Two-Dimensional Massless Dirac Fermions in Antiferromagnetic $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ $(A=\mathrm{Ba},\mathrm{Sr})$

We report infrared studies of $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ ($A=\mathrm{Ba}$, Sr), two representative parent compounds of iron-arsenide superconductors, at magnetic fields ($B$) up to 17.5 T. Optical transitions between Landau levels (LLs) were observed in the antiferromagnetic states of these two parent compounds. Our observation of a $\sqrt{B}$ dependence of the LL transition energies, the zero-energy intercepts at $B=0\mathrm{T}$ under the linear extrapolations of the transition energies and the energy ratio ($\sim 2.4$) between the observed LL transitions, combined with the linear band dispersions in two-dimensional (2D) momentum space obtained by theoretical calculations, demonstrates the existence of massless Dirac fermions in the antiferromagnet ${\mathrm{BaFe}}_2{\mathrm{As}}_2$. More importantly, the observed dominance of the zeroth-LL-related absorption features and the calculated bands with extremely weak dispersions along the momentum direction $k_z$ indicate that massless Dirac fermions in ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ are 2D. Furthermore, we find that the total substitution of the barium atoms in ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ by strontium atoms not only maintains 2D massless Dirac fermions in this system, but also enhances their Fermi velocity, which supports that the Dirac points in iron-arsenide parent compounds are topologically protected.

Figure 1

Magneto-optical response of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$. (a) False-color map of the $R(B)/R(B_0=0\mathrm{T})$ spectra. See the false-color map plotted as a function of $\sqrt{B}$ and energy in Fig. S4 of the Supplemental Material [43]. (b) Several representative $R(B)/R(B_0=0\mathrm{T})$ spectra and their fitting results (gray curves). The $R(B)/R(B_0=0\mathrm{T})$ spectra up to 120 meV are shown in Fig. S5 of the Supplemental Material [43]. (c) Upper panel: the reflectance spectra at $B=0$ and $B=17.5\mathrm{T}$, and their fitting results. Lower panel: the real part of ${\sigma }_{xx}(B,\omega )$ at $B=17.5\mathrm{T}$ and $\sigma (0\mathrm{T},\omega )$. The two black arrows in $\mathrm{Re}[{\sigma }_{xx}(B=17.5\mathrm{T},\omega )]$ indicate two additional peaks, $T_1$ and $T_2$. (d) Relative real part of the optical conductivity. The blue and red dashed lines are guides to the eye. Except $T_1$ and $T_2$, another peak emerges around 31 meV in (b) and (d) at $B>13\mathrm{T}$ (see its possible origin in the Supplemental Material [43], Sec. II, which includes Refs. [51, 52, 53]). The spectra in (b) and (d) are displaced from one another by 0.09 and 240 (${\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$) for clarity.

Figure 2

Landau level transitions and band dispersions of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ in the AFM state. (a) Observed LL transitions, $T_1$ and $T_2$, in an $E-\sqrt{B}$ plot. The inset shows the energy ratio of $T_1$ and $T_2$ as a function of $B$. The purple dashed line shows the theoretical energy ratio based on the MDF model. (b)–(d) Calculated band dispersions along $k_x$, $k_y$, and $k_z$ near $E_F$. The flat band in (d) represents the dispersion of MDF along $k_z$. In the AFM Brillouin zone of $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ ($A=\mathrm{Ba}$, Sr), $-\pi /a\le k_x\le \pi /a$, $-\pi /b\le k_y\le \pi /b$, and $-2\pi /c\le k_z\le 2\pi /c$, where $a$, $b$, and $c$ are the lattice constants. (e) Schematic of the LL transitions of 2D MDF in ${\mathrm{BaFe}}_2{\mathrm{As}}_2$. The green lines show the magnetic field dependence of the assumed $E_F$. (f) Left panel: LLs of 3D MDF. Right panel: DOS of 3D MDF. (g) Calculated relative real part of the optical conductivity of 3D MDF based on Eq. (15) of Ref. [61]. (h) $\mathrm{Re}[{\sigma }_{xx}(B)]-\mathrm{Re}[\sigma (0\mathrm{T})]$ of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ at $B=7\mathrm{T}$. (i) Left panel: LLs of 2D MDF. Right panel: Lorentz-shaped DOS of 2D MDF. (j) Calculated $\mathrm{Re}[{\sigma }_{xx}(B)]-\mathrm{Re}[\sigma (0\mathrm{T})]$ of 2D MDF based on Eq. (23) of Ref. [62]. When calculating the spectra in (g) and (j), we replaced the Kronecker delta functions in Eq. (15) of Ref. [61] and Eq. (23) of Ref. [62] by the Lorentzian functions with nonzero scattering rates.

Figure 3

Magneto-optical response and band dispersions of ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ in the AFM state. (a)–(c) Energy-momentum dispersions of MDF in ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ along $k_x$, $k_y$, and $k_z$. The flat band in (c) shows the dispersion of MDF along $k_z$. (d) False-color map of the $R(B)/R(B_0=0\mathrm{T})$ spectra of ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ as a function of $B$ and $E$. (e) Relative reflectance (upper panel) and relative optical conductivity (lower panel) spectra of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ at $B=17.5\mathrm{T}$. The inset shows the $\sqrt{B}$ dependence of the observed $T_1$ transition energy. The spectra of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ in (e) are displaced from those of ${\mathrm{SrFe}}_2{\mathrm{As}}_2$ by 0.023 and $412({\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1})$ for clarity.