Quantization and confinement effects in superconducting nanostructures

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Superlattices and Microstructures, Vol. 19, No. 3, 1996

Quantization and confinement effects in superconducting nanostructures V. V. M, M. B, V. V. M, E. R, M. V B, K. T, Y. B Laboratorium voor Vaste-Stoffysika en Magnetisme, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium

R. J Interuniversity Micro-Electronics Center, Kapeldreef 75, B-3001 Leuven, Belgium (Received 21 August 1995) The confinement of the flux lines by a lattice of submicron holes (‘antidots’) has been studied in nanostructured superconducting Pb/Ge multilayers. By introducing regular arrays of sufficiently large antidots, multi-quanta vortex lattices have been stabilized. Sharp cusp-like magnetization (M) anomalies, appearing at matching fields H \m/ /S in superconducting films m 0 explained. These anomalies with the antidot lattices having a unit cell area S, are successfully are, analogues of the well-known M(H) cusp at H , but for the onset of multi-quanta c1 (m]1)/ -vortices penetration at each subsequent matching field H . It is shown that the 0 M(H) curve between the matching fields H \H\H follows a msimple M©ln(H[H ) m`1 an unusual expansion of validity m dependence. These experimental observationsmhave revealed of the London limit in superconductors with lattices of relatively large antidots. The successful high quality fit of the M(H, T) curves convincingly demonstrates that a new type of the critical state B\const (‘single-terrace critical state’) can be realized in superconductors with the antidot lattices. ( 1996 Academic Press Limited

1. Introduction In superconducting nanostructures the confinement geometry for the condensate plays a role similar to that of the confinement potential in the quantum-mechanical problem ‘particle in a box’. Consequently, the phase boundary ‘field (H)[temperature (T)’ (which is in fact, the lowest Landau level of a nanostructure calculated for its boundary conditions) can be tuned by choosing the proper sample topology, like eigenvalues of a quantum particle can be tuned by varying the confinement potential. In our previous publications [1–3] we have shown that Al nanostructures all made from the same material but having different topology (lines, loops, dots) indeed have very different H[T boundaries which are fully governed by the quantization and confinement effects. Here, we are converting the confinement potential using antidots instead of dots and also periodic arrays instead of the individual nanostructures. We have focused on flux line (FL) confinement by antidots. We report on successful stabilization of the multi-quanta vortex lattices in superconducting films with regular arrays of relatively large antidots. 0749–6036/96/030183]08 $18.00/0

( 1996 Academic Press Limited

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˚ )/Ge(50 A ˚ )] multilayer with a triangular lattice of submicron holes Fig. 1. An AFM picture of a [Pb(100 A 2 (period d\1 lm, radius r[0.22 lm).

2. Multi-quanta vortex lattices In homogeneous superconductors the formation of the FL lattices consisting of multi-quanta vortices is energetically unfavorable [4,5]. Contrary to this, relatively large artificial pinning centers can stabilize multi-quanta vortices. The maximum possible number of the FL trapped by the single insulating inclusion with the radius r is determined by the saturation number n Vr/(2n(T)) [6]. For regular s arrays of artificial pinning centers the vortex-vortex interactions make the situation more complicated, but still multi-quanta FL lattices can be stabilized, provided that the radius r is sufficiently large [7]. Following the pioneering work of Hebard et al. [8], we have shown recently [9,10] that the regular arrays of submicron holes (‘antidots’, this definition has been borrowed from the publications [11] on similar semiconducting structures) can be successfully used to stabilize multi-quanta vortices in superconducting films with the antidot lattices. However, the qualitative explanation of the magnetization anomalies observed at matching fields H\H has not yet been found. The matching fields m H \m/ /S are determined by the unit cell area of the antidot lattice, S, and integer m. In the present m 0 paper we demonstrate that puzzling sharp cusp-like M(H) anomalies at H in films with the antidot m lattices are in fact, analogous to the M(H) cusp for bulk superconductors at lower critical field H , c1

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but for the onset of the penetration of multi-quanta vortices into microholes: 2/ at H\H , 3/ at 0 1 0 H\H , etc. Exactly at the matching fields H , magnetization is given by the relation M(H )© 2 m m m/ /k2(T), where k(T) is the temperature-dependent penetration depth. Between the matching fields 0 H \H\H , M(H) follows the logarithmic behavior M©ln(H[H ) typical for the system of strongly m m`1 m interacting FL. Therefore, the whole magnetization curve of superconducting films with the antidot lattice is now successfully described by the simple M(H) expression, derived for interacting multi-quanta vortices in the limit of constant magnetic induction B inside the specimen (‘single-terrace critical state’). We have carried out the M(H) measurements in non-perforated high quality reference Pb/Ge multilayers and in the same films with, by e-beam lithographically fabricated, antidot lattices (for details of the sample preparation see Bruynseraede et al. [12]. Figure 1 shows an atomic force microscopy (AFM) picture of a Pb/Ge multilayer with a triangular ‘antidot’ lattice. The distance d between the antidots of radius r\0.22 lm is 1 lm and the surface between them is quite flat, the root mean square roughness is 4 nm on a (0.5 lm)2 area. The matching fields H for these perforated films, with a triangular antidot m ˚ and k(0)\2600 A ˚, lattice, are m]23.9 G. The coherence length and penetration depth are n(0)\120 A respectively. Magnetization measurements were performed in a commercial Quantum Design SQUID-magnetometer with a scan length of 3 cm, corresponding to a field homogeneity better than 0.05%. Very high ‘contrast’ between M(H) curves for as-grown and perforated films (Fig. 2A) gives us the possibility to relate the M(H) values in perforated films directly to the contribution arising from the antidot lattice. For the temperature range used in our experiments, we see several distinct cusp-like anomalies at matching fields H\H . m The estimate of the temperature dependent penetration length K\2k2/t (where k is the penetration depth and t the total thickness of the film) shows that in our case K(T\6 K) V5.8 lmAd\1 lm. Since KAd, the approximation B\constant can be used to find the magnetization M of the superconductor with the antidot lattice in the field range H\H [5]: 1

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/ ba 0 ln l M(H)\[ 16n2j2 Jer

(1)

where a is the distance between / -vortices, b a numerical constant, for triangular vortex lattice in v 0 unperforated films b\0.381 [5]. Equation (1) has been obtained from the well-known result for a homogeneous superconductor [5] by a simple substitution of n(T) by r. First, we consider a very interesting situation, which occurs exactly at the matching fields H\H , when we expect that the FL lattice consists of only one type of vortices, namely, of m/ m 0 vortices forming a regular array coinciding with the antidot lattice. In this case in eqn (1) m/ should 0 be used instead of / and a \d: 0 v

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m/ bd m/ 0 ln 0 M(H )\[ P m 16n2j2 j2 Jer

(2)

The difference M(H )[M(H )©1/k2 is independent of m and is given only by k. All other parm m~1 ameters in eqn (2) are known. Therefore, at H\H a linear behavior of M(H ) as a function of the m m integer m should be seen. This behavior is in a very good agreement with the observed M(H ) m variation at different temperatures (see the dashed lines in Fig. 2A and Fig. 3). The slope of the dashed lines nicely follows the temperature variation of 1/k2©1[T/T (insert of Fig. 3). Two-fluid c model 1/k2©1[(T/T )4 also gives a good fit, but the Ginzburg Landau relation 1/j2P1[T/T proc c vides a better linearity of the slope DM/DH versus T (see insert of Fig. 3).

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˚ )/Ge(50 A ˚ )] multilayer with and without the Fig. 2. (A) Magnetization loop M(H) at T\6 K of a [Pb(100 A antidot lattice. The solid line is a fit with eqn (4) (see text). The dashed line is2 demonstrating the validity of the linear behavior of M(H ) (eqn (2)). The loops were measured for M[0 and symmetrized for clarity for M\0. m versus ln(H[H ) at T\6 K. The different slopes of the solid lines for the different (B) The magnetization M ˜ in eqn (4). periods are used to determine the effectivemflux / om

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˚ )/ Fig. 3. The upper right parts of the magnetization loops M(H) at different temperatures for the [Pb(100 A ˚ )] multilayer with an antidot lattice. The dashed lines, at different temperatures, clearly indicate the Ge(50 A 2 linear behavior of M(H ) as a function of the integer m. The insert presents the slopes DM/DH of the dashed m lines, as a function of 1[(T/T ). According to eqn (2) these slopes follow the temperature dependence of 1/k2. c fit. The solid line presents the linear

The cusps at H\H then can be considered as analogues of the well known M(H) anomaly m at H but this time for the onset of penetration of 2/ , 3/ , . . ., (m]1)/ -vortices at H , H , . . ., c1 0 0 0 1 2 H , respectively. Indeed, the expected lower critical fields H (m/ ) are m c1 0 m/ j H (m/ )\ 0 ln c1 0 4nj2 r

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Fig. 4. Schematic drawing of the field profile for (A) classical Bean critical state, (B) multi-terrace critical state [14] and (C) single-terrace critical state.

The estimate based on eqn (3) gives very low fields H (m/ )D10~2 Oe@H , which implies that the c1 0 m difference between H[H [H ((m]1)/ ) and H[H can be neglected and therefore one should m c1 0 m see the logarithmic behavior M(H)©ln(H[H ) in magnetic fields H \H\H , i.e. between the m m m`1 matching fields. Such behavior is fully supported by our experimental data (Figs 2A, B). This behavior is a result of the inserted relation a V(/ /(H[H ))1@2 into eqn (1), which is a good approxiv 0 m mation in fields H \H\H for a (m]1)/ vortex-vortex distance in superconductor with an m m`1 0 antidot lattice. In this interval the vortex-vortex interaction terms U in the Gibbs potential are ij represented by different contributions arising from the presence of two types of vortices: m/ and 0 (m]1)/ . These terms will contain instead of a usual / ]/ product [5], other products: 0 0 0 m/ ]m/ , m/ ](m]1)/ and (m]1)/ ](m]1)/ . Due to that, magnetization of the supercon0 0 0 0 0 0 ducting films with an antidot lattice should follow in the interval between the matching fields, H m and H , the logarithmic dependence: m`1 ˜ / b / 1@2 eff 0 M(H \H\H )V[ om ln (4) m`1 m 16n2j2 Jer H[Hm

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˜ is an effective flux and b an effective parameter appearing due to the summation of Here / om eff different terms in the vortex-vortex interactions U over vortex positions i and j on the antidot lattice. ij For H not far from H we may approximate /˜ as )m(m]1) / . m om 0 Plotting M versus ln(H[H ) (Fig. 3), from a logarithmic behavior of the magnetization (Fig. m ˜ . 2), we determined the slopes of the M versus ln(H[H ) lines which give us the effective flux / m om ˜ ˜ ˜ ˜ For a triangular antidot lattice we have / // \1; / // \1.79; / // \2.45; / // \2.72, for 01 0 02 0 03 0 04 0 H \H\H , H \H\H , H \H\H , H \H\H , respectively. The series 1.79, 2.45, 2.72 0 1 1 2 2 3 3 4 has to be compared with 1.41, 2.45, 3.46 estimated from ()m(m]1)/ )// . Therefore, besides a 0 0 ˜ convential logarithmic dependence, we see also a very reasonable variation of / in different field om intervals (H \H\H ). At higher temperatures the simple argument that /˜ should scale as m m`1 om )m(m]1)/ fails, since k(T) is spread over many periods and U terms, corresponding to 0 ij m/ ](m]1)/ and (m]1)/ ](m]1)/ , should be optimized numerically. 0 0 0 0 Finally, we would like to note that the stabilization of the FL by the antidot lattice has resulted in a remarkable collective M(H) behavior, typical for the presence of a m/ -FL lattice at H\H and 0 m a mixture of m/ - and (m]1)/ -vortices between the matching fields H \H\H . Due to the 0 0 m m`1 presence of the well defined pinning potential, we were able to obtain in superconductors with the antidot lattice the logarithmic irreversible magnetization behavior which is normally observed as a reversible magnetization of the Abrikosov FL lattice in homogeneous superconductors in fields H[H . The puzzling pronounced M(H) cusps at H\H have been interpreted as analogues of the c1 m usual H cusps, but for the onset of the penetration of multi-quanta vortices. c1

3. Conclusions These experimental observations have revealed an unusual expansion (up to H (T)!) of validity of c2 the London limit oWo\constant in superconductors with artificial lattices of relatively large antidots. The successful high quality fit of the M(H,T) curves with the minimum number of fitting parameters convincingly demonstrates that a new type of the artificial state B\constant (‘single-terrace critical state’ (Fig. 4), for terrace-critical state see Cooley and Grishin [13] can be realized in nanostructured superconductors at H\H . This novel single-terrace critical state (Fig. 4C) is, in fact, an ultimate m limit of the multi-terrace critical state (Fig. 4B), introduced recently by Cooley and Grishin [13]. Acknowledgements—This work is supported by the Belgian National Fund for Scientific Research (NFWO) and Concerted Action (GOA) Programs. We would like to acknowledge useful discussions with C. Van Haesendonck. M. Baert is a Postdoctoral Fellow supported by the IUAP programs and V. V. Metlushko supported by the University of Leuven. E. Rosseel is a Research Fellow of the Belgian Interuniversity Institute for Nuclear Sciences (I.I.K.W.) and M. Van Bael of the Belgian National Fund for Scientific Research (N.F.W.O.). K. Temst is a Postdoctoral Fellow supported by the Research Council of the K.U. Leuven.

References [1] V. V. Moshchalkov, L. Gielen, M. Dhalle´, C. Van Haesendonck and Y. Bruynseraede, Nature 361, 617 (1993). [2] V. V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck and Y. Bruynseraede, Nature 373, 319 (1995). [3] H. Vloeberghs, V. V. Moshchalkov, C. Van Haesendonck, R. Jonckheere and Y. Bruynseraede, Phys. Rev. Lett. 69, 1268 (1992). [4] A. A. Abrikosov, Soviet Phys.-JETP 5, 1174 (1957). [5] P. G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, New York: (1966).

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G. S. Mkrtchyan and V. V. Shmidt, Sov. Phys. JETP 34, 195 (1972). A. I. Buzdin, Phys. Rev. B 47, 11 416 (1993). A. F. Hebard, A. T. Fiory and S. Somekh, IEEE Transactions on Magnetics 1, 589 (1977). V. V. Moshchalkov et al., Physica Scripta T55, 168 (1994). M. Baert et al., Phys. Rev. Lett. 74, 3269 (1995). H. Fang, R. Zeller and P. J. Stiles, Appl. Phys. Lett. 55, 1433 (1989); K. Ensslin and P. M. Petroff, Phys. Rev. B41, 12 307 (1990). [12] Y. Bruynseraede et al., Physica Scripta T42, 37 (1992). [13] L. D. Cooley and A. M. Grishin, Phys. Rev. Lett. 74, 2788 (1995).