---
title: Local Anomalous Nernst Effect in Magnetic and 2D Systems
url: https://www.emergentmind.com/topics/local-anomalous-nernst-effect
type: topic
---

# Local Anomalous Nernst Effect in Magnetic and 2D Systems

Searching arXiv for recent and foundational papers on the local anomalous Nernst effect.
Local anomalous Nernst effect denotes the generation and detection of an anomalous Nernst voltage under spatially confined temperature gradients, typically on mesoscopic, micrometer, or nanometer length scales. In magnetic conductors, the effect is governed by the transverse response to the combination of magnetization $M$ and temperature gradient $\nabla T$, while in mesoscopic two-dimensional electron systems an anomalous Nernst component has been resolved after subtraction of a semiclassical background and linked to spin-correlated states [1801.04186], [1012.5724]. Experimental realizations include laser-heated magnetic tunnel junctions, locally heated multilayer Hall bars, near-field AFM-tip thermoelectric imaging, and FePt thin films with an additional spin-wave-mediated contribution [1801.04186], [1308.5485], [2407.13028], [1804.03751].

## 1. Constitutive relations and symmetry

In ferromagnets, the anomalous Nernst effect (ANE) converts a temperature gradient into a transverse electric field through the cross product with the magnetization. One form used for the local response is
$$
E_{ANE}=K_N\,(M\times \nabla T),
$$
and for a measured voltage along a detection axis $\hat n$,
$$
V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.
$$
Here $K_N$ is the anomalous Nernst coefficient in $\mathrm{V/(TK)}$ [1801.04186]. Equivalent constitutive forms also appear as
$$
E_N=N_A\,(M\times \nabla T)
$$
in multilayer films and
$$
E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)
$$
in near-field ANE imaging [1308.5485], [2407.13028].

The cross-product structure fixes the symmetry. Reversing either $M$ or $\nabla T$ reverses the sign of the measured ANE voltage. This sign inversion is a key diagnostic because quadratic thermomagnetic effects such as anisotropic magneto-thermopower (AMTP) and the planar Nernst effect (PNE) remain invariant under $M\rightarrow -M$ [1801.04186]. In FePt, reversing $\nabla T$ or flipping $M_z$ likewise changes the sign of $V_{ANE}$ [1804.03751].

A distinct but related use of anomalous Nernst terminology appears in a mesoscopic two-dimensional electron system (2DES). There, the measured coefficient is decomposed as
$$
N_{\mathrm{total}}(B,n_s,T)=N_{\mathrm{semi}}(B,n_s,T)+\delta N(B,n_s,T)+N_{\mathrm{an}}(B,n_s,T),
$$
where $N_{\mathrm{an}}$ is defined by subtracting the semiclassical contribution from the total Nernst signal [1012.5724]. No explicit microscopic theory for $N_{\mathrm{an}}$ is yet available in that system, although a phenomenological dependence on spontaneous magnetization and effective Berry curvature is discussed [1012.5724].

## 2. Locality in experiment: heating, detection, and spatial scales

Local ANE experiments rely on confining the thermal source, defining the voltage geometry so that the transverse response is isolated, and using modulation techniques to detect low signals. The principal implementations reported in the cited literature span several material platforms.

| System | Local thermal excitation | Measured local ANE signature |
|---|---|---|
| CoFeB/MgO/CoFeB MTJ | 638 nm laser, 2 $\mu$m FWHM, rastered in 1 $\mu$m steps | $\Delta V_{ANE}$ up to about $\pm 0.4\,\mu\mathrm{V}$ [1801.04186] |
| [CoSiB/Pt] multilayer Hall bar | $\sim 2\,\mu$m tungsten heater | $V_N(d)\propto 1/d^{1.3}$ [1308.5485] |
| CoFeB nanostructures | Near-field AFM-tip heating, Gaussian FWHM $\approx 20$ nm | spatial resolution about 80 nm [2407.13028] |
| Mesoscopic 2DES | ac heating current $I_h=1\,\mu\mathrm{A}$ at $f=11.3$ Hz | anomalous component $N_{an}$ after subtraction of $N_{semi}$ [1012.5724] |
| FePt thin film | end-heated Hall bar, $\Delta T$ up to 10 K over 6 $\mu$m | extra voltage superposed on conventional ANE [1804.03751] |

In CoFeB-based magnetic tunnel junctions, a tightly focused laser spot is scanned across a $30\,\mu\mathrm{m}\times 30\,\mu\mathrm{m}$ area, producing three-dimensional temperature gradients whose in-plane direction can be rotated continuously through $0^\circ$–$360^\circ$ [1801.04186]. In [CoSiB/Pt] multilayers, a local heater generates a nonuniform temperature field $T(x,y,z)$, and the lateral component $\nabla T_x$ is inferred from the ANE voltage rather than from a direct micro-thermometer [1308.5485]. In near-field ANE imaging, the heating source is confined by plasmonic coupling at the apex of a metallized AFM tip, enabling sub-diffraction thermal excitation [2407.13028].

This variety of geometries suggests that “local” is not tied to a single device class. A plausible implication is that the defining feature is the spatial confinement of the thermal perturbation and the corresponding ability to resolve transverse thermoelectric response on the length scale set by the active junction, voltage probes, or near-field thermal spot.

## 3. Magnetic tunnel junction realization and quantitative extraction

A systematic local ANE study was carried out on elliptical pseudo-spin-valve MTJs of size $6\,\mu\mathrm{m}\times 4\,\mu\mathrm{m}$ with the stack
Au(70 nm) / Ru(3 nm) / Ta(5 nm) / CoFeB(5.4 nm) / MgO(1.68 nm) / CoFeB(2.5 nm) / Ta(10 nm) on MgO(100) [1801.04186]. A continuous-wave diode laser with $\lambda=638$ nm, modulated at 77 Hz, was focused to a 2 $\mu$m full-width at half-maximum spot. Because the Au contact pads are much thicker than the optical penetration depth of about 15–20 nm, heating of the CoFeB layers is purely thermal [1801.04186].

Finite-element simulations using COMSOL show that a laser spot near one ellipse edge produces a dominant in-plane temperature gradient $\Delta T_{\mathrm{in\mbox{-}plane}}$ up to about 9 K across the 6 $\mu$m junction length, whereas a centrally located spot yields a temperature difference predominantly in the out-of-plane direction [1801.04186]. By scanning the spot around the ellipse, the in-plane gradient direction $\phi_{\nabla T}$ can be rotated continuously through the film plane.

The voltage is measured in the out-of-plane $z$ direction with a lock-in amplifier referenced to the laser modulation, while an in-plane external magnetic field $\mu_0 H$ is swept between $\pm 60$ mT to switch between parallel and antiparallel magnetization states [1801.04186]. In this geometry, any out-of-plane voltage arising from in-plane $\nabla T$ must be due to ANE, because AMTP and PNE generate electric fields in the $x$-$y$ plane and therefore do not project onto $z$ [1801.04186].

From each Seebeck loop $V(H)$ recorded at fixed laser position, two plateau voltages in the parallel states are identified, $V_{p1}$ for $H<0$ and $V_{p2}$ for $H>0$, and the ANE-induced shift is defined as
$$
\Delta V_{ANE}\equiv V_{p1}-V_{p2}.
$$
Mapping $\Delta V_{ANE}$ over laser position yields extrema of about $\pm 0.4\,\mu\mathrm{V}$ near the junction edges and a nodal line $\Delta V_{ANE}=0$ when the geometry produces no ANE projection [1801.04186]. The angular dependence follows
$$
\Delta V_{ANE}(\phi_{\nabla T})=A\sin(\phi_{\nabla T}-\phi_0)+V_{off},
$$
with fitted parameters
$$
A=(0.42\pm 0.04)\,\mu\mathrm{V},\qquad
\phi_0=(4\pm 5)^\circ,\qquad
V_{off}=(0.00\pm 0.03)\,\mu\mathrm{V}.
$$

Using an in-plane temperature difference $\Delta T\simeq 8$ K from COMSOL and saturation magnetization $M_s\simeq 1.6$ T, the anomalous Nernst coefficient was extracted as
$$
K_N\approx 1.6\times 10^{-8}\,\mathrm{V/(TK)}.
$$
The reported value is stated to compare well to ANE coefficients in ferromagnet/nonmagnet heterostructures in the range $10^{-6}\ldots 10^{-8}\,\mathrm{V/(TK)}$ [1801.04186]. The experiment therefore established a local Seebeck-based route to detect ANE with sub-$\mu$V sensitivity and without the need to disentangle the signal from inverse-spin-Hall-based spin Seebeck contributions [1801.04186].

## 4. Heat-flow geometry, spatial decay, and dimensional crossover

In [CoSiB/Pt] multilayer films with perpendicular magnetic anisotropy, the local ANE was measured in a Hall-bar geometry using a separate tungsten heater about 2 $\mu$m wide patterned across one end of the bar [1308.5485]. The active multilayer was $[\mathrm{CoSiB}(6\,\mathrm{\AA})/\mathrm{Pt}(14\,\mathrm{\AA})]_N$ on thermally oxidized Si with $N=7$ or 15; the Hall bar had width $W=10\,\mu$m and longitudinal voltage-lead spacing $L_y=20\,\mu$m [1308.5485].

For this geometry,
$$
V_N=E_N\cdot L_y=N_A\,[M_z\,(\partial T/\partial x)]\,L_y.
$$
The local heater dissipates power $P=R_{\mathrm{heater}}\,I_n^2$, heat spreads in three dimensions, and the measured saturated Nernst voltage scales with heater power with a slope about 1.15 in a $\log V_N^{sat}$ versus $\log P$ plot, supporting a thermal origin [1308.5485]. A separate calibration is provided by the coercive field $H_c$ extracted from $V_N(B)$ loops, which softens with increasing $P$ [1308.5485].

The principal spatial result is the dependence of the saturated ANE voltage on distance from the heater. Measurements at $d=60$, 90, 120, and 150 $\mu$m gave
$$
V_N(d)\propto \frac{1}{d^\alpha},
$$
with a fitted exponent
$$
\alpha=1.3\pm 0.1.
$$
This value lies between the limiting expectations $\alpha=1$ for pure two-dimensional spreading and $\alpha=2$ for pure three-dimensional spreading, indicating significant in-plane spreading together with non-negligible out-of-plane conduction into the substrate [1308.5485].

The magnetic signature is equally direct. For $N=7$ and $N=15$, the $V_N(B)$ loops display square hysteresis with coercive fields $H_c\simeq 220$ G and 150 G, respectively, matching anomalous Hall loops [1308.5485]. The identical loop shapes in $V_N(B)$ and $V_H(B)$ show that the local ANE signal tracks perpendicular magnetization reversal. In this setting, the local ANE functions simultaneously as a thermoelectric readout of magnetization and as an indirect probe of the spatial decay of the temperature gradient.

## 5. Near-field nano-imaging and mesoscopic anomalous components

Near-field ANE imaging extends the local response to nanostructures well below the optical diffraction limit. In the reported implementation, a 532 nm continuous-wave laser focused by an objective of numerical aperture 0.7 gives a far-field Gaussian spot with FWHM about 748 nm, while coupling the same laser to a metallized AFM tip produces a near-field heating spot with Gaussian FWHM about 20 nm [2407.13028]. COMSOL Multiphysics simulations of a CoFeB 15 nm/MgO stack used a heat source
$$
Q(r,z,t)= \frac{\alpha P}{2\pi \sigma^2 z_0}\exp\!\left[-\left(\frac{r^2}{2\sigma^2}+\frac{z}{z_0}\right)\right]\sin^2(2\pi f t),
$$
with optical skin depth $z_0\sim 10$ nm for CoFeB at 532 nm [2407.13028].

The key modeling result is that, besides out-of-plane temperature gradients, there are even larger in-plane temperature gradients. Specifically, the in-plane gradient $\nabla T_x(r)$ is antisymmetric about the beam center, has peak magnitude about $2\times \nabla T_z$, and has zero net average for the full spot [2407.13028]. This directly affects interpretation of ANE images. For a thin magnetic wire, the measured voltage between contacts along $y$ satisfies
$$
V_{ANE}\propto \mu_0 S_{ANE}\left[\int (m_x\nabla T_z)\,dx\,dy+\int (m_z\nabla T_x)\,dx\,dy\right]
\rightarrow
V_{ANE}\propto \mu_0 S_{ANE}(\nabla T_z M_x+\nabla T_x M_z).
$$

Experimentally, far-field SANE produced about $4\,\mu$V signals in a 10 $\mu$m wire at $P=5$ mW, whereas NF-SANE produced about 40 nV signals in a 2 $\mu$m wire with a noise floor of about 5–10 nV, yielding a signal-to-noise ratio of about 4–8 [2407.13028]. The spatial FWHM of $\nabla T_z$ extracted from a vortex-core line scan derivative was $84\pm 14$ nm, and domain-wall scans in perpendicular-anisotropy wires yielded a similar $66\pm 10$ nm spatial resolution; the work summarizes this as a spatial resolution of about 80 nm [2407.13028].

A different form of locality appears in the mesoscopic 2DES study. There, a $5\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$ gated region was defined in a Si $\delta$-doped GaAs/AlGaAs heterostructure, and two adjacent quantum point contacts served simultaneously as local thermometers, Hall/Nernst voltage probes, and isolation barriers [1012.5724]. A small ac heating current $I_h=1\,\mu$A at $f=11.3$ Hz established a local temperature rise $T_m\approx 242$ mK and a temperature difference $\Delta T\approx 9$ mK along $x$, while the transverse voltage $V_{xy}$ was recorded at $2f$ [1012.5724]. The Nernst coefficient was defined as
$$
N\equiv \frac{E_y}{\nabla_x T}=\frac{V_{xy}}{\Delta T\cdot L_y}.
$$

The semiclassical background was modeled as
$$
N_{\mathrm{semi}}\equiv S_{xy}^{\rm semi}
=
-\alpha\,\frac{\pi^2 k_B^2 T}{3|e|\,\epsilon_F}\,
\frac{\omega_c\tau}{1+\omega_c^2\tau^2},
$$
with fitted $\alpha\approx 1.9$, and equivalently through the Mott-type relation
$$
N_{\mathrm{semi}}=-\frac{\pi^2 k_B^2 T}{3e}\left(\frac{\partial \sigma_{yx}}{\partial \epsilon}\right)_{\epsilon_F}.
$$
After subtraction, the anomalous component $N_{an}$ exhibited a 2$\pi$ periodicity in $2k_F R$, while the anomalous Hall component $\Delta\gamma_H$ exhibited a $\pi$ periodicity [1012.5724]. The paper states that the anomalous Nernst signal therefore acts as a direct probe of the sign of the RKKY exchange, whereas the anomalous Hall effect, tracking $|J|$, cannot distinguish ferromagnetic from antiferromagnetic coupling [1012.5724].

## 6. Additional mechanisms, interpretive boundaries, and proposed uses

An important extension of local ANE physics is the spin-wave-mediated contribution reported in L1$_0$-ordered FePt thin films [1804.03751]. The starting point is an $s$-$d$ exchange Hamiltonian,
$$
H=p^2/(2m)-J_{sd}\,S\cdot M,
$$
combined with magnetization dynamics governed by the Landau-Lifshitz-Gilbert equation,
$$
\partial_t M=-\gamma\,M\times H_{eff}+\alpha\,M\times \partial_t M.
$$
Thermal fluctuations driven by $\nabla T$ excite spin waves, and through $s$-$d$ coupling generate a conduction-electron spin current. In the phenomenological form presented,
$$
j_s=\lambda_{sw}\,(\nabla T\times M),
$$
which is then converted by the inverse spin Hall effect,
$$
j_c=\theta_{SH}\,(j_s\times \hat \sigma),
$$
yielding an additional transverse voltage superposed on the conventional ANE [1804.03751].

The FePt experiments used 30 nm films patterned into Hall bars of 5 $\mu$m channel width, with a temperature gradient applied along $x$ and $\Delta T$ up to 10 K over 6 $\mu$m [1804.03751]. The transverse Seebeck coefficient $S_{xy}=E_y/\nabla_x T$ showed clear hysteresis with $H_z$. Below about 100 K, the ANE followed the Mott relation; above about 100 K, $S_{xy}$ was larger than the Mott prediction based on anomalous Hall data, and the extra spin-wave-mediated ANE part monotonically decreased with increasing uniaxial anisotropy $K_u$ [1804.03751]. This identifies a boundary condition for interpretation: a local anomalous Nernst voltage need not be purely the conventional band-structure ANE when thermally excited spin dynamics and internal spin-charge conversion are active.

Several recurrent misconceptions are explicitly addressed in the literature. First, in the out-of-plane voltage geometry of the CoFeB/MgO MTJ, an in-plane $\nabla T$ with in-plane $M$ isolates ANE because AMTP and PNE do not project onto $z$ and cancel under magnetization reversal [1801.04186]. Second, inverse-spin-Hall-based spin Seebeck signals are absent in that MTJ experiment because no normal-metal ISHE detector is used; the only voltage path is across the MTJ itself [1801.04186]. Third, near-field ANE imaging cannot be interpreted solely through $\nabla T_z$, because the finite-element modeling indicates even larger in-plane gradients $\nabla T_x$ [2407.13028]. Fourth, the mesoscopic 2DES study explicitly states that no microscopic theory for $N_{an}$ is yet available, so phenomenological links to spontaneous magnetization, effective Berry curvature, and RKKY exchange remain interpretive rather than derived from a complete theory [1012.5724].

The proposed uses of local ANE span memory, logic, thermometry, and imaging. In MTJs, the local ANE has been proposed for nonvolatile logic elements in which a programmed in-plane gradient and magnetic state jointly define a logic output voltage, and for direction-dependent thermometry that reports both the magnitude and orientation of $\nabla T$ [1801.04186]. In [CoSiB/Pt] multilayers, the local heating geometry was identified as a route toward nanoscale thermoelectric sensors for thermal microscopy and on-chip heat-to-voltage converters in spintronic devices [1308.5485]. Near-field ANE imaging was presented as relevant to antiferromagnetic spintronics, including Mn$_3$Sn, CuMnAs, and Mn$_2$Au, as well as racetrack memories with domain-wall readout at about 70–80 nm resolution [2407.13028]. In the mesoscopic 2DES, the ability of $N_{an}$ to resolve the sign of the RKKY interaction suggests a thermoelectric pathway to image spin textures and magnetic phases in low-dimensional conductors [1012.5724].

Taken together, these studies define the local anomalous Nernst effect as a family of transverse thermoelectric phenomena in which the decisive variables are the local structure of $\nabla T$, the vector orientation of $M$, and the measurement axis. The resulting voltages can encode magnetization reversal, heat-flow direction, exchange-sign information, or spin-wave-assisted spin conversion, depending on the material system and geometry [1801.04186], [1012.5724], [1308.5485], [2407.13028], [1804.03751].

Source: https://www.emergentmind.com/topics/local-anomalous-nernst-effect