Sliding tetralayer graphene

The device stacks two bilayer graphene sheets with a marginal twist. The stack relaxes into a patchwork of four-layer domains of two kinds: Bernal stacking, ABAB, and rhombohedral stacking, ABCA. The two differ by one slide between the second and third layers. Kelvin probe force microscopy shows the rhombohedral surface potential 22 mV below the Bernal one, with the Bernal domains the larger at charge neutrality.

Two gates set the carrier density n and the displacement field D independently. At D/ε₀ = −0.067 V/nm, sweeping from neutrality toward holes, the longitudinal resistance jumps at n = −15.5 × 10¹² cm⁻². On the way back the change persists until n = −4.1 × 10¹² cm⁻², where the resistance returns to its first value. The high-resistance state is the rhombohedral–Bernal mixed state, RB-mixed; the low one is the Bernal-dominant state, B-dominant. Short gate pulses switch between them again and again.

Carried to large positive D and small n, the two states differ in their magnetism. At 0.03 T the RB-mixed state shows a Hall resistance |R_yx| > 100 Ω with a hysteresis loop in the field, the anomalous Hall effect, and a Curie temperature around 8 K. The B-dominant state shows only the small ordinary Hall effect. Both readings come from exactly the same gate voltages: the sweep history alone selects the magnet or the non-magnet. At negative D the anomalous Hall sign reverses.

Core P-⁠5 states it directly: “A magnetic field is a macroscopic helical density gradient of the omnium: the bivector grade of the density field coherently aligned along an axis.” Imposing a geometry on a material is establishing a field. Here the geometry is the stacking of four layers, and the gate history sets it. In the RB-mixed state the stacking patterns carry a bivector grade coherently aligned along the stacking axis, which the Hall contacts read. In the B-dominant state that alignment is absent.

The record is a laboratory exhibit of geometric magnetism: a ferromagnet switched on and off by geometry alone, by electrostatic means, at fixed gate voltages and in a fixed applied field. Nothing magnetic is added or taken away. The slide between the second and third layers is the whole of the change.

The two states form a neighbour comparison in the sense of I-⁠3a. The same device, the same gates and the same applied field are read twice, once in each geometry, and the B-dominant history furnishes the reference. Everything the two readings share cancels in the difference, and what remains is the aligned bivector response of the RB-mixed stacking.

The sign reversal under reversed D reads the same way. Inversion-symmetric stackings give a response even in D; the odd part places the sign at the domain walls, where the displacement field and the wall geometry together set the sign of the aligned bivector response. This is an in-sample reading of the source’s domain-boundary picture and stays separate from the predictions below.

Two comparisons isolate the field that a physical twist establishes (P-⁠5). Three isolate the density channel that an isotope substitution adds at fixed geometry (I-⁠7). Each compares neighbours built so that the consensus difference cancels or is computed independently, and the analysis fixes for each a test point, a planned threshold and the rule that decides it. The amplitudes are conditional by construction: each rides on a declared response dictionary.

1. T-⁠1
   The hand-odd loop centre. Build mirror devices from one cut bilayer, with designed angles of 180° + 0.03° and 180° − 0.03°. P-⁠5 reads the physical twist as a field along the stacking axis, reversed with the hand. The raw anomalous Hall loop of the RB-mixed state at 1.5 K then centres at minus the twist field, with opposite signs for the two hands. In the consensus reading time reversal leaves the static twist untouched, and the centre is zero at each hand. At an illustrative twist field of 0.20 mT the hand-odd centre is −0.20 mT, against a planned threshold of 0.03 mT from twelve device pairs.
2. T-⁠2
   The field-odd switching boundary. On the same mirror devices at D/ε₀ = −0.067 V/nm, apply ±0.02 T. The switching density moves as k(B_a + B_tw)², so the two hands differ by 2kB_aB_tw, which is zero in the consensus reading. At k = 1 × 10¹⁴ cm⁻² T⁻² the contrast is 8 × 10⁸ cm⁻², against a threshold of 2 × 10⁸ cm⁻².
3. T-⁠3
   The isotope shift of the Curie point. Make the tetralayers from carbon-12, at 1% carbon-13, and from carbon-13, at 99%. At fixed geometry the heavier isotope reads a larger R_s and a higher exterior compactness, Δln α_ext = 0.08, and I-⁠7 reads its larger response as a higher Curie temperature. Against the consensus channels of zero-point volume, interlayer spacing and electron–lattice renormalisation, the illustrative log change is 0.08 with the density channel and −0.003 without it, with a planned threshold of 0.01.
4. T-⁠4
   The isotope contrast of the susceptibility. The field-even curvature of the switching boundary, k̂ = (n₊ + n₋ − 2n₀)/(2B_a²), measures the susceptibility difference between the two stackings. Across the two isotopes the heavier reads the larger magnitude: a log change of 0.08 with the density channel and −0.003 without it, with a threshold of 0.01.
5. T-⁠5
   The isotope contrast of the surface potential. Image rhombohedral–Bernal domain pairs by Kelvin probe in both compositions and correct each contrast for the tip’s lateral kernel. The 22 mV contrast grows in magnitude with the heavier isotope: 0.08 in the log with the density channel and −0.003 without it, with a threshold of 0.01.

Remove the twist field and the added density sensitivity, keep the source’s gate, stress and response dictionaries, and FDT returns the record as measured: the switching densities, the Hall signs and loops, the field-antisymmetrized Hall resistance and every printed bound. The gate dictionary is the source’s own:

With the same tight-binding set, work-function difference and energy offset, the free-energy calculation returns the same switching contours, an identity map that corroborates no invariant. The force law reduces exactly to Newton’s on its pair domain and recovers the weak-field data of general relativity at leading order. The five comparisons leave the common data only through the two additions FDT supplies: the field that a physical twist establishes, and the density-coupled isotope channel.