1. Introduction

Among the microwave and MM-wave solid-state devices,the impact Avalanche transit time (IMPATT) device has emerged as the most efficient and powerful source of MM-wave power. Most of the current research activities for MM-wave systems are focused on the design and development of IMPATT devices at MM-wave window frequencies,i.e.,35,94,140,220 GHz,where atmospheric attenuation is relatively low. These devices are finding important applications in tracking radars,missile guidance,battle field communication,collision avoidance systems and radiometers. IMPATT diodes based on Si,GaAs,InP have been experimentally realized to provide sufficient power at MM-wave frequencies^[1]. For realizing higher RF power ($P_{\rm RF})$ from an IMPATT device,one should choose a semiconductor material that has a higher value of critical electric field ($E_{\rm c})$,saturated drift velocity ($v_{\rm s})$ & thermal conductivity ($K$),since $P_{\rm RF}$ from an IMPATT device is proportional to $E_{\rm c}^{2}v_{\rm s}^{2}$. The excellent material properties of WBG semiconductors suggest that WBG-semiconductor-based IMPATT devices are the future MM-wave sources^[2]. With the advent of new technologies for growth of SiC crystals,researchers are showing renewed interest in exploring the possibilities of extracting more power from SiC-based IMPATT devices. The experimental research on the development of SiC-IMPATTS is underway^[3]. On the other hand,IMPATT device technology based on Si is well established over a wide frequency range. The authors have therefore chosen Si/SiC-based hetero-structure IMPATT diodes,to simulate the large signal properties of the device at W-band (75-110 GHz). The authors have developed a generalized technique based on a self-consistent model for large-signal simulation of SiC DDR IMPATT devices.

Several methods for the large-signal analysis of IMPATTs and other negative resistance devices are reported in References ^{[4, 5, 6, 7, 8]}. Earlier reported large-signal modelling is basically analytical modelling of read-diodes with some simplifications and restrictive assumptions,such as,equal carrier velocities,ionization rates of electrons and holes,punched through depletion layer boundaries,non-inclusion of mobile space charge effects^[9]. 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 the device impedance. In the present paper the authors have formulated a simple and generalized 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 obtained by considering the fundamental frequency and higher harmonic terms.

2. Modeling and simulation technique

The study of the large signal effects in IMPATT diodes is a significant aid in the accurate design and application of these devices as MM-wave oscillators. Thus the subject of large signal analysis of IMPATT diodes has received considerable attention over the years^{[4, 5, 6, 7, 8, 9]}. Analytical investigations of large signal performance are rendered difficult by the highly non-linear process occurring within the device and so numerical solution only could give an adequate treatment of large signal operation.

The purpose of this paper 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$^{+}$ pnn$^{+}$ type Si/SiC-based double drift IMPATT device,without any drastic assumption. In this self-consistent single frequency analysis of IMPATT diodes,the modified `field-maximum method'<sup>[10]</sup> is used to obtain the detailed `snap-shot' of the electric field,hole and electron current density as a function of the active region width during one complete cycle of steady-state oscillation. 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-dependent carrier ionization rates,mobile space charge effects,the behaviour of charge carriers and their interactions with electric field as well as most of the physical effects,such as elevated temperature effects,parasitic effects etc.,pertinent to IMPATT operation^[11]. The analysis takes into account the experimentally reported values of field-dependent carrier-ionization rates and drift velocities of charge carriers at 300 K $<$ $T$ $<$ 550 K^[12]. This method is applicable to all types of device structure,doping profile and different frequencies of operation of IMPATTs.

2.1 Details of large-signal simulation methodology

A drift-diffusion model has been used for the large-signal analysis. A sinusoidal electric field is assumed in the form,$E = E_{\rm m} + mE_{\rm m}$sin($\omega t$),where $E_{\rm m}$ is the DC peak field maximum,and $m$ is the modulation factor. The modulation factor is varied to study the effect of variation of RF voltage on the large-signal properties of the IMPATT diode. One dimensional large-signal simulation of a SiC IMPATT diode is presented in the paper assuming that the transport phenomena take place in the semiconductor bulk along the symmetry axis of the mesa structure of the device. The fundamental device equations i.e.,Poisson's equation,current density equation and continuity equation involving mobile space charge in the depletion layer are simultaneously solved under large signal conditions by using appropriate boundary conditions by using a double iterative field maximum computer method described^[11]. The equivalent circuit of DDR SiC IMPATT diode used for this simulation is shown in Figure 1. The analysis provides an electric field profile at various instances of time during a cycle of oscillation. A computer program for single frequency analysis of large-signal characteristics of the IMPATT diode is developed. In this method,impedance and admittance plane plots as a function of excitation frequency and RF voltage amplitude can be more compactly expressed and easily interpreted compared to the results obtained from the earlier analysis^{[4, 5, 6, 7, 8]}. The diode is biased with a DC voltage and driven by an AC voltage through a coupling capacitor,shown in Figure 2,i.e.,$V_{\rm D} = V_{\rm dc} + V_{\rm rf}$ $\times$ $\sin(\omega t +\phi)$,where $\omega$ is the operating frequency,and $\phi$ is initial phase angle. From this the current response is calculated and Fourier analyzed and the device impedance is simulated using the fundamental component of current. The total terminal current (AC plus DC) of an IMPATT diode is given by:

$I(t) = -C_{\rm d}\frac{{\rm d}V_{\rm t}}{{\rm d}t}+I_{\rm e}(t),$

(1)

where $C_{\rm d}$ is the depletion region capacitance of the diode,and $V_{\rm t}$ is the AC terminal voltage. For a sinusoidal voltage excitation,the first term is sinusoidal,the second term is far from sinusoidal for large signals. As the negative conductance of the device stems from the conductive component of the current,the second term has special interest.

Once the snap-shots of electric field and current profiles at different time & space over a complete cycle are obtained,the optimized values have been used for simulation of large-signal impedance and admittance characteristics of the device at W-band. The diode admittance (including the depletion region capacitance $C_{\rm d})$ at the oscillation frequency $\omega$,is obtained in terms of the fundamental frequency components of current & voltage and the terminal voltage $V(t)$. The final expression of the diode admittance $Y_{d}$ in terms of fundamental frequency component is as follows:

$Y_{\rm d} ={\rm j}\omega C_{\rm d}-\frac{I_{\rm e}(\omega)}{V_1(\omega)},$

(2)

where

$$\begin{split} {}& I_{\rm e}(\omega,t)=I_{\rm dc}+A_0\sin \omega \left(t+\frac{\tau_{\rm d}}{2}\right), \\& V_{\rm 1}(\omega,t)=\frac{A_0I_{\rm d}}{\varepsilon \omega A_{\rm r}}\cos \omega \left(t+\frac{\tau_{\rm d}}{2}\right)-\frac{2A_0I_{\rm d} \sin \left(\dfrac{\omega \tau_{\rm d}}{2} \right)}{\omega^2 \tau_{\rm d}\varepsilon A_{\rm r}} \\[2mm]& \quad \quad \quad -\left(\frac{\tau_{\rm a}E_{\rm c}I_{\rm d}}{2m} \right)\frac{A_0\omega \cos \omega \left(t+\dfrac{\tau_{\rm d}}{2} \right)} {I_{\rm dc}+A_0\sin \omega \left(t+\dfrac{\tau_{\rm d}}{2} \right)}. \end{split}$$

(3)

Also,$A_{0}=I_{\rm RF}\dfrac{\frac{\omega \tau_{\rm d}}{2}}{\sin \left(\frac{\omega \tau_{\rm d}}{2} \right)}.$ Finally,

$$\begin{split} Y_{\rm d} ={}& {\rm j}\omega C_{\rm d} \\& + {\rm j}\omega C_{\rm d} A_0I_{\rm RF} \exp(-{\rm j}\omega \tau_{\rm d}/2) \left\{A_0^2-A_0^2 \frac{\sin (\omega \tau_{\rm d}/2)}{(\omega \tau_{\rm d}/2)} \right. \\& \left. \exp (-{\rm j}\omega \tau_{\rm d}/2)-2I_{\rm dc}\frac{\omega^2}{\omega a^2} [I_{dc}-\sqrt{(I_{dc}^2-A_0^2)}]\right\}^{-1}. \end{split}$$

(4)

The symbols used in the equations are: $l_{\rm d}$ and $l_{\rm a}$ $=$ drift and Avalanche region lengths. $\tau_{\rm d}$ and $\tau_{\rm a}$ $=$ transit time of drift and avalanche region. $I_{\rm e}(t) = I_{\rm dc} + I_{\rm RF} \sin (\omega t +\phi').$ $A_{\rm r}$ $=$ diode area.

The large signal simulation is carried out by considering 500 space meshes and varying time meshes. The simulated results are presented in this paper. At peak operating frequency $f_{\rm P}$,the maximum power output $P_{\rm RF}$ from the device is obtained from the expression:

$P_{\rm RF}= (V_{\rm rf})^{2 } \cdot \vert -G_{\rm P}\vert \cdot (A_{\rm r}/2) ,$

(5)

where $G_{\rm P}$ is negative conductance at peak frequency.

At resonance,the reactance of the resonant cavity is mainly capacitive in nature. When the magnitude of negative conductance of the diode $\vert -G_{\rm P} \vert$ is equal to the load conductance $G_{\rm L}$,the condition of resonance is satisfied and as a result,power is absorbed in $G_{\rm L}$ and at the same time oscillation starts to build up in the circuit. Adlerstein \textit{et al}. developed a method for determining $R_{\rm S}$ from the threshold condition of IMPATT oscillation,under small-signal oscillation conditions^[14]. However,Adlerstein's approach involved several assumptions like equal ionization rates and equal drift velocities of charge carriers. In the present method the authors have determined the value of series resistance ($R_{\rm S})$ from the admittance characteristics using a realistic analysis of Gummel-Blue^[15] and Adlerstein et al.^[14]. The steady state condition for oscillation is given by:

$G_{\rm L} (\omega)=\vert -G (\omega) \vert - [B (\omega)]^{ 2} R_{\rm S} (\omega),$

(6)

where $G_{\rm L}$ is the load conductance. This relation provides minimum uncertainty in $G_{\rm L}$ at low power oscillation threshold. Under the small signal condition,$V_{\rm RF}$ (amplitude of the RF swing) is taken as assuming a small signal (2 %) modulation of the breakdown voltage $V_{\rm B}$. For such a small value of $V_{\rm RF}$,$R_{\rm S}$ can be calculated from Equation (6),considering the value of $G_{\rm L}$ nearly equal to the diode conductance ($-G$) at resonance. It may also be noted that -$G_{\rm P}$,$B_{\rm P}$ and $G_{\rm L}$ are normalized to the area of the diode.

3. Results and discussions

The optimized structural parameters of Si/SiC DD hetero-structure IMPATTs for operation at W-band are given in Table 1 for the bias current density of 8 $\times$ 10$^{8}$ A/m$^{2}$. Table 2 shows the values of series resistance ($R_{\rm S,total})$ of Si/SiC DD hetero-structure IMPATT at 94 GHz. It has been observed that the DC breakdown voltage is 207.6 V for the diode at current density 8.0 $\times$ 10$^{8}$ A/m$^{2}$. Figures 3(a)-3(c) denote the voltage and current waveform for the SiC IMPATT device operating under large-signal conditions. The plots depict the 180$^\circ$ phase-shift between voltage and terminal current,essential criteria for IMPATT oscillation. This proves the validity of the simulation software. Large-signal simulation provides the snap-shots of electric-field profiles at different phase angles as shown in Figures 4(a) to 4(e). The electric field increases from $t=$ 0 and attains its peak value in the positive half-cycle at $t =T/4 (E_{\rm max}$ $=$ 5 $\times$ 10$^{8}$ V/m) as shown in Figure 4(b). It then decreases and attains the same magnitude of negative peak in the negative half-cycle at $t = 3T/4,$ as shown in Figure 4(d). The program is also run for the second and consecutive cycles and it is observed that the above nature of variation of the electric field is repeated in each and every cycle.

The effect of voltage modulation on the large-signal negative resistance,reactance,RF power,efficiency,negative conductance and $Q$-factor of the device has been studied and the results are presented in this paper. Large-signal admittance plots (conductance versus susceptance) for different modulation factors are shown in Figure 5. It is observed that the magnitude of peak negative conductance decreases from 37.5 $\times$ 10$^{6}$~S/m$^{2}$ at 94 GHz to 5.0 $\times$ 10$^{6}$ S/m$^{2}$ when voltage modulation increases from 10 % to 50 %,i.e.,corresponding RF voltage increases from 20.0 to 104.0 V. In the limiting case of RF-voltage being very low,the large-signal peak negative conductance value should approach the small-signal value. When RF voltage modulation is very low,i.e.,2 % and the RF voltage amplitude is 4.0 V,the simulated large-signal negative conductance is 43.5 $\times$ 10$^{6}$ S/m$^{2}$. The authors have also carried out small-signal simulation of the Si/SiC device based on Gummel-Blue approach^[15] and obtained the admittance plot with the same design structural parameters to verify whether the peak negative conductance under small-signal conditions approaches that under large-signal conditions with negligibly small voltage modulation of 2 %. It is observed from Figure 5 that the large-signal admittance plot for lowest voltage modulation almost coincides with the simulated small-signal admittance plot which verifies the validity of the proposed large-signal modeling of the device.

Figure 6 shows the variation of RF output power with RF voltage. It is interesting to observe that under large-signal conditions RF power initially increases with the increasing voltage modulation,reaches a peak value at 50 % voltage modulation and then decreases with further increase of voltage modulation. Figure 6 shows the variation of efficiency with RF voltage. It is observed that the efficiency increases with increase in RF voltage,attains a peak value corresponding to 50 % voltage modulation and then starts decreasing. It is observed that,with the increasing amplitude of RF voltage from 21.0 to 104.0 V,the magnitude of negative resistance of the device decreases from 10.0 to 6.0 $\Omega$. The magnitude of negative reactance increases from 13.0 to 14.5 $\Omega$ when RF voltage increases from 10 % to 50 %. The variation of negative reactance with RF voltage is sharper than that of the negative resistance with RF voltage. At the 94 GHz window,there is an increase in $Q$-factor from 1.0 to 6.0 with change in RF voltage from 21.0 to 104 V,as expected. Large-signal $Q$-factor for a particular RF voltage indicates the overall RF performance of the device.

4. Conclusion

The authors have developed a generalized technique for large-signal simulation of a DDR SiC IMPATT diode. This simulator is applicable for other WBG-semiconductor-based IMPATTs and also for different structures and doping profiles of the device. The validity of the proposed technique is verified from the simulated small-signal admittance plot. The results show for DC breakdown voltage of 207.0 V the large-signal (for $\sim$ 50 % voltage modulation) power output and efficiency are 25.0 W and 15.0 %,respectively. To the best of the authors' knowledge,this is the first report on non-linear analysis of 4H-SiC IMPATTs at W-band.