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Hydrogenation induced deformation mode andthermal conductivity variations in graphene sheets

http://dx.doi.org/10.1016/j.carbon.2014.02.0010008-6223/� 2014 Elsevier Ltd. All rights reserved.

* Corresponding author.E-mail address: hzhao2@clemson.edu (H. Zhao).

Chengjian Li, Gang Li, Huijuan Zhao *

Mechanical Engineering, College of Engineering and Science, 100 Fluor Daniel EIB, Clemson University, Clemson, SC 29634-0921,

United States

A R T I C L E I N F O A B S T R A C T

Article history:

Received 11 September 2013

Accepted 1 February 2014

Available online 7 February 2014

In this study, we adopt molecular dynamics simulation and first principles theory calcula-

tion to investigate the deformation modes of graphene sheets induced by patterned hydro-

genation stripes, as well as thermal conductivity variation with respect to the doping

density and deformation modes. We demonstrate that the deformation modes can be con-

trolled by the hydrogenation patterning parameters. Both the doping density and morphol-

ogy contribute to the thermal conductivity variation of the graphene sheet. With the

control of hydrogenation patterning parameters, desired deformation modes and thermal

conductivity of graphene can be achieved.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

With its stable two dimensional honeycomb lattice structure

[1], graphene exhibits excellent mechanical [2,3], electronic

[4,5], thermal [6,7] and optical [8] properties. The structural

flexibility and superior material properties of graphene make

it a desirable material in the development of next generation

integrated circuits, electronics and ultra small sensors [9]. In

many graphene based applications, thermal transport prop-

erty of the material is of critical importance. For example, the

thermal conductivity directly determines the energy conver-

sion efficiency in graphene based thermoelectric materials

[10,11]; the high thermal conductivity of graphene enables a

faster heating rate and a more homogeneous temperature dis-

tribution in graphene-based transparent flexible heaters [12];

its low thermal boundary resistance with many materials can

help enhance the heat dissipation of high power density

electronic components [13]. Other applications where thermal

transport in graphene plays an important role include

graphene-based thermal interface materials [14], transparent

graphene electrodes in photovoltaic solar cells [15], and

graphene-based nanosensors [16], just to name a few. Recently,

chemical functionalization of graphene for controllable and

reversible tuning of its material properties has drawn tremen-

dous interests. Among various types of chemical functionali-

zation, hydrogenation of graphene has been investigated

extensively due to its relevance to hydrogen storage as well

as its scientific and practical importance for chemistry [17].

Theoretical and experimental studies have shown that elec-

tronic and magnetic properties of graphene can be modified

by using various hydrogenation approaches [18–22]. Since

then, significant progress has been made experimentally on

controlled hydrogenation of graphene, such as single-sided

hydrogenation of pristine graphene [23] and controlled hydro-

genation patterning of graphene [24]. The theoretical and

experimental progress on controlled hydrogenation of graph-

ene creates an opportunity for a more precise tuning of its

physical properties, including the thermal transport

properties.

In this letter, we demonstrate controllable thermal conduc-

tivity tuning through patterned hydrogenation of graphene.

We perform non-equilibrium molecular dynamics (MD)

186 C A R B O N 7 2 ( 2 0 1 4 ) 1 8 5 – 1 9 1

studies of hydrogenated graphene and show that single-sided

patterning of hydrogen on relaxed graphene produces global

deformation modes due to the local stress caused by sp3 C–C

bonding. Such deformation modes can be controlled by the

hydrogenation patterning parameters. Furthermore, first prin-

ciples theory calculations are performed to elucidate the rela-

tion between the deformation modes and local stress induced

by patterned hydrogenation. We show that both the doping

density and deformation mode contribute to the thermal con-

ductivity variation of the graphene sheet. Our work demon-

strates that, with the control of hydrogenation patterning

parameters, desired deformation mode and thermal conduc-

tivity of graphene can be achieved for applications such as

thermoelectric energy conversion or foldable electronics.

2. Method

From Fourier’s law, the thermal conductivity j is defined as

j ¼ �q=rT ð1Þ

where q is the heat flux and rT is the temperature gradient.

In carbon materials, such as graphene, heat conduction by

phonons is dominant due to the strong covalent sp2 bonding

between carbon atoms. In nanoscale, j is closely related to

size, boundary conditions and geometry of the atomistic

structures due to the ballistic transport of phonons and sig-

nificant or even dominant phonon-boundary scattering. Pre-

vious studies have shown that thermal conductivity of

graphene has significant size dependence and increases

quickly with the increasing length along the heat flux direc-

tion [25,26]. The thermal conductivity of graphene nanorib-

bon decreases remarkably with tensile strain [27,28] and

defects [29,30]. Effect of hydrogenation on the thermal con-

ductivity of graphene has also been investigated. It has been

shown that random hydrogenation or edge hydrogen passiv-

ation will bring a rapid drop of the graphene’s thermal con-

ductivity [31–33]. The thermal conductivity’s reduction path

is closely related with the direction of the patterned hydroge-

nation stripes, but not significantly related with the heat flux

direction [34]. Different from these studies, the single-sided

hydrogenation patterning considered in this work brings mor-

phology of graphene into the thermal transport tuning,

resulting in fundamentally different characteristics of the

thermal conductivity variation compared to those of random

or edge hydrogenation.

Non-equilibrium molecular dynamics simulations are per-

formed by using the molecular dynamics solver LAMMPS [35].

Although an optimized Brenner empirical potential has been

recently proposed for better approximation of thermal con-

ductivity of graphene [36], we adopt the adaptive intermolec-

ular reactive empirical bond order (AIREBO) potential with

REBO and Lenard Jones terms to describe the short range

and the long range of C–C, C–H interactions [37,27] for the fol-

lowing reasons: (1) the deformation modes predicted in this

study require validated approximation of graphene’s mechan-

ical properties, which have not been well tested with the opti-

mized Brenner potential; and (2) the torsional term of the

AIREBO potential is not included as it causes significant

underestimation of the thermal conductivity of graphene.

The graphene sheet is decomposed into cold, hot, and heat

flux regions along the zigzag direction as shown in Fig. 1.

The length of cold, hot and each of the heat flux regions are

5%, 5% and 45% of the graphene sheet length, respectively.

Periodic boundary condition (PBC) is applied to the simulation

box with a 20 nm thick vacuum space perpendicular to the

graphene sheet in order to prevent the non-bonding long

range interaction. With a time step of 0.5 fs, the graphene

sheet is first relaxed to the equilibrium state at 300 K for up

to 500 ps under the isothermal isobaric ensemble (NPT). Next,

with the microcanonical ensemble (NVE), a heat flux is im-

posed on the system by exchanging the velocity of the lowest

kinetic energy atom in the hot region with the velocity of the

highest kinetic energy atom in the cold region. After reaching

the non-equilibrium steady state, the exchanging process

continues for ts ¼ 100 ps in order to calculate the thermal con-

ductivity. Along the heat flux direction, we divide the graph-

ene sheet into bins with the width offfiffi3p

2 a, where a ¼ 1:42 A

is the lattice constant. The temperature profile can be calcu-

lated by averaging the kinetic energy of each bin as,

Ti ¼1

3NikB

XNi

j¼1

mv2j ; ð2Þ

where Ti is the temperature of the ith bin, Ni is the number of

carbon atoms in the ith bin, kB is Boltzmann’s constant, m is

the mass of carbon atom and vj is the velocity of atom j.

Meanwhile, the heat flux q is obtained as q ¼ DE=2Ats, where

A is the cross-sectional area of heat flux region with the thick-

ness of 3:4A; ts is the time period taken for statistical averag-

ing after the non-equilibrium steady state of the system is

reached, DE is the total swap energy during ts. Then, the ther-

mal conductivity can be calculated by using Eq. (1). To help

elucidate the effect of phonon spectrum on the thermal

transport, we calculate phonon density of states (PDOS)

through the Fourier transformation of the velocity auto-corre-

lation function as

DðxÞ ¼Z

e�ixtdthvðtÞ � vð0Þi; ð3Þ

where vðtÞ is the velocity at time t, and the angle brackets rep-

resent the velocity autocorrelation function. After reaching

the steady state, the total time period considered for PDOS

calculation is 10 ps.

3. Results and discussion

As a validation of the simulation method described above, we

first calculate the thermal conductivity of pristine graphene

as a function of size. It is found that thermal conductivity in-

creases significantly with the increasing length of the heat

flux region because of the maximum phonon wavelength

the system can carry is closely related to the heat flux length

and the periodic boundary condition that has been applied.

However, the size effect perpendicular to the heat flux direc-

tion is negligible. These simulation results are quite close to

those presented in Ref. [26,27]. For the sake of brevity, they

are not shown here. In the following simulations, the graph-

ene sheet size is defined as 5.10 nm �19:65 nm with the PBC

along the two in-plane directions.

Fig. 1 – Schematic diagram of a graphene sheet for the calculation of thermal conductivity. The graphene sheet has one cold

region (blue), two heat flux regions (green) and one hot region (red). (A color version of this figure can be viewed online.)

C A R B O N 7 2 ( 2 0 1 4 ) 1 8 5 – 1 9 1 187

3.1. Hydrogenation induced deformation modes

It has been found that thermal conductivity of graphene

greatly depends on the hydrogenation pattern and coverage

[31]. In this study, with the consideration of the flexibility of

graphene as a two dimensional material, we show that the

morphology of graphene can be manipulated by the hydroge-

nation pattern. We investigate two types of hydrogenation

pattern: double-side doping and single-side doping. Double-

side hydrogenation means that in identical hydrogenation-

graphene stripes, hydrogen atoms are bonded on both sides

of the graphene lattice in an alternative fashion, as shown

in Fig. 2(A). The doping density is defined as the ratio of

hydrogenation length W to stripe length H, i.e., W=H. We ap-

ply two hydrogenation stripes perpendicular (transverse

stripe) to and one hydrogenation stripe parallel (longitudinal

stripe) to the heat flux direction, respectively, and vary the

doping density. We observe that with the double-side hydro-

genation stripes, the relaxed graphene sheet remains planar

with a rippling structure.

Next we consider a single-side hydrogenation pattern with

identical hydrogenation-graphene stripes perpendicular to

the heat flux direction as shown in Fig. 2(B). The definition

of W;H and doping density are the same as in the double-side

hydrogenation case. In this study, we investigate four cases

with 4, 6, 8 and 10 transverse stripes along the zigzag direc-

Fig. 2 – (A) schematic diagram of patterned double-side hydroge

hydrogenation; (C–F) deformation modes induced by patterned

(E) aROL mode and (F) ROL mode. (A color version of this figure

tion of the graphene with H equal to 4.91 nm, 3.27 nm,

2.46 nm and 1.96 nm, respectively. We observe that the re-

laxed graphene structure no longer remains planar due to

the local stress concentration induced by the patterned

hydrogenation. Depending on H and W=H ratio, the graphene

sheet can be relaxed to the following deformation modes: tri-

angular mode (TRI), compact mode (COM), asymmetric rolling

mode (aROL) and symmetric rolling mode (ROL), as shown in

Fig. 2(C-F). The TRI mode includes symmetrically bent hydro-

genation regions and planar undoped graphene regions. As

depicted by the filled symbols in Fig. 3(A), the slope of the pla-

nar graphene regions, h, is linearly proportional to the hydro-

genation length W, and independent of the stripe length H.

The COM mode appears when the bending slope of the hydro-

genation region is over 90� or W > 0.65 nm, where the inter-

layer long range interaction between graphene segments

starts to play a role. If H is sufficiently large (>45 nm), further

increasing W would cause the graphene deform into an

unstable asymmetric mode (Fig. 2 (E)) with the ratio

0:7 < W=H < 0:8, and a stable symmetric mode (Fig. 2(F)) with

a higher ratio of W=H.

In order to fully understand the relation between the slope

h and the length W of the hydrogenation region in the TRI

mode, we perform first principles theory calculation with

the Vienna ab initio simulation package (VASP) [38,39]. The

electron–ionic core interaction is represented by the PAW

nation; (B) schematic diagram of patterned single-side

single-side hydrogenation: (C) TRI mode, (D) COM mode,

can be viewed online.)

Fig. 4 – (A) Thermal conductivity variation as a function of

doping coverage with longitudinal and transverse

hydrogenation patterns; (B) PDOS under various doping

coverage conditions with longitudinal and transverse

hydrogenation patterns; (C) acoustic phonon velocity

variation as a function of the transverse doping coverage. (A

color version of this figure can be viewed online.)

Fig. 3 – (A) slope h variation with hydrogenation length W;

(B–G) schematic diagrams of the beam bending model (E–G)

with the corresponding force and moment calculated from

the first principles theory (B–D). (A color version of this

figure can be viewed online.)

188 C A R B O N 7 2 ( 2 0 1 4 ) 1 8 5 – 1 9 1

pseudopotential and the electron exchange and correction is

treated by Perdew Burke Ernzerhof (PBE) [40] formulation.

The cutoff energy for the plane–wave is 500 eV. Along the out

of plane direction, a vacuum region of 5 nm is applied on each

side of the sheet in order to eliminate the interaction between

the graphene layers. A K-point mesh of 3� 3� 1 is used with

the Monkhorst–Pack sampling scheme. The width of the

supercell is 4:62A along the armchair direction. The length of

the supercell varies from 24.60 A to 49.19 A, along the zigzag

direction. Two stripes are considered in each supercell as

shown in Fig. 2(B). The bond length is relaxed to be 1.42 A.

We first fix the graphene sheet and relax the hydrogen atoms

in order to calculate the local forces due to the sp3 C–C bond-

ing. Then we fully relaxed the structure so that various defor-

mation mode can be observed. Same as the molecular

dynamics simulation, the TRI mode is observed with low

W=H ratio. The linear relation between the bending slope h

and the hydrogenation length W is shown as the dash lines

in Fig. 3(A). The slope variations match between first princi-

ples theory calculations and molecular dynamics simulations.

To elucidate how the bending deformation of the graphene

is induced by the hydrogenation within the stripe, we employ

a super cell of 4:62 A� 49:19 A with W ¼ 7:38 A as an example

to plot the force distribution on carbon atoms as shown in

Fig. 3(B). Each green circle represents two carbon atoms along

the width of the supercell. Each pink circle represents one

hydrogen atom bonded with the carbon atom next to it. The

red arrows represent the force acting on each of the carbon

atoms. Fig. 3(C) shows the net force along out of plane direc-

tion per unit width (4:62 A). It is clearly shown that a force

couple appears at the boundary of the doped region. As

shown in Fig. 3(E–G), considering the symmetry of the defor-

mation, the half of the hydrogenation region can be repre-

sented as a one-dimensional cantilever beam subject to a

constant bending moment M0 at a distance W=2 from the

fixed end of the beam. The bending moment is obtained from

the ab initio calculation as 1.54 ± 0.46 eV (Fig. 3(D)). For the

cantilever beam, the relation between the bending slope h

and the length of the beam can the obtained from classical

mechanics as h ¼ M0W=2EI, where M0 is the bending moment

and EI is the bending rigidity. The slope equation clearly

shows that the slope h is linearly proportional to the hydroge-

nation length W and independent of the beam length H. In our

calculation, @h=@W ¼ 0:62� 0:06 (rad/nm). The bending rigid-

ity EI can be derived as 4:97� 1:59 eV � A.

3.2. Thermal conductivity variation with double-sidehydrogenation patterns

With double-side hydrogenation patterns, we calculate the

effective thermal conductivity of the hydrogenated graphene

by defining the temperature gradient as ðTmax � TminÞ=L, where

Tmax;Tmin and L are the highest temperature, lowest tempera-

ture and length between Tmax and Tmin, respectively. With

longitudinal and transverse double-side hydrogenation pat-

terns, Fig. 4(A, B) show the thermal conductivity variation as

a function of doping coverage and the PDOS under various

doping coverage, respectively. With the increasing of the dop-

ing coverage, the thermal conductivity shows a decreasing

trend. Zero coverage of hydrogenation represents pristine

graphene. The phonon dispersion curve of pristine graphene

[41] indicates that the highest peak region (G-band) of the

PDOS contains optical phonons. The PDOS region immedi-

ately to the left of the G-band contains the longitudinal

acoustic phonons at the Brillouin zone boundary. A red shift

(shift to the left) or a decay of the G-peak simultaneously

represents a lowering of the longitudinal acoustic phonon

dispersion curve. Thus, the red shift of G-peak implies a

reduction of the phonon velocity. Graphene with 100% cover-

Fig. 5 – Thermal conductivity variation with hydrogenation

length W and stripe length H. The graphene sheet has the

size of 5.10 nm · 19.65 nm. The size of empty circles

represents the magnitude of thermal conductivity of the un-

relaxed graphene sheet with patterned hydrogenation. The

size and color of the filled circles represent the magnitude of

thermal conductivity and the deformation mode of the

relaxed graphene sheet with patterned hydrogenation. (A

color version of this figure can be viewed online.)

C A R B O N 7 2 ( 2 0 1 4 ) 1 8 5 – 1 9 1 189

age of hydrogenation is called graphane. Compared to graph-

ene, the phonon dispersion of graphane [42] indicates a

downward shift of the dispersion curves and a split of the lon-

gitudinal acoustic and optical phonon bands. With the PDOS

in Fig. 4(B), it is clear that when the hydrogen coverage in-

creases, (1) acoustic phonon dispersion curves are lowered;

and (2) phonons are redistributed among the bands. From

our MD simulations of graphene sheet with transverse hydro-

gen doping, we have calculated the phonon velocities of the

in-plane longitudinal (LA), in-plane transverse (TA) and out-

of-plane (ZA) phonon velocities as shown in Fig. 4(C). It shows

a reduction in LA velocity as the hydrogen coverage increases.

TA velocity also decreases, although more slowly. The ZA

velocity remains mostly the same. Note that, at 300 K, the

specific heat capacity is relatively insensitive to the decay of

the G-band [43]. In addition, it is well known that hydrogena-

tion reduces the phonon mean free path of graphene. Thus,

the reduction of the thermal conductivity of double-side

hydrogenation can be attributed to the combined effect of re-

duced phonon velocity and reduced phonon mean free path.

Although the PDOS are similar for longitudinal and trans-

verse doping patterns, the thermal conductivity variation

with doping coverage are different, which is due to the doping

direction. The longitudinal hydrogen doping only reduces the

coverage of sp2 C–C bonding without interrupting the phonon

transport path along the heat flux direction. The thermal con-

ductivity gradually decreases with the increasing of hydrogen

coverage. However, the transverse hydrogen doping creates

transverse barriers with sp3 C–C bonding and reduces the

phonon mean free path in the heat flux direction, causing a

large reduction in the thermal conductivity even at a small

hydrogen coverage. When the coverage increases beyond

85%, the transition thermal resistance regions of the two

neighboring hydrogenation boundaries start to overlap and

diminish when the coverage approaches 100%, resulting a

small increase of thermal conductivity as shown in Fig. 4(C).

3.3. Thermal conductivity variation with single-sidehydrogenation patterns

Since a graphene sheet with patterned single-side hydrogena-

tion stripes can be relaxed to different deformation modes

depending on both the size of H and its W=H ratio, it is impor-

tant to understand how the deformation modes will affect

the thermal conductivity. For a graphene sheet of the size

5.10 nm ·19.65 nm, we consider four different stripe lengths:

H = 4.91 nm, H = 3.28 nm, H = 2.46 nm and H = 1.97 nm, which

are corresponding to the 2, 3, 4, and 5 stripes on each heat flux

region, respectively. The hydrogenated graphene is then re-

laxed and the thermal conductivity is calculated. For compari-

son, we maintain the pre-stress along the heat flux direction

due to the patterned hydrogenation and keep the planar form

of the hydrogenated graphene, such that the deformation

modes shown in Fig. 2(C–F) do not occur (NON mode). Thermal

conductivity is then calculate for this unrelaxed configuration.

Clearly, the difference in the thermal conductivity of the

relaxed and unrelaxed graphene sheets represents the effect

of the global deformation modes on the thermal conductivity.

Fig. 5 shows the thermal conductivity of both relaxed and unre-

laxed graphene sheets with varying H and W. The thermal

conductivity of relaxed and unrelaxed graphene sheet is repre-

sented by filled and empty circles, respectively. The size of the

circles represents the magnitude of the thermal conductivity,

as shown on the right side of Fig. 5. The color of filled circles

represents the deformation mode. For unrelaxed graphene,

the thermal conductivity decreases monotonically with in-

crease of W or decrease of H. However, different trends are ob-

served for the relaxed graphene sheet. The TRI deformation

mode leads a higher decrement rate of the thermal conductiv-

ity. The COM deformation mode leads to a monotonically

increasing trend of the thermal conductivity due to the closer

packing of the carbon and hydrogen atoms as shown in Fig. 2.

The graphene’s thermal conductivity in the ROL modes mono-

tonically increases with the hydrogen coverage.The thermal

conductivity of aROL mode presents a relatively large fluctua-

tion closely related with the unstable deformation mode.

To further understand the effect of deformation mode on

the thermal conductivity variation, we compare the PDOS of

unrelaxed and relaxed graphene sheets with various hydroge-

nation length W and a given stripe length H = 4.91 nm, shown

in Fig. 6. The PDOS of the pristine graphene is plotted as green

solid line for reference. The black solid lines represent the

PDOS of unrelaxed graphene with different hydrogenation

length W. The blue, cyan, and magenta dashed lines represent

the TRI, COM and ROL deformation mode, respectively. In com-

parison of the relaxed and unrelaxed hydrogenated graphene,

while the thermal conductivity can be affected by the following

two factors: the position and the amplitude of the G-band, the

latter plays a more important role. In the TRI mode case

(W = 0.25 nm), the red-shift of the G-band for the unrelaxed

graphene sheet is due to the external tensile stress applied to

keep the graphene planar. Although the G-band for the relaxed

graphene sheet (TRI mode) does not shift to the left, the ampli-

tude decreases significantly. As discussed previously, the

decreasing amplitude is related with the left shift of acoustic

phonons, resulting in a lower thermal conductivity. For the

Fig. 6 – Comparison of PDOS between unrelaxed (solid line)

and relaxed (dash line) graphene with various

hydrogenation length W. The length of the stripe H is

4.91 nm. (A color version of this figure can be viewed

online.)

190 C A R B O N 7 2 ( 2 0 1 4 ) 1 8 5 – 1 9 1

COM case (W = 1.97 nm) and the ROL case (W = 4.67 nm), the

position of the G-bands are similar for both the unrelaxed

and relaxed graphene sheets. It shows that, for high doping

density cases, the hydrogen doping determines the PDOS and

the effect of stretching on the G-band position becomes insig-

nificant. However, the amplitude of the G-band is higher for the

relaxed graphene sheets, which implies a higher thermal con-

ductivity when they are in relaxed COM and ROL modes.

4. Conclusion

In summary, we investigate the thermal conductivity variation

of graphene with respect to patterned hydrogenation, doping

density and the deformation modes induced by hydrogenation

through non-equilibrium molecular dynamics simulations. By

relaxing the local stress in hydrogenated graphene sheets, var-

ious deformation modes are observed. First principles theory

calculations are performed to model and explain the relation

between the bending slope of the graphene sheet and the

length of the hydrogenation region. Our simulation results

show that, when the W=H ratio increases, the thermal conduc-

tivity decreases quickly under the TRI mode and exhibits an

increasing trend under the COM mode. This study illustrates

a new opportunity to control both the thermal conductivity

and the deformation mode of graphene through patterned

hydrogenation. Meanwhile, it demonstrates that great atten-

tion is necessarily to be paid to the thermal conductivity varia-

tions in graphene based foldable devices.

Acknowledgement

We gratefully acknowledge support from Clemson startup

fund and National Science Foundation support under Grant

No. CBET-0955096.

Appendix A. Supplementary data

Supplementary data associated with this article can be found,

in the onlineversion, at http://dx.doi.org/10.1016/j.carbon.2014

.02.001.

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