32952 | Phys. Chem. Chem. Phys., 2016, 18, 32952--32961 This journal is© the Owner Societies 2016

Cite this:Phys.Chem.Chem.Phys.,

2016, 18, 32952

Nanoscale thermal cloaking in graphene viachemical functionalization

Zhen-Qiang Ye and Bing-Yang Cao*

Macro-thermal cloaking is typically produced by coordinate transformations, but this method is unsuitable for

nanostructures. We designed a graphene-based nanoscale thermal cloak using a novel mechanism of

phonon localization. The nanocloak in graphene was produced via the chemical functionalization of

hydrogen, methyl and hydroxyl using molecular dynamics simulations. The cloaking performance was

quantified by the ratio of thermal cloaking (RTC). We found that the RTC correlated with the functionalization

fraction and it has a local maximum at a certain width, since the heat flux reduction in the exterior and the

protected region reversed if the width was excessive. The atomic mass of the functional group also

correlated with the RTC. Our simulations determined that phonon localization occurred due to sp2-to-sp3

bonding transitions, which caused the heat flux to avoid the transition region. Finally, the extent of phonon

localization was related to the cloaking performance.

1. Introduction

Growing interest in the manipulation of heat flux led researchersto design metamaterials,1–7 with certain properties. In thermalscience, metamaterials known as ‘‘thermal crystals,’’8 are used inthermal devices such as thermal diodes,9–11 heat concentrators,12

and thermal camouflage and thermal cloaks.13–19 However, mostof these devices are still in the early stages of development.A thermal cloak is a device that redirects the heat flow aroundan object and renders it thermally invisible, making it seeminvisible for heat flow.20 Cloaking is different than isolation.Isolation affects other parts of the object, while cloaking makes theobject seem invisible. Cloaking was first achieved by Pendry et al.,21

who used transformation optics to obtain an electromagneticcloak. Subsequently, other types of cloaks were22–26 proposed.Thermal cloaking is a fascinating device that may be used inthe future for thermal control and management.

Numerous studies on thermal cloaking have emerged sincethe pioneering theoretical work of Fan et al.,1 who first discoveredheat invisibility and reversed heat flux using the theory ofcoordinate transformation. Yang et al.14 further developed thescheme of arbitrary-shaped thermal cloaks. He et al.3 designed anopen cloak with finite material parameters. These early devicestypically required anisotropic and inhomogeneous thermal con-ductivities, which made their fabrication difficult. Therefore,a simplified scheme known as ‘‘multilayered metamaterials’’

was proposed,5,15,27 which discretized the inhomogeneousmaterials into a series of homogeneous materials. Experimentalthermal cloaks used two materials with different thermal proper-ties to form multilayered structures.13,20,28–30 Narayana et al.28

fabricated a micro-structured thermal cloak based on polyamide/copper multilayered structures. Xu et al.20 designed an ultrathinthree-dimensional thermal cloak. Dang et al.13 reported an activeand controllable thermal cloaking device that was more intelligentand flexible. In general, most existing studies are based on thecoordinate transformation approach and require inhomogeneousstructures. In practice, inhomogeneous structures are reduced tohomogeneous multilayered structures. However, this simplificationis unsuitable for nanoscale structures for two reasons. First, particlesize effects and interfacial thermal resistance cannot be neglectedfor nanoscale structures,31 and multilayers allow additionalthermal resistance and damage to their mechanical properties.Second, the system size is close to the phonon mean free pathsin the nanoscale structures, which suggests that the heattransport is nonlocal. Consequently, we cannot apply Fourier’slaw to define the thermal properties of the local region in thenanoscale structures.32 To date, no studies have reported onnanoscale thermal cloaking. However, growing interest in heattransfer at the nanoscale level, such as for large-scale integratedcircuits, calls for more research in nanoscale thermal cloaks forthe thermal protection of nanocomponents.

We developed a graphene-based thermal cloak via chemicalfunctionalization using a novel mechanism. Graphene,33 one ofthe most important and popular nanomaterials, is a promisingmaterial in nanoelectronics because of its excellent thermal,mechanical and electrical properties. In addition, its thermalproperties can be controlled by changing its edge, size, and

Key Laboratory for Thermal Science and Power Engineering of Ministry of

Education, Department of Engineering Mechanics, Tsinghua University,

Beijing 100084, P. R. China. E-mail: caoby@tsinghua.edu.cn;

Fax: +86 10 62794531; Tel: +86 10 62794531

Received 17th October 2016,Accepted 11th November 2016

DOI: 10.1039/c6cp07098a

www.rsc.org/pccp

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chemical functionalization.34–39 We used partial chemical func-tionalization, including hydrogen, methyl and hydroxyl ongraphene to form a thermal cloak. Molecular dynamics (MD)simulations were used to investigate thermal cloaking effectscaused by chemical functionalization on graphene. We focusedon the influence of the size fractions of functional groups,cloaking sizes, and functionalization types. Further, we used anovel mechanism based on phonon localization theory toexplain the thermal cloaking effects.

2. Molecular dynamics

The MD simulations were conducted using the LAMMPSpackages.40 We used an adaptive intermolecular reactive bondorder (AIREBO) potential to describe the C–C and C–H bondinginteractions,40–42 whose expression is

E ¼ 1

2

Xi

Xjai

EREBOij þ ELJ

ij þXkai; j

Xlai; j;k

ETORSIONkijl

" #; (1)

where E denotes the whole potential energy, and where thefirst term on the right side of the equation is the hydrocarbonREBO potential,42 the second term is the Lennard-Jonespotential, and the third term is an explicit four-body potentialcaused by various dihedral angle preferences in the hydrocarbonconfigurations. In addition, the C–O, CQO and O–H bonds weredefined by an OPLS-aa potential model.43 Fig. 1 shows that theschematic of graphene with a thermal cloak generated by hydro-gen, methyl and hydroxyl. To maintain structural stability, thefunctional groups were evenly distributed on both surfaces of thegraphene. The protected region was the center circle. The mostpopular design of the macroscale thermal cloak is a spheroid/cylinder shell.1,2,8 As such, the shape of this two dimensionalcloak was designed as a ring, and the heat flux in the center of thecircle was squeezed into the ring structure.

We used the nonequilibrium MD (NEMD) method to simulatethe system under a temperature gradient. The temperatureprofile and heat flux field were recorded to visualize thethermal cloaking effect, upon which we performed a quantita-tive analysis of the thermal cloaking. The system was separatedinto three regions: the fixed region, the thermostat region andthe free region. The X direction was applied a fixed boundarycondition, and the atoms of the two ends were fixed. The Y andZ directions were free boundary conditions. Next to the fixedregion was the thermostat region, with high and low tempera-tures on each side and with atoms calculated by Nose–Hooverthermostats as follows,44

d

dtpi ¼ Fi � gpi; (2.1)

d

dtg ¼ 1

t2TðtÞT0� 1

� �; (2.2)

TðtÞ ¼ 2

3NkB

Xi

pi2

2mi; (2.3)

where, pi, Fi and mi are the momentum, force and mass of atomi, g and t are the dynamic parameters and relaxation timesof the thermostat, kB is the Boltzmann constant, and N is theatomic number of the thermostats. The middle section, calledthe free region, was our primary focus, and the heat flux of eachatom was calculated by

Ji = eivi � Sivi, (3)

in which, Ji is the heat flux of atom i; e is the total energy,including both kinetic and potential energies, v is the velocityvector, and S is the stress tensor for each atom, which had ninecomponents: xx, yy, zz, xy, yx, xz, zx, yz, and zy. Therefore, theheat flux of three directions for each atom was expressed as

Jx = evx � (Sxxvx + Sxyvy + Sxzvz), (4.1)

Jy = evy � (Syxvx + Syyvy + Syzvz), (4.2)

Jz = evz � (Szxvx + Szyvy + Szzvz). (4.3)

For convenience, the subscript ‘‘i’’ was omitted. The time stepwas set at 0.5 fs. The system was first relaxed in the NVTensemble for 500 000 steps, and then in the NVE ensemblefor another 500 000 steps. Finally, the NEMD method was usedfor 1 000 000 steps, during which the heat flux of each atom inthe free region was recorded. The system size was 24.5 nm �21.2 nm. The cloaking region was placed in the center of thegraphene sheet with a diameter of 6 nm. The high temperaturethermostat was 350 K, and the low temperature thermostatwas 250 K.

3. Results and discussion

First, we used a hydrogenated cloak to demonstrate the thermalcloaking in the nanostructures. For ease of description, it iscalled ‘‘hydrogenated graphene.’’ The cloak had a width of1 nm and closely surrounded the cloaking region. To better

Fig. 1 (a) The schematic of graphene with thermal cloak, and partialenlarged view of the three types: (b) hydrogen, (c) hydroxyl and (d) methyl.

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illustrate the efficiency of our cloak, three specific conditions wereconsidered as references: perfect graphene, defective graphene,and hollow graphene. The perfect graphene was pristine graphenewithout a thermal cloak. For the defective graphene, the cloakwas generated with vacancy defects in which twenty carbonatoms around the cloaking region were removed. For the hollowgraphene, the cloaking region was entirely removed, whichequaled absolute isolation. Fig. 2(a–d) show the temperature

profiles of the four samples. To better visualize the temperatureprofiles, the temperature of each atom was the average of200 000 steps. Overall, the temperature profiles of thesegraphene sheets were not as regular as those of macro materials.In particular, the isotherms were less notable. We made threeobservations. First, we noticed a subtle outline of the cloaking regionin the defective graphene, in which the atom temperature wasrelatively uniform, indicating the occurrence of thermal cloaking.

Fig. 2 Temperature profiles of (a) perfect graphene, (b) defective graphene, (c) hollow graphene and (d) hydrogenated graphene; (e) the energyaccumulation applied to the thermostats of the four conditions, Zk is the normalized thermal conductivity.

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Second, for the hydrogenated graphene, the outline was moresignificant than that of the defective graphene, and the temperatureprofile of the exterior region was slightly affected. Third, thehollow graphene was equivalent to thermal isolation, and thetemperature profile was strongly influenced by the central hollowregion, which should be avoided. Luo et al.45 use the standarddeviation (STD) of the temperature difference of each small latticeto exam the thermal cloaking effectiveness of layered engineeringmaterials, the equation is

DT(i) = T(i) � TPendry(i), (5.1)

STD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi¼1ðDTðiÞÞ2

vuut ; (5.2)

in which, TPendry refers to the theoretical temperature based onthe transformation optics method. We separate the system into80 � 80 lattices, and calculate the temperature difference ofeach lattice. The results are shown in Table 1. Apparently, it isdifficult to preserve the exterior temperature profile, but thedefective graphene and hydrogenated graphene perform muchbetter than hollow graphene in this respect.

Fig. 2(e) shows the energy accumulation applied to the thermo-stats of each graphene sample. The slope reflects the relativevalues of thermal conductivity, and shows that the ranking orderof thermal conductivity is perfect 4 defective 4 hydrogenated 4hollow. So the hydrogenation can reduce the thermal conductivityof graphene, it agrees well with the literature.46 It is reported thatthe defect in the interface may enhance thermal conductance,47

but our case has a different mechanism. The introduction ofhydrogen can lead to the red shift in the phonon spectra, so thethermal conductivity decreases significantly.46 The thermalconductivities of the defective and hydrogenated graphenesamples were almost the same, but the hydrogenated samplehad remarkable superiority in thermal cloaking. Generally,the cloaks negatively impacted the thermal conductivities, butwere still acceptable.

Fig. 3(a–d) show the heat flux fields of the four samples. Theheat flux of each atom was the average value of 200 000 steps.As shown in Fig. 3(b), we focused on the middle section,which was separated into three typical regions. Consideringthe longitudinal symmetry, we focused on region A and regionB. The heat flux field of graphene was almost uniform, and itsorientation was parallel to the temperature gradient. For thedefective graphene, the heat flux of region B was much lessthan that of region A, suggesting that the heat flux avoided thecloaking region. In addition, the heat flux field in region A wasslightly heterogeneous, suggesting that it was affected by thecloak. An ideal thermal cloak should have little impact on otherregions. As such, the defective graphene was not an acceptablethermal cloak. The heat flux field was the standard of thermal

isolation for the hollow graphene, and showed that the heatflux of region A was considerably larger than that of the otherregions because of the isolation in the center. In addition, theregion near the inner ring where the direction of the heat fluxthat had a significant change was strongly influenced by thecloak. For the hydrogenated graphene, the orientation in regionA was very smooth. Moreover, the heat flux of region B inhydrogenated graphene was smaller than that of the defectivegraphene. Therefore, we arrived at a similar conclusion: theperformance of hydrogenated graphene was better than that ofdefective graphene.

We defined a new concept to quantify the performanceof thermal cloaking, which was called the ratio of thermalcloaking (RTC),

Z ¼ HFA

HFB

��������; (6)

where HFA and HFB is the mean heat flux in region A and region B.For a perfect cloak, there should be no heat flux in region B,such that HFB = 0, Z is infinity, while Z = 1 indicates no thermalcloaking. RTC was an index of cloaking performance, namely,the higher the Z, the better the cloaking performance. Table 2gives the RTC of perfect, defective and hydrogenated graphene.It shows that the RTC of hydrogenated graphene is 65% higherthan that of defective graphene. It should be emphasized thatour scheme is not a perfect thermal cloak, but it achievesheat flow manipulation in some sense. So we still call it‘‘thermal cloak’’.

Next, we investigated the influence of cloak width and hydrogenfractions. Fig. 4 shows the heat flux profiles of cloaks with differentwidths ranging from 0.5 nm to 2 nm. First, in each of the foursamples, the central region had yellow and gray arrows, indicatinga relatively low heat flux. Second, as the thermal cloakincreased in size, the blue region gradually increased andbecame more homogeneous, which indicated that the heat fluxmoved to the bottom and top regions. The middle section wasblocked by the cloak. Third, as shown in Fig. 4(d), the orangearrows for the 2 nm cloak increased significantly in the centralregion, indicating an increase of heat flux, but did not declinemonotonously as the thermal cloak increased. Although thethicker cloak had a stronger impact on the heat flux profile, itdid not result in a better cloaking performance.

Besides the cloak width, the hydrogen fraction was anotherimportant factor that affected the cloaking performance. Hydro-genated graphene is called graphane, and was first synthesizedin 2009 by Elias et al.48 Pure graphane does not exist becausecurrent hydrogenation methods cannot completely hydrogenategraphene. Thus, it is necessary to study the influence of thehydrogen fraction. Because of the stability of the structure, weobtained only three stable states: 100%, 25% and 12.5%. Fig. 5shows the heat flux profiles of the 12.5% and 25% cloaks, inwhich the width is 2 nm. Fig. 4(d) shows that, for the 100% cloak,as the hydrogen fraction decreased, a growing number of orangearrows appeared in the cloaking region, which indicated that theheat flux of the cloaking region was enhanced as the hydrogenfraction decreased, and that the cloaking performance declined.

Table 1 STD of perfect, defective, hollow and hydrogenated graphene

Perfect Defective Hollow Hydrogenated

STD, K 0 5.9 7.8 6.3

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Secondly, more blue arrows appeared in the top and bottomregions as the fraction increased, which showed that additionalflow flux avoided the cloak.

The RTCs of different widths and hydrogen fractions areshown in Fig. 6. The results agreed with the above analyses. Forthe fraction, RTC increased monotonously. For the cloak width,the RTC reached a maximum at 1.0 nm, but above 1.0 nm,thicker or thinner cloaks both had negative effects on thecloaking performance. The presence of a thermal cloak reducedthe heat flux of the whole system. The increased cloak widthnot only reduced the heat flux of the protected region, but alsoreduced the heat flux of the entire system. If the extent of thelatter is greater than that of the former, RTC will decrease withthe increasing width, otherwise, it will increase monotonously.Therefore, the RTC reached a maximum at a certain value thatwas dependent on the system size and the protected region size.

Besides hydrogenation, we used other functional groups todevelop thermal cloaks, such as methyl and hydroxyl. Our earlysimulations suggested that the mass of the functional group

had a positive impact on the cloaking performance. The relativeatomic mass of methyl is 15 and of hydroxyl is 17, both of whichare far greater than that of hydrogen. Therefore, we replacedhydrogen with methyl and hydroxyl to obtain enhanced thermalcloaking. We named the samples the methylic cloak and thehydroxy cloak. It should be noted that atoms of the functionalgroups will interfere with each other if the fraction is too highof methyl and hydroxyl. After several attempts, we obtained astable structure while the fraction was 12.5%. Fig. 7 shows thatthe enhanced thermal cloaks generated by methyl and hydro-xyl, in which the cloak width was 2 nm and the fraction was12.5%. The hydrogenated cloak with the same parameters isshown in Fig. 5(a). By comparison, we found that the cloakingperformance of methylic and hydroxy cloaks was significantlyimproved compared to that of hydrogenated cloak. In addition,the heat flux of the central region in the hydroxy cloak waslower than that in the methylic cloak, so the methylic cloak wassuperior to the hydroxyl cloak. Then, we calculated the RTCsbased on eqn (6), which were 1.714 for the methylic cloak and2.36 for the hydroxy cloak. The RTC of the hydrogenated cloakwith the same parameters was only 1.47, so the RTC increased16.5% and 60.5% with the methylic and hydroxyl cloaks. Weobserved that the RTC of the hydroxy cloak was even higherthan that of the 100% hydrogenated cloak, though the fractionof the hydroxyl was only 12.5%. Hydroxyl greatly improved the

Fig. 3 Heat flux fields of (a) perfect graphene, (b) defective graphene, (c) hollow graphene and (d) hydrogenated graphene.

Table 2 RTC of perfect, defective and hydrogenated graphene

Perfect graphene Defective Hydrogenated

Z 1.00 1.43 2.36

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cloaking performance compared to that of the methyl andhydrogen.

Prior studies that focused on the effect of hydrogenation onthermal conductivity of graphene found that the deteriorationof thermal conductivity was due to the sp2-to-sp3 bondingtransition, which softened the G-band phonon modes. Basedon this finding, we investigated the changes in the phononproperties in chemical functionalized graphene to determinethe underlying mechanism of the thermal cloak. The phonon

density of states (DOS) is the most representative property thatcan characterize the phonon states. The DOS was calculated bythe Fourier transform of the velocity autocorrelation function,

DOSðoÞ ¼ðe�iot vð0ÞvðtÞh idt; (7)

where o is the phonon frequency. Fig. 8 shows the four types oftypical DOS of carbon atoms in the cloak, including the normalsp2 carbon, the affected sp2 carbon, the normal sp3 carbon and

Fig. 4 Heat flux fields of hydrogenated graphene with a cloak width of (a) 0.5 nm, (b) 1 nm, (c) 1.5 nm and (d) 2 nm.

Fig. 5 Heat flux fields of the cloaks with hydrogen fraction of (a) 12.5% and (b) 25%. The cloak width is 2 nm.

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the affected sp3 carbon. The atoms in graphene are sp2 carbonatoms, but some will become sp3 atoms after chemical func-tionalization. Thus, both sp2 and sp3 carbon atoms existed inour samples. Fig. 8(a) shows the DOS of a normal sp2 carbon.A primary peak was observed at 48 THz, which was consistentwith that of perfect graphene,49 and thus we concluded thatthese atoms were not affected by the cloak. Fig. 8(b) shows theDOS of sp2 carbon atoms near the cloak. Compared to Fig. 8(a),two notable changes are depicted in Fig. 8(b). First, the peak at48 THz disappeared and was replaced by several small peaks,suggesting that high frequency states significantly decreased.This resulted from the atoms in the cloak having lower vibrationfrequencies, which affected the neighboring atoms. Second, themain peak shifted to 24 THz from 48 THz, so the cloak causedthe phonon spectra to shift to a low frequency region. The sametypes of atoms had different DOS values, which indicated thatthe phonon localization50 occurred near the cloak. A similarsituation occurred in Fig. 8(c) and (d). Fig. 8(c) shows the DOS ofnormal sp3 atoms. Both atoms were similar in structure to adiamond, and displayed a main peak at 37 THz.51 For Fig. 8(d),

the DOS had a slight change, in which the main peak shifted to30 THz. In general, the phonon localization phenomenonoccurred in this system, and was more significant for sp2 atoms.Therefore, the phonon localization of sp2 was our primary focusin determining the underlying mechanism of the thermal cloak.

The phonon localization can be visualized by the spatialdistribution of phonon modes. The spatial distribution wasequivalent to the weight distribution of the specific modes. Forthe ith atom, the weight of a specific range (L) of phononmodes on the whole spectral range was expressed as

pL;i ¼ÐLDOSidoÐ10 DOSido

: (8)

Since the changes of the sp2 atoms in the high frequencyregion were more significant, we showed the phonon (46 to50 THz) spatial distribution of sp2 atoms in Fig. 9. We used the12.5% hydrogenated cloak and the 12.5% hydroxy cloak todisplay the differences in phonon localization. Fig. 9(a) and(b) refer to the 12.5% hydrogenated and the 12.5% hydroxycloak. The red spheres indicated the highest spatial distribu-tion, then the yellow spheres with less spatial distribution, andfinally the blue spheres with the lowest spatial distribution. Wemade three observations. First, the edge atoms usually had alow spatial distribution. The phonon spectra of the edge atomsshifted to low frequencies because of the interaction betweensp3 and sp2 atoms, so the high frequency phonons hadrelatively low distributions at the edge. Second, more bluespheres appeared in Fig. 9(b) than in Fig. 9(a). This indicatedthat the extent of phonon localization was more prominent forthe hydroxy cloak, because the atomic mass of hydroxyl wasmuch larger than that of hydrogen, and therefore it had a largerimpact on its connected sp3 carbon atoms and indirectlyaffected the neighbor sp2 carbon atoms. Third, the top andbottom atoms usually had lower distributions. According to theheat flux profiles, the top and bottom regions had the highestheat flux, indicating that the interactions in this region weremore vigorous, and the phonon localization was more signifi-cant because of the stronger interactions.

Fig. 6 RTCs of different width and hydrogen fraction.

Fig. 7 Enhanced thermal cloak generated by (a) methyl and (b) hydroxy.

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To quantify the extent of the localization, Table 3 showsthe average phonon (46 to 50 THz) spatial distribution of theedge atoms. As a comparison, the distribution of a perfectgraphene is also given in Table 2. For the perfect graphene,high frequency phonons (46 to 50 THz) occupy almost 30% ofthe phonon states, but for the listed cloaks in Table 2, itdecreased to different extents. The 100% hydrogenated cloakreduced the most, to 13.7%, and the 12.5% hydrogenated cloakdeclined the least, to 17.8%. Generally, phonon localizationoccurs due to the sp2-to-sp3 bonding transition, which has agreat effect on the heat flux field. The heat flux automatically

avoids the transition region, so the central section is protectedfrom the heat flux. Our results also showed that the extent ofphonon localization was positively associated with the fractionand the mass of the functional group, and correlated withthe cloaking performance. Thus, we concluded that phononlocalization induced the thermal cloaking phenomenon.

4. Conclusions

In summary, we proposed a novel nanoscale thermal cloak andunderlying mechanism. Partial chemical functionalization ongraphene was used to form a heat flux channel that avoided thespecific region. MD simulations were used to determine thethermal cloaking effect caused by chemical functionalizationon graphene. First, we compared perfect graphene, defectivegraphene and hollow graphene, and found that hydrogenatedgraphene had a better cloaking performance than that of

Fig. 8 Four kinds of typical DOS of carbon atoms in the cloak, (a) normal sp2 carbon atom, (b) affected sp2 carbon atom, (c) normal sp3 carbon atom,and (d) affected sp3 carbon atom.

Fig. 9 Phonon spatial distribution of (a) 12.5% hydrogenated cloak and (b) 12.5% hydroxy cloak.

Table 3 Average phonon (46 to 50 THz) spatial distribution of the edgeatoms

100% H 25% H 12.5% H 12.5% OH 12.5% CH3 Perfect

13.7% 15.9% 17.8% 14.1% 16.1% 29.1%

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defective and hollow graphenes. This result indicated thathydrogenated graphene could cloak a subject and maintainthe original temperature field as long as possible. Hydroge-nated graphene also had a slight influence on the mechanicaland thermal properties of the system. Second, we investigatedthe influence of the chemical functionalization fraction andthe cloak width. The cloaking performance was quantified bythe RTC. A higher RTC indicated a better thermal cloakingperformance. The results showed that the RTC was positivelyassociated with the functional group fraction. For the 100%hydrogenated cloak, the RTC was 2.35, while it reduced to1.45 for the 12.5% hydrogenated cloak. The RTC reached alocal maximum when the cloak width was 1.0 nm, because thepresence of the thermal cloak reduced the heat flux of boththe exterior and the protected regions. The decreased extentreversed at a certain value, depending on the system size,indicating that the RTC reached its maximum. Third, we usedfunctional groups, methyl and hydroxyl, with higher atomicmasses to enhance the thermal cloaks. Although the fractionwas only 12.5%, we still observed a dramatic improvement ofthe cloaking performance by 16.5% and 60.5%, as compared to12.5% with the hydrogenated cloak. The 12.5% hydroxy cloakequaled the 100% hydrogenated cloak. Finally, we found themechanism of the thermal cloak based on the phonon localiza-tion theory. The connection between the thermal cloak andphonon localization was identified by the simulations. Theintroduction of chemical functional groups on graphene ledto a sp2-to-sp3 bonding transition, which further increasedphonon localization. The heat flux automatically avoided thetransition region because of phonon localization, and thecenter section was protected from the heat flux. Consideringthe popularity of graphene, our research not only has greatapplications for the thermal protection of graphene-baseddevices, but also has profound significance for the develop-ment of other nanoelectric devices.

Acknowledgements

This work was financially supported by the National NaturalScience Foundation of China (No. 51322603, 51676108,51628602, 51136001, and 51356001), the Science Fund forCreative Research Group (No. 51321002), and the TsinghuaNational Laboratory for Information Science and Technology ofChina (TNList).

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