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In most materials transport is well described by Ohm’s law, *V* = *I**R*_{0}, dictating that for small currents *I* the voltage drop across a material is proportional to a constant resistance *R*_{0}. Junctions that explicitly break inversion symmetry, for instance semiconductor *p**n* junctions, can produce a difference in resistance *R* as a current flows in one or the opposite direction through the junction, *R*( + *I*) ≠ *R*( − *I*); this difference in resistance is the key ingredient required to build a rectifier. A much greater degree of control over the rectification effect can be achieved when a similar non-reciprocity of resistance exists as a property of a material rather than a junction. However, to achieve such a non-reciprocity necessitates that the inversion symmetry of the material is itself broken. Previously, large non-reciprocal effects were observed in materials where inversion symmetry breaking resulted in strong spin–orbit coupling (SOC)^{6,7,8,9,10,11,12,14}. However, as SOC is always a very small energy scale, this limits the possible size of any rectification effect.

The non-reciprocal transport effect considered here is magnetochiral anisotropy (MCA), which occurs when both inversion and time-reversal symmetry are broken^{6,7,8,9,10,11,12,14}. When allowed, the leading order correction of Ohm’s law due to MCA is a term that is second order in current and manifests itself as a resistance of the form *R* = *R*_{0}(1 + *γ**B**I*), with *B* the magnitude of an external magnetic field and the rectification coefficient *γ* determines the size of the possible rectification effect. MCA may also be called bilinear magnetoelectric resistance^{9,15}. We note that non-reciprocal transport in ferromagnets^{3,4} does not allow the coefficient *γ* to be calculated and rectification of light into d.c. current due to bulk photovoltaic effects^{16,17,18} concerns much higher energy scales than those of MCA.

In heterostructures of topological materials it is possible to artificially break the inversion symmetry of a material^{19}; such an approach provides an unexplored playground to substantially enhance the size of non-reciprocal transport effects. In this context, quasi-one-dimensional (1D) bulk-insulating 3D TI nanowires^{19,20,21,22,23} are the perfect platform to investigate large possible MCAs due to artificial inversion symmetry breaking. In the absence of symmetry breaking, for an idealized cylindrical topological insulator (TI) nanowire—although generalizable to an arbitrary cross-section^{19,22}—the surface states form energy subbands of momentum *k* along the nanowire and half-integer angular momentum \(l =\pm \frac{1}{2},\frac{3}{2},\ldots \,\) around the nanowire, where the half-integer values are due to spin-momentum locking. The presence of inversion symmetry along a TI nanowire requires the subbands with angular momenta ±*l* to be degenerate. It is possible to artificially break the inversion symmetry along the wire, for instance, by the application of a gate voltage from the top of the TI nanowire^{19,21,23}. Such a gate voltage induces a non-uniformity of charge density across the nanowire cross-section, which breaks the subband degeneracy and results in a splitting of the subband at finite momenta^{19} (Fig. 1c). An additional consequence is that the subband states develop a finite spin polarization in the plane perpendicular to the nanowire axis (that is, the *y**z* plane) with the states with opposite momenta being polarized in the opposite directions, such that the time-reversal symmetry is respected. When a magnetic field is applied, the subbands can be shifted in energy via the Zeeman effect, which suggests that an MCA can be present in this set-up. Indeed, using the Boltzmann equation^{10,11,14} (Supplementary Note 4), we found an MCA of the vector-product type \(\gamma \propto {{{{{\mathbf{P}}}}}}\cdot (\hat{{{{{{\mathbf{B}}}}}}}\times \hat{{{{{{\mathbf{I}}}}}}})\) with the characteristic vector **P** in the *y**z* plane. For the rectification effect *γ*_{l}(*μ*) of a given subband pair *η* = ± labelled by *l* > 0, we found:

$${\gamma }_l={\gamma }_l^{+}+{\gamma }_l^{-}\approx \frac{{e}^{3}}{{({\sigma }^{(1)})}^{2}hB}\mathop{\sum}\limits_{\eta =\pm }{\tau }^{2}\left[{{{V}}}_l^{\eta }({k}_{l,{\rm{R}}}^{\eta })-{{{V}}}_l^{\eta }({k}_{l,{\rm{L}}}^{\eta })\right],$$

(1)

where *e* is the elementary charge, *h* is the Planck constant, *σ*^{(1)} is the conductivity in linear response, *τ* is the scattering time, \({{{V}}}_l^{\eta }(k)=\frac{1}{{\hslash }^{2}}{\partial }_{k}^{2}{\varepsilon }_l^{\eta }(k)\) in which \({\varepsilon }_l^{\eta }(k)\) describes the energy spectrum as a function of momentum *k* in the presence of symmetry breaking terms and the finite magnetic field *B* (Fig. 1c and Supplementary Note 4) and \({k}_{l,{\rm{R}}({\rm{L}})}^{\eta }\) is the right (left) Fermi momentum of a given subband (Fig. 1c). Owing to the non-parabolic spectrum of the subbands, differences in \({{{V}}}_{l }^{\eta }(k)\) are large for a TI nanowire, which results in the giant MCA. The quantities \({\gamma }_{l }^{+}\) and \({\gamma }_{l }^{-}\) are the contributions of the individual subbands. The behaviour of *γ*_{l} as a function of the chemical potential *μ* is shown in Fig. 1d. We found that, as the chemical potential is tuned through the subband pair, *γ*_{l} changes sign depending on the chemical potential. This makes the rectification effect due to the MCA highly controllable by both the magnetic field direction and the chemical potential *μ* within a given subband pair, which can be experimentally adjusted by a small change in gate voltage *V*_{g}. For reasonable experimental parameters, we predict that the theoretical size of the rectification can easily reach giant values of *γ* ≈ 5 × 10^{5} T^{−1} A^{−1} (Supplementary Note 5).

To experimentally investigate the predicted non-reciprocal transport behaviour, we fabricated nanowire devices^{24} of the bulk-insulating TI material BST, as shown in Fig. 1a by etching high-quality thin films grown by molecular beam epitaxy (MBE). The nanowires have a rectangular cross-section of thickness *d* ≈ 16 nm and width *w* ≈ 200 nm, with channel lengths up to several micrometres. The long channel lengths suppress coherent transport effects, such as universal conductance fluctuations, and the cross-sectional perimeter allows for the formation of well-defined subbands (Supplementary Note 8). An electrostatic gate electrode is placed on top of the transport channel for the dual purpose of breaking inversion symmetry and tuning the chemical potential. The resistance *R* of the nanowire shows a broad maximum as a function of *V*_{g} (Fig. 2a inset), which indicates that the chemical potential can be tuned across the charge neutrality point (of the surface-state Dirac cone; the dominant surface transport in these nanowires is further documented in Supplementary Note 7). Near the broad maximum (that is, around the charge neutrality point), the *V*_{g} dependence of *R* shows reproducible peaks and dips (Fig. 2a), which is a manifestation of the quantum-confined quasi-1D subbands realized in TI nanowires^{23}—each peak corresponds to the crossing of a subband minima, although the feature can be smeared by disorder^{23}. To measure the non-reciprocal transport, we used a low-frequency a.c. excitation current *I* = *I*_{0}sin*ωt* and probed the second-harmonic resistance *R*_{2ω}; here, *I*_{0} is the amplitude of the excitation current, *ω* is the angular frequency, and *t* is time. The MCA causes a second-harmonic signal that is antisymmetric with the magnetic field *B* and therefore we calculated the antisymmetric component \({R}_{2\omega }^{{\rm{A}}}\equiv \frac{{R}_{2\omega }({{{{{B}}}}})-{R}_{2\omega }(-{{{{{B}}}}})}{2}\), which is proportional to *γ* via \({R}_{2\omega }^{{\rm{A}}}=\frac{1}{2}\gamma {R}_{0}B{I}_{0}\approx \frac{1}{2}\gamma RB{I}_{0}\), where *R*_{0} is the reciprocal resistance (see Methods for details).

In our experiment, we observed a large \({R}_{2\omega }^{{\rm{A}}}\) for *V*_{g} ≳ 2 V with a magnetic field along the *z* axis. The \({R}_{2\omega }^{{\rm{A}}}({B}_{z})\) behaviour was linear for small *B*_{z} values (Fig. 2b) and \({R}_{2\omega }^{A}\) increased linearly with *I*_{0} up to ~250 nA (Fig. 2c), both of which are the defining characteristics of the MCA. The deviation from the linear behaviour at higher *B* fields is probably due to orbital effects (Supplementary Note 3). The magnetic-field-orientation dependence of *γ*, shown in Fig. 2d for the rotation in the *zx* plane, agrees well with *γ* ≈ *γ*_{0}cos*α*, with *α* the angle from the *z* direction and *γ*_{0} the value at *α* = 0; the rotation in the *yz* plane gave similar results, whereas MCA remained essentially zero for the rotation in the *xy* plane (Supplementary Note 10). This points to the vector-product type MCA, \({R}_{2\omega }^{{\rm{A}}}\propto {{{{{\mathbf{P}}}}}}\cdot ({{{{{\mathbf{B}}}}}}\times {{{{{\mathbf{I}}}}}})\), with the characteristic vector **P** essentially parallel to *y*, which is probably dictated by the large *g*-factor anisotropy^{25} (Supplementary Note 2). The maximum size of the ∣*γ*∣ in Fig. 2d reaches a giant value of ∣*γ*∣ ≈ 6 × 10^{4} A^{−1} T^{−1}. In addition, one may notice in Fig. 2b,d that the relative sign of *γ* changes for different *V*_{g} values, which is very unusual. We observed a giant MCA with a similarly large rectification *γ* in all the measured devices, some of which reached ~1 × 10^{5} A^{−1} T^{−1} (Supplementary Note 13). Note that in the MCA literature, *γ* is often multiplied by the cross-sectional area *A* of the sample to give *γ*′ (= *γ**A*), which is useful to compare the MCA in different materials as a bulk property. However, in nanodevices, such as our TI nanowires, the large MCA owes partly to mesoscopic effects and *γ*′ is not very meaningful. In fact, the large MCA rectification of ∣*γ*∣ ≈ 100 A^{−1} T^{−1} observed in chiral carbon nanotubes^{13} was largely due to the fact that a nanotube can be considered a quasi-1D system. In Supplementary Note 13, we present extensive comparisons of the non-reciprocal transport reported for various systems.

A unique feature of the predicted MCA is the controllability of its sign with a small change of *V*_{g}. To confirm this prediction, we measured detailed *V*_{g} dependences of \({R}_{2\omega }^{{\rm{A}}}\) in the *V*_{g} range of 5.1–5.5 V, in which the chemical potential appears to pass through two subband minima, because *R*(*V*_{g}) presents two peaks (Fig. 3a). We, indeed, observed the slope of \({R}_{2\omega }^{{\rm{A}}}({B}_{z})\) to change sign with *V*_{g} (Fig. 3b), and its zero-crossing roughly coincides with the peak or dip in the *R*(*V*_{g}) curve (compare Fig. 3a,b). A change in sign of the slope of \({R}_{2\omega }^{A}({B}_{z})\) on either side of the *R*(*V*_{g}) peaks was also observed in other devices (Supplementary Note 11). To obtain confidence in this striking observation, the evolution of the \({R}_{2\omega }^{{\rm{A}}}({B}_{z})\) behaviour on changing *V*_{g} is shown in Fig. 3c for many *V*_{g} values. This sign change on a small change of *V*_{g} also endows the giant MCA in TI nanowires with an unprecedented level of control. In addition, this *V*_{g}-dependent sign change of MCA gives a unique proof that the origin of the peak-and-dip feature in *R*(*V*_{g}) is, indeed, subband crossings.

The giant MCA observed here due to an artificial breaking of inversion symmetry in the TI nanowires not only results in a maximum rectification coefficient *γ* that is extremely high, but it is also highly controllable by small changes of chemical potential. Although rather different to the MCA of a normal conductor discussed here, we note that large rectification effects of a similar magnitude were recently discovered in non-centrosymmetric superconductor devices^{1,5} and in quantum anomalous Hall edge states^{4}, for which the controllability is comparatively limited. It is prudent to mention that the MCA reported here was measured below 0.1 K and it diminishes at around 10 K (Supplementary Note 12), which is consistent with the sub-bandgap of ~1 meV. As TI nanowire devices are still in their infancy^{24}, the magnitude and temperature dependence of the MCA could be improved with future improvements in nanowire quality and geometry; for example, in a 20-nm-diameter nanowire, the sub-bandgap would be ~10 meV, which enables MCAs up to ~100 K. The presence of the giant MCA provides compelling evidence for a large spin splitting of the subbands in TI nanowires with a broken inversion symmetry, which can be used for spin filters^{26,27}. Moreover, it has been suggested that the helical spin polarization and large energy scales possible in such TI nanowires with a broken inversion symmetry can be used as a platform for robust Majorana bound states^{19}, which are an integral building block for future topological quantum computers.

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