Tip-induced superconductivity

In the ballistic regime of transport for a N/S point contact, the differential conductance (AR) spectra is analysed under the BTK model [23]. The BTK model assumes the potential barrier as a delta function of the form H = V_o δ(x). In figure 3, the interface is placed at x = 0. The barrier strength is characterized by a dimensionless parameter, Z = $\frac{{V}_{o}}{\hslash {v}_{\mathrm{f}}}$, for mathematical convenience. This parameter originates primarily due to two reasons: (a) Fermi velocity mismatch between the materials forming the contact which suggests that the barrier can never be transparent for conduction and (b) due to presence of naturally occurring oxide layer on the surface of materials, the two materials forming a point contact will have an effective layer of oxide material acting as a barrier. Hence, the presence of a potential barrier presents a finite probability for the electrons to undergo normal reflection from the barrier along with AR. The expression for current through the point contact now requires inclusion of outgoing and incoming populations. The AR and normal reflection probabilities are represented by A(E) and B(E) respectively in the expression for current.

$$\begin{equation*}{I}_{\mathrm{N}\mathrm{S}}\propto N(0).{v}_{\mathrm{F}}{\int }_{-\infty }^{+\infty }[{f }_{0}(E-eV)-{f }_{0}(E)][1+A(E)-B(E)]\mathrm{d}E\end{equation*}$$

where, N(0) is the DOS at the Fermi level and v_F is the Fermi velocity. A(E) and B(E) can be easily calculated by matching the boundary conditions for a delta function potential with wave functions that satisfy the BdG equations.

$$\begin{equation*}i\hslash \frac{\partial f}{\partial t}=\left[-\frac{{\hslash }^{2}{\nabla }^{2}}{2m}-\mu (x)+V(x)\right]f(x,t)+{\Delta}(x)g(x,t),\end{equation*}$$

and

$$\begin{equation*}i\hslash \frac{\partial g}{\partial t}=-\left[-\frac{{\hslash }^{2}{\nabla }^{2}}{2m}-\mu (x)+V(x)\right]g(x,t)+{\Delta}(x)f(x,t)\end{equation*}$$

where, Δ(x) represents the superconducting energy gap, μ is the chemical potential, f(x, t) and g(x, t) are the elements of column vector ψ describing one type of BCS quasiparticle states. For constant μ(x), Δ(x) and V(x) we can use the following trial solutions f(x, t) = u e^{(ikx−iEt)/ℏ} and g(x, t) = v e(ikx − iEt)/ℏ

$$\begin{equation*}\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}\enspace V(x)=0,\qquad E=\sqrt{{\left[\frac{{\hslash }^{2}{k}^{2}}{2m}-\mu \right]}^{2}}.\end{equation*}$$

For energies greater than 0, we have

$$\begin{equation*}{u}^{2}=\frac{1}{2}\left[1{\pm}\frac{{({E}^{2}-{{\Delta}}^{2})}^{1/2}}{E}\right]=1-{v}^{2}.\end{equation*}$$

Now if the wave functions inside the normal metal and superconductor is represented by ψ_N and ψ_S respectively, it is possible to obtain the AR A(E) and normal reflection B(E) probabilities(provided in table below [25]) (table 1).

Table 1. Table showing Andreev and normal reflection probabilities for an unpolarized system.

Coefficient	E < Δ	E > Δ
A(E)	$\frac{{({\Delta}/E)}^{2}}{1-{\epsilon}{(1+2{Z}^{2})}^{2}}$	$\frac{{(uv)}^{2}}{{\gamma }^{2}}$
B(E)	1 − A(E)	$\frac{{[{u}^{2}-{v}^{2}]}^{2}{Z}^{2}(1+{Z}^{2})}{{\gamma }^{2}}$

Recently, PCAR technique has proved to be one of the widely used, efficient and reliable spectroscopic tool with a tremendous capability to investigate electronic properties of a variety of materials. In the previous section we have described the BTK formalism, utilized to analyse the PCAR spectrum, to extract energy and momentum resolved information. In practice, for a N/S point contact, the PCAR spectrum is fitted with a simulated curve obtained by utilizing the BTK formalism where, one can use the superconducting energy gap (Δ), the potential barrier strength (Z) and a broadening parameter (Γ) within some predefined constraints. PCAR spectroscopy can also be used to investigate magnetic materials to ascertain the degree of spin polarization. This gives us a new technique known as spin resolved PCAR.

Spin resolved PCAR is one of the most favoured techniques to measure the transport spin polarization of materials. This method is independent of sample geometry, has excellent energy resolution and does not require application of magnetic field. One may consider the possibility of surface modification, due to naturally occurring oxides at surfaces or due to chemical reactions at surfaces of superconductor and ferromagnet, as a disadvantage of the technique. However, since point contact is established after the tip penetrates the surface oxide layer, the effect it is probable that the effect of surface modification is negated and the results obtained arise purely from formation of contact between the intended material. A confirmation to this is provided by the good agreement of results obtained via other methods. The BTK formalism can be extended to analyse PCAR spectrum obtained for a ferromagnet/superconductor (FS) point contact where the presence of spin-polarized current necessitates the use of additional fitting parameter, namely the spin polarization (P_C). Now let us take a closer look at what goes on at a FS interface in terms of DOS. The Fermi level of a FS point contact is expected to be spin polarized i.e. at the Fermi level the DOS of down-spin (N_↓) and up-spin (N_↑) electrons are not equal. This implies that |N_↑-N_↓| electrons approaching the interface cannot undergo AR due to unavailability of accessible states in the opposite spin band (figure 4). This causes suppression of AR in FS point contacts. The BTK formalism in this case relies on determining this suppression in AR to extract the degree of spin polarization of the Fermi surface. The AR and normal reflection probabilities for this case in listed out in table 2 [25]. The steps followed to extract the absolution value of spin polarization of Fermi level are: (1) start with calculation of BTK current for zero spin polarization (I_BTKu) and for a fully (100%) spin polarized case (I_BTKp). (2) Then calculate the current for an intermediate spin polarized (P_t ) case by interpolating the current value between I_BTKu and I_BTKp by following the relation I_total = I_BTKu(1 − P_t ) + P_t I_BTKp [25]. (3) One just needs to take a derivative of I_total with respect to applied bias V to get the modified AR spectrum having signatures of spin polarization in the ferromagnet. Once again we emphasize that for fitting of such a spectrum under the modified BTK formalism [18, 39, 40] requires use of four fitting parameters Z, Δ, Γ and P_t .

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Table 2. Table showing the values of AR and normal reflection probabilities for a polarized system.

Coefficient	E < Δ	E > Δ
A(E)	0	0
B(E)	1	$\frac{{({{\epsilon}}^{1/2}-1)}^{2}+4{Z}^{2}{\epsilon}}{{({{\epsilon}}^{1/2}+1)}^{2}+4{Z}^{2}}$

The PCAR spectra obtained for point contact between normal metals and conventional superconductors are usually in good agreement with the BTK theory. However, the same cannot be said for certain point-contacts between (a) normal metals and unconventional superconductors and, (b) superconductors and ferromagnets where the experimentally obtained spectra is found to be broadened from the theoretical simulation. It is believed that this broadening occurs due to shortening of quasiparticle lifetime resulting due to inelastic scattering of charge carriers near the contact region. This warrants incorporation of this effect into the BTK formalism. The broadening is taken care of by including the inelastic term ${\Gamma}=\frac{\hslash }{\tau }$ in the BdG equations where, τ is the quasiparticle lifetime. The inclusion of the broadening parameter modifies the BdG equations which are

$$\begin{equation*}i\hslash \frac{\partial f(x,t)}{\partial t}=\left[\frac{{\hslash }^{2}{\nabla }^{2}}{m}+\mu +i{\Gamma}-V(x)\right]f(x,t)+{\Delta}g(x,t)\end{equation*}$$

$$\begin{equation*}i\hslash \frac{\partial g(x,t)}{\partial t}=\left[\frac{{\hslash }^{2}{\nabla }^{2}}{m}+\mu +i{\Gamma}-V(x)\right]g(x,t)+{\Delta}f(x,t).\end{equation*}$$

The trial wavefunction now contains a decaying term e^−Γt/ℏ and the pair occupation probabilities get modified to ${u}^{2}=\frac{1}{2}\left[1{\pm}\frac{{({(E+i{\Gamma})}^{2}-{{\Delta}}^{2})}^{1/2}}{E+i{\Gamma}}\right]=1-{v}^{2}$ resulting in modification of Andreev and normal reflection probabilities

$$\begin{equation*}A(E)=\frac{\sqrt{({\alpha }^{2}+{\eta }^{2})({\beta }^{2}+{\eta }^{2})}}{{\gamma }^{2}}\end{equation*}$$

and

$$\begin{equation*}B(E)={Z}^{2}\frac{{[(\alpha -\beta )Z-2\eta ]}^{2}+{[2\eta Z+(\alpha -\beta )]}^{2}}{{\gamma }^{2}}.\end{equation*}$$

Hence, Γ is incorporated in the BTK model as an additional fitting parameter to take care of broadening in the AR spectrum. (The modified DOS at energy E is given by $N(E)\sim \mathrm{R}\mathrm{e}\left(\frac{E+i{\Gamma}}{\sqrt{{(E+i{\Gamma})}^{2}-{{\Delta}}^{2}}}\right)$.

The BTK theory discussed above is based on a simple one-dimensional transport model. The extension of the moded to 2D and 3D becomes particularly necessary when (a) the superconducting order parameter is anisotropic in the momentum space and (b) where AR is not strictly a retro-reflection process. Such extensions were made by a number of groups in the context of the high T_C superconductors [27] and systems with linear Dirac bands [28–30]. In high T_C superconductors, where the pairing symmetry is d-wave due to the sign change in the order parameter along different momentum directions, zero energy Andreev bound states (ABS) emerge. In graphene, due to specular AR near the Dirac point, unusual suppression of AR happens. The details of such phenomena are beyond the scope of this review article.

3D Dirac semimetals, in general, are known to be stable compounds due to symmetry protected surface states. Another key feature of these materials is that the Dirac points exist close to the topological phase boundaries, as a result by breaking certain symmetries they can be driven into further exotic states such as TI, Weyl semimetals and topological superconductors. However, until the year 2014, this idea was not realized practically. In 2014, Dr Goutam Sheet's group began working on a II–V semiconductor Cd₃As₂. The existence of a stable 3D topological semimetallic phase in Cd₃As₂ was previously confirmed by angle resolved photoemission spectroscopy (ARPES) [41] and scanning tunnelling spectroscopy [42] experiments. It was further observed that the energy dispersion is linear along all three directions in momentum space. The group made a very surprising discovery, the stable 3D Dirac semimetal Cd₃As₂ could be easily driven to a superconducting phase by simple touch with a silver (Ag) needle. The superconducting phase was found to survive up to 7.5 K with an estimated superconducting gap amplitude of 6.5 meV [17]. Further, a thorough point contact spectroscopic study was conducted where the behaviour of the point contact was tested under different temperatures and applied magnetic field. The studies revealed that the gap energy is weakly dependent on temperature and survives up to 13 K [43]. This indicates presence of a pseudo-gap and that the induced superconductivity is unconventional in nature. Although, polycrystalline samples of Cd₃As₂ were used in this experiment, it is important to note that the measured value of superconducting energy gap is not necessarily the result of averaging. The gap value is often measured from a single grain due to small size of point contact. The measurement have been repeated over 100 times and every time the measured gap value turns out to be the same at a particular temperature indicating the existence of a nearly isotropic gap structure. Confirmation of this discovery was provided by Wang et al [44] where they repeated the same experiment on Cd₃As₂ single crystals.

A. Point-contact spectroscopy: different regimes of transport on Cd₃ As₂ . Contacts in the thermal regime of transport for Cd₃As₂/Ag exhibited a sharp resistive transition around 5.8 K (figure 10(a)). A similar transition is observed for point contacts on lead (Pb) as shown in previous sections. A magnetic field dependent study of the contact also showed a systematic decrease in the transition temperature with increasing field value (figure 10(b)) and is exactly same as expected for a superconducting point contact. Since neither the tip or the sample is superconducting alone, the emergence of a tip induced superconducting phase is indicated by this observation. A confirmation to this indication was provided by subsequent spectroscopic results.

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Figure 11(a) shows a normalised differential resistance spectrum ${(\mathrm{d}V/\mathrm{d}I)}_{\mathrm{N}}\left.\right)$ as a function of applied dc bias along with the I–V characteristics obtained for a thermal point contact between Cd₃As₂ and elemental superconductor lead (Nb) [17]. The spectrum present two sharp peaks at ±5 mV, which are typical for such point contacts [20, 26]. The I–V characteristics were also calculated for contact comprising only of Sharvin resistance following the BTK theory. This I–V characteristics was then subtracted from the characteristics obtained above to obtain I–V plot corresponding to purely Maxwell contribution. A clear dissipation less current state is seen where increase in current above the critical current value for the point contact leads to transition from superconducting state to the normal state as shown in figure 11(b).

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The authors performed point contact experiments in different regimes of transport by manipulating the contact diameter via physically engaging/withdrawing the tip from/onto the sample surface. Figure 12(a) shows differential conductance spectrum obtained for three different contacts corresponding to the ballistic regime of transport where, the Sharvin's resistance dominates. The hallmark signature of AR appear as two peaks which are symmetric about V = 0 in the spectra. Spectra ((dV/dI)_N vs V_dc) corresponding to the thermal regime showing the distinct critical current dominated peaks is shown in figure 12(b).

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Since neither the sample Cd₃As₂ nor the tip (silver (Ag)) are superconductors, and the spectra obtained for Cd₃As₂/Ag point contact resembles to that of a superconductor, hence it is the reasonable to compare them with spectra obtained for superconducting point contact. We find that the spectra has striking resemblance to those obtained for Pb/Ag point contact. From these results one may conclude that metallic point contacts on Cd₃As₂ are superconducting.

Since there is an indication that the point contact is superconducting, it is imperative to look at how the resistance of the contact changes with temperature. A clear transition from high resistance state to a lower resistance was observed (shown in figure 10(a)). The transition is strikingly similar to a superconducting transition as observed for superconducting point contacts. The transition begins at 5.8 K. Now that the variation of resistance with temperature is observed, it is important to look at the dependence on magnetic field. For a superconductor it is expected that as the applied magnetic field approaches the critical field value the superconducting transition temperature decreases and just above the critical field the material remains in its normal state. As expected, the resistivity plot evolves with the applied field and the at applied filed of 4.5 T the system only exhibits normal resistance state (shown in figure 10(b)). Now we focus on the effect of field on differential resistance spectrum and provide an estimation of point contact size.

(a). Magnetic field dependence: figure 13(a) shows the evolution of a critical current dominated spectra obtained in the thermal limit with applied magnetic field [17, 26]. The central dip is a result of superconducting transition below the critical current. The spectra evolved smoothly with field and the peaks and dips disappear at 4.5 T. The magnetic field dependence of the critical current dominated part of the I–V characteristics corresponding to R_M is shown in figure 13(b). The position of the peaks in the resistance spectrum gives the approximate value of the critical current for a given point-contact (inset of figure 13(b)). As expected for superconductors, the critical current decreases with increasing magnetic field.

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(b). Estimation of the point-contact size. The contact size can be estimated from the normal state resistance i.e. at high voltage bias by the Wexler's formula $\left[{R}_{\mathrm{P}\mathrm{C}},=,\frac{2h/{e}^{2}}{{(a{K}_{\mathrm{f}})}^{2}},+,{\Gamma},(l/a),\frac{\rho (T)}{2a}\right]$ where, Γ(l/a) is a numerical factor close to unity, a is the contact diameter and 2h/e² is quantum of resistance i.e. 50 kOhm, k_f is the magnitude of the Fermi wave vector which is 0.04 Å for Cd₃As₂ and ρ(T) is resistivity at any given temperature T. Estimation of contact diameter requires the contact to be approximated as circular in nature. The value of ρ at 1.5 K for Cd₃As₂ as measured by conventional four-probe method is 28 Ohm. By using this value the point-contacts are estimated to be approximately 40–400 nm for point-contact resistance R_PC 2–200 Ohm.

TaAs: the discovery of Weyl semimetals [76–87] facilitated the realization of Weyl fermions in condensed matter systems after more than 80 years from their theoretical discovery [91]. Weyl fermions were first shown by Hermann Weyl to emerge as solutions to the relativistic Dirac equation [78, 85] in quantum field theory. Weyl fermions remained illusive in nature until the discovery of TaAs as a Weyl semimetal [78, 79, 81–85]. Weyl semimetals belong to the class of topologically non-trivial materials known to demonstrate exotic quantum phenomena having unique surface states [88]. The bandstructure of Weyl semimetals contain a pair of Weyl nodes at the Fermi level where each node, in the momentum space, can be considered as a monopole/anti-monopole of Berry curvature [79, 89]. Each Weyl node is associated with a quantized chiral charge and the Weyl nodes are connected to each other through the boundary of the crystals through Fermi arcs [78, 81, 82, 85, 90]. Recent investigations of Weyl semimetal TaAs have revealed existence of highly spin polarized Fermi arcs lying in a completely 2D plane on the surface of the crystal [91]. Weyl semimetals are believed to possess a richer set of physical phenomena stemming from its exotic topological properties presenting a system which must be explored thoroughly (a) for gaining insight into the world of quantum mechanics (b) to find potential device applications. In this direction TISC phase realized on TaAs [18, 92] is a significant step forward.

In the subsequent section, the emergence of a tip-induced superconducting phase in mesoscopic contacts between silver (Ag) and a Weyl semimetal is discussed in detail pertaining to transport and magneto transport measurements in various regimes of mesoscopic transport. The experimental data obtained for Ag/TaAs point contacts were fitted under the modified BTK formalism that includes spin polarization as a fitting parameter. The experiments and fitting pointed towards indicated the coexistence of superconductivity and high transport spin polarization and hence indicated a flow of spin polarized super-current through TaAs point contacts. The TISC phase obtained in TaAs system did not show the possibility of an unconventional pairing mechanism, the spectra obtained in the ballistic or intermediate regime of transport did not show any ZBCP or any features resembling the pseudo-gap. This is in clear contrast to the TISC phase obtained on the 3D Dirac semimetal Cd₃As₂. The unique co-existence of superconductivity and high transport spin polarization in TaAs point contacts makes it an interesting candidate for spintronic applications.

Point contact spectroscopy in different transport regimes on TaAs: point contact spectroscopy was performed on single crystals of TaAs using elemental silver (Ag) tip [18]. Figure 20(a) shows a spectra obtained in the thermal regime of transport where a broad conductance peak can be seen along with critical current dominated dips indicating the existence of a superconducting phase [20, 26]. A temperature dependent study of the point contact resistance shows systematic evolution with applied magnetic field, presented in figure 20(b), consistent with expected signatures of superconductivity. The transition from normal state to the superconducting state occurred at 7.3 K in absence of magnetic field [18]. With application of field and increase in magnitude, as expected for materials exhibiting superconductivity, the transition temperature kept decreasing with increasing magnetic field. Though the above data hinted at existence of a superconducting phase in Ag/TaAs point contacts, additional data is required to in support of the superconducting phase. In this regard, the existence of superconductivity must be established by driving the point contact into the other regimes of transport where features related to AR can be observed.

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The presence of TISC phase was confirmed for Ag/TaAs point contact in the intermediate and ballistic regimes of mesoscopic transport. In figure 20(c) a spectrum obtained in the intermediate regime of transport is shown where in addition to the critical current dominated dips in conductance, two peaks symmetric about V = 0 corresponding to AR are observed [18]. The Ag/TaAs point-contact was driven to ballistic/diffusive regime of transport by reducing the contact diameter where the double peak structure associated with AR is clearly seen (figure 20(d)).

(a). Temperature and magnetic field dependence. From the data presented above it is clear that Ag/TaAS point-contacts exhibit superconducting phase. Now to get a complete picture, it is important to focus on the nature of superconductivity arising in the Ag/TaAs point contacts. In this section, data corresponding to the temperature and magnetic field dependent studies conducted on Ag/TaAs point contacts mainly in the ballistic regime of transport (figures 21(a) and (b)). In the figure, the dotted lines represent the experimental data points and the solid lines correspond to the fits within the modified BTK framework. In the presented case the experimental data points matched remarkably well with the theoretical fits. This is quite surprising due to superconducting phase being derived from a complex system, Weyl semimetal TaAs. The evolution of energy gap (Δ) with temperature, extracted from the data in figure 21(a), is shown in figure 21(c) along with the expected Δ vs T curve for a BCS superconductor.

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Superconducting energy gap determined through the BTK fits is 1.2 meV which systematically decreased with increasing temperature. However, the systematic evolution of energy gap with temperature do not coincide with the BCS prediction [20, 38]. Moreover, the gap disappeared completely at the observed T_c and unlike Cd₃As₂, any features associated with the pseudogap were absent [17]. All of the spectra were well fitted under the BTK model along with a high spin polarization indicating a contribution of s-wave component in the order parameter. Additionally, it was observed that the temperature dependence of Δ deviated from the BCS prediction. The deviation of certain spectra (ballistic/diffusive regime) from the theoretical fits was observed and considering the deviation of Δ vs T from BCS, indicates the possibility of a mixed angular momentum symmetry of the order parameter where an unconventional component is mixed with a strong s-wave component [93–95]. However, the possibility of existence of multiple gaps cannot be ruled out and further experimentation is necessary to reach a conclusion [96]. It is notable here that the extraction of spectroscopic information requires the analysis of spectra showing features associated with AR where critical current dips are absent and the normal state resistance remains independent of temperature. This is precisely what has been done in the presented case.

Now we focus on the variation of differential conductance spectra with applied magnetic field. The gap structure (double conductance peak in the differential conductance spectrum), as expected, decreases with increasing magnetic field. In the case of a point contact where the contact diameter is large enough to accommodate any vortices, the fitting of AR spectra becomes non-trivial. Moreover, since at present an exact theory describing the TISC remains to be unproven, the presented data cannot be analysed to confirm whether vortices can enter the point contact region nor if multiple vortices can exist there. All of the field dependent spectra were analysed and fitted under the BTK theory with inclusion of spin polarization. On the basis of analysis and fitting, Δ vs H plot was constructed (figure 21(d)) and it was found that Δ vanishes at 10 kG [18].

(b). Spin polarized surface states in TaAs. The spectra obtained for Ag/TaAs point contacts, when compared to the expected spectra for a simple elemental superconductor, showed a significant suppression of AR. It is known that the presence of spin polarized surface states causes suppression of AR [78].

Such spectra can be fitted within the BTK formalism-modified to include finite transport spin polarization [23, 39, 40]. As mentioned above, all the spectra obtained for the Ag/TaAs point contacts were a remarkable fit within the modified BTK formalism [45, 97] and indicated the presence of a large transport spin polarization of about 60%. Three such spectra (ballistic/diffusive) with a high spin polarization are illustrated in figures 22(a)–(c). A plot of the detected spin polarization against the barrier potential (Z) is shown in figure 22(d), where a decrease in transport spin polarization was observed with increasing Z. A linear extrapolation of the curve indicated an intrinsic transport spin polarization of 60 %. In presence of an external magnetic field, the measured transport spin polarization for a finite Z is seen to increase with increasing magnetic field (figures 22(e) and (f)). Previously spin polarization of TaAs was determined to be 80% through ARPES measurement. However, spin polarization determined by PCS is around 60% and lower than determined from ARPES measurement. This is due to the fact that PCS is a transport measurement where, the spin polarization of the transport current is measured rather than the absolute spin polarization [39, 97]. The presented results and presented analysis indicate the flow of highly spin polarized super-current through the TaAs point contact.

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(c). Anisotropic magnetoresistance. Apart from the temperature and field dependent evolution of the contact resistance, a field-angle dependent study of the point contact resistance of a ballistic contact was also explored. Here, the resistance of the contact was measured while the direction of magnetic field is rotated using a three-axis vector magnet with respect to the direction of injected current. A bias voltage of 13 mV, corresponding to the normal state of TISC, was applied across the contact and magnetic field was rotated. A large anisotropy in the magneto-resistance was observed which kept increasing with the increase in applied field strength (figure 23(a)). To explain the anisotropy in magneto-resistance, assume that the microconstriction is shaped as a nanowire and the field is rotated with respect to the direction of flow of current through the nanowire. A similar angular magnetoresistance has been previously observed in hybrid nano-structures involving materials whose surface states have complex spin texture [98, 99]. A repetition of the experiment at voltage bias (V = 0.3 mV) corresponding to the superconducting state of the TISC (figure 23(b)). The observed magneto-resistance remained anisotropic and equally noticeable as in the previous case. On the basis of these results and observations provide clear indication of coexistence of superconductivity and large spin polarization on TaAs point contacts [18].

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Further confirmation to the theory that spin polarized current causes emergence of AMR in the Ag/TaAs point contacts is provided from the fact that the AMR data could be fitted well using the typical cos²  θ dependence. The fits are shown in figure 23. It is important to consider that the AMR data presented were obtained for a point contact either in the ballistic or diffusive regimes of transport. For such contacts the resistance arises predominantly from the Sharvin's resistance which is independent of the bulk resistance of either of the materials forming the point contact. Hence, it was concluded that the large AMR observed in Ag/TaAs point contacts did not arise due to bulk TaAs [18].

With the data and discussion presented above, the authors attempt to explain the mechanism causing emergence of TISC phase in Ag/TaAs point contact. TISC phase may emerge due to local pressure, local doping and/or confinement effects. The TISC phase could not be realized at macroscopic interfaces of metallic silver films deposited on TaAs. Moreover, TaAS is not known to exhibit pressure induced superconductivity. Hence, combination of all three mechanisms mentioned above could be responsible for emergence of TISC on TaAs point contacts. To understand the exact mechanism facilitating the emergence of TISC additional experiments must be performed. One such experiment involves use of double-probe scanning tunnelling microscope capable of determining the exact symmetry of the order parameter. In the double-probe STM the first probe forms the contact with the sample to induce a TISC phase, the second probe is then used to perform spectroscopy in the TISC region i.e. close to the first tip attached to probe forming the contact. Such experiments can also probe the possibility of topological and FFLO superconductivity.

Recently, Xing-Yuan Hou et al realised two independent TISC phases with distinct behaviour of critical fields (H_C2) on TaAs [115]. The TISC phase has also been recently achieved on another Weyl semimetal TaP [131].

ZrSiS: ZrSiS is an abundant, non-toxic and highly stable material with a bandstructure that hosts several Dirac cones forming a Fermi surface with diamond shaped line of Dirac nodes [100–102]. The recently discovered Dirac semimetals exhibited Dirac cones with dispersion linear up to a small energy range, for example, for Cd₃As₂ the linearity exists only up to ∼200 meV. However for ZrSiS the linearity extends up to unusually high value of approximately 2 eV [102–104] making ZrSiS a fascinating topological system where lies an enormous possibility of finding anomalous properties such as large magnetoresistance [105] and quantum oscillations that survive even at high temperatures [106].

In the following section we present and discuss in detail, the emergence of a superconducting phase induced upon forming a contact between elemental silver (Ag) on crystalline ZrSiS. The Ag/ZrSiS point contacts showed a transition from normal resistance state to the superconducting state at 7.5 K with a superconducting energy gap determined from PCS to be approximately 1 meV [107]. ZrSiS exhibits a liner band dispersion over a large energy range resulting in a robust topological character against perturbation effects such as carrier doping, variation in stoichiometry etc and hence superconductivity in ZrSiS point contacts is an important discovery. Moreover, this implies that the topological properties of ZrSiS cannot be destroyed due to proximity to metallic tip forming the point contact and result in emergence of a superconducting phase i.e. the superconducting phase did not emerge at the expense of the topological nature of ZrSiS. Considering the facts presented above ZrSiS makes a promising candidate for realising topological superconductor.

Point-contact spectroscopy in different transport regimes on ZrSiS. In figure 24(a), the thermal regime PCS spectra obtained for ZrSiS/Ag point contact is shown. A striking resemblance is observed with PCS spectra obtained for known conventional superconductor Nb (figure 24(b)) with silver (Ag) as tip material. Therefore, it can be inferred that a TISC phase was obtained on ZrSiS/Ag point contact. The spectrum clearly shows critical current dominated conductance dips. Moreover, the spectrum was observed to evolve monotonically with increasing magnetic field. The evolution of contact resistance with temperature is shown in figure 24(c) where a clear superconducting transition is observed at 7.6 K. Furthermore, as expected for superconductors, the critical temperature decreased monotonically with increasing magnetic field. A H–T phase diagram was constructed from the field dependent R–T curves and is presented in figure 24(d). The expected empirical H–T phase diagram for a conventional superconductor is shown in figure 24(d) (blue dotted curve). From the H–T phase diagram it is predicted that the upper critical field for the TISC phase could be as high as 17 kG and is evident from the experiments where, the experimentally measured data points deviate slightly from the empirical expectation. Moreover, this deviation indicates the existence of an unconventional component in the TISC phase obtained for non-trivial ZrSiS point contacts.

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The authors using the BTK theory, simulated the I–V characteristics associated with the ballistic component (Sharvin's resistance part) of the point contact resistance. The zero resistance state was examined by subtracting the I–V of Sharvin's resistance part from the I–V curve associated with the total point-contact resistance, i.e. the I–V curve associated with the Maxwell contribution was obtained and analysed. The I–V curves and corresponding variation with magnetic field is shown in figures 24(e) and (f) for ZrSiS/Ag and Nb/Ag point contacts respectively. Temperature dependence of the PCS spectra for ZrSiS/Ag point contacts is shown in figures 25(a) and (b) and for Nb/Ag is shown in figures 25(c) and (d). Upon comparison it is confirmed that TISC phase was indeed realized on ZrSiS/Ag point contacts.

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On the basis of Wexler's formula, for a point contact in thermal regime, it is possible that a small but finite ballistic component may exist. A visual inspection of the thermal spectra reveals that the zero field spectra remains flat between ±0.8 meV. This could be a signature of AR for a contact where the barrier potential is extremely small (transparent barrier) and is consistent with BTK prediction. Hence, an estimation of approximated gap amplitude is possible for ZrSiS/Ag point contact. The approximated gap value is ∼0.8 meV for which the value of 2Δ/K_B T_c is 2.48. A value consistent with a weak-coupled superconducting phase emerging as a result of phonon mediated pairing of electrons. A first principles calculations of the electronic structure and electron–phonon coupling revealed an enormous increase in DOS at the Fermi level of ZrSiS arising due to formation of contact with metallic tips (figure 26(a) and (b)). However, a considerably smaller contribution of silver in terms of DOS at and around Fermi level indicated that a substantial increase in carrier density around E_F played an explicit role in emergence of TISC phase in ZrSiS point contacts.

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The electron–phonon coupling was also calculated for bulk ZrSiS and ZrSiS/Ag point contacts. The calculation correlated with the DOS at Fermi energy and it was observed that for ZrSiS formation of microscopic contact with silver leads to an almost two folds increase in Λ parameter than the bulk. However, the calculated T_C was two orders of magnitude smaller than the experimental value. Hence it was argued that conventional phonon mediated cooper pairing could not give rise to superconductivity. Moreover, the Dirac cone along ΓX disappeared upon formation of contact while the Dirac cones along MΓ and AZ lines were found to be protected along with appearance of extra nesting features (figure 26(b)) [107]. Therefore, it is possible that topological character of ZrSiS co-exists with TISC and provides strong indication towards the possibility of topological superconductivity at ZrSiS/Ag point contacts.

To note, ZrSiS shows a pressure induced quantum phase transition [128]. Whether the observed TISC in ZrSiS is related to that or not needs to be investigated. However, since, TISC is also seen to emerge under soft metallic tips, most likely the observed TISC in ZrSiS is not a consequence of the quantum phase transition. The TISC phase was recently achieved on other nodal semimetals like TaAs₂ and NbAs₂ [129]. Here, it may be noted that NbAs was to yield superconducting microstructures upon ion-sputtering [130]. Whether this is related to the TISC phase seen in NbAs₂ remains an open question.

Topological materials like TI and Topological crystalline insulators can be driven into novel phases of matter by introduction of perturbations in form of magnetic dopant, disorders, structural distortions etc [15, 53, 108–114, 116–122]. Using this idea, physicists worked tirelessly to achieve a topological superconducting phase in these materials. It is believed that such exotic superconducting phases may lead to detection of the elusive Majorana fermions [123–125]. Cu intercalated Bi₂Se₃, being a candidate material for topological superconductivity, has been extensively investigated in recent times [126]. One of the most compelling reasons for such a huge interest is the existence of spin-momentum locking in this system which results in superconductivity.

The protection mechanism of surface states in TCI's is fundamentally different from TIs. The surface states are protected by crystal symmetry in the case of TCI's [114, 118] whereas, TIs have time reversal symmetry protected surface states. TCIs possess highly tunable surface states, as compared to TIs, which can be tuned by introduction of perturbations [113]. Recently, Angle resolved photo-emission spectroscopy confirmed Pb_0.6Sn_0.4Te is a TCI. The authors (Shekar et al) demonstrated that a mesoscopic contact between Pb_0.6Sn_0.4Te and an elemental metal exhibits conventional superconductivity [127]. Moreover, the local superconducting phase was found to exhibit a contact dependent high transition temperature in the range of 3.7 K to 6.5 K. In the following sections, we discuss the experimental observations in detail.

Point contact spectroscopic investigation was performed on Pb_0.6Sn_0.4Te using elemental metallic tips, silver (Ag) and palladium (Pd). A differential conductance (dI/dV) was measured for the point contacts against an applied bias voltage (V) using a lock-in based modulation technique (figures 27(c) and (d)). The resulting spectra showed two sharp conductance dips which is similar to the conductance features obtained for Pb/Ag and Cu/Nb point contacts (figures 27(a) and (b)). These dips appear due to the critical current of superconductor. Additionally a zero bias peak also appears in the conductance spectrum which indicates that the superconducting part of the point contact undergoes a transition to the zero resistance state [26]. The inset of figures 27(c) and (d) shows the respective resistivity curves showing the transition temperatures of 5.5 K and 6.3 K for the Ag/Pb_0.6Sn_0.4Te and Pd/Pb_0.6Sn_0.4Te point contacts. The authors observed such critical current dominated features in dI/dV spectrum as well as a transition in the resistivity for a number of different point contacts. Hence, it was inferred that the point contacts on Pb_0.6Sn_0.4Te are superconducting. However, to affirm this claim one needs to provide additional data that validates superconductivity.

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The authors presented additional data and arguments supporting the three key aspects pertaining to superconductivity discussed below.

• (a)  
  R–T data showing systematic field dependence: in figures 27(e) and (f), the authors presented the resistance vs temperature characteristics, for the two point contacts formed on Pb_0.6Sn_0.4 with Ag and Pd tips respectively, in presence of various applied magnetic fields. A clear systematic evolution of superconducting transition temperature with applied magnetic field was observed. The authors observed that the transition temperature systematically decreased with increasing magnetic field, which is an exact behaviour expected for a superconductor. As discussed in previous sections, we remind you that though the overall resistance is zero for a bulk superconductor, in a point contact geometry the resistance is always finite. This is evident from the R–T data and corresponding differential conductance spectrum obtained for Pb/Ag point contact, presented in figure 27(a).
• (b)  
  Systematic temperature dependence of dI/dV spectra: in figure 28(a), we present the temperature dependence of differential conductance spectrum obtained for Pd/Pb_0.6Sn_0.4. A cursory look on the data reflects that the dI/dV spectrum smoothly evolves with temperature until they disappear completely at the critical temperature (6 K). Additionally, the critical current of the point contact can be estimated from the position of the conductance dips in the dI/dV. The estimated critical current is presented as a function of temperature in the inset of the figure 28(a). It is also clearly seen that the critical current decreases with increase in temperature. This temperature driven behaviour of critical current is consistent and expected for a superconducting point contact. Moreover, the contact diameter was estimated to be around 50 nm from the normal state resistance via use of Wexler's formula [19].
• (c)  
  Systematic field evolution of dI/dV spectra and disappearance at H_c: in figure 28(b), the field evolution of differential conductance spectra is presented. Here, it is clear from a visual inspection that the dI/dV spectra evolves smoothly with increasing magnetic field where the spectral features diminish monotonically and the spectral features (peaks and dips) completely vanish at a critical field (H_c) of 5 T. The field dependence of critical current is presented in the inset of figure 28(b).

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These results confirmed the existence of a superconducting phase in Pd/Pb_0.6Sn_0.4 point contacts. Now the authors focussed on determining the nature of superconducting phase in these point contacts for which they constructed that H–T phase diagram. The H–T phase diagram is presented in figures 29(a) and (b) for Ag/Pb_0.6Sn_0.4 and Pd/Pb_0.6Sn_0.4 point contacts respectively. The experimentally obtained H–T phase curves were observed to fall on the empirically expected H–T phase curve for conventional superconductors. This indicates that these point contacts exhibit a conventional superconducting phase. Pb_0.6Sn_0.4 is a topologically non-trivial material where we the authors realized a superconducting phase, hence, it is important to address the symmetry of superconducting order parameter where a p-wave symmetry may be possible [124]. This is infact due to the natural breaking of time reversal symmetry, result of p-wave symmetry, where superconducting behaviour is favoured due to proximity of a spin-polarized Fermi surface. This is at odds with conventional superconductors where proximity to a ferromagnet results in competition between the ferromagnetic order and superconducting order and in turn leads to suppression of the superconducting order [127].

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In the context of TISC it is also important to mention about a somewhat similar phenomenon known as tip-enhanced superconductivity (TESC). An enhancement in superconducting transition temperature was observed in superconductors under a metal tip [32, 47, 132–134]. However, the mechanism behind the origin of TESC like TISC remains unknown.

Therefore, in this review article, we discussed PCAR spectroscopy in different regimes of transport. We highlighted how such features can be used to detect and confirm the existence of a superconducting phase, called TISC, that appears only under a mesoscopic point contact between two non-superconducting materials, most of the times one of these being a topologically non-trivial system. We also discussed the possible factors behind the emergence of TISC phase. However, at present, the jury is still out on the origin of TISC phase in materials.