Impact of the two Fe unit cell on the electronic structure measured by ARPES in iron pnictides

In all iron pnictides, the positions of the ligand alternatively above and below the Fe plane create 2 inequivalent Fe sites. This results in 10 Fe $3d$ bands in the electronic structure. However, they do not all have the same status for an ARPES experiment. There are interference effects between the 2 Fe that modulate strongly the intensity of the bands and that can even switch their parity. We give a simple description of these effects, notably showing that ARPES polarization selection rules in these systems can not be applied by reference to a single Fe ion. We show that ARPES data for the electron pockets in Ba(Fe${}_{0.92}$Co${}_{0.08}$)${}_2$As${}_2$ are in excellent agreement with this model as well as with direct calculation of the spectral weight. We observe both the total suppression of some bands and the parity switching of some other bands. Once these effects are properly taken into account, the structure of the electron pockets, as measured by ARPES, becomes very clear and simple. By combining ARPES measurements in different experimental configurations, we clearly isolate each band forming one of the electron pockets. We identify a deep electron band along one ellipse axis with the $d_{xy}$ orbital and a shallow electron band along the perpendicular axis with the $d_{xz}$/$d_{yz}$ orbitals, in good agreement with band-structure calculations. We show that the electron pockets are warped as a function of $k_z$ as expected theoretically, but that they are much smaller than predicted by the calculation.

Figure 1

(a) Structure of one FeAs slab. The thick black line represents the 2D unit cell with inequivalent Fe at the center (A) and the corners (B) because of the position of the neighboring As. The dotted black line defines the basic Fe square. (b) Tetrahedral As environment of one Fe. (c) 2D Fermi surface expected in the 2 Fe BZ built on the unit cell. The color code for different orbital characters. The pockets at $\Gamma$ and $M$ are hole pockets, those at $X$ are electron pockets. The dotted square is the 1 Fe BZ built on the basic Fe square. (d) FS in the 1 Fe BZ. The additional pockets in the 2 Fe BZ can be obtained by folding with respect to the 2 Fe BZ boundaries or, equivalently, by translation by $q_F$. We follow here Ref. 13 for the definition of main and folded pockets; other authors reverse the $\Gamma$ and $M$ points (Ref. 14). (e) Sketch of the expected spectral distribution, following the analysis of Sec. 2.

Figure 2

(a) Sketch of the three $t_{2g}$ orbitals $d_{xy}$, $d_{xz}$, and $d_{yz}$. (b)–(e) Relative orientations of orbitals in real space for the following orbitals and points of the reciprocal space. (b) $d_{xy}$ at $\Gamma$, (c) $d_{xy}$* at $X(\pi /a,0)$, (d) $d_{xy}$* at $\Gamma$, (e) $d_{xz}$* at $X$. The blue line in (b) indicates the $xz$ mirror plane.

Figure 3

(a) Sketch of the bands forming the electron pockets. Thick and thin lines indicate main and folded bands in the sense of the 1 Fe BZ with glide-mirror symmetry. Solid and dotted lines represent the in-phase and out-of-phase orbitals. Colors indicate the main orbital character. (b) Sketch of the dispersion of the different bands along the path $\Gamma XM$. The color sketches the main orbital character, although there may be hybridization with other orbitals. (c) Symmetries of the different bands forming the electron pocket at $X$.

Figure 4

(a) Sketch of the Fermi surface expected at $k_z=1$. Thick and thin lines indicate main and folded bands in the sense of the 1 Fe BZ with glide-mirror symmetry. Solid and dotted lines represent the in-phase and out-of-phase orbitals. Only in-phase orbitals are expected to have sizable intensity in ARPES (see Sec. 2). (b) FS measured at $T=25$ K, $\hslash \omega =34$ eV, and polarization along $k_y$, except for the hole bands at $Z$, where it is in the $xz$ plane. (c) 3D BZ for BaFe${}_2$As${}_2$, indicating the $q$ wave vector for folding (Ref. 14). (d) Geometry of the experiment. The plane of light incidence is $xz$ and the analyzer slits are fixed along $k_y$. (e) FS crossings around extracted from $\alpha$ (green) and $\beta$ (blue). (f) Unfolded Fermi surfaces at $k_z=1$ in 1 Fe BZ with $d_{yz}$ spectral weight. (g) Same as (f) with all Fe $3d$ orbitals.

Figure 5

(a)–(c) Dispersion observed and calculated along the 3 cuts a, b, c, indicated in Fig. 4e. The first row corresponds to panel $\beta$ rotated by 90${}^{\circ }$, i.e., allowed bands are even/$xz$; the second row to $\alpha$, i.e., allowed bands are odd/$xz$; the third row to DFT calculations; the fourth and fifth rows to unfolded spectral function in 1 Fe BZ with only $d_{xz}$ and all $d$ orbitals, respectively. For (b4), because there is no $d_{xz}$ intensity in the original cut b, the spectral weight is plotted along a parallel $k$ path $0.04\times \frac{2\pi }{a}$ closer to $Z$ and enhanced by a factor of 20. For the DFT calculation (third row), the style of the lines are the same as in Fig. 3a. We show colors of the main orbital character for clarity, although there are non-negligible hybridization effects. White arrows in (c1) and (c2) indicate kinks at the band crossings.

Figure 6

(a) Weight for $d_{xy}$ and $d_{xz}$ in the deep electron band as a function of $k_z$ obtained in the calculation. (b) Intensity of the even deep electron band along $ZX$ integrated at the Fermi level, as a function of $k_z$. (c) Unfolded spectral function for the deep electron band shown in (b).

Figure 7

(a) Evolution of the electron pocket evolution expected as a function of $kz$. (b) Size of the electron pockets as a function of $k_z$. Points indicate $k_F$ along $k_x$ for the deep electron band (blue points) and along $k_y$ for the shallow one (green points). The points for the deep band were extracted from Fig. 5c. For the shallow electron band, three samples were measured for $20<\hslash \omega <45$ eV (circles) and $40<\hslash \omega <90$ eV (stars and squares). Thick lines correspond to values expected in calculations for BaFe${}_2$As${}_2$ doped by 10$\%$ Co. Thin lines are the contours calculated at $-0.11$ eV. (c) Shallow electron band at 39 eV ($k_z\left[X\right]=7$). (d) Shallow electron band at 27 eV ($k_z\left[X\right]=6$).