2. Modelling and Simulation Technique

1)!.+1/z+),10!.z/! z)+ !(/\_z0\$+/!z.!z,,!.! z%\*z0\$!z!.(%!.z(%0!.01.!/\_z)5z!z %2¥ ided into two main categories:


\$+1#\$z0\$!z"1((z/(!z/%)1(0%+\*z%/z,,(%(!z0+z3% !z2.%!05z+"z/0.101.!/z\* z0\$!5z\*z(¥ low for realistic physical processes at large RF amplitudes, they are difficult to program and time consuming to run on a routine basis. In addition to this, numerical instability appears to be a common problem with these simulations, particularly at larger RF amplitudes.

The purpose of this article is to describe an alternative to the generally unrealistic and/or time consuming and complicated approach of earlier analytical and numerical methods. In essence, the present full-scale program employs a generalized, non-linear analysis of a p+ p n n+ type SiC based Double Drift IMPATT device, without any drastic assumption. In this self- +\*/%/0!\*0z/%\*#(!z".!-1!\*5z\*(5/%/z+"z 
z %+ !/\_z0\$!z)+ %"%! zi"%!( w)4%)1)z)!0\$¥ od'is used to obtain the detailed 'snap-shot' of the electric field, hole and electron current !\*/%05z/z"1\*0%+\*z+"z0%2!z.!#%+\*z3% 0\$z 1.%\*#z+\*!z+),(!0!z5(!z+"z/0! 5w/00!z+/%((¥ tion. This program takes into account the non-linear model that contains the differential equations for the carrier concentrations, current density equations, un-equal values of field !,!\* !\*0z..%!.z%+\*%60%+\*z.0!/\_z)+%(!z/,!z\$.#!z!""!0/\_z0\$!z!\$2%+.z+"z\$.#!z..%¥ ers and their interactions with electric filed as well as most of the physical effects, such as elevated temperature effects, parasitic effects etc., pertinent to IMPATT operation. With this program it is possible to obtain large signal admittance and impedance of the diode, RF power output as well as other important properties of the device at larger amplitudes of RF signal. A drift-diffusion model has been used for the large-signal analysis. A time varying electric field is assumed in the form, iV DVi *<sup>p</sup>* BV' *<sup>x</sup>* i *<sup>p</sup>* -#(V Mq.NV BV' *<sup>x</sup> <sup>2</sup>* i *<sup>p</sup>* -#(V M:q.NV B*………*, where i *<sup>p</sup>* is the DC peak electric field and ' *<sup>x</sup>* is the modulation factor. The value of ' *<sup>x</sup>* can be varied to study the effect of field modulation on the large signal properties of IMPATT device. The physical phenomena take place in the semiconductor bulk along the symmetry axis of the mesa structure of IMPATT diodes.

Figure 1. One-dimensional model of DDR IMPATT device.

/\_z!-1(z..%!.z2!(+%0%!/\_z%+\*%60%+\*z.0!/z+"z!(!0.+\*/z\* z\$+(!/\_z,1\*\$! z0\$.+1#\$z !,(!¥ tion layer boundaries, non-inclusion of mobile space charge effects. Consequently in their analysis, the generated power increases monotonically with the increasing RF amplitudes i.e. those analyses do not exhibit saturation effects and this constitutes an important limitation in applying those type of models. The authors have assumed a sinusoidal current at the input of the Si/SiC double drift device and obtained the corresponding voltage response to calculate 0\$!z !2%!z%),! \*!^z \*z0\$!z,.!/!\*0z,,!.z0\$!z10\$+.z\$/z"+.)1(0! zz/%),(!z\* z#!\*!.(¥ ized method for large-signal simulation of Si/SiC IMPATTs at 94 GHz based on current excitation at the input of the device. The large-signal impedance as a function of frequency has been

1)!.+1/z+),10!.z/! z)+ !(/\_z0\$+/!z.!z,,!.! z%\*z0\$!z!.(%!.z(%0!.01.!/\_z)5z!z %2¥

1. \*(50%(z)+ !(%\*#z+"z! z %+ !/z3%0\$z/+)!z/%),(%"%0%+\*/z\* z.!/0.%0%2!z//1),¥ tions, such as, equal carrier velocities, ionization rates of electrons and holes, punched 0\$.+1#\$z !,(!0%+\*z(5!.z+1\* .%!/\_z\*+\*z%\*(1/%+\*z+"z)+%(!z/,!z\$.#!z!""!0/^z+\*¥ /!-1!\*0(5z%\*z 0\$!%.z \*(5/%/\_z 0\$!z #!\*!.0! z ,+3!.z%\*.!/!/z)+\*+0+\*%((5z3%0\$z 0\$!z%\*¥ creasing RF amplitudes i.e. those analyses do not exhibit saturation effects and this

2. The second type of large signal model is defined as 'full-scale-simulation' with again

\$+1#\$z0\$!z"1((z/(!z/%)1(0%+\*z%/z,,(%(!z0+z3% !z2.%!05z+"z/0.101.!/z\* z0\$!5z\*z(¥ low for realistic physical processes at large RF amplitudes, they are difficult to program and time consuming to run on a routine basis. In addition to this, numerical instability appears to be a common problem with these simulations, particularly at larger RF amplitudes.

The purpose of this article is to describe an alternative to the generally unrealistic and/or time consuming and complicated approach of earlier analytical and numerical methods. In essence, the present full-scale program employs a generalized, non-linear analysis of a p+

n+ type SiC based Double Drift IMPATT device, without any drastic assumption. In this self- +\*/%/0!\*0z/%\*#(!z".!-1!\*5z\*(5/%/z+"z 
z %+ !/\_z0\$!z)+ %"%! zi"%!( w)4%)1)z)!0\$¥ od'is used to obtain the detailed 'snap-shot' of the electric field, hole and electron current !\*/%05z/z"1\*0%+\*z+"z0%2!z.!#%+\*z3% 0\$z 1.%\*#z+\*!z+),(!0!z5(!z+"z/0! 5w/00!z+/%((¥ tion. This program takes into account the non-linear model that contains the differential equations for the carrier concentrations, current density equations, un-equal values of field !,!\* !\*0z..%!.z%+\*%60%+\*z.0!/\_z)+%(!z/,!z\$.#!z!""!0/\_z0\$!z!\$2%+.z+"z\$.#!z..%¥ ers and their interactions with electric filed as well as most of the physical effects, such as elevated temperature effects, parasitic effects etc., pertinent to IMPATT operation. With this program it is possible to obtain large signal admittance and impedance of the diode, RF

p n

obtained by considering the fundamental frequency and higher harmonic terms.

constitutes an important limitation in applying this type of model.

2. Modelling and Simulation Technique

ided into two main categories:

338 Physics and Technology of Silicon Carbide Devices

some assumption.

\$!z "1\* )!\*0(z !2%!z!-10%+\*/z%^!^\_z+%/+\*j/z!-10%+\*z c!-10%+\*z cDdd\_z+\*0%\*1%05z!-1¥ 0%+\*/zc!-10%+\*/zcEd\zcFddz\* z1..!\*0z !\*/%05z!-10%+\*/zc!-10%+\*/zcGd\zcHddz%\*2+(2%\*#z)+¥ bile space charge in the depletion layer are simultaneously solved under large-signal condition 5z1/%\*#z,,.+,.%0!z+1\* .5z+\* %0%+\*/z5z1/%\*#zz +1(!z%0!.0%2!z"%!( z)4%)1)z+)¥ puter method described below. The fundamental device equations are given below.

$$\frac{d\mathcal{L}\left(\mathbf{x},t\right)}{d\mathbf{x}} = \frac{q}{\varepsilon} \left(N\_D - N\_A + p\left(\mathbf{x},t\right) - n\left(\mathbf{x},t\right)\right) \tag{1}$$

$$\frac{\partial p\left(\mathbf{x},t\right)}{\partial t} = -\frac{\partial J\_{\rho}\left(\mathbf{x},t\right)}{\partial \mathbf{x}} + q\left(n\left(\mathbf{x},t\right)a\_{\mathbf{x}}\left(\mathbf{x},t\right)\mathbf{v}\_{\mathbf{x}}\left(\mathbf{x},t\right) + p\left(\mathbf{x},t\right)a\_{\rho}\left(\mathbf{x},t\right)\mathbf{v}\_{\rho}\left(\mathbf{x},t\right)\right) \tag{2}$$

$$\frac{\partial n\left(\mathbf{x},t\right)}{\partial t} = \frac{\partial J\_n\left(\mathbf{x},t\right)}{\partial \mathbf{x}} + q\left(n\left(\mathbf{x},t\right)a\_n\left(\mathbf{x},t\right)\mathbf{v}\_n\left(\mathbf{x},t\right) + p\left(\mathbf{x},t\right)a\_p\left(\mathbf{x},t\right)\mathbf{v}\_p\left(\mathbf{x},t\right)\right) \tag{3}$$

$$J\_{\rho} \left( \mathbf{x}, t \right) = q p \left( \mathbf{x}, t \right) \nu\_{\rho} \left( \mathbf{x}, t \right) - q D\_{\rho} \left( \frac{\left\| p \left( \mathbf{x}, t \right) \right\|}{\left\| \mathbf{x} \right\|} \right) \tag{4}$$

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) <sup>2</sup> 1

*x V t x t dx* 

*x*

*B*

contacts) by setting up the appropriate restrictions in equations (1) – (5).

*n p J xt J xt n xt p xt t t*

, ,0

, , <sup>0</sup>

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,

The magnitude of peak field at the junction (i *<sup>p</sup>*), the widths of avalanche and drift zones (*x <sup>A</sup>* and *x <sup>D</sup>*; where *x <sup>D</sup> = d <sup>n</sup> + d <sup>p</sup>*), breakdown voltage (*V <sup>B</sup>*) and the voltage drops across these zones (*V <sup>A</sup>*, *V <sup>D</sup>* = *V <sup>B</sup> - V <sup>A</sup>*) are obtained from double iterative DC simulation program at *t = 0*. \$!z /\*,w/\$+0/z +"z !(!0.%z "%!( z \* z 1..!\*0z !\*/%05z ,.+"%(!/z %\*z 0\$!z !,(!0%+\*z (5!.z +"z 
¥ PATT diodes can be obtained from the simultaneous numerical solution of the basic device equations (equations (1) – (5)) subject to appropriate boundary conditions (equations (6)& (7)). A software package has been developed for simultaneous numerical solution of the above said equations. Boundary conditions are imposed at the contacts (at the *n <sup>+</sup> -n*& *p <sup>+</sup> -p*

\$!z!2(10%+\*z+"z1..!\*0z\* z2+(0#!z%\*z0%)!z\z/,!z +)%\*z+"z0\$!z !2%!z\* z0\$!z!-1%2¥ lent circuit has been obtained through the developed simulation program. The total terminal

> ( ) ( ) *d e* ( ) *dV t It C I t dt*

Where, *C <sup>d</sup>* is the depletion region capacitance of the diode, *V(t)* is the AC terminal voltage and *I <sup>e</sup> (t)* is the terminal current induced by the transport of carriers through the diode (Fig. 2). The optimized design parameters are obtained after several computer runs. Once the /\*,w/\$+0/z+"z!(!0.%z"%!( z\* z1..!\*0z,.+"%(!/z0z %""!.!\*0z0%)!z\z/,!z+2!.zz+),(!0!z5¥
