- Email: [email protected]
Magnetic phase transitions of all-d metal Heusler type model
Journal Pre-proof Magnetic phase transitions of all-d metal Heusler type model Gülistan Mert PII:
S0925-8388(19)34545-1
DOI:
https://doi.org/10.1016/j.jallcom.2019.153299
Reference:
JALCOM 153299
To appear in:
Journal of Alloys and Compounds
Received Date: 31 July 2019 Revised Date:
4 December 2019
Accepted Date: 5 December 2019
Please cite this article as: Gü. Mert, Magnetic phase transitions of all-d metal Heusler type model, Journal of Alloys and Compounds (2020), doi: https://doi.org/10.1016/j.jallcom.2019.153299. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Author Contributions Section This study was designed, coded (in Fortran Language), analyzed, written and commented by G.M.
MAGNETIC PHASE TRANSITIONS OF ALL-D METAL HEUSLER TYPE MODEL Gülistan Mert* Department of Physics, Selçuk University, 42075, Konya, Turkey
ABSTRACT The magnetic phase transitions of all-d metal Heusler type model, Ni2MnZ (Z = Ti, Sc and V), are investigated by using the double-time Green`s Function Method within the random phase approximation. We assume that the magnetic moments of Ni, Mn, and Z in Ni2MnZ which crystallized in L21 structure is localized on their atom and the next-nearest neighbor interactions between Mn atoms are negative. While the Z atoms give only a small contribution to the total magnetic moment, it is emphasized that the importance of existence of antiferromagnetic interaction between Z atom and other atoms in observing first-order phase transitions. Ni2MnZ shows a ferromagnetic state depending the values of the Hamiltonian parameters. If any one of these interactions have negative value and the system is subject to an external magnetic field, it is observed that the magnetization curves exhibit to the first-order phase transitions. PACS: Keywords: All-d-Metal Heusler Structures; Ferromagnetism; Heisenberg Model; Green`s Function; Random Phase Approximation *
Email address: [email protected]
1. Introduction Heusler alloys with the general composition X2YZ were discovered at the beginning of the last century and reported the observation of ferromagnetic order in Cu2MnAl at room temperature although none of its constituent elements (Cu, Mn, or Al) shows magnetism [1]. Heusler alloys have attracted attention due to their various interesting technological applications such as magnetic shape memory effect [2-3], magnetoresistance [4-5], exchange bias [6], magneto-strain [7-8] and magnetocaloric effects [9-10]. Moreover, these alloys are known to exhibit first-order phase transitions [11-15]. Crystal structure of Heusler alloys with stoichiometric X2YZ are constructed from four interpenetrating fcc sublattices and crystallized in the L21 structure. Atoms on X are located at (0,0,0) and ( , , ) points, atoms on Y at ( , , ) point and atoms on Z at ( , , ) point. In classic Heusler structure, the X elements belong to noble or transition metal elements and the Z elements belong to main-group elements [16-17]. In the called all-d metal Heusler alloys which consist only 3d transition metal atoms, Z elements are also transition metals. These alloys have been investigated experimentally and theoretically in the last years. Wei et al reported the realization of ferromagnetic shape-memory alloys in all d-metal Heusler alloys, Ni-Mn-Ti [18] and Mn-Ni-Co-Ti [19]. Zhaoning Ni et al investigated the magnetic properties of Zn-Y-Mn (Y = Fe, Co, Ni, Cu) [20] and of Mn2NiTi [21] using first-principles calculations. Özdoğan et al studied the Heusler alloys Fe2CrZ and Co2CrZ (Z = Si, Ti V) by employing ab-initio electronic structure calculations [22]. Barocaloric properties of Ni-Mn-Ti alloy are studied by Azhar et al [23]. Yilin Han et al dealed with a series of all d-metal Heusler alloys X2MnV (X=Pd, Ni, Pt, Ag, Au, Ir, Co) [24] and Zn2MMn (M = Ru, Rh, Pd, Os, Ir) [25].
Many-body Green’s function theory is known as the standard method in theoretical studies of the magnetic systems [26]. This theory has been applied to many Heisenberg magnetic systems [27-32]. In this formalism, one obtains a nonlinear differential equation in which the higher-order Green’s functions are coupled with the lower order ones. Each of the higher-order Green’s function is again written down in the form of a nonlinear equation, and so on. To obtain tractable solutions, decoupling procedures are generally required to terminate the hierarchy of Green’s functions generated by the equations of motion. There are many methods to decouple the higher-order Green’s functions. We apply the random phase approximation (RPA) decoupling for the exchange interaction terms [33-34] and the Anderson-Callen decoupling for the anisotropy terms [35]. The RPA provides a simple enough way for giving results which are in good agreement with other approaches and experiments in a wide range of temperatures and magnetic fields. In our previous paper, we presented detail of the calculations for the four-sublattices Heisenberg model in two-dimensional lattice [36]. In this paper, we will present results for the four interpenetrating fcc lattices in three-dimension and apply the formalism of Green`s function formalism to calculate the magnetization of Heusler type model, Ni2MnZ, as a function of temperature. In Heisenberg model, the electrons are assumed to be localized around an atom. Therefore, in our systems, we assume that the magnetic moments of Ni, Mn, and Z is localized on their atom. While the Z atoms give only a small contribution to the total magnetic moment, it is emphasized the importance of negative exchange interactions between Z atom and other atoms in existence of the first order phase transitions. This paper is organized as follows. First, in Section 2, we introduce the model Hamiltonian considered in this paper and present the formalism of the Green's function method. In Section 3, the numerical results we have obtained are discussed. Finally, a summary is presented in Sec. 4. 2. Green’s function formalism and model
Figure 1. The schematic representation of Heusler alloy Ni2MnZ.
The crystal structure of the Heusler alloy Ni2MnZ with the ordered structure L21 is shown in Fig. 1. The Heusler unit cell is comprised of four interpenetrating fcc sublattices, which has Ni(A)(0,0,0),
2
Ni(B)( , , ), Mn( , , ), and Z( , , ) sites. We discuss the Heusler-type model using Green’s function theory. We deal with the following Hamiltonian. ∑
= −∑ , ,
Ni(B) ∑
(
∙
∙
−∑,
−∑,
) −ℎ ∑
∙
∙
+∑
−∑,
−∑
,
+∑
∙
+∑
∙
−∑
−∑
,
,
∙
,
∙
− ∑
,
− Ni(A) ∑ (
∙S −
) −
(1)
where the subscripts i, j, k and m label lattice sites for sublattices Ni(A), Ni(B), Mn and Z respectively, and the corresponding spin operators are Si, Sj, Sk and Sm which take the spin values 1, 1, 2 and 1/2, respectively. Jij ≡ JNi(A)-Ni(B), Jik ≡ J Ni(A)-Mn, Jim ≡ J Ni(A)-Z, Jjk ≡ J Ni(B)-Mn, Jjm ≡ J Ni(B)-Z, Jkm ≡ J Mn-Z represent the couplings between the nearest-neighboring sites and ≡ " (#)$" (#) , ≡ " (%)$" (%) , ≡ &'$&' represent the couplings between the next-nearest neighboring sites. JNi(A)-Ni(B) , JNi(A)-Mn , JNi(B)-Mn, " (#)$" (#) , " (%)$" (%) > 0, that is, these correspond to ferromagnetic interactions, JMn-Mn, JNi(A)-Z, JNi(B)-Z and JMn-Z < 0, that is, these correspond to antiferromagnetic interactions. Moreover, DNi(A) and DNi(B) are the single-ion anisotropy parameters for Ni(A) and Ni(B) sublattices and h is the applied external field which we assume to be along the +z direction. We apply the Double-time Green's functions method to deal with Hamiltonian in the Eq. (1). In order to evaluate sublattice magnetizations the following Green`s functions are introduced as follows: ⟨⟨
,⟨⟨ (()) = + ⟨⟨ *⟨⟨
.
()); 0 123
.
()); 0 123
.
.
4
()); 0 123
4 4
()); 0 123
4
$ 5 ⟩⟩ $ 5 ⟩⟩ 9 . $ 8 5 ⟩⟩ $ 5 ⟩⟩7
(2)
We derive the equation of motion of the Green`s function method following a standard method, after employing the Tyablikov decoupling and Anderson and Callen’s decoupling of the higher-order Green’s functions which appeared in the equation of motion [33-35], we obtain C⟨: B⟨: :;< − => ⋅ ((@) = B B⟨: A⟨:
.
, 0 123
.
, 0 123
.
.
4
, 0 123
4 4
, 0 123
4
$ 5 >⟩ F $ 5 >⟩ E , $ E 5 >⟩ E $ 5 >⟩D
(3)
where I is the four-dimensional unit matrix and P is 4×4 matrices. The elements of matrix P are as follows: G
G G G G G
= Ni(A)H# + 6 " (#)$" (%) J% + 4 " (#)$&' JL + 4 " (#)$M JN +12 " (#)$" (#) J# − 4 " (#)$" (#) J# γ + ℎ , = 2 " (#)$" (%) J# γ , = 4 " (#)$&' J# γ + Q γ , = 4 " (#)$M J# γ + Q γ , = 2 " (#)$" (%) J% γ , = Ni(B)H% + 6 " (#)$" (%) J# + 4 " (%)$&' JL + 4 " (%)$M JN
3
+12 " (%)$" (%) J% − 4 " (%)$" (%) J% γ + ℎ, G = 4 " (%)$&' J% γ − Q γ , G = 4 " (%)$" (M) J% (γ − Q γ ), G = 4 " (#)$&' JR (γ + Q γ ), G = 4 " (%)$&' JL (γ + Q γ ), G = 4 " (#)$&' J# + 4 " (#)$&' J% + 6 " (#)$" (%) JN +12 &'$&' JL − 4 &'$&' JL γ + ℎ, G = 2 &'$M JL γ , G = 4 " (#)$M JN (γ + Q γ ), G = 4 " (%)$M JN (γ − Q γ ), G = 2 &'$M JN γ , G = 4 " (%)$M J# + 4 " (%)$M J% + 6 &'$M JL + ℎ,
(4)
where mA, mB, mC and mD stand for the spontaneous magnetizations of the sublattices Ni(A), Ni(B), Mn and Z, respectively, the number i denotes imaginary unit and
γ γ γ γ
= cos VW cos VX cos V , = sin VW sin VX sin V , = cos 2VW cos 2VX + cos 2V cos 2VW + cos 2V cos 2VX , = cos 2VW + cos 2VX + cos 2V ,
(5)
where VZ = [ , (\ = ], ^, _), kx, ky and kz are components of a 3-dimensional wave vector k and the distance between the nearest neighbor sites is ɑ/2. Green’s functions can be obtained as following forms: 1
` 5 (a) =
` 5 (a) =
` 5 (a) =
4
⟨:2bc ,d ef3 23g >⟩ h
4 ⟨:2rc ,d ef3 23g >⟩
` 5 (a) =
h
4
ek en eo ep + l$m + l$m + l$m il$m q,
j
j
n
j
o
p
sk sn so sp + l$m + l$m + l$m il$m q,
⟨:2tc ,d ef3 23g >⟩ h
j
k
j
j
k
j
n
j
o
p
(6)
uk un uo up + l$m + l$m + l$m il$m q,
j
4 c ,d ef3 2 g >⟩ ⟨:2v 3
h
k
j
n
j
o
j
p
wk + wn + wo + wp q, il$m l$m l$m l$m
j
k
j
n
j
o
j
p
where Ei (i = 1, 2, 3, 4) are the roots of the polynomial ; + x; + y; + z; + { = 0. The constants K i (i = 1, 2, 3, 4) are m o .• m n .R m .•
}~ = (mb $m €)(mb$m €)(mb $m€ ) , b
r
b
t
b
v
(7)
being η = A, B, C and D. The coefficients a, b, c, d, y′, z ′ and {′ are the complicated functions of Hamiltonian and external parameters and described in Ref. 36. The self-consistent sublattice magnetizations are evaluated by means of the spectral theorem and the Callen`s technique [37]: J~ =
(2€ $‚€ )( .‚€ )nf€ ck .(2€ . .‚€ )‚€ nf€ ck ( .‚€ )nf€ ck $‚€ nf€ ck
,
(8)
4
where
ƒ~ = " ∑@ id „…k $ + d „…n $ + d „…o $ + d „…p $ q, j€k
j€n
j€o
j€p
(9)
where β = 1/kBT (kB is Botzmann’s constant and T is the absolute temperature). Here, the sum on k is over the reciprocal lattice vectors in the first Brillouin-Zone. In order to calculate the magnetization, it is necessary to find self-consistent solution to Eqs. (8) – (9) which cannot be solved analytically, so a numerical approach is necessary. The total magnetization of the system is the sum of the individual sublattice magnetizations: M = mA + mB + mC + mD. 3. Results and discussions Total magnetization is obtained by solving self consistently Equations (8) – (9) for each value of temperature, depending of values of parameters given in Hamiltonian. We will abbreviate JNi(A)-Ni(B) = J. We choose the other exchange interaction parameters to be J Ni(A)Mn/|J| = 5, JNi(B)-Mn/|J| = 2, JNi(A)-Ni(A)/|J| = 2, JNi(B)-Ni(B)/|J| = 2. We assume that the next-nearest neighbor interactions between Mn atoms are negative, JMn-Mn/|J|= – 0.5. Moreover, the single-ion anisotropy parameters of Ni(A) and Ni(B) atoms are fixed DNi(A)/|J| = DNi(B)/|J| = 5. We have considered firstly the case where the all nearest neighbor interaction constants of Z atom, (JNi(A)-Z, JNi(B)-Z and JMn-Z) have zero values. Figure 2(a) represents the temperature dependences of the magnetization at different values of magnetic field for JNi(A)-Z = JNi(B)-Z = JMn-Z = 0. As seen from Figure 2(a), for the case of h/|J| = 0.5,1 and 2, the transition temperatures shift towards the high temperature region and magnetization curves show the typical behavior of the system without the firstorder phase transition. Figure 2(b) shows the field dependences of the magnetization for various temperature values for JNi(A)-Z = JNi(B)-Z = JMn-Z = 0. As seen from the figure, decreasing magnetizations with increasing magnetic field does not match to experimental results. So, at least one of the exchange interactions between Ni(A)-Z, Ni(B)-Z and Mn-Z must be nonzero.
Figure 2. a) The temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = JNi(B)-Z = JMn-Z = 0.
In Figure 3, for JNi(A)-Z = – 0.5 and JNi(B)-Z = JMn-Z = 0, we plot temperature dependence of the magnetization for various magnetic field values and the field dependences of the magnetization for
5
various temperature values. As seen from Figure 3(a), the magnetization curve without an external magnetic field decreases to zero continuously with increasing the temperature and goes to zero at critical temperature. The transition becomes the second-order from ferromagnet to paramagnet. If the magnetic field is applied, contrary to that obtained in Figure 2(a), the magnetization curves exhibit the first-order phase transitions from ferromagnet to ferromagnet where it appears discontinuity. In this case, the ground state spin configuration is (+ + + −), while it happens (+ + + +) after this transition temperature. These discontinuities that we have observed in all-d metal Heusler type model Ni2MnZ (Z = Ti, Sc, V) are compatible with the results investigated experimentally in a series of Ni2Mn1xCrxGa Heusler alloys cited in Ref. [13]. This clearly shows that the presence of antiferromagnetic interaction between Z and Ni(A) atoms is important in observing these phase transitions. Şaşıoğlu et al emphasize that the importance of the sp atom (Z) play an important role in establishing magnetic properties in Ni2MnZ (Z = Ga, In, Sn, Sb) [38]. Figure 3(b) shows the field dependences of the magnetization for various temperature values. In Figure 4, for JNi(A)-Z = 0, JNi(B)-Z = – 0.5 and JMn-Z = 0, we plot temperature dependence of the magnetization for various magnetic field values and the field dependences of the magnetization for various temperature values. The behavior is similar to those obtained in Figure 3. Magnetization has a discontinuity at the first-order phase transition temperature. This clearly shows that the presence of antiferromagnetic interaction between Z and Ni(B) atoms is important in observing first-order phase transitions. These transition temperature decreases with increasing magnetic field. Figure 4(b) shows the field dependences of the magnetization for various temperature values.
Figure 3. a) The temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = – 0.5, JNi(B)-Z = 0 and JMn-Z = 0.
6
Figure 4. a) The temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = 0, JNi(B)-Z = – 0.5 and JMn-Z = 0.
In Figure 5, for JNi(A)-Z = JNi(B)-Z = 0 and JMn-Z = – 0.5, we plot temperature dependence of the magnetization for various magnetic field values and the field dependences of the magnetization for various temperature values. Magnetization curve behaves similar to those obtained in Figures 3 and 4. But, as it can be seen from Figure 5(a), at low temperatures, it appears diverse anomalies. This clearly shows that the presence of antiferromagnetic interaction between Z and Mn atoms is important in observing first-order phase transitions. Figure 5(b) shows the field dependences of the magnetization for various temperature values.
Figure 5. a) Temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = 0, JNi(B)-Z = 0 and JMn-Z = – 0.5.
7
4. Conclusions We present the results of the double-time Green`s function method within the random phase approximation for the temperature and field dependence of magnetization of the Heusler alloys which only consist of d-metal, Ni2MnZ (Z = Ti, Sc, and V). We analyzed the effects of the nearest-neighbor exchange interactions of Z atom on magnetization curves. Our detailed numerical calculations show that the first-order phase transitions depend sensitively on the antiferromagnetic interactions between Z atom and other atoms presented in the model. These results are in good agreement with available experimental results.
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9
Highlights
• • •
We investigated the magnetic phase transitions of all-d metal Heusler type model, Ni2MnZ We used the double-time temperature-dependent Green`s function technique. We discussed that the importance of existence of antiferromagnetic interaction between Z atom and other atoms in observing first-order phase transitions.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0925-8388(19)34545-1
DOI:
https://doi.org/10.1016/j.jallcom.2019.153299
Reference:
JALCOM 153299
To appear in:
Journal of Alloys and Compounds
Received Date: 31 July 2019 Revised Date:
4 December 2019
Accepted Date: 5 December 2019
Please cite this article as: Gü. Mert, Magnetic phase transitions of all-d metal Heusler type model, Journal of Alloys and Compounds (2020), doi: https://doi.org/10.1016/j.jallcom.2019.153299. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Author Contributions Section This study was designed, coded (in Fortran Language), analyzed, written and commented by G.M.
MAGNETIC PHASE TRANSITIONS OF ALL-D METAL HEUSLER TYPE MODEL Gülistan Mert* Department of Physics, Selçuk University, 42075, Konya, Turkey
ABSTRACT The magnetic phase transitions of all-d metal Heusler type model, Ni2MnZ (Z = Ti, Sc and V), are investigated by using the double-time Green`s Function Method within the random phase approximation. We assume that the magnetic moments of Ni, Mn, and Z in Ni2MnZ which crystallized in L21 structure is localized on their atom and the next-nearest neighbor interactions between Mn atoms are negative. While the Z atoms give only a small contribution to the total magnetic moment, it is emphasized that the importance of existence of antiferromagnetic interaction between Z atom and other atoms in observing first-order phase transitions. Ni2MnZ shows a ferromagnetic state depending the values of the Hamiltonian parameters. If any one of these interactions have negative value and the system is subject to an external magnetic field, it is observed that the magnetization curves exhibit to the first-order phase transitions. PACS: Keywords: All-d-Metal Heusler Structures; Ferromagnetism; Heisenberg Model; Green`s Function; Random Phase Approximation *
Email address: [email protected]
1. Introduction Heusler alloys with the general composition X2YZ were discovered at the beginning of the last century and reported the observation of ferromagnetic order in Cu2MnAl at room temperature although none of its constituent elements (Cu, Mn, or Al) shows magnetism [1]. Heusler alloys have attracted attention due to their various interesting technological applications such as magnetic shape memory effect [2-3], magnetoresistance [4-5], exchange bias [6], magneto-strain [7-8] and magnetocaloric effects [9-10]. Moreover, these alloys are known to exhibit first-order phase transitions [11-15]. Crystal structure of Heusler alloys with stoichiometric X2YZ are constructed from four interpenetrating fcc sublattices and crystallized in the L21 structure. Atoms on X are located at (0,0,0) and ( , , ) points, atoms on Y at ( , , ) point and atoms on Z at ( , , ) point. In classic Heusler structure, the X elements belong to noble or transition metal elements and the Z elements belong to main-group elements [16-17]. In the called all-d metal Heusler alloys which consist only 3d transition metal atoms, Z elements are also transition metals. These alloys have been investigated experimentally and theoretically in the last years. Wei et al reported the realization of ferromagnetic shape-memory alloys in all d-metal Heusler alloys, Ni-Mn-Ti [18] and Mn-Ni-Co-Ti [19]. Zhaoning Ni et al investigated the magnetic properties of Zn-Y-Mn (Y = Fe, Co, Ni, Cu) [20] and of Mn2NiTi [21] using first-principles calculations. Özdoğan et al studied the Heusler alloys Fe2CrZ and Co2CrZ (Z = Si, Ti V) by employing ab-initio electronic structure calculations [22]. Barocaloric properties of Ni-Mn-Ti alloy are studied by Azhar et al [23]. Yilin Han et al dealed with a series of all d-metal Heusler alloys X2MnV (X=Pd, Ni, Pt, Ag, Au, Ir, Co) [24] and Zn2MMn (M = Ru, Rh, Pd, Os, Ir) [25].
Many-body Green’s function theory is known as the standard method in theoretical studies of the magnetic systems [26]. This theory has been applied to many Heisenberg magnetic systems [27-32]. In this formalism, one obtains a nonlinear differential equation in which the higher-order Green’s functions are coupled with the lower order ones. Each of the higher-order Green’s function is again written down in the form of a nonlinear equation, and so on. To obtain tractable solutions, decoupling procedures are generally required to terminate the hierarchy of Green’s functions generated by the equations of motion. There are many methods to decouple the higher-order Green’s functions. We apply the random phase approximation (RPA) decoupling for the exchange interaction terms [33-34] and the Anderson-Callen decoupling for the anisotropy terms [35]. The RPA provides a simple enough way for giving results which are in good agreement with other approaches and experiments in a wide range of temperatures and magnetic fields. In our previous paper, we presented detail of the calculations for the four-sublattices Heisenberg model in two-dimensional lattice [36]. In this paper, we will present results for the four interpenetrating fcc lattices in three-dimension and apply the formalism of Green`s function formalism to calculate the magnetization of Heusler type model, Ni2MnZ, as a function of temperature. In Heisenberg model, the electrons are assumed to be localized around an atom. Therefore, in our systems, we assume that the magnetic moments of Ni, Mn, and Z is localized on their atom. While the Z atoms give only a small contribution to the total magnetic moment, it is emphasized the importance of negative exchange interactions between Z atom and other atoms in existence of the first order phase transitions. This paper is organized as follows. First, in Section 2, we introduce the model Hamiltonian considered in this paper and present the formalism of the Green's function method. In Section 3, the numerical results we have obtained are discussed. Finally, a summary is presented in Sec. 4. 2. Green’s function formalism and model
Figure 1. The schematic representation of Heusler alloy Ni2MnZ.
The crystal structure of the Heusler alloy Ni2MnZ with the ordered structure L21 is shown in Fig. 1. The Heusler unit cell is comprised of four interpenetrating fcc sublattices, which has Ni(A)(0,0,0),
2
Ni(B)( , , ), Mn( , , ), and Z( , , ) sites. We discuss the Heusler-type model using Green’s function theory. We deal with the following Hamiltonian. ∑
= −∑ , ,
Ni(B) ∑
(
∙
∙
−∑,
−∑,
) −ℎ ∑
∙
∙
+∑
−∑,
−∑
,
+∑
∙
+∑
∙
−∑
−∑
,
,
∙
,
∙
− ∑
,
− Ni(A) ∑ (
∙S −
) −
(1)
where the subscripts i, j, k and m label lattice sites for sublattices Ni(A), Ni(B), Mn and Z respectively, and the corresponding spin operators are Si, Sj, Sk and Sm which take the spin values 1, 1, 2 and 1/2, respectively. Jij ≡ JNi(A)-Ni(B), Jik ≡ J Ni(A)-Mn, Jim ≡ J Ni(A)-Z, Jjk ≡ J Ni(B)-Mn, Jjm ≡ J Ni(B)-Z, Jkm ≡ J Mn-Z represent the couplings between the nearest-neighboring sites and ≡ " (#)$" (#) , ≡ " (%)$" (%) , ≡ &'$&' represent the couplings between the next-nearest neighboring sites. JNi(A)-Ni(B) , JNi(A)-Mn , JNi(B)-Mn, " (#)$" (#) , " (%)$" (%) > 0, that is, these correspond to ferromagnetic interactions, JMn-Mn, JNi(A)-Z, JNi(B)-Z and JMn-Z < 0, that is, these correspond to antiferromagnetic interactions. Moreover, DNi(A) and DNi(B) are the single-ion anisotropy parameters for Ni(A) and Ni(B) sublattices and h is the applied external field which we assume to be along the +z direction. We apply the Double-time Green's functions method to deal with Hamiltonian in the Eq. (1). In order to evaluate sublattice magnetizations the following Green`s functions are introduced as follows: ⟨⟨
,⟨⟨ (()) = + ⟨⟨ *⟨⟨
.
()); 0 123
.
()); 0 123
.
.
4
()); 0 123
4 4
()); 0 123
4
$ 5 ⟩⟩ $ 5 ⟩⟩ 9 . $ 8 5 ⟩⟩ $ 5 ⟩⟩7
(2)
We derive the equation of motion of the Green`s function method following a standard method, after employing the Tyablikov decoupling and Anderson and Callen’s decoupling of the higher-order Green’s functions which appeared in the equation of motion [33-35], we obtain C⟨: B⟨: :;< − => ⋅ ((@) = B B⟨: A⟨:
.
, 0 123
.
, 0 123
.
.
4
, 0 123
4 4
, 0 123
4
$ 5 >⟩ F $ 5 >⟩ E , $ E 5 >⟩ E $ 5 >⟩D
(3)
where I is the four-dimensional unit matrix and P is 4×4 matrices. The elements of matrix P are as follows: G
G G G G G
= Ni(A)H# + 6 " (#)$" (%) J% + 4 " (#)$&' JL + 4 " (#)$M JN +12 " (#)$" (#) J# − 4 " (#)$" (#) J# γ + ℎ , = 2 " (#)$" (%) J# γ , = 4 " (#)$&' J# γ + Q γ , = 4 " (#)$M J# γ + Q γ , = 2 " (#)$" (%) J% γ , = Ni(B)H% + 6 " (#)$" (%) J# + 4 " (%)$&' JL + 4 " (%)$M JN
3
+12 " (%)$" (%) J% − 4 " (%)$" (%) J% γ + ℎ, G = 4 " (%)$&' J% γ − Q γ , G = 4 " (%)$" (M) J% (γ − Q γ ), G = 4 " (#)$&' JR (γ + Q γ ), G = 4 " (%)$&' JL (γ + Q γ ), G = 4 " (#)$&' J# + 4 " (#)$&' J% + 6 " (#)$" (%) JN +12 &'$&' JL − 4 &'$&' JL γ + ℎ, G = 2 &'$M JL γ , G = 4 " (#)$M JN (γ + Q γ ), G = 4 " (%)$M JN (γ − Q γ ), G = 2 &'$M JN γ , G = 4 " (%)$M J# + 4 " (%)$M J% + 6 &'$M JL + ℎ,
(4)
where mA, mB, mC and mD stand for the spontaneous magnetizations of the sublattices Ni(A), Ni(B), Mn and Z, respectively, the number i denotes imaginary unit and
γ γ γ γ
= cos VW cos VX cos V , = sin VW sin VX sin V , = cos 2VW cos 2VX + cos 2V cos 2VW + cos 2V cos 2VX , = cos 2VW + cos 2VX + cos 2V ,
(5)
where VZ = [ , (\ = ], ^, _), kx, ky and kz are components of a 3-dimensional wave vector k and the distance between the nearest neighbor sites is ɑ/2. Green’s functions can be obtained as following forms: 1
` 5 (a) =
` 5 (a) =
` 5 (a) =
4
⟨:2bc ,d ef3 23g >⟩ h
4 ⟨:2rc ,d ef3 23g >⟩
` 5 (a) =
h
4
ek en eo ep + l$m + l$m + l$m il$m q,
j
j
n
j
o
p
sk sn so sp + l$m + l$m + l$m il$m q,
⟨:2tc ,d ef3 23g >⟩ h
j
k
j
j
k
j
n
j
o
p
(6)
uk un uo up + l$m + l$m + l$m il$m q,
j
4 c ,d ef3 2 g >⟩ ⟨:2v 3
h
k
j
n
j
o
j
p
wk + wn + wo + wp q, il$m l$m l$m l$m
j
k
j
n
j
o
j
p
where Ei (i = 1, 2, 3, 4) are the roots of the polynomial ; + x; + y; + z; + { = 0. The constants K i (i = 1, 2, 3, 4) are m o .• m n .R m .•
}~ = (mb $m €)(mb$m €)(mb $m€ ) , b
r
b
t
b
v
(7)
being η = A, B, C and D. The coefficients a, b, c, d, y′, z ′ and {′ are the complicated functions of Hamiltonian and external parameters and described in Ref. 36. The self-consistent sublattice magnetizations are evaluated by means of the spectral theorem and the Callen`s technique [37]: J~ =
(2€ $‚€ )( .‚€ )nf€ ck .(2€ . .‚€ )‚€ nf€ ck ( .‚€ )nf€ ck $‚€ nf€ ck
,
(8)
4
where
ƒ~ = " ∑@ id „…k $ + d „…n $ + d „…o $ + d „…p $ q, j€k
j€n
j€o
j€p
(9)
where β = 1/kBT (kB is Botzmann’s constant and T is the absolute temperature). Here, the sum on k is over the reciprocal lattice vectors in the first Brillouin-Zone. In order to calculate the magnetization, it is necessary to find self-consistent solution to Eqs. (8) – (9) which cannot be solved analytically, so a numerical approach is necessary. The total magnetization of the system is the sum of the individual sublattice magnetizations: M = mA + mB + mC + mD. 3. Results and discussions Total magnetization is obtained by solving self consistently Equations (8) – (9) for each value of temperature, depending of values of parameters given in Hamiltonian. We will abbreviate JNi(A)-Ni(B) = J. We choose the other exchange interaction parameters to be J Ni(A)Mn/|J| = 5, JNi(B)-Mn/|J| = 2, JNi(A)-Ni(A)/|J| = 2, JNi(B)-Ni(B)/|J| = 2. We assume that the next-nearest neighbor interactions between Mn atoms are negative, JMn-Mn/|J|= – 0.5. Moreover, the single-ion anisotropy parameters of Ni(A) and Ni(B) atoms are fixed DNi(A)/|J| = DNi(B)/|J| = 5. We have considered firstly the case where the all nearest neighbor interaction constants of Z atom, (JNi(A)-Z, JNi(B)-Z and JMn-Z) have zero values. Figure 2(a) represents the temperature dependences of the magnetization at different values of magnetic field for JNi(A)-Z = JNi(B)-Z = JMn-Z = 0. As seen from Figure 2(a), for the case of h/|J| = 0.5,1 and 2, the transition temperatures shift towards the high temperature region and magnetization curves show the typical behavior of the system without the firstorder phase transition. Figure 2(b) shows the field dependences of the magnetization for various temperature values for JNi(A)-Z = JNi(B)-Z = JMn-Z = 0. As seen from the figure, decreasing magnetizations with increasing magnetic field does not match to experimental results. So, at least one of the exchange interactions between Ni(A)-Z, Ni(B)-Z and Mn-Z must be nonzero.
Figure 2. a) The temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = JNi(B)-Z = JMn-Z = 0.
In Figure 3, for JNi(A)-Z = – 0.5 and JNi(B)-Z = JMn-Z = 0, we plot temperature dependence of the magnetization for various magnetic field values and the field dependences of the magnetization for
5
various temperature values. As seen from Figure 3(a), the magnetization curve without an external magnetic field decreases to zero continuously with increasing the temperature and goes to zero at critical temperature. The transition becomes the second-order from ferromagnet to paramagnet. If the magnetic field is applied, contrary to that obtained in Figure 2(a), the magnetization curves exhibit the first-order phase transitions from ferromagnet to ferromagnet where it appears discontinuity. In this case, the ground state spin configuration is (+ + + −), while it happens (+ + + +) after this transition temperature. These discontinuities that we have observed in all-d metal Heusler type model Ni2MnZ (Z = Ti, Sc, V) are compatible with the results investigated experimentally in a series of Ni2Mn1xCrxGa Heusler alloys cited in Ref. [13]. This clearly shows that the presence of antiferromagnetic interaction between Z and Ni(A) atoms is important in observing these phase transitions. Şaşıoğlu et al emphasize that the importance of the sp atom (Z) play an important role in establishing magnetic properties in Ni2MnZ (Z = Ga, In, Sn, Sb) [38]. Figure 3(b) shows the field dependences of the magnetization for various temperature values. In Figure 4, for JNi(A)-Z = 0, JNi(B)-Z = – 0.5 and JMn-Z = 0, we plot temperature dependence of the magnetization for various magnetic field values and the field dependences of the magnetization for various temperature values. The behavior is similar to those obtained in Figure 3. Magnetization has a discontinuity at the first-order phase transition temperature. This clearly shows that the presence of antiferromagnetic interaction between Z and Ni(B) atoms is important in observing first-order phase transitions. These transition temperature decreases with increasing magnetic field. Figure 4(b) shows the field dependences of the magnetization for various temperature values.
Figure 3. a) The temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = – 0.5, JNi(B)-Z = 0 and JMn-Z = 0.
6
Figure 4. a) The temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = 0, JNi(B)-Z = – 0.5 and JMn-Z = 0.
In Figure 5, for JNi(A)-Z = JNi(B)-Z = 0 and JMn-Z = – 0.5, we plot temperature dependence of the magnetization for various magnetic field values and the field dependences of the magnetization for various temperature values. Magnetization curve behaves similar to those obtained in Figures 3 and 4. But, as it can be seen from Figure 5(a), at low temperatures, it appears diverse anomalies. This clearly shows that the presence of antiferromagnetic interaction between Z and Mn atoms is important in observing first-order phase transitions. Figure 5(b) shows the field dependences of the magnetization for various temperature values.
Figure 5. a) Temperature dependence of the magnetization for various magnetic field values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z = 0, JNi(B)-Z = 0 and JMn-Z = – 0.5.
7
4. Conclusions We present the results of the double-time Green`s function method within the random phase approximation for the temperature and field dependence of magnetization of the Heusler alloys which only consist of d-metal, Ni2MnZ (Z = Ti, Sc, and V). We analyzed the effects of the nearest-neighbor exchange interactions of Z atom on magnetization curves. Our detailed numerical calculations show that the first-order phase transitions depend sensitively on the antiferromagnetic interactions between Z atom and other atoms presented in the model. These results are in good agreement with available experimental results.
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Highlights
• • •
We investigated the magnetic phase transitions of all-d metal Heusler type model, Ni2MnZ We used the double-time temperature-dependent Green`s function technique. We discussed that the importance of existence of antiferromagnetic interaction between Z atom and other atoms in observing first-order phase transitions.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: