The many forms of carbon Carbon is not only the basis of life, it also provides an enormous variety of structures for nanotechnology. This versatility is connected to the ability of carbon to form two stable bonding configurations (sp 2, sp 3 ) with different bond geometry (planar, tetrahedral). sp 2 sp 3 + pzpz  -bonds  -bonds

 - and  - bonds  -bond:  -bond:  orbital  -orbital (bonding)  * orbital  *-orbital (antibonding) + -

Fullerene 0D Graphite, Graphene (= single sheet) 2D sp 2 Diamond 3D sp 3 Nanotube 1D

2D: Graphene, a single sheet of graphite 2010 Nobel Prize in Physics to Geim and Novoselov A graphene sheet can be obtained simply by multiple peeling of graphite with sticky tape. A single sheet is visible to the naked eye. Graphene is very strong, has high electron mobility, and provides a transparent conductor with possible applications in displays and solar cells. The E(p) relation is linear instead of quadratic.

Energy bands of graphite: sp 2 + p z Brillouin zone   M K E Fermi E k x,y  ** pzpz sp 2 s p x,y Graphite is a semi-metal, where the density of states approaches zero at E Fermi. The  -bands touch E Fermi only at the corners K of the Brillouin zone, while a free- electron band would form a Fermi circle. ** 

Two-dimensional  -bands of graphene Occupied Empty E Fermi E [eV] K  =0 M K Empty Occupied k x,y In two dimensions one has the quantum numbers E, k x,y. This plot of the energy bands shows E vertically and k x, k y in the horizontal plane.

“Dirac cones” in graphene A special feature of the graphene  -bands is the linear E(k) relation near the six corners (K) of the Brillouin zone (instead of the parabolic relation for free electrons). In a three-dimensional E(k x,k y ) plot one obtains cone-shaped energy band dispersions.

Topological Insulators A spin-polarized version of a “Dirac cone” occurs in “topological insulators”. These are insulators in the bulk and metals at the surface, because two surface bands bridge the bulk band gap. It is impossible topologically to remove the surface bands from the gap, because they are tied to the valence band on one side and to the conduc- tion band on the other. Hasan and Kane, Rev. Mod. Phys. The metallic surface state bands have been measured by angle- and spin-resolved photoemission (see Lecture 19).

1D: Carbon nanotubes Carbon nanotubes are grown using catalytic metal clusters (Ni, Co, Fe,…). (Lecture 7, Slide 7)

Indexing of Nanotubes armchair n=m zigzag m=0 chiral n  m Circumference vector c r = m a 1 + n a 2 Unwrap a nanotube into planar graphene

Energy bands of carbon nanotubes: Quantization along the circumference Analogous to Bohr’s quantization condition an integer number of electron wavelengths needs to fit around the circumference of the nanotube. Otherwise the electron waves would interfere destructively. This leads to a discrete number of allowed wavelengths n and wave vectors k n = 2  / n along the circum- ference. Along the axis of the nanotube the electrons cane move freely. One gets a one-dimensional band for each quantized state. The kinetic energy p 2 /2m of the motion along the tube is added to the quantized energy level.

Density of states of a single nanotube Calculated Density of States D(E) : Each peak shows the one-dimensional 1/  E singularity at the band edge (see density of states in Lecture 13). Scanning Tunneling Spectroscopy: dI/dV I/V  1 for V  0 when metallic  D(E)

“Two-dimensional” spectroscopy: Measure both photon absorption (x-axis) and photon emission (y-axis). Distinguishing nanotubes with different n and m

C 60 solution in toluene Buckminsterfullerene C 60 has the same hexagon + pentagon pattern as a soccer ball. The pentagons (highlighted) provide the curvature. 0D: Fullerenes Buckminster Fuller, father of the geodesic dome 1996 Nobel Prize in Chemistry to Curl, Kroto, Smalley

Fullerenes with increasing size Fewer pentagons produce less curvature. Symmetry

Mass spectrum showing the different fullerenes generated. Plasma generation of fullerenes in a Krätschmer-Huffman apparatus. Production of fullerenes

Molecular orbitals of C 60 The high symmetry of C 60 leads to highly degenerate levels. i.e., many distinct wave functions have the same energy. Up to 6 electrons can be placed into the LUMO of a single C 60, making it a popular electron acceptor in organic solar cells. The LUMO (lowest unoccupied molecular orbital) is located at the five-fold rings:

C 60 can be charged with up to 6 electrons The ability to take up that many electrons makes C 60 a popular electron acceptor for molecular electronics, for example in organic solar cells.

Empty orbitals of fullerenes from X-ray absorption spectroscopy LUMO, located at the strained five-fold rings C1s core level photon ** **