Efficient, broadband and wide-angle hot-electron transduction using metal-semiconductor hyperbolic metamaterials

Author’s Accepted Manuscript Efficient, Broadband and Wide-Angle Hot-Electron Transduction using Metal-Semiconductor Hyperbolic Metamaterials Maryam Sakhdari, Mehdi Hajizadegan, Mohamed Farhat, Pai-Yen Chen www.elsevier.com/locate/nanoenergy

PII: DOI: Reference:

S2211-2855(16)30159-8 http://dx.doi.org/10.1016/j.nanoen.2016.05.037 NANOEN1302

To appear in: Nano Energy Received date: 2 February 2016 Revised date: 19 April 2016 Accepted date: 20 May 2016 Cite this article as: Maryam Sakhdari, Mehdi Hajizadegan, Mohamed Farhat and Pai-Yen Chen, Efficient, Broadband and Wide-Angle Hot-Electron Transduction using Metal-Semiconductor Hyperbolic Metamaterials, Nano Energy, http://dx.doi.org/10.1016/j.nanoen.2016.05.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Efficient, Broadband and Wide-Angle HotElectron Transduction using Metal-Semiconductor Hyperbolic Metamaterials

Maryam Sakhdaria, Mehdi Hajizadegana, Mohamed Farhatb, and Pai-Yen Chena,* a

Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, U.S.A.

b

Division of Computer, Electrical, and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-69100, Saudi Arabia.

Abstract Hot-electron devices are emerging as promising candidates for the transduction of optical radiation into electrical current, as they enable photodetection and solar/infrared energy harvesting at sub-bandgap wavelengths. Nevertheless, poor photoconversion quantum yields and low bandwidth pose fundamental challenge to fascinating applications of hot-electron optoelectronics. Based on a novel hyperbolic metamaterial (HMM) structure, we theoretically propose a verticallyintegrated hot-electron device that can efficiently couple plasmonic excitations into electron flows, with an external quantum efficiency approaching the physical *

To whom correspondence should be addressed. E-mail: [email protected]

1

limit. Further, this metamaterial-based device can have a broadband and omnidirectional response at infrared and visible wavelengths. We believe that these findings may shed some light on designing practical devices for energyefficient photodetection and energy harvesting beyond the bandgap spectral limit.

Keywords:

hot

electrons,

internal

photoemission,

sub-bandgap

photovoltaics, photodetection, plasmonics, metamaterials

Introduction In the past decade, excitation of surface plasmon polaritons (SPPs) at the metaldielectric or metal-semiconductor interface has received tremendous attention for control of light-matter interactions [1] and manipulation of photon absorption [2],[3], emission [4], wavelength conversion [5],[6], localization [7]-[11], and energy transformation [12],[13] at the nanoscale, which may be beneficially used in a variety of domains, such as chemistry, biomedical, and information processing, nonlinear optics [5]-[11], and solar energy harvesting [2],[3]. Very recently, it has been demonstrated that hot electrons produced by surface plasmon decay in a metal nanostructures (e.g. gold nanoparticles) can generate a measurable photocurrent at the metal-semiconductor (MS) Schottky barrier [14][16], as illustrated in Fig. 1(a). This result was unexpected, given the rapid thermalization of charge carriers in the bulk metal. The decay of surface plasmons to highly energetic electrons (i.e. Landau damping) was thought as an adverse influence on plasmonic devices and inherently limits the lifetime and propagation length of SPP waves for certain optical applications. However, in nanostructures,

2

the momentum requirement at the interface that governs the electron emission (hot-electron escape cone) can be relaxed [14]-[16]. Following this initial discovery, some recent works have been directed towards designing plasmonic nanostructures that could extract “hot electrons” from metals via internal photoemission across the metal-semiconductor interface, and their useful works in photovoltaics [14]-[21] photodetection [15],[22]-[27], and photochemistry [19]. Since the spectral response of a hot-electron device does not depend on the optical bandgap of the semiconductor (but instead on the plasmonic resonance of the metal nano-structure), the photoconversion process is expected to be efficient, over a wide wavelength range, depending on the light (electromagnetic) management method. In this context, plasmonic nanostructures, emerging as a promising approach for light trapping and absorption enhancement [2],[3], may give large extinction cross-sections, absorbing a signification fraction of solar spectrum and offering high photoemission yields. In particular, hot-electron devices could, in principle, collect photons having energies lower than the bandgap energy of a semiconductor, thereby serving as the second junction in a tandem solar cell arrangement. However, their incident photon-to-electron conversion efficiencies have been low (typically below 20 % [25]-[29]). This could be attributed to several practical issues. First, the electron transport across a Schottky barrier is inefficient, rendered by a small electron escape cone [22]-[24]. In addition, a relatively thick metal layer, greater than the electron mean free-path, necessary for achieving high optical absorption, would cause hot electrons to lose most of their kinetic energy before reaching the Schottky barrier, thereby resulting

3

in a low internal quantum efficiency (IQE). On the other hand, although an ultrathin metal film could mitigate the IQE of hot-electron devices due to multipath scattering [22],[23],[30], it renders a very weak optical absorption and hence a low external quantum efficiency (EQE), as discussed in the following. Besides, the localized SPP resonance in nanopatterned metal films or metal nanoparticle clusters, although providing enhanced optical absorption, are normally narrowband and sensitive to the incident angle of light. In this work, we propose a novel hot-electron device hybridized with metamaterial structures, as shown in Fig. 1(b), which may enable substantially increased IQE and EQE by enhancing optical extinctions and by relaxing constraints on electron traversing rate for internal photoemission. We theoretically explore the possibility of reshaping multiple Schottky-barrier junctions into metamaterials or photonic nanostructures that could support the plasmonicallyenhanced light absorption over a wide spectral range. From the electromagnetic theory viewpoint, cascading multiple metal-semiconductor junctions with opposite signs in permittivity may create a hyperbolic metamaterial (HMM) substrate generating a broadband singularity in the photonic density of states [31][1]. A suitably patterned HMM substrate, although compact in size, may further support omnidirectional and broadband light absorption, due to the superposition of multiple slow-wave modes, so-called “rainbow trapping” effect [36]-[39]. In addition, the metamaterial-based hot-electron device shown in Fig. 1(b) can be seen as a multiple-stage stereo device, consisting of multiple hot-electron collectors (e.g. Schottky-barrier junctions) in a vertical integration manner, such

4

that the device can be operated in a highly efficient traveling-wave fashion. In this scenario, even though ultrathin metal films (i.e. thinner than electromagnetic skin depth) are used for enhancing the intrinsic IQE of device and the incident photon energy is less than the bandgap energy, a large EQE (or responsivity

   q/    EQE ) can be obtained, over broad wavelength and angular ranges, thereby being well-suitable for broadband photodetection and energy harvesting applications.

Photoconversion Quantum Yields The excited SPPs in metallic nano-structures could lose energy due to radiative damping (which accounts for light scattering) or non-radiative damping (which accounts for the generation of electron-hole pairs). The latter one may enable resonant light absorption, covering a broad range of the solar spectrum. In general, the hot-electron photoemission includes a series of consecutive steps [22],[23]: (1) photon absorption and hot electron generation due to non-radiative damping in metallic nano-structures, (2) hot electrons travel towards the metal-insulator interface with a probability of losing energy in inelastic collisions, (3) electron injection at the metal-semiconductor interface (Schottky barrier), with a probability based on the momentum escape cone, and (4) multiple (Lambertian) reflections of electrons at the interface, if the metal thickness is of the order of electron-electron scattering length. In step (1), the local generation rate of hot electrons is related to the optical absorptance. After a photon of energy  is absorbed, an electron in the conduction band of the metal is excited from an initial energy Ei ( EF   < Ei < EF ) to a higher energy state EF  Ed , where EF is 5

Fermi energy, Ed is the excess energy above the Fermi level ( 0  Ed   ). The distribution of electron energies can be described by the energy distribution of the joint density of states (EDJDOS). In the low photon energy regime (  < 3 eV), the EDJDOS is given by the product of the number of initial g  E    and final

g  E  electronic states in the metal that are available for direct transitions given the absorbed photon energy: (1)

where

f  E   1/ exp   E  EF  / KT   1

is

the Fermi-Dirac distribution

function determining the occupancy probability of an available energy level at temperature T (here we assume T = 300 K). For free-electron-like metals at low temperature, a parabolic band structure, as illustrated in Fig. 1(a), is obtained such that g  E   E is approximately valid. If the excited electrons have kinetic energy below the Schottky barrier energy  B , they cannot be emitted over the barrier. The IQE can therefore be calculated as the ratio of the population of hotelectrons with sufficient energy to be emitted over the total photoexcited electron population [17]-[18] :

(2)

6

where Pd  Ed  is the emission probability accounting for steps (2)-(4). In step (2), the hot electron may have sufficient energy to cross the Schottky barrier, provided that its excess energy is not lost to nearby electrons or atoms (mainly due to electron-electron scattering) before reaching the metal-semiconductor interface. In steps (3) and (4), at each metal-semiconductor interface, the transmission probability Pe  Ed  for hot-electrons with isotropically oriented momenta is determined by the kinetic energy and momentum matching conditions. Fowler’s theory [40] proclaims that the longitudinal component of kinetic energy must be at least equal to the barrier height such that hot electrons can be emitted across the barrier. In this scenario, the probability of electron emission Pe  Ed  is given by the ratio of the solid angle of the electron emission cone to the solid angle of the k-space sphere as: 2



  sin  d d 1 Pe ( Ed )  0 0  1  cos   , 4 2

(3)

where cos   2mB / k d  B / Ed for Ed   B (m and k d are electron’s mass and excess momentum). When the barrier height is vanishingly small, i.e.

 B  0, , the emission probability Pe  0.5, as the initial momentum distribution of hot electrons is isotropic and only half of them flow toward the interface. If a sufficiently thin metal is sandwiched between two semiconductors, an electron may travel back and forth within the metal film, and can be reflected at each interface for multiple times until its kinetic energy is reduced to  B , due to various scattering events and due to emission. The number of round trips can be

7

obtained as N  E  

 Le / tm   ln  E / B  ,

where tm is the thickness of the

metal film, Le is the phenomenological mean-free-path of electrons ( Le ~ 55 nm for Ag at energies close to the Fermi energy [41]-[43]). Based on Scale’s multiple-reflection model [22], for a double-Schottky-barrier device with two emission cones, the emission probability can be expressed as: N k-1 Pd ( Ek ) = Pe¢( Ed ) + å éê Pe¢( Ek ) Õ 1- Pe¢( Ek ¢ ) ùú, k ¢=0 k=1 ë û

( )

(4)

( )

where Ek  Ed e ktm / Le and Pe¢ Ed = 2Pe Ed , which is two times larger than (3) because a double-barrier structure allows two emission cones. The multiple reflections of electrons inside an ultrathin metal thinfilm may significantly increase the electron emission probability compared to a bulk metal. In the extreme case of an infinitesimally thin metal layer, i.e. tm / Le  0, one can obtain

Pd  1. Equations (2)-(4), considering all photoexcitation, transport, and emission processes, can be used to calculate IQE of a double-barrier hot-electron device, with its EQE related to IQE by: (5) where A is the optical absorptance ( 0
2 1 òV w Im[e ] E du ,  is the permittivity of the 2

metal, and E is the local electric field within the solid. Figure 2(a) shows IQE spectra of a TiO2-Ag-TiO2 double-barrier hot-electron device with metal thickness ranging from skin depth (  ~25 nm) to 6 nm. Here

8

the

dashed

line

represents

the

fundamental

limit

for

IQE:

IQEmax     B  / , which is obtained by assuming an infinitesimally thin Ag layer. It is seen from Fig. 2(a) that reducing the thickness of metal may substantially increase the intrinsic IQE value, due to multiple elastic and diffuse scattering of electrons at two metal-semiconductor interfaces, provided that the metal thickness is less than the electron mean-free-path. Figure 2(b) shows the corresponding EQE spectra of the same device. Here the optical absorption was calculated using the transfer matrix methods [44]. It is observed that EQE is much less than IQE, since the incident light is poorly absorbed by an ultrathin metal film. At higher photon energies, an ultrathin Ag film shows even a smaller EQE than the one with thickness comparable to the skin depth. Indeed, the Rozanov’s absorber theory [45] asserts that there exists a compromise between the amount of radiation that can be absorbed and the thickness of the absorber, which can also be applied in the case of thinfilm hot-electron devices. In the following, we will study the possibility of using the metamaterial approach to achieve the broadband optical absorption enhancement in few-nanometer Ag films, leading to high internal/external quantum efficiencies.

Hyperbolic Metamaterials Metamaterials are artificially engineered periodic structures with lattice constant much smaller than the wavelength of the propagating electromagnetic wave, and thus their repeating, scattering and resonant elements can be deliberately designed to manipulate the effective macroscopic quantities, such as permittivity, permeability, or both, of these effectively homogenous media. Here we will

9

explore the possibility of using the metamaterial approach to realize efficient hotelectron optoelectronics. The proposed device architecture consists of alternating metal and semiconductor layers (i.e. Schottky barrier junctions), effectively forming a HMM substrate. In such metal-semiconductor superlattices, the effective permittivity tensor can be described as [31]:

ˆ ˆ +   yy ˆˆ   ||zz ˆ ˆ) ,  eff   0 (  xx

(6a)

where N

     i i ,

(6b)

i=1

1

 ||     i1i  , N

i=1



(6c)

and  i and i are the relative permittivity and the volume fraction of the i-th constituent material, respectively. A multilayered Ag-TiO2 HMM can behave as a uniaxially anisotropic medium, provided that extreme material properties Re     Re ||   0 can be supported, such that the isofrequency surfaces become

open hyperboloids, as illustrated in Fig. 1(c). A HMM substrate has been shown to exhibit exotic optical properties, including negative refraction, hyperlensing, and Dyakonov plasmons [31]-[35]. In particular, HMMs can, in principle, achieve anomalously large density of photonic states (PDOS) in a broad spectral range [32], thus serving as an exciting platform for developing novel photonic and quantum-optical devices, such as non-resonant single photon sources and substrates that can mold spontaneous emission into directional beams [34],[35]. On the other hand, the use of nanopatterned HMM substrates may realize a

10

perfect absorber with broadband and omnidirectional optical absorption (“rainbow trapping” effect [36]-[39]). Tapered HMM-based waveguide arrays have shown potential for the broadband control of the velocity of light, due to the dramatic frequency and spatial dispersions of HMMs. This opens up exciting opportunities to achieve broadband and strong absorption of incoming light, due to the large attenuation of slow-wave modes that is inherently independent of the angle of incidence. By tailoring the size, geometry, and constituent metal/dielectric materials of the HMM waveguide array, the total absorption bandwidth can be effectively widened by incorporating multiple slow-wave modes excited in the tapered HMM waveguides. Quite recently, we proposed the use of HMM, comprising multiple metalinsulator-metal (MIM) junctions, for the efficient rectification of long-wavelength light (e.g. THz to mid-infrared). The multiphoton-assisted electron tunneling across the potential barrier of the MIM system introduces the second-order quantum conductivity responsible for nonlinear optical rectification, being greatly enhanced by the strong optical fields confined within the nano/subnano-insulators. Beyond these concepts, we propose here a new functionalized HMM substrate based on metal-semiconductor multilayers for realizing highly efficient hotelectron devices, which potentially allow for the broadband and omnidirectional transduction of infrared and visible light into useful electricity. We note that although the above two optoelectronic devices share similar HMM geometries, they are based on very different operation mechanisms, as well as electronic and photonic structures. Different from those photon-assisted tunneling devices

11

involved relatively thick metals and ultrathin insulators that allow large optical field localization and electron tunneling within the MIM junctions, the photoconversion efficiency of hot-electron devices depends on the excitation (related to optical absorption) and scattering properties of hot electrons inside the thin metal layers and its emission properties at the metal/semiconductor interface. In this context, each HMM unit cell of the hot-electron device is constituted by an Ag-TiO2 junction, effectively forming the Schottky-barrier device shown in Fig. 2. In addition, by nanopatterning a mirror-back HMM as a homogeneous slowlight medium [36]-[39], the rainbow trapping effect may allow scattering light into plasmonic junctions and trapping it there such that the strong optical fields can lead to a substantial increase in the light absorption in metals. Ideally, this HMM-based hot-electron device can be designed to absorb most of the incident electromagnetic energy in the near infrared and visible regions, and effectively collect plasmonically-excited hot electrons in metal layers, if the photon energy is greater than the Schottky barrier height. This vertically cascaded device may achieve a very high photoconversion efficiency, in analogy to those planar traveling-wave microwave and THz detectors [44], but with a much compact size due to the exotic slow-wave features. Figure 1(b) illustrates a plane wave with transverse-magnetic (TM) polarization incident along the zˆ -direction. The trenched HMM structure in Fig. 1(b) is, in some sense, similar to a periodic array of air/HMM/air waveguides, for which a guided mode propagating along the waveguide axis may be tailored to have a near zero group velocity, i.e. vg  ω / β  0, at certain photon energies, where ω is

12

the angular frequency and β is the modal wave vector. Based on Maxwell’s equations and periodic boundary conditions, the dispersion relation between w and β can be derived as: P W   tan  k02 ||    || /    β 2 2    ||

β 2  k02 ,

W  tan  β 2  k02   0 2 2 2  k0  ||    || /    β

,

(7)

where k0 is the free space wave number, P is the period of unit cells, and W is the width of air slots [see Fig. 1(b)]. Provided that the periodicity of each MIM unit cell is subwavelength ( P  0 ), all diffraction orders, except for the zero-th mode, are evanescent. Here we consider a HMM slab composed of vertically stacked Ag (6 nm) and TiO2 (60 nm) films, which is etched to form a periodic air/HMM/air waveguide array with P = 200 nm and W = 100 nm. Figure 3(a) shows the dispersion diagram for the eigenmodal propagation along this patterned HMM. In our calculation, we use realistic optical properties of materials extracted from previous experimental works. The permittivity of Ag, as a function of photon energy, can be modeled by a Drude-type dispersion as [46] :

(8)

where  0 is the free space permittivity, the static permittivity    5 ,

 p  2π  2175 THz, and   2π  4.35 THz. The refractive index of TiO2 is 2.5 [47], which is approximately valid in the near-infrared-visible range. It is seen

13

from Fig. 3(a) that near zero group velocity can be achieved at critical wavelengths (e.g.

 ~ 1eV and 1.5 eV ), implying that the incident light is

slowed down, trapped and absorbed inside the lossy and anisotropic HMM region. We have conducted the full-wave simulation based on the finite-integral technique [48] to validate the eigenmodal results. The bottom panel of Fig. 3(a) shows the contour plot of absorptance as a function of the photon energy  [eV] and the incident angle of illumination  for this HMM (5 MS superlattices) placed on top of a 50-nm-thick Ag film with near-perfect reflection. It is evidently seen that the enhanced absorption regimes (  ~ 1 eV and  ~ 1.5 eV) agrees excellently with the near-zero group velocity points in the dispersion diagram [Fig. 3(a)]. We note that total size of HMM remains subwavelength, since the large value of Re    suggests that the wavelength of incident light can be effectively shortened. Interestingly, since the operating wavelength simply depends on the eigenmode propagation inside each subwavelength air/HMM/air waveguide, as long as the array period is small (below the first Bragg resonance), the high optical absorption around the slow-wave regime is independent of the angle of incidence [36]-[39],[42]. This implies that the HMM-based hot-electron device is capable of highly efficient wide-angle operation. We have designed several HMM-based hot-electron devices shown in Fig. 1(b), with different numbers of MS superlattices (N), and their EQE calculated using Eq. (2) are presented in Fig. 4(a). Here design parameters are: Ag thickness tm = 6 nm, TiO2 thickness ts = 60 nm, W = 100 nm, and P = 200 nm, which yields a fundamental (first-order) slow-wave mode at photon energy of 1 eV, and the

14

mirror is made of 50 nm Ag thinfilm. It is seen that for moderate numbers of AgTiO2 superlattices, EQE can approach IQE (dashed line) around the slow-wave regimes, even though thin, lossy metal-films of only 6 nm are used. Moreover, the peak EQE values and bandwidth may increase with increasing the number of superlattices forming HMMs, particularly for those higher-order slow-wave modes. The high EQE of this HMM-based device can be attributed to the combined effect of the enhanced IQE in sandwiched “double-barrier” Ag ultrathin films with large emission probability Pd [22] and the near-perfect absorption in plasmonic slow-light structures. Figure 4(b) shows EQE spectra for the same HMM-based device in Fig. 4(a) (here N =5 pair Ag-TiO2), varying angles of incidence. It is surprisingly seen that the obtained high EQE is rather angle independent, consistent with results in Fig. 3(a). At high photon energies (  > 2 eV), the peak position of EQE is observed to be shifted with varying angles of incidence. This is because in the short wavelength (high photon energy) region, the optical absorption is due primarily to Fabry-Perot resonances. We note that EQE peak positions are tunable by tailoring slow-wave modes in the nanostructured HMM. This can be achieved, for instance, by changing the geometric parameters P and W, or the Ag/TiO2 filling ratio, as can be understood from Eq. (7). Figure 4(c) shows the EQE spectra for the same device in Fig. 4(b) under the normal incidence, while varying the periodicity P. It is seen that the position of EQE peak can be shifted over a wide range of photon energy by changing the geometric dimension, in a good agreement with the eigenmodal calculation results

15

using Eq. (7), which demonstrates an interesting wavelength-scalability for hot electron collections. As far as the fabrication process is concerned, the multilayered metal/dielectric thin films can be prepared by dual-depositing sputtering technique [49],[50] (which allows Ag film as thin as 6 nm, with a roughness below 1 nm and excellent mechanical flexibility [49],[50]) or by the state-of-the-art atomic-layer deposition [51]; both techniques enables ultrathin, smooth, and pinhole-free thinfilms. We have also considered the imperfection in the fabrication process and numerically studied the influence of surface roughness on the device performance. In our numerical simulations (not shown here for saving some space), the metal thickness was randomly varied by 20 % to mimic rough surfaces. We find that despite a 20 % fluctuations in the metal thickness, both homogeneous and inhomogeneous (discussed below) HMM devices exhibit only a small shift in the absorption spectrum (< 5 %) with the absorptance being essentially unchanged, within the spectral range of interest (1 eV-1.5 eV). The linear/tapered air trenches can be produced by advanced lithographic methods, including electron beam lithography,

laser

beam

direct

writing,

and

focused

ion

beam

milling/lithography/direct deposition [2]-[5],[37]-[39]. The vertical interconnect technology, as a mature process in integrated circuits (ICs), may be used to connect different superlattices for collecting hot electrons to loading circuit.

Thin-film Polaritonic Photonic Crystals Here we also compare the performance of metamaterial-based devices with alternative possible designs, such as the photonic crystal (PC) and the Fabry-Perot

16

plasmonic cavities (FPC). Figure 3(b) shows a one-dimensional (1D) PC composed of periodically laminated metal-semiconductor layers and its dispersion diagram. According to the Bloch-Floquet theorem [52]-[56], the dispersion relation for an infinite 1D PC under normal incident illumination can be derived as:

cosh ( b a ) = cos ( ksts ) cos ( kmtm

k )- (

/ e s ) + ( km / e m ) s 2

2ks km / e se m

2

sin ( ksts ) sin ( kmtm ) ,

(8)

where the lattice constant a=tm +ts , ks   s k0 and km   m k0 . High absorption may occur in the slow-wave regime (typically at the edge of plasmon-polariton bandgap), due to the large photon-matter interaction time emanating from a slow Bloch mode and a relatively long light path within the 1D PC. The bottom panel of Fig. 3(b) shows the contour of absorption for an Ag-TiO2 1D PC, varying the wavelength and incident angle of light; here design parameters for each unit cell are tm = 6 nm and ts = 75 nm, and totally 5 unit cells are placed on top of the reflective metal substrate. It is clearly seen that the absorption spectrum exhibits sharp peaks that are associated with the slow-wave modes at the edge of the bandgap (~1 eV) in the dispersion diagram [top panel of Fig. 3(b)]. The enhanced optical absorption is, however, rather insensitive to the incident angle q . Given by the Bloch theorem, for a finite-size 1D PC, the number of absorption peaks (resonant modes) within each photonic band would correspond to the number of the unit cells N [52]-[53]. We also find that the total absorption efficiency increases with increasing the MS superlattices. Figure 4(d) shows the

17

corresponding EQE spectra for the PC-based hot-electron devices of Fig. 3(b), with different number of layers. As expected, both the magnitude and the number of EQE peaks increase with increasing the number of Ag-TiO2 superlattices. Figure 4(e) shows the EQE spectrum for the PC-based devices with N = 5 (pair Ag-TiO2), under different illumination conditions. It is observed that EQE peak positons is angle-independent at the bandgap edge (1 eV), while becoming sensitive to the angle of incidence in the short-wavelength (high photon energy) region. Comparing Figs. 4(b) and 4(e) reveals that for the same number of superlattices, HMM-based devices may have larger EQE values than PC-based ones. Further, HMM-based devices show a much larger EQE in the shortwavelength region, when compared to PC-based devices. This can be explained from the dispersion diagram in Fig. 3, for which the HMM exhibits a dramatic dispersion and slow-wave modes at multiple wave vectors, while the PC-based devices have a rather smooth dispersion outside the slow-wave region. Therefore, for moderate number of superlattices, HMM-based devices may have larger EQE values, due to its highly dispersive nature. When a simple three-layer PC structure (N = 1) is used, the device comprising a thick TiO2 film sandwiched by two metal films may be reduced to a plasmonic Fabry-Perot cavity, which is normally used for controlling the optical absorption [22],[23]. In an FPC, the thick back-metal layer has a near-unity reflection, while the thin top-metal layer functions as a partially-reflective mirror, allowing the incident radiation to enter the cavity and get trapped inside. Figure 4(e) shows the EQE spectrum for a FPC-based device, under different illumination conditions;

18

here a cavity length is 120 nm that yields a fundamental FP mode at 1 eV. It is seen that the conventional FPC design has a much lower EQE value, when compared to HMM-/PC-based devices, due to the low optical absorption of this flat device.

Hot-Electron Devices Based on Tapered Hyperbolic Metamaterials Although a linear HMM-based hot-electron device can exhibit a high and angleindependent EQE, its bandwidth is limited around the slow-wave modes. Here we also study hot-electron devices based on tapered-HMM, as shown in Fig. 5(a), aimed at broadening the bandwidth of operation, necessary for solar energy harvesting and wideband photodetection applications. When the tapered geometry is used, the bandwidth is expected to be increased due to the superposition of multiple slow-wave modes. As can be understood from Fig. 4(c), the slow-wave modes sensitively respond to the period or the width of air trenches, and therefore a tapered waveguide array may support the slow-wave modes in a wide spectrum. Figure 5(b) shows the contour plot of absorptance as a function of the photon energy and the angle of incidence. It is evidently seen that a broadband (1 eV to 1.6 eV) and wide-angle (0o to grazing angle) optical absorption can be obtained using the tapered HMM shown in Fig. 5(a), backed by a metallic mirror (50 nm Ag thinfilm). The capability to effectively trap photons over a wide range of photon energies and illumination angles is of essential importance for making efficient hot-electron energy harvesters or photodetectors. The general limitation on the maximum bandwidth of a ground-backed absorber should always obey the passivity

and

causality

[45],

following

the

physical

bound

19

2 0 ln r    d   2  s,i ti / 0 , where r    is the reflection coefficient as a 

i

function of photon energy,  s,i and ti are the static permeability and the thickness of the i-th layer of the multilayered absorptive slab. For a non-magnetic effective medium,

we

can

define

a

Rozanov’s

limiting

factor

as Rozanov  0 ln r    d  2 2t , where t is the total thickness of the absorber. 

This factor must be less than unity, as a fundamental physical bound, and a larger

Rozanov value implies wider absorption bandwidth. For the tapered HMM in Fig. 5(a), our results show that Rozanov can be up to 40 %, which, although not optimized yet, is moderately high for TM polarized light [57],[58]. Figure 5(c) shows the EQE spectra for a hot-electron device based on the tapered HMM in Fig. 5(b), varying the angle of incidence. It is surprising to observe that high EQE values, close to the IQE bound, can be obtained from 1 eV to 1.6 eV, which is almost angle independent. Depending on the targeted application, a hotelectron device made of a linear/tapered HMM can be used to achieve a wavelength-selective operation (e.g. zero-power narrowband photodetectors or plasmonic sensors) or a broadband operation.

Conversion Efficiency in Energy Harvesting Applications So far, we have demonstrated that the optimal hot-electron devices can display high EQE and responsivities, as they may serve as good photodetectors that require high percentage of photons converted into electric currents under the short circuit condition. However, quantum efficiency alone does not reveal enough information for the energy conversion efficiency, i.e. the fraction of power

20

converted by the energy harvester. In this section, we also study the energy conversion efficiencies for the hot-electron devices discussed above and compare their respective efficiencies with fundamental physical limits. The current density (J)-voltage (V) characteristics of a hot-electron device can be determined by balancing the reverse photocurrent density due to hot electron injections Jphoto, the forward dark current density due to the thermionic emission Jdark (if T > 0 K), and the spectral irradiance distribution of the illumination source. The conversion efficiency can be expressed as:

c 

J photo  J dark V 

max min I ill ( )d 





max  min I ill ( )  (q /  )  EQE d   J dark (V ) V



max min I ill ( )d 

, (10)

where the integration in the denominator of Eq. (10) is the illumination irradiance over the wavelength range of interest (λmin, λmax). The forward current density is given by the Richardson’s formula as [57]:





J dark T ,V   ARDT 2eB / KBT eV / KBT  1 ,

where



ARD  qm*0 KB2 / 2 2

3



is

the

(11) Richardson

constant

(here

ARD  50 A cm2 K 2 for TiO2 [18]), m*0 is the effective electron mass, and K B is

the Boltzmann’s constant. The solar spectral irradiance can be described by Planck's blackbody radiation law as [57]: 2

 R  (2π )2 c 2 1 I ill   ,T    s  , 2 π /  K BT 5  e 1  rse 

(12)

where Rs is the sun’s radius and rse is the mean distance between the earth and sun. By assuming a 5800 K blackbody, Eq. (12) shows a good match with the shape and

irradiance of the ASTM standard AM0 spectrum, as shown in Fig. 6(a). The 21

optimum conversion efficiency is evaluated by sweeping the voltage in the J-V diagram (Eq. (10)) and recording the voltage that produces the maximum power delivered to a resistive load [57]. Figure 6(b) shows the conversion efficiency against the Schottky barrier height for an ideal double-barrier hot-electron device at different operating temperatures; here an infinitesimally-thin metal with a perfect absorption are assumed over the whole solar spectrum, resulting in the maximum quantum yield, i.e. EQE =

   B  /  . In our calculations, (λmin,

λmax) = (250 nm, 1500 nm), which includes most of the solar radiant energy [Fig. 6(a)]. Results in Fig. 6(b) may present the fundamental physical limits for metalsemiconductor (MS) hot-electron devices, which agree quantitatively with other theoretical works [17],[18]. We find that when hot-electron devices are used as solar energy harvesters, the conversion efficiency cannot generally exceed 7 %, due primarily to the existence of dark (thermionic emission) current that flows in a direction opposite to that of the photoexcited current. For those devices with small barrier energy (  B < 0.5 eV), the relatively large dark current may lead to a low conversion efficiency. However, if the barrier energy is too large, the conversion efficiency is also low, due to small quantum yield. The dashed lines in Fig. 6(b) present efficiencies with a 100 % EQE, which still show depreciation when the barrier energy is low. As can be understood from Eq. (11), the lowtemperature operation is necessary for reducing the dark current. It is seen from Fig. 6(b) that the conversion efficiency would increase with decreasing the device temperature (e.g. T = 77 K), which is in general not practical. A promising way to increase the conversion efficiency is to engineer the density of states in the

22

absorber using the quantum confinement effect [17], which is out of the scope of this work. Figure 6(c) shows the low-temperature conversion efficiency for 6 nm-Ag/TiO2 hot-electron devices structured into the tapered HMM [Fig. 5], PC [Fig. 4(d); N = 15], FPC [Fig. 4(f)], and Ag thin film [Fig. 2]. Here we consider a barrier height of 0.7 eV and the range of photon energy: min    1.5 eV . We should note that the work function of Ag typically ranges from 4.26 eV to 4.7 eV. The black dashed and solid lines in Fig. 6(c) represent the physical bound for an infinitesimally thin Ag film and a 6-nm-thick Ag film; both are assumed to have perfect light absorption. It is seen that among all devices, the HMM-based hotelectron device has the highest conversion efficiency close to the physical limit, which is more than an order of magnitude greater than an unstructured Ag film. Figure 6(d) is similar to Fig. 6(c), but for the room-temperature operation (T = 300 K). It is seen that with practical setups and material properties, the conversion efficiency is quite low at room temperature. Compared to the low-temperature operation, the conversion efficiency is reduced a lot. In short, a single hotelectron module may not be an efficient energy harvester due to the low conversion efficiency, limited intrinsically by the considerable dark current. On the other hand, the large zero-bias responsibility of metamaterial-based hotelectron devices may find interesting applications in photodetectors, which possess

several

advantages

including

sub-bandgap

detection

of

light,

polarization/bandwidth-selectivity, and enhanced light absorption due to plasmonic features.

23

Conclusions We have proposed the use of nanostructured hyperbolic metamaterials to realize a broadband, angle-independent hot-electron device, which, although operating at sub-bandgap photon energies, may exhibit high internal/external quantum efficiencies. Specifically, we have demonstrated that this stereo nano-device can be configured for wavelength-selective or broadband operation, depending on the cross-sectional geometry of the nanostructured HMM that has consequential influence on the excited slow-light modes. Our theoretical results also show that this metamaterial-based device may display a higher photoconversion yields compared to devices with conventional light-management structures, such as Fabry-Perot cavities and polaritonic photonic crystals, while enabling a more compact size and improved angular response. Although hot-electron devices, when used as a single solar cell module, show rather low conversion efficiencies, their peculiar features may still open exciting venues for high-speed, energyefficient photodetectors, sensitive spectrometers, and plasmonic sensors.

24

Figure 1 (a) (Left) Photoexcitation of electrons in the conduction band of metal to the unoccupied level above the Fermi energy; gray and red shadows represent the DOS in the metal’s conduction band and the distribution of hot-electron energies given by the EDJDOS. (a) (Right) Energy band diagram for the metalsemiconductor (Schottky) junction and schematics of hot-electron emissions. (b) Hot-electron devices based on the nanostructured hyperbolic metamaterial (HMM), effectively comprising multiple Ag-TiO2. (c) Isofrequency surfaces of extraordinary waves in normal dielectrics and HMMs.

25

Figure 2 (a)

IQE and (b) EQE for a double-barrier hot-electron device based on

Ag thinfilm with different thicknesses.

26

Figure 3 Dispersion diagram (top) and contours of absorption as a function of photon energy and incident angle (bottom) for the Ag-TiO2-based (a) HMM and (b) 1-D PC; here the wave number is normalized by the period of HMM or PC: a = tm+ts (tm and ts are Ag and TiO2 thicknesses).

27

Figure 4 (a) The EQE spectra for HMM-based hot-electron devices with different numbers of unit cells (N); here the thickness of Ag and TiO2 are tm = 6 nm and ts = 60 nm. (b) The EQE spectra for the HMM-based hot-electron device (5 unit cells, tm = 6 nm, and ts = 60 nm), under different angles of illumination (θ). (c) The EQE spectra for the HMM-based hot-electron device (5 unit cells, tm = 6 nm, and ts = 60 nm), with different periods of nanotrenches (P). (d) and (e) are similar to (a) and (b), but for the hot-electron device based on the thin-film photonic crystal (5 unit cells, tm = 6 nm, and ts = 75 nm). (f) is similar to (b), but for the hotelectron device based on the Fabry-Perot cavity. 28

Figure 5 (a) Snapshots of electric field distributions inside the tapered HMM at photon energies of 260 THz, 300 THz, and 320 THz (left to right). (b) Contour of absorptance as a function of photon energy and incident angle for the taperedHMM-based device in (a). (c) EQE spectra for the tapered-HMM-based device in (a), under different illumination angles.

29

Figure 6 (a) ASTM AM0 solar spectral irradiance distribution (red solid) and the fitted blackbody radiation spectrum (black dashed). (b) Physical limits (maximum conversion efficiency) against the barrier energy at operating temperature of 77 K and 300 K; dashed lines show the cases with 100 % EQE. Energy conversion efficiency for hot-electron devices hybridized with different photonic nanostructures at (c) 77 K and (d) 300K; the black dashed line shows the physical bound and the black solid line shows the maximum possible efficiency for a double-barrier, 6 nm-Ag hot-electron device with perfect absorption.

30

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Maryam Sakhdari is working toward her Ph.D. degree in the Department of Electrical and Computer Engineering, Wayne State University in Detroit. She received the M.Sc. and B.Sc. degree in electronic from Tarbiat Modares University (TMU) and Iran University of Science and Technology (IUST), Tehran, Iran, in 2013 and 2010, respectively. Her doctoral work mainly focuses on electromagnetic metamaterials and plasmonics, and their RF, THz, infrared and optical applications.

Mehdi Hajizadegan received the M.Sc. degree in electronic from Tarbiat Modares University, Tehran, Iran, in 2013. Since 2015 summer, he is a graduate research assistant in the Department of Electrical and Computer Engineering, Wayne State University in Detroit, where he is currently pursuing a Ph.D. in electrical engineering. His research mainly focuses on passive and active electromagnetic metamaterials and plasmonics, and their applications for energy harvesting. He is also actively engaged on graphene and other carbon-based biosensors.

Mohamed Farhat received his Ph.D. in Optics and Electromagnetism from Aix-Marseille University in 2010, where he obtained as well his Master degree in Theoretical Physics in 2006. He has been since postdoctoral fellow at University of Texas at Austin, University of Jena and KAUST. He has authored over 120 publications, including 50 journal papers, 4 book chapters, two international patents, and 60 conference papers, with over 1400 citations (h-index 21). He is associate editor for Advanced Electromagnetics. He has also organized many special sessions at the conference Meta and is active reviewer for many international journals (https://sites.google.com/site/mohameddfarhat/home).

Pai-Yen Chen is an Assistant Professor at the Wayne State University. He received his Ph.D. degree from the University of Texas at Austin in 2013. He was a Research Scientist at Intellectual Ventures' Metamaterial Commercialization Center during 2013-2014, and a Research Staff in the National Nano

38

Device Laboratory in Taiwan during 2006- 2009. He has been involved in multidisciplinary research on metamaterials, electromagnetics, plasmonics, nanophotonics, and nano-optoelectronics. He has published approximately 60 peer-review papers (4 journal covers), 55 conference proceedings, 4 book chapters, 1 co-edited book, and 10 US patents (http://pychen.eng.wayne.edu)

Research Highlights 

“stereo” hot-electron device based on hyperbolic metamaterials is presented.



Results show broadband and omnidirectional enhancement in external quantum efficiency.



Results show great potential in energy-efficient, sub-bandgap photodetectors, spectrometers, and plasmonic sensors.



Applicability of hot-electron devices to energy harvesting is evaluated

Graphical Abstract

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Efficient, Broadband and Wide-Angle Hot-Electron Transduction using Metal-Semiconductor Hyperbolic Metamaterials

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Maryam Sakhdari, Mehdi Hajizadegan, Mohamed Farhat, Pai-Yen Chen

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