Device preparation and electrical measurements
The procedure for device fabrication from PdGa crystal using Ga-ion-based focused-ion beam technique is as follows32. A lamella was milled out of the bulk crystal. Then the lamella was transferred in situ to the TEM grid for polishing till the desired thickness is achieved. The thickness of the lamella used to make devices in our study ranges from 1 μm to 4 μm. Then the lamella was microstructured into desired three-arm geometry on the TEM grid. Then the microstructured lamella was transferred in situ onto the SiO 2 substrate with prepatterned Au contact pads. The electrical contacts between the lamella and the contact pads were made by sputtering Ti–Au bilayers. Then the etching step was carried out to remove shorting between the electrical contacts. During the device preparation procedure, special care was taken to reduce the surface damage. For example, the transport channel in the device was never scanned with the Ga ion beam after the fine polishing step. However, surface damage of few tens of nanometres may still exists even after several fine polishing steps. Therefore, a dry etching step using low-energy Ar ion was performed to further reduce the thickness of this amorphous layer before the deposition of Ti–Au to make electrical contacts. The effect of surface damage on transport measurements was also studied by purposefully damaging the sample surface with the ion beam. The comparison of results before and after ion-irradiation were compared to study the influence of the amorphous layer on transport. As also observed in previous studies, the increase of the amorphous layer thickness did not negatively influence the transport response through topological states33.
The electrical measurements were primarily performed in a Bluefors LD−400 dilution refrigerator. Meanwhile, experimental data in Fig. 4 with the applied magnetic field of 2 T were measured in a PPMS DynaCool cryostat. The electrical measurements were carried out using Zurich Instruments lock-in amplifiers (MFLI) at 7.919 Hz and 13.333 Hz reference frequencies. The oscillator voltage amplitude was varied to sweep the applied current through the device (with a buffer resistance in series). The higher harmonic voltage responses from devices were simultaneously measured with the multi-demodulator option provided in an MFLI. Each demodulator produces two output signals: one corresponding to the in-phase component (X) and the other to the quadrature component (Y) relative to the reference signal. Special considerations were taken to increase the signal-to-noise ratio, such as using high-frequency electronic filters and avoiding ground loops. The magnetic field orientation was swept using a three-axis superconducting magnet from American Magnetics in a Bluefors refrigerator (Fig. 4b(ii)). An out-of-plane rotator puck was used in the PPMS case, in which the rotation axis is along the applied current under the magnetic field of 2 T (Fig. 4b(i),c). Two different configurations of electrode contacts with the electrical transport channel in a device were studied. It was observed that the measured third-order voltage responses were more prominent when the Ti–Au connected the top surface of the channel with the electrical contacts. The enhanced third-order response made it possible to measure the B θ dependence of the third-order response, as shown in Fig. 4b(ii),c(ii).
We have measured 23 devices fabricated in different geometries and crystal orientations. The five focal crystal orientations are presented in Extended Data Table 1. We can define four working states of the valve, depending on the relative magnitude of the chiral currents in the two arms. We have quantified the position of the valve using the term \(\phi ={\tan }^{-1}\frac{{{\bf{V}}}_{3\omega }^{{\rm{R}}}}{{{\bf{V}}}_{3\omega }^{\Gamma }}\), as shown in Extended Data Fig. 1. ϕ = 45° represents ‘valve on’ state, when there are equal magnitudes of the NLH currents from the Γ and R Fermi pockets in the right and left arms, respectively. ϕ = 0° represents ‘IΓ on’ state and ϕ =90° represents ‘IR on’ state, where NLH current in the right arm is predominantly generated due the Γ Fermi pocket and that in the left arm is predominantly generated due the R Fermi pocket. Finally, the ‘valve off’ state, in which chiral current generation is suppressed in both arms. We made four devices in a three-arm geometry near to the ‘valve on’ position. All four devices show the same experimental features discussed in the paper, namely, the appearance of distinct NLH responses of similar magnitude in different arms after the current threshold and their distinct symmetries of first-order responses with magnetic field orientations, as discussed in Fig. 3b(i),c(i). The absolute value of the nonlinear response varied with the dimensions of the device, as given in the Extended Data Table 2. The modulations of the third-order responses discussed with Fig. 3b(ii),c(ii) were observed in two of these devices. A good signal-to-noise ratio is needed to measure the nonlinear responses in the presence of the magnetic field, which we posit is a limiting factor in our PPMS system. We made three Mach–Zehnder interferometer (MZI) devices with a ‘valve on’ crystallographic position. All three showed oscillations in third-order response with applied current and magnetic field, as discussed in Fig. 5. The interference visibility ϑ for the MZI with ϕ = 49.4° and 38.6° were 0.86 ± 0.06 and 0.69 ± 0.1, respectively. ϑ was calculated using the relation \(\frac{{V}_{3\omega }^{\mathrm{amp}}-{V}_{3\omega }^{\mathrm{avg}}}{{V}_{3\omega }^{\mathrm{amp}}+{V}_{3\omega }^{\mathrm{avg}}}\), where \({V}_{3\omega }^{\mathrm{amp}}\) is the distance between the peak and crest of the oscillation, which is calculated using \(|{V}_{3\omega }^{\max }-{V}_{3\omega }^{\min }|\) and \({V}_{3\omega }^{\mathrm{avg}}\) is the mean offset of the oscillation given by \(|\frac{{V}_{3\omega }^{\max }+{V}_{3\omega }^{\min }}{2}|\). \({V}_{3\omega }^{\max }\) and \({V}_{3\omega }^{\min }\) are the minimum and maximum values of the V 3ω signal. We also found that ϑ was sensitive to the electronic properties of the conduction channel sidewall. The value of ϑ increased when the sidewall opposite to the voltage probe is not electrostatically screened by a presence of another electrode.
Passive and active control of the chiral fermionic valve
The chiral currents generated in the two arms of the device depend on the magnitudes of \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma }\) and \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{{\rm{R}}}\), as given in equation (1). We can control the relative projection of \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma }\) and \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{{\rm{R}}}\) along y, using the distinct symmetry of the Fermi pockets at the Γ and R points. Although the symmetry of a crystal is defined by its space group, the symmetry of a Fermi pocket is determined by the local symmetry of the band structure at a given k-point47. Therefore, Fermi pockets at the Γ and R points can locally exhibit distinct mirror-like symmetry \({{\mathcal{M}}}^{* }\), which maps \(({k}_{x},{k}_{y},{k}_{z})\,\longrightarrow \,({k}_{x},{-k}_{y},{k}_{z})\) (refs. 38,47). The projection of the net OAM along the y-axis is zero if a \({{\mathcal{M}}}^{* }\) exists along the xz-plane. \({{\mathcal{M}}}^{* }\) can selectively exist for only one of the Fermi pockets at Γ or R depending on the crystallographic direction of applied current. For example, when the current passes along [100] (z-axis) and the NLH-induced chiral current is collected along [010] (x-axis), \({{\mathcal{M}}}^{* }\) exists along the xz-plane for the Fermi pocket at Γ, whereas it is broken for the Fermi pocket at R. Thus, the OAM contribution from topological bands at R would predominantly exist along the y-axis. Extended Data Fig. 2a(i),(ii) shows the third-order and second-order responses in each arm in a device fabricated in the mentioned crystallographic directions. We can observe that the chiral current is primarily generated in the left arm of the device. It represents the valve in the ‘IR on’ state because the chiral current in the left arm preferentially exists in the Fermi pocket at R. The opposite is true when current is passed along the [100] (z-axis) and chiral current is collected along the [011] (x-axis). In this case, the OAM contribution from the topological bands at Γ would predominantly exist. Extended Data Fig. 2b(i),(ii) shows the nonlinear responses measured in a device made along these crystal directions. In this case, nonlinear currents are predominantly generated in the right arm of the device, thus representing the valve in the ‘IΓ on’ position. Finally, Extended Data Fig. 2c(i),(ii) shows the nonlinear responses in a device in the ‘valve off’ states where the generation of chiral current is suppressed in both arms. In this case, the current is passed along [011] and NLH currents were collected along [100]. We call this strategy to control the valve position ‘passive’ because the valve tunability is linked with the intrinsic quantum geometry of the topological bands rather than the experimentally controlled parameter.
We now discuss a proof-of-concept study to achieve active tunability of the valve. As discussed in the main text, the chiral fermionic current exists due to the preferential occupation of the R Fermi pocket in the left arm and Γ Fermi pocket in the right arm of the device. Thus, the magnitude of chiral currents can be controlled by tuning the occupational imbalance between two Fermi pockets. We fabricated a magnetic tunnel junction (MTJ) with in-plane magnetization on top of individual arms to locally probe the influence of the magnetic field on chiral currents. We developed a new fabrication technique to ensure the surface of the lamella is at a similar height to the substrate. Our strategy prevents any sudden height changes to ensure smooth deposition of the MTJ. Extended Data Fig. 3a shows the false-coloured SEM image of the prepared device with MTJ electrodes deposited on top of the left and right arms. The lamella was prepared in the ‘valve on’ position. In the first set of experiments, we studied the effects of the magnetization direction of the MTJ on the chiral current of both arms. M x and M z represent directions when the magnetization of MTJ is perpendicular and parallel to the current-induced orbital magnetization. Extended Data Fig. 3b(i),(ii) show the V 3ω responses at 77.77 Hz in the left and right arms, respectively, with M x and M z magnetizations. We can observe from Extended Data Fig. 3b that the chiral current has the same order of magnitude in the left and right arms for the M x orientation. However, the nonlinear response of the right arm switches sign when the MTJ magnetization was switched from M x to M z ; meanwhile, the V 3ω response in the left arm remains similar. Notably, these measurements were performed in the absence of an external magnetic field. The magnetic field was only used to switch the magnetization direction of the MTJ. Extended Data Fig. 3b shows a possible strategy to tune the valve state from ‘valve on’ to ‘IR on’, based on a MTJ magnetization switching mechanism. In the second experiment, we varied the frequency of the applied current to study the interaction of the chiral current with the magnetization of MTJ by measuring the inductive impedance. Extended Data Fig. 3c shows the dependence of V 3ω responses at different frequencies of the applied current with the MTJ in the M x configuration. We note from Extended Data Fig. 3b(ii),c(ii) that the impedance response of IΓ is not significantly changed when the frequency is varied between 77.77 Hz and 47 Hz, whereas the impedance response of IR is reduced by half. From Extended Data Fig. 3c(i), we see that the magnitude of IR is further reduced by an order of magnitude upon going to the frequency of 23 Hz. Meanwhile, the IΓ response, although diminished, has the same sign and the order of magnitude. Through Extended Data Fig. 3c, we show that the valve can be tuned from ‘valve on’ position to ‘IΓ on’ position by changing the frequency of the applied current from 77.77 Hz to 23 Hz with the MTJ in the M x configuration. Through these two experiments, we provide a first step to pursue active tunability of the chiral fermionic valve by its integration with an MTJ. However, a deeper understanding of the interaction between the chiral current and the magnetization dynamics is needed to further analyse these results. Also, a systematic study is crucial to analyse the switching reproducibility, fidelity and scalability of the MTJ-integrated device, which presents an important direction for future work.
Quantum Interference below threshold current
We have shown the appearance of the V 3ω responses after a certain current threshold in Fig. 2. Meanwhile, Fig. 5b shows clear oscillations of V 3ω with applied current even below the threshold current. As shown in Extended Data Fig. 4a, we could also observe weak V 3ω oscillation response below the threshold current in the three-arm geometry as well. The subsequent question is that why does the interference occur even below the threshold current. We can explain the observed phenomenon using the schematic shown in Extended Data Fig. 4b. We discussed in the main text that the fermions in different Fermi pockets gain a transverse velocity due to the presence of \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\). Equation (1) shows the magnitude of the current going into different arms of the device from different Fermi pockets is proportional to \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\). The appearance of a nonlinear response after the current threshold suggests that \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\) is non-zero only above it. However, \({{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\) does exist on individual topological Fermi pockets even when \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\) is zero. The presence of \({{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\) causes the fermions to contribute to the scattering equally in both arms of the device from both of the Fermi pockets. Therefore, the chiral current in each of the arms would not be preferentially carried by one of the Fermi pockets. However, the chiral currents in both of the arms do exist. But, the nonlinear responses of current with opposite chirality cancel out below the threshold current. \(\Delta {{\boldsymbol{\Omega }}}_{\varepsilon }^{\Gamma ,{\rm{R}}}\) becomes non-zero above the threshold current. Thereby, the fermions from the Fermi pockets at Γ and R, are preferentially scattered into the right and left arms, respectively. This creates an occupational imbalance in these arms, which leads to the observation of the chiral current response from the individual Fermi pockets. The presence of chiral current in both scenarios makes it possible to observe the quantum interference of the chiral current even below the threshold current.
Theoretical considerations for the NLH effect
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