Optical Properties of Bismuth Nanostructures Towards the Ultrathin Film Regime

Bulk bismuth presents outstanding optical properties, such as a giant infrared refractive index (n near 10) and a negative ultraviolet visible permittivity induced by giant interband electronic transitions. Although such properties are very appealing for applications in nanophotonics, the dielectric function of bismuth nanostructures has been scarcely studied. Here, we determine by spectroscopic ellipsometry the far infrared to ultraviolet dielectric function of pulsed laser deposited bismuth thin films with nominal thickness tBi varied from near 10 nm to several tens of nm. For tBi above 15 nm, the films display a continuous structure and their dielectric function is comparable with that of bulk bismuth. For tBi below 15 nm, the film structure is discontinuous, and the dielectric function differs markedly from that of bulk bismuth. It is proposed from FDTD simulations that this marked difference arises mainly from effective medium effects induced by the discontinuous film structure, where quantum electronic confinement does not play a dominant role. This suggests that ultrathin and continuous bismuth films should present the same outstanding optical properties as bulk bismuth for high performance nanophotonic devices.

. Far infrared -to -ultraviolet ellipsometry characterization of the films and of their dielectric function. (a) Spectra of the ellipsometric angles  and  at the angle of incidence of 70º: experimental spectra (dots) and the corresponding best-fit ones (lines). (b) Spectra of the real and imaginary part (1 and 2) of the best-fit dielectric function of each film. The model used for the ellipsometry analysis is shown on the left panel.
To determine the dielectric function  = 1 + i2 of each Bi film, its experimental  and  spectra measured at different angles of incidence were fitted simultaneously using a bilayer model. This model, which is depicted in Figure 1b and further detailed in Annex A2, Figure S2, consisted of a layer of dielectric function  covered by a roughness layer of dielectric function rough.  was described by a sum of Kramers Kronigconsistent oscillators. rough was described with a Bruggeman model mixing  and the dielectric function of air with an equal weight. The thicknesses of the two layers, t and trough were fixed at the values found from the structural characterization, as explained in Annex A1.
In short, trough is the average peak-to-valley roughness found by atomic force microscopy (AFM), and t is the remaining film thickness underneath this roughness as determined by combining AFM, RBS and scanning electron microscopy (SEM). The t and trough values are given in Annex A2, Table S1, together with the corresponding geometric thickness of the films: t + trough. With such thicknesses being fixed, the only parameters left free during the fit procedure were those of the Kramers Kronigconsistent oscillators describing the dielectric function  of the film.
An excellent agreement between the best-fit spectra and the experimental ones is found for all the films (Figure 1a and Annex A2, Figure S1). The corresponding best-fit spectra It is worth noting that the dielectric function of the Bi film with tBi = 17 nm is already comparable with that of bulk Bi, with peak 1 and 2 values reaching 70% of the bulk ones. Therefore, the dielectric function of very thin Bi films, say with tBi > 15 nm, is comparable with that of bulk Bi. In contrast, for thinner films (here, for tBi = 11 nm) the dielectric function departs strongly from the bulk one. This trend correlates with the film structure, which is continuous for tBi > 15 nm and discontinuous for tBi = 11 nm.

Structure of the Bi Films
To demonstrate this correlation, we show the structure of selected films (tBi = 11 nm, 21 nm and 78 nm) studied by SEM and AFM. The top-view SEM images presented in Figure   2a show that the film with tBi = 11 nm presents a discontinuous near-percolation structure, where voids with a near 10 nm width separate clusters of densely packed/coalesced nanoparticles. The in-plane size of these clusters ranges from 50 to 150 nm. For larger tBi values, the films present a continuous structure consisting of densely packed grains and few to no voids. Upon increasing tBi from 21 nm to 78 nm, the in-plane size of the grains increases from 50-150 nm to 100-200 nm while the few remaining voids are filled.   width is too small compared with the tip size. For the same reason, they might underestimate the depth of the contact region between densely packed/coalesced nanoparticles. Therefore, combining SEM and AFM measurements is necessary to provide a full picture of the film structure, especially for discontinuous near-percolation films such as that with tBi = 11 nm.
As summary, Figure 2c shows a cross-section schematic representation of the structure of the 3 films. For tBi = 11 nm, the film has a discontinuous near-percolation structure.
For tBi = 21 nm, the film has a continuous structure with few voids. For tBi = 78 nm, the film is also continuous yet without voids. These trends are a direct consequence of the growth mechanism of Bi on the surface-oxidized Si substrate, which follows a nucleation growthcoalescencepercolation scheme as tBi increases, in the same way as noble metals (Annex A3).
This growth mechanism impacts the optical properties of the film. Upon increasing tBi, the dielectric function of the Bi film gets closer to that of bulk Bi as the film becomes continuous and the density of voids decreases. It is especially remarkable that the strongest variation in 1 and 2 occurs when tBi increases from 11 to 17 nm and the film structure turns from discontinuous to continuous with a few voids. The variation in 1 and 2 is smaller when tBi increases above 17 nm and the few voids in the continuous film are gradually filled.

Discussion: Relation between the Structure and Optical Properties of the Bi Films
All the previous results point at a dominant effect of the Bi film discontinuity on the measured dielectric function when tBi = 11 nm. In order to investigate the origin of such effect, finite difference time domain (FDTD) simulations of the optical properties of a discontinuous Bi film were performed. To simplify the problem while including the main structural features of such film, it was considered as a square array of densely packed truncated nanospheroids. We assumed that the material constituting these nanospheroids has the same dielectric function as bulk Bi taken from ref. 1. The nanospheroid height H was 17 nm, the nanospheroid diameter D was varied between 20 and 50 nm, and the separation gap G between nanospheroids was varied between 0 and 10 nm, in accordance with the geometrical film thickness t + trough (Annex A2, Table S1), nanoparticle in-plane size and void width found for the film with tBi = 11 nm. Further technical details about the FDTD simulations are given in Annex A4 (Figure S4). In Figure 3a are shown the elementary cell used for the simulation with D = 50 nm and G = 10 nm and the corresponding electric field map at a photon energy of 1.5 eV. This map reveals mildly warm spots between the nanospheroids, which also present low internal and transmitted fields as a result of their strong absorption and dense packing.
FDTD simulations also provided the reflectance of the discontinuous film for the different values of D and G (Annex A5, Figure S5). The obtained reflectance values are much smaller than those of a continuous Bi film with the same layer thicknesses (t and trough) as the film with tBi = 11 nm. This implies that the simulated effective dielectric function of the discontinuous film also presents much smaller values than the bulk one, as shown in  Furthermore, simulations show that 2 peaks at a different photon energy (between 0.8 and 1.5 eV) depending on the values of D and G. Therefore, in a random array of nanospheroids characterized by a distribution of D and G, the 2 spectrum would display a broad band spreading in the near infrared and visible, as the one measured for the discontinuous Bi film with tBi = 11 nm (Figure 1b).
This leads us to propose that the dielectric function of the thinnest film studied here (tBi = 11 nm) is very different from that of bulk Bi mainly because of effective medium effects. The dielectric function measured for this film is an effective quantity resulting from the polarization of a Bi:void heterogeneous medium, where the Bi nanostructures are described by a dielectric function close to the one of bulk Bi. Such effective medium effects may also affect, yet more weakly, the dielectric function of thicker films still containing a few voids. This would explain their small difference with the dielectric function of bulk Bi.
In contrast, it seems that quantum electronic confinement does not have a dominant effect on the far infraredtoultraviolet dielectric function of even the thinnest film studied here, despite of the fact that it presents a discontinuous structure built from nanoparticles. This conclusion is in line with our previous work [5] in which we modeled satisfactorily the measured ultravioletvisible plasmonlike resonances of flattened Bi nanoparticles using classical electrodynamic calculations based on the dielectric function of bulk Bi. Therefore, a continuous 10 nmthick or even thinner Bi film might present a dielectric function comparable to the bulk one. A similar trend has been observed in the case of few nm Bi2Se3 films based on a careful material fabrication and spectroscopic ellipsometry characterization [20,21].

Conclusion
Summarizing, our results suggest that the outstanding optical properties of bulk Bi are shared by continuous Bi films down to the ultrathin film regime (thickness < 10 nm). This opens the way, for instance, to achieving a near total absorption of visible light with ultrathin and continuous Bi films. Also, they show that the dielectric function of bulk Bi can be used as input value for the rational design of flattened nanostructures such as nanocylinders or nanoflakes. Besides that, we also remark that the effective medium properties we put forward for the discontinuous Bi film are appealing for the fabrication of films with light trapping properties optimized by nanoscale design. Tailoring the film nanostructure [5,22] enables tuning its effective optical dispersion in a broad range. This is useful to meet the requirements for an optimal light harvesting [23] in Bi nanostructured materials, in particular for photocatalytic systems based on near-percolation Bi nanostructures [24][25].    Table S1. Nominal thickness tBi of the films, layer thicknesses trough and t used for the ellipsometry analysis, and geometrical thickness of the films, t + trough. The methods used to determine thicknesses are explained in Annex A1. Note that tBi is the geometrical thickness that a film would present if its Bi atoms were perfectly arranged, i.e. if it had a perfectly flat surface and a continuous structure with the atomic density of bulk Bi. However, in this work the films are not perfectly flat and the thinnest one has a discontinuous structure. Therefore, for these films tBi is smaller than the geometrical film thickness, t + trough.

A1. Details about the Fabrication and Characterization of the Bi Films
nm, coalescence events (step 3) make the nanoparticles grow laterally [5]. Percolation is reached for tBi above 10 nm for which the deposit consists of a discontinuous layer (step 4). In line with that, the film with tBi = 11 nm studied in this work displays a near-

A4. Details about the FDTD Simulations of Discontinuous Bi Films
3D-FDTD simulations were performed with the OptiFDTD32 software. A parallelepipedal elementary cell, with the geometry shown in Figures S4a and S4b, was used.  Wave Parameters and Reflectance Simulations. The incident electric field was xpolarized and incident from the air medium, downwards along the z axis, i.e. at normal incidence. Reflectance spectra were calculated by inverse Fourier transform of a light pulse, analyzed in the viewer plane located ~ 60 nm above the nanospheroid. This distance is sufficiently large so that the viewer plane lies in the far-field region. A mesh size of 0.5 nm was used for all the calculations and enabled obtaining converged reflectance spectra.

Effective Dielectric Function.
To determine the effective dielectric function  = 1 + i2 of the discontinuous Bi film formed by the square array of nanospheroids, its FDTDcalculated reflectance spectrum was fitted assuming a bilayer structure. This structure is shown in Figure S4c.  consisted of a sum of Kramers Kronig -consistent oscillators whose parameters were left free during the fit. rough was described by a Bruggeman model mixing  and the dielectric function of air with equal weight. The thicknesses of the bottom and top layer were set to 11 and 6 nm, respectively, to match with the layer thicknesses t and trough used to model the film with tBi = 11 nm and so that the corresponding geometric film thickness equals the nanospheroid height H. Figure S5. FDTD reflectance spectra for discontinuous Bi films consisting of a square array of Bi truncated nanospheroids, with different diameter D and separation gap G, and a height H = 17 nm. These spectra are compared with that of a continuous Bi film with the same layer thicknesses (t = 11 nm, trough = 6 nm) as the film with tBi = 11 nm.