Anti-Klein Tunneling in Graphene Systems

• Anti-Klein tunneling is a pseudospin- and chirality-governed phenomenon where electron transmission is suppressed at normal incidence, contrasting with perfect transmission in Klein tunneling.
• It is exemplified in bilayer graphene, where a sharp pn junction yields zero normal-incidence transmission, and can be modulated through gate-induced gaps and Berry-phase tuning.
• The concept extends beyond graphene to include multilayer systems, checkerboard lattices, and superconducting Dirac/Weyl junctions, highlighting its role in pseudospin-selective electron transport.

Anti-Klein tunneling denotes the suppression of transmission at particular incidence angles—most importantly at normal incidence—where Klein tunneling would instead yield perfect transmission. In graphene, the basic contrast is sharp: in monolayer graphene (MLG), a sharp $pn$ junction gives $T(0)=1$, whereas in Bernal bilayer graphene (BLG), for a sharp $pn$ step and normal incidence, the transmission is strongly suppressed and vanishes in the simplest model, $T(0)=0$. More broadly, anti-Klein tunneling is now understood as a pseudospin- and chirality-controlled reflection phenomenon appearing in multilayer graphene, gapped bilayers, anisotropic Dirac materials, checkerboard lattices, Weyl-based superconducting junctions, and several artificial-wave platforms (Elahi et al., 2022, Duppen et al., 2013, Huang et al., 2024, Betancur-Ocampo et al., 19 Dec 2025).

1. Microscopic definition and chiral origin

In graphene-based systems, anti-Klein tunneling is rooted in the pseudospin structure of low-energy Bloch states. For an $N$-layer Bernal-stacked graphene, the low-energy wavefunctions at momentum angle $\theta$ are two-component spinors of the form

$$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$

As $\theta$ winds around the Fermi surface, the pseudospin rotates by $N$ full turns, giving a Berry phase $N\pi$. The topology of this winding constrains the angular dependence of transmission and reflection across a $T(0)=1$0 junction (Elahi et al., 2022).

A compact statement of the odd–even distinction is that, for small incident angle $T(0)=1$1, the reflectivity behaves as

$T(0)=1$2

At normal incidence, $T(0)=1$3, odd $T(0)=1$4 therefore gives $T(0)=1$5 and $T(0)=1$6, while even $T(0)=1$7 gives maximal $T(0)=1$8 and minimal $T(0)=1$9. In MLG ($pn$0) this is Klein tunneling; in BLG ($pn$1) it is anti-Klein tunneling, namely perfect reflection at normal incidence in the ideal limit (Elahi et al., 2022).

The same logic was formulated for rhombohedral multilayers in terms of special angles

$pn$2

at which a component of pseudospin is conserved. For an $pn$3-layer rhombohedral stack at a symmetric $pn$4 junction, anti-Klein tunneling occurs at $pn$5 with $pn$6 even, while Klein tunneling occurs at $pn$7 with $pn$8 odd. Normal incidence is therefore KT for odd $pn$9 and AKT for even $T(0)=0$0 (Duppen et al., 2013).

2. Bilayer graphene, multilayers, and angular structure

The low-energy Hamiltonians make the distinction explicit. For MLG,

$T(0)=0$1

whereas BLG is effectively

$T(0)=0$2

The former describes massless Dirac fermions; the latter describes massive, chiral fermions with quadratic dispersion and double pseudospin winding. This change in chirality flips the normal-incidence behavior from KT to AKT (Elahi et al., 2022).

For Bernal-stacked $T(0)=0$3-layer graphene, a unified low-energy form is

$T(0)=0$4

Matching the two-component spinors at a sharp junction gives

$T(0)=0$5

with graphene Snell law

$T(0)=0$6

For BLG,

$T(0)=0$7

so that

$T(0)=0$8

This is the archetypal anti-Klein result (Elahi et al., 2022).

BLG also exhibits finite-angle transmission maxima. For even $T(0)=0$9, maxima occur at

$N$0

and for BLG ($N$1) this gives a single Brewster angle,

$N$2

The same condition can be parameterized as

$N$3

or equivalently $N$4. Thus BLG does not merely block head-on trajectories; it redirects transmission into oblique “sweet spots” (Elahi et al., 2022).

For arbitrary stacking, the low-energy theory can be decomposed into independent pseudospin doublets,

$N$5

with $N$6. Each doublet behaves like a rhombohedral subsystem with chirality $N$7, so KT and AKT angles follow the same parity rule with $N$8 (Duppen et al., 2013).

3. Experimental signatures and tunability in graphene

Direct transport evidence for anti-Klein tunneling was obtained in a circular “edgeless” Corbino geometry fabricated from gated graphene $N$9 junctions. In BLG $\theta$0–$\theta$1–$\theta$2 Corbino devices, the magnetoconductance $\theta$3 shows a pronounced dip at $\theta$4, because zero magnetic field selects nearly radial trajectories and therefore $\theta$5, where AKT predicts $\theta$6. As $\theta$7 increases, the trajectories bend toward the Brewster-like angle, producing local maxima or shoulders at finite $\theta$8. In MLG the trend is the opposite: $\theta$9–$$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$0–$$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$1 gives a peak at $$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$2, not a dip. In BLG the off-zero maxima migrate to higher fields as $$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$3 increases, in qualitative agreement with the analytical field scale

$$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$4

and with semi-classical ray-tracing plus Landauer-like conductance modeling (Elahi et al., 2022).

Anti-Klein tunneling in BLG is not immutable. In dual-gated BLG, a finite interlayer asymmetry $$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$5 opens a gap $$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$6, tilts pseudospin out of the plane, and breaks anti-Klein tunneling at normal incidence. For the $$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$7 junction considered in the theoretical analysis, the critical gap for perfect normal-incidence transmission is

$$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$8

and at $$\psi_C = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ e^{iN\theta}\end{pmatrix},\qquad \psi_V = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \ (-1)^{N+1}e^{iN\theta}\end{pmatrix}.$$9 the relevant pseudospin-polarization vectors satisfy

$\theta$0

This “evanescent-mode-assisted Klein tunneling” restores $\theta$1 even though the propagating pseudospins can remain anti-parallel. At the Klein point, the reflection phase shows a $\theta$2 jump at $\theta$3 (Huang et al., 27 Sep 2025).

Gate-controlled gap engineering in BLG also makes the Berry phase continuously tunable. In a high-quality Fabry–Pérot interferometer based on gapped BLG, the Berry phase was tuned from $\theta$4 down to $\theta$5, with a measured $\theta$6 point, and the corresponding normal-incidence transmission showed a transition from anti-Klein tunneling to nearly perfect Klein tunneling. In the phase-incoherent analysis, $\theta$7 reaches about $\theta$8 near $\theta$9 (Du et al., 2017). Earlier Fabry–Pérot measurements in a $N$0-long dual-gated BLG cavity had already shown that a gap opens in the device, that the gap destroys the perfect reflection for electrons traversing the barrier with normal incidence, and that the Berry phase is always involved in the oscillations regardless of magnetic field, in sharp contrast with single-layer graphene (Varlet et al., 2014).

4. Extensions beyond graphene

Once the reduced pseudospin rather than the microscopic material is taken as the organizing principle, anti-Klein tunneling appears in a wide range of electronic and wave systems. The recurring ingredients are chirality $N$1 or higher, strong anisotropy, mass generation, band inversion, or nontrivial interface matching.

In the deformed $N$2-$N$3 lattice, compressive uniaxial deformation merges the Dirac cones and opens a gap. For an $N$4 junction oriented perpendicular to the deformation direction, perfect Klein tunneling at normal incidence in the Dirac phase turns into anti-Klein tunneling in the gapped phase for all $N$5, and the super-Klein transparency of the dice limit $N$6 at $N$7 turns into anti-super-Klein tunneling, namely a totally opaque junction (Mandhour et al., 2020). In the 2D checkerboard lattice, the low-energy Hamiltonian

$N$8

has chirality $N$9; normally incident electron states can be perfectly reflected by a barrier with hole states inside, and a checkerboard nanotube can exhibit a perfect on-off ratio and act as a perfect “band filter” or “tunneling field effect transistor” (Hua et al., 2024).

Few-layer black phosphorus provides an anisotropic realization of the same logic. In FLBP $N\pi$0 junctions, electrons pass through the interface if it is oriented along the zigzag direction, but when the junction is rotated by $N\pi$1 and oriented along the armchair direction, the current is reflected completely for all angles of incidence and for a wide range of electron energies. This omni-directional total reflection is termed anti-super-Klein tunneling and is attributed to opposite pseudo-spins in the two regions of the junction; the effect persists even when oxidation strongly suppresses current in the top layer (Lizarraga-Brito et al., 7 Jul 2025).

A representative subset of non-graphene realizations is summarized below.

System	Mechanism	Signature
Deformed $N\pi$2-$N\pi$3 lattice	Cone merging and gap opening	KT $N\pi$4 AKT; dice limit gives anti-super-Klein (Mandhour et al., 2020)
2D checkerboard lattice	Chirality-2 fermions	Perfect normal reflection; perfect on-off ratio in nanotube (Hua et al., 2024)
Few-layer black phosphorus	Anisotropic pseudo-spin mismatch	Complete reflection for all angles in armchair-oriented $N\pi$5 junctions (Lizarraga-Brito et al., 7 Jul 2025)
Topological photonic crystals	Opposite Dirac masses / band inversion	Full reflection at the band edge $N\pi$6 for a trivial–topological interface at $N\pi$7 (Nakatsugawa et al., 2023)
Polariton graphene	TE–TM photonic SOC as emergent gauge field	Klein tunneling suppressed in a small energy range close to the Dirac points (Solnyshkov et al., 2015)

In topological photonic crystals, the language changes from charge transport to domain-wall transmission, but the structure remains recognizable. A trivial–topological interface, where the effective Dirac mass changes sign, shows full reflection at the band edge $N\pi$8 and $N\pi$9, not because propagating states are absent, but because band inversion changes the parity content of the states across the interface (Nakatsugawa et al., 2023). In polariton graphene, TE–TM photonic spin-orbit coupling can be written as an emergent gauge field and suppresses Klein tunneling in a small energy range close to the Dirac points; this is an anti-Klein-like regime generated by the SOC-induced deformation of the Dirac spectrum (Solnyshkov et al., 2015).

5. Superconducting and relativistic formulations

The anti-Klein concept also survives in superconducting Dirac and Weyl settings. In a Weyl-semimetal/superconductor junction, a single Weyl cone gives superconducting Klein tunneling: at normal incidence, Andreev reflection is perfect and the differential conductance doubles. If the single Weyl semimetal is replaced by a double Weyl semimetal, the behavior flips. In a DWSM/sDWSM junction, at normal incidence and for a strong interface barrier $T(0)=1$00,

$T(0)=1$01

so the incident electron is completely reflected as an electron. This is the superconducting analogue of anti-Klein tunneling in bilayer graphene and is tied to quadratic dispersion in two directions and double topological charge (Huang et al., 2024).

At the relativistic field-theory level, the relation between tunneling and pair creation becomes more subtle. For a Dirac step potential, the tunneling region $T(0)=1$02 splits into a Dirac tunneling subzone $T(0)=1$03, where the evanescent density under the step is electron-like, and a Klein tunneling subzone $T(0)=1$04, where the evanescent density is positron-like (Leo et al., 2013). A space-time resolved QFT treatment of a finite supercritical barrier then finds that no particle actually tunnels through the barrier, even in the case of resonant tunneling. Instead, the transmission is mediated by modulations in pair production rates at each edge of the barrier, caused by the incoming electron; for fermions this process saturates with barrier width, whereas for bosons it exhibits exponential superradiance (Alkhateeb et al., 2022). These results replace a literal through-barrier particle picture with an edge-mediated, pair-creation description.

6. General criterion, misconceptions, and current perspective

A general tight-binding framework places anti-Klein tunneling under a reduced-pseudospin criterion. In that formulation, interface scattering is encoded in effective two-component spinors $T(0)=1$05, and for a single interface the reflection coefficient can be written as

$T(0)=1$06

The relative pseudospin angle is

$T(0)=1$07

Klein tunneling corresponds to $T(0)=1$08, while anti-Klein tunneling corresponds to $T(0)=1$09: incident and transmitted reduced pseudospins are orthogonal, even though propagating modes may exist on both sides of the interface (Betancur-Ocampo et al., 19 Dec 2025).

This criterion corrects a common oversimplification. Anti-Klein tunneling is not merely total internal reflection or ordinary barrier opacity. In the systems above, the defining feature is pseudospin-enforced perfect or near-perfect reflection in the presence of allowed propagating states. Equally, AKT is not universal even within a fixed material class: in BLG it can be broken by a gate-induced gap, by Berry-phase tuning, or by evanescent-mode assistance, and in several anisotropic systems it can evolve into anti-super-Klein tunneling, where the entire angular sector is blocked (Huang et al., 27 Sep 2025, Du et al., 2017, Lizarraga-Brito et al., 7 Jul 2025).

Across the current literature, anti-Klein tunneling therefore functions as a unifying label for pseudospin-orthogonal interface scattering in low-dimensional materials and analog platforms. Its practical consequences are angular filtering, electron-optical mirrors, gate-tunable transmission gaps, and pseudospin-selective transport. The broader implication is that the same interface-matching principle now links bilayer graphene Corbino magnetoconductance, gapped Fabry–Pérot interferometers, checkerboard-lattice barriers, anti-super-Klein reflection in black phosphorus, and superconducting Weyl junctions into a single chiral-scattering taxonomy (Elahi et al., 2022, Du et al., 2017, Hua et al., 2024, Huang et al., 2024, Betancur-Ocampo et al., 19 Dec 2025).