Quantum numbers label eigenenergies and eigenfunctions of a Hamiltonian
Sommerfeld: -vector ( is momentum)
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.
Bloch: -vector ( is the crystal momentum ) and (the band index).
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 .
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: NOTE: Quantum mechanical definition of a mean velocity Bloch: The mean velocity of an electron in a level with band index and wave vector is: Conductivity of a perfect crystal:
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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
Fermi energy represents the sharp occupancy cut-off at T=0 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 Schr ö dinger 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.:
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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Schr ödinger 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.
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!
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:
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 of atoms In the plane perpendicular to atoms
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 :
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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