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PHYSICAL REVIEW B 73, 092512 ?2006?

Superconducting properties of thin mesoscopic rings with enhanced surface superconductivity

Guo-Qiao Zha, Shi-Ping Zhou, and Bao-He Zhu

Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, People’s Republic of China ?Received 7 January 2006; published 24 March 2006? The superconducting state of a thin mesoscopic superconducting ring surrounded by a medium which enhanced its superconductivity near the boundary is investigated by the phenomenological Ginzburg-Landau theory. The free energy, the Cooper-pair density, and the current density as well as the H – T phase diagram are investigated for a ring with different surface enhancement or for different rings with the same surface enhancement. It is also found that the stable multivortex state can occur in the small ring that we studied if the enhanced surface superconductivity is stronger, and the stable ?1 : L2? and ?2 : L2? states can exist as the ground states with increasing the inner radius. DOI: 10.1103/PhysRevB.73.092512 PACS number?s?: 74.20.De, 74.25.Op

A mesoscopic sample is such that its size is comparable to the magnetic ?eld penetration depth ? or the coherence length ?. The vortex properties of a mesoscopic superconductor are strongly in?uenced by the boundary condition besides its size and geometry. For a superconductor in contact with a medium with surface enhancement or suppression of superconductivity we have the general boundary condition1

? ?A ? ? 2? = ? ? ? ? ? ? 2 , ?? i? ? ?? ? ?A ? =? ? 2? j,

?2? ?3?

where the density of the superconducting current ? j given by 1 ? ? ? ?? ? ? *? ? ? ? ? 2A ?. ? j = ? ? *? 2i ?4?

? ?A ? ???? = i ??? , ? · ?? i? n s s b

?1?

? is ? is the unit vector normal to the sample surface, A where n the vector potential, ? is the order parameter, and b is the surface extrapolation length which is the effective penetration depth of the order parameter into the surrounding medium. For both the superconductor-vacuum and the superconductor-insulator boundary one has b → ?. The cases b ? 0 and b ? 0 correspond to surface suppression and enhancement of superconductivity, respectively. The superconducting states of thin mesoscopic disks and cylinders allowing for the enhanced surface superconductivity were studied in Refs. 2 and 3. The authors found that increasing the superconductivity near the surface leads to higher critical ?elds and critical temperatures. Moveover, the multivortex state can be stabilized by the surface enhancement of superconductivity and can be found as the ground state. In the present paper we investigate the effect of the enhancement of surface superconductivity on the critical ?eld and the critical temperature as well as the vortex state for thin mesoscopic superconducting rings with different inner radii Ri. We generalize the method of Ref. 2 for disks to the ring con?gurations in Ref. 4 and consider small mesoscopic rings with ?xed outer radius Ro = 2.0? and thickness d = 0.1? surrounded by a medium which enhances superconductivity at the sample surface. The ring has more than one boundary in comparison with a disk sample and more complex and interesting features are expected. Due to the thick? is uniform and ness d ? ? , ?, the external magnetic ?eld H directed normal to the rings plane. The Cooper pair density ???2 is determined from a solution of coupled nonlinear ???r ?? GL equations for the superconducting order parameter ??r ? ? ? ?? = ? ? A?r ?? and the magnetic ?eld h?r

1098-0121/2006/73?9?/092512?4?/$23.00

? = ?? , ? , z? and choose the We use the cylindrical coordinates r ? = ?H? / 2?e ? , where ? is the radial distance from the gauge A ? cylinder axis z, and ? is the azimuthal angle. The ring lies between z = d / 2 and z = ?d / 2. We measure the distance in units of the coherence length ?, and the magnetic ?eld in Hc2 = c? / 2e?2 = ??2Hc, and the superconducting current in j0 = cHc / 2??, where Hc is the thermodynamical critical ?eld and ? = ? / ? is the GL parameter. The free energy of the superconducting state, measured in F0 = H2 c V / 8? units, is expressed as

F= 2 V

?? ?

1 ?? ? A ? ??2 dV ? ???2 + ???4 + ?? i? 2

? ?r ? ?2 + 1 ?? ? H + ? 2? h b

? ? ?

dS???2 .

?5?

The last term in Eq. ?5? is the surface contribution. One can see that in the b ? 0 case this term reduces the free energy, implying the superconductivity enhanced effect. First, we investigate the giant vortex state whose vortexstructure possesses cylindrical symmetry. The ground-state free energy F of the giant vortex states with various angular quantum number L are shown in Fig. 1. For a ring with Ri = 0.6?, we can observe clearly that the superconductivity is enhanced ?i.e., a more negative free energy at H = 0? with increasing ?? / b?, and the superconducting/normal transition ?elds ?where the free energy equals zero? become higher and more L states are possible before superconductivity disappears. For a ring with larger inner radius but ?xed b value, the superconducting/normal transition ?elds also become higher and more L states appear. That is because the inside of the ring is also in?uenced by the surface enhancement effect

?2006 The American Physical Society

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PHYSICAL REVIEW B 73, 092512 ?2006?

FIG. 1. The ground-state free energy as a function of the applied magnetic ?eld H for superconducting rings with different ? / b or different Ri.

of superconductivity due to its small size. We then investigate how the Cooper-pair density and the current density vary with magnetic ?eld for different ring sizes under a ?xed surface enhancement of superconductivity. Figure 2 shows the magnetic-?eld dependence of the free energy ??a? and ?b??, the radial dependence of the Cooperpair density ???2 ??c? and ?e?? and the current density j ??d? and ?f?? at different magnetic ?elds for two rings with Ri = 0.3? and 0.6?, respectively. At low magnetic ?elds, the two samples are in the Meissner state. As long as the system is in the L = 0 state, there is no ?ux trapped in the hole and the superconductor induces a supercurrent only to expel the magnetic ?eld at the outside of the ring. From curves 1 and 2 in Figs. 2?d? and 2?f? we can get that the current ?ows in the

same direction throughout whole superconducting material, and the current densities both become more negative near the inner boundary and the outer boundary for the two rings with increasing the magnetic ?eld. So this will lead to a stronger depression of the Cooper-pair densities near the inner boundary and the outer boundary. But a comparison of curve 1 with curve 2 in Figs. 2?c? and 2?e? shows that the Cooper-pair density near the inner boundary increases for the ring with smaller inner radius. This can be explained as follows. Because of the small size of the hole, only the region near the outer boundary is in?uenced for low external magnetic ?eld and the Cooper-pair density near the outer boundary decreases. But because of the surface enhanced superconductivity near the inner boundary and the considerable suppression of the magnetic ?eld in the hole, the Cooper-pair density near the inner boundary becomes larger. Notice that further increasing the ?eld the Cooper-pair density near the inner boundary will also decrease for the ring in the Meissner state. When the ground state changes from the L = 0 to the L = 1 state, more ?ux is trapped in the hole and the local magnetic ?eld indicates a sharp peak at the inner boundary, which is larger than the external ?eld.5 Then the Cooper-pair density near the inner boundary has a minimum, whereas it has a maximum near the outer boundary. On this account, more current is needed to compensate the magnetic ?eld near the inner boundary than near the outer boundary and the sign of the current near the inner boundary becomes positive ?compare curve 2 with curve 3 from Figs. 2?c?–2?e??. Further increasing the external ?eld, we ?nd that the Cooper-pair densities near the inner boundary both increase for the two rings ?curves 3 and 4 in Figs. 2?c? and 2?e??. This case can also be found for a ring immersed in an insulating medium in Ref. 5, but the magnetic ?eld range within which the Cooperpair density near the inner boundary increases is very small. When the ground state changes from the L = 1 to the L = 2 state, extra ?ux is trapped in the hole, and the changes in the Cooper-pair density and the current density are analogous to that of the ground state from the L = 0 to the L = 1 state ?see curves 4 and 5?. Now we investigate the in?uences of the temperature on superconducting state. The temperature is included in ?, Hc2. Their temperature dependences are as follows:

??T? =

??1 ? T/Tc0? ,

??0?

Hc2?T? = Hc2?0??1 ? T/Tc0? ,

?6?

FIG. 2. The free energy as a function of the applied magnetic ?eld H for superconducting rings with Ri = 0.3? ?a? and Ri = 0.6? ?b? for the different L states ?dashed curves? and for the ground state ?solid curves?. The solid circles denote the free energies at different magnetic ?elds. The two samples have the same surface enhancement. The Cooper-pair density ?c? and the current density ?d? for the situations indicated by the solid circles in ?a? as a function of the radial position, and ?e? and ?f? for the situations in ?b?.

where Tc0 is the critical temperature at zero magnetic ?eld for the normal boundary condition, i.e., ???0? / b? = 0. We will use ??0? and Hc2?0? as the basis for our units. The H – T phase diagram for two superconducting rings with Ri = 0.3??0? and 0.6??0? for ? / b = 0 and ?0.02 is given in Fig. 3. The thick curves indicate the superconducting/normal transition and the corners indicate the transitions between the different L states. The thinner curves show the ground-state transitions between the giant vortex states with different vorticity L for the ring with Ri = 0.3??0?. It is clear that, with increasing ???0? / b?, the ground-state transition ?elds are almost the same for 0 ? 1 but increase for 1 ? 2 due to the

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FIG. 3. The H – T phase diagram for rings with Ri = 0.3??0? and 0.6??0? for ? / b = 0 and ?0.02. The thick curves indicate the superconducting/normal transition. The thinner curves indicate the ground-state transitions between the giant vortex states with different vorticity L for the ring with Ri / ??0? = 0.3 and different b values. The inset is an enlargement of the L = 0 state for the two rings with ? / b = ?0.02.

in?uence of the surface enhanced superconductivity on the con?ned ?ux. The critical temperature is very sensitive to the value of ??0? / b and the superconducting/normal transitions move to temperatures higher than Tc0. For rings with the same b value but different inner radii, we ?nd higher critical temperatures and higher critical ?elds with increasing the inner radius when L ? 0. While in the L = 0 state, the critical ?eld and the critical temperature decrease with increasing the inner radius for ??0? / b = 0 because the volume of the sample becomes smaller, and there is a crossover for ??0? / b = ?0.02 ?see the inset in Fig. 3? due to the stronger superconductivity with increasing the inner radius. In addition, we notice that the superconducting/normal transition shows an oscillatory behavior for the ring with larger inner radius like the Little-Parks oscillations which are a straightforward consequence of the ?uxoid quantization constraint.6 There are two kinds of ?meta-?stable vortex states in suf?ciently large disks: giant vortex states and multivortex states and the transition between such states is described by the saddle-point states which correspond to the energy barrier state between those states.2,7–10 In small rings immersed in an insulating medium, the con?nement effect dominates and only the giant vortex states are stable.5 Because of the enhanced surface superconductivity, the multivortices can become stable with increasing ?? / b?.2 Figure 4 shows the free energies of the ground states and the stable multivortex states for different superconducting rings with different b values. We only consider the case of L ? 7 and the multivortex states are denoted as ?L1 : L2?, where L1 and L2 are the angular momentum values of which the multivortex states are composed. The different giant vortex states are shown by the solid curves and the stable ?0 : L2?, ?1 : L2?, and ?2 : L2? multivortex states by dash-dotted, dashed, and dotted curves, respectively. The solid circles indicate the transitions from a multivortex state to a giant vortex state. For a ring with Ri = 0.3? and ? / b = ?0.1 ?Fig. 4?a??, we ?nd that the ?1 : L2? states can be found for 5 ? L2 ? 7 besides the ?0 : L2? states comparing with a disk.2 That is to

FIG. 4. The free energy of the ground state and the stable multivortex state as a function of the applied magnetic ?eld H for superconducting rings with Ro = 2.0?, d / ? = 0.1 and ?a? Ri = 0.3? and ? / b = ?0.1; ?b? Ri = 0.5? and ? / b = ?0.1; ?c? Ri = 0.3? and ? / b = ?0.2; and ?d? Ri = 0.5? and ? / b = ?0.2. The different giant vortex states are shown by the solid curves. The stable ?0 : L2?, ?1 : L2?, and ?2 : L2? multivortex states are shown by dash-dotted, dashed, and dotted curves, respectively. The solid circles indicate the transitions from a multivortex state to a giant vortex state.

say, for small Ri the con?nement effect is not dominant and the hole in the center of the ring can trap ?uxons and stabilize the multivortex states. Increasing Ri to 0.5? ?Fig. 4?b??, the ?2:7? state occurs but the ?0:4? state disappears. That is because a larger hole can trap more vortices. Notice that these stable multivortex states can only be found as the metastable states. For ? / b = ?0.2, we ?nd that, besides the ?0 : L2? states and the ?1 : L2? states, the ?2 : L2? states can exist for the ring with Ri = 0.3? and the ?1 : L2? states with L2 ? 5 can occur in very

FIG. 5. Contour plot of the Cooper-pair density for the ?1:5? multivortex state in a ring with Ro = 2.0? and Ri = 0.3? at magnetic ?eld H = 2.96Hc2 for ? / b = ?0.1 ?a? and ? / b = ?0.2 ?b? as well as at H = 3.26Hc2 for ? / b = ?0.1 ?c? and ? / b = ?0.2 ?d?. Light and dark regions correspond to low and high Copper-pair density, respectively.

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large magnetic ?eld ranges ?Fig. 4?c??. Moreover, the ?1 : L2? states can exist in the ground states, but the ?0 : L2? and the ?2 : L2? states are only found as the metastable. The reason is that increasing ?? / b? corresponds to an enhancement of superconductivity near the boundary and the magnetic ?eld is more dif?cult to penetrate. As an example, Fig. 5 shows the Cooper-pair density of the ?1:5? multivortex state ?see Figs. 4?a? and 4?c?? at magnetic ?eld H = 2.96Hc2 for ? / b = ?0.1 ?a? and ? / b = ?0.2 ?b? as well as at magnetic ?eld H = 3.26Hc2 for ? / b = ?0.1 ?c? and ? / b = ?0.2 ?d?. We notice that with increasing ?? / b? the vortices are pushed towards the outer boundary at the ?xed ?eld. Increasing the ?eld, the vortices move towards each other and merge together for ? / b = ?0.1 ?Fig. 5?c?? while clearly separated for ? / b = ?0.2 ?Fig. 5?d?? at H = 3.26Hc2. Thus a larger transition ?eld between the multivortex state and the giant vortex state is needed for the stronger surface enhancement. Increasing the inner radius to 0.5? ?Fig. 4?d??, we ?nd that the ?0 : L2? states are only found for a larger L2 but the ?1 : L2? state and ?2 : L2? state can occur for a smaller L2, comparing with Fig. 4?c?. Furthermore, notice that the ?2:7? state also can be found as the ground state

and its magnetic ?eld range becomes very large. In conclusion, we studied the superconducting state of thin mesoscopic rings with enhanced surface superconductivity by the phenomenological Ginzburg-Landau theory. We found that the superconductivity is enhanced with increasing ?? / b? and the inner radius. For a ring with small inner radius, the Cooper-pair density near the inner boundary increases with increasing the ?eld when the system is in the Meissner state. According to the H – T phase diagram, the surface enhancement of superconductivity can signi?cantly increase the critical temperature and the critical ?eld, and there is an oscillatory behavior like the Little-Parks effect for the ring with large inner radius. Finally, for the stable multivortex states we found that the ?1 : L2? states and the ?2 : L2? states can exist as the ground states with increasing the inner radius and their magnetic ?eld ranges become very large. This work was supported by the National Natural Science Foundation of China ?Grant No. 60371033? and by Shanghai leading academic discipline program, China.

1 P.

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6 W.

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