# Exchange interactions and temperature dependence of the magnetization in half-metallic Heusler alloys

###### Abstract

We study the exchange interactions in half-metallic Heusler alloys using first-principles calculations in conjunction with the frozen-magnon approximation. The Curie temperature is estimated within both mean-field (MF) and random-phase-approximation (RPA) approaches. For the half-Heusler alloys NiMnSb and CoMnSb the dominant interaction is between the nearest Mn atoms. In this case the MF and RPA estimations differ strongly. The RPA approach provides better agreement with experiment. The exchange interactions are more complex in the case of full-Heusler alloys CoMnSi and CoCrAl where the dominant effects are the inter-sublattice interactions between the Mn(Cr) and Co atoms and between Co atoms at different sublattices. For these compounds we find that both MF and RPA give very close values of the Curie temperature slightly underestimating experimental quantities. We study the influence of the lattice compression on the magnetic properties. The temperature dependence of the magnetization is calculated using the RPA method within both quantum mechanical and classical approaches.

###### pacs:

75.50.Cc, 75.30.Et, 75.30.Ds, 75.60.-d^{†}

^{†}preprint: submitted to the Physical Review B

## I Introduction

During the last decade the half-metallic ferromagnets have become one of the most studied classes of materials. The existence of a gap in the minority-spin band structure leads to 100% spin-polarization of the electron states at the Fermi level and makes these systems attractive for applications in the emerging field of spintronics. Zutic In half-metals the creation of a fully spin-polarized current should be possible that should maximize the efficiency of magnetoelectronics devices.deBoeck

The half-metallicity was first predicted by de Groot and collaborators in 1983 when studying the band structure of a half-Heusler alloy NiMnSb.deGroot They found that the spin-down channel is semiconducting. In 2002 Galanakis et al. have shown that the gap arises from the interaction between the -orbitals of Ni and Mn creating bonding and antibonding states separated by a gap. GalanakisHalf Ishida and collaborators have proposed that also the full-Heusler compounds of the type CoMnZ, where Z stands for Si and Ge, are half-metals.Ishida In these compounds the origin of half-metallicity is more complex than in the half-Heusler alloys because of the presence of the states located entirely at the Co sites.GalanakisFull Several other Heusler alloys have been predicted to be half-metals.various Akinaga and collaborators Akinaga were able to crystallize a CrAs thin film in the zinc-blende structure, that is similar to the lattice of the Heusler alloys. The magnetic moment per formula unit was found to be close to 3 that corresponds to the integer value characteristic for half-metals. A number of further half-metallic materials are CrO in a metastable cubic phase, FeO, the manganites (e.g LaSrMnO)Soulen , the diluted magnetic semiconductors (e.g. Mn impurities in Si or GaAs). Freeman ; Akai

Besides strong spin polarization of the charge carriers in the ground state the spintronics materials must possess a high Curie temperature to allow the applications in the devices operating at room temperature. Available experimental information shows that the Heusler alloys are promising systems also in this respect. Webster Up to now the main body of the theoretical studies was devoted to the properties of the half-metallic gap.Nanda Recently, Chioncel and collaborators studied the influence of the correlation effects on the electron structure of CrAs.Chioncel They found that the spin-magnon interaction leads to the appearance of non-quasiparticle states in the spin-minority channel. The states are shown to lie above the Fermi level and to be sensitive to the value of the lattice constant. For a number of Heusler alloys it was shown that half-metallicity is preserved under tetragonalization of the crystal lattice Block and application of the hydrostatic pressure Picozzi-Gala . Mavropoulos et al. studied the influence of the spin-orbit coupling on the spin-polarization at the Fermi level and found the effect to be very small Mavropoulos that is in agreement with a small orbital moment calculated by Galanakis. GalanakisOrbit Larson et al.Larson have shown that the structure of Heusler alloys is stable with respect to the interchange of atoms and Orgassa and collaborators and Picozzi and collaborators have demonstrated that a small degree of disorder does not destroy the half-metallic gap.Orgassa ; Picozzi2004 Dowben and Skomski have shown that at non-zero temperatures the spin-wave excitations lead to the presence at the Fermi level of the electron states with opposite spin projections leading to decreasing spin-polarization of the charge carriers. Dowben

Despite very strong interest to the half-metallic ferromagnetism in Heusler alloys the number of theoretical studies of exchange interactions and Curie temperature in Heusler alloys is still very small. The first contribution to the density functional theory of the exchange interactions in these systems was made in an early paper by Kübler et al. Kubler83 where the microscopic mechanisms of the magnetism of Heusler alloys were discussed on the basis of the comparison of the ferromagnetic and antiferromagnetic configurations of the Mn moments. Recently, the studies of the inter-atomic exchange interactions in several Heusler compounds were reported by the present authors and Kurtulus et al. Sasioglou2004 ; Sasioglou2005 ; Kurtulus . Şaşıog̃lu et al. studied the exchange interactions in non-half-metallic NiMnZ (Z=Ga,In,Sn,Sb) and half-metallic MnVZ (Z=Al,Ge). The importance of the inter-sublattice exchange interaction has been demonstrated. For example, in the case of MnVZ (Z=Al,Ge) it was shown that the antiferromagnetic coupling between the V and Mn moments stabilizes the ferromagnetic alignment of the Mn moments. KüblerKubler2003 estimated of NiMnSb to be 601 K to 701 K depending on the approach used in the calculations. These values are in good correlation with experimental value of 730 K.Webster

The main task of the present contribution is the study of the exchange interactions in both half- and full-Heusler alloys. We use the calculated exchange parameters to estimate the Curie temperature in both the random phase () and the mean field approximations (). In Section II we briefly discuss the formalism employed in the calculations. In Section III we present the results on the spin magnetic moments and the density of states (DOS) for four compounds studied: NiMnSb, CoMnSb, CoMnSi and CoCrAl. In Section IV we discuss the calculated exchange interactions and Curie temperatures. Section V is devoted to the consideration of the temperature dependence of magnetization. The films of Heusler alloys grown on different substrates can have different lattice parameters and, as a result, noticeable variation of the electron structure. Section V contains the summary. In the Appendix, we present the formalism for the calculation of the Curie temperature of a multi-sublattice ferromagnet within the framework of the random phase approximation.

## Ii Calculational Method

Half- and full-Heusler alloys crystallize in the and structures respectively (see Fig. 1). The lattice consists from 4 interpenetrating fcc lattices. In the case of the half-Heusler alloys (XYZ) one of the four sublattices is vacant. The Bravais lattice is in both cases fcc. In full-Heusler alloy the atomic basis consists of four atoms. For example, in CoMnSi the positions of the basis atoms in Wyckoff coordinates are the following: Co atoms at and

The calculations are carried out with the augmented spherical waves method (ASW)asw within the atomic-sphere approximation (ASA).asa The exchange-correlation potential is chosen in the generalized gradient approximation. gga A dense Brillouin zone (BZ) sampling is used. The radii of all atomic spheres are chosen equal. In the case of half-Heusler alloys we introduce an empty sphere located at the unoccupied site.

### ii.1 Exchange parameters

The method for the calculation of exchange constants has been presented elsewhere.Sasioglou2004 Here we give a brief overview.

We describe the interatomic exchange interactions in terms of the classical Heisenberg Hamiltonian

(1) |

In Eq. (1), the indices and number different sublattices and and are the lattice vectors specifying the atoms within sublattices, is the unit vector pointing in the direction of the magnetic moment at site .

We employ the frozen-magnon approach to calculate interatomic Heisenberg exchange parameters.magnon The calculations involve few steps. In the first step, the exchange parameters between the atoms of a given sublattice are computed. The calculation is based on the evaluation of the energy of the frozen-magnon configurations defined by the following atomic polar and azimuthal angles

(2) |

The constant phase is chosen equal to zero. The magnetic moments of all other sublattices are kept parallel to the z axis. Within the Heisenberg model (1) the energy of such configuration takes the form

(3) |

where does not depend on q and the Fourier transform is defined by

(4) |

In the case of the sum in Eq. (4) does not include . Calculating for a regular -mesh in the Brillouin zone of the crystal and performing back Fourier transformation one gets exchange parameters for sublattice . The determination of the exchange interactions between the atoms of two different sublattices and is discussed in Ref. Sasioglou2004, .

### ii.2 Curie temperature

The Curie temperature is estimated within two different approaches: the mean–field approximation (MFA) and random phase approximation (RPA). The MFA for a multi-sublattice material requires solving the system of coupled equationsSasioglou2004 ; Anderson

(5) |

where is the average component of and . Eq. 5 can be represented in the form of eigenvalue matrix-problem

(6) |

where , is a unit matrix and is the vector of . The largest eigenvalue of matrix gives the value of .Anderson

A more consequent method for the study of the thermodynamics of Heisenberg systems is provided by the RPA approach. tyablikov ; Callen The RPA technique is intensively used for studies of both single-sublattice pajda ; bouzerar and multi-sublatticeantiferro ; ferri ; magnetite ; Azaria ; t_layer ; m_layer_1 ; m_layer_2 ; Nolting ; Turek2005 systems. In the case that only the exchange interactions within one sublattice are important the Curie temperature within the RPA is given by the relationpajda

(7) |

We use the RPA approach to study the temperature dependence of the magnetization in the temperature interval from 0 K to . The RPA technique for a multi-sublattice system is briefly presented in Appendix.

Compound | a(Å) | X | Y | Z | Void | Total |
---|---|---|---|---|---|---|

NiMnSb - a | 5.93 | 0.20 | 3.85 | -0.09 | 0.04 | 4.00 |

NiMnSb - a | 5.68 | 0.32 | 3.68 | -0.05 | 0.05 | 4.00 |

NiMnSb - a | 5.62 | 0.33 | 3.64 | -0.04 | 0.05 | 3.97 |

CoMnSb - a | 5.87 | -0.32 | 3.41 | -0.11 | 0.02 | 3.00 |

CoMnSb - a | 5.22 | 0.45 | 2.57 | -0.06 | 0.04 | 3.00 |

CoMnSb - a | 5.17 | 0.48 | 2.52 | -0.05 | 0.04 | 2.99 |

CoCrAl - a | 5.74 | 0.62 | 1.83 | -0.08 | - | 3.00 |

CoCrAl - a | 5.55 | 0.69 | 1.68 | -0.06 | - | 3.00 |

CoCrAl - a | 5.49 | 0.69 | 1.66 | -0.05 | - | 2.99 |

CoMnSi - a | 5.65 | 0.93 | 3.21 | -0.06 | - | 5.00 |

CoMnSi - a | 5.49 | 0.97 | 3.10 | -0.04 | - | 5.00 |

CoMnSi - a | 5.43 | 0.97 | 3.01 | -0.04 | - | 4.97 |

## Iii DOS and magnetic moments

### iii.1 NiMnSb and CoMnSb

In this section we report the calculation of DOS and magnetic moments at different lattice parameters for NiMnSb and for CoMnSb compound that has one electron per formula unit less than NiMnSb. The electronic structure of both compounds has been extensively studied earlier and the reader is referred to the review GalanakisReview, and references therein for detailed discussion. Here we present a brief description of the calculational results aiming to provide the basis for further considerations and to allow the comparison with previous work.

In Table 1 we collect the atomic and total spin moments for three different lattice parameters. The investigation of the influence of the value of the lattice parameter on the properties of the Heusler alloys is important since the samples grown on different substrates can have different lattice spacings. The first calculation is performed for the experimental bulk lattice constant.Webster The calculated densities of states (DOS) for this case are presented in the upper panel of Fig. 2. For both NiMnSb and CoMnSb the Fermi level lies in the low-energy part of the half-metallic gap. The compression of the lattice pushes the majority states to higher energies that results in increased energy position of the Fermi level with respect to the half-metallic gap. At the lattice parameter the Fermi level coincides with the upper edge of the gap (Fig. 2). In the next step we further contracted the lattice constant by 1% (lattice parameter , bottom panel in Fig. 2). In this case the Fermi level is slightly above the gap and the total spin moment is slightly smaller than the integer values of 3 and 4 for CoMnSb and NiMnSb respectively.

The contraction of the lattice leads to an increase of the hybridization between the orbitals of different transition-metal atoms. This results in a decrease of the spin moment of Mn. In the case of NiMnSb this change is small: the reduction of the Mn spin moment under lattice contraction from the experimental lattice parameter to is 0.2 . The Ni spin moment increases by about the same value to preserve the integer value of the total spin moment of 4 .

In CoMnSb, the half-metallic gap is larger than in NiMnSb. As a result, the transition of the Fermi level to the upper gap-edge requires a large lattice contraction of 11% (Table 1). This leads to a strong decrease of the Mn moment by 0.84. To compensate this decrease the Co moment changes its sign transforming the magnetic structure from ferrimagnetic to ferromagnetic.

The influence of the lattice contraction on the exchange interactions and Curie temperature is discussed in the next Section.

### iii.2 CoCrAl and CoMnSi

The second group of materials studied in the paper is formed by the full-Heusler compounds CoMnSi and CoCrAl. The electronic structure of these systems has been studied earlier. GalanakisFull Compared to half-Heusler systems, the presence of two Co atoms per formula unit results in an increased coordination number of Co atoms surrounding Mn atoms (eight instead of four in CoMnSb). This leads to an increased hybridization between the orbitals of the Mn and Co atoms. The spin moment of Co in CoMnSi is about 1 that is considerably larger than the Co moment in CoMnSb. In CoCrAl the Co moment is about 1/3rd smaller than in CoMnSi that reflects a smaller value of the Cr moment compared to the Mn moment (Table 1).

As in the case of the half-Heusler compounds discussed above, the variation of the lattice parameter leads to the change in the position of the Fermi level. At the experimental lattice parameter the Fermi level of CoCrAl lies in the lower part of the half-metallic gap while for CoMnSi it is close to the middle of the gap (Fig. 2b). The contraction of the lattice needed to place the Fermi level at the upper edge of the gap is smaller than for CoMnSb. As a result, the change in the magnetic moments is also relatively weak (Table 1).

## Iv Exchange parameters and Curie temperature

### iv.1 NiMnSb and CoMnSb

In Fig. 3 we present the exchange constants calculated for various lattice spacings. The Co-Co, Ni-Ni exchange interactions as well as the exchange interactions between the moments of the 3d atoms and the induced moments of Sb atoms are very weak and are not shown. The weakness of the effective Co-Co and Ni-Ni exchange interactions can be explained by a relatively large distance between atoms (Fig. 1) and relatively small atomic moments.

On the other hand, each Ni(Co) atom is surrounded by four Mn atoms as nearest neighbors that results in strong Mn-Ni(Co) exchange interaction (Fig. 3). Also the exchange interaction between large Mn moments is strong.

The ferromagnetic Mn-Mn interactions are mainly responsible for the stable ferromagnetism of these materials. For both systems and for all lattice spacings studied the leading Mn-Mn exchange interaction is strongly positive. In NiMnSb, the Mn-Ni interaction of the nearest neighbors is positive for all three lattice parameters leading to the parallel orientation of the spins of the Mn and Ni atoms. In CoMnSb the situation is different. At the experimental lattice parameter the leading Mn-Co interaction is negative resulting in the ferrimagnetism of the system. For the contracted lattices the interaction changes sign resulting in the ferromagnetic ground state of the alloy.

The analysis of the strength of the exchange interaction as a function of the lattice parameter shows that in CoMnSb the contraction leads to a strong increase of both leading Mn-Co and Mn-Mn interactions. On the other hand, in NiMnSb the increase of the Mn-Ni interaction is accompanied by a decrease of the leading Mn-Mn interaction. Simultaneously, the interaction between the second-nearest Mn atoms increases with contraction in the case of NiMnSb staying almost unchanged in CoMnSb. This complexity of the behavior reflects the complexity of the electronic structure of the systems.

The interatomic exchange parameters are used to evaluate the Curie temperature within two different approaches: MFA and RPA. In Table 2 we present the values of the Curie temperature obtained, first, by taking into account the Mn-Mn interactions only and, second, with account for both Mn-Mn and Mn-Ni(Co) interactions. The contribution of the inter-sublattice interactions to the Curie temperature appears to be less than 5 % for both compounds and the Curie temperature is mainly determined by the intra-sublattice Mn-Mn interaction.

The MFA and RPA estimations of the Curie temperature differ rather strongly (Table 2). The relative difference of two estimations is about 20%. The reason behind this difference will be discussed in the following section. For the systems considered here the RPA estimations of the Curie temperatures are in good agreement with the experiment, somewhat overestimating the experimental values.

(K) | MFA-Y | RPA-Y | MFA-all | RPA-all | Exp. |
---|---|---|---|---|---|

NiMnSb - a | 1096 | 880 | 1112 | 900 | 730 |

NiMnSb - a | 1060 | 853 | 1107 | 908 | - |

NiMnSb - a | 1008 | 802 | 1063 | 869 | - |

CoMnSb - a | 785 | 619 | 815 | 671 | 490 |

CoMnSb - a | 1185 | 940 | 1276 | 1052 | - |

CoMnSb - a | 1140 | 893 | 1252 | 1032 | - |

CoCrAl - a | 148 | 141 | 280 | 270 | 334 |

CoCrAl - a | 168 | 159 | 384 | 365 | - |

CoCrAl - a | 164 | 154 | 400 | 379 | - |

CoMnSi - a | 232 | 196 | 857 | 740 | 985 |

CoMnSi - a | 142 | 118 | 934 | 804 | - |

CoMnSi - a | 110 | 75 | 957 | 817 | - |

Recently Kübler Kubler2003 reported estimations of the Curie temperature of NiMnSb. His approach is based on the evaluation of the non-uniform magnetic susceptibility on the basis of the Landau-type expansion for the free energy. Within some approximations the parameters used in the study of the thermodynamical properties can be expressed in terms of the quantities evaluated within the first-principles DFT calculations. The estimated values of the Curie temperature are 601 K for a static approach and 701 K if the frequency dependence of the susceptibility is taken into account. These estimations are somewhat lower than the value of 880 K given by the RPA approach (Table 2). A detailed comparative analysis of the two calculational schemes is needed to get an insight in the physical origin of this difference.

The contraction of the lattice in the case of the NiMnSb compound leads to an increase of the Mn-Ni interactions (Fig. 3). This results in increased difference between the Curie temperatures calculated with the Mn-Mn interactions only and with both Mn-Mn and Ni-Mn interactions taken into account (Table 2). For CoMnSb, the leading exchange interactions of both Mn-Mn and Mn-Co types increase in the value under transition from the experimental lattice constant to (Fig. 3). As a result, the Curie temperature increases with contraction by about 50%.

### iv.2 CoCrAl and CoMnSi

The presence of an extra Co atom in the full-Heusler alloys makes the interactions more complex than in the case of the half-Heusler alloys. In CoMnSb the important interactions arise between nearest Mn atoms (Mn-Mn interactions) and between nearest Mn and Co atoms (Mn-Co interaction). In the case of CoMnSi (Fig. 4) the interactions between Co atoms at the same sublattice (Co-Co) and between Co atoms at different sublattices (Co-Co) must be taken into account. The cobalt atoms at different sublattices have the same local environment rotated by 90 about the [001] axis. The leading interaction responsible for the stability of the ferromagnetism is the Mn-Co interaction between Mn atoms and eight nearest Co atoms (Fig. 4). This interaction changes weakly with the contraction of the lattice. Our exchange parameters agree well with the parameters of Kurtulus et al. (Fig. 4) who also found the Co-Mn exchange interaction to be leading. Kurtulus

The interaction between nearest Co atoms at different sublattices (empty squares in Fig. 4) favors the ferromagnetism also and is stronger than the ferromagnetic interaction between the nearest Mn atoms (filled spheres). Although the spin moment of Mn atoms is larger than the moment of Co atoms (Table 1) the opposite relation between exchange parameters can be the consequence of the smaller distance between the Co atoms: between the Co atoms and between the Mn atoms. An interesting feature of the intra-sublattice Mn-Mn and Co-Co interactions is different signs of the exchange parameters for different distances between atoms. This leads to a RKKY-like oscillations of the parameters (Fig. 4).

In CoCrAl the leading Cr-Co interactions (filled triangles) are much smaller than corresponding Mn-Co interactions in CoMnSi. On the other hand, the leading inter-sublattice ferromagnetic Co-Co interactions are comparable in both systems. The compression of the lattice leads to an increase of the magnitude of the inter-sublattice Co-Cr and Co-Co coupling. The intra-sublattice Cr-Cr and Co-Co interactions oscillate with varying inter-atomic distances.

The difference in the properties of the exchange parameters of the half- and full-Heusler alloys is reflected in the calculated Curie temperatures (Table 2). In contrast to CoMnSb where the Mn-Mn exchange interactions are dominant, in CoMnSi they play a secondary role. The T calculated taking into account these interactions only is much smaller than the Curie temperature calculated with all inter-atomic exchange interactions taken into account (Table 2). The same conclusion is valid for CoCrAl where the Cr-Cr interactions give about half of the Curie temperature obtained with all interactions included into consideration.

A striking feature of the full Heusler compound CoCrAl that differs it strongly from the half-Heusler systems considered in the previous Section is a very small difference between the values calculated within the MFA and RPA approaches. A similar behavior was obtained for the Curie temperatures of the zincblende MnSi and MnC.stability_zb In CoMnSi, the relative difference of the MFA and RPA estimations assumes an intermediate position between the half-Heusler systems and CoCrAl.

To understand the origin of the strong variation of the relative difference of the MFA and RPA estimations of the Curie temperature we compare in Fig. 5 the frozen magnon dispersions for two compounds. The magnons correspond to the Mn sublattice in the case of NiMnSb and to the Cr sublattice in the case of CoCrAl. As seen from Table 2 the MFA and RPA estimations obtained with the use of these dispersions differ by 20% for NiMnSb and by 5% for CoCrAl.

The Curie temperature is given by the average value of the magnon energies. In MFA this is the arithmetic average while in RPA this is harmonic average. Therefore we need to understand why for CoCrAl these two averages are much closer than for NiMnSb. The following properties of the averages are important for us. The arithmetic average takes all the magnon values with equal weight whereas in the harmonic average the weight decreases with increasing energy of the magnon.pajda ; bouzerar ; stability_zb ; above_room ; sabiryanov It is an arithmetic property that the MFA estimation is larger than the RPA one or equal to it if all numbers to be averaged are equal to each other. In terms of magnon energies, is equal to in the case that the magnon spectrum is dispersion-less.

Considering the frozen-magnon dispersions from the viewpoint of these properties we indeed can expect that the arithmetic and harmonic averages will be closer for CoCrAl. In Fig. 5 both curves are scaled to have almost the same maximal value. It is seen that the CoCrAl dispersion has smaller relative contribution of the low-energy magnons because of the steeper increase of the curve at small wave vectors. It has also smaller contribution of the magnons with the largest energies because the maxima have the form of well-defined peaks opposite to NiMnSb where we get a plateau. Thus the main contribution in the case of CoCrAl comes from intermediate energies that makes the MFA and RPA estimations closer.

In Fig. 6 we present the calculated spin-wave spectra for NiMnSb and CoCrAl. The spin-wave energies are obtained by the diagonalization of the matrix of exchange parameters that contains all important intra- and inter-sublattice interactions. The number of branches in the spectrum is equal to the number of magnetic atoms in the unit cell: two in NiMnSb and three in CoCrAl. One of the branches is acoustic and has zero energy for zero wave vector. Also in the spin-wave spectra, we see strong difference between two systems. In NiMnSb, the acoustic branch is predominantly of the Ni type stemming from the weak interaction between Mn and Ni magnetic moments (see Fig. 3). On the other hand, the optical branch is of predominantly the Mn type. The strong hybridization between two sublattices is obtained only about . In CoCrAl, the energy scale of the branches differs much smaller and the hybridization between sublattices is stronger than for NiMnSb.

Coming back to the considerations of the Curie temperatures, we conclude that, in general, the Curie temperatures of CoMnSi and CoCrAl calculated within both MFA and RPA are in good agreement with experiment while the MFA values in the case of NiMnSb and CoMnSb overestimate the Curie temperature strongly.

The lattice contraction leads in both compounds to an enhancement of the Mn-Co(Cr-Co) exchange constants that results in an increase of the Curie temperature.

Kurtulus and collaborators have calculated the Curie temperature for CoMnSi within MFA and found the value of 1251 K that is considerably larger than our MFA estimate of 857 K. This difference is unexpected since the values of the exchange parameters obtained by Kurtulus et al. agree well with our parameters (Fig.4). To reveal the origin of the discrepancy we performed the MFA calculation of the Curie temperature with the exchange parameters of Kurtulus et al. and obtained the value of 942 K which is in reasonable agreement with our estimate. Apparently the reason for the inconsistency is in the procedure of the solving of the multiple-sublattice MFA problem used by Kurtulus et al. that should deviate from the standard one. Anderson

## V Temperature dependence of the magnetization

The study of the temperature dependence of the magnetic properties of itinerant ferromagnets is one of the fundamental problems of ongoing researches. Although density functional theory can formally be extended to the finite temperatures Mermin , it is rarely used because of the lack of suitable exchange-correlation potentials for magnetic systems at finite temperatures. Statistical mechanics treatment of model Hamiltonians is usually employed. In this section we will present the results of the calculation of the temperature dependence of magnetization that is based on the consideration of the Heisenberg hamiltonian with exchange parameters calculated within a parameter-free DFT approach (Sect. II).

To calculate the temperature dependence of the magnetization we use the RPA method as described in appendix. We consider both classical-spin and quantum-spin cases.

In the classical-spin calculations the calculated values of the magnetic moments (Table 1) are used. To perform quantum-mechanical RPA calculation we assign integer values to the atomic moments. In the semi Heusler compounds we ignore the induced moments on Ni and Co atoms and assign the whole moment per formula unit to the Mn atom: 4 () in NiMnSb and 3 () in CoMnSb. In CoMnSi we take the values of 3 () and 1 () for Mn and Co atoms respectively. This assignment preserves the value of the total spin moment per chemical unit. In CoCrAl we use in the quantum-RPA calculations the atomic moment of 2 () for Cr and 1 () for Co.

In Fig. 7(a), we present in the normalized form the calculated temperature dependence of the magnetization for both families of Heusler compounds. The calculations are performed for the experimental lattice parameter. For comparison, the experimental curves are presented. The nature of the spin (quantum or classical) influences the form of the curves considerably. The classical curve lies lower than the quantum one. This results from a faster drop of the magnetization in the low-temperature region in the case of classical spins. In general, the quantum consideration gives better agreement of the form of the temperature dependence of the magnetization with experiment.

In Fig. 7(b) we present the temperature dependence of the magnetization of individual sublattices. As expected from the previous discussions in half-Heusler systems the main contribution to the magnetization comes from the Mn sublattice while for the full-Heusler systems both 3d atoms contribute substantially.

Considering the calculated Curie temperatures we notice that the value of calculated within the quantum-mechanical RPA is substantially larger than the corresponding classical estimation (see Fig. 7). This property is well-known and has its mathematical origin in the factor entering the RPA expression for the Curie temperature (Eq. 21). In Fig. 8 we show the dependence of the Curie temperature calculated within the quantum mechanical RPA approach on the value of . The exchange parameters are kept unchanged in these calculations. We see that the dependence has a monotonous character tending to a classical limit for large .

Presently we do not have an explanation why quantum-mechanical calculations give better form of the temperature dependence while the classical calculation provides better value of the Curie temperature. We can suggest the following arguments. The quantum treatment is more appropriate than the classical one in the low-temperature region. At high temperatures characterized by strong deviation of the atomic spins from the magnetization axis the quantum treatment gives too slow decrease of the magnetization. It is worth noting that the consequent theory should take into account not only the orientational disorder of the atomic moments but also the single-particle (Stoner-type) excitations leading to the decrease of atomic moments. Another important aspect is related to the fact that the exchange parameters used in the calculations are estimated within the picture of classical atomic moments described above. It is possible that the values of the exchange parameters must be modified for the use in the quantum-mechanical case. These questions belong to fundamental problems of the quantum-mechanical description of the magnetic systems with itinerant electrons.

## Vi Summary and conclusions

We studied the electronic structure of several Heusler alloys using the augmented spherical waves method in conjunction with the generalized gradient approximation to the exchange and correlation potential. Using the frozen-magnon approximation we calculated inter-atomic exchange parameters that were used to estimate the Curie temperature. The Curie temperature was estimated within both mean-field and random-phase approximation techniques.

For the half-Heusler alloys NiMnSb and CoMnSb the dominant interaction is between the Mn atoms. The lattice compression results in considerable change of the exchange parameters and Curie temperature.

The magnetic interactions are more complex in full-Heusler alloys CoMnSi and CoCrAl. In both cases the ferromagnetism is stabilized by the inter-sublattice interactions between the Mn(Cr) and Co atoms and between Co atoms belonging to different sublattices. Both the random phase and mean field approximations slightly underestimate the values of the Curie temperature. Compression of the lattice constant has little effect on the magnetic properties of the full-Heusler alloys.

We study the temperature dependence of the magnetization within the quantum mechanical and classical RPA. The quantum-mechanical approach gives the form of the temperature dependence that is in good agreement with experiment. The value of the Curie temperature is, however, overestimated in the quantum-mechanical calculation.

###### Acknowledgements.

The financial support of Bundesministerium für Bildung und Forschung is acknowledged. I.G. is a fellow of the Greek State Scholarship Foundation. We thank the authors of Ref. Turek2005, for making the manuscript available before publication.## Appendix A The random phase approximation for multi-sublattice Heisenberg Hamiltonian

The Green function approach is a powerful tool in the study of the magnetism of complex systems. (See, e.g., the application of the method to antiferromagnetantiferro , ferrimagnetsferri ; magnetite , random alloysAzaria , layered systems t_layer ; m_layer_1 ; m_layer_2 , disordered dilute magnetic systems Nolting , multi-sublattice ferromagnets Turek2005 .) In this appendix we briefly overview the formalism to study the temperature dependence of the magnetization of multi-sublattice systems within the random phase approximation.

We start with the Heisenberg Hamiltonian for quantum spins

(8) |

where is the normalized spin operator corresponding to site ().

In terms of the creation and destruction operators the Hamiltonian can be written in the form

(9) |

where .

Following Callen Callen let us introduce Green function

(10) |

where is a parameter, is the step function ( for ), denotes the commutator and is the thermal average over the canonical ensemble, ie., with

Writing the equation of motion for we obtain

(11) | |||||

The last commutator term in Eq. (11) generates higher-order Green functions. These functions can be reduced to lower-order functions by using Tyablikov decoupling (random phase approximation) schemetyablikov :

(12) |

Applying this decoupling procedure to Eq. (11) we get

(13) | |||||

After a Fourier transformation to energy and momentum space [] we obtain

(14) | |||||

Eq. (14) can be written in a compact matrix form

(15) |

where is a symmetric square matrix, I is a unit matrix and the inhomogeneity matrix is expressed by

(16) |

matrix is defined by

(17) |

Next, we introduce the transformation which diagonalizes matrix :m_layer_2

(18) |

where is the diagonal matrix whose elements give the spin wave energies . The number of branches in the spin wave spectrum is equal to the number of magnetic atoms in the unit cell. The transformation matrix and its inverse are obtained from the right eigenvectors of as columns and from the left eigenvectors as rows, respectively.

Using the spectral theorem and Callen’s technique Callen one obtains the thermal averages of the sublattice magnetizations:

(19) |

where is an auxiliary function given by

(20) |

In Eq. (20), is the number of points in the first BZ.

Eq. (19) is the central equation for the calculation of the sublattice magnetizations. It must be solved self-consistently. The Curie temperature is determined as the point where all sublattice magnetizations vanish.

Near ( and ) Eq. (19) can be simplified. Expanding in and using Eq. (20) one obtains

(21) |

From Eq. (21), it follows that for spin-independent Heisenberg exchange parameters [Eq. (8)] the dependence of the Curie temperature on the spin value is defined by the factor .

The classical limit can be obtained by letting in Eqs. (19) and (21).Turek2005 Factor in Eq. (21) becomes in this case unity. The temperature dependence of the magnetization can be calculated using a semiclassical analog of Eq. (19) given by Turek2005 ; Callen2

(22) |

where is the Langevin function and is the angular momentum vector of size one.

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