1.
Introduction

Since graphene was successfully found in 2004^[1], it has attracted a lot of interest due to its fascinating physical and chemical properties. The two-dimensional lattice and Dirac-like electronic spectrum lead to unique physical properties^{[2, 3]}, such as the quantum Hall effect^[4], optical absorption^[5] and so on. Those interesting features suggest that graphene is a promising candidate for electronic and thermoelectric devices, such as field-effect transistors^[6], photodetectors^[7], transparent electrodes for displays^[8] and so on.



Double-layer graphene consists of two monolayers of graphene. Pronounced negative differential conductance (NDC) has been found in double-layer graphene^[9]. NDC has also been predicted in diodes with graphene/boron nitride heterochannels^[10], in double-barrier resonant tunneling diodes^[11], and in graphene nanoribbon superlattices with resonant tunneling through the minibands and the Wannier–Stark ladder regime^[12]. NDC stimulates numerous potential applications in graphene-based frequency multipliers, fast switches, and, especially, terahertz frequency oscillators.



While linear phenomena in graphene are a focus of current research, nonlinear transport effects are much less studied. The redistribution of the charge carriers in the momentum and energy space induced by the radiation incident on the device results in nonlinear effects and causes time- and space-dependent oscillations as well as steady-state and spatially homogeneous ones. These effects are very important and informative when studying nonequilibrium optical and electronic processes and provide information about their fundamental properties. Naturally, nonlinear effects in graphene have attracted attention^{[13, 14]}, where a number of phenomena, including second and third harmonic generation^[15], frequency mixing^[16], photon drag effects^[17], photocurrents in graphene p–n junctions^[18], and the optical Kerr effect^[19] etc, have been observed and studied. Compared to ordinary semiconductor systems, the microscopic mechanisms of nonlinear effects in graphene are quite different. These effects can be strongly enhanced due to its high electron velocity and the linear dispersion in graphene. The study of nonlinear transport provides access to various properties of graphene. These make graphene a promising material for the development of novel electronic devices for terahertz applications. In this paper, we present a study of the nonlinear dynamics of the n⁺nn⁺ double-layer graphene diode driven by a DC bias and a THz signal, based on the time-dependent hydrodynamic equations.

2.
Time-dependent hydrodynamic equations for double-layer graphene diode

For spatially inhomogeneous systems, electron transport can be investigated with the following hydrodynamic equations^[20]:

$frac{{partial n}}{{partial t}} = frac{1}{q}frac{{partial {j_{ m{n}}}}}{{partial x}},$

(1)

${j_{ m{n}}} = q{mu _{ m{n}}}left(frac{{{K_{ m{B}}}T}}{q}frac{{partial n}}{{partial x}} + frac{{{K_{ m{B}}}n}}{q}frac{{partial {T_{ m{e}}}}}{{partial x}} - nfrac{{partial phi }}{{partial x}} ight),$

(2)

$frac{{partial (nvarepsilon )}}{{partial t}} = - frac{{partial (nS)}}{{partial x}} - {j_{ m{n}}}frac{{partial phi }}{{partial x}} - nW,$

(3)

and the Poisson equation:

$frac{{{partial ^2}phi }}{{partial {x^2}}} = - frac{q}{epsilon }left(n - {N_{ m{d}}} ight).$

(4)

in which $n$ is the electron density,${j_{
m{n}}}$ is the current density, $epsilon $ is the static dielectric constant, $phi $ is the electrostatic potential, ${mu _{
m{n}}} = {V_{
m{d}}}/E$ is the electron mobility, ${N_{
m{d}}}$ is the doping concentration, $varepsilon $ is the electron energy related to the electron temperature ${T_{
m{e}}}$, $S$ is the energy flow density, and $W$ is the energy loss rate. The boundary conditions for $phi (x,t)$ and $n(x,t)$ are as follows:

$phi ({x_{ m{L}}},t) = 0,quad phi ({x_{ m{R}}},t) = V(t),$

(5)

$n(x_{ m{L}},t) = N_{ m{d}}(x_{ m{L}}),quad n(x_{ m{R}},t) = N_{ m{d}}(x_{ m{R}}),$

(6)

$V(t) = {V_{{ m{dc}}}} + {V_{{ m{ac}}}}sin (2pi {f_{{ m{ac}}}}t),$

(7)

where ${V_{{
m{ac}}}}$ is the driving amplitude and ${f_{{
m{ac}}}}$ is the driving frequency of the AC voltage. By numerically solving the time-dependent hydrodynamic equations, we can find $phi (z,t),n(z,t)$ and ${T_{
m{e}}}(z,t)$, and all other useful quantities such as $E(z,t)$, $v(z,t)$ and $j(z,t)$. The total current density is the sum of the conduction current density and the displacement current density

$J(t) = j(z,t) + {epsilon _{ m{s}}}frac{{partial E(z,t)}}{{partial t}}.$

(8)

Here we present the detailed calculation for double-layer graphene diode by solving the hydrodynamic balance equations. The n-base length of the n⁺nn⁺ double-layer graphene diode is set to be $l = 0.5$ μm and the lattice temperature is $T = 300$ K.

3.
Transitions between periodic and chaotic states in double-layer graphene

Double-layer graphene has shown pronounced NDC effects, as shown in Fig. 1. When double-layer graphene is biased, the electrons acquire drift velocity. Above a threshold value of the field, the current decreases when the electric field increases. NDC varies with the voltage or current applied to the device. With applied DC voltage, n⁺nn+ graphene diodes exhibit current self-oscillation with frequencies up to terahertz. With realistic scattering mechanisms, which are more sophisticated than the drift-diffusion model, we examine the transitions between the periodic and chaotic states with different chaos-detecting methods, such as the Poincaré bifurcation diagram and the first return map.






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Figure1.
(Color online) Analytical fit for the velocity-field relation of double-layer graphene at lattice temperature T = 300 K.




When the double-layer graphene diode is subjected to both DC and AC biases, the system is typically nonlinear, far from the equilibrium state. The NDC effect is indispensable for producing a spatiotemporal travelling domain. Under periodic THz radiation, self-oscillating systems represent one important nonlinear dynamical system. The spatiotemporal behavior is strongly dependent on the frequency and amplitude of the external signal, i.e., coupling of the DC and AC biases.



First, we choose the ac amplitude ${V_{{
m{ac}}}}$ as the control parameter to study the nonlinear dynamics of the double-layer graphene diode. In the simulations, ${f_{{
m{ac}}}} = G ;{f_{
m{s}}}$, in which $G$ is the inverse golden mean $[ equiv (1 + sqrt 5 )/2 = 1.618 ldots ]$, and ${f_{
m{s}}}$ is the self-oscillating frequency. Under this ratio, the driving frequency is incommensurate and various nonlinear dynamics with different mixing modes may appear^[21–23]. Here we investigate the nonlinear dynamics in a double-layer graphene diode at ${V_{{
m{dc}}}} = 0.88$ V, and ${V_{{
m{dc}}}} = 1.2$ V, respectively. When the n⁺nn⁺ double-layer graphene is subjected to a periodic voltage, a variety of spatiotemporal behaviors appear, and are affected by the dc bias ${V_{{
m{dc}}}}$ and ac amplitude ${V_{{
m{ac}}}}$.



Fig. 2 shows the current response to varying AC amplitude under a fixed DC bias ${V_{{
m{dc}}}} = 0.88$ V in the double-layer graphene diode. In this case, the self-oscillating frequency ${f_{
m{s}}}$ is 0.462 THz. The AC frequency is fixed to be ${f_{{
m{ac}}}} = 0.748$ THz, and the AC amplitude ${V_{{
m{ac}}}}$ changes from 0 to 0.15 V with a step of 0.0001 V. To clarify the periodic and chaotic regions, we refer to the Poincaré bifurcation diagram. The Poincaré mapping is defined as follows: Let ${T_{{
m{ac}}}} = 1/{f_{{
m{ac}}}}$ be the period of the AC driving signal. For each value of the AC amplitude ${V_{{
m{ac}}}}$, we calculate the current density at times ${t_{{m}}} = m{T_{{
m{ac}}}} $$(m = 1,2,3, ldots )$. The calculated Poincaré bifurcation diagram is shown in Fig. 2(a), which indicates that there exist 1- and 2-periodic windows and beyond all the periodic windows the spatiotemporal solutions become chaotic. When ${V_{{
m{ac}}}} < $0.0554 V, for each ${V_{{
m{ac}}}}$ there are a number of points in the bifurcation diagram and the solutions of the system are chaotic. As ${V_{{
m{ac}}}}$ approaches 0.0554 V, the chaotic behavior disappears abruptly and the solutions are attracted to one periodic orbit. When ${V_{{
m{ac}}}} > 0.1268$ V, the solutions become two-periodic. Furthermore, the widths of the windows become narrower and narrower with increasing ${V_{{
m{ac}}}}$. To further distinguish the chaotic state from the periodic one, the first return maps are used for three specific AC voltages ${V_{{
m{ac}}}}$ = 0.035, 0.1, and 0.1355 V, shown in Figs. 2(b), 2(c) and 2(d). The attractors for the chaotic solutions are infinite random points while those for m-periodic solutions are just the m separate points.






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Figure2.
(a) Poincaré bifurcation diagram and first return maps at (b) ${V_{{
m{ac}}}} = 0.035$ V, (c) 0.1 V, (d) 0.1355 V, respectively, for the NDC n⁺nn⁺ double-layer graphene at ${V_{{
m{dc}}}} = 0.88$ V.




We have also studied the current response of the double-layer graphene diode biased at ${V_{{
m{dc}}}} = 1.2$ V. This DC voltage leads to a self-oscillation frequency at ${f_{
m{s}}}$ = 0.35 THz and the driving frequency is fixed at ${f_{{
m{ac}}}} = G ; {f_{
m{s}}} = 0.566$ THz. The current response to varying AC amplitude is shown in Fig. 3(a). From the Poincaré bifurcation diagram we can see that the oscillation alternates its behavior between the periodic and chaotic states. When ${V_{{
m{ac}}}} > 0.207$ V, the solutions become 1-periodic. The first return maps are used for three specific AC voltages ${V_{{
m{ac}}}} = 0.015$, 0.12 and 0.245 V, shown in Figs. 3(b), 3(c) and 3(d). From the first return map we can see that the solutions are quasiperiodic when ${V_{{
m{ac}}}}$ is 0.015 and 0.12 V, since the first return map is a closed smooth curve with regular distribution of the points.






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Figure3.
(a) Poincaré bifurcation diagram and first return maps at (b) ${V_{{
m{ac}}}} = 0.015$ V, (c) 0.12 V, (d) 0.245 V, respectively, for the NDC n⁺nn⁺ double-layer graphene at ${V_{{
m{dc}}}} = 1.2$ V.




We would like to mention further theoretical and experimental aspects of the present time-dependent hydrodynamic simulation. As we well know, the numerical Monte-Carlo technique requires a very heavy computational cost for the implementation of a realistic numerical calculation, although it is generally believed to be capable of producing a rather accurate semiclassical description of nonstationary and nonlocal phenomena in transport and dealing with the detailed band structure of semiconductors. Many different approaches, such as hydrodynamic models, have been used to simulate carrier conduction in semiconductor devices with characteristic size in the deep submicron region due to their high accuracy/cost ratio. Compared to ordinary semiconductor systems, the microscopic mechanisms of nonlinear effects in the present double-layer graphene system are quite different. These nonlinear effects can be strongly enhanced due to the high electron velocity in graphene. New physical phenomena such as gate-controlled resonant tunneling^[9] have been experimentally observed in double-layer graphene. Recent studies have shown how to achieve gate-controlled resonant tunneling and pronounced negative differential velocity (NDV) in graphene-based multilayers^[9]. By using this experimental NDV in the hydrodynamic equations, the results for a realistic n⁺nn⁺ double-layer graphene diode would be promising because of the self-consistent calculation of the collision terms without invoking empirical approximations.

4.
Conclusions

In conclusion, by using the time-dependent hydrodynamic balance equations, we have theoretically studied nonlinear dynamics in an n⁺nn⁺ double-layer graphene diode subjected to a DC bias and a THz electromagnetic radiation. The NDC effect in a double-layer graphene diode results in a cooperative nonlinear oscillatory mode which leads to a complicated chaotic dynamics. The double-layer graphene diode shows typical nonlinear dynamical behavior with the DC bias, AC amplitudes and the AC frequency as the control parameters. The present study provides access to various properties of graphene, which, as a nonlinear element, is a promising material for the development of novel electronic devices for terahertz applications.