T - Academia Sinica

-TIGP course Dec. 2007
Wei-Li Lee, IoP, Academia Sinica

-TIGP TIGP course on carbon nanostructure
Dec. 15 th - Lecture Ι : 0D system, carbon-based buckyballs ( fullerene)
Dec. 22 st - Lecture ΙΙ : 2D system, emerging material : graphene
D 29th Dec. 29 L t ΙΙΙ 1D t b t b
th - Lecture ΙΙΙ : 1D system, carbon nanotube
Jan. 5 th at 3 pm : Study group oral presentation
Jan. 12 th - Final exam

4 topics :
Lecture - Monday 2pm - 5pm
-TIGP course on carbon nanostructure
20 minutes break from 3:20pm to 3:40 pm
Study group – 2-3 2 3 people a group
each person, 15 minutes oral presentation, 5 for questions
0D system : Practical application of fullerene
1D system : Recent progress of carbon nanotube composites as a space elevator
2 D system : Graphene electronics : advantage and disadvantage
3D system : Quantum information using diamond nanocrystals
Reference Books:
Introduction to Nanotechnology by Poole and Owens
Science of Fullerenes and carbon nanotubes by Dresselhaus and Eklund
Solid state Physics by Ashcroft/ Mermin

WMAP image of CMB (3 Kelvin) Angular spectrum
WMAP
Others
Carbon
Oxygen
Baryonic ayo cmatter attein Universe U vese
4600
Helium
Hydrogen
0 200000 400000 600000 800000
parts per million by mass

Dave Wilkinson
James Peebles

Others
Nitrogen
Hydrogen
Carbon
Human Body
OOxygen
0 20 40 60 80
mass percentage

• Carbon : atomic number 6, 1s 2 [2s 2 2px 1 2py 1 ]
Orbitals used
for bond
Molecular orbital :
ψ
∑
= i
C
X
j
ij i
i
Xi : atomic orbitals
Shape &
bond angle g
Examples:
Sp s, px Digonal 180 o C 2H 2 Acetylene
S 2 Sp Ti l G hit C H
2 s, px, py Trigonal
120o Graphite, C2H4 Ethylene
sp 3 s, px, py, pz Tetrahedra Diamond, CH 4
109 o 28’ Methane

• Many stable and known forms at RR.T. T
Examples: diamond, graphite, amorphous carbon, fullerene, carbon
nanotube and nanobud…etc
• TTwo main i structures t t : one with ith sp 3 hhybrid b id bbonds d (di (diamond) d) and d th the
other with sp2 hybrid bonds (graphite, fullerene, nanotube and
nanobud)
• Dramatic different properties between diamond and graphite
Diamond Graphite
Electric Insulator Conductive
Hardness 10 (Mohs scale) 11-2 2
Appearance Transparent Opaque (black)
Value Expensive cheap

• European Football like molecule containing 60 carbons
- 12 pentagonal and 20 hexagonal faces
- Double bond length 1.4 Å and single bond 1.46 Å
• NNamed d after ft architect hit t R. R Buckminster B k i t Fuller F ll (1895 (1895-1983) 1983)
Geodesic dome
• Geometrically-allowed y fullerene C20+h*2 20+h 2
12 pentagonal faces + arbitrary number
of hexagonal faces (h)
• Smallest Fullerene C 20
• Smallest isolated (stable) Fullerene C 60

• “An An idea from outer space” space : 5.6 5 6 eV optical extinction
• Robert Kurl, Harold Kroto and Richard Smalley
-- Nobel Prize laureates in Chemistry 1996

• Carbon ball with a metal core
• Mass production of Fullerene by astrophysicists D.
R. Huffmann and W. Krätschmer
• CCarbon b nanotube t b – special i l electric l t i and d
mechanical properties
•New superconducting p g crystals y M3C 3 60

• quasi-0D system : d < ℓ mfp
• Di Discrete t energy llevel l resembles bl th the atomic t i llevel l of f a ffree atom t
- ex : 3-D infinite rectangular square well
2 2
ππ h 2 2 2
2
En = ( )( nx
+ ny
+ nz
) = E0n
2ma
6 degeneracy (including
spin) at ground state
QQuantum t number b = 0,1,2… 012 level Eo

LUMO
• Shell model in a free fullerene : symmetry-based model
• 60 πelectrons filling the molecular level
Gap ~ 1.92 eV
HOMO
• Free C60 molecule should be a good insulator

• Molecular crystal : BCC stacking structure
• Grown by slow evaporation from benzene solution
filled with C60 molecules
• Band calculation :
LDA + Gaussian orbital basis set
• Useful information :
Insulator with direct band gap ~ 1.5 eV band width ~ 0.4 eV
charge density map suggest weak coupling
b/w fullerene molecule

• Two standard orientations of fullerene molecule
TTwo fold f ld sym. axis i
Two standard orientations
• Merohedral disorder : random choice b/w one of the two possible standard
orientations ( lack of four fold symmetry point)
relative orientation b/w adjacent C 60 can affect its physical properties.

• Transport charges from doping alkali metal element M
• Two competing process due to doping effect:
- decrease of C 60 wave func. overlap
- increase of the D.O.S
• Best conductivity occurs at x ~ 3 (half filled band)
-available available sites (octahedral and tetrahedral) all filled
• undoped fullerene ρ 300K~ 10 8 Ωcm
• Single crystal K K3CC 60 ρ ρ300K ~5mΩcm ~ 5 mΩcm
•Strongly correlated electronics system
-k Fℓ

• Discovered in K 3C 60 by Hebard (Bell lab, 1991) T c ~ 19.8 K
• Highest Tc ~ 33K in Cs2RbC 2 60 60
• The larger the radius of the dopant alkali atom the higher the Tc

• In weak-coupling limit ( ג

• Hall coefficient : R H = 1/ne
Sign Sign change at 200K
Both electron and hole like pockets
•Transverse magnetoresistance :
ρ ρ ρ C L,
I
ρ ρ ρ
Δ
Δ Δ
= +
0
Classical orbital contribution :
positive and quadratic in H
0
0
Weak localization and ee-e e interaction:
Negative, H 2 at low H and H 1/2 in high H

• Appeared In disordered and time reversal symmetric system
• Negative MR : Strongly suppressed by applying magnetic field
• Merohedral disorder and also missing alkali ion at the tetrahedral and
octahedral sites
origin
Beff ⊙
impurity
∫ ∫ ⋅
Constructive |2A| 2 localization
B⋅
dA = A dl
: additional phase change

(Ι) Pauli susceptibility
B
• DOS D.O.S. :
Spin down
E
K 3C 60 : 28 states/eV-C60
Rb Rb3CC 60 : 38 states/eV states/eV-C60
C60
E F
D.O.S.
Spin up
1
n = ∫ d dε
ε N ( εε
+ μμ
B H ) f ( εε
)
↑ ∫ 2
M = μ ( n − n )
P
B
2
B
↑
χ = μ N(
E
F
↓
)

(ΙΙ) Specific Heat
• In crystals with free electron gas model at low T
C = Ce
+ C p
C
e
⎡ ⎡π
= ⎢
⎣ 3
2
k
2
B
⎤
N(
ε F ) ⎥T
,
⎦
C
p
=
3
4
12 12π
⎛ T ⎞
n
5 ⎜ ik
B
θ D
• In amorphous system and also fullerene solids at low T
Einstein mode (1907) ( )p plays y an important p role
C = C
e
+ C
p
+ C
2
C = pnik
B
E B
ω / k T
E
( hω
/ k
h E ( e
E B
T ) e
−1)
h hωω
/ k T
E
2
B
⎝
⎟ ⎟
⎠

• From BCS theory :
⎡C sup . − Cnorm
. ⎤
⎢ ⎥
⎣ ⎦ C T
norm
Cnorm
N(E ( F) F) ~ 12 states /eV-C 60
= 1 1.
43
3 fold smaller than that from Pauli
susceptibility measurement
Many-body effect and spin fluctuation
enhancement in χ pauli measurement
(ΙΙ) Specific Heat
E : Einstein term inter(intra)-molecule
vibration modes with θ E~ 34K
D : DDebye b tterm gives i θ θD ~70K 70K
TL : electronic term

(ΙΙΙ) Thermopower
• Current density
r r t
J e = σ E + α(
−∇T
) , α : peltier cond. tensor,
turn on − ∇ T,
and in steady state J = J = 0 , − ∇ T = 0,
assume
⇒
e
N
Thermopower
= ρα
x
isotropic ,
xy
− ρSσ
xy
,
S
• Sign convention of S
Ex
α xx ≡ = ,
∇ T σσ
x
Nernst
signal
Positive for hole like carrier
Negative for electron-like
e
N
x
≡
E
y
y
− ∇
x
z
T
x
,
y
y
−∇
H
x
T
θ
α( ∇T
)
t
α(
−∇T
)
r
E r t
σ
θα

(ΙΙΙ) Thermopower
• Semiclassical theory
S
d
2
π
= −
2
k
e
B
⎛ k ⎞ BT
⎜
⎟
⎟,
linear in T
⎝ ε F ⎠
• Determination of E F from the slope dS/dT
K 3C 60 : E F ~ 0.32 eV 5 states/eV-C 60-spin
Rb 3C 60 : E F ~ 0.2 eV 9 states/eV-C60-spin
•In
K3C
60,
S
p
S = S + S
: phonon drag effect
d
p
contribution

• Fullerene structure : C 20+h*2
• An example of strongly correlated electronic system
Insulator – undoped C60 Metallic – Alkali-doped C60 SSuperconductivity d ti it – A A3CC 60 (A (A=K, K Rb,CsK,RbCs)
Rb C K RbC )
•T c increase linearly with lattice constant : BCS theory prediction
• Reduced Hall coefficient and sign change at 200K : both electron and
hole pocket
• Weak localization effect associated with Merohedral disorder and
missing alkali ions.
• D.O.S. at Fermi Level in K K3C 3C60: 60:
Pauli susceptibility : 28 states/eV-C60(spin fluctuation enhancement
Thermopower S : 11 states/eV-C60
states/eV C60
Specific heat : 12 states/eV-C60