Magnetism and magnetoresistance in the critical region of a dilute ferromagnet

We present detailed experimental measurements and simulations of the field-dependent magnetization and magnetoresistance in the vicinity of the Curie temperature in the highly disordered dilute ferromagnetic semiconductor (Ga,Mn)As. The observed dependence of the magnetization on external magnetic field and temperature is consistent with three-dimensional Heisenberg equation of state calculations including a narrow distribution of critical temperatures. The magnetoresistance shows a peak at the Curie temperature due to the suppression of magnetic scattering in an applied magnetic field, which is well-described by considering changes in the square of the magnetization induced by the magnetic field.


Scientific Reports
| (2021) 11:2300 | https://doi.org/10.1038/s41598-021-81893-2 www.nature.com/scientificreports/ Magnetization and its temperature derivative. Detailed magnetometry measurements around the Curie temperature were performed on a 4-mm-sized sample. The sample was first field cooled to 2 K to form a single domain state with saturation magnetization (M s ) of 65 emu/cm 3 . To compare with the simulation result, the magnetometry data are presented in dimensionless quantities of reduced magnetization m = M/M s , reduced magnetic field h = μ 0 μ B H/k B T C and reduced temperature t. Here μ 0 is the vacuum permeability, μ B is the Bohr magneton, and k B is the Boltzmann constant. Figure 1a shows the temperature-dependent reduced magnetization (m vs. t) curves with external magnetic field up to 20 Oe applied along the magnetic easy axis. The magnetization does not drop to zero sharply but has a tail around T C . The size of the tail is field-dependent and can extend more than 1% t (~ 2 K) above the real T C when only a few Oe field is applied. Figure 1b shows the temperature derivative of m (dm/dt vs. t) calculated by numerical differentiation. The dm/dt vs. t curve has a sharp minimum around T C . The peak becomes broader and moves to the right as the magnetic field increases (up to 0.4% of t, for H = 20 Oe).  www.nature.com/scientificreports/ Within the critical region, the relation of m, h, and t can be described by an approximate equation of state for the limit of small t, m, and h/m 26 : with model-dependent critical exponents β, γ and amplitudes a, b. We first performed the numerical simulation to obtain the m vs. t curves around T C using the mean field model with critical exponents β = 0.5, γ = 1 and amplitudes a = 1, b = 1/3 for the simulation. We also introduced a rectangular distribution for T C of 0.5 K (0.28% or ± 0.14% of T C ) to simulate a sample with a linear variation of T C from one side to the other. The calculations ( Fig. 1c-f) indicate a much weaker effect of the external magnetic field compared to the experimental result. For a uniform sample, no shift in the minimum of the simulated dm/dt is observed as the field increases (Fig. 1d). Even with the inclusion of the 0.5 K T C broadening, the dm/dt minimum still only shifts by a maximum of 0.05% (Fig. 1f). Hence, the critical behavior of the highly disordered (Ga,Mn)As system is not well described by the mean field equation of state.
Using three-dimensional Heisenberg critical exponents (β = 0.369, γ = 1.396) and amplitudes (a = 0.82, b = 0.4), the simulated m vs. t curves show similar shape, size and field-dependence of the tail when compared with the experimental data ( Fig. 1g-j). The shift of the minimum of dm/dt with increasing field is also quantitatively consistent with the experiment. These results indicate that the field-and temperature-dependence of the magnetization in the critical region is well-described by the three-dimensional Heisenberg model.

Magnetoresistance. Magnetoresistance measurements were conducted on two 285 μm × 45 μm Hall bars
with current channels along the easy [110] and hard [110] crystalline axes, respectively. The devices were initially field-cooled from room temperature to 2 K to form a single domain state. Then, the longitudinal resistance (R xx ) was measured in zero field as the temperature was increased to above 200 K (Fig. 2a). The Curie temperatures of 178.8 ± 0.2 K and 176.1 ± 0.2 K were estimated from the peak positions in the temperature derivative of the resistivity curves 27 . The reduction of T C compared to the magnetometry sample is because the devices were fabricated from different part of the wafer, and possibly also due to the effect of photolithography and etching.
Magnetoresistance measurements were performed at temperatures around T C with up to 15 Oe field applied along the [110] crystal direction (Fig. 2b,c). The magnetoresistance is defined as the change of resistance when applying a magnetic field: where R xx (H) is the longitudinal resistance in an applied magnetic field of strength H. As temperature approaches T C from below, the MR becomes more and more negative. The MR then becomes less negative with a modified shape of the MR curve after the temperature rises above T C . The most negative MR curve corresponds to a temperature which is slightly higher than the T C obtained from the peak in the temperature derivative of the resistivity. Small differences are observed between the two devices.
We distinguish between anisotropic magnetoresistance (AMR), due to the dependence on the direction of the magnetization relative to the current and crystal, and isotropic MR, which below we relate to the suppression of magnetization fluctuations by the magnetic field. To remove the AMR contribution, we take the average of the |MR| for the [110] and [110] oriented Hall bars at each reduced temperature t (temperature normalized using Curie temperatures obtained previously) where the external magnetic field is respectively parallel and perpendicular to the current. Figure 3a,b show the resulting |MR| as well as |MR|/H versus reduced temperature for different values of H. The curves show a clear peak at t = 0, which broadens and shifts slightly to higher temperatures with increasing magnetic field. The peak is particularly pronounced for the |MR|/H curves, with a full width at half maximum of around 0.5% (1 K) for the lowest magnetic fields used.
Magnetization fluctuations in a ferromagnet close to T C result in a contribution to the electrical resistivity, which in the lowest order approximation is proportional to [1 − m(H,T) 2 ] 18,20-23 . Hence, the relationship between the magnetoresistance and the magnetization in the critical region can be described as: where m 2 = m 2 (h) − m 2 (0) . Figure 3c,d show the plots of δ(m 2 ) and δ(m 2 )/h vs. t, obtained from the magnetometry data. Similar to the magnetoresistance behavior, the curves are sharply peaked at t = 0, with a crossover of the δ(m 2 )/h curves at t > 0. Simulated curves using the three-dimensional Heisenberg equation of state ( Fig. 3e-h) are consistent with both the magnetoresistance and magnetometry experimental results. They also show a clear effect of inhomogeneity, with significant broadening and shift of the peak when a 0.5 K rectangular broadening of T C is included.

Discussion
Our results demonstrate that the magnetoresistance in the critical region of (Ga,Mn)As is well-described by considering only the magnetic field-induced suppression of magnetization fluctuations. The very sharp cusp of MR/H, due to its dependence on the square of the magnetization at a given magnetic field, provides a unique probe of the ferromagnetic phase transition. It can be used for accurate determination of T C , which is an important benchmark property of dilute ferromagnetic semiconductors 5,9,10,19,24,27,28 . www.nature.com/scientificreports/ Calculations using an approximate equation of state with three-dimensional Heisenberg-like critical exponents are in good agreement with the measured magnetic field-dependent magnetization in the critical region, and by extension also with the magnetoresistance. In contrast, mean-field critical exponents give a substantial reduction of the field-induced magnetization at temperatures around T C , compared to the measured data. The mean-field model underestimates the suppression of magnetization fluctuations by an applied magnetic field around T C

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. As a result, the mean-field calculated field-dependence of the peak position of dm/dt is rather weak. The observed agreement of field-dependent magnetization of (Ga,Mn)As in the critical region around T C is well-described by calculations with the 3D Heisenberg model. This is consistent with our previous study of the thermoremanent magnetization in a similar set of samples, for which 3D Heisenberg-like critical exponents were obtained using Kouvel-Fisher analysis 16 . However, due to the intrinsic disorder present in randomly substituted dilute ferromagnetic semiconductors such as (Ga,Mn)As, the accessible critical region is limited to reduced temperatures |t|< 0.025. Any long-range inhomogeneities in the samples will further limit the critical region and may be responsible for some apparently contradictory results in earlier studies [11][12][13][14][15] . Hence, the observed good agreement of the field-induced magnetization with the three-dimensional Heisenberg model, as shown in Fig. 1, provides an important validation of the results of Ref. 16 . This does not rule out the possibility that other universality classes may be found by tailoring the properties of the material, for example by inducing a strong uniaxial magnetic anisotropy by growth on relaxed (In,Ga)As buffer 30 . The dependence of T C (x) in (Ga 1-x ,Mn x ) As (shown in e.g. Ref. 24 ) points to a long-range character of the spin-spin interactions.
The magnetization and magnetoresistance measurements and calculations further demonstrate the marked effect of even a moderate broadening of ∆T C = 0.5 K = ± 0.14%. The peak in the calculated δ(m 2 )/h becomes broader and shifts to higher temperature when the broadening is included. The effect of broadening due to inhomogeneity can also be observed in the experimental magnetoresistance traces; for example, the largest |MR| is observed at a higher temperature than the T C estimated from the dR xx /dT curves.  Magnetometry measurement. The 4-mm-sized sample was measured using a Quantum design MPMS system. The sample first has been measured at 2 K to obtain the saturation magnetization. Then the system has been carefully demagnetized to reduce the remanent field within magnet coil to less than 0.5 Oe, before performing the detailed measurements around Curie temperature.