---
title: Hall-Memresistance in Spintronic Memory
url: https://www.emergentmind.com/topics/hall-memresistance
type: topic
---

# Hall-Memresistance in Spintronic Memory

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Hall-memresistance denotes a class of history-dependent transport phenomena in which a Hall voltage, a Hall-derived longitudinal voltage, or a Hall-type resistance level is coupled to an internal state variable and therefore encodes memory. In the literature considered here, that state variable is variously the magnetic flux in a feedback giant-magnetoresistive spin valve, the perpendicular magnetization of engineered Hall bars, the quantized Hall state of a quantum anomalous Hall device, trap occupation in a topological field-effect memristor, or the Néel vector in an antiferromagnet. The unifying feature is architectural rather than microscopic: the Hall channel either updates the internal state, reads it out, or does both, so the measured response depends on prior excitation [1104.5518; 2109.04820; 2011.00170; 2507.18388].

## 1. Conceptual scope and formal definitions

The earliest explicit theoretical formulation in this set is a generalized memristive system built from controlled spin polarizations in a giant-magnetoresistive material and a feedback loop based on the classical Hall Effect. That work already departs from the strict homogeneous memristor picture by allowing a pinched hysteretic loop whose self-crossing knot is not located at the origin, and by showing that passive memory systems not subject to Ohm’s Law can display such non-origin-crossing dynamics [1104.5518].

Within that formalism, a *homogeneous* memristive system has
$$
u=g(x,t)\,i,
$$
so every \(u\)–\(i\) loop must cross the origin. The broadened *heterogeneous* form is
$$
u = g(x,i,t)\,i + \tilde g(x,i,t), \qquad \dot x = f(x,i,t),
$$
where the nonzero \(\tilde g\) permits off-origin pinched loops. The same work further enlarges the framework to *compound memory electronic systems*,
$$
H(\{u,i,\dots\},x,t)=0, \qquad \dot x = F(\{u,i,\dots\},x,t),
$$
for situations in which resistive, capacitive, and inductive memory effects are tightly combined [1104.5518].

Later uses of the term are more device-specific. In mixed-symmetry Hall devices, Hall-memresistance appears as a longitudinal readout voltage whose odd-in-field component tracks magnetization because the Hall coefficient is engineered to vary along the current path. In antiferromagnetic Hall-memristors, it is defined directly as a transverse Hall resistance that depends on the history of the applied electric field via a slowly varying Néel vector. In quantum anomalous Hall memory, the stored bit is the sign of a quantized Hall resistance level. This suggests that Hall-memresistance is best understood as a family of memory-resistance effects mediated by Hall physics rather than as a single constitutive law.

## 2. Feedback spin-valve origin and generalized memristive dynamics

The feedback spin-valve realization is built from a CPP-stack giant-magnetoresistive spin valve consisting of two ferromagnetic layers separated by a thin nonmagnetic spacer. When the two magnetizations are parallel the resistance is low, \(R_{\uparrow\uparrow}\); when antiparallel it is high, \(R_{\uparrow\downarrow}\). The memristive behavior is created by driving the GMR stack with a current \(i(t)\), using the classical Hall voltage generated in the same conductor to drive a feedback coil, and letting the resulting magnetic flux \(\phi_m\) bias the ferromagnetic layers and hence modify the instantaneous GMR resistance \(R(\phi_m)\) [1104.5518].

The empirical GMR law is
$$
R(\phi_m)=R_{\uparrow\uparrow}+\bigl(R_{\uparrow\downarrow}-R_{\uparrow\uparrow}\bigr)\,\mathrm{sech}(\phi_m/\phi_{m0}),
$$
with \(\phi_{m0}\) a normalization flux. The Hall electromotive force is
$$
\varepsilon_H=\frac{1}{n e d}\,B\,i = D_0\,\phi_m\,i, \qquad D_0\equiv \frac{1}{n e d S},
$$
using \(B S=\phi_m\). Together with the coil relation \(\phi_m=L i_H\), the dynamics reduce to
$$
u(t)=R(\phi_m)\,i+\frac{D_0}{L}\phi_m^2,
$$
$$
\dot \phi_m = D_0\,\phi_m\,i-\frac{R_L}{L}\phi_m.
$$
The quadratic term \((D_0/L)\phi_m^2\) is the heterogeneous contribution that breaks strict \(u\propto i\) homogeneity [1104.5518].

Under a sinusoidal drive \(u(t)=u_0\cos(\omega t)\), numerical simulation gives a pinched hysteresis loop in the \(u\)–\(i\) plane, but the self-crossing knot lies off the origin. Physically, even when \(u=0\), the flux \(\phi_m\) need not vanish, so
$$
R(\phi_m)\,i+\frac{D_0}{L}\phi_m^2=0
$$
can be satisfied by nonzero \(i\). In the limit \(R_L/L\to 0\), corresponding to large inductance, the \(\phi_m\)-dynamics decouple and the system recovers a homogeneous memristor \(u=R(\phi_m)i\) with \(\phi_m^2\propto q\), so the pinch point returns to the origin. If the feedback-inductance term dominates, the hysteresis collapses into a single loop with no self-crossing [1104.5518].

The same analysis provides a sufficient stability condition using a Floquet-exponent estimate. Writing \(\phi_m=\phi_m^*+\eta\) around a limit cycle and bounding the periodic coefficient with the fact that \(x\,\mathrm{sech}\,x\,\tanh x\) is bounded by \(\Gamma_0\approx0.577\), the exponent satisfies
$$
\lambda \le \frac{2u_0 D_0(M_2-M_1)}{\omega}-\frac{2\pi R_L}{\omega L},
$$
so asymptotic stability is ensured if
$$
D_0(M_2-M_1)<\frac{\pi R_L}{u_0 L}.
$$
The significance of this result is conceptual as much as dynamical: it establishes that non-origin pinching is compatible with passive memory behavior when the constitutive law is heterogeneous rather than homogeneous.

## 3. Mixed-symmetry Hall devices and magnetization memory

A distinct Hall-memresistance concept was developed in conducting films with a Hall coefficient that varies along the current trajectory. In such directionally inhomogeneous media, a longitudinal voltage acquires an antisymmetric, odd-in-field component. The underlying mechanism is not a violation of Onsager reciprocity, but the superposition of Ohmic and Hall voltages in a medium whose Hall response is spatially nonuniform [2109.04820].

Two Co\(_{20}\)Pd\(_{80}\) implementations were fabricated. The three-terminal Hall bar uses a single ferromagnetic stripe in which contacts “a” and “c” source the current and a third contact “b” lies downstream on the same edge as “a”. The partitioned FM–NM Hall bar divides a contiguous bar into a ferromagnetic segment with large extraordinary Hall coefficient \(R_{EHE}\) and a thick, low-resistance normal-metal segment with negligible ordinary Hall response. In both cases the effective Hall coefficient profile is discontinuous along \(x\), with a finite Hall response in the FM section and zero response either at the reference contact or in the NM section [2109.04820].

In the circuit model, the local Hall voltages at two cross-sections are
$$
V_{xy,ab}=R_H(ab)\,I, \qquad V_{xy,cd}=R_H(cd)\,I,
$$
and the edge voltages become
$$
V_{ac}=I\,R+\tfrac12\bigl(V_{xy,ab}-V_{xy,cd}\bigr), \qquad
V_{bd}=I\,R-\tfrac12\bigl(V_{xy,ab}-V_{xy,cd}\bigr).
$$
In the ideal three-terminal or FM–NM geometries, one cross-section has zero Hall coefficient or zero local magnetization, so the odd component is maximal:
$$
V_{xx}^{\rm odd}\approx \tfrac12\,R_{EHE}\,M_z\,I.
$$
The measured longitudinal voltage in the three-terminal device can therefore be written as
$$
V_{xx}(m_z)=I\,R_{xx}+\tfrac12\,R_{EHE}\,m_z\,I,
$$
so reversing \(m_z=\pm1\) shifts the readout by \(\Delta V=R_{EHE}\,I\) [2109.04820].

The memory mechanism relies on perpendicular anisotropy and square hysteresis loops in the ferromagnetic sections. Magnetization is reversed by sweeping an external field \(B_z\) past the local coercive field \(H_c\). In a partitioned FM–NM–FM structure with two FM bars of different thicknesses, \(H_{c1}<H_{c2}\) and \(R_{EHE,1}\neq R_{EHE,2}\), so pulses satisfying
$$
H_{c1}<|B_z|<H_{c2}
$$
flip only one FM segment, while larger fields flip both. The four combinations \((m_{z1},m_{z2})\in\{(+,+),(-,-),(+,-),(-,+)\}\) then yield four distinct zero-field voltages
$$
V_{xx}=I\,R_{xx}+\tfrac12\bigl[R_{EHE,1}m_{z1}+R_{EHE,2}m_{z2}\bigr]\,I.
$$
This is an explicit multi-bit Hall-memory readout based on static magnetic states rather than dynamic resistive switching [2109.04820].

Experimentally, all voltages were measured with a dc current \(I=100\,\mu\mathrm{A}\). In the three-terminal 20 nm Co\(_{20}\)Pd\(_{80}\) device, the antisymmetric detection efficiency \(E=V_{xx}^{\rm odd}/V_{xx}^{\rm even}\) was approximately \(0.15\%\) at 300 K and \(0.4\%\) at 77 K. In the partitioned 20 nm FM + 100 nm CuAu device, \(E\approx0.8\%\) at room temperature and \(E\approx2.1\%\) at 77 K, with total resistance reduced by a factor \(5.6\) relative to a single FM. In the FM\(_1\)–NM–FM\(_2\) cell, the efficiency
$$
E=\frac{V_{xx}(\text{up-up})-V_{xx}(\text{down-down})}{V_{xx}^{\rm even}}
$$
was approximately \(1.25\%\), the minimum spacing between adjacent read levels was on the order of tens of microvolts at \(100\,\mu\mathrm{A}\), and no drift was observed over days. The antisymmetric spikes scaled strictly linearly with \(I\), reversed sign under \(I\to -I\), and flipped sign when measured along the opposite edge, confirming a mixed-symmetry Hall origin rather than a true nonreciprocal effect [2109.04820].

## 4. Quantum anomalous Hall and topological memristive implementations

A quantum anomalous Hall realization stores non-volatile binary information directly in quantized Hall resistance levels. In a twisted-bilayer graphene on hBN moiré stack near three-quarter filling, spontaneous ferromagnetism yields a zero-field QAHE with first Chern number \(v=\pm1\) and
$$
R_{xy}=\frac{h}{v e^2},
$$
so the two logic states are
\[
\text{logic “1”}: R_{xy}=+h/e^2, \qquad
\text{logic “0”}: R_{xy}=-h/e^2.
\]
The four-terminal cell writes by controlled hysteretic switching between the two Hall states using nanoampere currents of opposite polarities, and reads non-destructively by sensing the sign of the transverse Hall voltage at a current \(I_{\rm read}\) chosen inside the no-switching window \(I_c^-<I_{\rm read}<I_c^+\) [2011.00170].

The reported thresholds at 4 K and zero magnetic field are \(I_c^-\approx-4\,\mathrm{nA}\) and \(I_c^+\approx+0.1\,\mathrm{nA}\). A practical read bias is \(I_{\rm read}\approx-2\,\mathrm{nA}\), for which \(V_{xy}=I_{\rm read}R_{xy}\) is positive for the stored “1” state and negative for the stored “0” state. The excitation gap is described by
$$
R_{xy}(T)=\frac{h}{e^2}\left[1-\exp\!\left(-\frac{\Delta}{k_B T}\right)\right],
$$
with \(\Delta\approx 26\,K\cdot k_B\). At the architecture level, each QAHE cell is placed in series with a two-terminal mixed-ionic-electronic-conduction selector with ON/OFF ratio \(\sim10^6\) and leakage \(<10\,\mathrm{pA}\), enabling a 3D cross-point array with \(V/2\) biasing. Reported cell metrics are a cell area of \(\sim130\,\mathrm{nm}^2\), write powers of approximately \(0.96\,\mathrm{pW}\) for “0” and \(0.24\,\mathrm{pW}\) for “1”, read power \(\approx0.1\,\mathrm{pW}\), and latency of \(\sim5\)–\(10\,\mathrm{ns}\) [2011.00170].

A separate topological route combines floating-gate memristive behavior with a six-terminal Hall bar in the quantum spin Hall regime. The device is an inverted InAs/GaInSb/InAs trilayer quantum well with intrinsic floating-gate behavior arising from charge traps in a SiO\(_2\)/SiN dielectric. When the drain and top-gate are shorted, trap charge \(Q_t\) shifts the gate potential by
$$
\Delta V_{\rm TG}=Q_t/C_{\rm eff},
$$
and the normalized trap occupation
$$
w(t)\equiv Q_t(t)/Q_{\max}\in[0,1]
$$
acts as the internal state variable. The memristor model is
$$
V(t)=R\bigl(w,I(t)\bigr)\,I(t),
$$
with
$$
\frac{dw}{dt}=
\begin{cases}
\alpha_+\,I(t)\,(1-w), & I>0,\\[4pt]
-\alpha_-\,|I(t)|\,w, & I<0.
\end{cases}
$$
In this platform, one resistance state is governed by dissipationless helical edge channels with
$$
R_{\rm edge}=\frac{h}{2e^2}\approx 12.9\,\mathrm{k\Omega},
$$
while the other is an incoherent bulk-conduction state with typical differential resistance \(R_{\rm bulk}^{\rm diff}\sim2\)–\(10\,\mathrm{k\Omega}\) for \(\pm10\) V sweeps outside gap alignment [2511.17090].

The memristive loop is observed for current sweeps of \(\pm10\,\mu\mathrm{A}\) at \(T=4.2\) K, with two distinct voltages at \(I=0\) corresponding to bulk and edge conduction. A linearized fit
$$
R(w)=R_{\rm bulk}+\bigl(R_{\rm edge}-R_{\rm bulk}\bigr)w
$$
gives \(\alpha_+\approx10^{-6}\,\mathrm{s/A}\) and \(\alpha_-\approx3\times10^{-7}\,\mathrm{s/A}\). By tuning the back gate, the high-resistance plateau crosses zero bias near \(V_{\rm BG}\approx-8\) V, and the maximum high/low resistance ratio reaches \(22.5\) for \(\pm10\) V sweeps. In transistor mode, \(R_{xx}\approx100\,\mathrm{k\Omega}\) appears when the Fermi level lies in the gap; in the same regime \(R_{xy}\) remains zero at small magnetic fields and no broken-symmetry quantum Hall plateaus appear up to \(1\) T, consistent with time-reversal-invariant topological transport. The floating-gate state is stable over \(10^4\) s and over more than \(10^4\) full \(\pm10\) V sweeps, with typical power \(\sim10\,\mu\mathrm{W}\) in the edge state and up to \(100\,\mu\mathrm{W}\) in the bulk state [2511.17090].

These two implementations represent different limits of Hall-based memory. The QAHE device uses quantized transverse Hall resistance as the stored variable itself, whereas the topological field-effect memristor uses a Hall-bar topological transport platform in which memory is carried by trap occupation and read out through a history-dependent switch between coherent edge transport and incoherent bulk transport.

## 5. Antiferromagnetic Hall-memristors

In antiferromagnetic Hall-memristors, Hall-memresistance is defined explicitly as a transverse Hall resistance that depends on the history of the applied electric field through a slowly varying internal antiferromagnetic order parameter, namely the Néel vector. Under a fixed longitudinal current \(I\), the Hall voltage \(V_H\) therefore remembers prior current or field pulses. The microscopic mechanism combines a nonlinear Hall effect, which generates a transverse current density quadratic in the applied field, with a nonlinear Edelstein effect, which uses the same quadratic field dependence to induce a nonequilibrium spin polarization that slowly tilts the Néel vector [2507.18388].

The constitutive relations are
$$
J^{y}=\sigma_{y,xx}\bigl(\mathbf n\bigr)\,E_x^2,
$$
$$
\delta s_x=\chi_{x,xx}\,E_x^2,
$$
and, using \(w\equiv n_x\) as the state variable,
$$
J^y(t)=\sigma\!\bigl(w(t)\bigr)\,E_x^2(t), \qquad
\frac{dw}{dt}=g\!\bigl(w(t),E_x^2(t)\bigr).
$$
For CuMnAs, the symmetry analysis starts from space group \(P4/nmm\): each Mn sublattice is locally noncentrosymmetric even though combined \(\mathcal{PT}\) symmetry remains. Under the site point group \(C_{4v}\), the allowed intrinsic second-order tensors are
$$
\sigma_{y,xx}=-\sigma_{x,yy}\neq0, \qquad
\chi_{x,xx}=-\chi_{y,yy}\neq0.
$$
Thus a pulse \(E_x\) writes \(n_x\) via the nonlinear Edelstein effect, and a low-amplitude read field probes the stored state through the nonlinear Hall response [2507.18388].

The paper develops both a tilted massive Dirac toy model and a CuMnAs lattice model. The toy Hamiltonian is
$$
\mathcal H(\mathbf k)=v_F(k_x\sigma_y-k_y\sigma_x)+\boldsymbol\Delta\!\cdot\!\boldsymbol\sigma+v_t k_y \sigma_0,
$$
with \(\boldsymbol\Delta=(\Delta_x,0,\Delta_z)\). In the lattice realization, the \(4\times4\) antiferromagnetic Hamiltonian contains
$$
\epsilon_0(\mathbf k)=-t(\cos k_x+\cos k_y), \qquad
V_{AB}=-2\tilde t\cos\frac{k_x}{2}\cos\frac{k_y}{2},
$$
and staggered spin-orbit-coupled fields
$$
\mathbf h_A(\mathbf k)=-\mathbf h_B(\mathbf k)
=\bigl(-h^x_{\rm AFM}-\alpha_R\sin k_y,\ \alpha_R\sin k_x,\ h^z_{\rm AFM}\bigr).
$$
The nonlinear Hall conductivity is then obtained from an intrinsic Kubo-type expression involving Berry connections, and numerically shows strong dependence on \(h^x_{\rm AFM}\) and chemical potential [2507.18388].

The four-terminal geometry separates write and read paths. Terminals \(L,R\) inject a longitudinal current \(I_x\) or field \(E_x\); terminals \(T,B\) measure the transverse Hall voltage \(V_H=V_T-V_B\). Writing uses pulses of amplitude \(E_0\) and duration \(\Delta t\), which tilt the Néel vector by \(\delta n_x\sim\chi_{x,xx}E_0^2\Delta t\). Reading uses a smaller field \(E_{\rm read}\) to avoid disturbing the state. The Hall-memristance is defined by
$$
R_H^{\rm mem}(w)=\frac{V_H}{I_x}=M_H\!\bigl(w,E_{\rm read}\bigr).
$$
The state evolution is modeled by a Landau–Khalatnikov-type equation,
$$
\frac{dw}{dt}=-\frac{1}{\tau}\Bigl[w-w_0-J\,\chi_{x,xx}E_x^2\Bigr],
$$
which yields hysteresis in \(J^y\) versus \(E_x\) [2507.18388].

For CuMnAs, the material parameters used include \(t=0.08\) eV, \(\tilde t=1\) eV, \(\alpha_R=0.8\) eV, \(\alpha_D=0\), \(J=16\,\mathrm{eV\cdot\AA^2}/\hbar\), and pulse amplitude \(E_0=0.01\,\mathrm{eV/\AA}\). The nonlinear Hall-memconductance \(\sigma_{y,xx}\) reaches up to \(\sim10^{-7}\,\Omega^{-1}\mathrm{cm}^{-1}\) per \((\mathrm{V/cm})^2\) near avoided-crossing hot spots, and the normalized hysteresis-loop area peaks at \(\omega\tau\approx1\), indicating robust nonvolatile behavior for a broad range of GHz–THz writing frequencies. CuMnAs has \(T_N\approx480\) K, implying room-temperature stability of the stored state. Although exact switching energy and retention time are not yet measured, the authors estimate a switching energy in the fJ–pJ range and retention times \(\sim\tau\) that can be engineered into the ns–\(\mu\)s window [2507.18388].

## 6. Related Hall-derived magnetoresistive phenomena and recurrent misconceptions

Several recurring interpretive issues accompany Hall-memresistance. One is the assumption that every pinched hysteresis loop must cross the origin. The feedback spin-valve analysis shows that this is true for homogeneous memristive systems but not for generalized heterogeneous memristive systems, where a nonzero additive term \(\tilde g\) permits off-origin self-crossing. Another is the assumption that odd-in-field longitudinal signals in mixed-symmetry Hall devices represent genuine nonreciprocal transport. In the engineered CoPd structures, the observed effect is instead explained as the superposition of Ohmic and Hall voltages in a directionally inhomogeneous conductor, and the strict linearity in current, sign reversal under \(I\to-I\), and edge dependence all support that interpretation [1104.5518; 2109.04820].

A related but distinct line of work is spin Hall magnetoresistance in heavy-metal/magnetic-insulator bilayers. In Pt/Ba\(_2\)CoGe\(_2\)O\(_7\), the resistance modulation arises from the spin Hall effect, inverse spin Hall effect, and spin-current dissipation at the interface. When the spin accumulation \(\mu_s\) is parallel to the magnetic order parameter \(m\), reflection dominates and the resistance is low; when \(\mu_s\perp m\), spin-transfer torque or other spin-dissipation channels increase absorption and the resistance is high. The standard SMR ratio is
$$
\Delta R/R_0 \equiv (R_\perp-R_\parallel)/R_0.
$$
At \(T=2\) K and \(\mu_0H=1.9\) T, the maximum in-plane SMR ratio is approximately \(+2.5\times10^{-4}\) for \(j_c\parallel[100]\) and about \(+1.2\times10^{-4}\) for \(j_c\parallel[1\bar10]\), with angular dependence \(\rho_{xx}(\phi)/\rho_0-1\propto-(\Delta\rho/\rho_0)\cos 2\phi\) and a current-direction anisotropy following \(\simeq-A_2\cos2\alpha-A_4\cos4\alpha\) [2501.19198].

The Pt/Ba\(_2\)CoGe\(_2\)O\(_7\) study is not presented as Hall-memresistance, but it is relevant because it extends the Hall-derived magnetoresistance landscape into anisotropic altermagnetic interfaces. The work rules out several alternative explanations for the anisotropy, including domain-population differences, electric-polarization coupling, crystalline anisotropic magnetoresistance of Pt, and magnetic proximity effects, and proposes that the data may be understood through anisotropic altermagnetic ordering. This suggests that Hall-memresistance sits within a broader hierarchy of Hall-mediated spintronic responses in which symmetry, interfacial spin conversion, topological transport, and slow magnetic state variables can all shape the observable resistance [2501.19198].

Taken together, the available literature spans classical Hall feedback, engineered Hall-coefficient inhomogeneity, quantized Hall state storage, topological edge-state memristance, and nonlinear Hall/Edelstein antiferromagnetic memory. Hall-memresistance is therefore best characterized as a research program in which Hall physics is promoted from a passive probe of electronic structure to an active ingredient of memory-resistive functionality.

Source: https://www.emergentmind.com/topics/hall-memresistance