Transcript Solid State Physics - Florida State University

Molecules and Solids
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Topics

Quantum Theory of Conduction

Band Theory of Solids

Summary
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Quantum Theory of Conduction
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Quantum Theory of Conduction
In an infinitely large, perfect, crystal calculations
show that the free electrons suffer
no scattering.
In real crystals, electrons scatter
off lattice imperfections and
the thermal vibrations of the
lattice ions. The average distance
between collisions is called the
mean free path, λ.
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Quantum Theory of Conduction
Mean Free Path – Consider a box of length L and
cross-sectional area A that contains n ions per unit
volume. Suppose each ion presents a cross sectional
area of size a.
What is the probability of a collision
between an electron and an ion
within the box?
L
a(nAL)
Pr 
 naL
A
A
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Quantum Theory of Conduction
Mean Free Path – A collision is guaranteed to occur
when Pr = 1, that is, when L = λ, the mean free path.
This yields
1

 v 
na
where <v> is the average speed of
the electron and τ is the average
time between collisions.
L
A
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Quantum Theory of Conduction
Resistance – For a wire of length L and cross
sectional area A, the electrical resistance, R, can
be written as
L
R
A
where ρ, the resistivity, is
inversely proportional to the
mean free path ρ = C / λ.
L
A
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Quantum Theory of Conduction
Classically, lattice ions are modeled
as spheres of cross-sectional area
a = πr2. In the quantum theory, we
model ions as points vibrating in
three dimensions with an average
cross section of
a   r2
where <r2> is the average oscillation
amplitude of the ions.
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Quantum Theory of Conduction
Let’s model the ions as simple harmonic
oscillators with potential energy
1 2 1
2 2
U  Kr  M  r
2
2
If we assume the equipartition theorem
holds, then the average potential energy
of a ion is given by
1
U  M  2 r 2  kT
2

r 2  2kT / M  2
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Quantum Theory of Conduction
The mean free path is therefore
1
1
M 2 1
 

2
na n r
2 nk T
So, in this simple model, quantum
theory predicts that the resistivity is
proportional to the temperature
 1 /   T
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Extra Credit
R / R0   / 0  r 2 / r02
As can be seen from the
graph, the prediction
 T
U
1
M  2 r 2  kT
2
fails at very low
temperatures.
Extra Credit: create a
better model!
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Extra Credit
Derive the temperature
R / R0   / 0  r / r
dependence of R/R0 by
computing the average potential energy <U>
of a lattice ion
assuming that the energy
level of the nth vibrational
r 2 U 122M Ur  kT/ M  2
state is given by
2
2
2
0
2
En  (n  12 )
and assuming the ions are
identical, but distinguishable.
Due: November 30
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Band Theory of Solids
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Band Theory of Solids
So far we have neglected the lattice of positively
charged ions.
We have also ignored the Coulomb repulsion between
the electrons and the attraction between the lattice
and the electrons.
The band theory of solids takes into account the
interaction between the electrons and the lattice ions.
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Band Theory of Solids
Consider the potential energy of a 1-dimensional solid
approximated by the Kronig-Penney Model
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Band Theory of Solids
The task is to compute the quantum states and
associated energy levels of this simplified model
by solving the Schrödinger equation
1
2
3
d 2 ( x)

 U ( x) ( x)  E ( x)
2
2m dx
2
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Band Theory of Solids
For periodic potentials, Felix Bloch showed that
the solution of the Schrödinger equation must
be of the form
ikx and the wavefunction must
 ( x)  uk ( x)e reflect the periodicity of
the lattice:
 ( x  n(a  b))
1
2
3
  ( x)e
ikn ( a b )
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Band Theory of Solids
By requiring the wavefunction and its derivative
to be continuous everywhere, one finds energy
levels that are grouped into bands separated by
energy gaps. The gaps occur at
ka   n
1
2
3
The energy gaps
are energy levels that
cannot occur in the
solid.
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Band Theory of Solids
Completely free
electron
p 2 (hk)2
E

2m
2m
electron in a
lattice
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Band Theory of Solids
When, ka   n the wavefunctions become
standing waves. One wave peaks at the lattice
sites, and another peaks between them. Ψ2 has
lower energy
than Ψ1. Moreover, there is a jump in energy
between these states, hence the energy gap.
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Band Theory of Solids
The allowed ranges of the wave vector k are
called Brillouin zones.
zone 1: -π/a < k < π/a; zone 2: -2π/a < k < - π/a;
zone 3: π/a < k < 2π/a, etc.
The theory can explain why some
substances are conductors, some
insulators and
others
semi
conductors
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Conductors, Insulators, Semiconductors
Sodium (Na) has one electron in the 3s state, so
the 3s energy level is half-filled. Consequently, the
3s band, the valence band, of solid
sodium is also half-filled. Moreover,
the 3p band, which for Na is
the conduction band, overlaps with
the 3s band.
So valence electrons can easily be
raised to higher energy states.
Therefore, sodium is a good
conductor.
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Conductors, Insulators, Semiconductors
NaCl has a band gap of 2 eV, which is much larger than
the thermal energy of ~ 1/40 eV at T = 300K
Consequently, only a tiny fraction of electrons are
in the conduction band and so NaCl is an insulator.
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Conductors, Insulators, Semiconductors
Silicon and germanium have band gaps of 1 eV and
0.7 eV, respectively. At room temperature, a small
fraction of the electrons are in the conduction
band. Si and Ge are therefore semiconductors.
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Summary

A simple quantum model predicts a resistivity
proportional to temperature, which is accurate except
at low temperatures.

Energy gaps arise in solids because the wavefunctions
are standing wave states.

The size of the energy gap between the valence and
conduction bands determines whether a substance is a
conductor, an insulator or a semiconductor.
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