Physics PY4118
Physics of Semiconductor Devices
Hybrid Bonds
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
2.1
Why?
Orbitals?
They explain the subsequent crystal structure
Crystal Structure?
This is important in generating band structure
The crystal also has interesting symmetry
Symmetry & Band Structure?
Leads to physical properties
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.2
Hybrid Orbitals
One might expect the number of bonds formed by an
atom would equal its unpaired electrons
Chlorine, for example, generally forms one bond as it has
one unpaired electron - 1s22s22p5
Oxygen, with two unpaired electrons, usually forms two
bonds - 1s22s22p4
However, Carbon, with only two unpaired electrons,
generally forms four (4) bonds
C (1s22s22p2) [He] 2s22p2
The four bonds come from the 2 (2s) paired electrons
and the 2 (2p) unpaired electrons
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.3
Hybrid Orbitals
Linus Pauling proposed that the valence atomic orbitals in
a molecule are different from those of the isolated atoms
forming the molecule
Quantum mechanical computations show that if specific
combinations of orbitals are mixed mathematically,
“new” atomic orbitals are obtained
The spatial orientation of these new orbitals lead to more
“stable” bonds and are consistent with observed
molecular shapes
These new orbitals are called: “Hybrid Orbitals”
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.4
Hybrid Orbitals
Types of Hybrid Orbitals
Each type has a unique geometric arrangement
The hybrid type is derived from the number of s, p, d
atomic orbitals used to form the Hybrid
Hybrid Orbitals
(Hybridization)
Geometric
Arrangements
Number of Hybrid
Orbitals
Formed by
Central Atom
Example
sp
Linear
2
Be in BeF2
sp2
Trigonal planar
3
B in BF3
sp3
Tetrahedral
4
C in CH4
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.5
sp Hybrid Orbitals
SP Hybridization
2 electron groups surround central atom
o
Linear shape, 180 apart
VB theory proposes the mixing of two nonequivalent
orbitals, one “s” and one “p”, to form two equivalent
“sp” hybrid orbitals
Orientation of hybrid orbitals extend electron density
in the bonding direction
Minimizes repulsions between electrons
Both shape and orientation maximize overlap between
the atoms
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.6
“sp” Hybrid Orbitals
Ex: BeCl2
The Be-Cl bonds in BeCl2 are neither
spherical (s orbitals) nor dumbbell (p
orbitals)
hybrid
orbitals
Beryllium Hybrid Orbital Diagram
The Be-Cl bonds have a hybrid shape
In the Beryllium atom the 2s orbital
and one of the 2p orbitals mix to form
2 sp hybrid orbitals
Each Be Hybrid sp orbital overlaps a
Chlorine 3p orbital in BeCl2
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
orbital box diagrams
PY4118 Physics of
Semiconductor Devices
2.7
“sp2” Hybridization
sp2 - Trigonal Planar geometry
(Central atom bonded to three ligands)
The three bonds have equivalent hybridized shapes
The sp2 hybridized orbitals are formed from:
1 “s” orbital and 2 “p” orbitals
Note: Of the 4 orbitals available (1 s & 3 p) only the s
orbital and 2 of the p orbitals are used to form
hybrid orbitals
Note: Unlike electron configuration notation, hybrid
orbital notation uses superscripts for the number of
atomic orbitals of a given type that are mixed, NOT for
the number of electrons in the orbital, thus,
sp2 (3 orbitals), sp3 (4 orbitals)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.8
“sp2” Hybridization
Hybrid Orbital Diagram
BF3
The 3 B-F bonds are neither
spherical nor dumbell shaped
They are all of identical shape
Boron (B) 1s22p1
In Boron, the “2s” orbital and two of
the “2p” orbitals mix to form 3 sp2
hybrid orbitals, each containing one
of the 3 total valence electrons
Forms 3 sp2 hybrid orbitals
Each of the Boron hybrid sp2 orbitals
overlaps with a 2p orbital of a
Fluorine atom
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
BF3
PY4118 Physics of
Semiconductor Devices
2.9
sp3 Hybrid Orbitals
sp3 (4 bonds, thus, Tetrahedral geometry)
The sp3 hybridized orbitals are formed from:
1 “s” orbital and 3 “p” orbitals
Example”
Carbon is the basis for “Organic Chemistry”
Carbon is in group 4 of the Periodic Chart and has 4 valence
electrons – 2s22p2
The hybridization of these 4 electrons is critical in the formation
of the many millions of organic compounds and as the basis of
life as we know it
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.10
sp3 Hybrid Orbitals
2p
2p
Energy
2s
This structure implies
different shapes and
energies for the “s” and
“p” bonds in carbon
compounds.
2s
Observations indicate
that all fours bonds are
equivalent
1s
1s
C atom (ground state)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
C atom (promoted)
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.11
Hybridization of Carbon in CH4
4 sp3 orbitals formed
sp3
2p
sp3
Energy
C-H bonds
2s
1s
C atom
(ground state)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
1s
1s
C atom
(hybridized state)
ROINN NA FISICE
Department of Physics
C atom
(in CH4)
PY4118 Physics of
Semiconductor Devices
2.12
Oxygen Atom Bonding in H2O
4 sp3 Hybridized Orbitals
2p
sp3
sp3
Energy
2s
1s
O
Central Atom
(ground state)
Tetrahedral
1s
O atom
(hybridized state)
lone
pairs
O-H
bonds
1s
O atom
(in H2O)
PY4118 Physics of
Semiconductor Devices
Spatial Arrangement of
sp3 Hybrid Orbitals
Shape of sp3 hybrid orbital
different than either s or p
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.14
sp3 orbitals - details
1. sp3 = ½ s - ½ px - ½ py + ½ pz
2. sp3 = ½ s - ½ px + ½ py - ½ pz
3. sp3 = ½ s + ½ px - ½ py - ½ pz
4. sp3 = ½ s + ½ px + ½ py + ½ pz
Linear Combination of Atomic Orbitals
Scalar product:
(n.sp3; m.sp3)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
=0
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.15
Hybridization visualisation
http://www.mhhe.com/physsci/chemistry/esse
ntialchemistry/flash/hybrv18.swf
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.16
Physics PY4118
Physics of Semiconductor Devices
Semiconductor Crystals
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
2.17
Crystals
The language used for crystals is required:
To describe semiconductors
To understand details of band structure
To design certain types of devices
Will revisit during discussion on symmetry
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.18
Valence e’s for “main group” elements
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
PY4118 Physics of
Semiconductor Devices
Group 4 Semiconductors
4 sp3 orbitals formed
sp3
3p
sp3
Energy
Si-Si bonds
3s
3s23p2
2s
Si atom
(ground state)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
2s
2s
Si atom
(hybridized state)
ROINN NA FISICE
Department of Physics
Si crystal
PY4118 Physics of
Semiconductor Devices
2.20
III-V Semiconductors
3 sp3 orbitals formed
4 sp3 orbitals formed
sp3
sp3
sp3
Ga-As bonds
Energy
4s24p1
3s
Ga atom
(hybridized state)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
4s24p3
3s
As atom
(hybridized state)
ROINN NA FISICE
Department of Physics
3s
GaAs crystal
PY4118 Physics of
Semiconductor Devices
2.21
Zincblende (diamond structure)
Si, Ge, GaAs, InP, etc.
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
Diamond: The 2 atoms are the same.
Zincblende: The 2 atoms are different.
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.22
Wurtzite (hexagonal structure)
GaN, AlN etc. (semiconductors for blue lasers)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.23
Survey of Solid State Physics
Taken from:
EP364 Solid State Physics
Prof.Dr. Beşire GÖNÜL
A link is provided, please look over for a
description of the crystal lattice
Now a very short summary…
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.24
CRYSTAL LATTICE
What is crystal (space) lattice?
In crystallography, only the geometrical properties of the crystal
are of interest, therefore one
replaces each atom by a
geometrical point located at the equilibrium position of that atom.
Platinum
Platinum surface
Crystal lattice and
structure of Platinum
25
Crystal Structure
Crystal structure can be obtained by attaching atoms, groups
of atoms or molecules which are called basis (motif) to the
lattice sides of the lattice point.
Crystal Structure = Crystal Lattice
+ Basis
26
A two-dimensional Bravais lattice with
different choices for the basis
27
Types Of Crystal Lattices
1) Bravais lattice is an infinite array of discrete points with an
arrangement and orientation that appears exactly the same,
from whichever of the points the array is viewed. Lattice is
invariant under a translation.
Nb film
28
Translational Lattice Vectors – 2D
A space lattice is a set of points such that
a translation from any point in the lattice
by a vector;
P
Rn = n1 a + n2 b
Point D(n1, n2) = (0,2)
Point F (n1, n2) = (0,-1)
Coláiste na hOllscoile Corcaigh,
Éire University College Cork,
Ireland
locates an exactly equivalent point, i.e. a
point with the same environment as P .
This is translational symmetry. The
vectors a, b are known as lattice vectors
and (n1, n2) is a pair of integers whose
values depend on the lattice point.
ROINN NA FISICE Department of Physics
29
Lattice Vectors – 2D
The two vectors a and b
form a set of lattice vectors
for the lattice.
The choice of lattice
vectors is not unique.
Thus one could equally well
take the vectors a and b’ as
a lattice vectors.
30
Lattice Vectors – 3D
An ideal three dimensional
crystal is described by 3
fundamental translation vectors a, b and c. If there is a lattice
point represented by the position vector r, there is then also a
lattice point represented by the position vector where u, v and w
are arbitrary integers.
r’ = r + u a + v b + w c
31
Unit Cell in 2D
The smallest component of the crystal (group of atoms, ions
or molecules), which when stacked together with pure
translational repetition reproduces the whole crystal.
b
S
S
S
S
S
S
S
S
S
S
S
S
S
S
a
S
32
Unit Cell in 2D
The smallest component of the crystal (group of atoms, ions
or molecules), which when stacked together with pure
translational repetition reproduces the whole crystal.
S
The choice of
unit cell
is not unique.
S
b
S
S
a
33
2D Unit Cell example -(NaCl)
We define lattice points ; these are points with identical
environments
34
Choice of origin is arbitrary - lattice points need not be
atoms - but unit cell size should always be the same.
35
This is also a unit cell it doesn’t matter if you start from Na or Cl
36
- or if you don’t start from an atom
37
This is NOT a unit cell even though they are all the
same - empty space is not allowed!
38
Unit Cell in 3D
39
Three common Unit Cell in 3D
40
UNIT CELL
Primitive
Single lattice point per cell
Smallest area in 2D, or
Smallest volume in 3D
Simple cubic(sc)
Conventional = Primitive cell
Conventional & Non-primitive
More than one lattice point per cell
Integral multiples of the area of
primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
41
The Conventional Unit Cell
A unit cell just fills space when
translated through a subset of
Bravais lattice vectors.
The conventional unit cell is
chosen to be larger than the
primitive cell, but with the full
symmetry of the Bravais lattice.
The size of the conventional cell is
given by the lattice constant a.
42
Primitive and conventional cells of FCC
43
Primitive Unit Cell and vectors
A primitive unit cell is made of primitive
translation vectors a1 ,a2, and a3 such that
there is no cell of smaller volume that can
be used as a building block for crystal
structures.
A primitive unit cell will fill space by repetition
of suitable crystal translation vectors. This
defined by the parallelpiped a1, a2 and a3.
The volume of a primitive unit cell can be
found by
V = a1.(a2 x a3)
(vector products)
Cubic cell volume = a3
44
Primitive Unit Cell
The primitive unit cell must have only one lattice point.
There can be different choices for lattice vectors , but the
volumes of these primitive cells are all the same.
a1
P = Primitive Unit Cell
NP = Non-Primitive Unit Cell
45
Wigner-Seitz Method
A simply way to find the primitive
cell which is called Wigner-Seitz
cell can be done as follows;
1. Choose a lattice point.
2. Draw lines to connect these
lattice point to its neighbours.
3. At the mid-point and normal to
these lines draw new lines.
The volume enclosed is called as a
Wigner-Seitz cell.
46
Wigner-Seitz Method
47
Wigner-Seitz Cell - 3D
48
Crystal Directions
We choose one lattice point on the line as
an origin, say the point O. Choice of origin
is completely arbitrary, since every lattice
point is identical.
Then we choose the lattice vector joining O
to any point on the line, say point T. This
vector can be written as;
R = n1 a + n2 b + n3 c
To distinguish a lattice direction from a
lattice point, the triple is enclosed in square
brackets [ ...] is used.[n1n2n3]
[n1n2n3] is the smallest integer of the same
relative ratios.
Fig. Shows
[111] direction
49
Examples
210
X=1,Y=½,Z=0
[1 ½ 0]
[2 1 0]
X=½ ,Y=½,Z=1
[½ ½ 1]
[1 1 2]
50
Examples of crystal directions
X=1,Y=0,Z=0
[1 0 0]
X = -1 , Y = -1 , Z = 0
[110]
_
Negative One =1
51
Miller Indices
Miller Indices are a symbolic vector representation for the orientation of an
atomic plane in a crystal lattice and are defined as the reciprocals of the
fractional intercepts which the plane makes with the crystallographic axes.
To determine Miller indices of a plane, take the following steps;
1) Determine the intercepts of the plane along each of the three
crystallographic directions
2) Take the reciprocals of the intercepts
3) If fractions result, multiply each by the denominator of the smallest fraction
52
Example-1
(1,0,0)
Axis
X
Y
Z
Intercept
points
1
∞
∞
Reciprocals
1/1
Smallest
Ratio
1
Miller İndices
1/ ∞ 1/ ∞
0
0
(100)
53
Example-2
(0,1,0)
Axis
X
Y
Z
Intercept
points
1
1
∞
Reciprocals
1/1
1/ 1
1/ ∞
Smallest
Ratio
1
1
0
Miller İndices
(110)
(1,0,0)
54
Example-3
(0,0,1)
(0,1,0)
(1,0,0)
Axis
X
Y
Z
Intercept
points
1
1
1
Reciprocals
1/1
1/ 1
1/ 1
Smallest
Ratio
1
1
1
Miller İndices
(111)
55
Example-4
Axis
X
Y
Z
Intercept
points
1/2
1
∞
Reciprocals
(0,1,0)
(1/2, 0, 0)
Smallest
Ratio
1/(½) 1/ 1 1/ ∞
2
Miller İndices
1
0
(210)
56
Example-5
Axis
a
b
c
Intercept
points
1
∞
½
Reciprocals
1/1
1/ ∞
1/(½)
Smallest
Ratio
1
0
2
Miller İndices
(102)
57
Example-6
Axis
a
b
c
Intercept
points
-1
∞
½
Reciprocals
1/-1
1/ ∞
1/(½)
Smallest
Ratio
-1
0
2
Miller İndices
(102)
58
Miller Indices
[2,3,3]
2
c
Plane intercepts axes at
Reciprocal numbers are:
3a , 2 b , 2 c
1 1 1
, ,
3 2 2
Indices of the plane (Miller): (2,3,3)
b
2
Indices of the direction: [2,3,3]
a
3
(200)
(110)
(100)
(111)
(100)
59
60
TYPICAL CRYSTAL STRUCTURES
3D – 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEM
There are only seven different shapes of unit cell which can be
stacked together to completely fill all space (in 3 dimensions)
without overlapping. This gives the seven crystal systems, in
which all crystal structures can be classified.
Cubic Crystal System (SC, BCC, FCC)
Hexagonal Crystal System (S)
Triclinic Crystal System (S)
Monoclinic Crystal System (S, Base-C)
Orthorhombic Crystal System (S, Base-C, BC, FC)
Tetragonal Crystal System (S, BC)
Trigonal (Rhombohedral) Crystal System (S)
61
The only semiconductor
crystals we are interested in
62
Coordinatıon Number
Coordinatıon Number (CN) : The Bravais lattice points closest
to a given point are the nearest neighbours.
Because the Bravais lattice is periodic, all points have the
same number of nearest neighbours or coordination number.
It is a property of the lattice.
A simple cubic has coordination number 6; a body-centered
cubic lattice, 8; and a face-centered cubic lattice,12.
63
Atomic Packing Factor
Atomic Packing Factor (APF) is defined as the
volume of atoms within the unit cell divided by
the volume of the unit cell.
Volume of Atoms in the Unit Cell
APF =
Volume of the Unit Cell
64
1-CUBIC CRYSTAL SYSTEM
a- Simple Cubic (SC)
Simple Cubic has one lattice point so its primitive cell.
In the unit cell on the left, the atoms at the corners are cut
because only a portion (in this case 1/8) belongs to that cell. The
rest of the atom belongs to neighboring cells.
Coordinatination number of simple cubic is 6.
b
c
a
65
Atomic Packing Factor of SC
66
b-Body Centered Cubic (BCC)
BCC has two lattice points so BCC is
a non-primitive cell.
BCC has eight nearest neighbors.
Each atom is in contact with its
neighbors only along the bodydiagonal directions.
Many metals (Fe,Li,Na..etc), including
the alkalis and several transition
elements choose the BCC structure.
b
c
a
67
Atomic Packing Factor of BCC
APF BCC
V atoms
=
= 0.68
V unit cell
Can you work this out?
68
c- Face Centered Cubic (FCC)
There are atoms at the corners of the unit cell and at the center of
each face.
Face centered cubic has 4 atoms so its non primitive cell.
Many of common metals (Cu,Ni,Pb..etc) crystallize in FCC
structure.
69
Atomic Packing Factor of FCC
APF FCC
BCC
V atoms
0.74
=
= 0.68
V unit cell
Can you work this out?
70
Unit cell contents
Counting the number of atoms within the unit cell
Atoms
corner
face centre
body centre
lattice type
P
I
F
Shared Between:
8 cells
2 cells
1 cell
Each atom counts:
1/8
1/2
1
cell contents
1
[=8 x 1/8]
2
[=(8 x 1/8) + (1 x 1)]
4
[=(8 x 1/8) + (6 x 1/2)]
71
72
2 - HEXAGONAL SYSTEM
A crystal system in which three equal coplanar axes
intersect at an angle of 60 , and a perpendicular to the
others, is of a different length.
73
Hexagonal Close-packed Structure
Bravais Lattice : Hexagonal Lattice
He, Be, Mg, Hf, Re (Group II elements)
ABABAB Type of Stacking
a=b a=120, c=1.633a,
basis : (0,0,0) (2/3a ,1/3a,1/2c)
74
Packing
Close packed
A
A
A
A
B
B
B
B
A
B
A
B
C
C
A
B
A
B
C
C
A
A
B
C
A
A
C
A
B
A
C
A
A
A
B
B
A
A
A
A
C
C
C
Sequence ABABAB..
-hexagonal close pack
Sequence ABCABCAB..
-face centered cubic close pack
B
A
B
A
A
A
B
A
A
Sequence AAAA…
- simple cubic
Sequence ABAB…
- body centered cubic
75
Hexagonal
Closest
Packing
76
Cubic
Closest
Packing
77
4 - Diamond Structure
The diamond lattice is consist of two interpenetrating face
centered bravais lattices.
There are eight atom in the structure of diamond.
Each atom bonds covalently to 4 others equally spread
about atom in 3d.
78
4 - Diamond Structure
The coordination number of diamond
structure is 4.
The diamond lattice is not a Bravais lattice.
Si, Ge and C crystallizes in diamond
structure.
79
5- Zinc Blende
Zincblende has equal numbers of zinc and sulfur
ions distributed on a diamond lattice so that
each has four of the opposite kind as nearest
neighbors. This structure is an example of a
lattice with a basis, which must so described
both because of the geometrical position of the
ions and because two types of ions occur.
AgI,GaAs,GaSb,InAs,
80
Zincblende (ZnS) Lattice
Zincblende Lattice
The Cubic Unit Cell.
81
Physics PY4118
Physics of Semiconductor Devices
Crystal Symmetry
Coláiste na hOllscoile Corcaigh, Éire University
College Cork, Ireland
ROINN NA FISICE Department of Physics
2.82
Symmetry?
This is actually really important for some
semiconductor devices, especially:
Inversion Symmetry:
This is (not) required for:
Second harmonic generation
The electro-optic effect
Piezo-electric effect
etc.
Coláiste na hOllscoile Corcaigh,
Éire University College Cork,
Ireland
ROINN NA FISICE Department of Physics
3.83
ELEMENTS OF SYMMETRY
Each of the unit cells of the 14 Bravais lattices has one or
more types of symmetry properties, such as inversion,
reflection or rotation,etc.
SYMMETRY
INVERSION
REFLECTION
ROTATION
84
Lattice goes into itself through
Symmetry without translation
Operation
Element
Inversion
Point
Reflection
Plane
Rotation
Axis
Rotoinversion
Axes
85
Reflection Plane
A plane in a cell such that, when a mirror reflection in this
plane is performed, the cell remains invariant.
86
Rotation Axis
90°
120°
180°
This is an axis such that, if the cell is rotated around it
through some angles, the cell remains invariant.
The axis is called n-fold if the angle of rotation is 2π/n.
87
Symmetry
The geometry of the crystal lattice can be
described by its symmetry
The symmetry is key in understanding certain
physical properties of material
Let’s see how and why
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.88
Crystal Optics - susceptibility
For a real material, the dielectric susceptibility is
not necessarily the same in all directions, thus:
 Px 
 χ11
P  = ε χ
0  21
 y
 Pz 
 χ 31
χ12
χ 22
χ 32
χ13   E x 
χ 23   E y 
 
χ 33   E z 
The dielectric susceptibility is a type 2 tensor
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.89
Type 2 Tensor
A thermodynamic argument shows that the tensor
is symmetric:
 χ11 χ12 χ13 
χ ij = χ ji → χ =  χ12

 χ13
Here is the proof:
χ 22
χ 23
Link 1
χ 23 

χ 33 
Link 2
Let’s now consider symmetry operations…
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.90
Symmetry Operations
A vector can be rotated, mirrored, inverted etc., using a
matrix operator. For example
0
− 1 0
The inversion matrix is: i =  0 − 1 0 


0 − 1
 0
0  x  − x 
− 1 0






Thus:  0 − 1 0   y  = − y 
0 − 1  z   − z 
 0
Link to a list of more symmetry operators
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.91
Symmetry Operations
1 0 0 
Reflection in x-y plane: σ z = 0 1 0 


0 0 − 1


 2π 
 2π 
 cos n  sin n  0






2π 
2π  



Rotation around z: Cn (z ) = − sin
 cos
 0

 n 
 n  

0
0
1




Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.92
Symmetry Operations
So a 4-fold rotation symmetry, will be:


 2π 
 2π 
 cos 4  sin 4  0





  0 1 0
 2π 
 2π   


C4 (z ) = − sin
cos
0
=
−
1
0
0





 4 
 4   

0
0
1  0 0 1




 0 1 0  x   y 






=
−
−
1
0
0
y
x
Thus: 
   
 0 0 1  z   z 
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.93
Symmetry Operations
The symmetry operators can also be used to rotate the matrix
T
′
T = MTM
NOTE
So a 4-fold rotation of our tensor is:
0
− 1

 0
0
= − 1

 0
1 0  χ11 χ12 χ13  0 − 1 0





0 0 χ12 χ 22 χ 23 1 0 0



0 1  χ13 χ 23 χ 33  0 0 1
1 0  χ12 − χ11 χ13   χ 22 − χ12
0 0  χ 22 − χ12 χ 23  =  − χ12 χ11

 
0 1  χ 23 − χ13 χ 33   χ 23 − χ13
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
χ 23 
− χ13 

χ 33 
2.94
Symmetry Operations
Thus, we can then equate the rotated and non-rotated
tensors:  χ11 χ12 χ13   χ 22 − χ12 χ 23 
χ
 12
 χ13
χ 22
χ 23


χ 23 = − χ12
 
χ 33   χ 23
Thus: χ11 = χ 22
χ11
− χ13
 χ11

χ12 = − χ12 = 0
→χ= 0

 0
χ13 = χ 23 = − χ13 = 0

− χ13

χ 33 
0
χ11
0
0 

0

χ 33 
If cubic, then there will be 4-fold rotation along other axis
as well, in which case χ11 = χ 22 = χ 33
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics
PY4118 Physics of
Semiconductor Devices
2.95