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Surface Josephson plasma waves in layered superconductors

Sergey Savel’ev,1 Valery Yampol’skii,1, 2 and Franco Nori1, 3

1

arXiv:cond-mat/0508716v1 [cond-mat.supr-con] 30 Aug 2005

Frontier Research System, The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama, 351-0198, Japan 2 A. Ya. Usikov Institute for Radiophysics and Electronics, NASU, 61085 Kharkov, Ukraine 3 Physics Department, MCTP, CSCS, University of Michigan, Ann Arbor, MI 48109-1040, USA (Dated: February 2, 2008) We predict the existence of surface waves in layered superconductors in the THz frequency range, below the Josephson plasma frequency ωJ . This wave propagates along the vacuum-superconductor interface and dampens in both transverse directions out of the surface (i.e., towards the superconductor and towards the vacuum). This is the ?rst prediction of propagating surface waves in any superconductor. These predicted surface Josephson plasma waves are important for di?erent phenomena, including the complete suppression of the specular re?ection from a sample (Wood’s anomalies) and a huge enhancement of the wave absorption (which can be used as a THz detector).

PACS numbers: 74.72.Hs

High temperature Bi2 Sr2 CaCu2 O8+δ superconductors have a layered structure of superconducting CuO2 planes with Josephson coupling between them [1]. This structure favors the propagation of electromagnetic waves, called Josephson plasma waves (JPW) [2–4], through the layers. These waves attract considerable interest because of their Terahertz frequency range (from 300 GHz to 30 THz corresponding to 1000-90 ?m (wavelength)), which is still hardly reachable for both electronic and optical devices [6]. During the last decade there have been many attempts to push THz science and technology forward [6] because of many important applications in physics, astronomy, chemistry, biology and medicine, including THz imaging, spectroscopy, tomography, medical diagnosis, health monitoring, environmental control, as well as chemical and biological identi?cation. The unusual optical properties of layered superconductors, including re?ectivity and transmissivity, caused by the JPW excitation were studied in, e.g., Refs. [2, 7–10]. In particular, Ref. [8] demonstrated that the spectrum of JPW consists of two branches due to a peculiar effect [11] of dynamical breaking of charge neutrality and, therefore, the transmissivity exhibits a sharp peak at frequencies just above the Josephson plasma frequency ωJ . All previous works on this problem have focused on running waves in the frequency range above the gap of the JPW spectrum, i.e., above the Josephson plasma frequency ω > ωJ . A similar gapped spectrum also appears in solid state plasma [12]. In such situations the presence of the sample boundary can produce a new branch of the wave spectrum inside the gap, i.e., a surface plasmon [12, 13]. In general, surface waves play a very important role in many fundamental resonant phenomena, such as the “Wood anomalies” in the re?ectivity [13, 14] and transmissivity [15] of periodically corrugated metal and semiconducting samples, and are employed in many devices. Therefore, it is important to know if surface waves can exist in layered superconductors. Here, we prove that surface Josephson plasma waves

can propagate along the surface between the superconductor and the vacuum in a wide frequency range below ωJ . We derive the dispersion relation for these waves and propose ways to excite these. We show that these surface waves play an important role in the absorption and re?ection of electromagnetic waves, including their resonance dependence on the incident angle θ. The studied resonant absorbability could be experimentally observed by measurement of the surface impedance of a sample or dc resistivity. The predicted phenomena are potentially useful for designing THz detectors. Surface Josephson plasma waves.— We consider a semi-in?nite layered superconductor in the geometry shown in the inset in Fig. 1. Using a standard approximation, the spatial variations inside the very thin superconducting layers are neglected in the direction perpendicular to the layers. The crystallographic ab-plane coincides with the xy-plane and the c-axis is along the z-axis. Superconducting layers are numbered by the integer l ≥ 0. We consider a surface p–wave with the electric, E = {Ex , 0, Ez }, and magnetic, H = {0, H, 0}, ?elds damped away from the interface z = 0, H, Ex , Ez ∝ exp(?iωt + iqx ? klD) inside a layered superconductor, z < 0, and

vac vac H vac , Ex , Ez ∝ exp(?iωt + iqx ? kvac z),

(1)

(2)

in vacuum, for z > 0 and q > ω/c. Here D is the spatial period of the layered structure. The Maxwell equations for waves (2), qH vac = vac vac vac ?(ω/c)Ez , kvac H vac = ?(iω/c)Ex , iqEz + vac vac kvac Ex = ?(iω/c)H , provide both the usual dispersion relation, kvac = q 2 ? ω 2 /c2 , for the wave in vacuum and the ratio of amplitudes for tangential electric and magnetic ?elds at the interface z = +0 (i.e., right above the sample surface)

vac ic Ex = H vac ω

q 2 ? ω 2 /c2 .

(3)

2 Inside the layered superconductor, where z < 0, the gauge invariant phase di?erence is described by a set of coupled sine-Gordon equations. For Josephson plasma waves, these non-linear equations can be linearized and rewritten in terms of the magnetic ?elds H l between the lth and (l + 1)th layers, 1? λ2 2 ab ? D2 l ?H l ?2H l 2 + ωr + ωJ H l ?t2 ?t c2 ? 2 H l = 0. ε ?x2 layer. We ignore the displacement current in Eq. (6) because it is proportional to a small parameter (λab ω/c)2 . Besides, we neglected the contribution of the gaugeinvariant scalar potential into the x-component of the electric ?eld since the Debye length is much shorter than the wave length q ?1 . In order to eliminate H 0 , we use the relation H vac ? H 0 = H vac [1 ? exp(?kD)], that follows from Eq. (1) with l = 0 and l = 1. Now, using Eq. (6), we obtain the ratio between electric and magnetic ?elds at z = ?0 (i.e., right below the sample surface) Ex iωλ2 ab = [1 ? exp(?kD)]. H vac cD (7)

2 ?αωJ ?l2 H l ?

(4)

Here λab is the London penetration depth in the zdirection, the operator ?l2 is de?ned as ?l2 fl = fl+1 + fl?1 ? 2fl , ωJ = 8πeDJc /? ε is the Josephson plasma h frequency determined by the maximum Josephson current Jc and dielectric constant ε, and ωr is the relaxation frequency. The e?ect of breaking charge neutrality [11], which is crucial for our analysis, is taken into account in Eq. (4). The constant α characterizing this e?ect was estimated, e.g., in Ref. [9], α ? 0.05–0.1 for Bi-2212 or Tl-2212 crystals. Substituting the wave (1) in Eq. (4), we obtain the implicit equation for the damped-wave transverse wave vector k(q, ω) λ2 q 2 ω2 c =1+ 2 ωJ 1 ? (4λ2 /D2 ) sinh2 [k(q, ω)D/2] ab ?4α sinh2 [k(q, ω)D/2], λ2 = c c2 2 ωJ ε (5)

Using the continuity conditions for the tangential components of electric ?eld at the surface and Eqs. (3), (5), and (7), we obtain the dispersion relation for two branches of the surface wave corresponding to two solutions of Eq. (5). For (1 ? ω/ωJ ) ? α/εD/λab ≈ 5 · 10?4 , this spectrum can be written as κ? (?) = ?; κ+ (?) = ? 1 + β 2 ?2 Γ?

1/2 1/2

× 1 + 2Γ? 1 ? 1 + Γ?1 ?

(8)

for two branches “?”. Here we introduce the dimensionless variables: κ = cq/ωJ , ? = ω/ωJ , β = 2λ2 ωJ /cD, and Γ? = (1 ? ?2 )/4α. The value of the paab rameter β for Bi-2212 is about 1.4. The spectra Eqs. (8) are shown in Fig. 1. Both branches merge in a narrow frequency region below ωJ , i.e., (1??) ? (α/ε)1/2 (D/λab ).

for ωr → 0. A similar spectrum, but for running Josephson plasma waves (with imaginary k(q, ω)), was earlier obtained in Ref. [9]. At ω < ωJ , solving Eq. (5) with respect to sinh2 [k(q, ω)D/2] results in two branches of positive transverse spatial decrement, k± (q, ω) > 0. It is important to stress that Eq. (5) is not a spectrum of the studied longitudinal surface waves. Indeed, Eq. (5) has two free parameters, ω and q. We need to obtain the q(ω) dispersion relation for our surface waves. This dispersion relation q(ω) can be obtained by joining ?elds (1) in a superconductor and (2) in vacuum at the sample surface via the boundary conditions. Thus, in order to ?nd the spectrum of the surface Josephson plasma waves, we can derive the ratio Ex /H in the superconductor at the sample surface and equate this ratio to Eq. (3). The di?erence between the magnetic ?eld H vac in vacuum and the value H 0 between the 0th and the 1st superconducting layers is described by the London equation, Ax0 ?ic H vac ? H 0 ≈ 2 ≈ 2 Ex0 . D λab λab ω (6)

0.8

?

κ

z x

l=0 l=1 l=2 l=3

κ+

Vacuum

0.4

y

0 0.4

FIG. 1: (Color online) Spectra of the surface Josephson plasma waves κ? for the parameters α = 0.1, and β = 1.4, standard for Bi-2212. The inset shows the geometry used.

κ

Layered superconductor

0.8

Here Ax0 and Ex0 are the x-components of the vector potential and electric ?eld in the ?rst superconducting

The surface mode κ? (?) attenuates into the vacuum at very large distances, of about cλab /ωD, and seems to be di?cult to observe experimentally. Another mode,

3 κ+ (?), dampens on scales ? c/ω and is of signi?cant interest. As we show below, it plays an important role in the transmissivity and re?ectivity properties of layered superconductors at frequencies ω < ωJ . Excitation of the surface waves; resonant electromagnetic absorption.— One of the ways to excite surface waves is via externally applied electromagnetic waves on a sample having spatially modulated parameters. Thus, we consider a weak modulation of the maximum current density Jc via, say, creation of pancake vortices by the out-of-plane magnetic ?eld. This can result in the modulation of ωJ . For simplicity, we assume that

2 ωJ (x)2 = ωJ 1 + 2? cos

To ?rst order approximation, the solution of Eq. (4) with ωJ (x) inside the superconductor consists of both the forced component, Hforced exp iq1 x ? iωt ? κ? ω sin θ , ω Dl , c (13)

which attenuates with the same decrement as H0 , and free oscillations Hsurf exp {iq1 x ? iωt ? κ+ (q1 , ω) Dl} . (14)

2πx a

, ? ? 1,

(9)

The amplitude Hforced is de?ned by Eq. (4) itself, whereas the amplitudes, Hsurf (in dotted magenta) and corresponding vacuum mode

vac 2 Hsurf exp iq1 x ? iωt ? q1 ?

where a is a spatial period. An electromagnetic wave with ω < ωJ incident at an angle θ with respect to the sample surface generates modes having longitudinal wave vectors qm = ω sin θ/c + 2πm/a, with integer m. Almost all of these modes for m = 0 are weak, because ? ? 1. However, one of these modes (e.g., for m = 1) can be excited with large amplitude at resonance, i.e., when the wave vector q1 = (ω/c) sin θ + 2π/a is close to the wave vector q+ = ωJ κ+ (?)/c of the surface wave (8). This corresponds to the incident angle θ close to the resonance angle θ0 de?ned by 2πc ? sin θ0 + = κ+ (?). aωJ (10)

ω2 c2

1/2

z

,

(15)

are determined by the boundary conditions for harmonics q = q1 of H and Ex at z = 0. Solving the corresponding set of nonhomogeneous linear equations for Hsurf and vac Hsurf we obtain the amplitude of the resonance wave:

vac vac Hsurf = Hin 2 1 2?q0 2 2 · (1 ? ?2 )(q1 ? q0 ) R

(16)

with the denominator R = X + iY = 2κ+ (?) cos θ0 (θ ? θ0 ) β? κ2 (?) ? ?2 + + iωr α?3 . ωJ (1 ? ?2 )3

Because of this resonance, the amplitude of the wave with q = q1 can be of the same order, or even higher than, vac the amplitude Hin of the incident wave. In resonance, the mode with q = q1 is actually the surface Josephson plasma wave discussed above. Let us discuss the mechanism of excitation of surface vac waves. The incident Hin (red arrow in Fig. 2) and vac specularly-re?ected Hsp (blue dashed arrow in Fig. 2) electromagnetic waves

vac Hin;sp exp i

(17) The resonance is characterized by the Lorentz form 1/R = 1/[X(θ) + iY ]. The excitation of the surface wave results in the resonant peak in the electromagnetic absorption,

2 2 Absorption(θ) ∝ σ⊥ Ez + σ Ex ∝

1 , (18) X 2 (θ) + Y 2

x sin θ ± z cos θ ?t ω c

(11)

generate the wave (in green, Fig. 2) damped inside the superconductor, H0 exp i ω sin θ x sin θ ? t ω ? κ? , ω Dl , (12) c c

having a lower κ? (?), compared to κ+ (?). Indeed, as was shown in Ref. [9], the longer wave is mainly genα/εD/λab ≈ 5 · 10?4 . erated at (ω ? ωJ )/ωJ ? Moreover, at low frequencies, this longer wave exhibits a correct limiting behavior, producing magnetic ?elds described by the well-known London equation. The waves vac vac vac vac Hin , Hsp ≈ Hin , and H0 ≈ 2Hin represent the solution of the problem in the zero approximation with respect to ?.

shown in Fig. 2. The resonance in the absorption can be observed by measuring the dependence of the surface impedance on the angle θ. Alternatively, the peak in absorption produces a temperature increase, resulting in a sharp increase of the DC resistance or even the transition of the sample to the normal state at θ = θ0 . Naturally, the absorption peak is accompanied by the vac resonant decrease of the amplitude Hsp of the specularly re?ected wave. Even though this e?ect is of secondorder with respect to ?, it can result in a signi?cant (up vac to complete) suppression of Hsp at θ = θ0 due to the resonance denominator R(θ). In the optical range, this phenomenon is known as Wood’s anomalies [14] of re?ectivity and has been used for several di?raction devices. THz detectors.— These e?ects could be potentially useful for the design of THz detectors, an important current goal of many labs worldwide. The simplest design could be a spatially modulated Bi2212 sample ?xed on

4 tained their dispersion relation. The absorption of the incident electromagnetic wave can strongly increase at certain incident angles due to the resonant generation of the predicted surface waves. This is the ?rst prediction of propagating surface waves in any superconductor. We propose a way to experimentally observe these surface waves. We acknowledge partial support from the NSA and ARDA under AFOSR contract No. F49620-02-1-0334, and by the NSF grant No. EIA-0130383.

incident wave θ = θ0

specular wave

(a)

Absorbtion, arb. units

9

δ = 0.01 δ = 0.05

(b)

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6

3

0 -0.1

0

θ ? θ0

0.1

FIG. 2: (Color online) (a) Schematic diagram showing the mechanism of excitation of the surface waves along the superconductor-vacuum interface. To zero-order approximation, with respect to the amplitude ? of the spatial modulations in a superconducting sample, an incident wave (shown as a solid red arrow) re?ects as a specular wave (the straight dashed blue arrow), producing a damped wave (green wave) inside superconductors. To ?rst order approximation, the very intense surface wave (dotted magenta wave) can be excited at a certain resonant angle between the incident wave and sample surface. (b) Absorption obtained using 2 Eq. (18) for di?erent e?ective dampings δ = βωr α?4 (k+ ? 2 1/2 2 3 ?1 ? ) [2ωJ (1 ? ? ) k+ cos θ0 ] .

a precisely rotated holder and attached by contacts to measure its resistance. Spatial modulations in the sample could be fabricated by either using ion irradiation of the sample covered by periodically modulated mask [16], or even mechanically [17]. When rotating the sample, the incident THz radiation can produce a surface wave at certain angles. This results in a strong enhancement of absorption associated with increasing of temperature in the sample and, thus, its resistance. The relative positions of the resonance peaks (the set of angles) allows to calculate the angle and the frequency of the incident THz radiation, while the relative heights of the resistance peaks can be used to estimate the intensity of the incident radiation. Conclusions.— We have derived the surface Josephson plasma waves in layered superconductors and ob-

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