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

1Department of Physics, Faculty of Science, Kagoshima University, Kagoshima, 890-0065 Japan 2The Graduate School of=College of Arts and Sciences, The University of Tokyo, Tokyo, 153-0041 Japan

In the Ni2MnGa 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 TMðe=aÞ strongly depends on e=a and increases with increasing e=a. In this paper, to examine the effect of Xatom on the phase transformation in Ni–Mn–Ga–Xalloys, 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 ÁEðe=aÞ between two phases among four phases were calculated as a function of e=a. The variations of TMðe=aÞ were predicted by the difference ÁEðe=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 TMðe=aÞ reflect the changes of the density of states of Xatoms.

(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 martensitic phase in the range e=a < 7:62, (II) paramagnetic parent phase , (ferromagnetic parent phase) , ferromag- Many researchers have reported on the crystal structures netic martensitic phase in the range 7:62 < e=a < 7:65 and and the phase transformations of the Ni–Mn–Ga alloys. The (III) paramagnetic parent phase , paramagnetic martensitic tetragonal structure was observed in the martensitic phase phase , ferromagnetic martensitic phase in the range around valence electron concentration e=a ¼ 7:50 (stoichio- 7:65 < e=a.12) The symbol ‘‘,’’ denotes the process of metric Ni2MnGa). On the other hand, the orthorhombic and transformation between two phases. monoclinic structures were observed.1–4) For example, the Previously, paying attention to only two systems in a orthorhombic structure was observed at the e=a ¼ 7:635 and ferromagnetic state, the authors calculated total energy the monoclinic structures at e=a ¼ 7:64, 7.67, 7.72 and differences ÁE between the cubic and monoclinic structures 7.78.2,3) It was also reported that the tetragonal phase in the as a function of e=a and related the ÁE with the e=a 5) 13) lower e=a can be suppressed by Ni excess. Furthermore, it is dependence of TM. It was found that ÁEðe=aÞ changes like also predicted theoretically that the tetragonal and orthor- a straight line in the range e=a ¼ 7:50{7:625 for the case hombic structures may be metastable and the monoclinic where Xatoms occupy Ni sites, while like a parabolic line in structure is the most stable state among these structures for the range e=a ¼ 7:625{7:77 for the case where Xatoms 6) Ni2:17Ni0:83Ga and Ni2(Pd0:17Ni0:83)Ga. Thus, it is possible occupy Mn sites. The characteristic behavior of ÁEðe=aÞ is for the monoclinic structure to appear in martensitic phase in similar to the behavior of TMðe=aÞ of Ni2:16ÀxCoxMn0:84Ga, wide e=a range. Ni2:20ÀzFezMn0:80Ga and Ni2:16Mn0:84ÀyCoyGa. However, for The martensitic transformation temperature TM and the Ni2þxMn1ÀxGa, the correspondence between ÁEðe=aÞ and Curie temperature Tc for Ni2MnGa (e=a ¼ 7:50) are 202 and TMðe=aÞ is good in the range e=a ¼ 7:50{7:625 but not good 7) 376 K, respectively. The various values of TM were in the range e=a > 7:625. observed in the wide range from 175 to 626 K in the range In this paper, new four systems and a paramagnetic state 8–10) e=a ¼ 7:45{8:10. For Ni–Mn–Ga alloys, Tc decreases will be considered to investigate in more detail the effect of X 5,9) and TM increases with increasing e=a. They merge in the atom on the phase transformation in Ni–Mn–Ga–Xsystems. 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 2. Crystal Structure and Method of Calculation 5,9) e=a. Moreover, Xin et al. reported that the values of TM for ferromagnetic alloys Ni–Mn–Ga are higher than 300 K and As described in the previous section, it was reported lower than Tc and that TM is represented by the equation theoretically that the monoclinic structure is the most stable 11) TM ¼ 702:5ðe=aÞÀ5067 K as a function of e=a. These among the cubic, tetragonal, orthorhombic and monoclinic results indicate that the martensitic transformation occurs in a structures for Ni2:17Ni0:83Ga and Ni2(Pd0:17Ni0:83)Ga. Then, ferromagnetic phase for the lower e=a and in a paramagnetic we consider the cubic structure and the monoclinic structure state for the higher e=a. as the parent phase and the martensitic phase, respectively. Moreover, Tsuchiya et al. reported three types of trans- The symmetry of the monoclinic structure is lower than that formations: (I) paramagnetic parent phase , ferromagnetic of the cubic structure. The cubic structure is treated as a parent phase , intermediate phase , ferromagnetic monoclinic structure with an angle of shown in Fig. 1 to calculate under the same condition. The angle is 71.565  *Graduate Student, Kagoshima University. and 98.461 or the cubic structure and the monoclinic Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X205

(a) Monoclinic Structure Ni y=1 plane Mn Ga

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

2 Cubic Structure 3 1 4 3 y=1/2 plane Mn(2) 3 1 y=0 plane 1 y=1/4 Ni(1) Mn(1) (b) y z Monoclinic Structure x

Mn, Ga : y=1/2 Fig. 2 Monoclinic structure. The monoclinic structure has twenty-four Ni : y=3/4 atoms in the unit cell, which corresponds to the observed monoclinic structure having the shuffling of 6 layers of (2 2 0) planes.

2 Cubic Structure 4 2 circles, solid circle and circle with slants denote Ni, Mn and 4 3 Ga, respectively. The Mn(1), Mn(2) and Ni(1) are the sites 4 where Xatoms occupy. Recently, it was confirmed that the 1 monoclinic structure is equivalent to the tetragonal structure 14) 2 of the named of 2M. The alloy where a sixth of Mn atoms of Ni2MnGa were Fig. 1 Relation between the cubic and monoclinic structures of Ni–Mn– replaced with Ni atoms was described as Ni2:17Ni0:83Ga in the Ga–Xalloy. The constituent atoms on the y ¼ 0 and y ¼ 1=4 planes are 6) shown in (a) and ones on y ¼ 1=2 and y ¼ 3=4 planes in (b). The numbers previous papers. In this paper, the alloy is described as denote the atomic sites in the monoclinic structure. The cubic structure is Ni2(X1=6Mn5=6)Ga where the Ni atoms at the Mn(1) sites are treated as the monoclinic structure with an angle of ¼ 71:565. described in parentheses with the Mn atoms. Here, we consider six systems of Ni–Mn–Ga–Xalloys where Ni or Mn atoms in Ni2MnGa or Ni2(Ni1=6Mn5=6)Ga are replaced with 6) structure of Ni2:17Ni0:83Ga. When we assume that y-axis is other transition element. They are listed in Table 1, where the vertical to this paper, Mn and Ga atoms are located on the name of the systems, the molecular formula and atoms at the y ¼ 0 (or 1) and 1=2 planes and Ni atoms on the y ¼ 1=4 and Mn(1), Mn(2) and Ni(1) sites are shown. For example, in 3=4 planes shown in Fig. 1. Each of nickel, manganese and (Ni5=6X1=6)2(Ni1=6Mn5=6)Ga, Ni atoms are replaced with X gallium in Ni2MnGa has the four different atomic sites in the atoms and Mn(1) atoms with Ni atoms. The s-Nm1-n1 and s- unit cell with the P2/m symmetry of the tenth space group. Nm1-m2 are new notation for sys-N1 and sys-M2 in the For example, the sites of Ni and Mn atoms are distinguished previous paper, respectively.13) When transition elements are by such as the symbols of Ni(1), Mn(1) and Mn(2). The chosen as the Xatoms, these alloys are in the range of Ni(1), Mn(1) and Mn(2) are located at the 2j, 1a and 1f sites. e=a ¼ 7:50{7:77. When we cannot choose a real element as The monoclinic structure has twenty-four atoms in the unit the Xatom for the special value of e=a, we adopt an artificial cell, which corresponds to the observed monoclinic structure atom. For example, the artificial atom is described like Z27.5 having the shuffling of 6 layers of (2 2 0) planes.2) This where the number of 27.5 means the atomic number and the monoclinic structure is shown in Fig. 2 and the symbols open number of electrons.

Table 1 Six systems classified by the site of Xatom (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 Constituent atom Replaced of system Molecular formula Mn(1) Mn(2) Ni(1) site

s-m1 Ni2(X1=6Mn5=6)Ga XMn Ni

s-m2 Ni2(Mn5=6X1=6)Ga Mn XNi Mn s-Nm1-m2 Ni2(Ni1=6Mn4=6X1=6)Ga Mn XNi

s-Cm2-m1 Ni2(X1=6Mn4=6Co1=6)Ga XCo Ni

s-Nm1-n1 (Ni5=6X1=6)2(Ni1=6Mn5=6)Ga Ni Mn X Ni s-Nm12-n1 (Ni5=6X1=6)2(Ni1=6Mn4=6Ni1=6)Ga Ni Ni X 206 K. Yamaguchi, S. Ishida and S. Asano

Band calculations were carried out self-consistently by the 0.80 15) LMTO-ASA method. The exchange correlation potential X at Mn site (a) s-Cm2-m1 Co was treated within the framework of the local-spin-density Ni -1 0.76 16) Fe (LSD) approximation. Co Mn Fe 0.72 Ni 3. Results and Discussion unit-cell s-m1 Co s-Nm1-m2 / aJ Fe Ni Mn

E 0.68

3.1 Total energy and valence electron concentration , s-m2 X at Mn(1) or Mn(2)

E 7.625 In a previous paper, total energy differences ÁE between Mn 0.64 cubic and monoclinic structures were calculated for s-Nm1- E =E cub.-E mono. n1 (old notation: sys-N1) and s-Nm1-m2 (old notation: sys- M2).13) Here, ÁE were newly calculated for four systems 0.60 listed in Table 1 except for above two systems. To calculate 7.49 7.54 7.59 7.64 7.69 7.74 7.79 7.84 Valence Electron Concentration, e/a the e=a dependence of ÁE, transition elements were chosen as Xatoms such as Mn, Fe, Z26.5, Co, Z27.5 and Ni for s-m1 0.80 where the value of e=a changes from 7.50 to 7.625. In this (b) X at Ni site 0.76 study, a paramagnetic state is newly considered. Therefore, -1 Ni band calculations were performed for four phases; paramag- Z27.5 netic cubic (PC), ferromagnetic cubic (FC), paramagnetic 0.72 Co unit-cell monoclinic (PM) and ferromagnetic monoclinic (FM) pha- Ni s-Nm12-n1

aJ Z27.5 ses. The obtained total energies of six systems are the lowest 0.68 Co for FM phase among four phases. E, E / s-Nm1-n1 X at Ni(1) 7.625 At first, we consider the transformation in the ferromag- 0.64 netic state, that is, the transformation between FC and FM E =E cub.-E mono. phases. The total energy differences ÁE between FC and FM 0.60 phases is described as ÁEFC-FM. The curves of ÁEFC-FMðe=aÞ 7.49 7.54 7.59 7.64 7.69 7.74 7.79 7.84 are shown in Fig. 3(a) for four systems where Xatoms Valence Electron Concentration, e/a 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) Fig. 3 Valence electron concentration (e=a) dependence of difference and s-Nm1-n1 are equivalent to the case X= Mn in s-Nm1- (ÁE) of total energies between the cubic and monoclinic structures in a m2 which correspond to e=a ¼ 7:625. Also, the case of X= ferromagnetic state. In (a), the solid curves with solid circles, open Co in s-m1 (s-m2) and the case of X= Ni in s-Cm2-m1 are 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 equivalent to those of X= Mn in s-Cm2-m1 and X= Co in s- solid circles and a broken line with open circles distinguish s-Nm1-n1 and Nm1-m2. The curves for s-m1, s-m2 and s-Cm2-m1 are s-Nm12-n1. 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 by solid circles. The experimental values of TM for hand, the curves of s-Nm12-n1 is similar to that of s-Nm1-n1 Ni2:16ÀxCoxMn0:84Ga, Ni2:20ÀzFezMn0:80Ga and in Fig. 3(b) and the ÁEFC-FMðe=aÞ increases linearly with Ni2:16Mn0:84ÀyCoyGa observed by Khovailo are plotted by increasing e=a. open triangle, open square and open circle in the Fig. 4(a), 18) Thus, the change of ÁEFC-FMðe=aÞ depends on the site of X respectively. The values of TMðÁE) refer to the left (right) atom (Mn or Ni site) and the value is not unique for e=a. axis. The values of ÁEFC-FMðe=aÞ is plotted so that the values Now, we will discuss the relation between ÁEFC-FMðe=aÞ and ÁEFC-FMðe=aÞ of the case X= Ni in the s-Nm1-n1 and X= the martensitic transformation temperature TM. Mn in the s-Nm1-m2 are superposed on the values of TM at 17) Chernenko et al. have measured the temperature e=a ¼ 7:625. The values of TM are distributed near the dependence of the transformation stress to be ÁEFC-FMðe=aÞ line for s-Nm1-n1 in the range d=dT ¼ 13 MPa/K for the alloy Ni–23.5Mn–23.9Ga. Tsu- e=a ¼ 7:54{7:625, while along the ÁEFC-FMðe=aÞ curve for 12) chiya et al. have studied the e=a dependence of TM and Tc s-Nm1-m2 in the range e=a ¼ 7:625{7:71, as described in the 13) for Ni–Mn–Ga alloys and estimated the transformation previous paper. Thus, the values of TM for 18 entropy ÁS to be 48:2  10 aJ/molK, using the value Ni2:16ÀxCoxMn0:84Ga and Ni2:20ÀzFezMn0:80Ga correspond d=dT ¼ 13 MPa/K. When the e=a changes from 7.50 to to those of ÁEFC-FMðe=aÞ for s-Nm1-n1 and the values of TM 7.625, corresponding change in TM was observed to be about for Ni2:16Mn0:84ÀyCoyGa correspond to those ÁEFC-FMðe=aÞ 100 K. In the same interval, the increase of ÁEFC-FMðe=aÞ is for s-Nm1-m2. 5:71  1021 aJ/mol. The value is converted to the increase in In the Fig. 4(b), the experimental values for 18 TM to be 118 K, using ÁS ¼ 48:2  10 aJ/molK. Thus, the Ni2þxMn1ÀxGa are plotted by crosses for TM and by open 12) correspondence between the variation of the ÁEFC-FMðe=aÞ diamonds for the Curie temperature Tc. The TM increases and that of TM is fairly well. along the ÁEFC-FMðe=aÞ line for s-Nm1-n1 with increasing The six curves of ÁEFC-FMðe=aÞ shown in Fig. 3 are again e=a, while the Tc decreases in the range e=a ¼ 7:50{7:65. shown by the solid and broken lines in Fig. 4. The theoretical And the TM and Tc are nearly equal in the range values ÁEFC-FMðe=aÞ near the experimental values are plotted e=a ¼ 7:65{7:71. The values of TMðe=aÞ distribute along Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X207 K /

800 800 c /K (a) 7.625 7.71 T (b) 7.625 7.71 , M M T 700 0.780 700 0.780

s-Nm1-m2 T -1 600 -1 600

500 0.732 500 0.732 unit-cell unit-cell 400 400 / aJ / aJ E E s-Nm12-n1 , 300 0.684 , 300 0.684 E s-Nm1-n1 E s-Nm1-n1 200 200 100 100 Transformation Temperature, 0.639 0.639

7.48 7.53 7.58 7.63 7.68 7.73 7.78 Transformation Temperatures, 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 ÁE to the right axes. The solid and broken curves are the curves of ÁE shown in Fig. 3. The solid curves with solid circles show the curves of ÁEFC-FMðe=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 Ni2:16ÀxCoxMn0:84Ga, 18) Ni2:20ÀzFezMn0:80Ga and Ni2:16Mn0:84ÀyCoyGa, respectively. In (b), crosses and diamonds indicate the values of TM and Tc for 12) Ni2þxMn1ÀxGa, respectively.

ÁEðe=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 T do not distribute near M 2.0 the broken curve for s-Nm1-m2 but the curve for s-Nm12-n1. (a) X at Mn site E=E-E FM Thus, it was found that the ÁEðe=aÞ for s-Nm12-n1 1.7 corresponds to the TMðe=aÞ of Ni2þxMn1ÀxGa in the range -1 e=a ¼ 7:65{7:71. 1.4 E PC-FM

unit-cell 1.1 3.2 Intermediate state 7.625 aJ

In the previous section, we considered above the transfor- / 0.8

mation in the ferromagnetic state and also we will consider E FC-FM the paramagnetic state in followings. The differences E, E 0.5 (ÁE ¼ E À EFM) of total energies between the FM phase 0.2 with the lowest total energy and the other phase are plotted as E PM-FM a function of e=a in Figs. 5(a) and (b). The differences ÁE are -0.1 shown in Fig. 5(a) for the case where Mn atoms are replaced 7.49 7.53 7.57 7.61 7.65 7.69 7.73 7.77 7.81 7.85 with Xatoms and in Fig. 5(b) for the case that Ni atoms are Valence Electron Concentration, e/a replaced with Xatoms. In Fig. 5(a), the three curves with solid symbols in the 2.0 (b) X at Ni site range e=a ¼ 7:50{7:625 and with open symbols in the range E=E-E FM 1.7 e=a ¼ 7:625{7:77 are drawn for s-m1 and s-Nm1-m2, -1 respectively. The symbols ‘‘triangle’’, ‘‘circle’’ and ‘‘square’’ 1.4 correspond to ÁEPC-FMðe=aÞ, ÁEFC-FMðe=aÞ and E PC-FM

unit-cell 1.1 ÁEPM-FMðe=aÞ, respectively. In Fig. 5(b), the differences 7.625 aJ ÁE are drawn by the three straight lines for s-Nm1-n1 and s- 0.8 Nm12-n1 as in Fig. 5(a). E FC-FM Here, we refer to the experimental results that the E, E / 0.5 E martensitic transition occurs in the ferromagnetic state for PM-FM 0.2 the lower e=a. In the range e=a ¼ 7:50{7:70, the ÁEFC-FMðe=aÞ varies like a parabolic or straight line as -0.1 described above, while the ÁEPC-FMðe=aÞ and ÁEPM-FMðe=aÞ 7.49 7.53 7.57 7.61 7.65 7.69 7.73 7.77 7.81 7.85 decrease with increasing e=a. The increase of ÁEFC-FMðe=aÞ Valence Electron Concentration, e/a corresponds to the increase of TM and the decrease of ÁEPC-FMðe=aÞ and ÁEPM-FMðe=aÞ corresponds to the decrease Fig. 5 The e=a dependence of the total energy difference ÁE for four phases; PC, FC, PM and FM phases. The differences (ÁE ¼ E À EFM) of Tc. between the FM phase with the lowest total energy and the other phase are Our results show that the total energy becomes lower in shown for the case (a) where Mn atoms are replaced with Xatoms for s-m1 order of PC, FC, PM and FM phases and suggest the and s-Nm1-m2 and the case (b) where Ni atoms are replaced with Xatoms possibility of four kinds of transitions as follows: for s-Nm1-n1 and s-Nm12-n1. 208 K. Yamaguchi, S. Ishida and S. Asano

(Trans.1) PC ! FC ! FM, because the difference of e=a and systems is due to the X (Trans.2) PC ! PM ! FM, atoms. It is expected that the change in the total density of (Trans.3) PC ! FM and state (DOS) due to the difference of e=a and system mainly (Trans.4) PC ! FC ! PM ! FM. comes from the change in the local DOS of Xatom (X-DOS). Thus, there is the possibility that FC and PM phases become The change in the DOS affects the change of total energy the intermediate phase between PC and FM phases. Now, we differences ÁE between cubic and monoclinic structures. compare our results with those of Tsuchiya et al.12) As Then, we pay attention to the relation between the ÁEðe=aÞ described in the introduction, they reported three types of and the X-DOS for case that the X atom occupies the Mn transitions in three regions: (I) 7:5 > e=a, (II) 7:62 < e=a < sites. The X-DOS curves for s-m1 are shown in Fig. 6 where 7:65 and (III) 7:65 < e=a. We guess that Trans.1 correspond the curves of the cubic structure are shown for the majority to the transition in the range e=a < 7:65 where their observed and minority spins in Figs. 6(a) and (b) and those of the intermediate state may be a ferromagnetic phase with a monoclinic structure in Figs. 6(c) and (d). The variation in the structure different from the monoclinic structure, Trans.2 X-DOS for X = Mn, Fe, Co and Ni atoms is quite similar to does to the transition in the range e=a > 7:65 and Trans.3 that of s-Nm1-m2 (old notation: sys-M2).13) The vertical line does to the transition in the range 7:62 < e=a < 7:65. The denotes the Fermi energy. Since the Xatom in the cubic magnetic transition is not natural in the Trans.4 among our structure is surrounded by eight Ni atoms, the X-DOS has the four types of transitions. Therefore, Trans.4 may be not characteristics of the bcc structure, that is, the X-DOS is observed. We have to consider entropy in order to discuss composed of two large peaks. The large valley between the transitions accurately. two peaks disappears in the monoclinic structure and the occupied states generally move to the states with the lower 3.3 Density of states energy. Therefore, we can guess that the band energy for the In the previous section, it was found that curves of the monoclinic structure is lower than that of the cubic structure. differences ÁEðe=aÞ are similar for the four systems of s-m1, In the minority spin states, the X-DOS curve shifts from the s-m2, s-Nm1-m2 and s-Cm2-m1 where the Xatoms occupy higher energy region to the lower energy region beyond the the Mn sites. The similarities are also seen in the curves of s- Fermi energy, when the Xatom changes from Mn to Ni in the Nm1-n1 and s-Nm12-n1 where the Xatoms occupy the Ni order of Mn, Fe, Co and Ni. On the other hand, in the majority sites. It is natural to pay attention to Xatoms in considering spin state, the two large peaks under the Fermi energy shift to energy differences due to differences of e=a and systems, the higher energy region with increasing e=a except for the

50 50 (a) Ni2(X1/6Mn5/6)Ga (c ) Ni (X Mn )Ga -1 Mn -1 2 1/6 5/6 Mn 40 X at Mn(1) Fe 40 X at Mn(1) Fe Co Co Ni Ni

atom spin 30 atom spin 30

-1 Cub. -1 Mono.

/ aJ 20 / aJ 20 n n

10 10 States, States,

0 0 50 50 Mn Mn -1 (b) Ni2(X1/6Mn5/6)Ga -1 (d) Ni (X Mn )Ga Fe 2 1/6 5/6 Fe 40 X at Mn(1) Co 40 X at Mn(1) Co Ni Ni 30

atom spin atom spin 30 Mono.

-1 Cub. -1

/ aJ 20 / aJ 20 n n 10 10 State, State,

0 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Energy, E / aJ unit-cell-1 Energy, E / aJ unit-cell-1

Fig. 6 Local DOS of Xatoms in s-m1. Four curves distinguish the cases of X= Mn, Fe, Co and Ni in Ni 2(X1=6Mn5=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. Valence Electron Concentration and Phase Transformations of Shape Memory Alloys Ni–Mn–Ga–X209

-1 30 30 -1 (a) X=Co FC (a) X=Co FC (Ni X ) (Ni Mn )Ga FM FM 5/6 1/6 2 1/6 5/6 (Ni5/6X1/6)2(Ni1/6Mn5/6)Ga 20 20 atom spin atom spin -1 -1 / aJ / aJ n 10 n 10 States, States, 0 0

-1 30 30 -1 (b) X=Z27.5 FC (b) X=Z27.5 FM FC (Ni X ) (Ni Mn )Ga 5/6 1/6 2 1/6 5/6 (Ni5/6X1/6)2(Ni1/6Mn5/6)Ga FM 20 20 atom spin atom spin -1 -1 / aJ / aJ n

10 n 10 States, States, 0 0 30 30 -1 -1 (c) X=Ni FC (c) X=Ni FC FM FM (Ni5/6X1/6)2(Ni1/6Mn5/6)Ga (Ni5/6X1/6)2(Ni1/6Mn5/6)Ga 20 20 atom spin atom spin -1 -1 / aJ / aJ 10

n 10 n States,

States, 0 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 Energy, E / aJ unit-cell Energy, E / aJ unit-cell-1 Fig. 7 Local DOS of Xatoms in s-Nm1-n1. The DOS of X= Co, Z27.5 Fig. 8 Local DOS of Xatoms in s-Nm1-n1. The DOS of X= Co, Z27.5 and Ni in (Ni5=6X1=6)2(Ni1=6Mn5=6) Ga are shown in (a), (b) and (c), and Ni in (Ni5=6X1=6)2(Ni1=6Mn5=6) Ga are shown in (a), (b) and (c), respectively. The DOS curves for the majority spin state in FC and FM 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 phases are drawn by the solid and dotted lines, respectively. The vertical line shows the Fermi energy. line shows the Fermi energy. case X= Mn. We can roughly guess from these changes that 4. Conclusion the difference of band energy between cubic and monoclinic structures becomes larger with increasing atomic number of To investigate in more detail the effect of Xatom on the Xatom. Therefore, the ÁE increases with increasing atomic phase transformation in Ni–Mn–Ga–Xsystems, the electro- number. nic structures were calculated for six Ni2MnGa based Next, we turn our attention to the case that the Xatoms systems listed in Table 1. The total energies were also occupy the Ni sites. The X-DOS curves for s-Nm1-n1 are calculated for four phases, which are PC, FC, PM and FM shown in Figs. 7 and 8. The curves of Co, Z27.5 and Ni in FC phases. Since the total energies become lower in order of the and FM phases are compared for the majority and minority PM, FC, PM and FM phases, there is possibility that the FC spins in Figs. 7 and 8, respectively. The change of the X- and PM phases become an intermediate phase between PC DOS due to the difference of Xatom is small in the majority and FM phases. spin for both of the FC and FM phases. On the other hand, the For the six systems treated in this paper, the total energy difference of the X-DOS between the FC and FM phases differences ÁEFC-FMðe=aÞ between FC and FM phases becomes larger in the minority spin, when Xatom changes calculated by changing the Xatom in Ni–Mn–Ga–Xalloys from X= Co to Ni. The changes bring the linear increase in from Mn to Ni among transition elements. The ÁEFC-FMðe=aÞ ÁEFC-FMðe=aÞ. have a similar e=a dependence if the Xatom 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 210 K. Yamaguchi, S. Ishida and S. Asano from 7.50 to 7.625 is comparable to that of TM which is (2000) 3027–3038. 4) U. Stuhr, P. Vorderwisch and V. V. Kokorin: J. Phys. Condens. Matter. converted from the increase of ÁEFC-FMðe=aÞ. Therefore, the 12 (2000) 7541–7545. e=a dependence of TMðe=aÞ corresponds to that of 5) A. N. Vasil’ev, A. D. Bozhko, V. V. Khovailo, I. E. Dikshtein, V. G. ÁEFC-FMðe=aÞ for s-Nm1-n1 in the range e=a < 7:625 and Shavrov, V. D. Buchelnikov, M. Matsumoto, S. Suzuki, T. Takagi and those for s-Nm1-m2 and s-Nm12-n1 in the range J. Tani: Phys. Rev. B59 (1999) 1113–1120. e=a > 7:625. Thus, the TMðe=aÞ may be predicted from the 6) S. Ishida, M. Furugen and S. Asano: Int. J. Appl. Electr. Mechan. 12 ÁEðe=aÞ which is not unique against the value of e=a. The (2000) 41–48. 7) P. J. Webster, K. R. A. Ziebeck, S. L. Town and M. S. Peak: Philos. variation of ÁEðe=aÞ due to the difference of the Xatoms Mag. 49B (1984) 295–310. mainly comes from the variation of the X-DOS in Ni–Mn– 8) V. A. Chernenko, V. A. L’vov, M. Pasquale, S. Besseghini, C. Sasso Ga–Xalloys. and D. A. Polenur: Int. J. Appl. Electro. Mechan. 12 (2000) 3–8. 9) V. A. Chernenko: Scr. Mater. 40 (1999) 523–527. Acknowledgments 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. The authors wish to thank Professor Koichi Tsuchiya of Phys. 91 (2002) 8222–8224. Toyohashi University of Technology for giving information 12) K. Tsuchiya, A. Tsutsumi, H. Nakayama, S. Ishida, H. Ohtsuka and M. and significant discussions. This work was supported by a Umemoto: J. Phys. IV, in press. Grant-in-Aid (13640638) for scientific Research from the 13) K. Yamaguchi, S. Ishida and S. Asano: Mater. Trans. 43 (2002) 846– 851. Ministry of Education, Science and Culture of Japan. 14) K. Inoue: Private communication. 15) K. Andersen, O. Jepsen and D. Glotzel: Proc. Int. School of Physics REFERENCES ‘‘Enrico Fermi’’ Course 89, ed. by F. Bassami, F. Fumi and M. P. Tosi (North-Holland, Amsterdam, 1985) pp. 59–176. 1) V. A. Chernenko, C. Segui, E. Cesari, J. Pons and V. V. Kokorin: Phys. 16) V. L. Moruzzi, J. F. Janak and A. R. Williams: Calculated Electronic Rev. B57 (1998) 2659–2662. Properties of Metals, (Pergamon, New York, 1978) pp. 1–22. 2) K. Inoue, K. Enami, M. Igawa, K. Inoue, Y. Yamaguchi and K. 17) V. A. Chernenko, V. V. Kokorin, O. M. Babii and I. K. Zasimchuk: Ohoyama: Proc. Int. Conf. on Solid-Solid Phase Transformations Intermetalics 6 (1998) 29–34. (1999) pp. 1120–1123. 18) V. Khovailo: Proc. Seminar. on Shape Memory Alloys Related 3) J. Pons, V. A. Chernenko, R. Santamarta and E. Cesari: Acta Mater. 48 Technology, (Inst. Fluid Science, Tohoku University, 1999) pp. 30–33.