Si is an integral part of all electronic devices. It plays an important role in optoelectronic devices such as photodetectors and photovoltaic devices^{1, 2}. However, since it is an indirect bandgap material, phonon assistance is required in the photoexcitation process to compensate the wavevector difference between the valence band maximum and the conduction band minimum of holes (Γ point) and electrons (X point), respectively (Fig. 1^{3}). Therefore, although Si dominates the photonics devices, a field that is better known as Si photonics^{4}, it is a substrate for the light emitting materials^{5, 6}. Moreover, it can be used for a much wider variety of applications if its optical absorption in the telecommunication wavelength range is improved^{7}. In this article, some important recent studies directed towards improving the performance of Si photonic devices have been reviewed.
Theoretical advancesOptical nearfield (ONF) effect can potentially improve the carrier excitation in indirect bandgap materials. This is attributed to the fact that ONF is expected to have large wave vector components (Δk), which implies very small Δx, due to the field localization. This was validated in a study which demonstrated that ONFs, as a dipolemode plasmon, generate carriers directly^{8}. It was shown that in small particles, the plasmons induce an electric field that exhibits Fourier components with large wave numbers. Furthermore, it was demonstrated that such an electric field generates carriers without phonon assistance. Further investigation of the ONF excitation was performed in a system comprising of metallic nanospheres embedded in crystalline Si^{9}. Here, the excitation was evaluated under the framework of linear perturbation theory, and it was found that the ONF effect is too weak to be measured in a real system. However, these calculations were based on analytical model systems, and not on the real carrier excitations.
To this end, Yamaguchi et al.^{10} evaluated the carrier dynamics using firstprinciples calculations, where they calculated the electron excitation by considering the indirect interband transition between states with different Bloch wave numbers. Instead of using the Bloch condition, which cannot describe the indirect interband transitions, they implemented the Bornvon Kármán boundary condition^{11, 12}. Further, they also performed a timedependent calculation of the ONF excitation using onedimensional KronigPenny (1DKP) model (Figs. 2(a) and 2(b)). They considered the dispersion relation with an indirect band gap (Fig. 2(c)). As shown in Figs. 2(d) and 2(e), it was observed that the absorption spectra for farfield excitation decreases at the bandgap energy (E_{d}), while the absorption spectrum for nearfield excitation decreases at the bandgap energy (E_{g}) (much smaller than E_{d}) which was realized by the direct excitation. Furthermore, they clearly showed that the nearfield component of the dipole radiation (r^{3}) is the origin of ONF excitation^{13}.
A more detailed investigation of the ONF excitation in realistic three dimensional Si systems (Fig. 3(a)) was reported recently^{14}. Here, a realtime and realspace gridbased timedependent densityfunctional theory (DFT) approach^{1517} was used. A supercell method along with the Bornvon Kármán periodic condition was implemented to calculate the wavevector excitation^{11}, where the vertical band gap energy at the Γ point was 2.0 eV. In this system, the source of the ONF was a ypolarized point dipole with the nearfield component (r^{3}), which was set 5 Å apart from the surface of Si. Thus, realistic metallic structures, which could generate plasmon, were not treated explicitly in this study. The potentials of the ONF were described by
$ {V_{{\rm{near}}}}(r, t) =  \frac{{C(y  {y_{\rm{p}}})}}{{{r^3}}}\sin (\omega t){\sin ^2}(\frac{{{\rm{ \mathsf{ π} }}t}}{T}), $  (1) 
where ω is the frequency of the oscillating dipole, C is a constant, and T = 30 fs is the pulse duration. The electric field distribution induced by the oscillating dipole, as described by Eq. (1) is nonuniform in spatial domain (see Fig. 3(b)). Therefore, it results in the generation of large components of the Fourier domain as shown in Fig. 3(c), which implies that the ONF has large wave vectors. Based on these observations, the absorption intensity was obtained by calculating the transition probability. As shown in Fig. 3(d), the absorption intensity for the farfield excitation (blue solids circles and blue line) decreases around 2.0 eV, which corresponds to the band gap energy at the Γ point. On the contrary, the ONF induces a sufficiently large absorption intensity, which causes a red shift of the absorption spectra (red solid circles and red line in Fig. 3(d)). Furthermore, direct optical transitions between different wave vectors were also confirmed. Figure 3(e) shows the absorption intensity as a function of the variation in wave vector for the electronic excitation between secondhighest valence band and the lowest conduction band in the ONF excitation at 1.6 eV. Here, the absorption intensity is maximum at ∆k ~4 nm^{1}. This was the direct evidence of the wavevector excitation induced by the ONF. In other words, the direct ONF excitation does not require phonon assistance and can be realized with photons exclusively.
Experimental advances Scattering effect: Black SiA simple yet effective way to improve the optical absorption in the solar cells is to increase the optical path length through the depletion layer between the pn junction. To improve the light scattering efficiency, the scatterers have been fabricated at the surface of the detector, such as pyramidal Si hillocks made by the anisotropic etching^{18}, vertically aligned single crystalline Si nanowire array^{19}, or needles made by the laser irradiation with the ultrashort pulse width^{20}. In particular, the latter (Si surface with needles) works as an extremely good scatterer because of its small sharp tip with no reflection from the surface. Consequently, it appears black, and is therefore called a black Si. This black Si has been used to develop highly efficient Si solar cells with 22.1% efficiency^{21, 22}.
Plasmonic effectTo improve the scattering efficiency, many researchers utilized the field enhancement using the plasmon resonance^{23, 24}. The scattering and absorption crosssections are given by point dipole model^{25}:
$ {C_{{\rm{sca}}}} = \frac{{{k^4}}}{{6{\rm{ \mathsf{ π} }}}}{\left \alpha \right^2} = \frac{{8{\rm{ \mathsf{ π} }}}}{3}{\left( {\frac{{2{\rm{ \mathsf{ π} }}}}{\lambda }} \right)^4}{a^6}{\left {\frac{{\varepsilon  {\varepsilon _{\rm{m}}}}}{{\varepsilon + 2{\varepsilon _{\rm{m}}}}}} \right^2}, $  (2) 
$ {C_{{\rm{abs}}}} = k{\mathop{\rm Im}\nolimits} \left[ \alpha \right] = 4{\rm{ \mathsf{ π} }}\left( {\frac{{2{\rm{ \mathsf{ π} }}}}{\lambda }} \right){a^3}{\mathop{\rm Im}\nolimits}\left[ {\frac{{\varepsilon  {\varepsilon _{\rm{m}}}}}{{\varepsilon + 2{\varepsilon _{\rm{m}}}}}} \right], $  (3) 
where α is the polarizability, a is the radius of the particle, ε is the dielectric function of the particle, ε_{m} is the dielectric function of the surrounding media, and λ is the wavelength. It is evident from Eq. (2) that ε should be negative (2ε_{m}), i.e., the scatterer should be a metal, to utilize the plasmon resonance for the light scattering. Further from Eq. (3), it is clear that the plasmon induced light scattering is indispensable for a strong optical absorption. For a more effective trapping of the scattered light at the metallic nanoparticles on the surface, the scattered light excites the waveguide mode using a thin Si substrate of Sioninsulator (SOI)^{26, 27} wafer. Furthermore, a higher coupling efficiency can be realized by controlling the size and the shape of the metallic nanoparticle^{28, 29}. In particular, it was observed that smaller size, as well as cylindrical and hemispherical shapes of spheres lead to a longer path length in the substrate. Further improvement of the carrierexcitation was realized by introducing a photonic design to induce light trapping with threedimensional structures consisting of nanowires and nanoparticles^{30, 31}. Since the plasmon resonance for the Au sphere occurs at ~520 nm, ellipsoid shape^{32}, chains of nanoparticles^{33}, or elongated shapes such as nanorod^{34} with longer resonance wavelength have been implemented to obtain larger absorbance in this wavelength range. In addition, the periodic structure of the metal, i.e., metamaterialplasmonic absorbers^{35, 36}, has been investigated to enhance the optical absorption. The periodic structure works as a grating coupler of light with a normal incident angle. Consequently, it results in a strong absorption exceeding 80%. We investigated the selfassembly method^{37} to improve the efficiency of nanoparticles deposition. Using the laserassisted deposition of the electrode metal with a reverse biased pn junction, we realized a selective photocurrent generation in the transparent wavelength range^{38}. We observed a drastic change in the surface morphology of the metal, and confirmed the increase in the photocurrent at the wavelength that is close to that used during the electrode deposition. Since Au has high absorption coefficient, alternative materials, such as transparent conducting oxides were considered to achieve the resonance at longer wavelengths^{39, 40}.
Hot electronsThe internal photoemission on a Schottky barrier (Fig. 4) has been investigated to utilize the scenarios where the photon energy is lower than the band gap energy of Si (E_{C}E_{V} = 1.1 eV)^{4143}. Since the Schottky barrier between Au and Si (ϕ_{B} ~ 0.5 eV) is lower than the band gap energy of Si, the infrared wavelength region that is longer than the bandgap wavelength (1100 nm) can be used. The carriers of electron (hot electrons) were excited to obtain energies that are higher than the Schottky barrier. By optimizing the plasmon resonator, the responsivity of 4.5×10^{4} A W^{1} was obtained at nearly 1600 nm (~0.77 eV). Using a three dimensional plasmonic resonator, a photocapacitance structure was developed to detect the charge generation at the Schottky barrier with higher efficiency (Fig. 4(b))^{44}.
Nearfield effectAs discussed in the previous section, the ONF effect can improve the absorption efficiency at the bandgap wavelength. This implies that the localized field can induce large wave vector components, which in turn leads to direct excitation in indirect bandgap semiconductors (Fig. 1(b)). To corroborate this effect, we fabricated a Si photodetector with a lateral pn junction (Fig. 5(a))^{3}. We avoided the field enhancement due to the Au nanoparticles^{24, 26, 27} in this device by introducing their nearfield sources with extremely low coverage ~2 % (Figs. 5(b) and 5(c)). Consequently, we observed a 40 % increase in the photosensitivity as compared to that without Au nanoparticles (Figs. 5(d) and 5(e)). As shown in Fig. 5(e), the photosensitivity rate increased at longer wavelengths near the bandgap wavelength. This dependence has not been reported for devices that utilized plasmon resonance^{23, 24, 26, 27}, where the enhancement in photosensitivity exhibits a peak around the plasmon resonance, which is determined by the materials and shapes. To find the origin of this dependence, we evaluated the wavelength dependence under the assumption that the ONF effect causes a direct excitation in the indirect bandgap semiconductor. The absorption coefficients of the indirect (α_{I}) and the direct (α_{D}) bandgap materials near the energy band gap E_{g} are described as^{45}
$ {\alpha _{\rm{I}}} \propto {(h\nu  {E_{\rm{g}}})^2}/h\nu , $  (4) 
$ {\alpha _{\rm{D}}} \propto {(h\nu  {E_{\rm{g}}})^{1/2}}/h\nu , $  (5) 
where h is Planck constant and ν is frequency of light. Since the bias voltage was 0, the photocurrent becomes the short circuit current described as^{46}
$ {I_{{\rm{SC}}}} = Q(1  R)\{ 1  \exp (  \alpha l)\} en, $  (6) 
where Q is the collection efficiency, R is the reflection coefficient, l is the absorbing layer thickness, e is the electron charge, and n is the number of photons per second per unit of the pn junction. The ONF can induce a direct transition by the inclusion of Au nanoparticles. Thus, the photocurrent for this device is the same as that of the direct transition (I_{SC_D}). However, since the coverage of the Au nanoparticles A is not 100 %, the photocurrent shows an increase in the regions where Au nanoparticles existed (the ratio was A), while it remained the same (the ratio was (1A)) in other regions. Thus, the increased rate in the device can be expressed as follows:
$ \begin{array}{l} \frac{{A{I_{{\rm{SC\_D}}}} + (1  A){I_{{\rm{SC\_I}}}}}}{{{I_{{\rm{SC\_I}}}}}}\\ = \frac{{A\left( {1  \exp (  {\alpha _{\rm{D}}}l)} \right) + (1  A)\left( {1  \exp (  {\alpha _{\rm{I}}}l)} \right)}}{{1  \exp (  {\alpha _{\rm{I}}}l)}}\\ \approx \frac{{\left( {A{\alpha _{\rm{D}}} + (1  A){\alpha _{\rm{I}}}} \right)l}}{{{\alpha _{\rm{I}}}l}}\\ = C\frac{{A{{(h\nu  {E_{\rm{g}}})}^{1/2}} + (1  A){{(h\nu  {E_{\rm{g}}})}^2}}}{{{{(h\nu  {E_{\rm{g}}})}^2}}}, \end{array} $  (7) 
where C is the proportional constant. As shown in Fig. 5(e), the solid curves represent the calculated increased rates, where blue, red, and black curves correspond to N = 10, 5, and 1, respectively. The experimental data confirmed that the observation of increased rate near the bandgap energy supports the possibility of the direct transition by the ONF. Further improvement of the increased rate can be realized by implementing a larger coverage of the Au nanoparticles and a larger depletion area by introducing the intrinsic Si layer between the p and ntype layers, i.e., pin diode structure.
Figure 6(a) shows the variation of increased rate as a function of size of the Au nanoparticles. We observe that the increased rate is maximum for Au nanoparticle with a diameter of 100 nm. To quantify this size dependence, numerical calculations were performed using a finite difference time domain (FDTD) method^{47}. The calculated field distributions (Fig. 6(b)) were then used to obtain the Fourier spectra (Fig. 6(c)), where the spectra were obtained from the crosssectional profile of the field distribution along the xaxis and they were averaged along yaxis 0 ≤ y ≤ D/2 (D is the diameter of Au nanoparticle). Since the wave number difference between the Γ and X points (k_{x}_{_}_{ΓX}) is 4.92 nm^{1} ^{48}, we studied the diameter dependence of the power spectrum at k_{x}_{_}_{ΓX} (F(E) _{ΓX}^{2}, red circles in Fig. 6(d)). It is observed that F(E) _{ΓX}^{2} also attains a maximum at D = 100 nm. The calculated size dependence is found to be in good agreement with the experimental results (Fig. 6(a)). The effect of generating a larger kcomponent by the ONF was then investigated by determining the normalized power spectra. This normalization was performed using the square of the volume of Au nanoparticles (open blue circles in Fig. 6(d))^{49}. The normalized power spectra increase with a decrease in size. These calculations confirm that, a greater Δk is generated when the size of Au nanoparticles decreases. In other words, the efficiency of the direct optical transition by the ONF increases with a decrease in size due to large components of the wave number.
Although the theoretical results of the large Δk generated using the firstprinciple calculation^{14} supported the experimental results^{3} qualitatively, the enhancement value was not explained quantitatively. The disagreement might be due to the disagreement of the materials and the size. For example, in the firstprinciple calculation, the point dipole was used as a source of the ONF. More detailed calculations using the real material in a real system will be required to discuss quantitatively. In addition, the limitation of the material size in the firstprinciple calculation results in the disagreement of the system size. To resolve this disagreement, a smaller size of the source for ONF generation should be used, such as porous Si with several nanometer scale^{5052}.
Summary and future directionsSi is an indirect bandgap semiconductor and therefore exhibits poor optical absorption efficiency. This review presents an overview of various theoretical and experimental efforts to improve the performance of Sibased photodetectors and photovoltaic devices. The theoretical framework based on first principle calculations along with experiments including scattering effect, plasmon effect, hot electrons, and nearfield effect were discussed. The ONF effect permits the generation of a large Δk by field localization, which in turn can improve the absorption efficiency of these devices. The ONF effect can intrinsically induce a direct transition due to nonuniformity of the optical field. In addition to the reviewed topics, we confirmed that nearfield could enhance the optical absorption by the field enhancement effect^{53}, which does not utilize plasmon resonance. The field enhancement by ONF was confirmed both in the firstprinciple calculation and in the experimental absorption spectra of the metal complex for CO_{2} reduction. Therefore, if the nearfield source is placed with appropriate position, further enhancement of absorption is expected. Further, since the ONF has nonuniform optical field distribution in nanoscale, the ONF can generate even harmonics in materials with inversion symmetry^{54}. Yamaguchi et al. have showed that the ONF inherently realizes strong second harmonic generation (SHG) and suggested ways to improve the efficiency of this process^{16, 5557}. Further improvements of Sibased optoelectronic devices may be achieved by combining and optimizing these effects. Recently, the band engineering method was proposed to enhance the absorption in another indirect bandgap material (Ge)^{58}. Similar investigations have been made in other indirect bandgap materials including InSe^{5961} and MoS_{2}^{62, 63}. The enhancement was achieved by introducing a strain in the crystal, which changed the band diagram to the direct bandgap structure. Overall, indirect bandgap materials open new avenues for various applications, which are not limited to photodetectors^{64, 65, 66} and photovoltaic devices^{1, 2,} but also include other realms such as lightemitting devices^{67, 68}, watersplitting^{62, 69}, etc.
AcknowledgementsThe author wishes to express special thanks to Drs. Kenji Iida (Institute for Molecular Science), Masashi Noda (University of Tsukuba), Maiku Yamaguchi (University of Tokyo), Prof. Katsuyuki Nobusada (Institute for Molecular Science) and Kazuhiro Yabana (University of Tsukuba) for their active support and discussions. This work was partially supported by JSPS KAKENHI (Nos. JP18H01470, JP18H05157), a MEXT as a social and scientific priority issue (Creation of new functional devices and highperformance materials to support nextgeneration industries: CDMSI) to be tackled by using postK computer (ID: hp170250), Asahi Glass Foundation, and Research Foundation for OptoScience and Technology.
Competing interestsThe author declares no competing financial interests.
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