电子掺杂石墨烯中的电子—声子耦合和电子特性
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摘要
石墨烯作为未来重要的电子材料,它吸引了人们的许多实验和理论的研究。石墨烯是由碳原子组成的单层蜂窝结构,它的一个重要的特性是它的荷动力学不是由薛定谔方程,而是由外尔方程,或可以说成是无质量狄拉克方程来决定。从理论上预言的石墨烯中电子的这种奇特的能带结构已经可以从角分辨光电子谱实验测量中定量确认。角分辨光电子谱是一个测量二维晶体中电子谱函数的强大的实验技术。特别地,角分辨光电子谱可以直接研究电子与晶格振动的相互作用(即电子-声子相互作用)。
在石墨烯的角分辨光电子谱的测量中,在费米面印以下200meV的地方发现了一个kinko并且对于不同的掺杂浓度,这个kink的位置是不变化的。为了解释这个特征,人们提出了许多理论。其中,理评论快报中的一篇论文[Phys. Rev. Lett.99,236802(2007)]提出了一种可以解析计算石墨烯中电子-声子相互作用对电子特性重整的模型。这篇文章的作者声称用电子-声子相互作用可以解释被观测到的kink。但是他们得到的kink结构与角分辨光电子谱实验和第一性原理计算得到的结果不符合。在本文中,我们用他们的模型解析地和数值地计算了电子-声子相互作用对准粒子能谱的重整。
本论文的第一章中我们概括性的介绍了石墨烯的特点和基本结构,角分辨光电子谱实验的原理和数据,以及本论文的主要研究内容。第二章我们介绍了石墨烯在紧束缚近似下的电子结构,以及哈密顿量对角化变换矩阵。另外,我们还引入了电子-声子相互作用哈密顿量,并且获得了电子-声子相互作用顶角矩阵。第三章中我们则详细的计算了电子-声子相互作用下自能的实部和虚部。第四章中我们讨论了石墨烯在电子-声子相互作用下的准粒子的自能,能量分布曲线,费米能级的重整化,质量重整参数λeff以及准粒子色散。第五章中,我们还计算了铍中由于电子-声子相互作用造成的化学势的重整和掺杂浓度的关系。我们数值地计算了在几种不同的掺杂浓度下,铍中二维电子-声子相互作用系统的强耦合自洽方程。我们得到了在电子-声子相互作用下的准粒子的自能,谱函数,能谱,最后我们求得准粒子化学势的重整和粒子数分布函数。第六章则为我们对本论文的简单总结以及对石墨烯中电子-声子相互作用研究的展望。本论文的主要研究工作如下：
1.我们计算了石墨烯在电子-声子相互作用下的自能的实部和虚部,发现我们得到的自能虚部于物理评论快报中的一篇论文得到的一样,但是自能实部比他们得到的多出一项,而这一项在掺杂浓度比较大时不可以忽略。然后我们计算了在掺杂浓度分别为n=1.0,4.5,12.0×1013cm-2时的准粒子自能,准粒子的能量分布曲线,费米能级的重整化,质量重整参数λeff,以及分别在温度T=OK和T=25K下的准粒子色散。我们发现在低掺杂浓度n=1.0×1013cm-2时,只能看到一个微小的kink,这与物理评论快报中的论文结果不符合。我们计算了在掺杂浓度n=4.5×1013cm-2和n=12.0×1013cm-2时的准粒子能谱,发现掺杂浓度越大,则电子-声子相互作用对准粒子能谱的影响越大,这种情况与角分辨光电子谱实验和第一性原理计算结果一致。我们的计算表明了电子-声子相互作用产生的扭结在费米面以下的175到220meV之间,这很好的符合了实验结果。我们计算了费米能的重整化并且发现,费米能级∈F在360meV时相对于零级的费米能级被提高了5meV,而在1200meV时相对于零级的费米能级被提高了20meV,我们的结果比一些文章中得到的要小。我们得到的当∈0F=0时,有效电子-声子耦合参数λeffph=0.0084.而在掺杂浓度n=1.0×1013—12.0×1013cm-2的范围内,质量重整参数Aeff～10—13%,这比以上提到的文章中得到的质量重整参数要小。最后,我们计算了在温度T=25K,掺杂浓度n=12.0×1013cm-2时的准粒子能谱。发现谱函数的宽度由于温度的增加而变大,并且我们得到的kink的形状,位置,尖锐度与第一性原理计算结果一致。
2.我们数值地解得了在几种不同的掺杂浓度下,铍中二维电子-声子相互作用系统的强耦合自洽方程。我们发现电子-声子相互作用在电子填充为半满时对准粒子能谱影响最大。而化学势越偏离零,也就是电子填充越偏离半满,电子-声子相互作用对准粒子能谱影响越小。或可以表达为化学势为零时kink结构最明显,而化学势偏离零越大则此结构越不明显。在ω=0处,当自能实部Re∑(iη)≠0时,会导致化学势的重整,化学势偏离零越大则此重整越大,而在μ=0.30eV处重整大约为10%。电子-声子相互作用造成电子的激发,使在零温时电子数的分布与有限温时的电子数的分布相似。
Graphene has attracted a great deal of experimental and theoretical activity for its po-tential role as major future electronics material. Graphene consists of a single layer of car-bon atoms with honeycomb lattice structure. An important aspect of the charge dynamics of graphene is that it is governed by a Weyl's equation or a Dirac equation with vanishing rest mass rather than a Schrodinger equation. This peculiar band structure of electrons in graphene predicted theoretically have been qualitatively confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements. ARPES is a powerful experimental technique for probing the electronic spectral function A(k, ω) in two-dimensional crys-tals. In particular, ARPES allows studies of the interaction between electrons and lattice vibrations (electron-phonon interactions).
In the ARPES measurements, a kink was observed at about200meV below the Fermi level∈F. Its relative position with respect to∈F is unchanged for different doping level of graphene. In order to explain this feature, many theories have been proposed. A theory of analytical calculation of electron-phonon interaction effects on the electronic properties of graphene was developed by the paper in the Physical Review Letters [Phys. Rev. Lett.99,236802(2007)]. They claimed that the main features of the observed ARPES spectra could be explained by the electron-phonon interaction. But their kink structure was not consistent with the experimental ARPES studies and the result from first-principle calculations of the graphene electronic spectra. In this paper, we revisit this electron-phonon interaction approach both analytically and numerically.
In Chapter1, we introduce the characteristics and the basic structure of graphene, the principle and the data of ARPES experimenta, and outline our main work and the content. In chapter2, we introduce the electronic structure of graphene in tight binding model, the diagonalizable matrix of Hamiltonian. We also introduce the electron-phonon interaction Hamiltonian, and obtain the electron-phonon coupling matrix. In chapter3, we calculate the real and imaginary parts of the self-energy with the electron-phonon in-teraction. In chapter4, we calculate the mass-renormalization parameter λeff, the Fermi level renormalizations and the quasiparticle dispersions. Then, we compare our results with the ARPES experimental data and the first-principle calculations. In chapter5, we also study the relation between the renormalization of the chemical potential due to mul-tiphonon effects at the surface of Be(0001) and doping. we solve the strong-coupling self-consistent equations of a two-dimensional (2D) electron-phonon interaction system of Be(0001) numerically at some different doping levels. We present the self-energy of quasiparticle with electron-phonon interacting. Then, we calculate the quasiparticle spec-tral function, the quasiparticle dispersion. Finally, we calculate the renormalization of the chemical potential and the quasiparticle distribution function. In chapter6, we summary the total topic and look forward to the future of the study of electron-phonon interaction in graphene.
The main work in this thesis are listed as follows:
1. We evaluate analytically the imaginary parts of the self-energy and find the same results as the imaginary parts of the self-energy worked out in the paper as men-tioned above. However, the real parts of the self-energy we found have an extra term which does not appear in the real parts of the self-energy in the above men-tioned paper. And for the case that∈E0cannot be ignored as compared with the band cutoff, we must consider the extra term when we calculate the real parts of the self-energy. Then we calculate the energy distribution curves, the Fermi-level renormalization, the mass-renormalization parameter λeff, and the quasiparticle dispersion. Our results show that the electron-phonon interaction has a slight effect on the band-structure renormalization at lower doping level n=1.0×1013m-2. And we calculate the renormalized conduction-band energy spectrum at higher dop-ing n=4.5,12×1013cm-2, and find that the electron-phonon interaction effect on the band-structure renormalization becomes larger with increasing doping. This situation is in agreement with the ARPES data and the first-principles calculations. Our calculations reveal phonon-induced kinks near the Fermi energy at binding energies between175and220meV, in good agreement with experimental photoe-mission maps. Then we calculate the Fermi-level renormalizations, and find that the Fermi level∈F is shown to be lifted by5-20meV from the Fermi level∈F0without electron-phonon interaction in the range360-1200meV upon doping. Our results are smaller than that calculated in some paper. The effective electron-phonon cou-pling parameter is found to be λeffph=0.0084for∈F. The mass-renormalization pa-rameter λeff～10-13%within the doping range n=1.0×1013-12.0×1013cm-2. The λeff is smaller than calculated in the paper as mentioned above. We also carry out calculation of the spectral function at finite temperature (T=25K) and find that the shape and position of the kink at doping n=12.0x1013cm-2is in accord with the first-principles calculations.
2. We have solved the strong-coupling self-consistent equations of a2D elec-tron-phonon interaction system numerically for different chemical potentials. We have also obtained self-energy, momentum distribution curves, electron distribution function at a finite temperature, and the relationship between chemical potential and the electron density. We find that the effect of electron-phonon interaction on electron structure is strongest at the half filling, but it has no effect on the chemical potential. However, the chemical potential shows distinct renormalization effects away from half filling due to the electron-phonon interaction. The renormalization of the chemical potential is about10%at μ=0.30eV. And we find that the electron distribution functions with electron-phonon interaction at zero temperature are very similar to the electron distribution function at a finite temperature.
在石墨烯的角分辨光电子谱的测量中,在费米面印以下200meV的地方发现了一个kinko并且对于不同的掺杂浓度,这个kink的位置是不变化的。为了解释这个特征,人们提出了许多理论。其中,理评论快报中的一篇论文[Phys. Rev. Lett.99,236802(2007)]提出了一种可以解析计算石墨烯中电子-声子相互作用对电子特性重整的模型。这篇文章的作者声称用电子-声子相互作用可以解释被观测到的kink。但是他们得到的kink结构与角分辨光电子谱实验和第一性原理计算得到的结果不符合。在本文中,我们用他们的模型解析地和数值地计算了电子-声子相互作用对准粒子能谱的重整。
本论文的第一章中我们概括性的介绍了石墨烯的特点和基本结构,角分辨光电子谱实验的原理和数据,以及本论文的主要研究内容。第二章我们介绍了石墨烯在紧束缚近似下的电子结构,以及哈密顿量对角化变换矩阵。另外,我们还引入了电子-声子相互作用哈密顿量,并且获得了电子-声子相互作用顶角矩阵。第三章中我们则详细的计算了电子-声子相互作用下自能的实部和虚部。第四章中我们讨论了石墨烯在电子-声子相互作用下的准粒子的自能,能量分布曲线,费米能级的重整化,质量重整参数λeff以及准粒子色散。第五章中,我们还计算了铍中由于电子-声子相互作用造成的化学势的重整和掺杂浓度的关系。我们数值地计算了在几种不同的掺杂浓度下,铍中二维电子-声子相互作用系统的强耦合自洽方程。我们得到了在电子-声子相互作用下的准粒子的自能,谱函数,能谱,最后我们求得准粒子化学势的重整和粒子数分布函数。第六章则为我们对本论文的简单总结以及对石墨烯中电子-声子相互作用研究的展望。本论文的主要研究工作如下：
1.我们计算了石墨烯在电子-声子相互作用下的自能的实部和虚部,发现我们得到的自能虚部于物理评论快报中的一篇论文得到的一样,但是自能实部比他们得到的多出一项,而这一项在掺杂浓度比较大时不可以忽略。然后我们计算了在掺杂浓度分别为n=1.0,4.5,12.0×1013cm-2时的准粒子自能,准粒子的能量分布曲线,费米能级的重整化,质量重整参数λeff,以及分别在温度T=OK和T=25K下的准粒子色散。我们发现在低掺杂浓度n=1.0×1013cm-2时,只能看到一个微小的kink,这与物理评论快报中的论文结果不符合。我们计算了在掺杂浓度n=4.5×1013cm-2和n=12.0×1013cm-2时的准粒子能谱,发现掺杂浓度越大,则电子-声子相互作用对准粒子能谱的影响越大,这种情况与角分辨光电子谱实验和第一性原理计算结果一致。我们的计算表明了电子-声子相互作用产生的扭结在费米面以下的175到220meV之间,这很好的符合了实验结果。我们计算了费米能的重整化并且发现,费米能级∈F在360meV时相对于零级的费米能级被提高了5meV,而在1200meV时相对于零级的费米能级被提高了20meV,我们的结果比一些文章中得到的要小。我们得到的当∈0F=0时,有效电子-声子耦合参数λeffph=0.0084.而在掺杂浓度n=1.0×1013—12.0×1013cm-2的范围内,质量重整参数Aeff～10—13%,这比以上提到的文章中得到的质量重整参数要小。最后,我们计算了在温度T=25K,掺杂浓度n=12.0×1013cm-2时的准粒子能谱。发现谱函数的宽度由于温度的增加而变大,并且我们得到的kink的形状,位置,尖锐度与第一性原理计算结果一致。
2.我们数值地解得了在几种不同的掺杂浓度下,铍中二维电子-声子相互作用系统的强耦合自洽方程。我们发现电子-声子相互作用在电子填充为半满时对准粒子能谱影响最大。而化学势越偏离零,也就是电子填充越偏离半满,电子-声子相互作用对准粒子能谱影响越小。或可以表达为化学势为零时kink结构最明显,而化学势偏离零越大则此结构越不明显。在ω=0处,当自能实部Re∑(iη)≠0时,会导致化学势的重整,化学势偏离零越大则此重整越大,而在μ=0.30eV处重整大约为10%。电子-声子相互作用造成电子的激发,使在零温时电子数的分布与有限温时的电子数的分布相似。
Graphene has attracted a great deal of experimental and theoretical activity for its po-tential role as major future electronics material. Graphene consists of a single layer of car-bon atoms with honeycomb lattice structure. An important aspect of the charge dynamics of graphene is that it is governed by a Weyl's equation or a Dirac equation with vanishing rest mass rather than a Schrodinger equation. This peculiar band structure of electrons in graphene predicted theoretically have been qualitatively confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements. ARPES is a powerful experimental technique for probing the electronic spectral function A(k, ω) in two-dimensional crys-tals. In particular, ARPES allows studies of the interaction between electrons and lattice vibrations (electron-phonon interactions).
In the ARPES measurements, a kink was observed at about200meV below the Fermi level∈F. Its relative position with respect to∈F is unchanged for different doping level of graphene. In order to explain this feature, many theories have been proposed. A theory of analytical calculation of electron-phonon interaction effects on the electronic properties of graphene was developed by the paper in the Physical Review Letters [Phys. Rev. Lett.99,236802(2007)]. They claimed that the main features of the observed ARPES spectra could be explained by the electron-phonon interaction. But their kink structure was not consistent with the experimental ARPES studies and the result from first-principle calculations of the graphene electronic spectra. In this paper, we revisit this electron-phonon interaction approach both analytically and numerically.
In Chapter1, we introduce the characteristics and the basic structure of graphene, the principle and the data of ARPES experimenta, and outline our main work and the content. In chapter2, we introduce the electronic structure of graphene in tight binding model, the diagonalizable matrix of Hamiltonian. We also introduce the electron-phonon interaction Hamiltonian, and obtain the electron-phonon coupling matrix. In chapter3, we calculate the real and imaginary parts of the self-energy with the electron-phonon in-teraction. In chapter4, we calculate the mass-renormalization parameter λeff, the Fermi level renormalizations and the quasiparticle dispersions. Then, we compare our results with the ARPES experimental data and the first-principle calculations. In chapter5, we also study the relation between the renormalization of the chemical potential due to mul-tiphonon effects at the surface of Be(0001) and doping. we solve the strong-coupling self-consistent equations of a two-dimensional (2D) electron-phonon interaction system of Be(0001) numerically at some different doping levels. We present the self-energy of quasiparticle with electron-phonon interacting. Then, we calculate the quasiparticle spec-tral function, the quasiparticle dispersion. Finally, we calculate the renormalization of the chemical potential and the quasiparticle distribution function. In chapter6, we summary the total topic and look forward to the future of the study of electron-phonon interaction in graphene.
The main work in this thesis are listed as follows:
1. We evaluate analytically the imaginary parts of the self-energy and find the same results as the imaginary parts of the self-energy worked out in the paper as men-tioned above. However, the real parts of the self-energy we found have an extra term which does not appear in the real parts of the self-energy in the above men-tioned paper. And for the case that∈E0cannot be ignored as compared with the band cutoff, we must consider the extra term when we calculate the real parts of the self-energy. Then we calculate the energy distribution curves, the Fermi-level renormalization, the mass-renormalization parameter λeff, and the quasiparticle dispersion. Our results show that the electron-phonon interaction has a slight effect on the band-structure renormalization at lower doping level n=1.0×1013m-2. And we calculate the renormalized conduction-band energy spectrum at higher dop-ing n=4.5,12×1013cm-2, and find that the electron-phonon interaction effect on the band-structure renormalization becomes larger with increasing doping. This situation is in agreement with the ARPES data and the first-principles calculations. Our calculations reveal phonon-induced kinks near the Fermi energy at binding energies between175and220meV, in good agreement with experimental photoe-mission maps. Then we calculate the Fermi-level renormalizations, and find that the Fermi level∈F is shown to be lifted by5-20meV from the Fermi level∈F0without electron-phonon interaction in the range360-1200meV upon doping. Our results are smaller than that calculated in some paper. The effective electron-phonon cou-pling parameter is found to be λeffph=0.0084for∈F. The mass-renormalization pa-rameter λeff～10-13%within the doping range n=1.0×1013-12.0×1013cm-2. The λeff is smaller than calculated in the paper as mentioned above. We also carry out calculation of the spectral function at finite temperature (T=25K) and find that the shape and position of the kink at doping n=12.0x1013cm-2is in accord with the first-principles calculations.
2. We have solved the strong-coupling self-consistent equations of a2D elec-tron-phonon interaction system numerically for different chemical potentials. We have also obtained self-energy, momentum distribution curves, electron distribution function at a finite temperature, and the relationship between chemical potential and the electron density. We find that the effect of electron-phonon interaction on electron structure is strongest at the half filling, but it has no effect on the chemical potential. However, the chemical potential shows distinct renormalization effects away from half filling due to the electron-phonon interaction. The renormalization of the chemical potential is about10%at μ=0.30eV. And we find that the electron distribution functions with electron-phonon interaction at zero temperature are very similar to the electron distribution function at a finite temperature.
引文
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