Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni Mn Ga X

Materials Transactions, Vol. 44, No. 1 (2003) pp. 204 to 210 #2003 The Japan Institute of Metals EXPRESS REGULAR ARTICLE Valence Electron Concentrati...
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Materials Transactions, Vol. 44, No. 1 (2003) pp. 204 to 210 #2003 The Japan Institute of Metals EXPRESS REGULAR ARTICLE

Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X Kenichi Yamaguchi1; *, Shoji Ishida1 and Setsuro Asano2 1 2

Department of Physics, Faculty of Science, Kagoshima University, Kagoshima, 890-0065 Japan The Graduate School of=College of Arts and Sciences, The University of Tokyo, Tokyo, 153-0041 Japan

In the Ni2 MnGa based alloys with additions of transition element Ni–Mn–Ga–X, the martensitic transformation temperature TM was observed as a function of the valence electron concentration per atom e=a. The TM ðe=aÞ strongly depends on e=a and increases with increasing e=a. In this paper, to examine the effect of X atom on the phase transformation in Ni–Mn–Ga–X alloys, the electronic structures for six systems were calculated for four phases, that is, the paramagnetic cubic, the paramagnetic monoclinic, the ferromagnetic cubic and the ferromagnetic monoclinic phases. Moreover, the total energy differences Eðe=aÞ between two phases among four phases were calculated as a function of e=a. The variations of TM ðe=aÞ were predicted by the difference Eðe=aÞ between the cubic and monoclinic structures in a ferromagnetic state. It was found that their correspondence is good for some systems and that the features of TM ðe=aÞ reflect the changes of the density of states of X atoms. (Received October 9, 2002; Accepted December 13, 2002) Keywords: shape memory, martensitic transformation temperature, valence electron concentration, electronic structure, total energy, nickel, manganese, gallium, Curie temperature

1.

Introduction

Many researchers have reported on the crystal structures and the phase transformations of the Ni–Mn–Ga alloys. The tetragonal structure was observed in the martensitic phase around valence electron concentration e=a ¼ 7:50 (stoichiometric Ni2 MnGa). On the other hand, the orthorhombic and monoclinic structures were observed.1–4) For example, the orthorhombic structure was observed at the e=a ¼ 7:635 and the monoclinic structures at e=a ¼ 7:64, 7.67, 7.72 and 7.78.2,3) It was also reported that the tetragonal phase in the lower e=a can be suppressed by Ni excess.5) Furthermore, it is also predicted theoretically that the tetragonal and orthorhombic structures may be metastable and the monoclinic structure is the most stable state among these structures for Ni2:17 Ni0:83 Ga and Ni2 (Pd0:17 Ni0:83 )Ga.6) Thus, it is possible for the monoclinic structure to appear in martensitic phase in wide e=a range. The martensitic transformation temperature TM and the Curie temperature Tc for Ni2 MnGa (e=a ¼ 7:50) are 202 and 376 K, respectively.7) The various values of TM were observed in the wide range from 175 to 626 K in the range e=a ¼ 7:45{8:10.8–10) For Ni–Mn–Ga alloys, Tc decreases and TM increases with increasing e=a.5,9) They merge in the range e=a ¼ 7:635{7:70. It was also shown that TM is lower than Tc in the lower e=a and higher than Tc in the higher e=a.5,9) Moreover, Xin et al. reported that the values of TM for ferromagnetic alloys Ni–Mn–Ga are higher than 300 K and lower than Tc and that TM is represented by the equation TM ¼ 702:5ðe=aÞ  5067 K as a function of e=a.11) These results indicate that the martensitic transformation occurs in a ferromagnetic phase for the lower e=a and in a paramagnetic state for the higher e=a. Moreover, Tsuchiya et al. reported three types of transformations: (I) paramagnetic parent phase , ferromagnetic parent phase , intermediate phase , ferromagnetic *Graduate

Student, Kagoshima University.

martensitic phase in the range e=a < 7:62, (II) paramagnetic parent phase , (ferromagnetic parent phase) , ferromagnetic martensitic phase in the range 7:62 < e=a < 7:65 and (III) paramagnetic parent phase , paramagnetic martensitic phase , ferromagnetic martensitic phase in the range 7:65 < e=a.12) The symbol ‘‘,’’ denotes the process of transformation between two phases. Previously, paying attention to only two systems in a ferromagnetic state, the authors calculated total energy differences E between the cubic and monoclinic structures as a function of e=a and related the E with the e=a dependence of TM .13) It was found that Eðe=aÞ changes like a straight line in the range e=a ¼ 7:50{7:625 for the case where X atoms occupy Ni sites, while like a parabolic line in the range e=a ¼ 7:625{7:77 for the case where X atoms occupy Mn sites. The characteristic behavior of Eðe=aÞ is similar to the behavior of TM ðe=aÞ of Ni2:16x Cox Mn0:84 Ga, Ni2:20z Fez Mn0:80 Ga and Ni2:16 Mn0:84y Coy Ga. However, for Ni2þx Mn1x Ga, the correspondence between Eðe=aÞ and TM ðe=aÞ is good in the range e=a ¼ 7:50{7:625 but not good in the range e=a > 7:625. In this paper, new four systems and a paramagnetic state will be considered to investigate in more detail the effect of X atom on the phase transformation in Ni–Mn–Ga–X systems. 2.

Crystal Structure and Method of Calculation

As described in the previous section, it was reported theoretically that the monoclinic structure is the most stable among the cubic, tetragonal, orthorhombic and monoclinic structures for Ni2:17 Ni0:83 Ga and Ni2 (Pd0:17 Ni0:83 )Ga. Then, we consider the cubic structure and the monoclinic structure as the parent phase and the martensitic phase, respectively. The symmetry of the monoclinic structure is lower than that of the cubic structure. The cubic structure is treated as a monoclinic structure with an angle of  shown in Fig. 1 to calculate under the same condition. The  angle is 71.565 and 98.461 or the cubic structure and the monoclinic

Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X

205

(a) Monoclinic Structure

Ni Mn Ga

y=1 plane

y=3/4 plane

Mn,Ga : y=0 or 1 Ni : y=1/4 2 3 4

Cubic Structure 1

3

y=1/2 plane

Mn(2)

3

1

y=0 plane

1

y=1/4 Ni(1)

(b)

x

Monoclinic Structure

Mn, Ga : y=1/2 Ni : y=3/4 2 4

z

Fig. 2 Monoclinic structure. The monoclinic structure has twenty-four atoms in the unit cell, which corresponds to the observed monoclinic structure having the shuffling of 6 layers of (2 2 0) planes.

Cubic Structure 2

3

4

Mn(1)

y

4 1 2

Fig. 1 Relation between the cubic and monoclinic structures of Ni–Mn– Ga–X alloy. The constituent atoms on the y ¼ 0 and y ¼ 1=4 planes are shown in (a) and ones on y ¼ 1=2 and y ¼ 3=4 planes in (b). The numbers denote the atomic sites in the monoclinic structure. The cubic structure is treated as the monoclinic structure with an angle of  ¼ 71:565 .

structure of Ni2:17 Ni0:83 Ga.6) When we assume that y-axis is vertical to this paper, Mn and Ga atoms are located on the y ¼ 0 (or 1) and 1=2 planes and Ni atoms on the y ¼ 1=4 and 3=4 planes shown in Fig. 1. Each of nickel, manganese and gallium in Ni2 MnGa has the four different atomic sites in the unit cell with the P2/m symmetry of the tenth space group. For example, the sites of Ni and Mn atoms are distinguished by such as the symbols of Ni(1), Mn(1) and Mn(2). The Ni(1), Mn(1) and Mn(2) are located at the 2j, 1a and 1f sites. The monoclinic structure has twenty-four atoms in the unit cell, which corresponds to the observed monoclinic structure having the shuffling of 6 layers of (2 2 0) planes.2) This monoclinic structure is shown in Fig. 2 and the symbols open

circles, solid circle and circle with slants denote Ni, Mn and Ga, respectively. The Mn(1), Mn(2) and Ni(1) are the sites where X atoms occupy. Recently, it was confirmed that the monoclinic structure is equivalent to the tetragonal structure of the named of 2M.14) The alloy where a sixth of Mn atoms of Ni2 MnGa were replaced with Ni atoms was described as Ni2:17 Ni0:83 Ga in the previous papers.6) In this paper, the alloy is described as Ni2 (X1=6 Mn5=6 )Ga where the Ni atoms at the Mn(1) sites are described in parentheses with the Mn atoms. Here, we consider six systems of Ni–Mn–Ga–X alloys where Ni or Mn atoms in Ni2 MnGa or Ni2 (Ni1=6 Mn5=6 )Ga are replaced with other transition element. They are listed in Table 1, where the name of the systems, the molecular formula and atoms at the Mn(1), Mn(2) and Ni(1) sites are shown. For example, in (Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 )Ga, Ni atoms are replaced with X atoms and Mn(1) atoms with Ni atoms. The s-Nm1-n1 and sNm1-m2 are new notation for sys-N1 and sys-M2 in the previous paper, respectively.13) When transition elements are chosen as the X atoms, these alloys are in the range of e=a ¼ 7:50{7:77. When we cannot choose a real element as the X atom for the special value of e=a, we adopt an artificial atom. For example, the artificial atom is described like Z27.5 where the number of 27.5 means the atomic number and the number of electrons.

Table 1 Six systems classified by the site of X atom (Mn or Ni site) in the shape memory alloys Ni–Mn–Ga–X. The symbols, the molecular formula used in this paper are listed. The atoms at Mn(1), Mn(2) and Ni(1) sites are also shown and the other atoms occupy the regular sites. Symbol of system s-m1

Constituent atom Molecular formula

Replaced

Mn(1)

Mn(2)

Ni(1)

Ni2 (X1=6 Mn5=6 )Ga

X

Mn

Ni

Ni2 (Mn5=6 X1=6 )Ga

Mn

X

Ni

s-Nm1-m2 s-Cm2-m1

Ni2 (Ni1=6 Mn4=6 X1=6 )Ga Ni2 (X1=6 Mn4=6 Co1=6 )Ga

Mn X

X Co

Ni Ni

s-Nm1-n1

(Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 )Ga

Ni

Mn

X

s-Nm12-n1

(Ni5=6 X1=6 )2 (Ni1=6 Mn4=6 Ni1=6 )Ga

Ni

Ni

X

s-m2

site

Mn

Ni

K. Yamaguchi, S. Ishida and S. Asano

Results and Discussion

3.1 Total energy and valence electron concentration In a previous paper, total energy differences E between cubic and monoclinic structures were calculated for s-Nm1n1 (old notation: sys-N1) and s-Nm1-m2 (old notation: sysM2).13) Here, E were newly calculated for four systems listed in Table 1 except for above two systems. To calculate the e=a dependence of E, transition elements were chosen as X atoms such as Mn, Fe, Z26.5, Co, Z27.5 and Ni for s-m1 where the value of e=a changes from 7.50 to 7.625. In this study, a paramagnetic state is newly considered. Therefore, band calculations were performed for four phases; paramagnetic cubic (PC), ferromagnetic cubic (FC), paramagnetic monoclinic (PM) and ferromagnetic monoclinic (FM) phases. The obtained total energies of six systems are the lowest for FM phase among four phases. At first, we consider the transformation in the ferromagnetic state, that is, the transformation between FC and FM phases. The total energy differences E between FC and FM phases is described as EFC-FM . The curves of EFC-FM ðe=aÞ are shown in Fig. 3(a) for four systems where X atoms occupy Mn sites and in Fig. 3(b) for two systems where X atoms occupy Ni sites. The cases of X = Ni in s-m1 (s-m2) and s-Nm1-n1 are equivalent to the case X = Mn in s-Nm1m2 which correspond to e=a ¼ 7:625. Also, the case of X = Co in s-m1 (s-m2) and the case of X = Ni in s-Cm2-m1 are equivalent to those of X = Mn in s-Cm2-m1 and X = Co in sNm1-m2. The curves for s-m1, s-m2 and s-Cm2-m1 are similar to that of s-Nm1-m2 shown in the previous paper13) and the curve of s-m2 overlaps with that of s-m1 each other. Their shapes are like a parabola with a top at Co. On the other hand, the curves of s-Nm12-n1 is similar to that of s-Nm1-n1 in Fig. 3(b) and the EFC-FM ðe=aÞ increases linearly with increasing e=a. Thus, the change of EFC-FM ðe=aÞ depends on the site of X atom (Mn or Ni site) and the value is not unique for e=a. Now, we will discuss the relation between EFC-FM ðe=aÞ and the martensitic transformation temperature TM . Chernenko et al.17) have measured the temperature dependence of the transformation stress to be d=dT ¼ 13 MPa/K for the alloy Ni–23.5Mn–23.9Ga. Tsuchiya et al.12) have studied the e=a dependence of TM and Tc for Ni–Mn–Ga alloys and estimated the transformation entropy S to be 48:2  1018 aJ/molK, using the value d=dT ¼ 13 MPa/K. When the e=a changes from 7.50 to 7.625, corresponding change in TM was observed to be about 100 K. In the same interval, the increase of EFC-FM ðe=aÞ is 5:71  1021 aJ/mol. The value is converted to the increase in TM to be 118 K, using S ¼ 48:2  1018 aJ/molK. Thus, the correspondence between the variation of the EFC-FM ðe=aÞ and that of TM is fairly well. The six curves of EFC-FM ðe=aÞ shown in Fig. 3 are again shown by the solid and broken lines in Fig. 4. The theoretical values EFC-FM ðe=aÞ near the experimental values are plotted

-1

(a)

E , E / aJ unit-cell

3.

0.80

Mn

0.72 s-m1 Fe

0.68 Mn

Co

X at Mn site

Co

s-Cm2-m1 Fe

0.76

0.64

Ni

Fe Ni

Co

s-Nm1-m2

Mn

s-m2 7.625

Ni

X at Mn(1) or Mn(2) E =E cub.-E mono.

0.60 7.49

7.54

7.59 7.64 7.69 7.74 7.79 Valence Electron Concentration, e/a

7.84

0.80 X at Ni site

(b) -1

Band calculations were carried out self-consistently by the LMTO-ASA method.15) The exchange correlation potential was treated within the framework of the local-spin-density (LSD) approximation.16)

E, E / aJ unit-cell

206

0.76 Ni Z27.5

0.72

Co

s-Nm12-n1

Z27.5 Ni 0.68

Co s-Nm1-n1

0.64

X at Ni(1)

7.625

E =E cub.-E mono. 0.60 7.49

7.54

7.59

7.64

7.69

7.74

7.79

7.84

Valence Electron Concentration, e/a Fig. 3 Valence electron concentration (e=a) dependence of difference (E) of total energies between the cubic and monoclinic structures in a ferromagnetic state. In (a), the solid curves with solid circles, open diamonds and crosses distinguish s-m1, s-m2 and s-Cm2-m1, respectively. A broken line with open circles is for s-Nm1-m2. In (b), a straight line with solid circles and a broken line with open circles distinguish s-Nm1-n1 and s-Nm12-n1.

by solid circles. The experimental values of TM for Ni2:16x Cox Mn0:84 Ga, Ni2:20z Fez Mn0:80 Ga and Ni2:16 Mn0:84y Coy Ga observed by Khovailo are plotted by open triangle, open square and open circle in the Fig. 4(a), respectively.18) The values of TM ðE) refer to the left (right) axis. The values of EFC-FM ðe=aÞ is plotted so that the values EFC-FM ðe=aÞ of the case X = Ni in the s-Nm1-n1 and X = Mn in the s-Nm1-m2 are superposed on the values of TM at e=a ¼ 7:625. The values of TM are distributed near the EFC-FM ðe=aÞ line for s-Nm1-n1 in the range e=a ¼ 7:54{7:625, while along the EFC-FM ðe=aÞ curve for s-Nm1-m2 in the range e=a ¼ 7:625{7:71, as described in the previous paper.13) Thus, the values of TM for Ni2:16x Cox Mn0:84 Ga and Ni2:20z Fez Mn0:80 Ga correspond to those of EFC-FM ðe=aÞ for s-Nm1-n1 and the values of TM for Ni2:16 Mn0:84y Coy Ga correspond to those EFC-FM ðe=aÞ for s-Nm1-m2. In the Fig. 4(b), the experimental values for Ni2þx Mn1x Ga are plotted by crosses for TM and by open diamonds for the Curie temperature Tc .12) The TM increases along the EFC-FM ðe=aÞ line for s-Nm1-n1 with increasing e=a, while the Tc decreases in the range e=a ¼ 7:50{7:65. And the TM and Tc are nearly equal in the range e=a ¼ 7:65{7:71. The values of TM ðe=aÞ distribute along

7.71

s-Nm1-m2

0.780 -1

600 500

0.732

400 300 s-Nm1-n1

0.684

200 100 0.639 7.48 7.53 7.58 7.63 7.68 7.73 7.78

800 7.625

(b)

700

7.71 0.780

600

-1

700

7.625

207

0.732

500 400 s-Nm12-n1

300

0.684

E , E / aJ unit-cell

(a)

Transformation Temperatures, TM, Tc /K

800

E , E / aJ unit-cell

Transformation Temperature, TM /K

Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X

s-Nm1-n1

200

100 0.639 7.48 7.53 7.58 7.63 7.68 7.73 7.78 Valence Electron Concentration, e/a

Valence Electron Concentration, e/a

Fig. 4 Comparison between phase transformation temperatures and total energy differences. The values of martensitic transformation temperature TM refer to the left axes and those of total energy difference E to the right axes. The solid and broken curves are the curves of E shown in Fig. 3. The solid curves with solid circles show the curves of EFC-FM ðe=aÞ which are comparable with the experimental values. In (a), the open squares, open triangles and the open circles indicate the values of TM for Ni2:16x Cox Mn0:84 Ga, Ni2:20z Fez Mn0:80 Ga and Ni2:16 Mn0:84y Coy Ga, respectively.18) In (b), crosses and diamonds indicate the values of TM and Tc for Ni2þx Mn1x Ga, respectively.12)

(a)

X at Mn site

E=E-E FM

E, E / aJ unit-cell

-1

1.7 1.4

E PC-FM 1.1

7.625

0.8 E FC-FM

0.5 0.2

E PM-FM

-0.1 7.49 7.53 7.57 7.61 7.65 7.69 7.73 7.77 7.81 7.85 Valence Electron Concentration, e/a 2.0

(b)

X at Ni site

1.7

E=E-E FM

-1

3.2 Intermediate state In the previous section, we considered above the transformation in the ferromagnetic state and also we will consider the paramagnetic state in followings. The differences (E ¼ E  EFM ) of total energies between the FM phase with the lowest total energy and the other phase are plotted as a function of e=a in Figs. 5(a) and (b). The differences E are shown in Fig. 5(a) for the case where Mn atoms are replaced with X atoms and in Fig. 5(b) for the case that Ni atoms are replaced with X atoms. In Fig. 5(a), the three curves with solid symbols in the range e=a ¼ 7:50{7:625 and with open symbols in the range e=a ¼ 7:625{7:77 are drawn for s-m1 and s-Nm1-m2, respectively. The symbols ‘‘triangle’’, ‘‘circle’’ and ‘‘square’’ correspond to EPC-FM ðe=aÞ, EFC-FM ðe=aÞ and EPM-FM ðe=aÞ, respectively. In Fig. 5(b), the differences E are drawn by the three straight lines for s-Nm1-n1 and sNm12-n1 as in Fig. 5(a). Here, we refer to the experimental results that the martensitic transition occurs in the ferromagnetic state for the lower e=a. In the range e=a ¼ 7:50{7:70, the EFC-FM ðe=aÞ varies like a parabolic or straight line as described above, while the EPC-FM ðe=aÞ and EPM-FM ðe=aÞ decrease with increasing e=a. The increase of EFC-FM ðe=aÞ corresponds to the increase of TM and the decrease of EPC-FM ðe=aÞ and EPM-FM ðe=aÞ corresponds to the decrease of Tc . Our results show that the total energy becomes lower in order of PC, FC, PM and FM phases and suggest the possibility of four kinds of transitions as follows:

2.0

E, E / aJ unit-cell

Eðe=aÞ line of s-Nm1-n1 in the range e=a ¼ 7:50{7:65. In the range e=a > 7:625, the values of TM do not distribute near the broken curve for s-Nm1-m2 but the curve for s-Nm12-n1. Thus, it was found that the Eðe=aÞ for s-Nm12-n1 corresponds to the TM ðe=aÞ of Ni2þx Mn1x Ga in the range e=a ¼ 7:65{7:71.

1.4

E PC-FM

1.1

7.625 0.8 0.5

E FC-FM E PM-FM

0.2 -0.1 7.49 7.53 7.57 7.61 7.65 7.69 7.73 7.77 7.81 7.85 Valence Electron Concentration, e/a Fig. 5 The e=a dependence of the total energy difference E for four phases; PC, FC, PM and FM phases. The differences (E ¼ E  EFM ) between the FM phase with the lowest total energy and the other phase are shown for the case (a) where Mn atoms are replaced with X atoms for s-m1 and s-Nm1-m2 and the case (b) where Ni atoms are replaced with X atoms for s-Nm1-n1 and s-Nm12-n1.

208

K. Yamaguchi, S. Ishida and S. Asano

(Trans.1) PC ! FC ! FM, (Trans.2) PC ! PM ! FM, (Trans.3) PC ! FM and (Trans.4) PC ! FC ! PM ! FM. Thus, there is the possibility that FC and PM phases become the intermediate phase between PC and FM phases. Now, we compare our results with those of Tsuchiya et al.12) As described in the introduction, they reported three types of transitions in three regions: (I) 7:5 > e=a, (II) 7:62 < e=a < 7:65 and (III) 7:65 < e=a. We guess that Trans.1 correspond to the transition in the range e=a < 7:65 where their observed intermediate state may be a ferromagnetic phase with a structure different from the monoclinic structure, Trans.2 does to the transition in the range e=a > 7:65 and Trans.3 does to the transition in the range 7:62 < e=a < 7:65. The magnetic transition is not natural in the Trans.4 among our four types of transitions. Therefore, Trans.4 may be not observed. We have to consider entropy in order to discuss transitions accurately. 3.3 Density of states In the previous section, it was found that curves of the differences Eðe=aÞ are similar for the four systems of s-m1, s-m2, s-Nm1-m2 and s-Cm2-m1 where the X atoms occupy the Mn sites. The similarities are also seen in the curves of sNm1-n1 and s-Nm12-n1 where the X atoms occupy the Ni sites. It is natural to pay attention to X atoms in considering energy differences due to differences of e=a and systems,

because the difference of e=a and systems is due to the X atoms. It is expected that the change in the total density of state (DOS) due to the difference of e=a and system mainly comes from the change in the local DOS of X atom (X-DOS). The change in the DOS affects the change of total energy differences E between cubic and monoclinic structures. Then, we pay attention to the relation between the Eðe=aÞ and the X-DOS for case that the X atom occupies the Mn sites. The X-DOS curves for s-m1 are shown in Fig. 6 where the curves of the cubic structure are shown for the majority and minority spins in Figs. 6(a) and (b) and those of the monoclinic structure in Figs. 6(c) and (d). The variation in the X-DOS for X = Mn, Fe, Co and Ni atoms is quite similar to that of s-Nm1-m2 (old notation: sys-M2).13) The vertical line denotes the Fermi energy. Since the X atom in the cubic structure is surrounded by eight Ni atoms, the X-DOS has the characteristics of the bcc structure, that is, the X-DOS is composed of two large peaks. The large valley between the two peaks disappears in the monoclinic structure and the occupied states generally move to the states with the lower energy. Therefore, we can guess that the band energy for the monoclinic structure is lower than that of the cubic structure. In the minority spin states, the X-DOS curve shifts from the higher energy region to the lower energy region beyond the Fermi energy, when the X atom changes from Mn to Ni in the order of Mn, Fe, Co and Ni. On the other hand, in the majority spin state, the two large peaks under the Fermi energy shift to the higher energy region with increasing e=a except for the

50

States, n / aJ

States, n / aJ

Cub. 20 10

Mn Fe Co Ni

30

Mono. 20 10

40

50 -1

Mn Fe Co Ni

(b) Ni2(X1/6Mn5/6)Ga X at Mn(1)

atom spin

-1

50

30

State, n / aJ

20 10 0 -0.8

40

Mn Fe Co Ni

(d) Ni2(X1/6Mn5/6)Ga X at Mn(1)

30

Mono.

-1

Cub.

-1

atom spin

40

0

0

State, n / aJ

(c ) Ni2(X1/6Mn5/6)Ga X at Mn(1)

-1

30

atom spin

40

Mn Fe Co Ni

-1

atom spin

-1

(a) Ni2(X1/6Mn5/6)Ga X at Mn(1)

-1

50

20 10 0

-0.6

-0.4

-0.2

0

0.2 -1

Energy, E / aJ unit-cell

0.4

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-1

Energy, E / aJ unit-cell

Fig. 6 Local DOS of X atoms in s-m1. Four curves distinguish the cases of X = Mn, Fe, Co and Ni in Ni2 (X1=6 Mn5=6 )Ga, respectively. The DOS for the cubic structure are shown in (a) and (b) and for the monoclinic structure in (c) and (d). The upper ((a) and (c)) and the lower ((b) and (d)) are of the majority and the minority spin. The vertical line shows the Fermi energy.

atom spin

(a) X=Co (Ni5/6X1/6)2(Ni1/6Mn5/6)Ga

FC FM

20

10

30

-1

(b) X=Z27.5

atom spin

-1

30 FC FM

(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga 20

(b) X=Z27.5 20

FC FM

(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga

States, n / aJ

-1

-1

States, n / aJ atom spin

20

FC FM

(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga

0

10

10

0

0

30 -1

30

atom spin

20

FC FM

(c) X=Ni (Ni5/6X1/6)2(Ni1/6Mn5/6)Ga

(c) X=Ni 20

FC FM

(Ni5/6X1/6)2(Ni1/6Mn5/6)Ga

States, n / aJ

-1

-1

-1

(a) X=Co

-1

States, n / aJ

10

0

States, n / aJ atom spin

209

30

-1

30

-1

States, n / aJ atom spin

-1

Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X

10

0 -0.8

-0.6

-0.4

-0.2

0

0.2

10

0 -0.8

0.4

-1

Energy, E / aJ unit-cell

Fig. 7 Local DOS of X atoms in s-Nm1-n1. The DOS of X = Co, Z27.5 and Ni in (Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 ) Ga are shown in (a), (b) and (c), respectively. The DOS curves for the majority spin state in FC and FM phases are drawn by the solid and dotted lines, respectively. The vertical line shows the Fermi energy.

case X = Mn. We can roughly guess from these changes that the difference of band energy between cubic and monoclinic structures becomes larger with increasing atomic number of X atom. Therefore, the E increases with increasing atomic number. Next, we turn our attention to the case that the X atoms occupy the Ni sites. The X-DOS curves for s-Nm1-n1 are shown in Figs. 7 and 8. The curves of Co, Z27.5 and Ni in FC and FM phases are compared for the majority and minority spins in Figs. 7 and 8, respectively. The change of the XDOS due to the difference of X atom is small in the majority spin for both of the FC and FM phases. On the other hand, the difference of the X-DOS between the FC and FM phases becomes larger in the minority spin, when X atom changes from X = Co to Ni. The changes bring the linear increase in EFC-FM ðe=aÞ.

-0.6

-0.4

-0.2

0

Energy, E / aJ unit-cell

0.2

0.4

-1

Fig. 8 Local DOS of X atoms in s-Nm1-n1. The DOS of X = Co, Z27.5 and Ni in (Ni5=6 X1=6 )2 (Ni1=6 Mn5=6 ) Ga are shown in (a), (b) and (c), respectively. The DOS curves for the minority spin state in FC and FM phases are drawn by the solid and dotted lines, respectively. The vertical line shows the Fermi energy.

4.

Conclusion

To investigate in more detail the effect of X atom on the phase transformation in Ni–Mn–Ga–X systems, the electronic structures were calculated for six Ni2 MnGa based systems listed in Table 1. The total energies were also calculated for four phases, which are PC, FC, PM and FM phases. Since the total energies become lower in order of the PM, FC, PM and FM phases, there is possibility that the FC and PM phases become an intermediate phase between PC and FM phases. For the six systems treated in this paper, the total energy differences EFC-FM ðe=aÞ between FC and FM phases calculated by changing the X atom in Ni–Mn–Ga–X alloys from Mn to Ni among transition elements. The EFC-FM ðe=aÞ have a similar e=a dependence if the X atom occupies the same atomic site (Ni or Mn site). It was shown that the increase of the martensitic transformation temperature TM due to the increase of e=a

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K. Yamaguchi, S. Ishida and S. Asano

from 7.50 to 7.625 is comparable to that of TM which is converted from the increase of EFC-FM ðe=aÞ. Therefore, the e=a dependence of TM ðe=aÞ corresponds to that of EFC-FM ðe=aÞ for s-Nm1-n1 in the range e=a < 7:625 and those for s-Nm1-m2 and s-Nm12-n1 in the range e=a > 7:625. Thus, the TM ðe=aÞ may be predicted from the Eðe=aÞ which is not unique against the value of e=a. The variation of Eðe=aÞ due to the difference of the X atoms mainly comes from the variation of the X-DOS in Ni–Mn– Ga–X alloys. Acknowledgments The authors wish to thank Professor Koichi Tsuchiya of Toyohashi University of Technology for giving information and significant discussions. This work was supported by a Grant-in-Aid (13640638) for scientific Research from the Ministry of Education, Science and Culture of Japan. REFERENCES 1) V. A. Chernenko, C. Segui, E. Cesari, J. Pons and V. V. Kokorin: Phys. Rev. B57 (1998) 2659–2662. 2) K. Inoue, K. Enami, M. Igawa, K. Inoue, Y. Yamaguchi and K. Ohoyama: Proc. Int. Conf. on Solid-Solid Phase Transformations (1999) pp. 1120–1123. 3) J. Pons, V. A. Chernenko, R. Santamarta and E. Cesari: Acta Mater. 48

(2000) 3027–3038. 4) U. Stuhr, P. Vorderwisch and V. V. Kokorin: J. Phys. Condens. Matter. 12 (2000) 7541–7545. 5) A. N. Vasil’ev, A. D. Bozhko, V. V. Khovailo, I. E. Dikshtein, V. G. Shavrov, V. D. Buchelnikov, M. Matsumoto, S. Suzuki, T. Takagi and J. Tani: Phys. Rev. B59 (1999) 1113–1120. 6) S. Ishida, M. Furugen and S. Asano: Int. J. Appl. Electr. Mechan. 12 (2000) 41–48. 7) P. J. Webster, K. R. A. Ziebeck, S. L. Town and M. S. Peak: Philos. Mag. 49B (1984) 295–310. 8) V. A. Chernenko, V. A. L’vov, M. Pasquale, S. Besseghini, C. Sasso and D. A. Polenur: Int. J. Appl. Electro. Mechan. 12 (2000) 3–8. 9) V. A. Chernenko: Scr. Mater. 40 (1999) 523–527. 10) V. A. Chernenko, E. Cesari, V. V. Kokorin and I. N. Vitenko: Scr. Metall. Mater. 33 (1995) 1239–1244. 11) X. Jin, M. Marioni, D. Bono, S. M. Allen and R. C. O’Handley: J. Appl. Phys. 91 (2002) 8222–8224. 12) K. Tsuchiya, A. Tsutsumi, H. Nakayama, S. Ishida, H. Ohtsuka and M. Umemoto: J. Phys. IV, in press. 13) K. Yamaguchi, S. Ishida and S. Asano: Mater. Trans. 43 (2002) 846– 851. 14) K. Inoue: Private communication. 15) K. Andersen, O. Jepsen and D. Glotzel: Proc. Int. School of Physics ‘‘Enrico Fermi’’ Course 89, ed. by F. Bassami, F. Fumi and M. P. Tosi (North-Holland, Amsterdam, 1985) pp. 59–176. 16) V. L. Moruzzi, J. F. Janak and A. R. Williams: Calculated Electronic Properties of Metals, (Pergamon, New York, 1978) pp. 1–22. 17) V. A. Chernenko, V. V. Kokorin, O. M. Babii and I. K. Zasimchuk: Intermetalics 6 (1998) 29–34. 18) V. Khovailo: Proc. Seminar. on Shape Memory Alloys Related Technology, (Inst. Fluid Science, Tohoku University, 1999) pp. 30–33.

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