Propagation of TE-Surface Waves on Semi-Bounded Quantum Plasma

The propagation of the TE-surface waves on a semibounded quantum plasma is investigated by using the system of generalized quantum hydrodynamic (QHD) model and Maxwell’s equations. The dispersion relations for these surface waves on quantum electron plasma in the presence of external magnetic field which is parallel to the wave propagation are derived. The perturbation of electron density and the electric fields of the TE-surface waves are also obtained. However, it was found that quantum effects (Bohm potential and statistical) have no remarkable action on the electric and magnetic field components in the case of unmagnetized plasma. But, it was found that the dispersion relation of surface modes depends significantly on these effects in the case of electrostatic or unmagnetized plasma.


Introduction
In recent years, quantum effects in plasmas and in electronic devices have attracted a lot of interest, due to their references therein, in microelectronics and nanotechnologies, for example, for the resonant tunnel diode [1], nanoelectron tubes (nanotriode) [2], as well as in dense laser produced plasmas [3].Also, the quantum plasma has received great attention due to their theoretical relevance in many of the astrophysical plasmas [4].So, the classical transport models are not sufficient to describe the plasma behavior in such devices.In general, the quantum effects become important in plasma, when De Broglie wavelength associated with the particles is comparable to the dimension of the system and the temperature is lower than the Fermi temperature.
The Wigner model is often used in quantum kinetic plasma research.It describes the statistical behavior of quantum plasma in velocity space by using the Wigner-Poisson equations.The quantum hydrodynamic (QHD) model is derived by taking moments of the Wigner equation as in the classical fluid model.This model consists of a set of equations describing the transport of charges, momentum, and energy in a charge particle system interacting through a self-electrostatic potential.The QHD model generalizes the fluid model for plasmas with the inclusion of quantum correction term also known as Bohm potential [5].The latter is responsible for the electron tunneling at nanoscales as well as for introducing new types of plasma waves in dense quantum plasmas [6].
In addition, the surface plasma waves [7,8] propagating along the plasma-vacuum interface have attracted much attention since the frequency spectra have wide applications in many areas such as laser physics, plasma spectroscopy, plasma technology, and surface science.The nonlinear propagation of the surface waves on a cold plasma half-space has been investigated [9].Also, the study of surface waves propagating in uniform quantum plasma seems promising from the outlook of their use in many applications like microwave electronics [10,11].
Quantum effects on the dispersion relation of linear waves, describing Langmuir oscillations, are investigated by Chang and Jung [12].Lazar et al. [13] studied the dispersion relation of surface plasmons that can exist on a dense quantum plasma half-space.
In this paper, we investigate propagation of the transverse electric (TE) surface modes on semibounded quantum International Journal of Plasma Science and Engineering plasma in the presence of external magnetic field.The dispersion relations of the TE-modes are derived by using the quantum hydrodynamic model with Maxwell's equations.The dispersion properties of these modes would provide a useful tool for investigating the physical properties of quantum plasmas.

Assumptions and Equations
We consider a uniform quantum plasma half-space x > 0, and the interface vacuum-plasma is located at x = 0. We assume the surface waves perturbations propagating along an external magnetic field B o = B o e y .In what follows, we will, due to the large inertia, neglect the quantum corrections to the ion motion.The basic quantum fluid model describing the dynamics of the electron plasma is [14] where, n, − → v , and m are the number density, the velocity, and the mass of electrons, respectively.The first term on the right-hand side of ( 2) is the force due to the quantum statistical pressure in dense Fermi-Dirac plasmas.It can be written (for 1D case) as 2m); T Fe and v Fe are the electron Fermi temperature and speed.The quantum diffraction effect (Bohm potential) contained in the D-dependent term is in (2).Here, the quantum collisionless regime can be considered due to the highdensity plasma which can be easily understood by Pauli's exclusion [15].Also, within the single-electron model in thermodynamic equilibrium, the spin effects are ignored for a slowly varying magnetic field perturbation [16,17].
Without any loss of generality, we study the possibility of propagation of the symmetric TE-modes with the field components (B x , B y , E z ) (i.e., propagation of surface modes with electric field perturbation perpendicular to the wave propagation) which can be governed by Maxwell's equations ( We assume that every physical quantity φ (representing where φ o is the unperturbed value, and φ 1 is a small perturbation with φ 1 φ o , and For plasma equilibrium, it is also assumed that , and k y is the component of the wave vector along the y-axis which is the direction of the dense plasma-vacuum interface and also in the same direction of external magnetic field.
For very slow nonlocal variation [13], the electron perturbation density obeys the following equation: where ω c = eB o /m is the electron Larmor frequency, and Besides, the magnetic field components and the wave equation for the electric field components E z of the TEsurface modes can be obtained as follows where One can see that the quantum effects does not have any action on the electric and magnetic field components in the case of unmagnetized plasma ω c = 0.

Surface Waves on Quantum Plasma
The wave equation for the electron density perturbation equation ( 6) can be solved in quantum magnetized plasma to obtain the following form where, A is a constant and We can see that the perturbation of plasma density depends essentially on both quantum effects.But, in the case of unmagnetized plasma, the only statistical quantum effect (Fermi temperature) must be considered to obtain the electron perturbation density.Also, the TE-surface waves are excited only in this condition International Journal of Plasma Science and Engineering 3 The electric field component of these surface waves can be derived by solving the wave equation ( 8) in the vacuum and plasma regions as follows where D 1 and D 2 are integration constants, and 1/2 for vacuum.Imposing the continuity condition of the electric field E z at x = 0, together with the boundary condition of the electron velocity v x = 0 (at x = 0) [13], one can obtain the amplitudes of the generated field in ( 14) After some algebraic manipulations, we also obtain the dispersion relation for the surface modes including the quantum effects in the semibounded quantum plasmas in the following form: Equation ( 16) has two solutions.The first one is simply Taking into account the expressions of γ p and α p in ( 7) and ( 12), we obtain the following dispersion relationship: where λ q = (D 2 /4m 2 ω 2 p ) 1/4 is the quantum wavelength of electrons.In the case of classical plasma (i.e., λ q , v Fe → 0), there is only the oscillations by the Larmor frequency ω = ω c .But, when the quantum diffraction effects are only neglected (in the case of v Fe ≈ v Te ), we have It is also noticed that the dispersion relation becomes ω = k y v Fe for unmagnetized plasma, which is corresponding to the dispersion of Langmuir wave in classical plasma.Now, we return to the second solution Again, by inserting the expressions γ p and α p in ( 7) and ( 12) into the above formula, one can get where ω 2 H = ω 2 p + ω 2 c is the upper hybrid frequency.Equation (21) shows the dispersion relation of TEsurface modes with quantum effects corrections in magnetized plasma.Again, when all the quantum effects' are ignored, one can obtain the same result of the first solution (i.e., Larmor oscillations).But, if the quantum diffraction effect is only ignored, (viz., λ 4 q k 4 y → 0), (21) will be immediately reduced to the following relation: International Journal of Plasma Science and Engineering For the electrostatic modes (i.e., c → 0), (21) becomes and at unmagnetized plasma ω c = 0, the dispersion relation is One can see that the dispersion relations of surface modes (( 23) and ( 24)) in both cases of electrostatic and unmagnetized plasma are significantly affected by the quantum effects.

Conclusion
In this paper, the behavior of surface waves on quantum plasma half-space, magnetized or not, has been investigated.The effects of a quantum statistical Fermi electron temperature and the quantum electron tunneling are included.The dispersion relations for these surface waves on quantum electron plasma in the presence of external magnetic field which is parallel to the wave propagation are derived by using the quantum hydrodynamic model (QHD) with Maxwell's equations.The perturbation of electron density and the electric fields of the TE-surface waves are also obtained.Our analysis investigates that by neglecting the quantum effects, there is only Larmor oscillations without propagation as in the case of classical plasma (here, quantum effects play the same role of thermal effects to facilitate the Langmuir oscillations propagation).Furthermore, the dispersion relations (ω = k y v Fe and (24)) depend essentially on quantum effects in the case of unmagnetized plasma.