Flat-band-induced itinerant ferromagnetism in ${\mathrm{RbCo}}_2{\mathrm{Se}}_2$

$A{\mathrm{Co}}_2{\mathrm{Se}}_2$ ($A$=K, Rb, Cs) is a homologue of the iron-based superconductor $A{\mathrm{Fe}}_2{\mathrm{Se}}_2$. From a comprehensive study of ${\mathrm{RbCo}}_2{\mathrm{Se}}_2$ via measurements of magnetization, transport, neutron diffraction, angle-resolved photoemission spectroscopy, and first-principles calculations, we identify a ferromagnetic order accompanied by an orbital-dependent spin splitting of the electronic dispersions. Furthermore, we identify the ordered moment to be dominated by a $d_{x^2-y^2}$ flat band near the Fermi level, which exhibits the largest spin splitting across the ferromagnetic transition, suggesting an itinerant origin of the ferromagnetism. In the broader context of the iron-based superconductors, we find this $d_{x^2-y^2}$ flat band to be a common feature in the band structures of both iron chalcogenides and iron pnictides, accessible via heavy electron doping.

Figure 1

FM characterization. (a) Crystal and magnetic structure of ${\mathrm{RbCo}}_2{\mathrm{Se}}_2$ with magnetic moments indicated by arrows. (b) BZ notations. (c) Temperature-dependent magnetization at 1000-Oe external magnetic field. (d) Field-dependent magnetization at 2 K. (e) Resistance as a function of temperature. (f) Neutron diffraction measurements of the integrated intensity of the (0,0,4) magnetic Bragg peak as a function of temperature.

Figure 2

Fermi surfaces across $T_c$. (a) FS mapping at 150 K $> T_c$. The $k_y>0$ ($k_y<0$) region shows the raw data (2D curvature). Markers indicate MDC-fitted FSs. (b) MDC at $E_F$ along cut 1 in (a) fitted with four Lorentzians and a constant background. The blue arrows indicate the $\beta$ band. (c) and (d) Same as (a) and (b), except taken at 30 K in the FM state.

Figure 3

Electronic structure across $T_c$. (a) Spectral image along cut 1 in Fig. 2 with MDC-fitted dispersions. The $k_F$ value of $\varepsilon$ is obtained from fitting the MDC at the equivalent $X_2$ point. (b) Same as (a), except taken at 30 K in the FM state. (c) Energy distribution curves (EDCs) at the $X_1$ point. The markers indicate the $\beta$ band bottom obtained by fitting the corresponding data points in (a) and (b) with a parabolic curve. Fitting uncertainty gives a $\pm 10$-meV error bar. (d) DFT band calculations in the paramagnetic phase renormalized by a factor of 2.9. (e) Same as (d), except in the FM state.

Figure 4

Flat band across $T_c$. (a) and (b) Spectra images along cut 2 in Fig. 2 at 150 and 30 K, respectively. Markers are the corresponding fitted peak positions in (c). (c) EDC stacks of (a) and (b). Red markers indicated peak positions from fitting the EDCs with two Lorentzians and a constant background multiplied by the Fermi-Dirac function. (d) Temperature-dependent EDCs at the momentum indicated by the blue dashed line in (b).

Figure 5

Temperature dependence of band splittings. (a)–(d) Temperature-dependent spectra along cut 1 marked in Fig. 2. The left (right) half shows the raw data (second momentum derivative). (e) Fitted $k_F$ of ${\alpha }_1, {\alpha }_2$, and ${\beta }_1$ bands as a function of temperature. Error bars are obtained from the fitting uncertainty. (f) Temperature dependence of $\alpha$-band $\mathrm{\Delta }{k}_F$ and the shift of the flat band from fitting the peak position in the EDCs and maximum positions in Fig. 4.

Figure 6

Calculated electronic structures. (a) Calculated PDOS projected onto Co $d$ orbitals in the nonmagnetic state. (b)–(d) Calculated orbital-resolved band dispersion in the nonmagnetic, spin-majority, and spin-minority states, respectively. The marker size represents the spectral weight.

Figure 7

Electronic phases in iron-based superconductors. (a) and (b) Band structure calculations of FeSe [32] and ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ [33]. Common features along $\mathrm{\Gamma }-X$ are highlighted by dominant orbitals. Horizontal lines represent different chemical potentials tunable via electron doping as exemplified by FSs of (c) Fe(Se,Te), (d) high-$T_c {\mathrm{RbFe}}_2{\mathrm{Se}}_2$, and (e) flat band itinerant FM ${\mathrm{RbCo}}_2{\mathrm{Se}}_2$.