Classical Gradual-Channel Modeling of Graphene FieldEffect Transistors (FETs) 

by Frank Crowne 

ARL-TR-5281 August 2010 

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ARL-TR-5281 August 2010 

Classical Gradual-Channel Modeling of Graphene FieldEffect Transistors (FETs) 

Frank Crowne 

Sensors and Electron Devices Directorate, ARL 

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14. ABSTRACT 

This technical report describes initial efforts, as part of the new Strategic Technology Initiative (STI) on carbon electronics, to 

model and simulate the performance of graphene field-effect transistors (FETs) using macroscopic descriptions that are 

classical for semiconductor devices. It is argued that the underlying physics that differentiates these devices from their normal 

semiconductor-based counterparts is most clearly revealed by non-computer-intensive descriptions that allow the designer to 

compare their behavior with that of their well-studied semiconductor counterparts. Because it admits a reasonable description of 

both the lateral and vertical field and transport functionality of the FET structure, the gradual-channel approximation is key to 

this approach. The availability of closed-form solutions to the problem within this approach, in turn, allows small-signal and 

microwave parameters to be calculated quickly and transparently. 

15. SUBJECT TERMS

Graphene, FET, gradual channel, I-V curve, stability 

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Frank Crowne 

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iii

Contents 

List of Figures iv

1. Introduction 1

2. Description of Gradual-Channel Modeling 2

3. Modeling Methodology 4

3.1 Channel Carrier Density ..................................................................................................4

3.2 Drift Velocity ..................................................................................................................6

3.2.1 The “Simulator” Model .......................................................................................6

3.2.2 Electron Heating Due to Phonons .......................................................................7

4. Gradual-Channel Models for Graphene 8

4.1 Modeling by Meric et al. .................................................................................................9

4.2 Corrected Version of Simulator Model .........................................................................16

5. Conclusions 24

7. References 25

Appendix A. Derivation of I-V Curves From Velocity-Field Relations 27

Appendix B. Stability of Meric’s I-V Curve Without Access Resistance 33

Appendix C. Small-Signal Parameters From Meric’s I-V Curve Without Access Resistance37

Appendix D. Stability of Meric’s I-V Curve With Access Resistance 39

Appendix E. Small-Signal Parameters From Meric’s I-V Curve With Access Resistance 45

Appendix F. Modeling of Graphene FETS Using Meric et al. With Corrections (No Access 

Resistance) 51

Appendix G. Evaluation of Integrals in Electrical Parameters of Graphene FETS Using 

The Corrected Model of Meric et al. (No Access Resistance) 57
iv

Appendix H. Small Signal Parameters for Graphene FETS Using the Corrected Model of 

Meric et al. (No Access Resistance) 61

List of Symbols, Abbreviations, and Acronyms 67

Distribution List 68 
iv

List of Figures 

Figure 1. Typical FET geometry (gallium arsenide [GaAs] MESFET). .........................................2

Figure 2. Relation between internal potential and terminal voltages. The red line indicates the 

FET channel, the blue line the gate. ...........................................................................................3

Figure 3. Lossy transmission line model of FET channel. Resistors I R are functions of the 

local potential difference between channel and gate. ................................................................4

Figure 4. Dual-gate graphene FET structure of Meric et al. ............................................................9

Figure 5. (a) ID vs. Vds for the simulator model and (b) ID vs. Vgst for the simulator model. .........13

Figure 6. (a) sd vs. Vds for the simulator model and (b) |gm| vs. Vds for the simulator model. .......13

Figure 7. (a) ID vs. Vds for the simulator model with access resistances and (b) ID vs. Vgst for 

the simulator model with access resistances. ...........................................................................15

Figure 8. (a) sd vs. Vds for the simulator model and (b) |gm| vs. Vds for the simulator model. ........16

Figure 9. (a) ID vs. Vds in the corrected simulator model and (b) ID vs. Vgst in the corrected 

simulator model. ......................................................................................................................18

Figure 10. ID vs. Vds for Vgst = 3.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................19

Figure 11. ID vs. Vds for Vgst = 3.58 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................19

Figure 12. ID vs. Vds for Vgst = 4.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................20

Figure 13. ID vs. Vds for Vgst = 4.58 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................20

Figure 14. ID vs. Vds for Vgst = 5.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................20

Figure 15. ID vs. Vds for Vgst = 5.58 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................21

Figure 16. ID vs. Vds for Vgst = 7.58 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................21

Figure 17. gm vs. Vds for Vgst = 3.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................22

Figure 18. gm vs. Vds for Vgst = 3.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................22

Figure 19. gm vs. Vds for Vgst = 4.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................22

Figure 20. sd vs. VD for Vgst = 3.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................23
v

Figure 21. sD vs. Vds for Vgst = 3.58 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................23

Figure 22. sD vs. Vds for Vgst = 4.18 V, blue curve-uncorrected model, red curve-corrected 

model........................................................................................................................................24

Figure A-1. Plots of the functions f  x x   and g  x a bx    for (a) a = 3, b = 2 and a = 3, b = 

0.4 and (b) a = 3, b = 2 and a = 3, b = 0.4. ....................................................................29
1

1. Introduction 

Over the past 60 years, a large number of papers have been written about the properties of 

graphene (1), due not only to its defining characteristic as the ultimate two-dimensional (2-D) 

electronic system but also to its unique band structure and the strongly covalent features of its 

atomic bonding, properties that it inherits from the semimetal graphite. Indeed, the earliest 

papers on graphene (2) treat the material as an idealized version of graphite, i.e., a perfect 

monolayer of carbon with a hexagonal graphite structure. Despite the apparent simplicity of this 

“single-layer graphite” model, the picture of graphene that evolved from these papers is one of 

extreme complexity. This is because the relative simplicity of its constituents―atoms of a lightatomic-number element, carbon, in a simple high-symmetry lattice―favors the application of a 

high-powered many-body analysis of the electronic degrees of freedom, which, in turn, predicts 

ferromagnetism, superconductivity, charge-density waves, quantum Hall effects, and other even 

more novel phenomena. However, in view of the extreme difficulty of creating such monolayer 

structures, many of the theoretical results obtained from this extensive research have been 

considered irrelevant by technologists. As has often happened in the past, experimental advances 

have now overcome the fabrication difficulties to the point that the reality of graphene as an 

object of laboratory study is beyond question, and so these theoretical predictions can be put to 

the test. 

On the other hand, the technological implications of graphene’s physics―its monatomic nature, 

excellent electron/hole transport properties, and mechanical strength among others―have driven 

an effort to create devices that (it is claimed) could rival silicon-based transistors, and possibly 

harness the various exotic properties of the material that have been foreseen by theorists. At this 

time, there are still formidable obstacles to making graphene devices and circuits on the scale of 

those based on any standard semiconductor-based material, let alone highly evolved silicon 

devices. However, interest in graphene devices has now reached a point where models of such 

devices are needed that describe the terminal characteristics of circuit components without the 

need to track the interior details of the device function. The availability of fast computer 

simulations of ordinary semiconductor devices has motivated efforts to incorporate such details 

in the device computations, but the problem of long running times for codes based on these 

simulations has not been solved. 

In this report, I argue that, rather than imitating those models of standard semiconductor-based 

devices that incorporate massive ab initio computer calculations―metal-oxide semiconductor 

field-effect transistors (MOSFETs), high electron mobility transistors (HEMTs), metal 

semiconductor field-effect transistors (MESFETs), and their ilk―the development of graphene 

device models should be evolutionary, i.e., the models should first mimic the simplest standard 

models. As a first step in this program, I discuss the simplest semiconductor device―the field-
2

effect transistor (FET)―treated at the lowest level of device modeling, i.e., Shockley’s gradualchannel model. My choice of FETs is driven by ongoing interest in radio frequency (RF) devices 

here at the U.S. Army Research Laboratory (ARL) and the mature understanding of the highfrequency behavior of FETs in general. In the course of this analysis, I highlight places where the 

unique properties of graphene affect the predicted terminal characteristics and S-parameters of a 

possible graphene FET. 

2. Description of Gradual-Channel Modeling 

This type of device modeling, which was developed by Shockley (3) in the pre-computer era, is 

based on highly idealized pictures of the interior fields and currents of the device under study. 

For a FET, the standard geometry is shown in figure 1. The device consists of an insulating 

substrate for electrical isolation and mechanical support; a conducting “active” layer of width h 

whose conductance can be modulated by varying the 2-D charge density and drift velocity of 

carriers in the layer; “source” and “drain” electrodes that contact the active layer; a dielectric or 

depleted non-conducting layer of width d on top of the active layer; and a metallic gate electrode 

on top of this last layer. Like any transistor, the FET has three terminals, but the most common 

configuration for its use is as a circuit two-port, i.e., two of the electrodes (usually the source and 

gate) are connected together. In this configuration, the FET converts an input current-voltage 

(I-V) pair (gate voltage, gate current) into an output I-V pair (drain voltage, drain current). 

Hence, the modeler’s task is to determine two equations for the output electrical quantities, 

referred to as “terminal relations.” 

Figure 1. Typical FET geometry (gallium arsenide [GaAs] MESFET). 

Substrate (undoped)

Source Drain

V=0 V=VD

V=VG

Ohmic

contact

Ohmic

contact

Depletedlayer

Gate

L

ND

ND

+

Undepleted layer

(Channel)

- device active region
3

When voltages are applied to the electrodes, a highly non-uniform electric field exists inside the 

dielectric layer and a current flows within the active layer. The device operation is predicated on 

the degree of control each electrode voltage has over the electric field and current: the charge 

density in the channel is enhanced or suppressed by that “portion” of the electric field controlled 

by the gate electrode, while the drain electrode controls the carrier velocities through its 

“portion.” While the concept of a “portion” is necessarily vague, it can be made more precise by 

introducing mutual capacitances (active and passive) between the electrodes. Because the 

electrodes are asymmetrically placed, it is expected that the current, like the field, is nonuniform. The distribution of potential within the structure is schematically shown in figure 2. 

Figure 2. Relation between internal potential and terminal voltages. The red line indicates the FET channel, the 

blue line the gate. 

In figure 2, V x( ) denotes the voltage at point x along the graphene layer and VV V , , and S G D

denote the voltages on the source, gate, and drain electrodes, respectively. 

The guiding physical principle behind gradual-channel FET modeling is that the current flowing 

from source to drain be one-dimensional. Charge conservation then forces this current to be 

uniform everywhere. When this is so, the constant source-drain current I

D Is given by 

I ZhJ Zqh V E     D    (1) 

where Z is the width of the FET channel, h is its thickness, and the (fluid-mechanical) carrier 

current density J qV E    . In this expression, E is the electric field along the channel, q is 

the electron charge,   E is the drift velocity of the carriers in the field E,   V is the threedimensional (3-D) number density (algebraic) of charge carriers in the channel, and V is the 

potential “associated” with E. For extremely thin channels, it is correct to introduce a 2-D 

density of charge carriers in the channel nV h V      , so that the channel current of the FET is 

I Zq E n V     D   . 

It is clear that the field E and potential V are defined in a markedly inconsistent way. For the 

current to be constant, there must be no forces perpendicular to channel, implying that the field E

must point only from source to drain inside the active layer. However, if this E is the gradient of 

V(0)=VG =VS 

VG 

V x( ) V(L)=VD 

x=0 x x=L
4

some potential V, the latter cannot depend on the voltage applied to the gate electrode, which 

contradicts the ability of the gate to control the channel charge. This difficulty is avoided in the 

gradual-channel model by assuming that the internal electric field changes direction abruptly at 

the boundary between the active layer and the dielectric/depleted layer. The interior field/current 

configuration is most easily represented by the circuit diagram shown in figure 3, which is that of 

a lossy and non-uniform transmission line. 

Figure 3. Lossy transmission line model of FET channel. Resistors I R are functions of the local potential difference 

between channel and gate.

Although this picture of the internal electric field and carrier transport in a FET is fundamentally 

unrealistic, it has proved quite successful in reproducing the FET terminal characteristics, which 

points to considerable insensitivity of these characteristics to the detailed internal device 

dynamics. For this reason, most of the device models described in Sze’s book on semiconductor 

devices (4) start with this model, which makes it an excellent template for comparing graphene 

devices with their better-known counterparts. 

3. Modeling Methodology 

Once the general concepts of the gradual-channel approach are accepted, the physics of the 

device operation are needed, that is, we must specify the functions n V  and   E . The former 

is derived primarily from the geometric structure of the device, the latter from microscopic 

features of the carrier transport. 

3.1 Channel Carrier Density 

Since V varies along the channel, modeling n V  involves the dielectric properties of the device 

interior. The capacitor-transmission line picture suggests that for a dielectric layer between gate 

and active layer, 

        q 1 nV V V nV V V G G th th d qd



   



 (2) 

C C C C C C C C C C C C C C C C C

R R R R R R R R R R10 R11 R12 R13 R14 R15 R16
5

where d is the dielectric layer thickness and Vth is a threshold voltage below which there are no 

carriers in the channel. In the MOSFET literature, this is simply known as the “linear model.” Its 

use implies that the terminal characteristics exhibit no nonlinear behavior with drain voltage, and 

indeed the MOSFET at this level is simply a gate-controlled resistor. In an effort to explain 

current saturation in MOSFETs, Hofstein and Heiman (5) added a variation of n V  with the 

channel voltage V: 

        q 1 nV V V V nV V V V G G th th d qd



     



 (3) 

Within the gradual channel approximation this assumption leads to the I-V curve 

0 1 2

2

Z

I V VV V D Ld G th D D

              (4) 

Here we note the appearance of a rather embarrassing feature of the gradual-channel model: for 

values of VD greater than V V G th  , the current I

D decreases and eventually changes sign at 

2  V V G th  . Because such a decrease in current with drain voltage under DC conditions is a sign 

of thermodynamic instability, its appearance here must be unphysical. It is tempting to assume 

that it arises solely from the poor description of the internal field in the gradual channel model, 

which should be fully 2-D, i.e., a solution of the 2-D Poisson equation; however, at least one of 

the many gradual channel models for MOSFETs―the Pao-Sah model (6)―avoids this problem 

completely without introducing 2-D effects. In this model, the functional form of n V  is quite 

complicated: for a p – MOSFET with semiconductor 3-D bulk charge densities 0 pp (majority) 

and 0 n

p (minority), 

 

 

 

 

1 2

2 , , 0

1/2

, , 1 1

V e d nV n q L p D F V

s

V V FV e e e e

         

                          

 (5) 

where 1

k TB

  , 0

0

n

p

pp

  , 2 L

D q b

  

 is the Debye length for the bulk material, 

0 p b p   is the majority-carrier bulk charge density, and s  is the surface potential, which is 

evaluated from the gate voltage equation: 
6

1/2

2

1 1 s s s V V VV e e e e G s FB s s q LD

                             

 (6) 

Since its introduction in 1966, this model has regarded by designers of silicon-based device as 

too awkward and time-consuming to use. However, from the standpoint of designing graphenebased devices, its formulation includes two interesting features: in-channel ambipolarity, i.e., 

majority and minority carriers switch places within a pinched-off channel; and diffusion, which 

may be the key to its success in avoiding the need for fully 2-D computations. Unfortunately, the 

treatment of carriers in the channel is based on the Boltzmann equation for a gas of classical 

charged particles, which makes it unsuitable for the analysis of a truly 2-D channel, e.g., that of a 

HEMT. 

3.2 Drift Velocity 

At low electric fields, the drift velocity E of current carriers in a semiconductor can easily be 

derived from Ohm’s law, i.e.,   E E , where the mobility  is taken to be a material constant. 

However, because most useful devices, including FETs, must operate under large-signal 

conditions, this linear result must be generalized to include large fields. One way to do this is to 

write   E E E , where   E is derived from transport theory in bulk material. Because the 

derivation of E is a difficult problem even in this case, it is common practice to specify it 

empirically (7). From the standpoint of device design, the most important requirement for such 

an empirical description is that   E incorporate the experimentally observed saturation of the 

carrier drift velocity   E in most semiconductor materials. For graphene FETs, this exercise is 

highly useful. Several types of drift velocity relations are considered here, among them 

3.2.1 The “Simulator” Model 

  0

0 1

E

E

E

sat



 







 (7) 

This expression (7) is commonly used for silicon (Si) devices and was in fact used recently by 

Meric et al. (8) to model graphene. In appendix A, it is shown that it leads to the I-V relation 

  0

0

0 1

V Zq D

n V dV

L

I

D

V

L D

sat





 





 (8) 
7

Using the MOSFET linear model for the channel, we can find the I-V relation explicitly. 

    1 2

2 0 0

0 1 2

2

0 1

V V D D

n V dV V V dV V V V

qd qd G G D D

Z

VV V

Ld G D D

I

D

V

L D

sat

            

      

   







 (9) 

Like the MOSFET model of Hofstein and Heiman, this model exhibits the problem that for 

values of VD greater than VG the current I

D decreases, and eventually changes sign at 2VG . 

This shows that the introduction of velocity saturation is not an automatic cure for the problem of 

thermodynamic instability. 

The expression for   E given previously is often used in numerical simulations, but its 

empirical nature suggests that is has very little physical content. Device models based on hotelectron transport, of which we will discuss two, are much more interesting: 

3.2.2 Electron Heating Due to Phonons 

The following velocity-field relation, which is derived from inelastic scattering by optical 

phonons in Conwell (9), is occasionally used in more detailed numerical simulations: 

  0

2

0 1

E

E

E

sat



 

  

     

 

 (10) 

A derivation of the I-V curve that corresponds to this expression is given in appendix A . This 

expression differs markedly from the “simulator” result: 

2

2 ( ) 0 0

VD I

D I L Zq n V dV D Zq sat

 

         

 (11) 

A related but slightly different velocity-field relation, derived from inelastic scattering by 

acoustic phonons: 
8

  0

2

1 1 0

2 2

E

E

E

sat



 

  

      

 

 (12) 

Despite the similarity of equations 11 and 12, the latter gives a rather strange result for the I-V 

curve: 

 

   

2

1 2 0 4

2 0 0 0

2 0

V V D D dV I L n V dV L D V n V D dV sat

n V Zq sat

                    

                



 (13) 

These two expressions will be discussed in a separate report. 

4. Gradual-Channel Models for Graphene 

With regard to device modeling in general, there are four peculiarities of graphene that 

distinguish it from bulk materials: 

1. two-dimensionality 

2. ambipolarity 

3. zero band gap 

4. a conical 2-D band structure 

In this section of the report, I examine the impact of these features on three kinds of FET models, 

which differ in the way velocity saturation is modeled. My analysis focuses on the stability of the 

I-V curves (10), which is an important criterion for the usefulness of any FET model. I define a 

DC I-V curve as stable if it has no regions where drain current decreases with drain voltage. 

Stated another way, a stable DC I-V curve should exhibit no regions with negative AC output 

conductance. Since this parameter, along with the transconductance, are needed to specify the 

full low-frequency AC response of the FET and the AC equivalent circuit, stability of the I-V 

curve impacts predictions of microwave performance, i.e., the S-parameters of the device. 
9

4.1 Modeling by Meric et al. 

The first step to modeling a real FET structure is to incorporate a varying channel voltage into 

the expression for the channel charge density n V  . In their paper (8), Meric et al. provide such 

an expression and use it in their initial effort to model the microwave response of a graphene 

FET. Because the authors wanted to study the role of ambipolarity in the functioning of their 

device, they equipped it with both a top and a back gate (figure 4), where the latter is used to 

“dope” the FET by electrostatically generating either n- or p-type graphene concentrations 

analogous to MOSFET channels. In what follows, I will discuss this FET model using data from 

their paper, denoting voltages on the top and back gates by Vgst and Vgsb, respectively. 

Figure 4. Dual-gate graphene FET structure of Meric et al. 

In providing an expression for the channel carrier density n V  , Meric et al. incorporate an 

additional peculiarity exhibited by graphene films: the existence of a “minimum conductivity” in 

the graphene channel, due apparently to ambipolarity. They do this by assuming that the 

minimum conductivity corresponds to a minimum carrier density 0n , and then using the 

following (empirical) expression for the function n V  , which is now the total number of mobile 

carriers: 

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
10

    

 

2 2 2 / 0

0 0

nV n C q V V V gst gst th

Cgsb VV V V gst th gsb gsb Cgst

  

 

 (14) 

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, but this is legitimate if electrons and holes have identical 

mobilities, which may be the case in graphene. The expression contains as fitting parameters two 

threshold gate voltages 0 0 V V , gsb gst at which the minimum carrier density 0

n is reached as the 

back and top gates are varied separately. The design numbers for this FET are 

2.1 m

1.0 m

2 cm 550 0 V s

8 1. 10 cm/s

11 2 5 10 cm 0

Z

L

F

n

 

 

 

 

  

 

2 552 nF/cm

0 1.45 V

2 12 nF/cm

0 2.7 V

Cgst

Vgt

Cgsb

Vgb









 (15) 

where Z is the top gate width, L its length, 0 the channel mobility, F the Fermi velocity for 

an infinite graphene sheet, 0

n the minimum channel density, Cgst the capacitance between the 

channel and top gate, and C

gsb the capacitance between the channel and back gate. Using these 

numbers and a back gate voltage of  40 V, they arrive at a threshold voltage for the channel of 

V 2.38 V th  . 

To model velocity saturation, they use the simulator expression, which leads to the following 

formula for I

D derived in appendix A: 

0 ( )

0

0 1

V Zq ds

n V dV

L

I

D

V

L ds

sat





 





 (16) 
11

Although they have written several papers relating to this modeling effort, Meric et al. are 

unclear as to the value of the saturation velocity sat  , although they mention the number 

7 5.5 10 cm/s  (8). They justify their choice of the simulator expression (equation 16) implicitly 

by referencing to reference 7, but presumably simplicity was a factor as well. However, because 

this expression was developed for bulk-semiconductor devices, its use for graphene is open to 

criticism. One consequence of its use is the prediction of I-V curves that have a point of 

inflection, or “kink,” when the biases drive the Dirac point into the channel. Their experimental 

curves do indeed show such a kink; however, the extent to which the latter can be explained by a 

gradual-channel model needs testing. 

Because the expression for the I-V curves is explicit, it is easy to apply a stability test to it by 

computing the differential output conductance. From appendix B, we find that 

   

 

   

 

0 0 1

0

2

0 1

0

0 0 1

0

2

0 1

Vds

V n V n V dV

L L ds ds

sat sat Vds

V

L ds dI Zq D sat

dV L V ds ds

V n V n V dV

L L ds ds

sat sat Vds

V

L ds

sat

            

    

     

       

                  

     

    

 (17) 

where I

D is given by equation 16. In appendix B, it is shown that the condition that this quantity 

always be positive is that 

 

 

0

0

V Zq n V I ds ds sat D

V Zq n V I ds ds sat D

  

   

 (18) 

It is clear that if this inequality is violated, the violation will occur within a certain voltage 

interval   V V  ,   bounded by points where 0

dID

dVds V V ds



 

, points that mark a peak and 

an adjacent trough in the S-shaped I-V curve. As the gate voltage is varied, this interval shrinks 
12

to zero, at which point = V V   . Denoting the common value of V and V by V

c , we see that 

two equations 

0

dID

dVds V V ds c





 (19) 

2

0 2

d ID

dVds V V ds c





 (20) 

are satisfied there simultaneously, so that the curve becomes almost flat. The second equation is 

the condition for a point of inflection in the I-V curve, i.e., a “kink.” It should be emphasized 

that equation 19 and equation 18 can hold at the same time; indeed, these conditions are required 

for the presence of a physically acceptable kink. If V V    there will be two points where 

0

dID

dVds

 . 

Let us demonstrate this for the expression used in reference 8. Since 

   

2

2

1 0

0

Cgst nV n V V V ds ds th gst n q

 

        

, (21) 

we find explicitly that 

  0 cosh sinh cosh sinh

2 0

VD n

n V dV gs gs gs gd gd gd

           

     (22) 

where 

0

Cgst

n q

  and 

    1 1

gd ds th th sinh sinh V VV V V gst gst gs           (23) 

Figure 5 shows plots of I

D versus Vds and Vgst for three values of top-gate voltage for the 

transistor specified by the numbers given in reference 8 and listed above, along with a (rather 

low) value of 7 2.5 10 cm / s  for sat  . For this transistor the threshold top-gate voltage for 

negative conductance is 3.58 V. The blue curves are below the threshold voltage for negative 
13

conductance, and show a kink with positive slope; the red curves are for a device gated at 

threshold, where the kink has zero slope; and the yellow curves are for a device gated above 

threshold and have a region of negative conductance. The output conductance 

dID s

d dVds

 and 

transconductance 

dID g

m dV

gst

 are shown in figure 6; the absolute value of g

m is plotted in 

accordance with reference 8. 

 (a) (b) 

Figure 5. (a) ID vs. Vds for the simulator model and (b) ID vs. Vgst for the simulator model.

 (a) (b) 

Figure 6. (a) sd vs. Vds for the simulator model and (b) |gm| vs. Vds for the simulator model. 

For the real devices Meric et al. investigated, it was necessary to include series “access” 

resistances associated with the source and drain contacts, leading to the following modified I-V 

expression (11): 
14

  0

0 1 2

V RI Zq ds s D

n V dV

L R Is D I

D

V RI

L ds s D

sat

 



 

 



 (24) 

which is now an implicit equation for the current. From appendix C, we find that 

 

      

 

      

2

0 0 1 4

0

2

0 0 1 4

Zq n V R I I sat ds s D D V RI Zq ds s D

V RI R nV RI n RI dI L L ds s D s ds s D s D

D sat

dV L ds sat Zq n V R I I sat ds s D D V RI Zq ds s D

V RI R nV RI nRI

L L ds s D s ds s D s D

sat

                   

            

                    

       (25) 

Then the condition that this quantity always be positive is that 

 

 

2

2

V R I Zq n V R I I ds ds sat sD sD D

V R I Zq n V R I I ds ds sat sD sD D

   

    

 (26) 

which clearly reduces to equation 18 for no access resistances. 

Using the expression introduced in reference 8: 

  2 2

1 0 nV RI n V V V RI ds ds th gst sD sD       , (27) 

we again find explicitly that 
15

 

  

  

1 cosh sinh cosh sinh 0 2

2 2 1 cosh 0 0

2 2 1 cosh 0 0

V RI ds s D

n V dV n gss gss gss gds gds gds R Is D

nV RI n V V RI V n ds ds th gds gst sD sD

nRI n V RI V n gst gss sD sD th



           

    

       

     

 (28) 

and 

    1 1 sinh sinh V V RI V V RI V gst gst gss gds ds th th sD sD

           (29) 

Figure 7 again shows plots of I

D versus V

ds and V

gst for the same three values of top-gate 

voltage, for the same transistor and numbers given above, with a value of 700  for the access 

resistance R

s

. Since the expression for I

D involves an implicit function for nonzero R

s

, 

evaluation of I

D requires a numerical calculation; once it is computed, however, it can be 

inserted into the expressions for

dID

dVds

. The output conductance 

dID s

d dVds

 and 

transconductance 

dID g

m dV

gst

 are shown in figure 8. 

 

 (a) (b) 

Figure 7. (a) ID vs. Vds for the simulator model with access resistances and (b) ID vs. Vgst for the simulator model with 

access resistances. 
16

 (a) (b) 

Figure 8. (a) sd vs. Vds for the simulator model and (b) |gm| vs. Vds for the simulator model. 

Let us compare the results of this calculation with the results for no access resistance. Note that 

the overall current output is suppressed, which is unsurprising; in addition, the conductance is 

much larger, a more obscure consequence of the additional 1400  of access resistance. The 

threshold top gate voltage for kink formation is found to be 4.01 V, an increase over the value for 

no access resistance. Figure 8(b) shows curves similar to the experimental curves given in 

reference 12, including curvature to the left of the peak that is absent from the curves for 

transistors without access resistance. 

4.2 Corrected Version of Simulator Model 

The authors of reference 12 mention (and then seemingly discard) an interesting feature of 

carrier transport in graphene: the saturation velocity appears to be a function of carrier density, a 

phenomenon that has never been reported in ordinary FETs. To incorporate this effect, they 

make use of a semi-empirical dependence   n sat F n

         

, where  and  are fitting 

parameters (=0.07). The use of this function in simulations is problematic, since it allows sat 

to become negative. The proper form of the I-V curve with this feature included is derived in 

appendix B, where we also replace the semi-empirical dependence with the following more 

reasonable approximation: since 

lim 1

0

1

n

n n n n n n n

     

                      

, (30) 

we can replace   n sat F n

          by  

1

F n sat

n n



    

     

 with little error. This 

procedure gives us a well-behaved dependence on n everywhere except at very low carrier 

densities. 
17

The derivation given in appendix A is correct until we reach equation A-4. In order to include the 

density dependence of the saturation velocity, we need the expression 

   

0 ( )

0

0 1

0

V Zq ds

n V dV

L

I

D Vds dV

L n V sat









 



 (31) 

Using the “repaired” expression  

1

F n sat

n n



    

     

 gives 

0

1

0 1 1 2

Zq

H

L I

D

H H

L F



  

 

 

 (32) 

where 

  1

0

Vds

H n V dV     2

0

Vds

H n V dV   (33) 

Since    2 2 1 0 nV n V V V gst th     , we can evaluate these integrals. The first one was 

evaluated in the previous section: 

   2 2 1 1 0

0 0

1 2 cosh sinh cosh sinh 0 2

V V ds ds

H n V dV n V V V dV gst th

q n

C gs gs gs gd gd gd gst

      

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

 (34) 
18

where     1 1 sinh , sinh V V V VV gst gst gs gd th th ds           . The second integral is 

more challenging, but it too can be evaluated in closed form: 

   

1/4 2 1/2 2 1 2 0 0 0

sinh cosh sinh cosh

1 3/2

3 0 1 11 1 1 1 cn sech | cn sech |

2 2 2 2

V V ds ds

H n V dV n V V V dV gst th

gs gs gd gd q n

Cgst gs gd

            

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

 (35) 

where 1 1 cn | 2

x        is the inverse function of the Jacobi elliptic cosine 1 cn | 2x

     . 

For the stability check, we take the same derivative as before: 

                0

0 0 1 1 12 12

I I D D nV nV nV nV nV nV dI Zq ds ds ds ds ds ds Zq Zq DF F V V ds ds dV L ds HH HH

L L F F

           

                                    

 (36) 

If we calculate the I-V curves of a graphene transistor using the parameters of Meric et al. and 

the corrected model, we get the curves shown in figure 9. 

 (a) (b) 

Figure 9. (a) ID vs. Vds in the corrected simulator model and (b) ID vs. Vgst in the corrected simulator model. 
19

We find that while a kink is still present, there is no trace of negative resistance in the voltage 

intervals shown. This feature of the corrected model is easier to see from a detailed comparison 

of the I-V curves shown in figures 10–16, where the corrected model predicts a lower current at a 

given voltage as the top-gate bias increases, with negligible negative-slope tendencies. Further 

computations indicate that the onset of an unphysical region in the I-V curve is postponed to a 

higher voltage (5.21 V) for the corrected model, but the actual negative slope is still extremely 

small for voltages as high as 7.58 V (figure 8a). This indicates that the corrected version is a 

more reliable guide to the transistor behavior over the range of drain and gate voltages of 

interest.

Figure 10. ID vs. Vds for Vgst = 3.18 V, blue curve-uncorrected model, red 

curve-corrected model.

Figure 11. ID vs. Vds for Vgst = 3.58 V, blue curve-uncorrected model, red 

curve-corrected model. 
20

Figure 12. ID vs. Vds for Vgst = 4.18 V, blue curve-uncorrected model, red 

curve-corrected model 

Figure 13. ID vs. Vds for Vgst = 4.58 V, blue curve-uncorrected model, red 

curve-corrected model. 

 

Figure 14. ID vs. Vds for Vgst = 5.18 V, blue curve-uncorrected model, red 

curve-corrected model. 
21

Figure 15. ID vs. Vds for Vgst = 5.58 V, blue curve-uncorrected model, red 

curve-corrected model. 

 

Figure 16. ID vs. Vds for Vgst = 7.58 V, blue curve-uncorrected model, red 

curve-corrected model. 

Also of interest are the output conductance and transconductance predicted by the corrected 

model compared to those of the uncorrected model, which are shown in figures 17–22. For the 

transconductance the difference between the two models is not large, but the suppression of 

negative conductance is clearly seen in the output conductance for the corrected model. The 

more rapid decrease in the conductance away from the origin for the corrected model is also 

apparent. 
22

Figure 17. gm vs. Vds for Vgst = 3.18 V, blue curve-uncorrected model, red 

curve-corrected model. 

 

Figure 18. gm vs. Vds for Vgst = 3.58 V, blue curve-uncorrected model, red 

curve-corrected model. 

Figure 19. gm vs. Vds for Vgst = 4.18 V, blue curve-uncorrected model, red 

curve-corrected model. 
23

Figure 20. sd vs. VD for Vgst = 3.18 V, blue curve-uncorrected model, red 

curve-corrected model. 

Figure 21. sD vs. Vds for Vgst = 3.58 V, blue curve-uncorrected model, red 

curve-corrected model. 
24

Figure 22. sD vs. Vds for Vgst = 4.18 V, blue curve-uncorrected model, red 

curve-corrected model. 

5. Conclusions 

It is interesting that the simulator model used by Meric et al. produces a negative resistance as 

soon as the channel current exceeds the maximum allowable current Zq n V sat   ds  at the drain 

end of the device. In standard two-region FET models, it is customary when this current is 

reached in part of the channel (the “saturated velocity region”) to “pin” the variation of the 

channel density with respect to Vds beyond this point. In such “two-region” models (13), 

increasing Vds then increases the portion of the channel where the velocity is saturated, which 

increases the current slowly, and the device is defined to be “pinched off” once the entire 

channel is saturated and the current is a function only of the gate voltage. The saturation current 

that flows is referred to as I

dss. Two-region models do not exhibit negative resistance, but this is 

because of the ad-hoc division of the channel into segments. 

Thanks to the papers by Meric et al., it is clear that simple modeling of the DC and microwave 

responses of graphene FETs is feasible. It is, however, important to choose a model that includes 

as many of the peculiarities of graphene as possible, in order to avoid a misleading picture of 

device function leading to false predictions and design errors. This report will be followed by a 

“physics-based” modeling effort, in which the model described here will be “fleshed out” with 

more realistic features. 
25

7. References 

1. See the bibliography of the article by Castro Neto, A. H.; Guinea, F.; Peres, N.M.R.; 

Novoselov, K. S.; Geim, A. K. The Electronic Properties of Graphene. Reviews of Modern 

Physics 2009, 81, 109. 

2. Wallace, P. R. Phys. Rev. 1947, 71, 622. 

3. Shockley, W. A Unipolar Field-effect Transistor. Proc. IRE 1952, 40, 1365–1376. 

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

2nd ed. 

5. Hofstein, S. R.; Heiman, F. P. Proc. IEEE 1963, 51, 1190. 

6. Pao, H. C.; Sah, C. T. Solid-State Electronics 1966, 9, 927; IEEE Trans. Electron Devices

1966, 13, 393. 

7. Canali, C.; Majni, G.; Minder, R.; Ottaviani, G. IEEE Trans. Electron Devices 1975, ED-22, 

1045. 

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

Nanotechnology (Letters) 2008, 3, 654. 

9. Cronwell, E. High Field Transport in Semiconductors, suppl. 9 in the series Solid State 

Physics, F. Seits, D. Turnbull, and H. Ehrenreich eds. (Academic Pres, New York, 1967). 

10. Balkan, N.; Ridley, B. K.; Vickers, A. J. Negative Differential Resistance and Instabilities in 

2-D Semiconductors, page 2, Springer, 1993. 

11. Meric, I.; Han, M.; Young, A. F.; Ozyilmaz, B.; Kim, P.; Shepard, K. L. supplement to 

Nature Nanotechnology (Letters) 2008, 3, 654. 

12. 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. 

13. Pucel, R. A.; Haus, H. A.; Statz, H. Signal and Noise Properties of Gallium Arsenide 

Microwave Field-effect Transistors. Adv. Electr. Electron. Phys. 1974, 38, 195. 
26

INTENTIONALLY LEFT BLANK. 
27

Appendix A. Derivation of I-V Curves From Velocity-Field Relations 

Three types of drift velocity relations are considered here 

A-1 The “Simulator” Form 

  0

0 1

E

E

E

sat



 







. (A-1) 

includes an absolute value function, which is an attempt to reflect the vector character of the drift 

velocity. In order to use this expression, we first write the local current in the channel at point x

as 

I  x Zq E x n V x        D   (A-2) 

Three electrical quantities – field, potential, and current – are connected by this expression, but a 

second relation is needed. To derive it, we first solve equation A-2 for E. The absolute value in 

the denominator makes this slightly tricky. 

   

 

0 0 1 0

0 1

0 0 1 1 0

E E

I Zq n V I Zq E n V D D E sat

sat

E E

Zq n V E I I D D

sat sat

  

     

      



  

      

     

 (A-3) 

since the denominator in equation A-1 is always positive. Then 
28

   

 

0 0

0 0

0

0

E I

D Zq n V E I I E Zq n V I DD D

sat sat

I

D E

I

D Zq n V

sat

   

       

   

 



 



 (A-4) 

From this we get 

 

 

   

0

0 0 1 11 1

0 0 0 1

0 0

I E Zq n V D I sat D

I I E Zqn V sat D D sat

Zq n V Zq n V

sat sat sat



  

    

    

  

  

 (A-5) 

We now integrate equation A-1 over the distance x from source to drain, whose separation we 

call L: 

      0 1 0 00 0 0 1

LL L E I

D I dx Zq n V dx Zq n V Edx D E Zqn V sat

sat

         

   

  



 (A-6) 

But the constancy of I

D says that the left side of this equation is just I L D . As for the right side, 

note that dV E Edx dV

dx

  , so that 

   

 

1 0 0

0 0

0 0

V

ds I

D I L Zq n V dV D Zqn V sat

V V Zq ds ds dV I n V dV I D L L D

sat

           

 

   



 (A-7) 

where V

ds is the source-drain voltage. 

Let us label the two integrals   0

0

V Zq ds

a n V dV

L



  and 0

0

V

ds dV b

L sat



 



, so that the equation 

to solve is 

I a bI D D   (A-8) 
29

The integrals a and b have positive integrands; hence, they have the same sign as the drain 

voltage V

ds . Figure A-1 shows plots of the functions f I I D D   and g I a bI   D D   . 

a b

Figure A-1. Plots of the functions f  x x   and g  x a bx    for (a) a = 3, b = 2 and a = 3, b = 0.4 and 

(b) a = 3, b = 2 and a = 3, b = 0.4. 

For 0 V

ds  (figure A-1(a)) and 0 Vds  (figure A-1b); the intersection of these functions 

defines the solutions to equation (A-3). It is clear from figure A-1a that equation A-8 always has 

a positive solution, and in addition a negative solution if b > 1; likewise, figure A-1b shows that 

equation A-8 always has a negative solution, plus a positive solution if b > 1. We reject the 

additional solutions as unphysical, in which case we easily find that 

 0

1

1 0

1

a V

b ds a I I D D a b V

b ds

   

  

    

 (A-9) 

and so the I-V curve is 

  0

0

0 1

0

V Zq ds

n V dV

L

I

D V

ds dV

L sat









 



 (A-10) 

In a bulk semiconductor the saturation velocity is a material quantity and therefore constant with 

voltage. Then it comes outside the integral, and so we have simply 
30

  0

0

0 1

V Zq ds

n V dV

L

I

D

V

L ds

sat





 





 (A-11) 

where the nonanalyticity at V 0 ds  is worth noting. 

A-2 Hot-electron Physics of the Channel 

Hot-electron physics of the channel leads to the following expression, associated with scattering 

by optical phonons: 

  0

2

0 1

E

E

E

sat



 

  

     

 

 (A-12) 

This resulting I-V curve differs markedly from the form derived in section A-1. Again we solve 

for E: 

 

 

2 2 22 2 0

0 2 2

0 0

E I

D I I Zq n V E E D D

sat Zq n V ID sat

  

                       

 (A-13) 

and as before, 

     

2

1 1 1

2 2

1 0 0 1 2 2

( ) 0 0

I

D

Zqn V sat E I sat D sat

Zq n V ID sat

            



        

 (A-14) 

so that 

2

0 1 () ( ) 0

V Zq ds I

D I n V dV D L Zqn V sat

            

 (A-15) 

A-3 Drift Velocity Associated with Acoustic Phonon Scattering 

For the drift velocity associated with acoustic phonon scattering: 
31

 

 

0

1 1 2

0 2 2

E

E

E sat



 

  

 (A-16) 

solving for E gives 

 

 

 

   

     

       

 

     

 

     

 

0

1 1 2

0 2 4

2

1 12 2 0 0

2 2 2 2 2 2 0 0

2 2 22 2 24 4 0 0 0

0

2 2 2

0 0

1 0

2 2 2 0 1

E

I Zq E n V Zq n V D

E sat

I E Zq En V D sat

I I E Zq En V I DD D sat

I E E Zq n V E Zq n V I E D D sat

Zq n V ID E

Zq n V I sat D

Zqn V I E Zqn V D

Zqn V I sat sat D



 

  

               

     

    



 

  



 

        2 2

I

sat D

Zqn V I sat D



  (A-17) 

Setting I Zqn V  sat sat

  , we find that 

 

 

2 2

2 11 1 0 1 1

24 2 2 2

2 2 2 2 2 22 4 2 2 2

1 22 2 22 2 2

2 2 2

11 1 2 2

E I I sat D

I I sat sat D

I I II II I I sat sat D D sat sat D D

II II sat sat D D I I sat D

II I sat sat D

I I sat D

                                

        

    

 

    

    

2

2 2

I

D

I I sat D





 (A-18) 

Since 2 2 I I sat D  , we have 
32

2 2 2 2 2 2

11 1 22 22 22

22 2

1 2 1

2 2 2 2

1 1 0 2

1 1 2 4 2 2

II I I I sat sat D D sat

I I I II I sat sat sat D DD

II I sat D D

I I E sat sat I I sat D

sat I I sat D

              

    

   

                

 (A-19) 

Then 

 

   

 

     

 

     

0

0 11 11 2 2 0

0 0 24 24

2

2 1 0 00 2 2 0 00

0 0 2 2

0 2 2 0 00

V E ds n V dV

I Zq n V I L Zq D D

E E sat sat

V VV ds ds ds I

D n V

Zq n V dV Zq n V dV Zq I dV D Isat Zqn V sat

V VV ds ds ds dV dV Zq n V dV I I D D nV nV Zq Zq sat sat



   

     

 

         

      

          

   

 

 

   

0 0

2

1 2 0 4

2 0 0 0

2 0

Vds

I L Zq n V dV D

V V ds ds dV I L n V dV L D V n V ds dV sat

n V Zq sat

  

                                      



 (A-20) 
33

Appendix B. Stability of Meric’s I-V Curve Without Access Resistance 

First, we calculate the output conductance: 

   

 

00 0 () () ()

00 0

00 0 11 1

0 0 () ()

0 0

0 0 1 1

VV V Zq Zq Zq ds ds ds

n V dV n V dV n V dV

LL L

I VV D ds ds

VV V

LL L ds ds ds

sat sat sat

V V Zq Zq ds ds

n V dV n V dV

dI d d L L D Vds dV dV dV ds ds ds V V

L L ds ds

sat sat

 

 

    

 

 

 

       

         

   

 

   

0 0 () ()

0 0

0 0 1 1

Vds

V V Zq Zq ds ds

n V dV n V dV

d d L L

V V ds ds dV dV ds ds V V

L L ds ds

sat sat









 

 

 

         

             

     (B-1) 

     

   

0 0 0

0 0

2 0 0 0 1 1 0 1

0 0 1

0 0

2

0 1

V Zq ds Zq n V dV n V V L ds L Zq d ds L sat n V dV

dV L ds V V

L L ds ds V

sat sat L ds

sat

Vds

V n V n V dV

Zq L L ds ds

sat sat

L

V

L ds

sat

   

 

        

    

  

           

       

       

    





 (B-2) 
34

   

   

0 0 1

0 0 2

0 1

0

0 0 1

0 0 2

0 1

Vds

V n V n V dV

L L ds ds

sat sat Vds

V

L ds dI Zq D sat

dV L V ds ds

V n V n V dV

L L ds ds

sat sat Vds

V

L ds

sat

                               

               

            

 (B-3) 

 

 

 

 

0 0

 0 0

0 0 1 1

0

0 0

 0

 0 0 1

0 1

Zq

nV I

L L ds D Zqn V I sat ds sat D V V ds ds

V V

L L ds ds

sat sat

Zq L sat Zqn V I nV I ds sat D ds D V L L ds sat Vds V

L ds

V sat

L ds

sat

  

                     

     

               























 (B-4) 

Then for 0 Vds  we get stability when 

   

 

 

0 0 1

0 0 0 2

0 1

0 0

0 1

Vds

V n V n V dV

Zq L L ds ds dID sat sat

dV L ds

V

L ds

sat

Vds

n V dV Zq

Zq n V I sat ds ds L

V

L ds

sat

        

      

     

  

 

  







 (B-5) 

while for 0 Vds  we have 
35

   

 

 

0 0 1

0 0 0 2

0 1

0 0

0 1

Vds

V n V n V dV

Zq L L ds ds dID sat sat

dV L ds

V

L ds

sat

Vds

n V dV Zq

Zq n V I sat ds ds L

V

L ds

sat

        

      

        

 

   



 

 (B-6) 
36

INTENTIONALLY LEFT BLANK. 
37

Appendix C. Small-Signal Parameters From Meric’s I-V Curve Without 

Access Resistance 

C-1 Output Conductance 

 

 

 0

0 1

0

 0

0 1

Zqn V I ds sat D Vds

Vds dI L D sat

s

d dV L ds sat Zqn V I ds sat D Vds

V

L ds

sat

        

      

   

   

 

  

 (C-1) 

C-2 Transconductance 

In this section, we make the dependence of n on both gate and channel voltages explicit, i.e., we 

write n V  as nVV, gst

     . Then 

 

0 0 , ,

0 0

0 0 1 1

2 2

2 , / ,, 0

0

V V Zq ds ds Zq d

n V V dV n V V dV

gst L dV gst L dIds gt I g ds m dV

gst V V

L L ds ds

sat sat

d d nVV n C q V V V nVV nVV gst gst gst gst gt th dV dV gt

Zq d n V

dI L dV D gm dV

gst

               

  

 

 

 

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



  

   

, , 0,

0 0

0 0 1 1

2 2 2

2 22 / / 0 0 0

0 1

Vds

V dV

gst Zq nV V n V ds gst gst

L

V V

L L ds ds

sat sat

n C qV n C q V V V Zq gt gt th th ds gst

L

V

L ds

sat

                           

 

   

     

          



  

 (C-2) 
38

INTENTIONALLY LEFT BLANK. 
39

Appendix D. Stability of Meric’s I-V Curve With Access Resistance 

Again, we calculate the output conductance: 

 

 

   

 

   

0

0 1 2

0 0

2 2

0 0 12 12

V RI Zq ds s D

n V dV

L R Is D I

D

V RI

L ds s D

sat

V RI V RI Zq Zq ds ds sD sD

n V dV n V dV

L L RI RI sD sD V RI V RI ds ds s D sD

V RI V RI

L L ds ds sD sD

sat sat

 



 

 



   

 

     

 

 

 

(D-1) 

 

 

 

 

 

 

0 2

0 1 2

0 1

0 0 12 12

V RI ds s D

n V dV

dI Zq R I D d s D V RI ds s D dV L dV ds ds V RI

L ds s D

sat

V RI

d ds s D

n V dV

dV V R I Zq R I ds ds s D s D d n V dV

L dV R I ds V RI V RI s D L L ds s D ds s D

sat sat

d

dV

  

  



        



 

          

    

                    

     

0 0

V RI V RI RI ds s D ds s D s D d d n V dV n V dV n V dV

dV dV ds ds ds R I

s D

 

   

(D-2) 
40

   

 

     

 

1

1 lim 

0 0

1

1

0

1

dID V RI R V

V RI ds s D s ds dV ds s D ds d n V dV n V dV

dV V ds ds V V RI ds ds s D

dID R nV RI s ds s D dVds

dID RI R V

R I dV s D s ds

s D ds dI d D n V dV n V dV R n R I

dV V dV s sD

ds ds ds R Is D

V RI

d ds s D

n V dV R

dVds R Is D

 

         

  

   

 

       

 

    



      

    

dI dI D D nV RI R nRI s ds s D s s D dV dV ds ds

dID nV RI R nV RI nRI ds s D s ds s D s D dVds

         

         

 (D-3) 

   

    

 

 

1 1 0 1 2 2 0 1 2 0 1 2

0 1 2

0

1

0 1 2

dI d D R

dV L dV s

ds ds sat V RI

L ds s D V RI

sat L ds s D

sat

dID nV RI R nV RI n RI ds s D s ds s D s D dVds

V RI dI Zq L ds s D D sat

dV L ds V RI ds s D

n V dV

R Is D V RI

L ds s

sat

       

          

    

  

       



 

 

 

       

    

  

0 1 2 2

dID R

L dV s

sat ds

D

 

 

 

                

   

 (D-4) 
41

    

 

 

 

  

0 1 2

0 0

0 1 2

0

0

0 0 1 2

dI dI D D L nV RI R nV RI nRI I R

Zq ds s D s ds s D s D D s dV Zq L dV ds ds sat

L

V RI

L ds s D sat

L nV RI I ds s D D Zq L sat

dI Zq D V RI R nV RI nRI

L dV L ds s D s ds s D s D

sat ds

            

                           



   

   

        

  

    

 

0 2

0

0 00 2

0 0

0 00 2

dID

dVds

dI L D I R

Zq L dV D s

sat ds

Zq dI L L D nV RI I RI R nV RI n RI

L Zq L Zq L dV ds s D D s D s ds s D s D sat sat ds

Zq Z

nV RI I RI

L LL ds s D D s D

sat sat

 

 

        





   

                  

              

 

   

     0 q dID R nV RI nRI

L dV s ds s D s D

ds

                        

 (D-5) 

    

 

 

  

0 00 12 2

0 0

0 1 4

0

0

Zq dID V RI RI R nV RI nRI

L ds s D s D s ds s D s D L L dV sat sat ds

Zq

nV RI I

L L ds s D D

sat

V RI ds s D dI Zq L sat D nV R

dV L ds Zq ds R nV RI nRI

L s ds s D s D

                

          

         

    

  

   

 

 

          

  0 I I sD D L sat



 

 (D-6) 

 

    

0

0 0 1 4

dI Zq n V R I I D sat ds s D D

dV L Zq ds sat V RI R nV RI n RI

L L ds s D s ds s D s D

sat

  

 

        

  

 (D-7)
42

 

 

 

 

 

 

0 2

0 1 2

0 1

0 0 12 12

V RI ds s D

n V dV

dI Zq R I D d s D V RI ds s D dV L dV ds ds V RI

L ds s D

sat

V RI

d ds s D

n V dV

dV V R I Zq R I ds ds s D s D d n V dV

L dV R I ds V RI V RI s D L L ds s D ds s D

sat sat

d

dV

  

  



         

 

          

    

                    

      

   

    

1 1 0 1 2 2 0 1 2 0 1 2

0

V RI ds s D dID n V dV n V R I R n V R I n R I ds s D s ds s D s D dV ds ds R Is D

dI d D R

dV L dV s

ds sat ds V RI

L ds s D V RI

sat L ds s D

sat

dID nV RI R nV RI n RI ds s D s ds s D s D

dI Zq D

dV L ds



          

          

                     

       



 

 

 

 

    

0 1 2

1 0 1 2 2

0 1 2

0 0

dVds

V RI

L ds s D

sat

V RI ds s D dID n V dV R

L dV s R I sat ds s D V RI

L ds s D

sat

dID L nV RI R nV RI n RI I

Zq ds s D s ds s D s D D dV Zq ds

L

                                                                  

      

     

     



 

0 1 2

0 1 2

dID R

L dV s

sat ds

V RI

L ds s D

sat

                               

 (D-8) 
43

 

 

  

 

0

0

0 0 1 2

0 2

0

0 00 0 2

L nV RI I ds s D D Zq L sat

dI dI Zq D D V RI R nV RI nRI

L dV L dV ds s D s ds s D s D

sat ds ds

dI L D I R

Zq L dV D s

sat ds

Zq Zq

nV RI I RI R nV RI

L LL L ds s D D s D s ds s

sat sat

      

 

   

           

        



    

  

    

     dID nRI D sD dVds

            

 (D-9) 

    

 

    

 

0 00 12 2

0 0

0 0 1 4

0

Zq dID V RI RI R nV RI n RI

L L L dV ds s D s D s ds s D s D

sat sat ds

Zq

nV RI I

L L ds s D D

sat

Zq dID V RI R nV RI nRI

L L dV ds s D s ds s D s D

sat ds

Zq

nV RI I

L ds s D D

                

          

         

    

   

        

      



 

 

    

0

0

0 0 1 4

L sat

dI Zq n V R I I D sat ds s D D

dV L Zq ds sat V RI R nV RI n RI

L L ds s D s ds s D s D

sat

      

    

  

 

        

     (D-10) 

Final Form: 

 

      

 

      

2

0 0 1 4

0

2

0 0 1 4

Zq n V R I I sat ds s D D V RI Zq ds s D

V RI R nV RI n RI dI L L ds s D s ds s D s D

D sat

dV L ds sat Zq n V R I I sat ds s D D V RI Zq ds s D

V RI R nV RI n RI

L L ds s D s ds s D s D

sat

                   

            

                    

      

 (D-11) 
44

INTENTIONALLY LEFT BLANK. 
45

Appendix E. Small-Signal Parameters From Meric’s I-V Curve With Access 

Resistance 

E-1 Output Conductance 

 

      

 

      

2

0 0 1 4

0

2

0 0 1 4

Zq n V R I I sat ds s D D V RI Zq ds s D

V RI R nV RI n RI dI L L ds s D s ds s D s D

D sat

dV L ds sat Zq n V R I I sat ds s D D V RI Zq ds s D

V RI R nV RI n RI

L L ds s D s ds s D s D

sat

                   

            

                    

      

 (E-1) 

E-2 Transconductance 

 

 

,

0 2

0 1 2

,

0 1 ,

0 12 1

V RI ds s D

n V V dV

gst dI Zq R I D d s D V RI ds s D dV L dV

gst gst V RI

L ds s D

sat

V RI

d ds s D

n V V dV

dV V R I gst Zq R I gt ds s D s D d n V V dV

L dV gst R I gst V RI s D L ds s D

sat

  

         

        



 



                       

   

   0 V RI 2

L ds s D

sat

                   

 (E-2) 
46

, ,,

0 0

1 , , lim 0 0 0

dID V RI ds s D d

V RI V RI RI

ddd ds s D ds s D s D

n V V dV n V V dV n V V dV

dV dV dV gst gst gst

gst gst gst R Is D

V

V gst

V RI gst

d ds s D n V V V dV n V V dV gst gst dV V gst V gst gst gst

n V

 

 

            

   

                      

 ,

0

, ,

1 0

lim 

0

,

V RI ds s D

dID V RI ds s D d

dID V RI ds s D d

V RI ds s D

V dV

gst

V

V gst

gst

n V V V n V V dV

gst gst gst

V V

gst gst V

V gst

gst

n V V dV

gst



 

 



                            

      

                               

         

, ,

0

,, ,

0 0

,

V RI ds s D dI d D n V V dV R n V R I V

dV dV gst s ds s D gst

gst gst

RI RI sD sD dI d d D n V V dV n V V dV R n R I V

dV dV dV gst gst s s D gst

gst gst gst

V RI ds s d n V V dV

dV gst

gst R Is D















  

 



           

               



      

,

, ,

V RI ds s D d n V V dV

dV gst D R I gst s D

dID R nV RI V n RI V s ds s D gst s D gst dV

gst

  

        

   

                      

(E-3) 
47

   

1 1 0 2 2 0 1 2 0 1 2

, ,,

0 1 2

0

dI d D R

dV L dV s gst gst sat V RI

L ds s D V RI

sat L ds s D

sat

V RI ds s D dI d D n V V dV R n V R I V n R I V

dV dV gst s ds s D gst s D gst R I gst gst s D

V R dI Zq L ds D sat

dV L

gst

      

                



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



     

 

 

 

0 2

, 2

0 1 2

, 0 0 2

0

0 1 2

I

s D

V RI R

ds s D s L dI sat D n V V dV

gst dV R I gst s D V RI

L ds s D

sat

V RI ds s D d Zq n V V dV

dV gst I nV Zq R I D ds gst L L s D sat

L

V RI

L ds s D sat

                                                            



             



    

, ,

0 1 2

RI V n RI V s D gst s D gst dID Rs dV

gst V RI

L ds s D sat

                



  

(E-4) 

 

 

0 0 12 ,

0 0 2 ,,

0 00 1 22

V RI dID Zq ds s D d V R I n V V dV

L dV L dV ds s D gst sat gst gst R Is D

Zq dID I nV RI V n RI V R D ds s D gst s D gst s L L dV sat gst

Zq

V RI RI R nV

L LL ds s D s D s ds

sat sat

    

         

    

                             

 

    

 

 

, ,

0 ,

0 ,

0 0 14 , ,

dID RI V n RI V s D gst s D gst dV

gst

V RI Zq ds s D d n V V dV

L dV gst R I gst s D

V RI Zq ds s D d n V V dV

L dV gst dI R I gst D s D

dV Zq gst V RI R nV RI V n RI V

L L ds s D s ds s D gst s D gst sat

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

         

        

 

         

  

 

, , 0

0 0 14 , ,

Zq nV RI V n RI V ds s D gst s D gst

g

m L Zq

V RI R nV RI V n RI V

L L ds s D s ds s D gst s D gst sat

           

  

     

    

                     

(E-5) 
48

 

 

,

0 2

0 1 2

,

0 1 ,

0 12 1

V RI ds s D

n V V dV

gst dI Zq R I D d s D V RI ds s D dV L dV

gst gst V RI

L ds s D

sat

V RI

d ds s D

n V V dV

dV V R I gst Zq R I gst ds s D s D d n V V dV

L dV gst R I gst V RI s D L ds s D

sat

  

         

         

 



                          

   0 V RI 2

L ds s D

sat

                    

(E-6) 

,, ,

0

,, ,

0 0

,

V RI V RI ds s D ds s D dI d d D n V V dV n V V dV R n V R I V

dV dV dV gst gst s ds s D gst

gst gst gst R Is D

RI RI sD sD dI d d D n V V dV n V V dV R n R I V

dV dV dV gst gst s s D gst

gst gst gst

d nVV d

dV gst

gst

 

               

               

       , ,,

V RI V RI ds s D ds s D dI d D V n V V dV R n V R I V n R I V

dV dV gst s ds s D gst s D gst RI RI gst gst sD sD

 

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

(E-7) 

   

1 1 0 2 2 0 1 2 0 1 2

, ,,

0 1 2

0

dI d D R

dV L dV s gst gst sat V RI

L ds s D V RI

sat L ds s D sat

V RI ds s D dI d D n V V dV R n V R I V n R I V

dV dV gst s ds s D gst s D gst R I gst gst s D

V R dI Zq L ds D sat

dV L

gst

      

                



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



     

 

 

 

0 2

, 2

0 1 2

, 0 0 2

0

0 1 2

I

s D

V RI R ds s D s L dI sat D n V V dV

gst dV R I gst s D V RI

L ds s D

sat

V RI ds s D d Zq n V V dV

dV gst I nV Zq R I gst D d L L s D sat

L

V RI

L ds s D sat

                                                            



              



    

, ,

0 1 2

RI V n RI V s s D gst s D gst dID Rs dV

gst V RI

L ds s D sat

                



  

(E-8) 
49

 

 

0 0 12 ,

0 0 2 ,,

0 00 1 22

V RI dID Zq ds s D d V R I n V V dV

L dV L dV ds s D gst sat gst gst R Is D

Zq dID I nV RI V n RI V R D ds s D gst s D gst s L L dV sat gst

Zq

V RI RI R nV

L LL ds s D s D s ds

sat sat

    

         

    

                               

 

   

 

 

, ,

0 ,

0 ,

0 0 14 , ,

dID RI V n RI V s D gst s D gst dV

gst

V RI Zq ds s D d n V V dV

L dV gst R I gst s D

V RI Zq ds s D d n V V dV

L dV gst dI R I gst D s D

dV Zq gst V RI R nV RI V n RI V

L L ds s D s ds s D gst s D gs sat

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

         

        

 

         

  

 

, , 0

0 0 14 , ,

t

Zq nV RI V n RI V ds s D gst s D gst

g

m L Zq

V RI R nV RI V n RI V

L L ds s D s ds s D gst s D gst sat

           

  

     

    

                     

(E-9) 

Final form: 

   

 

, ,

2

0 0 14 , ,

0

, ,

0 0 14 ,

nV RI V n RI V ds s D gst s D gst V RI Zq ds s D

V RI R nV RI V n RI V Zq L L ds s D s ds s D gst s D gst sat

g

m L nV RI V n RI V ds s D gst s D gst

Zq

V RI R nV RI V

L L ds s D s ds s D

sat

            

                                      

   

  2

,

V RI ds s D

n RI V gst s D gst

                                           

 (E-10) 
50

INTENTIONALLY LEFT BLANK. 
51

Appendix F. Modeling of Graphene FETS Using Meric et al. With Corrections 

(No Access Resistance) 

The derivation used in appendix A is correct until we reach equation A-10. For tracking the 

density dependence of the saturation velocity, we need the following expression: 

   

0 ( )

0

0 1

0

V Zq ds

n V dV

L

I

D Vds dV

L n V sat









 



 (F-1) 

Using the “repaired” expression  

1

F n sat

n n



       

  

 gives 

 

   

0 0

1 0

0 0 1 1 2 1

0 0

V Zq ds Zq n V dV

L H

L I

D V V ds ds H H n V dV n V dV L L F F

  

    

          

 (F-2) 

 

   

 

   

 

 

   

 

0

0

0 0 1

0 0 0

0 0 1

0 0 0

0 1

0 0

V Zq ds

n V dV

L

V

V V ds

V ds ds Zq ds n V dV n V dV n V dV L dI L F D d d

dV dV dV V V V ds ds ds ds ds Zq ds

n V dV n V dV n V dV

L L F V

V V ds

ds ds

n V dV n V dV

L F

                                     

            

                       

























(F-3) 
52

 

 

   

 

 

   

0

0

0 1

0 0

0

0

0 1

0 0

V Zq ds

n V dV

d L

Vds dV V V ds ds ds

n V dV n V dV

L F

V Zq ds

n V dV

d L

Vds dV V V ds ds ds

n V dV n V dV

L F

     



        

              

  



                       

 (F-4) 

 

   

 

   

 

     

   

0 0

0

0 0 1 1

00 00

0 1

0 0 0

0

0 1

0 0

V Zq ds Zq n V dV

L n V d L ds

dV VV VV ds ds ds ds ds

n V dV n V dV n V dV n V dV

L L F F

V V ds ds d n V dV n V dV

V L dV Zq ds F ds

n V dV

L V V ds ds

n V dV n V dV

L F

  

                            

                    

           



 

   

   

   

 

2

0 0

1 2

0 1 0 1

0 0 0 0

0 0

1

0 1 1 2

nV nV Zq n V ds ds ds H

L L V V ds ds F V V ds ds n V dV n V dV

L n V dV n V dV

F L F

n V Zq n Vds H

L L F H H

L F

     

     

                                                                         

 

      

      

   

2

0 1 1 2

n V ds ds

H H

L F

 

   

  

                      

(F-5) 
53

     

     

0

1 0 0

1 2 0 0 1 1 0 2 1 12 1 1 2

0 0

0 0 1 1 12 12

Zq

H nV nV Zq n V ds ds L ds I H D L L F H H HH

L L H H

F F L F

nV nV Zq n V ds ds ds

L L F HH HH

L L F F

 

   

                                                     



                            

     

     

 

0

1

0 1 1 2

0 0

0 0 1 1 12 12

0

0 1 1 2

0

0

1

Zq

H

L

H H

L F

nV nV Zq n V ds ds ds I

L L D

F HH HH

L L F F

I

D nV nV nV Zq ds ds ds Zq F

L

H H

L F

V Zq ds

n V dV

d L

dVds

  



     

         



            

         

                        

        







    

      0

0 0 1 1 2

0 0

I

D nV nV nV Zq ds ds ds Zq F

V V L ds ds H H n V dV n V dV L L F F

         

                 

      

                 

(F-6)
54

 

   

 

   

 

      

   

0 0

0

0 0 1 1

00 00

0 1 1

0 0 0

0

0 1

0 0

V Zq ds Zq n V dV

L n V d L D

dV VV VV ds ds ds ds ds

n V dV n V dV n V dV n V dV

L L F F

V V ds ds d n V dV n V dV

V L dV Zq ds F D

n V dV

L V V ds ds

n V dV n V dV

L F

  

                            

                    

           



 

   

   

   

 

2

0 0

1 2

0 1 0 1

0 0 0 0

0 0

1

0 1 1 2

nV nV Zq n V ds ds ds H

L L V V ds ds F V V ds ds n V dV n V dV

L n V dV n V dV

F L F

n V Zq n Vds H

L L F H H

L F

     

     

                                                                         

 

 

    

      

   

2

0 1 1 2

n V ds ds

H H

L F

 

   

  

                      

(F-7) 
55

     

     

0

1 0 0

1 2 0 0 1 1 0 2 1 12 1 1 2

0 0

0 0 1 1 12 12

Zq

H nV nV Zq n V ds ds L ds I H D L L F H H HH

L L H H

F F L F

nV nV Zq n V ds ds ds

L L F HH HH

L L F F

 

   

                                                     



                            

     

     

 

0

1

0 1 1 2

0 0

0 0 1 1 12 12

0

0 1 1 2

0

0

1

Zq

H

L

H H

L F

nV nV Zq n V ds ds ds I

L L D

F HH HH

L L F F

I

D nV nV nV Zq ds ds ds Zq F

L

H H

L F

V Zq ds

n V dV

d L

dVds

  



     

         



     

      

         

                        

        







    

      0

0 0 1 1 2

0 0

I

D nV nV nV Zq ds ds ds Zq F

V V L ds ds H H n V dV n V dV L L F F

         

                 

      

                  (F-8) 

       

       

0 1 1 2

0

0 1 1 2

I

D nV nV nV ds ds ds Zq F Vds

H H

dI L Zq D F

dV L I ds D nV nV nV ds ds ds Zq F Vds

H H

L F

               

   

   

   

   

    

         

   

        

 (F-9)
56

INTENTIONALLY LEFT BLANK. 
57

Appendix G. Evaluation of Integrals in Electrical Parameters of Graphene 

FETS Using The Corrected Model of Meric et al. (No Access Resistance) 

Since       2 2 2 / 0 0 nV n C q V V gst   , we can evaluate the integrals that appear in the 

FET electrical parameters: 

   

 

   

2

2 2

1 0

2 2 2

2 0 0 2 22 sinh sinh 0 0

2

2 2 2 22 2 1 sinh cosh cosh 0 00 0

0

V V b b Cgst H n V dV n V V dV th q V V a a

nq nq C C gst gst V V n VV n th th C q qC gst gst

Cgst n n n VV n th q

n q

dV

Cgs

 

          

   

           

   

 

              

  

 

2 22 1 cosh cosh 1 cosh 2 10 0 2

1 1 2 2 sinh cosh sinh cosh sinh cosh 0 0 2 2

1 sinh

1 2

0 2

b b q q d Hn d n d

C C t gst gst a a

q q b n n

C C ba b b a a

gst a gst

Cgs

q n

Cgst

                 

    

  

               

    



 

      

 

 

2

2

1

00 0

2

2 1 sinh 1

00 0

C C t gst gst VV VV VV b th b th b th nq nq nq

CC C gst gst gst VV VV VV aa a th th th nq nq nq

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

    

                     

 (G-1) 

Likewise, we have 

   

 

1/4 2

2 2

2 0

2 2 2

2 0 0 2 2 2 22 sinh sinh cosh 00 0

0 0 1/2 cosh 1 0

V V b b Cgst H n V dV n V V dV th q V V a a

nq nq C C gst gst VV n VV n n th th C q qC gst gst

nq nq

dV d H n

C C gst gst

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

   

            

   

 

         

3/2 cosh

b

d

a

           

(G-2) 
58

2 4 cosh 2 sinh 1 

cosh 4 4 2 3/2 cosh 2 2

4 44 1 11 cosh

x xdx d x d

b b b xdx t dt t dt d d

x tt a a a

      

  

       

    

 (G-3) 

 

  

 

 

      

    

  

 

1 11 sn | dn | 1 1 cn | cn | 2 , cn | cn | ,

4 4 cn | 1 1

44 2 4 2 22 1 cn | 1 cn | 1 cn | cn | 1 cn | 1 cn |

4 1 sn | dn |

4 2 2 1 cn | sn | 2dn | cn |

xm xm

t dt dx x m m a b xm t x m a b

t x m

t xm xm xm x m xm xm

t dt xm xm

d

t xm xm xm xm

                    

 



 

           

 

  

   

   

1

4 2 cn |

sn dn sn dn 1 11 21 2

2 2 4 2 3 3 3 cn cn 1 cn cn

sn dn sn dn 1 1 21 2

3 3 3 2 2 cn cn cn 1

sn dn 11 2

1 2 2 32 4 3 cn cn

dx

x

x m

b b d d bb aa m m b a

m

a a b a

b bb aa d

m m a

m b

b a a

b d b b m m

a



 

                 

       

  

            

     



  

   

     sn dn 1

3 2 cn

a a

a b

b a

         

  

(G-4) 

   

1 cn sech , , ,

2 1 sn 1 cn 1 1 sech , , 2 ,

,

2 2 1 11 1 dn cn 1 cn 1 1 sech ,1 , , 2 , 2 22 ,

a b a b

a b

ab ab a b

a b

m m ab ab ab a b

a b

  



     



                    

 (G-5) 
59

  1 1 2 sn dn 1 sech tanh , , , , 2 2

1

sn dn tanh , , 2 , 1 sinh cosh 3 , , cn sech sech 2 , , ,

2 sinh cosh 2 sinh cosh

1/2 1/2 0 0 1

20 0 4 3 2 1 cn cn sech

ab ab ab ab

a b ab ab

ab ab

ab ab a b

nq nq b a a d b b

Hn n

C C gst gst a

      

  

   

  

       

            1 cn sech

sinh cosh sinh cosh

1 1/2 0

3 0 1 1 1 1 cn sech cn sech

2 2

b a

a a n q b b

n

Cgst b a

            

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

(G-6) 
60

INTENTIONALLY LEFT BLANK. 
61

Appendix H. Small Signal Parameters for Graphene FETS Using the 

Corrected Model of Meric et al. (No Access Resistance) 

We already have the output conductance: 

       

       

     

0 1 1 2

0

0 1 1 2

1

1

2 1

0

I

D nV nV nV ds ds ds Zq F VD

H H

dI L Zq D F

s

d dV L I ds D nV nV nV ds ds ds Zq F VD

H H

L F

Zq n V n V n V I F ds ds ds D

L F H H

               

   

   

       

    

         

   

        

  

          

 

 

 (H-1) 

For the transconductance we proceed as before: 

0 ,

0

0 1, ,

0 0

V Zq ds

n V V dV

L gst

I

D V V ds ds

n V V dV n V V dV

L gst gst F

        

   

         

 (H-2) 

0 ,

0

0 1, ,

0 0

V Zq ds

n V V dV dI L gst D d

g

m dV dV V V

gst gst ds ds

n V V dV n V V dV

L gst gst F

         

   

         

 (H-3) 
62

 

 

 

0 ,

0

0 1, ,

0 0

0

0

0 1, ,

0 0

V Zq ds

n V V dV

L gst

V

V V ds

ds ds

n V V dV n V V dV

L gst gst F d

dV V

gst Zq ds

n V dV

L

V

V V ds

ds ds

n V V dV n V V dV

L gst gst F

           

         

                    

    

                             

























 (H-4) 

 

 

0 ,

0

0 1, ,

0 0

0 ,

0

0 1, ,

0 0

V Zq ds

n V V dV

L gst d Vds dV V V

gst D D

n V V dV n V V dV

L gst gst F

V Zq ds

n V V dV

L gt d Vds dV V V

gst ds ds

n V V dV n V V dV

L gst gst F

           

        

                  

       

                         













    

 

 (H-5) 

0 0 , , 0 0

0 0 1, , 1, ,

0 0 0 0

0 ,

V

Zq V Zq ds ds d n V V dV n V V dV L dV gst L gst d gst

dV VV V V

gst D D ds ds

n V V dV n V V dV n V V dV n V V dV

L gst gst L gst gst F F

Zq

nVV

L gst

                

                                              

    



1

0 0 1, ,

0 0

Vds d dV

dV V V

gst ds ds

n V V dV n V V dV

L gst gst F

 

                        



 (H-6) 
63

0 ,

0

0 1, ,

0 0

0 , ,

0 0 0 ,

0

0 1

V Zq ds d n V V dV

L dV gst

gst

V V ds ds

n V V dV n V V dV

L gst gst F

V V ds ds d d n V V dV n V V dV

V L dV dV gst gst Zq ds F gst gst n V V dV

L gst

n

L F

       

        

               

                               







 2

, ,

0 0

V V ds ds

V V dV n V V dV

gst gst

                          

(H-7) 

  

(, ) (, )

2 2 2 (, ) / 0

(, ) (, )

d d nV V nV V

dV dV gst gst gst nV V n C q V V V gst gst gst th d d nV V nV V

dV dV gst gst gst

   

     

   



(H-8) 

0 0 , ,

0 0

0 0 1 , ,1 , ,

00 00

0 ,

V V Zq Zq ds ds d n V V dV n V V dV

L L dV gst gst d

dV VV VV

gst ds ds ds ds

n V V dV n V V dV n V V dV n V V dV

L L gst gst gst gst F F

Zq

nVV

L gst

                                                    

   



0 , ,

0 0

2 0

0 1, ,

0 0

V V ds ds d d n V V dV n V V dV

V L dV dV gst gst ds F

dV

V V ds ds

n V V dV n V V dV

L gst gst F

                         

                              



(H-9) 
64

0 , 0,

0 ,

0 0 1, ,

0 0

0 , 0, , 0,

Zq

nV V n V V ds gst gst Zq ds L n V V dV

V V L gst ds ds

n V V dV n V V dV

L gst gst F

nV V n V nV V n V

L ds ds gst gst gst gst F

                                                   

                               2

0 1, ,

0 0

0 , 0,

0 1, ,

0 0

0 ,

V V ds ds

n V V dV n V V dV

L gst gst F

Zq

nV V n V

L ds gst gst

V V ds ds

n V V dV n V V dV

L gst gst F

n V

L ds

F

         

                             

                                          







0, , 0,

0 1, ,

0 0

0 , 0, , 0,

V n V nV V n V I

gst gst gst gst ds D

V V ds ds

n V V dV n V V dV

L gst gst F

nV V n V nV V n V

L ds ds gst gst gst gst F

                                        

                       

               

        

0 , 0,

0 1 2 1

Zq

I nV V n V D L ds gst gst

H H

L F

                                 

           

 (H-10) 

0 0 , , 0 0

0 0 1 , ,1 , ,

00 00

0 ,

V

Zq V Zq ds ds d n V V dV n V V dV L dV gst L gst d gst

dV VV VV

gst ds ds ds ds

n V V dV n V V dV n V V dV n V V dV

L L gst gst gst gst F F

Zq

nVV

L gst

                                                        

   

1

0 0 1, ,

0 0

Vds d dV

dV V V

gst ds ds

n V V dV n V V dV

L gst gst F

                           



 (H-11) 
65

0 ,

0

0 1, ,

0 0

0 , ,

0 0 0 ,

0

0 1

V Zq ds d n V V dV

L dV gst

gst

V V ds ds

n V V dV n V V dV

L gst gst F

V V ds ds d d n V V dV n V V dV

V L dV dV gst gst Zq ds F gst gst n V V dV

L gst

n

L F

       

        

               

              

                







 2

, ,

0 0

V V ds ds

V V dV n V V dV

gst gst

                          

 (H-12) 

  

(, ) (, )

2 2 2 (, ) / 0

(, ) (, )

d d nV V nV V

dV dV gst gst gst nV V n C q V V V gst gst gst th d d nV V nV V

dV dV gst gst gst

   

     

   



 (H-13) 

0 0 , ,

0 0

0 0 1 , ,1 , ,

00 00

0 ,

V V Zq Zq ds ds d n V V dV n V V dV

L L dV gst gst d

dV VV VV

gst ds ds ds ds

n V V dV n V V dV n V V dV n V V dV

L L gst gst gst gst F F

Zq

nVV

L gst

                                                    

   



0 , ,

0 0

2 0

0 1, ,

0 0

V V ds ds d d n V V dV n V V dV

V L dV dV gst gst ds F

dV

V V ds ds

n V V dV n V V dV

L gst gst F

                         

                              



 (H-14) 
66

0 , 0,

0 ,

0 0 1, ,

0 0

0 , 0, , 0,

Zq

nV V n V V ds gst gst Zq ds L n V V dV

V V L gst ds ds

n V V dV n V V dV

L gst gst F

nV V n V nV V n V

L ds ds gst gst gst gst F

                                                   

                               2

0 1, ,

0 0

0 0 , , 0,

0 1, ,

0 0

V V ds ds

n V V dV n V V dV

L gst gst F

Zq n V nV V n V L ds L ds gst gst F

V V ds ds

n V V dV n V V dV

L gst gst F

         

                             

                                             

0, , 0,

0 1, ,

0 0

0 , 0, , 0,

V n V nV V n V I

gst gst gst gst ds D

V V ds ds

n V V dV n V V dV

L gst gst F

nV V n V nV V n V

L ds ds gst gst gst gst F

                                        

                       

                        

0 , 0,

0 1 2 1

Zq

I nV V n V D L ds gst gst

H H

L F

                                  

           

 (H-15) 

so the final form is 

 

0 0 , 0, , 0, , 0,

0 1 2 1

0 , 0, ,

Zq

nV V n V nV V n V I nV V n V

L L ds ds ds gst gst gst gst gst gst D

F Vds

H H

L F

gm

nV V n V nV V

L ds ds gst gst F

                                                         



                              

0 0, , 0,

0 1 2 1

1 , 0, , 0,

0

Zq

n V I nV V n V

gst gst gst gst D L ds

Vds

H H

L F

nV V n V nV V n V ds ds gst gst gst gs

L F

                                                                                   

                      

, 0,

0 1 2 1

I Zq n V V n V t gst gst D F ds

H H

L F

                                   

      (H-16) 
67

List of Symbols, Abbreviations, and Acronyms 

2-D two-dimensional 

3-D three-dimensional 

AC alternating current, nonzero frequency 

ARL U.S. Army Research Laboratory 

DC direct current, zero-frequency 

FET field-effect transistor 

HEMT high electron mobility transistor 

I-V current versus voltage 

MESFET metal semiconductor field-effect transistor 

MOSFET metal-oxide semiconductor field-effect transistor 

RF radio frequency 
68

NO. OF 

COPIES ORGANIZATION 

1 ADMNSTR 

ELECT DEFNS TECHL INFO CTR 

 ATTN DTIC OCP 

 8725 JOHN J KINGMAN RD STE 0944 

 FT BELVOIR VA 22060-6218 

1 CD OFC OF THE SECY OF DEFNS 

 ATTN ODDRE (R&AT) 

 THE PENTAGON 

 WASHINGTON DC 20301-3080 

1 US ARMY RSRCH DEV AND ENGRG 

 CMND 

 ARMAMENT RSRCH DEV & ENGRG 

 CTR 

 ARMAMENT ENGRG & TECHNLGY 

 CTR 

 ATTN AMSRD AAR AEF T J MATTS 

 BLDG 305 

 ABERDEEN PROVING GROUND MD 

 21005-5001 

1 PM TIMS, PROFILER (MMS-P) 

 AN/TMQ-52 

 ATTN B GRIFFIES 

 BUILDING 563 

 FT MONMOUTH NJ 07703 

1 US ARMY ARDEC 

 ATTN AMSRD AAR AEE P 

 C HAINES 

 BLDG 25 

 PICATINNY ARSENAL NJ 07806-5000 

1 US ARMY ARDEC 

 ATTN AMSRD AAR AEM C 

 A MARSTON 

 BLDG 61S 

 PICATINNY ARSENAL NJ 07806-5000 

3 US ARMY ARDEC 

 ATTN AMSRD AAR AEM L 

 S GILMAN 

 ATTN AMSRD AAR AEP F 

 O NGUYEN 

 ATTN AMSRD AAR AEP F S SHRI 

 BLDG 65S 

 PICATINNY ARSENAL NJ 07806-5000 

1 US ARMY INFO SYS ENGRG CMND 

 ATTN AMSEL IE TD A RIVERA 

 FT HUACHUCA AZ 85613-5300 

NO. OF 

COPIES ORGANIZATION 

1 COMMANDER 

 US ARMY RDECOM 

 ATTN AMSRD AMR 

 W C MCCORKLE 

 5400 FOWLER RD 

 REDSTONE ARSENAL AL 35898-5000 

1 US GOVERNMENT PRINT OFF 

 DEPOSITORY RECEIVING SECTION 

 ATTN MAIL STOP IDAD J TATE 

 732 NORTH CAPITOL ST NW 

 WASHINGTON DC 20402 

1 DEPT OF PHYSICS AND GEOLOGY 

 UNIV OF TEXAS - PAN AMERICAN 

 ATTN S TIDROW 

 1201 W UNIVERSITY DR 

 EDINBURG TX 78541 

1 US ARMY RSRCH LAB 

 ATTN RDRL CIM G T LANDFRIED 

 BLDG 4600 

 ABERDEEN PROVING GROUND MD 

 21005-5066 

37 US ARMY RSRCH LAB 

 ATTN IMNE ALC HRR MAIL & 

 RECORDS MGMT 

 ATTN RDRL SER L O NAYFEH 

 ATTN RDRL CIM L TECHL LIB 

 ATTN RDRL CIM P TECHL PUB 

 ATTN RDRL D 

 ATTN RDRL SE J PELLEGRINO 

 ATTN RDRL SED E D KATSIS 

 ATTN RDRL SED E K A JONES 

 ATTN RDRL SEE I S SVENSSON 

 ATTN RDRL SEE O W M GOLDING 

 ATTN RDRL SER E B HUEBSCHMAN 

 ATTN RDRL SER E E VIVEIROS 

 ATTN RDRL SER E F CROWNE 

 (10 COPIES) 

 ATTN RDRL SER E G BIRDWELL 

 ATTN RDRL SER E R DEL ROSARIO 

 ATTN RDRL SER E T IVANOV 

 ATTN RDRL SER J MAIT 

 ATTN RDRL SER L B NICHOLS 

 ATTN RDRL SER L C ANTON 

 ATTN RDRL SER L L CURRANO 

 ATTN RDRL SER L M CHIN 

 ATTN RDRL SER L M DUBEY 

 ATTN RDRL SER L M ERVIN 

 ATTN RDRL SER L R POLCAWICH 
69

NO. OF 

COPIES ORGANIZATION 

 ATTN RDRL SER L S KILPATRICK 

 ATTN RDRL SER M J TATUM 

 ATTN RDRL SER P AMIRTHARAJ 

 ATTN RDRL SER U C FAZI 

 ADELPHI MD 20783-1197 

TOTAL: 51 (49 HCS, 1 ELECT, 1 CD) 
77

INTENTIONALLY LEFT BLANK.