Editing Superlattice (section)

== Other dimensionalities ==

Soon after two-dimensional electron gases ([[2DEG]]) had become commonly available for experiments, research groups attempted to create structures<ref>{{Cite journal | last1 = Heitmann | first1 = D. | last2 = Kotthaus | first2 = J. R. P. | doi = 10.1063/1.881355 | title = The Spectroscopy of Quantum Dot Arrays | journal = Physics Today | volume = 46 | issue = 6 | pages = 56 | year = 1993 | bibcode = 1993PhT....46f..56H }}</ref> that could be called 2D artificial crystals. The idea is to subject the electrons confined to an [[Heterojunction|interface between two semiconductors]] (i.e. along ''z''-direction) to an additional modulation potential {{as written|''V''(''x'',''y'')}}. Contrary to the classical superlattices (1D/3D, that is 1D modulation of electrons in 3D bulk) described above, this is typically achieved by treating the heterostructure surface: depositing a suitably patterned metallic gate or etching. If the amplitude of ''V''(''x'',''y'') is large ({{as written|take <math>V(x,y)=-V_0(\cos 2\pi x/a+\cos 2\pi y/a), V_0>0</math>}} as an example) compared to the Fermi level, <math>|V_0|\gg E_f</math>, the electrons in the superlattice should behave similarly to electrons in an atomic crystal with square lattice (in the example, these "atoms" would be located at positions ({{as written|''na'',''ma''}}) where ''n'',''m'' are integers).

The difference is in the length and energy scales. Lattice constants of atomic crystals are of the order of 1Å while those of superlattices (''a'') are several hundreds or thousands larger as dictated by technological limits (e.g. electron-beam lithography used for the patterning of the heterostructure surface). Energies are correspondingly smaller in superlattices. Using the simple quantum-mechanically [[Particle in a box|confined-particle]] model suggests <math>E\propto 1/a^2</math>. This relation is only a rough guide and actual calculations with currently topical [[graphene]] (a natural atomic crystal) and artificial graphene<ref>{{Cite journal | last1 = Kato | first1 = Y. | last2 = Endo | first2 = A. | last3 = Katsumoto | first3 = S. | last4 = Iye | first4 = Y. | title = Geometric resonances in the magnetoresistance of hexagonal lateral superlattices | doi = 10.1103/PhysRevB.86.235315 | journal = Physical Review B | volume = 86 | issue = 23 | pages = 235315 | year = 2012 |arxiv = 1208.4480 |bibcode = 2012PhRvB..86w5315K | s2cid = 119289481 }}</ref> (superlattice) show that characteristic band widths are of the order of 1 eV and 10 meV, respectively. In the regime of weak modulation (<math>|V_0|\ll E_f</math>), phenomena like commensurability oscillations or fractal energy spectra ([[Hofstadter's butterfly|Hofstadter butterfly]]) occur.

Artificial two-dimensional crystals can be viewed as a 2D/2D case (2D modulation of a 2D system) and other combinations are experimentally available: an array of quantum wires (1D/2D) or 3D/3D [[photonic crystal]]s.