Suppression of superconductivity and enhanced critical field anisotropy in thin flakes of FeSe

FeSe is a unique superconductor that can be manipulated to enhance its superconductivity using different routes while its monolayer form grown on different substrates reaches a record high temperature for a two-dimensional system. In order to understand the role played by the substrate and the reduced dimensionality on superconductivity, we examine the superconducting properties of exfoliated FeSe thin flakes by reducing the thickness from bulk down towards 9 nm. Magnetotransport measurements performed in magnetic fields up to 16T and temperatures down to 2K help to build up complete superconducting phase diagrams of different thickness flakes. While the thick flakes resemble the bulk behaviour, by reducing the thickness the superconductivity of FeSe flakes is suppressed. In the thin limit we detect signatures of a crossover towards two-dimensional behaviour from the observation of the vortex-antivortex unbinding transition and strongly enhanced anisotropy. Our study provides detailed insights into the evolution of the superconducting properties from three-dimensional bulk behaviour towards the two-dimensional limit of FeSe in the absence of a dopant substrate.

FeSe is a unique superconductor that can be manipulated to enhance its superconductivity using different routes while its monolayer form grown on different substrates reaches a record high temperature for a twodimensional system. In order to understand the role played by the substrate and the reduced dimensionality on superconductivity, we examine the superconducting properties of exfoliated FeSe thin flakes by reducing the thickness from bulk down towards 9 nm. Magnetotransport measurements performed in magnetic fields up to 16 T and temperatures down to 2 K help to build up complete superconducting phase diagrams of different thickness flakes. While the thick flakes resemble the bulk behaviour, by reducing the thickness the superconductivity of FeSe flakes is suppressed. In the thin limit we detect signatures of a crossover towards two-dimensional behaviour from the observation of the vortex-antivortex unbinding transition and strongly enhanced anisotropy. Our study provides detailed insights into the evolution of the superconducting properties from three-dimensional bulk behaviour towards the two-dimensional limit of FeSe in the absence of a dopant substrate.
Amongst iron-based superconductors, FeSe has the simplest stoichiometric crystal structure, making it an ideal candidate to study the mechanisms of superconductivity [1]. Twodimensional FeSe has attracted much interest due to the discovery of superconductivity above 65 K in monolayer FeSe grown on SrTiO 3 [2][3][4], which is the highest critical temperature of all iron-based superconductors. Additionally, due to the weak van der Waals bonding of the FeSe layers, the material cleaves readily and has potential applications in heterostructure devices [5,6]. It is therefore important to understand any changes in the properties of the material as it is thinned towards the monolayer limit.
Previous studies examining the thickness dependence of FeSe have been limited to measurements on thin films grown using techniques such as molecular beam expitaxy [7], pulsed laser deposition [8], and DC sputtering [9], all of which require well-optimised growth protocols. The resulting thin films are strongly susceptible to interaction with the growth substrate, due to factors such as strain and charge transfer. This can lead to effects such as the enhancement or suppression of the superconducting and structural transition temperatures when compared to single crystals [8]. Additionally, as the thickness of the films is reduced towards the single layer limit, superconductivity is observed to be systematically suppressed, often resulting in a superconductor-insulator transition [10].
An alternative to the growth of thin films is to create devices by mechanical exfoliation of high quality single crystals. This has proven extremely successful in the case of the layered superconductors NbSe 2 [11], TaSe 2 [12], and Bi 2 Sr 2 CaCu 2 O 8+δ [13], in which the inherent thicknessdependence of the superconducting transition has has been measured down to a monolayer. Recently, exfoliated FeSe devices have been realised [14][15][16], with samples displaying superconducting behaviour at a thickness where thin films of FeSe are typically insulating [9]. However, these samples ex-hibit a suppressed superconducting critical temperature when compared to the bulk crystals from which they were exfoliated. One possible factor in suppressing superconductivity is the sample degradation caused by multiple fabrication steps, as well as long term exposure to air [17]. It is therefore important that any study of the thickness dependence of superconductivity of FeSe utilise a fabrication method free from the harsh chemicals and high temperatures involved in traditional lithographic processing.
In this work we present a detailed study of the nature of superconductivity in ultra-thin flakes of FeSe fabricated utilising a deterministic transfer method [18]. We use magnetotransport measurements in high magnetic fields up to 16 T to investigate the effect of thickness on this materials superconducting properties. As the thickness is reduced from 100 nm towards 9 nm, we detect a crossover towards a two-dimensional character in superconductivity that manifests as a significant enhancement in the anisotropy of the upper critical field. Our results give important insights into the intrinsic nature of FeSe superconductivity in the thin limit, unaffected by substrate interactions or other external effects.
The effect of reducing sample thickness on the transport behaviour Fig. 1a) shows the typical temperature-dependence of the normalised resistance (R(T)/R(300 K) for a bulk crystal and five thin flake devices with thickness t in the range 9-100 nm (see Fig. S2 in Supplemental Material (SM) for additional devices). We observe significant changes in the transport behaviour of our devices which are highly dependent on the flake thickness. Firstly, the thick flake devices with t ≥ 58 nm are of highly quality with large RRR (see Fig. S3). They display similar transport behaviour to bulk FeSe [21], in which the nematic phase transition occurs around T s ∼ 89 K accompanied by a superconducting transition at T c ∼ 9 K (see Fig.  1b)). We notice that T c of all thick flake devices is slightly lower than in bulk, with a maximum of T c ∼ 7 K for the 100 nm flake despite a relatively high residual resistance ra- FIG. 1. a) Temperature dependence of the normalised resistance R(T )/R(300 K) for a bulk crystal and five different thin flakes (t=9-100 nm). b) The high-temperature temperature dependence of the normalised derivative of resistance, dR/dT , for the bulk sample and four thin flakes in a). The arrows indicate the position of the structural transition at Ts. The curves in a) and b) are shifted vertically for clarity. c) The low temperature superconducting transitions as in a) but scaled to R(T )/R(15 K). d) Thickness dependence of the superconducting critical temperature for several thin flake samples measured in this study, along with data from Refs. [14,17]. The solid line is a fit to the Cooper model [19,20]. The inset shows an optical image of a 14 nm FeSe device capped with a thin layer (∼20 nm) of h-BN. The scale bar corresponds to 10 µm.
tio of 16 and a sharp transition width, ∆T c of 0.3 K. Next, in thinner flake devices we observe a systematic suppression of superconductivity, accompanied by a broadening of the resistive transition width, as shown in Fig. 1c). Lastly, the thinnest device reported here with t = 9 nm displays an anomalous upturn in resistance at low temperature, before a sharp decrease near 3 K, indicating that a superconducting phase may only be stabilised below the experimental temperature limit of 2 K. Another important signature in the transport data of FeSe is the emergence of the nematic phase that triggers a tetragonal to orthorhombic structural transition at T s and causes significant in-plane distortion of the Fermi surface [21]. Fig. 1b) shows that the nematic transition has a sharp anomaly identified by the minimum in dR/dT that is slightly suppressed for thick flakes (t > 50 nm), as compared with bulk single crystals of FeSe. However, in thinner samples (t < 50 nm) this transition is ill-defined and appears to be significantly reduced, as shown in Fig. 1b) and Fig. S1 in SM. This behaviour is reminiscent of that found in polycrystalline samples of FeSe [1] or Cu-doped FeSe [22] in which the RRR is reduced, as the degree of disorder and local inhomogeneity is much higher (Fig. S1 in SM), in comparison to high quality single crystals of FeSe in which quantum oscillations have been observed [23].
A summary showing how the superconductivity of thin flakes of FeSe is affected by the thickness reduction is shown in Fig. 1d). While T c remains relatively constant for thicker flakes (50 -100 nm), a sharp decrease in superconductivity occurs for the thinnest flakes (t < 25 nm). We can describe the observed superconducting behaviour using the Cooperlaw given by T c ∼ exp(−t m /t) , where t m = 2a/(N 0 V ), a is the Thomas Fermi screening length and N 0 V is the bulk pairing potential [19,20]. Since a is inversely proportional to the square root of the density of states at the Fermi energy, this behaviour is expected for systems with very small Fermi energy, as found in FeSe [21]. The Cooper-law is commonly used to describe superconducting thin films but the trends observed in our data are in qualitative agreement with those found in thin films of FeSe [9], thin flakes of FeSe 0.3 Te 0.7 [24] and nanoflakes devices of FeSe fabricated using alternate device fabrication techniques, reported in Refs. [14,17].
BKT transition in thin flakes of FeSe. Next, we focus on other manifestations of superconductivity in thin flake devices of FeSe. The appearance of a Berezinskii-Kosterlitz-Thouless (BKT) transition in a material is a signature of a 2D superconducting state [25,26]. This arises due to the thermal nucleation of vortex-antivortex pairs in the absence of an external magnetic field. Vortex-antivortex unbinding gives rise to dissipation, which results in a resistive transition even when the temperature is below the mean field pairing temperature. Just above the critical current, I c , the IV curves follow the V ∝ I α(T ) dependence, where α is a temperature-dependent exponent. At T BKT the critical exponent, α, abruptly increases from 1 at higher temperatures, due to flux flow of thermally dissociated vortex-antivortex pairs to 3 at lower temperatures due to the current-driven dissociation of vortex-antivortex pairs [27,28].
In order to determine whether BKT physics plays an important role in the observed suppression of superconductivity in FeSe thin flakes, we investigate the temperature-dependent current-voltage characteristics for two devices with t =14 nm and t = 100 nm at temperatures near T c , as shown in Fig.  2a,b). We find a non-linear behaviour in the high current regime suggesting a current-induced vortex-antivortex depairing, as expected for a BKT transition, without displaying the sudden jump in α(T ). The value of the exponent reaches the critical value of α = 3 at T BKT = 2.9 K for the t = 14 nm sample and T BKT = 6.67 K for the t = 100 nm sample, as shown in Fig. 2c,d). In both cases, the calculated T BKT lies below the temperature (see also Fig. S9 in SM) at which the resistance is 1% of the normal state value, suggesting that BKT physics is not the cause of the suppression of superconductivity in thin flakes. This appearance of a possible BKT transition is in qualitative agreement with previous reports on monolayer FeSe/SrTiO 3 [29], and thick films of FeSe [30], supporting a scenario that superconductivity in FeSe is quasi-two dimensional. The lack of the sudden jump in α at T BKT and the non-linear IV behavior has been found in other thin films of conventional superconductors, where the disorder smears out the sharp features [31]. In our FeSe devices this disorder may be caused by the formation of structural domains at temperatures below the nematic transition which create local inhomogeneities, leading to an increase in the width of the superconducting transition.
Upper critical field in thin flakes of FeSe A key feature of two-dimensional superconductivity is the significant enhancement of anisotropy in the angular dependence of the upper critical field H c2 [11]. To investigate the suppression of superconducting behaviour in magnetic field we have measured the resistance of several devices for different orientations of applied field. Figures 3a,b) show the temperature dependence of the resistive transition of a thin flake device with t=14 nm in constant magnetic fields up to 0.8 T with H||c, and in fields up to 8 T with H||(ab). Additional transport measurements as a function of magnetic field performed at different fixed temperatures, as well as at fixed angles θ are shown in Fig. S6 in SM. Based on these measurements, one can construct the phase diagram of the upper critical field for several different devices, as summarised in Fig. 3d).
Additionally, to assess the changes in the superconducting anisotropy at the lowest experimental temperature we have performed an angle-dependent study of the upper critical field H c2 (θ) for two devices with t = 14 nm and t = 24 nm at T = 2 K, as presented in Fig. 3c) using a methodology presented in Fig. S8. To be able to compare the two different devices, we plot the ratio H c2 (θ)/H c2 (θ = 0)), as shown in Fig. 4c). Using the anisotropic Ginzburg-Landau (GL) theory [32], we can extract the anisotropy parameter Γ defined by the ratio between H c2 when H||(ab) to H c2 for H||c. We find that Γ increases significantly from 2.4 for the t=24 nm device to >10 when t=14 nm. This indicates a significant increase in anisotropy as the flakes become thinner and closer to the twodimensional limit, as shown in Fig. 4b) (additional data for a t=16 nm flake is shown in Fig. S7). The value for the thicker flake device is comparable to the value of 1.8 observed in bulk FeSe crystals [33], while the thinner flake device has a large anisotropy, comparable to that observed in FeS crystals [34]. This suggests that the enhanced anisotropy can be linked to an increase in two-dimensionality of the Fermi surface.
As the superconducting anisotropy, Γ, is strongly temperature dependent, we analyse in detail the complete superconducting phase diagram as a function of magnetic fields parallel and perpendicular to the conducting planes for different devices, as shown in Fig. 3d). We find that the standard threedimensional Werthamer-Helfand-Hohenberg (WHH) model [35], with the inclusion of spin paramagnetism and spin-orbit scattering, describes the temperature dependence of the upper critical field of the thick 100 nm flake device. A list of all obtained parameters can be found in Table I in SM. Orbital pair breaking alone accounts for the temperature dependence of H c2 for H||c, as shown in Fig.3d). However, when the magnetic field is aligned along the conducting (ab) plane, a Pauli pair breaking contribution has to be included which reduces the orbital-limited critical field by µ 0 H P = µ 0 H orb c2 / 1 + α 2 M , where α M is the Maki parameter. The extracted Maki parameter α M is 2.4 for thick flakes (t=54 and 100 nm), close to the value of α M = 2.1 found for bulk single crystals [33] For a thinner flake (t=24 nm), the Maki parameter increases to 4.15 (as shown in Table I in SM). To describe the data fully the spin-orbit scattering constant needs to be included and varies from λ SO ∼ 0.2 − 0.35 (see Fig. S5 in SM). Our WHH fitting parameters are close to the values obtained for FeSe 0.6 Te 0.6 single crystals where α M ∼5.5 and λ SO ∼ 1 suggesting that the upper critical field is dominated by Pauli paramagnetic effects [36].
In stark contrast to the behaviour found in thick devices, the thinnest measured device with t=14 nm exhibits a drastically different temperature dependence of the upper critical field for H c2 ||(ab), reaching a relatively high value of ∼12 T at 2 K despite the strongly suppressed T c ∼ 3.63 K. As a result, the slope close to T c increases dramatically, predicting extremely large orbital limiting field H orb c2 = −0.69|dH c2 /dT | T =Tc T c (94 T as shown in Table I  (2D-GL) theory [37], which predicts a square root temperature dependence of the in-plane H c2 close to T c . This accurately describes the observed behaviour of the t=14 nm device, as shown in Fig. 3d) by the solid line and in Fig. S5 in SM. This finding further emphasizes the change in character of superconductivity of FeSe in the thin limit, becoming more two-dimensional.
Discussion In order to compare the effect of thickness on the upper critical field of FeSe devices we investigate a reduced upper critical field phase diagram, by normalizing the upper critical field as h = H c2 /H P (0) against the reduced temperature T /T c of each device, as shown in Fig. 4a). Here, the BCS Pauli paramagnetic limit is defined in the weak coupling limit as H P (0) = 1.85T c [38]. Interestingly, for thick devices we observe a similar temperature dependence of the reduced upper critical field for each orientation that can be well described by the WHH model. Furthermore, the in-plane upper critical field at zero temperature exceeds the Pauli limit, H c2 (0) ∼ 1.6H P (0) for thick flakes and increases above 2 for the thinnest t=14 nm flake (Fig. 4a). This suggests that spin paramagnetic effects play an important role in determining the upper critical field of these thin flakes, and indicate that FeSe thin flake may provide a possible route towards unconventional triplet pairing [39]. Furthermore, exceeding the Pauli paramagnetic limit coupled with a large value of the Maki parameter without a finite λ SO would lead to a first-order transition at low temperature, known as a FFLO state [40,41], however this in not observed in our devices. In monolayer systems such NbSe 2 [11] and ion gated Mos 2 [42] intrinsic spinorbit interaction effects lead to Ising superconducting and a significant increase of H c2 ||(ab). As FeSe retains inversion symmetry at all thickness, this mechanism cannot explain the enhancement of H c2 (ab)/H P (0) in the t=14 nm device. Moreover, in a multi-band system like FeSe the Pauli limit can exceed the single-band estimate since there are several gaps but a single T c [43] and one could expect that the largest gap sets the Pauli limit [44,45]. Fig. 4b) summarises the thickness dependence of the superconducting critical temperature T c and the upper critical field anisotropy parameter Γ at T = 0.9 T c . We observe that while Γ increases, the critical temperature T c decreases suggesting the evolution towards two-dimensional superconductivity in the thin limit of FeSe flakes. To understand this further, we use the Ginzburg-Landau formalism to estimate the coherence lengths from the slope of the upper critical value near T c for the two magnetic field orientations in Fig. 3d). Fig. 4c) shows that the coherence length in the (ab) plane, ξ ab , exponentially increases from ∼ 4 nm to a value of ∼ 10 nm in the thin limit. These coherence lengths were estimated using the 3D GL and are a factor of 2 larger than the values extracted using 2D GL in the thin limit (see Fig. 4c)). In contrast, ξ c decreases significantly as the flakes get thinner, to ξ c ∼0.6 nm when t =14 nm, much smaller than the bulk value of ξ c ∼1-2 nm [46], and is comparable in length to the c-axis lattice constant of ∼ 0.55 nm, providing further evidence that the superconductivity is becoming increasingly two-dimensional, by confining the order parameter in one unit cell of FeSe. In this case, the weak Josephson coupling of the (ab) planes strongly reduces the role of orbital pair-breaking effects on H c2 . This result is somewhat surprising, as a t=14 nm flake is composed of approximately 25 individual FeSe layers, well above the FeSe monolayer limit and not comparable in thickness to the bulk value of ξ c .
The superconductivity in two-dimensional superconductors can also be suppressed by disorder. In the 2D limit, conduction electrons can be easily localised due to the quantum interference effect in the presence of disorder that give rise to Anderson localization [47]. As the degree of disorder increases, the superconductivity can be destroyed due to the suppression of amplitude of the superconducting order parameter or when the phase fluctuates strongly and its coherence is lost. Despite a reduced T c and lower RRR when compared to bulk, the normal state sheet resistance of the t =14 nm device is ∼ 100 Ω/ and remains well below the quantum resistance (R Q = h/4e 2 = 6.45 kΩ/ ) at which a superconductor- insulator transition is expected to occur [48]. This indicates that the suppression of T c is not driven by disorder, as was previously reported in amorphous thin films [30]. In order to ensure that the observed suppression of superconductivity and broadening effects are not extrinsic, we have examined the effect of air exposure on T c , shown in Fig. S2 in SM. We find that the encapsulated FeSe thick flakes are quite robust to air exposure whereas the thinner ones are more sensitive. However, the timescale required to significantly reduce T c is much longer than that used in our study (which was less than 1 hour). Another important parameter that can affect the superconducting and transport behaviour of thin flakes of FeSe is the strain induced by the substrate and its changes with temperature. Recent work on thin films of FeSe showed that positive in-plane strain enhances T c , whilst reducing the structural transition at T s [8]. This indicates that the suppression of both superconductivity and the structural transition in flakes cannot be solely attributed to in-plane strain effects from the substrate. However, the substrate inherently affects the thin flakes and can play a role in determining the local microstructure of the nucleated twin domain structure, and may lead to broader superconducting transitions in thinner samples.
The superconducting anisotropy of almost 10 detected in FeSe thin flakes is large compared with bulk FeSe and flakes of FeTe 0.55 Se 0.45 of similar thickness. However, large anisotropy is also found in the ultra thin limit of a 1 nm FeSe EDLT device with a large T c ∼ 40 K [49], suggesting that the character of the two-dimensional superconductivity is not changed by gating and doping of charge carriers.
The reduced dimensionality of thin flakes together with the the short coherence lengths can enhance the thermal fluctuations of the superconducting order parameter near T c , in comparison to classical superconductors [50]. In thin flakes of FeSe the type of fluctuations described by the Ginzburg number (that can be also related to (k B T c /E F ) 4 ) can be large due to the small Fermi energy of FeSe [21]. This number increases upon reduction of the flake thickness, approaching values similar to those found in cuprates [50]. The presence of these fluctuations coupled with the observation of the BKT transition in the thinnest flakes supports the idea that by thinning down FeSe, one stabilises a fluctuating two-dimensional and highly anisotropic superconductor. The suppression of superconductivity can be either linked to strong fluctuations or potentially to the loss of Josephson coupling between conducting layers. As flakes become thinner, screening of the Coulomb interaction becomes weaker and eventually the superconductivity is destroyed. For a system with a very small Fermi energy, such as FeSe, this mechanism is expected to be particularly pronounced. It remains to be understood how this type of superconductor interacts with a substrate to drive high-T c superconductivity towards the single atomic layer limit. The interface between the FeSe monolayer and the SrTiO 3 substrate also plays an important a role in superconductivity due to the strain caused by the lattice mismatch, enhancement of electron-phonon coupling, polaronic effects associated with the high dielectric constant of the substrate, and carrier doping from the interface [51].
Conclusions In summary, we have investigated the evolution of superconductivity in high quality FeSe thin flakes devices as a function of thickness. We have identified a strong change in the character of superconductivity from the thick limit, in which samples show similar behaviour to that of bulk FeSe, to the thin limit where superconductivity is strongly suppressed and highly anisotropic. Our studies indicate that in the thin limit and in the absence of a dopant substrate, the superconductivity of FeSe still exhibits a twodimensional character. This supports the premise that enhanced two-dimensionality could be one of the key components of a high-temperature superconductor. Future studies are needed to assess independently the role of strain and carrier doping in stabilising the robust high-temperature superconducting state in FeSe.
Acknowledgments. We thank Lara Benfatto for very helpful comments on our manuscript. The research was funded by the Oxford Centre for Applied Superconductivity (CFAS) at Oxford University. We also acknowledge financial support of the John Fell Fund of the Oxford University. This work was partly supported by EPSRC (EP/I004475/1, EP/I017836/1). LF is supported by the Bath/Bristol Centre for Doctoral Training in Condensed Matter Physics, under the EPSRC (UK) Grant No. EP/L015544. Part of this work was supported by HFML-RU/FOM and LNCMI-CNRS, members of the European Magnetic Field Laboratory (EMFL) and by EPSRC (UK) via its membership to the EMFL (EP/N01085X/1) AAH acknowledges the financial support of the Oxford Quantum Materials Platform Grant (EP/M020517/1). AIC acknowledges an EPSRC Career Acceleration Fellowship (EP/I004475/1).
Methods Thin FeSe flakes were mechanically exfoliated from high quality single crystals onto silicone elastomer polydimethylsiloxane (PDMS) stamps. Flakes of suitable geometry and thickness were then transferred onto Si/SiO 2 (300 nm oxide) substrates with pre-patterned Au contacts using a dry transfer set-up housed in a nitrogen glovebox with an oxygen and moisture content < 1 ppm. To minimise environmental exposure a thin capping layer (∼20 nm) of hexagonal boron nitride (h-BN) was then transferred on top of the FeSe flake, encapsulating the sample. An optical image of a typical sample is shown in the inset of Fig. 1d). The thickness of each sample was accurately determined by atomic force microscopy (AFM) after all measurements had been completed. Magneto-transport measurements at temperatures down to 2 K and magnetic fields up to 16 T were performed using a Quantum Design Physical Property Measurement System (PPMS), with an additional sample measured at temperatures down to 0.37 K and magnetic fields up to 37.5 T at the High Field Magnet Laboratory (HFML-EMFL) in Nijmegen (shown in Fig. S4 in Supplemental Material (SM)). The Hall and longitudinal resistivity contributions were separated by (anti)symmetrizing the data using 4-point measurements obtained under negative and positive magnetic fields. The de-vices presented are of high quality having a relatively high residual resistance ratio (RRR), R(300 K)/R(15 K) ∼ 6 − 16, as detailed in Fig. S3 in SM. The superconducting critical temperature, T c , and upper critical field, H c2 was normally defined as the position at which the resistance reached 50% of its normal state value or the maximum in its derivative. The upper critical field was measured for two different orientations of the conducting (ab) plane with respect to the applied magnetic field (either parallel to the conducting plane, H||(ab) (θ=90 • ) or perpendicular to it, and parallel to the crystallographic caxis, H||c (θ=0 • )). Angular-dependent studies were also performed at 2 K.
Additional Information Correspondence and requests for materials should be addressed to LF (lsf24@bath.ac.uk) and A.I.C (amalia.coldea@physics.ox.ac.uk).

Upper critical field and relevant parameters
In an conventional BCS superconductor, an external magnetic field can destroy the Cooper pairs either due to a) the orbital pair breaking due to the Lorentz force acting on the charge of the paired electrons, known as the orbital limit, ∆ ∼ µ 0 µ B H orb c or b) due to the Pauli paramagnetic pair breaking as a result of the Zeeman energy that leads to the alignment of the two opposite spins of the two electrons forming the singlet state with the applied field, called the Pauli paramagnetic limit [52].

WHH model
The single-band model by WHH provides predictions for the upper critical field, H c2 of type-II superconductors as a function of temperature, T in the dirty limit given by [35]: where t = T /T c , and h = 4H c2 /[π 2 T c (dH c2 /dT ) T =Tc . λ so = /(3πk B T c τ so ) accounts for spin-orbit and spin-flip scattering with τ so as the mean free scattering time.
In the absence of spin paramagnetic effect (α M = 0), the upper critical field is restricted by orbital pair breaking effect. In the weak-coupling case, assuming λ so = 0): The Maki parameter, α M = √ 2H orb c2 /H pm c2 indicates which of the orbital or spin-paramagnetic effects is more dominant in determining the upper critical field. When α M > 1 the paramagnetic effects become essential. This condition can be easily satisfied for materials with low Fermi energies and high T c .
In a single-band system, in the clean limit the parameter α M for H||c is given by [45]: and for H||(ab) by where where v ab F and v c F are Fermi surface velocities. For H||c, the FFLO instability occurs at α M > 1.8, where for a single-band spherical Fermi surface which implies ∆ > E F , which can hardly happen in a single-band conventional superconductor with a low effective mass m * ∼ m e [44]. The criterion α M > 1.8 can be satisfied more easily in strongly anisotropic materials in a magnetic field parallel to the layers, H||ab, in which case α M is enhanced by the large v ab F /v c F ratio. A FFLO state would be easier to stabilize for H||ab, in strongly correlated materials in presence of strong correlations where m * m e [44]. Coherence lengths.
The superconducting coherence length quantifies the size of Cooper pairs. In a clean metal with a large Fermi velocity, it is the strong overlap of Cooper pairs that provides the superconducting phase coherence and the coherence length is given by: where v F is the Fermi velocity and ∆ the energy gap. In a dirty superconductor, the mean free path is much shorter than the coherence length which is given by: where v F is the Fermi velocity and ∆ the energy gap.
The upper critical field perpendicular to the conducting planes, H c2⊥ is determined by vortices whose screening currents flow parallel to the planes.
The coherent lengths ξ ab and ξ c were estimated from the single band anisotropic Ginzburg-Landau equations and The anisotropy ratio Γ is given by Superconductivity in anisotropic superconductors is confined to the two-dimensional planes. The three-dimensional phase coherence is provided by Josephson current between planes.
2D Ginzburg-Landau model A 2D Ginzburg-Landau (2D-GL) theory predicts that the upper critical field follows the following relationship: where φ 0 is the flux quantum, t is the thickness of the superconducting layer, and ξ GL is the Ginzburg-Landau coherence length. The 2D-GL behaviour exhibits a square root temperature dependence of H c2 ab close to T c which accurately describes the observed behaviour of the flake with t= 14 nm.
Angular dependence of the upper critical field The anisotropic Ginzburg-Landau (GL) theory [32] describes the TABLE S1. Summary of superconducting parameters of the FeSe thin flake devices for different thickness t. The listed parameters contain beside superconducting critical temperature, Tc, the WHH fitting parameters related to the the upper critical field slope of near Tc, (dHc2/ dT)T =Tc , the orbital upper critical field, µ0H orb c2 , the experimental upper critical field µ0Hc2. The extracted Maki parameter, αM, and the spin-orbit scattering constant, λSO, are listed.   58 nm in c). The low temperature resistive part is fitted to R = Rne a−bt −1/2 , where Rn is the normal state resistance [30]. The dashed lines are fits to the high-temperature regime above Tc, a and b are constants and t = (T − TBKT)/TBKT is the reduced temperature. b), d) The dependence ln(R/Rn) against t −1/2 for the two different samples from a) and c). A linear dependence is evidence that the lowest temperature finite resistance region can be described in terms of BKT effects.