Boosting the voltage gain of graphene FETs through a differential amplifier scheme with positive feedback

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by  Roberto Grassi

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Solid-StateElectronics,[VOL100,(October2014)]DOI10.1016/j.sse.2014.07.003

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Boosting the voltage gain of graphene FETs through a
differential amplifier scheme with positive feedback
R.Grassia,1,∗,A.Gnudia,V.DiLeccea,E.Gnania,S.Reggiania,G.Baccarania
aE.DeCastroAdvancedResearchCenteronElectronicSystems(ARCES),UniversityofBologna,Vialedel
Risorgimento2,40136Bologna,Italy
Abstract
We study a possible circuit solution to overcomethe problemof low voltage gain of
short-channelgrapheneFETs. Thecircuitconsistsofafullydifferentialamplifierwith
aloadmadeofacross-coupledtransistorpair. Startingfromthedevicecharacteristics
obtainedfrom self-consistentballistic quantumtransportsimulations, we explore the
circuitparameterspaceandevaluatetheamplifierperformanceintermsofdcvoltage
gainandvoltagegainbandwidth.Weshowthatthedcgaincanbeeffectivelyimproved
bythenegativedifferentialresistanceprovidedbythecross-coupledpair.Contactresis-
tanceisthemainobstacletoachievinggainbandwidthproductsintheterahertzrange.
Limitationsoftheproposedamplifierareidentifiedwithitspoorlinearityandrelatively
largeMillercapacitance.
Keywords: Differentialamplifier,gainenhancement,grapheneFET,positive
feedback,radio-frequencyoperation
1. Introduction
Graphene has received considerable interest in recent years for radio-frequency
(RF) applications[1, 2]. Itshighintrinsic carriermobility[3] andlargeFermiveloc-
ity [4] promise a high device transconductanceg , which should in principle enable
m
∗Correspondingauthor
Emailaddresses:[email protected](R.Grassi),[email protected](A.Gnudi),
[email protected](V.DiLecce),[email protected](E.Gnani),
[email protected](S.Reggiani),[email protected](G.Baccarani)
1Phone:[phone];Fax:[phone]
PreprintsubmittedtoElsevier August26,2014
deviceoperationupto theterahertzrangeoffrequencies. Integratedcircuitsmadeof
grapheneFETs (GFETs) have already been demonstrated [5, 6] and cut-off frequen-
ciesofhundredsofgigahertz,competingwiththeonesofIII-VHEMTs,haveindeed
beenreported[7–10]. Despite these progresses,challengesstillremain, inparticular,
inachievingmaximumoscillationfrequenciescomparablewithcut-offfrequenciesfor
poweramplifierapplications[11, 12] andintrinsicgainsg /g (g is the devicedrain
m d d
conductance)suitableforvoltageamplifierapplications[11,13–15]. Thelatterprob-
lem is due to the poor currentsaturation (i.e., large g ) in monolayerGFETs related
d
to the absence of a band gap[16, 17]. Valuesof voltage gain up to 14 have actually
beenreportedforgrapheneinvertersmadeoflong-channelmonolayerdevices(ofthe
orderof1µm)[15,18]. However,asthechannellengthisscaleddown,thegaingets
muchworse[11]sincevelocitysaturationduetocarrierscattering,whichhelpscurrent
saturation,getssuppressed.
The use of bilayer graphene has been proven to effectively improve the gain of
GFETs, leading to g /g values as high as 35 thanks to a band gap opening effect
m d
[19,20]. However,thiskindofdeviceinherentlyrequireslargeelectricfieldsandlarge
voltage dropsin the verticaldirection. In a recentpaper of ours[21] we have inves-
tigatedadifferentapproachtowardtheimprovementofthevoltagegain,whichrelies
ontheeffectofnegativedifferentialresistance(NDR)occurringintheoutputcharac-
teristicsofmonolayerdevicesunderspecificbiasconditions.Againenhancementcan
be obtained thanks to a cancellation effect between the negative g and the positive
d
conductanceoftheload. TheadvantageoverthebilayerGFETisthatthepresenceof
a back gate with a large applied voltage is not essential, althoughit can be usefulin
order to provide electrostatic doping of the source and drain access regions. On the
other hand, the major drawback is represented by the difficulty in controlling circuit
stability,whichrequiresapropercompensationatboththeinputandtheoutputports
andlimitsthescopeofpossibleapplications.
In the context of standard CMOS analog circuit design, a technique based on a
similarconceptofconductancecancellation(named“positive-feedback”or“negative
conductance”technique)hasbeenproposedtoenhancethegainandgain-bandwidth-
product(GBW)ofamplifiers[22–24]. Inthiscase,theoutputconductanceispositive
3
anda feedbackfromtheoutputnodeisusedto generateanegativeloadconductance
thatcompensatesitsvalue,therebyachievinghighgains.Suchtechniquecouldalsobe
appliedtoGFET-basedamplifiers.Sincethedeviceswouldoperateinthestandardnon-
NDRregime,theaforementionedstabilityproblemswouldbeavoided.Itisthepurpose
ofthisworktoinvestigatesuchanidea,throughanumericalstudyofthesmall-signal
performanceofaspecificpositive-feedbackamplifiermadebyshort-channelGFETs.
The paperis organizedas follows. Section 2 deals with the referenceGFET that
isusedasbuildingblockoftheamplifier: thedevicestructure,simulationmodel,and
device characteristics are briefly discussed. In Section 3 the amplifier circuit is pre-
sented. Therein, a detailed circuitanalysis, based on the simulated characteristics of
thesingledeviceandfocusedontheamplifierdcgainandgainbandwidth,isreported.
ConclusionsarefinallydrawninSection4.
2. Structure,model,andsimulationresultsoftheintrinsicdevice
Thedeviceisidenticaltotheoneconsideredinourpreviouswork[21]. Thestruc-
tureisadual-gateGFET,withatopgatelengthof20nm(Fig.1a). Thetopdieletricis
a1.2-nm-thicklayerofAl O (effectiveoxidethicknessof0.5nm),whereastheback
2 3
dielectricissiliconoxidewiththicknessof10nm. Thebackgateisusedasameansto
electrostaticallydopethegrapheneaccessregionsbetweenthetopgateandthesource
anddraincontacts, anditcouldbereplacedbyasecondtopgatecoveringtheaccess
regionsalone.
Thedccharacteristicsofthedevicearecomputedwithanin-housedevelopedcode
forGFETs, basedontheself-consistentsolutionofthe2-DPoissonequationandthe
ballistic non-equilibrium Green’s function (NEGF) equations [25], with a p tight-
z
bindingHamiltonian(moredetailscanbefoundin[21]).
The small-signal frequency analysis is based on a small-signal equivalent circuit
(see[21, Fig.2])whoseintrinsicpartisderivedfrom[26]. Theintrinsicsmall-signal
circuit parameters are extracted from the simulated dc I–V and Q–V characteristics
usingfinitedifferences. Asforthe parasitics, we considerthe possibilityofnon-zero
sourceanddraincontactresistances;sensitivitytogateresistanceandparasiticcapaci-
4
(a)
(b) (c) 4
12 V = -0.3 to 0.4 V V = -0.3 to 0.4 V
GS GS
∆V = 0.1 V ∆V = 0.1 V P1
10 GS GS
3
P
m] 8 VGS 1
µ d
[mA/D 6 P2 g/gm2 P2
I 4 V
1 GS
2
0 0
0 0.2 0.4 0.6 0 0.2 0.4 0.6
V [V] V [V]
DS DS
Figure1: (a)Longitudinal cross-sectionofthesimulatedGFET.(b)Outputcharacteristics atVBS = 9V
showingn-type-FEToperation. (c)Correspondingintrinsicvoltagegain. Azerotop-gate/graphene work-
functiondifferenceΦmgisassumed. P1andP2arespecificoperatingpoints. Figs. (a–b)areadaptedfrom
[21].
tancesisconsideredforonespecificcase.
Fig.1bshowsthedeviceoutputcharacteristicsobtainedwithaback-gate-to-source
voltage V = 9 V and assuming a zero metal-grapheneworkfunctiondifferencefor
BS
both gates(in this paper the back-gate/grapheneworkfunctiondifference will always
betaken tobe zero). Suchvalueof V correspondsto a heavyn-typedopingof the
BS
sourceanddrainregions.Inalltherangeoftop-gate-to-sourcevoltageV considered
GS
intheplot,thechanneldopingisalson-typeandthedeviceoperatesasaconventional
n-type FET, although showing only a weak (or “quasi-”) saturation. The resulting
intrinsicgainislimitedtoabout3.4(Fig.1c). Thankstothesymmetryofthegraphene
bandstructureand the use of an electrostatic doping, the same device exhibitsp-type
5
(a)
(b)
Y =[Y ] +[Y ] +
L 2 11 2 22
−[Y ] −[Y ]
2 12 2 21
Figure2: (a)SchematicoftheCSdifferentialamplifier. TransistorsT1andT2areidenticaltoT3andT4,
respectively.Resistorsrepresentsourceanddraincontactresistances.(b)Correspondingsmall-signalmodel,
where[Y1]([Y2])istheY-matrixoftheextrinsictransistorT1(T2).YLiscalculatedfrom[Y2]asindicated.
RAistheoutputresistanceoftheinputvoltagesource.
characteristics completely symmetrical to the ones in Fig. 1b if the polarities of all
voltages,includingV ,arereversed(notshown).
BS
3. Analysisofthe“positive-feedback”amplifier
Theamplifiercircuitconsideredhere(Fig.2a)issimilartothefirstoneproposedin
theliteratureonpositive-feedbackamplifiers(see[23,Fig.1a]).Itisacommon-source
(CS)fullydifferentialamplifierwithn-typedrivertransistorsandanactiveloadmade
ofcross-coupledp-typetransistors. AlthoughdifferentsymbolsareusedinFig.2ato
indicaten-andp-typetransistors,theyactuallysharethesamestructure: asexplained
in Section 2, one or the other type of transistor is obtained by changing the polarity
of V . The circuit is symmetric, in the sense that transistors T1 and T2 are respec-
BS
tivelyidenticaltoT3andT4.Ontheotherhand,T1andT2mayhavedifferentchannel
widths(W andW ,respectively)ordifferenttopgatematerials,resultingindifferent
1 2
top-gate/grapheneworkfunction differences (Φ and Φ , respectively, with Φ
mg1 mg2 mg1
assumedzero).Contactresistancesaretreatedasfixedparametersindependentofbias.
Weconsideracommonvalueforthesourceanddraincontactresistancesofthesame
transistor(becauseofthesymmetryofthedevicestructure),andweassumeittoscale
6
withtheinverseofthechannelwidth(1/R isthecontactconductanceperunitwidth).
C
Unprimed/primedsymbolsareusedtoindicateintrinsic/extrinsicdeviceterminals,as
done in Fig. 2a for T1 (the blue box represents the “extrinsic” device T1). The cir-
cuitequationsaresolvedusingasplineinterpolationofthedeviceI–V characteristics
obtainedinSection2.
Thankstothesymmetryoftheamplifier,asimplesmall-signalequivalentcircuitis
derivedforthecaseofpurelydifferential(zerocommonmode)signals(Fig.2b),which
involves only the Y-matrices of transistors T1 and T2 ([Y ] and [Y ], respectively),
1 2
which,inturn,areeasilycalculatedfromthedevicesmall-signalequivalentcircuit.
AccordingtothesymboldefinitionsinFig.2b,theamplifiervoltagegainisgiven
byA =v /v =−[Y ] /([Y ] +Y )andtakesasimpleformatfrequency f =0:
v o i 1 21 1 22 L
g˜
A =− m1 , (1)
v0 g˜ +g˜ −g˜
d1 d2 m2
whereg /g˜ =g /g˜ =[1+(R /W )(2g +g +g )],withg ,g ,andg
m1 m1 d1 d1 C 1 d1 m1 mb1 d1 m1 mb1
thedrainconductance,top-gatetransconductance,andback-gatetransconductanceof
T1,respectively(similardefinitionsholdforT2).Eq.1explainsthegain-enhancement
effect: thetermG = g˜ −g˜ ,whichistheloadconductanceduetoT2,canbeneg-
L d2 m2
ative and compensate g˜ . The gain is theoretically infinite if the matching is exact.
d1
However,thecircuitbecomesunstableforg˜ +G <0(orA >0)2. Ifconventional
d1 L v0
well-saturatedMOSFETswereused,g ≫g ,henceadditionaldiode-connectedtran-
m d
sistorsinparalleltothecross-coupledloadtransistorswouldberequiredtoallowcir-
cuitstability[23,Fig.1a]. ThesearenotnecessaryinourGFETimplementationsince
g ∼g (Fig.1c).
m d
3.1. Dcvoltagegainandvoltagegainbandwidth
Thecircuitoperatingpointisdeterminedbyaconsiderablenumbersofvariables/pa-
rameters,e.g.,V,V ,V ,W /W ,Φ andR (seeFig.2a;W canbetakenasthe
i BB DD 2 1 mg2 C 1
2Wearereferringheretothestabilityoftheamplifierconnectedtoashortcircuitattheinputportand
toanopencircuitattheoutputport. Thestabilityinothersource/loadconditionscanbeevaluatedwiththe
standardstability-circletechnique[27].
7
5 20
14 Φ [eV]
mg2
0 10
12 0.1 4
m] 10 0.2 0
µW [mA/2 68 RC = 0 W/W2123 Av0--2100
I/2 4 -30
1
2 -40
(a) (b) (c)
0 0 -50
0 0.2 0.4 0 0.2 0.4 0 0.2 0.4
V =-V [V] V [V] V [V]
L DS2 L L
Figure3: AnalysisofthedcvoltagegainasafunctionofVL = VDD−Vo. TheoperatingpointofT1is
heldconstantatP1andazerocontactresistanceisassumed. (a)I–Vcharacteristicsofdiode-connectedT2
fordifferentvaluesoftop-gate/grapheneworkfunctiondifferenceΦmg2.ThedottedlinesshowtheT2output
characteristics atconstantVGS2+Φmg2from-0.4V(topcurve)to0.3V(bottomcurve)instepsof0.1V.
(b)CorrespondingratiobetweenthewidthsofT1andT2,whichisdeterminedbytheconstraintI1=I2.(c)
Correspondingdcvoltagegain.Positivevaluesindicateanunstablecircuit.
referencewidth).Inordertosimplifytheanalysis,wechoosetofixtheoperatingpoint
ofT1,forexample,atthepointP ofmaximumg /g ofFigs.1b–c,which,according
1 m d
to (1), should favor high amplifier gains since it also corresponds to high g (≃ 17
m
mS/µm).Inthisway,foragivenvalueofR ,wefixV,V ,aswellastheoutputvolt-
C i BB
ageV andthecurrentI thatflowsthroughT1. ForagivenvalueofΦ , arelation
o 1 mg2
betweenV (orV = V −V )andW isobtainedfromthe I–V characteristicsof
DD L DD o 2
T2andtheconstraintI = I .
2 1
The above procedure is illustrated in Fig. 3, where we take the bias point of T1
equaltoP ,R = 0,andweconsiderthreevaluesofΦ . ForeachvalueofV ,we
1 C mg2 DD
first compute the currentper unit width throughT2, I /W , by imposing its top-gate
2 2
and drain voltages to be equal (Fig. 3a). Then, we calculate W /W from W /W =
2 1 2 1
(I /W )/(I /W ) (Fig. 3b). Finally, having set the operating points of both T1 and
1 1 2 2
T2, we obtain the amplifier gain A using (1) (Fig. 3c). It can be observed that, if
v0
Φ = 0andV islimitedto0.5V,themaximumgainisonly12,whichmeansthat
mg2 L
G is not sufficiently negative. In order to further decrease G , which is reduced to
L L
G = g −g inthecaseofR = 0,oneneedstoincreasetheratiog /g ,i.e.,to
L d2 m2 C m2 d2
8
5
4 Φmg2 [eV] 4
0
0.4 4 2
m] 3 0.7 0
µ
A/ W13
W [m22 W/22 Av0 --42
I/2 (a) -6
1 1
R = 100 -8
C Ω⋅µm (b) (c)
0 0 -10
0 0.5 1 0 0.5 1 0 0.5 1
V =-V [V] V [V] V [V]
L D’S’2 L L
Figure4: SameasinFig.3butforRC =100Ω·µm. Here,VBB =9.79V.Thedottedlinesin(a)arethe
outputcharacteristicsoftheextrinsictransistorT2atconstantVGS′2+Φmg2from-0.5V(topcurve)to0.1V
(bottomcurve)instepsof0.1V.
movetheoperatingpointofT2intothequasi-saturationregion. Thiscanbeachieved
by choosing a suitable top-gate material of T2 that provides Φ > 0, as shown in
mg2
Fig.3a,wheretheT2outputcharacteristicsatconstantV +Φ arealsoshownfor
GS2 mg2
comparison. The consideredvalues of Φ are within the range of workfunctionof
mg2
severalmetals[28]. IncreasingΦ alsoleadstotheaforementioneddivergenceand
mg2
changeofsignof A ,whichmarkthebeginningoftheinstabilityregion(Fig.3c). It
v0
shouldbenotedthattheresultingvaluesofW /W arereasonableinallthreecases.
2 1
ThesameprocedurehasbeenrepeatedforthecaseofR = 100Ω·µm(Fig.4),a
C
valuewhichiswithinreachofpresentgraphenetechnology[29].Itisseenthat,givena
maximumV voltageof1V,itisstillpossibletoachievehighgains,butalargerΦ
L mg2
isrequiredtomovethebiaspointofT2intothequasi-saturationregion(Φ ≃0.7eV
mg2
couldbeachievedwith,forexample,Au,Pd,orPt[28]).
Theamplifierbandwidthisevaluatedbynumericallycomputingthefrequency f
p
of the dominant pole of A (s) = v (s)/v(s). In the case of R = 0, an analytical
v o i C
expressionisavailable:
1 g +g −g
f = d1 d2 m2, (2)
p 2π C +C
dd1 L
whereC =C +C +C +C (thesubscripts“1”and“2”referrespectivelyto
L gg2 dd2 gd2 dg2
T1andT2asusual),andthecapacitancesaredefinedasCkk =∂QK/∂VK andCkl|k,l =
9
(a) 2 40
P, R =0 (d)
1 C
f [THz] 1GBWfp 20|A|v0 PPP122, par. 4405
P1, Y=0
0 0 L 35
0 0.1 0.2 0.3
(b11f [10Hz]) 24 GPB1W, RC=f1p00 Ω⋅µm 2400|A|v0 filled: ReCm=1p0ty0: ΩR⋅Cµ=m0223050|A| [dB]v
0 0 15
0.4 0.5 0.6
(c) 2 30 10
P, R =0
2 C
THz] 1 fp GBW 15A|v0 5
f [ | 100 101 102 103 1004
0 0 f [GHz]
0 0.2 0.4 0.6
V [V]
L
Figure5:(a–c)Dcvoltagegain,dominantpolefrequency,andgainbandwidthproductvs. VLfordifferent
operatingpointsofT1andvaluesofcontactresistance(seelegend).ValuesofΦmg2in(a–c)are0.2,0.7,and
0.2eV,respectively.(d)FrequencymagnituderesponseofvoltagegainattheVLpointsindicatedbyvertical
dottedlinesin(a–c). TheperformanceoftransistorT1withopencircuitload(YL = 0)isalsoshownfor
comparison.Thecurvelabeled“P2,par.”isobtainedattheoperatingpointP2includinggateresistanceand
parasiticcapacitancesasdescribedinthetext.
−∂Q /∂V (k,l∈{g,d,s,b}),withthechargesQ computedasdescribedin[21]. The
K L K
gain-bandwidth-productisdefinedasGBW=|A |f .
v0 p
The plots of f and GBW for the cases of R = 0 and Φ = 0.2 eV and of
p C mg2
R = 100Ω·µmandΦ = 0.7eVareshowninFigs.5a–b,respectively,wherewe
C mg2
also replot the respective dc gain from Figs. 3–4. Contact resistance has a dramatic
effectonboth f andGBW,reducingthembyalmostoneorderofmagnitude.
p
ItisinterestingtoseehowtheRFperformanceisaffectedbychoosingadifferent
biaspointforT1,forexample,thepointP ofFigs.1b–c,whichcorrespondstoalower
2
g (≃ 11.9mS/µm). AsshowninFig.5cforthecase ofR = 0, theperformanceis
m C
actuallyimproved. HighervaluesofGBW(exceeding1THz)canbeachieved,which
means that the lower g gets compensated by smaller capacitances. In addition, the
m
dcgainisbounded,whichindicatesthatthecircuitisstableoverthewholeconsidered
rangeofV .
L
10