Abstract
The strength of lightmatter interaction is of central importance in photonics and optoelectronics. For many widely studied twodimensional semiconductors, such as MoS_{2}, the optical absorption due to exciton resonances increases with thickness. However, here we will show, fewlayer black phosphorus exhibits an opposite trend. We determine the optical conductivity of fewlayer black phosphorus with thickness down to bilayer by infrared spectroscopy. On the contrary to our expectations, the frequencyintegrated exciton absorption is found to be enhanced in thinner samples. Moreover, the continuum absorption near the band edge is almost a constant, independent of the thickness. We will show such scenario is related to the quanta of the universal optical conductivity of graphene (σ_{0} = e^{2}/4ħ), with a prefactor originating from the band anisotropy.
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Introduction
In recent years, twodimensional (2D) materials including graphene, transition metal dichalcogenides (TMDCs) and black phosphorus (BP), are at the forefront of scientific research. Strong lightmatter interactions have been demonstrated in these atomically thin materials^{1,2,3,4,5,6,7,8,9,10,11,12}, holding great promise for photonic and optoelectronic applications. Optical absorption in 2D materials is a fundamental lightmatter interaction process, basically governed by the optical sheet conductivity σ(ħω). Graphene is a wellknown 2D example, which exhibits a universal conductivity σ_{0} = e^{2}/4ħ in a broad frequency range, where e is the electron charge and ħ is the reduced Plank constant^{1,2}. Moreover, Nlayer graphene exhibits an optical conductivity of Nσ_{0}, showing quantized optical transparency^{1,13}. In semiconducting TMDCs, such as MoS_{2}, the optical conductivity due to Kpoint exciton increases with layer number^{14}. However, in this paper, we will show that excitons in thinner samples absorb more light in fewlayer BP. Meanwhile, the absorption due to the electronhole continuum near its own edge is almost the same for each thickness, with a value close to that in monolayer graphene.
Fewlayer BP is an elemental 2D semiconductor beyond graphene, with a strongly layerdependent direct bandgap^{15,16,17,18}, offering an ideal platform to probe the layerdependent properties and dimensional crossover from 3D to 2D. Moreover, the intrinsic inplane band anisotropy definitely distinguishes fewlayer BP from other widely studied 2D materials, such as graphene, TMDCs and InSe. Along with the moderate bandgap and large tunability, fewlayer BP is unique and promising in polarized IR detectors and emitters. From this point of view, a quantitative determination and thorough understanding of the optical absorption in fewlayer BP is in great demand. The optical absorption of fewlayer BP is found to be dominated by robust excitons^{10,11,12}. In previous optical studies^{10,15,16}, the absorption intensity is not well quantified. In other words, quantitative insight of the absolute absorption is yet to be gained, though it is highly desirable for future optoelectronic applications, such as evaluating the quantum efficiency of photoluminescence (PL) and photocurrent generation. Our study resolves this issue and provides new insights into lightmatter interaction in anisotropic 2D gapped materials.
Results
Sample preparation and IR characterization
A Fourier transform infrared (FTIR) spectrometer was used to obtain the extinction spectra (1 − T/T_{0}) of fewlayer BP on polydimethylsiloxane (PDMS) or quartz substrates (see Methods for sample preparation and IR characterization), where T and T_{0} denote the light transmittance of the substrate with and without BP samples, respectively, as illustrated in Fig. 1d. For atomically thin materials on a thick transparent substrate, like in our case, when the optical conductivity is not large, the extinction (1 − T/T_{0}) is approximately proportional to (the real part of) the optical conductivity^{2,4}: σ(ħω) = (1 − T/T_{0}) · (n_{s} + 1) · c/8π, where n_{s} is the refractive index of the substrate (n_{s} = 1.39 for PDMS in the measured IR range^{19}) and c is the speed of light.
We systematically measured IR conductivity σ(ħω) spectra (in the unit of σ_{0}) for 2–7 L BP in a broad range of 0.4–1.36 eV at room temperature, as shown in Fig. 2. The incident light is normal to the layer plane and polarized along the armchair (AC) direction. The spectra for zigzag (ZZ) polarization are featureless^{15}, hence not discussed here. Previous studies have revealed quantized subband structures of fewlayer BP^{15,16}, due to the quantum confinement in the outofplane direction and considerable interlayer interactions, in analogy to traditional quantum wells (QWs). In symmetric QWs with normal light incidence, optical transitions obey the ∆j = 0 selection rule (j is the subband index)^{20,21}. E_{jj} denotes the exciton resonance associated with the optical transition between the jth pair of subbands (v_{j} → c_{j}) at Γ point of the 2D Brillouin zone, as illustrated in Fig. 1b. As seen from Fig. 2, the E_{11} resonance exhibits a very narrow linewidth even at room temperature, especially for thicker BP. In addition, the Stokes shift of fewlayer BP is almost negligible (see Supplementary Fig. 1 as an example of 3 L), indicating good sample quality. The IR conductivity σ(ħω) at the E_{11} resonance reaches 6.6σ_{0} in 2 L BP, directly translated to a light absorption of 15% in freestanding case. This suggests very strong lightmatter interactions in this atomically thin material. Moreover, we can observe a spectrally flat and broad response above the exciton energy, attributed to the continuum of bandtoband transitions. Steplike features of continuum absorption underline the steplike 2D joint density of states (DOS) in QWlike structures, as sketched in Fig. 1c, this still holds in anisotropic fewlayer BP^{22}.
Layerdependent exciton absorption
The σ(ħω) spectra can provide a wealth of information on the exciton oscillator strength, which is directly related to the frequencyintegrated conductivity (or absorption) of excitons^{23,24}, as indicated by the shaded areas in Fig. 2. The peak height of the exciton feature is not as informative, since it is sensitive to the sample quality. Figure 3a shows the integrated conductivity σ_{I} of the ground state (1 s) exciton of fewlayer BP as a function of layer number N. The details for spectral fitting and extraction of σ_{I} are presented in Methods. For each layer thickness, at least three samples were measured, with the error bar defined as the spread of the data. From Fig. 3a, one can find that thinner BP has larger absorption. In other words, it manifests a fact that less material absorbs more light at exciton resonances. This remarkable result is in sharp contrast to widely studied 2D semiconducting TMDCs, in which the exciton absorption increases with layer number (see Supplementary Fig. 2 for the absorption of 1–4 L MoS_{2} and also ref. ^{14}). To be more quantitative, the integrated absorption due to excitons is directly proportional to \(L_z \cdot \left {\varphi _{ex}(0)} \right^2\), where L_{z} is the thickness of the sample, φ_{ex}(0)^{2} is the modular square of the exciton wavefunction at origin, describing the probability to find the electron and the hole at the same location^{23,24,25}. Certainly, the smaller the thickness is, the larger the confinement in the z direction is and hence the larger φ_{ex}(0)^{2} is. Theory has predicted that φ_{ex}(0)^{2} ~ 1/L_{z}^{2} for QWs^{26}, if the penetration of the electron and hole wavefunctions into the barriers is negligible^{27,28}. This gives the integrated absorption ~1/L_{z} (L_{z} ~ N for BP), which means stronger absorption for thinner samples. This argument correctly predicts the trend but the fitting with such a relation does not work well. The prediction shows a much steeper decrease than what we observed, as shown in Fig. 3a (black dashed curve), which calls for a more refined model.
We can tackle this problem from another perspective. In the 2D hydrogen model for excitons, we find that the integrated absorption for the 1 s exciton is proportional to the exciton binding energy (E_{b}) (see Supplementary Note 1 for details). This is a very reasonable conclusion, since the larger the binding energy is, the closer the electron is to the hole, which favors light absorption at exciton resonances. We adopt this result for our analysis, even though excitons are not ideally 2D in fewlayer BP. Olsen et al. proposed a simple screened hydrogen model for 2D excitons^{29}, in which E_{b} can be analytically expressed as a function of the reduced effective mass μ and the 2D sheet polarizability χ. Under the condition of 32πμχ/3 >> 1, E_{b} can be simplified as E_{b} ≈ 3/4πχ, which is proved to be valid for fewlayer BP^{10}. The sheet polarizability χ relates to the dielectric screening of electronhole interactions. In the case of Fig. 2, fewlayer BP is supported by a PDMS substrate, which introduces additional dielectric screening. Thus, both of the substrate and the material itself contribute to the screening. In view of this, the sheet polarizability χ is replaced by an effective value for Nlayer BP, expressed as χ_{eff} = χ_{0} + Nχ_{1}, with χ_{0} = 6.5 Å and χ_{1} = 4.5 Å describing the screening effect from the PDMS substrate and singlelayer BP, respectively, based on our previous study^{10}. Thus, we have σ_{I} ∝ 1/(χ_{0} + Nχ_{1}). With this relation, it is clear that the exciton absorption increases as the layer number decreases, though it is not as dramatic as ~1/N. We use this relation to fit the data for E_{11}, as shown in Fig. 3a, the overall agreement is much better than the 1/N scaling. The basic behavior is well captured by the modified model, though the agreement with experimental data is still not so excellent. The deviation is mainly caused by the experimental uncertainty in determining the optical conductivity, especially for thinner BP samples, which are expected to be more susceptible to the environment. The substrate plays a role in reducing the exciton binding energy, hence the integrated absorption. For freestanding BP, this model also gives a scaling of 1/N (ref. ^{10}), consistent with the previous prediction for QWs. For E_{22} resonances, the exciton absorption increases with the decrease of layer number as well, as also shown in Fig. 3a. It should be noted that, the thickness dependence of exciton absorption in BP is even qualitatively quite different from that in traditional QWs. For the latter, the absorption reaches maximum at certain thickness and decreases again with decreasing thickness, due to the exciton wavefunction leakage into the barriers in the ultrathin limit^{27,28}. Such scenario becomes more evident in shallow QWs. The hard confinement in atomically thin BP definitely distinguishes it from traditional QWs. From this perspective, atomically thin BP provides us unique opportunities to examine this new type of QWs.
The frequencyintegrated conductivity of excitons is robust against temperature, although the linewidth is typically vulnerable to various factors. Since the aforementioned PDMS substrate thermally expands (contracts) during the heating (cooling) process, significant strain effects can be introduced to fewlayer BP samples during temperaturedependent measurements, overwhelming the pure temperature effect. Therefore, we transferred fewlayer BP samples to quartz substrates with a much smaller thermal expansion coefficient, so that the strain effect can be ignored^{30}. Fig 4a, d show the IR extinction spectra of a 3 L and 7 L BP at varying temperatures from 10 K to 300 K, respectively. To improve the ratio of signaltonoise for such measurements, the incident IR beam size was set to be larger than the sample size. Thus, the absolute intensity of exciton absorption is underestimated, but it’s legitimate to compare the integrated area of exciton peaks for the same sample at different temperatures, which is directly related to the exciton oscillator strength. To quantitatively probe the temperature effect, the integrated area and linewidth of exciton peaks are extracted from spectral fitting using Eq. (3) in Methods. Clearly, the exciton width increases with temperature both for E_{11} and E_{22} as expected (Fig. 4c, f). Such scenario is common in semiconductors, which is attributed to the enhanced electronphonon scattering channels at elevated temperature^{31,32}. The integrated areas of exciton peaks, on the other hand, are nearly independent on the temperature, as shown in Fig. 4b, e. The slight deviation from a constant may arise from the experimental uncertainty and data fitting procedure. A recent study shows that for typical semiconductors, in the incoherent region of lightmatter interactions, the integrated exciton absorption only depends on the radiative decays, rather than the scattering decays, hence the absorption is independent of temperature^{33}. This is exactly the case for our observations. Since the integrated exciton intensity is proportional to the exciton binding energy, our results indicate that the exciton binding energy is almost a constant within the tested temperature range.
Continuum absorption
Next, we will focus on the layer dependence of the absorption due to the continuum transition. As seen in Fig. 2, a relatively flat response can be observed in the σ(ħω) spectra above the ground state exciton energy, mainly associated with the continuum bandtoband transitions (labeled as T_{jj}) and possibly excited excitonic states^{10}. A closer examination of Fig. 2 shows that the continuum absorption is not strictly a constant but decreases gradually with increasing photon energy. For simplicity, we focus on the continuum absorption close to the band edge (ħω ≈ E_{g}), indicated by the red (blue) arrows for T_{11} (T_{22}) transitions in Fig. 2. The results are summarized in Fig. 3b as a function of layer number N, they approximately follow a constant value despite of the experimental uncertainty. This means that the absorption due to the bandgap continuum transitions is almost the same, regardless of the thickness of BP samples.
According to the Fermi’s golden rule, if a light beam with frequency ω and linear polarization along \(\vec e_0\) is normally incident to a directgap 2D system, the dimensionless absorption A(ħω) due to a pair of conduction and valence bands can be expressed as (in SI units)^{34}
where ε_{0} is the vacuum permittivity, m_{0} is the free electron mass, and A_{r} is the area of the 2D material. The momentum matrix element is defined as \(\overrightarrow P _{cv}(\overrightarrow k ) = \big\langle {c,\overrightarrow k } \big\overrightarrow e _0 \cdot \overrightarrow P \big {v,\overrightarrow k } \big\rangle\), with \(\overrightarrow P =  i\hbar \nabla\) as the momentum operator. \(E_{cv}(\vec k) = E_{c\vec k}  E_{v\vec k}\) and \(\vec k\) is the wave vector.
It first looks like that the absorption is strongly band parameter dependent, since the terms of \(\overrightarrow P _{cv}(\vec k)\) and DOS are involved in Eq. (1). Surprisingly, the absorption in isotropic graphene^{1,2} and InAs QWs^{35} are found to be universal, indicating that there is a cancelling mechanism between these two terms. For massless graphene, which has a unique linear energy dispersion near the Dirac point \(E_{cv}(k) = \pm\! {\hbar} v_Fk\) (v_{F} is the Fermi velocity), the cancelation is exact with no frequency dependence, leading to a welldefined universal absorption quanta πα, where α = e^{2}/4πε_{0}ħc is the fine structure constant (~1/137)^{36}. For anisotropic 2D massive semiconductors (see Supplementary Note 2 for details), based on the \(\vec k \cdot \vec p\) perturbation theory^{34}, it is obtained that \(E_{cv}({\vec{k}}) = E_g + \frac{{{\hbar}^2}}{2}\left( {\frac{{k_x^2}}{{{\mu} _x}} + \frac{{k_y^2}}{{{\mu} _y}}} \right)\), and \(\frac{1}{{{\mu} _x}} = \frac{{4\big {{\overrightarrow {P}} _{cv}} \big^2}}{{m_0^2 E_g}}\), where μ_{x(y)} is the reduced effective mass in the AC (ZZ) direction. With these, and also taking into account the degrees of spin and valley degeneracy of g_{s} and g_{v}, we thus have
where Θ(ħω − E_{g}) is the step function, describing the 2D joint DOS, and E_{g} is the bandgap. This indicates that most of the band parameter dependent terms are perfectly canceled out, leaving behind the frequency dependent term ω and the band anisotropy term \(\sqrt {\frac{{\mu _y}}{{\mu _x}}}\). Our observations in fewlayer BP can be well explained by Eq. (2): (i) above the bandgap ħω > E_{g}, the absorption gradually decreases as the photon energy increases; (ii) near the bandgap ħω ≈ E_{g}, with g_{s} = 2, g_{v} = 1 for BP^{20}, Eq. (2) gives a value of \(\pi \alpha \sqrt {\frac{{\mu _y}}{{\mu _x}}}\) (or equals to the optical conductivity \(\sigma _0\sqrt {\frac{{\mu _y}}{{\mu _x}}}\)), slightly deviating from the universal value πα. The prefactor \(\sqrt {\frac{{\mu _y}}{{\mu _x}}}\) originates from the band anisotropy, which is nearly layerindependent for 2–7 L BP^{17}. When μ_{x} = μ_{y}, it collapses to the isotropic case^{36}; (iii) with the photon energy higher than the second subband transition (T_{22}), additional contribution in absorption shows up with another step height of \(\pi \alpha \sqrt {\frac{{\mu _y}}{{\mu _x}}}\).
Discussion
As a simple explanation of what we observed for both exciton and electronhole continuum transitions, we can revisit the 2D band structure of fewlayer BP. Due to the strong coupling between layers, the conduction and valence bands split into multiple 2D subbands with sizable energy spacing (for the case with sample thickness below 10 L)^{15}. If we focus on the bandgap continuum transition, no matter the thickness or how many subbands in total, the relevant ones are only c_{1} and v_{1} subbands. As a consequence, one can not expect more absorption with photon energy right above the bandgap when the layer number increases, given that only the first pair of 2D subbands is involved and the 2D joint DOS typically barely vary. Therefore the increase of sample thickness does not cause enhanced absorption for the continuum. However, on the other hand, the excitonic effect weakens with increasing layer number due to weaker confinement in the z direction, which actually decreases the exciton absorption. This can qualitatively explain the findings in Fig. 3. This argument applies to other 2D materials as well. For MoS_{2}, the electronic bands associated with directgap exciton at K point show almost no splitting when the layer number increases from 1 to 2 (or other thickness)^{37}. The degenerate subbands certainly double the DOS and the exciton absorption can increase accordingly, giving an opposite trend compared to BP. As for Nlayer graphene, in spite of the complexity of the very low energy band structure, in the range from visible to nearIR, N pairs of subbands are involved in the optical absorption and each pair contributes an absorption quanta πα^{1,2}. While for Nlayer BP, if we only focus on the bandgap region, only one pair of subbands is involved and one absorption step is contributed, regardless of the thickness. Of course, if we increase the photon energy, more and more subbands are counted in and the absorption increases step by step. Eventually the continuum absorption is comparable to that of Nlayer graphene. Therefore, counting the 2D subbands does give us a good estimation of absorption in different 2D systems^{36}.
In summary, we have determined the optical conductivity of 2–7 L BP using IR absorption spectroscopy. Our results reveal that the exciton absorption increases as the layer number decreases, a direct consequence of enhanced excitonic effects in reduced dimensionality. Moreover, the absorption from the continuum states near the band edge exhibits a layerindependent value. Fewlayer BP provides us an ideal platform to probe the dimensional effect on the strength of optical absorption from bound (exciton) and unbound (continuum) states in the same material. The highly enhanced exciton absorption of atomically thin BP, which tends to host high density of optical excitations, may open up new possibilities for applications in nonlinear optics and quantum optics. Our results are expected to stimulate further theoretical interests in anisotropic 2D materials.
Methods
Sample preparation
Fewlayer BP samples were prepared by a modified mechanical exfoliation method^{38}. Firstly, a piece of bulk BP crystal (HQ Graphene Inc.) was cleaved several times using a Scotch tape. Secondly, the tape containing BP flakes was slightly pressed against a PDMS substrate of low viscosity and then peeled off rapidly. Some thin BP flakes with relatively large area and clean surface were left on the PDMS substrate. To achieve high optical quality, the samples on PDMS were directly used for roomtemperature IR measurements, without any additional sample transfer process. Supplementary Fig. 1 shows an example of a 3 L BP on PDMS. The negligible Stokes shift indicates good sample quality. To avoid sample degradation in air, the samples were prepared in a N_{2} glove box with O_{2} and H_{2}O levels lower than 1 ppm, and then loaded into a cryostat purged with N_{2} for subsequent IR measurements. The cryostat is only for sample protection instead of temperature control. The layer number and crystal orientation of BP were readily identified using polarized IR spectroscopy^{15}.
Polarized IR spectroscopy
The IR extinction (1 − T/T_{0}) spectra of fewlayer BP were measured using a Bruker FTIR spectrometer (Vertex 70 v) equipped with a Hyperion 2000 microscope, as illustrated in Fig. 1d. A tungsten halogen lamp was used as the light source to cover the broad spectral range of 0.4–1.36 eV with 1 meV resolution, in combination with a liquid nitrogen cooled MercuryCadmiumTelluride (MCT) detector. The lower bound cutoff photon energy is restricted by the substrate. The linearly polarized incident light was obtained after the IR beam passing through a broadband ZnSe grid polarizer, then was focused on BP samples using a ×15 IR objective. For roomtemperature measurements, the aperture size was set to be smaller than the sample size to make sure the IR beam all goes through the sample. A spectrum was typically acquired over 1000 averages to improve the signaltonoise ratio.
Low temperature IR measurements
For low temperature measurements, fewlayer BP samples were transferred from PDMS to quartz substrate, to avoid the significant strain effect of PDMS during the heating (cooling) process, since the latter has a much smaller thermal expansion coefficient. Then, the samples were loaded into a liquid He cooled cryostat, with the temperature range from 10 K (or lower) to 300 K. To improve the signaltonoise ratio, the aperture size was set to be larger than the sample size.
Data analysis
Generally, homogeneous broadened lineshape is characterized by a Lorentzian function. However, for the E_{11} peaks in Fig. 2, attempts failed to fit the lineshape using a single Lorentzian function, since the highenergy tail deviates the spectrum from the ideal lineshape, especially for thinner BP. Similar observations have been reported in QWs^{39,40,41}, the exciton absorption (or emission) lineshape is slightly asymmetric, exhibiting a Lorentzian and an exponential slope at the low and highenergy side, respectively. This is caused by the socalled “exciton localization effect”, as a consequence of spatial inhomogeneity^{39}. Nevertheless, such imperfection will not affect our determination of the integrated areas of exciton peaks.
To determine the integrated conductivity (σ_{I}), the E_{11} peaks in Fig. 2 are fitted using a documented model, which was first proposed by Schnabel et al.^{39}, and has been successfully applied to QWlike structures^{40,41}. This model is analytically expressed as:
The first part is a Lorentzian function, describing the symmetric lineshape with area of L_{S} and linewidth of γ_{L}. ∆ = ħω − ħω_{0}, with ħω_{0} denoting the average transition energy between the conduction and valence subbands. The second part describes the asymmetric lineshape, γ is the linewidth and η describes the asymmetric broadening by exciton localization, L_{A} is the spectral weight. We use Eq. (3) to fit the E_{11} peaks. As shown in Fig. 2 (red curves), the general agreement is excellent, except for a small discrepancy at the highenergy end of the peaks, given that the excited excitonic states (2 s or 3 s states) and the continuum absorption are also involved in this spectral region. While E_{22} peaks (blue curves) can be well fitted only using the symmetric part (Lorentzian function). The integrated conductivities shown in Fig. 3a are extracted from the fitted peaks. In addition, as indicated by the extracted parameter η for each layer thickness (Supplementary Fig. 4), we can see that thinner BP exhibits larger asymmetry. This is reasonable, since thinner BP is more sensitive to the underlying substrate and environment.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
H.Y. is grateful to the financial support from the National Young 1000 Talents Plan, National Science Foundation of China (grants: 11874009, 11734007), the National Key Research and Development Program of China (grants: 2016YFA0203900 and 2017YFA0303504), Strategic Priority Research Program of Chinese Academy of Sciences (XDB30000000), and the Oriental Scholar Program from Shanghai Municipal Education Commission. M.H. is grateful to the financial support from the Initiative Funds for Shenzhen High Class University (grant: G02206301). G.Z. acknowledges the financial support from the National Natural Science Foundation of China (grant: 11804398), Natural Science Basic Research Program of Shaanxi (grant: 2020JQ105), Open Research Fund of State Key Laboratory of Surface Physics and the Fundamental Research Funds for the Central Universities. Part of the experimental work was carried out in Fudan Nanofabrication Lab.
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H.Y. and G.Z. initiated the project and conceived the experiments. G.Z. prepared the samples and performed the roomtemperature IR measurements. G.Z., M.H., and S.H. performed the low temperature IR measurements with assistance from F.W., Q.X., C.S., C.W., and Y.L. G.Z., M.H., and H.Y. performed the data analysis and cowrote the manuscript. H.Y. supervised the whole project. All authors commented on the manuscript.
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Zhang, G., Huang, S., Wang, F. et al. The optical conductivity of fewlayer black phosphorus by infrared spectroscopy. Nat Commun 11, 1847 (2020). https://doi.org/10.1038/s41467020156997
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DOI: https://doi.org/10.1038/s41467020156997
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