Lattice constant

Alattice constantorlattice parameteris one of the physical dimensions and angles that determine the geometry of theunit cellsin acrystal lattice, and is proportional to the distance between atoms in the crystal. Asimple cubiccrystal has only one lattice constant, the distance between atoms, but in general lattices in three dimensions have six lattice constants: the lengthsa,b, andcof the three cell edges meeting at a vertex, and the anglesα,β, andγbetween those edges.
The crystal lattice parametersa,b, andchave the dimension of length. The three numbers represent the size of theunit cell, that is, the distance from a given atom to an identical atom in the same position and orientation in a neighboring cell (except for very simple crystal structures, this will not necessarily be distance to the nearest neighbor). TheirSI unitis themeter, and they are traditionally specified inangstroms(Å); an angstrom being 0.1nanometer(nm), or 100picometres(pm). Typical values start at a few angstroms. The anglesα,β, andγare usually specified indegrees.
Achemical substancein the solid state may formcrystalsin which theatoms,molecules, orionsare arranged in space according to one of a small finite number of possiblecrystal systems(lattice types), each with fairly well defined set of lattice parameters that are characteristic of the substance. These parameters typically depend on thetemperature,pressure(or, more generally, the local state ofmechanical stresswithin the crystal),electricandmagnetic fields, and itsisotopiccomposition.The lattice is usually distorted near impurities,crystal defects, and the crystal's surface. Parameter values quoted in manuals should specify those environment variables, and are usually averages affected by measurement errors.
Depending on the crystal system, some or all of the lengths may be equal, and some of the angles may have fixed values. In those systems, only some of the six parameters need to be specified. For example, in thecubic system, all of the lengths are equal and all the angles are 90°, so only thealength needs to be given. This is the case ofdiamond, which hasa= 3.57Å= 357pmat 300K. Similarly, inhexagonal system, theaandbconstants are equal, and the angles are 60°, 90°, and 90°, so the geometry is determined by theaandcconstants alone.
The lattice parameters of a crystalline substance can be determined using techniques such asX-ray diffractionor with anatomic force microscope. They can be used as a natural length standard of nanometer range.In theepitaxial growthof a crystal layer over a substrate of different composition, the lattice parameters must be matched in order to reduce strain and crystal defects.
The volume of the unit cell can be calculated from the lattice constant lengths and angles. If the unit cell sides are represented as vectors, then the volume is thescalar triple productof the vectors. The volume is represented by the letterV. For the general unit cell
For monoclinic lattices withα= 90°,γ= 90°, this simplifies to
For orthorhombic, tetragonal and cubic lattices withβ= 90°as well, then
Matching of lattice structures between two differentsemiconductor materialsallows a region ofband gapchange to be formed in a material without introducing a change in crystal structure. This allows construction of advancedlight-emitting diodesanddiode lasers.
For example,gallium arsenide,aluminium gallium arsenide, andaluminium arsenidehave almost equal lattice constants, making it possible to grow almost arbitrarily thick layers of one on the other one.
Typically, films of different materials grown on the previous film or substrate are chosen to match the lattice constant of the prior layer to minimize film stress.
An alternative method is to grade the lattice constant from one value to another by a controlled altering of the alloy ratio during film growth. The beginning of the grading layer will have a ratio to match the underlying lattice and the alloy at the end of the layer growth will match the desired final lattice for the following layer to be deposited.
The rate of change in the alloy must be determined by weighing the penalty of layer strain, and hence defect density, against the cost of the time in the epitaxy tool.
For example,indium gallium phosphidelayers with aband gapabove 1.9 eV can be grown ongallium arsenidewaferswith index grading.