Theory of the evolution of magnetic order in ${\mathrm{Fe}}_{1+y}\mathrm{Te}$ compounds with increasing interstitial iron

We examine the influence of the excess of interstitial Fe on the magnetic properties of ${\mathrm{Fe}}_{1+y}\mathrm{Te}$ compounds. Because in iron chalcogenides the correlations are stronger than in the iron arsenides, we assume in our model that some of the Fe orbitals give rise to localized magnetic moments. These moments interact with each other via exchange interactions as well as phonon-mediated biquadratic interactions that favor a collinear double-stripe state, corresponding to the ordering vectors $(\pm \pi /2,\pm \pi /2)$. The remaining Fe orbitals are assumed to be itinerant, giving rise to the first-principles derived Fermi surface displaying nesting features at momenta $(\pi ,0)/(0,\pi )$. Increasing the amount of itinerant electrons due to excess Fe, $y$, leads to changes in the Fermi surface and to the suppression of its nesting properties. As a result, due to the Hund's coupling between the itinerant and localized moments, increasing $y$ leads to modifications in the local moments' exchange interactions via the multiorbital generalization of the long-range Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. By numerically computing the RKKY corrections and minimizing the resulting effective exchange Hamiltonian, we find, in general, that the excess electrons introduced in the system change the classical magnetic ground state from a double-stripe state to an incommensurate spiral, consistent with the experimental observations. We show that these results can be understood as a result of the suppression of magnetic spectral weight of the itinerant electrons at momenta $(\pi ,0)/(0,\pi )$, combined with the transfer of broad magnetic spectral weight from large to small momenta, promoted by the introduction of excess Fe.

Figure 1

Fermi surfaces of the TBM describing the itinerant electrons obtained for different values of the chemical potential $\mu$: (a) $\mu =0$ eV, (b) $\mu =0.4$ eV, (c) $\mu =0.8$ eV. The holelike and the electronlike pockets are shown by yellow (light) and purple (dark) regions, correspondingly. The primary contribution to the electron pockets is from the $xz$ and $yz$ orbitals, whereas the hole pocket at the corner of the Brillouin zone is mainly of $xy$ character.

Figure 2

(a) The dependence of the chemical potential shift $\mu$ on the concentration of excess iron $y$ when each excess iron provides six electrons (orange) and eight electrons (purple). (b) The orbitally resolved electron occupation numbers $N$ as functions of $y$. The occupation of $3z^2-r^2$, $xz$, $yz$, $xy$, and $x^2-y^2$ orbitals are shown by red, green, brown, blue, and black lines, respectively. The $xz$ and $yz$ orbitals have the same electron occupation numbers due to the tetragonal symmetry of the system.

Figure 3

(a)–(c) The bare static spin susceptibility ${\chi }^0(\mathbf{q},0)$ and (d)–(f) the RPA spin susceptibility ${\chi }^{\mathrm{RPA}}(\mathbf{q},0)$ calculated using the TBM after projecting out the $x^2-y^2$ and $3z^2-r^2$ orbitals. The concentration of excess Fe is (a) and (d) $y=0.0$; (b) and (e) $y=0.07$; (c) and (f) $y=0.15$. The RPA spin susceptibility is calculated for $U=1.0$ eV and $J_H=U/5$.

Figure 4

(a) Cut of the bare spin susceptibility along a high-symmetry path. Red, green, and blue lines correspond to $y=0.0$, $0.07$, and $0.15$, respectively. (b) Cut of the RPA enhanced spin susceptibility along the same high-symmetry path. The legend provides the $y$ values of each line. We use the following notations: $\Gamma =(0,0)$, $X=(\pi ,0)$, and $M=(\pi ,\pi )$. The RPA spin susceptibility is calculated for $U=1.0$ eV and $J_H=U/5$.

Figure 5

(a) Static magnetic susceptibility $\chi (q_x,0)$ of the toy model (15) for $\alpha =0.2$ (magenta) and $\alpha =0.6$ (blue). (b) $J^{\mathrm{RKKY}}$ (in units of $J_H^2{\chi }_0$) of the same toy model as function of the $(\pi ,0)/(0,\pi )$ peak intensity $\alpha$. $J_1^{\mathrm{RKKY}}$, $J_2^{\mathrm{RKKY}}$, and $J_3^{\mathrm{RKKY}}$ are shown by red, green, and blue lines, respectively.

Figure 6

(a) The evolution of $J_1^{\mathrm{RKKY}}$ (red solid line), $J_2^{\mathrm{RKKY}}$ (green solid line), $J_3^{\mathrm{RKKY}}$ (blue solid line) with increasing concentration of excess of interstitial Fe atoms $y$. The other neighbor interactions are $J_4^{\mathrm{RKKY}}$ (brown dashed line), $J_5^{\mathrm{RKKY}}$ (orange dashed line), $J_6^{\mathrm{RKKY}}$ (purple dashed line). (b) The evolution of the effective exchange couplings $J_{ij}^{\mathrm{eff}}$, where $J_1^{\mathrm{eff}}=-3.4+J_1^{\mathrm{RKKY}}\left(y\right)$ (red line), $J_2^{\mathrm{eff}}=11.6+J_2^{\mathrm{RKKY}}\left(y\right)$ (green line), and $J_3^{\mathrm{eff}}=15.1+J_3^{\mathrm{RKKY}}\left(y\right)$ (blue line). All interactions are given in meV/$S^2$.

Figure 7

Magnetic phase diagram of the effective low-energy model (10) as a function of Fe excess $y$ and the ratio between the first- and second-neighbor biquadratic exchanges $\frac{K_1}{K_2}$, computed with (in meV/$S^2$ units) $J_1^{\mathrm{eff}}\left(y\right)=-3.4+J_1^{\mathrm{RKKY}}\left(y\right)$, $J_2^{\mathrm{eff}}\left(y\right)=11.6+J_2^{\mathrm{RKKY}}\left(y\right)$, $J_3^{\mathrm{eff}}\left(y\right)=15.1+J_3^{\mathrm{RKKY}}\left(y\right)$. The RKKY interactions $J_{ij}^{\mathrm{RKKY}}$ are shown in Fig. 6. We set $K_2=3$.

Figure 8

Schematic representations of the spin configurations in the ground state obtained by the minimization of the classical energy with respect to ${\phi }_1,{\phi }_2,{\phi }_3,q_x$, and $q_y$: (a) ${\phi }_1=0,{\phi }_2=\pi ,{\phi }_3=\pi ,q_x=0,q_y=0$ gives the single-stripe phase, (b) ${\phi }_1=0,{\phi }_2=0,{\phi }_3=\pi ,q_x=\pi /2,q_y=\pi /2$ gives the double-stripe phase, (c) ${\phi }_1=\pi /2-\delta ,{\phi }_2=\pi -2\delta ,{\phi }_3=\pi /2-\delta ,q_x=\pi /2-\delta ,q_y=\pi /2-\delta$ gives the incommensurate spiral phase. (d), (e) The structure factors computed for the magnetic orders displayed in (a)–(c), respectively. Bright spots correspond to the sharp peaks that appear at the corresponding ordering wave vectors.

Figure 9

Most relevant elastic modes in ${\mathrm{Fe}}_{1+y}\mathrm{Te}$. (a) The uniform monoclinic mode $u_{xy}$. (b) The nonuniform mode corresponding to $u_5^x=u_5^y$. (c) The nonuniform mode corresponding to the $u_6$ distortion. The mode corresponding to the $u_7$ distortion would display the same configuration but rotated by ${90}^{\circ }$. We use the following convention: red bonds are lengthened with respect to the tetragonal lattice, yellow bonds are shortened, blue bonds remain of the same length. Green (red) sites have spins ferromagnetically (antiferromagnetically) aligned with each other.