Velocity-Tunable Klein Gap in Dirac Systems

• Velocity-tunable Klein gap is a tunable energy window in quantum Dirac systems where controlled carrier velocity modulates tunneling behavior.
• This phenomenon leverages electrostatic gating, space–time modulation, and lattice engineering to adjust pseudospin, Berry phase, and interface sharpness.
• Its precise control enables novel device applications such as electron-optic switches, tunable transistors, and advanced quantum simulators.

A velocity-tunable Klein gap refers to an energy or momentum (and thus velocity) window, determined by electrostatic gating, band structure engineering, space/time potential modulation, or analogous mechanisms, within which Klein tunneling transitions from perfect transmission to pronounced suppression and vice versa. In quantum Dirac systems such as graphene, bilayer graphene, designer photonic and phononic lattices, and engineered space–time modulated materials, the phenomenon is rooted in the interplay between carrier velocity, pseudospin polarization, Berry phase, and interface sharpness or anisotropy. Control over this gap, and thus over transmission properties, emerges as a central paradigm for both fundamental transport studies and advanced nanoelectronic, optoelectronic, and quantum device applications.

1. Fundamental Mechanism and Theoretical Framework

Klein tunneling is characterized by perfect transmission of Dirac-like quasiparticles through potential barriers at normal incidence, even when the barrier is classically impenetrable. In pristine monolayer graphene, this is a direct consequence of chirality and pseudospin conservation in the Dirac Hamiltonian, whereby the wavefunctions on opposite sides of a sharp barrier preserve overlap at normal incidence.

The concept of a velocity-tunable Klein gap is realized when transmission probability, $T(\theta)$, is modulated as a function of incident particle velocity or its angular momentum component. Mechanisms that induce such a gap include:

• Electrostatic gating and local bandgap opening in graphene or bilayer graphene, which can modify the velocity–momentum relation and alter pseudospin alignment, leading to a tunable window of suppressed transmission at specific densities, momenta, or velocities (Cheraghchi et al., 2013, Du et al., 2017, Huang et al., 27 Sep 2025).
• In fiber Bragg gratings, the mapping of the Dirac equation to coupled optical waves creates a photonic stop band whose shape, location, and transmission window are controlled by the local “group velocity” (chirp) and index profile, allowing analogues of velocity-tunable Klein gaps (Longhi, 2010).
• Space–time modulated potentials in relativistic quantum systems, where an interface moving at subluminal speed introduces a finite-width Klein gap whose position and width are explicit functions of the modulation velocity (Ok et al., 24 Oct 2025).
• Lattice engineering and deformations (e.g., in the $\alpha$-T$_3$ model) that drive transitions from Dirac-like (linear) to massive (quadratic) dispersion along selected directions, directly affecting velocity selectivity in transmission (Mandhour et al., 2020).
• Photonic and phononic crystals or micro/nanoelectromechanical systems, where the local “Fermi velocity” or group velocity can be engineered spatially, producing clearly tunable windows for perfect or suppressed Klein transmission (Zhang et al., 4 Mar 2024, Lee et al., 8 Aug 2024).

2. Model Systems and Experimental Realizations

Several systems exemplify velocity-tunable Klein gaps:

Platform	Tunability Parameter	Key Klein Gap Feature
Graphene/Bilayer Graphene	Gate bias, Fermi energy, interface type	Transmission window set by pseudospin/velocity
Fiber Bragg Gratings	Chirp length, local group velocity	Transmission window in stop band
Dirac Photonic/Phononic Lattice	Effective “mass” and group velocity	Frequency-dependent perfect transmission
Dual-gated BLG	Bandgap Δ (via gate), Berry phase	Switch from anti-Klein to Klein tunneling
Space–time modulated Dirac	Modulation speed $v_m$, scalar/vector potential	Velocity-tunable Klein gap with reduced thresholds (Ok et al., 24 Oct 2025)

In graphene p–n junction devices, velocity tuning is achieved by adjusting the carrier density profile, which sets the local Fermi wavevector $k_F$ and also controls the critical angle for transmission through Snell’s law analogues:

$$\sin \phi_1 = \nu \sin \theta_1, \quad \nu = -\sqrt{n/p}$$

and the critical angle: $\theta_c = \arcsin \sqrt{p/n}$ For incident angles $|\theta_1| > \theta_c$, total internal reflection yields a suppression (“Klein gap”) in transmission, tunable continuously by carrier density (Wilmart et al., 2014, Dauber et al., 2020).

In bilayer graphene, independent top/bottom gates modulate the band gap $\Delta$ and thus the pseudospin orientation. When the propagating and evanescent pseudospin polarization vectors at the n-p interface become orthogonal, perfect transmission ("revival" of Klein tunneling) is restored, at a tunable critical $\Delta_c$: $\Delta_c = E \cos[\arctan(E/(V-E))]$ The Berry phase also becomes a tunable function of junction parameters (Huang et al., 27 Sep 2025), transitioning between $2\pi$, $\pi$, and $0$ (Du et al., 2017, Park et al., 2011).

In photonic and phononic lattices, group velocity engineering via refractive index or structure design enables transmission windows (“Klein gaps”) whose location and width are frequency- or group velocity-selective (Longhi, 2010, Lee et al., 8 Aug 2024, Zhang et al., 4 Mar 2024).

3. Mathematical Description and Transmission Formulas

Transmission through a potential barrier in a Dirac system with different local velocities (or effective mass terms) is generally given by:

$$T = \frac{\cos^2\phi\, \cos^2\theta} {\cos^2\phi\, \cos^2\theta \cos^2(q_x D) + \sin^2(q_x D)[1 + \sin\phi\,\sin\theta]^2}$$

where group velocities, incidence angle $\phi$, and transmission angle $\theta$ are set by the local Fermi velocities and potential profiles (Lee et al., 8 Aug 2024).

In graphene p–n–p structures or Klein-tunneling transistors, the angular filtering can be modeled as:

$$T(\phi) \simeq \exp\left[ -\frac{\pi d k_p^2 \sin^2 \phi}{(k_p + k_n)} \right]$$

and current density: $$\frac{I}{W} = \frac{4e^2}{h} \frac{k_p}{\pi} T V_{ds}$$ where $d$ is the interface width, and $k_p, k_n$ are the Fermi wavevectors in p- and n-like regions (Wilmart et al., 2014).

In space–time modulated Dirac systems (Ok et al., 24 Oct 2025), Klein gap boundaries are determined by the relation: $$\left[ (E_i - qV_2 ) - v_m (p_i - qA_2 ) \right]^2 - (m/\gamma_m )^2 = 0$$ with gap width

$\Delta_V = (2m/\gamma_m)(1 - v_m r_A)$

and threshold voltage lowered to

$q\Delta V^{(th)} = 2m e^{-\omega_g}$

when $v_m$ approaches the group velocity $v_g$ ($\omega_g = \text{arctanh} \, v_g$).

4. Physical Interpretation and Role of Pseudospin, Berry Phase, and Interface

The core physical mechanism of velocity-tunable Klein gaps stems from the interplay of:

• Pseudospin orientation: In graphene and BLG, the alignment or mismatch of pseudospin polarization vectors (propagating/evanescent, left/right of a junction) dictates the probability of transmission at normal or tilted incidence. Gate-tunable gaps arise from manipulating these vectors via external bias or layer asymmetry (Huang et al., 27 Sep 2025, Park et al., 2011, Du et al., 2017).
• Berry phase: The Berry phase, varying as a function of Fermi energy and gap, modulates interference and Klein/anti-Klein tunneling transitions (Du et al., 2017, Park et al., 2011). Control over this phase by gating corresponds to dynamic tuning of the Klein gap.
• Interface geometry/sharpness: The width and abruptness of a potential interface modulate the tunneling regime (sharp: Klein regime; smooth: suppressed tunneling/“gap”), with the effective tuning parameter being the normalized width $d_{pn} = w_{pn} / \lambda_F$ (Schelter et al., 2010, Dauber et al., 2020).
• Lattice effects: At the atomic scale, discrete lattice hopping and interface phase factors shift the Klein transmission peak from normal incidence to a finite (“tilted”) angle, thus enabling both gap opening and velocity selectivity via tuning of chemical potential or atomic-scale interface (Zhang et al., 2018).

5. Device Architectures, Applications, and Experimental Control

Device designs exploiting velocity-tunable Klein gaps include:

• Graphene Klein tunneling transistors (GKTFETs) and logic switches, which employ collimating and analyzing p–n junctions to create gate-tunable angular filtering and transmission gaps (Tan et al., 2017, Elahi et al., 2018).
• Dual-gated BLG devices that realize electrical control from anti-Klein to Klein tunneling, suitable for electron-optic switches or interferometers (Huang et al., 27 Sep 2025, Du et al., 2017).
• Photonic and phononic analogues—fiber Bragg gratings or nanoelectromechanical metamaterials—with controllable bandgaps and transmission windows in the optical or acoustic domain (Longhi, 2010, Lee et al., 8 Aug 2024).
• Space–time modulated Dirac systems, which enable Klein tunneling below the static (Schwinger) threshold and are promising for ultra-fast, low-threshold electronic devices or quantum simulators (Ok et al., 24 Oct 2025).

Experimental tunability is achieved via global or local gate voltages, modulation of barrier width or height, spatial/temporal interface engineering, electrostatic carrier injection, strain, or dynamic group velocity control (optical, phononic, or electronic).

6. Broader Implications, Limitations, and Future Directions

Velocity-tunable Klein gaps provide a powerful paradigm for tailoring quantum transport, collimation, and beam steering without sacrificing high carrier mobility—enabling ballistic transistors with high ON/OFF ratios, narrow-band filters, and phase-coherent electron or wave manipulation.

Key limitations and challenges include:

• Edge and interface roughness-induced leakage in electron-optics devices, which remap an ideal gap to a pseudogap and degrade ON/OFF ratios (Elahi et al., 2018).
• Precise control of space–time modulation and velocity matching for relativistic kinematic regimes (Ok et al., 24 Oct 2025).
• Realization in systems with finite disorder or scaling to high carrier densities where lattice or many-body effects become essential (Zhang et al., 2018, Zhang et al., 4 Mar 2024).

Future research will likely expand to:

• Quantum information devices requiring coherent phase manipulation via Berry-phase engineering.
• Valleytronic architectures exploiting valley-selective Klein gaps (An et al., 2019).
• Advanced photonic, phononic, and hybrid classical/quantum platforms, where effective particle velocity and Klein gap are dynamically reconfigurable.
• Theoretical exploration of multi-gap, nonreciprocal, and topological variants under strong space–time modulation or many-body interaction.

• For details of the tight-binding and recursive Green’s function formalism in graphene rings and control of AB oscillations: (Schelter et al., 2010)
• For fiber Bragg grating analogues and the role of group velocity and Compton-like length: (Longhi, 2010)
• For Berry phase tunability and Klein/anti-Klein transitions in bilayer graphene: (Du et al., 2017, Park et al., 2011)
• For device-level engineering in bilayer graphene via velocity or gate control: (Cheraghchi et al., 2013, Huang et al., 27 Sep 2025, Wilmart et al., 2014, Tan et al., 2017)
• For semiclassical, wave-packet, and Berry curvature effects: (Puetter et al., 2012)
• For velocity-tunable Klein gaps via space–time modulation of potentials: (Ok et al., 24 Oct 2025)
• For lattice-induced tilted Klein tunneling: (Zhang et al., 2018)
• For direct measurement of Klein tunneling efficiency in interferometric settings: (Dauber et al., 2020)
• For phononic analogues and group velocity engineering: (Lee et al., 8 Aug 2024)
• For quantized Klein tunneling and hybrid spinor states in photonic Dirac chains: (Zhang et al., 4 Mar 2024)
• For high-temperature superconducting pair tunneling via Klein processes: (Perconte et al., 2019)

This body of research demonstrates that the Klein gap—far from fixed by underlying material symmetry—can be dynamically tuned by external parameters controlling velocity, bandgap, pseudospin, or spatial/temporal modulation, suggesting broad prospects for fundamental studies and next-generation device engineering.