A DETAILED ANALYTICAL STUDY OF NON-LINEAR SEMICONDUCTOR DEVICE MODELLING

This paper presents a detailed analytical study of Gunn, SCR, and p-n junction and of the physical processes that occur inside. Based on the properties of these devices, models for Gunn, SCR, and p-n 
junction diode have been developed. The results of computer simulated examples have been presented in each case. The non-linear lumped model for Gunn is a unified model as it describes the diffusion effects as the-domain traves from cathode to anode. An additional feature of this model is that it describes the domain extinction and nucleation phenomena in Gunn dioder with the help of a simple timing circuit. The non-linear lumped model for SCR is general and is valid under any mode of operation in any circuit environment. The memristive circuit model for p-n junction diodes is capable of simulating realistically the diode’s dynamic behavior under reverse, forward and sinusiodal operating modes. The model uses memristor, the charge-controlled resistor to mimic various second-order effects due to conductivity modulation. It is found that both storage time and fall time of the diode can be accurately predicted.


INTRODUCTION
Non linearity is an inherent property of semiconductor devices.Modern devices like the Gunn, SCR, IMA, and IMPATT devices, etc., are strongly non-linear.These devices exhibit complex dynamic behaviors.In order to design a circuit involving these devices, it does not suffice just to know the terminal behavior of the device.It becomes all the more necessary to design the device or model it properly.Though no general theory on device modelling is available, we can establish a set of systematic approximations to proceed from a description of the general subregion to a lumped model for the subregion.In the present analysis, first a general subregion is selected.Next, it is determined what physical processes and morphologic characteristics dominate the behavior of the subregion.Next, the form of equations is postulated that define the dominant factors on distributed basis, that is, with independent space and time variables.The postulated equations are combined.Approximations are then made, which in essence permit the removal of dependence on continuous space variables.The result is that the device is now approximated by a set of differential equations with independent variable time.The techniques available for network analysis and synthesis can be used to generate a symbolic model that will be network-like in form and therefore amenable to all the analysis and physical reasoning used with networks.
(/) Device physics analysis and partitioning It consists of identifying the important physical variables, phenomena, and operating mechanism through a careful analysis of the device physics.This information can be used to partition the internal structure of the device into as many geometrically distinct regions as possible, such that each unit can be modelled separately.
(ii) Physical equation formulation It consists of formulating the relevant physical equations that relate the internal physical variables in the partitioned units to each other and to the external terminal currents and voltages.
(iii) Equation simplification and solution It consists of solving the equations formulated in step (ii).Since these equations are generally non-linear partial differential equations, no explicit closed form solution is generally possible.Lumped approximations are obtained by reducing the partial differential equations to ordinary dif- ferential equations.
(iv) Non-linear network synthesis If step (iii) cannot be reduced to a lumped model, then step (iv) consists of synthesising a circuit model that uses both lumped and distributed (transmission lines) circuit elements.
(B) Black Box Approach If device physics and operating mechanisms are not well understood or when the device is so complex that a physical approach is not possible, we resort to this approach.The following basic steps are involved in the Black Box Approach.
(i) Experimental observation (ii) Mathematical modelling (iii) Model validation (iv) Non-linear network synthesis.The present study is restricted to physical approach only.

CONCEPT OF A DEVICE MODELITS CHARACTERISTICS
Let D describe a device.It is physically impossible to measure the generally infinite collection F(D) of admissible voltage--current signal pairs.Our approach is to develop a device model M(D) made of some well defined set of ideal circuit elements so that each admissible voltage current pair associated with M(D) represents a good approximation to a corresponding (v(t), i(t)) measured from D.
A circuit model of D is hence not an equivalent circuit of D because no physical device can be exactly mimicked by a circuit or mathematical model.
In fact, depending on the application (e.g., amplitude and frequency of operation) a given device may have many distinct models.The best model in the given situation is the simplest model capable of yielding realistic solutions.

Two Terminal Elements and Their Constitutive Relations
Although it is impossible to measure and store all admissible voltage-current signal pairs f(d) associated with a device D, we can define a hypothetical or ideal circuit element whose associated collection f() of admissible voltage-current pairs is generated by a prescribed algorithm hereforth called the constitutive relation of .
The constitutive relationship of a two-terminal element may be expressed in any convenient compact form.The constitutive relation of the three standard linear elements (R,L and C) can be expressed by non-linear current-controlled resistor can be described by v v(i) A non-linear voltage-controlled resistor can be described by i(v) or implicitly by fR (v,i)= 0 The nonlinear inductor and capacitor can be described by fL(*,i)=0 and f(q,)=0 For more complex elements, the constitutive relation will generally consist of a system of non-linear, algebraic, differential, (ordinary or partial) and/or integral equations involving not only the variable v and i, their higher order derivatives, and higher order integrals but also internal variable, their derivatives, and integrals.

Lumped Versus Distributed Element
A two-terminal element is said to be lumped if its constitutive relation can be expressed by a finite number of equations involving only algebraic, ordinary differential and integral operation on the instantaneous values of the terminal variables (v,i) and/or a finite number of additional internal variables (X1, X 2 Xn).Otherwise is said to be distributed.
In non-linear network synthesis the specification is the prescribed constitutive relation and the solution is a network having 2-external terminals made of a prescribed set of building blocks.

Four Basic Elements
A two-terminal element is called a 2-terminal Resistor, Inductor, Capacitor, Memristor if its constitutive relation can be expressed by an algebraic relationship denoted symbolically by fR(u,i,t) 0 v-i;R fl(,i,t) 0 -i;L f(q,v,t) 0 q-v;C fm(q,0,t) 0 q-qJ;M Memristor--The Mixing Element A IV element called memristor has been postulated to fill the missing link in the 4-element diagram.A charge controlled memristor can also be described by v R(q) where d(q) R(q) dt is called the memristor.

NON-LINEAR LUMPED CIRCUIT MODEL FOR GUNN
Introduction to Gunn and its various modes In a voltage-controlled DNR like Gunn, the decrease in current density with an increase in electric field is brought about by decreasing drift velocity of electrons by increasing the field.
The current density is proportional to the density of carriers as well as their drift velocity (J nr Vd).The mechanism that causes a fall in drift felocity of electrons with the rise in electric field across the sample as it crosses over a limit of some thousand volts/cm is a well known RWH mechanism.In a Gunn, when field exceeds certain threshold value, DNR is observed.
The product noL gives the condition for formation of stable domains.The criterion for different modes of operation is (i) if noL < 1012, table domains are not formed, though space charge accumula- tions are formed.This is called space charge accumulation mode.
(ii) noL > 1012, if a resonant circuit is connected across the Gunn, various other modes of operation are possible.
(a) Transit Time Mode When the fL product is approximately equal to the average carrier drift velocity (fL is 10 7 cm/sec) the sample operates in this mode.In this mode, the high field domain is nucleated at the cathode and travels the full lengths of samples to the anode.
Frequency of oscillation is given by f Udm L* L* effective length of sample (b) Quenched Domain Mode (fL > 2 10 7 cm/sec) A bulk Ga-As oscillator can oscillate at frequencies higher than the transit time frequency if the high field domain is quenched before it reaches the anode.Dipole domain quenching occurs when the bias voltage across the samples is reduced below Vs.The frequency of oscillation is frequency of the resonant circuit.
(c) Inhibited Mode If the total voltage across an RWH device is below threshold at the instant a dipole disappears at the anode, the formation of a new dipole is delayed until the voltage rises above threshold.If the new domain is inhibited from starting for a time equal to the domain transit time, then the waveform will be approximately a square wave.The frequency of resonant circuit is tuned to half the transit time frequency.

(d) LSA Mode
In a uniformly doped semiconductor without any internal space charges, the internal field would be uniform and simply proportional to the applied voltage.The current would be proportional to the drift velocity at this field level.Hence, current-voltage characteristics of the device would be the same in shape as the velocity field curve.If this is coupled to an external resonant circuit, this could excite oscillations at the frequency of a resonant circuit.In this mode, the electric field across the diode rises from below threshold and falls back again so quickly that space charge distribution associated with high-field domain does not have enough time to form.Only primary layers form near the cathode, while the rest of the samples remains fairly homogeneous.

Assumptions made in the Non-linear Model developed for Gunn
This model is a verfied circuit model.It however, makes the following assumptions (i) The parameters associated with the model are fixed for each device, regardless of the mode of operation or external circuits.(ii) There exists a high field domain with a single local maximum that propagates without change of shape with a domain velocity VD(t) from cathode to anode.
(iii) Quasi-Static assumption We introduce a new variable y as the distance measured in a system of coordinates moving in the direction of the electron flow with the domain velocity VD(t) Let us take a 1-D structure of length L, area A, dielectric constant , and a uniform donor concentration no (Fig. l(a)), noL > 1012 enables support of high-field domains consisting of an accumulation layer with carrier concentration n > no and a depletion layer with carrier concentration n < no (Fig. l(b)).The corresponding dipole-induced field distribution is shown in (Fig. l(c)).These figures give shapes of n(x) and F(x) at one instant of time.As the domain grows in size, it propagates from the cathode (x = O) to the anode (x L) with an instantaneous velocity vD(t).Electric field distribution inside Gunn Diode at one instant of time.
Shapes of n(x) and ,E(x) at different instant of time are governed by complex dynamics and affect device current and voltage as we shall see n(x) na(X One can write, na=na(E (Eo, Em) and n d n0(E) (E o, E m as single valued function of E. Description of the Model--Evaluation of Associated Model Parameters and Lt fo l x2(t) Xl(t) Idt T ---) oo (2) Terminal current ID of the controlled current source depends on the instantaneous value of capacitor voltage Vl(t) and V2(t) along with the device terminal current I(t) in accordance with the following non-linear relationship (5) na(E), nd(E), Em,D(E)are found for each value of (,2)as: Accumulation layer carrier concentration na(E) dE (7) For each given value of (V1,V2) na(E) is obtained by solving the following scalar non-linear ordinary differential equation where, Na(na, E;VI,VD) A na[V(E VD ] + n o q (n a no)naD'(E) / q/E (n a no)D(E) (8(d)) Since 8(a) is a non-linear scalar first-order differential equation, 8(c) and 8(d) cannot be satisfied simultaneously for arbitrary values of VD.Consequently, to solve for n=n(E) we must choose an optimum value for VD in order that 8(c) and 8(d) are satisfied.Hence, we must solve a two-point boundary problem giving both VD and n(E) as its solution.
To solve this two-point boundary value problem, we would use a 'shooting method', na(E) na(E, Vl, Em) na(Vl, V2) for E m Era(V1, V2) Depletion Layer conc.nd (E): For each given value of (Vl, V2) nd(E) V obtained by solving following the 'initial value problem' V D VD(VI,V2) (9d) V D is already obtained from equation 8(a) for given (V, V2).n d depends on both V1 and E so, nd(E) rid(E; V1; Em) Equations ( 7), ( 8) and (9) are coupled together; hence, we must solve them by iterative methods (as simultaneous equations) Equation ( 7) is only a scalar algebraic equation and can be solved by Newton- Raphson or the Secant method.Once na(E),no(E) and E m are found, F(V, W2) can be found.

Derivation of the Model
Let us call region O < x x x 2 < I as a domain region that contains a high-field domain at any instant t.The remaining region where n no and E Eo is referred to as the 'outside region'.Current density J2 (inside the domain) J1 (outside the region) 0E 0 0Eo qn v(E) / E Ot -q xx D (E)n q noV(Eo) / E Ot (10) Integrating both sides of (11) from X to X 2 (12 Let E has a single max at X X 3 of E Em value using Poisson's equation:- Let us derive an expression relating the external device current I to the internal field.Total current outside the domain, I Aq noV (Eo) + EA dEo dt (16) Corresponding voltage across the device is given by These equations 15, 16, and 17 completely describe the dynamics of high-field domain in terms of external device current I and V.
Applying KCL at upper node, I Aq noV (Vl/L) -EA dv L dt At lower node, Substituting V1/L for Eo we get Hence, our model is consistent with device physics.Also, Finally n, n d satisfy following condition, n(X1) na (Eo) no n(X3) na (Em) no n(X2) na(Eo) n o Since the high field domain grows as it propagates from the cathode (X O) to anode (X L) with an instantaneous velocity VD(t), the electric field E and the electron density n depend upon both 'time' t and 'space' variable x.The exact form of n a, rid, and E must be obtained by solving the partial differential equation.Here we would make some simplifying assumptions.For the general field--dependent diffn, case, the external device current I can be expressed as, Let us introduce, y & X ft Vo (t) dt, t' t O This was our initial assumption (ii) Aq q aY a(Dn) E aE ay nv(E) nov(V1/L) V D q (18) 0E 0E 0X q (n-no) This completes our derivation of model.

Derivation of Simplified Models
Let us recast (20a) into (1 Integrating both sides from E o to E we get, Accumulation layer: Substracting for n-nd in we can find E m by which gives F(V1,V2) and thus I D 2. Zero diffusion case D(E) 0.
We can solve for G(Em;V1,V2)= 0. This gives E m as Em -- Complete Lumped Circuit Model Including Domain Extinction (at the anode) and

Nucleation
The Model developed so far is valid as long as the domain is extinguished by the external circuit constraints before it reaches the anode such as in the quenched domain mode or when a nature domain does not exist such as in LSA mode.In the transit time mode or delayed domain mode, the above is valid only during the time the domain has not reached the anode.
In order to model the domain extinction at the anode, we devised a timing cirucit that tracks the domain motion and causes the domain capacitor to discharge quickly, wherever the domain reaches the anode.This timing cirucit is added to the original Gunn Model.

Timing Circuit
It has one non-linear-controlled current source I c and three voltage-controlled resistors RI(V), R2(V2) and R3(V3).These resistors behave like relays and should be described by a separate sub-routine using logic statements.Hence, considrable computer time can be saved.
I c Aq noVD (Vl, V2) Average time constant for a domain to collapse.
These components are defined below along with their relationships in terms of the junction voltage (a) Diffusion current components: Idl Is11 (e v/vT 1) hole injected from J1.
into N and which diffused across N 1.
into P2 and which diffused across P2.
into N 2 and which diffused across N 2.
The parameters Iii (j 1 to 6) are constants that depend on the intrinsic parameters of the device.
(b) Recombination current components outside the depletion region: Io 'Ylldl fraction of holes injected from J, which recombine with electrons in N1.
Io 2 "y lid2 fraction of holes injected from J2, which recombine with electrons in N.
A memristor can be considered therefore as a charge controlled linear resistor.
W Rm(qm The controlled current source Vj -1 + 11 (i,ij,vj,qm) Ill p(i) + I1. c(Vi) max (Va,Vb)   Iqml + Is'r q(Vi) 1 + (lil / I)'r p(-Vi) % max (%, d) Cd(V) + Q (v i) Let us consider the diode circuit shown in Fig. 4a.The switch S is thrown from right to left at t=to and before to the diode is in steady state with a current i(t) IF,  Let us consider a p-n junction (N A > > ND, step junction).Cj is used to represent effective transition layer capacitance, il is used to simulate the leakage carriers through the layer.R m is used to simulate the conductance modulation of neutral region i2 is used to represent the recombination of carriers Non-linear junction capacitance Ci(Vi) Ka

T v)
K a A(2q ND) 1/2 Memristor R m (qm) N A > > ND so conductance of p-region is much greater than that of n-type, i.e., resistance of diode is mainly contributed by the n-type region (the base region), conductivity tr (x) of base region is given by r(x) qnxo + lapp (x) Let us consider the one dimensional steady-state diffusion equation p,2 P'n (x) P'n (x)   OtX 2 Lp 2 region.
O, P', (x) P,(xl) pno excess hole concentration in n-type For large S'p solution of steady state diffusion, the equation is Ph The stored excess minority charge is W Aq P(x)dx Aq P(o) Lp Cosh (Wn/L)-1 Sin h (Wn/Lp) q =a P(x)= AqLp Sinh (Wn/LI,) The stored excess minority charge in steady state may be q' but in a non-steady state it is qm when q'm :/: O. qm explains the conductivity modulation in base region.P(x) q'P ( Sinh (Wn/Lt,) _1) [ Cosh ( x ) The controlled current source 2 12 (qm) from equation (41): q Aq p(o) Lp Sinh (Wn/Lp)

Lp
Coth -p where we have used the identity L 2 Dp'rp.On the other hand, the diode current P is given by -Aq Dp jpn(X) Using the model and circuit parameters given in Fig. 9 for the quenched domain, the steady-state voltage waveforms V(t), I(t), Vl(t), VE(t) and Va(t) are shown in Figs. 10 (a), (b), (c), (d) and (e).
The operation of the quenched domain mode can be illustrated with the help of where the domain voltage VE(t) drops to zero, thereby switching RE(V2) to a short circuit.This causes Va(t) to drop instantaneously to zero and remain so until Vl(t)   rises above the normalized threshold voltage V1 1 at point (4), thereby switching RI(V1) to an open circuit.Thereafter, the domain voltage VE(t) begins to build up while the time-base voltage Va(t) rises to keep track of its motion.In sharp contrast with the operation of the preceding examples, however, observe that the domain voltage VE(t) becomes zero at point (1) before the domain reaches the anode.Consequently, R3(V3) remains an open-circuit for all times and the timing circuit therefore plays no role in determining the waveforms in the quenched domain mode.The term 'quench' is used to emphasize that the high-field domain in this case is extinguished by the waveforms (V(t)) and not by discharging at the anode.

LSA Mode
The waveforms corresponding to the preceding examples for the LSA mode are shown in Fig. 11.Here we observe that Vl(t), because V2(t) 0 everywhere except over a very small time interval where V2(t) increases to no longer than 0.01.Consequently, R2(V2) becomes a short circuit most of the time, and the domain can be assumed to be almost non-existing.Since the time-base voltage V3(t) is also almost zero, R3(V3) remains an open circuit for all times in the LSA mode.
(B) For SCR- Fig. 12 shows a family of IA versus VA curves computed numerically for a typical SCR.They resemble the basic static characteristics.current IG(t) shown in Fig ( 13 c).The associated IG versus vG relationship is shown in Fig ( 13 d).Note that it is a multi-valued curve.This multi-value relationship is seen to persist even at extremely low operating frequencies.In other words, the SCR is truly a "dynamic" device.The model parameters chosen for this example Multivalued VG-IG relationship.
FIGURE 1 (a): A one-dimensional Gunn Diode (b): Carrier concentration at one instant of time (c);

Functions
and a voltage v(t) Ef.Reverse transient waveform is shown in Fig 4(a) and (b).

Fig. 10 .
Fig. 10.Unlike the preceding examples, let us start with point (1) in Fig (10.C),where the domain voltage VE(t) drops to zero, thereby switching RE(V2) to a short circuit.This causes Va(t) to drop instantaneously to zero and remain so untilVl(t)

FIGURE 9
FIGURE9 Complete Gunn-Diode model for zero diffusion.

FIGURE 10
FIGURE 10 Waveforms associated with Gunn-Diode operating under the Quenched Domain mode.
Memristive diode model is prescribed by p physical parameter { to, , ND, an, Pp, lno, pno, A, W n, Dp, "rp, V, I } and two empirical parameters Dynamic behavior of junction diodes during reverse transients The