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Dimensional crossover and finite size effects in superconducting films
Ph'YSICA
Physica C 179 (1991) 125-130 North-Holland
Dimensional crossover and finite size effects in superconducting films T. Schneider a n d J.-P. L o c q u e t IBM Research Division, Zurich Research Laboratory, CH-8803 Riischlikon, Switzerland
Received l0 June 1991
Using the Ginzburg-Landau description of dirty and anisotropic superconductors, the dependence of Tc and of the upper critical field H~2 on film thickness are traced back to a finite size effect, corresponding to a dimensional crossover. Since the strength of fluctuations increases with reduced dimensionality, their effect is to decrease the transition temperature. If the film thickness exceedsthe low temperature value of the perpendicular correlation length, the crossovercan also be seen in the temperature dependence o f H ¢ 1 2 . Our analysisof the experimental data for Fe/V/Fe sandwiches and Pb/Ge superlanices clearly reveals this crossoverphenomenon.
1. Introduction Although superlattice [ 1,2 ] and layered structures [ 3 ] have become a promising tool in the search for new superconducting phenomena, thin films [ 4 ] and ultrathin crystal properties are a crucial aspect of the understanding of superconductivity in superlattices and layered bulk materials. In fact, the physical properties near a continuous phase transition do indeed have a marked dependence on dimensionality. This can be explored with samples where one of the physical dimensions becomes comparable to the correlation length. In the infinite system the correlation length diverges by approaching the transition temperature from below as ~. =bj_[(Tc-T)/T~]
-v3 ,
(l)
where v3 is the critical exponent characterizing the divergence of the correlation length in the three-dimensional bulk system. In a film of finite thickness d, however, ~l cannot exceed d. According to finite size scaling, the dependence of T¢ on film thickness is then given by [5] T c ( d ) / T ¢ = 1 - ( b ± / d ) t/~3 .
(2)
The term b . provides an estimate of the zero temperature correlation length and also of the range of the interaction driving the transition. It is also de-
pendent on the geometry and boundary conditions. Clearly, eq. (2) is strictly valid only close to To. Otherwise one expects correction-to-scaling terms to contribute. From eq. (2) it can be seen that T c ( d ) decreases with decreasing d, This decrease of T¢ is a dimensional crossover phenomenon, driven by the strength of the order parameter fluctuations, which increases with reduced dimensionality. The fall of To, caused by reducing the dimensionality in terms of thickness d has been observed in 4He films [6], superconductor/insulator superlattices [ 1,2,7 ] at the limit where the coupling between the superconducting layers becomes negligible, in ultrathin crystals of layered superconductors [ 8 ] and in superconducting films [ 9, l 0 ]. Moreover, extensive studies of this phenomenon have also been performed in model systems, including the Ising, Heisenberg and xy-model [ 5 ]. In fig. 1 we sketched the phase diagram of these model systems and 4He. The shaded region is the region where the quasi-twodimensional fluctuations predominate. In other words, by approaching T o ( d ) from above or below, quasi-two-dimensional critical behavior appears in the shaded region and dominates close to T o ( d ) . In 4He and superconductors, the order parameter is a two-component vector, and the bulk Tc is expected to be an isolated point connected to a line Kosterlitz-Thouless transitions, where even below Tc no
0921-4534/91/$03.50 © 1991 ElsevierScience Publishers B.V. All fights reserved.
T. Schneider, J.-P. Locquet / Dimensional crossoverandfinite size effects
126
p
I
rc
o
1
lO
20 M
Fig. 1. Phase diagram of a slab or film of M atomic layers along the z-axis and infinite in the two remaining directions, d=Ma is the film thickness where a denotes the separation of the atomic layers. Tcis the bulk transition temperature (M= ~ ). The shaded region is the region where two-dimensional fluctuations predominate. long-range order exists for any finite thickness [ 11 ]. The observation of the associated phenomena requires d values of the order of b±. Thus, in 4He bz ~, 3.5 A and cuprate superconductors with b~ ~ c, where c denotes the lattice constant perpendicular to the layers, ultrathin films are required. In conventional superconductors including the dirty limit, b± is at least one order of magnitude larger. However, in clean and conventional superconductors with a nearly spherical Fermi surface, the reduction of film thickness might lead to a regime where the conduction bands, due to the discreteness of the wave vector k l normal to the film plane, break up into subbands (approximately nn/d). This quantum size effect will lead to an oscillatory dependence of the density of states and other properties on film thickness [12,13]. In this paper we examine a superconducting slab of thickness d. The slab can also be realized in superconductor/insulator superlattices in the limit where the coupling between the superconducting layers is negligible. We consider the dirty limit where quantum size effects are suppressed. In section 2 we sketch the theoretical treatment using the Ginzburg-Landau approach for dirty and an-
isotropic superconductors. Anisotropy is introduced to account for the diffusion in the restricted geometry. Moreover, we assume that the order parameter vanishes at the film surfaces. In agreement with finite size scaling, the fall of Tc with decreasing film thickness is traced back to the dimensional crossover, which is controlled by the perpendicular diffusion coefficient. The upper critical field Hc~ turns out to be independent of the film thickness d, while H~2 exhibits a pronounced d-dependence. In the limit d > ~l the temperature dependence of H~2 is identical to that of an anisotropic bulk superconductor, while for d < ~± the Tinkham formula [ 14] describing the two-dimensional behavior of ultrathin films is recovered. Our treatment, however, describes the full crossover. In section 3 we compare our theoretical results with recent experiments on F e / V / F e sandwiches and P b / Ge superlattices, where the Ge layers are sufficiently thick for the Pb sheets to be treated as independent isolated superconducting slabs. The essential features of the experimental results for To(d) and the upper critical fields Hc~2 and H¢~, including the dimensional crossover, are in remarkable agreement with our theoretical description.
2. Theoretical treatment
Close to the bulk Tc the linearized Ginzburg-Landau equation is expected to be a reliable approximation. In the dirty limit it reads [ 2 ]
h ( V - 2ieA~D ( V ~-c ]
2ieA~ ij\ - ~-c , / ~ , = a ~ ,
(3)
where A = (0, x cos O - z sin O, 0 ) H ,
(4)
H = (sin O, 0, cos O ) H , and
:°, D,° °0)
D=~
0
a=ot( T - Tc) .
(5)
D~_
Tc denotes the bulk transition temperature, Dll and D± the diffusion constants parallel (x, y) and perpendicular (z) to the layers. For very thick films,
T. Schneider, J.-P. Locquet / Dimensionalcrossoverandfinite size effects diffusion becomes isotropic, D,=D± and independent of d. We impose boundary conditions so that the order parameter vanishes at z = 0 and z=d. Next we consider the zero field case. The transition temperature is then given by the ground state of eq. (3) satisfying the boundary conditions. The solution is u( z ) = Uosin k l z with
a=a(Te(d)-Tc)=
hD ± x 2 d2
(6)
-l
~=(~)2(
1
(~(d))2=
- -1otd2 Tc
'
(7)
Te~d))
hD, aTe(d) "
(12)
In the case of a parallel field (0= ~/2), eqs. (3-5) give
(
hD. ~z 2 -hDll
H222 u ( z ) = a u ( z ) . k'/~O /
for the ground state. Accordingly,
T~(d)_l Tc
127
(13)
/
An approximate solution for the ground state that satisfies the boundary conditions u (z) = 0 at z = 0 and z = d is readily obtained by the substitution [ 15 ]
where z
2__[d'X 2 2[n ) ~) cot~z ,
(14)
(~o)2= ~D± aTe
(8)
is the amplitude of the correlation length in the dirty limit. Equation (4) fully agrees with the finite size scaling expression (2), because 1/v3= 2 in the mean field treatment. Moreover, the amplitude b is expressed in terms of the perpendicular correlation length. It is important to emphasize that eq. (7) is valid only for d> n~° .
(9)
In conventional and dirty bulk superconductors, ~ is of the order of 500 ~. Thus, to satisfy the condition of eq. (9), rather thick films are needed, where ~± = ~li= ~. By reducing the film thickness, however, the restricted geometry leads to anisotropic diffusion of the carriers in the normal state, and the mean free path perpendicular to the slab cannot exceed the thickness of the film. Thus for d £ x~o one expects D± ~ d and according to eq. (8) ~o2 ~d, so that 1
Tc(d) Te
1 d"
(10)
The upper critical field H ~ ( O = 0 ) is fully equivalent to that of anisotropic superconductors and yields H
in the potential term. The resulting equation is exactly solvable and gives for the ground state energy
[15] h 2 n 2 D ± ( l + /16d4H2DH + 1 ) 2d 2 q
a=
and in turn 0o (1 + d2~ '/2 H~2=2d~,\ x2~2} ,
with
(11)
(16)
where
~ (T) for ~± (T) < d ,
(¢°(d))2 1 Te )
for~± ( T ) > d . (17) For d,> n~.L we recover the expression for anisotropic bulk superconductors H~2 -
¢o
2n~ ±~ll
°° 0
- 2x~ ° ~
- 2~°~,t~,
(15)
rc-~)
"
(18)
For d 2 / ( ~ o )2,~ 1 - T/Te (d), however, there is a region close to Te where
128
T. Schneider, J.-P. Locquet / Dimensional crossoverand finite size effects r
H~2~
1
,
I
(19) 7.o
and the 2D behavior dominates. This expression is close to the Tinkham formula [14 ] for thin films. Thus, the replacement (14) is correct in the two limits. In between it provides an approximate description of the dimensional crossover. In fact, for suitable values of d and ~o and sufficiently close to Tc(d), nlcl2 will exhibit two-dimensional behavior (eq. (19) ). By lowering the temperature, however, it will cross over to the temperature dependence of the 3D bulk system (eq. (19)). The full crossover including the limiting behavior is controlled, at least approximately, by eq. (16). From eqs. ( 11 ) and (16) we obtain H~2
x~lJ l +
d2
(20)
~ 6.8 6.6 6.4 6.2 o
I
I
lOO
200 d (A)
Fig. 2. Tcvs. d for Pb/Ge superlanices with dGe= 30/k according to ref. [ 7 ]. 008
0.06
as the ratio of the critical fields. 0.04
3. Comparison with experiment In this section we compare the main theoretical results with experimental data. From eq. (7) it is seen that the fall of Tc with decreasing thickness provides in principle an estimate of the correlation length ~o. In conventional and dirty superconductors, however, ~0 is of the order of 500 A. As a consequence, the asymptotic behavior given by eq. (7) will dominate for very thick films only. In fig. 2 we depicted the Pb data of ref. [ 7 ] for thin slabs. From the dependence of T~ on the thickness of the Ge layers it is known that T~ saturates for doe > 30 A. Accordingly, the data corresponds closely to an isolated Pb slab of thickness d. Since the correlation length of the bulk exceeds the thickness of these slabs, eq. (7) is no longer applicable. D becomes d-dependent and eq. (10) is expected to apply. Indeed, from the plot 1 - T ~ ( d ) / T e versus 1/d, shown in fig. 3, it is clear that for d < 245/k, (~o2)2~ d. Here one enters the regime where the mean free path is limited by the film thickness and n~ ° becomes smaller than d. According to eqs. (18) and (19), this is simply the condition for a dimensional crossover in H~2 as the temperature is lowered. This expectation is fully
0.02 I
0
I
0.005 1/d (~,-')
I
omo
0.015
Fig. 3. 1-Tc(d)/T¢ vs. 1/d for the data shown in fig. 2. The straight line is a guide to the eye. confirmed by the measured temperature dependence of H~2 in Pb slabs, as shown in fig. 4, where we plotted (nlcl2) 2 versus 1 - T / T c ( d ) . For comparison with eq. (16), we determined ~ from the H ~ data and ~o ~ 62/k from H~2. Apparently, eq. (16) leads to excellent agreement over the entire temperature range, including the crossover, as the temperature is lowered. As expected linearity occurs close to Tc and remains valid up to T / T ¢ = 0 . 9 . In this temperature interval the 2D fluctuations dominate, as indicated by the shaded area in fig. 1. For much lower temperatures, the perpendicular correlation length becomes smaller than the thickness and 3D behavior appears. Systems where our theoretical treatment applies more directly are sandwiches like F e / V / F e , because the Fe layers act as strong pair-breakers. Here our
129
T. Schneider, J.-P. Locquet / Dimensional crossover and finite size effects
2.0.10 3
I
I
I
I
I
-V-. 06
1'5"103
, = ~
l,
• Ill•
0.4
•
•Q
I
lo.lo°
0.2 0
0.5=103
0
I
0.002
I
I
0.006
I
I
0.010
1/M
0.2
0.4 1-T / Tc(d)
016
0.8
Fig. 4. (Hlcl2)2 vs. l - T / T c ( d ) of dvb= 175 A, taken from a Pb/ Ge supedanice with dc~=30 A. (A) experimental data taken from ref. [ 7 ], ( ) temperature dependence according to eq. ( 16 ) with C° = 62 A, ~ = 245 A and Tc(d) = 6.929 K,(- - - ) asymptotic linear slope according to eq. (19).
I
I
I
I
I
I
I
6
O•
~4
• O°°
2 0
I
0
200
I
600 M
Fig. 6. (1-Tc(d)/Tc) '/2 vs. I/M for the data shown in fig. 5. The straight line indicates the estimated asymptotic slope yielding n~° ~364 a~777 ,/L fig. 7 for slabs o f thickness d ranging from M = 153 to 1011. Here we plotted the data ofref. [ 10] in terms o f (H~2)2 versus 1 - T / T ~ ( M ) . Below M < 314 the temperature dependence is essentially linear and corresponds to two-dimensional behavior. For M > 314 the temperature interval around T~ where linearity holds is seen to shrink and for lower temperature, the upturn signals the crossover to threedimensional behavior, dominated by the second term in eq. (16).
I
1000
Fig. 5. Tc vs. M, the number of atomic V layers in the Fe/V/Fe sandwich, according to ref. [10]. d=Ma=M×2.135 A is the thickness. boundary condition, requiring a vanishing order parameter outside the superconducting slab, is certainly justified. The experimental results in ref. [ 10] for the dependence o f T~ on the thickness o f the V slab are shown in fig. 5, revealing again the expected fall o f Tc with decreasing thickness. To estimate the bulk correlation length we plotted the data in fig. 6 in terms o f ( 1 - T ~ ( M ) / T c ) 1/2 versus I / M . M denotes the n u m b e r o f atomic V layers in the slab with spacing 2.135 A. The straight line provides an estimate, apparently a lower bound, for C° yielding n~ ° = 3 6 4 a ~ 777 A. Thus, dimensional crossover in the temperature dependence o f HIcl2 can be expected for films thicker than M > 365. This expectation is nicely confirmed by the experimental data shown in
4. Conclusions Adopting an anisotropic description o f the superconducting properties o f a thin slab, we have shown that the fall o f Tc with reduced thickness can be understood in terms o f a boundary effect and the thickness dependence o f the perpendicular diffusion coefficient. This leads to a thickness dependence o f the perpendicular correlation length, which controls the fall o f Tc and the temperature dependence o f the parallel upper critical field. However, in the evaluation o f this field the boundary effect turns out to be crucial and leads to two distinct regimes depending on the magnitude o f d / ~ ° . For large d / ~ ° values the system is essentially three-dimensional, while for small values the two-dimensional behavior, reminiscent o f an ultrathin slab, is recovered. The comparison with experimental data revealed remarkable agreement in the limiting behavior including the crossover region. In the F e / V / F e sandwich the crossover was seen as a function o f thick-
130
T. Schneider, J.-P. Locquet / Dimensional crossover and finite size effects 300
I
r
J
{
I
I
M = 153
/
250
173
=
244
~.~ 150 ~o
m314 loo
o
•
0
0.1
0.2
0.3
o
0.4
1-T/Tc
/
0.5
0.6
0.7
(M)
Fig. 7. (H~2)2 vs. 1 - T/To(M) for F e / V / F e sandwiches of thickness d=Ma. Data taken from ref. [ 10].
ness. In Pb, however, the crossover was seen in the temperature dependence, since ~o ~ d < ~ . This example shows that a film which is two-dimensional in the classical sense ( d < ~ ) can show three-dimensional behavior if ~o ~ d as the temperature is lowered. This crossover cannot be explained by the standard theory of thin slabs, where the crossover between the two limits, anisotropic bulk and ultrathin slab is not included.
Acknowledgements We acknowledge stimulating discussions with G. Deutscher, J.G. Bednorz, K.A. Miiller and A. Schmidt.
References [ 1] S.T. Ruggiero and M.R. Beasley, in: Synthetic Modulated Structures, eds. L.L. Chang and B.C. Giessen (Academic Press, New York, 1985) p. 435. [2] B.Y. Jin and J.B. Ketterson, Adv. Phys. 38 (1989) 189. [3] D.E. Prober, R.E. Schwall and M.R. Beasley, Phys. Rev. B 21 (1980) 2717.
[4] D.L. Miller, M. Strongin and O.F. Kammerer, Phys. Rev. B13 (1976) 4834; M. Strongin and O.F. Kammerer, J.E. Crow, R.D. Parks, D.H. Douglas Jr. and M.A. Jensen, Phys. Rev. Lett. 21 (1968) 1320. [ 5 ] M.N. Barber, in: Phase Transitions and Critical Phenomena, vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983). [6] Y.Y. Yu, D. Finotello and F.M. Gasparini, Phys. Rev. B 10 (1989) 6519. [7]J.-P. Locquet, D. Neerinck, H. Vanderstraeten, W. Sevenhans, V. Van Haesendonck, Y. Bruynseraede, H. Homma and I.K. Schuller, Jpn. J. Appl. Phys. 26 (1987) 1431; J.-P. I.x~cquet, in: Struktuur en supergeleiding van artificieel gelaagde systemen, Ph.D. Thesis, K.U. Leuven, Belgium (1989). [8] R.F. Frindt, Phys. Rev. Lett. 28 (1972) 299. [9] W. Buckel, Supraleitung (Physik Verlag, Weinheim, 1972 ). [10] H.K. Wong and J.B. Ketterson, J. Low Temp. Phys. 63 (1986) 139. [ 11 ] V. Ambegaokar, B.I. Halperin, D.R. Nelson and E.D. Siggia, Phys. Rev. B 21 (1980) 1806. [ 12] J.M. Blatt and C.J. Thompson, Phys. Rev. Lett. 10 (1963) 332. [13] D.S. Falk, Phys. Rev. 132 (1963) 1576. [ 14] M. Tinkham, Introduction to Superconductivity (Krieger, Malabar, 1980). [15] D. ter Haar, Selected Problems in Quantum Mechanics (Infsearch limited, London ( 1964 ).
Physica C 179 (1991) 125-130 North-Holland
Dimensional crossover and finite size effects in superconducting films T. Schneider a n d J.-P. L o c q u e t IBM Research Division, Zurich Research Laboratory, CH-8803 Riischlikon, Switzerland
Received l0 June 1991
Using the Ginzburg-Landau description of dirty and anisotropic superconductors, the dependence of Tc and of the upper critical field H~2 on film thickness are traced back to a finite size effect, corresponding to a dimensional crossover. Since the strength of fluctuations increases with reduced dimensionality, their effect is to decrease the transition temperature. If the film thickness exceedsthe low temperature value of the perpendicular correlation length, the crossovercan also be seen in the temperature dependence o f H ¢ 1 2 . Our analysisof the experimental data for Fe/V/Fe sandwiches and Pb/Ge superlanices clearly reveals this crossoverphenomenon.
1. Introduction Although superlattice [ 1,2 ] and layered structures [ 3 ] have become a promising tool in the search for new superconducting phenomena, thin films [ 4 ] and ultrathin crystal properties are a crucial aspect of the understanding of superconductivity in superlattices and layered bulk materials. In fact, the physical properties near a continuous phase transition do indeed have a marked dependence on dimensionality. This can be explored with samples where one of the physical dimensions becomes comparable to the correlation length. In the infinite system the correlation length diverges by approaching the transition temperature from below as ~. =bj_[(Tc-T)/T~]
-v3 ,
(l)
where v3 is the critical exponent characterizing the divergence of the correlation length in the three-dimensional bulk system. In a film of finite thickness d, however, ~l cannot exceed d. According to finite size scaling, the dependence of T¢ on film thickness is then given by [5] T c ( d ) / T ¢ = 1 - ( b ± / d ) t/~3 .
(2)
The term b . provides an estimate of the zero temperature correlation length and also of the range of the interaction driving the transition. It is also de-
pendent on the geometry and boundary conditions. Clearly, eq. (2) is strictly valid only close to To. Otherwise one expects correction-to-scaling terms to contribute. From eq. (2) it can be seen that T c ( d ) decreases with decreasing d, This decrease of T¢ is a dimensional crossover phenomenon, driven by the strength of the order parameter fluctuations, which increases with reduced dimensionality. The fall of To, caused by reducing the dimensionality in terms of thickness d has been observed in 4He films [6], superconductor/insulator superlattices [ 1,2,7 ] at the limit where the coupling between the superconducting layers becomes negligible, in ultrathin crystals of layered superconductors [ 8 ] and in superconducting films [ 9, l 0 ]. Moreover, extensive studies of this phenomenon have also been performed in model systems, including the Ising, Heisenberg and xy-model [ 5 ]. In fig. 1 we sketched the phase diagram of these model systems and 4He. The shaded region is the region where the quasi-twodimensional fluctuations predominate. In other words, by approaching T o ( d ) from above or below, quasi-two-dimensional critical behavior appears in the shaded region and dominates close to T o ( d ) . In 4He and superconductors, the order parameter is a two-component vector, and the bulk Tc is expected to be an isolated point connected to a line Kosterlitz-Thouless transitions, where even below Tc no
0921-4534/91/$03.50 © 1991 ElsevierScience Publishers B.V. All fights reserved.
T. Schneider, J.-P. Locquet / Dimensional crossoverandfinite size effects
126
p
I
rc
o
1
lO
20 M
Fig. 1. Phase diagram of a slab or film of M atomic layers along the z-axis and infinite in the two remaining directions, d=Ma is the film thickness where a denotes the separation of the atomic layers. Tcis the bulk transition temperature (M= ~ ). The shaded region is the region where two-dimensional fluctuations predominate. long-range order exists for any finite thickness [ 11 ]. The observation of the associated phenomena requires d values of the order of b±. Thus, in 4He bz ~, 3.5 A and cuprate superconductors with b~ ~ c, where c denotes the lattice constant perpendicular to the layers, ultrathin films are required. In conventional superconductors including the dirty limit, b± is at least one order of magnitude larger. However, in clean and conventional superconductors with a nearly spherical Fermi surface, the reduction of film thickness might lead to a regime where the conduction bands, due to the discreteness of the wave vector k l normal to the film plane, break up into subbands (approximately nn/d). This quantum size effect will lead to an oscillatory dependence of the density of states and other properties on film thickness [12,13]. In this paper we examine a superconducting slab of thickness d. The slab can also be realized in superconductor/insulator superlattices in the limit where the coupling between the superconducting layers is negligible. We consider the dirty limit where quantum size effects are suppressed. In section 2 we sketch the theoretical treatment using the Ginzburg-Landau approach for dirty and an-
isotropic superconductors. Anisotropy is introduced to account for the diffusion in the restricted geometry. Moreover, we assume that the order parameter vanishes at the film surfaces. In agreement with finite size scaling, the fall of Tc with decreasing film thickness is traced back to the dimensional crossover, which is controlled by the perpendicular diffusion coefficient. The upper critical field Hc~ turns out to be independent of the film thickness d, while H~2 exhibits a pronounced d-dependence. In the limit d > ~l the temperature dependence of H~2 is identical to that of an anisotropic bulk superconductor, while for d < ~± the Tinkham formula [ 14] describing the two-dimensional behavior of ultrathin films is recovered. Our treatment, however, describes the full crossover. In section 3 we compare our theoretical results with recent experiments on F e / V / F e sandwiches and P b / Ge superlattices, where the Ge layers are sufficiently thick for the Pb sheets to be treated as independent isolated superconducting slabs. The essential features of the experimental results for To(d) and the upper critical fields Hc~2 and H¢~, including the dimensional crossover, are in remarkable agreement with our theoretical description.
2. Theoretical treatment
Close to the bulk Tc the linearized Ginzburg-Landau equation is expected to be a reliable approximation. In the dirty limit it reads [ 2 ]
h ( V - 2ieA~D ( V ~-c ]
2ieA~ ij\ - ~-c , / ~ , = a ~ ,
(3)
where A = (0, x cos O - z sin O, 0 ) H ,
(4)
H = (sin O, 0, cos O ) H , and
:°, D,° °0)
D=~
0
a=ot( T - Tc) .
(5)
D~_
Tc denotes the bulk transition temperature, Dll and D± the diffusion constants parallel (x, y) and perpendicular (z) to the layers. For very thick films,
T. Schneider, J.-P. Locquet / Dimensionalcrossoverandfinite size effects diffusion becomes isotropic, D,=D± and independent of d. We impose boundary conditions so that the order parameter vanishes at z = 0 and z=d. Next we consider the zero field case. The transition temperature is then given by the ground state of eq. (3) satisfying the boundary conditions. The solution is u( z ) = Uosin k l z with
a=a(Te(d)-Tc)=
hD ± x 2 d2
(6)
-l
~=(~)2(
1
(~(d))2=
- -1otd2 Tc
'
(7)
Te~d))
hD, aTe(d) "
(12)
In the case of a parallel field (0= ~/2), eqs. (3-5) give
(
hD. ~z 2 -hDll
H222 u ( z ) = a u ( z ) . k'/~O /
for the ground state. Accordingly,
T~(d)_l Tc
127
(13)
/
An approximate solution for the ground state that satisfies the boundary conditions u (z) = 0 at z = 0 and z = d is readily obtained by the substitution [ 15 ]
where z
2__[d'X 2 2[n ) ~) cot~z ,
(14)
(~o)2= ~D± aTe
(8)
is the amplitude of the correlation length in the dirty limit. Equation (4) fully agrees with the finite size scaling expression (2), because 1/v3= 2 in the mean field treatment. Moreover, the amplitude b is expressed in terms of the perpendicular correlation length. It is important to emphasize that eq. (7) is valid only for d> n~° .
(9)
In conventional and dirty bulk superconductors, ~ is of the order of 500 ~. Thus, to satisfy the condition of eq. (9), rather thick films are needed, where ~± = ~li= ~. By reducing the film thickness, however, the restricted geometry leads to anisotropic diffusion of the carriers in the normal state, and the mean free path perpendicular to the slab cannot exceed the thickness of the film. Thus for d £ x~o one expects D± ~ d and according to eq. (8) ~o2 ~d, so that 1
Tc(d) Te
1 d"
(10)
The upper critical field H ~ ( O = 0 ) is fully equivalent to that of anisotropic superconductors and yields H
in the potential term. The resulting equation is exactly solvable and gives for the ground state energy
[15] h 2 n 2 D ± ( l + /16d4H2DH + 1 ) 2d 2 q
a=
and in turn 0o (1 + d2~ '/2 H~2=2d~,\ x2~2} ,
with
(11)
(16)
where
~ (T) for ~± (T) < d ,
(¢°(d))2 1 Te )
for~± ( T ) > d . (17) For d,> n~.L we recover the expression for anisotropic bulk superconductors H~2 -
¢o
2n~ ±~ll
°° 0
- 2x~ ° ~
- 2~°~,t~,
(15)
rc-~)
"
(18)
For d 2 / ( ~ o )2,~ 1 - T/Te (d), however, there is a region close to Te where
128
T. Schneider, J.-P. Locquet / Dimensional crossoverand finite size effects r
H~2~
1
,
I
(19) 7.o
and the 2D behavior dominates. This expression is close to the Tinkham formula [14 ] for thin films. Thus, the replacement (14) is correct in the two limits. In between it provides an approximate description of the dimensional crossover. In fact, for suitable values of d and ~o and sufficiently close to Tc(d), nlcl2 will exhibit two-dimensional behavior (eq. (19) ). By lowering the temperature, however, it will cross over to the temperature dependence of the 3D bulk system (eq. (19)). The full crossover including the limiting behavior is controlled, at least approximately, by eq. (16). From eqs. ( 11 ) and (16) we obtain H~2
x~lJ l +
d2
(20)
~ 6.8 6.6 6.4 6.2 o
I
I
lOO
200 d (A)
Fig. 2. Tcvs. d for Pb/Ge superlanices with dGe= 30/k according to ref. [ 7 ]. 008
0.06
as the ratio of the critical fields. 0.04
3. Comparison with experiment In this section we compare the main theoretical results with experimental data. From eq. (7) it is seen that the fall of Tc with decreasing thickness provides in principle an estimate of the correlation length ~o. In conventional and dirty superconductors, however, ~0 is of the order of 500 A. As a consequence, the asymptotic behavior given by eq. (7) will dominate for very thick films only. In fig. 2 we depicted the Pb data of ref. [ 7 ] for thin slabs. From the dependence of T~ on the thickness of the Ge layers it is known that T~ saturates for doe > 30 A. Accordingly, the data corresponds closely to an isolated Pb slab of thickness d. Since the correlation length of the bulk exceeds the thickness of these slabs, eq. (7) is no longer applicable. D becomes d-dependent and eq. (10) is expected to apply. Indeed, from the plot 1 - T ~ ( d ) / T e versus 1/d, shown in fig. 3, it is clear that for d < 245/k, (~o2)2~ d. Here one enters the regime where the mean free path is limited by the film thickness and n~ ° becomes smaller than d. According to eqs. (18) and (19), this is simply the condition for a dimensional crossover in H~2 as the temperature is lowered. This expectation is fully
0.02 I
0
I
0.005 1/d (~,-')
I
omo
0.015
Fig. 3. 1-Tc(d)/T¢ vs. 1/d for the data shown in fig. 2. The straight line is a guide to the eye. confirmed by the measured temperature dependence of H~2 in Pb slabs, as shown in fig. 4, where we plotted (nlcl2) 2 versus 1 - T / T c ( d ) . For comparison with eq. (16), we determined ~ from the H ~ data and ~o ~ 62/k from H~2. Apparently, eq. (16) leads to excellent agreement over the entire temperature range, including the crossover, as the temperature is lowered. As expected linearity occurs close to Tc and remains valid up to T / T ¢ = 0 . 9 . In this temperature interval the 2D fluctuations dominate, as indicated by the shaded area in fig. 1. For much lower temperatures, the perpendicular correlation length becomes smaller than the thickness and 3D behavior appears. Systems where our theoretical treatment applies more directly are sandwiches like F e / V / F e , because the Fe layers act as strong pair-breakers. Here our
129
T. Schneider, J.-P. Locquet / Dimensional crossover and finite size effects
2.0.10 3
I
I
I
I
I
-V-. 06
1'5"103
, = ~
l,
• Ill•
0.4
•
•Q
I
lo.lo°
0.2 0
0.5=103
0
I
0.002
I
I
0.006
I
I
0.010
1/M
0.2
0.4 1-T / Tc(d)
016
0.8
Fig. 4. (Hlcl2)2 vs. l - T / T c ( d ) of dvb= 175 A, taken from a Pb/ Ge supedanice with dc~=30 A. (A) experimental data taken from ref. [ 7 ], ( ) temperature dependence according to eq. ( 16 ) with C° = 62 A, ~ = 245 A and Tc(d) = 6.929 K,(- - - ) asymptotic linear slope according to eq. (19).
I
I
I
I
I
I
I
6
O•
~4
• O°°
2 0
I
0
200
I
600 M
Fig. 6. (1-Tc(d)/Tc) '/2 vs. I/M for the data shown in fig. 5. The straight line indicates the estimated asymptotic slope yielding n~° ~364 a~777 ,/L fig. 7 for slabs o f thickness d ranging from M = 153 to 1011. Here we plotted the data ofref. [ 10] in terms o f (H~2)2 versus 1 - T / T ~ ( M ) . Below M < 314 the temperature dependence is essentially linear and corresponds to two-dimensional behavior. For M > 314 the temperature interval around T~ where linearity holds is seen to shrink and for lower temperature, the upturn signals the crossover to threedimensional behavior, dominated by the second term in eq. (16).
I
1000
Fig. 5. Tc vs. M, the number of atomic V layers in the Fe/V/Fe sandwich, according to ref. [10]. d=Ma=M×2.135 A is the thickness. boundary condition, requiring a vanishing order parameter outside the superconducting slab, is certainly justified. The experimental results in ref. [ 10] for the dependence o f T~ on the thickness o f the V slab are shown in fig. 5, revealing again the expected fall o f Tc with decreasing thickness. To estimate the bulk correlation length we plotted the data in fig. 6 in terms o f ( 1 - T ~ ( M ) / T c ) 1/2 versus I / M . M denotes the n u m b e r o f atomic V layers in the slab with spacing 2.135 A. The straight line provides an estimate, apparently a lower bound, for C° yielding n~ ° = 3 6 4 a ~ 777 A. Thus, dimensional crossover in the temperature dependence o f HIcl2 can be expected for films thicker than M > 365. This expectation is nicely confirmed by the experimental data shown in
4. Conclusions Adopting an anisotropic description o f the superconducting properties o f a thin slab, we have shown that the fall o f Tc with reduced thickness can be understood in terms o f a boundary effect and the thickness dependence o f the perpendicular diffusion coefficient. This leads to a thickness dependence o f the perpendicular correlation length, which controls the fall o f Tc and the temperature dependence o f the parallel upper critical field. However, in the evaluation o f this field the boundary effect turns out to be crucial and leads to two distinct regimes depending on the magnitude o f d / ~ ° . For large d / ~ ° values the system is essentially three-dimensional, while for small values the two-dimensional behavior, reminiscent o f an ultrathin slab, is recovered. The comparison with experimental data revealed remarkable agreement in the limiting behavior including the crossover region. In the F e / V / F e sandwich the crossover was seen as a function o f thick-
130
T. Schneider, J.-P. Locquet / Dimensional crossover and finite size effects 300
I
r
J
{
I
I
M = 153
/
250
173
=
244
~.~ 150 ~o
m314 loo
o
•
0
0.1
0.2
0.3
o
0.4
1-T/Tc
/
0.5
0.6
0.7
(M)
Fig. 7. (H~2)2 vs. 1 - T/To(M) for F e / V / F e sandwiches of thickness d=Ma. Data taken from ref. [ 10].
ness. In Pb, however, the crossover was seen in the temperature dependence, since ~o ~ d < ~ . This example shows that a film which is two-dimensional in the classical sense ( d < ~ ) can show three-dimensional behavior if ~o ~ d as the temperature is lowered. This crossover cannot be explained by the standard theory of thin slabs, where the crossover between the two limits, anisotropic bulk and ultrathin slab is not included.
Acknowledgements We acknowledge stimulating discussions with G. Deutscher, J.G. Bednorz, K.A. Miiller and A. Schmidt.
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[4] D.L. Miller, M. Strongin and O.F. Kammerer, Phys. Rev. B13 (1976) 4834; M. Strongin and O.F. Kammerer, J.E. Crow, R.D. Parks, D.H. Douglas Jr. and M.A. Jensen, Phys. Rev. Lett. 21 (1968) 1320. [ 5 ] M.N. Barber, in: Phase Transitions and Critical Phenomena, vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983). [6] Y.Y. Yu, D. Finotello and F.M. Gasparini, Phys. Rev. B 10 (1989) 6519. [7]J.-P. Locquet, D. Neerinck, H. Vanderstraeten, W. Sevenhans, V. Van Haesendonck, Y. Bruynseraede, H. Homma and I.K. Schuller, Jpn. J. Appl. Phys. 26 (1987) 1431; J.-P. I.x~cquet, in: Struktuur en supergeleiding van artificieel gelaagde systemen, Ph.D. Thesis, K.U. Leuven, Belgium (1989). [8] R.F. Frindt, Phys. Rev. Lett. 28 (1972) 299. [9] W. Buckel, Supraleitung (Physik Verlag, Weinheim, 1972 ). [10] H.K. Wong and J.B. Ketterson, J. Low Temp. Phys. 63 (1986) 139. [ 11 ] V. Ambegaokar, B.I. Halperin, D.R. Nelson and E.D. Siggia, Phys. Rev. B 21 (1980) 1806. [ 12] J.M. Blatt and C.J. Thompson, Phys. Rev. Lett. 10 (1963) 332. [13] D.S. Falk, Phys. Rev. 132 (1963) 1576. [ 14] M. Tinkham, Introduction to Superconductivity (Krieger, Malabar, 1980). [15] D. ter Haar, Selected Problems in Quantum Mechanics (Infsearch limited, London ( 1964 ).