Flux-pinning mediated superconducting diode effect in NbSe₂/CrGeTe₃ heterostructure

Type II superconductors (type II SCs) are distinguished by their characteristic mixed state in the phase diagram due to the appearance of superconducting vortices [1]. The dynamics of these Abrikosov vortices can be modified by applied current, external magnetic field, sample geometry/confinement [2, 3], and magnetic proximity effect [4–6], thereby opening various possibilities for controlling device properties, for instance, the superconducting diode effect (SDE) [7–9]. The appearance of nonreciprocal current-voltage characteristics in a superconductor has received revived interest following the observation of field-free superconducting rectification effects in van der Waals heterostructures (vdWHs) and multilayered Rashba superconductors [10, 11]. These SDEs were reported to arise from breaking inversion symmetry and time-reversal symmetry [12]. Asymmetric critical currents have also been observed in other works using conformally mapped nanoholes [13], size confinement in constrictions [2], and Josephson junctions [14, 15]. It is plausible that asymmetric I–V characteristics can arise from extrinsic properties such as interface or edge roughness in lithographically defined constrictions [16] or other intrinsic mechanisms such as asymmetric vortex-flow energy barriers [1, 17, 18], magnetic flux penetration [19, 20], flux pinning effects [21, 22], vortex limited critical currents [23] and avalanches [24, 25]. Hence, identifying the key driving mechanisms of the SDE and establishing their relations to other nonreciprocal transport effects is of paramount importance for future superconducting electronic as well as spintronic devices. Atomically flat interfaces in artificially engineered vdWHs, which have been integral in the discovery of various emergent quantum effects in spintronic, thermoelectric, superconducting, and optical applications [26–28], could offer unique possibilities for a more detailed understanding of the mechanisms of SDEs by excluding extrinsic interfacial mixing or disorder.

In this work, we demonstrate the observation of SDE in the vdWHs of NbSe₂/CrGeTe₃ (CGT), where the magnetic domain walls in CGT offer a unique possibility for guiding superconducting vortices [29, 30] essential for magnetically controlled pinning of vortices at the NbSe₂/CGT interface [21]. NbSe₂ is an anisotropic layered superconductor extensively studied for its unique superconducting and spintronic properties in thicknesses down to the monolayer regime [2, 31–35]. CGT is a 2D magnetic semiconductor (E_g $\sim$0.38 eV) [36] with a Curie temperature of approximately 65 K and exhibits diverse thickness-dependent magnetic domain transformations [37, 38]. The magnetic coupling of superconducting NbSe₂ with CGT allows us to exploit its topological vortex phase transitions by modifying the confinement potentials [39] of Abrikosov vortices and, in turn, controlling the SDE via asymmetric flux-pinning at the NbSe₂/CGT and NbSe₂/SiO₂ interfaces [23].

Figure 1(a) shows a schematic diagram of our NbSe₂/CGT vdWH device, along with the electrical configuration for longitudinal resistance measurement. The NbSe₂ crystals ($\unicode{x2A7E}$99.999% purity) were obtained from Ossila Ltd. The CGT ($\unicode{x2A7E}$99.99% purity) crystals were grown using the flux method as described in [40]. Figure 1(b) shows an example micrograph of one of the measured devices (named device B) comprising adjacent NbSe₂/CGT and bare NbSe₂ devices, which we measured at the same time under identical experimental conditions in a Cryogenic Ltd cryostat. We adopted a top-down device fabrication approach where the vdWH (NbSe₂/CGT) was transferred onto a pre-patterned Si/SiO₂ substrate with Ti/Au (45 nm) electrical contacts patterned using standard UV lithography and e-beam evaporation. In the first step of our device fabrication, NbSe₂ and CGT crystals were mechanically exfoliated onto two different polydimethylsiloxane stamps (from Teltec Ltd) under ambient conditions. After identifying homogenous flakes, we first transferred the NbSe₂ flake on the contacts, followed by the CGT flake transfer on one side of the NbSe₂ flake. We fabricated and studied several devices with NbSe₂ thicknesses ranging from 5 nm up to 45 nm, while device degradation due to the oxidation of flakes was minimized by completing flake-transfer processes in under an hour.

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We performed temperature-dependent resistance measurements (RT) of both the bare and capped NbSe₂ sides. The current was applied along the x-direction (as shown in figure 1(a)) parallel to the ab-plane of NbSe_2, while an out-of-plane (OOP) external magnetic field was applied along the z-direction parallel to the c-axis of the flake. Figure 1(c) shows the RT of a device (labeled device A) for various applied magnetic fields. The zero-field critical temperature of the bare NbSe₂ (T_C ∼ 6.93 K) is just slightly larger than the CGT-capped NbSe₂ critical temperature (T_C ∼ 6.91 K); otherwise, both T_C values are close to that of bulk NbSe₂, indicating the high quality of our devices, as confirmed by the residual resistivity ratios [RRR = R (room temperature)/R (above T_c)] ranging from 10 to 20, typical for exfoliated NbSe₂ flakes [32, 35]. Figure 1(d) depicts the field dependence of the T_C extracted based on the 80% line of the normalized resistance plots in figure 1(c). Here, we can observe that for B < 200 mT, the critical temperature of NbSe₂/CGT is slightly higher than that of bare NbSe₂ (see the inset in figure 1(d)). On the other hand, when B > 200 mT and CGT is saturated, the T_C of the bare NbSe₂ is larger than that of the NbSe₂/CGT. This corresponds to the magnetization saturation field of the bulk CGT crystal (later shown in figure 4(c)). For the bare side, in the low-field range, the transition is accompanied by the appearance of intermediate kinks previously attributed to phase slips or inhomogeneous strain distribution [41, 42]. To illustrate this further, we plotted $\Delta T_{\text{C}} = T_{\text{C}}^{{\text{NbS}}{{\text{e}}_2}} - T_{\text{C}}^{{\text{NbS}}{{\text{e}}_2}/{\text{CGT}}}$ in figures 1(e) and (f), where a clear sign change of Δ${T_{\text{C}}}$ at B = 200 mT corresponds to the magnetization saturation point of CGT. It is also worth noting that for Δ${T_{\text{C}}}$ < 0 values in figure 1(f), the minimum is located in the field range of 80–100 mT, which, as we show later, plays a major role in the modulation of the critical current via the magnetic domain reorientation of CGT or flux-pinning [38, 43].

Next, we present magnetoresistance (MR) measurements of our devices with an OOP external magnetic field (see figure 2(a)). The CGT/NbSe₂ MR curve exhibits hysteresis with different H_C values for trace and retrace measurements. This hysteretic MR feature, which is completely absent in the bare NbSe₂ flakes, can be attributed to flux pinning in NbSe₂ by the CGT domains [21] and corresponds to the field regime where a sign change in ${{\Delta }}$ T_C was observed in figures 1(e) and (f). To identify the asymmetry in the trace and retrace H_C values, we define $H_{\text{C}}^{\,{\text{tr}}1}$ ($H_{\text{C}}^{\,{\text{tr}}2}$) as the positive (negative) H_C value when sweeping the field from 0.5 T to −0.5 T, as shown in figure 2(a), where the solid curve represents the trace measurements. Similar parameters are defined on the retrace measurement (dashed curve) from −0.5 T to 0.5 T. When tracing/retracing the field from higher magnitudes toward 0 T, there is a sharp transition from the mixed state to the Meissner phase; however, the transition is smoother when increasing the field value from 0 T to higher magnitudes, indicative of stronger vortex pinning. To observe how these lower critical fields develop by increasing the applied current, we performed MR measurements for varying currents. The current dependence of $\left| {H_{\text{C}}^{{\text{tr}}2}} \right|$ and $\left| {H_{\text{C}}^{{\text{retr}}2}} \right|$, obtained from the extrema of the first field derivative of MR and plotted in figure 2(b), does not show an appreciable difference over the measured field range. Nevertheless, based on this observation and by comparing several I–V measurements collected in the flux-pinning window, we can estimate the strengh of the pinned flux in NbSe₂. We consider a series of I–V measurements collected at 100 mT while the field is set to this value through two different scenarios: first, by coming from the higher field of +150 mT (solid black line in figure 2(c) noted as V_HL), and second, by coming from the lower field of 0 T (dashed red line, V_LH for lower to higher). As shown in the inset here (adapted from figure 2(a)), one of these two I–V sweeps represents the Meissner phase, and the other is performed in the dissipative phase of the Abrikosov lattice inside the hysteretic flux-pinning regime. The observed opening between the two I–V curves arises at currents above 0.5 mA, which also agrees with the appearance of hysteresis in the MR measurements. We can estimate the amount of this trapped flux to be approximately 20 mT by observing the relative overlap of the I–V curve at an applied field of 120 mT, which was set to this value by coming from an initial 0 T (blue dashed line in figure 2(c)). Figure 2(d) depicts the difference between V_HL and V_LH, which shows how this difference changes the sign for field values inside and outside of this pinning window resulting from the modulation of the critical current (the inset shows the MR for another device).

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After establishing the impact of the CGT magnetization on the superconductivity of NbSe₂, we next discuss the characterizations of the SDE in the NbSe₂/CGT heterostructure side of device B. This device comprises a thin NbSe₂ flake (5 nm) and exhibits smaller critical currents, as expected for thinner superconductors [44]. As such, we limited the maximum applied current to below 2 mA to maintain high device quality and temperature-dependent stability. Figure 3(a) illustrates a typical trace and retrace I–V characterization of NbSe₂/CGT at an applied out-of-plane field of 20 mT. The trace curves (solid black) were recorded by sweeping current first from zero to the maximum positive current and second from zero to the maximum magnitude of the negative current. The retrace measurements (dashed red) from the maximum magnitudes back to 0 are not the focus of this paper but represent the retrapping current [9, 45]. The positive critical currents (here: $I_{\text{C}}^ +$ = 1208 ±1 μA) and negative critical currents ($I_{\text{C}}^ -$ = −1200 ±1 μA) on positive and negative sweeps are different, indicating nonreciprocal transport of this SDE device with a nonreciprocity window of nearly 8 μA. Figure 3(b) depicts this window more clearly through the reflection (through the origin) of the negative critical current to the first quadrant. As mentioned above, these two curves are derived from the trace measurements where the current has been swept from lower to higher current magnitudes; thus, heat dissipation processes are relatively identical. As a proof of concept for SDE, we show the device rectification measurements in figure 3(c), in which the input signal is a current pulse with symmetric positive and negative values of approximately $\pm$1.2 mA that fit in the nonreciprocity window shown in figure 3(b), and the measured output voltage plotted with red dashed lines clearly demonstrates the rectifying behavior.

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Figure 4(a) shows the dependence of the positive critical currents ($I_{\text{C}}^ +$) and the absolute value of the negative critical currents ($I_{\text{C}}^ -$) on the applied OOP magnetic field at 4 K. In this exemplary case, the nonreciprocity is especially pronounced for fields below 40 mT, with the inset showing a wider magnetic field range of ±150 mT. We note a change in the overall slope of the resulting graph with the kinks at approximately ±30 mT, which marks a transition between a fast and slow decay of I_C values. Figure 4(b) presents the positive critical currents of the heterostructure for different temperatures from 2 K to 5 K. With increasing temperature, we observe a decrease in the field range where the slope increases. In this figure, the corresponding negative critical currents are not depicted in the interest of distinguishing the plotted curves. It is also observed here that the maximum magnitude of the critical currents at each temperature decreased from 1.25 mA at 2 K to 0.8 mA at 5 K. We note that our maximum rectification ratio ($Q \equiv 2\left( {I_{\text{C}}^ + - \left| {I_{\text{C}}^ - } \right|} \right)/(I_{\text{C}}^ + + \left| {I_{\text{C}}^ - } \right|$)) was ∼5% observed at 2 K. In addition, we have also observed that the Q-factor shows oscillatory behavior at some data points, which can be attributed to CGT magnetic domain reorientation or phase transitions of the finite-momentum pairing states [46]. Figure 4(c) shows the magnetization measurement of our CGT crystal for magnetic fields applied parallel to the ab-plane (black curve) and c-axis (red curve). The magnetization measurement along the c-axis saturates at a lower field, suggesting an easy axis along this crystallographic direction.

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When NbSe₂ is magnetically coupled to a CGT flake, dipolar fields from the CGT magnetic texture (B < 50 mT) can control the formation and dynamics of Abrikosov vortices in NbSe₂. Due to the asymmetric boundaries of NbSe₂, which is interfaced with CGT from one surface but with SiO₂ at the other surface, the forces at the vortex ends are not equal. This may cause a tilted core with stronger dynamics at the top NbSe₂/CGT interface than at the bottom NbSe₂/SiO₂. This asymmetry can be considered in explaining the nonreciprocal current-voltage characteristics [4]. As shown in figure 4(a), we have observed a similar asymmetry between the two curves, likely due to the CGT magnetic domains that have not yet fully aligned with the applied out-of-plane field. To shed more light on this pinning mechanism, we consider a simple bilayer structure (figure 4(d)) and estimate the critical-current asymmetry induced by magnetic domains. In the asymmetric magnetic environment, vortices are tilted, and their length can be estimated as $L = {d_{{\text{SC}}}}\sqrt {1 + B_x^2\left( 0 \right)/B_z^2}$, where ${d_{{\text{SC}}}}$ is the thickness of the superconducting NbSe₂; ${B_z}$ is the external magnetic field; and ${B_x}\left( 0 \right)$ is the field component due to the asymmetric magnetic environment (dipolar field), which depends on the sweep direction of ${B_z}$ and applied bias current $I$. Stray fields, which are generated by the stripe magnetic domains in the CGT, close inside the superconducting NbSe₂ flake thereby generating a non-zero in-plane field component ${B_x}\left( {I = 0} \right)$ at zero applied current $I$. The sign and magnitude of ${B_x}\left( 0 \right)$ is determined by the random distribution of domains and the imbalance between the number of up and down domains by the applied magnetic field along the z-axis. The Lorentz force is proportional to the applied current $I$ and ${d_{{\text{SC}}}}$ and can therefore be given as ${F_L} = \alpha {d_{{\text{SC}}}}I$. However, the pinning force is proportional to the number of pinning centers that can trap the vortex line. If the pinning centers are homogeneously distributed in the sample (no asymmetry) with an average distance of $a$ between them, then the pinning force is ${F_p} = \beta L/a$. In ${F_L}$ and ${F_p}$, $\alpha$ and $\beta$ are the magnetic-field independent constants. In the presence of a bias current, the magnetic-environment field component ${B_x}$ changes proportionally to the current as ${B_x} = {B_x}\left( 0 \right) + \gamma I$, where $\gamma$ is a material-specific constant. The physical origin of the in-plane field ${B_x}\left( 0 \right)$ generated by the charge current occurs due to the requirement of Ampere's law: $I \sim \Delta {B_z}/{d_{{\text{SC}}}} - \Delta {B_x}/{\delta _W}$ with $\Delta {B_z}{\text{ }}$ and $\Delta {B_x}{\text{ }}$ being the changes in the out-of-plane and in-plane magnetic field components, respectively, while ${\delta _W}$ is the characteristic stripe domain wall width. By estimating the critical current ${I_{\text{C}}}$ through the balance of the pinning force and Lorentz force at $I = {I_{\text{C}}}$, we obtain the equation $\alpha {d_{{\text{SC}}}}{I_{\text{C}}} = \left( {\beta /a} \right){d_{{\text{SC}}}}\sqrt {1 + {{\left( {{B_x}\left( 0 \right) + \gamma {I_{\text{C}}}} \right)}^2}/B_z^2}$, whose solution is not symmetric under current reversal $\left( {I_{\text{C}}^ + \ne I_{\text{C}}^ - } \right)$. This simple explanation captures the observed asymmetry in an asymmetric magnetic environment ${B_x}\left( 0 \right) \ne 0$ even if pinning centers are homogeneously distributed. To obtain a deeper phenomenological understanding of the SDE, combining the transport measurements (as in this work) with direct microscopic imaging of vortex dynamics will be useful to reveal the actual operation mechanism at play [47]. We also anticipate that employing additional constrictions of varying dimensions to exploit extrinsic pinning, as in [2], will offer pathways to further increase the Q-factors.

In conclusion, we have reported the observation of the SDE in a NbSe₂/CrGeTe₃ vdWH, where the size and asymmetry of the critical currents are modulated by the applied magnetic field, the magnetic domain texture of the CGT and inherent flux pinning in the NbSe₂/CrGeTe₃ hetrostructure. Furthermore, this pinning effect, due to the asymmetric magnetic environments and magnetic domain transformations, can provide a means to tune and utilize the SDE for energy harvesting from ambient radiation. More studies are required to improve the rectification efficiency and relation between intrinsic and extrinsic pinning effects using nanofabricated NbSe₂/CrGeTe₃ devices with constrictions, for instance, by modifying the shape anisotropy of CGT as well as the critical currents of NbSe₂. It is also interesting to look at the effects of the CGT thickness on the rectification behavior since ultrathin CGT flakes (<10 nm) exhibit very low demagnetization fields. Our results lay the groundwork for exploring the possibility of the field-programmable SDE and its manipulation through magnetic textures in two-dimensional systems.

We thank Akashdeep Kamra for the fruitful discussion. The research conducted at Loughborough University received support from the School of Science strategic fund and EPSRC, UK, through a Grant for multiuser equipment EP/P030599/1. We wish to acknowledge the support of the Henry Royce Institute for Advanced Materials for A M through the Student Equipment Access Scheme enabling access to the cleanroom facilities at the University of Leeds; EPSRC Grant Number EP/R00661X/1. The work at the University of Warwick was supported by EPSRC, UK, through Grants EP/T005963/1 and EP/N032128/1.

All data that support the findings of this study are included within the article (and any supplementary files).