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 nanocrystals since surface to nanocrystal 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 nanometre). Lattice constants can be determined using techniques such as Xray diffraction or with an atomic force microscope.
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.
Contents

Volume 1

Lattice matching 2

Lattice grading 3

List of Lattice Constants at 300K 4

References 5
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 .^{[3]}
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 lightemitting 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)

^{[4]}

C (graphite)

2.461(a); 6.708(c)

Hexagonal


Si

5.431

Diamond (FCC)

^{[5]}

Ge

5.658

Diamond (FCC)

^{[5]}

AlAs

5.6605

Zinc blende (FCC)

^{[5]}

AlP

5.4510

Zinc blende (FCC)

^{[5]}

AlSb

6.1355

Zinc blende (FCC)

^{[5]}

GaP

5.4505

Zinc blende (FCC)

^{[5]}

GaAs

5.653

Zinc blende (FCC)

^{[5]}

GaSb

6.0959

Zinc blende (FCC)

^{[5]}

InP

5.869

Zinc blende (FCC)

^{[5]}

InAs

6.0583

Zinc blende (FCC)

^{[5]}

InSb

6.479

Zinc blende (FCC)

^{[5]}

MgO

4.212

Rocksalt (FCC)

[2]

SiC

3.086(a); 10.053 (c)

Wurtzite

^{[5]}

CdS

5.8320

Zinc blende (FCC)

^{[4]}

CdSe

6.050

Zinc blende (FCC)

^{[4]}

CdTe

6.482

Zinc blende (FCC)

^{[4]}

ZnO

4.580

Rocksalt (FCC)

^{[4]}

ZnS

5.420

Zinc blende (FCC)

^{[4]}

PbS

5.9362

Rocksalt (FCC)

^{[4]}

PbTe

6.4620

Rocksalt (FCC)

^{[4]}

BN

3.6150

Zinc blende (FCC)

^{[4]}

BP

4.5380

Zinc blende (FCC)

^{[4]}

CdS

4.160(a); 6.756(c)

Wurtzite

^{[4]}

ZnS

3.82(a); c=6.26(c)

Wurtzite

^{[4]}

AlN

3.112(a); 4.982(c)

Wurtzite

^{[5]}

GaN

3.189(a); 5.185(c)

Wurtzite

^{[5]}

InN

3.533(a); 5.693(c)

Wurtzite

^{[5]}

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

^{[6]}

Fe

2.856

BCC

^{[6]}

Ni

3.499

FCC

^{[6]}

Cu

3.597

FCC

^{[6]}

Mo

3.142

BCC

^{[6]}

Pd

3.859

FCC

^{[6]}

Ag

4.079

FCC

^{[6]}

W

3.155

BCC

^{[6]}

Pt

3.912

FCC

^{[6]}

Au

4.065

FCC

^{[6]}

Pb

4.920

FCC

^{[6]}

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

^{[7]}

ZrC_{0.97}

4.698

Rocksalt

^{[7]}

HfC_{0.99}

4.640

Rocksalt

^{[7]}

VC_{0.97}

4.166

Rocksalt

^{[7]}

NC_{0.99}

4.470

Rocksalt

^{[7]}

TaC_{0.99}

4.456

Rocksalt

^{[7]}

Cr_{3}C_{2}

11.47(a); 5.545(b); 2.830(c)

Orthorombic

^{[7]}

WC

2.906(a);2.837(c)

Hexagonal

^{[7]}

ScN

4.52

Rocksalt

^{[8]}

LiNbO_{3}

5.1483(a);13.8631(c)

Hexagonal

^{[9]}

KTaO_{3}

3.9885(a)

Cubic perovskite

^{[9]}

BaTiO_{3}

3.994(a);4.034(c)

Tetragonal perovskite

^{[9]}

SrTiO_{3}

3.98805(a)

Cubic perovskite

^{[9]}

CaTiO_{3}

5.381(a);5.443(b);7.645(c);

Orthorhombic perovskite

^{[9]}

PbTiO_{3}

3.904(a);4.152(c);

Tetragonal perovskite

^{[9]}

EuTiO_{3}

7.810(a)

Cubic perovskite

^{[9]}

SrVO_{3}

3.838(a)

Cubic perovskite

^{[9]}

CaVO_{3}

3.767(a)

Cubic perovskite

^{[9]}

BaMnO_{3}

5.673(a);4.71(c)

Hexagonal

^{[9]}

CaMnO_{3}

5.27(a);5.275(b);7.464(c);

Orthorhombic perovskite

^{[9]}

SrRuO_{3}

5.53(a);5.57(b);7.85(c);

Orthorhombic perovskite

^{[9]}

YAlO_{3}

5.179(a);5.329(b);7.37(c);

Orthorhombic perovskite

^{[9]}

References

^ and angles between the sides given by α, β, γc, b, aUnit cell definition using parallelepiped with lengths

^ Mudar A. Abdulsattar, Solid State Sci. 13, 843 (2011).

^ 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.

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} Davey, Wheeler (1925). "PRECISION MEASUREMENTS OF THE LATTICE CONSTANTS OF TWELVE COMMON METALS". Physical Reviews 25: 753.

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Toth, L.E. (1967). Transition Metal Carbides and Nitrides. New York: Academic Press.

^ Saha, B. (2010). "Electronic structure, phonons, and thermal properties of ScN, ZrN, and HfN: A firstprinciples study". Journal of Applied Physics 107: 033715.

^ ^{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 perovskiterelated structure, Table 2 Part 1". SpringerMaterials  The LandoltBörnstein Database.
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