PHYSICAL REVIE% B VOLUME 25, NUMBER 7 1 APRIL 1982

Anomalous flux entry into NbSe2

M. %. Denhoff and Suso GygaxDepartment of Physics, Simon Fraser University, Burnaby, British Columbia, Canada VSA 1S6

(Received 26 October 1981)

Measurements of superconducting transitions of NbSe2 single crystals were performed. They

allow the determination of the critical field for flux entry H,„and its angular dependence. An

anomalous cusplike feature was observed which cannot be explained by anisotropic Ginzburg-

Landau theory. A model is proposed which assumes easy directions for entering flux lines.

Agreement with the measurements is good.

I. INTRODUCTION

Anisotropic superconductors have been the objectof much interest in recent years. The most common-ly studied anisotropic superconductors are the natur-ally occurring layered transition-metal dichal-cogenides' which can be intercalated by a number ofmetallic elements and organic molecules. ' Anotherclass are filamentary superconductors' and Chevrelphase materials. ~ More recently there has been agrowing number of investigations of artificial layeredcomposites involving both soft superconductors' andthe high-T, transition-metal superconductors. Com-mon to all these materials is a high upper critical fieldH, 2 and a strong anisotropy of the superconductingparameters. Anistropy in H, 2 is particularly pro-nounced and seems to be well understood either interms of an anisotropic effective mass in Ginzburg-Landau theory' or a more sophisticated model involv-ing Josephson coupling. Only recently ho~ever hasthere been a theoretical treatment of the lo~er criticalfield H, i.9 No experimental determination of its an-isotropy has been undertaken up to now. For anideal type-II superconductor knowledge about H, i im-plies information about H, 2. It is therefore temptingto use low-field measurements to determine super-conducting parameters.

There are a number of problems associated with adetermination of H, i. %hat can be measured withconfidence is the entrance field, i.e., the applied fieldH,„at which flux starts to enter a superconductor.This field is not necessarily equal to H, ~. In addition,since H, ~ refers to the internal field, a demagnetiza-tion correction has to be applied in most cases. Nosuch complication arises at H, 2. This is part of thereason why very little attention has been given toH, ~. A more serious problem is because of the pres-ence of flux pinning in most materials. This effectwill delay substantial flux entry to higher fields, To alesser extent a surface barrier'0 at shiny surfacesparallel to the applied field can also pose a problem.

%C have measured low-field transitions on careful-

ly chosen NbSe2 crystals and determined the field atwhich flux first starts to enter the sample. Thesemeasurements were performed as a function of theangle of the applied field. Sin1ilarly shaped samplesof an isotropic material, Nb52Ti4S, were also measuredto distinguish anisotropy from demagnetization.

In Sec. II we briefly describe the apparatus. Theproblem of demagnetization is discussed in Sec. III A,and it is shown how it can be taken into accountquantitatively, Section III 8 presents the methodsused to determine the flux entrance field H,„. Theresults are discussed and compared with theory inSec. IV. Results for the principal orientations per-pendicular and parallel to the layers are presentedfirst (Sec. IV A). The full angular dependence of 0„is compared with theory in Sec. IVB where a newmodel involving easy axes for entering flux lines ispresented. Section IV C discusses preliminary mea-surements of trapped flux in connection with pinning.A short summary is given in Sec. V. Preliminaryresults have already been published. "

II. APPARATUS

Our mcasuremcnts were made in a supcl'-conducting-quantum-interference-device (SQUID)magnetometer" which detects the flux because of themagnetization of a sample. It can be operated with afixed applied field, ~here the temperature of thesample is varied, or the temperature can be held con-stant and the field varied. Nearly all the measure-ments were done in the first mode, where the sampleis cooled in zero field to well below T, then magnet-ized and warmed up. The resulting transition curvesin constant fields form the basis of our measure-ments.

A. Magnetometer

The magnetometer consists of a superconductingfield coil with persistent mode switch and a pair of

Oc1982 The American Physical Society

TABLE I. Geometrical parameters (thickness t, area A, and measured demagnetization factors I., M, and N) of the NbSe2and Nb52Ti48 samples, and measured temperature derivatives of H,„and of the corresponding internal field 8,„; at 8, =0' and90',

SampledH, „(0)/dT dH, „(90)/dT dH, „,(0)/dT dH, „,(90)/dT

(mm) (mm ) (1 —L) ' (1 —M) ' (1 —N) ' (Oe/K) (Oe/K) (Oe/K) (Oe/K)

NbSe2 No. 1 0.0413 0.877No. 2 0.0142 0.27No. 3 0.0276 3.1

1.041.021.01

1.031.02

12.8219.229

—11.5—8.08—6.04

—47.0—48.5—42.5

—48.9—49.4—42.9

Nb52Ti48 0.128 1.95

coaxial astatic pick-up loops connected to a SQUIDdetector by a superconducting flux transformer. Thepick-up loops are wound on a glass-filled-epoxy tubewhich forms the lo~er part of the sample space which

contains the sample stick. The sample stick is insert-ed through an air lock at the top of the cryostat. Ex-change gas in the sample space allows rapid initial

cooldown, afterwards it is pumped away. A thermalbridge makes thermal contact between a copper blockon the sample stick and the helium bath. A set of in-

terchangeable fixed-angle holders milled fromoxygen-free high-conductivity (OFHC) copper slide

over a sapphire rod at the bottom of the evacuated

sample stick. These determine the angle of the sam-

ple relative to the applied field, Apiezon %grease is

used to hold the sample and the copper angles in

place. The sapphire rod separates the sample, placed

in the upper pick-up loop, from a OFHC copper piececontaining a rnanganin heater and a calibrated carbon

glass thermometer. The electrical leads and a shortcupronickel tube between this copper piece and the

thermal bridge, together with some residual exchange

gas, determine the thermal time constant for heating

and cooling the sample.

S. Samples

The samples were grown by the iodine vapor trans-

port method. " They were carefully chosen single

crystals with smooth surfaces and uniform thickness.Each sample was weighed on an electrobalance. Thedimensions of the broad surfaces were measuredunder a microscope. Using the measured area andmass of the crystal its thickness was calculated, taking

the density of NbSe2 to be 6.45 g/cm'.In order to investigate the effects of demagnetiza-

tion and to clarify the role of anisotropy, a large K,

isotropic superconductor was measured as well. Wechose a NbTi alloy. The material was obtained fromSupercon' and was in the form of a bar with nominal

concentration of N152Ti48. It was annealed in an in-

duction oven in vacuum at 1200'C for 4 h. Samples

in the shape of oblate spheroids were then machinedfrom this material and smoothed with a file and em-ery paper. After a final etching in a fresh mixture oftwo parts H2S04, one part HF, and one part 30/o

H202 for 5 min the samples which now had smoothbut not shiny surfaces were ready for measurements.Table I lists the dimensions of the samples whichwere studied most extensively.

A. Role of demagnetization

Since the important quantity is the internal fieldrather than the applied field, demagnetization effectshave to be taken into account. Disregarding the an-isotropy of the penetration depth, a superconductorin its Meissner state is an isotropic material withX= —I/4m. The NbSe2 samples are in the form ofthin disks. We therefore consider an oblate spheroidwith its symmetry axis in the z direction (for theNbSe2 crystals this ~ould correspond to the crystal-line c axis). The usual relationship between appliedfield H, and internal field HI.'HJ=H, i(I +4wDJX) '

introduces the demagnetization coefficients Dialong the principal axes, which we label" L,M = Land S, where L +M + W =1. If the applied fieldis taken to lie in the x,z plane and pointing at anangle 8, with respect to the z axis we immediatelyfind

H;(8, ) =H, [sin'8, (1+LX) ' +co' s8(1 +N )X']'r'

(1)

If the thermodynamic lower critical field H, i is isotrop-ic, the angular dependence of the applied fieldH t (8 ) marking this transition becomes

H, i, (8,)= Hi[ is'n(8l L) '+cos'8, (l——N) '] 'i'

Note that H, i, (0) =H, t(1 N) and H, (9t0)—

25 ANOMALOUS FLUX ENTRY INTO NbSe2 4481

=H, ~(1 L—) as expected. Our apparatus measuresflux. For an ellipsoid of isotropic susceptibility withthe applied field along its z axis the exact flux linkedto the pick-up loop can be calculated. The field dis-tribution outside such an ellipsoid in a uniform fieldcan be found using ellipsoidal coordinates in thesame way as demagnetization coefficients are found,The integral of field over the area of the pick-up loopcan be done analytically and has been published else-where. " For other angles between the c axis and theapplied field the flux integral cannot be done analyti-cally and numerical methods have to be used. How-

ever, for a sample which is much smaller than thepick-up loop the linked flux approaches that of a di-

pole (spherical sample). The situation is illustrated in

Fig. 1. The ellipsoidal sample with semiaxes b, b andc has a magnetization in its Meissner state ofM = H&/4' —and therefore a dipole momentm =MY, where Vis the sample volume. Its com-ponent m, perpendicular to the pick-up loop of radius8 is m, = m cos(8I —8,), and produces a flux

@=2m ' =2m —cos(8; —8, )R R

Since

to the transition height S(90) for a field along the b

axis is S(0)/S(90) =(1—L)/(1 N—). SinceI. +M +N =1, the demagnetization factors for theprincipal axes can be determined. They are listed in

Table I. The agreement with the demagnetizationfactors calculated" from sample dimensions assumingellipsoidal shapes is very good for the smaller sam-

ples. For the larger samples the exact flux calculationfor 8 =0' gives a somewhat larger value than the di-

pole approximation of Eq. (4). Figure 2 shows themeasured angular dependence S(8,) of the transitionheight for NbSe2 No. 1, together with the angulardependence of Etl. (4). There is fairly good agree-ment between the measurements and the calcula-tions. The measurements do fall slightly below thedipole curve indicating that for larger samples the di-

pole approximation starts to fail.

%e conclude then that the demagnetization effectscan be taken into account quantitatively for all anglesof the applied field, not only for the isotropic, ellip-soidal Nb52Ti48 samples, but also for the anisotropic,flat, very-thin NbSe2 crystals. This is possible be-cause (a) the Meissner state is isotropic and (h) thesamples are much smaller than the pick-up loop area.

rg8;=H„/H&=(1 —1V)(1 L) 'rg8, —

one finds from trigonometry

cos(8; —8,) = [sin 8, (1 —L) '+cos 8, (1 —W) ']H, /H;

and finally for the angular dependence of the fluxlinked

$(8,) = — H, [sin'8, (1 —L) '+cos28, (1 —N) ']

The height S of the total magnetic transition fromMeissner state to normal state can now be used tofind the demagnetization of the sample. The ratio ofthe transition height S(0) for a field along the c axis

p00 30

ga (deg)

I

60I

90

FIG. 1. Spheroidal sample in an applied field H, , P—P isthe plane of the pickup loop, H; and m are the directions ofthe internal field and magnetization, respectively.

FIG. 2. Measured (dots) and calculated transition heightsS as a function of angle of applied field for NbSe2 No. 1.

4482 M. %. DENHOFF AND SUSO GYGAX

8. Determination of H, ~

A set of normalized transition curves for a fixedangle Hg ls shown ln Flg. 3. The parameter ls the ap-plied field and the transitions are obtained while in-creasing the temperature. The arrow in Fig. 3 indi-cates the temperature T(H,„) at which flux firststarts to enter the sample for 8,=4.25 Oe. It corre-sponds to the point where a transition starts to breakaway from the curves in lower fields. The relation-ship between H,„and T is linear.

%e have investigated the possibility that the en-trance field H,„so defined ~ould represent an arbi-trary choice. %e demonstrate in the following thatthis is not so and that moreover H,„is a bulk proper-ty. Besides the transition curves in constant field wehave occasionally used a field s~eep method, In thiscase the sample was again cooled in zero field to atemperature below T,. It is then held at a fixed tem-perature while a graduaIly increasing field is applied.This corresponds to a measurement of a magnetiza-tion curve. Since there was considerable more noisein the SQUID signal, we did not make extensivemeasurements. Moreover, the empty magnetometerproduced a field-dependent background signal due tonearby construction materials. Nevertheless, goodmagnetization curves can be obtained as shown in

Fig. 4. Ideally, for low fields, when the sample is inthe Meissner state, the magnetization curve shouldbe linear. %e have used the curve for the lowesttemperature as a base and note the field ~here themagnetization curves start to break away. Thesefields are within error the same as 0,„. %e have also

FIG. 4, Magnetization curves for NbSe2 No. 1.

used an ac susceptibihty method at 8, =0' and 90'where no copper angles in the sample holder werenecessary. An additional solenoid allo~ed the simul-taneous Rppllcatlon of 8 small Rc field ln RddltloQ tothe steady dc field. In-phase and out-of-phase mea-surements of the ac SQUID signal were performed.The quadrature signal showed distinct onset of lossesat the same T (H,„) as found in the dc measure-ments. More details of this method is given else-~here. "

The entrance field we measure with these differentmethods could be due to preliminary flux entry atsome sample irregularities both in shape and compo-

06.9 7.0 7.l

0—0 30

FIG. 3. Temperature-induced magnetic transition forNbSe2 No. 3. Arrows indicate the temperatures T(H,„)and T(Hen ~ for HN 4.25 Oe as discussed ln the tex

FIG. 5. Angular dependences for H,„and 0,„for NbSe2No. 1 at T, —T =0.05 K.

25 ANOMALOUS FLUX ENTRY INTO NbSeq 4483

sition. But an indication of how the flux enters thebulk of the sample is found by considering the mainregion of the transition, ~here a large quantity offlux is entering. This part of the transition is linearand can be extrapolated back to where it crosses alow-field curve (see Fig. 3). This defines a newquantity T(H,'„). H,'„ is again linear in temperaturebut has a larger slope. Finally, Fig. 5 shows the fullangular dependences of both H,„and H,'„ for one ofthe samples. It is quite evident that the data for H,'„show the same features as H,„, only it is larger byabout a factor of 1.4. This would inidcate that theangular dependence of H,„, particularly the cusplikefeature around 8, =80', is a bulk property and not

(a)

because of some inhomogeneities in composition orirregularities in the shape of the sample. This canalso be seen by comparing the internal fields H, „&

corresponding to the measured entrance fields H,„atthe extremal angles. Table I lists the slopes dH, „t/dTfor the thoroughly studied samples at 8, =0 and 90'.For the isotropic N15qTi48 alloy one finds an isotropicvalue as expected. For NbSeq, dH, „;/dT is anisotropicand the variation from crystal to crystal is quite smallconsidering the differing sample sizes and shapes.This indicates again that the measured entrance fieldH,„is a bulk property. Figure 6 shows a summary ofthe angular dependences of H,„. In Fig. 6(a) wedisplay the behavior of a N15~Ti48 sample. The solidline is the theoretical curve as explained in Sec. IV.Figure 6(b) shows the results for the NbSeq samples.Here the solid lines are drawn to help distinguish thedifferent samples.

—60—hC

I40—

0COT

20—

IV. DISCUSSION OF RESULTS

A. Perpendicular and parallel

orientations

In the following we make a comparison betweenthe measured entrance fields H,„and the lower criti-cal field H, » for 8, =0' and 90', although due to fluxpinning no direct measurements of H, » seem possi-ble. Ginzburg-Landau theory" predicts that for ahigh K material H, » and H, & are connected by

'0 I

30 608, (deg)

90 H, t=-, Hg(in~+0. 49 7)x ' . (5)

(b) 60-

& 40-II—0Co 20-T0

00 30ea (deg}

60 90

FIG. 6. Angular dependence of dH, „/dT. (a) Nb5~Ti48No. 2. The solid line is the angular dependence of Eq. (2).(b) NbSeq sample No. 1 (Cl ), No. 2 (b, ), No. 3 ( ~). Thesolid lines are drawn as an aid to distinguish between thedifferent samples.

Some measurements of H, ~ exist" for the isotropicalloy system Nb» Ti„. They are for x =37% a valueof dH, q/d T= —15.4 kOe/K and for x =56% a value of—25.9 kOe/K. Interpolating between these resultsgives for our alloy with x =48% a value of dH, q/dT=—20 kOe/K. To find ~ we have measured thenormal-state resistivity p with a four-probe techniqueon a very long-thin cylinder and find p =6.5 x 10 '0 cm. In the dirty limit one has'

»/rdH, p

dr

This gives K=39. From Eq. (5) we than estimatedH, &/dT =27 Oe/K. As can be seen from Table Ithe slopes of the internal field H,„;corresponding tothe measured entrance field H,„are in good agree-ment with each other, for both orientations and bothsamples. The value dH, „I/dT =72 Oe/K is howevermuch larger than the estimated dH, t/dT =27 Oe/K.We have to conclude that flux pinning is responsiblefor this. It is evident however, that our samplepreparation resulted in samples with isotropic fluxpinning.

For NbSeq there exist "measurements of the an-

M. %. DENHOFF AND SUSO GYGAX 25

isotropy of H, 2 and estimates of K. Taking the mostrecent values" of dH, 1(0)/dT =6.45 kOe/K,dH, 1(90)/dT = 25.7 kOe/K, Kg=13.5, and Ks=54,one expects dH, 1(0)/dT =55 Oe/K and dH, t(90)/dT =20 Oe/K. A comparison with the slopes of theH,„; listed in Table I shows that the valuesdH, „;(0)/dT =155 Oe/K and dH, „;(90)/dT =50Oe/K are larger by a factor of 2.8 and 2.5, respective-ly. Again the discrepancy can be blamed on flux pin-

ning, this time anisotropic.

S. Angular dependence of H,~

As can be seen from Fig. 6 there is a markeddifference in the angular dependence of H,„betweenthe isotropic alloy N152Ti48 and the anisotropic crys-tals of NbSe2. The former is well understood. As-suming isotropic pinning, the angular dependence ofH,„is just the influence of demagnetization on theisotropic H, ~. The applied field corresponding to H, I

should be given by Eq. (2). This is exactly the angu-lar dependence of H,„(8,) given by the solid line inFlg. 6(a).

For NbSe2 the situation is much more complicated.If there were again isotropic pinning, the angulardependence of H,„should reflect the angular depen-dence of H, I. Theoretical calculations of H, I for an-

isotropic materials have recently been performed byKlemm and Clem. 9 They were done in the frame-work of Ginzburg-Landau theory using an anisotropiceffective mass. Assuming an infinitely large aniso-tropic type-II superconductor they showed thatthrough suitable coordinate transformations it was

possible to bring the Ginzburg-Landau free energyinto an isotropic form. But the new Ginzburg-Landau parameter ~ depends now not only on theeffective-mass tensor but also on the directionalcosines of the magnetic induction with respect to thecrystal lattice. Specifically, for a layered material with

an anisotropy in the effective mass of e = m„/m„onefinds

47re1=2 '~1/OH, (ink+0. 497) K (7)

K = K1(COS Hs + 6S111 Hs)

where ~j is the Ginzburg-Landau parameter for fields

applied perpendicular to the layers, and 8g is the an-

gle of 8 with respect to the crystalline c axis. 22 Forlarge values of ~j the line energy ~~ of an isolatedvortex has the same form as in an isotropic type-IIsuperconductor:

der to get some insight into the peculiar angulardependence of H, ~ we use a different but equivalentapproach. %e extend the familiar argument'~ for aninfinitely large type-II superconductor to an oblate el-lipsoidal specimen. Near H, I where we can neglectflux line interaction and 8' terms we can write forthe Gibbs free energy in the mixed state

0 4 16 =I' —=—B ~ H =F~+n&I — B ~ H;4m 4m

I'~ is the free energy in the Meissner state andI1 =8/$0 is the fiux line density. Introducing thedemagnetization tensor D and eliminating H; onefinds

6 = F1r + n et —H, 8/4n ( I —D)

Since G~ =F~ in the Meissner state, H, ~ is nowdetermined by

H, I, ~ B4mn~) =

1 —D

As before we introduce a coordinate system with thez axis in the symmetry axis of an ellipsoidal sample.The applied field H, is taken to lie in the x-z planemaking an angle 8, with the z axis. Due to the sym-metry 8 lies also in the x-z plane at an angle 8~ with

respect to the z axis. One finds then

41l'6& slnH~ slnHS cosHq cosHScia

Po 1 —L 1 —W

Apart from the denominator, this expression is thesame as for an infinite superconductor, The denomi-nator takes care of demagnetization effects. In con-trast to an isotropic superconductor the line energy elgiven by Eq. (7) now depends through K in Eq. (6)on 8~. Putting everything together one finds

H =2-'~'H (inK+0 497)t -1

sin8, sin8g cos8, cos8~X —+

1 —L 1 —X

The direction of 8 is found by minimizing Eq. (8)with respect to 8~ and one obtains

( )t slnH~ 6 cosH~

(1 —I.) stnHS (1 —W) cosHS

1

slnH slnHS cosH cosHS+ 1 —e1 —L

But the calculation of H, I is complicated by the factthat H and 8, which is the direction of the first fewflux lines, are not parallel except for fields applied in

the symmetry directions. For finite samples one hasto consider demagnetization effects. This has beendone by Klemm in a subsequent publication. 2' In or-

This equation must be solved numerically for eachfixed angle 8, of the applied field. The results areidentical to the previous calculations with thedemagnetization correction applied. ' Figure 7 shows

H, I, plotted against 8, for a sample with ~~=13.5and the demagnetization factors appropriate for

ANOMALOUS FLUX ENTRY INTO NbSe2

90

50

00 30e~ (degI

60 90 0

e~ (deg)

FIG. '7. Theoretical angular dependence of H, ~, for anoblate spheroid with (1 —L) '=1.04, (1 —S) '=12.82, andx~=13.5. Solid line: anisotropy a=m„/m, =0.0625; dashedline: ~ =0.0375. The data points are those of NbSe2 No. 1.

FIG. 8. Theoretical relationship between 8~ and 8~. Thesame parameters as in Fig. 7 have been used. The dot-dashline corresponds to an isotropic sample of the same dimen-'sions, where 8~ = le,. 8«corresponds to the critical angledefined by Eq. (10).

NbSe2 No. 1 for two anisotropies ~ =0.0625 and0.0375. For a mildly anisotropic superconductor thecurve is continuous, but a break appears for higheranisotropies. The latter curve has some resemblanceto the measured angular dependence of H,„. Thesolid line has the anisotropy appropriate for NbSe2, asdetermined by Schwall et a/. ' If a higher anisotropyis chosen, such that the calculated curve goesthrough the data at 8, =90, one finds ~ =0.0375,which is too low for NbSe2. Moreover the break oc-curs at the wrong angle and the curve does not fit thedata. Changing xj has no great effect. %e must con-clude then that the measured angular dependence ofH,„cannot be explained by the H, j calculations ofKlemm and Clem. 9 Since they do predict a break, itis nevertheless instructive to see what happens at thatparticular angle. %e show in Fig. 8 the relationshipbetween Hg and 8, for the same parameters as usedin Fig. 7. The dot and dash line corresponds to anisotropic material where Hg = 8;. %e can see that inanisotropic materials the initial flux line is not parallelto the internal field, i.e., 8~ is always larger than 8~

and flux lines tend to line up closer to the layerplanes. Moreover, for anisotropies beyond ~ =0.0375the curves become discontinuous and there is a rangeof angles Hg in which the initial flux lines never lie.This discontinuity of 8~ marks the position of thebreak in H, ~,.

The Fig. 8 suggests that in layered anisotropic su-perconductors, according to these calculations, theentering flux lines tend to line up close to the crystalplanes. In the presence of strong anisotropy there isa critical angle beyond which a sudden jump in fluxline alignment occurs. The point is however that

NbSe2 is too weakly anisotropic to show this discon-tinuity. Kogan has pointed out recently that in theanisotropic effective-mass model the current distribu-tion of a flux line should be tilted with respect to itsaxis. Since the favorable directions for current trans-port are in the crystal planes, his model suggests asecond easy direction for entering flux lines, i.e., per-pendicular to the layers.

%e have gone one step further and postulate thatthe entering flux lines can only lie either perpendicu-lar or parallel to the layers. The actual configurationwill depend on the respective internal entrance fieldsin these directions, H,„;(0)and H,„;(90), The x andz components of H& cannot exceed these values. Thisleads directly to the conditions for H,„,(e;):

H,„;cos8; = H,„i(0)

H, „;sinei =H,„(90')

whichever is smaller. Transforming to H,„and 8„these conditions become the smaller of

H,„(e.}=H,„(0)icose. ,

or

H,„(e.) =H,„(90)/sine. .

This angular dependence can also be obtained fromEq. (g) by substituting H,„for H, i, and imposing theconditions Hg =0' or 90 rather than minimizing withrespect to es. The angular dependence of Eq. (9)based on our simple parallel and or perpendicularmodel is a good fit to the data as can be seen fromFig. 9. Not only does it folio~ the shape of the mea-

M. W. DENHOFF AND SUSO GYGAX

60-

I0

e 20-0

~ R~ R

030 60

ea (degj9Q I t

30 60ea(degj

90

FIG. 9. Angular dependence of dH, „/dT for all the NbSe2samples compared with the parallel and or perpendicularmodel of Eq. (9) (solid lines). Sample No. 1 (0), No. 2(b, ), and No. 3 (O).

FIG. 10. Angular dependence of trapped flux f, forNbSe2No. 3 (0) and Nb52Ti48No. 2 (4). The lines aredrawn to connect the data points. 0,=1 Oe.

sured curve, it also predicts the position of the break.The critical angle 8„is given by

8„=arctan[(H„(90) /H„(0) 1

%e like to stress that this simple model is notnecessarily a model of H, ~ in anisotropic supercon-ductors like NbSe2. Although it takes over somefeatures of the theoretical H, ~ calculations ' themodel takes into account the inherent discreteness ofthe layered structure rather than averaging it out intoa homogeneous, although anisotropic, medium.Since it describes the angular variation of the en-trance field rather than H, ~ the effects of pinning isalready contained in it.

C. Trapped flux

Pinning seems to play a major role in our measure-ments of H,„. This is evident in the discrepanciesbetween H, „~ and H, ~ along the principal directions(see Table I). For the NbTi alloy H„I is isotropic asis H, t. The ratio H, „&/H, t =2.67 is large, implyingconsiderable but isotropic pinning. For NbSe2 this ra-tio is 2.8 and 2.5 for 0' and 90', respectively. This isagain large with a 12% anisotropy.

In an effort to clarify the effects of pinning wehave measured the angular dependence of trappedflux. The data were taken by cooling the sample in anapplied field through its transition temperature. Thesignal due to incomplete flux expulsion was com-pared with the full Meissner transition. This ratio,f„was measured as a function of field and angle for

both the N152Ti4S alloy and the NbSe2 crystals. Acomplete description and analysis of the data will begiven in a later publication. In Fig. 10 we comparethe angular dependencies of f,(8,) for the two ma-terials. Again we observe a striking differencebetween the isotropic and the anisotropic supercon-ductor. For N152Ti48 with its isotropic pinning theangular dependence of f,(8,) is a shape effect. InNbSe2 there is an additional very pronounced dip inthe curve. It occurs at the same angle H„where thebreak in the entrance field H,„(8,) shows up. Herethe pinning is obviously weakest. In comparing theangular dependencies of H,„and f, one has to keepin mind that the entrance field reflects the pinning ofthe first few flux lines, whereas the trapped flux re-flects the pinning of a large quantity of flux. Thefact that both H„and f& have breaks at the same an-gle suggests however that the two are closely related.It remains to be seen if the idea of easy axes appliedto the critical state model will give the observed an-gular dependence of trapped flux.

V. CONCLUSIONS

%e have shown that through proper analysis ofsimple magnetic transition curves on superconductorsone can obtain a value of the entrance field H,„which is a bulk property. Demagnetization effectscan be taken into account in a simple way for bothisotropic and anisotropic superconductors as long asthe sample dimensions are small compared to the sizeof the pickup loop.

A comparison of H,„and H, ~ in NbSe2 and an iso-tropic NbTi alloy shows that H,„is always larger than

H, ~ as expected in the presence of pinning, Thesample preparation of NbTi resulted in isotropic pin-ning, ~hereas in NbSe2 pinning is stronger for fieldsperpendicular to the layers.

The angular dependence of H,„shows an unusualcusplike feature for all the NbSe2 samples studied,whereas it is smooth for the NbTi. Comparing thisbehavior with recent calculations9 of the angulardependence of H, ~ in anisotropic superconductorsone finds that they cannot explain the experimentalresults. The angular dependence of H,„in NbTi onthe other hand follows the expected angular depen-dence of H, ~ with isotropic pinning. One could con-clude then that anisotropic pinning effects are respon-sible for the observed H„(,8,) relationship in NbSe2.%e believe however that thc H, ~ calculations point toa simple model of flux entry into anisotropic super-conductors. Proposing easy axes for entering fluxlines along directions perpendicular and parallel tothe layers seems to account well for our observations.

Long a feature in ferromagnetism, easy axes havenot been proposed for anisotropic superconductorsbefore. They take into account the microscopic lay-ered structure rather than assigning an anisotropic ef-fective mass to an otherwise homogeneous medium.The role of pinning in such a model still has to beworked out in detail, preferably along the lines of thecritical state model. Preliminary measurements oftrapped flux on the same materials show similarfeatures as the entrance field, pointing to a relation-ship between the two.

ACKNOVfLEDGMENTS

%e thank J. R. Long for valuable help in the pre-liminary stages of assembling the apparatus, R. F.Frindt for providing us with thc NbSe2 crystals, andR. A. Klemm, J. R. Clem, and V. G. Kogan for help-ful discussions on theory. The work wss supportedby a grant from the National Science snd EngineeringResearch Council of Canada.

'J. S. %ilson and A. D. Yoffe, Adv. Phys. 18, 193 (1969).R. F. Gamble, J. H. Osiecki, M. Cais, R. Pisharody, F. J.

DiSalvo, and T. H. Geballe, Science 174, 493 (1971).R. L. Greene, G. B. Street, and L. J. Suter, Phys. Rev.

Lett. 34, 577 (1975).~Ternary Superconductors, edited by G. K. Shenoy, 8, D.

Dunlap, and F. Y. Fradin (North-Holland, New York,1981).

5C. R. Spencer, P. Martinoli, E. D. Gibson, J. D. Verhoe-ven, and D. K. Finnemore, Phys. Rev. 8 18, 1216 (1978};T. %. Haywood and D. G. Ast, ibid. 18, 2225 (1978).

6S. T. Ruggiero, T. %. Barbee, Jr., and M. R. Beasley, Phys.Rev. Lett. 45, 1299 (1980).

7L. P. Gorkov and T. K. Melik-Barkhudarov, Zh. Eksp.Teor. Fix. 45, 1493 (1964) [Sov. Phys. JETP 18, 1031(1964))', D. R. Tilley, Proc. Phys. Soc. 85, 1177 (1965);86, 289 (1965); R. C. Morris, R. V. Coleman, and R.Bhandari, Phys. Rev. 8 5, 895 (1972).

8%. Lawrence and S. Doniach, in Proceedings of the 12th In-ternational Conference on Low Temperature Physics, editedby E. Kanda (Academic, Kyoto, 1971), p. 361; N. Boc-cara, J. P, Carton, and G. Sarma, Phys. Lett. 49A, 165(1974); R. A. Klemm, A. Luther, and M. R. Beasley,Phys. Rev. 8 12, 877 {1975);D. E. Prober, R. E. Schwall,and M, R. Beasley, ibid. 21, 2717 (1980).

9R. A. Klemm and J. R. Clem, Phys. Rev. 8 21, 1868(1980).

~OA. S. Joseph and %. J. Tomash, Phys. Rev. Lett. 12, 219(1964); C. P. Bean and J. D. Livingston, Ibid. 12, 14(1964).

~~M. Denhoff and S. Gygax, Physica (Utrecht) 1078+C,299 (1981).

'2S Gygax and M. Denhoff (unpublished}.3R. Kershow, M. Vlasse, and A. %old, Inorg. Chem. 6,

1599 (1967).'4Norton, Supercon Division, 9 Erie Dr. , Natick, Mass.' J. A. Osborn, Phys. Rev. 67, 351 (1945).~6M. Denhoff, S. Gygax, and J. R. Long, Cryogenics 21, 400

(1981).See A. L. Fetter and P. C. Hohenburg, in Superconductivi-

ty, edited by R. D. Parks (Dekker, New York, 1969), Vol.2, Chap. 14.

'SL. J. Neuringer and Y. Shapira, Phys. Rev. A 140, 1638(1965).

~ T. P. Orlando, E. J. McNiff, Jr., S. Foner, and M. R. Beas-ley, Phys. Rev. 8 19, 4545 (1979).

20R. C. Morris, R. V. Coleman, and B. Bhandari, Phys. Rev.8 5, 895 (1972); R. C. Morris and R. V. Coleman, ibid. 7,991 (1973); P. deTrey, S. Gygax, and J.-P. Jan, J. LowTemp. Phys. 11, 421 (1973);Y. Muto, N. Toyota, K.Noto, and A. Hoshi, Phys. Lett. 45A, 99 (1973);N. Toy-ota, H. Nakatsuji, K. Noto, A. Hoshi, N. Kobayashi, andY. Muto, J, Low Temp. Phys. 22, 485 (1976).

2~R. E. Schmall, G. R. Stew&art, and T. H. Geballe, J. LowTemp. Phys. 22, 557 (1976).Note that some authors define the anisotropy parameter as(m„/rn, ) ~~2, The parameter x& used in Ref. 9 is equal toour Kg.

23R. A. Klemm, J. Low Temp. Phys. 39, 589 {1980).24Ref. 17, p. 830.25V. G. Kogan, Phys. Rev. 8 24, 1572 (1981).

C. P. Bean, Phys. Rev. Lett. 8, 250 (1962); Y. B. Kim,and M. J. Stephen, in Superconductivity, edited by R. D.Parks (Dekker, New York, 1969), Vol. 2, p. 1145.