Local Anomalous Nernst Effect in Magnetic and 2D Systems

• Local anomalous Nernst effect is a thermoelectric phenomenon where a confined temperature gradient and magnetization produce a transverse voltage.
• Experimental setups use laser heating, near‐field AFM, and patterned heaters to isolate and measure the effect on mesoscopic, micrometer, and nanometer scales.
• Quantitative analyses extract key parameters, such as the anomalous Nernst coefficient and spatial decay exponents, advancing nanoscale thermometry and spintronic applications.

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 (Martens et al., 2018, Goswami et al., 2010). 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 (Martens et al., 2018, Kelekci et al., 2013, Pandey et al., 2024, Mizuguchi et al., 2018).

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 $\mathrm{V/(TK)}$ (Martens et al., 2018). 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 (Kelekci et al., 2013, Pandey et al., 2024).

The cross-product structure fixes the symmetry. Reversing either $M$ or $\nabla T$0 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 $\nabla T$1 (Martens et al., 2018). In FePt, reversing $\nabla T$2 or flipping $\nabla T$3 likewise changes the sign of $\nabla T$4 (Mizuguchi et al., 2018).

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

$\nabla T$5

where $\nabla T$6 is defined by subtracting the semiclassical contribution from the total Nernst signal (Goswami et al., 2010). No explicit microscopic theory for $\nabla T$7 is yet available in that system, although a phenomenological dependence on spontaneous magnetization and effective Berry curvature is discussed (Goswami et al., 2010).

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 $\nabla T$8m FWHM, rastered in 1 $\nabla T$9m steps	$E_{ANE}=K_N\,(M\times \nabla T),$0 up to about $E_{ANE}=K_N\,(M\times \nabla T),$1 (Martens et al., 2018)
[CoSiB/Pt] multilayer Hall bar	$E_{ANE}=K_N\,(M\times \nabla T),$2m tungsten heater	$E_{ANE}=K_N\,(M\times \nabla T),$3 (Kelekci et al., 2013)
CoFeB nanostructures	Near-field AFM-tip heating, Gaussian FWHM $E_{ANE}=K_N\,(M\times \nabla T),$4 nm	spatial resolution about 80 nm (Pandey et al., 2024)
Mesoscopic 2DES	ac heating current $E_{ANE}=K_N\,(M\times \nabla T),$5 at $E_{ANE}=K_N\,(M\times \nabla T),$6 Hz	anomalous component $E_{ANE}=K_N\,(M\times \nabla T),$7 after subtraction of $E_{ANE}=K_N\,(M\times \nabla T),$8 (Goswami et al., 2010)
FePt thin film	end-heated Hall bar, $E_{ANE}=K_N\,(M\times \nabla T),$9 up to 10 K over 6 $\hat n$0m	extra voltage superposed on conventional ANE (Mizuguchi et al., 2018)

In CoFeB-based magnetic tunnel junctions, a tightly focused laser spot is scanned across a $\hat n$1 area, producing three-dimensional temperature gradients whose in-plane direction can be rotated continuously through $\hat n$2–$\hat n$3 (Martens et al., 2018). In [CoSiB/Pt] multilayers, a local heater generates a nonuniform temperature field $\hat n$4, and the lateral component $\hat n$5 is inferred from the ANE voltage rather than from a direct micro-thermometer (Kelekci et al., 2013). 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 (Pandey et al., 2024).

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 $\hat n$6 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) (Martens et al., 2018). A continuous-wave diode laser with $\hat n$7 nm, modulated at 77 Hz, was focused to a 2 $\hat n$8m 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 (Martens et al., 2018).

Finite-element simulations using COMSOL show that a laser spot near one ellipse edge produces a dominant in-plane temperature gradient $\hat n$9 up to about 9 K across the 6 $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$0m junction length, whereas a centrally located spot yields a temperature difference predominantly in the out-of-plane direction (Martens et al., 2018). By scanning the spot around the ellipse, the in-plane gradient direction $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$1 can be rotated continuously through the film plane.

The voltage is measured in the out-of-plane $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$2 direction with a lock-in amplifier referenced to the laser modulation, while an in-plane external magnetic field $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$3 is swept between $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$4 mT to switch between parallel and antiparallel magnetization states (Martens et al., 2018). In this geometry, any out-of-plane voltage arising from in-plane $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$5 must be due to ANE, because AMTP and PNE generate electric fields in the $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$6-$V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$7 plane and therefore do not project onto $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$8 (Martens et al., 2018).

From each Seebeck loop $V_{ANE}=K_N\,(M\times \nabla T)\cdot \hat n.$9 recorded at fixed laser position, two plateau voltages in the parallel states are identified, $K_N$0 for $K_N$1 and $K_N$2 for $K_N$3, and the ANE-induced shift is defined as

$K_N$4

Mapping $K_N$5 over laser position yields extrema of about $K_N$6 near the junction edges and a nodal line $K_N$7 when the geometry produces no ANE projection (Martens et al., 2018). The angular dependence follows

$K_N$8

with fitted parameters

$K_N$9

Using an in-plane temperature difference $\mathrm{V/(TK)}$0 K from COMSOL and saturation magnetization $\mathrm{V/(TK)}$1 T, the anomalous Nernst coefficient was extracted as

$\mathrm{V/(TK)}$2

The reported value is stated to compare well to ANE coefficients in ferromagnet/nonmagnet heterostructures in the range $\mathrm{V/(TK)}$3 (Martens et al., 2018). The experiment therefore established a local Seebeck-based route to detect ANE with sub-$\mathrm{V/(TK)}$4V sensitivity and without the need to disentangle the signal from inverse-spin-Hall-based spin Seebeck contributions (Martens et al., 2018).

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 $\mathrm{V/(TK)}$5m wide patterned across one end of the bar (Kelekci et al., 2013). The active multilayer was $\mathrm{V/(TK)}$6 on thermally oxidized Si with $\mathrm{V/(TK)}$7 or 15; the Hall bar had width $\mathrm{V/(TK)}$8m and longitudinal voltage-lead spacing $\mathrm{V/(TK)}$9m (Kelekci et al., 2013).

For this geometry,

$E_N=N_A\,(M\times \nabla T)$0

The local heater dissipates power $E_N=N_A\,(M\times \nabla T)$1, heat spreads in three dimensions, and the measured saturated Nernst voltage scales with heater power with a slope about 1.15 in a $E_N=N_A\,(M\times \nabla T)$2 versus $E_N=N_A\,(M\times \nabla T)$3 plot, supporting a thermal origin (Kelekci et al., 2013). A separate calibration is provided by the coercive field $E_N=N_A\,(M\times \nabla T)$4 extracted from $E_N=N_A\,(M\times \nabla T)$5 loops, which softens with increasing $E_N=N_A\,(M\times \nabla T)$6 (Kelekci et al., 2013).

The principal spatial result is the dependence of the saturated ANE voltage on distance from the heater. Measurements at $E_N=N_A\,(M\times \nabla T)$7, 90, 120, and 150 $E_N=N_A\,(M\times \nabla T)$8m gave

$E_N=N_A\,(M\times \nabla T)$9

with a fitted exponent

$$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$0

This value lies between the limiting expectations $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$1 for pure two-dimensional spreading and $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$2 for pure three-dimensional spreading, indicating significant in-plane spreading together with non-negligible out-of-plane conduction into the substrate (Kelekci et al., 2013).

The magnetic signature is equally direct. For $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$3 and $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$4, the $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$5 loops display square hysteresis with coercive fields $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$6 G and 150 G, respectively, matching anomalous Hall loops (Kelekci et al., 2013). The identical loop shapes in $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$7 and $$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$8 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 (Pandey et al., 2024). COMSOL Multiphysics simulations of a CoFeB 15 nm/MgO stack used a heat source

$$E_{ANE}=\mu_0 S_{ANE}\,(\nabla T\times M)=-\,N\,(M\times \nabla T)$$9

with optical skin depth $M$0 nm for CoFeB at 532 nm (Pandey et al., 2024).

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 $M$1 is antisymmetric about the beam center, has peak magnitude about $M$2, and has zero net average for the full spot (Pandey et al., 2024). This directly affects interpretation of ANE images. For a thin magnetic wire, the measured voltage between contacts along $M$3 satisfies

$M$4

Experimentally, far-field SANE produced about $M$5V signals in a 10 $M$6m wire at $M$7 mW, whereas NF-SANE produced about 40 nV signals in a 2 $M$8m wire with a noise floor of about 5–10 nV, yielding a signal-to-noise ratio of about 4–8 (Pandey et al., 2024). The spatial FWHM of $M$9 extracted from a vortex-core line scan derivative was $\nabla T$00 nm, and domain-wall scans in perpendicular-anisotropy wires yielded a similar $\nabla T$01 nm spatial resolution; the work summarizes this as a spatial resolution of about 80 nm (Pandey et al., 2024).

A different form of locality appears in the mesoscopic 2DES study. There, a $\nabla T$02 gated region was defined in a Si $\nabla T$03-doped GaAs/AlGaAs heterostructure, and two adjacent quantum point contacts served simultaneously as local thermometers, Hall/Nernst voltage probes, and isolation barriers (Goswami et al., 2010). A small ac heating current $\nabla T$04A at $\nabla T$05 Hz established a local temperature rise $\nabla T$06 mK and a temperature difference $\nabla T$07 mK along $\nabla T$08, while the transverse voltage $\nabla T$09 was recorded at $\nabla T$10 (Goswami et al., 2010). The Nernst coefficient was defined as

$\nabla T$11

The semiclassical background was modeled as

$\nabla T$12

with fitted $\nabla T$13, and equivalently through the Mott-type relation

$\nabla T$14

After subtraction, the anomalous component $\nabla T$15 exhibited a 2$\nabla T$16 periodicity in $\nabla T$17, while the anomalous Hall component $\nabla T$18 exhibited a $\nabla T$19 periodicity (Goswami et al., 2010). 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 $\nabla T$20, cannot distinguish ferromagnetic from antiferromagnetic coupling (Goswami et al., 2010).

6. Additional mechanisms, interpretive boundaries, and proposed uses

An important extension of local ANE physics is the spin-wave-mediated contribution reported in L1$\nabla T$21-ordered FePt thin films (Mizuguchi et al., 2018). The starting point is an $\nabla T$22-$\nabla T$23 exchange Hamiltonian,

$\nabla T$24

combined with magnetization dynamics governed by the Landau-Lifshitz-Gilbert equation,

$\nabla T$25

Thermal fluctuations driven by $\nabla T$26 excite spin waves, and through $\nabla T$27-$\nabla T$28 coupling generate a conduction-electron spin current. In the phenomenological form presented,

$\nabla T$29

which is then converted by the inverse spin Hall effect,

$\nabla T$30

yielding an additional transverse voltage superposed on the conventional ANE (Mizuguchi et al., 2018).

The FePt experiments used 30 nm films patterned into Hall bars of 5 $\nabla T$31m channel width, with a temperature gradient applied along $\nabla T$32 and $\nabla T$33 up to 10 K over 6 $\nabla T$34m (Mizuguchi et al., 2018). The transverse Seebeck coefficient $\nabla T$35 showed clear hysteresis with $\nabla T$36. Below about 100 K, the ANE followed the Mott relation; above about 100 K, $\nabla T$37 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 $\nabla T$38 (Mizuguchi et al., 2018). 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$39 with in-plane $\nabla T$40 isolates ANE because AMTP and PNE do not project onto $\nabla T$41 and cancel under magnetization reversal (Martens et al., 2018). 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 (Martens et al., 2018). Third, near-field ANE imaging cannot be interpreted solely through $\nabla T$42, because the finite-element modeling indicates even larger in-plane gradients $\nabla T$43 (Pandey et al., 2024). Fourth, the mesoscopic 2DES study explicitly states that no microscopic theory for $\nabla T$44 is yet available, so phenomenological links to spontaneous magnetization, effective Berry curvature, and RKKY exchange remain interpretive rather than derived from a complete theory (Goswami et al., 2010).

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$45 (Martens et al., 2018). In [CoSiB/Pt] multilayers, the local heating geometry was identified as a route toward nanoscale thermoelectric sensors for thermal microscopy and on-chip heat-to-voltage converters in spintronic devices (Kelekci et al., 2013). Near-field ANE imaging was presented as relevant to antiferromagnetic spintronics, including Mn$\nabla T$46Sn, CuMnAs, and Mn$\nabla T$47Au, as well as racetrack memories with domain-wall readout at about 70–80 nm resolution (Pandey et al., 2024). In the mesoscopic 2DES, the ability of $\nabla T$48 to resolve the sign of the RKKY interaction suggests a thermoelectric pathway to image spin textures and magnetic phases in low-dimensional conductors (Goswami et al., 2010).

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$49, the vector orientation of $\nabla T$50, 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 (Martens et al., 2018, Goswami et al., 2010, Kelekci et al., 2013, Pandey et al., 2024, Mizuguchi et al., 2018).