---
title: Odd Parity Magnetoresistance
url: https://www.emergentmind.com/topics/odd-parity-magnetoresistance-omr
type: topic
---

# Odd Parity Magnetoresistance

Searching arXiv for papers on odd-parity magnetoresistance and closely related odd-parity magnetoconductivity to ground the article in the cited literature.
Odd parity magnetoresistance (OMR) is a longitudinal magnetotransport response in which the resistance or resistivity acquires a component that is antisymmetric under magnetic-field reversal. In conventional magnetoresistance, Onsager reciprocity enforces an even-in-field response when time-reversal symmetry (TRS) is unbroken; OMR appears only when TRS is broken, whether by intrinsic magnetization, a field-selected magnetic domain, or magnetic proximity. Across the literature, OMR has evolved from a symmetry-allowed odd-in-$B$ correction in pyrochlore iridates and ferromagnets to a gate-tunable response in proximitized semiconductor channels and magnetized bilayer graphene, culminating in a proximity-induced topological heterostructure, $\alpha$-Sn/(In,Fe)Sb, where the odd component reaches $1{,}150\%$ at $1\,\mathrm{T}$ [1508.01318] [2003.11731] [2407.14071] [2507.23166].

## 1. Definition and reciprocity

Magnetoresistance is commonly written as
$$
MR(B)=\frac{R(B)-R(0)}{R(0)}.
$$
With unbroken TRS, Onsager reciprocity requires $R(B)=R(-B)$, so $MR(B)$ is an even function of $B$. The odd component is extracted as
$$
MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},
$$
or, equivalently in resistivity notation,
$$
\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.
$$
By construction, $MR_{\mathrm{odd}}(-B)=-MR_{\mathrm{odd}}(B)$ and $\rho_{xx}^{\mathrm{odd}}(-B)=-\rho_{xx}^{\mathrm{odd}}(B)$, whereas the conventional magnetoresistive contribution is the symmetric part [1508.01318] [2407.14071] [2507.23166].

A more general reciprocity statement in magnetic matter is
$$
\sigma_{ab}(B,M)=\sigma_{ba}(-B,-M),
$$
so an odd longitudinal response is allowed only when the material possesses an intrinsic order parameter $M$ that is itself odd under TRS. This distinguishes OMR from ordinary even-parity magnetoresistance and from the anomalous Hall effect (AHE): OMR is diagonal in the transport tensor and odd in the applied field, whereas the AHE is off-diagonal [1904.00330] [2509.13277].

In several experimentally relevant models, TRS breaking is necessary but not always sufficient. In one-dimensional InAs edge channels magnetized by a proximity effect from (Ga,Fe)Sb, simultaneous breaking of TRS and spatial inversion symmetry (SIS) is required to prevent cancellation of the odd contribution in the Boltzmann transport integrals [2003.11731].

## 2. Symmetry structure and tensor formulations

In ferromagnets with a fixed intrinsic magnetization $\mathbf{M}$, the resistivity tensor can be expanded to linear order in magnetic field as
$$
\rho_{ij}(\mathbf{H},\mathbf{M})
= \rho_{0}\delta_{ij}
+ \rho_{A}\epsilon_{ijk}M_k
+ \rho_M M_i M_j
+ \rho_1(\mathbf{M}\!\cdot\!\mathbf{H})\delta_{ij}
+ \rho_2\epsilon_{ijk}H_k
+ \rho_3\epsilon_{ikl}\epsilon_{jmn}M_l H_n
+ \rho_4(M_i H_j+H_i M_j)+\dots.
$$
Within this structure, the leading odd-parity longitudinal terms arise from $\rho_1(\mathbf{M}\!\cdot\!\mathbf{H})\delta_{ij}$ and from the $\rho_3$ contribution associated with $(\mathbf{M}\times\mathbf{E})\times\mathbf{H}$. In the canonical geometry with $\mathbf{E}\parallel \hat x$, $\mathbf{M}\parallel \hat z$, and $\mathbf{H}$ rotated in the $x$–$z$ plane, the observed longitudinal and transverse odd responses take the forms
$$
\Delta \rho_{xx}(\theta)=\lambda M H \cos\theta,\qquad
\rho_{xz}(\theta)=\eta M H \sin\theta,
$$
linking OMR and the odd-parity planar Hall effect within a common tensorial framework [1904.00330].

In all-in/all-out antiferromagnets, the symmetry description is naturally expressed through a rank-3 tensor $\chi_{ijk}$ coupling $E_j B_k$ to $J_i$:
$$
J_i=\dots+\chi_{ijk}E_j B_k+\dots.
$$
For Eu$_2$Ir$_2$O$_7$ thin films, this permits an odd-in-$B$ longitudinal contribution once the all-in/all-out order breaks TRS while preserving inversion and cubic symmetries on average. Projected onto the longitudinal channel, the result is $\Delta \rho_{xx}\propto B$, consistent with a linear odd magnetoresistance below the magnetic transition [1508.01318].

A later general symmetry analysis formulated the same issue in terms of odd-parity magnetoconductivity (OMC). In that treatment, the conductivity tensor is expanded as
$$
\sigma_{ij}(B)=\sigma^{(0)}_{ij}+\sigma^{(\mathrm{odd})}_{ij}B+\sigma^{(\mathrm{even})}_{ij}B^2+\cdots.
$$
Longitudinal OMC follows the same point-group constraints as the anomalous Hall conductivity, whereas symmetric transverse OMC obeys distinct rules. This places OMR within a broader band-geometric transport classification rather than treating it only as a resistive anomaly [2509.13277].

## 3. Experimental platforms and phenomenology

OMR has been realized in several distinct material classes, with widely different magnitudes, field scales, and control parameters. Representative cases are summarized below [1508.01318] [1904.00330] [2003.11731] [2407.14071] [2507.23166].

| Platform | Control or geometry | Reported odd response |
|---|---|---|
| Eu$_2$Ir$_2$O$_7$ thin films | Cooling-field selection of all-in/all-out domains | $0.4\%$ at $9\,\mathrm{T}$ |
| SmCo$_5$ | Fixed intrinsic magnetization | $0.05\%$ at $0.5\,\mathrm{T}$ |
| InAs/(Ga,Fe)Sb | Gate control of magnetic proximity and edge asymmetry | $13.5\%$ at $10\,\mathrm{T}$; $27\%$ upon full field reversal |
| Magnetized bilayer graphene | Electrostatic tuning near band edges | $\sim 40\%$ at $14\,\mathrm{T}$ |
| $\alpha$-Sn/(In,Fe)Sb | In-plane $B\parallel I$ in a proximitized ultrathin film | $1{,}150\%$ at $1\,\mathrm{T}$, $2\,\mathrm{K}$ |

The Eu$_2$Ir$_2$O$_7$ case established a domain-controlled odd linear term: after zero-field cooling, $\rho_{xx}(B)$ is symmetric; after cooling in $\pm 9\,\mathrm{T}$, a clear asymmetric term appears, $\rho_{xx}^{\mathrm{asym}}(B)\simeq \alpha B$, with the sign of $\alpha$ determined by the A- or B-domain selected during cooling. The coefficient saturates once $|B_{\mathrm{cool}}|\gtrsim 3\,\mathrm{T}$ and vanishes above $T_M\approx 105\,\mathrm{K}$, confirming the direct relation to all-in/all-out order [1508.01318].

In InAs/(Ga,Fe)Sb, the odd component originates in one-dimensional edge channels rather than the bulk two-dimensional channel. Four-terminal measurements show a large $R_{\mathrm{odd}}(B)$ linear up to $\pm 10\,\mathrm{T}$, persisting to room temperature. Opposite edges yield equal-magnitude OMR of opposite sign, while a two-terminal geometry across the full width produces nearly zero $R_{\mathrm{odd}}$ because the two edge contributions cancel. Gate voltages can turn the effect on and off, and can reverse the sign of the uncompensated response [2003.11731].

In bilayer graphene proximitized by Cr$_2$Te$_2$Ge$_6$, the longitudinal antisymmetric response is absent in a BLG/hBN control device but substantial in BLG/CGT. The odd part is linear in $B$ up to quantum oscillations, peaks near the band edges, is reduced at higher carrier density, remains essentially unchanged up to $T\approx 250\,\mathrm{K}$, and vanishes near $T_c\approx 360\,\mathrm{K}$ with a fitted critical form $\alpha(T)\simeq \alpha_0(1-T/T_c)^\beta$, $\beta\approx 0.20$ [2407.14071].

## 4. Microscopic mechanisms

Two microscopic mechanisms were isolated in ferromagnetic systems with pinned magnetization. One is spin-polarization-dependent scattering on a Zeeman-shifted Fermi surface: unequal relaxation times for the two spin populations imply that a small field-induced change in spin polarization generates a longitudinal term proportional to $(\mathbf{M}\!\cdot\!\mathbf{H})$. The other is an anomalous-velocity mechanism usually associated with Berry-curvature physics. Starting from
$$
\frac{m}{\tau}\,v = eE + \frac{e}{c}\,v\times H + a\,M\times E,
$$
iteration to first order in $H$ produces a term proportional to $(\mathbf{M}\times \mathbf{E})\times \mathbf{H}$, which contributes both to odd longitudinal magnetoresistance and to the odd planar Hall effect [1904.00330].

In one-dimensional InAs edge channels, the effective Hamiltonian is
$$
H_{\mathrm{id}}(k_x)=\frac{\hbar^2 k_x^2}{2m^*}\,1 + (A_{\mathrm{side}}k_x+\Delta_z)\sigma_z + A_{\mathrm{top}}k_x \sigma_y.
$$
Here $A_{\mathrm{side}}$ and $A_{\mathrm{top}}$ encode Rashba terms from side-wall and top-surface electric fields, while $\Delta_z=g\mu_B B_z$ is the Zeeman term. Under the assumption $A_{\mathrm{top}}\ll A_{\mathrm{side}}$, solving the linearized Boltzmann equation with distinct relaxation times $t_+\neq t_-$ for the two Rashba branches yields an odd term in $\sigma_{xx}$ proportional to $(1-a)\Delta_z$, with $a=t_+/t_-$. In this framework, giant OMR requires the simultaneous presence of magnetic proximity, edge-induced inversion breaking, and asymmetric scattering between the Rashba branches [2003.11731].

Berry curvature and orbital magnetic moment provide a more general band-geometric description. In magnetized bilayer graphene, the leading odd contribution to the longitudinal conductivity is
$$
\sigma_{xx}^{\mathrm{odd}}(B)
= \frac{e^3\tau}{\hbar}\sum_n \int \frac{d^2k}{(2\pi)^2}
\,[\Omega_{n,z}(k)\,m_{n,z}(k)]\,(-\partial_\varepsilon f_0)\,B+\dots,
$$
so the response is finite only when exchange coupling breaks TRS and generates net $\Omega_z$ and $m_z$ [2407.14071]. The later general OMC theory extends this result by showing that $\sigma_{ab}^{(\mathrm{odd})}$ contains three distinct band-geometric contributions: terms involving $v_a^m v_b^0+v_a^0 v_b^m$, a term containing $\partial_\varepsilon(m_z \partial_\varepsilon f^0)$, and a Berry-curvature term proportional to $\Omega_z v_a^0 v_b^0$. Inverting conductivity to resistivity gives
$$
\rho_{xx}^{(\mathrm{odd})} = -\,\frac{\sigma_{xx}^{(\mathrm{odd})}}{(\sigma_{xx}^{(0)})^2},
$$
which makes explicit the conductivity origin of the odd resistive signal [2509.13277].

A distinct antecedent framework was developed for hopping transport through non-zero-angular-momentum impurity orbitals in organic semiconductors and HOPG graphite. In that theory, TRS preserves the degeneracy between $m$ and $-m$ orbitals, so the leading low-field correction is quadratic. Broken TRS lifts the $m\leftrightarrow -m$ degeneracy and produces a linear negative hopping contribution, written as
$$
MR_h \approx -\,|m|\,\kappa r\,b = -\,\frac{|m|\,B}{B_h},
$$
with the linear term interpreted as a signature of TRS breaking in hopping-mediated transport [1208.2866].

## 5. Proximity-induced topological states in $\alpha$-Sn/(In,Fe)Sb

The $\alpha$-Sn/(In,Fe)Sb heterostructure provides the largest reported OMR in the supplied literature. The system is grown on InSb(001), with a $7\,\mathrm{nm}$ ferromagnetic semiconductor (In,Fe)Sb bottom layer, $T_C\approx 300\,\mathrm{K}$, and a $3\,\mathrm{nm}$ ($\approx 18$ monolayers) $\alpha$-Sn top layer grown by MBE. Standard Hall bars of $200\,\mu\mathrm{m}\times 50\,\mu\mathrm{m}$ are used, with a dc current $I=10\,\mu\mathrm{A}$ along $x$ and an in-plane magnetic field rotated parallel or antiparallel to the current so that the ordinary Hall contribution is nulled. At $2\,\mathrm{K}$ and $B\parallel I$, the odd magnetoresistance reaches up to $1{,}150\%$ at $1\,\mathrm{T}$, grows roughly linearly with field with slight nonlinearity at higher $B$, follows the projection of $B$ onto the current direction with a $\cos\theta$ law, vanishes when $B\perp I$, and disappears above approximately $100\,\mathrm{K}$ [2507.23166].

The striking magnitude is not attributed to ordinary size effects in a trivial ultrathin semiconductor. Although $\alpha$-Sn at this thickness is a trivial narrow-gap semiconductor, perpendicular-field measurements reveal large even magnetoresistance, about $2\times 10^6\%$ at $2\,\mathrm{K}$, together with Shubnikov–de Haas oscillations. The FFT of $\Delta G_{xx}$ versus $1/B$ yields a single frequency $f=10\,\mathrm{T}$, implying a small two-dimensional Fermi pocket; the temperature dependence gives $m^*\approx 0.16\,m_0$; the Landau-fan intercept gives a Berry phase $\phi_B\approx 0.65\pi$; and the angle dependence follows $f(\theta)=f(0)/\cos\theta$ up to $\theta\approx 50^\circ$, confirming the two-dimensional character of the pocket before a secondary three-dimensional InSb pocket appears at larger tilt. These observations were interpreted as evidence for linear Dirac/Weyl-type dispersion localized at the interface [2507.23166].

First-principles calculations connect these oscillation data to magnetic proximity. Bulk $\alpha$-Sn under small biaxial compressive strain is a Dirac semimetal with inverted $s/p$ ordering, but an $18$-ML free-standing film becomes trivial because quantum confinement reverses the inversion and opens a small gap. In a slab model with Fe atoms on one surface, Fe $d$–Sn $p$ hybridization closes the gap and restores linear dispersion around $E\approx -0.25\,\mathrm{eV}$. The resulting bands along $\Gamma$–X and $\Gamma$–Y are inequivalent, and each Dirac or Weyl cone is slightly tilted in the film plane, with $\beta_x\neq \beta_y$ [2507.23166].

The semiclassical interpretation uses oppositely tilted Weyl cones. Near node $\chi=\pm 1$,
$$
H_\chi(k)=\chi v_F\,\sigma\!\cdot\!k + \beta\!\cdot\!k,
$$
with dispersion
$$
\varepsilon_{\chi,\pm}(k)=\pm v_F |k| + \beta\!\cdot\!k.
$$
A nonzero tilt breaks $k\to -k$ inversion for each cone and, through Berry curvature, generates a current correction linear in $B$,
$$
j_{\mathrm{odd}} \propto \sum_\chi \chi\,\beta\,\frac{e^3\tau^2}{\hbar}\,(\mathbf{E}\!\cdot\!\mathbf{B}).
$$
For $\mathbf{E}\parallel x$ and $\mathbf{B}\parallel x$, the total odd component obeys $\sigma_{xx}^{\mathrm{odd}}\propto \beta_x B_x$; because the two cones have opposite chirality and opposite tilt, their contributions add rather than cancel. A more complete fit gives
$$
j_{\mathrm{odd}} \propto A\beta_x B_x + C\beta_y B_y,
$$
and the full three-dimensional angular dependence at $B=0.2\,\mathrm{T}$ is reproduced with $\beta_x/\beta_y\approx 4.3$, consistent with the anisotropic tilts extracted from DFT [2507.23166].

## 6. Significance, applications, and unresolved issues

OMR has become a transport probe of intrinsic TRS breaking in systems where a purely even longitudinal response would otherwise be expected. In BLG/CGT, the odd longitudinal signal is presented as a significant probe of TRS breaking in quantum materials in which the crystal symmetries preclude the appearance of anomalous Hall effect, or where the AHE is too small to detect [2407.14071]. A later crystalline-symmetry analysis states, however, that longitudinal OMC follows the same point-group constraints as the anomalous Hall conductivity, whereas symmetric transverse OMC obeys different rules [2509.13277]. This suggests that comparisons across the OMR literature must distinguish carefully between conductivity and resistivity formulations, between longitudinal and transverse odd responses, and between bulk symmetry arguments and device-level measurement geometries.

The device implications are correspondingly diverse. In $\alpha$-Sn/(In,Fe)Sb, the odd resistance change is both large and linear, so the device behaves as a “Hall-like” sensor. With sensitivity defined as $S=V_{\mathrm{signal}}/(B\cdot V_{\mathrm{drive}})$, the reported value is $S_{\mathrm{OMR}}\approx 11.5\,\mathrm{mV}/\mathrm{mT}/\mathrm{V}$, compared with $1.2$–$7\,\mathrm{mV}/\mathrm{mT}/\mathrm{V}$ for the best commercial InSb Hall sensors. The same work identifies ultrasensitive magnetic-field detectors, low-power spintronic read heads, and on-chip magnetometry as plausible applications [2507.23166]. In InAs/(Ga,Fe)Sb, the ability to tune magnetic proximity and edge asymmetry independently enables ON/OFF switching and polarity reversal of OMR; in BLG/CGT, the response is field- and gate-programmable and remains operative at room temperature [2003.11731] [2407.14071].

A recurrent misconception is to equate OMR with any linear magnetoresistance. In the modern transport literature, OMR is specifically the antisymmetric longitudinal component extracted by $[R(B)-R(-B)]/2$ or $[\rho_{xx}(B)-\rho_{xx}(-B)]/2$, and it should vanish when the relevant TRS-breaking order disappears or when opposite odd contributions cancel. Eu$_2$Ir$_2$O$_7$ is symmetric after zero-field cooling and loses its odd term above $T_M\approx 105\,\mathrm{K}$; two-terminal InAs measurements across both edges show nearly zero odd response because the two edge contributions cancel; and a BLG/hBN control sample exhibits a purely even $R_{xx}(B)$ [1508.01318] [2003.11731] [2407.14071].

Taken together, the available results identify OMR as a sharply symmetry-constrained yet microscopically diverse transport phenomenon. Its realizations span domain-selected antiferromagnets, high-coercivity ferromagnets, inversion-broken one-dimensional channels, magnetized graphene, and proximity-induced topological states. The most extreme case so far, in $\alpha$-Sn/(In,Fe)Sb, indicates that a small Fermi surface in a linear band, strong tilt induced by magnetic-proximity-reinvoked topology, and additive odd responses from two cones can push OMR beyond $1{,}000\%$ at low field, far above the modest values reported previously [2507.23166].

Source: https://www.emergentmind.com/topics/odd-parity-magnetoresistance-omr