Download Electronic States in Solids. What will be covered? 1. Electronic

Electronic States in Solids.
What will be covered?
1. Electronic States in Solids – Basic concepts
2. Energy Bands.
3. Density of States
Why do we need to know this material?
This forms the basic for understanding many optical, electrical and magnetic
properties of materials and will assist in the future section on semi-conductors.
Overview
The states of electrons in solids have an important similarity to the states of electrons of
the free atoms because the interactions between electrons on adjacent atoms can
never completely destroy the basic structure of the electronic levels of the individual
atoms.
On the other hand, the interactions between atoms are strong enough to cause
serious perturbation of the levels of the free atoms.
The most important consequence of this perturbation is the splitting of the valence
energy levels of the free atoms into nearly continuous energy bands.
This phenomenon is the origin of many electrical, magnetic and optical properties of
solids.
The results obtained from band theory are more general than than those for simple
coulombic interactions and include covalent and metallic bonding as special cases.
This generality provides a basis for the classification of solids and for the prediction of
certain of their intrinsic properties.
When the atoms are separated widely separated, the 1s electrons in each atom "see"
only the nucleus of the atom in which they reside; the allowed energy levels are simply
the atomic energy levels.
The energies of the two atoms are the same, regardless of the spin state (±1/2).
As the electrons are allowed to come into close proximity, the electron clouds overlap
and the electrons begin to "see" the potential fields of both nuclei.
The wave mechanical treatment of this state is complicated, but the results of the
analysis may be simply stated.
When the electron clouds overlap, the 1s atomic energy level subdivides into one of
two separate levels depending on the relative spins of the two electrons.
This means that the energy of the molecule may have one of two values depending
on the spins of the electrons.
When the spins of the two electrons are antiparallel, the energy of the molecule is
lower than the energies of the two isolated atoms, i.e. a stable situation.
The physical meaning of this lower energy state is that when the atoms are close
together the electrons may spend much of their time between the two positively
charged nuclei.
The wave mechanical treatment also shows that when the spins of the electrons are
parallel, the corresponding energy of the molecule is higher than the energies of the
two isolated atoms, i.e. an unstable situation.
Energy Bands.
The formation of energy bands in a solid can be demonstrated easily by considering
changes in electronic energy levels, which occur when a group of identical atoms
with wide distances of separation are brought together to form a solid.
Let us first consider what happens when two hydrogen atoms are brought together.
Schematic representation of energy levels for the H2 molecule as a function of atomic
separation r. ro is the equilibrium separation distance.
The energy of a hydrogen molecule as a function of internuclear separation.
Hence, when the atoms come close together the outer electrons begin to interact
and the electronic energy levels subdivide, i.e. the 1s atomic energy level splits into
two separate and distinct energy levels.
Now let N identical atoms be placed on a lattice with interatomic spacing so large
that the atoms have virtually no interaction with one another.
For this system, the energy level diagram of the solid is precisely the same as that of a
single atom.
If the lattice spacing of the ensemble of atoms is decreased, the atoms come within
interaction range of one another and the free-atom energy levels change.
What was a single level for the solid with large lattice spacing broadens out into a
large number of closely spaced levels with smaller lattice spacing.
Features of the Energy Band
The binding energy of the solid as a whole must be caused by shifts in the levels of the
electrons just as for chemical bonding between two atoms.
Thus, as the solid is formed, the energy levels must shift downward on average.
The electrons which are most affected are the valence electrons since they are
closest to neighboring ions.
The equilibrium lattice spacing must be at an energy minimum with the energy levels
rising at a closer distance of separation.
Each atomic level splits into N molecular levels, which can accommodate all the
electrons that formerly were in similar single atomic levels, i.e. N atoms result in N levels.
If N is sufficiently large, as in a real crystal, a quasicontinuum of levels result.
The bandwidths increase very slowly as N increases; the width of the outer band
increases very slowly as N increases; the width of the outer band increases slightly
more than that of the next inner band.
The inner completed levels are very tightly bound to their nuclei and are virtually
unaffected by the presence of other ions in the lattice.
The outermost bands whether they overlap or whether they are completely filled or
not, provide a basis for the prediction of the properties of solids and for their
classification.
Such prediction and characterization depends on the extent to which the outermost
band is filled and whether or not it overlaps other bands.
The width of any gaps, or energy intervals, between the outer bands also plays a
significant role.
The energy bands in a solid can be classified as follows:
•
The highest energy band that is at least partially occupied is known as the
valence band.
•
All the bands below the valence band are core bands.
•
The energy band above the valence band is the conduction band.
•
The term band gap refers to the magnitude of the forbidden energy range
between the valence and conduction bands.
The electrons in a band of energy levels are mobile and are not localized on individual
atoms.
An electron that is at one instant in orbit in a particular atom has a finite probability of
being captured by a neighboring atom.
The greater the amount of overlap, the greater the probability that transfer occurs
from one atom to another.
Such electrons belong jointly to the entire collection of atoms of the crystal rather than
to particular atoms.
Let us now consider qualitatively the electron distribution in a solid with a partially filled
valence band.
At 0K, the electrons are distributed in the lowest possible energy levels consistent with
the Pauli exclusion principle.
As the temperature increases, electrons gain energy and some of them are promoted
to higher energy levels.
Electron distributions within a partially filled valence band as a function of
temperature: (a) at 0 K, (b) at T1 > 0 K, and (c) at T2 > T1.
We can make the discussion more quantitative by evaluating the probability that an
energy level is occupied according to the Fermi - Dirac distribution function:
where f(E) is the probability that an energy level E is fully occupied and Ef is the Fermi
energy.
At 0 K when all of the electrons have occupied the lowest possible energy levels, the
energy of the most energetic electron is the Fermi energy.
Note that at 0 K, all energy levels below Ef are filled and all those above Ef are empty.
For T>0 K, there is a nonzero probability that any level above Ef may be occupied while
any level below Ef may be unoccupied.
Perhaps, the most important aspect of the Fermi-Dirac distribution function is that the
probability of occupancy for any level above Ef increases with temperature.
(a) The change in the probability of occupation, as described by the Fermi-Dirac
distribution function f(E), as a function of temperature, and (b) the probability that
energy level E* in part (a) is occupied, f(E*), as a function of temperature.
A comparison of the qualitative and quantitative representations of the electron
distribution within a partially filled valence band at several temperatures is given
below:
The relationship between the occupied energy levels and f(E) as a function of
temperature for solids with a partially filled energy band: (a) at 0 K, (b) at T > 0 K, and
(c) at T2 > T1.
We must now consider the temperature dependence of the electron distribution in
band gap materials.
For an electron to occupy an energy level, two criteria must be satisfied:
z
At the temperature of interest, f(E) must yield a reasonable probability of
occupation; and
z
The energy level of interest must be located within an allowed energy range, i.e.
not in an energy gap.
The occupied energy levels were not limited by the second constraint for solids with a
partially filled valence band; however, in solids with completely filled energy bands at
0 K the second constraint is important.
The next figure shows the relationship between the occupied energy levels, f(E) and
temperature for solids with an energy gap.
In such solids, Ef is located in the center of the energy gap.
At 0 K f(E) predicts that levels in the lower half of the band gap are occupied;
however, since this energy region is forbidden, these levels remain unoccupied.
At higher temperatures, f(E) assumes a sigmoidal shape. For the situation in part (b), all
levels above Ef are still in the band gap and hence unoccupied.
Only when the temperature reaches that of part (c) do the lowest levels of the
conduction ban have a small probability of being occupied.
The
relationship between the occupied energy levels and f(E) as a function of
temperature for solids with a band gap: (a) at 0 K, (b) at T > 0 K, and (c) at T2 > T1.
Density of States.
An energy band exists corresponding to each valence energy level, or state, in the
isolated atom, and each valence band has as many individual states as there are
interacting atoms in the system.
However, the states are not uniformly distributed across the energy band.
It can be demonstrated that the density of states, i.e. the number of states per unit
energy N(E), is greatest in the center of the energy band.
In addition, N(E) in the lower part of the band increases parabolically with energy such
that N(E) is proportional to E1/2.