Plasmon confinement in fractal quantum systems
Recent progress in the fabrication of materials has made it possible to create arbitrary non-periodic two-dimensional structures in the quantum plasmon regime. This paves the way for exploring the plasmonic properties of electron gases in complex geometries such as fractals. In this work, we study the plasmonic properties of Sierpinski carpets and gaskets, two prototypical fractals with different ramification, by fully calculating their dielectric functions. We show that the Sierpinski carpet has a dispersion comparable to a square lattice, but the Sierpinski gasket features highly localized plasmon modes with a flat dispersion. This strong plasmon confinement in finitely ramified fractals can provide a novel setting for manipulating light at the quantum scale.
Nowadays, different experimental techniques allow for the creation of arbitrary non-periodic two-dimensional (2D) lattices. For example, artificial lattices Gibertini et al. (2009); Polini et al. (2013), systems consisting of quantum dots that can be arranged in any custom shape, have attracted a lot of attention lately. More generally, nanolithography methods can be used to make high-quality 2D structures of arbitrary shape with a resolution in the order of tens of nanometers Scarabelli et al. (2015). Other methods, such as molecular self-assembly Newkome et al. (2006); Shang et al. (2015) have been used to grow Sierpinski gaskets. This presents an opportunity to experimentally study complex 2D systems, such as fractals.
Fractals have no translational invariance, so where a Bloch description is natural in the case of lattices, here it is not possible. Still, the Schrödinger equation has been solved analytically on some simple fractals with finite ramification Domany et al. (1983). For others, like the Sierpinski carpet, no analytical expressions for eigenenergies and eigenstates have been found yet. The latter systems are better tackled numerically Perreau et al. (1996). It has been shown that the quantum conductance of Sierpinski carpets exhibits fractal fluctuations van Veen et al. (2016) and that their optical conductivity features sharp peaks due to electronic state pairs at characteristic length scales present within the carpet van Veen et al. (2017). However, its plasmonic properties have not been investigated yet.
Historically, in most plasmonic devices, the Fermi wavelength of the electrons was much smaller than the plasmon wavelength which is of the order of the geometric size of the system for standing waves. In other words, the characteristic plasmon wave vector , where is the Fermi wave vector. In this regime, plasmons can be described classically and there is no need to use a quantum mechanical approach Nozières and Pines (1999); Platzman and Wolff (1973); Vonsovsky and Katsnelson (1989); Giuliani and Vignale (2005).
Recently, due to the progress in nanodevice fabrication, the quantum regime for plasmons has been reached Scholl et al. (2012); Tame et al. (2013). In this regime, localized surface plasmons make it possible to confine light to scales much smaller than the scales of conventional optics, and as such provide a unique way for light manipulation on scales below the diffraction limit. Surface plasmons have found applications in surface-enhanced spectroscopy Haynes et al. (2005); Pipino et al. (1994), biological and chemical sensing Yonzon et al. (2005), lithographic fabrication Srituravanich et al. (2004), and photonics Brongersma and Kik (2007).
However, the theory of inhomogeneous quantum electron plasma, even in the simplest random-phase approximation (RPA) Nozières and Pines (1999); Platzman and Wolff (1973); Vonsovsky and Katsnelson (1989); Giuliani and Vignale (2005), is quite complicated due to the essential nonlocality of the dielectric function Vonsovsky and Katsnelson (1989). Recently, a rigorous scattering theory of plasmons by obstacles was built Torre et al. (2017), but finding plasmon eigenmodes of inhomogeneous quantum systems still remains a challenge. As a matter of fact, this problem is very old, starting with the early considerations Bloch (1933); Jensen (1937) of “atomic plasmons” Ishmukhametov (1971); Sen (1973); Gadiyak et al. (1975); Ishmukhametov and Katsnelson (1975); Amusia and Ivanov (1978) which eventually turned out to not exist Verkhovtseva et al. (1976); Ishmukhametov et al. (1981). Previous attempts use additional uncontrollable approximations such as truncation of quantum states Amusia and Ivanov (1978), semi-classical Ishmukhametov (1971); Ishmukhametov and Katsnelson (1975); Ishmukhametov et al. (1981) or even classical Gadiyak et al. (1975) approaches.
Here we will present the results of accurate, straightforward calculations of plasmon spectra in an inhomogeneous quantum system with nontrivial geometry, namely Sierpinski carpets and gaskets, two prototypic examples of infinitely and finitely ramified fractals, respectively. These two types of fractals can have widely different properties. For example, it has been found that infinitely ramified fractals exhibit phase transitions not present in finitely ramified fractals Gefen et al. (1984).
In this letter first we outline the methods used and present a numerical method for calculation of plasmonic properties of systems with no translational invariance that is applicable to arbitrary geometries. Then, we discuss the results of these calculations on fractal systems. We compare the plasmon dispersions of the Sierpinski carpet and gasket to those of a square and triangle, respectively.
We consider a system described by a tight-binding Hamiltonian
where is the hopping parameter. Here, we have taken the on-site potential to be zero and only consider nearest-neighbor hoppings. The two systems of interest are illustrated in Fig. 1.
Fractals are made using an iterative process. For example, to make the Sierpinski carpet, a previous iteration (indicated in red in Fig. 1) is copied times to make a next iteration that is times wider. With each fractal we can associate a Hausdorff dimension, given by , as a measure for how space-filling its structure is. For the carpet , for the gasket .
Moreover, for each fractal we can define a ramification number, giving a measure of how connected it is. The Sierpinski carpet is infinitely ramified: as a higher iteration is taken, the number of bonds that need to be cut to separate it from a lower iteration goes to infinity. In contrast, the Sierpinski gasket is finitely ramified.
We use a hopping parameter and a lattice constant . These are the parameters for graphene, and they are representative for 2D systems in general. Choosing a different lattice constant will lead to a different plasmon spectrum, but the same qualitative behaviour.
Using this tight-binding model we obtain the exact eigenstates with corresponding eigenenergies , to use for the calculation of the dielectric function.
The dielectric function operator , by definition, relates the external potential to the total potential :
is the dimension of our problem. For the systems considered here . Treating as a perturbation, within RPA, the dielectric function may be expressed as follows Vonsovsky and Katsnelson (1989):
denotes a position eigenvector; is the Coulomb interaction potential; is the polarizability function; is the inverse relaxation time; is spin degeneracy; is ’th energy level occupational number according to the Fermi-Dirac distribution
We used room temperature and an inverse relaxation time .
Eqs. (3) allow us to exactly calculate the full dielectric function of any tight-binding system without translational invariance. The open source project documentation Westerhout (2017) lists the computational techniques employed which, despite the algorithmic complexity, make calculations possible for systems of up to several thousands of sites.
To visualise the plasmon modes in a quantum mechanical system Wang et al Wang et al. (2015) introduced the following method. Consider the dielectric function in its spectral decomposition:
In this method, for each we consider only the eigenvalue that has the highest value of , which gives us the plasmon eigenmode that contributes most to the loss function.
However, it is not clear how to access these plasmon modes experimentally. Currently, the standard way of probing plasmon properties of small quantum mechanical systems is EELS. The fact that we calculate the full dielectric function gives us the possibility to calculate the following Fourier transform, which distinguishes this study from others:
The loss function is then directly measurable using EELS techniques Nozières and Pines (1999); Platzman and Wolff (1973); Vonsovsky and Katsnelson (1989); Giuliani and Vignale (2005); Lu et al. (2009).
Formally, there are two ways of identifying plasmons. A plasmon frequency is either given by a local maximum of the loss function , or by a frequency at which . These frequencies are not exactly equal due to Landau damping, which is quantified by Andersen et al. (2012).
The real-space loss function of the highest contributing plasmon mode is shown in Fig. 2. It shows that there is a large number of plasmon frequencies, and that the associated losses increase with increasing frequency. At each discontinuity in a different mode is found to be the highest contributor to the loss function. Such a discontinuity is not associated with a plasmon, even though switches sign.
The real part of the highest contributing plasmon eigenmodes for both the carpet and gasket are shown in Fig. 3. For further analysis, the inverse participation ratio can give us a measure of localization. The average IPR of was found to be an order of magnitude higher for the gasket than for the carpet. This can be seen as a consequence of the finite ramification of the gasket, i.e. the fact that it is less connected, and therefore the electrons are more confined and exhibit more localized plasmon eigenmodes. Fig. 3(d) shows an example of such a highly localized mode.
We now turn to the Fourier transform of the real-space loss function in order to make a comparison to EELS experiments. Fig. 4 shows the loss function as function of both and .
There is a close resemblance between the carpet (Fig. 4(a)) and a square sample (Fig. 4(b)). The dispersion of the carpet has extra broadening, similar to the broadening found in systems with disorder Jin et al. (2015). However, generally speaking, both curves look like a regular dispersion relation for surface plasmons Giuliani and Vignale (2005). The carpet exhibits no translational invariance, i.e. is not actually a good quantum number, so this behavior is quite remarkable. The dispersion of the fourth iteration Sierpinski carpet is already very close to the third iteration dispersion. This convergence indicates that the result is representative for the real fractal at infinite iteration.
For the Sierpinski gasket (Fig. 4(c)), we observe different behavior. This fractal does not closely follow the dispersion relation of a triangle built out of a triangular lattice (Fig. 4(d)). Instead, we can clearly see the formation of multiple localized modes with near flat dispersion. Again, this result is reasonably converged.
Concluding, in this work we have calculated the plasmon dispersion for the Sierpinski carpet and Sierpinski gasket. The Sierpinski carpet has a plasmon dispersion comparable to the dispersion of a square lattice, whereas the gasket exhibits highly localized plasmon modes. More generally, a finitely ramified fractal can exhibit strong plasmon confinement, providing a novel setting for the manipulation of light at the quantum scale. With current experimental techniques, these results can be probed experimentally. Moreover, we have presented a rigorous approach for calculating plasmonic properties of generic tight-binding systems, published as an open source software project Westerhout (2017). We believe that this code can be very useful for future projects relating to plasmonic properties of non-translationally invariant systems.
This work was supported by the National Science Foundation of China under Grant No. 11774269 and by the Dutch Science Foundation NWO/FOM under grant No. 16PR1024 (S.Y.), and by the European Research Council Advanced Grant program (contract 338957) (M.I.K.). Support by the Netherlands National Computing Facilities foundation (NCF), with funding from the Netherlands Organisation for Scientific Research (NWO), is gratefully acknowledged.
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