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Phase boundaries of superconducting mesoscopic square loops
Physica B 249—251 (1998) 476—479
Phase boundaries of superconducting mesoscopic square loops V.M. Fomin!,",*, V.R. Misko!,#, J.T. Devreese!, V.V. Moshchalkov$ ! Theoretische Fysica van de Vaste Stof, Universiteit Antwerpen (U.I.A.), Universiteitsplein 1, B-2610 Antwerpen, Belgium " Department of Theoretical Physics, State University of Moldova, str. Mateevici 60, MD-2009 Kishinev, Moldova # Institute of Applied Physics, str. Academiei 5, MD-2028 Kishinev, Moldova $ Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium
Abstract The superconducting state is described and the phase boundary is determined for mesoscopic superconducting square loops of intermediate sizes comparable with the coherence length and the penetration depth. The calculated phase boundaries are in good agreement with the experimental data obtained for the Al mesoscopic superconducting square loops. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Mesoscopic superconductor
1. Introduction The recent experiments [1,2] carried out on superconducting mesoscopic aluminum loops with sizes smaller than the temperature-dependent coherence length m(¹) and the penetration depth k(¹) reveal the influence of the sample topology on the superconducting critical parameters, such as critical temperature ¹ as a function of the magnetic # field H. In a mesoscopic square loop, the Little—Parkstype oscillatory H—¹ phase boundary has been detected; it is related to the effect of fluxoid quantization in the loop [1]. To interpret these oscillations, the authors of Ref. [1] applied the Tinkham formula [3] which describes the superconducting * Corresponding author. Tel.: #32 3 8202460; fax: #32 3 8202245; e-mail: [email protected].
transition in a perpendicular magnetic field for a circular loop with emphasis on the consequences of fluxoid quantization. Although the observed ¹ (H) oscillations can be # interpreted in the afore-mentioned way by choosing some effective radii of cylinders, nevertheless the investigation of magnetic properties of the mesoscopic superconducting structures of realistic shape (e.g., square loops) requires a more adequate theoretical description. The purpose of the present work (see also the preliminary report [4]) is to analyze the superconducting properties of squareshaped loops, relevant to the available experiments.
2. The boundary problem The Ginzburg—Landau (GL) equations for the order parameter t and the vector potential A of
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 6 8 - 9
V.M. Fomin et al. / Physica B 249 —251 (1998) 476—479
respective set of the boundary conditions at the external boundary results:
a magnetic field H"rot A are [5—7] 2e 1 (!i++! A)2t#at#bDtD2t"0, c 2m
(1)
4nie+ 16pe2 *A" (t*+t!t+t*)# ADtD2. mc2 mc
Lt (x, y) a Lx
(2)
A A
The geometry of a real superconducting structure enters the problem via the boundary condition
A
BK
2e n ) !i++t! At c
"0, (3) "06/$!3: where n is the unit vector normal to the boundary. The leads are supposed to have the same width as the loop. This corresponds to the shape of the real structure which has been studied in the experiments reported in Ref. [1]. We take the following sizes of the loop: the width d"150 nm, the outer side Q "1000 nm (the side of the opening, there% fore, is Q "700 nm) and the variable length of the * leads is Q . The magnetic field H is applied in the 0 z-direction, i.e., perpendicularly to the sample which lies on the xy-plane. In our calculations, we will consider all the relevant physical quantities, namely, the magnetic field inside the structure H, the phase / and the squared amplitude t2 of the ! order parameter, as functions of x and y whereas the z-dependence will be neglected. Further, Eqs. (1) and (2) are transformed (see Refs [5,6]) to dimensionless variables. The temperature dependences of the coherence length m(¹), the magnetic field penetration depth j(¹) as well as the coefficient a are the following:
A A
B B
A
477
B
¹ ~1@2 ¹ ~1@2 m"m 1! , j"j 1! , 0 0 ¹ ¹ # # ¹ a"a 1! (4) 0 ¹ # with the coefficients given in Ref. [4]. The GL parameter is defined as i"j/m. The order parameter t is a complex-valued function of coordinates and can be presented in a general form as t(x, y)"t (x, y)e*((x,y). (5) a Substituting Eqs. (4) and (5) to the GL Eqs. (1) and (2), we obtain the set of equations in partial derivatives to determine t (x, y), /(x, y) and H(x, y). The !
K
K
Lt (x, y) ! "0, "0, Ly %95, x/#0/45 %95, y/#0/45 L/(x, y) i # H "0, Lx 2J2 0y %95, x/#0/45 L/(x, y) i ! H "0, Ly 2J2 0x %95, y/#0/45 H(x, y)D "H . (6) %95 0 The similar set of the boundary conditions at the internal boundary is not given here for brevity. The magnetic field inside the opening H depends on the * applied magnetic field H and is determined from 0 the integral relation (see, e.g., Ref. [7]) c 4p
Q
BK BK
c rot H ) dl! (U !U )"0, * 4pj2 &
(7)
i with the condition of the fluxoid quantization U "¸U , ¸"0, 1, 2,2. (8) & 0 Here U "ch/2e is the flux quantum and U " 0 * { A ) dl is the magnetic flux through the opening of * the loop. The integration runs over the internal boundary of the loop.
3. Distributions of the order parameter and of the magnetic field As a result of a self-consistent numerical solution of the Ginzburg—Landau equations for a mesoscopic superconducting square loop with leads of finite width, the distribution of the amplitude of the order parameter in the loop is demonstrated to exhibit peculiarities in the corners of the loop and in the vicinity of the midpoints of the sides linked to the leads, see Fig. 1. The critical magnetic field H (0) of bulk Al at zero temperature is used as # a unit of the magnetic field. At low values of the applied magnetic field, negative or positive, the peaks of the order parameter appear in the corners of the loop. On increasing the applied magnetic field, first, those peaks split and broaden; second, the components move towards
478
V.M. Fomin et al. / Physica B 249—251 (1998) 476—479
Fig. 2. Regions of the thermodynamically stable states for different ¸ for the superconducting loop. Every curve represents the boundary of the area in which t2(x,y)'0 for a given ¸. The ! following denotations of states are used: diamonds for ¸"0, pluses for ¸"1, squares for ¸"2, crosses for ¸"3.
Fig. 1. Evolution of the magnetic field and of the superconducting phase in the Al mesoscopic square loop with leads is shown at the temperature ¹/¹ (0) "0.94 for the fixed orbital quantum # number ¸"1.
the midpoints of the external sides of the loop, ultimately forming superconducting “islands”; third, the superconducting phase “flows” from the outer to the inner sides of the loop and reveals local features of cylindrical symmetry. This distribution is shown to be thermodynamically stable. A further increase of the applied magnetic field suppresses the superconducting state at a given number ¸ of magnetic flux quanta in the opening of the loop, and the
minimum of the free energy passes on to the state with ¸#1 magnetic flux quanta. For the structures under consideration, when the width of the loop and leads relate to the external side of the loop as 1.5 : 10, it is revealed that the presence of leads does not strongly affect the magnetic field and the order parameter distributions in the loop itself. Leads distort the patterns of the distributions obeying the C -symmetry peculiar for 4 an isolated square loop and reduce their symmetry to C . 2 We have plotted schematically the regions of the thermodynamically stable states for different ¸ for the superconducting loop under consideration in Fig. 2. The envelope of all the curves shown in Fig. 2 forms, obviously, the phase boundary which includes different numbers of magnetic flux quanta ¸ according to the requirement of the thermodynamical stability.
4. Comparison with the experiment Thus, interplay of the symmetries of the magnetic field and of the structure is shown to lead to different oscillatory superconducting phase boundaries ¹ (H) related to various phase boundary defini# tions. As a realistic criterion of the phase boundary,
V.M. Fomin et al. / Physica B 249 —251 (1998) 476—479
479
has been recently observed in these structures [2], is interpreted in the framework of the “multifilamentary superconductor” approach. With this purpose, the superconductor is split into a set of current-carrying filaments which have the form of a straight strip in each of the leads, the length of the straight strips being determined by the coherence length. Acknowledgements
Fig. 3. Phase boundaries calculated using the criterion of the appearance of the “superconducting path” in a superconducting square loop, in comparison with the experimental data [1]. The calculation was performed for the parameters taken from experiment: m "100 nm, i"0.9, Q "1000 nm and Q "700 nm. 0 % *
the appearance of a single-connected region with non-zero order parameter between the links of the leads to the loop (a so-called “superconducting path”) is proposed [4]. This definition of the phase boundary allows to obtain a good agreement between the calculated ¹ (H) curve and the experi# mental data for superconducting mesoscopic aluminum structures with leads, see Fig. 3. It is worth to be emphasized, that when applying the model of a circular loop [3] for explanation of the aforementioned experiment, one treats both internal and external radii as fitting parameters. As distinct from the model of a circular loop, the present approach contains no fitting parameters and relates the observed phase boundary [1] to the realistic distribution of the order parameter in a superconducting mesoscopic square loop. The effect of non-locality of the superconducting state in the leads on the phase boundaries, which
This work has been supported by the F.W.O.-V. projects Nos. G.0287.95, G.0232.96 and the W.O.G. 0073.94N; Inter-University Poles of Attraction Programme — Belgian State, Prime Minister’s Office — Federal Office for Scientific, Technical and Cultural Affairs; Bijzonder Onderzoeksfonds (BOF); PHANTOMS Research Network. V.M.F. is supported by BOF NOI 97 “Thermodynamica van interagerende identieke deeltjes met behulp van padintegralen”. V.R.M. acknowledges the Visiting Research Fellowship of the F.W.O.-V. References [1] V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature 373 (1995) 319. [2] C. Strunk, V. Bruyndoncx, V.V. Moshchalkov, C. Van Haesendonck, Y. Bruynseraede, Phys. Rev. B 54 (1996) R12701. [3] M. Tinkham, Phys. Rev. 129 (1963) 2413. [4] V.M. Fomin, V.R. Misko, J.T. Devreese, V.V. Moshchalkov, Solid State Commun. 101 (1997) 303. [5] V.L. Ginzburg, L.D. Landau, Zh. Eksp. Teor. Fiz. 20 (1950) 1064. [6] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 9 (Statistical Physics, vol. 2), Pergamon, Oxford, 1989. [7] P.G de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, Reading MA, 1989.
Phase boundaries of superconducting mesoscopic square loops V.M. Fomin!,",*, V.R. Misko!,#, J.T. Devreese!, V.V. Moshchalkov$ ! Theoretische Fysica van de Vaste Stof, Universiteit Antwerpen (U.I.A.), Universiteitsplein 1, B-2610 Antwerpen, Belgium " Department of Theoretical Physics, State University of Moldova, str. Mateevici 60, MD-2009 Kishinev, Moldova # Institute of Applied Physics, str. Academiei 5, MD-2028 Kishinev, Moldova $ Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium
Abstract The superconducting state is described and the phase boundary is determined for mesoscopic superconducting square loops of intermediate sizes comparable with the coherence length and the penetration depth. The calculated phase boundaries are in good agreement with the experimental data obtained for the Al mesoscopic superconducting square loops. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Mesoscopic superconductor
1. Introduction The recent experiments [1,2] carried out on superconducting mesoscopic aluminum loops with sizes smaller than the temperature-dependent coherence length m(¹) and the penetration depth k(¹) reveal the influence of the sample topology on the superconducting critical parameters, such as critical temperature ¹ as a function of the magnetic # field H. In a mesoscopic square loop, the Little—Parkstype oscillatory H—¹ phase boundary has been detected; it is related to the effect of fluxoid quantization in the loop [1]. To interpret these oscillations, the authors of Ref. [1] applied the Tinkham formula [3] which describes the superconducting * Corresponding author. Tel.: #32 3 8202460; fax: #32 3 8202245; e-mail: [email protected].
transition in a perpendicular magnetic field for a circular loop with emphasis on the consequences of fluxoid quantization. Although the observed ¹ (H) oscillations can be # interpreted in the afore-mentioned way by choosing some effective radii of cylinders, nevertheless the investigation of magnetic properties of the mesoscopic superconducting structures of realistic shape (e.g., square loops) requires a more adequate theoretical description. The purpose of the present work (see also the preliminary report [4]) is to analyze the superconducting properties of squareshaped loops, relevant to the available experiments.
2. The boundary problem The Ginzburg—Landau (GL) equations for the order parameter t and the vector potential A of
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 1 6 8 - 9
V.M. Fomin et al. / Physica B 249 —251 (1998) 476—479
respective set of the boundary conditions at the external boundary results:
a magnetic field H"rot A are [5—7] 2e 1 (!i++! A)2t#at#bDtD2t"0, c 2m
(1)
4nie+ 16pe2 *A" (t*+t!t+t*)# ADtD2. mc2 mc
Lt (x, y) a Lx
(2)
A A
The geometry of a real superconducting structure enters the problem via the boundary condition
A
BK
2e n ) !i++t! At c
"0, (3) "06/$!3: where n is the unit vector normal to the boundary. The leads are supposed to have the same width as the loop. This corresponds to the shape of the real structure which has been studied in the experiments reported in Ref. [1]. We take the following sizes of the loop: the width d"150 nm, the outer side Q "1000 nm (the side of the opening, there% fore, is Q "700 nm) and the variable length of the * leads is Q . The magnetic field H is applied in the 0 z-direction, i.e., perpendicularly to the sample which lies on the xy-plane. In our calculations, we will consider all the relevant physical quantities, namely, the magnetic field inside the structure H, the phase / and the squared amplitude t2 of the ! order parameter, as functions of x and y whereas the z-dependence will be neglected. Further, Eqs. (1) and (2) are transformed (see Refs [5,6]) to dimensionless variables. The temperature dependences of the coherence length m(¹), the magnetic field penetration depth j(¹) as well as the coefficient a are the following:
A A
B B
A
477
B
¹ ~1@2 ¹ ~1@2 m"m 1! , j"j 1! , 0 0 ¹ ¹ # # ¹ a"a 1! (4) 0 ¹ # with the coefficients given in Ref. [4]. The GL parameter is defined as i"j/m. The order parameter t is a complex-valued function of coordinates and can be presented in a general form as t(x, y)"t (x, y)e*((x,y). (5) a Substituting Eqs. (4) and (5) to the GL Eqs. (1) and (2), we obtain the set of equations in partial derivatives to determine t (x, y), /(x, y) and H(x, y). The !
K
K
Lt (x, y) ! "0, "0, Ly %95, x/#0/45 %95, y/#0/45 L/(x, y) i # H "0, Lx 2J2 0y %95, x/#0/45 L/(x, y) i ! H "0, Ly 2J2 0x %95, y/#0/45 H(x, y)D "H . (6) %95 0 The similar set of the boundary conditions at the internal boundary is not given here for brevity. The magnetic field inside the opening H depends on the * applied magnetic field H and is determined from 0 the integral relation (see, e.g., Ref. [7]) c 4p
Q
BK BK
c rot H ) dl! (U !U )"0, * 4pj2 &
(7)
i with the condition of the fluxoid quantization U "¸U , ¸"0, 1, 2,2. (8) & 0 Here U "ch/2e is the flux quantum and U " 0 * { A ) dl is the magnetic flux through the opening of * the loop. The integration runs over the internal boundary of the loop.
3. Distributions of the order parameter and of the magnetic field As a result of a self-consistent numerical solution of the Ginzburg—Landau equations for a mesoscopic superconducting square loop with leads of finite width, the distribution of the amplitude of the order parameter in the loop is demonstrated to exhibit peculiarities in the corners of the loop and in the vicinity of the midpoints of the sides linked to the leads, see Fig. 1. The critical magnetic field H (0) of bulk Al at zero temperature is used as # a unit of the magnetic field. At low values of the applied magnetic field, negative or positive, the peaks of the order parameter appear in the corners of the loop. On increasing the applied magnetic field, first, those peaks split and broaden; second, the components move towards
478
V.M. Fomin et al. / Physica B 249—251 (1998) 476—479
Fig. 2. Regions of the thermodynamically stable states for different ¸ for the superconducting loop. Every curve represents the boundary of the area in which t2(x,y)'0 for a given ¸. The ! following denotations of states are used: diamonds for ¸"0, pluses for ¸"1, squares for ¸"2, crosses for ¸"3.
Fig. 1. Evolution of the magnetic field and of the superconducting phase in the Al mesoscopic square loop with leads is shown at the temperature ¹/¹ (0) "0.94 for the fixed orbital quantum # number ¸"1.
the midpoints of the external sides of the loop, ultimately forming superconducting “islands”; third, the superconducting phase “flows” from the outer to the inner sides of the loop and reveals local features of cylindrical symmetry. This distribution is shown to be thermodynamically stable. A further increase of the applied magnetic field suppresses the superconducting state at a given number ¸ of magnetic flux quanta in the opening of the loop, and the
minimum of the free energy passes on to the state with ¸#1 magnetic flux quanta. For the structures under consideration, when the width of the loop and leads relate to the external side of the loop as 1.5 : 10, it is revealed that the presence of leads does not strongly affect the magnetic field and the order parameter distributions in the loop itself. Leads distort the patterns of the distributions obeying the C -symmetry peculiar for 4 an isolated square loop and reduce their symmetry to C . 2 We have plotted schematically the regions of the thermodynamically stable states for different ¸ for the superconducting loop under consideration in Fig. 2. The envelope of all the curves shown in Fig. 2 forms, obviously, the phase boundary which includes different numbers of magnetic flux quanta ¸ according to the requirement of the thermodynamical stability.
4. Comparison with the experiment Thus, interplay of the symmetries of the magnetic field and of the structure is shown to lead to different oscillatory superconducting phase boundaries ¹ (H) related to various phase boundary defini# tions. As a realistic criterion of the phase boundary,
V.M. Fomin et al. / Physica B 249 —251 (1998) 476—479
479
has been recently observed in these structures [2], is interpreted in the framework of the “multifilamentary superconductor” approach. With this purpose, the superconductor is split into a set of current-carrying filaments which have the form of a straight strip in each of the leads, the length of the straight strips being determined by the coherence length. Acknowledgements
Fig. 3. Phase boundaries calculated using the criterion of the appearance of the “superconducting path” in a superconducting square loop, in comparison with the experimental data [1]. The calculation was performed for the parameters taken from experiment: m "100 nm, i"0.9, Q "1000 nm and Q "700 nm. 0 % *
the appearance of a single-connected region with non-zero order parameter between the links of the leads to the loop (a so-called “superconducting path”) is proposed [4]. This definition of the phase boundary allows to obtain a good agreement between the calculated ¹ (H) curve and the experi# mental data for superconducting mesoscopic aluminum structures with leads, see Fig. 3. It is worth to be emphasized, that when applying the model of a circular loop [3] for explanation of the aforementioned experiment, one treats both internal and external radii as fitting parameters. As distinct from the model of a circular loop, the present approach contains no fitting parameters and relates the observed phase boundary [1] to the realistic distribution of the order parameter in a superconducting mesoscopic square loop. The effect of non-locality of the superconducting state in the leads on the phase boundaries, which
This work has been supported by the F.W.O.-V. projects Nos. G.0287.95, G.0232.96 and the W.O.G. 0073.94N; Inter-University Poles of Attraction Programme — Belgian State, Prime Minister’s Office — Federal Office for Scientific, Technical and Cultural Affairs; Bijzonder Onderzoeksfonds (BOF); PHANTOMS Research Network. V.M.F. is supported by BOF NOI 97 “Thermodynamica van interagerende identieke deeltjes met behulp van padintegralen”. V.R.M. acknowledges the Visiting Research Fellowship of the F.W.O.-V. References [1] V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature 373 (1995) 319. [2] C. Strunk, V. Bruyndoncx, V.V. Moshchalkov, C. Van Haesendonck, Y. Bruynseraede, Phys. Rev. B 54 (1996) R12701. [3] M. Tinkham, Phys. Rev. 129 (1963) 2413. [4] V.M. Fomin, V.R. Misko, J.T. Devreese, V.V. Moshchalkov, Solid State Commun. 101 (1997) 303. [5] V.L. Ginzburg, L.D. Landau, Zh. Eksp. Teor. Fiz. 20 (1950) 1064. [6] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 9 (Statistical Physics, vol. 2), Pergamon, Oxford, 1989. [7] P.G de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, Reading MA, 1989.