Tunable intraband optical conductivity and polarization-dependent epsilon-near-zero behavior in black phosphorus

Optical and electrical characterization of multilayer BP field-effect heterostructure

We used Fourier-transform IR micro-spectroscopy to measure reflection spectra of multilayer BP structures. A typical field-effect heterostructure, schematically illustrated in side view in Fig. 1B, constructed using van der Waals assembly technique, is shown in Fig. 1C, where the BP is 18.7 nm and the top and bottom hBNs are 36.8 and 36.4 nm, respectively. The carrier density in BP was tuned by applying a gate voltage across the bottom hBN and SiO₂ (285 nm). Such a geometry allowed independent electrical and optical characterization of the induced 2DEG in BP. Polarized Raman spectroscopy was used to identify the AC and ZZ axes of BP as indicated in the optical image of the device. All optical and electrical measurements were performed in ambient at room temperature.

The reflection spectrum, shown in Fig. 1D, for the same device was measured at the minimal conductance point (MCP), confirmed from two-terminal electrical measurements as shown in Fig. 1 (E to G), with light polarized along the AC direction. This spectrum is normalized to that of optically thick Au (approximately 500 nm) evaporated on the same sample as a reference surface. Three prominent features dominate the spectrum—a narrow hBN phonon around 1370 cm⁻¹, a broad dominant SiO₂ phonon around 1100 cm⁻¹ [recent studies (15) show multiple phonon contributions in SiO₂], and the beginning of band edge absorption around 3000 cm⁻¹ convoluted with an interference dip coming from the entire stack. In addition, from our transport measurements, a hole mobility of 1107 cm²/V·s and an electron mobility of 412 cm²/V·s were obtained at low doping levels, corresponding to scattering rates on the order of approximately 5 to 10 meV. A separate figure in the Supplementary Materials shows the spectral features from interband absorption for multiple subbands. As shown in Fig. 1 (H and I), these reflection spectra can be heavily modified under positive or negative gate voltages.

Modeling the optical conductivity of the BP electron/hole gas allows us to gain an understanding of the quasiparticle dynamics under applied voltage. Our BP flakes are between 10 and 20 nm thick and described by a sheet conductivity σ since the effective modulation is confined to only 2 to 3 nm from the interface of BP/b-hBN. The thickness of this modulated region was estimated from the results of band bending calculations, which are detailed in the Supplementary Materials. This sheet conductivity has contributions from both interband and intraband processes, given as σ = σ_interband + σ_intraband = σ₁(ω) + iσ₂(ω). The interband contribution accounts for absorption above the band edge, including all subbands, while the intraband part accounts for free-carrier response. One can explicitly calculate for optical conductivity using the Kubo formalism as follows (48)σinterband=−igsℏe2(2π)2Σss′jj′∫dk f(Esjk)−f(Es′j′k′)Esjk−Es′j′k′*⟨ϕsjk∣να̂∣ϕs′j′k′⟩⟨ϕs′j′k′∣νβ̂∣ϕsjk⟩Esjk−Es′j′k′+ℏω+iη (1)σintraband,j=iDjπ(ω+iηℏ),Dj=πe2Σi=1Nnimi,j*(2)

Here, να,β̂ is the velocity operator defined as ℏ⁻¹∂_kα,β H, g_s = 2 is used to denote the spin degeneracy, and f(E) is the Fermi-Dirac distribution function; the indices s(s′) refer to conduction (valence) bands and the indices j(j′) refer to the subbands. H is the low-energy in-plane Hamiltonian around the Γ point, and E_sjk and ϕ_sjk are the eigenenergies and eigenfunctions of H; mi,j* represents the effective mass of carriers in each subband (i) along a specific crystal orientation j, n_i represents the charge density in each subband, and η is a phenomenological damping term.

Electrostatic doping of BP primarily brings about two fundamental changes in the optical response: the emergence of a strong intraband component in the mid- to far IR and a shift of the optical gap (interband transitions), shown schematically in Fig. 1J. A combination of multiple electro-optical effects at the band edge has been shown to explain the observed modulation (more details are available in the Supplementary Materials). All of the observed reflection modulation spectra exhibit strong anisotropy with respect to the BP crystal axes under AC- and ZZ-polarized illumination. This strong anisotropy is predicted by theory and results from the puckered honeycomb lattice crystal structure of phosphorene (51). Our results in Fig. 1 (H and I) indicate strong optical modulation in the 2DEG and are in excellent agreement with results from previous studies (40–43).

Low-energy doping–dependent intraband response in multilayer BP

We now turn our attention to the low–photon energy regime, which is dominated by the intraband conductivity of BP. Figure 2 describes this response, the understanding of which is a central result of this paper. Figure 2 (A and B) shows reflectance spectra (normalized as before) for light polarized along the AC and ZZ axes, respectively. Both electron and hole doping can modify the free-carrier response of the 2DEG. As doping increases, a strong spectral feature is observed to appear around the characteristic hBN (~1360 cm⁻¹) and SiO₂ (~1100 cm⁻¹) intrinsic phonon peaks with both electron and hole doping. We propose that this feature results from an increase in the free-carrier density, which increases the Drude conductivity and thus modifies the optical properties of BP. This broad intraband modulation interferes with the previously described hBN/SiO₂ phonons, giving rise to an absorption line shape with a Fano-like modulation in the hBN/SiO₂ phonon regime. We hypothesize that this asymmetric Fano-like resonance shape (52, 53) indicates optical coupling between the narrow phonon resonances and the weak free-carrier absorption continuum. To better understand the nature of the line shape, we performed thin-film transfer matrix calculations to fit the spectra and account for multiple reflections and interferences in the heterostructure stack. Our model incorporates a Drude-like function for the intraband optical conductivity of the BP 2DEG, given by Eq. 2, with which we are able to extract the Drude weight as a function of doping. Assuming a simple parallel plate capacitor model, we can estimate the doping density at each gate voltage. For undoped BP, we assume a charge density of 10¹¹/cm² to account for the finite MCP response (coming from any defects or trapped charges). The contribution to the linewidth of the imaginary component of Drude conductivity from dephasing associated with finite scattering times was assumed to be on the order of that obtained from DC transport measurements (approximately 5 meV), which is a valid approximation in the energy ranges considered here. The possible sources of scattering include electron-phonon coupling, electron-electron repulsion, and interaction with defects and impurities. Studies have shown that crystals of layered materials on substrates with strong phonons can also show losses from electron-surface polar phonon coupling (54). There have also been reports of DC transport mobilities that are not well correlated with optical scattering times, or which even show an anti-correlation (55). Further fundamental spectroscopic studies at far-IR (terahertz) frequencies will be required to further elucidate these low-energy scattering mechanisms in BP as a function of doping and temperature. Figure 2 (C and D) summarizes the fitted results without any offset to better understand the impact of doping on the lineshape of reflectance modulation. Excellent agreement between experimental data and transfer matrix simulations is visible in Fig. 2 (A and B), which indicates that the intraband (Drude) model suffices to explain the reflectance modulation observed at photon energies well below the band edge. Figure 2 (E and F) also shows in false colors the changes in the reflection modulation for AC and ZZ polarization as a function of electron/hole doping density assuming a constant effective mass for BP. Reflection/transmission in highly sub-wavelength BP films is mostly dominated by the losses in the material, and thus, it is important to note that we do not incorporate the interband region in our Drude modeling of the sub-bandgap response because we are working much below the (even the Stark shifted) bandgap, where the influence of interband losses is almost negligible to first order. Similar assumptions have been experimentally validated for studies on graphene (49, 56). Also, it should be noted that while the interband anisotropy is primarily governed by the parity of wave functions and subsequent selection rules coming from dipole matrix elements in BP, intraband anisotropy stems from the difference in fermionic effective mass along the two crystallographic axes.

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Measurement of the multilayer BP complex permittivity and tunable ENZ and hyperbolicity

Figure 3 (A to D) illustrates the experimental real and imaginary parts (denoted as ϵ₁ and ϵ₂) of the dielectric function [obtained as ϵAC/ZZ(ω)=ϵ∞(AC/ZZ)+iσAC/ZZ(ω)tϵ0ω,where t=thickness of the 2DEG (2.9 nm) and ϵ∞(AC/ZZ) accounts for oscillators not captured in our Drude spectral window] for BP at different doping densities under polarized excitation conditions along the AC and ZZ directions. At higher energies, the dielectric function is dominated by subband transitions whose oscillator strength diminishes upon doping primarily due to Pauli blocking, along with the aforementioned electro-optic effects. The lower-energy response is mostly dominated by free carriers. We observe a strong modulation of the dielectric function below the bandgap with doping density, indicating that free-carrier response is a significant effect in the mid-IR range. An important finding of our study is the appearance of an ENZ in BP for higher gate voltages/charge densities along the AC direction, where the real part of the permittivity transitions from positive to negative. A false-color plot showing the variation of the modeled real part of the dielectric permittivity with doping density along the AC direction is shown in Fig. 3E. The ENZ region is seen to systematically shift to higher photon energies with increased doping. Here, the effective mass of BP is assumed to be 0.14 m₀, independent of doping density, for the sake of simplicity. No such negative permittivity region was identified for the ZZ direction measurements, implying extreme bianisotropy and the possibility to generate surface plasmon modes and in-plane hyperbolic photonic dispersion in BP. We further calculate the isofrequency contours for in-plane plasmons (TM-polarized surface modes) (10) for two different doping densities (one on the electron side and one on the hole side) at 750 cm⁻¹ to show that the hyperbolic dispersion is electrically tunable. Electrically tunable hyperbolic dispersion, illustrated in Fig. 3F, has intriguing implications, suggesting opportunities for active hyperbolic plasmonics and photonics. We note that the doping density achieved here is modest (~7 × 10¹² cm⁻²), and higher doping densities with larger κ dielectrics may enable the hyperbolic dispersion regime to move to shorter wavelengths. In the frequency regime accessible in our measurements, we do not observe a negative real permittivity along the ZZ direction, and we expect it to occur at much lower frequencies (<500 cm⁻¹), which indicates that any surface plasmon modes at frequencies above 500 cm⁻¹ will inherit a hyperbolic dispersion (as shown in Fig. 3F), whereas those below will inherit an elliptical dispersion, thereby undergoing a topological transition in photonic dispersion. It should be possible to electrically tune the transition point, as indicated by our results. In addition, a Dirac-like dispersion [also known as a Dirac plasmonic point (57)] can be engineered in the system at slightly lower frequencies than the spectral window accessed in our measurements and has been discussed in section S14. Theoretical studies of surface plasmons in BP and their corresponding dispersion relations have been discussed elsewhere (8); however, our results provide a concrete step in that direction.

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Determination of carrier effective masses in a multilayer BP 2DEG

Last, we use our experimental results to obtain carrier effective masses that can be compared with results from theory as shown in Fig. 4 (A and B). We see qualitatively good agreement with theory (m_eff ≈ 0.14 m₀) for our results along the AC axis; our extracted fermionic effective mass is slightly heavier than the previously theoretically calculated results. We speculate that this could be an interplay of two effects. First, in our BP thin films, the 2DEG is highly confined, and thus, the band dispersion is modified, leading to heavier confined fermions (28). In addition, when the Fermi level moves into either the conduction or valence band with gating, we access not only the minima and maxima of the first subbands in the conduction and valence band, respectively, but also the higher subbands (because of the broad Fermi-Dirac tail at 300 K). In BP, for higher lying subbands along the AC axis, the effective mass increases gradually, as given by mACi=ℏ22γ2δi+η, where δⁱ is the subband transition energy, γ denotes the effective coupling between the conduction and valence bands, and η is related to the in-plane dispersion of the bands (47, 48). It is possible that the carriers participating in the intraband transitions come from a mixture of the different subbands, thus leading to an overall lower perceived effective mass. For the ZZ axis (m_eff ≈ 0.71 m₀), we see a slightly larger variation in the extracted effective mass between the electron and hole side. Electronic confinement leads to heavier fermions along the ZZ direction, which is in accordance with our optical measurements for hole conductivity. It is possible that non-parabolic band effects give rise to a slightly lower effective mass for electrons, but further detailed analysis is needed to resolve the observed electron-hole asymmetry. It should be noted that for the ZZ direction, the effective mass is not expected to depend on the subband index. As expected from theory, the ZZ carriers are found to be much heavier than carriers associated with transport along the AC direction. To verify all of our results, we performed similar measurements with an unpolarized beam on the same sample (presented in the Supplementary Materials). We see highly consistent and reproducible interband conductivity modulation effects for light polarized along the AC or ZZ axis. Below the band edge, we see a systematic strengthening of the Fano-like response for both electron and hole doping in the wavelength range near the hBN and SiO₂ phonons. Fits to those datasets are in excellent agreement with our theoretical analysis. In addition, we find that the effective mass obtained is approximately the average of the in-plane effective masses along the AC and ZZ directions. This indicates that carriers along both the axes participate in the intraband response and is consistent with BP transport measurements. Subtle deviations are expected to occur in the estimation of the effective mass because of the nonsymmetric nature of the unpolarized beam in our setup (it is slightly elliptically polarized) and possibly because of complex cross-scattering mechanisms between the two crystallographic directions. Overall, both polarized and unpolarized measurements are in good agreement with theory. We note that it is difficult to eliminate small conductivity changes due to local fluctuations in the charge density arising from charge puddles (bubbles/grain boundaries/defects) in heterostructures since our measurements probe a large area (approximately 2500 μm²) (58). In addition, studies have shown some systems to have an energy-dependent quasiparticle scattering rate owing to strong electron-electron and electron-phonon interactions (59, 60). Such interactions can cause additional broadening in both the interband and the intraband absorption features and have not been considered in our data analysis. We also note that in thick BP films (>5 nm) like those measured here, excitonic effects are negligible and hence have not been taken into account. We performed measurements on three other BP heterostructures and saw very similar behavior.

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