Isotropic in-plane quenched disorder and dilution induce a robust nematic state in electron-doped pnictides

The phase diagram of electron-doped pnictides is studied varying the temperature, electronic density, and isotropic in-plane quenched disorder strength and dilution by means of computational techniques applied to a three-orbital ($xz,yz,xy$) spin-fermion model with lattice degrees of freedom. In experiments, chemical doping introduces disorder but in theoretical studies the relationship between electronic doping and the randomly located dopants, with their associated quenched disorder, is difficult to address. In this publication, the use of computational techniques allows us to study independently the effects of electronic doping, regulated by a global chemical potential, and impurity disorder at randomly selected sites. Surprisingly, our Monte Carlo simulations reveal that the fast reduction with doping of the Néel $T_N$ and the structural $T_S$ transition temperatures, and the concomitant stabilization of a robust nematic state, is primarily controlled in our model by the magnetic dilution associated with the in-plane isotropic disorder introduced by Fe substitution. In the doping range studied, changes in the Fermi surface produced by electron doping affect only slightly both critical temperatures. Our results also suggest that the specific material-dependent phase diagrams experimentally observed could be explained as a consequence of the variation in disorder profiles introduced by the different dopants. Our findings are also compatible with neutron scattering and scanning tunneling microscopy, unveiling a patchy network of locally magnetically ordered clusters with anisotropic shapes, even though the quenched disorder is locally isotropic. This study reveals a remarkable and unexpected degree of complexity in pnictides: the fragile tendency to nematicity intrinsic of translational invariant electronic systems needs to be supplemented by quenched disorder and dilution to stabilize the robust nematic phase experimentally found in electron-doped 122 compounds.

Figure 1

Internal structure of dopant sites. Sketch shows the location of a dopant where the magnitude of the localized spin, $S_{\mathrm{I}}$, is reduced from the original value $S$. In addition, the neighboring localized spins are also assumed to be affected by the presence of the dopant. The four immediate nearest neighbors have a new localized spin magnitude $S_{\mathrm{NN}}$, while the four next-nearest neighbors have a new localized spin magnitude $S_{\mathrm{NNN}}$, such that $S_{\mathrm{I}}\le S_{\mathrm{NN}}\le S_{\mathrm{NNN}}\le S$ ($S$ is the undoped localized spin magnitude, assumed to be 1 in this publication unless otherwise stated).

Figure 2

Clean limit and effect of Co doping. The clean limit results (open and solid red points) indicate that $T_S=T_N$ and both are approximately constant in the range studied. For Co doping, the Néel temperature $T_N$ (open circles and black dashed line) and the structural transition temperature $T_S$ (filled circles and black solid line) vs the percentage of impurities $x$ are shown. The on-site disorder is $I_{\mathrm{I}}=-0.1$ and the off-diagonal disorder is determined by $S_{\mathrm{I}}=0, S_{\mathrm{NN}}=S/4$, and $S_{\mathrm{NNN}}=S/2$. For both sets of curves, i.e., with and without quenched disorder, the density of doped electrons equals $x$ to simulate Co doping. The cluster used has a size $64\times 64$.

Figure 3

The magnetic susceptibility (open black symbols) and the lattice susceptibility (filled red symbols) vs temperature. The sharp peaks indicate the Néel temperature $T_N$ and the structural transition temperature $T_S$ for the case of 5% Co doping. The on-site disorder is $I_{\mathrm{I}}=-0.1$ and the off-diagonal disorder is defined by $S_{\mathrm{I}}=0, S_{\mathrm{NN}}=S/4$, and $S_{\mathrm{NNN}}=S/2$. The cluster used is $64\times 64$.

Figure 4

Real-space spin-spin correlation functions vs distance on a $64\times 64$ lattice; (a) corresponds to $T=120$ K ($T>T_S$) in the paramagnetic regime, (b) to $T=95$ K ($T_N<T<T_S$) in the nematic state, and (c) to $T=80$ K ($T<T_N$) in the long-range ordered magnetic state. The AFM correlations along $x$ are indicated with solid circles while the FM correlations along $y$ are denoted with open circles. The results are for 5% Co doping with off-diagonal disorder set by $S_{\mathrm{I}}=0, S_{\mathrm{NN}}=S/4$, and $S_{\mathrm{NNN}}=S/2$.

Figure 5

Contrast of effects of Cu and Co doping. The Néel temperatures $T_N$ (dashed lines) and the structural transition temperatures $T_S$ (solid lines) for Co doping (black open and solid circles) and for Cu doping (blue open and solid triangles) are shown. Results are presented first (a) vs the impurity density $x$ and second (b) vs the added electronic density $n$. The off-diagonal disorder is set at $S_{\mathrm{I}}=0, S_{\mathrm{NN}}=S/4$, and $S_{\mathrm{NNN}}=S/2$. The cluster size is $64\times 64$.

Figure 6

Dependence of results with impurity characteristics. The Néel transition temperature $T_N$ (dashed lines) and the structural transition temperature $T_S$ (solid lines) vs the percentage of impurities $x$ for different settings of the off-diagonal disorder. Case I corresponds to the clean limit with no impurity sites (red squares). Case II has $S_{\mathrm{I}}=S/2$ and $S_{\mathrm{NN}}=S_{\mathrm{NNN}}=S$ untouched (blue triangles). This case may be sufficient for Ru doping, which is magnetic. Case III has $S_{\mathrm{I}}=0$ and $S_{\mathrm{NN}}=S_{\mathrm{NNN}}=S$ untouched (green diamonds). Case IV has $S_{\mathrm{I}}=S/2, S_{\mathrm{NN}}=0.7S$, and $S_{\mathrm{NNN}}=0.9S$ (purple upside-down triangles). Finally, Case V has $S_{\mathrm{I}}=0, S_{\mathrm{NN}}=S/4$, and $S_{\mathrm{NNN}}=S/2$ (black circles). Case V appears to be the best to describe experiments for nonmagnetic doping. The density of doped electrons equals $x$ as in Co doping. In all cases, the on-site disorder potential is kept fixed at $I_{\mathrm{I}}=-0.1$. The lattice size is $64\times 64$.

Figure 7

Magnetic and nematic order in the paramagnetic regime. The results are for 5% Co doping at $T=120$ K ($T>T_S$) and using a $64\times 64$ lattice. (a) The magnetic structure factor $S\left(\mathbf{k}\right)$, showing that the wave vectors $(\pi ,0)$ and $(0,\pi )$ have similar intensity. (b) Monte Carlo snapshot of the spin-nematic order parameter with approximately the same amount of positive (green) and negative (orange) clusters. The impurity sites are indicated by black dots.

Figure 8

Magnetic and nematic order in the nematic regime. The results are for 5% Co-doping at $T=95$ K ($T_N<T<T_S$) and using a $64\times 64$ lattice. (a) The magnetic structure factor $S\left(\mathbf{k}\right)$ is shown, with clear dominance of the $(\pi ,0)$ state. (b) Monte Carlo snapshot of the spin-nematic order parameter. Impurity sites are indicated by black dots. A positive nematic order (green) dominates, but there are still small pockets of negative order (orange). (c) Monte Carlo snapshot displaying the on-site component along the easy axis, $S_e$, of the localized spin multiplied by the factor ${(-1)}^{{\mathbf{i}}_x}$, with ${\mathbf{i}}_x$ the $x$-axis component of the location of site $\mathbf{i}$. Both the dominant blue and red clusters indicate regions with $local(\pi ,0)$ order, but shifted by one lattice spacing. This shift suppresses long-range order when averaged over the whole lattice, but short-range order remains. Impurity sites are denoted as black dots.