Keywords

1 Introduction

Impact ionization avalanche transit time (IMPATT) diode is used to generate the high power at microwave, millimeter wave and sub-millimeter wave regions [1]. It has high power capability compared to other diodes [2, 3]. Its operation provides a phase shift through avalanche and drift delays [4,5,6]. Phase noise due to phase shift causes a negative resistance in IMPATT. Operations at these regions are highly disturbed due to phase noise [7]. The negative resistance is arising from these delays as reported in [4,5,6, 8,9,10,11,12]. These delays are indirectly affecting the phase noise through negative resistance, and the phase noise degrades the conversion efficiency. But, the phase noise and DC-to-RF conversion efficiency mainly depend on base material [13]. Therefore, here, we have taken Si, Ge, GaAs, InP and WzGaN as base substrate material to study the conversion efficiency and the delay amount. Under actual operating condition, considerable amount of DC current flow through the diode and the space charges modifies the electric field distribution which in turn affects the current density profiles of electrons and holes by changing the magnitude of ionization rates. So, through simultaneous solution of the Poisson and current continuity equations (Eqs. 1 and 2) [14,15,16,17,18], using the computer simulation program, we can accurately investigate the conversion efficiency, delay amount, inductance-capacitance values in avalanche region and resonant frequencies of IMPATT diode. IMPATT has been modeled as shown in Fig. 1 [18,19,20,21] at 0.094–30 THz. In Fig. 1, field maximum (Em) is at X0, W comprises of drift layer width for electrons (dn), drift layer width for holes (dp) and avalanche layer width (XA), and J0 represents total current density.

Fig. 1
figure 1

Mathematical model of DDR IMPATT

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The Poisson’s equation and current continuity equation are, respectively, given by

$$\frac{\partial E\left( x \right)}{\partial x} = \frac{q}{\varepsilon }\left[ {N_{\text{D}} - N_{\text{A}} + p\left( x \right) - n\left( x \right)} \right]$$
(1)

and

$$- \frac{{\partial J_{n} }}{\partial x} = \frac{{\partial J_{p} }}{\partial x} = \alpha_{n} J_{n} + \alpha_{p} J_{p}$$
(2)

where \(E(x)\) and q are electric field profile and electronic charge, \(\varepsilon = \varepsilon_{0} \varepsilon_{\text{s}}\) is permittivity of semiconductor, \(N_{\text{D}}\) and \(N_{\text{A}}\) are donor and acceptor densities, \(n\) and p for electron and hole, and J and \(\alpha\) are current density and ionization rate [14].

2 Numerical Method and Design Parameters

To find the conversion efficiency, the numerical method is initiated from X0 with Em. Em and X0 are initially chosen suitably for the fitted doping profile and current density. Then, Eqs. (1) and (2) are solved simultaneously by taking space steps of very small width (0.001 nm). Iteration over Em and X0 is carried out till boundary conditions are being satisfied at x = 0 and W. Thus, electric field and current distribution are obtained from the computations. The method described above gives the avalanche breakdown characteristics and the avalanche layer width XA. Computer simulation using MATLAB is carried out, and the width of the epilayers is accordingly chosen using the transit time formula, Wn,p = ((0.5 × Vns,ps)/f) with π transit angle, where Vs and f are saturation velocity and frequency [17]. The breakdown voltage (VB) and avalanche drop (VA) are found using Eq. (3) [17, 22,23,24,25,26,27,28].

$$V_{\text{B}} = \int\limits_{0}^{W} {E(x){\text{d}}x \,{\text{and}}\,V_{\text{A}} = \int\limits_{{X{\text{A}}_{1} }}^{{X_{{{\text{A}}_{2} }} }} {E(x){\text{d}}x} }$$
(3)

Then, the drift voltage drop is VD = VB − VA. The DC-to-RF conversion efficiency is

$$\eta \left( \% \right) = \frac{{2mV_{\text{D}} }}{{\pi V_{\text{B}} }}$$
(4)

where m is the modulation index.

The avalanche response time is obtained by solving Eq. (5)

$$\tau_{\text{A}} \frac{\partial J}{\partial t} = - (J_{p} - J_{n} )|_{0}^{{X_{\text{A}} }} + 2 J\int\limits_{0}^{{X_{\text{A}} }} {\alpha {\text{d}}x}$$
(5)

where \(\tau_{\text{A}} = X_{\text{A}} /v_{s}\) is avalanche transit time with saturation velocity \((v_{s} )\) and J = \({\text{J}}_{n} + J_{p}\) [7].

Equivalent circuit of IMPATT is shown in Fig. 2, where G, B and Rs are conductance, susceptance and series resistance of diode, respectively. Further internal circuitry in terms of inductance and capacitance effect of the IMPATT in avalanche and drift region is shown in Fig. 3. The inductive and capacitive natures in the avalanche region are having parallel effect, but the inductance in the drift region is in series with drift region capacitance [12]. However, the effect of this series inductance is less as compared to drift capacitive impedance and can be neglected.

Fig. 2
figure 2

Equivalent circuit of IMPATT diode with packaging

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Fig. 3
figure 3

Inductive and capacitive nature at avalanche and drift region of IMPATT

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The avalanche inductance and capacitance are

$$\begin{aligned} L_{\text{A}} & = \frac{{\tau_{\text{A}} }}{{2 J_{0 } \alpha^{\prime}A}} \\ & = \frac{{\tau_{\text{A}} }}{{2 J_{s} \alpha^{\prime}A}}\left( {1 - \int\limits_{0}^{W} {\left\langle \alpha \right\rangle {\text{d}}x} } \right) \\ & = \frac{{\tau_{\text{A}} }}{{2 J_{s} \left( {\frac{\partial \alpha }{\partial E}} \right)A}}\left( {1 - \int\limits_{0}^{W} {\left\langle \alpha \right\rangle {\text{d}}x} } \right) \\ \end{aligned}$$
(6)
$$C_{\text{A}} = \frac{{\varepsilon_{s} A}}{{X_{\text{A}} }}$$
(7)

and the resonant frequency of this L-C combination is given by

$$\omega_{r} = 2 \pi f_{r} = \sqrt {\frac{{2 \alpha^{\prime}v_{s} J_{0} }}{{\varepsilon_{s} }}}$$
(8)

where \(J_{0 }\) is the DC current density, \(\alpha^{\prime} = (\partial \alpha /\partial E),\) A and \(\alpha\) are diode area and ionization integrand, respectively [7]. Drift response time is given by

$$\tau_{\text{D}} = \frac{{\left( {W - X_{\text{A}} } \right)}}{{v_{s} }}$$
(9)

and the drift capacitance is

$$C_{\text{Drift}} = \frac{{A \varepsilon_{s} }}{{\left( {W - X_{\text{A}} } \right)}}$$
(10)

Recently reported design parameters have been used in our numerical study [17].

3 Results and Discussion

The frequency, where maximum efficiency is obtained, is termed here as peak frequency. IMPATT’s efficiencies by using different materials are given in Table 1. The highest efficiency of 14.12% is obtained from InP at 20 THz, which is the maximum among all. InP has better efficiency at all corresponding frequencies. GaAs has the closest value to InP. Si and Ge have moderate level, whereas WzGaN has very low efficiency. Peak frequencies for GaAs, Si, Ge and WzGaN are 20 THz, 20 THz, 22 THz and 22 THz, respectively.

Table 1 Efficiency values over THz frequencies
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3.1 Avalanche Response Time Determination

The avalanche response time \((\tau_{\text{A}} )\) is determined for n+-n-p-p+ DDR IMPATT based on Si, Ge, WzGaN, GaAs and InP at window frequencies of 0.094–30 THz. Dependency of avalanche response time in frequencies is shown in Fig. 4. \(\tau_{\text{A}}\) value of InP is much higher than that of Si, Ge, GaAs and WzGaN. Si- and WzGaN-based IMPATTs are having lowest \(\tau_{\text{A}}\) values and both are close to each other in the range of 0.5–0.9 ps at the corresponding frequencies. GaAs and Ge are having in the range of 1.3–1.8 ps. As Si and WzGaN are having the lowest values, the charge carriers are quickly generating more number of carriers and can produce high frequency power. Due to higher avalanche response time, InP cannot produce high frequency power. However, InP is having highest DC-to-RF conversion efficiencies as shown in Table 1.

Fig. 4
figure 4

Avalanche response time as a function of frequency

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3.2 Drift Response Time Calculation

A comparative drift response time with frequency variation is shown in Fig. 5. In the drift region, InP-based charge carriers are taking less time to reach the n-p junction as compared to its Si, Ge, WzGaN and GaAs counterparts. From the obtained results, it can be understood that though WzGaN and Si have higher drift response time, still those values are close to that of others.

Fig. 5
figure 5

Dependency of drift response time on frequency

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3.3 Drift Capacitance and Resonant Frequency

In the THz band frequencies, drift capacitance and resonant frequency in IMPATT are computed and listed in Tables 2 and 3. The drift capacitance is increasing with frequency for all the materials. Ge-based IMPATT has higher drift capacitance variation from 0.0685  to 0.1113 F, and its WzGaN counterpart is having lower variation from 0.0061 to 0.0142 F. InP is having lower resonant frequencies as compared to others at the corresponding frequencies. From these variations of drift capacitance, we can get an idea about the time constant of the device.

Table 2 Computed drift capacitance
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Table 3 Computed resonant frequency
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4 Conclusion

Maximum conversion efficiencies are obtained from the InP-based IMPATT as compared to Si, Ge, GaAs and WzGaN at the THz band of 0.094–30 THz. Since no experimental results are available based on response time determination, inductance and capacitance profiles measurement and efficiency calculation over the entire frequency range of THz band for all the materials, no comparison could be made. However, these results can be helpful for the practical realization of IMPATT and to improve the IMPATT performance based on doping, avalanche and drift widths selection with proper semiconducting material.