Leapfrogging solitary waves in coupled traveling-wave field-effect transistors

Abstract

Leapfrogging solitary waves are characterized in two capacitively coupled traveling-wave field-effect transistors (TWFETs). The coupling implies that a nonlinear solitary wave moving on one of the devices is bounded with the wave moving on the other one, which results in the periodic amplitude/phase oscillation called leapfrogging. In conservative systems, the oscillation energy of leapfrogging is converted to the generation of radiative waves; hence, leapfrogging is shown to cease in finite duration. This study investigated the possibility of amplification of moving solitary waves to stabilize leapfrogging in coupled TWFETs. First, coupled Korteweg–de Vries equations with perturbation were derived to verify the limit-cycle dynamics of the soliton’s amplitude and phase corresponding to the stable leapfrogging. We then numerically solved the transmission equations of the coupled TWFETs to validate stable leapfrogging in practical situations.

Appendix: Derivation of the coupled KdV equations

In this Appendix, we obtained the coupled KdV equations with the perturbing terms through the reductive perturbation method [24]. First, long-wavelength approximation is applied to the transmission equations for T$_{1}$ and $T_{2}$, resulting in

$$\begin{aligned}&L_g\{C_{gd}+C_{gs}(V_1)+C_m\}\frac{\partial ^2V_1}{\partial t^2}\nonumber \\&\qquad +\,L_g\frac{dC_{gs}(V_1)}{dV_1}\left( \frac{\partial V_1}{\partial t}\right) ^2\nonumber \\&\qquad +\,R_g\{C_{gd}+C_{gs}(V_1)+C_m\}\frac{\partial V_1}{\partial t}\nonumber \\&\qquad -\,R_gC_{gd}\frac{\partial W_1}{\partial t} - L_gC_{gd}\frac{\partial ^2W_1}{\partial t^2}\nonumber \\&\qquad -\,C_mL_g\frac{\partial ^2 W_2}{\partial t^2} - C_m R_g\frac{\partial W_2}{\partial t}\nonumber \\&\quad =\frac{\partial ^2 V_1}{\partial x^2}+\frac{1}{12}\frac{\partial ^4 V_1}{\partial x^4} + \frac{1}{360}\frac{\partial ^6 V_1}{\partial x^6}, \end{aligned}$$

(26)

$$\begin{aligned}&L_d\{C_{gd}+C_{ds}(W_1)\}\frac{\partial ^2W_1}{\partial t^2}\nonumber \\&\qquad +\,L_d\frac{\mathrm{d}C_{ds}(W_1)}{\mathrm{d}W_1}\left( \frac{\partial W_1}{\partial t}\right) ^2\nonumber \\&\qquad +\,R_d\{C_{gd}+C_{ds}(W_1)\}\frac{\partial W_1}{\partial t}\nonumber \\&\qquad -\,L_dC_{gd}\frac{\partial ^2V_1}{\partial t^2}\nonumber \\&\qquad -\,R_dC_{gd}\frac{\partial V_1}{\partial t}+L_d\frac{\mathrm{d}I_{\mathrm{d}s}(V_1,W_1)}{dV_1}\frac{\partial V_1}{\partial t}\nonumber \\&\qquad +\,L_d\frac{\mathrm{d}I_{ds}(V_1,W_1)}{\mathrm{d}W_1}\frac{\partial W_1}{\partial t}+R_dI_{ds}(V_1,W_1)\nonumber \\&\quad =\frac{\partial ^2 W_1}{\partial x^2}+\frac{1}{12}\frac{\partial ^4 W_1}{\partial x^4} + \frac{1}{360}\frac{\partial ^6 W_1}{\partial x^6}. \end{aligned}$$

(27)

We introduce new coordinates $(\tau , \xi )$ as $\tau =\epsilon ^{3/2}t$ and $\xi =\epsilon ^{1/2}(x-ut)$ and expand $V_i$ ($i = 1, 2$) as

$$\begin{aligned} V_{i}= & {} V_{G0}+\sum _{n=1}^\infty \epsilon ^nv_{i}^{(n)}, \end{aligned}$$

(28)

$$\begin{aligned} W_{i}= & {} V_{D0}+\sum _{n=1}^\infty \epsilon ^nw_{i}^{(n)}. \end{aligned}$$

(29)

The transmission equations are then expanded in series with respect to $\epsilon$. We assume herein that $R_{g, d}$ and $I_d$ are $\epsilon ^{3/2}$ order parameters and $C_m$ is an $\epsilon$ order. By $O(\epsilon ^2)$ terms, u is shown to satisfy the condition

$$\begin{aligned} u=u_\pi \equiv \sqrt{\frac{X_2-\sqrt{X_2^2-4L_gL_dX_1}}{2L_gL_dX_1}}, \end{aligned}$$

(30)

where $X_1=C_{gd}C_{GS0}+C_{GS0}C_{\mathrm{DS}0}+C_{\mathrm{DS}0}C_{gd}$ and $X_2=(C_{\mathrm{DS}0}+C_{gd})L_d+(C_{GS0}+C_{gd})L_g$. Moreover, $v_{i}^{(1)}$ ($i=1, 2$) is shown to be proportional to $w_{i}^{(1)}$, i.e.,

$$\begin{aligned} v_{i}^{(1)}=A_\pi w_{i}^{(1)}, \end{aligned}$$

(31)

where $A_\pi =C_{gd}L_gu_\pi ^2/\{(C_{gd}+C_{GS0})L_gu_\pi ^2-1\}$. The model transmission equation is straightforwardly obtained by the $O(\epsilon ^3)$ terms. The resulting equation for $\psi _i^\prime =\epsilon w_i$ ($i=1, 2$) is

$$\begin{aligned}&\frac{\partial \psi _1^\prime }{\partial t}+u_\pi \frac{\partial \psi _1^\prime }{\partial x}-\frac{\alpha u_c^2u_\pi ^5L_gL_dC_{gd}}{2A_\pi (u_c^2-u_\pi ^2)} \psi _1^\prime \frac{\partial \psi _1^\prime }{\partial x}\nonumber \\&\qquad +\,\frac{u_\pi }{24}\frac{\partial ^3 \psi _1^\prime }{\partial x^3}\nonumber \\&\quad =\frac{u_c^2u_\pi ^4L_gL_dC_{gd}}{2A_\pi (u_c^2-u_\pi ^2)}I_{ds}(V_{G0}+A_\pi \psi _1^\prime ,V_{D0}+\psi _1^\prime )\nonumber \\&\qquad -\,\left( \frac{u_c^2u_\pi ^2C_{gd}L_dA_\pi }{u_c^2-u_\pi ^2}\frac{L_gR_d-L_dR_g}{2L_gL_d}+\frac{R_d}{2L_d}\right) \psi _1^\prime \nonumber \\&\qquad +\,\frac{C_mC_{gd}u_\pi ^2X_1^{-3/2}}{2u_c(u_c^2-u_\pi ^2)\sqrt{L_gL_d}}\left( \frac{\partial \psi _2^\prime }{\partial x}-A_\pi \frac{\partial \psi _1^\prime }{\partial x}\right) ,\nonumber \\ \end{aligned}$$

(32)

where

$$\begin{aligned} u_c= & {} \sqrt{(X_2+\sqrt{X_2^2-4L_gL_dX_1})/2L_gL_dX_1}, \end{aligned}$$

(33)

$$\begin{aligned} \alpha= & {} \frac{mA_\pi ^3C_{GS0}}{V_{G0}-V_\mathrm{SDG}+V_{J}}+\frac{mC_{\mathrm{DS}0}}{V_{\mathrm{DS}0}-V_\mathrm{SDD}-V_{J}}.\nonumber \\ \end{aligned}$$

(34)

We simplify the coefficients by introducing other coordinates $s=u_\pi t/2$ and $z=12^{1/3}(x-u_\pi t)$ and a scaled variable $\psi _i = \nu _0\psi _i^\prime$ ($i = 1, 2$), where

$$\begin{aligned} \nu _0 = \frac{12^{1/3}\alpha u_c^2u_\pi ^4L_gL_dC_{gd}}{6A_\pi (u_c^2-u_\pi ^2)}. \end{aligned}$$

(35)

We then obtain the following from Eq. (32)

$$\begin{aligned} \frac{\partial \psi _1}{\partial s}-6\psi _1\frac{\partial \psi _1}{\partial z}+\frac{\partial ^3\psi _1}{\partial z^3}=F_1(\psi _1,\psi _2), \end{aligned}$$

(36)

where $F_1(.)$ represents the effect of the drain current and line resistances to the wave transmission. The explicit form is given by

$$\begin{aligned}&F_1(\psi _1,\psi _2)\nonumber \\&\quad =\frac{12^{1/3}\alpha u_\pi }{6}\left\{ \frac{u_c^2u_\pi ^3L_gL_dC_{gd}}{A_\pi (u_c^2-u_\pi ^2)}\right\} ^2\nonumber \\&\quad \quad \times I_{ds}\left( V_{G0}+\nu _0^{-1} A_\pi \psi _1,V_{D0}+\nu _0^{-1}\psi _1\right) \nonumber \\&\quad \quad -\,\left( \frac{u_c^2u_\pi C_{gd}A_\pi }{u_c^2-u_\pi ^2}\frac{L_gR_d-L_dR_g}{L_g}+\frac{R_d}{u_\pi L_d}\right) \psi _1\nonumber \\&\quad \quad +\,\frac{12^{1/3}C_mC_{gd}u_\pi X_1^{-3/2}}{u_c(u_c^2-u_\pi ^2)\sqrt{L_gL_d}}\left( \frac{\partial \psi _2}{\partial z}-A_\pi \frac{\partial \psi _1}{\partial z}\right) .\nonumber \\ \end{aligned}$$

(37)

Similarly, the transmission equations of $T_2$ are given by

$$\begin{aligned}&L_g\{C_{gd}+C_{gs}(V_2)\}\frac{\partial ^2V_2}{\partial t^2}+L_g\frac{dC_{gs}(V_2)}{dV_2}\left( \frac{\partial V_2}{\partial t}\right) ^2\nonumber \\&\qquad +\,R_g\{C_{gd}+C_{gs}(V_2)\}\frac{\partial V_2}{\partial t}-L_gC_{gd}\frac{\partial ^2W_2}{\partial t^2}\nonumber \\&\qquad -\,R_gC_{gd}\frac{\partial W_2}{\partial t}\nonumber \\&\quad =\frac{\partial ^2 V_2}{\partial x^2}+\frac{1}{12}\frac{\partial ^4 V_2}{\partial x^4}+\frac{1}{360}\frac{\partial ^6 V_2}{\partial x^6}, \end{aligned}$$

(38)

$$\begin{aligned}&L_d\{C_{gd}+C_{ds}(W_2)+C_m\}\frac{\partial ^2W_2}{\partial t^2}+L_d\frac{\mathrm{d}C_{ds}(W_2)}{\mathrm{d}W_2}\nonumber \\&\qquad \times \left( \frac{\partial W_2}{\partial t}\right) ^2+R_d\{C_{gd}+C_{ds}(W_2)+C_m\}\frac{\partial W_2}{\partial t}\nonumber \\&\qquad -\,L_dC_{gd}\frac{\partial ^2V_2}{\partial t^2}-R_dC_{gd}\frac{\partial V_2}{\partial t}-C_mL_d\frac{\partial ^2 V_1}{\partial t^2}\nonumber \\&\qquad -\,C_m R_d\frac{\partial V_1}{\partial t}+L_d\frac{\mathrm{d}I_{ds}(V_2,W_2)}{\mathrm{d}V_2}\frac{\partial V_2}{\partial t}\nonumber \\&\qquad +\,L_d\frac{\mathrm{d}I_{ds}(V_2,W_2)}{\mathrm{d}W_2}\frac{\partial W_2}{\partial t}+R_dI_{ds}(V_2,W_2)\nonumber \\&\quad =\frac{\partial ^2 W_2}{\partial x^2}+\frac{1}{12}\frac{\partial ^4 W_2}{\partial x^4}+\frac{1}{360}\frac{\partial ^6 W_2}{\partial x^6}. \end{aligned}$$

(39)

The same procedure gives

$$\begin{aligned}&\frac{\partial \psi _2^\prime }{\partial t}+u_\pi \frac{\partial \psi _2^\prime }{\partial x}-\frac{\alpha u_c^2u_\pi ^5L_gL_dC_{gd}}{2A_\pi (u_c^2-u_\pi ^2)} \psi _2^\prime \frac{\partial \psi _2^\prime }{\partial x}+\frac{u_\pi }{24}\frac{\partial ^3 \psi _2^\prime }{\partial x^3}\nonumber \\&\quad =\frac{u_c^2u_\pi ^4L_gL_dC_{gd}}{2A_\pi (u_c^2-u_\pi ^2)}I_{ds}(V_{G0}+A_\pi \psi _2^\prime ,V_{D0}+\psi _2^\prime )\nonumber \\&\qquad -\,\left( \frac{u_c^2u_\pi ^2C_{gd}L_dA_\pi }{u_c^2-u_\pi ^2}\frac{L_gR_d-L_dR_g}{2L_gL_d}+\frac{R_d}{2L_d}\right) \psi _2^\prime \nonumber \\&\qquad +\,\frac{C_mC_{gd}u_\pi ^2X_1^{-3/2}}{2A_\pi u_c(u_c^2-u_\pi ^2)\sqrt{L_gL_d}}\left( A_\pi \frac{\partial \psi _1^\prime }{\partial x}-\frac{\partial \psi _2^\prime }{\partial x}\right) .\nonumber \\ \end{aligned}$$

(40)

Using coordinates s and z and scaled variables, we obtain

$$\begin{aligned} \frac{\partial \psi _2}{\partial s}-6\psi _2\frac{\partial \psi _2}{\partial z}+\frac{\partial ^3\psi _2}{\partial z^3}=F_2(\psi _1,\psi _2), \end{aligned}$$

(41)

where the source-term function $F_2(.)$ is defined as

$$\begin{aligned} F_2(\psi _1,\psi _2)= & {} \frac{12^{1/3}\alpha u_\pi }{6}\left\{ \frac{u_c^2u_\pi ^3L_gL_dC_{gd}}{A_\pi (u_c^2-u_\pi ^2)}\right\} ^2\nonumber \\&\times I_{ds}\left( V_{G0}+\nu _0^{-1} A_\pi \psi _2,V_{D0}\right. \nonumber \\&\left. +\,\nu _0^{-1}\psi _2\right) \nonumber \\&-\,\left( \frac{u_c^2u_\pi C_{gd}A_\pi }{u_c^2-u_\pi ^2}\frac{L_gR_d-L_dR_g}{L_g}\right. \nonumber \\&\left. +\,\frac{R_d}{u_\pi L_d}\right) \psi _2\nonumber \\&+\,\frac{12^{1/3}C_mC_{gd}u_\pi X_1^{-3/2}}{A_\pi u_c(u_c^2-u_\pi ^2)\sqrt{L_gL_d}}\left( A_\pi \frac{\partial \psi _1}{\partial z}\right. \nonumber \\&\left. -\,\frac{\partial \psi _2}{\partial z}\right) . \end{aligned}$$

(42)

For symmetrical TWFETs, we set $L_g = L_d\equiv L_0$, $C_{GS0} = C_{\mathrm{DS}0} \equiv C_0$, and $R_g=R_d = R_0$. In addition, both $C_{gs}$ and $C_{ds}$ are biased at a common reverse bias $V_B$, i.e., $V_{G0}-V_\mathrm{SDG} = V_\mathrm{SDD}-V_{\mathrm{DS}0} \equiv V_B$. Subsequently, $u_c$, $u_\pi$, $\alpha$, and $A_\pi$ become $1/\sqrt{L_0C_0}$, $1/\sqrt{L_0(C_0+2C_{gd})}$, $-2mC_0/(V_B+V_J)$, and $-1$, respectively. Equations (37) and (42) are simplified to

$$\begin{aligned}&F_1(\psi _1,\psi _2)=-\frac{mC_0\sqrt{L_0}}{12^{2/3}(C_0+2C_{gd})^{3/2}(V_B+V_J)}\nonumber \\&\quad \times I_{ds}\Biggl (V_{G0}-\frac{12^{2/3}(C_0+2C_{gd})(V_B+V_J)}{2mC_0}\psi _1,\nonumber \\&\quad V_{D0}+\frac{12^{2/3}(C_0+2C_{gd})(V_B+V_J)}{2mC_0}\psi _1\Biggr )\nonumber \\&\quad -\,R_0\sqrt{\frac{C_0+2C_{gd}}{L_0}}\psi _1\nonumber \\&\quad +\,\left( \frac{3}{2}\right) ^{1/3}\frac{C_m}{C_0+2C_{gd}}\left( \frac{\partial \psi _1}{\partial z}+\frac{\partial \psi _2}{\partial z}\right) , \end{aligned}$$

(43)

$$\begin{aligned}&F_2(\psi _1,\psi _2)=F_1(\psi _2,\psi _1). \end{aligned}$$

(44)