Nonquasi-Static Effects and the Role of Kinetic Inductance in Ballistic Carbon-Nanotube Transistors

IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 9, NO. 4, JULY 2010 449

Nonquasi-Static Effects and the Role of KineticInductance in Ballistic Carbon-Nanotube Transistors

Navid Paydavosi, Mohammad Meysam Zargham, Kyle David Holland, Curtis Michael Dublanko,and Mani Vaidyanathan, Member, IEEE

Abstract—Nonquasi-static effects in ballistic carbon-nanotube(CN) FETs (CNFETs) are examined by solving the Boltzmanntransport equation self-consistently with the Poisson equation. Webegin by specifying the proper boundary conditions that shouldbe employed in time-dependent simulations at high speeds; theseare the proper boundary conditions for a characterization of theso-called intrinsic transistor, i.e., the internal portion of the devicethat is unaffected by the source and drain contacts. A transmission-line model that includes both the kinetic inductance (LK ) andquantum capacitance (CQ ) is then analytically developed from theBoltzmann and Poisson equations, and it is shown to representthe intrinsic transistor’s behavior at high frequencies, including acorrect prediction of resonances in the transistor’s y-parameters.Finally, we show how to represent LK using lumped elements inthe transistor’s traditional quasi-static equivalent circuit, and wedemonstrate that the resulting circuit is capable of modeling theintrinsic behavior of a ballistic CNFET, including the observed res-onances, to frequencies beyond the unity-current-gain frequencyfT . External parasitics can be easily added for an overall compactmodel of ballistic CNFET operation.

Index Terms—Carbon-nanotube (CN) FET, CN transistor,equivalent circuit, high-frequency behavior, kinetic inductance,nonquasi-static effects, quantum capacitance, RF behavior, time-dependent transport, two-port parameters.

I. INTRODUCTION

CARBON-NANOTUBE (CN) FETs (CNFETs) arepresently being explored for prospective applications in

RF analog electronics [1], [2]. Extensive experimental [3]–[20]and modeling [21]–[30] work has been performed over the pastseveral years, leading to demonstrations of basic radio systemsbuilt with CNFETs [1], [31].

Essential to the continued progress of RF electronics withCNFETs is a compact circuit model that can accurately predictall of the 2 × 2 small-signal parameters [10], [17], at least up tothe transistor’s current-gain cutoff frequency (fT ), and prefer-ably, somewhat beyond. As far as we know, the most extensivecompact model for CNFETs is the one proposed by Deng andWong [32], [33]. While such a model is comprehensive and pow-erful, it is also quite involved, and it would be beneficial to havean alternative way to describe and understand basic CNFETbehavior at radio frequencies, including nonquasi-static effects

Manuscript received October 29, 2008; revised April 9, 2009 and August20, 2009; accepted August 31, 2009. Date of publication September 29, 2009;date of current version July 9, 2010. This work was supported by the NaturalSciences and Engineering Research Council of Canada. The review of this paperwas arranged by Associate Editor R. Lake.

The authors are with the Department of Electrical and Computer Engineering,University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TNANO.2009.2032918

and the role of the transistor’s kinetic inductance LK . While therole of kinetic inductance in nanotubes used as interconnects hasbeen studied quite extensively [21]–[23], [34]–[36], a completedescription of its role in transistors has not yet been performed.In our present study, we extend the investigation that we beganin [30] with the primary aim of clarifying the impact of kineticinductance on high-frequency transistor behavior.

As in [30], we use a collisionless Boltzmann–Poisson formal-ism, and focus on the intrinsic performance of “MOSFET-like”CNFETs to obtain an initial understanding of LK . The mainoutcomes of this paper are a step-by-step development, illus-trating the significance of LK , and its inclusion in a classicalquasi-static circuit [37, Fig. 9.5] that is commonly employedto predict FET behavior; we demonstrate that this circuit cansuccessfully predict the behavior of the intrinsic transistor forfrequencies up to and beyond fT , including resonances in thetransistor’s y-parameters that we first observed in [30]. Extrin-sic parasitics can easily be added to obtain an overall first-cutmodel for ballistic transistor behavior, including nonquasi-staticeffects.

Section II of this paper briefly reviews the Boltzmann–Poisson approach, with a focus on the proper boundary con-ditions required to simulate intrinsic transistor behavior undertime-dependent conditions; readers who prefer to focus on theresults can skip this section without a loss of continuity. InSection III, we follow an approach similar to the one orig-inally used in [34] to develop a transmission-line model fora CNFET working under quasi-equilibrium conditions (whenthe drain–source bias voltage vDS is zero); the transmissionline incorporates both kinetic inductance and quantum capac-itance, which are unconventional transistor elements, and wedemonstrate that it agrees with results obtained from the self-consistent Boltzmann–Poisson simulations. In Section IV, weextend the transmission-line approach for transistors under nor-mal operating bias (vDS �= 0), and in Section V, we exploit thedevelopment and results from the earlier sections to representthe effects of LK with lumped elements in the transistor’s tra-ditional quasi-static equivalent circuit [37, Fig. 9.5]; this circuitis then shown to predict the transistor’s y-parameters, as deter-mined by the Boltzmann–Poisson approach, for frequencies upto fT and beyond. The conclusions of our study are summarizedin Section VI.

II. BOLTZMANN–POISSON APPROACH AND TIME-DEPENDENT

BOUNDARY CONDITIONS

The study in this paper is based on the Boltzmann–Poissonapproach outlined in [30, Sec. II]. Here, we will focus on the

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450 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 9, NO. 4, JULY 2010

Fig. 1. (a) Illustration of the nin CN structure employed in this study; botha pictorial view of the structure and an axial cross-section for the top half(r ≥ 0) are shown. (b) Transport simulation space and boundary conditions.(c) Electrostatic simulation space and proper boundary conditions for time-dependent simulations. Further details on the specific device employed in thisstudy are available in Appendix III.

key points, particularly, the electrostatic boundary conditions.The readers are referred to [30] for the remaining details.

A. Approach

We consider a coaxially gated nin nanotube, as sketched inFig. 1(a). Such a structure accounts only for intrinsic transistorperformance, in the sense that the details of real source and draincontacts, which must feed the two n regions, are omitted; realcontacts will create parasitic resistances and capacitances thatshould be added to the intrinsic model that we have developedin this paper. Our focus on the intrinsic transistor is consistentwith the historical approach toward transistor modeling, suchas with traditional MOSFETs [37] and bipolar transistors [38],where transport within an internal transistor is used to create acore model, and external parasitics (which will vary with overalldevice structure) are then added around this core.

Transport along the nanotube is described by a 1-D Boltzmanntransport equation (BTE), written for a conduction subband as

∂f

∂t+ v

∂f

∂z− qEz

h

∂f

∂k= Sopf (1)

where t is the time, z is the axial coordinate along the length ofthe tube, k is the electron wave vector, measured from the sub-band minimum, v ≡ v(k) is the electron velocity for the state k,specified by v(k) = (1/h)[dE(k)/dk], where E(k) refers to theE–k dispersion relation, Ez ≡ Ez (z, t) is the z-component of theelectric field along the surface of the tube at an axial coordinatez and a time t, available from the gradient of the electrostaticpotential found from a solution of Poisson’s equation, discussedshortly, f ≡ f(z, k, t) is the distribution function, q is the mag-nitude of the electronic charge, Sop is the collision operator, and

h ≡ h/(2π), with h being Planck’s constant. As in [30], in thispaper, we set Sopf = 0.

Electrostatics are described by Poisson’s equation as

−∇ · ε0εr (r)∇V (r, t) = ρV (r, t) (2)

where ε0 is the permittivity of free space, εr (r) is the relativedielectric constant at a point r, V (r, t) is the vacuum electro-static potential, and ρV (r, t) is the volume charge density. Asspecified in [30], the conduction-subband edge at a point z anda time t is then

EC (z, t) = −qV (rT , z, t) − χCNT (3)

where χCNT is the electron affinity of the nanotube andV (rT , z, t) is the vacuum potential along the nanotube surface.

B. Boundary Conditions

The boundary conditions for (1) and (2) are illustrated inFigs. 1(b) and (c); these conditions are the same as those speci-fied in [30], except that we have now modified the electrostaticboundary conditions at the axial endpoints, z = 0 and z = zT .

As discussed in [30], the usual practice of using aVon-Neumann boundary condition at z = 0 and z = zT , wherethe potential V is allowed to float, but the normal componentof the electric field is set to zero [39], [40], corresponding to aflat conduction-subband edge, should be modified under high-speed, time-dependent conditions. This follows because the fi-nite time it takes for the band edge to relax causes it to be strictlysloped at the ends of the nin structure [30, Fig. 7], such thatthe normal component of the electric field is strictly nonzerounder time-dependent conditions. The effect is not pronouncedfor the device and excitation frequencies that we considered, butto be as correct as possible, it is important to specify and usethe proper boundary conditions over the frequency range wherethe BTE is valid. The BTE is valid for those signal frequenciesthat are sufficiently low for the wave packet defining a semi-classical state [41, pp. 215–216] to be considered stationary:ω <∼ (E − EC )/h. Under bias, we found E − EC ∼ 5kB T foran average carrier in the channel of the device, where E − EC

takes on its lowest values, which yields f <∼ 30 THz, withf ≡ ω/(2π).

To allow both the potential and the normal component ofthe electric field to remain unspecified at the axial endpoints,we view the regions to the left and right of z = 0 and z = zT ,respectively, as behaving electrostatically like “perfect electricconductors” (PECs). Correspondingly, we employ PEC bound-ary conditions, which allow both the potential and the normalcomponent of the electric field to float, and demand only that thetangential (radial) component of the field be zero. Such bound-ary conditions will introduce a surface charge on the radialplanes at z = 0 and z = zT when the band edge is significantlysloped, but we found that this charge was negligible for thedevice and frequencies considered in this paper.

From a transport perspective, the regions to the left and rightof the axial endpoints are each presumed to supply an inbounddistribution function that is thermalized. Hence, at z = 0, we

PAYDAVOSI et al.: NONQUASI-STATIC EFFECTS AND THE ROLE OF KINETIC INDUCTANCE IN BALLISTIC CN TRANSISTORS 451

can specify that for k > 0

f(0, k, t) ≡ f1(k, t)

=1

1 + exp{[EC (0, t) + E(k) − EF + qvS (t)]/kB T} (4)

and at z = zT , we can specify that for k < 0

f(zT , k, t) ≡ f2(k, t)

=1

1+ exp{[EC (zT , t)+E(k)−EF + qvD (t)]/kB T} (5)

where EF is the equilibrium Fermi level, kB is Boltzmann’sconstant, T is the temperature, and vS (t) and vD (t) are thesource and drain voltages, respectively.

Thus, for the purposes of simulating an intrinsic nin struc-ture, the two n regions are presumed to be fed by idealizedcontacts, which act like PECs from an electrostatic viewpoint,and which are essentially in equilibrium from a transport view-point, supplying thermalized electrons to the internal transistor.We found that this perspective was supported by simulation ofa “fictitious” extended five-region n+ninn+ structure, whichshowed conditions at points in the n regions consistent withwhat we have assumed; we say “fictitious” because the dopinglevels required in the outer n+ regions would be much higherthan what could practically be realized in a CN, so that theseregions serve only to emulate idealized contacts feeding theinternal n regions.

Finally, while the source and drain voltages vS (t) and vD (t)enter through (4) and (5), it is useful to note that the gate voltageenters through the electrostatic boundary condition at the gateterminal. Assuming a zero flat-band voltage for simplicity, the(Dirichlet) boundary condition applied at the gate terminal (i.e.,for r = rG and zS ≤ z ≤ zD ) is [42, eq. (3)]

V = vG (t) − φG/q (6)

where vG (t) is the gate voltage and φG is the work function ofthe gate metal.

In what follows, we compare simulation results obtained fromthe commercial solver COMSOL [43] and the aforementionedframework with analytical results developed from (1) and (2),and we focus on isolating and modeling the role of the transis-tor’s kinetic inductance LK .

III. TRANSMISSION-LINE MODEL UNDER

QUASI-EQUILIBRIUM CONDITIONS

Consider a CNFET working under quasi-equilibrium con-ditions, i.e., with a nonzero applied gate bias (vGS �= 0), buta zero drain bias (vDS = 0). In this situation, it is possible toshow that (1) and (2) are equivalent to two transmission-lineequations, and can hence be represented by a transmission-linecircuit.

Note that it is not essential to follow the full development ona first reading. It suffices to note the two line equations, and thedefinitions of the line elements; the appropriate relations havebeen boxed for convenience. In addition, while the developmentis similar to that used for interconnects in [34], and while thetransmission line bears a resemblance to those presented in the

past for interconnects [21], [22], [34], our approach (here and,subsequently, in Section IV) distinguishes itself by clarifyingthe assumptions involved, and by demonstrating how the ideasare generalized to apply to transistors.

A. Kinetic Inductance and the First Line Equation

To begin, when vDS = 0, the distribution function can beapproximated by a local quasi-equilibrium form

f(z, k, t) ≈ 11 + eε(z ,k ,t)/kB T

(7)

where, at any time t,

ε(z, k, t) = E(k) + EC (z, t) − µ(z, t) (8)

is an energy variable representing the difference between thetotal energy E(k) + EC (z, t) of an electron in a state k at apoint z, and the local quasi-Fermi level (or chemical potential)µ(z, t).

As shown in Appendix I, the BTE (1) with Sopf = 0 can nowbe rewritten in the following form:

∂f

∂t+ v

(−∂f

∂ε

)∂µ

∂z= 0. (9)

Multiplying both sides of (9) by −qv/L, where L is a normal-ization length, and summing over k, it is then easy to obtain thefirst moment of the BTE as

∂I

∂t=

q2

L

1kB T

∑k

v2f(1 − f)∂(µ/q)

∂z(10)

where I(z, t) ≡ (1/L)∑

k −qvf(z, k, t) is the current, and theidentity

∂f

∂ε≡ −1

kB Tf(1 − f) (11)

from (7) must be used.If we now assume a signal perturbation at the radian frequency

ω (arising from a perturbation in one or more of the externalvoltages), then the distribution function f , current I , and quasi-Fermi level µ can all be split into static (dc) and dynamic (ac)parts as

f(z, k, t) = f(z, k) + f(z, k)ejωt (12)

I(z, t) = I + I(z)ejωt (13)

µ(z, t) = µ(z) + µ(z)ejωt . (14)

These equations can now be substituted into the first moment(10); doing this, disregarding higher order terms in ω, and rec-ognizing that (10) implies ∂µ/∂z = 0, one finally obtains thefirst transmission-line equation

jωI =1

LK(z)∂(µ/q)

∂z(15)


where LK(z) is the “local kinetic inductance per unit length,”defined by the relation

1LK(z)

=q2

L

1kB T

∑k

v2 f(1 − f). (16)

An alternative form of this relation, found by using the identity(11), is also useful

1LK(z)

=−q2

L

∑k

v2 ∂f

∂ε

∣∣∣∣∣o

(17)

where the notation “|o” is used to emphasize that the derivativeis to be evaluated at the transistor’s operating point [37, p. 442].

B. Electrostatic and Quantum Capacitances, and the SecondLine Equation

1) Electrostatic Capacitance: Now consider the situationwhere a transistor biased at a (quasi-equilibrium) operating pointhas a small (but constant) change in one or more of its exter-nal voltages. The laws of electrostatics will correspondinglydemand a small change in charge and potential throughout thedevice. The change in potential can be written as a sum of thechanges in the Laplace and Poisson parts of the solution toPoisson’s equation. If we view the solution along the nanotubesurface as being represented by EC in (3), then EC + ∂EC rep-resents the solution after the change, and the quantity ∂EC /(−q)can be expressed in the form

∂EC (z)−q

≡ ∂UL (z)−q

+∂ρ(z)CE (z)

(18)

where ∂UL/(−q) is the change in the Laplace part of the po-tential and ∂ρ/CE is the change in the Poisson part of thepotential. Here, on the right side of (18), based on the discus-sion of Poisson’s equation in Section II, where the only Dirich-let boundary condition is the application of vG at the gate, itis possible to write ∂UL (z)/(−q) ≡ ∂vG for all z, i.e., anychange in the Laplace part of the potential can arise only dueto a change in gate voltage; however, we will retain the sym-bol UL in the equations for physical clarity. Additionally, onthe right side of (18), ∂ρ(z) is the local change in charge perunit length along the nanotube surface (viewed conceptuallyas an integration of ρV (r) in (2) over a thin ring representingthe finite thickness of the nanotube in the radial direction), andCE (z) is a solution coefficient known as the “local electrostaticcapacitance per unit length.” CE thus characterizes how theelectrostatics requires changes in charge and potential to be in-terrelated, and rearranging (18) yields its defining equation asfollows:

CE (z) ≡ −q∂ρ(z)∂[EC (z) − UL (z)]

∣∣∣∣o

. (19)

Given the changes in the numerator and denominator on theright side of (19), evaluated at the transistor’s operating point,one can compute CE (z) on the left side. Of course, such changes

must be found in a manner that is consistent with the laws oftransport.

2) Quantum Capacitance: The laws of transport will alsodemand that a change in the external voltages be accompaniedby a change in charge and potential. In the present formulation,the transport charge is determined by summing the distributionfunction f in (7). If the changes in charge and potential aresmall, then the transport restrictions on charge and potential canbe found by appropriately linearizing (7) around the operatingpoint [44], [45, p. 173], to write

∂ρ(z) = −q∂

[1L

∑k

f

]

= −q1L

∂ [∑

k f ]∂(µ − EC )

∂(µ − EC )

≡ CQ (z)[−1/q][∂µ − ∂EC ] (20)

where

CQ (z) ≡ q2

L

∑k

∂f

∂(µ − EC )

∣∣∣∣∣o

=−q2

L

∑k

∂f

∂ε

∣∣∣∣∣o

(21)

is the “local quantum capacitance per unit length,” with the“|o” notation again being used to emphasize that the derivativeson the right side should be evaluated at the transistor’s operat-ing point. CQ thus characterizes how the transport interrelateschanges in charge and potential. By exploiting the identity in(11), it is possible to express CQ in the following alternativeform, which is also useful:

CQ (z) =q2

L

1kB T

∑k

f(1 − f). (22)

3) Second Line Equation: In a given situation, the electro-statics and transport must agree, i.e., the solutions for charge andpotential must be self-consistent. Thus, equating ∂ρ(z) from(18) and (20), one obtains the following relation:

CE (∂EC − ∂UL ) = CQ (∂µ − ∂EC ) (23)

or

CE (∂EC − ∂UL ) = C(∂µ − ∂UL ) (24)

where

C =CE CQ

CE + CQ(25)

is an overall capacitance representing the series combination ofCE and CQ .


Fig. 2. Transmission-line circuit representation of the transistor’s ac behav-ior under quasi-equilibrium conditions (vDS = 0), following from the lineequations (15) and (28). As discussed in the text below (18), based onthe boundary conditions for Poisson’s equation in Section II, we have setUL (z)/(−q) ≡ vG at all z. As also discussed in the text, the contact resis-tances are RC 1 = RC 2 ≈ 1/8G0 .

The second line equation can now be obtained by startingfrom the continuity equation (the zeroth moment of the BTE)

∂ρ

∂t= −∂I

∂z. (26)

Replacing ∂ρ from (18), and then employing (24), the continuityequation becomes

C(z)−(∂µ/q − ∂UL/q)

∂t= −∂I

∂z. (27)

Under dynamic conditions, we can employ (13) and (14) alongwith UL (z, t) = UL (z) + UL (z) exp(jωt) in this result, leadingto the second transmission-line equation

−jωC(z) × [µ/q − UL/q] = −∂I

∂z. (28)

C. Transmission-Line Model

1) Equivalent Circuit: The line equations (15) and (28) canconveniently be represented in the form of an equivalent circuit,as shown in Fig. 2. In the figure, the line current is I(z) and theline voltage is φ(z) = −µ(z)/q, as marked. At each point, theseries capacitance network of CE (z) and CQ (z) is connectedto the local Laplace potential −UL (z)/q, consistent with (28).The start and end of the intrinsic nin transistor correspond tothe potential points φ(0) = −µ(0)/q and φ(zT ) = −µ(zT )/q,respectively. While the inbound electronic distributions at thesepoints are governed by the source and drain chemical potentials,which determine f1 and f2 in (4) and (5), the overall chemi-cal potentials are not equal to those of the source and drain,since the outbound distributions are unknown. The source anddrain chemical potentials only apply deep within the (fictitious)contacts that are presumed to supply f1 and f2 to the intrinsictransistor. The intrinsic transistor hence connects to these knownpotential points (represented by the generators vS ≡ −µS /q andvD ≡ −µD /q in Fig. 2) through contact resistances. The contact

resistances can be found by perturbing the system through an in-cremental drain voltage ∂vD (with the source and gate voltagesheld fixed), observing the incremental drain current ∂iD thatresults, and then setting RC 1 = RC 2 = (1/2)(∂vD /∂iD ). OurBTE–Poisson approach yielded values RC 1 = RC 2 ≈ 1/8G0 ,where G0 = q2/h is the quantum of conductance; these valuescan hence be used to a good approximation.

2) Line Elements: As suggested in Appendix II, at each z,the relative importance of a state k in carrying ac informationis determined by the value of (∂f/∂ε)|o at that k, and it is easyto show that this quantity peaks for those states k whose totalenergy at the operating point is equal to the local quasi-Fermilevel: E(k) + EC (z) = µ(z). From this perspective, 1/LK(z)in (17) is a measure of the kinetic energy (proportional to v2)stored in such states, and CQ (z) in (21) is a measure of thecharge stored in such states. The stored energy and charge leadto inertial effects on the dynamic behavior, represented by LKand CQ , respectively.

In Fig. 3, we have plotted values of LK , CQ , and the electro-static capacitance CE versus z for a sample nin structure; de-tails on the structure are provided in Appendix III. Also shown isthe total series capacitance, C = CE CQ/(CE + CQ ). All datawere taken from our BTE–Poisson solutions, with LK and CQ

being evaluated from (16) and (22), and CE being found fromits defining relation (19) after a perturbation in gate voltage.As illustrated, all element values are most pronounced withinthe i (“channel”) region. CE is negligible in the n regions, andapproaches the classical gate-dielectric capacitance value for acoaxial structure under the gate, i.e., 2πεr ε0/ ln(rG/rT ), whererG and rT refer to the outer and inner radii, respectively, asshown in Fig. 1. CQ is comparable to CE in the channel, andhence, cannot be neglected. LK is also significant, with a valueof about 6 nH/µm in the channel, which is on the same order ofmagnitude as that found in the recent modeling [34], [35] andmeasurement [36] of nanotube interconnects.

3) Drain Current: Fig. 4 shows simulation results for thenormalized magnitude |ıD /vG | of the ac drain current, obtainedwith an applied ac gate voltage vG , while the ac source anddrain voltages vS and vD were set to zero. As illustrated, thetransmission-line model of Fig. 2 is able to match the BTE–Poisson results, predicting the resonant peak in the current. Thegood agreement in the figure suggests that the transmission-lineapproach holds promise for modeling a transistor under normalbias conditions (when vDS �= 0), and that it can be used as ameans of introducing LK into the transistor’s traditional quasi-static equivalent circuit [37, Fig. 9.5].

IV. TRANSMISSION-LINE MODEL UNDER BIAS

A. Idea

Under normal bias conditions (i.e., when vDS �= 0), it is alsopossible to represent the transistor by a transmission-line model.The approach is to begin with the known form of the solutionto the BTE under time-independent conditions, in which case,it is possible to write the solution as the sum of two parts,f = f11 + f22 , where f11 and f22 are defined as follows: f11represents the solution to the BTE when the inbound distribution


Fig. 3. Transmission-line element values LK , CQ , CE , and C versus positionz for an nin transistor structure working under quasi-equilibrium conditions(vDS = 0). (a) Inductance LK . (b) Capacitances. The element values wereevaluated as described in the text, and the transistor’s parameters are detailed inAppendix III. The gate-source bias is vGS = 0.5 V.

at the source (z = 0) is retained, while the inbound distributionat the drain (z = zT ) is set to zero (i.e., the time-independentversion of f1 from (4) is retained, while f2 in (5) is replacedby f2 ≡ 0); f22 represents the solution to the BTE when theinbound distribution at the source (z = 0) is set to zero, whilethe inbound distribution at the drain (z = zT ) is retained (i.e.,f1 in (4) is replaced by f1 ≡ 0, while the time-independentversion of f2 in (5) is retained). Mathematically, based on thework in [46], each of the parts f11 and f22 can be expressed in aquasi-equilibrium form within complementary (noninteracting)portions of phase space, with energy arguments ε11 and ε22determined by the source and drain Fermi levels µS and µD ,respectively, as detailed by (44)–(47) in Appendix IV; of course,the overall distribution f = f11 + f22 is highly nonequilibriumin form.

For the present discussion, the details in Appendix IV are lessimportant than the idea of writing the overall distribution as a

Fig. 4. Normalized magnitude |ıD /vG | of the ac drain current from thetransmission-line model of Fig. 2 (solid line) and the BTE–Poisson simula-tions (dashed line), found with an applied ac gate voltage vG , while the acsource and drain voltages vS and vD were set to zero. The gate-source bias isvGS = 0.5 V and the drain-source bias is vDS = 0. Details on the transistorare available in Appendix III.

sum of two quasi-equilibrium parts f11 and f22 , each solvinga BTE where one of the inbound distributions has been setto zero. This approach can be exploited under time-dependentconditions, where we assume that such a sum continues to hold,and that the quasi-equilibrium forms for the components f11 andf22 continue to be valid, except that we now let the respectiveenergy arguments become time-dependent

ε11(z, k, t) = E(k) + EC (z, t) − µ11(z, t) (29)

ε22(z, k, t) = E(k) + EC (z, t) − µ22(z, t) (30)

where µ11(z, t) and µ22(z, t) are now (unknown) local quasi-Fermi levels. If desired, the precise mathematical forms off11(z, k, t) and f22(z, k, t) can be found from (44) and (45)in Appendix IV with ε11(z, k, t) and ε22(z, k, t) replacingε11(z, k) and ε22(z, k), respectively.

B. Line Equations

Since the components f11(z, k, t) and f22(z, k, t) each have aquasi-equilibrium form, and since each solves a BTE, it followsthat the development used in Section III can now be appliedto each BTE separately, leading to two transmission lines. Thisapproach neglects the fact that, at every z, the quasi-equilibriumforms for f11 and f22 only apply over (noninteracting) portionsof k-space (above and below ktop in Appendix IV); however,the development of Section III can be viewed as applying withinthese domains individually. Later, we will augment the approachto account for the fact that f11 and f22 do not each have a quasi-equilibrium form over all k. We will first state the governingline equations, and then examine an equivalent circuit that in-corporates the lines.


Fig. 5. Partial-circuit representation of the transistor’s ac behavior under nor-mal bias (vDS �= 0), following from the line equations (31)–(33). As discussedin the text below (18), based on the boundary conditions for Poisson’s equationin Section II, we have set UL (z)/(−q) ≡ vG at all z. The values of the contactresistances RC , RC 11 , and RC 22 are discussed in the text.

The first set of line equations have a form identical to (15)developed in Section III

jωI11 =1

LK 11(z)∂(µ11/q)

∂z

jωI22 =1

LK 22(z)∂(µ22/q)

∂z

(31)

where I11 and I22 are the line currents, and LK 11(z) andLK 22(z) are the per-unit-length line inductances, specified by(16) with f appropriately replaced by f11 or f22 .

The second set of line equations are similar in form to (28),but now written in terms of local quantum capacitances, insteadof an overall capacitance

−jωCQ11(z) × [µ11/q − EC /q] = −∂I11

∂z

−jωCQ22(z) × [µ22/q − EC /q] = −∂I22

∂z

(32)

where CQ11(z) and CQ22(z) are the per-unit-length, local quan-tum capacitances, specified by (22) with f appropriately re-placed by f11 or f22 .

The electrostatic capacitance then enters by demanding thatthe total transport charge be equal to the electrostatic charge,which leads to a third line equation

CQ11(z) × [µ11 − EC ] + CQ22(z) × [µ22 − EC ]

= CE (z) × [EC − UL ].(33)

C. Partial Equivalent Circuit and Line Elements

The line equations (31)–(33) lead directly to the partial-circuit form illustrated in Fig. 5. The relevant line currents and

Fig. 6. Transmission-line element values CQ 11 and CQ 22 versus positionz for an nin transistor structure working under normal bias (vDS �= 0). Theelectrostatic capacitance CE is also shown. The element values were evaluatedas described in the text, and the transistor’s parameters are detailed in AppendixIII. The gate-source and drain-source bias voltages are both 0.5 V (vGS =vDS = 0.5 V).

potentials are marked on the diagram. The transmission line de-rived from f11 is excited by a generator vS ≡ −µS /q, while theline derived from f22 is excited by a generator vD ≡ −µD /q,consistent with the boundary conditions presumed to determinethese components of the overall distribution. Each generatorconnects to its line through the expected contact resistanceRC ≈ 1/8G0 . The values of the contact resistances RC 11 andRC 22 on the opposing ends of the lines, and the subsequent ter-minations at these ends, will be left unspecified for now. First,we will examine the values of the various line elements.

Figs. 6 and 7 display plots of the line elements as a func-tion of position z for the nin structure described in AppendixIII. Fig. 6 displays the source- and drain-line quantum capac-itances CQ11 and CQ22 , along with the electrostatic capaci-tance CE , while Fig. 7 displays the kinetic inductances LK 11and LK 22 , along with an overall kinetic inductance LK , de-fined as the parallel combination of LK 11 and LK 22 . All valueswere evaluated using our BTE–Poisson results, with f11 andf22 appropriately replacing f in (16) and (22), and with CE

appropriately evaluated from (19) after a perturbation in gatevoltage.

As expected, the electrostatic capacitance CE is negligible inthe n regions, being appreciable only in the channel region underthe gate; however, unlike in the equilibrium case (see Fig. 3),the presence of a nonzero drain bias causes the values to deviatefrom the classical dielectric capacitance of a coaxial structure.The source-line quantum capacitance CQ11 is comparable toCE under the gate, and hence, cannot be neglected there. Onthe other hand, the drain-line quantum capacitance CQ22 isnegligible under the gate; in fact, it is negligible everywhere tothe left of the channel-drain (in) barrier region, which simplyreflects the fact that the drain injects negligible charge into thechannel under typical bias conditions.

The source-line kinetic inductance LK 11 peaks within thechannel region, taking on a value of about 10 nH/µm,


Fig. 7. Transmission-line element values LK 11 , LK 22 , and LK versus po-sition z for an nin transistor structure working under normal bias (vDS �= 0).(a) Source-line element LK 11 along with the overall kinetic inductance LK ,where LK is defined as the parallel combination of LK 11 and LK 22 at each z.(b) Drain-line element LK 22 , and a log vertical axis is employed to display thelarge range of values encountered along the length of the device. The elementvalues were evaluated as described in the text, and the transistor’s parametersare detailed in Appendix III. The gate-source and drain-source bias voltages areboth 0.5 V (vGS = vDS = 0.5 V).

comparable to values predicted [34], [35] and measured [36]for nanotube interconnects. The drain-line kinetic inductance iscomparable only in the n region near the drain, where there isappreciable drain charge, so that the overall kinetic inductanceLK is essentially equal to the source-determined value LK 11over most of the device.

D. Complete Equivalent Circuit

The partial circuit in Fig. 5 is a passive network, and hencecannot be a complete representation of an active device, suchas a transistor. One reason that the circuit misses activity isbecause the derivation of the line equations (31)–(33) ignoresthe fact that the distribution-function components f11 and f22

Fig. 8. (a) Traditional small-signal circuit from [37, Fig. 9.5]. (b) Definitionsof element values in terms of appropriate derivatives of charge or current withrespect to voltage, where all derivatives are to be evaluated at the dc operatingpoint; the symbols have their usual meanings. The symbol χ represents thepartitioning factor for the channel charge, as discussed in [30].

have quasi-equilibrium forms only in certain regions of k-space;moreover, these regions are dynamic (determined by the strictlytime-dependent quantity ktop in Appendix IV), which is anotherfact rejected by the derivation. To properly account for transistoraction, we will hence complete the circuit phenomenologically;this can be done by comparing the form of the network in Fig. 5to the usual circuit employed for FETs.

Fig. 8(a) shows the well-known circuit for FETs [37, Fig. 9.5],where the symbols have their usual meanings and are spec-ified in Fig. 8(b). Simple network manipulation allows thecircuit to be redrawn in the form shown in Fig. 9(a), whereıS ≡ ıS1 + ıS2 and ıD ≡ ıD1 + ıD2 , and the external excita-tion voltages vG , vS , and vD are explicitly shown. The passivenetwork contained within nodes 1–5 is now removed, and thepartial transmission-line model of Fig. 5 is inserted in its place,with RC 11 ≡ 1/gm − 1/8G0 and RC 22 ≡ 1/gsd − 1/8G0 . Theresulting circuit, in Fig. 9(b), then provides a representation ofboth transistor action and charging effects, and it is intuitivelysatisfying in several ways. For example, the path between nodes3 and 4 in the original circuit of Fig. 9(a) represents transistoraction through the resistive element 1/gm and the capacitanceCm , and Cgs between nodes 1 and 3 represents gate-sourcecharging effects. In Fig. 9(b), these elements are effectively re-placed by a subcircuit involving the transmission line derivedfrom the source-injected electrons, which are those responsiblefor transistor action under typical bias and those responsiblefor gate-source charging effects; moreover, the terminated linereduces to a resistance of 1/gm between nodes 3 and 4 at lowfrequencies (ω → 0), consistent with Fig. 9(a). Similarly, thepath between nodes 2 and 5 in Fig. 9(a) represents the effectsof the output conductance gsd and the output capacitance Csd ,while gate-drain charging effects are taken into account throughCgd between nodes 1 and 5; these elements are then replacedby a subcircuit that involves the drain transmission line between


Fig. 9. (a) Redrawn version of the traditional small-signal circuit fromFig. 8(a). (b) Redrawn circuit with the passive network contained withinnodes 1–5 removed and replaced with the partial circuit of Fig. 5; the con-tact resistances are chosen to be RC = 1/8G0 , RC 11 = 1/gm − 1/8G0 , andRC 22 = 1/gsd − 1/8G0 , as discussed in the text.

the same set of nodes, and the terminated line reduces to 1/gsdbetween nodes 2 and 5 at low frequencies. Overall, the transmis-sion lines in Fig. 9(b) can be viewed as providing a more carefulrepresentation of distributed charging effects than the lumpedapproach in Fig. 9(a), with the terminations and topology beingchosen to retain transistor action.

One consequence of the phenomenological approach isthat the line potentials φ11(z) ≡ µ11(z)/(−q) and φ22(z) ≡µ22(z)/(−q) in Fig. 9(b) lose their physical significance, i.e.,they can no longer be directly linked to the distribution-functioncomponents f11 and f22 discussed earlier; however, this incon-sistency can be overlooked, since the circuit of Fig. 9(b) is verysuccessful in modeling the device behavior, and because it pro-vides an indication of how the usual circuit of Fig. 8(a) needs tobe modified, as will be explained in Section V.

Fig. 10. Forward common-source y-parameters versus frequency from theBTE–Poisson simulations and from the circuit of Fig. 9(b). (a) Magnitude.(b) Phase. The transistor’s fT is 3.4 THz. Other details on the transistor areavailable in Appendix III. The gate-source and drain-source bias voltages areboth 0.5 V (vGS = vDS = 0.5 V).

E. y-Parameters

At high frequencies, the circuit of Fig. 9(b) is capable of mod-eling the dynamic terminal behavior of the transistor, includingresonances that we first observed in [30]. This is illustrated inFigs. 10 and 11, which show plots of both the forward andreverse common-source y-parameters from the circuit and theBTE–Poisson simulations. While phenomenologically derived,the transmission-line representation in Fig. 9(b) can hence beused to guide the introduction of kinetic inductance into theusual circuit of Fig. 8(a).

V. KINETIC INDUCTANCE IN THE TRANSISTOR

EQUIVALENT CIRCUIT

A. Circuit Topology

While the circuits in Fig. 9(a) and (b) both account for transis-tor action (by design), the gate-source and gate-drain charging


Fig. 11. Reverse common-source y-parameters versus frequency from theBTE–Poisson simulations and from the circuit of Fig. 9(b). (a) Magnitude.(b) Phase. The transistor’s fT is 3.4 THz. Other details on the transistor areavailable in Appendix III. The gate-source and drain-source bias voltages areboth 0.5 V (vGS = vDS = 0.5 V).

networks in the circuit of Fig. 9(a) are purely capacitive (repre-sented by Cgs and Cgd , respectively), whereas those in Fig. 9(b)involve distributed RLC networks. This suggests that the usualcircuit for field-effect devices in Fig. 8(a), from which Fig. 9(a)is derived, and where the gate-source and gate-drain chargingnetworks are purely capacitive, should be modified such that thecharging networks involve RLC components; R should equalthe usual contact resistance RC = 1/8G0 , L should be relatedto the total kinetic inductance of the transistor, and the Cs canbe left as the usual gate-source and gate-drain capacitances Cgsand Cgd .

One possible modification is illustrated in Fig. 12(a). In thiscircuit, the gate-drain charging network is now represented bythe combination of RC , LKd , and Cgd , while the gate-sourcecharging network is represented by the combination of RC , LKs ,and Cgs . The insertion of the two RC elements in series with

Fig. 12. (a) Modified form of the traditional small-signal circuit from [37,Fig. 9.5]. The elements g ′m and g ′sd are modified values of the transconductanceand output conductance, as specified by (34) and (35). The contact resistancesare RC = 1/8G0 , and two kinetic-inductance components LKd and LKs areemployed. The remaining symbols have the same meanings and definitions asin Fig. 8(b). (b) Alternative form that places the contact resistances and kinetic-inductance components directly in series with the drain and source leads. Thevalues of g ′′m and g ′′sd are given by (39) and (40), and the usual circuit for theintrinsic transistor [37, Fig. 9.5] is shown boxed.

the transconductance generator and output conductance requiresmodified values for the latter quantities

g′m = gm /(1 − gm RC ) (34)

g′sd = gsd/(1 − gsdRC ). (35)

Use of these modified values preserves the correct low-frequency response. To a first approximation, all other elementvalues can be found in the usual way, i.e., as originally specifiedin Fig. 8(b); here, it may be helpful to note that the value of thecharge-partitioning factor χ is greater than or equal to unity fora ballistic CNFET, as discussed in [30].


B. Choice of Kinetic Inductance Values

The choice of the drain kinetic-inductance component LKd inFig. 12 can be made by noting that the RC –LKd–Cgd networkbetween the gate and drain is responsible for the resonances inthe y-parameters. Since Cgd is found in the usual way, one canchoose LKd by fitting it to the known resonant frequency

ω0 =1√

LKdCgd. (36)

The value of LKs can then be assigned as

LKs = LKK − LKd (37)

where LKK is the total kinetic inductance of the device, equalto the integrated value of LK(z), which was shown in Fig. 7

LKK ≡∫ zT

0LK(z) dz. (38)

The primary need for LKs in the circuit is to model the degra-dation in the magnitude response of y11 and y21 , and in theirsum −(y11 + y21), at high frequencies, where −(y11 + y21) isindicative of the source current in common-source operation.

In more general cases, the values of LKd and LKs can befound by estimating the total kinetic inductance LKK , and thenappropriately splitting it between the drain and source, in muchthe same way that Cgd and Cgs represent a split of the totalgate capacitance Cgg = Cgd + Cgs , and the charge-partitioningfactor χ splits the channel charge between the drain and sourceterminals [30], [37, Ch. 7].

C. Comparison With BTE–Poisson Simulations

Figs. 13 and 14 show a comparison of the y-parameters fromthe BTE–Poisson simulations with those from the circuit ofFig. 12(a). The good agreement demonstrates that the circuit ofFig. 12(a) is a valid representation of the intrinsic transistor, i.e.,that it can predict all the details of the response, including reso-nances in the y-parameters, for frequencies up to the transistorfT and beyond.

As an alternative, we have presented a somewhat simplertopology in Fig. 12(b), where the contact resistances and kinetic-inductance components have been placed directly in series withthe drain and source leads, and the usual circuit for the intrinsictransistor is shown boxed. Here, all the element values are iden-tical to those in the circuit of Fig. 12(a), except for g′′m and g′′sd ,which should take on the values

g′′m =gm

(1 − gm RC − 2gsdRC )(39)

g′′sd =gsd

(1 − gm RC − 2gsdRC )(40)

to preserve the correct low-frequency response. While there issome error in the value of the resonant peaks of the forwardy-parameters, we found that this circuit is otherwise equallysuccessful in modeling the ac behavior of the intrinsic transistor,and can hence be used whenever a simpler topology is desired.

Fig. 13. Forward common-source y-parameters from the BTE–Poisson sim-ulations and from the circuit of Fig. 12(a). (a) Magnitude. (b) Phase. Thetransistor’s fT is 3.4 THz. Other details on the transistor are available inAppendix III.

VI. CONCLUSION

The following conclusions can be drawn from this studyof nonquasi-static effects and kinetic inductance in ballisticCNFETs.

1) A characterization of the intrinsic (contact-independent)behavior of a ballistic CNFET in time- or frequency-dependent simulations requires an update of the tradi-tional electrostatic floating boundary condition [39], [40];the new boundary condition leaves both the potential andthe normal component of the electric field unspecified atthe source and drain endpoints.

2) By starting from the Boltzmann and Poisson equations un-der quasi-equilibrium conditions (when the drain–sourcebias voltage vDS is zero), a transmission-line represen-tation of an intrinsic transistor containing both the dis-tributed kinetic inductance and the distributed quantumcapacitance can be developed.


Fig. 14. Reverse common-source y-parameters from the BTE–Poisson sim-ulations and from the circuit of Fig. 12(a). (a) Magnitude. (b) Phase. Thetransistor’s fT is 3.4 THz. Other details on the transistor are available inAppendix III.

3) The transmission-line approach can be extended to nor-mal bias conditions (when vDS �= 0), and can successfullyrepresent the intrinsic behavior of a ballistic CNFET forfrequencies up to and beyond fT .

4) By appropriately introducing two lumped kinetic-inductance components (LKs and LKd ) into the traditionalquasi-static equivalent circuit used for FETs [37, Fig. 9.5],along with an appropriate choice for the so-called charge-partitioning factor χ, a circuit topology is obtained thatis able to predict intrinsic CNFET behavior for frequen-cies up to and beyond fT , including resonances in they-parameters.

External parasitics can easily be added to the final circuittopologies emerging from this paper (see Fig. 12) for an overallrepresentation of ballistic CNFET operation.

APPENDIX

I. BTE IN NEW FORM

To obtain (9), first combine (7) and (8) to write

v∂f

∂z= v

∂f

∂ε

∂(EC − µ)∂z

(41)

and

−qEz

h

∂f

∂k= −∂EC

∂z

∂f

∂(hk)

= −∂EC

∂z

∂f

∂ε

∂ε

∂(hk)

= −v∂f

∂ε

∂EC

∂z. (42)

Then, use (41) and (42) on the left side of (1) with Sopf = 0.

II. SIGNIFICANCE OF (∂f/∂ε)|oUnder dynamic excitation, it is possible to write the ac part

of the distribution function appearing in (12) as

f(z, k) =∂f

∂ε

∣∣∣∣o

(z, k)[EC (z) − µ(z)] (43)

where EC (z) and µ(z) refer to the ac amplitudes of theconduction-subband edge and quasi-Fermi level, respectively, atpoint z. This means that at each z, the k dependence of f(z, k),and hence the k dependence of all ac information, which is foundby appropriately summing f(z, k) over k, will be determined bythe k dependence of (∂f/∂ε)|o . In other words, ac informationwill primarily be carried by those electronic states k for which(∂f/∂ε)|o is maximum. It is easy to show that such states arethose for which the total energy at the operating point is equalto the local quasi-Fermi level: E(k) + EC (z) = µ(z).

III. nin TRANSISTOR PARAMETERS

The device used for all results reported in this paper has thefollowing properties: a zigzag (16,0) structure with a diameterof 1.25 nm and an electron affinity of χCNT = 4.2 eV [47],source and drain n regions that each have a doping level ofND = 109 m−1 and a width of 50 nm, an undoped i region ofwidth 20 nm, a gate oxide with a thickness of 3 nm and a relativedielectric constant of εr = 16, and a gate metal with a workfunction of φG = 4.5 eV. A sketch of the device is available inFig. 1(a).

IV. MATHEMATICAL FORMS OF f11 AND f22

Under time-independent conditions, the two parts of the so-lution to the BTE f11(z, k) and f22(z, k) can be expressed in aquasi-equilibrium form as follows [46]:

f11(z, k) =

11 + exp[ε11(z, k)/kB T ]

, if k > ktop(z)

0, if k ≤ ktop(z)

(44)


Fig. 15. (a) Illustration of the BTE phase space, showing the regions where thedistribution-function components f11 and f22 are nonzero. (b) Band diagramfor the transistor under normal bias (vGS = vDS = 0.5 V), illustrating keyparameters appearing in the equations discussed in the text.

and

f22(z, k) =

0, if k > ktop(z)

11 + exp[ε22(z, k)/kB T ]

, if k ≤ ktop(z)

(45)where ktop(z) will be defined below, and the energy argumentsε11(z, k) and ε22(z, k) are

ε11(z, k) = E(k) + EC (z) − µS (46)

and

ε22(z, k) = E(k) + EC (z) − µD (47)

with µS ≡ −qvS and µD ≡ −qvD . Here, at each z, the valueof ktop(z) delineates the points in phase space above and be-low which f11(z, k) and f22(z, k) are nonzero, respectively, assketched in Fig. 15(a). If required, an equation for ktop(z) can

be found by setting the total carrier energy at z equal to theenergy of the top of the barrier

E[ktop(z)] + EC (z) = EC top (48)

where EC top is labeled in the band diagram of Fig. 15(b).When the E(k) relation is known, (48) can be inverted to solvefor ktop(z). By employing E(k) as specified in [30, eq. (8)], theresult for a zigzag nanotube is

ktop(z) = ∓

√(EC top − EC (z)

2a0tE /2+

23d

)2

−(

23d

)2

(49)

where, for the purposes of delineating between f11 and f22 ,the negative solution applies for points z up to ztop and thepostive solution applies thereafter, d is the tube diameter, anda0 ≈ 1.42 A and tE ≈ 3 eV are the carbon–carbon bond lengthand energy, respectively. Under time-dependent conditions, ktopstrictly becomes time-dependent; the ramifications of this timedependence for the small-signal conditions of primary interestin this paper are discussed in Section IV-B and IV-D.

V. ACKNOWLEDGMENT

The authors would like to thank Dr. D. Kienle of SandiaNational Laboratories (USA) for very helpful discussions ontime-dependent transport, Dr. R. Karumudi and Dr. V. Vanof the University of Alberta for very helpful discussions onthe electrostatic boundary conditions, and P. Greidanus andDr. T. Zhou of the Center of Excellence in Integrated Nanotools,University of Alberta, for indispensable computer support.

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Navid Paydavosi received the B.A.Sc. degree in elec-trical engineering from Shahid Beheshti University,Tehran, Iran, in 2005. He is currently working to-ward the Ph.D. degree at the University of Alberta,Edmonton, AB, Canada.

His research interests include the theory andmodeling of future alternatives to ordinary silicontransistors, including carbon-based devices, with anemphasis on studying the high-frequency electricalcapabilities.

Mohammad Meysam Zargham received the B.Sc.degree from Sharif University of Technology, Tehran,Iran, in 2005, and the M.Sc. degree in electrical en-gineering from the University of Alberta, Edmonton,AB, Canada, in 2008. He is currently working to-ward the Ph.D. degree at the University of Toronto,Toronto, ON, Canada.

He was involved in many different projects in avariety of groups, including the design of analog low-density parity-check decoders, microfluidic lab-on-a-chip design, and the modeling of carbon-nanotube

transistors. His research interests include CMOS-integrated circuits for bacteriadetection.


Kyle David Holland received the B.Sc. degree in2009 in engineering physics (nanoengineering op-tion) from the University of Alberta, Edmonton, AB,Canada, where he is currently working toward thePh.D. degree in electrical engineering.

His research interests include quantum simula-tion of carbon-based nanoelectronics, with an em-phasis on modeling the high-frequency performanceof graphene devices.

Curtis Michael Dublanko was born in Leduc,Alberta in 1985. He received his B.Sc. degree in Elec-trical Engineering with distinction in 2009 from theUniversity of Alberta. He is currently pursuing anM.Sc. degree in Electrical Engineering at the Uni-versity of Alberta in VLSI digital-circuit design. Hisresearch interests include VLSI circuit layout anddesign, circuit characterization and testability, andVHDL design.

Mani Vaidyanathan (S’95–M’99) received theB.A.Sc. degree in computer engineering and theM.A.Sc. degree in electrical engineering from theUniversity of Waterloo (UW), Waterloo, ON, Canada,in 1990 and 1993, respectively, and the Ph.D. degreefrom the University of British Columbia (UBC), Van-couver, BC, Canada, in 1999.

He was with Nortel Networks, IBM, and Gene-sis Microchip. From 1993 through 1994, he was anAdjunct Lecturer and a Research Assistant at UW,where he was engaged in teaching courses and per-

forming research in the area of semiconductor devices. In 1999, he joined theDepartment of Electrical and Computer Engineering, University of Californiaat San Diego (UCSD), La Jolla, as a Postdoctoral Fellow, where he was anAssistant Research Scientist during Fall 2001. During 2002–2003, he was anVisiting Assistant Professor at Purdue University, West Lafayette, IN, on leavefrom UCSD. In Fall 2004, he joined the faculty of the University of Alberta, Ed-monton, AB, Canada, where he is currently an Assistant Professor. His researchinterests include theory and modeling of semiconductor devices, where he hasworked on topics ranging from studying carrier transport in small-dimensiondevices to modeling high-frequency distortion for wireless applications.

Dr. Vaidyanathan received Postgraduate Scholarships from the Natural Sci-ences and Engineering Research Council (NSERC) of Canada for the M.A.Sc.and Ph.D. degrees, as well as an NSERC Postdoctoral Fellowship. He was aKillam Scholar and a Governor-General’s Gold Medal nominee at UBC. He isthe inaugural recipient of the University of Alberta’s Provost’s Award for EarlyAchievement of Excellence in Undergraduate Teaching.

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Nonquasi-Static Effects and the Role of Kinetic Inductance in Ballistic Carbon-Nanotube Transistors

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