Physics-based Modeling of a Graphene-loaded Capacitor

by Frank J. Crowne

ARL-TR-5481 March 2011

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Army Research Laboratory

Adelphi, MD 20783-1197

ARL-TR-5481 March 2011

Physics-based Modeling of a Graphene-loaded Capacitor

Frank J. Crowne

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited. 
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Physics-based Modeling of a Graphene-loaded Capacitor

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Frank J. Crowne

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14. ABSTRACT

In this report a mathematical model is developed for a capacitor containing a monolayer of graphene. The model can be used 

in a zero-order description of the terminal structure of a graphene field-effect transistor in which the gate-source and gatedrain connections are replaced by capacitances, a standard approximation used in the modeling of ordinary MOSFETs. The 

analysis leads to mathematical expressions for the charge density in the graphene channel. Whereas in a previous report 

(ARL-TR-5281) the author finessed this aspect of the transistor modeling by using an empirical model for the channel charge 

density, the density calculated here includes atomistic properties of the graphene layer. The treatment also allows nonlinear

effects—i.e., varactor action—to be studied, which indicate that the graphene-loaded capacitor is an interesting device in its 

own right.

15. SUBJECT TERMS

Graphene, ambipolarity, varactor

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Frank J. Crowne

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iii

Contents

List of Figures iv

1. Introduction 1

2. Modeling Methodology 2

3. Carrier Statistics In Graphene 5

3.1 Zero Temperature ............................................................................................................5

3.2 Nonzero Temperature......................................................................................................6

4. Charge Dynamics 8

4.1 Zero Temperature ............................................................................................................9

4.2 Nonzero Temperature....................................................................................................10

5. Nonlinear Response and Varactor Action 13

6. Conclusions 18

7. References 19

Appendix. Varactor Calculations 21

Distribution List 27
iv

List of Figures

Figure 1. Dual-gate graphene FET structure...................................................................................1

Figure 2. Graphene-loaded capacitor with a positive bias on the top gate. ....................................4

Figure 3. Graphene-loaded capacitor with a negative bias on the top gate. ...................................9

Figure 4. Fermi level versus back gate voltage Vbg.....................................................................12

Figure 5. Electron and hole densities versus back gate voltage Vbg............................................13

Figure 6a. 2nd harmonic surface charge density vs. ac voltage.....................................................15

Figure 6b. 3nd harmonic surface charge density vs. ac voltage.....................................................16

Figure 6c. 4th harmonic surface charge density vs. ac voltage. ....................................................16

Figure 6d. 5th harmonic surface charge density vs. ac voltage. ....................................................17

Figure 6e. 6th harmonic surface charge density vs. ac voltage. ....................................................17
1

1. Introduction

The standard approach to modeling transport-based semiconductor devices such as transistors, 

charge-coupled devices, etc. (1) begins with analysis of the device response under open-circuit 

conditions in the output circuit—i.e., with either an open source or an open drain contact. In the 

absence of lateral transport from a source to drain, these devices can be described by one-port 

circuits. In the case of field-effect transistors (FETs), the lack of direct current (dc) gate current 

makes the transistor essentially a capacitor, which greatly simplifies the full two-port device 

equivalent circuit. More importantly, the open-circuit response lets us find the charge density 

in the channel as a function of gate voltage. Knowledge of  is sufficient to determine many 

other fundamental features of the transistor characteristics, for example the transistor’s

alternating current (ac) input impedance.

The approximate description of the aforementioned transistor can be formulated by first 

analyzing a purely capacitive structure in which a monolayer of graphene is embedded in the 

dielectric layer of a simple parallel-plate capacitor. This approach to modeling is especially 

fruitful in the case of graphene-based transistor structures such as the one shown in figure 1 (2), 

which shows a graphene device structure in current use. Unlike a normal field-effect transistor, 

this structure includes two gates, a top gate and a bottom (back) gate, both of which control the 

channel charge density .

Figure 1. Dual-gate graphene FET structure.

V=0 V=Vds

V=Vgst

Ohmic

contact

Ohmic

contact

Graphene

layer

Dielectric layer 1 (HfO2)

Top

Gate

- Device active region

LS LD

Dielectric layer 2 (SiO2)

Doped Si back gate

V=Vgsb
2

Since graphene is ambipolar, both electrons and holes are present in the channel and contribute 

to the current; hence,  must include both species of carrier. The back gate electrode makes it 

possible to ―dope‖ the FET channel n- or p-type by electrostatically generating the appropriate

―inversion layers‖ using MOSFET terminology.

In a series of papers (3–5), Meric et al. specify the ―total‖ charge density  by the following 

empirical expression:

      V V C V V C V V gst gst gst gst gsb gsb gsb gsb

 

             

, (1)

where 

V V and gst gsb

are voltages applied to the top and back gates, respectively. This 

expression incorporates a peculiarity exhibited by graphene films: the existence of a ―minimum 

conductivity‖ in the graphene channel (6), apparently associated with ambipolarity. Meric et al. 

argue in (3–5) that this minimum conductivity corresponds to a minimum charge density, 



, in 

the channel. The expression equation 1 contains five fitting parameters: the minimum charge

density 





, the capacitances per unit area

C C and gst gsb

between the two gates and the 

monolayer, and threshold gate voltages 

0 0 V V , gsb gst

at which the minimum charge density 





is reached as the back and top gates are varied separately.

The fact that this expression has a minimum at some voltage obscures the fact that neither the 

number of electrons nor the number of holes has such a minimum. It, therefore, sidesteps much 

of the discussion of ambipolar transport. However, as we shall see, this issue must be addressed 

for the graphene-loaded capacitor.

2. Modeling Methodology

A rigorous expression for the channel charge density 

  V V gst gsb  

can be obtained by 

modeling the multilayer structure shown in figure 2. This structure, which is related to figure 1 

by a clockwise rotation of 90, consists of a graphene monolayer located at the interface between

two different dielectrics (chosen to be 

SiO2

and 

HfO2

by Meric et al.), with dielectric constants 

1



and 

2

 . There are two electrodes that act as gates, a back gate on the left-hand dielectric and a 

top gate on the right-hand dielectric. When voltages

V

gb

and 

Vgt

are applied to these gates, the 

internal potential energy 

U x qV x      

between them (where q is the charge of a single 

electron/hole) is a piecewise-linear function of the distance x through the capacitor along an axis 

from the back gate to the top gate. When the graphene layer is uncharged, U x consists of two 
3

straight line segments with a jump in slope at the interface between the dielectrics, due to the 

differing dielectric constants of the layers. A charge  on the graphene layer causes a further 

change in the slope. Let us write 

U x 

as follows:

 

 ,

2 2

 ,

2 2

L L U R x x d gb

U x

L L U S x x d gt

     

            

 

    



           

,

where the left electrode is at 

2

L

x  

, the right electrode is at 

2

L

x  

, and d is the position of 

the graphene monolayer. In this expression, 

U qV gb gb  

is the value of the potential energy at 

2

L

x  

and

U qV gt gt  

is its value at 

2

L

x  

, while R and S are constants to be determined.

At the monolayer, we have two boundary conditions:

(1) Continuity of 

U x x d   at 

:

2 2 2 2

L L L L U S d U R d U R d S d gt gb G

                                  

,

where 

U U U G gt gb    .

(2) A jump in the derivative of 

U x x d   at 

, determined by the Poisson equation 

         

2

' '

2 1 2 1

d dU x q x d U d U d q S R q p n

dx dx

                  ,

where p and n are the 2D hole and electron (number) densities in the monolayer.
4

Figure 2. Graphene-loaded capacitor with a positive bias on the top gate.

Let 

1 2 2 2

L L d d                   

. Then solving these two equations for B and C gives

 

 

 

1 2

 , 2 2 2 2

1 2

 , 1 2 2 2

L L L U U d q p n x x d gb g

U x

L L L U U d q p n x x d gt g

                                    

 

                         

         

.

At x = d we find that

   

2 2 1 2

1 2 2 2 4

L L L q U d d U d U d p n gb gt

       

                            

.

Let 

1

2

C

gsb L

d







and 

2

2

Cgst L

d







be the capacitances per unit area between the gates and 

the monolayer. Then it is easy to show that 

2

2

4

C C gsb gst

L

d



 



and 

1

1 2 2 2

C U C U L L gsb gb gst gt

d U d U gb gt C C gsb gst



     

                   

Metal

top 

gate

Metal

top

gate

1 2

Metal

back 

gate

graphene

x = L/2 x = d x = L/2

SiO2 HfO2

F

1 2

Metal

back 

gate

Metal 

top 

gate

graphene

x = L/2 x = d x = L/2

SiO2

F

HfO2

1 2

Metal

back 

gate

graphene

x = L/2 x = d x = L/2

SiO2

F

HfO2 U = U gb U = U gt U = U gb U = U gb

U = U gt

U = U gt

(a) (b) (c)
5

so that

   

C U C U 2

gsb gb gst gt q

U d p n

C C C C gsb gsb gst gst



  

 

.

3. Carrier Statistics In Graphene

Because both electrons and holes are simultaneously present in graphene, let us pause to discuss 

the nature of transport in a graphene FET. Although the large-scale graphene band structure is 

complex, in discussing transport we may confine our attention to the conduction and valence 

bands of the material. In addition to being two-dimensional (2-D), the bands are conical in 

shape—i.e., the energy of an electron in the conduction band is

 

2 2 E k k k k , C x y x y F

   ,

while the energy of an electron in the valence band is

 

2 2 E k k k k ,

V F x y x y     .

Note that these bands ―touch‖ at a single point in k-space

 k k x y , 0,0    

; in the literature, this 

is referred to as the Dirac point. Contrast this to the case of an ordinary three-dimensional (3-D)

direct-gap semiconductor, for which the energy of an electron in the conduction band is

   

2

2 2 2 1

, ,

2 2

E k k k k k k E C G x y x y z z

mC

    ,

while the energy of an electron in the valence band is

   

2

2 2 2 1

, ,

2 2

E k k k k k k E V x y x y z z

m G

V

     ,

where 

mC

, mVare the conduction-band and valence-band effective masses. 

3.1 Zero Temperature

The energy distribution functions for electrons and holes at zero temperature are simply

   

   

fe e e F

f

h h h F

     

     

,
6

regardless of the dimensionality of the system. The 2-D nature of the graphene band structure 

enters the problem when we consider the Fermi distribution in k-space. For a 3-D system, the 

zero-temperature electron/hole distribution is a filled sphere of radius

k

F

, with

k

F

the Fermi 

wave vector. Because the electron/hole density (n or p) is the integral over this distribution, we 

find for 3-D bands that

 

3

2 2

4

3 2 2 0 3

k

F k n F k dk

p



    

  

For a 3-D electron gas, the Fermi energy measured from the bottom of the conduction band is 

given by 

2 2

2

k

F

F m

 

, implying that 

2/3 2/3

n p or F  . In contrast, the distribution at zero 

temperature for a 2-D system is a filled disk of radius

k

F

. Because the electron/hole density (n or

p) is the integral over this distribution, we find that for a 2-D system

 

2

4

2

2

2 0

k

F k n F kdk

p



    

  

For a typical III-V 2D electron gas, e.g., the channel of a GaAs/AlGaAs HEMT, the Fermi 

energy is again given by 

2 2

2

k

F

F m

 

, implying that 

n p or F  . In contrast to these 

―normal‖ systems, the conical nature of the band structure for a graphene layer gives 

k

F F F   

, so that 

   

1/2 1/2

n p or F F F       .

3.2 Nonzero Temperature

Both carrier species are present at finite temperatures, due to the vanishing band gap. Let us 

introduce the inverse thermal energy,

1

k TB

  , and the usual temperature-dependent carrier 

energy distributions: for electrons we have (7)

 

 

 

     

 

4 4 2 2

2 2 2 2

1

2 2 2

2 2 1

0 0 1

1

kdk n f k d k e

e F e

kdk udu

e

k n F F F u k

e F F F e F F

  

   

       



 

         

              

,
7

where 

 

1

F x

is the order-1 Fermi integral. Note the factor 4, which takes into account both the 

spin degeneracy of the electrons and the presence of two ―cones‖ at different points in k-space 

with identical parameters—a case of the so-called ―valley degeneracy‖ and yet another 

peculiarity of graphene. Likewise, for holes we have (5)

 

 

   

   

   

   

 

4 2

2

2

1 1 , 1 1

1 1

1 2 2

2

0

1 1

2 2

2 2 1

0 1

p f k d k h

k f k f k F e h h k k F F F F e e

kdk kdk

p

k k F F F F e e

udu F

u F

e F

F F



 

 

        

        

 

       

         



 



   

 

      

.

As noted in (8), the temperature dependence of ―intrinsic‖ carriers in graphene takes a peculiar 

form: since the equilibrium densities of electrons and holes are the same, the Fermi level must be 

at zero for all temperatures. Then 

 

 

 

 

 

 

 

0

2 1

2

2

0

2 1

2

2 1

2

2 1

2 2

F

F

F n p

F

F

F

F

F

F

F

n p np n

  

  

















  



 

  

 

  

   

 

 

.

Since 

 

12

2

0

1



F 

, we find that 

2

6 



















 

F

kT

n p



, i.e., the densities decrease slowly with 

decreasing temperature according to a square law, in contrast to the decrease predicted by the 

expression 

















  

kT

G

E

i

n p n

2

exp

in an ordinary semiconductor with bandgap 

G

E

and intrinsic 

carrier concentration 

3/2

2

2

2



























kT Vm

C

m

i

n

, which is exponentially rapid.
8

4. Charge Dynamics

Assume that the back gate is shorted to the monolayer so that charge can flow between them. 

Figure 2(a) shows the case of no applied voltage, for which both the Fermi level, 

F

 , and the 

Dirac point are at the same voltage as the electrodes. Note that this situation is rather artificial 

since the Dirac point should lie within the energy gaps of both the insulator cladding layers—i.e., 

below the conduction bands (which may themselves be offset) by some ―confinement energy‖ 

determined by differences in work functions. This implies that its position will be offset relative 

to the applied voltages, as well. Because this should only cause an uninteresting dc shift in the 

device response, we will ignore it here. 

Let us now apply a negative voltage (positive 

Ugt

) to the top gate, thereby creating a nonzero 

potential 

U x 

(figure 2(b)). If the monolayer and the back gate are not connected, the field is 

nonzero everywhere, and the Dirac point is always at a potential 

U d   0

. However, if the back 

gate and the monolayer are connected, the applied voltage will cause charge to flow, creating a 

counterfield that increases until the Fermi level is constant and equal to

Ugb

everywhere to the 

left, resulting in the potential curve shown in figure 2(b). As a result of screening by the charge 

in the monolayer, the field in the region to the left is now zero. However, the following argument

shows that the left-hand field is actually nonzero—i.e., the screening will be incomplete. This is 

true for the following reason: as the charge on the monolayer increases, the counter-field drives 

the potential at the monolayer (and, hence, the Dirac point) downward. But the charge added or 

subtracted to the monolayer also drives the Fermi level away from the Dirac point, as shown in 

figure 2(c). Eventually 

F



reaches zero when enough electrons are drained out of the 

monolayer, leaving it with a positive hole charge. However, the Fermi level and the potential are 

no longer the same, due to the kinetic energy  inherent in the filled Fermi sea. Since the charge 

that caused the original change in the discontinuity in the voltage derivative is the same charge 

that produces the Fermi level displacement, it can be calculated by virtue of the relation between 

the Fermi level and the monolayer charge density:

U d  0

F

    

The same argument holds for holes. Figure 3(a) shows the case of no applied voltage, for which 

both the Fermi level

F



and the Dirac point are again at the same voltage as the electrodes. 

Applying a positive voltage (negative 

Ugt

) to the top gate creates a nonzero potential 

U x 

(figure 3(b)). If the monolayer and the back gate are not connected, the field is nonzero 

everywhere and the Dirac point is always at a potential 

U d   0. Connecting the back gate and

the monolayer causes charge to flow, creating a counterfield that drives the Fermi level upward 
9

until it equals 

Ugb

everywhere to the left, resulting in the potential curve shown in figure 3(b).

Again, as the charge on the monolayer increases, the counter-field drives the potential at the 

monolayer (and, hence, the Dirac point) upward and drives the Fermi level away from the Dirac 

point, as shown in figure 3(c). 

F



becomes zero when enough electrons are added to the 

monolayer, leaving it with a negative electron charge. Again, the Fermi level and the potential 

are no longer the same, due to the kinetic energy inherent in the filled Fermi sea. The added

charge can be calculated by virtue of the relation between the electron Fermi level and the 

monolayer charge density.

Figure 3. Graphene-loaded capacitor with a negative bias on the top gate.

4.1 Zero Temperature

When 

U U U 0 g gt gb     , figure 2 clearly shows that

  U d —i.e., the monolayer is 

charged with holes. At T = 0 there are no thermal electrons in the monolayer, and so 

 

1/2

p

F

    , which gives 

 

2

1/2 ( ) 0

C U C U gb gb gt gt q

U d p p F C C C C gb gb gt gt



       

 

.

Metal

top

gate

1

Metal

back 

gate

graphene

x = L/2 x = d x = L/2

SiO2

2

HfO2

F

U = U gb

U = U gt

U = U gt

metal

metal

graphene

F

1

Metal

back 

gate SiO2

U = U gb

2

HfO2

Metal

top

gate

x = L/2 x = d x = L/2

metal

metal

graphene

F

1

Metal

back 

gate SiO2

U = U gb

2

HfO2

U = U gt

Metal

top

gate

x = L/2 x = d x = L/2

 (a) (b) (c)
10

From the last result we can solve for p to obtain

2

1 2

p A B A          

,

where

    2 2 2

F A C C B C U C U gb gb gb gt gt gt

q q

  

    .

In contrast, when 

U U U 0 g gt gb    

we clearly have 

 U d( )—i.e., the monolayer is 

charged with electrons. At T = 0 there are now no holes in the monolayer, and so 

 

1/2

n

F

    

, and

 

2

1/2 ( ) 0

C U C U gb gb gt gt q

U d n n F C C C C gb gb gt gt



        

 

,

from which we can solve for n to obtain

2

1 2

n A A B         

.

4.2 Nonzero Temperature

At finite temperatures, both carrier species are present due to the vanishing band gap. Let us 

introduce the inverse thermal energy,

1

k TB

  , and the temperature-dependent carrier 

distributions previously introduced. With both charges present, we write the charge density

 

 

    

2

2 1 1

q

q p n F F

F

      

  

.

Note that at low temperatures, we have

 

 

 

 

2

2 1 1 2

0

2 2

2

2 1 1 2

0

2 2

e n

F

F

e p

F

F



      

                            

    

                                   

,
11

which recovers the zero-temperature result. Once more we can write

 

   

2

2

2 1 1

q

F F U

C C gt gb F

       



 

  

,

where 

C U C U gt gt gb gb U

C C gt gb







is an ―average‖ potential with respect to the dielectric 

capacitances. Once this expression is solved numerically for the Fermi energy, this energy can be 

used to find n and p separately:

 

   

 

   

2

2 1

2

2 1

p F p U

F

n F n U

F

  

  

  

  

.

Figure 4 is a plot of the Fermi level versus back-gate voltage for the following parameters (2–4):

dielectric layers:

SiO2

layer width: 61 nm

SiO2

dielectric constant (

1



): 3.9

HfO2

layer width: 253 nm

HfO2

dielectric constant (

1



): 16

graphene layer

temperature: 300 K

Fermi velocity:

8

1. 10 cm/s 

In figure 5, the corresponding electron and hole densities are plotted versus the back gate 

voltage. Note the changeover from electron to hole conduction at a back gate voltage of about 

5 V.
12

Figure 4. Fermi level versus back gate voltage Vbg.
13

(a)

(b)

Figure 5. Electron and hole densities versus back gate voltage Vbg.

5. Nonlinear Response and Varactor Action

To find the capacitance of a graphene-loaded capacitor at finite temperatures, we need its chargevoltage relation. This follows from the relation between the Fermi level and the voltage:

 

   

2

2

2 1 1

q

F F U

C C gt gb F

       



 

  

.
14

Writing this in dimensionless form gives

   

     

 

1 1

 , 1 1

1

F F U

z U z F z F z f z

z f

           

                 

   

,

where 

 

2

2

2

q

C C gt gb F

  



  

. Assume that a dc voltage is applied to the back gate of the 

varactor and a single-frequency ac component is applied to its top gate, i.e., 

 

 

0

cos 0 1 cos 1

U U C C gb gb gt U U U t

U U t C C C C gt gt gt gb gb

 



      

      

.

Let

 Z cos cos     0 0 1 1

C C gb gt U U t t

C C C C gt gt gb gb

            

 

,

where 

1 0    . Since the net charge per unit area is

 

 

           

2 1 1

2 1 1

q

q p n F F C C U C C z gt gt gb gb q q

F

                   

  

,

we can find the response once we know the function 

z.

The inversion of the implicit function relation 

    1 1        z F z F z    

can be done by 

power series. First, we write 

  0

z a A Z  

, i.e., the dc and ac parts of z are formally separated. 

Then, assuming that the function 

A Z 

is small compared to the dc component 

0

a

, we have the 

following Taylor series expansion:

 

                 

           

0

1 1 2 3 ' '' ''' 0 0 0 0 0 2 6

1 1 1 4 5 6

0 0 0 24 120 720

Z f z

z a A Z f z f a A Z f a A Z f a A Z f a

iv v vi A Z f a A Z f a A Z f a

    

      

   

The derivatives of 

f z 

are given in the appendix, which also contains mathematical details of 

the calculation. Expanding the function 

A Z 

as a power series in Z with unknown coefficients 
15

and matching coefficients on both sides of the expression 

  0

   Z f z

leads to an expression 

for the function 

z

in powers of 



, which can then be rewritten as a Fourier series for the 

charge density of the form

 

6

cos

0

z c m t

m

m

    



.

This in turn can be inserted into the charge density expression to give the varactor response:

     

6 1

cos cos 0 1 0 1 2C C c c t c m t gt gb q m

m

 

                   

       

The lowest-frequency nonlinear effect is a shift in the operating point—i.e., the self-biasing 

effect. The ―true" ac nonlinearities start with second-harmonic generation, as measured by the 

coefficient 

2

c .

Figures 6a–6e illustrate the variation of the induced surface charge density versus ac voltage at 

the second through sixth harmonic frequencies for an operating-point voltage 

0

V 

2 V. It is 

worth noting that the charges at the even harmonics are considerable larger than those at the odd 

harmonics ̧ despite the antisymmetrical shape of the overall charge-voltage curve. The 

explanation for this is the peculiar shape of that curve at large voltages, which is asymptotically 

quadratic for both signs of the voltage, but with a negative coefficient in the negative direction 

and a positive coefficient in the positive direction to ensure the antisymmetry. As to the numbers, 

since current density is the time derivative of the charge density and the harmonic densities are

measured in nanocoulombs per square centimeter, we can cautiously infer that a 

2

1 cm

capacitor 

driven at a frequency of 10 MHz could generate milliamps of ac current.

Figure 6a. 2nd harmonic surface charge density vs. ac voltage.
16

Figure 6b. 3nd harmonic surface charge density vs. ac voltage.

Figure 6c. 4th harmonic surface charge density vs. ac voltage.
17

Figure 6d. 5th harmonic surface charge density vs. ac voltage.

Figure 6e. 6th harmonic surface charge density vs. ac voltage.
18

6. Conclusions

The graphene system possesses four unusual physical features:

 two-dimensionality

 ambipolarity

 zero band gap

 a conical 2-D band structure

all of which impact the response of any device based on graphene. For this reason, analogies 

with other devices without all of these features can be extremely misleading. Thus, treating the 

graphene as if it were 3-D alters the quantum statistics, while the conical shape of the band 

structure (which can occur in 3-D systems, e.g., InSb) has specific implications for the energy 

density of states and, therefore, the charge-voltage relation. The unusual feature of ambipolarity 

complicates the discussion even more.

The analysis given here is of limited value as a guide to the time-dependent behavior of the 

device because of its quasi-dc nature: since changes in charge density cannot be instantaneous, 

but rather are limited by the RC time constant of the graphene material, the real device response 

must include discussion of the mobility of carriers in the channel, which is known to be 

determined by substrate material (i.e., the cladding dielectrics in this capacitor structure), as well 

as defect and phonon scattering in the graphene layer itself.

A further limitation on the varactor analysis is the power-series treatment of the charge-voltage 

relation. Fermi integrals have always been notoriously hard to deal with mathematically: precomputer calculations tended to founder on the inability to match power series and asymptotic 

series for these integrals due to the ―elbow bend‖ that is so apparent in their plots. This means 

that the nonlinear response of a device whose physics requires their use will only be welldescribed by global computations. Such computations will be the subject of future research.
19

7. References

1. Sze, S. M. Physics of Semiconductor Devices; Wiley-Interscience, New York: Wiley, 1981, 

2nd ed.

2. Crowne, F. Classical Gradual-Channel Modeling of Graphene Field-Effect Transistors 

(FETs); ARL-TR-5281; U.S. Army Research Laboratory: Adelphi, MD 20783, August 

2010.

3. Meric, I.; Han, M.; Young, A. F.; Ozyilmaz, B.; Kim, P.; Shepard, K. L. Nature 

Nanotechnology (Letters) 2008, 3, 654.

4. Meric, I.; Han, M.; Young, A. F.; Ozyilmaz, B.; Kim, P.; Shepard, K. L. Nature 

Nanotechnology (Letters) 2008, 3, 654.

5. Meric, I.; Baklitskaya, N.; Kim, P.; Shepard, K. L. RF Performance of Top-gated, Zerobandgap Graphene Field-effect Transistors. 2008 IEEE International Electron Devices 

Meeting (IEEE, Piscataway, 2008), paper no. 4796738, pp. 1–4.

6. Adam, S.; Hwang, E. H.; Galitski, V. M.; Das Sarma, S. PNAS November 2007, 104 (47), 

18392.

7. Fang, T.; Konar, A.; Xing, H. L.; Jena, D. Appl. Phys. Lett. 2007, 91, 092109.

8. Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions (Nat. Bureau of 

Standards Applied Mathematics Series 55, US Gov. Printing Office, Wash. DC, 1972), ch. 

22.
20

INTENTIONALLY LEFT BLANK.
21

Appendix. Varactor Calculations

The derivatives of the function 

f z 

are

     

 

 

 

 

 

 

 

 

 

 

 

' 1 ln 1 ln 1

1

''

1

''' 2 2

1

1

2

3

1

2

4 1

2

4

1

3 2 11 11 1

2

5

1

z z f z e e

z

e

f z z

e

z

e

f z

z

e

z

iv z e

f z e

z

e

z z

e e

v z f z e

z

e

z z z

e e e

vi z f z e

z

e



          



 



 





 



 

  



  

 



Let us expand the function 

A Z 

in a power series in Z with unknown coefficients:

 

   

 

 

2 3 4 5 6

1 2 3 4 6 5

2 3 4 5 '

0 0 1 2 3 4 6 0 5

2 1 2 2 3 4 5 '' 1 2 3 4 6 0 5 2

3 1 3 2 3 4 5 ''' 1 2 3 4 6 0 5 6

1 4 2 3 4

1 2 3 4 5 24

A Z a Z a Z a Z a Z a Z a Z

Z f a Z a a Z a Z a Z a Z a Z f a

Z a a Z a Z a Z a Z a Z f a

Z a a Z a Z a Z a Z a Z f a

Z a a Z a Z a Z a Z

      

                

            

            

      

 

 

4

5

6 0

5 1 5 2 3 4 5

1 2 3 4 6 0 5 120

6 1 6 2 3 4 5

1 2 3 4 6 0 5 720

iv a Z f a

v

Z a a Z a Z a Z a Z a Z f a

vi Z a a Z a Z a Z a Z a Z f a

       

            

             
22

We obtain the ―operating point‖ of the varactor (not the dc component of z, due to self-bias 

effects) by numerically solving the implicit relation 

  0 0   f a

, which gives the first term 

0

a

in the dc charge density. We will look for frequency multiplication up to 

6

using this power 

series, i.e., 

   

6

cos cos 0 0 1 0Z t z c m t

m

m

                



where the 

c

m

are unknown amplitudes. Then the charge density is

         

6 1 1

cos cos 0 1 0 1 2C C z C C c c t c m t gt gt gb gb q q m

m

 

                       

        

Collecting powers of Z and matching coefficients on both sides of the expansions (with the help 

of Mathematica) lead to the following coefficient set 

a

m

for the series representation of 

A Z 

out to 6th order in

1



:

 

 

   

 

 

     

 

     

     

1

'

1 1 0 1 '

0

3

1 1 1 2 2 0 ' '' '' 2 0 1 0 2 1 0 2 2 '

0

5

1 1 1 2

3 3 0 ' '' ''' 3 '' ' ''' 3 0 1 2 0 1 0 3 1 0 0 0 6 6 '

0

1 1 1 2 2 5 0 ' '' ''' 4 0 2 1 3 0 1 2 0 0 2 6 24

a f a a

f a

a f a a f a a f a

f a

a f a a a f a a f a a f a f a f a

f a

a f a a a a f a a a f a a f



   

           

 

                    

 

          

 

           

             

 

       

0

7

1 1 3 2 4

15 '' 10 ' '' ''' ' 4 1 0 0 0 0 0 0 24 '

0

1 1 1 2 2 3 5 0 ' '' ''' 5 0 2 3 1 4 0 1 2 1 3 0 1 2 0 1 0 2 6 120

9 4 2

105 '' 105 ' '' ''' 1 5 1 0 0 0 0

5 1 120 '

0

iv

a

iv a f a f a f a f a f a f a

f a

iv v a f a a a a a f a a a a a f a a a f a a f a

f a f a f a f a

a

f a

                   

      

    

    

 

 

   

         

2 2

10 ' ''' 0 0

2 3

15 ' '' ' 0 0 0 0 0

f a f a

iv v f a f a f a f a f a

 

 

   
23

     

     

 

       

 

1 1 1 2 3 2 0 ' '' ''' 6 0 3 2 4 1 0 2 1 2 3 1 4 0 5 2 6 2

1 1 1 1 1 2 2 2 3 4 6

1 4 1 2 1 3 0 1 2 0 1 0 2 4 6 24 720

5 3

945 '' 1260 ' '' ''' 0 0 0 0

11 2

280 ' 1 6 1 0

6 1 720 '

0

a f a a a a a a f a a a a a a a f a

iv v vi a a a a a a f a a a f a a f a

f a f a f a f a

f a f

a

f a

   

              

 

        



   

     

     

         

           

   

2 2 2

'' ''' 210 ' '' 0 0 0 0 0

3 3

35 ' ''' 21 ' '' 0 0 0 0 0 0

4

'

0 0

iv a f a f a f a f a

iv v f a f a f a f a f a f a

vi f a f a

 

 



 



 

If the dimensionless voltage Z is a single frequency sinusoid, powers of Z are expressible as 

harmonics of the fundamental frequency via the relation

cos cos n t T n t    n  

where 

T x n  

are Chebyshev polynomials of the first kind, the first seven of which are (8)

 

 

 

 

 

 

 

1

0

1

2

2 1 2

3

4 3 3

4 2 8 8 1 4

5 3 16 20 5 5

6 4 2 32 48 18 1 6

T x

T x x

T x x

T x x x

T x x x

T x x x x

T x x x x





 

 

  

  

   

Inverting these equations allows us to express simple powers as sums of Chebyshev 

polynomials:
24

 

 

   

   

     

     

       

1

0

1

2 1 1

0 2 2 2

3 3 1

1 3 4 4

4 3 1 1

0 2 4 8 2 8

5 5 5 1

1 3 5 8 16 16

6 5 15 3 1

0 2 4 6 16 32 16 32

T x

x T x

x T x T x

x T x T x

x T x T x T x

x T x T x T x

x T x T x T x T x





 

 

  

  

   

Then

 

 

1

2

2 3 4 5 6 3

0 1 2 3 4 6 0 1 2 3 4 6 5 5

4

5

6

1 0 0 0 0 0 0

0 1 0 0 0 0 0

1 1 0 0 0 0 0

2 2

3 1 0 0 0 0 0

4 4

0 1 2 3 4 6 5

3 1 1 0 0 0 0

8 2 8

5 5 1 0 0 0

8 16 16

5 15 3 1 0 0 0

16 32 16 32

x

x

a a x a x a x a x a x a x a a a a a a a x

x

x

x

a a a a a a a

   

 

 

 

 

        

 

 

 

     

   







 

0

1

2

3

4

5

6

T

T

T

T

T

T

T

   

 

 

 

 

 

 

 

 

      



and so
25

 

1 0 0 0 0 0 0

0 1 0 0 0 0 0

1 1 0 0 0 0 0

2 2

3 1 0 0 0 0 0

4 4

0 1 2 3 4 6 5

3 1 1 0 0 0 0

8 2 8

5 5 1 0 0 0 0

8 16 16

5 15 3 1 0 0 0

16 32 16 32

1 3 5 3 5 1 1 15 1 5 1 3 1 1

0 2 4 6 1 3 2 4 6 3 4 6 6 5 5 5 2 8 16 4 8 2 2 32 4 16 8 16 16 32

a a a a a a a

a a a a a a a a a a a a a a a a

   

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

          

 

defines the ―conversion coefficients‖ for harmonic generation amplitudes:

1 3 5

0 0 2 4 6 2 8 16

3 5

1 1 3 5 4 8

1 1 15

2 2 4 6 2 2 32

1 5

3 3 5 4 16

1 3

4 4 6 8 16

1

5 5 16

1

6 6 32

c a a a a

c a a a

c a a a

c a a

c a a

c a

c a

   

  

  

 

 





For the dc component and linear susceptance we find explicitly that

       

   

 

1 3 5

0 0 0 0 0 1 0 1 0 0 2 4 6 0 0 2 8 16

1 3 5

1 0 1 0 2 4 6 2 8 16

0 0 ln 1 ln 1

3 5 1 3 5 1

1 1 1 3 1 3 1 5 5 4 8 4 8 '

0 0 1 ln 1

f a c f a c a F a F a a a a a

F a F a a a a

a a

e e

c a a a a a

f a a

e

                    

          

      

                                             

     



 

3 5

1 3 5 4 8 0 ln 1

a a

a

e

                      

The frequency multiplication signals are 

 

1

m m C C c gt gb q

  



with m > 1. Then
26      

2

2 2 2

2 2 2

2

2

0

C C

q q q q gt gb

C C C C q gt gt gb gb F F F

q kT cm

m mc

F





    

          

 

      

      

are the ac charge-density components.
27

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