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Investigation of I–V characteristics of Bi-2212 intrinsic Josephson junctions for THz-wave generation
Physica C 469 (2009) 1604–1607
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Investigation of I–V characteristics of Bi-2212 intrinsic Josephson junctions for THz-wave generation T. Tachiki *, A. Sugawara, T. Uchida Department of Electrical and Electronic Engineering, National Defense Academy, Hashirimizu 1-10-20, Yokosuka, Kanagawa 239-8686, Japan
a r t i c l e
i n f o
Article history: Available online 31 May 2009 PACS: 85.25.j 85.25.Cp Keywords: Josephson junctions Terahertz Bi-based cuprates Sine-Gordon equation
a b s t r a c t Voltage jumps have been observed in the quasiparticle branch I–V characteristics of Bi2Sr2CaCu2O8+x intrinsic Josephson junctions fabricated as 60 lm 300 lm 0.5 lm mesas. The Josephson frequency at the highest jump voltage was estimated to be 0.65 THz, close to the fundamental cavity resonance frequency of the transverse Josephson plasma mode. Numerical simulations using coupled sine-Gordon equations demonstrated voltage jumps similar to the above experimental results, which appeared in the I–V characteristics, and the electric field distribution at the highest jump voltage exhibited resonance in the junctions. Moreover, the maximum radiation power, which was obtained in the vicinity of the highest voltage, was proportional to the square of the number of junctions. This can be explained by reduction of the impedance mismatch between the junctions and free space with increasing the number of junctions. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction It is well known that a Josephson junction can generate electromagnetic waves by the AC Josephson effect. However, the radiated power of these electromagnetic waves is usually small, typically on the order of pW. Since the discovery of the intrinsic Josephson effect [1,2], it has been expected that intrinsic Josephson junctions (IJJs) would be utilized for electromagnetic wave oscillators, especially as terahertz (THz) sources, because IJJs have a natural junction-array structure suitable for generation of strong radiation. After several reports on the detection of THz signals radiated from high-Tc cuprates Bi2Sr2CaCu2O8+x (Bi-2212) IJJs [3–6], Ozyuzer et al. observed 0.5 lW THz waves at frequencies of 0.3–0.85 THz from IJJs biased by DC current only [7]. In addition, radiation emission peaks were detected at voltage jumps observed in the I–V characteristics (IVCs) of the IJJs. However, the origin of these voltage jumps remains unclear. In this paper, we report experimental and theoretical investigations of the IVCs of IIJs to clarify the relationship between voltage jumps and THz-wave radiation. 2. Observations of voltage jumps in Bi-2212 IJJs As in [7], a Bi-2212 mesa with a junction area of 60 lm 300 lm and a height of approximately 0.5 lm (320 junctions
* Corresponding author. Tel.: +81 46 841 3810; fax: +81 46 844 5903. E-mail address: [email protected] (T. Tachiki). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.214
0.5 lm/1.5 nm per junction) was fabricated on a Bi-2212 single crystal. Fig. 1 shows an SEM image of the fabricated device. A part of the mesa was covered with a CaF2 thin film to avoid a short circuit between the top electrode and the side of the mesa. The inset in Fig. 2 shows the IVCs of the fabricated mesa at 40 K. The numerous quasiparticle branches imply that the mesa contained over 300 junctions. The main figure shows the last quasiparticle branch in the IVCs of the positive bias region. This branch was traced when the current sweep decreased from maximum bias to 0 A. The figure shows several voltage jumps, which are indicated by arrows. We believe that these jumps indicate THz radiation from the mesa, since the emission peaks of THz radiation observed by Ozyuzer et al. [7] occur at similar voltage jumps to those observed in this study. From now on, we define a jump voltage by the midpoint voltage between the start and end point of the jump. The Josephson frequency at the highest jump voltage of approximately 0.43 V was estimated to be 0.65 THz, assuming that all of the junctions produced the highest voltage. This is close to the fundamental resonant frequency (0.72 THz) of a transverse Josephson plasma in a cavity with a junction width of 60 lm, which can be pffiffiffiffi expressed as f0 ¼ c=ð2L er Þ, where c is the speed of light in vacuum, L = 60 lm is the cavity length, er 12 is the relative dielectric constant of Bi-2212 [8]. 3. Numerical simulations for IVCs of IJJs Numerical simulations were performed using coupled sine-Gordon equations (CSGEs) based on the inductive coupling model
T. Tachiki et al. / Physica C 469 (2009) 1604–1607
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Fig. 3. Schematic of the simulation model used to calculate IVCs of IJJs.
Fig. 1. SEM image of a 60 lm 300 lm 0.5 lm Bi-2212 mesa.
E0 = V0/s and P 0 ¼ /20 fp =ð2pl0 sÞ, respectively. Typically, s = 1.5 nm, kc = 150 lm, fp = 100 GHz, V0 = 0.2 mV, the critical current density Jc = 770 A/cm2 and P0 = 40 lW. A boundary condition connecting the electromagnetic fields inside the junction to radiated waves ac at the junction edge is given by Eac i;i1 ð0; tÞ ¼ Ei;i1 ð0; tÞ, Hi;i1 ð0; tÞ ¼ ac þ ac þ ð0; tÞ=Z , E ðl; tÞ ¼ E ðl; tÞ, H ðl; tÞ ¼ E ðl; tÞ=Z E 0 0 , where i;i1 i;i1 i;i1 i;i1 i;i1 ac Eac i;i1 and Hi;i1 are the AC components of electric and magnetic fields between the (i1)th and ith superconducting layer, E i;i1 and H i;i1 are the fields of the radiated waves propagating in the direction of ±x, l is the junction length, and Z0 is the characteristic impedance of free space. Under this boundary condition, IVCs were calculated for N = 2–18. A two-dimensional cavity resonance [11] is excited in IJJs biased at a certain voltage. The resonant frequency fmn in the normalized unit is written as
fmn ¼
mpcn ; l
ð1Þ
where m and n are the mode numbers in the x-direction and zdirection, respectively, and cn is the following velocity from the Josephson plasma mode [12]:
1 cn ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o ; np 1 þ 2f 1 cos Nþ1
Fig. 2. IVCs of the fabricated mesa at 40 K. The inset: IVCs traced for many current sweeps. The main figure: the last quasiparticle branch in the positive bias region.
[9,10]. Fig. 3 shows a schematic of our two-dimensional model with x- and z- coordinates parallel to the Poynting vector S of the radiated waves and the c-axis of Bi-2212, respectively. x, z, time t, bias current I, and voltage Vj across the jth junction were normalized by the c-axis London penetration depth kc, the interlayer spacing s, the inverse of the Josephson plasma angular frequency 2pfp, the critical current Ic, and V0 = /0 fp (/0 is the flux quantum), respectively. By using these normalized units, the Josephson relation can be written as Vj = f. The electric field E inside the junction and power P radiated from the junctions were normalized by
ð2Þ
where f is the magnetic coupling parameter between the layers, typically f = 10,000 [10]. n = 1 for the in-phase plasma mode. In this paper, we focused on the fundamental resonance in both x- and zdirections, i.e. m = n = 1, because the radiation power was expected to be very high in these resonances. In order to select an identical f11 of 1.57 for different N, l was varied from 0.02 to 0.12. As a typical simulation result, the IVCs for N = 2 are shown in Fig. 4. The horizontal axis shows the average voltage P V j =N. Voltage jumps similar to the experimental results V ave ¼ shown in Fig. 2 were clearly observed in the quasiparticle branch. The jump in the high bias region appeared at Vave 1.57, corresponding to f11, whereas the jumps in the low bias region appeared in the vicinity of Vave = 0.90 corresponding to f12. Therefore, the two distinct branches separated by these jumps indicate different plasma modes. The distribution of electromagnetic fields provides further support. Fig. 5 shows the electric field distribution in the top and bottom junctions at (a) I = 1.05 and (b) I = 0.60 that are typical bias currents in the high and low bias regions, respectively. The solid and broken lines indicate the distributions at t = 0 and t = 2.0 (t = 4.8) in Fig. 5a (Fig. 5b), respectively. The time interval between the two lines is one half-period of the standing waves. All distribu-
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Fig. 4. IVCs and bias voltage dependence of radiation power for N = 2, obtained from the simulation.
tions showed a fundamental resonance in the x-direction. Moreover, Fig. 5a shows that the standing waves in the top and bottom junctions were in-phase, while Fig. 5b shows them out-of-phase. Therefore, the branch in the high bias region of Fig. 4 indicates the in-phase plasma mode, while the branch in the low bias region represents the out-of-phase mode. Next, we investigate the electromagnetic power radiated from the junctions. The radiation power P is defined as the sum of Poynting vectors at the junction edges (see Fig. 3). As shown in Fig. 4, the maximum power Pmax was obtained at Vave 1.57 in the inphase branch. For other values of N, Pmax was also obtained at nearly the same voltage, where the fundamental resonance of the in-phase mode was excited in the junctions. Finally, the relationship between Pmax and N2 is shown in Fig. 6. Pmax was proportional to N2. Ozyuzer et al. found experimentally that the radiation power produced by Bi-2212 IJJs was proportional to the square of the number of junctions activating the radiation [7]. This was interpreted as coherent radiation, as from a laser. However, there was a discrepancy between their experiment and the simulation in this study: in the experiment, the number of
Fig. 6. Relationship between the maximum radiation power and the number of junctions, obtained from the simulation.
active junction varied while N did not, while in the latter case N was variable. The relationship shown in Fig. 6 can be explained as follows: an impedance mismatch between the junctions and free space decreases with increasing N, since the highest mode velocity c1 (Eq. (2)) increases to approach the speed of light in the medium with increasing N; the power P0 for an individual junction is proportional to N because of the mismatch reduction. Therefore, the total power Pmax = NP0 is proportional to N2. 4. Conclusions To observe THz-wave radiation from Bi-2212 IJJs, we investigated IVCs as follows: from experimental measurements of fabricated Bi-2212 IJJs, the Josephson frequency corresponding to the highest jump voltage was 0.65 THz. This agrees with the frequency of the fundamental cavity resonance with transverse mode. Numerical simulations demonstrated that the electric field distribution at the highest jump voltage showed resonance in the junctions. Moreover, the maximum radiation power was obtained close
Fig. 5. Electric field distributions at (a) I = 1.05 Ic and (b) I = 0.60 Ic, obtained from the simulation.
T. Tachiki et al. / Physica C 469 (2009) 1604–1607
to the highest voltage and was proportional to N2 because of reduction of the impedance mismatch with increasing N. References [1] R. Kleiner, F. Steinmeyer, G. Kunkel, P. Müller, Phys. Rev. Lett. 68 (1992) 2394. [2] G. Oya, N. Aoyama, A. Irie, S. Kishida, H. Tokutaka, Jpn. J. Appl. Phys. 31 (1992) L829. [3] K. Lee, W. Wang, I. Iguchi, M. Tachiki, K. Hirata, T. Mochiku, Phys. Rev. B 61 (2000) 3616. [4] I.E. Batov, X.Y. Jin, S.V. Shitov, Y. Koval, P. Müller, A.V. Ustinov, Appl. Phys. Lett. 88 (2006) 262504.
1607
[5] K. Kadowaki, I. Kakeya, T. Yamamoto, T. Yamazaki, M. Kohri, Y. Kubo, Physica C 437–438 (2006) 111. [6] M.-H. Bae, H.-J. Lee, J.-H. Choi, Phys. Rev. Lett. 98 (2007) 027002. [7] L. Ozyuzer, A.E. Koshelev, C. Kurter, N. Gopalsami, Q. Li, M. Tachiki, K. Kadowaki, T. Yamamoto, H. Minami, H. Yamaguchi, T. Tachiki, K.E. Gray, W.–K. Kwok, U. Welp, Science 308 (2007) 1291. [8] M.B. Gaifullin, M. Matsuda, N. Chikumoto, J. Shimoyama, K. Kishio, Phys. Rev. Lett. 84 (2000) 2945. [9] S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. [10] M. Tachiki, M. Iizuka, S. Tejima, H. Nakajima, Phys. Rev. B 71 (2005) 134515. [11] R. Kleiner, Phys. Rev. B 50 (1994) 6916. [12] S. Sakai, A.V. Ustinov, H. Kohlstedt, A. Petraglia, N.F. Pedersen, Phys. Rev. B 50 (1994) 12905.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Investigation of I–V characteristics of Bi-2212 intrinsic Josephson junctions for THz-wave generation T. Tachiki *, A. Sugawara, T. Uchida Department of Electrical and Electronic Engineering, National Defense Academy, Hashirimizu 1-10-20, Yokosuka, Kanagawa 239-8686, Japan
a r t i c l e
i n f o
Article history: Available online 31 May 2009 PACS: 85.25.j 85.25.Cp Keywords: Josephson junctions Terahertz Bi-based cuprates Sine-Gordon equation
a b s t r a c t Voltage jumps have been observed in the quasiparticle branch I–V characteristics of Bi2Sr2CaCu2O8+x intrinsic Josephson junctions fabricated as 60 lm 300 lm 0.5 lm mesas. The Josephson frequency at the highest jump voltage was estimated to be 0.65 THz, close to the fundamental cavity resonance frequency of the transverse Josephson plasma mode. Numerical simulations using coupled sine-Gordon equations demonstrated voltage jumps similar to the above experimental results, which appeared in the I–V characteristics, and the electric field distribution at the highest jump voltage exhibited resonance in the junctions. Moreover, the maximum radiation power, which was obtained in the vicinity of the highest voltage, was proportional to the square of the number of junctions. This can be explained by reduction of the impedance mismatch between the junctions and free space with increasing the number of junctions. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction It is well known that a Josephson junction can generate electromagnetic waves by the AC Josephson effect. However, the radiated power of these electromagnetic waves is usually small, typically on the order of pW. Since the discovery of the intrinsic Josephson effect [1,2], it has been expected that intrinsic Josephson junctions (IJJs) would be utilized for electromagnetic wave oscillators, especially as terahertz (THz) sources, because IJJs have a natural junction-array structure suitable for generation of strong radiation. After several reports on the detection of THz signals radiated from high-Tc cuprates Bi2Sr2CaCu2O8+x (Bi-2212) IJJs [3–6], Ozyuzer et al. observed 0.5 lW THz waves at frequencies of 0.3–0.85 THz from IJJs biased by DC current only [7]. In addition, radiation emission peaks were detected at voltage jumps observed in the I–V characteristics (IVCs) of the IJJs. However, the origin of these voltage jumps remains unclear. In this paper, we report experimental and theoretical investigations of the IVCs of IIJs to clarify the relationship between voltage jumps and THz-wave radiation. 2. Observations of voltage jumps in Bi-2212 IJJs As in [7], a Bi-2212 mesa with a junction area of 60 lm 300 lm and a height of approximately 0.5 lm (320 junctions
* Corresponding author. Tel.: +81 46 841 3810; fax: +81 46 844 5903. E-mail address: [email protected] (T. Tachiki). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.214
0.5 lm/1.5 nm per junction) was fabricated on a Bi-2212 single crystal. Fig. 1 shows an SEM image of the fabricated device. A part of the mesa was covered with a CaF2 thin film to avoid a short circuit between the top electrode and the side of the mesa. The inset in Fig. 2 shows the IVCs of the fabricated mesa at 40 K. The numerous quasiparticle branches imply that the mesa contained over 300 junctions. The main figure shows the last quasiparticle branch in the IVCs of the positive bias region. This branch was traced when the current sweep decreased from maximum bias to 0 A. The figure shows several voltage jumps, which are indicated by arrows. We believe that these jumps indicate THz radiation from the mesa, since the emission peaks of THz radiation observed by Ozyuzer et al. [7] occur at similar voltage jumps to those observed in this study. From now on, we define a jump voltage by the midpoint voltage between the start and end point of the jump. The Josephson frequency at the highest jump voltage of approximately 0.43 V was estimated to be 0.65 THz, assuming that all of the junctions produced the highest voltage. This is close to the fundamental resonant frequency (0.72 THz) of a transverse Josephson plasma in a cavity with a junction width of 60 lm, which can be pffiffiffiffi expressed as f0 ¼ c=ð2L er Þ, where c is the speed of light in vacuum, L = 60 lm is the cavity length, er 12 is the relative dielectric constant of Bi-2212 [8]. 3. Numerical simulations for IVCs of IJJs Numerical simulations were performed using coupled sine-Gordon equations (CSGEs) based on the inductive coupling model
T. Tachiki et al. / Physica C 469 (2009) 1604–1607
1605
Fig. 3. Schematic of the simulation model used to calculate IVCs of IJJs.
Fig. 1. SEM image of a 60 lm 300 lm 0.5 lm Bi-2212 mesa.
E0 = V0/s and P 0 ¼ /20 fp =ð2pl0 sÞ, respectively. Typically, s = 1.5 nm, kc = 150 lm, fp = 100 GHz, V0 = 0.2 mV, the critical current density Jc = 770 A/cm2 and P0 = 40 lW. A boundary condition connecting the electromagnetic fields inside the junction to radiated waves ac at the junction edge is given by Eac i;i1 ð0; tÞ ¼ Ei;i1 ð0; tÞ, Hi;i1 ð0; tÞ ¼ ac þ ac þ ð0; tÞ=Z , E ðl; tÞ ¼ E ðl; tÞ, H ðl; tÞ ¼ E ðl; tÞ=Z E 0 0 , where i;i1 i;i1 i;i1 i;i1 i;i1 ac Eac i;i1 and Hi;i1 are the AC components of electric and magnetic fields between the (i1)th and ith superconducting layer, E i;i1 and H i;i1 are the fields of the radiated waves propagating in the direction of ±x, l is the junction length, and Z0 is the characteristic impedance of free space. Under this boundary condition, IVCs were calculated for N = 2–18. A two-dimensional cavity resonance [11] is excited in IJJs biased at a certain voltage. The resonant frequency fmn in the normalized unit is written as
fmn ¼
mpcn ; l
ð1Þ
where m and n are the mode numbers in the x-direction and zdirection, respectively, and cn is the following velocity from the Josephson plasma mode [12]:
1 cn ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o ; np 1 þ 2f 1 cos Nþ1
Fig. 2. IVCs of the fabricated mesa at 40 K. The inset: IVCs traced for many current sweeps. The main figure: the last quasiparticle branch in the positive bias region.
[9,10]. Fig. 3 shows a schematic of our two-dimensional model with x- and z- coordinates parallel to the Poynting vector S of the radiated waves and the c-axis of Bi-2212, respectively. x, z, time t, bias current I, and voltage Vj across the jth junction were normalized by the c-axis London penetration depth kc, the interlayer spacing s, the inverse of the Josephson plasma angular frequency 2pfp, the critical current Ic, and V0 = /0 fp (/0 is the flux quantum), respectively. By using these normalized units, the Josephson relation can be written as Vj = f. The electric field E inside the junction and power P radiated from the junctions were normalized by
ð2Þ
where f is the magnetic coupling parameter between the layers, typically f = 10,000 [10]. n = 1 for the in-phase plasma mode. In this paper, we focused on the fundamental resonance in both x- and zdirections, i.e. m = n = 1, because the radiation power was expected to be very high in these resonances. In order to select an identical f11 of 1.57 for different N, l was varied from 0.02 to 0.12. As a typical simulation result, the IVCs for N = 2 are shown in Fig. 4. The horizontal axis shows the average voltage P V j =N. Voltage jumps similar to the experimental results V ave ¼ shown in Fig. 2 were clearly observed in the quasiparticle branch. The jump in the high bias region appeared at Vave 1.57, corresponding to f11, whereas the jumps in the low bias region appeared in the vicinity of Vave = 0.90 corresponding to f12. Therefore, the two distinct branches separated by these jumps indicate different plasma modes. The distribution of electromagnetic fields provides further support. Fig. 5 shows the electric field distribution in the top and bottom junctions at (a) I = 1.05 and (b) I = 0.60 that are typical bias currents in the high and low bias regions, respectively. The solid and broken lines indicate the distributions at t = 0 and t = 2.0 (t = 4.8) in Fig. 5a (Fig. 5b), respectively. The time interval between the two lines is one half-period of the standing waves. All distribu-
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Fig. 4. IVCs and bias voltage dependence of radiation power for N = 2, obtained from the simulation.
tions showed a fundamental resonance in the x-direction. Moreover, Fig. 5a shows that the standing waves in the top and bottom junctions were in-phase, while Fig. 5b shows them out-of-phase. Therefore, the branch in the high bias region of Fig. 4 indicates the in-phase plasma mode, while the branch in the low bias region represents the out-of-phase mode. Next, we investigate the electromagnetic power radiated from the junctions. The radiation power P is defined as the sum of Poynting vectors at the junction edges (see Fig. 3). As shown in Fig. 4, the maximum power Pmax was obtained at Vave 1.57 in the inphase branch. For other values of N, Pmax was also obtained at nearly the same voltage, where the fundamental resonance of the in-phase mode was excited in the junctions. Finally, the relationship between Pmax and N2 is shown in Fig. 6. Pmax was proportional to N2. Ozyuzer et al. found experimentally that the radiation power produced by Bi-2212 IJJs was proportional to the square of the number of junctions activating the radiation [7]. This was interpreted as coherent radiation, as from a laser. However, there was a discrepancy between their experiment and the simulation in this study: in the experiment, the number of
Fig. 6. Relationship between the maximum radiation power and the number of junctions, obtained from the simulation.
active junction varied while N did not, while in the latter case N was variable. The relationship shown in Fig. 6 can be explained as follows: an impedance mismatch between the junctions and free space decreases with increasing N, since the highest mode velocity c1 (Eq. (2)) increases to approach the speed of light in the medium with increasing N; the power P0 for an individual junction is proportional to N because of the mismatch reduction. Therefore, the total power Pmax = NP0 is proportional to N2. 4. Conclusions To observe THz-wave radiation from Bi-2212 IJJs, we investigated IVCs as follows: from experimental measurements of fabricated Bi-2212 IJJs, the Josephson frequency corresponding to the highest jump voltage was 0.65 THz. This agrees with the frequency of the fundamental cavity resonance with transverse mode. Numerical simulations demonstrated that the electric field distribution at the highest jump voltage showed resonance in the junctions. Moreover, the maximum radiation power was obtained close
Fig. 5. Electric field distributions at (a) I = 1.05 Ic and (b) I = 0.60 Ic, obtained from the simulation.
T. Tachiki et al. / Physica C 469 (2009) 1604–1607
to the highest voltage and was proportional to N2 because of reduction of the impedance mismatch with increasing N. References [1] R. Kleiner, F. Steinmeyer, G. Kunkel, P. Müller, Phys. Rev. Lett. 68 (1992) 2394. [2] G. Oya, N. Aoyama, A. Irie, S. Kishida, H. Tokutaka, Jpn. J. Appl. Phys. 31 (1992) L829. [3] K. Lee, W. Wang, I. Iguchi, M. Tachiki, K. Hirata, T. Mochiku, Phys. Rev. B 61 (2000) 3616. [4] I.E. Batov, X.Y. Jin, S.V. Shitov, Y. Koval, P. Müller, A.V. Ustinov, Appl. Phys. Lett. 88 (2006) 262504.
1607
[5] K. Kadowaki, I. Kakeya, T. Yamamoto, T. Yamazaki, M. Kohri, Y. Kubo, Physica C 437–438 (2006) 111. [6] M.-H. Bae, H.-J. Lee, J.-H. Choi, Phys. Rev. Lett. 98 (2007) 027002. [7] L. Ozyuzer, A.E. Koshelev, C. Kurter, N. Gopalsami, Q. Li, M. Tachiki, K. Kadowaki, T. Yamamoto, H. Minami, H. Yamaguchi, T. Tachiki, K.E. Gray, W.–K. Kwok, U. Welp, Science 308 (2007) 1291. [8] M.B. Gaifullin, M. Matsuda, N. Chikumoto, J. Shimoyama, K. Kishio, Phys. Rev. Lett. 84 (2000) 2945. [9] S. Sakai, P. Bodin, N.F. Pedersen, J. Appl. Phys. 73 (1993) 2411. [10] M. Tachiki, M. Iizuka, S. Tejima, H. Nakajima, Phys. Rev. B 71 (2005) 134515. [11] R. Kleiner, Phys. Rev. B 50 (1994) 6916. [12] S. Sakai, A.V. Ustinov, H. Kohlstedt, A. Petraglia, N.F. Pedersen, Phys. Rev. B 50 (1994) 12905.