Carbon geometries as optimal configurations
Ulisse Stefanelli
In collaboration with
Edoardo Mainini, Hideki Murakawa, Paolo Piovano
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
1 / 18
Carbon structures
Graphene
Nanotubes
Fullerenes
local minimizers
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
2 / 18
Quantum ?
Schr¨
odinger
60 carbon atoms ⇒ solve Schr¨
odinger in R1080
(R ∼ {x1 , . . . , x10 }
Ulisse Stefanelli (Vienna)
101080 grid points)
ESI Vienna, 17.10.2014
3 / 18
Configurational energy
Nuclei: {x1 , . . . , xn } with charges {z1 , . . . , zn }
Electrons: {y1 , . . . , ym }
Time-independent, Born-Oppenheimer
temperature → 0
n
o
H1
Admissible waves: A = ψ : (R3 ×Z2 )m −→ C, kψkL2 = 1, antisymm.
Electronic hamiltonian
X
X zi zj
X zi
1
1X 2
H=
+
−
∂yα +
2 α
|yα −yβ |
|xi −xj |
|yα −xi |
α6=β
i6=j
i,α
Energy
Z
E (x1 , . . . , xn ) = min
ψ∈A
Ulisse Stefanelli (Vienna)
ψ ∗ (y , s)H(x, y )ψ(y , s) dy ds
(R3 ×Z2 )m
ESI Vienna, 17.10.2014
4 / 18
Carbon nanostructures
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
5 / 18
Carbon nanostructures
[Tersoff 89]
E=
1X
1 X
v2 (|xi −xj |)+
v3 (θijk )
2
2
i6=j
ijk∈NN
xi
θijk
xk
xj
Two-body interactions
v2
Three-body interactions
v3
1
[E & Li 09]
2π/3
−1
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
4π/3
6 / 18
2D Crystallization
Three-body-interactions
I
I
I
[E & Li 09] thermodynamic limit
[Mainini & S. 14] finite crystallization
[Davoli, Piovano, & S. 1?] Wulff shape, isoperimetric
Two-body-interactions
I
I
I
I
triangular lattice
[Heitman & Radin 80] sticky potentials
[Radin 81] [Wagner 83] soft potentials
[Theil 06] long-range interactions
[Au Yeung, Friesecke, & Schmidt 12], [Schmidt 13] Wulff shape
Three-body-interactions
I
hexagonal lattice
square lattice
[Mainini, Piovano, & S. 14] finite crystallization, Wulff shape
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
7 / 18
Graphene
E=
1X
1 X
v2 (|xi −xj |) +
v3 (θijk )
2
2
i6=j
ijk∈NN
Ground states are honeycomb in 2D
p
En = −b3n/2 − 3n/2c
(exact surface energy
geometry)
Grounds states are perimeter-minimizers
Differ from Wulff by O(n3/4 ) atoms
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
8 / 18
Rolling up
newly activated bonds = λ
θnanotube = 2π/3 + cw −1
v3 (θnanotube ) = cw −2
number of angles = cw λ
λ
w
Enanotube = Egraphene − #(new bonds) + #(angles)v3 (θnanotube )
1
= Egraphene − λ + cλw 2 < Egraphene
w
Aspect ratio: n = w λ, minimize −λ + c
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
λ2
⇒ λ ∼ n, w ∼ c
n
9 / 18
Rolling up
Rolling up a nanotube
sufficiently large diameter needed
armchair
b
pa+qb
zigzag
a
Growing a nanotube
Ulisse Stefanelli (Vienna)
constant diameter by adding atoms
ESI Vienna, 17.10.2014
10 / 18
Fullerenes
Local minimality
v3 strictly convex and decreasing around 3π/5
⇒ C20 and C60 strict local minimizers
[video]
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
11 / 18
Fullerenes
5
6
1 XX
1 XX
E˜ ≥ −#(bonds) +
v3 (πi ) +
v3 (hi )
2 pent
2
i=1
hex i=1
!
!
5
6
X
X
1
1X
1X
1
v3
πi +
v3
hi
≥ −#(bonds) +
2 pent
5
2
6
i=1
≥ −#(bonds) +
Ulisse Stefanelli (Vienna)
hex
i=1
1
1
· 12 · 5v3 (3π/5) + · 20 · 6v3 (2π/3) = E (C60 )
2
2
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Fullerenes
! Planarity of faces needed
corannulene
Ulisse Stefanelli (Vienna)
X Planarity ⇒ stability?
[Kamatgalimov et al. 10]
pentaindenocorannulene
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graphene
13 / 18
Nanotubes
α
β
α
Two competing periodic models:
Rolled-up
α = 2π/3 > β
[Dressehaus et al. 95]
Polyhedral
α = β < 2π/3
[Cox-Hill 07]
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
14 / 18
Nanotubes
Minimize α 7→ E3 (α) = 2v3 (α) + v3 (β(α, γ))
γ
α
β
0.1
2v3( )+v3( ( ))
Rolled-up
Cox-Hill
0.004725
β(α)
0.00472
0.05
0.004715
2.0795
αp
Ulisse Stefanelli (Vienna)
α∗
αru
0
2.06
ESI Vienna, 17.10.2014
2.08
2.0805
2.08
2.081
2.1
2.12
15 / 18
Nanotubes
C (α∗ ) is a strict local minimizer
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
16 / 18
Nanotubes
Local minimality
v3 strictly convex at 2π/3 ⇒ C (α∗ ) is a strict local minimizer
E˜3,density =
X
2v3 (αi ) + v3 (βi ) /#(atoms)
≥ 2v3 (αmean ) + v3 (βmean )
(convexity of v3 )
≥ 2v3 (αmean ) + v3 (β(αmean , γ∗ ))
(if γmean ≤ γ∗ )
≥ 2v3 (αmean ) + v3 (β(αmean , γmean ))
≥ 2v3 (b
α) + v3 (β(b
α, γ∗ ))
(numerically checked)
∗
≥ 2v3 (α∗ ) + v3 (β(α∗ , γ∗ )) = E3 (α )
Ulisse Stefanelli (Vienna)
(concavity of β)
ESI Vienna, 17.10.2014
(minimality)
17 / 18
Nanotubes
0.12
F*
Random
Modified
0.11
0.1
E3
0.09
0.08
0.07
0.06
0.05
1.4137
1.4137
1.41371
1.41371
1.41372
1.41372
1.41373
1.41373
1.41374
http://www.mat.univie.ac.at/∼stefanelli
Ulisse Stefanelli (Vienna)
ESI Vienna, 17.10.2014
18 / 18

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Metal-Carbon Nanotube Contacts - NCCAVS

Geometry I