Theoretical study of anisotropy and ultra-low thermal conductance of porous graphene nanoribbons

Wu Cheng-Wei; Ren Xue; Zhou Wu-Xing; Xie Guo-Feng

doi:10.7498/aps.71.20211477

Abstract

The thermal transport properties of porous graphene nanoribbons are studied by the non-equilibrium Green's function method. The results show that owing to the existence of nano-pores, the thermal conductance of porous graphene nanoribbons is much lower than that of graphene nanoribbons. At room temperature, the thermal conductance of zigzag porous graphene nanoribbons is only 12% of that of zigzag graphene nanoribbons of the same size. This is due to the phonon localization caused by the nano-pores in the porous graphene nanoribbons. In addition, the thermal conductance of porous graphene nanoribbons has remarkable anisotropy. With the same size, the thermal conductance of armchair porous graphene nanoribbons is about twice higher than that of zigzag porous graphene nanoribbons. This is because the phonon locality in the zigzag direction is stronger than that in the armchair direction, and even part of the frequency phonons are completely localized.

Keywords:

PACS:

78.67.Rb	(Nanoporous materials)
65.80.Ck	(Thermal properties of graphene)
63.20.Pw	(Localized modes)
65.80.-g	(Thermal properties of small particles, nanocrystals, nanotubes, and other related systems)

Authors and contacts

1.
School of Materials Science and Engineering, Hunan University of Science and Technology, Xiangtan 411201, China
2.
Hunan Provincial Key Laboratory of Advanced Materials for New Energy Storage and Conversion, Hunan University of Science and Technology, Xiangtan 411201, China

Corresponding author: Zhou Wu-Xing, wuxingzhou@hnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12074115, 11874145).

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1. 引　言

石墨烯优良的电学、力学、导热等性能, 使其成为理想纳米电子器件候选材料, 但由于石墨烯的零带隙电子结构, 限制了其在纳米电子器件领域的应用^[1,2]. 因此, 近年来, 人们尝试了很多使得石墨烯打开带隙的方法^[3-10]. 研究发现, 引入纳米孔洞是一种调控二维材料的电学、机械、光学等性能的有效方法^[11-20]. 在石墨烯中引入孔洞缺陷, 实验上可以通过自上而下的方法, 在完美石墨烯薄膜上刻蚀孔洞, 也可以通过自下而上的方法, 通过小分子自组装合成具有周期性孔洞的二维多孔石墨烯材料. 2018年, Moreno 等^[21]利用二苯基-10, 10'-二溴-9, 9'-联二蒽 (diphenyl-10, 10′- dibromo-9, 9′-bianthracene, DP-DBBA) 合成了周期性纳米多孔石墨烯 (Nano-porous graphene). 研究发现, 由于孔洞的引入, 使得零带隙的石墨烯转变为固有带隙约为1 eV的多孔石墨烯. 更有趣的是, 多孔石墨烯中纳米级的孔洞有望用于分子筛分, 并被预测为可同时进行分子筛分和电传感的高效材料.

多孔石墨烯中的纳米孔洞除了改变石墨烯的电子结构以外, 对机械性能、光学性质以及热输运等性质都会造成影响. Mortazavi等^[22]采用第一性原理研究了多孔石墨烯的力学、光学、电学以及晶格热导率. 研究发现, 多孔石墨烯薄膜可以吸收可见光、近红外以及远红外光, 在纳米光学具有很好的应用前景. 另外, 大量孔洞的出现, 多孔石墨烯的弹性模量只有石墨烯的0.3—0.5倍, 扶手椅方向为174 N/m, 锯齿方向为144 N/m, 仍具有较高的力学强度. 并且相较于石墨烯, 多孔石墨烯的热导率降低了两个数量级. 另外, Hu等^[19]在研究中用非平衡格林函数方法和分子动力学方法研究了石墨烯声子晶体结构的热输运, 结果都表明孔洞会显著的降低石墨烯的热导率^[23,24]. Singh等^[25]利用第一性原理计算方法预测了多孔石墨烯的热电输运性能, 发现多孔石墨烯在室温下具有极大的塞贝克系数1662.59 μV/K, 以及热电优值ZT =1.13, 并且表现为各向同性, 说明多孔石墨烯在热电领域也具有良好的应用前景.

然而, 对于先前的研究大多集中在二维多孔石墨烯材料的电学、光学以及机械性能, 热输运性质的研究还非常少, 尤其是一维的多孔石墨烯纳米带的热输运性质还没有被报道过. 鉴于多孔石墨烯纳米带在各个方面的潜在应用, 本文采用非平衡格林函数(NEGF) 方法, 系统地研究了多孔石墨烯纳米带的热输运性质. 通过与完美石墨烯纳米带热输运性质的对比, 研究了纳米孔洞的引入对热输运性质的影响.

2. 模型与方法

多孔石墨烯的结构如图1(a)所示, 蓝色虚线框所框选的区域为多孔石墨烯的单胞. 每个单胞包括80个C原子和20个H原子, 多孔石墨烯具有纳米级尺寸的单胞(a = 0.8534 nm, b = 3.2383 nm). 考虑到多孔石墨烯是有DP-DBBA分子聚合而成, 故按照DP-DBBA分子的聚合的规律^[21]裁剪出多孔石墨烯纳米带(nano-porous graphenenanoribbons), 如图1(b), (c)所示. 其中, 图1(b) 分别为宽度为1和2的锯齿型多孔石墨烯纳米带, 分别简写为NPZGNR-1和NPZGNR-2. 同理, 图1(c)为扶手椅型多孔石墨烯纳米带(NPAGNR-1 和NPAGNR-2).

$图 1 (a) 多孔石墨烯结构图, 红色代表C原子, 白色代表H原子; (b) 纳米带NPZGNR-1和NPZGNR-2的结构图; (c) 纳米带NPAGNR-1和NPAGNR-2的结构图\r\nFig. 1. (a) Structures of nano-porous graphene, red atoms represent C, white atoms represent H; (b) structures of NPZGNR-1 and NPZGNR-2; (c) structures of NPAGNR-1 and NPAGNR-2.$

图 1 (a) 多孔石墨烯结构图, 红色代表C原子, 白色代表H原子; (b) 纳米带NPZGNR-1和NPZGNR-2的结构图; (c) 纳米带NPAGNR-1和NPAGNR-2的结构图

Fig. 1. (a) Structures of nano-porous graphene, red atoms represent C, white atoms represent H; (b) structures of NPZGNR-1 and NPZGNR-2; (c) structures of NPAGNR-1 and NPAGNR-2.

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在纳米尺度下, 体系的声子输运呈量子输运特性, 此时声子输运表现为弹道输运, 而NEGF方法能精准地描述弹道输运行为^[26]. 结合声子的NEGF方法^[26,27-31], 研究体系被分为3个区域: 左热库(L), 中间散射区(C), 右热库(R). 左右热库为中间散射区提供温度差∆T = (T_R– T_L), 且热流密度 ${J_{\text{C}}}$ 可由(1)式 ^[32-33]求得:

${J_{\text{C}}} = \frac{\hbar }{{2{\text{π }}}} \int_0^\infty {\omega \cdot } {T_{{\text{ph}}}}(\omega ) \cdot \left[ {f(\omega ,{T_{\text{R}}}) - f(\omega ,{T_{\text{L}}})} \right]{\rm{d}}\omega ,$

(1)

其中, T_R, T_L分别为右热库温度和左热库温度, $\hbar$ 为约化普朗克常数, $\omega$ , ${T_{{\text{ph}}}}$ 分别为频率和声子透射系数, 玻色-爱因斯坦分布函数 $f(\omega , T) =$ ${\left[ {\exp \left( {\hbar \omega /{k_{\text{B}}}T - 1} \right)} \right]^{ - 1}}$ . 再结合傅里叶导热定律 $\kappa =$ ${{{J_{\text{C}}}} \mathord{\left/ {\vphantom {{{J_{\text{C}}}} {\Delta T}}} \right. } {\Delta T}}$ , 当∆T足够小时, 特定温度T下的热导 $\kappa$ 可由Landauer公式^[34]计算得出:

$\kappa = \frac{\hbar }{{2{\text{π }}}}\int_0^\infty {\omega \cdot } {T_{{\text{ph}}}}(\omega ) \cdot \frac{{\partial f(\omega ,T)}}{{\partial T}}{\rm{d}}\omega ,$

(2)

由(2)式可知, 研究体系 $\kappa$ 的关键在于获得 ${T_{{\text{ph}}}}$ . 基于NEGF方法, 体系的滞后格林函数^[35]表示为

${{\text{G}}^{{r}}}(\omega ) = \bigg[(\omega + {\text{i}}\eta )^2 {\boldsymbol I} - {\boldsymbol K} - \sum\limits_{{\text{L}}}^r {(\omega )} - \sum\limits_{{\text{R}}}^r {(\omega )} \bigg]^{ - 1},$

(3)

其中 ${\boldsymbol{I}}$ 为单位矩阵, ${\boldsymbol{K}}$ 力常数矩阵, 在此工作中, 采用晶格动力学的方法, 结合Gulp程序中的Brenner经验势得到纳米带的 ${\boldsymbol{K}}$ , Brenner势可准确的描述碳材料的动力学行为, 其准确性在先前的研究中以得到充分验证^[36-40]. $\displaystyle\sum\nolimits_{{\text{L}}/{\text{R}}}^r {(\omega )}$ 为左右热库的滞后自能项, 且左右热库与中心散射区的相互作用可被表示为 ${\varGamma _{{\text{L/R}}}} = {\text{i}}\left( {\displaystyle\sum\nolimits_{{\text{L/R}}}^r {(\omega )} - \displaystyle\sum\nolimits_{{\text{L/R}}}^a {(\omega )} } \right)$ , 那么 ${T_{{\text{ph}}}}$ 可由Caroli公式^[35]求得:

${T_{{\text{ph}}}}(\omega ) = {\text{Tr}}\left[ {{{\text{G}}^{{r}}}(\omega ){\varGamma _{\text{L}}}(\omega ){{\text{G}}^{{a}}}(\omega ){\varGamma _{\text{R}}}(\omega )} \right],$

(4)

其中, 超前格林函数 ${{\text{G}}^{{a}}}(\omega ) = {\left[ {{{\text{G}}^{{r}}}(\omega )} \right]^\dagger }$ . 通过NEGF方法, 中间散射区的声子局域态密度(LDOS)被定义为^[29,30]

${\rm{LDOS}}(\omega ) = \frac{{ - {{\rm{Im}}} \left[ {{{\text{G}}^{ {r}}}(\omega )} \right]}}{{\text{π }}},$

(5)

LDOS反映了体系中声子在实空间的分布, 可以更直观地得出声子的行为信息对热导的影响.

3. 分析与讨论

首先研究了纳米孔洞对石墨烯纳米带热输运的影响, 通过对比相同宽度的完美石墨烯纳米带和多孔石墨烯纳米带的热导, 发现多孔石墨烯纳米带的热导远低于完美的石墨烯纳米带, 本文中完美石墨烯纳米带的边缘都采用H原子钝化. 在图2(a), (c)中, 分别给出了NPAGNR-2, NPZGNR-2以及对应相同宽度的完美石墨烯纳米带 (AGNR-2, ZGNR-2) 的热导随温度变化的关系. 从图中可知, NPGNR-2以及对应相同宽度的石墨烯纳米带的热导都随温度的升高而升高, 这是因为随着温度升高, 更高频率的声子模式被激发, 参与热输运的声子模数增加. 同时, 还发现相同温度下, NPGNR-2的热导都要远低于GNR-2. 室温下NPAGNR-2的热导 $\kappa$ = 1.13 nW/K, NPZGNR-2的热导 $\kappa$ = 0.45 nW/K, 相较于完美石墨烯纳米带, 分别下降了66%和88%, 且差距会随着温度的升高进一步的增大. 为了解纳米孔洞导致热导降低的原因, 在图2(b), (d)中给出了对应结构的声子透射谱. 从图中可以看出, 石墨烯纳米带的透射谱曲线呈阶梯状, 是典型的量子化热导特征, 其每一支声子模的透射系数都为1. 且在低频 (0—100 cm^–1)下, 石墨烯纳米带的透射系数随着温度的升高, 呈台阶状升高^[27,28]. 虽然NPGNR-2同样属于理想晶格, 但是从图中可以看出, 低频下NPGNR-2的透射谱曲线并未像石墨烯纳米带一样呈完美的台阶状升高, 而是透射系数在整个频率范围内被抑制. 这是因为周期性纳米孔洞会阻止一定频率的声子传播, 从而造成声子局域. 周期性纳米孔洞不仅可以阻止高频光学声子模的传播, 也可以阻止低频声子模的传播^[19,30], 因此, 在低温(50 K)下NPGNR-2的热导就明显低于GNR-2. 随着温度升高, GNR-2更高频率的声子模被激发而参与导热, 而NPGNR-2中的声子模式被局域, 所以孔洞对热导的影响就更加明显^[41]. 另外, 从图2(b), (d)中可以看出, NPZGNR-2的透射曲线被抑制的程度更大, 且在整个频率范围内出现多个零输运点, 这说明孔洞对锯齿方向输运的声子模影响更大.

$图 2 AGNR-2和NPAGNR-2的热导和温度的关系(a), 以及对应的声子透射谱图(b); ZGNR-2和NPZGNR-2的热导和温度的关系(c), 以及对应的声子透射谱图(d)\r\nFig. 2. (a) Thermal conductance of AGNR-2 and NPAGNR-2 with different temperatures; (b) phonon transmission spectrumof AGNR-2 and NPAGNR-2; (c) thermal conductance of ZGNR-2 and NPZGNR-2 with different temperatures; (d) phonon transmission spectrum of ZGNR-2 and NPZGNR-2.$

图 2 AGNR-2和NPAGNR-2的热导和温度的关系(a), 以及对应的声子透射谱图(b); ZGNR-2和NPZGNR-2的热导和温度的关系(c), 以及对应的声子透射谱图(d)

Fig. 2. (a) Thermal conductance of AGNR-2 and NPAGNR-2 with different temperatures; (b) phonon transmission spectrumof AGNR-2 and NPAGNR-2; (c) thermal conductance of ZGNR-2 and NPZGNR-2 with different temperatures; (d) phonon transmission spectrum of ZGNR-2 and NPZGNR-2.

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为更直观地理解纳米孔洞对整个频率范围内声子模的影响, 图3给出了ZGNR-2和NPZGNR-2的3个典型的频率 (50 cm^–1, 500 cm^–1, 815 cm^–1)下的声子LDOS. 从图中可以看出, NPZGNR-2的孔洞破坏了完美石墨烯纳米带的声子输运通道, 使声子局域化, 而局域声子模在输运方向没有产生声子输运通道. 因此, 多孔石墨烯纳米带的热导远低于石墨烯纳米带是源于纳米孔洞的存在, 致使声子产生强烈的局域化, 尤其是低频声子的局域化导致的.

$图 3 ZGNR-2和NPZGNR-2在频率50 cm–1, 500 cm–1, 815 cm–1下的局域声子态密度图\r\nFig. 3. LDOS of ZGNR-2 and NPZGNR-2 at 50 cm–1, 500 cm–1 and 815 cm–1.$

图 3 ZGNR-2和NPZGNR-2在频率50 cm^–1, 500 cm^–1, 815 cm^–1下的局域声子态密度图

Fig. 3. LDOS of ZGNR-2 and NPZGNR-2 at 50 cm^–1, 500 cm^–1 and 815 cm^–1.

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此外, 在图4(a), (b) 中, 分别给出了宽度对NPAGNR和NPZGNR热导的影响. 图中可以看出, 两种纳米带的热导都随着宽度的增大而单调增大, 这一规律和先前类似研究中出现的规律是一致的^[42]. 这是因为宽度的增大使得原子总数目增加, 参与传热的声子模式也随之增加, 声子输运系数增大, 导致更高的热导.

$图 4 (a) NPAGNR-1, NPAGNR-2, NPAGNR-3的热导与温度的关系图; (b) NPZGNR-1, NPZGNR-2, NPZGNR-3的热导与温度的关系图\r\nFig. 4. (a) Thermal conductance of NPAGNR-1, NPAGNR-2 and NPAGNR-3 with different temperatures; (b) thermal conductance of NPZGNR-1, NPZGNR-2 and NPZGNR-3 with different temperatures.$

图 4 (a) NPAGNR-1, NPAGNR-2, NPAGNR-3的热导与温度的关系图; (b) NPZGNR-1, NPZGNR-2, NPZGNR-3的热导与温度的关系图

Fig. 4. (a) Thermal conductance of NPAGNR-1, NPAGNR-2 and NPAGNR-3 with different temperatures; (b) thermal conductance of NPZGNR-1, NPZGNR-2 and NPZGNR-3 with different temperatures.

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最后讨论不同手性对多孔石墨烯纳米带热输运性质的影响. 考虑到纳米尺度下尺寸效应的影响, 选取横向(与输运方向垂直)尺寸相近的NPAGNR-2和NPZGNR-3两种纳米带(纳米带宽度分别为3.2 nm和2.9 nm), 来比较锯齿和扶手椅两个方向上的热输运性质. 从图5(a) 可以看出, 相同温度下NPAGNR-2的热导要大于NPZGNR-3的, 室温下NPAGNR-2的热导是NPZGNR-3的2倍, 这说明多孔石墨烯纳米带的热输运具有显著的各向异性特征, 并且扶手椅型多孔石墨烯纳米带的热输运能力高于锯齿型多孔石墨烯纳米带. 而且这种各向异性特征随着温度升高而变得更加显著, 当温度升高至800 K时, NPAGNR-2的热导是NPAGNR-3的2.15倍. 为了解释多孔石墨烯纳米带显著的各向异性热输运特性, 图5(b) 给出了NPAGNR-2和NPZGNR-3的声子透射谱曲线, NPAGNR-2的透射系数整体高于NPZGNR-3的透射系数, 且未出现零输运频率点, 而NPZGNR-3的透射系数不仅整体低于NPAGNR-2, 而且从低频50 cm^–1到高频1000 cm^–1范围内出现多个零输运点, 从而导致在锯齿型方向上多孔石墨烯纳米带具有较低的热导.

$图 5 AGNR-2和NPZGNR-3的热导与温度的关系图(a) 以及对应的声子透射谱图(b)\r\nFig. 5. (a) Thermal conductance of NPAGNR-2 and NPZGNR-3 with different temperatures; (b) the corresponding phonon transmission spectrum.$

图 5 AGNR-2和NPZGNR-3的热导与温度的关系图(a) 以及对应的声子透射谱图(b)

Fig. 5. (a) Thermal conductance of NPAGNR-2 and NPZGNR-3 with different temperatures; (b) the corresponding phonon transmission spectrum.

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为了能更直观地理解多孔石墨烯纳米带热输运的各向异性特性, 从NPZGNR-3的多个零输运点中挑选了低、中、高频零输运点, 画出了NPAGNR-2和NPZGNR-3在这3个频率下的声子局域态密度. 如图6所示, 在频率为80 cm^–1, 315 cm^–1, 725 cm^–1下, NPAGNR-2的声子分布虽然存在局域现象, 但是在输运方向仍形成了一些扩展的输运通道. 然而, 在这3个频率点下, NPZGNR-3的声子分布严重局域化, 未形成扩展的声子输运通道. 所以, 扶手椅型多孔石墨烯纳米带的热输运能力强于锯齿型多孔石墨烯纳米带是由于锯齿型多孔石墨烯纳米带的声子局域性明显强于扶手椅型多孔石墨烯纳米带导致的, 并且当温度持续升高时, 扶手椅型多孔石墨烯纳米带中更多的声子模被激发参与导热, 而锯齿型多孔石墨烯纳米带中大部分声子完全局域, 对热导没有贡献.

$图 6 NPAGNR-2和NPZGNR-3在频率80 cm–1, 315 cm–1, 725 cm–1下的声子局域态密度\r\nFig. 6. LDOS of NPAGNR-2 and NPZGNR-3 at 80 cm–1, 315 cm–1 and 725 cm–1.$

图 6 NPAGNR-2和NPZGNR-3在频率80 cm^–1, 315 cm^–1, 725 cm^–1下的声子局域态密度

Fig. 6. LDOS of NPAGNR-2 and NPZGNR-3 at 80 cm^–1, 315 cm^–1 and 725 cm^–1.

下载: 全尺寸图片幻灯片

4. 结　论

利用非平衡格林函数方法研究了纳米孔洞的引入、宽度以及手性对多孔石墨烯纳米带热输运性质的影响. 发现由于纳米空洞的引入, 多孔石墨烯纳米带的热导远低于石墨烯纳米带的热导, 通过分析声子透射谱和声子局域态密度发现这是由于纳米孔洞会阻止部分频率的声子传播, 引起声子局域, 从而大幅度降低热导. 另外, 随着纳米带宽度的增大, 原子数目增加, 使得更多的声子模参与导热, 导致扶手椅型多孔石墨烯纳米带和锯齿型多孔石墨烯纳米带的热导都随着宽度的增大而单调增大. 最后, 通过比较NPAGNR-2和NPZGNR-3的热导, 发现扶手椅型多孔石墨烯纳米带的热导是锯齿型多孔石墨烯纳米带热导的2倍以上, 表明多孔石墨烯纳米带的热输运具有显著的各向异性特性. 通过分析声子透射谱和声子局域态密度可知: 在锯齿型方向, 整个频率范围内, 声子局域化更为严重, 导致更低的热导.

References

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其他类型引用(4)

图 1 (a) 多孔石墨烯结构图, 红色代表C原子, 白色代表H原子; (b) 纳米带NPZGNR-1和NPZGNR-2的结构图; (c) 纳米带NPAGNR-1和NPAGNR-2的结构图

Figure 1. (a) Structures of nano-porous graphene, red atoms represent C, white atoms represent H; (b) structures of NPZGNR-1 and NPZGNR-2; (c) structures of NPAGNR-1 and NPAGNR-2.

DownLoad: Full-Size Img PowerPoint

图 2 AGNR-2和NPAGNR-2的热导和温度的关系(a), 以及对应的声子透射谱图(b); ZGNR-2和NPZGNR-2的热导和温度的关系(c), 以及对应的声子透射谱图(d)

Figure 2. (a) Thermal conductance of AGNR-2 and NPAGNR-2 with different temperatures; (b) phonon transmission spectrumof AGNR-2 and NPAGNR-2; (c) thermal conductance of ZGNR-2 and NPZGNR-2 with different temperatures; (d) phonon transmission spectrum of ZGNR-2 and NPZGNR-2.

DownLoad: Full-Size Img PowerPoint

图 3 ZGNR-2和NPZGNR-2在频率50 cm^–1, 500 cm^–1, 815 cm^–1下的局域声子态密度图

Figure 3. LDOS of ZGNR-2 and NPZGNR-2 at 50 cm^–1, 500 cm^–1 and 815 cm^–1.

DownLoad: Full-Size Img PowerPoint

图 4 (a) NPAGNR-1, NPAGNR-2, NPAGNR-3的热导与温度的关系图; (b) NPZGNR-1, NPZGNR-2, NPZGNR-3的热导与温度的关系图

Figure 4. (a) Thermal conductance of NPAGNR-1, NPAGNR-2 and NPAGNR-3 with different temperatures; (b) thermal conductance of NPZGNR-1, NPZGNR-2 and NPZGNR-3 with different temperatures.

DownLoad: Full-Size Img PowerPoint

图 5 AGNR-2和NPZGNR-3的热导与温度的关系图(a) 以及对应的声子透射谱图(b)

Figure 5. (a) Thermal conductance of NPAGNR-2 and NPZGNR-3 with different temperatures; (b) the corresponding phonon transmission spectrum.

DownLoad: Full-Size Img PowerPoint

图 6 NPAGNR-2和NPZGNR-3在频率80 cm^–1, 315 cm^–1, 725 cm^–1下的声子局域态密度

Figure 6. LDOS of NPAGNR-2 and NPZGNR-3 at 80 cm^–1, 315 cm^–1 and 725 cm^–1.

DownLoad: Full-Size Img PowerPoint

[1]	Balandin A A, Ghosh S, Bao W Z, Calizo I, Teweldebrhan D, Miao F, Lau C N 2008 Nano Lett. 8 902 Google Scholar
[2]	Xu X, Pereira L F, Wang Y, Wu J, Zhang K, Zhao X, Bae S, Tinh Bui C, Xie R, Thong J T, Hong B H, Loh K P, Donadio D, Li B, Ozyilmaz B 2014 Nat. Commun. 5 3689 Google Scholar
[3]	Gao H, Wang L, Zhao J, Ding F, Lu J 2011 J. Phys. Chem. C 115 3236 Google Scholar
[4]	Sławińska J, Zasada I, Klusek Z 2010 Phys. Rev. B 81 155433 Google Scholar
[5]	Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard K L, Hone J 2010 Nat. Nanotechnol. 5 722 Google Scholar
[6]	Zhang Y, Tang T T, Girit C, Hao Z, Martin M C, Zettl A, Crommie M F, Shen Y R, Wang F 2009 Nature 459 820 Google Scholar
[7]	Zhou S Y, Gweon G H, Fedorov A V, First P N, de Heer W A, Lee D H, Guinea F, Castro Neto A H, Lanzara A 2007 Nat. Mater. 6 770 Google Scholar
[8]	Giovannetti G, Khomyakov P A, Brocks G, Kelly P J, van den Brink J 2007 Phys. Rev. B 76 073103 Google Scholar
[9]	Ohta T, Bostwick A, Seyller T, Horn K, Rotenberg E 2006 Science 313 951 Google Scholar
[10]	Jeon K J, Lee Z, Pollak E, Moreschini L, Bostwick A, Park C M, Mendelsberg R, Radmilovic V, Kostecki R, Richardson T J, Rotenberg E 2011 ACS Nano. 5 1042 Google Scholar
[11]	Kaur S, Narang S B, Randhawa D K K 2017 J. Mater. Res. 32 1149 Google Scholar
[12]	Du L, Nguyen T N, Gilman A, Muniz A R, Maroudas D 2017 Phys. Rev. B 96 245422
[13]	Zhao Y, Yang L, Kong L, Nai M H, Liu D, Wu J, Liu Y, Chiam S Y, Chim W K, Lim C T, Li B, Thong J T L, Hippalgaonkar K 2017 Adv. Func. Mater. 27 1702824 Google Scholar
[14]	Sadeghzadeh S, Rezapour N 2016 Superlattice. Microst. 100 97 Google Scholar
[15]	Nemnes G A, Visan C, Manolescu A 2017 J. Mater. Chem. C 5 4435
[16]	Sadeghi H, Sangtarash S, Lambert C J 2015 Sci. Rep. 5 9514 Google Scholar
[17]	Hu S, An M, Yang N, Li B 2016 Nanotechnology 27 265702 Google Scholar
[18]	Baskin A, Kral P 2011 Sci. Rep. 1 36 Google Scholar
[19]	Hu S, Zhang Z, Jiang P, Ren W, Yu C, Shiomi J, Chen J 2019 Nanoscale 11 11839 Google Scholar
[20]	Xiao Y, Chen Q Y, Ma D K, Yang N, Hao Q 2019 arXiv preprint arXiv: 1910.04913
[21]	Moreno C, Vilas-Varela M, Kretz B, Garcia-Lekue A, Costache M V, Paradinas M, Panighel M, Ceballos G, Valenzuela S O, Peña D, Mugarza A 2018 Science 360 6385
[22]	Mortazavi B, Madjet M E, Shahrokhi M, Ahzi S, Zhuang X, Rabczuk T 2019 Carbon 147 377 Google Scholar
[23]	Hu S, Zhang Z, Jiang P, Chen J, Volz S, Nomura M, Li B 2018 J. Phys. Chem. Lett. 9 3959 Google Scholar
[24]	Feng T, Ruan X 2016 Carbon 101 107 Google Scholar
[25]	Singh D, Shukla V, Ahuja R 2020 Phys. Rev. B 102 075444 Google Scholar
[26]	陈晓彬, 段文晖 2015 物理学报 64 186302 Google Scholar Chen X B, Duan W H 2015 Acta Phys. Sin. 64 186302 Google Scholar
[27]	吴宇, 蔡绍洪, 邓明森, 孙光宇, 刘文江 2018 物理学报 67 026501 Google Scholar Wu Y, Cai S H, Deng M S, Sun G Y, Liu W J 2018 Acta Phys. Sin. 67 026501 Google Scholar
[28]	吴宇, 蔡绍洪, 邓明森, 孙光宇, 刘文江, 岑超 2017 物理学报 66 116501 Google Scholar Wu Y, Cai S H, Deng M S, Sun G Y, Liu W J, Cen C 2017 Acta Phys. Sin. 66 116501 Google Scholar
[29]	姚海峰, 谢月娥, 欧阳滔, 陈元平 2013 物理学报 62 068102 Google Scholar Yao H F, XieY E, Ouyang T, Chen Y P 2013 Acta Phys. Sin. 62 068102 Google Scholar
[30]	Zhou W X, Chen K Q 2015 Carbon 85 24 Google Scholar
[31]	Qian X, Zhou J, Chen G 2021 Nat. Mater. 20 1188 Google Scholar
[32]	Yamamoto T, Watanabe K 2006 Phys. Rev. Lett. 96 255503 Google Scholar
[33]	Mingo N, Yang L 2003 Phys. Rev. B 68 245406 Google Scholar
[34]	Sevinçli H, Sevik C, Çaın T, Cuniberti G 2013 Nature. Sci. Rep. 3 1228
[35]	Peng Y N, Yu J F, Cao X H, Wu D, Jia P Z, Zhou W X, Chen K Q 2020 Physica E:Low-Dimens. Syst. Nanostruct. 122 114160 Google Scholar
[36]	Gale J. D. 1997 J. Chem. Soc. , Faraday Trans. 93 629 Google Scholar
[37]	Lu Y, Guo J 2012 Appl. Phys. Lett. 101 043112 Google Scholar
[38]	Lindsay L, Broido D A 2010 Phys. Rev. B 81 205441 Google Scholar
[39]	Khare R, Mielke S L, Paci J T, Zhang S, Ballarini R, Schatz G C, Belytschko T 2007 Phys. Rev. B 75 075412 Google Scholar
[40]	Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni B, Sinnott S B 2002 J. Phys.: Condens. Matter 14 783 Google Scholar
[41]	Chen X K, Hu X Y, Jia P, Xie Z X, Liu J 2021 Int. J. Mech. Sci. 206 106576 Google Scholar
[42]	Li D, Wu Y, Kim P, Shi L, Yang P, Majumdar A 2003 Appl. Phys. Lett. 83 2934 Google Scholar

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