A charge control model for III-V HEMTs using a self-consistent numerical solution of the Schrodinger and Poisson equations

Rochester Institute of Technology

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Theses

Thesis/Dissertation Collections

2006

A charge control model for III-V HEMTs using a

self-consistent numerical solution of the

Schrodinger and Poisson equations

Melissa Manney

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RDDD6484D67

A Charge Control Model for III-V HEMTs Using a Self-Consistent Numerical

Solution of the Schrodinger and Poisson Equations

by

Melissa A. Manney

A Thesis Submitted in Partial Fulfillment of the

Requirements for the degree

0

f Masters

0

f Science

ill

Electrical Engineering

Approved by:

Professor _ _ _ _ _ _ _ _ _ _ _

_

(Dr. Syed Islam -Advisor)

Professor

_ _ _ _ _ _

--: _ _

--:-:--:-_

(Dr. James Moon -Committee Member)

Professor

-(Dr. Sannasi Ramanan - Committee Member)

Professor

-(Dr. Robert Bowman - Department Head)

DEP ARTMENT OF ELECTRICAL ENGINEERING

COLLEGE

OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

THESIS RELEASE PERMISSION

DEP ARTMENT OF ELECTRICAL ENGINEERING

COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

Title of Thesis:

A Charge Control Model for 111-V HEMTs Using a

Self-Consistent Numerical Solution of the Schrodinger and Poisson

Equations

I, Melissa A. Manney, hereby grant permission to Wallace Memorial Library of the

Rochester Institute of Technology to reproduce my thesis

in

whole or

in

part. Any

reproduction

will

not be for commercial use or profit.

Signature _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

5""+-I ...!..../5:=..J.

1

-"

O

:::..s.

6

L - _

Acknowledgements

I cannot remember_exactlywhenIfoundmyself

thinking

thatI would neverfinish

my

thesis,

butIcan_{surely say}thatI startedto believe it. Everyonewouldtellmein

responsethatitwillbeoverbeforeyouknow it. Ididn't believethatatthe_{time, but here}

I am. Finished. In completingthisdegree IdidwhatI set outto do whenI started. I had

thechanceto delvedeeper intotopics thatwere_{only minimally}covered

during

_my

undergraduate studies.Despitesome oftheissuesthatcome with_working fulltimeand

workingon adegreepart

time,

thishasbeenagreat experience andI woulddo itagainin

aheartbeat.

I wouldliketo thank_myadvisor,Dr. Islam forhis

help

andsupport

-notto

mentionhispatience

-in_completingthis thesis. I wouldalso liketo thankDr. Moonand

Dr. Ramanan for participatingon_{my defense}committee.

Fromthe time that I startedthis

degree,

I realizedthatIhave agreater support

systemthanI everknew. OfcourseI haveto thank_mymotherfor herencouragement.

Also,

mybandofcheerleadersthatseemsto expand_{every day! To my}friendsand co

workersbothold and_new,congratulations! Youdon't haveto _worryabout me_goingbald

from_{pulling my hair}out while_wrestlingwith_myschoolwork anymore. Thankyou all. I

don't know howto beginto _repayyouforall ofthesupportthatyou'vegiven me atthis

Abstract

Withthe

increasing

demand forsmallerdevicesthatcanperformfasterand more

reliablythere isa needtoexplore options thatallowforthiscombinationinspite ofthe

factthatsmaller sizetends to leadtoareductionin reliabilityand overalldevice

performance. Highelectron_mobilitytransistors

(HEMTs)

haveprovensuperiorto

MOSFETS andBJTs intheareas of voltage andtemperaturerangesinwhich

they

can

operate as well astheirhigh breakdown

fields,

carrier

density

and _velocitysaturation.

Studying

HEMTsand_{understanding}theirconstraints andhow

they

respondtovariations

intheir_makeupand intheenvironment inwhich

they

willbeused can

help

tomakethe

bestpossible choices in

fabricating

thesedevicesas well as_pushingtheenvelopetomake

thembetter.

Inanattemptto _{study III-V}heterojunctions ingeneral,andAlGaN/GaN

heterojunctionsinparticular, anumericalsolutionto thecoupledPoisson-Schrddinger

equationswillbepresented.

Thefocusofthisworkwasthe AlGaN/GaN heterostructurewhereanextraboost

inchannelcarrier concentrations is achievedthroughpolarization comparedto

AIGaAs/GaAs heterostructures. Thegoalisto _eventuallymakethis modelapplicableto

any III-V heterojunction

by

adjustingthematerialparameterssuch as

doping

concentrations,buffer layerandspacerlayerthicknessesand latticeconstants.At this

point, the modelhasbeen builtaroundandtestedforbothAlGaN/GaN andAlGaAs

devices.

Once thecoupledPoisson-Schrddingerequationsare solvedthecharge control

characteristics ofthe

device,

including

carrier concentrations and

distribution,

gate

voltageforthe given surface_potential,andthresholdvoltagecanbecalculated andthe

charge controlcharacteristics canthenbe obtained.

Temperaturevariationwas also addedto themodeltoobservehowthedevice

reactsfora given_change,which canbeexperienced fromenvironmentas wellasfrom

self-heatingeffects.

TableofContents

Acknowledgements

Abstract

TableofContents

ListofFigures

ListofTables

ListofAcronyms

ListofSymbols

Chapter 1: Introduction

1.1 Introduction

1.2 Nitrides

1.2.1 PropertiesofGaN

1.2.2Polarization

1.3 Literature Review

1.4 Contributions

1.5 ThesisLayout

Chapter 2:

Methodology

2.1 Introduction

2.2 The Poisson Equation

2.2.1 TheFinite-Difference Solution

2.2.2 Piezoelectricand Spontaneous Polarization

2.3 The Schrddinger Equation

2.4 I-VPlots

2.5 Sheet CarrierConcentrationvs. GateVoltage

2.6Temperature Dependence

Chapter3: The Modeland DeviceProperties

Chapter 4: Results and Discussion

4.1 Introduction

4.2 The Poisson Equation

4.3 Schrddinger's Equation

4.4 Sheet Carrier Concentrationvs. Gate Voltage

4.5 Current Voltage Results

4.6 Temperature Dependence

Chapter 5: Conclusions andFuture Work

5.1 Conclusions

5.2Future Work

References

25

25

27

30

33

33

34

36

37

43

45

51

51

52

63

ListofFigures

1.1a HEMTcross-section 1

1.1b HEMT band diagram 1

1.2 Sacconiconductionband diagramvariation with polarization 7

2.1

Self-consistant

Poisson-Schrddingerequation solution

loop

15

2.2 HEMTmesh orientation 21

4.1 Calculatedconductionbandprofile 35

4.2 Calculatedwavefunctions 36

4.3 Scaledwavefunctions 37

4.4 Carrierconcentrationfora molefractionof0. 1,

0.2,

and0.3 38

4.5 AlGaAs/GaAscarrier profile 39

4.6 AlGaAs/GaAsnsvs.

VGB

40

4.7 AlGaAs/GaAsrigvs.

Vgb

as afunctionofbuffer layerthickness 41

4.8 AlGaN/GaN_nsvs.

Vgb

42

4.9 AlGaN/GaN

IDs

vs.

VDs

characteristics 44

4. 1 0

Ids

vs.

VGs

fortheAlGaN/GaN heterojunction 45

4.11 GaN

bandgap

as a functionoftemperature 46

4. 12 GaNnfconcentrationas afunctionoftemperature 46

4. 13 GaN latticeconstantas afunctionoftemperature 47

4.1 4 Chargecharacteristicsas afunctionoftemperature 48

4. 1 5 Current-_{Voltage characteristics}as afunctionoftemperature 49

4.16

Ids

vs.

Vgs

as afunctionoftemperature 49

ListofTables

1.1

Comparison

of semiconductor compound properties 3

3.1 AlGaAs/GaAsheterojunctionmaterial properties 3 1

3.2 AlGaN/GaN heterojunctionmaterialproperties 32

4.1 Physical Constants 34

ListofAcronyms

Acronym

Meaning

2DEG Two-DimensionalElectron Gas

Al Aluminum

AlGaAs Aluminum Gallium Arsenide

AlGaN AluminumGallium Nitride

A1N Aluminum Nitride

BSBH BareSurface Barrier Height

FET FieldEffect Transistor

GaAs Gallium Arsenide

GaN Gallium Nitride

HEMT High Electron

Mobility

Transistor

HFET Heterojunction Field-Effect Transistor

InGaAs Indium Gallium Arsenide

InP Indium Phosphate

MOS Metal Oxide Semiconductor

MOSFET Metal Oxide Semiconductor Field Effect Transistor

TWA Triangular Well Approximation

UID Unintentional

Doping

List ofSymbols

Sym bol _Meaning

6 Spacer Layer Thickness

Es Material Perm _itivity

Es1 Perm_itivity of the Buffer Layer

s2 Perm_itivity ofthe Channel Layer

v Mobility

a Total Polarization

O(x) Position Dependent Potential

0B Barrier Potential

>gi(x) Position Dependent Potential _Along the Buffer Layer

*i Wave Function _{Corresponding} to the jth Eigen _Energy Level

a Lattice Constant

D _{D ensity} of States

E0+ Electric Field at the Buffer/Channel Layer Interface on the Channel

Layer S ide

AEC Conduction Band _{Discontinuity} E, Ferm _{i E nergy Level}

Eg

Material _Bandgap

E, Intrinsic _Energy Level

Ej

jth

Eigen _Energy Level

'ds Drain-Source Voltage

k Boltzmann's Constant

K Kelvins

m* Effective Mass

n(x) Position Dependent Carrier Concentration

Na Acceptor Atom

Na

+

Ionized Acceptor Atom

Nc Conduction Band _Density of States

Nd Donor Atom

Nd

+

Ionized Donor Atom

ni Intrinsic carrier concentration

n. Sheet Carrier Concentration

Nv Valence Band Dencity of States

P(x) Position Dependent Total Polarization

PRE Piezoelectric Polarization

Psp Spontaneous Polarization

q E lectron Charge T Tern perature

V(x) Position Dependent Potential Vgb Gate Voltage

X Mesh Position

y Channel W idth (Perpendicular to the paper)

z Channel depth (From the top of the conduction band discontinuity to

Chapter1: Introduction

1.1 Introduction

1.2 Nitrides

1.2.1 PropertiesofGaN

1.2.2 Polarization 1.3 Literature Review 1.4 Contributions

1.5 Thesis Layout

1.1 Introduction

Every day

intheworld ofelectronics,devices are_getting smaller andfaster. The

task nowisto

try

toachieve bothofthese goals while_gettingthebestpossible

performancefromthedevice.Asolutionto _{achieving both}ofthesegoalsisthe

heterojunction field-effecttransistor

(HFET),

onetype of whichis knownasthehigh

electron_mobilitytransistor

(HEMT)

forreasonsthatwillbeexplainedbelow. HEMTs

areformed

by

_sandwichinganundoped spacerlayer betweenadoped layerofthesame

materialthatwill provide carriersto thechannelforconductionand an undoped channel

layerofadifferent material(Figure 1.1).

2 DEO{ ^y

fiouicc^yy^

y^a^cy^y ^yDmu

y

Cap

Buffer Lnycr (dopedii !,!>

p SpncciLnyer(UID) Chraincl Layer (HID) Nnitc.iin>iiLnycr

".miNrnl.it Buffer

Figur

tends

carrie unint

dopar

e1. la: HEMTcross-section.The bufferla\

tobe dopedn-typesothatelectrons arethe

rs.Thespacer and channellayersare

intionallydoped(UID).Gettingsomelevel tatomsintotheselayersisunavoidable.

er

of

[image:13.533.290.487.462.638.2]

Buffer Lnycr

Figure 1.1b: HEMTbanddiagram.S isthespacer

[image:13.533.77.260.467.646.2]

The band_gapofthebuffer layeris

typically

largerthan thatofthechannel

layer,

which aidsinthe

formation

oftheconduction channel.

Making

thespacerlayertobeof

the same material asthebuffer layerensuresthatthe quantumwelldoesnot formatthe

junctionwiththebuffer layer. Whenthematerials are

joined,

aquantumwellformsatthe

junctionofthespacerlayerandthechannellayerasthestructure seekstofind

equilibrium[1]. Carriers fromthebuffer layerwillflowto thiswellwhere

they

willthen

beavailableforconduction. Thespacerlayer_separatingthebufferandchannellayers

protectsthecarriersfrom

impurity

scattering. Themore

highly

amaterialis

doped,

the

more carrierstherewill beavailableforconduction.

However,

ifthecarriersinhabitthe

same space asthedopantatoms, morecarriersinthearea alsotranslates tomore

scattering. InaHEMT structure, wherethecarriersare removedandkeptseparatefrom

thedopantatoms, theamount of_{scattering in}thechannelisgreatlyreduced. Inthis_way theelectrons willtravelfartherandbuild upmore speedthan

they

wouldbeabletodo

withoutthisspacer.Nowthemajor cause of_scattering willbephonon_{scattering induced}

by

temperature fluctuations [1]. The factthatcarriers can reach

top

speedsin HEMT

devicesmakesthese structures_veryattractiveforRFapplications, andisalso thereason

behindthestructure's name.

Withinthequantumwell, discrete energylevels limittheenergylevelsthat the

electronscanhave[21]. The highertheenergy

level,

thefewerthenumberof electrons

thatwill resideatthatlevel because goingto ahigher energylevelwould requirethe

levels,

thatiswheretheelectronswillgo.

Any

energy levelthatfallsbelowtheFermi

level,

Ef,

willbe_nearly

completely

filled,

whilethoseabove

Ef

willbe_{only partially}

filled.

Electrons inthechannellayerare confinedto atwo-dimensional

(2D)

motion

traveling

alongthey-_{and z-axis(as}

definedinFigure

Lib),

butnotinthexdirection

because

thecarriers are confinedinthewell_alongthisaxis. For_{this reason, the}carriers

have beentermedtwo-dimensionalelectron gas(2DEG).

1.2Nitrides

1.2.1Properties ofGaN

While HEMTdevicesare notnew, theuse of nitride-based_{HEMTs is relatively}

new. GaAsandInPare_verypopular compoundsthatare_widelyusedatspeedsof

anywherefrom 800MHzto 100 GHz

[2]

whileGaNhasbeenreportedto show cutoff

frequenciesof121 GHz.

Table1.1: ComparisonofSemiconductorCompoundProperties_{[23, 25]}

Parameter GaN SiC Si GaAs InP

Breakdown Field _(Vcm"1) 5x10b 1x106 3*lOb

4x10b 5x10b

Bandgap (eV)

3.42 2.36 1.11 1.43 1.344

j

Electron

Mobility

_(cm2A/s) 1100 900 1400 8500 5400

Relative Dielectric Constant 10 9.66 11.7 12.5 12.5

vSat (x107

cm/sec) 1.45 2.5 1 0.7 1.5

Thermal _{Conductivity (W/cmK)} 1.7 4.9 1.3 0.54 0.68

Evenso, GaNisnot yet_commonlyusedbecause it isstill_{veryearly in}its

development

and it isnotyet practical

financially

or asregardstheoverall_{understanding} oflong-term

stabilityand_reliabilityofdevices usingthismaterial. GaNHEMTs holdseveral

[image:15.533.74.490.423.518.2]

breakdown

fields1,

_{very high}carrier

density

andhigh saturationvelocity. Thewide

bandgap

ofthematerial allows for highervoltages andtemperatures tobe appliedto the

device,

openingtherange of applications forwhichitcanbeused.

However,

there are

drawbacksas _well,as mentioned in [2]. Oneofthesedrawbacksiscurrent slump. Current

slumpiswhere channelcharge is

lowered,

and_subsequently lowersRFpowerand

efficiency. Thisis_presumablycaused

by

thereduction ofcarriersdueto

deep

trapsinthe

buffer layer [3]. Another seemingdisadvantage oftheGaN systemisthat the _mobilityis

notashighasthatoftheGaAssystem, asseenin Table 1. Thisis due bothto thelarger

numberof carriers andthelargernumber ofdefects inthenitride-based material [27].

An

interesting

aspect ofnitride-baseddevices inparticularisthateven without

doping

thebuffer

layer,

a2DEG will stillbegenerated. Giventhis

feature,

itisclearthat

GaN deviceswouldgeneratemorecarriersthanadevicecomposed of a non-nitride-based

compound. The sourceofthis2DEG is not

fully

understood, thoughsome _saythat it is

theresult of surface states

[2,

4].

Koyley

and Spenser

[4]

approachedthisissue

by

concurrentlyobserving thechanges inthe Bare Surface Barrier Height

(BSBH),

or, the

barrierheight inthearea ofthesurfacebetweenthedevicecontacts, andthe2DEGwitha

varying AlGaN buffer layerthickness.

They

flashed UV light onthedevice and readthe

transientresponse toobtaintheirmeasurements.

They

foundthatBSBH and2DEG grew

linearly

withbothsaturatingwhenthebuffer layerreached about200

A

thickness.

Increasing

thebuffer layerthickness lowerstheFermi levelandthereforeempties more

surface states,whichthencontributeto the2DEG

density

[1,4].

Passivating

thebare

1

See Table 1 The breakdownfield,VBd,forGaNisafullorderof magnitude abovethatof eitherGaAsor

surface wasfoundto decreasethe

BSBH,

but

increase

the2DEGconcentration. This is

important

inthe structuredesign becausethespacerlayer shouldbethickenoughto

create a separationbetweenthebuffer layerandthe2DEGinthechannellayerto

optimizetheperformance of_{the gas,} andthinenoughto ensurethat thecarrierscan

actuallymakeitto thechannellayeras opposedto

forming

a parasiticchannel withinthe

spacerlayer. Thoughthestructure observed

by [4]

didnot utilize adoped buffer

layer,

as

one was not_necessaryto getthe2DEG concentration_{to appear,} similar considerations

should_applywhenthespacer layerappearsbetweenabuffer layerand a channellayer.

1.2.2 Polarization

AnotherconsiderationofAlGaNmaterialsisthe contribution of piezoelectricand

spontaneous polarization ofthejunctionto thedeviceproperties. Thesetwo effectsadd

extrafieldterms to thestandardPoisson

equation2

sothata strongerfieldwillbe formed

forthese devicesoverthose thatdo nothave thisadded source of charge [5].

Theseeffects are most significantwhenthe structureisgrown onthe

[0001]

directionasthisisthegrowthdirection forwhichthe heterojunction hasa wurtzite

structure_{and, subsequently,} whereboththespontaneousandpiezoelectricpolarization

are_{strongest,allowing} forthefull benefitsof_usingaAlGaN/GaNheterojunction

[5, 6,

7]. Itis thelowsymmetryofthewurtzitestructurethatgives riseto thestrain

withinthe structure.Thisstrainisspontaneous polarization and willbepresent whether

piezoelectric polarizationis presentinthesystem or not. Forthezincblende growth

[111],

spontaneous polarization will notbe_{very strong} at all.

It should also benotedthatnot_onlydoesthegrowthdirection [1 1

1]

vs.

[0001]

affecttheproperties of_{the material,}butthat thefinal faceputonthelayer doesaswell.

Differentresults willbeobtained_{from Ga-faced} structures(thosewithGaasthe

top

layer

oftheGaN layerto interfacewiththeAlGaNandthatare referredto when_specifying

[0001]

grown_materials) as opposedto aN-facedstructurewithnitrogen atomsasthe

interfacing

layer,

whichisgrowninthe

[0001]

direction. Theseeffects aredescribed in

[28].

Piezoelectric polarizationarises fromthestrainbetweenthematerials. Twotypes

of strain were calledout in [27]. One ofthesesources resultsfrom latticemismatch

betweenmaterials.Thematerialschosenandthemolefractionofthedopantsusedwill

affectthis source ofstrain.

InanAlGaN/GaN system,ifthebuffer layerhasa mole fractionof0.05then the

buffer layerwillbemore_closely matchedto thechannellayerandstrainwillbekepttoa

minimum. For highervaluesofthemole

fraction,

thestrainwillincrease

leading

toa

higherfieldandthusproduce morebenefits.

However,

oncethemolefractionreaches

approximately 0.35 to

0.4,

thenumber ofdefectspresentinthematerialbeginstoriseto a

point that it isno longer beneficialtodeviceperformanceand so further raisingthemole

fractionpastthispoint beginsto work againstthedeviceperformance.

[27,

28].

Assuming

agrowth methodthatminimizesthenumber ofdefects inthematerialis_used,

structure willbe closerto idealwithregardstodefects andthe experimentaland modeled

structures willthenbe more_closelymatched.

The second source oflatticemismatch would resultfromthermalmismatch

between_{the materials,} which causes a strainbetweenthematerials whenthestructure

cools(or alternatelyheatsup). ThiswouldbeseenattheGaN/nucleation layerandthe

AlGaN/GaN interfaces. Temperatureeffects willbetakeninto accountinthismodel.

Ascanbe seenfrom Sacconi'sconductionband plots as represented

by

Figure 1.2

[5]

addingthepolarizationtermsis_veryimportantinthissystem. Ifnot, calculations

doneonthis system would_grosslymisrepresentthesystem. Withouttheadded

field,

the

carrier concentration wouldbeunderrated andthesolutionto thePoissonequation would

produce awell moreshallowthan itshould

be,

_or, asshown in Figure 1.2, notactually

present at all.

0 100 200 300 400 500 600 700 800

Device Depth [Angstroms]

Figure1.2Changeintheconductionbandplotbasedon

polarization representedinthePoissonequation.

Thesolidlinerepresentstheconductionbandwithbothspontaneousand piezoelectric polarization

accountedfor.The dottedlineaccountsfortheconductionbandlotwhen spontaneous polarizationisnot

accountedfor.Thedashed/dottedlineshowstheconductionbandwhere neitherspontaneous nor

[image:19.533.108.457.398.598.2]

Fromthe Poisson_{equation, the}conductionbandwould notbewellrepresentedandso

Schrddinger'sequation would not_effectivelypredicttheeigenfunctionsandeigenvalues

ofthesystem. Because of_{these errors, the}carrier calculationwouldbe farofffromwhat

couldbeexpected andthussubsequent calculations ofI-V andC-Vcharacteristics would

not_{be accurately}represented forthesystem. As_{mentioned, the} polarizationchargewill

beadded as an additionalfieldterminthePoissonequation. Thedetailsofthiswillbe

further investigated inChapter 2.

Bernardi

[6]

laidout fouradvantagesof_{using GaN} over otherIII-Vmaterialsin

heterojunction devices:

1. Thepiezoelectric polarizationin GaNjunctionscanbeasmuchas 1 Oxthat

inotherIII-V

junctions3;

2. Spontaneouspolarizationinnitridecompoundsis very

large;

again, this

addsto theoverall chargeinthe system;

3. Unlikeother III-Vcompounds, nitride compoundshavealarger

internal-strainionic termthanthedamped ionterm;and

4. Nitridecompoundshaveamuchlargerpiezoelectric responsethanother

III-Vcompounds.

3

1.3 Literature Review

Evenwiththese

benefits,

attemptshave been madeto getevenmore

improvements

out ofthese systems. One_waytoboostthe advantages ofthenitride-based

structure wasto delta dopeit.

Kahnetal.

[8]

foundthat the physicaldeviceperformance fellshort ofthe

expected performance as a result of resistanceinthephysical

device,

whichlimitedthe

availabilityofsheet carriers. As_such,

they

used delta

doping

tocombatthis. Inthiscase

they

chosetodopethechannellayer.Nomention was made of

impurity

_{scattering in}the

region as a result ofthis

decision,

but

they

didmentionthatthecarriers were abletoreach

saturation velocity.Thismethod allowedforahighersheet charge product and ahigher

breakdownvoltage.

Cheng

etal.

[9]

attempteddelta

doping

toimprovethedeviceperformancein power applications.

They

boostedperformance evenfarther

by

testing

notjust delta

doping,

but delta

doping

inadouble junctionwhere

they

choseto putthedopedlayerin

thebuffer layerratherthaninthe channel.Thesetwo improvementscombined canleadto

a

doubling

ofthecarrier

density

oversinglejunction HEMTs.

They

obtainedahigher

draincurrent withasmallerthreshold voltage, butthetradeoffwasthat thehighdelta

doping

thatleadto these improvementsreducedthetransconductance. In_{the end,}

they

hadto findtheoptimal

doping

levelsforthe two delta-doped layersto get optimal performance fromthedevice.

Finally, Fu, Wang,

and Willander

[10]

useddelta

doping

with aGaAs

small_areas, ontheorder of nanometersin

diameter,

thatare etchedinto amaterialto

confine carriers. Inthesame_waythata quantum well reducesthecarrier_energyspectrum

fromathree

dimensional

continuumto two

dimensional

discretized_{energy levels}where movementisconfined_{to the well,} quantumdotsconfine electron"movement"to the

small space ofthequantumdot with zerodegreesoffreedom [26].

Fu, Wang,

and

Willanderetchedtrenchesinto theirheterojunctionstructure and used multiple gates aroundthe intended"dot"

wherethedotisthenrealized

by

_applyingnegativevoltage on all ofthegates around theetched area. Thisallowedthem tocontrolboththesize ofthe

quantumdotandthenumber of carriersinthechannel. To get evenmoreimproved results incarrier concentration and_mobilitythedelta doped layerwascombined with an InGaAs layerwhichcreated a"bowl"intheconductionband allowing for increased

electron confinementthus

increasing

carriers_contributingto2DEGconcentration. This

makes sense asit has beenobservedthat theshapeofthewell will_playa rolein

deterrnining

thecarrier

density

therein[11].

Delta doped

HEMTs,

though

they

offerimprovedperformance,will notbe

consideredhere. Thiswork willfocuson aHEMTofthestructurepresentedin Figure 1.1a. Sacconietal.

[5]

considered suchastructure with respecttothe calculationofthe

self-consistentSchrddinger-Poisson equation,which

they

thenusedtocalculatetheI-V

characteristicsoftheHEMT.

They

didnotdelvedeeperinto temperaturedependenceof

the structure, whichcanlead tosignificant changes suchas areductionin2DEG

mobility, reducingcarriervelocityandthestructure'scurrent andtransconductance [3].

They

also didnotinvestigatetheeffectsofpolarizationon other system parameters.

Trellakiset al.

[12]

usedanapproximationschemeto solvefor_{ris, the}sheet

carrier

density,

while_savingcalculationtime. Theirworkformedabasis fromwhich

[5]

structuredtheircalculations. Withcalculationtime

being

adifficultconstraint in solving

thissystem of_equations, it'snot_surprisingthatothershavealso investigatedwaysto

speed_upthisprocess.

Cole, Boettcher,

andSnowden

[13]

created an approximationto

speed _upthecalculations and reduce_{the effort, computationally,}of_solvingthecoupled

equations. Meanwhile LuiandFukuma

[14]

used a series of matricestosolve

Schrddinger'sequation analytically. Thismethod would savetimeand effortin solving

theSchrddngerequationforaHEMTwithapotential wellwhoseapproximated shape

hasaknown

Airy

function.

Aconcernthatwas addressed

by

Ando et al.

[15]

wastheproblem of_potentially

divergentwavefunctionsolutions.

They

useda costfunction in_{their analysis,}which

would ensurethat thenumerical solutionto Schrodingerwould converge. Thismethod

was nottested on anitridecompound or on a quasi-triangular_well, soits_{reliability in}

thosecases cannotbe vouchedfor here. Eigenfunction divergenceis aproblemthatwas

encounteredinthemodeltobepresentedas well.

Maetal.

[16]

focused onthe_validityofthe triangularwellapproximation

(TWA),

whichtends tobeusedforanalyticalmodels. Though

they

tested thison a

different structure, HEMTsandMOSFETsshare _manyphysicalproperties andthe

assumptionhereis thattheconclusionsreachedin

[16]

will also_applyto theHEMT

structure.

They

foundthat assuming theTWAworkedwellfor predictingthesurface

electronconcentration, ns, andsurfacepotential,

*PS>

ifall parameters are chosen_well, but

theanalytical modelfailedin

finding

thecarrierdistributionprofile,andtheinversion

layer_centroid,andforuse at flat-bandvoltage.

AnotherMOSbased_study

involving

theself-consistent Schrddinger-Poisson

equation wasthatofJanikandMajkusaik [17].

They

foundthatquantization_strongly

affectstheelectrondistributionatthesurface.

Classically

thedistributionwould peakat

the

interface,

butthequantum mechanical analysis putsthepeak somedistance away

fromtheinterface. _{This seemingly}smalldifferenceaffectstheamountofinversion

charge, the threshold voltage,band

bending,

andthemobility. Withall ofthese things

depending

onthepresenceorlackofdiscretized_energy

levels,

quantumanalysiswould

seeminglygive amore accurateresultovertheclassical approach.

Finally,

Chang

and Fetterman

[18]

proposedan analytic modelthatincludeda2D

solutionto thePoissonequationand parasitic resistancesinthechannellayerwith_very

good agreementto experimentaldata.

They

focusedonthe GaAsstructure,

however,

so

theresultsdidnot_{verify how}wellthismethod would agreewithexperimentaldata fora

GaNstructure.

1.4 Proposed Work

As devicesize continuesto get smallerandsmaller inphysical

dimension,

the

quantum effectsbecomemore andmoreimportant [17]. As has been stated, these

quantum effects canhave a

big

impactontheoutcome of someparameters and so inthis

work, thejunctionwillbeconsideredthroughanalysis ofthecoupled self-consistent

Schrddinger-Poisson

equations. Thisworkwillhave abase in

[23]

andexpand onit

by

solving numericallyto includethequantum effectsintheequations.

Solving

theseequations _numericallywilltake_{away any inaccuracies}that come

about as a result oftheapproximations usedinanalytical solutions.

Also,

there shouldbe

fewerconditions

binding

thismodelthan thereareforthoseusedto make classical

solutions valid. For_{this reason, the} model shouldbeapplicabletoa_varietyof cases.

Themodels presented_primarilyfocusedon oneaspect oftheheterojunctionor

another.

Pulling

fromtheobservationsmade

by

othersof waysof_approaching

Schrddinger'sandPoisson'sequations orontheeffectsoftemperature, or evenhow

differentmaterials_{may have differentproperties, the}proposalisto tietogetherafewof

theseaspects. Themodelwilltie theeffects oftemperature changes andpolarization

variabilityintothenumericallysolved self-consistentSchrddinger-Poissonequation.

Temperature can affect_manyaspects of a

junction,

fromthe strainbetweenthematerials

to the

density

and_mobilityofthecarriers. Assuch, thiswouldbeacrucial parameterto

include intheanalysis. Polarizationeffects will_{vary from}materialtomaterialand are

based onthephysical structure ofthedevice. Giventhatpolarization strength willaffect

the

density

ofcarriers availablefor conduction,thiswould also beanimportant parameter

to includewhen_{considering the}behaviorofthedevice.

Themodel will beverifiedwithexperimentaldataavailablethroughliterature.

1.5Thesis Layout

Chapter 1 was an overview ofthebasic informationonHEMT devicesandnitride

material propertiesinparticular. Chapter 2willcovertheanalysis ofthe system. Chapter

3 willdescribethemodelthatwas createdtorepresentthedevice. Chapter 4will show

theresults andChapter 5 willcoverconclusions and opportunitiestoexpandonthis

thesis.

Chapter 2: Methodology

2.1. Introduction

2.2 ThePoissonEquation

2.2. 1 The Finite-Difference Solution

2.2.2 Piezoelectricand SpontaneousPolarization

2.3. The SchrddingerEquation 2.4. I-V Plots

2.5. Sheet CarrierConcentrationvs. Gate Voltage

2.6. Temperature Dependence

Starl

!

Poisson

Equation

2.1. Introduction

The numerical solutionsto theSchrddingerandPoissonequationswillbeusedto

findtheconductionbandandsurfaceelectron

density

distribution. Theequationsare

coupled _{self-consistently}sothat

they

update

each other untiltheelectron concentration

converges. Thisprocess is shownin Figure 2.1.

Thissystem of equations will_ultimatelybe

investigated in depthwith anAlGaN/GaN

heterojunction inorderto takeadvantage ofthe

additional polarizationsources inthatmaterial.

To start, though,AlGaAs/GaAswillbe

modeled. Froma_workingGaAsmodel, the

GaN model willbe created

by

_making

adjustmentsto thePoissonequationto account

Schrddinger

Equation

1

;; In( v\dx

Has

co live

[image:27.533.299.482.348.628.2]

Enc

Figure 2. 1; Coupledself-consistent Schr6dinger-Poissonequation system

ofequations.

fortheadditional charge fromthepolarization ofthenitridecompound. Theeffects of

thissmall change willfilterthroughtherestoftheequations.

2.2. The Poisson Equation

Tostartthis systemof_equations,a surface potentialisplugged intothePoisson

Equation

(2.1)

[1,

19].

-f=L-_^p_ll+N:_N:)

(2J)

ax dx _ss

Inthis equation,qis theelectron_charge,esisthe _permitivityof_{the material,p is}the

density

of

holes,

nisthe

density

of electronsin_{the material,} and

N/

and

Na~

arethe

ionizeddonorand acceptor_atoms,respectively. Theassumptionisthat thebuffer layeris

n-type andthechannellayeris_{unintentionally doped slightly}p-type. Becausetherewill

notbea significant sourceofionizeddonors inthep-typechannel

layer,

andthePoisson

equationisobserved fromthechannel_{material, the}

Nj

termwillfallout oftheequation.

As_{such, the}Poissonequationinthechannellayer becomes:

C^r

=_-(p-n-N;)

(2.2)

dx'

ss

The holeconcentrationis represented

by

theconcentration oftheunintentional

doping

in

thechannel layerunder_{the quasi-neutrality}approximation. Thefirsttime through the

coupledequations, thesheet carrier concentrationistaken tobetheclassical

approximationasexpressedinEquation(2.3).

//,=

(2.3)

5

V,

where_n;istheintrinsiccarrierconcentrationand

Na

is theconcentration ofunintentional

doping. Thisnsvalue is_onlyused as aplaceholderthefirsttime through thecycle. Once

thenumerical calculations are madeforthe surface carrier concentration_and,

subsequently,thecarrier

distribution,

thisvalueisusedinplace ofthe_classically

calculated value.

The Poissonequationwillbecalculated_manytimesoverthroughout thelengthof

device inorderto get position-specific measurementsto

develop

theband diagram. To do

thisa mesh needstobe created. Thefirst _{step is}todeterminethe_{necessary GaN layer}

thickness. Athicknessoftwo times the depletionregion width was chosenbecausethis

widthwould ensurethat theGaN bulk wouldbeneutraland unaffected

by

_anyofthe

charge

balancing

thatwilloccur atthefrontendofthedevice. Thismodeldoesnottake

the

body

connectionintoconsideration,and_ensuringthat theGaNlayer isthickenough

helpsto avoidthatcomplication. TheGaN layer isthendivided into Nequal parts. The

largerNthesmootherthe_resultingplots, butthatalso increasesthetimeittakes the

calculationsto run. A balancemust bestruckbetweenspeed and accuracy.

The Fermi level

(Ef)

isrelatedto theelectronandholedistributionthrough

Equation (2.4).

e

EfEi qijiU)

kT =_e kT

(2.4)

where

,

isthe intrinsicenergy

level,

kis Boltzmann'sconstant, Tisthe temperature in

degrees

Kelvin,

and^(x)isthepotentialforagiven

location,

x,intothestructure. Herex

willbe

determined

by

_thesize ofthemesh.

Considering

_this,theelectronconcentration

fora specific point wouldberepresented

by

n2 EfE' 2 #w

n(x)=_-^e kT

=-i-e kT

(2.5)

whiletheholeconcentrationfora specific pointisrepresented

by

p=NaekT _=Nae '

(2.6)

Plugging

Equations

(2.5)

and

(2.6)

intoEquation

(2.2),

it isclear, as seenin

Equation

(2.7),

that thePoissonsolution will goto

infinity

_once^(x)has_anysignificant

value whetherthepotentialispositiveornegative.

d2V(x) a -^

if qJ^

dx~

ss

Na

This

did,

in

fact,

proveto bea problemwhen

doing

the initial Poissonequation

calculations. Whilethefirsttwomesh points calculated provided goodresults, as verified

by

the_resultingcarrier concentration vs. gatevoltageplots, subsequent pointsincreased

without bound.

Thenext attemptwasto solvethePoissonequationthrougha method outlinedin

[19]

as

describedbelow.

2.2.1. TheFinite-DifferenceSolution

To plot aproper conductionband

diagram,

thedesire wastofinda methodthat

would notdivergeforlargervaluesof surface potential. ThePoisson_equation,as a

second orderdifferential equation,canbe solved

by

thefinite-difference methodas

shownin Equation2.8

[19,

20].

d2V(x)_<t>(xl_l)

+_{<f>(x,+l)-2<fi(xl)}

dx2 (Ax:)2

(2.8)

Thiscanthenberepresented

by

thematrix equationAO=

R(<D)

where, fora structure

withn meshpoints,A isa square

(nxn)

matrix_representingthecoefficientsfortheright

handside of

(2.8),

O isamatrixofthepotential experienced at agivenpoint_alongthe

structure, andR isrepresented

by

R=

^-(p(x)

-n(x)-N-a)=

^~

s.

Ne kT

2 i]</H.v)

* v TV

By

thenatureofthefinite differenceequation,

(2.8)

becomes

(2.9)

1 0 0 0

1 -2 1 0

0 1 -2 1

0 0 1 -2

0 0 0 0 0 1 <j>(xx)

to)

=R

to,)

toJ

J/

(2.10)

withthe

boundary

conditionsthat </>

(xi)

=

R(xi)

=

<j>(x=

0)

sothatatthesurfaceofthe

device,

thesurface potentialis

fully

felt,

and </>

(x)

=

R(xn)

=

<t>(x>

2w)

where wisthe

depletionregionwidth, placing thelast mesh point wellinto thebulkwherethesurface

potentialwouldhavenoeffect onthe device.

Using

thismethodRneed_onlybecalculated_{explicitlyonce,} fortheinitialpoint

one removed fromthesurface. Eachsubsequent mesh point iscalculatedbasedonthe

result ofthepointsbefore andafterthepoint under consideration. Thesolutionisno

longer basedon an equationthatdependsonthoseexponentialtermspasttheinitial

calculation, sotheissue ofthesolution

diverging

ratherthan_convergingis correctedin

thisway.

Theclassical solution sufficedto _accuratelyrepresentthepotential and_resulting

conductionbandprofile forthebufferlayer [1].

q<t>GAx)=

-q2Ndx2

|

qss2E0y

[

fNdSx

q2NdS2

| ^ <

q2NdS2

2Ss.

Ssl Ssl

2s.

2Ss\

(2.11)

where

Nd

istheconcentration ofdopantatomsinthebuffer

layer,

d isthe thicknessofthe

spacer

layer,

_esiand_S2 arethe_permitivityofthebufferand channel

layers,

_{respectively,}

Eo+

istheelectric fieldonthechannelside ofthe

junction,

and

AEC

isthe conduction

band_{gap between}the twomaterials. In_{this way, the}righthand quadrantsofFigure 2.2

willbe calculated_numerically

by

Equation2.8 andtheleft handquadrants willbe

calculated_classically

by

Equation 2.11.

Figure2 2 Band diagramplot.Allpointsalongdiepositive \ a\is willbecalculated_numericallywhile allpointsalong ihenegative n axiswillhecalculated_classicallysincethereare noquantumeffects inthispartofthedevice

2.2.2 Piezoelectricand SpontaneousPolarization

Toaccountforthepolarizationfactor inthePoissonequation

(2.1)

becomes

dE d

(

d

1W ^ ni >

=

ss }

(x)

+

P(x)

dx dx[ dx

d2V(x)

dP(x)

dx'

dx

(2.12)

where

P(x)

isthetotalpolarization, spontaneous and_{piezoelectric,}atlocationx. Thetotal

polarizationinducedatthejunctionisthedifference inthespontaneous

(Psp)

and

piezoelectric

(Ppe)

polarizationsbetweenthetwomaterialsperEquation

(2.13)

[28,29].

ff=

(P*y,

<"0

~

***>

H

+

[P^

0>

~

PPEaM

(m))

(2. 1

3)

where misthealuminummole fractionofthedoped materialinthesystem. Giventhat

theGaN materialinthedevice under considerationis_verythick, andthe GaNlayer

is,

therefore, consideredtoberelaxed, theGaNpiezoelectricpolarizationtermiseliminated

so thatEquation

(2.13)

becomes [image:33.533.154.393.53.245.2]

Plol=

\P(x)dx

=

Psp

-Psp

+Ppr

""

J *>'_{MOM '"GolV} I I-A

(2.14)

where

Psp

.

(w)

=(-0.052m

-0.029)

[C/m2]

/>., =-0.29

[C/m2]

^a(0)-a(m)^

a(m)

e,3(/)xC,,(/??)

[C/m2].

(2.15)

(2.16)

(2.17)

EachoftheAlGaNproperties was calculated as aninterpolationofthephysical

propertiesofGaNandA1Nper [28]. Here a(0) isthelatticeconstant wherethematerialis

undoped and_a(m)isthelatticeconstantforthedopedmaterialwhere_{a(m) is}calculated

as

a(m)=(-0.077_m+2.1

89)

xlO'10

[m]

(2.18)

Thepiezoelectric constantsarerepresentedin Equation

(2.17)

by

e3i and_e32calculated

by

g31(w)=(-0.11m

-0.49)

[C/m2]

(2.19)

and

e3i(m)=(0.73m₊

0.73)

[C/m2].

(2.20)

The terms

C)3

and

C33

in Equation

(2.17)

arerepresentativeoftheelasticconstants

calculatedas

Cl3(m)

=

(5mxl05)

[GPa]

(2.21)

and

C33=(-32m+

405)

[GPa]

(2.22)

Onceo is_{calculated, it is}convertedto

Ptot

throughthecalculation

PM=-[1/m3]

(2.23)

q

which expresses carrierspercubic meter and canbeconvertedfrom hereto

cm"3 to fit

into thePoissonequation_alongwiththe terms forcarriers present atthejunction.

Thispolarizationtermwas added_onlynearthejunctionofthe

device,

as opposed

toit

being

accountedforinthebulkofthematerial or over alargearea.

2.3. Schrddinger's Equation

OncethePoissonequationhas been_solved,other attributes ofthe structurecanbe

found,

such astheelectricfield andconductionbandprofile. Thenext_stepthrough the

cycleisto solve Schrddinger'sequation,which

[15]

and

[19]

show

being

donewith

matrices, inmuchthesame_waythatthePoissonequation was solvedhere.

However,

here Schrddingerwas solved_numerically

by

_wayofthe _shootingmethod [20]. The mesh

forthisportionofthemodel was changed becausethequantumcalculations requireda

smallermeshthan thePoissonequation portion. Themeshwastaken tobe 1

A

increments fromx=₀

untilthepointat whichtheheightoftheconductionband

surpassedtheheight oftheconductionband discontinuity.

The eigenvalues, orwavefunctions

(*Pj),

werecalculated

by

xj(x)

=

m*(x)

^f-(V(x)-EJ)

+

-^

+

^-T

lr m

(x)

m

(x-\)

4MX-2)

m

(x-\)

(2.24)

where

T/xi)

= ₀

and

lF/x2)

=

1,

and istheeffective massofanelectronatpoint x

alongthechannel.

Eigenvalues,

orthe_energylevelspresent inthepotential well

(EJ),

arecalculated

by

searching fora valuethatcausesthewavefunctiontoconverge_starting fromaninitial

value and_makingsmall adjustmentsto thisvalueuntil convergenceis achieved.The

resultingwavefunctionsare normalized as shownin Equation

(2.25)

[20,

22].

if/(x)

Thisnormalization_step ensuresthatthe wavefunctions satisfytherequirementthat

\i//*(x)xy/(x)dx

=\ [21,22].

Having

solved Schrddinger'sequationto findall wavefunctions and_energy

levelsforthequantum well predictedfromthe Poisson equation, n(x) canbe found

by

[5,10,12,24].

(2.25)

'K*)=E'^kw|:=mrI> l+e kT

j(x)f

(2.26)

whereD isthe

density-of-states,

as represented

by

D=

,

%

representsthewave

jth

h'

wave functionofthewell, and

Ej

istheeigenvalue associated with each wave function.

This n(x) isthenpluggedinto thePoissonequationasthenumberofelectronsinthe

channelat location*. Thiscycle continues inthiswayuntil

^n(x)dx

ceasestochange

significantly. Atthispoint,nshas convergedandthebanddiagramisdeterminedasare

the_{energy levels}ofthequantumwell.

2.4.Current-_{Voltage Plots}

Thenext stepsinthecoupled

Poisson-Shrddinger

solutionwill leadto thefinal

parametersto

develop

thecurrent-_voltage

(I-V)

characteristicplots. Thegatevoltageand

sheet carrier concentration willberefined and willthenbeusedinconjunctionwiththe

modeldevelopedin

[23],

with some minor adjustmentstoplottheI-V characteristicsof

thedevice.The inclusionof polarization_alongwithtemperature changes provide alook

atthefullrange oftheperformance ofthedevice giventhesechanges.

2.5. SheetCarrier Concentrationvs. Gate Voltage

Thenumerical model presented will_stepthroughthe

loop

presented in Figure 2. 1

until a solution converges. Oncethecarrier concentrationceasestochange_{significantly,}

the

loop

will stop. Astheprogramcontinuesto theprocesssteps,bothnsand

Vgb

willbe

refined untiltheproper valuesarefound.

Gate voltageis calculatedas

VQB

=<t>B-<l>C,\ +

rEf-AEc

^

(2.27)

where <j> Bisthebarrierpotential which canbecalculatedas

tB(m)

=(1.3m₊

0.84),

(2.28)

AEC

istheconductionband

discontinuity

and^oi isthepotentialdifferenceoftheAlGaN

layerat itsinterface withthegatecontactandattheAlGaN/GaN

interface,

whichis

expressed inEquation 2. 1 1 in itspositiondependent

form,

butwillbetakenasthe

difference between

thepotential atthe two specifiedlocations forthepurposeofthis

calculation.

The Fermi _energy,

Ef,

fromthebottomofthe conductionband onthe GaN side of

thestructure attheAlGaN/GaN

interface,

is

being

calculated as

Ef=t,-<PB-

(E.

\

Son

\ 2

where </>sisthesurface _potential,<pBis

being

calculated as

(2.29)

kT <pB = _xln

9

rN\

k". j

(2.30)

and E isthe

bandgap

of galliumnitride. The resultinggatevoltage fora given surface

potentialwillthenbepluggedinto thecurrent-voltage equation.

For everygiven surface potentiala_{corresponding} sheetcarrier_{concentration, ns,}

willbe calculated aswell. Therearetwopossible waystocalculatethisvalue.

Onemethod involves simply

looking

attheslope ofthepotential well. Inthis

way, nscanbe calculated

by

,= x^L

(2.31)

q

where E, istheelectric fieldattheinterface

ontheGaNside ofthejunctionand_sGnis

the_permitivityofthegallium nitride.

Anothermethod wouldbetousethedefinitionthat states,

ns = \n(x)dx

(2.32)

where_n(x) isthecarrierdistributioninthechannelthatis calculatedbasedonthe

Schrddinger

equation's_resultingwave

functions

as calculatedinEquation (2.26).

2.6. Temperature Dependence

Temperatureeffects are_veryimportantto consider giventhat theresistancesin

thematerials alone can cause _{self-heating,} which can raisethe temperatureofthedevice

enoughto alter material parameters.

Knowing

howa givendevicewillrespondto these

changeswillensurethatit is designed toperform as expectedon paperaswellasin its

physicalform. To accountfortemperaturedependenceseveral ofthe material parameters

were adjusted fromconstants to theirtemperature-dependent form.

The intrinsiccarrier concentrationis_overtlyafunctionoftemperature

by

E.

,=W

2kT}-

(2-33)

Beyondtheexplicittemperaturedependenceofthe temperature intheexponential ofthe

equation, severalofthe termswithintheequationaretemperature-dependent aswell.

Per

[25],

the

bandgap

ofeach materialchangeswithtemperature _accordingto

T2

E(T)

=

EQ

(0)-7.7xl0^x

[eV]

(2.34)

gy ' sowv

r+600

fortheGaN material where

Eg(0)

=3.47 eV forwurtzite

GaN,

andforA1N

T2

E

(T)

=E

(0)-1.799x10"'

x [eV].

(2.35)

*"y ' _s

7+1462

Eg(0)

was not given_explicitlyfor

A1N,

soitwastakenas6.2219 eVinthemodel

being

presented, asthatvalue providestheproper

bandgap

forthematerialat300Kperthe

equation.

It shouldbementionedherethatwhen_obtainingcertainmaterialproperties, such

as

bandgap,

of

AlGaN,

Vegard'sLawisused

by

which alinear interpolationbased inthe

percentage ofA1NandGaN isusedto determinedwhatpercentageof each material

contributedto thegiven property.

Using

Vegard's

Law,

thetemperature dependent

calculationforthe

bandgap

for AlGaNwouldbecalculated as

^,

(D

=

*(^ <?))

+

a

-*x^ (D)

(2.36)

Othermaterial parameters wouldbecalculated inthesame manner.

Theconductionbandandvalenceband

density

ofstates

(Nc

and

Nv

_{respectively)}

are also temperaturedependentasdefined

by

GaN:

Nc

xT'2

[cm"3]

(2.37)

Nv=S.9xlOl5xT'2

[cm"3]

(2.38)

AN: Nc=1.2x10l'xr2

[cm"3]

(2.39)

Vv=9.4xlOl6x7^

[cm"3].

(2.40)

Again,

Vegard's Law isusedhereto determinethematerial propertiesofAlGaN.

The conductionband

discontinuity

betweentheGaNandAlGaNmaterialswill

varywithtemperaturewhenit istakenas

AE=0.7(Ee

e v _Saigon -Ee_Sgh

)

'

(2.41)

v '

per

[28]

and[23].

The equation usedto calculate_mobilityinthismodel wasextractedfroma

temperaturedependent plotin [33].

Only

a small portion ofthe _mobilityplotwasneeded

and

limiting

theinformationusedtoextract an equation servedto refinethe final

expression and ensuredthatitwould produce relevant results. Equation

(2.42)

isthe

resultingequationthatwillbeusedtorepresent_{mobility in}themodel.

//(r)=(3xl08)xr21154

(2.42)

There isa_relationshipforthe temperaturedependenceofthelattice constant as

well, as

[30]

_shows,

however,

no _source,

including

_{this one,explicitly}gavethat

relationshipforGaNinparticular.

However,

anequation wasextractedfroma plotin

[25]4.

Thisplot wasreproducedin Figure 4.13.

Chapter 3: The Modeland Device Properties

The model presented wasfirst developed foranAlGaAs/GaAs heterojunction.

This GaAs basedmodel wasthenupdated withthe_necessarychangestocreate an

AlGaN/GaN junction. Theproperties of each ofthesedevicesare shownin Table 3.1 and

3.2.

The basicstructure ofthedevice isthesameforeach oftheheterojunctions. As

depicted in Figure 1.la, asingle wellheterojunction isconsidered. Ifthere weretobe

more_{wells, there}wouldbe multiplelevelsof_{alternating buffer layers}andchannellayers

untilthedesirednumber ofwellswas achieved.

The AlGaAsstructure wasdevelopedpertheparameters specifiedin [31]. The

device inquestion waspresentedfor bothanaluminummolefractionof

0.26,

the3468

device,

and

0.28,

the3469 device. Thefocusforthepurposeof

building

thepresent

model wasthemodelbuiltonthe3469 device.

The buffer layerthicknesswasset at400

A

witha spacerlayerthickness of65

A.

Though inthe idealcasetherewouldbeno

doping

to the channel

layer,

eveninthebest

processes, therewillbe somelevelof

doping

to thechannellayer

by

nature ofthe

material growth process. Thisunintentional

doping (UID)

levelwas notspecifiedinthe

paper;

however,

one was selectedbasedonthevaluethatalignedthecalculated results

withthosepresentedin [31]. Alsonotethatthisunintentional

doping

concentrationis

usedforboththespacerlayerandthechannellayer. The

doping

levelofthebufferlayer

wasspecifiedat0.6x

1018

cm"3. Forthemolefractionof

0.28,

a conductionband

discontinuity

of280meV was specified. Itshouldbenotedthatachannellayerthickness

was called outto be

8000A; however,

ratherthan_usingtheassigned_{value, the}model

calculateda channellayerthicknessbasedonthe depletionregion ofthechannellayeras

[image:43.533.63.470.197.368.2]

described

in Chapter 2.

Table 3.1: AlGaAs/GaAs heterojunctionmaterial properties

Material

Property

AlGaAs/GaAs Heterojunction

Aluminum MoleFraction 0.28

Doping

Concentration

[cm3]

0.6xl018

Unintentional

Doping

Concentration

[cm3]

lxlO11

Buffer Layer Thickness

[A]

400

Spacer Layer Thickness

[A]

65

Channel Layer Thickness

[A]

8000

Conduction Band

Discontinuity [meV]

280

The AlGaN/GaN heterojunctionmaterial characteristics werebasedonthose

provided in

[5],

whicharelisted in Table 3.2. Sacconiet altested theirmodelfor

aluminum mole fractionsof

0.1, 0.2,

0.3 and0.4. Themodelto bepresented_onlyused

the firstthreeastheserepresentthemorecommonrange of molefractionsusedin

devicesto ensurethefewest defects inthedevice. The buffer layerwas set at 150

A,

with

aspacerlayerof50

A.

The

doping

concentrationofthe AlGaNlayeris 1x1018 cm"3 with

anunintentional

doping

concentrationof1xIO17cm"3. AswiththeGaAs-based

heterojunction,

thechannellayerthicknesswas calculatedbasedonthedepletionregion

depth. Inthecase of

[5]

no channellayerdepthwas specifiedexceptto _saythat itwas

sufficientlythicksothatall potential effects were confinedinthe channel

layer,

so this [image:44.533.65.463.149.324.2]

assumption seemedtobe justified.

Table 3.2: AlGaN/GaN heterojunctionmaterial properties

Material

Property

AlGaN/GaNHeterojunction

AluminumMole Fraction 0.1,0.2,0.3

Doping

Concentration

[cm3]

lxlO18

Unintentional

Doping

Concentration

[cm"3]

lxlO17

Buffer Layer Thickness

[A]

150

SpacerLayer Thickness

[A]

50

Channel Layer Thickness

[A]

2xXdepl

Conduction Band

Discontinuity

[meV]

170,330,510

Finally,

theconductionband

discontinuity

wasset at 170meV, 330meV, and

510 meVforaluminum molefractionsof

0.1, 0.2,

and

0.3,

respectively. Theassumption

forthesedevicesisthat thejunctionisabrupt. Fermi-Diracstatisticsare usedto calculate

thecarrierdistribution. The caplayerpresentin Figure 1.1awasnotincluded inthe

calculationsforthemodel.

Also,

as_previouslymentioned, themodelassumesno _activity

ontheback endofthe

device,

thoughthatcanbechanged

by

_{adjusting Equation}

(2.27)

to

includebiaseffects onthebody.

Chapter4: Results andDiscussion

4.1.

Introduction

4.2.The Poisson Equation

4.3.

Schrddinger's

Equation

4.4. Sheet Carrier Concentrationvs. _{Gate Voltage}

4.5. Current-_{Voltage Results}

4.6.

Temperature

Dependence

4.1.Introduction

Thischapter will showthecalculated results asthemodelprogressesthrough the

system of equations as outlinedinFigure 2. 1. Thischapterwill firstexploretheplotted

solutionsforthePoisson_equation,followed

by

thosefor Schrddinger'sequation.Next

the_resultingsheet carrier concentration andthe gate voltagewillbereviewed;after

whichthe_mobilityandtheI-Vresultswillbepresented.

Finally,

temperaturedependence

willbe discussed.

Whileamodel_representingthebehaviorof aGaN-based heterojunctionwas

sought, thebestplaceto start seemedto bea_simpler,yet similar_{problem; that}of

developing

amodelforaGaAs-basedmodel. Theconstantsusedinthemodels are

outlinedinTable 4.1.

[image:46.533.84.440.70.244.2]

Table 4.1: PhysicalProperties

Constant Value Reference

Temperature,

T

[K]

300 Assigned

Electron

Charge,

_q

[C]

1.6el9

[1],

[19]

Boltzmann's

Constant,

k

[J/K]

1.3805e-23

[1],

[19]

Plank's

Constant,

h

[J-s]

6.63e-34

[1],

[19]

ReducedPlank's

Constant,

h

[J-s]

1.05458e-34

[1],

[19]

Free Electron

Mass,

mo

[kg]

9.11e-31

[1], [19]

Density

of

States,

D

[cm2 J"1

]

1.743e32

While AlGaAs-specificparameters, suchas

bandgap

and conductionband

discontinuity

have groundedequations,as mentionedin Chapter

2,

Vegard's Lawwas usedtocalculate

thoseparametersforthe AlGaN/GaNheterojunction.

4.2. The Poisson Equation

Tostartoffthissystem ofequations,thesize andshape ofthequantum well must

bedetermined. Equations

(2.9)

and

(2.10)

providethemeanstodo this.Theresults are

showninFigure 4. 1.

ConductionBand

0.8

c H 0.6

3

I

04

0.2

0

-500 0 500 1000 1500

[image:47.533.135.392.56.236.2]

Channel Depth(Angstroms)

Figure 4. 1Conduction bandprofilefromthebuffermaterialthrough thebulk

ofthechannel materialforanAlGaN/GaNheterojunction device.

The numerical solutionfortheGaNsideofthejunctionrepresentsthe diagram

fromtheconductionband

discontinuity

intothebulk ofthe material. The AlGaNsideof

thejunctionwas solved withtheclassical equation.

Initially,

thefactthatitdidnothave

the"bowl"shapethat_verycommonlyappearsinthe representationofthewide

bandgap

material portionofthebanddiagram foraheterojunctionwasa concern.

However,

a

comparisontotheplotin

[5]

provedthatitwasa good match. Theslope ofthe AlGaN

conductionband is_{very steep}as comparedto themore curved version obtained inthe

AlGaAsversion oftheplot.Thiscanbeattributedto the_{higher field resulting}fromthe

polarization charges.

Notealso that thedivergence ofthebulkmaterial conductionbandplotthatwas

mentioned earlierwas, in

fact,

solved

by

_usingthematrixformofthePoissonequation.

4.3.

Schrddinger's

Equation

Oncetheband diagram is _established,thewell canthenbeusedto solve

Schrddinger's equation perEquation

(2.24)

to calculatethewavefunctionsand

eigenvalues ofthe system.Figure4.2 showsthe_resultingwavefunctions foragiven

temperature and voltage.

Boundary

conditions were sethere

by forcing

_\|/(1)=₀

and_\(/(2)= ₁

. To ensure

that the system_converged,themodel checkstheprogressoftheequationupdates until

thefinalpoint reaches some pre-defined valuethatisconsideredto_{be sufficiently}small.

Using

thelastpointofthemesh provedtobeproblematic insome casesbecausewhile

the lastmeshpointdidnot_converge,in_{these cases,} pointsbeforethatdid. Analternative

method wastoviewthewavefunctionsfromsomedistance in fromthelastmesh point.

Distancesanywherefrom 50

A

to 100

A

were usedto test thismethod. Thismethoddid [image:48.533.90.432.415.616.2]

work,butuponfurther refining Schrddinger'sequation,itproved unnecessary.

Figure 4.2Eigenfunctionsplottedthrough SchrOdingerequationwhenfedtheconductionbandprofile.

Eachcurve correspondstoanenergy levelinthequantumwell.

Figure4.2 showsthe_resultingwave

functions

foreach ofthe three_{energy levels in}

thiswell. Noticethateach subsequent wave

function

is smallerthan theone beforeitwith

thefirstwave function

being

muchlargerthan_anyoftheothers. This isanindicationof

therelative concentration of carriers at each_energylevel. Thewavefunctionsarethen

scaled perEquation

(2.25)

as showninFigure 4.3. [image:49.533.89.436.196.422.2]

Device Depth(A)

Figure 4.3 Scaledwavefunctionplot.Thevaluesfromthesescaledwavefunctionswilldeterminethecarrier

concentrationinthechannelperEquation (2.26).

This scaling makesthewave functionsall_{approximately the}sameinmagnitude and

ensuresthat

they

fulfilltherequirementthat

jV*

(x)

xy/(x)dx=1

. From

here,

carrier

distributionand sheet carrier

density

canbe found.

4.4. Sheet CarrierConcentration vs.Gate Voltage

ThecarrierdistributionfoundthroughEquation

(2.26)

isshowninFigure 4.4.

9E-H9 8E+19 7E+19 6E+19 5E+19 4E+19 3E+19 2E+19 1E+19 0

A

m=_0.3

\

y

Wy

m=_0.2

m=_0.1

1/ \ vC

It \ _^J.^w^ %, V

50 0 60 100 150 200

[image:50.533.130.393.58.217.2]

Channel Depth(A)

Figure 4.4. Carrierconcentration as afunctionof molefractionform=_{0. 1}

,0.2,

and0.3.Asthemolefractionrises,sodoesthe totalnumber of carriers.

Ascanbe seen

here,

theplots are ingood agreement withthose obtainedin [5].

The classical approachto carrierdistributiontakes the carrier concentration peakright at

theAlGaN/GaNinterface. This distribution isobtainedthroughFermi-Dirac statistics and

as such showsthedistributionof carriersbasedonthe_probabilityof anelectron

being

present at locationx. Thecarriers are_stronglyconfinedto thewellbutthere isan

increasing

probability,moving fromthe gate/bufferinterfacetowards the spacer/channel

interface,

of

finding

an electroninthebuffer layer.

The factthat itwouldbepossibleto findelectronsinthebufferlayerside ofthe

junction highlightstheimportanceof_ensuring that thespacerlayeristhinenoughto

allowtheelectronsto passthroughto thechannel. ThoughtheslopeoftheAlGaN

conductionband_maynot support_confining electronstoa_{secondarywell, this}wouldbea

general concernfor_{any buffer}materialband structure

having

that"bowl"shapetoit. In

that case,anyelectronsthat_mayexist outsideofthequantum wellcould gettrappedin

thatbowlandformaparasitic_channel, thus

taking

power_awayfromthedevice.

By

comparison, Figure4.5 showsthe electronconcentrationforadevice dopedto

1x10 with a mole

fraction

of0.28. Inthiscasethedeviceparameterswere changed

to matchthoseused forthe

AlGaN/GaN

device forabettercomparison.

2.5E+19

2E+19

^1.5E+19 E

u, 1E+19

ry 5E+18

* ₀

c0 100 150 200 250 300

Device Depth _(Angstroms)

Figure4.5. Carrierconcentration ofAlGaAs/GaAsconcentration wherethebuffer

materialis dopedto1xIO111cm"3

andthemolefractionis 0.28. Buffer layerthicknessis

150Aandthespacerlayerthicknessis 50A.

Thoughthemolefractionis0.02belowthe comparableplot forthe0.3 curvefor

GaN,

it isapparentthat thereisquite alarge difference betweenthe twoheterojunctions.

Where GaN reachedapeak of_{approximately}

8><1019

cm"3, theGaAs device onlymadeit

to 2xIO19

cm"3. The extracarriersintheGaN based devicecanbeattributedto theextra

polarization chargeaswellasthedeeperquantum welldueto theconductionband

discontinuity

(0.51 for GaNvs. 0.28 for GaAsfortheirrespective molefractions). These

extra carrierswould lendthemselvesto supportingtheseresults.

To test the relationshipbetweensheetcarrierconcentration and gate voltagefor

GaAs,

[31]

wasused_{to verify the} results. Figure 4.6 showsthe results. [image:51.533.114.409.167.314.2]

[image:52.533.86.434.55.294.2]

Gate Voltage

(V)

Figure 4.6. Sheetcarrier concentration vs. gate voltagefortheAlGaAs/GaAs devicepertheparameters

specified_byVinter[31]. Thesolidlinerepresents adevicean aluminumconcentration of_0.26,anAlGaN

thicknessof550A,and a spacerlayerthicknessof75A,whilethedashedlineisadevicewith an

aluminum concentration of0.28,anAlGaNthicknessof400A,and a spacerlayerthicknessof65A.

Thoughthere are somediscrepancies betweentheplotsin Figure 4.6andthosein

[31]

towardsthelower nsvalues, particularlyinthe3468

device,

thedataismuch closerfor

the largernsvalues. Ascanbe_{seen, the}modelisnot_{very strong in}thesub-threshold

region.

Also,

fromtheplot in

[29]

_showinghowthethicknessofthe buffer layeraffects

the relationshipbetweensheet carrier

density

andgate_voltage,a similar plot wasdone

fortheGaAs deviceasshown in Figure 4.7.

St) In c SJ H O u -7E+11 6E+11 5E+11 4E+11 3E+11 2E+11 1E+11 0 -1E+11

tb= _150Angstroms

tb= _{200 Angstroms}

tb= ₂₅₀

Angstroms

tb= _{300 Angstroms}

-* tb= _350Angstroms - tb= _400Angstroms

******tHi"*P*f><>

5 -1 -0.5 0

VGB(V)

Figure4.7Carrierconcentration vs.Vgbwith_varyingthicknessforthe3469GaAs device describedin Figure 4.6.

Theseresults cannotbecompared_{quantitatively}withtheplot in

[29],

as

they

are

madefromtwodifferent materials;

however,

qualitatively itis apparentthat theresults

hold. Asthebuffer layerthicknessgets

larger,

thegate voltagefora given surface

potential getssmaller. Thiswouldindicatethatittakeslessvoltageto generate_carriers,

which makes sense becausethere wouldbemorecarriers availablefor conductionwith a

larger buffer layer.

Physically,

increasing

thebuffer layerthickness willaffect_{q&Gi, the}

difference betweenthepotentialatthe gateandthatatthe AlGaAs/GaAsinterface. This

change in thepotentialdifference gets larger- because

thegate voltage wouldhave less

effectonthejunction interface potential

-causingthegate voltagetobemore negativeto

createaspecificconcentrationofcarriersasthebufferlayerthicknessincreases.

Note,

[image:53.533.91.454.66.335.2]

too,

that althoughthe_step size ofthebuffer layerthickness increaseisconstant, the

curves experience alargershift asthethicknessget larger.

The nsvs.

VGB

plot forthe_{AlGaN/GaN junction is}shownin Figure 4.8. [image:54.533.93.