This article provides a pedagogical review on Klein tunneling in graphene, i.e. the peculiar tunneling properties of two-dimensional massless Dirac electrons. We consider two simple situations in detail: a massless Dirac electron incident either on a potential step or on a potential barrier and use elementary quantum wave mechanics to obtain the transmission probability. We emphasize the connection to related phenomena in optics, such as the Snell-Descartes law of refraction, total internal reflection, Fabry-Perot resonances, negative refraction index materials (the so called meta-materials), etc. We also stress that Klein tunneling is not a genuine quantum tunneling effect as it does not necessarily involve passing through a classically forbidden region via evanescent waves. A crucial role in Klein tunneling is played by the conservation of (sublattice) pseudo-spin, which is discussed in detail. A major consequence is the absence of backscattering at normal incidence, of which we give a new shorten proof. The current experimental status is also thoroughly reviewed. The Appendix contains the discussion of a one-dimensional toy model that clearly illustrates the difference in Klein tunneling between mono- and bi-layer graphene.

https://materias.df.uba.ar/e2a2014v/files/2014/02/epjb301.pdf

Klein tunneling in graphene: optics with massless electrons

The analogies between optical and electronic Goos–Hanchen effects are established based on electron wave optics in semiconductor or graphene-based nanostructures. In this paper, we give a brief overview of the progress achieved so far in the field of electronic Goos–Hanchen shifts, and show the relevant optical analogies. In particular, we present several theoretical results on the giant positive and negative Goos–Hanchen shifts in various semiconductor or graphene-based nanostructures, their controllability, and potential applications in electronic devices, e.g. spin (or valley) beam splitters.

https://iopscience.iop.org/article/10.1088/2040-8978/15/3/033001/pdf

Electronic analogy of the Goos–Hänchen effect: a review*

We investigate electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice. It is found that such structure can possess a zero-k¯ gap which exists in all Fibonacci sequences. Different from Bragg gap, zero-k¯ gap associated with Dirac point is less sensitive to the incidence angle and lattice constants. The defect mode appeared inside the zero-k¯ gap has a great effect on transmission, conductance, and shot noise, which can be applicable to control the electron transport.

Electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice

We have studied the electronic properties in aperiodic graphene superlattices of Thue-Morse sequence. Although the structure is aperiodic, an unusual Dirac point (DP) does exist and its location is exactly at the energy corresponding to the zero-averaged wave number (zero-k¯). Furthermore, the zero-k¯ gap associated with the DP is robust against the lattice constants and the incident angles, and multi-DPs may appear under the suitable conditions. A resultant controllability of electronic transport in Thue-Morse sequence is predicted, which may facilitate the development of many graphene-based electronics.

Electronic band gaps and transport in aperiodic graphene superlattices of Thue-Morse sequence

We investigate electronic band gap and transport in Fibonacci quasi-periodic graphene superlattice It is found that such structure can possess a zero-$\bar{k}$ gap which exists in all Fibonacci sequences Different from Bragg gap, zero-$\bar{k}$ gap associated with Dirac point is less sensitive to the incidence angle and lattice constants The defect mode appeared inside the zero-$\bar{k}$ gap has a great effect on transmission, conductance and shot noise, which can be applicable to control the electron transport

We have investigated the transmission in monolayer graphene barrier at nonzero angle of incidence. Taking the influence of parallel wave vector into account, the transmission as the function of incidence energy has a gap due to the evanescent waves in two cases of Klein tunneling and classical motion. The modulation of the transmission gap by the incidence angle, the height, and width of potential barrier may lead to potential applications in graphene-based electronic devices.

Design of electron wave filters in monolayer graphene by tunable transmission gap

The quantum Goos-Hanchen effect in graphene is found to be the lateral shift of Dirac fermions on the total reflection at a single p-n interface. In this paper, we investigate the lateral shifts of Dirac fermions in transmission through a monolayer graphene barrier. Compared to the smallness of the lateral shifts in total reflection, the lateral shifts can be enhanced by the transmission resonances when the incidence angle is less than the critical angle for total reflection. It is also found that the lateral shifts, as the function of the barrier's width and incidence angle, can be negative and positive in the cases of Klein tunneling and classical motion. The modulation of the lateral shifts can be realized by changing the electrostatic potential and induced gap, which gives rise to some applications in graphene-based devices.

Goos-Hanchen-like shifts for Dirac fermions in monolayer graphene barrier

We investigate the Goos-H\"{a}nchen-like shifts for Dirac fermions in transmission through a monolayer graphene barrier. The lateral shifts, as the functions of the barrier's width and the incidence angle, can be negative and positive in Klein tunneling and classical motion, respectively. Due to their relations to the transmission gap, the lateral shifts can be enhanced by the transmission resonances when the incidence angle is less than the critical angle for total reflection, while their magnitudes become only the order of Fermi wavelength when the incidence angle is larger than the critical angle. These tunable beam shifts can also be modulated by the height of potential barrier and the induced gap, which gives rise to the applications in graphene-based devices.

Jia-Wei Tao

Papers

Design of electron wave filters in monolayer graphene by tunable transmission gap

Goos-Hanchen-like shifts for Dirac fermions in monolayer graphene barrier

Design of electron wave filters in monolayer graphene by tunable transmission gap

Goos-H\"{a}nchen-like shifts for Dirac fermions in monolayer graphene barrier