Lattice constant
A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the
The crystal lattice parameters a, b, and c have the dimension of length. The three numbers represent the size of the
Introduction
A
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 the
The lattice parameters of a crystalline substance can be determined using techniques such as
Volume
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 the scalar triple product of the vectors. The volume is represented by the letter V. For the general unit cell
For monoclinic lattices with α = 90°, γ = 90°, this simplifies to
For orthorhombic, tetragonal and cubic lattices with β = 90° as well, then[6]
Lattice matching
Matching of lattice structures between two different
For example, gallium arsenide, aluminium gallium arsenide, and aluminium arsenide have almost equal lattice constants, making it possible to grow almost arbitrarily thick layers of one on the other one.
Lattice grading
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,
List of lattice constants
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] |
References
- ^ "Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α, β, γ". Archived from the original on 4 October 2008.
- ISSN 0034-6748.
- S2CID 119191299.
- ^ Dept. of Crystallography & Struc. Biol. CSIC (4 June 2015). "4. Direct and reciprocal lattices". Retrieved 9 June 2015.
- ^ a b c d e f g h i j k l "Lattice Constants". Argon National Labs (Advanced Photon Source). Retrieved 19 October 2014.
- ^ a b c d e f g h i j k l m n o "Semiconductor NSM". Retrieved 19 October 2014.
- ^ "Fundamental physical constants". physics.nist.gov. NIST. Retrieved 17 January 2020.
- ^ "Substrates". Spi Supplies. Retrieved 17 May 2017.
- ^ Hadis Morkoç and Ümit Özgur (2009). Zinc Oxide: Fundamentals, Materials and Device Technology. Weinheim: WILEY-VCH Verlag GmbH & Co.
- ^ .
- ^ a b c d e f g h Toth, L.E. (1967). Transition Metal Carbides and Nitrides. New York: Academic Press.
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- ^ a b c d e f g h i j k l m Goodenough, J. B.; Longo, M. "3.1.7 Data: Crystallographic properties of compounds with perovskite or perovskite-related structure, Table 2 Part 1". SpringerMaterials - The Landolt-Börnstein Database.