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Thickness dependence of perpendicular critical fields in superconducting films of In, Pb and Sn
Physica 132B (1985) 217-222 North-Holland, Amsterdam
T H I C K N E S S D E P E N D E N C E O F P E R P E N D I C U L A R C R I T I C A L FIELDS IN S U P E R C O N D U C T I N G F I L M S O F In, Pb AND Sn S. O N O R I and A. R O G A N I Physics Laboratory, lstituto Superiore di SanitY, Viale Regina Elena 299, 00161, Rome, Italy Received 23 November 1984 Revised 19 February 1985 Transition from mixed- to intermediate-states in In, Pb and Sn films, studied with a microwave phase-sensitive technique, is treated. Values obtained for the critical thicknesses d~ separating intermediate- and mixed-state behaviour confirm the theoretical predictions and the results of the direct observation of the magnetic structure but are much lower than the ones deduced on the basis of critical field measurements. The question is discussed. Perpendicular critical magnetic field values obtained in the 65 nm-200 ~m thickness range are also reported. Results obtained are shown to be in good agreement with theoretical estimates for thin (d < d±) and thick (d ~>~:) films. A semiempirical functional expression is proposed for critical field in the intermediate thickness range, where no theoretical estimates yet exist.
1. Introduction
The magnetic behaviour of T y p e - I superconductors in a perpendicular magnetic field depends on sample thickness. As T i n k h a m [1] has pointed out, very thin films in a perpendicular magnetic field assume an A b r i k o s o v vortex state similar to the mixed state in T y p e - I I superconductors. T h e perpendicular critical field decreases with increasing film thickness and a second-order phase transition can be expected at critical field He2. As thickness increases, transition f r o m mixed- to intermediate-state occurs. For samples with thicknesses much greater than the coherence length ~, L a n d a u ' s [2] model for the intermediate state foresees that critical field H_t increases with increasing film thickness and that it depends on the surface energy factor (A/d) in, where A is the surface energy parameter. F r o m Lasher's [3] and Maki's [4] theoretical studies on solutions to the G i n z b u r g - L a n d a u (GL) equations for second-order phase transition, critical thickness d±, which separates mixed- and intermediate-state behaviour, can be estimated. Nevertheless, for d ~> d l , neither the structure of intermediate state nor the thickness d e p e n d e n c e of H i has yet been fully understood.
Others [5-9] have already used different techniques to experimentally study the magnetic properties of T y p e - I superconducting films and foils in perpendicular fields. Considerable interest has been devoted to the determination of the critical thickness d±, nevertheless reported values for d± differ appreciably. In particular, direct observation of microscopic magnetic state [8] gives values for d I in agreement with microscopic theories [3, 4] but in contradiction with results from critical field transition analysis. Moreover, values obtained by others [5-7] for perpendicular critical field as a function of thickness confirm that theoretical expressions, which are c o m m o n l y u s e d to fit experimental data, are inadequate over most of the sample thickness range. In a preceding p a p e r [10] we described a test for a microwave phase-sensitive technique that had been designed to study the transition from T y p e - I to T y p e - I I behaviour in T y p e - I superconducting films in a perpendicular magnetic field and at various temperatures. The m e t h o d described in that p a p e r was capable of distinguishing between mixed and intermediate state behaviours. F u r t h e r m o r e , the values obtained for d I confirmed Lasher and Maki's theoretical estimates [3, 4].
0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
218
This paper considers the matter in greater detail and presents the perpendicular critical field data that we obtained for films and foils of Sn, Pb and In by using our previously tested microwave differential technique. Hi_ thickness dependence for thin (d < d;) and for thick (d-> A) films is in good agreement with theoretical estimates [3,4]. A semiempirical functional expression is also proposed to estimate perpendicular critical field for thicknesses d > d±.
2. Background According to Tinkham's theory [1], the perpendicular critical field for very thin films (d < dz) can be expressed by
with eq. (2) for all thicknesses may produce considerable errors in the determination of d l and A. In fact, critical thickness d~ was obtained by equating eq. (2) for the intermediate-state solution to Tinkham's mixed-state solution, eq. (1). Moreover, it must be stressed that, for d > d±, it is not possible to overlook the thickness dependence of parameter A. Thus, even supposing Landau's laminar model valid down to d d~, A thickness dependence must be taken into account in eq. (2). The surface parameter for t = 0 has been calculated by Bardeen [13] using the two-fluid model and can be expressed b y
A (0, d) = A (0, d_.~)g(s),
(3)
s
Hz(t, d ) = H¢2 4zrh2(0' d)H2(O) (1 - t2] =
(1)
¢o
where ¢0 is the flux quantum, He(t) is the thermodynamic critical field, A(t, d) is the penetration depth and t = T/T c is the reduced temperature. The thickness dependence of the critical field is introduced through A(0, d). For thicknesses that are much greater than the surface energy parameter A (t), Guyon et al. [11] computed the perpendicular critical field for the Landau laminar model [2] of the intermediate state, obtaining
H±(t, d)= H~(t, d)[1- (~-~)'r2] ,
(2)
where C = 0.88. Similar expressions have also been determined [11, 12] for other intermediate state models in the large thickness range (d -> A). Even films with thicknesses d ~> d± exhibit a superconductivity transition in perpendicular field that goes to intermediate state. Experimental data show that films of such thicknesses have a Type-I behaviour and that critical field increases with increasing film thickness. Some authors [5--7] have used eq. (2) to fit H± experimental data. However, eq. (2) gives a proper fit to the data only over a rather limited range of thicknesses (d-> a). Attempting to fit the data
where s = (eHc/hc)A 2 and g(s) is a decreasing function of s. However, between s = 0.1 and 1.0, g(s) varies almost linearly with s and is approximately equal to: g(s)~ 1.1-1.6s. Calculated values of A are however larger than observed. Probably the theoretical value can be regarded as an upper limit attainable only for an ideal specimen. In all cases, d~ values obtained from critical field data depend on the Hx thickness dependence used to fit experimental data in mixedand intermediate-state. Hence it is fundamental to know actual H± thickness dependence for
d > d I. 3. Experimental procedure Our measurements were taken working with pure specimens of Sn, In and Pb in a 65 nm200/zm thickness range. Source materials were obtained commercially and of 5N nominal purity. Films were shaped rectangularly (4 mm × 2 mm) and obtained by electron-gun evaporation (VT114 B Varian evaporator) on quartz substrates kept at room temperature under a less than 10-7Torr pressure. Evaporation rate was approximately 5 nm/s. Film thicknesses were monitored during evaporation by using a quartz
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
oscillator and were then measured with a M20 Wild interferometer. Foils were obtained by the usual rolling technique and had the same geometry and dimensions as the films. Experiments performed with a microwave-absorption spectrometer working in the X band (Strandeberg Labs. Corporation Model 602 A/K) enabled us to measure surface resistance derivative dRJdH as a function of the magnetic field parallel and/or perpendicular to sample surface. To obtain these measurements, samples were placed in the middle of a cylindrical cavity with TE011 excitation mode at a resonance frequency of 9.4 GHz. Results were obtained by modulating microwave power with a 30 Hz magnetic field and observing the a.c. component of the reflected power by means of a phase-sensitive technique. This technique is described extensively elsewhere [10, 14]. The cryogenic method enabled us to take measurements down to 1.5 K.
4. Results and discussion
4.1. Critical thickness determination Fig. 1 shows two typical differential superconductivity transition curves for Sn films as functions of perpendicular magnetic field at reduced temperature t = 0.62. Curve (a) corresponds to a sample of thickness d = l l 0 n m and curve (b) to a sample of thickness d = 3400 nm. As previously observed [10] the superconductivity transition patterns for thin films are symmetrical but become markedly asymmetrical with increasing film thickness. Different curve patterns correspond to different microscopic states. As a matter of fact we observed, also for the samples analyzed in this work, that for symmetrical superconductivity transition patterns like the one shown in fig. la, the H i value for the maximum of each curve is in good agreement with theoretical predictions for thin films, eq. (1). In addition, the ratio H,±/H± (where H,~ is the magnetic field at which superconductivity curve
dRs/dH (au,
IH,L //~llOnm
Sn t-0.62 1
219
d= 3400 nrn
IH~L
0.~
10
2010" HiT
HIT
Fig. 1. Superconductivity transition curves for two Sn films vs. magnetic field perpendicular to sample surface for t = 0.62: a) d = 110 nm, b) d = 3 4 0 0 n m .
drops to zero) equals 1.7 and H,± thereby assumes the physical meaning of surface critical field/-/ca in agreement with the Saint-James and de Gennes theory [15]. Moreover, for very thick samples (d >>A (t)) showing asymmetrical superconductivity patterns like the one reported in fig. lb, the H~ value for the maximum of each curve agrees well with Landau's model, eq. (2). On the basis of these observations the symmetrical transition pattern was considered as typical of mixed- to normal-state transition, while the asymmetrical curve as representative of intermediate- to normal-state transition. Then the analysis of the change in line shape allows to study the passage from intermediate- to mixedstate behaviour: the largest film thickness for which we observe a symmetrical curve pattern is assumed as critical thickness d± [10]. Fig. 2 compares our main findings for Sn, In and Pb films at different temperature values (from t = 0.5 to t = 0.85) with the Lasher theory previsions for a sample characterized by the G.L. parameter k(t, d), critical field H i and thickness d. The solid line represents the boundary between mixed- and intermediate-state regions as deduced from Lasher's theory. Symbols I and M come from observations of intermediate- or mixed-state behaviours, respectively, as derived from an asymmetry analysis of superconductivity transition curves. Fig. 2 clearly shows a good agreement between Lasher's previsions and our data obtained by the microwave phase-sensitive technique. Moreover, it may be seen that temperature does not seem to affect
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
220
210
2k2
MIXED STATE (M)
0.7-
210
~ ~
-
.o
o.s-
~o
®
~'u'
~
,3o ,~o
A~
I® 1_6o ~ ~ ~ , {[]
o.1~ - f
~
INTERMErlIATE STATE(I)
[] , []L
0
,0
0.2
Sn
Pb A
, 0.4
,
l 0,6
t
0.8
~
1
10
~
t
_1.2
dVH/4%
Fig. 2. Comparison between observations of mixed (M) or intermediate (I) state behaviours deduced from analysis of superconductivity transition curve asymmetry and Lasher's previsions. So]id line is Lasher's boundary between intermediate- and mixed-states.
critical thickness d± significantly. In additon, table I reports our d L values, deduced as previously described, values obtained by other authors as a result of critical field experiments and of direct observation of microscopic state. Table I also reports Lasher's theoretical estimates.
[16]. For d >dL, the ratio HJHc increases with increasing thickness. However, our experimental results cannot be succesfully accounted for by the simple relation between critical field and thickness that follows from Landau's model. The solid line in fig. 3 was deduced from eq. (2) by assuming A (t)= A (0)/(1- t4) 1/2 where A (0) is the
4.2. Criticalfields data
Table I Values for critical thicknesses obtained with microwave phase-sensitive technique, those obtained from critical field data, from magnetic structure observations and from Lasher's previsions
Fig. 3 reports the ratio HA ~He for Sn films as a function of d at reduced temperature t = 0.55. H A is the field value corresponding to the maximum of each superconductivity transition curve determined to within 2%. For thin films, perpendicular critical field decreases with increasing thickness and our experimental data are in good agreement with Thinkham's expression, eq. (1), up to d = 190 nm. The short dashed line was deduced from eq. (1) by using the empirical expression for A(0, d) obtained from the parallel critical field analysis performed on the same samples following the procedure described in
Film
d t a) (nm)
dj. b) (nm)
dl c) (rim)
d.Ld) (rim)
Sn (2.1 K) In (2.1 K) Pb (4.2 K)
190 110 280
520 600 1000
180 80 250
180 90 300
")Thickness of thickest film for which mixed state was observed. b) From macroscopic critical field results [5, 6]. =)From magnetic structure observations [8]. o) From Lasher's theory [3].
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields 1 Sn
t =055
~Hc
"
221
Sn ~,,,,,""~
"
t-0.85
0.6 - - , -
0.8-
0.2
0.6-
t, I
0.4-
//
..." -
I__
I 102
103
r
I
,~ f
,\ , ~ ~ " ' " ~
0.2101
J I
104
d/nm
1__ t=0.7
0.6
105
Fig. 3. Ratio H±IHc for Sn at t = 0.55vs. thickness: . . . . from ref. 1; - and . . . . from ref. 11; w _ _ _ _ computed from eq. (2) of text using Bardeen's calculation of surface energy parameter [13].
0.2
,
,,f
1
I
,
I
I
/
~'\
,
t -0.55
0.6
value obtained from fitting H~ data vs. t for sufficiently thick samples, i.e. A (0) = 205 nm. Eq. (2) may be seen to provide a good fitting for the data over a rather limited range of thicknesses, thus supporting the theoretical restriction that eq. (2) is valid only for d >>A (t). Thus the question arises as to whether the Landau laminar model is adequate to describe the entire intermediate thickness region or whether, as some results show [9], a thickness dependent energy parameter A(t, d) should be taken into account. We have compared our experimental data with expressions similar to eq. (2) for other intermediate state models. One such comparison was that with the dotted line in fig. 3, which corresponds to the Caroli et al. model [11] in which elliptic cylinder structures were investigated rather than laminae. As a further step, we also modified eq. (2) to allow for the A (0, d) thickness dependence deduced from Bardeen's previsions [13] (dashed line in fig. 3). Nevertheless, as can easily be seen, these other expressions only provide a good fitting for the experimental data in the large thickness range also taking into account the introduction of a thickness dependent energy parameter. So far we have found a semiempirical expression valid for any thickness d > d± at the same time taking into account the asymptotic behaviour of eq. (2). Fig. 4 shows ratio vs.
H±/Hc
0.2-
lOT
102
14
105
d/nm
Fig. 4. Ratio H±/Hc for Sn vs. thickness for three different temperatures.
d for Sn films at three temperatures. The solid line that describes our experimental results best was obtained from eq. (2) which was modified as follows:
HJHc= l-[O'8dA(t)exp(-A(d) + fl)]~a,
(4)
where A (0) = 205 nm. Coefficient fl was obtained as a parameter by interpolating data with a nonlinear least-square fitting technique. The values obtained in the reduced temperature range, 0.50.85, fall b e t w e e n - 0 . 2 and -0.3. The dashed line represents eq. (1). We should like to emphasize that agreement between eq. (4) and our experimental data is rather remarkable at all temperatures and for all other samples analysed (In and Pb). 5. Conclusions
We want here to summarize some interesting data emerged from experimental results:
222
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
i) The values of the critical thickness dz for Sn, In and Pb films obtained by the microwave phase-sensitive technique are in good agreement with the microscopic theory [3, 4] and with direct observation of magnetic flux structure [8]. ii) The d± values obtained by others [5-7] in critical field measurements from the intersection of the theoretical expressions for thin and thick films region, eqs. (1) and (2), exceed the values predicted by the microscopic theory. Nevertheless, our experimental results (fig. 3) clearly show that expressions like eq. (2) should be used only in the limit of very thick samples. An inappropriate use of the theory of the intermediate state can lead to a fictitious increase in d~. Then the disagreement between microscopic theory and results from critical field analysis could be due to the inadequacy of the expressions like eq. (2) to describe the entire intermediate thickness region. iii) The measured critical magnetic field values, /-/1, are in good agreement with Tinkham's theory for thin films and with Landau's model for thick films. For films with thicknesses d ~>A no theoretical estimate yet exists. A better understanding of intermediate-state structure in this region would be useful but difficult. We have proposed a semiempirical functional expression which fits all the experimental data concerning H± for any thickness d > d I. What is more, the intersection of this expression with the theoretical one for thin films, eqs. (4) and (1), is for a di
value in agreement with theoretical estimates [3, 41.
Acknowledgements We wish to thank M. Flamini for his technical assistance.
References [1] [2] [3] [4] [5]
M. Tinkham, Phys. Rev. 129 (1963) 2413. L.D. Landau, Phys. Z. Sowiet 11 (1937) 129. G. Lasher, Phys. Rev. 154 (1966) 345. G. Maki, Ann. Phys. (New York) 34 (1965) 363. R.E. Miller and G.D. Cody, Phys. Rev. 173 (1968) 494, 481. [6] B.L. Bran&, R.D. Park and R.D. Chaudhari, J. Low Temp. Phys. 4 (1971) 41. [7] M,D. Maioney, F. de la Cruz and M. Cardona, Phys. Rev. B5 (1972) 3558. [8]' C.J. Dolan and J. Silcox, Phys. Rev. Lett. 30 (1973) 603. [9] T. Miyazaki, B. Shinozaki, R. Aoki, M. Katayama and S. Yamashita, J. Phys. Soc. Japan 45 (1978) 1487. [10] M. Flamini, S. Onori and A. Rogani, I! Nuovo Cimento
2D (1983) 1497. [11] E. Guyon, C. Caroli and A. Martiner, ft. Phys. (Paris) 25
(1964) 683. [12] E.A. Davies, Proc. R. Soc. London, A255 (1960) 407. [13] J. Bardeen, Phys. Rev. 94 (1954) 554. [14] E.Di Crescenzo, M. Flamini, P.L. Indovina, S. Onori and A. Rosati, Ann. Ist. Sup. San 7 (1971) 560. [15] P.G. de Gennes and D. Saint-James, Phys. Lett. 7 (1963)
306. [16] S. Onori and A. Rogani, PhysicaB100 (1980)93.
T H I C K N E S S D E P E N D E N C E O F P E R P E N D I C U L A R C R I T I C A L FIELDS IN S U P E R C O N D U C T I N G F I L M S O F In, Pb AND Sn S. O N O R I and A. R O G A N I Physics Laboratory, lstituto Superiore di SanitY, Viale Regina Elena 299, 00161, Rome, Italy Received 23 November 1984 Revised 19 February 1985 Transition from mixed- to intermediate-states in In, Pb and Sn films, studied with a microwave phase-sensitive technique, is treated. Values obtained for the critical thicknesses d~ separating intermediate- and mixed-state behaviour confirm the theoretical predictions and the results of the direct observation of the magnetic structure but are much lower than the ones deduced on the basis of critical field measurements. The question is discussed. Perpendicular critical magnetic field values obtained in the 65 nm-200 ~m thickness range are also reported. Results obtained are shown to be in good agreement with theoretical estimates for thin (d < d±) and thick (d ~>~:) films. A semiempirical functional expression is proposed for critical field in the intermediate thickness range, where no theoretical estimates yet exist.
1. Introduction
The magnetic behaviour of T y p e - I superconductors in a perpendicular magnetic field depends on sample thickness. As T i n k h a m [1] has pointed out, very thin films in a perpendicular magnetic field assume an A b r i k o s o v vortex state similar to the mixed state in T y p e - I I superconductors. T h e perpendicular critical field decreases with increasing film thickness and a second-order phase transition can be expected at critical field He2. As thickness increases, transition f r o m mixed- to intermediate-state occurs. For samples with thicknesses much greater than the coherence length ~, L a n d a u ' s [2] model for the intermediate state foresees that critical field H_t increases with increasing film thickness and that it depends on the surface energy factor (A/d) in, where A is the surface energy parameter. F r o m Lasher's [3] and Maki's [4] theoretical studies on solutions to the G i n z b u r g - L a n d a u (GL) equations for second-order phase transition, critical thickness d±, which separates mixed- and intermediate-state behaviour, can be estimated. Nevertheless, for d ~> d l , neither the structure of intermediate state nor the thickness d e p e n d e n c e of H i has yet been fully understood.
Others [5-9] have already used different techniques to experimentally study the magnetic properties of T y p e - I superconducting films and foils in perpendicular fields. Considerable interest has been devoted to the determination of the critical thickness d±, nevertheless reported values for d± differ appreciably. In particular, direct observation of microscopic magnetic state [8] gives values for d I in agreement with microscopic theories [3, 4] but in contradiction with results from critical field transition analysis. Moreover, values obtained by others [5-7] for perpendicular critical field as a function of thickness confirm that theoretical expressions, which are c o m m o n l y u s e d to fit experimental data, are inadequate over most of the sample thickness range. In a preceding p a p e r [10] we described a test for a microwave phase-sensitive technique that had been designed to study the transition from T y p e - I to T y p e - I I behaviour in T y p e - I superconducting films in a perpendicular magnetic field and at various temperatures. The m e t h o d described in that p a p e r was capable of distinguishing between mixed and intermediate state behaviours. F u r t h e r m o r e , the values obtained for d I confirmed Lasher and Maki's theoretical estimates [3, 4].
0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
218
This paper considers the matter in greater detail and presents the perpendicular critical field data that we obtained for films and foils of Sn, Pb and In by using our previously tested microwave differential technique. Hi_ thickness dependence for thin (d < d;) and for thick (d-> A) films is in good agreement with theoretical estimates [3,4]. A semiempirical functional expression is also proposed to estimate perpendicular critical field for thicknesses d > d±.
2. Background According to Tinkham's theory [1], the perpendicular critical field for very thin films (d < dz) can be expressed by
with eq. (2) for all thicknesses may produce considerable errors in the determination of d l and A. In fact, critical thickness d~ was obtained by equating eq. (2) for the intermediate-state solution to Tinkham's mixed-state solution, eq. (1). Moreover, it must be stressed that, for d > d±, it is not possible to overlook the thickness dependence of parameter A. Thus, even supposing Landau's laminar model valid down to d d~, A thickness dependence must be taken into account in eq. (2). The surface parameter for t = 0 has been calculated by Bardeen [13] using the two-fluid model and can be expressed b y
A (0, d) = A (0, d_.~)g(s),
(3)
s
Hz(t, d ) = H¢2 4zrh2(0' d)H2(O) (1 - t2] =
(1)
¢o
where ¢0 is the flux quantum, He(t) is the thermodynamic critical field, A(t, d) is the penetration depth and t = T/T c is the reduced temperature. The thickness dependence of the critical field is introduced through A(0, d). For thicknesses that are much greater than the surface energy parameter A (t), Guyon et al. [11] computed the perpendicular critical field for the Landau laminar model [2] of the intermediate state, obtaining
H±(t, d)= H~(t, d)[1- (~-~)'r2] ,
(2)
where C = 0.88. Similar expressions have also been determined [11, 12] for other intermediate state models in the large thickness range (d -> A). Even films with thicknesses d ~> d± exhibit a superconductivity transition in perpendicular field that goes to intermediate state. Experimental data show that films of such thicknesses have a Type-I behaviour and that critical field increases with increasing film thickness. Some authors [5--7] have used eq. (2) to fit H± experimental data. However, eq. (2) gives a proper fit to the data only over a rather limited range of thicknesses (d-> a). Attempting to fit the data
where s = (eHc/hc)A 2 and g(s) is a decreasing function of s. However, between s = 0.1 and 1.0, g(s) varies almost linearly with s and is approximately equal to: g(s)~ 1.1-1.6s. Calculated values of A are however larger than observed. Probably the theoretical value can be regarded as an upper limit attainable only for an ideal specimen. In all cases, d~ values obtained from critical field data depend on the Hx thickness dependence used to fit experimental data in mixedand intermediate-state. Hence it is fundamental to know actual H± thickness dependence for
d > d I. 3. Experimental procedure Our measurements were taken working with pure specimens of Sn, In and Pb in a 65 nm200/zm thickness range. Source materials were obtained commercially and of 5N nominal purity. Films were shaped rectangularly (4 mm × 2 mm) and obtained by electron-gun evaporation (VT114 B Varian evaporator) on quartz substrates kept at room temperature under a less than 10-7Torr pressure. Evaporation rate was approximately 5 nm/s. Film thicknesses were monitored during evaporation by using a quartz
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
oscillator and were then measured with a M20 Wild interferometer. Foils were obtained by the usual rolling technique and had the same geometry and dimensions as the films. Experiments performed with a microwave-absorption spectrometer working in the X band (Strandeberg Labs. Corporation Model 602 A/K) enabled us to measure surface resistance derivative dRJdH as a function of the magnetic field parallel and/or perpendicular to sample surface. To obtain these measurements, samples were placed in the middle of a cylindrical cavity with TE011 excitation mode at a resonance frequency of 9.4 GHz. Results were obtained by modulating microwave power with a 30 Hz magnetic field and observing the a.c. component of the reflected power by means of a phase-sensitive technique. This technique is described extensively elsewhere [10, 14]. The cryogenic method enabled us to take measurements down to 1.5 K.
4. Results and discussion
4.1. Critical thickness determination Fig. 1 shows two typical differential superconductivity transition curves for Sn films as functions of perpendicular magnetic field at reduced temperature t = 0.62. Curve (a) corresponds to a sample of thickness d = l l 0 n m and curve (b) to a sample of thickness d = 3400 nm. As previously observed [10] the superconductivity transition patterns for thin films are symmetrical but become markedly asymmetrical with increasing film thickness. Different curve patterns correspond to different microscopic states. As a matter of fact we observed, also for the samples analyzed in this work, that for symmetrical superconductivity transition patterns like the one shown in fig. la, the H i value for the maximum of each curve is in good agreement with theoretical predictions for thin films, eq. (1). In addition, the ratio H,±/H± (where H,~ is the magnetic field at which superconductivity curve
dRs/dH (au,
IH,L //~llOnm
Sn t-0.62 1
219
d= 3400 nrn
IH~L
0.~
10
2010" HiT
HIT
Fig. 1. Superconductivity transition curves for two Sn films vs. magnetic field perpendicular to sample surface for t = 0.62: a) d = 110 nm, b) d = 3 4 0 0 n m .
drops to zero) equals 1.7 and H,± thereby assumes the physical meaning of surface critical field/-/ca in agreement with the Saint-James and de Gennes theory [15]. Moreover, for very thick samples (d >>A (t)) showing asymmetrical superconductivity patterns like the one reported in fig. lb, the H~ value for the maximum of each curve agrees well with Landau's model, eq. (2). On the basis of these observations the symmetrical transition pattern was considered as typical of mixed- to normal-state transition, while the asymmetrical curve as representative of intermediate- to normal-state transition. Then the analysis of the change in line shape allows to study the passage from intermediate- to mixedstate behaviour: the largest film thickness for which we observe a symmetrical curve pattern is assumed as critical thickness d± [10]. Fig. 2 compares our main findings for Sn, In and Pb films at different temperature values (from t = 0.5 to t = 0.85) with the Lasher theory previsions for a sample characterized by the G.L. parameter k(t, d), critical field H i and thickness d. The solid line represents the boundary between mixed- and intermediate-state regions as deduced from Lasher's theory. Symbols I and M come from observations of intermediate- or mixed-state behaviours, respectively, as derived from an asymmetry analysis of superconductivity transition curves. Fig. 2 clearly shows a good agreement between Lasher's previsions and our data obtained by the microwave phase-sensitive technique. Moreover, it may be seen that temperature does not seem to affect
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
220
210
2k2
MIXED STATE (M)
0.7-
210
~ ~
-
.o
o.s-
~o
®
~'u'
~
,3o ,~o
A~
I® 1_6o ~ ~ ~ , {[]
o.1~ - f
~
INTERMErlIATE STATE(I)
[] , []L
0
,0
0.2
Sn
Pb A
, 0.4
,
l 0,6
t
0.8
~
1
10
~
t
_1.2
dVH/4%
Fig. 2. Comparison between observations of mixed (M) or intermediate (I) state behaviours deduced from analysis of superconductivity transition curve asymmetry and Lasher's previsions. So]id line is Lasher's boundary between intermediate- and mixed-states.
critical thickness d± significantly. In additon, table I reports our d L values, deduced as previously described, values obtained by other authors as a result of critical field experiments and of direct observation of microscopic state. Table I also reports Lasher's theoretical estimates.
[16]. For d >dL, the ratio HJHc increases with increasing thickness. However, our experimental results cannot be succesfully accounted for by the simple relation between critical field and thickness that follows from Landau's model. The solid line in fig. 3 was deduced from eq. (2) by assuming A (t)= A (0)/(1- t4) 1/2 where A (0) is the
4.2. Criticalfields data
Table I Values for critical thicknesses obtained with microwave phase-sensitive technique, those obtained from critical field data, from magnetic structure observations and from Lasher's previsions
Fig. 3 reports the ratio HA ~He for Sn films as a function of d at reduced temperature t = 0.55. H A is the field value corresponding to the maximum of each superconductivity transition curve determined to within 2%. For thin films, perpendicular critical field decreases with increasing thickness and our experimental data are in good agreement with Thinkham's expression, eq. (1), up to d = 190 nm. The short dashed line was deduced from eq. (1) by using the empirical expression for A(0, d) obtained from the parallel critical field analysis performed on the same samples following the procedure described in
Film
d t a) (nm)
dj. b) (nm)
dl c) (rim)
d.Ld) (rim)
Sn (2.1 K) In (2.1 K) Pb (4.2 K)
190 110 280
520 600 1000
180 80 250
180 90 300
")Thickness of thickest film for which mixed state was observed. b) From macroscopic critical field results [5, 6]. =)From magnetic structure observations [8]. o) From Lasher's theory [3].
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields 1 Sn
t =055
~Hc
"
221
Sn ~,,,,,""~
"
t-0.85
0.6 - - , -
0.8-
0.2
0.6-
t, I
0.4-
//
..." -
I__
I 102
103
r
I
,~ f
,\ , ~ ~ " ' " ~
0.2101
J I
104
d/nm
1__ t=0.7
0.6
105
Fig. 3. Ratio H±IHc for Sn at t = 0.55vs. thickness: . . . . from ref. 1; - and . . . . from ref. 11; w _ _ _ _ computed from eq. (2) of text using Bardeen's calculation of surface energy parameter [13].
0.2
,
,,f
1
I
,
I
I
/
~'\
,
t -0.55
0.6
value obtained from fitting H~ data vs. t for sufficiently thick samples, i.e. A (0) = 205 nm. Eq. (2) may be seen to provide a good fitting for the data over a rather limited range of thicknesses, thus supporting the theoretical restriction that eq. (2) is valid only for d >>A (t). Thus the question arises as to whether the Landau laminar model is adequate to describe the entire intermediate thickness region or whether, as some results show [9], a thickness dependent energy parameter A(t, d) should be taken into account. We have compared our experimental data with expressions similar to eq. (2) for other intermediate state models. One such comparison was that with the dotted line in fig. 3, which corresponds to the Caroli et al. model [11] in which elliptic cylinder structures were investigated rather than laminae. As a further step, we also modified eq. (2) to allow for the A (0, d) thickness dependence deduced from Bardeen's previsions [13] (dashed line in fig. 3). Nevertheless, as can easily be seen, these other expressions only provide a good fitting for the experimental data in the large thickness range also taking into account the introduction of a thickness dependent energy parameter. So far we have found a semiempirical expression valid for any thickness d > d± at the same time taking into account the asymptotic behaviour of eq. (2). Fig. 4 shows ratio vs.
H±/Hc
0.2-
lOT
102
14
105
d/nm
Fig. 4. Ratio H±/Hc for Sn vs. thickness for three different temperatures.
d for Sn films at three temperatures. The solid line that describes our experimental results best was obtained from eq. (2) which was modified as follows:
HJHc= l-[O'8dA(t)exp(-A(d) + fl)]~a,
(4)
where A (0) = 205 nm. Coefficient fl was obtained as a parameter by interpolating data with a nonlinear least-square fitting technique. The values obtained in the reduced temperature range, 0.50.85, fall b e t w e e n - 0 . 2 and -0.3. The dashed line represents eq. (1). We should like to emphasize that agreement between eq. (4) and our experimental data is rather remarkable at all temperatures and for all other samples analysed (In and Pb). 5. Conclusions
We want here to summarize some interesting data emerged from experimental results:
222
S. Onori and A. Rogani / Thickness dependence of perpendicular critical fields
i) The values of the critical thickness dz for Sn, In and Pb films obtained by the microwave phase-sensitive technique are in good agreement with the microscopic theory [3, 4] and with direct observation of magnetic flux structure [8]. ii) The d± values obtained by others [5-7] in critical field measurements from the intersection of the theoretical expressions for thin and thick films region, eqs. (1) and (2), exceed the values predicted by the microscopic theory. Nevertheless, our experimental results (fig. 3) clearly show that expressions like eq. (2) should be used only in the limit of very thick samples. An inappropriate use of the theory of the intermediate state can lead to a fictitious increase in d~. Then the disagreement between microscopic theory and results from critical field analysis could be due to the inadequacy of the expressions like eq. (2) to describe the entire intermediate thickness region. iii) The measured critical magnetic field values, /-/1, are in good agreement with Tinkham's theory for thin films and with Landau's model for thick films. For films with thicknesses d ~>A no theoretical estimate yet exists. A better understanding of intermediate-state structure in this region would be useful but difficult. We have proposed a semiempirical functional expression which fits all the experimental data concerning H± for any thickness d > d I. What is more, the intersection of this expression with the theoretical one for thin films, eqs. (4) and (1), is for a di
value in agreement with theoretical estimates [3, 41.
Acknowledgements We wish to thank M. Flamini for his technical assistance.
References [1] [2] [3] [4] [5]
M. Tinkham, Phys. Rev. 129 (1963) 2413. L.D. Landau, Phys. Z. Sowiet 11 (1937) 129. G. Lasher, Phys. Rev. 154 (1966) 345. G. Maki, Ann. Phys. (New York) 34 (1965) 363. R.E. Miller and G.D. Cody, Phys. Rev. 173 (1968) 494, 481. [6] B.L. Bran&, R.D. Park and R.D. Chaudhari, J. Low Temp. Phys. 4 (1971) 41. [7] M,D. Maioney, F. de la Cruz and M. Cardona, Phys. Rev. B5 (1972) 3558. [8]' C.J. Dolan and J. Silcox, Phys. Rev. Lett. 30 (1973) 603. [9] T. Miyazaki, B. Shinozaki, R. Aoki, M. Katayama and S. Yamashita, J. Phys. Soc. Japan 45 (1978) 1487. [10] M. Flamini, S. Onori and A. Rogani, I! Nuovo Cimento
2D (1983) 1497. [11] E. Guyon, C. Caroli and A. Martiner, ft. Phys. (Paris) 25
(1964) 683. [12] E.A. Davies, Proc. R. Soc. London, A255 (1960) 407. [13] J. Bardeen, Phys. Rev. 94 (1954) 554. [14] E.Di Crescenzo, M. Flamini, P.L. Indovina, S. Onori and A. Rosati, Ann. Ist. Sup. San 7 (1971) 560. [15] P.G. de Gennes and D. Saint-James, Phys. Lett. 7 (1963)
306. [16] S. Onori and A. Rogani, PhysicaB100 (1980)93.