Magnetic phase transitions of all-d metal Heusler type model

Magnetic phase transitions of all-d metal Heusler type model

Gülistan Mert

Department of Physics, Selçuk University, Konya, 42075, Turkey

a r t i c l e i n f o

Article history: Received 31 July 2019 Received in revised form 4 December 2019 Accepted 5 December 2019 Available online 9 December 2019 Keywords:

All-d-metal Heusler structures Ferromagnetism

Heisenberg model Green’s function

Random phase approximation

a b s t r a c t

The magnetic phase transitions of all-d metal Heusler type model, Ni2MnZ (Z ¼ Ti, Sc and V), are investigated by using the double-time Green’s Function Method within the random phase approxima-tion. 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 observingfirst-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 magneticfield, it is observed that the magnetization curves exhibit to the first-order phase transitions.

© 2019 Elsevier B.V. All rights reserved.

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

tempera-ture 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], ex-change bias [6], magneto-strain [7,8] and magnetocaloric effects [9,10]. Moreover, these alloys are known to exhibitfirst-order phase transitions [11e15].

Crystal structure of Heusler alloys with stoichiometric X2YZ are

constructed from four interpenetrating fcc sublattices and crystal-lized in the L21structure. Atoms on X are located at (0,0,0) and (12;12; 1

2) points, atoms on Y at (14;14;14) point and atoms on Z at (34;34;34) 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 experimen-tally 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 MneNieCoeTi [19]. Zhaoning Ni et al. investigated the magnetic properties of ZneYeMn (Y ¼ Fe, Co,

Ni, Cu) [20] and of Mn2NiTi [21] usingfirst-principles calculations.

€Ozdogan et al. studied the Heusler alloys Fe2CrZ and Co2CrZ (Z¼ Si,

Ti V) by employing ab-initio electronic structure calculations [22]. Barocaloric properties of NieMneTi alloy are studied by Azhar et al. [23]. Yilin Han et al. dealed with a series of all d-metal Heusler alloys X2MnV (X¼ Pd, Ni, Pt, Ag, Au, Ir, Co) [24] and Zn2MMn

(M¼ 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 [27e32]. 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 func-tion is again written down in the form of a nonlinear equafunc-tion, 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 interac-tion 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 E-mail address:gmert@selcuk.edu.tr.

Contents lists available atScienceDirect

Journal of Alloys and Compounds

j o u rn a l h o m e p a g e :h t t p : / / w w w . e l s e v i e r . c o m / l o c a t e / j a l c o m

https://doi.org/10.1016/j.jallcom.2019.153299

interpenetrating fcc lattices in three-dimension and apply the formalism of Green’s function formalism to calculate the magne-tization of Heusler type model, Ni2MnZ, as a function of

tempera-ture. 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 exis-tence of thefirst 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 Section3, the numerical results we have ob-tained are discussed. Finally, a summary is presented in Section4.

2. Green_{’s function formalism and model}

The crystal structure of the Heusler alloy Ni2MnZ with the

or-dered 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), Ni(B) (1

2;12;12), Mn(14;14;14), and Z(34;34;34) sites. We discuss the

Heusler-type model using Green’s function theory. We deal with the following Hamiltonian.

H ¼ X i;j JijSi, Sj X i;k J_ikS_i, S_kX i;m JimSi, Sm X j;k J_jkS_j, S_k X j;m JjmSj, Sm X k;m JkmSk, Sm X i;i’ Jii’Si, Si’ X j;j’ Jij’Sj, Sj’ X k;k’ Jkk’Sk, Sk’ DNiðAÞ X i  Sz_i2 DNiðBÞ X j  Sz_j2  hX i Sz_iþX j Sz_jþX k Sz_kþX m Sz_m; (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, Skand Smwhich take the spin values 1, 1, 2 and

1/2, respectively. Jij≡ JNi(A)-Ni(B), Jik≡ JNi(A)-Mn, Jim≡ JNi(A)-Z, Jjk≡ J Ni(B)eMn, Jjm≡ JNi(B)-Z, Jkm≡ JMn-Zrepresent the couplings between

the nearest-neighboring sites and Jii’≡J_{NiðAÞNiðAÞ}; J_jj’≡J_{NiðBÞNiðBÞ}; J_kk’≡ JMnMn represent the couplings between the next-nearest

neigh-boring sites. JNi(A)-Ni(B), JNi(A)-Mn,JNi(B)eMn, JNiðAÞNiðAÞ; JNiðBÞNiðBÞ> 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 theþz direction.

We apply the Double-time Green’s functions method to deal with Hamiltonian in Eq.(1). In order to evaluate sublattice mag-netizations the following Green’s functions are introduced as follows: GðtÞ ¼ 0 B B B B B B B @ CSþ_i ðtÞ; eaSz l_S lD CSþ_j ðtÞ; eaSz lS lD CSþ_kðtÞ; eaSz l_S lD CSþ_mðtÞ; eaSz l_S l D 1 C C C C C C C A : (2)

We derive the equation of motion of the Green’s function method following a standard method, after employing the Tya-blikov decoupling and Anderson and Callen’s decoupling of the higher-order Green’s functions which appeared in the equation of motion [33e35], we obtain

½

u

I P , GðkÞ ¼ 2 6 6 6 6 6 6 6 4 ChSþ_i; eaSz lS l i D ChSþ_j; eaSz l_S l i D ChSþ_k; eaSzl_S l i D ChSþ_m; eaSz lS l i D 3 7 7 7 7 7 7 7 5 ; (3)

where I is the four-dimensional unit matrix and P is 4 4 matrices. The elements of matrix P are as follows:

þ12JNiðAÞNiðAÞmA 4JNiðAÞNiðAÞmA

g

3þ h;

P12¼ 2JNiðAÞNiðBÞmA

g

4

P13¼ 4JNiðAÞMnmAð

g

1þ i

g

2Þ;

P14¼ 4JNiðAÞZmAð

g

1þ i

g

2Þ;

P21¼ 2JNiðAÞNiðBÞmB

g

4;

P22¼ DNiðBÞ

t

Bþ 6JNiðAÞNiðBÞmAþ 4JNiðBÞMnmCþ 4JNiðBÞZmD

þ12JNiðBÞNiðBÞmB 4JNiðBÞNiðBÞmB

g

3þ h;

P23¼ 4JNiðBÞMnmBð

g

1 i

g

2Þ;

P24¼ 4JNiðBÞNiðZÞmBð

g

1 i

g

2Þ;

P31¼ 4JNiðAÞMnmcð

g

1þ i

g

2Þ;

P32¼ 4JNiðBÞMnmCð

g

1þ i

g

2Þ;

P33¼ 4JNiðAÞMnmAþ 4JNiðAÞMnmBþ 6JNiðAÞNiðBÞmD

þ12JMnMnmC 4JMnMnmC

g

3þ h;

P34¼ 2JMnZmC

g

4 Fig. 1. The schematic representation of Heusler alloy Ni2MnZ.

P41¼ 4JNiðAÞZmDð

g

1þ i

g

2Þ

P42¼ 4JNiðBÞZmDð

g

1 i

g

2Þ;

P43¼ 2JMnZmD

g

4;

P44¼ 4JNiðBÞZmAþ 4JNiðBÞZmBþ 6JMnZmCþ h;

P11¼ DNiðAÞ

t

Aþ 6JNiðAÞNiðBÞmBþ 4JNiðAÞMnmCþ 4JNiðAÞZmD

þ 12JNiðAÞNiðAÞmA 4JNiðAÞNiðAÞmA

g

3þ h; P12 ¼ 2JNiðAÞNiðBÞmA

g

4P13¼ 4JNiðAÞMnmA 

g

₁þ i

g

₂P14 ¼ 4JNiðAÞZmA 

g

₁þ i

g

₂P21¼ 2JNiðAÞNiðBÞmB

g

4P22

¼ DNiðBÞ

t

Bþ 6JNiðAÞNiðBÞmAþ 4JNiðBÞMnmCþ 4JNiðBÞZmD

þ 12J_{NiðBÞNiðBÞ}mB 4JNiðBÞNiðBÞmB

g

3þ h; P23 ¼ 4J_NiðBÞMnmB 

g

₁ i

g

₂P24¼ 4JNiðBÞZmB 

g

₁ i

g

₂P31 ¼ 4J_NiðAÞMnmC 

g

₁þ i

g

₂P32¼ 4JNiðBÞMnmC 

g

₁þ i

g

₂P33

¼ 4J_NiðAÞMnm_Aþ 4J_NiðAÞMnmBþ 6JNiðAÞNiðBÞmD

þ 12JMnMnmC 4JMnMnmC

g

3þ h; P34 ¼ 2JMnZmC

g

4P41¼ 4JNiðAÞZmD 

g

₁þ i

g

₂P42 ¼ 4JNiðBÞZmD 

g

₁ i

g

₂P43¼ 2JMnZmD

g

4P44 ¼ 4JNiðBÞZmAþ 4JNiðBÞZmBþ 6JMnZmCþ h; (4)

where mA, mB, mCand mDstand for the spontaneous

magnetiza-tions of the sublattices Ni(A), Ni(B), Mn and Z, respectively, the number i denotes imaginary unit and.

g

4¼ sin

a

xsin

a

ysin

a

z;

g

₄¼ cos 2

a

xcos 2

a

yþ cos 2

a

zcos 2

a

xþ cos 2

a

zcos 2

a

y;

g

4¼ cos 2

a

xþ cos 2

a

yþ cos 2

a

z;

g

1¼ cos

a

xcos

a

ycos

a

z;

g

2¼ sin

a

xsin

a

ysin

a

z;

g

3

¼ cos 2

a

xcos 2

a

yþ cos 2

a

zcos 2

a

xþ cos 2

a

zcos 2

a

y;

g

4

¼ cos 2

a

xþ cos 2

a

yþ cos 2

a

z;

(5)

where

a

n ¼akn

2; ð

n

¼ x; y; zÞ, kx, kyand kzare components of a

3-dimensional wave vector k and the distance between the nearest neighbor sites isɑ/2.

Green’s functions can be obtained as following forms:

GjlðkÞ ¼ ChSþ_j; eaSzlS l i D 2

p

 K_b1

u

 E1þ K_b2

u

 E2þ K_b3

u

 E3þ K_b4

u

 E4  ; G_klðkÞ ¼C h Sþ_k; eaSz lS l i D 2

p

 Kc1

u

 E1þ Kc2

u

 E2þ Kc3

u

 E3þ Kc4

u

 E4  ; GmlðkÞ ¼ ChSþ_m; eaSz lS l i D 2

p

 K_d1

u

 E1þ K_d2

u

 E2þ K_d3

u

 E3þ K_d4

u

 E4  ; G_ilðkÞ ¼C h Sþ_i; eaSz lS l i D 2

p

 K_A1

u

 E1þ K_A2

u

 E2þ K_A3

u

 E3 þ KA4

u

 E4  ; G_jlðkÞ ¼C h Sþ_j; eaSz lS l i D 2

p

 KB1

u

 E1þ KB2

u

 E2þ KB3

u

 E3 þ KB4

u

 E4  ; GklðkÞ ¼C h Sþ_k; eaSz lS l i D 2

p

 KC1

u

 E1þ KC2

u

 E2þ KC3

u

 E3 þ KC4

u

 E4  ; G_mlðkÞ ¼C h Sþ_m; eaSz lS l i D 2

p

 KD1

u

 E1þ KD2

u

 E2þ KD3

u

 E3þ KD4

u

 E4  ; (6)

where Ei(i ¼ 1, 2, 3, 4) are the roots of the polynomial

u

4þ a

u

3þ

b

u

2_{þ c}

_u

_{þ d ¼ 0. The constants K} i(i ¼ 1, 2, 3, 4) are Khi¼ Ei 3_{þ bh’E} i2þ ch’Eiþ dh’  Ei Ej  ðEi EkÞðEi EmÞ (7) being

h

¼ A, B, C and D. The coefficients a, b, c, d, b’_hc’_{hand d}’_hare

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]:

mh¼ðSh

F

hÞð1 þ

F

hÞ 2Shþ1_{þ ðS} hþ 1 þ

F

hÞ

F

h2Shþ1 ð1 þ

F

hÞ2Shþ1

F

h2Shþ1 ; (8) where

F

h¼4 N X k  Kh1 ebE1 1þ Kh2 ebE2 1þ Kh3 ebE3 1þ Kh4 ebE4 1  ; (9)

where

b

¼ 1/kBT (kBis 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 tofind self-consistent solution to Eqs.

(8) e (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 ¼ mAþ mBþ mCþ mD.

3. Results and discussions

Total magnetization is obtained by solving self consistently Eqs.

(8) e (9)for each value of temperature, depending of values of parameters given in Hamiltonian.

We will abbreviate JNi(A)-Ni(B)¼ J. We choose the other exchange

interaction parameters to be JNi(A)-Mn/|J| ¼ 5, JNi(B)eMn/|J| ¼ 2, J Ni(A)-Ni(A)/|J| ¼ 2, JNi(B)eNi(B)/|J| ¼ 2. We assume that the next-nearest

neighbor interactions between Mn atoms are negative, JMn-Mn/|

J| ¼ e 0.5. Moreover, the single-ion anisotropy parameters of Ni(A) and Ni(B) atoms arefixed DNi(A)/|J| ¼ DNi(B)/|J| ¼ 5.

We have considered firstly the case where the all nearest neighbor interaction constants of Z atom, (JNi(A)-Z, JNi(B)-Zand JMn-Z)

of the magnetization at different values of magneticfield for J Ni(A)-Z¼ JNi(B)-Z¼ JMn-Z¼ 0. As seen fromFig. 2(a), for the case of h/|

J| ¼ 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 first-order phase transition.

Fig. 2(b) shows the field dependences of the magnetization for various temperature values for JNi(A)-Z¼ JNi(B)-Z¼ JMn-Z¼ 0. As seen

from the figure, decreasing magnetizations with increasing Fig. 2. a) The temperature dependence of the magnetization for various magneticfield values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z¼ JNi(B)-Z¼ JMn-Z¼ 0.

Fig. 3. a) The temperature dependence of the magnetization for various magneticfield values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z¼ e 0.5, JNi(B)-Z¼ 0 and JMn-Z¼ 0.

Fig. 4. a) The temperature dependence of the magnetization for various magneticfield values and b) The field dependences of the magnetization at various temperature values for JNi(A)-Z¼ 0, JNi(B)-Z¼ e 0.5 and JMn-Z¼ 0.

magneticfield 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.

InFig. 3, for JNi(A)-Z ¼ e 0.5 and JNi(B)-Z¼ JMn-Z¼ 0, we plot

temperature dependence of the magnetization for various mag-neticfield values and the field dependences of the magnetization for various temperature values. As seen from Fig. 3(a), the magnetization curve without an external magneticfield 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 magneticfield is applied, contrary to that obtained inFig. 2(a), the magnetization curves exhibit thefirst-order phase transitions from ferromagnet to ferromagnet where it appears discontinuity. In this case, the ground state spin configuration is (þþ þ ), while it happens (þþ þ þ) after this transition temperature. These discontinuities that we have observed in all-d metal Heusler type model Ni2MnZ

(Z¼ Ti, Sc, V) are compatible with the results investigated experi-mentally in a series of Ni2Mn1-xCrxGa 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. S¸as¸ıoglu et al. emphasize that the impor-tance of the sp atom (Z) play an important role in establishing magnetic properties in Ni2MnZ (Z¼ Ga, In, Sn, Sb) [38].Fig. 3(b)

shows the field dependences of the magnetization for various temperature values.

InFig. 4, for JNi(A)-Z¼ 0, JNi(B)-Z¼ e 0.5 and JMn-Z¼ 0, we plot

temperature dependence of the magnetization for various mag-neticfield values and the field dependences of the magnetization for various temperature values. The behavior is similar to those obtained inFig. 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 observingfirst-order phase transitions. These transition temperature decreases with increasing magnetic_field.

Fig. 4(b) shows the field dependences of the magnetization for various temperature values.

InFig. 5, for JNi(A)-Z ¼ JNi(B)-Z¼ 0 and JMn-Z ¼ e 0.5, we plot

temperature dependence of the magnetization for various mag-neticfield values and the field dependences of the magnetization for various temperature values. Magnetization curve behaves similar to those obtained inFigs. 3 and 4. But, as it can be seen from

Fig. 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. Fig. 5(b) shows the field dependences of the magnetization for various temperature values.

4. Conclusions

We present the results of the double-time Green’s function method within the random phase approximation for the temper-ature andfield dependence of magnetization of the Heusler alloys which only consist of d-metal, Ni2MnZ (Z ¼ Ti, Sc, and V). We

analyzed the effects of the nearest-neighbor exchange interactions of Z atom on magnetization curves. Our detailed numerical calcu-lations show that thefirst-order phase transitions depend sensi-tively on the antiferromagnetic interactions between Z atom and other atoms presented in the model. These results are in good agreement with available experimental results.

Author contributions section

This study was designed, coded (in Fortran Language), analyzed, written and commented by G.M.

Declaration of competing interest

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.

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