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The superconducting state in square mesoscopic samples with two and four antidots
Physica C 404 (2004) 56–60 www.elsevier.com/locate/physc
The superconducting state in square mesoscopic samples with two and four antidots G.R. Berdiyorov *, B.J. Baelus, M.V. Milosevic, F.M. Peeters Departement Natuurkunde, Condensed Matter Theory Group, Universiteit Antwerpen (Campus Drie Eiken), Universiteitsplein 1, B-2610 Antwerpen, Belgium
Abstract The superconducting state of mesoscopic square samples with two and four antidots is studied using the nonlinear Ginzburg–Landau (GL) theory. We found a qualitative difference in the nucleation of the superconducting state in samples with different number of antidots, which is related to the interplay of the different types of symmetry. The H –T phase diagram of these structures reveals an oscillatory behavior caused by the formation of different stable vortex configurations in these small clusters of pinning centers. A comparison with the experiment of H –T phase diagram is given. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Mesoscopic superconductor; Antidot; Vortex
1. Introduction Recent technical and theoretical advances have led to a revival of the interest in the magnetic properties of superconducting networks and artificially structured superconducting films [1–3], which have been proposed as potential new components for low temperature electronics. The advantage of such a system is that the different vortex states can be studied on a macroscopic level and they can even be visualized [4]. For mesoscopic samples the nucleation of the superconducting state depends strongly on the boundary conditions imposed by the sample shape [5], i.e. on the topology of the system. *
Corresponding author. Tel.: +32-3-820-2865; fax: +32-3820-2245. E-mail address: [email protected] (G.R. Berdiyorov).
In this work we investigated the superconducting state of thin (d n) square samples containing two and four antidots (Fig. 1) in the presence of a uniform perpendicular magnetic field H0 . Following the numerical approach of Schweigert and Peeters [6] we solved the system of GL equations for the given structures. The details of our approach can be found in Ref. [7]. In order to see how the vortex configuration and the critical parameters are influenced by the sample geometry we compare our results to the one without holes (WH) [7]. The theoretically calculated H –T phase diagram will be compared with the experimental results [2].
2. Free energy and magnetization Figs. 1(a) and (b) show the free energy for a square mesoscopic superconductor with two and
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.09.111
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
16 18
14
-0.2 7
8
9 -6
F/F0
F/F0(10-2)
6
-0.4 4 5
-0.6
3
12
-8 11
-10 -12
-0.8
L=10
2
-1.0 L=0 0.0
1.8
1
W0
-0.2
(b) 16 14 15 13
Wi 11
-0.6 8
-0.8
6
0.5
4
7
9
0
F/F0 (10 )
10
0.0
19
12
-0.4
-2
F/F0
W
1 2 -1.0 L=0
2.0 2.2 H0/Hc2
-2 -3 L=17
-4
5
2.8
3
1.0
1.5 2.0 H0/Hc2
19
18
-1
2.5
3.0 3.2 H0/Hc2
3.0
3.5
Fig. 1. The free energy as a function of the applied magnetic field for two- (a) and four-antidot (b) samples. The open circles indicate continuous transitions between different vortex states and open squares show the S/N transition field. The insets show the geometry of the samples and the free energy for higher vorticity. W ¼ 7:0n is the size of the sample, Wi ¼ 2:0n is the size of the antidot and W0 ¼ 1:0n is the distance between antidots, j ¼ 0:28 and the sample thickness equals d ¼ 0:1n.
four antidots, respectively, as a function of the applied magnetic field. The insets show the geometry of the investigated samples and an enlargement of the free energy of the states with large vorticity (L). In the two-antidot sample (Fig. 1(a)) vortex states up to L ¼ 18 with a superconducting/normal (S/N) transition field of Hc3 =Hc2 ¼ 3:22 can nucleate, while for the WH case Lmax ¼ 11 and the S/N transition occurs at H0 =Hc2 ¼ 2:01 (see Ref. [7]). Contrary to the case of the WH, where only DL ¼ 1 transitions are present, in this case we found with increasing magnetic field DL ¼ 2 transitions such as L ¼ 2 ! L ¼ 4,
L ¼ 6 ! L ¼ 8, L ¼ 12 ! L ¼ 14, L ¼ 14 ! L ¼ 16, and L ¼ 16 ! L ¼ 18. Notice that the last three transitions are continuous. The energy of all superconducting states is lower and the ground state transitions occur at lower magnetic fields as compared to the WH case. In the case of the fourantidot sample (Fig. 1(b)) each of the vortex states have a larger stability region, the energy of the different superconducting states are lower as compared to the two-antidot sample, the transitions between different L states occur at lower magnetic fields and all thermodynamic equilibrium transitions are discontinuous with DL ¼ 1. Vortex states with vorticity up to L ¼ 19 can be nucleated and the S/N transition field is Hc3 =Hc2 ¼ 3:32. Figs. 2(a) and (b) show the magnetization of the two- and four-antidot samples, respectively, when the magnetic field is averaged over W W region, as a function of the applied magnetic field. For the WH sample [7] the maximum in the magnetization curve decreases with increasing L. For the twoantidot sample the largest flux expulsion is reached for L ¼ 2 and for the four-antidot sample for the states with L ¼ 0 and L ¼ 4. There is also a clear bunching of the magnetization curves of the samples with antidots, which is absent in the WH case. The number of curves, which are bunched together increases with the number of antidots. The Meissner state, i.e. L ¼ 0 state becomes less stable when antidots are present. For the full square superconductor a paramagnetic response [8] (i.e. M < 0) can be realized in a small magnetic field region of the L ¼ 1, 4, 5 states. It should be
2 8 4 L=0 3 L=0 1 56 8 12 4 6 910 7 3 5 67 11 1213 8 4 14 910 16 1112 2 17 18 14 16 18 0 15 19 -2 -4 (a) (b) -6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 H0/Hc2 H0/Hc2
-M/Hc2 (10-3)
(a)
0.0
57
Fig. 2. The magnetization as a function of the applied magnetic field for the two- (a) and four-antidot (b) samples. The vertical lines show the ground state transitions between different vortex states.
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
stressed that the superconductor is in a metastable state when such a paramagnetic response is found. For the samples with antidots many vortex states exhibit paramagnetic response and also there are ground states transitions to states with M < 0.
4 (a)
(b)
(c)
(d)
(e)
(f)
2 y/ξ
58
0 -2 -4 2
y/ξ
3. Stability of vortex states Fig. 3 shows the magnetic field range DHs (triangles) over which the vortex state with vorticity L is stable and the magnetic field region DHg (circles) over which each of the vortex states are the ground state as a function of the vorticity L for the two(solid curve) and four-antidot (dashed curve) samples. For the WH sample DHs decreases with increasing L, except for the L ¼ 4 state, which exhibits an enhanced stability [7]. Contrary to these results, for the superconductors with two and four antidots, the vortex states with even vorticity are more stable than the ones with odd vorticity. The ground state region DHg also shows similar features as the stability region DHs . The difference is that DHs exhibits an overall decrease with increasing L in the larger magnetic field region which is not present in DHg . In all cases, the vortex states show enhanced stability for commensurate vorticity, namely when the number of vortices is a multiple of the number of antidots. However, for
0 -2 -4 -4
-2
0 x/ξ
2
-4
-2
0 x/ξ
2
-4
-2
0 x/ξ
2
4
Fig. 4. The phase of the order parameter for the two-antidot sample for the states with L ¼ 5 (a) and 6 (b), and the magnetic field distribution for the four-antidot sample for the states L ¼ 1 (c), 2 (d), 3 (e) and 4 (f). Phases near zero are given by light gray regions and phases near 2p by dark gray regions. High magnetic field is given by dark gray regions.
higher magnetic field these commensurability effects disappear or are less pronounced. The latter can be explained as follows: for the two-antidot sample after the L ¼ 5 state, vortices are mainly located along the diagonal (Figs. 4(a,b)), forming a normal pass. But for the four-antidot sample vortices up to the L ¼ 16 state are located in the antidots (Figs. 4(c–f)).
4. S/N phase diagram 2 antidots 4 antidots , ∆Hs , ∆Hg
1.0
∆H/Hc2
0.8 0.6 0.4 0.2 0.0
0
2
4
6
8
10 12 14 16 18 20 L
Fig. 3. The stability region DHs (triangles) and ground state region DHg (circles) as a function of the vorticity L, for the two(solid curve) and four-antidot (dashed curve) samples.
Next, we investigate the influence of temperature on the superconducting state in the square sample with four antidots. The distances are now expressed in units of nð0Þ, magnetic field in Hc2 ð0Þ and temperature will be rescaled by the critical temperature Tc0 at zero magnetic field. In order to compare our results with the experimental ones, we used the parameters from Ref. [2]. They found the coherence length nð0Þ ¼ 92 nm, and the penetration depth kð0Þ ¼ 140 nm for a full square superconductor as well as for the microsquares with antidots. For the value of the coherence length nð0Þ ¼ 92 nm our theoretical results show a good qualitative agreement. But the theoretical predicted S/N transition at a fixed L
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
occurs at higher temperatures than observed experimentally. Also the transitions between the successive L states appear at slightly larger fields in our calculations. This quantitative disagreement may be due to the uncertainties in the dimensions of the sample (holes), or the assumed value of the coherence length at zero temperature. To explore the latter possibility, we repeated the calculation and varied nð0Þ keeping all other parameters fixed. The H –T phase diagram for the four-antidot superconductor is shown in Fig. 5 for nð0Þ ¼ 120 nm (dashed curve). The H –T phase diagram shows clear oscillations in the S/N state boundary. Moreover, the period of the oscillation and the peak amplitude for the state with vorticity L ¼ 4 is larger than for the other states, which is due to a commensurability effect. For this value of the coherence length the correspondence is obviously much better. Still, a small difference in the transition fields exists. This can be explained by the different criteria for the determination of the S/N transition. Namely, in the experiment, one assumes that superconductivity is destroyed when the region between the contacts becomes normal. In our model, for the same magnetic field, superconducting regions would still be present in the corners of the sample.
5. Conclusion We investigated theoretically the influence of the topology of mesoscopic superconducting samples on the vortex configuration and critical parameters. The insertion of antidots in the sample increases the critical field and decreases the free energy for a fixed L and ground state transitions occur at lower magnetic fields. The Meissner state becomes less stable for samples with antidots. For superconductors with antidots, the states with even vorticity are more stable than the ones with odd vorticity. For all the considered structures the vortex states show enhanced stability for commensurate values of the vorticity. We found that the magnetization is strongly influenced by the presence of the antidots. More vortex states exhibit paramagnetic response and ground state transitions to states with M < 0 are also possible. The calculated H –T phase diagram shows clear oscillations in the S/N boundary. Contrary to the WH case in the four-antidot sample the period of the oscillations and the peak amplitude is not the same for all vortex states, which was explained by the stability of the different vortex states. The theoretically calculated H –T diagram shows a good agreement with the experimental results.
Acknowledgements
5 experiment GL theory, (0)=120 nm
6 4
µ0 H (mT)
59
5 4
3
2
3 2
This work was supported by the Flemish Science Foundation (FWO-Vl), the Belgian Inter-University Attraction Poles (IUAP), the ‘‘Onderzoeksraad van de Universiteit Antwerpen’’ (GOA), and the ESF programme on ‘‘Vortex matter’’. One of us (BJB) is a post-doctoral researcher with the FWO-Vl.
1
1
L=0
References 0
0.95
0.96
0.97 0.98 T/Tc(0)
0.99
1.00
Fig. 5. The H –T phase diagram for the four-antidot sample for nð0Þ ¼ 120 nm (dashed curve). Parameters of the superconductor are (see Fig. 1) W ¼ 17:0nð0Þ, Wi ¼ 3:83nð0Þ, W0 ¼ 3:83nð0Þ, j ¼ 1:17, d ¼ 0:2nð0Þ. The solid curve is the experimental measured result.
[1] T. Puig, E. Rosseel, L. Van Look, M.J. Van Bael, V.V. Moshchalkov, Y. Bruynseraede, R. Jonckheere, Phys. Rev. B 58 (1998) 5744. [2] V. Bruyndoncx, J.G. Rodrigo, T. Puig, L. Van Look, V.V. Moshchalkov, R. Jonckheere, Phys. Rev. B 60 (1999) 4285. [3] L. Van Look, B.Y. Zhu, R. Jonckheere, B.R. Zhao, Z.X. Zhao, V.V. Moshchalkov, Phys. Rev. B 66 (2002) 214511.
60
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
[4] K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, V.V. Moshchalkov, Science 274 (1996) 1167. [5] V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature 373 (1995) 319.
[6] V.A. Schweigert, F.M. Peeters, Phys. Rev. B 57 (1998) 13817. [7] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515. [8] A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peeters, V.A. Schweigert, Nature (London) 407 (2000) 55.
The superconducting state in square mesoscopic samples with two and four antidots G.R. Berdiyorov *, B.J. Baelus, M.V. Milosevic, F.M. Peeters Departement Natuurkunde, Condensed Matter Theory Group, Universiteit Antwerpen (Campus Drie Eiken), Universiteitsplein 1, B-2610 Antwerpen, Belgium
Abstract The superconducting state of mesoscopic square samples with two and four antidots is studied using the nonlinear Ginzburg–Landau (GL) theory. We found a qualitative difference in the nucleation of the superconducting state in samples with different number of antidots, which is related to the interplay of the different types of symmetry. The H –T phase diagram of these structures reveals an oscillatory behavior caused by the formation of different stable vortex configurations in these small clusters of pinning centers. A comparison with the experiment of H –T phase diagram is given. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Mesoscopic superconductor; Antidot; Vortex
1. Introduction Recent technical and theoretical advances have led to a revival of the interest in the magnetic properties of superconducting networks and artificially structured superconducting films [1–3], which have been proposed as potential new components for low temperature electronics. The advantage of such a system is that the different vortex states can be studied on a macroscopic level and they can even be visualized [4]. For mesoscopic samples the nucleation of the superconducting state depends strongly on the boundary conditions imposed by the sample shape [5], i.e. on the topology of the system. *
Corresponding author. Tel.: +32-3-820-2865; fax: +32-3820-2245. E-mail address: [email protected] (G.R. Berdiyorov).
In this work we investigated the superconducting state of thin (d n) square samples containing two and four antidots (Fig. 1) in the presence of a uniform perpendicular magnetic field H0 . Following the numerical approach of Schweigert and Peeters [6] we solved the system of GL equations for the given structures. The details of our approach can be found in Ref. [7]. In order to see how the vortex configuration and the critical parameters are influenced by the sample geometry we compare our results to the one without holes (WH) [7]. The theoretically calculated H –T phase diagram will be compared with the experimental results [2].
2. Free energy and magnetization Figs. 1(a) and (b) show the free energy for a square mesoscopic superconductor with two and
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.09.111
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
16 18
14
-0.2 7
8
9 -6
F/F0
F/F0(10-2)
6
-0.4 4 5
-0.6
3
12
-8 11
-10 -12
-0.8
L=10
2
-1.0 L=0 0.0
1.8
1
W0
-0.2
(b) 16 14 15 13
Wi 11
-0.6 8
-0.8
6
0.5
4
7
9
0
F/F0 (10 )
10
0.0
19
12
-0.4
-2
F/F0
W
1 2 -1.0 L=0
2.0 2.2 H0/Hc2
-2 -3 L=17
-4
5
2.8
3
1.0
1.5 2.0 H0/Hc2
19
18
-1
2.5
3.0 3.2 H0/Hc2
3.0
3.5
Fig. 1. The free energy as a function of the applied magnetic field for two- (a) and four-antidot (b) samples. The open circles indicate continuous transitions between different vortex states and open squares show the S/N transition field. The insets show the geometry of the samples and the free energy for higher vorticity. W ¼ 7:0n is the size of the sample, Wi ¼ 2:0n is the size of the antidot and W0 ¼ 1:0n is the distance between antidots, j ¼ 0:28 and the sample thickness equals d ¼ 0:1n.
four antidots, respectively, as a function of the applied magnetic field. The insets show the geometry of the investigated samples and an enlargement of the free energy of the states with large vorticity (L). In the two-antidot sample (Fig. 1(a)) vortex states up to L ¼ 18 with a superconducting/normal (S/N) transition field of Hc3 =Hc2 ¼ 3:22 can nucleate, while for the WH case Lmax ¼ 11 and the S/N transition occurs at H0 =Hc2 ¼ 2:01 (see Ref. [7]). Contrary to the case of the WH, where only DL ¼ 1 transitions are present, in this case we found with increasing magnetic field DL ¼ 2 transitions such as L ¼ 2 ! L ¼ 4,
L ¼ 6 ! L ¼ 8, L ¼ 12 ! L ¼ 14, L ¼ 14 ! L ¼ 16, and L ¼ 16 ! L ¼ 18. Notice that the last three transitions are continuous. The energy of all superconducting states is lower and the ground state transitions occur at lower magnetic fields as compared to the WH case. In the case of the fourantidot sample (Fig. 1(b)) each of the vortex states have a larger stability region, the energy of the different superconducting states are lower as compared to the two-antidot sample, the transitions between different L states occur at lower magnetic fields and all thermodynamic equilibrium transitions are discontinuous with DL ¼ 1. Vortex states with vorticity up to L ¼ 19 can be nucleated and the S/N transition field is Hc3 =Hc2 ¼ 3:32. Figs. 2(a) and (b) show the magnetization of the two- and four-antidot samples, respectively, when the magnetic field is averaged over W W region, as a function of the applied magnetic field. For the WH sample [7] the maximum in the magnetization curve decreases with increasing L. For the twoantidot sample the largest flux expulsion is reached for L ¼ 2 and for the four-antidot sample for the states with L ¼ 0 and L ¼ 4. There is also a clear bunching of the magnetization curves of the samples with antidots, which is absent in the WH case. The number of curves, which are bunched together increases with the number of antidots. The Meissner state, i.e. L ¼ 0 state becomes less stable when antidots are present. For the full square superconductor a paramagnetic response [8] (i.e. M < 0) can be realized in a small magnetic field region of the L ¼ 1, 4, 5 states. It should be
2 8 4 L=0 3 L=0 1 56 8 12 4 6 910 7 3 5 67 11 1213 8 4 14 910 16 1112 2 17 18 14 16 18 0 15 19 -2 -4 (a) (b) -6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 H0/Hc2 H0/Hc2
-M/Hc2 (10-3)
(a)
0.0
57
Fig. 2. The magnetization as a function of the applied magnetic field for the two- (a) and four-antidot (b) samples. The vertical lines show the ground state transitions between different vortex states.
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
stressed that the superconductor is in a metastable state when such a paramagnetic response is found. For the samples with antidots many vortex states exhibit paramagnetic response and also there are ground states transitions to states with M < 0.
4 (a)
(b)
(c)
(d)
(e)
(f)
2 y/ξ
58
0 -2 -4 2
y/ξ
3. Stability of vortex states Fig. 3 shows the magnetic field range DHs (triangles) over which the vortex state with vorticity L is stable and the magnetic field region DHg (circles) over which each of the vortex states are the ground state as a function of the vorticity L for the two(solid curve) and four-antidot (dashed curve) samples. For the WH sample DHs decreases with increasing L, except for the L ¼ 4 state, which exhibits an enhanced stability [7]. Contrary to these results, for the superconductors with two and four antidots, the vortex states with even vorticity are more stable than the ones with odd vorticity. The ground state region DHg also shows similar features as the stability region DHs . The difference is that DHs exhibits an overall decrease with increasing L in the larger magnetic field region which is not present in DHg . In all cases, the vortex states show enhanced stability for commensurate vorticity, namely when the number of vortices is a multiple of the number of antidots. However, for
0 -2 -4 -4
-2
0 x/ξ
2
-4
-2
0 x/ξ
2
-4
-2
0 x/ξ
2
4
Fig. 4. The phase of the order parameter for the two-antidot sample for the states with L ¼ 5 (a) and 6 (b), and the magnetic field distribution for the four-antidot sample for the states L ¼ 1 (c), 2 (d), 3 (e) and 4 (f). Phases near zero are given by light gray regions and phases near 2p by dark gray regions. High magnetic field is given by dark gray regions.
higher magnetic field these commensurability effects disappear or are less pronounced. The latter can be explained as follows: for the two-antidot sample after the L ¼ 5 state, vortices are mainly located along the diagonal (Figs. 4(a,b)), forming a normal pass. But for the four-antidot sample vortices up to the L ¼ 16 state are located in the antidots (Figs. 4(c–f)).
4. S/N phase diagram 2 antidots 4 antidots , ∆Hs , ∆Hg
1.0
∆H/Hc2
0.8 0.6 0.4 0.2 0.0
0
2
4
6
8
10 12 14 16 18 20 L
Fig. 3. The stability region DHs (triangles) and ground state region DHg (circles) as a function of the vorticity L, for the two(solid curve) and four-antidot (dashed curve) samples.
Next, we investigate the influence of temperature on the superconducting state in the square sample with four antidots. The distances are now expressed in units of nð0Þ, magnetic field in Hc2 ð0Þ and temperature will be rescaled by the critical temperature Tc0 at zero magnetic field. In order to compare our results with the experimental ones, we used the parameters from Ref. [2]. They found the coherence length nð0Þ ¼ 92 nm, and the penetration depth kð0Þ ¼ 140 nm for a full square superconductor as well as for the microsquares with antidots. For the value of the coherence length nð0Þ ¼ 92 nm our theoretical results show a good qualitative agreement. But the theoretical predicted S/N transition at a fixed L
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
occurs at higher temperatures than observed experimentally. Also the transitions between the successive L states appear at slightly larger fields in our calculations. This quantitative disagreement may be due to the uncertainties in the dimensions of the sample (holes), or the assumed value of the coherence length at zero temperature. To explore the latter possibility, we repeated the calculation and varied nð0Þ keeping all other parameters fixed. The H –T phase diagram for the four-antidot superconductor is shown in Fig. 5 for nð0Þ ¼ 120 nm (dashed curve). The H –T phase diagram shows clear oscillations in the S/N state boundary. Moreover, the period of the oscillation and the peak amplitude for the state with vorticity L ¼ 4 is larger than for the other states, which is due to a commensurability effect. For this value of the coherence length the correspondence is obviously much better. Still, a small difference in the transition fields exists. This can be explained by the different criteria for the determination of the S/N transition. Namely, in the experiment, one assumes that superconductivity is destroyed when the region between the contacts becomes normal. In our model, for the same magnetic field, superconducting regions would still be present in the corners of the sample.
5. Conclusion We investigated theoretically the influence of the topology of mesoscopic superconducting samples on the vortex configuration and critical parameters. The insertion of antidots in the sample increases the critical field and decreases the free energy for a fixed L and ground state transitions occur at lower magnetic fields. The Meissner state becomes less stable for samples with antidots. For superconductors with antidots, the states with even vorticity are more stable than the ones with odd vorticity. For all the considered structures the vortex states show enhanced stability for commensurate values of the vorticity. We found that the magnetization is strongly influenced by the presence of the antidots. More vortex states exhibit paramagnetic response and ground state transitions to states with M < 0 are also possible. The calculated H –T phase diagram shows clear oscillations in the S/N boundary. Contrary to the WH case in the four-antidot sample the period of the oscillations and the peak amplitude is not the same for all vortex states, which was explained by the stability of the different vortex states. The theoretically calculated H –T diagram shows a good agreement with the experimental results.
Acknowledgements
5 experiment GL theory, (0)=120 nm
6 4
µ0 H (mT)
59
5 4
3
2
3 2
This work was supported by the Flemish Science Foundation (FWO-Vl), the Belgian Inter-University Attraction Poles (IUAP), the ‘‘Onderzoeksraad van de Universiteit Antwerpen’’ (GOA), and the ESF programme on ‘‘Vortex matter’’. One of us (BJB) is a post-doctoral researcher with the FWO-Vl.
1
1
L=0
References 0
0.95
0.96
0.97 0.98 T/Tc(0)
0.99
1.00
Fig. 5. The H –T phase diagram for the four-antidot sample for nð0Þ ¼ 120 nm (dashed curve). Parameters of the superconductor are (see Fig. 1) W ¼ 17:0nð0Þ, Wi ¼ 3:83nð0Þ, W0 ¼ 3:83nð0Þ, j ¼ 1:17, d ¼ 0:2nð0Þ. The solid curve is the experimental measured result.
[1] T. Puig, E. Rosseel, L. Van Look, M.J. Van Bael, V.V. Moshchalkov, Y. Bruynseraede, R. Jonckheere, Phys. Rev. B 58 (1998) 5744. [2] V. Bruyndoncx, J.G. Rodrigo, T. Puig, L. Van Look, V.V. Moshchalkov, R. Jonckheere, Phys. Rev. B 60 (1999) 4285. [3] L. Van Look, B.Y. Zhu, R. Jonckheere, B.R. Zhao, Z.X. Zhao, V.V. Moshchalkov, Phys. Rev. B 66 (2002) 214511.
60
G.R. Berdiyorov et al. / Physica C 404 (2004) 56–60
[4] K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, V.V. Moshchalkov, Science 274 (1996) 1167. [5] V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature 373 (1995) 319.
[6] V.A. Schweigert, F.M. Peeters, Phys. Rev. B 57 (1998) 13817. [7] B.J. Baelus, F.M. Peeters, Phys. Rev. B 65 (2002) 104515. [8] A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peeters, V.A. Schweigert, Nature (London) 407 (2000) 55.