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PII: DOI: Reference:

S0375-9601(15)00177-2 10.1016/j.physleta.2015.02.020 PLA 23098

To appear in:

Physics Letters A

Received date: 19 November 2014 Revised date: 15 January 2015 Accepted date: 11 February 2015

Please cite this article in press as: A. Moradi, Quantum effects on propagation of bulk and surface waves in a thin quantum plasma film, Phys. Lett. A (2015), http://dx.doi.org/10.1016/j.physleta.2015.02.020

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Highlights

• New bulk and surface plasma dispersion relations due to quantum effects are derived, in a thin quantum plasma film. • It is found that quantum effects become important for a thin quantum film of small thickness.

Quantum eﬀects on propagation of bulk and surface waves in a thin quantum plasma ﬁlm Afshin Moradi1,2∗

1

Department of Engineering Physics, Kermanshah University of Technology, Kermanshah, Iran 2

Department of Nano Sciences, Institute for Studies in Theoretical Physics and Mathematics

(IPM), Tehran, Iran Abstract The propagation of bulk and surface plasma waves in a thin quantum plasma ﬁlm is investigated, taking into account the quantum eﬀects. The generalized bulk and surface plasma dispersion relation due to quantum eﬀects is derived, using the quantum hydrodynamic dielectric function and applying appropriate additional boundary conditions. The quantum mechanical and ﬁlm geometric eﬀects on the bulk and surface modes are discussed. It is found that quantum eﬀects become important for a thin ﬁlm of small thickness.

Keywords: Thin quantum plasma ﬁlm; Surface plasmon; Dispersion relation Proposed PACS numbers: 73.20.Mf

∗

E-mail: [email protected]; Correspondence address: Department of Engineering Physics, Kermanshah

University of Technology, Kermanshah, Iran. Tel.: +989183312692.

1

1

Introduction

Recently, there has been a great interest in physical characteristics and properties of the surface plasmon polriton (SPP) and surface plasmon (SP) waves on a quantum plasma half-space. The propagation of surface waves on a unmagnetized quantum plasma half-space including the quantum eﬀects, quantum statistical Fermi electron temperature, and the quantum electron tunnelling has investigated by Lazar et al [1], using the quantum hydrodynamic (QHD) model and Maxwell-Poisson equations. Chang and Jung [2] studied the quantum eﬀects on the propagation of surface Langmuir oscillations in semi-bounded quantum plasmas, by means the specular reﬂection method. Marklund et al [3] studied the properties of quantum surface plasmon polaritons at the interface between an electron quantum plasma and a dielectric material and discussed the importance of the inﬂuence of the quantum broadening of the transition layer. They derived the well-known electrostatic dispersion relation when the transition layer thickness goes to zero. Using quantum magnetohydrodynamic (QMHD) theory in conjunction with Maxwell equations, the propagation of surface waves in a semi-bounded plasma was studied by Mohamed [4]. He found that the external magnetic ﬁeld which is parallel to the propagation waves has a stronger eﬀect on wave decay in the quantum regime compared with the classical case. The dispersion properties of electrostatic surface waves propagating along the interface between a quantum magnetoplasma composed of electrons and positrons, and vacuum, have been studied by Misra et al. [5]. Also, Misra [6] investigated the propagation of SPP waves along a uniform magnetic ﬁeld in a quantum electron-hole semiconductor plasma half-space. On the other hand, by employing the QMHM model, Abdel Aziz [7] obtained the dispersion relation of a transverse electric (TE) surface waves propagating along the interface between a magneto-quantum plasma-relativistic beam system and vacuum. SPPs in a semi-bounded degenerate plasma were studied by Tyshetskiy et al [8], using the quasi-classical mean-ﬁeld kinetic model, taking into account the spatial dispersion of the plasma (due to quantum degeneracy of electrons) and electron-ion (electron-lattice, for metals) collisions. Also, Khorashadizadeh et al [9] and Niknam et al [10], investigated the propagation of surface waves on a semi-bounded quantum plasma in the presence of the external magnetic ﬁeld and collisional eﬀects, by means

2

of the QMHM model. Zhu et al [11] presented a theoretical formalism on the propagation of SPP waves on the relativistic quantum plasma half-space. They show that the frequency of SPPs has a blue-shift. Furthermore, Tyshetskiy et al [12] considered SPs in a semi-bounded quantum plasma with degenerate electrons and discussed some interesting consequences of electron Pauli blocking for the SP dispersion and temporal attenuation. Recently, the propagation of surface electromagnetic waves on the plane between a vacuum and a warm quantum magnetized plasma, have been studied by Li et al [13], by means the QMHD model includes quantum diﬀraction eﬀect (Bohm potential), and quantum statistical pressure. More recently, the dispersion relations of SPPs has been derived by Zhu [14], using QMHD model with quantum eﬀects due to the Bohm potential, Fermi degenerate pressure and electron spin. Also, Chandra et al [15], studied the nonlinear self-interaction of an electrostatic surface wave on a semi-bounded quantum plasma with relativistic degeneracy, by means QHD model and the Poisson equation with appropriate boundary conditions. Nevertheless, the quantum mechanical and geometric eﬀects on the surface quantum plasma waves in a thin quantum plasma ﬁlm have been studied by Jung and Hong, only [16]. They obtained the symmetric and anti-symmetric dispersion modes of the quantum surface waves using the quantum plasma dielectric function in conjunction with the kinetic dispersion model [17]. Moreover, they found that the group velocity of the anti-symmetric mode in small wave numbers is negative so that the corresponding wave is propagating in backward direction in the quantum plasma ﬁlm. In the present work, following the previous papers [1-15] and motivated by the recent work given in Ref. 16, we want to study the quantum eﬀects on propagation of bulk and surface waves in a thin quantum plasma ﬁlm. We obtain the generalized bulk and surface plasma dispersion relation due to quantum eﬀects, using the QHD dielectric function and applying appropriate additional boundary conditions.

3

2

Theory

Let us consider the system consisting of a dielectric medium characterized by a real, positive dielectric constant ε1 in the region z > d, a thin quantum plasma ﬁlm characterized by a longitudinal quantum dielectric function ε2L and a transverse dielectric function ε2T in the region 0 < z < d, and a dielectric medium characterized by a real, positive dielectric constant ε3 in the region z < 0. The SPP waves are supposed to propagate parallel to the interface z = 0 and z = d along the x-direction. The quantum eﬀects can be accounted by employing the QHD model [18], and this is the approach adopted here. In this way, we employ the usual, Drude type, transverse dielectric function ε2T = εother −

ωp2 , ω2

(1)

where εother in general is frequency-dependent and describing the remaining dielectric response of the system [19]. On the other hand, using the Poisson equation with the Fourier transformation in ω − k space, the longitudinal plasma dielectric function [18] in a quantum plasma is found to be ε2L = εother −

ωp2 , ω 2 − α2 kL2 − β 2 kL4

(2)

where ω is the frequency, kL is the quantum nonlocal longitudinal wave vector, ωp = (4πe2 n0 /me )1/2

is the classical plasma frequency in homogeneous electron quantum plasma, α =

3/5vF ,

β = aB vB /2, with the Fermi speed vF = h ¯ (3π 2 n0 )1/3 /me , the Bohr radius aB = h ¯ 2 /e2 me , and the Bohr speed vB = e2 /¯h, respectively. We note that the second term in denominator of Eq. (2) is regarded as the quantum statistical eﬀect caused by the internal interactions in the electrons species and the third term regarded as quantum diﬀraction eﬀect comes from the quantum pressure. The longitudinal quantum plasma waves obey the dispersion relation ε2L (kL , ω) = 0,

(3)

and the transverse electromagnetic waves satisfy the relation kT2 = ε2T 4

ω2 , c2

(4)

where c is the light speed in vacuum. The propagation constants of the longitudinal and transverse modes that are excited in thin ﬁlm are obtained from Eqs. (3) and (4), respectively. From Eq. (3), the expression of kL2 can be written as

kL2

ω2 α2 α2 4β 2 = − 2 ± 2 1 + 4 ω2 − p 2β 2β α εother

1/2

.

(5)

Let us note when β → 0, the solution of kL2 should be consistent with previous results in the absence of the quantum diﬀraction eﬀect [20-24]. This means that the symbol ” + ” in Eq. (5) may be the expression of which consistent with previous works, in the limit of the low value of β [23]. We consider the case of the TM polarization that propagates parallel to the interface z = 0 and z = d along the x-direction. In a wave of this polarization the magnetic vector is perpendicular to the plane of incidence, where it deﬁned by the direction of propagation and the normal to the surface [25]. We note for the TE polarization there can be no interaction between transverse and longitudinal waves in isotropic media [26]. The components of the magnetic ﬁeld in the system that describe a TM electromagnetic wave propagating in the x-direction and localized to each of its two interfaces, can be written in the following forms: H1 (x, z, t) = (0, A1 , 0) e−κ1 z eikx−iωt ,

z ≥ d,

H2 (x, z, t) = 0, A2 eκ2T z + A3 e−κ2T z , 0 eikx−iωt , H3 (x, z, t) = (0, A4 , 0) eκ3 z eikx−iωt , where κ1 = (k 2 − ε1 ω 2 /c2 )

1/2

, κ2T = (k 2 − kT2 )

1/2

(6)

0 ≤ z ≤ d,

(7)

z ≤ 0,

and κ3 = (k 2 − ε3 ω 2 /c2 )

(8) 1/2

determine the

decay of the electromagnetic ﬁeld with increasing distance from the surfaces z = 0, and z = d. Using the Maxwell equation E = (−c/iωε) ∇ × H, the electric ﬁeld components of the TM wave in each medium can be written in the following forms: E1 (x, z, t) = − E2 (x, z, t) = −

c A1 (κ1 , 0, ik) e−κ1 z eikx−iωt , iωε1

z ≥ d,

c κ2T −A2 eκ2T z + A3 e−κ2T z , 0, ik A2 eκ2T z + A3 e−κ2T z eikx−iωt , iωε2T

(9) 0 ≤ z ≤ d, (10)

5

E3 (x, z, t) = −

c A4 (−κ3 , 0, ik) eκ3 z eikx−iωt , iωε3

z ≤ 0.

(11)

Also in the thin quantum plasma ﬁlm, at the same frequency ω, there is a longitudinal wave (bulk plasmon) with described by the following electric ﬁelds

EL (x, z, t) = ik A5 eκ2L z + A6 e−κ2L z , 0, κ2L A5 eκ2L z − A6 e−κ2L z

eikx−iωt ,

0 ≤ z ≤ d, (12)

where κ2L = (k 2 − kL2 )

1/2

. The coeﬃcients A1 − A6 in the above equations can be determined

from the matching boundary conditions of the ﬁelds at z = 0 and z = d. The usual two boundary conditions at the plasma-dielectric interface require the continuity of the tangential components of the electric and magnetic ﬁelds across the interfaces. We note that in the thin ﬁlm both the transverse and longitudinal (usually neglected) waves give a contribution to the value of electric ﬁeld, we have H1y (x, z, t)|z=d = H2y (x, z, t)|z=d ,

(13)

H3y (x, z, t)|z=0 = H2y (x, z, t)|z=0 ,

(14)

E1x (x, z, t)|z=d = E2x (x, z, t)|z=d + ELx (x, z, t)|z=d ,

(15)

E3x (x, z, t)|z=0 = E2x (x, z, t)|z=0 + ELx (x, z, t)|z=0 .

(16)

and

Now, we use the additional boundary condition of the continuity of the normal component of the displacement ﬁeld, was derived by Yan et al [19]. The third boundary condition gives ε1 E1z (x, z, t)|z=d = εother E2z (x, z, t)|z=d + εother ELz (x, z, t)|z=d ,

(17)

ε1 E3z (x, z, t)|z=0 = εother E2z (x, z, t)|z=0 + εother ELz (x, z, t)|z=0 .

(18)

and

The boundary conditions [Eqs. (13)-(18)] yield a system of six linear equations for the coeﬃcients A1 − A6 . Equating to zero the determinant of this system yields the following dispersion relation of the SPP waves

−κ2L d κ2T d Γ− + Γ+ κ2L + Λ eκ2T d − e−κ2L d k 2 32 e 12 e

6

× =

×

κ2L d −κ2T d Γ+ + Γ− κ2L + Λ eκ2L d − e−κ2T d k 2 32 e 12 e

−κ2L d −κ2T d Γ+ + Γ− κ2L + Λ e−κ2T d − e−κ2L d k 2 32 e 12 e

κ2L d κ2T d Γ− + Γ+ κ2L + Λ eκ2L d − eκ2T d k 2 , 32 e 12 e

(19)

where Γ± 12 =

κ1 κ2T κ3 κ2T ± , Γ± ± , 32 = ε1 ε2T ε3 ε2T 1 1 − . Λ= εother ε2T

This is the generalized bulk and surface waves dispersion relation due to quantum eﬀects in a thin quantum plasma ﬁlm, using the QHD dielectric function and applying appropriate additional boundary conditions.

3

Numerical Result and Discussion

In this section, with the above equation, we present the simulation results and discussion of the problem. We see from Eq. (19) that in the limit as d → ∞ yields the pair of equations

κ1 κ2T + ε1 ε2T κ3 κ2T + ε3 ε2T

κ2L +

1 εother

κ2L +

1 εother

−

1 ε2T

−

1 ε2T

k 2 = 0,

(20)

k 2 = 0,

(21)

that show the dispersion relations for SPPs at the dielectric-metal interface. An another interesting special case of Eq. (19) is the one where both superstrate and substrate of thin quantum plasma ﬁlm are same, i.e., ε1 = ε3 and κ1 = κ3 . In this case Eq. (19) can be rearranged into two categories: even and odd modes. For an even mode, we have:

κ1 κ2T κ2T d + coth ε1 ε2T 2

κ2L d κ2L + coth 2

1 εother

−

1 ε2T

−

1 ε2T

k 2 = 0,

(22)

k 2 = 0,

(23)

and for an odd mode, we obtain:

κ1 κ2T κ2T d + tanh ε1 ε2T 2

κ2L d κ2L + tanh 2

7

1 εother

In the following, we restrict our attention to the electrostatic surface waves only [1]. For large wave numbers the frequencies of the even and odd modes deﬁned by Eqs. (22) and (23) are

2 ωeven κ2L d 1 k −kd = 1 − e 1+ coth 2 ωp 2 κ2L 2

2 κ2L d ωodd 1 k −kd = 1 + e 1+ tanh 2 ωp 2 κ2L 2

,

(24)

,

(25)

when the free electron form [Eq. (1)] for ε2T and ε1 = εother = 1 are assumed. For mathematical purposes it is convenient to transform the dispersion relations Eqs. (24) and (25) in the more suitable forms by introducing the following dimensionless variables: Ω = ω/ωp , K = kd, η = ωp d/α, and H = βωp /α2 . Thus, in terms of these variables we can write Eqs. (24) and (25) in the forms

χ 1 K = 1 − e−K 1 + coth 2 χ 2

,

(26)

Ω2odd

χ 1 K = 1 + e−K 1 + tanh 2 χ 2

,

(27)

Ω2even

where χ = κ2L d = K 2 +

η2 2H 2

−

η2 2H 2

[1 + 4H 2 (Ω2 − 1)]

1/2 1/2

. We note, χ real corresponds

to surface modes, while χ imaginary corresponds to volume modes. Also, we see the above result shows a general scaling property of Ω = ω/ωp , in terms of the variable K = kd. The ”plasmonic” coupling parameter, i.e., H, used in Eqs. (26) and (27), describes the ratio of the plasmonic energy density to the electron Fermi energy density. In the absence of the quantum diﬀraction eﬀect (Bohm potential), i.e., H → 0, Eqs. (26) and (27) becomes identical to the resonant plasmon frequency equation obtained by Dharamvir et al [27] for the surface electrostatic waves on a thin thermal plasma ﬁlm. Let us now discuss the behavior of the surface and bulk modes in a thin quantum plasma ﬁlm for diﬀerent values of the parameter H, K and η. In Fig. 1, we plot dimensionless frequency Ω, versus dimensionless variable η, for diﬀerent values of H, when K = 1. Panels (a1) and (a2) represent even modes and panels (b1) and (b2) represent odd modes. As we show in Fig. 1, for each category of modes there is only one surface mode and an inﬁnite series of bulk modes. From panels (a1) and (a2) for even modes, and panels (b1) and (b2) for odd modes, it is easy to ﬁnd by increasing the plasmonic coupling parameter H, the number of bulk mode branches 8

5

5

(b1) H=0

(a1) H=0 4

4

Bulk modes

Bulk modes 3

3

2

2

Surface mode

1

1

Surface mode Even modes for K=1

Odd modes for K=1

0

0 2

4

6

8

Η

10

12

14

2

4

6

8

Η

10

12

14

10

12

14

5

5

(b2) H=0.3

(a2) H=0.3 4

4

3

3

2

2

1

1

0

0 2

4

6

8

Η

10

12

14

2

4

6

8

Η

Figure 1: The dimensionless plasmon dispersion curves Ω = ω/ωp of a thin quantum plasma ﬁlm, versus the dimensionless variable η = ωp d/α, for H = 0 and 0.3, when K = 1. Panels (a1) and (a2) represent even modes and panels (b1) and (b2) represent odd modes.

9

5

5

(a1) H=0

(b1) H=0

4

4

Bulk modes Bulk modes

3

3

2

2

Surface mode

Surface mode

1

1

Odd modes for Η=6

Even modes for Η=6 0

0 0

2

4

6

8

10

12

14

0

2

4

6

K

8

10

12

14

10

12

14

K 5

5

(a2) H=0.4

(b2) H=0.4 4

4

3

3

2

2

1

1

0

0 0

2

4

6

8

10

12

14

0

K

2

4

6

8

K

Figure 2: The dimensionless plasmon dispersion curves Ω = ω/ωp of a thin quantum plasma ﬁlm, versus the dimensionless variable K = kd, for H = 0 and 0.4, when η = 6. Panels (a1) and (a2) represent even modes and panels (b1) and (b2) represent odd modes.

10

decreases, and their separations increase. Also, we can conclude that increasing H can grow the frequencies of the bulk modes. On the other hand, it can be seen that the frequencies of the even and odd surface and bulk modes will depend very much on the thickness of the thin ﬁlm (parameter η). From all panels of Fig. 1, one can observe, by increasing the thickness of the system, the number of bulk mode branches increases, and their separations decreases, while the surface mode frequencies are red-shifted. Furthermore, for a ﬁxed value of wave number, the plasmonic coupling parameter H play an important role on the frequencies of the even and odd surface modes in the low range of parameter η. In Fig. 2, we show dimensionless frequency Ω, versus dimensionless variable K, for diﬀerent values of H, when η = 4. It is clear that the even mode of the surface wave monotonically increases while the odd mode of the surface wave is red-shifted ﬁrst and then blue-shifted, after the local minimum. Also, for a ﬁxed value of thin ﬁlm thickness, we ﬁnd plasmonic coupling parameter H can play an important role on the frequencies of the even and odd surface modes in the high range of wave number. Finally, we note in the present work, the processes such as the nonlinear coupling between high frequency surface waves and low frequency ion oscillations on a quantum plasma surface [28] and drift ion acoustic waves [29] are not discussed. Also, a future study may be focused on the role of quantum eﬀects on the anomalous reﬂection of surface plasma waves [30].

4

Conclusion

To summarize, we have studied the propagation of bulk and surface waves in a thin quantum plasma ﬁlm, taking into account the quantum eﬀects. We have obtained a analytical expression of the dispersion relation for the bulk and surface wave oscillations of the system, using QHD theory. We have found that by increasing the quantum eﬀects, the number of bulk mode branches decreases, and their separations increase. Generally, increasing the quantum eﬀects can grow the frequencies of the bulk modes. However, the behaviors of the even and odd modes of the surface quantum plasma wave are diﬀerent. We have shown quantum eﬀects become important for surface modes of a thin ﬁlm of small thickness, when the wave number is ﬁxed. 11

On the other hand, for a ﬁxed value of thin ﬁlm thickness, when wave number of the surface modes becomes smaller, the quantum eﬀects will be weak.

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