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Supplementary material
Simultaneous measurement of intrinsic electrical and thermal
conductivities of suspended monolayer graphene
Haidong Wang1, Kosaku Kurata1, Takanobu Fukunaga1, Hiroki Ago2, Hiroshi Takamatsu*1,
Xing Zhang*3, Tatsuya Ikuta4, Koji Takahashi4, Takashi Nishiyama4, Yasuyuki Takata5
1Department
2Institute
for Materials Chemistry and Engineering, Kyushu University
3Department
4Department
5International
of Mechanical Engineering, Kyushu University
of Engineering Mechanics, Tsinghua University
of Aeronautics and Astronautics, Kyushu University
Institute for Carbon-Neutral Energy Research, Kyushu University
*Corresponding Author,
E-mail: [email protected], Fax: +81 92-802-3123
*Corresponding Author,
E-mail: [email protected], Fax: +86 10-62781610
1. Brief introduction to the sample fabrication process
In order to perform electrothermal measurement for graphene, a wet etching method
using buffered hydrofluoric acid is normally used for suspending graphene sample.
However, it is difficult to suspend the graphene membrane with a width of several
micrometers because the etched distance between the graphene and substrate is limited
to only hundreds of nanometers. In order to fabricate micrometer-sized suspended
graphene samples, we developed a new fabrication method as following.
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Fig. S1 Flow-chart of the fabrication process of graphene sample.
Figure S1 shows the flow-chart of the fabrication process: (1) the monolayer
graphene grown by chemical vapor deposition method was transferred onto a SiO2/Si
substrate; (2) 300 nm thick EB-resist was spin-coated on the surface of graphene and
patterned into micrometer wide ribbons; (3) the graphene was cut into ribbons by O2
plasma etching; (4) another EB-resist layer was patterned into a desired shape for
making metallic electrode pads and micro-beam sensor; (5) a metallic thin film of 10nm
Cr and 100nm Au was deposited by using a physical vapor deposition method. The
electrode pads and sensor were created after a lift-off process; (6) a protection layer for
graphene was made from the patterned EB-resist layer; (7) the SiO2 not covered by
2
EB-resist was removed by reactive ions etching; (8) the Si substrate exposed to XeF2
gas was etched isotropically in depth and the whole graphene device was suspended; (9)
the EB-resist and SiO2 were removed separately. Then, the suspended graphene device
was dried using a super-critical point dryer.
2. Polymeric residues on the electrode pads
(a)
(b)
Fig. S2 Comparison between the suspended graphene with and without polymeric
residues.
Figure S2 shows the SEM images of suspended graphene with and without polymeric
residue layer. We found that the proper temperature and immersion time are important
for removing the polymeric residues. If the temperature is too low or the immersion
time is too short, some polymeric residues may be left on the graphene after drying, as
shown in Fig. S2 (a). In contrast, Fig. S2 (b) shows a clean suspended graphene used for
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measurement.
Fig. S3 Random particles and polymeric residues on the electrode pad (the particles and
polymeric residues are marked by the red and yellow arrows, respectively).
Figure S3 shows the suspended graphene sample in the main text. Some impurity
particles and polymer residues are found on the electrode pad. The impurity particles
were randomly generated during the metal film deposition process. There are no such
particles found on the suspended graphene sample.
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Fig. S4 Zoom-in SEM image of the polymeric residues left on the electrode pad.
Figure S4 shows the polymeric residues left on the electrode pad. In the upper figure,
the yellow area was covered by EB-resist layer in order to protect the graphene during
the reactive ion etching process. The green area was not covered by any polymer layer.
In the figure below, the polymeric residue in line-shape was the edge of the protection
layer, which became insoluble after ions bombardment. Fortunately, the rest of the
polymer layer could be removed by dipping into warm ZDMAC solution. The insoluble
polymeric residues are far from the target graphene ribbon and cause no influences to
the sample quality.
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It should be noted that to our knowledge, there is still no method that can completely
remove all the polymeric residues on the graphene. Some nanoscale residues may be left
on the suspended graphene beyond the SEM imaging. This is one important factor that
suppresses the thermal conductivity of graphene. Comparing with the simple fabrication
process of the suspended graphene for Raman measurement, the sample preparation for
electrothermal measurement is much more complicated. Thus, there is a bigger chance
to leave more residues on the graphene sample. This explains why the thermal
conductivity of graphene measured by using micro-electronic device is usually smaller
than that measured by Raman method. On the other hand, the Raman spectrum of our
suspended graphene sample shows a good quality, same as the sample prepared for
Raman measurement. Thus, our results are closer to the experimental data measured by
the Raman method.
3. Lattice dynamics theory for predicting the thermal conductivity
Based on a lattice dynamics theory, the thermal conductivity of monolayer graphene
can be calculated using the next equation [1, 2]:
1
=
4 k BT 2
where kB,
 
s  TA,LA,ZA,
TO,LO,ZO
qmax
qmin

exp  s (q) / k BT 
2

 s (q)vs (q)   s (q)
2
 exp  s (q) / kBT   1




q dq ,


(1)
, ωs, τs, q and T are the Boltzmann constant, reduced Planck constant,
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phonon frequency, relaxation time, wave vector and temperature, respectively. δ = 0.35
nm is the interplanar spacing of graphite. vs = dωs /dq is the group velocity. The
subscript s stands for six different phonon polarization branches, including three
acoustic branches (TA, LA, ZA) and three optical branches (TO, LO, ZO). A full
phonon dispersion relation for all six phonon branches is used to calculate the thermal
conductivity of graphene.
The relaxation time τs is given as:
1
 1
1 
 s (q)  

 ,
 U , s (q)  B , s (q) 
(2)
where τU,s and τB,s are the relaxation times of phonon Umklapp scattering and boundary
scattering, given as:
 U , s (q) 
1 M vs2 s ,max
,
 s2 kBT s2 (q)
(3)
d 1 p
,
vs (q ) 1  p
(4)
 B ,s (q) 
where γs, vs and M are the Gruneisen parameter, average phonon velocity and mass of a
graphene unit cell, respectively. ωs,max = ωs (qmax) is the maximum cut-off frequency. d
is the width of SLG ribbon. p is a specularity parameter describing the roughness at the
edges. γLA and γTA are chosen to be 1.80 and 0.75, respectively, as recommended in the
literature [1].
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As discussed in the main text, the phonons of in-plane LA and TA modes are
important energy carriers in graphene. Several parameters, such as width, specularity
parameter and cut-off frequencies of phonons, play important roles in deciding the final
thermal conductivity of graphene. We found that the minimum cut-off frequency ωs,min
is particularly different for the monolayer graphene. If the graphene is supported by
substrate or has interactions with neighboring layers, more long wave-length phonons
will be restricted and ωs,min will be increased accordingly. In an ideal case without any
outside influences, ωs,min tends to zero. However, the defects and residues are
unavoidable for the prepared suspended monolayer graphene. Therefore, the minimum
cut-off frequency is larger than zero, but still much less than the bulk value. This is one
of the important reasons that the monolayer graphene has much higher thermal
conductivity than graphite and multilayer graphene.
References
[1] Nika, D. L. and Balandin, A. A. Two-dimensional phonon transport in graphene. J.
Phys.: Condens. Matt. 24, 233203 (2012).
[2] Nika, D. L., Pokatilov, E. P., Askerov, A. S. and Balandin, A. A. Phonon thermal
conduction in graphene: role of Umklapp and edge roughness scattering. Phys. Rev.
B 79, 155413 (2009).
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