Intrinsic Nonlinear Valley Nernst Effect

• Intrinsic nonlinear valley Nernst effect is a second-order phenomenon where a temperature gradient produces a transverse valley-odd current in 2D nonmagnetic crystals.
• It originates from quantum geometric properties, such as the quantum metric and Berry connection polarizability, enabling valley currents even when linear responses vanish.
• This effect is observable via second-harmonic thermoelectric and nonlocal transport measurements in materials like tilted Dirac semimetals and strained bilayer graphene.

The intrinsic nonlinear valley Nernst effect is a second-order thermoelectric transport phenomenon in which a longitudinal temperature gradient generates a transverse, valley-odd current in a nonmagnetic two-dimensional crystal with two time-reversal-related valleys $K$ and $K'$. In the notations used across the recent literature, the response is written either as $j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$ or $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$. Its defining feature is that it survives in situations where the linear valley Nernst effect is forbidden because Berry-curvature contributions vanish under combined symmetry constraints, while the nonlinear response remains allowed through quantum-geometric quantities such as the quantum metric or the Berry connection polarizability dipole. In this sense, intrinsic NVNE extends valley caloritronics into a regime governed by second-order semiclassical dynamics and band quantum geometry rather than by linear Berry curvature alone (Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025, Sharma et al., 27 Jan 2025).

1. Definition and physical content

In two-dimensional honeycomb metals such as graphene, electrons occupy two inequivalent valleys $K$ and $K'$. By analogy with the anomalous Nernst effect in ferromagnets, a valley Nernst effect describes the generation of a transverse valley current $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$ perpendicular to an applied temperature gradient. When both time-reversal symmetry $\mathcal T$ and inversion symmetry $\mathcal P$ are preserved, the Berry curvature vanishes globally and the linear valley Nernst effect is forbidden. The intrinsic nonlinear valley Nernst effect arises by going to second order in $\partial T$, and in that regime a pure valley current can appear even in centrosymmetric, nonmagnetic crystals (Wu et al., 2 Sep 2025).

The recent formulations agree on several core properties. First, the response is valley odd and transverse. Second, the intrinsic contribution is independent of the relaxation time $K'$0. Third, its microscopic origin is quantum geometric rather than Drude-like. One formulation states that the effect has a quantum origin from the quantum metric, while another states that it owes microscopically to the Berry connection polarizability dipole of valley electrons; these descriptions are consistent with the use of interband Berry-connection matrix elements in both derivations (Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025, Sharma et al., 27 Jan 2025).

A central conceptual point is that the intrinsic NVNE does not require a finite linear anomalous response. This addresses a common misconception imported from linear valley transport: the disappearance of Berry-curvature-driven linear currents under $K'$1 and $K'$2 does not preclude a second-order valley thermoelectric current. The nonlinear effect is explicitly permissible in both inversion-symmetric and inversion-asymmetric materials, provided the relevant tensor and valley symmetries allow it (Zhang et al., 27 Aug 2025).

2. Semiclassical and quantum-geometric formulation

A semiclassical derivation starts from the Boltzmann equation in the relaxation-time approximation,

$K'$3

with $K'$4. In steady state with no electric or magnetic field, $K'$5 and $K'$6. Expanding the distribution up to second order in the temperature gradient gives

$K'$7

with

$K'$8

The second-order thermoelectric current separates into three parts,

$K'$9

where $j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$0 is intrinsic, $j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$1 is Berry-curvature driven, and $j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$2 is Drude-like (Wu et al., 2 Sep 2025).

The intrinsic coefficient takes the compact form

$j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$3

with

$j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$4

and

$j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$5

Because $j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$6 localizes the response to a Fermi-surface integral, the intrinsic NVNE is controlled by quantum geometry near the Fermi level rather than by the entire filled band structure (Wu et al., 2 Sep 2025).

An equivalent quantum-geometric formulation uses the Berry connection polarizability tensor

$j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$7

For a single valley $j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$8, the intrinsic nonlinear valley Nernst tensor is

$j_a^{v}=\beta^v_{abc}\,\partial_bT\,\partial_cT$9

and the valley-odd response is $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$0. The tensor is antisymmetric in its first two indices, $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$1, which encodes the transverse character of the effect (Zhang et al., 27 Aug 2025).

3. Symmetry structure and allowed tensor components

The symmetry logic of intrinsic NVNE differs sharply from that of linear valley thermoelectricity. Under $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$2, $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$3, $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$4, and $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$5. Under $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$6, $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$7, $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$8, and $$j_a^{v(2)}=[\alpha^{v,\mathrm{in}}_{abc}]\,\partial_bT\,\partial_cT$$9. Combined $K$0 and $K$1 force $K$2, eliminating linear Berry-curvature-driven valley currents, but they do not eliminate the quantum metric. Consequently, a nonlinear valley current remains symmetry-allowed even when the linear response vanishes (Wu et al., 2 Sep 2025).

Whether a specific tensor element survives is then decided by the local point-group symmetry around each valley. In two-dimensional systems, the local largest symmetry near the valleys for a nonvanishing intrinsic NVNE is a single mirror symmetry. More specifically, a single surviving mirror plane $K$3, with the temperature gradient along $K$4, suffices to allow $K$5 (Wu et al., 2 Sep 2025).

A complementary classification uses the antisymmetric second-rank tensor $K$6, which transforms under a point-group operation $K$7 as

$K$8

where $K$9 for valley-preserving operations and $K'$0 for valley-switching operations. For the two independent components $K'$1 and $K'$2, common mirrors, rotations, and operations combined with time reversal or inversion select which component is allowed. This formalism makes explicit that intrinsic NVNE is symmetry-allowed in both inversion-symmetric and inversion-asymmetric settings, provided the relevant valley tensor transforms appropriately (Zhang et al., 27 Aug 2025).

4. Model realizations in tilted Dirac systems and bilayer graphene

A prototypical $K'$3-symmetric anisotropic tilted Dirac semimetal is described near valley $K'$4 by

$K'$5

Its eigenenergies are

$K'$6

In this model the band-resolved polarizability tensor is even under $K'$7 and independent of $K'$8, while the thermoelectric correction to the orbital magnetization contains a factor $K'$9. For the valence band,

$j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$0

so the sign reverses between $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$1 and $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$2. In the limit $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$3 and zero temperature broadening, the dominant tensor element is

$j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$4

and in the gapless limit $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$5 it saturates at $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$6 (Sharma et al., 27 Jan 2025).

Uniaxially strained, gapless AB-stacked bilayer graphene provides a distinct realization. Near valley $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$7, the low-energy two-band Hamiltonian is

$j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$8

with strain parameter $j_a^{(v)}\propto \alpha^{v}_{ab}\partial_bT$9 and $\mathcal T$0 eV. When $\mathcal T$1 along $\mathcal T$2, the threefold local $\mathcal T$3 symmetry near each valley is reduced to a single mirror $\mathcal T$4, which is exactly the minimal point group needed for $\mathcal T$5. The corresponding valley NVNE conductivity is

$\mathcal T$6

and because $\mathcal T$7 flips sign under $\mathcal T$8, the response is a pure valley current with zero net charge flow (Wu et al., 2 Sep 2025).

System	Symmetry ingredient	Reported NVNE behavior
Anisotropic tilted Dirac semimetal	$\mathcal T$9 tilt in a $\mathcal P$0-symmetric model	Valley-contrasting orbital magnetization and dominant $\mathcal P$1
Strained gapless bilayer graphene	Uniaxial strain reduces local $\mathcal P$2 to single mirror $\mathcal P$3	$\mathcal P$4, pure valley current, sign reversal from compressive to tensile strain
$\mathcal P$5-bilayer WTe$\mathcal P$6	Point group $\mathcal P$7 with mirror $\mathcal P$8 only	Only $\mathcal P$9 is allowed

In strained bilayer graphene, typical parameters $\partial T$0 and $\partial T$1 m/s give a characteristic energy $\partial T$2 meV. Numerical evaluation shows that $\partial T$3 has a pronounced peak when $\partial T$4 lies near the Dirac point, where $\partial T$5 is largest, and that the peak sharpens and grows in magnitude as $\partial T$6. Varying $\partial T$7 through the Lifshitz transition points $\partial T$8 and $\partial T$9 produces two extrema in $K'$00. Crucially, $K'$01 changes sign when the strain is tuned from compressive $K'$02 to tensile $K'$03, providing a direct experimental signature (Wu et al., 2 Sep 2025). In the tilted-Dirac illustration used for the BCP-dipole theory, numerical calculations with $K'$04 m/s, $K'$05, and $K'$06 eV show peaks near band edges, sign reversals across the gap, and excellent agreement with the low-temperature Mott relation discussed below (Zhang et al., 27 Aug 2025).

5. Orbital magnetization, measurement protocols, and nonlocal transport

A distinctive feature of the intrinsic NVNE literature is the role assigned to thermoelectric corrections to orbital magnetization. In addition to the semiclassical velocity term, the current contains an orbital contribution, and in second order one finds a thermoelectric correction to the orbital magnetization $K'$07. Integrating $K'$08 up to the Fermi surface gives a net orbital magnetization $K'$09 whose valley contrast distinguishes $K'$10 from $K'$11 in the nonlinear current. This valley-contrasting orbital magnetization is described as playing essentially the same role as orbital angular momentum in the linear valley Hall and valley Nernst effects, but it survives even when Berry curvature and linear response vanish due to $K'$12 symmetry (Sharma et al., 27 Jan 2025).

Direct detection schemes proposed for the intrinsic NVNE emphasize second-harmonic thermoelectric measurements. One route is a Hall-bar geometry under an AC thermal drive, where a nonlinear second-harmonic transverse valley response is isolated from linear backgrounds. Another route is valley-resolved Kerr or Faraday rotation, which probes the accompanying valley-contrasting orbital magnetization arising from a temperature-gradient-induced correction to the orbital magnetic moment $K'$13 (Wu et al., 2 Sep 2025).

A more developed transport theory predicts a nonlocal second-harmonic signal. In a strip of width $K'$14, a local transverse valley current generated by a temperature difference $K'$15 is detected at distance $K'$16 as a nonlocal voltage $K'$17. Two leading processes contribute at order $K'$18: nonlinear VNE in the generation region followed by linear inverse VHE in the detection region, and linear VNE followed by nonlinear inverse VHE. In the notation of that theory,

$K'$19

and

$K'$20

with $K'$21 the valley diffusion length, $K'$22 the resistivity, and $K'$23 the Seebeck coefficient. In the low-temperature regime, for $K'$24, the dominant contribution scales as

$K'$25

This $K'$26 scaling is proposed as a distinct nonlocal signature of intrinsic NVNE (Zhang et al., 27 Aug 2025).

First-principles results for $K'$27-bilayer WTe$K'$28 provide a concrete target. With point group $K'$29, only $K'$30 is allowed. DFT+mBJ+SOC with Wannier interpolation yields at $K'$31 K a peak $K'$32 nA nm/K$K'$33 at $K'$34 eV. Using $K'$35, $K'$36S, $K'$37m, and $K'$38 K/$K'$39m gives a predicted $K'$40 nV, while the ratio $K'$41 remains $K'$42, indicating that the direct nonlinear VNE term dominates over the Seebeck-driven contribution (Zhang et al., 27 Aug 2025).

6. Relation to nonlinear valley Hall transport, intrinsic character, and extrinsic corrections

The intrinsic nonlinear valley Nernst tensor is connected to the intrinsic nonlinear valley Hall conductivity through a generalized Mott relation. Writing

$K'$43

with

$K'$44

one obtains at low temperature

$K'$45

This is presented as the nonlinear analogue of the Wiedemann-Franz law and provides a direct bridge between nonlinear thermal and nonlinear electrical valley responses (Zhang et al., 27 Aug 2025).

The designation “intrinsic” refers specifically to the $K'$46 character of the response. In the semiclassical decomposition, this contribution is insensitive to disorder scattering rates, unlike the Berry-curvature piece $K'$47 and the Drude-like piece $K'$48 (Wu et al., 2 Sep 2025). The same distinction appears in the nonlocal measurement theory, where an extrinsic, scattering-dependent $K'$49 term arises from the second-order distribution-function correction,

$K'$50

For WTe$K'$51, taking $K'$52 fs makes this extrinsic contribution one order smaller than the intrinsic one (Zhang et al., 27 Aug 2025).

Within the current literature, the intrinsic NVNE is therefore positioned as a robust quantum-geometric mechanism for generating pure valley currents in systems where linear Berry-curvature thermoelectricity is absent or strongly constrained. The same body of work points to nonlinear valley caloritronics, strain-sensitive transport in bilayer graphene, nonlocal second-harmonic detection, nonreciprocal directional dichroism through the orbital magnetic quadrupole, and valley pumping in inversion-asymmetric cases as natural extensions of the phenomenon (Wu et al., 2 Sep 2025, Zhang et al., 27 Aug 2025).