Giant gate-controlled odd-parity magnetoresistance in one-dimensional channels with a magnetic proximity effect

According to Onsager’s principle, electrical resistance R of general conductors behaves as an even function of external magnetic field B. Only in special circumstances, which involve time reversal symmetry (TRS) broken by ferromagnetism, the odd component of R against B is observed. This unusual phenomenon, called odd-parity magnetoresistance (OMR), was hitherto subtle (< 2%) and hard to control by external means. Here, we report a giant OMR as large as 27% in edge transport channels of an InAs quantum well, which is magnetized by a proximity effect from an underlying ferromagnetic semiconductor (Ga,Fe)Sb layer. Combining experimental results and theoretical analysis using the linearized Boltzmann’s equation, we found that simultaneous breaking of both the TRS by the magnetic proximity effect (MPE) and spatial inversion symmetry (SIS) in the one-dimensional (1D) InAs edge channels is the origin of this giant OMR. We also demonstrated the ability to turn on and off the OMR using electrical gating of either TRS or SIS in the edge channels. These findings provide a deep insight into the 1D semiconducting system with a strong magnetic coupling.

Investigation of new magnetoresistance (MR) phenomena is an important issue in condensed matter physics, magnetism, and spintronics. For example, the discovery of giant MR 1,2 and tunneling MR 3,4 paved the way to the creation of non-volatile storage and memory devices. Generally, these MRs are even functions of external magnetic field B according to Onsager's principle 5 . However, it may not be the case when time reversal symmetry (TRS) is broken by magnetism in the system. The odd-parity MR (OMR) in a linear-response regime has been observed in systems where TRS is violated [6][7][8][9] . (See also Supplementary  Table 1). To explain these OMR phenomena, various possible origins were proposed, including non-trivial Berry curvature, magnetic moments, side jump mechanism 10 , and coexistence of spin-orbit interaction (SOI) and ferromagnetic coupling in a helical magnet 11 . Even in such rare systems, the OMR magnitude is typically very subtle (the magnitude reported thus far is at most 2%). In addition, these systems reported thus far are metallic, which hinders the control of OMR by external means such as electrical gate voltage.
In this Article, we report a giant and gate-controlled OMR in the edge transport channels of an InAs thin film interfaced with a ferromagnetic semiconductor (FMS) (Ga,Fe)Sb [12][13][14] layer (see Fig. 1a). The OMR is found to be unprecedently large; the resistance change is 27% of the total resistance when the B direction is reversed between ±10 T at I = 1 μA. This is striking, considering that the SOI of InAs is much smaller than other materials such as SmCo 5 and pyrochlores in which OMR was observed. We argue that this originates from the simultaneous breaking of both TRS and spatial inversion symmetry (SIS), which is entwined with both a strong magnetic proximity effect (MPE) from the underlying (Ga,Fe)Sb 15 and a Rashba SOI effect at the InAs edges. Using field-effect transistor structures, we demonstrate electrical control of the OMR by individually tuning the TRS or SIS in the system. The unprecedented strong OMR with gate controllability in mainstream semiconductors such as InAs is ideal not only for elucidating the crucial roles of the TRS and SIS breakings in solid-state physics but also for providing pathways to electronic device applications.

Results
Magnetoresistance and its current dependence in InAs/ (Ga,Fe)Sb The structure examined in this study consists of, from top to bottom, InAs (thickness 15 nm)/(Ga 1−x ,Fe x )Sb (Fe content x = 20%, 15 nm)/AlSb (300 nm)/AlAs (15 nm)/GaAs (100 nm) on semi-insulating GaAs (001) substrates grown by molecular beam epitaxy (See Fig. 1a). We utilize two samples A and B with the same heterostructure in this study (see Methods in detail). In this structure, the InAs layer is a nonmagnetic quantum well (QW) that is responsible for over 99% of the electron transport because all the other layers underneath are highly resistive 15 . (Ga,Fe)Sb is an FMS with a high Curie temperature over 300 K [12][13][14] . The preparation and characterization of the samples are explained in ref. 15. Due to the high crystal quality and staggered band profile at the InAs/(Ga,Fe)Sb interface, in which the conduction band bottom of InAs is at lower energy than the valence band top of (Ga,Fe)Sb, the electron wavefunction in the InAs QW significantly penetrates into the ferromagnetic (Ga,Fe)Sb layer. This induces a large MPE and spindependent scattering in the nonmagnetic InAs electron channel 15 .
We pattern the InAs/(Ga,Fe)Sb bilayers into 100 × 600 μm 2 Hall bars with electrodes labeled "1" to "6", as shown in Fig. 1b. We drive a DC current I from "1" to "4" and measure the voltage differences V ij = |V i − V j | (i, j = 1, 2, 3, 4, 5, 6), from which we obtain the resistances R ij = V ij /I. A magnetic field B is applied perpendicular to the film plane (B//z). As shown in Fig. 1c, d, the B dependence of the four-terminal resistance R 23 measured at I = 1 μA shows (i) a very large odd-function MR, (ii) a large negative MR, and (iii) Shubnikov-de Haas (SdH) oscillations. The last two phenomena ((ii) and (iii)), which are even functions of B, are characteristics of the two-dimensional (2D) electron transport with an MPE in the InAs thin film, as thoroughly discussed in our previous work 15 . Also for (iii), the angular dependence of B also reveals that the SdH oscillations originate from the 2D transport (See Supplementary Fig. 1). In contrast, the large odd-function component, extracted as R 23 odd shows a linear dependence on B with SdH oscillations (see Supplementary Note 1) over the full range of magnetic field (|B | < 10 T) and persists up to 300 K (lower panel of Fig. 1d). R 23 odd (B) is 2.0 kΩ at B = 10 T and 2.5 K, corresponding to 13.5% of the total resistance, and the resistance R 23 (B) is changed by 27% of the zero-field resistance R 23 (0) upon reversing B from 10 T to −10 T. This is the largest OMR observed thus far. The OMR magnitude remains almost constant in the whole range of 240 nA <I < 100 μA, drops suddenly to one-third of its magnitude at I C =~200 nA, then remains at this low magnitude when I is decreased further to the lower measurable limit at 50 nA (See Fig. 2a and Supplementary Fig. 3). Even when we reverse the current direction, the OMR remains unchanged ( Supplementary Fig. 4). These features indicate that the OMR presented here occurs in a linear transport regime. The reason for the sudden drop at I C is discussed in Supplementary Note 2. Also, we find that the OMR magnitude depends on the crystallographic orientation, which may be due to the non-uniformity of Fe atoms in (Ga,Fe)Sb (See Supplementary Note 3).
One-dimensional (1D) transport in InAs and the origin of OMR An important observation, obtained by comparing R 23 and R 65 in Fig. 2b, is that the sign of the OMR flips when we switch the voltage terminals contacting the side edge while maintaining the same measurement setup. Given that B and I are fixed in the same directions, this observation suggests that the OMR originates from the electrical transport along the side edges of the InAs thin film, where the SIS is broken by the opposite polarities, as discussed in the next paragraph. This argument is further supported by the disappearance of the OMR in our two-terminal resistance measured between electrodes 1 and 4 (R 14 ), where the positive and negative OMR components from the two side edges of the InAs thin film exactly cancel (see Fig. 2c and Supplementary Note 4). We note that, however, a large OMR was observed when we measured the resistance only along one edge by the twoterminal method (see Supplementary Note 5).
Two types of edge transport are known to occur in InAs/GaSb bilayers. One involves a non-trivial quantum spin Hall edge state 16-18 , which is formed at the edge of the InAs/GaSb interface when a   topological gap is opened due to the inverted band structure (the valence band top of GaSb is at higher energy than the conduction band bottom of InAs) and SOI. However, because this topological gap is very small (~4 meV), the non-trivial edge state cannot survive at high temperatures, which contradicts our observation of the OMR up to room temperature. The other involves a trivial edge state formed at the edge of the InAs layer due to the pinning of the Fermi level at the top and side vacuum surfaces, which is located as high as 0.1−0.3 eV above the conduction band bottom [19][20][21][22][23][24] . As a result, the conduction band potential of InAs is strongly bent downward at the surfaces, which we confirmed using Kelvin force microscopy measurements (See Methods and Supplementary Fig. 10). The effect is two-fold: First, the electron carriers accumulate more at the edges than in the center of the InAs film; thus, two 1D edge channels and one 2D transport channel coexist. This fact is confirmed by the transport measurements on devices with different sizes, which is discussed in Supplementary Note 6. Second, the SIS is broken at the side edges due to the resulting built-in electric field. Since we define the directions of I and B in our measurements as the x and z directions, respectively, as shown in Fig. 1a, the built-in electric field E sur pushes the electron carriers outward along the y direction. The directions of E sur in the two edge channels are opposite, which explains the opposite signs of the OMRs in R 23 and R 65 . As shown in Fig. 2d, the OMR almost disappears when we apply B parallel to the current I direction (the x-axis) or the E sur direction (the y-axis) (see Supplementary Fig. 12). This indicates that OMR can only be induced when B, I, and E sur are mutually orthogonal. This is also because the MPE from (Ga,Fe)Sb, which breaks the TRS in InAs, is only effectively induced by the z-component of the magnetization of (Ga,Fe)Sb 15 .

Control of SIS and TRS breaking via gate voltage
To examine our scenario, we apply an electrical gate voltage to individually tune the TRS and SIS breakings in the edge transport of InAs and evaluate their impacts on the OMR. We fabricated two field-effect transistor devices, D1 and D2; one (D1) has a single gate electrode, G, that controls the whole InAs Hall-bar (Fig. 3a), and the other (D2) has two separate gate electrodes, G 1 and G 2 , that control the conduction of each edge independently (Fig. 3b). In device D1, a negative (positive) voltage applied to G push the electron wavefunctions in InAs towards the (Ga,Fe)Sb (top surface) side, which effectively enhances (suppresses) the MPE 15 . As shown in Fig. 3c 3.05 negative gate voltage V g from 0 to −5 V on G, with which the MPE is enhanced, the OMR intensity strongly increases by more than threefolds (2.5 to 8%, respectively). Meanwhile, when applying a positive V g from 0 to 5 V on G, which effectively suppresses the MPE, the OMR intensity decreases and almost vanishes at V g = −5 V. These results clearly demonstrate the important role of TRS breaking in inducing the OMR. This fact is also confirmed by the small OMR magnitude (=1.8% at 14 T) in an InAs/GaSb reference sample, where there is no FM coupling, as shown in Supplementary Fig. 13. On the other hand, in device D2, by applying a voltage in one of these two gates (for example, G1), we modulate the band profile in one edge of the InAs layer (the edge along terminals 2 and 3). This enhances the OMR in one edge more than another and results in an appearance of OMR even in the magnetoresistance measured between terminals 1 and 4. Figure 3d shows the magnetoresistance characteristics measured between terminals 1 and 4 when we applied V g1 = 7 and −7 V on G1. One can see that a negative (positive) OMR is induced at V g1 = 7 V (−7 V), as expected. This can be understood because a positive (negative) V g1 enhances (suppresses) the E sur of the right edge relative to that of the left edge. Therefore, the important role of SIS breaking at the edge channels is clearly demonstrated by these results.

Theoretical analysis
Finally, we discuss the theoretical model to explain the OMR in InAs/ (Ga,Fe)Sb. If we temporarily neglect the MPE from the (Ga,Fe)Sb layer, the Hamiltonian of the 1D edge channel of InAs can be described as where k x is the wavenumber along the x-direction, m* is the effective mass of electrons, Λ top(side) (= ħλ top(side) ) is the effective Rashba SOI due to the built-in potential at the top (side edge) surface, ħ is the Dirac's constant, Δ z (=gμ B B z ) is the Zeeman splitting due to an applied magnetic field along the z-axis (B z ), σ i (i = x, y, z) are the elements of the Pauli matrix that acts on the electron spin degree of freedom, and σ 0 is the identity matrix. The energy dispersion from Eq. (1) can be described as where s = +/− denotes the upper and lower bands E + and E − , as depicted in Fig. 4a, respectively. Here we define the energy band bottom of E − as E = 0. It is important to note that due to the Rashba SOI (Λ top and Λ side ), the spin components σ y and σ z are locked to the momentum k x in opposite directions between the bands E + and E − . Thus the + andsubscripts also indicate the difference in "chirality" of these bands, which are shown as green and pink lines, respectively, in the right-side graph of Fig. 4a. We solve Boltzmann's equations and obtain the electrical conductivity σ xx by summing the conductivities of all the bands that cross the Fermi level (E F ) (see Supplementary Note 7), where e is the elementary charge, τ s is the relaxation time, h is Planck's constant, and E F is the Fermi energy. Reflecting the breaking of the SIS at the side surface edges, we assume Λ top ≪ Λ side , which indicates that the electric field at the side edges is much larger than that at the top surface 25 . From Eq. (3) and Fig. 4a, the odd-order B zdependent conductivity can be non-zero in the case that E F crosses only the lower band shown in region (II) of Fig. 4a. However, this case is unlikely because the gap Δ g (B) is only 24 and 44 meV at B = 0 and 14 T, respectively, obtained by using the parameters of an InAs nanowire of m*/m 0 = 0.08 26 , g = 18 27 , m*λ 2 side = 0.45 meV 28 , and m*λ 2 top = 0.027 meV 28 . Due to the Fermi level pinning at the edge surface, E F in the edge channel lies in the region (I) of Fig. 4a, where the odd-order B z -dependent conductivities from the upper and lower bands cancel each other out, and thus, no OMR should be expected.
However, the OMR can be induced if the relaxation times in the E + and E − bands are different (τ + ≠ τ − ), which results from the MPE and the Rashba SOI in the 2D and 1D channels of InAs as explained in the following. Our analysis of the transport data (see Supplementary Notes 2, 6) indicates that the MPE mainly affects the 2D channel, inducing a splitting energy gap Δ 2D between 2D bands of opposite σ z components (indicated by purple arrows in Fig. 4b). Therefore, the MPE affects the 1D channel only indirectly via electron scattering between the 1D and 2D channels. Considering that σ y is locked to k x because of Λ top in both the 1D and 2D channels, the lower and upper bands in each channel (1D and 2D) have different chiralities, as indicated by the green and pink colors in Fig. 4b. The relaxation time of E + and E − (τ + and τ − , respectively) in the 1D channel thus are mainly determined by scattering events between bands with the same chirality (blue and red arrows in Fig. 4b). The difference in the density of states (DOS) at the Fermi level of the two 2D bands (pink and green bands in Fig. 4b)  and E − in the 1D edge channel, and different values of τ + and τ − (see Supplementary Note 8 and 9 for detailed discussions). Consequently, the linear-response conductivity σ xx is rewritten as Here, we set the phenomenological parameter α as τ − = ατ + to express the different relaxation times of electron carriers in the E + and E − states. Under the influence of the strong MPE and chirality-dependent scattering at the interface (α ≪ 1), the linear B z -dependent MR appears in the conductivity σ xx due to the contribution of the last term in the brackets of Eq. (4). Using α = 0.1, E F = 100 meV, m*λ 2 side = 0.45 meV 28 , and Δ z /B z = 0.52 meV/T for g = 18 27 , the OMR is clearly reproduced by Eq. (4), as shown in Fig. 4c. The different signs of R 23 odd and R 65 odd shown in Fig. 2b are explained by the different signs of the Rashba parameter λ side (blue and red lines) between the two side edges. The dependences of the OMR ratio ΔR/R 0 on α and E F are shown in Fig. 4d and e, respectively. A large difference in the relaxation time of the spin channels, which means a small α, produces a large OMR ratio. This indicates the important role of MPE at the InAs/(Ga,Fe)Sb interface in inducing the large OMR. This conclusion is also supported by the fact that the OMR magnitude ΔR/R 0 , where ΔR = R 23 odd (10 T) and R 0 = R 23 (0 T), is enhanced with decreasing temperature T as ln(1/T) (see the inset of Fig. 1d). This behavior is characteristic of the Kondo-effect-related transport coming from the spin-dependent scattering at the InAs/ (Ga,Fe)Sb interface. Another important result is that a smaller E F leads to a larger OMR. If we set E F at approximately 24 meV, which is the same as Δ g (0 T), then Eq. (4) can reproduce the experimental value (ΔR/R 0 = 13.5%), as shown in Fig. 4e.
In conclusion, we found the giant odd-parity magnetoresistance in the 1D edge channels of the InAs/(Ga,Fe)Sb heterostructure, and demonstrated the ability to electrically turn on and off the effect using field-effect transistor structures. Our results highlight the abundance of new physics in solid-state systems when TRS and SIS are simultaneously broken, even in well-known materials such as InAs. The linear OMR presented in this work can be applied to magnetic field sensors, which provide a large dynamic range (0−10 T) owing to its linearity. This new type of sensor can work at room temperature, requires only   27 , m*λ 2 side = 0.45 meV 28 , and m*λ 2 top = 0.027 meV 28 . In the case of B = 14 T, E + and E − can be labeled by chirality (pink and green lines, respectively) which is determined by the SIS breaking in the z-direction. Two different regions (I) and (II) can be observed, defined by whether the Fermi energy E F crosses only one or two dispersion branches E + and E − . Δ g is the energy gap between the minima of E + and E − . b Schematic energy dispersion (upper) and its Fermi surface (lower) of the 1D (right) and 2D (left) channels. The purple dashed line indicates the Fermi level E F . Due to the Rashba SOI in the z direction in both channels, the y spin component (σ y ) of electrons is locked to the momentum k x in opposite directions between the green and pink bands. Electron scattering between the 1D and 2D channels occurs mainly between bands with the same chirality, as indicated by the red and blue arrows. In the 2D channel, MPE opens the gap (Δ 2D ) between two bands with opposite z spin components (σ z ). Different density of states between the two (pink and green) 2D bands leads to different relaxation times τ + and τ − in the 1D channel. c Calculated results of the OMR using Eq. (4) with α = 0.1, E F = 100 meV, m*λ 2 side = 0.45 meV 28 , and Δ z /B z = 0.52 meV/T for g = 18 27 . The sign of the Rashba parameter λ side determines the polarity of the OMR component in the 1D system. d, e OMR as functions of α (with E F = 100 meV) and E F (with α = 0.1), respectively. α (= τ − /τ + ) represents the different relaxation times of electron carriers in the E − and E + states; α is small in the case of a strong MPE. A strong MPE and a small α lead to a large OMR. simple DC measurements for detection, and its sensitivity can be further enhanced by material engineering, such as optimizing the carrier concentration and SOI strength.

Sample preparation and characterization
We grew heterostructures consisting of InAs (thickness 15 nm)/ (Ga,Fe)Sb (15 nm, Fe 20%, T C > 300 K)/AlSb (300 nm)/AlAs (15 nm)/ GaAs (100 nm) on semi-insulating GaAs (001) substrates by molecular beam epitaxy (MBE). The growth temperature (T S ) was 550°C for the GaAs and AlAs layers, 470°C for the AlSb layer, 250°C for the (Ga,Fe)Sb layer, and 235°C for the InAs layer. We also grew a nonmagnetic InAs/GaSb heterostructure as a reference, whose structure is the same as the sample mentioned above, except for the lack of Fe doping. The top two layers (InAs and GaSb) of this sample were grown at 470°C, while the other layers were grown under the same conditions as the Fe-doped samples. The in situ reflection high energy electron diffraction (RHEED) patterns of InAs and (Ga,Fe)Sb are bright and streaky, indicating good crystal quality and a smooth surface (see Supplementary Fig. S2b in ref. 15). In this paper, we used two different samples A and B of InAs/(Ga,Fe)Sb heterostructures with sheet carrier concentrations of 2.0 × 10 12 cm −2 and 1.8 × 10 12 cm −2 , and electron mobilities of 9.4 × 10 2 cm 2 /Vs and 1.9 × 10 3 cm 2 /Vs, respectively. Also, the quantum mobility of sample A is estimated to be 2070 cm 2 /Vs from the SdH oscillations (See Supplementary Fig. 14).
The mobility difference between samples A and B suggests that the static electric fields, which determine the confinement potential in the 1D and 2D channels of InAs, in the two samples are different. The confinement potential sensitively affects the strength of both the proximity magnetoresistance 15 and the Rashba SOI in the InAs channel, and consequently yields different OMR in these samples A and B. We note that the different confinement potentials may originate from different surface pinning effects at the top and side surfaces of the devices, which depend on the detailed conditions during device fabrication.

Fabrication process of the Hall bar devices and transport measurement
The samples were patterned into 100 × 600 μm 2 Hall bars using standard photolithography and Ar ion milling down to the AlSb buffer layer. The etched surface was passivated by depositing a thin SiO 2 layer. Then electrodes were formed by electron-beam evaporation and lift-off of Au (50 nm)/Cr (5 nm) films. Figure 1b shows an optical microscopy image of the Hall bar device examined in Figs. 1, 2, and 3c, d. For the field-effect transistor (FET) devices in Fig. 3c, d, we deposited a 50-nm-thick Al 2 O 3 layer as a gate insulator by atomic layer deposition. Figure 3a, b show optical microscopy images of the Hall bar FET device examined in Fig. 3c (D1) and d (D2), respectively. Magnetotransport measurements were conducted using a Quantum Design physical property measurement system (PPMS) by a standard 4-terminal method, except for R 14 , which was measured by a two-terminal method. We use a DC current for I > 1μA, and an AC current with a lock-in amplifier (lock-in frequency is 5261 Hz) for lower I. Supplementary Fig. 10a shows the atomic force microscope (AFM) images of the InAs/(Ga,Fe)Sb sample before (left) and after (right) etching using Ar ion milling, similar to the fabrication process of the field-effect transistors in our work. The root mean square (RMS) of the roughness is only 0.4 nm, which is less than two monolayers of InAs and remains almost unchanged after the etching. Also, no apparent etch pit is observed in these AFM images. This result indicates that our films are homogeneous and smooth, which is not affected by the device fabrication process.
Work function measurements by Kelvin probe force microscopy (KFM) We investigated distribution of the surface potential on the InAs/ (Ga,Fe)Sb by KFM in vacuum conditions (~10 −5 Pa) at room temperature. In KFM, an AC bias at frequency f (=1 kHz in our case) and a DC bias are applied between the tip and the sample under noncontact operation in atomic force microscopy (AFM) (see Supplementary  Fig. 10b and c in Supplementary Information). When the tip approaches the sample surface in the z-direction, the electric bias induces an electrostatic force F expressed as where C is the capacitance between the tip and the sample, V dc is the DC bias voltage, V ac is the AC voltage magnitude, and 4ϕ is the work function difference between the tip and the sample. Similar to AFM measurements, the force F is deduced from the shift of the cantilever oscillation frequency. V dc is adjusted using a feedback control so that the f-frequency component in F, which is measured using a lock-in amplifier, is nullified. Then V dc gives the value of Δϕ=e according to Eq. (5). Therefore, we can obtain 4ϕ and consequently, the work function distribution on the sample. We note that the potential profile at the topmost InAs surface detected by KFM might be different from the one at several-atomiclayer depth below the surface. This is because of a screening effect from a large amount of charged surface states (top and side surfaces), which are common at InAs surfaces. Thus, the potential profile change along the y direction measured by the KFM tip is much milder than the real confinement potential at the edge of the InAs channel 29 . As a result, the potential profile at the top surface shown in Supplementary  Fig. 10c might largely exaggerate the width of the triangular potential at the bulk InAs side surface, which should be much less than 2 µm. Therefore, we consider that the static electric field in the y and z directions should have the same order of magnitude.