Artificial Graphene
I have been actively working in projects regarding the nanofabrication and optical study of semiconductor-based replica of graphene, the so-called "artificial graphene". In the following, I introduce the topic I have been experimentally involved in.
In 2009 C.H. Park and S.G. Louie (C. Park, Nanolett. 9, 1793 (2009)) and M. Gibertini et al. independently published a paper on the possibility of recreating the band structure of graphene in a two-dimensional electron gas (2DEG) confined in a barrier of AlGaAs/GaAs.
They showed that the modulation of the 2DEG due to a periodic potential with hexagonal geometry and large lattice constant (about 100 nm) leads to the creation of isolated massless Dirac points, with a tunable Fermi velocity.
Consider in fact a 2DEG confined in a quantum well of AlGaAs/GaAs. The electrons have an effective mass mb = 0.067m, with m the mass of an electron in vacuum. The introduction of an external periodic potential with large lattice constant compared to that of GaAs suggests the use orthogonalized expansion in plane waves (OPW) for the calculation of the band structure.
They have been obtained assuming as radius of the muffin-tin disks r = 52.5nm, lattice constant a = 150nm and for three different values of V0: V0 = +1.0; -0.125; -0.8 meV. The energy minibands calculated by numerical simulations with the OPW method are shown in Fig 1 for V0=-0.8 meV, a = 150nm, r = 52.5nm.
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As can be seen from the figures, in all three cases Dirac points with linear dispersion occur. For attractive potentials, Gibertini et al. found that there is a potential threshold above which the obtained Dirac points are isolated. However, the effects of electron-electron correlation in the confined 2DEG inhibit the formation of massless Dirac fermions, making their observation a goal yet to be achieved.
In 2011, however, A. Singha et al. have shown that these structures are very suitable for the study of Mott-Hubbard many-body physics. From spectroscopic measurements of these patterns emerged a collective excitation which was unprecedented in an unmodulated 2DEG, which is attributed to the Mott-Hubbard excitation energy gap.
The modulation of the external potential, chosen to have hexagonal geometry as in the work of Gibertini et al., was achieved by digging a periodic pattern on the sample surface, with depth of about 55nm, with the technique of dry etching.
In detail, first it was first processed, by means of electron beam lithography, the pattern of disks with honeycomb geometry with pitch a = 130nm and radius r = 90nm (see Figure 2). Subsequently, nickel was evaporated to the exposed surface to create a mask for the dry etching. The artificial lattice covered an area of approximately 100μm x 100μm.
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The modulation of the potential is achieved due to a folding of the bands of the crystal: in the etched regions, the distance of the well from the surface is drastically reduced and the potential which is generated at the interface between the semiconductor and the air, weakly variable on the scale of nm, penetrates into the quantum well, raising all the bands of a constant amount of the order of meV. Thus, a periodic potential with the desired geometry is formed in the well.
The value of minimum V0 of the potential induced by the nanofabrication can be tuned experimentally by varying the depth of etching. A. Singha et al. have estimated the value of V0 to be about 4 meV.
The spectra of inelastic light scattering in presence of the magnetic field of the 2DEG showed, in addition to the existence of excitations with energy ∝B (due to transitions between Landau levels in the 2DEG), also excitations to lower energy, whose collective character is suggested by the intensity and spectral width of the peak, with a sublinear dependence on the magnetic field (√B), as shown in Fig 3. These modes have been identified as Hubbard modes and their nature was then attributed to on-site correlation effects of the electrons.
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