Nonreciprocal superconducting NbSe2 antenna

The rise of two-dimensional (2D) crystalline superconductors has opened a new frontier of investigating unconventional quantum phenomena in low dimensions. However, despite the enormous advances achieved towards understanding the underlying physics, practical device applications like sensors and detectors using 2D superconductors are still lacking. Here, we demonstrate nonreciprocal antenna devices based on atomically thin NbSe2. Reversible nonreciprocal charge transport is unveiled in 2D NbSe2 through multi-reversal antisymmetric second harmonic magnetoresistance isotherms. Based on this nonreciprocity, our NbSe2 antenna devices exhibit a reversible nonreciprocal sensitivity to externally alternating current (AC) electromagnetic waves, which is attributed to the vortex flow in asymmetric pinning potentials driven by the AC driving force. More importantly, a successful control of the nonreciprocal sensitivity of the antenna devices has been achieved by applying electromagnetic waves with different frequencies and amplitudes. The device’s response increases with increasing electromagnetic wave amplitude and exhibits prominent broadband sensing from 5 to 900 MHz.

C rystal symmetry plays an essential role in condensed matter physics. Breaking inversion symmetry in materials profoundly changes the electronic ground states of materials and thus brings about many novel physical properties and functionalities, such as nonlinear Hall effect in Weyl semimetals 1,2 , chiral optical responses in semiconductors 3,4 and ferroelectricity 5 . One particular example is the nonreciprocal charge transport in systems with both broken inversion and timereversal symmetries 6 , where the electrical resistivity of a conductor is expected to vary depending on the current and magnetic field direction. Experimentally, nonreciprocal charge transport has been demonstrated in Bi helix 7 , chiral magnet 8,9 , Rashba semiconductor 10 , LaAlO 3 /SrTiO 3 oxide interface 11 and in various superconducting systems. These include superconducting noncentrosymmetric gated-MoS 2 12 , Bi 2 Te 3 /FeTe heterostructures 13 and MoGe/Y 3 Fe 5 O 12 bilayers 14 , where the nonreciprocal response is markedly enhanced by several orders of magnitude compared to non-superconducting systems due to the large energy scale difference between the Fermi energy and the superconducting gap 12,15 . Apart from being a powerful tool to study the interplay between superconductivity and chirality in non-centrosymmetric superconductors 6,16 , nonreciprocal charge transport also promises great potential in superconducting device applications such as vortex diodes 17 and flux lenses 18 , both of which are in great demand for future electrical circuits.
Atomically thin NbSe 2 is an emerging non-centrosymmetric superconductor possessing unique intrinsic Ising-type spin-orbit coupling, in which the electron spin is locked to the out-of-plane direction 19,20 . Accordingly, many exotic superconducting characteristics arise, for example, extremely large upper critical fields exceeding the Pauli limit 19,21,22 and an unusual continuous paramagnetic-limited superconductor-normal metal transition 20 . Meanwhile, as the thickness is reduced to the atomic scale where the fluctuation and disorder begin to play roles, 2D NbSe 2 becomes very sensitive to environmental perturbations 23 . Based on the aforementioned good merits, 2D NbSe 2 provides an ideal platform for exploring new mechanisms of nonreciprocal charge transport in non-centrosymmetric superconductors 15,24 and further device applications 25 . In particular, compared with conventional diodes that utilize a built-in electric field in semiconductor junctions, rectifiers based on atomically thin superconductors using the intrinsic electronic properties of quantum crystals pave the way towards the realization of highfrequency sensors and detectors for next-generation wireless networks 26 . However, practical sensing devices based on layered superconductors are still lacking.
Here we report the observation of nonreciprocal charge transport in atomically thin 2D NbSe 2 and the demonstration of successful manipulation of nonreciprocal sensitivity in atomically thin NbSe 2 antenna devices. The second harmonic magnetoresistance of few-layer NbSe 2 exhibits multiple antisymmetric reversals when the temperature is below the superconducting transition temperature T C , manifesting itself as a feature of reversible nonreciprocal charge transport due to the broken inversion symmetry. Utilizing the reversible nonreciprocal charge transport in NbSe 2 , we have built superconducting antenna devices that exhibit a strong reversible nonreciprocal sensitivity to the applied alternating current (AC) electromagnetic waves. Furthermore, we find that the response of the antenna increases monotonically with the increasing amplitude of the electromagnetic waves and that the devices show prominent broadband sensing from 5 to 900 MHz. Our research not only demonstrates the exotic physics in 2D NbSe 2 but also establishes it to be a promising platform for radio-frequency energy-harvesting, sensing, and identification applications.

Results
Nonreciprocal charge transport in 2D NbSe 2 . Figure 1a shows the typical crystal structure of NbSe 2 . It has a hexagonal lattice structure within the a-b plane and crystalizes with the P63/mmc space group 27 . Monolayer NbSe 2 consists of a sublayer of Nb atoms sandwiched between two sublayers of Se atoms in the trigonal prismatic structure 19,28 . Spatial inversion symmetry is broken in monolayer NbSe 2 because the Nb and Se sites are not equivalent. Various thicknesses of atomically thin NbSe 2 were obtained via exfoliation of bulk crystals onto SiO 2 /Si substrates (see Methods for details). Figure 1b displays an optical image of an exfoliated few-layer NbSe 2 flake where the number of layers is marked. Figure 1c shows an atomic-resolution transmission electron microscopy (TEM) image taken from an atomically thin exfoliated NbSe 2 flake. In perfect regions of the crystal, the atomic arrangement agrees well with the expected 2H crystal structure of NbSe 2 29 . However, Fig. 1c shows that some point defects (highlighted by circles) can be observed locally, even in high-quality exfoliated materials. This kind of point defects can act as asymmetric pinning potentials in superconducting regimes 30-32 which we will discuss later. Figure 1d displays the selected-area electron diffraction (SAED) pattern, confirming that the exfoliated flake is a single crystal and its dominant surface is {0001}.
As is typical for systems with broken inversion symmetry, nonreciprocal charge transport, so-called magneto-chiral anisotropy, will appear when time-reversal symmetry is broken by an external magnetic field 24 direction and can be phenomenologically described as 7,12,24,34 Here, R 0 represents the resistance at zero magnetic field, I is the electrical current, B is the external magnetic field and γ is a coefficient representing the strength of the magneto-chiral anisotropy effect(see Supplementary Note 1 for detail). Based on Eq. (1), we have carried out nonreciprocal charge transport experiments using a typical device structure shown in Fig. 2a with the optical image in Fig. 2b inset. The first and second harmonic magnetoresistances were measured simultaneously. The temperature-dependent normalized resistance R/R 300 K of a fivelayer device is shown in Fig. 2b. The sample exhibits a metallic behavior upon cooling and becomes superconducting at T C = 6.5 K (T C is defined as the temperature corresponding to 50% of the resistance above the superconducting transition R N ). Figure 2c illustrates the temperature-dependent first harmonic resistance R ω with a perpendicular magnetic field from −7 to 7 T, in which each R ω (B)-T curve overlaps with R ω (-B)-T curve. Interestingly, for the temperature-dependent R 2ω as depicted in Fig. 2d, the R 2ω (B)-T and R 2ω (-B)-T curves are symmetric with respect to the x-axis. These phenomena are also consistent with the behaviors of first harmonic and second harmonic magnetoresistance isotherms shown in Fig. 2e, f, where R ω -B and R 2ω -B curves are respectively symmetric and antisymmetric with respect to the y-axis. The antisymmetric feature of R 2ω -B curve is consistent with Eq. (1) which unambiguously suggests the existence of the magnetochiral anisotropy in 2D NbSe 2      . e, Magnetoresistance isotherm of the device with the temperature changing from 2 to 7 K, which are symmetric with respect to the y-axis. f, R 2ω -B curves of the device at temperatures of 2 to 7 K, showing antisymmetric behavior at T < 7 K, which is consistent with the first harmonic signal in (e). g, The extracted maximum value of R 2ω -B curves R 2ω MAX as a function of temperature. The pink squares R 2ω MAX1 stands for the peak values in the larger magnetic field regime, the purple triangles R 2ω MAX2 stands for the peak values in the small magnetic field regime at B ≥ 5.25 K. h, Calculated temperature-dependent γ and │γ′│, where γ ¼ respectively. γ and γ′ have an opposite sign due to the opposite nonreciprocity. i, Temperature-magnetic field phase diagram of the NbSe 2 device. The dark pink dots stand for crossover between the vortex solid (glass) state and vortex liquid state at which R 2ω goes to zero. The dark blue dots show the crossing point of the vortex liquid state and normal state. R 2ω goes to zero when further increasing the magnetic field. The yellow dots are the crossing point where R 2ω changes sign, defining the boundary between the activated pinned vortex states (light blue area) and vortex liquid state at T M ≥ 5.25 K. Here T M is the melting temperature above which the vortex solid melts.
Furthermore, the R 2ω -B curve shows one pair of peaks at low temperatures in the first and third quadrant (T ≤ 5 K) and as the temperature increases further (T > 5 K), another pair of peaks emerge in the second and fourth quadrant (also see Supplementary Fig. 2). The extracted peak values of the R 2ω -B curve R 2ω are shown in Fig. 2g, demonstrating that both are greatly enhanced at T < T C and R 2ω

MAX1
saturates at T ≤ 4.5 K. The deduced γ and γ´values as a function of temperature are shown in Fig. 2h, here γ and γ´are defined as , respectively (the corresponding values of B and R ω were used to calculate γ and γ′). The maximum of γ and γ′ are 6.53 × 10 2 T −1 A −1 and 3.43 × 10 4 T −1 A −1 , respectively. Both of them are higher than those reported in other non-superconducting systems such 10 and LaAlO 3 /SrTiO 3 oxide interface 11 (γ ∼ 10 2 A −1 T −1 ). The normalized coefficient value which defined as γ N ¼ γA and γ 0 N ¼ γ 0 A (A here is the cross-sectional area of device) are 1.44 × 10 −11 T −1 A −1 m 2 and 7.55 × 10 −10 T −1 A −1 m 2 , respectively. Both are also higher than that observed in LaAlO 3 /SrTiO 3 oxide interface 11 (∼1.17 × 10 −11 T −1 A −1 m 2 ). The large enhancement of the nonreciprocity below the superconducting transition temperature is due to the reduction of the energy denominator from the Fermi energy (∼100 meV) to the superconducting gap (∼1 meV) 12,13,15 . Note that R 2ω appears only in the resistive state (because the vortex flow in 2D NbSe 2 causes dissipation, see Supplementary Note 2 for details). In other words, R 2ω is nonzero only when NbSe 2 is in the vortex flow regime, signaling the close relationship between R 2ω and the vortex motion. Recent theory has revealed that nonreciprocal charge transport occurs in non-centrosymmetric superconductors when vortices driven by the external charge current move among the asymmetric pinning potentials in the vortex flow regime 15 . In 2D NbSe 2 , the asymmetric pinning potentials naturally appear as a consequence of disorder 15 such as defects [30][31][32] in 2D crystals with inversion symmetry breaking, as shown in Fig. 1c. Accordingly, we attribute the emergence of another pair of R 2ω peaks at T > 5 K to the melting of the vortex solid (glass) state into the activated pinned vortex states at relatively high temperatures as shown in Fig. 2i. In the lowtemperature regime (T ≤ 5 K), as the magnetic field increases, NbSe 2 transforms from the non-resistive vortex solid state into the resistive vortex liquid state, in which R 2ω reaches its maximum value. Further increasing the magnetic field will quench the NbSe 2 into the normal state, giving rise to a nondetectable R 2ω . While in the high temperate regime (T ≥ T M = 5.25 K), the vortex solid state will melt into the resistive activated pinned vortex states 35 where R 2ω also appears. Then the system undergoes a transition from activated pinned vortex states to vortex liquid states as the magnetic field increases. Note that the vortex in activated pinned vortex states will be thermally activated and will jump 36 between asymmetric pinning barriers, which is different from that in vortex liquid states where the vortex moves more freely. We then infer that the nonreciprocity of the vortex motion in activated pinned vortex states and vortex liquid states are opposite, leading to the two pairs of antisymmetric peaks in the high-temperature regime (also see Supplementary Note 2).
Next, we try to measure the second harmonic magnetoresistance of NbSe 2 at various applied AC currents with different amplitudes I 0 (I 0 is the effective value of the AC current, see Methods). Figure 3a depicts the first harmonic magnetoresistance R ω of the device measured at T = 2 K with I 0 changing from 5 to 100 μA. In the small current regime (I 0 ≤ 15 μA), the R ω -B curves almost overlap with each other, indicating a negligible effect of the applied current on the superconducting states of NbSe 2 . While in the larger current regime (I 0 ≥ 20 μA), the nonzero region (0 < R ω < R N ) of the R ω -B curves expands as the current further increases. In other words, the larger the current is, the easier the magnetic field will bring the system into resistive states. Correspondingly, the second harmonic magnetoresistance firstly increases then decreases as I 0 increases as shown in Fig. 3b. We extract the maximum value of R 2ω MAX versus I 0 in Fig. 3c. In the small current regime (I 0 ≤ 15 μA), the maximum value of R 2ω MAX increases linearly with the increase of I 0 , consistent with Eq. (1). As I 0 further increases (I 0 ≥ 20 μA), the rectification effect of vortex motion will be decreased by the relative weakening of the pinning potentials 30,37,38 . Also, the quenching of superconductivity in NbSe 2 can no longer be neglected. As a result, R 2ω MAX decreases as I 0 increases at I 0 ≥ 20 μA.
Reversible nonreciprocal DC sensitivity in atomically thin NbSe 2 antenna. Having understood the nonreciprocal charge transport in 2D NbSe 2, we next explore its direct current (DC) sensitivity and the relationship with nonreciprocal charge transport. We first built a NbSe 2 antenna device in order to investigate whether or not it can respond to externally applied electromagnetic waves. The device structure is illustrated schematically in Fig. 4a with the corresponding optical image in Fig. 4b. Here an AC electromagnetic wave is applied to the resistor fabricated on the same substrate. The resistance of the resistor is intentionally designed to be ∼50 Ω so as to give an impedance matching the AC signal. Figure 4c shows the R 2ω (B)-T curve of a three-layer NbSe 2 device. The R 2ω (B)-T and R 2ω (-B)-T curves are symmetric with respect to the x-axis, consistent with the previous five-layer device in Fig. 2d. Figure 4d is the DC response of the device with the AC signal applied across the resistor under various magnetic fields (V P-P = 200 mV, f IN = 5.52 MHz, see Supplementary Figs. 6, 7 for additional data). The device gives a prominent DC response as the temperature drops below T C. Surprisingly, the V DC (B)-T and V DC (-B)-T curves are symmetric with respect to the x-axis, same as the R 2ω (B)-T curve, suggesting a close relationship between the second harmonic signal and the DC response. As shown in Fig. 4e, f, the similarity of R 2ω and V DC can also be observed in second harmonic magnetoresistance and V DC isotherms with the temperature varying from 2 to 6.75 K. Note that the device also gives an antisymmetric DC response to the environmental fluctuation when there is no AC signal applied to the resistor (Fig. 4f inset and Supplementary Fig. 7b, the environmental fluctuations are mainly a few MHz electromagnetic waves in a cryostat 14,23 ).
The similarity of R 2ω and V DC can be explained using Eq. (1) which describes the nonreciprocal charge transport in 2D NbSe 2 due to the vortex in asymmetric pinning potentials. If we apply an AC excitation current of I ¼ ffiffi ffi 2 p I 0 sin ωt to the device, then the voltage of the device can be expressed as 13,15,39 where the first term is the first harmonic term, the second term is the second harmonic term and the third term is the DC term (see Supplementary Note 1 for a detailed derivation of the equation). Equation (2) above suggests that if we apply an AC current to the device with inversion symmetry breaking, a DC current will be generated. Then the device can be viewed as a p-n junction or rectifier where their asymmetry can convert an AC current passing through the device into a DC current 14 . The difference is that the device here has both an AC and DC current. In Eq. (2), the second term and the third DC term carry the same γB term, indicating that the second harmonic signal and the DC response should have the same relationship with the applied magnetic field. This explains the phenomena that we see in our experiments; that the V DC curves of the device have a similar shape as the R 2ω curves. Furthermore, from the point of view of vortex motion, the asymmetric pinning potential in superconducting NbSe 2 exerts a counterforce with different magnitudes to each direction of the AC driving force 40,41 (introduced by the electromagnetic waves radiated onto NbSe 2 ). As a result, the vortices acquire a net velocity, generating the DC voltage 17,42 (see Supplementary Note 2). Also, because the trilayer NbSe 2 here exhibits a much lower melting temperature than the five-layer device above, there is no vortex solid state for the device when T ≥ 2 K (see Supplementary Fig. 3). Consequently, the NbSe 2 changes from the activated pinned vortex state to the vortex liquid state as the magnetic field increases, leading to two pairs of antisymmetric peaks in both R 2ω -B and V DC -B curves at T ≥ 2 K (see Supplementary Note 2). It should also be noted that compared to conventional ratchets composed of artificial structures which rectify AC-driven vortices into a DC electric field without sensing ability 17,42,43 , our device utilizes the intrinsic inversion symmetry breaking in few-layer NbSe 2 and provides extreme sensitivity in the superconducting regime, thus realizing nonreciprocal sensing in 2D NbSe 2 .
Manipulating nonreciprocal sensitivity in NbSe 2 antenna. To control the performance of the NbSe 2 antenna device, we then try to change the frequency and the amplitude of the applied AC signal. Figure 5a displays the color plot of V DC as a function of the frequency f IN and magnetic field of the NbSe 2 antenna device at V P-P = 1 V and T = 2 K. The device shows prominent broadband sensing from 5 to 900 MHz(see Supplementary Figures 8, 9 for additional data) and the maximum response of the device appears at f IN ∼ 200 MHz. Also, the color plot is symmetric with respect to y-axis and the sign of V DC changes 3 times as the magnetic field increases from −3.5 to 3.5 T, indicating the nonreciprocal sensitivity of NbSe 2 antenna device. Figure 5b shows the Color plot of V DC as a function of frequency f IN and V P-P of the device at B = 0.12 T and T = 2 K. The entire spectrum agrees with Fig.5a and the device's response increases with the increase of the V P-P , suggesting that the device provides an increased DC response as the applied power of the resistor increases (see Supplementary  Fig. 8). This is also confirmed in Fig. 5c, which shows the DC response of the device under various AC amplitudes at f IN = 200 MHz. Accordingly, as the amplitude increases from 0 to 1 V (considering the resistor value of ∼50 Ω, the applied power P IN increases from 0 to 2.5 mW as V P-P changes from 0 to 1 V, see Supplementary Fig. 8), the maximum value V MAX1 increases from 3.5 to 5.9 μV and V MAX2 increases from 0.5 to 22 μV (Fig. 5d, e). Fig. 5a. The larger variation of V MAX2 than V MAX1 in response to increased applied power is due to the larger γ´value than γ (see Supplementary Note 2). All these results suggest that we have successfully pushed the relative low-frequency rectification in nonreciprocal charge transport into the radio-frequency sensitivity of the antenna device. It should be noted that the maximum response frequency of each NbSe 2 device varies. This may due to the different levels of disorder and defects in each device (see Supplementary  Figs. 11-15 for additional antenna devices with different thicknesses). Additionally, the device does not show a cutoff behavior at the maximum frequency that our equipment can reach (900 MHz). Note this value is much larger than the resonant frequency of the vortex moving in the artificial periodic potential (∼100 MHz) 18 . This may due to the much smaller size (atomic level) of the asymmetric pinning potential in 2D NbSe 2 (see Supplementary Note 4). Thus, the NbSe 2 sensing antenna has great potential to realize sensitive detection at much higher frequencies in the future.
To probe the stability of the device, we first measure the dynamic behavior of the NbSe 2 antenna device. In Fig. 5f, the DC response of the device was monitored with the AC signal (f IN = 200 MHz, V P-P = 1 V) being switching on and off. The device exhibits a stable and repeatable response to the AC signal with different magnetic fields and the on/off voltage ratio reaches ∼17. We then turn to measure the retention characteristic of the NbSe 2 antenna. As shown in Fig. 5g, we keep the magnetic field at the peak position value (P 1 -P 4 inset Fig. 5g) and measure the DC response of the device with the AC wave (F IN = 200 MHz, V P-P = 1 V) applied on the resistor. The DC response of the device was monitored for 4300 s, during which the generated DC voltage is very stable in all four states. This nonreciprocal multi-states of the antenna device suggest its potential applications in radiofrequency information storage and identification 44 . As shown in Fig. 5h, we have also connected a 100 Ω load resistor to the device (Fig. 5h inset) and measured the power on it (P L ) for 4300 s. The generated power P L by the NbSe 2 antenna is very stable during   monolayer rather than bulk material in which inversion symmetry is preserved. As a result, we can observe the nonreciprocal charge transport and DC sensitivity in few-layer NbSe 2 which is not present in the bulk material. We believe that the nonreciprocity still dominates the vortex motion due to the weaker interlayer coupling constant compared to the spin splitting energy in few-layer NbSe 2 (see Supplementary Note 7 for a detailed discussion). However, the underlying exquisite physics needs further theoretical and experimental investigation. This study has suggested that low-dimensional crystalline superconductors are promising systems to realize the control of vortex motion and further device applications. Our work highlights NbSe 2 as a model system for exploring nonreciprocal charge transport and controlling nonreciprocal sensitivity in its antenna devices, paving the way to understanding the exotic physics in layered lowdimensional crystalline superconductors and allowing their integration into new functionality devices.

Methods
Sample growth. High-quality 2H-NbSe 2 single crystals were synthesized using the Chemical Vapor Transport (CVT) method. The stoichiometric-ratio of Nb and Se powders with 0.2% excess of Se and 0.1 g iodine were evacuated and sealed in the quartz tube. The sealed tube was then placed in a double zone furnace horizontally and grown for 2 weeks in a temperature gradient of 730 to 770°C. After that, the single crystals of 2H-NbSe 2 were formed at the low-temperature end.
Device fabrication. Different thicknesses of NbSe 2 were obtained through mechanical exfoliation of bulk single crystals onto pre-patterned SiO 2 (285 nm)/Si  substrates using polydimethylsiloxane (PDMS) stamps in the glove box. Multiterminal electrical contacts were fabricated by standard EBL process using Polymethylmethacrylate/Methyl methacrylate bilayer polymer and subsequent deposition of Ti/Au (5 nm/80 nm) by magnetron sputtering. For the antenna device with the resistor, the resistance of the resistor was intentionally set to be ∼50 Ω by controlling the thickness of the gold and the length and width of the resistor. We limited the time that the devices were exposed to the air to less than 3 min to minimize the environmental effects on NbSe 2 .
Transport measurements. Four-terminal temperature-dependent magnetotransport and two-terminal DC measurements were carried out in a Physical Property Measurement System (PPMS) system (Quantum Design). Both the firstand secondharmonic signals of the AC resistance were measured by means of lock-in amplifiers (SR830) by applying an AC current I ¼ ffiffi ffi 2 p I 0 sintωt. The measured first and second harmonic resistance is defined as R ω = V ω /I 0 , R 2ω = V 2ω /I 0 , here I 0 is the effective value of the applied AC current, V ω and V 2ω are the measured first harmonic and second harmonic voltage drop. During the AC resistance measurements, the applied current frequency is between 10-100 Hz. The phase between the first and second harmonic signal was set to be π/2. The DC response of the NbSe 2 sensing device was collected using Keithley 2182 A and the high-frequency sine wave AC signal was applied to the resistor using Zurich Instruments UHFLI and Keithley 3390.

Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.