JOURNAL OF APPLIED PHYSICS
`
`VOLUME 87, NUMBER 9
`
`1 MAY 2000
`
`The effect of field cooling and field orientation on the martensitic phase
`transformation in a Ni2MnGa single crystal
`S.-Y. Chu, A. Cramb, M. De Graef, D. Laughlin, and M. E. McHenrya)
`Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh,
`Pennsylvania 15213
`The temperature and field dependence of the magnetization in a Ni2MnGa single crystal was
`measured using a magnetometer with an applied field oriented in the @001# and @011# directions,
`respectively, of the parent cubic phase. It was found that the magnetic field magnitude and direction
`could be used to determine the magnetization of the sample during a thermal transformation from
`the austenitic phase to the martensite phase. This is explained in terms of a magnetic field induced
`growth of the twin variant having a favorable orientation to the external magnetic field. A model to
`interpret the magnetic response in terms of aligned twin variants in the shape memory material is
`discussed. © 2000 American Institute of Physics. @S0021-8979~00!93208-0#
`
`I. INTRODUCTION
`
`A new mechanism for magnetically driven actuation has
`been suggested in Ni2MnGa materials in terms of the choice
`of twin variants of a transformed martensitic phase. It has
`been previously reported that the twin variants of the mar-
`tensite can be reoriented or aligned by an external magnetic
`field or stress.1–4 The largest magnetostrictive strains in the
`tetragonal martensitic phase are predicted when a single twin
`variant exists with its c axis normal to the direction of the
`external magnetic field. However, typical experimental data
`shows only a small fraction of the lattice constant change
`(Dc/c526.56%) due to the strain accommodation by dif-
`ferent twin variant orientations.
`It is well known that a martensitic phase transformation
`can be accompanied by the relaxation of strain associated
`with the formation of twin variants that choose configura-
`tions that minimize the strain energy. Each variant of the
`martensite has a unique value of its projected magnetization
`related to the angle between its c axis and the direction of the
`applied field. When an external magnetic field is applied, the
`total energy in the system can minimized by either nucleat-
`ing favorable variants or increasing the volume fraction of
`the favorable variants present, through detwinning and/or
`twin boundary motion.
`In comparison with the isotropic parent cubic phase, the
`tetragonal martensitic phase has a strong magnetocrystalline
`anisotropy.5 This magnetic anisotropy energy density may
`play an important role in the nucleation of particular variants
`during a cooling procedure. In this work we explore this
`issue through the study of the field, field history, and tem-
`perature dependence of the magnetization near the martensi-
`tic phase transition temperature. When more than one variant
`exists in the sample determination of the anisotropy con-
`stants is more difficult, requiring more involved data analy-
`sis. We illustrate the use of the so-called ‘‘singular point
`detection’’ ~SPD! technique to estimate the anisotropy con-
`
`a!Electronic mail: mm7g@andrew.cmu.edu
`
`stants and the volume fraction of the twin variants with dif-
`ferent orientations in Ni2MnGa martensitic phase.
`
`II. EXPERIMENTAL PROCEDURES
`
`Crystals of Ni2MnGa were grown by melting the pure
`elements in an evacuated flat-bottom quartz tube back-filled
`with a low partial pressure of Ar. The single crystal (m
`523.4 mg) employed here was cut and polished into a disk
`shape. Both circular faces were aligned parallel to the ~100!
`plane of the parent cubic L2 1 room temperature structure.
`The ratio of diameter to thickness of the crystal was about
`10. Clear evidence of the cubic to tetragonal martensitic
`phase transformation is seen in thermal hysteresis of the
`fixed field magnetization at a temperature T m;190 K as
`shown in Fig. 1.
`device
`quantum interference
`A superconducting
`~SQUID! magnetometer ~Quantum Design, MPMSR2! was
`used to measure the dc magnetization of the sample in fields
`up to 5 T. The applied magnetic field, which was parallel to
`the both circular faces, was oriented along the @001# or @011#
`directions, respectively, of the sample at room temperature.
`We refer to these field orientations as the @001# orientation
`
`FIG. 1. Temperature hysteresis of the magnetization in a single crystal of
`Ni2MnGa in a fixed external field of 1.1 kOe oriented along the @110# axis of
`the cubic lattice.
`
`0021-8979/2000/87(9)/5777/3/$17.00
`
`5777
`
`© 2000 American Institute of Physics
`
`TDK Corporation Exhibit 1008 Page 1
`
`

`
`5778
`
`J. Appl. Phys., Vol. 87, No. 9, 1 May 2000
`
`Chu et al.
`
`FIG. 2. Magnetization curves, M (H), of the single crystal Ni2MnGa in the
`martensitic phase.
`
`FIG. 3. Field dependence of the first and the second derivatives of the
`magnetization M (H), with respect to H, dM /dH, and d 2M /dH 2.
`
`and @011# orientation. At low temperature T,T m , these ar-
`rangement would lead to different field orientations with the
`tetragonal crystal lattice for each individual twin variant in
`the sample. For our convention for labeling the different ori-
`entations we always let the @001# direction coincide with the
`short axis ~i.e., c’5.44 Å! and the @100# or @010# with either
`of the long axes ~i.e., a5b.5.90 Å!.
`To prepare the initial state of the transformed martensite
`and its twin variant distribution, two experimental processes
`were employed:
`~a! for a so-called zero field cooling ~ZFC! measure-
`ment, the sample was first cooled from its Curie temperature,
`T 0;380 K, to 170 K in zero field (H,1 Oe). Subsequently,
`the sample was saturated in an applied field of 5 T. Its dc
`magnetic moment was then measured as a function of de-
`creasing field;
`~b! for a so-called field cooling ~FC! process, after de-
`creasing the temperature from 380 to 350 K in zero field, the
`sample was slowly cooled to 170 K in a field of 5 T. After
`cooling the moment was measured as the applied field was
`decreased.
`
`III. RESULTS AND DISCUSSIONS
`
`ZFC and FC magnetization curves for the @001# and
`@011# field orientations for the martensitic Ni2MnGa are il-
`lustrated in Fig. 2. The amplitude of the field shown here is
`the internal magnetic field, which has been corrected for de-
`magnetization effects based on the sample’s geometry. The
`is 650
`the sample, M s ,
`saturation magnetization of
`emu cm23 for both, in fields of H;1 T. However, the shapes
`of these magnetization curves are obviously different. Sev-
`eral observations can be made:
`~1! It can be seen that the FC curves always lie above the
`ZFC curves regardless of the field arrangement.
`~2! The difference between the FC and ZFC curve for the
`field oriented along the @001# direction is much larger
`than that for the @011# orientation.
`
`The difference in the anisotropy energy density, required
`saturating the sample in different field orientations, cannot be
`interpreted in terms of an anisotropy change associated with
`the sample’s shape. The largest field induced strain is
`26.56%. Even if this strain was along one direction, the
`elliplicity of the sample caused by the strain would only
`change the demagnetization factor, D, by about 0.02%. Cor-
`responding to this small change of the D factor, the energy
`density difference from that calculated using the original
`value of D is less than 0.5%. Our interpretation to the ex-
`perimental result is that the crystal anisotropy energy of the
`martensite contributes to the field orientation dependence of
`the magnetization.
`It has been indicated by a Ullakko et al.’s experiments
`that an external magnetic field can be used to align some of
`the twin variants in single crystal of Ni2MnGa. 1,2 As com-
`pared with zero field cooling procedure, the field cooling
`induced a larger negative strain in the martensite phase with
`decreasing temperature. This implies that the favorable vari-
`ants formed during the martensitic transformation are those
`for which the c axis is parallel to the direction of the applied
`magnetic field. The difference of the FC and the ZFC curves
`with @001# field orientation illustrated in the Fig. 2 supports
`the claim that the c axis of the tetragonal phase is the easy
`axis of magnetization.
`It is well known that the magnetization curve, M (H),
`for a single crystal with its field, H, oriented in a hard direc-
`tion has a singularity at the saturation field, i.e., where the
`applied field, H5H A , the anisotropy field. The singularity is
`a consequence of
`the intersection of
`the two different
`branches of the magnetization curve. For H,H A , M de-
`pends on H, while as H.H A , M (5M s) is a constant. The
`formula:
`~1!
`H A5~2K 114K 2!/M s
`expresses the anisotropy field in terms of the first and second
`order anisotropy constants, K 1 and K 2 , respectively, of the
`single crystal. When a magnetic field is a applied to a poly-
`crystalline magnet, or to one with several twin variants, a
`
`TDK Corporation Exhibit 1008 Page 2
`
`

`
`J. Appl. Phys., Vol. 87, No. 9, 1 May 2000
`
`Chu et al.
`
`5779
`
`FIG. 4. Calculated magnetization curves ~solid line! for fitting the experi-
`mental data to yield the crystal anisotropy constants and the volume frac-
`tions of the different twin variants in the sample. Inset shows the calculated
`magnetization curve for a single tetragonal variant.
`
`smoother M (H) behavior is observed. However, due to the
`contribution of the twinned regions oriented with their hard
`directions nearly parallel to H, the singularity in M (H) can
`still be observed but at fields less than the saturating field.
`The sharpness of the change of M (H) at H A depends on the
`distribution of hard axes, and the ratio of the first to second
`order anisotropy constants.6,7 Because of the static equilib-
`rium achieved by contacting twinned regions along their
`edges, the most probable direction of the c axes of the vari-
`ants is along one of the three primary orthogonal axes of the
`cubic lattice of the untransformed single crystal. The distri-
`bution of the three twin variants should reflect equal prob-
`ability of the three orthogonal orientations in the absence of
`a field. Thus the volume fraction of each of the variants
`should equal one third for a single crystal that undergoes
`zero field cooling. However, application of a field will result
`in a distribution of the variants that minimizes the total en-
`ergy, which now includes anisotropy and Zeeman energy
`terms.
`We assume for our analysis that the transformed marten-
`site has a small deformation ~in comparison with the large
`strain in each of the individual variants! and that each variant
`has a unique projected magnetization. A singular point on
`the magnetization curve, M (H), of our sample can be de-
`tected by observing successive derivatives of M with respect
`to the internal field, H. The first and the second derivatives,
`dM /dH and d 2M /dH 2, shown in Fig. 3 respectively, illus-
`trate such a singular point detection. The peak position in
`dM 2/dH 2 as a function of H corresponds to the anisotropy
`field, H A , of the martensite phase. A theoretical prediction is
`that the amplitude of the peak in the second derivative,
`d 2M /dH 2, as a function of H, should be proportional to the
`volume fraction of crystallites oriented with their easy axes
`perpendicular to H. Our results indicate that the field cooling
`process produces an increased volume fraction of variants
`with c axes parallel to the direction of the field.
`
`FIG. 5. Schematic of the distribution of twin variants with different orien-
`tations in the martensitic Ni2MnGa material.
`
`Considering Zeeman ~field! and anisotropy energy den-
`sities, for a variant with it’s c axis perpendicular to the field,
`the magnetization curve, M (H) can be described by
`~2!
`HM s5@~2K 114K 2~ M /M s!2#~ M /M s!
`for H<H A . The best fits to the experimental M (H) curve,
`using expression ~2! are shown in Fig. 4, for which the first
`and second order anisotropy constants are K 152.03106 and
`K 250.503106 erg cm23, respectively. The derived value of
`H A59.5 kOe is in good agreement with the observation of
`the SPD technique shown in Fig. 3. The volume fraction of
`each variant in the sample in initial state (H;H A) has been
`schematically illustrated in Fig. 5.
`
`ACKNOWLEDGMENTS
`
`This work is supported by the Air Force Office of Sci-
`entific Research, Air Force Materiel Command, USAF, un-
`der Grant No. F49620-96-1-0454. The authors acknowledge
`the efforts of C. Kline in orienting the single crystals.
`Note added in proof: After submission of this paper it came
`to our attention that R. Tickle and R. D. James reported
`K n52.453106 erg cm23 for a single variant of Ni2MnGa
`~Ref. 8!.
`
`1 K. Ullakko, J. K. Huang, C. Kantner, R. C. O’Handley, and V. V. Koko-
`rin, Appl. Phys. Lett. 69, 1966 ~1996!.
`2 K. Ullakko, J. K. Huang, V. V. Kokorin, and R. C. O’Handley, Scr. Mater.
`36, 1133 ~1997!.
`3 R. Tickle, R. D. James, T. Shield, P. Schumacher, M. Wutting, and V. V.
`Kokorin ~unpublished!.
`4 A. DeSimone and R. D. James, J. Appl. Phys. 81, 5706 ~1997!.
`5 P. J. Webster, K. R. A. Ziebeck, S. L. Town, and M. S. Peak, Philos. Mag.
`B 49, 295 ~1984!.
`6 G. Asti, F. Bolzoni, and L. Pareti, J. Magn. Magn. Mater. 83, 270 ~1990!.
`7 G. Asti, R. Cabassi, F. Bolzoni, S. Wirth, D. Eckert, P. A. P. Wendhausen,
`and K. H. Muller, J. Appl. Phys. 76, 6268 ~1994!.
`8 R. Tickle and R. D. James, J. Magn. Magn. Mater. 193, 627~1999!.
`
`TDK Corporation Exhibit 1008 Page 3