Abstract
When electric conductors differ from their mirror image, unusual chiral transport coefficients appear that are forbidden in achiral metals, such as a nonlinear electric response known as electronic magnetochiral anisotropy (eMChA)^{1,2,3,4,5,6}. Although chiral transport signatures are allowed by symmetry in many conductors without a centre of inversion, they reach appreciable levels only in rare cases in which an exceptionally strong chiral coupling to the itinerant electrons is present. So far, observations of chiral transport have been limited to materials in which the atomic positions strongly break mirror symmetries. Here, we report chiral transport in the centrosymmetric layered kagome metal CsV_{3}Sb_{5} observed via secondharmonic generation under an inplane magnetic field. The eMChA signal becomes significant only at temperatures below \({T}^{{\prime} }\approx \) 35 K, deep within the chargeordered state of CsV_{3}Sb_{5} (T_{CDW} ≈ 94 K). This temperature dependence reveals a direct correspondence between electronic chirality, unidirectional charge order^{7} and spontaneous timereversal symmetry breaking due to putative orbital loop currents^{8,9,10}. We show that the chirality is set by the outofplane field component and that a transition from left to righthanded transport can be induced by changing the field sign. CsV_{3}Sb_{5} is the first material in which strong chiral transport can be controlled and switched by small magnetic field changes, in stark contrast to structurally chiral materials, which is a prerequisite for applications in chiral electronics.
Main
The role that symmetries play in determining the properties of matter can hardly be overstated. Two opposite extremes are particularly interesting in crystalline solids. Higher symmetries constrain emergent degrees of freedom to mimic free particles—creating, for instance, massless Dirac or Weyl fermions that recover, at low energies, almost the full Lorentz group of free space^{11,12,13,14}. A second approach is to study lowsymmetry systems with novel responses. Among these, asymmetric systems characterized as ‘chiral’ play a special role across biology, chemistry and physics^{15,16}. Crystals are structurally chiral if they possess no mirror, inversion or rotoinversion symmetry, giving rise to left and righthanded enantiomers. This chirality can be imprinted on the crystals’ emergent excitations, which are then also characterized by a definite handedness. The interaction between structural chirality and the breaking of timereversal symmetry (TRS) is of particular interest, as it links the static chirality to temporal processes, such as growth, catalysis and wave propagation^{17}. Response functions that jointly arise because of chirality and TRS breaking are called magnetochiral anisotropies^{18}. Specifically, in metals, one studies the electronic magnetochiral anisotropy (eMChA), which opens up possibilities to detect, manipulate and utilize chiral properties in electronics^{1,2,3,4,5,6}.
eMChA usually refers to a change in resistance R due to an applied current I and external magnetic field B that is conventionally expressed as R(B,I) = R_{0}(1 + μ^{2}B^{2} + γ^{±}B⋅I)^{1} (see Fig. 1). Timereversal symmetry in nonmagnetic metals enforces a magnetoresistance even infield, which usually takes the semiclassical form μ^{2}B^{2}, with μ being the mobility. The scalar product B⋅I is only allowed in chiral crystals without spacereflection symmetries, and hence eMChA appears. Its strength is described by the coupling constant, γ^{±}, which takes opposite sign for the two enantiomers and is tensorial in anisotropic conductors. eMChA is most commonly detected by the associated secondharmonic voltage generation under lowfrequency a.c. currents, 4V_{2ω}/V_{ω} = ΔR/R, where ΔR = R(B,I) − R(B,−I) denotes the odd incurrent and R = R(B,I) + R(B,−I) denotes the even incurrent contribution to the resistivity.
To display eMChA, a conductor must break inversion symmetry, which can occur as a weak effect in any metal when its macroscopic shape is chiral^{2,3}, for example, in a coil (Fig. 1). Alternatively, materials with chiral crystal structure^{1,4} generally show eMChA in any conductor shape. We note that the ‘chiral electronic structure’ of the symmetrybroken phase mentioned here does not necessarily have to have the symmetries of a chiral space group. In a layered, quasitwodimensional compound one refers to a structure as ‘chiral’ when the inplane mirror symmetries are broken, whereas the M_{z} mirror symmetry might still be intact. However, this lack of inplane mirror symmetry is enough to enable the observation of eMChA in the geometry of our measurements. eMChA expresses an imbalance between scattering processes of different handedness, which can occur either from the intrinsic handedness of the carriers in chiral crystals, or extrinsically from chiral defects, as in plastically twisted conductors. When electronic interactions form ordered phases within chiral materials, as, for example, in chiral magnets, eMChA can be further amplified via scattering off, for example, an emergent chiral spin texture^{5,6}.
In this work, we demonstrate eMChA in a rectangular bar of CsV_{3}Sb_{5}, a layered metal in which vanadium atoms form kagome nets. In this system, a cascade of correlated symmetrybreaking electronic phases emerges at low temperatures^{7,10,19,20,21,22}, including a chargedensity wave (CDW) state below T_{CDW} ≈ 94 K and superconductivity below T_{c} ≈ 2.5 K (refs. ^{19,23,24,25,26,27}). Experimental evidence mounts for a further transition within the chargeordered phase at \({T}^{{\prime} }\approx 35\) K, accompanied by an additional 4a_{0} unidirectional ordering vector^{7} and timereversal symmetry breaking^{8,9,10}. The sudden onset of unexpectedly strong chiral transport at \({T}^{{\prime} }\) is our main observation. Crucially, this system is centrosymmetric at high temperatures, yet the relevant mirror symmetries are spontaneously broken by correlated phases of the itinerant carriers (Fig. 1). Reversible chirality of the electronic structure within the CDW phase has been observed in scanning tunnelling microscopy (STM) experiments^{28}. Note that the accompanying crystal distortion is so weak that the lowtemperature crystal structure remains actively debated^{7,24,29,30}. In contrast to structurally chiral crystals that strongly differ from their mirror image, here the differences between the enantiomers are subtle and test the limits of experimental resolution. Hence, the observation of eMChA itself in this compound points to its novel origin. As a consequence, the material’s chirality itself can be switched, which leads to fieldswitchable chiral transport in CsV_{3}Sb_{5}.
To truly obtain symmetry lowering from spontaneous symmetry breaking, it is critical to avoid any accidental strain fields that may break the symmetry explicitly. To do so, we decouple the crystalline bar mechanically as much as possible from its supporting substrate^{31} (Fig. 2a). This structure is mechanically supported by goldcoated SiN_{x} membranebased (200 nm thick) springs, and the differential thermal contraction strain is estimated to be less than 20 bar. Any signatures of chiral transport vanish in a reference experiment with even modest strain fields caused by stiff substrate coupling (see Methods), evidencing a strong coupling between the charge order and lattice distortions, which is not surprising in CDW systems^{32}. This provides a natural explanation for the opposing STM experiments^{28,33}.
Observation of eMChA in CsV_{3}Sb_{5}
Our main observation is the appearance of a sizeable secondharmonic response, V_{2ω}, to a lowfrequency transport current (7 Hz), which clearly evidences the diodelike behaviour due to chiral transport within the chargeordered state at low temperatures (Fig. 2). First, we discuss outofplane currents under an approximately inplane magnetic field, which is purposely misaligned by 0.5° with respect to the kagome planes. At zero field and T = 5 K, just above the superconducting transition, no second harmonic is observed, yet the signal quickly grows with increasing magnetic field. Its field dependence is well described by V_{2ω} ∝ B^{3} up to 18 T, the highest fields accessible to the experiment. This is a striking departure from the behaviour of structurally chiral materials, such as αTe (ref. ^{1}), where V_{2ω} displays a linear field dependence, ΔR/R = γ^{±}B⋅I (ref. ^{1}). This suggests that the magnitude of eMChA itself is field dependent, given by γ^{±}(B).
eMChA depends on the relative direction of the field and the current, and hence, even in a nonlinear scenario, V_{2ω}(B) must change sign when the field polarity reverses, as is observed experimentally. This antisymmetric field dependence provides strong evidence against a putative thermal origin of secondharmonic voltage generation by Joule heating, as the linear magnetoresistance is even in the magnetic field (see Methods). Pronounced quantum oscillations are also observed above B = 10 T, demonstrating an influence of Landau quantization on eMChA. This behaviour is observed consistently in two devices with different mechanical mounting approaches, rendering potential torque artefacts due to the softmounted structure unlikely. An identically shaped sample probing inplane transport does not show secondharmonic generation at any field configuration, demonstrating that eMChA is only relevant in the interplane transport (see Methods).
To further characterize eMChA and elucidate its origin in this nearly centrosymmetric material, we next turn to the temperature dependence of V_{2ω}. Figure 2b displays the raw V_{2ω}(B) without antisymmetrization. Yet, at elevated temperatures the weak thermal secondharmonic generation can obscure the chiral transport signatures; therefore, we focus on the antisymmetric component ΔV_{2ω} = V_{2ω}(18 T) − V_{2ω}(−18 T) (see Methods for full data). At high temperatures above T_{CDW}, no ΔV_{2ω} is observed, as expected. The transition into the CDW state is clearly evident as a sharp spike in ΔV_{2ω} at T_{CDW}, which we associated with the nonanalyticity of R(T). A continuous antisymmetric secondharmonic signal only emerges at temperatures below 70 K. Its slow increase with decreasing temperature suddenly accelerates at \({T}^{{\prime} }\approx \) 35 K, which is apparent as a change in slope on the logarithmic scale. At lower temperatures, ΔV_{2ω} increases significantly and saturates at its maximum value below 3 K. Although our observations only evidence the absence of chiral scattering and do not exclude a chiral order at T_{CDW} that merely does not affect transport, our results are suggestive of a secondary transition or crossover at lower temperatures of \({T}^{{\prime} }\). In particular, this temperature dependence agrees well with both the Fourier transformation intensity of the 4a_{0} CDW vector (q_{4a0}) obtained from STM experiments^{7} and the large anomalous Nernst effect^{34}. Such correspondence demonstrates the direct connection between the unidirectional charge order, electronic chirality and hidden magnetic flux. This consistency is further supported by the results of muonrelaxation experiments, which suggest the onset of TRS breaking at around 70 K and a subsequent rearrangement of local field distribution at 30 K (refs. ^{8,9,10}).
Fieldswitchable electronic chirality
The unusual nature of the eMChA in CsV_{3}Sb_{5} becomes apparent when the field orientation is varied with respect to the kagome planes (θ = 0° denotes the inplane field orientation; Fig. 3). No V_{2ω} is observed at large field angles (θ > 10°). Only within a narrow angle range, θ = ±10°, does V_{2ω} quickly grow as the fieldangle approaches θ = 0°. It reaches a maximum around θ ≈ 0. 5°, the configuration discussed previously in Fig. 2. For smaller θ, V_{2ω} rapidly decreases and vanishes for fields within the kagome planes (θ = 0°). On further rotation, to small negative θ, the signal repeats but with the opposite sign. This marks a most striking aspect of the data: tilting the field across the kagome nets changes the handedness of the material. Rotating the field by 1° barely changes B, hence an abrupt sign change of V_{2ω} implies a transition into the opposite enantiomer. Furthermore, the signal’s magnitude strongly reduces on raising the temperature or lowering the field strength, whereas the angular extent and the sharp anomaly at the inplane field persists. At temperatures above 35 K the peak is hardly observable, and the faint residual anomaly reflects the exponential drop of V_{2ω} above \({T}^{{\prime} }\) (Fig. 2d). The rotation curves are slightly hysteric; however, given the sharpness of the steep transition, it was impossible to distinguish an intrinsic hysteresis from the mechanical backlash of the rotator.
The possibility and ease of magnetic manipulation of the electronic chirality presents a unique electromagnetic response of CsV_{3}Sb_{5}. It suggests that the lowtemperature state differs from a simple chiral charge redistribution, as observed, for example, in the 3q chiral CDW^{35} state of TiSe_{2}. Such a static charge redistribution only couples to magnetic fields via higher order interactions, and its involved lattice response renders it unlikely to be easily manipulated at temperatures well below T_{CDW}. Instead, the experimental situation in CsV_{3}Sb_{5} points to coupled TRS breaking, including the concomitant magnetic anomalies at T_{CDW}, the field tunability, as well as muon spectroscopy experiments^{8,9}. As a microscopic picture for this correlated state, an orbital loopcurrent phase in the kagome planes has been proposed, which is consistent with these experimental observations^{22,36,37}.
Analysis of eMChA strength
Despite its exotic properties, the eMChA in CsV_{3}Sb_{5} can be rationalized within the existing theoretical framework. The magnetoresistance of CsV_{3}Sb_{5} is approximately linear in B for small angles θ at high magnetic fields (see inset of Fig. 3d). Such behaviour is indeed not unexpected for a material with densitywave order^{38} and has also been observed in many semimetals^{39,40}. This marks a crucial difference to previous eMChA studies, in which the conventional resistance R(B,I) + R(B,−I) ≈ 2R_{0} remains approximately field independent. This, and the strong θ dependence of the eMChA coefficient γ, means that eMChA cannot be characterized by a constant tensor, as is common practice in the literature for conventional eMChA materials. Yet, we can gain some insights about the magnitude of eMChA by computing ΔR/(R∣B∣∣J∣) = 4V_{2ω}/(V_{ω}∣B∣∣J∣) for given magnetic field strength B and current density J (see Fig. 3d)^{1}, for quantitative comparisons to other systems. The quantity ΔR/(R∣B∣∣J∣)equals the constant γ when it is a fieldindependent parameter in chiral materials commonly used in the literature. At B = 18 T and θ = 0.5° we find ΔR/(R∣B∣∣J∣) ≈ 2.4 × 10^{−11} m^{2} T^{−1} A^{−1}. In comparison, this value is smaller than its record observations in tTe (ref. ^{1}) (10^{−8} m^{2} T^{−1} A^{−1}) and TTFClO_{4} (ref. ^{4}) (10^{−10} m^{2};T^{−1} A^{−1}), for which the distinct structural chirality results in relatively large eMChA, whereas it is larger than that of chiral magnets, such as CrNb_{3}S_{6} (ref. ^{5}) (10^{−12} m^{2} T^{−1} A^{−1}) and MnSi (ref. ^{6}) (10^{−13} m^{2} T^{−1} A^{−1}), in which the chiral spin texture plays a major role in eMChA.
As the conventional eMChA analysis is only applicable for materials with negligible magnetoresistance, a description in terms of the conductance is more appropriate to further capture the lowest order fieldtuned behaviour of the response in CsV_{3}Sb_{5} (see Methods). For purely longitudinal transport and negligible Hall resistivity, the conductance is the inverse of the resistance, such that in analogy to the usual analysis of eMChA, for the conductance we write σ + Δσ ≈ 1/R − ΔR/2R^{2} (see Methods). We can thus extract \(\Delta \sigma \propto {V}_{2}\omega \,/\,{V}_{\omega }^{2}\), where V_{ω} is now linear in B for large fields. For a field applied approximately inplane, Δσ is thus approximately linear in B, which is the lowest order coupling between magnetic field and current (see Methods) and naturally explains the B^{3} dependence of V_{2ω}. The linear field dependence of Δσ yields a fieldindependent firstorder derivative ∂(Δσ)/∂B (see Fig. 4). The sudden sign reversal of ∂(Δσ)/∂B for small θ then suggests that the outofplane component of the field, B_{z}, has a nonperturbative effect on the system and we treat it separately, whereas the inplane component is a perturbation to linear order. In other words, we write \(\Delta \sigma ({\bf{B}},{I}_{z})=\widetilde{\sigma }({B}_{z}){B}_{x}{I}_{z}\). Note that such a coupling is only allowed for a system that breaks the y ↦ −y mirror symmetry. With Δσ(B,I_{x}) vanishingly small, no similar conclusion can be drawn for the mirror symmetry z ↦ −z.
Theoretical modelling
The behaviour of Δσ seen in our experiment demonstrates that the charge order in CsV_{3}Sb_{5}: (1) breaks inplane mirror symmetries, at least below \({T}^{{\prime} }\approx \) 35 K and (2) can be manipulated by an outofplane magnetic field in the same temperature regime. We thus establish that the tunability of the chirality of charge order in CsV_{3}Sb_{5}, previously seen in STM experiments, is a macroscopic bulk property of the unconventional charge order.
We further propose the following qualitative scenario, which would be consistent with the full θ dependence of our experimental observations and calls for confirmation by localprobe techniques (Fig. 4). B_{z} is the natural tuning parameter in kagomenet physics, as evidenced by our experiments as well as STM experiments. Akin to a soft ferromagnet, large values of B_{z} (large θ) induce a fully polarized, monochiral state. In this polarized state, only intrinsic chiral scattering processes induce eMChA, which commonly are weak. As the field is tilted towards the plane, B_{z} is reduced and domains of opposite chirality appear, which act as ideal chiral scattering centres. Hence, domainwall scattering leads to strong extrinsic eMChA. Naturally, a local probe, such as STM, would observe a chiral structure and occasionally the required domain boundaries between them, as indeed is the experimental situation^{28,33}. At even smaller B_{z} for fields very close to the planes, both chiralities appear symmetrically and hence a globally averaging probe, such as transport, observes a macroscopically symmetric conductor with vanishing eMChA. A fully symmetric process appears if the field is turned further, yet with inverted roles of majority and minority chirality.
In this scenario, the chirality switching is driven by B_{z} independent of the inplane field, in particular it would also occur for fully outofplane fields, where no eMChA is observed in our experiment. Yet, unlike structurally chiral systems, here the magnetic field plays a dual role. Whereas B_{z} sensitively changes the sign of ∂(Δσ)/∂B, the large inplane field is essential to observe finite eMChA, as Δσ ∝ I_{z}∣B∣. Given the close relationship between the fieldswitchable chiral transport and the chiral domains in CsV_{3}Sb_{5}, it is worth exploring its generality in other materials with suspected chiral orbital loop current.
Outlook
Although the small magnitude and extreme environmental conditions probably preclude direct applications of CsV_{3}Sb_{5}, it showcases that spontaneous symmetry breaking can be used to transform small changes in external fields into singular changes in the response functions of chiral conductors. Given the subtle deviation from centrosymmetry of the chargeordered phase, the emergence of eMChA in correlated states calls for new theoretical approaches to identify the microscopic mechanisms. The multitude of competing ground states in correlated materials gives rise to their versatility and tunability, which now presents a new approach towards chiral transport. In this direction, the fieldswitchable chiral transport adds a new aspect to the emergent picture of a highly frustrated, strongly interacting electron system on the kagome planes of CsV_{3}Sb_{5}. Although the magnitude of eMChA is unexpectedly large, these results link well with recent works that have shown the chargeordered state to be chiral and TRS breaking ^{8,9,10,28}. Akin to the theme of coupled orders in multiferroics, a series of new response functions emerges in materials such as CsV_{3}Sb_{5}, with multiple intertwined order parameters.
Methods
Crystal synthesis and characterization
CsV_{3}Sb_{5} crystallizes in the P6/mmm space group, which features a layered structure of kagome planes formed by the V atoms (Extended Data Fig. 1). The single crystals were grown by the selfflux method^{24}. Hexagonal plateshaped crystals with typical dimensions of 2 × 2 × 0.04 mm^{3} were obtained. The crystals were characterized by Xray diffraction off the maximum surface on a PANalytical diffractometer with CuKα radiation at room temperature. As shown in Extended Data Fig. 1, all the peaks in the Xray diffraction pattern can be identified as the (00l) reflections of CsV_{3}Sb_{5}.
On the basis of the crystalline structure, we have calculated the band structure of CsV_{3}Sb_{5} by density functional theory using the Quantum Espresso package^{41}, the details of which can be found in ref. ^{31}. The obtained electronic structure features multiple Dirac nodal lines lifted by spinorbit coupling, leaving only symmetryprotected Dirac nodes at L points. These results are consistent with previous reports^{23,24}.
Magnetoresistivity measurements were performed with electric current and magnetic field applied along outofplane (z) and inplane (x) directions, respectively. The magnetoresistance displays a quasilinear field dependence up to B = 18 T, whereas Hall resistivity is almost negligible compared with that. This is expected, as the electrical current is applied along the outofplane direction the Hall resistivity should vanish for such a quasitwodimensional material, with the Brillouin zone dominated by the large cylindrical Fermi surfaces.
Angular dependence of magnetoresistance and its relation to V _{2ω}
At low temperature, angledependent magnetoresistivity displays a strong peak when the magnetic field direction rotates across the kagome plane (Extended Data Fig. 2). With increasing temperature this peak gradually transitions to a broad hump at T = 50 K. This strong enhancement of magnetoresistivity at inplane fields is probably a feature of openorbit magnetotransport, which is expected for a metal featuring cylindrical Fermisurface sheets^{42}. The smearing of the spike at high temperatures therefore demonstrates the reduction of carrier mobility with increasing temperature.
In the meantime, the strong increase of magnetoresistivity naturally leads to enhancement of V_{2ω}. Combining the angular dependence of both leads to a clear quadratic relationship between V_{2ω} and ρ_{c}, which provides further evidence for the chiral conductance analysis.
I–V characteristics of eMChA
Secondharmonic voltage generation due to eMChA is expected to display a quadratic current dependence. Here we present the I–V characteristics of both first and secondharmonic voltages measured with a 7 Hz a.c. current (Extended Data Fig. 3). For V_{ω} the relationship depends linearly on current, which corresponds to the firstorder resistance term. On the other hand, the secondharmonic voltage shows a clear quadratic current dependence, which is an expected signature of eMChA. These results again demonstrate that the observed V_{2ω} originates from the chiral correction of conductivity due to the electronic chirality of CsV_{3}Sb_{5}.
Examination of Joule heating effect
Joule heating is a natural extrinsic origin of higher harmonic voltage generation^{43,44}. Applying an a.c. electric current, I_{ω}, must result in an oscillating temperature with a frequency of 2ω. Therefore if the electrodes of the device are strongly imbalanced in contact resistance, an extrinsic V_{2ω} can be observed. To further check the influence of Joule heating in the measurements of eMChA in CsV_{3}Sb_{5}, we performed systematic currentdependent V_{2ω} measurements under different thermal conditions (Extended Data Fig. 4). By controlling the helium gas pressure of the sample space, the thermal link between the device and the sample chamber can be easily tuned. Within a low current regime (below 0.12 mA), the collapse of all curves measured at different conditions suggest the insignificance of the Joule heating effect. With further increasing current, Joule heating inevitably grows and becomes detectable. To avoid any disturbance due to Joule heating, all measurements of eMChA have been performed with a relatively low a.c. current of 0.1 mA and high gas pressure (p_{s} ≈ 600 mbar) of the sample space, providing the maximal cooling power.
Reproducibility of eMChA with two different devices
To show the reproducibility of the secondharmonic voltage generation due to eMChA in CsV_{3}Sb_{5}, we have measured two membranebased devices with different mounting techniques/geometries (Extended Data Fig. 5). For device S1, the sample was completely suspended by soft Aucoated membrane springs. In comparison, device S2 was attached to the membrane only on one side, and the other side of the sample was welded directly to the Si substrate by focus ion beam (FIB)assisted Pt deposition. Device S2 displays a slightly broader CDW transition than S1 in the temperature dependence of resistivity across T_{CDW}, yet the transition temperatures are exactly the same. This suggests a marginally larger strain gradient across the device due to thermal contraction for device S2, which is compatible with the estimated strain value presented in the next section. The secondharmonic voltage was consistently observed among the two devices, with a similar value as well as an almost identical angular spectrum. These results demonstrate the clear consistency among different lowstrain samples, and therefore evidence that the observed eMChA in CsV_{3}Sb_{5} is an intrinsic material property. These devices further differ in their coupling strength between the substrate and the device. In view of the much stiffer coupling in S2, the similarity of the data speaks against magnetic torque induced angle changes as a putative error source.
Estimation of strain due to differential thermal contraction
To obtain the tensile strain applied to the sample, we first need to estimate the total displacement of the samples and substrates used due to different thermal contraction coefficients. On cooling from 300 K to 4 K, the integrated thermal contraction coefficients of SiN_{x} (ε_{SiN}) and Si (ε_{Si}) were 0.0342% and 0.0208%, respectively. For the sample itself we assumed a typical thermal contraction coefficient for alkali metal of ε_{Sample} ≈ 0.1%, which provides a conservative, upper bound. On the basis of these parameters and the actual device geometry illustrated in Extended Data Fig. 6, the total displacement can be easily obtained as:
The spring constant of the SiN_{x} spring for device S1 is estimated as 125 N m^{−1} from finite element simulations^{31} (COMSOL), the total pressure can be calculated as:
where A is the crosssection of the spring.
Meanwhile, for device S2, the pressure can be calculated using the same process:
In both cases the pressure is less than 20 bar. Taking the typical Young’s modulus of alkali metals (≈5 GPa), the strain applied on the sample is estimated to be ≈0.04%, which quantifies the lowstrain nature of these devices.
Strain effect on eMChA
The necessity of lowstrain mounting was revealed by a comparative study of device S4, which features a sample that is glued down to a sapphire substrate. Here the device is structured into an L shape with two long beams along both the a and c directions (Extended Data Fig. 7). As the sample and substrate are mechanically coupled via the glue droplet, the thermal contraction difference between them results in a tensile strain along the beam direction. This tensile strain not only shifts the CDW transition of device S4 to a higher temperature compared to the strainfree S1, but also suppresses the superconducting transition down to lower temperatures. Most importantly, no meaningful secondharmonic voltage has been observed for device S4. These observations suggest the importance of c axis tensile strain, which in defining an extrinsic, longrange domain structure, is unable to be switched or tuned. This observation suggests that residual strain fields would provide a natural explanation for the contradictory STM experimental results^{28,33}.
Fieldsymmetry analysis of secondharmonic voltage
To further demonstrate the origin of secondharmonic voltage generation, we also measured the temperaturedependent V_{2ω} at B = 18 and −18 T (Extended Data Fig. 8). By taking the sum and difference of these two results we obtained both the fieldsymmetric and asymmetric components of V_{2ω}. It is clear that the antisymmetric component dominates the total signal at low temperatures, whereas the symmetric component, which is probably due to Joule heating at the electric contacts, is merely a minor part.
Theoretical considerations
In this section, we discuss the magnetoresistance of a singleband model to illustrate the appearance of the various contributions to the linear magnetoresistance, and to the secondorder response discussed in the main text. In particular, we are interested in the effect of a CDW on the magnetotransport, when the CDW not only breaks translation, but also breaks timereversal and several mirror symmetries. Note that we focus here on intrinsic contributions that enable us to explain the abrupt switching of the secondorder response at small θ. To model the full θ dependence, extrinsic contributions would have to be included as well, as discussed in the main text.
In most metals, the transverse magnetoresistance scales quadratically with magnetic field, \({\rho }_{zz}({B}_{x})\propto {B}_{x}^{2}\). However, this behaviour can change to linear with B if there are small Fermi surfaces or Fermi surfaces with sharp corners^{45}. Although the conditions for such Blinear behaviour are probably not satisfied in the normal state of CsV_{3}Sb_{5}, the Fermisurface reconstruction due to the CDW instability is expected to result in new, smaller Fermi surfaces, such that a linear magnetoresistance, as observed, can be explained. Note that linear magnetoresistance in densitywave materials has indeed been observed and discussed in the context of Fermisurface reconstruction by Feng and coworkers^{38}.
When the densitywave instability breaks additional symmetries, we find further contributions to the magnetoresistance, or, equivalently, to the conductivity. To see this, we use Boltzmann transport theory in the relaxationtime approximation, in which the conductivity is given by
with e the electron charge, τ the scattering time and
the velocity for electrons with dispersion ξ_{k}. We assume in the following that we are in the symmetrybroken chargeordered phase and the dispersion is given by \({\xi }_{{\bf{k}}}^{{\rm{CDW}}}\) at zero magnetic field. If we minimally couple the vector potential to the momentum, we first find the dominant term \({\sigma }_{zz}^{0}\propto 1/ B \), as discussed above.
In addition to minimal coupling, a magnetic field can couple to the electron dispersion directly, if its Bloch states have an orbital magnetic moment M(k). In this case, we can write
Note that M(k) is a pseudovector. If TRS is present, M(k) is an odd function of k. If in addition inversion symmetry is present, we find M(k) ≡ 0. This should be the case above the CDW transition temperature. In the CDW phase, by contrast, the various broken symmetries enable M(k) to be nonvanishing. For concreteness, we will now only discuss the case of broken x mirror symmetry, which is sufficient to explain the experimentally observed response. In this case, we find to lowest order in k that M_{x}(k) ∝ k_{z}. If, further, this symmetry breaking is due to an ordered phase with order parameter Δ^{CDW}, we can expand in both k and Δ^{CDW} and write \({M}_{x}({\bf{k}})\approx \widetilde{M}{\Delta }^{{\rm{CDW}}}{k}_{z}\) with \(\widetilde{M}\) a modeldependent constant.
We can now use eqns (8) and (9) to calculate the velocity in the z direction,
(More generally, M_{x}(k) will be an odd function of k_{z}, such that the additional contribution to the velocity will be even.) Finally, we find for the conductivity in the z direction (dropping, for simplicity, the subscripts zz)
up to the order of B_{x}(Δ^{CDW})^{2}. Here we have used the stationary Fermi distribution to calculate the current carried by the system. We thus find an additional contribution to the conductivity Δσ ∝ B_{x}I_{z}, which will result in a secondharmonic signal in an a.c. electric field applied in the z direction.
To further support the above arguments for the case of CsV_{3}Sb_{5}, we have performed a tightbinding calculation, which shows, in the chiral CDW phase, that: (1) the Fermisurface structure does indeed become more structured with sharp corners, and (2) a finite orbital magnetic moment M_{x} arises, see Extended Data Fig. 9.
Note that, experimentally, we find that a small magnetic field in the z direction can change the sign of the observed signal. This implies that a magnetic field B_{z} couples linearly to the order parameter Δ^{CDW}, which in turn implies that the order breaks TRS in addition to the mirrors M_{x} and M_{y}. This is in agreement with the experimental findings in ref. ^{28}. In terms of a simple Landau theory,
with α < 0, β > 0 and γ ≪ ∣α∣, β, the CDW order parameter is \({\Delta }^{{\rm{C}}{\rm{D}}{\rm{W}}}\approx {\rm{s}}{\rm{i}}{\rm{g}}{\rm{n}}({B}_{z})\sqrt{\alpha /\beta }={\rm{s}}{\rm{i}}{\rm{g}}{\rm{n}}(\theta ){\Delta }_{0}\), with \({B}_{z}= {\bf{B}} \sin \theta \), implying its sign change with B_{z}.
In summary, for the slope of the chiral conductivity, we find
For small angles θ off the basal plane, this yields a step function in good qualitative agreement with the experimentally extracted form for small angles θ. Moreover, combining the angular dependence of both the theoretically predicted chiral conductivity and the experimentally measured magnetoresistance, we also derived the secondharmonic voltage V_{2ω} as a function of angle (Extended Data Fig. 10), which is also consistent with the experimental results.
Finally, if we want to compare the conductivity calculated above to the experiment and the standard eMChA literature, we need to express the conductivity in eqn (12) as a resistance. Namely, one usually writes R(B,I) = R + ΔR/2, where in general R can depend on B, and in particular here we have R ≈ ∣B∣. We thus find
This yields Δσ ≈ −ΔR/2R^{2} and in turn, we expect
By applying a lowfrequency a.c. current I_{ω} = I_{0}sin(ωt), the generated electric voltage can be expressed as:
here V_{DC} stands for the d.c. background voltage. This yields:
which also suggests:
exactly in line with our experimental data.
Data availability
Data that support the findings of this study are deposited to Zenodo with the access link: https://doi.org/10.5281/zenodo.6787797.
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Acknowledgements
This work was funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MiTopMat, grant agreement no. 715730, and PARATOP, grant agreement no. 757867). This project received funding by the Swiss National Science Foundation (grant no. PP00P2_176789). M.G.V., I.E. and M.G.A. acknowledge the Spanish Ministerio de Ciencia e Innovacion (grant PID2019109905GBC21). M.G.V., C.F. and T.N. acknowledge support from FOR 5249 (QUAST) lead by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). This work has been supported in part by Basque Government grant IT97916. This work was also supported by the European Research Council Advanced Grant (no. 742068) ‘TOPMAT’, the Deutsche Forschungsgemeinschaft (ProjectID no. 247310070) ‘SFB 1143’ and the DFG through the Würzburg–Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147, ProjectID no. 390858490).
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Crystals were synthesized and characterized by D.C. and C.F. The experiment design, FIB microstructuring, the magnetotransport measurements and the secondharmonic voltage measurements were performed by C.G., C.P., S.K., X.H. and P.J.W.M. M.H.F. and T.N. developed and applied the general theoretical framework, and the analysis of experimental results has been done by C.G., C.P. and P.J.W.M. Band structures were calculated by M.G.A., I.E. and M.G.V. All authors were involved in writing the paper.
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Extended data figures and tables
Extended Data Fig. 1 Basic properties of CsV_{3}Sb_{5}.
a, Crystal structure of CsV_{3}Sb_{5}. b, XRD pattern of the (001) facet of a CsV_{3}Sb_{5} crystal. c, Band structure of CsV_{3}Sb_{5} calculated by density functional theory (DFT) using the Quantum Espresso package (QE)^{41}. d, Field dependence of magnetoresistivity and Hall resistivity measured at T = 5 K. A large quasilinear magnetoresistance is observed up to B = 18 T. In comparison, the Hall resistivity is almost negligible.
Extended Data Fig. 2 Angular dependence of magnetoresistivity and eMChA.
a, Angledependent magnetoresistivity of CsV_{3}Sb_{5} measured with B = 18 T at various temperatures. A strong spike can be observed around θ = 0 deg which becomes broader with increasing temperatures. b, Correspondingly, V_{2ω} also gets pronounced within the same angle range. c, The V_{2ω} depends linearly on the square of caxis resistivity, which demonstrates the direct connection between magnetoresistivity and eMChA.
Extended Data Fig. 3 Current dependence of first and second harmonic voltage.
a, b, Currentdependence of both V_{ω} and V_{2ω}. As expected V_{ω} depends linearly on current while V_{2ω} displays a quadratic current dependence. c, Summary ofthe relation between V_{2ω} and V_{ω}, which shows a parabolic dependence.
Extended Data Fig. 4 Influence of Joule heating effect.
Currentdependence of V_{2ω} measured at B = 18 T and T = 2 K with varying levels of helium exchange gas pressure in the cryostat. The curves differ only at currents above 0.12 mA, suggesting that the heat generation and accumulation at lower currents is not a dominant factor. eMChA was measured at lower values of 0.1 mA.
Extended Data Fig. 5 Reproducibility of eMChA with two different devices.
a, Scanning electron microscope (SEM) images of devices S1 and S2. b, Temperaturedependent resistivity of S1 and S2 from 300 K to 1.6 K. c, Angledependent second harmonic voltage measured at B = 18 T and T = 2 K.
Extended Data Fig. 6 Illustration of device configuration for both S1 and S2.
While S1 is completely suspended by the membrane springs, S2 is attached to the Sisubstrate frame on one side and membrane springs on the other side.
Extended Data Fig. 7 Strain effect on eMChA.
a, SEM image of device S4. The sample is attached to the sapphire substrate via a glue droplet. The thin beam along the caxis allows us to measure the electric response with both current and tensile strain applied along the caxis. b, Temperature dependence of resistivity for S1 and S4. The CDW transition is enhanced to a higher temperature with tensile strain along the caxis (S4), while T_{c} is reduced to lower temperature. c, Angular spectrum of second harmonic voltage (V_{2ω}). Clearly V_{2ω} is completely suppressed with tensile strain along caxis.
Extended Data Fig. 8 Fieldsymmetry analysis of second harmonic voltage.
a, Temperature dependence of V_{2ω} measured at B = ± 18 T respectively. b, Temperaturedependent fieldsymmetric and asymmetric part of V_{2ω}.
Extended Data Fig. 9 Tightbinding model on the stacked triangular lattice with 2 × 2 × 2 chiral CDW order.
a, The unit cell of the model consists of 8 lattice sites (black) on two layers, each harbouring a single electronic orbital. The model is defined by the hoppings as indicated, where ω = e^{i2π/3} (all complex hoppings are oriented to the right). All dashed lines are associated with real hopping amplitude t_{z}. We choose numerical values t = 1, t_{z} = 0.2, Δ = 0.05. b, Fermi surface of the model for chemical potential μ = 2.4 (black lines) and orbital magnetic moment component M_{x} of the states closest to the Fermi level. We observe that M_{x} is not only finite, but also large for states on the Fermi surface. c, Band structure along k_{z} for a particular choice of k_{x} and k_{y}, coloured by the orbital magnetic moment component M_{x}, demonstrating that M_{x} is an odd function of k_{z} as required for the measured nonlinear response to be nonvanishing.
Extended Data Fig. 10 Comparison between experimental results and theoretical prediction.
a, Experimentally measured and b, theoretically predicted angular dependence of second harmonic voltage V_{2ω} with B = 18 T and T = 2 K.
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Guo, C., Putzke, C., Konyzheva, S. et al. Switchable chiral transport in chargeordered kagome metal CsV_{3}Sb_{5}. Nature 611, 461–466 (2022). https://doi.org/10.1038/s41586022051279
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DOI: https://doi.org/10.1038/s41586022051279
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