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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 7
ONE BLOCH AT A TIME
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Last time
1. Vector space (expectation values measure the projection
on different eigenvectors)
2. Eigenvalues and eigenstates as a linear algebra problem
3. Variational principle
4. Its application to a H atom (atomic units)
5. Hamiltonian for a molecular system; bonding and
antibonding states
6. Potential energy surface of a molecule
7. Vibrations at equilibrium; quantum harmonic oscillator
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Study
• Chapter 2 of Singleton textbook – “Band
Band
theory and electronic properties of solids”
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Dynamics, Lagrangian style
• First construct L=T
L=T-V
V
• Then, the equations of motion are given by
d ⎛⎜ ∂L
dt ⎜⎝ ∂q& j
⎞ ∂L
⎟−
=0
⎟ ∂q
j
⎠
(the dot is a time derivative)
• Why ? We can use generalized coordinates.
Also, we only need to think at the two
scalar functions T and V
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
Newton’s second law, too
• 1-d,
1 d 1 particle: T=1/2 mv2, V=V(x)
d ⎛⎜ ∂L
dt ⎜⎝ ∂q& j
⎞ ∂L
⎟−
=0
⎟ ∂q
j
⎠
⎛ ⎛1 2⎞⎞
&
∂
mx
⎜
⎟
⎟
d ⎜ ⎝2
⎠ ⎟ + ∂V = 0
⎜
dt ⎜
∂x&
⎟ ∂x
⎜
⎟
⎝
⎠
∂V
d
( mx& ) = −
∂x
dt
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Hamiltonian
• We could use it to derive Hamiltonian
dynamics (twice the number of differential
equations, but all first order). We introduce
a Legendre transformation
pi =
∂L
∂q&i
q&i =
H (q, p,t) = ∑ q&i pi − L(q, q& ,t)
i
∂H
∂pi
− p& i =
∂H
∂qi
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
1-dimensional monoatomic chain
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
4
Properties
• Unique solutions for k in the first BZ
us
us+1
• Phase velocity and group velocity
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Properties
• Standing waves
• Long wavelength limit
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
5
Ring geometry
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1-dimensional diatomic chain
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
6
Image removed due to copyright restrictions.
Please see Fig. 22.10 in Ashcroft, Neil W., and N. David Mermin. Solid State Physics. Belmont,
CA: Brooks/Cole, 1976. ISBN: 9780030839931.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Translational Symmetry
3 Dim
1 Dim
t1
a
2 Dim
t2
Double
t1
t2 Triple
Single
t1
t1
Cl-
Cu+
t2
Figure by MIT OpenCourseWare.
7
Bravais Lattices
• Infinite arrayy of points with an arranggement and orientation that
appears exactly the same regardless of the point from which the
array is viewed.
r r
r
r
R = la1 + ma2 + na3 l,m and n integers
r r
r
a1 , a2 and a3 primitive lattice vectors
• 14 Bravais lattices exist in 3 dimensions (1848)
• M. L. Frankenheimer in 1842 thought they were 15. So, so naïve…
7 Crystal Classes
4 Lattice Types
Bravais
Lattice
Parameters
Triclinic
a1 = a2 = a3
a 12 = a 23 = a 31
Monoclinic
a1 = a2 = a3
a 23 = a 31 = 900
a 12 = 900
Orthorhombic
a1 = a2 = a3
a 12 = a 23 = a 31 = 900
Tetragonal
a1 = a2 = a3
a 12 = a 23 = a 31 = 900
Trigonal
a1 = a2 = a3
a 12 = a 23 = a 31 < 1200
Cubic
a1 = a2 = a3
a 12 = a 23 = a 31 = 900
Hexagonal
a1 = a2 = a3
a 12 = 1200
a 23= a 31= 900
Simple
(P)
a3
Volume
Centered (I)
Base
Centered (C)
Face
Centered (F)
a2
a1
Figure by MIT OpenCourseWare.
8
Symmetry
• Symmetry operations: actions that
transform an object into a new but
undistinguishable configuration
• Symmetry elements: geometric entities
(axes, planes, points…) around which we
carry out the symmetry operations
Figure by MIT OpenCourseWare.
9
Symmetry elements and their corresponding operations
Symmetry operations
Symmetry elements
leave molecule unchanged
E
Identity
E
Cn
n-Fold rotation axis
Cn, Cn2,....., Cnn rotate about axis by 360o /n 1, 2, .... , n times (indicated by superscript)
σ
Mirror plane
σ
reflect through the mirror plane
i
Inversion center
i
(x, y, z)
Sn
n-Fold rotation-reflection axis
Sn
rotate about axis by 360o /n, and reflect through a plane perpendicular to axis.
(-x, -y, -z)
Figure by MIT OpenCourseWare.
Group Therapy…
A group G is a finite or infinite set of elements A, B, C, D…together
with an operation “☼” that satisfy the four properties of:
1. Closure: If A and B are two elements in G, then A☼B is also in G.
2. Associativity: For all elements in G, (A☼B) ☼C==A☼ (B☼C).
3. Identity: There is an identity element I such that I☼A=A☼I=A for
every element A in G.
4. Inverse: There is an inverse or reciprocal of each element.
Therefore, the set must contain an element B=inv(A) such
that A☼inv(A)=inv(A) ☼A=I for each element of G.
10
Examples
• Integer numbers, and addition
• Integer numbers, and multiplication
• Real numbers,, and multip
plication
• Rotations around an axis by 360/n
C2v
Figure by MIT OpenCourseWare.
11
Symmetries
of H2O
Figure by MIT OpenCourseWare.
Symmetries
of H2O
Figure by MIT OpenCourseWare.
12
The 4 symmetry operations of H2O
form a group (called C2v)
1.
2.
3.
4.
Closure: A☼B is also in G.
Associativity: (A☼B) ☼C=A☼ (B☼C)
Identity: I☼A=A☼I
Inverse: A☼inv(A)=inv(A) ☼A=I
Second
Operation
First Operation
E
C2
σv
σ'v
E
E
C2
σv
σ'v
C2
C2
E
σ'v
σv
σv
σv
σ'v
E
C2
σ'v
σ'v
σv
C2
E
Figure by MIT OpenCourseWare.
Ten crystallographic point groups in 2d
1
2
3
4
6
C1
C2
C3
C4
C6
m
2mm
Cs
C2ν
3m
C3ν
4mm
6mm
C4ν
C6ν
The ten crystallographic plan point groups. Upper symbol,
international notation; lower symbol, Schoenflies notation
(see text).
Figure by MIT OpenCourseWare.
13
The Crystallographic Point Groups and the Lattice Types.
Crystal System
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Hexagonal
Cubic
Schoenflies
Symbol
C1
Ci
C2
Cs
C2h
D2
C2v
D2h
C4
S4
C4h
D4
C4v
D2d
D4h
C3
C3i
D3
C3v
D3d
C6
C3h
C6h
D6
C6v
D3h
D6h
T
Th
O
Td
Oh
Hermann-Mauguin
Symbol
1
1
2
m
2/m
222
mm2
mmm
4
4
4/m
422
4mm
42m
4/m mm
3
3
32
3m
3m
6
6
6/m
622
6mm
6m2
6/m mm
23
m3
432
43m
m3m
Order of the
group
1
2
2
2
4
4
4
8
4
4
8
8
8
8
16
3
6
6
6
12
6
6
12
12
12
12
24
12
24
24
24
48
Laue Group
1
32 crystallographic
point groups in 3d
2/m
mmm
4/m
4/m mm
3
3m
6/m
6/m mm
m3
m3m
Figure by MIT OpenCourseWare.
Crystal Structure = Lattice + Basis
Crystal Structure = Lattice + basis
Lattice
Basis
Figure by MIT OpenCourseWare.
14
Primitive unit cell and conventional unit cell
Figure by MIT OpenCourseWare.
Periodic boundary conditions
for the ions (i.e. the ext. potential)
• Unit cell = Bravais lattice =
space filler
• Atoms in the unit cell +
infinite periodic replicas
15
Reciprocal lattice (I)
• Let’s start with a Bravais lattice, defined in
terms of its primitive lattice vectors…
r
a3
r
a2
r
a1
r r
r
r
R = la1 + ma2 + na3
l , m, n integer
numbers
r
R = ( l , m, n )
Reciprocal lattice (II)
• …and then let’s take a plane wave
r r
r
Ψ (r ) = A exp[i (G ⋅ r )]
16
Reciprocal lattice (III)
• What are the wavevectors for which our
plane wave has the same amplitude at all
lattice points ?
r r
r r r
exp[i(G ⋅ r )] = exp[i (G ⋅ (r + R))]
r r
exp[i(G ⋅ R )] = 1
r r
r
r
exp[[i(G ⋅ (la1 + ma2 + na3 ))] = 1
r r
r
a1 , a2 and a3 define the
primitive unit cell
r r
Gi ⋅ a j = 2πδ ij
r r
r
G1 , G2 and G3 define the
reciprocal space Brillouin Zone
Reciprocal lattice (IV)
r r
Gi ⋅ a j = 2πδ ij n integer is satisfied by
r r
r
r
G = hb1 + ib2 + jb3 with h, i, j integers,
provided
r r
a2 ×a3
r
b1 = 2π r r r
a1⋅ a2 ×a3
(
)
r r
a3 ×a1
r
b2 = 2π r r r
a1⋅ a2 ×a3
(
)
r r
a1×a2
r
b3 = 2π r r r
a1⋅ a2 ×a3
(
)
r
G = ( h, i, j ) are the reciprocal-lattice vectors
17
Examples of reciprocal lattices
Direct lattice
Reciprocal lattice
Simple cubic
Simple cubic
FCC
BCC
BCC
FCC
Orthorhombic
Orthorhombic
r r
a2 × a3
r
b1 = 2π r r r
a1⋅ a2 × a3
(
)
18