Hindawi Publishing CorporationActive and Passive Electronic ComponentsVolume 2011, Article ID 830182, 8 pagesdoi:10.1155/2011/830182

Research Article

Modelling of an Esaki Tunnel Diode in a Circuit Simulator

Nikhil M. Kriplani, Stephen Bowyer, Jennifer Huckaby, and Michael B. Steer

Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695-7914, USA

Correspondence should be addressed to Nikhil M. Kriplani, nkriplani@ncsu.edu

Received 15 January 2011; Accepted 26 February 2011

Academic Editor: G. Ghibaudo

Copyright © 2011 Nikhil M. Kriplani et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A method for circuit-level modelling a physically realistic Esaki tunnel diode model is presented. A paramaterisation techniquethat transforms the strongly nonlinear characteristic of a tunnel diode into two relatively modest nonlinear characteristics isdemonstrated. The introduction of an intermediate state variable results in a physically realistic mathematical model that is notonly moderately nonlinear and therefore robust, but also single-valued.

1. Introduction

Tunnelling here refers to the phenomenon of movement ofcharge across a potential barrier with the result that there is anonunique relationship between current and applied voltage.The effect occurs in semiconductor-based tunnel diodes,molecular diodes, and in current leakage through the thinoxides of advanced metal-oxide-metal (MOS) transistors,see Olivo et al. [1]. Tunnel diodes, see Hall [2], have beenused in fast-switching electronic circuits ever since theirdiscovery, Esaki [3]. While tunnelling is a critical effect, itcan be difficult to model in a circuit simulator because of themultivalued flux-potential relationship which is also stronglynonlinear. While this paper focuses on the modelling oftunnel diodes, the technique presented can be used to modelmany types of tunnelling effects.

One unique feature of an Esaki tunnel diode is theNegative Differential Resistance (NDR) region of its current-voltage (I-V) characteristic. Under certain circuit condi-tions, this creates a bistable or multistable circuit, andswitching between the two stable states can be rapid. Inother situations, tunnelling is undesirable and sets limitson performance. Determining the onset of tunnelling isimportant in designing integrated circuits using sub-65 nmMOS transistors where the gate oxide is necessarily thin.The typical current-voltage (I-V) characteristic of an Esakitunnel diode can be divided into three regions in thedirection of increasing voltage, see Figure 1. The first region

involves an increase in the output current until it reaches apeak value and represents a Positive Differential Resistance(PDR). The region following this involves a decrease incurrent with increasing voltage and this corresponds to theNDR region. Here the current decreases until it reaches aminimum or valley value. The third region corresponds toconduction of a forward-biased p-n junction diode and isthe second PDR region. All three regions are represented byexponential functions and are therefore strongly nonlinear.These exponential functions are derived from a quantum-physical description of the diode, Sze [4], and the totalcurrent is a sum of these exponential functions. In orderto simulate circuits consisting of Esaki tunnel diodes, itis essential to be able to accurately model the nonlinearbehaviour of the I-V characteristic as well as the negativeresistance. Ideally, one would like to model the Esaki tunneldiode in a circuit simulator with minimal changes to thequantum-physical representation of the device.

An Esaki tunnel diode is not directly modelled incommercial circuit simulators. Three strategies have beenused to mitigate the bistable and/or strongly nonlinearphysical description of the diode. Techniques used include:macromodelling approximation by curve-fitting to a rela-tively smooth function, and modifications to the underlyingalgorithms that solve a circuit containing a tunnel diode. Themacro-modelling was used in Kuo et al. [5] wherein a tunneldiode was represented as a set of linear current sources whichwere appropriately switched on or off depending on bias

2 Active and Passive Electronic Components

Valley

I

Peak

Jtotal

JTH

JX

V

JT

Figure 1: Esaki tunnel diode I-V characteristic.

conditions. This approach uses combinations of switches,resistors, capacitors, voltage sources, and voltage-controlledcurrent sources arranged such that they will approximatelymimic the tunnel diode I-V characteristic. Curve-fitting hasalso been used extensively wherein the I-V characteristic wasapproximated by a piecewise-linear model, see Mohan etal. [6], or by a piecewise nonlinear model utilising a diodefor each of the PDR regions, see Neculoiu and Tebeanu[7]. Similarly, polynomial curve-fitting techniques, Changet al. [8], have also been used. Macro-modelling and curvefitting are essentially nonphysical techniques introducingwell-behaved functions to model a device whose operationis quite complex.

Physics-based modelling is essential when a strong cor-relation between simulation and measurement is desired—particularly with steadily decreasing device geometries. Aphysics-based description of the diode was used in Bhat-tacharya and Mazumder [9] where the iterative techniqueused to solve the nonlinear set of differential equationsdescribing a circuit was modified. In particular, algorithmicmodifications were made to the Newton-Raphson iterativemethod and implemented in a circuit simulator. Basedon the onset of certain conditions during simulation, thealgorithm switches from voltage-based iterations to current-based iterations. Since the I-V characteristic of the tunneldiode can have multiple values of voltage for a single value ofcurrent, the authors proposed a technique to approximatelyassign a single value of voltage and resolve this uncertainty. Inorder to calculate this voltage value, the authors introducedadditional parameters in the diode model. Other techniquesproposed to model circuits with tunnel diodes can bethought of as variations to homotopy-based methods asdescribed in Melville et al. [10], and in general require asteep learning curve on the part of the user to be able tounderstand and implement the model of a nonlinear device.Also, the generality of the modified iterative techniques andthe homotopy methods is limited.

In this paper, a technique for modelling a physically basedtunnel diode is demonstrated that makes no changes to thebasic equations describing the operation of a device andrequires no modification to the underlying algorithms of acircuit simulator. A similar approach can be used for otherstrongly nonlinear devices.

2. Parameterised Device Modelling withState Variables

Simulating circuits with strongly nonlinear devices can oftenbe plagued by nonconvergence. In the well-established circuitsimulator tradition, modifications are made at the individualelement level to obtain convergence. This philosophical pro-cedure is used in the SPICE simulator to obtain convergencefor circuits with junction diodes that have exponential char-acteristics. In SPICE, heuristics are used including limitingthe fraction by which current and voltage can change fromone iterative step to the next. This cannot be extended tohandling tunnel diodes. In essence, the technique describedin this paper introduces an intermediate state variablebetween the current and voltage variables in order to mitigatethe nonlinear diode I-V characteristics. The iterative solverthen acts on this intermediate state variable while trying tominimize the error function associated with the circuit underconsideration. As outlined in Christofferssen et al. [11], whenan element is modelled using state variables, the current andvoltage are expressed as a function of these state variables.This is,

v(t) = v(

x(t),dxdt

, . . . ,dmxdtm

, xD(t))

,

i(t) = i(

x(t),dxdt

, . . . ,dmxdtm

, xD(t))

,

(1)

where v(t) and i(t) are vectors of voltages and currents atthe ports of the nonlinear device, x(t) is a vector of statevariables, and xD(t) is a vector of time-delayed state variables,that is, [xD(t)]k = xk(t − τk). Equation (1) defines themodelling scope. This can be a problem if the describingequations are strongly nonlinear as local convergence con-trol cannot be used. However, avoiding the use of localconvergence control, and the ad-hoc modifications of theiterative algorithm required, enables state-of-the-art, off-the-shelf numerical libraries to be employed.

Consider the equation for a microwave diode whoseI-V relationship is described by an exponential function.Once the diode is forward biased, a small change in inputvoltage can produce a large change in output current. Thisnonlinear behaviour can be a source of serious simulationerrors that can manifest themselves as an incomplete or non-convergent solution, or worse still, an incorrect solution. Inorder to continue to use the physics-based equations fora nonlinear device and overcome the problem of strongnonlinear behaviour, the technique of parameterisation wasintroduced in Rizzoli and Neri [12]. This is illustrated byconsidering the example of a microwave diode.

The conventional current equation for a diode isexpressed as

i(t) = Is[exp(αv(t))− 1

], (2)

where v(t) is the voltage across the diode, Is is the reversesaturation current, and α is a parameter representing theslope of the conductance curve of the diode characteristic.Exponential relationships of this type are responsible formost of the numerical instabilities in circuit simulation. To

Active and Passive Electronic Components 3

−1 −0.5 0 0.5 1 1.5 2 2.5−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

X (Volts)

I(A

mp

eres

)

Figure 2: Parameterised I-X relationship for a diode.

−1 −0.5 0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

X (Volts)

V(V

olts

)

Figure 3: Parameterised V-X relationship for a diode.

circumvent this, the state variable in (2) is changed fromvoltage to a fictitious nonconventional variable, x(t). Thisstate variable is identical to the junction voltage below somenonzero threshold value, say V0, and is defined as a linearfunction of the current in (2) above V0. Care is taken toensure that the current and its derivatives are continuous atx = V0. The equations for diode voltage and current are nowexpressed as

v(t) =⎧⎪⎨⎪⎩V0 +

1α

ln{1 + α[x(t)−V0]}, V0 ≤ x(t),

x(t), x(t) ≤ V0,

i(t) =⎧⎨⎩Is exp(αV0){1 + α[x(t)−V0]} − Is, V0 ≤ x(t),

Is[exp(αv(t))− 1

], x(t) ≤ V0,

(3)

Itotal

Itotal

JVj

+

Vtotal

Vtotal

Cj

R0

−

+

−

+

−

Figure 4: Esaki tunnel diode circuit model.

V

I

CjdV/dx

IC = dq/dt = CJ (dV/dx)(dx/dt)

Guess x

Guess dx/dt

VR = ItotalRS

RS

Vtotal

Itotal

Figure 5: Evaluation of total current and voltage as a function ofthe state variable.

where (3) is the exact parametric representations of the diodeI-V characteristic. The highly nonlinear I-V characteristicis now transformed into current-state-variable (I-X) andvoltage-state-variable (V-X) characteristics that are not asstrongly nonlinear, without any changes to the essentialmathematical model of the device. The solution is wellbehaved and local convergence or heuristics within the diodemodel are not required. After the transformations of (3),the reduced nonlinear nature of the relationship is illustratedin Figure 2, the I-X relationship, and in Figure 3, the V-Xrelationship, with Is = 100 aA, α = 38.686, and V0 = 0.8 V.

3. Modelling the Esaki Tunnel Diode

The intrinsic Esaki tunnel diode model consists of an idealtunnel diode a nonlinear capacitor and a nonlinear resistor,see Figure 4. There are two controlling variables in thismodel, which are an as yet unspecified time-dependent statevariable, and its time derivative. The ideal tunnel diode ismodelled using current and voltage as a function of theprimary state variable x. The capacitance is modelled byusing the evaluated voltage and the time derivative of thestate variable, dx/dt, to calculate the derivative of charge as afunction of time or dq/dt. From this, the current through thecapacitor can be determined. The resistor is modelled usingthe evaluated voltage and the total current. An outline of themodel calculation of the total current and voltage is shownin Figure 5.

4 Active and Passive Electronic Components

The tunnel diode current density is typically described asthe sum of three exponential functions derived from quan-tum mechanical considerations. This formulation appearsin Sze [4], although here the physics is limited only tothe forward-bias direction. Referring to Figure 1, this isexpressed as

Jtotal = V(t)VP

JT + JX + JTH, (4)

where

JT = JP exp(

1− V(t)VP

), (5)

JX = JV exp(A2(V(t)−VV )), (6)

JTH = JS

(exp

(qV(t)kT

)− 1

). (7)

The first term is a closed-form expression of the tunnellingcurrent density which describes the behaviour particularto the tunnel diode. This includes the negative resistanceregion which captures the core functionality of the tunneldiode. The second term describes the excess tunnellingcurrent density while the third term is the normal diodecharacteristic. In (5), JP is the peak current density and VP

is the corresponding peak voltage. In (6), JV is the peakcurrent density and VV is the corresponding peak voltage.The parameter A2 represents an excess current prefactor.Finally, in (7), JS is the saturation current density, q is thecharge of an electron, k is Boltzmann’s constant, and T is thetemperature in degrees Kelvin.

As mentioned in the previous section, the exponentialfunctions can lead to a rapid change in current for arelatively small change in voltage which can create conver-gence issues for the simulator’s nonlinear solver. The tunneldiode equation cannot be solved for current as a singleand unique function of voltage. Recognising that the threemajor components of current density can be thought ofas three diodes with exponential I-V characteristics, theyare parameterised as previously described. After individualparameterisation of the three regions based on equatingcurrent density and first derivatives at threshold points, thetunnel diode equations become

JT(t) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

JP exp(

1− VT

VP

)×(

1− 1VP

(xT(t)−VT))

,

xT(t) ≤ VT ,

JP exp(

1− xT(t)VP

), xT(t) > VT ,

V(t) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

VT −VP × ln(

1− 1VP

(xT(t)−VT))

,

xT(t) ≤ VT ,

xT(t), x(t) > VT ,

JX(t) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

JV exp(A2(xX(t)−VV )), xX(t) ≤ VX ,

JV exp(A2(VX(t)−VV ))

×(1 + A2(xX(t)−VX)), xX(t) > VX ,

V(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

xX(t), xX(t) ≤ VX ,

VX +1A2× ln(1 + A2(xX(t)−VX)),

xX(t) > VX ,

JTH =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

JS exp(q xTH

kT

)− JS, xTH ≤ VTH,

JS exp(q VTH

kT

)×(

1 +q

kT(xTH −VTH)

)− JS,

xTH(t) > VTH,

V(t) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

xTH(t), xTH ≤ VTH,

VTH +kT

qln(

1 +q

kT(xTH(t)−VTH)

),

xTH > VTH.

(8)

As can be seen from the expressions above, each current isa function of a different x(t). Since the voltage across eachcurrent source is the same, the voltage equations can be setequal to each other to obtain a relationship between xT(t),xX(t), and xTH(t). The linear version of JT is used for xT(t) ≤VT while the linear versions of JX and JTH are respectivelyused for xX(t) > VX and xTH(t) > VTH. Therefore, whencombining the terms as a function of a single state variablex(t), they should be a function of xT(t) for xT(t) ≤ VT and afunction of xX(t) or xTH(t) for xX(t) > VX and xTH(t) > VTH,respectively.

Careful study of the various possible relationshipsbetween VT , VX , and VTH reveals that the transition betweenthe various regions will be continuous as long as VT ischosen to be less than or equal to VX and VTH. Using thisconstraint, continuous equations for the current densitiesand voltage as a function of a single x(t) are derived. Notethat min{VX ,VTH} has been used as a criteria to determinewhether the xX(t) or xTH(t) should be used for large x(t)where VX and VTH are calculated based on the first derivativeof the current. The expressions for the diode current densityand voltage as a function of a single state variable x(t) areexpressed as

JT(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

JP exp(

1− VT

VP

)×(

1− 1VP

(x(t)−VT))

,

x(t) ≤ VT ,

JP exp(

1− x(t)VP

), VT < x(t) ≤ VXTH,

AT2 × (1 + B(x(t)−VXTH))CT , x(t) > VXTH,

Active and Passive Electronic Components 5

JX(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

JV exp(A2(VT −VV ))

×(

1− 1VP

(x(t)−VT))−A2VP

, x(t) ≤ VT ,

JV exp(A2(x(t)−VV )), VT < x(t) ≤ VXTH,

AX2(1 + B(x(t)−VXTH))CX , x(t) > VXTH,

JTH(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

JS exp(qVT

kT

)

×(

1− 1VP

(x(t)−VT))−qVP /kT

− JS,

x(t) ≤ VT ,

JS exp

(qx(t)kT

)− JS, VT < x(t) ≤ VXTH,

ATH2 × (1 + B(x(t)−VXTH))CTH − JS,

x(t) > VXTH,

V(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

VT −VP × ln(

1− 1VP

(x(t)−VT))

,

x(t) ≤ VT ,

x(t), VT < x(t) < VXTH,

VXTH +1B× ln(1 + B(x(t)−VXTH)),

x(t) > VXTH,

(9)

which include a number of constant terms. The expressionsfor the voltage thresholds, VT , VX , and VTH, define theboundary between the current regions. These are derived bytaking the first derivative of the current equation and solvingfor voltage as a function of slope while maintaining currentcontinuity:

VT = VP −VP ln(−mTVP

JP

), slope mT < 0,

VX = VV +1A2

ln(

mX

A2JV

), slope mX > 0,

VTH = kT

qln

(mTHkT

JSq

), slope mTH > 0.

(10)

When VX ≤ VTH the other constants are,

VXTH = VX ,

AT2 = JP exp(

1− VX

VP

),

AX2 = JV exp(A2(VX −VV )),

ATH2 = JS exp(qVX

kT

),

B = A2,

CT = − 1A2VP

,

CX = 1,

CTH = q

kTA2,

(11)

and when VX > VTH,

VXTH = VTH,

AT2 = JP exp(

1− VTH

VP

),

AX2 = JV exp(A2(VTH −VV )),

ATH2 = JS exp(qVTH

kT

),

B = q

kT,

CT = − kT

qVP,

CX = A2kT

q,

CTH = 1.

(12)

The tunnel diode model is completed by incorporatingthe diode’s junction capacitance Cj . Referring to the param-eter values for the diode shown in Table 1, the model ofCj depends on whether the zero-bias depletion capacitanceparameter ��� is set. If it is set, then

Cj =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

���(1−Vj/��

)���� , −Vj > 0,

���+ Vj

(k1 + ���Vj

)

1 + k3+k2 k3

1 + k3, −Vj < 0,

(13)

where k1 = ��� × ����/��, k2 = ���/����0.2, k3 =exp(10(Vj − 0.8��)), and Vj is the diode junction voltage.If the depletion capacitance parameter, ��, is set, then Cj isincremented by the amount ��× exp(����Vj).

The parameterised I-V relationship using the constantslisted in Table 1 are compared with the original I-V equationand in Figure 6. For the sake of brevity, only the totalcurrent equation is plotted instead of the individual currentcomponents. As can be seen, the results are identical and theparameterisation process has not affected the accuracy of theunderlying equations for the diode.

The benefits of parameterisation are demonstrated byplotting total current and voltage with respect to the suitablyintroduced state variable x. As can be seen in Figure 7, whichplots tunnel diode voltage with respect to a state variable,and Figure 8, which plots tunnel diode current with respectto x, the highly nonlinear relationship between current andvoltage is replaced by two milder nonlinearities.

6 Active and Passive Electronic Components

Table 1: Esaki tunnel diode parameter table.

Parameter Description Default Units

�� Saturation current 1× 10−16 A

��� Zero bias depletion capacitance 0 F

�� Built-in barrier potential 0.8 V

� Capacitance power law parameter 0.5 —

��� Zero bias diffusion capacitance 0 F

�� Slope factor of diffusion capacitance 38.696 1/V

�� Series resistance in forward bias 0 Ω

� Intrinsic depletion layer time constant 0 sec

�� Area multiplier 1 —

�� Valley current 1× 10−4 A

�� Peak current 1× 10−3 A

�� Valley voltage 0.5 V

��� Peak voltage 0.1 V

� Excess current prefactor 30 —

� Tunnel current slope factor −1 —

� Excess current slope factor 1 —

�� Thermal current slope factor 1 —

��� Temperature 300◦ K

−0.2 0 0.2 0.4 0.6 0.8−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

V (Volts)

I(A

mpe

res)

Original current versus voltageParameterized current versus voltage

Figure 6: Original Esaki tunnel diode I-V characteristic comparedwith the parameterised I-V characteristic.

4. Simulation and Results

The parameterised Esaki tunnel diode model describedabove was implemented in an open-source circuit simulator(http://www.freeda.org/). Simulated I-V characteristics ofthe tunnel diode using the parameter values in Table 1are generated by sweeping the voltage from −0.05 voltsto 0.6 volts. This voltage range is the typical range ofoperation for this device. The simulated curve is comparedwith measured I-V characteristics and this is displayed inFigure 9. The diode used in measurements is a general-purpose germanium diode with a nominal peak voltage

−3 −2 −1 0 1 2 3−0.5

0

0.5

1

X (Volts)

V(V

olts

)

Figure 7: Esaki tunnel diode voltage with respect to state variable.

Vp of 105 mV, valley voltage Vv of 380 mV peak-to-valleycurrent ratio of 8. A match obtained between simulationand measurement thereby validates the effectiveness of thismodelling technique. In other words, it is possible to easilyand efficiently implement a highly nonlinear device keepingthe physics of the device intact while requiring no change tothe underlying algorithm that solves the nonlinear equationsassociated with the circuit.

A transient simulation was also performed on a canonicaloscillator circuit containing the tunnel diode to demonstratethe switching characteristics of the device. The circuit usedalong with the values of the components is shown inFigure 10. A snapshot of the simulated output voltageversus time in Figure 11 shows the circuit oscillating at

Active and Passive Electronic Components 7

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

X (Volts)

I(A

mpe

res)

Figure 8: Esaki tunnel diode current with respect to state variable.

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6−3

−2

−1

0

1

2

3

4

5

6×10−4

V (Volts)

I(A

mpe

res)

Simulated DC characteristicsMeasured DC characteristics

Figure 9: Simulated I-V characteristics of the Esaki tunnel diodecompared with measurement.

approximately 150 MHz. The values of the diode parametersused for this simulation that differ from those in Table 1 areindicated in Table 2.

5. Conclusions

Tunnelling is an important aspect of charge transport insemiconductor and molecular devices. While a semiconduc-tor tunnel diode with characteristics described by sums ofexponential functions was considered, the parameterisationtechnique introduced can be used with any analytic expres-sion. The central result is transcribing a circuit simulationproblem from one dealing with strong nonlinearities to oneworking with well-behaved, moderately nonlinear element

284

1 nF10 K16VDC 1 nH

R1

R2 R3 C3L3

−

+

Esaki diode

Figure 10: Oscillator circuit used for transient simulation.

2.9 2.92 2.94 2.96 2.98 3

×10−6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

V(V

olts

)

Figure 11: Simulated transient response of oscillator circuit.

Table 2: Nondefault Esaki tunnel diode parameters used intransient simulation.

Parameter Value

�� 2× 10−12

�� 1.1× 10−4

�� 0.35

��� 0.065

� 10

��� 0.03× 10−12

� 0.19

�� 0.0368

�� 1.5

characteristics. This is achieved without an increase inproblem size. The introduction of an alternative state (orstate variable) enables the strongly nonlinear character oftunnelling to be modelled, with full accuracy, by inter-mediate equations which are relatively well behaved andmoderately nonlinear. Thus, off-the-shelf numerics can beused in a circuit simulator and there is no need forlocal heuristics, homotopy, or functional approximation toobtain convergence. The fundamental requirement is that thecircuit simulator support state variables. The state variablesreplace the (usual) nodal voltages as the unknowns and theerror function becomes the energy norm rather than solelybased on Kirchoff ’s Current Law while the total number ofunknowns remains unchanged.

8 Active and Passive Electronic Components

References

[1] P. Olivo, T. N. Nguyen, and B. Ricco, “High-field-induceddegradation in ultra-thin SiO2 films,” IEEE Transactions onElectron Devices, vol. 35, no. 12, pp. 2259–2267, 1988.

[2] R. N. Hall, “Tunnel diodes,” IRE Transactions on ElectronDevices, vol. 7, no. 1, pp. 1–9, 1960.

[3] L. Esaki, “New phenomenon in narrow germanium p-njunctions,” Physical Review, vol. 109, no. 2, pp. 603–604, 1958.

[4] S. M. Sze, Physics of Semiconductor Devices, John Wiley & Sons,New York, NY, USA, 2nd edition, 1981.

[5] T. H. Kuo, H. C. Lin, U. Anandakrishnan, R. C. Potter,and D. Shupe, “Large-signal resonant tunneling diode modelfor SPICE3 simulation,” in Proceedings of the InternationalElectron Devices Meeting, pp. 567–570, Washington, DC, USA,December 1989.

[6] S. Mohan, J. P. Sun, P. Mazumder, and G. I. Haddad, “Deviceand circuit simulation of quantum electronic devices,” IEEETransactions on Computer-Aided Design, vol. 14, no. 6, pp.653–662, 1995.

[7] D. Neculoiu and T. Tebeanu, “SPICE implementation ofdouble barrier resonant tunnel diode model,” in Proceedingsof the International Semiconductor Conference (CAS ’96), pp.181–184, Bucharest, Romania, October 1996.

[8] C. E. Chang, P. M. Asbeck, K. C. Wang, and E. R. Brown,“Analysis of heterojunction bipolar transistor/resonant tun-neling diode logic for low-power and high-speed digitalapplications,” IEEE Transactions on Electron Devices, vol. 40,no. 4, pp. 685–691, 1993.

[9] M. Bhattacharya and P. Mazumder, “Augmentation of SPICEfor simulation of circuits containing resonant tunnelingdiodes,” IEEE Transactions on Computer-Aided Design ofIntegrated Circuits and Systems, vol. 20, no. 1, pp. 39–50, 2001.

[10] R. C. Melville, L. Trajkovic, S.-C. Fang, and L. T. Watson,“Artificial parameter homotopy methods for the DC operatingpoint problem,” IEEE Transactions on Computer-Aided Designof Integrated Circuits and Systems, vol. 12, no. 6, pp. 861–877,1993.

[11] C. E. Christoffersen, U. A. Mughal, and M. B. Steer, “Objectoriented microwave circuit simulation,” International Journalof RF and Microwave Computer-Aided Engineering, vol. 10, no.3, pp. 164–182, 2000.

[12] V. Rizzoli and A. Neri, “Expanding the power-handlingcapabilities of harmonic-balance analysis by a parametricformulation of the MESFET model,” Electronics Letters, vol. 26,no. 17, pp. 1359–1361, 1990.

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