Magnetic order and fluctuations in the quasi-two-dimensional planar magnet Sr(${\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x{)}_2{\mathrm{As}}_2$

We use neutron scattering to investigate spin excitations in $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$, which has a $c$-axis incommensurate helical structure of the two-dimensional (2D) in-plane ferromagnetic (FM) ordered layers for $0.013\le x\le 0.25$. By comparing the wave vector and energy dependent spin excitations in helical ordered $\mathrm{Sr}{\left({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}\right)}_2{\mathrm{As}}_2$ and paramagnetic ${\mathrm{SrCo}}_2{\mathrm{As}}_2$, we find that Ni doping, while increasing lattice disorder in $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$, enhances quasi-2D FM spin fluctuations. However, our band structure calculations within the combined density functional theory and dynamic mean field theory ($\mathrm{DFT}+\mathrm{DMFT}$) failed to generate a correct incommensurate wave vector for the observed helical order from nested Fermi surfaces. Since transport measurements reveal increased in-plane and $c$-axis electrical resistivity with increasing Ni doping and associated lattice disorder, we conclude that the helical magnetic order in $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$ may arise from a quantum order-by-disorder mechanism through the itinerant electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions.

Figure 1

(a) Crystal structure of $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$. (b) Schematics of spin fluctuations in reciprocal space and the Co lattice in real space. Red (blue) areas represent FM (AF) spin fluctuations. (c) Helical magnetic structure in the $x=0.1$ compound. (d) Neutron diffraction intensity in the $[H,H,L]$ plane. The red arrows indicate the helical wave vector. (e) Magnetic wave vectors and 3D transition temperatures in $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$. (f), (g) Electronic band structures and the Fermi surfaces for $k_z=0$ in $x=0.1$. Green (red) color represents the $d_{x^2-y^2}$ ($d_{z^2}$) orbital and blue is the contribution from the $t_{2g}$ ($d_{xz}, d_{yz}, d_{xy}$) orbitals. Yellow is the mixture of the red ($d_{z^2}$) and green ($d_{x^2-y^2}$). (h), (i) Calculated 3D Fermi surfaces of the $x=0.1$ compound.

Figure 2

Neutron diffraction results for $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$ single crystals. (a)–(d) Scan along the $[0,0,L]$ direction for $x=0.05,0.125$, 0.15, and 0.2, respectively. (e)–(h) Temperature dependence of the magnetic order parameter, where solid lines are guides to the eye.

Figure 3

(a)–(d) Constant energy slices of spin fluctuations of $\mathrm{Sr}\left({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}\right){\mathrm{As}}_2$ within the $[H,K]$ plane at $E=12\pm 2,17\pm 3,35\pm 5$, and $52\pm 7$ meV, respectively. (e)–(h) The dynamic spin susceptibility from the DFT $+$ DMFT calculations at the corresponding energies for $\mathrm{Sr}{\left({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}\right)}_2{\mathrm{As}}_2$. (i)–(l) The same constant energy slices in pure ${\mathrm{SrCo}}_2{\mathrm{As}}_2$.

Figure 4

(a)–(c) Constant energy slices of spin fluctuations of $\mathrm{Sr}{\left({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}\right)}_2{\mathrm{As}}_2$ within the $[\mathrm{H}$, H, $\mathrm{L}]$ plane at $E=3\pm 0.75,7\pm 1$, and $16\pm 1.5$ meV, respectively. The corresponding 1D cuts along the $[0,0,L]$ direction are shown in the bottom panel. (d)–(f) 1D cuts of spin fluctuations along the $[H,H]$ direction at different $L$ and energies. (g)–(i) Corresponding 1D cuts along the $[H,0]$ direction. The solid lines are Gaussian fits to the cuts.

Figure 5

(a), (b) Transverse cuts of the AF spin fluctuations in $\mathrm{Sr}{\left({\mathrm{Co}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{As}}_2$ with $x=0,0.1$ at $E=10\pm 2$ and $40\pm 5$ meV, respectively. The solid lines are Gaussian fits to the data. (c) Energy dependence of the line widths of AF spin fluctuations along the $[1,K]$ direction. (d) The longitudinal scans of the spin fluctuations in the $x=0$ and 0.1 at $E=12\pm 3$ and $46\pm 5$ meV. (e) Temperature dependence of the normalized electrical resistivity ${\rho }_c\left(T\right)/{\rho }_c\left(300\mathrm{K}\right)$ along the $c$ axis for $x=0,0.1,0.2,$ and 0.5.

Figure 6

(a) Schematics of spin fluctuations in the $[H,K]$ two-dimensional reciprocal space, where the red (blue) areas represent FM (AF) spin fluctuations. The black diamond marks the integration region for calculating energy-dependent local dynamic susceptibility. The red and blue squares are the integration region for calculating energy-dependent FM and AF local dynamic susceptibility. (b) Energy-dependent local susceptibility ${\chi }^{\prime \prime }\left(E\right)$ of $\mathrm{Sr}{\left({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}\right)}_2{\mathrm{As}}_2$ in absolute units normalized by using a vanadium standard. The black line is a guide to the eye. The purple line is the result of ${\mathrm{SrCo}}_2{\mathrm{As}}_2$ from our previous work [18]. (c) Energy dependence of the integrated ${\chi }^{\prime \prime }\left(E\right)$ of FM (red) and AF (blue) spin fluctuations of $\mathrm{Sr}{\left({\mathrm{Co}}_{0.9}{\mathrm{Ni}}_{0.1}\right)}_2{\mathrm{As}}_2$.