2570 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 7, JULY 2014

Simulation of DC and RF Performanceof the Graphene Base Transistor

Stefano Venica, Francesco Driussi, Pierpaolo Palestri, David Esseni, Sam Vaziri, and Luca Selmi

Abstract— We examined the DC and RF performance of thegraphene base transistor (GBT) in the ideal limit of unitycommon base current gain. To this purpose, we developed amodel to calculate the current–voltage characteristics of GBTswith semiconductor or metal emitter taking into account spacecharge effects in the emitter–base and base–collector dielectricsthat distort the potential profile and limit the upper value of fT .Model predictions are compared with available experiments. Weshow that, in spite of space charge high current effects, optimizedGBT designs still hold the promise to achieve intrinsic cutofffrequency in the terahertz region, provided that an appropriateset of dielectric and emitter materials is chosen.

Index Terms— Graphene, modeling, RF performance.

I. INTRODUCTION

THE outstanding carrier mobility of graphene couldpossibly lead to high-performance graphene-based

devices, especially for analog applications [1], [2]. Amongthe proposed device concepts, the graphene base transistor(GBT) has recently gained interest [3]–[7]. The GBT (Fig. 1)is very different from the graphene field effect transistor(GFET) [2]: two insulating layers sandwich the graphene base,thus forming a vertical stack that separates the emitter fromthe collector. GBTs operate similarly to hot electron transis-tors [8]: electrons travel from the emitter toward the collector,by tunneling through the emitter–base insulator (EBI) and thebase–collector insulator (BCI) [4]. Hence, the charges crossperpendicularly the graphene sheet [9], [10]. In GBTs, the baseis ultimately thin (one atomic layer), but the high mobility ofgraphene is expected to reduce the base resistance.

The DC functionality of GBTs has been proven experimen-tally [3], [5], but at the moment, no RF measurements areavailable and the potential for RF operation can be investigatedby simulations only. In [6], we already suggested that the

Manuscript received February 19, 2014; revised April 16, 2014; acceptedMay 14, 2014. Date of publication June 3, 2014; date of current versionJune 17, 2014. This work was supported in part by the European Unionthrough the FP7 STREP Project GRADE under Grant 317839 and in part bythe Italian Ministry of Education, University and Research through the BasicResearch Investment Fund Project under Grant RBFR10XQZ8. The reviewof this paper was arranged by Editor G. L. Snider.

S. Venica, F. Driussi, P. Palestri, D. Esseni, and L. Selmi are with theDepartment of Electrical and Management and Mechanical Engineer-ing, University of Udine, Udine 33100, Italy (e-mail: stevenica@alice.it;francesco.driussi@uniud.it; palestri@uniud.it; esseni@uniud.it; luca.selmi@uniud.it).

S. Vaziri is with the School of Information and Communication Technology,KTH Royal Institute of Technology, Kista SE-164 40, Sweden (e-mail:vaziri@kth.se).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2014.2325613

Fig. 1. Sketch of the GBT structure. Electrons leave the emitter by tunnelingthrough the EBI, cross the graphene base, and then travel inside the conductionband of the BCI.

intrinsic device may be capable of reaching cutoff frequencies( fT ) in the terahertz range, whereas the predicted outputresistance and intrinsic gain are larger than for GFETs [2].These predictions were confirmed in [7]. Therefore, it isexpected that GBTs may overcome the typical limitations ofGFETs.

In this paper, we have further developed the model originallyproposed in [6] in several ways. First, we have extendedour analysis to GBTs with Si-emitter, which allowed us tovalidate our model against available experimental data [3].Then, we included typical effects of high current operationthat usually reduce the maximum fT value in heterojunctionbipolar transistors (HBT). We have, therefore, studied thespace charge effects in both EBI and BCI, to verify ifthey prevent achieving terahertz operation. Finally, we havecompared the RF performance of GBTs with either metal orSi emitter.

II. MODELING OF THE GBT WITH Si-EMITTER

Experimental I–V curves of GBTs with a n-dopedSi-emitter have been recently published [3], [5]. Hence, weextended the model of [6] to GBTs with a Si-emitter andvalidated the modeling approach against experiments of [3].

The 1-D model solves the electrostatics in the x-directionof the GBT (Fig. 1) and calculates the tunneling current andthe fT [6]. Concerning the currents, since the physical originof the base current is still unclear and debated [5], in thefollowing we assume a perfectly transparent graphene layerand we neglect the base current [6]. Hence, we assume a priorithat the collector current JC is the current due to electronsinjected from the emitter (JE = JC ) and coherently crossingthe whole device.

A. Model Description of the GBT With Si-Emitter

Fig. 2 shows the conduction band (CB) diagram of a GBTwith a n-doped Si-emitter [3]. To solve the electrostatics of thedevice, the total potential drop in the Si-emitter (�VE ) and the

0018-9383 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

VENICA et al.: SIMULATION OF DC AND RF PERFORMANCE OF THE GRAPHENE BASE TRANSISTOR 2571

Fig. 2. CB diagram of a GBT with Si-emitter. The band bending in theSi-emitter is calculated by the model self-consistently with the potential alongthe GBT structure. Arrows: positive quantities.

Fermi level in the graphene base (ef ) have to be determined.Note that ef is referred to the Dirac point (DP) and it is takenpositive when the Fermi level lies below the DP, as it happensin normal GBT operation [6]. As for the emitter–base section,by considering the applied base–emitter voltage VBE, we write(Fig. 2)

qVBE + EFR + χE = q�VE + q�V1 + �G + ef (1)

where �V1 is the potential drop in the EBI, whereas χE

and �G are the Si and graphene affinities, respectively.Furthermore, EFR is the Fermi level referred to the CB inthe Si-emitter and, again, we take it positive when the Fermilevel is below the CB bottom. In particular, by assuming aFermi–Dirac distribution in the Si-emitter, we have

EFR = −kB TF−11/2(ND/NC ) (2)

where ND is the Si doping concentration, NC is the effectivedensity of states for electrons, kB is the Boltzmann constant,and T is the temperature. Fi (γ ) is the Fermi integral of order iaccording to [11].

From Fig. 2, we obtain also the equation for thebase–collector section

�G + ef + qVCB = q�V2 + �M2 (3)

where �M2 is the work function of the metal collector,1 �V2is the potential drop on the BCI, and VCB is the appliedcollector–base voltage (Fig. 2).

1Henceforth, to keep the notation as light as possible, we indicate thequantities related to the emitter section (x < 0−) and collector section(x > 0+) with the subscripts 1 and 2, respectively.

The base charge density QB(ef ) is calculated as the dif-ference between the electron and hole charges, hence it isrelated to the Fermi level position in graphene. Equation (4),reported at the bottom of the page, directly expresses QB(ef )[6], [12], where e = E − q�VE − q�V1 − �G is the energyreferred to the DP (for graphene, we considered a Fermi–Diracdistribution for carriers and negligible chemical doping). Now,Gauss law across the graphene layer yields

QB(ef ) = �V1Cox1 − �V2Cox2 = �V1k1ε0

tox1− �V2

k2ε0

tox2(6)

where Cox1 and Cox2 are the capacitance, k1 and k2 are therelative permittivity, and tox1 and tox2 are the thickness of theEBI and BCI, respectively. We obtain

�V2 = 1

Cox2(Cox1�V1 − QB(ef )). (7)

It is worth noting that, in the derivation of (6) and (7), wedo not consider the electrostatic effect of electrons travelinginside the EBI and BCI during the GBT conduction. This pointwill be addressed in Section III-A.

The continuity of the electric displacement field at theSi-emitter/EBI interface gives

�V1

tox1k1ε0 = εE ES (8)

where εE is the permittivity of the emitter material (εE = εSiin our case) and ES is the electric field at the emitter sideof the interface. By following a similar approach as in [13],we express ES as a function of the potential drop �VE , thusobtaining (5), as shown at the bottom of the page. The exactdefinition of KFn and KFp and the complete derivation of (5)are reported in the Appendix.

By inserting (8) into (1) and (7) into (3), we can cast theequations for the emitter and collector regions as⎧⎨

⎩

qεECox1

ES(�VE ) + q�VE + ef = qVBE + EFR + χE − �G

qεECox2

ES(�VE ) − qCox2

QB(ef ) − ef = �G + qVCB − �M2

(9)

where ES is expressed as a function of �VE by (5) andQB(ef ) is given by (4). Equation (9) is a nonlinear systemof equations in the unknowns ef and �VE . After numericallysolving the system, it is possible to calculate �V1 and �V2by means of (5), (7), and (8).

QB(e f ) = 2q

πv2F h̄2

[ +∞∫

0

e · exp [(ef − e)/(kBT )]1 + exp [(ef − e)/(kB T )]de −

0∫

−∞|e| · de

1 + exp [(ef − e)/(kB T )]]

(4)

ES =√

2kB T

εE

[

− NV

(

F3/2

(KFn

kB T− F−1

1/2(ND/NC ) + KFp

kB T

)

− F3/2

(KFn

kB T− F−1

1/2(ND/NC ) − �VE

Vth+ KFp

kB T

))

− NC

(

F3/2(F−1

1/2(ND/NC )) − F3/2

(

F−11/2(ND/NC ) + �VE

Vth

))

− �VE

Vth(ND − NA)

]1/2

(5)

2572 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 7, JULY 2014

Once the GBT electrostatic is obtained, the GBT currentdensity can be calculated. The energy spectrum of JE is [14]

d JE = qmE kB T

2π2h̄3 ln

[

1 + exp

(E − EF

kB T

)]

· TWKB(E)d E

(10)

where EF = χE +EFR is the Fermi level in the Si-emitter withrespect to the vacuum level (see Fig. 2) and mE is the electronmass in the emitter (for Si, we consider sixfold degenerate CBminimum and 〈100〉 crystal orientation, see Table I). Then, wecalculate the GBT current by integrating d JE over energy andcross-sectional area. In (10), we neglect quantization effectsat the Si/EBI interface. Indeed, they may induce small shiftsalong the VBE axis in the simulated JE (VBE) curves [15], butwe consider them less relevant than other model uncertainties.

In (10), TWKB(E) is the transmission coefficient fromthe emitter to the collector for the electrons tunneling atthe energy E (defined as in Fig. 2), which we computethrough the WKB approximation. We consider both direct andFowler–Nordheim tunneling and leave the possibility to spec-ify different tunneling masses in the EBI and BCI (mI 1 andmI 2, respectively). Furthermore, by denoting with χ1 and χ2,the EBI and BCI affinities, respectively, TWKB depends on�B1 = (χE −χ1) and �B2 = (�M2 −χ2) that are the energybarriers between the emitter and EBI and between the collectorand BCI, respectively.

Concerning transport through graphene, since the DP is atthe K-points [12], whereas the energy at the �-point is larger,the assumption of momentum conservation would result in atunneling barrier, which is not the CB profile of Fig. 2 but,instead, presents an additional barrier at the position of thegraphene layer [4]. However, in computing TWKB, we con-struct the tunneling barrier using the minima of the graphene atthe K-point (Fig. 2), i.e., neglecting momentum conservation,since scattering due to disorder at the EBI/graphene interfacecan occur. In any case, we have verified that this assumption isnot a critical choice for the JE calculation, since the thicknessof this additional barrier would be very thin (≈0.3 nm).

The model for Si-emitter GBTs [(9) and (10)] tends tothe model for metal emitter presented in [6] when assuming�VE = 0, ES(�VE ) = 0 and considering EF = �M1, where�M1 is the work function of the metal emitter [6].

B. Comparison With Experiments

In [3], a GBT with 5-nm SiO2 EBI, 16-nm Al2O3 BCIand Ti collector has been fabricated and characterized inDC. The emitter is n-doped Si with a very low and farfrom optimum dopant concentration (ND = 1015 cm−3). Thematerial parameters used in the simulations are listed in TableI.

Fig. 3 compares the available experimental data [3] to thecurrent calculated with our model. Since, at present, our modelneglects the base current, we have JC = JE , whereas JC � JE

in the experiments. We see that, above the noise floor (i.e.,for VBE > 4.5 V), the experimental and simulated emittercurrents are in fairly good agreement, validating our model.The measured JC , instead, is much smaller than the modelprediction because of the large base current in the experiments.

TABLE I

PARAMETERS USED IN THE SIMULATIONS OF FIG. 3

Fig. 3. Comparison between the experimental [3] and simulated currentdensities as a function of VBE. Model parameters are reported in Table I.

Although the base current cannot be neglected and itsorigin needs further investigation in order to be included inthe model, the good agreement between the measured andsimulated JE indicates that the base current is mostly dueto the capture in graphene of the electrons injected from theemitter rather than the hole injection from the base.

III. HIGH CURRENT EFFECTS

In general, the high current effects impacting the maximumfT of bipolar devices are due to the electrostatic feedbackgenerated by the mobile charge traveling through the collectorhigh field region [18]. To investigate if similar effects arerelevant also in GBTs, we included in the electrostatics thecharge traveling in the EBI and BCI.

A. Model Description

Since in the GBT current we consider only electrons movingfrom the emitter to collector, thus neglecting possible holetunneling components, the electron density in the EBI is

n1 = JE

qvd1(11)

where vd1 is the average electron drift velocity in the EBI(assumed constant in space). A similar expression relatesn2 and vd2 in the BCI. Hereafter, vd1 and vd2 are set to arealistic value of 107 cm/s [19] and hence n1 = n2 = n.Since the carrier density is assumed constant, the CB profilebecomes parabolic inside the EBI and BCI (Fig. 4).

VENICA et al.: SIMULATION OF DC AND RF PERFORMANCE OF THE GRAPHENE BASE TRANSISTOR 2573

Fig. 4. CB diagram across a GBT with metal emitter when applying VBE andVCB. The figure reports the physical quantities used to derive the electrostaticmodel. Arrows: positive quantities.

With reference to metal emitter GBT, we rewrite (1) as

�M1 + qVBE = q�V1 + �G + ef (12)

while the potential profile in the BCI is still described by (3).The negative uniform charge (ρ = −qn) due to the electrons

traveling across the EBI and BCI enters the Poisson’s equation,which provides the parabolic potential profile in the dielectrics

ϕ = ax2 + bx + c = qn

2εx2 + bx + c (13)

where ε is equal to k1ε0 for EBI and to k2ε0 for BCI, and b andc are two constants to be determined. �V1 is the differencebetween the potential at the graphene interface (x = 0) and atthe emitter interface (x = −tox1, Fig. 4). From (13), we get

�V1 = ϕ(0) − ϕ(−tox1) = − qn

2ε1t2ox1 + b1tox1 (14)

hence we can express the parameter b1 as a function of �V1

b1 = 1

tox1

[

�V1 + qn

2ε1t2ox1

]

. (15)

Next, we find the electric field E1(0−) at x = 0−

E1(0−) = −dϕ

dx

∣∣∣∣x=0−

= −[�V1

tox1+ qn

2ε1tox1

]

. (16)

A similar approach is used for the BCI. By assuming nopotential drop inside the graphene layer (Fig. 4), we obtain

E2(0+) = −dϕ

dx

∣∣∣∣x=0+

= −[�V2

tox2− qn

2ε2tox2

]

. (17)

The base charge is QB(ef ) = ε2 E2(0+) − ε1 E1(0−) and,by using (16) and (17), as well as �V1 and �V2 defined in(12) and (3), we obtain (18). Now, using (4) in (18), as shownat the bottom of the page, we derive a nonlinear equation inthe only unknown ef , whose solution provides the potentialprofile once JE is given.

To calculate the GBT current density, we again integrateover energy (10), by assuming mI 1 = mI 2 = 0.25m0,EF = �M1, and mE = m0 as the electron mass in the

Fig. 5. Cutoff frequency calculated with the model of [6] (open symbols)and of this paper (filled circles). For VBE < 1.5 V, the latter agrees withthe old model of [6], where τD accounts for the electron drift across thedielectrics (open circles). For VBE > 1.5 V, the electrostatic feedback ofthe traveling charge further reduces fT with respect to the model of [6].In this model, neglecting τD results in very high fT values (open squares).�B1 = �B2 = �B . k is the relative permittivity of both EBI and BCI andwe assumed �M1 = 4.50 eV and m E = m0 [6].

metal emitter. Now, by combining (10), (11), and (18), thecomplete system of nonlinear equations for the device elec-trostatics and the transport is obtained. To solve the system,we implement an iterative, selfconsistent procedure that, ateach iteration, calculates the potential profile and then JE .This latter is used to evaluate the charge inside the dielectrics[n, through (11)] that becomes an input for the solution of thepotential at the next iteration (18). A damping algorithm pro-hibits collector current changes >10% between two iterations,thus ensuring convergence.

The model also estimates the cutoff frequency of the intrin-sic device, by means of a quasi-static approach, that is

fT = 1

2πτC= 1

2π d Q Bd JE

. (19)

Differently from [6], (19) does not include the transit time ofelectrons through the dielectrics (τD = (tox1 + tox2)/(2vd ),Ramo’s theorem [20]), because it is inherently accountedfor in the electrostatics, where the traveling charge and itsmodulation with bias are included. This is demonstrated byFig. 5; for VBE < 1.5 V, the new model (black circles) agreeswith the model in [6], which calculates fT in (19) by addingthe term τD to τC (open circles). Instead, the two modelslargely differ if τD is neglected (open squares), thus confirmingthat, in the new model, the delay due to the traveling time ofelectrons is already included. For VBE > 1.5 V, the electrosta-tic feedback of the electrons in the insulators induces holes inthe base and reduces the transconductance with respect to thenonself-consistent simulations, because the field in the EBIchanges. Consequently, fT is smaller than in the old model(also when including τD).

B. Impact of High Current Effects on GBT Performance

Fig. 6 shows JE as a function of VBE for a few val-ues of the EBI thickness. For VBE < 1 V, the JE values

QB(ef ) = qn

2(tox1 + tox2) + k1ε0

tox1(φM1 + VBE − φG − ef ) − k2ε0

tox2(φG + ef + VCB − φM2) (18)

2574 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 7, JULY 2014

Fig. 6. Current density versus VBE for different tox1. The inclusion of highcurrent effects (solid lines) leads to a saturation of JE at the larger VBEvalues. �B1 = �B2 = �B . k = k1 = k2.

Fig. 7. Cutoff frequency versus JE for a few dielectric constantsk1 = k2 = k. Comparison of the results obtained without (dashed lines) orwith (solid lines) the inclusion of high current effects. Reference data for SiGeHBT devices are reported for comparison (symbols) [21], [22]. Simulationsof a GBT with very thick BCI are also reported (red line). We assumed�B1 = �B2 = �B = 0.5 eV.

calculated, including the injected charge in the electrostatics(solid lines) are close to those obtained in [6] (dashed lines).For larger VBE, instead, the current in the device increases andthe traveling charge limits JE . Indeed, the charge in the oxides(mainly in the BCI) opposes the further injection of electronsfrom the emitter, thus leading to saturation of JE . Note that JE

at high VBE is insensitive to tox1, because JE (VBE) is limitedby the charge in the BCI, in agreement with [7].

The model makes it possible to estimate the maximumfT and compare it with recently reported HBT data. Fig. 7shows the GBT cutoff frequency as a function of JE fora few k = k1 = k2 values. The fT drop typical of highcurrent conditions can be observed (solid lines). In spite of thereduced performance, the maximum fT for GBT overcomesthe 1-THz threshold. High current effects appear to be less alimiting factor in GBTs than in SiGe HBTs. A maximum fT

comparable with that of HBTs would be reached only for aGBT design with a very thick BCI (tox2 = 50 nm, red line).Indeed such BCI thickness would be close to the typicalbase–collector depletion region width in HBTs [18], thusexplaining why this case shows similar maximum fT .

However, the maximum fT for GBTs is obtained forJE ≈ 0.1 A/μm2; at present, it is not clear if any dielectric cansustain such high JE . Reliability of SiO2 has been studied for

Fig. 8. Cutoff frequency versus VBE for a few values of k1 = k2 = k forGBTs with (a) metal- and (b) Si-emitter. The opposite fT dependence on thek value is explained by the CB diagrams shown in Fig. 9.

Fig. 9. CB diagrams calculated for GBTs with (a) metal- and (b) Si-emitter.Note the opposite dependence of the EBI barrier on the k1 = k2 = k.

current densities approximately up to 10−7 A/μm2 [23] and,consequently it is hard to extrapolate a time-to-breakdown cor-responding to the current densities in Fig. 7. On the other hand,tunneling current densities in the range 10−3/10−2 A/μm2

have been measured across boron nitride layers sandwichedbetween two graphene electrodes [24].

IV. OPTIMIZATION OF THE METAL-EMITTER

AND Si-EMITTER GBTs

By extending the model in Section II-A similarly to whatdone in Section III-A (thus, including high current effectsalso in the Si-emitter case), we could compare the DC andRF performance of GBTs with different emitter materials.Fig. 8 reports fT versus VBE curves for GBTs with metal-and Si-emitters. In both cases, the model predicts that terahertzoperation may be possible, even in presence of high currentseffects (solid lines).

However, the devices show markedly different fT depen-dence on the permittivity of EBI and BCI (k1 = k2 = k) thussuggesting different optimization strategies. With the metalemitter, at VBE � 1.5 V, the smaller the k value is, thethinner the EBI barrier for electrons is [Fig. 9(a)]. In thiscase, the charge in the insulators is small and high currenteffects negligible. A thinner energy barrier leads to larger JE

and hence larger fT values [Fig. 8(a)] [6].For VBE � 1.5 V, instead, when high current effects are

accounted for, smaller k values result in a larger impact of n

VENICA et al.: SIMULATION OF DC AND RF PERFORMANCE OF THE GRAPHENE BASE TRANSISTOR 2575

on the electrostatics (13) that reduces JE and hence fT . Thisreduction has a twofold origin. First, the charge inside the BCIinduces holes in the graphene that increases ef and reduces�V1 (for a given VBE), thus lowering the tunneling current.Second, it raises the BCI energy barrier [Fig. 9(a), black line].Hence, a tradeoff exists, whereby a low k increases injectionfrom emitter, but also space charge effects. As a result, theVBE at maximum fT changes appreciably but the maximumfT only slightly increases when reducing k [Fig. 8(a), solidlines].

For the Si-emitter, instead, fT increases when increasing k[Fig. 8(b)]. Indeed, Fig. 9(b) shows that larger k values resultin lower EBI barriers with respect to the Si Fermi level (dashedline). This is explained by (8): a larger k increases ES and theSi band bending, thus lowering down the bottom of the Si CBat the interface with respect to the Fermi level. Consequently,the energy barrier for the electrons tunneling at the Fermienergy decreases, thus increasing the current. Furthermore, thelarger the k is, the lower the insulator charge related effectsare, as discussed above. Finally, the sudden drop of fT atVBE ≈ 2 V [Fig. 8(b), solid lines] is due to the fact thatthe electrons at the Si CB bottom start to see the top of theparabolic CB profile of the BCI, pushed up by electron chargebuild-up. This effect strongly reduces TWKB and JE .

In Fig. 8(b), we assumed ND = 1019 cm−3. However, themaximum fT of Si-emitter GBTs is weakly dependent on ND

(not shown). Indeed, smaller ND leads to larger voltage drop inthe Si, but once the emitter is brought in strong accumulation(high VBE), GBTs with different Si doping show similar JE

and similar fT . Therefore, a lower ND will translate in a largerVBE to reach high currents, but the intrinsic device maximumfT is essentially unchanged.

To further evaluate the RF performance of GBTs, we alsoestimated the unity power gain frequency ( fmax) as [18]

fmax =√

fT

8π RBCBC(20)

where CBC is estimated considering the BCI capacitance only,that is, CBC = W · L · ε2/tox2, where W · L is the areaof a rectangular GBT. RB is the graphene base resistance,calculated as sum of the intrinsic base resistance (Rint_B ,Fig. 1) and the contact resistance between the graphene layerand the base contact (RCONT [ · μm]), hence

RB = Rint_B + RCONT

W= L

12W· 1

QB(ef )μGp

+ RCONT

W

(21)

considering the graphene base contacted at both sides [18](L is the distance between contacts; W is the contact width).The intrinsic base resistance depends on the effective holedensity in the base and on the graphene hole mobility (μG

p ).Fig. 10 reports the simulated fT and fmax as functions of

VBE in the case of metal- and Si-emitter devices. We eval-uated fmax by assuming a realistic mobility value of μG

p =1000 cm2/(Vs) [25] and setting either RCONT = 0 μm(triangles up) or RCONT = 500 μm (triangles down) [26].As expected, fmax is lower than fT and degrades when

Fig. 10. Comparison between fmax and fT versus VBE for (a) metal- and(b) Si-emitter devices and for two values of RCONT. We set k1 = k2 = 10,whereas hole mobility in graphene is 1000 cm2/(Vs) [25]. W = L = 1 μm.

increasing RB . The contact resistance RCONT has a strongimpact on fmax and this indicates that special care should betaken in the fabrication of the contacts to the graphene. Notethat, even in the worst case, the fmax values are well above100 GHz and they can reach 1 THz for very small RCONT.

V. CONCLUSION

We developed a model for the GBT that calculates self-consistently the electrostatic and the current accounting forthe charge in the graphene base and the electrons traveling inthe EBI and BCI. The model can handle silicon as well asmetal emitters. Comparison with available experiments showsreasonably good agreement with JE , suggesting that the basecurrent is mainly given by electrons captured by the graphene.

We estimated the maximum fT , showing that high currenteffects are not very critical for GBTs, mainly because of thelimited BCI thickness. Terahertz operation can be reached bythe intrinsic device within a relatively large design parame-ter range.

GBTs with metal- and Si-emitters have been compared,showing the different impact of design parameters (especiallythe EBI and BCI k value) on the two devices. The fmax hasalso been studied, showing the important role of contact resis-tance to the graphene layer in defining the RF performance.A low RCONT is essentially to retain high fmax.

APPENDIX

To derive an explicit expression for ES as a function of theapplied potentials, we start by defining the difference betweenthe Si intrinsic Fermi level and the CB minimum as

KFn = − EGAP

2+ kB T

2ln

(NV

NC

)

(A.22)

where NV is the density of states for holes. Similarly, thedistance between the top of the valence band and the Siintrinsic Fermi level is

KFp = −[

EGAP

2+ kB T

2ln

(NV

NC

)]

. (A.23)

Consequently, by assuming Fermi–Dirac distribution for thecarriers in Si, we write the electron and hole density as

n = NCF1/2

(

u(x) + KFn

kB T

)

(A.24)

2576 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 7, JULY 2014

ES(�VE ) =√

2kB T

εE

[

− NV

(

F3/2

(

− u B + KFp

kB T

)

− F3/2

(

− uS + KFp

kB T

))

− NC

(

F3/2

(

u B + KFn

kB T

)

− F3/2

(

uS + KFn

kB T

))

+ (ND − NA)(u B − uS)

]1/2

(A.28)

p = NV F1/2

(

− u(x) + KFp

kB T

)

(A.25)

where u(x) = q/(kBT )ϕ(x) is the normalized (dimensionless)potential profile along the Si.

With the definitions above, the Poisson’s equation becomes

d2u(x)

dx2 = q2

kB T εE

[

NCF1/2

(

u(x) + KFn

kB T

)

− NV F1/2

(

− u(x) + KFp

kB T

)

−ND +NA

]

= q2

kB T εES(KFn, KFp). (A.26)

Let us indicate with u B and uS the potentials in the deepbulk (x → −∞) and at the Si/EBI interface (x = −tox1),respectively, so (du/dx)(−∞) = 0 and (du/dx)(−tox1) =(q ES/kB T ), where ES is again the electric field at the EBIinterface. By defining y = (du/dx)2 and following the methodin [13], we obtain

∫ 0

(qESkB T

)2 dy =∫ u B

uS

2q2

kB T εES(KFn, KFp)du. (A.27)

Keeping in mind the definition of S(KFn, KFp) in (A.26) andthe general propriety of the Fermi integrals (dF j (η)/dη =F j−1) [11], the solution of (A.27) is (A.28), as shown at thetop of the page.

By considering the electron density in the deep bulk equalto the Si doping (n(−∞) = ND), we can calculate the valueof dimensionless potential u B [by inverting (A.24)] and uS

(as a function of the potential drop in Si) as

u B = − KFn

kBT+ F−1

1/2(ND/NC ), uS = u B + �VE

Vth. (A.29)

Now, by substituting (A.29) in (A.28), we obtain (5).

ACKNOWLEDGMENT

The authors would like to thank Prof. M. Lemme,Prof. M. Östling, and C. Macuz for their support and helpfuldiscussions.

REFERENCES

[1] S. O. Koswatta, A. Valdes-Garcia, M. B. Steiner, Y.-M. Lin, andP. Avouris, “Ultimate RF performance potential of carbon electronics,”IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2739–2750,Oct. 2011.

[2] A. Paussa, M. Geromel, P. Palestri, M. Bresciani, D. Esseni, andL. Selmi, “Simulation of graphene nanoscale RF transistors includingscattering and generation/recombination mechanisms,” in IEEE IEDMTech. Dig., Dec. 2011, pp. 11.7.1–11.7.4.

[3] S. Vaziri et al., “A graphene-based hot electron transistor,” Nano Lett.,vol. 13, no. 4, pp. 2370–2375, 2013.

[4] W. Mehr et al., “Vertical graphene base transistor,” IEEE Electron DeviceLett., vol. 33, no. 5, pp. 691–693, May 2012.

[5] C. Zeng et al., “Vertical graphene-base hot-electron transistor,” NanoLett., vol. 13, no. 4, pp. 1435–1439, 2013.

[6] F. Driussi, P. Palestri, and L. Selmi, “Modeling, simulation and designof the vertical graphene base transistor,” Microelectron. Eng., vol. 109,pp. 338–341, Sep. 2013.

[7] V. Di Lecce, R. Grassi, A. Gnudi, E. Gnani, S. Reggiani, andG. Baccarani, “DC and small-signal numerical simulation of graphene-base transistor for terahertz operation,” in Proc. ESSDERC, 2013, p. 314.

[8] M. Heiblum, “Tunneling hot electron transfer amplifiers (theta): Ampli-fiers operating up to the infrared,” Solid-State Electron., vol. 24, no. 4,pp. 343–366, 1981.

[9] L. Britnell et al., “Field-effect tunneling transistor based on verticalgraphene heterostructures,” Science, vol. 335, no. 6071, pp. 947–950,2012.

[10] S. B. Kumar, G. Seol, and J. Guo, “Modeling of a vertical tunnel-ing graphene heterojunction field-effect transistor,” Appl. Phys. Lett.,vol. 101, no. 3, p. 033503, 2012.

[11] J. S. Blakemore, “Approximations for Fermi-Dirac integrals, especiallythe function F1/2(η) used to describe electron density in a semicon-ductor,” Solid-State Electron., vol. 25, no. 11, pp. 1067–1076, 1982.

[12] H.-S. P. Wong and D. Akinwande, Carbon Nanotube and GrapheneDevice Physics. Cambridge, U.K.: Cambridge Univ. Press, 2011.

[13] E. H. Nicollian and J. R. Brews, MOS (Metal Oxide Semiconductor)Physics and Technology. New York, NY, USA: Wiley, 1982.

[14] S. Takagi, N. Yasuda, and A. Toriumi, “A new I-V model for stress-induced leakage current including inelastic tunneling,” IEEE Trans.Electron Devices, vol. 46, no. 2, pp. 348–354, Feb. 1999.

[15] J. Suñé, P. Olivo, and B. Riccó, “Quantum-mechanical modeling ofaccumulation layers in MOS structure,” IEEE Trans. Electron Devices,vol. 39, no. 7, pp. 1732–1739, Jul. 1992.

[16] Y. T. Hou, M.-F. Li, H. Y. Yu, and D. L. Kwong, “Modeling of tunnelingcurrents through HfO2 and (HfO2)x(Al2O3)1−x gate stacks,” IEEEElectron Device Lett., vol. 24, no. 2, pp. 96–98, Feb. 2003.

[17] S. Spiga et al., “Experimental and simulation study of the program effi-ciency of HfO2 based charge trapping memories,” in Proc. ESSDERC,2010, pp. 408–411.

[18] S. M. Sze, Physics of Semiconductor Devices. New York, NY, USA:Wiley, 1981.

[19] P. Palestri, L. Selmi, E. Sangiorgi, and M. Pavesi, “Cathode hot electronsand anode hot holes in tunneling MOS capacitors,” in Proc. ESSDERC,2000, pp. 296–299.

[20] R. G. Meyer and R. S. Muller, “Charge-control analysis of the collector-base space-charge-region contribution to bipolar-transistor time con-stant τT ,” IEEE Trans. Electron Devices, vol. 34, no. 2, pp. 450–452,Feb. 1987.

[21] B. Heinemann et al., “SiGe HBT technology with fT / fmax of300 GHz/500 GHz and 2.0 ps CML gate delay,” in IEEE IEDM Tech.Dig., Dec. 2010, pp. 688–691.

[22] B. Geynet et al., “SiGe HBTs featuring fT � 400 GHz at room tem-perature,” in Proc. Bipolar/BiCMOS Circuits Technol. Meeting, 2008,pp. 121–124.

[23] S. Lombardo, J. H. Stathis, B. P. Linder, K. L. Pey, F. Palumbo, andC. H. Tung, “Dielectric breakdown mechanisms in gate oxides,” J. Appl.Phys., vol. 98, no. 12, p. 121301, 2005.

[24] L. Britnell et al., “Electron tunneling through ultrathin boron nitridecrystalline barriers,” Nano Lett., vol. 12, no. 3, pp. 1707–1710, 2012.

[25] J. Lee, L. Tao, Y. Hao, R. S. Ruoff, and D. Akinwande, “Embedded-gategraphene transistors for high-mobility detachable flexible nanoelectron-ics,” Appl. Phys. Lett., vol. 100, no. 15, p. 152104, 2012.

[26] A. Hsu, H. Wang, K. K. Kim, J. Kong, and T. Palacios, “Impactof graphene interface quality on contact resistance and RF deviceperformance,” IEEE Electron Device Lett., vol. 32, no. 8, pp. 1008–1010,Aug. 2011.

Authors’ photographs and biographies not available at the time of publication.