单壁碳纳米管的金属—半导体相变
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摘要
自从Iijima1991年发现碳纳米管以来,由于其在电子器件中的广泛应用前景,理论和实验方面的研究都取得了快速的进展。就其结构而言,单壁碳纳米管(SWCNTs)可以看成由单层石墨(石墨烯)卷曲而成,其电子能带结构也可以从石墨烯电子能带在布里渊区折叠近似得到。考虑到石墨烯价带顶和导带底在Dirac点相接触,能隙为零,石墨烯具有金属导电性。当石墨烯卷曲成SWCNTs时,环状周期边界条件导致该方向上电子波矢的量子化。环向波矢通过Dirac点的单壁碳管为金属,否则为半导体。Saito由此总结出1/3单壁碳管为金属,2/3为半导体的简单分类。然而这种分类与实验测量结果不符,在管径很小的碳纳米管中差别尤其显著。考虑到碳管的曲率效应对碳原子π-σ轨道交叠积分的影响,名义上为金属的大多数SWCNTs确实能打开一定的能隙,但扶手椅型SWCNTs不受曲率的影响,仍然保持为金属。这个结论与新近的实验测量结果严重不符。扫描隧道显微镜电性测量表明扶手椅型SWCNTs不仅有能隙,而且能隙即使在外加磁场扫描情况下仍然存在。这种能隙被冠以“真实”能隙。显然这种“真实”能隙是不能由目前流行的物理机制加以解释的。首先,根据电子能带的折叠理论,即使环状边界条件在零磁场下不包含Dirac点为其量子化波矢,通过调节外加磁场附加的位相可以移动量子化波矢至Dirac点,从而促使能隙的消失,因而折叠理论不能解释“真实”能隙；其次,理论上预言扶手椅型碳纳米管的金属性不依赖曲率效应,也与实验严重不符。为此Deshpande等人认为“真实”能隙只能来源于电子的强关联效应。目前碳纳米管的强关联效应由于小量子系统的充电效应难以区分也没有证实。
本论文的研究内容就单层碳纳米管的“真实”能隙的产生机理展开讨论。虽然在强关联机理被提出来之前,各种物理机制已经得到了广泛的讨论,尤其是晶格畸变导致的Peierls相变和二聚化效应。例如,SWCNTs的Kekule碳管结构和各种碳管的形变模式被广为讨论,这些结构畸变模式也确实能打开能隙,但是却不能破坏碳管上的两类碳原子的化学等价性。这类能隙并不对应实验得到的“真实”能隙,因为磁场调节可以消除这样的能隙。为了得到“真实”能隙,必须破坏两类碳原子在空间结构上的等价性。碳纳米管的褶皱结构应该是一个很好的设想。褶皱模型虽然在Viet等人的唯象模型中曾经提到,但是这种褶皱结构是否在能量上稳定,褶皱的幅度与碳纳米管半径和手性之间的关系,尤其是电子能带并没有从第一性原理计算上加以证实。
本论文首先采用第一性原理计算方法分析了扶手椅型褶皱SWCNTs结构的稳定性。通过计算两类碳原子沿径向不同的褶皱度和键长下的碳纳米管的能量,证实碳纳米管的结构确实收敛到褶皱结构,从而证明了褶皱结构的稳定性。尤为重要的是,原先无褶皱的碳纳米管结构其实并不稳定,甚至都算不上亚稳态,在无褶皱和有褶皱碳纳米管结构间不存在势垒。褶皱结构的稳定性和褶皱度随碳纳米管半径减小而迅速增大,而当半径趋于无穷时,结构回归到没有褶皱的平面石墨烯。由于褶皱结构和曲率效应一起破坏了两类碳原子的结构对称性,导致了碳纳米管的金属半导体相变以及“真实”能隙的产生。根据第一性原理计算所得的结构和能带参数,我们进一步构造了褶皱碳纳米管的紧束缚哈密顿。在此基础上讨论了不同手性SWCNTs的二维和一维电子能带结构,具体分析了折叠模型,曲率模型以及褶皱模型下碳纳米管的能隙和“真实”能隙随纳米管半径的依赖关系。我们的结论是碳纳米管的能隙与碳管类型密切相关,但是“真实”能隙主要由半径和褶皱度决定,随着半径的增大迅速减小。
Since the carbon nanotube was discovered by Iijima in1991, tremendous progress-es have been made both experimentally and theoretically for its potential application in electronic devices. A single-wall carbon nanotube can be seen as rolled up from a graphene sheet, and this construction suggests that its electronic properties can be deduced from that of graphene under the simple zone-folding scheme. For an ideal graphene, the valence and conduction bands touch each other at Dirac points and form a gapless metal. When folding a graphene sheet into a carbon nanotube, wavevectors along the circumferential direction are quantized due to the circumferential boundary conditions. If the wavevectors pass through the Dirac points, a SWCNT is metallic, otherwise it is a semiconductor, according to which, Saito et al. arrived at a simple categorization that one-third SWCNTs are metal, and the other two-third are semicon-ductors. This categorization does not agree with the recent experiments well, in par-ticular for carbon nanotubes with small diameters. Considering the hopping integrals corrections induced by π-σ hybridization from the curvature effect on the tube, band-gaps open for most nominally metallic SWCNTs, while armchair SWCNTs remain metallic and are protected from such a curvature effect. But this conclusion contradict-s with recent experiment seriously. Band-gaps were observed for armchair SWCNTs by scanning tunneling microscopy, and those band-gaps would not vanish as external magnetic field varies. The band-gaps are call true band-gaps, and it is obviously this true band-gaps can not be explained by the popular physics mechanism. Firstly, if the Dirac points are not in the permitted quantized wavevectors by simply zone-folding method, the Dirac points would move into the quantized wavevectors by a phase sup-plied by an external magnetic field, and the band-gaps would vanish. The zone-folding theory cannot explain the true band-gaps. Secondly, the zero band-gaps for armchair SWCNTs are supposedly protected from the curvature effect, which also differs from the experiment. Deshpande et al. ascribed the true band-gaps to the strongly correlat-ed electronic correlations. At present, experimental observation of strong correlations remains a challenge since their signature is masked by charging effect for the small quantum system of carbon nanotubes.
We investigate the true band-gaps opening mechanism for SWCNTs in this thesis. Before the strong correlated model was proposed, other possibilities were explored ex-tensively, especially Peierls transition and dimerization induced by lattice deformation such as the Kekule structure. Those deformation do create band-gaps, but do not create true band-gaps, for that those deformation cannot destroy the chemical identicalness of two different types of carbon atoms on the tube, and with the applied magnetic field, the band-gaps will vanish. To have the true band-gaps, the structural symmetry of the two types of carbons should be broken. Corrugated structure of carbon nanotube is a good candidate. Such a model has been proposed for carbon nanotubes in a phe-nomenological model by Viet et al., but the stability of corrugated structures has not been confirmed by ab initio calculations, and the relationship among the band-gaps, corrugation length, radius, and chirality of SWCNTs has not been investigated up to now.
In this thesis, with ab initio calculations, firstly, we analyze the stability of the cor-rugated structures of armchair SWCNTs. The total energies of SWCNTs are calculated for different corrugation length and bond length with two types of carbon atoms dis-placed from each other radially. The structure do converge to corrugated structure, which verifies the stability of corrugated structure. It is especially important that the noncorrugated SWCNT structure is not stable, not even metastable, since no potential barrier exists between the structures with and without corrugation. The stability and corrugation length increase rapidly as tube's radius decreases. When radius approaches to infinity, corrugation vanishes for flat graphene. Combined with curvature effect, the corrugation breaks the local symmetry of two types of carbon atoms, which induces the metal-semiconductor transition and true band gaps emerge for carbon nanotubes. With the corrugation length and energies from the first-principles calculation, the tight- binding Hamiltonian is constructed for corrugated SWCNTs. The two dimensional and one dimensional electronic band structures are discussed for various types of SWCNTs. We analyze the band-gaps and true band gaps for zone-folding model, pure curvature model and corrugation model, respectively. Our conclusion is that the band-gaps sen-sitively depend on the chirality, while the true band-gaps, independent of SWCNTs' chirality, mainly depend on the corrugation and curvature effect, and increase rapidly with the decease of radius.
本论文的研究内容就单层碳纳米管的“真实”能隙的产生机理展开讨论。虽然在强关联机理被提出来之前,各种物理机制已经得到了广泛的讨论,尤其是晶格畸变导致的Peierls相变和二聚化效应。例如,SWCNTs的Kekule碳管结构和各种碳管的形变模式被广为讨论,这些结构畸变模式也确实能打开能隙,但是却不能破坏碳管上的两类碳原子的化学等价性。这类能隙并不对应实验得到的“真实”能隙,因为磁场调节可以消除这样的能隙。为了得到“真实”能隙,必须破坏两类碳原子在空间结构上的等价性。碳纳米管的褶皱结构应该是一个很好的设想。褶皱模型虽然在Viet等人的唯象模型中曾经提到,但是这种褶皱结构是否在能量上稳定,褶皱的幅度与碳纳米管半径和手性之间的关系,尤其是电子能带并没有从第一性原理计算上加以证实。
本论文首先采用第一性原理计算方法分析了扶手椅型褶皱SWCNTs结构的稳定性。通过计算两类碳原子沿径向不同的褶皱度和键长下的碳纳米管的能量,证实碳纳米管的结构确实收敛到褶皱结构,从而证明了褶皱结构的稳定性。尤为重要的是,原先无褶皱的碳纳米管结构其实并不稳定,甚至都算不上亚稳态,在无褶皱和有褶皱碳纳米管结构间不存在势垒。褶皱结构的稳定性和褶皱度随碳纳米管半径减小而迅速增大,而当半径趋于无穷时,结构回归到没有褶皱的平面石墨烯。由于褶皱结构和曲率效应一起破坏了两类碳原子的结构对称性,导致了碳纳米管的金属半导体相变以及“真实”能隙的产生。根据第一性原理计算所得的结构和能带参数,我们进一步构造了褶皱碳纳米管的紧束缚哈密顿。在此基础上讨论了不同手性SWCNTs的二维和一维电子能带结构,具体分析了折叠模型,曲率模型以及褶皱模型下碳纳米管的能隙和“真实”能隙随纳米管半径的依赖关系。我们的结论是碳纳米管的能隙与碳管类型密切相关,但是“真实”能隙主要由半径和褶皱度决定,随着半径的增大迅速减小。
Since the carbon nanotube was discovered by Iijima in1991, tremendous progress-es have been made both experimentally and theoretically for its potential application in electronic devices. A single-wall carbon nanotube can be seen as rolled up from a graphene sheet, and this construction suggests that its electronic properties can be deduced from that of graphene under the simple zone-folding scheme. For an ideal graphene, the valence and conduction bands touch each other at Dirac points and form a gapless metal. When folding a graphene sheet into a carbon nanotube, wavevectors along the circumferential direction are quantized due to the circumferential boundary conditions. If the wavevectors pass through the Dirac points, a SWCNT is metallic, otherwise it is a semiconductor, according to which, Saito et al. arrived at a simple categorization that one-third SWCNTs are metal, and the other two-third are semicon-ductors. This categorization does not agree with the recent experiments well, in par-ticular for carbon nanotubes with small diameters. Considering the hopping integrals corrections induced by π-σ hybridization from the curvature effect on the tube, band-gaps open for most nominally metallic SWCNTs, while armchair SWCNTs remain metallic and are protected from such a curvature effect. But this conclusion contradict-s with recent experiment seriously. Band-gaps were observed for armchair SWCNTs by scanning tunneling microscopy, and those band-gaps would not vanish as external magnetic field varies. The band-gaps are call true band-gaps, and it is obviously this true band-gaps can not be explained by the popular physics mechanism. Firstly, if the Dirac points are not in the permitted quantized wavevectors by simply zone-folding method, the Dirac points would move into the quantized wavevectors by a phase sup-plied by an external magnetic field, and the band-gaps would vanish. The zone-folding theory cannot explain the true band-gaps. Secondly, the zero band-gaps for armchair SWCNTs are supposedly protected from the curvature effect, which also differs from the experiment. Deshpande et al. ascribed the true band-gaps to the strongly correlat-ed electronic correlations. At present, experimental observation of strong correlations remains a challenge since their signature is masked by charging effect for the small quantum system of carbon nanotubes.
We investigate the true band-gaps opening mechanism for SWCNTs in this thesis. Before the strong correlated model was proposed, other possibilities were explored ex-tensively, especially Peierls transition and dimerization induced by lattice deformation such as the Kekule structure. Those deformation do create band-gaps, but do not create true band-gaps, for that those deformation cannot destroy the chemical identicalness of two different types of carbon atoms on the tube, and with the applied magnetic field, the band-gaps will vanish. To have the true band-gaps, the structural symmetry of the two types of carbons should be broken. Corrugated structure of carbon nanotube is a good candidate. Such a model has been proposed for carbon nanotubes in a phe-nomenological model by Viet et al., but the stability of corrugated structures has not been confirmed by ab initio calculations, and the relationship among the band-gaps, corrugation length, radius, and chirality of SWCNTs has not been investigated up to now.
In this thesis, with ab initio calculations, firstly, we analyze the stability of the cor-rugated structures of armchair SWCNTs. The total energies of SWCNTs are calculated for different corrugation length and bond length with two types of carbon atoms dis-placed from each other radially. The structure do converge to corrugated structure, which verifies the stability of corrugated structure. It is especially important that the noncorrugated SWCNT structure is not stable, not even metastable, since no potential barrier exists between the structures with and without corrugation. The stability and corrugation length increase rapidly as tube's radius decreases. When radius approaches to infinity, corrugation vanishes for flat graphene. Combined with curvature effect, the corrugation breaks the local symmetry of two types of carbon atoms, which induces the metal-semiconductor transition and true band gaps emerge for carbon nanotubes. With the corrugation length and energies from the first-principles calculation, the tight- binding Hamiltonian is constructed for corrugated SWCNTs. The two dimensional and one dimensional electronic band structures are discussed for various types of SWCNTs. We analyze the band-gaps and true band gaps for zone-folding model, pure curvature model and corrugation model, respectively. Our conclusion is that the band-gaps sen-sitively depend on the chirality, while the true band-gaps, independent of SWCNTs' chirality, mainly depend on the corrugation and curvature effect, and increase rapidly with the decease of radius.
引文
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