# Inhomogeneous Superconductivity in Organic and Related Superconductors

## Abstract

**:**

_{p}. Methods for predicting the existence of inhomogeneous superconductivity are shown to work for the organic superconductors, and then used to suggest new materials to study.

## 1. Introduction

- $\kappa $-(BEDT-TTF)${}_{2}$Cu(NCS)${}_{2}$:$\kappa $ET-CuNCS,
- $\lambda $-(BETS)${}_{2}$GaCl${}_{4}$:$\lambda $BETS-GaCl,
- $\beta $${}^{\prime}$${}^{\prime}$-(ET)${}_{2}$SF${}_{5}$CH${}_{2}$CF${}_{2}$SO${}_{3}$:$\beta $${}^{\prime}$${}^{\prime}$ET-SF5,
- $\kappa $-(ET)${}_{2}$Cu[N(CN)${}_{2}$]Br:$\kappa $ET-Br.

#### 1.1. Inhomogeneous Superconductivity

_{p}, and properly called the Chandrasekhar–Clogston Pauli paramagnetic limit [32]. We will refer to this level of magnetic field as the paramagnetic limit for short.

_{p}. This new superconducting state is the FFLO state and is characterized by Cooper pairs with non-zero momentum, and a spatially modulated order parameter [1,2].

#### 1.2. Critical Parameters

_{p}, and Pauli paramagnetism will dominate, favoring inhomogeneous superconductivity. The other parameter r, defined above, is a measure of how clean the system is. It measures the ability for the material to support an extended wave function, necessary for the long range order of the FFLO state [39], although there are claims that clean ordered systems are not absolutely necessary [50]. These parameters come from routine measurements. In the ratio $r=\ell /\xi $, ℓ comes from the scattering time, which is related to the Dingle temperature, ${T}_{D}$, and the Fermi velocity, ${v}_{F}=\sqrt{2{E}_{F}/{m}^{*}}$, where ${E}_{F}$ is the Fermi energy and ${m}^{*}$ is the effective mass. The Fermi energy, effective mass, and ${T}_{D}$, can all be measured via Shubnikov–de Haas (SdH) or de Haas–van Alphen oscillations. The other necessary parameter for finding r is $\xi $, which is found from the measurement of the superconducting critical field versus temperature and the formula ${H}_{c2}={\Phi}_{0}/2\pi {\mu}_{0}{\xi}^{2}$, or estimated from the size of the superconducting energy gap, $\Delta $, and the formula $\xi =\hslash {v}_{F}/2\Delta $. Note that, for the same Fermi energy, a higher effective mass results in a smaller ${v}_{F}$ and a smaller $\xi $, and hence a higher ${H}_{orb}^{0}$. For this reason, heavy fermion superconductors should favor the FFLO state. In general, ${H}_{orb}^{0}$ comes from critical field measurements and H

_{p}can be found by analyzing specific heat measurements and determining the superconducting energy gap, as is described in detail below.

## 2. Materials and Methods

#### 2.1. The Paramagnetic Limit

_{p}. As we showed in a previous paper [6], following Clogston [31], we can find the critical magnetic field associated with the quenching of superconductivity by estimating the superconducting energy gap by analyzing specific heat data and setting this energy equal to the gain in free energy in a metal with susceptibility ${\chi}_{e}$. More specifically, we equate the superconducting condensation energy

_{p}as the designation of the Chandrasekhar–Clogston Pauli paramagnetic limit as is common in most articles.

#### 2.2. Specific Heat, Energy Gap, and Other Parameters

_{p}is then calculated from Equation (6). H

_{p}is important for the study of the FFLO state because it determines the magnetic field that separates the vortex state of superconductivity from the FFLO state. Given that the superconducting energy gap does not change much in the lower half of the temperature range of the superconducting phase diagram, H

_{p}and hence the vortex state-FFLO state phase line should be virtually temperature independent [76].

_{p}are shown for a number of materials in Table 1. We have also collected a number of other useful parameters for the study of inhomogeneous superconductors. Most of the compounds are from the family of organic crystalline superconductors. We have also added the heavy fermion CeCoIn${}_{5}$ and the pnictides KFe${}_{2}$As${}_{2}$ and LiFeP as examples of other materials where the FFLO state has been claimed to be found.

_{p}, as determined by ${\alpha}_{M}$. It was determined theoretically that, above the critical value of ${\alpha}_{M}$ = 1.8, the FFLO state could be stablilzed [91] in a clean material (r > 2). The search for the FFLO state involves careful measurements of some parameter that can be probed in the superconducting state around the value of H

_{p}, such as specific heat, penetration depth or NMR, in order to find evidence of a phase transition.

## 3. Results and Discussion

_{p}, that was calculated with the specific heat data via the Alpha Model, and also that the phase line has near zero slope, consistent with the superconducting energy gap, which is almost temperature independent below ${T}_{c}/2$.

_{p}phase line is a first order transition based on hysteresis in the specific heat as the transition is crossed in up and down sweeps of the magnetic field [3]. An example of the hysteresis in the specific heat is shown in Figure 5a along with a TDO measurement that shows the same hysteresis. The TDO measurement is a much simpler measurement, and the idea that it also can provide evidence of a first order transition is useful for future experiments. The magneto-caloric measurements, shown in Figure 5b, also provided the important evidence that the higher field state has a greater entropy than the vortex state, consistent with the fact that the FFLO state is less ordered than the uniform superconducting state, because of the unpaired electrons [3,94,95]. Another piece of information that can be gathered from the phase diagram is that the enhancement of the ultimate critical field ${H}_{c2}$ is ≈1.4H

_{p}, consistent with predictions [20,36] for a 2D material. One difference that can be found between this data and the theory is that T*, the place where the FFLO phase starts is at ${T}_{c}/3$, much lower than ≈${T}_{c}/2$ as most theories predict.

_{p}and ${T}_{c}$. In this figure, the three materials scale relatively well, although the upper critical field for $\lambda $BETS-GaCl is not quite as high as the others. This would suggest that it is not as two-dimensional as the other superconductors [20], however, we have used the vortex-FFLO phase line as the H

_{p}for this figure. It is also possible that impurities or spin-orbit scattering could have raised the value of H

_{p}[32,48].

_{p}, but surprisingly an identical ${H}_{c2}$. The drop in the value of H

_{p}is expected if it is dependent on similar pair breaking mechanisms as ${H}_{c2}$ [32,48]. In particular, less spin-orbit scattering should lower the critical field. The surprise is that the upper critical field for the FFLO state, where the material crosses into the normal state, is robust with respect to the degree of scattering, unlike H

_{p}, the upper critical field for the vortex state. In the traditional superconducting state, H${}_{c2}$ is very sensitive to scattering. This may be understandable in the context of ${H}_{c2}$ as calculated by [32,48] where similar changes in the ${H}_{c2}$ phase line are found for greater orbital effects (vortices) or more spin orbit scattering. However, spin-orbit scattering can directly modify H

_{p}, but vortex effects will not. In addition, if orbital (vortex) effects are suppressed, as they are in the highly anisotropic organics, ${H}_{c2}$ must be the result of the magnetic energy in the system, which is more immune to scattering events than orbital effects. Therefore, the vortex FFLO phase line should follow H

_{p}, which is dependent on the energy gap and spin orbit scattering, and H${}_{c2}$ will be insensitive to scattering. In many of these materials, it may be important to always adjust H

_{p}to account for the degree of scattering and spin-orbit scattering to properly interpret the results.

#### 3.1. Other Materials

_{p}from Table 1. We note that $\alpha $ for CeCoIn${}_{5}$ was calculated based on two sets of specific heat data [83,84] with nearly identical results. The parameter $g*/g$ was more difficult to determine, and for this study we relied on $\gamma $ and $\chi $ measurements [83] averaged over temperatures below ${T}_{c}$ and used Wilson’s ratio [69]. What is surprising is that although the vortex-FFLO phase line found by specific heat and torque does not line up with H

_{p}, and it also wouldn’t even have the right slope for the H

_{p}line, the phase line found by TDO rf penetration measurements [64] matches the value of H

_{p}as calculated from the specific heat. Adding to this evidence for a phase line at 9.2 T, Koutroulakis et al. [100] identified a phase they call the exotic superconducting state, or ESC that starts at a minimum magnetic field of 9.2 T. It may be worth looking at the CeCoIn${}_{5}$ specific heat data to see if there is any indication of this lower phase line. In any case, it can be seen that the Q state, as it is called, is not a simple FFLO state, or at least not the same as found in the organic and pnictide superconductors.

_{p}between 4.45 and 5.23 T (the result in Table 1 is the average). For convenience, we used the value from Figure 4 in Cho et al. [107] to scale their phase diagram and superimpose it on the the phase diagram from the organic conductors. The result is in Figure 10. This diagram and the KFe${}_{2}$As${}_{2}$ specific heat data raise a number of questions about the claim of the FFLO state in this material.

_{p}in $\kappa $ET-CuNCS is 1.35, and in KFe${}_{2}$As${}_{2}$ it is 1.24 if you extrapolate to zero temperature. This is consistent with the difference in anisotropy. According to Matsuda [20], the enhancement in an ideal 2D and 3D system should be 1.4 and 1.2 respectively. The ratio of parallel and perpendicular critical fields at zero temperature is one indication of the anisotropy of a material, and this value is ~4–5 for both KFe${}_{2}$As${}_{2}$ [86] and for $\kappa $ET-CuNCS, yet this is misleading because $\kappa $ET-CuNCS is strongly paramagnetically limited. Using the initial slopes of the critical field lines in different orientations as described in Section 1.2, Equation (2) determines ${H}_{orb}^{0}$. ${H}_{orb}^{0}$ is a much better measure of the superconducting anisotropy because it is based on the anisotropy of the vortices, and hence the superconducting coherence lengths, $\xi $. This anisotropy measurement is reliable as long as the diameter of the vortices is larger than the interlayer spacing, a limit that is always true close to ${T}_{c}$ where $\xi $ diverges. Using these slopes, the anisotropy of KFe${}_{2}$As${}_{2}$ is still 4.5 but for $\kappa $ET-CuNCS the anisotropy is ≈21, if the slope of ${H}_{c2}$ is properly measured [93]. Thus, this anisotropy explains the difference in critical field enhancement, but the anisotropy of KFe${}_{2}$As${}_{2}$ is inconsistent with the onset of the FFLO state in a multiband superconductor. According to Gurevich [42], the lower anisotropy of the pnictides should result in the FFLO state starting at $t<0.5$, below the results found by Cho et al., and certainly below the results of the highly anisotropic organic conductors. Zocco et al. [88] has a higher temperature onset of the FFLO state according to their calculations, but a much smaller critical field enhancement than the data.

_{p}. As expected, to first order, this line should have no slope if H

_{p}is proportional to the energy gap, given that the BCS superconducting energy gap is almost constant for $t<0.4$. Within experimental error, this is almost the case for $\kappa $ET-CuNCS. Doing a fit to our largest set of data points, from the TDO measurements, we find a slope of <0.05 T/K. The slope of this phase line for KFe${}_{2}$As${}_{2}$ is 0.3 T/K. The increased slope could be due to the presence of vortices, as expected for the pnictides, which are less anisotropic than the organic superconductors. Another interpretation is that this line is really ${H}_{c2}$ and the bump in specific heat at a higher field is not due to the FFLO state. Given the form of the specific heat data in Agosta et al. [3] and the calculations of Ptok [106], a small jump in specific heat at ${H}_{c2}$ followed by a large jump in specific heat at the FFLO to uniform superconductivity phase line does not make sense. In Ptok, these jumps are calculated as similar in size, and given that ${H}_{c2}$ is a transition into a bulk superconducting state, with the formation of Cooper pairs throughout the sample, the resulting drop in entropy should produce a robust specific heat peak at ${H}_{c2}$, and as calculated in Ptok and in many publications for single band superconductors [109]. Furthermore, without more angular data to see if the specific heat signature of the FFLO state slowly changes when the sample is rotated with respect to the magnetic field, the data is less compelling. Many of the other studies of KFe${}_{2}$As${}_{2}$ that have been done, as referenced in the above paragraph, and in particular the one by Zocco et al., do not see any indication of a FFLO state. There still is hope to find indications of the FFLO state in KFe

_{2}As

_{2}and other pnictides, but the present claim in Cho et al. although exciting, needs more evidence.

#### 3.2. New Materials

_{p}for the largest gap, all Cooper pairs will be subject to pair breaking via the Zeeman energy. The data claiming inhomogeneous superconductivity in KFe${}_{2}$As${}_{2}$ [107], if it is correct, shows the H

_{p}phase line at the value of the higher gap energy consistent with the argument above. For this reason, the parameters for both of the pnictides in Table 1 correspond to the higher energy of the two gaps found in each compound. The energy gaps and ratio $\alpha $ for these materials were not calculated from our version of the alpha model but came from the papers referenced in the table, and use a two gap model. It is also important to note that ${g}^{*}/g$ for LiFeP is not known, so it was set to equal one. Given what is known, the value of H

_{p}in LiFeP is above H${}_{c2}$ and no FFLO state should exist. It is possible that a weak variation of the FFLO state could exist when the magnetic field was greater than H

_{p}corresponding to the lower energy gap, but it is difficult to search for evidence of such a weak FFLO state, if it exists at all.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Color online. From Wosnitza [35] On the left, the traditional BCS Cooper pair is represented. On the right, the energy of the up and down electrons have been shifted by the magnetic field. A momentum $\mathbf{q}$ can be added to one of the electrons to create a zero momentum center of mass Cooper pair. In the diagram, the initial momentum of the electrons is isotropic. In real material, the shape of the Fermi surface could create more complex diagrams.

**Figure 2.**Color online, from [3]. Cartoon of the FFLO state showing the nodes in the order parameter as horizontal planes where we estimate the spin-polarization to be ≈10% at 25 T in the low temperature limit. In the diagram the black arrow labeled

**B**represents the applied magnetic field, and the red arrows represent the net spin polarization. Although the diagram is schematic, all of the lengths are to scale; small boxes represent the unit cells of $\kappa $ET-CuNCS, yellow slabs represent the least conducting layers of the crystal, and red rectangles represent Josephson core-less vortices at about the right distance apart in a 25 T field. The full height of the crystal is ≈20 nm.

**Figure 3.**The angle dependence of the penetration depth near the parallel orientation for $\kappa $ET-CuNCS. The lowest trace is exactly parallel, or 90° in our absolute coordinates. The other traces are in order of increasing angle. The traces are vertically shifted to aid visualization. It is truly remarkable how all the vortex details are absent at the exactly parallel orientation.

**Figure 4.**Color online. From Agosta et al. [3] Phase diagram of $\kappa $ET-CuNCS for parallel magnetic field ($\theta =0$). Solid black circles represent calorimetric observations of the phase transitions between the lower and higher field superconducting phases at H

_{p}, and squares, the normal and superconducting state at ${H}_{c2}\left(T\right)$. Points from an earlier calorimetric determination of ${H}_{c2}\left(T\right)$ [93] are shown as open blue squares. Also included are determinations of both the ${H}_{c2}$ and H

_{p}phase boundaries by means of rf penetration measurements (green) [6] and NMR measurements [7,8] (open purple and red symbols, respectively).

**Figure 5.**Color online. (

**a**) specific heat and TDO up and down field traces as the measurements cross the vortex-FFLO phase transition showing the hysteresis. This is direct evidence of a first order transition; (

**b**) from Agosta et al. [3], the magneto caloric effect, which shows the direction of the flow of latent heat as the vortex-FFLO phase line is crossed. This information can be used to show that the higher field state, in this case the FFLO state, is of higher entropy than the vortex state, consistent with what is expected for this phase transition [3,94,95].

**Figure 6.**Color online. Phase diagram of $\kappa $ET-CuNCS, $\lambda $BETS-GaCl and $\beta $″ET-SF5 each normalized using the phase line between the vortex and FFLO states. The BETS does not seem to have as much enhancement of the upper critical field as the other two superconductors, which scale identically.

**Figure 7.**We obtained new samples from Kobayashi that are higher quality samples (more pure) based on Shubnikov–de Haas oscillations. The higher quality samples show on an absolute scale that the upper critical field of the clean and dirty samples is the same, but H

_{p}is very sensitive to impurities. The new data was taken in a dilution refrigerator and we are looking forward to extending the data to higher temperatures in the near future.

**Figure 8.**Color online. Example of spectrum simulation compared to recorded spectrum (blue) from Koutroulakis et al. [12]. The simulation is a sum (green) of four Gaussian-broadened contributions (red, orange) arising from a single-Q sinusoidal modulation of the SC order parameter.

**Figure 9.**Color online. Phase diagram of CeCoIn${}_{5}$ with $\kappa $ET-CuNCS as our baseline FFLO phase diagram. If the magnetic field is normalized by H

_{p}as calculated by Equation (6), the TDO measurement [64] (blue squares) finds a phase line at H

_{p}. NMR experiments [100] confirm a change of the material properties at this field value too, suggesting that there may be a FFLO type transition at the calculated H

_{p}. The more vertical phase line discovered by specific heat [54] (black triangles) at about t = 0.1 could be related to the SDW.

**Figure 10.**Color online. The phase diagram of $\kappa $ET-CuNCS superimposed with the KFe${}_{2}$As${}_{2}$ data from Cho et al. [107], normalized to ${T}_{c}$ and H

_{p}for each sample respectively. We note the two main differences, the slope of the vortex-FFLO transition, and the overall enhancement of the FFLO state over H

_{p}.

**Table 1.**Parameters that are useful in the study of inhomogeneous superconductivity. The ratio $\alpha $ is determined from the specific heat data in the reference next to $\alpha $. The other parameters are calculated from values in the table or references. The ${T}_{c}$ values come from the specific heat data referenced for $\alpha $.

Material | $\mathit{\alpha}$ | ${\mathit{g}}^{*}/\mathit{g}$ | ${\mathit{T}}_{\mathit{c}}$(K) | ${\mathit{H}}_{\mathit{p}}$(T) | ${\mathit{\alpha}}_{\mathit{M}}$ | ${\mathit{H}}_{\mathbf{orb}}^{0}$(T) | $\mathit{\xi}$(Å) | ℓ(Å) |
---|---|---|---|---|---|---|---|---|

$\kappa $-(BEDT-TTF)${}_{2}$Cu(NCS)${}_{2}$ | 3.0 [72] | 1.26 [69] | 9.6 | 21.6 | 4.9 | 130 [35] | 13 | 900 [77] |

${\beta}^{\prime \prime}$-(BEDT-TTF)${}_{2}$SF${}_{5}$CH${}_{2}$CF${}_{2}$SO${}_{3}$ | 1.94 [78] | 1.0 [69] | 4.5 | 9.2 | 3.9 | 75 [35] | 21 | 520 |

$\alpha $-(ET)${}_{2}$NH${}_{4}$Hg(SCN)${}_{4}$ | 1.76 [79] | 0.86 [69] | 0.96 | 2.1 | 5.5 | 8.1 [76] | 53 | 681 [76] |

$\lambda $-(BETS)${}_{2}$GaCl${}_{4}$ | 1.83 [80] | 1.0 [69] | 4.3 | 8.3 | 3.9 | 23.1 [10] | 31.5 | 170 [56] |

$\kappa $-(ET)${}_{2}$Cu[N(CN)${}_{2}$]Br | 2.77 [72] | 1.4 [69] | 11.5 | 23.8 | 9.6 | 161 [81] | 12 | 260 [82] |

CeCoIn${}_{5}$ | 3.03 [83,84] | 0.73 [83] | 2.16 | 9.44 | 6.5 | 43.5 [85] | 23 | 810 |

KFe${}_{2}$As${}_{2}$ | 1.75 [86] | 1.3 [87] | 3.14 | 4.84 | 2.9 | 9.9 | 48 | 1770 [88] |

LiFeP | 1.89 [89] | 1.0 | 17.6 | 34.9 | 2.1 | 51 [90] | 21 | 5500 [90] |

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Agosta, C.C. Inhomogeneous Superconductivity in Organic and Related Superconductors. *Crystals* **2018**, *8*, 285.
https://doi.org/10.3390/cryst8070285

**AMA Style**

Agosta CC. Inhomogeneous Superconductivity in Organic and Related Superconductors. *Crystals*. 2018; 8(7):285.
https://doi.org/10.3390/cryst8070285

**Chicago/Turabian Style**

Agosta, Charles C. 2018. "Inhomogeneous Superconductivity in Organic and Related Superconductors" *Crystals* 8, no. 7: 285.
https://doi.org/10.3390/cryst8070285