Temperature dependence of atomic vibrations in monolayer graphene. Supplementary Information.
Christopher S. Allen*1, Emanuela Liberti1, Judy S. Kim1, Qiang Xu2, Ye Fan1, Kuang He1, Alex W.
Robertson1, Henny Zandbergen3, Jamie H. Warner1, Angus I. Kirkland1.
1. University of Oxford, Department of Materials, Parks Road, Oxford, U.K. OX1 3PH
2. DENs Solutions, Delftechpark 26, 2628 XH Delft, The Netherlands
3. Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands.
Electron diffraction pattern simulations
To verify that scattering in mono-layer graphene can be regarded as kinematic, we have compared
electron diffraction patterns calculated under dynamic and kinematic scattering assumptions.
Calculations were performed using a multi-slice C++ program, which implements the Lobato
parameterization of the electron scattering factor [32]. To include dynamical scattering, the specimen
potential was divided in 0.1 Å thick slices, perpendicular to the electron beam. The scattered electron
wave was then transmitted and propagated between the slices using a generalized transmission function
and Fresnel propagator. In the kinematic case, the specimen was assumed to be a pure phase object. In
this case, only one slice is required for the calculation and propagation was ignored. After scattering,
the transmitted electron wave accumulates a phase change, proportional to the projected potential of
the specimen. In all calculations, microscope effects were included using partially coherent transfer
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functions. Thermal vibrations were also included, by implementing the frozen phonon approach, which
uses the Einstein model with 500 phonon configurations. The Debye-Waller factor was taken to be the
experimental value determined from the analysis of the diffraction patterns. A numerical real space grid
of 4096 x 4096 pixels was used for a super cell size of 60 x 60 Å (14 x 8 units in sample box),
corresponding to a sampling resolution of 0.015 Å/pixel.
a.
b.
Supplementary Figure 1. Total number of counts as a function of scattering angle for peaks
extracted from diffraction patterns simulated for (a) kinematic and (b) dynamical scattering.
The red lines show fits to (1).
Supplementary figure 1 shows fits of (1) to the kinematic and dynamic diffraction simulations using
the approach described in the main text. At 313 K we calculate identical values for the in-plane
displacements for dynamical and kinematic scattering. At T = 1273 K, corresponding to the highest
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experimental temperature studied, we have found that the in-plane displacements calculated using the
phase object approximation and using a full multi-slice calculation differ by only 0.2 pm, below the
error in our experimentally measured values. We therefore conclude that electron scattering from
mono-layer graphene can be regarded as kinematic across the temperature range pertinent to the
experiments described.
Supplementary Table 1: Temperature dependence of 𝑢𝑝2
Temperature (±30 K)
𝑢𝑝2 (pm2)
103
15 ± 3
293
13 ± 4
313
15 ± 1
373
16 ± 1
473
19 ± 2
573
24 ± 2
673
24 ± 2
773
30 ± 2
873
30 ± 2
973
32 ± 2
3
1073
39 ± 2
1173
40 ± 2
1273
42 ± 2
Supplementary Figure 2. Temperature dependence
of the planar (𝑢𝑝2 ) mean square displacements for
four different mono-layer graphene samples (all
prepared using the procedure outlined in [17]).
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Supplementary Figure 3. High resolution TEM
image of pristine mono-layer graphene recorded at
∼1070 K. At temperatures in excess of ≳ 800 K
the graphene surface is largely free of
contamination.
Derivation of 𝒖𝟐𝒛
The mean square atomic displacement can be written [33]
𝑢𝑧2 =
1
∑⟨(𝒖𝑠 (𝒌))2 ⟩
2𝑁
𝒌,𝑠
with
ℏ
𝒖𝑠 (𝒌) = 𝜺𝑠 (𝒌)√
(𝑐 (𝒌) + 𝑐𝑠† (−𝒌))
2𝑀𝜔𝑠 (𝒌) 𝑠
giving
𝑢𝑧2 =
ℏ 1
𝜀𝑠 (𝑘)2
1
∑
[𝑛(𝜔(𝑘)) + ]
2𝑀 𝑁
𝜔(𝑘)
2
𝑘,𝑠
As there is only one phonon branch for flexural phonons in mono-layer graphene:
∑ 𝜀𝑠 (𝑘) = 1
𝑠
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Writing the sum over all 𝒌 states as an integral over spatial (𝑝) and momentum (𝑞) co-ordinates (two
dimensional Bohr-Somerfield):
𝜕 2𝑝 𝜕 2𝑞
∑=∬
ℎ2
𝑘
gives
𝑢𝑧2 =
ℏ 1
𝜕 2𝑝 𝜕 2𝑞 1
1
1
∬
[
+ ]
2
2𝑀 𝑁
ℎ
𝜔(𝑘) exp(𝛽ℏ𝜔(𝑘)) − 1 2
Further, assuming a quadratic dispersion for flexural phonons in mono-layer graphene:
𝜔(𝑘) = 𝛼𝑘 2
with 𝛼 = 6.2 × 10−7 m2/s.
Noting that ∫ 𝜕 2 𝑝 is simply an area, 𝐴 (in 𝑘 space) and converting to polar co-ordinates, (i.e.∫ 𝜕 2 𝑞 =
ℏ𝑘𝑑𝑘𝑑𝜃 ) gives
𝑢𝑧2 = 𝜎 −1
ℏ
1 1 𝑑𝜃𝑑𝑘
1
1
∫ 2 [
+ ]
2
2
2𝑀 (2𝜋) 𝛼
𝑘
exp(𝛽ℏ𝛼𝑘 ) − 1 2
with 𝜎 the density of states 𝑁/𝐴 and 𝛽 = 1/𝑘𝐵 𝑇.
2𝜋
Evaluating the polar integral ∫0 𝑑𝜃 = 2𝜋 and writing 𝑥 = 𝛽ℏ𝛼𝑘 2 (giving 𝑑𝑥 = 2𝛽ℏ𝛼𝑘 𝑑𝑘) gives
𝑢𝑧2 = 𝜎 −1
ℏ
𝑑𝑥
1
1
∫ [
+ ]
8𝑀𝜋𝛼 𝑥 exp(𝑥) − 1 2
For phonons in two dimensions
𝑘𝐷2
𝜎=
4𝜋
where 𝑘𝐷 is the Debye wave-vector.
As in the case for planar phonons we introduce a smallest phonon wave-vector 𝑘𝑠 in order to avoid the
divergence of the function at any finite temperature.
This analysis finally yields (7)
𝑥 1
ℏ
1
1
𝑢𝑧2 = 2𝑀𝑘 2 𝛼 ∫𝑥 𝐷 𝑥 [exp(𝑥)−1 + 2] 𝑑𝑥
𝐷
𝑠
for which there is no analytical solution to the integral which must therefore be solved numerically.
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