A two dimensional Heusler alloy model

Solid State Communications 149 (2009) 73–77

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Solid State Communications journal homepage: www.elsevier.com/locate/ssc

A two dimensional Heusler alloy model R. Rodríguez-Alba a , F. Aguilera-Granja b , J.L. Morán-López c,∗ a

Department of Physics and Mathematics, Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico

b

Instute of Physics, Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico

c

Center for Computational Materials, Institute for Computational Engineering and Science, University of Texas at Austin, Austin, TX 78712-0027, United States

article

info

Article history: Received 6 March 2008 Received in revised form 1 August 2008 Accepted 7 October 2008 by R. Merlin Available online 19 October 2008 PACS: 75.70.Ak 64.60.De 64.60.Ej

a b s t r a c t A two dimensional version of a Heusler alloy X2 Mnc Z1−c is presented. The Hamiltonian includes chemical interactions between nearest neighbors and magnetic interactions between first, second and third neighbors. The ground state phase diagrams at zero magnetic field and their range of stability with regard to the chemical and magnetic interactions are calculated by using the method of linear inequalities. The unit used in the calculation is a five point cluster, which allows describing an ordered alloy with the Mn atoms forming decorated ferromagnetic, antiferromagnetic, superantiferromagnetic and other more complex arrangements. Results for c = 1/2 and 3/4 are presented. © 2008 Elsevier Ltd. All rights reserved.

Keywords: A. Magnetic material A. Low dimensional systems A. Heusler alloys D. Ground states

The Heusler alloys, are ternary systems X2 MnZ that have been know since 1903 [1]. These systems have manganese as one of the main components and show a rich variety of magnetic phases, depending on the two other chemical components and on the temperature [2]. These alloys looked very promising for applications since the manganese atom has a magnetic moment close to 4µB . These ternary systems crystallize with L21 structure, and in general the X element is a noble or transition metal and the Z element has s and p valence electrons. According to a previous calculation [3] the role of the X atoms is to determine the lattice constant and the Z atoms mediate the interactions between Mn atoms. There are more complex Heusler alloys in which the element X is also magnetic. Recently these kind of systems have been intensively studied owing to great potential for spintronics [4,5], magnetically driven actuators [6] and shape memory materials [7]. Two of those systems are Co2 MnGa and Ni2 MnGe, in which the magnetic properties of Co and Ni make the alloy more complex but at the same time richer in magneto-electronic behavior. In

∗ Corresponding address: Universidad Politécnica de San Luis Potosí, Iturbide 140 Zona Centro, 78000 San Luis Potosi, Mexico. Tel.: +52 444 8342012; fax: +52 444 8342010. E-mail address: [email protected] (J.L. Morán-López). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.10.011

particular, the Co alloys has a density of states that show halfmetallicity. i.e. the majority and minority spin bands show a metallic and semiconductor character. This makes this alloys attractive for applications in spintronics where the capability to inject electrically spin-polarized carriers into unpolarized semiconductors [8–10] is the key element. On the other hand, the Ni alloys are important as magnetic shape memory materials. Here we restrict to Heusler alloys with Mn as its only magnetic component. This paper has a two-fold motivation. On one hand, a two dimensional model of a Heusler alloy may be more tractable than the three dimensional version and may serve to identify key parameters and recognize interesting features. In addition, due to the complexity of the three-dimensional Heusler alloys, the interplay of magnetism and chemical order, has been addressed only in a reduced number of theoretical studies [11,12]. The simplification to a two-dimensional system, let us study further the chemical and magnetic-order interplay, which rules the properties of these systems. Aware of the development of sophisticated techniques which allow one to deposit multiple chemical elements, it might be possible to grow in the future such two-dimensional systems. Here, for the first time, we present calculations of the ground states of a two dimensional version of a Heusler alloy within a phenomenological model in which only pairwise interactions,

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cluster shown in Fig. 1(b). We denote the sites of the square vertex βi , i = 1, 2, 3, 4, and the one in the middle by γ . One can notice that in this cell, the number of first, second, and third neighbors are z1 = 4, z2 = 4, and z3 = 2, respectively. Since each site can be occupied by Mn↑ , Mn↓ or Z, the total number of configurations is 35 = 243, however, many of them are degenerate with a multiplicity λr . Here, we consider only ordering alloys and by taking into account the symmetry of the cluster one finds that the total number of different configurations reduces to 34. If we denote the probability to find the Xr configuration by xr , it follows that 34 X

λr xr = 1.

(2)

r =1

There is a second constraint that has to be observed and involves the nominal concentration of Mn atoms in the binary system 34 X

cr λr xr = c

(3)

r =1

Fig. 1. (a) The two-dimensional Heusler crystalline structure showing the various sublattices occupied by the X, Mn, and Z elements. (b) The decimated lattice containing only the sites occupied by Mn and Z elements. The five-point cluster showing the βi and the γ sublattices are also displayed.

chemical and magnetic, are included. The chemical interactions between nearest neighbors of type I and J are denoted by VI ,J . The magnetic interaction between the nth neighbor manganese atoms are denoted by Jn . In Fig. 1(a), we show the two dimensional Heusler lattice model considered here. It is a square lattice in which the four interpenetrating lattices are also square; two of them are occupied by the X atoms, and the other two by the Mn and the Z components. Among the rich variety of behaviors, it has been found [2], there are alloys in which, in a wide range of temperatures, the X element does not interchange sites with the other components (Pd2 MnIn, Pd2 MnSn). Thus, the element X just provides the skeleton and one can ignore the two sublattices occupied by the X element and decimate these sets of sites. By applying this procedure, we obtain the lattice shown in Fig. 1(b), where we show only the two square interpenetrating lattices occupied by the elements Mn and Z. In terms of the chemical and magnetic interactions, the total internal energy of the system can be written as E=−

X I ,J

N I ,J V I ,J −

3 X X (σi σj )n Jn , n

(1)

i ,j

where I and J denote the Mn and Z atoms, σi and σj are the magnetic spins of Mn with orientation up (↑) or down (↓), and n = 1, 2, 3. At low temperatures, positive values of V1 = VMnMn + VZZ − 2VMnZ drive the alloy to an ordered array while negative ones tend the alloy to separate into two phases. Furthermore, positive (negative) values of Jn favor a ferromagnetic (antiferromagnetic) alignment between the n-th Mn neighbors. To calculate the ground states that can be attainable with the interactions considered in our Hamiltonian, we take the five-point

where cr is the concentration of Mn atoms in the cluster r. Furthermore, this cluster allows to describe ordered arrangements corresponding to the concentrations c = 1, 7/8, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8, and 0. In this communication we only report the cases of c = 1/2 and 3/4. In Fig. 2 we present the different arrangement possible within the five-point cluster in the case of the equiatomic alloy. All the figures represent a complete ordered alloy with the Mn atoms arranged with different magnetic patterns. It is important to notice that in this case there are no pairs of Mn atoms as nearest neighbors. In Fig. 2(a), the Mn atoms order ferromagnetically (F) and the phase is characterized by J2 > 0, J3 > 0. In Fig. 2(b) the manganese atoms are arranged in alternating diagonals with ferromagnetic and antiferromagnetic coupling (F–AF). This pattern has an equal number of ferromagnetic and antiferromagnetic second and third neighbor pairs. Thus the magnetic contribution to the energy is zero. A superantiferromagnetic (SAF) pattern is shown in Fig. 2(c). This phase has an equal number of ferromagnetic and antiferromagnetic second neighbor pairs and the third neighbor pairs are coupled antiferromagnetically (J3 < 0). Finally, Fig. 2(d), represents the manganese atoms with antiferromagnetic order (AF). This phase is characterized by J2 < 0 and J3 > 0. Now we proceed to calculate the ground state of the system as a function of the energy parameters. Since E is a linear function of the configurational parameters xi , all possible ordered states are located inside a convex polyhedron in configurational space. The range of stability with respect to the interaction parameters is given by an hypercone with extreme rays defined by the normals to all phases of the configurational polyhedral converging to the vertex in question [13]. The results for the ground states for the equiatomic system depend on the energy parameters VMnZ , J2 and J3 . In Fig. 3(a) we show the results in the V12 = VMnZ /J2 versus J32 = J3 /J2 space and assuming positive values for J2 . The only arrangements possible in this part of the phase space are the ferromagnetic state F, and the superantiferromagnetic state SAF. Since J2 > 0 the states with ferromagnetic arrangements between second nearest neighbors are the ones that are stable. As mentioned above, the SAF is stabilized only by negative values of J3 . In the hatched area the energy is positive; i.e. the magnetic energy does not exceed the negative chemical energy that tends the system to form a segregated alloy. We show in Fig. 3(b) the phase diagram in the J3V = J3 /VMnZ versus J2V = J2 /VMnZ , for the case VMnZ < 0. The possible stable states are the ferromagnet F, the antiferromagnet AF, and

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(a) Ferromagnet.

(b) Ferro–antiferromagnet.

(c) Superantiferromagnet.

(d) Antiferromagnet.

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Fig. 2. The various magnetic phases of a Heusler alloy with c = 1/2.

Fig. 3. (a) The ground state phase diagram in the V12 (=VMnZ /J2 ) versus J32 (= J3 /J2 ) parameter space for systems with J2 > 0, for an alloy with cMn = 0.5. In the hatched areas there are no stable solutions. (b) The ground state phase diagram in the J3V (=J3 /VMnZ ) versus J2V (=J2 /VMnZ ) parameter space for systems with VMnZ < 0, for an alloy with cMn = 0.5. In the hatched area there are no stable solutions.

the superantiferromagnet SAF. There is also a part in the center of the phase diagram in which there are no solutions. This is due to the small value of magnetic interactions as compared with the negative values of VMnZ that do not favor chemical ordering. We now consider the case of an ordering alloy with Mn concentration c = 3/4. In Fig. 4 we show the six different phases: (a) ferro-diagonal, characterized by positive values of J1 , J2 and J3 , (b) ferro-square, also characterized by positive values of the magnetic interactions but with different number of magnetic

pairs. In Fig. (c) we show a superantiferromanetic phase, stabilized by J1 < 0, J2 > 0, and J3 > 0. In (d) an antiferro-square arrangement is shown, with similar magnetic interactions as the previous case but different number of pairs. In Fig. (e) we present a ferro–antiferro-diagonal pattern in which the number of ferro and antiferro first magnetic neighbors is the same (leading to zero contribution to the energy) and J2 > 0, J3 > 0. Finally, Fig. (f) shows a ferro–antiferro-p4 phase in which an antiferromagnetic periodicity every four second nearest neighbors

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(a) Ferro-diagonal.

(b) Ferro-square.

(c) Superantiferro.

(d) Antiferro-square.

(e) Ferro–antiferro-diagonal.

(f) Ferro–antiferro-p4.

Fig. 4. The various magnetic phases of a Heusler alloy with c = 3/4.

Fig. 5. (a) The ground state phase diagram in the J31 (=J3 /J1 ) versus J21 (=J2 /J1 ) parameter space for systems with J1 < 0, for an alloy with cMn = 0.75. In the hatched area there are no stable solutions. (b) The ground state phase diagram in the J32 (=J3 /J2 ) versus J12 (=J1 /J2 ) parameter space for systems with J2 < 0. The values for the chemical interactions were taken as VMnMn = 3, and VMnZ = 1. In the hatched area there are no stable solutions.

is present. This phase is characterized by an equal number of ferro and antiferro magnetic first neighbors and positive magnetic interactions between second and third magnetic neighbors. It is important to notice that the superantiferromagnetic phase is different from the one for c = 1/2 (Fig. 2(c)). In contrast to the system with c = 1/2, in this case there are magnetic first neighbor pairs, but no Z–Z first neighbors. Thus, there is now a contribution to the energy coming from the J1

magnetic and VMnMn chemical interactions. In order to simplify the number of parameters and therefore the analysis, we assumed the following chemical interactions VMnMn = 3 and VMnZ = 1, which in the absence of magnetic interactions lead to spatial order. In Fig. 5(a) we show the phase diagram in the J31 = J3 /J1 versus J21 = J2 /J1 parameter space, assuming J1 < 0. The two stable phases are the antiferromagnetic-square and the superantiferromagnetic. In the hatched area there are no ordered solutions, mainly due to the

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fact that all the six phases are stabilized by positive values of the magnetic interaction between third neighbors. The phase diagram in the J32 = J3 /J2 versus J12 = J1 /J2 , parameter space is presented in Fig. 5(b) for negative values for J2 . The possible states are superantiferromagnet, antiferro-square, ferro-square, and ferro-diagonal. Again one finds a large area with no stable solutions. Positive values of J32 imply that both J2 and J3 are negative, i.e. antiferromagnetically coupled second and third neighbors. As mentioned above, there are no states with those characteristics. In summary, we have presented the analysis of the stability of the ground states for a two dimensional Heusler alloy model, in which the only magnetic component is Mn. We ignored the X element and considered only the part Mnc Z1−c . We analyzed the cases in which the concentrations of the magnetic component are c = 1/2 and c = 3/4. We have modeled the system by including in our Hamiltonian, chemical as well as magnetic interactions and the number of sites in the cluster unit taken exactly into account is five. Even in this simplified model, our results show a rich variety of decorated magnetic phases. A full analysis with other concentrations of the magnetic element will be published elsewhere. We expect that with the implementation of recent techniques for the synthesis of ultra thin films, a new area of twodimensional magnetic systems is now open and new applications can be found.

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Acknowledgments J.L. M.-L. acknowledges the support of the J. Tinsley Oden Faculty Fellowship Research Program in the Institute for Computational Engineering and Sciences at The University of Texas at Austin and the partial support by CONACYT under grant 61417 S-3131. Useful discussions with A. Zayak are kindly acknowledged. References [1] [2] [3] [4] [5] [6] [7]

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