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MINDMAP 6 SOLID STATE PHYSICS, 1D region of space • A particle …
MINDMAP 6
SOLID STATE PHYSICS
FREE ELECTRON IN METALS I
Structure of atom :
Resistance :
• measure of the opposition to current flow in an electrical circuit
• Unit
(Ω)
.
Conductance :
• Reciprocal of resistance
• Measure of the ease with which current is established
• Unit (
siemens
)
Ohm’s Law
• electric current is proportional to voltage and inversely proportional to resistance.
Material Conductivity
Semiconductors
• Materials which have a conductivity between conductors and nonconductors or insulators
• Ex:
silicon, germanium, gallium arsenide
Conductivity between 10-8 < σ < 104 (S/m)
• Conductivity lies between of conductors and semiconductor
Comparison
Insulators
• Have high resistance so current doesn’t flow in them
• Most insulator are compound of several elements
• Ex : Glass,ceramic,plastic&wood
• Conductivity between 10-16 < σ < 10-8 (S/m)
• A good insulator has more than 4 electron valences
• Do not allow current to flow through them
Conductors
• Have low resistance so electrons flow through them with ease
• Ex:
metals and alloys
• Metal have conductivity between 1010 < σ < 104 (S/m)
• Best element for conductor :
Copper,silver,gold,aluminium and nickel
• Allow current to flow through them
3 stages of electron theory
S1:
The Classical Free Electron Theory
(free electrons obey the laws of classical mechanics)
S2 :
The Quantum Free Electron Theory
(free electrons obey quantum laws)
S3 :
The Band Theory or Zone Theory
(free electrons move in a periodic field provided by lattice)
Classical Free Electron Theory
a
. Large number of free electrons moving freely within the metals. The electrons revolve around the nucleus in an atom.
b
.The free electrons are assumed to behave like gas molecules
c
. Electric conduction is due to motion of free electrons only. The +ve ion cores are at the fixed positions. The free electrons undergo incessant collisions with the ion core.
d.
When an electric field is applied to the metals, the free electrons are accelerated in the direction opposite to the direction of applied electric field.
e. These electron is called as
"Conduction Electron"
Success of classical free electron
theory:
•
It verifies Ohm’s Law.
•
It explains the electrical and thermal conductivities of metals
•
It derives Wiedemann – Franz Law
.
Failure of classical free electron
theory:
1.Lorentz number
2.Heat capacity
-Metal specific heat :
•C contributed by electron is much smaller than the calculated results.
•Drude model fails to predict C for metal
3.Failure to correctly account for the T dependence of resistivity in metals
FREE ELECTRON IN METALS 2
Fermi–Dirac distribution function
•Fermi function f(E) specifies how many of the existing states at the energy E will be filled with electrons.
• Free electrons possess different energies.
• The first and next electron dropped occupy the lowest available energy, E0
• The third electron dropped would occupy the next energy level E1 (> E0) - Pauli’s exclusion principle.
• If metal contains N (even) number of electrons, they will be distributed in the first N/2 energy levels →the higher energy levels will be empty.
• The highest filled level known as
Fermi level
and the energy in this level called as
Fermi energy (EF)
•At 0 K→ Fermi energy EF is represented as EF0
As temperature of the metal is increased from 0K to TK :
•
The electrons below KBT
from Fermi level, will not take thermal energies → they will not find vacant electron states.
•
The electrons above KBT
from Fermi level,may take thermal energies equal to KBT and occupy higher energy levels
From
Temperature dependence of Fermi-Dirac distribution
At T = 0 K
• All states having energies less than the Fermi energy are occupied
• All states having energies greater than the Fermi energy are vacant
At T > 0 K
• As T increases, the distribution rounds off slightly
• Due to thermal excitation, states below EF lose population & states above EF gain population.
Quantum-based free-electron theory of metals
• The outer-shell electrons are free to move through the metal but trapped within a 3D box formed by metal surfaces.
• Each electron represented as a particle in a box
• Particles in a box are restricted to quantized energy levels.
• By
quantum statistics
- each state of the system can be occupied by only 2 electrons (1 spin up & 1 spin down)
3D region of space
•φ(x, y, z) = 0 at the boundaries of the metal.
•Energy for electron
•me is the mass of the electron and nx, ny, and nz are quantum numbers
•Energy are quantized.
Energy of particle
Density of states function
• We treat quantum numbers as continuous variables
Sommerfeld’s Model
Electronic properties
•
Drude model
- applied kinetic theory to the electrons in a solid.
• It’s able to explain electrical and thermal conductivity although it overestimated the electronic heat capacity
• Electrons are modelled as a Fermi gas, which obey the quantum mechanical Fermi-Dirac statistics.
•
Improvement to the Drude’s model
- Applied Pauli’s exclusion principle
• Electrons cannot all be in the lowest energy state
Schrödinger equation
• It recognize Quantized nature of energy,Wave-particle duality.
• Applied concept of energy conservation to observe the behavior of an electron bound to a nucleus.
(Kinetic Energy + Potential Energy = Total Energy)
Hamilton’s representation of classical mechanics
• Schrödinger used Hamilton’s equation as the basis of quantum mechanics
Motion of an electron in free space
•For electron traveling in +x direction in 1D,V=0 :
•For 3D, the motion of an electron in free space,V=0 :
• Energy of an electron in 3D free space
• wave number k is related to the wavelength λ
The equation can be decomposed into 3 independent equations involving x, y or z by setting (reduction of 3D to 1D case):
(
The total wave function is now expressed as a linear combination of eight different plane waves:
Free electron in a box
•An application of the boundary condition 𝜓(0) = 0 and 𝜓(L) = 0 to Schrodinger’s equation for a single free electron
• Without e-core /e-e interactions, the conduction electrons behave like independent free electrons
• An extension to 3D space leads to wave function
• Energy of free electron confined in a cube with edge length L
The Fermi sphere
•The surface of the Fermi sphere represent the boundary between occupied and unoccupied k states
at absolute zero
for the free electron gas.
•Fermi radius kF:
•Energy of a free electron with Fermi radius kF
Band Theory
Addition of the periodic potential
• PE of the electron around an isolated atom
• When N atoms are arranged to form the crystal, then there is an overlap of individual electron PE functions
• PE of the electron, V(𝑥), inside the crystal is periodic with a period 𝑎.
• The electron PE, V(𝑥), inside the crystal is periodic with the same periodicity as that of the crystal, 𝑎.
Bloch’s Waves
• The mathematical representation of the potential is a periodic function with a period 𝑎
•The wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as Ψ(𝑥) =
𝑒^(𝑖𝑘𝑥)𝑢(𝑥)
where 𝑢(𝑥) is a periodic function which satisfies
𝑢(𝑎 + b) = 𝑢(𝑥)
.
Kronig-Penney Model
• Assuming a periodic lattice structure
• Electrons are subjected to these periodic potentials
•
Scattering power of the potential barrier
(measure of the strength with which electrons in a crystal are attracted to the ions on the crystal lattice sites):
Plot of E vs k
• Discontinuities occur at k = ±π/a, ±2π/a, ±3π/a,
• Extended zone representation
• -π/a<k< +π/a is defined as BZ
• Since k’= k+G
•
P→0 ;Weak barrier
• E vs k follow free electron theory→parabolic
•
P→∞;large barrier
• At large P,electron are tightly bound
• The difference in energy between these two standing waves leads to a discontinuity in energy at k=±nπ/a
Differences of bands and gaps between metals, semiconductors, insulators
Insulators
• Fermi energy EF is at the midpoint between the valence band and the conduction band.
•At T=0, the valence band is filled and the conduction band is empty.
•The band gap energy Eg between the two is relatively large (~ 10 eV)
•Very few electrons in the conduction band and the electrical conductivity is low
Semiconductors
• Fermi energy EF is at the midpoint between the valence band and the conduction band.
•At T=0, the valence band is filled and the conduction band is empty.
•The band gap energy is relatively small (1-2 eV)
•the electrical conductivity of semiconductors is poor at low T but increases rapidly with temperature.
Metals
• The Fermi energy EF is inside the conduction band
•At T=0, all levels in the conduction band below EF are filled with electrons while those above EF are empty.
•For T>0, some electrons can be thermally excited to energy levels above EF
•Electrons can be easily excited to levels above EF by applying a (small) electric field to the metal.
• Metals have high electrical conductivity
Effective Mass
Mass electron:
m from eq above is
effective mass
Holes in semiconductor
• It is produced due to electron movement in a periodic potential
• It can be considered as a hole with a +ve charge.
• The acceleration of an electron:
• Rate of current change when applied with an electric field E
• If one electron in the band
• If the bands are full, then the total effective electron is zero
• If one of the electron marked as j, is jump to the upper bands.
• This phenomena is referred as current
• this current is caused by a positive charged particle known as “hole”
1D region of space
• A particle confined to a 1D region of space, called as particle-in-a-box
• A particle is bouncing back and forth along x axis between two impenetrable walls separated by distance L
wave function φ(x) must be zero for x < 0 and x > L.
Number of electrons per unit volume
MOHAMAD FARID IZUAN BIN AZMI
A19SC0169
