Inverse Mean Valley

• Inverse Mean Valley describes reciprocal processes across disciplines, such as converting valley imbalances into transverse electrical signals in 2D materials like MoS₂.
• The topic encompasses nonlinear valley responses where distinct symmetry characteristics yield higher-harmonic voltage signals, advancing electrical detection and signal engineering.
• Applications include inverse design in valley-Hall photonic crystals, inverse mean molecular-weight trends in exoplanet atmospheres, and inverse mean value properties in PDE analysis and sensing.

“Inverse Mean Valley” is an ambiguous expression used in several technically unrelated literatures. In electronic valleytronics, it denotes the inverse valley Hall effect (IVHE), the reciprocal conversion of a valley imbalance into a transverse electrical signal. In nonlinear valley transport, it extends to nonlinear inverse valley responses with symmetry properties distinct from the direct nonlinear valley Hall effect. In topological photonics, the phrase arises through inverse design of valley-Hall photonic crystals with prescribed pseudospin states. In exoplanet science, it describes an inverse relationship between planet size and atmospheric mean molecular weight across the radius valley. In analysis, it evokes inverse mean value properties that characterize domains through harmonic or panharmonic test functions. In deep sub-electron read-noise sensing, it appears as an “Inverse Mean Valley” ratio derived from valley-peak modulation.

1. Electronic inverse valley Hall conversion in monolayer MoS$_2$

In monolayer MoS$_2$, the valley degree of freedom is associated with the inequivalent $K$ and $K'$ extrema of the Brillouin zone. Broken inversion symmetry produces finite Berry curvature near these valleys, while time-reversal symmetry enforces the valley-contrasting relation $\Omega(K)=-\Omega(K')$. Under an in-plane electric field, the anomalous velocity is

$$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$

so carriers from $K$ and $K'$ acquire equal and opposite transverse velocities. The result is a charge-neutral valley current transverse to the applied field, i.e. the valley Hall effect (VHE). The inverse process, IVHE, converts a diffusing valley imbalance or valley current into a transverse electric field and a measurable nonlocal voltage. The intrinsic Hall conductivity is written as

$$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$

and the valley Hall angle is

$\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$

The nonlocal detection geometry used for monolayer MoS$_2$0 implements electrical generation and electrical detection without optical pumping. A representative Hall-bar device on $_2$1 nm SiO$_2$2/Si has $_2$3, $_2$4, $_2$5, and injector–detector separation $_2$6, with Ti/Au $_2$7 contacts. In the Type I nonlocal configuration, a DC source–drain bias $_2$8 drives a local charge current at the injector, the VHE launches a transverse valley current, and IVHE at the detector converts the arriving valley imbalance into an open-circuit nonlocal voltage $_2$9. In the small-angle, narrow-arm limit, a widely used scaling form is

$K$0

where $K$1 is the diffusion constant and $K$2 the intervalley relaxation time. The full self-consistent expression used in the experiment includes finite $K$3 and $K$4 and the feedback of VHE/IVHE at injector and detector (Hung et al., 2018).

The reported signals are much larger than the Ohmic background. Monolayer devices exhibit $K$5 at $K$6 and $K$7 at $K$8, detected $K$9 from the injector. A SPICE resistor-network using measured $K'$0 and $K'$1 predicts an Ohmic nonlocal voltage of $K'$2 for the same geometry and $K'$3, consistent with multilayer control devices that show only $K'$4–$K'$5. The extracted valley Hall angle is $K'$6 at $K'$7. The valley diffusion length grows as temperature decreases, with $K'$8 experimentally in the high-$K'$9 regime, consistent with an analytic phonon-scattering result $\Omega(K)=-\Omega(K')$0, and reaches $\Omega(K)=-\Omega(K')$1 at low temperature. In-plane magnetic fields up to $\Omega(K)=-\Omega(K')$2 do not affect $\Omega(K)=-\Omega(K')$3, ruling out spin Hall contributions. A common misconception is that valley polarization in MoS$\Omega(K)=-\Omega(K')$4 must be optically generated; this work establishes purely electrical VHE generation and IVHE detection at room temperature (Hung et al., 2018).

2. Nonlinear inverse valley responses and harmonic nonlocal transport

For linear nonlocal valley transport, the direct and inverse processes are reciprocal. In a strip geometry, the linear VHE and linear IVHE are written as

$\Omega(K)=-\Omega(K')$5

and Onsager reciprocity gives $\Omega(K)=-\Omega(K')$6. The nonlinear case is different. The nonlinear valley Hall effect (NVHE) is valley-odd,

$\Omega(K)=-\Omega(K')$7

whereas the nonlinear inverse valley Hall effect needed for nonlocal voltage generation must be valley-even: $\Omega(K)=-\Omega(K')$8 This distinction is not a minor formal point. The direct nonlinear coefficient $\Omega(K)=-\Omega(K')$9 and the inverse nonlinear coefficient $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$0 have distinct symmetry character, distinct microscopic mechanisms, and are not reciprocal to one another. In the symmetry analysis of two-valley, time-reversal-invariant systems, $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$1 is forbidden by inversion $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$2 and the twofold rotation $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$3, while the relevant NVHE tensor components can remain allowed even when linear VHE is symmetry-forbidden (Cao et al., 24 Feb 2025).

The nonlocal transport theory predicts distinct harmonic content under low-frequency AC drive. Linear valley transport appears at $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$4. Mixed nonlinear/linear pathways appear at $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$5 and $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$6. Signals involving nonlinear inverse conversion appear at $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$7 and $$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$8. The theory gives, for example,

$$\mathbf{v}_a=-\frac{e}{\hbar}\,\mathbf{E}\times \boldsymbol{\Omega}_n(\mathbf{k}),$$9

and

$K$0

The change from $K$1 to $K$2 is a diagnostic consequence of the nonlinear inverse stage. The same framework predicts sizable $K$3 and $K$4 nonlocal voltages in bilayer $K$5-WTe$K$6, with $K$7 and $K$8 for detector spacing $K$9 under the parameter set quoted for that material (Cao et al., 24 Feb 2025).

The experimental observation of giant nonlinear valley Hall transport in graphene–hBN moiré superlattices supplies the corresponding nonlocal phenomenology. In a monolayer graphene strip of width $K'$0, with ultrahigh mobility $K'$1, AC frequency $K'$2, and injector–detector spacing $K'$3, the first-, second-, third-, and fourth-harmonic nonlocal voltages satisfy

$K'$4

The corresponding nonlocal resistances scale as $K'$5, $K'$6, $K'$7, and $K'$8. $K'$9 exhibits opposite-signed peaks on the two sides of the Dirac gap, and $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$0 exceeds $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$1 near Dirac points for $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$2. The measured valley diffusion length is $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$3. A common misconception is that a nonlinear inverse response should be the direct nonlinear response run backward; the combined theory and experiment show that the nonlinear inverse channel is symmetry-distinct, and that its signatures are encoded in the higher harmonics of the nonlocal voltage (He et al., 5 Mar 2025).

3. Inverse design in valley-Hall topological photonic crystals

In valley photonic crystals (VPCs), the relevant “valley” structure is photonic rather than electronic. Two-dimensional photonic crystals with inequivalent extrema at $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$4 and $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$5 can acquire localized Berry curvature when inversion symmetry is broken in lattices with $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$6 symmetry. A domain wall between two VPCs with opposite valley topological phases supports topological kink states. In this context, the “inverse” aspect is not inverse Hall conversion but inverse design: the target is to engineer unit cells whose band-edge pseudospin states and valley topological phases are specified in advance.

The framework is formulated in the frequency domain. The electromagnetic eigenproblem is

$$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$7

with Bloch boundary conditions

$$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$8

For the 2D TM formulation, the driven Helmholtz equation for $$\sigma_{xy}(E_F)=\frac{e^2}{\hbar}\sum_n \int \frac{d^2\mathbf{k}}{(2\pi)^2}\,\Omega_n(\mathbf{k})\, f(E_{n,\mathbf{k}}),$$9 is used, and the in-plane magnetic field follows from

$\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$0

The design variable is a density field $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$1, with

$\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$2

where $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$3 and $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$4. Pseudospin is enforced through the normalized transverse spin angular momentum

$\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$5

with $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$6. The complete optimization uses $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$7, $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$8, and $\theta_v \equiv \frac{\sigma_{xy}}{\sigma_{xx}}.$9, together with driven solves at $_2$00, $_2$01, $_2$02, and $_2$03, near $_2$04, $_2$05, and $_2$06. COMSOL Multiphysics and globally convergent MMA are used, with PDE-based filtering, Heaviside projection, minimum-length-scale constraints, and artificial damping $_2$07 with $_2$08 (Sato et al., 28 Mar 2025).

Two optimized unit cells illustrate arbitrary pseudospin assignment across multiple bands: PC1 has target pseudospins $_2$09, and PC2 has $_2$10. Both exhibit a lower band gap near $_2$11 and a higher band gap near $_2$12. Berry curvature, computed from discrete Wilson loops on a $_2$13 $_2$14-grid, is localized near $_2$15 and $_2$16; PC1 and PC2 share the same valley phase in the first band but differ in the fourth. Domain-wall supercells then realize selective kink states: PC1$_2$17PC2 supports kink states only in the higher gap, whereas PC2$_2$18PC1′ supports them only in the lower gap. In a composite router, an input at $_2$19 activates the upper path with transmittance $_2$20, while an input at $_2$21 activates the lower path with transmittance $_2$22. The significance of this literature is terminological rather than reciprocal: “inverse” denotes inverse design, and “valley” refers to photonic $_2$23 band topology rather than electronic IVHE (Sato et al., 28 Mar 2025).

4. Radius-valley chemistry and inverse mean molecular-weight trends

In exoplanet science, “inverse mean valley” refers to a gradient of atmospheric mean molecular weight across the radius valley. The radius valley, or Fulton gap, is the dearth of planets around $_2$24–$_2$25 Earth radii separating rocky super-Earths from sub-Neptunes that retain significant H/He envelopes. The CHOSSi outgassing model links radius-valley constraints to atmospheric chemistry by combining carbon–hydrogen–oxygen–sulfur–silicon equilibrium chemistry, non-ideal fugacities at high pressure, and dissolution of water and hydrogen into melt. The underlying readout from radius-valley population inference is that, for $_2$26, atmospheric surface pressures are $_2$27–$_2$28 bar and melt temperatures are $_2$29–$_2$30 K, both decreasing with radius (Heng et al., 3 Apr 2025).

The chemical system includes gas-phase species H$_2$31, H$_2$32O, CH$_2$33, CO, CO$_2$34, O$_2$35, S$_2$36, SO$_2$37, H$_2$38S, SiO, and SiH$_2$39. Equilibrium is solved through Gibbs free-energy minimization and equilibrium constants $_2$40, while non-ideal behavior is encoded through fugacity $_2$41. Water and hydrogen dissolution are modeled with

$_2$42

The mean molecular weight is

$_2$43

with He neglected as a minor correction. Lower $_2$44 and lower retained H/He shift equilibrium toward heavier oxidized molecules and away from H$_2$45 and CH$_2$46, especially at higher oxygen fugacity $_2$47.

The predicted trend is explicitly inverse: $_2$48. On the large side of the valley, $_2$49, atmospheres remain nearly H$_2$50-dominated with $_2$51–$_2$52. On the small side and across the valley, $_2$53, $_2$54–$_2$55 is common, and the upper envelope reaches $_2$56–$_2$57 under oxidized conditions and low $_2$58/high $_2$59. The crucial control is oxygen fugacity, not carbon enrichment: the strength of the $_2$60 gradient is primarily driven by $_2$61, whereas the upper $_2$62 envelope shows minimal correlation with C/H. This directly addresses a common misconception that “metallicity” is the primary driver. Observationally, increasing $_2$63 from $_2$64 to $_2$65 reduces the scale height $_2$66 and the amplitudes of transmission features by $_2$67, implying an inverse-amplitude trend across the radius valley. The paper identifies CH$_2$68/CO$_2$69 as especially sensitive to $_2$70 and proposes JWST and ARIEL retrievals as tests of the predicted inverse mean-molecular-weight valley (Heng et al., 3 Apr 2025).

5. Inverse mean value properties in elliptic PDE

In analysis, the relevant concept is “inverse mean value,” not a valley degree of freedom. For harmonic functions, the classical mean value property states that if $_2$71 in a domain $_2$72, then

$_2$73

for every admissible ball $_2$74. Inverse mean value theorems reverse this logic: if a domain-wide averaging identity holds for every function in an appropriate PDE class, then the domain must be geometrically rigid, typically a ball. The survey literature reviews such results for harmonic and panharmonic functions and extends them to annuli, strips, and cylinders in the harmonic case (Kuznetsov, 2022).

For the modified Helmholtz equation,

$_2$75

the ball mean value formula acquires a Bessel weight. Writing

$_2$76

one has

$_2$77

whenever $_2$78 solves the modified Helmholtz equation and $_2$79. The function $_2$80 is monotone increasing on $_2$81, satisfies $_2$82, and diverges as $_2$83. A canonical positive radial solution is

$_2$84

with $_2$85 and $_2$86 (Kuznetsov, 2021).

The central Kuran-type theorem states that if $_2$87, $_2$88, is bounded, its complement is connected, $_2$89 satisfies $_2$90, and there exists $_2$91 such that

$_2$92

for every positive function $_2$93 satisfying $_2$94 in the dilated domain $_2$95, then—together with the additional volume condition when $_2$96—one must have $_2$97. The proof compares $_2$98 and $_2$99 in the two geometric cases $K$00 and $K$01, using the strict radial monotonicity of $K$02. The survey further notes that Kuran’s original harmonic theorem, Bennett’s equality of volume and surface means, and inverse results for annuli and strips belong to the same family, while panharmonic analogues for annuli and strips remain open in that account (Kuznetsov, 2021, Kuznetsov, 2022).

6. Valley-peak modulation, phase-space invariance, and inverse read-noise mapping

In deep sub-electron read-noise CMOS sensing, “Inverse Mean Valley” is a reparameterization of valley-peak modulation (VPM). The standard amplitude-domain Poisson–Gaussian model is

$K$03

where $K$04, $K$05, $K$06 is the conversion gain, $K$07 is the DC offset, and $K$08 is the input-referred read noise. A representative valley-peak modulation metric is

$K$09

with $K$10 and $K$11 the two tallest neighboring peaks and $K$12 the intervening valley. In the deep sub-electron read-noise regime, Starkey and Fossum derived exposure-independent approximations such as

$K$13

but the exact amplitude-domain quantity still depends on both $K$14 and $K$15 (Hendrickson et al., 1 Mar 2026).

The phase-space construction removes exposure exactly by quotienting out the integer photoelectron lattice. Defining

$K$16

the phase variable has a wrapped-Gaussian density independent of $K$17: $K$18 The phase-space peak and valley heights are

$K$19

so the exposure-invariant phase-space VPM is

$K$20

Using the inverse elliptic nome, the read-noise inversion is

$K$21

with $K$22 the complete elliptic integral of the first kind.

The “Inverse Mean Valley” form is then simply the reciprocal peak-to-valley normalization. In the amplitude domain,

$K$23

and in phase space,

$K$24

This clarifies the relation between the historical IMV terminology and the newer phase-space formulation: the “inverse” quantity is not a different observable, but a reciprocal reparameterization of the same valley–peak structure, with the advantage that phase-space VPM is exactly invariant to quanta exposure (Hendrickson et al., 1 Mar 2026).