Electrons in Solids Carbon as Example Electrons are characterized by quantum numbers which can be measured accurately, despite the uncertainty relation.

Electrons in SolidsCarbon 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: px,y,z

• E is related to the translation symmetry in time (t),

px,y,z to the translation symmetry in space (x,y,z) .

• Symmetry in time allows t = E = 0 (from E · t ≥ h/4 )Symmetry in space allows x = p = 0 (from p · x ≥ h/4 )

• The quantum numbers px,y,z live in reciprocal space since p = ћ k .

Likewise, the energy E corresponds to reciprocal time. Therefore, one

needs to think in reciprocal space-time, where large and small are

inverted (see Lecture 6 on diffraction).

Use a single crystal to simplify

calculations

Unit cell

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.

Two-dimensional energy bands of graphene

Occupied

Empty

EFermi

E

kx,y

K =0 M K

M

Empty

OccupiedIn 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)

Brillouin

zone

M

K

EFermi

E

kx,y

*

s

px,y

pz

pz

sp2

*

The graphite energy bands resemble those of graphene, but the , * bands broaden due to the interaction between the graphite layers (via the pz orbitals).

Energy bands of carbon nanotubes A) Indexing of the

unit cell

21 aaC mnr

armchairn=m

zigzagm=0

chiralnm0

Circumference

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

k

(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 .

Energy bands of carbon nanotubes

C) Relation to graphene

(5,5) Nanotube

folded in half

EF

Graphene

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.

K M (for the dashed band)

Two classes of solids Carbon nanotubes cover

both

• Energy levels are continuous . • Electrons need very little energy

to move electrical conductor

Metals

Semiconductors, Insulators

• Filled and empty energy levels are separated by an energy gap.

• Electrons need a lot of energy

to move poor conductor .

Energy

filled levels

empty levels

empty levels

filled levels

gap

Measuring the quantum numbers of electrons in a solid

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

− Photon energy

= Electron energy inside the solid

Photon in Electron outside

Electron inside

Ohta et al., Phys. Rev. Lett, 98, 206802 (2007)

Energy bands of graphene from photoemission

In-plane k-components

(single layer graphene) Evolution of the perpendicular band dispersion

with the number of layers: N

monolayers produce N discrete k-points.

Evolution of the in-plane band dispersion with the number of layers

The density of states D(E)

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.

Tailoring the Density of States

by Confinement to Nanostructures

Adjust Potential Wellvia d

1D

3D EFermi d

Energy

Densityof States

Cees Dekker, Physics Today, May 1999, p. 22.

Density of states of a single nanotube from scanning tunneling

spectroscopy

Calculated Density of States

Scanning Tunneling Spectroscopy (STS)

Bachilo et al., Science 298, 2361 (2002)

Optical spectra of nanotubes

with different diameter, chirality

Simultaneous data for fluorescence (x-axis) and absorption (y-axis) identify the nanotubes

completely.

O'Connell et al, Science 297, 593 (2002)

Need to prevent nanotubes from touching each other for sharp levels

Sodium Dodecyl Sulfate (SDS)

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 = 2 e2/h 1 / 13k

(Factor of 2 for spin , )

Each wave function = band =

channel contributes G0 . Expect

2G0 = 4 e2/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.

Quantized Conductance

Conductance per channel: G = G0•T (G0 = 2 e2/h, transmission T1)

Energy to switch one bit: E = kBT • ln2

Time to switch one bit: t = h / E

Energy to transport a bit: E = kBT • d/c (distance d,

frequency )

Limits of Electronics from Information Theory

Birnbaum and Williams, Physics Today, Jan. 2000, p. 38.

Landauer, Feynman Lectures on Computation .

Energy scales in carbon nanotubes

~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)