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RF impedance of intrinsic Josephson junction in flux-flow state with a periodic pinning potential
Physica C 468 (2008) 1295–1297
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
RF impedance of intrinsic Josephson junction in flux-flow state with a periodic pinning potential Y. Yamada a,*, K. Nakajima b, K. Nakajima c a
Department of Electrical and Computer Engineering, Oyama National College of Technology, Oaza-Nakakuki 771, Oyama 323-0806, Japan Graduate School of Science and Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan c Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan b
a r t i c l e
i n f o
Article history: Available online 23 May 2008 PACS: 74.25.Nf 74.25.Qt Keywords: Josephson junction Vortex Flux-flow Sine-Gordon equation Simulation
a b s t r a c t We have investigated the dynamics of Josephson vortices interacting with electromagnetic waves in Bi2Sr2CaCu2O8+y intrinsic Josephson junction (IJJ) stacks by means of millimeter wave irradiation and numerical simulations based on coupled sine-Gordon equations while taking into account a sinusoidal form of the periodic pinning potential. The numerical simulation results for the influence of the electromagnetic waves on the flux-flow properties reveal that the periodic pinning potential induces the inphase motion of Josephson vortices over the junctions. In order to prove from another viewpoint, we investigate RF impedance of IJJ in flux-flow state in this study. A remarkable negative real part region appears at 1st harmonic step, it means that the IJJ in flux-flow state acts as an oscillator at the negative real part region. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Josephson vortex dynamics in a stack of inductively-coupled Josephson junctions attracts our attentions because the stack models highly anisotropic high-Tc superconductors represented by Bi2Sr2CaCu2O8+y (BSCCO). In BSCCO-type high-Tc superconductors, superconducting CuO2 planes are stacked along the c-axis by interposing insulating layers consisting of BiO and SrO, and Josephson effects govern the current transport along the c-axis. Since the discovery of the intrinsic Josephson effects of BSCCO [1] arises from such a stack structure, much attention has been paid to the mechanism and the potentials of intrinsic Josephson junctions (IJJ) as possible high-frequency devices capable of electromagnetic wave generation up to the THz region [2], exceeding the conventional low-Tc flux-flow oscillators (FFO) [3]. The IJJ is also expected to be superior to the conventional one, since output power of the IJJ grows as the square of the number of junctions in the stack [4]. Such superior performances of IJJ would be achieved by inphase motion of Josephson vortices in stacked junctions. We have studied the effect of a periodic pinning potential induced by the pancake vortices upon Josephson vortices. From the experimental results [5], we assume that the periodic pinning potential plays
* Corresponding author. Tel.: +81 285 20 2234. E-mail address: [email protected] (Y. Yamada). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.014
an important role for inducing the in-phase vortex motion. The Josephson vortex dynamics in the BSCCO-type stack containing N junctions is expressed by N coupled sine-Gordon equations. We describe approaches to introduce the periodic pinning potential in an ordinary numerical simulation by using the coupled sine-Gordon equation [6]. The numerical simulation results for the influence of the electromagnetic waves on the flux-flow properties reveal that the periodic pinning potential induces the in-phase motion of Josephson vortices over the junctions. In order to prove from another viewpoint, we investigate RF impedance of IJJ in flux-flow state in this study. The negative real part of the RF impedance of the Josephson junction appears under certain condition, it means that the Josephson junction acts as an oscillator at negative real part region [7]. Thus we calculate the negative real part of the RF impedance of IJJ in detail. 2. Numerical simulation procedure The numerical model developed for a system of stacked Josephson junctions [8] has proven to be a very useful tool for understanding the Josephson vortex dynamics of BSCCO-type IJJ. We consider a stack of long Josephson junction with a length L, a width W and a junction number N. The model for a stack of N Josephson junctions consists of N inductively-coupled sine-Gordon equations. Assuming identical parameters for each junction and coupling
1296
Y. Yamada et al. / Physica C 468 (2008) 1295–1297
parameter S between the junctions, we obtain simultaneous equations in following form [8]:
0
1
1
S
0
BS /1 B B . C B B . C B . C B B C B o2 B B /i C ¼ B C B B 2 ox B C B B .. C B @ . A B B @ /
1
S
0
1
S ..
1
0
.. 0
.
S
.
N
0
0 1 C jz1 C CB . C CB . C CB . C CB z C CB j C 0 CB i C CB . C CB . C C@ . A C S A jz N S 1
ð1Þ
z
where /i is the phase difference across each junction and ji is the current in the z-direction, which is the stack direction correspondz ing to the c-axis direction of the BSCCO unit cell. The current ji can be expressed as z ji
¼
1 c2SW
o2 /i o/ þC i ot ot2
! þ
1 k2J
ðsin /i cB Þ;
ð2Þ
where C = G/C and cB is the normalized bias current of each junction. C and G are the capacitance and the conductance of a unit area, respectively. cSW and kJ correspond to the Swihart velocity and the Josephson penetration depth, respectively. The coupling parameter, S, is defined as
S¼
s 0 d
ð3Þ
for 0
d ¼ dI þ 2kL coth
dS kL
ð4Þ
and
s¼
kL ; sinhðdS =kL Þ
ð5Þ
where dS and dI are the thickness of a superconducting layer and an insulating layer, respectively. kL is a London penetration length. When dS is much smaller than kL, S approaches 0.5, which means a very strong coupling between junctions. An applied magnetic field Bdc parallel to the junctions does not appear in the equations, but is provided by the boundary conditions at the junction edges, x = 0, L, where L is the length of the junction. The applied electromagnetic wave is considered to be an alternating magnetic field parallel to
Fig. 1. Assumed circuit model for the RF impedance calculation.
the junctions at one of the junctions edges and is provided by the boundary condition at one of the junction edges, x = 0. Then, the boundary conditions are expressed as
0 o/i 2eðd þ 2sÞ ¼ ðBdc þ Brf sin 2pfrf t Þ; h ox x¼0 0 o/i 2eðd þ 2sÞ Bdc ; ¼ h ox
ð6Þ ð7Þ
x¼L
where Brf and frf are an amplitude and a frequency of the alternating magnetic field, respectively. The applied electromagnetic wave induces an alternating current in the Josephson junction stack. Therefore we consider that the alternating current source Irf is connected to the edge of the junction (x = 0). Fig. 1 shows the circuit model. The RF impedance calculation procedure is as follows: first, fast Fourier transform of ii and vi are taken, then Zrf (=vrf/irf) is calculated, where Zrf, vrf, irf are frf component of impedance, voltage, current, respectively. In order to take into account the effects of a periodic pinning potential for the Josephson vortex dynamics governed by the above coupled sine-Gordon equations, we assume an additional current 2 2 ai d /d =dx in the z-direction to introduce the periodic pinning potential into Eq. (2) as follows: Z ji
¼
1 c2SW
o2 /i o/ þC i ot ot 2
! þ
1
2
d /d ðsin /i cB Þ ai ; 2 k2J dx
Fig. 2. (a) The I–V and I–rrf characteristics for Bdc = 0.55 T, Brf = 0.11 T and a = 0.5. (b) The enlarged I–rrf characteristics of the ‘‘1st step region”.
ð8Þ
Y. Yamada et al. / Physica C 468 (2008) 1295–1297
where /d is a phase difference across the imaginary junction. ai is the coefficient that expresses the strength of pinning in each real junction. It is proportional to the diagonal element of the inverse matrix of the matrix in Eq. (1) when the maximum pinning force in each junction becomes equal. We assume d/d/dx to be a sinusoidal function [9], which is proportional to the pinning potential. Therefore, /d is assumed as follows:
/d ¼
0 2eðd þ 2sÞ aL 2mp Bdc sin x ; h 2mp L
ð9Þ
where m is the number of pancake vortex rows in one superconducting layer and a represents the pinning strength.
1297
Fig. 2a shows the I–V characteristics calculated for Bdc = 0.55 T, Brf = 0.11 T and a = 0.5. As shown with arrow, a current step appears at a voltage satisfying the Josephson relation V = NU0 frf, where N is the number of junctions (N = 3), that is 1st harmonic step. Fig. 2a and b show the dependencies of the real part of the RF impedance on the bias current of first junction (i = 1) and second junction (i = 2). As shown with arrows, remarkable negative resistance regions appear at 1st step. The negative real part of the impedance which occurs at 1st step means that the Josephson junction acts as an oscillator generating power at the frequency frf. 4. Conclusion
3. Results and discussion Numerical simulations are carried out for a stack of three Josephson junctions of junction length L = 10 lm, junction width W = 2 lm, superconductor layer thickness dS = 3 Å, insulating layer thickness dI = 12 Å, and critical current Jc = 100 A/cm2. Under these conditions, we obtain the coupling strength parameter S = –0.5, the McCumber parameter bc = 29, the plasma frequency fp = 32 GHz, the Josephson penetration depth kJ = 3.3 lm, and the anisotropy parameter c 2000. In our experiments [5], the c-axis direction magnetic field to transit into pancake state is about 3 mT. According to a rough estimate of the relations between the pancake vortices and the Josephson vortices [10], it seems to be ‘‘1d-chain” state. Therefore, m is given by 0
2mp 2pBdc ðd þ 2sÞ ; ¼ L U0
ð10Þ
where U0 is the flux quantum. The parameters that represent the pinning strength at each junction are assumed as follows: a1 = 3, a2 = 4, and a3 = 3.
Numerical simulations of the negative resistance in the fluxflow properties were carried out based on the inductively-coupled sine-Gordon equations by taking into account a sinusoidal form of the pinning potential induced by the pancake vortices. Our results reveal that the RF power can be generated by in-phase flux-flow state with a periodic pinning potential. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. Kleiner, P. Müller, Phys. Rev. B 49 (1994) 1327. M. Machida, K. Koyama, A. Tanaka, M. Tachiki, Physica C 330 (2000) 85. T. Nagatsuma, K. Empuku, F. Irie, K. Yoshida, J. Appl. Phys. 63 (1988) 1130. R. Monaco, S. Pagano, G. Costabile, Phys. Lett. A 131 (1988) 122. Y. Yamada, K. Nakajima, T. Yasuda, T. Yamashita, Koji Nakajima, IEEE Trans. Appl. Supercond. 15–2 (2005) 1028. Y. Yamada, Koji Nakajima, K. Nakajima, J. Kor, Phys. Soc. 48 (2006) 1053. F. Auracher, T. Van Duzer, J. Appl. Phys. 44 (1973) 848. S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. A.E. Koshelev, Phys. Rev B 68 (2003) 094520. A. Grigorenko, S. Bending, T. Tamegai, S. Ooi, M. Henini, Nature (London) 414 (2001) 728.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
RF impedance of intrinsic Josephson junction in flux-flow state with a periodic pinning potential Y. Yamada a,*, K. Nakajima b, K. Nakajima c a
Department of Electrical and Computer Engineering, Oyama National College of Technology, Oaza-Nakakuki 771, Oyama 323-0806, Japan Graduate School of Science and Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan c Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan b
a r t i c l e
i n f o
Article history: Available online 23 May 2008 PACS: 74.25.Nf 74.25.Qt Keywords: Josephson junction Vortex Flux-flow Sine-Gordon equation Simulation
a b s t r a c t We have investigated the dynamics of Josephson vortices interacting with electromagnetic waves in Bi2Sr2CaCu2O8+y intrinsic Josephson junction (IJJ) stacks by means of millimeter wave irradiation and numerical simulations based on coupled sine-Gordon equations while taking into account a sinusoidal form of the periodic pinning potential. The numerical simulation results for the influence of the electromagnetic waves on the flux-flow properties reveal that the periodic pinning potential induces the inphase motion of Josephson vortices over the junctions. In order to prove from another viewpoint, we investigate RF impedance of IJJ in flux-flow state in this study. A remarkable negative real part region appears at 1st harmonic step, it means that the IJJ in flux-flow state acts as an oscillator at the negative real part region. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Josephson vortex dynamics in a stack of inductively-coupled Josephson junctions attracts our attentions because the stack models highly anisotropic high-Tc superconductors represented by Bi2Sr2CaCu2O8+y (BSCCO). In BSCCO-type high-Tc superconductors, superconducting CuO2 planes are stacked along the c-axis by interposing insulating layers consisting of BiO and SrO, and Josephson effects govern the current transport along the c-axis. Since the discovery of the intrinsic Josephson effects of BSCCO [1] arises from such a stack structure, much attention has been paid to the mechanism and the potentials of intrinsic Josephson junctions (IJJ) as possible high-frequency devices capable of electromagnetic wave generation up to the THz region [2], exceeding the conventional low-Tc flux-flow oscillators (FFO) [3]. The IJJ is also expected to be superior to the conventional one, since output power of the IJJ grows as the square of the number of junctions in the stack [4]. Such superior performances of IJJ would be achieved by inphase motion of Josephson vortices in stacked junctions. We have studied the effect of a periodic pinning potential induced by the pancake vortices upon Josephson vortices. From the experimental results [5], we assume that the periodic pinning potential plays
* Corresponding author. Tel.: +81 285 20 2234. E-mail address: [email protected] (Y. Yamada). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.014
an important role for inducing the in-phase vortex motion. The Josephson vortex dynamics in the BSCCO-type stack containing N junctions is expressed by N coupled sine-Gordon equations. We describe approaches to introduce the periodic pinning potential in an ordinary numerical simulation by using the coupled sine-Gordon equation [6]. The numerical simulation results for the influence of the electromagnetic waves on the flux-flow properties reveal that the periodic pinning potential induces the in-phase motion of Josephson vortices over the junctions. In order to prove from another viewpoint, we investigate RF impedance of IJJ in flux-flow state in this study. The negative real part of the RF impedance of the Josephson junction appears under certain condition, it means that the Josephson junction acts as an oscillator at negative real part region [7]. Thus we calculate the negative real part of the RF impedance of IJJ in detail. 2. Numerical simulation procedure The numerical model developed for a system of stacked Josephson junctions [8] has proven to be a very useful tool for understanding the Josephson vortex dynamics of BSCCO-type IJJ. We consider a stack of long Josephson junction with a length L, a width W and a junction number N. The model for a stack of N Josephson junctions consists of N inductively-coupled sine-Gordon equations. Assuming identical parameters for each junction and coupling
1296
Y. Yamada et al. / Physica C 468 (2008) 1295–1297
parameter S between the junctions, we obtain simultaneous equations in following form [8]:
0
1
1
S
0
BS /1 B B . C B B . C B . C B B C B o2 B B /i C ¼ B C B B 2 ox B C B B .. C B @ . A B B @ /
1
S
0
1
S ..
1
0
.. 0
.
S
.
N
0
0 1 C jz1 C CB . C CB . C CB . C CB z C CB j C 0 CB i C CB . C CB . C C@ . A C S A jz N S 1
ð1Þ
z
where /i is the phase difference across each junction and ji is the current in the z-direction, which is the stack direction correspondz ing to the c-axis direction of the BSCCO unit cell. The current ji can be expressed as z ji
¼
1 c2SW
o2 /i o/ þC i ot ot2
! þ
1 k2J
ðsin /i cB Þ;
ð2Þ
where C = G/C and cB is the normalized bias current of each junction. C and G are the capacitance and the conductance of a unit area, respectively. cSW and kJ correspond to the Swihart velocity and the Josephson penetration depth, respectively. The coupling parameter, S, is defined as
S¼
s 0 d
ð3Þ
for 0
d ¼ dI þ 2kL coth
dS kL
ð4Þ
and
s¼
kL ; sinhðdS =kL Þ
ð5Þ
where dS and dI are the thickness of a superconducting layer and an insulating layer, respectively. kL is a London penetration length. When dS is much smaller than kL, S approaches 0.5, which means a very strong coupling between junctions. An applied magnetic field Bdc parallel to the junctions does not appear in the equations, but is provided by the boundary conditions at the junction edges, x = 0, L, where L is the length of the junction. The applied electromagnetic wave is considered to be an alternating magnetic field parallel to
Fig. 1. Assumed circuit model for the RF impedance calculation.
the junctions at one of the junctions edges and is provided by the boundary condition at one of the junction edges, x = 0. Then, the boundary conditions are expressed as
0 o/i 2eðd þ 2sÞ ¼ ðBdc þ Brf sin 2pfrf t Þ; h ox x¼0 0 o/i 2eðd þ 2sÞ Bdc ; ¼ h ox
ð6Þ ð7Þ
x¼L
where Brf and frf are an amplitude and a frequency of the alternating magnetic field, respectively. The applied electromagnetic wave induces an alternating current in the Josephson junction stack. Therefore we consider that the alternating current source Irf is connected to the edge of the junction (x = 0). Fig. 1 shows the circuit model. The RF impedance calculation procedure is as follows: first, fast Fourier transform of ii and vi are taken, then Zrf (=vrf/irf) is calculated, where Zrf, vrf, irf are frf component of impedance, voltage, current, respectively. In order to take into account the effects of a periodic pinning potential for the Josephson vortex dynamics governed by the above coupled sine-Gordon equations, we assume an additional current 2 2 ai d /d =dx in the z-direction to introduce the periodic pinning potential into Eq. (2) as follows: Z ji
¼
1 c2SW
o2 /i o/ þC i ot ot 2
! þ
1
2
d /d ðsin /i cB Þ ai ; 2 k2J dx
Fig. 2. (a) The I–V and I–rrf characteristics for Bdc = 0.55 T, Brf = 0.11 T and a = 0.5. (b) The enlarged I–rrf characteristics of the ‘‘1st step region”.
ð8Þ
Y. Yamada et al. / Physica C 468 (2008) 1295–1297
where /d is a phase difference across the imaginary junction. ai is the coefficient that expresses the strength of pinning in each real junction. It is proportional to the diagonal element of the inverse matrix of the matrix in Eq. (1) when the maximum pinning force in each junction becomes equal. We assume d/d/dx to be a sinusoidal function [9], which is proportional to the pinning potential. Therefore, /d is assumed as follows:
/d ¼
0 2eðd þ 2sÞ aL 2mp Bdc sin x ; h 2mp L
ð9Þ
where m is the number of pancake vortex rows in one superconducting layer and a represents the pinning strength.
1297
Fig. 2a shows the I–V characteristics calculated for Bdc = 0.55 T, Brf = 0.11 T and a = 0.5. As shown with arrow, a current step appears at a voltage satisfying the Josephson relation V = NU0 frf, where N is the number of junctions (N = 3), that is 1st harmonic step. Fig. 2a and b show the dependencies of the real part of the RF impedance on the bias current of first junction (i = 1) and second junction (i = 2). As shown with arrows, remarkable negative resistance regions appear at 1st step. The negative real part of the impedance which occurs at 1st step means that the Josephson junction acts as an oscillator generating power at the frequency frf. 4. Conclusion
3. Results and discussion Numerical simulations are carried out for a stack of three Josephson junctions of junction length L = 10 lm, junction width W = 2 lm, superconductor layer thickness dS = 3 Å, insulating layer thickness dI = 12 Å, and critical current Jc = 100 A/cm2. Under these conditions, we obtain the coupling strength parameter S = –0.5, the McCumber parameter bc = 29, the plasma frequency fp = 32 GHz, the Josephson penetration depth kJ = 3.3 lm, and the anisotropy parameter c 2000. In our experiments [5], the c-axis direction magnetic field to transit into pancake state is about 3 mT. According to a rough estimate of the relations between the pancake vortices and the Josephson vortices [10], it seems to be ‘‘1d-chain” state. Therefore, m is given by 0
2mp 2pBdc ðd þ 2sÞ ; ¼ L U0
ð10Þ
where U0 is the flux quantum. The parameters that represent the pinning strength at each junction are assumed as follows: a1 = 3, a2 = 4, and a3 = 3.
Numerical simulations of the negative resistance in the fluxflow properties were carried out based on the inductively-coupled sine-Gordon equations by taking into account a sinusoidal form of the pinning potential induced by the pancake vortices. Our results reveal that the RF power can be generated by in-phase flux-flow state with a periodic pinning potential. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. Kleiner, P. Müller, Phys. Rev. B 49 (1994) 1327. M. Machida, K. Koyama, A. Tanaka, M. Tachiki, Physica C 330 (2000) 85. T. Nagatsuma, K. Empuku, F. Irie, K. Yoshida, J. Appl. Phys. 63 (1988) 1130. R. Monaco, S. Pagano, G. Costabile, Phys. Lett. A 131 (1988) 122. Y. Yamada, K. Nakajima, T. Yasuda, T. Yamashita, Koji Nakajima, IEEE Trans. Appl. Supercond. 15–2 (2005) 1028. Y. Yamada, Koji Nakajima, K. Nakajima, J. Kor, Phys. Soc. 48 (2006) 1053. F. Auracher, T. Van Duzer, J. Appl. Phys. 44 (1973) 848. S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. A.E. Koshelev, Phys. Rev B 68 (2003) 094520. A. Grigorenko, S. Bending, T. Tamegai, S. Ooi, M. Henini, Nature (London) 414 (2001) 728.