Order 1252715: Condensed Matter physics
tutorthammy
Lect-11-Tight-Binding.pdf
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Tight Binding Electron Model
In this approach we consider how the electronic configuration
of the free atoms change as they are brought together to form
a solid.
Assumes electrons sufficiently tightly bound to the atoms of
the solid to be identified with quantum states in the free atom.
To a good approximation, valid for core electrons, not for
valence electrons.
Ch 10 Ashcroft & Mermin
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Tight Binding Electron Model
Non-degenerate electronic
levels in an atomic potential
Energy levels for N such atoms
in a periodic array, plotted as inverse
interatomic spacing. Atoms far apart,
the levels nearly degenerate, closer together,
the levels broaden into bands.
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Tight Binding Electron Model
Two Na atoms,
interatomic
spacing apart (3.7 Ang)
r
Radial wave
functions x r
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Tight Binding Model -1D
In one-dimension:
Consider a single orbital on atom n, called
Assume N sites, and periodic boundary conditions
(so site N is same as site 0)
Assume orthogonal to each other
Take a general trial wave function:
Effective Schrodinger equation can
be written as:
where the matrix element is:
Ashcroft & Mermin
Ch 10
(1)
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Tight Binding Model – 1D
Write Hamiltonian as
where K is the kinetic energy
and Vj is the Coulomb interaction
of the electron with the nucleus at site j
First term on RHS is Hamiltonian for only a single nucleus;
Take the tight-binding orbitals to be the atomic orbitals:
is the energy of an electron on nucleus m in absence of other nuclei
p is the “crystal
momentum”
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Tight Binding Model – 1D
Can then obtain
Need to find second term on RHS. It describes an interaction such
that an electron on the mth atom can be transferred to the nth atom.
Generally only possible if n and m very close in energy.
Can write:
“tight binding chain” well studied, t known as “hopping term” has dimensions of energy – large when orbitals close, small when far away
Here, no electron transferred,
Just shift energy on site
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Tight Binding Model
The bandwidth depends on magnitude of t, which depends on
distance between nuclei. To the right there are N states
each belonging to an atomic orbital. On left, these N states form a band.
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Tight Binding Model – 1D
Solution: Ansatz for normalization, N sites
in system
Substitute into LHS of SE (1)
(1)
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Tight Binding Model – 1D
Solution: Ansatz for normalization, N sites
in system
Substitute into LHS of SE (1)
Is equal to the RHS of SE (1)
Obtain:
- Has a maximum and
a minimum energy
- Electrons only have eigenstates within
a certain energy band
Energy difference between top and bottom
of band called the “band width”
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Tight Binding Model
Bottom of band of similar form to
that for free electrons:
Near bottom band, dispersion parabolic:
Expanding for small k
View bottom of band as almost
being like free electrons, except
define new mass
(Nothing to do with actual mass of
electron, depends on t )
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Tight Binding Model – Filling bands
Assume each atom (of N atoms)
contributes one electron into the band
(atom has one valence electron). N
possible k-states in band, means
band ½ occupied (two electrons per
state, spin-up + spin-down)
If small electric field applied,
costs a small energy to shift the
Fermi surface, populating few
k-states moving right;
depopulating some moving left –
current induced; System metal
If atoms donate two electrons, band entirely full – small
electric field, no freedom to populate states; carries no current
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Tight Binding Model – Multiple bands
Consider one atom per unit cell, but with two orbitals
If system has one electron per cell (one orbital occupied), lower orbital filled
and upper one empty. As atoms come together lower band remains filled,
upper empty, till bands overlap when both partially filled and becomes a metal.
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Tight Binding
Model – Two atoms per cell, each with
one orbital
Now two possible
energy eigenvalues
for each k – two bands
If each atom is
monovalent, lower
band completely filled,
upper empty.
If each atom divalent
both bands
completely filled
Energy Bands – 1D
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Two orbitals per unit cell, with 2 electrons per atom:
exactly enough electrons to fill lower band – two possibilites:
Get insulator with “direct band gap”
(valence band maximum and
conduction band minimum at same k)
Get metal - two bands both partially
filled. If greater separation of bands,
would get insulator again with indirect
band gap.
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Energy Bands – 2D
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• Consider square lattice of monovalent atoms. The Brillouin
zone is square. Since one electron per atom, enough electrons
to half fill BZ.
• If no periodic potential, get circle. With periodic
Potential gaps open at BZ boundaries. Fermi surface
of copper,
monovalent
• Square lattice of divalent atoms – enough electrons to fill BZ
Leftmost: No periodic potential; Right: with strong periodic potential
all states filled (gap at zone boundary) Intermediate strength; 2 electrons
Tight Binding Model – 2D
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Similar to expectation
for band-gap
opening at zone-
boundary from
nearly-free electron
model
For single atomic orbital on a square lattice
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Direct and Indirect Electron Transitions
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Band-gap determines minimum
energy excitation
If k for valence band
maximum same as for
conduction band minimum,
called “direct band-gap”
If not, called “indirect band-gap”
Relationship – Band structure and Density of
states
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Metals and insulators / semiconductors
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Metals and insulators / semiconductors
• A metal has a finite density of states at the chemical
potential (Fermi energy).
• A semiconductor must have an absolute gap in its band
structure (only necessary
criterion, not sufficient).
• The number of electrons per unit cell must be such
that all the bands are
exactly filled up to this gap.
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Angle-resolved photoemission (ARPES)
• Measure the energy and emission angle of the photoemitted electrons outside the surface.
• Calculate the energy and the k-vector outside the surface.
• Infer the energy and the k-vector inside the solid, i.e. the bands.
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Band structure of Al
measure E, k outside the solid
deduce E(k) inside the solid
