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AID:24624 /SCO Doctopic: Plasma and fluid physics

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Propagation of electrostatic surface waves in a thin degenerate plasma ﬁlm with electron exchange–correlation effects

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a

A. Abdikian , Zahida Ehsan

b,∗

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Department of Physics, Malayer University, 65719-95863 Malayer, Iran b Department of Physics, COMSATS Institute of Information Technology (CIIT), Defence Road, Off Raiwind Road, 54000, Lahore, Pakistan

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a r t i c l e

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Article history: Received 17 June 2016 Received in revised form 9 July 2017 Accepted 14 July 2017 Available online xxxx Communicated by F. Porcelli Keywords: Quantum surface wave Thin plasma ﬁlm External magnetic ﬁeld Symmetric and anti-symmetric modes

a b s t r a c t

87 88

Propagation of an electrostatic surface wave in a thin degenerate Fermi plasma ﬁlm in the presence of constant external magnetic ﬁeld is studied here. Dispersion relations for the symmetric and antisymmetric modes have been derived and studied quantitatively with the exchange–correlation effects. It has been studied that with the increase in the strength of magnetic ﬁeld, phase velocity of the waves decreases. Also electron exchange–correlation effects signiﬁcantly modify the behavior of the surface waves such as frequency of surface wave is found to be downshifted by these effects. Moreover it has been studied that the group velocity of the anti-symmetric mode is greater than the symmetric mode for the whole wave numbers; however, these modes merge into a single mode with the increase of the wave number. © 2017 Elsevier B.V. All rights reserved.

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In plasmas with low temperature and high electron density, Pauli’s principle restricts the electrons not to be more than one in each quantum state; such a system is known as quantum or degenerate plasma [1–3]. In such a plasma system, electron number density for instance in white dwarfs where they exist can be of the order of 1030 cm−3 and even more, and in the interior of neutron stars it can be up to 1036 cm−3 [4]. In recent years, however, a huge interest has been developed in the area of quantum plasmas which is motivated by its potential applications in modern technology, e.g., metallic and semiconductor nanostructures, metal clusters, thin metal ﬁlms, spintronics, nanotubes, quantum well and quantum dots, quantum X-ray free-electron lasers, etc. [5–9] in addition to their applications in dense astrophysical plasma environment [10]. Also thanks to the development of intense laser systems, able to attain the multi-Petawatt domain, which will prove a step forward in the laboratory astrophysics [11]. When density of plasma particles is too high, the thermal de Broglie wavelength associated with each plasma particle may exceed the average inter-particle distance, in this situation wave functions associated with particles may overlap and the particles are no longer distinguishable. The particles with antisymmetric wave function are known as Fermi particles, i.e., the particles with spin 1/2, 3/2, 5/2 . . . ; Tsintsadze et al. 2009 derived a novel quan-

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*

Corresponding author. E-mail address: [email protected] (Z. Ehsan).

http://dx.doi.org/10.1016/j.physleta.2017.07.020 0375-9601/© 2017 Elsevier B.V. All rights reserved.

tum kinetic equation for such particles and obtained a set of ﬂuid equations describing the quantum plasmas [12]. In dense quantum plasmas like systems, the interaction between electrons due to the electrostatic potential of the total electron density is represented by the Hartree and also exchange– correlation terms. The latter, however, is due to the spin effects. These effects embody a short-range electric potential, which depends only on the number density of the Fermi particles. Since electron exchange–correlation effects are very weak therefore they are usually ignored in many investigations. However, it has been studied that these effects become signiﬁcant for the high density and low temperature plasmas such as in the ultra small electronic devices [13–15]. For degenerate plasma, these have been discussed in detail by Landau and Lifshitz in book “Statistical Physics” [16]. Also exchange correlations for proton interaction have been calculated by Tsintsadze et al. [17]. The electron exchange–correlation potential is a complex function of electron density and is given by

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V xc = V x + V c

= −0.985

102

n1/3 e 2 m

1+

0.034 a B n1/3

1/3 ln 1 + 18.37n a B ,

121

(1)

which can be obtained via the adiabatic local-density approximation, where a B = εh¯ 2 /me e 2 is the Bohr radius and ε = 4πε0 is the effective dielectric permeability of material. V x and V c represent the exchange and correlation effects, respectively [6,13,32,34] and are given as

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AID:24624 /SCO Doctopic: Plasma and fluid physics

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V x = 0.985

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A. Abdikian, Z. Ehsan / Physics Letters A ••• (••••) •••–•••

2

n1/3 e 2

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(2)

m

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and

V c = −0.034

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e2 ma B

ln 1 + 18.37a B n

1/3

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(3)

Above equation shows that as density increases, the exchange effects increase, while correlation effects decrease. The inclusion of exchange–correlation effects in the momentum equation gives rise to an additional force on electrons. These effects were ﬁrst considered in quantum hydrodynamic by Crouseilles et al. [6] and can somehow retrievable from the density functional theory (DFT). Wide ranges of applications in plasma diagnostics, laser physics and atomic spectroscopy, inertial conﬁnement fusion, solar corona, etc., have made surface waves investigation one of the important research areas [19–24]. These waves propagate along the interface between a plasma and dielectric material and damp in the perpendicular direction; their ﬁeld amplitude is maximum at the plasmadielectric interface and decay away from the interface boundary. Plasma torus emanating from Io in the Jovian magnetosphere, and the occurrence of high structured ﬂux tubes in the presence of an external magnetic ﬁeld are few of the examples of surface waves in space plasmas [25]. Recently, the surface waves made inroads from classical to quantum (degenerate) plasmas primarily in connection with their possible role in the astrophysical objects, the white dwarf stars magnetars, the plasma processing technologies, microelectronics, semiconductor systems, nanotechnologies, metallic nanostructures, nanotubes the intense laser-solid density plasma experiments, the interface of plasmon, the temperature and density variations in a plasma half space, and the instability due to collisional effects [24]. Mohamed and Aziz [26] have investigated the propagation of the TE modes of surface waves on a semi-bounded quantum plasma. They used quantum hydrodynamic model and Maxwell equations to study the behavior of surface waves in the quantum plasma in the absence and presence of magnetic ﬁeld and found that quantum effects play a signiﬁcant role in the dispersion relation of surface waves in the case of electrostatic or unmagnetized plasma. Moreover, the propagation of surface waves on semi-bounded quantum plasma is applicable to computer chips for faster data transfer. Shokri and Rukhadze have shown that surface waves are unstable on a thin two-component plasma layer, whereas are damped in one-component plasma [27]. Lazar et al. [28] using the quantum hydrodynamic (QHD) model derived dispersion relation for the surface waves in un-magnetized quantum electron plasma half-space. They showed that at room temperature and standard metallic densities, like dense gold metallic plasma, the dispersion relation of electrostatic surface waves mainly depends on the quantum effects, whereas Chang and Jung studied the propagation of the surface Langmuir oscillations in a semi-bounded quantum plasma [29], authors also investigated geometric effects on the symmetric and anti-symmetric modes of the surface plasma wave in thin ﬁlms [30]. Misra et al. investigated the propagation of electrostatic surface waves along the interface of the quantum magneto plasma and vacuum [31]. Moradi has recently investigated quantum effects on the propagation of bulk and surface plasma waves in a thin quantum plasma ﬁlm [32]. However despite the aforesaid investigations, propagation of surface waves in plasma slabs have not been been given much attention; this is important because the actual plasmas in laboratory are ﬁnite and space plasmas often take slab structures. On the other hand surface waves are the subject of active investigation for these have potential applications in plasma processing and diagnostics, laser physics, plasma technologies, and

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Fig. 1. Schematic drawing of the setup.

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astrophysics [19–23]. These waves propagate along the interface between a plasma and dielectric material and damp in the perpendicular direction. Shokri and Rukhadze have shown that surface waves are unstable on a thin two-component plasma layer, whereas are damped in one-component plasma [27]. Lazar et al. [28] using the quantum hydrodynamic (QHD) model derived dispersion relation for the surface waves in un-magnetized quantum electron plasma half-space, whereas Chang and Jung studied the propagation of the surface Langmuir oscillations in a semibounded quantum plasma [29], authors also investigated geometric effects on the symmetric and anti-symmetric modes of the surface plasma wave in thin ﬁlms [30]. Misra et al. investigated the propagation of electrostatic surface waves along the interface of the quantum magneto plasma and vacuum [31]. Moradi has recently investigated quantum effects on the propagation of bulk and surface plasma waves in a thin quantum plasma ﬁlm [32]. However very few studies deal with the effects of the electron exchange–correlation [33] on the propagation of the surface waves on a thin quantum plasma ﬁlm. Therefore here we aim at discussing the propagation of electrostatic surface waves by using the set of QHD equations in the presence of a static and constant magnetic ﬁeld and for the quantitative analysis of the results we will chose typical parameters of the gold metallic plasma, needless to mention such investigation relations may also be useful in studying surface waves in a slab geometry such as in planar magnetrons. We consider propagation of electrostatic surface waves in a degenerate plasma ﬁlm, the plasma sheet is located in z = 0 and its sides are bounded by a dielectric medium characterized by a real, positive dielectric constant εd . The constant external, mag 0 = B 0 zˆ is supposed to be perpendicular to the sheet netic ﬁeld B (see Fig. 1). The thin quantum plasma ﬁlm occupies the region 0 < z < d. The ﬂuid equations for such a system are given below:

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∂n + ∇ · (nv) = 0, ∂t ∂ v e 0] − ∇ P + v × B + (v · ∇)v = − [−∇Φ ∂t me me n 2√ h¯ 2 1 ∇ n V xc , ∇ √ ∇ − +

111

2me

∇ · E = −

e

ε0

(n − n0 )

n

me

(4)

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(5)

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(6)

where n, v, Φ , P and V xc represent the number density, ﬂuid velocity, electrostatic potential, ﬂuid quantum pressure, and exchange correlation effect respectively are given by Eqs. (1)–(3). The parameters e, me , n0 , ε0 and h¯ are the electric charge, mass, uniform density, vacuum permittivity and Planck’s constant divided by 2π . For such quantum plasma, the ﬂuid quantum pressure is deﬁned as 2 3 2 P = me V Fe n /(3n0 ), in which V Fe = 2k B T Fe /me is the Fermi thermal speed, T Fe is the particle temperature, k B is the Boltzmann constant. We assume that the ﬁelds and the perturbed densities associated with the surface wave with the wave number k and the frequency ω vary as φ = φ( z) exp[i (kx x − ωt )], so we obtain

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h¯ 2 2 β2 − ∇ 2n + ω2 − ωc2 − ω2p n = 0 ∇ 2

αe = 0.985(n10/3 e2 /me ε), and ηe = 1/ 3 e2 n 1 + (18.376ne a B )me ε . ωc = e B 0 /me and ω2p = ε m0 are the cy0 e 2 where β 2 = 13 (3V Fe − αe − 2ηe ),

clotron and plasma frequencies respectively. By substituting ∇ =

13

where

k2z = k2x +

ω +ω −ω 2 p

n( z ) =

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(9)

.

h¯ 2 k2x 2me2

0,

z ≥ d , z ≤ 0,

C exp(−k z z),

0 ≤ z ≤ d.

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where A 1 , A 2 , A 3 , A 4 , F 1 and F 2 are the coeﬃcients and the amplitudes of electrostatic potentials. Before ﬁnding the values of the coeﬃcients, we obtain the transverse component of the electron velocity using Eqs. (5) and (6),

vz =

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A1e

−k z d

−kx A 1 e

+ A2e

kz d

−k z d

+ A3e

+ kz A2e

(13)

−kx d

+ A4e

kx d

= F 1e

−kx d

,

kz d

− kx A 3 e

−kx d

+ kx A 4 e

kx d

(16) (17)

whereas the third boundary condition is that the normal velocity component is zero, therefore, we are left with:

1−

Q 2β 2 2 k z − k2x k z − A 1 e −k z d + A 2 ek z d 2

ωp

−kx d

kx d

= kx A 3 e − kx A 4 e , Q 2β 2 2 2 1− k − k z x k z (− A 1 + A 2 ) = k x A 3 − k x A 4 , 2

ωp

where Q 2 = 1 +

h¯ 2 k2x 4me2 β 2

.

(18) (19)

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Q β

2

ω2p

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k2z − kx

k z cosh(k z l) + εd kx sinh(k z l) − A 1 + A 2

= kx cosh(kxl) + εd kx sinh(kxl) A 3 − A 4 ,

k z sinh(k z l) + εd kx cosh(k z l) A 1 + A 2

= − kx sinh(kxl) + εd kx cosh(kxl) A 3 + A 4 .

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(22)

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ωp

= −kx sinh(kxl) A 3 + A 4 .

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cosh(k z l) − A 1 + A 2

= kx cosh(kxl) A 3 − A 4 , Q 2β 2 2 2 kz 1 − k − k sinh(k z l) A 1 + A 2 z x 2

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(23)

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(24)

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(25)

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After some algebraic steps, we obtain expressions for the symmetric and anti-symmetric dispersion relations, for the electrostatic surface waves propagating in the degenerate plasma on thin ﬁlms.

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εd tanh(k z l) = 1 + εd tanh(kxl), γ kz γ 1 kx εd + coth(k z l) = 1 + εd coth(kxl), γ kz γ

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1

+

kx

(26)

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(27)

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where

γ =1−

Q 2β2

ω2p

(k2z − k2x ).

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When l → ∞, εd = 1 and ωc = 0, one could obtain the dispersion relation of the surface electrostatic wave on the quantum plasma half-space which is given as:

(14) (15)

−k z A 1 + k z A 2 − kx A 3 + kx A 4 = F 2kx ,

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∂ 2 ∇ Φ . ∂z

= − F 1kx e −kx d ,

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h¯ 2 kx ∂Φ − β2 + ∂z 2me2

The other boundary condition which assumes the normal component of the displacement vector should be continuous, gives us:

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ωme

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A1 + A2 + A3 + A4 = F 2.

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ie

Using the boundary conditions of a slab geometry (z = 0 and z = d), the coeﬃcients can be obtained, so from the continuity of the tangential component of the electric ﬁeld

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(12)

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We consider d = 2l, and introduce the new constants A 1 = A 1 e −k z l , A 2 = A 2 ek z l , A 3 = A 3 e −kx l and A 4 = A 4 ekx l . Then, adding and subtracting Eqs. (18) and (19) we have

⎧ z ≥ d, Φ1 ( z) = F 1 e −kx z , ⎪ ⎪ ⎨ ( z) = A 1 e −k z z + A 2 ek z z Φ 2 Φ( z) = ⎪ + A 3 e −kx z + A 4 ekx z , 0 ≤ z ≤ d, ⎪ ⎩ z ≤ 0, Φ3 ( z) = F 3 ekx z ,

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∂ 2Φ e − k2x Φ = n ε0 ∂ z2

General solutions for the differential equation (11) for three regions are as follows:

(20)

(21)

Similarly one could obtain:

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= A 3 (−εd kx − kx ) + A 4 (εd − εd kx ).

Eqs. (6) and (8) give the following equation

(11)

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A 1 (k z + εd kx ) + A 2 (εd kx − k z )

(10)

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= A 3 e −kx d (εd kx − kx ) + A 4 ekx d (εd kx + kx ),

kz 1 −

2

Solution of the above equation can be given as:

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2 c

β2 +

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(8)

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A 1 e −k z d (k z − εd kx ) − A 2 ek z d (k z + εd kx )

and neglecting the very slow, the

2

12

15

− k2x 4

last term containing (∂ 4 /∂ z ) of above equation, we obtain

∂ n − k2z n = 0 ∂ z2

11

∂2 ∂ z2

2

From Eqs. (14)–(17), we obtain:

(7)

4me

3

2 p

2

2 p

k x ω = 2ω − ω k z ,

(28)

which agrees with the equation derived by Lazar et al. [28]. For a cold, classical plasma (¯h → 0), ( V Fe → 0), with no exchange correlations, Eq. (28) gives the equation for surface plasmons ωsp = √ ω p / 2. For the quantitative analysis of the above dispersion relations (26) and (27), we such as d¯ = d/λq , k¯ x = kx λq , k¯ z = k z λq , ¯ c = ωc /ω p and ω¯ = ω/ω p . λq = V¯ Fe = V Fe /λq ω p , β¯ = β/ V Fe , ω

h¯ 2mω p

represents the quantum wavelength.

Figs. 1 and 2 are the plots between the normalized frequency and the normalized wavenumber for both symmetric and anti¯ c = 0.5 the symmetric modes respectively, for the medium (ω ¯ c = 1.5 the dark line) values of the exdashed lines) and strong (ω ternal magnetic ﬁeld. We also chose four different values of the exchange correlation effects, i.e., (β¯ = 0, β¯ = 0.2, β¯ = 0.6, and β¯ = 0.8). The normalized thickness of thin ﬁlms and the normalized Fermi thermal speed are arbitrarily chosen as d¯ = 20 and V¯ Fe = 1, respectively [30]. It is obvious from Fig. 1 that although their behavior is the same, the phase velocity of the curve with weak

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¯ in terms of k¯ x for Fig. 2. Schematic of the frequency of the symmetric mode ω different values of the magnetic ﬁeld and the exchange correlation effects.

Fig. 4. The group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave.

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k2z =

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¯ in terms of k¯ x for Fig. 3. Schematic of the frequency of the anti-symmetric mode ω the different values of the magnetic ﬁeld and the exchange correlation effects. (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

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magnetic ﬁeld is greater than the other cases. The group velocities of these modes are plotted in Fig. 3, the blue (red) curves are for the symmetric (anti-symmetric) modes with different strength ¯ c = 0.5 and ω¯ c = 1.5). We note that the group of magnetic ﬁeld (ω velocity of the symmetric mode for the strong magnetic ﬁeld (the blue solid curve) is greater than in case of the medium magnetic ﬁeld (the blue dashed curve). However this is opposite for the anti-symmetric mode i.e. the group velocity of the anti-symmetric mode for the strong magnetic ﬁeld (the red solid curve) is smaller than that of the medium magnetic ﬁeld (the red dashed curve). Fig. 3 also points out that the group velocity of the symmetric mode for small wave numbers is negative which shows that this mode is propagating as a backward wave whereas the antisymmetric mode propagates as a forward wave as the group velocity of the anti-symmetric mode is found to be positive for all wave numbers. 0 = B 0 zˆ Here we consider a special case when by choosing B i.e. when the magnetic ﬁeld is parallel to the sheet, one can get

(ω2p + k2x β 2 Q 2 − ω2 )(ω2 − ωc2 )

ωc2 ω2p − (ω2 − ω2p )(ω2 − ωc2 ) + k2x β 2 Q 2 ω2

90 91 92

(29)

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And by following the above steps, one can derive the symmetric and anti-symmetric dispersion relations for the electrostatic surface waves propagating in the degenerate plasma on thin ﬁlms, as follows

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εd tanh(k z l) = 1 + εd tanh(kxl) γ kz γ 1 kx εd + coth(k z l) = 1 + εd coth(kxl) γ kz γ

99

1

34

45

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a relation similar to the Eq. (8), but with new deﬁnition for k2z as follows

24

44

87

+

kx

(30)

96 97 98 100 101

(31)

Here we have studied the propagation of an electrostatic surface wave in a thin degenerate plasma ﬁlm in the presence of external magnetic ﬁeld and exchange correlations effects. The dispersion relations for the symmetric and anti-symmetric modes have been examined quantitatively for typical parameters n0 = 5.9 × 1022 cm−3 , ωpe = 1.37 × 1016 s−1 and V Fe = 1.4 × 108 cm/s of the gold metallic plasma at room temperature are chosen [28] (see Fig. 4). The importance of the applied magnetic ﬁeld for such thin ﬁlms can be clearly seen in Figs. 1, 2. Overall the frequency of surface plasmon wave is downshifted by the exchange–correlation effects. In addition, the group velocity of the anti-symmetric mode was found to propagate as a forward wave. The symmetric mode may be considered as a backward wave because its group velocity is negative in small wave numbers. Conclusively the external magnetic ﬁeld and correlations effects are found to play a crucial role in the propagations of symmetric and anti-symmetric dispersion modes of the surface quantum wave in a thin degenerate plasma ﬁlm. (See also Figs. 5–7.) Results obtained here are in particular useful in nanotechnology research. One of us (Z.E.) is grateful to Professor Nodar Tsintsadze for the stimulating discussions which helped to understand this problem better. We are grateful to the anonymous referees for constructive suggestions to improve the manuscript.

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Uncited references

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[18]

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¯ in terms of kx for the Fig. 5. Schematic of the frequency of the symmetric mode ω different values of the magnetic ﬁeld and the exchange correlation effects.

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¯ in terms of k¯ x for Fig. 6. Schematic of the frequency of the anti-symmetric mode ω the different values of the magnetic ﬁeld and the exchange correlation effects.

References [1] [2] [3] [4] [5]

G. Manfredi, Fields Inst. Commun. 46 (2005) 263. F. Haas, Europhys. Lett. 77 (2007) 45004. P.K. Shukla, B. Eliasson, Phys. Rev. Lett. 99 (2007) 096401. Y.D. Jung, Phys. Plasmas 8 (2001) 3842. H.G. Craighead, Science 290 (2000) 1532; M. Shahid, G. Murtaza, Phys. Plasmas 20 (2013) 082124. [6] N. Crouseilles, P.A. Hervieux, G. Manfredi, Phys. Rev. B 78 (2008) 155412. [7] R.S. Fletcher, X.L. Zhang, S.L. Rolston, Phys. Rev. Lett. 96 (2006) 105003.

Fig. 7. The group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave.

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