石墨纳米条带的输运性质
摘要
本论文采用紧束缚近似方法(tight-banding)和Landau-Buttiker公式计算了石墨条带的电子输运性质。研究了门压、磁场对不同边界石墨纳米条带体系的电导性质的影响。并用Dirac方程求解了石墨纳米条带的波函数。
首先,从理论角度用紧束缚近似方法和Dirac方程求解了石墨二维无限大体系和armchair边界与zigzag边界下无限长纳米条带的色散关系和波函数。通过比较这两种近似方法计算的结果,讨论了两种方法各自的优缺点。
其次,研究了门压和垂直磁场对电子输运的影响。研究发现,在门压的作用下,石墨条带的能带会发生移动,但碳原子之间的隧穿积分并不会发生变化。而在垂直磁场作用下,能带和隧穿积分都会改变,隧穿积分将由无磁场作用时的t_(ij)变为,其中A为矢势,积分沿着所选六个碳原子组成的蜂巢形区域的输运路径进行。
最后,我们利用以上理论讨论了不同边界的石墨纳米条带在门压和磁场作用下电导谱图的变化。通过理沦推导和数值计算对石墨纳米条带的输运性质有了更深刻的认识。更重要的是,本文研究得到的一些定性的结果将为石墨单层(Graphene)在纳米器件上的应用提供理论指导。
Carbon is the most common element in nature. In recent years,the discover of C60 and successful preparation of carbon nanotubes exploited the new configuration of carbon, and the findings of magnetism and superconductivity for carbon compound have attracted much interests for this common element and even some people give so-called“carbon electronics”. Scientists are still investigating the new configuration of carbon, so that much more new phenomena can be found. However, conventional configuration of carbon is often ignored. Graphene, which is one of carbon configurations and belongs to carbon electronics field, is layer structure and each carbon atom of interlayer is sp2 hybridized orbital with neighboring three carbon atoms form three equivalent distanceσbond, therefore conform carbon atomic infinite planar layer, and also, every carbon atom is vertical with this pz orbital of planar and they overlapped each other to form delocalizationπbond. Electron presents metallic conductivity property in inner planar, whereas, high resistance when it is vertical with the planar. Thus, graphene is considered planar conductor and belongs to conventional semi-metallic materials whose transportation character is intervenient of metal and semiconductor.
Nano graphene layer may be a new generation electrics basal materials based on design and preparation of graphene nanoribbons. These basic functional apparatus consists of rectifier, PN junctions, field effect transistor and so on. We must make a systemic study for each kind of grapheme in order to observe most valuable physical phenomena, which could make the application of graphene on nano-apparatus to be realized. The theoretic investigation which is not restricted by laboratorial techniques, provided some instructional information for experimental study. It can afford credible physical law for establishing and enriching principle of relativity condensed matter physics basal theory system. It provided valuable theoretic information for apparatus design of orgin from granpene nano-electrics. Graphene open a wide filed for future application of electrical apparatus, such as ballistic transistor. Planar graphene material could become basal material of nano-electrics since electron of graphene keep high mobility even at room temperature and high carrier concentration and it could be cut into diversified nanostructures.
Different from other semiconductor system, description of graphene should adopt Dirac equation which was used to describe relativistic particles, not Schrodinger equation which was used to describe non- relativism. Equations in this system consists of two kinds of non- equivalent A atom and B atom, form two energy band and intersect in Brillouin zone boundary neighborhood and come into being conical type energy spectroscopy. Thus the quasiparticle of graphene existed a linear dispersive connection. It is just this linear spectrum that we may think that quasiparticle character of graphene is different from our familiar particle properties of metallic and semiconductor.
Transport properties of electrons are usually calculated by density functional theory (DFT) and nonequilibrium Green function. We simulate nanostructure system and electrical perproties of nano-apparatus and transport properties of quanta. Based on electronic calculations such as pseudopotential method and linear combination of atomic orbital, electronic transport properties of nano-device under external voltage and graphene under external magnetic field were disposed by using nonequilibrium Green’s function method and transport theory method. We will discuss the effect of size of boundary and device zone on conductivity.
In this thesis, electronic transport properties of graphene ribbons were calculated through tight-banding calculation and Landau-Buttiker formula. Exoteric factors such as voltage, magnetic field and their corporate influence on graphene ribbons conductivity were investigated. Wave function of nanoribbons was processed through Dirac equation.
First of all, we settled graphene planar infinity system, energy and wave function of armchair and zigzag bondary infinity nanoribbons. The merit and limitation of two kinds of methods were compared through the results of the two approximate calculations.
Secondary, electronic transport under voltage and vertical magnetic field was investigated through transport theory. Interestingly, energy band of graphene ribbons can move up and down, however, the tunnel integral among carbon atoms hasn’t changed under voltage influence. Under vertical magnetic field influence, energy band and tunnel integral are also changed and tunnel integral varied as exponential.
In the end, we discussed the conductivity spectrum change under the influence of graphene ribbons different bondary under the voltage and magnetic field through above theory. Through theory and numerical value calculation, we have a more clear understanding. More important result is that our theory studying will provide directive information for application of graphene on nano-device.
首先,从理论角度用紧束缚近似方法和Dirac方程求解了石墨二维无限大体系和armchair边界与zigzag边界下无限长纳米条带的色散关系和波函数。通过比较这两种近似方法计算的结果,讨论了两种方法各自的优缺点。
其次,研究了门压和垂直磁场对电子输运的影响。研究发现,在门压的作用下,石墨条带的能带会发生移动,但碳原子之间的隧穿积分并不会发生变化。而在垂直磁场作用下,能带和隧穿积分都会改变,隧穿积分将由无磁场作用时的t_(ij)变为,其中A为矢势,积分沿着所选六个碳原子组成的蜂巢形区域的输运路径进行。
最后,我们利用以上理论讨论了不同边界的石墨纳米条带在门压和磁场作用下电导谱图的变化。通过理沦推导和数值计算对石墨纳米条带的输运性质有了更深刻的认识。更重要的是,本文研究得到的一些定性的结果将为石墨单层(Graphene)在纳米器件上的应用提供理论指导。
Carbon is the most common element in nature. In recent years,the discover of C60 and successful preparation of carbon nanotubes exploited the new configuration of carbon, and the findings of magnetism and superconductivity for carbon compound have attracted much interests for this common element and even some people give so-called“carbon electronics”. Scientists are still investigating the new configuration of carbon, so that much more new phenomena can be found. However, conventional configuration of carbon is often ignored. Graphene, which is one of carbon configurations and belongs to carbon electronics field, is layer structure and each carbon atom of interlayer is sp2 hybridized orbital with neighboring three carbon atoms form three equivalent distanceσbond, therefore conform carbon atomic infinite planar layer, and also, every carbon atom is vertical with this pz orbital of planar and they overlapped each other to form delocalizationπbond. Electron presents metallic conductivity property in inner planar, whereas, high resistance when it is vertical with the planar. Thus, graphene is considered planar conductor and belongs to conventional semi-metallic materials whose transportation character is intervenient of metal and semiconductor.
Nano graphene layer may be a new generation electrics basal materials based on design and preparation of graphene nanoribbons. These basic functional apparatus consists of rectifier, PN junctions, field effect transistor and so on. We must make a systemic study for each kind of grapheme in order to observe most valuable physical phenomena, which could make the application of graphene on nano-apparatus to be realized. The theoretic investigation which is not restricted by laboratorial techniques, provided some instructional information for experimental study. It can afford credible physical law for establishing and enriching principle of relativity condensed matter physics basal theory system. It provided valuable theoretic information for apparatus design of orgin from granpene nano-electrics. Graphene open a wide filed for future application of electrical apparatus, such as ballistic transistor. Planar graphene material could become basal material of nano-electrics since electron of graphene keep high mobility even at room temperature and high carrier concentration and it could be cut into diversified nanostructures.
Different from other semiconductor system, description of graphene should adopt Dirac equation which was used to describe relativistic particles, not Schrodinger equation which was used to describe non- relativism. Equations in this system consists of two kinds of non- equivalent A atom and B atom, form two energy band and intersect in Brillouin zone boundary neighborhood and come into being conical type energy spectroscopy. Thus the quasiparticle of graphene existed a linear dispersive connection. It is just this linear spectrum that we may think that quasiparticle character of graphene is different from our familiar particle properties of metallic and semiconductor.
Transport properties of electrons are usually calculated by density functional theory (DFT) and nonequilibrium Green function. We simulate nanostructure system and electrical perproties of nano-apparatus and transport properties of quanta. Based on electronic calculations such as pseudopotential method and linear combination of atomic orbital, electronic transport properties of nano-device under external voltage and graphene under external magnetic field were disposed by using nonequilibrium Green’s function method and transport theory method. We will discuss the effect of size of boundary and device zone on conductivity.
In this thesis, electronic transport properties of graphene ribbons were calculated through tight-banding calculation and Landau-Buttiker formula. Exoteric factors such as voltage, magnetic field and their corporate influence on graphene ribbons conductivity were investigated. Wave function of nanoribbons was processed through Dirac equation.
First of all, we settled graphene planar infinity system, energy and wave function of armchair and zigzag bondary infinity nanoribbons. The merit and limitation of two kinds of methods were compared through the results of the two approximate calculations.
Secondary, electronic transport under voltage and vertical magnetic field was investigated through transport theory. Interestingly, energy band of graphene ribbons can move up and down, however, the tunnel integral among carbon atoms hasn’t changed under voltage influence. Under vertical magnetic field influence, energy band and tunnel integral are also changed and tunnel integral varied as exponential.
In the end, we discussed the conductivity spectrum change under the influence of graphene ribbons different bondary under the voltage and magnetic field through above theory. Through theory and numerical value calculation, we have a more clear understanding. More important result is that our theory studying will provide directive information for application of graphene on nano-device.
引文
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[52]MitsutakaFujata,Katsunori Wakabayashi,Kyoko Nakada and Koichi Kuskabe . Journal of Physical Society of Japan Vol.65,No.7,July,1996,pp.1920-1923.
[53]M. I. Katsnelson arXiv:cond-mat/0512337 v3 11 Jun 2006.
[54]Kikuo Harigaya Atsushi Yamashiro Yukihiro Shimoi Katsunori Wakabayashi.Current Applied Physics 4 (2004) 587–590.
[55] 量子力学导论,曾谨言.
[2]Novoselov, K. S., et al., Nature (2005) 438, 197.
[3]Zhang, Y., et al., Nature (2005) 438, 201.
[4]G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).
[5]Y. Zhang et al, Nature 438, 201 (2005).
[6]Claire Berger et al. Science, 312, 1191(2006) .
[7]Zheng Y. and Ando T. Phys. Rev. B 65, 245420 (2002).
[8]Gusynin V. P. and Sharapov S. G. Phys. Rev. Lett. 95, 146801 (2005).
[9]Novoselov K. S. et al Nature Phys. 2, 177 (2006).
[10]Y. Zhang et al. Phys. Rev. Lett. 96, 136806 (2006).
[11]Y.-W. Tan et al. cond-mat/0707.1807 (2007).
[12]Katsnelson M. I. et al Nature Phys. 2, 620 (2006).
[13]Milton Pereira, Jr., J., et al., Phys. Rev. B (2006) 74, 045424 .
[14] V. V. Cheianov and V. I. Fal’ko, Phys. Rev. B 74, 041403 (2006).
[15]Shon N. H. and Ando T. J. Phys. Soc. Jpn. 67, 2857 (1998) .
[16]Katsnelson M. I. Eur. Phys. J. B 51, 157 (2006)
[17]Morozov S. V. et al, Phys. Rev. Lett. 97, 016801 (2006).
[18]McCann E. et al Phys. Rev. Lett. 97, 146805 (2006) .
[19]Morpurgo A. F. and Guinea F. Phys. Rev. Lett. 97, 196804 (2006) .
[20]Khveshchenko D. V. Phys. Rev. Lett. 97, 036802 (2006).
[21]Berger C. et al, J. Phys. Chem. B 108, 19912 (2004) .
[22]Ohta T. et al, Science 313, 951 (2006).
[23]Brey L. and Fertig H. A. Phys. Rev. B 73, 235411(2006).
[24]Silvestrov P. G. and Efetov K. B. Phys. Rev. Lett. 98, 016802 (2007).
[25]Son Y. W. et al, Phys. Rev. Lett. 98, 97, 216803 (2006) .
[26]Areshkin D. A. et al, Nano Letters 7, 204 (2007).
[27]K. Nakada et al.,Phys. Rev. B54, 17954 (1996) .
[28]M. Fujita et al., J. Phys. Soc. Jpn., 65, 1920 (1996) .
[29]M. Ezawa, Phys. Rev. B 73, 045432 (2006).
[30]F. Munoz-Rojas et al., Phys. Rev. B. 74, 195417 (2006) .
[31]Zheng,HX. et al.Phys.Rev.B,75,165414(2007) .
[32]Wakabayashi.K. et al.Phys.Rev.B,59,8271(1999) .
[33]Ando,T. J.Phys.Soc.Jpn.74,777(2005) .
[34]Sasaki,K. et al. J.Phys.Soc.Jpn.75,074713(2006) .
[35] M.S. Dresselhaus, G. Dresselhaus, K. Sugihara, I.L. Spain, H.A. Goldberg, Graphite Fibers and Filaments, (Springer-Verlag, 1988).
[36]Sasaki,K. et al. Appl.Phys. Lett. 88,113110(2006) .
[37]Yousuke Kobayashi, Ken-ichi Fukui, Toshiaki Enoki, Koichi Kusakabe and Yutaka Kaburagi, Phys.Rev. B 71, 193406 (2005) .
[38]Tsuneya ANDO and Takeshi NAKANISHI1 Journal of the Physical Society of Japan Vol. 67, 1704-1713(1998) .
[39] L. Brey and H. A. Fertig, Phys. Rev. B 73, 195408 (2006) .
[40] J. Tworzyd lo, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker. arXiv:cond-mat/0603315 v3 10 Apr 2006.
[41] L. Brey and H. A. Fertig, Phys.Rev.B 75, 125434 (2007) .
[42] Chunxu Bai and Xiangdong Zhang, Phys.Rev.B 76, 075430 (2007) .
[43] N. M. R. Peres, A. H. Castro Neto, and F. Guinea Phys.Rev. B 73, 195411 (2006).
[44]N. M. R. Peres, A. H. CastroNeto, and F. Guinea Phys. Rev. B 73, 239902 (2006) .
[45] K. Wakabayashi and M. Sigrist, Phys. Rev. Lett.84, 3390.
[46] K. Wakabayashi, Phys. Rev. B64 125428 (2001) .
[47] S. Datta, Electronic Transport in Mesoscopic Systems(CambridgeUniversity Press, Cambridge, 1995).
[48] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).
[49] Y. Meir and N. S. Wingreen, Phys. Rev. Lett 68, 2512.
[50] .D.H.Lee and J.D.Joannnopulos,phys.Rev.B23 4988 (1981) .
[51]KATSUNORI WAKABAYASHI andTAKASHI AOKI International Journal of Modern Physics B, Vol. 16, No. 32 (2002) 4897-4909.
[52]MitsutakaFujata,Katsunori Wakabayashi,Kyoko Nakada and Koichi Kuskabe . Journal of Physical Society of Japan Vol.65,No.7,July,1996,pp.1920-1923.
[53]M. I. Katsnelson arXiv:cond-mat/0512337 v3 11 Jun 2006.
[54]Kikuo Harigaya Atsushi Yamashiro Yukihiro Shimoi Katsunori Wakabayashi.Current Applied Physics 4 (2004) 587–590.
[55] 量子力学导论,曾谨言.