Twist and density in a current-driven helicene rotor

The molecule is dithioacetyl[13]helicene, a heterocyclic helicene of thirteen fused rings wound into two turns, built by elongating shorter helicenes with benzofuran scaffolds and isolated as a racemic mixture. Bulky tert-butyl side groups shield its nitrogen and oxygen lone pairs from the undercoordinated gold atoms of the tips. In a mechanically controllable break junction at 4.2 K in cryogenic vacuum, a notched gold wire is broken into two atomically sharp tips whose separation a piezoelectric element sets with sub-ångström resolution. The molecules are sublimed into the junction from a local heated source, and a junction is measured once its conductance holds near the helicene’s, between about 1 × 10⁻³ and 2 × 10⁻² 𝒢₀, with low noise for one second. The source’s calculated junctions bind the molecule to each electrode through a sulfur atom.

At low bias the current follows the cubic voltage dependence of off-resonant tunnelling. Held at a fixed bias, it switches among discrete levels, which the source labels by rank: A the highest, B the intermediate, C the lowest. In the junction of the source’s Figure 1 the current jumps A → B → C → A at 420 mV, and at −420 mV the order is A → C → B → A. Of 316 junctions that switch among two to five levels above about |200| mV, 122 resolve three or more, the number a direction needs, and in every one of them every observed jump follows a single cyclic order: a directionality of 100%, the percentage of counted jumps in that order. In one junction at 600 mV the dwell times in A, B and C are exponentially distributed, with means of 118, 19 and 7.6 ms.

The rate ℛ of complete cycles grows exponentially with the bias. The source fits ℛ = ℛ₀ exp[−(E_b − 𝗀_we𝒱)/k_BT] with ℛ₀ fixed at 10¹⁰ s⁻¹: five data sets return the dimensionless barrier E_b/k_BT from 32.1 to 67 and the slope e𝗀_w/k_BT from 32 to 63 V⁻¹, and their ratio E_b/(e𝗀_w), 0.77 to 1.53 V, needs no temperature. At an assumed 4.2 K, the bath’s, the barriers read 12 to 24 meV, and at an assumed 20 K, 55 to 115 meV.

Above an onset current, defined where the measured current departs from the cubic fit by more than a predefined 10%, the current first rises moderately above the cubic trend, then sharply. The onset current is lowest at an intermediate electrode displacement, in three junctions followed through the displacement and in histograms of 321; the source’s calculated torque work per rotation, divided by the barrier, is largest at an intermediate displacement, and the source reads the opposite trends as linking the current rise to rotation. A sweep halted below about 52 µA leaves the molecule in place, its zero-bias conductance the helicene’s and its inelastic steps molecular vibrations; halted above, it leaves a gold atomic junction of 1 to 5 𝒢₀ with the steps of gold phonons. The conductance estimated at the maximum current of an intact junction stays below 0.78 𝒢₀. Junctions of a linear anthracene molecule of comparable conductance and length show neither the cyclic jumps nor the sharp rise.

The source’s calculations are built on the rotation it attributes. Density functional theory gives barriers of several hundred meV every 120° in a hollow–hollow junction and a few meV in a top–top one, and at the angles where the barriers peak the transmission rises: the HOMO and HOMO−1, built mainly on the p states of the terminal sulfur atoms, couple more strongly to the electrodes and move toward the Fermi energy. In the source’s four-level model only the two sulfur–gold couplings 𝗍_L and 𝗍_R vary with the angle; the work of the current-induced torque over one rotation is a line integral around the closed contour the two couplings trace, equal to the area integral of the curl of the torque flow it encloses, which the source evaluates as three times the area integral over its shaded region, small differences between A, B and C ignored; the contour collapses to a line of zero area for a non-chiral molecule and is traversed the other way by the opposite hand. In a tilted-washboard Langevin model that nonconservative “wind” torque tilts a periodic potential, stepwise hopping gives way to continuous rotation when its work per cycle exceeds the friction’s dissipation, and the simulated current rises by more than an order of magnitude at the crossover.

There is one substance, the omnium, and the junction is a driven region of it. The helicene’s two turns are a helical geometry, and imposing a physical twist on a material is establishing a magnetic field: a macroscopic helical density gradient of the omnium, the bivector grade of the density field coherently aligned along an axis (P-⁠5). The helicene carries its own twist field, written 𝖡_tw. Mechanical vorticity and magnetic moment are two grade projections of one density-gradient topography, and no carrier is exchanged between them (I-⁠2). Mirror images carry opposite twist fields; the kernel that sizes the field and the map from a circulating charge to an I-⁠1 input are to be derived (O-⁠6).

Every density cone carries a flow direction, the sense in which its helical geodesics wind about its axis, intrinsic and fixed once for all, and chirality is one carrier of a flow direction (P-⁠4). The helicene’s hand is that carrier. P-⁠4’s attraction–repulsion rule is not applied to the molecule and the current: its magnitudes are I-⁠1 derivations with a declared kernel, and how the system-wide settlement signs a pair it does not join is to be derived (O-⁠5).

The current is electrons, closures of strand count 3, crossing the junction on helical geodesics of truncated density cones: quantum mechanics is helical geodesics on density cones, and the wavefunction is the projection of a helix onto the plane normal to the cone axis (I-⁠4, I-⁠5). The current counts the closures that cross per unit time, and the rotor is an extended pattern turning about its axis. The source treats the angle classically and the electrons coupled to it quantum-mechanically; the classical angle and the quantum-mechanically treated electrons are one statement read in two limits (I-⁠4), and the source’s estimate of the rotor’s angular momentum, about a hundred times ħ, places the turning deep in the classical limit.

The electrodes’ phonons and the molecule’s vibrations, which dominate the source’s friction, are one carrier at different density and geometry regimes: photon, graviton, gluon, phonon and magnon are the same spin-1 stroke of the density field (I-⁠2). The wind torque is nonzero only out of equilibrium, and it vanishes at zero bias as every driven observable vanishes at zero drive (I-⁠1). Force times length is energy, so a torque integrated over an angle is a work: the torque work per cycle is that integral. The rotation rate is a rate, never a force, and the source’s conservative torque from the equilibrium potential is no I-⁠1 pair force; at condensed-matter compactness the bare law returns a small number, a correct reading, and a barrier scale is reached only through a declared response kernel.

The orbitals that carry the current, the HOMO and HOMO−1 on the terminal sulfur atoms, are nodal patterns of the driven density field, and their geometry is set by the clamps, the gold apex and the sulfur nuclei, not by the medium (P-⁠2, P-⁠3). A rotation that rearranges the clamps shifts the resonances, as the source’s calculation moves them up in energy between 0° and 60°. Each clamp nucleus carries two charts: an exterior compactness of order 10⁻³⁸ at its own charge radius, and the skin value α_core → 1⁻ of its interior chart, with no map between them.

The rotor’s gravitational window is its own compactness on the A-⁠1 chart, α_ext = 𝓜c²/(2F_maxr_f) = 2.5 × 10⁻⁴² for the bound molecule at the declared r_f = 1.0 nm: a small number and a correct reading, with the curvature of general relativity (0.1c, I-⁠1, I-⁠3). Every channel of the junction enters its total gravitating energy 𝓜 once (I-⁠7):

with M the rest mass of the rotor and the electrode apices, E_rot the rotational kinetic energy, E_heat the heat the wind work leaves in the electrodes, phonons and electron–hole pairs, E_bias the electrostatic energy of the bias, and the thermal term of the declared junction volume, (2 nm)³, 2.1 × 10⁻⁵⁶ kg at 4.2 K. The wind work and its heat are one channel: the heat is the work’s destination, never a second entry.

A pattern whose rest energy depends on where it sits in a twist takes that energy as its clock factor, the helical window (0.1c):

with 𝗆 the held moments, 𝖡 the twist’s magnetic field where they sit, ℰ_ind the energy of the moments the twist induces, the work of inducing them included, ℰ₀ the rest energy outside every twist and z the coordinate along the twist’s gradient. The gradient of ℰ along the twist pushes, and patterns whose held and induced moments, summed per unit ℰ, differ along ∂_z𝖡 fall differently where they sit; a uniform twist turns a held moment and pushes nothing. The universality of free fall is the gravitational window’s, and it stands.

The junction is deterministic. The spread of dwell times is the spread of initial data that no apparatus fixes (I-⁠4), and each hop is a system-wide settlement of the state it starts from, in which events do not commute (0.1a).

Each test compares two arms in one common design: sixteen cooldowns of the source’s break-junction setup, eight per arm, each cooldown loading one arm from the local heated source, the arm of each cooldown fixed by a balanced random permutation drawn before the first cooldown. That randomisation makes the permutation test exact. The statistic, the difference of the arms’ mean cooldown values, is set against all 12,870 balanced relabellings: a two-sided test at 0.0027 rejects when the observed statistic is among the 34 largest in magnitude, exact size 0.00264, and a one-sided test at 0.00135 when it is among the 17 most extreme in its direction, exact size 0.00132. Every executed rate below comes from 2 × 10⁴ Monte Carlo trials of the full design.

Two tests read the twist field (P-⁠5). Their arms are enantiopure (P)- and (M)-dithioacetyl[13]helicene, resolved from the source’s racemate by chiral chromatography, five junctions per cooldown. The observable is the asymmetry of the zero-bias differential conductance 𝒢, measured by the source’s lock-in method, between the two settings of a reversed field or magnetisation, taken at the bias where the junction’s DC current vanishes; the statistic is its mean over the (P) cooldowns less its mean over the (M) cooldowns. The twist field reverses with neither the solenoid nor the tips, and mirror images carry opposite twist fields, so the asymmetry has a hand-odd part. Both tests are existence tests and two-sided: the kernel that sizes the hand-odd part is to be derived (O-⁠6).

1. T-⁠1
   A field-odd zero-bias conductance with a hand-odd part. An axial field of ±1 T along the electrode axis from a superconducting solenoid, reversed forty times per junction in ABBA order, gives 𝒴_𝖡 = [𝒢(+𝖡_app) − 𝒢(−𝖡_app)]/[𝒢(+𝖡_app) + 𝒢(−𝖡_app)] and its hand contrast Δ_𝖡. The comparator is zero. In the consensus account two-terminal reciprocity keeps the linear conductance unchanged when every magnetic field is reversed, junction by junction; the mirror carries a (P) junction at +𝖡_app onto an (M) junction at −𝖡_app; field-odd instrumental offsets are common to both hands and cancel; and the reading at zero DC current removes the magnetochiral second-order term, odd in both field and hand. At the declared noise the hand contrast has a standard deviation of 7.6 × 10⁻⁴, so the detection threshold, three standard deviations, sits near 2.3 × 10⁻³; the executed false-rejection rate is 0.00245.
2. T-⁠2
   A magnetisation-odd zero-bias conductance with a hand-odd part. The break junction is cut from a notched nickel wire; both tips are magnetised together along the wire axis by a ±0.5 T pulse and the applied field returned to zero, twenty reversals per junction in ABBA order, and 𝒴_𝔐 compares the tips’ two remanent magnetisations ±𝔐. The injected electrons’ held moments, polarised by the tips, cross the twist field where it changes along the junction axis at the molecule’s two ends, and patterns whose held and induced moments, summed per unit ℰ, differ along ∂_z𝖡 fall differently where they sit (0.1c). In the consensus account reciprocity with every field and magnetisation reversed together makes the conductance even in the tips’ magnetisation; the laboratory field, which the pulse does not reverse, adds a magnetisation-odd term even in the hand, common to both hands like the remanent-domain differences. The standard deviation is 1.4 × 10⁻³, the detection threshold near 4.2 × 10⁻³ and the false-rejection rate 0.00250.

Detection is rejection at 0.0027. A hand contrast of 6 × 10⁻³ in T-⁠1, or 8 × 10⁻³ in T-⁠2, is excluded when one-sided permutation tests at 0.00135 find the measured contrast below zero with each (P) cooldown lowered by that size and above zero with each raised by it; with no hand effect the rules exclude at 0.99950 and 0.901. A detection is read as the twist field when the material left in each cooldown’s heated source, recovered at warm-up, shows an enantiomeric excess of at least 95%, when the reversal-even part of the conductance agrees between the hands within three times the error of its hand difference, and, in T-⁠1, when the contrast at 0.5 T is half that at 1 T within three times the error of their difference.

Three tests read the density channel (I-⁠7). Their arms are perdeuterated dithioacetyl[13]helicene, every hydrogen position replaced by deuterium at an isotopic purity of at least 98%, and the protiated molecule. The density input is the rotor’s own compactness on the A-⁠1 chart, moved by the substitution at fixed pressure, radius and packing: for the bound rotor’s sixty-four deuterons, Δln α_ext = Δln 𝓜 = 0.0626. I-⁠7 fixes the input’s sign: for a purely density-coupled equilibrium observable a heavier isotope gives a larger R_s, a higher α and a larger response, opposite in sign to reduced-mass scaling. The fitted barrier, the conductance peak and the shoot-up voltage lie outside that scope. Each observable’s sign follows the product of the sensitivities S_i = ∂ln O/∂ln x_i; the density channel’s sensitivity in each is to be derived, and the analysis reads the larger response as a larger E_b/k_BT, a higher peak conductance and a larger barrier 𝔙₀. Each test is set at the residual its exclusion rule names, above the comparator’s upper end.

1. T-⁠3
   The isotope shift of the fitted rotation barrier. Fifteen junctions per cooldown with complete rate fits by the source’s cycle counting and ℛ₀ = 10¹⁰ s⁻¹ give O₃ = E_b/k_BT, the negative of the fitted intercept, at the 4.2 K bath, and the statistic Δln O₃, perdeuterated less protiated. The comparator holds the Born–Oppenheimer barrier and adds the channels the substitution moves: the attempt frequency’s inertia, which raises the fitted E_b/k_BT by ½Δln ℐ to Δln ℐ, with Δln ℐ from 0.06 to 0.14, and so Δln O₃ by +0.0004 to +0.0044 over the source’s barriers of 32.1 to 67; the barrier’s zero-point change, ±0.01; the friction prefactor, ±0.002; and the effective temperature, sensitivity −1, ±0.05. Joined and rounded outward, the interval is −0.07 to +0.07. With the density channel at the test-point residual of 0.35, Δln O₃ = 0.42 against the comparator’s upper end, 0.07: a difference of 0.35, beyond the threshold of 0.14 above that end.
2. T-⁠4
   The isotope shift of the molecular conductance peak. O₄ = 𝒢_peak/𝒢₀, the most probable conductance of the molecular peak in a 10,000-trace histogram of the source’s pull–push cycles without data selection, at 100 mV, the bias of the source’s Fig. S5 caption; its Supplementary text gives 50 mV. The Born–Oppenheimer transmission is unchanged, and its zero-point and vibronic-averaging change bounds the comparator at −0.02 to +0.02. The test point is 0.32 against the upper end, 0.02: a difference of 0.30, beyond the threshold of 0.12.
3. T-⁠5
   The isotope shift of the shoot-up voltage. 𝒱_shootup, the bias at which the current shoots up, as the source defines it, and per cooldown the median of ln 𝒱_shootup over thirty junctions. In the source’s critical relation the critical torque scales as η(𝔙₀/ℐ)^½; the density channel is read as moving the barrier 𝔙₀, sensitivity +½ in the critical torque. The comparator adds the inertia channel, −½Δln ℐ times the shoot-up voltage’s sensitivity to the critical torque, between 0 and 1; the friction’s isotope change, ±0.05 times that sensitivity; and the zero-point change of 𝔙₀, ±0.01, the inertia and friction channels carrying one sensitivity, taken at the same value in the join: −0.13 to +0.03. The test point is 0.23 against the upper end, 0.03: a difference of 0.20, beyond the threshold of 0.097.

Each isotope test is a one-sided rejection at 0.00135 after the comparator’s upper end is subtracted from the perdeuterated cooldowns, with executed false-rejection rates of 0.00115, 0.00150 and 0.00150 there. An FDT-sign residual of 0.35, 0.30 or 0.20 above that end is excluded when the one-sided test in the negative direction rejects; with no density channel the rules exclude at 0.99910, 0.99865 and 0.979. A detection is read as the density channel when the perdeuterated material meets its purity, each cooldown’s 1 𝒢₀ gold peak agrees between the compositions within 2%, the C–D stretch step replaces the C–H step in the inelastic spectra, and forty-eight further cooldowns of tert-butyl-d36 against the protiated molecule return an excess within three combined errors of the perdeuterated excess times the two rotors’ Δln 𝓜 ratio, 0.570, and more than three from it times each comparator channel’s ratio between the two rotors.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. The limiting map is the identity on the source’s own inputs and dictionary: given its biases, counts, fits, calculated geometries and model parameters, FDT returns the values it prints: set 1’s barrier, 32.1 k_B × 4.2 K, is the 11.62 meV of Supplementary Table 2, and its ratio, 32.1/41.6, the printed 0.77 V. The returned values are the counts of Supplementary Table 1 and the cyclic order of the levels with its inversion, the Arrhenius fits and Supplementary Table 2, the dwell fits, the backward-hop estimate, 10⁻²³ times the unbiased ratio at 420 mV, the critical torque-to-friction ratio of 0.42, the model’s junction-temperature estimate of 42 K, the calculated barriers, transmissions and torques, and the simulated crossover and current rise. Two prints differ from their partners. The simulated steady rotation is printed as an angular velocity of about 71 GHz and elsewhere as a frequency of about 70 GHz; read as an angular velocity, it is a frequency of 11 GHz. The main text’s slope range begins at 23 V⁻¹, a value Supplementary Table 2 does not print: its least slope is 32 ± 1 V⁻¹. In the gravitational window containment holds: for the same stress tensor and data FDT returns the curvature of general relativity (I-⁠3), and the rotor’s window is a small and correct number (I-⁠1). The mirror tests, with the twist field at its consensus value, zero, return zero hand contrast, and the isotope tests, with the density channel at zero, return each comparator interval; the predicted differences come from the twist field of P-⁠5 and the density channel of I-⁠7 alone, never from α, a chart or a branch tag.