Magnetization of low-κ superconductors II. The lower critical field Hc1

Physica C 161 (1989) 287-293 North-Holland

M A G N E T I Z A T I O N O F L O W - t o S U P E R C O N D U C T O R S II.

THE L O W E R C R I T I C A L F I E L D He1 H.W. WEBER, E. SEIDL, M. BOTLO, C. L A A a n d H.P. W I E S I N G E R Atominstitut der Osterreichischen Universitdten, A-1020 Wien, Austria

J. R A M M E R Physikalisches lnstitut der Universitdt, D-8580 Bayreuth, Fed. Rep. of Germany

Received 3 July 1989

In this second part of a series of three publications on the magnetization of low-x superconductors, we present experimental and theoretical results on the lower critical field Hcl. We concentrate on highly pure niobium and vanadium samples. Both materials are type-II superconductors with a first order transition at H¢~. After a brief description of experimental techniques, the theoretical methods used in the present study are outlined. We show that a successful theoretical analysis of experimental data requires the inclusion of both strong-coupling and anisotropy effects. Because of the first order transition at H¢I, the thermodynamics of the interacting vortex lattice has to be solved in order to obtain H¢~.In this way, we achieve excellent agreement between experiment and theory, consistent with our results on other characteristic features of the magnetization in these superconductors. Near To, the validity of Ginzburg-Landau theory is confirmed with high accuracy.

I. Introduction In the first part [ 1 ] o f this series o f three publications, we have p o i n t e d out that the p r i m a r y objective o f our experimental p r o g r a m was to establish the phase b o u n d a r i e s between different types o f magnetization curves observed experimentally in conventional superconductors, which characterize type-I, type-II / 1 a n d type-II / 2 superconductivity. In the first case, the magnetization is d e t e r m i n e d entirely by the Meissner effect or, in sample with finite d e m a g n e t i z a t i o n factors D, by the Meissner a n d the i n t e r m e d i a t e states, which are separated at the field (1 -D)H~. In the second and third case, the m i x e d state is f o r m e d at the lower critical field Hcl, but the nature o f the phase transition at H ~ d e p e n d s on the magnitude o f the G i n z b u r g - L a n d a u p a r a m e t e r K(Tc). It is o f first order ( t y p e - I I / 1 ) for 1/X/~ Kcr.

The experimental technique used p r i m a r i l y for these investigations was selected on the basis o f therm o d y n a m i c considerations for superconductors with 0921-4534/89/$03.50 © Elsevier Science Publishers B.V. ( North-Holland )

finite d e m a g n e t i z a t i o n factors published by Kerchner et al. [ 2 ], who have shown that the derivative o f the magnetization, dM/d/~oH, is most sensitive for detecting the nature o f the phase transition at the lower critical field. In the case o f a long cylinder magnetized p e r p e n d i c u l a r to its long axis, the following shapes o f the "differential" magnetization curve can be observed, depending on the type o f superconductivity. F o r type-I superconductors the signal in the Meissner state is constant and equal to - 1/ ( l - D ) ; at the field ( 1 - D ) # 0 H c , the transition to the i n t e r m e d i a t e state occurs a n d the signal will be again constant a n d equal to 1/D. In the case o f typeII superconductors, the signal in the Meissner state will be the same. Then, d e p e n d i n g on the m a g n i t u d e o f x, the material will either show an i n t e r m e d i a t e m i x e d state ( t y p e - I I / 1 b e h a v i o r ) or a pure m i x e d state ( t y p e - I I / 2 ) . In the first case, sharp transitions o f the signal from - 1 / ( 1 - D ) to 1/D will occur at the field poH~ = ( 1 --D)goHc~ a n d again at the field ].~oH2 = ( l - - D )~oHc i "~ DBo, where the magnetization changes from the i n t e r m e d i a t e m i x e d state to the usual m i x e d state, which is governed by a continuous

288

H. W Weberet al. /Magnetization oflow-x superconductors H

decrease of the flux line spacing with increasing magnetic field. In the second case, the plateau region characterizing the intermediate mixed state will be absent and only one sharp change of signal at #oHm= (1-D)/zoH¢~ followed by a continuous decrease will be observed. Finally, in all cases a discontinuity will occur at the transition field to the normal conducting state (/~/¢,/toH¢2). Regarding an experimental study of the lower critical field H¢l, the magnetization of type-II/1 superconductors offers distinct advantages, because H¢l can be related to H~, //2 and the magnetization M ( H a ) measured in the plateau region of the intermediate mixed state (I.toH¢l=ltoH2-DM(H2)). However, from an experimental point of view, H d is still the most difficult quantity to be evaluated unambiguously from the magnetization curve. This is caused by possible small superheating effects at Hi or supercooling effects at H2, residual effects of flux pinning, the existence of surface barriers or the presence o f " e n d " effects resulting from a not completely ideal sample geometry, which will all result in the appearance of slight hysteresis effects, especially near H¢I. Although we have adopted certain evaluation procedures and kept them unchanged throughout the whole study, the absolute values of H¢~ can still be affected by systematic errors amounting to a few percent. In the following, we will present the evaluation of H¢~ from the differential magnetization curve (section 2), discuss the results for Nb and V in terms of various theoretical treatments (section 3 ) and draw our conclusions in section 4.

2. Experimental Details of the sample preparation and their characterization in terms of basic superconducting properties have been presented in ref. [ 1 ]. A brief summary of data pertaining to materials to be discussed in the present paper is given in table I. The magnetization measurements were made by placing the cylindrical samples, with their long axis perpendicular to the external magnetic field direction, into the center of a pick-up coil in Helmholtz configuration and sweeping the external field as slowly as possible, in order to achieve thermody-

namic equilibrium conditions [ 1,3 ]. Since the whole measuring system is computer controlled, large quantities of data points could be collected and processed numerically for the data evaluation. The typical result of such a magnetization cycle is shown in fig. 1, where the derivative of the magnetization, dM/d/toH, is plotted versus the external field for the field cycles 0--,He2 and Hc2~0. It will be noted that the magnetization in the mixed state is completely reversible, but small irreversibilities appear at both ends of the intermediate mixed state. In general, both the transition fields H~ a n d / / 2 are larger along the initial magnetization than in decreasing fields. If the same numerical evaluation procedures are applied to both branches of the magnetization curve, we typically obtain differences A H / H of about 1.5% in H¢I and of about 2.5% in He. Because of the better definition of the transition fields, only the data pertaining to the initial magnetization have been used for the final evaluation. This was also motivated by the observation that the amount of irreversibility varied slightly with temperature and would have led to additional error sources, if an averaging procedure between the two branches had been applied. Concerning the definition of the transition fields themselves, the following procedures were adopted. Firstly, H ~ = ( 1 - D ) H ¢ ~ was defined as the field where d M / d / ~ r / = 0 . This definition was preferred to all other options, because it can be calculated from an interpolation of the data (cf. fig. 2a) with high accuracy, remains valid in exactly the same form, if the sample does not form an intermediate mixed state, and finally reflects the fact that the slopes in the Meissner ( - 1/( 1 - D ) ) and intermediate mixed states ( 1/D) are equal for D = 1/2 (infinitely long cylinder magnetized perpendicular to its long axis) or roughly equal for our samples with D ~ 0.48. This definition, therefore, represents the maximum of a "rounded off" magnetization curve and corresponds to a linear extrapolation of the slopes in the Meissner and intermediate mixed states (cf. the insert of fig. 1 ). Secondly, the field/-/2 was defined as the intersection point of two tangents calculated on both sides of the transition (figs. 2b, 2c). This field is related to the induction jump Bo through the relation #oH2= ( 1 - D ) p o H c l + D B o . Thirdly, the magnetization M in the "plateau" region of the intermediate mixed state could be used to calculate H¢~ from the

H. W Weber et al. /Magnetization oflow-x superconductors H

289

Table I Material parameters of the superconductors discussed in the present paper. sample

crystal

Nb 8

single

cr~ (at%)

RRR

p.(4.2 K) (nflm)

Tc (K)

T* (K)

x(Tc)

ot

-

2080

0.07

9.301

9.059

0.720

0.011

0.01

385

0.26

9.289

8.960

0.725

0.020

-

626

0.40

5.453

4.737

0.810

0.045

[1101 Nb 7

poly

V lWO

single [110]

~H2 b l ~

Nb7 T:3.399K

"

!

HC2

~

H.I

¢ "0

" I 100

i

~,oe (,-T)

1 200

300

Fig. 1. Differential magnetization curve of sample Nb 7 in increasing (0~Hc2) and decreasing (Hc2--,0) fields. The insert shows the shape of the integrated magnetization curve near the field H~.

relation aoHc~ =floH2-DM(H2). However, in view of the non-constant behavior of M within the intermediate mixed state, which is caused predominantly by the electronics of our equipment, H~ was preferred for the evaluation of lower critical fields throughout this study. In summary, the experimental conditions selected in the present work are believed to come close to a situation which enables us to deduce Hc~ or Hc~/Hc with some confidence. In addition to the fact that the sample preparation quite successfully avoided the

formation of pinning centers, the major profit is drawn from the existence of a large demagnetization factor. For D = 0, d M / d p o H would become very large at H~ and the signal processing has to be done with settings on the lock-in amplifier, which are unfavorable for the entire remaining magnetization curve. Of course, because of H, =HE, an identification of the Bo-jump is almost impossible. For large demagnetization factors, on the other hand, the signal at H~ is strongly reduced. In addition, if an intermediate mixed state is formed, the magnitude of D can be

290

H. W. Weber et al. /Magnetization oflow-r superconductors H

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Fig. 2. Experimental data illustrating the evaluation procedures of the fields H 1 ( d M / d '~oH=0, (a)) and//2 (tangents below (b) and above (c) H2). determined experimentally from Ht,//2 and perhaps from M(H2), which enables us to minimize the errors associated with the calculation of absolute values for the lower critical field H~].

3. Theoretical considerations and results As discussed extensively in the preceding paper [ l ], type-II/1 superconductivity is characterized by a discontinuous magnetic transition at Hal. This peculiar behavior is associated with flux line attraction at low inductions [4,5], and consequently with a first order phase transition at H~. As usual, such a phase transition can be "superheated". The "superheating field", to be called H ~ in the following, is always larger than Hex. Upon increasing the applied magnetic field, the Meissner state will be thermodynamically stable at fields below Hal, and unstable in the region from H c l up to H c ~ . At fields exceeding H ~ , the Meissner state will become unstable for

thermodynamic reasons, and flux lines will penetrate. However, because of the unstable character of the Meissner state in the field region H¢1 < H < H ~ , flux penetration will already occur at the field He1 under appropriate experimental conditions. It is noted that in the case of ordinary type-II (type-II/ 2) behavior, where a second order transition occurs at Hcl, we have H ~ - H e , . Generally, the lower critical field of a type-II superconductor is defined as the external magnetic field at which the mixed state becomes thermodynamically favorable compared to the Meissner state. In the type-II/2 case, the mixed state will be established in the form of a vortex lattice with (ideally) infinite vortex spacing, and the vortex distance will increase monotonically with increasing applied field. The lower critical field is, therefore, determined by the condition that the Gibbs free energies of the Meissner state and the mixed state with one isolated vortex line are equal [6,7]. In the type-II/l case, on the other hand, the mixed state will be established in the

H. W.. Weberet al. /Magnetization of low-x superconductorsH form of a vortex lattice with finite spacing. In this situation, the above definition (involving isolated vortices) does not yield H~t, because the vortices are not isolated at the transition. Instead, the above definition evidently yields the superheating field H ~ in the type-II/1 case (also called "asymptotic lower critical field" ). The symbol " ~ " in H ~ refers to the fact that an infinite flux line spacing is assumed. The calculation of H ~ proceeds along the lines sketched in the preceding paper [1 ]. Specifically, H ~ is computed from the solution of the Gor~kovEilenberger equations in cylindrical symmetry, with sufficiently large vortex radii, including strong coupling and anisotropy effects (see section 4.1 of ref. [ 1 ] ). The (measured) lower critical field, H~, has to be computed from the thermodynamics of the interacting vortex lattice, with finite flux line distance. These calculations yield magnetization curves such as the ones shown in fig. 10 of ref. [ 1 ]. In the typeII/1 case, the magnetization is metastable at low inductions. From these curves, both Hc~ and the magnitude of the discontinuity at Hc~, the induction jump Bo, can be determined using a Maxwell construction, as depicted in that figure. (The induction jump Bo was discussed in detail in ref. [ 1 ].) Of course, H ~ is given by the field, where the magnetization curves reach B = 0 (i.e., B ( H = H ~ ) = 0 ) . According to the above discussion, the type-II/1 phase is also characterized by the fact that Hcl is always smaller than the asymptotic lower critical field H¢~. This effect is rather pronounced especially for clean low-x type-II superconductors "deep" in the type-II/1 region, which display huge metastable parts in the magnetization. In fig. 3 we show our results for H ~ and H~1 in the framework of different theoretical approaches, for pure vanadium ( x = 0 . 8 1 ) and niobium ( x = 0 . 7 2 ) . First, it is noted that the temperature dependence of H ~ / H ~ is altered in a characteristic way by strong coupling: at temperatures below Tc, the corresponding curves are lowered, as compared to the weak-coupling theory. The anisotropy effect on H ~ / H c , on the other hand, is not significant and is, therefore, not shown. Also shown in fig. 3 are our theoretical results for Hd/H¢ in the framework of the weak-coupling isotropic and the strong-coupling anisotropic theories, along with our experimental data. The figure reveals very good agreement between the results of the

291

strong-coupling anisotropic theory and experiment. We have used the same type of anisotropy as in ref. [1], with an anisotropy parameter o~4=2.3. It is noted that the inclusion of both strong coupling and anisotropy is vital for achieving this excellent agreement, in particular for niobium. We also see from fig. 3 that all curves approximately meet at Tc, as required by Ginzburg-Landau theory. As a final remark, we wish to come back to the experimental uncertainties associated with the evaluation of Hc~/Hc. First of all, we have used a "prescription" to determine Hc~ (initial magnetization, dM/d/~oH=0) which may be subject to systematic errors, but provides us with relatively low statistical errors ( g +0.5%). Secondly, we need the thermodynamic critical field He, which is affected by the largest error bars ( + 1.5%) of all the parameters deduced from a magnetization measurement on a typeII superconductor. Hence, each data point in fig. 3 has an error margin of roughly + 1.6%. The systematic errors, on the other hand, may cancel partly in the ratio Hcl/Hc as pointed out in section 2. Lastly, the excellent agreement between theory and experiment, which is achieved without changing any parameters in the theory as compared to the evaluation of the phase boundaries and the magnitude of the Bojump discussed in ref. [ 1 ], provides rather convincing evidence for the validity of the present approach. Further evidence will be presented in the following paper [8 ], where the generalized Ginzburg-Landau parameters x~ and x2 will be discussed in detail.

4. Conclusions We have studied the lower critical field Hc~ of pure niobium and vanadium. On the experimental side, special care has been taken to select the most favorable measuring conditions (transverse geometry) and to minimize errors arising from residual flux pinning and superheating/supercooling effects in these typeII/1 superconductors. Under certain precautions, this seems to provide us with a reliable set of data on the temperature dependence of Hci in Nb and V. Regarding the theoretical evaluation of our results, we have solved the Gor'kov-Eilenberger and the Eliashberg equations including the model anisotropy introduced in refs. [ 1 ] and [ 7 ], and discussed the ef-

292

H. IV.. Weber et al. / Magnetization of low-x superconductors H 1.0(

V w e a k - iso

0.9!

go

o o ~' ~ ' Hcl

~

strong - aniso

O.gO I

I

I

I

I

I

I

I

I

I

weak-iso

s t r o n g - iso

1.00

~

*

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0.95

°(~ ° ° °

0

°°

o

st r o n g - aniso

O O o~"

0.90

0

I

0.1

I

0.2

I

0.3

I

0.4

I

I

0.5

0.6

I

0.7

I

0.8

I

0.9

I

1.0

t Fig. 3. Temperature dependence of Hc~/Hc in V and Nb. The solid curves refer to the theoretical solutions for H~] in the isotropic weakand strong-coupling cases, and for Hc~ in the weak-coupling isotropic and strong-coupling anisotropic cases.

fects of strong coupling and anisotropy on Hc~. It turns out that both real metal effects are vital for achieving very good agreement between theory and experiment, the cotlpling correction being more important than anisotropy in the case of H~,. Finally, the validity of Ginzburg-Landau theory near Tc has been confirmed with high accuracy. This refers also to the functional form of the generalized Ginzburg-Lan-

dau parameter K3, which is directly related to the ratio Hcl/Hc.

Acknowledgements We wish to thank Mr. H. Niedermaier for his help with the experiments and F.A. Schmidt, Ames, for

H. IV.. Weber et al. /Magnetization o f low-x superconductors H

p r o v i d i n g us with the v a n a d i u m single crystal. Valuable discussions with Dr. F.M. Sauerzopf, Wien, Dr. U. Klein, Linz, a n d Profs. W. Pesch, D. R a i n e r a n d L. K r a m e r , Bayreuth, are gratefully acknowledged. T h i s work was s u p p o r t e d in part b y D e u t s c h e Fors c h u n g s g e m e i n s c h a f t a n d by F o n d s z u r F 6 r d e r u n g der W i s s e n s c h a f t l i c h e n F o r s c h u n g , W i e n , u n d e r contract # 3973 a n d 5032.

293

References [ 1] H.W. Weber, E. Seidl, M. Botlo, C. Laa, E. Mayerhofer, F.M. Sauerzopf, R.M. Schalk, H.P. Wiesinger and J. Rammer, PhysicaC 161 (1989) 272. [2] H.R. Kerchner, D.K. Christen and S.T. Sekula, Phys. Rev. B 21 (1980) 86. [3] P. Hahn and H.W. Weber, Cryogenics 23 ( 1983 ) 87. [4] L. Kramer, Z. Phys. 258 (1973) 367. [ 5 ] M.C. Leung, J. Low-Temp. Phys. 12 ( 1973 ) 215. [6 ] W. Pesch and L. Kramer, J. Low Temp. Phys. 15 (1974) 367. [7] J. Rammer, W. Pesch and L. Kramer, Z. Phys. B 68 (1987) 49. [8] E. Seidl, C. Laa, H.P. Wiesinger, H.W. Weber, J. Rammer and E. Schachinger, Physica C 161 (1989) 294.