\section{INTRODUCTION}

The $122$-type cobalt pnictides $A$Co$_{2}Pn_{2}$ ($A=$~Ca, Sr, Eu, $Pn=$~As, P) with the tetragonal ThCr$_{2}$Si$_{2}$-type structure (space group $I4/mmm$) \cite{Johnston_2010, Hoffman_1985, Jia_2009, Pandey_2013, Anand_2014, Anand_2014b, Sefat_2009, Imai_2017,Jayasekara_2013} are metals with fascinating properties due to magnetic frustration within their square Co layers \cite{Sapkota_2017} and $Pn$-$Pn$ hybridization-driven magnetoelastic interactions \cite{Hoffman_1985,Reehuis_1998}.  In particular, CaCo$_{1.86}$As$_{2}$ shows evidence of Stoner-enhanced ferromagnetism (FM), and SrCo$_{2}$As$_{2}$  \cite{Pandey_2013, Anand_2014}  harbors itinerant antiferromagnetic (AF) fluctuations \cite{Jayasekara_2013} centered at neutron-momentum transfers $\mathbf{Q}$ corresponding to the stripe-type AF found in various $122$-type Fe-pnictide superconductors \cite{Lynn_2009, Johnston_2010, Paglione_2010, Canfield_2010, Stewart_2011}.  These fluctuations exist despite SrCo$_{2}$As$_{2}$ remaining paramagnetic (PM) down to a temperature of at least $T=0.05$~K \cite{Li_un}.  Here, we present results from neutron diffraction experiments made on the series Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ which detail the microscopic changes to the AF order as Ca is replaced  by Sr.

CaCo$_{1.86}$As$_{2} $ has A-type AF order below a N\'{e}el temperature of $T_{\text{N}} = 52$~K, which consists of FM aligned square Co layers stacked AF along the crystalline $\mathbf{c}$ direction (${+}{-}{+}{-}$ structure) and an ordered magnetic moment $\bm{\mu}$ lying parallel to $\mathbf{c}$ \cite{Quirinale_2013, Jayasekara_2017}, as shown in Fig.~\ref{Fig1} \cite{Sangeetha_2017, Momma_2011}.  Despite the A-type order, its spin fluctuation spectrum indicates strong itinerant fluctuations within the extremely magnetically frustrated Co planes and only weak coupling along $\mathbf{c}$ \cite{Sapkota_2017}.  Creating a slight imbalance of the frustrated intralayer interactions of the effectively $2$D FM order may drive the ground state towards stripe-type AF \cite{Sapkota_2017}.  Thus, studying Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ provides a possible route to find quantum critical behavior between $2$D FM and stripe-type AF ground states.  

\begin{figure}[]
	\centering
	\includegraphics[width=1.0\linewidth]{./Fig1_bgu_v2.pdf}
	\caption{  \label{Fig1} Chemical and magnetic structures of Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$, and the magnetic phase diagram based on magnetization data from Ref.~[\onlinecite{Sangeetha_2017}]. The AF$1$ phase  has A-type antiferromagnetic (AF) order characterized by  an AF propagation vector of $\bm{\tau}=(0,0,1)$, and consists of ferromagnetic Co layers stacked AF along the crystalline $\mathbf{c}$ axis (${+}{-}{+}{-}$). The ordered magnetic moment $\bm{\mu}$ lies parallel to $\mathbf{c}$. The UNK phase has a  ${+}{+}{-}{-}$ structure with $\bm{\tau}=(0,0,\frac{1}{2})$ and $\bm{\mu}\parallel\mathbf{c}$. The AF$2$ phase has $\bm{\tau}=(0,0,\frac{1}{2})$ but $\bm{\mu}\perp\mathbf{c}$.  Its order is either ${+}{+}{-}{-}$ (left), or a clock-type AF structure (right). The occurrence of magnetic domains prevents us from distinguishing between these structures.  Similarly, an amplitude-modulated spin-density wave cannot be ruled out.  PM stands for paramagnetic and $\mu$ is given per Co atom.  The diagrams were created with \textsc{vesta} [\onlinecite{Momma_2011}].}
\end{figure}

Magnetization data for the Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ series reveal multiple magnetic transitions \cite{Ying_2013,Sangeetha_2017} between $x=0$ and $1$, and similar data exist for Ca$_{1-x}$Sr$_{x}$Co$_{2}$P$_{2}$ \cite{Jia_2009}.  Recent reports for the arsenide series find that the $x=0$ A-type phase (AF$1$) transitions to an unknown magnetic phase (UNK) for $0.2 \alt x \alt 0.3$, into an AF phase (AF$2$) with $\bm{\mu}\perp\mathbf{c}$  for $0.3 \alt x\alt 0.5$, and finally into a PM state for $x\agt0.5$.  The effective magnetic anisotropy has different signs in the AF$1$ and AF$2$ phases, with $\bm{\mu}$ oriented parallel or perpendicular to $\mathbf{c}$, respectively.   In the UNK phase, $\bm{\mu}\parallel\mathbf{c}$ but the magnetic anisotropy is essentially zero \cite{Sangeetha_2017}.  The microscopic details of the magnetic order in the UNK and AF$2$ phases were previously unknown.

In this paper, we show that the square Co layers of Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ remain FM aligned in the UNK and AF$2$ phases, and that the AF stacking of the layers changes with $x$.  Previous research has shown that materials with such coupled FM aligned planes may be described using a one-dimensional ($1$D) Heisenberg model  \cite{Majlis_2000} in which tuning the interlayer coupling strengths gives rise to a variety of collinear and noncollinear magnetic ground states \cite{Orbach_1958, Yang_1966a, Yang_1966b, Yang_1966c, Baxter_1972}.  We show that this is the case here, and reveal that the evolution of magnetic order in Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ can be understood in terms of the above $1$D Heisenberg model with nearest-layer (NL) and next-nearest-layer (NNL) exchange interactions and single-ion magnetic anisotropy.  Using neutron diffraction data and analytical calculations,  we show that the AF ordered phases for $0.2\alt x\alt0.3$ and $0.3\alt x\alt0.5$ \cite{Sangeetha_2017} both have an AF propagation vector of $\bm{\tau}_{\sfrac{1}{2}}\equiv(0,0,\frac{1}{2})$, which requires relatively large NNL exchange over much of the phase diagram.   For $0.2\alt x\alt0.3$ the FM NL exchange is partially frustrated by the AF NNL exchange, which may explain the occurrence of substantial FM correlations \cite{Sangeetha_2017} and a small $\mu$.  


\section{EXPERIMENT}
We synthesized plate-like single crystals of Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ by solution growth using Sn flux, and confirmed their stoichiometry via energy-dispersive x-ray spectroscopy measurements.  We found no evidence for vacancies of the Co sites in the $x=0.24$ and $0.44$ samples used for the neutron diffraction measurements within an uncertainty of $\approx4.5\%$ (i.e.\ $y=0\pm4.5\%$).  We previously discussed that the presence of vacancies and/or the growth technique used may lead to the different observed values of $T_{\text{N}}$ for $x=0$, ranging from $T_{\text{N}}=52$ to $76$~K \cite{Jayasekara_2017}.  Nevertheless, this level of vacancies does not affect the occurrence of A-type AF order \cite{Jayasekara_2017,Quirinale_2013,Anand_2014,Cheng_2012, Ying_2013, Ying_2012}.  

Neutron diffraction experiments were performed with the HB-$1$A fixed-incident-energy triple-axis spectrometer at the High Flux Isotope Reactor, using a fixed neutron energy of $14.6$~meV.  Effective collimations of $40^{\prime}{-}40^{\prime}{-}40^{\prime}{-}80^{\prime}$ were utilized and pyrolitic graphite  filters were placed before the sample. Single crystals with $x=0.24(3)$ and $0.44(7)$ and masses of $21.9$ and $20.4$~mg, respectively, were measured with their ($H~H~L$) reciprocal-lattice planes coincident with the scattering plane, and cooled down to $T=5$~K using a He closed-cycle refrigerator.  A high-energy x-ray diffraction measurement was made as described in Ref.~[\onlinecite{Jayasekara_2017}] on a $0.6$~mg single-crystal of Ca$_{0.60(2)}$Sr$_{0.40(2)}$Co$_{1.93(3)}$As$_{2}$  at station $6$-ID-D at the Advanced Photon Source to confirm that the sample retained $I4/mmm$ symmetry down to $T=5$~K.  In this report, we express $\mathbf{Q}$ in reciprocal-lattice units (r.l.u.).  

\section{Results}

\begin{figure*}[]
	\centering
	\includegraphics[width=1.0\linewidth]{./Fig2_v2.pdf}
	\caption{  \label{Fig2} (a-c) Neutron diffraction data for $x=0.24$ from scans along $(1~1~L)$ taken at $T=6$~K (a), and from longitudinal ($\theta$-$2\theta$) scans for $(0~0~\frac{3}{2})$ (b) at $T=6$~K and $(1~1~{-\frac{1}{2}})$ (c) at $T=6$ and $90$~K.  The inset to (a) shows a reciprocal-space map which indicates the scan (grayed areas) or position corresponding to each panel.  (d) Temperature dependence of the integrated intensity of the $x=0.24$ sample's $(1~1~{-\frac{1}{2}})$ magnetic Bragg peak. (e-g) Neutron diffraction data for $x=0.44$ from scans along $(0~0~L)$ taken at $T=5$ and $60$~K (e), and from longitudinal ($\theta$-$2\theta$) scans for $(0~0~\frac{5}{2})$ (f) and $(1~1~{-\frac{1}{2}})$ (g) performed at various temperatures.  (h) Temperature dependence of the intensity of the $x=0.44$ sample's $(1~1~\frac{5}{2})$ magnetic Bragg peak.  Arrows in (a) and (e) point from structural to magnetic Bragg peaks.  Lines in (b), (c), (f), and (g) show fits to a Gaussian lineshape with a constant background, and lines in (d) and (h) are guides to the eye.  }
\end{figure*}

Neutron diffraction data for $x=0.24$ and $0.44$ are shown in Figs.~\ref{Fig2}(a)--\ref{Fig2}(d) and \ref{Fig2}(e)--\ref{Fig2}(h), respectively.  The arrows in Figs.~\ref{Fig2}(a) and \ref{Fig2}(e) point from structural to magnetic Bragg peaks and demonstrate that the AF order is characterized  by $\bm{\tau}_{\sfrac{1}{2}}=(0,0,\frac{1}{2})$ for both compounds.  This propagation vector corresponds to a periodicity along $\mathbf{c}$ of four square Co layers, which differs from the A-type AF order found for $x=0$ with $\bm{\tau_{\text{A}}}=(0,0,1)$ (alternating square Co layers) \cite{Quirinale_2013}, and from the stripe-type AF order found in many $122$-type Fe-pnictide superconductors and the stripe-type spin fluctuations in SrCo$_{2}$As$_{2}$ with $\bm{\tau_{\text{st}}}=(\frac{1}{2},\frac{1}{2},1)$ \cite{Jayasekara_2013}.   The widths of the magnetic and structural Bragg peaks are similar, which attests to the presence of long-range AF order.  We find no evidence for magnetic Bragg peaks in data measured at reciprocal-lattice positions corresponding to $\bm{\tau_{\text{A}}}$ and $\bm{\tau_{\text{st}}}$.

Figures~\ref{Fig2}(b) and \ref{Fig2}(e), respectively, illustrate that magnetic Bragg peaks are absent at $\mathbf{Q}=(0,0,\frac{L}{2})$, $L=$ odd integer, positions for $x=0.24$, but that they occur for $x=0.44$.  Since neutron diffraction is sensitive to the component of $\bm{\mu}\perp\mathbf{Q}$, these data indicate that $\bm{\mu}\parallel\mathbf{c}$ for $x=0.24$, whereas $\bm{\mu}$ has a component in the $\mathbf{ab}$ plane for $x=0.44$.  These results agree with the conclusions from magnetization data that $\bm{\mu}\parallel\mathbf{c}$ for $x=0.24$ and $\bm{\mu}\perp\mathbf{c}$ for $x=0.44$ \cite{Sangeetha_2017}.

Figures~\ref{Fig2}(c), \ref{Fig2}(f), and \ref{Fig2}(g) show detailed views of select magnetic Bragg peaks at various temperatures.  The peaks are quite weak, which implies small ordered moments.  Using the magnetic structures shown in Fig.~\ref{Fig1} for the UNK and AF$2$ phases, we find  $\mu=0.08(1)~\mu_{\text{B}}/$Co for $x=0.24$ at $T=6$~K and $0.27(9)~\mu_{\text{B}}/$Co for $x=0.44$ at $5$~K.  Three magnetic Bragg peaks were used to determine $\mu$ for $x=0.24$ and six were used for $x=0.44$.  More details concerning the magnetic structures are given below.

The temperature dependence of the magnetic order parameter for $x=0.24$ and $0.44$ is presented in Figs.~\ref{Fig2}(d) and \ref{Fig2}(h), respectively.  Upon cooling, a magnetic diffraction signal first appears at $T_{N}\approx80$ and $50$~K for $x=0.24$ and $0.44$, respectively, which are above the values of $T_{\text{N}}\approx67$ and $37$~K expected from magnetization data \cite{Sangeetha_2017}.  This may be due to differences in the level of vacancies between the samples used for the neutron diffraction experiments and those used for magnetization \cite{Sangeetha_2017}.  In particular, the level of Co vacancies for the neutron diffraction samples is $y=0.00(4)$, whereas the magnetization samples with $x=0.25$ and $0.45$ have $y=0.10(5)$ and $0.08(2)$, respectively. Samples with Co vacancies seem to have lower values for $T_{\text{N}}$, with $T_{\text{N}}=52(1)$~K for $y=0.14$ \cite{Quirinale_2013} and $T_{\text{N}}=76$~K for $y=0$ \cite{Cheng_2012}.  On the other hand, it has been suggested that vacancies alone may not explain the differing values of $T_{\text{N}}$ from different reports, and that different growth conditions may also be responsible \cite{Zhang_2015}.



\section{Discussion}
We capture the observed magnetic ordering behaviors using the classical Heisenberg Hamiltonian:
\begin{align}
\mathcal{H}&=\mathcal{H}_{\text{in-plane}} + J_{z}\sum_{\mathbf{R}}{\mathbf S}_{\mathbf{R}}\cdot{\mathbf S}_{\mathbf{R}+\mathbf{d}}+J_{z}^{\prime}\sum_{\mathbf{R}}{\mathbf S}_{\mathbf{R}}\cdot{\mathbf S}_{\mathbf{R}+2\mathbf{d}}\nonumber\\
&-D_{z}\sum_{\mathbf{R}}\left( S^z_{\mathbf{R}}\right)^2-D_{xy}\sum_{\mathbf{R}}\left[\left( S^x_{\mathbf{R}}\right)^4+\left( S^y_{\mathbf{R}}\right)^4\right] \,.
\label{eq:Hamiltonian}
\end{align}
Here, $\mathcal{H}_{\text{in-plane}}$ contains competing FM and AF interactions between Heisenberg spins within a square layer, $J_{z}$ ($J_{z}^{\prime}$) is the effective NL (NNL) magnetic exchange interaction along $\mathbf{c}$, $D_{z}$ ($D_{xy}$) is the single-ion magnetic anisotropy along $\mathbf{c}$ (within the $\mathbf{ab}$ plane), and $\mathbf{d}=d\mathbf{\hat{c}}$ where $d$ is the distance between neighboring Co layers.  We regard each FM-aligned Co layer as a single localized Heisenberg spin ${\mathbf S}_{\mathbf{R}}$ at position $\mathbf{R}$ and consider the layers' relative orientations along $\mathbf{c}$ in terms of a $1$D model.  For helical AFs, this is a common model denoted as the $J_{0}$-$J_{1}$-$J_{2}$ model \cite{Johnston_2012, Johnston_2015}.

\begin{figure}[]
	\centering
	\includegraphics[width=1.0\linewidth]{./Figure_Theory-NoLine.pdf}
	\caption{  \label{Fig3} (a) Zero-temperature magnetic phase diagram for varying nearest-layer exchange $J_{z}$ and single-ion magnetic anisotropy $D_{z}$ along $\mathbf{c}$, and fixed antiferromagnetic (AF) next-nearest-layer exchange $J^{\prime}_{z}$ ($J^{\prime}_{z} >0$).  The lower bound of  $D_{z}$, $D_{z}^{\text{lb}}$, and anisotropy within the $\mathbf{ab}$ plane, $D^{\text{lb}}_{xy}$, necessary to suppress helical-AF order are indicated by black and magenta lines, respectively. Arrows illustrate the magnetic order, and numbers in ovals show estimated locations for three values of $x$. (b) Calculated spin-flop ($h_{\text{sf}}$) and saturation ($h_{\text{sat}}$) magnetic fields along $\mathbf{c}$ versus $J_{z}/J^{\prime}_{z}$ for $D_{z}/J^{\prime}_{z} = 1$ and $D_{xy}=0$.  The fields correspond to using $J^{\prime}_{z} = 3 J_{z}$ for $x=0$. The inset shows the calculated magnetization $m$ versus magnetic field $h$ curves for $J_{z}/J_{z}^{\prime} = -1$~(yellow) and $3$~(red). (c) $D^{\text{lb}}_{z}$ and $D^{\text{lb}}_{xy}$ in units of $J_{z}^{\prime}$ and  $J_{z}^{\prime}/S^{2}$, respectively, versus $J_{z}/J^{\prime}_{z}$.}
\end{figure}

We analytically calculate the classical ground-state energies in units of $J_{z}^{\prime}$, which we assume to be AF ($J_{z}^{\prime}>0$), and find the phase diagram given in Fig.~\ref{Fig3}(a).  More details of our calculations are given in Appendix~\ref{AppA}.  In the absence of anisotropy, the ground state is either an  A-type AF, a single-${\mathbf{Q}}$ helix with a turn angle of $\phi = \cos^{-1}[-J_{z}/(4J^{\prime}_{z})]$, or FM, with phase boundaries at $J_{z}/J_{z}^{\prime}=4$ and $-4$, respectively \cite{Johnston_2012,Johnston_2015, Johnston_2017}.  AF order with a propagation vector of $\bm{\tau}_{\sfrac{1}{2}}$ occurs only at $J_{z}/J_{z}^{\prime}=0$.

On the other hand, both neutron diffraction and magnetization data require that $D_{z}>0$ for  $x \alt 0.3$ and $D_{z}<0$ for $0.3\alt x \alt 0.5$, and it turns out that magnetic anisotropy suppresses helical AF order in favor of regions with either $\bm{\tau}_{\sfrac{1}{2}}$-type AF, A-type AF, or FM order.  In the case of $D_{z}>0$, $D_{z}$ must be greater than a lower bound of $D_{z}^{\text{lb}}$ to suppress helical AF order.  For $D_{z}<0$, anisotropy that picks a specific direction within the $\mathbf{ab}$ plane, $D_{xy}$, must be included and must be greater than a lower bound of $D_{xy}^{\text{lb}}$ to suppress helical AF order.  We determined $D_{z}^{\text{lb}}$ and  $D_{xy}^{\text{lb}}$ by comparing the energy of a helical AF state at finite $D_{z}>0$ or $D_{xy}$ (with $D_{z}<0$) to the state found at large $D_{z}$ or $D_{xy}$ (with $D_{z}<0$).  The computed boundaries are plotted in Fig.~\ref{Fig3}(c) and are included in Fig.~\ref{Fig3}(a), where the helix region corresponds to coplanar helical AF order.  

The top part of Fig.~\ref{Fig3}(a) shows that for $D_{z}>D_{z}^{\text{lb}}$, the value of $\lvert J_{z} \rvert/J_{z}^{\prime}$ and the sign of $J_{z}$ determine the stacking of the FM layers.  The ground state is ${+}{+}{-}{-}$ ($\bm{\tau}_{\sfrac{1}{2}}$) for $\lvert J_{z} \rvert/J_{z}^{\prime} < 2$ and either FM (${+}{+}{+}{+}$) or A-type AF (${+}{-}{+}{-}$) for $\lvert J_{z}\rvert/J_{z}^{\prime} >2$.  Note that half of the NL interactions are frustrated for $\lvert J_{z} \rvert/J_{z}^{\prime}<2$ whereas the NNL interactions are frustrated for $\lvert J_{z} \rvert/J_{z}^{\prime}>2$.  In other words, $J_{z}^{\prime}$ dominates $J_{z}$ for $\lvert J_{z} \rvert/J_{z}^{\prime} < 2$ and $J_{z}$ dominates $J_{z}^{\prime}$ for $\lvert J_{z} \rvert/J_{z}^{\prime} > 2$.  Figure~\ref{Fig3}(a) shows that the phase diagram looks quite similar for $D_{z}<0$ and $D_{z}>0$. 

Our theory predicts that $D_{z}>D_{z}^{\text{lb}}$ for $x=0.24$ and that the corresponding AF structure is ${+}{+}{-}{-}$ with $\bm{\mu}\parallel \mathbf{c}$, as shown in Fig.~\ref{Fig1} for the UNK phase.  For  $x=0.44$, which has $D_{z}<0$, we predict $D_{xy} > D^{\text{lb}}_{xy}$ and that the AF order is either the ${+}{+}{-}{-}$ or the $4$-state clock structure shown for the AF$2$ phase in Fig.~\ref{Fig1}.  Both of these  magnetic structures correspond to $\bm{\tau}_{\sfrac{1}{2}}$ and produce similar neutron diffraction patterns due to the presence of magnetic domains. We cannot differentiate between them using our data.  Similarly, we cannot rule out an amplitude-modulated spin-density wave for either $x=0.24$ or $0.44$.  The absence of evidence for a distortion away from tetragonal symmetry in our high-energy x-ray diffraction data for $x=0.40$ may favor the $4$-state clock structure for AF$2$.

Figure~\ref{Fig3}(a) also illustrates that close to $J_{z}/J^{\prime}_{z} = \pm 2$ the degree of anisotropy needed to suppress helical AF order becomes quite significant.  In particular, for $D_{z}<0$ and a weak dependence of $D_{xy}$ on $x$,  we predict distorted helix states to emerge  when $J_{z}/J_{z}^{\prime}$ falls outside the window where $\bm{\tau}_{\sfrac{1}{2}}$-type order is stabilized. These distorted helix states are multi-${\mathbf Q}$ states, and have a turn angle which varies along $\mathbf{c}$ as the spins are canted towards $\mathbf{a}$ and $\mathbf{b}$ by $D_{xy}$.  The precise form of the distorted helix order, which is not observed for $x=0.44$, may be determined numerically, as done for helical AFs in Refs.~[\onlinecite{Johnston_2017}] and [\onlinecite{Johnston_arxiv_2018}], or by classical Monte-Carlo simulations.

We further test our model by analytically determining the spin-flop and saturation magnetic fields, $h_{\text{sf}}$ and $h_{\text{sat}}$, respectively, for the A-type and $\bm{\tau_{\sfrac{1}{2}}}$-type ground states with $D_{z}>D_{z}^{\text{lb}}$, and compare its predictions to magnetization $M$ versus magnetic field $H$ data \cite{Sangeetha_2017}.  We assume that the $\mathbf{ab}$ component of the flopped spins have helical AF order similar to that for $D_{z}<0$ and negligible $D_{xy}$, and plot the results in Fig.~\ref{Fig3}(b).  In our calculations, $h=g\mu_{\text{B}}H$ and we set $\mu=g\mu_{\text{B}}S=1$, where $g$ is the spectroscopic splitting factor.

We find that our model predicts the observed spin flop in the A-type AF phase \cite{Sangeetha_2017} for $2 <J_{z}/J_{z}^{\prime} < 4$ with
\begin{equation}
h_{\text{sf}}=\frac{SJ_{z}^{\prime}}{4}\sqrt{8d_{z}- (j_{z}-4)^2} \sqrt{-8d_{z}+ (j_{z}+4)^2}\,,
\end{equation}
where $j_{z} = J_{z}/J_{z}^{\prime}$ and $d_{z} = D_{z}/J_{z}^{\prime}$.  A spin flop occurs only if $j_{z} > \sqrt{8\left(d_{z}-2\right)}$; otherwise the compound directly saturates with increasing $H$.  For $-2 < j_{z} < 2$, the spin-flop field is
\begin{equation}
h_{\text{sf}} =\frac{S J_{z}^{\prime}}{4}\sqrt{8d_{z}-j_{z}^2} \sqrt{-8d_{z}+(j_{z}+4)^2} \,, 
\end{equation}
and a spin flop occurs only for $j_{z} > -2+2\sqrt{2 d_{z}-1}$.

For the saturation fields, we find ${h_{\text{sat}}=\frac{SJ_{z}^{\prime}}{4}\left[-8d_{z}+(j_{z}+4)^2\right]}$ for $-2 <J_{z}/J_{z}^{\prime} < 4$.  If the system directly saturates without a spin flop, we find $h_{\text{sat}}=2SJ_{z}$ for $2 <J_{z}/J_{z}^{\prime} < 4$ and $h_{\text{sat}}=S(J_{z}+2J_{z}^{\prime})$ for $-2 <J_{z}/J_{z}^{\prime} < 2$.  

Using the expressions for $2 <J_{z}/J_{z}^{\prime} < 4$, the experimental $M(H)$ data for $x=0$ \cite{Sangeetha_2017} and the experimentally determined value of $\mu=0.43~\mu_{\text{B}}/\text{Co}$ \cite{Jayasekara_2017}, we estimate that $J_{z} \approx 0.24$~meV and $D_{z} \approx 0.08$~meV for $x=0$ and place it on the phase diagram in Fig.~\ref{Fig3}(a) by arbitrarily assuming that $J_{z}/J_{z}^{\prime}=3$.  We place the $x=0.24$ compound in the $\bm{\tau}_{\sfrac{1}{2}}$-type AF ordered region of Fig.~\ref{Fig3}(a) corresponding to FM $J_{z}$ and $D_{z}>D_{z}^{\text{lb}}$ based on the observed AF propagation vector for $x=0.24$ and the fact that magnetization data find evidence for strong FM correlations along $\mathbf{c}$ coexisting with the AF order \cite{Sangeetha_2017}.  In this region, $J_{z}$ is partially frustrated, and may cause the strong FM correlations and a value of $\mu$ much lower than that found for either $x=0$ or $0.44$.  Magnetization data for $x=0.44$ do not show evidence for strong FM correlations along $\mathbf{c}$ \cite{Sangeetha_2017}.  Hence, we place it on the positive side of the $J_{z}/J^{\prime}_{z}$ axis in Fig.~\ref{Fig3}(a).

For Ca$_{1-x}$Sr$_{x}$Fe$_{2}$As$_{2}$, stripe-type AF order persists across the series with an almost constant $\bm{\mu}$ despite $T_{\text{N}}$ increasing by $\approx48\%$ between $x=0$ and $0.3$ \cite{Kirshenbaum_2012}.   The change in $T_{\text{N}}$ is tied to changes in the chemical unit cell size and structure.  This behavior is distinct from our observations for Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$.  Nevertheless, the  crossover from a collapsed-tetragonal to tetragonal phase in Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ and the associated large increase in $c$ and changes to other unit-cell parameters with increasing $x$ \cite{Sangeetha_2017} likely play a role in the variation of $J_{z}$ and $D_{z}$ with composition.

A recent report on electronic-band-structure calculations for $A$Co$_{2}$As$_{2}$, $A=$~Ca, Sr, Ba, find that the overall electronic structures are similar for all three compounds, with only the proximity to a van Hove singularity of a flat band associated with the Co $e_{g}$ orbitals responsible for FM fluctuations within the $\mathbf{ab}$ plane differing between them \cite{Mao_2018}.  The flat band lies just below the van Hove singularity for $A=$~Ca, causing an enhancement of the dynamical susceptibility which may lead to the observed A-type AF order.  The flat band lies further away from the van Hove singularity for $A=$~Sr and Ba, which have not been observed to magnetically order.   The closer proximity of the flat band to the van Hove singularity for $A=$~Ca is tied to a larger Co-Co bond length and lower As height above a Co layer.   It would be interesting to observe how the enhancement in the dynamical susceptibility seen for $A=$~Ca changes as Ca is systematically replaced by Sr.

\section{Conclusion}
Our results highlight the manifestation of highly-tunable and analytically-determinable magnetic ground states in Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$ in the presence of frustrated NL or NNL exchange between FM-aligned square Co layers and magnetic anisotropy.  More generally, we have found that the cobalt-arsenide system manifests strong magnetic frustration both within its square layers and between them.  The origins of frustration within the layers likely trace back to flat electronic bands associated with Stoner-like ferromagnetism \cite{Mao_2018}, whereas here we highlight a different kind of frustration: frustration between FM-aligned layers.  Future band structure calculations and inelastic neutron scattering experiments can provide detailed information on the magnetic state of the layers themselves, and determine whether or not the itinerant FM fluctuations present for $x=0$ persist into the UNK and AF$2$ phases, and if the stripe-type fluctuations found in SrCo$_{2}$As$_{2}$ \cite{Jayasekara_2013} exist in Ca$_{1-x}$Sr$_{x}$Co$_{2-y}$As$_{2}$.  Such work should also result in a better understanding of the microscopic origin of the compositional changes to our Heisenberg model's parameters, as well as the limits of our $1$D local-moment model.

\begin{acknowledgments}
We are grateful for assistance from D.~Robinson with performing the x-ray experiments, and helpful conversations with T. W. Heitmann, D.~H.~Ryan, L.~Ke, and D.~Vaknin.  Work at the Ames Laboratory was supported by the U.~S.~Department of Energy (DOE), Basic Energy Sciences, Division of Materials Sciences \& Engineering, under Contract No.~DE-AC$02$-$07$CH$11358$. A portion of this research used resources at the High Flux Isotope Reactor, a U.~S.~DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.  This research used resources of the Advanced Photon Source, a U.~S.~DOE Office of Science User Facility operated for the U.~S.~DOE Office of Science by Argonne National Laboratory under Contract No.~DE-AC$02$-$06$CH$11357$. Y.~S. and P.~P.~O. acknowledge support from Iowa State University Startup Funds.
\end{acknowledgments}

