Electronic Band Structure of Solids

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Electronic Band Structure of Solids

• Introduction to Solid State Physics
  http//www.physics.udel.edu/bnikolic/teaching/phy
  s624/phys624.html

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What are quantum numbers?

• Quantum numbers label eigenenergies and
  eigenfunctions of a Hamiltonian

Sommerfeld -vector ( is momentum)
Bloch -vector ( is the crystal
momentum) and (the band index).

• The Crystal Momentum is not the Momentum of a
  Bloch electron the rate of change of an electron
  momentum is given by the total forces on the
  electron, but the rate of change of electronic
  crystal momentum is
• where forces are exerted only by the external
  fields, and not by the periodic field of the
  lattice.

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Semiclassical dynamics of Bloch electrons

• Bloch states have the property that their
  expectation values of and , follow
  classical dynamics.  The only change is that now
  (band structure) must be used
  
• A perfectly periodic ionic arrangement has zero
  resistance. Resistivity comes from imperfections
  (example a barrier induces a reflected and
  transmitted Bloch wave), which control the
  mean-free path. This can be much larger than the
  lattice spacing.
• A fully occupied band does not contribute to the
  current since the electrons cannot be promoted to
  other empty states with higher . The current
  is induced by rearrangement of states near the
  Fermi energy in a partially occupied band.
• Limits of validity

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What is the range of quantum numbers?

• Sommerfeld runs through all of k-space
  consistent with the Born-von Karman periodic
  boundary conditions

Bloch For each , runs through all wave
vectors in a single primitive cell of the
reciprocal lattice consistent with the Born-von
Karman periodic boundary conditions runs
through an infinite set of discrete values.
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What are the energy levels?
Sommerfeld

Bloch For a given band index n, has no
simple explicit form. The only general property
is periodicity in the reciprocal space
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What is the velocity of electron?
Sommerfeld The mean velocity of an electron in
a level with wave vector is

Bloch The mean velocity of an electron in a
level with band index and wave vector
is Conductivity of a perfect crystal
NOTE Quantum mechanical definition of a mean
velocity
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What is the Wave function
Sommerfeld The wave function of an electron
with wave vector is

Bloch The wave function of an electron with
band index and wave vector
is where the function has no
simple explicit form. The only general property
is its periodicity in the direct lattice (i.e.,
real space)
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Sommerfeld vs. Bloch Density of States
Sommerfeld ? Bloch

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Bloch van Hove singularities in the DOS

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Bloch van Hove singularities in the DOS of
Tight-Binding Hamiltonian

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Sommerfeld vs. Bloch Fermi surface

• Fermi energy
  represents the sharp occupancy cut-off at T0 for
  particles described by the Fermi-Dirac statitics.
• Fermi surface is the locus of points in
  reciprocal space where

No Fermi surface for insulators!
Points of Fermi Surface in 1D
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Sommerfeld vs. Bloch Fermi surface in 3D
Sommerfeld Fermi Sphere
Bloch Sometimes sphere, but more likely anything
else

For each partially filled band there will be a
surface reciprocal space separating occupied from
the unoccupied levels ? the set of all such
surfaces is known as the Fermi surface and
represents the generalization to Bloch electrons
of the free electron Fermi sphere. The parts of
the Fermi surface arising from individual
partially filled bands are branches of the Fermi
surface for each n solve the equation
in variable.

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Is there a Fermi energy of intrinsic
Semiconductors?

• If is defined as the energy separating
  the highest occupied from the lowest unoccupied
  level, then it is not uniquely specified in a
  solid with an energy gap, since any energy in the
  gap meets this test.
• People nevertheless speak of the Fermi energy
  on an intrinsic semiconductor. What they mean is
  the chemical potential, which is well defined at
  any non-zero temperature. As , the
  chemical potential of a solid with an energy gap
  approaches the energy of the middle of the gap
  and one sometimes finds it asserted that this is
  the Fermi energy. With either the correct of
  colloquial definition, does not
  have a solution in a solid with a gap, which
  therefore has no Fermi surface!

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DOS of real materials Silicon, Aluminum, Silver

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Colloquial Semiconductor Terminology in Pictures
?PURE

DOPPED?
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Measuring DOS Photoemission spectroscopy

Fermi Golden Rule Probability per unit time of
an electron being ejected is proportional to the
DOS of occupied electronic states times the
probability (Fermi function) that the state is
occupied
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Measuring DOS Photoemission spectroscopy

Once the background is subtracted off, the
subtracted data is proportional to electronic
density of states convolved with a Fermi
functions.
We can also learn about DOS above the Fermi
surface using Inverse Photoemission where
electron beam is focused on the surface and the
outgoing flux of photons is measured.
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Fourier analysis of systems living on periodic
lattice
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Fouirer analysis of Schrödinger equation
Potential acts to couple with its
reciprocal space translation
and the problem decouples into N independent
problems for each within the first BZ.
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Fourier analysis, Bloch theorem, and its
corollaries

• Each zone n is indexed by a vector and,
  therefore, has as many energy levels as there are
  distinct vector values within the Brillouin
  zone, i.e.

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Free Bloch electrons?

• Really free electrons ? Sommerfeld
  continuous spectrum with infinitely
  degenerate eigenvalues.
• does not mean
  that two electrons with wave vectors and
  have the same energy, but that
  any reciprocal lattice point can serve as the
  origin of .

• In the case of an infinitesimally small periodic
  potential there is periodicity, but not a real
  potential. The function than is
  practically the same as in the case of free
  electrons, but starting at every point in
  reciprocal space.

Bloch electrons in the limit
electron moving through an empty lattice!
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Schrödinger equation for free Bloch electrons
Counting of Quantum States Extended Zone
Scheme Fix (i.e., the BZ) and then count
vectors within the region corresponding to
that zone. Reduced Zone Scheme Fix in any
zone and then, by changing , count all
equivalent states in all BZ.
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Free Bloch electrons at BZ boundary
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Free Bloch electrons at BZ boundary

• Second order perturbation theory, in crystalline
  potential, for the reduced zone scheme

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Free Bloch electrons at BZ boundary

• For perturbation theory to work, matrix elements
  of crystal potential have to be smaller than the
  level spacing of unperturbed electron ? Does not
  hold at the BZ boundary!

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Extended vs. Reduced vs. Repeated Zone Scheme

• In 1D model, there is always a gap at the
  Brillouin zone boundaries, even for an
  arbitrarily weak potential.
• In higher dimension, where the Brillouin zone
  boundary is a line (in 2D) or a surface (in 3D),
  rather than just two points as here, appearance
  of an energy gap depends on the strength of the
  periodic potential compared with the width of the
  unperturbed band.

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Fermi surface in 2D for free Sommerfeld electrons

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Fermi surface in 2D for free Bloch electrons

• There are empty states in the first BZ and
  occupied states in the second BZ.
• This is a general feature in 2D and 3D Because
  of the band overlap, solid can be metallic even
  when if it has two electrons per unit cell.

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Fermi surface is orthogonal to the BZ boundary

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Tight-binding approximation
?Tight Binding approach is completely opposite to
free Bloch electron Ignore core electron
dynamics and treat only valence orbitals
localized in ionic core potential.

There is another way to generate band gaps in the
electronic DOS ? they naturally emerge when
perturbing around the atomic limit. As we bring
more atoms together or bring the atoms in the
lattice closer together, bands form from mixing
of the orbital states. If the band broadening is
small enough, gaps remain between the bands.
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Constructing Bloch functions from atomic orbitals
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From localized orbitals to wave functions overlap

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Tight-binding method for single s-band
?Tight Binding approach is completely opposite to
free Bloch electron Ignore core electron
dynamics and treat only valence orbitals
localized in ionic core potential.

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One-dimensional case
?Assuming that only nearest neighbor orbitals
overlap

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One-dimensional examples s-orbital band vs.
p-orbital band

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Wannier Functions
?It would be advantageous to have at our disposal
localized wave functions with vanishing overlap
Construct Wannier functions
as a Fourier transform of Bloch wave functions!

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Wannier functions as orthormal basis set

1D example decay as power law, so it is not
completely localized!
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Band theory of Graphite and Carbon Nanotubes
(works also for ) Application of TBH
method

• Graphite is a 2D network made of 3D carbon atoms.
  It is very stable material (highest melting
  temperature known, more stable than diamond). It
  peels easily in layers (remember pencils?).
• A single free standing layer would be hard to
  peel off, but if it could be done, no doubt it
  would be quite stable except at the edges
  carbon nanotubes are just this, layers of
  graphite which solve the edge problem by curling
  into closed cylinders.
• CNT come in single-walled and multi-walled
  forms, with quantized circumference of many
  sizes, and with quantized helical pitch of many
  types.

Lattice structure of graphite layer There are
two carbon atoms per cell, designated as the A
and B sublattices. The vector connects the
two sublattices and is not a translation vector.
Primitive translation vectors are
.
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Chemistry of Graphite hybridization,
covalent bonds, and all of that

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Truncating the basis to a single orbital
per atom

• The atomic orbitals as well as the
  atomic carbon functions form strong
  bonding orbitals which are doubly occupied and
  lie below the Fermi energy. They also form
  strongly antibonding orbitals which are high up
  and empty.
• This leaves space on energy axis near the Fermi
  level for orbitals (they point
  perpendicular to the direction of the bond
  between them)
• The orbitals form two bands, one
  bonding band lower in energy which is doubly
  occupied, and one antibonding band higher in
  energy which is unoccupied.
• These two bands are not separated by a gap, but
  have tendency to overlap by a small amount
  leading to a semimetal.

Eigenstates of translation operator
Bloch eigenstates
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Diagonalize 2 x 2 Hamiltonian
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Band structure plotting Irreducible BZ
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Graphite band structure in pictures

• Plot for some special directions
  in reciprocal space there are three directions
  of special symmetry which outline the
  irreducible wedge of the Brillouin zone. Any
  other point of the zone which is not in this
  wedge can be rotated into a k-vector inside
  the wedge by a symmetry operation that leaves the
  crystal invariant.

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Graphite band structure in pictures
Pseudo-Potential Plane Wave Method
Electronic Charge Density
In the plane perpendicular to atoms
In the plane of atoms
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Diamond vs. Graphite Insulator vs. Semimetal
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Carbon Nanotubes

• Mechanics Tubes as ultimate fibers.
• Electronics Tubes as quantum wires.
• Capillary Tubes as nanocontainers.

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From graphite sheets to CNT

• Single-wall CNT consists of rolling the honeycomb
  sheet of carbon atoms into a cylinder whose
  chirality and the fiber diameter are uniquely
  specified by the vector

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Metallic vs. Semiconductor CNT

• The 1D band on CNT is obtained by slicing the 2D
  energy dispersion relation of the graphite sheet
  with the periodic boundary conditions
• Conclusion
• The armchair CNT are metallic
• The chiral CNT with
  are moderate band-gap semiconductors.

Metallic 1D energy bands are generally unstable
under a Peierls distortion ? CNT are exception
since their tubular structure impedes this
effects making their metallic properties at the
level of a single molecule rather unique!
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CNT Band structure