Symmetry of reentrant tetragonal phase in ${\mathrm{Ba}}_{1-x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$: Magnetic versus orbital ordering mechanism

Magnetostructural phase transitions in ${\mathrm{Ba}}_{1-x}{A}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ (A = K, Na) materials are discussed for both magnetically and orbitally driven mechanisms, using a symmetry analysis formulated within the Landau theory of phase transitions. Both mechanisms predict identical orthorhombic space-group symmetries for the nematic and magnetic phases observed over much of the phase diagram, but they predict different tetragonal space-group symmetries for the newly discovered reentrant tetragonal phase in ${\mathrm{Ba}}_{1-x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ $(x\sim 0.24–0.28)$. In a magnetic scenario, magnetic order with moments along the $c$ axis, as found experimentally, does not allow any type of orbital order, but in an orbital scenario, we have determined two possible orbital patterns, specified by $P4/mnc{1}^{\prime }$ and $I{4221}^{\prime }$ space groups, which do not require atomic displacements relative to the parent $I4/mmm{1}^{\prime }$ symmetry and, in consequence, are indistinguishable in conventional diffraction experiments. We demonstrate that the three possible space groups are, however, distinct in resonant x-ray Bragg diffraction patterns created by Templeton & Templeton scattering. This provides an experimental method of distinguishing between magnetic and orbital models.

Figure 1

(a) Parent $I4/mmm{1}^{\prime }$ crystal structure with atoms in the positions Ba/A $2a(4/mmm{1}^{\prime })$, Fe $4d\left(\overline{4}m{21}^{\prime }\right)$, As $4e\left(4mm{1}^{\prime }\right)$. (b) Orthorhombic $Fmmm{1}^{\prime }$ structure of the nematic phase with atoms in the positions Ba/A $4a\left(mmm{1}^{\prime }\right)$, Fe $8f\left({2221}^{\prime }\right)$, As $8i\left(mm{21}^{\prime }\right)$. (c) Orthorhombic magnetic $C_{A}mca$ structure with atoms in the positions Ba/A $4a\left(m{m}^{\prime }{m}^{\prime }\right)$, Fe $8f\left(2^{\prime }{22}^{\prime }\right)$, As $8g\left(2m^{\prime }{m}^{\prime }\right)$ [6]. The displayed coordinate system does not correspond to the standard setting of $C_{A}mca$ (see Table 1 for the standard setting).

Figure 2

(a) $a$ and $b$ unit cell parameters of ${\mathrm{Ba}}_{0.76}{\mathrm{Na}}_{0.24}{\mathrm{Fe}}_2{\mathrm{As}}_2$ as a function of temperature (parameter for the $I4/mmm{1}^{\prime }$ phase is multiplied by $\sqrt{2})$. (b) Phase fractions of the orthorhombic and tetragonal phases as a function of temperature. (c) Temperature dependence of integrated intensity of the magnetic $(-1/2,1/2,1)$ reflection. Inset shows the temperature dependence of the unit cell parameter $c$. (d) Rietveld refinement of the neutron powder diffraction pattern collected on the WISH diffractometer (ISIS). The cross symbols (black) and solid line (red) represent the experimental and calculated intensities, respectively, and the line below (blue) is the difference between them. Tick marks indicate the positions of Bragg peaks. The first two rows from the top correspond to the nuclear $I4/mmm{1}^{\prime }$ and $Fmmm{1}^{\prime }$ phases; the second two rows represent their magnetic counterparts. Inset shows the patterns collected at 1.5 K and 100 K, at a vicinity of the strongest magnetic peaks. (e) Ordered moment as a function of temperature in the orthorhombic and reentrant tetragonal phases.

Figure 3

(a) Tetragonal magnetic structure with the $P_{C}4/mbm$ space group, involving two propagation vectors ${\mathit{k}}_1=(-1/2,1/2,0)$ and ${\mathit{k}}_2=(1/2,1/2,0)$ and atoms in the positions Ba/A1 $2a(4^{\prime }/m{m}^{\prime }m)$, Ba/A2 $2b(4^{\prime }/m{m}^{\prime }m)$, Fe $8i(2.mm)$, As1 $4g\left(4^{\prime }{m}^{\prime }m\right)$, As2 $4h\left(4^{\prime }{m}^{\prime }m\right)$ [6]. Only the unit cell of the parent $I4/mmm{1}^{\prime }$ structure is displayed (see Table 1 for the choice of the magnetic cell). (b) Atomic displacements (and orbital ordering) with $P4/mmm{1}^{\prime }$ symmetry, allowed in the magnetic $P_{C}4/mbm$ space group as a secondary order parameter, which contribute to the $h+k+l$ odd reflections with $l\ne 0$.

Figure 4

Tetragonal magnetic structure with the $P_C{4}_2/ncm$ space group, involving two propagation vectors ${\mathit{k}}_1=(-1/2,1/2,0)$ and ${\mathit{k}}_2=(1/2,1/2,0)$ and atoms in the positions Ba/A $4c(m^{\prime }.m{m}^{\prime })$; Fe1 $4e\left({\overline{4}}^{\prime }{2}^{\prime }m\right)$, the site with zero magnetic dipole moment; Fe2 $4f\left(\overline{4}{2}^{\prime }{m}^{\prime }\right)$, the site with nonzero magnetic dipole moment; As $8i(2.mm)$ [6]. Only the unit cell of the parent $I4/mmm{1}^{\prime }$ structure is displayed (see Table 2 for the choice of the magnetic cell).

Figure 5

(a) Orbital ordering associated with the $M_3^{+}$ irrep resulting in the tetragonal $P4/mnc{1}^{\prime }$ space group. The crystallographic positions occupied by atoms in $P4/mnc{1}^{\prime }$ are Ba/A $2a(4/m..1^{\prime })$, Fe $4d(2.{221}^{\prime })$, As $4e(4..1^{\prime })$. (b) Combination of the $M_3^{+}$ orbital ordering with the $m{X}_3^{+}({\mu }_3\ne 0,{\mu }_4=0)$ magnetic order, resulting in the orthorhombic $P_{C}ccn$ magnetic space group with atoms in the positions Ba/A $4e(..2^{\prime }/m^{\prime })$, Fe1 $4a\left(2^{\prime }{2}^{\prime }2\right)$, Fe2 $4b\left(2^{\prime }{2}^{\prime }2\right)$, As $8k(..2^{\prime })$ [6].

Figure 6

(a) Orbital ordering associated with the ${\Gamma }_1^{-}$ irrep resulting in the tetragonal $I{4221}^{\prime }$ space group. The crystallographic positions occupied by atoms in $I{4221}^{\prime }$ are Ba/A $2a\left({4221}^{\prime }\right)$, Fe $4d(2.{221}^{\prime })$, As $4e(4..1^{\prime })$. (b) Combination of the ${\Gamma }_1^{-}$ orbital ordering with the $m{X}_3^{+}({\mu }_3\ne 0,{\mu }_4=0)$ magnetic order, resulting in the orthorhombic $C_A{222}_1$ magnetic space group with atoms in the positions Ba/A $4a\left({22}^{\prime }{2}^{\prime }\right)$, Fe1 $4c\left(2^{\prime }{22}^{\prime }\right)$, Fe2 $4d\left(2^{\prime }{22}^{\prime }\right)$, As $8f(.2^{\prime }.)$ [6].

Figure 7

The plane of scattering spanned by primary $\left(\mathit{q}\right)$ and secondary $\left({\mathit{q}}^{\prime }\right)$, and the Bragg wave vector $\mathit{\tau }\left(hkl\right)=\mathit{q}-{\mathit{q}}^{\prime }$. Polarization labeled $\sigma$ and ${\sigma }^{\prime }$ is normal to the plane, and polarization labeled $\pi$ and ${\pi }^{\prime }$ lies in the plane of scattering. The beam is deflected through an angle $2\theta$. In the nominal setting of the crystal, the $b$ axis and $c$ axis are parallel with $\mathit{q}+{\mathit{q}}^{\prime }$ and $\sigma$ polarization, respectively.