# Spin-Phonon interaction in quasi 2D- $Cr_{2}Te_{3}$

Gurupada Ghorai  School of Physical Sciences, National Institute of Science
Education and Research (NISER) Bhubaneswar, An OCC of Homi Bhabha National
Institute, Jatni-752050, Odisha, India.  Kalyan Ghosh  School of Physical
Sciences, National Institute of Science Education and Research (NISER)
Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni-752050, Odisha,
India.  Abhilash Patra  School of Physical Sciences, National Institute of
Science Education and Research (NISER) Bhubaneswar, An OCC of Homi Bhabha
National Institute, Jatni-752050, Odisha, India.  Prasanjit Samal  School of
Physical Sciences, National Institute of Science Education and Research
(NISER) Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni-752050,
Odisha, India.  Kartik Senapati  School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni, Odisha - 752050, India.  Center for Interdisciplinary
Sciences (CIS), NISER Bhubaneswar, HBNI, Jatni-752050, Odisha, India.  Pratap
K. Sahoo<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni, Odisha - 752050, India.  Center for Interdisciplinary
Sciences (CIS), NISER Bhubaneswar, HBNI, Jatni-752050, Odisha, India.

(August 28, 2024)

###### Abstract

Spin-phonon interaction plays an important role in 2D magnetic materials and
motivates the development of next-generation spin- and charge-dependent
microelectronic devices. Understanding the spin-phonon interaction by tuning
the growth parameter of single crystal Cr${}_{2}Te_{3}$, a robust quasi-2D
room temperature magnetic material, is crucial for spintronic devices. The
synthesis of single crystal 2D Cr${}_{2}Te_{3}$ flakes on a Si substrate from
co-deposited thin film by plasma annealing techniques is a significant
achievement. The temperature dependence and polarization-resolved Raman
spectroscopy with support of density functional theory classified lattice
symmetry operations were used to identify the phonon modes to investigate the
spin/electron-phonon interactions in Cr${}_{2}Te_{3}$. The mean-field theory
model in single crystal Cr${}_{2}Te_{3}$ is employed to quantify the spin-
phonon interaction and correlate with in-plane and out-of-plane magnetic
behavior. The observation of a positive correlation between phonon mode
frequency and spin-phonon interaction strength in single crystal
Cr${}_{2}Te_{3}$ can be a potential candidate for spintronic applications.

Keyword

2D-Cr22Te3, TEM, SQUID, Raman spectroscopy, DFT.

##  I INTRODUCTION

The thermal transport and spin properties of the 2D magnetic materials are
crucial for spintronic devices[1]. Although spin-phonon, electron-phonon, and
phonon-phonon interactions are offered for study for high-performance spin-
based thermal management devices[2, 3]. This recommends establishing the
influence of spin-phonon interaction on the physical phenomena of the 2D
magnet. The spin-phonon solid interaction in the 2D $CrSiTe_{3}$ was proposed
from the temperature-dependent I-R spectroscopy and super-exchange mechanism
influenced in the Si-Te stretching and Te displacement in temperature-
dependent frequency study[4, 5]. The unconventional spin-phonon interaction
vis Dzyaloshinskii–Moriya interaction(DMI) in $Y_{2}Ir_{2}O_{7}$ crystal by
temperature-dependent infrared spectroscopy was also reported[6]. The spin-
phonon interaction in the bulk crystal of yttrium iron garnet (YIG) with
enhanced interaction strength at higher phonon frequency was investigated [7].
However, spin-phonon coupling is a primary need to tackle how spin affects
phonon thermal transport. To address these coupling issues, we are interested
in exploring the spin-phonon interaction of $Cr_{2}Te_{3}$. The 2D magnetic
order in chromium trihalide materials was first discovered[8]. The 2D magnetic
materials could have been lead to one of the spintronics materials by
utilizing charge and spin for applications ranging from molecular quantum
devices and sensing to ultrathin high-density data storage devices[9, 10]. In
a similar line, the intrinsic 2D layer-dependent ferromagnetism properties in
CrGe2 were also investigated [11]. The structural properties using scanning
tunneling microscopy, bandgap, and electronic properties were calculated using
DFT calculation of $CrGeTe_{3}$ materials in ref[12]. Although three.
Theoretically, it has been suggested that layered materials like
$CrXTe_{3}(X=Si,Ge,Sn)$ show 2D ferromagnetic or antiferromagnetic behavior
dependent on their exchange or superexchange interaction strength[13, 14]. It
has been proposed that $Cr_{2}Te_{3}$ is a tunable curie temperature up to
room temperature and suitable for magnetic memory device applications[15]. So,
for magnetic-based applications of $Cr_{2}Te_{3}$, there should be clear
evidence of the magnetic properties of such material, which are influenced by
defect spins in the lattice. It has been demonstrated that the 2D
$Cr_{2}Te_{3}$ materials show ferrimagnet or ferromagnet properties, which are
dependent on the strength of the nearest neighbor exchange interaction[16].
The (001) oriented crystal grown at a temperature of 400∘C at a rate of 1.7
nm/min on Al${}_{2}O_{3}$(0001) using molecular beam epitaxial(MBE) reveals
the local electronic and magnetic structure of Cr ions, which was demonstrated
the edge of energy of Cr L2,3 using X-ray magnetic circular dichroism (XMCD)
and soft x-ray absorption spectroscopy[17]. However, the magnetic structure of
$Cr_{2}Te_{3}$ is not yet fully understood, because of the complicated
coupling of the spins in three distinct Cr sites. Curie temperature and
magnetic properties vary with various defect sites in the 2D magnetic
materials.

To understand the spin-phonon interaction, we have synthesized a single
crystal $Cr_{2}Te_{3}$, a quasi 2D magnetic material by co-deposited Cr and Te
thin film followed by thermal annealing. The $A_{g}$ and three $E_{g}$ phonon
modes were analyzed from the temperature-dependent Raman study. Different
vibrational modes and frequencies were obtained using DFT calculations and
compared with the experimental results. The anharmonicity of Raman modes can
be well-fitted with Balkanski’s model above the critical temperature. The
ferromagnetic transition with temperature matched well with the Raman
anharmonicity, which can be explained in terms of spin-phonon interaction. The
results can be understood in terms of magnon-phonon coupling.

##  II Experimental method

The commercially available high-purity (99.999$\%$, Sigma Aldrich) metal
pieces of Chromium(Cr) and Tellurium(Te) are co-evaporated using the e-beam
evaporation techniques and thermal evaporation technique in a base vacuum of
2$\times 10^{-7}$ mbar to produce thin films of CrxTey on the clean Si
substrate. The deposition rate of Cr and Te were 0.1 Å/s and 0.4Å/s,
respectively, and the thickness of 40 nm was measured by a quartz crystal
thickness monitor attached to the deposition chamber. The Co-deposited samples
were annealed at 500∘C inside of quartz tube of the plasma-enhanced chemical
vapor deposition (PECVD) system at 120 W plasma power with the ambient of
forming gas for two hours. The surface micrographs were collected using a
Field emission of scanning electron microscopy (FESEM) system (SIGMA ZEISS),
and the elemental mapping and percentage of composition were estimated using
Apex energy dispersive spectroscopy (EDS) attached to the FESEM. The XRD
pattern of the $Cr_{2}Te_{3}$ crystalline phases was obtained using a Rigaku
diffractometer with $CuK_{\alpha}$ radiation ($\lambda=1.54056$ Å) at room
temperature with a grazing incidence X-ray diffraction (GIXRD) approach at a
grazing angle of 1.0 degree with a step size of 0.02deg. The local crystalline
structural properties of the 2D-$Cr_{2}Te_{3}$ were verified using a high-
resolution transmission electron microscope (HRTEM) of a 200 KV energetic
electron source. The high-resolution confocal Raman scattering with a spatial
resolution of 2-4 microns was used to understand the vibrational modes of
$Cr_{2}Te_{3}$. The temperature-dependent Raman spectrum of $Cr_{2}Te_{3}$
flake was measured using a Linkam cooling stage. The specimens were mounted on
a sample holder fixed to a cold head in a vacuum with a constant liquid
nitrogen flow to cool the chamber down to 80K. A half-wave plate in an
incident light path was fixed to study the polarization-dependent Raman
spectra. We also calculated the Raman peaks position, and modes of vibration
were obtained theoretically using density functional theory (DFT). The
temperature dependence of the Raman spectrum of $Cr_{2}Te_{3}$ flakes was
measured using a Horiba Raman microscope equipped with a Linkam cooling stage.
The specimens were mounted on a non-background sample holder fixed to a cold
head in a vacuum, and the liquid nitrogen was used to cool the chamber up to
80K. The superconducting quantum interference device (SQUID) system of Quantum
Design was used to investigate the magnetic properties of $Cr_{2}Te_{3}$ in
the temperature range of 5-250K.

##  III Results and discussions

The GIXRD measurement at a grazing angle of 1∘ was used to obtain the
crystalline properties and the orientation of the $Cr_{2}Te_{3}$ growth. The
GIXRD spectra of the pristine and 120W plasma, 500∘C annealed samples are
shown in figure 1(a). The pristine sample shows a weak XRD peak at 27.56∘,
corresponding to the lattice plane of Te (101) (JCPDS # 36-1452)[18, 19]. The
120W plasma annealed sample shows three XRD peak positions at 2$\theta$ values
of 14.54∘, 29.68∘, and 44.52∘ corresponding to the lattice plane of (002),
(004), and (006), respectively for the crystalline plane of $Cr_{2}Te_{3}$.
The above lattice plans describe the Trigonal family of $Cr_{2}Te_{3}$, with
the space group of 163 $(P\bar{3}1c)$, with lattice parameters of a=6.78 Å,
b=6.78 Å, c=12.06 Å, $\alpha$=90, $\beta$=90 and $\gamma$=120 (PDF#71-2245),
which matches well with an earlier report[20, 17]. Figure 1(b) and (c) shows
the FESEM surface morphology of the pristine and 120 W annealed sample. The
pristine sample shows a very smooth surface. It has been observed that after
annealing the pristine at 500∘C in 120W plasma for 2 hours, the CrTe thin
films grew as individual flakes with clear grain boundaries, as shown in
figure 1(c), which is a 2D $Cr_{2}Te_{3}$. To further clarify the crystalline
growth of quasi 2D $Cr_{2}Te_{3}$, we have prepared a lamella for TEM study
from a single flake. The lamella was prepared using a focused ion beam (FIB)
system using 30 keV Ga beam followed by in situ polishing by low energy Ar
ion. The lamella provides the cross-sectional view of 40 nm thin flake as
shown in figure 1(d), which is basically the c-axis of the 2D-$Cr_{2}Te_{3}$
crystalline plane. The high-resolution TEM with lattice arrangements and
corresponding selected area electron diffraction (SAED) patterns of
2D-Cr${}_{2}Te_{3}$ is shown in figure 1(e). The yellow parallel line shows
the (001) plane of the Cr${}_{2}Te_{3}$ with interplanar separation d001 of
3.03 Å. The schematic image along the (010) zone axis and the lattice spacing
of d001 is shown in figure 1(f). The XRD and HRTEM images strongly suggest the
crystalline growth of quasi-2D $Cr_{2}Te_{3}$ along the (001) plane.

Figure 1:  (a) GIXRD data of the pristine, applied 120 W plasma power samples.
(b) Schematic illustration of crystal structure of $Cr_{2}Te_{3}$. (c) FESEM
image of co-deposited films, (d) Surface morphology image of $Cr_{2}Te_{3}$
sample after annealed $500^{0}C$ with the presence of 120 W plasma power.

###  III.1 Raman spectra of Cr2Te3 from density functional theory

Raman spectroscopy is a versatile tool to understand the electron-phonon or
spin-phonon interaction in 2D materials. Depending on the crystal orientation
the vibrational modes can be suppressed or enhanced which may affect the spin-
phonon interaction. The observation of Raman modes in the case of
Cr${}_{2}Te_{3}$ strongly depends on the crystalline quality and synthesis
process [xx-xx]. In order to identify different Raman modes of
Cr${}_{2}Te_{3}$ , we have used the density functional theory[21, 22] using
the Vienna Ab-initio Simulation Package (VASP).[23, 24, 25, 26]. We have used
the recommended Perdew-Burke-Ernzerhof[27] (PBE) PAW pseudo potentials in all
the calculations along with the plane wave basis cutoff of 500 eV and set the
value of EDIFF = $1.0\times 10^{-8}$ eV to represent an energy convergence
criterion. For the Brillouin zone integration, a $\Gamma$-centered ($12\times
12\times 6$) k-point sampling is used. The Hubbard correction method within
semilocal DFT[28], i.e., GGA(PBE)+U is used for both structure optimization
and studying the Raman spectra. The value of U=3.0 eV[29] for Cr atoms is set
to take into account the orbital localization. The unit cell of Cr2Te3 with
P-31c space group is optimized with the above given set up and obtained the
lattice constants as a=b=6.99 Å, and c=12.65 Å, respectively. To get the Raman
active modes, the dielectric matrix is calculated for each mode using the
density functional perturbation theory(DFPT) as implemented in the VASP
package. The intensity of each mode is calculated following the Ref. [30] and
using the raman-sc[31] python package. The Raman spectra of Cr2Te3 is shown in
figure 2(a). The Lorentzian broadening width of 5 cm-1 is used to match with
the experimental spectra. Six dominating peaks corresponding to more intense
peaks are found at 40, 73, 102, 123, 144, and 190 cm-1. For comparison, the
experimental Raman peaks are shown in figure 2(b). The peaks at 102, 123, and
144 cm-1 are closer to experimental peaks at 106, 128, and 146 cm-1. One can
observe that two modes at 40 and 190 cm-1 are not observed in the experimental
Raman spectra. The Raman modes observed from the experiment and DFT
calculation are tabulated in Table 1.

Figure 2:  (a) Polarisation dependent Raman at the temperature 150K where blue
color code for unpolarised(XX) condition and red color curve for polarised(XY)
Raman spectrum. (b) The theoretical DFT calculation of the Raman spectrum of
$Cr_{2}Te_{3}$ single crystal. The atomic displacements of the Raman active
modes (the color code of atoms is the same as Fig 2(a)). The out-of-plane
$A_{g}$ vibrational modes are depicted on the left panel (top view and side
view), while the in-plane$E_{g}$ vibrational modes are on the right (top view
only).

It should be noted that Cr2Te3 belongs to the point group of D3d(-3m). Based
on the point groups of the character table, the deduced Raman active modes for
Cr2Te3 are 4A1g+9Eg. The Raman-active mode intensity is related to the Raman
tensor and the direction of incident and scattered light as:
$I_{m}^{R}=\frac{d\sigma}{d\Omega}\propto\sum_{m}|\vec{e}_{s}\cdot
Rt_{m}\cdot\vec{e}_{i}|^{2}$. Where Rtm is the Raman tensor of mth active mode
and $\vec{e}_{s}$ ( $\vec{e}_{i}$) is polarization vectors of scattered
(incoming) light[32, 33]. The Raman tensors correspond to active Raman modes
A1g, and Eg are as follows,

A${}_{1g}(D{{}_{3d}}(-3m)):$ $\begin{pmatrix}a&0&0\\\ 0&a&0\\\
0&0&b\end{pmatrix}$ E${}_{g}(D{{}_{3d}}(-3m)):\begin{pmatrix}c&0&0\\\
0&-c&d\\\ 0&d&0\end{pmatrix},\begin{pmatrix}0&-c&-d\\\ -c&0&0\\\
-d&0&0\end{pmatrix}.$

We use the notation XX and YX to represent unpolarized and polarized
scattering configurations, respectively. In the case of unpolarized
configuration XX, both the incoming and scattering polarization vectors are
$\vec{e_{i}}$=(1, 0, 0)=$\vec{e_{s}}$ and for YX, $\vec{e_{i}}$=(1, 0, 0),
$\vec{e_{s}}=(0,1,0)$. So, under unpolarized configuration XX, I(A1g) = a2,
and I(Eg) = c2. Under polarized condition YX, I(A1g) = 0, and I(Eg) = c2. So,
in cross polarization condition, the intensity of A1g mode vanishes, and Eg
peaks remain unchanged.

Table 1: The experimental and theoretical frequencies, and vibrational modes of strain-free R3 Cr${}_{2}Te_{3}$. The ‘$||$’ and ‘$\perp$’ represent the parallel and cross polarization conditions, respectively. The ‘$-$’ represents peaks that were not clearly observed. mode | $A_{g}$ | $E_{g}$ | $A_{g}$ | $E_{g}$ | $E_{g}$ | $A_{g}$
---|---|---|---|---|---|---
EXPT‘$||$’ | - | 96 | 106 | 128 | 146 | -
EXPT‘$\perp$’ | - | 96 | - | 128 | 146 | -
Theory | 40 | 73 | 102 | 123 | 144 | 190

The Raman active modes for $Cr_{2}Te_{3}$ belong to the trigonal lattice
structure with 163 space group[34]. For this space group, two types of active
Raman vibration modes are observed, i.e non-degenerate $E_{g}$ and degenerate
$A_{g}$[34]. So the $A_{g}$ mode is the out-plane lattice vibration and
$E_{g}$ is in-plane atomic vibration of the lattice.

We have studied the polarization Raman spectra at low temperatures (80K) to
best match the theoretical Raman modes. Compared to theoretical parameters
where unpolarized configuration XX, I(A1g) = a2, and I(Eg) = c2, the
experimental spectra shows four Raman peaks at $96cm^{-1}$, $106cm^{-1}$,
$128cm^{-1}$ and $146cm^{-1}$. For the theoretical polarized condition YX,
I(A1g) = 0, and I(Eg) = c2, the experimental spectra shows three Raman peaks
at $96cm^{-1}$, $128cm^{-1}$ and $146cm^{-1}$. The above observation clearly
illustrates the condition and provides the information that the $A_{g}$ is
suppressed and other peaks correspond to $E_{g}$ Raman modes. This procedure
quantifies the modes of vibration in $Cr_{2}Te_{3}$ quasi 2D materials. The
schematic configuration of the atomic vibration in the lattice atoms of
$Cr_{2}Te_{3}$ is shown in figure 2(c, d, e, f, g, h). The direction of
vibration is shown in red arrows and the Cr and Te atoms are shown in magenta
and yellow, respectively. Figure 2(c, d, e) shows the three configurations as
A1g, A2g, A3g for out-of-plane vibration in the c-axis, in which the light
interact ‘$||$’ to the lattice plane. Figure 2(f, g, h) represents three
configurations as E1g, E2g, E3g for in-plane vibration in the XY plane, in
which polarized light interacts $\perp$ to the lattice plane.

###  III.2 Temperature dependent Raman study:

The interaction of phonon with spin and magnon can be understood from the
temperature dependant Raman spectroscopy. The interpretation of Raman spectra
of soft modes in a ferromagnetic material can be understood from its over-
damped line shape and the temperature dependence near the transition
temperature ($T_{N}$).It has been observed that Raman spectroscopy a unique
tool to probe spin-phonon coupling(s) which is strongly influenced by
elementary excitations in multiferroic materials [7]. For a better
understanding of the phonon modes in Cr2Te3, systematic Raman spectra were
collected in the temperature range of 80-300 K and analyzed their evolution
using Lorentzian multifunction. The temperature dependant Raman spectra of
Cr2Te3, sample synthesized at 120 W is shown in Fig. 3(a). Two distinct peaks
at 125 cm-1 and 144 cm-1 corresponding to $E_{2g}$ and $E_{3g}$, respectively,
are observed with peak intensities varying with temperature. A 2D contour plot
shown in Fig 3(b) indicates the distribution of Raman peak intensities within
the frequency range of 80-165 cm-1. One can clearly perceive that both $E_{g}$
mode intensities are maximum within the temperature range of 100-165 K and
then reduced at the higher temperature. The evolution of parameters like peak
positions, full width at half maximum (FWHM), and integrated intensity as a
function of temperature are evaluated to understand the strength of the spin-
phonon interaction and the phonon dynamics of the system.

Figure 3: (a) Temperature dependent Raman spectrum of 120 W plasma power
samples in the temperature range of 82K to 273K. (b) 2D contour, in the range
of 80 to 270 cm-1.  Figure 4: Magnetisation measurement (a) The MH loop
measurement parallel the ab plane configuration of 120 W plasma sample. (b)
The MH loop parallel to c axis of the 120 W plasma sample. (c) The MT
measurement with applied 300 Oe field parallel and perpendicular to the ab
plane of 120 W plasma used sample.(d) are the Exchange bias field of the above
four configurations.

###  III.3 Magnetization study:

In the magnetically ordered system, both Raman line shift and magnetic moments
deviate at similar temperatures due to the strong spin-phonon interaction. In
order to correlate that, the magnetic properties of the material, hysteresis
loop, zero field cooled (ZFC), and field cooled (FC), measurements were
performed with both perpendicular and parallel applied external magnetic
fields with respect to the sample plane. In Figure 4(a), the magnetic moments
(ZFC and FC) are plotted as a function of temperature for the 120 W plasma
power sample. The brown curve represents the case of a perpendicular applied
field of 300 Oe to the ab plane, while the pink curve represents the case of
an applied field parallel to the ab plane. The inset of Figure 4 shows the
first order derivative of M-T, and the TC is determined to be $\approx 162K$.
However, the ZFC curve of the parallel external magnetic field for the 120 W
plasma power samples shows an increase in the magnetic moment at the
transition temperature. This suggests the presence of a well-defined
paramagnetic to AFM phase transition. In the case of $Cr_{2}Te_{3}$, the
magnetic behavior is complex due to the coexistence of strong FM and weak AFM
interactions within the system. The system exhibits spin frustration at lower
temperatures, leading to the emergence of a re-entrant spin glass phase.
Previous studies by Roy et al. and Hashimoto et al. have reported the presence
of both FM and AFM interactions in $Cr_{2}Te_{3}$, with the stronger FM
interactions dominating above the TC (165K)[35]. The existence of a more
complex magnetic order below the TC has also been predicted. The magnetic
field-dependant magnetization M-H measurement of the 120W plasma power sample
was performed to confirm the FM and perpendicular anisotropy. The M-H loops
are shown in Figure 4(b) and (c). Figure 4(d) shows the exchange bias field
from the M-H loop as a function of temperature.

It has been observed that the exchange field decreases with increasing
temperature, indicating the presence of FM and AFM coupling in the sample. The
observation of exchange bias in $Cr_{2}Te_{3}$ coupled with the ferromagnet
CdTe suggests the presence of an AFM component in $Cr_{2}Te_{3}$, likely
arising from c-axis Cr atom defects in the lattice[36]. The existence of both
ferromagnet or ferrimagnet properties in $Cr_{2}Te_{3}$ dependent on the
nearest neighbor spin coupling energy of the Cr sites [16]. It has been
reported that the ferromagnetic order enhanced perpendicular magnetic
anisotropy. Depending on the spin orientations by the applied magnetic field,
the net magnetization in both directions is different. So, the magnetization
property will depend on the direction of the applied magnetic field to the
c-axis of the $Cr_{2}Te_{3}$. The unit cell of $Cr_{2}Te_{3}$ contains
alternating layers of Cr and Te atoms and has CrI vacancies in every second
metal layer. The CrI atoms in the partially occupied layers are sandwiched
between the CrIII layers and have no close-by neighbors in the a-b plane,
whereas the CrII and CrIII atoms make up the fully occupied metal layers. The
crystal growth resembles a hexagonal lattice in the (001) plane, indicating
high crystalline quality as observed from XRD measurements.

###  III.4 Spin-phonon interaction in $Cr_{2}Te_{3}$

The Raman shift of the most prominent peaks, $E_{2g}$(125 cm-1) and
$E_{3g}$(144 cm-1), as a function of temperature are shown in Figure 5(a) and
(b). The Balkanski model fits the data well, indicating the presence of strong
electron-phonon interactions. The FWHM and integrated intensity of the
$E_{2g}$(125 cm-1) and $E_{3g}$(144 cm-1) peaks were studied as a function of
temperature, as depicted in Figure 5(c) and (d). Anomalous behavior near the
transition temperature of 166 K indicates that all Raman peaks correspond to
the quasi-2D $Cr_{2}Te_{3}$ film. Magnetization measurement of $Cr_{2}Te_{3}$
confirms the presence of a strong ferromagnetic component with a weak
antiferromagnetic component. The temperature-dependent behavior of the
magnetic moment suggests the induction of a magnetic moment below the
transition temperature, owing to the influence of nearest neighboring spin-
spin interactions. The excellent agreement between the temperature-dependent
Raman spectroscopy and magnetization measurements, and theoretical
calculations provides a comprehensive understanding of the spin-phonon
interaction in $Cr_{2}Te_{3}$, which is rucial for potential applications in
spintronics and contributes to the understanding of the material’s behavior
and properties.

Figure 5: Temperature dependent Raman spectrum, (a) The Raman peak position of
E2g (125 cm-1) and (b) E3g (145 cm-1) mode of vibration with temperature and
fitted equation(1) for showing the anharmonicity. (c) FWHM with the function
of temperature of two most prominent peaks of E2g and E3g modes, respectively.
(d) Integrated intensity of Raman peaks as a function of temperature of two
most prominent peaks of E2g and E3g modes. The extra Raman peak shift below
the transition temperature of both peaks (e) E2g and (f) E3g and fitting
function equation 4.

In the absence of spin order above the transition point, the temperature-
induced phonon position shift follows a three-phonon scattering model that
takes into account the anharmonicity contributions leading to softening or
hardening of some Raman modes. The occupation probabilities of the phonons are
responsible for the temperature dependence Raman shift and linewidth
broadening. In the case of the phonon decay process, the optical phonon at
$\gamma$ point with energy $\hbar\omega_{c2}(0)$ decays into two acoustic
phonons from the same branch, keeping both energy and momentum conserved. So
the decay phonon has $\hbar\omega_{c2}(0)$/2 energy with equal and opposite
momentum. This process can be expressed as the temperature dependence Raman
shift for the optical phonon frequency $\omega_{c2}(T)$ as follows (Klemens’s
model):

$\omega_{c2}(T)=\omega_{0}-C(1+\frac{2}{\exp\frac{\hbar\omega}{k_{B}T}-1})$
(1)

Here, $\omega_{0}$ represents the phonon frequency at 0 K, $\hbar$ is the
reduced Planck constant, $k_{B}$ is the Boltzmann constant, T is the
temperature, and C is the three phonon interaction constant. The first term
represents the initial phonon frequency, while the second term accounts for
anharmonic contributions due to phonon decay. This can be understood by
fitting parameters such as C and $\omega_{0}$. It has been observed that the
temperature dependence Raman shift fitted well with Klemens’s model up to
162K. The anomaly below 162K does not follow the three-phonon decay process.
According to Klemens’s model, the decayed acoustic phonon should have a
constant velocity below the critical temperature depending on the thermal
conductivity of the materials. However, the upturn behavior indicates the
increase of the phonon velocity. By further decreasing the temperature below
162 K the Raman shift deviats from the expected anharmonic tendency indicating
an additional scattering mechanism involved. Such a mechanism can be
attributed to the coupling of the lattice to spin fluctuations. e.g., spin-
phonon interaction. Revisiting the M-T data of $Cr_{2}Te_{3}$, it has been
observed that the paramagnetic to FM transition occurs at 162K, indicating the
role of spin below TC of 162K. Fennie et al. suggested that for an FM material
spin coupled with optical phonon below transition temperature [37]. The
optical phonon frequency $\omega{{}^{\prime}}$ can be considered as the
contribution of the anharmonic behavior and an extra term due to the deviation
due to spin-phonon interaction ($\lambda_{sp}$), which can be written as :

$\omega{{}^{\prime}}=\omega_{c2}(T)+\Delta{\omega_{sp}}$ (2)

Where $\omega_{c2}(T)$ is the Raman frequency shift as mentioned in the
Klemens model and $\Delta\omega_{sp}$ is the deviation due to spin-phonon
interaction from the Klemens model fitting. Finally, the spin-phonon coupling
parameter $\lambda_{sp}$ for $Cr_{2}Te_{3}$ can be calculated using this
method. This effect is similar to that observed in other magnetic materials,
and it is associated with phonon renormalization induced by magnetic ordering
due to the coupling between magnetic ordering and crystal lattice. Considering
the N-N Heisenberg spin system, the magnetic energy of the FM system can be
correlated with the frequency shift of lattice vibrations by introducing the
proportional constant as spin-phonon coupling parameter $\lambda_{sp}$, which
depends on the derivatives of the exchange constants with respect to the
coordinates of the magnetic ions. In the absence of magnetostriction effects
and electronic states renormalization, the contribution of spin-phonon
coupling $(\lambda_{sp}$ of the $k^{th}$ position is approximately given by[7,
38]:

$(\Delta{\omega_{k}})_{sp}=-\sum_{i,j}\lambda_{sp}<S_{i}.S_{j}>$ (3)

where $<S_{i}.S_{j}>$ is the spin correlation function. This contribution,
considering only first-neighbor interactions and a molecular field
approximation, can be considered in the case of the lattice vibration, as

$(\Delta{\omega_{p}})=-\lambda_{sp}S^{2}\phi(T)$ (4)

where $\Delta\omega_{p}$ indicates the deviation of Raman frequency,
$\lambda_{sp}$ is the coefficients of the spin-phonon coupling,
$<S_{i}.S_{i+1}>$ denotes the average for adjacent spins (where S can be taken
as 3/2 for Cr spins); and $\phi$(T) is the order parameter given by
$\phi(T)=1-(T/T_{\theta})^{\gamma}$ where $T_{\theta}$ is the transition
temperature. The temperature dependence function of $\phi(T)$ is the
normalized order parameter, which is proportional to the square of the average
magnetization M(T).

Figure 6: (a) Above the TC the lattice vibration are observed due to Laser
shining. (b) Below the transition temperature lattice vibration perturbed the
exchange interaction.

The quasi-2D ferromagnetism in $Cr_{2}Te_{3}$ is intriguing due to its high
spin-phonon interaction, especially above its paramagnetic region. Klemens’s
model well fits the temperature dependence Raman frequency till TC,
$\approx$162K and anomalies below that, as shown in Figure 5(a) and (b),
indicating anharmonicity and spin-phonon interactions. Figure 5(e) and (f)
fitted with the equations 3 and 4 in the temperature range of 80 – 160 K. The
spin-phonon interaction parameter ($\lambda_{sp}$) is calculated from the
fitting and found to be 0.47$\pm$0.03 cm-1. This value confirms the presence
of a strong spin-phonon interaction, which is responsible for the large phonon
dispersion. Below the transition temperature, the order parameter is well-
fitted, and the $\gamma$ value is determined to be 2.61$\pm$0.04. The value of
$\lambda_{sp}$ is comparable with the literature and indicates the colinear
spin arrangement describes the magnetostructural transition below TC. The
temperature-dependent Raman concluded that the anomalies below TC is the
signature of spin-phonon interaction arising from the paramagnetic to FM
transition. The larger spin-phonon interaction implies that atoms with
stronger bonds have a greater influence on the magnetic ordering with colinear
spin states. We attribute that below TC the structural fluctuation rearranges
the spins and tends to frustrate spin states with larger magnetization. A
schematic model, as shown in Figure 6(a) shows the normal lattice vibration
and Figure 6(b) illustrates the lattice vibration with spin-spin interaction
and distortion of the exchange coupling due to spin-phonon interaction due to
laser shining. Both the Raman modes and the magnetic moment as a function of
temperature strongly suggest the interaction of spin momentum with lattice
phonons below TC due to nearest neighbor spin-spin interactions.

##  IV Conclusions

The quasi 2D FM $Cr_{2}Te_{3}$ flakes were synthesized and optimized from co-
evaporated thin films and subsequent plasma treatment. The FESEM and EDX
analysis confirms the surface flake-like structures. The XRD and HRTEM confirm
the (001) oriented single crystal $Cr_{2}Te_{3}$ flakes with c-axis van-der-
wall gap of 2.95 Å. The magnetic behavior of the 2D $Cr_{2}Te_{3}$ has been
investigated using SQUID measurements with both perpendicular and parallel
applied magnetic fields to the c-axis of the flakes. A mixed magnetic phase
with strong FM and weak AFM components has been confirmed from M-H and M-T
measurements. The polarization-dependent Raman analysis of the thin flakes
provides insights into the modes of lattice vibration. Additionally, the DFT
calculations support the experimental frequency of the Raman modes. This
strengthens the identification of the Raman modes and supports the reliability
of the experimental observations. The observed Raman frequency, FWHM, and
integrated intensity from the temperature-dependent Raman measurements
indicate the transition temperature. The anomalies below the TC suggest the
strong spin-phonon interaction parameter ($\lambda_{sp}$) of 0.47$\pm$0.03
cm-1. The combination of experimental techniques, along with theoretical
calculations, has provided a comprehensive understanding of the structural,
magnetic, and vibrational properties of the 2D $Cr_{2}Te_{3}$ flakes and
strengthened the understanding of strong spin-phonon interaction mechanisms.
The spin-phonon phenomena of 2D ferromagnetic $Cr_{2}Te_{3}$ thin flakes can
be suitable for 2D magnetic spintronic device applications.

###### Acknowledgements.

The authors acknowledge the National Institute of Science Education and
Research Bhubaneswar, DAE, India for supporting this work through the project
RIN-4001.

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