# Physical properties of the layered $f$-electron van der Waals magnet Ce2Te5

Yu Liu, M. M. Bordelon, A. Weiland, P. F. S. Rosa, S. M. Thomas, J. D.
Thompson, F. Ronning, and E. D. Bauer  MPA-Q, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545, USA

###### Abstract

We report a detailed study of the magnetic, transport, and thermodynamic
properties of Ce2Te5 single crystals, a layered $f$-electron van der Waals
magnet. Four consecutive transitions at $\sim$ 5.2, 2.1, 0.9, and 0.4 K were
observed in the $ac$-plane electrical resistivity $\rho$(T), which were
further confirmed in specific heat $C_{\textrm{p}}$(T) measurements. Analysis
of the magnetic susceptibility $\chi$(T), the magnetic-field variation of
$\rho$(T), and the increase of the first transition temperature
($T_{\textrm{c}}\sim$ 5.2 K) with applied magnetic field indicates
ferromagnetic order, while the decrease of the other transitions with field
suggests different states with dominant antiferromagnetic interactions below
$T_{2}\sim$ 2.1 K, $T_{3}\sim$ 0.9 K, and $T_{4}$ = 0.4 K. Critical behavior
analysis around $T_{\textrm{c}}$ that gives critical exponents
$\beta=0.31(2)$, $\gamma=0.99(2)$, $\delta=4.46(1)$, $T_{\textrm{c}}=5.32(1)$
K indicates that Ce2Te5 shows a three-dimensional magnetic critical behavior.
Moreover, the Hall resistivity $\rho_{\textrm{xy}}$ indicates that Ce2Te5 is a
multi-band system with a relatively high electron mobility $\sim 2900$ cm2 V-1
s-1 near $T_{\textrm{c}}$, providing further opportunities for future device
applications.

##  I INTRODUCTION

Layered van der Waals (vdW) materials have attracted widespread attention due
to the exotic quantum states they exhibit, such as correlated insulating,
ferromagnetic, and superconducting states in “twisted” bilayer graphene CaoY1
; CaoY2 ; Matthew , or quantum criticality in twisted transition metal
dichalcogenides Ghiotto . The discovery of intrinsic long-range magnetic order
in monolayer CrI3 and bilayer CrGeTe3 has opened up new avenues of research
into magnetism in the two-dimensional (2D) limit Huang ; Gong  as well as
integration of 2D magnetic layers for control of magnetism by gating or other
electrical means in devices Ralph2019 ; Burch2018 ; Song2018 ; ACSNANO ; YL ;
YL001 ; YL002 . Accordingly, several other $3d$-electron vdW magnets, such as
FePS3, Fe3GeTe2, VSe2, VI3, CrTe2, and MnSe2, have been extensively
investigated Lee ; Deng ; Bonilla ; Zhang ; Hara ; VI3 ; Fe3GeTe2 . In
contrast, very few $f$-electron vdW magnets have been studied (e.g., CeSiI
CeSiI , EuC6 EuC6_1 ; EuC6_2 ; EuC6_3 , and GdTe3 Lei ). Many $f$-electron
materials exhibit significant hybridization between the $f$-electrons and
conduction electrons, leading to highly correlated quantum states with narrow
$f$-bands near the Fermi level; thus, 2D $f$-electron vdW materials may be
highly tunable with modest amounts of pressure, uniaxial strain, or magnetic
field, making them promising candidates for discovering and exploring unusual
quantum states.

The family of rare-earth telluride RTex (R = rare-earth element; $x$ = 2, 2.5,
and 3) adopts a layered crystal structure, consisting of square planar Te
layers and corrugated RTe slabs. The RTe slabs are semiconducting and
responsible for magnetism, while the Te layers form 2D conducting bands; thus
RTex exhibits highly anisotropic transport and magnetic properties Min ; Kwon
; Shin ; Iyeiri ; Ru ; Ru1 . Among this series, CeTe2 and CeTe3 crystallize in
the layered Cu2Sb-type tetragonal (space group: $P4/nmm$) structure and
NdTe3-type weakly orthorhombic (space group: $Cmcm$) structure, respectively,
with localized Ce3+ magnetic moments. CeTe2 contains a single layer of Te and
undergoes an antiferromagnetic (AFM) transition at $T_{\textrm{N}}$ = 4.3 K
Kwon1 ; Min1 ; Jung ; Jung1 . At 2 K, a metamagnetic transition to a field-
induced ferromagnetic (FM) state with an easy $c$ axis occurs at a small
magnetic field of 0.06 T. The resistivity shows a sharp peak at $T_{\rho}$ = 6
K well above $T_{\textrm{N}}$ with a large negative magnetoresistance (MR)
Jung ; Jung1 ; Jung2 ; Kasuya ; Min2  arising from magnetic-polaron and/or
short-range FM ordering. Neutron diffraction measurements indicate a down-up-
up-down AFM configuration along the $c$ axis with FM Ce double layers above
and below the Te layer in CeTe2 Park ; Park1 ; Stowe ; Liu1 ; Shim . CeTe3
contains double layers of Te connected via weak vdW force; it exhibits two AFM
transitions at $T_{\textrm{N1}}$ = 3.1 K and $T_{\textrm{N2}}$ = 1.3 K with
non-parallel easy axes that are perpendicular to the layer stacking direction,
i.e., strongly easy-plane character Deguchi ; Zocco ; Okuma ; Watanabe .

Layered Ce2Te5 can be considered as a combination of CeTe2 and CeTe3 (Fig. 1),
consisting of alternating single and double Te layers stacked along the $b$
axis of the orthorhombic unit cell and separated by CeTe slabs Chen . Ce2Te5
crystallizes in a weakly orthorhombic structure, similar to CeTe3, with two Ce
sites either adjacent to the double layers of Te (Ce1) or to the monolayer of
Te (Ce2). Chen $et$ $al.$ reported three magnetic transitions at 5.1, 2.3, and
0.9 K in Ce2Te5 single crystals Chen .

In this study we report the physical properties of single crystals of Ce2Te5,
including magnetic susceptibility, magnetization, specific heat, and
longitudinal and Hall resistivity measurements. An additional magnetic
transition at $\sim$ 0.4 K was observed, where the resistivity features a weak
kink and the specific heat exhibits a peak. When magnetic field is applied
along the $b$ axis, the resistivity shows that the first transition
$T_{\textrm{c}}\sim 5.2$ K broadens and shifts to higher temperatures,
consistent with FM ordering; however, the second and third transitions at
$T_{2}$ = 2.1 K and $T_{3}$ = 0.9 K move to lower temperatures. The critical
exponents obtained around $T_{\textrm{c}}$ indicates that Ce2Te5 shows a
three-dimensional magnetic critical behavior. Furthermore, the Hall effect
suggests that Ce2Te5 is a multi-band system with a relatively high electron
mobility around $T_{\textrm{c}}$.

##  II METHODS

###  II.1 Experimental details

Single crystals of Ce2Te5 with typical dimensions of $3\times 3\times 0.1$ mm3
were grown by a RbCl/LiCl flux Chen . The crystallographic structure of Ce2Te5
was verified at room temperature by a Bruker D8 Venture single-crystal X-ray
diffractometer equipped with Mo radiation. X-ray diffraction analysis shows
that Ce2Te5 crystallizes in the orthorhombic space group $Cmcm$ (No. 63) with
lattice parameters $a\sim c\approx$ 4.42 Å and $b\approx$ 44.17 Å, in
agreement with a previous report Chen .

The magnetization was measured in a Quantum Design Magnetic Property
Measurement System (MPMS) from 2 to 350 K up to magnetic fields $\mu_{0}H$ = 6
T and $\mu_{0}$ is magnetic permeability in vacuum. For critical analysis, the
reported internal magnetic field ($\mu_{0}H_{\textrm{int}}$) has been
corrected, $\mu_{0}H_{\textrm{int}}=\mu_{0}H-NM$, where $\mu_{0}H$ is the
applied magnetic field, $M$ is the measured magnetization, and $N\sim 0.95$ is
the demagnetization factor. The specific heat was measured using a Quantum
Design Physical Property Measurement System (PPMS) from 0.35 to 20 K that
utilizes a quasi-adiabatic thermal relaxation technique. The longitudinal and
Hall resistivity were measured in a PPMS using standard four-probe
configurations with the current flowing in the $ac$-plane and the magnetic
field applied along the $b$-axis. The Hall resistivity
$\rho_{\textrm{xy}}(\mu_{0}H)$ was calculated by the difference of transverse
resistivity measured at positive and negative fields, i.e.,
$\rho_{\textrm{xy}}(\mu_{0}H)=(\rho_{\textrm{H+}}-\rho_{\textrm{H-}})/2$, so
as to effectively eliminate the longitudinal resistivity contribution due to
voltage probe misalignment.

###  II.2 Scaling analysis

A second-order phase transition around the Curie temperature $T_{\textrm{c}}$
is characterized by a set of interrelated critical exponents $\beta$,
$\gamma$, $\delta$ and a magnetic equation of state Stanley . The critical
exponents $\beta$ and $\gamma$ are associated with the spontaneous
magnetization $M_{\textrm{s}}$ and the inverse initial susceptibility
$\chi_{\textrm{ini}}^{-1}$, below and above $T_{\textrm{c}}$, respectively,
while $\delta$ is the critical isotherm exponent. The definitions of $\beta$,
$\gamma$, $\delta$ from magnetization measurement are given below:

$M_{\textrm{s}}(T)=M_{0}(-\varepsilon)^{\beta},\varepsilon<0,T<T_{\textrm{c}},$
(1)
$\chi_{\textrm{ini}}^{-1}(T)=(\mu_{0}h_{0}/m_{0})\varepsilon^{\gamma},\varepsilon>0,T>T_{\textrm{c}},$
(2) $M=D(\mu_{0}H_{\textrm{int}})^{1/\delta},T=T_{\textrm{c}},$ (3)

where $\varepsilon=(T-T_{\textrm{c}})/T_{\textrm{c}}$ is the reduced
temperature, and $M_{0}$, $\mu_{0}h_{0}/m_{0}$ and $D$ are critical amplitudes
Fisher .

The magnetic equation of state in the critical region ($\varepsilon\leq 0.1$)
can be expressed as

$M(\mu_{0}H_{\textrm{int}},\varepsilon)=\varepsilon^{\beta}f_{\pm}(\mu_{0}H_{\textrm{int}}/\varepsilon^{\beta+\gamma}),$
(4)

where $f_{-}$ for $T<T_{\textrm{c}}$ and $f_{+}$ for $T>T_{\textrm{c}}$,
respectively, are regular functions. Eq.(4) can be further written in terms of
scaled magnetization
$m\equiv\varepsilon^{-\beta}M(\mu_{0}H_{\textrm{int}},\varepsilon)$ and scaled
field $\mu_{0}h\equiv\varepsilon^{-(\beta+\gamma)}\mu_{0}H_{\textrm{int}}$ as

$m=f_{\pm}(\mu_{0}h).$ (5)

This suggests that for true scaling relations and the right choice of $\beta$,
$\gamma$, $\delta$ values, the scaled $m$ and $\mu_{0}h$ will fall on
universal curves above $T_{\textrm{c}}$ and below $T_{\textrm{c}}$,
respectively.

##  III RESULTS AND DISCUSSION

Figure 1: (Color online). Crystal structure of the van der Waals (vdW) magnet
Ce2Te5 with alternating stacking of CeTe2 and CeTe3 along the $b$-axis.

Figure 2: (Color online). Temperature dependence of (a) magnetic
susceptibility $\chi$(T), defined as $M/\mu_{0}H$, and (b) inverse magnetic
susceptibility $1/\chi$(T) of Ce2Te5 measured in magnetic field of $\mu_{0}H$
= 0.1 T applied parallel and perpendicular to the $b$-axis in zero-field-
cooled (ZFC) and field-cooled (FC) modes. The solid lines are linear fits to
the data. (c) The low temperature $\chi$(T) measured in a low magnetic field
of $\mu_{0}H$ = 1 mT. (d) Field dependence of magnetization $M$($\mu_{0}H$) of
Ce2Te5 measured at $T$ = 2 K.

Figure 2(a) shows the temperature dependence of magnetic susceptibility
$\chi$(T) measured in $\mu_{0}H$ = 0.1 T applied parallel and perpendicular to
the $b$-axis. A rapid upturn in $\chi$(T) at low temperature is observed for
both field directions, indicating a FM transition. In this field, the zero-
field-cooled (ZFC) and field-cooled (FC) data overlap well for each
orientation. The temperature dependence of inverse susceptibility $1/\chi$(T)
is plotted in Fig. 2(b). A linear fit from 200 to 300 K to a Curie-Weiss form,
$\chi=C/(T-\theta)$, where $C$ is the Curie constant and $\theta$ is the
paramagnetic Curie-Weiss temperature, gives $\theta$ = 9.6 K for $H\parallel
b$ and -18.0 K for $H\perp b$, respectively. The positive value of $\theta$
for $H\parallel b$ is consistent with a dominant FM interaction, while the
negative $\theta$ for $H\perp b$ suggests a dominant AFM interaction. The
derived effective moment $\mu_{\textrm{eff}}$ = 2.56 $\mu_{\textrm{B}}/$Ce for
$H\parallel b$ and 2.52 $\mu_{\textrm{B}}/$Ce for $H\perp b$, respectively,
are very close to Hund’s value for Ce3+ of 2.54 $\mu_{\textrm{B}}$. It should
be noted that the high-temperature anisotropy and deviation from Curie-Weiss
behavior with decreasing temperature may be attributed to a crystalline
electric field (CEF) effect as explained below. Figure 2(c) shows the low-
temperature $\chi$(T) measured in a small field of $\mu_{0}H$ = 1 mT. In the
ordered state, $\chi$(T) for $H\parallel b$ is 20 times larger than that of
$H\perp b$, indicating a large magnetic anisotropy. The bifurcation of ZFC and
FC curves below $T_{\textrm{c}}\approx 5.2$ K is likely due to a FM domain
effect. Figure 2(d) displays the isothermal magnetization measured at 2 K. The
magnetization $M(H\parallel b)$ rapidly saturates to $\sim$ 0.5
$\mu_{\textrm{B}}/$Ce at 40 mT, whereas $M(H\perp b)$ gradually increases. It
is interesting that $M(H\perp b)$ increases up to a higher value than
$M(H\parallel b)$ above 2.5 T [Fig. 2(d)], in line with the previous results
Chen , indicating that the magnetic order is more complex than simple
ferromagnetism in Ce2Te5. A similar feature was also observed in bulk CrI3
YL001 . As shown in the inset of Fig. 2(d), a clear hysteresis loop with
coercive field $\mu_{0}H_{\textrm{c}}\approx 20$ mT is observed for
$H\parallel b$, indicating soft ferromagnetism with an easy $b$ axis.

Figure 3: (Color online). Temperature dependence of $1/(\chi-\chi_{0})$(T) of
Ce2Te5 with a crystalline electric field (CEF) model fit (solid lines) as
explained in the main text.

As shown in Fig. 3, the anisotropic magnetic susceptibilities can be modeled
with a CEF Hamiltonian for Ce3+ of total angular momentum $J=5/2$. In Ce2Te5,
due to $a$ and $c$ being accidentally degenerate in the orthorhombic $Cmcm$
crystal structure, the local Ce point group has $C_{4v}$ symmetry with the
four-fold rotational axis along the crystallographic $b$-axis. The resultant
CEF Hamlitonian contains three CEF parameters $B_{n}^{m}$ with corresponding
Steven’s operators $\hat{O}_{m}^{n}$ StevensOps  as

$H_{\textrm{CEF}}=B_{2}^{0}\hat{O}_{2}^{0}+B_{4}^{0}\hat{O}_{4}^{0}+B_{4}^{4}\hat{O}_{4}^{4},$
(6)

which produces three Kramers doublets. In the $C_{4v}$ point group and
$J,m_{j}$ basis, these doublets are labeled $\Gamma_{6}$ with $m_{j}=\pm 1/2$
components or $\Gamma_{7}^{1}$/$\Gamma_{7}^{2}$ with mixed $m_{j}=\pm 5/2$ and
$m_{j}=\pm 3/2$ components. Magnetic susceptibility of the CEF Hamiltonian was
calculated using Mantid Plot MP  with an additional temperature independent
$\chi_{0}$ term and effective mean-field exchange interactions
$\Theta_{\perp}$ and $\Theta_{\parallel}$. An effective susceptibility was
calculated as
$\chi_{\textrm{eff}}=\chi_{\textrm{CEF}}^{\textrm{calc}}/(1-\Theta\chi_{\textrm{CEF}}^{\textrm{calc}})$
and compared to the observed $\chi_{\textrm{obs}}=\chi-\chi_{0}$, where
$\chi_{0}$ is a temperature-independent contribution. The overall fit was
determined by minimizing
$X^{2}=(\chi_{\textrm{calc}}-\chi_{\textrm{obs}})^{2}/\chi_{\textrm{calc}}$
with a final $X^{2}=15.8$ and extracted parameters $B_{2}^{0}=-0.39003$ meV,
$B_{4}^{0}=0.06595$ meV, $B_{4}^{4}=-0.21987$ meV, $\chi_{0}=-0.000195$ emu
mol-1, $\Theta_{\perp}=24.74$ K, and $\Theta_{\parallel}=18.81$ K. The ground
state doublet is $\Gamma_{7}^{1}=0.396|\pm 5/2\rangle+0.918|\mp 3/2\rangle$,
the first excited state doublet $\Gamma_{7}^{2}=0.918|\pm 5/2\rangle-0.396|\mp
3/2\rangle$ is at 16.23 meV, and the second excited state doublet
$\Gamma_{6}=|\pm 1/2\rangle$ is at 24.67 meV. The ground state doublet has
projected $g$ factors $g_{\perp}=1.393$ and $g_{\parallel}=1.497$,
respectively. Taking $J_{\textrm{eff}}=1/2$ for the ground state doublet, the
expected saturated moment $gJ\mu_{\textrm{B}}$ is 0.697 $\mu_{\textrm{B}}$/Ce
for $H\perp b$ and 0.749 $\mu_{\textrm{B}}$/Ce for $H\parallel b$, in
reasonable agreement with the magnetization at 2 K and 6 T $M_{H\perp b}$ =
0.87 $\mu_{\textrm{B}}$/Ce and $M_{H\parallel b}$ = 0.76
$\mu_{\textrm{B}}$/Ce.

Figure 4: (Color online). (a) Arrott plots of $M^{2}$ vs
$\mu_{0}H_{\textrm{int}}/M$ around $T_{c}$ for Ce2Te5 with $H\parallel b$. (b)
Temperature dependence of the spontaneous magnetization $M_{\textrm{s}}$
(left) and the inverse initial susceptibility $\chi_{\textrm{ini}}^{-1}$
(right) with fitted curves for Ce2Te5 explained in the main text. (c) Isotherm
in log10M-log10($\mu_{0}$$H_{\textrm{int}}$) measured at $T_{\textrm{c}}$ =
5.3 K, along with a linear fit to the data. (d) Kouvel-Fisher plots of
$M_{\textrm{s}}(dM_{\textrm{s}}/dT)^{-1}$ (left) and
$\chi_{\textrm{ini}}^{-1}(d\chi_{\textrm{ini}}^{-1}/dT)^{-1}$ (right), along
with linear fits to the data.

In order to understand the nature of the FM transition in Ce2Te5, one approach
is to study in detail the critical exponents around $T_{\textrm{c}}$.
Magnetization isotherms along the easy $b$-axis were measured from 4.6 to 6 K
at intervals of 0.1 K. An Arrott plot of $M^{2}$ vs
$\mu_{0}H_{\textrm{int}}/M$ at various temperatures is displayed in Fig. 4(a).
In the mean field description of the magnetization near $T_{\textrm{c}}$,
curves in the Arrott plot should be a series of parallel straight lines with
the one passing through the origin indicating the $T_{\textrm{c}}$ Arrott1 ;
Banerjee ; Arrott2 . It is clear that mean field critical exponents do not
work for Ce2Te5, as illustrated by a set of curved lines shown in Fig. 4(a).
According to the Arrott-Noaks equation of state
$(\mu_{0}H_{\textrm{int}}/M)^{1/\gamma}=a\varepsilon+bM^{1/\beta}$ Arrott2 ,
$a$ and $b$ are constants, a modified Arrott plot should be used to obtain the
critical exponents. $\beta$ and $\gamma$ can be obtained self consistently.
After selecting $\beta$ and $\gamma$, the linear extrapolation from the high
field region to the intercepts with the axes $M^{1/\beta}$ and
$(\mu_{0}H_{\textrm{int}}/M)^{1/\gamma}$ yields the values of
$M_{\textrm{s}}$(T) and $\chi_{\textrm{ini}}^{-1}$(T). A new set of $\beta$
and $\gamma$ can be obtained by fitting data following Eqs. (1) and (2), which
can be used to reconstruct a new modified Arrott plot. This procedure is then
repeated until the values of $\beta$ and $\gamma$ are stable Pramanik .

Figure 4(b) presents the final $M_{\textrm{s}}$(T) and
$\chi_{\textrm{ini}}^{-1}$(T) with the fitted curves. Critical exponents
$\beta=0.30(2)$, $\gamma=0.99(5)$, and $T_{\textrm{c}}=5.32(1)$ K are
obtained. Figure 4(c) exhibits the field dependence of magnetization of Ce2Te5
at $T_{\textrm{c}}=5.3$ K in a log10M-log10($\mu_{0}$$H_{\textrm{int}}$) plot,
yielding $\delta$ = 4.46(1) from Eq. (3). In comparison to the theoretical
prediction based on the Widom relation Kadanoff ,

$\delta=1+\frac{\gamma}{\beta}$ (7)

the derived $\delta$ = 4.3(1) is close to that obtained in Fig. 4(c). In
addition, critical exponents can also be determined according to the Kouvel-
Fisher (KF) method Kouvel :

$\frac{M_{\textrm{s}}(T)}{dM_{\textrm{s}}(T)/dT}=\frac{T-T_{\textrm{c}}}{\beta}$
(8)
$\frac{\chi_{\textrm{ini}}^{-1}(T)}{d\chi_{\textrm{ini}}^{-1}(T)/dT}=\frac{T-T_{\textrm{c}}}{\gamma}$
(9)

$M_{\textrm{s}}(T)/[dM_{\textrm{s}}(T)/dT]$ and
$\chi_{\textrm{ini}}^{-1}(T)/[d\chi_{\textrm{ini}}^{-1}(T)/dT]$ are linear
functions of temperature with slopes of $1/\beta$ and $1/\gamma$,
respectively. As shown in Fig. 4(d), the linear fits give $\beta=0.31(2)$,
$\gamma=0.99(2)$, and $T_{\textrm{c}}=5.32(1)$ K, which are consistent with
those generated by the modified Arrott plot.

Figure 5: (Color online). (a) The scaling plots of normalized magnetization
$m=M|\varepsilon|^{-\beta}$ as a function of normalized field
$\mu_{0}h=\mu_{0}H_{\textrm{int}}|\varepsilon|^{-(\gamma+\beta)}$ below and
above $T_{c}$ for Ce2Te5. (b) Scaled magnetization
$M(\mu_{0}H_{\textrm{int}})^{-1/\delta}$ vs scaled field
$\varepsilon(\mu_{0}H_{\textrm{int}})^{-1/\beta\delta}$. (c) The magnetic
entropy change $-\Delta S_{M}$ obtained from magnetization at various magnetic
fields along the $b$-axis. (d) Field dependence of the maximum magnetic
entropy change $-\Delta S_{M}^{max}$ with a power law fit.

Following Eq. (5), the scaled $m$ vs scaled $\mu_{0}h$ is plotted in Fig.
5(a). All the data reasonably well collapse into two separate branches: one
below $T_{\textrm{c}}$ and another above $T_{\textrm{c}}$. The scaling
equation of state also takes another form:

$\frac{\mu_{0}H_{\textrm{int}}}{M^{\delta}}=k[\frac{\varepsilon}{(\mu_{0}H_{\textrm{int}})^{1/\beta}}]$
(10)

where $k(x)$ is the scaling function. Figure 5(b) shows
$M(\mu_{0}H_{\textrm{int}})^{-1/\delta}$ vs
$\varepsilon(\mu_{0}H_{\textrm{int}})^{-1/(\beta\delta)}$ for Ce2Te5, where
the experimental data collapse reasonably onto a single curve, and
$T_{\textrm{c}}$ locates at the zero point of the horizontal axis. The well-
scaled curves confirm reliability of the obtained critical exponents. Figure
5(c) presents the derived magnetic entropy change $-\Delta
S_{\textrm{M}}=\int_{0}^{\mu_{0}H}\left[\partial M(T,\mu_{0}H)/\partial
T\right]_{\mu_{0}H}d(\mu_{0}H)$ Amaral , which shows a broad peak centered
near $T_{\textrm{c}}$. The peak value monotonically increases with increasing
field, reaches 4.7(1) J kg-1 K-1 in 6 T. The field dependence of $-\Delta
S_{\textrm{M}}^{\textrm{max}}$ follows a power law $-\Delta
S_{\textrm{M}}^{\textrm{max}}\propto(\mu_{0}H)^{n}$ with
$n=1+(\beta-1)/(\beta+\gamma)$ VFranco ; YLMC1 ; YLMC2 ; YLMC3 ; YLMC4 .
Fitting of $-\Delta S_{\textrm{M}}^{\textrm{max}}$ gives $n=0.49(1)$, which is
close to the calculated value of 0.46(1), further verifies the reliability of
the obtained critical exponents.

Table 1: Comparison of critical exponents of Ce2Te5 with different theoretical models. MAP, KFP, and CI represent the modified Arrott plot, the Kouvel-Fisher plot, and the critical isotherm, respectively. | Reference | Technique | $T_{\textrm{c}}$ | $\beta$ | $\gamma$ | $\delta$
---|---|---|---|---|---|---
Ce2Te5 | This work | MAP | 5.32(1) | 0.30(2) | 0.99(5) | 4.30(5)
|  | KFP | 5.32(1) | 0.31(2) | 0.99(2) | 4.2(1)
|  | CI | 5.3 |  |  | 4.46(1)
2D Ising | Widom  | Theory |  | 0.125 | 1.75 | 15.0
Mean field | Kouvel  | Theory |  | 0.5 | 1.0 | 3.0
3D Heisenberg | Kouvel  | Theory |  | 0.365 | 1.386 | 4.8
3D Ising | Huang3  | Theory |  | 0.325 | 1.24 | 4.82
3D XY | Phan  | Theory |  | 0.345 | 1.316 | 4.81
Tricritical mean field | Fisher1972  | Theory |  | 0.25 | 1.0 | 5.0

Taroni _et al._ pointed out that the value of $\beta$ for a 2D magnet should
be within a window $0.1\leq\beta\leq 0.25$ Taroni . The value of $\sim$ 0.31
obtained here indicates a clear 3D behavior in Ce2Te5. As we can see, the
critical exponent $\beta$ of Ce2Te5 is close to the theoretical value
($\beta=0.325$) of the 3D Ising model (Table I), consistent with the large
anisotropy in magnetization below $T_{\textrm{c}}$ [Fig. 2(d)]. However, the
value of $\gamma\sim 1.0$ of Ce2Te5 deviates from $\gamma=1.24$ of the 3D
Ising model, which might be arising from the long-range Ruderman-Kittel-
Kasuya-Yosida (RKKY) interactions.

Figure 6: (Color online). Temperature dependence of (a) in-plane electrical
resistivity $\rho$(T) and (b) specific heat $C_{\textrm{p}}/T$ of Ce2Te5 in
zero field. Inset in (b) shows the electronic entropy $S_{4f}$(T)/Rln2 of
Ce2Te5.

Having delineated salient features of the FM state below $T_{\textrm{c}}$, we
now turn to an investigation of electrical transport properties. Figure 6(a)
shows the temperature dependence of $ac$-plane resistivity $\rho$(T) of
Ce2Te5, a typical metallic behavior with slope change around 100 K. This might
be related to the hybridization between local moments and conduction
electrons, and the presence of a CEF doublet near 190 K. The low-temperature
$\rho$(T) of Ce2Te5 displays four consecutive transitions at 5.2, 2.1, 0.9,
and 0.4 K [inset in Fig. 6(a)]. The first transition temperature corresponds
well to the FM transition ($T_{\textrm{c}}$) in $\chi$(T) [Fig. 2(a)]. Below
$T_{\textrm{c}}$ and $T_{2}$, $\rho$(T) decreases more rapidly due to a
decrease of spin disorder scattering; however, $\rho$(T) shows a rapid upturn
below $T_{3}$ and a weak kink at $T_{4}$ that suggest the opening of a small
gap possibly due to a spin-density-wave. The specific heat divided by
temperature, $C_{\textrm{p}}/T$ [Fig. 6(b)], shows peaks at 5.2 K, 2.1 K, 0.9
K, and an upturn toward 0.4 K that are consistent with the four transitions
observed in $\rho$(T). The electronic entropy $S_{4f}$ [inset in Fig. 6(b)] is
determined by subtracting a $\beta T^{3}$ phonon contribution from
$C_{\textrm{p}}/T$ (obtained from a linear fit to
$C_{\textrm{p}}/T=\gamma+\beta T^{2}$ from 9 to 10 K), and integration of the
resulting $4f$ contribution to the specific heat $S_{4f}=\int{(C_{4f}/T)}dT$.
The value of $S_{4f}$ approaches $R$ln2 at 10 K, indicating a crystal field
doublet ground state in Ce2Te5. About 75% of $R$ln2 of entropy is released
below $T_{\textrm{c}}$ and about 30% of $R$ln2 is released below $T_{2}$. One
plausible scenario is that one Ce sublattice orders ferromagnetically below
$T_{\textrm{c}}$, while the other Ce sublattice orders antiferromagnetically
below $T_{2}$. This scenario is consistent with the roughly equal amount of
entropy released below $T_{\textrm{c}}$ and $T_{2}$ [Fig. 6(b) inset], the
soft ferromagnetic response of $M$(H) at $T$ = 2 K below $T_{2}$, the field
dependence of the ordering temperatures discussed below, and the competition
of magnetic interactions in Ce2Te5. Alternatively, magnetic coupling between
the two magnetic Ce sublattices may give rise to interesting magnetically
ordered states, such as ferrimagnetism below $T_{c}$, coexistent FM and AFM
states at 2 K (see Fig. 2d), or incommensurate ordered states. Further neutron
scattering $\mu$SR measurements will be useful to determine the nature of the
magnetically ordered states and their underlying magnetic interactions in
Ce2Te5.

With increasing magnetic fields applied along the $b$ axis, as shown in Fig.
7(a), the first transition $T_{\textrm{c}}$ broadens and shifts to higher
temperature, confirming it is a FM transition. $T_{2}$ and $T_{3}$, however,
move to lower temperature and are finally suppressed in high fields,
indicating an AFM character. Figure 7(b) shows the field dependence of
longitudinal magnetoresistance (MR) measured at 0.5, 1.5 and 4 K. The MR at 4
K shows a relatively large positive value of 75% in $\mu_{0}H$ = 9 T. In
addition, an inflection was observed at low temperatures. As depicted in the
inset of Fig. 7(b), a peak in $d$MR/$d(\mu_{0}H)$ indicates that a critical
field of $\sim$ 4.1 T at 0.5 K and 3.5 T at 1.5 K, respectively, which are
associated with the $T_{2}$ and $T_{3}$ transitions. Based on these transport
measurements, the magnetic phase ($T-\mu_{0}H$) diagram of Ce2Te5 is
summarized in Fig. 7(c). The contribution of the interlayer magnetic
interactions between CeTe2 and CeTe3 likely generates complex magnetic
interactions in Ce2Te5. The nature of $T_{4}$ at the lowest temperature needs
further investigation, and neutron scattering measurements are required to
determine the CEF levels and the magnetic structure of Ce2Te5.

Figure 7: (Color online). (a) Magnetic-field variations of in-plane electrical
resistivity $\rho$(T) for $I\perp b$ and $H\parallel b$ for a single crystal
of Ce2Te5. The lines are guides to the eye. (b) The longitudinal
magnetoresistance (MR) of Ce2Te5 at indicated temperatures. Inset shows
$d$MR/$d(\mu_{0}H)$ curves at 0.5 and 1.5 K. (c) The magnetic phase diagram
constructed from transport measurements (solid symbols from $\rho$; open
symbols from MR). PM, FM, and AFM represent paramagnetic, ferromagnetic, and
antiferromagnetic phases, respectively. The lines are guides to the eye.

Figure 8: (Color online). The field dependence of Hall resistivity
$\rho_{xy}$($\mu_{0}H$) at various (a) high and (b) low temperatures with
$I\perp b$ and $H\parallel b$ for a single crystal of Ce2Te5. The lines in (a)
and (b) are fits of the two-band model to the data as explained in the main
text. The derived (c) electron concentration $n_{e}$ and hole concentration
$n_{h}$, (d) electron mobility $\mu_{e}$ and hole mobility $\mu_{h}$.

To shed light on the transport carriers in Ce2Te5, we further measured the
field dependence of Hall resistivity $\rho_{\textrm{xy}}$($\mu_{0}H$) of
Ce2Te5 with current flowing in the $ac$-plane and the field applied along the
$b$-axis at various temperatures, as shown in Figs. 8(a) and 8(b). The Hall
coefficient $\rho_{\textrm{xy}}$/($\mu_{0}H$) is negative at high
temperatures, indicating dominant electron-like carriers. With decreasing
temperature, $\rho_{\textrm{xy}}$ exhibits nonlinear behavior below 60 K, and
the shape of $\rho_{\textrm{xy}}$ changes significantly below 40 K, becoming
parabolic at low temperatures. These observations suggest that Ce2Te5 is a
multi-band system, as has been observed in a related compound GdTe3 Lei .
Assuming a two-band model, the Hall resistivity is expressed as,

$\rho_{\textrm{xy}}=\frac{\mu_{0}H}{e}\frac{(n_{h}\mu_{h}^{2}-n_{e}\mu_{e}^{2})+(n_{h}-n_{e})(\mu_{e}\mu_{h}\mu_{0}H)^{2}}{(n_{h}\mu_{h}+n_{e}\mu_{e})^{2}+(n_{h}-n_{e})^{2}(\mu_{e}\mu_{h}\mu_{0}H)^{2}}$
(11)

where $e$ is the elementary charge; $n_{e}$ and $n_{h}$ are the electron and
hole carrier concentrations, respectively; $\mu_{e}$ and $\mu_{h}$ are the
electron and hole carrier mobilities, respectively. The derived fitting
parameters $n_{e}$, $n_{h}$, $\mu_{e}$ and $\mu_{h}$ are plotted in Figs. 8(c)
and 8(d), respectively, where the electron and hole carriers are almost
compensated at high temperatures and become uncompensated below 60 K. The
carrier concentration is of the order of 1020 \- 1021 cm-3 [Fig. 7(c)]. The
hole mobility $\mu_{h}$ is $\sim$ 350 - 930 cm2/V$\cdot$s and is weakly
temperature-dependent; however, $\mu_{e}$ increases abruptly below 60 K and
features a relatively large value of $\sim$ 2900 cm2/V$\cdot$s near
$T_{\textrm{c}}$. This high electron mobility in Ce2Te5, a vdW layered and
magnetically ordered material, is larger than that of some other rare-earth
materials such as YbMn(Bi,Sb)2 and EuMnBi2 Wang ; Wang1 ; May , and provides
further opportunities for future device applications.

##  IV CONCLUSIONS

In summary, we studied the magnetic, transport, and thermodynamic properties
of the $f$-electron vdW magnet Ce2Te5 and summarized its magnetic phase
diagram. Four magnetic transitions were observed at $T_{\textrm{c}}$ = 5.2,
$T_{2}$ = 2.1, $T_{3}$ = 0.9, and $T_{4}$ = 0.4 K. Critical exponents in the
vicinity of the FM transition are determined to be $\beta=0.31(2)$,
$\gamma=0.99(2)$, $\delta=4.46(1)$, $T_{\textrm{c}}=5.32(1)$ K, indicating
that Ce2Te5 shows a three-dimensional magnetic critical behavior. A crystal
electric field model of the magnetic susceptibility suggests that the ground
state $\Gamma_{7}^{1}$ doublet has a dominant $|3/2\rangle$ character with
excited $\Gamma_{7}^{2}$ and $\Gamma_{6}$ states at $\sim$ 16 and 25 meV,
respectively. Further neutron scattering or X-ray absorption spectroscopy
measurements will be needed to confirm this CEF scheme. The Hall resistivity
analysis indicates that Ce2Te5 is a multi-band system with a relatively high
electron mobility around $T_{\textrm{c}}$. Furthermore, with rapid
developments in the field of 2D materials, we expect our experimental work to
stimulate broad interest for exploring its magnetic and transport properties
in the 2D limit.

## Acknowledgements

Work at Los Alamos National Laboratory was performed under the auspices of the
U.S. Department of Energy, Office of Basic Energy Sciences, Division of
Materials Science and Engineering under project “Quantum Fluctuations in
Narrow-Band Systems”. Y.L., M.M.B., and A.W. acknowledges the Director’s
Postdoctoral Fellowship through the Laboratory Directed Research and
Development program.

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