
Citation: | Long CHEN, Zuojun CUI, Weifu GAO, Ping DUAN, Zichen KAN, Congqi TAN, Junyu CHEN. Effect of ion stress on properties of magnetized plasma sheath[J]. Plasma Science and Technology, 2024, 26(2): 025001. DOI: 10.1088/2058-6272/ad0d4f |
In the plasma sheath, there is a significant gradient in ion velocity, resulting in strong stress on ions treated as a fluid. This aspect has often been neglected in previous sheath studies. This study is based on the Braginskii plasma transport theory and establishes a 1D3V sheath fluid model that takes into account the ion stress effect. Under the assumption that ions undergo both electric and diamagnetic drift in the presheath region, self-consistent boundary conditions, including the ion Bohm velocity, are derived based on the property of the Sagdeev pseudopotential. Furthermore, assuming that the electron velocity at the wall follows a truncated Maxwell distribution, the wall floating potential is calculated, leading to a more accurate sheath thickness estimation. The results show that ion stress significantly reduces the sheath thickness, enhances ion Bohm velocity, wall floating potential, and ion flux at the wall. It hinders the acceleration of ions within the sheath, leading to notable alterations in the particle density profiles within the sheath. Further research indicates that in ion stress, bulk viscous stress has the greatest impact on sheath properties.
With the development of modern science and technology, terahertz (THz) waves that have not been fully exploited and utilized have attracted more and more people's attention [1, 2]. Today, THz technology is widely used in medical diagnosis [3, 4], space literature [5], security detection [6–8] and military fields [9]. However, a great leap forward in THz technology can only be achieved by finding efficient THz radiation sources and detectors [10, 11]. In a recent paper by Wang et al, theoretical research and analysis based on the nonlinear optical difference frequency principle show that the THz radiation source with optical parameter effect can effectively generate the THz signal [12]. As early as nearly 30 years ago, Dyakonov and Shur et al had shown that stable DC currents in ballistic field-effect transistors (FETs) could be unstable against the generation of high-frequency plasma waves due to the wave reflection from the device boundaries, which provides a new mechanism of electromagnetic radiation [13]. Aizin et al extended the hydrodynamic model to the case considering stubs, which proved that the instability of plasma in FETs can be controlled by optimizing boundary conditions, the velocity of plasma waves, etc [14].
Plasma is an ionized gas composed of charged particles as well as neutral particles. The plasma has a very high electrical conductivity, and a strong coupling effect with the electromagnetic field. Plasma waves are caused by a wide variety of collective motion modes. There are elastic restoring forces, thermal pressure, electrostatic force and magnetic field force in plasmas. Because of the interaction among different forces, plasma waves include electromagnetic waves (shear waves) caused by electromagnetic force, static waves (longitudinal waves) caused by electrostatic force, and sound waves caused by thermal pressure. The plasma wave traveling in the FET channel is reflected and amplified when it spreads to the drain. Then the plasma wave becomes unstable, which leads to electromagnetic radiation. When the gate length reaches to submicron, the frequencies of electromagnetic radiation belong to the THz range [12].
Spin is an intrinsic motion caused by the intrinsic angular momentum of particles, which is an intrinsic property of particles [15]. The contribution of intrinsic spin to the ponderomotive force was explored in a magnetized plasma by using an expression containing both the classical as well as the spin-induced ponderomotive force and the results show that a strong spin polarization in a plasma can be induced by the spin-ponderomotive force [16]. In a recent paper by Brodin [17], the properties of solitary structures were investigated taking into account the microscopic spin properties of the electrons and it is found that the quantum spin effects will induce the solitary structures. Lin et al analyzed the effect of the spin effect on the propagation of electromagnetic waves in any direction in magnetized plasmas, and found that the spin effect has a significant effect on low-frequency modes [18]. A fully consistent quantum-mechanical description of a spin-polarized plasma was constructed by using a generalized matrix Wigner function [19]. There has been many investigations on single-particle spin effects in external field models, for example, intense laser field [20]. In recent years, the study on the excitation of collective modes in spin systems has also aroused curiosity in the scientific community [21, 22]. It is shown by previous studies that there are many novel features and certain advantages in THz radiation in the presence of the spin effect of electrons in the channel of nano-FET.
Experimental and numerical research has demonstrated that the THz plasma waves become unstable in the presence of the asymmetric boundary conditions that the impedance between the gate and the drain approaches infinity, and the impedance between the gate and the source tends to zero Ohm. In fact, there are some difficulties in the realization of asymmetric boundary conditions, which leads to the attention of other boundary conditions. The instability of THz plasma waves was developed using a compact model with tunable narrow-channel regions with an increased width, and the plasmonic boundary conditions are matched at the interfaces between different regions in the channel [14]. In 2018 [23], by linearization of hydrodynamic equations describing the collective behavior of two-dimensional electron gas in the channel of high electron mobility transistor, the THz radiation was firstly explored with the impact of non-ideal termination impedances at the source and the drain, and numerical results showed that the DS instability would be affected by non-ideal boundary conditions. However, to the authors' best knowledge, very few publications address the issue of the characteristics of THz plasma waves traveling in the channel of FETs with non-ideal boundary conditions, spin effects and quantum effects.
In our work, the instability of THz plasma waves in FETs under the non-ideal boundary conditions is investigated with the joint action of spin effects, quantum effects and the external magnetic field. In order to study the features of the THz plasma waves, we started from the quantum fluid dynamics equation and spin equation, and obtained dispersion relation with non-ideal boundary conditions and quantum effects. The influences of drain capacitance, source capacitance, spin effects, quantum effect, channel width and electron drift velocity on the instability of plasma THz wave in FETs under the joint action of non-ideal boundary conditions are discussed in detail.
With the rapid development of nanotechnology, the characteristic size of electronic devices is becoming smaller and smaller, that is, from the micron to nanometer. The characteristic size of traditional semiconductor devices is in the order of microns, and the particle property of electrons is dominant. When the characteristic size of the device is equal to or smaller than the de Broglie wavelength of electrons or the mean free path of electrons, and the volatility of electrons is dominant, and then we get all sorts of quantum effects. At Fermi temperature, the electron state changes from the Maxwell–Boltzmann distribution to the Fermi–Dirac distribution [24]. In this case, the motion of the electron must be described using the equations of quantum fluid mechanics with quantum effects. So, in order to investigate the instability of THz plasma waves with spin effects and quantum effects, the collective behavior of electron fluid is governed by the quantum fluid dynamics equations [25, 26] with spin effects [27, 28] and Maxwell's equations:
∂n∂t+∇·(nv⃑ | (1) |
(2) |
(3) |
(4) |
(5) |
(6) |
where
The evolution equation describing the spin effects in quantum plasma is as follows:
(7) |
Because the scale of the electron is larger than the Larmor radius of the electron in the magnetohydrodynamic model, we can ignore the quadratic term of the spin
(8) |
here,
To study the stability of THZ plasma waves, we set
(9) |
(10) |
(11) |
(12) |
(13) |
where
To study the features of THz plasma waves simply, the linearized expressions of equations (9)–(13) are as follows,
(14) |
(15) |
(16) |
(17) |
(18) |
where
According to equations (14)–(18), we obtain the dimensionless dispersion relation:
(19) |
In equation (19),
From equation (19), we can obtain the seven roots of
(20) |
(21) |
here,
DS instable theory indicated that the THz plasma waves become unstable in the presence of the asymmetric boundary conditions that the impedance between the gate and the drain approaches infinity, and the impedance between the gate and the source tends to zero Ohm. Then, Mona et al have come up with other boundary conditions that source capacitance and drain capacitance are attached in high electron mobility transistors [23]. These boundary conditions agree with equations
(22) |
(23) |
here
According to equations (20)–(23), we obtain:
(24) |
We shall write the above expression as:
(25) |
here,
In order to investigate the propagation of THz plasma waves in the channel of FET with the combined contribution of spin effects, the quantum effects and non-ideal boundary conditions, we numerically solve equations (19) and (25). The features of instability of THz plasma waves are discussed in two different cases.
Case 1 Source capacitance is greater than the drain capacitor
Figure 1 shows the trend of the oscillation frequency
The characteristics of FET will change in the presence of the quantum effects. Figure 2 shows the effect of quantum effects
Figure 3 shows the effect of the channel width
Figure 4 shows the effect of the spin effect on the oscillation frequency and the instability gain when the source capacitance is larger than the drain capacitor. We can see from figure 4 that the oscillation frequency and instability gain with the spin effect are larger than that without the spin effect. That is, the spin can not only enhance the oscillation frequency but also increase the instability gain. Obviously, the presence of the spin reduces the energy loss during the plasma oscillation and amplification.
Since
Case 2 Drain capacitor is greater than the source capacitance.
Figures 5–8 give an example of the instability of THz plasma waves in the channel of FET with spin effects, quantum effects and non-ideal boundary conditions when the source capacitance
Similarly, as shown in figures 7(b) and 8(b), the effects of the channel width
Under asymmetric boundary conditions, the source impedance
In conclusion, the linear characteristics of THz plasma waves in the channel of FET with non-ideal boundary conditions are considered by linearizing Maxwell's equations and the quantum fluid dynamics equations containing the spin effect and quantum effect. Summing up the results, it can be concluded that in a FET, if the source capacitance is greater than the drain capacitance, the THz plasma waves become unstable, which indicates that there is THz emission in this case. When the drift velocity is fixed, the oscillation frequency increases with the increase in quantum effect, and decreases with the increase of source capacitance and drain capacitance, while the instability gain increases with the increase of source capacitance and quantum effect, and decreases with the increase of drain capacitance. In contrast, if the drain capacitance is greater than the source capacitance, the value of the instability gain is less than zero, that is, the THz plasma waves are stable. Meanwhile, the quantum effect and spin effect enhance the emission of THz waves, and the boundary conditions can affect the oscillation frequency and the instability gain of plasma THz waves. In [30], the effects of spin, quantum effects, drift velocity and other factors on the oscillation frequency and instability gain of THz plasma waves are investigated under asymmetric boundary conditions. Compared with the results in [30], we found that the influence of boundary conditions is very important. The oscillation frequency with asymmetric boundary conditions is smaller than that under non-ideal boundary conditions, but the instability gain of THz plasma waves becomes lower under non-ideal boundary conditions. The findings suggest that this study could also be useful for the development of THz source and detector.
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