first principle studies on graphene fet

A first principles theoretical examination of graphene-based field effecttransistorsJames G. Champlain

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A first principles theoretical examination of graphene-based fieldeffect transistors

James G. Champlaina)

Electronics Science and Technology Division, Naval Research Laboratory, Washington, D.C. 20375, USA

(Received 6 October 2010; accepted 5 March 2011; published online 22 April 2011)

This paper presents an in-depth theoretical examination of graphene-based field effect transistors,

looking at thermal statistics, electrostatics, and electrodynamics. Using a first principles approach,

the unique behavior observed in graphene-based field effect transistors, such as the V-shaped

transfer characteristic, limited channel pinch-off, and lack of off-state (under gate modulation), are

described. Unlike previous attempts, a description of both drift and diffusion currents in the device

is presented. The effect of external resistance on steady-state and high-frequency performance is

examined. Comparisons of the theoretical results to experimental results are made and show good

agreement. Finally, the theoretical work in this paper is used as a basis to discuss the possible

source of some observed behavior in practical graphene-based field effect transistors. VC 2011American Institute of Physics. [doi:10.1063/1.3573517]

I. INTRODUCTION

There has been much interest as of late in graphene and

its application in electronic devices, specifically field effect

transistors (FETs). Within the past decade there has been

increased work in the area of general thermal statistics and

fundamental transport for graphene.17 Also within the past

few years there has been initial work in the description and

modeling of device physics.812 Some of this work has incor-

rectly interpreted or applied the fundamental transport

physics to the conductivity of graphene sheets, particularly

when applied to gated structures, such as FETs, resulting in

the incorrect interpretation of device behavior, which will be

discussed.

This paper presents an in-depth theoretical examination

of the behavior of graphene-based FETs (gFETs) using pre-

vailing theory regarding fundamental graphene charge carrier

transport and first principles solid-state physics, including

thermal statistics, electrostatics, and electrodynamics. A de-

scription of current transport in a gFET is presented along

with standard FET parameters and characteristics, such as the

transfer characteristic, transconductance, drain characteristic,

and drain conductance, with discussions on several unique

phenomena observed in graphene (e.g., V-shaped transfer

characteristic, limited channel pinch-off, lack of an off-state).

The frequency performance and limits of these devices, in the

form of short-circuit current-gain cutoff frequency, are exam-

ined. The influence of external resistance and the observed

effects on gFET steady-state behavior (e.g., sublinear transfer

characteristic and transconductance suppression at high drain

currents) and high-frequency performance is explored. The

validity of the work presented in this paper is discussed, with

reference to experimental and other theoretical work. Specifi-

cally, comparisons to measured device data are made, show-

ing good agreement with theory, and the nature of the

conductivity of the graphene channel and its correct represen-

tation are discussed. Finally, the theoretical work in this paper

is used as a basis to discuss the possible sources of some

observed behavior in practical graphene-based FETs not

directly described by this work, such as some observed asym-

metric transfer characteristics and device performance.

II. GRAPHENE THERMAL STATISTICS

The thermal statistics of graphene have been previously

developed and presented.13,7,13 It is offered here as a foun-

dation and reference for the work within this paper.

As a result of graphenes two-dimensional (2D), hexag-

onal lattice structure, its band structure is unique among

semiconductors (Fig. 1). Some of graphenes distinctive fea-

tures are its lack of a bandgap (i.e., the so-called zero

bandgap) and linear energymomentum dispersion relation

about the zero momentum point,1,3,7,13

e eD 6hmFjkj: (1)Here, e is the energy of the particular momentum state withwave number k, eD is the Dirac energy of the graphene sheet(i.e., the zero-momentum energy, where conduction band

and valence band meet, eD ec ev), h is the reducedPlanks constant, and mF 106 m/sec is the Fermi velocityof carriers in graphene.2,3,7 The positive sign in Eq. (1) leads

to energies in the conduction band (i.e., e eD), and the neg-ative sign results in energies in the valence band (i.e.,

e eD). Arguably, graphene sits in a unique positionbetween a classical semiconductor (eg > 0) and a semimetal(ec < ev).

For such a band structure, it has been shown that the

conduction band and valence band density of states are linear

with energy:

DCe gsgv2p

e eDhvF 2

; (2a)a)Electronic mail: [email protected].

0021-8979/2011/109(8)/084515/19/$30.00 VC 2011 American Institute of Physics109, 084515-1

JOURNAL OF APPLIED PHYSICS 109, 084515 (2011)


DV e gsgv2p

eD ehvF 2

; (2b)

with gs 2 and gv 2 being the spin and valley degenera-cies, respectively.3,7 The electron (sheet) concentration, n,and hole (sheet) concentration, p, are given by

n 1eD

DC e f e de NGF 1 gF ; (3a)

p eD1

DV e 1 f e de NGF 1 gF : (3b)

f e is the FermiDirac distribution, which dictates the occu-pational probability of a state depending on its displacement

in energy from the Fermi energy, eF. NG is the effective gra-phene (sheet) density of states given by

NG gsgv2p

kBT

hmF

2; (4)

with kB being Boltzmanns constant and T being the absolutetemperature of the system in Kelvin. F 1 6gF is the com-plete FermiDirac integral with index i 1:

F i 6gF 1

C i 1 10

ui

1 egFeu du; (5)

C is the gamma function. gF, introduced in Eq. (3), is arelative measure of the Fermi energy, eF, with respect to theDirac energy, eD, normalized to kBT:

gF eF eDkBT

: (6)

The complete FermiDirac integral is commonly approxi-

mated by

F i 6gF 1

C i 1 10

uie6gFeudu e6gF : (7)

This is because, in most cases, the Fermi energy lies well

outside the energetic range of states of concern (e.g., the

semiconductor is nondegenerate). In such instances, the

Fermi-Dirac distribution function can be replaced by

the simpler MaxwellBoltzmann distribution function. In the

case of graphene, the Fermi energy lies near to or within the

energetic states of concern. Therefore, except in specific

cases (e.g., very low temperature, very high carrier concen-

trations, very low carrier concentrations) the complete

FermiDirac integral as presented in Eq. (3) cannot be

replaced by approximations.

Figure 2 shows the electron concentration, n, and holeconcentration, p, dependence on Fermi energy, eF.

III. GRAPHENE FET ELECTROSTATICS

In practice, there have been two basic device layouts

employed in the fabrication of gFETs.1416 One is akin to a

classic silicon (Si) MOSFET layout, in which the gate metal

and oxide cover the entire length of the channel from the

source contact to the drain contact.14,16 The other is similar

to standard IIIV MESFET/HEMT layouts, in which the

gate metal sits between the source and drain contacts with an

amount of access region on either side of the gate. In the lat-

ter case, the gate oxide tends to cover the entire expanse

between the source and drain contacts.14,15 The specific lay-

out of the gFET is not critical to the work presented in this

paper.

Additionally, three methods have emerged as the domi-

nant means of growing or producing graphene for device

applications: silicon sublimation from silicon carbide (SiC),

graphene exfoliation from a graphite source, and graphene

chemical vapor deposition (CVD), commonly onto a metal

substrate.6,1422 The first method results in a graphene sheet

in intimate contact with a SiC substrate.6,14 The second and

third methods require minor subsequent processing but com-

monly result with a graphene sheet resting on a dielectric,

usually silicon dioxide (SiO2) on a Si substrate. In many

cases, the Si substrate is highly doped and used as a back

gate for the gFET.15,16,21,22 For the bulk of this paper, gra-

phene on SiC is considered. In general, this impacts only the

final numerical results, the physics governing current trans-

port and overall behavior of the gFET remain essentially

unchanged (See Appendix A for a treatment of graphene on

a dielectric substrate with back gate).

FIG. 2. (Color online) Plot of electron concentration (n) and hole concentra-tion (p) versus Fermi energy relative to Dirac energy (eF eD). The dashedlined corresponds to the electron concentration using the MaxwellBoltz-

mann distribution function, Eq. (7).

FIG. 1. (Color online) Unique band structure (e k diagram) of graphene,highlighting the linear energy-momentum dispersion of the material near

zero momentum.

084515-2 James G. Champlain J. Appl. Phys. 109, 084515 (2011)


Figure 3 shows the physical structure of the gFET, under

the gate, used in this paper. As shown, the graphene sheet,

which constitutes the channel of the FET, lies within the

xz-plane. The source edge of the gate metal is located atx 0, the drain edge is located at x L. The width of thegate metal is W. tox is the thickness of the gate oxide. Thegate is treated as an equipotential with voltage VG. The volt-age at the drain edge (x L) of the graphene sheet (channel)is VD; at the source edge (x 0), VS.

In examining the electrostatics and electrodynamics of

the gFET, the gradual channel approximation (i.e., @Dy=@y @Dx=@x) is applied. This includes the assumption that allproperties of the gFET are uniform along the width of

the gate (i.e., z-direction).23,24 Therefore, Gausss law(r D q) becomes separable in all three dimensions and

dDxdx

0; (8a)

dDydy

q x; y ; (8b)

dDzdz

0: (8c)

Here D is the displacement field vector with Dx, Dy, and Dzbeing its components in the x-, y-, and z-directions, respec-tively. q x; y is the volume charge density at locationx; y. A band diagram and charge profile for a vertical slice(in the yz-plane) across the metal-oxide-graphene structureof Fig. 3, located at x, is presented in Fig. 4.

In the band diagram of Fig. 4, Vch Vch x is the (local)voltage in the graphene channel. As defined earlier,

eF qVch is the (local) Fermi energy within the channel(with q being the elementary electric charge) and eD eD x is the (local) Dirac energy. /mo vox vm and/so vox vsc are the metal-oxide and the semiconductor-oxide offsets, respectively, with vox, vm, and vsc being the ox-ide, metal, and semiconductor (graphene) electron affinities,

respectively. Eox Eox x is the (local) electric field withinthe oxide.

In the charge profile of Fig. 4, Qm Qm x is the chargesheet density in the metal, n n x is the electron concentra-tion within the graphene channel with qn being its associ-ated charge sheet density, p p x is the hole concentrationwithin the graphene channel with qp being its associatedcharge sheet density, and Qf is the fixed charge sheet densityrepresenting any fixed charge within the graphene sheet and/

or oxide, imaged to the graphene sheet. Qnet Qnet x q p n is simply the net mobile charge sheet densitywithin the graphene channel.

For the structure to be charge neutral (i.e., the electric

field goes to zero outside the structure), the total charge must

sum to zero:

Qm Qnet Qf 0: (9)To find a relationship between the mobile charges in the

channel and the applied voltages, a voltage (potential

energy) loop around the metal-oxide-graphene band diagram

is completed. Beginning at the gate metal and proceeding

around the loop,

q VG Vch /mo qEoxtox /so eD eF 0:(10)

The electric field in the oxide, Eox, can be found by integrat-ing Eq. (8b) across the gate metal-oxide interface:0

eoxEox

dDy toxtox

Qmd y tox dy: (11)

FIG. 3. Physical structure of the gFET examined in this paper. In addition

to the various spatial dimensions used in the analysis of the device, the

applied voltages (VG, VD, VS) and associated current densities (JT , Jn, Jp,see Sec. IV) are indicated. The band diagram and charge profile along the

section line a a0 is presented in Fig. 4.

FIG. 4. Band diagram (top) and charge profile (bottom) of the gFET metal-

oxide-graphene structure corresponding to the section line a a0 in Fig. 3.



Here, d y tox is a Dirac delta function locating the chargein the metal, Qm, at the gate metal-oxide interface, and t

ox

and tox are positions just to either side of the interface aty tox. eox er; oxeo is the permittivity of the oxide layerwhere er; ox is the relative permittivity of the oxide and eo isthe free-space permittivity. The result of Eq. (11), substitut-

ing Qm Qnet Qf

from Eq. (9), is

Eox 1eox Qm 1

eoxQnet Qf

: (12)

Recalling gF eF eD =kBT from Eq. (6) and with somealgebraic manipulation, Eq. (10) becomes the charge-voltage

(QV) relationship for the gFET,

VG Vch 1Cox

Qnet kBTq

gF 1

CoxQf 1

q/mo /so 0:

(13)

Here, Cox eox=tox is the areal capacitance of the gate oxide.Equation (13) gives a unique relationship between the

voltages VG and Vch and the normalized, relative position ofthe Fermi energy with respect to the Dirac energy, gF; which,through Eq. (3), gives the electron, n, and hole, p, concentra-tions within the graphene channel. Because an assumption

on the hole and electron concentrations with relation to the

Fermi energy cannot be made (see Sec. II), the expressions

for n and p in Eq. (3), and associatively Qnet, cannot bereduced, leaving Eq. (13) transcendental.

The electron and hole concentrations and normalized,

relative Fermi energy (gF) are plotted as a function ofgate voltage, VG, with the channel grounded (VD VS Vch 0 x L 0), for various fixed charge densities(Qf), in Fig. 5. Taking a Ti/Au gate metal and Al2O3 gate ox-ide,14,22,25 the values used for the various device parameters

are listed in Table I.

A. Metal-oxide-graphene capacitive network

The metal-oxide-graphene structure as shown in Figs. 3

and 4 corresponds to an equivalent capacitive network,

which is a useful tool in analyzing the devices behavior.

Figure 6 shows the capacitive network with CT @Qm=@VG; ch 1=Cox 1=CQ

1being the total capacitance of

the structure, Cox @Qm=@VG; gr eox=tox being the capaci-tance of the gate oxide, and CQ @Qnet=@Vch; gr Cq F 0 gF F 0 gF being the quantum capacitance ofthe graphene sheet (with Cq q2NG=kBT); withVG; ch VG Vch as the voltage across the entire metal-ox-ide-graphene structure, VG; gr VG Vgr (with qVgr eD)as the voltage across the gate oxide, and Vch; gr Vch Vgr kBT=q gF as the voltage across the gra-phene sheet.

IV. GRAPHENE FET ELECTRODYNAMICS

The total electron current density (Jn) and total hole cur-rent density (Jp) are defined using the following equations:

Jn lnnreF; (14a)

Jp lppreF: (14b)

And the total current density is

JT Jn Jp: (15)

In Eq. (14), ln and lp are the electron and hole mobilitieswithin the graphene channel, respectively.

Under nonequilibrium conditions, a single Fermi energy

(eF), as in Eq. (14), cannot strictly be assumed. Instead, theelectron and hole concentrations and currents should follow

their respective and distinct quasi-Fermi energies (eF; n andeF; p). However, a general analysis with two quasi-Fermi

FIG. 5. (Color online) Plot of electron concentration (n), hole concentration(p), and normalized, relative Fermi energy (gF) versus applied gate voltage(VG) for fixed charge densities (Qf =q) of (a) 0, (b) 5 1012 cm2, and (c)5 1012 cm2.

TABLE I. Device parameters and dimensions used in this paper, unless oth-

erwise noted.

Parameter Value

er; ox er;Al2O3 a 10tox (A) 200

vTi (eV)b 4.33

vgraphene (eV)c 4.5

Qf =q (cm2) 0

vF (m/sec) 106

L (lm) 1ln (cm

2/Vsec) 5000

lp (cm2/Vsec) 5000

aReferences 26 and 27.bReference 28.cReferences 29 and 30.



energies can quickly become intractable. Alternatively, if it

is assumed that the conduction band and valence band states

are in intimate communication (i.e., high generation/recom-

bination rate), which is highly possible in a zero bandgap

semiconductor where recombination lifetimes are very

short,3133 then the electron and hole quasi-Fermi energies

will deviate only slightly from each other. The assumption of

a single Fermi energy allows the analysis to become manage-

able while introducing only minimal error.

By using the definition that current density is propor-

tional to the gradient of the Fermi energy (J / reF), currentresulting from electric fields (i.e., drift current) and from

concentration gradients (i.e., diffusion current) are captured

within one expression.34 As will be shown in Sec. V, diffu-

sion current can comprise an appreciable portion of the total

current flowing through the channel of a gFET, and has been

commonly left unaddressed in previous discussions of cur-

rent transport.8,9,11

Within this paper, constant mobility is assumed. Present

theory takes transport in graphene to be limited by scattering

from impurities and phonons, with impurity scattering being

the dominant scattering mechanism.47 The scattering time

has been found to be inversely proportional to the impurity

concentration, which subsequently leads to a conductivity

proportional to the carrier concentration and inversely

proportional to the impurity concentration. Expressing the

conductivity as a product of the mobility and charge concen-

tration (r qln) results in the mobility being a constant fora specific impurity concentration. A continued discussion of

mobility and conductivity is presented in Sec. IX.

Due to the 2D nature of graphene, current only flows

within the plane of the graphene sheet (xz-plane), ignoringgate leakage. Additionally, since the gFET is uniform along

the gate width (z-direction), the gradients in Eq. (14) reduceto one-dimensional (1D) derivatives in the x-direction:

JT Jn Jp lnndeFdx

lppdeFdx

: (16)

Recalling that eF qVch, the total current density may bewritten as

JT Jn Jp qlnn dVchdx

qlpp

dVchdx

: (17)

Assuming no sources or sinks of charge within the channel

(r JT @q=@t 0), the total (steady-state) current den-sity must be constant. Therefore, multiplying Eq. (17) by dx,integrating from source (x 0) to drain (xL), and dividingby the gate length (L) results in

JT 1L

L0

Jndx 1L

L0

Jpdx Jn Jp; (18)

where Jn and Jp are the mean values of the total electron cur-rent and total hole current along the channel, respectively,

which upon evaluation are given by

Jn 1L

L0

Jndx 1LIngDgS; (19a)

Jp 1L

L0

Jpdx 1LIpgDgS; (19b)

with gS gF x 0 and gD gF x L as the normalized,relative Fermi energy at the source and drain edges of the

graphene channel, respectively, and

I n I n gF lnkBTNGF 2gF

CqCox

F 1gF2

gFF 2gF F 3gF; (20a)

I p Ip gF lpkBTNGF 2 gF

CqCox

F 1 gF 2

gFF 2 gF F 3 gF ; (20b)

with Cq q2NG=kBT (as defined in Sec. III A).Defining the the drain current (density), JD, as the cur-

rent that flows directed from drain to source, assuming

VD > VS, the drain current is then

JD JT Jn Jp

JD JD; n JD; p; (21)

with JD; n Jn 1=L I ngSgD

and JD; p Jp 1=L I pgSgD. The associated transconductance, gm, is given by

gm @JD@VGS

VDS

1Lqln nS nD qlp pS pD ; (22)

with VGS VG VS, VDS VD VS, nD n x L , nS n x 0 , pD p x L , and pS p x 0 . The drain con-ductance, gd, is

gd @JD@VDS

VGS

1L

qlnnD qlppD

: (23)

By means of the QV relationship in Eq. (13), to determine gFwith respect to VG and Vch and Eqs. (20) and (21), the draincurrent density in the gFET channel can be solved numeri-

cally as a function of applied biases VG, VD, and VS. Thetransconductance and drain conductance can be similarly

numerically solved.

Using the values in Table I (e.g., L 1 lm,ln lp 5; 000 cm2/Vsec) and Qf 0, the transfer

FIG. 6. Metal-oxide-graphene capacitive network.



characteristic and transconductance and drain characteristic

of a gFET are shown in Figs. 7 and 8, respectively.

In Fig. 7(a), the characteristic V-shape of the gFET

transfer characteristic can clearly be seen. VDirac, the so-called Dirac voltage (i.e., the voltage associated with mini-

mum current, zero transconductance),20,35,36 given by

VDirac VGSjgm0 1

1 ln=lpVDS 1

CoxQf 1

q/mo /so ;

(24)

is not constant and shifts depending on bias conditions. Also,

as can be clearly seen, material related characteristics, such

as Qf, /mo, and /so, directly impact the value of VDirac. Thedirect dependence of VDirac on the fixed charge density (Qf),which is in essence the impurity concentration within the

structure, has been experimentally observed.5,10 Addition-

ally, Eq. (24) suggests that work function engineering (i.e.,

selection of gate metal based on its work function, /mo, orelectron affinity, vm) is a possible method for controlling theDirac voltage.

The transconductance (Fig. 7(b)) asymptotically

approaches a maximum value of gm;max lnCox=L VDSand a minimum value of gm;min lpCox=L

VDS. The

slope of the transconductance at the Dirac voltage is

@gm=@VGSjVGSVDirac ln lp

Cox=L. Though not a standard

figure of merit, @gm=@VGSjVDirac is useful as it is independentof drain-source bias (VDS) and gives insight into some physi-cal parameters (e.g., ln, lp) of the gFET. The derivationof the preceding equations (Eqs. (18)(24), gm;max, gm;min,@gm=@VGSjVDirac ) can be found in Appendix B.

In the drain characteristics of Fig. 8(a), a current satura-

tion is observed. This saturation is not a result of velocity

saturation within the graphene channel, which has not been

included in this work (see Sec. VIII), but the creation of a

highly resistive region within the graphene channel associ-

ated with the pinch-off (following Shockleys terminology)23

and transition of the majority charge carrier channel. Further

discussion is found in Sec. V.

In Fig. 9, the transfer characteristic is replotted, explic-

itly showing the relative electron and hole contributions to

the drain current. As has been previously theorized, at gate-

source voltages above the Dirac voltage (VGS > VDirac), thetotal current is dominated by electron current; below

the Dirac voltage (VGS < VDirac), the total current is domi-nated by hole current. Around VGS VDirac, the total currentis a combination of electron and hole currents with the

electron and hole current contributions being equal at

VGS VDirac.2,7,17,37

V. GRAPHENE FET ELECTROSTATIC ANDELECTRODYNAMIC PROFILES

As the gFET exhibits behavior uncommon in traditional

FETs, insight into this behavior can be gained by examining

the devices electrostatic and electrodynamic profiles along

FIG. 7. (Color online) (a) Plot of the transfer characteristics (JD versus VGS)for a gFET at three drain-source voltages (VDS 0:1; 1; 2:5V). (b) Plot ofthe associated transconductance (gm). The dashed line represents the asymp-totic behavior of the transconductance for the VDS 2:5 V curve, asdescribed by gm;max, gm;min, and @gm=@VGSjVGSVDirac .

FIG. 8. Plots of the drain characteristics (JD versus VDS) for the gFET: (a)VGS 0 to 4 V at 0.5 V steps, (b) VGS 0 to 4 V at 0.5 V steps. Theheavy line in both plots indicates the VGS 0 V curve. The dashed line in(a) corresponds to VGS 1:5 V, the gate-source bias used for Figs. 12and 13.



the length of the channel. Although there is not a direct rela-

tion between the position within the channel and the various

device properties (as dervied in this paper), it is possible to

parametrically relate and plot the various properties with

respect to position within the channel as a function of the

local normalized, relative Fermi energy, gF.Given drain and source voltages (VD and VS, respec-

tively), the channel voltage (Vch) varies between these twovalues: Vch VD; VS . With the addition of gate voltage(VG), the normalized, relative Fermi energy at the drain andsource (gD and gS, respectively) may be found via Eq. (13)with gF in the channel varying between the two:gF gD; gS . The position along the channel, 0 x L,can be found, given gF, by

x gF LI n Ip gF

gS

I n I p gD

gS

: (25)

As previously defined, the Fermi energy in the channel is

directly related to the channel voltage, with the Dirac energy

related to the Fermi energy via gF: eF qVch andeD eF kBTgF.

The electron (n) and hole (p) concentrations, as func-tions of gF, are found using Eq. (3). The drain currentthroughout the channel is constant with respect to position

and given by Eqs. (20) and (21). The total electron and hole

components of the drain current are, respectively,

JD; n gF JDlnn

lnn lpp; (26a)

JD; p gF JDlpp

lnn lpp: (26b)

And the electron diffusion and drift current components are

JdiffD; n gF JD; n1

1 CQ=Cox ; (27a)

JdriftD; n gF JD; n JdiffD; n JD; n1

1 Cox=CQ ; (27b)

with n replaced by p for the hole components. Recall,CQ Cq F 0 gF F 0 gF is the quantum capacitance ofthe graphene sheet (see Sec. III A).

The electric field along the channel, Ech Ex, is

Ech gF JdriftD; nqlnn

JdriftD; p

qlpp: (28)

Note, the negative sign in Eq. (28) indicates that the field is

directed from drain to source, as expected for VD > VS. Thederivation of the preceding equations can be found in

Appendix C.

In Fig. 10, the band diagram, electron and hole concen-

trations, total electron and hole current densities, and electric

field in the channel versus position within the gFET channel

for a constant drain-source voltage (VDS 1V) and varyinggate-source voltage (VGS 0:5; 0; 0:33; 0:6; 1:25V; iv inFig. 9) are plotted. The evolution from an essentially p-chan-nel (Fig. 10, column i), to a mixture of p- and n-channels(Fig. 10, columns iiiv), to a n-channel (Fig. 10, column v) isclearly seen in the plots of band diagram, carrier concentra-

tion, and currents as the gate-source voltage swings from

less than the Dirac voltage (VGS < VDirac), to around theDirac voltage (VGS VDirac), to greater than the Dirac volt-age (VGS > VDirac).

The electric field within the channel (Ech) is always neg-ative (i.e., directed from drain to source). An interesting

behavior is how the peak electric field is located at the drain

edge for completely n-channel biases and located at thesource edge for completely p-channel biases, both of whichare expected, but the peak electric field is located at the loca-

tion where the Fermi energy crosses the Dirac energy

(eF x eD) for biases resulting in a mixed channel and fol-lows this point as it moves between drain and source. Such

behavior has also been proposed and experimentally

observed by Freitag et al. by monitoring the thermal emis-sions from a graphene channel under bias.38

Figure 11 shows an expanded view of the electron and

hole currents, with their drift and diffusion components, of

the current plots from Fig. 10 (row c). As can be seen in thefigure, although the majority of the current is comprised of

drift current, diffusion current comprises a nonnegligible

portion of the total current, across all biases.

In Fig. 12, the band diagram, electron and hole con-

centrations, total electron and hole current densities, and

electric field in the channel versus position within the gFET

channel for a constant gate-source voltage (VGS 1:5V, thedashed line in Fig. 8(a)) and varying drain-source voltage

(VDS 0:1; 2; 4V) are plotted. The saturation observed inFig. 8(a) is due to the pinch-off and transition of the majority

charge carrier channel, which eventually leads to a mixed

carrier channel. As a result of the pinch-off and transition of

the majority carrier channel, a high resistivity region within

the channel is formed. It is this high resistivity region that

leads to the observed saturation.

FIG. 9. (Color online) Plot of the transfer characteristic for the gFET

(VDS 1V) explicitly showing the (mean value) electron current (JD; n) andhole current (JD; p) contributions to the drain current (JD). The triangles (~)with labels indicate the bias points used for Figs. 10 and 11.



Specifically, for a given gate-source voltage, as the

drain-source voltage is increased, the majority carrier concen-

tration decreases (electrons in Fig. 12(b), columns i and ii),accompanied by a slight increase in minority carrier

concentration (holes), eventually leading to a situation similar

to pinch-off in classical FETs (Fig. 12(b), column ii; nD inFig. 13(a); and JD; n in Fig. 13(b)).

23 Associated with this

pinch-off of the majority carrier channel is a decrease in the

drain conductance (gd) to its minimum (Fig. 13(c)). As thedrain-source voltage is further increased, the majority and

FIG. 10. (Color online) Plots of the (a) band diagram (eD: solid line, eF: dashed line), (b) electron and hole concentrations (n: solid symbols, p: hollow sym-bols), (c) electron and hole current densities (JD; n: solid symbols, JD; p: hollow symbols), and (d) electric field in the channel (Ech) versus position within thechannel for a constant VDS 1 V and (i) VGS 0:5V, (ii) VGS 0V, (iii) VGS VDirac 0:33V, (iv) VGS 0:6V, and (v) VGS 1:25V, indicated in Fig. 9by the triangles (~). All axes are linear.

FIG. 11. (Color online) Plots of the electron and hole current densities

(JD; n: solid circles, JD; p: hollow circles) and their respective drift (squares)and diffusion current (triangles) components. The bias conditions corre-

spond to those for the first three columns in Fig. 10: (a) VGS 0:5V, (b)VGS 0V, and (c) VGS 0:33V, with VDS 1 V for all plots.

FIG. 12. (Color online) Plots of the (a) band diagram (eD: solid line, eF:dashed line), (b) electron and hole concentrations (n: solid symbols, p: hol-low symbols), (c) electron and hole current densities (JD; n: solid symbols,JD; p: hollow symbols), and (d) electric field in the channel (Ech) versus posi-tion within the channel for a constant VGS 1:5 V (corresponding to thedashed line in Fig. 8(a)) and (i) VDS 0:1V, (ii) VDS 2V, and (iii)VDS 4V. All axes are linear.



minority carriers exchange roles and a new majority car-

rier channel is formed (Fig. 12(b), column iii; pD in Fig.13(a); and JD; p in Fig. 13(b)), alleviating the observed satura-tion (JD in Fig. 13(b)) and resulting in an associated increasein drain conductance, gd (Fig. 13(c)). A similar explanationfor this observed saturation for gFETs has been postulated

before by Meric et al.9

VI. INFLUENCE OF RESISTANCE

The general influence of external drain and source resist-

ance on the performance of the gFET will be examined in

this section. The specific source of the resistance is not con-

sidered here. A discussion of possible sources of resistance

is discussed in Sec. VIII.

Figure 14 shows a schematic of a gFET with external

drain and source resistances. In general, the addition of

external resistance affects the performance of gFETs in the

same manner as conventional FETs in other material sys-

tems, that is, it reduces the voltage drop across the intrinsic

portion of the device through simple voltage division, effec-

tively shifting the internal bias point of the device.

Using an iterative method to solve for the internal

biases, Fig. 15 shows the effect of various drain and source

resistances on the transfer characteristic of the gFET. As

expected, in both cases, the drain current is reduced for a

given drain-source (VDS) and gate-source (VGS) bias point.Sub-linear transfer characteristics have been observed exper-

imentally,15,22,39,40 and their basis as a resistively limited

current provides an uncomplicated explanation for the

observed behavior, without the need to invoke new physics

(e.g., alternative scattering methods).

The extrinsic transconductance (gm; ext) can be deter-mined using the classic, small-signal FET model (Fig. 16).

Evaluating at low frequency (f ! 0), the extrinsic transcon-ductance is

gm; ext gm1 gmRS gd RS RD : (29)

Plots of the extrinsic transconductance versus gate-source

voltage with different drain and source resistances are shown

in Fig. 17. As with the transfer characteristic, the transcon-

ductance is essentially reduced for a given bias point.

FIG. 13. (Color online) Plots of the (a) electron (nD) and hole (pD) concen-tration at the drain, (b) drain current (JD) and its (mean value) electron(JD; n) and hole (JD; p) components, and (c) drain conductance (gD) versusdrain-source voltage (VDS) for VGS 1:5 V.

FIG. 14. Schematic diagram of a gFET with extrinsic drain (RD) and source(RS) resistances. The portion bounded by the dashed line corresponds to theintrinsic gFET.

FIG. 15. (Color online) Plots of the gFET transfer characteristic

(VDS 2:5V) under the influence of various (a) drain and (b) source resis-tances. In both cases, the resistance is increased as 0.01 X mm, 0.1 X mm,and 1 X mm.

FIG. 16. Classic, small-signal model for a FET. The portion bounded by the

dashed line corresponds to the intrinsic FET. It should be noted that the vol-

tages in this figure are small-signal (ac) voltages, not bias-point (dc) voltages

(e.g.,VG t VG ~vG t ).



Transconductance curves of this form have also been

observed experimentally,15,22,40 with the collapse of the ex-

trinsic transconductance at higher gate-source voltages being

a direct result of the high drain conductance (gd) seen atthese biases. As the drain conductance increases at higher

gate-source and/or drain-source biases, it effectively shorts

out the intrinsic device, leading toward the FET appearing

simply as the series combination of the drain and source

resistances, with the influence of the gate voltage on the

drain current removed.

VII. FREQUENCY PERFORMANCE

Using the same small-signal model for a FET (Fig. 16),

the intrinsic short-circuit current-gain cut-off frequency

(fs; int) is commonly given as fs; int gm=2p Cgs Cgd

gm=2pCg, where Cgs is the gate-source capacitance, Cgd isthe gate-drain capacitance, and Cg Cgs Cgd is the totalgate capacitance. In defining fs; int for the gFET, the moreappropriate form of

fs; int 12p

gmCg

2s jgmj

2pCg(30)

is used with the (intrinsic) transconductance (gm) as definedin Eq. (22). In the derivation of the common form of fs; int, itis assumed that the transconductance is always positive,

allowing the transconductance to be moved outside of the

square root in Eq. (30) without taking its absolute value.

Since graphene-based FETs have demonstrated both positive

and negative transconductance, this simplification cannot be

made, giving fs; int the form shown in Eq. (30).The corresponding extrinsic short-circuit current-gain

cut-off frequency (fs; ext) (assuming Cgd Cgs Cg) isgiven by

fs; ext jgmj2pCg 1 gd RS RD

fs; int1 gd RS RD ; (31)

with the drain conductance (gd) as defined in Eq. (23)The total (linear) gate capacitance (Cg), in Eqs. (30) and

(31), is defined as

Cg @Qgate@VG

VD;VS

@Qgate@VGS

VDS

; (32)

with Qgate L0Qmdx. Therefore,

Cg @Qgate@VGS

VDS

L0

@Qm@VG; ch

dx L0

CTdx; (33)

with CT 1=Cox 1=CQ 1

being the total (areal) capaci-

tance associated with the metal-oxide-graphene structure

(Sec. III A).

Performing the integration of Eq. (33) gives

Cg q2N2gJT

lnFn lpFpgDgS (34)

with

Fn Fn gF F 1 gF 2 gFF 2 gF F 3 gF ; (35a)Fp Fp gF F 1 gF 2

gFF 2 gF F 3 gF : (35b)

A plot of Cg versus gate-source voltage (VGS) is shownin Fig. 18. The minimum gate capacitance occurs at the

Dirac voltage, VDirac, and approaches its maximum value(Cg;max CoxL) for VGS VDirac and VGS VDirac.

Figure 19 shows a plot of fs; int and fs; ext versus drain cur-rent (JD) (RS RD 0:05 X mm for fs; ext). As can be seenin the plot, fs; int and fs; ext both track the transconductance.

FIG. 17. (Color online) Plots of extrinsic transconductance (gm; ext) versusgate-source voltage (VGS) under the influence of various (a) drain and (b)source resistances. In both cases, VDS 2:5V and the resistance is increasedas 0.01 X mm, 0.1 X mm, and 1 X mm.

FIG. 18. (Color online) Total gate capacitance (Cg) (normalized to its maxi-mum value, Cg;max CoxL) versus gate-source voltage (VGS) forVDS 0:1; 1; and 2:5 V.



The maximum values of fs; int (for a given drain-sourcevoltage, VDS) are

f maxs;int

gm;min 2pCg;max

VGS VDiracgm;max 2pCg;max

VGS VDirac

8>>>>>:

1

2pLlp

VDSL

VGS VDirac

1

2pLln

VDSL

VGS VDirac

8>>>>>: (36)

fs; int has two maximum values as the electron and holemobilities, and subsequently jgm;minj lpCox=L

VDS and

jgm;maxj lnCox=L VDS, are not necessarily equal. Fig-ure 20 shows a plot of the maximum intrinsic short-circuit

current-gain cut-off frequency (f maxs; int) versus gate length (L).

Assuming that the physically maximum (group) velocity

achievable in the channel is the Fermi velocity, then the min-

imum transit time under the gate is sf ;min L=vF. The asso-ciated short-circuit current-gain frequency is

fs; vF 1

2psf ;min vF

2pL: (37)

For fs; vF values of 100 GHz and 1 THz, Eq. (37) suggests agate length no greater than L 1:6 lm and L 0:16 lm,respectively. Clearly, Eq. (37) represents an upper asymptote

for f maxs; int (Fig. 20), with actual achievable values fallingbelow fs; vF , as the saturation velocity within graphene mustbe below the Fermi velocity (vsat < vF).

VIII. THEORY VERSUS EXPERIMENT

Plots of the modeled transconductance versus gate-

source voltage and drain characteristics, along side the ex-

perimental results of Moon et al.,40 are shown in Fig. 21.Most of the values used for the model (Table II) were ei-

ther taken directly from Ref. 40 (e.g., L, W, tox) or by extrac-tion. For example, Eq. (24), with minor manipulation, may

be used to determined the electron to hole mobility ratio:

DVDirac 1= 1 ln=lp

DVDS. In conjunction with theexpression for @gm=@VGSjVDirac ln lp

Cox=L, the indi-

vidual electron and hole mobilities may be extracted.

FIG. 19. (Color online) fs; int (solid lines) and fs; ext (dashed lines) versusdrain current (JD) for VDS 0:1; 1; and 2:5 V. For fs; ext, RS RD 0:05 Xmm. The absolute value of the (intrinsic) transconductance ( gmj j) forVDS 2:5 V is plotted, illustrating the dependence of fs; int and fs; ext on gm.

FIG. 20. (Color online) f maxs; int versus gate length (L) forVDS 0:1; 1; and 2:5 V. The shaded region indicates frequencies over fs; vF .

FIG. 21. (Color online) (a) Plot of modeled transconductance (gm) versusgate-source voltage (VGS) (solid lines) and (b) modeled drain characteristic(solid lines) with the experimental results from Moon et al. (symbols withline, Ref. 40).



As can be seen in the figures, the theoretical develop-

ment given in this work matches the experimental behavior

well. In the case presented here, the largest deviation from

the experiment is for biases resulting in p-type channels(e.g., the dashed lines in Fig. 21(a)). A possible source of

this deviation is the effect of type (i.e., n-type, p-type) transi-tion outside of the gate region. This phenomenon, in addition

to drift current saturation, has not been included in the theo-

retical work presented in this paper, however both are dis-

cussed in the following sections.

A. Graphene-type transition outside the gate region

It has been well established that the native doping or

type of graphene is heavily dependent upon not only the

method for producing the graphene but any possible process-

ing the graphene is exposed to, resulting in n-type graphene,p-type graphene, or any degree in between.6,10,41 Addition-ally, the choice of metal for contacting graphene has been

shown to effectively dope the graphene beneath, or in

other words, set its type.30,42

With regard to a gFET, this leads to the possibility of

the access region and/or contacts outside of the gate region

being of an incompatible type, as compared to the region

under the gate, resulting in an overall increased extrinsic re-

sistance dependent upon the relative type of the region under

the gate to the region outside the gate, which could appear as

a variable bias-dependent extrinsic resistance.

For example, taking the region outside of the gate to be

natively n-type (due to whatever growth or processing con-siderations), then for a gate, drain, and source bias resulting

in a n-channel underneath the gate, the gFET sees extrinsicresistances simply associated with the conduction through

the access region (Fig. 22, column i). As the drain bias takesthe channel into a mixed channel, one can envision the for-

mation of a low conductivity (high resistivity) region just to

the exterior of the gate (see Fig. 22, column ii). This lowconductivity region outside the gate region is identical to

that observed under the gate for bias schemes at or near

channel pinch-off (see Fig. 12). The resultant effect is a

higher observed external resistance for drain biases leading

to mixed channels (i.e., higher drain biases). Similarly, for

gate biases that take the entire channel into a p-channel re-gime, two low conductivity regions are formed to either side

of the gate, also resulting in a higher observed external re-

sistance as compared to biases resulting in an n-channel (Fig.22, column iii). Behavior of this form has been shownthrough simulation by Ancona.12 This example was for a

natively n-type channel, but the same could be said for anatively p-type channel.

This bias-dependent external resistance could explain

some of the asymmetry observed in the transfer characteris-

tics in some practical gFETs.14,22,39,43 And it is believed to

be the reason that the model over predicts the total current as

compared to that observed by Moon et al. at biases in thetransfer characteristic and drain characteristic associated

with mixed or p-channel operation (Fig. 21).14 Of course, asimilar argument is plausible for MOSFET-like gFETs

where the type beneath the contact (dictated by the contact

metal) differs from that in the channel.

This behavior would also suggest that care must be

taken in the choice of contact metal and/or native gra-

phene type with regard to the operation of the gFET. For

example, if one wishes the gFET to operate equally well

in both n-channel and p-channel biases then this wouldsuggest that an access region with an intrinsic-like typing,

and/or contacts that result in a similar intrinsic-typing, are

desirable. Although, such a typing would lead to a higher

overall external resistance compared to a strongly n-typeor p-type access region, the resistance presented to theintrinsic gFET would be essentially equivalent, regardless

of the channel being n-type, p-type, or mixed. On the otherhand, if the gFET is to be used primarily with one type of

channel (n or p), an access region and contact of similartype is desirable in order to reduce the overall external

resistance.

TABLE II. Device parameters and dimensions used for Fig. 21.

Parameter Value

vTi (eV) 4.33vgraphene (eV) 4.5vF (m/sec) 10

6

er; ox 3.9tox (A) 275

L (lm) 2.94W (lm) 4ln (cm

2/Vsec) 6000

lp (cm2/Vsec) 9000

RS (X mm) 0RD (X mm) 0.9Qf =q (cm

2) in Fig. 21(a) 3 1011Qf =q (cm

2) in Fig. 21(b) 1.2 1012

FIG. 22. (Color online) Illustrative plots of the (a) band diagrams (eD: solidline, eF: dashed line), (b) electron and hole concentrations (n: solid symbols,p: hollow symbols), and (c) conductivity (r) in the channel versus position,in the region underneath the gate and just outside the gate.



B. Drift current saturation

Present thought is that drift current experiences a satura-

tion due to (relatively) high energy phonon scattering

encountered under high electric field strengths.9,11,44 To

date, though, no clear evidence of this current (or velocity)

saturation has been observed in graphene. In the work pre-

sented in this paper, in which drift and diffusion currents areconsidered, the electric field and any electric field dependent

phenomenon are not determinable until the total current in

the channel is solved (see Secs. IV and V). Without prior

knowledge or the simultaneous solution of the electric field,

drift current saturation can not be directly incorporated into

the solution of the total current in the channel.

Some papers have included drift current saturation, but,

in doing so, ignore diffusion current (i.e., assume drift cur-

rent comprises the total of the current in the channel).9,11 As

has been shown in this paper, diffusion current is a nonnegli-

ble portion of the total current in the channel. Given a spe-

cific current density within the channel, the assumption that

the total current is solely comprised of drift current leads to

an overestimation of the electric field within the channel.

This in turn leads to an early onset of drift current saturation.

In order to maintain a good fit, parameters used to describe

the drift current saturation must be altered. The process

becomes a mathematical fitting procedure, counter to the

original intent for including drift current saturation: the first

principles description of gFET behavior.

Clearly, future work must look to correctly incorporate

drift current saturation in the presence of both drift and diffu-

sion currents.

IX. CONDUCTIVITYAND MOBILITY

The conductivity of graphene, under the relaxation time

approximation, has been given by multiple sources as having

the general form of4,5,7,45

rn JE q

2s

ph2eF eD : (38)

Under specific conditions (e.g., T 0, n p), the conduc-tivity, due to impurity scattering, may be approximated as4,7

rn 2q2

h

p hvF 2u2o

n; (39)

with h being Planks constant; uo q4nimp=16e2r e

2o

p, the

strength of the scattering potential resulting from charged

impurities; and nimp, the impurity sheet density. Note,q nimp should not be confused with Qf, the fixed chargesheet density. Qf clearly includes charges that affect mobility(e.g., charges within or within proximity of the graphene

sheet) but also other fixed charges within the device structure

that do not have any real influence on mobility (e.g., charges

deep in the gate insulator).

By equating Eq. (39) to

rn 2q2

h

p hvF 2u2o

n qlnn; (40)

the mobility can be equated to

ln 2q

h

p hvF 2u2o

16e2r e

2ohv

2F

q31

nimp/ 1

nimp: (41)

The mobility in Eq. (41) is inversely proportional to the

impurity density, nimp, which has been experimentallyobserved;6 and is the premise for the assumption of constant

mobility, for a given impurity density, in the work presented

in this paper.

Many papers incorrectly apply the conductivity of

Eq. (39) in analyzing the behavior of graphene.4,8,11,20,35 The

conductivity of Eq. (39) was derived for a single carrier type

(e.g., electrons) in a single band of graphene (e.g., the con-

duction band, e > eD).4 However, in subsequent application,

the conductivity has been set proportional to the net carrierconcentration,4,8,11,20,35 which, using the terminology of this

paper, is

r lQnet ql n p : (42)This leads to two distinct conclusions for the conductivity:

(1) r 0 for Qnet 0 and (2) r < 0 (i.e., negative) for pre-dominately p-type conditions, which is nonphysical and hasnot been experimentally observed in graphene.1,2,7,10

In order to correct this, some authors have artificially

replaced the net carrier concentration by its absolute value

and introduced a residual carrier concentration (n), whichprovides for a nonzero conductivity when Qnet ! 0. Thisresults in a piecewise equation for the conductivity of

graphene:

r qln n pj j n < n

ql n pj j n pj j n > n

(; (43)

where n is an impurity concentration-dependent constantrelated to the Dirac voltage.4 Alternatively, some suggest

adding a minimum charge density to the net carrier concen-

tration (e.g., n pj j no).11 Expressions like Eq. (43) havebeen further extended by relating the net carrier concentra-

tion to a voltage within the structure (e.g., gate voltage,

channel voltage) via a capacitance (e.g., gate capacitance,

quantum capacitance), commonly under the restriction

jVch Vgrj jeF eDj=q kBT=q, with subsequent incor-rect application to biases outside this restriction:10,11,20,35

r Kq2

h

1

nimp

Coxq

VG n |{z}

n

n > n

Kq2

h

1

nimpn n < n

8>>>>>>>>>:

; (44a)

or

r 12lCQ Vch Vgr

r0; (44b)where K is a unitless constant associated with the strength ofthe scattering potential and r0 is a constant, in line with n,used to maintain a nonzero conductivity.11,35 Finally, in



cases where behavior corresponding to different mobilities

for electrons and holes is observed, authors replace l by lnfor n p > 0 and lp for n p < 0.10

Although these equations have been shown to approxi-

mately fit the behavior of graphene fairly well, especially at

large gate voltages,11,20 they offer poor physical insight. In

fact, due to the proportionality to the net carrier concentra-

tion of Eq. (43), the conductivity leads to the nonphysical

interpretation that the electron and hole currents oppose

each other under a single driving force (e.g., electric field,

Fermi energy gradient), or, alternatively stated, the electron

and holes fluxes flow in the same direction under a singledriving force. Such phenomena has been attributed to ambi-

polar transport, which, stated simply, only describes the

motion of excess charge carriers under the condition oflocal (quasi-)neutrality (i.e., Qnet Qf 0 for all x and z).This is not the case here, as it has been shown theoretically

in this paper and experimentally elsewhere (Ref. 38) that

gFETs are not locally neutral under operation (i.e.,

Qnet Qf Qm 6 0).Instead, beginning with the original derivation of con-

ductivity for a single carrier type, expressed as a constant

mobility, the total current is a sum of the currents resulting

from each carrier type, as in Eq. (15). Rewriting Eq. (15),

using Eq. (14),

JT qlnn qlpp reF=q: (45)

Recognizing that reF=q can be interpreted as an electricfield associated with an applied bias, then the conductivity is

r qlnn qlpp; (46)

the classic definition of conductivity for a system of elec-

trons and holes.

In essence, this definition of conductivity is the basis for

the work presented in this paper. With the addition of ther-

mal statistics, electrostatics, and electrodynamics, the unique

behaviors of graphene have been described and character-

ized, without the need for artificial corrections.

For example, this can be seen in the drain conductance

derived earlier. If the drain conductance is simply defined as

the conductivity of the channel (referenced to the drain) mul-

tiplied by the appropriate geometric ratio, we have

Gd WLrjxL

W

LqlnnD qlppD

: (47)

The drain conductance per unit width is then

gd GdW 1

LqlnnD qlppD

; (48)

which is identical to the drain conductance of Eq. (23),

derived in Appendix B.

X. CONCLUSIONS

This paper has presented an in-depth theoretical exami-

nation of the graphene-based field effect transistor. Begin-

ning with the prevailing theory regarding the fundamental

charge carrier transport in graphene and using a first princi-

ples approach, the unique behaviors observed in graphene-

based FETs were described. Expressions describing the

current in the channel (JD), transconductance (gm), drain con-ductance (gd), and Dirac voltage (VDirac) were developed. Thebehavior under bias of the electron (n) and hole (p) carrierconcentrations, electron and hole currents (JD; n, J

driftD; n , J

diffD; n

and JD; p, JdriftD; p , J

diffD; p), and the electric field in the channel

(Ech), along the length of the channel, were described andexamined. The characteristic V-shaped transfer characteristic

observed in graphene-based FETs was shown to be due to gra-

phene channel being capable of supporting both an n- and p-channel, with biases around the Dirac voltage resulting in a

mixed carrier channel. In essence, the graphene channel is

never fully depleted of mobile charge carriers. This character-

istic of graphene also directly leads to the lack of an off-state

(under gate modulation) observed in these devices.

The frequency performance of these devices, in terms of

the short-circuit current-gain cutoff frequency (fs), was alsoexamined. The intrinsic and extrinsic fs was found to beheavily influenced by the transconductance (gm) of thegFET. The maximum intrinsic fs (f

maxs; int) was found to have an

absolute upper bound limited by the Fermi velocity (vF) ofgraphene (f maxs; int < fs; vF vF=2pL).

The influence of external resistance on the steady-

state and frequency performance of the graphene-based

FET was also examined. As in nongraphene-based FETs,

external resistance was found to reduce the current in the

channel and transconductance for a given set of external

biases. Consequently, fs under the influence of externalresistance (fs; ext) is reduced as compared to the intrinsiccase (fs; int).

The theoretical work developed in this paper was found

to agree well with observed experimental results. By means

of this work, various device and material parameters, such as

electron and hole mobilities and fixed charge density, can be

extracted from experimental results, aiding in device and

material analysis.

Finally, behaviors not directly predicted by the theory

presented here, such as carrier type transition outside of the

gate region, were discussed with concern to possible implica-

tions on device performance.

APPENDIX A: QV RELATIONSHIP FOR GRAPHENE ONA DIELECTRIC SUBSTRATE WITH BACK GATE

A band diagram and charge profile for graphene on a

dielectric substrate with back gate and top gate is shown in

Fig. 23.

All terms are as defined earlier for the case of graphene

on SiC (see Sec. III) with the addition of the subscript T forterms referring to the top gate and B for terms referring tothe bottom gate. The bottom gate, commonly highly doped

Si, is assumed to be an equipotential. Proceeding with the

same method as presented in Sec. III, two voltage (potential

energy) loops around the structure are completed. One from

the top gate to the graphene channel:



q VG; T Vch /mo; T qEox; Ttox; T /so; T eD eF 0; (A1a)

and one from the graphene channel to the back gate:

eD eF /so;B qEox;Btox;B /mo;B q VG;B Vch

0: (A1b)A third voltage loop exists from the top gate to the back

gate, but is unnecessary for the analysis here.

In order to maintain charge neutrality, Qm; T QnetQf Qm;B 0. By Gausss law, Eox;B Qm;B=eox;B andEox; T Qnet Qf Qm;B

=eox; T . Solving Eq. (A1b) for

Qm;B, using the preceding relations, and plugging the resultinto Eq. (A1a) gives the QV relationship for graphene on adielectric substrate with back gate:

VG; T Cox;BCox; T

VG;B 1 Cox;BCox; T

Vch 1

Cox; TQnet

1 Cox;BCox; T

kBT

qgF

1

Cox; TQf 1

q/mo; T /so; T

1q

/mo;B /so;B Cox;B

Cox; T 0: (A2)

The various device characteristics for a gFET on a dielectric

substrate with back gate can now be solved, following the

same methods used in this paper.

APPENDIX B: FULL DERIVATION OF THE DRAINCURRENT, TRANSCONDUCTANCE, AND DRAINCONDUCTANCE

Beginning with Eq. (17), both sides are multiplied by

dx:

JTdx Jndx Jpdx qlnn dVch qlpp dVch : (B1)

As before, it is assumed there are no sources or sinks of

charge within the channel (r JT @q=@t 0), thereforethe total current density (JT) is constant. Integrating Eq. (B1)gives

JTx x0

Jndxx0

Jpdx

Vch x VS

qlnn dVch Vch x VS

qlpp dVch : (B2)

The electron (n) and hole (p) concentrations cannot bealgebraically related to the voltage in the channel (Vch).However, they are related by the normalized, relative

Fermi energy (gF) through the QV relationship of thegFET, established in Eq. (13). By differentiating the QVrelationship with respect to gF, with some algebraicmanipulation,

dVchdgF

1Cox

dQnetdgF

kBTq

; (B3)

recognizing that

dQnetdgF

ddgF

q p n

ddgF

q NGF 1 gF NGF 1 gF

qNG F 0 gF F 0 gF kBT

q

q

kBTqNG F 0 gF F 0 gF

dQnetdgF

kBTq

CQ; (B4)

and multiplying both sides of Eq. (B3) by dgF,

dVch kBTq

1 CQCox

dgF; (B5)

dVch in Eq. (B2) can be replaced:

JTx x0

Jndxx0

Jpdx

Vch x VS

qlnn dVch Vch x VS

qlpp dVch

JTx I ngF x gS

I pgF x gS

; (B6)

FIG. 23. Band diagram (top) and charge profile (bottom) of a gFET struc-

ture with top and back gates.



with gS gFx 0 and

I ngg I n gF

gg

gg

lnkBTn 1CQCox

dgF; (B7a)

I pgg I p gF

gg

gg

lpkBTp 1CQCox

dgF: (B7b)

Evaluating the integrals of Eqs. (B7a) and (B7b) gives

In gF and I p gF :

In gF lnkBTNGF 2 gF

CqCox

F 1 gF 2

gFF 2 gF F 3 gF ; (B8a)

I p gF lpkBTNGF 2 gF

CqCox

F 1 gF 2

gFF 2 gF F 3 gF ; (B8b)

with Cq q2NG=kBTFinally, the total current density (at x) is found by divid-

ing Eq. (B6) by x,

JT 1x

x0

Jndx 1x

x0

Jpdx

1xI njgF x gS

1

xI pjgF x gS : (B9)

Solving Eq. (B9) at x L (gF L gD) leads to the resultspresented in Eq. (18),

JT 1L

L0

Jndx 1L

L0

Jpdx Jn Jp; (B10)

and Eq. (19),

Jn 1LI njgDgS ; Jp

1

LI pjgDgS : (B11)

Defining the drain current (density), JD, as the current thatflows directed from drain to source, assuming VD > VS, thedrain current is then

JD JT Jn Jp JD; n JD; p; (B12)

with simply JD; n Jn 1=L I njgSgD and JD; p Jp 1=L I pjgSgDThe transconductance is defined as

gm @JD@VGS

VDS

: (B13)

Using Eqs. (B11) and (B12), Eq. (B13) may be written as

gm @JD@VGS

@@VGS

1

LIngSgD 1LI pgSgD

gm 1L

@I n@VGS

gSgD

@I p@VGS

gSgD

!: (B14)

The derivatives in Eq. (B14) may be written as

@I n@VGS

@I n@gF

@gF@VGS

;@Ip@VGS

@I p@gF

@gF@VGS

: (B15)

Recall Eq. (B7),

I njgg gg

lnkBTn 1CQCox

dgF

In g In g

dI n; (B16a)

I pjgg gg

lpkBTp 1CQCox

dgF

Ip g Ip g

dI p: (B16b)

Therefore,

@I n@gF

lnkBTn 1CQCox

; (B17a)

@I p@gF

lpkBTp 1CQCox

: (B17b)

By differentiating the QV relationship from Eq. (13) withrespect to gF, following a similar procedure as in Eqs. (B3)and (B4) gives

dVGSdgF

kBTq

1 CQCox

: (B18)

Eq. (B15) then becomes

@I n@VGS

@I n@gF

@gF@VGS

@I n@gF

@VGS@gF

1

lnkBTn 1CQCox

kBT

q1 CQ

Cox

1@I n@VGS

qlnn; (B19a)

@Ip@VGS

@Ip@gF

@gF@VGS

@I p@gF

@VGS@gF

1

lpkBTp 1CQCox

kBT

q1 CQ

Cox

1@Ip@VGS

qlpp: (B19b)



Finally, evaluating Eq. (B14) using Eq. (B19) results in the

transconductance being

gm 1LqlnnS qlnnD qlppS qlppD

1Lqln nS nD qlp pS pD ; (B20)

with nD n gD , nS n gS , pD p gD , and pS p gS The Dirac voltage is defined as VDirac VGSjgm0. Set-

ting the transconductance (gm, Eq. (B20)) equal to zeroresults in

ln nS nD lp pD pS : (B21)

From Fig. 9, it is clear that VDirac corresponds to a bias con-dition resulting in a mixed carrier channel, specifically with

an n-channel toward the source and a p-channel toward thedrain (Fig. 11(c)). Therefore, assuming gD gF x L 0and gS gF x 0 0, then nD pS 0 and Eq. (B21)becomes

lnnS lppD; (B22)

with

nS NG g2S

2

Coxq

VG VS 1

CoxQf 1

q/mo /so

; (B23a)

pD NG g2D

2

Coxq

VD VS VG VS 1Cox

Qf 1q

/mo /so

:

(B23b)

Recognizing that VG VS VGS VDirac (and VD VS VDS) in Eq. (B23) and applying Eq. (B23) to Eq. (B22),the Dirac voltage, with some algebraic manipulation, is

VDiracVGSjgm0 1

1 ln=lpVDS 1

CoxQf 1

q/mo /so :

(B24)

The maximum transconductance occurs for VGS VDirac,under which gD; gS 0 and pD; pS 0:

gm;max 1L

qln nS nD

qlnL

Coxq

VG VS 1

CoxQf 1

q/mo /so

Coxq

VG VD 1

CoxQf 1

q/mo /so

gm;max lnCox

LVD VS lnCox

LVDS: (B25)

The minimum transconductance occurs for VGS VDirac,under which gD; gS 0 and nD; nS 0:

gm;min 1Lqlp pS pD

qlpL

Coxq

VS VG 1

CoxQf 1

q/mo /so

Coxq

VD VG 1

CoxQf 1

q/mo /so

gm;min lpCoxL

VS VD lpCoxL

VDS: (B26)

The derivative of the transconductance with respect to the

gate-source voltage (@gm=@VGS) is given by

@gm@VGS

1L

qln@ nS nD

@VGS qlp

@ pS pD @VGS

: (B27)

Solving at the Dirac voltage (VGS VDirac), again gD 0,gS 0, and nD pS 0. With nS and pD given by Eq.(B23), the derivatives in Eq. (B27) can easily be solved

and

@gm@VGS

VGSVDirac

CoxL

ln lp

: (B28)

The drain conductance is defined as

gd @JD@VDS

VGS

@@VDS

1

LI ngSgD 1LI pgSgD

: (B29)

The derivative of any independent function by definition is

zero. Because gS is independent of the drain voltage,@I n gS =@VDS and @I p gS =@VDS are zero and

gd 1L @In gD

@VDS @I p gD

@VDS

1L @In gD

@gD

@gD@VDS

@Ip gD @gD

@gD@VDS

gd 1L @I n gD

@gD

@VDS@gD

1 @I p gD

@gD

@VDS@gD

1" #:

(B30)

Again, following a similar procedure as in Eqs. (B3) and

(B4) and evaluating Eq. (B17) at gF gD (Vch VD), thederivatives in Eq. (B30) can be replaced, and the drain con-

ductance becomes

gd 1L

qlnnD qlppD

: (B31)

APPENDIX C: DERIVATION OF POSITIONDEPENDENCYOF DEVICE PROPERTIES

The various device properties of the gFET (e.g., carrier

concentration, current density, electric field in the channel)

cannot be directly related to the position within the channel,

but can be related indirectly though the normalized, relative

Fermi energy, gF. The normalized, relative Fermi energy, gF,which varies between its value at the drain (gD) and source



(gS), is determined via the QV relationship for the graphenechannel (Eq. (13)) from the applied biases: VG, gate voltage;VD, drain voltage; and VS, source voltage.

The position within the channel for the various device

properties can be found by rewriting Eq. (B6):

x gF x0Jndx

x0Jpdx

JT

InjgFgS I pjgFgSJT

x gF LI n I pgFgSIn I pgDgS

: (C1)

The Fermi energy in the channel is simply the channel volt-

age multiplied by a negative electron charge: eF qVch.The Dirac energy, in essence the band edge for both the

conduction band and valence band, is related to the

Fermi energy by the normalized, relative Fermi energy, gF:eD eF kBTgF.

The total current (density), JT, throughout the entirechannel is constant with position, ignoring gate leakage, and

is given by Eqs. (18)(20) or Eqs. (B8), (B10), and (B11),

knowing the drain and source normalized, relative Fermi

energies (gD and gS, respectively). The total current is equalto the sum of the total electron (Jn) and hole (Jp) current den-sities, Eq. (16). Taking the ratio of the total electron current

density to the total current density,

JnJT lnn deF=dx

lnn deF=dx lpp deF=dx lnn

lnn lpp; (C2)

and rewriting

Jn gF JTlnn

lnn lpp: (C3)

gives the total electron current density in the channel as a

function of gF, via n and p.By expanding the original definition of the total electron

current from Eq. (16), the electron drift and diffusion current

densities are found:

Jn lnndeFdx

lnnd

dxeD kBTgF

lnnd

dxqVgr lnkBTn dgFdx

qlnnEch lnkBTndgFdx

Jn Jdriftn Jdiffn ; (C4)

with qVgr eD and Ech Ex dVgr=dx.The derivative of the normalized, relative Fermi energy

(gF) with respect to x can be found by inverting the deriva-tive of x (Eq. (C1)) with respect to gF:

dgFdx

dxdgF

1

1JT

d

dgFI n IpgFgS

1: (C5)

As the normalized, relative Fermi energy at the source (gS) isfixed for a given bias condition, its derivative with respect to

gF is zero and Eq. (C5) reduces to

dgFdx

JT dIn gF dgF

dI p gF dgF

1: (C6)

The derivatives in Eq. (C6) have already been evaluated in

Eq. (B17). Therefore,

dgFdx

dxdgF

1 JT

lnn lpp

kBT 1 CQ=Cox : (C7)

The electron diffusion current density is then

Jdiffn gF lnkBTndgFdx

JT lnnlnn lpp1

1 CQ=CoxJdiffn gF Jn

1

1 CQ=Cox : (C8)

Rewriting Eq. (C4) gives the electron drift current density:

Jdriftn gF Jn Jdiffn Jn 1 1

1 CQ=Cox

Jdriftn gF Jn1

1 Cox=CQ : (C9)

Following a similar procedure, the hole current density, the

hole diffusion current density, and the hole drift current den-

sity are, respectively,

Jp gF Jdriftp Jdiffp JTlpp

lnn lpp(C10a)

Jdiffp gF Jp1

1 CQ=Cox (C10b)

Jdriftp gF Jp1

1 Cox=CQ (C10c)

The electric field in the channel is found by rewriting the

expression for the drift current from Eq. (C4):

Ech gF Jdriftnqlnn

Jdriftp

qlpp: (C11)

Finally, as the drain current (density) is defined as flowing

opposite to the total current density, JD JT , the variousproperties referenced to the drain current become



JD; n gF Jn gF ; (C12a)

JdiffD; n gF Jdiffn gF ; (C12b)

JdriftD; n gF Jdriftn gF ; (C12c)

Ech gF JdriftD; nqlnn

; (C12d)

with p replacing n for the hole components.

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