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ger de and ludwig graf de 5 bragg and williams 6 and bethe 7 theories were based on the transition of arrangement of atoms in crystal lattices from disordered state to an ordered state mechanical properties edit j s koehler theoretically predicted 8 that by using alternate nano layers of materials with high and low elastic constants shearing resistance is improved by up to 100 times as the frank read source of dislocations cannot operate in the nanolayers the increased mechanical hardness of such superlattice materials was confirmed firstly by lehoczky in 1978 on al cu and al ag 9 and later on by several others such as barnett and sproul 10 on hard pvd coatings semiconductor properties edit if the superlattice is made of two semiconductor materials with different band gaps each quantum well sets up new selection rules that affect the conditions for charges to flow through the structure the two different semiconductor materials are deposited alternately on each other to form a periodic structure in the growth direction since the 1970 proposal of synthetic superlattices by esaki and tsu 11 advances in the physics of such ultra fine semiconductors presently called quantum structures have been made the concept of quantum confinement has led to the observation of quantum size effects in isolated quantum well heterostructures and is closely related to superlattices through the tunneling phenomena therefore these two ideas are often discussed on the same physical basis but each has different physics useful for applications in electric and optical devices semiconductor superlattice types edit superlattice miniband structures depend on the heterostructure type either type i type ii or type iii for type i the bottom of the conduction band and the top of the valence subband are formed in the same semiconductor layer in type ii the conduction and valence subbands are staggered in both real and reciprocal space so that electrons and holes are confined in different layers type iii superlattices involve semimetal material such as hgte cdte although the bottom of the conduction subband and the top of the valence subband are formed in the same semiconductor layer in type iii superlattice which is similar with type i superlattice the band gap of type iii superlattices can be continuously adjusted from semiconductor to zero band gap material and to semimetal with negative band gap another class of quasiperiodic superlattices is named after fibonacci a fibonacci superlattice can be viewed as a one dimensional quasicrystal where either electron hopping transfer or on site energy takes two values arranged in a fibonacci sequence semiconductor materials edit gaas alas superlattice and potential profile of conduction and valence bands along the growth direction z semiconductor materials which are used to fabricate the superlattice structures may be divided by the element groups iv iii v and ii vi while group iii v semiconductors especially gaas al x ga 1 x as have been extensively studied group iv heterostructures such as the si x ge 1 x system are much more difficult to realize because of the large lattice mismatch nevertheless the strain modification of the subband structures is interesting in these quantum structures and has attracted much attention in the gaas alas system both the difference in lattice constant between gaas and alas and the difference of their thermal expansion coefficient are small thus the remaining strain at room temperature can be minimized after cooling from epitaxial growth temperatures the first compositional superlattice was realized using the gaas al x ga 1 x as material system a graphene boron nitride system forms a semiconductor superlattice once the two crystals are aligned its charge carriers move perpendicular to the electric field with little energy dissipation h bn has a hexagonal structure similar to graphene s the superlattice has broken inversion symmetry locally topological currents are comparable in strength to the applied current indicating large valley hall angles 12 production edit superlattices can be produced using various techniques but the most common are molecular beam epitaxy mbe and sputtering with these methods layers can be produced with thicknesses of only a few atomic spacings an example of specifying a superlattice is fe 20 v 30 20 it describes a bi layer of 20å of iron fe and 30å of vanadium v repeated 20 times thus yielding a total thickness of 1000å or 100 nm the mbe technology as a means of fabricating semiconductor superlattices is of primary importance in addition to the mbe technology metal organic chemical vapor deposition mo cvd has contributed to the development of superconductor superlattices which are composed of quaternary iii v compound semiconductors like ingaasp alloys newer techniques include a combination of gas source handling with ultrahigh vacuum uhv technologies such as metal organic molecules as source materials and gas source mbe using hybrid gases such as arsine ash 3 and phosphine ph 3 have been developed generally speaking mbe is a method of using three temperatures in binary systems e g the substrate temperature the source material temperature of the group iii and the group v elements in the case of iii v compounds the structural quality of the produced superlattices can be verified by means of x ray diffraction or neutron diffraction spectra which contain characteristic satellite peaks other effects associated with the alternating layering are giant magnetoresistance tunable reflectivity for x ray and neutron mirrors neutron spin polarization and changes in elastic and acoustic properties depending on the nature of its components a superlattice may be called magnetic optical or semiconducting x ray and neutron scattering from the fe 20 v 30 20 superlattice miniband structure edit the schematic structure of a periodic superlattice is shown below where a and b are two semiconductor materials of respective layer thickness a and b period d a b displaystyle d a b when a and b are not too small compared with the interatomic spacing an adequate approximation is obtained by replacing these fast varying potentials by an effective potential derived from the band structure of the original bulk semiconductors it is straightforward to solve 1d schrödinger equations in each of the individual layers whose solutions ψ displaystyle psi are linear combinations of real or imaginary exponentials for a large barrier thickness tunneling is a weak perturbation with regard to the uncoupled dispersionless states which are fully confined as well in this case the dispersion relation e z k z displaystyle e_ z k_ z periodic over 2 π d displaystyle 2 pi d with over d a b displaystyle d a b by virtue of the bloch theorem is fully sinusoidal e z k z δ 2 1 cos k z d displaystyle e_ z k_ z frac delta 2 1 cos k_ z d and the effective mass changes sign for 2 π d displaystyle 2 pi d m ℏ 2 2 e k 2 k 0 displaystyle m frac hbar 2 partial 2 e partial k 2 _ k 0 in the case of minibands this sinusoidal character is no longer preserved only high up in the miniband for wavevectors well beyond 2 π d displaystyle 2 pi d is the top actually sensed and does the effective mass change sign the shape of the miniband dispersion influences miniband transport profoundly and accurate dispersion relation calculations are required given wide minibands the condition for observing single miniband transport is the absence of interminiband transfer by any process the thermal quantum k b t should be much smaller than the energy difference e 2 e 1 displaystyle e_ 2 e_ 1 between the first and second miniband even in the presence of the applied electric field bloch states edit for an ideal superlattice a complete set of eigenstates states can be constructed by products of plane waves e i k r 2 π displaystyle e i mathbf k cdot mathbf r 2 pi and a z dependent function f k z displaystyle f_ k z which satisfies the eigenvalue equation e c z z ℏ 2 2 m c z z ℏ 2 k 2 2 m c z f k z e f k z displaystyle left e_ c z frac partial partial z frac hbar 2 2m_ c z frac partial partial z frac hbar 2 mathbf k 2 2m_ c z right f_ k z ef_ k z as e c z displaystyle e_ c z and m c z displaystyle m_ c z are periodic functions with the superlattice period d the eigenstates are bloch state f k z ϕ q k z displaystyle f_ k z phi _ q mathbf k z with energy e ν q k displaystyle e nu q mathbf k within first order perturbation theory in k 2 one obtains the energy e ν q k e ν q 0 ϕ q k ℏ 2 k 2 2 m c z ϕ q k displaystyle e nu q mathbf k approx e nu q mathbf 0 langle phi _ q mathbf k mid frac hbar 2 mathbf k 2 2m_ c z mid phi _ q mathbf k rangle now ϕ q 0 z displaystyle phi _ q mathbf 0 z will exhibit a larger probability in the well so that it seems reasonable to replace the second term by e k ℏ 2 k 2 2 m w displaystyle e_ k frac hbar 2 mathbf k 2 2m_ w where m w displaystyle m_ w is the effective mass of the quantum well wannier functions edit by definition the bloch functions are delocalized over the whole superlattice this may provide difficulties if electric fields are applied or effects due to the superlattice s finite length are considered therefore it is often helpful to use different sets of basis states that are better localized a tempting choice would be the use of eigenstates of single quantum wells nevertheless such a choice has a severe shortcoming the corresponding states are solutions of two different hamiltonians each neglecting the presence of the other well thus these states are not orthogonal creating complications typically the coupling is estimated by the transfer hamiltonian within this approach for these reasons it is more convenient to use the set of wannier functions wannier stark ladder edit applying an electric field f to the superlattice structure causes the hamiltonian to exhibit an additional scalar potential eφ z efz that destroys the translational invariance in this case given an eigenstate with wavefunction φ 0 z displaystyle phi _ 0 z and energy e 0 displaystyle e_ 0 then the set of states corresponding to wavefunctions φ j z φ 0 z j d displaystyle phi _ j z phi _ 0 z jd are eigenstates of the hamiltonian with energies e j e 0 jefd these states are equally spaced both in energy and real space and form the so called wannier stark ladder the potential φ 0 z displaystyle phi _ 0 z is not bounded for the infinite crystal which implies a continuous energy spectrum nevertheless the characteristic energy spectrum of these wannier stark ladders could be resolved experimentally transport edit overview of the different standard approaches for superlattice transport the motion of charge carriers in a superlattice is different from that in the individual layers mobility of charge carriers can be enhanced which is beneficial for high frequency devices and specific optical properties are used in semiconductor lasers if an external bias is applied to a conductor such as a metal or a semiconductor typically an electric current is generated the magnitude of this current is determined by the band structure of the material scattering processes the applied field strength and the equilibrium carrier distribution of the conductor a particular case of superlattices called superstripes are made of superconducting units separated by spacers in each miniband the superconducting order parameter called the superconducting gap takes different values producing a multi gap or two gap or multiband superconductivity recently felix and pereira investigated the thermal transport by phonons in periodic 13 and quasiperiodic 14 15 16 superlattices of graphene hbn according to the fibonacci sequence they reported that the contribution of coherent thermal transport phonons like wave was suppressed as quasiperiodicity increased other dimensionalities edit soon after two dimensional electron gases 2deg had become commonly available for experiments research groups attempted to create structures 17 that could be called 2d artificial crystals the idea is to subject the electrons confined to an interface between two semiconductors i e along z direction to an additional modulation potential 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 take v x y v 0 cos 2 π x a cos 2 π y a v 0 0 displaystyle v x y v_ 0 cos 2 pi x a cos 2 pi y a v_ 0 0 as an example compared to the fermi level v 0 e f displaystyle v_ 0 gg e_ f 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 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 confined particle model suggests e 1 a 2 displaystyle e propto 1 a 2 this relation is only a rough guide and actual calculations with currently topical graphene a natural atomic crystal and artificial graphene 18 superlattice show that characteristic band widths are of the order of 1 ev and 10 mev respectively in the regime of weak modulation v 0 e f displaystyle v_ 0 ll e_ f phenomena like commensurability oscillations or fractal energy spectra 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 crystals applications edit the superlattice of palladium copper system is used in high performance alloys to enable a higher electrical conductivity which is favored by the ordered structure further alloying elements like silver rhenium rhodium and ruthenium are added for better mechanical strength and high temperature stability this alloy is used for probe needles in probe cards 19 see also edit cu pt type ordering in iii v semiconductor tube based nanostructures wannier function references edit johansson linde 1925 the x ray determination of the atomic arrangement in the mixed crystal series gold copper and palladium copper annalen der physik 78 21 439 bibcode 1925anp 383 439j doi 10 1002 andp 19253832104 bradley albert james jay a h 2 may 1932 the formation of superlattices in alloys of iron and aluminium proceedings of the royal society of london series a containing papers of a mathematical and physical character 136 829 210 232 doi 10 1098 rspa 1932 0075 issn 0950 1207 gorsky 1928 x ray investigations of transformations in the cuau alloy z phys 50 1 2 64 81 bibcode 1928zphy 50 64g doi ...