Revisit to terahertz wave emission with motions of Josephson vortices

Physica C 491 (2013) 40–43

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Revisit to terahertz wave emission with motions of Josephson vortices H. Matsumoto a,⇑, T. Koyama a, M. Machida b, Y. Ota c, S. Yamada b a

Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CCSE, Japan Atomic Energy Agency, Kashiwa, Ibaraki 277-8587, Japan c Riken Advanced Science Institute, Wako, Saitama 351-0198, Japan b

a r t i c l e

i n f o

Article history: Accepted 18 December 2012 Available online 29 December 2012 Keywords: Terahertz wave emission Intrinsic Josephson junctions Josephson vortex

a b s t r a c t By use of a recently developed multi-scale simulation method, we re-investigate terahertz wave emission with motions of Josephson vortices, as a possible mechanism for enhancing the emitted power. Results show that there appears a strong wave emission around the n = 2 cavity mode (one-wavelength mode), when motions of vortices and cavity modes are resonate. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Since the observation of a strong terahertz (THz) electromagnetic wave (EMW) emission from mesa samples of Bi2Sr2CaCu2O8 without an applied magnetic field [1], many experimental and theoretical studies have been performed to clarify the mechanism of the wave emission and to obtain further stronger emission. It has been turned out that a strong emission is obtained at resonant frequencies of cavity modes and it is pointed out that strong emissions are related to the p-kink state (PKS), which are solitonic excitations in intrinsic Josephson junctions (IJJ’s) [2,3]. We have studied the THz wave emission by a simulation method to treat the system of IJJ’s, electrodes and surrounding vacuum, simultaneously [4]. Recently we have developed a multi-scale simulation method [5,6] to overcome numerical difficulties in spatial scales of large differences among the thickness of the layers d (10 Å), the Josephson wave length kJ ( lm) and the London penetration depth along the c-axis kc (102 lm). We have clarified the phase states at the fundamental cavity mode (p-mode); the phase state is a kind of PKS mixed with small waves of kJ. We will call this mixed state as p-kink like state (PKLS). The phase change of PKLS is as 0 ? p ? 0, while that of PKS is as 0 ? p. PKLS is naturally created even from the initial condition of the in-phase state under a homogeneous bias current, though an infinitesimally small spatial asymmetry (for example, small defect of the critical current jc at the edge) is necessary to induce a transition to PKLS. It should be noted that the transition from the in-phase state to PKLS becomes possible because of the existence of two scales, kJ and kc. Actually, ⇑ Corresponding author. Address: Institute for Materials Research, Tohoku University, Sendai 980-8756, Japan. Tel.: +81 22 215 2008; fax: +81 22 215 2006. E-mail address: [email protected] (H. Matsumoto). 0921-4534/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physc.2012.12.010

as precursors of this transition, standing waves with the wave length kJ are excited, and amplified as a cavity resonance, reaching to PKLS [5]. However, PKLS is inevitably accompanied by small waves of kJ, which causes reduction of the emitted power. Therefore if small waves are removed, a stronger emission is obtainable. In order to reduce kJ-scale waves and to achieve more coherent PKLS, we have studied the system of IJJ’s with a dielectric cover [7]. The emitted power increases at a certain condition. A new situation appears; a higher harmonics (2p-mode) is also excited in PKLS. This indicates that an outer EMW couples with the inside phase motion, excites a solitonic motion and leads to a stronger emission. Here the electromagnetic interface plays an important role to modify the inside phase state. Then we are motivated to re-investigate THz wave emission in an applied magnetic field, since the simplest electromagnetic interface may be achieved by applying a magnetic field, which may modify the boundary conditions between the inside and outside EMW and may affect excitations of cavity modes by vortex motions. In high Tc superconductors, the idea to excite THz EMW by vortex motions was first studied in Ref. [8]. Detailed numerical studies were performed in rather high magnetic field region. Machida et al. [9] showed that the motion of vortices in a square lattice leads to a strong emission, while Tachiki et al. [10] claimed that vortices form not a rigid vortex lattice but distribute randomly and form a certain bunching wave, producing a strong emission in certain frequency region. However, so far a strong emission by vortex motions in high field region has not been observed experimentally, though such an indication was reported [11]. The reasons may be that the field strength is weakened by averaged spatial variation in high magnetic field and also that a formation of a triangular vortex lattice is more plausible in layered systems. Therefore we seek the possibility in a low magnetic field region. In a low applied magnetic field

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H. Matsumoto et al. / Physica C 491 (2013) 40–43

and in the electrodes and vacuum

8 > < ðjext =jc ÞðLx =2kc Þ þ H=Bc By0 ðx; zÞ=Bc ¼ ðjext =jc Þðx=kc Þ þ H=Bc > : ðjext =jc ÞðLx =2kc Þ þ H=Bc

ðx > Lx =2Þ ðLx =2 < x < Lx =2Þ : ðx < Lx =2Þ ð2Þ

Physical scales are the plasma angular frequency xp, and Ep = ⁄xp/2e, Bc = (⁄c/2e)(1/kcd). Define u = /  /0, by = By  By0, ez = EZ and ex = Ex. The scaled equations for the IJJ’s are

ð@=xp @tÞuðt; x; ‘Þ ¼ ð1  aD2z Þðez ðt; x; ‘Þ=Ep Þ

ð3Þ

ð@=xp @tÞðez ðt; x; ‘Þ=Ep Þ ¼ jext =jc þ kc rx ðby ðt; x; ‘Þ=Bc Þ Fig. 1. Systems of junctions, electrodes and vacuum.

 sinðuðt; x; ‘Þ þ u0 ðx; ‘ÞÞ  bðez ðt; x; ‘Þ=Ep Þ by ðt; x; ‘Þ=Bc ¼ ð1  g

region, the emitted power for the fundamental cavity mode in mesa samples has been investigated in Ref. [12] experimentally and in Ref. [13] theoretically. The experimental result shows the increase and decrease of the emitted power, while the theoretical result shows generally decrease. Since the dynamical boundary condition was used in the analysis of Ref. [13], we restudy this problem by treating the systems of junctions, electrodes and the surrounding vacuum, simultaneously. We will show that there is the region, where a strong EMW emission is realized by the vortex motion.

D2z Þ1 kc

rx uðt; x; ‘Þ

ð4Þ ð5Þ

with D2z being the second order difference in ‘, a and g being interlayer coupling constants and b = r/Epjc (r: conductivity). For electrodes and vacuum, the Maxwell equations are

ð@=xp @tÞð0 ez ðt; x; zÞ=Ep Þ ¼ jext ðx; zÞ=jc þ kc rx ðby ðt; x; zÞ=Bc Þ  bðxÞðez ðt; x; zÞ=Ep Þ

ð6Þ

ð@=xp @tÞð0 ex ðt; x; zÞ=Ep Þ ¼ kc rz ðby ðt; x; zÞ=Bc Þ  bðxÞðex ðt; x; zÞ=Ep Þ

ð7Þ

ð@=xp @tÞðby ðt; x; zÞ=Bc Þ ¼ kc rx ðez ðt; x; zÞ=Ep Þ 2. Formulation

 kc rz ðex ðt; x; zÞ=Ep Þ;

We consider the xz-model, which is a two-dimensional model composed of IJJ’s, electrodes and the surrounding vacuum enclosed by the perfectly matched layers (PML) as illustrated in Fig. 1. Two kinds of the vacuum and PML are introduced to perform the multiscale simulation [6]. The z-direction is along the c-axis, the junction-planes are parallel to the x- and y-directions. The spatial variation along the y-direction is neglected. The bias current jext is injected along the z-axis, homogeneously at the top and bottom surfaces of electrodes. The external magnetic field H is applied along the y-axis. The length of the junctions is Lx. The equations for the xz-model are given in the junctions by the gauge-invariant phase difference /(t, x, ‘), the electric fields Ez(t, x, ‘) and magnetic field By(t, x, ‘) with ‘being labels for junctions, and in electrodes and vacuum by the electromagnetic fields Ex(t, x, z), Ez(t, x, z) and By(t, x, z). The bias current and the applied magnetic field produce static field inside junctions as

where 0 and  are dielectric constants of the vacuum and junctions, respectively, jext(x, z) = jext and b(x, z) = bL inside the electrodes, and jext(x) = 0 and b(x) = 0 in the vacuum. Quasi-Maxwell equations for PML are found in Ref. [6,7] as well as the treatment of the boundary conditions.

/0 ðx; ‘Þ ¼ ðjext =2jc Þðx=kc Þ2 þ ðH=Bc Þðx=kc Þ; By0 ðx; ‘Þ=Bc ¼ ðjext =jc Þðx=kc Þ þ H=Bc ;

ð1Þ

(×10-5) 3

S/Sp

2.5

3. Results Using the set of equations in the previous section, we have performed numerical simulation. The used parameters are Lx/kc = 0.5, b = 0.05, a = 0.1, g = 10,000, bL = 100,  = 10, 0 = 1 and the position of observation is Lobs/kc = 3.0. The thicknesses of the electrode and the junctions are wd/kc = 0.018 and wz/kc = 0.09, respectively. In IJJ’s, actual number of layers is about 1000. We approximate the system by the periodic one with Nz = 20 layers along the z-axis [6]. We take the multi-scale spatial spacings. in the outer vacuum dx‘/kc = 0.1, in the inner vacuum and electrodes dx/kc = dxl/11 and in the junctions dxs = dx. Small defect at the edge is introduced in jc. The emitted power is evaluated by the time average of the

(×10-5) 3

(×10-5) 3

(a) H/2πBc=0

2.5

(b) H/2πBc=1.0 (1,0,0) (-1,0,0)

2.5

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0

(c) H/2πBc=2.0

2

2

0.5 (1,0,0)

ð8Þ

(1,0,0) (-1,0,0)

2 4 6 8 10 12 14 16 18020 2 4 6 8 10 12 14 16 18020 2 4 6 8 10 12 14 16 18 20

V/Vp

V/Vp Fig. 2. Voltage-dependences of the emitted power.

V/Vp

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H. Matsumoto et al. / Physica C 491 (2013) 40–43

(×10-5)

4

(1,0,0) (-1,0,0) (1,0,0) (-1,0,0)

3.5 3

2π-mode S/Sp

2.5 2 1.5 1 0.5 0

π-mode 3π-mode 0.5

1

1.5

2

2.5

3

3.5

4

H/2πBc Fig. 3. Magnetic field dependence of the emitted power.

10

20 15 (a)H/2πBc=0, 10 V/Vp~2π 5 0 -5 -10 -15 -20 10 -0.2 -0.1 0

5 0

10

(c)H/2πBc=0.375, V/Vp~2π

5

(e)H/2πBc=0.75 V/Vp~2π

-5 -10

0

-15 0.1

(b)H/2πBc=0 8 V/Vp~4π

ϕ (x)

ϕ (x)

Poynting vector [6]. The initial condition is chosen as u(t, x, ‘) = 0 at jext/jc = 1, and jext/jc is reduced with a step djext/jc = 0.01. In Fig. 2 the voltage-dependences of the emitted power in xand (x)-direction (red circles and blue triangles) are plotted for H/2pBc = (a) 0, (b) 1, (c) 2, which correspond to a zero, half and single flux in junctions, respectively. Here Vp = Epd, Sp = EpBc. There

appear strong emissions at the voltages of the cavity modes (V/ 2Vp  p, 2p, 3p). The peak intensity for the p-mode changes with H. The calculation shows the emission in the z-direction has the strongest power in the p-mode. Since the tendency of H-dependence is same, we have plotted those of the x and (x)-directions for comparison with the 2p-mode. Though 2p-mode has a weak emission at zero field, it starts to show a strong emission from a certain H (Fig. 2b and c). The calculation shows that, in this case, the emission to the z-direction is very weak, suggesting a dipolelike emission. In Fig. 2c, the peak for the 3p-mode is also seen. In Fig. 3, we plot the magnetic field dependences of the emitted powers for the cavity modes, p-, 2p- and 3p-modes. The p-mode shows a decrease at the beginning, then shows three peaks up to H/2pBc = 1.5, which roughly correspond to the half-flux state inside the junctions. The 2p-mode shows a strong emission from a certain magnetic field (H/2pBc  0.375) and forms several regions of the emitted power, suggesting a kind of phase transition in phase states. The strength diminishes rapidly around H/2pBc  4.0. The emission from the 3p-mode appears from around H/2pBc = 1.5. We also notice that the 2p-mode gives the strongest emitted power. In Fig. 4 we plot snapshots of u(x, ‘) for the p-mode and 2pmode at H/2pBc=0.0, 0.375. 0.75, and in Fig. 5 snapshots of u(x, ‘) and sin(/(x, ‘)) for the 2p-mode at H/2p Bc = 1.5, 2.0, 2.5. The results of 20 layers are plotted in the same figure, and the red bold solid lines are for the first junction. Note that u(Lx/2) = u(Lx/2), guaranteeing the total flux obtained from /0(x).

0.2

-208

-0.2 -0.1 0 0.1 6 (d)H/2πBc=0.375 4

6

0.2

V/Vp~4π

2

4 5.88 2 5.84 0 5.8

0 -2

-0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2

-4

x/λ c

-0.2 -0.1

0

0.1

-54 -0.2 -0.1 0 0.1 2 (f)H/2πBc=0.75 V/Vp~4π 0 -2 -4 -6 -8

0.2

-0.2 -0.1

x/λ c

0

0.1

0.2

0.2

x/λ c

Fig. 4. Snapshots of u(x, ‘) for the p-mode and for the 2p-mode.

12

sin (φ (x))

ϕ (x)

8 4

12

(a)H/2πBc=1.5 V/Vp~4π

4

0

0

-4

-4

-8

-8

-12 2.5 -0.2 -0.1 0 0.1 2 (b)H/2πB =1.5 c 1.5 V/Vp~4π 1 0.5 0 -0.5 -1 -0.2 -0.1 0 0.1

x/λc

8 (e)H/2πB =2.5 c 4 V/Vp~4π

8 (c)H/2πBc=2.0

0.2

0.2

V/Vp~4π

0 -4 -8

-12 2.5 -0.2 -0.1 0 0.1 2 (d)H/2πB =2.0 c 1.5 V/Vp~4π 1 0.5 0 -0.5 -1 -0.2 -0.1 0 0.1

0.2

0.2

0.6 0.5 0.4 0.3 0.2

-12 -0.2 -0.1 0 0.1 0.2 2.5 -0.2 -0.1 0 0.1 0.2 2 (f)H/2πB =2.5 c 1.5 V/Vp~4π 1 0.5 0 -0.5 -1 -0.2 -0.1 0 0.1 0.2

x/λc

Fig. 5. Snapshots for u(x, ‘) and sin(/(x, ‘) for the 2p-mode.

x/λc

H. Matsumoto et al. / Physica C 491 (2013) 40–43

The spatial distributions of the phases for the p-mode in Fig. 4 show PKLS-behaviors with small waves of the wavelength pffiffiffi kJ ð kc = gÞ. The strong emission is obtained when phase moves coherently as in Fig. 4e. The magnetic field adjusts to synchronize phase motions, leading to stronger emission peaks as is seen in Fig. 3. The experiment of Ref. [12] shows roughly three peaks and is consistent with the present theoretical result (Note 2pBc  69Oe for d = 15 Å, kc = 200 lm). Fig. 4d and f show clear PKLSbehaviors; (d) shows a complete alternative pattern in layers. Depending on the magnitude of H, their solitonic structures change and the emitted power in Fig. 3 shows jumps according to the change of solitonic structures. Compared to the case of the p-mode, the spatial distributions are rather smooth and waves of kJ-scale are not seen. As for the p-mode, the spatial distribution of u(x) is in general similar to the case of Fig. 4, that is, PKLS and at the position of stronger emitted power, phases are synchronized. On the other hand, the 2p-mode shows a little different behavior around H/ 2pBc = 2. The power S/Sp in Fig. 3 in the region 2 6 H/2pBc 6 2.75 keeps a constant large value. There are recognizable differences of the emitted power to x- and (x)-direction. In Fig. 5, u(x, ‘) change from PKLS-solitonic behavior to a simple in-phase one. The inset in Fig. 5e shows a simple trigonometric behavior. Note that the amplitude of the inset in Fig. 5e is roughly ten times larger than that in Fig. 4b. The Josephson current sin(/(x)) in Fig. 5f shows a clear coherent motion of the Josephson vortices, whose centers are identified as the zero point of sin/(x, ‘) crossing from 1 to 1. The results show that the 2p-mode is strongly excited by applying the magnetic field H. In lower H, a strong emission is realized as PKLS, but around H of a single flux, a motion of an in-phase vortex lattice produces a strong emission due to a synchronization between the vortex motion and the cavity mode.

4. Concluding remarks We have re-investigated the magnetic field dependence of the THz EMW emission from IJJ’s. Differently from the old works, low field region is investigated.

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In the p-mode emission, there appears three peak structure in the region 0.5 6 H/2pBc 6 1.5, where a half flux of the vortex flows in junctions. At higher field, the power is diminished, though certain increase is observed around H/2pBc  3.0. In the 2p-mode emission, a stronger emission is obtained. There are several transitions of phase motions, which induces power jumps in H. At lower field region, the phase state is PKLS, and its solitonic structure changes with increasing field. When the field strength reaches the region of one flux in junctions, 2 6 H/2pBc 6 2.75, the phase motion is in an in-phase state of a simple square vortex lattice, and spatial distribution of phase-difference is trigonometric. The results show that one can obtain stronger emitted power by synchronizing the motion of the Josephson vortices and the cavity modes. Acknowledgment This work was supported by the Grant-in-Aid for Scientific Research (C) 23500056. References [1] L. Ozyuzer, A.E. Koshelev, C. Kurter, N. Gopalsami, Q. Li, K. Kadowaki, T. Yamamoto, H. Minami, H. Yamaguchi, T. Tachiki, K.E. Gray, W.K. Kwok, U. Welp, Science 318 (2007) 1291. [2] S.-Z. Lin, X. Hu, Phys. Rev. Lett. 100 (2008) 247006. [3] X. Hu, S.-Z. Lin, Supercond. Sci. Technol. 23 (2010) 053021. [4] T. Koyama, H. Matsumoto, M. Machida, K. Kadowaki, Phys. Rev. B79 (2009) 104522. [5] T. Koyama, H. Matsumoto, Y. Ota, M. Machida, Physica C471 (2011) 1202. [6] T. Koyama, H. Matsumoto, M. Machida, Y. Ota, Supercond. Sci. Technol. 24 (2011) 085007. [7] T. Koyama, H. Matsumoto, M. Machida, Y. Ota, Physica C (this volume). [8] T. Koyama, M. Tachiki, Solid State Commun. 95 (1995) 367. [9] M. Machida, T. Koyama, M. Tachiki, Physica C362 (2001) 16. [10] M. Tachiki, M. Iizuka, K. Minami, S. Tejima, H. Nakamura, Phys. Rev. B71 (2005) 134515. [11] K. Kadowaki, I. Kakeya, T. Yamamoto, T. Yamazaki, M. Kohri, Y. Kubo, Physica C437–438 (2006) 111. [12] K. Yamaki, M. Tsujimoto, T. Yamamoto, H. Minami, K. Kadowaki, Physica C470 (2010) S804. [13] Y. Nonomura, Physica C470 (2010) S824.