Nature of itinerant ferromagnetism of ${\mathrm{SrRuO}}_3$: A DFT+DMFT study

We have investigated the temperature-dependent evolution of electronic structures and magnetic properties of an itinerant ferromagnet ${\mathrm{SrRuO}}_3$, employing the combined scheme of density functional theory and dynamical mean-field theory (DFT+DMFT). The inclusion of finite dynamical correlation effects beyond DFT well describes not only the incoherent hump structure observed in the photoemission experiment but also the temperature-dependent magnetic properties in accordance with experiments. We have shown that the magnetization of ${\mathrm{SrRuO}}_3$ evolves with the Stoner behavior below the Curie temperature $T_c$, reflecting the weak itinerant ferromagnetic behavior, but the local residual magnetic moment persists even above $T_c$, indicating the local magnetic moment behavior. We suggest that the ferromagnetism of ${\mathrm{SrRuO}}_3$ has a dual nature of both weak and local moment limits, even though the magnetism of ${\mathrm{SrRuO}}_3$ is more itinerant than that of Fe.

Figure 1

(top) Band structures obtained with DFT (left) and DFT+DMFT at $T=100$ K (right). The size of the red (blue) dot in the DFT plot denotes the amount of the $t_{2g} \left(e_g\right)$ component in the wave function. (bottom) DFT FSs (left) and DFT+DMFT FSs at $T=100$ K (right). Up and Dn denote spin up and down, respectively. The first BZ is provided in Appendix pp1.

Figure 2

Ru $4d$ partial density of states (PDOS) from DFT and DFT+DMFT at different $T$. Red solid and blue dashed lines represent $t_{2g}$ and $e_g$ projected PDOSs, respectively. Calculated magnetic moment $M_{Ru}$ at each $T$ is provided. For DFT+DMFT at $T=100$ K, the coherent and incoherent parts are indicated.

Figure 3

The Ru $4d$ PES spectrum (black solid line) [10, 17] is compared with the calculated PDOSs in DFT and DFT+DMFT (100 K). Blue dotted and red dashed lines represent the DFT and DFT+DMFT PDOSs, respectively, which are broadened with a Gaussian function (0.10 eV FWHM). The coherent part at $-0.25$ eV, the dip at $-0.73$ eV, and the incoherent part at $-1$ eV in the PES data are indicated.

Figure 4

(a) The DFT+DMFT results for the Ru magnetic moment $M_{Ru}$ in ${\mathrm{SrRuO}}_3$ vs $T$. The blue line represents $M_{Ru}\left(T\right)$ with the Stoner behavior below $T_c=160\mathrm{K}[M_{Ru}\left(0\right)=1.06{\mu }_B$ from the DFT]. We assign the Stoner regime for $T<200$ K and the residual moment regime for $T>200$ K. In the former, $M_{Ru}\left(T\right)$ shows the fast drop with $T$, while in the latter, it shows the slow drop. We used averaged $M_{Ru}$ values from the DMFT iterations, and the error bar corresponds to the standard deviation from the averaged value. The inset presents the $T$-dependent exchange splitting ${\mathrm{\Delta }}_{ex}$ between spin-up and -down bands, which shows behavior similar to $M_{Ru}\left(T\right)$. ${\mathrm{\Delta }}_{ex}$ is defined as a splitting between peak positions of spin-up and -down parts in the $t_{2g}$ PDOS. (b) The reduction of the scaled magnetic moment of DFT+DMFT $\left(1-\frac{M_{\mathrm{DMFT}}}{M_{\mathrm{DFT}}}\right)$ vs the square of scaled temperature [$\left(T/T_c\right){}^2]$ in the Stoner regime. The blue linear line is from the least-squares fitting of data, including the origin. We employed $M_{\mathrm{DFT}}$ as $M_{Ru}(T=0)$ since the experimental magnetic moment of ${\mathrm{SrRuO}}_3$ is in good agreement with that in DFT [9, 14]. (c) The scaled magnetic moment of the DFT+DMFT $\left(\frac{M_{\mathrm{DMFT}}}{M_{\mathrm{DFT}}}\right)$ vs the scaled temperature $\left(T/T_c\right)$ in the residual moment regime. The blue linear line is from the least-squares fitting. (d) The DFT+DMFT results for the quasiparticle weight $Z$ of $t_{2g}$ orbital vs $T$. Blue circles and red squares represent $Z$'s for spin-up and -down electrons, respectively.

Figure 5

(a) The crystal structure of ${\mathrm{SrRuO}}_3$. The local axes ($x^{\prime }$ and $z^{\prime }$), which are used for the orbital projection in Fig. 7, are depicted. The $y^{\prime }$ axis is nearly parallel to the $b$ axis. We have adopted the experimental structure of the Pnma space group [29]. (b) The first BZ of ${\mathrm{SrRuO}}_3$. High-symmetry points $\mathrm{\Gamma },Z,T$, and $Y$ for the band structure plots are depicted.

Figure 6

The orbital-projected DFT band structures of ${\mathrm{SrRuO}}_3$. The size of the red, green, and blue dots denotes the amount of the $x^{\prime }{y}^{\prime },y^{\prime }{z}^{\prime }$, and $x^{\prime }{z}^{\prime }$ components in the wave function, respectively. Up and Dn denote spin-up and -down bands. Note that the spin-up band at the $Z$ point near $E_F$ has $x^{\prime }{y}^{\prime }$ and $y^{\prime }{z}^{\prime }$ orbital characters.

Figure 7

The self-energies of ${\mathrm{SrRuO}}_3$ in DFT+DMFT for $T=100$, 150, and 200 K (top, middle, and bottom). Left (right) panels are for real (imaginary) self-energies. Blue dashed (red solid) lines present spin up (spin down).