Disordered spin gapless semiconducting CoFeCrGa Heusler alloy thin films on Si (100): experiment and theory

Abstract

Spin gapless semiconductors (SGSs) are an intriguing class of quantum materials that bridge the gap between half-metallic ferromagnets and semiconductors. The presence of a semiconducting bandgap for one spin channel and zero band gap for other spin channels, together with the possibility of four different band structure configurations, makes them one of the most desirable candidates to be used in tunable spin transport based spintronics devices. Here, we have performed various structural, magnetic and transport measurements on an optimized CoFeCrGa (CFCG) Heusler alloy thin film (∼50 nm) grown over a Si(100) substrate using an industry-viable magnetron sputtering technique. The grown film showed B2-ordering under the given set of X-ray diffraction measurement conditions with a saturation magnetization (M_s) of 1.86μ_B per f.u. (at 5 K) and a Curie temperature of ∼595 K. Nearly linearly varying longitudinal resistivity with a negative temperature coefficient was observed. A fitted longitudinal conductivity curve through a “two-carrier model” shows a slight band overlap in the gapless channel for one spin channel and a small energy gap (ΔE) of 167 meV for other spin channels. A negative and linear out-of-plane magnetoresistance response was observed in these films. The temperature dependent anomalous Hall effect measurement gives nearly temperature independent carrier concentration (and/or) mobility with an anomalous Hall conductivity of 91.35 S cm⁻¹ at 5 K. The first principles calculations have also been performed for bulk and (220) CFCG surfaces to correlate the various structural, electronics and magnetic properties of the optimized CFCG Heusler alloy thin film. The DFT derived results, viz. lattice parameter and M_S exhibit a good match with the experimentally observed results. All these properties collectively imply that the grown film possesses disordered-SGS like behaviour. It is remarkable to note that CFCG films with the (022) surface possess a very high electronic spin polarization of 91%. The results of the study suggest that CFCG is a potential candidate to be used in spintronics-based devices such as spin-injectors.

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1. Introduction

Based on the electronic band structure, materials in nature are broadly classified into metals, semiconductors/insulators and semi-metals with the energy band schematics as shown in Fig. 1(a), (b), and (c), respectively. In the current era, spintronics has captivated immense attention in the field of research from both applied and fundamental points of view. The presence of spin-polarization at the Fermi level is the key requirement for a material to be used for spin manipulation in spintronics-based devices. Half-metallic ferromagnets (HMFs) are thus considered as an ideal candidate for spintronics applications as they possess 100% spin polarization at the Fermi level [as shown in Fig. 1(d)] and hence are capable of producing considerable spin polarized current.^1–3 The current spintronics technology can potentially use HMFs in devices like magnetic tunnel junctions (MTJs), non-local spin-valve (NLSV), current perpendicular to plane giant magnetoresistance (CPP-GMR), spin-transfer torque (STT) oscillators,^4–8etc.

Fig. 1 The schematic of density of states n(E) as a function of energy (E) corresponding to a (a) metal, (b) semiconductor (small band gap)/insulator (large band gap), (c) semi-metal, (d) half-metal and (e) spin-gapless semiconductor. The up and down arrows represent the states corresponding to the majority and minority spins.

A very new class of materials namely spin-gapless semiconductors (SGSs) in the family of HMFs has recently come into existence which is anticipated to play a crucial role in the semiconductor spintronics industry.^9–11 The SGS behaviour was first predicted in Co-doped PbPdO₂ through first principles electronic and band structure calculations using density functional theory.⁹ These materials possess a semiconducting bandgap for the minority spin channel and a zero bandgap for the majority spin channel at the Fermi level [see Fig. 1(e)] and hence act as a bridge between HMFs and semiconductors.⁹ Their unique band structure leads to some specific properties like generation of spin-polarized current from both electrons and holes, ability to switch from n-type to p-type spin polarized current by inducing some external field and need for almost negligible energy for exciting electrons from a valence band to a conduction band. The unique band structure of SGS materials thus makes them an ideal replacement for diluted magnetic semiconductors (DMS) with a Curie temperature (T_C) in excess of 300 K. The presence of the SGS characteristic in Heusler alloys makes them more prominent over DMS because of their relatively high T_C.¹²

Recently, many Heusler compounds have been reportedly found to exhibit the SGS characteristic as confirmed by ab initio calculations.^11,13–15 However, very few of them have been confirmed experimentally, e.g. Mn₂CoAl, CoFeMnSi, Ti₂MnAl and CoFeCrGa.^10,16–19 CoFeCrGa (CFCG) is an alloy, an equiatomic quaternary Heusler alloy (EQHA), with a 1 : 1 : 1 : 1 stoichiometry. These compounds have the general formula AA′BC and interestingly, the spin diffusion length of such compounds is remarkably higher than those of pseudo ternary alloys of the chemical formula A₂A′_1−xB_xC, hence resulting in low dissipation power based spintronics devices.^20,21

Gao et al. first investigated the half-metallic nature of the CFCG alloy through first principles calculations²² and then Bainsla et al. probed its SGS characteristic theoretically (through first principles calculations) and experimentally (in a polycrystalline bulk CFCG alloy grown by arc melting).¹⁹ Rani et al. have further studied the effect of excess Co-atom concentration (replacing the Fe atom) on the structural, magnetic and magnetotransport properties of bulk Co_1+xFe_1−xCrGa samples grown by arc melting.²³ However, the literature still lacks reports on the thin film growth of the CFCG alloy, which is the need of the hour. Moreover, the literature also misses the detailed study of magnetotransport properties like the anomalous Hall effect (AHE) and the linear magnetoresistance (MR) of this particular quaternary alloy. These factors motivated us to grow CFCG thin films and study their detailed structural, magnetic and magnetotransport properties.

In the present work, we have grown ordered CFCG thin films over a Si(100) substrate through an industry-friendly magnetron sputtering technique. The structural, morphological, magnetic and transport (electron-transport and magneto-transport) measurements are performed in detail for the investigation of SGS behaviour and the quality of thin CFCG films. DFT calculations are also performed for one of the most stable structures among three different types of CFCG structures for both bulk and thin film morphology. These first principles DFT results have been correlated with the experimentally obtained results. The presence of SGS like characteristics in CFCG thin films indicates that this material can be used as a promising candidate for future spintronics-based devices.

2. Experimental methodology

2.1. Sample preparation

A set of four different CFCG thin films namely B-250, B-350, B-450 and B-550 of optimal thickness (∼50 nm) were grown over Si(100) substrates using a pulsed DC magnetron sputtering system. Here, the number attached to the sample names represents the post annealing temperature, T_A = 250 °C, 350 °C, 450 °C and 550 °C, for the films, respectively. Before loading to a vacuum chamber, all the substrates were cleaned with acetone and isopropyl alcohol using an ultrasonication method. Prior to the deposition process, the chamber was pumped to the base pressure less than 2 × 10⁻⁷ Torr by employing a cryo pump, a turbomolecular pump and a rotary-pump. The deposition process was carried out at a fixed working pressure of 3 × 10⁻³ Torr by applying a pulsed-DC power of 90 Watt to the composite CFCG target (diameter 2′′) at a fixed argon gas flow rate of 64 sscm. Prior to starting the deposition, the Si(100) substrates were pre-heated in a UHV chamber at 650 °C for 1 hour. All of the layers were deposited at room temperature (RT) and then post-annealed in situ for 2 hours at the respective T_A at 2 × 10⁻⁷ Torr. The structural features of the CFCG films were then investigated. The magnetic and transport characteristics for only the structurally refined CFCG optimal phase (B-550) were studied.

2.2. Film characterization

The crystal structures of all the film samples and the thickness, roughness and density of the optimized samples were investigated by performing X-ray diffraction (XRD) and X-ray reflectivity (XRR) measurements, respectively, using a PANalytical make model – X'pert-PRO-diffractometer equipped with a Cu-K_α (l = 1.541 Å) X-ray source. All further measurements were performed over optimized CFCG film samples. Atomic force microscopy (AFM) was used to probe the surface morphology and roughness of the CFCG films using a Bruker make (model – Dimension Icon) microscope. The elemental composition was determined by Energy Dispersive X-ray analysis (EDAX) measurement performed in a Hitachi High Tecnol (Model-TM3000). The XRR and AFM measurements are described in ESI section S1† while EDAX measurement is described in section S2.† Magnetization measurements were performed using a superconducting quantum interference device (SQUID) magnetometer. All the transport measurements (temperature-dependent longitudinal resistivity (ρ_xx), the anomalous Hall effect and magnetoresistance) were performed in the DC transport module of the Quantum Design Inc. make PPMS (model Evercool-II).

3. Results and discussion

3.1 Structural characterization

3.1.1 X-Ray diffraction. The EQHAs with the AA′BC structure can possibly exhibit a number of chemical orders/disorders viz. intermixing of A and A′ results in an L2₁ structure (Cu₂MnAl-type) of the composition (AA′)BC, intermixing of B and C results in a B2 structure (CsCl-type) and complete randomization between A, A′, B and C results in a fully disordered A2 structure (Tungsten-type). The explicit determination of crystallinity is the key measure to evaluate/quantify the degree of ordering. The structure factor of a CFCG Heusler alloy for the most stable Type-I structure (described in section 3.4.2) with Co, Fe, Cr and Ga occupying the Wyckoff positions (0.5, 0.5, 0.5), (0, 0, 0), (0.25, 0.25, 0.25) and (0.75, 0.75, 0.75), respectively, can be written as:^23,24where f_Co, f_Fe, f_Cr and f_Ga denote the atomic scattering factors corresponding to Co, Fe, Cr and Ga atoms, respectively. The absolute values for (111) and (200) superlattice reflections and (220) fundamental reflections are thus expressed as:

F₁₁₁ = 4[(f_Fe − f_Co)² + (f_Cr − f_Ga)²]^1/2 (2)

F₂₀₀ = 4|(f_Co + f_Fe) − (f_Cr + f_Ga)| (3)

F₂₂₀ = 4(f_Co + f_Fe + f_Cr + f_Ga) (4)

For the Cu-K_α radiation source, the atomic scattering factors for Co and Fe atoms are almost similar (f_Co ∼ f_Fe) and hence it is challenging to distinguish between Y and L2₁ structures. Thus, the structure factor for (111) superlattice reflection, in this case, reduces to

F_hkl = 4|(f_Cr − f_Ga)| (5)

From eqn (2)–(5), we can assess that the complete randomization between Co and Fe as well as Cr and Ga elements results in the vanishing of (111) superlattice reflection and therefore the resulting structure corresponds to full B2 disorder. On the other hand, the complete randomization among Co, Fe, Cr and Ga elements i.e., each element occupies all the four Wyckoff positions with the equal probability, leads to the A2 disordered structure with the absence of both (111) and (200) reflections. Moreover, in EQHAs, the disorder only among tetrahedral site atoms (i.e., Co and Fe) gives an L2₁ disordered structure. It is worth noting that even the presence of (111) and (200) peaks do not guarantee the complete B2 or L2₁ ordering. With finite A2 disordering the intensities of these peaks are expected to be lowered accordingly. The presence of fundamental (220) reflection is, however, independent of the L2₁, B2 or A2 structure.

Fig. 2(a) shows the grazing incidence XRD (GIXRD) spectra of all samples at room temperature, collected at a fixed glancing angle (GA) of 1° together with the simulated XRD curve (using Powder Cell, version 2.4) for the CFCG Type-I structure (shown by a solid blue line). The details regarding the different types of CFCG Heusler structures depending on the occupancy of various Wyckoff positions by different elements are mentioned in the theoretical section 3.4.1.2. The intensities of diffraction peaks [Fig. 2(a)] and crystallite sizes [Fig. 2(b)] both increase with the T_A, owing to the agglomeration of nucleated stable islands via enhanced thermal surface diffusivity with increased T_A.^25,26 The small intensity of XRD reflections and lesser crystallite size for T_A ≤ 350 °C infer the nanocrystalline film growth at lower T_A. Both the B-450 and B-550 films exhibit three fundamental reflections ((022), (004), and (224)) and one impurity reflection at ∼36°. The intensities of all these reflections are relatively stronger in the B-550 film due to the improvement in the crystallinity. The impurity reflection is attributed to the formation of an oxide of Fe (Fe₃O₄) and/or Co (Co₃O₄) elements as the peak position of this reflection matches well with the (311) reflection of the oxides.^27,28 The (111) and (002) superlattice reflections, which are the typical characteristics of a well ordered quaternary Heusler structure, are absent in the data obtained from conventional X-ray diffractometers. This could be either due to nearly similar scattering factors of the constituent elements Co, Fe, Cr and Ga, or the very low intensity of the (111) and (002) superlattice reflections compared to the fundamental reflections (see the simulated curve in Fig. 2(a)) or the presence of other antisite disorders (B2, DO₃ or A2). Deepika et al. and Bainsla et al. also failed to observe (200) and (111) reflections even for bulk CFCG samples through a conventional diffractometer. While Rani et al. succeeded in observing these reflections through synchrotron radiation based XRD, Bainsla et al. established the claim of L2₁ ordering based on differential thermal analysis.²⁹ Therefore, it should be noted that our B-550 sample exhibits the A2-structrure based on the limited measurement capability of the diffractometer. Thus, the possibility of partial crystallization of the L2₁ structure is not fully ruled out. The well-known Debye–Scherrer's relationship, , was used to calculate the crystallite size (D) of the samples, where β and θ are the full width of half maxima (FWHM) and Bragg's angle, respectively, corresponding to the principal (220) reflection and λ (= 1.5406 Å) is the wavelength of the Cu-K_α radiation X-ray source. Whereas the lattice parameter (a) was evaluated using Bragg's diffraction formula, 2d sin θ = nλ with the interplanar spacing which for B-250, B-350, B-450 and B-550 samples are found to be 5.779 (±0.012) Å, 5.723 (±0.007) Å, 5.720 (±0.002) and 5.721 (±0.001) Å, respectively, and these values are very close to the experimentally and theoretically reported values for the CFCG alloy as shown by the dashed lines in Fig. 2(c).^24,30

Fig. 2 (a) Room-temperature GIXRD patterns, and variation of the (b) crystallite size and (c) lattice parameter with T_A for B-250, B-350, B-450 and B-550 CFCG films.

3.2 Magnetization measurements

Temperature-dependent isothermal magnetization hysteresis (MH) curves traced at different temperatures (T = 5 K, 50 K, 200 K and 300 K) for the B-550 film sample under an applied magnetic field parallel to the film plane are shown in Fig. 3(a). In Fig. 3(b), the temperature dependences of the coercivity (H_C) and saturation magnetization (M_S) inferred from the MH-loops are plotted. From Fig. 3(b), it is evident that the H_C decreases from 13.8 mT (at 5 K) to 5 mT (at 300 K), suggesting that the film is magnetically soft over the entire temperature range. The decrease in H_C with temperature is accompanied by a reduction in the anisotropy field with temperature and also in part due to the weakening of the strength of pinning of the domain walls in the intergranular regions, which is typical in polycrystalline films, like the one in the present study.³¹ The value of M_S is found to vary from 1.86μ_B per f.u. (at 5 K) to 1.81μ_B per f.u. (at 300 K, i.e., room temperature) as shown in Fig. 3(b). The experimentally obtained values of M_S here are slightly smaller than the earlier reported values for bulk samples (2.1μ_B per f.u.) and also the theoretically predicted value (1.97μ_B per f.u.).^23,29,30 Some of the Co-atoms occupying the tetrahedral vacancy and/or the swapping of the atoms between Co–Fe and Co–Cr might be causing this observed drop in the overall magnetic moment.^24,32

Fig. 3 (a) In-plane isothermal magnetization curves recorded at different temperatures (T = 5 K, 50 K, 200 K and 300 K), variation of (b) H_C and M_S with temperature and (c) the normalized thermal magnetization curve for the B-550 sample with the fitted line (shown by the red line) extrapolated to the zero-magnetization value.

The thermal magnetization curve [M(T)] for the B-550 sample was recorded within the temperature range of 5–300 K as shown in Fig. 3(c). The sample was first cooled to 5 K in the presence of an applied magnetic field (μ₀H) of 500 mT applied parallel to the film and then the data were recorded during heating while maintaining the magnetic field at a constant value (i.e., the field cool warming procedure). Various half-metallic Heusler compounds have been reported to have T² dependent magnetization.^33–35 In view of this, we used the following empirical formula for an itinerant type FM material to fit the experimentally observed M(T) curve, as illustrated by the solid red line in Fig. 3(c):

Here M(0) is the zero-temperature value of magnetization which is here considered to be the same as the value at 5 K. The fitted curve was extrapolated to the zero-magnetization value so as to infer the T_C, which is found to be ∼595 K. This value is less than the earlier reported value for bulk samples (690 K).²³ The key parameters that influence a material's T_C are the 3d exchange interaction between the transition metal elements and the lattice parameter in crystallites.³⁶ Moreover, the properties of the specimen, such as magnetic moment, exchange coupling, and crystal anisotropy, are greatly affected during the transition from 3D (bulk) to 2D (thin film), resulting in a weakening of interatomic exchange interactions and, as a result, a lower ordering temperature (T_C) in thin films compared to the bulk sample.¹⁰

3.3 Transport measurement

3.3.1 Longitudinal resistivity measurement. Fig. 4 presents the variation of longitudinal conductivity (σ_XX) in the main panel and longitudinal resistivity (ρ_XX) in the inset for the B-550 thin film sample with T in the range of 5 K to 300 K, under a zero applied magnetic field. A negative temperature coefficient of resistivity (dρ/dT < 0) indicates the non-metallic nature of the film. Such a kind of behaviour has reportedly been observed in various SGS materials, e.g. CoFeMnSi, Ti₂MnAl, CFCG, Mn₂CoAl and CrVTiAl.^{10,16–19,31,37} It can be seen that the resistivity decreases nearly linearly with temperature which is very different from the case of classical semiconductors, where the resistivity decreases exponentially with temperature. A similar behaviour of resistivity with a negative temperature coefficient has also been observed in transition metal alloys, which is reportedly caused by strong impurity scattering.³⁸ Moreover, there are some disordered-SGS materials, e.g. Mn₂CoAl³⁹ and CoFeCrAl,⁴⁰ which have shown a similar behaviour of resistivity. Surprisingly, Bainsla et. al have observed a positive temperature coefficient of resistivity in slightly disordered arc-melted bulk CoFeCrAl,⁴¹ opposite to the case of disordered CoFeCrAl thin films, where a negative temperature coefficient of resistivity was observed.⁴² These results indicated that SGS materials are fragile towards the disorder present in the system. We have also observed that in the given temperature range (5–300 K), the value of σ_xx swings from 5705 S cm⁻¹ to 7127 S cm⁻¹, which is comparable to the room temperature value reported for the CoFeMnSi thin SGS film (5785 S cm⁻¹) and slightly higher than that of the bulk SGS Heusler CFCG (4545 S cm⁻¹).^31,40

Fig. 4 (a) Longitudinal conductivity variation with the temperature under a zero external field together with the fit (shown by the solid red line) using the two carrier model [employing eqn (7)] for the CFCG B-550 thin film. The inset depicts the variation of longitudinal resistivity with temperature. (b) Schematic of the density of states for disordered-SGSs. The band edges of the diagram follow the 3D Van Hove singularity.⁴⁴

In order to explain the temperature dependence of resistivity we have considered CFCG here as a disordered-SGS and adopted the two-carrier model used by Kharel et al.⁴⁰ Kharel et al. modelled the conductivity of the majority spin channel for spin narrow-gap semiconductors, spin gapless semiconductors and disordered spin-gapless semiconductors. The model assumes that one spin channel for disordered-SGSs has a band overlap while another channel has a semiconducting band gap, the schematic of which is shown in Fig. 4(b). The conductivity for the disordered SGS according to this model is thus expressed as a combination of conductivity through the overlapped channel (σ_SGS) and the semiconducting channel (σ_SC),^37,43i.e.

where σ₀, g, k_B, σ_SC, and ΔE are the zero-temperature conductivity of the gapless component, band overlap between the gapless channels (called the gap-width parameter), Boltzmann's constant, the zero-temperature contribution of conductivity to the semi-conducting component and the carrier activation energy, respectively. The value of g from the fitted curve is found to be ∼−15 meV, which indicates that there is a small band overlap present in the gapless channel, suggesting that the present system (B-500) probably contains some amount of structural disorder in the film. The fitted curve yields the value of ΔE ∼ 167 meV, which lies in the semiconducting regime.

3.3.2 Magnetoresistance measurements. The magnetoresistance curves were recorded for the B-550 sample in the out-of-plane geometry (see the inset of Fig. 5) at different temperatures varying from 5 K to 300 K by sweeping the magnetic field from −2 T to 2 T as shown in the main panel of Fig. 5. All the curves show a linear trend up to 2 T similar to the case of bulk Mn₂CoAl and other gapless materials.^10,45 It was observed that the MR value was small and negative at all the investigated temperatures without any change in the sign of MR, similar to those reported for the thin Mn₂CoAl films.⁴⁶ The MR% increases with an increase in measurement temperature, as shown in the inset in the bottom left of Fig. 5, which is attributed to the increased impurity scattering with increasing temperature.⁴⁶

Fig. 5 Magnetoresistance curves for the B-550 sample traced at different measurement temperatures. The direction of the applied magnetic field is perpendicular to the film surface. The top inset shows the geometry of the measurement and the bottom inset shows the variation of MR% (at 2 T) with the measurement temperature.

3.3.3 Anomalous Hall-effect measurements. The temperature-dependent AHE measurements were performed to identify the temperature dependence of the carrier concentration (n) and mobility (μ) and to quantify the different scattering factors responsible for the magnetotransport behaviour of the B-550 sample. The transverse resistivity (ρ_XY) for a FM material can be expressed as a collective effect of the ordinary Hall resistivity contribution (ρ^OHE_XY) and the anomalous Hall resistivity contribution (ρ^AHE_XY),⁴⁷

ρ_XY = ρ^OHE_XY + ρ^AHE_XY = R₀H + R_AHEM_Z (8)

where R₀, R_AHE, and M_Z refer to the ordinary Hall coefficient, the anomalous Hall coefficient and the perpendicular component of magnetization with respect to the surface of the sample, respectively [see eqn (8)]. The Hall resistivity curves were traced at different temperatures (5 K, 20 K, 50 K, 100 K, 150 K, 200 K, 250 K and 300 K) by varying the magnetic field, applied perpendicular to the film surface, in the 0 to 5 T range, as shown in Fig. 6(a). The data were recorded by applying a DC current (1 mA) source in a four probe geometry as depicted in the inset of Fig. 6(a). It can be visualized from the Hall data that all ρ_XY(T) curves contain both the AHE contribution (in the lower field regime: 0 T ≤ μ₀H ≤ 1 T) and the OHE contribution (in the higher field regime: 1 T ≤ μ₀H ≤ 5 T). The slope of the OHE curves is positive and linear with a positive R₀ for T ≤ 250 K which confirms the p-type transport. However, the OHE slope shifts from a positive to negative value for T = 300 K, which is ascribed to the change in the carrier type, i.e. p-type to n-type, and a change in n (and/or μ) as a result of the Fermi level (E_f) shifting with temperature.³⁶ All the OHE curves were fitted linearly and extrapolated to zero field [shown by red lines in Fig. 6(a)] and the intercept value of the extrapolated fitted curves yielded us the value of the anomalous Hall effect resistivity (ρ_AH˙). The n and μ values for the fitted curves are calculated using the following pair of equations:

n(T) = 1/eR₀ and μ(T) = R₀/ρ_XX (9)

Fig. 6 (a) Variation of the Hall resistivity (ρ_XY) with the magnetic field (μ₀H) traced at different temperatures ranging from 5 K to 300 K. The solid red lines show the linearly fitted OHE contribution to transverse resistivity, the intercept of which gives the value of the AHE resistivity, i.e., ρ_AH and (b) the corresponding plot of carrier concentration (and/or mobility) at the respective temperatures, obtained through the fitted OHE curves. (c) ρ_AH variation with ρ²_XX fitted with eqn (11) as shown by the solid red line. (d) Plot of the Hall conductivity curve for the B-550 sample at 5 K. The value of σ_AH, obtained by extrapolating the OHE data to the zero field, is shown in the main panel.

The plot of the variation of n and μ with respect to the temperature is shown in Fig. 6(b) through which it can be observed that these parameters are nearly temperature-independent up to the temperature range of 5 K to 250 K, which is typically a characteristic of SGSs. However, at 300 K, an increase in the value of μ while a drop in the value of n are observed, which is attributed to the change from p-type to n-type carriers as mentioned above. The values of n[μ] at 5 K and 300 K are found to be (3.92 ± 0.21) × 10²¹ cm⁻³ [(9.15 ± 0.48) cm² V⁻¹ s⁻¹] and (0.99 ± 0.01) × 10²¹ cm⁻³ [(44.37 ± 0.28) cm² V⁻¹ s⁻¹], respectively, which are quite higher than those expected for normal semiconductors. This is expected in the SGSs since they possess zero band gap near the Fermi level of the majority spin channel. Due to this, carriers are readily available to conduct even at a very low temperature similar to the case of semimetals and metals. Several reports confirmed the range of carrier concentration of SGS materials.^16,31,48

The AHE resistivity, i.e., ρ_AH majorly arises because of two contributions: (I) intrinsic contribution and (II) extrinsic contribution. The intrinsic contribution particularly relies upon the band structure of the system and is highly independent of impurity induced or SOC-induced scattering, while the reverse is true for the extrinsic contribution.⁴⁷ The intrinsic contribution to ρ_AH(ρ_int) was first explained theoretically by Karplus and Luttinger where they considered that the spin orbit interaction (SOI) and inter-band mixing effectively induce an anomalous velocity contribution to the carriers in the direction perpendicular to the applied field, according to which ρ_intαρ²_XY.⁴⁹ Lately, this theory has been revised by considering the Berry phase as an essential parameter.⁵⁰ During an adiabatic motion of an electron (in which the electron cannot interchange its band), its wave function is associated with an extra phase part, called the Berry phase. This phase depends on the curvature of the band and associates with something called the Berry curvature (b_n). The Berry curvature is basically a pseudo magnetic field whose magnitude depends on the Berry phase or band curvature. By the effects of an electric field and this pseudo magnetic field, the conduction electrons experience an anomalous velocity perpendicular to both the electric field and Berry curvature giving rise to the AHE. On the other hand, the extrinsic contribution to ρ_AH(ρ_ext) is explained by the (I) skew scattering⁵¹ and (II) side-jump scattering⁵² mechanisms. The skew scattering arises due to the asymmetric scattering of a conduction electron because of an effective spin orbit interaction. According to Smit, this SOI results in a transverse spin orbit force.⁵¹ In a pure periodic crystal, this force can only polarize the charge distribution but cannot accelerate the electron as the periodicity of the transverse force and the crystal are identical. But in the presence of disorder or impurity, the periodicity is disturbed, and the transverse force can slightly scatter the electron. As the electron is scattered slightly, the direction of its velocity changes which in turn changes the direction of the resulting relativistic magnetic field. The dependency of scattering cross section on the transverse component of spin and orbital magnetic moments of the electron results in the asymmetric scattering of the electron.

The side-jump phenomenon is however explained by the local distortion of the wavefunction by an impurity and creation of a local current density. This contribution to the AHE can be understood in terms of localized electrons or wave packets (but not in terms of plane waves or momentum representation) where the center of mass of the conduction electron undergoes a sideways displacement (Δy) on scattering by an impurity or a phonon in the presence of spin–orbit interaction.⁵³ This displacement is also spin dependent i.e., an electron with opposite spin will experience displacement along the opposite direction.

We employed here an expanded scaling model proposed by Tian et al.⁵⁴ to estimate the extrinsic and intrinsic contributions to the AHE. According to this model, the ρ_AH is expressed as

ρ_AH(T) = αρ_XX0 + βρ²_XX0 + bρ²_XX(T) (10)

After merging the temperature independent term, the above equation reduces to

ρ_AH(T) = a′ρ_XX0 + bρ²_XX(T) (11)

where a′ = α + βρ_XX0, which represents the extrinsic contribution from side jump impurity scattering and skew scattering, b is the parameter representing the contribution from intrinsic scattering and is related to the Berry curvature and ρ_XX0 as the residual resistivity derived by extending the longitudinal resistivity curve to 0 K (see the inset of Fig. 4(a)). Fig. 6(c) shows the variation of ρ_AH with ρ²_XX fitted with eqn (11) (shown by the solid red line) for the B-550 sample. The fitted curve gives the values of the parameters a′ and b as ∼2.08 × 10⁻³ and ∼80.5 S cm⁻¹, respectively.

3.4 Electronic-structure analysis

3.4.1 Bulk calculations.
3.4.1.1 Computational methodology. To gain more insight into the structural, electronic, and magnetic properties of the CFCG film, density functional theory (DFT) calculations were performed as implemented within the QUANTUM ESPRESSO code with a projected augmented wave (PAW) basis set.^55,56 To deal with the electronic exchange and correlation interactions, we have used the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA).⁵⁷ For all the calculations, a plane wave energy cut-off of 350 Ry and a 15 × 15 × 15 k-mesh are used. For the Brillouin zone integration, the optimized tetrahedron method is used.⁵⁸ The SCF calculations were stopped when the variation in the total energy was less than 10⁻⁶ Ry. All the crystal structures for CFCG were generated using the VESTA program.⁵⁹
3.4.1.2 Structural properties. The EQHAs are found to exist in an ordered cubic structure of space group No. 216 with the LiMgPdSn prototype (Y-type),⁶⁰ with four formula units (f.u.) per unit cell. The constituent atoms can occupy different Wyckoff positions 4a (0, 0, 0), 4b (0.5, 0.5, 0.5), 4c (0.25, 0.25, 0.25) and 4d (0.75, 0.75, 0.75). Therefore, here for the CFCG alloy, we have considered three non-degenerate atomic configurations depending on the arrangement of these four kinds of elements (Co, Fe, Cr and Ga) randomly occupying the Wyckoff positions (4a, 4b, 4c and 4d). The three possible atomic arrangements are noted as Type-I (Fe at 4a, Co at 4b, Cr at 4c and Ga at 4d), Type-II (Cr at 4a, Co at 4b, Fe at 4c and Ga at 4d) and Type-III (Cr at 4a, Fe at 4b, Co at 4c and Ga at 4d). The 16 atoms’ conventional unit-cells for all different kinds of arrangements are shown in Fig. 7.

Fig. 7 The crystal structure for the (a) Type-I, (b) Type-II, and (c) Type-III CFCG alloy.

To have an idea about the ground state structural properties, we have calculated the total energy per formula unit (f.u.) vs. volume curves for all kinds of arrangements, as shown in Fig. 8, which shows that CFCG is structurally more stable with the Type-I structure. To determine the optimized lattice parameter and ground state energy, total energy per f.u. vs. lattice volume has been fitted using the Murnaghan equation of state.⁶¹

Fig. 8 Total energy as a function of the unit cell volume of the CFCG alloy with Type-I, Type-IIandType-III structures. The coloured filled circles represent the calculated data, and the solid lines are the fit corresponding to all four structures.

All the fitted parameters – the optimized lattice parameter, minimum total energy, bulk modulus (B₀) and its derivative (B′₀) for all the structures are tabulated in Table 1. The optimized lattice parameter for the Type-I structure, which is the most stable structure, is found to be 5.72 Å, similar to the earlier reported value⁶² and also agrees with the experimental value of 5.721 (±0.001) Å obtained from the XRD data of Fig. 2(c).

Table 1 The optimized volume, optimized lattice parameter and total energy per f.u. for Type-I, Type-II, and Type-III structures for the CFCG alloy

Structure	Optimized volume (Å^3)	Optimized lattice parameter (Å)	Total energy (Ry per f.u.)	B 0 (GPA)	B′0
Type-I	187.46	5.72	−1232.1698	182.5	5.63
Type-II	193.93	5.79	−1232.1696	162.0	5.04
Type-III	202.92	5.88	−1232.1270	150.4	2.27

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3.4.1.3 Electronic structure and magnetic properties. We have calculated the spin-resolved density of states (DOS) and the band structure for the Type-I CFCG structure, as shown in Fig. 9. The calculated total and atomic magnetic moments for CFCG are given in Table 2. From the DOS and band structure, it is obvious that for the spin up electrons, valence band maxima and conduction band minima touch each other, and for the spin down electrons, there is a finite gap. As seen from Table 2, the calculated magnetic moment on the unit cell (M_tot) for CFCG is 2μ_B per f.u., which is consistent with the Slater Pauling Rule, which is one of the prerequisites for SGS.⁶⁰ Therefore, CFCG is a spin gapless semiconductor in nature. The major contribution to the M_tot of CFCG comes from the Cr and Co atomic moments, while the Fe atoms are antiferromagnetically coupled with the Cr and Co moments. Ga atoms have a negligible moment, as also reported in another reference.⁶²

Fig. 9 Spin resolved band structures for (a) the majority and (c) minority spins and (b) DOS per f.u. for the Type-I CFCG alloy.

Table 2 Total calculated magnetic moment (in units of μ_B) per f.u. for CFCG QHA, and atomic magnetic moments for Co, Fe, Cr and Ga atoms

	M tot (μB per f.u.)	M Co (μB)	M Fe (μB)	M Cr (μB)	M Ga (μB)	Nature
This work	2.00	0.99	−0.70	1.68	−0.03	SGS
Shi et al.^62	1.97	0.94	−0.74	1.81	−0.04	SGS

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Although the calculated lattice parameter is matched well with the experimental lattice parameter, the calculated magnetic moment, however, exceeds the experimentally measured value. Possibly, this discrepancy could be due to the consideration of CFCG in the bulk geometry in the DFT calculations. In the next section, therefore, we present the results of DFT calculations for the case of the CFCG alloy in the thin film form where the film surface, as opposed to bulk, dominantly governs the physical properties of the thin films.

3.4.2 Surface calculations. To understand the surface electronic properties of the optimized B-550 film, we have performed DFT calculations for the (022) surface, which is the most prominent surface observed from XRD measurements (see Fig. 1(a)). The (022) surface has been modelled using an asymmetric slab comprising 4 (bottom) layers fixed and 10 Å vacuum. For all the surface calculations, an 11 × 15 × 1 k-point mesh was chosen, and atoms were allowed to relax until the forces are ≤10⁻² Ry per Bohr to obtain the optimized atomic geometry for the corresponding slab. To ensure that the selected thickness is sufficient to study the surface properties, convergence with respect to surface energy was calculated with a threshold of 1 meV per atom per Ų using the relaxed slabs. Fig. 10(b) shows that the slabs with ≥12 monolayers meet the criteria; therefore, further calculations are done for slabs with 14 monolayers as shown in Fig. 10(a).

Fig. 10 (a) A slab with 14 monolayers for electronic calculations. (b) Convergence of surface energy with respect to the number of layers in the CFCG slab for the (022) surface.

In Fig. 11(a)–(c), the plots of the band structure and density of states are presented. As observed from these band structure and DOS plots for the CFCG thin film geometry, the spin up channel is metallic with several electronic states crossing E_F; whereas in the spin down channel, there are very small and negligible electronic states (∼0.13486 states per eV per f.u.). As a result, it is concluded that the SGS nature of CFCG is lost for the (022) surface, and such a slab of CFCG with the (022) surface is expected to have nearly half-metallic nature, which is in contrast to the experimentally observed non-metallic behavior in the transport measurements (see Fig. 4).

Fig. 11 (a) Spin-up band structure, (b) density of states and (c) spin-down band structure plots for a slab with 14 monolayers of the (022) CFCG alloy.

The electronic spin polarization is calculated from the DOS plot (Fig. 11(b)). It is quite remarkable to note that the (022) surface of the CFCG thin film possesses a very high spin polarization of 91%, in contrast to other Co-based equiatomic quaternary Heusler alloys which show a low value for electronic spin polarization in thin films.^22,63,64 Thus, it turns out that CFCG thin films could have several potential spintronics applications.

On the other hand, the total magnetic moment for the (022) surface is found to be 1.75μ_B per f.u. which is much closer to the experimentally measured value (∼1.86μ_B per f.u.), compared to the magnetic moment value for the bulk structure (2μ_B per f.u.). The difference in the experimentally observed magnetic moment and electronic transport from the DFT findings can be attributed to the presence of other (hkl) planes on the surface and also the point-defects (site and antistites disorders, and vacancies) present in the thin films.^65–69 Consideration of such disorder for DFT-calculations for the thin film geometry in the CFCG alloy is beyond the scope of the present investigations.

4 Conclusion

The polycrystalline thin films of the CoFeCrGa (CFCG) Heusler alloy were grown over Si(100) substrates using the pulsed DC magnetron sputtering technique. The various structural, magnetic and transport measurements were performed to characterize the optimized B-550 CFCG thin film. The structural measurement revealed that the grown film possesses A2-ordering under the limit of measurement. However, the possibility of B2 and L2₁-ordering still persists as the (200) and (111) superlattice reflections can be traced by implying a higher resolution X-ray source such as synchrotron radiation. The magnetic measurements give saturation magnetization and Curie temperature values of 1.81μ_B per f.u. and ∼595 K, respectively. An almost linear variation of longitudinal resistivity (and/or conductivity) with temperature was observed with a negative temperature coefficient. Fitting of the conductivity curve through a two carrier model reveals that the gapless channel of the CFCG B-550 film has some band overlap which corresponds to the presence of some structural disorder in the system. The isothermal magnetoresistance curves recorded at different temperatures (5–300 K) are found to exhibit negative and linear behaviour up to 2 T magnetic field. The temperature-dependent anomalous Hall effect measurement gives a nearly temperature-independent carrier concentration (mobility) with a room temperature value of 3.9 ± 0.04 × 10²¹ cm⁻³ (9.15 ± 0.48 cm² V⁻¹ s⁻¹). The expanded scaling mechanism performed over the AHE data reveals that both extrinsic and intrinsic factors contribute to the AHE; however, the phonon-induced skew scattering has a greater extrinsic contribution compared to impurity induced skew scattering and also the intrinsic Berry phase has a considerable contribution to the AHE. The positive slope of high field AHE curves up to 250 K suggests that holes are the majority carriers which agrees with the results of resistivity measurement. The anomalous Hall conductivity obtained by extrapolating the AHE curve at a cryogenic temperature (T = 5 K) is found to be 91.35 S cm⁻¹. Irrespective of the absence of B2 and L2₁-ordering, our experimental findings such as a positive temperature coefficient of resistivity, a nearly temperature independent carrier concentration (and/or mobility) and linear MR justify the presence of SGS like behaviour within our film. To correlate the obtained structural, electronic, and magnetic properties, DFT calculations were performed for bulk and the thin film CFCG alloy. The structural and magnetic properties are in well agreement with the experimentally obtained results, while the electronic nature is not consistent and needs further investigation. The experimental and theoretical studies suggest that the CFCG alloy is a desirable candidate for spintronics based devices and has further scope to be studied for dynamic and device-based studies.

Author contributions

Vireshwar Mishra: conceptualization (lead), data curation (lead), formal analysis (lead), investigation (lead), methodology (lead), validation (lead), visualization (lead), writing original draft (lead), and writing – review and editing (lead). Amar Kumar: conceptualization (supporting), data curation (equal), formal analysis (supporting), methodology (supporting), writing original draft (supporting) and writing – review and editing (supporting). Lalit Pandey: conceptualization (supporting), data curation (supporting), formal analysis (supporting), and methodology (supporting). Nanhe Kumar Gupta: conceptualization (supporting) and formal analysis (supporting). Soumyarup Hait: data curation (supporting), formal analysis (supporting), and writing – review and editing (supporting). Vineet Barwal: data curation (supporting) and formal analysis (supporting). Nikita Sharma: data curation (supporting) and formal analysis (supporting). Nakul Kumar: data curation (supporting), formal analysis (supporting), and methodology (supporting). Sharat Chandra: data curation (supporting), formal analysis (supporting), and validation (supporting). Sujeet Chaudhary: conceptualization (equal), formal analysis (equal), funding acquisition (lead), investigation (equal), project administration (lead), validation (equal), and writing – review and editing (equal).

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge the Central Research Facility (CRF), IIT Delhi for EDAX and VSM measurements, IIT Delhi HPC facility for computational resources and the Ministry of Education, Government of India and the Council of Scientific and Industrial Research (CSIR) India for financial assistance throughout this work.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2nr03424g

‡ These authors contributed equally.

§ Present Address: Centre for Magnetic and Spintronics Materials (CMSM), National Institute for Materials Science (NIMS), 3050047 (Japan).

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