Twist-Controlled Tunneling in 2D Heterostructures

• Twist-controlled tunneling is a transport phenomenon where a relative twist between layers alters momentum alignment and local registry to modulate tunneling.
• It leverages momentum conservation and moiré-induced hybridization to yield resonant, suppressed, or spin-selective tunneling in 2D heterostructures.
• Experimental platforms such as graphene/hBN devices and STM studies reveal moiré-modulated local density of states and tunable tunneling responses.

Twist-controlled tunneling denotes a class of transport phenomena in which a relative twist degree of freedom governs the amplitude, selectivity, direction, or spectral structure of tunneling. In van der Waals heterostructures, the operative twist variable is usually the relative crystallographic angle between adjacent layers, which shifts valleys or Dirac cones in momentum space, generates moiré patterns, and reorganizes local stacking, hybridization, and screening. In that setting, tunneling can become resonant, suppressed, angle-selective, spin-selective, or many-body-enhanced depending on whether energy, in-plane momentum, chirality, spin-valley quantum numbers, or collective order are compatible across the junction. Closely related uses of the term also appear in optical and ultrafast scanning-probe contexts, where a geometric twist or waveform asymmetry controls coupling phases or current polarity rather than crystallographic registry (Mishchenko et al., 2014).

1. Conceptual basis and scope

In its most established condensed-matter form, twist-controlled tunneling arises because a relative rotation between two crystalline layers displaces their low-energy states in momentum space and changes their local registry. For small twist angles in graphene-based systems, this displacement is commonly expressed as $\Delta K \approx K\theta$ or related small-angle forms, making the twist angle a direct control parameter for the kinematic conditions of tunneling (Mishchenko et al., 2014). When a crystalline tunnel barrier such as hBN preserves in-plane translational symmetry sufficiently well, energy and in-plane momentum conservation become effective selection rules, and the bias or gate voltages can then be used to compensate the twist-induced mismatch.

A second, equally important mechanism is moiré-modulated local hybridization. In monolayer-on-metal or strongly coupled bilayer systems, twist changes the moiré wavelength and the distribution of local registries, which in turn modulate band edges, local density of states, charge transfer, and effective barrier profiles. In exfoliated single-layer MoS$_2$ on Au(111), for example, the local gap is defined as

$$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$

and both band edges and $\Delta(\vec r)$ oscillate across the moiré unit cell with a twist-angle-dependent amplitude (Pushkarna et al., 2023).

A broader use of the term encompasses systems in which the relevant “twist” is not a lattice rotation. In a twisted multicore fiber, uniform mechanical twisting generates a synthetic gauge phase that controls optical tunneling between cores (Parto et al., 2019). In attosecond STM, the paper on two-color laser control explicitly interprets “twist” as waveform asymmetry: changing the relative delay between a fundamental and its second harmonic “twists” the sub-cycle field and thereby controls the net tunneling direction (Davidovich et al., 14 Jul 2025).

2. Kinematic, orbital, and symmetry mechanisms

The most elementary twist-controlled tunneling mechanism is momentum matching. In graphene/hBN/graphene vertical junctions, the top and bottom Dirac cones are shifted by the twist angle, and resonant tunneling is strongest when bias compensates the twist-induced mismatch so that energy and in-plane momentum can be conserved simultaneously. In the small-angle regime, the main resonance occurs when

$\Delta U = \hbar v_F \Delta K,$

at which the cone intersection becomes effectively a straight line in momentum space and the current is maximized; beyond this point the available phase space collapses and negative differential conductance (NDC) appears (Mishchenko et al., 2014).

Orbital character strongly modifies this kinematic picture. In MoS$_2$/Au(111), the proposed coupling is mainly between sulfur $p$-orbitals and gold $d$-orbitals, strongest when the bottom sulfur atoms sit directly above Au atoms. Because the valence band has stronger out-of-plane orbital character than the conduction band, the valence-band maximum shifts more strongly than the conduction-band minimum, so the moiré-scale modulation of the tunneling spectra is especially pronounced on the valence side (Pushkarna et al., 2023). The same study emphasizes that hybridization is not fixed solely by nearest-neighbor distance; it also depends on local atomic environment and screening, motivating an effective distance $d_{\text{eff}}$ and an average charge-transfer estimate

$$\overline{\Delta Q}=\frac{1}{N}\sum_{i=1}^{N} f(d_{\text{eff}}(\vec r_i)).$$

Symmetry and internal quantum numbers introduce further selection rules. In WSe$_2$0-hBN-WSe$_2$1, a $_2$2 relative twist aligns equal valleys and therefore equal spins, enabling spin-conserving resonant tunneling near zero interlayer bias. By contrast, a $_2$3 twist aligns each valley with its time-reversed partner, so momentum can match geometrically while the low-energy spins do not, suppressing the zero-bias resonance (Kim et al., 2018). In biased twisted bilayer graphene, a perpendicular interlayer bias opens a gap and rotates the pseudospin texture sufficiently to suppress normal-incidence transmission, producing near-total reflection at low energy for small twist angles (Benlakhouy et al., 27 Jul 2025).

The same general logic extends beyond electrons. In a four-core twisted optical fiber, the nearest-neighbor coupling acquires a geometric phase,

$_2$4

and at the special value $_2$5 the amplitudes reaching the opposite core interfere destructively, yielding Aharonov–Bohm suppression of tunneling (Parto et al., 2019).

3. Resonant vertical transport in layered electronic heterostructures

The canonical device realization is the graphene/hBN/graphene resonant tunneling transistor. When two graphene electrodes are aligned to within about $_2$6 across a thin hBN barrier, the device exhibits a strong current peak, a pronounced NDC region, and tunable radio-frequency oscillations. The resonance is momentum-sensitive: applying an in-plane magnetic field adds

$_2$7

shifts the resonance, and confirms that the peak originates from momentum-conserving tunneling rather than disorder-assisted transport (Mishchenko et al., 2014).

Monolayer–bilayer graphene junctions generalize the same principle while enriching the resonance structure through the multiband spectrum of Bernal bilayer graphene. In graphene/hBN/bilayer-graphene transistors with top and back gates, the twist angle changes the momentum-conservation condition, moves resonant loci in the gate-bias plane, and can eliminate some features from the low-energy window. The current is then controlled simultaneously by bias $_2$8, back gate $_2$9, top gate $$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$0, and the twist-dependent momentum shift $$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$1 (Lane et al., 2015). A related monolayer-graphene/Bernal-bilayer-graphene vertical transistor shows sharp elastic resonances, weaker inelastic resonances, and NDC most clearly associated with a band-nesting-like resonance; in-plane magnetic field shifts the elastic resonances but hardly moves the inelastic ones, underscoring their different microscopic origins (Ghazaryan et al., 2021).

Twist can also restructure the nonlinear dynamics of sequential tunneling. In a twisted graphene trilayer with a floating middle layer, resonant tunneling between the source and middle layer competes with ordinary momentum-nonconserving tunneling between the middle and drain layers. The self-consistency condition

$$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$2

admits multiple steady solutions, producing intrinsic bistable $$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$3–$$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$4 characteristics controlled by the twist-dependent resonance scale $$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$5 and by interlayer geometry (Rodriguez-Nieva et al., 2015).

Transition-metal dichalcogenide heterostructures add spin-valley selection to this framework. In WSe$$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$6-hBN-WSe$$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$7, the $$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$8 devices show a sharp differential conductance peak at zero interlayer bias with a full width at half maximum of about $$\Delta(\vec r)=\mathrm{CBm}(\vec r)-\mathrm{VBM}(\vec r),$$9 mV at $\Delta(\vec r)$0 K, together with adjacent NDC. The same zero-bias resonance is negligible in the $\Delta(\vec r)$1 device, while higher-bias features remain because the relevant spin-split bands can align away from zero bias (Kim et al., 2018).

4. Moiré-resolved tunneling spectroscopy and spatially varying barriers

Scanning tunneling microscopy and spectroscopy have made twist-controlled tunneling directly visible in real space. In exfoliated single-layer MoS$\Delta(\vec r)$2 on Au(111), constant-current STM together with $\Delta(\vec r)$3 and $\Delta(\vec r)$4 maps show that the local density of states is modulated at the moiré wavelength, the brightest moiré maxima coincide with the strongest electronic effect, and the $\Delta(\vec r)$5 contrast can invert at certain biases. That inversion rules out a purely structural origin of the moiré pattern and identifies the contrast as fundamentally electronic. As the twist angle increases, the moiré period shrinks, the local registries become more homogeneous after screening, and the amplitudes of both the VBM and CBm modulations decrease monotonically and eventually vanish (Pushkarna et al., 2023).

Twisted bilayer graphene STM/STS reveals a complementary but more strongly correlated phenomenology. Across a continuous twist-angle range from $\Delta(\vec r)$6 to $\Delta(\vec r)$7, spanning the third, second, and first magic angles, the low-energy flat-band peaks evolve, merge, broaden, and redistribute spectral weight, while the remote conduction-band peak positively correlates with twist angle and is insensitive to strain. The flat-band spectral weight resides mainly on AA sites, whereas remote conduction-band features appear mainly on AB/BA domains and domain walls. Near the first magic angle, the anomalous transfer of spectral weight from FB1 to FB2 in the range $\Delta(\vec r)$8–$\Delta(\vec r)$9 suggests strong inter-flat-band interactions beyond simple single-particle or Hartree–Fock descriptions (Yu et al., 2024).

Twist inhomogeneity itself can become the tunneling object. In moiré graphene with twist-angle domains, a domain wall between regions of twist angles $\Delta U = \hbar v_F \Delta K,$0 and $\Delta U = \hbar v_F \Delta K,$1 acts as an effective step barrier with height

$\Delta U = \hbar v_F \Delta K,$2

where $\Delta U = \hbar v_F \Delta K,$3 is the van Hove singularity energy. This produces a zero-bias sub-meV transport gap that scales with the twist mismatch, suppresses Klein tunneling near the $\Delta U = \hbar v_F \Delta K,$4 point, and yields a differential Fano factor peak whose position and height measure the degree of twist-angle inhomogeneity (Padhi et al., 2020).

Electrostatic barriers in twisted bilayer graphene provide another spatially resolved scattering geometry. In a dual-gated model with representative twist angles $\Delta U = \hbar v_F \Delta K,$5, $\Delta U = \hbar v_F \Delta K,$6, and $\Delta U = \hbar v_F \Delta K,$7, perpendicular interlayer bias $\Delta U = \hbar v_F \Delta K,$8 suppresses normal-incidence transmission by opening a gap and generates strong angle-dependent and valley-dependent asymmetries in the transmission. Fabry–Pérot-like resonances shift in energy and intensity with bias, and the exact barrier transmission and reflection follow transfer-matrix expressions involving $\Delta U = \hbar v_F \Delta K,$9, $_2$0, and the evanescent component $_2$1 (Benlakhouy et al., 27 Jul 2025). A finite TBG superlattice with a defect barrier extends this picture further: decreasing twist changes the number, depth, and position of transmission gaps and resonance peaks, while the defect creates tunneling states inside transmission gaps whose energy can be tuned by the well width (Bahlauoi et al., 25 Dec 2025).

5. Many-body coherence, magnetism, and collective tunneling anomalies

Twist-controlled tunneling is not restricted to single-particle band alignment. In graphene double layers aligned to within about $_2$2, interlayer tunneling in the quantum Hall regime develops sharp zero-bias conductance peaks near $_2$3, $_2$4, and related imbalanced half-integer combinations corresponding to total filling $_2$5 or $_2$6. The peaks are narrower than the zero-field resonance, remain pinned to zero interlayer bias despite variations in layer filling factors, weaken with increasing temperature, and vanish by about $_2$7 K in the reported data. These properties are interpreted as signatures of interlayer phase coherence rather than ordinary density-of-states alignment (Lin et al., 2022).

Moiré magnetism introduces a different collective channel. In twisted double-bilayer CrI$_2$8, a twist near $_2$9 creates a moiré landscape of alternating ferromagnetic and antiferromagnetic interlayer exchange, generating frustrated noncollinear textures. Tunneling magnetoresistance then reads out the spin texture because conductance increases as spins become more parallel. Two distinct non-volatile zero-field textures—an out-of-plane domain state and an in-plane domain state—can have nearly the same net out-of-plane magnetization, so photocurrent magnetic circular dichroism alone cannot distinguish them, whereas TMR does. The experimental resistance difference between the two zero-field states is on the order of a few percent, while simulations predict about $p$0 contrast (Yang et al., 2024).

In twisted cuprate bilayers near $p$1, the tunneling structure controls whether a spontaneously time-reversal-breaking superconducting state also becomes fully gapped and topological. The disorder-mediated incoherent-tunneling model replaces strict momentum conservation by a random interlayer coupling with finite momentum transfer, yet still supports a fully gapped chiral phase with Chern number $p$2 for parameters argued to be relevant to Bi$p$3Sr$p$4CaCu$p$5O$p$6. Compared with the coherent, symmetry-constrained limit, increasing incoherence narrows the angular extent of the $p$7-broken wedge around $p$8 but does not immediately destroy it (Haenel et al., 2022).

A common misconception is that twist-controlled tunneling is exhausted by single-particle resonance geometry. The systems above show otherwise: zero-bias anomalies can reflect interlayer coherence, spin-filtering can resolve nearly degenerate magnetic textures, and incoherent tunneling can preserve rather than eliminate a topological phase (Lin et al., 2022).

6. Ultrafast, optical, and photonic extensions

The concept has been exported beyond electronic interlayer transport. In a uniformly twisted four-core optical fiber, the coupling Hamiltonian acquires complex nearest-neighbor phases, and the output intensity in the opposite core is governed by interference between supermodes. The result is a photonic realization of Aharonov–Bohm suppression of tunneling: at the special twist-induced phase $p$9, the opposite core remains dark for any propagation length, and the cancellation persists in nonlinear and multimode regimes (Parto et al., 2019).

A microwave analog appears in dual-layer wire metasurfaces. Two identical wire arrays separated by a small gap exhibit strong twist-angle dependence: the original resonance around $d$0–$d$1 MHz splits, new resonances emerge at lower frequencies, and field localization becomes moiré-like. Experimentally, the resonance shifts from about $d$2 MHz at $d$3 to about $d$4 MHz at $d$5, to resonances near $d$6 MHz and $d$7 MHz at $d$8, and to a lowest resonance near $d$9 MHz at $d_{\text{eff}}$0. The paper explicitly frames this as a photonic analog of twistronics, in which twist controls effective interlayer electromagnetic coupling rather than quantum-mechanical electron hopping (Torres et al., 2024).

Ultrafast STM introduces a temporal version of twist control. With a Pt:Ir nanotip over a gold substrate at zero bias, two-color laser pulses at $d_{\text{eff}}$1 nm with a second-harmonic intensity ratio of about $d_{\text{eff}}$2 synthesize an asymmetric electric field whose relative delay is modulated as

$d_{\text{eff}}$3

Changing that delay flips the dominant electron flow between tip $d_{\text{eff}}$4 sample and sample $d_{\text{eff}}$5 tip. The lock-in current oscillates with a $d_{\text{eff}}$6 fs period, the underlying mechanism is identified as non-adiabatic tunneling near Keldysh parameter $d_{\text{eff}}$7, and the current-burst duration is on the order of $d_{\text{eff}}$8 as. Directionality is quantified by

$d_{\text{eff}}$9

which reaches $$\overline{\Delta Q}=\frac{1}{N}\sum_{i=1}^{N} f(d_{\text{eff}}(\vec r_i)).$$0 at strong driving (Davidovich et al., 14 Jul 2025).

7. Experimental diagnostics, applications, and interpretive boundaries

Across platforms, twist-controlled tunneling has become a diagnostic as well as a control modality. In strained twisted bilayer graphene, the energy of the first remote conduction-band peak provides a one-to-one positive correlation with local twist angle and is reported to be insensitive to strain, allowing local twist-angle mapping in a single device (Yu et al., 2024). In twist-angle-domain transport, the differential Fano factor peak near threshold provides a global noise-based measure of local twist disorder (Padhi et al., 2020). In resonant graphene and WSe$$\overline{\Delta Q}=\frac{1}{N}\sum_{i=1}^{N} f(d_{\text{eff}}(\vec r_i)).$$1 tunnel junctions, in-plane magnetic field introduces a controlled momentum mismatch, shifting or suppressing resonances and thereby verifying the role of momentum conservation (Mishchenko et al., 2014).

The application space is correspondingly diverse. Graphene/hBN/graphene resonant tunneling diodes support NDC and MHz-range oscillations in an LC circuit, with the authors noting that much higher frequencies should be possible with reduced parasitic capacitance (Mishchenko et al., 2014). MoS$$\overline{\Delta Q}=\frac{1}{N}\sum_{i=1}^{N} f(d_{\text{eff}}(\vec r_i)).$$2/Au(111) suggests that twist angle can act as a tuning knob for hybridization and doping in monolayer-on-metal heterostructures, enabling spatially periodic electronic potentials and charge-transfer patterns (Pushkarna et al., 2023). Twisted CrI$$\overline{\Delta Q}=\frac{1}{N}\sum_{i=1}^{N} f(d_{\text{eff}}(\vec r_i)).$$3 provides non-volatile resistance states associated with distinct moiré spin textures (Yang et al., 2024). In attosecond STM, directional control of current bursts points toward triggering and detecting ultrafast charge dynamics at nanometer scales (Davidovich et al., 14 Jul 2025).

At the same time, the literature establishes clear interpretive boundaries. Twist does not always act monotonically. In hBN bilayers with embedded quantum emitters, the transition energy depends on both twist angle and local stacking, and the spectral shifts can change sign because different moiré sites generate different electrostatic environments (Gale et al., 6 Mar 2026). More generally, “twist-controlled tunneling” does not denote a single mechanism but a family of angle- or phase-controlled coupling phenomena. In some platforms the central variable is momentum mismatch, in others moiré hybridization, spin-valley locking, magnetic texture, incoherent pair tunneling, synthetic gauge phase, or temporal waveform asymmetry. This suggests that the unifying principle is not any one microscopic model, but the use of a twist degree of freedom to reprogram the tunneling selection rules and the resulting transport response.