Electrons in Solids Carbon as Example. Electrons are characterized by quantum numbers which can be measured accurately, despite the uncertainty relation. In a solid these quantum numbers are: Energy: E Momentum: p x,y,z E is related to the translation symmetry in time (t),
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In a solid these quantum numbers are:
px,y,zto the translation symmetry in space (x,y,z) .
Symmetry in space allows x=p=0(from p ·x ≥ h/4)
Instead of calculating the electrons for an infinite crystal,
consider just one unit cell (which is equivalent to a molecule).
Then add the couplings to neighbor cells via hopping energies.
The unit cell in reciprocal space is called the Brillouin zone.
K =0 M K
In two dimensions one has the quantum numbers E,px,y.
Energy band dispersions (or simply energy bands) plot E vertical and kx ,ky horizontal.
Energy bands of graphite (including bands)
The graphite energy bands resemble those of graphene, but the,* bands broaden due to the interaction between the graphite layers (via thepzorbitals).
Energy bands of carbon nanotubes A) Indexing of the unit cell
Energy bands of carbon nanotubes B) Quantization along the circumference
Analogous to Bohr’s quantization condition one requires that an integer number n of electron wavelengths fits around the circumference of the nanotube. (Otherwise the electron waves would interfere destructively.)
This leads to a discrete number of allowed wavelengths n and k values kn=2/n .(Compare the quantization condition for a quantum well, Lect. 2, Slide 9). Two-dimensional k-space gets transformed into a set of one-dimensional k-lines (see next).
1/r Quantization along the circumference
(A) Wrapping vectors (red) and allowed wave vectors kn (purple) for (3,0) zigzag, (3,3) armchair, and (4,2) chiral nanotubes. If the metallic K-point lies on a purple line, the nanotube is metallic, e.g. for (3,0) and (3,3). The (4,2) nanotube does not contain K, so it has a band gap. All armchair nanotubes (n,n) are metallic, since the purple line through contains the two orange K-points. Note that the purple lines are always parallel to the axis of the nanotube, since the quantization occurs in the perpendicular direction around the circumference.
(B) Band structure of a (6,6) armchair nanotube, including the metallic K-point (orange dot). Each band corresponds to a purple quantization line. Their spacing is Δk = 2/1= 2/circumference= 1/ radius.
Quantization along the circumference K M (for the dashed band)
Energy bands of carbon nanotubesC) Relation to graphene
folded in half
Two steps lead from the - bands of graphene to those of a carbon nanotube:
1) The gray continuum of 2D bands gets quantized into discrete 1D bands.
2) The unit cell and Brillouin zone need to be converted from hexagonal (graphene) to rectangular (armchair nanotube. Thereby, energy bands become back-folded.
empty levels Quantization along the circumference
filled levelsTwo classes of solids Carbon nanotubes cover both
to move electrical conductor
to move poor conductor.
The quantum numbers E and k can both be measured by angle-resolved photoemission. This is an elaborate use of the photoelectric effect which was explained as quantum phenomenon by Einstein:
Energy and momentum of an emitted photoelectron are measured.
Use energy conservation to get the electron energy:
Electron energy outside the solid
= Electron energy insidethe solid
Photon inElectron outside
Energy bands of graphene from photoemission Quantization along the circumference
Evolution of the in-plane band dispersion with the number of layers
(single layer graphene)
Evolution of the perpendicular band dispersion with the number of layers: Nmonolayersproduce Ndiscrete k-points.
Ohta et al., Phys. Rev. Lett, 98, 206802 (2007)
The density of states D(E) Quantization along the circumference
D(E) is defined as the number of states per energy interval. Each electron with a distinct wave function counts as a state.
D(E) involves a summation over k, so the k-information is thrown out.
While energy bands can only be determined directly by angle-resolved photoemission, there are many techniques available for determining the density of states.
By going to low dimensions in nanostructures one can enhance the density of states at the edge of a band (E0). Such “van Hove singularities” can trigger interesting pheno-mena, such as superconductivity and magnetism.
Density of states of a single nanotube from scanning tunneling spectroscopy
Calculated Density of States
Scanning Tunneling Spectroscopy (STS)
Cees Dekker, Physics Today, May 1999, p. 22.
Optical spectra of nanotubes with tunneling spectroscopydifferent diameter, chirality
Simultaneous data for fluorescence (x-axis) and absorption (y-axis) identify the nanotubes completely.
Bachilo et al., Science 298, 2361 (2002)
Sodium Dodecyl Sulfate (SDS)
O'Connell et al, Science 297, 593 (2002)
Quantized Conductance levels
Dip nanotubes into a liquid metal (mercury, gallium). Each time an extra nanotube reaches the metal. the conductance increases by the same amount.
The conductance quantum: G0= 2e2/h 1/13k
(Factor of 2 for spin ,)
Each wave function = band = channel contributes G0. Expect 2G0=4e2/h for nanotubes, since 2 bands cross EF at the K-point. This is indeed observed for better contacted nanotubes (Kong et al., Phys. Rev. Lett. 87, 106801 (2001)).
Cees Dekker, Physics Today, May 1999, p. 22.
Conductance per channel: G= G0•T (G0=2e2/h, transmission T1)Energy to switch one bit: E = kBT•ln2Time to switch one bit: t = h/EEnergy to transport a bit: E = kBT•d/c(distance d, frequency )
Birnbaum and Williams, Physics Today, Jan. 2000, p. 38.
Landauer, Feynman Lectures on Computation .
Energy scales in carbon nanotubes levels~20 eV Band width (+* band)~ 1 eV Quantization along the circumference (k=1/r)~ 0.1 eV Coulomb blockade (charging energy ECoul=Q/eC ) Quantization energy along the axis (k||=2/L)~ 0.001 eV Many-electron effects (electron holon + spinon)