Grafeno, una hamaca de cristal M. Ruiz-Garc´ıa1 1
´ Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics Gregorio Millan Universidad Carlos III de Madrid
´ Barbany, 3 de febrero de 2016 Seminario Junior del Instituto Gregorio Millan
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
1
Outline
1
From the beginning Carbon Carbon materials
2
Graphene General properties Mechanical properties
3
Model 1D Model 2D Model
4
Conclusions
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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From the beginning
Carbon
The Carbon atom, 1s2 2s2 2p2
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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From the beginning
Carbon
Carbon atoms in a material
sp2 hybridized orbitals
sp3 hybridized orbitals
One hybridized orbital is perpendicular to the other three
Four hybridized orbitals forming a tetrahedron
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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From the beginning
Carbon materials
Old friends Diamond
Graphite
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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From the beginning
Carbon materials
New friends
Fullerenes
Carbon nanotube
It includes hexagons and pentagons Discovered in 1985
Discovered in 1991
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
General properties
Last but not least... Graphene
Geim and Novoselov, 2004
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
General properties
Why is it so popular?
Strongest material ’ever measured’ Stiffest known material Most stretchable and pliable crystal Record thermal conductivity (suspended graphene)
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
General properties
Graphene dispersion relation
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
Mechanical properties
Ripples
A flat (2D) crystal is not thermodynamically stable (Landau-Peierls- Mermin-Wagner). But, is it really flat?
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
Mechanical properties
Ripples Nanometric ripples
Meyer J C, et al. 2007 Nature 446 60 Atomic-length ripples
A flat (2D) crystal is not thermodynamically stable (Landau-Peierls- Mermin-Wagner). But, is it really flat?
Mao Y, et al. 2011 ACS Nano. 5 1395 M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
Mechanical properties
Buckling Buckling of suspended graphene induced by STM Reversible deformation
Xu P, et al. 2012 Phys. Rev. B 85, 121406(R) (a) Copper grid substrate (b) Constant-current data (c) Sketch of the deformation
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Graphene
Mechanical properties
Buckling Buckling of suspended graphene induced by STM Irreversible deformation
Reversible deformation
Xu P, et al. 2012 Phys. Rev. B 85, 121406(R) (a) Copper grid substrate (b) Constant-current data (c) Sketch of the deformation
Schoelz J K, et al. 2015 Phys. Rev. B 91, 045413 At Vc = 2.5V a sudden height increase is found (buckling). Once the layer is buckled, this state is permanent and independent of V.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
Goals
Features of our model Clamped at the boundaries. Temperature dependent.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
Goals
Features of our model Clamped at the boundaries. Temperature dependent.
Expected results To exhibit ripples of I I
nanoscale size atomic size
To buckle
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
8
Spin-String Model
Dx
σj = ±1 (Ising Spins). uj height of each atom. N 2 X pj k H= + (uj+1 − uj )2 − fuj σj 2m 2 j=0
pj linear momentum. −fuj σj Spin-String coupling term. uj = 0 for j = 0, N + 1, BC.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Critical temperature
H=
N 2 X pj k + (uj+1 − uj )2 − fuj σj 2m 2 j=0
In equilibrium, the probability of one specific configuration is h i exp −H k T B
Z and the partition function is Z=
Z Y N j=1
M. Ruiz-Garcia (UC3M)
dpj duj
X σ1 =±1,σ2 =±1,...
Introduction to Graphene
−H exp kB T
14
Model
1D Model
Summing for all different spin configurations: N P − fu σ j j X −fu1 σ1 X −fu2 σ2 X j=0 exp exp ... exp = kB T kB T kB T σ =±1 σ =±1 σ =±1,σ =±1,... 1
1
2
2
the partition function take the form Z=
Z Y N j=1
M. Ruiz-Garcia (UC3M)
+ V eff 2m kB T
fuj kB T
j=0 dpj duj exp −
where Veff =
N P
k (uj+1 − uj )2 − kB T ln cosh 2
Introduction to Graphene
pj2
+ kB T ln(2),
15
Model
1D Model
Minimizing the effective potential, k (uj+1 + uj−1 − 2uj ) + f tanh
fuj kB T
= 0,
and solving for small uj a critical temperature appears T0 =
f 2 KN2 , k
1.2
KN ∼
N , π
T
1.0
u
0.8 0.6 0.4 0.2
T>T0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x Details: Bonilla L L et al. 2012 Phys. Rev. E 85 031125 M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Free energy: Antiferromagnetic interaction
Including the antiferromagnetic interaction, the hamiltonian is H=
N 2 X pj k + (uj+1 − uj )2 − fuj σj + Jσj+1 σj 2m 2 j=0
8
when N 1
Dx
In ∆x we can suppose uj ∼ cte
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Free energy: Antiferromagnetic interaction In the continuum limit (N 1) and summing over the spin configurations: F , P[u] ∝ exp − θ " # 2 Z 1 u κ 1 ∂u F[u] = N dx − θ ln ζ , , 2π 2 ∂x θ θ 0 | {z }
Canonical distribution Free Energy
f (u, du ) dx
Where ζ is the partition function of Ising spins: u κ κ u κ ζ , = exp − cosh + exp θ θ θ θ θ with κ =
s
u 4κ 1 + exp − sinh2 . θ θ
T J , θ= . T0 T0
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Minimizing the free energy
1.2
T
1.0 0.8
T > T0 flat string T < T0 buckled
u
Case κ = 0
0.6 0.4 0.2
T>T0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Minimizing the free energy 1.2
T
1.0 0.8
T > T0 flat string
u
Case κ = 0
0.6 0.4
T < T0 buckled
0.2
T>T0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x 1.2
1.0
Case κ 6= 0 and T < T0
Antiferromagnetic Ferromagnetic
u
0.8
κ ∼ 0.15
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Minimizing the free energy 1.2
T
1.0 0.8
T > T0 flat string
u
Case κ = 0
0.6 0.4
T < T0 buckled
0.2
T>T0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x 1.0
0.8
Case κ 6= 0 and T < T0 −
κ → 0.3
u
0.6
0.4
Antiferromagnetic Ferromagnetic
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Minimizing the free energy 1.2
T
1.0 0.8
T > T0 flat string
u
Case κ = 0
0.6 0.4
T < T0 buckled
0.2
T>T0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x 1.0
0.8
Case κ 6= 0 and T < T0 0.6
u
κ > 0.3−
0.4
Antiferromagnetic
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Keeping the spins
H=
N 2 X pj k + (uj+1 − uj )2 − fuj σj + Jσj+1 σj 2m 2 j=0
to understand stability of the previous solutions we can integrate Z=
X σ1 =±1,σ2 =±1,...
Z Y N j=1
dpj duj exp
−H kB T
=
X σ1 =±1,σ2 =±1,...
exp
−A kB T
and restricting our study to configurations that are antiferromagnetic at the boundaries and ferromagnetic in the centre h i 2 A(σna ) = − (na − 1) −3N(na + 1) + 4na2 + na + 24KN2 κ + 7 , 3 where na is the number of antiferromagnetic spins at each side.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
1D Model
Increasing the antiferromagnetic interaction κ: 350 300 250 200 150 100 50 0
800
1.2
1.0
0.8
400 200 0 1
N8
N4
N2
u
A
A
600
0.4
0.2
1
N8
N4
N2
0.0 0.0
0.2
0.4
Σna
Σna
0.6
0.8
1.0
0.6
0.8
1.0
0.6
0.8
1.0
x 1.0
0 100
0.8
-50 0.6
50
u
A
A
0.6
-100
0
0.4
0.2
-150
-50 1
N8
N4
N2
1
N8
Σna
N4
N2
-100
-400 N4
Σna M. Ruiz-Garcia (UC3M)
0.4
x 0.8
0.6
N2
u
A
A
-300 N8
0.2
1.0
0 -100 -200 -300 -400 -500 -600
1
0.0
Σna
0
-200
0.0
0.4
0.2
1
N8
N4
N2
Σna Introduction to Graphene
0.0 0.0
0.2
0.4
x 21
Model
2D Model
2D Hamiltonian æ
æ æ
æ
=
+
+
æ æ
æ
æ
• nearest-neighbors • next-nearest-neighbors #
"
H
æ æ
pij2 − fuij σij + J 0 σij (σi−1,j−1 + σi,j−2 + σi+1,j−1 ) 2m ij ( i X k h (uij − ui+1,j )2 + (uij − ui,j−1 )2 + (uij − ui,j+1 )2 2 |i−j|=even )
X
Jσij (σi+1,j + σi,j−1 + σi,j+1 ) ,
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Dynamics ¨ij − KN2 (ui+1,j + ui,j−1 + ui,j+1 − 3uij ) = σij , u | {z }
Equation of motion
discrete laplacian
δ ωij (σ|u) = (1 − γij σij ), 2
Transition rate for spins
γij depends on the nearest-neighbors (κ) on the next-nearest-neighbors (λ) and on uij . In the continuum limit, a2 2 ui+1,j + ui,j−1 + ui,j+1 − 3uij → (∂x u + ∂y2 u), 4 √ 2 −1 3n − 2 KN = a = √ ∝ n, π 6π n number of rows.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Phase diagram Parameters used to create the phase diagram:
Parameters DL =
1 3N
X
[3 + σij (σi+1,j + σi,j−1 + σi,j+1 )],
Domain-Wall Length
|i−j|=even
X 1 M = σij , N ij p F = h(∆∗ e)2 i,
Magnetization ∆∗ e = (e − hei).
Fluctuations
where e is the energy per atom.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Phase diagram Parameters used to create the phase diagram:
Parameters DL =
1 3N
X
[3 + σij (σi+1,j + σi,j−1 + σi,j+1 )],
Domain-Wall Length
|i−j|=even
X 1 M = σij , N ij p F = h(∆∗ e)2 i,
Magnetization ∆∗ e = (e − hei).
Fluctuations
where e is the energy per atom.
Simulations The membrane is flat and at rest at t = 0. Random initial values for the spins. Temperature is fixed to θ = 0.01.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Phase diagram
Domain-Wall Length: DL
Magnetization: M
Fluctuations: F M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
A
Phase diagram
B
C
Pointing: • up • down M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
A Phase diagram
B
C
B → C: 1st order phase transition Rippling → buckling transition ? Pointing: • up • down M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
F
Phase diagram
I
J
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Outlook
To look for a transition from a rippled state to a buckled one. In experiments, the STM heat up the graphene layer causing the transition. Phys. Rev. B 91 (2015), 045413
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Outlook
To look for a transition from a rippled state to a buckled one. In experiments, the STM heat up the graphene layer causing the transition. Phys. Rev. B 91 (2015), 045413 C
M. Ruiz-Garcia (UC3M)
B
Introduction to Graphene
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Model
2D Model
Outlook To look for a transition from a rippled state to a buckled one. In experiments, the STM heat up the graphene layer causing the transition. Phys. Rev. B 91 (2015), 045413 We perform new simulations where: The simulation starts with the system at a rippled state. There is a source of heat in the center of the surface. The boundary is at room temperature. DL Parameter
Magnetization
Central Atom Height 1.2
0.7 0.6
1.0 0.6
0.4
0.8
0.5
0.6
0.4
0.4
0.2 0.2
0.3 100
200
300
400
500
time
100
200
300
400
500
time
100
200
300
400
500
time
- 0.2
Temperature starts rising at t = 100. M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Model
2D Model
Increasing temperature we get a transition from a rippled state to a buckled one, through a nucleation process:
Pictures for t = 100, 360, 362, 370.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Conclusions
Model features 1.2
1.0
0.8
u
Analytical solutions for the 1D problem.
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Conclusions
Model features 1.2
1.0
0.8
u
Analytical solutions for the 1D problem.
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x
2D model simulations presenting ripples of different scales.
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Conclusions
Model features 1.2
1.0
0.8
u
Analytical solutions for the 1D problem.
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x
2D model simulations presenting ripples of different scales. Rippled to buckled transition increasing the temperature. DL Parameter
Magnetization
Central Atom Height 1.2
0.7 0.6
1.0 0.6
0.4
0.8
0.5
0.6
0.4
0.4
0.2 0.2
0.3 100
200
300
400
500
time
100
200
300
400
500
time
100
200
300
400
500
time
- 0.2
For further reading: Ruiz-Garcia M, Bonilla L L and Prados A, 2015, J. Stat. Mech. P05015 (arXiv:1503.03112)
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Conclusions
Minimizing free energy For low temperature, θ 1, we get the approximate equation for u: 1 d 2u = −sgn(u) Θ(|u| − u0 ) π 2 dx 2 κ u0 = 2κ + θ log[1 + exp(−2 )] θ For a specific value of u0 the following solutions appear 3 1.0
0.23 < κ < 0.3. The three solutions exist. 1 is the stable one. κ > 0.3. 1 is the only solution.
0.8
u
κ < 0.23. The three solutions exist. 1 and 2 are metastable solutions whereas 3 is stable.
2
0.6
u0 0.4
0.2
1
0.0 0.0
0.2
0.4
0.6
0.8
1.0
x M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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Conclusions
H
Phase diagram
E
G
M. Ruiz-Garcia (UC3M)
Introduction to Graphene
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