Heusler compounds have drawn much attention in recent years due to their high Curie temperature (T_C)^{[1, 2]} and flexible tunability^[3–6], making them complex and diversified, and thus bringing a lot of interesting phenomena^[3]. Among these materials, Co–Mn–Al ternary Heusler alloy is an attractive system, especially in the cases of Co₂MnAl as a half-metal^[7] and Mn₂CoAl as a typical spin gapless semiconductor^[8–10]. The Fermi levels in both cases lie in the gaps of their minority-spin bands. But in their majority-spin bands, Co₂MnAl acts like a metal, while Mn₂CoAl is close to a zero-gap semiconductor^[11] featuring a semiconducting-like transport behavior^{[8, 10]}.

Different from Co₂MnAl or Mn₂CoAl, we have synthesized Co_1.65Mn_1.35Al (CMA), an off-stoichiometric alloy Co₂MnAl, in which perpendicular magnetic anisotropy (PMA) emerges as a result of residual strain. This material provides a material platform for studying the effect of strain on spin-orbit coupling related phenomena such as anomalous Hall effect (AHE), and could also be a promising candidate for potential applications in spintronic devices due to its PMA characteristic^[12]. Therefore, it is meaningful to investigate the physical properties, especially the magneto-transport behavior of the CMA film systematically.

In this work, we report the temperature T dependence of the longitudinal resistivity ρ_xx and the negative magneto-resistance (MR) at different temperatures of a 20 nm-thick CMA film. The semiconducting-like transport character in the T range between 5 and 300 K implies the existence of an activation energy E_a. It is found that ρ_xx can be described by a simple activation model with varying E_a in different T regions. The T dependences of the magnetization M are also different correspondingly. The first transition temperature at around 110 K could be ascribed to a transition from localized to itinerant ferromagnetism. We measure the Hall resistivity at several temperatures and obtain the intrinsic anomalous Hall conductivity of ~55 Ω^–1cm^–1, which is almost twenty times smaller than that of Co₂MnAl^[13].

The single-crystalline film of CMA was epitaxially grown on GaAs (001) substrate by molecular-beam epitaxy (MBE). After the GaAs de-oxidation and buffer layer deposition, a 20 nm-thick CMA film was grown at 200 °C, capped with 2 nm-thick aluminum for protection from oxidation. The sample was patterned to a standard Hall bar, with the channel along the GaAs [110] direction for transport measurements. The magnetic field H was applied perpendicular to the film plane, i.e., parallel with the GaAs [001] direction. The transport data of the film were collected with T between 5 and 300 K and H up to 50 kOe using a physical property measurement system (Quantum Design, PPMS-9). The magnetic properties of CMA film were measured using a superconducting quantum interference device (SQUID) down to 5 K.

Fig. 1(a) shows the dependence of ρ_xx on T varying from 5 to 300 K with a zero magnetic field. ρ_xx increases with decreasing temperature in the whole range of T in our experiment. Different from Co₂MnAl^[13], ρ_xx of the CMA film resembles that of Mn₂CoAl films following an activation-type dependence^{[8, 10]}, which is usually described by a simple activation model^{[10, 14]}: σ_xx(T) = 1/ρ_xx(T) = σ₀ + σ_aexp(–E_a/k_BT), where σ₀ denotes the non-stoichiometric contributions at the low temperature and σ_a is a coefficient with the same dimension as conductivity, respectively. As shown in Fig. 1(a), an obvious deviation from the fitting curve to ρ_xx(T > 110 K) is found at temperatures lower than 110 K, which could result from the change of the E_a in the equation. Thus we learn this transition between different T regions divided by the T points at 35 and 110 K. The best fits of the data yield the critical parameters of E_a1 = 34 meV, σ₀₁ = 2.97 × 10^–3μΩ^–1cm^–1 (110 to 300 K), E_a2 = 15 meV，σ₀₂ = 2.93 × 10^–3μΩ^–1cm^–1 (35 to 110 K), and E_a3 = 1.4 meV, σ₀₃ = 2.89 × 10^–3μΩ^–1cm^–1 (5 to 35 K), respectively. To clearly show the change of E_a, we plot the relations of ln(Δσ_xx) = ln(σ_xx – σ₀) versus 1/T in the inset of Fig. 1(a), which depict good linear dependencies and different slopes in three T regions. According to these results, the CMA film is discovered to have a decreasing E_a when T is lowered, which could be attributed to a band shift with T.

The relations of MR with H are selectively presented in Fig. 1(b). The MR is defined as (ρ_xx– ρ_xx0)/ρ_xx0 × 100%, where ρ_xx0 = ρ_xx(H = 0). The negative MR is observed over the whole T range, which is usually ascribed to the spin-dependent scattering in similar systems^{[10, 15]}. And it means that this scattering, which is inevitably affected by magnetism, dominates MR compared to the contributions of the cyclotron effect in our measurement with H lower than 50 kOe^[16]. There are two major changes of MR with T andH, respectively. One is that MR (T > 200 K) clearly changes much more slowly than those at low temperatures when H is not larger than 30 kOe. This variation could be an evidence for different activation modes in different T regions, consistent with the implications from ρ_xx(T). Another difference is that the MR approaches a linear trend when it reaches about 30 kOe and higher. This linear dependency between MR and H has been reported in the Mn₂CoAl and some other spin gapless systems^{[8, 17]}. Moreover, the MR of the CMA film shows a maximum in the temperatures between 50 and 100 K as shown in the inset of Fig. 1(b). The maximum value of MR here could possibly result from the change of the band structure rather than the combined influence of the impurity and the spin-dependent scattering reported in Mn₂-CoAl^{[8, 9]}.

In order to learn whether a similar transition happens for the ferromagnetism of this sample, we carried out the magnetic measurements using a SQUID magnetometer. The sample was cooled down to 5 K with H = 0 and then saturated by a magnetic field as high as 50 kOe. After H was set at 20 Oe, the T dependence of M along the [001] direction was measured by heating the sample from 5 to 280 K. The results are shown in Fig. 2(a). For our CMA film, a linear dependence of M² on T² is observed with T ranging from about 110 to 280 K, which is ordinary in itinerant ferromagnetic materials such as MnSi^[18] and ZrZn₂^[19]. Therefore, M(T) can be fitted by the formula of M(T) = M(0)×[1 – (T/T_C)²]^1/2. The best fit yields the parameters of M₁(0) = 292 emu/cm³ and T_C = 640 K, close to 693 K of Co₂MnAl^[2] and 720 K of bulk Mn₂CoAl^[8]. However, an obvious deviation from this T² behavior is observed when T is lower than 110 K. In the T range between 43 and 110 K, the more rapid variation of M versus T can be described by the classical Bloch T^3/2 law: M(T) = M(0)(1 – ηT^3/2), where η is related to the spin-wave stiffness constant and exchange coupling energy. It means that the variation of M(T) in this T range could be attributed to collective spin wave excitations^[17]. The best fit of the curve yields the M₂(0) and η values of 294 emu/cm³ and 1.8 × 10^–5 K^–3/2, respectively. Furthermore, an enhanced changing rate of the M(T) curve occurs in the lowest T range between 5 and 35 K. Although the picture of this anomalous behavior is still unclear in our system, the M(T) curve may be described by an exponential dependence: M(T) = M(0)(1 – αexp(–Δ/T)), where the coefficients α = 1.64 × 10^–2 and Δ = 3.74 × 10^–2 K are obtained. To show the change of M(T) near 110 K more clearly, the dependences of ln(1 – M(T)²/M(0)²) (T ~ 110–280 K) and ln(1 – M(T)/M(0)) (T ~ 43–110 K) on T are plotted in Fig. 2(b), showing good linearity in both regions. Combining the above results of transport and magnetic measurements, the transition of ρ_xx and M could be originated from the same contributions because similar changes take place near 35 and 110 K.

Fig. 3(a) shows the measured Hall resistivity ρ_xy varying with H and T after the subtraction of the background signals coming from ρ_xx. ρ_xy is dominated by the anomalous Hall signal, which reflects the ferromagnetism corresponding to the hysteresis loop shown in the inset. The Hall resistivity is generally described by the formula: ρ_xy(H) = R_oH + R_AM(H), where R_o and R_A are the ordinary and anomalous Hall coefficient, respectively. When H is large enough to saturate a ferromagnetic material, the slope of ρ_xy(H) is mainly determined by R_o = 1/(nq) based on the one band model, where n is the carrier concentration and q equals the charge of an electron. However, in the CMA film, we notice that the slope of ρ_xy(H) gradually becomes larger when H is increased, which seems to be contradicted with the saturation state shown in the hysteresis loop. Considering the different behaviors of MR when H is lower and higher than 30 kOe, M could also change gradually with H which is not evident compared to the remanent magnetization. We then obtain ρ_AH as the value of ρ_xy(H = 0) for estimating the anomalous Hall conductivity (AHC). In general, the anomalous Hall resistivity ρ_AH can be described in terms of ρ_xx in the form of ρ_AH = βρ_xx + γρ_xx², where β characterizes the extrinsic skew scattering and γ denotes the intrinsic origin or the extrinsic side-jump scattering^[20]. The ρ_xx and ρ_xy are measured simultaneously in our experiment and ρ_AH shows a linear dependence on ρ_xx² as shown in Fig. 3(b). The γ can be described by the slope value of ρ_AH versus ρ_xx², which is estimated to be ~55 Ω^–1cm^–1 in favor of a topological origin from Berry phase effect in the case of ρ_xx ~ 300–350 μΩ·cm^{[9, 20, 21]}. Therefore, the intrinsic AHC of the CMA film is almost twenty times smaller than that of Co₂MnAl film^[14] but relatively larger than that of bulk Mn₂CoAl^[8].

In conclusion, we have systematically investigated the magneto-transport properties of a 20 nm-thick CMA film. The film exhibits semiconducting-like transport behavior below 300 K. Its longitudinal resistivity ρ_xx(T) is properly described by a simple activation model with different activation energies in three T regions separated by 110 and 35 K. The activation energies are 34, 15 and 1.4 meV in the temperature range of 110–300, 35–110, and 5–35 K, respectively. Similar transitions at nearly the same temperature in M(T) have also been discovered, implying a same origin of the transitions of ρ_xx(T) and M(T), which might probably be related to the bands’ shift. The Curie temperature is estimated to be ~640 K, which is a typical value among full-Heusler alloys. An intrinsic AHC of about 55 Ω^–1cm^–1 is obtained from the linear dependence of ρ_AH versus ρ_xx². Further experiments and calculations are needed to understand the relations between the transitions and the band structures of the film.

This work was supported by the Ministry of Science and Technology under Grant Nos. 2015CB921500, 2017YFB0405701 and the National Natural Science Foundation of China under Grant Nos. U1632264 and 11704374.