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Quantum and geometric effects on the symmetric and anti-symmetric modes of the surface plasma wave
Physics Letters A 377 (2013) 560–563
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Quantum and geometric effects on the symmetric and anti-symmetric modes of the surface plasma wave Young-Dae Jung a,b,∗ , Woo-Pyo Hong c a b c
Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407, USA Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea Department of Electronics Engineering, Catholic University of Daegu, Hayang, 712-702, South Korea
a r t i c l e
i n f o
Article history: Received 4 October 2012 Received in revised form 22 November 2012 Accepted 28 December 2012 Available online 3 January 2013 Communicated by C.R. Doering
a b s t r a c t The propagation of the surface quantum plasma waves is investigated in a thin quantum plasma slab. The symmetric and anti-symmetric dispersion modes of the quantum surface wave are obtained by the plasma dielectric function with the kinetic dispersion model for the slab geometry. The quantum mechanical and slab geometric effects on the symmetric and anti-symmetric modes are also discussed. © 2013 Elsevier B.V. All rights reserved.
Keywords: Symmetric and anti-symmetric modes Surface plasma waves Quantum plasmas
Recent years, there has been a great interest in physical characteristics and properties of various quantum plasmas [1–10] since the quantum plasmas are ubiquitous and have been found in numerous modern sciences and technologies such as semiconductor devices, nano-wires, quantum dot, and also laser produced dense plasmas. In addition, the physical characteristics of quantum plasmas have been extensively investigated including the Bohm potential term as well as the Fermi–Dirac distribution [6,7,10] since the de Broglie wavelength of the plasma particle would be comparable to the atomic scale in quantum plasmas. In addition, the propagation and dispersion properties of surface plasma waves on the bounded and semi-bounded plasmas have drawn much interest in numerous areas such as laser physics, materials science, nanotechnology, plasma spectroscopy, space science, etc. [11–14]. Moreover, the investigations for the propagation of surface waves on a plasma slab have been found in several excellent papers [15–17]. However, the quantum mechanical and geometric effects on the surface quantum plasma waves in a thin plasma slab have not been investigated as yet. It is quite obvious that the investigation of the dispersion properties of the surface wave in a quantum plasma slab would be a useful tool for investigating the physical characteristic of quantum waves and physical properties of the bounded quantum plasmas. Thus, in this Letter, we investigate the quantum
*
Corresponding author at: Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea. Tel.: +82 31 400 5477; fax: +82 31 400 5457. E-mail address: [email protected] (Y.-D. Jung). 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.12.035
and geometric effects on the symmetric and anti-symmetric modes of the surface plasma waves in a thin quantum plasma slab with the kinetic dispersion model for the slab geometry. In the nonrelativistic quantum hydrodynamic (QHD) model [10], the continuity equation and the momentum equation for a quantum plasma are given by
∂ nα + ∇ · (nα vα ) = 0, ∂t ∂ vα + vα · ∇ vα ∂t =−
qα mα
∇ϕ −
1 mα nα
(1)
∇ pα +
h¯ 2
∇ 2
2mα
√ ∇ 2 nα , √ nα
(2)
and Poisson’s equation is
∇ 2 ϕ = −4π
q α nα ,
(3)
α
where nα , vα , qα , mα , and p α are, respectively, the number density, the velocity, the charge, the mass, and the pressure for a species α , ϕ is the electric potential and h¯ is the rationalized Planck constant. It has been shown that the QHD equation are valid [18] when the plasma energy is smaller or comparable to the electron Fermi energy and the electron–ion relaxation time is greater than the oscillation period of the electron plasma. In addition, it is shown that the QHD model would be useful to investigate the transport of charge, momentum, and energy in nanoscience and semiconductor physics [7]. In Eq. (2), the pressure
Y.-D. Jung, W.-P. Hong / Physics Letters A 377 (2013) 560–563
term represents the quantum statistical effect due to the fermionic behavior of the plasma particles and the h¯ -dependent term is known as the Bohm potential term due to the quantum diffraction effect [19]. Using the Poisson equation with the Fourier transformation in ω–k space, the plasma dielectric function [19] in a quantum plasma is found to be
ε(ω, k) = 1 −
ω2pe ω2 − k2 V F2 − h¯ 2k4 /4me2
(4)
,
where ω is the frequency, k is the wave vector, ω pe [= (4π ne e 2 / me )1/2 ] is the electron plasma frequency, e is the charge of the electron, V F [= (2k B T F /me )1/2 ] is the electron Fermi velocity, k B is the Boltzmann constant, and T F is the Fermi temperature. It has been shown that the kinetic dispersion relation of semibounded plasmas would be obtained by the specular reflection condition; however, the investigation of kinetic dispersion relations of surface waves in plasma slab geometries is rather few [20]. In a semi-bounded quantum plasma, the quantum effects are investigated on the propagation of the surface Langmuir oscillation including the Bohm potential term [14]. Here, we consider a quantum plasma slab (0 x L ) bounded by a vacuum (x < 0 and x > L). Using the appropriate physical boundary conditions for the plasma distribution and the electric E and magnetic B fields at the plasma–vacuum interfaces and the Fourier transform method with extending of the plasma electric field out of the plasma region since the plasma solutions and the vacuum solutions should be matched at the interfaces x = 0 and x = L, Lee and Lim [17] have obtained the closed form of the kinetic dispersion relation for surface electromagnetic waves propagating in the z-direction in the plasma slab between the interfaces at x = 0 and x = L such as
∞
dkx
εt − c 2k2 /ω2
k2
−∞
+i
∞
c2
ω
k2x
+
k2z
1 ∓ e ikx L 1 ± e ikx L
εl
dkx kx (k2z − ω2 /c 2 )1/2
2
−∞
εt − c 2k2 /ω2
= 0,
(5)
where k = k2x + k2z , kx (= k⊥ ) and k z (= k ) are, respectively, the perpendicular and parallel components of the wave vector k, εt and εl are the transverse and longitudinal components of the plasma dielectric function, the upper and lower signs in the integrand correspond to the anti-symmetric and symmetric modes, respectively. In the evaluation of Eq. (5), the singularities associated with 1 ± e ikx L = 0 should simply be ignored in the contour integration and, however, other singular points of the denominator in the integrand are to be substituted into the exponential terms. In the electrostatic wave modes, i.e., taking the limit c → ∞, the kinetic dispersion relation [17] for electrostatic waves for the plasma slab in the region 0 x L is given by 2
∞
π+ −∞
dkx
kz
1 ∓ e ikx L
(k2x + k2z ) εl (ω, kx , k z ) 1 ± e ikx L
= 0.
(6)
When the thickness of the plasma slab is extended to infinity (L → ∞), the exponential terms e ikx L in Eq. (6) vanish due to rapid oscillations so that Eq. (6) reduces to the well-known expression (ABR) formula [21], of∞the Alexandrov–Bogdankevich–Rukhadze 2 −∞ dk x k z /(π k εl ) + 1 = 0, for the kinetic dispersion relation of the surface waves in semi-infinite plasmas. Since the singularities associated with 1 ± e ikx L = 0 should be ignored in the contour integration, using Jordan’s lemma with Eqs. (4) and (6) in the
561
upper-half domain of the complex k¯ x -plane, the dispersion relations for the symmetric and anti-symmetric modes of the surface plasma wave in the quantum plasma slab bounded by the interfaces at x = 0 and x = L are, respectively, obtained by the following contour integrations:
∞ −∞
dkx k z
1 + e ikx L
(k2x + k2z )εl 1 − e ikx L
=i
dk¯ x k¯ z cot(k¯ x L¯ /2) 1 2 2 ¯ ¯ (kx + k z ) 1 − 2 2 2
ω¯ 2 −(k¯ x +k¯ z ) V¯ F −(k¯ 2x +k¯ 2z )2
= −π , ∞
(7)
dkx k z
−∞
(k2x
+ k2z ) l
ε
= −i
1−e
ik x L
1 + e ikx L
dk¯ x k¯ z tan(k¯ x L¯ /2) 1 2 2 ¯ ¯ (kx + k z ) 1 − 2 2 2
ω¯ 2 −(k¯ x +k¯ z ) V¯ F −(k¯ 2x +k¯ 2z )2
= −π ,
(8)
where k¯ x ≡ kx λq , λq (≡ h¯ 2 /4me2 ω2pe )1/4 is the quantum wave
¯ ≡ ω/ω pe , and V¯ F ≡ V F /λq ω pe . The length, k¯ z ≡ k z λq , L¯ ≡ L /λq , ω expressions of Eqs. (7) and (8), would be quite useful to investigate various physical properties of the symmetric and anti-symmetric modes of the electrostatic surface quantum plasma wave in a specific slab plasma geometry. In the case of L → ∞, it is interesting to note that the symmetric and anti-symmetric modes are merged into the surface quantum wave mode of the semi-bounded quantum plasma. For a very thin quantum plasma slab, the volume density n(x) would be replaced by σ δ(x) where σ is the surface particle number density [22]. This mathematical treatment would be possible only when the thickness of the plasma slab is much smaller than the wave length of the surface wave and quantum shielding distance. Even though the delta-function type δ(x) plasma density is legitimate in mathematically proposed plasma geometries, we shall not consider here the case of zero slab thickness since the plasma slab would not be shrunk to zero in real plasma situations. The detailed theoretical investigation on the surface waves on the thin plasma sheet was given in the limit of zero slab thickness [22]. In addition, Eqs. (7) and (8) correspond to the transverse magnetic mode dispersion relations obtained by Gradov and Stenflo [15] based on the fluid descriptions. As it is seen, the three singularities in the upper-half domain of the complex k¯ x -plane would provide the contribution to the contour integrations in Eqs. (7) and (8) among the total ¯2 + six singularities: k¯ 2x + k¯ 2z = 0 (primary poles) and (k¯ 2x + k¯ 2z )2 ω (k¯ 2x + k¯ 2z ) V¯ F2 − ω¯ 2 + 1 = 0 (quantum poles). Hence, the dispersion properties of the symmetric and anti-symmetric modes of the surface quantum plasma in a quantum plasma slab would be obtained by using the contour integrations with one primary pole and two quantum poles. The spatial distribution of the wave fields would be important for investigating the physical characteristics of the surface wave. However, the dispersion relation ω = ω(k) would be enough to explore the physical properties of the electrostatic surface wave since the field propagation can be determine by the Fourier component exp[−i (k · r + ω(k))t ]. Very recently, an excellent work has provided the Lennard-Jones type new attractive interaction potential between ions in degenerate electron quantum plasmas including the influence of the quantum statistical pressure and quantum Bohm terms as well as the electron exchange and correlations due to the influence of the electron spin [23]. Hence, the investigation of propagating quantum surface waves in a degenerate quantum plasma slab in the framework
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Y.-D. Jung, W.-P. Hong / Physics Letters A 377 (2013) 560–563
of the quantum hydrodynamical description will be treated elsewhere. In Eqs. (7) and (8), it is found that the unstable modes due to the imaginary solutions of the modulation frequency are quite small and negligible compared to the stable mode solutions so that the stability of the modes is satisfied. However, in future work we shall investigate the effect of the instability on the propagation of the surface wave in a quantum plasma slab including the quantum statistical pressure and quantum Bohm effects as well as the electron exchange and correlation effects due to the influence of the electron spin [23] since the criteria for the wave instability would be a crucial condition for maintaining the bounded plasma system such as the thin quantum plasma slab. In quantum plasma [24] composed of electrons and ions, the plasma number density n and the temperature T are about 1020 –1024 cm−3 and 5 × 104 –106 K, respectively. The characteristics of the quantum plasma [24] can also be represented by the coupling parameter Γ [= e 2 /ak B T ], the degeneracy parameter Θ (= k B T / E F ), and the density parameter r s (= a/a0 ), where a is the average interparticle distance in a plasma. Recently, the excellent investigations have provided the useful effective potential model in dense semiclassical plasmas taking into accounts the quantum diffraction, symmetry, and degeneracy effects as well as quantum shielding effect. Hence, the additional correction to the propagation of the surface quantum plasma waves in a quantum plasma slab including the influence of the effective Ramazanov– Dzhumagulova pseudopotential [24–26] will be treated elsewhere. Recently, the excellent works have provided the detailed investigations on the dispersion properties of electromagnetic [27] and electrostatic [28] surface waves propagating along the interface between a magnetized quantum plasma and vacuum. Then, the propagation of the surface quantum plasma waves in a magnetized quantum plasma slab will also be treated elsewhere. In addition, a recent excellent work [29] has shown the importance of the influence of the quantum broadening of the transition layer on the properties of quantum surface plasmon polaritons at the interface between an electron quantum plasma and a dielectric material, or vacuum. Hence, the propagation of the surface quantum plasma waves in a quantum plasma slab including the effect of quantum broadening of the transition layer will also be treated elsewhere. In order to investigate the physical characteristics of the sur¯ pe = 4 and 8, where h¯ ω¯ pe ≡ face quantum plasma, we consider h¯ ω h¯ ω pe /Ry and Ry (= me e 4 /2h¯ 2 ≈ 13.6 eV) is the Rydberg constant. Since the physical behaviors of the surface wave can be more easily understood by the graphical illustration using the dimensionless or scaled variables, we use the dimensionless frequencies and wave numbers in figures with various plasma parameters. Fig. 1(a) shows the geometric effects on the two dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The symmetric or anti-symmetric mode of the surface quantum plasma wave corresponds to the case of symmetric or anti-symmetric with respect to the slab axis in the z-direction. As shown in this figure, the behaviors of the symmetric and antisymmetric modes of the surface quantum plasma wave are quite different. From this figure, we have found that the frequency of the symmetric mode of the surface quantum plasma wave decreases with an increase of the wave number in small wave number region and then increases with increasing wave number after the local minimum. However, it is found that the anti-symmetric mode of the surface quantum plasma wave monotonically increases with an increase of the wave number. It is interesting to find out that the phase velocity of the anti-symmetric mode of the surface quantum plasma wave is always greater than that of the symmetric mode of the surface quantum plasma wave for a given wave number. It is found that an increase of the slab thickness suppresses the phase velocity of the anti-symmetric mode of the surface
Fig. 1. (a) The geometric effects on the dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The anti-symmetric mode for L¯ = 10, Γ Θ r s = 8, and V¯ F = 0.5 (red solid curve) and L¯ = 20, Γ Θ r s = 8, and V¯ F = 0.5 (blue solid curve). The symmetric mode for L¯ = 10, Γ Θ r s = 2, and V¯ F = 1.0 (red dotted curve), L¯ = 20, Γ Θ r s = 2, and V¯ F = 1.0 (blue dotted curve). (b) The quantum effects on the dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The anti-symmetric mode for L¯ = 20, Γ Θ r s = 2, and V¯ F = 1.0 (red solid curve) and L¯ = 20, Γ Θ r s = 8, and V¯ F = 0.5 (blue solid curve). The symmetric mode for L¯ = 20, Γ Θ r s = 2, and V¯ F = 1.0 (red dotted curve) and L¯ = 20, Γ Θ r s = 8, and V¯ F = 0.5 (blue dotted curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
quantum plasma wave. However, we have found that the phase velocity of the symmetric mode of the surface quantum plasma wave increases with an increase of the slab thickness. Hence, it would be expected that the group velocities of the symmetric mode of the surface quantum plasma are greater than those of the anti-symmetric mode of the surface quantum plasma in a thin quantum plasma slab. However, it is found that the geometric thickness effect on both the symmetric and anti-symmetric modes of the surface quantum plasma decreases with increasing wave number. It is interesting to note that the symmetric and anti-symmetric modes of the surface quantum plasma merge into a single mode with an increase of the slab thickness. This converging single mode corresponds to the ABR mode of the surface wave of the semi-bounded quantum plasma since the exponential terms (1 ∓ e ikx L )/(1 ∓ e ikx L ) in Eq. (6) would be unity as L¯ 1. Fig. 1(b) represents the quantum effects on the dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. As it is seen in this figure, an increase of the Fermi velocity suppresses the phase velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. Hence, we have found that the quantum effects diminish the frequencies of the symmetric and anti-symmetric modes of the surface quantum plasma wave in large wave numbers. It is also found that the quantum effect on the surface quantum plasma wave increases with increasing wave number. It is then found that the quantum effect on the surface quantum plasma wave is more significant than the geometric effect on the surface quantum plasma wave in large wave numbers. However, the geometric effect on the surface quantum plasma wave is found to be more significant than the quantum effect on the surface quantum plasma wave in small wave numbers. Fig. 2 shows the group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave in the quantum plasma slab. As we can see from this figure, 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 slab. However,
Y.-D. Jung, W.-P. Hong / Physics Letters A 377 (2013) 560–563
Fig. 2. The group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The symmetric mode for L¯ = 10, Γ Θ r s = 8, and V¯ F = 0.5 (red solid curve) and L¯ = 50, Γ Θ r s = 8, and V¯ F = 0.5 (blue solid curve). The anti-symmetric mode for L¯ = 10, Γ Θ r s = 8, and V¯ F = 0.5 (red dotted curve) and L¯ = 50, Γ Θ r s = 8, and V¯ F = 0.5 (blue dotted curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
the group velocity of the symmetric mode is found to be positive in all wave numbers so that the symmetric mode propagates as a forward wave. The backward phenomenon has been also found in surface wave propagations in a dielectric planar waveguide geometry [20]. It is also found that the group velocity decreases with an increase of the thickness of the plasma slab in small wave numbers. Hence, we can expect that the geometric effect suppresses the group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. To summarize, we have investigated the quantum mechanical and geometric effects on the surface quantum plasma wave in a thin plasma slab. Using the plasma dielectric function with the kinetic dispersion model, the symmetric and anti-symmetric dispersion modes of the quantum surface wave are obtained for the slab geometry. The results show that the frequency of the antisymmetric mode of the surface quantum plasma wave decreases with an increase of the wave number in small wave number region and then increases with increasing wave number after the local minimum. However, we have shown that the symmetric mode of the surface quantum plasma wave monotonically increases with an increase of the wave number. We also found that the phase velocity of the anti-symmetric mode of the surface quantum plasma is always greater than that of the symmetric mode of the surface quantum plasma in a thin quantum plasma slab. In addition, it is found that the geometric thickness effect on both the symmetric and anti-symmetric modes of the surface quantum plasma decreases with increasing wave number. It is also found that the quantum effect suppresses the frequencies of the symmetric and anti-symmetric modes of the surface quantum plasma wave in large wave numbers. Hence, we have shown that the quantum effect on the surface quantum plasma wave is more significant than the geometric effect on the surface quantum plasma wave in large wave numbers and, however, the geometric effect on the surface quantum plasma wave is found to be more significant than the quantum effect on the surface quantum plasma wave in small wave numbers. Moreover, it is 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 slab. However, the group velocity of the symmetric mode is found to be positive in all wave numbers so that
563
the symmetric mode propagates as a forward wave. In addition, it is found that the geometric effect suppresses the group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. These results would be applied to quantum plasma systems in nanodevices for investigating the physical characteristics and properties of the quantum plasma slab containing the quantum pressure and Bohm potential terms. In this work, it is important to find out that the existence and physical behaviors of the symmetric and anti-symmetric modes of the surface quantum plasma wave would be caused by the quantum statistical effect due to the fermionic behavior of the plasma particles and the Bohm potential term due to the quantum diffraction effect as well as the geometrical effect of the quantum plasma slab. From these results, we have shown that the quantum mechanical and geometric effects play crucial roles in the propagations of the symmetric and anti-symmetric dispersion modes of the surface quantum surface wave in a quantum plasma slab. These results would be useful for understanding the physical and dispersion properties of the surface quantum wave in a slab geometry of the quantum plasma. Acknowledgements The authors gratefully acknowledge Prof. H.J. Lee and Prof. Y.K. Lim for providing highly useful references and valuable discussions. One of the authors (Y.-D.J.) gratefully acknowledges Dr. M. Rosenberg for useful discussions and warm hospitality while visiting the Department of Electrical and Computer Engineering at the University of California, San Diego. This research was initiated while one of the authors (Y.-D.J.) was affiliated with UCSD as a Visiting Professor. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-001493). References [1] T.S. Ramazanov, K.N. Dzhumagulova, Phys. Plasmas 9 (2002) 3758. [2] P.K. Shukla, M. Marklund, Phys. Scr. T 113 (2004) 36. [3] T.S. Ramazanov, K.N. Dzhumagulova, Y.A. Omarbakiyeva, Phys. Plasmas 12 (2005) 092702. [4] T.S. Ramazanov, K.N. Dzhumagulova, A.Zh. Akbarov, J. Phys. A 39 (2006) 4335. [5] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78 (2006) 591. [6] P.K. Shukla, L. Stenflo, Phys. Plasmas 13 (2006) 044505. [7] H. Ren, Z. Wu, P.K. Chu, Phys. Plasmas 14 (2007) 062102. [8] P.K. Shukla, Nat. Phys. 5 (2009) 92. [9] N. Shukla, P.K. Shukla, B. Eliasson, L. Stenflo, Phys. Lett. A 374 (2010) 1749. [10] P.K. Shukla, B. Eliasson, Rev. Mod. Phys. 83 (2011) 885. [11] H.S. Hong, H.J. Lee, Phys. Plasmas 6 (1999) 3422. [12] L. Stenflo, P.K. Shukla, M.Y. Yu, Phys. Plasmas 7 (2000) 2731. [13] H.J. Lee, Phys. Plasmas 12 (2005) 094701. [14] I.-S. Chang, Y.-D. Jung, Phys. Lett. A 372 (2008) 1498. [15] O.M. Gradov, L. Stenflo, Phys. Rep. 94 (1983) 111. [16] L. Stenflo, Phys. Scr. T 63 (1996) 59. [17] H.J. Lee, Y.K. Lim, J. Korean Phys. Soc. 50 (2007) 1056. [18] F. Haas, Quantum Plasmas, Springer, Berlin, 2011. [19] S. Ali, P.K. Shukla, Phys. Plasmas 13 (2006) 102112. [20] Yu.M. Aliev, H. Schlüter, A. Shivarova, Guided-Wave Produced Plasmas, Springer, Berlin, 2000. [21] A.F. Alexandrov, L.S. Bogdankevich, A.A. Rukhadze, Principles of Plasma Electrodynamics, Springer, Berlin, 1984. [22] H.J. Lee, J. Korean Phys. Soc. 43 (2003) 1033. [23] P.K. Shukla, B. Eliasson, Phys. Rev. Lett. 108 (2012) 165007. [24] T.S. Ramazanov, K.N. Dzhumagulova, M.T. Gabdullin, Phys. Plasmas 17 (2010) 042703. [25] T.S. Ramazanov, Zh.A. Moldabekov, K.N. Dzhumagulova, M.M. Muratov, Phys. Plasmas 18 (2011) 103705. [26] Zh.A. Moldabekov, T.S. Ramazanov, K.N. Dzhumagulova, Contrib. Plasma Phys. 52 (2012) 207. [27] A.P. Misra, Phys. Plasmas 14 (2007) 064501. [28] A.P. Misra, N.K. Ghosh, P.K. Shukla, J. Plasma Phys. 76 (2010) 87. [29] M. Marklund, G. Brodin, L. Stenflo, C.S. Liu, Europhys. Lett. 84 (2008) 17006.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Quantum and geometric effects on the symmetric and anti-symmetric modes of the surface plasma wave Young-Dae Jung a,b,∗ , Woo-Pyo Hong c a b c
Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407, USA Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea Department of Electronics Engineering, Catholic University of Daegu, Hayang, 712-702, South Korea
a r t i c l e
i n f o
Article history: Received 4 October 2012 Received in revised form 22 November 2012 Accepted 28 December 2012 Available online 3 January 2013 Communicated by C.R. Doering
a b s t r a c t The propagation of the surface quantum plasma waves is investigated in a thin quantum plasma slab. The symmetric and anti-symmetric dispersion modes of the quantum surface wave are obtained by the plasma dielectric function with the kinetic dispersion model for the slab geometry. The quantum mechanical and slab geometric effects on the symmetric and anti-symmetric modes are also discussed. © 2013 Elsevier B.V. All rights reserved.
Keywords: Symmetric and anti-symmetric modes Surface plasma waves Quantum plasmas
Recent years, there has been a great interest in physical characteristics and properties of various quantum plasmas [1–10] since the quantum plasmas are ubiquitous and have been found in numerous modern sciences and technologies such as semiconductor devices, nano-wires, quantum dot, and also laser produced dense plasmas. In addition, the physical characteristics of quantum plasmas have been extensively investigated including the Bohm potential term as well as the Fermi–Dirac distribution [6,7,10] since the de Broglie wavelength of the plasma particle would be comparable to the atomic scale in quantum plasmas. In addition, the propagation and dispersion properties of surface plasma waves on the bounded and semi-bounded plasmas have drawn much interest in numerous areas such as laser physics, materials science, nanotechnology, plasma spectroscopy, space science, etc. [11–14]. Moreover, the investigations for the propagation of surface waves on a plasma slab have been found in several excellent papers [15–17]. However, the quantum mechanical and geometric effects on the surface quantum plasma waves in a thin plasma slab have not been investigated as yet. It is quite obvious that the investigation of the dispersion properties of the surface wave in a quantum plasma slab would be a useful tool for investigating the physical characteristic of quantum waves and physical properties of the bounded quantum plasmas. Thus, in this Letter, we investigate the quantum
*
Corresponding author at: Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea. Tel.: +82 31 400 5477; fax: +82 31 400 5457. E-mail address: [email protected] (Y.-D. Jung). 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.12.035
and geometric effects on the symmetric and anti-symmetric modes of the surface plasma waves in a thin quantum plasma slab with the kinetic dispersion model for the slab geometry. In the nonrelativistic quantum hydrodynamic (QHD) model [10], the continuity equation and the momentum equation for a quantum plasma are given by
∂ nα + ∇ · (nα vα ) = 0, ∂t ∂ vα + vα · ∇ vα ∂t =−
qα mα
∇ϕ −
1 mα nα
(1)
∇ pα +
h¯ 2
∇ 2
2mα
√ ∇ 2 nα , √ nα
(2)
and Poisson’s equation is
∇ 2 ϕ = −4π
q α nα ,
(3)
α
where nα , vα , qα , mα , and p α are, respectively, the number density, the velocity, the charge, the mass, and the pressure for a species α , ϕ is the electric potential and h¯ is the rationalized Planck constant. It has been shown that the QHD equation are valid [18] when the plasma energy is smaller or comparable to the electron Fermi energy and the electron–ion relaxation time is greater than the oscillation period of the electron plasma. In addition, it is shown that the QHD model would be useful to investigate the transport of charge, momentum, and energy in nanoscience and semiconductor physics [7]. In Eq. (2), the pressure
Y.-D. Jung, W.-P. Hong / Physics Letters A 377 (2013) 560–563
term represents the quantum statistical effect due to the fermionic behavior of the plasma particles and the h¯ -dependent term is known as the Bohm potential term due to the quantum diffraction effect [19]. Using the Poisson equation with the Fourier transformation in ω–k space, the plasma dielectric function [19] in a quantum plasma is found to be
ε(ω, k) = 1 −
ω2pe ω2 − k2 V F2 − h¯ 2k4 /4me2
(4)
,
where ω is the frequency, k is the wave vector, ω pe [= (4π ne e 2 / me )1/2 ] is the electron plasma frequency, e is the charge of the electron, V F [= (2k B T F /me )1/2 ] is the electron Fermi velocity, k B is the Boltzmann constant, and T F is the Fermi temperature. It has been shown that the kinetic dispersion relation of semibounded plasmas would be obtained by the specular reflection condition; however, the investigation of kinetic dispersion relations of surface waves in plasma slab geometries is rather few [20]. In a semi-bounded quantum plasma, the quantum effects are investigated on the propagation of the surface Langmuir oscillation including the Bohm potential term [14]. Here, we consider a quantum plasma slab (0 x L ) bounded by a vacuum (x < 0 and x > L). Using the appropriate physical boundary conditions for the plasma distribution and the electric E and magnetic B fields at the plasma–vacuum interfaces and the Fourier transform method with extending of the plasma electric field out of the plasma region since the plasma solutions and the vacuum solutions should be matched at the interfaces x = 0 and x = L, Lee and Lim [17] have obtained the closed form of the kinetic dispersion relation for surface electromagnetic waves propagating in the z-direction in the plasma slab between the interfaces at x = 0 and x = L such as
∞
dkx
εt − c 2k2 /ω2
k2
−∞
+i
∞
c2
ω
k2x
+
k2z
1 ∓ e ikx L 1 ± e ikx L
εl
dkx kx (k2z − ω2 /c 2 )1/2
2
−∞
εt − c 2k2 /ω2
= 0,
(5)
where k = k2x + k2z , kx (= k⊥ ) and k z (= k ) are, respectively, the perpendicular and parallel components of the wave vector k, εt and εl are the transverse and longitudinal components of the plasma dielectric function, the upper and lower signs in the integrand correspond to the anti-symmetric and symmetric modes, respectively. In the evaluation of Eq. (5), the singularities associated with 1 ± e ikx L = 0 should simply be ignored in the contour integration and, however, other singular points of the denominator in the integrand are to be substituted into the exponential terms. In the electrostatic wave modes, i.e., taking the limit c → ∞, the kinetic dispersion relation [17] for electrostatic waves for the plasma slab in the region 0 x L is given by 2
∞
π+ −∞
dkx
kz
1 ∓ e ikx L
(k2x + k2z ) εl (ω, kx , k z ) 1 ± e ikx L
= 0.
(6)
When the thickness of the plasma slab is extended to infinity (L → ∞), the exponential terms e ikx L in Eq. (6) vanish due to rapid oscillations so that Eq. (6) reduces to the well-known expression (ABR) formula [21], of∞the Alexandrov–Bogdankevich–Rukhadze 2 −∞ dk x k z /(π k εl ) + 1 = 0, for the kinetic dispersion relation of the surface waves in semi-infinite plasmas. Since the singularities associated with 1 ± e ikx L = 0 should be ignored in the contour integration, using Jordan’s lemma with Eqs. (4) and (6) in the
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upper-half domain of the complex k¯ x -plane, the dispersion relations for the symmetric and anti-symmetric modes of the surface plasma wave in the quantum plasma slab bounded by the interfaces at x = 0 and x = L are, respectively, obtained by the following contour integrations:
∞ −∞
dkx k z
1 + e ikx L
(k2x + k2z )εl 1 − e ikx L
=i
dk¯ x k¯ z cot(k¯ x L¯ /2) 1 2 2 ¯ ¯ (kx + k z ) 1 − 2 2 2
ω¯ 2 −(k¯ x +k¯ z ) V¯ F −(k¯ 2x +k¯ 2z )2
= −π , ∞
(7)
dkx k z
−∞
(k2x
+ k2z ) l
ε
= −i
1−e
ik x L
1 + e ikx L
dk¯ x k¯ z tan(k¯ x L¯ /2) 1 2 2 ¯ ¯ (kx + k z ) 1 − 2 2 2
ω¯ 2 −(k¯ x +k¯ z ) V¯ F −(k¯ 2x +k¯ 2z )2
= −π ,
(8)
where k¯ x ≡ kx λq , λq (≡ h¯ 2 /4me2 ω2pe )1/4 is the quantum wave
¯ ≡ ω/ω pe , and V¯ F ≡ V F /λq ω pe . The length, k¯ z ≡ k z λq , L¯ ≡ L /λq , ω expressions of Eqs. (7) and (8), would be quite useful to investigate various physical properties of the symmetric and anti-symmetric modes of the electrostatic surface quantum plasma wave in a specific slab plasma geometry. In the case of L → ∞, it is interesting to note that the symmetric and anti-symmetric modes are merged into the surface quantum wave mode of the semi-bounded quantum plasma. For a very thin quantum plasma slab, the volume density n(x) would be replaced by σ δ(x) where σ is the surface particle number density [22]. This mathematical treatment would be possible only when the thickness of the plasma slab is much smaller than the wave length of the surface wave and quantum shielding distance. Even though the delta-function type δ(x) plasma density is legitimate in mathematically proposed plasma geometries, we shall not consider here the case of zero slab thickness since the plasma slab would not be shrunk to zero in real plasma situations. The detailed theoretical investigation on the surface waves on the thin plasma sheet was given in the limit of zero slab thickness [22]. In addition, Eqs. (7) and (8) correspond to the transverse magnetic mode dispersion relations obtained by Gradov and Stenflo [15] based on the fluid descriptions. As it is seen, the three singularities in the upper-half domain of the complex k¯ x -plane would provide the contribution to the contour integrations in Eqs. (7) and (8) among the total ¯2 + six singularities: k¯ 2x + k¯ 2z = 0 (primary poles) and (k¯ 2x + k¯ 2z )2 ω (k¯ 2x + k¯ 2z ) V¯ F2 − ω¯ 2 + 1 = 0 (quantum poles). Hence, the dispersion properties of the symmetric and anti-symmetric modes of the surface quantum plasma in a quantum plasma slab would be obtained by using the contour integrations with one primary pole and two quantum poles. The spatial distribution of the wave fields would be important for investigating the physical characteristics of the surface wave. However, the dispersion relation ω = ω(k) would be enough to explore the physical properties of the electrostatic surface wave since the field propagation can be determine by the Fourier component exp[−i (k · r + ω(k))t ]. Very recently, an excellent work has provided the Lennard-Jones type new attractive interaction potential between ions in degenerate electron quantum plasmas including the influence of the quantum statistical pressure and quantum Bohm terms as well as the electron exchange and correlations due to the influence of the electron spin [23]. Hence, the investigation of propagating quantum surface waves in a degenerate quantum plasma slab in the framework
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of the quantum hydrodynamical description will be treated elsewhere. In Eqs. (7) and (8), it is found that the unstable modes due to the imaginary solutions of the modulation frequency are quite small and negligible compared to the stable mode solutions so that the stability of the modes is satisfied. However, in future work we shall investigate the effect of the instability on the propagation of the surface wave in a quantum plasma slab including the quantum statistical pressure and quantum Bohm effects as well as the electron exchange and correlation effects due to the influence of the electron spin [23] since the criteria for the wave instability would be a crucial condition for maintaining the bounded plasma system such as the thin quantum plasma slab. In quantum plasma [24] composed of electrons and ions, the plasma number density n and the temperature T are about 1020 –1024 cm−3 and 5 × 104 –106 K, respectively. The characteristics of the quantum plasma [24] can also be represented by the coupling parameter Γ [= e 2 /ak B T ], the degeneracy parameter Θ (= k B T / E F ), and the density parameter r s (= a/a0 ), where a is the average interparticle distance in a plasma. Recently, the excellent investigations have provided the useful effective potential model in dense semiclassical plasmas taking into accounts the quantum diffraction, symmetry, and degeneracy effects as well as quantum shielding effect. Hence, the additional correction to the propagation of the surface quantum plasma waves in a quantum plasma slab including the influence of the effective Ramazanov– Dzhumagulova pseudopotential [24–26] will be treated elsewhere. Recently, the excellent works have provided the detailed investigations on the dispersion properties of electromagnetic [27] and electrostatic [28] surface waves propagating along the interface between a magnetized quantum plasma and vacuum. Then, the propagation of the surface quantum plasma waves in a magnetized quantum plasma slab will also be treated elsewhere. In addition, a recent excellent work [29] has shown the importance of the influence of the quantum broadening of the transition layer on the properties of quantum surface plasmon polaritons at the interface between an electron quantum plasma and a dielectric material, or vacuum. Hence, the propagation of the surface quantum plasma waves in a quantum plasma slab including the effect of quantum broadening of the transition layer will also be treated elsewhere. In order to investigate the physical characteristics of the sur¯ pe = 4 and 8, where h¯ ω¯ pe ≡ face quantum plasma, we consider h¯ ω h¯ ω pe /Ry and Ry (= me e 4 /2h¯ 2 ≈ 13.6 eV) is the Rydberg constant. Since the physical behaviors of the surface wave can be more easily understood by the graphical illustration using the dimensionless or scaled variables, we use the dimensionless frequencies and wave numbers in figures with various plasma parameters. Fig. 1(a) shows the geometric effects on the two dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The symmetric or anti-symmetric mode of the surface quantum plasma wave corresponds to the case of symmetric or anti-symmetric with respect to the slab axis in the z-direction. As shown in this figure, the behaviors of the symmetric and antisymmetric modes of the surface quantum plasma wave are quite different. From this figure, we have found that the frequency of the symmetric mode of the surface quantum plasma wave decreases with an increase of the wave number in small wave number region and then increases with increasing wave number after the local minimum. However, it is found that the anti-symmetric mode of the surface quantum plasma wave monotonically increases with an increase of the wave number. It is interesting to find out that the phase velocity of the anti-symmetric mode of the surface quantum plasma wave is always greater than that of the symmetric mode of the surface quantum plasma wave for a given wave number. It is found that an increase of the slab thickness suppresses the phase velocity of the anti-symmetric mode of the surface
Fig. 1. (a) The geometric effects on the dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The anti-symmetric mode for L¯ = 10, Γ Θ r s = 8, and V¯ F = 0.5 (red solid curve) and L¯ = 20, Γ Θ r s = 8, and V¯ F = 0.5 (blue solid curve). The symmetric mode for L¯ = 10, Γ Θ r s = 2, and V¯ F = 1.0 (red dotted curve), L¯ = 20, Γ Θ r s = 2, and V¯ F = 1.0 (blue dotted curve). (b) The quantum effects on the dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The anti-symmetric mode for L¯ = 20, Γ Θ r s = 2, and V¯ F = 1.0 (red solid curve) and L¯ = 20, Γ Θ r s = 8, and V¯ F = 0.5 (blue solid curve). The symmetric mode for L¯ = 20, Γ Θ r s = 2, and V¯ F = 1.0 (red dotted curve) and L¯ = 20, Γ Θ r s = 8, and V¯ F = 0.5 (blue dotted curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
quantum plasma wave. However, we have found that the phase velocity of the symmetric mode of the surface quantum plasma wave increases with an increase of the slab thickness. Hence, it would be expected that the group velocities of the symmetric mode of the surface quantum plasma are greater than those of the anti-symmetric mode of the surface quantum plasma in a thin quantum plasma slab. However, it is found that the geometric thickness effect on both the symmetric and anti-symmetric modes of the surface quantum plasma decreases with increasing wave number. It is interesting to note that the symmetric and anti-symmetric modes of the surface quantum plasma merge into a single mode with an increase of the slab thickness. This converging single mode corresponds to the ABR mode of the surface wave of the semi-bounded quantum plasma since the exponential terms (1 ∓ e ikx L )/(1 ∓ e ikx L ) in Eq. (6) would be unity as L¯ 1. Fig. 1(b) represents the quantum effects on the dispersion relations of the symmetric and anti-symmetric modes of the surface quantum plasma wave. As it is seen in this figure, an increase of the Fermi velocity suppresses the phase velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. Hence, we have found that the quantum effects diminish the frequencies of the symmetric and anti-symmetric modes of the surface quantum plasma wave in large wave numbers. It is also found that the quantum effect on the surface quantum plasma wave increases with increasing wave number. It is then found that the quantum effect on the surface quantum plasma wave is more significant than the geometric effect on the surface quantum plasma wave in large wave numbers. However, the geometric effect on the surface quantum plasma wave is found to be more significant than the quantum effect on the surface quantum plasma wave in small wave numbers. Fig. 2 shows the group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave in the quantum plasma slab. As we can see from this figure, 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 slab. However,
Y.-D. Jung, W.-P. Hong / Physics Letters A 377 (2013) 560–563
Fig. 2. The group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. The symmetric mode for L¯ = 10, Γ Θ r s = 8, and V¯ F = 0.5 (red solid curve) and L¯ = 50, Γ Θ r s = 8, and V¯ F = 0.5 (blue solid curve). The anti-symmetric mode for L¯ = 10, Γ Θ r s = 8, and V¯ F = 0.5 (red dotted curve) and L¯ = 50, Γ Θ r s = 8, and V¯ F = 0.5 (blue dotted curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)
the group velocity of the symmetric mode is found to be positive in all wave numbers so that the symmetric mode propagates as a forward wave. The backward phenomenon has been also found in surface wave propagations in a dielectric planar waveguide geometry [20]. It is also found that the group velocity decreases with an increase of the thickness of the plasma slab in small wave numbers. Hence, we can expect that the geometric effect suppresses the group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. To summarize, we have investigated the quantum mechanical and geometric effects on the surface quantum plasma wave in a thin plasma slab. Using the plasma dielectric function with the kinetic dispersion model, the symmetric and anti-symmetric dispersion modes of the quantum surface wave are obtained for the slab geometry. The results show that the frequency of the antisymmetric mode of the surface quantum plasma wave decreases with an increase of the wave number in small wave number region and then increases with increasing wave number after the local minimum. However, we have shown that the symmetric mode of the surface quantum plasma wave monotonically increases with an increase of the wave number. We also found that the phase velocity of the anti-symmetric mode of the surface quantum plasma is always greater than that of the symmetric mode of the surface quantum plasma in a thin quantum plasma slab. In addition, it is found that the geometric thickness effect on both the symmetric and anti-symmetric modes of the surface quantum plasma decreases with increasing wave number. It is also found that the quantum effect suppresses the frequencies of the symmetric and anti-symmetric modes of the surface quantum plasma wave in large wave numbers. Hence, we have shown that the quantum effect on the surface quantum plasma wave is more significant than the geometric effect on the surface quantum plasma wave in large wave numbers and, however, the geometric effect on the surface quantum plasma wave is found to be more significant than the quantum effect on the surface quantum plasma wave in small wave numbers. Moreover, it is 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 slab. However, the group velocity of the symmetric mode is found to be positive in all wave numbers so that
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the symmetric mode propagates as a forward wave. In addition, it is found that the geometric effect suppresses the group velocities of the symmetric and anti-symmetric modes of the surface quantum plasma wave. These results would be applied to quantum plasma systems in nanodevices for investigating the physical characteristics and properties of the quantum plasma slab containing the quantum pressure and Bohm potential terms. In this work, it is important to find out that the existence and physical behaviors of the symmetric and anti-symmetric modes of the surface quantum plasma wave would be caused by the quantum statistical effect due to the fermionic behavior of the plasma particles and the Bohm potential term due to the quantum diffraction effect as well as the geometrical effect of the quantum plasma slab. From these results, we have shown that the quantum mechanical and geometric effects play crucial roles in the propagations of the symmetric and anti-symmetric dispersion modes of the surface quantum surface wave in a quantum plasma slab. These results would be useful for understanding the physical and dispersion properties of the surface quantum wave in a slab geometry of the quantum plasma. Acknowledgements The authors gratefully acknowledge Prof. H.J. Lee and Prof. Y.K. Lim for providing highly useful references and valuable discussions. One of the authors (Y.-D.J.) gratefully acknowledges Dr. M. Rosenberg for useful discussions and warm hospitality while visiting the Department of Electrical and Computer Engineering at the University of California, San Diego. This research was initiated while one of the authors (Y.-D.J.) was affiliated with UCSD as a Visiting Professor. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-001493). References [1] T.S. Ramazanov, K.N. Dzhumagulova, Phys. Plasmas 9 (2002) 3758. [2] P.K. Shukla, M. Marklund, Phys. Scr. T 113 (2004) 36. [3] T.S. Ramazanov, K.N. Dzhumagulova, Y.A. Omarbakiyeva, Phys. Plasmas 12 (2005) 092702. [4] T.S. Ramazanov, K.N. Dzhumagulova, A.Zh. Akbarov, J. Phys. A 39 (2006) 4335. [5] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78 (2006) 591. [6] P.K. Shukla, L. Stenflo, Phys. Plasmas 13 (2006) 044505. [7] H. Ren, Z. Wu, P.K. Chu, Phys. Plasmas 14 (2007) 062102. [8] P.K. Shukla, Nat. Phys. 5 (2009) 92. [9] N. Shukla, P.K. Shukla, B. Eliasson, L. Stenflo, Phys. Lett. A 374 (2010) 1749. [10] P.K. Shukla, B. Eliasson, Rev. Mod. Phys. 83 (2011) 885. [11] H.S. Hong, H.J. Lee, Phys. Plasmas 6 (1999) 3422. [12] L. Stenflo, P.K. Shukla, M.Y. Yu, Phys. Plasmas 7 (2000) 2731. [13] H.J. Lee, Phys. Plasmas 12 (2005) 094701. [14] I.-S. Chang, Y.-D. Jung, Phys. Lett. A 372 (2008) 1498. [15] O.M. Gradov, L. Stenflo, Phys. Rep. 94 (1983) 111. [16] L. Stenflo, Phys. Scr. T 63 (1996) 59. [17] H.J. Lee, Y.K. Lim, J. Korean Phys. Soc. 50 (2007) 1056. [18] F. Haas, Quantum Plasmas, Springer, Berlin, 2011. [19] S. Ali, P.K. Shukla, Phys. Plasmas 13 (2006) 102112. [20] Yu.M. Aliev, H. Schlüter, A. Shivarova, Guided-Wave Produced Plasmas, Springer, Berlin, 2000. [21] A.F. Alexandrov, L.S. Bogdankevich, A.A. Rukhadze, Principles of Plasma Electrodynamics, Springer, Berlin, 1984. [22] H.J. Lee, J. Korean Phys. Soc. 43 (2003) 1033. [23] P.K. Shukla, B. Eliasson, Phys. Rev. Lett. 108 (2012) 165007. [24] T.S. Ramazanov, K.N. Dzhumagulova, M.T. Gabdullin, Phys. Plasmas 17 (2010) 042703. [25] T.S. Ramazanov, Zh.A. Moldabekov, K.N. Dzhumagulova, M.M. Muratov, Phys. Plasmas 18 (2011) 103705. [26] Zh.A. Moldabekov, T.S. Ramazanov, K.N. Dzhumagulova, Contrib. Plasma Phys. 52 (2012) 207. [27] A.P. Misra, Phys. Plasmas 14 (2007) 064501. [28] A.P. Misra, N.K. Ghosh, P.K. Shukla, J. Plasma Phys. 76 (2010) 87. [29] M. Marklund, G. Brodin, L. Stenflo, C.S. Liu, Europhys. Lett. 84 (2008) 17006.