Universal features of the optical properties of ultrathin plasmonic films

Igor V. Bondarev; Vladimir M. Shalaev

doi:10.1364/OME.7.003731

1. Introduction

Current development of nanofabrication techniques makes it possible to design advanced plasmonic nanomaterials with optical properties on-demand [1–5]. One type of such advanced nanomaterials are optical metasurfaces (see, e.g., Ref. [6] for review). Metasurfaces are often based on thin quasi-two-dimensional (2D) plasmonic films, which enable new physics and phenomena that are distinctly different from those observed for their 3D counterparts [6–20]. Nowadays, a careful control of the geometry, structural dimensions, and material composition allows one to produce thin and ultrathin metasurfaces for applications in optoelectronics, ultrafast information technologies, microscopy, imaging, and sensing as well as for probing the fundamentals of the light-matter interactions at the nanoscale [21–25]. A key to realizing these applications is the ability to fabricate metallic films of precisely controlled thickness down to a few monolayers, which also exhibit desired optical properties [3–5]. As the film thickness decreases, the strong electron confinement could lead to new confinement related and dimensionality related effects [6–20], which require theory development to understand their role in the light-matter interactions and optical response of thin and ultrathin plasmonic films.

We develop a quasiclassical theory for the electron confinement effects and their manifestation in the optical response of thin plasmonic films of finite variable thickness. We start with the Coulomb interaction potential in the confined planar geometry to obtain the equations of motion and the conditions for the in-plane collective electron motion. The plasma frequency thus obtained, while being constant for relatively thick films, acquires spatial dispersion typical of 2D materials and gradually shifts to the red as the film thickness decreases. As a consequence, the complex-valued dynamical dielectric response function shows the gradual red shift of its epsilon-near-zero point with the dissipative loss decreasing at any fixed frequency and gradually going up at the plasma frequency as it shifts to the red with the film thickness reduction. We argue that these are the universal features peculiar to all plasmonic thin films, which can be controlled not only by varying the thickness and material composition of the film but also by choosing deposition substrates and coating layers appropriately. We conclude with a brief comparison of our results against those reported by others for similar systems.

2. Theory

The Coulomb interaction in thin films (see Fig. 1) increases strongly with the film thickness reduction if the film dielectric constant is much larger than those of the film surroundings [26]. In general, any kind of spatial confinement results in the increase of the Coulomb interaction between charges confined as the characteristic confinement size decreases, provided that the outside environment has a lower dielectric constant than that of the bounded region. This is because the field produced by the confined charges outside of the bounded region begins to play a perceptible role with its size reduction. If the surrounding medium dielectric constant is much less than that of the bounded medium, then the increased ’outside’ contribution makes the Coulomb interaction between the charges confined stronger than that in a homogeneous medium with the dielectric constant of the bounded region. Specific examples to confirm this fact, theoretical and experimental ones, can be found in the literature both for quasi-1D and for quasi-2D confined geometries [27–29].

Fig. 1 (a,b) Schematic to show the geometry notations and the normalized electrostatic (Keldysh) potential for the Coulomb interaction of the two point charges e and e′ confined in a planar thin film of finite thickness.

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For the thin film of thickness d with the background dielectric constant ε, surrounded by media with the dielectric constants ε₁ and ε₂ as shown in Fig. 1 (a), the Coulomb potential between the two confined charges e and e′ loses its z-coordinate dependence to turn into a pure in-plane 2D potential when ε_1,2 ≪ ε and the in-plane inter-charge distance ρ ≫ d. The potential takes the form (first reported by L.V. Keldysh [26])

V (ρ) = \frac{π e e^{'}}{ε d} [H_{0} (\frac{ε_{1} + ε_{2}}{ε} \frac{ρ}{d}) - N_{0} (\frac{ε_{1} + ε_{2}}{ε} \frac{ρ}{d})],

where N₀(x) and H₀(x) are the Neumann and Struve functions, respectively. This (Keldysh) potential can be further expanded to give two different asymptotic expressions as follows

V (ρ) \approx V_{1} (ρ) = \frac{2 e e^{'}}{ε d} [\ln (\frac{2 ε}{ε_{1} + ε_{2}} \frac{d}{ρ}) - C],

(C ≈ 0.577 is the Euler constant) if ε/(ε₁ + ε₂) ρ/d ≫ 1, and

V (ρ) \approx V_{2} (ρ) = \frac{2 e e^{'}}{(ε_{1} + ε_{2}) ρ},

if ρ/d ≫ ε/(ε₁ + ε₂) 1. This latter expression includes no parameters to represent the thin film itself, and it shows no screening at all for the film in air (ε₁ = ε₂ = 1), which is quite an interesting result.

Figure 1(b) presents the normalized Keldysh potential [the ratio V (ρ)/V₂(ρ) with V (ρ) and V₂(ρ) given by Eqs. (1) and (3), respectively] as a function of ρ/d and the relative dielectric constant (ε₁ + ε₂)/ε. The potential is seen to vary drastically in the domain where ε ≫ ε₁ + ε₂ and d ≪ ρ, which is just the parameter range for thin plasmonic films [4–6, 9]. The drastic change of the Coulomb interaction potential in this domain comes from Eq. (2), which varies much faster than any Coulomb type (~ 1/ρ) potential does — a solely confinement related effect associated with the strong spatial dispersion of the thin-film dielectric response function, the dielectric permittivity.

One can easily evaluate the plasma frequency spatial dispersion in finite-thickness plasmonic films (ε) sandwiched [Fig. 1 (a)] between dielectric substrates (ε₁) and superstrates (ε₂). Using the in-plane (2D) Fourier expansion of the Keldysh potential in Eq. (1), the Coulomb potential energy of the quasi-free outer-shell electron located at the point ρ_j = (ρ_j, φ_j) of the lattice site j, which interacts with the other electrons of the finite-thickness thin film — all immersed in the positive background of the film material (“jellium” model first proposed by Bohm and Pines [30–32]), takes the form

V (ρ_{j}) = \frac{4 π e^{2}}{ε L^{2}} \sum_{l, k}^{'} \frac{\exp (i k \cdot ρ_{j l})}{k [k d + (ε_{1} + ε_{2}) / ε]} .

Here, ʃ dk exp[ik · (ρ − ρ′)]/(2π)² = δ(ρ − ρ′) is used as the normalization condition for the Fourier expansion basis function set, with k representing the in-plane electron quasimomentum (k_x,y = 2πn_x,y/L with n_x,y = 0, ±1, …; L ≫ d stands for the square-sized film length), k = |k|, and ρ_jl = ρ_j − ρ_l. The summation sign is primed to indicate that the terms with l = j and k = 0 associated with the electron self-interaction and with the all-together electron displacement, respectively, must be dropped.

For simplicity, we proceed with the quasiclassical description of the electron behavior following Pines and Bohm [31]. The quantum description does not change the result. Using Eq. (4) along with the electron kinetic energy $K = \sum_{l} m^{*} {\dot{ρ}}_{l}^{2} / 2$ , where m^∗ is the electron effective mass, one arrives at the individual electron equations of motion

{\ddot{ρ}}_{j} = - i \frac{4 π e^{2}}{ε m^{*} L^{2}} \sum_{l, k}^{'} \frac{k \exp (i k \cdot ρ_{j l})}{k [k d + (ε_{1} + ε_{2}) / ε]} .

To obtain the equations of motion for the density of electrons, we introduce the local surface electron density

n (ρ) = \sum_{l} δ (ρ - ρ_{l}) = \sum_{k} n_{k} \exp (i k \cdot ρ)

with the Fourier components

n_{k} = \frac{1}{L^{2}} \sum_{l} \exp (- i k \cdot ρ_{l}), n_{k = 0} = N_{2 D}

(N_2D being the equilibrium surface electron density), and use Eq. (5) to get the second time derivative in the form

{\ddot{n}}_{k} = - \frac{4 π e^{2}}{ε m^{*}} \sum_{q}^{'} \frac{(k \cdot q) n_{q} n_{k - q}}{q [q d + (ε_{1} + ε_{2}) / ε]} - \frac{1}{L^{2}} \sum_{l} {(k \cdot {\dot{ρ}}_{l})}^{2} \exp (- i k \cdot ρ_{l}) .

This can now be simplified in the random phase approximation (RPA) by dropping alternating-sign terms (q ≠ k) in the sum over q (a detailed analysis of the RPA validity range can be found in Ref. [31]), to obtain after using N_2D of Eq. (6) the expression

{\ddot{n}}_{k} + ω_{p}^{2} n_{k} = - \frac{1}{L^{2}} \sum_{l} {(k \cdot {\dot{ρ}}_{l})}^{2} \exp (- i k \cdot ρ_{l})

with

ω_{p} = ω_{p} (k) = \sqrt{\frac{4 π e^{2} k N_{2 D}}{ε m^{*} (k d + (ε_{1} + ε_{2}) / ε)}} .

Equation (8) is seen to turn into the oscillator equation provided

k^{2} ≪ k_{c}^{2} = ω_{p}^{2} / υ_{0}^{2}

with υ₀ given by

m^{*} υ_{0}^{2} / 2 = E_{F}

, or k_BT for the degenerate and non-degenerate electron gas system, respectively [31]. When electron wave vectors are much less than the cut-off vector k_c, the right hand side of Eq. (8) becomes much less than

k_{c}^{2} υ_{0}^{2} n_{k} = ω_{p}^{2} n_{k} s

, yielding the thin-film electron density coherent oscillations with the plasma frequency featuring the spatial dispersion given by Eq. (9), as opposed to bulk (3D) plasmonic materials in which case the plasma frequency is the k-independent constant

ω_{p}^{3 D} = \sqrt{\frac{4 π e^{2} N_{3 D}}{ε m^{*}}}

with N_3D representing the volumetric electron density.

For thin enough plasmonic films, one has N_3Dd = N_2D, so that Eq. (9) can be written as

ω_{p} = ω_{p} (k) = \frac{ω_{p}^{3 D}}{\sqrt{1 + (ε_{1} + ε_{2}) / ε k d}} .

If (ε₁ + ε₂)/εkd ≪ 1 (relatively thick film), then

ω_{p} = ω_{p}^{3 D}

of Eq. (10), whereas one has

ω_{p} = ω_{p}^{2 D} (k) = \sqrt{\frac{4 π e^{2} N_{2 D} k}{(ε_{1} + ε_{2}) m^{*}}}

if (ε₁ + ε₂)/εkd ≫ 1 (ultrathin film), which agrees precisely with the plasma frequency spatial dispersion of the 2D electron gas in air (see, e.g., Ref. [33]), but does show the explicit dependence on bottom (ε₁) and top (ε₂) surrounding materials.

The ratio $ω_{p} / ω_{p}^{3 D}$ of Eq. (11), considered as a function of the dimensionless variables kd and (ε₁ + ε₂)/ε, represents a universal conversion factor to relate the plasma frequency parameter in quasi-2D electron gas systems (thin finite-thickness plasmonic films [4, 5], graphene [34], and related 2D materials [7, 10, 29]) to that in bulk plasmonic materials. The ratio is shown in Fig. 2(a). The regimes of the relatively thick and ultrathin films [Eqs. (10) and (12), respectively] are separated by the vertical plane kd = (ε₁ + ε₂)/ε. The ratio $ω_{p} / ω_{p}^{3 D}$ is nearly constant in the domain kd ≫ (ε₁ + ε₂)/ε, while being strongly dispersive in the domain kd ≪ (ε₁ + ε₂)/ε. In this latter case, ω_p of Eq. (11) goes down with the film thickness as $\sqrt{d}$ at all fixed k, which agrees with the recent plasma frequency ellipsometry measurements done on ultrathin TiN films of controlled variable thickness [5]. Figure 2(b) shows the contour plot of $ω_{p} / ω_{p}^{3 D}$ as a function of kd obtained by cutting Fig. 2(a) with parallel vertical planes of constant (ε₁ + ε₂)/ε. We see the gradual graph profile change in the direction of the (ε₁ + ε₂)/ε increase (shown by the thick vertical arrow), offering a controllable way to adjust the spatial dispersion and related optical properties of plasmonic thin films and metasurfaces, in particular, not only by varying their thickness [5] and material composition [6], but also by choosing the deposition substrates (ε₁) and coating layers (ε₂) appropriately.

Fig. 2 (a) The ratio $ω_{p} / ω_{p}^{3 D}$ given by Eq. (11) as a function of the dimensionless variables kd and (ε₁ + ε₂)/ε. (b) The contour plot of the same ratio as a function of kd obtained by cutting the graph in (a) with parallel vertical planes of constant (ε₁ + ε₂)/ε. The thick vertical blue arrow shows the direction of the (ε₁ + ε₂)/ε increase.

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With the plasma frequency dispersion (11) in hand, it is straightforward to obtain the complex-valued dynamical dielectric response function, the dielectric permittivity, for the electron gas confined in the finite-thickness ultrathin plasmonic films. The starting point and main ingredient of the theory is the Fourier-transform of the Coulomb potential energy in Eq. (4). With losses taken into account phenomenologically, the isotropic RPA (or Lindhard [35]) low-momentum high-frequency dielectric response function ε(k, ω) (commonly referred to as the Drude response function [9]) takes the well known form

\frac{ε (k, ω)}{ε} = 1 - \frac{ω_{p}^{2}}{ω (ω + i γ)},

where γ is the phenomenological inelastic electron scattering rate and ω_p is given by Eq. (11). This expression is normally used to describe the contribution of the outer-shell (s-band) electrons in metals [9], with ε assigned to be responsible for the positive background of the ions screened by the remaining inner-shell electrons. In many cases, however, it needs to be supplemented with an extra term (Drude-Lorentz response function [4, 5]) to account for interband electronic transitions absent from Eq. (13).

Expressing all frequency parameters of Eq. (13) in units of $ω_{p}^{3 D}$ , one obtains the universal complex-valued function to feature the dielectric response of the quasi-2D electron gas in the finite-thickness plasmonic films. Figures 3(a) and 3(b) show its real (ε′/ε) and imaginary (ε″/ε) parts as functions of the dimensionless variables $ω_{p} / ω_{p}^{3 D}$ and (ε₁+ ε₂)/εkd, and we also show in Fig. 4 the plasmon peak behavior given by the energy-loss function −Im[ε/ε(k, ω)] in terms of the same variables. All graphs are calculated with a moderate parameter ratio $γ / ω_{p}^{3 D} = 0.1$ . In Figure 3(b) we see the approach of ε″/ε to the horizontal axis and the shift of the zero point of ε′/ε from unity at (ε₁ + ε₂)/εkd ≪ 1 (relatively thick film) towards values lower than unity as (ε₁+ ε₂)/εkd increases to approach the ultrathin film limit at (ε₁ + ε₂)/εkd ≫ 1. These correspond to the dissipative loss being decreased at a fixed frequency and the plasma frequency being red shifted to go lower than $ω_{p}^{3 D}$ with the film thickness reduction, which agrees well with the recent measurements done on ultrathin TiN films of controlled variable thickness [5]. At the same time, the red shift of the plasma frequency is accompanied by the gradual increase of the dissipative loss at the plasma frequency. This is clearly seen in Fig. 3(b) as the ε″/ε magnitude rise in the zeros of the respective ε′/ε as one moves along the blue arrow, resulting in the plasmon peak red shift and broadening with increasing (ε₁ + ε₂)/εkd as Fig. 4 shows. One can also see in Fig. 4 that the energy-loss function does decrease at the plasma frequency as the plasmon peak broadens with the film thickness reduction. We stress that all these features described are universal, peculiar to all plasmonic thin films. Their specific manifestation in real experimental systems is controlled by the film thickness d, by the plasma frequency $ω_{p}^{3 D}$ , and by the relative dielectric constant (ε₁ + ε₂)/ε.

Fig. 3 (a) Real (red) and imaginary (green) parts of Eq. (13) as functions of the dimensionless variables $ω_{p} / ω_{p}^{3 D}$ and (ε₁ + ε₂)/εkd. (b) The contour plot one obtains by cutting the graph in (a) with parallel vertical planes of constant (ε₁ + ε₂)/εkd. The thick horizontal blue arrow shows the direction of the (ε₁ + ε₂)/εkd increase.

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Fig. 4 Plasmon resonance peak behavior given by −Im[ε/ε(k, ω)] of Eq. (13) as a function of the dimensionless variables $ω_{p} / ω_{p}^{3 D}$ and (ε₁ + ε₂)/εkd.

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3. Conclusion

We predict universal confinement related effects in the optical response of thin plasmonic films as their thickness decreases. The effects we report on originate from the simple fact that in finite-thickness films the Fourier transform of the inter-electron Coulomb potential (Keldysh potential) goes as 4πe²/k[kd + (ε₁+ ε₂)/ε] with k being the in-plane electron quasimomentum, and not as 4πe²/k² commonly believed, where k is the 3D quasimomentum in bulk materials. Vertical confinement causes effective dimensionality reduction and changes the way in which the electrons confined interact with each other. As a result, while being constant for relatively thick films, the plasma frequency acquires spatial dispersion ~[εkd/(ε₁ + ε₂)]^1/2 typical of 2D materials such as graphene [34], gradually shifting to the red at all fixed k as the film thickness decreases. The dielectric dissipative loss, while decreasing at any fixed frequency, gradually goes up at the plasma frequency as it shifts to the red with the film thickness reduction. This latter circumstance turns out to be favorable, leading to the effective decrease of the energy-loss function of ultrathin-film plasmonic devices at the plasma frequency, which makes them very promising candidates for a variety of practically useful nanophotonics applications.

On another note, plasma frequency spatial dispersion and associated complex dielectric response function nonlocality we report here for the ultrathin plasmonic films are quite natural for systems of reduced dimensionality. For example, our Eq. (9) can also be obtained from the plasmonic spectra of finite-thickness Dirac systems such as bilayer graphene [7] and thin films of topological insulators [8] in the limit of small in-plane electron momenta, neglecting the terms of the infinitesimal order higher than linear in k (see Eqs. (41) and (12) in Refs. [7] and [8], respectively). For this very reason, our predicted confinement related dielectric response nonlocality has nothing to do with that one obtains within the framework of hydrodynamical Drude models [36,37], where it comes about due to the pressure term ~k² in degenerated electron gas systems [32, 36]. That is why new nonlocal effects reported recently using the hydrodynamic approach for similar planar finite-thickness plasmonic systems [16–20] should be corrected appropriately to account for the plasma frequency red shift with the thickness reduction, which was lately observed experimentally for ultrathin TiN films of controlled variable thickness [5] and is derived herein theoretically.

We emphasize once again that the optical response features we report herewith are peculiar to all plasmonic thin films and so are important to know about as they offer ways to optimize the spatial dispersion and related (nonlocal) optical properties of plasmonic thin films and metasurfaces, in particular, not only by varying their material composition but also by precisely controlling their thickness and by choosing surrounding substrate and superstrate materials appropriately.

Funding

I.V.B is supported by NSF-ECCS-1306871. V.M.S is supported by NSF-DMR-1506775.

Acknowledgments

We acknowledge fruitful discussions with Alexandra Boltasseva and Harsha Reddy, College of Engineering and Birck Nanotechnology Center at Purdue University.

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Universal features of the optical properties of ultrathin plasmonic films

Abstract

1. Introduction

2. Theory

3. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (4)

Equations (14)

Optical Materials Express