石墨烯纳米带

石墨烯纳米带是指大概宽度小于50 nm的石墨烯条带。其理论模型最初于1996年提出^[1]^[2]^[3]。

电子结构

為了要賦予單層石墨烯某種電性，會按照特定樣式切割石墨烯，形成石墨烯纳米带（Graphene nanoribbon）。切開的邊緣形狀可以分為鋸齒形和扶手椅形。採用緊束縛近似模型做出的計算，預測鋸齒形具有金屬鍵性質，又預測扶手椅形具有金屬鍵性質或半導體性質；到底是哪種性質，要依寬度而定^[4]。可是，近來根據密度泛函理論計算得到的結果，顯示出扶手椅形具有半導體性質，其能隙與納米帶帶寬成反比^[5]。實驗結果確實地展示出，隨著納米帶帶寬減小，能隙會增大^[6]。但是，直至2009年, 尚沒有任何測量能隙的實驗試著辨識精確邊緣結構。通過施加外磁場，石墨烯納米帶的光學響應也可以調整至太赫茲頻域 ^[7]。

石墨烯納米帶的結構具有高電導率、高熱導率、低雜訊，這些優良品質促使石墨烯納米帶成為積體電路互連材料的另一種選擇，有可能替代銅金屬。有些研究者試著用石墨烯納米帶來製成量子點，他們在納米帶的某些特定位置改變寬度，形成量子禁閉（quantum confinement）^[8]。

石墨烯納米帶的低维結構具有非常重要的光电性能：粒子數反轉和寬帶光增益。這些優良品質促使石墨烯納米帶放在微腔或纳米腔体中形成激光器 ^[9] 和放大器。根据2012年10月的一份研究表明有些研究者試著将石墨烯納米帶应用于光通信系统，发展石墨烯納米激光器 ^[10]。

光學性質

最早的石墨烯奈米帶光學性質的數值結果是林與徐於2000年預測^[4]^[11]。扶手椅和鋸齒形邊緣的石墨烯奈米帶中光學躍遷具有的不同選擇規則。這些結果在2007年得到了Hsu和Reichl的研究支持，其對鋸齒形石墨烯奈米帶與單壁扶手椅碳奈米管進行比較研究^[12]。結果表明，鋸齒形石墨烯奈米帶的選擇規則與碳奈米管的選擇規則不同，鋸齒帶的本徵態可分為對稱或反對稱。此外，也預測邊緣態應該在鋸齒形石墨烯奈米帶的光學吸收中發揮重要作用。邊緣態和體態之間的光學躍遷應造成低能量區域的吸收光譜 (<3 eV)。於2011年，對光學躍遷的特性出現理論與數值上的分析推導，並明確了選擇規則^[13]^[14]^[4]，當入射光的極化方向平行於鋸齒形石墨烯奈米帶時，其光學選擇規則是 $\Delta J=J_{2}-J_{1}$ 要是奇數，其中 $J_{1}$ 和 $J_{2}$ 為能帶編號。而對於極化方向垂直於鋸齒形石墨烯納米帶時， $\Delta J=J_{2}-J_{1}$ 是偶數。

對於扶手椅石墨烯奈米帶，當入射光的極化方向平行於扶手椅石墨烯奈米帶時，其光學選擇規則為 $\Delta J=J_{2}-J_{1}=0$ ^[13]^[14]^[4]。類似於碳奈米管中的電子躍遷，扶手椅石墨烯奈米帶禁止子帶間躍遷。儘管單壁扶手椅碳納奈管和鋸齒形石墨烯奈米帶的選擇規則不同，但吸收峰間的隱藏相關性已被預測^[15]。管晶胞中的原子數 $N_{t}$ 與鋸齒形帶晶胞中的原子數 $N_{r}$ 有如下的條件: $N_{t}=2N_{r}+4$ (這就是所謂的匹配條件週期性和硬壁邊界條件)。在最近鄰緊束縛模型的中獲得的這些結果已被考慮到交換和相關效應的第一性原理密度泛函理論計算所支持^[16]。

包含了準粒子校正和多體效應的第一原理計算研究了石墨烯基材料的電子和光學特性^[17]。通過GW計算，可以準確研究石墨烯基材料的特性，包括石墨烯奈米帶^[18]^[19]^[20]、邊緣和表面功能化的扶手椅石墨烯奈米帶^[21]以及扶手椅石墨烯奈米帶中的縮放特性^[22]。

參見

参考文献

[1]
Fujita M., Wakabayashi K., Nakada K. and Kusakabe K. Peculiar Localized State at Zigzag Graphite Edge. Journal of the Physics Society Japan. 1996, 65 (7): 1920. Bibcode:1996JPSJ...65.1920F. doi:10.1143/JPSJ.65.1920.
[2]
Nakada K., Fujita M., Dresselhaus G. and Dresselhaus M.S. Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Physical Review B. 1996, 54 (24): 17954. Bibcode:1996PhRvB..5417954N. doi:10.1103/PhysRevB.54.17954.
[3]
Wakabayashi K., Fujita M., Ajiki H. and Sigrist M. Electronic and magnetic properties of nanographite ribbons. Physical Review B. 1999, 59 (12): 8271. Bibcode:1999PhRvB..59.8271W. arXiv:cond-mat/9809260 . doi:10.1103/PhysRevB.59.8271.
[4]
Chung, H. C.; Chang, C. P.; Lin, C. Y.; Lin, M. F. Electronic and optical properties of graphene nanoribbons in external fields. Physical Chemistry Chemical Physics. 2016, 18 (11): 7573–7616. doi:10.1039/C5CP06533J.
[5]
Barone, V., Hod, O., and Scuseria, G. E. Electronic Structure and Stability of Semiconducting Graphene Nanoribbons. Nano Lett. 2006, 6 (12): 2748. PMID 17163699. doi:10.1021/nl0617033.
[6]
Han., M.Y., Özyilmaz, B., Zhang, Y., and Kim, P. Energy Band-Gap Engineering of Graphene Nanoribbons. Phys. Rev. Lett. 2007, 98: 206805. doi:10.1103/PhysRevLett.98.206805.
[7]
Junfeng Liu, A. R. Wright, Chao Zhang, and Zhongshui Ma. Strong terahertz conductance of graphene nanoribbons under a magnetic field. Appl Phys Lett. 29 July 2008, 93: 041106–041110. doi:10.1063/1.2964093.
[8]
Wang, Z. F., Shi, Q. W., Li, Q., Wang, X., Hou, J. G., Zheng, H.; et al. Z-shaped graphene nanoribbon quantum dot device. Applied Physics Letters. 2007, 91: 053109. doi:10.1063/1.2761266.
[9]
Shan, G.C.; et al. Nanolaser with a Single-Graphene-Nanoribbon in a Microcavity. Journal of Nanoelectronics and Optoelectronics. 2011, 6: 138-143. doi:10.1166/jno.2011.1148.
[10]
Shan, G.C.,Shek, C.H.,. Modeling an Electrically Driven Graphene-Nanoribbon Laser for Optical Interconnects. IEEE Conference. 2012. doi:10.1109/PGC.2012.6458072.
[11]
Lin, Ming-Fa; Shyu, Feng-Lin. Optical Properties of Nanographite Ribbons. Journal of the Physical Society of Japan. 2000-11-15, 69 (11): 3529–3532. doi:10.1143/JPSJ.69.3529.
[12]
Hsu, Han; Reichl, L. E. Selection rule for the optical absorption of graphene nanoribbons. Physical Review B. 2007-07-19, 76 (4): 045418. doi:10.1103/PhysRevB.76.045418.
[13]
Chung, H. C.; Lee, M. H.; Chang, C. P.; Lin, M. F. Exploration of edge-dependent optical selection rules for graphene nanoribbons. Optics Express. 2011, 19 (23): 23350–63. Bibcode:2011OExpr..1923350C. PMID 22109212. arXiv:1104.2688 . doi:10.1364/OE.19.023350.
[14]
Sasaki, K.-I.; Kato, K.; Tokura, Y.; Oguri, K.; Sogawa, T. Theory of optical transitions in graphene nanoribbons. Phys. Rev. B. 2011, 84 (8): 085458. Bibcode:2011PhRvB..84h5458S. S2CID 119091338. arXiv:1107.0795 . doi:10.1103/PhysRevB.84.085458.
[15]
Saroka, V. A.; Shuba, M. V.; Portnoi, M. E. Optical selection rules of zigzag graphene nanoribbons. Phys. Rev. B. 2017, 95 (15): 155438. Bibcode:2017PhRvB..95o5438S. arXiv:1705.00757 . doi:10.1103/PhysRevB.95.155438.
[16]
Payod, R.B.; Grassano, D.; Santos, G.N.C.; Levshov, D.I.; Pulci, O.; Saroka, V. A. 2N+4-rule and an atlas of bulk optical resonances of zigzag graphene nanoribbons. Nat. Commun. 2020, 11 (1): 82. Bibcode:2020NatCo..11...82P. PMC 6941967 . PMID 31900390. doi:10.1038/s41467-019-13728-8 .
[17]
Onida, Giovanni; Rubio, Angel. Electronic excitations: Density-functional versus many-body Green's-function approaches. Rev. Mod. Phys. 2002, 74 (2): 601. Bibcode:2002RvMP...74..601O. doi:10.1103/RevModPhys.74.601. hdl:10261/98472 .
[18]
Prezzi, Deborah; Varsano, Daniele; Ruini, Alice; Marini, Andrea; Molinari, Elisa. Optical properties of graphene nanoribbons: The role of many-body effects. Physical Review B. 2008-01-07, 77 (4): 041404. doi:10.1103/PhysRevB.77.041404.
[19]
Yang, Li; Cohen, Marvin L.; Louie, Steven G. Magnetic Edge-State Excitons in Zigzag Graphene Nanoribbons. Physical Review Letters. 2008-10-28, 101 (18): 186401. doi:10.1103/PhysRevLett.101.186401.
[20]
Yang, Li; Cohen, Marvin L.; Louie, Steven G. Excitonic Effects in the Optical Spectra of Graphene Nanoribbons. Nano Letters. 2007-10-01, 7 (10): 3112–3115. doi:10.1021/nl0716404.
[21]
Zhu, Xi; Su, Haibin. Excitons of Edge and Surface Functionalized Graphene Nanoribbons. J. Phys. Chem. C. 2010, 114 (41): 17257. doi:10.1021/jp102341b.
[22]
Zhu, Xi; Su, Haibin. Scaling of Excitons in Graphene Nanoribbons with Armchair Shaped Edges. Journal of Physical Chemistry A. 2011, 115 (43): 11998–12003. Bibcode:2011JPCA..11511998Z. PMID 21939213. doi:10.1021/jp202787h.

外部連結

Graphene nanoribbons on arxiv.org

[Edgestate-1] [1]
Fujita M., Wakabayashi K., Nakada K. and Kusakabe K. Peculiar Localized State at Zigzag Graphite Edge. Journal of the Physics Society Japan. 1996, 65 (7): 1920. Bibcode:1996JPSJ...65.1920F. doi:10.1143/JPSJ.65.1920.

[Estatenakada-2] [2]
Nakada K., Fujita M., Dresselhaus G. and Dresselhaus M.S. Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Physical Review B. 1996, 54 (24): 17954. Bibcode:1996PhRvB..5417954N. doi:10.1103/PhysRevB.54.17954.

[MagPro-3] [3]
Wakabayashi K., Fujita M., Ajiki H. and Sigrist M. Electronic and magnetic properties of nanographite ribbons. Physical Review B. 1999, 59 (12): 8271. Bibcode:1999PhRvB..59.8271W. arXiv:cond-mat/9809260 . doi:10.1103/PhysRevB.59.8271.

[ReviewPCCPChung2016-4] [4]
Chung, H. C.; Chang, C. P.; Lin, C. Y.; Lin, M. F. Electronic and optical properties of graphene nanoribbons in external fields. Physical Chemistry Chemical Physics. 2016, 18 (11): 7573–7616. doi:10.1039/C5CP06533J.

[ArmchRibb-5] [5]
Barone, V., Hod, O., and Scuseria, G. E. Electronic Structure and Stability of Semiconducting Graphene Nanoribbons. Nano Lett. 2006, 6 (12): 2748. PMID 17163699. doi:10.1021/nl0617033.

[EgEngGNR-6] [6]
Han., M.Y., Özyilmaz, B., Zhang, Y., and Kim, P. Energy Band-Gap Engineering of Graphene Nanoribbons. Phys. Rev. Lett. 2007, 98: 206805. doi:10.1103/PhysRevLett.98.206805.

[7] [7]
Junfeng Liu, A. R. Wright, Chao Zhang, and Zhongshui Ma. Strong terahertz conductance of graphene nanoribbons under a magnetic field. Appl Phys Lett. 29 July 2008, 93: 041106–041110. doi:10.1063/1.2964093.

[8] [8]
Wang, Z. F., Shi, Q. W., Li, Q., Wang, X., Hou, J. G., Zheng, H.; et al. Z-shaped graphene nanoribbon quantum dot device. Applied Physics Letters. 2007, 91: 053109. doi:10.1063/1.2761266.

[9] [9]
Shan, G.C.; et al. Nanolaser with a Single-Graphene-Nanoribbon in a Microcavity. Journal of Nanoelectronics and Optoelectronics. 2011, 6: 138-143. doi:10.1166/jno.2011.1148.

[10] [10]
Shan, G.C.,Shek, C.H.,. Modeling an Electrically Driven Graphene-Nanoribbon Laser for Optical Interconnects. IEEE Conference. 2012. doi:10.1109/PGC.2012.6458072.

[11] [11]
Lin, Ming-Fa; Shyu, Feng-Lin. Optical Properties of Nanographite Ribbons. Journal of the Physical Society of Japan. 2000-11-15, 69 (11): 3529–3532. doi:10.1143/JPSJ.69.3529.

[12] [12]
Hsu, Han; Reichl, L. E. Selection rule for the optical absorption of graphene nanoribbons. Physical Review B. 2007-07-19, 76 (4): 045418. doi:10.1103/PhysRevB.76.045418.

[#1-13] [13]
Chung, H. C.; Lee, M. H.; Chang, C. P.; Lin, M. F. Exploration of edge-dependent optical selection rules for graphene nanoribbons. Optics Express. 2011, 19 (23): 23350–63. Bibcode:2011OExpr..1923350C. PMID 22109212. arXiv:1104.2688 . doi:10.1364/OE.19.023350.

[#2-14] [14]
Sasaki, K.-I.; Kato, K.; Tokura, Y.; Oguri, K.; Sogawa, T. Theory of optical transitions in graphene nanoribbons. Phys. Rev. B. 2011, 84 (8): 085458. Bibcode:2011PhRvB..84h5458S. S2CID 119091338. arXiv:1107.0795 . doi:10.1103/PhysRevB.84.085458.

[15] [15]
Saroka, V. A.; Shuba, M. V.; Portnoi, M. E. Optical selection rules of zigzag graphene nanoribbons. Phys. Rev. B. 2017, 95 (15): 155438. Bibcode:2017PhRvB..95o5438S. arXiv:1705.00757 . doi:10.1103/PhysRevB.95.155438.

[16] [16]
Payod, R.B.; Grassano, D.; Santos, G.N.C.; Levshov, D.I.; Pulci, O.; Saroka, V. A. 2N+4-rule and an atlas of bulk optical resonances of zigzag graphene nanoribbons. Nat. Commun. 2020, 11 (1): 82. Bibcode:2020NatCo..11...82P. PMC 6941967 . PMID 31900390. doi:10.1038/s41467-019-13728-8 .

[17] [17]
Onida, Giovanni; Rubio, Angel. Electronic excitations: Density-functional versus many-body Green's-function approaches. Rev. Mod. Phys. 2002, 74 (2): 601. Bibcode:2002RvMP...74..601O. doi:10.1103/RevModPhys.74.601. hdl:10261/98472 .

[18] [18]
Prezzi, Deborah; Varsano, Daniele; Ruini, Alice; Marini, Andrea; Molinari, Elisa. Optical properties of graphene nanoribbons: The role of many-body effects. Physical Review B. 2008-01-07, 77 (4): 041404. doi:10.1103/PhysRevB.77.041404.

[19] [19]
Yang, Li; Cohen, Marvin L.; Louie, Steven G. Magnetic Edge-State Excitons in Zigzag Graphene Nanoribbons. Physical Review Letters. 2008-10-28, 101 (18): 186401. doi:10.1103/PhysRevLett.101.186401.

[20] [20]
Yang, Li; Cohen, Marvin L.; Louie, Steven G. Excitonic Effects in the Optical Spectra of Graphene Nanoribbons. Nano Letters. 2007-10-01, 7 (10): 3112–3115. doi:10.1021/nl0716404.

[21] [21]
Zhu, Xi; Su, Haibin. Excitons of Edge and Surface Functionalized Graphene Nanoribbons. J. Phys. Chem. C. 2010, 114 (41): 17257. doi:10.1021/jp102341b.

[22] [22]
Zhu, Xi; Su, Haibin. Scaling of Excitons in Graphene Nanoribbons with Armchair Shaped Edges. Journal of Physical Chemistry A. 2011, 115 (43): 11998–12003. Bibcode:2011JPCA..11511998Z. PMID 21939213. doi:10.1021/jp202787h.

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