1. Introduction

Novel materials platforms could provide a route to room-temperature infrared detectors with improved performance. Black phosphorus (BP) has recently been investigated for this purpose, due to ease of integration with silicon and high mobility [1–5]. Several BP – based photodetectors and diodes have been demonstrated with high responsivity and low switching times [3,6–15]. In addition, BP has an electrostatically tunable bandgap, providing spectral selectivity. However, for bandgap tuning to be effective, the film thickness must be reduced below 50 nm, greatly reducing absorption. To alleviate this concern, we introduce a nanostructured device design that uses light-trapping techniques [16,17] to boost absorption in thin BP films.

Light trapping has previously been studied for photovoltaics [18,19] and photodetection [20–23]. The basic design principle is to create localized, electromagnetic resonance modes with high field intensity, enhancing absorption [24,25]. Absorption enhancement designs for black phosphorus have used a variety of nano- and microstructures [26–31], including structures that support surface plasmon modes [32–48]. However, previous work has largely focused on black phosphorus as a 2D material, i.e. an atomic monolayer described by a metallic permittivity. For infrared photodetector applications, it is desirable to exploit the properties of black phosphorus thin films. For thin films, which are many atomic layers thick, the material acts as a semiconductor. Due to the physically different nature of the dielectric function, the light trapping strategies used for black phosphorus monolayers cannot directly be applied. Moreover, for infrared photodetector applications such as focal-plane arrays, it is desirable to develop light-trapping designs for which electrical contacts can easily be integrated.

Here we show that a metal-insulator-metal (MIM) design provides large absorption enhancement in a black phosphorus thin film, while simultaneously providing a convenient method for vertical contact integration. A MIM device [49–57] is comprised of a dielectric layer sandwiched between an array of nanoscale metal elements and a metal back reflector. This structure creates a lateral resonance mode, increasing absorption in the dielectric layer [56]. Here, we focus on the 3–5 µm wavelength range, which is of particular interest for infrared imaging [58,59] and sensing [60–62]. We design gold – black phosphorus MIM absorbers resonating at a wavelength of 4 µm for BP layer thicknesses ranging from 5 to 50 nm. In order to test the light-trapping efficiency of our absorbers, we compute their enhancement factor and study its dependence on the BP layer thickness. By optimizing our design, we show that the absorption enhancement exceeds the conventional 4n² [63] light-trapping limit for any BP layer thickness under 50 nm. We further show that the peak absorption wavelength and amplitude is robust to angle of incidence.

A notable feature of our approach is that the top metallic grating in the MIM structure can also serve as an electrical contact. The other contact is provided by the bottom metal layer, which is in direct contact with the BP active region. This is in contrast to previous work in the literature, done in the visible and near infrared. These designs either used a single metal top layer [39,44] or had a metal back reflector that was separated from the BP active layer by an insulating region [33]. Our results suggest that by using light-trapping techniques, very thin, black-phosphorus layers can be used for effective, spectrally selective photodetection in mid-IR applications.

2. Methods

Figure 1(a) shows a schematic of the structure we use for absorption enhancement. A thin layer of black phosphorus with thickness d_BP lies on a gold back reflector with thickness d_Au = 50 nm. A gold grating with thickness d_grating = 0.1 µm, stripe length L, and period a lies on top of the black phosphorus layer. The coordinate axes x, y, and z align with the BP crystal axes c, a, and b respectively. The thickness of the back reflector is chosen to be greater than the skin depth of gold in the infrared (∼ 10 nm), to ensure that there is no transmission through the structure. The entire structure is supported on a 5 µm thick SiO₂ substrate. We refer to the structure of Fig. 1(a) as a Metal-Dielectric Nanocavity Absorber (MDNA).

 

Fig. 1. (a) Absorption enhancement structure. (b) Unit cell of the structure depicting the waveguide picture. The magenta curve shows the magnitude of the z component of the electric field associated with the TM₀ waveguide mode in Region 1.

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The working principle of the MDNA can be understood by viewing the unit cell as a combination of two waveguide regions [56]. This situation is depicted in Fig. 1(b). In a particular wavelength range of interest, Region 1 supports a TM₀ mode propagating in the + x direction. The magenta curve in Fig. 1(b) shows E_z for the TM₀ mode, calculated from Lumerical MODE solver for a wavelength of 4 µm. We note that the MIM structure supports propagating modes even for insulating layer thicknesses much less than the wavelength [64]. Region 2 does not support a propagating mode. This leads to an impedance mismatch at the boundary between Regions 1 and 2, resulting in the reflection of the TM₀ mode back and forth between the ends of Region 1. The superposition between forward and backward propagating modes leads to the formation of a standing wave, localized below the metal stripe. We call the standing wave a lateral Fabry-Perot (LFP) resonance.

We visualize the standing wave in our black phosphorus MDNA using Lumerical FDTD solutions. We choose a grating periodicity of 0.6 µm and adjust L to place the fundamental LFP resonance at 4 µm. The optical constants of gold and SiO₂ are taken from Palik et al., while those of black phosphorus are obtained from the pseudopotential calculation data published by Morita [65]. Figure 2(a) shows the electric field profile for the fundamental LFP mode for a BP layer thickness of 20 nm. Most of the field is concentrated in the BP layer. Field intensities are high near the Au stripe edges and drop off towards the center, indicating a node. We note that the stripe length L required to create an LFP mode is significantly smaller than the wavelength, a result of high effective index in the MIM waveguide [56]. For the structure of Fig. 2(a), we calculated n_eff of 5.29 + 0.19i using the Lumerical MODE solver.

 

Fig. 2. (a) Electric field profile for an MDNA unit cell with d_BP = 20 nm. (b) Absorption spectrum of an MDNA with d_BP = 20 nm.

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3. Results

We calculate the wavelength-dependent absorptivity from the FDTD simulation. We simulate a single unit cell of our MDNA and apply periodic boundary conditions at the left and right edges to account for the periodicity (see Fig. 1). The structure is illuminated by a plane wave source and reflection/transmission are recorded by placing power flux monitors above the MDNA and at the top and bottom of the BP layer. In order to spatially resolve the thin BP layer, we use a mesh with grid resolutions of 5 and 0.5 nm along the x and z directions respectively (Fig. 1(b)). The total absorption is given by 1 minus the reflection from the top of the MDNA. The absorption in the BP layer is given by the difference in flux at the top and bottom of the layer. In Fig. 2(b), we show the wavelength-dependent absorptivity for the entire MDNA with d_BP = 20 nm (black, dashed line). The absorptivity is strongly peaked, with a center wavelength of 3.95 µm. The absorptivity in the BP layer alone is shown by the red, solid line. The difference between the red, solid and black, dashed lines is due to absorption in the gold. For comparison, we also present the single-pass absorptivity of an isolated 20 nm-thick BP layer, shown by the blue, solid line. The MDNA design provides significant absorption enhancement over a single layer of black phosphorus. At a wavelength of 4 µm, the single pass absorptivity in an isolated 20 nm thick BP layer is 3.8 × 10⁻³. The absorptivity in the BP layer of the MDNA is 0.6, corresponding to an enhancement factor of 166.

Next, we investigate the dependence of absorption enhancement on the BP layer thickness.

To keep the fundamental LFP resonance wavelength fixed at 4 µm, the metal stripe length L must be adjusted as the BP layer thickness is varied. Figure 3(a) shows the variation of the resonance wavelength λ_max with L and d_BP. The data was plotted by performing simulations of the MDNA, with L varied in steps of 0.1 µm and d_BP in steps of 5 nm. We observe that λ_max increases with stripe length L, as expected for a lateral Fabry-Perot cavity. λ_max decreases with layer thickness d_BP. This is due to a decrease in the effective index of the waveguide mode with d_BP (verified via separate mode solver simulations). For the LFP mode, a decrease in effective index decreases the resonant wavelength [56], in analogy to a standard Fabry-Perot cavity. Using MATLAB’s linear interpolation routine, we determine L corresponding to λ_max = 4 µm for each d_BP. These values of L are indicated by solid green circles in Fig. 3(a). The underlying solid black line serves as a guide to the eye to identify the approximate constant wavelength contour.

 

Fig. 3. (a) Colormap showing the variation of λ_max with L and d_BP for a = 0.6 µm. The values of L corresponding to λ_max = 4 µm are indicated by solid green circles and the corresponding constant wavelength contour is shown by the underlying black line. (b) Wavelength-dependent absorptivity in the BP layer of an MDNA for a few values of d_BP. (c) Variation of spectrally averaged absorption in the BP layer of an MDNA with the layer thickness. (d) Variation of absorption enhancement factor at a wavelength of 4 µm with the BP layer thickness for our MDNA design. The conventional 4n² limit is indicated by the solid red line.

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Using the values of L obtained from Fig. 3(a), we calculate the wavelength-dependent absorptivity in the BP layer, A_{BP, MDNA}, for several values of d_BP. The results are shown in Fig. 3(b). In each case, the absorptivity peaks near 4 µm, as desired. In Fig. 3(c), we plot the spectrally-averaged absorption in the BP layer over the 3–5 µm range, $\bar{A}_{BP,\ MDNA}$, as a function of layer thickness. The value is plotted as a percentage (out of 100%). As suggested from Fig. 3(b), the absorption increases with layer thickness.

4. Discussion

We quantify the absorption enhancement offered by an MDNA relative to a single pass through a layer of black phosphorus. The wavelength-dependent enhancement factor F is defined as:

An important benchmark for absorption enhancement is the classical 4n² limit [63]. For a textured layer of isotropic, homogeneous dielectric with refractive index n, absorption enhancement results from total internal reflection. In this case, statistical ray optics indicates an upper limit of 4n² for the enhancement factor [63,66,67]. We will refer to this limit as F_c. Literature on light-trapping using microphotonic structures has shown that this limit can be exceeded in the wave optics regime. These works have largely focused on photovoltaic applications in the visible and near-infrared range, rather than on the mid-infrared range studied here. We calculate F_c for a BP layer by using the refractive index along its b axis and neglecting dispersion. Taking n = 2.9, we get F_c= 34.

Figure 3(d) shows the enhancement value F for our MDNA design, as a function of d_BP. For each d_BP, the value of F is calculated at the resonance wavelength (approximately 4 µm in all cases). We observe that an enhancement of F = 425 can be obtained by using an MDNA with d_BP = 5 nm. This is significantly greater than the classical limit, F_c = 34. Moreover, F > F_c for all layer thicknesses. This highlights the utility of our MDNA design in enhancing the absorption of ultra-thin dielectric films.

The enhancement factor of a given MDNA can be further increased by optimizing the gold grating period. Figure 4(a) shows the absorption spectrum for a few values of the grating period a. The peak wavelength of the resonance does not shift with grating period, as expected for uncoupled resonators [56]. For fixed gold stripe length, the magnitude of absorption decreases as the grating period increases. This result can be explained by a simple physical picture. In Fig. 2(a) above, we have shown that the fields associated with the LFP mode are confined in the region under the gold stripe. Figure 4(a) suggests that as the width of the unit cell increases relative to the stripe, the fraction of incident power coupled into the resonance decreases. This results in a reduction of absorption in the black phosphorus layer.

 

Fig. 4. (a) Wavelength dependent absorptivity in the BP layer of the MDNA with d_BP= 5 nm for a few different values of a. (b) Variation of F with a.

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Figure 4(b) shows the variation of the enhancement factor with grating period. F increases sharply with decreasing grating period over the range shown. A maximum of 561 is obtained for a grating period equal to 0.3 µm. In principle, it is possible to reduce the grating period below 0.3 µm. However, that would significantly reduce the gap between successive metal stripes and make their fabrication difficult.

A further advantage of the optimized MDNA design is strong absorption at off-normal angles of incidence. In simulations, we vary the angle of incidence θ from 0 to 60 degrees and observe the variation in the wavelength-dependent absorptivity. From Fig. 5(a), it can be observed that λ_max does not vary significantly with change in θ . This observation is consistent with the discussion by Todorov et al. about the existence of flat dispersion for MDNAs with dielectric layer thicknesses much smaller than the wavelength of operation [56]. In Fig. 5(b), we show that the spectrally averaged absorption in the BP layer increases with an increase in θ. This can be attributed to enhanced coupling of incident radiation to the waveguide 1 region at large angles of incidence.

 

Fig. 5. (a) Wavelength dependent absorptivity in the BP layer of the MDNA with d_BP= 5 nm and a = 0.3 µm for a few different values of θ. (b) Variation of the spectrally averaged absorption in the BP layer of the MDNA with θ.

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5. Conclusion

We designed infrared absorbers based on a gold – black phosphorus periodic nanostructure to achieve enhanced absorption in the 3–5 µm wavelength range. By utilizing the lateral Fabry Perot effect, we were able to achieve strong field confinement in a thin BP layer sandwiched between a gold stripe grating and a back reflector. We used an interpolation-based numerical technique to design absorbers with peak absorption at 4 µm for various BP layer thicknesses. For a 5 nm thick BP layer, we were able to achieve an absorption enhancement of 425 at a wavelength of 4 µm. This is significantly greater than the conventional 4n² limit. We showed that by suitably optimizing the grating period for this particular structure, the enhancement factor can be increased to 561.

The design we present will be useful for the realization of mid-IR photodetectors based on black phosphorus. The metal grating used for light-trapping and absorption enhancement can potentially be used as an electrical top contact for the structure, while the gold back-reflector serves as the bottom contact.