## Abstract

Random field induced spontaneous excess moments appear in field cooled single crystals of diluted Ising antiferromagnets. Here we report results from low temperature measurements of field cooled (including zero field) magnetic hysteresis loops parallel and perpendicular to the *c*-axis of a single crystal of composition Fe_{0.6}Zn_{0.4}F_{2}. We find that weak static ferromagnetic excess moments attained on field cooling give rise to an apparent exchange bias of the magnetic hysteresis loops, whose magnitude is controlled by temperature and the strength and direction of the cooling field. Random field induced temporal excess moments only become observable in cooling fields larger than 1 T applied along the *c*-axis direction of the Fe_{0.6}Zn_{0.4}F_{2} single crystal.

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## Introduction

The antiferromagnet FeF_{2}^{1} is a physical realization of a model 3d Ising system^{2} as well as an important spintronic material^{3,4}. At temperatures below the Néel temperature, T_{N} = 78.4 K, of FeF_{2} an excess magnetic moment develops, that near T_{N} decays with a critical exponent characteristic of 3d Ising systems^{5}. This excess moment gives rise to an apparent exchange bias associated with the vertical shift of the hysteresis loops occurring when cooling the sample through T_{N} in a finite magnetic field^{6}. The excess moment is rigidly locked to the cooling field direction and is virtually unaffected by any magnetic field changes in the antiferromagnetic state. This is reflected in a field dependent rapidly saturating thermo-remnant magnetization (TRM) and zero isothermal remnant magnetization (IRM) at all accessible fields and temperatures below T_{N}^{6}. A similar behavior of the TRM and IRM has been observed in antiferromagnetic Co_{3}O_{4} nanowires^{7,8}, results that are discussed and compared to corresponding results on the dilute antiferromagnet Fe_{1-x}Zn_{x}F_{2}^{9}.

A dilute Ising antiferromagnet in a homogeneous magnetic field (DAFF)^{10,11,12} is a replica of the random field Ising model (RFIM)^{13}. FeF_{2} diluted by diamagnetic dopant Zn (Fe_{1-x}Zn_{x}F_{2} (FZF)) has been extensively used as experimental model systems of a random exchange Ising model (REIM) system in zero applied field and RFIM system in an applied field^{14,15,16,17,18}. Measurements of the field and time dependence of the remnant magnetization of FZF^{18,19,20,21} have revealed characteristics of RFIM systems superposed on an additional low field static remnant moment comparable to that of FeF_{2}.

In this article, we examine the temperature and field dependence of the parallel and perpendicular magnetization of a Fe_{0.6}Zn_{0.4}F_{2} single crystal with special emphasis on apparent exchange bias and the effects of random fields by comparing the parallel and the perpendicular magnetization behaviors.

## Results and discussion

Figure 1a shows the temperature dependence of the ZFC parallel and perpendicular susceptibility at different high magnetic fields. The Néel temperature is found at the temperature where d(M(T)/H)/dT has a maximum, see Fig. 1b. The observed low field T_{N} ~ 48 K and the decrease of T_{N} with increasing field accords with earlier findings on samples of the same composition^{22}. The H dependence of T_{N} stems from the random fields effect induced in DAFF^{23}. It is interesting to note that the differential susceptibility (Δχ = χ_{||} − χ_{⊥}) has somewhat larger relative amplitude than that of pure FeF_{2} (see Fig. 7 in Ref.^{1}) and that the Néel temperature decreases linearly from that of FeF_{2} (T_{N} = 78.4 K) with increasing *x* as has been observed earlier^{24,25}.

Figure 1c,d shows the temperature dependence of the low field magnetic susceptibility χ = M/H recorded after ZFC and FC protocol in H = 5 Oe and 100 Oe for two different orientations, parallel (||*c*) (Fig. 1c) and perpendicular (⊥*c*) to the *c*-axis (Fig. 1d) for Fe_{0.6}Zn_{0.4}F_{2} single crystal. The overall susceptibility χ(T) behavior at low fields is quite similar to that of the FeF_{2} single crystal reported in Ref.^{6}. The irreversibility between the ZFC and FC curves below T_{N} provides the signature of the excess moments associated with uncompensated antiferromagnetic sub-lattices. In the case of FeF_{2}, the excess moment supposedly arises from a distortion of the antiferromagnetic domain structure of piezo magnetic origin, which also has been suggested as the origin of the low field frozen in excess moments of FZF^{19}.

Even though we have used the ultra-low field option before performing all these measurements, a small stray/remnant field was still present in our experiment. Because of this, the observed ZFC (cooled in a stray magnetic field) curves contain a finite excess moment directed along the stray field (see also Fig. SM7 in supplementary material of Ref.^{6}). This effect is demonstrated in Fig. 2a, where the temperature dependence of M(T) was recorded after cooling the sample in a low negative magnetic field H_{FC} = − 5 Oe and measured under fields H = 10 Oe and 100 Oe during the heating and cooling cycle for H||*c*-axis. When cooled in – 5 Oe, the excess moment (M_{exc}) is almost saturated and directed in the negative magnetization direction. Applying a positive field does not change the direction of M_{exc}. Thus, in the M(T) experiments on increasing temperature, the measured M_{FC}(T) = χ_{||}H + M_{exc}(T) in the FC case and M_{-5Oe}(T) = χ_{||}(T)H − M_{exc}(T) when cooled in − 5 Oe. The difference between the cooling and heating curve at a particular temperature is then equal to 2 × M_{exc}(T). This is demonstrated in Fig. 2b, where (M_{FC}(T) − M_{-5Oe}(T))/2 is plotted together with the measured M_{TRM}(T) after FC in 100 Oe. The somewhat smaller magnitude of the derived M_{exc}(T) compared to M_{TRM}(T) is due to the fact that the excess moment is only almost saturated when field cooling in − 5 Oe.

The signatures of random fields are observed in the temperature dependence of thermo-remnant magnetization M_{TRM}(T). The M_{TRM}(T) curve for H (≥ 1 T) applied along the *c*-axis (H||*c*-axis) shows a continuous non-linear increase at low temperatures (T < T_{N}) due to cumulative effect of low field static excess moments and random field induced remnant moments (Fig. 3a). Whereas, for low fields (H = 100 Oe) and for all fields in the case of H⊥*c*-axis only the saturated static excess moment is observed and M_{TRM}(T) curves (Fig. 3a–c) are quite similar to the corresponding curves of FeF_{2} (inset of 3c)^{6}. See supplementary Fig. SM2 for M_{TRM}(T) curves represented in µ_{B}/Fe for two selected H_{FC} = 10 mT and 4 T. A comparison of M_{TRM} curves for both orientations under H_{FC} = 4 T are shown in Fig. 3d. The logarithmic dependence of ΔM_{TRM}(= M_{TRM}(H_{FC}, 10 K) − M_{TRM}(H_{FC} = 1 T, 10 K)) on log(H_{FC}) for H||*c*-axis (shown in the inset of Fig. 3a) follows a linear behavior with a slope of ~ 2.8, which is in agreement with the predicted behavior due to random field effects and in agreement with earlier findings^{18}.

Figure 4a shows the M(H) curves recorded at T = 10 K with field sweep 0 → + 1 T → − 1 T → + 1 T after ZFC for H||*c*-axis and H⊥*c*-axis. A shift in the hysteresis loops (apparent exchange bias^{6}) is observed for both orientations when the sample is cooled from T = 60 K to T = 10 K in H_{FC} = 100 Oe as shown in Fig. 4b. The observed shift |H_{EB}| is 36 Oe and 170 Oe for ⊥ c and || c orientations, respectively. The effect of temperature on the FC M(H) curves are shown in Fig. 5. The M(H) curves are recorded after FC in H_{FC} = 100 Oe from 60 K down to different temperatures with field sweep 100 Oe → + 1 T → − 1 T → + 1 T. Zoomed views of the M(H) curves at low fields are shown in Fig. 5c,d. A significant decrease in the slope χ = dM/dH and increase in the apparent exchange bias H_{EB} are observed as the measuring temperature decreases from T = 60 K → 2 K, for H||*c*-axis. However, for H⊥*c*-axis, these changes are very subtle as compared to H||*c*-axis. In Fig. 6 M(H) curves recorded after cooling the sample form 60 K to 2 K under different cooling fields with field sweep + H_{FC} → 5 T → − 5 T → 5 T are shown for (a) H||*c*-axis and (b) H⊥*c*-axis (see supplementary Fig. SM4 for the corresponding T = 10 K data). In contrast to H⊥*c*-axis, M(H) curves for H||*c*-axis show non-linear asymmetric behavior. This behavior becomes more prominent with increasing H_{FC} and decreasing T. This trend can be clearly noticed from the absolute value of M at maximum fields (± 5 T) for T = 2 K and 10 K (see inset of supplementary Fig. SM4b). The difference plot of the M(H) curves recorded after FC in H_{FC} = 5 T and H_{FC} = 1 T (∆M = M(H_{FC} = 5 T,H) − M(H_{FC} = 1 T,H)) shown in inset of Fig. 6a confirm the upward shift of the 5 T curve, compared to the 1 T curve, due to random field induced excess moments. Interestingly, it also implies an enhanced overall response to magnetic field changes of the sample when cooled in 5 T, compared to the M vs H curve of the sample when cooled in a lower field (1 T), where random field induced excess moments are hidden by the larger static excess moments. Another observation is that temperature dependent relaxation effects yield intrinsic coercivity changes with increasing H_{FC}, as illustrated in Fig. 6c,d (note the direction of the arrows marking the magnetic field sweeps in those panels in comparison to the low-field results presented in Fig. 4). The observed temporal intrinsic coercivity H_{C} is 330 and 140 Oe at 10 K and 2 K, respectively, for H_{FC} = 5 T. Whereas, the small spurious coercivity noticed in ZFC and H_{FC} = 1 T M(H) curves is associated with the artifact due to the field history dependent residual field in the superconducting magnet of the SQUID magnetometer (see Fig. 4 and supplementary Fig. SM5). Neither non-linearities nor intrinsic coercivity enhancement is observed in undoped FeF_{2} in the same magnetic field range, as illustrated in Fig. 6e. The results of ZFC M vs. H measurements H||*c*-axis up to fields of 18 T on Fe_{0.6}Zn_{0.4}F_{2}^{22} indicate that the hysteresis properties of a sample that has been field cooled in very high magnetic field (e.g. 18 T) should show interesting new features compared to the ZFC sample.

The linear M vs H curves observed for fields H⊥*c*-axis and low fields H||*c*-axis can be described by the equation: M = χH + M_{R}, yielding H_{EB} = − M_{R}/χ at M = 0. For comparison to this simple description, the parameters H_{EB}, M_{R} and χ(inset) extracted from Figs. 5 and 6 are plotted in Fig. 7 for both H||- and ⊥*c*-axis. The behavior of M_{R}/χ is consistent with the observed values of the apparent H_{EB}. For H||*c*-axis the M_{R} dependence at higher cooling fields accords with that predicted for RFIM systems^{18}. As seen in Fig. 7b, H_{EB} value approaches ~ 5 kOe at 2 K (H_{FC} = 5 T). These observations are summarized in Fig. 8. Figure 8 shows the temperature dependence of the ratio of M_{TRM}(H_{FC}) (from Fig. 3a) divided by the susceptibility M_{FC}/H recorded at H = 1 T (Fig. 1a) for H||*c*-axis. For comparison |H_{EB}| determined from the M(H) curves (Fig. 6a and supplementary Fig. SM4a) is shown by square symbols. The inset shows a comparison of M_{TRM}/(M_{FC}/H), |H_{EB}| and M_{R}/χ for H_{FC} = 100 Oe.

## Conclusions

The field and temperature dependence of the magnetization measured parallel and perpendicular to the *c*-axis of the dilute Ising antiferromagnet Fe_{0.6}Zn_{0.4}F_{2} reveal apparent exchange bias effects governed by static excess moments, which at higher fields in the case of H||*c*-axis are enhanced by random field induced temporal excess moments. H_{EB} amounts to ~ 5kOe at 2 K (H_{FC} = 5 T). The observed hysteresis behavior with field H⊥*c*-axis and at low fields H||*c*-axis is similar to that observed in FeF_{2}^{6}, indicating that the origin of static excess moments is the same in pure and diluted samples. Irrespective of origin of the excess moments, the vertical shift (apparent exchange bias) of the hysteresis curves is directly reflected in the field dependence of the TRM(H) and IRM(H) curves^{7,8,9}; where IRM(H) = 0 at all accessible field, i.e. no remnant magnetization is attained on a zero field cooled sample. Systems with such properties exhibit magnetic hysteresis curves with apparent exchange bias that are fully controlled by the cooling field and the measurement temperature. The temporal excess moment give rise to cooling field and field sweep rate dependent intrinsic coercivity that of the order of ~ 330 Oe at 10 K.

## Methods

The temperature and field dependent magnetization measurements for Fe_{0.6}Zn_{0.4}F_{2} single crystal were performed using a superconducting quantum interference device (SQUID) magnetometer from Quantum Design Inc. (Model: XL). The same single crystal was used in Ref.^{5}. The magnetic field H was applied along two different orientations of the single crystal (*i*) parallel to the *c*-axis (H||*c*-axis) and (*ii*) perpendicular to the *c*-axis (H⊥*c*-axis). The temperature dependence of the magnetization M(T) was recorded in zero field cooled (ZFC) and field cooled (FC) conditions in different magnetic fields. The thermo-remnant magnetization (TRM) was recorded on warming in zero magnetic field, after cooling the sample from 60 K down to 2 K in presence of an applied magnetic field. The schematic of M(T) measurement is shown in supplementary Fig. SM1. The field dependence of the magnetization M(H) was recorded at a temperature T after zero-field cooling from 60 K down to low temperatures (ZFC M(H)) and after field cooling in H_{FC} (FC M(H)). For the FC M(H) measurements, the field is increased from H_{FC} to the maximum field (H_{max}) then decreased to − H_{max} and finally increased back to H_{max}. Before performing all of the ZFC or FC measurements with H_{FC} ≤ 100 Oe, the background magnetic field of the magnet was reset to zero by using the ultra-low field option.

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## Acknowledgments

We thank the Stiftelsen Olle Engkvist Byggmästare and the Swedish Research Council (VR) for financially supporting this work.

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Open Access funding provided by Uppsala University.

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D.C.J., P.N., and R.M. planned and performed the experiments and jointly wrote the final manuscript.

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Joshi, D.C., Nordblad, P. & Mathieu, R. Random fields and apparent exchange bias in the dilute Ising antiferromagnet Fe_{0.6}Zn_{0.4}F_{2}.
*Sci Rep* **10**, 14588 (2020). https://doi.org/10.1038/s41598-020-71533-6

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DOI: https://doi.org/10.1038/s41598-020-71533-6

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