THz Cherenkov radiation of Josephson vortex

Physics Letters A 372 (2008) 712–715 www.elsevier.com/locate/pla

THz Cherenkov radiation of Josephson vortex A.S. Malishevskii ∗ , V.P. Silin, S.A. Uryupin, S.G. Uspenskii P.N. Lebedev Physics Institute, Russian Academy of Sciences, Moscow 119991, Russia Received 10 July 2007; accepted 16 July 2007 Available online 23 August 2007 Communicated by V.M. Agranovich

Abstract It is shown that Josephson vortices travelling in sandwich embedded in dielectric media radiate electromagnetic waves with THz frequencies. This phenomenon is caused by the Cherenkov effect and takes place if vortex velocity exceeds the speed of light in dielectric. © 2007 Elsevier B.V. All rights reserved. PACS: 74.50.+r Keywords: Josephson junction; THz Cherenkov radiation; Vortex

Different aspects of electromagnetic waves radiation by Josephson junctions (JJ) attract attention of researchers for a long time (see, e.g., the recent review [1] and references therein). Many articles have been devoted to microwave radiation into waveguides and transmissions lines (see, e.g., Ref. [2]). In connection with the problem of THz radiation generation the significant attention was paid to experimental study of radiation emission by layered high-temperature superconductors, which can treated as multilayered Josephson structure [3–9]. Theoretical conceptions on radiation generation by single JJ are based on the possibility of Swihart waves emission through edges of junction with finite length in the direction of vortex motion [10–14]. Another possibility of Cherenkov radiation generation by moving vortex is presented in this Letter. This possibility realizes in the Josephson sandwich embedded into dielectric media. It is accepted that junction size is unlimited in the direction of vortex motion. Such situation can be simulated in the annular sandwich with the rather great radius. The possibility of Cherenkov radiation generation appears in the case of vortex velocity is greater than the speed of light in the dielectric. Cherenkov radiation is emitted from the surfaces of superconducting electrodes of the sandwich. The diagram of ra* Corresponding author. Tel.: +7 495 1357808; fax: +7 495 1357880.

E-mail address: [email protected] (A.S. Malishevskii). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.07.084

diation emission depends on vortex velocity and on the relation between sandwich Swihart velocity and the speed of light in the dielectric. Radiation appearing at rather high vortex velocities has high enough frequencies. For typical sandwiches these frequencies fall on THz domain. The system under consideration consists of plane Josephson sandwich located at the region −L − d < x < d + L and placed into nonconducting and nonmagnetic media with dielectric constant m . Sandwich is formed by two identical superconducting electrodes with thickness L. Electrodes are separated by thin nonsuperconducting layer −d < x < d. Josephson current with the critical density jc can flows through this nonsuperconducting layer. The equation for phase difference ϕ = ϕ(z, t) of superconducting order parameters of electrodes at different sides of tunnel layer in the case of sandwich placed in vacuum was derived in Ref. [15]. In our case instead of equation of Ref. [15] there is the following equation: ∂ 2 ϕ(z, t) ωj2 sin ϕ(z, t) + ∂t 2  ∂ ∂ϕ(z , t  ) dz dt  Q(z − z , t − t  ) = , (1) ∂z ∂z √ where ωj ≡ 4π cjc d/φ0  is the Josephson plasma frequency, c is the speed of light in vacuum, φ0 is the magnetic flux quantum,  is the dielectric constant of the tunnel layer. Fourier

A.S. Malishevskii et al. / Physics Letters A 372 (2008) 712–715

transform of the kernel Q(z, t) is given by (cf. Refs. [15,16]): Q(k, ω) ≡ vs2 th(L/λ)

2 κ − λω2 cth(L/λ) cm , 2 κ − λω2 th(L/λ) cm

(2)

√ where vs ≡ c d/λ is the Swihart velocity in the case of bulk √ electrodes, λ is the London penetration depth, cm ≡ c/ m is the speed of light in dielectric media,     2  Θ c2 k 2 − ω2 κ ≡ k 2 − ω2 /cm m    2 2 − iΘ ω2 − cm (3) k sign ω , where Θ(x) is the Heaviside function. It is accepted in Eq. (2), that typical scale of phase difference variation is much greater than London depth λk  1. The thickness of tunnel layer is assumed much less than λ th(L/λ). Such assumptions have not impose any serious restrictions on system parameters because typical values of 2d are usually about few nanometers, while L and λ are about 100 nm. Using Fourier transform of the phase difference satisfying Eq. (1) one can find electromagnetic fields outside sandwich for |x| > d + L. Fourier transforms of electric and magnetic fields have the form:    H (x, k, ω) = H (±d ± L, k, ω) exp κ −|x| + L + d , Ex (x, k, ω) = (kc/ωm )H (x, k, ω), Ez (x, k, ω) = (ic/ωm )∂H (x, k, ω)/∂x,

(4)

where H (±d ± L, k, ω)

  2 −1 = −i(φ0 /4πλ) th(L/λ) − cm κ/ω2 λ × ch−1 (L/λ)kϕ(k, ω).

(5)

According to Eqs. (3) and (4) the sign of κ imaginary part corresponds to radiation propagation from sandwich deep into nonconducting media. Let us consider the radiation of vortex travelling in Josephson sandwich in the case of characteristic frequencies and wave numbers satisfy the following condition:  2 k 2 c /ω2 λ  th(L/λ). ω 2 − cm (6) m In that case Q(k, ω)  Us2 +Qrad (k, ω), where Us ≡ vs cth1/2 × (L/λ), and 2 k2 ω 2 − cm cm Us2 . Qrad (k, ω) ≡ −2i (7) λ sh(2L/λ) ω|ω| The Swihart velocity Us differs from the well-known Swihart velocity Vs ≡ vs th1/2 (L/λ) of sandwich with electrodes of thickness L in the quasimagnetostatic limit [17,18]. The Swihart velocity modification is due to the strong interaction between Swihart wave and electromagnetic wave in the dielectric surrounding sandwich. Strong interaction takes place in the case of inequality (6), which provides closeness of phase velocities of Swihart wave ω/k and electromagnetic wave in the dielectric cm . The Swihart velocity increases from Vs to Us in cth(L/λ)

713

times. Such increasing especially important in the case of thin electrodes, when L  λ and cth(L/λ)  λ/L  1. According to Eq. (7), Eq. (1) for phase difference takes the form: ∂ 2ϕ ∂t 2  2 ∂ ∂ϕ(z , t  ) 2∂ ϕ dz dt  Qrad (z − z , t − t  ) = Us 2 + . ∂z ∂z ∂z (8) Integral term in the right-hand side of Eq. (8) describes the phase difference dependence on the radiation field inside the dielectric surrounding sandwich. In the assumption of small radiation losses the vortex structure energy per unit length of y-axis is given by the formula (cf. Ref. [19]):

 φ02 dz ωj2 (1 − cos ϕ) W (t) = 32π 3 c2 d   1 2 ∂ϕ 2 1 ∂ϕ 2 + . + Us (9) 2 ∂z 2 ∂t

ωj2 sin ϕ +

Taking into account Eq. (8), for the small radiation losses of energy per unit time we find:  φ02 ∂ϕ(z, t) ∂ ˙ dz (W )rad = ∂t ∂z 32π 3 c2 d  ∂ϕ(z , t  ) . × dz dt  Qrad (z − z , t − t  ) (10) ∂z Uniform motion of vortex with constant velocity v > 0 in the absence of losses is described by the equation ωj2 sin ψ = (Us2 − v 2 )ψ  , where ψ = ψ(ζ ) ≡ ϕ(z, t), ζ ≡ z − vt. For v < Us this equation has solution describing the elementary vortex:   ψ0 = 4 arctg exp(−kj ζ ) , (11) where kj ≡ ωj / Us2 − v 2 . The characteristic wave number in the Fourier transform of the function (11) approximately equals to kj . Therefore in the case of uniformly travelling vortex the condition (6) has the form: 2 U 2 − v2 v 2 − cm λ Us s  th3/2 (L/λ) , (12) λj cm v2 where λj ≡ vs /ωj is the Josephson length. Energy of elementary vortex is W = φ02 cth(L/λ)kj /4π 3 λ (cf. Ref. [19]). According to Eq. (10), small losses of vortex energy per unit of time due to electromagnetic waves radiation from sandwich surface (x = ±(d + L)) into dielectric media (|x| > d + L) are equal to  φ2 kj cm 2. v 2 − cm (W˙ )rad = − 30 2 2 (13) 4π λ sh (L/λ) v Expression (13) is valid in the case of energy losses at the distance of about vortex size ∼ 1/kj are small in comparison with total vortex energy. The vortex passes the distance 1/kj during the time 1/vkj . Thus the condition of losses smallness has the form (W˙ )rad /(vkj )  W . This condition is fulfilled automatically if fulfilled inequality (12).

714

A.S. Malishevskii et al. / Physics Letters A 372 (2008) 712–715

According to Eq. (13) Cherenkov losses due to electromagnetic waves emission from sandwich into dielectric media are possible if vortex velocity is greater than the light speed in the dielectric: v > cm . At the same time, velocity of the elementary vortex (11) is smaller than the renormalized Swihart velocity of the sandwich: v < Us . Thereby, it is meaningful to discuss radiation losses only in the systems for which Us > cm , or cth(L/λ) > λ/m d. At L > λ radiation losses (13) are exponentially small. Assuming d  L ∼ λ, from the condition Us > cm we find that dielectric constant of surrounding media must be much greater than dielectric constant of tunnel layer m > (λ/d) th(L/λ) ∼ L/d  . For example, if  = 4, L/d = 25, than m > 100. In the system under consideration beside radiation losses there are vortex energy losses caused by normal electrons of superconducting electrodes and caused by finite conductivity of tunnel layer. For rather low temperatures number of normal electrons is small and losses in the electrodes are small. Following to Ref. [20] one can see that power of losses due to finite conductivity σ of tunnel layer is described by the expression (W˙ )α = −(φ02 /4π 3 λλ2j )αv 2 / Us2 − v 2 , with where α ≡ 4πσ/ωj  1. Comparing this formula 2 > Eq. (13), we see that radiation losses are greater if v 2 − cm 2 3 the case of thin electrodes α(λ/λj )(v /cm vs ) sh (L/λ). In 2 > c α(λ/λ ). AsL ∼ λ and for v ∼ vs ∼ cm we get v 2 − cm m j suming that α ∼ 10−2 , λ/λj ∼ 10−2 , we can see that radiation losses becomes leading if vortex velocity exceeds slightly cm . Creating by vortex electromagnetic field outside the sandwich follows from Eqs. (4) and (5): φ0 ψ 4πλ sh(L/λ) 0     × ζ + (v/cm )2 − 1 |x| − d − L ,

H =−

Ex = (c/m v)H,  Ez = −(c/m v) (v/cm )2 − 1 sign x · H.

λVs2 ω|ω| . 2 k2 cm sh(2L/λ) ω2 − cm

(17)

In this case the equation for phase difference is Eq. (8) where one must substitute Vs instead of Us and use new kernel Qrad (17). The vortex energy per unit of y-axis length is given by Eq. (9) with Vs instead of Us . Energy losses per unit time due to radiation is given by Eq. (10) with kernel (17). Uniform motion with constant velocity v is described by equation ωj2 sin ψ = (Vs2 − v 2 )ψ  ,which has the solution of a type (11), where kj ≡ ωj / Vs2 − v 2 . The solution in the form of elementary vortex exists if fulfilled |Qrad (k, ω)|  Vs2 and reverse Eq. (6), where ω = kv and k ∼ kj : 2 V 2 − v2 v 2 − cm λ Vs s  th1/2 (L/λ) 2 λj cm v   × max 1, sh−2 (L/λ) . (18)

(14)

(15)

(W˙ )rad = −

φ02

2 cm 64π 3 λ2 sh2 (L/λ) v     2 × ψ0 ζ + (v/cm )2 − 1 |x| − d − L   × (v/cm )2 − 1 sign x · ex + ez .

Qrad (k, ω) ≡ −2i

According to Eqs. (10) and (17), energy losses of vortex per unit of time due to radiation of electromagnetic waves to the dielectric media (|x| > d + L) are given by expression

Corresponding to this field the Poynting vector in dielectric media S ≡ (c/4π)[EH] is given by formula: S=

It is clear from Eq. (12) that solution given by Eqs. (11), (14) and (15), where kj depends on Us , is valid in narrow velocity domain. That leads to narrow spectral line of radiation emission. The next point for consideration is a radiation from Josephson sandwich with characteristic frequencies and wave numbers satisfying the condition reverse to Eq. (6). In this case interaction between Swihart wave with electromagnetic wave in dielectric is rather small and does not lead to the Swihart velocity modification and we obtain from Eq. (2) Q1 (k, ω)  Vs2 + Qrad (k, ω),

(16)

Integrals of Sx taken on any surfaces |x| − d − L = const > 0 describes radiation losses according to Eq. (13). From Eq. (16) 2  c radiation flux propagates one can see that for v 2 − cm m mainly along the sandwich electrodes. From the Fourier transform of fields (14) and (15) follows that cyclic frequencies of Cherenkov radiation are given by formula ω = kv, where k ∼ kj . As λkj  1, than characteristic frequencies are located in the interval ωj cm /Us  ωj v/Us  ω  v/λ. Assuming L ∼ λ ∼ 100 nm, λj ∼ 104 nm and v ∼ cm ∼ 3 × 109 cm/sec, we obtain that frequency ν ≡ ω/2π belongs to the THz range of frequencies: 0.5 THz  ν  50 THz.

v 3 kj3 . 2 12π 3 ch2 (L/λ) cm v 2 − cm φ02

(19)

The vortex energy is equal to W = φ02 th(L/λ)kj /4π 3 λ (cf. Ref. [19]). Then one has (W˙ )rad /W vkj  1 in the conditions (18). Magnetic field in dielectric (|x| > d + L) created by the vortex is given by formula: H=

v2 φ0 ψ  2 0 4π ch(L/λ) cm v 2 − cm     × ζ + (v/cm )2 − 1 |x| − d − L ,

(20)

while electric field is connected with H by formulae (15). Corresponding to these fields Poynting vector is equal to: S=

φ02 2



v3

2 64π 3 λ2 ch (L/λ) cm v 2 − cm     2 × ψ0 ζ + (v/cm )2 − 1 |x| − d − L   × (v/cm )2 − 1 sign x · ex + ez .

(21)

A.S. Malishevskii et al. / Physics Letters A 372 (2008) 712–715

Integrals of Sx on an arbitrary surfaces |x| − d − L = const > 0 gives Eq. (19). As before, spectral composition of Cherenkov radiation from the sandwich is described by formula ω = kv. However, now we have k ∼ kj ∼ ωj / Vs2 − v 2 and possible velocities are determined by inequalities (18). Characteristic cyclic frequencies of radiation belong to the domain ωj cm /Vs < ωj v/Vs  ω  v/λ. For parameters of the system, that were accepted above, frequencies of radiation ν again belong to THz range from 0.5 THz to 50 THz. For typical sandwiches vortices with velocities from interval given by Eq. (18) generate radiation with rather wide spectral line than vortices with v from region given by Eq. (12). If Vs  cm , than vortex velocity v can be much greater than cm . According to Eq. (21) at v  cm radiation has 2  c radiation propawide radiation diagram. For v 2 − cm m gates mainly along electrodes surface. In summary, we have shown that vortices, travelling in the Josephson sandwich with velocities greater than the speed of light in the external dielectric, generate the THz electromagnetic radiation. This radiation is due to the Cherenkov effect. Acknowledgements This work was supported by RFBR (Project No. 05-0217547), the President Programs in Support of Leading Scientific Schools and Young Scientists (Projects Nos. NSh4122.2006.2 and MK-868.2007.2) and by “Dynasty” Foundation.

715

References [1] I.K. Yanson, Low Temp. Phys. 30 (2004) 516. [2] V.P. Koshelets, S.V. Shitov, Supercond. Sci. Tech. 13 (2000) R53. [3] G. Hechtfischer, R. Kleiner, A.V. Ustinov, P. Müller, Phys. Rev. Lett. 79 (1997) 1365. [4] J.H. Lee, Y. Chong, S. Lee, Z.G. Khim, Physica C 341–348 (2000) 1079. [5] I.E. Batov, X.Y. Jin, S.V. Shitov, et al., Appl. Phys. Lett. 88 (2006) 262504. [6] K. Kadowaki, I. Takeya, T. Yamamoto, et al., Physica C 437–438 (2006) 111. [7] M.-H. Bae, H.-J. Lee, IEICE Trans. Electron. E89-C (2006) 106. [8] M.-H. Bae, H.-J. Lee, Appl. Phys. Lett. 88 (2006) 142501. [9] M.-H. Bae, H.-J. Lee, J.-H. Choi, Phys. Rev. Lett. 98 (2007) 027002. [10] D.N. Langenberg, D.J. Scalapino, B.N. Taylor, R.E. Eck, Phys. Rev. Lett. 7 (1965) 294. [11] I.O. Kulik, I.K. Yanson, The Josephson Effect in Superconductive Tunneling Structures, Keter Press, Jerusalem, 1972. [12] M. Tachiki, M. Iizuka, K. Minami, Phys. Rev. B 71 (2005) 134515. [13] L.N. Bulaevskii, A.E. Koshelev, Phys. Rev. Lett. 97 (2006) 267001. [14] L.N. Bulaevskii, A.E. Koshelev, J. Supercond. Nov. Magn. 19 (2006) 349. [15] K.N. Ovchinnikov, V.P. Silin, S.A. Uryupin, Phys. Met. Metallogr. 83 (1997) 468. [16] A.S. Malishevskii, V.P. Silin, S.A. Uryupin, Phys. Solid State 41 (1999) 1055. [17] J.C. Swihart, J. Appl. Phys. 32 (1961) 461. [18] S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. [19] P. Lebwohl, M.J. Stephen, Phys. Rev. 163 (1967) 376. [20] D.W. McLaughlin, A.C. Scott, Phys. Rev. A 18 (1978) 1652.