# Fluctuation effects in the organic superconductor K-(BEDT-TTF)2Cu(NCS)2

Denis Jerome, Claude R. Pasquier, Y. Loirat, S. Friemel

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Denis Jerome, Claude R. Pasquier, Y. Loirat, S. Friemel. Fluctuation effects in the organic superconductor K-(BEDT-TTF)2Cu(NCS)2. Physica C: Superconductivity and its Applications, 1996, 10.1016/0921-4534(95)00809-8. [hal-05025873](https://hal.science/hal-05025873v1)

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# Fluctuation effects in the organic superconductor $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$

S. Friemel, C. Pasquier, Y. Loirat, D. Jérôme

Laboratoire de Physique des Solides, URA 2 CNRS, Université Paris-Sud, F-91405 Orsay-Cedex, France

Received 26 October 1995

## Abstract

The resistivity data of $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ single crystals show large superconducting fluctuation effects in the presence of a magnetic field applied perpendicular to the conducting planes. These effects manifest themselves a) in a broadening of the transition into the superconducting state and b) in a melting transition of the vortex lattice into a vortex liquid. The former is investigated by employing a scaling analysis, resulting in temperature-dependent values for the upper critical field. The latter is confirmed by identifying the line of vanishing resistivity with the melting line.

## 1. Introduction

The organic superconductors [1] of the $\kappa$-(BEDT-TTF)$_2$X family present a strong anisotropy in their transport properties related to their chemical structure, comparable to the copper oxide superconductors. In analogy to the superconducting CuO$_2$-planes of the high-$T_c$ superconductors, the delocalized central C=C double bindings of the BEDT-TTF molecule form highly conducting layers. The remaining parts of the BEDT-TTF molecules and the polymerized anions X yield insulating regions [2]. One member of this family is $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$. This compound has a critical temperature of about 10 K. With an upper critical magnetic field at zero temperature for the field applied perpendicular to the layers, $H_{c2\perp}(0) \approx 50$ to 100 kOe, the in-plane coherence length is small, $\xi_{\parallel} \approx 7.5$ nm. The coherence length perpendicular to the planes is even smaller, $\xi_{\perp} \approx 0.5$ nm [3], and lies below the layer distance $s \approx 1.5$ nm. Moreover $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ is a highly type-II superconductor, since the value of the in-plane penetration

depth at zero temperature is about $\lambda_{\parallel}(0) \approx 1$ µm [4-6], which results in a Ginzburg-Landau parameter $\kappa_{\parallel}$ of the order of 100. As a result of these properties, the effect of fluctuations in the superconducting order parameter $\Psi(r) = |\Psi(r)| \exp[i\phi(r)]$ should be large. A measure of the strength of the fluctuations is the Ginzburg number

$$G_i = \frac{1}{2} \frac{(k_B T_C)^2}{(H_c^2(0) \xi_{\parallel}^2 \xi_{\perp})^2},$$

where $H_c(0)$ denotes the thermodynamic critical field at zero temperature and $k_B$ the Boltzmann constant. $G_i$ is small for bulk conventional superconductors ($G_i \lesssim 10^{-5}$), but enhanced for $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ to $G_i \approx 10^{-2}$. This value is comparable to the values for high-$T_c$ superconductors, which show large fluctuation effects.

In this paper, we will demonstrate the influence of fluctuations on the electrical transport properties in the superconducting state in presence of a magnetic field. On one hand, these superconducting fluctuations
broaden the transition from the normal to the superconducting state, which is analyzed employing a scaling theory. On the other hand, the topological arrangement of the vortices in the mixed state is also influenced by thermal fluctuations, which results in a melting transition of the vortex lattice into a vortex fluid state.

## 2. Experimental details

The single crystals of κ-(BEDT-TTF)₂Cu(NCS)₂ investigated in this paper were grown in a standard electrocrystallization process described elsewhere [7]. Their sizes are about 200 × 350 × 10 µm³ (sample #1) and 125 × 400 × 20 µm³ (sample #2), respectively. They were grown independently in two different batches of the electrocrystallization process. On top of the plate-like samples, four gold contacts were evaporated using a shadow mask technique. The two outer pads (current contacts) also cover the two outer sides of the crystals to short-circuit the superconducting planes and ensure a homogeneous current flow. The magnetic field was applied perpendicular to the planes of highest conductivity and perpendicular to the current. The resistance was measured in a standard four-probe setup, employing low-frequency lock-in technique. For temperature measurements, a carbon-glass resistor was used, which was carefully calibrated against shifts in presence of a magnetic field.

## 3. Results and discussion

Fig. 1 shows the measured resistivity versus temperature curves for both crystals in different magnetic fields. It can be noticed that even small fields broaden the transition from the normal to the superconducting state substantially. For higher fields, the onset temperature of superconductivity shifts also to lower temperatures.

### 3.1. Broadening of the superconducting transition

In this paragraph, we investigate the broadening of the transition by employing a scaling analysis according to the theory of Ullah and Dorsey [8]. This theory

<figure><img src="image_2.png" /><figcaption>Fig. 1. Measured resistivity versus temperature curves for sample #1 (upper panel) and sample #2 (lower panel). The magnetic field is applied perpendicular to the superconducting planes.</figcaption></figure>

was first applied to measurements on YBa₂Cu₃O₇ single crystals by Welp et al. [11]. Ullah and Dorsey calculated the effect of correlated thermal fluctuations of the phase and the amplitude of the superconducting order parameter in presence of a sufficiently large magnetic field applied perpendicular to the conducting planes. Their calculations imply scaling formulas for the superconducting fluctuation conductivity around the transition into the superconducting state depending on the dimensionality, 2D or 3D, of the system:

$$ \sigma_{fl}(H)_{2D} = \left[ \frac{T}{H} \right]^{1/2} F_{2D} \left[ A_1 \frac{T - T_c(H)}{(TH)^{1/2}} \right], \quad (1) $$

$$ \sigma_{fl}(H)_{3D} = \left[ \frac{T^2}{H} \right]^{1/3} F_{3D} \left[ A_2 \frac{T - T_c(H)}{(TH)^{2/3}} \right], \quad (2) $$

where $T_c(H)$ is the meanfield transition temperature for a given magnetic field, and $F_{2D}$, $F_{3D}$ are universal scaling functions. $A_{1,2}$ were taken independent on $T$ in all scaling analyses reported so far [11,12], although the original theory of Ullah and Dorsey imposes a proportionality $A_{1,2} \sim dH_{c2\perp}/dT$, which is temperature dependent. For the high-$T_c$ superconductors, the approximation of constant $A_{1,2}$ does not imply a considerable error, because the investigated mag-
netic fields are small compared to $H_{c2\perp}(0)$, so that the $H_{c2\perp}(T)$-line is not expected to have a large curvature in the probed region in the $H-T$-plane. However, for the compound studied here, with an upper critical field of about $H_{c2\perp}(0) \lesssim 100$ kOe, the $H_{c2\perp}(T)$-line may deviate from the linear behavior within the experimentally accessible region. This possibility has to be taken into account when performing the scaling analysis.

To analyze our data, we first calculate the fluctuation conductivity $\sigma_{fl}(T)$ from the measured conductivity by subtracting the normal state conductivity. The normal state resistivity is obtained from a parabolic fit to a high-field $\rho(T)$-curve for each crystal independently. These fits result in $\rho_n(T) = (51.8 + 0.65T^2) \mu\Omega\text{cm}$ for crystal #1 and $\rho_n(T) = (172.6 + 0.9T^2) \mu\Omega\text{cm}$ for crystal #2, respectively. These values are similar to the values obtained in literature [9,10].

According to Eqs. (1) and (2) the scaled $\sigma_{fl}(T; H)$-curves should collapse on one single curve. Whether Eq. (1) or Eq. (2) results in a better collapsing of the data may depend on whether the fluctuations are more 2D- or 3D-like. We performed the scaling for both situations for conductivity curves between 9.6 kOe and 36 kOe. The free parameters are the values for $T_c(H)$, which were chosen independently in the fitting procedure. No linear dependence of $T_c$ on $H$ was assumed. Instead, the value of $dH_{c2\perp}/dT$ was calculated for each curve in respect to the neighboring points in the $H-T$-plane.

The results of the fitting procedures are shown in Fig. 2 and Fig. 3. The scaling holds down to about 15% of the normal resistance, a value, which is not far away from the value found for $YBa_2Cu_3O_7$ by Welp et al. [11]. The scaling is about equally good for the 2D and the 3D case. From this scaling behavior it cannot be decided, whether the fluctuations are more 2D or 3D in nature within the probed field range. So, according to this analysis, $\kappa$-(BEDT-TTF)$_2Cu(NCS)$_2$ seems to be located at the threshold from a highly anisotropic 3D to a 2D superconductor, a conclusion, which was also drawn from a similar analysis of magnetization data by Lang et al. [13].

The $H_{c2\perp}(T)$-lines resulting from our analysis are shown in Fig. 4. They are, within the error bars, in rather good agreement with the results of Lang et al. They exhibit, although not assumed a priori, a linear dependence on temperature. The extrapolated values

<figure><img src="image_3.png" /><figcaption>Fig. 2. 3D scaling of high-field conductivity curves for sample #1. Inset: 2D scaling behavior.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. 3D scaling of high-field conductivity curves for sample #2. Inset: 2D scaling behavior.</figcaption></figure>

for $H_{c2\perp}(0)$ are 50 kOe and 100 kOe for sample #1 and #2, respectively. It can be noticed, that these values differ considerably. This may be due to different sample qualities, i.e. the amount of impurities and crystal faults, which seem to be larger for sample #2. This also manifests itself in the different residual resis-
tivities and hence in different elastic mean free paths, $l_1 \approx 24$ nm and $l_2 \approx 7$ nm, where the indices refer to the sample number.

## 3.2. Melting transition

Next, we investigate the effect of fluctuations deep within the mixed state. Here, fluctuations in the amplitude of the order parameter become less important. But nevertheless, fluctuations in the phase of the order parameter should still play a somewhat important role. In a different picture, these fluctuations can be interpreted as thermal fluctuations of the positions of the vortices. In this picture, the vortices can form a well-ordered lattice at sufficiently low temperatures. This lattice melts at a field-dependent temperature due to thermal excitations of the vortices from their equilibrium lattice positions. This results in a vortex state without long-range positional order, the so-called vortex liquid, for temperatures above the melting transition. In a classical picture, the vortex lattice melts at a certain temperature, when the root mean square of the vortex displacements from their equilibrium position is larger than a certain fraction of the lattice constant, $d(T) \succeq c_L a$, where $d(T)$ is the temperature-dependent root mean square displacement, a the lattice parameter of the vortex lattice and $c_L$ the Lindemann number, a constant typically of order 0.1-0.2. This melting criterion is called the Lindemann criterion. Houghton et al. [14] derived the temperature and field-dependent root mean square displacements of the vortices in a nonlocal elastic theory. We will use their analytical approximation for the melting criterion:

$$ \left[ \frac{t}{(1-t)^{1/2}} \right] \left[ \frac{h^{1/2}}{(1-h)} \right] \times \left[ 4\left(\sqrt{2}-1\right)/(1-h)^{1/2} + 1 \right] \succeq 2\pi G_i^{-1/2} c_L^2 \quad (3) $$

where $t = T/T_c$ and $h = H/H_{c2\perp}(T)$.

We determine the points of the melting transition in the H-T-plane from our experimental data by taking the zero-resistance temperatures for each magnetic field, i.e. the temperature, where the sample resistance drops below our experimental sensitivity ($10^{-5}$ Ω). This identification can be justified as follows: below the melting temperature exists a well-defined vortex lattice, which is pinned by disorder within the real

<figure><img src="image_5.png" /><figcaption>Fig. 4. Temperature dependence of the upper critical field for sample #1 and sample #2. The horizontal bars indicate the range, for which similar scaling qualities can be achieved. Also shown are the points of the melting transition for both crystals. The fits result from employing Eq. (3) for sample #1 (solid line) and sample #2 (dashed line), respectively.</figcaption></figure>

crystal lattice. Above the melting temperature the situation changes. The vortex fluid is less effectively pinned, which, in presence of a transport current, results in movement of vortices and hence in an increasing transport resistivity.

Fig. 4 shows the melting line for sample #1 and #2 determined in the way described above. The solid curve is a fit using Eq. (3), where $H_{c2\perp}(T)$ is taken from the results of our scaling analysis. The only free parameter is the Lindemann number; the Ginzburg number $G_i$ can be estimated: We first assume that $\xi_{\perp}(0) \approx 0.5$ nm is independent on the amount of disorder and related only to the superconducting properties of the material. This assumption is justified by the idea, that $\xi_{\perp}$ corresponds to the thickness of the superconducting layers, which is not influenced by the amount of disorder. The values for $\xi_{\parallel}(0)$ are deduced from $H_{c2\perp}(0)$ employing $\xi_{\parallel}(0) = (\Phi_0/2\pi H_{c2\perp}(0))^{1/2}$, where $\Phi_0$ is the flux quantum. This results in $\xi_{\parallel1}(0) \approx 8.1$ nm and $\xi_{\parallel2}(0) \approx 5.7$ nm. The values of the effective penetration depth $\lambda_{\parallel}(0)$ can be deduced from the London penetration depth $\lambda_L(0) \approx 0.52$ µm [4,5] and the ratio $\xi_{\parallel}(0)/l$ using $\lambda_{\parallel}(0) \approx (1 + \xi_{\parallel}(0)/l)^{1/2}\lambda_L(0)$. We get $\lambda_{\parallel1}(0) = 0.6$ µm and $\lambda_{\parallel2}(0) = 0.7$ µm and accordingly $\kappa_1 \approx 74$ and $\kappa_2 \approx 122$, which finally results in $G_{i1} \approx 1.4 \ 10^{-2}$ and $G_{i2} \approx 2.3 \ 10^{-2}$. The best fits to the transition line correspond to a Lindemann number of 0.14 and 0.11 for sample #1 and sample #2, re-
spectively, which lie well within the predicted range. These results are a strong indication that the transition from zero to nonzero resistivity is actually the melting transition. In literature, the melting transition for high-$T_c$ superconductors is often identified with the irreversibility line, i.e. the crossover from reversible high-temperature to irreversible low-temperature behavior of certain sample properties, for example the magnetization. This is often justified by the argument that the vortices in the lattice are effectively pinned, which results in a hysteresis due to the unpinning of the lattice, whereas above the irreversibility line the vortex liquid is unpinned. In comparison to the irreversibility line found in magnetization measurements on $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$ [13], the melting line found here differs widely. This may be due to different properties of the sample used by Lang et al. and our samples, although the melting lines of our samples do not differ considerably in contrast to their otherwise rather different properties. A possible alternative explanation of the difference between the magnetization data and our data may be, that the melting line cannot be identified with the irreversibility line. This was recently also found for $Bi_2Sr_2CaCuO_8$ single crystals by Majer et al. [15]. This explanation is also supported by the fact, that we were not able to find any irreversibility effects in our electrical transport measurements, though this may also be due to an insufficient experimental sensitivity.

## 4. Conclusions

In summary we have shown the effect of fluctuations of the superconducting order parameter for $\kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$. By a scaling analysis of the broadened transition into the superconducting state, the temperature-dependent $H_{c2\perp}$-line was determined. From this, a value of $H_{c2\perp}(0) = 50$ to 100 kOe can be estimated, which depends strongly on crystal quality. These values imply values for the superconducting coherence length $\xi_\parallel$ of 6 to 8 nm. Additionally, we investigated the region below the transition where resistivity approaches zero. The line of approximately vanishing resistivity can be identified as the vortex melting line, which separates a vortex lattice phase below the field-dependent melting temperature from a vortex liquid at higher temperatures.

## Acknowledgements

We thank P. Batail for providing the samples. We acknowledge fruitful discussions with H. Mayaffre, P. Wzietek, M. Gabay, and P. Lederer. We also acknowledge M. Nardone for technical help. One of us (C. P.) was supported by DRET under contract n° 94C8024.

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