Magnetic properties of CrSb compounds with zinc-blende and wurtzite structures

During recent years half-metallic ferromagnetic systems have attracted much attention because of their possible application as materials for spintronics. In particular, they are supposed to be applicable as a source of spin-polarized current for spin injection. A crucial feature required for this purpose is a high degree of spin polarization $\frac{{n}_{\uparrow }({E}_{\mathrm{F}})-{n}_{\downarrow }({E}_{\mathrm{F}})}{{n}_{\uparrow }({E}_{\mathrm{F}})+{n}_{\downarrow }({E}_{\mathrm{F}})}$, where n_↑(E_F) and n_↓(E_F) are the densities of states at the Fermi level of the electrons with up- and down-spin orientation. With a finite value of n_σ(E_F) for one spin channel and with n_−σ(E_F) zero for the other, spin polarization in half-metallic materials is expected to be 100%. This behavior has been found for various classes of materials, like Heusler alloys [1], metal oxides [2], perovskite manganites [3], and diluted magnetic semiconductors (DMSs) [4]. However, practical application of half-metallic ferromagnets (HMFs) meet some problems either because of problems with the matching of HMF and semiconductor crystal structures (heterostructures) or because of a low Curie temperature as for DMSs, which is far below room temperature. A new type of half-metallic ferromagnetic system, CrAs with zinc-blende (ZB) structure, was predicted theoretically and synthesized experimentally using molecular beam epitaxy on a GaAs substrate [5]. The remanent magnetization in this system was observed up to 400 K, that allows us to conclude that the value of Curie temperature is well above room temperature. These properties look promising for future applications of these materials, in particular in spintronics. Therefore, since that time many investigations have been performed on various compounds of transition metal alloys having ZB structure synthesized on different substrates [6–13]. The features of the electronic structure responsible for the half-metallic behavior of transition metal (TM) compounds having ZB structure were discussed in detail by Galanakis and Mavropoulos [6]. Sanyal et al [7], using the results of ab initio electronic structure of these compounds, have also calculated the inter-atomic exchange interactions. Their results show in particular that the strongest FM interaction between the first-neighboring TM atoms corresponds to that within Cr compounds. Very detailed investigations of the exchange coupling parameters in Cr based alloys with ZB structure with corresponding analysis of stability of their magnetic structure and Curie temperature have been made recently by Bose and Kudrnovský [14]. Xie et al in their studies on half-metallic properties of TM pnictides and chalcogenides have shown the existence of an energy gap at the Fermi level for minority-spin states, also for compounds with wurtzite structure [15], showing a similar behavior as the compounds with ZB structure. Here it is important to note that the electronic and magnetic properties of deposited films, i.e. half-metallicity and FM order, can be modified in the surface and interface regions. This is of great importance for application in spintronics and therefore motivates further investigations [16–20].

In the present work we focus on the CrSb system with ZB and wurtzite structures. Artificial CrSb films with ZB structure were grown first on GaAs, AlGaSb, and GaSb substrates using molecular beam epitaxy techniques [11, 21]. However, in these experiments only very thin films of a few nanometres have been grown successfully, which obviously was not enough to obtain the half-metallic ferromagnetic behavior expected for a bulk system. Recently, thicker films have been fabricated using the pulse laser deposition technique on Si(100) [12], and deposition on KCl(100) substrates by magnetron sputtering [13].

The present calculations of the electronic structure and magnetic properties have been performed for CrSb with different lattice parameters assuming that it is fabricated epitaxially on a substrate adopting its structure and lattice parameter. In addition, calculations have been performed for thin films deposited on III–V semiconductors having various lattice parameters to study the electronic structure and magnetic properties at the interface and at the surface.

The electronic structure calculations for ZB and wurtzite structures have been done by means of the spin-polarized fully relativistic KKR (Korringa–Kohn–Rostoker) Green's function method [22, 23]. Exchange and correlation were treated within the framework of the local spin density approximation (LSDA) for density functional theory (DFT) using the parametrization of Vosko, Wilk and Nusair [24]. All calculations have been done using the atomic sphere approximation (ASA). Because of open ZB and wurtzite structures so-called empty spheres have been added, which allow us to minimize the overlap between the atomic spheres corresponding to the atoms in the unit cell (see, e.g., [25, 26]). For the angular momentum expansion of the Green's function a cutoff of ℓ_max = 3 was applied. For the evaluation of the number of occupied states the Lloyd's formula has been used. For the Brillouin zone (BZ) integration we have used a regular $\vec{k}$-mesh of 30×30×30 points in the full 3D BZ for the bulk calculations and 30×30 points in the full 2D BZ for the film calculations. The magneto-crystalline anisotropy (MCA) was investigated by performing calculations for the magnetic torque using the expression [27]:

$$\begin{eqnarray} &&{T}_{\hat {u}}(\theta ,\phi )=-\partial E(\vec{M}(\theta ,\phi ))/\partial \theta . \end{eqnarray} \tag{ 1 }$$

Here, the vector $\hat {u}$ specified by the angles θ and ϕ lies within the surface plane and is perpendicular to the direction of the magnetic moment ${\hat {e}}_{\mathrm{M}}$. Within the present work the main interest is whether the easy magnetization axis is in the plane or out of the plane of the magnetic films. To deal with this, a special geometry can be used which gives a simple relationship between the magnetic torque and the energy difference between the in-plane and out-of-plane magnetization directions. Setting θ = π/4, the torque component ${T}_{\hat {u}}$ gives the ϕ dependent energy difference ${T}_{\hat {u}}(\theta =\pi /4,\phi )={E}_{\parallel }(\phi )-{E}_{\perp }$ [27–29].

The KKR formalism allowed us to study the electronic, structural and magnetic properties of the ground state of CrSb with different structures at T = 0 K. Finite temperature magnetic properties have been determined performing Monte Carlo (MC) simulations based on the classical Heisenberg model with the exchange coupling parameters J_ij calculated within the magnetic force theorem approximation [30].

The present work is devoted to the investigations of magnetic properties of CrSb with metastable ZB and wurtzite structures. The origins of the half-metallic properties of the ZB structure were analyzed by Galanakis and Mavropoulos [6] on the basis of electronic structure calculations. They showed that for this structure the tetrahedral coordination of the TM element by an sp element (Sb in the present case) plays the central role leading to the opening of an energy gap due to strong hybridization of TM t_2gd-like states, with the p states of surrounding sp atoms of group V and VI elements. The position of the gap with respect to the Fermi energy for different spin channels depends on the exchange splitting of 3d states of the TM atoms. For the wurtzite structure the TM atoms have a very similar tetrahedral environment [15], that also leads to hybridization of their d states with p states of sp atoms creating an energy gap in full analogy with the ZB structure.

The present calculations have been performed for bulk systems with different lattice parameters, assuming that because of the metastability of the ZB and wurtzite structures these systems can be synthesized as films by growing on different types of substrates leading to films thick enough to neglect the effect of the surface and interface. Figure 1 represents the spin projected DOS for CrSb with ZB (left panel) and wurtzite (right panel) structures. One can see for most of these compounds that the Fermi energy, E_F, is positioned within the energy gap of the minority-spin states. The DOS of the majority-spin states at the Fermi level is always finite, leading to a pronounced spin polarization of the electronic states at E_F. Upon increase of the lattice parameter the Cr 3d states become more localized, leading to a decrease of their bandwidth and increase of the exchange splitting of the up- and down-spin channels. As a result, the spin-down d band moves up with respect to the Fermi energy (see figure 1). At the same time, the occupation of the states for both spin channels remains unchanged. One can also see that the energy gap increases due to the increase of the lattice parameter leading to a localization of Cr 3d states. Similar trends are also observed for the DOS of CrSb in the wurtzite structure, that is presented in the right panel of figure 1.

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Table 1 shows the element projected spin magnetic moments in CrSb with ZB and wurtzite structures obtained in the present work for different lattice parameters. As emphasized by other authors [8, 31], the total spin magnetic moment ${m}_{\mathrm{spin}}^{\mathrm{tot}}$ for the half-metallic state of CrSb should be integer and equal to 3 μ_B per unit cell. This was ensured within the present KKR calculations by making use of the so-called Lloyd's formula (see for example the discussion by Galanakis et al [31, 32]). In addition, the rather high cutoff value l_max = 5 was used for the angular momentum expansion. This results in the total magnetic moments close to integer value in the case of half-metallic behavior of the system.

Table 1.  Total and element projected Cr and Sb spin magnetic moments (μ_B/atom) in compounds with ZB and wurtzite structure having different lattice parameters, calculated at l_max = 5.

	alat (au)	${m}_{\mathrm{spin}}^{\mathrm{tot}}~({\mu }_{\mathrm{B}}/\mathrm{atom})$	${m}_{\mathrm{spin}}^{\mathrm{Cr}}~({\mu }_{\mathrm{B}}/\mathrm{atom})$	${m}_{\mathrm{spin }}^{\mathrm{Sb }}~({\mu }_{\mathrm{B }}/\mathrm{atom })$
		ZB/wurtzite	ZB/wurtzite	ZB/wurtzite
InN	9.41	1.049/1.536	1.017/1.3994	0.024/0.080
GaP	10.30	2.450/2.658	2.381/2.5093	−0.021/0.035
GaAs	10.68	2.893/2.969	2.823/2.8616	−0.065/−0.019
InP	11.09	2.993/2.987	3.020/3.0063	−0.149/−0.113
InAs	11.45	2.994/2.979	3.138/3.1226	−0.233/−0.205
GaSb	11.53	2.995/2.978	3.165/3.1455	−0.252/−0.223
InSb	12.25	2.999/2.875	3.411/3.3491	−0.434/−0.443

The induced magnetic moments of Sb atoms are negative except for the case with smallest lattice parameters. At the same time, the magnitude of the Cr and Sb magnetic moments increases upon increase of the CrSb lattice parameters. This is a consequence of the increase of localization of the electronic states and their exchange splitting mentioned above. The energetic position of the Cr majority-spin states and, as a result, their hybridization with Sb p states remains nearly unchanged upon increase of the lattice parameter. However, hybridization of the minority Cr states, moving up in energy, with corresponding Sb p states becomes weaker, thus increasing the non-compensated negative spin density on Sb atoms.

It should be noted that the calculations discussed above are simplified by assuming that the structure of the CrSb film follows exactly the structure of the substrate without any relaxation. Because of the lattice mismatch of the equilibrium CrSb lattice with ZB structure and that of the substrate, the film undergoes relaxation, modifying the structure parameters to minimize the total energy. Without performing total energy calculations we assume that the compound adopts new structure parameters while keeping its volume unchanged, leading to a tetragonal distortion. The corresponding value for the equilibrium volume was taken from the literature [20]. The tetragonal distortion results in the modification of the electronic structure as shown in figure 1, displaying two main effects: an energy shift of the electronic states because of the change of the inter-atomic distances and a splitting of the electronic states due to symmetry breaking in the system. As a consequence, the tetragonal distortion should lead to an increase of magneto-crystalline anisotropy (MCA) in the system. It should be noted that, within such a simplified consideration, the c/a ratio becomes significant already at a = 11.09 (c/a = 1.14) and a = 12.25 au (c/a = 0.85), that can result in a loss of half-metallicity due to the electronic structure modification mentioned.

We have investigated in the present work the dependence of MCA of the CrSb film on the type of substrate, performing magnetic torque calculations. The result is shown in figure 3. One can see a nearly perfect linear MCA dependence on the tetragonality, leading to an in-plane MCA at c/a > 1 and to the out-of-plane MCA at c/a < 1. This behavior is governed by electronic structure modifications, discussed already in the literature [16, 33]. As soon as the MCA is determined by the maximal energy gain for a certain direction of the magnetization due to the SOC of the occupied and unoccupied states with corresponding symmetry, the tetragonal distortion results in different shifts of the electronic states originating from t_2g-like as well as e_g-like states. This way the distance in energy between the SOC-coupled occupied and unoccupied states is changed, leading to the in-plane or out-of-plane MCA, depending on which type of occupied and unoccupied state becomes closer under variation of c/a ratio.

The stability of the magnetic structure was analyzed in the present work on the basis of calculations of the Cr–Cr exchange coupling parameters for ZB and wurtzite structures of CrSb. Calculations have been performed for different lattice parameters within the limits of a = 9.41 and 12.25 au. In all cases (except of wurtzite structure with a = 12.25 au) the exchange interactions between the first-neighbor Cr atoms have strong FM character. At larger distances the Cr–Cr exchange interactions can be both positive and negative, but their magnitude is rather small, leading to a weak impact on the magnetic order. This can be seen in figure 2, that shows the Cr–Cr exchange interactions calculated for the CrSb alloy with ZB structure with lattice parameter a = 10.68 au (GaAs) and a = 11.45 au (InAs), respectively. The corresponding exchange interactions obtained in the case of tetragonal distortion are presented by the open symbols. In this case one has to distinguish between two types of exchange interaction: Cr1–Cr1 between the atoms which belong to the planes parallel to the substrate, and Cr1–Cr2 interactions between the atoms belonging to the neighboring planes. For the systems represented by figure 2 the distance between the Cr1 atoms is compressed, that leads to a crucial decrease of the exchange interactions at a = 10.68 au. On the other hand, the interactions between Cr1 and Cr2 atoms do not change essentially.

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To investigate the finite temperature magnetic structure and critical temperatures, MC simulations have been performed. According to these, the CrSb compounds have a ferromagnetic ground state for all considered lattice parameters, both, in the case of wurtzite and ZB structures. Only in the case of wurtzite structure with a = 9.41 au was a non-collinear magnetic structure obtained. Figure 4 shows the dependence of the Curie temperature on the lattice parameter. For both structures the variation of the Curie temperature with lattice parameters is similar. In particular, one can see a maximum of T_C for the lattice parameter a = 10.68 au in the case of ZB structure and for a = 11.08 au in the case of wurtzite structure.

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This behavior correlates with the behavior of the exchange interactions between the first-neighbor Cr atoms being primarily responsible for the FM order in the system. One can make a qualitative analysis to understand the behavior of the Curie temperature under variation of lattice parameter. Considering the half-metallic systems, it has already been discussed in the literature that in this case the RKKY interaction cannot be considered as exclusively responsible for the magnetic order [34, 35]. In the case of the half-metallic state the FM state is stabilized due to strong hybridization of Cr d states and Sb p states discussed above, that leads to the indirect exchange interactions between Cr atoms, mediated by the sp electrons of the Sb atoms. Within a simple tight-binding consideration (see, e.g., [36, 37]), one can show the stabilizing role of the sp electrons of Sb for the FM state of the compound as well as to get some qualitative arguments for dependence of T_C on the lattice parameter. The highest T_C was obtained in the present calculations for structure parameters close to the equilibrium, obtained within the total energy calculations [20]. A decrease of the lattice parameter results in a decrease of the exchange splitting of Cr d states with opposite spin directions, and as a consequence to a decrease of local magnetic moments and Cr–Cr exchange interactions. When the lattice parameter increases, in spite of the increase of the exchange splitting of Cr d states, the hybridization of Cr d states and Sb p states becomes weaker, leading to a decrease of the Cr–Cr exchange interactions mediated by the sp electrons of Sb. In both cases the Curie temperature in the system decreases. When the system loses its half-metallic behavior and becomes metallic, the RKKY interaction becomes more pronounced and can lead to an oscillating behavior of the exchange interactions with the distance, that can lead in turn to an AFM ground state or to non-collinear magnetic structure.

The discussion of the half-metallic FM properties in the bulk ZB and wurtzite CrSb compounds has to be complemented by a discussion of corresponding properties of thin films. In particular, the properties at the interface with a substrate and at the surface are of great importance for practical applications. Because of the different number of Cr neighbors at the surface or different types of atom around Cr at the interface, the properties in these regions can differ substantially from the bulk properties. Accordingly, there is no guarantee that the half-metallicity found for the bulk is present also at the interface and at the surface.

Another characteristic of HM materials interesting for an application in spin injecting devices for spintronics is a magnetic structure, which can be non-collinear or AFM at the interface and surface in spite of clear FM order in the bulk. This feature also needs further detailed investigations although the importance of these properties and their investigations for some systems have been already discussed to some extent in the literature (e.g. [18–20]).

Here we present the results for 5 ML CrSb films with ZB structure deposited on the (001) surface of a II–VI and III–V semiconductor substrate. The calculations have been performed assuming ideal bulk-like positions of the atoms in the whole film, i.e. no relaxation effects at the interface and at the surface have been taken into account. To separate the influence of the lattice parameter and chemical environment at the interface, two types of calculation have been performed: with and without buffer layer. Assuming that the buffer layer (GaSb in our present calculations) adopts the structure of the substrate, variation of its lattice parameter allows us to focus on the influence of this parameter on half-metallicity and magnetic order in the vicinity of the interface. Calculations without the buffer layer exhibit in addition the 'chemical' influence of the substrate on these properties.

Figure 5 represents the electronic DOS for the atoms in the vicinity of the surface, calculated for a 5 ML CrSb film deposited on InAs substrate as well as on GaSb, considered as a buffer layer with the lattice parameter of bulk InAs (a = 11.45 au). In the center of the films the electronic structure is in rather good agreement with the electronic structure obtained for corresponding bulk systems. Two different surface terminations are considered here, with Cr ((b), (d)) and Sb ((a), (c)) atoms in the topmost surface layer. The results are in qualitative agreement with previous results of other authors obtained for similar compounds (e.g. [8, 6]). This means that in the case of a Cr terminated surface the half-metallicity is not destroyed either within the surface Cr layer or in the next to surface Sb layer. In the case of a Sb terminated surface, however, the Sb dangling bonds create new states within the energy gap, breaking the half-metallicity for the surface Sb layer. As a consequence, a finite DOS also appears for the minority-spin channel on Cr atoms adjacent to the surface Sb layer.

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Another case considered here is a system with its Fermi energy at the border of the energy gap. This takes place when the CrSb film is deposited on a GaAs substrate (a = 10.68 au). Figure 6 represents the corresponding results for the DOS of electrons in the surface layers that were obtained for films with and without a buffer layer. Similarly to the previous case discussed above, the calculations have been performed for two different surface terminations. Despite a different position of the Fermi level and a smaller energy gap, the system exhibits the same features as for the bigger lattice parameter. This means that the half-metallicity is destroyed only in the vicinity of the surface with Sb atoms in the topmost layer.

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Similar to the situation at the surface, half-metallic properties can also be destroyed at the substrate–film interface. For technological applications this feature is crucial and requires detailed investigations. Figure 7(a) shows the electronic DOS for the interface (i.e. Sb/Cr) Cr layer in the CrSb film deposited on top of a GaSb (buffer) film with InAs lattice parameter (a = 11.45 au). As can be seen, the Fermi level is positioned well in the middle of the energy gap. The same holds for the DOS of Sb atoms within the layers attached to the Cr layer (figure 7(b)). This behavior is similar to that of the bulk system and, therefore, one cannot see any features destroying the half-metallicity. The DOS for the Cr interface layer for the CrSb film on GaSb substrate with GaAs lattice parameter (a = 10.68 au) is represented in figure 8(a). The energy gap for the spin-down channel within the interface Cr layer is nearly closed. A similar trend can be seen for the DOS for the Sb layer next to the Cr interface layer (figure 8(b)). Thus, the energy gap for the Sb/Cr/Sb type of interface is closed, leading to a finite DOS at E_F. However, if the energy gap is not too small and the Fermi level is positioned away from its boundaries, half-metallic properties intrinsic for a bulk system can also survive at the interface.

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When the CrSb film is deposited on a GaAs substrate (figure 5, thin line), the electronic states are slightly shifted towards higher binding energy, resulting in a shift of the energy gap with respect to E_F both at the interface and at the surface. Considering the CrSb film deposited on the InAs substrate, one finds that the energy gap in InAs is located slightly below the Fermi energy. Therefore, the Cr minority-spin DOS at the As/Cr interface is finite at E_F (figure 8), while for the surface layer the use of different substrates does not result in a noteworthy modification of the electronic structure. However, pronounced changes of the DOS occur at the interface when the CrSb film is grown with an As/Sb/Cr interface (figure 9). In this case states with a rather sharp DOS maximum appear within the energy gap. This maximum presumably indicates the instability of this configuration of the interface leading to a structural relaxation.

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Another crucial point for the surface and interface regions, as was mentioned in the introduction, is their magnetic order, which can be different when compared to the magnetic order in the bulk. As was shown above, in the case of bulk compounds the Cr–Cr exchange interactions have rather strong FM character between the first neighbors, while the magnitude of interactions with the atoms at longer distances is much smaller. This behavior is reproduced quite well in the middle of the 5 ML film, although with some modification because of finite film thickness. The biggest changes occur for the interface and surface layers. Figure 10 shows the exchange coupling parameters calculated for 5 ML CrSb films on the GaSb substrate with the lattice parameter 10.68 au, for different crystal terminations at the surface and interface. The most pronounced exchange interactions in the films have FM character. However, one can see that in the case of Cr termination the exchange interactions between the first-neighbor Cr atoms within the surface layer have both FM as well as AFM character. Within the interface Cr layer (Sb/Cr interface, see figure 10(b)) the most pronounced Cr–Cr exchange interactions have FM character. Very similar interactions take place within the Cr layer at the As/Cr interface in the case of a CrSb film deposited GaAs substrate.

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However, it should be noted that deposition of a film creating an As/Sb/Cr interface leads to exchange interactions within the interface Cr layer having both FM and AFM character (figure 11(a)), that can result in a magnetic structure different from ferromagnetic. Using the exchange interaction parameters calculated for the film, MC simulations at T = 1 K have been performed to obtain the equilibrium magnetic structure. Figure 11(b) shows the result of MC simulations for a 5 ML CrSb film on GaAs with As/Sb/Cr interface and with Cr the topmost surface layer. One can see that AFM interactions between the interface Cr atoms lead to an AFM structure within the layer, while the non-collinear structure at the surface is a result of a competition between FM and AFM Cr–Cr exchange interactions.

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Deposition of CrSb film on a GaSb substrate with larger lattice parameter leads to pronounced FM Cr–Cr interaction in the interior as well as at the surface and at the interface. However, in the case of an InAs substrate the exchange interactions between neighboring Cr atoms within the interface Cr layer (As/Cr/Sb) are positive and negative depending on direction and are similar in magnitude, leading to a non-collinear magnetic structure within the layer (figure 12).

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In summary, half-metallicity has been found for bulk CrSb with ZB and wurtzite structure in a certain region of the lattice parameter. These compounds clearly exhibit FM order. The maxima of the Curie temperature correspond to those compounds which have the Fermi level positioned close to the middle of the energy gap of the minority-spin states. The calculations performed for CrSb layers deposited on different substrates show that half-metallic and FM properties observed for bulk can be destroyed at the film/substrate interface and at the surface of the film and even can lead, in the case of very thin CrSb films of a few monolayer (1–2 ML) thickness, to a complete loss of the properties attractive for technological applications. Because of the finite DOS at E_F in these cases the RKKY mechanism for Cr–Cr exchange interactions becomes more pronounced and can lead to the increase of AFM interactions between Cr atoms at certain distances and as a result to AFM or non-collinear magnetic structure at the surface or interface. As soon as such a behavior depends in an appreciable way on the structure parameters, it is very important to account for the lattice relaxation. This applies especially in those cases when the lattice mismatch between the film and substrate is rather large. However, investigations of relaxation effects in the case of deposited films have not yet been made in the present work.

Financial support by the Deutsche Forschungsgemeinschaft within the framework of the priority program (DFG-Schwerpunktprogramm 1415) Kristalline Nichtgleichgewichtsphasen (KNG)—Präparation, Charakterisierung und in situ-Untersuchung der Bildungsmechanismen is gratefully acknowledged.