Terahertz plasmon and surface-plasmon modes in hollow nanospheres
- Yiming Xiao^{1},
- Wen Xu^{1, 2}Email author,
- Yaya Zhang^{1} and
- Jiaguang Hu^{1, 3}
https://doi.org/10.1186/1556-276X-7-578
© Xiao et al; licensee Springer. 2012
Received: 19 July 2012
Accepted: 8 October 2012
Published: 23 October 2012
Abstract
We present a theoretical study of the electronic subband structure and collective electronic excitation associated with plasmon and surface plasmon modes in metal-based hollow nanosphere. The dependence of the electronic subband energy on the sample parameters of the hollow nanosphere is examined. We find that the subband states with different quantum numbers l degenerate roughly when the outer radius of the sphere is r_{2} ≥ 100 nm. In this case, the energy spectrum of a sphere is mainly determined by quantum number n. Moreover, the plasmon and surface plasmon excitations can be achieved mainly via inter-subband transitions from occupied subbands to unoccupied subbands. We examine the dependence of the plasmon and surface-plasmon frequencies on the shell thickness d and the outer radius r_{2} of the sphere using the standard random-phase approximation. We find that when a four-state model is employed for calculations, four branches of the plasmon and surface plasmon oscillations with terahertz frequencies can be observed, respectively.
Keywords
Hollow nanosphere Electronic subband structure Collective excitation modes Terahertz radiationBackground
In recent years, there has been a great interest in the investigation of metal-based hollow nanostructures because of their unique characteristics such as low density, large specific area, mechanical and thermal stability, and surface permeability. These advanced materials have been widely applied in catalysis[1], drug delivery[2, 3], food and cosmetic industries[4], fuel cell[5, 6], biotechnology[7], lubricant[8], sensing[9], photonic devices[10], micro/nanoreactors[11], etc. In particular, metal-based hollow nanospheres[12] can be realized via using polystyrene (PS) latex particles as templates[13]. Such structures have intriguing features of surface plasmon resonance[14]. The collective oscillations of the conducting electrons in response to optical excitation, such as plasmon and surface plasmon excitations, affect strongly the optical properties of metal hollow nanospheres. At present, it has become possible to fabricate metal hollow nanosphere structures in which the radius and shell thickness of the sphere can be controlled artificially. Such structures have been widely applied to realize terahertz (10^{12} Hz or THz) plasmonic devices[15]. Hence, it is of great importance and significance to study the electronic subband structure and corresponding collective electronic excitations from these advanced nanomaterial systems. In conjunction with recent experimental achievement in the field, in this article, we would like to develop a simple theoretical approach to study the electronic subband structure and plasmon and surface-plasmon modes in a hollow nanosphere. The aim of this study is to examine how sample parameters affect the electronic subband energy and the plasmon and the surface-plasmon modes in the device systems.
Methods
Theoretical approach
Electronic subband structure
where k = nΠ/d.
E_{n 1} and E_{n 2} are determined numerically via solving respectively Equations (8) and (10).
Electron-electron interaction
${\mathbf{c}}^{\mathbf{k}}\mathbf{(}{\mathit{N}}^{\mathbf{\prime}}\mathbf{,}\mathit{N})$
l _{ N } | ${\mathit{l}}_{{\mathit{N}}^{\mathbf{\prime}}}$ | m _{ N } | ${\mathit{m}}_{{\mathit{N}}^{\mathit{\prime}}}$ | c^{0}(N^{ ′ },N) | c^{1}(N^{ ′ },N) | c^{2}(N^{ ′ },N) |
---|---|---|---|---|---|---|
s | s | 0 | 0 | 1 | ||
p | p | 0 | 0 | 1 | 4 | |
s | p | 0 | 0 | 1 |
l_{ N } and m_{ N } are angular momentum quantum number and magnetic quantum number for a quantum state N, respectively, whereas${l}_{{N}^{\prime}}$ and${m}_{{N}^{\prime}}$ for a quantum state N^{ ′ }. c^{0}(N^{ ′ },N), c^{1}(N^{ ′ },N), and c^{2}(N^{ ′ },N) are angle factors for$\mathbb{k}=0,1,2$ defined by Equations (13) and (15).
Plasmon and surface-plasmon modes
is the pair bubble (or density-density correlation function) in the absence of e-e coupling with g_{ s }= 2, counting for spin degeneracy, and f(E_{ N }) =${[1+{e}^{({E}_{N}-{E}_{F})/{\mathbb{k}}_{B}T}]}^{-1}$ being the Fermi-Dirac function.
Results and discussion
It should be noted that when r_{2} > 100 nm and d ∼ 10 nm, E_{ nl }≃ E_{n 0}, and plasmon and surface-plasmon frequencies in a hollow nanosphere are determined mainly by transition events between E_{2l}and E_{1l}. This implies that although only four electronic subbands are included within current calculations, the obtained results should be very much similar to the case where more electronic states are considered when r_{2} > 100 nm and d ∼ 10 nm. The results obtained from this study indicate that the electronic subband energy and the plasmon and surface-plasmon modes in hollow nanospheres are determined mainly by sample parameters such as the diameter of the sphere r_{2} and the shell thickness d. When r_{2} > 100 nm, the energy levels depend very weakly on inner or outer radius (i.e., r_{1} or r_{2}) at a fixed d. Thus, the shell thickness affects more strongly the electronic subband energies in a hollow nanosphere. We find that when d ∼ 10 nm and r_{2} ≥ 100 nm, the energy spacing between E2_{ l }and E1_{ l } states is about 10 meV or about 2.4 THz. The frequencies of plasmon and surface-plasmon modes in the structure are also in the THz bandwidth. The plasmon and surface-plasmon modes depend sensitively on the geometrical parameters such as the outer radius r_{2} and shell thickness d. These effects imply that metal-based hollow nanosphere structures can be applied as THz materials or devices in which THz optical absorption and excitation can be achieved via inter-subband electronic transitions. It is known that THz technology is of great potential to impact many interdisciplinary fields such as telecommunication, biological science, pharmaceutical technology, anti-terrorist, etc.[23]. The application of nanostructure in THz technology has become a fast growing field of research in recent years. The theoretical findings from this work confirm that hollow gold-nanosphere structures are indeed the THz plasmonic materials which can be applied as frequency-tunable THz optoelectronic devices.
Conclusions
In this study, we have examined theoretically the electronic subband structure and the plasmon and surface-plasmon modes of hollow nanosphere structures. We have found that when the diameter of the sphere r_{2} > 100 nm and the shell thickness d ∼ 10 nm, the energy levels for different l states roughly degenerate. In such a case, the electronic subband energy, E_{ nl }≃ E_{n 0} = ℏ^{2}Π^{2}n^{2}/2μ d^{2}, does not depend on r_{2}. When r_{2} < 200 nm, the plasmon and surface-plasmon modes induced by different electronic transition channels have significantly different frequencies. When r_{2} > 200 nm, the plasmon and surface-plasmon frequencies approach roughly to Ω^{ p }∼ Ω^{ s }∼ (E_{20} − E_{10})/ℏ, which depend largely on d and depend very little on r_{2}.
It should be noted that at present, little research work has been carried out to look into the electronic subband structure of the hollow nanosphere structures using more powerful theoretical tools such as the first principle calculations which require large scale numerical computations and are CPU-consuming. The simple analytical results obtained from this study can be applied further to study the electronic and optoelectronic properties of the hollow nanosphere structures. We have found that the plasmon and surface-plasmon excitations can be achieved via inter-subband electronic transition channels in the hollow nanospheres. In particular, we have demonstrated that in metal hollow nanospheres, the energy difference between E1_{ l } and E2_{ l } states, and the plasmon and surface-plasmon frequencies are all in the THz bandwidth. This can lead to an application of metal hollow nanosphere structures in THz optics and optoelectronics.
Author’s information
WX is the distinguished professor at Yunnan University and research professor at the Institute of Solid State Physics, Chinese Academy of Sciences. YX and YZ are post-graduate students at Yunnan University. JH is a PhD student at Yunnan University.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant no.: 10974206), the Ministry of Science and Technology of China (grant no.: 2011YQ130018), the Department of Science and Technology of Yunnan Province, and by the Chinese Academy of Sciences.
Authors’ Affiliations
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