Anisotropic light scattering by prismatic semiconductor nanowires

1. Introduction

The interaction of light with micrometric or nanometric sized particles or other nanostructures, made of dielectric materials with high refractive index, is a very rich and active field of research [1]. Part of the interest is driven by the phenomenology associated to the scattering of light in the optical range, i.e., of wavelength comparable to the size of the scatterers, without significant losses, unlike the scattering on metallic systems of similar size [2–4]. This interest includes the study of fundamental effects of the light-matter interaction at nanoscale, like the presence of nonconservative forces related to the momentum transfer [5–8] and the coupling of nanoparticles and light propagation due to the near-field effects of the scattered optical field [9,10]. The material and the geometry of the nanoparticles allow for the tunability of the electric or magnetic resonances, which often coexist, making them interesting components of optical devices to manipulate visible light at nanoscale [11–16].

The classical textbook example of light scattering on spherical nanoparticles can be analyzed with the old Mie theory [17]. However, the experimental evidence of specific resonances is still an ongoing research subject [18–20]. More recently a variety of objects of nanometric size have been considered for light scattering, like nanoblocks, nanopyramids [21,22], or nanowires [23–25]. This wide range of nanostructures shows complex phenomenology related to geometry, internal composition and material properties, which includes directional scattering through the interference of different resonant modes [26–29], second and third-harmonic generation [30], or Fano resonances [31].

Semiconductor based nanowires can be produced with various cross section geometries which have an influence on their interaction with light. Nanowires of III-V materials produced by bottom-up techniques grow typically as hexagonal columns [32–35]. Other prismatic geometries, although less common, have also been obtained, like square [36] or triangular [37–40]. A special interest lies in the fabrication of core-shell nanowires, i.e., radial heterojunctions of two different materials. Elaborated bottom-up growth methods can lead to combined structures such as a hexagonal core surrounded by a triangular shell [41,42]. Another interesting aspect is that the core may be etched out leaving behind a tubular nanowire with vacuum inside [33,43].

The fabrication of nanowires with a well defined geometry and/or internal structure has recently motivated experimental and theoretical research on their electronic transport properties [44–47], on Majorana based devices [48,49] and on their potential for optoelectronics [50–52]. However, to the best of our knowledge, the interaction of light with prismatic semiconductor nanowires has not received much attention.

In our present work we consider nanowires with polygonal cross sections and explore the effects of the prismatic geometry on the scattering when the direction of light incidence is varied, from being perpendicular to the sides to being perpendicular to the edges of the prism. The scattering cross section is weakly dependent on the illumination direction, only for certain modes a significant difference is observed. For these modes, the radiation patterns show both quantitative and qualitative variations. In this context, the control of the resonances and the tunable directional scattering have already been demonstrated to allow for interesting applications, e.g., in sensing and switching, both with single nanoparticles [53–56] or with dimers [57–60]. Tunable directivity in rectangular nanowires has also been demonstrated by changing the polarization of the incident light or by changing the illumination direction from transversal to longitudinal incidence [61]. Usually, either a change in the geometry of the structure or in the characteristics of the incident light is needed for the tuning of the directivity. The possibility of controlling light scattering by a rotation has only recently been considered in pyramidal silicon nanostructures [62]. In this work, we demonstrate that, due to the natural anisotropic scattering which results from their geometry, prismatic nanowires allow for in-situ tuning the directivity. This is possible by means of a simple rotation, either of the nanowire or the direction of the incident light, and thus with a flexibility that offers potential advantages functioning as switches or for sensing purposes.

2. Model

We consider prismatic nanowires of hexagonal, square, and triangular cross section, as shown in Fig. 1, with radii between 100-250 nm, in the optical/near-infrared range, where the wavelength is several times larger. The length of the nanowires varies from values comparable with the radius up to several times larger. We calculate numerically the scattering cross section using finite differences in COMSOL Multiphysics [63]. The simulation domain consists of a sphere with 800 nm radius which has an outer perfectly-matched layer of 400 nm thickness, with the nanowire located at the center of the sphere.

 

Fig. 1. Sketch of the two polarizations (TE and TM) of the incident light, for example on a nanowire with square cross section, and the different angles of incidence studied in each polygonal geometry: hexagonal, square, and triangular.

Download Full Size | PPT Slide | PDF

In all the studied cases, the nanowires have the same refractive index $n = 3.5$, which could correspond to GaP at $\lambda = 530$ nm [64] and is similar to other semiconductor materials like Si, Ge or III-V compounds in the optical and infrared range. Since we are only interested in the effects of the geometry, the absorption is neglected in all cases. We note, however, that our results would still hold if absorption was taken into account, as the scattering spectra can be red-shifted to a wavelength range where it becomes negligible, by means of changing the wavelength and the dimensions accordingly to Maxwell’s equation scaling. The surrounding medium is air, i.e., refractive index $n = 1$. The meshing maximum size is chosen to be a sixth of the wavelength in each material, $\lambda /6n$, and the Maxwell equations are solved for the scattered field in the far-field approximation.

Using the computational method described above we have recently reproduced the scattering spectra of cylindrical nanowires, including Fano resonances [65]. For prismatic nanowires the spectra are expected to differ, due to the change of geometry and the decrease of the effective scattering volume. Another variable in the present case is the incidence angle of the light, as indicated by the colored arrows in Fig. 1.

3. Results

We first study the influence of the incidence angle, both in the scattering spectra and in the radiation patterns, for short ($L =$ 200 nm) nanowires (or nanoparticles) and long nanowires ($L =$ 600 nm). The scattering cross section is normalized by dividing, in each case, by the geometrical cross section of a cylindrical nanowire of the same dimensions (radius and length).

The geometrical cross section, i.e., the effective surface of the prismatic nanowire perpendicular to the direction of the incoming light, and the scattering conditions depend on the angle of incidence. Thus, a variation of the scattering spectra is expected to appear as the angle of incidence changes. Because of the geometric symmetries this angle has limited ranges, between $0-{30}^\circ$ for the hexagonal, $0-{45}^\circ$ for the square, and $0-{60}^\circ$ for the triangular case (Fig. 1). The geometrical cross section seen by the light varies over these incidence intervals by a factor of $2/\sqrt {3}\approx 1.15$ for the hexagonal and triangular cases, and by $\sqrt {2}\approx 1.41$ for the square, indicating that the square geometry is expected to be more effective. The anisotropy is also expected to increase with the radius of the nanostructures, as the absolute differences in the geometrical cross sections under a change in the incidence angle would also increase, getting closer to the incident light wavelength.

In particular, for triangular nanowires, a rotation of ${60}^\circ$ leaves the geometrical cross section unchanged, but due to the lack of an inversion center, it leads to different scattering conditions when the light impinges along a symmetry axis from opposite directions. This effect has already been reported in a recent paper for Si nanopyramids [66]. For the same reason, even if within a specific range, the scattering spectra show almost no difference for different angles of incidence, there may still be a difference in the contribution from each individual mode [67]. We also note that, under TM polarization the radiation pattern is no longer symmetric when the light incides along one of the sides of the triangle.

3.1 Short nanowires

We consider light in the near infrared range ($\lambda = 600-1200$ nm). The radii of the nanowires are $R = 175, 200, 250$ nm for the hexagonal, square, and triangular profile, respectively and the length is $L = 200$ nm. Here by radius we mean the distance between the center of the polygon and the corners. The reduction of the number of sides leads to a decrease of the volume (keeping the same radius), thus both a blue-shift and a decrease of amplitude of the scattering cross section follows. To compensate this effect and to be able to compare the results for the three different geometries, the radii are chosen in such a way that the low order modes, i.e., electric (ED) and magnetic (MD) dipoles and magnetic quadrupole (MQ), are the main contributors to the total scattering within the same range. In Fig. 2 we show the scattering cross section for the three prismatic geometries, with different angles of incidence and in Fig. 3 we show radiation patterns for the selected wavelengths.

 

Fig. 2. The scattering cross section vs. wavelength, for different incidence angles of light, for short nanoprisms with length $L = 200$ nm, of hexagonal (a,b), square (c,d), and triangular (e,f) cross section, with radii $R = 175, 200, 250$ nm, respectively. Figures a, c, e correspond to TE polarization and b, d, f to TM polarization. The colors of the curves match the colors of the incidence angle in Fig. 1: the red and blue curves correspond to the scattering cross section obtained with the light incident along the corner-center direction and incident perpendicular to one of the facets of the prism, respectively. In the triangular case the total scattering cross section with the light inciding at ${60}^\circ$ is indistinguishable from the case corresponding to ${0}^\circ$ and the dashed curves correspond to the individual contributions of the dipole moments (ED and MD). The vertical lines correspond to the wavelength values for which the directivity patterns are shown in Fig. 3.

Download Full Size | PPT Slide | PDF

 

Fig. 3. 2D radiation patterns for the hexagon (a,b), square (c,d) and triangular (e,f) nanoprisms. ${0}^\circ$ corresponds to forward scattering. The colors of the figures again match the colors of the incidence angle shown in Fig. 1. For the hexagonal nanoparticle the wavelength is $\lambda = 540$ nm and TE polarization in (a) and $\lambda = 636$ nm and TM polarization in (b). For the square $\lambda = 636$ nm and TE in (c) and $\lambda = 816$ nm and TM in (d). For the triangular case $\lambda = 640$ nm and TE (e) and $\lambda = 1148$ nm and TM where the Kerker condition is satisfied (f). The units for the far-field norm are $10^{-7}$ V/m.

Download Full Size | PPT Slide | PDF

The scattering in the hexagonal prism shows practically no sign of anisotropy on a large interval of wavelengths. Only for the magnetic hexapole (MH) a remarkable difference appears when rotating the angle of incidence, as can be seen in Fig. 2(a) and 2(b). In Fig. 3(a) and 3(b) we show the radiation patterns corresponding to the wavelengths marked by the blue lines in Fig. 2. The radiation patterns for the hexagonal nanoparticle under the MH modes reveal a clear difference resulting from this anisotropy, as the MH is either diminished under TE or red-shifted under TM conditions. This suggests that the number of sides/corners strongly affects the modes whose symmetry matches the geometry of the sample.

Indeed, for the square geometry, a significant difference now in the MQ is observed in Fig. 2(c) and 2(d). For $\lambda = 636$ nm, under TE polarization, we observe an interesting situation (Fig. 3(c)) in which the forward scattering is almost completely cancelled when the light impinges perpendicular to one of the corners, as the scattering cross section is almost zero for that wavelength. For TM polarization a considerable red-shift of the MQ mode is observed, and the directivity pattern shown in Fig. 3(d) clearly reflects this remarkable change. Due to their matching modes, both the square and the hexagonal geometry allow for a high degree of tunability through the rotation of the incidence angle.

As we pointed out previously, for the wavelength intervals for which the scattering spectra are not sensitive to the angle of incidence, one may still expect a difference in the contribution from each individual mode, and thus different radiation patterns. However, in all the cases tested for the square and the hexagon, the directivities are almost completely insensitive. In Fig. 4 we show the asymmetry parameter corresponding to the nanoprisms in the studied wavelengths range. As can also be observed, only a slight degree of anisotropy occurs aside the aforementioned modes. Only for the triangular case this effect has a significant magnitude, in Figs. 2(e) and 2(f) we show the individual contributions of the dipole moments, where the underlying anisotropy, despite the invariance in the total scattering cross section is revealed. This is also distinctly reflected in the differences of its asymmetry parameter, specially under TE polarization (Fig. 4(e)) over the whole range of wavelengths. Remarkably, as observed at $\lambda \approx 750$ nm, the scattering direction can be switched from being predominantly forward to backward by a rotation of ${60}^\circ$, as the asymmetry parameter changes its sign. We also note that the MQ mode is clearly diminished under TE polarization as the mode no longer matches the geometry of the sample. Instead, for TM, as the mode is longitudinal, the MQ remains unaffected. The radiation patterns for the MQ are shown in Fig. 3(e), revealing a highly anisotropic scattering as expected by its asymmetry parameter.

 

Fig. 4. Asymmetry parameter as a function of the wavelength for the hexagonal (a,b), square (c,d), and triangular (e,f) nanoprisms as in Fig 2. The blue lines also correspond to the same wavelength for which the directivity patterns where obtained in Fig. 3

Download Full Size | PPT Slide | PDF

When the magnitude of the ED and the MD are equal (situation known as the first Kerker condition) the backward scattering is cancelled via destructive interference. In Fig. 3(f), the directivity pattern when this condition is satisfied, for the triangular nanowire (under TM polarization), shows a slight difference, also observable in Fig. 4(f), due the dependence of the individual dipole modes to the illumination direction.

Finally, the electric field configurations inside the nanoparticles for all the cases considered here are shown in Fig. 5. As expected, taking into account the previous results, the electric field inside the particles is rather sensitive to the angle of incidence for the cases where we observe anisotropic scattering. In particular, we note that when the light impinges the nanoparticles with TM polarization, the shift of the MH ($\lambda = 636$ nm) and the MQ ($\lambda = 816$ nm) for the hexagon and the square, respectively, is clearly reflected in the electric field configurations. We observe how both completely disappear, and how the magnitude of the electric field is remarkably diminished, when the incidence is rotated from being perpendicular to one edge to being perpendicular to one side, in accordance to the results shown in Fig. 2. The anisotropic scattering of the triangular nanoparticle, despite having the same value of the scattering cross section, is also reflected under TE polarization for the MQ ($\lambda = 652$ nm). We observe how the lack of an inversion center results in completely different field configurations inside the nanoparticle when the light impinges perpendicular to one facet or perpendicular to one edge. For TM, in the Kerker condition ($\lambda = 1156$ nm), the field configuration shows little difference, leading to isotropic scattering in accordance to the directivity patterns shown in Fig. 3

 

Fig. 5. Electric field configurations inside the nanoparticles. The direction of incidence of the light is parallel to the $x$ axis with TE or TM polarization depending on the orientation of the nanoparticles. The units of the electric field in the color scale are V/m.

Download Full Size | PPT Slide | PDF

3.2 Long nanowires

Longer nanowires have richer and more complex scattering spectra. Mie resonances now coexist with longitudinal guided modes that develop as the length increases. The guided modes are the result of Fabry-Perot interferences resulting from the longitudinal bouncing of the modes along the nanowire. This overlapping of transverse and longitudinal modes leads to sharp Fano resonances which we can observe in our spectra.

We consider now light in the range of $\lambda = 400-1100$ nm to also include higher-order modes. For these modes the wavelength is closer to the geometrical cross section difference of the structure under rotation and thus, they are much more sensitive to the incidence angle.

In Fig. 6 we show the scattering cross section dependence on the angle of incidence for the nanowires with radii of $R = 100, 135, 175$ nm for the hexagonal, square, and triangular profiles, respectively, and in Figs. 7 and 8 we show the corresponding directivity patterns and asymmetry parameters. The length is now $L = 600$ nm in all cases. The guided sharp TE$_{01}$ modes are associated to the reflections of the purely transversal MD and overlap with the broad MD background leading to the sharp Fano resonances. The guided modes for TM polarization are also associated with the MD mode, which has now longitudinal symmetry, and the reflections along the length of the nanowire lead to hybrid Fabry-Perot modes (HE$_{11}$).

 

Fig. 6. As in Fig. 2 for nanowires with radii of $R = 100, 135, 175$ nm for the hexagonal, square, and triangular geometry, respectively, and $L = 600$ nm.

Download Full Size | PPT Slide | PDF

 

Fig. 7. 2D radiation patterns for the hexagonal nanowires for TE at $\lambda = 692$ nm (a) and TM at at $\lambda = 784$ nm (b). Square nanowire for TE at $\lambda = 470$ nm (c) and for TM at $\lambda = 562$ nm (d). (e) and (f) for the triangular nanowires at $\lambda = 504$ nm and $\lambda = 444$ nm for TE and TM polarizations, respectively. The color code is the same as in Fig. 3.

Download Full Size | PPT Slide | PDF

 

Fig. 8. Asymmetry parameter as a function of the wavelength as in Fig. 4 for long nanowires.

Download Full Size | PPT Slide | PDF

The dimensions (radii and length) of the nanowires are chosen in such a way that we obtain the resonances studied in [31] and [65], which we treat as a reference point. The main Fano resonances are seen in Fig. 6(a) and 6(b), coming from the overlapping of the $TE^3_{01}$ and $HE^5_{11}$ modes, respectively, with the MD. A more detailed analysis of these modes can be found in [31] for cylindrical nanowires.

In our case, we notice a nearly isotropic scattering cross section for the hexagonal nanowire in Figs. 6(a) and 6(b), both under TE and TM, also reflected in the corresponding asymmetry parameter in Fig. 8. The radiation patterns for the $TE^3_{01}$ and $HE^5_{11}$ modes are shown in Figs. 7(a) and 7(b), respectively, and reveal a field distribution insensitive to the incidence angle. As the radius is smaller in this case, the geometrical cross section difference under a rotation of the angle of incidence is also smaller. Thus, the anisotropy is still expected to appear but for shorter wavelengths and, again, especially in those modes whose geometry matches the sample, i.e., magnetic hexapole and its associated longitudinal modes.

On the contrary, the square nanowire, Figs. 6(c) and 6(d), shows remarkable anisotropic scattering cross sections and asymmetry parameters within a wide range of wavelengths. Consequently, a rich set of directivity patterns with flexible tunability is allowed. These high-order modes are rather complex, and their interference and overlapping makes their classification almost unviable. The directivity patterns shown in Fig. 7 correspond to wavelengths where interesting features appear. A nearly complete suppression of the forward scattering (Fig. 7(c)) is still possible through the interference of high-order modes, but now a much sharper directivity is achieved. In Fig. 7(d) we observe a situation similar to the one discussed in the previous section. The radiation pattern reveals a MQ-like behavior, highly anisotropic as it matches the geometry of the square nanowire.

Interestingly, the triangular nanowire also shows a highly anisotropic directional scattering despite the negligible differences in the scattering cross section spectra, both under TE and TM polarization, as seen in the asymmetry parameter in Figs. 8(e) and 8(f). The high anisotropy also occurs within a wide range of wavelengths, and, to some degree, it also extends to longer ones. In this case, a diminishment of the forward scattering is also achievable (Fig. 7(e)) at $\lambda = 504$ nm under TE polarization when the light impinges perpendicular to one corner. Because of its asymmetric scattering under TM polarization, the angle of maximum scattering could be rotated up to $1{5}^\circ$ (Fig. 7(f)).

As for the previously discussed short nanowires, we show in Fig. 9 the electric field configurations for all the cases considered in this subsection. Here, the field configurations are much more complex as more modes, both transversal and longitudinal, contribute and overlap. It is still possible to identify the key features. We can clearly observe that the longitudinal modes are the result of the bouncing along the length of the nanowires of the transversal ones. For the hexagonal nanowire, the $TE^3_{01}$ ($\lambda = 692$ nm) and $HE^5_{01}$ ($\lambda = 784$ nm) modes change only very slightly when rotating the angle of incidence, as expected from the isotropic scattering shown in Fig. 7. For the cases of the square and, especially, triangular nanowires, the anisotropy is clearly reflected through the changes in the electric field inside the nanowires. In particular, we remark how anisotropic the electric field is in the triangular case, even when the scattering cross section shows almost negligible variations, contrarily to the hexagonal case, which is in accordance with the directivity patterns of Fig. 7.

 

Fig. 9. Electric field configurations inside the nanowires, as in Fig. 5.

Download Full Size | PPT Slide | PDF

4. Conclusions

The scattering of light by prismatic nanowires depends on the illumination direction. When the difference of the geometrical cross section of the structures under a rotation is comparable to the incident light wavelength a strongly anisotropic scattering occurs. The effect is less pronounced in hexagonal nanowires, as they are more symmetric, i.e., closer to a cylinder, but it is especially remarkable for the square and triangular geometries. In general, the modes that are more sensitive to the direction of incidence are the ones whose symmetry matches the geometry of the structure, i.e., the magnetic hexapole and quadrupole and its associated longitudinal modes for the hexagon and square, respectively. Because of the lack of an inversion center, triangular nanowires show unique features, such as anisotropic scattering despite negligible differences in the scattering cross section spectra. This anisotropy allows for the possibility of a flexible in-situ tuning of the directivity by a simple change of the incidence angle or by changing the light polarization. Forward scattering may be considerably diminished this way, allowing for interesting applications in switching or sensing.

The scattering spectra can be shifted in multiple ways by scaling either the radius, the length, or the aspect ratio of the nanowires. In particular, we note that the high-order resonant modes that we included in Fig. 7 would easily vanish from the visible domain because of absorption, which we have neglected. Still, as we mentioned, they would be achievable in larger samples, due to Maxwell’s equations scaling, by red-shifting them to the spectral region where the absorption is negligible for most of semiconductor materials. Apart from the geometric variables, the refractive index of the material is another parameter that can shift the resonances of the scattering spectra. In this sense, core-shell structures offer an additional degree of freedom for the tunability of the scattering characteristics and proposed core-shell structures and aggregates have also been demonstrated to allow for highly directional light scattering [68–75]. Therefore, we emphasize the advantage of a core-shell nanowire where, to a first approximation, a blue or a red shift can be achieved, also shown by a recent study of the high-order modes in core-shell spherical particles [76].