Band Theory for Metals
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What might the MO picture for a bulk
metal look like?
For n AOs, there will be n MOs
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When a sample contains a very large
number of Li atoms (e.g. 6.022×1023
atoms in 6.941 g), the MOs produced
will be so close in energy that they
form a band of energy levels.
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Bands are named for the AOs from
which it was made (e.g. 2s band)
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Band Theory
When large no. of atomic orbitals overlap a large no. of closely
spaced molecular orbitals will be formed and they form a band.
So in metals there are 1s band, 2s band , 2p band etc.. Each
energy level in an s band is associated with a maximum of two
quantum states & that in a p band with maximum of six
quantum states. Thus a metallic crystal with N no. of atoms will
have 2N quantum states associated with s band & 6N states with
p band.
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Bands
The band formed by the combination of valence orbitals of
the atoms is called valence band and that formed by the combination
of next lying empty orbitals is called conduction band. For example,
3s valence atomic orbitals of sodium metal overlaps to form valence
3s band . There is no need to consider 1s, 2s 2p electrons of sodium,
because these electrons are strongly bound to individual atoms and do
not contribute significantly to bonding. 3p orbitals overlap to form
conduction band .The number of individual molecular orbitals is equal
to the number of contributing atomic orbitals .
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22S2
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Band Theory for Metals (and Other Solids)
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In an alkali metal, the valence s band is only half full.
e.g. sodium
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If there are N atoms of sodium in a sample, there
will be 2N electrons in 3s orbitals.
There will be N states made from 3s orbitals, each
able to hold two electrons.
As such, of the states in the 3s band will be full
and states will be empty (in ground state Na).
Like all other alkali metals, sodium conducts
electricity well because the valence band is
only half full. It is therefore easy for electrons
in the valence band to be excited into empty
higher energy states.
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Methods of conduction : Band structure of conductors
Conduction occurs because there is no energy gap between filled & unfilled
M.Os .
1.Alkali metals : Each atom has one s’ electron , therefore the valence s band
contain n electrons. But an s band can accommodate 2n no. of electrons ie, for
alkali metals the valence s band is only half filled .When a potential difference
is applied, an electron in a partly filled band can jump to a vacant M.O in the
same band . Very little energy is required to do this because of the continuous
nature of the band . This electron is free to move throughout the band & thus
conduction occurs.
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Band Theory for Metals (and Other Solids)
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In an alkaline earth metal, the valence s band is full.
e.g. beryllium (band structure shown at right)
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If there are N atoms of beryllium in a sample, there
will be 2N electrons in 2s orbitals.
There will be N states made from 2s orbitals, each
able to hold two electrons so all states in the 2s band will be
and
will be empty (in ground state Be).
So, why are alkaline earth metals conductors?
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Recall that the energy difference between 2s and 2p AOs is
is lower for elements on the LHS of the periodic table. So,
the 2s band in beryllium overlaps with the empty p band.
Electrons in the valence band are easily excited into the
conduction band.
In beryllium, the conduction band (band containing
the lowest energy empty states) is the 2p band.
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2.Alkaline earth metals: Since each atom has 2 electrons , the
valence s band contains 2n no .of electrons & is filled. But the
alkaline earth metals are conductors. This is because the valence s
band and the next conduction p band overlap , giving a partially
filled s & p bands. This overlap of bands occurs because of small
separation between valence s & p bands in atoms at the left side
of the periodic table. The resulting mobility of electrons make the
alkaline earth metals good conductors of electricity.
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Band Theory for Metals (and Other Solids)
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Consider an insulator e.g. diamond (band structure shown below)
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If there are N atoms of carbon in a sample, there will be 2N valence electrons.
The valence orbitals of the carbon atoms will combine to make two bands, each
containing 2N states.
The lower energy band will therefore be the valence band, containing 4N electrons
(in ground state diamond).
The higher energy band will be the conduction band, containing no electrons (in
ground state diamond).
The energy gap between the valence band and the conduction band is big enough that
it would be difficult for an electron in the valence band to absorb enough energy to
be excited into the conduction band.
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Band structure of insulators: The valence band in insulators is full. It
is separated by a large, forbidden gap from the next conduction band
which is empty. Diamond is an excellent insulator with a band gap of 6
e.v Electrons cannot be promoted to an empty level. Hence no
conductivity occurs. The size of the band gap depends upon the atoms
involved. For elements ,the gap show a general decrease down the group
in the periodic table following the general trend towards metallic
behavior For example, diamond ( 6 e.v) , silicon( 1.2 e.v), Germanium (
0.8 e.v) , tin (0.1 e.v ).
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Band Theory for Metals (and Other Solids)
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Materials will exhibit a range of band gaps determining whether
they are conductors, insulators or semi-conductors.
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Our measuring stick is the temperature-dependent kB·T
-23 J/K
 kB is the Boltzmann constant: 1.38065 × 10
 T is the temperature in Kelvin
kB·T is a measure of the average thermal energy of particles in a
sample
As a rule of thumb:
 If the size of the band gap is much larger than kB·T, you have an
insulator. e.g. diamond: ~200×kB·T
 If the size of the band gap is smaller than (or close to) kB·T, you
have a conductor. e.g. sodium: 0×kB·T, tin: 3×kB·T
 If the size of the band gap is about ten times larger than kB·T, you
have a semiconductor. e.g. silicon: ~50×kB·T
Band gaps can be measured by absorption spectroscopy. The
lowest energy light to be absorbed corresponds to the band gap.
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Band structure of semiconductors : Semiconductors have a similar band
structure to insulators, but the band gap is not very large. Hence at room
temperature thermal energies are large enough to promote electrons to
conduction band .
This generate two partially filled bands electronic
conduction is possible. Si & Ge have completely filled v.b & would be
expected to be an insulator. The number of mobile electrons can be
increased by two ways ie, two types of conduction mechanisms may be
distinguished in semiconductors.
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Band Gap
The conducting behaviour of a crystal is determined by the nature of the
valence band & its separation from the empty conduction band(energy gap or
band gap) . The width of the bands & the size of the band gap between V.B
& C.B is proportional to the degree of interaction between at. orbitals on
adjacent atoms . The stronger the interaction , wider the band & larger the
band gap. The degree of interaction follows the order 2p – 2p > 3p – 3p >4p
– 4p > 5p – 5p . So the band gap is largest for diamond & smallest for lead.
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The size of the band gap depends upon the atoms involved. For
elements ,the gap show a general decrease down the group in the
periodic table following the general trend towards metallic behavior
For example, diamond ( 6 e.v) , silicon( 1.2 e.v), Germanium ( 0.8
e.v) , tin (0.1 e.v ). Hence Diamond is an insulator, Si is a semi
conductor, Ge is a better semi conductor, Sn is a conductor & lead
is a metallic conductor
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