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Numerical research on RF induced zero-crossing steps of stacked Josephson junction in flux-flow state
Physica C 471 (2011) 820–823
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Numerical research on RF induced zero-crossing steps of stacked Josephson junction in flux-flow state Y. Yamada a,⇑, K. Nakajima b a b
Oyama National College of Technology, Nakakuki 771, Oyama 323-0806, Japan Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan
a r t i c l e
i n f o
Article history: Available online 13 May 2011 Keywords: Josephson junction Vortex Flux-flow Sine-Gordon equation Simulation
a b s t r a c t RF responses of intrinsic Josephson junction stacks in the flux-flow state have been studied to explore vortex motion. We have reported various RF induced effects on flux-flow of Bi2Sr2CaCu2O8+y intrinsic Josephson junctions. In a pinning free state, a remarkable zero-crossing step appears at a certain voltage on I–V curve, which is closed to the voltage V = NU0f, where N is the number of junction, U0 is the flux quantum, f is the RF frequency, respectively. It is shown that vortex motion phase-locked to external microwaves play an important role in the responses. In this report, we have carried out numerical simulation by using the coupled sine-Gordon equations in order to compare with the experimental results. The numerical simulations reveal that the alternating magnetic field of microwaves drives vortices into the stack and generates the zero-crossing step. Ó 2011 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 the Josephson effect governs 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 for potentials of intrinsic Josephson junctions (IJJ) as possible high-frequency devices capable for 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 phaselocking motion of Josephson vortices in stacked junctions. The phase-locking Josephson vortices motion interacting with a variety of electromagnetic modes is an important factor in generating effective emission from the intrinsic junction stack. In this regard, studies on RF responses of the intrinsic junction stacks in the fluxflow state are useful in analyzing the phase-locking phenomena. In ⇑ Corresponding author. Address: Department of Electrical and Computer Engineering, Oyama National College of Technology, Oaza-Nakakuki 771, Oyama 3230806, Japan. Tel.: +81 285 20 2234; fax: +81 285 20 2885. E-mail address: [email protected] (Y. Yamada). 0921-4534/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2011.05.064
this paper we present numerical simulation results of microwave responses in the flux-flow state of the intrinsic Josephson junction stacks. 2. Numerical simulation procedure The numerical model developed for a system of stacked Josephson junctions [5] 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 parameter S between the junctions, we obtain simultaneous equations in following form [5]:
0 B B 2 B @ B B @x2 B B B @
0
1
S
0
BS /1 B B .. C C B . C B C B B /i C C¼B C .. C B B . A B B @ /N
1
S ..
0
0
S
1
S ..
1
. 0 .
0
S
1 0 1 C jz1 C CB . C CB . C CB . C CB z C CB j C CB i C CB . C CB . C C@ . A C S A jz 1
ð1Þ
N
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
821
Y. Yamada, K. Nakajima / Physica C 471 (2011) 820–823
Fig. 1. The I–V characteristics for Bdc = 0.2 T, Brf = 0.55 T and frf = 50 GHz. Remarkable zero-crossing steps appear at voltages satisfying the Josephson relation Va = 2U0frf = 207 lV (the step A) and V1 = NU0frf = 310 lV, where N is the number of junctions (N = 3).
Fig. 4. The I–V characteristics for Bdc = 0.2 T, Brf = 0.5 T and frf = 50 GHz. The zerocrossing step does not exist.
z
ji ¼
1 c2SW
@ 2 /i @/ þC i @t @t 2
! 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 Fig. 2. The snapshot of Josephson currents distribution at the bias point in the zerocrossing step (I = 0 lA and V = V1 = 310 lV). Centers of Josephson vortices are marked by circles.
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
Fig. 3. The time evolution of the phase sin / of 1st and 2nd junction at the bias point in the zero-crossing step (I = 0 lA and V = V1 = 310 lV). That of 3rd junction is omitted because it is the same motion as that of 1st junction. (a) The 1st junction (i = 1). (b) The 2nd junction (i = 2).
822
Y. Yamada, K. Nakajima / Physica C 471 (2011) 820–823
Fig. 5. The time evolution of the phase sin / of 1st and 2nd junction at the zero bias point (i = 1, 2). That of 3rd junction is omitted because it is the same motion as that of 1st junction. (a) The 1st junction (i = 1). (b) The 2nd junction (i = 2).
Fig. 6. The Brf dependencies of upper side and lower side of 1st step and the step at the voltage Va for Bdc = 0.2 T and frf = 50 GHz.
wave is considered to be an alternating magnetic field parallel to 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 @/i 2pðd þ 2sÞ ¼ ðBdc þ Brf sin 2pfrf tÞ; @x x¼0 U0 0 @/i 2pðd þ 2sÞ ¼ Bdc ; @x U x¼L
ð6Þ ð7Þ
0
where Brf and frf are an amplitude and a frequency of the alternating magnetic field, respectively. 3. Results and discussion Numerical simulations are carried out for a stack of three Josephson junctions of junction length L = 15 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. Fig. 1 shows the I–V characteristics for Bdc = 0.2 T, Brf = 0.55 T and frf = 50 GHz. Remarkable RF induced steps appear at voltages satisfying the Josephson relation Vn = nNU0frf, where N is the
number of junctions (N = 3), n is the degree of these steps. The 1st step in the positive voltage region is zero-crossing. At the zero-crossing bias point, where I = 0 lA and V = V1 = 310 lV, the Josephson vortex motion is phase-locked motion [6] and Josephson vortex lattice has a rectangular arrangement. Fig. 2 is snapshot of vortex motion in this bias point. Fig. 3a and b show the time evolution of the phase sin / of 1st and 2nd junction at the bias point in the zero-crossing step (i = 1, 2), respectively. That of 3rd junction (i = 3) is omitted because it is the same motion as that of 1st junction. The vortices induced at an edge of the junction (x = 0), then the vortices flow to opposite edge at same velocity in every junction although the dc bias current does not exist. In addition, the step at voltage satisfying the Josephson relation Va = 2U0 frf = 207 lV is also zero-crossing (the step A in Fig. 1). This is result from the phase-locked motion of 1st and 3rd junctions, and that of 2nd junction is not caused in this step. Fig. 4 shows the I–V characteristics for Bdc = 0.2 T, Brf = 0.5 T and frf = 50 GHz. In this condition, the zero-crossing step does not exist. Fig. 5a and b show the time evolution of the phase of 1st and 2nd junction at the zero bias point, respectively. That of 3rd junction is omitted by same reason. Under this condition, the vortices do not flow to opposite edge without dc bias current. Fig. 6 shows the Brf dependencies of upper side and lower side of steps for Bdc = 0.2 T and frf = 50 GHz. The zero-crossing step appears at 0.55 6 Brf 6 0.75. The condition that the zero-crossing
Y. Yamada, K. Nakajima / Physica C 471 (2011) 820–823
steps appear in whole junctions is only Brf = 0.55 T. The others are the condition that the step at the voltage Va appears, which is the zero-crossing steps in only junctions i = 1 and i = 3. These results can explain that the zero-crossing step in the experimental results [7] has not clear edges. It is supposed that the phase-locking of the vortex motion in the intrinsic junction stack has not been perfected because of many junctions. 4. Conclusion Numerical simulations of the flux-flow properties were carried out based on the inductively-coupled sine-Gordon equations. The zero-crossing steps appear at V = NU0frf under a certain condition. Under the condition, Josephson vortices induced at an edge of the
823
junction, then the vortices flow to opposite edge at same velocity in every junction although the dc bias current does not exist. In addition, Josephson vortices align vertically along the stack direction as well. References [1] [2] [3] [4] [5] [6] [7]
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. 58 (1985) 441. R. Monaco, S. Pagano, G. Costabile, Phys. Lett. A 131 (1988) 122. S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. A.V. Ustinov, J. Mygind, V.A. Oboznov, J. Appl. Phys. 72 (1992) 1203. K. Nakajima, Y. Yamada, T. Yamashita, Physica C 468 (2008) 660.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Numerical research on RF induced zero-crossing steps of stacked Josephson junction in flux-flow state Y. Yamada a,⇑, K. Nakajima b a b
Oyama National College of Technology, Nakakuki 771, Oyama 323-0806, Japan Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan
a r t i c l e
i n f o
Article history: Available online 13 May 2011 Keywords: Josephson junction Vortex Flux-flow Sine-Gordon equation Simulation
a b s t r a c t RF responses of intrinsic Josephson junction stacks in the flux-flow state have been studied to explore vortex motion. We have reported various RF induced effects on flux-flow of Bi2Sr2CaCu2O8+y intrinsic Josephson junctions. In a pinning free state, a remarkable zero-crossing step appears at a certain voltage on I–V curve, which is closed to the voltage V = NU0f, where N is the number of junction, U0 is the flux quantum, f is the RF frequency, respectively. It is shown that vortex motion phase-locked to external microwaves play an important role in the responses. In this report, we have carried out numerical simulation by using the coupled sine-Gordon equations in order to compare with the experimental results. The numerical simulations reveal that the alternating magnetic field of microwaves drives vortices into the stack and generates the zero-crossing step. Ó 2011 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 the Josephson effect governs 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 for potentials of intrinsic Josephson junctions (IJJ) as possible high-frequency devices capable for 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 phaselocking motion of Josephson vortices in stacked junctions. The phase-locking Josephson vortices motion interacting with a variety of electromagnetic modes is an important factor in generating effective emission from the intrinsic junction stack. In this regard, studies on RF responses of the intrinsic junction stacks in the fluxflow state are useful in analyzing the phase-locking phenomena. In ⇑ Corresponding author. Address: Department of Electrical and Computer Engineering, Oyama National College of Technology, Oaza-Nakakuki 771, Oyama 3230806, Japan. Tel.: +81 285 20 2234; fax: +81 285 20 2885. E-mail address: [email protected] (Y. Yamada). 0921-4534/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2011.05.064
this paper we present numerical simulation results of microwave responses in the flux-flow state of the intrinsic Josephson junction stacks. 2. Numerical simulation procedure The numerical model developed for a system of stacked Josephson junctions [5] 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 parameter S between the junctions, we obtain simultaneous equations in following form [5]:
0 B B 2 B @ B B @x2 B B B @
0
1
S
0
BS /1 B B .. C C B . C B C B B /i C C¼B C .. C B B . A B B @ /N
1
S ..
0
0
S
1
S ..
1
. 0 .
0
S
1 0 1 C jz1 C CB . C CB . C CB . C CB z C CB j C CB i C CB . C CB . C C@ . A C S A jz 1
ð1Þ
N
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
821
Y. Yamada, K. Nakajima / Physica C 471 (2011) 820–823
Fig. 1. The I–V characteristics for Bdc = 0.2 T, Brf = 0.55 T and frf = 50 GHz. Remarkable zero-crossing steps appear at voltages satisfying the Josephson relation Va = 2U0frf = 207 lV (the step A) and V1 = NU0frf = 310 lV, where N is the number of junctions (N = 3).
Fig. 4. The I–V characteristics for Bdc = 0.2 T, Brf = 0.5 T and frf = 50 GHz. The zerocrossing step does not exist.
z
ji ¼
1 c2SW
@ 2 /i @/ þC i @t @t 2
! 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 Fig. 2. The snapshot of Josephson currents distribution at the bias point in the zerocrossing step (I = 0 lA and V = V1 = 310 lV). Centers of Josephson vortices are marked by circles.
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
Fig. 3. The time evolution of the phase sin / of 1st and 2nd junction at the bias point in the zero-crossing step (I = 0 lA and V = V1 = 310 lV). That of 3rd junction is omitted because it is the same motion as that of 1st junction. (a) The 1st junction (i = 1). (b) The 2nd junction (i = 2).
822
Y. Yamada, K. Nakajima / Physica C 471 (2011) 820–823
Fig. 5. The time evolution of the phase sin / of 1st and 2nd junction at the zero bias point (i = 1, 2). That of 3rd junction is omitted because it is the same motion as that of 1st junction. (a) The 1st junction (i = 1). (b) The 2nd junction (i = 2).
Fig. 6. The Brf dependencies of upper side and lower side of 1st step and the step at the voltage Va for Bdc = 0.2 T and frf = 50 GHz.
wave is considered to be an alternating magnetic field parallel to 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 @/i 2pðd þ 2sÞ ¼ ðBdc þ Brf sin 2pfrf tÞ; @x x¼0 U0 0 @/i 2pðd þ 2sÞ ¼ Bdc ; @x U x¼L
ð6Þ ð7Þ
0
where Brf and frf are an amplitude and a frequency of the alternating magnetic field, respectively. 3. Results and discussion Numerical simulations are carried out for a stack of three Josephson junctions of junction length L = 15 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. Fig. 1 shows the I–V characteristics for Bdc = 0.2 T, Brf = 0.55 T and frf = 50 GHz. Remarkable RF induced steps appear at voltages satisfying the Josephson relation Vn = nNU0frf, where N is the
number of junctions (N = 3), n is the degree of these steps. The 1st step in the positive voltage region is zero-crossing. At the zero-crossing bias point, where I = 0 lA and V = V1 = 310 lV, the Josephson vortex motion is phase-locked motion [6] and Josephson vortex lattice has a rectangular arrangement. Fig. 2 is snapshot of vortex motion in this bias point. Fig. 3a and b show the time evolution of the phase sin / of 1st and 2nd junction at the bias point in the zero-crossing step (i = 1, 2), respectively. That of 3rd junction (i = 3) is omitted because it is the same motion as that of 1st junction. The vortices induced at an edge of the junction (x = 0), then the vortices flow to opposite edge at same velocity in every junction although the dc bias current does not exist. In addition, the step at voltage satisfying the Josephson relation Va = 2U0 frf = 207 lV is also zero-crossing (the step A in Fig. 1). This is result from the phase-locked motion of 1st and 3rd junctions, and that of 2nd junction is not caused in this step. Fig. 4 shows the I–V characteristics for Bdc = 0.2 T, Brf = 0.5 T and frf = 50 GHz. In this condition, the zero-crossing step does not exist. Fig. 5a and b show the time evolution of the phase of 1st and 2nd junction at the zero bias point, respectively. That of 3rd junction is omitted by same reason. Under this condition, the vortices do not flow to opposite edge without dc bias current. Fig. 6 shows the Brf dependencies of upper side and lower side of steps for Bdc = 0.2 T and frf = 50 GHz. The zero-crossing step appears at 0.55 6 Brf 6 0.75. The condition that the zero-crossing
Y. Yamada, K. Nakajima / Physica C 471 (2011) 820–823
steps appear in whole junctions is only Brf = 0.55 T. The others are the condition that the step at the voltage Va appears, which is the zero-crossing steps in only junctions i = 1 and i = 3. These results can explain that the zero-crossing step in the experimental results [7] has not clear edges. It is supposed that the phase-locking of the vortex motion in the intrinsic junction stack has not been perfected because of many junctions. 4. Conclusion Numerical simulations of the flux-flow properties were carried out based on the inductively-coupled sine-Gordon equations. The zero-crossing steps appear at V = NU0frf under a certain condition. Under the condition, Josephson vortices induced at an edge of the
823
junction, then the vortices flow to opposite edge at same velocity in every junction although the dc bias current does not exist. In addition, Josephson vortices align vertically along the stack direction as well. References [1] [2] [3] [4] [5] [6] [7]
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. 58 (1985) 441. R. Monaco, S. Pagano, G. Costabile, Phys. Lett. A 131 (1988) 122. S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. A.V. Ustinov, J. Mygind, V.A. Oboznov, J. Appl. Phys. 72 (1992) 1203. K. Nakajima, Y. Yamada, T. Yamashita, Physica C 468 (2008) 660.