1. Introduction

Photon energy conversion utilizing hot carriers generated in metals has gained tremendous interest in recent years. Research in this area has been mainly focused on photovoltaics [1–8] and photodetection [9–16]. In this aspect, surface plasmon-enhanced absorption via non-radiative decay process has been demonstrated particularly in plasmonic noble metals whose interband absorption thresholds are typically larger than 2 eV. Depending on the transparency to incident photons, different materials such as $\textrm{Al}_2\textrm{O}_3$ [1,11,16] and $\textrm{TiO}_2$ [6,8,13,15] for visible frequencies and $n$-type silicon (Si) [9,10,12] for infra-red operations have been used to form Schottky junctions for extraction of hot electrons. For noble metals, plasmon-induced hot electrons make some processes a reality that would otherwise not possible with photo-generated electrons.

While plasmonic noble metals such as cold (Au) and silver (Ag) have been widely utilized in surface-plasmon-enhanced hot carrier generations to overcome poor photon absorption within the metals, hot electron emitters using aluminum (Al) that fully take advantages of its quite complex band structure are relatively less explored. Part of the reason may be that Al allows direct interband transitions for a whole range of incident photon energies particularly greater than 1.5 eV, making it a less attractive metal in the field of plasmonics to the exclusion of photoexcited hot carriers. However, the difficulty of separating photoexcited (interband transition) and plasmon-induced (intraband transition) electrons also provides unique opportunities. Because of its unique band structure, Al has a broadband response to plasmon resonances, though remarkably sensitive to the degree of oxidation [17], that ranges from ultraviolet to much of the visible frequencies due to its relatively high bulk plasmon frequency. With a proper Schottky barrier and structure-dependent optical and plasmonic designs of the device, hot electrons produced in Al due to interband and intraband transitions may both contribute significantly to the photocurrent.

In this work, we theoretically investigate and experimentally demonstrate photon energy conversion concurrently utilizing photonic and plasmonic approaches driven by quasi-localized plasmon resonance (QLPR). The connection between the QLPR in a two-dimensional (2D) Al grating and the strong optical absorption in a contiguous planar-Al film due to diffraction is unraveled. On the other hand, theoretical efficiency limits in the framework of direct and fully-nondirect transitions are obtained for the first time on length scales ranging from a few to hundreds of nanometers, combining rigorous electromagnetic (EM) computations for optical absorption, the full electronic band structure for carrier generation, electron-electron (e-e) scattering for 3D carrier transport, and tunneling probabilities incorporated with momentum matching upon emission for collection. The prototype of the proposed device is fabricated and characterized at visible frequencies. The measured dark current that is often not reported in hot electron-based photodetection is also provided in an attempt to justify the photodetection process.

2. Fundamentals of photonic-plasmonic Schottky photodetector

The platform devised for investigating concurrent photonic and plasmonic energy conversion is shown in Fig. 1, where the upper Al structure consists of four nano-hexagonal prisms of two different radii (for broadband absorption) on a planar Al film. Titanium dioxide (TiO$_2$) as an insulator for free-space wavelength $\lambda _0>430$ nm separates the upper Al structure from the bottom Ag layer. Silicon nitride (Si$_3$N$_4$) existing among nano-prisms is used to minimize the device reflectance. The $x$- and $y$-periods of nano-prisms are set equal ($\Lambda _x=\Lambda _y=\Lambda$), thus forming a 2D square lattice in the $xy$-plane. With light illumination on the Al side and Ag as the bottom metal (whose interband absorption threshold is around 4 eV), the reverse electron flow is minimized.

 

Fig. 1. (a) Schematic of one primitive cell in the proposed photonic-plasmonic Schottky photodetector (PD) in multilayered metal-dielectric configuration. (b) The scanning electron microscope image of fabricated nano-hexagonal prisms of radii 85 nm (Hex. 2) and 115 nm (Hex. 1).

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Because of the hetero-interfaces of the TiO$_2$ film that is only a few nanometers thick, we observe unique guidance characteristics of the surface plasmon polariton (SPP) mode sustained in this novel device. Figure 2(a) depicts the constituent planar geometries of the proposed device: the air/Al/TiO$_2$/Ag and the air/Si$_3$N$_4$/Al/TiO$_2$/Ag. Both the air and bottom Ag regions are assumed semi-infinite for simplicity. As a representative combination among others, the thickness of the Al film $t_{\textrm{planar-Al}}$ is 60 nm (10 nm) in geometry 1 (geometry 2), while that of the Si$_3$N$_4$ is 50 nm in geometry 2. Their dispersion relations with the harmonic time dependence $e^{j\omega t}$ were rigorously formulated based on the transverse resonance condition [18] and are depicted in Fig. 2(b).

 

Fig. 2. Guidance characteristics of the lowest-order SPP modes sustained in planar metal-dielectric multilayered geometries: (a) the geometries considered: air/Al/TiO$_2$/Ag (Geom. 1) and air/Si$_3$N$_4$/Al/TiO$_2$/Ag (Geom. 2) with the TiO$_2$ thickness $t_d=5$ nm and (b) complex dispersion curves $\gamma (\omega )=\beta (\omega )-j\alpha (\omega )$ that are degenerate at Al-TiO$_2$ and TiO$_2$-Ag interfaces.

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With $t_d=5$ nm, the SPP modes supported at the Al-TiO$_2$ and TiO$_2$-Ag interfaces in both geometries are found to be degenerate, exhibiting identical $\beta$- and $\alpha$-spectral behaviors over a broad range of frequencies. The degeneracy remains for the normalized frequency $n_dt_d/\lambda _0<0.17278$ such that the phase factor introduced by the TiO$_2$ film of index $n_d$ is negligibly small in the $z$ direction. Moreover, the complex propagation constant $\gamma$ of the bound SPP mode in geometry 2 is found to be nearly independent of the Si$_3$N$_4$ thickness $t_{{\textrm{Si}}_3\textrm{N}_4}$ for $t_{\textrm{planar-Al}}\geq 10$ nm. Thus nearly identical dispersion behaviors of the SPP modes sustained in both geometries are observed for $\omega <4.7\times 10^{15}$ rad/s.

While the bound SPP mode determines the combination of the diffracted order $i$ and the period $\Lambda$ via the phase matching condition, no simple equations are available offering physical insight into the connections between strong absorption preferably in the planar-Al film and the plasmonics excitation. A theoretical analysis facilitating our understanding of such connections is of primary importance. As will be elaborated later, the delocalized resonant fields revealed in Section 2 are similar to and facilitate our understanding of those associated with 2D periodic hexagonal (hex.) prisms electrically-connected to the planar-Al film.

3. Exact formulation of 1D metallic grating in multilayered configuration

The theoretical formulation is based on the rigorous transmission-line network approach for the transverse magnetic (TM) wave. The electric $\textbf {E}$ and magnetic $\textbf {H}$ fields of a TM wave inside a general 1D grating [Fig. 3(a)] may be expressed in terms of the modal fields as [19]

In Eqs. (2)–(4), $\underline {\textbf {I}}(z)$, $\underline {\textbf {G}}(z)$, and $\underline {\textbf {V}}(z)$ are $N_t\times 1$ column vectors with $N_t$ being the total number of space harmonics (diffracted order) inside (outside of) the grating, whereas $\underline {\underline {\boldsymbol {\sigma }}}_x$ (a diagonal matrix) and $\underline {\underline {\textbf {C}}}$ (a square matrix) are $N_t\times N_t$ with elements $\sigma _{x,ii}=\textbf {k}_g\cdot \hat {\textbf {x}}+(N_t-i+1)|\textbf {K}|$ and $c_{ij}$, $i-j=p$, denoting the Fourier coefficients $c_p$’s associated with the grating profile.

 

Fig. 3. One-dimensional $\textrm{Si}_3\textrm{N}_4$-filled Al grating in multilayered configuration and the associated transmission-line network model for unraveling the mechanism behind strong absorption in the planar-Al film.

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Combining Eqs. (2)–(4) leads to the second-order differential matrix equation for the grating region

Applying eigen-decomposition to $\underline {\underline {\boldsymbol {{\Omega }}}}$ and solving Eq. (5), we obtain the modal current $\underline {\textbf {I}}(z)$ and the modal voltage $\underline {\textbf {V}}(z)$ waves in column vector form as follows:

The power absorption in an arbitrary layer (uniform or periodic) must be expressed in terms of its associated modal voltage and current waves that also depend on the structure underneath it. Such dependence is fully described by the input-output relations of $\underline {\textbf {V}}(z)$ and $\underline {\textbf {I}}(z)$ across each layer. In this respect the local coordinate $z'$ is used for easy cascading of transmission-line networks [see Fig. 3(b)], where $z'=0$ ($z'=-t$) coincides with the bottom (top) interface of each layer of thickness $t$. Key physical quantities in the formulations include the input impedance matrix $\underline {\underline {\textbf {Z}}}_{\textrm{in}}(z'=-t)$ seen looking from $z'=-t$ toward the end of the transmission line network at $z'=0$, the load impedance matrix $\underline {\underline {\textbf {Z}}}_{L,\,i}(z'=0)$ at $z'=0$, the reflection coefficient matrix $\underline {\underline {\boldsymbol {\Gamma }}}_{L}$ at $z'=0$, and the transfer matrices of modal voltages and currents, $\underline {\underline {\textbf {T}}}_{V}$ and $\underline {\underline {\textbf {T}}}_{I}$ associated with each layer.

We start from the formulation of $\underline {\underline {\textbf {Z}}}_{\textrm{in}}$ as follows. The continuity of modal voltage and current waves demands $\underline {\textbf {V}}_{i\,}(z'=0)=\underline {\textbf {V}}_{i+1}(z'=-t)$ and $\underline {\textbf {I}}_{i}(z'=0)=\underline {\textbf {I}}_{i+1}(z'=-t)$ at the interface between the $i$th and the $(i+1)$th layers, giving rise to $\underline {\underline {\textbf {Z}}}_{L,\,i}(z'=0)=\underline {\underline {\textbf {Z}}}_{\textrm{in},\,i+1}(z'=-t)$. For a uniform layer of thickness $t_u$,

Using $\underline {\textbf {V}}(z)$ and $\underline {\textbf {I}}(z)$ equations dedicated to a uniform or a periodic layer, we obtain the general expressions for the input-output relations across one layer of thickness $t$:

For a uniform layer, the transfer matrices are given by

For the grating region, we obtain

The power absorption within the $i$th layer is the difference in total time-average powers at the input and load ends of the associated lines:

Figure 4 shows the calculated absorptance in the planar-Al film $A_{\textrm{planar-Al}}$ and the 1D Si$_3$N$_4$-filled Al grating $A_{\textrm{grating}}$ as a function of grating thickness $t_g$ and filling factor $FF$ at $\lambda _0=638.9$ nm for two representative periods. Note that both periods can produce propagating SPPs at this wavelength. As observed, $A_{\textrm{planar-Al}}$ exhibits a strong selectivity for $t_g$, as opposed to its relatively slow varying behavior over the $FF$. In the case of $\Lambda =600$ nm, the highest $A_{\textrm{planar-Al}}$ and the highest $A_{\textrm{grating}}$ occur at nearly the same $FF\approx 0.34$. Furthermore, we noticed that strong absorption in the planar Al film is typically accompanied with strong absorption in the grating region, both occurring over a similar range of the grating thickness.

 

Fig. 4. Absorptance studies of a Si$_3$N$_4$-filled 1D Al grating in multilayered configuration [Fig. 3(a)] on a 500-$\mu$m-thick Si substrate at $\lambda _0=638.9$ nm: (a) and (b) are associated with $\Lambda =240$ nm, while (c) and (d) are with $\Lambda =600$ nm. Layer thicknesses are set to $(t_{\textrm{planar}-\textrm{Al}},t_d,t_{\textrm{Ag}},t_{\textrm{Si}})=(10,5,50,5\times 10^{5})$ nm. The number of space harmonics is 201.

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The concurrent strong absorption in the grating and the planar-Al film is found to be caused by resonant absorption, power penetrations of delocalized edge resonance into the film, and conventional wave diffraction. In Figs. 5(a), (c), and (e) where the fields and the $z$-directed time-average Poynting vector $P_z$ along the top boundary of the 1D Al grating are depicted, the $FF$ at which both $A_{\textrm{planar-Al}}$ and $A_{\textrm{grating}}$ are significant [see Fig. 4(c) and (d)] corresponds to the Al strip width that produces resonant superposition of all weighted space harmonics of the $E_x$ field at the strip edges. The resonant $\textrm{Re}[E_x]$ field is strongest at the top ends of the strip edges and diminishes at different space rates of change as it propagates downward, depending on the spatial distribution of diffraction efficiencies associated with all space harmonics. Both $E_x$ and $H_y$ fields subsequently penetrate into the planar-Al film, causing appreciable $P_z$ spikes in the planar-Al film underneath the strip edges [Figs. 5(b), (d), and (f)].

 

Fig. 5. Normalized $E_x$ and the corresponding $H_y$ (A/m) and $P_z$ (W/m$^{2}$) along the top (a) and bottom (b) interfaces of the 1D Al grating region shown in Fig. 3(a) with $\Lambda =600$ nm, $FF=0.34$, and $t_g=34$ nm at $\lambda _0=638.9$ nm. (c) and (d) [(e) and (f)] are the counterparts of (a) and (b), respectively, but with $(FF,t_g)=(0.34,60\textrm{ nm})$ [$(FF,t_g)=(0.60,34\textrm{ nm})$]. Layer thicknesses are identical to those given in Fig. 4. Dashed lines indicate the Al strip edges.

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The fact that the delocalized resonant field’s contribution to $A_{\textrm{planar-Al}}$ depends on the diffraction efficiencies of space harmonics are expatiated further below. For $FF=0.34$ and $t_g=34$ nm, Fig. 5(b) reveals that at the bottom interface of the grating region, $\textrm{Re}[E_x]$ is strongest midway between Al strips where $P_z$ is also greatest. Wave diffraction into the planar-Al film is therefore the major contributor to $A_{\textrm{planar-Al}}$ for this case. The situation is quite the opposite in Fig. 5(d) with $(FF,t_g)=(0.34,60\textrm{ nm})$, where $-z$-directed $P_z$ spikes almost at the bottom ends of the strip edges and is much smaller elsewhere. Since diffraction is minimum in this case, $A_{\textrm{planar-Al}}$ is largely contributed by delocalized resonant field penetration. With the right grating thickness that results in quite efficient grating diffraction [Fig. 5(f)], we still observe appreciable power penetration associated with the delocalized resonant field produced above the Al film.

4. Schottky PD with 2D periodic Al hexagonal prisms

Referring to Figs. 1 and 6, a more practical, polarization-independent, sub-optimum design (for relative ease of fabrication) is developed based on the work presented above. As observed in Fig. 6(b), the absorptance within the upper Al structure is larger than $80\%$ for $\lambda _0=[512,643]$ nm. Moreover, we see that the introduction of hexagonal prisms enhances the absorptance within the whole planar Al film [Fig. 6(c)] with each radius covering different spectral regions, despite the fact that they do present as obstacles to light incidence into the preferred Al film.

 

Fig. 6. Spectral behaviors of the proposed photonic-plasmonic PD in Fig. 1: (a) reflectance of the PD and absorptance spectra of one Hex. 1, one Hex. 4, and the planar Al film, (b) the absorptance spectrum of the upper Al structure as a function of polarization angle $\phi$ ($\phi =0^{\circ }$: $x$-polarized, $\phi =90^{\circ }$: $y$-polarized) at normal incidence, and (c) the absorptance enhancement factor (normalized to $A_{\textrm{planar-Al}}$ of an identical planar geometry without nano prisms) associated with different regions in the planar-Al film.

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At wavelengths around the peak of $A_{\textrm{planar-Al}}$, the propagating resonant fields described earlier exhibit field patterns similar to those observed in non-propagating localized plasmon resonance (LPR) around metallic nanoparticles. This is observed in Fig. 7. The $\textbf {E}$ field components resonate with symmetry or antisymmetry about the axial center of the prism. It is worth mentioning that, at resonance, a hexagonal prism exhibits a larger energy absorption density compared to a circular nanodisk of identical radius owing to stronger electric field intensities in the vicinities of vertical edges. As shall be seen later, the diffracted wave of order 10 for $\Lambda =600$ nm is phase matched to the SPP mode at the Al-TiO$_2$ interface. The dispersion relation in the grating region thus demands a small normalized $z$-directed wavenumber $\textrm{Re}[k_z/k_0]=0.27073$, corresponding to a longer effective wavelength. Since the resonance bear a strong resemblance to non-propagating LPR but remain slow propagating in the $-z$ direction, it may be termed as quasi-localized plasmon resonance (QLPR).

 

Fig. 7. Spatial distributions of normalized electric field (real part) in the $xy$-plane across the middle of the grating region at $\lambda _0=615.4$ nm under normal incidence of an $x$-polarized plane wave.

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The way the QLPR contributes to $A_{\textrm{planar-Al}}$ is very similar to that of the delocalized resonant fields in the 1D case. As observed in Figs. 8(a), (e) and (f), the power absorption density $P_{\textrm{abs}}(\textbf {r},\lambda _0)$ is relatively strong on account of the QLPR around the peripheries [points $a$, $e$, and $f$ in Fig. 9(b)] near the prism bottoms, causing strong absorption in the planar-Al film. In addition, relatively broadband absorption with a FWHM bandwidth of about 170 nm ($\lambda _0\approx [526, 696]$ nm) is readily observed. Their absorption spectra extend to longer wavelengths, covering nearly identical absorption bandwidths at points $c$, $d$, and $h$ where the strongest absorption occurs at the Al-TiO$_2$ interface. On the other hand, strong absorption in the planar-Al film between hexagonal prisms shown in Figs. 8(i) $-$ (l) is attributed to conventional grating diffraction.

 

Fig. 8. Power absorption spectra $P_{\textrm{abs}}(\textbf {r},\lambda _0)$ (W/m$^{3}$) as a function of $z$ position at some representative points in one primitive cell due to normal incidence of a plane wave polarized at $45^{\circ }$ with respect to the $x$-axis. Subplots (a) $-$ (h) [(i) $-$ (l)] correspond to points $a-h$ ($i-l$) inside (between) the hex. prisms shown in Fig. 9(b), respectively. In (a) $-$ (h), the horizontal white lines from the top are air-prism, prism-planar Al film, planar Al film-TiO$_2$, and TiO$_2$-Ag boundaries, while those in (i) $-$ (l) are Si$_3$N$_4$-planar Al film, planar Al film-TiO$_2$, and TiO$_2$-Ag boundaries, respectively.

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Fig. 9. Interference of SPP waves at the Al-TiO$_2$ interface of the proposed PD under normal incidence of an $x$-polarized plane wave of wavelength 615.4 nm: (a) the $E_z$ field (real part) and (b) the grating vectors $\textbf {K}$’s resulting in the interference pattern shown in (a). Also shown in (b) are some representative points $a-l$ at which the $z$-dependence of $P_{\textrm{abs}}(\textbf {r},\lambda _0)$ is investigated (see Fig. 8).

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The degenerate SPP waves are excited by the penetration of the $E_z$ field associated with the QLPR. They subsequently produce interference patterns at the Al-TiO$_2$ interface shown in Fig. 9(a). By treating points along the periphery of a hexagonal prism as source points of scalar spherical waves, the associated interference pattern may be written as

5. Theoretical efficiency estimates

Given the photonic and plasmonic processes driven by the QLPR in the Al-based photodetection, it is fundamentally important to quantify the efficiency limits of the proposed device. Theoretical formulation combining the spatial distribution of excited electrons, 3D carrier transport, and momentum matching upon emission at the planar Schottky junction is presented in this section. In this regard, excited electrons due to photons and non-radiative plasmon decay concurrently existing in metallic regions are practically indistinguishable. On the other hand, although localized electronic states exist in metals at nanoscale, giving rise to geometry-assisted intraband transitions [20] also termed plasmon-induced carrier generation [21], here direct and fully-nondirect transitions based on detailed electronic states obtained at the macroscopic scale are separately considered for simplicity, thus temporarily avoiding a multi-scale treatment for carrier generation.

The spatial distribution of all excited electrons is directly determined by the net $\textbf {E}$ field distribution within the metals. The per-unit-volume generation rate (in s$^{-1}$m$^{-3}$) is therefore given by $G_p(\textbf {r}',\nu )=\eta _{\textrm{IQE}}P_{abs}(\textbf {r}',\nu )/(h\nu )$, where $\textbf {r}'$ denotes the position at which upward transitions occur, $\eta _{\textrm{IQE}}(\nu )$ the internal quantum efficiency (assumed position-independent), $h$ the Planck constant, and $P_{\textrm{abs}}(\textbf {r}',\nu )=0.5\epsilon _0\varepsilon _m''\omega |\textbf {E}(\textbf {r}',\nu )|^{2}$ the absorbed power per unit volume with $\omega =2\pi \nu$ and $\epsilon _0$ and $\varepsilon _m''(\nu )$ the free-space permittivity and the relative permittivity (imaginary part) of the metal, respectively.

While $G_p(\textbf {r}',\nu )$ scales with the net $\textbf {E}$ field distribution, how the excited electrons distribute along the energy axis determines the probability that an electron is to be excited to a specific energy state. In the context of vertical (i.e. direct) transitions, this is described by the energy distribution of joint density of states (EDJDOS) [22,23] representing the number of allowed stimulated transitions per unit volume per unit energy squared due to $h\nu$ [23]:

The EDJDOS and $P_{\textrm{exc}}(E_f,h\nu )$ functions are quantified based on the realistic energy band structures of Al and Ag. Detailed descriptions can be found in [23].

The probability that an excited electron generated at some position $\textbf {r}'$ reaches and emits across the Al-TiO$_2$ Schottky junction may be described by the reaching-emission probability $P_{r\textrm{-}e}(\textbf {r}',E_f)$, taking into account electron-electron scattering and momentum matching upon emission. As $G_p(\nu ,\textbf {r}')$ is position-dependent, $P_{r\textrm{-}e}(\textbf {r}',E_f)$ is handled fully numerically in this work. Assuming isotropic carrier transport at zero bias, the paths an excited electron traveled during multiple reflections starting from an initial direction $\textbf {a}_{k_e}$ are numerically computed, as depicted in Fig. 10(a). Note that a total internal reflection and a zero photoemission probability are assumed at the Al-Si$_3$N$_4$ boundary because of the high Schottky barrier $\Phi _B$ of about $2.13$ eV. The allowed number of reflections within the upper Al structure is limited by the mean free path (MFP) $l_{e\textrm{-}e}$ largely determined by inelastic electron-electron (e-e) scattering [24,25]. Thus the equation for $P_{r\textrm{-}e}(\textbf {r}',E_f)$ may be written as

 

Fig. 10. Reaching-emission probability $P_{r\textrm{-}e}(\textbf {r}',E_f)$ computations: (a) illustration of a possible path traveled by an excited electron within the upper Al structure prior to emitting across the Al-TiO$_2$ Schottky junction, (b) convergence test and verification of the numerically-computed $P_{r\textrm{-}e}$ with a planar Si$_3$N$_4$-Al-TiO$_2$ geometry with $t_{\textrm{Al}}=50$ nm, $E_{ex}=3$ eV, $\Phi _B=0.39$ eV, $l_{e\textrm{-}e}=50$ nm, and $N_{\textrm{refl},m}=5$, and (c) the spatial distribution of $P_{r\textrm{-}e}$ for $\Phi _B=0.39$ eV and $E_{ex}=1.41$ eV in an irreducible region of one primitive cell.

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In Eq. (24) the tunneling probability $P_{\textrm{tunl}}(\textbf {r}',E_f,\textbf {a}_{k_e})$ is described by the Wentzel-Kramers-Brillouin approximation [26]

Figure 10(c) depicts the spatial distribution of $P_{r\textrm{-}e}(\textbf {r}',E_f)$ for $E_{ex}=1.41$ eV and $\Phi _B=0.39$ eV as a representative case. The total number of mesh points is 1,678,126 in one primitive cell. As observed, electrons excited within the planar-Al film, except for the region directly underneath the hexagonal prisms, have higher $P_{r\textrm{-}e}$ values up to $39\%$ owing to total reflections at the Al film-Si$_3$N$_4$ interface and shorter traveling distances prior to emission. In contrast, at points within prisms $P_{r\textrm{-}e}$ is $<15\%$. The decrease in $P_{r\textrm{-}e}$ with increasing $z$ position becomes more pronounced as the excess energy of a hot electron increases, making the MFP shorter and allowing fewer multiple reflections. Note that to fully exploit the impinging photon energy and to minimize losses during carrier transport, here the upper Al geometry is structured to inherently possess high absorptance and a relatively higher average reaching-emission probability.

Using the expression for $G_p(\textbf {r}',\nu )$ and Eqs. (23) and (24), the photocurrent produced in one primitive cell $I_{ph,\,\Lambda }$ may be obtained as follows:

The theoretical efficiency limits quantified in terms of the emission efficiency $\eta _{\textrm{emi}}$ and net quantum yield (QY) $\eta _{\textrm{QY, net}}$ are shown in Fig. 11. In arriving at these results the standard AM1.5G solar irradiance is used and the Schottky barrier energies are assumed to be $\Phi _{B,\textrm{top}}=0.89$ eV and $\Phi _{B,\textrm{bot}}=0.81$ eV based on the theoretical fit to the measured dark current [Fig. 13(a)], which are larger than those formed by bulk materials (0.38 eV and 0.36 eV at Al-TiO$_2$ and Ag-TiO$_2$ junctions, respectively). $\eta _{\textrm{emi}}$ is defined as the ratio of emitted electrons to excited electrons. Despite strong absorption of $>30\%$ for $\lambda _0\approx [348,651]$ nm there, nano prisms have a lower $\eta _{\textrm{emi}}$ than the planar-Al film does as a result of smaller $P_{r\textrm{-}e}$’s imposed by larger degrees of freedom and associated longer traveling distances during transport.

 

Fig. 11. Theoretical efficiency estimates: (a) emission efficiency $\eta _{\textrm{emi}}$ of the proposed PD and (b) comparisons of net QY spectra between the proposed PD and an identical, planar Si$_3$N$_4$/Al/TiO$_2$/Ag geometry (w/o periodic nano prisms). The net QY is about $4.841\%$ at $\lambda _0\approx 612.62$ nm in the context of direct transitions. Also shown in (b) are the calculation results in the fully nondirect approximation where the restriction on the conservation of momentum is lifted.

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On the other hand, comparative results of $\eta _{\textrm{QY, net}}$ in direct and fully nondirect transitions for planar and nanostructured geometries are given in Fig. 11(b). We define $\eta _{\textrm{QY, net}}$ as the ratio of the net collected electron flux ($\Phi _{e,\,\textrm{Al}}-\Phi _{e,\,\textrm{Ag}}$) to the incident photon flux. With constant matrix elements assumed and apart from exact momentum eigenstates, results from fully-nondirect transitions might approximate only to some degree the plasmon-induced carrier generation. The peak in $\eta _{\textrm{QY, net}}$ spectra at $\lambda _0\approx 391$ nm is produced solely by the QLPR. Compared with the planar multilayered geometry of identical layer thicknesses, the proposed PD exhibits enhanced $\eta _{\textrm{QY, net}}$ spectra over a major portion of the spectrum of interest as a result of enhanced absorption owing to the QLPR and diffracted waves into the planar-Al film.

6. Experimental demonstrations and discussions

The proof-of-concept prototype of the proposed design was fabricated [see Fig. 1(b) and Fig. 12(a)] and experimentally demonstrated on a Si substrate. The large-scale active region is about 500 $\mu$m $\times$ 500 $\mu$m in area with $\textrm{Si}_3\textrm{N}_4$, planar-Al film, TiO$_2$, and bottom Ag thickness of 50, 10, 5, and 50 nm, respectively. The TiO$_2$ film was grown by atomic layer deposition at $90^{\circ } \textrm{C}$. Al and Ag films were deposited using electron-beam evaporation at about 0.9 $\mathring {\textrm{A}}/\textrm{s}$ under the base pressure of $4\times 10^{-6}$ torr or smaller. Nano-hexagonal prisms were fabricated using standard electron-beam lithography, anisotropic etching of $\textrm{Si}_3\textrm{N}_4$, e-beam evaporation, and lift-off techniques.

 

Fig. 12. (a) Optical microscope image (at $500\times$) of the fabricated Al-based PD and (b) the measured reflectance spectrum $R$ with varying polarization angles ranging from $-90^{\circ }$ to $90^{\circ }$.

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Figure 12(b) shows the measured reflectance spectrum normalized to that of a high-reflectance silver mirror as a function of rotation angle $\varphi$ ranging from $-90^{\circ }$ to $+90^{\circ }$. The measurements were conducted using a 6-around-1 (each with a 100-$\mu$m core diameter) reflection fiber probe (Ocean Optics) precisely aligned to the rotation centers of the active area and the motorized stage, thus translating the rotation angle $\varphi$ to the polarization angle $\phi$ of the incident light. With six-fold symmetry of the hexagonal prisms, the range of $\varphi$ would cover all polarization angles relative to the grating vector $\textbf {K}_x$, even if the incident $\textbf {E}$ vector is not perfectly parallel to $\textbf {K}_x$ at $\varphi =0^{\circ }$. The measured reflectance is shown to be constantly $<10\%$ for $\lambda _0\approx [550,750]$ nm and is $<20\%$ for $\lambda _0\approx [400,950]$ nm across the angular spectrum.

Figure 13(a) shows the current-voltage characterizations of the dark current $I_{\textrm{dark}}$ and the photocurrent $I_{ph}$ under light illumination at $\lambda _0=638.9$ nm conducted within a shielding box at room temperature. The device was configured with the planar-Al film connected to the positive terminal of a Keithley 6430 source meter. In Fig. 13(a) each curve is the average of at least two runs of $I$-$V$ measurements; each current value is the mean of 10 measurement readings at each applied voltage $V_a$. The dark current is typically in the order of sub-nano ampere for $V_a=[-0.265,0.5]$ V and its asymmetric dependence on $V_a$ may be attributed to unequal Schottky barriers at metal-TiO$_2$ boundaries. In the present case, we speculate $\Phi _{B,\,\textrm{top}}>\Phi _{B,\,\textrm{bot}}$ since a positive bias is required to raise the quasi-Fermi level of Ag so as to balance thermal electron flows produced on both sides of the TiO$_2$ film. On the other hand, the photocurrent measured under a constant optical power of $663.25\times 10^{-3}$ W exhibits a diode-like behavior and increases as the reverse bias is increased. With the negative bias applied to the planar-Al film, the band edge of TiO$_2$ is tilted down toward the Ag side as a result of lifting the quasi-Fermi level of Al relative to that of Ag [see the left inset in Fig. 13(b)], thus effectively reducing the barrier thickness and increasing tunneling probabilities of hot electrons generated in the upper Al structure.

 

Fig. 13. Dependence of measured dark current $I_{\textrm{dark}}$ and photocurrent $I_{ph}$ on applied bias (both with error bars of one standard deviation) (a) and the corresponding responsivity and EQE (b) of the fabricated device.

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The dependence of the responsivity $\mathcal {R}_\lambda$ and the external quantum efficiency (EQE) (defined as the ratio of the number of electrons in the photocurrent to the number of incident photons) on the applied bias voltage is shown in Fig. 13(b). The responsivity in $\mu$A/mW (responsivity per unit area in $\mu$A/mW-mm$^{2}$) is 0.0011 (0.0043), 0.2414 (0.9655), 1.5199 (6.0794), and 4.1889 (16.7556) at $2.9564\times 10^{-5}$V, $-0.1$ V, $-0.3$ V, and $-0.485$ V, respectively, while the corresponding EQE is $2.0649\times 10^{-4}\%$, $0.0468\%$, $0.2949\%$, and $0.8129\%$. It is five orders of magnitude larger than the propagating-SPP-based planar MIM geometry operating at 633 nm reported previously [1,16].

7. Conclusions

Photon energy conversion concurrently utilizing photonic and plasmonic approaches in Al-based multilayered configuration with periodic nano-hexagonal prisms has been proposed, theoretically investigated and experimentally demonstrated. Strong absorption preferably in the planar-Al film contiguous to the Al grating is found to occur when the Al grating produces delocalized resonant superposition of all weighted space harmonics of the transverse $\textbf {E}$ field component at the strip edges. The spatial distribution of diffraction efficiencies of these space harmonics greatly affects plasmonics contributions to absorption in the planar Al film. In the proposed device, the situation is generalized to the slow-propagating QLPR with small longitudinal phase constants associated with dominant diffracted orders existing around the peripheries of nano-hexagonal Al prisms. EM power penetration associated with the QLPR, optical wave diffraction, and interference of SPP waves (excited by the QLPR and are degenerate at hetero-interfaces of 5-nm-thick TiO$_2$ film) all contribute to strong photon energy absorption in the upper Al structure.

Theoretical efficiency limits have been quantified in the frame work direct and fully-nondirect transitions based on 3D EM computations together with realistic energy band structures of Al and Ag, 3D carrier transport, and momentum matching at the Schottky junction. The results indicate that the emission efficiency and in turn the net QY are largely limited by the spatial degrees of freedom and the associated traveling distances of hot electrons during carrier transport. A nanostructured metallic photon absorber inherently minimizing these limitations may largely increase its conversion efficiency. The theoretical QY of the proposed PD at zero bias is about $4.5461\%$ at $\lambda _0=638.9$ nm. The fabricated device is demonstrated to be polarization insensitive and the measured dark current is in the order of sub-nano ampere for $V_a=[-0.265,0.5]$ V. With the upper Al structure designed to combine strong power absorption with a high average reaching-emission probability for hot electrons, the responsivity and the EQE at $V_a=-0.485$ V are up to 4.1889 $\mu$A/mW (or 16.7556 mA/W-mm$^{2}$) and $0.8129\%$, respectively. Our results provide physical insights into related metallic, nanostructure-dependent problems and the demonstrated approach may offer a route to more efficient, hot-carrier based photoelectric conversion devices operating at visible and infrared frequencies.