A First-Principles Exploration - ACS Publications - American Chemical

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Cite This: J. Am. Chem. Soc. 2017, 139, 17181-17185

Two-Dimensional Boron Polymorphs for Visible Range Plasmonics: A First-Principles Exploration Yuefei Huang,‡ Sharmila N. Shirodkar,‡ and Boris I. Yakobson* Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: Recently discovered two-dimensional (2D) boron polymorphs, collectively tagged borophene, are all metallic with high free charge carrier concentration, pointing toward the possibility of supporting plasmons. Ab initio linear response computations of the dielectric function allow one to calculate the plasmon frequencies (ω) in the selected example structures of boron layers. The results show that the electrons in these sheets indeed mimic a 2D electron gas, and their plasmon dispersion in the small wavevector (q) limit accurately follows the signature dependence ω ∝ √q. The plasmon frequencies that are not damped by single-particle excitations do reach the nearinfrared and even visible regions, making borophene the first material with 2D plasmons at such high frequencies, notably with no necessity for doping. The existence of several phases (polymorphs), with varying degree of metallicity and anisotropy, can further permit the fine-tuning of plasmon behaviors in borophene, potentially a tantalizing material with utility in nanoplasmonics.

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INTRODUCTION Plasmons as collective excitations of electrons are an important ingredient of a material’s optical properties. Plasmons restricted to two dimensions only, in surfaces or layers, exhibit ultrasubwavelength confinement with phase velocities reduced up to ∼100 times relative to the speed of light1 and enhanced light− matter interaction.2 This plasmon restriction to the nanometer scale has been explored for a wide range of applications of twodimensional (2D) materials, such as enhanced sensing of biomolecules,3,4 nanoimaging,5 waveguides,6 tunable lasing,7 light harvesting,8 and nanoantennas.9,10 Not only does reduction in dimensionality change the plasmon confinement, but it also qualitatively changes the plasmon dispersion, to ∼√q (q is plasmon wavevector, q = |q|) as opposed to nondispersive plasmons in 3D.11 This trait of 2D electron gas12 has been observed in several 2D materials, viz., graphene,13−16 black phosphorus,17 and transition metal dichalcogenides (TMDCs).18 The plasmon frequency in these materials scales as √n with their free carrier concentration n (with the exception of graphene’s ∼n1/4 dependence,14,16 due to its linear dispersion at the Dirac point), implying that plasmon frequencies are tunable by the carrier concentration. All these materials are either semiconducting or metallic with low n, requiring a gate- or chemical-doping to increase the carrier concentration to tune the plasmonic response. Doping via gating has been reported to induce carriers to densities of ∼1013 cm−2 (1017 m−2) in graphene15,19 and black phosphorus,17 reaching plasmonic frequencies in the mid infrared. The plasmonic response of metallic TaSe2 compound18 is also restricted to the near-infrared regime (NIR), with carrier concentrations of ∼1 × 1015 cm−2 (1 × 1019 m−2). To date, © 2017 American Chemical Society

there has only been one report of 2D plasmons in the visible frequency range20 at a very high level of doping in a Ligraphene intercalation compound. Boron is the neighbor of carbon in the periodic table and has shown similarity to carbon in forming planar clusters,21−23 cage-like fullerenes,24−26 and 1D nanotubes.27−29 The boron analogue of graphene,30 2D boron (borophene), has been recently explored, and its properties have been probed theoretically as well.31−33 First-principles studies predict 2D boron polymorphism with large number of possible low-energy structures.34 These polymorphs differ from their parent structure, the buckled triangular boron, via concentration of vacant B sites. This rich polymorphism in 2D boron has been confirmed by experimental synthesis of triangular boron by Mannix et al.35 and two other polymorphs, β12 and χ3, by Feng et al.36,37 on the Ag(111) surface by molecular beam epitaxy. Unlike β12, free-standing triangular boron and χ3 have been predicted to be dynamically unstable.38 However, the adhesion that is accompanied by charge transfer39 on the Ag(111) surface makes them thermodynamically stable and experimentally synthesizable, with a stability order of χ3 > β12 > triangular boron.36 We would like to note that although triangular boron is metastable/energetically higher than the other polymorphs, it can be synthesized under specific conditions.35,36 It has been shown that 2D boron is a metal,35−37,40 in contrast to its 3D crystal, being a semiconductor. It is one of the very few 2D materials that are intrinsically metallic/ semimetallic such as the 1T polymorphs of MX2 comReceived: September 27, 2017 Published: November 1, 2017 17181

DOI: 10.1021/jacs.7b10329 J. Am. Chem. Soc. 2017, 139, 17181−17185

Article

Journal of the American Chemical Society

calculations50 as implemented in the GPAW code, the microscopic dielectric function ϵG,G′(ω, q) is determined in the Fourier space as a function of wavevector q and frequency ω of the incident wave, for the reciprocal lattice vectors G and G′. It is calculated as ϵRPA GG′ (q, ω) = δGG′ − (4π/|q + G|2)χ0GG′(q, ω), under the random phase approximation (RPA), where χ0GG′(q, ω) is the Fourier transform of the noninteracting density response function χ0GG′(r, r′, ω). We use a planewave cutoff of 100 eV for reciprocal lattice vectors. All the other parameters were converged within 0.1 eV of plasmon frequencies calculated using methodology developed by Anderson et al.51 At the frequency of a self-sustained charge-density oscillationa plasmonthe real part of the dielectric function must go to zero from a negative value, Re[ϵ(ω = ωp)] = 0. The plasmon resonance, at frequency ωp, can also be identified as the local maximum/peak appearing in the loss function, −Im[ϵ(ω, q)−1], while the area under the loss function peak represents the strength of the plasmon mode.17,18 The ωp determined in our materials appears independent of the two methods described above, since Im[ϵ] is negligibly small at the zeros of Re[ϵ]. An important measure of a plasmon in a material is the wave localization or shrinkage, quantified as λair/λp (λair is the electromagnetic wavelength at frequency ωp in a vacuum, and λp = 2π/q). This estimates the confinement of the wavelength of electromagnetic radiation in a plasmon. A plasmon is damped by single-particle excitations (SPEs), i.e., formation of electron−hole pairs. These SPEs are determined as poles (peaks) in the real (imaginary) part of the response function χ, and the threshold non-negligible transition momentum is chosen as >0.1 au The damping regions due to SPEs occurring at ωSPE frequencies can be approximated as bounded by ℏωSPE ± ℏνFq + ℏ2q2/2m, where νF is the Fermi velocity14,52 (as q → 0, ℏ2q2/2m is negligible as compared with ℏνFq).

pounds.18,41−43 Moreover, 2D boron has a much higher density of electronic states at the Fermi level,38 compared to 1T- MX2 compounds,44,45 eliminating the need to dope or gate it to increase charge carrier density. Interestingly, the 2D boron form has been theoretically38 as well as experimentally shown to exhibit Dirac cones37 and has also been predicted to possibly become superconducting around ∼10−20 K.38 The existence of several polymorphs, structural anisotropy, and inherent metallic conductivity of 2D boron motivate us to explore if plasmons are feasible, especially in the visible range of frequencies. Using first-principles calculations we explore the plasmonic properties of three example polymorphs of 2D boron. Our calculations show that the free charge density in 2D boron behaves like a 2D gas of electrons and exhibits anisotropic plasmons with √q dispersion at small q and long wavelength. Most importantly, we find that at larger q values plasmons in 2D boron do reach the infrared to visible range frequencies, depending on the structure of the specific polymorph.

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COMPUTATIONAL DETAILS

We carry out density functional theory (DFT) calculations as implemented in the GPAW electronic structure code.46 The interaction between the ions and electrons is described by the projector augmented wave method,47 and the exchange−correlation energy of electrons is approximated by the Perdew−Burke−Ernzerhof (PBE) functional.48 The 2D layers are modeled in periodic cells with 10 Å vacuum in the direction normal to the layer, to reduce the interaction between periodic images (see Supporting Information for details). The Kohn−Sham wave functions are represented using a plane wave basis set with an energy cutoff of 340 eV. The 2D Brillouin zone (BZ) is sampled using 256 × 128 × 1 k-points for B△ and 256 × 256 × 1 k-points for B□ and for B◇ (Figure 1a−c). At the nanometer length scale, the local electromagnetic field effects are non-negligible, affecting the material response considerably, which is not captured by the Drude model. A full quantum mechanical description of the dielectric function, i.e., microscopic dielectric function, accounting for the local field effects, is necessary to accurately describe plasmons in 2D materials. 49 Using linear response

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RESULTS AND DISCUSSION A 2D boron sheet exhibits various polymorphs, conveniently described as a pseudoalloy, B1−xVx for x = 0.1−0.15, where V is a vacant site in the underlying triangular B lattice.34 Among the large number of such polymorphs, we select only examples that are energetically preferable or have fewer atoms per unit cell, for computational feasibility. Hence, we explore the same three structures of single-layer 2D boron, B△, B□ (known as β12), and B◇ (known as χ3) (Figure 1), which were studied by Penev et al.38 Even if only B□ is stable in free-standing form, the other two polymorphs are stabilized by the Ag(111) substrate35,36 and hence is instructive to study the properties of all three polymorphs, as they provide crucial insights into the potential applicability of 2D boron in nanoplasmonics. B△ (x = 0) is a buckled triangular lattice (Figure 1a) with GGA-optimized lattice constants a = 1.62 Å and b = 2.85 Å and atomic coordination of 6. The distance between the two sublayers (degree of buckling) is 0.86 Å, greater than the buckling in 2D silicene (0.44 Å) and germanene (0.64 Å).53 Its electronic band structure is shown in Figure 1d, with the smallest allowed SPE marked by the green arrow. In comparison to the band structure of VS2 or TaSe218 where only one band crosses the Fermi level, we find that B shows many more crossings by highly dispersed (almost parabolic) bands.38 This suggests that 2D boron has a larger free charge carrier concentration as compared to metallic MX2 compounds. Removing some B atoms, up to x = 1/6 (B□ in Figure 1b, with maximum atomic coordination of 6), turns the lattice planar and its primitive cell larger, with lattice constants a = 2.92 Å and b = 5.08 Å (GGA values). This is known as the most stable structure, free-standing34 as well as grown on a silver substrate.35,37 We note that the band structure (Figure 1e) has several band crossings at the Fermi level. Interestingly, the Dirac cone appears at ∼2 eV above the Fermi level along the

Figure 1. Atomic structures (top view) (a)−(c) and electronic band structures (d)−(f) of B△, B□, and B◇, respectively. At the bottom of (a) is the side view of buckled B△. The red arrows in the BZ represent the direction of q of the plasmon modes in our calculations. The green vertical arrows in the band structures show the minimum energy SPEs (with non-negligible threshold transition momentum >0.1 au) that damp plasmons. 17182

DOI: 10.1021/jacs.7b10329 J. Am. Chem. Soc. 2017, 139, 17181−17185

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Journal of the American Chemical Society

Figure 2. Plasmon dispersion of (a) B△, (b) B□, and (c) B◇. The gray areas are the regions damped by single-particle excitations (SPEs). Blue (circles) and red (squares) curves are for plasmon dispersion (i.e., q) along the Γ−X and Γ−Y directions, respectively. Fitting to classical 2D electron gas plasmon dispersion relation 2 is shown by solid lines. The strength of the plasmon mode is directly proportional to the thickness of the circles/ squares. The SPEs are at 3.4, 1.7, and 0.95 eV for B△, B□, and B◇, respectively.

Figure 3. Wave shrinkage (a−c) of B△, B□, and B◇, respectively. The shaded regions correspond to areas that are damped by interband and intraband (SPEs) transitions. Blue (circles) and red (squares) curves are for plasmon dispersion (i.e., q) along the Γ−X and Γ−Y directions. Blueand red-shaded regions correspond to damping regions for plasmons along the Γ−X and Γ−Y directions (estimated as the lowest frequency at which plasmon dispersion cuts the damping region in Figure 2), respectively.

Γ−X direction in the BZ (while free-standing), which has been reported to get filled by electrons donated by the Ag substrate.37 The last polymorph, B◇, which corresponds to x = 1/5 (Figure 1c), is a planar structure with a maximum atomic coordination of 5. The primitive cell, however, is nonorthogonal, with lattice constants a = b = 4.55 Å and an angle of 141.7° between them. Similar to the above-discussed polymorphs, the band structure of B◇ (Figure 1f) shows highly dispersed bands crossing the Fermi level. We calculated the plasmon dispersion along the high-symmetry lines denoted by red arrows in the BZ in Figure 1d−f, i.e., the plasmon wavevector q parallel to these directions. The plasmon dispersion curves for B△, B□, and B◇ are shown in Figure 2. We note that the plasmons differ along different directions of the BZ; this anisotropy can be traced back to the anisotropy in the electronic structure and of the lattice. For all three structures, the dispersion looks similar to that of 2D electron gas,54 ωp =

σe 2q 2m * ϵ 0

electron mass): for B△ m* = 1.4m0 and 5.2m0, and for B□ m* = 3.5m0 and 3.7m0 along the Γ−X and Γ−Y directions, respectively, whereas for B◇ m* = 1.4m0 and 3.8m0 along the Γ−X and Γ−Y directions, respectively. As we see, the plasmon’s anisotropy in each structure is also reflected in the effective mass differences of the carriers, similar to anisotropic plasmons in black phosphorus.17 We note that the deviation of the computed plasmon dispersion from that of a 2D electron gas at larger q values is related to nonparabolicity of bands and screening by interband transitions,18 although distinguishing between these different effects is nontrivial. As mentioned before, plasmons are damped by SPEs, which are both intraband and interband (also known as Landau damping). Damping regions in Figure 2 are determined by the methods discussed above. The poles in the response function (SPEs) were identified at 3.4, 1.7, and 0.95 eV for B△, B□, and B◇, respectively. These symmetry-allowed interband excitations between the occupied and unoccupied states, shown by green arrows in the band structure in Figure 1d−f, damp the plasmons in borophene. To map the damping regions, the Fermi velocity (νF) on the Fermi surface (in Γ−X and Γ−Y) of all three phases was calculated, and the maximum value of νF was chosen. Our estimates of maximum νF in B△, B□, and B◇ are 1.4 × 106, 0.9 × 106, and 1.3 × 106 m/s in Γ−X and 0.95 × 106, 0.86 × 106, and 0.95 × 106 m/s in Γ−Y, respectively. Due to damping, only the plasmons with frequencies up to 3.0 eV and q < 0.05 Å−1 along Γ−X and up to 2.3 eV for q < 0.21 Å−1 along Γ−Y survive in B△. Notably, this appears to be the first material with 2D plasmons in the visible range with no need of doping. In B□ the plasmons are undamped until 1.5 eV (q < 0.03 Å−1) along Γ−X and Γ−Y and 0.9 eV (q < 0.01 Å−1) for B◇ along both Γ−X and Γ−Y directions. Note that the jump in the

, where σ is the number of charge carriers

per unit area and m* is the effective mass of the carriers. The outer electronic configuration of B is 2s2, 2p1, and assuming one free charge carrier from the p orbital per B atom, the free carrier density per unit area for B△, B□, and B◇ is 4.3 × 1015 cm−2 (4.3 × 1019 m−2), 3.4 × 1015 cm−2 (3.4 × 1019 m−2), and 3.3 × 1015 cm−2 (3.3 × 1019 m−2), respectively, all much higher than any of the free carrier densities previously reported for doped or undoped 2D materials. Since distinguishing the electronic excitations that contribute to the plasmonic resonance is nontrivial (due to the large number of band crossings), assigning the m* values for those particular bands may also be ambiguous. Instead, we proceed by fitting an expression for 2D electron gas (solid lines in Figure 2) to our computed plasmon dispersion data and evaluating m* as a fitting parameter (m0 is 17183

DOI: 10.1021/jacs.7b10329 J. Am. Chem. Soc. 2017, 139, 17181−17185

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Journal of the American Chemical Society plasmon dispersion around q = 0.05 Å−1 along Γ−X in B◇ is probably due to the coupling to another plasmon mode. Hence, we find that the plasmon response of the experimentally synthesized35,37 polymorph of 2D boron B□ can reach the NIR range. We also notice that the SPE energies decrease with a larger number of vacant B-sites in the 2D boron “alloy”, perhaps because the number of states and symmetry-allowed transitions across the Fermi level increase, thus increasing the chance of having SPE at lower energy. Accordingly, increasing the B-vacancy concentration (x) tends to lower the frequency range of undamped plasmons. Although the main loss mechanism is Landau damping due to SPE discussed above, one should keep in mind other everpresent losses caused by defects and electron−electron and electron−phonon scattering, which introduce finite relaxation time, τ. Its value, not available for boron, can be, for the sake of estimates, based on graphene conductivity measurements. In the Supporting Information we present propagation length, commonly used to quantify the overall losses. The maximum wave shrinkage λair/λp (Figure 3a−c) is ∼30 for B△ and B□ and ∼15 for B◇ (by extrapolation) for undamped plasmons determined at the border of damped and undamped regions for Γ−X, whereas the maximum subwavelength confinement (wave shrinkage) for Γ−Y is 150, 50, and 25 for B△, B□, and B◇, respectively. We note that these wave shrinkage values are larger for Γ−Y than for Γ−X; that is, the plasmonic properties improve for plasmons with smaller frequencies or larger effective masses. Large shrinkage finds applications in subdiffraction-limited imaging and drastic miniaturization of plasmonic devices. In comparison, graphene exhibits a maximum wave shrinkage of 100−20055 for doping of 1012 cm−2 (1016 m−2, EF = 0.135 eV), which is considerably larger than in 2D boron, since the plasmon in graphene is also at a lower frequency. However, the wave shrinkage in 2D boron could be increased by application of a gate,1 improving its performance in nanoplasmonics. As already mentioned, the plasmons in 2D semiconducting materials, graphene and black phosphorus, appear in the mid/ near-infrared frequencies and only with doping. Black phosphorus exhibits plasmons up to 150 meV if doped to n = 5 × 1013 cm−2 (5 × 1017 m−2),17 while the experimental doping level is n ≈ 5.5 × 1012 cm−2 (5.5 × 1016 m−2),56 implying that plasmons must actually be below 150 meV. Graphene experimentally shows undamped plasmons up to its Fermi level,13,14 typically 100 meV.15 A recent theoretical study by Shirodkar et al.20 showed undamped plasmons in the visible frequencies, for multilayered graphene intercalated with lithium, at large doping concentrations. On the other hand, one metallic 2D TMDC TaSe218 exhibits plasmons in the NIR regime, due to smaller charge carrier concentration (≈1 × 1015 cm−2, 1019 m−2). We note that in all these instances of 2D plasmons (except for TaSe2) the material is doped in order to increase the carrier concentration. Chemical doping increases disorder and gate doping may cause leakages in devices, which can alter the plasmonic properties of the material. In contrast, 2D boron is intrinsically metallic with a higher concentration of carriers up to 4 × 1015 cm−2 (4 × 1019 m−2), enabling the plasmons in the visible range for B△ and NIR regions for B□ and B◇, respectively.

directions for three polymorphs of 2D boron, B△, B□, and B◇. We find that 2D boron has a strong potential to exhibit plasmons in the visible and NIR range of frequencies with no need of doping. Further, the plasmons display strong anisotropy, a result of difference in effective mass of the charge carriers in the different directions of BZ, hence indicating that the frequency of plasmons is a function of the polarization of exciting radiation. We also note that the plasmons in 2D boron do not show a subwavelength confinement as high as that in graphene. Recent experiments have reported substrate doping where silver (substrate) transfers electrons to the 2D boron; in such a case, due to increased free charge carrier concentration, one expects that the plasmon frequencies in 2D boron may further increase. As for the special case of B□, if the Fermi level shifts by ∼2.0 eV up in energy, which fills the previously unoccupied linearly dispersive bands, the Dirac point appears 0.25 eV below the Fermi level;37 this way, B□ may exhibit graphene-like plasmons in the mid infrared range of frequencies, with larger subwavelength confinement (wave shrinkage) as compared to the undoped structure. Our results show that the metastable B△ polymorph exhibits visible frequency plasmons, whereas the other two reach the NIR. Since our findings for 2D boron look promising, it is very likely that other polymorphs may support plasmons in the visible range. Overall, with such a high frequency range for plasmonic response with essentially no doping, 2D boron emerges as a strong candidate for exploration and possibly applications in photonic and plasmonic devices. One should of course appreciate that to realize this tantalizing potential, borophene should be transferred or grown on insulating substrates, which is a subject of focused effort in experimental laboratories.57

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jacs.7b10329. Cohesive energies and optimized coordinates of B△, B□, and B◇; convergence of plasmon dispersion with respect to vacuum length; variation in plasmon propagation vs relaxation time (PDF)

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AUTHOR INFORMATION

Corresponding Author

*[email protected] ORCID

Sharmila N. Shirodkar: 0000-0002-9040-5858 Author Contributions ‡

Y. Huang and S. N. Shirodkar contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Army Research Office Grant W911NF-16-1-0255 and by the Robert Welch Foundation (C1590). We thank Alex Kutana and Henry Yu for helpful discussions. We used computational resources on the DAVinCI cluster at Rice University.

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CONCLUSIONS In conclusion, using first-principles calculations we have determined the plasmon dispersion along the high symmetry

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DOI: 10.1021/jacs.7b10329 J. Am. Chem. Soc. 2017, 139, 17181−17185