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Solid State Physics
Bands & Bonds
PROBABILITY
DENSITY
The quantum
world is governed
by statistics.
The probability density P(x,t) is information that tells us
something about the likelihood of obtaining one of a
variety of possible results in a position measurement.
As particles move through space, their wavefunction
evolves with time and the probability density changes
(along with the position information it contains).
Postulates of
Quantum Mechanics
1. Every physically-realizable state of a system is described
by a state function  that contains all accessible physical
information about the system in that state.
2. If a system is in a quantum state , then
is the probability that in a position measured at time t the
particle will be detected in an infinitesimal volume dv.
3. In quantum mechanics, every observable is expressed by
an operator that is used to obtain physical information
about the observable from state functions. For an
observable from state functions.
Observable
Position
Symbol
Operation
Multiply by x
momentum
Hamiltonian
Kinetic energy
Potential energy
Multiply by V(x,t)
Postulates of
Quantum Mechanics
4. The time development of
the state functions of an
isolated quantum system is
governed by the timedependent Schrodinger
equation (TDSE).
Classical versus Quantum
The TDSE is the “equation of motion” in quantum mechanics
just as Newton’s 2nd is the equation of motion in classical
mechanics.
In the classical limit, the
laws of quantum
mechanics reduce to
Newtonian mechanics.
Classical versus Quantum
In classical physics a
macroscopic particle moves
through 2D space along a
trajectory.
In quantum physics a
microscopic particle moves
through 1D space and is
represented by a complex
state function, the real part
of which is shown.
Translation
Under translation, a macroscopic free particle must not alter
its probability density, even though the state function will be
altered.
Wave Packets
To construct a wave packet with a single region of enahnced amplitude, we must
superpose an infinite number of plane wave functions with infinitesimally differing
wave numbers.

i ( kx t )
1
 ( x, t ) 
2
Note: although this is
developed for the free
particle, it is applicable to
many one-dimensional
quantum systems.


A(k )e
dk
1
( x, 0) 
2



A(k )eikx dk
The amplitude function A(k) is the Fourier transform of the state function at
time t = 0.
Time-independent Schrödinger
In solid state physics, we’ll use the time-independent equation
because in the study of most material properties solids are
modeled assuming stationary nuclei at their equilibrium
positions in the solid and other properties are statistically
uniform.
d  ( x)

 V ( x) ( x)  E ( x)
2
2m dx
2
Particle in a Box
0
L
Energies
Why do atoms bond to each other?
When they are bound together, the energy
of the system is lower than when the
atoms are isolated.
 In fact, most electrons occupy energy
states that are lower than the ground state
of an isolated atom.

Conductors, Insulators, and
Semiconductors
Consider the available energies for electrons in the materials.
As two atoms are brought
close together, electrons
must occupy different
energies due to Pauli
Exclusion principle.
Instead of having
discrete energies as in
the case of free atoms,
the available energy
states form bands.
Wave Equation for an Electron in a
Periodic Potential
We’ve been considering electrons as moving freely in a box potential. In
reality, the periodicity of the crystal leads to a periodic potential.
Bonding Mechanisms
Can be qualitatively distinguished on the basis of electron distribution.

Covalent


Metallic


Electron concentration is greater in the neighborhood
along the line “joining” two atoms together.
Electron concentration is uniform throughout the
spaces in between atoms.
Ionic

An electron can be transferred from the neighborhood
surrounding one atom to that of another and the
atoms can then be viewed as bonded through
Coulomb interaction.
Quantum Mechanical View
Electron wave functions overlap – forming a new wave function
Calculation of the total energy of a solid begins with
finding solutions to the Schrödinger equation for the
electron energies and wavefunction.
 2
 (r, t )

  (r, t )  U (r) (r, t )  i
2m
t
2
2
2


(
r
,
t
)


(
r
,
t
)

 (r, t )
2
where   (r, t ) 


2
2
x
y
z 2
Quantum Mechanical View


The wave function
can be written in
terms of its spatial
part and its temporal
part.
The temporal part has
the form…
 (r, t )   (r) f (t )
f (t )  e
 it
Quantum Mechanical View

The real and imaginary parts of the wave
function oscillate sinusoidally with time, but the
probability density is independent of time.
 ( r, t )   (r )
2

2
The angular frequency of the wave function is
related to the energy of the electron.
 it
E   , so with (r, t )   (r)e
we get...
 2

  (r)  U (r) (r)  E (r)
2m
Quantum Mechanical View



The wave function
and the probability
density depend on
the potential
energy function for
the electron.
The individual e
are replaced by a
continuous
distribution.
Hartree Approach
electron - nuclei interactio ns
e2
U en (r)  
40

i
Zi
r Ri
electron - electron interactio ns
e2
U ee (r) 
40

n(r )
d 
r  r
where n(r )   i (r, )
i
2
Hydrogen Bonding
H2 molecule
electron
r
R
r
R
Proton B
Proton A
1
1
U (r)  
 
4 0  r r  R
e
2



Antibonding States
Not all molecular states result in a lowering of
energy.
States



When atoms are brought close together, an
electron state is created for each core state of
an atom – some bonding (lower energy) and
some antibonding (higher energy).
These states are all occupied. Good thing we
have the Pauli exclusion principle.
Atoms are attracted by the outer electrons,
occupying bonding states, and repelled mostly
by ion core electrons.
Covalent Bonding
Let’s develop a bonding wave function through
linear combination of localized atomic orbitals for
each atom a:
 a (r)   Aan an (r)
n
where
n runs over all atomic states
 an is an atomic orbital, different for each type of atom
Aan is a constant, different for each type of atom
Covalent Bonding

For two atoms:
 (r)  Ca  an (r  R a )  Cb  bn (r  R b )
where
R a , R b are the atomic positions
Aan , Abn , Ca , Cb chosen to satisfy th e Schrodinge r Eqn.

Lots of solutions, but the lowest energy states
have coefficients that make the overlap large.
Covalent Bonding
There are two types of covalent bonds : sigma and
pi. When the covalent bonds are linear or aligned
along the plane containing the atoms, the bond is
known as sigma () bond. Sigma bonds are strong
and the electron sharing is maximum. Methane
CH4 is a good example for sigma bond and it has
four of them.
Covalent Bonds
When the electron orbitals overlap laterally, the
bond is called the pi () bond. In pi bonds, the
resulting overlap is not maximum and these bonds
are relatively weak.
Covalent Bonds have
well-defined directions
in space. Attempting to
alter those directions is
resisted – making them
both hard and brittle.
Ionic Bonding
Consider two different atoms, such as Ga and As joined by a covalent
bond. The  functions come from different orbitals. The C constants
would be different. The wave function still spreads between the
atoms, but the probability is that the shared electron will be nearer to
one atom than another.
Atoms in this situation act like oppositely charged ions that are
attracted by the Coulomb force.
Ionic Bonding
Ionic Bonding
+1
1
Ionic Bonding
Bonding may be evaluated energetically via
calorimetric measurements.
1.
2.
3.
measure a-a bond energy in pure a material
measure b-b bond energy in pure b material
measure a-b material – This will be greater
than the average of a-a and b-b bonds.
The difference is the energy of the ionic
bond.
van der Waals Bonding
An atom has an electric dipole moment
when the average position of its electrons
does not coincide with that of the nucleus.
 The dipole moment produces an electric
field, even if the atom is neutral.
 This field induces dipole moments in other,
nearby atoms.

van der Waals Bonding
p2
E
R
E
1


3
p

R̂
R̂

p
1
1
3
1
4 0 R
p1
van der Waals Bonding

 1
p2   E 
3
p

R̂
R̂

p
1
1
3
4 0 R
The potential energy of
a dipole in an electric
field is...



1
2
2
U   p2  E 
3
(
p

R̂
)

p
1
1
2 6
(4 0 ) R

Graphite
A
B
3.4 Å
A
1.42 Å
Metallic Bonds

The outer electrons of metallic atoms are loosely
bound. When a solid formed, the potential
energy barrier is substantially reduced.

Metals typically found in columns IA, IIA, IB, IIB
of the periodic table have cohesive energies of
around 1 to 5 ev/atom (for comparison,
convalent bonds are in the 3 to 10 eV/atom
range).