The so-called Klein paradox—unimpeded penetration of relativistic particles through high and wide potential barriers—is one of the most exotic and counterintuitive consequences of quantum electrodynamics. The phenomenon is discussed in many contexts in particle, nuclear and astro-physics but direct tests of the Klein paradox using elementary particles have so far proved impossible. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment using electrostatic barriers in single- and bi-layer graphene. Owing to the chiral nature of their quasiparticles, quantum tunnelling in these materials becomes highly anisotropic, qualitatively different from the case of normal, non-relativistic electrons. Massless Dirac fermions in graphene allow a close realization of Klein’s gedanken experiment, whereas massive chiral fermions in bilayer graphene offer an interesting complementary system that elucidates the basic physics involved.

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Chiral tunnelling and the Klein paradox in graphene

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New approaches to nanofabrication: molding, printing, and other techniques.

The properties of quasi-two-dimensional semiconductor quantum dots are reviewed. Experimental techniques for measuring the electronic shell structure and the effect of magnetic fields are briefly described. The electronic structure is analyzed in terms of simple single-particle models, density-functional theory, and "exact" diagonalization methods. The spontaneous magnetization due to Hund's rule, spin-density wave states, and electron localization are addressed. As a function of the magnetic field, the electronic structure goes through several phases with qualitatively different properties. The formation of the so-called maximum-density droplet and its edge reconstruction is discussed, and the regime of strong magnetic fields in finite dot is examined. In addition, quasi-one-dimensional rings, deformed dots, and dot molecules are considered. (Less)

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Electronic structure of quantum dots

The observation of oscillations in the conductance characteristics of narrow graphene p–n-junctions confirms their ability to collimate ballistic carriers. Moreover, the phase of these oscillations at low magnetic field suggests the occurrence of the perfect transmission of carriers normal to the junction as a direct result of the Klein effect. The observation of quantum conductance oscillations in mesoscopic systems has traditionally required the confinement of the carriers to a phase space of reduced dimensionality1,2,3,4. Although electron optics such as lensing5 and focusing6 have been demonstrated experimentally, building a collimated electron interferometer in two unconfined dimensions has remained a challenge owing to the difficulty of creating electrostatic barriers that are sharp on the order of the electron wavelength7. Here, we report the observation of conductance oscillations in extremely narrow graphene heterostructures where a resonant cavity is formed between two electrostatically created bipolar junctions. Analysis of the oscillations confirms that p–n junctions have a collimating effect on ballistically transmitted carriers8. The phase shift observed in the conductance fringes at low magnetic fields is a signature of the perfect transmission of carriers normally incident on the junctions9 and thus constitutes a direct experimental observation of ‘Klein tunnelling’10,11,12.

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Quantum interference and Klein tunnelling in graphene heterojunctions

I. Introduction (Preface, Nanostructures in Si Inversion Layers, Nanostructures in GaAs-AlGaAs Heterostructures, Basic Properties). 
II. Diffusive and Quasi-Ballistic Transport (Classical Size Effects, Weak Localization, Conductance Fluctuations, Aharonov-Bohm Effect, Electron-Electron Interactions, Quantum Size Effects, Periodic Potential). 
III. Ballistic Transport (Conduction as a Transmission Problem, Quantum Point Contacts, Coherent Electron Focusing, Collimation, Junction Scattering, Tunneling). 
IV. Adiabatic Transport (Edge Channels and the Quantum Hall Effect, Selective Population and Detection of Edge Channels, Fractional Quantum Hall Effect, Aharonov-Bohm Effect in Strong Magnetic Fields, Magnetically Induced Band Structure).

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Quantum Transport in Semiconductor Nanostructures

We have succeeded in fabricating a two-dimensional electron gas (2DEG) on the cleaved (110) edge of a GaAs wafer by molecular beam epitaxy (MBE). A (100) wafer previously prepared by MBE growth is reinstalled in the MBE chamber so that an in situ cleave exposes a fresh (110) GaAs edge for further MBE overgrowth. A sequence of Si-doped AlGaAs layers completes the modulation-doped structure at the cleaved edge. Mobilities as high as 6.1×10^5 cm^2/V s are measured in the 2DEG at the cleaved interface.

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Formation of a high quality two-dimensional electron gas on cleaved GaAs

We introduce an electrostatic lens for ballistic electrons in two‐dimensional (2D) systems and demonstrate its focusing action in very high mobility GaAs‐(AlGa)As heterostructures. This is the first refractive element for the control of 2D electrons. It exemplifies the close analogy between ballistic propagation in 2D electron systems and traditional geometrical optics.

Electron focusing in two‐dimensional systems by means of an electrostatic lens

We have been able to fabricate a two‐dimensional electron gas containing an atomically precise, lateral Kronig–Penney potential of 102 A periodicity. The structure was formed by modulation‐doped molecular beam epitaxy overgrowth on the cleaved edge of a 71 A GaAs/31 A AlGaAs compositional superlattice. Low‐temperature magnetotransport reveals clear quantum Hall characteristics. From the onset of the Shubnikov–de Haas oscillations at 0.25 T we deduce a lower limit of the mobility of μ≳40 000 cm2/V s at an electron density of 3.0×1011 cm−2 and infer that the carriers are crossing more than 200 GaAs/AlGaAs interfaces without losing phase coherence.

Atomically precise superlattice potential imposed on a two-dimensional electron gas

Ballistic electron transport beyond 100 microm IN 2D electron systems

A refractive electrostatic prism is used to switch a beam of ballistic electrons between different collectors in the two‐dimensional electron gas of an AlGaAs‐GaAs heterostructure. This represents a new concept in electronic switching which utilizes the electrostatically controlled refraction of ballistic electrons in high‐mobility two‐dimensional systems.

J. Spector

Papers

Formation of a high quality two-dimensional electron gas on cleaved GaAs

Electron focusing in two‐dimensional systems by means of an electrostatic lens

Atomically precise superlattice potential imposed on a two-dimensional electron gas

Ballistic electron transport beyond 100 microm IN 2D electron systems

Refractive switch for two‐dimensional electrons