The symmetry and geometry of native crystals are part of the field of ... These and other material properties are discussed in the monograph by Nye. The special.
L Crystal SYmmetrY A. Crystal Symmetry and Bravais Lattice B. Primitive vectors, voiumes C. Reciprocal lattice
D. Extended Applications
A. Crystal Classification The symmetry and geometry of native crystals are part of the field of mathematics crystallography and are formally catagonzed by group theory. That field of and leads is concerned with the symmetry properties of point groups, as objects in space,
point to a classification system based on the set of symmetry operations which leave the group invariant. These operations, which include rotations, inversions and reflections but not translations, form a unique set for a given point group'
In
1848, Bravais, a Russian mathematician, showed that there are only fourteen
possible simplified space groups which can be used to characterize crystals. The regular,
periodic space array of atoms which is associated with a crystal can be reduced to simplifed space array of lattice points known possiblities are shown in Fig.
as the
a
Bravais lattice. The fourteen
I-1. Each lattice point in the Bravais lattice represents some
the Bravais abstract space point comprised of one or more real atoms. Because of this,
lattice catagorizes the skeletal part of an actual lattice' On a coarser scale, there are seven crystallographic systems which represent the These types of geometric shapes, or morphology, associated with the Bravais lattices. angles systems are based on simple parallelpipeds of sides a, b, and c with inclination s.
=
lab,
g=
lbc,
and
y=
lca.
On a finer scale, the slmmetry properties of an actual
lattice fall into one of 32 crystallographic point groups, or classes, designated by a has standardized alpha-numeric label known as the Hermann-Maugin symbols. Thus, one
the following classification in tabular form.
t-2
System
Paralielpiped
Bravais Lattices
Point Groups
Examples
e class
Cubic
a:b:c
J
5
isotropic
I
5
NaCl, GaAs Si, C, ZNS LiNbO3
L
7
BaTiO3
uniaxial
I
7
ZnS, SiO2
uniaxial
4
J
CaCO3, TII
biaxial
2
3
biaxial
1
2
biaxial
14
32
Ct:g:y:90o
a:b:c
Trigonal
uniaxial
cr=8ry*90o Tetragonal
a:bx
Ct:B=y:90o Hexagonal
a:b* cr=B=9oo
Orthorhombic
y:720"
a4*
o=B=Y:90o
Monoclinic
artx
cr=B:90o*?
a4*
Triclinic
d*Bry 7
GaAs is an example of a cubic crystal. The lattice structure shown in Fig. I-2 is
known as the zinc blende lattice appropriate to III-V and certain II-VI crystals or the diamond lattice appropriate to column IV crystals such as Si and Ge. The lattice is comprised of two interpenetrating face-centered-cubic Bravais lattices (fcc) so that cubic,
fcc cubic @ravais lattice) or the appropriate Hermann-Maugin symbols label the crystal in increasing levels of sophistication. A knowledge of the symmetry properties of the point
group is most useful in that the symmetry dictates the
slT
nmetry properties of any
measurable physical coefficient. For example, all cubic crystals are characterized by
having the same dielectric constant, e, along the crystal axis (istotropic) in contrast to
tetragonal and hexagonal crystals for which one dielectric constant is different (uniaxial). These and other material properties are discussed in the monograph by case of the
432
Nye. The
special
class appropriate to zinc blende structures is treated in part D.
As seen in this table, a given material such as ZnS may crystallize in two forms: cubic or zinc blende 14lm1and hexagonal or wurtzite (6mm) structures. Most of the
III-
V alloys of interest crystallize only in the zinc blende form. However, strain, such as a
I-3
compressive force applied along a crystallographic directior1 can distort the lattice creating a tetragonal structure which is manifested optically by an anisotropy in the index
of refraction. Intentionally strained systems are employed in both electronic and optical devices.
B. Primitive vectors, volumes The lattice possesses the property of translational invariance which means that there are sets of translational displacements which, when invoked, replicate the original space array. These displacements are most conveniently represented in vector form as shown in Fig.
I-3.
Thus, labelling one lattice point as an origin, any other lattice point is
givenby F= nrdr+nrdr+n3d3, n,
=*i""'.
The setofvectors,
d,,
arecalled
primitive vectors and are not uniquely defined. As an example, Fig. I-4a) shows
a
fcc
Bravais lattice along with one choice for the set of primitive vectors. The symbol, a, labels the length of one side of the cube and is the familiar lattice constant. Denoting unit
vectors with a carot, one has that
ar=|a1i+fi where
?
,
j,
and
i
"r"
along the x-
y-
a2=la1f
+i1
ar=|a1i +i)
and z-axis, respectively. The primitive unit vectors
themselves can be used to define a volume known asthe primitive unit cel/ which, in
general, is a parallelpiped with a physical volume of d, . d2 x d3. The primitive unit cell has the characteristic of bounding one
in Fig.
I-4.
full lattice point, some examples of which are
seen
The more familiar conventional unit cell is the volume associated with the
Bravais lattice. The conventional unit cell of the fcc Bravais lattice bounds four lattice points whereas this same cell bounds eight atoms in an actual 43m, or zinc blende crystal.
The Wigner-Seitz cell is a third volume which finds great use in our understanding
of electronic bands. This volume is constructed from planes whose normals are along the direction lines connecting lattice points and which are located midway between the lattice
points. The Wigner-Seitz cell for the simple cubic Bravais lattice in Fig.
I-l is clearly a
l-4
cube centered on one lattice
point. The Wigner-Seitz cells for the fcc-
and bcc-cubic
Bravais lattices are shou,n in Fig. I-5
C. Reciprocal lattice Because of the periodicity of the lattice, any attempt at expressing a periodic
function associated with the lattice will be reduced to considering a complex exponential
eii.;
wnere
x
ls some three dimensional position vector. Figure I-6 illustrates one
situation for which point-a and point-b are indistinguishable. Because of this, one must
have ,ii.r -
,i8.1;+R.1o, ,,[.F =
1
Since E is generated from primitive vectors, there must be a set of vectors,
E.R
=2rm
wherem is an integer. The set of vectors,
i,
i
,such that
also span aBravais lattice
which will be called the reciprocal lattice. To avoid confusion, the original lattice is now
If 6, areused to denote the primitive vectors of the reciprocal lattice, one simply requires that D-, 'd j = 2n5 ii' with this and E = mtEt * *2i2 + *16, ' called the direct lattice.
then
[
Since
. R. D-,
'd
= Zx(ryn1*
j
= 2nd
i,
m2n2
+ m3ry)
as
required. The 6,
are
constructed as follows'
then
6i=cdjxdt i*j*k where c is some constant found by forming
d,'6,. Hence
di'6,=2r=cdi'drxdo or
This leads to the formal relationship between the primitive vectors bt
_ Zndzxdt
\.d2
xd3
= ---.+
2ndt x dt a2'a3 xa1
-
?---
2ndt x d'r a3'd1xa2
Note that the denominators are all equal and define the volume of the primitive unit cell of the direct lattice, V6r. The volume of the primitive cell of the reciprocal lattice,
I-5
Vr, = 6r.6r
*
Er, is simply
found tobe Vo =
q2n13 t
L'0,
As an example, using the
primitive vectors for the fcc Bravais lattice, one finds
6r=1i+j-ilzxta 6r=1i-j+ilznta 6r=1-i+j+ilznta and are visualized in Fig.
I-4b).
These set of primitive vectors span a body-centered-
cubic, bcc, Bravais lattice with a conventionalunit cell of side 4nla. Later, use will be made of the Wigner-Seitz cell, the most important of which is that associated with a fcc-
cubic direct lattice. It is suggested that this be marked for future recall.
D. Extended Applications As an example of the use of the symmetry properties to deduce properties of measurable coefficients, we shall treat the case of
the 43rr. point group. First the
symmetry operations are deduced and then applied to linear and quadratic relations. The symmetry of this crystal class is the same as the molecule methane, CH4, and is seen in the shaded five atom group in Fig.
I-2. If we choose a coordinate
axis to be parallel with the
axis of the conventional unit cell, then using the center atom as an origin, a projection onto the plane containing the center atom reveals part of the symmetry and is shown in Fig. I-6. The plus and minus symbols refer to atoms which are above or below the projection plane. The important symmetry operations for this class start with the inversion operation.
Inversion An inversion operation replaces all coordinates by their negative. In matrix form, the inversion operator is the negative of the identity operator.
If
after an
inversion operation, the new space array of atoms matches the original space array, the space array is said to posses a center of inversion and to be centosymmetric. Mentally,
applyrng the inversion operation to the five atom group in either Fig.l-2 or Fig. I-6, one sees
that the new space array is rotated 90o from the original so that the group is not
[-6
centosymmetric. ln contrast, the elementals, C, Si, and Ge, are cubic crystal in the point group m3m wtnch is centosymmetric.
7:
This symbol refers to a symmetry axis for which a90' (2114) rotation plus an
inversion replicates the space array. For Fig. I-6, a 90o rotation about the z-axis partially replicates the space array but leaves some of the atoms in the wrong z-position; a subsequent inversion corrects for
this. Denoting the operation as If, in matrix form
,'[l] and that
Qil4
[i] L: : lll
is the identity matrix. Because of the equivalence of the
i00, 010 and
directions, there are three sets of.Ia operations corresponding to rotations in the and
001
x-!,!-z
z-x planes. .3: This symbol denotes a symmetry axis about which a rotation of 120o (2n13)
leaves the space array unchanged. This axis is parallel to the bond angles in Fig. I-2 which
form six equivalent I 1 1 directions which comprise the body diagonals of cubes. The matrix representation will not be given.
z: This symbol denotes mirror planes parallelto
the/4 axis. Two of these are
shown in Fig. I-6 as dotted lines. In matrix form, the mirror operations are
"[l]
:L: L;]
: ll Lll "'[l []l [ : :
ILI]
and are found from a l80o rotation about the dotted lines followed by the replacement
of
-zby z.
To deduce information about the physical properties of materials,Neumann's principle asserts that the symmetry of a physical property of a crystal must include the symmetry properties of the point group of that crystal since it is the response of the atoms in the crystalto an external stimulus which determine the material coefficients. The
l-7
is far simpler to material symmetry properties can be found from first principles but is
material apply the symmetry properties to the stimulus and response to deduce the responses. symmetry properties. These will be illustrated for linear and quadratic
Linear dielectric coefficients An example of
a linear case is the normal
relationship between displacement vector and electric field
D, = e 3Erwhere 1,2,
and3 refer to the
x,y,
D, = EIE* Dy = e2E,
and
andz components in a cOordinate system
two fields parallel to the conventional unit cell. We apply the syrnmetry operation to the material coefficients' and then use the original relations to deduce information about the
Applying
,Ioz
results in the mapping
D, )
-D,
D, -+ D*
and
D,
) D,
The same
transformed mapping also apply to the electric field components. Thus the original and equations become
Dr = !1E* - -D, - -!tEy - -!zEy
D,=e2Er1 D,=!2E,=ttE, D, = e 3E, -+ D, = e3E, from which one concludes that E1=e2' Applying either lvo or
{'
which involve rotations
all linear of the z-component, would reveal that e3:e1. Thus for the Z3z system,
isotropic' coefficients are independent of crystal direction and e is said to be quadratic coefficient can be visualized as a euadratic dielectric coefficients A
Taylor series expansion of
a general materialresponse as
+""'
Considering only * in E2 +terms in E3 = 6 oE, * P, = e oE* + so/1Er. terms E,E r' For. the latter, no the quadratic ternr, there are nine products of the form E,E, and
D,
special significance can be placed on EiE
i
relative to E rE, so the two are grouped
multiplying together resulting in only six terms per D component. The material coemcient these terms would be subscripted
with lij which is awkward to use. Instead, the ij
with the Voigt combination is replaced by a single contracted number, q, in accordance notation which is shown below
I-8
and is universally understood. We shall denote the material coefficientas dn
O
so that
the quadratic terms my be written as
P!' = drrd
+ dr2E2,
*
drrt
+
droErE, + drrErE, + druE,E,
Pi' = drtd
+ drrEf,
* drrd
+
droErE, + drrE,E, + d'uE'E,
P!'
=
drrd * dy4 + drrEl + dtoErE, + drrE,E, + druE,E,
where the superscnpt nl reminds us that this is a nonlinear polarization. One immediate
result can be found for centosymmetric systems such as Si. For this case, all components are replace by their negative value leading to the conclusion that
all diq:'diqso all d1O are
zero. Thus quadratic nonlinearities are only seen in non-centosymmetric crystals' Returning to the 43n class, applying the I; operation to Pll results in
-P!t = dvEtr + fi2Ef, + fi3E| - d34ErE, + d35ErE, - d36E'E, =
-dt4? - dvEzy - dnE? - fiaErE2 - d35EvE7 - d36E7sEv
from which d33:d34:d35:0 and
dlf--dZZ. Applying the liand mirror operations to the
at this other pair of components results in dp=d25, d15='d24 and all other d1O=0' Thus values: stage, of the 18 possible coefficients, only five have possibly non-zero
Plt=dpEyET+d6ErE*
P!'
= -d6ErE2
Plt
=
Because of the equivalence of x,
+dpErE,
dtEtr - dvEl + d36ErE,
y,
and z (or by using the symmetry operations in the other
two directions), by inspection d$:d31:0 and d36=dl4leaving
pi' = dyErEr, and Pll = dgErEr. for this crystal class are
Pll = dsErEr,
Thus through second order in E,the D-E relations
I-9
D* = eoE, + e6f 1E, + dlaErE, = ElEx + dlaErE, elEr+d14ErEr. Dy=
Dz=
elEr*dgEyE,
Phenomena assiciated with the second order non-linear coefficients include dc-field
induced shifts in optical dielectric constants (Pockel's effect) used for optical phase
modulators, optical second harmonic generation and rectification. The quadratic relation is also used to describe piezoelectricity. These and other phenomena are discussed in the monograph by Nye.
Elastic constanls
A second example of the application of the symmetry properties of a material in determining the material coefficients is that of the elastic properties. For small distortions,
all solids obey Hookes law governed by an elastic force law of the form F:k6x. The dimensions of the spring constant k are Nlm so that dividing the force equation by an area
A, we have FiA=(k/distance) 6x/dislance. The term FiA is denoted as a stress (o), ft/distance is the elastic stiffness (c) and \x/distance is the strain (e=fractional length
change). The units of stress and the elastic constants are the same as pressure or for cel area (N
n\ : note 1 Mbar-l
5
Mpsi=LO} GP a=lOl I 1,J7*2=1 gl2 4,ne /cmz
.
For the three dimensional body shown below, the components of stress, oy , c&n be normal (compressive or tensile) as well as tangential (shear). The most general
relationship between stress and strain,
ep would involve
a notation denoting the face and
the direction of the stress/strain. Because the elastic constants can have directional dependence the most general relationship would be o,7
=
cij4ekt where
ii, k and/ take on
values of 1,2 or 3 corresponding to the crystallographic x, y, and z-axis. As suggested by this result, there are 34=81 possible elastic coefficients which can be reduced to 36 by the
following line of reasoning. A body under equilibrium must have balanced forces which implies that some of the strain components must be inter related to prevent a net rotation
I-10
on.
10,
or translation of the body. As a consequence of this one must have
o, = oir and
e,i = e ii which leads to a new elastic constants defined in terms of combinations of the original elastic constants using the Voigt notation. The new constants are defined by (contracted indicies in parenthesis)
c(i)(k|):cm,
when m and
n are L,2 or
3
2c1ii1@D=c*n when either m ot n are 4,5 or 6 acq41gt1=c*r, when both lrl and n are 4,5 or 6 which reduces the number of elastic coefficients to 36. The number are further reduced.by requiring that the symmetry of the crystal be replicated in the coefficient matrix. For cubic crystals, this results rn 12 non-zero coefficients. In matrix form we thus have o1
o2 o3
o4 o5
o6
cll ct1 clZ ctt ct2 ct7
ctz
clz
ctl
c,q Tii ct+
r--.,|[::
An inverse relationship can be found between e and o (e=so) involving an s-matnx (compliance) which has the same form as the c-matrix. For cubic crystals,
I-11
ctt * clz
(cll
-
cn)Gt1+2cP)
-
-ctz cn)@t1+ZcP)
tL
(crr
s44
=l I c44
Poisson's ratio, v, is defined
aS
the negative ratio of the transverse strain to the
in the strain perpendicular to axial strain and characterizes the sign and magnitude change
ratio of stress some stress direction. Young's modulus, E, is defined to be the along a direction. For the cubic crystal class
to strain
E=llsgand v:-s12ls11, both in the [100]
Poisson's ratio in direction. The general directional dependence to Young's modulus and
cubic crystals has been derived (Brantley, Bell Labs' 1973)
(GPa) Table I Elastic constants for some zinc blende-diamond materials.
ctl
ctz
cqq
Elrool
v11o0l
C
1076
t25.
576 8
t049
.10
Si
t65.64
63 94
79 51
t30
.28
Ge
t28.9
48.3
103
.27
AlP
l,l42
=-74
=38
91
.34
AlAs
120.2
57.
s8.9
84
.32
Alsh
89.4
44.3
41 6
60
.JJ
GaP
t41.2
62.53
70 47
103
.31
GaAs
118.8
53.8
59.4
85
.31
GaSh
88.4
403
43.2
63
.31
InP
102.2
57.6
46
61
.36
InAs
83.29
45.26
39.59
51
.35
InSb
66.7
36.5
30.2
41
.35
Material
67
1
l-12 References
For a very elementary discussion of crystals and point groups see.F. H. pough,l Field Guide to Rocks and Minerals, (Irufifflirq Boston, 1956). For an introductory text on crystallography see: C. S. Hurlbut, Jr., Dana,s Manual of Mineralogt, (Wiley, New york, l97l). For an application of group theory to the determination of the symmetry properties of measurable coefficients, such as conductivity, see: J. Nye, Physical Properties of Crystals, (Oxford University, Oxford, 1957).
A very readable discussion on crystal lattices is contained in: N. W. Ashcroft and N. D. Mermin, solid state Physics, (Holt, Rinehart and winston, New york, 1976), chs. 4 & 5. If you insist on purchasing a text, this is recommended. W.A. Brantley, "Calculated elastic constants for stress problems associated with semiconductor devices", J. Appl. Phys., 44,534-535 (1973). R. M. Martin, "Elastic Properties of ZnS Structure Semiconductors", phys. Rev. B 1, 4005-401 I (1970) "Stresses in semiconductors. Ab initio calculations on Si, Ge, and GaAs", Phys. Rev. 8,32,pp.3792-3905 (1995).
o.H. Nielson and R.M. Martin,
fa l"l.i llli t_F-l/ L-= lrrJric
L)/
|,(orectnic
flfl@@
DD
Orthorhornbic
\ fl)ft) ryry &( Tetragonal
a L-Y
Hexagonal
I
Cubic
Fig. I-1
The fourteen Bravais lattices'
_:-a:
Fig' l'2
JDiamond lanice unit cell, showing rhe four nearest neigh-
bor structure. (From Electrons and Holes in Semiconductors by W. Shockley. O 1950 by Litton Educational Publishing Co., lnc.; by permission of Van Nostrand Beinhold Co.. lnc')
(
lllllltto fllaltttl Itrltllll ,Z/
cHttttrt at
tlaltrlat Fig. I-3
Two dimensional lattice showing one choice for the primitive vectors.
()
tltlltlal P 7 Point I t I t I t I tta'f I t t\-R-r I I t I \ r Ftrnt b t I t t I I I h-4'l r ftttlltll a
Fig. I-6 Part of an infinite two dimensional Lattice for point a and point b must be equivalent.
(
which
4'-)
L)
Fig. 7-4 a) Conventional, left, and primitive, right, unit ce1ls for the fcc Bravais lattice. AIso shown are one choice for the primitive vectors. b) Conventional, left, and prirnitive, right, unit eells for the bcc Bravais lattice and one ehoiee for the primitive vectors. (From Ashcroft and Mermin)
Fig. I-5
Wigner-Seitz eell for the bcc, left, and fec, right, Bravais lattice. For the fcc case, the cube shown in this figure is comprised of two half conventional ce11s (From Ashcroft and Mermin).
v
(Ib\
6 a
X
X /+ Fig. I-5 Projection of the shaded five
atom group in Fig. I-2 ont'o a plane containing the central atom. The.central atom is shown as a square and the atoms above (below) the plane are shown as solid (open) circles with + (-) symbols to remind us of their z-position. Ihe dashed lines indieate mirror planes parallel to the z-axis and the circle denotes the arc for a rotacion about z.
I
HW ChaPter I a) The lattice constant, a in Fig. l-2, for the elementals C, Si and Ge is 3.567, 5 -43 I and 5.646 A respectively. If we think of the lattice in terms of packed hard spheres,
.
determine the three atomic radii from the lattice constant. b) If every other Si atom in a Si lattice is replaced by a Ge aton! estimate the lattice constant of this alloy
radii. From this construct
Sig.5Geg.5 from the atomic
a function for the lattice constant
of the alloy Si1-*Ge* 0
