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

Authors: B. F. Mohamed; M. Abdel Aziz
Journal: International Journal of Plasma Science and Engineering
 (2010)

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## Abstract

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.


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## Body

## 1. 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 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.


## 2. Assumptions and Equations

We consider a uniform quantum plasma half-spacex>0, and the interface vacuum-plasma is located at x=0. We assume the surface waves perturbations propagating along an external magnetic field Bo=Boêy. 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]
(1)∂n∂t+∇·(nv⃗)=0,(2)∂v⃗∂t=-∇Pmn-em[E⃗+v⃗∧B⃗]+ℏ22m2∇(∇2nn),
where, n, v⃗, 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 P=2TFen(TFe=ℏ2(3π2n)2/3/2m); TFe and vFe are the electron Fermi temperature and speed. The quantum diffraction effect (Bohm potential) contained in the ℏ-dependent term is in (2). Here, the quantum collisionless regime can be considered due to the high-density 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(Bx,By,Ez) (i.e., propagation of surface modes with electric field perturbation perpendicular to the wave propagation) which can be governed by Maxwell's equations
(3)∇∧E⃗=-1c∂B∂t,∇∧B⃗=1c∂E⃗∂t+4πcJ⃗.We assume that every physical quantityϕ (representing n, v⃗, E⃗, B⃗) has the following form
(4)ϕ=ϕo+ϕ1,
where ϕo is the unperturbed value, and ϕ1 is a small perturbation with ϕ1≪ϕo, and
(5)ϕ1∝ϕ1(x)exp(ikyy-iωt).For plasma equilibrium, it is also assumed thatE⃗o=vo⃗=0, and ky 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:
(6)∂2n∂x2-γp2n=-iωcωp24πωvFe2D∂E∂x,
where ωc=eBo/m is the electron Larmor frequency, and
(7)γp2=ω2-ωc2Dω2[ky2-ω2vFe2],(8)D=1+ℏ2ky24m2vFe2(1+ω2-ωc2ω2).Besides, the magnetic field components and the wave equation for the electric field componentsEz of the TE-surface modes can be obtained as follows
(9)Bx=ckyωEz,By=icω∂Ez∂x,∂2Ez∂x2-℘p2Ez=-4πieωωcc2(ω2-ωc2)(vFe2+ℏ2ky24m2)(∂n∂x),
where
(10)℘p2=ky2-ω2c2(1-ωp2ω2-ωc2).One can see that the quantum effects does not have any action on the electric and magnetic field components in the case of unmagnetized plasmaωc=0.


## 3. 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
(11)n=Aexp[-(γp2-αp2)1/2x],
where, A is a constant and
(12)αp2=ωp2ωc2c2(ω2-ωc2)D(1+ℏ2ky24m2vFeFe2).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(13)ky2≥ω2vFe2+ω2ωp2ωc2c2(ω2-ωc2)2(1+ℏ2ky24m2vFeFe2).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
(14)Ez=D1e℘νx,x<0,Ez=D2e℘px-(4πieAωcωc2(ω2-ωc2))(γp2-αp2)1/2[℘p2-γp2+αp2]×exp[-(γp2-αp2)1/2x],x>0,
where D1 and D2 are integration constants, and ℘ν=(ky2-ω2/c2)1/2 for vacuum.Imposing the continuity condition of the electric fieldEz at x=0, together with the boundary condition of the electron velocity vx=0 (at x=0) [13], one can obtain the amplitudes of the generated field in (14)
(15)D1=-(4πieAωωcc2(ω2-ωc2))(γp2-αp2)1/2(℘p+℘ν)[℘p+(γp2-αp2)1/2],D2=(4πieAωωcc2(ω2-ωc2))(γp2-αp2)1/2(℘p+℘ν)[℘p-(γp2-αp2)1/2].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:(16)(γp2-αp2)1/2(1+ℏ2ky24m2vFeFe2)×{1+[℘p+(γp2-αp2)1/2]/(℘ν+℘p)[℘p2-(γp2-αp2)]1+ωp2ωc2c2(ω2-ωc2)×1+[℘p+(γp2-αp2)1/2]/(℘ν+℘p)[℘p2-(γp2-αp2)]}=0.Equation (16) has two solutions. The first one is simply
(17)γp2-αp2=0.Taking into account the expressions ofγp and αp in (7) and (12), we obtain the following dispersion relationship:
(18)(ω2-ωc2)(ω2-ky2vFe2)+ωp2ωc2c2[vFe2+λq4ky2ωp2]=0,
where λq=(ℏ2/4m2ωp2)1/4 is the quantum wavelength of electrons. In the case of classical plasma (i.e., λq,vFe→0), there is only the oscillations by the Larmor frequency ω=ωc. But, when the quantum diffraction effects are only neglected (in the case of vFe≈vTe), we have
(19)ω2(ω2-ωc2)-ky2vFe2(ω2-ωc2+ωp2ωc2c2ky2)=0.It is also noticed that the dispersion relation becomesω=kyvFe for unmagnetized plasma, which is corresponding to the dispersion of Langmuir wave in classical plasma.Now, we return to the second solution(20)[℘p2-(γp2-αp2)]+ωp2ωc2c2(ω2-ωc2)×[1+[℘p+(γp2-αp2)1/2](℘ν+℘p)]=0.Again, by inserting the expressionsγp and αp in (7) and (12) into the above formula, one can get
(21)(ω2-ωc2)2+ω2ky2vFe2D(ω2-ky2vFe2)×{1+[℘p+(γp2-αp2)1/2]/(℘ν+℘p)[℘p2-(γp2-αp2)](ω2-ωc2)-ω2c2ky2(ω2-ωH2)+ωp2ωc2c2ky2D×[1+λq4ky2ωp2vFe2+D[2℘p+℘ν+(γp2-αp2)1/2](℘p+℘ν)]}=0,
where ωH2=ωp2+ωc2 is the upper hybrid frequency.Equation (21) shows the dispersion relation of TE-surface 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., λq4ky4→0), (21) will be immediately reduced to the following relation:
(22)(ω2-ωc2)2+ω2ky2vFe2(ω2-ky2vFe2)×{1+[℘p+(γp2-αp2)1/2]/(℘ν+℘p)[℘p2-(γp2-αp2)](ω2-ωc2)-ω2c2ky2(ω2-ωH2)+ωp2ωc2c2ky2[2℘p+℘ν+(γp2-αp2)1/2](℘p+℘ν)}=0.
For the electrostatic modes (i.e., c→0), (21) becomes
(23)(ω2-ωc2)+ky2vFe2[ωc2c2+λq4ky2ωp2vFe2(1+ω2-ωc2ω2)]=0,
and at unmagnetized plasma ωc=0, the dispersion relation is
(24)ω2[1-vFe2c2+2λq4ky2ωp2c2]+2λq4ky4ωp2=0.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.


## 4. 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=kyvFe and (24)) depend essentially on quantum effects in the case of unmagnetized plasma.



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*Source: 693049-2010-08-09.xml*