Effect of Magnetic Vacancies on the Spontaneous Spin-Reorientation Transition in HoFe$_{1-x}$Al$_x$O$_3$ Single Crystals

Abstract: In this work, we report the first growth of single crystals of the substitution series HoFe$<em>{1-x}$Al$_x$O$_3$ with aluminium concentrations up to $x=0.2$ and investigate the evolution of their spontaneous spin-reorientation transition (SRT). Among rare-earth orthoferrites, HoFeO$_3$ exhibits a distinctive sequence of magnetic phases ($Γ_4$-$Γ</em>{24}$-$Γ<em>{12}$-$Γ_2$). This complex sequence arises from the competition between the $K</em>{ac}$ and $K_{ab}$ anisotropies associated with the Ho$<sup>{3+}$ ions and the effective magnetic field produced by the weak ferromagnetic moment of the canted Fe$<sup>{3+}$ sublattice. Introducing magnetic vacancies perturbs the antiferromagnetic compensation of the Fe$<sup>{3+}$ subsystem in the $ab$ plane and generates an additional effective magnetic field acting on the Ho$<sup>{3+}$ ions. This field alters the balance between the $K_{ac}$ and $K_{ab}$ anisotropies within the SRT temperature range and thereby broadens the stability range of the $Γ_{12}$ phase in the magnetic phase diagram.

• The paper demonstrates that Al substitution in floating-zone-grown HoFe₁₋ₓAlₓO₃ crystals reshapes the spin-reorientation phase diagram by expanding the intermediate Γ₁₂ phase while suppressing Γ₂.
• The study finds that each 5% increase in Al raises the spin-reorientation temperature by about 2.5 K, lowers the Néel temperature, narrows Γ₂₄, and reduces spontaneous magnetization through Fe-sublattice dilution.
• A statistical local-environment model estimates an additional vacancy-induced canting angle of approximately 1.77° at x = 0.05, while neutron or resonant X-ray measurements remain necessary to distinguish Γ₁₂ from Γ₁ order.

Overview

This paper reports the first growth of HoFe$_{1-x}$Al$_x$O$_3$ single crystals across a controlled substitution series ($x = 0$, 0.05, 0.1, 0.2) and investigates how nonmagnetic vacancies in the iron sublattice modify the spontaneous spin-reorientation transition (SRT) (2608.16348). The parent compound HoFeO$_3$ is unusual among rare-earth orthoferrites in exhibiting a four-phase sequence $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$ on cooling, driven by competition between the $K_{ac}$ and $K_{ab}$ anisotropy contributions of the Ho$^{3+}$ ions and the effective field generated by the weak ferromagnetic moment of the canted Fe$^{3+}$ sublattice. The central finding is that magnetic vacancies perturb the antiferromagnetic compensation of the Fe subsystem in the $_x$0 plane, generating an additional effective field on the Ho$_x$1 ions that rebalances $_x$2 and $_x$3 and broadens the stability range of the intermediate $_x$4 phase.

The work extends an earlier program on HoFe$_x$5Mn$_x$6O$_x$7, where Mn substitution raised the SRT temperature from 58 K to 294 K at $_x$8 and changed the transition type from II ($_x$9) to I ($_3$0). The only prior study of Al-substituted crystals, from the early 1990s, used flux-grown samples with composition uncertainty of about 4%, motivating the use of the optical floating-zone method here, which fixes the reagent charge error to no more than 0.05%.

Crystal growth and structural characterization

Polycrystalline charges of stoichiometrically mixed Ho$_3$1O$_3$2, Fe$_3$3O$_3$4, and Al$_3$5O$_3$6 were sintered at 1450 °C and grown in an optical floating-zone furnace in air, with growth rates of 1–3 mm/h depending on the Fe:Al ratio. Powder X-ray diffraction confirms single-phase perovskite samples, with lattice parameters and unit-cell volume decreasing monotonically with $_3$7, consistent with the smaller ionic radius of Al$_3$8 relative to Fe$_3$9:

$x = 0$0	$x = 0$1 (Å)	$x = 0$2 (Å)	$x = 0$3 (Å)	$x = 0$4 (Å$x = 0$5)
0.00	5.2921	5.6029	7.6151	225.80
0.05	5.2701	5.5760	7.5862	222.93
0.20	5.2563	5.5429	7.5527	220.05
1.00	5.1820	5.3240	7.3700	203.33

Laue diffraction patterns along all three principal planes confirm high crystal quality. Magnetization was measured between 4.2 and 350 K (PPMS) and 400–1000 K (VSM); characteristic temperatures and magnetization values were fully reproducible after thermal cycling to 1000 K.

Field-dependent magnetization

Isothermal $x = 0$6 curves at 4.2 K along the $x = 0$7, $x = 0$8, and $x = 0$9 axes reveal strongly anisotropic behavior characteristic of Ho$_3$0 as an Ising-like ion, valid within the quasispin-1/2 approximation below 10 K. From near-saturation data at 9 T, the angle between the Ho$_3$1 moment and the $_3$2 axis is $_3$3–65°, and the total moment lies between 8.8 and 9.1 $_3$4, close to the free-ion value of 10 $_3$5. This establishes that the Ising character of the rare-earth ion survives substitution at all studied concentrations.

With increasing $_3$6, spin-flop transitions emerge for fields along $_3$7 and $_3$8, and the magnetization along $_3$9 increases. These features are attributed to formation of the intermediate $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$0 phase, consistent with the temperature-dependent data discussed next.

Temperature-dependent magnetization and the phase diagram

High-temperature measurements show two systematic effects of substitution: the Néel temperature decreases with $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$1, reflecting dilution of the iron exchange network, and the spontaneous magnetization decreases monotonically, as expected when nonmagnetic Al replaces Fe.

Low-field $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$2 measurements (with a field-history protocol ensuring a single-domain state above the coercive field) yield the SRT evolution. In parent HoFeO$$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$3, the phase boundaries are approximately $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$4–$$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$5 at 60 K, $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$6–$$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$7 at 50 K, and $$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$8–$$\Gamma_4 \rightarrow \Gamma_{24} \rightarrow \Gamma_{12} \rightarrow \Gamma_2$$9 at 40 K; notably, the $K_{ac}$0 phase produces no signature in magnetization for the pure compound. Substitution produces three trends:

• SRT temperature shift: the transition moves to higher temperature by roughly 2.5 K per 5% Al — substantially weaker than the ~40 K per 5% Mn effect in HoFe$K_{ac}$1Mn$K_{ac}$2O$K_{ac}$3.
• Narrowing of $K_{ac}$4: the mixed $K_{ac}$5 phase occupies a progressively smaller temperature interval.
• Emergence of $K_{ac}$6: a minimum in $K_{ac}$7 near 40 K appears for $K_{ac}$8 and broadens with $K_{ac}$9, marking the enlarged $K_{ab}$0 stability range that is invisible in the parent compound's magnetization.

The resulting concentration–temperature phase diagram shows that magnetic vacancies suppress the low-temperature $K_{ab}$1 phase in favor of the mixed $K_{ab}$2 phase below the SRT. The authors attribute this to enhanced competition between $K_{ab}$3 and $K_{ab}$4, which stabilize the $K_{ab}$5 and $K_{ab}$6 tendencies respectively; since vacancies saturate $K_{ab}$7 faster than $K_{ab}$8, the balance tips toward larger deviation of the antiferromagnetic vector G toward the $K_{ab}$9 axis.

Canting-angle analysis via statistical configurations

To quantify the magnetization enhancement in the $^{3+}$0 region, the authors model the local environment statistically: each Fe site has six nearest neighbors, and the probability of $^{3+}$1 Al neighbors follows a binomial distribution $^{3+}$2. For $^{3+}$3, $^{3+}$4 (six Fe neighbors) and $^{3+}$5 accounts for configurations with one or more Al neighbors, weighted by a factor 6/7 for the fraction of magnetic sites among seven positions.

Taking the DMI canting angle of the parent compound as $^{3+}$6 (from $^{3+}$7 per formula unit at 52 K and $^{3+}$8), the excess magnetization observed upon substitution corresponds to an additional canting angle $^{3+}$9 for Fe sites adjacent to vacancies. This analysis rests on two assumptions stated explicitly by the authors: that the weak ferromagnetic moment in the $^{3+}$0 phase equals its saturated low-$^{3+}$1 value (all Fe moments polarized), and that $^{3+}$2 is monotonic within the $^{3+}$3 phase for $^{3+}$4. Both conditions fail for $^{3+}$5 and 0.2, which show a broad maximum above the SRT, so the binomial decomposition is applied only to the lowest substituted composition.

Limitations and open questions

Several caveats bear directly on the interpretation. First, magnetization alone cannot unambiguously distinguish the mixed $^{3+}$6 phase from a pure $^{3+}$7 phase in the substituted crystals; the authors label it $^{3+}$8 while acknowledging it may be $^{3+}$9. Neutron diffraction or resonant X-ray scattering would be required to resolve the actual orientation of the G vector. Second, the quantitative canting-angle model is restricted to $_x$00 and cannot be extended to higher concentrations without additional assumptions. Third, the microscopic mechanism by which vacancies generate an effective field on the Ho$_x$01 sublattice — presumably through modified superexchange paths and altered DMI at vacancy-adjacent bonds — remains inferred rather than directly measured. Finally, whether the $_x$02 range continues to expand beyond $_x$03, eventually stabilizing a full $_x$04 ground state as in the Mn-substituted series, is left open.

Conclusion

This work establishes a controlled series of HoFe$_x$05Al$_x$06O$_x$07 single crystals grown by the optical floating-zone method and demonstrates that nonmagnetic vacancies in the iron sublattice qualitatively restructure the SRT phase diagram: they lower the Néel temperature, shift the SRT upward by ~2.5 K per 5% Al, narrow the $_x$08 window, and stabilize an expanded $_x$09 (possibly $_x$10) phase at the expense of $_x$11. A binomial local-environment analysis quantifies the vacancy-induced increase in DMI canting angle ($_x$12 at $_x$13). The results confirm that magnetic vacancies act on orthoferrite magnetism primarily through the anisotropy balance of the rare-earth subsystem rather than through exchange dilution alone, and they identify direct structural determination of the low-temperature magnetic order as the key missing measurement.