Massive Dirac quasiparticles in the optical absorbance of graphene, silicene, germanene, and tinene

It is well known that graphene is atomically flat, whereas silicene, germanene, and tinene should be buckled [27]. Using DFT-GGA for the ionic relaxation, we calculated the buckling height Δ and the in-plane nearest-neighbor distance d, which is related to the lattice constant a of the hexagonal lattice via $a=\sqrt{3}d$. The results are summarized in table 1. There are clear chemical trends. The characteristic lateral geometry parameters a and d and the buckling height Δ increase along the row C, Si, Ge, Sn. The increasing buckling amplitude indicates the deviation from the pure sp² bonding and the formation of mixed sp²–sp³ bonding. The sp³ character also increases along the row C, Si, Ge, Sn. Indeed the buckling amplitude Δ approaches more and more the value ${\Delta }_{{\mathrm{sp}}^{3}}=a/2\sqrt{6}$, 0.50 (C), 0.79 (Si), 0.82 (Ge), or 0.95 (Sn) Å, for complete sp³ bonding.

The band structures obtained from Kohn–Sham eigenvalues of the DFT are well known for graphene, silicene, and germanene [7, 9, 27]. For tinene the bands _ν(k) are plotted in figure 1. Within the DFT-GGA framework, tinene shows a (zero-gap) semiconducting character in contrast to the α-Sn crystallizing in diamond structure [28] with a L_6c band below the Fermi level. The apex of the valence band Dirac cone around K and K' points in tinene fixes the Fermi energy as a shortcoming of the method. Also with the HSE06 functional, we find a zero-gap semiconductor, but with an increased direct bandgap at the Γ point. For all 2D crystals under investigation, the hybrid functional HSE06 without spin–orbit coupling (SOC) leads to zero-gap semiconductors with isotropic Dirac cones at the K and K' points of the hexagonal BZ. Only the band dispersion is modified with respect to the GGA calculation. In table 1 we compare the Fermi velocities obtained with both functionals for the four 2D honeycomb crystals. The Fermi velocity decreases along the row C, Si, Ge, and Sn independent of the functional. The quasiparticle corrections are zero at the Fermi level [10] thus, in zero-gap semiconductors, quasiparticle corrections do not open a bandgap. Only the dispersion of the conduction and valence bands increases and, consequently, leads to an increase of the Fermi velocity when using HSE06 compared to DFT-GGA (see figure 1 for tinene). The Fermi velocity of freestanding graphene has been determined experimentally. Its value is about 1.1 × 10⁶ m s⁻¹ [29], and thus well predicted by the HSE06 functional (see table 1). The QP value v_F = 1.15 × 10⁶ m s⁻¹ using the GW approximation is slightly larger [9]. Larger values measured by means of angle-resolved photoemission spectroscopy (ARPES) for silicene [2] than those in table 1 are probably a consequence of the substrate influence. We conclude that the HSE functional simulates important features of the QP approximation. The energy gaps outside K or K' are significantly opened while the wavevector dispersion near K or K' is increased with respect to standard semilocal DFT calculations.

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We investigated the influence of SOC on the quasiparticle band structure, in particular close to the Dirac points. This requires much numerical effort. The convergence of the electronic bandgap induced by SOC as a function of the number of k points is very slow when using hybrid functionals. The regular 2D k-point meshes have to be carefully chosen since the convergence is strongly enhanced if the K and K' points are excluded from the regular k-point mesh [30]. Consequently, we applied a 32 × 32 × 1 Γ-centered Monkhorst–Pack k points mesh [31] for all materials. The HSE06 band structures of the four 2D honeycomb crystals along high-symmetry lines in the hexagonal BZ are plotted in figure 2. For germanene and tinene the inset also shows the linear dispersion around the K or K' point obtained without SOC in comparison to the dispersion with SOC. It can be seen that SOC induces a fundamental gap E_g and, hence, massive Dirac particles with effective masses m* close to the conduction and valence band edges near K (or K') points. We do not find a k-induced splitting of the twofold degeneracy due to SOC, i.e. away from Γ. At least along the high-symmetry lines we do not observe Rashba [32] or Dresselhaus [33, 34] components in the band structure, in agreement with other studies [17, 18]. At the Γ point a similar splitting of the fourfold degenerate uppermost valence bands into two twofold degenerate valence bands occurs as in the bulk. For graphene, the smallness of the SOC-induced E_g makes its precise HSE06 computation difficult. Since the size of the estimated bandgap E_g is about 0.02 meV, we only discuss the effect of SOC for the other group IV materials. The electrons ν =+  and holes ν =−  in silicene, germanene, and tinene in the vicinity of K or K' can be approximately described by a dispersion relation of massive relativistic Dirac particles [35]

$$\begin{eqnarray} &&{\epsilon }_{\pm }(\mathbf{k})=\pm \sqrt{\left (\frac{{E}_{\mathrm{g}}}{2}\right )^{2}+(\hbar {v}_{\mathrm{F}}\Delta \mathbf{k})^{2}} \end{eqnarray} \tag{ 1 }$$

with Δk as the deviation of the 2D wavevector from its value at a K or K' point. The SOC-induced gaps E_g are also listed in table 1. They follow the same clear chemical trend as the spin–orbit splittings Δ_SO in the diamond structures of the group IV materials mentioned above. The E_g values are, however, by one order of magnitude smaller. Thereby, the spin–orbit effects using non-local HSE06 XC potentials are significantly larger than those obtained within DFT-GGA as a consequence of the more localized quasiparticle states. Our findings in table 1 are in agreement with E_g values in other recent DFT computations. Values E_g = 1.55 (Si), 23.9 (Ge), and 73.5 (Sn) meV have been given in the literature [16, 35, 36]. The finite gap gives rise to massive Dirac (electron and hole) particles near a K or K' point with the effective mass

$$\begin{eqnarray} &&{m}^{\ast }=\frac{{E}_{\mathrm{g}}}{2{v}_{\mathrm{F}}^{2}}. \end{eqnarray} \tag{ 2 }$$

Its values in table 1 also show a clear increase with the atomic number of the group IV element.

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The absorbance of group IV honeycomb crystals exhibits the peculiarity that the optical absorption is related to the fine structure constant independent of the group IV atom, the sheet buckling, and the orbital hybridization in the limit of vanishing photon frequencies [6, 7]. On the other hand, one knows that semilocal DFT functionals rather accurately predict ground state properties, but fail to predict excited state properties such as the optical bandgap in non-metals and the correct frequency dependence of the dielectric function. However, we deal here with zero-gap semiconductors (at least without SOC). Since in zero-gap semiconductors with Dirac cones only the Fermi velocity changes due to quasiparticle effects, it should be possible to determine the optical absorbance in the low-frequency regime without applying more sophisticated approaches to the electron–electron interaction, such as the many-body perturbation theory within the GW approximation, or the Bethe–Salpeter equation to account for excitonic effects.

In contrast, the increasing importance of SOC on the electronic properties for heavy group IV atoms and the deformation of the Dirac cones near the K and K' points ask for a more detailed investigation of the impact of the SOC on the optical properties, therefore, we studied the optical absorbance A(ω). This is connected to the frequency-dependent imaginary part of the dielectric function ε(ω) of the 3D supercell with lattice constant L by [7]

$$\begin{eqnarray} &&A(\omega )=\frac{\omega }{c}L \hspace{0.167em} \mathrm{Im}\hspace{0.167em} \varepsilon (\omega ). \end{eqnarray} \tag{ 3 }$$

The optical absorbance resulting from the independent-quasiparticle computations for all four group IV honeycomb crystals as a function of the photon energy is shown in figure 3. Without SOC the low-frequency absorbance approaches the well-known limit A(ω → 0) = πα as demonstrated earlier for independent Kohn–Sham electrons [7]. However, using the HSE approach, the optical absorption peaks are blueshifted due to quasiparticle corrections. In a wide frequency range the optical absorbance taking into account SOC shows a very good overall agreement with the absorbance obtained without SOC. However, a more detailed analysis reveals a remarkable modification of the optical absorbance close to the fundamental absorption edge. For ħω < E_g the absorption vanishes, while for ħω > E_g the absorbance is significantly increased for ħω → E_g. This behavior is particularly pronounced for germanene (figure 3(c)) and tinene (figure 3(e)). Within our numerical accuracy we estimate that the absorbance is increased by a factor of two at ħω = E_g due to SOC.

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This peculiar lineshape of the absorbance in the vicinity of the fundamental absorption edge asks for a detailed explanation. In general, it is a consequence of the relativistic band dispersions (1) and the optical matrix elements (see equation (7) in [7]) modified due to the inclusion of SOC when calculating the imaginary part of the dielectric function. The modified band dispersion is illustrated in figure 2. The strength of the corresponding interband transitions is displayed in figure 4. As an example, we study the optical matrix elements between the highest occupied and the lowest unoccupied Bloch bands of germanene along the two high-symmetry directions KΓ and KM. The optical transition matrix elements resulting within the longitudinal approximation for Pauli spinors and germanene near a K or K' point are displayed in figure 4 for in-plane light polarization.

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Without SOC, the squares of the momentum-matrix elements D(k) = ∑_j=x,y|〈ck|p_j|vk〉|² reduce to ${m}^{2}{v}_{\mathrm{F}}^{2}$ at a Dirac point ${\mathbf{k}}_{0}~\hat {=}~\mathrm{K},{\mathrm{K}}^{\prime}$ with a weak, almost linear, Δk dependence, whereas the matrix elements D(k) including SOC are significantly enhanced by a factor of two near ${\mathbf{k}}_{0}~\hat {=}~\mathrm{K},{\mathrm{K}}^{\prime}$ with a pronounced k dependence. Further away from the Dirac point the matrix elements with and without SOC are practically identical.

With SOC the matrix elements display an additional Lorentzian Δk dependence near a Dirac point. This can be represented analytically by

$$\begin{eqnarray} D({\mathbf{k}}_{0}+\Delta \mathbf{k})&=&{m}^{2}{v}_{\mathrm{ F}}^{2}\left (1+\frac{1}{1+b \Delta {k}^{2}}\right ),\nonumber\\ b&=&\frac{2{\hbar }^{2}}{{E}_{\mathrm{g}}{m}^{\ast }}=\left (\frac{2\hbar {v}_{\mathrm{F}}}{{E}_{\mathrm{g}}}\right )^{2}. \end{eqnarray} \tag{ 4 }$$

The quantity b is determined by exhausting the oscillator-strength sum rule on the dielectric function in the low-frequency regime. We have to stress that the agreement of equation (4) with the numerical curve in figure 4 is excellent. The same holds for silicene and tinene (not shown). Including the corrected matrix elements (4) the optical absorbance is readily calculated by means of the analytical method described in a previous work [7]. Taking into account the modified band dispersion equation (1) due to the massive Dirac particles in the presence of SOC we derive

$$\begin{eqnarray} &&A(\omega )=\pi \alpha \left [1+\left (\frac{{E}_{\mathrm{g}}}{\hbar \omega }\right )^{2}\right ]\Theta (\hbar \omega -{E}_{\mathrm{ g}}). \end{eqnarray} \tag{ 5 }$$

Since the joint density of states near the Dirac points ∼ħωΘ(ħω − E_g) is still linear in the photon energy, the strong modification of the absorbance is mainly a consequence of the modified transition matrix elements (4). The result recovers important features encountered in the numerical calculations. First of all, for photon energies that are large (but much below the optical interband transitions near critical points in the band structure in figures 1 and 2) compared to the electronic bandgap, the absorbance reduces to the well-known limit A(ω ≫ E_g/ħ) = πα. Furthermore, for photon energies in the vicinity of the fundamental absorption edge the absorbance increases to A(ω ≈ E_g/ħ) = 2πα. In between these two limits the absorbance follows the power law as predicted by equation (5). For vanishing SOC, i.e. E_g → 0, the previous analytical description is recovered.

The previous findings, in particular equations (4) and (5), also hold for GGA calculations, if the proper band gaps E_g and Fermi velocities v_F are taken from table 1.