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Effect of order parameter fluctuations on the specific heat of superconductor films
Solid State Communications,
Vol. 10, pp. 567—570, 1972.
Pergamon Press.
Printed in Great Britain
EFFECT OF ORDER PARAMETER FLUCTUATIONS ON THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS *
L. Gunther Department of Physics, Tufts University, Medford, Massachusetts 02155 and L.W. Gruenbergt Francis Bitter National Magnetic Laboratory M.I.T. Cambridge, Massachusetts 02139 (Received 20 Lecember 1971 by P.G. de Gennes)
We study the effect of fluctuations of the order parameter on the specific heat of one, two, and three dimensional superconductors using a Hartree approximation in the free energy functional. In all three dimensions, as the temperature is lowered the specific heat rises continuously to a peak value of the bulk discontinuity in the absence of fluctuations. Our results are compared with recent specific heat data on superconductor films. While a two-parameter fit yields excellent agreement with experiment, the experimental width of the transition decreases (rather than increases as theory indicates) with the resistance per square R0.
THE QUESTION of whether superconductivity can occur in a sample greatly restricted in one or its dimensions at more great of length these past has few been years.investigated While one cannot have Off-Diagonal Long Range Order (ODLRO)1 in one or two dimensions, there still remains the question of whether physical properties of superconductors can be exhibited in samples of finite size. Experimentalists have already observed vanishingly small resistivity in thin films2’6 and theories3 ~have appeared to explain these results.
One of the approaches we have taken to the problem has been a Hartree-like 4 A approximation similar approach in the free-energy functionaL has been taken by S. Mar~eljaet al.5’6 While the basis for the approximation has not yet been clearly defined, it is the simplest approximation that can be made in order to examine the fluctuation effects of the quartic term in the freeenergy functional, and which leads to physical predictions which we believe have a grain of qualitative truth? in fact, we will show that the specific heat predicted by the theory is in excellent agreement with recent experiments on superconductor films.
_____________
*
Partially supported by National Science
We assume that the probability of having an
Foundation Grants GP—6474 and GP—16025.
. order parameter ~s proportional to P W’~ = exp [— ~F h,LII1
Supported by the United States Office of Naval Research and the National Science Foundation. 567
THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS
• 568
where for small
Vol. 10, No. 6
we can write: F1VJ~=$dic~*G~[a+ bI~)!2/2 _th2/2m)V21c1j(~). (1)
three-dimensions. We furthermore introduce the Ginzburg—Landau length ~GL= [Ti2(T— T coherence 2, which equals 0)/2maT0 P’ 0.72 ~ in the clean limit and 0.85~I(el)in the
The Hartree approximation consists ci replacing ~2 in the fourth order term by its thermal
dirty limit. Then, our results can be sumarized as follows: There exists a critical region about the temperature 1~ whose width i~is given by
~,,
average plus a small fluctuation, and neglecting terms of second order in the fluctuation: thus,
Clean
F~çl~-~ F 2/2m) V2iç~) 11 ~ =$d~*(~)[a+CT_(h [1a2/2b
(kF
>
(bkT’cl) ~
(a
±
0d)~ (k~.ldY 2~3 (k.S21/~
3 1
D
-
(k~~
=
-_~<~i
a
(k~
2— D (7)
D
—
0i~’
>
~ =
3
(2)
where fi is the sample volume and the self-energy a is determined self-consistently by the2 equation 2 b a~b
Dirty (k: l3~T1
~~)4
(3)
— y~2q2/2rnY’.
In performing the q-space integrations it is sometimes necessary to introduce a cutoff which we will call q~. For a pure superconductor q~ ~ the Pippard (temperature-independent) coherence lengthy For a dirty superconductor the corresponding cutoff is on the order of the
where d is the film thickness and S the crosssectional area of the wire. In the dirty, twodimensional case, use of the phenomenological relation: conductivity ne2 l/mvF e2k,~I /3~h (8) =
=
leads to the relation 2/h) R~,
T 0
where R
(9)
0.009 (e
=
0 is the normal state resistance per square. Quartic terms in FH lc~become important
inverse mean free path 1’.
in this region.
*
The correlation function g (~) = <~ G’) ~,~(0)> has the Ornstein—Zernike form for large r: in three-dimensions,
The specific heat is obtained from the thermodynamic relation:
-~
g(r)
=
IrnkT\ exp[—r/~(T)1 __________ \27T~2)
(4)
r
C
where the coherence length c~(T)is given by 2/2m (a -i- a)J’72. (5) ~(T) = [1i In three-dimensions there is a phase
a
i =
-
2a T aT — (~F)J~
(10)
exp [—/3FHIcO~I.
(11)
where
exp (—~3F)
=
transition — indicated by the divergence of — at a temperature which differs from the bulk B.C.S. transition temperature T 0 byorder a of niiniscule amount i~ 1, which is on the ~ /(k~)2in the clean limit and TO/(kFlf in
Close to T~,the internal energy is given by
the dirty limit. k~is the Fermi wave vector,
which should be compared with the Landau theory expression, wherein is replaced by the square of the non-fluctuating order parameter g~:~Li~ — a/b (h2 /2m~Lb) T when T< T 0 and 1j,~ 0 when T>
It is convenient to define a reduced temperature r~(T — T~)/
2 ~Ij
2m~L
=
(6)
=
—
=
where 7 T~ in the case of one-or twodimensions and T~ T0 — A 7~ in the case of =
=
In all three dimensionalities the specific
(12)
Vol. 10, No. 6
THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS
heat rises continuously over a temperature range on the order of several times ‘i~from zero to a value Ac given by the bulk specific heat discontinuity in the absence of fluctuations. We find
r(T)
C(T)/AC =
ii
GL
+
rt
3
—
D (T> T) c
2I2}~ 2
—
D
ie~ /e(T)r~
(13)
/~(T)i-~’2I2 ~-‘ ~ D
+ [~
=
1’21 0
/e(T)T
+ [~
—
These results are in disagreement with those reference. 6, wherein specific heat has a of sharp peak near 7~in allthethree dimensionalities. COMPARISON WITH EXPERIMENT 9(Z--M) have recently Zallyresults and Mochel reported for the specific heat of films of BiSb alloy for which I <<~, so that we are in the dirty limit. To compare the Hartree approximation specific heat with experiment we now present further details of our results for twodimensional samples.
value of AC used in plotting was theoretically determined by Z—M9 to be 0.0143 ergs/°K ±10%, The constants A and B were determined by fitting the experimentally determined f[C(T) Ito a straight line, with the result that A = 322.6/°K and B = 669.3. The value of T 0 should be about 2.1°K,which corresponds to a value of To of about 1.6 xT i0~. Since ‘r~”isextremely close to B, and 0 iseGL not)2~~%3~ precisely we cannot estimate lntq~ Theknown, experimental points are in excellent agreement with the theoritical curve. Similar a~reementis obtained in the case of Z—M9’s 800 A film of BiSb 043 with a distinct decrease in This decrease in9 is in disagreement with our theory since Z—M found R 0 to be larger for the 800 A sample [cf.equation (9) 1. ;.
%
DISCUSSION V/e must admit that the agreement between theory and experiment is much better than we had anticipated hence the long delay in publishing our theoretical results. The specific heat singularities obtained using Ornstein—Zernike —
We introduce the parameter
~
(14) whose temperature dependence in the Hartree approximation is determined by the equation 7)
T+T
~o02F
~°
OIO~
:
\
(15)
2~~1’
0
lnlI1+(q~~GL~
U0iY~4’-
By combining equations (13), (14), and (15) and noting that (q~~ 0~/i~ ~kF~od>> 1 in both
~ o~L I 2.060
the clean and dirty limits we obtain f(C)
[~‘—iI + In [c’—iI = AT
where A (T0 iY~ and B
B (16)
—
2/T)(17) i~— in
569
[(q C ~GL )
0
In Fig. 1 we compare the experimental results of a 1350 A film of BiSbo. 40 with our theory. The
2.060 2.00 2 20 TEYPERATURE (K)
2J40
FIG. 1. Comparison of theory with experiment. The dots are the experiipental values of the specificagainst plotted heat of temperature. a 1350 A sample The of solid BiSb060 curve is the theoretical curve of specific heat versus temperature determined by the equation f(C/0.0143) (O.O143/C—1)+ln(0.O143/C--l) 322.6 T
—
669.3.
• 570
THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS
theory1°or the Thouless theory8 are highly exaggerated. Fluctuations are saturated, as in the Hartree approximation, due to the quartic terms in the free-energy funtional. They are unbounded in the other theories. We believe, though, that the Hartree approximation is inadequate for a three dimensional superconductor in that the saturation effect is overly exaggerated and the actual specific heat might retain a singularity. Finally, the fact that the net change in the specific heat is AC, independent of the dimensionality is just another example of a
Vol. 10, No. 6
property associated with a phase transition into the superfluid state which does not require the existence of ODLRO.1’ This result should be even less surprising in the case of the specific heat, which is explicitly dependent only upon short-range correlations. Acknowledgements — We are indebted to Dr.Y.Imry for discussions we have had concerning this work. We are extremely grateful to Dr.J.Mochel for sending us his specific heat data prior to publication and for numerous enlightening conversations concerning the significance of his results.
REFERENCES 1.
BYERS N. and YANG C.N., Phys.Rev.Lext. 7, 46 (1961).
2.
See, for example, GLOVER RE., III, Phys.Letx. 25A,
3.
See, for example, ASLAMAZOV L.G. and LARKIN A.!., Fiz.Tverd.TeIa ~0, 1104 (1968); FERRELL R.A., Proc Batsheva Seminar on Quantum Fluids,(edited by WISER N. and AMIT D.J.) Gordon & Breach, New York (1970); and more recently, KELLER J. and KORENMAN V., Phys.Rev.Leti 27, 1270 (1971) and PATTON B., Phys.Rev.L ext .27, 1273 (1971).
4.
GUNTHER L., Phys.Lext. 28A, 102 (1969).
5.
MARCELJA S., Phys.L ett. 28A, 180 (1968), MARCELJA S., MASKER W.E. and PARKS R.D., Phys. Rev.Lett. 22, 124 (1969).
6.
MASKER WE., MARCELJA S. and PARKS RD., Phys.Rev. 188, 745 (1969).
7.
The theoretical results of this paper were first presented a few years ago. See GUNTHER L. and GRUENBERG L.W., Bull. Am.Phys.Soc. 14, 633 (1969).
8.
See THOULESS D., Ann.Phys.(N.Y.) 10, 553 (1960).
9.
ZALLY G.D. and MOCHEL 3M., Phys.Rev.L ext. 27, 1710 (1971). Equation (4) is a misprint.
542
(1967).
10.
See,,for example, BROUT R., Phase Transitions, Benjamin, New York (1965).
11.
See, for example, GUNTHER L. and IMRY Y., Solid State Commun. 7, 1391 (196~,where it is shown that one can observe flux quantization in a thin hollow cylinder in the absence of ODLRO. Nous étudions l’effet de fluctuations du paramètre d’ordre sur la chaleur spéciiique de supraconducteurs a un, deux, et trois dimensions, en employant une approximation de Hartree dans la fonctionnelle de l’énergie libre. Pour ces trois cas, quand la temperature s’abaisse la chaleur spCcifique croft continuement jusqu’I Ia valeur maximum de la discontinuité d’un échantillon massif en ijabsence de fluctuations. Nous comparons nos résultats avec des experiences récentes sur la chaleur spécifique de films du supraconducteur BiSb. Un ajustement a deux paramétres nous dorine un accord excellent avec les experiences. Mais, Ia largeur expérimentale de la transition décrott avec la résistance-par-carré R 0, et Ia théorie indique une dépendance contraire.
Vol. 10, pp. 567—570, 1972.
Pergamon Press.
Printed in Great Britain
EFFECT OF ORDER PARAMETER FLUCTUATIONS ON THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS *
L. Gunther Department of Physics, Tufts University, Medford, Massachusetts 02155 and L.W. Gruenbergt Francis Bitter National Magnetic Laboratory M.I.T. Cambridge, Massachusetts 02139 (Received 20 Lecember 1971 by P.G. de Gennes)
We study the effect of fluctuations of the order parameter on the specific heat of one, two, and three dimensional superconductors using a Hartree approximation in the free energy functional. In all three dimensions, as the temperature is lowered the specific heat rises continuously to a peak value of the bulk discontinuity in the absence of fluctuations. Our results are compared with recent specific heat data on superconductor films. While a two-parameter fit yields excellent agreement with experiment, the experimental width of the transition decreases (rather than increases as theory indicates) with the resistance per square R0.
THE QUESTION of whether superconductivity can occur in a sample greatly restricted in one or its dimensions at more great of length these past has few been years.investigated While one cannot have Off-Diagonal Long Range Order (ODLRO)1 in one or two dimensions, there still remains the question of whether physical properties of superconductors can be exhibited in samples of finite size. Experimentalists have already observed vanishingly small resistivity in thin films2’6 and theories3 ~have appeared to explain these results.
One of the approaches we have taken to the problem has been a Hartree-like 4 A approximation similar approach in the free-energy functionaL has been taken by S. Mar~eljaet al.5’6 While the basis for the approximation has not yet been clearly defined, it is the simplest approximation that can be made in order to examine the fluctuation effects of the quartic term in the freeenergy functional, and which leads to physical predictions which we believe have a grain of qualitative truth? in fact, we will show that the specific heat predicted by the theory is in excellent agreement with recent experiments on superconductor films.
_____________
*
Partially supported by National Science
We assume that the probability of having an
Foundation Grants GP—6474 and GP—16025.
. order parameter ~s proportional to P W’~ = exp [— ~F h,LII1
Supported by the United States Office of Naval Research and the National Science Foundation. 567
THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS
• 568
where for small
Vol. 10, No. 6
we can write: F1VJ~=$dic~*G~[a+ bI~)!2/2 _th2/2m)V21c1j(~). (1)
three-dimensions. We furthermore introduce the Ginzburg—Landau length ~GL= [Ti2(T— T coherence 2, which equals 0)/2maT0 P’ 0.72 ~ in the clean limit and 0.85~I(el)in the
The Hartree approximation consists ci replacing ~2 in the fourth order term by its thermal
dirty limit. Then, our results can be sumarized as follows: There exists a critical region about the temperature 1~ whose width i~is given by
~,,
average plus a small fluctuation, and neglecting terms of second order in the fluctuation: thus,
Clean
F~çl~-~ F 2/2m) V2iç~) 11 ~ =$d~*(~)[a+CT_(h [1a2/2b
(kF
>
(bkT’cl) ~
(a
±
0d)~ (k~.ldY 2~3 (k.S21/~
3 1
D
-
(k~~
=
-_~<~i
a
(k~
2— D (7)
D
—
0i~’
>
~ =
3
(2)
where fi is the sample volume and the self-energy a is determined self-consistently by the2 equation 2 b a~b
Dirty (k: l3~T1
~~)4
(3)
— y~2q2/2rnY’.
In performing the q-space integrations it is sometimes necessary to introduce a cutoff which we will call q~. For a pure superconductor q~ ~ the Pippard (temperature-independent) coherence lengthy For a dirty superconductor the corresponding cutoff is on the order of the
where d is the film thickness and S the crosssectional area of the wire. In the dirty, twodimensional case, use of the phenomenological relation: conductivity ne2 l/mvF e2k,~I /3~h (8) =
=
leads to the relation 2/h) R~,
T 0
where R
(9)
0.009 (e
=
0 is the normal state resistance per square. Quartic terms in FH lc~become important
inverse mean free path 1’.
in this region.
*
The correlation function g (~) = <~ G’) ~,~(0)> has the Ornstein—Zernike form for large r: in three-dimensions,
The specific heat is obtained from the thermodynamic relation:
-~
g(r)
=
IrnkT\ exp[—r/~(T)1 __________ \27T~2)
(4)
r
C
where the coherence length c~(T)is given by 2/2m (a -i- a)J’72. (5) ~(T) = [1i In three-dimensions there is a phase
a
i =
-
2a T aT — (~F)J~
(10)
exp [—/3FHIcO~I.
(11)
where
exp (—~3F)
=
transition — indicated by the divergence of — at a temperature which differs from the bulk B.C.S. transition temperature T 0 byorder a of niiniscule amount i~ 1, which is on the ~ /(k~)2in the clean limit and TO/(kFlf in
Close to T~,the internal energy is given by
the dirty limit. k~is the Fermi wave vector,
which should be compared with the Landau theory expression, wherein
It is convenient to define a reduced temperature r~(T — T~)/
2 ~Ij
2m~L
=
(6)
=
—
=
where 7 T~ in the case of one-or twodimensions and T~ T0 — A 7~ in the case of =
=
In all three dimensionalities the specific
(12)
Vol. 10, No. 6
THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS
heat rises continuously over a temperature range on the order of several times ‘i~from zero to a value Ac given by the bulk specific heat discontinuity in the absence of fluctuations. We find
r(T)
C(T)/AC =
ii
GL
+
rt
3
—
D (T> T) c
2I2}~ 2
—
D
ie~ /e(T)r~
(13)
/~(T)i-~’2I2 ~-‘ ~ D
+ [~
=
1’21 0
/e(T)T
+ [~
—
These results are in disagreement with those reference. 6, wherein specific heat has a of sharp peak near 7~in allthethree dimensionalities. COMPARISON WITH EXPERIMENT 9(Z--M) have recently Zallyresults and Mochel reported for the specific heat of films of BiSb alloy for which I <<~, so that we are in the dirty limit. To compare the Hartree approximation specific heat with experiment we now present further details of our results for twodimensional samples.
value of AC used in plotting was theoretically determined by Z—M9 to be 0.0143 ergs/°K ±10%, The constants A and B were determined by fitting the experimentally determined f[C(T) Ito a straight line, with the result that A = 322.6/°K and B = 669.3. The value of T 0 should be about 2.1°K,which corresponds to a value of To of about 1.6 xT i0~. Since ‘r~”isextremely close to B, and 0 iseGL not)2~~%3~ precisely we cannot estimate lntq~ Theknown, experimental points are in excellent agreement with the theoritical curve. Similar a~reementis obtained in the case of Z—M9’s 800 A film of BiSb 043 with a distinct decrease in This decrease in9 is in disagreement with our theory since Z—M found R 0 to be larger for the 800 A sample [cf.equation (9) 1. ;.
%
DISCUSSION V/e must admit that the agreement between theory and experiment is much better than we had anticipated hence the long delay in publishing our theoretical results. The specific heat singularities obtained using Ornstein—Zernike —
We introduce the parameter
~
(14) whose temperature dependence in the Hartree approximation is determined by the equation 7)
T+T
~o02F
~°
OIO~
:
\
(15)
2~~1’
0
lnlI1+(q~~GL~
U0iY~4’-
By combining equations (13), (14), and (15) and noting that (q~~ 0~/i~ ~kF~od>> 1 in both
~ o~L I 2.060
the clean and dirty limits we obtain f(C)
[~‘—iI + In [c’—iI = AT
where A (T0 iY~ and B
B (16)
—
2/T)(17) i~— in
569
[(q C ~GL )
0
In Fig. 1 we compare the experimental results of a 1350 A film of BiSbo. 40 with our theory. The
2.060 2.00 2 20 TEYPERATURE (K)
2J40
FIG. 1. Comparison of theory with experiment. The dots are the experiipental values of the specificagainst plotted heat of temperature. a 1350 A sample The of solid BiSb060 curve is the theoretical curve of specific heat versus temperature determined by the equation f(C/0.0143) (O.O143/C—1)+ln(0.O143/C--l) 322.6 T
—
669.3.
• 570
THE SPECIFIC HEAT OF SUPERCONDUCTOR FILMS
theory1°or the Thouless theory8 are highly exaggerated. Fluctuations are saturated, as in the Hartree approximation, due to the quartic terms in the free-energy funtional. They are unbounded in the other theories. We believe, though, that the Hartree approximation is inadequate for a three dimensional superconductor in that the saturation effect is overly exaggerated and the actual specific heat might retain a singularity. Finally, the fact that the net change in the specific heat is AC, independent of the dimensionality is just another example of a
Vol. 10, No. 6
property associated with a phase transition into the superfluid state which does not require the existence of ODLRO.1’ This result should be even less surprising in the case of the specific heat, which is explicitly dependent only upon short-range correlations. Acknowledgements — We are indebted to Dr.Y.Imry for discussions we have had concerning this work. We are extremely grateful to Dr.J.Mochel for sending us his specific heat data prior to publication and for numerous enlightening conversations concerning the significance of his results.
REFERENCES 1.
BYERS N. and YANG C.N., Phys.Rev.Lext. 7, 46 (1961).
2.
See, for example, GLOVER RE., III, Phys.Letx. 25A,
3.
See, for example, ASLAMAZOV L.G. and LARKIN A.!., Fiz.Tverd.TeIa ~0, 1104 (1968); FERRELL R.A., Proc Batsheva Seminar on Quantum Fluids,(edited by WISER N. and AMIT D.J.) Gordon & Breach, New York (1970); and more recently, KELLER J. and KORENMAN V., Phys.Rev.Leti 27, 1270 (1971) and PATTON B., Phys.Rev.L ext .27, 1273 (1971).
4.
GUNTHER L., Phys.Lext. 28A, 102 (1969).
5.
MARCELJA S., Phys.L ett. 28A, 180 (1968), MARCELJA S., MASKER W.E. and PARKS R.D., Phys. Rev.Lett. 22, 124 (1969).
6.
MASKER WE., MARCELJA S. and PARKS RD., Phys.Rev. 188, 745 (1969).
7.
The theoretical results of this paper were first presented a few years ago. See GUNTHER L. and GRUENBERG L.W., Bull. Am.Phys.Soc. 14, 633 (1969).
8.
See THOULESS D., Ann.Phys.(N.Y.) 10, 553 (1960).
9.
ZALLY G.D. and MOCHEL 3M., Phys.Rev.L ext. 27, 1710 (1971). Equation (4) is a misprint.
542
(1967).
10.
See,,for example, BROUT R., Phase Transitions, Benjamin, New York (1965).
11.
See, for example, GUNTHER L. and IMRY Y., Solid State Commun. 7, 1391 (196~,where it is shown that one can observe flux quantization in a thin hollow cylinder in the absence of ODLRO. Nous étudions l’effet de fluctuations du paramètre d’ordre sur la chaleur spéciiique de supraconducteurs a un, deux, et trois dimensions, en employant une approximation de Hartree dans la fonctionnelle de l’énergie libre. Pour ces trois cas, quand la temperature s’abaisse la chaleur spCcifique croft continuement jusqu’I Ia valeur maximum de la discontinuité d’un échantillon massif en ijabsence de fluctuations. Nous comparons nos résultats avec des experiences récentes sur la chaleur spécifique de films du supraconducteur BiSb. Un ajustement a deux paramétres nous dorine un accord excellent avec les experiences. Mais, Ia largeur expérimentale de la transition décrott avec la résistance-par-carré R 0, et Ia théorie indique une dépendance contraire.