Ab initio calculation of Ni50−xFexTi50

Journal of Physics and Chemistry of Solids 66 (2005) 1748–1754 www.elsevier.com/locate/jpcs

Ab initio calculation of Ni50KxFexTi50 D. Go´ra *, K. Parlinski Institute of Nuclear Physics PAN, ul.Radzikowskiego 152, 31-342 Cracow, Poland Received 21 June 2004; revised 14 July 2005; accepted 9 August 2005

Abstract The structure, lattice dynamics and electronic band structure of Ni43.75Fe6.25Ti50 were obtained using ab initio calculations. The phonon dispersion relations and phonon density of states were calculated using the direct method. The stability of Ni50KxFexTi50 structure for xZ0.0, 6.25, 12.5, 25 has been investigated and shown that the orthorhombic structure is the most stable phase for xZ25. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.15.Mb; 63.20.Mk Keywords: C. Ab initio calculation; D. Phonons; D. Electronic structure

The NiTi intermetallic compound has been used for more than three decades as shape memory alloy and it has attracted a great deal of attention due to its important technological applications.  At high temperature, this material have the austenite cubic Pm3m phase, also called B2 phase. As temperature is lowered, the austenite NiTi phase undergoes a structural phase transformation via incommensurate and an intermediate R-phase, and at about 273 K it undergoes a martensitic transformation to a monoclinic P21/m structure, known as B19 0 [1–3]. The crystal structure of R-phase has been investigated by a  several researchers. They first reported the P31m space group [4,5], but later results based on the electron-diffraction and powder X-ray-diffraction methods suggest P3 symmetry [6]. The calculation indicates that the transition from austenite to R-phase is proceeded by softening of a transverse phonon mode close to the wave vector (1/3, 1/3,0) and is due to a strong electron–phonon coupling at the nesting areas of the Fermi surfaces [7]. Measurements of phonon dispersion relations in the austenite phase [6,8] found this phonon anomaly. Some first-principle studied of the martensitic phase transformation have also been carried out. For example Zhang and Guo [9] have optimized the cubic and hypothetical orthorhombic structures of the parent and martensitic phases. Probing the ground state energy of NiTi against Bain strain, which is a homogeneous tetragonal distortion, these authors * Corresponding author. Tel.: C48 12 662 8348; fax: C48 12 662 8012. E-mail address: [email protected] (D. Go´ra).

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.08.001

have found some total-energy minima. This indicates the possibility of existing of an unstable intermediate orthorhombic phase in the austenite-martensitic transformation. Also ab initio calculations performed in Ref. [3] indicated the possibility of existence of orthorhombic structure, Pmcm phase. The symmetry of this phase follows directly from the soft mode observed in austenite phase and from the group theory analysis. This orthorhombic intermediate phase of stoichiometric NiTi remains metastable with respect to the monoclinic martensitic phase. Experimentally, it is established that the orthorhombic phase of NiTi is not stable. But any alternation of electronic structure of NiTi by doping, with other transitions metals such as Fe and Cu, has a significant effect on the stability of R-phase and in consequence on the martensitic transformation. For example, it is well known that the transition temperature depends on stoichiometry and strongly decreases by alloying NiTi with small amounts of Fe [4,10]. Recently, it has been found by [11] that martensitic structure of NiTi is unstable relative to based centered orthorhombic structure (space group Pmma). However, the Pmma orthorhombic structure is not observed experimentally because monoclinic martensitic structure of NiTi is stabilized by a wide range of applied or residual internal stresses. In this paper, we present the ab initio calculations of Ni50Kx FexTi50 structure. We have calculated the phonon dispersion curves of pure NiTi and Ni43.75Fe6.25Ti50 phases using the direct method [12,13]. We also studied the influence of iron doping on the stability of cubic, orthorhombic and martensite phases. It is shown that the orthorhombic structure is the most stable phase for Fe concentration xZ25.

D. Go´ra, K. Parlinski / Journal of Physics and Chemistry of Solids 66 (2005) 1748–1754

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Table 1 Structural data obtained from the present ab initio calculations on the 1!1!1 supercell of NiTi Phase

Space group

Z

Lattice constants ˚) (A

Lattice angles

Volume V/Z

Atomic positions

Austenite

 Pm3m

1

acZ3.011

acZ908

27.287

Orthorhombic

Pmcm

2

P21/m

2

aoZ908 boZ908 goZ900 amZ908 bmZ908 gmZ82.638

27.171

Martensite

aoZ2.800 boZ4.631 coZ4.190 amZ2.877 bmZ4.686 cmZ4.099

Ni:(0.0; 0.0; 0.0) Ti:(0:5; 0:5; 0:5) Ni:(0.0, 0.7800, 0.25) Ti: (0.5, 0.3161, 0.25)

The ab initio calculations of pure and doped NiTi structures have been performed using the Vienna ab initio simulation program (VASP) [14] with the planewave basis, and the fully nonlocal Vanderbilt-type ultrasoft pseudopotentials [15]. The exchange-correlation term, as defined by Perdew and Zunger [16], and the general gradient approximation of Perdew and Wang [17] were applied. The pseudopotentials for Ni, Ti and Fe atoms represent 3d94s1, 3d34s1 and 3d74s1 electronic configurations, respectively. In the cases of stoichiometric NiTi, we have used primitive cubic, orthorhombic and monoclinic unit cells with 2, 4 and 4 atoms, respectively. The monoclinic lattice vectors (am, bm, cm) are oriented with respect to the cubic lattices vectors (ac, bc, cc) as am xac , bm xbc Kcc and cm xbc C cc . Through the phase transition the orthorhombic unit cell (ao, bo, co) retains the same orientation as the monoclinic one, i.e. ao xam , bo x bm and co xcm , but goZ908, and gm!908. The calculated unit cell parameters and atomic positions are shown in Table 1. Inpffiffithe case iron NiTi, we have used 2!2!2, ffi p ffiffiffi of p ffiffiffi doped pffiffiffi 2 ! 2 ! 2, 2 ! 2 ! 2 supercells for cubic, orthorhombic and monoclinic phases, respectively. The schematic picture of studied supercells and relation between supercell’s and unit cell basic vectors are presented in Fig. 1. These supercells contained 16 atoms and possessed approximately the same shape and volume. The Brillouin zone integration was performed with 6!6!6 k wave vector mesh for cubic phase, and 6!8!6 for orthorhombic and monoclinic phases. First, the ground-state properties of the austenite Ni50K xFexTi50 structure for iron concentration xZ0.0, 6.25, 12.5, 25 were obtained by minimizing of the total energy with respect to the unit cell volumes. For that we have build a 2! 2!2 supercell of NiFeTi consisting of 16 atoms with periodic boundary conditions and having the symmetry of the space  group Pm3m. The resulting lattice constants and unit cell volumes for different concentrations x are collected in Table 2. For pure cubic NiTi the calculated lattice constant is aZ ˚ and it agrees very well with experimental 3.014 A ˚ 3.011 A [18]. For iron doped NiTi, we observe about 1% of decrease of lattice constants and consequently of supercell volume. This also agrees with experimental measurements [19]. Note that during the system relaxation, the Ti atoms decrease slightly their distances to Fe:(0.5,0.5,0.5) atoms. The Ti atoms shift

27.401

Ni:(0:9256; 0:7832; 0:0) Ti:(0:5374; 0:3242; 0:0)

from (0.25,0.25,0.25) to (0.2522,0.2522,0.2522) for concentration xZ6.25. The phonon frequencies were determined by the direct method using the PHONON code [20] for the doped NiTi with iron concentration xZ6.25. For that the Hellmann–Feyman forces were computed for positive displacements with the ˚ . The force constants were used to amplitude of 0.03 A construct the dynamical matrix, to diagonalize it and to find the phonon frequencies. The phonon dispersion curves are shown in Fig. 3. The path in reciprocal space along which the phonon branches are calculated is presented in Fig. 2. For pure NiTi structure, the direct method provides exact phonon frequencies at G:(0,0,0), (X):(1/2,0,0), (M):(1/2, 1/2, 0), (R):(1/2, 1/2, 1/2), (M):(1/2, 0, 1/2). For doped NiTi structures, the number of high-symmetry points is the same, but the volume of the Brillouin zone reduces twice due to two times larger size of doped NiTi cubic

Fig. 1. (a) Orientation relationship of the orthorhombic (index iZo) and monoclinic (index iZm) supercell’s basic vectors (ais, bis, cis) with respect to the orthorhombic and monoclinic unit cell vectors (ai, bi, ci), respectively. In the case of cubic structure, supercell’s basic vectors are acsZ2ac, bcsZ2bc, ccsZ2cc. Angles a, b, g are lattice angles of studied supercells, see Tables 1, 3 and 4 for more details. Schematic plot of the supercells of considered phases: (b) austenite phase; (c) orthorhombic phase; (d) monoclinic phase.

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D. Go´ra, K. Parlinski / Journal of Physics and Chemistry of Solids 66 (2005) 1748–1754

Table 2 Structural data obtained from present the ab initio calculations of 2!2!2 supercell of Ni50KxFexTi50; Iron atomic positions listed in column five indicate only initial atomic positions expressed in fractional coordinates with respect to the supercell x

˚) Lattice constants (A

Lattice angles

˚ 3) Volume V (A

Atomic positions

0.00

acZ6.022

acZ908

218.38

6.25

acZ6.007

acZ908

216.69

12.5

acZ5.991

acZ908

215.04

25.0

acZ5.960

acZ908

211.75

Ni:(0:0; 0:0; 0:0) Ti:(0:25; 0:25; 0:25) Ni:(0:0; 0:0; 0:0) Fe:(0:5; 0:5; 0:5) Ti:(0:25; 0:25; 0:25) Fe:(0:5; 0:5; 0:5) Fe:(0:5; 0:5; 0:0) Fe:(0:5; 0:5; 0:0) Fe:(0:0; 0:5; 0:5) Fe:(0:5; 0:0; 0:5) Fe:(0:5; 0:5; 0:5)

interaction of iron with Ni and Ti atoms. The curves of Fig. 3a and b are calculated along the same high-symmetry lines of reciprocal space. In brackets, we quote the corresponding high-symmetry point of the austenite Brillouin zone, from which a given high symmetry point of lowsymmetry phases arises. Such a presentation of the dispersion curves allows comparison of phonons in different reciprocal lattices. Also a soft mode at G(M) point is seen, but its frequency is higher by about 0.5 THz then in NiTi. Note that the observed soft mode at (M) in Fig. 3a now exists in all G points of iron doped NiTi structure. Also a soft mode along (G)–(R) line for pure NiTi is now seen along G(G)–R and and R–G(R) directions of iron doped NiTi, see point K in Fig. 3b.

(a)

10

FREQUENCY (THZ)

supercell. In this case, the wave numbers are G:(0,0,0), X:(1/ 4,0,0), M:(1/4, 1/4, 0), R:(1/4, 1/4, 1/4), M:(1/4, 0, 1/4). From Fig. 2, it is also seen that all exact points of austenite Brillouin zone are now G points of doped NiTi Brillouin zone. In brackets, we quote the corresponding high-symmetry points of the austenite Brillouin zone, while without brackets we denote the high symmetry points of low-symmetry phase. In the case of NiTi structure, the phonon branches spread up to 7.4 THz and show considerable dispersions, Fig. 3a. There is a doubly degenerate soft mode at (M): kZ(1/2, 1/2, 0) reciprocal lattice wave vector [21]. The volume in the reciprocal space, where the soft mode branch is imaginary, is quite large and surrounds the reciprocal-lattice point (M). It is so wide that the imaginary mode seen close to kZ(1/3, 1/3, 1/ 3) wave vector labeled by K in Fig. 3a along (G)–(R) line, still belongs to the same domain of the soft mode at (M) point. The soft mode at (M) point generates unstable intermediate orthorhombic phase which drives the austenite to a stable martensitic monoclinic phase [3]. In the case of iron doped NiTi structure, Fig. 3b, the phonon dispersion relations are much more complex due to the

8

(Γ)

(X)

(M)

(Γ)

(R)

6 4 2 0 K

–2 Γ(Γ) X Γ(X)

FREQUENCY (THZ)

(b)

(M)

M

Γ(M)

Γ(Γ)

Γ(R)

R

Γ(M)

10 8 6 4 2 0

K

K

–2 WAVE VECTOR Fig. 2. The high symmetry points of austenite structure and NiFeTi. The path in reciprocal space along which the phonon branches are calculated for pure NiTi and Ni43.75Fe6.25Ti50 is marked by solid thick line.

Fig. 3. (a) Calculated phonon dispersion relations of NiTi for the austenite phase. (b) Calculated phonon dispersion relations of Ni43.75Fe6.25Ti50. In (a) and (b) the path in reciprocal space, along the phonon branches are the same.

FORCE CONSTANTS (N/m)

D. Go´ra, K. Parlinski / Journal of Physics and Chemistry of Solids 66 (2005) 1748–1754 100 Fe 50 Ni

Ni

0 Ni –50

Ni

Ni

Ti

–100 Ti 0

1

2

3

4

5

6

DISTANCE (Angstroms) Fig. 4. Force constants as a function of distance for the Fe atom, Ni43.75Fe6.25Ti50.

Assuming that the value of the soft mode indicates the depth of the atomic potential, one may expect that iron decreases the transition temperature from the austenite to martensite phase. More precisely the value of negative soft mode at G(M) changes from K2.91 THz for iron doped NiTi to K2.45 THz for pure NiTi. It means that the depth of atomic potential for iron doped NiTi is also about 0.5 THz smaller then for pure NiTi. Therefore, the transition temperature from the austenite to martensite phase for doped NiTi should be smaller than

transition temperature for pure NiTi. This conclusion agrees with Ref. [10], where it was shown that by alloying NiTi with a small amount of Fe the transition temperature to martensitic phase strongly decreases, and simultaneously it leads to an increase of the temperature range, where the intermediate incommensurate (R-phase) phase exists [22]. For example, the transition temperature to the martensitic phase changes from 273 to 24 K going from xZ0 to 3.2 [4]. Applying the procedure of direct method [12] parameters of the force constants are calculated for iron doped NiTi. Knowledge of force constants allows to estimate the range of potential interaction between atoms. As an example, in Fig. 4, we show force constants as a function of distance for the Fe atom of doped NiTi. It is easily seen from this figure that coupling between Fe atom to Ti atoms are stronger than ˚ , which is between Fe to Ni atoms. For example, at 2.58 A shortest distance from the Fe atom to the nearest-neighbors Ti atoms, the element of force constant equals about 117 N/m. At larger distances, the magnitude of force constants between ˚ to 8 N/m at Fe–Ni diminishes from 29 N/m at distance 3.00 A ˚ distance 5 A. Since the largest elements of force constants decide about the general behavior of phonons, so we expect that Fe–Ti coupling will influence phonon spectra and density of states. Fig. 5 shows the total and partial phonon density of states for Ni, Ti and Fe atoms. There are three wide bands. The first one

0.5

0.4

DENSITY OF STATES (1/THz)

DENSITY OF STATES (1/THz)

0.5

Ni

0.3 0.2 0.1 0

0.4

Ti

0.3 0.2 0.1 0

–2

0

2

4

6

8

–2

FREQUENCY (THz)

0

2

4

6

8

FREQUENCY (THz) 0.5

DENSITY OF STATES (1/THz)

0.1

DENSITY OF STATES (1/THz)

1751

Fe

0.05

0

0.4

Total

0.3 0.2 0.1 0

–2

0

2

4

6

FREQUENCY (THz)

8

–2

0

2

4

6

8

FREQUENCY (THz)

Fig. 5. Total and partial phonon density of states for Ni, Ti and Fe atoms, Ni43.75Fe6.25Ti50 (solid line) and pure NiTi (dashed line).

D. Go´ra, K. Parlinski / Journal of Physics and Chemistry of Solids 66 (2005) 1748–1754

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Fig. 6. Calculated electronic band structure of (a) austenite, and (b) Ni43.75Fe6.25Ti50.

(K2–3 THz), corresponding to the soft mode, is mainly reflecting the vibrations of Ni atoms; the second (3–6 THz) originates from Ti and Fe atomic vibrations, and the last (7–10 THz) is occupied only by Ti atoms. The Ni atoms vibrate

preferentially at lower frequencies although the mass of Ni is only slightly larger than that Fe. On Fig. 5, the partial phonon density of states for Ni and Ti atoms of pure NiTi is also shown (dashed lines). It is seen, that iron doping only slightly affect the partial phonon density for Ni atoms. On the other hand, the partial phonon density of states for Ti atoms changes significantly. In this case doped iron causes splitting of Ti spectrum (dashed line) on two bands, see vibriational states with frequencies between 3–6 THz and the 7–10 THz for iron doped NiTi (solid line). This effect is a direct consequence of stronger Ti–Fe coupling, and weaker coupling of Fe to Ni atoms, as shown in Fig. 4. Optimization of the ground-state energy requires to calculate the energy of electronic levels. Fig. 6 shows the calculated electronic band structure for the austenite NiTi and iron doped NiTi. The nested Fermi surface region is near the (M)KG line and is labeled by N in Fig. 6. For the austenic phase very similar results were obtained by selfconsistent augmented plane wave calculations fitted to a tight-binding Hamiltonian [7,23]. The band structure of iron doped NiTi is much more complicated due to the interaction of iron with Ti and Ni atoms, but again we see electron nesting pocket along G(M)–G(G) line close to G(M) point. Finally, we studied the influence of iron doping on the stability of cubic, orthorhombic and martensitic phases. The NiTi structure doped with one, two and four iron atoms are accompanied by the lattice distortions. They involve variations of the unit cell parameters, Tables 2–4. Notice that volume of the cell decreases with increasing of iron concentration and ˚ 3 at concentrations reaches the same value of about 211.8 A 25% for all studied structures. The individual lattice constants change also and achieve the same values; a x3:0 and b xc x 4:2 AA for xZ25. Since simultaneously the lattice angles became aZbZgZ908, so the orthorhombic structure is

Table 3 Structural data obtained from the present ab initio calculations of the orthorhombic Ni50KxFexTi50 structure x

Lattice constants ˚) (A

Lattice angles

Lattice constants ˚) (A

Lattice angles

˚ 3) Volume V (A

Atomic positions

6.25

aoZ2:8004

aoZ908

aosZ5.6008

aZ85.198

216.18

Fe:(0.5, 0.2689, 0. 5189)

12.5

boZ4.5816 coZ4.2123 aoZ2.8645

boZ908 goZ908 aoZ908

bosZ6.2237 cosZ6.2237 aosZ5.7290

bZ908 gZ908 aZ88.128

214.93

boZ4.4122

boZ908

bosZ5.9518

bZ908

Fe:(0.5, 0.2515, 0. 5142) Fe:(0.5, 0.7485, 0. 4858)

coZ4.2642 aoZ3.0259

goZ908 aoZ908

cosZ6.3068 aosZ6.0519

gZ908 aZ89.998

Fe:(0.5, 0.2665, 0. 5165) boZ4.1830 boZ908 bosZ5.9155 bZ908 Fe:(0.5, 0.7335, 0. 4835) coZ4.1827 goZ908 cosZ5.9155 gZ908 Fe:(0.5, 0.2335, 0. 9835) Fe:(0.5, 0.7665, 0. 0165) pffiffiffi pffiffiffi Lattice constants defined as aoZaos/2, boZ(bosCcos)/2, coZ(cosCbos)/2, where aos, bos, cos are the calculated lattice vectors of 2 ! 2 ! 2 supercell for orthorhombic Ni50KxFexTi50 structure; x is the iron concentration. Optimized atomic positions of iron expressed in fractional coordinates with respect to the supercell.

25.0

211.77

D. Go´ra, K. Parlinski / Journal of Physics and Chemistry of Solids 66 (2005) 1748–1754

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Table 4 pffiffiffi pffiffiffi Structural data obtained from the present ab initio calculations of 2 ! 2 ! 2 monoclinic Ni50KxFexTi50 supercell; amZams/2, bmZ(bmsCcms)/2, cmZ(cmsCbms) /2, where ams, bms, cms are the lattice vectors obtained from minimization of monoclinic supercell for different iron concentration x x

Lattice constants ˚) (A

Lattice angles

Lattice constants ˚) (A

Lattice angles

˚ 3) Volume V (A

Atomic positions

6.25

amZ2.8230

aoZ908

amsZ5.6477

aZ83.908

216.84

Fe:(0.5151,0.8485, 0.3485)

12.5

bmZ4.6319 cmZ4.1630 amZ2.8789

boZ908 goZ84.598 aoZ908

bmsZ6.2277 cmsZ6.2277 amsZ5.7577

bZ94:018 gZ94:018 aZ82.468

217.51

bmZ4.6782

boZ908

bmsZ6.2206

bZ82.498

Fe:(0.4706, 0.8449, 0.3449) Fe:(0.9706, 0.8449, 0.3449)

cmZ4.1000 amZ3.0269

goZ79.988 aoZ908

cmsZ6.2206 amsZ6.0537

gZ82.498 aZ908

211.81

bmZ4.1825

boZ908

bmsZ5.9151

bZ908

cmZ4.1827

goZ908

cmsZ5.9151

gZ908

Fe:(0.5000, 0.8750, 0.3750) Fe:(0.5000, 0. 87500, 0.8750) Fe:(0.5000, 0.3750, 0.3750) Fe:(0.5000, 0.3751, 0.8751)

25.0

As before in Table 3, optimized atomic positions of iron expressed in fractional coordinates with respect to the supercell.

electronic band structure calculations show nested Fermi region, which is responsible for the premartensitic transformation of the cubic iron doped NiTi structure. The influence of iron on the stability of cubic, orthorhombic and martensite phases from small (6%) to large (25%) iron concentrations was studied. Typical experiments are made only for iron doping of about 3%. We showed that the orthorhombic structure is the most stable phase for iron concentration of xZ25. The possibility of the existence of a metastable orthorhombic phase instead of martensitic structure agrees with the last theoretical calculations [9,11], where the authors suggest instability of martensitic structure with respect to orthorhombic structure. The authors would like to thank J. Lal.zewski, and P.T. Jochym for fruitful discussions. The calculations were performed at ACK-Cyfronet, grant SGF2800/IFJ/102/2001.

–10 –20

∆Ex (meV)

formed, even if the optimization started from the monoclinic phase. Fig. 7 shows the DEx of the total ground state energy per two atoms as a function of concentration x. The DExZEi(x)KEc(x) is defined as the difference of ground state energy of a given phase Ei(x) minus the ground state energy of the cubic phase Ec(x), with the same concentration x. The comparison is done between fully converged results. The total energy was converged numerically to less than 10K5 meV per atom with respect to electronic, ionic and unit cell degrees of freedom. It is worthwhile to notice that the comparison of the ground state energies DEx can be made only for systems of the same concentrations of iron. The ground state energies of different concentrations cannot be compared due to possible different definitions of the pseudopotentials. The DEx between orthorhombic and cubic phases is smaller than the DEx between monoclinic and cubic phases for all x, except xZ25, where the orthorhombic structure has the lowest energy. Additional optimization test with different position of two iron atoms leads to a small deviations of ground state energy by about 5 meV per two atoms. These deviations are smaller than changes of ground state energies corresponding to difference between monoclinic and orthorhombic structures. Also optimization test with different four iron positions starting from monoclinic phase leads to the orthorhombic symmetry with only slightly different total ground state energy. We must stress that we always obtained the orthorhombic structure from the optimization of the monoclinic structure with xZ25. It seems that iron doping at the level of xZ25 stabilizes orthorhombic structure. In summary, the ab initio calculations of pure NiTi and Ni50KxFexTi50 structure are presented. Lattices vectors agree quite well with the measured results [18,19]. The phonon dispersion curves for pure NiTi and first time for Ni43.75Fe6.25Ti50 structures, using direct method, are obtained. Our

–30 –40 –50 –60

Eo(x)-Ec(x)

–70

Em(x)-Ec(x)

–80 –90 0

5

10

15

20

25

x Fig. 7. The DEx of the total ground state energy per two atoms of a given phase with respect to the cubic Ni50KxFexTi50 ground state energy. Note, that for xZ 25, the Em(25) is not the total ground state energy of monoclinic structure but the ground state energy of orthorhombic phase obtained from optimization of monoclinic structure, see text for more details.

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