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Fluctuation effects in quasi-one-dimensional superconductors
Physica 108B {1981) 1179-1180 North-Holland Publishing Company
SC 1
FLUCTUATION EFFECTS IN QUASI-ONE-DIMENSIONAL SUPERCONDUCTORS
H.J. Schulz
I. Institut f~r Theoretische Physik, Universit~t Hamburg, 2000 Hamburg 36, West Germany The influence of the strong anisotropy of a quasi-one-dimensional metal on superconducting fluctuation properties is investigated using time-dependent Ginzburg-Landau theory. We find that the longitudinal fluctuation conductivity is largely enhanced by the anisotropy, whereas there is no enhancement of the transverse conductivity or the fluctuation induced diamagnetism. These results may explain the unusual behaviour of the longitudinal conductivity of the quasi-one-dimensional superconductor (TMTSF)2PF 6 above its transition temperature.
|.
INTRODUCTION
In strictly one-dimensional systems strong fluctuations destroy long-range order at any finite temperature. In quasi-one-dimensional metals the weak coupling between adjacent conducting chains makes phase transitions at a finite temperature T possible, however, above • C . T one still expects fluctuatlon effects to be m~ch stronger than in isotropic systems. Here we investigate fluctuation effects above the superconducting transition temperature. Our theory may be applied to organic superconductors like (TMTSF)pPF 6 [I]. These compounds are characteriz~d by-a 6onductivity anisotropy of the order IO ~ [2], and corresponding anisotrpies are also found in the plasma edge [3] and in the upper critical fields of the superconducting state [4]. In the following Chapter we use time-dependent Ginzburg-Landau theory to treat fluctuation effects in quasi-one-dimensional superconductors. Though this approach suffers from serious limitations, especially due to the competing superconducting and charge (or spin) density wave instabilities of quasi-one-dimensional metals, it provides a starting point for the understanding of the influence of strong electronic anisotropy on fluctuation effects. The application of the theory to (TMTSF)2PF 6 is discussed at the end of the paper. 2. GINZBURG-LANDAU THEORY OF SUPERCONDUCTING FLUCTUATION EFFECTS
To derive the coefficients of the GinzburgLandau theory microscopically we use as a model system a square array of parallel Conducting chains with the possibility of electron tuneling between adjacent chains. The single electron propagator is G(q,~ n) =
!
,
1[~n+Slgn(~n)~]-
e(q)
I,
(I)
where the free-electron energy is given by
(~)
I
2
= 2-~qmz + J(cos qx a +cos qya) - cF .
(la)
0378.4363/81/0000~)000/$02.50 © North-HollandPublishingCompany
J/2 and a are the transverse tunneling integral and lattice constant respectively, and T is the single electron lifetime due to impurity scattering within a single chain. The calculations follow closely ref. 5, and the free energy is given to lowest order in the Fourier components ~k of the superconducting order parameter by 2 j2 F = [(a+q~+ ---=-(2-cosq a-cosq a ) ) I ~ [ 2 (2) " L ~ 2vZm x y ~ ' F with q 8~2T2 IT a'l T (2a) C~
=
=
7~(3)mvF2x((m~TT) -I) •
•
,
rO
T°
•
O
where v F is the Ferml veloclty, T the meanfield transition temperature and ~(x) Gorkov's x-function [5]. From eq.(2) the longitudinal and transverse correlation lengths follow as ~II(T) = (2m~)-I/2, ~I(T ) =
Ja
~ (T)
. (3)
d'v F It is interesting to note that the t~ansverse coupling in (2) is proportional to J , i.e. to the square of the transverse electronic effective mass at the Fermi level. This is a consequence of the weak interchain coupling, which may be considered as Josephson coupling, and therefore the anisotropy of the order parameter fluctuations as given by ~ii/~l is equal to the~anisotropy of the electronic spectrum. The JZ-dependence is valid as long as the transverse coupling is smaller than the Fermi energy, i.e. as long as the Fermi surface is open in the transverse directions.On the other hand, if the Fermi surface becomes closed (J~eF) , the the transverse coupling of the order parameter is linear in J, so that the order parameter anisotropy is proportional to the square root of the anisotropy of the electronic spectrum, as is the case in effective mass theory [6]. However, in the organic superconductors single-electron properties [2-4] indicate that the weak coupling condition is well satisfied.
1179
1180
The longitudinal and transverse supercurrents are me ~ q z l ~ i 2 2eaj2 ~sinqxal~l 2 (4) JU =~-' J~ 2 + " q mvF q For the calculation of the fluctuation conductivity, a knowledge of the order parameter dynamics is necessary [7]. In time-dependent Ginzburg-Landau theory [8] this is given by the equation of motion ~a' 3~k 6F 8T ~t + ~
verse conductivity is dominated by singleelectron conduction. Another quantity of interest is the fluctuation induced diamagnetim. Including a vector potential in the free energy (2) [9], it is straightforward to calculate the fluctuation induced diamagnetism [IO]: kT e 2 ~(T) ×~ 6~ c 2 ~U (r) 2 ' (9a) kT e 2 $I (T) (~(r)2+ __)I/2
= O
(5) ×U
Following closely the method of ref.7, we obtain 2 VF e2a' [m__]1/2 [ a 1 °i| jma2 16 ~2aJ l- (a2+4~2)i/2 (a2+4~) , (6a)
2 II1J2 m
o~.
e a = -16
~ [l 3a a3 ] L 2 (a2+4~12) 1/2 + 2 ( a 2 + ~ [2)3/2 ' (6b) for the contribution of the fluctuations to the longitudinal and transverse conductivity. . . . . . o We now conslder two llmltlng cases:(1) Near T , • , c a becomes sufflclently small, so that the transverse correlation length is larger than the lattice spacing. Then, the fluctuations are essentially three-dimensional, and (6) becomes 3D VF 2 e2a,[m )I/2 3D j2a2 3D ~
°ll •
j- a2 r /r j
=--r°,
.
vF
.
o
-1/2
.
This shows the typical (T-T) divergence of • , c • a three-dimensional superconductor [7] in both the longitudinal and transverse conductivity, however, the longitudinal value is enhanced over its valu@ i~ @ n isotropic superconductor by a factor VF/J~a-, which may be very large in organic conductors due to the small interchain tunneling integral. This enhancement is due to the small transverse correlation length compared to 6|| which leads to a large mean square amplitude of the superconducting order parameter fluctuations. (ii) At temperature sufficiently above T ° $~.becomes smaller than the interchain • c dlstance, i.e. fluctuations on adjacent chains are essentially uncorrelated and can therefore be considered as one-dimensional. In this limit one has ID ~e 2 ~' ID 3~2e2j 4 a' o||
16a2 (2m 3) 1 / 2 ,
o l"
32v4
(2m3a5) l / 2
lD . (8) l| i s t h e w e l l known r e s u l t f o r a o n e - d i m e n sional superconductor [7]. A transverse supercul"rent between adjacent chains requires a well d e v e l o p e d o r d e r p a r a m e t e r on b o t h c h a i n s . As t h e propability for this to happen decr~ses with increasing temperature for ~j
6~ c 2 ~(T)
,
(9b)
for a field applied perpendicular and parallel to the chain direction, respectively. Both expressions apply in the low field limit. Eq.(9a) shows that the fluctuation diamagnetism ~s not enhanced over its value in an isotropic superconductor. This result may be understood noting that the circular Meissner currents which are at the origin of the diamagnetism have to pass between adjacent chains, and that these currents are strongly suppressed by the weak interchain coupling (eq.(4)). This effect compensates the large amplitude of the fluctuations. 3. DISCUSSION Eq. (6a) shows that the longitudinal fluctuation conductivity is largely enhanced in quasi-onedimensional superconductors.Using experimental 3 results [2-4] an enhancement factor of |O ..IO results for (TMTSF)2PF 6. Indeed 6 for a mean free path £ of about 20A and using T =5.6K good numerical agreement can be foun~ between eq.8 and experimental results up to 4OK [ll]. Two points should be noted:(|) Strong fluctuations lead to a considerable depression of T below T ° however, in the one-dimensional fluctuation c' regime an effective T ° can be used [ll].(ii) The • e Maki-Thompson dlagrams [12] further increase the fluctuation conductivity, so that even for considerably larger values of £ than the somewhat small value of 2OA superconducting fluctuations can explain the resistivity of (TMTSF)2PF 6. [|] JErome, D. et al.,J.Phys.Lett.41(1980),L95. [2] Bechgaard, K. et al., Solid State Commun. 33(|980), 1119. [3] Jacobsen,C.S. et al., preprint. [4] D. Mailly, M. Ribault, to be published; D. JErome, H.J. Schulz, in J. Bernasconi, T. Schneider(eds.), Physics in One Dimension (Springer, Berlin, 1 9 8 1 ) , p.239. [5] Gorkov, L.P., Sov. Phys. JETP IO(1960),998. [6] Tilley, D.R., Proc. Phys. Soc.86(1965),289. [7] Schmidt, H., Z. Phys. 216(1968), 336. [g] Schmid, A.,Phys.Kondens.Mat. 5(1966), 302. [9] Turkevich, L.A., Klemm, R.A., Phys. Rev. Bl9 (1979), 2520. [IO] Schmid, A., Phys. Rev. 180 (1969), 527. [ll] Schulz, H.J. et al., J. Phys., in press. [12] Maki, K., Progr. Theor. Phys. 40(1968),193; Thompson, R.S., Phys. Rev. Bl (1970), 327.
SC 1
FLUCTUATION EFFECTS IN QUASI-ONE-DIMENSIONAL SUPERCONDUCTORS
H.J. Schulz
I. Institut f~r Theoretische Physik, Universit~t Hamburg, 2000 Hamburg 36, West Germany The influence of the strong anisotropy of a quasi-one-dimensional metal on superconducting fluctuation properties is investigated using time-dependent Ginzburg-Landau theory. We find that the longitudinal fluctuation conductivity is largely enhanced by the anisotropy, whereas there is no enhancement of the transverse conductivity or the fluctuation induced diamagnetism. These results may explain the unusual behaviour of the longitudinal conductivity of the quasi-one-dimensional superconductor (TMTSF)2PF 6 above its transition temperature.
|.
INTRODUCTION
In strictly one-dimensional systems strong fluctuations destroy long-range order at any finite temperature. In quasi-one-dimensional metals the weak coupling between adjacent conducting chains makes phase transitions at a finite temperature T possible, however, above • C . T one still expects fluctuatlon effects to be m~ch stronger than in isotropic systems. Here we investigate fluctuation effects above the superconducting transition temperature. Our theory may be applied to organic superconductors like (TMTSF)pPF 6 [I]. These compounds are characteriz~d by-a 6onductivity anisotropy of the order IO ~ [2], and corresponding anisotrpies are also found in the plasma edge [3] and in the upper critical fields of the superconducting state [4]. In the following Chapter we use time-dependent Ginzburg-Landau theory to treat fluctuation effects in quasi-one-dimensional superconductors. Though this approach suffers from serious limitations, especially due to the competing superconducting and charge (or spin) density wave instabilities of quasi-one-dimensional metals, it provides a starting point for the understanding of the influence of strong electronic anisotropy on fluctuation effects. The application of the theory to (TMTSF)2PF 6 is discussed at the end of the paper. 2. GINZBURG-LANDAU THEORY OF SUPERCONDUCTING FLUCTUATION EFFECTS
To derive the coefficients of the GinzburgLandau theory microscopically we use as a model system a square array of parallel Conducting chains with the possibility of electron tuneling between adjacent chains. The single electron propagator is G(q,~ n) =
!
,
1[~n+Slgn(~n)~]-
e(q)
I,
(I)
where the free-electron energy is given by
(~)
I
2
= 2-~qmz + J(cos qx a +cos qya) - cF .
(la)
0378.4363/81/0000~)000/$02.50 © North-HollandPublishingCompany
J/2 and a are the transverse tunneling integral and lattice constant respectively, and T is the single electron lifetime due to impurity scattering within a single chain. The calculations follow closely ref. 5, and the free energy is given to lowest order in the Fourier components ~k of the superconducting order parameter by 2 j2 F = [(a+q~+ ---=-(2-cosq a-cosq a ) ) I ~ [ 2 (2) " L ~ 2vZm x y ~ ' F with q 8~2T2 IT a'l T (2a) C~
=
=
7~(3)mvF2x((m~TT) -I) •
•
,
rO
T°
•
O
where v F is the Ferml veloclty, T the meanfield transition temperature and ~(x) Gorkov's x-function [5]. From eq.(2) the longitudinal and transverse correlation lengths follow as ~II(T) = (2m~)-I/2, ~I(T ) =
Ja
~ (T)
. (3)
d'v F It is interesting to note that the t~ansverse coupling in (2) is proportional to J , i.e. to the square of the transverse electronic effective mass at the Fermi level. This is a consequence of the weak interchain coupling, which may be considered as Josephson coupling, and therefore the anisotropy of the order parameter fluctuations as given by ~ii/~l is equal to the~anisotropy of the electronic spectrum. The JZ-dependence is valid as long as the transverse coupling is smaller than the Fermi energy, i.e. as long as the Fermi surface is open in the transverse directions.On the other hand, if the Fermi surface becomes closed (J~eF) , the the transverse coupling of the order parameter is linear in J, so that the order parameter anisotropy is proportional to the square root of the anisotropy of the electronic spectrum, as is the case in effective mass theory [6]. However, in the organic superconductors single-electron properties [2-4] indicate that the weak coupling condition is well satisfied.
1179
1180
The longitudinal and transverse supercurrents are me ~ q z l ~ i 2 2eaj2 ~sinqxal~l 2 (4) JU =~-' J~ 2 + " q mvF q For the calculation of the fluctuation conductivity, a knowledge of the order parameter dynamics is necessary [7]. In time-dependent Ginzburg-Landau theory [8] this is given by the equation of motion ~a' 3~k 6F 8T ~t + ~
verse conductivity is dominated by singleelectron conduction. Another quantity of interest is the fluctuation induced diamagnetim. Including a vector potential in the free energy (2) [9], it is straightforward to calculate the fluctuation induced diamagnetism [IO]: kT e 2 ~(T) ×~ 6~ c 2 ~U (r) 2 ' (9a) kT e 2 $I (T) (~(r)2+ __)I/2
= O
(5) ×U
Following closely the method of ref.7, we obtain 2 VF e2a' [m__]1/2 [ a 1 °i| jma2 16 ~2aJ l- (a2+4~2)i/2 (a2+4~) , (6a)
2 II1J2 m
o~.
e a = -16
~ [l 3a a3 ] L 2 (a2+4~12) 1/2 + 2 ( a 2 + ~ [2)3/2 ' (6b) for the contribution of the fluctuations to the longitudinal and transverse conductivity. . . . . . o We now conslder two llmltlng cases:(1) Near T , • , c a becomes sufflclently small, so that the transverse correlation length is larger than the lattice spacing. Then, the fluctuations are essentially three-dimensional, and (6) becomes 3D VF 2 e2a,[m )I/2 3D j2a2 3D ~
°ll •
j- a2 r /r j
=--r°,
.
vF
.
o
-1/2
.
This shows the typical (T-T) divergence of • , c • a three-dimensional superconductor [7] in both the longitudinal and transverse conductivity, however, the longitudinal value is enhanced over its valu@ i~ @ n isotropic superconductor by a factor VF/J~a-, which may be very large in organic conductors due to the small interchain tunneling integral. This enhancement is due to the small transverse correlation length compared to 6|| which leads to a large mean square amplitude of the superconducting order parameter fluctuations. (ii) At temperature sufficiently above T ° $~.becomes smaller than the interchain • c dlstance, i.e. fluctuations on adjacent chains are essentially uncorrelated and can therefore be considered as one-dimensional. In this limit one has ID ~e 2 ~' ID 3~2e2j 4 a' o||
16a2 (2m 3) 1 / 2 ,
o l"
32v4
(2m3a5) l / 2
lD . (8) l| i s t h e w e l l known r e s u l t f o r a o n e - d i m e n sional superconductor [7]. A transverse supercul"rent between adjacent chains requires a well d e v e l o p e d o r d e r p a r a m e t e r on b o t h c h a i n s . As t h e propability for this to happen decr~ses with increasing temperature for ~j
6~ c 2 ~(T)
,
(9b)
for a field applied perpendicular and parallel to the chain direction, respectively. Both expressions apply in the low field limit. Eq.(9a) shows that the fluctuation diamagnetism ~s not enhanced over its value in an isotropic superconductor. This result may be understood noting that the circular Meissner currents which are at the origin of the diamagnetism have to pass between adjacent chains, and that these currents are strongly suppressed by the weak interchain coupling (eq.(4)). This effect compensates the large amplitude of the fluctuations. 3. DISCUSSION Eq. (6a) shows that the longitudinal fluctuation conductivity is largely enhanced in quasi-onedimensional superconductors.Using experimental 3 results [2-4] an enhancement factor of |O ..IO results for (TMTSF)2PF 6. Indeed 6 for a mean free path £ of about 20A and using T =5.6K good numerical agreement can be foun~ between eq.8 and experimental results up to 4OK [ll]. Two points should be noted:(|) Strong fluctuations lead to a considerable depression of T below T ° however, in the one-dimensional fluctuation c' regime an effective T ° can be used [ll].(ii) The • e Maki-Thompson dlagrams [12] further increase the fluctuation conductivity, so that even for considerably larger values of £ than the somewhat small value of 2OA superconducting fluctuations can explain the resistivity of (TMTSF)2PF 6. [|] JErome, D. et al.,J.Phys.Lett.41(1980),L95. [2] Bechgaard, K. et al., Solid State Commun. 33(|980), 1119. [3] Jacobsen,C.S. et al., preprint. [4] D. Mailly, M. Ribault, to be published; D. JErome, H.J. Schulz, in J. Bernasconi, T. Schneider(eds.), Physics in One Dimension (Springer, Berlin, 1 9 8 1 ) , p.239. [5] Gorkov, L.P., Sov. Phys. JETP IO(1960),998. [6] Tilley, D.R., Proc. Phys. Soc.86(1965),289. [7] Schmidt, H., Z. Phys. 216(1968), 336. [g] Schmid, A.,Phys.Kondens.Mat. 5(1966), 302. [9] Turkevich, L.A., Klemm, R.A., Phys. Rev. Bl9 (1979), 2520. [IO] Schmid, A., Phys. Rev. 180 (1969), 527. [ll] Schulz, H.J. et al., J. Phys., in press. [12] Maki, K., Progr. Theor. Phys. 40(1968),193; Thompson, R.S., Phys. Rev. Bl (1970), 327.