Effects of Rashba-spin–orbit coupling on superconducting boron-doped nanocrystalline diamond films: evidence of interfacial triplet superconductivity

The microstructure of poly and nanocrystalline diamond films has been extensively studied with much of the research directed towards the properties of the grain boundaries [31]. It has been established that the surface termination of the diamond grain results in the formation of an extendedπ^* orbital configuration due to hybridization of dangling bonds [32]. It has also been shown that heightened stress at such grain boundary regions leads to surface states through a modification of the electronic energy, thus forming frontier electronic orbitals [33]. Such findings are used to explain the many reports on the grain boundary conduction of nanocrystalline diamond systems [34]. Furthermore, there is also much evidence suggesting that carbon can form complex superlattice-like structures through alternating sp² and sp³ bonding [35, 36], thus supporting the interpretation as an array of S-I-S junctions (figure 1(h)) this becomes particularly important when evaluating the superconducting boron doped diamond system as the granular nature ensures the occurrence of tunnelling and thus enhanced Coulomb repulsion between grains [36, 37], leading to Cooper pair confinement [18–20]. All these studies indicate that the electronic transport in granular diamond systems is greatly dependent on the grain boundary region.

In figures 1(a)–(d), high-angle annular dark field STEM imaging of the boron doped diamond has been used to image the nanocrystalline diamond films. As shown in figure 1(a), the individual grains are predominantly columnar, where single grains can extend throughout the entire film thickness. At higher magnification, the boundary regions become more prominent (figure 1(b)). In figure 1(c), we demonstrate that the system is composed of highly ordered crystalline grains separated from each other by a sharp boundary region. The lattice miss match between such regions can be large, leading to stark crystal symmetry breaking, providing evidence of an interfacial (2D) structure between grains. In figures 1(d) and (e), it is shown that within the grains twinning frequently occurs. In order to investigate the effects of the microstructure on the transport, angle dependent MR is measured. As shown in figures 1(f) and (g) the resistance of the granular diamond films has a pronounced dependence on the angle of the applied magnetic field (rotation angle Θ = 0 when applied normal to the film), not only does the resistance value oscillate when rotating the sample in a magnetic field of 0.1 T, but also demonstrates a shifting in the critical temperature (figure 1(f)). Such features have been widely investigated in superconducting spin valve systems and are a result of the magnetization non-collinearity as Cooper pairs tunnel into a spin active domain. Similar features have recently been reported [27] and there also related to confinement along the grain boundaries.

The observation here indicates not only the interference between the applied magnetic field and spin active interfaces but also demonstrate the intimate link between the microstructure and charge carrier transport. As argued before, when considering the confinement effects of the grain boundary region, the system behaves as an array of isolated grains where the Cooper pair wave function is restricted to individual grains by a confining potential related to the asymmetric charging that occurs between grains due to variation in the density of states (see figure 1(h)). This charging effect along with the sharp inversion symmetry breaking at the grain boundary ensures that charge carriers conducting along the boundary region will be influenced by SOC.

The superconducting phase transition shows a broad temperature dependence, this is known for granular superconducting systems as the coupling between grains is reduced due to enhanced Coulomb repulsion leading to a more resistive transition than what is observed in clean single crystal superconducting materials. In order to shed light on the microscopic character of the normal state charge carriers responsible for this broadening of the transition temperature, MR measurements are conducted at temperatures near the critical point. As shown in figure 2(a) (figure 2(b)), the low field MR (magnetoconductance) shows a complex series of transitions, at temperatures within the superconducting (zero resistance) phase; (i) the applied field firstly breaks the superconducting state leading to an initial increase in resistance which (ii) peaks then decreases steeply before (iii) reaching a temperature dependent minimum. By further increasing the field the system then only shows increasing resistance in accordance to field induced pair breaking. When increasing the temperature, the zero-field resistance can be seen to invert, and finally demonstrates a sharp peak centred at zero magnetic field.

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Such non-monotonic resistance features are not at all expected for superconducting systems showing purely field induced Cooper pair breaking but are rather more likely to result from WAL due to the presence of spin–orbit coupling, which has been widely observed in diamond interfaces [15, 38, 39]. The observation of the WAL effect is significant as it is a hallmark feature of spin–orbit coupling in a 2D system, and considering the confinement as well as charging effects intrinsic to the nanocrystals that compose the film we attribute this observation to the interfacial region between individual grains [shown in figures 1(a)–(c)]. To date, a large number of reports exist on the observation of the WAL effect in diamond surfaces [15, 38, 39], where this effect has been reported to exist due to a Rashba spin–orbit coupling (RSOC) that results from the structural symmetry breaking at the diamond edge and unbalanced confinement potential, both of which are intrinsic properties of the grain boundary in our nanocrystalline films this if schematically indicated in figure 2(c).

In order to deeper investigate this phenomenon, we utilize conventional analysis relying on the Hikami–Larkin–Nagaoka (HLN) formalism [40]:

$$\begin{equation}\begin{aligned}\hfill {\Delta}\sigma & =\sigma \left(B\right)-\sigma \left(0\right)\hfill \\ \hfill & =\frac{\alpha {\mathrm{e}}^{2}}{2{\pi }^{2}\hslash }\left[\mathrm{ln}\left(\frac{\hslash }{4Be{L}_{\varphi }^{2}}\right)-\psi \left(\frac{1}{2}+\frac{\hslash }{4Be{L}_{\varphi }^{2}}\right)\right]\hfill \end{aligned}\end{equation} \tag{ 2 }$$

as well as its modified version which has been specifically applied to hole type carriers in diamond surfaces [41]:

$$\begin{equation}\begin{aligned}\hfill {\Delta}\sigma & =-\frac{{\mathrm{e}}^{2}}{2{\pi }^{2}\hslash }\left[\psi \left(\frac{1}{2}+\frac{{B}_{\varphi }+{B}_{\text{SO}}}{B}\right)\right]+\frac{1}{2}\psi \left(\frac{1}{2}+\frac{{B}_{\varphi }+2{B}_{\text{SO}}}{B}\right)-\frac{1}{2}\psi \left(\frac{1}{2}+\frac{{B}_{\varphi }}{B}\right)\hfill \\ \hfill & \quad -\mathrm{ln}\left(\frac{{B}_{\varphi }+{B}_{\text{SO}}}{B}\right)-\frac{1}{2}\left(\frac{{B}_{\varphi }+2{B}_{\text{SO}}}{B}\right)+\mathrm{ln}\left(\frac{{B}_{\varphi }}{B}\right).\hfill \end{aligned}\end{equation} \tag{ 3 }$$

In the first equation the only fitting parameters are the pre-factor α and L_φ, the coherence length whereas the second equation defines the phase coherence (B_φ) and spin coherence fields (B_SO). The pre-factor α takes on values of between 0.5 and 1 and allows us to determine an overall validity of our interpretation of the data. As shown in figure 2(b) the second equation captures the low field features of the magnetoconductance where temperature dependence of the fitting parameters are plotted in figures 2(d) and (e). Following the previous reports of WAL in diamond systems, we assume a Rashba splitting [15] and from the fit we can extract the temperature dependence of the phase and spin coherence lengths which are given by ${L}_{\varphi }={\left(\hslash /4\mathit{\text{e}}{B}_{\varphi }\right)}^{0.5}$ and ${L}_{\text{SO}}={\left(\hslash /4\mathit{\text{e}}{B}_{\text{S}\text{O}}\right)}^{0.5}$, respectively. As shown in figure 2(d), the phase coherence length shows a much stronger temperature dependence when compared to the spin coherence length, which is nearly temperature independent in the evaluated temperature range.

One additional observation is that the crossover of the WAL to a WL observed in the MR corresponds to the temperature where the spin coherence length exceeds the phase coherence length. This is an indication of the spin precession destroying the phase coherence of the hole carriers [15, 42–44]. It is also possible to interpret this crossover in the MR as evidence of the system moving from a regime dominated by surface conduction (or interfacial grain boundary transport) to one where charge carriers of the bulk dominate at elevated temperature. This interpretation is supported by the recent report from Zhang et al [27] where the anomalous angle dependent MR demonstrated carrier localization at the grain boundary. From the extraction of the spin coherence field, it is also possible to determine the spin relaxation time τ_SO = ℏ/4eDB_SO, (where D is the diffusion constant and related to the hole carrier density). Using this value, we determine the spin–orbit splitting given by the following relationship: ${{\Delta}}_{\text{SO}}=\hslash /{\left(2\tau {\tau }_{\text{SO}}\right)}^{0.5}$. The temperature dependence of this spin–orbit splitting is plotted in figure 2(e) and although the values are smaller than what has been reported for gated diamond surfaces [38], the values are within the range reported for other quantum well structures [45, 46]. Although this is the first report of the observation of the WAL to WL crossover in nanocrystalline diamond films, the data does neatly follow the expected theory and yields realistic parameters in accordance to what has been cited in literature [38, 45, 46]. These findings thus further establish the significance of the diamond grain boundary in understanding the transport properties of this system. In particular it highlights the possibility of RSOC as an alternative explanation for the origin of spin active grain interfaces as recently reported for hydrogenated granular diamond systems [21].

Although much work has been directed at investigating the order parameter of this system through STM measurements [22–26], up to now many anomalous features reported in these works have not been reconciled. Although STM imaging results in high resolution spectra with a well-defined theoretical backing, it is only a local probe and thus is not ideal when considering dynamical processes such as those present when Cooper pairs tunnel through the grain boundaries [47]. This is particularly important in the present system as inversion symmetry breaking at the boundary leading to SOC at grain interface are expected to allow for spin mixing and triplet formation and can show ZBCP features.

In order to investigate spin-triplet and RSOC effects on superconductivity, we rely on transport four-probe voltage biased measurement through the diamond films. As shown in figure 3(a), the differential conductance obtained by measuring current transport through the film does indeed show pronounced ZBCP's. As indicated in the inset of figure 4(a), this ZBCP is only observed at temperature below the superconductivity onset and thus as such features have not been seen in single crystal samples or in the STM spectra of granular diamond samples, the most plausible explanation is that such features arise from the intergranular transport through the diamond films.

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The formation of such ZBCP is well documented in superconducting systems that deviate from conventional s-wave singlet superconductivity and are a result of the formation of so-called mid gap bound states [48–50]. These bound states are caused by constructive interference between incident and Andreev reflected quasiparticles along a grain boundary, essentially surface states, and are intimately linked to the properties of the order parameter [50]. In the boron doped diamond films, these ZBCP have shown a pronounced dependence on temperature and applied field (figure 3(a) and figure 4(a), respectively). As the ZBCP is a type of resonance mode, by extracting the full width half maximum (FWHM) of the peak it is possible to determine the lifetime of the bound states that give rise to this feature. This is shown in figure 4(b) where the extracted lifetime can be seen to decay exponentially with increasing temperature, reaching a minimum value within the ps regime. This is significant as from the MR data the spin coherence lifetime assumed from the spin active interface can also found to be limited to the ps regime, an indication that the two are intimately linked. As can be seen in the inset figure 3(b) the ZBCP height decreases with a power law scaling as temperature is increased and the FWHM shows a broadening with increased temperature (inset figure 3(b)). The peak is only observed at temperatures below the superconducting phase transition onset and quickly vanishes above this temperature.

Further support linking a triplet state to the observed ZBCP is the field dependence. Upon application of a magnetic field the peak height is reduced as shown in figure 4(a), indicating suppression of the superconducting phase. Another interesting observation is the robust nature of the zero-energy peak towards increasing magnetic field. As shown in figures 4(a) and (b) the zero-energy peak does not show any splitting with increasing field, a hallmark feature of spin triplet superconductivity [51]. Upon increasing the applied field strength, the peak height does however decrease until it is completely supressed at approximately 1.5 T, after which further increase of the field leads to a pronounced dip or V-shape in the differential conductance. Such a magnetic field dependent resistive behaviour has been well documented in various systems, either intrinsic or in heterostructures exhibiting triplet superconductivity [52–55].

A further observation is that the magnetic field strength where the peak is almost completely suppressed, corresponds to a crossing point in the high field MR isotherms as indicated in figure 4(b). From these transport features it is apparent that the complex structure of the nanocrystalline diamond system, most notably the grain boundary interface, has a significant influence on the properties of the superconducting phase. Moreover, when considering the multilayer system with broken inversion symmetry, i.e. superlattice, along with possible RSOC as mentioned above, it is possible for the predicted spin mixing effect to occur. As there are several mechanisms describing the observation of ZBCP we further probe this feature by applying a magnetic field to rule out competing mechanisms, a thorough discussion is included in the supplementary information (https://stacks.iop.org/NJP/22/093039/mmedia) section.

Although the influence of Rashba induced spin polarization of Andreev bound states is an actively researched area, it is still not experimentally known to what extant Rashba SOC can influence these resonance modes and as shown in recent publications [56] the effect of spin–obit coupled layers in superconducting heterostructures can greatly influence the transport. We have highlighted several anomalous features linked to the possibility of a triplet component in the boron doped diamond system, however further investigation will be required to fully understand the dynamics of this system

In this Letter analysis of these data firmly establishes the effect of Rashba-spin–orbit coupling on superconductivity in the boron doped diamond system. The SOC is expected to result from the broken inversion symmetry and asymmetric confining potential between crystal grains. We go on to show the spin coherence lifetime (τ_SO) and triplet resonance lifetimes (τ) are connected, indicating they share the same origin. The effect of the SOC on superconductivity in such as low dimensional superconducting system has huge implications for the nature of the superfluid condensate. As predicted by theory in such a system a single/triplet mixing occurs, this has been verified through the observation of pronounced ZBCP in the differential conductance which is robust to applied fields. These findings help us unify the previously reported variation in STM measurements and further establish the nanocrystalline boron doped diamond system as an unconventional superconductor with a triplet component.