Semiconductors Subvolume B II-VI and I-VII Compounds; Semimagnetic Compounds 2

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Ref. p. 45] Gemeral Introduction { Structure and Symmetry 1
General Introduction
Semiconductor properties and their de(cid:12)nitions
0. Structure and Symmetry
Intrinsic semiconductors are characterized by a single-electron energy spectrum in which, at zero
temperature, the topmost occupied (or valence band) state is separated by an energy gap
E from the lowest unoccupied (or conduction band) state [76A, 91S, 93I, 96K1, 96Y]. The
g
energy gaps of semiconductors range between zero and a few eV, thus covering the spectral range
from the far-infrared to the near ultra-violet. The electronic properties of semiconductors can
be modi(cid:12)ed by lattice defects. In particular impurities, which replace the host atoms, become
e(cid:11)ective by creating electronic states in the energy gap from which carriers can be thermally
excited to the conduction or valence band. The energy gap and the possibility to manipulate the
electronic properties by doping are essential for the optic and electronic device applications of
semiconductors.
This characteristic single-particle spectrum of semiconductors is a consequence of closed-shell
con(cid:12)gurationswhicharerealizedinavarietyofcrystalstructureswithatomiccoordinationdepend-
ingonthevalenceelectrons. Aprominentexampleisthefour-foldortetrahedral coordination
withcovalentbondsintheelement-IVsemiconductorsthatgiverisetothediamondstructure. But
also its isoelectronic analogues,the III-V, II-VI, and I-VII compound semiconductors,form stable
crystalstructureswithfour-foldcoordinationyetwithincreasingadmixtureofheteropolarbinding
which can be quanti(cid:12)ed in di(cid:11)erent schemes [93G]. Tetrahedral coordination not only allows for
cubiccrystalstructuresasinthediamondorzinc blendelatticesbutalsoforuniaxialstructures
like the tetragonal chalcopyrite or the hexagonalwurtzite lattices.
Besides in tetrahedral coordination closed-shell con(cid:12)gurations are possible in quite di(cid:11)erent
atomic arrangements. They give rise to a variety of crystal structures covering almost all crystal
classes. Among the elemental semiconductors, for example, Se and Te form trigonal lattices, P
and S are found in orthorhombic and monoclinic phases, and B, As, and Bi prefer rhombohedral
lattices. But also in the cubic and hexagonal crystal classes, besides the four-fold coordinated
structures, one (cid:12)nds semiconductors in the rocksalt structure (e.g. PbS and MgO) with six-fold
coordination, and in the CsCl structure (e.g. TlCl) with eight-fold coordination. Most of the
physical properties of the semiconductors listed in the tables are influenced by or related to the
symmetry of the underlying crystal structure. Therefore, a brief description of these structures,
theirsymmetries,andsomegeneralconsequenceswithrespecttomaterialparametersshouldstand
at the beginning of this volume.
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2 Gemeral Introduction { Structure and Symmetry [Ref. p. 45
Crystal structures and Wigner-Seitz cells
A crystal structure is the periodic arrangement of single atoms or groups of atoms attached to
the points of a Bravais lattice. The Bravais lattice is de(cid:12)ned by lattice vectors
Rn =n1a1+n2a2+n3a3 (1)
withintegerni;i=1;2;3andprimitivetranslationsa1;a2,anda3 [57K2,76A,93I,96K1]. Inthree
dimensions 14 di(cid:11)erent Bravais lattices can be realized. A mono-atomic lattice is obtained by
attaching a single atom to each point of the Bravais lattice, it is completely characterized by the
primitive translations. For a lattice with basis, with a group of atoms attached to each lattice
point, in addition to the primitive translations the positions t of the atoms in the basis relative
(cid:20)
to the lattice point mustbe de(cid:12)ned. For agivenBravaislattice the whole spacecanbe thoughtto
be (cid:12)lled without voids by identical volumes around each lattice point or by replica of this volume
obtained from translations by lattice vectors Rn. There is no unique way to de(cid:12)ne this volume or
primitivecell(orunit cell). AspecialchoicereflectingthesymmetryoftheBravaislatticeisthe
Wigner-Seitzcell. Itisobtainedbyconsideringthevolumearoundalatticepointwhichiscloser
to this one than to any other lattice point. The Wigner-Seitz cell is enclosed by the symmetry
planes of the lines connecting a lattice point with its nearest and next nearest neighbors.
c
b a
g
a
b
Fig. 0.1. Crystal axes a;b, and cand themeaning of lattice parameters.
Another choice of the unit cell is that of a parallelepiped spanned by three independent vectors
a;b, and c which de(cid:12)ne the crystal axes. With respect to these the lattice is characterized by
lattice parameters which are the lengths a;b, and c and the angles (cid:11);(cid:12), and γ between pairs
of these vectors (see Fig. 0.1). Depending on the lattice parameters the Bravais lattices fall into
seven lattice systems (Fig. 0.2), which can be distinguished also by their symmetry. The seven
lattice systems are:
(1) cubic with a=b=c and (cid:11)=(cid:12) =γ =90(cid:14),
(2) tetragonal with a=b6=c and (cid:11)=(cid:12) =γ =90(cid:14),
(3) orthorhombic with a6=b6=c and (cid:11)=(cid:12) =γ =90(cid:14),
(4) monoclinic with a6=b6=c and (cid:11)=γ =90(cid:14) 6=(cid:12),
(5) triclinic with a6=b6=c and (cid:11)6=(cid:12) 6=γ,
(6) trigonal with a=b=c and (cid:11)=(cid:12) =γ <120(cid:14) but not 90(cid:14),
(7) hexagonal with a=b6=c and (cid:11)=(cid:12) =90(cid:14) and γ =120(cid:14).
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Ref. p. 45] Gemeral Introduction { Structure and Symmetry 3
c
b
a
simple cubic body - centered cubic face - centered cubic
c
b
a
simple tetragonal body - centered tetragonal
c
b
a
simple orthorhombic one - face - centered body - centered orthorhombic all - face - centered
orthorhombic
orthorhombic
c
b
a
simple monoclinic one - face - centered
monoclinic
c
c
b
b
a c
a b
a
triclinic trigonal hexagonal
rhombohedral
Fig. 0.2. The14 Bravais lattices and theirunit cells spanned by thevectors a;b, and c.
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4 Gemeral Introduction { Structure and Symmetry [Ref. p. 45
ForsomeoftheselatticesystemsmorethanoneBravaislatticeispossibleleadingtothe14Bravais
lattices shown in Fig. 0.2. Their primitive translations a1;a2, and a3, which in some cases are
identical with the vectors a;b, and c spanning the unit cell, are given here with their Cartesian
coordinatestogetherwiththenamesandfrequentlyusedabbreviationsfortheselattices. Forsome
of them a notation is in use also which refers to the lattice type of a representative species [58P]:
1. simple cubic (sc): the three primitive translations are parallel to the cubic crystal axes,
they are orthogonalto each other and have the length of the elementary cube:
a1 =a(1;0;0); a2 =a(0;1;0); a3 =a(0;0;1)
2. body-centered cubic(bcc;A2): thethreeprimitivetranslationspointtothreeofthebody-
centers around an edge of the elementary cube:
a1 =a=2(−1;1;1); a2 =a=2(1;−1;1); a3 =a=2(1;1;−1)
3. face-centered cubic (fcc;A1): the three primitive translations point to three of the face-
centers nearest to an edge of the elementary cube:
a1 =a=2(0;1;1); a2 =a=2(1;0;1); a3 =a=2(1;1;0)
4. simple tetragonal (st): correspondsto the simple cubic lattice but with di(cid:11)erent lengthof
a3:
a1 =a(1;0;0); a2 =a(0;1;0); a3 =c(0;0;1)
5. body-centered tetragonal (bct;A5): corresponds to the body-centered cubic lattice but
with di(cid:11)erent length of the third Cartesian coordinate:
a1 =(−a=2;a=2;c=2); a2 =(a=2;−a=2;c=2); a3 =(a=2;a=2;−c=2)
6. simple orthorhombic: corresponds to the simple cubic lattice but with di(cid:11)erent lengths
of the three primitive translations:
a1 =a(1;0;0); a2 =b(0;1;0); a3 =c(0;0;1)
7. one-face centered orthorhombic: two of the primitive translations point to two face-
centers in the base-plane, the third one is orthogonalto this plane:
a1 =(a=2;b=2;0); a2 =(a=2;−b=2;0); a3 =(0;0;c)
8. body-centered orthorhombic: corresponds to the body-centered cubic lattice but with
three di(cid:11)erent lengths of the Cartesian coordinates:
a1 =(−a=2;b=2;c=2); a2 =(a=2;−b=2;c=2); a3 =(a=2;b=2;−c=2)
9. all-face centered orthorhombic: corresponds to the face-centered cubic lattice but with
three di(cid:11)erent lenghts of the Cartesian coordinates:
a1 =(0;b=2;c=2); a2 =(a=2;0;c=2); a3 =(a=2;b=2;0)
10. simple monoclinic: correspondstothesimple orthorhombiclatticebutwitha1 anda2 not
being orthogonalto each other
11. one-face centered monoclinic: correspondsto the one-face centeredorthorhombiclattice
but with a1 and a2 not being orthogonalto each other
12. triclinic: thethreeprimitivetranslationshavedi(cid:11)erentlengthsandincludepairwisedi(cid:11)erent
angles
13. trigonal: the three primitive translations have the same lengths and include pairwise the
same angle <120(cid:14) but not 90(cid:14)
14. hexagonal: two of the primitive translations have the same length and include an angle of
120(cid:14), the third one is orthogonalto them:
a1 =a(1=2;−1=2;0); a2 =a(1=2;1=2;0); a3 =(0;0;c):
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Ref. p. 45] Gemeral Introduction { Structure and Symmetry 5
Crystal symmetries
Due to the periodic arrangementof the atoms the crystalstructure is invariantunder translations
by lattice vectors Rn (which form the translation group) but also under rotations, reflections,
and- in some cases- under inversion(point group) [57K2,63K].Eachcrystalstructure that can
be realized in three dimensions belongs to one of the 32 possible point groups or crystal classes.
Elements of the translation group and of the point group can be combined to form symmetry
operations of the space group. The space group is called symmorphic if these are all possible
spatial symmetry operations of the crystal structure; it is called non-symmorphic if elements of
the point group(e.g. the inversionin the diamond structure)combined with translations di(cid:11)erent
fromEq.(1)leavethe crystalinvariant. There are73symmorphicand157non-symmorphicspace
groups. Two notations are in use for the space groups and point groups: the Schoenflies and the
International(full andshort) notations [63K,93I] (the notations for the 32 point groups are given
in Table 0.1). In contrast to the Schoenflies notation the International notation uses symbolic
expressions for the symmetry operations.
Table 0.1. The 32 point groups in International and Schoenflies notation (from [63K]).
International International
Schoen- Schoen-
flies flies
Short Full Short Full
1. 1 1 C1 17. (cid:22)3 (cid:22)3 C31(S6)
2. (cid:22)1 (cid:22)1 C1(S2) 18. 32 32 D3
3. 2 2 C2 19. 3 m 3 m C3v
4. m m C3(C1h) 20. (cid:22)3 m (cid:22)3 m2 D3d
2
5. 2/m m C2h 21. 6 6 C6
6. 222 222 D2(V) 22. (cid:22)6 (cid:22)6 C3h
6
7. m m 2 m m 2 C2v 23. 6/m m C6h
2 2 2
8. m m m m m m D2h(Vh) 24. 622 622 D6
9. 4 4 C4 25. 6 m m 6 m m C6v
10. (cid:22)4 (cid:22)4 S4 26. (cid:22)6 m 2 (cid:22)6 m 2 D3h
4 6 2 2
11. 4/m m C4h 27. 6/mmm m m m D6h
12. 422 422 D4 28. 23 23 T
13. 4 m m 4 m m C4v 29. m 3 m2 (cid:22)3 Th
14. (cid:22)4 2 m (cid:22)4 2 m D2d(Vd) 30. 432 432 O
15. 4/mmm m4 m2 m2 D4h 31. (cid:22)4 3 m (cid:22)4 3 m Td
16. 3 3 C3 32. m 3 m m4 (cid:22)3 m2 Oh
Electronic or vibrational eigenstates of the crystal Hamiltonian are classi(cid:12)ed by exploiting the
crystal symmetry [78M]. As far as translational symmetry is concerned this is done by using the
concept of Brillouin zones and of the reciprocal lattice [57K2, 75J, 76A, 96K1]. The latter is
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6 Gemeral Introduction { Structure and Symmetry [Ref. p. 45
de(cid:12)ned by reciprocal lattice vectors
Gm =m1b1+m2b2+m3b3 (2)
with integer m ;i=1;2;3, and primitive reciprocal lattice vectors
i
a (cid:2)a
b =2(cid:25) j k (3)
i a1(cid:1)(a2(cid:2)a3)
where i;j;k, are cyclic permutations of 1,2, and 3. The Wigner-Seitz cell of the reciprocal lattice
isthe (cid:12)rst Brillouin zone (Fig. 0.3). As thereciprocallattice isde(cid:12)nedbythe Bravaislattice in
realspace,there areas manydi(cid:11)erent Brillouinzones asBravaislattices. The primitive reciprocal
lattice vectors for the 14 Bravais lattices can be obtained by applying Eq.(3) to the primitive
translations a1;a2, and a3. It turns out (and is to be expected from the fact, that the reciprocal
lattices follow the same mathematicalrule as the Bravaislattices), that the geometryof the recip-
rocal lattices repeats that of the Bravais lattices (e.g. the reciprocal lattice of the face-centered
cubic Bravais lattice is a body-centered cubic lattice and vice versa). Thus the Brillouin zones in
Fig. 0.3 can be looked at also as being the Wigner-Seitz cells of the 14 Bravais lattices.
k k
z z
R
L
T S
G D X
P
S Z ky
M F
kx L D D H
G S G ky
k N
simple cubic x
k
z
body - centered cubic
L
L U
G D Q S
X k
S Z y
W
K
k
x
face - centered cubic
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k k
z z
Z U R U Z V
S A T
L N
V G
W T D
S ky
G D X X S
S Y ky kx
M
k
x
body - centered tetragonal (c>a)
simple tetragonal
k
k z
z Z B T
B T
Z A
A G L H ky R L ED H ky
U R E D Y S G C Y
G S D Q C
kx X D S kx S
simple orthorhombic one - face centered orthorhombic
k k
z z
Z A Z B
R L S H
L
D
E S
X G Y k
G S G C y
D D
k
x
kx X
T
k
y all - face - centered orthorhombic
body - centered orthorhombic
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8 Gemeral Introduction { Structure and Symmetry [Ref. p. 45
k
z
k
z
Z A
E
C
L
D
G
G
L
A
Y
B k Y
y
k
y
k
x k
x
simple monoclinic
one - face - centered monoclinic
k k
z z
Z F
B
A
S R
L H L
A
G S' D
U
D S D CY ky P T S G ky
k K T' M
x
k
x
trigonal (c>a)
(rhombohedral)
hexagonal
Fig. 0.3. The Brillouin zones of the 14 Bravais lattices with points and lines of high symmetry. Not
shown is theBrillouin zone of thetriclinic lattice.
Classi(cid:12)cationofeigenstateswithrespecttotranslationalsymmetryisunequivocalforwavevectors
k belonging to the (cid:12)rst Brillouin zone. For a given wave vector further classi(cid:12)cation is possible
due to point group operations that leave k invariant and de(cid:12)ne the group of the wave vector.
Evidentlythe centerofthe Brillouinzone,k=(0;0;0)orΓpoint,is invariantunderalloperations
of the crystal point group. Points and lines of high symmetry, i.e. with groups of the wave
vector k containing more elements than just the identity, are denoted by capital Greek (Roman)
letters if they are inside (on the surface) of the (cid:12)rst Brillouin zone as shown in Fig. 0.3. The
crystal eigenstates at a given k can be classi(cid:12)ed according to the irreducible representations
of the group of the wave vector, which for the 32 possible point groups are given in [63K].
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Ref. p. 45] Gemeral Introduction { Structure and Symmetry 9
Semiconductors with tetrahedral coordination
As a special case of crystal structures those of semiconductors with tetrahedral coordination (to
which all those of volumes LB41A and LB41B belong) are presented here in detail. Of these the
cubic structures derive from the face-centered cubic (fcc or A1) Bravais lattice with the lattice
constant a (see Fig. 0.2). By attaching to each lattice point Rn a base of two atoms at
a
t1 =(0;0;0) and t2 = (1;1;1) (4)
4
one obtains a crystal structure consisting of two interpenetrating fcc lattices (Fig. 0.4), which
is the diamond structure (A4), if both atoms in the base are identical (as for the element-IV
semiconductors), or the zinc blende structure (B2), if the basis consists of two di(cid:11)erent atoms
(as for compound semiconductors of the III-V, II-VI, and I-VII family). The Wigner-Seitz cell
of these structures (which is the Brillouin zone of the body-centered cubic Bravais lattice, Fig.
0.3) contains two atoms. Evidently diamond and zinc blende structure have the same translation
group as they derive from the same Bravais lattice. The point group of the diamond structure
(O in the Schoenflies notation, m3m in the International (short) notation [63K, 93I]) consists
h
of 48 elements, half of them (including the inversion) interchange the two interpenetrating fcc
lattices and are elements of the space group only in combination with a translation t2 −t1, i.e.
the space group O7(Fd3m) of the diamond structure is nonsymmorphic. These nonsymmorphic
h
symmetry operations and the corresponding elements of the point group are missing in the zinc
blendestructure. Therefore,itsspacegroupT2(F(cid:22)43m)issymmorphicandthepointgroupT ((cid:22)43m)
d d
consists of 24 elements.
Fig. 0.4. Diamond lattice (if all atoms shown are
equal),zincblendelattice(ifthetwointerpenetrating
fcc lattices are occupied by di(cid:11)erent atoms). The
a lattice constant a is indicated.
A di(cid:11)erent way of looking at semiconductor crystal structures with tetrahedral coordination is
that of stacking sequences. Along the body diagonal(or [111]direction) the zinc blende structure
correspondsto a repeated stacking sequence ABC (or 3C in the Ramsdell notation [47R], where 3
referstothenumberoflayersandCtothecubiccrystalstructure)ofthreebilayers,eachconsisting
of the two di(cid:11)erent base atoms. The A, B, and C layers di(cid:11)er by their projection onto the plane
perpendicular to the cubic [111]direction [93I, 96K1,97W1]. The shortest stacking sequence that
leads to a crystal structure with tetrahedral coordination is AB or 2H (Ramsdell notation) which
formsthe wurtzite structure (B4). Itisa hexagonallattice withacrystalaxiscorrespondingto
the cubic [111] direction (Fig. 0.5). It derives from the hexagonal close-packed structure (hcp or
A3)consistingoftwopinterpenetratinghexagonalBravaislatticeswithabasisoftwoidenticalatoms
at(0,0,0)and(a=2; 3a=6;c=2). The wurtzite structureis made up oftwohexagonalclose-packed
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10 Gemeral Introduction { Structure and Symmetry [Ref. p. 45
structures with a basis
t1 =(0;0;0) and t2 =(0;0;c=2) (5)
occupied by di(cid:11)erent atoms,thus the Wigner-Seitz cell contains 4 atoms. Assuming ideal tetrahe-
dronsin thepwurtzite structure the lattice constantsa andc wouldbe relatedto eachother by the
ratio c=a= 8=3. In realized crystal lattices of the wurtzite type deviations from this ideal ratio
are observed. The space group of the wurtzite structure C64v(P63mc) is nonsymmorphic, half of
thesymmetryoperationsarecombinedwiththe nonprimitivetranslationt2−t1. The pointgroup
is C6v(6mm) and consists of 12 elements. The Brillouin zone of the wurtzite structure is that of
the hexagonalBravais lattice (Fig. 0.3).
Besidestheshorteststackingsequences,AB(2H)leadingtothewurtzitestructureandABC(3C)
leading to the zinc blende structure, some compound semiconductors (e.g. SiC and ZnS) are
realized in a variety of crystal structures with longer stacking sequences (e.g. 4H, 6H or 15R,
where R stands for rhombohedral), known as polytypes. Polytypism is connected with ground
state energies that di(cid:11)er only slightly for the di(cid:11)erent crystal structures. A list of polytypes and
their stacking sequences is presented in Section 1.5 (SiC).
c
Fig. 0.5. Wurtzite lattice, composed of two inter-
penetrating hexagonal close-packed lattices occupied
bydi(cid:11)erent atoms. Thelattice constants a andc are
a
indicated.
Another variety of crystal structures with tetrahedral coordination is realized in ternary semi-
conductors deriving from the III-V family by replacing the four group-III atoms of a tetrahedron
by each two atoms of the neighboring group-II and group-IV elements. It is the chalcopyrite
structure, which derives from the body centered tetragonal Bravais lattice with a basis of four
atoms (Fig. 0.6). Usually the c=a ratio deviates slightly from the ideal one of c=a=2. The space
groupof the chalcopyritestructure is D21d2(I(cid:22)42d), the correspondingpoint groupD2d (or (cid:22)42m) has
8 elements. The reciprocal lattice is a tetragonalface centered lattice.
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