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Random fields and apparent exchange bias in the dilute Ising antiferromagnet Fe0.6Zn0.4F2

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 Fe0.6Zn0.4F2. 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 Fe0.6Zn0.4F2 single crystal.

. 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. 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  www.nature.com/scientificreports/ 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)    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. 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 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   www.nature.com/scientificreports/

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.  www.nature.com/scientificreports/

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⊥caxis). 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.