Reentrant Phase Coherence in Superconducting Nanowire Composites

TL;DR

This paper demonstrates that inhomogeneous nanofilament composites can exhibit reentrant phase coherence, overcoming phase fluctuation issues in low-dimensional superconductors, with potential for more resilient superconducting materials.

Contribution

It introduces a new approach using nanofilament composites to achieve reentrant phase coherence in low-dimensional superconductors, enhancing their stability against fluctuations.

Findings

Reentrant phase coherence observed upon increasing temperature, magnetic field, or current.

Peak in Josephson energy correlates with reentrant phase coherence.

Na$_{2- ext{delta}}$Mo$_6$Se$_6$ serves as a blueprint for resilient nanofilamentary superconductors.

Abstract

The short coherence lengths characteristic of low-dimensional superconductors are associated with usefully high critical fields or temperatures. Unfortunately, such materials are often sensitive to disorder and suffer from phase fluctuations in the superconducting order parameter which diverge with temperature $T$, magnetic field $H$ or current $I$. We propose an approach to overcome synthesis and fluctuation problems: building superconductors from inhomogeneous composites of nanofilaments. Macroscopic crystals of quasi-one-dimensional Na$_{2-\delta}$Mo$_6$Se$_6$ featuring Na vacancy disorder ($\delta\approx$~0.2) are shown to behave as percolative networks of superconducting nanowires. Long range order is established via transverse coupling between individual one-dimensional filaments, yet phase coherence remains unstable to fluctuations and localization in the zero-($T$,$H$,$I$) limit. However, a region of reentrant phase coherence develops upon raising ($T$,$H$,$I$). We attribute this phenomenon to an enhancement of the transverse coupling due to electron delocalization. Our observations of reentrant phase coherence coincide with a peak in the Josephson energy $E_J$ at non-zero ($T$,$H$,$I$), which we estimate using a simple analytical model for a disordered anisotropic superconductor. Na$_{2-\delta}$Mo$_6$Se$_6$ is therefore a blueprint for a future generation of nanofilamentary superconductors with inbuilt resilience to phase fluctuations at elevated ($T$,$H$,$I$).

Equations

$$E_J\propto \frac{1}{R_T}△\tanh \left(\frac{△}{2k_{B}T}\right)$$

Full text

KEYWORDS: superconductivity $\cdot$ nanofilaments $\cdot$ quasi-one-dimensional $\cdot$ reentrance

Composite nanofilamentary systems offer a unique environment in which to study the effects of dimensionality and phase fluctuations on the superconducting transition 1. Such materials may be described as “quasi-one-dimensional” (q1D), since they exist in bulk macroscopic form yet exhibit an intense uniaxial anisotropy in their physical properties. Modulating the transverse (inter-filamentary) coupling in these materials is of fundamental interest due to an expected dimensional crossover, whose impact on the electronic properties remains unclear 2. From a practical perspective, one can also imagine synthetic “ropes” of coupled one-dimensional (1D) nanowires as future superconducting cables 3. Aside from the obvious morphological advantage, there are powerful incentives to develop functional superconductors from nanoscale q1D building blocks. Firstly, orbital limiting is suppressed in q1D superconductors 4, thus increasing the critical field $H_{c2}$ to the Pauli limit (and in principle, a q1D superconductor with triplet pairing would be completely immune to magnetic fields). Secondly, various mechanisms for enhancing the critical temperature $T_{c}$ exist in superconducting nanostructures, including tuning the density of states through van Hove singularities, shape resonances 5, strain-induced renormalization of the electronic structure 6 and shell effects 7. Coupled arrays of nanowires therefore represent an attractive and realistic route towards developing new functional superconductors, as well as exploring the role of dimensionality in correlated electron systems.

Although fabrication techniques for such nanofilamentary composites are in their infancy, we may hasten their development by studying quasi-one-dimensional (q1D) crystals featuring chain-like structures, which behave as weakly-coupled arrays of parallel nanowires. Within the superconducting phase, coupling between nanowires is expected to occur via the Josephson effect, i.e. phase-coherent Cooper pair tunnelling 8. An identical process is responsible for establishing long-range order between crystal planes in highly two-dimensional (2D) superconductors, including cuprates 9 and pnictides 10. However, the superconducting transitions in q1D materials exhibit important differences compared with 2D or 3D superconductors, which we summarize in Fig. 1(a). In a q1D system, transverse Josephson coupling (i.e. phase-coherent inter-chain Cooper pair tunnelling) is a second order process 2 which occurs below temperature $T_{J}\!\geq\!t_{\perp}^{2}/t_{/\!/}$, where $t_{\perp}$ and $t_{/\!/}$ are the single-particle hopping energies perpendicular and parallel to the structural chains. For most q1D superconductors, $T_{c}~{}{\ll}~{}t_{\perp}^{2}/t_{/\!/}$ and there is a direct transition from the normal state to a quasi-three-dimensional (q3D) superconducting phase (a detailed discussion is provided in section I of the Supporting Information (SI)). However, if the anisotropy and the pairing interaction are both sufficiently large, the onset temperature for superconducting fluctuations $T_{ons}$ may be greater than $t_{\perp}^{2}/t_{/\!/}$: in this case, a “2-step” superconducting transition occurs. Below $T_{ons}$, 1D superconductivity develops within individual chains but topological defects (phase slips) locally suppress the amplitude of the superconducting order parameter $\left|\Psi\right|$ to zero, creating a resistive state identical to that of a single nanowire 11, 12, 13. Subsequently, a 1D$\rightarrow$q3D dimensional crossover occurs within the superconducting state: phase coherence (and hence long range order) are established at $T_{J}<T_{ons}$.

The $M_{2}$Mo6Se6 family 14 ($M$ = Tl, In, Na, K, Rb, Cs) are archetypal q1D materials, in which infinite-length (Mo6Se6)∞ chains are aligned along the $c$ axis of a hexagonal lattice (Fig. 1(b)). $M$ ions are intercalated between the chains and act as a charge reservoir for a single 1D conduction band of predominant Mo $d_{xz}$ character. Transverse coupling occurs via the $M$ ions and the electronic anisotropy may hence be tuned by selection of $M$. Furthermore, $M_{2}$Mo6Se6 crystals typically display small $M$ ion deficiencies 15, constituting an intrinsic disorder. These $M$ vacancies reduce the coupling between (Mo6Se6)∞ chains and break them electronically into finite-length segments. Tl2Mo6Se6 and In2Mo6Se6 are already known to exhibit superconducting ground states 16, 17, 18; $M_{2}$Mo6Se6 are therefore ideal materials in which to investigate the behavior of future superconducting composites made from coupled nanowire arrays.

In this work, we show that disordered single crystals of superconducting Na2-δMo6Se6 display “reentrant” characteristics, where phase coherence is stabilized by an increased inter-filamentary coupling within a region of non-zero $(T,H,I)$ phase space. The ability to control the transverse phase coherence by modulating $(T,H,I)$ arises from the sensitivity of the inter-filamentary coupling to electron localization, which is gradually suppressed as $(T,H,I)$ increase. Reentrant phase coherence is a highly desirable property, since maintaining phase stiffness 19 at elevated $(T,H,I)$ is perhaps the greatest challenge to the development of new functional superconductors 20. Our results pave the way towards synthesizing nanofilamentary materials whose superconducting properties are enhanced rather than destroyed within the high $(T,H,I)$ domain.

1 RESULTS

1.1 Electronic and crystal structure of Na2-δMo6Se6

Our identification of Na2-δMo6Se6 as a model nanofilamentary superconductor was motivated by a combination of electronic structure calculations and crystal growth considerations. Firstly, strong crystalline anisotropy (i.e. a weak transverse coupling) is an essential prerequisite for accurately simulating a nanofilamentary superconductor. Using ab initio density functional theory, we have calculated transverse Josephson coupling temperatures $t_{\perp}^{2}/t_{/\!/}=$ 4.4 K, 3.0 K, 1.0 K for Tl2Mo6Se6, In2Mo6Se6 and Na2Mo6Se6 respectively (see SI section I for details). The ground state of Na2-δMo6Se6 has until now remained unknown; however the extremely weak transverse coupling implies that any superconducting order parameter in this material will be highly anisotropic and exhibit a low phase stiffness. Secondly, the small Na atomic radius and elevated growth temperature (1750 *∘*C) are expected to increase the Na ion mobility during crystal synthesis, resulting in a substantially larger Na deficiency than the usual $\sim 2.5-5$% observed in Tl2Mo6Se6 15. Increasing the Na vacancy concentration (and hence the disorder) will result in crystals which are more realistic analogies to an inhomogeneous nanofilamentary array.

We therefore synthesized a series of needle-like Na2-δMo6Se6 single crystals with typical lengths 2-3 mm (see Methods for details). Synchrotron X-ray diffraction measurements indicate a typical 10% Na deficiency, i.e. $\delta\approx$ 0.2, although the (Mo6Se6)∞ crystal superstructure remains highly ordered. [A complete structural refinement is included in the Supporting Information.] The influence of the Na vacancy-induced disorder can clearly be seen in the electrical resistivity $R(T,H,I)$, which rises due to localization at low temperature (Fig. S6) before the crystals undergo a transition to a superconducting ground state (Figs. 2,3). Here we focus on the extent and control of phase-coherent superconductivity as a function of $(T,H,I)$ alone. A brief discussion of the possibility of tuning the superconducting ground state by varying the disorder level may be found in SI section VIII.

1.2 Dimensional crossover in the superconducting transition

We first demonstrate that Na2-δMo6Se6 displays the 2-step transition outlined in Fig. 1(a). Figure 2 shows the superconducting transitions in $R(T)$ for a typical Na2-δMo6Se6 single crystal with $T_{ons}=$ 2.7 K. Na vacancy disorder will reduce the coherence length $\xi$ and hence the energy barrier to topological defect formation 12, 13 in the 1D regime ($T_{J}<T<T_{ons}$). We therefore anticipate an important contribution to $R(T)$ from thermally-activated phase slips (TAPS) along individual (Mo6Se6)∞ nanowires within this temperature range. To model our data, we adapt the well-known Langer-Ambegaokar-McCumber-Halperin (LAMH) TAPS model to describe an array of superconducting nanowires and proceed to fit the $R(T)$ superconducting transitions (Fig. 2). A detailed discussion of the LAMH model may be found in SI section IIIB,C. Directly below $T_{ons}$, our model accurately reproduces $R(T)$ independently of the excitation $I$: this indicates a universal onset of fluctuating 1D superconductivity. To the best of our knowledge, Na2-δMo6Se6 is the first bulk q1D superconductor to be accurately described by any 1D phase slip theory.

However, 1D LAMH theory can only describe our data over a finite temperature range $\sim$ 0.4 K. As the temperature is reduced further, an anomaly appears in each $R(T)$ curve whose position is displaced to lower temperature as $I$ increases. For $I\leq 0.1$ mA, the LAMH regime in $R(T)$ is terminated by a sharp peak: this corresponds to a suppression of single-particle tunnelling between phase-incoherent superconducting filaments, followed by the onset of phase coherence (i.e. inter-chain Cooper pair tunnelling) at lower temperature. $R(T)$ subsequently forms a finite-resistance plateau instead of falling to zero, which we attribute to isolated barriers such as micro-cracks and twin boundaries separating macroscopic phase-coherent superconducting regions (SI sections IIID, VIII). Eventually $R(T)$ rises again as $T\rightarrow 0$: this is the first experimental signature of reentrance. In contrast, for $I>0.1$ mA the resistance begins to diverge from the LAMH model around $T\sim 1.8$ K but continues to fall without forming a plateau, and eventually saturates with no upturn in the $T\rightarrow 0$ limit. The sharp peak is smeared into a broad hump (Fig. S3), which we attribute to pair-breaking effects from the increased current. Together, these features indicate the emergence of a phase-coherent q3D superconducting state composed of coupled 1D filaments.

Our electronic structure calculations predict that a 1D$\rightarrow$q3D dimensional crossover for two-particle hopping (i.e. Josephson coupling) should occur at temperature $t_{\perp}^{2}/t_{/\!/}=$ 1.0 K (SI Section I). However, the anomaly in the $R(T)$ data and the deviation from LAMH fits suggest that transverse coupling develops at higher temperature 1.4 K $\lesssim T_{J}\lesssim$ 2.0 K. This increase in $T_{J}$ relative to our theoretical expectations may be attributed to two factors: firstly, any defects (including Na vacancies) strongly reduce the effective $t_{/\!/}$ due to the ease of blocking electron motion along a single (Mo6Se6)∞ filament. Although it may initially seem counter-intuitive for inter-chain defects to influence intra-chain transport, each Na vacancy not only removes one electron from the conduction band (which is predominantly of Mo $d_{xz}$ character), but also locally modifies the crystal field. In such intensely anisotropic materials, even minor crystal field inhomogeneities can lead to an enhanced back-scattering at low temperature 21. Secondly, $T_{J}$ is believed to be enhanced to higher temperatures by increasingly strong electron-electron interactions 2, although the behavior of q1D electron liquids below the single-particle dimensional crossover temperature ($T_{x}\leq t_{\perp}\sim$ 120 K in Na2-δMo6Se6) remains to be completely understood.

Interestingly, our experimental $R(T)$ and voltage-current $V(I)$ data share all the features of the well-known Berezinskii-Kosterlitz-Thouless (BKT) transition, which establishes long-range order in 2D materials. This suggests that an exponential divergence in the transverse phase correlation length occurs close to $T_{J}$ in q1D materials. The similarity between 2D systems exhibiting BKT transitions and q1D superconductors becomes apparent upon considering the phase of the order parameter on each 1D filament (Fig. S4(a)). In the plane perpendicular to the filaments, the spatial variation of the phase satisfies 2D $XY$ symmetry. However, the validity of BKT physics in q1D materials is neither obvious nor trivial, since it would imply that phase fluctuations parallel to the filaments do not influence the onset of transverse phase coherence. Nevertheless, a BKT-type analysis (SI section IIIE) of our transport data yields $T_{J}=$ 1.71 K, in good agreement with the anomalies which we observe in our $R(T)$ data.

Combining our LAMH fitting parameters and $T_{J}$ enables us to estimate a typical filament diameter $\sim$ 0.4-0.7 nm (SI section IV). This corresponds closely to the (Mo6Se6)∞ chain diameter of 0.60 nm and the hexagonal lattice parameter $a=$ 0.86 nm, indicating that single (Mo6Se6)∞ chains behave as 1D superconducting nanowires. Our analysis also indicates that current flows inhomogeneously through Na2-δMo6Se6 and is supported by simulations of a disordered q1D conductor with an anisotropic random resistor network (SI section V). We attribute the inhomogeneous flow to the enhanced influence of disorder in 1D materials: above $T_{J}$, defects (e.g. Na vacancies) restrict transport along individual (Mo6Se6)∞ chains, forcing the current to follow a highly percolative route.

1.3 Reentrant phase coherence

Aside from the 2-step 1D $\rightarrow$ q3D transition, the major feature of interest in Fig. 2 is the rise in resistance as $T\!\rightarrow\!0$ for $I\leq 0.1$ mA. This reversion to a fluctuation-dominated state suggests that the transverse phase coherence is fragile and reentrant. We track the evolution of the reentrance with current in Fig. 3(a), where three important trends may be identified. Firstly, the superconducting transition is conventionally suppressed to lower temperature as $I$ increases. However, the temperature dependence of the critical current $I_{c}(T)$ does not follow the standard Bardeen relation derived for bulk superconductors 22 (Fig. 3(a) inset), remaining unusually large at high temperature. Secondly, for $T<1$ K the resistance falls as $I$ rises. This indicates that elevated currents induce reentrant phase coherence even in the $T\rightarrow 0$ limit. Thirdly, the resistance rises (signalling a loss of phase coherence) upon reducing the temperature for $I\leq 0.1$ mA. Long range superconducting order is therefore only stable within a well-defined region of non-zero $(T,I)$ phase space.

To determine whether the phase coherence is also reentrant in magnetic fields, we measure the magnetotransport perpendicular and parallel to the $c$ axis (Fig. 3(b-e)). A clear dichotomy is observed between data acquired using low (Fig. 3(b,d)) and high (Fig. 3(c,e)) currents. The $R(T)$ transitions in $I_{\mathrm{high}}\equiv$ 0.6 mA resemble those of a conventional superconductor (albeit substantially broadened) and no trace of reentrance is visible. In this case, transverse phase coherence has already been stabilized by the large current and hence the magnetic field exhibits a purely destructive effect on superconductivity. In contrast, for $I_{\mathrm{low}}\equiv$ 1 $\mu$A, $R(T)$ rises at low temperature, indicating field-induced reentrance. For $H_{\perp}=$ 0.75 T, $R(T)$ falls again below $T=$ 0.5 K, indicating double-reentrant behavior 23, 24. Crucially, both phase coherence and double-reentrance are absent as $T\rightarrow 0$ in our zero-field data acquired at low current. This implies that a sequence of superconducting fluctuations, suppression of quasiparticle tunnelling and eventual macroscopic phase coherence (which has been suggested to create “double dips” in $R(T\rightarrow 0)$ in other inhomogeneous superconductors 25, 26) cannot be responsible for these data. Instead, the double-reentrance may be a signature of a divergent $H_{c2}$ caused by mesoscopic fluctuations 27.

Regardless of the applied current, the magnetotransport varies strongly with the field orientation, as expected for a q1D superconductor. We quantify this anisotropy via the temperature dependence of the upper critical fields $H_{c2\perp,/\!/}(T)$, which we plot in Fig. 3(f). Using the Werthamer-Helfand-Hohenberg (WHH) model to estimate $H_{c2/\!/}(T\!=\!0,I_{\mathrm{small}})$ and subsequently applying anisotropic GL theory (SI Section VI), we extract coherence lengths 14.4 nm $\leq\xi_{/\!/}(0)\leq$ 21.0 nm, 4.28 nm $\leq\xi_{\perp}(0)\leq$ 4.58 nm and an anisotropy $\epsilon\equiv\xi_{/\!/}/\xi_{\perp}$ of 3.14 $\leq\epsilon\leq$ 4.90. This value is lower than the 12.6 reported for Tl2Mo6Se6 17, in spite of the increased electronic anisotropy: $t_{/\!/}/t_{\perp}=$ 86 for Na2-δMo6Se6, versus 31 for Tl2Mo6Se6. Two factors are responsible for this: firstly, disorder from the high Na vacancy density strongly suppresses $\xi_{/\!/}$ and hence $\epsilon$. This is exposed by the orbitally-limited values for $H_{c2\perp}$ in Tl2Mo6Se6 and In2Mo6Se6: 0.47 T and 0.25 T respectively 17, an order of magnitude lower than the 3.7-5 T measured for Na2-δMo6Se6. [Note that this reduction in $\epsilon$ also supports the previously-discussed enhancement of $T_{J}$ above $t_{\perp}^{2}/t_{/\!/}\equiv$ 1.0 K.] Secondly, our estimated $H_{c2/\!/}\approx$ 16-18 T exceeds the weak-coupling BCS Pauli limit $H_{P}\equiv 1.84T_{ons}=$ 5.0 T. This means that paramagnetic rather than orbital limiting is likely to suppress superconductivity for $H/\!/c$ and hence GL theory may only provide an upper limit for $\xi_{\perp}$.

A striking divergence of $R(T)$ is observed at low current for $T\lesssim 0.8$ K and $H_{\perp}\gtrsim 2.5$ T (Fig. 3(d)). This is a signature of magnetic field-induced Anderson localization, predicted to occur in q1D materials when a field is applied perpendicular to the high-conductivity axis 28. In the normal state, the conditions for localization are $\hbar\omega_{c}\gg t_{\perp}$ and $k_{B}T\ll t_{\perp}$ (where $\omega_{c}$ is the cyclotron frequency $\mu_{0}He/m_{e}$). Since electrons are paired for $T<T_{ons}$, Josephson tunnelling supplants single-particle hopping as the transverse coupling mechanism and we replace $t_{\perp}$ in the above conditions with $k_{B}T_{J}$. Cooper pair localization is therefore expected for $H_{\perp}\gg$ 1.3 T and $T\ll$ 1.7 K, in good agreement with our data. The resistance does not diverge for high currents (Fig. 3(e)) due to the pair-breaking effect of the increased current: in this case, field-induced localization is only expected for $\mu_{0}H>t_{\perp}m_{e}/\hbar{e}\equiv 90$ T.

1.4 Experimental phase diagram

The reentrance visible within our $R(T)$ data in Figs. 2,3 may be summarized by independently scanning $R(T)$, $R(H_{\perp,/\!/})$ and $R(I)$ (Fig. 4(a)). In the superconducting phase of Na2-δMo6Se6, $R$ is always minimized at non-zero $(T,H,I)$: this is in direct contrast to the behavior of a conventional bulk superconductor, where phase fluctuations in $R$ are invariably minimized as $(T,H,I)\rightarrow 0$. We highlight the fact that the $R(H_{/\!/,\perp})$ and $R(I)$ curves were acquired at $T=0.1$ K, thus confirming that transverse phase coherence is reentrant even as $T\rightarrow 0$ for sufficiently large magnetic fields or currents. We also note that the magnetoresistance (MR) is initially positive for $R(H_{\perp,/\!/})$, before falling steeply to its minimum value as transverse coupling is established. Conversely, quasiparticle tunnelling contributions to the transport would be expected to yield a gradual, monotonic negative MR prior to reentrance at low temperature 29. The absence of such a feature from our data provides further evidence that quasiparticle tunnelling does not play a major role in the reentrant behavior of Na2-δMo6Se6.

We map the extent of phase coherence in Na2-δMo6Se6 by assembling our experimental data into a single phase diagram (Fig. 4(b)). $H_{c2}(T)$ and $I_{c}(T)$ (circles) from Figs. 2,3 accurately describe the evolution of the superconducting transition, but do not capture the loss of phase coherence at lower temperature. To track this loss of coherence, we therefore define a “reentrance threshold” temperature $T_{R}(H,I)$ (stars) using the minima in $R(T,H)$ from Figs. 3,4(a). $T_{R}$ varies from zero to 1.6 K, illustrating how the reentrant regime spans a broad temperature range as $(H,I)$ are tuned. At temperatures below $T_{R}(H,I)$, neighboring filaments are phase-incoherent and Na2-δMo6Se6 exhibits the finite resistance characteristic of a fluctuating 1D superconductor. The volume enclosed by $H_{c2}(T)$, $I_{c}(T)$ and $T_{R}(H,I)$ in $(T,H,I)$ phase space hence contains a reentrant “shell” of phase-coherent superconductivity (dark red shading in Fig. 4(b)), which surrounds a phase-incoherent regime as $(T,H,I)\rightarrow 0$ (pink shading in Fig. 4(b)). Even if two of $(T,H,I)$ fall to zero, reentrance can still occur if the third parameter is sufficiently large: for example, phase coherence is still reentrant as $(T,H)\rightarrow 0$ above a threshold $I\sim 0.1$ mA.

1.5 Mechanisms for reentrance

In an inhomogeneous superconductor, reentrance may occur if the Josephson energy $E_{J}(T,H,I)$ rises with respect to the thermal energy $k_{B}T$ (or the Coulomb energy $E_{C}$ in granular materials) 23, 30, 31, 32, 33, 34. $E_{J}$ is a measure of the phase stiffness (and hence the coupling strength) between neighboring superconducting regions: microscopically, $E_{J}$ is proportional to the spatial overlap of the Cooper pair wavefunctions from each region. A rise in $E_{J}$ increases the energy cost of creating phase discrepancies, hence facilitating Cooper pair (Josephson) tunnelling between the regions. Once $E_{J}$ exceeds a threshold value of the order of $k_{B}T+E_{C}$, global phase coherence is established. Previous experimental observations of reentrance attributed to Josephson effects have generally occurred in amorphous, ultra-thin or granular films 24, 26, 29, 34, 35, 36, 37, 38, 39, 40.

Na2-δMo6Se6 is not an inhomogeneous or granular superconductor in the traditional sense, where phase coherence is determined by the ratio of $E_{J}$ to $E_{C}$ for individual grains 31, 32. Instead, it is a crystalline superconductor whose normal state is a localized metal, due to the combination of Na vacancy disorder and intense 1D anisotropy. Signatures of localization are clearly visible in the normal-state resistance $R_{\mathrm{NS}}$, which diverges as $T\!\rightarrow\!0$ and displays a large negative magnetoresistance (Fig. S6). Localization causes electronic states which lie close in energy to become widely separated in space 41, leading to a characteristic activation energy 42 $E_{a}(T,H)$. Although we deduce the presence of localization from the normal-state transport, its influence persists within the superconducting phase, where pairing occurs between localized electrons (provided that $\xi_{/\!/}$ remains shorter than the localization length 41). As the temperature falls, the electron wavefunctions become increasingly localized and the inter-filamentary pair hopping energy begins to fall below the $t_{\perp}^{2}/t_{/\!/}$ limit imposed by the electronic anisotropy. This implies a progressive reduction in the wavefunction overlap between neighbouring filaments: localization is suppressing the transverse coupling and hence $E_{J}$. Eventually, $E_{J}(T,H,I)$ falls below the threshold for phase coherence ${\sim}k_{B}T$ at $T=T_{R}(H,I)$. The influence of localization is accentuated by an emergent spatial inhomogeneity in the superconducting order parameter 43, 44, 45 which may locally suppress pairing, leading to further reductions in $E_{J}$. To achieve reentrant phase coherence, it is necessary to increase $E_{J}$ by delocalising the electrons. In principle, this may be achieved by thermal activation (raising $T$), reducing the barrier height between localized states (raising $I$) or Zeeman-splitting localized energy levels 46 (raising $H$).

Let us now attempt to model the effects of localization on the transverse phase coherence. The $(T,H,I)$ evolution of the inter-filamentary pair hopping energy cannot be determined experimentally: even if the transverse resistance $R_{\perp}(T,H,I)$ could be accurately measured (an extremely challenging task due to the crystal morphology, anisotropy and disorder), it would contain inseparable contributions from single-particle and pair hopping, and fall to zero (depriving us of information) upon establishing phase coherence. Instead, we estimate the $(T,H,I)$ dependence of $E_{J}$ within an analytical framework originally proposed by Belevtsev et al. for inhomogeneous superconductors 34:

[TABLE]

where $R_{T}$ is the resistance between superconducting filaments and $\triangle$ is the pairing energy. For reentrant superconductivity, a peak is expected to form in $E_{J}$ at finite $(T,H,I)$. Since $\triangle\tanh\left(\triangle/2k_{B}T\right)$ falls monotonically to zero as $(T,H,I)$ increase, a peak in $E_{J}$ at non-zero $(T,H,I)$ can only develop if the drop in $\triangle\tanh\left(\triangle/2k_{B}T\right)$ is initially compensated by a larger reduction in $R_{T}$ (see Fig. S7). In the granular superconductors for which equation (1) was originally derived, the origins of this reduction are well understood. Energy levels in individual grains are quantized, creating an activation energy for electron transfer: $R_{T}$ therefore falls exponentially as $T$ rises. Increasing $I$ also reduces $R_{T}$, since the associated increase in voltage diminishes the effective barrier height between grains. If $R_{T}$ furthermore exhibits negative magnetoresistance, then reentrance may occur upon raising $T$, $H$ or $I$. Applying a similar scenario in Na2-δMo6Se6, we extract $E_{a}(T,H)$ from $R_{\mathrm{NS}}(T,H)$ and treat this term analogously to the granular activation energy discussed above. We do not consider any Coulomb contribution to the reentrance, since Na2-δMo6Se6 is crystalline and $R_{\mathrm{NS}}(T)$ does not obey the Efros-Shklovskii hopping law 42 indicative of strong Coulomb repulsion. Using a similar current dependence for the inter-filamentary electron transfer rate to that in granular superconductors (see SI Section VII), we may then utilize the framework of equation 1 to estimate the evolution of $E_{J}(T,H,I)$.

Although we cannot calculate absolute values of $E_{J}$ (since several scaling parameters remain unknown), we may nevertheless establish the existence and location of any peaks in $E_{J}(T,H,I)$. The resultant curves are plotted above the corresponding $R(T,H,I)$ data in Fig. 4(a). Independently of changing $T$, $H$ or $I$, a peak appears in $E_{J}$. The peak positions approximately correspond to the resistance minima and hence the reentrance threshold $T_{R}(H,I)$. An exception to this trend occurs for $H/\!/c$, where the minimum in $R(H_{/\!/})$ occurs at a lower field than the peak in $E_{J}(H_{/\!/})$. This may be caused by the WHH model overestimating the true $H_{c2/\!/}$ for Na2-δMo6Se6 due to paramagnetic limiting. Finally, we simulate our experimental phase diagram, plotting the theoretical $\triangle(T,H,I)$ instead of $H_{c2}(T),I_{c}(T)$ and calculating $T_{R}(H,I)$ by evaluating the locus of the peaks in $E_{J}$ within the $(H,T)$ and $(I,T)$ planes. The results are shown as an inset to Fig. 4(b): a clear agreement is visible between our experimental and simulated phase diagrams.

2 DISCUSSION

Our data indicate that phase coherence in superconducting Na2-δMo6Se6 is stabilized by a large reentrant coupling between electron-doped (Mo6Se6)∞ chains. The reentrance is a direct consequence of electron localization induced by Na vacancy disorder. To clarify this mechanism, we sketch an inhomogeneous q1D superconductor composed of finite-length dirty nanofilaments in Fig. 5(a). The conduction band electrons become localized at low temperature, leading to a rise in $R(T)$ and a negative MR (Fig. 5(b)). In the $(T,H,I)\rightarrow 0$ limit, Josephson coupling between the nanofilaments is suppressed, resulting in phase-incoherent fluctuating superconductivity. Here we must point out a flaw in our model, which predicts $R_{T}\rightarrow\infty$ as $T\rightarrow 0$, independently of $(H,I)$. This would imply a vanishing Josephson energy $E_{J}\rightarrow 0$ and an invariable loss of phase coherence as $T\rightarrow 0$. In contrast, our experiments suggest that phase coherence remains stable at elevated $(H,I)$, even at $T=0$ (Fig. 4(b)). $R_{T}$ must therefore remain finite (i.e. metallic) at $T=0$.

This discrepancy between data and model is linked to the nature of the disorder-induced superconductor-insulator transition in Na2-δMo6Se6. It is possible that the typical disorder level in our crystals is sub-critical, i.e. $R(T\!=\!0)\ll\infty$ and the q1D hopping model which reproduces our experimental $R(T)$ data (Fig. S6, SI Section VII) is merely valid over a finite temperature range. The superconductor-insulator transition may also be replaced by a superconductor-metal-insulator transition for sufficiently large $(H,I)$, as suggested by the finite resistance which we measure as $T\rightarrow 0$ in Fig. 3(e). We note that zero-temperature metallic states have been predicted 47 and observed 48, 45 in Josephson-coupled superconducting arrays as well as amorphous NbxSi1-x 49. However, the physical concept which underlies our model (i.e. the formation of peaks in $E_{J}$ at non-zero $(T,H,I)$) remains valid regardless of the zero-temperature state of Na2-δMo6Se6: equation (1) continues to yield values for $T_{R}(H,I)$ in good agreement with our experimental data, even as our calculated $E_{J}$ values become vanishingly small in the $T\rightarrow 0$ limit.

Upon increasing $(T,H,I)$, the electrons are delocalized due to thermal activation, Zeeman level splitting and reduced barrier heights. Cooper pairs begin to tunnel between the (Mo6Se6)∞ filaments and phase coherence is initially stabilized rather than destroyed (Fig. 5(c)). Provided that the order parameter exhibits $s$-wave symmetry, we conclude that disorder in a nanofilamentary composite is beneficial to superconductivity, shortening $\xi_{/\!/}$ (thus raising $H_{c2\perp}$) while facilitating reentrant phase coherence. Although we acknowledge that $T_{ons}$ is low ($\lesssim 3$ K) in Na2-δMo6Se6 (due to the combination of a small density of states at the Fermi level and weak electron-phonon coupling), the electron delocalization mechanism responsible for reentrance remains active at temperatures at least an order of magnitude higher (see Fig. S6). We stress that in the presence of a pairing interaction, there is no obvious thermal limitation to this reentrance mechanism: negative MR and $dR/dT<0$ can both persist up to room temperature in disordered nanomaterials 50. Furthermore, we anticipate that competing instabilities (such as density waves partially gapping the Fermi surface) which generate similar normal-state transport properties in other q1D superconductors may also enable reentrant phenomena to develop.

In addition to the transverse coupling detailed above, we cannot rule out some contribution from intra-filamentary defects - i.e. Josephson coupling across barriers cutting (Mo6Se6)∞ chains - to the observed reentrance in Na2-δMo6Se6. Such defects are certain to be present in our samples, and we believe that they share responsibility for the finite-resistance plateaus which form in $R(T)$ at low current (Figs. 2, S8; SI Section VIII). However, supercurrents can percolate around such barriers without large resistive losses, provided that the chains are phase-coherent: this explains why $R(T)$ remains small and approximately constant in the plateau region. Since phase coherence is established at $T_{J}>T_{R}$, the clear rise in the resistance for $T<T_{R}(H,I)$ must correspond to the loss of transverse phase coherence. This is an inter- rather than intra-filamentary effect. The key role of transverse coupling in establishing phase coherence is confirmed by Fig. 3(d), in which 1D localization of Cooper pairs completely suppresses transverse coupling between (Mo6Se6)∞ chains for sufficiently large transverse magnetic fields and low temperatures. If the transverse coupling were not the principal factor controlling the resistance below $T_{J}$, $R(T)$ would not diverge as $T\rightarrow 0$, in direct contrast with our data.

In summary, we have demonstrated that Na2-δMo6Se6 single crystals behave as ideal inhomogeneous nanofilamentary superconductors, in which a 1D$\rightarrow$q3D dimensional crossover occurs via transverse Josephson coupling. Inhomogeneity and disorder result in electron localization, which is evident from the normal-state magnetotransport: the superconducting nanofilaments consequently become decoupled at low temperatures. However, transverse phase coherence is reentrant upon increasing $(T,H,I)$, since the electrons are progressively delocalized and hence the Cooper pair wavefunction overlap rises between neighbouring filaments. This reentrance constitutes a key advantage over homogeneous materials, whose superconducting properties generally deteriorate at elevated $(T,H,I)$ due to phase fluctuations. Together with recent work indicating giant $T_{c}$ enhancements in superconducting nanoparticles 5, 6, 7, 51, 52, this inbuilt resilience to phase fluctuations supports the assembly of dirty nanowire arrays as an attractive route towards synthesizing new functional superconductors. Moreover, Na2-δMo6Se6 and similar q1D filamentary materials provide unrivalled opportunities for investigating dimensional crossover and its influence on emergent electronic order: a field of key relevance to low-dimensional materials and nanostructures.

3 Methods

3.1 Crystal growth and characterization

Na2Mo6Se6 precursor powder was prepared using a solid-state ion exchange reaction technique 53. Na2-δMo6Se6 single crystals of mass approximately 150 $\mu$g and diameter 100-200 $\mu$m were grown by heating this cold-pressed powder in a sealed Mo crucible at 1750 ∘C for 3 hours. A full structural (X-ray) characterization may be found in a .cif file attached to the Supporting Information.

3.2 Transport measurements

Crystals were initially cleaned using sequential baths of hydrochloric acid, an ethanol/acetone mixture and distilled water. Subsequently, four Au pads of thickness 20 nm were sputter-deposited onto the surface, two at each end of the crystal ($I_{+/-}$) and two closer to the centre ($V_{+/-}$). Electrical contacts were made to these pads using 50 $\mu$m Au wire and Epotek E4110 Ag-loaded epoxy. Resistivity measurements were performed using a standard ac four-probe method ($\nu=470$ Hz) using two separate systems: a Quantum Design Physical Property Measurement System (PPMS) with a 14 T magnet and a cryogen-free dilution refrigerator equipped with a 9 T/4 T vector magnet. RF noise was removed from our measurement cables using ferrite filters prior to entering the dilution refrigerator. Inside the refrigerator, all signals were carried by stainless steel microcoaxial cables. To remove blackbody radiation, the cables were thermally anchored at numerous points (including the mixing chamber, i.e. the coldest part of the refrigerator) before reaching the sample. The standard inbuilt ac transport hardware was used in the PPMS, while $R(T)$ measurements in the dilution refrigerator were performed using a Keithley 6221 ac current source and a Stanford SRS830 lock-in amplifier. Both methods provided identical and reproducible data. Importantly, no phase shift was observed by the lock-in, implying that no extrinsic capacitive effects are present in our data. The $V(I)$ curves in Fig. 2 were acquired using a pulsed dc technique with a Keithley 6221 current source and 2182A nanovoltmeter. The crystals are fragile and highly sensitive to thermal cycling from 0.05-300K. They therefore exhibit a finite experimental lifetime, at the end of which $R_{NS}$ exhibits small irreversible jumps after each subsequent thermal cycle. We exclude such “end of lifetime data” from our analysis. The data which we plot in Figs. 2-4 are directly obtained from the raw voltage output of the lock-in using $R=V/I_{ac}$: no further data-processing is performed.

{acknowledgement}

We thank A. Chang and M. Croitoru for enlightening discussions, and Alexei Bosak (Beamline ID28, ESRF Grenoble) for assistance with data collection and processing. This work was supported by the National Research Foundation, Singapore, through Grant NRF-CRP4-2008-04.

This document is the unedited Author’s version of a Submitted Work that was subsequently accepted for publication in ACS Nano, copyright $\copyright$ American Chemical Society after peer review. To access the final edited and published work see DOI: 10.1021/acsnano.5b05450.

{suppinfo}

Supplementary data including electronic structure calculations, X-ray diffraction, modelling of q1D superconducting transitions, calculation of the filamentary diameter, crystal simulation using an anisotropic random resistor network, Werthamer-Helfand-Hohenberg fits for $H_{c2}$($T$), Josephson energy simulation, a discussion of additional reentrance mechanisms and the impact of tuning the disorder level. (PDF)

Crystallographic Information File (CIF)