1. Introduction

Semiconductor nanowires (NWs) have recently attracted significant interest because of their potential for the miniaturization of low-power electronic and optoelectronic devices. Because subwavelength NWs cannot support photonic Fabry-Perot resonances, their hybrid integration in optical nanocavities presenting a high quality factor (${\text{Q}}_{\text{c}}$) on mode volume (${\text{V}}_{\text{m}}$) ratio is of particular importance for the development of nanophotonic devices. This has already led to the demonstration of the Purcell effect in photonic crystal (PhC) nanocavities [1] and the realization of plasmonic [2–4] or PhC [5] nanolasers. To circumvent fabrication difficulties, most studies have reported the integration of NWs in plasmonic cavities [2–4], but in order to minimize optical losses, one would rather embed NWs in dielectric PhC nanocavities [1, 5–8]. In particular, semiconductor NWs positioned in grooved silicon PhCs have recently been shown to be a promising platform to achieve high-Q nanocavities at telecommunication wavelengths [1]: as opposed to more classical structures, such a nanocavity is not induced by a structural modulation of the PhC but rather by the sole presence of the NW in the grooved PhC, which makes it a highly versatile and spatially tunable design. To extend this hybrid approach to the visible and ultra-violet (UV) range and allow for the coupling with semiconductor NWs emitting in this range (e.g. CdS, CdSe, GaN, InGaN, ZnO, GaP, diamond), we here propose to use an alternative material, namely silicon nitride, which has the advantage of circumventing the absorption cutoff of silicon, while being easily processable. Let us note that silicon nitride has already been shown to be suitable for the fabrication of bare PhC nanocavities with quality factors up to $5.5\times {10}^4$ at λ = 624nm [9] and that hybrid integration with diamond nanocrystals [10] or CdSe NWs [6] has already been demonstrated in the red and near infra-red ranges.

In the present paper we show by means of three-dimensional finite-difference time-domain (3D-FDTD) calculations that the hybrid approach of [1] is significantly more challenging when using a lower refractive index material such as SiN (${\text{n}}_{\text{r}}^{\text{SiN}}=2$). We thus propose various strategies to circumvent such difficulties and demonstrate that the hybrid approach of [1] can be extended to single NWs based on semiconductors emitting at shorter wavelengths e.g. diamond, ZnO or GaN (${\text{n}}_{\text{r}}^{\text{NW}}~2.4$). In particular, we show that the size of the mode-gap as well as symmetry issues are key to the formation of NW-induced cavities based on two-dimensional (2D) PhC cavities. Furthermore, inspired by works demonstrating that one-dimensional (1D) PhCs can be beneficial for the formation of high-Q cavities in low refractive index materials [7,11], we dedicate a large part of our study to investigating the effect of PhC dimensionality by looking at NW-induced 1D PhC nanocavities, a novel design that was yet to be investigated. We also discuss how the quality factor and mode volume of such nanocavities are influenced by critical design parameters such as PhC geometry, groove dimensions as well as NW geometry and position. Finally, we find that with realistic design parameters optimized for the UV range, quality factors as high as ${\text{Q}}_{\text{c}}=2.5\times {10}^4$ can be obtained in grooved 2D PhC line-defect waveguides with a mode volume ${\text{V}}_{\text{m}}=5.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$, where $\text{λ}$ is the resonant wavelength. In grooved nanobeam PhCs, quality factors can even reach ${\text{Q}}_{\text{c}}=5.1\times {10}^4$ with a much smaller mode volume ${\text{V}}_{\text{m}}=1.8{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$.

2. Definitions and calculation parameters

In this work, we consider NWs with a refractive index ${\text{n}}_{\text{r}}^{\text{NW}}=2.4$ embedded in SiN PhCs with a refractive index ${\text{n}}_{\text{r}}^{\text{SiN}}=2.0$. This PhC refractive index is much smaller than in previously reported NW-induced nanocavities based on grooved silicon PhCs [1,8]. As a result, the photonic bandgap (PBG) as well as the design parameter space allowing for light confinement are significantly reduced. The PhC slab is oriented parallel to the xz plane, its thickness is t, its lattice constant is a and we assume an infinite groove oriented along the z axis with a rectangular section defined by its height ${\text{h}}^{\text{groove}}$ and width ${\text{w}}^{\text{groove}}$. The NWs are positioned inside the groove along the z axis, their length is ${\text{L}}^{\text{NW}}$ and they have a rectangular section in the xy plane defined by its height ${\text{h}}^{\text{NW}}$ and width ${\text{w}}^{\text{NW}}$. We also consider circular NW sections with diameter ${\text{δ}}^{\text{NW}}$, which were not investigated in previous studies dealing with NW-induced nanocavities in grooved PhCs [1,11]. In both cases, the section is small enough so that the NW cannot support photonic Fabry-Perot modes in the considered wavelength range. In the following, we will sometimes refer to both the groove and the NW as the “core” of the structure.

The PhC properties are investigated by 3D-FDTD using a $\text{a}/20$ grid-spacing. For 2D PhCs, the calculation domain corresponds to 50a, 20a and 80a respectively along the x, y, and z axes. For 1D PhCs, the calculation domain corresponds to 20a, 20a and 100a respectively along the x, y, and z axes. As boundary conditions, we use 8 perfectly matched layers in every direction. Band-diagrams are obtained by a Fourier transform of the magnetic field component $\left|{\text{H}}_{\text{y}}\right|$, using a broadband excitation source in a TE-like configuration. Cavity field distributions and energy decays are then obtained using a narrow-band excitation source, the wavelength of which matches the cavity resonance. Quality factors of cavity modes are extracted from the energy decay and mode volumes are obtained from the electric field E distribution, following ${\text{V}}_{\text{m}}={{\displaystyle \sum }}^{\text{}}\frac{{\text{ε}}_{\text{r}}{\left|\text{E}\right|}^2}{{\text{ε}}_{\text{r}}{\left|{\text{E}}_{\text{max}}\right|}^2}$. Further details about the design and confinement principles of NW-induced cavities based on 2D and 1D PhCs are given in the following Sections. The key parameters and figure of merits of all investigated designs are referenced throughout the paper and summed up at the end of each Section in Tables 1 and 2 respectively for 2D and 1D PhCs.

Table 1. Summary of the cavity designs investigated in Section 3.

View Table | View all tables in this article

Table 2. Summary of the cavity designs investigated in Section 4.

View Table | View all tables in this article

3. Two-dimensional PhC cavities

3.1 Cavity design

As represented in Fig. 1(a), NW-induced cavities based on 2D PhCs are constituted of a hexagonal lattice of air holes in a SiN slab. The structural parameters of the cavity design are represented in Fig. 1(b). The slab thickness is $\text{t }=\text{a}$ and the radius of circular holes is $\text{r}=0.25\text{a}$, allowing for the opening of a narrow PBG [Fig. 1(c)]. A row of holes is omitted along the ΓK direction to create a line-defect waveguide. The width of this waveguide is defined as the distance between the centre of the nearest neighbour holes on both sides of the line-defect ${\text{w}}^{\text{WG}}=0.85{\text{w}}^0$, where ${\text{w}}^0=\text{a}\sqrt{3}$. Unless specified otherwise, the core is placed at the centre of the line defect. As represented schematically in Fig. 1(d) and numerically in Figs. 1(c) and 1(e) for ${\text{h}}^{\text{core}}={\text{w}}^{\text{core}}=0.45\text{a}$, varying the refractive index of the line-defect core ${\text{n}}_{\text{r}}^{\text{core}}$ modifies the dispersion of the mode guided in the line-defect, creating a mode-gap between modes guided in cores of different refractive indices. The light confinement of our NW-induced cavities is based on the presence of a single NW inside the groove that induces a cavity mode surrounded by such a mode-gap [8], the grooved waveguide acting as a barrier. We here define the refractive index of the barrier core as with ${\text{n}}_{\text{r}}^{\text{barrier}}={\text{n}}_{\text{r}}^{\text{core}}=1$. But because of the small refractive index of SiN, the parameter space within which such a confinement occurs remains narrow. Indeed, for both guided modes to exist, the core width ${\text{w}}^{\text{core}}$ cannot be larger than $0.75a$ [Fig. 1(f)], the waveguide width ${\text{w}}^{\text{WG}}$ has to stay within the $0.8{\text{w}}^0$ to $0.9{\text{w}}^0$ range [Fig. 1(g)] and the slab thickness t cannot be smaller than $0.85a$ [Fig. 1(h)], other parameters being equal. Within this narrow parameter space, we successfully found nanocavity mode formation, as represented in Fig. 2(a) for a square-section NW, ${\text{h}}^{\text{core}}={\text{w}}^{\text{core}}=0.45\text{a}$, ${\text{n}}_{\text{r}}^{\text{NW}}=2.4$ and ${\text{n}}_{\text{r}}^{\text{barrier}}=1$. For a short nanowire length ${\text{L}}^{\text{NW}}=15\text{a}$ (design #1), the design however leads to a rather low quality factor ${\text{Q}}_{\text{c}}=3.4\times {10}^3$ with ${\text{V}}_{\text{m}}=4.7{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3=2.7{(\text{λ}/{\text{n}}_r^{SiN})}^3$. A first way to boost the nanocavity performance while preserving the mode-gap design is to consider longer NWs (designs #1 to #4) [Figs. 2(a) and 2(b)]: it tends to increase the quality factor while only modestly increasing the mode volume, which results in higher ${\text{Q}}_{\text{c}}/{\text{V}}_{\text{m}}$ ratios and better light-matter interaction prospects. Because of its simplicity and good ${\text{Q}}_{\text{c}}/{\text{V}}_{\text{m}}$ ratio, this design is already appealing, but in the following Subsection we propose to engineer the mode-gap in order to further improve the cavity performance.

 

Fig. 1 (a) Schematic view of a grooved 2D PhC embedding a single NW. (b) Representation of the structural parameters of the NW-induced 2D PhC cavity. (c) Band-diagrams of a PhC slab, a grooved PhC waveguide (${\text{n}}_{\text{r}}^{\text{core}}=1.0$) and a NW PhC waveguide (${\text{n}}_{\text{r}}^{\text{core}}=2.4$) as calculated by 3D-FDTD for $\text{t }=\text{a}$, $\text{r}=0.25\text{a}$ and ${\text{h}}^{\text{core}}={\text{w}}^{\text{core}}=0.45\text{a}$. Insets represent the calculated structures. The white dashed lines correspond to the light line. (d) Representation of the band diagram as a function of z for the 2D PhC structure shown in (a) and reproduced in the bottom inset: the dashed green (resp. purple) line represents the edge of the mode guided in a line-defect PhC with ${\text{n}}_{\text{r}}^{\text{core}}=1.0$ (resp. ${\text{n}}_{\text{r}}^{\text{core}}=2.4$). (e) Edge of the mode guided in a PhC waveguide as a function of the core refractive index, calculated by 3D-FDTD for ${\text{w}}^{\text{WG}}=0.85{\text{w}}^0$ and ${\text{h}}^{\text{core}}={\text{w}}^{\text{core}}=0.45\text{a}$. (f) Edge of the guided modes as a function of the core width ${\text{w}}^{\text{core}}$ for a given height ${\text{h}}^{\text{core}}=0.45\text{a}$: open circles correspond to ${\text{n}}_{\text{r}}^{\text{core}}=1$, and closed circles correspond to ${\text{n}}_{\text{r}}^{\text{core}}=2.4$. (g) Edge of the guided modes as a function of the waveguide width ${\text{w}}^{\text{WG}}$ for ${\text{h}}^{\text{core}}={\text{w}}^{\text{core}}=0.45\text{a}$. (h) Edge of the guided mode as a function of the slab thickness for ${\text{w}}^{\text{WG}}=0.85{\text{w}}^0$.

Download Full Size | PDF

 

Fig. 2 (a) Magnetic and electric field xz cross-sections of the fundamental NW-induced cavity mode obtained for a square-section NW, ${\text{L}}^{\text{NW}}=17.5\text{a}$, ${\text{w}}^{\text{core}}={\text{h}}^{\text{core}}=0.45\text{a}$, ${\text{n}}_{\text{r}}^{\text{NW}}=2.4$ and ${\text{n}}_{\text{r}}^{\text{barrier}}={\text{n}}_{\text{r}}^{\text{groove}}=1$ (design #3). For this mode, $\text{a}/\text{λ}=0.391$, ${\text{Q}}_{\text{c}}=2.5\times {10}^4$ and ${\text{V}}_{\text{m}}=5.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (b) Cavity ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of the NW length (designs #1 to #4). Other parameters are same as in (a). (c) Cavity ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of the barrier refractive index and for ${\text{n}}_{\text{r}}^{\text{NW}}=2.4$ and ${\text{L}}^{\text{NW}}=17.5\text{a}$ (designs #3 and #5 to #15). The vertical dashed line highlights the NW refractive index. Other parameters are same as in (a). (d) Cavity ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of the core width and for ${\text{h}}^{\text{NW}}={\text{h}}^{\text{groove}}=0.45\text{a}$ and ${\text{n}}_{\text{r}}^{\text{barrier}}=1$ (designs #3 and #16 to #22). Other parameters are same as in (c).

Download Full Size | PDF

3.2 Importance of the size of the mode gap

Similarly to other mode-gap cavities [12,13], one expects that the confinement of light is greatly influenced by the size of the mode-gap. The mode-gap size can for example be tailored by modifying the refractive index difference between the NW and the groove material: the normalized frequency of the edge of the guided mode is maximal for ${\text{n}}_{\text{r}}^{\text{core}}=1$ but can be reduced by increasing ${\text{n}}_{\text{r}}^{\text{core}}$ [Fig. 1(d)]. Increasing the refractive index ${\text{n}}_{\text{r}}^{\text{barrier}}$ of the barrier core (designs #5 to #15) thus shrinks the mode-gap and drastically changes ${\text{Q}}_{\text{c}}$ with an optimum ${\text{Q}}_{\text{c}}=7\times {10}^4$ reached for ${\text{n}}_{\text{r}}^{\text{barrier}}=2.2$ [Fig. 2(c)]. This boost occurs because reducing the size of the mode-gap tends to lower the impedance mismatch between both guided modes, hence decreasing the scattering losses at the NW facets. It goes along with a marginal increase of ${\text{V}}_{\text{m}}$ from $5.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ up to $5.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ for ${\text{n}}_{\text{r}}^{\text{barrier}}=2.2$. For larger refractive indices, the mode-gap becomes too small for an effective light confinement, so that ${\text{Q}}_{\text{c}}$ drops while ${\text{V}}_{\text{m}}$ shoots up. Let us note that this index optimization is not a purely abstract concept as it can be practically implemented by filling the groove a posteriori with ionic liquids (e.g. 1-ethyl-3-methylimidazolium heptaiodide leading to ${\text{n}}_{\text{r}}^{\text{barrier}}=2.01$ [14], or 1-butyl-3-methylimidazolium triiodide leading to ${\text{n}}_{\text{r}}^{\text{barrier}}=1.78$ [15], or just by omitting the groove surrounding the NW (${\text{n}}_{\text{r}}^{\text{barrier}}=2$). Similarly, reducing the groove (resp. NW) dimensions lowers (resp. increases) the normalized frequency of the guided mode edge and reduces the mode-gap [Fig. 1(e)]. Thus, reducing the core width down to ${\text{w}}^{\text{core}}=0.05\text{a}$ (designs #16 to #22) also translates in a ${\text{Q}}_{\text{c}}$ increase up to $4\times {10}^4$ [Fig. 2(d)]. However, as ${\text{w}}^{\text{core}}$ becomes smaller, the field also tends to spread out of the NW in the surrounding SiN, which significantly increases ${\text{V}}_{\text{m}}$ [Fig. 2(d)]. Finally, if we consider a circular NW section with ${\text{δ}}^{\text{NW}}=0.45\text{a}$ (design #23) instead of a square section with ${\text{w}}^{\text{NW}}={\text{h}}^{\text{NW}}=0.45\text{a}$, the mode-gap is marginally reduced because the effective index of the core is smaller. As a result, this improves ${\text{Q}}_{\text{c}}$ up to $3.1\times {10}^4$ while increasing the mode volume up to ${\text{V}}_{\text{m}}=6.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$.

We should note that in addition to light confinement optimization, reducing the size of the mode-gap also expands the space parameters in which the confinement remains possible. For example, the methods presented above allow for the use of thinner slabs that will ultimately shrink the cavity mode volume ${\text{V}}_{\text{m}}$.

3.3 Polarization and confinement factor

Let us now investigate the electric field distribution in the simpler structure described in Subsection 3.1 (design #3). As opposed to previous calculations made for silicon-based NW-induced cavities [8], we find that the cavity mode is slightly polarized in the z direction with $\left|{\text{E}}_{\text{z}}\right|/\left|{\text{E}}_{\text{x}}\right|=1.18$ [Fig. 3(a)]. However, because the z-component of the electric field $\left|{\text{E}}_{\text{z}}\right|$ is concentrated at the SiN/Air interface, it does not couple efficiently with the NW. Conversely, the x-component $\left|{\text{E}}_{\text{x}}\right|$ penetrate significantly the core of the NW, so that the part of the electric field coupled to the NW is more strongly x-polarized with $\left|{\text{E}}_{\text{z}}\right|/\left|{\text{E}}_{\text{x}}\right|=0.53$. Over all, for a square core section with ${\text{w}}^{\text{core}}=0.45\text{a}$, 16.5% of the electric field is confined into the NW, i.e. the confinement factor is $\text{Γ}=16.5\text{\%}$ with $\text{Γ}=\frac{{{\displaystyle \sum }}_{NW}\left|E\right|}{{{\displaystyle \sum }}_{domain}\left|E\right|}$. As mentioned in Subsection 3.2, reducing the core width down to ${\text{w}}^{\text{core}}=0.25\text{a}$ (design #20) tends to spread the field outside the NW and degrades the confinement factor down to $\text{Γ}=5.5\text{\%}$ [Fig. 3(b)]. Furthermore, considering a circular NW section with ${\text{δ}}^{\text{NW}}={\text{w}}^{\text{groove}}={\text{h}}^{\text{groove}}=0.45a$ (design #23) tends to increase the intensity of the electric field x-component at the SiN/Air interface and thus reduces the confinement factor down to $\text{Γ}=5.8\text{\%}$ with a significantly larger part of the field lying in air [Fig. 3(c)]. As a result, we find that in order to maximize the confinement factor, one should maximize the core size while the groove and NW geometries should be as close as possible from each other. It is finally worth noting that because the field is mostly confined within the cavity length ${\text{L}}^{\text{NW}}$ [Fig. 2(a)], the confinement factor is mainly determined by the xy section geometry of the cavity and has negligible dependence on the NW length or the refractive index of the barrier core.

 

Fig. 3 (a) xy section of the normalized electric field x and z components for design #3 investigated in Fig. 2(a). $\text{a}/\text{λ}=0.391$, ${\text{Q}}_{\text{c}}=2.5\times {10}^4$ and ${\text{V}}_{\text{m}}=5.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (b) Same as (a) with ${\text{w}}^{\text{core}}=0.25\text{a}$ (design #20).$\text{a}/\text{λ}=0.394$, ${\text{Q}}_{\text{c}}=2.1\times {10}^4$ and ${\text{V}}_{\text{m}}=6.8{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (c) Same as (a) for a circular NW section (design #23). $\text{a}/\text{λ}=0.397$, ${\text{Q}}_{\text{c}}=3.1\times {10}^4$ and ${\text{V}}_{\text{m}}=6.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. Lower frames show barrier and cavity cross-sections in the xy plane.

Download Full Size | PDF

3.4 Impact of asymmetry

In fabricated structures, several sources of asymmetry may influence the light confinement of NW-induced cavities: misalignment of the groove from the centre of the line defect and misalignment of the NW from the centre of the groove. Considering the cavity described in Section 3.1, a misalignment of the groove along the x axis up to ${\Delta }^{\text{groove}}=0.25\text{a}$ (designs #24 to #26) leads to a decrease of ${\text{Q}}_{\text{c}}$ from $2.5\times {10}^4$ down to $6.2\times {10}^3$, with no major impact on ${\text{V}}_{\text{m}}$ [Fig. 4(a)]. Considering now a NW with ${\text{w}}^{\text{NW}}=0.35\text{a}$ positioned in a ${\text{w}}^{\text{groove}}=0.45\text{a}$ groove (design #27), a small misalignment of the NW from the centre of the groove along the x axis ${\Delta }^{\text{NW}}=0.05\text{a}$ (design #28) leads to a significant decrease of the quality factor from ${\text{Q}}_{\text{c}}=1.4\times {10}^4$ down to ${\text{Q}}_{\text{c}}=4.9\times {10}^3$ with little impact on ${\text{V}}_{\text{m}}$ [Fig. 4(b)]. The effect of such extrinsic asymmetry can be explained by looking at the cavity mode in the wavevector space, using the Fourier transform of $\left|E_x\right|$. One can for example observe an increase of the field components overlapping with the light cone when shifting the groove [Fig. 4(c) and 4(d)], which indicates larger out-of-plane losses. Let us note that in order to limit the misalignment of the NW from the centre of the groove and minimize related optical losses, one would need to fabricate grooves with width as close as possible from the NW diameter.

 

Fig. 4 (a) ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of ${\Delta }^{\text{groove}}$ for the cavity described in Figs. 2(a) and 3(a) (designs #3 and #24 to #26). (b) ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of ${\Delta }^{\text{NW}}$ and for ${\text{L}}^{\text{NW}}=17.5\text{a}$, ${\text{h}}^{\text{core}}=0.45\text{a}$, ${\text{w}}^{\text{groove}}=0.45\text{a}$ and ${\text{w}}^{\text{NW}}=0.35\text{a}$. (c) Fourier transform of $\left|E_x\right|$ for the fundamental NW-induced cavity mode described in Figs. 2(a) and 3(a) with ${\Delta }^{\text{groove}}=0$ (design #3). (d) Same as (c) with ${\Delta }^{\text{groove}}=0.05a$ (design #24). (e) xy section of the normalized magnetic field of design #3 investigated in Figs. 2(a) and 3(a). $\text{a}/\text{λ}=0.397$, ${\text{Q}}_{\text{c}}=2.5\times {10}^4$ and ${\text{V}}_{\text{m}}=5.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (f) Same as (e) with a core positioned at the center of the slab (design #31). $\text{a}/\text{λ}=0.390$, ${\text{Q}}_{\text{c}}=3.6\times {10}^4$ and ${\text{V}}_{\text{m}}=6.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (g) Same as (f) for ${\text{n}}_{\text{r}}^{\text{barrier}}=2$ (design #32). $\text{a}/\text{λ}=0.390$, ${\text{Q}}_{\text{c}}=1.4\times {10}^5$ and ${\text{V}}_{\text{m}}=6.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ Lower frames show barrier and cavity cross-sections in the xy plane.

Download Full Size | PDF

In addition to asymmetry in the xz plane induced by misalignments, one must also consider asymmetries due to NW fabrication issues such as the presence of defects on the NW sidewalls and the existence of a taper along the NW axis. We consider the effect of such asymmetries on design #23 featuring a NW with a circular section and a constant diameter ${\text{δ}}^{\text{NW}}=0.45a$. Adding a taper that varies the NW diameter from ${\text{δ}}^{\text{NW}}=0.45a$ at its base to ${\text{δ}}^{\text{NW}}=0.25a$ at its tip (design #29) tends to reduce the quality factor from ${\text{Q}}_{\text{c}}=3.1\times {10}^4$ down to ${\text{Q}}_{\text{c}}=2.4\times {10}^3$ whereas the mode volume decreases from ${\text{V}}_{\text{m}}=6.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ down to ${\text{V}}_{\text{m}}=4.9{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. Adding a spherical defect with radius $0.25a$ randomly positioned on the NW sidewall (design #30) reduces the quality factor from ${\text{Q}}_{\text{c}}=3.1\times {10}^4$ down to ${\text{Q}}_{\text{c}}=1.3\times {10}^4$ with little effect on the mode volume. Such NW fabrication defects would thus not prevent the nanocavity to be formed but the impact on the optical confinement is significant, so that a careful choice of individual NWs and/or fabrication methods should be made to minimize the presence of NW defects.

Besides asymmetry induced by fabrication disorder, our cavity design presents an intrinsic asymmetry in the xy plane because the NW is positioned away from the slab centre [Fig. 4(e)]. As a result, despite its versatility, the grooved structure is not optimized in terms of light confinement and its asymmetry brings about scattering losses. We assess the losses induced by such asymmetry by positioning the core at the centre of the slab (design #31) [Fig. 4(f)]. It results in a higher symmetry of the field distribution [Fig. 4(f)] and leads to an increase of the quality factor from ${\text{Q}}_{\text{c}}=2.5\times {10}^4$ up to ${\text{Q}}_{\text{c}}=3.6\times {10}^4$, confirming that asymmetry has here an impact, albeit limited, on optical losses. Although such a structure including a buried air core is difficult to realize, it is possible to bury the NW at the center of the SiN slab, the cavity barrier being constituted of a SiN line defect waveguide with no groove (design #32). This actually combines the symmetrization of the photonic structure together with the mode-gap size optimization evidenced in Section 3.2, which drastically boosts the quality factor up to ${\text{Q}}_{\text{c}}=1.4\times {10}^5$ for a mode volume ${\text{V}}_{\text{m}}=6.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ [Fig. 4(g)]. Although such a buried NW design loses the versatility of the grooved structure, a similar approach has been chosen in [6] with CdSe NW emitting in the red. In the latter case however, the calculated quality factor of the buried NW design is lower by more than an order of magnitude as compared to the one investigated in this manuscript. There are various differences between the design parameters of our buried NW design and the one described in [6]. CdSe NWs have a significantly larger refractive index ${\text{n}}_{\text{r}}^{\text{NW}}=2.85$ which induces stronger constraints on the geometry of the NW, but the NW used in [6] have a diameter as small as $0.2a$ which considerably lowers the size of the mode-gap. The reason for the significantly smaller quality factor must thus lie in the length of the CdSe NWs which is significantly shorter (${\text{L}}^{\text{NW}}=5\text{a}$) and induces larger scattering losses.

4. One-dimensional PhC cavities

4.1 Cavity design

As represented in Fig. 5(a), NW-induced cavities based on 1D PhCs are constituted of an air-bridge nanobeam PhC that presents a groove embedding a single NW with ${\text{n}}_{\text{r}}^{\text{NW}}=2.4$. The structural parameters of the cavity design are represented in Fig. 5(b). The nanobeam width is ${\text{w}}^{\text{NB}}$ and the holes are rectangular with respective dimensions ${\text{d}}^{\text{x}}$ and ${\text{d}}^{\text{z}}$ along the x and z axes. In 1D PhC cavities the light confinement is usually obtained by a structural modulation of e.g. the hole size, the nanobeam width, the lattice constant and/or the width of a slot [7, 9, 11, 16–19]. Instead, in the present work the presence of the single NW in the grooved structure locally modulates the refractive index, which induces a change in the band structure and redshifts the PBG, as represented in Fig. 5(c). As a result, providing the lowest guided mode of the NW PhC located at the upper edge of the NW PBG lies within the groove PBG, light confinement can be successfully realized. This essential condition is not as trivial as it seems because the large refractive index difference ${\text{n}}_{\text{r}}^{\text{NW}}-{\text{n}}_{\text{r}}^{\text{groove}}$ induces a significant PBG shift.

 

Fig. 5 (a) Schematic view of a grooved 1D PhC embedding a single NW. (b) Representation of the structural parameters of the NW-induced 1D PhC cavity. (c) Representation of the PBG as a function of z for the 1D PhC structure shown in (a) and reproduced in the bottom inset. (d) Upper and lower edges of the groove PBG (open circles) and lowest guided mode of the NW PhC (closed circles) as a function of ${\text{w}}^{\text{core}}$ for the following parameters: $\text{t}=0.67\text{a}$, ${\text{w}}^{\text{NB}}=3\text{a}$, ${\text{d}}^{\text{x}}=0.75\text{w}$, ${\text{d}}^{\text{z}}=0.25\text{a}$, a rectangular NW section and ${\text{h}}^{\text{NW}}={\text{h}}^{\text{groove}}=0.6\text{a}$ . The left (resp. right) inset shows the magnetic field $\left|{\text{H}}_y\right|$ of the grooved nanobeam with (resp. without) an embedded NW and for ${\text{w}}^{\text{core}}=0.6\text{a}$. (e) Same as (d) as a function of the slab thickness t and for ${\text{h}}^{\text{core}}={\text{w}}^{\text{core}}=0.6\text{a}$. (f) Same as (e) as a function of ${\text{w}}^{\text{NB}}$ and for $\text{t}=0.67\text{a}$.

Download Full Size | PDF

As can be seen in Figs. 5(d)-5(f), the resulting parameter space fulfilling the condition is narrow. One way to limit the PBG shift is to reduce the size of the core as exemplified in Fig. 5(d) where the light confinement condition is fulfilled for ${\text{w}}^{\text{core}}<0.5a$. In the same spirit, considering a circular NW section instead of a square section, reduces the PBG shift. However, because the NW geometry is often a fabrication constraint, one would rather tailor the nanobeam geometry to independently modify the grooved PhC and the NW PhC band structures. To that end we should note that in both cases the electric field is essentially confined in the highest index material [inset of Fig. 5(d)], that is to say inside the core for the NW PhC and inside the SiN nanobeam for the grooved PhC. As a result, by changing the thickness or the width of the nanobeam, one can strongly modify the groove PBG while marginally altering the NW PBG [Figs. 5(e) and 5(f)].

Structures meeting the PBG condition mentioned above can present high quality factors and mode volumes even smaller than in the 2D PhC cavities investigated in Section 3. For example, a NW-induced cavity with a rectangular section NW, ${\text{w}}^{\text{core}}=0.4\text{a}$, ${\text{h}}^{\text{core}}=0.6\text{a}$, ${\text{w}}^{\text{NB}}=3\text{a}$,${\text{d}}^{\text{x}}=0.83\text{w}$, ${\text{d}}^{\text{z}}=0.25\text{a}$, and ${\text{L}}^{\text{NW}}=20.5\text{a}$ (design #33) supports a resonant mode at $\text{a}/\text{λ}=0.406$ with a quality factor ${\text{Q}}_{\text{c}}=1.4\times {10}^4$ and a mode volume ${\text{V}}_{\text{m}}=2.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ [Fig. 6(a)]. As opposed to 2D PhC cavities, reducing the core width (designs #33 to #36) tends to degrade the quality factor [Fig. 6(b)]. The mode volume, on the other hand, follows a similar trend as the one seen in Section 3.2 with the field spreading outside the NW as its section shrinks [Fig. 6(b)]. The length of the nanowire has the expected influence already highlighted in Section 3.2 (designs #33 and #37 to #39): longer NWs brought about an increase of both ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ [Fig. 6(c)].

 

Fig. 6 (a) Magnetic and electric field xz cross-sections of the fundamental NW-induced cavity mode obtained for a rectangular section NW, ${\text{L}}^{\text{NW}}=20.5\text{a}$, ${\text{w}}^{\text{core}}=0.4\text{a}$, ${\text{h}}^{\text{core}}=0.6\text{a}$,${\text{w}}^{\text{NB}}=3\text{a}$,${\text{d}}^{\text{x}}=0.83\text{w}$, ${\text{d}}^{\text{z}}=0.25\text{a}$, $\text{t}=0.67\text{a}$ (design #33). For this mode, $\text{a}/\text{λ}=0.406$, ${\text{Q}}_{\text{c}}=1.4\times {10}^4$ and ${\text{V}}_{\text{m}}=2.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (b) Cavity ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of the core width (designs #33 to #36). Other parameters are same as in (a). (c) Cavity ${\text{Q}}_{\text{c}}$ and ${\text{V}}_{\text{m}}$ as a function of the NW length (design #33 and #37 to #39). Other parameters are same as in (a).

Download Full Size | PDF

4.2 Polarization and confinement factor

Similarly to 2D PhC cavities, we find fundamental 1D PhC cavity modes polarized in the x direction, with $\left|{\text{E}}_{\text{z}}\right|/\left|{\text{E}}_{\text{x}}\right|=0.23$ for design #33 presented in Figs. 6(a) and 7(a). This rectangular section NW design presents a limited part of its electric field confined in the NW, with a confinement factor close to $\text{Γ}=10\%$ [Fig. 7(a)]. Since the field tends to spread outside of the NW as the NW width shrinks (see Section 4.1), one would like to expand the NW width in order to optimize the confinement factor. However if one wants to keep at the same time an efficient light confinement, we showed in Section 4.1 that it is necessary to increase either the nanobeam width ${\text{w}}^{\text{NB}}$ or its thickness t [Figs. 5(e) and 5(f)]. For example, a NW-induced cavity with a larger NW ${\text{w}}^{\text{core}}={\text{h}}^{\text{core}}=0.6\text{a}$, and with ${\text{w}}^{\text{NB}}=4.5\text{a}$,${\text{d}}^{\text{x}}=0.88\text{w}$, ${\text{d}}^{\text{z}}=0.25\text{a}$, and ${\text{L}}^{\text{NW}}=26.5\text{a}$ (design #40) supports a fundamental mode at $\text{a}/\text{λ}=0.370$ with ${\text{Q}}_{\text{c}}=8.9\times {10}^4$ and a mode volume ${\text{V}}_{\text{m}}=3.5{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ [Fig. 7(b)]. Such a design results in a significant increase of the confinement factor up to $\text{Γ}=31\%$. Finally, considering a circular NW section with ${\text{δ}}^{\text{NW}}={\text{w}}^{\text{groove}}={\text{h}}^{\text{groove}}=0.6a$ (design #41) reduces the confinement factor down to $\text{Γ}=13\%$ [Fig. 7(c)]. This is because, as already found for 2D PhC cavities, matching the NW geometry to the groove geometry maximizes the amount of electric field confined into the NW. Overall it is worth noting that the confinement factors are significantly larger for 1D PhC cavities, thus such designs show better prospects for an efficient light-matter interaction.

 

Fig. 7 (a) Electric field xz cross-sections of the fundamental NW-induced cavity mode featured in Fig. 6(a) (design #33). (b) Electric field xz cross-sections of the fundamental NW-induced cavity mode obtained for a square section NW, ${\text{L}}^{\text{NW}}=26.5\text{a}$, ${\text{w}}^{\text{core}}={\text{h}}^{\text{core}}=0.6\text{a}$,${\text{w}}^{\text{NB}}=4.5\text{a}$,${\text{d}}^{\text{x}}=0.88\text{w}$, ${\text{d}}^{\text{z}}=0.25\text{a}$, $\text{t}=0.67\text{a}$ (design #40). For this mode $\text{a}/\text{λ}=0.37$, ${\text{Q}}_{\text{c}}=8.9\times {10}^4$ and ${\text{V}}_{\text{m}}=3.5{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (c) Electric field xz cross-sections of the fundamental NW-induced cavity mode obtained for a circular section NW, ${\text{L}}^{\text{NW}}=20\text{a}$, ${\text{w}}^{\text{groove}}={\text{h}}^{\text{groove}}={\delta }^{NW}=0.6\text{a}$,${\text{w}}^{\text{NB}}=4\text{a}$,${\text{d}}^{\text{x}}=0.88\text{w}$, ${\text{d}}^{\text{x}}=0.88\text{w}$, $\text{t}=0.67\text{a}$ (design #41). For this mode $\text{a}/\text{λ}=0.396$, ${\text{Q}}_{\text{c}}=5.1\times {10}^4$ and ${\text{V}}_{\text{m}}=1.8{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$.

Download Full Size | PDF

4.3 Asymmetry and NW positioning

Like the 2D PhC cavities studied in Section 3, asymmetry arising from fabrication imperfections also has an impact on 1D NW-induced cavities. For example, as represented in Fig. 8(a), a misalignment of the groove along the x axis ${\Delta }^{\text{groove}}$ up to $0.25\text{a}$ (design #42 to #44) leads to ${\text{Q}}_{\text{c}}$ dropping from $8.9\times {10}^4$ down to $5.7\times {10}^3$, while ${\text{V}}_{\text{m}}$ stays stable at $3.5{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. Once again, looking at the cavity mode in the wavevector space gives us further insight into the effect of asymmetry: the relative increase of the field components $\left|E_x\right|$ overlapping with the light cone when shifting the groove [Figs. 8(b) and 8(c)] indicates larger out-of-plane losses. The drastic effect of the groove shift shows that the cavity design is particularly sensitive to extrinsic asymmetries. Besides, let us note that 1D PhC cavities also show intrinsic asymmetry in the xy plane. However, unlike 2D PhC cavities, this cannot be solved by burying the NW in the SiN slab because the NW intersects the holes of the nanobeam.

 

Fig. 8 (a) ${\text{Q}}_{\text{c}}$ and ${\text{V}}_m$ as a function of ${\Delta }^{\text{groove}}$ for the cavity described in Fig. 7(b) (designs #40 and #42 to #44). (b) Fourier transform of $\left|E_x\right|$ for the fundamental NW-induced cavity mode described in Fig. 7(b) (design #40). (c) Same as (b) for a NW shifted by ${\Delta }^{\text{groove}}=0.15a$ along the x axis (design #43). $\text{a}/\text{λ}=0.370$, ${\text{Q}}_{\text{c}}=1.4\times {10}^4$ and ${\text{V}}_{\text{m}}=3.5{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. (d) Same as (b) for a NW shifted by $\text{a}/2$ along the z axis (design #45). $\text{a}/\text{λ}=0.370$, ${\text{Q}}_{\text{c}}=3.9\times {10}^4$ and ${\text{V}}_{\text{m}}=3.5{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$.

Download Full Size | PDF

In addition to asymmetry issues, one must also consider the z-axis positioning of the NW. In conventional nanobeam cavities, the impedance mismatch between the barriers and the cavity is smoothed out through some form of design gradient in order to minimize out-of-plane losses [16–19]. In NW-induced cavities, not only the impedance mismatch at the NW end-facets is rather abrupt and cannot be easily smoothed out, but out-of-plane losses will drastically depend on how the NW end-facets match the nanobeam interfaces. For end-facets matching the SiN/Air interface of the nanobeam rectangular holes [inset of Fig. 8(b)], scattering losses are minimized. This is typically the case of designs #33 and #40 investigated in Figs. 7(a) and 7(b). However when the NW is shifted by $\text{a}/2$ along the z axis (design #45), the end-facets match the Air/SiN interface of the nanobeam rectangular holes [inset of Fig. 8(d)] and scattering losses are significantly higher. For the design investigated in Fig. 7(b), such a shift brings down the quality factor from ${\text{Q}}_{\text{c}}=8.9\times {10}^4$ to ${\text{Q}}_{\text{c}}=3.9\times {10}^4$ with a similar mode volume ${\text{V}}_{\text{m}}=3.5{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$. The Fourier transform of $\left|E_x\right|$ exhibits a larger overlap of the electric field with the light cone in the wavevector space when the NW is shifted by $\text{a}/2$, indicating larger out-of-plane losses [Figs. 8(b) and 8(d)]. It is worth mentioning that, although out-of-plane losses are also significant in the 2D cavities studied in Section 3 [Figs. 4(c) and 4(d)], this issue does not arise because the NW does not intersect the PhC holes.

5. Conclusion

After investigating the influence of key design parameters such as NW and PhC geometry, PhC dimensionality and design asymmetries, we have found that NW-induced nanocavities based on both 1D and 2D grooved SiN PhCs can support high-quality factor resonances with relatively small mode volumes. For both kind of cavities we have found ways to optimize the design e.g. through the barrier index or the core geometry for 2D PhC cavities, and the nanobeam width and thickness as well as the core geometry for 1D PhC cavities. In both cases, we have also found that asymmetry is detrimental to the quality factor of the resonant mode and that matching the geometry of the NW to the geometry of the groove is key to maximize the confinement factor and hence optimize light-matter coupling. It is worth mentioning that one of the important differences reported in our work between 1D and 2D PhC cavities is the fact that this confinement factor is larger for 1D PhC cavities by a factor ~2. Furthermore, the mode volumes of 1D PhC cavities appear to be significantly smaller. This theoretically implies better prospects for such nanocavities, but their design seems less robust as they are sensitive to both fabrication-induced asymmetries and NW positioning. In addition, their fabrication might be more challenging because they are intrinsically more fragile. Overall, the best 1D grooved PhC design (design #39) shows a quality factor as high as ${\text{Q}}_{\text{c}}=5.1\times {10}^4$ for a mode volume ${\text{V}}_{\text{m}}=1.8{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ and a normalized frequency $\text{a}/\text{λ}=0.396$ [Fig. 6(c)], while the best 2D grooved PhC design (design #3) presents a quality factor as high as ${\text{Q}}_{\text{c}}=2.5\times {10}^4$ for a mode volume ${\text{V}}_{\text{m}}=5.1{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ and a normalized frequency $\text{a}/\text{λ}=0.391$ [Fig. 2(a)]. In the 2D case, using the buried NW design mentioned in Subsection 3.4 (design #30) can improve the cavity performance with a quality factor as high as ${\text{Q}}_{\text{c}}=1.4\times {10}^5$ for a mode volume ${\text{V}}_{\text{m}}=6.6{(\text{λ}/{\text{n}}_{\text{r}}^{\text{NW}})}^3$ and a normalized frequency $\text{a}/\text{λ}=0.390$ [Fig. 4(g)].

SiN PhC cavities with resolution limited quality factors ranging from $5.9\times {10}^3$ at λ = 590nm [19] up to $5.5\times {10}^4$ at λ = 624nm [9] have already been realized. It shows that from the PhC processing perspective there should not be any major problem implementing our nanocavity designs with e.g. diamond NWs emitting in this range. To couple such nanocavities with e.g. ZnO NWs emitting around 380 nm would require PhC lattice constants around $\text{a }=150 \text{nm}$ that are yet to be realized in SiN membranes. However, such short lattice constant PhCs have already been fabricated in other kind of materials such as group-III nitrides that are significantly more challenging to process and with quality factors in the high ${10}^3$ range [20,21]. Implementing our NW-induced nanocavity designs in SiN PhCs operating down to the near-UV range thus appears to be realistic.

Finally, as explained at length throughout this paper, our nanocavity design imposes some constraints on the geometry of the NW, especially regarding diameter, length and symmetry. Nevertheless, a large variety of NW materials and fabrication methods remain compatible with such constraints. For example, out-of-plane diamond NWs can be obtained by etching methods with high flexibility on the NW geometry [22]: NWs with diameters as small as 170 nm, lengths of several microns and little defects can be obtained but such NWs are somewhat tapered. An etching method for the fabrication of in-plane diamond structures [23] should overcome the latter drawback but the realization of NWs with such a method remains to be demonstrated. Emitting at shorter wavelengths, GaN/InGaN NWs with diameters as small as 50 nm, with lengths of several microns, little defects and no taper can be grown by molecular beam epitaxy [24]. Besides, a recent selective area sublimation method [25] with no constraint on the NW diameter seems also to be promising for the implementation of GaN/InGaN NWs. In addition, ZnO NWs with diameters as small as 60 nm, with lengths of several microns, little defects and no taper are commercially available. As a result, various kind of materials emitting at short wavelengths can be practically employed to implement our NW-induced nanocavity designs. It confirms the potential of grooved SiN PhCs as a versatile platform for the realization of novel nanophotonic devices operating across the near-UV and visible ranges.