Alattice constant orlattice parameter is one of the physical dimensions and angles that determine the geometry of theunit cells in acrystal lattice, and is proportional to the distance between atoms in the crystal. Asimple cubic crystal has only one lattice constant, the distance between atoms, but, in general, lattices in three dimensions have six lattice constants: the lengthsa,b, andc of the three cell edges meeting at a vertex, and the anglesα,β, andγ between those edges.
The crystal lattice parametersa,b, andc have 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 unit is 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, andγ are usually specified indegrees.
Achemical substance in the solid state may formcrystals in which theatoms,molecules, orions are 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 stress within the crystal),[2]electric andmagnetic fields, and itsisotopic composition.[3] 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 thea length needs to be given. This is the case ofdiamond, which hasa = 3.57Å = 357pm at 300 K. Similarly, inhexagonal system, thea andb constants are equal, and the angles are 60°, 90°, and 90°, so the geometry is determined by thea andc constants alone.
The lattice parameters of a crystalline substance can be determined using techniques such asX-ray diffraction or with anatomic force microscope. They can be used as a natural length standard of nanometer range.[4][5] In theepitaxial growth of 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 product of 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[6]
Matching of lattice structures between two differentsemiconductor materials allows a region ofband gap change to be formed in a material without introducing a change in crystal structure. This allows construction of advancedlight-emitting diodes anddiode lasers.
For example,gallium arsenide,aluminium gallium arsenide, andaluminium arsenide have 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 phosphide layers with aband gap above 1.9 eV can be grown ongallium arsenidewafers with index grading.
| Material | Lattice constant (Å) | Crystal structure | Ref. |
|---|---|---|---|
| C (diamond) | 3.567 | Diamond (FCC) | [7] |
| C (graphite) | a = 2.461 c = 6.708 | Hexagonal | |
| Si | 5.431020511 | Diamond (FCC) | [8][9] |
| Ge | 5.658 | Diamond (FCC) | [8] |
| AlAs | 5.6605 | Zinc blende (FCC) | [8] |
| AlP | 5.4510 | Zinc blende (FCC) | [8] |
| AlSb | 6.1355 | Zinc blende (FCC) | [8] |
| GaP | 5.4505 | Zinc blende (FCC) | [8] |
| GaAs | 5.653 | Zinc blende (FCC) | [8] |
| GaSb | 6.0959 | Zinc blende (FCC) | [8] |
| InP | 5.869 | Zinc blende (FCC) | [8] |
| InAs | 6.0583 | Zinc blende (FCC) | [8] |
| InSb | 6.479 | Zinc blende (FCC) | [8] |
| MgO | 4.212 | Halite (FCC) | [10] |
| SiC | a = 3.086 c = 10.053 | Wurtzite | [8] |
| CdS | 5.8320 | Zinc blende (FCC) | [7] |
| CdSe | 6.050 | Zinc blende (FCC) | [7] |
| CdTe | 6.482 | Zinc blende (FCC) | [7] |
| ZnO | a = 3.25 c = 5.2 | Wurtzite (HCP) | [11] |
| ZnO | 4.580 | Halite (FCC) | [7] |
| ZnS | 5.420 | Zinc blende (FCC) | [7] |
| PbS | 5.9362 | Halite (FCC) | [7] |
| PbTe | 6.4620 | Halite (FCC) | [7] |
| BN | 3.6150 | Zinc blende (FCC) | [7] |
| BP | 4.5380 | Zinc blende (FCC) | [7] |
| CdS | a = 4.160 c = 6.756 | Wurtzite | [7] |
| ZnS | a = 3.82 c = 6.26 | Wurtzite | [7] |
| AlN | a = 3.112 c = 4.982 | Wurtzite | [8] |
| GaN | a = 3.189 c = 5.185 | Wurtzite | [8] |
| InN | a = 3.533 c = 5.693 | Wurtzite | [8] |
| LiF | 4.03 | Halite | |
| LiCl | 5.14 | Halite | |
| LiBr | 5.50 | Halite | |
| LiI | 6.01 | Halite | |
| NaF | 4.63 | Halite | |
| NaCl | 5.64 | Halite | |
| NaBr | 5.97 | Halite | |
| NaI | 6.47 | Halite | |
| KF | 5.34 | Halite | |
| KCl | 6.29 | Halite | |
| KBr | 6.60 | Halite | |
| KI | 7.07 | Halite | |
| RbF | 5.65 | Halite | |
| RbCl | 6.59 | Halite | |
| RbBr | 6.89 | Halite | |
| RbI | 7.35 | Halite | |
| CsF | 6.02 | Halite | |
| CsCl | 4.123 | Caesium chloride | |
| CsBr | 4.291 | Caesium chloride | |
| CsI | 4.567 | Caesium chloride | |
| Al | 4.046 | FCC | [12] |
| Fe | 2.856 | BCC | [12] |
| Ni | 3.499 | FCC | [12] |
| Cu | 3.597 | FCC | [12] |
| Mo | 3.142 | BCC | [12] |
| Pd | 3.859 | FCC | [12] |
| Ag | 4.079 | FCC | [12] |
| W | 3.155 | BCC | [12] |
| Pt | 3.912 | FCC | [12] |
| Au | 4.065 | FCC | [12] |
| Pb | 4.920 | FCC | [12] |
| V | 3.0399 | BCC | |
| Nb | 3.3008 | BCC | |
| Ta | 3.3058 | BCC | |
| TiN | 4.249 | Halite | |
| ZrN | 4.577 | Halite | |
| HfN | 4.392 | Halite | |
| VN | 4.136 | Halite | |
| CrN | 4.149 | Halite | |
| NbN | 4.392 | Halite | |
| TiC | 4.328 | Halite | [13] |
| ZrC0.97 | 4.698 | Halite | [13] |
| HfC0.99 | 4.640 | Halite | [13] |
| VC0.97 | 4.166 | Halite | [13] |
| NbC0.99 | 4.470 | Halite | [13] |
| TaC0.99 | 4.456 | Halite | [13] |
| Cr3C2 | a = 11.47 b = 5.545 c = 2.830 | Orthorhombic | [13] |
| WC | a = 2.906 c = 2.837 | Hexagonal | [13] |
| ScN | 4.52 | Halite | [14] |
| LiNbO3 | a = 5.1483 c = 13.8631 | Hexagonal | [15] |
| KTaO3 | 3.9885 | Cubic perovskite | [15] |
| BaTiO3 | a = 3.994 c = 4.034 | Tetragonal perovskite | [15] |
| SrTiO3 | 3.98805 | Cubic perovskite | [15] |
| CaTiO3 | a = 5.381 b = 5.443 c = 7.645 | Orthorhombic perovskite | [15] |
| PbTiO3 | a = 3.904 c = 4.152 | Tetragonal perovskite | [15] |
| EuTiO3 | 7.810 | Cubic perovskite | [15] |
| SrVO3 | 3.838 | Cubic perovskite | [15] |
| CaVO3 | 3.767 | Cubic perovskite | [15] |
| BaMnO3 | a = 5.673 c = 4.71 | Hexagonal | [15] |
| CaMnO3 | a = 5.27 b = 5.275 c = 7.464 | Orthorhombic perovskite | [15] |
| SrRuO3 | a = 5.53 b = 5.57 c = 7.85 | Orthorhombic perovskite | [15] |
| YAlO3 | a = 5.179 b = 5.329 c = 7.37 | Orthorhombic perovskite | [15] |
