structural and thermodynamic properties of fe1.12te with multiple phase transitions
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Structural and thermodynamic properties of Fe1.12Te with multiple phase transitionsDona Cherian, S. Rößler, C. Koz, A. A. Tsirlin, U. Schwarz, S. Wirth, and Suja Elizabeth Citation: Journal of Applied Physics 115, 123912 (2014); doi: 10.1063/1.4870233 View online: http://dx.doi.org/10.1063/1.4870233 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Structural transitions and unusual magnetic behavior in Mn-doped Bi1−xLaxFeO3 perovskites J. Appl. Phys. 112, 084102 (2012); 10.1063/1.4759435 The effect of strontium doping on the structural and magnetic transition of YMnO3 AIP Conf. Proc. 1447, 1013 (2012); 10.1063/1.4710349 Structural and magnetic phase transition of mixed olivines Li x Fe1−y Ni y PO4 by lithium deintercalation J. Appl. Phys. 111, 07D722 (2012); 10.1063/1.3678468 Anomalous heat capacity and x-ray photoelectron spectroscopy of superconducting FeSe1/2Te1/2 J. Appl. Phys. 109, 07E122 (2011); 10.1063/1.3556682 Magnetic properties of Bi 2 FeMnO 6 : A multiferroic material with double-perovskite structure Appl. Phys. Lett. 97, 122502 (2010); 10.1063/1.3490221
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Structural and thermodynamic properties of Fe1.12Te with multiple phasetransitions
Dona Cherian,1,a) S. R€oßler,2 C. Koz,2 A. A. Tsirlin,2,3 U. Schwarz,2 S. Wirth,2
and Suja Elizabeth1
1Department of Physics, Indian Institute of Science, Bangalore 560012, India2Max Planck Institute for Chemical Physics of Solids, N€othnitzer Straße 40, 01187 Dresden, Germany3National Institute of Chemical Physics and Biophysics, 12618 Tallinn, Estonia
(Received 12 February 2014; accepted 21 March 2014; published online 31 March 2014)
The parent compound of iron chalcogenide superconductors, Fe1þyTe, with a range of excess Fe
concentrations exhibits intriguing structural and magnetic properties. Here, the interplay of
magnetic and structural properties of Fe1.12Te single crystals have been probed by low-temperature
synchrotron X-ray powder diffraction, magnetization, and specific heat measurements.
Thermodynamic measurements reveal two distinct phase transitions, considered unique to samples
possessing excess Fe content in the range of 0:11 � y � 0:13. On cooling, an antiferromagnetic
transition, TN � 57 K is observed. A closer examination of powder diffraction data suggests that
the transition at TN is not purely magnetic, but accompanied by the commencement of a structural
phase transition from tetragonal to orthorhombic symmetry. This is followed by a second
prominent first-order structural transition at TS with TS < TN , where an onset of monoclinic
distortion is observed. The results point to a strong magneto-structural coupling in this material. VC 2014AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4870233]
I. INTRODUCTION
Simple crystal structure, unique magnetic ground
states, intriguing electronic properties, structural transi-
tions, and superconductivity in substituted and pristine Fe
chalcogenides make these compounds interesting candi-
dates for the studies of condensed matter physics.1–4
Fe1þyTe is an antiferromagnetic compound whereas FeSe,
the other binary system, is a superconductor with a super-
conducting transition temperature TC � 8 K.5,6 Substitution
with other chalcogenides (Se/S) at the Te site in Fe1þyTe
induces superconductivity7,8 and the maximum TC of
15 K was observed for about 50% Se substitution.7,9,10
Even though different Fe pnictides share several common
properties,11 Fe chalcogenides possess rather different elec-
tronic and magnetic behavior.12 The parent compound,
Fe1þyTe is distinguished in its Fermi surface topology,
magnetic ground states, and structural properties.5,13
Fe1þyTe with y¼ 0.06 exhibits a prominent magneto-
structural transition at 70 K below which an antiferromag-
netic order sets.5 For y increasing up to below 0.11, the mate-
rial undergoes a first-order transition from paramagnetic
tetragonal to a commensurate antiferromagnetic monoclinic
phase. The magneto-structural transition occurs in a broad
range of temperature, approximately from 70 K to 57 K
depending on the amount of excess Fe. As Fe content
increases beyond y¼ 0.11, the paramagnetic tetragonal
phase changes into an incommensurate antiferromagnetic
orthorhombic phase14,15 at about 57 K. The magnetic propa-
gation vector is temperature dependent until a lock-in
transition occurs at a lower temperature. For y� 0.14, a sin-
gle continuous phase transition to an incommensurate
antiferromagnetic phase with orthorhombic symmetry has
been observed.16,17 The temperature-composition phase dia-
gram portrays the dependence and sensitivity of transition
temperatures to the excess Fe concentration.17,18
Although the Fe1þyTe system is relatively well investi-
gated by now, one of the issues that remains particularly
controversial is the exact nature of the magnetic and struc-
tural transitions and couplings in the intermediate composi-
tional range of 0:11 � y � 0:13. For instance, in Fe1.13Te,
the two phase transitions occur at 57 K and 46 K. In a previ-
ous report on Fe1.13Te single crystals,19 we presented pre-
liminary low-temperature synchrotron powder diffraction
data which suggested a symmetry-breaking transition
occurring only at the second transition (46 K). These data,
however, were limited to only four temperature points.
Using synchrotron powder diffraction, the composition of
the same crystal is now carefully redetermined as Fe1.12Te.
Around the same time, a neutron scattering work20 on
Fe1.10Te with similar thermodynamic properties like those
of our Fe1.12Te single crystals, reported a structural distor-
tion at 63 K followed by a magnetic transition at 57.5 K.
Later investigations17,21 on polycrystalline Fe1.12Te indi-
cated a two-step transition from a tetragonal-orthorhombic
followed by an orthorhombic-monoclinic transition upon
cooling. These results, combined with the neutron scatter-
ing data,16 suggest that the two transitions have both mag-
netic and structural components. Now, the different results
found in the polycrystalline Fe1.12Te and the powdered sin-
gle crystals of the same composition could be either due to
microstructural effects or due to the coarse temperature
grid used in Ref. 19. With the intention to resolve this issue,
here we re-examine the same sample that was used in Ref.
19 with diffraction data taken at a temperature interval of
2 K in the temperature range of 70–30 K. Further, wea)donacherian@physics.iisc.ernet.in
0021-8979/2014/115(12)/123912/6/$30.00 VC 2014 AIP Publishing LLC115, 123912-1
JOURNAL OF APPLIED PHYSICS 115, 123912 (2014)
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present a detailed analysis of the specific heat and report
the Sommerfeld coefficient, Debye temperature, and the
change in entropies at the phase transition.
II. EXPERIMENTAL
Single crystals of Fe1.12Te were grown by a modified
horizontal Bridgman method. The details of crystal growth
can be found elsewhere.19 The grown crystals were charac-
terized using Laue back scattering and powder X-ray diffrac-
tion (XRD). A composition analysis was made to verify the
homogeneous nature of the sample over a finite area by scan-
ning the surface using an electron probe microanalyzer,
JEOL-JXA-8530F. Powder XRD patterns were obtained in
the temperature range of 30 K–300 K using synchrotron
source at ID 31 at European Synchrotron Radiation Facility
(ESRF). Temperature dependent XRD patterns were recorded
by cooling the powdered crystal sample in a liquid-helium
cryostat. Measurements were made from 30 K to 70 K at inter-
vals of 2 K from a synchrotron source of wavelength
0.430459 A. Data between 75 K and 300 K were collected
using a source of wavelength 0.399928 A. XRD patterns were
refined and structural parameters and occupancy were esti-
mated. Resistivity and specific heat measurements were carried
out using a Quantum Design physical property measurement
system (PPMS). The dc magnetization was measured in a
SQUID magnetometer (Quantum Design).
III. RESULTS AND DISCUSSION
The layered structure and the inherent anion-anion
repulsion enable the as-grown Fe chalcogenide crystals to be
cleaved along the ab plane. The Laue photograph (Figure
1(a)) of cleaved Fe1.12Te illustrates the single crystalline na-
ture of the as-grown crystal. Since physical properties are
extremely sensitive to the concentration of Fe, it is necessary
to check the chemical homogeneity of the crystal employed
for this study. The sample surface was scanned using wave-
length dispersive X-ray (WDX) spectroscopy to confirm the
chemical homogeneity. The WDX surface image is pre-
sented in Figure 1(b), which shows a uniform distribution of
Fe. The lines on the top left corner are cracks formed while
cleaving. Similar colored spots are due to some pits on the
surface.
Synchrotron powder XRD data of Fe1.12Te crystals are
analyzed and the crystal structure is refined by Rietveld
method22 using FullProf code assuming a starting model
from a previous report on Fe1.04Te0.5Se0.5 (Ref. 10).
Composition of the as-grown crystals used in Ref. 19 was
also redetermined using XRD from synchrotron data and
was obtained as Fe1.12Te with a standard deviation of 0.002.
The crystals used for XRD possess two separate transitions
as discussed in Ref. 19. The transition at 57 K corresponds to
a second order transition referred to as TN and the one at
46 K is a first order transition referred to as TS. The transition
temperatures are determined from the specific heat measure-
ments (presented later). Figure 2 illustrates the refined pow-
der pattern at 70 K. The structure is tetragonal and is refined
in P4/nmm space group. The inset of Figure 2 shows the tet-
ragonal crystal structure. The structure consists of layers of
Fe square lattice with Te occupying checker board positions
above and below the Fe layer. Excess Fe occupies the 2cposition in the Te plane whose magnetic interactions play a
crucial role in lattice distortion.23 The strong magnetoelastic
coupling in Fe1þyTe and its distinguishable behavior at the
critical composition substantiate the role of subtle changes in
Fe(y). XRD data above and below TN are carefully analyzed
using Rietveld refinement in P4/nmm space group and ortho-
rhombic Pmmn space group. For 70 K, which is above TN,
the fitted peak intensities and residual of P4/nmm are consid-
erably better than Pmmn with corresponding v2 values of
1.59 and 2.1, respectively. The pattern at 54 K is better
refined in Pmmn ðv2 ¼ 1:39Þ than in P4/nmm ðv2 ¼ 1:8Þ.The matching of the fitted peaks to the observed peaks is
also taken into account. The distinction between the tetrago-
nal and orthorhombic phases becomes clearer when the tem-
perature is lowered to 50 K. At 48 K, Pmmn yields v2 ¼ 1:71
and P21/m yields v2 ¼ 1:8 indicating that Pmmn is a better
model for the crystal structure for temperatures between TN
and TS. Below TS, a monoclinic distortion should be intro-
duced in order to describe the broadening of the (101) reflec-
tion. However, the monoclinic phase alone does not provide
a complete description of the intensity at (101) and (�101).
This missing intensity is well described by a small amount
(7%) of an orthorhombic phase. For data below TS, the
refinement is performed in monoclinic phase alone and also
in a combination of P21/m and Pmmn. The latter yields a
FIG. 1. (a) Laue photograph showing diffraction spots and four fold symme-
try. (b) Fe Composition mapping done by WDX spectroscopy on a 2 mm �1.5 mm cleaved surface.
FIG. 2. XRD crystal structure refinement of 70 K data. Global v2 ¼ 1:65
and Bragg R-factor¼ 1.94. Inset shows the tetragonal crystal structure.
123912-2 Cherian et al. J. Appl. Phys. 115, 123912 (2014)
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better reliability factor and residual (for example, 44 K data
yields v2 ¼ 1:48 for the combination model, whereas a pure
monoclinic model gives v2 ¼ 1:79). The coexistence of the
orthorhombic and monoclinic phases is not unexpected,
according to the earlier observation of the phase separation
in polycrystalline Fe1.12Te.17
A closer look at the diffraction patterns reveals that
below TN, broadening of the (200) peak starts. This is sug-
gestive of the onset of an orthorhombic distortion within the
tetragonal lattice. As the temperature further falls to 52 K,
the split in peaks become more evident and the structure
transforms to the orthorhombic space group Pmmn. On fur-
ther lowering the temperature below TS, the crystal exhibits a
monoclinic distortion. The orthorhombic splitting and mono-
clinic broadening of different Bragg peaks as the temperature
is varied across the transition, are shown in Figure 3. The
(200) Bragg peak starts broadening at 54 K and splits into
(200) and (020) as the temperature is lowered. The splitting
observed below TN clearly indicates the appearance of the
orthorhombic phase below TN. Similarly, peak splittings are
observed in other sets of Bragg peaks, (101) and (011), (211)
and (121), (301) and (031), (311) and (131), and (312) and
(132). When the temperature is below TS, monoclinic broad-
ening starts for some of the above peaks, (�101), (�211)
and (�312). To account for the peak broadening of the (101)
Bragg plane, a monoclinic (�101) Bragg plane has to be
considered. The intensity mismatch between (101) and (011)
peaks also calls for a mixed phase of P21/m and Pmmn for
refinement. However, there is no clear splitting of the (112)
Bragg peak that is characteristic of a pure monoclinic phase.
This is better understood from the analysis of the full
width at half maxima (FWHM) of different Bragg peaks as a
function of temperature in Figure 4. In case of split transitions,
FWHM is estimated as the width of the overall profile com-
prising the split peaks, at half intensity of the higher peak.
Sizable changes are perceptible across the phase transition.
Below TN, FWHM of (200) and (020) increases in magnitude,
indicating a phase transition. Below TS, the peaks are well sep-
arated. Similar inference is made for the set of (110) planes.
Even if no clear split is seen in the (112) peak, there is a signif-
icant increase in the calculated FWHM. The values remain
more or less the same till TS. Within the discussed temperature
range below TS, the values increase progressively, showing a
tendency towards a monoclinic distortion. Thus, the tempera-
ture evolution of Bragg peaks indicates the multiple structural
phase transitions occurring in Fe1.12Te. This is significant as
the cross over region in temperature-composition phase dia-
gram shows coexistence of both low temperature crystal struc-
tures (P21/m and Pmmn) as found for lower and higher Fe
content, respectively. The monoclinic distortion is not com-
plete at 36 K from FWHM values. Consequently, we consider
that the low temperature XRD data below TS signals the onset
of a monoclinic distortion.
The lattice parameters and bond angles are plotted as a
function of temperature in Figure 5. The phase transition at
TN is explicitly marked by the drastic change in lattice pa-
rameters. The region above TN is a tetragonal phase while
the shaded regions characterize the temperature range of the
orthorhombic peak broadening. The region below TS is
where the monoclinic distortion is observed. As a conse-
quence of multiple structural phase transition, a single Fe1-
Te distance splits into two at the orthorhombic transition
(TN) and at even lower T, splits into three at the onset
of monoclinic transition (TS). Similar is the case with the
two distinct bond angles of FeTe4 tetrahedra. The
Fe1-Te1-Fe1(1 and 2) bond angles split into two only after
the monoclinic distortion whereas Fe1-Te1-Fe1(3 and 4)
FIG. 3. (a) Splitting of the (101),(�101), and (011) peaks. (b) Splitting of
the (200) and (020) peaks. (c) Peak splitting corresponding to (211),(�211)
and (121), (�121). (d) Peak splitting corresponding to (301) and (031),
(311) and (131), (312), and (�312) and (132). The black lines are the calcu-
lated intensity from refinement. The vertical green lines indicate the possible
Bragg peaks corresponding to the monoclinic (upper row) and orthorhombic
(lower row) space groups for the given 2h range.
FIG. 4. (a) FWHM (with error bars) as a function of temperature. The verti-
cal lines indicate the temperatures at phase transitions as observed in the
thermodynamic data.
123912-3 Cherian et al. J. Appl. Phys. 115, 123912 (2014)
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split at the orthorhombic phase (numbers refer to the inset to
Figure 5(b)). The temperature evolution of bond length and
bond angles gives a vivid picture of the lattice distortion
around the phase transition. The crystal composition investi-
gated here lies near the tricritical point where a mixed mag-
netic phase is predicted.19 The multiple structural transitions
have to be seen in conjunction with the occurrence of mag-
netic phases and its association to different crystal symme-
tries. A detailed study of the magnetic ground states of the
system is relevant as commensurate and incommensurate
AFM ground states are strongly related to the crystal struc-
ture. In a recent study on polycrystalline Fe1þyTe, the transi-
tions are examined as a function of excess Fe(y).17 The
phase diagram provides regions of mixed crystallographic
phases of monoclinic and orthorhombic structure. The com-
position we refer to, falls at the boundary of this mixed phase
region.
Careful estimation and analysis of the unit cell volume
are regarded as crucial in understanding the magneto-elastic
effects. The cell volume in the temperature range from 300 K
to 36 K is estimated from XRD. The unit cell volume in the
high temperature nonmagnetic region is described by a
Gr€uneisen approximation using the Debye equation for the in-
ternal energy, U(T).24 This is expressed as VðTÞ ¼ CUðTÞ=B0 þ V0, where UðTÞ ¼ 9NkBTðT=HDÞ3
ÐHD=T0
x3=ðex� 1Þdx.
C in this context is the Gr€uneisen parameter, B0 is the bulk
modulus, V0 is the zero temperature volume, HD is the Debye
temperature, and N is the number of atoms in the specimen.
The experimental data of the unit cell volume can be
well described by the expression in the nonmagnetic high
temperature region of Figure 6. The fit yields HD
¼ 252:7K; C=B0 ¼ 0:63� 10�10 Pa�1, and V0¼90.73 A3.
However, there are significant deviations at lower tempera-
ture, close to the phase transitions. These deviations become
apparent in the inset to Figure 6 where the high temperature
fit has been extrapolated to below TS. We note that the esti-
mated value of HD will be later compared to a more accurate
value obtained from specific heat data.
Specific heat measurements (Cp) also show two well
separated transitions as already evident in magnetization and
resistivity measurements.19 The second-order lambda-like
transition from paramagnetic to antiferromagnetic phase is
clearly visible at TN¼ 57 K in Figure 7(a). The prominent
peak at 46 K is due to the first-order transition, TS. From syn-
chrotron measurements, we infer that the monoclinic distor-
tion commences at TS. The clear peaks in the specific heat
data corroborate the results of our synchrotron data. The
magnitude of Cp at high temperature falls close to the limit-
ing value given by the Dulong-Petit law, Cp¼ 3rR, where R
is the universal gas constant and r is the number of atoms per
molecule. To evaluate the thermodynamic parameters, the
specific heat is analyzed using the relation Cp ¼ cT þ b1T3
in the ordered region well below the transitions. Figure 7(b)
gives a decent fit in the low temperature region considering
the electronic contribution (linear term) and the phononicFIG. 5. (a) Lattice parameters. (b) Bond angles and (c) bond distances. Bond
distances and bond angles are calculated between the Fe in the square lattice
(Fe1) and Te(Te1). Below TS, the bond angles and bond lengths are calcu-
lated by considering a monoclinic distortion.
FIG. 6. Unit cell volume as a function of temperature. Open circles with the
error bars represent the experimental data and red line represents fit. Inset
gives a magnified low temperature region where the fit deviates.
123912-4 Cherian et al. J. Appl. Phys. 115, 123912 (2014)
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term (T3). HD obtained from the fit is �245.5 K. The HD val-
ues obtained from the specific heat and unit cell volume fit
match very closely. Cp in the nonmagnetic state is modeled
using a combination of Debye and Einstein models25 in the
temperature region 2 K to 300 K using the relation
CðTÞ ¼ cT þ CDebyeðTÞ þ CEinsteinðTÞ;
CDebye ¼ 9rR=x3D
ðxD
0
x4DexD
ðexD � 1Þ2dxD;
CEinstein ¼ 3rRX
i
ai
�x2
EniexEni
ðexEni � 1Þ2�;
(1)
where xD ¼ �hxD=kBT ¼ HD=T and xEni ¼ �hxEni=kBTi
¼ HEni=T. The model is fitted to the data excluding the tran-
sition region from 45 K to 60 K in Figure 7(a). The c and HD
values obtained from the low temperature fit is used for the
entire range. Initially, c is kept fixed to obtain the remaining
parameters. Later, c is varied as a free parameter together
with the other obtained fitting parameters. The lattice terms
with Debye term and two Einstein terms provide the lattice
specific heat. The Sommerfeld coefficient, c obtained from
the fit is 34.1 mJ/mol K2 which is comparable with the previ-
ous reports.3 In Figure 7(c), linear and lattice contributions
are subtracted from the total specific heat to calculate the
excess specific heat Cexe, across the transitions. The first
order transition at TS and the k-like transition at TN are
clearly visible in the Cexe vs T plot. Change in entropy across
the two transitions is obtained from Sexe ¼Ð T
30ðCexe=TÞ dT as
DSexe ¼ 1:3 J=mol K. The change in excess entropy across
the transitions is very small in comparison with Rln2, the
reason for which is attributed to the weak long range mag-
netic ordering.20
Specific heat measurements were also performed at
applied fields 0 T, 5 T, and 9 T across the magnetic and struc-
tural transitions, given in Figure 7(d). The anomaly in Cp
observed at TN does not show any field dependence up to
9 T. On the contrary, TS shifts to lower temperature
ðDT � 1 KÞ, with slight reduction in magnitude of the anom-
aly in Cp. In previous reports, a slight shift is observed in the
coupled magneto-structural transition with applied magnetic
field.26 In our samples, we observe that TS alone shows a
shift in transition temperature. Magnetization data at
1000 Oe in the field cooled warming (FCW) and field cooled
cooling (FCC) protocol and resistivity data are plotted in
Figure 8(a). Magnetization measurements show a similar
trend as observed in polycrystalline sample, with a wide hys-
teresis region.17 Two transition temperatures are pronounced
which are characterized by a sudden drop in magnetization
and resistivity. Temperature derivative of magnetization in
FCC and FCW measurements plotted together in Figure 8(b)
gives a clear picture of the two transitions present. In the de-
rivative plot of magnetization, both TS and TN can well be
identified. On close observation we can see that a hysteresis
starts near 54 K just below TN, where a clear spitting of (200)
diffraction peak occurs due to the orthorhombic structure. The
hysteresis lasts until 30 K, a broad temperature range over
which the first order transition occurs. In the derivative plot of
resistivity (Figure 8(c)), TS clearly stands out as a pronounced
peak whereas TN is seen as a small broad hump.
FIG. 7. (a) Different contributions to specific heat in the temperature range
of 3 K–300 K. (b) Electronic and lattice specific heat fit to the low tempera-
ture Cp. (c) Excess specific heat obtained by subtracting the fit from the total
specific heat. Change in entropy (excess) across the transitions is also plot-
ted. (d) Cp variation in applied magnetic field.
FIG. 8. (a) Magnetization and resistivity plotted as a function of tempera-
ture. (b) Temperature derivative of magnetization in the heating and cooling
measurements showing the two well resolved peaks. A thermal hysteresis
starts below TN where an orthorhombic distortion is observed. (c)
Temperature derivative of resistivity.
123912-5 Cherian et al. J. Appl. Phys. 115, 123912 (2014)
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IV. CONCLUSION
We investigated the structural and magnetic phase tran-
sitions in Fe1.12Te single crystals grown using Bridgman
method. From the synchrotron powder XRD as well as ther-
modynamic and resistivity measurements, the occurrence of
two phase transitions, a magnetic one at TN and a structural
one at TS was confirmed. Upon cooling, the crystal symmetry
of the material is lowered in two steps. Below TN, the crystal
structure adopts an orthorhombic Pmmn symmetry. The
orthorhombic phase partially transforms into a monoclinic
P21/m phase below TS. The present work independently sup-
ports previously published results obtained on polycrystal-
line samples.17 This agreement is remarkable, specifically
when considering the non-stoichiometry of Fe1þyTe and a
random distribution of excess Fe in the lattice. However, a
comparison of the fractions of the Pmmn and P21/m phases
at low temperature may indicate that the single crystals have
a larger volume fraction of the monoclinic phase. The com-
patibility of the commensurate and incommensurate antifer-
romagnetic order associated with the P21/m and Pmmnphases, respectively, is not well understood. The results, in
any case, ascertain the presence of strong magneto-structural
couplings in this system. At present, it is not clear whether
the orthorhombic and monoclinic phases are completely sep-
arated from each other or they rather form an intergrowth.
Consequently, the low-temperature microstructure of
Fe1þyTe appears to be highly complex. A local probe such as
transmission electron microscopy or scanning tunneling mi-
croscopy would be necessary to shed some light on the
microstructural properties of Fe1þyTe at low temperatures.
ACKNOWLEDGMENTS
We acknowledge support by A. Fitch and Y. Watier
at beamline ID31, at ESRF Grenoble, during the experi-
ments based on Proposal No. HS4825. DC thanks S.
Harikrishnan for fruitful discussions. The authors thank
DST (India) and DAAD (Germany) for financial support
and travel grant. The project was partially supported by
DFG SPP 1458. AAT thanks the funding from the ESF
through MTT77.
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