Cite this Article
Feng Wei. Hydrodynamic simulation of chaotic dynamics in InGaAs oscillator in terahertz region. Chinese Physics B, 2020, 29(4): 047302  
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Hydrodynamic simulation of chaotic dynamics in InGaAs oscillator in terahertz region
Feng Wei †
Department of Physics, Jiangsu University, Zhenjiang 212013, China

 

† Corresponding author. E-mail: wfeng@ujs.edu.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11604126) and China Scholarship Council (Grant No. 201808695016).

Abstract

Hydrodynamic calculations of the chaotic behaviors in n+nn+ In0.53Ga0.47As devices biased in terahertz (THz) electric field have been carried out. Their different transport characteristics have been carefully investigated by tuning the n-region parameters and the applied ac radiation. The oscillatory mode is found to transit between synchronization and chaos, as verified by the first return map. The transitions result from the mixture of the dc induced oscillation and the one driven by the ac radiation. Our findings will give further and thorough understanding of electron transport in In0.53Ga0.47As terahertz oscillator, which is a promising solid-state THz source.

PACS: ;73.61.Ey;;73.50.Fq;;85.30.Fg;;85.30.De;
Keyword:;terahertz;;chaotic;;current oscillation;
1. Introduction

Devices operating in the THz frequency region have attracted a great deal of attention due to their wide applications, such as security inspection, imaging, and radar.[1–4] Compact and coherent solid-state THz sources working at room temperature are fundamental and important for such applications. Various devices have been investigated theoretically and experimentally as THz sources, from both optical and electronic aspects. As a typical optical device, the quantum cascade laser (QCL) represents an efficient coherent THz source,[5] but its low operating temperature is a major limitation. As typical electronic devices, resonant tunneling diodes (RTDs), silicon complementary metal oxide semiconductors (Si CMOS), hetero-bipolar transistors (HBTs), and high electron-mobility transistors (HEMTs) are being studied intensively as THz sources.[6–8] Besides, negative differential velocity (NDV) oscillators are good candidates as room-temperature THz sources.

Many recent efforts have been devoted to the NDV effect, as originally proposed by Krömer, to generate and amplify the current oscillation.[9] There are two NDV mechanisms. One is the Ridley–Watkins–Hilsum (RWH) mechanism. In a multi-valley semiconductor like GaAs and InGaAs,[10,11] the transfer of carriers into a higher energy state with larger effective mass and lower velocity results in the NDV effect. The other one is the negative effective mass (NEM) mechanism due to the nonparabolic band dispersion like in InN.[12] In a single nonparabolic band, when the electron transits through the inflection point, its effective mass changes sign to be negative, leading to a different NDV effect. Various materials have been explored as NDV THz oscillators.[13,14] In 2007, the first hetero-structure planar Gunn diode based on AlGaAs/GaAs was demonstrated with operating frequency above 100 GHz.[15] However, the frequency of the device, as limited by GaAs’s saturation velocity, cannot be increased efficiently to high enough. To overcome this disadvantage, In0.53Ga0.47As layer, which has a good lattice match to the InP substrate, was proposed to replace GaAs. In 2013, the first 164 GHz In0.53Ga0.47As diode was achieved.[16] To increase the frequency further, a submicron In0.53Ga0.47As diode in a planar layout was proposed. Compared to the linear optical and electronic responses in In0.53Ga0.47As, which have been extensively studied, fewer investigations are based on nonlinear transport effects. The external radiation on the device is expected to result in complicated transport of electrons and the resulting oscillations are time- and space-dependent. Thus, nonlinear transport effects are very essential and efficient for nonequilibrium optical and electronic processes. They are crucial for a thorough understanding of fundamental properties of the In0.53Ga0.47As oscillator.

In the present work, we apply a hydrodynamic model to investigate nonlinear dynamics in n+nn+ In0.53Ga0.47As diode. Although the Monte–Carlo method has been widely used in the theoretical investigations of the lattice-matched In0.53Ga0.47As for Gunn oscillators,[14,17,18] the hydrodynamic model has higher efficiency by exploiting some reasonable approximations. Using the hydrodynamic model, we have previously studied the current self-oscillation in the n+nn+ In0.53Ga0.47As diode under dc bias. Here we continue to investigate the chaotic behaviors under THz radiation by carefully tuning the parameters of dc bias, doping concentration, and intensity of radiation.

2. Hydrodynamic simulation of chaotic dynamics in InGaAS THz oscillator

InGaAs pronounced NDV feature was reported theoretically and experimentally.[11] Figure 1 (a) shows the relation of the drift velocity Vd to applied electric field E. The inset shows the schematic diagram of the n+nn+ InGaAs diode.[19] At room temperature, for an doped In0.53Ga0.47As diode (Nd = 1018 cm−3), the velocity first increases, reaching a pronounced peak value of 1.93 × 105 m/s at E = 3.9 kV/cm, and then decreases, which forms the NDV region. The fitting curve for the Vd–E relation can be expressed as

which is in a very good agreement with the experiment. When the diode is applied with a dc bias, self-current oscillation shows up, as seen in Fig. 1(b). Different dc biases will lead to current oscillations with different frequencies.

Fig. 1. (a) Fitting Vd–E curve of InGaAs at room temperature in comparison with experiment. (b) Self-current oscillations under two different dc biases.

To get a better understanding of the oscillations in n+nn+ InGaAs THz oscillator, we next investigate the oscillatory modes of the diode under the influences of both self-sustained electric field and external THz radiation in the form of

where Es is the self-sustained electric field, and Es = Vdc/l with Vdc the dc bias and l the n region length which is set to be 0.5 μm here. Eext is the external THz radiation expressed as Eext(t) = Vac cos (2πfact)/l, with ac intensity Vac and frequency fac. We apply the hydrodynamic equations[20–25] to study the nonlinear dynamics in the In0.53Ga0.47As diode

Here, jn is the current density, μn = Vd/E is the mobility, ε is the electron energy, S is the energy flow density, and W is the energy loss rate. The electron density n is related to the electrostatic potential φ through the Poisson equation

where ε is the static dielectric constant and Nd is the doping concentration. Any small doping inhomogeneity in the NDV region can lead to the formation of domain and oscillations.[20] We assume that there is a slight doping fluctuation near each end of the diode. This fluctuation results in the formation of electric field domain and hence current oscillation. By applying the transient hydrodynamic model to n+nn+ In0.53Ga0.47As diodes at room temperature, we carry out the simulation of the time-dependent current density in the diodes, which can be expressed as

The boundary conditions for the time-dependent and space-dependent electrostatic potential φ(x, t) and election density n(x, t) are set to be

When both a dc bias Vdc and a sinusoidal ac voltage Vac are applied to the InGaAs diode, nonlinear chaotic dynamics are observed to be dependent on many factors, such as Vdc, Vac, and fac. The results are shown in Figs. 2 and 3. Here in the simulations shown in Fig. 2, Vdc is 0.5 V and Nd is 5 × 1016 cm−3 at room temperature. At Vdc = 0.5 V, the current oscillates with frequency fs = 0.093 THz. In the simulations, we set the ac frequency as fac = G × fs ≈ 0.15 THz. The G [≡ (1 + 5)/2 = 1.618...] is the inverse golden mean, and this ratio is regarded as the most irrational of all irrational numbers and frequently used in nonlinear systems.[26,27] Figure 2(a) shows the Poincaré bifurcation diagram with Vac as the controlling parameter changing from 0 to 0.07 V. When Vac is smaller than 0.055 V, the system exhibits a complicated and chaotic behavior. When increasing the ac amplitude to be larger than 0.055 V, the system exhibits a periodic behavior. Here we have applied the first return maps to distinguish periodic states from chaotic ones. The first return maps under two specific ac voltages Vac = 0.012 V and 0.065 V are shown in Figs. 2(b) and 2(c). The first return maps provide us with a clear picture to distinguish the chaotic and periodic states, where m separate points indicate the system to be m-periodic while infinite random points imply a chaotic system. As shown in Fig. 2 for Vac = 0.012 V, there are infinite random points in the first return map, which indicates that the system exhibits a complicated and chaotic behavior. While for Vac = 0.065 V, there is only one point in the first return map, which indicates one periodic system.

Fig. 2. Nonlinear and dynamic oscillatory modes of In0.53Ga0.47As diode at Vdc = 0.5 V and Nd = 5 × 1016 cm−3. (a) The bifurcation diagram, (b) the first return map at Vac = 0.012 V, and (c) the first return map at Vac = 0.065 V.
Fig. 3. Nonlinear and dynamic oscillatory modes of In0.53Ga0.47As diode at Vdc = 0.55 V and Nd = 5 × 1016 cm−3. (a) The bifurcation diagram, (b) the first return map at Vac = 0.04 V, (c) the first return map at Vac = 0.075 V.

The effect of the dc voltage on the current has also been considered and simulated. Figure 3 shows the bifurcation diagram and first return maps with Nd = 5 × 1016 cm−3 and Vdc = 0.55 V at room temperature. Vac changes from 0 to 0.12 V. The bifurcation diagram (Fig. 3(a)) shows a different complex nonlinear dynamics from the oscillatory modes in Fig. 2. When Vac is smaller than 0.0662 V, the results have lots of points representing a chaotic state. As Vac is 0.0662 V, a 3-periodic pattern appears abruptly. With Vac further increasing and approaching 0.08 V, the periodic behavior disappears and the solutions become chaotic again. Furthermore, when Vac increases beyond 0.092 V, the system changes to single-periodic. With tuning Vac, the cooperative nonlinear oscillatory mode transits between the chaotic and periodic patterns. In Figs. 3(b) and 3(c), the first return maps for Vac = 0.04 V and 0.075 V are displayed, respectively. When Vac = 0.04 V, there are infinite random points verifying the chaotic state, while when Vac = 0.075 V there are 3 separate points verifying the 3-periodic state. These transitions originate from the competition between the intrinsic self-oscillation and the external THz radiation. For the periodic states, the oscillatory modes are the linear mixture of the external radiation induced oscillations with fac, while for the chaotic ones, the oscillatory modes result from both intrinsic self-oscillation and the external THz radiation with fs and fac.

3. Conclusion

To summarize, by applying hydrodynamic equations, we have simulated the nonlinear dynamics in n+nn+ In0.53Ga0.47As diodes subjected to both dc and ac signals. The dependence of the cooperative nonlinear oscillatory mode on th edc bias, doping concentration, and ac intensity has been carefully analyzed. The transitions between periodic and chaotic patterns are revealed and originate from the mixture and competition between the intrinsic oscillation and the ac-induced one. Our findings will provide further fundamental understanding of In0.53Ga0.47As diodes, applicable as a promising solid-state source for THz applications.

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