Tuning interchain ferromagnetic instability in $A_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ ternary arsenides by chemical pressure and uniaxial strain

We analyze the effects of chemical pressure induced by alkali-metal substitution and uniaxial strain on magnetism in the $A_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ ($A$=Na, K, Rb, and Cs) family of ternary arsenides with quasi-one-dimensional structure. Within the framework of the density functional theory, we predict that the nonmagnetic phase is very close to a three-dimensional collinear ferrimagnetic state, which realizes in the regime of moderate correlations, such tendency being common to all the members of the family with very small variations due to the different interchain ferromagnetic coupling. We uncover that the stability of such interchain ferromagnetic coupling has a nonmonotonic behavior with increasing the cation size, being critically related to the degree of structural distortions which is parametrized by the Cr-As-Cr bonding angles along the chain direction. In particular, we demonstrate that it is boosted in the case of the Rb, in agreement with recent experiments. We also show that uniaxial strain is a viable tool to tune the nonmagnetic phase towards an interchain ferromagnetic instability. The modification of the shape of the Cr triangles within the unit cell favors the formation of a net magnetization within the chain and of a ferromagnetic coupling among the chains. This study can provide relevant insights about the interplay between superconductivity and magnetism in this class of materials.

Figure 1

(a) Cr triangles belonging to the $[{\left({\mathrm{Cr}}_3{\mathrm{As}}_3\right)}^{2-}{]}_{\infty }$ subnanotubes of the $A_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ compounds. The deviation from the ideal hexagonal structure is parametrized by the angles ${\alpha }_1$ and ${\alpha }_2$. Blue and green circles denote Cr and As atoms, respectively. (b),(c) Cr-As-Cr bonding angles along different chain direction (see the reported axes).

Figure 2

Cr-spin configurations investigated in the paper, defined as (a) the ferromagnetic state (FM), (b) the interlayer antiferromagnetic state (AFM), (c) the up-up-down/up-up-down ($\uparrow \uparrow \downarrow \text{−}\uparrow \uparrow \downarrow$) stripe state, and (d) the up-up-down/down-down-up ($\uparrow \uparrow \downarrow \text{−}\downarrow \downarrow \uparrow$) zigzag state.

Figure 3

Energies per Cr atom of the FM, AFM, zigzag, and NM states measured with respect to the stripe state energy, plotted as functions of the Coulomb repulsion for the compound ${\mathrm{Cs}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 4

Angle ${\alpha }_1$ as a function of the Coulomb repulsion for the compound ${\mathrm{Cs}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 5

Magnetic moment of the ${\mathrm{Cr}}_1$ ion as a function of the Coulomb repulsion for the compound ${\mathrm{Cs}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 6

Difference between the energies per Cr atom of the zigzag and of the stripe state as a function of the Coulomb interaction for the compounds of the family $A_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ ($A$=Na, K, Rb, and Cs).

Figure 7

Angle ${\alpha }_1$ in the ground state as a function of the Coulomb repulsion for the compounds of the family $A_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ ($A$=Na, K, Rb, and Cs).

Figure 8

Magnetic moment $m_1$ in the ground state as a function of the Coulomb repulsion for the compounds of the family $A_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ ($A$=Na, K, Rb, and Cs).

Figure 9

Energy difference between the NM and the interchain FM state (blue line, left axis) and bonding angle ${\mathrm{Cr}}_6\text{−}{\mathrm{A}}_6\text{−}{\mathrm{Cr}}_6$ (red line, right axis) for different choices of the cation. The value of the Coulomb repulsion is $U=0.75\mathrm{eV}$.

Figure 10

Behavior of the four bonding angles Cr-As-Cr for different choices of the cation. The value of the Coulomb repulsion is $U=0.75\mathrm{eV}$.

Figure 11

Graphic representation of the strain $\varepsilon$ applied orthogonally to the basis of the isosceles triangles in the chain. A negative value of $\varepsilon$ indicates a compressive strain of the $y$ component of the ${\mathbf{R}}_2$ vector, while a positive one indicates a tensile strain of the $y$ component of the ${\mathbf{R}}_2$ vector.

Figure 12

Energies per Cr atom of the FM, AFM, zigzag, and NM states measured with respect to the stripe state energy, plotted as functions of the Coulomb repulsion for the compound ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$. A compressive strain $\varepsilon =-0.03$ is applied to the system. At low values of $U$ some data are missing due to the lack of convergence of the numerical procedure in that regime.

Figure 13

Comparison of the magnetic moments of the ${\mathrm{Cr}}_1$ atom in the ground state for different choices of compressive/tensile strain applied to ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 14

Comparison of the ${\alpha }_1$ angle in the ground state for different choices of compressive/tensile strain applied to ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 15

Band structure of ${\mathrm{Na}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ near the Fermi level for $U=0.3\mathrm{eV}$ (left), together with the corresponding orbitally resolved partial densities of states (right). Red and green lines refer to the ${\mathrm{Cr}}_1$ atom at the basis of the triangle and denote the DOSs projected onto orbitals symmetric with respect to the basal plane $d_{xy}$, $d_{x^2-y^2}$, and $d_{z^2}$ and onto orbitals antisymmetric with respect to the basal plane $d_{yz}$ and $d_{xz}$, respectively. Blue and purple lines refer to the ${\mathrm{Cr}}_2$ atom at the vertex of the triangle and denote the DOSs projected onto the symmetric and antisymmetric orbitals, respectively.

Figure 16

Same as in Fig. 15 for ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 17

Same as in Fig. 15 for ${\mathrm{Rb}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 18

Same as in Fig. 15 for ${\mathrm{Cs}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 19

Same as in Fig. 15 for ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ when a tensile strain $\varepsilon =+0.06$ is applied to the system.

Figure 20

Same as in Fig. 15 for ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ when a compressive strain $\varepsilon =-0.06$ is applied to the system. In panel (a) we show the spin-up channel, while panel (b) shows the spin-down channel.

Figure 21

Energies per Cr atom of the FM, AFM, zigzag, and NM states measured with respect to the stripe state energy, plotted as functions of the Coulomb repulsion for the compound ${\mathrm{Na}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$. At low values of $U$ some data are missing due to the lack of convergence of the numerical procedure in that regime.

Figure 22

Same as in Fig. 21 for the compound ${\mathrm{Rb}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 23

Angle ${\alpha }_1$ as a function of the Coulomb repulsion for the compound ${\mathrm{Na}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$. At low values of $U$ some data are missing due to the lack of convergence of the numerical procedure in that regime.

Figure 24

Same as in Fig. 23 for the compound ${\mathrm{Rb}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 25

Magnetic moment of the ${\mathrm{Cr}}_1$ ion, as a function of the Coulomb repulsion for the compound ${\mathrm{Na}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$. At low values of $U$ some data are missing due to the lack of convergence of the numerical procedure in that regime.

Figure 26

Same as in Fig. 25 for the compound ${\mathrm{Rb}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$.

Figure 27

Arrangements of the Cr spin of the magnetic configuration investigated in addition to the previous ones discussed in the main text in order to obtain the magnetic exchanges. This new magnetic configuration, which we name “stripe2” for further reference, has the Cr atoms at the basis with opposite spin direction. $J_a$ is the magnetic coupling between the Cr atoms at the basis and those at the apex of the isosceles triangles, as for example, ${\mathrm{Cr}}_1$ and ${\mathrm{Cr}}_2$; $J_b$ is the magnetic coupling between the two Cr atoms at the basis, as ${\mathrm{Cr}}_1$ and ${\mathrm{Cr}}_3$. $J_c$ is the interplane coupling constant, relating, for example, to atoms ${\mathrm{Cr}}_1$ and ${\mathrm{Cr}}_4$.

Figure 28

Magnetic exchanges for ${\mathrm{Na}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ as functions of the Coulomb repulsion $U$. $J_a$, $J_b$, and $J_c$ are three magnetic exchanges defined in the text. The dashed line indicates the extrapolation in the region where the mapping is not possible.

Figure 29

Magnetic exchanges for ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ as functions of the Coulomb repulsion $U$. $J_a$, $J_b$, and $J_c$ are three magnetic exchanges defined in the text. The dashed line indicates the extrapolation in the region where the mapping is not possible.

Figure 30

Magnetic exchanges for ${\mathrm{Rb}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ as functions of the Coulomb repulsion $U$. $J_a$, $J_b$, and $J_c$ are three magnetic exchanges defined in the text. The dashed line indicates the extrapolation in the region where the mapping is not possible.

Figure 31

Magnetic exchanges for ${\mathrm{Cs}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ as functions of the Coulomb repulsion $U$. $J_a$, $J_b$, and $J_c$ are three magnetic exchanges defined in the text. The dashed line indicates the extrapolation in the region where the mapping is not possible.

Figure 32

Magnetic exchanges of ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ as functions of the strain for $U=0.85$ eV and $J_a$ in the case of $U=0.75$ eV. $J_a$, $J_b$, and $J_c$ are three magnetic exchanges defined in the text.