Introduction

Our recent study revealed chiral orbital currents (COC) in a colossal magnetoresistance (CMR) material, ferrimagnetic Mn3Si2Te6 (Fig. 1a–c)1. CMR is conventionally dictated by a spin polarization that drastically reduces spin scattering and thus electric resistance and is insensitive to applied electric currents. However, the CMR in Mn3Si2Te6 occurs only when a spin polarization is absent2 and is unprecedentedly current-sensitive1. The intriguing phenomena are explained in terms of a state of intra-unit-cell, ab-plane chiral orbital currents or ab-plane COC that generate net c-axis orbital magnetic moments (MCOC) which couple with the simultaneously ferrimagnetically ordered Mn spins1. In essence, the COC circulate along the edges of MnTe6 octahedra to underpin an astonishing 107-CMR that occurs without a net magnetic polarization along the magnetic hard axis (Fig. 1f)1,2,3. Note that a COC state was initially proposed and investigated in studies of high-TC cuprates, and later other materials [1, references therein].

Fig. 1: Key structural and physical properties.
figure 1

a The crystal and magnetic structure of Mn3Si2Te61. The colored circles and vertical arrows indicate the ab-plane COC and induced MCOC, respectively; different colors indicate different magnitudes of the ab-plane COC and MCOC; the green triangles denote off-ab-plane COC that are insignificant1; the faint cylindrical arrows are Mn spins. b The COC circulating in the honeycomb lattice in the ab plane1. c The canted Mn spins in the ab plane8. Schematics of the ab-plane COC at H = 0 (d) and H||c > 0 (e). f The magnetic field dependence of the a-axis magnetoresistance ratio [ρxx(H)-ρxx(0)]/ρxx(0) and the magnetization M (dashed lines, right scale) for H || c axis (thick blue curve) and H || a axis (thin red curve)2. g The magnetic field dependence of the Hall resistivity ρxy (H | |c) (thick blue curve) and ρxy (H | |b) (thin red curve) at T = 30 K and I = 2 mA; the black arrow marks a critical field HC. The insets are configurations for sample (S) measurements of ρxy (H | |c) (lower inset) and ρxy (H | |b) (upper inset).

Full size image

Ferrimagnetic Mn3Si2Te6 with trigonal symmetry (P-31c)4,5,6,7 orders at a transition temperature TC = 78 K, with the magnetic easy axis along the a axis and the magnetic hard axis along the c axis (Fig. 1f)1,2,3,4,5,6,7,8,9,10. There are two inequivalent Mn1 and Mn2 sites in the unit cell. The MnTe6 octahedra form a honeycomb sublattice of Mn1 ions in the ab plane (Fig. 1b), whereas the MnTe6 octahedra form a triangular sublattice of Mn2 ions sandwiched between the honeycomb layers (Fig. 1a, b)4.

A recent neutron diffraction study reveals a noncollinear magnetic structure with the magnetic space group C2’/c’, where the Mn spins lie predominantly within the ab plane but tilt toward the c axis by ~10 degree in ambient conditions (Fig. 1c)8, which simultaneously breaks mirror and time reversal symmetries8,9. Such a noncollinear magnetic structure is essential for the COC to form below TC [1, Methods]. The COC circulate on the edges of MnTe6 octahedra but predominantly within the ab plane (Fig. 1a, b), and therefore generate orbital moments MCOC primarily oriented along the c axis (Fig. 1a)1. (Although the orbital moments could interact with each other at H = 0, this causes no long-range order, likely due to thermal fluctuations already evident in previous studies2,8). The MCOC is estimated to be on the order of 0.1 μB1, and is coupled with the Mn spins, which yields an unusual spin-orbit effect that produces a large anisotropy field of 13 T (note that the orbital angular momentum is zero for the Mn2+ (3d5) ion with a half-filled 3d shell)1,2,4. In the absence of a magnetic field H || c axis, the net circulation of the COC is zero since it can circulate both clockwise and counterclockwise (Fig. 1d). This results in disordered circulation domains that cause strong scattering and high resistance. However, application of H || c axis favors only one direction of circulation (i.e., either clockwise or counterclockwise) and expands its domains, concurrently reducing and eventually suppressing other domains with the opposite direction of circulation (Fig. 1e). The increased size of the preferred COC domains (and concurrent decrease in domain wall volume) leads to a sharp reduction in electron scattering, and thus the 107-CMR (Fig. 1f)1. The COC as intrinsic currents are unusually susceptible to externally applied currents I that disrupt and eventually “melt” the COC state ΨC, resulting in a first-order transition to a trivial state ΨT when I exceeds a critical threshold1. (Note that ΨC refers to the COC state below TC at ambient conditions and the metallic state in the presence of H || c axis; ΨT represents the trivial state above TC and a state where the COC are destroyed by applied currents1).

The interaction between COC and I presents new, intriguing physics that needs to be understood. We have applied the Hall effect as a fundamental, powerful probe of this interaction. There are a number of diverse models that have been formulated to explain the transverse conductivity σxy of a variety of material types under various experimental conditions. The ordinary Hall effect (OHE) is attributed to the transverse emf proportional to H resulting from the Lorentz force on electrons. The OHE may be accompanied by an anomalous Hall effect (AHE) that is present in a ferromagnetic state with broken time-reversal symmetry11,12,13. The Hall resistivity ρxy is thus anticipated to be proportional to the magnetization M12,13,14,15,16,17. More recent studies indicate that an intrinsic AHE (independent of scattering) can occur in a noncollinear antiferromagnet with a strong spin-orbit interaction (SOI) so long as mirror and time reversal symmetries both are broken18,19. In other helical magnets such as MnSi, ρxy exhibits an unusual stepwise field profile that is attributed to an effective magnetic field due to chiral spin textures20. Moreover, the Berry phase13, which acts as an internal magnetic field21,22, has been proposed as a source of a topological Hall effect (THE) observed in certain topological semimetals with a strong SOI [e.g.16,17,18,19,20,21,22,23,24,25].

Here, we show a strongly-current-sensitive Hall effect in ferrimagnetic Mn3Si2Te6 that exhibits the following novel behaviors: (1) A distinct, sharp peak in the field dependence of ρxy is a sensitive function of I (Fig. 1g) (so is the carrier density n) and (2) A scaling relation σxy ∝ σxxα is obeyed with α-values ranging between 3 and 5, which are unprecedentedly large compared to α ≤ 2 typical of all solids17, and sensitively depend on I. In addition, the Hall angle (given by the ratio of the Hall conductivity σxy to the longitudinal conductivity σxx) reaches up to 0.15 which is comparable to values reported in magnets having a giant Hall effect18,19,24,26. An exceptionally large α indicates that in the COC state, σxy rises with H much faster than σxx. We argue that the c-axis orbital moments MCOC induced by the COC produce a real-space magnetic field bc that adds to an applied field H || c axis, i.e., H + bc; as such the charge carriers gain an additional transverse velocity that generates the giant, current-sensitive Hall effect. This current-sensitive Hall effect shows no simple correlation with the magnetization M, or resemblance to conventional AHE (Fig. 1f, g), as predicted by the Karplus-Luttinger theory12, nor does it behave as observed or expected in other materials18. This unique Hall effect is a clear manifestation of the existence of the COC state and an intriguing interaction between the intrinsic and extrinsic currents.

Experimental details, including measurement techniques and processes, and additional data are described in Methods and Supplementary Figs. 1–8. All data reported here are reproduced in a dozen different samples (average sample size = 1.0 × 1.0 × 0.3 mm3). Note that Joule heating is inconsequential as it is discussed in1 and confirmed by additional measurements specifically designed to investigate Joule heating in this material [Methods and Supplementary Figs. 4,5].

Results

Magnetic field dependence of current-sensitive Hall resistivity

We first focus on ρxy as a function of H || c axis at T = 30 K as an example (Fig. 2). Note that ρxy exhibits a sharp peak at a critical field HC that marks an onset of the COC state ΨC. The peak, a hallmark of the COC state, is then followed by a rapid decrease of ρxy by up to two orders of magnitude (Fig. 2a–e). (Note that I = 1 mA corresponds to a modest current density J ≈ 1 A/cm2 in the samples measured; for clarity we use I in the discussion.) The peak shifts to higher fields with increasing I, revealing a sharp switching at I = 3 mA and 4.5 mA (Fig. 2c, d) before evolving into a broader peak at I = 5 mA (Fig. 2e), which signals a vanishing COC state ΨC and an emerging trivial state ΨT. This behavior indicates that the COC weaken as I is increased, and thus stronger H || c axis are required to offset the disruption of the COC state caused by I. The correlation between I and HC at T = 30 K is illustrated in Fig. 2f. The peak at HC (an indicator for ΨC) can persist up to TC = 78 K so long as I is small (e.g., 1 mA; see Fig. 2g–i for selected temperatures T). (Note that a strong hysteresis in ρxy and ρxx is seen between H ramping up and down [see Supplementary Fig. 1], as well as in previous studies1,2, consistent with the presence of COC domains discussed above.) However, larger I exceeding a certain threshold value IC can readily suppress the COC and recover ΨT even well below TC; this happens, for example, at 30 K when I ≥ 5 mA. In these cases, the field dependence of ρxy exhibits a behavior similar to that at 100 K and I = 1 mA, which is a benchmark for ΨT that is signaled by only a broad or suppressed peak in the field dependence of ρxy (Fig. 2j).

Fig. 2: The Hall effect as functions of magnetic field and external current.
figure 2

a–e The magnetic field dependence of ρxy (H | |c) at 30 K for selected currents I; the black arrows mark the critical field HC. f The correlation between HC and I at T = 30 K; g–j The magnetic field dependence of ρxy (H | |c) at selected temperatures and currents. k The zoomed-in ρxy (H | |c) at T = 30 K (a–e) in a higher-field regime of 9-14 T. l–o The carrier density n estimated from the data of ρxy (H | |c) for selected temperatures; the yellow and gray shaded areas are the COC state ΨC and trivial states ΨT, respectively. Note that at I = 5 mA, n = − 9.9 × 1022/m3 and − 3.7 × 1023/m3 A at T = 50 and 70 K, respectively.

Full size image

The rapid decrease in ρxy at H > HC indicates a fully developed ΨC that is therefore much more conductive, i.e., ρxy (H > HC) << ρxy (HC), provided I ≤ 3 mA. ρxy (H > HC) increases slowly and linearly with H (see left scale in Fig. 2k). However, with increasing I, ΨC weakens and eventually transitions to ΨT, and accordingly, the slope of ρxy evolves from positive to negative (right scale in Fig. 2k). Remarkably, at low T and increasing I, n(I) decreases by 3 orders of magnitude with a sign change at I > IC (Fig. 2l–o) that marks a change of state. For example, at T = 50 K (70 K), n(I) is on the order of 1025/m3 (1026/m3) and remains positive in ΨC (I < 3 mA) and becomes negative, -1022/m3 (-1023/m3) in ΨT (I > 3 mA) (Fig. 2m, n). At T = 100 K > TC, n(I) stays negative (Fig. 2o). These observations allow us to conclude that the charge carriers are primarily holes in ΨC and electrons in ΨT between TC and 120 K [Methods].

In sharp contrast to ρxy for H || c axis, ρxy for H || b axis exhibits a field dependence that is more consistent with an ordinary Hall effect (e.g., Fig. 1g and Supplementary Fig. 3) with no discernible evidence of an AHE, despite the fact that the a axis is the magnetic easy axis, and the a-axis magnetization Ma is fully saturated, reaching 1.6 μB/Mn at μoH < 0.1 T (Fig. 1c)1,2. The corresponding n(I) is on the order of 1023/m3, comparable to that for ΨT with H || c axis (Fig. 2o). The contrasting Hall responses observed for H || c axis and H || a axis signal a highly anisotropic band structure10, and more generally the novelty of the observed Hall effect.

Current-sensitive scaling relation σ xy ∝ σ xx α and Hall angle

Moreover, for H || c axis and T < TC, the Hall angle, defined as the ratio of the Hall conductivity \({\sigma }_{{xy}}\left(=\frac{{\rho }_{{xy}}}{{\rho }_{{xx}}^{2}+{\rho }_{{xy}}^{2}}\right)\) to the longitudinal conductivity \({\sigma }_{{xx}}\left(=\frac{{\rho }_{{xx}}}{{\rho }_{{xx}}^{2}+{\rho }_{{xy}}^{2}}\right)\), or σxy/σxx, rises drastically when H enters a certain crossover region. As shown in Fig. 3a, b, σxy/σxx initially increases slowly with H and remains smaller than 0.01 below 9 T. However, above the crossover region marked by the gray band in Fig. 3a, b, σxy/σxx rapidly rises, reaching up to 0.15, which is among large values reported in magnets having a giant Hall effect18,19,24,26; the estimated mobility of charge carriers μ is on the order of 100 cm2/V•s [Methods], one order of magnitude smaller than those of the forementioned magnets [e.g.19,26].

Fig. 3: The Hall angle and scaling relation as a function external current.
figure 3

The magnetic field dependence of the Hall angle σxy/σxx at I = 2 mA (a) and I = 3 mA (b) for selected temperatures; the gray band marks a crossover region between ΨC and ΨT. Note that the gray area is approximately defined, and mainly serves as a guide to the eye. c–i The scaling relation σxy ∝ σxxα at T = 30 K for selected currents I. The shaded areas indicate a regime of ΨT having a lower value of the exponent α; the arrows indicate the cutoff field. j The scaling relation σxy ∝ σxx1.8 at T = 100 K and I = 1 mA. k σxx (left scale) and σxy (right scale) as a function of H || c axis at T = 30 K and I = 2 mA; the marked field 8.3 T indicates a crossover field above which σxy rises much faster than σxx. l The phase diagram of the exponent α as a function of external current I. Note that ΨC features 3 ≤ α = αHF < 5 (yellow shaded area) whereas ΨT 1 ≤ α = αLF < 2.2 (gray shaded area). Note that αc corresponds that for ΨC, and αT for ΨT.

Full size image

The large values of σxy/σxx indicates that σxy increases much faster than σxx (~ μ) with increasing H when the COC state is fully developed in higher fields ( > 6 T), giving rise to the further enhanced MCOC or bc that produces the additional transvers velocity of charge carriers. With vanishing ΨC at T > TC, σxy/σxx expectedly rises with H only slightly (Fig. 3a, b). Remarkably, near the crossover region, σxy/σxx exhibits a brief, yet prominent inverted peak at I = 2 mA when T = 70 K (Fig. 3a) and at I = 3 mA when T = 50 K (Fig. 3b). This peak persistently occurs whenever the system approaches the vicinity of the transition between ΨC and ΨT and is discussed further below.

We now examine the behavior of the scaling relation σxy ∝ σxxα for a few representative I and T. Below TC, σxy scales with σxx and generates two different values of the exponent, namely αHF (obtained at higher fields) and αLF (obtained at lower fields), which define two distinct regions corresponding to a fully developed ΨC state and ΨT or a mixed state of ΨC and ΨT, respectively. A cutoff field that separates αHF and αLF falls in the crossover region marked in Fig. 3a, 3b. (Note that the gray area is approximately defined, and mainly serves as a guide to the eye.) An unanticipated, novel feature of this scaling relation is that the exponent αHF is both unprecedentedly large and sensitive to I. Specifically, αHF reaches 5 at I = 1 mA, as shown in Fig. 3c where the data above the cutoff field (9.5 T) perfectly trace the scaling relation σxy ∝ σxx5. With increasing I, αHF reduces to 4 and 3 at I = 2 mA and 3 mA, respectively (Fig. 3e, g), suggesting that ΨC gets weakened with increasing I. Below the cutoff field (shaded regions in Fig. 3c, e, g), σxy conforms to a scaling relation where α = αLF = 2.2, 2.1 and 1.8 at I = 1 mA, 2 mA and 3 mA, respectively (Figs. 3d, f, h), indicating a vanishing ΨC and an emerging ΨT. Applying I = 5 mA suppresses ΨC and generates a linear scaling relation σxy ∝ σxx at 30 K (Fig. 3i). Similarly, a scaling relation with αLF ≤ 2 is seen at T = 70 K and I = 3 mA [Supplementary Fig. 8], and T = 100 K and I = 1 mA (Fig. 3j). Importantly, σxy and σxx exhibit distinct H dependences at higher H. For example, at I = 2 mA and T = 30 K, σxy rises rapidly and linearly whereas σxx exhibits a tendency of saturation above H > 8.3 T (Fig. 3k); therefore, the increase of σxy or [σxy(14 T) - σxy(8.3 T)]/σxy(8.3 T) = 440% above 8.3 T, but the value for σxx is merely 83%. This implies that an additional driving force strongly affects the transverse current (discussed below) and explains the unusual scaling relation with the exceptionally large α.

A phase diagram generated from the data illustrates that a hallmark of a fully developed ΨC is a strongly current-dependent scaling relation σxy ∝ σxxα with an unprecedented range of 3 ≤ α = αHF < 5 (Fig. 3l). (Note that αc corresponds that for ΨC, and aT for ΨT.) However, when ΨC is less robust or suppressed, this unique scaling relation is supplanted by another with 1 ≤ α = αLF < 2.2, which is qualitatively similar to the range 1.6 ≤ α ≤ 2 commonly observed in insulators and bad metals having strong disorder17,25.

Temperature dependence of current-sensitive Hall resistivity

We now turn to the Hall effect as a function of T. Both ρxx and ρxy peak at T = TP well above TC (marked by hollow and solid arrows in Fig. 4a, b). The peak at TP is due to the broadening of the ferrimagnetic transition2 by H (in this case, μoH||c = 7 T). Both TP and TC progressively shift to lower T with increasing I, but the temperature difference, ΔT, remains essentially unchanged, i.e., ΔT = TP - TC ≈ 32 K at 7 T. As T decreases, ρxx drops rapidly over the ΔT interval and reaches its lowest value slightly below TC (Fig. 4a). (Note that at I ≥ 5 mA, TC is suppressed, thus ΨT emerges.) On the other hand, ρxy at I = 1 mA drops more than one order of magnitude from TP to TC (blue curve in Fig. 4b). With increasing I, ρxy undergoes a rapid sign change from positive to negative to positive again over the span of ΔT. This change results in a sharp inverted peak that progressively amplifies as I increases (Fig. 4b). This peak could be a consequence of a transient crystal and/or band structure change driven by I, which could strongly affect the COC27,28,29. However, the original crystal and/or band structure can be quickly recovered via either further increasing H or decreasing T (see Fig. 3a, b and Fig. 4b).

Fig. 4: The Hall effect and resistivity as functions of temperature and external current.
figure 4

The temperature dependence of (a), ρxx and (b), ρxy at μoH || c = 7 T for selected currents I; the hollow and solid arrows mark the peak temperature TP and the Curie temperature TC, respectively. TP -TC ≈ 32 K for all I. c The schematic of the COC Hall effect.

Full size image

Discussion

While the microscopic origin of the observed Hall effect is yet to be established, we argue that the c-axis orbital moments MCOC induced by the COC play an essential role in this Hall effect. The high sensitivity of the observed Hall effect to small I suggests a very delicate nature of the COC circulating along the edges of MnTe6 octahedra. As already recognized1, application of H || c axis expands the ab-plane COC domains with one direction of circulation and concurrently shrinks the COC domains with the opposite direction of circulation (Fig. 1d, e). The expanded ab-plane COC domains in turn generate stronger c-axis MCOC, which render a magnetic field bc aligned along the c axis and proportional to H. This induced field bc couples with H || c axis to yield an enhanced effective magnetic field H + bc acting on itinerant holes in the COC state. The itinerant holes are strongly deflected by H + bc and thereby gain a significant, additional transverse velocity, which is reflected by the greatly enhanced Hall current/conductivity schematically illustrated in Fig. 4c. This scenario qualitatively explains the key observations of this study. The sharp peak at HC in the field dependence of ρxy signals the emergence of bc, and the rapid decrease in ρxy at H > HC is a consequence of an added Hall current generated by H + bc as ΨC is fully developed (Fig. 2). When higher I is applied, a stronger HC is needed (Fig. 2f) to offset the damage done to the COC in order to stabilize or further enhance the COC domains, thus bc. The unprecedentedly large α in the current-sensitive scaling relation σxy ∝ σxxα (Fig. 3) along with the large Hall angle can be ascribed to the additional transverse velocity of the holes (∝ H + bc) that drives an extraordinarily strong increase in σxy in the high-field regime where bc (∝ H) gets further strengthened; in contrast, σxx (~ μ) in this high-field regime tends to saturate (Fig. 3k). This explains that the scaling relation with 3 ≤ α = αHF < 5 is operative at much higher H only when the COC are fully established. In summary, the current-sensitive Hall effect is a novel transport phenomenon with great fundamental and technological promise, and merits extensive future investigations.

Methods

Experimental details and data processing

The Hall resistivity is measured with a Quantum Design DynaCool 14 T PPMS. A standard 4-wire configuration for the Hall coefficient measurements is adopted. The applied current is supplied by a Keithley 6220 precision current source, which is paired with a Keithley 2812 A nanovoltmeter to measure the Hall voltage using the delta mode.

The measured voltage, Vmeas, is a superposition of the Hall voltage, Vxy, and the longitudinal voltage, Vxx, due to the inevitably misaligned voltage leads (similarly the measured resistivity ρmeas is a superposition of ρxy and ρxx). To eliminate Vxx, a field H sweep is performed and Vxy is determined from the antisymmetrization of Vmeas:

$${V}_{{xy}}\left(H\right)=\frac{1}{2}\left[{V}_{{meas}}\left({{{{{\bf{H}}}}}}\right)-{V}_{{meas}}\left(-{{{{{\bf{H}}}}}}\right)\right]$$
(1)

Similarly, Vxx is determined from the symmetrization of Vmeas:

$${V}_{{xx}}\left(H\right)=\frac{1}{2}\left[{V}_{{meas}}\left({{{{{\bf{H}}}}}}\right)+{V}_{{meas}}\left(-{{{{{\bf{H}}}}}}\right)\right]$$
(2)

Because of the nature of the COC domains in Mn3Si2Te6, a hysteresis in the Hall measurements is observed between the field ramping up data and the field ramping down data (Supplementary Fig. 1). To eliminate the effect of hysteresis in the antisymmetrization of Vmeas, we process the data in the following way: The data as a function of H are grouped into 4 parts: 14 T → 0 T (positive H, |H| decreasing), 0 T → -14 T (negative H, |H| increasing), -14 T → 0 T (negative H, |H| decreasing), and 0 T → 14 T (positive H, |H| increasing). The two sets of |H| decreasing data and the two sets of |H| increasing data are regrouped and Vxy is obtained from the antisymmetrization of either group. The Hall voltage Vxy retrieved this way is proven consistent, independent of the hysteresis.

The Hall voltage Vxy is then normalized to the Hall resistivity \({\rho }_{{xy}}={V}_{{xy}}\cdot t/I\), where I is the current and t is the thickness of the sample. The longitudinal resistivity ρxx is determined in a similar way. The longitudinal conductivity σxx and the Hall conductivity σxy are thus given by \({\sigma }_{{xx}}={\rho }_{{xx}}/({\rho }_{{xx}}^{2}+{\rho }_{{xy}}^{2}){{{{{\rm{and}}}}}}\,{\sigma }_{{xy}}={\rho }_{{xy}}/({\rho }_{{xx}}^{2}+{\rho }_{{xy}}^{2})\).

Noncolinear magnetic structures

Some topological Hall effect originates from noncollinear magnetic structures18,19; however, this type of Hall effect is never found to be highly sensitive to small currents. Moreover, the non-collinear magnetic structure, which persists up to high magnetic fields applied, is also essential for the formation of the COC1. Below TC ( = 78 K), the ferrimagnetic order is observed with magnetic symmetry group C2’/c’ (No. 15.89, BNS setting)8. This symmetry group allows certain configurations of Te orbital currents circulating within the unit cell. This symmetry-preserving COC state yields currents circulating on octahedral top/bottom faces. The resulting magnetic moments are all oriented exactly along the c axis and can produce a nonzero net c-axis orbital moments MCOC, which couple to the ferrimagnetic order due to Mn ions.

Inconsequential Joule heating

Joule heating effects cause a continuous drift in local temperature. They are generally isotropic or diffusive and vary continuously with changing current. Such behavior is ruled out in the present study: The peak in the field dependence of ρxy occurs and shifts abruptly with increasing external current I (Fig. 2a–e). On the other hand, the peak in the field dependence of ρxy at a constant current remains essentially unshifted with increasing temperature, as shown in Supplementary Fig. 2. The contrasting behaviors in Fig. 2a–e and Supplementary Fig. 2 are inconsistent with self-heating effects and rule out thereof. The absence of Joule heating is also confirmed in the data of the field dependence of ρxy at H || b axis, as shown in Supplementary Fig. 3, where the field dependence of ρxy shows no significant change with increasing I.

Furthermore, we have conducted measurements to directly measure sample temperature T at different magnetic fields, using a Cernox thermometer attached to a single-crystal sample. As shown in Supplementary Fig. 4, the sample temperature is measured via the Cernox while a current is applied to the sample. We have applied the current up to 5 mA, which is the highest current used in our studies including the Hall effect study, and the magnetic field up to 14 T. Representative data are illustrated in Supplementary Fig. 5. For example, at T = 30 K, the sample temperature increase, ΔT, can be up to 6 K at I = 5 mA and H = 0; this value decreases to 3 K at 3 T and to 1-2 K at H > 3 T. Similarly, ΔT at 5 mA is approximately 6 K at 10 K, and 2 K at 50 K and 70 K, as shown in Supplementary Fig. 5.

We would like to point out that all the COC phenomena reported in1 and here occur at currents smaller than 3 mA. According to the data in Supplementary Fig. 5, the sample temperature increase is no more than 3 K or ΔT ≤ 3 K. In short, Joule heating does exist but causes no more than 6 K increase in sample temperature; its impact is therefore inconsequential for the phenomena reported here and1.

Roles of electric currents and electric field

From the data presented in Supplementary Fig. 6, 7, we can infer that the observed current-sensitive phenomena are primarily driven by electric current rather than electric field.

Our dielectric measurements using a Quad-Tech LCR meter reveal a large dissipation factor, DF, even at low temperatures, as shown in Supplementary Fig. 6. The large DF indicates a large leakage current, which therefore prevents a robust electric field from being established in this material, particularly at higher temperatures, say, 30 K at which most of our data is collected.

We have measured a dozen samples with varying sample size for this study. Compiling these data, we have found a converging behavior in these samples: The critical magnetic field Hc (defined by the peak that occurs in ρxy (H) (Fig. 2)) essentially follows the same current-density J dependence, as shown in Supplementary Fig. 7, which contains data obtained from four different samples. This strongly indicates that the Hall response depends on applied currents.

Additional data for scaling relation

A scaling relation with αLF ≤ 2 is observed at T = 70 K and I = 3 mA, as shown in Supplementary Fig. 8 This behavior provides additional evidence that an external current exceeding a critical current IC can suppresses the COC state below TC ( = 78 K).

Additional notes on the charge carriers

The charge carriers are primarily holes in ΨC and electrons in ΨT between TC and 120 K. Above 120 K, the charge carriers become holes again, as indicated by this and our previous studies2.

Estimate of mobility of charge carriers

$${\sigma }_{{xx}}={e}^{2}{\tau }n/m \sim {10}^{4}{\left(\Omega {{{{{\rm{m}}}}}}\right)}^{-1},\,{{{{{\rm{and}}}}}}\,{n} \sim {10}^{25}/{{{{{{\rm{m}}}}}}}^{3}\,{{{{{\rm{for}}}}}}\,{I}\le 3\,{{{{{\rm{mA}}}}}}$$
$${\mu }=e{\tau }/m={\sigma }_{{xx}}/{en} \sim {10}^{-2}\,{{{{{{\rm{m}}}}}}}^{2}/{{{{{\rm{Vs}}}}}}=100\,{{{{{{\rm{cm}}}}}}}^{2}/{{{{{\rm{Vs}}}}}}$$