Planar narrowband Tamm plasmon-based hot-electron photodetectors with double distributed Bragg reflectors

Hot-electron photodetectors (HE PDs) are attracting a great deal of attention from plasmonic community. Many efficient HE PDs with various plasmonic nanostructures have been demonstrated, but their preparations usually rely on complicated and costly fabrication techniques. Planar HE PDs are viewed as potential candidates of cost-effective and large-area applications, but they likely fail in the simultaneous achievement of outstanding optical absorption and hot-electron collection. To reconcile the contradiction between optical and electrical requirements, herein, we propose a planar HE PD based on optical Tamm plasmons (TPs) consisted of an ultrathin gold film (10 nm) sandwiched between two distributed Bragg reflectors (DBRs). Simulated results show that strong optical absorption (>0.95) in the ultrathin Au film is realized. Electrical calculations show that the predicted peak photo-responsivity of proposed HE PD with double DBRs is over two times larger than that of conventional single-DBR HE PD. Moreover, the planar dual-DBR HE PDs exhibit a narrowband photodetection functionality and sustained performance under oblique incidences. The optical nature associated with TP resonance is elaborated.


Introduction
Energetic carriers (i.e., hot electrons and holes) generated in metal nanoparticles and nanostructures due to the optical absorption promise considerable benefits in a variety of applications, such as nanoplasmonic sensing [1,2], photocatalysis [3,4], solar energy harvesting [5,6], surface imaging [7,8], and photodetection [9][10][11][12][13][14][15]. During the past decade, hot-electron photodetection has been extensively studied due to the capability of band gap-free detection [16], high tunability of working wavelength [17], and the possibility of room-temperature operation [18]. Hot-electron photodetectors (HE PDs) collect the hot electrons with sufficient energy that surmount the Schottky barrier in the metal-semiconductor (M/S) junction, forming a steady-state photocurrent. Given the extremely low internal quantum efficiency (IQE), several strategies have been proposed to boost IQE of HE PDs, such as adoption of multi Schottky junctions [19], reduction of Schottky barrier height [20], manipulation of interface roughness [21], use of bias voltage [22], and desings of various plasmonic nanostructures [23]. Indeed, a number of patterned metal nanostructures [24][25][26][27][28][29] that support the excitation of surface plasmons have been integrated within HE PDs to improve the performance of devices because enhanced localized electric field around M/S interface associated with surface plasmons can boost hot-electron generation efficiency [30].
In practice, despite the great progresses in nanofabrication technologies, the fabrication of HE PDs with patterned metal nanostructures is accompanied by complicated (usually costly) preparation procedures and difficulties of large-area application. The planar HE PDs, on the other hand, suffer from either weak photon Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. absorption or poor hot-electron collection. For a planar HE PD, as the thickness of metal film increases, more photons would be absorbed in metal in the visible and near infrared regimes, whereas the thermalization loss inevitably raises during the hot-electron transport process, resulting in the reduction of hot-electron collection efficiency. In other words, the dilemma for enhancing planar hot-electron photodetection lies in the lack of successful light trapping approaches for ultrathin metal films. Additionally, narrowband photoelectric response is highly desired in some applications that involve hot-electron photodetection, such as hot electron-assisted sensors [1,2] and spectrally selective plasmonic nanodiodes [31]. However, the full width at half maximum (FWHM) of photoelectric response for planar HE PDs is relatively large compared with that of HE PDs with patterned metal nanostructures [2,32]. In order to promote the development of planar hot-electron photodetection, both drawbacks must be addressed.
In this work, we propose a design of planar narrowband HE PDs based on Tamm plasmon (TP) resonance which FWHM is less than 20 nm. It is reported that the mean free path of hot electrons in Au is about 20 nm [33]. However, previously reported TP-based HE PDs with single DBR used a relative thick Au film with thickness larger than 20 nm to realize strong optical absorption [32][33][34][35][36]. Therefore, hot-electron loss during the transport process is considerable because of the probability of a hot electron reaching the metal-semiconductor interface being reduced to a value less than 37%. In order to suppress transport loss, the thickness of Au film needs to be reduced further. Meanwhile, in order to guarantee the reliability of experimental preparation and characterization, the thickness of a continuous and ultrasmooth film Au film should be larger than 5 nm [37,38]. Our device employs an ultrathin gold (Au) film (10 nm) sandwiched between two properly engineered distributed Bragg reflectors (DBRs), simultaneously guaranteeing prominent photon absorption and hotelectron collection. Benefitting from strong optical absorption (>0.95) in the ultrathin Au film, the peak photoresponsivity (PR) of planar HE PD with double DBRs at the TP resonance is significantly improved from that of conventional single-DBR HE PD. Detailed optical evaluations were performed to unveil the physics behind the TP resonance.

Simulation method and theoretical calculation
With the optical constants from Palik [39], optical responses of the designed device were examined by solving Maxwell's equations in a finite-element platform (COMSOL Multiphysics) [40]. Considering the planar symmetry geometry of the planar HE PDs, two-dimensional (2D) computational domains with bounded boundaries were modelled to reduce the simulation time. The propagation of the incident signal wave is set along the negative direction of z axis. Two perfectly matched layers were used along the positive and negative directions of z axis to truncate the simulation region. Adopting the assumption of the infinite planes for the device, the periodic boundary condition was used along x direction. The triangles with characteristic size of 5 nm are used to mesh the 2D simulation region. After convergence test for the simulated results, we obtained the optical absorption in the ultrathin Au film.
Based on the simulated optical results, the performance of the proposed HE PD could be assessed by the probability-based analytic calculations. Several theoretic studies have been addressed how to calculate the PR spectrum of a planar HE PD [17,34,35]. These calculations usually evaluate four quantities that quantify hotelectron dynamic process within M/S junction, namely, hot-electron energy distribution, spatial hot-electron generation rate, hot-electron transport probability, and energy-dependent injection probability. The detailed descriptions and formula about these quantities are present in the following.
Upon the photon absorption, a hot electron is excited with relative energy E−E F over the Fermi energy level (E F ). Hot-electron energy distribution D(E) is calculated and normalized [41].
where l(E) is hot-electron mean free path that determined by electron-electron and electron-phonon scatterings, j is the moving angle, and d(z) is the distance from the hot-electron generation position to the M/S interface. After arriving at M/S interface, the hot electron has an injection probability P inj to cross the interface which is calculated by [18] ( ) where m s and m m are the effective electron masses in the semiconductor and metal, respectively. It is noted that the calculation of P inj considers the hot-electron interfacial reflection and the continuity of tangential hotelectron momentum component parallel to the M/S interface [18]. In our calculations, the power of the incident light for any wavelength was normalized to be 1 W. Therefore, we obtain wavelength-dependent PR(λ) of the proposed HE PDs that can be written as where A(λ) is the simulated absorption spectrum, h is the Planck constant, e is the electron charge, and c is the light speed.  Figure 1(b) shows that the operation of HE PDs relies on three consecutive electronic processes that occur within the Au-TiO 2 junction: hot-electron generation, transport, and injection. When the device is illuminated by a light signal, photon energies are deposited in Au film through the excitation of TPs. Consequently, electrons below Fermi level are excited to higher unoccupied levels. The generated hot electrons with relative energy of E−E F diffuse to Au-TiO 2 interface, experiencing electronelectron and electron-phonon scatterings. Upon arriving at the Au-TiO 2 interface, the over-barrier hot electrons can be injected into TiO 2 with interfacial reflections. Below-barrier hot electrons are blocked by Schottky barrier Besides the numbers of both DBR pairs, the dependences of optical absorption on d 1 and d 2 were also investigated, as depicted in figures 2(a) and (b), respectively. Figure 2(a) shows that: (1) as d 1 increases from 10 to 70 nm, resonance wavelength displays a blue shift; (2) with an appropriate d 1 (10-20 nm), almost all the incident light energy can be absorbed by the thin Au film; (3) further increasing d 1 , most of the incident light is directly reflected by the top DBR and Au film. For the tunability of the working wavelength, we break the constraint that the optical thickness of TiO 2 layer adjacent to Au is a quarter of λ central . It is clear from figure 2(b) that, as d 2 increases from 20 nm to 400 nm, device shows three absorption bands from visible to near infrared band and device keeps strong optical absorption at TP resonance. This multiband response of the device increases the flexibility of the device preparation because more than one values of d 2 can realize a TP resonance at a targeted wavelength. Overall, it is convenient for one to obtain targeted optical responses by adjusting the planar thicknesses. To get more insight into TP resonance, phase accumulation (P PD ) at the interface between Au+top-DBR and bot-DBR+substrate was examined by using the optical transfer-matrix method [17]. P PD can be expressed as.

Results and discussion
where P 1 (P 2 ) is the phase shift due to the reflection for the wave incident on Au+top-DBR (bot-DBR+substrate) from TiO 2 medium. P PD was normalized by 2π. Figures 2(c) and (d) depict the maps of wavelength-dependent P PD as a function of d 1 and d 2 , respectively. One can see in figure 2(c) that there is a contour lines corresponding to P PD =0. Moreover, in figure 2(d) there are three contour lines corresponding to P PD =0, π, and 2π. The absorption profiles presented in figures 2(a) and (b) indicate that TP resonances of the device occurs when the phase matching condition (i.e., P 1 +P 2 =2 πm, in which m=0, 1, 2K) is satisfied [36]. The absorption band in figure 2(a) presents zero-order TP resonance. The absorption bands in figure 2(b) present zero-, first-, and second-order TP resonances.
To explore the electrical responses of proposed device, three electronic processes described above were quantitatively analyzed [34], as presented in the following discussions. Figure 3(a) plots the spatial distributions of G and P tran in the ultrathin Au layer (d 1 =10 nm) with λ TP of 936 nm (photon energy of 1.325 eV). It is found that G achieves the peak value at the Au-TiO 2 interface (i.e., z=0) due to the excitation of TP resonance at the interface between Au and bot-DBR. When z further increases (i.e., towards Au-SiO 2 interface), G decreases because the spatial optical absorption associated with TP resonance reduces [44]. The generated hot electrons diffuse to the Au-TiO 2 and Au-SiO 2 interfaces. But the hot electrons that reach the Au-SiO 2 interface will not have enough energy to surmount the barrier of Au-SiO 2 contact with a value of 4.35 eV, i.e., the difference of work function (W=5.1 eV) of Au and the electron affinity (χ=0.75 eV) of SiO 2 [45]. Only generated hot electrons that reach the Au-TiO 2 have a chance to be collected. The blue curve in figure 3(a) shows that P tran also achieves its peak value of 0.5 at Au-TiO 2 interface. Considering the adopted ultrathin Au layer, these favourable hot-electron spatial distributions are expected to significantly relieve thermalization loss and therefore to realize boosted photocurrent output. Red curve in figure 3(b) shows the calculated population distribution of generated hot electrons (N gen ) with wavelength of 936 nm. It is found that the populations of hot electrons with energy larger than 1.325 eV are negligible. The ultrafast relaxation of generated hot electrons during the hot-electron transport process usually degrades the performance of HE PDs [41]. Blue curve in figure 3(b) depicts the population distribution (N tran ) of hot electrons that reach the Au-TiO 2 interface. It is suggested that the transport loss of above-barrier hot electrons is larger than that of below-barrier hot electrons because both lifetime and mean free path of hot electrons in metals are inversely proportional to hot-electron energy [46]. The adoption of ultrathin Au film with thickness of 10 nm aims to slows down the relaxation of short-lived abovebarrier hot electrons for collection. Figure 3(c) depicts the energy-dependent populations of collected hot electrons (N col ). It is considered that only over-barrier hot electrons contribute to device electrical output. Moreover, the over-barrier hot electrons experience interfacial reflection [18], leading to the further reduction of collected hot electrons. These three losses, namely, transport, barrier, and reflection losses result in an extremely low IQE of 0.00187 at 936 nm. In order to investigate the impact of working wavelength on the device performance, we defined α as the ratio of the populations between over-barrier hot electrons and generated hot electrons. The calculation results show that α reduces linearly with the red shift of λ TP , as depicted in figure 3(d). We obtained the external quantum efficiency (EQE TP ) of device with the different TP resonance wavelengths. Red line in figure 3(e) shows that EQE TP decreases with λ TP . It is known that EQE describes the overall efficiency with which the device converts incident photons to collected hot electrons. However, EQE convolutes the effects of optical absorption with the subsequent three hot-electron electronic processes. Blue solid line in figure 3(e) shows that device maintains strong optical absorption with the red shift of λ TP . Therefore, considering the relation between EQE and IQE (i.e., EQE TP =A TP ×IQE TP ), the reduction of EQE TP with red shift of λ TP should originate from the dependence of IQE TP on λ TP . Solid line in figure 3(f) shows that the calculated IQE TP of the planar HE PD is strongly related with λ TP . It is considered that the reduction of α with λ TP is responsible for the IQE TP reduction with λ TP . Moreover, the calculated IQE TP was fitted with Fowler equation where η is the coefficient that depends on device-specific details [10]. It is found that the fitted (dashed) line with coefficient of η=0.0227 agrees well with the calculated (solid) line, suggesting the rationality of our electrical evaluations.
We also investigated, for comparison purposes, the performances of the planar HE PDs with and without (w/o) top DBR. First, the device with parameters of d 2 =45 nm and d 1 =10 nm were exemplified. Figures 4(a) and (b) show the calculated A PD and PR spectra, respectively. It was found that: (1) the peak absorption of dual-DBR design (i.e., with top DBR) is enhanced by over 2-fold compared with that of single-DBR design (i.e., without top DBR) with a blue shift (∼10 nm) in position; (2) the peak photo-responsivity (PR∼2.35 mA W −1 at λ TP =843 nm) of the device with top DBR increased by over 2-fold compared with that of the device without top DBR; (3) the FWHM of PR spectrum of dual-DBR HE PD is less than 20 nm, which is prominently smaller than that of single-DBR HE PD. For the sake of generality, the peak photo-responsivity (PR TP ) and spectral width (FWHM PD ) versus d 2 of the designs (d 1 =10 nm) with and without top DBR are shown in figures 4(c) and (d), respectively. Obviously, both optical and electrical performances of dual-DBR system are better than those of the single-DBR design. It is noted that FWHM of dual-DBR HE PDs is only 11 nm at λ TP =938 nm when d 2 =80 nm, which is comparable to the reported FWHM (∼10 nm) of HE PDs with patterned metal nanostructures [2].
Finally, we examined the angular performances of the proposed dual-DBR HE PD (d 1 =10 nm), with which both the transverse electric (TE) and transverse magnetic (TM) incidences must be considered. As one can see in figures 5(a) and (b), with the increase of incident angle (θ), TP resonance wavelength has a blue shift, but the amount of blue shift under TM incidence is larger than that under TE incidence. Besides the angular optical absorption, the angular electrical responses were also investigated. For TE illuminations, as shown in figure 5(c), PR at TP resonance increases with θ even if the value of A PD reduces slightly with θ. For TM illumination, figure 5(d) shows that the high PR is maintained even when θ is as large as 60°, while the value of A PD decreases with θ. The distinct behaviours between angular optical and electrical responses are ascribed to that the PR of a HE PD, or equivalently its EQE, is determined not only by optical absorption but also by IQE which is a function of working wavelength, as shown in equation (9). As on can see in figures 5(a) and (b), peak optical absorption reduces with the increase of θ. However, the PR at TP resonance can be somewhat compensated for with the enhanced IQE due to the blue shift of λ TP .

Conclusion
In summary, we have demonstrated a planar narrowband HE PD formed by inserting an ultrathin Au film into two DBRs. Our design can circumvent the conventional trade-off between optical absorption and hot-electron transport for planar hot-electron photodetection. Simulated results show that significantly enhanced optical absorption (>0.95) in ultrathin Au film is realized compared to that of single-DBR structure. Transfer-matrix method was employed to reveal that TP resonances occur when the phase accumulation at the interface between top-DBR+Au and bot-DBR+substrate is an integer multiple of 2π. Analytical probability-based electrical analysis demonstrated that compared with the single-DBR counterpart, boosted PR (∼2.35 mA W −1 at λ=843 nm when d 2 =45 nm) with narrowband spectral width (less than 20 nm) is obtained for the planar HE PDs with double DBRs. Furthermore, these outstanding optical and electrical responses are found to sustain over a broad range of the incident angles. Our work is expected to facilitate the efficient, low-cost, and large-area photodetection.