Abstract
We present a theoretical study of the electronic subband structure and collective electronic excitation associated with plasmon and surface plasmon modes in metalbased 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 intersubband transitions from occupied subbands to unoccupied subbands. We examine the dependence of the plasmon and surfaceplasmon frequencies on the shell thickness d and the outer radius r_{2} of the sphere using the standard randomphase approximation. We find that when a fourstate 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 metalbased 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, metalbased 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 surfaceplasmon 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 surfaceplasmon modes in the device systems.
Methods
Theoretical approach
Electronic subband structure
In this study, we consider an air/metalshell/airbased hollow nanosphere structure. The inner radius of the structure is r_{1}, the outer radius or the diameter of the sphere is r_{2}, and the metal shell thickness is d = r_{2} − r_{1}. Such structure can be realized experimentally by selectively removing the hard spherical core (e.g., PS latex particles) in the fabrication process [16]. For a case where the electrons in the metal shell are not tunneling or hopping into the core and outside air, the confining potential for electrons in the structure can be modeled simply as
Thus, the corresponding Schrödinger equation takes a form
Here, P = (p_{x},p_{y},p_{z}) is the momentum operator, μ is the effective mass for an electron in the structure, R = (x,y,z) = (r,θ,ϕ), and N stands for all quantum numbers. The solution of Equation (2) is , where N = (nlm),
and R_{nl}(r) is determined by
Here, l = 0,1,2,⋯ is the angular momentum quantum number, m = l,l−1,⋯,−l is the magnetic quantum number, is the associated Legendre function, and l must be a positive integer in the range l ≥ m. Letting E_{N }= ℏ^{2}k^{2}/2μ, x = kr, and , the radial equation, Equation (4), for r_{1} ≤ r ≤ r_{2} becomes a Bessel equation with a general solution: , where J_{α}(x) is a Bessel function and is a normalization factor. Considering the boundary conditions: R(r_{1}) = 0 and R(r_{2}) = 0, we have
which is applied to determine the energy spectrum of the sample structure. Thus, the electron wave function becomes
For case of l = 0, we obtain E_{n0} = ℏ^{2}Π^{2}n^{2}/2μd^{2} with n = 1,2,⋯. The radial eigenfunction is
where k = nΠ/d.
For l = 1, we have
and the radial eigenfunction becomes
For l = 2, we get
E_{n1} and E_{n2} are determined numerically via solving respectively Equations (8) and (10).
Electronelectron interaction
The matrix element for the bare electronelectron (ee) interaction can be obtained by applying the electron wave function to the interaction Hamiltonian induced by the Coulomb potential [17], which reads
with κ being the highfrequency dielectric constant of the shell material. It can be simplified as
where as shown in Equation (3), and
where R_{<} (R_{>}) is the smaller (bigger) value of {R_{1},R_{2}}. In order that c^{k} can have a nonzero value, must satisfy the conditions
Table 1 gives the values of for s and p electrons in case of m = 0.
l_{N} and m_{N} are angular momentum quantum number and magnetic quantum number for a quantum state N, respectively, whereas and for a quantum state N^{′}. c^{0}(N^{′},N), c^{1}(N^{′},N), and c^{2}(N^{′},N) are angle factors for defined by Equations (13) and (15).
Plasmon and surfaceplasmon modes
From electron energy spectrum obtained from the solution of the Schrödinger equation, we can derive the retarded and advanced Green’s function for electrons. Applying these Green’s functions and V_{αβ} with β =N^{′}N to the diagrammatic techniques to derive effective ee interaction under the random phase approximation, the element of the dielectric function matrix is obtained as [17,18]
where
is the pair bubble (or densitydensity correlation function) in the absence of ee coupling with g_{s }= 2, counting for spin degeneracy, and f(E_{N}) = being the FermiDirac function.
In a hollow nanosphere system described by quantum number N = (nlm), the electronic subband energy depends only on n and l quantum numbers, namely E_{N }= E_{nl}. In this study, we consider that n = 1 states with many l and m numbers are occupied, and n = 2 states with any l and m numbers are unoccupied. For simplicity, we take a fourstate model (FSM) to calculate the dielectric function matrix. We consider that two lowest electronic states for n = 1, E_{10}, and E_{11} are occupied, and two lowest electronic states for n = 2, E_{20}, and E_{21} are unoccupied, as shown in Figure 1. Because the electronic subband energy in a hollow sphere does not depend on the quantum number m, we take m = 0 in the calculations. On the basis that all electronic states in a hollow nanosphere are quantized, intrasubband transitions do not contribute to dielectric function. Moreover, the transitions within the occupied and within the unoccupied states do not contribute to the dielectric function as well. Thus, as shown in Figure 1, there are eight possible transition channels induced by intersubband transitions from occupied (unoccupied) states to unoccupied (occupied) states in this FSM. Setting the electronic state index as 1 = (100), 2 = (110), 3 = (200), and 4 = (210), the dielectric function of a hollow nanosphere in the FSM is a 16×16 matrix and can be obtained from Equation (16). The determinant of the dielectric function matrix then is
with a_{αβ }= −V_{αβ}π_{β}(Ω). The plasmon and surfaceplasmon modes are determined by Reε(Ω)→0 and Reε(Ω)→−1, respectively. In this study, we employ a matrix to present the dielectric function in a multienergy level system such as a hollow nanosphere structure. Such an approach was applied to study the plasmon excitations in semiconductorbased twodimensional electron gas systems [19] and Rashba spintronic systems [18]. We note that in the present study, we consider a simple model to calculate the electronic subband structure of a hollow nanosphere. The effect of the spinorbit interaction in the system is not included.
Figure 1. The fourstate model for electronic transitions in a hollow nanosphere. E_{10} and E_{11} are occupied states, and E_{20} and E_{21} are unoccupied states. E_{F} is the Fermi energy. Here, we consider the electronic states in the case of m = 0. The possible transition channels are indicated.
Results and discussion
In numerical calculations, we take the effective mass for an electron to be about the rest electron mass, i.e., μ ≃ 0.99m_{e}, and the highfrequency dielectric constant κ = 1.53 for gold shell [2022]. In Figure 2, we show the electronic subband energy for E1_{l} and E2_{l}(l = 0, 1, and 2) as a function of outer radius r_{2} of the hollow nanosphere with a fixed shell thickness d = 10 nm. We see that the energy levels with different l quantum numbers roughly degenerate when r_{2} > 100 nm. In such a case, the subband energy depends very little on r_{2}and E_{nl }≃ E_{n0} = ℏ^{2}Π^{2}n^{2}/2μd^{2}. In Figure 3a, the electronic subband energies for E1_{l }and E2_{l}(l = 0, 1, and 2) are shown functions of shell thickness d at a fixed outer radius r_{2} = 100 nm of the hollow nanosphere. The results for different l states coincide roughly. The subband energy decreases with increasing shell thickness as E_{nl }≃ E_{n0}∼d^{−2}. In Figure 3b, the electronic subband energies for E1_{l}and E2_{l}(l = 0, 1, and 2) are shown functions of shell thickness d for a fixed outer radius r_{2} = 25 nm of the hollow nanosphere. The subband energies degenerate roughly at small shell thickness and show difference with increasing shell thickness d as shown in the inset in Figure 3b. We know that the energy for the n^{th} subband at l = 0 is determined only by the shell thickness d = r_{2} − r_{1} of a hollow nanosphere. The results shown in Figures 2 and 3 indicate that different l states degenerate at a fixed n quantum number when r_{2} > 100 nm. This feature is mainly induced by the symmetry of the confining potential for electrons, given as Equation (1). However, when r_{2} is relatively small (see Figure 2 and Figure 3b), the electronic subband energy depends on quantum number l for a fixed quantum number n and E_{nl}>E_{n0} (here l > 0). This suggests that the stronger quantum effect can be achieved in smaller sample structures. Such an effect can be understood by the fact that when r_{2}→∞, the energies determined by Equations (8) and (10) approach E_{nl }→ E_{n0} = ℏ^{2}Π^{2}n^{2}/2μd^{2} and when r_{2} takes a finite value E_{nl }> E_{n0}.
Figure 2. The subband energies vary with r_{2} at a fixed d. Electronic subband energy, E1_{l }and E2_{l}for l = 0, 1, and 2, as a function of outer radius r_{2} of hollow nanosphere at a fixed shell thickness d = 10 nm.
Figure 3. The subband energies vary with d at fixed r_{2}. Electronic subband energy, E1_{l }and E2_{l}for l = 0, 1, and 2, as a function shell thickness d of hollow nanosphere for outer radius r_{2} = 100 nm in (a) and r_{2} = 25 nm in (b). The inset in (b) shows the energy difference in different l states.
In Figure 4, the plasmon and surfaceplasmon frequencies of hollow nanosphere are shown as a function of outer radius r_{2} at a fixed shell thickness d. Using the FSM, there are four modes for both plasmon and surfaceplasmon excitation from a hollow nanosphere. We see that (1) the plasmon and surfaceplasmon frequencies decrease with increasing r_{2} when r_{2} < 200 nm. When r_{2} > 200 nm, the plasmon and surfaceplasmon frequencies approach approximately to the energygap between E_{20} and E_{10}; (2) the frequencies of all these modes are in the THz regime; (3) the plasmon frequency with being a surfaceplasmon frequency. This is the primary relationship between plasmon and surfaceplasmon modes; (4) the surfaceplasmon frequency is slightly higher than plasmon frequency ; and (5) and .
Figure 4. Dependence of plasmon and surfaceplasmon frequencies on outer radius r_{2} at a fixed shell thickness d = 10 nm.
In Figure 5, the plasmon and surfaceplasmon frequencies are shown as functions of shell thickness d at a fixed r_{2}. The plasmon and surfaceplasmon frequencies decrease with increasing shell thickness. The frequency difference between different excitation modes gets wider with increasing d.
Figure 5. Dependence of plasmon and surfaceplasmon frequencies on the shell thickness d at a fixed outer radius r_{2} = 100 nm.
It should be noted that when r_{2} > 100 nm and d ∼ 10 nm, E_{nl }≃ E_{n0}, and plasmon and surfaceplasmon 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 surfaceplasmon 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 surfaceplasmon modes in the structure are also in the THz bandwidth. The plasmon and surfaceplasmon modes depend sensitively on the geometrical parameters such as the outer radius r_{2} and shell thickness d. These effects imply that metalbased hollow nanosphere structures can be applied as THz materials or devices in which THz optical absorption and excitation can be achieved via intersubband electronic transitions. It is known that THz technology is of great potential to impact many interdisciplinary fields such as telecommunication, biological science, pharmaceutical technology, antiterrorist, 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 goldnanosphere structures are indeed the THz plasmonic materials which can be applied as frequencytunable THz optoelectronic devices.
Conclusions
In this study, we have examined theoretically the electronic subband structure and the plasmon and surfaceplasmon 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_{n0} = ℏ^{2}Π^{2}n^{2}/2μd^{2}, does not depend on r_{2}. When r_{2} < 200 nm, the plasmon and surfaceplasmon modes induced by different electronic transition channels have significantly different frequencies. When r_{2} > 200 nm, the plasmon and surfaceplasmon 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 CPUconsuming. 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 surfaceplasmon excitations can be achieved via intersubband 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 surfaceplasmon frequencies are all in the THz bandwidth. This can lead to an application of metal hollow nanosphere structures in THz optics and optoelectronics.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
WX proposed and supervised the research work. YX carried out the analytical and numerical calculations. YZ and JH participated in the discussions and analyzes of the obtained results. All authors read and approved the final manuscript.
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 postgraduate students at Yunnan University. JH is a PhD student at Yunnan University.
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.
References

Shchukin DG, Caruso RA: Template synthesis and photocatalytic properties of porous metal oxide spheres formed by nanoparticle infiltration.
Chem Mater 2004, 16:2287. Publisher Full Text

Zhu YF, Shi JL, Shen WH, Dong XP, Feng JW, Ruan ML, Li YS: Stimuliresponsive controlled drug release from a hollow mesoporous silica sphere/polyelectrolyte multilayer coreshell structure.
Angew Chem Int Ed 2005, 44:5083. Publisher Full Text

Chen JF, Ding HM, Wang JX, Shao L: Preparation and characterization of porous hollow silica nanoparticles for drug delivery application.
Biomaterials 2004, 25:723. PubMed Abstract  Publisher Full Text

Shchukin DG, Sukhorukov GB, Möhwald H: Smart inorganic/organic nanocomposite hollow microcapsules.
Angew Chem Int Ed 2003, 42:4472. Publisher Full Text

Liang HP, Zhang HM, Hu JS, Guo YG, Wan LJ, Bai CL: Pt hollow nanospheres: facile synthesis and enhanced electrocatalysts.
Angew Chem Int Ed 2004, 43:1540. Publisher Full Text

Yang J, Lee JY, Too HP, Valiyaveettil S: A bis(psulfonatophenyl)phenylphosphinebased synthesis of hollow Pt nanospheres.
J Phys Chem B 2006, 110:125. PubMed Abstract  Publisher Full Text

Mathiowitz E, Jacob JS, Jong YS, Carino GP, Chickering DE, Chaturvedi P, Santos CA, Vijayaraghavan K, Montgomery S, Bassett M, Morrell C: Biologically erodable microspheres as potential oral drug delivery systems.
Nature 1997, 386:410. PubMed Abstract  Publisher Full Text

Rapoport L, Bilik Y, Feldman Y, Homyonfer M, Cohen SR, Tenne R: Hollow nanoparticles of WS2 as potential solidstate lubricants.
Nature 1997, 387:791. Publisher Full Text

Zhang Q, Zhang TR, Ge JP, Yin YD: Permeable silica shell through surfaceprotected etching.
Nano Lett 2008, 8:2867. PubMed Abstract  Publisher Full Text

Prodan E, Radloff C, Halas NJ, Nordlander P: A hybridization model for the plasmon response of complex nanostructures.
Science 2003, 302:419. PubMed Abstract  Publisher Full Text

Yin YD, Rioux RM, Erdonmez CK, Hughes S, Somorjai GA, Alivisatos AP: Formation of hollow nanocrystals through the nanoscale Kirkendall effect.
Science 2004, 304:711. PubMed Abstract  Publisher Full Text

Caruso F, Caruso RA, Möhwald H: Nanoengineering of inorganic and hybrid hollow spheres by colloidal templating.
Science 1998, 282:1111. PubMed Abstract  Publisher Full Text

Duan GT, Lv FJ, Cai WP, Luo YY, Li Y, Liu GQ: General synthesis of 2D ordered hollow sphere arrays based on nonshadow deposition dominated colloidal lithography.
Langmuir 2010, 26:6295. PubMed Abstract  Publisher Full Text

Lin A, Son DH, Ahn IH, Song GH, Han WT: Visible to infrared photoluminescence from gold nanoparticles embedded in germanosilicate glass fiber.
Optics Express 2007, 15:6374. PubMed Abstract  Publisher Full Text

Polyushkin DK, Hendry E, Stone EK, Barnes WL: THz generation from plasmonic nanoparticle arrays.
Nano Lett 2011, 11:4718. PubMed Abstract  Publisher Full Text

Lou XW, Archer LA, Yang Z: Hollow micro/nanostructures: synthesis and applications.
Adv Mater 2008, 20:3987. Publisher Full Text

Condon E, Shortley GH: The Theory of Atomic Spectra. London: Cambridge University Press; 1959,.
p. 174

Xu W: Plasmons of a twodimensional electron gas in the presence of spin orbit interaction.
Appl Phys Lett 2003, 82:724. Publisher Full Text

Das Sarma S, Madhukar A: Collective modes of spatially separated, twocomponent, twodimensional plasma in solids.
Phys Rev B 1981, 23:805. Publisher Full Text

Szczyrbowski J: A new simple method of determining the effective mass of an electron or the thickness of thin metal films.
J Phys D: Appl Phys 1986, 19:1257. Publisher Full Text

Johnson PB, Christy RW: Optical constants of the noble metals.
Phys Rev B 1972, 6:4370. Publisher Full Text

Etchegoin PG, Le Ru EC, Meyer M: An analytic model for the optical properties of gold.
J Chem Phys 2006, 125:164705. PubMed Abstract  Publisher Full Text

Siegel PH: Terahertz technology.
IEEE Trans Microwave Theory Tech 2002, 50:910. Publisher Full Text