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Introduction

A chemical substance in the solid state may form crystals in which the atoms, molecules, or ions are arranged in space according to one of a small finite number of possible crystal systems (lattice types), each with fairly well defined set of lattice parameters that are characteristic of the substance. These parameters typically depend on the temperature, pressure (or, more generally, the local state of mechanical stress within the crystal),¹ electric and magnetic fields, and its isotopic composition.² 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 the cubic system, all of the lengths are equal and all the angles are 90°, so only the a length needs to be given. This is the case of diamond, which has a = 3.57 Å = 357 pm at 300 K. Similarly, in hexagonal system, the a and b constants are equal, and the angles are 60°, 90°, and 90°, so the geometry is determined by the a and c constants alone.

The lattice parameters of a crystalline substance can be determined using techniques such as X-ray diffraction or with an atomic force microscope. They can be used as a natural length standard of nanometer range.³⁴ In the epitaxial 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.

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

V = a b c 1 + 2 cos ⁡ α cos ⁡ β cos ⁡ γ − cos 2 ⁡ α − cos 2 ⁡ β − cos 2 ⁡ γ . {\displaystyle V=abc{\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 ⁡ β . {\displaystyle V=abc\sin \beta .}

For orthorhombic, tetragonal and cubic lattices with β = 90° as well, then⁵

V = a b c . {\displaystyle V=abc.}

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

Lattice constants for various materials at 300 K

Material	Lattice constant (Å)	Crystal structure	Ref.
C (diamond)	3.567	Diamond (FCC)	6
C (graphite)	a = 2.461c = 6.708	Hexagonal
Si	5.431020511	Diamond (FCC)	78
Ge	5.658	Diamond (FCC)	9
AlAs	5.6605	Zinc blende (FCC)	10
AlP	5.4510	Zinc blende (FCC)	11
AlSb	6.1355	Zinc blende (FCC)	12
GaP	5.4505	Zinc blende (FCC)	13
GaAs	5.653	Zinc blende (FCC)	14
GaSb	6.0959	Zinc blende (FCC)	15
InP	5.869	Zinc blende (FCC)	16
InAs	6.0583	Zinc blende (FCC)	17
InSb	6.479	Zinc blende (FCC)	18
MgO	4.212	Halite (FCC)	19
SiC	a = 3.086c = 10.053	Wurtzite	20
CdS	5.8320	Zinc blende (FCC)	21
CdSe	6.050	Zinc blende (FCC)	22
CdTe	6.482	Zinc blende (FCC)	23
ZnO	a = 3.25c = 5.2	Wurtzite (HCP)	24
ZnO	4.580	Halite (FCC)	25
ZnS	5.420	Zinc blende (FCC)	26
PbS	5.9362	Halite (FCC)	27
PbTe	6.4620	Halite (FCC)	28
BN	3.6150	Zinc blende (FCC)	29
BP	4.5380	Zinc blende (FCC)	30
CdS	a = 4.160c = 6.756	Wurtzite	31
ZnS	a = 3.82c = 6.26	Wurtzite	32
AlN	a = 3.112c = 4.982	Wurtzite	33
GaN	a = 3.189c = 5.185	Wurtzite	34
InN	a = 3.533c = 5.693	Wurtzite	35
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	36
Fe	2.856	BCC	37
Ni	3.499	FCC	38
Cu	3.597	FCC	39
Mo	3.142	BCC	40
Pd	3.859	FCC	41
Ag	4.079	FCC	42
W	3.155	BCC	43
Pt	3.912	FCC	44
Au	4.065	FCC	45
Pb	4.920	FCC	46
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	47
ZrC0.97	4.698	Halite	48
HfC0.99	4.640	Halite	49
VC0.97	4.166	Halite	50
NbC0.99	4.470	Halite	51
TaC0.99	4.456	Halite	52
Cr3C2	a = 11.47b = 5.545c = 2.830	Orthorhombic	53
WC	a = 2.906c = 2.837	Hexagonal	54
ScN	4.52	Halite	55
LiNbO3	a = 5.1483c = 13.8631	Hexagonal	56
KTaO3	3.9885	Cubic perovskite	57
BaTiO3	a = 3.994c = 4.034	Tetragonal perovskite	58
SrTiO3	3.98805	Cubic perovskite	59
CaTiO3	a = 5.381b = 5.443c = 7.645	Orthorhombic perovskite	60
PbTiO3	a = 3.904c = 4.152	Tetragonal perovskite	61
EuTiO3	7.810	Cubic perovskite	62
SrVO3	3.838	Cubic perovskite	63
CaVO3	3.767	Cubic perovskite	64
BaMnO3	a = 5.673c = 4.71	Hexagonal	65
CaMnO3	a = 5.27b = 5.275c = 7.464	Orthorhombic perovskite	66
SrRuO3	a = 5.53b = 5.57c = 7.85	Orthorhombic perovskite	67
YAlO3	a = 5.179b = 5.329c = 7.37	Orthorhombic perovskite	68

External links

• How to Find Lattice Constant

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