology,Atlanta,GA.

During1990,he was with the Italian Army for mandatory military service. From1991to1993,he was with Schlumberger Industries,Italy.In1994he joined AMRT,Advanced Meter Reading Technology,a joint venture partnership between Schlumberger Industries and Motorola,Inc.Norcross,GA.From June 1999until August2000,he was a Research Engineer with the Computational Electronics Group,Georgia Institute of Technology.Since then,he has become an Assistant Professor of Electrical Engineering at Boston University,Boston, MA.His research interests are in the area of semiconductor electronic structure calculation,physics and transport properties of wide bandgap semiconductors, and computational electronics.

Kevin F.Brennan(SM’90)received the B.S.degree in physics from the Mass-achusetts Institute of Technology,Cambridge,and the M.S.degree in physics and Ph.D.degree in electrical engineering from the University of Illinois at Ur-bana-Champaign.

He is a Professor of electrical and computer engineering at the Georgia In-stitute of Technology,Atlanta,GA,where he has been since1984.His research interests lie in the general area of semiconductors and microelectronics,with a particular emphasis on the physics and device application of emerging semicon-ductor materials for future high-power,high-frequency and photonic detection applications.His research experience includes the design,evaluation and opti-mization of infrared,optical and ultraviolet photonic detectors and emitters.He has published numerous papers in scientific peer reviewed journals,has five U.S. patents,and is the author of The Physics of Semiconductors with Applications to Optoelectronic Devices(Cambridge,U.K.:Cambridge University Press,1999). Michele Goano(M’98)received the Laurea and Ph.D.degrees in electronics engineering from Politecnico di Torino,Torino,Italy in19and1993,respec-tively.

In1994and1995,he was a Postoctoral Fellow in the Departement de Genie Physique,Ecole Polytechnique de Montreal,Montreal,QC,Canada.Since 1996,he has been a Research Assistant in the Dipartimento di Elettronica, Politecnico di Torino.From September1998to February1999,and from November1999to February2000,he was a Visiting Scholar at the School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta.From November2000to January2001,he was a Visiting Scholar at the the Department of Electrical and Computer Engineering,Boston University,Boston,MA.He has been engaged in modeling of semiconductor optical components and Monte Carlo simulation of quantum-well devices, and is currently involved in research on coplanar components,traveling wave modulators,and wide bandgap semiconductors.

Enrico Ghillino received the Laurea degree in electronics engineering from Politecnico di Torino,Torino,Italy in2000.

From June to December,2000,he was at the Department of Information Tech-nology,Mid-Sweden University,Sundsvall,Sweden.He is currently engaged in research on the electrical and transport properties of wide bandgap semiconduc-tors.542IEEE TRANSACTIONS ON ELECTRON DEVICES,VOL.48,NO.3,MARCH2001

Giovanni Ghione(SM’94)graduated in electronics engineering from Politec-nico di Torino,Torino,Italy in1981.

From1983to1987,he was a Research Assistant with Politecnico di Torino. From1987to1990,he was an Associate Professor with Politecnico di Milano, Milano,Italy.In1990he joined the University of Catania,Catania,Italy as Full Professor of Electronics.Since1991,he has been a Full Professor at II Faculty of Engineering,Politecnico di Torino.Since1981,he has been engaged in Italian and European research projects(ESPRIT255,COSMIC and MANPOWER)in the field of active and passive microwave CAD.His present research interests concern the physics-based simulation of active microwave and optoelectronic devices,with particular attention to noise modelling,thermal modelling,active device optimization.His research interests also include several topics in compu-tational electromagnetics,including coplanar component analysis.He has pub-lished more than150papers and book chapters in the above fields.

Prof.Ghione is member of the Associazione Elettrotecnica Italiana.He is an Editorial Board member of the IEEE T RANSACTIONS ON M ICROWA VE T HEORY AND T ECHNIQUES.

John D.Albrecht received the Ph.D.degree in electrical engineering from the University of Minnesota,Minneapolis,in1999.

In1999,he joined the Electronic Science and Technology Division,Naval Re-search Laboratory,Washington,DC,where he is currently a National Research Council Post-Doctoral Fellow.His research includes studies of carrier transport in wide bandgap semiconductors,device physics,and computational models of mesoscopic composites.P.Paul Ruden(M’88–SM’92)received the Ph.D.degree in physics from the University of Stuttgart,Germany,in1982.

He worked on semiconductor superlattices at the Max Planck Institute for Solid State Research in Stuttgart,West Germany,from1979to1983.From1983 to1985,he was with the Electronics Science and Technology Division,Naval Research Laboratory,Washington,DC,and also with the Department of Elec-trical and Computer Engineering,North Carolina State University,Raleigh.In 1985,he joined the Corporate Research Laboratory,Honeywell Inc.,Bloom-ington,MN,where he led research efforts in the areas of semiconductor pho-todetectors and III–V heterostructure field effect transistor technology.Since 19,he has been on the faculty of the Department of Electrical and Computer Engineering,University of Minnesota,Minneapolis,where he is currently Pro-fessor of electrical engineering.His research focuses on modeling of semicon-ductor materials and devices.He has published over100papers in technical journals in these fields.

Dr.Ruden is a member of the American Physical Society and the Materials Research Society.