Absorption through a coupled optical resonance in a horizontal InP nanowire array

Ali Hosseinnia; Nicklas Anttu

doi:10.1364/PRJ.3.000125

Abstract

We study through electromagnetic modeling the absorption of light of a given wavelength in an array of horizontal InP nanowires of diameter less than 100 nm. Such absorption is performed most efficiently by using polarized light and by exciting a coupled optical resonance in a sparse array. In that case, we excite a resonance in the individual nanowires and couple the resonances in neighboring nanowires through a lattice resonance of the periodic array. At such a resonance, an array with nanowires of 80 nm in diameter can absorb more than eight times more strongly than a tight-packed array, despite containing a seven times smaller amount of the absorbing InP material.Nanophotonics and photonic crystals

Keywords

nanostructures Absorption Subwavelength structures

III–V semiconductor nanowires are popular for many optoelectronic applications ^[1,2], including photodetection ^[3,4]. Nanowires allow for miniaturized single-nanowire detectors as well as for large-area detectors when the nanowires are placed next to each other ^[3,5]. Furthermore, small-diameter nanowires allow, due to strain relaxation in the radial direction, for a much larger freedom in the combination of lattice mismatched materials than in a conventional thin-film layer ^[6–9]. Such material freedom allows, through ternary III–V compounds ^[10], for the engineering of the bandgap in photodetection applications. The detection of photons is at the heart of current high-speed optical communication. There, a matching of the bandgap wavelength to the communication wavelength is desirable. Such a matching typically leads to a higher signal-to-noise ratio in the detection by suppressing the dark current in the photodetector ^[11].

Thus, we would like to enhance the absorption of light in the vicinity of the bandgap wavelength in small-diameter nanowires. For that purpose, we use here three properties common for nanostructures. First, single nanostructures can show optical resonances whose wavelength position depends on the size and shape of the structure ^[12–19]. Second, periodic nanostructures can show period-dependent lattice resonances that originate from the diffractive coupling between the constituent structures ^[12,13,20]. Third, coupling between these two types of resonances can show up, which has been identified in metallic systems ^[13,21].

However, such coupling of resonances has not been studied in absorbing horizontal nanowire arrays ^[22–24]. Here, we show that the optical resonances in individual horizontal nanowires ^[1] can couple with lattice resonances, leading to absorption peaks. We use an array of InP nanowires embedded in $glass / {SiO}_{2}$ as a model system. We show that the best way to absorb photons of a given wavelength in nanowires of diameter less than 100 nm is to use polarized light and to excite a coupled resonance. For a diameter of 80 nm, the nanowires can absorb 30% of the incident photons when the coupled resonance is excited. In this case, the nanowires absorb eight times more strongly than a tight-packed array of nanowires, despite consisting of a seven times smaller amount of InP. The resonance becomes stronger and narrower when the diameter decreases toward 50 nm, for which an absorptance of 50% can be reached. In case a higher absorptance is required, we need to turn to nanowires with diameter exceeding 100 nm.

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We consider InP nanowires of diameter $D$ placed in an array of period $p$ and surrounded by glass of refractive index $n_{glass} = 1.5$ (Fig. 1). Light is incident at a normal angle to the array to maximize the projected area of the array to the incident light. TM (TE) polarized incident light has the electric field parallel (perpendicular) to the axis of the nanowires. We perform the modeling with a scattering matrix method ^[25] that solves the Maxwell’s equations for the light scattering for a given wavelength $λ$ . To describe the optical response of the InP material in the nanowires, we use tabulated values ^[26] for the complex-valued refractive index $n_{InP} (λ)$ whose imaginary part gives rise to absorption in the material. We calculate the reflectance $R$ and the transmittance $T$ of the array. From these, the absorptance $A = 1 - R - T$ of the nanowires can be obtained.

$Periodic array of horizontal InP nanowires embedded in a glass matrix of refractive index nglass=1.5. The period is p and the nanowire diameter is D. Light is incident at a normal angle to the array and either TE or TM polarized. Notice that we show here four of the infinitely many nanowires of the array. Furthermore, we assume in the modeling that each nanowire is infinitely long.$

Figure 1.Periodic array of horizontal InP nanowires embedded in a glass matrix of refractive index $n_{glass} = 1.5$ . The period is $p$ and the nanowire diameter is $D$ . Light is incident at a normal angle to the array and either TE or TM polarized. Notice that we show here four of the infinitely many nanowires of the array. Furthermore, we assume in the modeling that each nanowire is infinitely long.

We consider light of $λ = 920 nm$ in wavelength, which is in the vicinity of 925 nm, the bandgap wavelength ^[10] of InP. Note that we use $n_{InP} (λ = 920 nm) = 3.40 + i 0.059$ ^[26] in the modeling. We start with the case of a rather dense array with a fixed diameter to period ratio of $p / D = 3$ (Fig. 2). We find that TM polarized light is absorbed considerably more strongly than TE polarized light. This polarization behavior can actually be understood from an electrostatic model. The incident electric field of TE polarized light of $| E_{inc} |$ is screened from the interior of each nanowire by a factor of $β = | 2 n_{glass}^{2} / (n_{glass}^{2} + n_{InP} {(λ = 920 nm)}^{2}) | = 0.326$ ^[4,27]. Thus, the electric field in the interior of each nanowire shows the magnitude of $| E_{NW} | = β | E_{inc} |$ . In contrast, TM polarized light does not experience such screening, and in this case $| E_{NW} | = | E_{inc} |$ ^[4]. Since the local absorption in each nanowire is proportional to ${| E_{NW} |}^{2}$ , we expect a factor of $1 / β^{2} \approx 9$ stronger absorption for TM polarized light. Actually, we can calculate analytically the absorptance of the nanowire array under this electrostatic approximation. Notice that the irradiance of the incident light is given by $I_{inc} = c n_{glass} ε_{0} {| E_{inc} |}^{2} / 2.0$ ^[27]. Therefore, the power that is incident into a unit cell of period $p$ is given by $P_{inc} = p I_{inc} = p c n_{glass} ε_{0} {| E_{inc} |}^{2} / 2.0$ . Next, the local absorption, that is, the ohmic heating, is given by $P_{local} = 2 π c ε_{0} Re (n_{InP}) Im (n_{InP}) {| E_{NW} |}^{2} / λ$ ^[27]. Therefore, the absorption in a unit cell, which occurs in a nanowire of cross-section area $A_{c s} = π {(D / 2)}^{2}$ , is given by $a_{NW} = A_{c s} P_{local} = π {(D / 2)}^{2} 2 π c ε_{0} Re (n_{InP}) Im (n_{InP}) {| E_{NW} |}^{2} / λ$ . Consequently, the absorptance in this electrostatic approximation is given by $A = a_{NW} / P_{inc}$ . We find here for $p / D = 3$ good agreement between the full electromagnetic modeling and this electrostatic approximation (solid versus dashed lines in Fig. 2). Notice that even for the stronger absorbed TM polarized light, the absorptance reaches values of just 5% at a diameter of 100 nm.

Figure 2.Absorptance at $λ = 920 nm$ in a horizontal array of InP nanowires for a fixed period to diameter ratio of $p / D = 3$ . Here, results from full electromagnetic modeling (solid lines) as well as from an electrostatic approximation (dashed lines) are shown for both TE and TM polarized incident light.

However, we know from previous studies that individual nanowires of $D < 100 nm$ should show a ${TM}_{01}$ resonance for TM polarized incident light at $λ = 920 nm$ ^[1]. By solving for the eigenmodes ^[1] in individual nanowires, we expect the ${TM}_{01}$ mode to show up for $λ = 920 nm$ at $D \approx 80 nm$ . Indeed, when we fix the period to $p = 900 nm$ to study a sparse array where we expect to possibly see the response of the individual nanowires, we find an absorption peak at $D \approx 80 nm$ for TM polarized light (Fig. 3). Regarding the behavior of the eigenmodes, notice that the ${TM}_{01}$ mode in individual, isolated nanowires shows symmetry in the polar direction (inset in Fig. 3). This follows from the fact that the first index in the mode labeling, $m$ in ${TM}_{m n}$ or ${TE}_{m n}$ , defines the polar dependence of the form $\cos (m θ)$ ^[1]. Furthermore, in the periodic array, we find an absorption peak for both TM and TE polarized light at $D \approx 180 - 200 nm$ . These peaks are caused by the ${TM}_{11}$ and ${TE}_{01}$ eigenmodes in the individual nanowires, which are expected to show up for $D \approx 190 nm$ from our calculation on individual, isolated nanowires. Notice that the ${TE}_{01}$ mode will not show any dependence in the polar direction. In contrast, the ${TM}_{11}$ mode will show two nodes along the polar direction due to the $\cos (θ)$ dependence.

Figure 3.Absorptance at $λ = 920 nm$ in a horizontal nanowire array of period $p = 900 nm$ for both TE and TM polarized incident light. The arrows indicate peaks that arise from resonances in the individual nanowires. The inset shows the field distribution of the ${TM}_{01}$ eigenmode in an individual nanowire, as calculated also in [1]. The eigenmode shows up for $D = 74 nm$ , and the dashed circle denotes the nanowire.

Thus, to investigate if we can increase the absorptance beyond the 5% value for $p / D = 3$ (Fig. 2), we choose $D = 80 nm$ and vary $p$ (Fig. 4). For TE polarized light, $A$ drops with increasing $p$ . This drop is a signature that the collection area of incident light is limited for each nanowire, and more light “slips” through the array with increasing $p$ , as seen in the inset of Fig. 4, where $T \to 1$ and $R \to 0$ with increasing $p$ . In strong contrast, when $p$ increases from 80 to 400 nm for TM polarized light, the absorptance does not drop from the value of 4%. Even more interestingly, the absorptance shows a very pronounced peak of $A = 31 %$ at $p = 574 nm$ . Thus, the nanowire array absorbs eight times more strongly in this sparse configuration compared to the case of a tight-packed array of $p = D = 80 nm$ , where the array contains seven times more InP. We note here that the reflectance of TM polarized light shows a peak at $p = 574 nm$ and a dip at $p = 613 nm$ (see inset in Fig. 4). Such a peak–dip pair resembles the behavior of a high-contrast grating where a Fano resonance, characterized by a peak–dip pair, shows up due to the interference between the normally incident light and an in-plane resonance of the grating ^[28,29]. Thus, we have reason to believe that the absorption peak in Fig. 4 arises due to the excitation of an in-plane resonance in the nanowire array.

Figure 4.Absorptance for both TE and TM polarized incident light at $λ = 920 nm$ in a horizontal nanowire array with nanowires of diameter $D = 80 nm$ . The inset shows the reflectance $R$ and transmittance $T$ for both polarizations.

To understand the behavior of this absorption peak at $p = 574 nm$ , we show in Fig. 5(a) the dependence of the absorptance on both $p$ and $D$ . We find that the peak shifts to a smaller diameter with increasing period. Thus, by tuning the period, we find an absorption peak even for nanowires of just 50 nm in diameter. Actually, the resonance becomes stronger at $D = 50 nm$ compared to at $D = 80 nm$ , and $A$ increases from 30% to 50%. We assign this absorption peak to a coupled resonance between the ${TM}_{01}$ resonance and a lattice resonance. The lattice resonance shows here up at a period of $p \approx λ / n_{glass}$ where the first diffracted order couples the light scattered between neighboring nanowires. We note that we have found similar, but weaker, peaks whenever $p \approx m λ / n_{glass}$ , where the $m$ th diffracted order couples neighboring nanowires (not shown).

Figure 5.Absorptance as a function of nanowire diameter $D$ and array period $p$ for (a) TM and (b) TE polarized light at $λ = 920 nm$ . The dashed lines denote the diameter where the ${TM}_{01}$ , ${TM}_{11}$ , and ${TE}_{01}$ resonances [1] show up in a single nanowire for $λ = 920 nm$ .

We show in Fig. 6(a) the electric field distribution for $D = 80 nm$ and $p = 400 nm$ and in Fig. 6(b) for $D = 80 nm$ and $p = 574 nm$ , which is at the coupled mode resonance. For $p = 400 nm$ , we see predominantly the excitation of the ${TM}_{01}$ mode in the individual nanowires, without noticeable coupling between the nanowires [see Fig. 6(c) for a schematic]. The field pattern here follows closely that of the ${TM}_{01}$ eigenmode solved for an individual, isolated nanowire (inset in Fig. 3). In contrast, at $p = 574 nm$ , we see how light is scattered between neighboring nanowires through the lattice resonance [see Fig. 6(d) for a schematic]. Due to this coupling of the ${TM}_{01}$ resonance between neighboring nanowires, the intensity in each nanowire is enhanced by an order of magnitude over that in the case of $p = 400 nm$ above, leading consecutively to the enhancement in absorption by a factor of 8 in Fig. 4.

Figure 6. ${| E |}^{2}$ for TM polarized light in the $x - z$ plane at $λ = 920 nm$ for a period of (a) $p = 400 nm$ and (b) $p = 574 nm$ . Here, three nanowires of 80 nm in diameter are shown (dashed circles). Light is incident from the top side with ${| E_{inc} |}^{2} = 1$ and a region of 600 nm in length both above and below the nanowire array is shown. Notice the difference in the color scale by a factor of 10 between (a) and (b). (c) Schematic of the ${TM}_{01}$ resonance in the individual nanowires in (a). (d) Schematic of the coupled resonance between the ${TM}_{01}$ resonance and the lattice resonance in (b).

Regarding the coupling between nanowires, actually, at $p = λ / n_{glass} = 613 nm$ , we find that $A \approx 0$ (Fig. 4) for TM polarized light. Further analysis showed that $T \approx 1$ here (see inset in Fig. 4). Thus, the nanowires can become transparent to the incident light. This Wood’s anomaly ^[20] shows up as follows. Part of the incident light scatters from the nanowires into the first diffracted order, which propagates in the plane of the array, that is, in the $x$ -direction, when $p = λ / n_{glass}$ . Also this first diffracted order is TM polarized, and it is excited in such a way that it is out of phase with the incident light at the location of the nanowires (not shown here). Therefore, the electric field at the location of the nanowires reaches low values, leading to $A \to 0$ and $T \to 1$ .

We notice that there is a peak in the absorptance for TM polarized light also at $D \approx 190 nm$ [Fig. 5(a)], but this peak does not shift noticeably with period $p$ . We attribute this peak to the ${TM}_{11}$ mode and the weak dependence on the period to a weaker scattering of the ${TM}_{11}$ mode compared to the ${TM}_{01}$ mode in individual nanowires (not shown), which leads to weaker excitation of the lattice resonance.

We note that no peaks show up in $A$ for TE polarized light for $D < 100 nm$ [Fig. 5(b)]. For this polarization, to maximize $A$ , it is best to use a tight-packed array with $p = D$ , but even then, the absorptance is limited to $A < 4 %$ . However, if we allow for $D > 100 nm$ , we find also for TE polarized light an absorption resonance that shifts toward smaller diameters with increasing $p$ . We attribute this peak to coupling between the ${TE}_{01}$ mode, which is expected at $D \approx 190 nm$ , and the lattice resonance. In this case $A \approx 50 %$ can be reached.

We note that our results on the coupled mode resonance are not exclusive for a wavelength in the immediate vicinity of the bandgap wavelength. We have found similar results for InP nanowires at for example $λ = 600$ , 700, and 800 nm. Also, we expect the coupled resonance to be a property of the nanowire geometry and the array period, and therefore not strongly dependent on the exact choice of the direct bandgap III–V semiconductor material. Furthermore, we have found that when the nanowires are placed on a glass substrate, the coupled mode resonance shows up. In this case, the resonance shows increasing strength as the nanowires are covered by a glass layer of increasing thickness to resemble the case of nanowires fully embedded in glass, as considered above.

We note that in experiments, nanowires of a finite length are considered. A detailed study of the impact of the nanowire length on the coupled resonance is beyond the scope of this study. However, we have found indication that the optical response of nanowires of length equal to five times the wavelength starts to resemble closely the response of infinitely long nanowires.

In conclusion, we have shown in the linear optical regime that the optimum way to absorb light of a given wavelength in an array of horizontal InP nanowires of diameter less than 100 nm is to use coupled optical resonances. By carefully tuning the nanowire diameter and the array period, resonances in the individual nanowires can couple between neighboring nanowires through a lattice resonance.

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