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-TIGP course Dec. 2007<br />
Wei-Li Lee, IoP, <strong>Academia</strong> <strong>Sinica</strong>
-TIGP TIGP course on carbon nanostructure<br />
Dec. 15 th - Lecture Ι : 0D system, carbon-based buckyballs ( fullerene)<br />
Dec. 22 st - Lecture ΙΙ : 2D system, emerging material : graphene<br />
D 29th Dec. 29 L t ΙΙΙ 1D t b t b<br />
th - Lecture ΙΙΙ : 1D system, carbon nanotube<br />
Jan. 5 th at 3 pm : Study group oral presentation<br />
Jan. 12 th - Final exam
4 topics :<br />
Lecture - Monday 2pm - 5pm<br />
-TIGP course on carbon nanostructure<br />
20 minutes break from 3:20pm to 3:40 pm<br />
Study group – 2-3 2 3 people a group<br />
each person, 15 minutes oral presentation, 5 for questions<br />
0D system : Practical application of fullerene<br />
1D system : Recent progress of carbon nanotube composites as a space elevator<br />
2 D system : Graphene electronics : advantage and disadvantage<br />
3D system : Quantum information using diamond nanocrystals<br />
Reference Books:<br />
Introduction to Nanotechnology by Poole and Owens<br />
Science of Fullerenes and carbon nanotubes by Dresselhaus and Eklund<br />
Solid state Physics by Ashcroft/ Mermin
WMAP image of CMB (3 Kelvin) Angular spectrum<br />
WMAP<br />
Others<br />
Carbon<br />
Oxygen<br />
Baryonic ayo cmatter attein Universe U vese<br />
4600<br />
Helium<br />
Hydrogen<br />
0 200000 400000 600000 800000<br />
parts per million by mass
Dave Wilkinson<br />
James Peebles
Others<br />
Nitrogen<br />
Hydrogen<br />
Carbon<br />
Human Body<br />
OOxygen<br />
0 20 40 60 80<br />
mass percentage
• Carbon : atomic number 6, 1s 2 [2s 2 2px 1 2py 1 ]<br />
Orbitals used<br />
for bond<br />
Molecular orbital :<br />
ψ<br />
∑<br />
= i<br />
C<br />
X<br />
j<br />
ij i<br />
i<br />
Xi : atomic orbitals<br />
Shape &<br />
bond angle g<br />
Examples:<br />
Sp s, px Digonal 180 o C 2H 2 Acetylene<br />
S 2 Sp Ti l G hit C H<br />
2 s, px, py Trigonal<br />
120o Graphite, C2H4 Ethylene<br />
sp 3 s, px, py, pz Tetrahedra Diamond, CH 4<br />
109 o 28’ Methane
• Many stable and known forms at RR.T. T<br />
Examples: diamond, graphite, amorphous carbon, fullerene, carbon<br />
nanotube and nanobud…etc<br />
• TTwo main i structures t t : one with ith sp 3 hhybrid b id bbonds d (di (diamond) d) and d th the<br />
other with sp2 hybrid bonds (graphite, fullerene, nanotube and<br />
nanobud)<br />
• Dramatic different properties between diamond and graphite<br />
Diamond Graphite<br />
Electric Insulator Conductive<br />
Hardness 10 (Mohs scale) 11-2 2<br />
Appearance Transparent Opaque (black)<br />
Value Expensive cheap
• European Football like molecule containing 60 carbons<br />
- 12 pentagonal and 20 hexagonal faces<br />
- Double bond length 1.4 Å and single bond 1.46 Å<br />
• NNamed d after ft architect hit t R. R Buckminster B k i t Fuller F ll (1895 (1895-1983) 1983)<br />
Geodesic dome<br />
• Geometrically-allowed y fullerene C20+h*2 20+h 2<br />
12 pentagonal faces + arbitrary number<br />
of hexagonal faces (h)<br />
• Smallest Fullerene C 20<br />
• Smallest isolated (stable) Fullerene C 60
• “An An idea from outer space” space : 5.6 5 6 eV optical extinction<br />
• Robert Kurl, Harold Kroto and Richard Smalley<br />
-- Nobel Prize laureates in Chemistry 1996
• Carbon ball with a metal core<br />
• Mass production of Fullerene by astrophysicists D.<br />
R. Huffmann and W. Krätschmer<br />
• CCarbon b nanotube t b – special i l electric l t i and d<br />
mechanical properties<br />
•New superconducting p g crystals y M3C 3 60
• quasi-0D system : d < ℓ mfp<br />
• Di Discrete t energy llevel l resembles bl th the atomic t i llevel l of f a ffree atom t<br />
- ex : 3-D infinite rectangular square well<br />
2 2<br />
ππ h 2 2 2<br />
2<br />
En = ( )( nx<br />
+ ny<br />
+ nz<br />
) = E0n<br />
2ma<br />
6 degeneracy (including<br />
spin) at ground state<br />
QQuantum t number b = 0,1,2… 012 level Eo
LUMO<br />
• Shell model in a free fullerene : symmetry-based model<br />
• 60 πelectrons filling the molecular level<br />
Gap ~ 1.92 eV<br />
HOMO<br />
• Free C60 molecule should be a good insulator
• Molecular crystal : BCC stacking structure<br />
• Grown by slow evaporation from benzene solution<br />
filled with C60 molecules<br />
• Band calculation :<br />
LDA + Gaussian orbital basis set<br />
• Useful information :<br />
Insulator with direct band gap ~ 1.5 eV band width ~ 0.4 eV<br />
charge density map suggest weak coupling<br />
b/w fullerene molecule
• Two standard orientations of fullerene molecule<br />
TTwo fold f ld sym. axis i<br />
Two standard orientations<br />
• Merohedral disorder : random choice b/w one of the two possible standard<br />
orientations ( lack of four fold symmetry point)<br />
relative orientation b/w adjacent C 60 can affect its physical properties.
• Transport charges from doping alkali metal element M<br />
• Two competing process due to doping effect:<br />
- decrease of C 60 wave func. overlap<br />
- increase of the D.O.S<br />
• Best conductivity occurs at x ~ 3 (half filled band)<br />
-available available sites (octahedral and tetrahedral) all filled<br />
• undoped fullerene ρ 300K~ 10 8 Ωcm<br />
• Single crystal K K3CC 60 ρ ρ300K ~5mΩcm ~ 5 mΩcm<br />
•Strongly correlated electronics system<br />
-k Fℓ
• Discovered in K 3C 60 by Hebard (Bell lab, 1991) T c ~ 19.8 K<br />
• Highest Tc ~ 33K in Cs2RbC 2 60 60<br />
• The larger the radius of the dopant alkali atom the higher the Tc
• In weak-coupling limit ( ג
• Hall coefficient : R H = 1/ne<br />
Sign Sign change at 200K<br />
Both electron and hole like pockets<br />
•Transverse magnetoresistance :<br />
ρ ρ ρ C L,<br />
I<br />
ρ ρ ρ<br />
Δ<br />
Δ Δ<br />
= +<br />
0<br />
Classical orbital contribution :<br />
positive and quadratic in H<br />
0<br />
0<br />
Weak localization and ee-e e interaction:<br />
Negative, H 2 at low H and H 1/2 in high H
• Appeared In disordered and time reversal symmetric system<br />
• Negative MR : Strongly suppressed by applying magnetic field<br />
• Merohedral disorder and also missing alkali ion at the tetrahedral and<br />
octahedral sites<br />
origin<br />
Beff ⊙<br />
impurity<br />
∫ ∫ ⋅<br />
Constructive |2A| 2 localization<br />
B⋅<br />
dA = A dl<br />
: additional phase change
(Ι) Pauli susceptibility<br />
B<br />
• DOS D.O.S. :<br />
Spin down<br />
E<br />
K 3C 60 : 28 states/eV-C60<br />
Rb Rb3CC 60 : 38 states/eV states/eV-C60<br />
C60<br />
E F<br />
D.O.S.<br />
Spin up<br />
1<br />
n = ∫ d dε<br />
ε N ( εε<br />
+ μμ<br />
B H ) f ( εε<br />
)<br />
↑ ∫ 2<br />
M = μ ( n − n )<br />
P<br />
B<br />
2<br />
B<br />
↑<br />
χ = μ N(<br />
E<br />
F<br />
↓<br />
)
(ΙΙ) Specific Heat<br />
• In crystals with free electron gas model at low T<br />
C = Ce<br />
+ C p<br />
C<br />
e<br />
⎡ ⎡π<br />
= ⎢<br />
⎣ 3<br />
2<br />
k<br />
2<br />
B<br />
⎤<br />
N(<br />
ε F ) ⎥T<br />
,<br />
⎦<br />
C<br />
p<br />
=<br />
3<br />
4<br />
12 12π<br />
⎛ T ⎞<br />
n<br />
5 ⎜ ik<br />
B<br />
θ D<br />
• In amorphous system and also fullerene solids at low T<br />
Einstein mode (1907) ( )p plays y an important p role<br />
C = C<br />
e<br />
+ C<br />
p<br />
+ C<br />
2<br />
C = pnik<br />
B<br />
E B<br />
ω / k T<br />
E<br />
( hω<br />
/ k<br />
h E ( e<br />
E B<br />
T ) e<br />
−1)<br />
h hωω<br />
/ k T<br />
E<br />
2<br />
B<br />
⎝<br />
⎟ ⎟<br />
⎠
• From BCS theory :<br />
⎡C sup . − Cnorm<br />
. ⎤<br />
⎢ ⎥<br />
⎣ ⎦ C T<br />
norm<br />
Cnorm<br />
N(E ( F) F) ~ 12 states /eV-C 60<br />
= 1 1.<br />
43<br />
3 fold smaller than that from Pauli<br />
susceptibility measurement<br />
Many-body effect and spin fluctuation<br />
enhancement in χ pauli measurement<br />
(ΙΙ) Specific Heat<br />
E : Einstein term inter(intra)-molecule<br />
vibration modes with θ E~ 34K<br />
D : DDebye b tterm gives i θ θD ~70K 70K<br />
TL : electronic term
(ΙΙΙ) Thermopower<br />
• Current density<br />
r r t<br />
J e = σ E + α(<br />
−∇T<br />
) , α : peltier cond. tensor,<br />
turn on − ∇ T,<br />
and in steady state J = J = 0 , − ∇ T = 0,<br />
assume<br />
⇒<br />
e<br />
N<br />
Thermopower<br />
= ρα<br />
x<br />
isotropic ,<br />
xy<br />
− ρSσ<br />
xy<br />
,<br />
S<br />
• Sign convention of S<br />
Ex<br />
α xx ≡ = ,<br />
∇ T σσ<br />
x<br />
Nernst<br />
signal<br />
Positive for hole like carrier<br />
Negative for electron-like<br />
e<br />
N<br />
x<br />
≡<br />
E<br />
y<br />
y<br />
− ∇<br />
x<br />
z<br />
T<br />
x<br />
,<br />
y<br />
y<br />
−∇<br />
H<br />
x<br />
T<br />
θ<br />
α( ∇T<br />
)<br />
t<br />
α(<br />
−∇T<br />
)<br />
r<br />
E r t<br />
σ<br />
θα
(ΙΙΙ) Thermopower<br />
• Semiclassical theory<br />
S<br />
d<br />
2<br />
π<br />
= −<br />
2<br />
k<br />
e<br />
B<br />
⎛ k ⎞ BT<br />
⎜<br />
⎟<br />
⎟,<br />
linear in T<br />
⎝ ε F ⎠<br />
• Determination of E F from the slope dS/dT<br />
K 3C 60 : E F ~ 0.32 eV 5 states/eV-C 60-spin<br />
Rb 3C 60 : E F ~ 0.2 eV 9 states/eV-C60-spin<br />
•In<br />
K3C<br />
60,<br />
S<br />
p<br />
S = S + S<br />
: phonon drag effect<br />
d<br />
p<br />
contribution
• Fullerene structure : C 20+h*2<br />
• An example of strongly correlated electronic system<br />
Insulator – undoped C60 Metallic – Alkali-doped C60 SSuperconductivity d ti it – A A3CC 60 (A (A=K, K Rb,CsK,RbCs)<br />
Rb C K RbC )<br />
•T c increase linearly with lattice constant : BCS theory prediction<br />
• Reduced Hall coefficient and sign change at 200K : both electron and<br />
hole pocket<br />
• Weak localization effect associated with Merohedral disorder and<br />
missing alkali ions.<br />
• D.O.S. at Fermi Level in K K3C 3C60: 60:<br />
Pauli susceptibility : 28 states/eV-C60(spin fluctuation enhancement<br />
Thermopower S : 11 states/eV-C60<br />
states/eV C60<br />
Specific heat : 12 states/eV-C60