Strain-induced dimensionality crossover of precursor modulations in
Ni2MnGa
1. Experimental Methods
A near-stoichiometric Ni2MnGa single crystal was prepared using a floating-zone method.
The phase transition properties of the single crystal were examined by using a
superconducting-quantum-interference device (SQUID) in the temperature interval of 25
- 300 K. The thermo-magnetization curves were recorded at a magnetic field of 100 Oe,
with a temperature heating and cooling rate of 3 K min−1. The premartensitic transition
and MT temperatures were detected to be ~230 K and ~175 K, respectively.
Synchrotron high-energy x-ray diffraction (HEXRD) was used to investigate thermo and
uniaxial stress induced martensitic transition in Ni2MnGa single crystals. The HEXRD
experiments (115 keV) were performed at the beamline 11-ID-C of Advanced Photon
Source (APS), Argonne National Laboratory (ANL). Synchrotron angle-dispersive x-ray
diffraction (ADXRD) was utilized to investigate the hydrostatic pressure induced
martensitic transition in the sample. A symmetrical diamond anvil cell (DAC) was used to
generate static quasi-hydrostatic pressure. The ADXRD experiments were performed at
the beamline 16-BM-D of the High Pressure Collaborative Access Team (HPCAT), ANL.
The used photon energy was of 30 keV (wavelength 0.376 Å). CeO2 powders were used
as standard sample in the synchrotron-based x-ray diffraction experiments for the
calibration of detector-to-sample distance and tile angles of the detector to the incident
x-ray beam. The software package FIT2D was used for diffraction data analysis. The
detailed procedures for in-situ x-ray diffraction experiments are shown specifically in the
following aspects:
(1) Thermo induced martensitic transition. The incident x-ray beam was cut to 0.8 × 0.8
1
mm2 by sample slits. A single crystal with dimensions of 0.5 × 0.5 × 0.5 mm3 was
studied here, which was mounted on a goniometer head with alignment of <010>P
parallel to incident beam and <001>P along vertical axis. P donates the parent. An
Oxford Cryostream System (80 to 500 K) was used for temperature control from 160
K to 300 K. Two-dimensional (2D) scattering patterns were collected by a Mar345
image plate detector. At each temperature, the sample was rotated ±10° around
<001>P direction while exposing to cover a large reciprocal space.
(2) Uniaxial stress induced martensitic transition. The used beam size was 0.3 × 0.3 mm2.
A single crystal with dimensions of 2 × 2 × 4 mm3 was studied here. Uniaxial stress
was conducted at room temperature with a strain-control mold along <001>P direction
on the single crystal. The incident beam was applied parallel to <010>P. The sample
was rotated ±10° around the loading direction (LD) while exposing at each stress state.
A PerkinElmer α-Si flat-panel large-area detector was used to collect 2D diffraction
patterns. The data collection rate was fast, up to 15 frames per second.
(3) Hydrostatic pressure induced martensitic transition. The incident x-ray beam was
focused to 5 μm (horizontal) × 15 μm (vertical) by KB-type mirrors. A symmetrical
diamond anvil cell (DAC) was used to generate quasi-hydrostatic pressure to 19 GPa.
The Ni2MnGa single crystal used at aspect (1) was crushed into fine powder. One
compressed powder pellet with size of ~ 50 × 50 × 20 μm3 was loaded into a
pre-indented T301 stainless steel gasket hole of ~ 150 μm diameters. Two thin ruby
chips were loaded around the sample as a pressure calibrant and silicone oil was used
as the pressure transmitting medium. The pressure was determined from the ruby
fluorescence spectra. The specimen was rotated around the horizontal axial over an
angle of ±5° around vertical while exposing to cover a larger interval of reciprocal
space. 2D diffraction patterns were collected by a Mar345 image plate detector.
2. Phonon Dispersion Calculations
2
First-principles calculations in the present work are performed by the VASP code [1]
together with the ion-electron interaction described by the projector augmented wave
(PAW) method [2] and the exchange-correction functional depicted by the generalized
gradient
approximation
of
Perdew-Burke-Ernzerhof
(GGA-PBE)
[3].
Phonon
calculations are performed by using the supercell method [4] with the force constants
calculated directly by VASP [1] and the phonon properties predicted by Yphon code [5].
Here, the 96- and 64-atom supercells are employed for Ni2MnGa with lattice parameters
ratios c/a  1 and > 1, respectively. The following deformation matrices (D96 and D64) are
used to generate the supercell lattice vectors matrices R via R  DR ,
D96
1.5  1.5 0
  1
1
0
 0
0
2
and
D64
1
1
1

  1  1 1  ,
 1  1  1
(1)
where R is the original lattice vectors matrix. In an effort to predict well the imaginary
phonon modes in Ni2MnGa for the cases of c/a  1, the D96 matrix is selected in order to
generate a supercell possessing a longer length along the [110] direction [6]. The
recommended electronic configurations by VASP are adopted for each element in the
present work, i.e., 13 electrons used for the valence for Ga (3d104s24p1), 7 for Mn
(3d54s2), and 10 for Ni (3d84s2). In VASP calculations, the 4×4×4 and 3×4×3 k-points
meshes (for 64- and 96-atom supercells, respectively), as well as a 350 eV energy cutoff
of the wave function are employed. The energy convergence criterion of the electronic
self-consistency is chosen as 10-5 eV per supercell for all the calculations. The
reciprocal-space energy integration is performed by the Methfessel-Paxton technique [7]
for structure relaxations and the calculations of forces acting on the atoms in the
perturbed supercells. Due to the ferromagnetic natures of Mn and Ni, all the calculations
are performed within the spin-polarized approximation. In addition, the theoretical
equilibrium volume for each Ni2MnGa structure is used in phonon calculations. For more
details of phonon calculations using VASP and Yphon codes, see e.g. [5, 8-12].
3
3. Reconstruction of three-dimensional reciprocal space
Ewald construction gives a geometrical description of diffraction that satisfies Bragg's
Law. The 2D scattering signal collected by the image plate detector is a plane projection
of reciprocal space which happens to fall on the surface of Ewald sphere, containing rich
information on Bragg reflection and diffuse scattering of crystal lattice. The reciprocal
lattice will go through the surface of Ewald sphere by means of rotating the crystal
around a particular axis. A series of 2D patterns collected while the crystal is rotated
stepwise are used here to reconstruct the 3D spatially distribution of the reciprocal lattice
surrounded with diffuse scattering. The spatial relation between Ewald sphere and
scattering pattern in 3D reciprocal space is schematically shown in Figure S1. The
collected 2D pattern can be described by a discrete matrix (2θ, φ, I) using polar
coordinates, where 2θ is the reflection angle, φ is the azimuth angle and I is the scattering
intensity. 2D pattern (2θ, φ, I) is point-to-point corresponding to the surface of Ewald
sphere (x, y, z, I), where x, y and z are using rectangular coordinates with the center of
sphere as origin. The mathematic translation of (2θ, φ, I) to (x, y, z, I) is represented by
the following equations.
x 
1

cos2 


sin2  sin    .

2


1

z  sin2  cos   

2

y  
1
(2)
A reciprocal lattice vector Hhkl is expressed in terms of the reciprocal vectors b1b2b3 and
the miller indices as Hhkl = hb1 + kb2 + lb3, which can be obtained through normalizing
xyz coordinates using parameters of a certain hkl-plane, i.e., d400 and (x, y, z)400, where b1
// x, b2 // y and b3 // z with the 2D detector center (or beam center) as origin. Finally, a
series of 2D diffraction patterns of the crystal rotated with 0.2° step from -10° to 10°
around <001>P direction were used to reconstruct the spatial distribution of the diffuse
4
scattering signals and satellite spots around (400)P in three-dimensional (3D) reciprocal
space.
Figure S1 Geometry relation between Ewald Sphere and detector plate in reciprocal
space.
4. References
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[4]
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Liu, J. Phys.-Condes. Matter 22, 202201 (2010).
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M. A. Uijttewaal, T. Hickel, J. Neugebauer, M. E. Gruner, and P. Entel, Physical
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Review Letters 102, 035702 (2009).
[7]
M. Methfessel, and A. T. Paxton, Phys. Rev. B 40, 3616 (1989).
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