Hyperspectral absorption of semiconductor monolayer crystals

Strongly absorbing semiconductors provide one possible route to realize high quantum efficiency photodetectors,^1,2 solar cells,^3–6 and artificial photosynthesis devices.^7,8 Currently, the record silicon⁹ and III–V multi-junction¹⁰ solar cells have around 26% and 38% energy conversion efficiency, respectively, while they need to be few (GaAs) to hundred (Si) μm thick to absorb most of the sunlight. In this connection, thin nanostructures could significantly enhance conversion efficiency through improved carrier transport at the nanoscale. Moreover, they could become efficient photosynthetic devices that transform solar energy, water, and CO₂ to chemicals and fuels, at an ideal photovoltage (1.7–2 V) required to overcome the uphill potential barrier in chemical reactions while providing sufficient current density under sunlight illumination. In this regard, conventional high-efficiency solar cell designs, such as multi-junction devices, are not suited for the emerging artificial photosynthesis applications: they often provide an overly large photovoltage for the chemical reactions at the expense of significantly reduced photocurrent density. For these applications, it is therefore highly desired to develop new light absorbers that can offer broadband light absorption and efficient charge-carrier separation while maintaining relatively high photovoltage. Here, we focus on hyperspectral absorption that we quantify to be 50% or more over spectral range covering several eV.

In semiconductors, light absorption creates electrons to the conduction band and holes (vacancies) to the valence band. The Coulomb attraction can bind them into excitons, with the 1s exciton having the lowest energy. After their optical excitation, coherent 1s excitons decay radiatively via light–matter coupling or nonradiatively via disorder-, Coulomb-, and phonon-interaction induced dephasing γ. The radiative decay $Γ∝1/(aB)2$ is inversely proportional to exciton radius a_B squared for effectively two-dimensional excitons.¹¹ Generally, the ratio of γ and Γ defines the strength of the excitonic absorption, and the maximum absorption is detected when γ and Γ are equal.¹¹ The specific material determines Γ and it often is too small to realize strong-absorption condition $γ=Γ$ although γ can be tuned with either the temperature or electron–hole density. Since Γ increases with shrinking confinement, $Γ→γ$ is easiest to reach at the limit of a single atomic layer, monolayer (ML), which already is standard for transition metal dichalcogenides (TMDCs).¹² Indeed, a hBN-encapsulated TMDC monolayer has been demonstrated to produce a radiative-limited linewidth of few meV.^13–15 Nearly 100%^16,17 excitonic absorption has been measured for the MoSe₂ monolayer with a mirror next to monolayer, but only close to a 2 meV-wide absorption peak, far from being hyperspectral. We have recently predicted¹⁸ and demonstrated^19,20 a new class of room-temperature quantum materials—GaN monolayers that exhibit exciton binding comparable to TMDCs while being the only known materials whose energy band edges straddle water redox potentials. This makes nitride monolayers an exciting candidate for efficient solar-fuel generation.

In this Letter, we present strategies to extend high-absorption bandwidth from few meV to 2 eV based on TMDC and III-nitride monolayers as well as GaAs quantum wells to cover a broad range of design possibilities. We will introduce monolayer arrays that consist of different monolayers separated by barriers with thickness L [Fig. 1(a)] and position the structure in front of a mirror. By changing the exciton energy of each monolayer to be partially overlapping, we expect to increase the bandwidth of the collective absorption as illustrated in Fig. 1(b). Since this involves a changing exciton energy with position, we call this approach as color chirping. Our results suggest that the excitonic absorption can be combined with multi-layer-enhanced continuum absorption to produce hyperspectral absorption without the need to compromise the ideal photovoltage.

We fully describe many-body aspects of monolayers with the semiconductor Bloch equations systematically coupled to the Maxwell equations, following the discussion in Refs. 11 and 21, as summarized in the supplemental material. Since the sunlight is weak in terms of semiconductor nonlinearities, it is sufficient to analyze the linear response defined by a dimensionless susceptibility $ξ(ω)$⁠. This defines the absolute absorption of light, $α(ω)=2Im[ξ(ω)]/|1−iξ(ω)|2$⁠, for a free-standing monolayer.¹¹ The properties of 1s-exciton absorption alone can be understood from its linear susceptibility¹¹ 

where $Γ1s$ defines the radiative decay of 1s excitons having energy $E1s=ℏω1s$⁠. The full many-body description introduces Coulomb- and phonon-interaction induced dephasing $γ(ω)$ that eventually depends on exciton state and light frequency.¹¹ At the 1s resonance $ω=ω1s$⁠, the linear susceptibility reduces to $ξ(ω1s)=iΓ1s/γ≡i/r$⁠, which yields 50% peak absorption at $γ=Γ1s$⁠, following¹¹ from $αpeak=2γΓ1s(γ+Γ1s)2=2r(1+r)2$ for a general ratio r of nonradiative and radiative dephasing. The 1s energy defines a wavelength $λ1s$⁠, and adding a mirror at $λ1s/4$ distance from the monolayer doubles the radiative decay $Γeff=2Γ$⁠, halves $r=γ/Γeff$⁠, and doubles $αpeak=4r(1+r)2$ peaking at 100% for $γ=Γeff$⁠, as measured recently.^16,17

Since Coulombic enhancement of light–matter interaction is weak for excitonic continuum states, their radiative coupling is equally weak. The corresponding ratio $rcont≡Re[i/ξ(ωcont)]∝1/α(ωcont)$ is large producing only a weak continuum absorption $α(ωcont)$⁠, evaluated at $ℏωcont$ matching the energy of the continuum states. In typical semiconductors, $ξ(ωcont)$ and thus a single layer $α(ωcont)$ are one-to-two orders of magnitude smaller than for 1s. However, it is well known that radiative coupling among multiple quantum wells strongly modifies the absorption^22,23 through significantly altered effective r. Additionally, radiative coupling among multiple quantum wells can produce new spectral resonances²⁴ not describable by the Beer's law, as verified in the supplementary material.

We first analyze multiple monolayers, all positioned at the same location, $λ1s/4$ spacing away from a mirror. This artificial study approximates²³ monolayer arrays with inter-monolayer spacing much smaller than the resonance wavelength $L≪λ1s$⁠. Our system has an exciton binding energy E_B = 267 meV and radiative coupling $Γ1s=0.9$ meV matching a typical TMDC monolayer.^13–15,25 The total dephasing is the sum of Coulomb- and phonon-scattering induced dephasing, and we analyze a realistic case with $γ1s=2$ meV and $γλ≠1s=4γ1s$⁠, due to the diffusive nature Coulomb scattering.²⁶ We also add $γ=γλ+10$ meV due to room-temperature phonons.²⁷ Figure 2(a) shows the absorption spectra of 1 (shaded area), 10 (black line), and 120 (red line) TMDC-type monolayers as a function of photon detuning $Δ≡ℏω−E1s$ with respect to the 1s energy $E1s$⁠. We see that while the absorption near the 1s energy increases from 1 to 10 monolayers, it significantly drops for 120 monolayers. At the same time, the continuum absorption increases to reach nearly perfect absorption over a broadband.

As discussed above, the 1s and continuum absorption have a very different radiative $Γλ$ and thus $rλ=γ/Γλ$ that defines the corresponding peak absorption $αpeak$⁠. Having N monolayers at the same position increases $Γλ(N)=ΓλN$ linearly as a function of N while a more general monolayer spacing introduces additional coupling effects analyzed later in Figs. 3 and 4. The pure $Γλ(N)$ effects always decrease $rλ$ for increasing N, which produces $r(N)=γNΓλ=r(1)N∝1N$ such that we can determine the peak absorption from an effective $αpeak$ vs $r(N)=γ/Γ(N)$ curve shown in Fig. 2(b). As discussed after Eq. (1), perfect absorption emerges at r(N) = 1 implying that a system having

monolayers produces perfect absorption. Our TMDC example has $r1s(1)=6.7$ and $rcont(1)=125$⁠, yielding a high $αpeak=0.45$ (red solid line) and low $αpeak=0.03$ (blue solid line) for the $1s$ and continuum states, respectively. Since r(N) decreases with increasing N, both cases can be brought toward the perfect absorption by increasing N toward $ND=r(1)$⁠. However, if N is increased further, the absorption starts to droop, producing a significant decrease in absorption as N is elevated further. For this system, we find the droop-point N_D = 6.7 for the 1s resonance and N_D = 125 for the continuum states. These numbers explain why 1s absorption in Fig. 2(a) increases from N = 1 to N = 10 and then droops for the N = 120 whereas continuum absorption continuously improves all the way to N = 120.

Reaching hyperspectral absorption through 1s absorption is limited because while $1s$ absorption is strong, it remains spectrally narrow and droops faster for elevated N. At the same time, continuum absorption intrinsically covers a broad spectral range and its absorption level can be increased continuously up to a very large N, to eventually reach hyperspectral absorption. As a further advantage, light absorption to continuum states creates unbound electron–hole pairs which can be directly harvested as free charges, unlike excitons created by the 1s absorption.²⁶ Since excitons do not participate in currents, they must be dissociated into free carriers before being useful for solar-cell or photodetection applications, which adds a new design complication. We therefore conclude that hyperspectral absorption can and should be optimized with respect to continuum absorption.

Since the continuum-absorption-based (CAB) hyperspectral absorbers typically need a large N, it is unlikely that the total thickness can be made much smaller than λ. This inevitably introduces coupling effects beyond modifying only $Γλ(N)$⁠. To identify the effect of nontrivial coupling effects on CAB absorbers, we compare N = 120 monolayer absorption when they are positioned at L = 0, L = 10 nm, and L = $λ1s$/2 spacing in Figs. 3(a)–3(c), respectively. While setting L = 0 eliminates nontrivial coupling effects, it is not experimentally feasible, except as a limiting case having L much smaller than the wavelength. Materials parameters are the same as the TMDC-type monolayers studied in Fig. 2. The shaded areas show the absorption from all states. We see that while a 1s droop appears for L = 0, the 1s absorption becomes large and extended-band for L = 10 nm and splits for L = $λ1s/2$⁠. The 1s droop for L = 0 results from the too strong radiative coupling among all monolayers as discussed above, which is also true for L = $λ1s/2$ at $ℏω$ = $E1s$ where all monolayers are coupled in phase. For L = 10 nm spacing, the radiative coupling changes it nature because most monolayers couple out of phase and in-phase monolayers form a subset of N. This reduces the number of optimally coupled monolayers and the droop. These phase aspects split the 1s resonance into 90 meV-wide miniband for L = 10 nm, although this is not enough to extend high absorption all the way to the continuum, above detuning $Δ≡ℏω−E1s>$ 267 meV. At the same time, the continuum absorption itself remains high in all three cases.

To quantify the effect of 1s state on continuum absorption, we compute absorption without the 1s refractive-index $n1s≡Re[ξ1s]$ (red line) or without the 1s absorption $α1s≡Im[ξ1s]$ (black line) contributions in the total ξ, shown in Fig. 3. For L = 0, the $n1s$ is essential for reaching nearly perfect absorption in continuum because the absorption without $n1s$ (red line) is 15% lower than the full absorption (shaded area) while eliminating only the $α1s$ (black line) reproduces the actual continuum absorption. To explain this unexpected 1s contribution to the continuum absorption, we analyze Eq. (1) further. Since $ξ1s$ contributes nonresonantly to continuum frequencies as $ξ1s∝−1Δ+iγ1s$⁠, $n1s≡Re[ξ1s]$ contribution is $Δ/γ1s$ times larger than $α1s≡Im[ξ1s]$⁠. As we have $Δ/γ1s=22$ at Δ  = 267 meV, the $n1s$ part dominates the 1s contribution for the continuum. At the same time, negative $n1s∝−1Δ$ means that the 1s state reduces the total refractive index $ncont$ at the continuum, further enhanced by the monolayer number N. Since the actual radiative coupling has a $Γeff∝1/ncont$ dependence,¹¹ reducing $ncont$ by $n1s$ increases continuum's radiative coupling, which explains the enhanced continuum absorption for L = 0. When the monolayer spacing L becomes sufficient, the out-of-phase coupling tends to average out the $n1s$ contributions. Indeed, $n1s$ has very little effect on the continuum part for L = 10 nm and L=$λ1s/2$ spacing.

These results identify a straightforward strategy to realize hyperspectral absorption, from 1s to continuum: we combine the N-enhancement of radiative coupling for continuum states with color-chirping [Fig. 1(b)] to improve the bandwidth of 1s absorption. Here, we consider color-chirped crystals where the 1s energy of each monolayer changes by about $ΔE1s=2 Γ1s$ with respect to the previous one. The crystal itself is positioned $λ1s/4$ away from a mirror to enhance absorption further. To cover the design possibilities for different materials, we compute TMDC, nitride, and GaAs representing strong to weak Coulomb interaction. The TMDC parameters are chosen as in Figs. 2 and 3. We choose a typical GaAs quantum well having 8 nm thickness, 10 meV binding energy, and $Γ1s=25 μ$eV,²⁴ as well as nitride monolayers having 200 meV binding energy^18–20 and $Γ1s=0.16$ meV.

With these materials, our designs lead to color-chirped structures with $ΔE1s=$ 2, 0.4, and 0.05 meV for TMDC, nitride, and GaAs-based crystals, respectively. The TMDC crystals have 4 nm thick hBN spacers between each layer. The nitride crystal has 2 nm spacers since eight monolayers of the AlN barrier are sufficient to form the confinement.¹⁸ The GaAs crystals have 20 nm spacers to accommodate reasonably thick quantum wells. To quantify hyperspectral absorption, we determine an average absorption

over a hyperspectral energy window defined by $Ehyper$⁠. For all cases, we use $Ehyper=2$ eV above the lowest 1s absorption resonance. Figure 4(a) shows $α¯$ for TMDC (red line), GaAs (black line), and nitride (blue line) crystals as a function of N. For all cases, $α¯$ increases monotonically as a function of N. Onset to above 80% (50%) $α¯$ is reached at N = 120 (60), 240 (100), and 700 (300) for TMDC, GaAs, and nitride-based crystals, respectively. We notice that absorption per monolayer, $α¯/N$⁠, is weakest for nitrides and strongest for TMDCs. This order might be unexpected by comparing the relative strengths of $Γ1s$ of these materials (⁠$Γ1s(GaAs)≪Γ1s(GaN)$⁠). However, while nitrides have a much stronger Coulomb enhancement for the 1s state than GaAs, they become weaker for the continuum state. As a result, we find the continuum droop points N_D = 290, 560, and 1670 for TMDC, GaAs, and nitride crystal, respectively. Although nitrides require a large N to reach hyperspectral absorbers, they can be tightly packed with a small L due to large barrier provided, e.g., by AlN spacers. With the used example, $α¯=80% (50%)$ is reached with 1.5 (0.7) μm thick nitride crystals compared to 0.6 (0.35) μm TMDC and 6.7 (2.8) μm GaAs crystals, respectively. Since trivial reflection from a sample surface can be eliminated with broadband antireflection coating,⁹ we have assumed perfect antireflection coating and have verified that a simple single-layer coating reduces $α¯$ only by 2% for monolayer crystals. At the same time, we have on purpose included continuum absorption of only two bands and expect that these thicknesses are an overestimate because continuum absorption from additional bands should lower the thickness threshold for hyperspectral absorption.

Figures 4(b)–4(d) show the absolute absorption spectrum for TMDC, nitride, and GaAs crystals, respectively. Each case compares color-chirped (solid line) and fixed-energy (shaded area) monolayers to assign the benefits of color-chirping. For TMDC and nitride, color-chirping creates an extended 1s flatband and reduces the droop. For GaAs structures, the 1s absorption is smaller due to weaker Coulomb and resembles more continuum absorption than for TMDCs and nitrides. These results show that color chirping is only needed for systems with a strong $Γ1s$⁠, whereas CAB-effects yield hyperspectral absorbers for all materials. For the artificial photosynthesis applications, one can grow a p-type doping gradient^7,8 to create a built-in electric field that can yield an efficient charge carrier extraction along the lateral dimension of each monolayer. As such, relatively high photovoltage can be maintained for photogenerated charge carriers in each monolayer.

In conclusion, we identified a clear design strategy to realize hypersepctral absorbers with semiconductor monolayers. By color-chirping the 1s resonance, strong and peaked exciton resonances can be converted into a broad flatband absorption. When combined with continuum-absorption based (CAB) optimization, monolayer crystals can reach high absorption bandwidth of several eV. In III-nitride-based systems, color-chirping can be realized with In and Al alloys to essentially cover the spectral range of sunlight. Our analysis shows that such quantum structures could be two orders of magnitude thinner than those used in existing solar-fuel applications, which should significantly improve the total photon-to-electron conversion efficiency.

See the supplementary material for the many-body theory of excitonic absorption and light propagation.

This work was supported by ARO funding through Award Nos. W911NF1810299 and W911NF19P0025 as well as the College of Engineering Blue Sky Research Program.