Lattice constant

Unit cell definition using parallelopiped with lengths a, b, c and angles between the sides given by α, β, γ^[1]

The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c. However, in the special case of cubic crystal structures, all of the constants are equal and we only refer to a. Similarly, in crystal structures, the a and b constants are equal, and we only refer to the a and c constants. A group of lattice constants could be referred to as lattice parameters. However, the full set of lattice parameters consist of the three lattice constants and the three angles between them.

For example, the lattice constant for diamond is a = 3.57 Å at 300 K. The structure is equilateral although its actual shape cannot be determined from only the lattice constant. Furthermore, in real applications, typically the average lattice constant is given. Near the crystal's surface, lattice constant is affected by the surface reconstruction that results in a deviation from its mean value. This deviation is especially important in nano-crystals since surface to nano-crystal core ratio is large.^[2] As lattice constants have the dimension of length, their SI unit is the meter. Lattice constants are typically on the order of several angstroms (i.e. tenths of a nano-metre). Lattice constants can be determined using techniques such as X-ray diffraction or with an atomic force microscope. Lattice constant of a crystal can be used as a natural length standard of nanometer range.^[3][4]

In epitaxial growth, the lattice constant is a measure of the structural compatibility between different materials. Lattice constant matching is important for the growth of thin layers of materials on other materials; when the constants differ, strains are introduced into the layer, which prevents epitaxial growth of thicker layers without defects.

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 dot product of one vector with the cross product of the other two vectors. The volume is represented by the letter V. For the general unit cell $V = a b c \sqrt{1+2\cos(\alpha)\cos(\beta)\cos(\gamma)-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)}.$ For monoclinic lattices with α = 90°, γ = 90°, this simplifies to $V = a b c \sin(\beta).$ For orthorhombic, tetragonal and cubic lattices with β = 90° as well, then $V = a b c$.^[5]

Lattice matching

Matching of lattice structures between two different semiconductor materials allows a region of band gap change to be formed in a material without introducing a change in crystal structure. This allows construction of advanced light-emitting diodes and diode lasers.

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, indium gallium phosphide layers with a band gap above 1.9 eV can be grown on gallium arsenide wafers with index grading.

List of Lattice Constants at 300K

Material	Lattice Constant (Å)	Crystal Structure	Ref
C (diamond)	3.567	Diamond (FCC)	[6]
C (graphite)	2.461(a); 6.708(c)	Hexagonal
Si	5.431	Diamond (FCC)	[7]
Ge	5.658	Diamond (FCC)	[7]
AlAs	5.6605	Zinc blende (FCC)	[7]
AlP	5.4510	Zinc blende (FCC)	[7]
AlSb	6.1355	Zinc blende (FCC)	[7]
GaP	5.4505	Zinc blende (FCC)	[7]
GaAs	5.653	Zinc blende (FCC)	[7]
GaSb	6.0959	Zinc blende (FCC)	[7]
InP	5.869	Zinc blende (FCC)	[7]
InAs	6.0583	Zinc blende (FCC)	[7]
InSb	6.479	Zinc blende (FCC)	[7]
MgO	4.212	Rocksalt (FCC)
SiC	3.086(a); 10.053 (c)	Wurtzite	[7]
CdS	5.8320	Zinc blende (FCC)	[6]
CdSe	6.050	Zinc blende (FCC)	[6]
CdTe	6.482	Zinc blende (FCC)	[6]
ZnO	3.25 (a); 5.2 (c)	Wurtzite (HCP)	[8]
ZnO	4.580	Rocksalt (FCC)	[6]
ZnS	5.420	Zinc blende (FCC)	[6]
PbS	5.9362	Rocksalt (FCC)	[6]
PbTe	6.4620	Rocksalt (FCC)	[6]
BN	3.6150	Zinc blende (FCC)	[6]
BP	4.5380	Zinc blende (FCC)	[6]
CdS	4.160(a); 6.756(c)	Wurtzite	[6]
ZnS	3.82(a); c=6.26(c)	Wurtzite	[6]
AlN	3.112(a); 4.982(c)	Wurtzite	[7]
GaN	3.189(a); 5.185(c)	Wurtzite	[7]
InN	3.533(a); 5.693(c)	Wurtzite	[7]
LiF	4.03	Rocksalt
LiCl	5.14	Rocksalt
LiBr	5.50	Rocksalt
LiI	6.01	Rocksalt
NaF	4.63	Rocksalt
NaCl	5.64	Rocksalt
NaBr	5.97	Rocksalt
NaI	6.47	Rocksalt
KF	5.34	Rocksalt
KCl	6.29	Rocksalt
KBr	6.60	Rocksalt
KI	7.07	Rocksalt
RbF	5.65	Rocksalt
RbCl	6.59	Rocksalt
RbBr	6.89	Rocksalt
RbI	7.35	Rocksalt
CsF	6.02	Rocksalt
CsCl	4.123	Cesium Chloride
CsI	4.567	Cesium Chloride
Al	4.046	FCC	[9]
Fe	2.856	BCC	[9]
Ni	3.499	FCC	[9]
Cu	3.597	FCC	[9]
Mo	3.142	BCC	[9]
Pd	3.859	FCC	[9]
Ag	4.079	FCC	[9]
W	3.155	BCC	[9]
Pt	3.912	FCC	[9]
Au	4.065	FCC	[9]
Pb	4.920	FCC	[9]
TiN	4.249	Rocksalt
ZrN	4.577	Rocksalt
HfN	4.392	Rocksalt
VN	4.136	Rocksalt
CrN	4.149	Rocksalt
NbN	4.392	Rocksalt
TiC	4.328	Rocksalt	[10]
ZrC0.97	4.698	Rocksalt	[10]
HfC0.99	4.640	Rocksalt	[10]
VC0.97	4.166	Rocksalt	[10]
NC0.99	4.470	Rocksalt	[10]
TaC0.99	4.456	Rocksalt	[10]
Cr3C2	11.47(a); 5.545(b); 2.830(c)	Orthorombic	[10]
WC	2.906(a);2.837(c)	Hexagonal	[10]
ScN	4.52	Rocksalt	[11]
LiNbO3	5.1483(a);13.8631(c)	Hexagonal	[12]
KTaO3	3.9885(a)	Cubic perovskite	[12]
BaTiO3	3.994(a);4.034(c)	Tetragonal perovskite	[12]
SrTiO3	3.98805(a)	Cubic perovskite	[12]
CaTiO3	5.381(a);5.443(b);7.645(c);	Orthorhombic perovskite	[12]
PbTiO3	3.904(a);4.152(c);	Tetragonal perovskite	[12]
EuTiO3	7.810(a)	Cubic perovskite	[12]
SrVO3	3.838(a)	Cubic perovskite	[12]
CaVO3	3.767(a)	Cubic perovskite	[12]
BaMnO3	5.673(a);4.71(c)	Hexagonal	[12]
CaMnO3	5.27(a);5.275(b);7.464(c);	Orthorhombic perovskite	[12]
SrRuO3	5.53(a);5.57(b);7.85(c);	Orthorhombic perovskite	[12]
YAlO3	5.179(a);5.329(b);7.37(c);	Orthorhombic perovskite	[12]

References

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2. Mudar A. Abdulsattar, Solid State Sci. 13, 843 (2011).
3. R. V. Lapshin (1998). "Automatic lateral calibration of tunneling microscope scanners" (PDF). Review of Scientific Instruments (USA: AIP) 69 (9): 3268–3276. doi:10.1063/1.1149091. ISSN 0034-6748.
4. R. V. Lapshin (2015). "Drift-insensitive distributed calibration of probe microscope scanner in nanometer range: Approach description" (PDF). Applied Surface Science (Netherlands: Elsevier B. V.) 359: 629–636. doi:10.1016/j.apsusc.2015.10.108. ISSN 0169-4332.
5. Dept. of Crystallography & Struc. Biol. CSIC (4 June 2015). "4. Direct and reciprocal lattices". Retrieved 9 June 2015.
6. "Lattice Constants". Argon National Labs (Advanced Photon Source). Retrieved 19 October 2014.
7. "Semiconductor NSM". Retrieved 19 October 2014.
8. Hadis Morkoç and Ümit Özgur (2009). Zinc Oxide: Fundamentals, Materials and Device Technology. Weinheim: WILEY-VCH Verlag GmbH & Co.
9. Davey, Wheeler (1925). "PRECISION MEASUREMENTS OF THE LATTICE CONSTANTS OF TWELVE COMMON METALS". Physical Review 25: 753–761. Bibcode:1925PhRv...25..753D. doi:10.1103/PhysRev.25.753.
10. Toth, L.E. (1967). Transition Metal Carbides and Nitrides. New York: Academic Press.
11. Saha, B. (2010). "Electronic structure, phonons, and thermal properties of ScN, ZrN, and HfN: A first-principles study". Journal of Applied Physics 107: 033715. Bibcode:2010JAP...107c3715S. doi:10.1063/1.3291117.
12. 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.