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EVOLUTION OF THE ELECTRON MODEL - Coggle Diagram
EVOLUTION OF THE ELECTRON MODEL
FREE ELECTRON THEORY IN METALS
Resistance
:star: measure of how easily charge flows through circuit.
:star: unit is ohm (Ω)
Conductance
:star: reciprocal of resistance.
:star: measure of ease with which current is established.
:star: G = 1/R
:star: unit is siemens (S)
Ohm's Law
:star: voltage across a conductor is proportional to current flowing through it at constant temperature.
:star: or V ∝ I, constant T
:star: V = IR
Classification of Materials
:star: conductors - has valence valence electron
:star: semiconductors - has 4 valence electrons
:star: insulators - atoms are tightly bound
CLASSICAL FREE ELECTRON THEORY OF METALS
also
Drude-Lorentz Theory
Assumptions
:star: large number of free electrons
move freely within metals
. they revolve around a nucleus.
:star: free electrons assumed to behave like gas molecules. thus,
obeys law of kinetic theory of gas
and mean KE of free electron = mean KE of gas molecules at the same temperature.
:star: motion of free electron = electric conduction.
free electrons go through incessant collision with fixed positive ion cores
.
:star: electric field due to ion core is constant.
electron repulsion is negligible
.
:star: free electron
accelerated in the direction opposite to applied electric field
.
Electron Gas Model in Metals
where -eZ is valence electrons, -e(Za - Z) is core electrons and eZa is the nucleus
:star: nucleus and ion cores retain configuration in free atoms.
:star: valence electrons leave to form electron gas in free atoms.
e.g : Sodium (Na) configuration,
where 3s¹ is the valence electron ( = conduction electron ) and the rest are core electrons.
figure above shows equilibrium positions of atomic cores on crystal lattices and surrounded by conduction electrons.
Basic Approximations
(1) between collisions
:star: neglect
electron-ion core
interaction [
free electron approximation
]
:star: neglect
electron-electron
interaction [
independent electron approximation
]
(2) during collisions
:star: assumes electron bounce off the ion core.
:star: assuming some form of scattering.
(3) relaxation time approximation
:star: collision mean free time, 𝜏 is
independent of electron position and velocity
.
(4) the collisions are assumed to maintain thermal equilibrium
.
SUCCESS OF CLASSICAL FREE ELECTRON THEORY
verifies Ohm's law
explains the relationship between electrical conductivity, σ and thermal conductivity, κ
in a uniform metallic rod containing free electrons,
we get,
then,
since,
derives Wiedemann-Franz law
since
and KE of electron,
thus,
which yields the classical lorenz number,
FAILURE OF CLASSICAL FREE ELECTRON THEORY
lorenz number
based on classical theory, lorenz number is given by
which yields
but experimentally, the value should be
specific heat capacity
electron specific heat,
then
theoretically
, if there is a metal with electron density n, its specific heat will be,
but
experimentally
, specific heat contributed by electrons is
only 1% of theoretical value
.
explanation for temperature dependence of resistivity in metals
theoretically,
which yields,
however, for most metals, experimentally the relationship is as such,
explanation for temperature dependence of thermal conductivity in metals
since,
which yields,
then since,
thus we get,
and graphically,
however, experimentally, the relationship is more complicated,
QUANTUM-BASED FREE ELECTRON THEORY IN METALS
:star: fixes shortcomings of the classical model by taking into account the
wave nature of electrons
.
:star: outer shell electrons are free to move through metal but are
trapped within 3D box formed by surface
.
:star: each electrons are represented as
particle in a box
and restricted to
quantized energy levels
.
Fermi Dirac Distribution Function
the probability a particular state with energy E is occupied by one of the electrons in a solid (under equilibrium condition),
which can be viewed as dropping electrons into a potential well,
About Potential Wells
:star: follow
Pauli's exclusion principle
.
:star:
fermi level
- top of the collection of electron energy levels at 0 K.
:star:
fermi energy
- highest energy possessed by an electron at 0 K.
:star:
at absolute zero
- free electrons occupy different energy levels continuously without any vacancies between states.
:star:
when 0 K → T K
electrons
within thermal energy KT of fermi energy
will have its thermal energy = KT and occupy higher energy levels.
electrons below KT of fermi energy
will have its thermal energy ≠ KT
Considering T = 0 K
:star: f(E<EF) = 1
:star: f(E=EF) = 1/2
:star: f(E>EF) = 0
Considering T > 0 K
:star: fermi energy (EF) dependence on temperature is weak in metals.
:star: f(E ≥ EF + 3KT),
:star: f(E ≤ EF + 3KT),
:star: based on thermal excitation theory,
near + below EF = population loss
near + above EF = population gain
Region of Space
one dimensional
:star: [classical viewpoint] particle bounce back and forth along x-axis between 2 impenetrable walls separated by distance L
:star: for 0 < x < L, φ(x) = 0
where energy of particle,
with n = 1,2,3,...
three dimensional
:star: becomes particle in a box with edge length of L
:star: φ(x,y,z) = 0
:star: energy of particle becomes,
Density of States Function
:star: macroscopic size L leads to electron energy levels being close together.
:star: quantum numbers become continuous variables.
:star: number of allowed states per unit volume with energies between E and E + dE is given by,
Number of Electrons Per Unit Volume
given by N(E) = g(E)*f(E) dE
thus N(E) is,
graphically, the relationship is as such,
SOMMERFELD'S MODEL
:star: recognizes that Pauli's principle needs to be utilized (electrons cannot all be in lowest energy state).
:star: electrons are modelled after fermi gas.
:star: provides improved predictions for heat capacity of metals.
Assumptions
:star: mostly similar to Drude model.
:star:
free and independent
electrons. no assumption on nature of scattering.
:star: begins at
TISE
:star: ignores that electrons
move through atomic periodic potential
.
from
to
Schrodinger's Equation
:star: potential differential equation.
:star: for motions of
microscopic particles
.
:star: uses concept of
energy conservation
to obtain
behavior of electron bound to nucleus
.
:star: able to recognize,
quantized nature of energy
wave-particle duality
[where classical mechanics failed]
Hamiltonian's Representation of Classical Mechanics
where the
kinetic energy operator
is given by,
in one dimensional
considering V(x) = 0, we will get,
where
in three dimensional
considering V(x) = 0, for EΨ(x,y,z) we will get,
through decomposition we get,
thus total wavefunction will be,
energy of electron becomes,
and energy of moving electron is,
free electron in a box
:star: suppose that
V(x,y,z) = 0
everywhere inside a cube of length L but infinite at each phase which gives
Ψ(x,y,z) = 0.
:star: without electron-core and electron-electron interaction, conduction electrons
behave like independent electrons
.
:star: applying
boundary condition
into Schrodinger, Ψ(0) = 0 and Ψ(L) = 0,
:star: any function with
periodic lattice
can be written as a
fourier series of the reciprocal lattice
and when periodic boundary condition is applied over translation vector
L
,
extension to 3D space
energy of free electron confined in a cube becomes,
construction of fermi sphere
1.
2 electrons with spin up and down go into lowest energy state where nx = ny = nz = 0 or (0,0,0).
2.
12 electrons go into next lowest energy states: (1,0,0), (0,1,0), (0,0,1), (-1,0,0), (0,-1,0) and (0,0,-1).
3.
process continues with electrons up to 6.02E23 fill reciprocal space.
4.
sphere in reciprocal space obtained.
5.
electron sphere = fermi sphere, electron surface = fermi surface.
then assuming total number of free electron = N0 and radius of fermi sphere = KF,
from that,
thus,
BAND THEORY
addition of periodic potential
potential energy around an
isolated electron
gives,
but when we consider the effects due to
periodic crystal structure
,
we will get
and the potential energy of electrons inside a periodic crystal a becomes,
bloch waves
:star: the wavefunction solution of schrodinger's equation when a
potential is periodic
is
where u(x) = u(a+b)
:star: only needs to find a
solution for single period
.
:star: should be
continuous
smooth
kronnig-penney model
:star: to find u(x) in each region electron wavefunction has to be manipulated.
assumptions
:star: periodic lattice structure.
:star: there is a change in periodic potential surrounding ion cores forming potential wells.
outside the barrier, V = 0 and thus,
within the barrier, E < V0 which gives,
and both of these can be solved with
by assuming potential in periodic delta function, w ➝ 0, V0 ➝ ∞ as long as strength barrier, wV0 = constant
and will yield,
width gap is governed by P
P ➝ 0; very weak barrier
cos (αa) = cos (ka)
α = k
thus,
E = ћ²k² / 2m
and E vs K graph is
parabolic
[follows free electron theory]
P ➝ ∞; large barrier
solution restricted to sin (αa) = 0 and thus, αa = ±nπ
cos (αa) = cos (ka)
cos (nπ) = cos (ka)
nπ = ka
k = nπ/a
thus,
E = ћ²(nπ/a)² / 2m
and E vs K graph has
forbidden gaps at k = ±nπ/2
for metals
T = 0; levels below fermi energy are filled with electrons while above fermi energy is empty.
T > 0; some electrons are thermally excited to above fermi level but not much difference than when T = 0.
electrons are easy to be thermally excited = high electrical conductivity.
for insulators
T = 0; valence band filled. conduction band empty.
T > 0; big band gap energy = few electrons in conduction band = low electrical conductivity
for semiconductors
T = 0; valence band filled. conduction band empty.
T > 0; small band gap = a lot of electrons in conduction band = ↑ T, ↑ electrical conductivity
