# Antiferromagnetism and large magnetoresistance in GdBi single crystal

Gourav Dwari, Souvik Sasmal, Bishal Maity, Vikas Saini, Ruta Kulkarni,
Arumugam Thamizhavel  Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400 005, India

###### Abstract

Single crystal of the binary equi-atomic compound GdBi crystallizing in the
rock salt type cubic crystal structure with the space group $Fm\bar{3}m$ has
been grown by flux method. The electrical and magnetic measurements have been
performed on well oriented single crystals. The antiferromagnetic ordering of
the Gd moments is confirmed at $T_{\rm N}=27.5$ K. The magnetization
measurement performed at $2$ K along the principal crystallographic direction
[100] did not show any metamagnetic transition and no sign of saturation up to
$7$ T. Zero field electrical resistivity reveals a sharp drop at $27.5$ K
suggesting a reduction in the spin disorder scattering due to the
antiferromagnetic alignment of the Gd moments. The residual resistivity at $2$
K is 390 n$\Omega$cm suggesting a good quality of the grown crystal. The
magneto resistance attains a value of $1.0~{}\times~{}10^{4}\%$ with no sign
of saturation, in a field of $14$ T, at $T=2$ K. Shubnikov de Hass (SdH)
oscillations have been observed in the high field range of the
magnetoresistance with five different frequencies corresponding to the
extremal areas of the Fermi surface. Analysis of the Hall data revealed a near
compensation of the charge carriers accounting for the extremely large
magnetoresistance.

##  I Introduction

The recent focus in condensed matter physics is on the observation of a new
state of topological quantum matter in novel materials that have interesting
band structures [1, 2, 3]. Dirac and Weyl semimetals that have gained
importance these days fall in this category. The Dirac point has a linear
dispersion and can also be considered as a pair of Weyl points in $k$-space
that are protected by both crystalline inversion $\mathcal{I}$ and time
reversal $\mathcal{T}$ symmetries. When either of the symmetries is broken the
Dirac semimetal evolves into a Weyl semimetal [4, 5]. The prototypic Dirac
semimetals Cd3As2 and Na3Bi have been theoretically predicted as Dirac
semimetals and later realized in experiments [6, 7, 8, 9]. One of the
interesting features of these Dirac semimetals is that they exhibit extremely
large magnetoresistance, ultra-high mobility and depict chiral anomaly,
quantum Hall effect etc., due to the symmetry protected band crossings. The
extremely large magnetoresistance (XMR) has been observed in several binary
intermetallic compounds like WTe2, MoSi2, WSi2, NbP, MoP2, WP2 etc. Ultra-high
mobility ($\approx 10^{4}$ cm2/V s) and electron-hole resonance with
relatively lesser carrier concentration are the reasons for the XMR in these
compounds [10, 11, 12, 13, 14]. Although, such XMR is observed in non-magnetic
intermetallic compounds, the natural extension of these studies is to combine
the topological aspects with the strong electronic correlations. The strong
electronic correlation is observed in $f$ electron systems and the rare-earth
monopnictides $RX$, where $R$ is a rare-earth element and $X$ is Sb or Bi,
crystallizing in the simple rock-salt type cubic crystal structure was the
default choice. To start with, the non-magnetic LaBi and LaSb have been
studied and both the compounds were showing XMR of the order of $10^{5}\%$
[15, 16]. The field dependence of the electrical resistivity of these two
compounds showed an upturn and plateau region which were observed in several
semimetallic compounds and hence led to the construction of a universal
triangular phase diagram [15, 12]. Several of the RBi compounds like PrBi,
SmBi, ErBi, HoBi etc., have been reported recently also exhibit large MR [17,
18, 19, 20]. It is interesting to note that in spite of the increasing $f$
electron count the XMR is still observed which reveals that the band structure
may remain unchanged due to the highly localized nature of the $f$ electrons.
PrBi does not show any magnetic ordering due to the singlet ground state of
the crystal electric field split $(2J+1)=9$-fold degenerate of Pr3+ ion [17].
The higher rare-earths HoBi and ErBi show magnetic ordering at 5.9 K and 3.6
K, respectively [20, 19]. Furthermore, Li et al [21] predicted GdBi as a
compensated semimetal with non-trivial band topology in its antiferromagnetic
state.

In this work, we present a systematic investigation on the transport and
magnetic properties of GdBi single crystal. The magnetic measurement depict a
long range antiferromagnetic ordering of Gd3+ moments at $T_{\rm N}=27.5$ K.
The magnetic ordering is further confirmed from the electrical resistivity and
specific heat studies. The electrical resistivity displayed a typical
semimetallic character in zero field measurements. With the application of
magnetic field the resistivity revealed an upturn at low temperature similar
to a metal-insulator-like transition. At sufficiently high fields in the range
$10-14$ T, the quantum oscillations are observed in magnetoresistance. Five
different frequencies have been observed in Shubnikov-de Haas (SdH)
oscillations.

##  II Experimental Methods

Although, GdBi melts congruently at $1770~{}^{\circ}$C [22], the high melting
temperature precludes its growth by Czochralski method due to the high vapour
pressure of Bi at such high temperature. Hence, the single crystal of GdBi has
been grown by self flux method using molten Bi as flux. High purity Gd ingots
(99.9%, Alfa Aesar) and Bi lumps (99.998%, Alpha Aesar) were packed into a
baked round bottomed alumina crucible in the molar ratio of Gd:Bi = 20:80 and
sealed under vacuum in a quartz tube. The quartz tube was subsequently placed
in a box-type resistive heating furnace and the sample was heated to 1100 ∘C
with a heating rate of 50 ∘C/h and held at this temperature for about 12 h for
homogenization. Then the furnace was cooled down to 920 ∘C at a rate of 1 ∘C/h
followed by 3 days of annealing. We centrifuged the excess Bi-flux at $920$ ∘C
in order to avoid the formation of GdBi2 which crystallizes below 910 ∘C.
Cubic single crystals of GdBi with typical dimensions of $\approx$ 2 $\times$2
$\times$ 2 mm3 were obtained. The compositional analysis of the grown crystals
was performed using Energy Dispersive X-ray Spectroscopy (EDX). The phase
purity of the crystals were confirmed from x-ray diffraction (XRD) performed
in a PANalytical x-ray diffractometer equipped with a monochromatic
Cu-$K_{\alpha}$ x-ray source ($\lambda=1.5406$ Å) and the crystals were
oriented along the principal crystallographic direction using Laue diffraction
using a polychromatic x-ray source. The crystals were cut into desired shapes
using a spark erosion electric discharge machine (EDM). Magnetic measurements
were performed in a SQUID magnetometer (MPMS, Quantum Design, USA) and the
electrical and heat capacity measurements were performed in a physical
property measurement system (PPMS, Quantum Design, USA).

##  III Results and Discussion

###  III.1 X-ray diffraction

The crystal structure of GdBi is shown in Fig. 1(a). A small piece of the as
grown crystal was subjected to XRD at 300 K with $2\theta$ scan ranging from
10 to 90∘, peaks corresponding to $(h00)$-planes are observed at Bragg angles
28.2∘ and 58.44∘ thus confirming the flat plane of the crystal to be
(100)-plane. According to the previous report, GdBi crystallizes in NaCl type
structure with space group Fm-3m (No. 225) [23]. The Laue pattern
corresponding to (100) and (111) planes are shown Fig. 1(c) and (d). Well
defined circular spots together with the four fold symmetry in (100)-plane
confirmed the good quality of single crystal. The composition of grown crystal
was confirmed from EDX measurement.

Figure 1:  (a) Crystal structure of GdBi. (b) Room temperature XRD pattern of
a single crystal of GdBi, (inset) as grown single crystal of GdBi. Laue
pattern of GdBi for (c) (100) (or equivalent) plane and (d) (111) (or
equivalent) plane.

###  III.2  Magnetic properties

The temperature dependence of magnetic susceptibility $\chi(T)$ measured in
applied magnetic field $(B)$ of 0.1 T, parallel to [100] direction in the $T$
range $2-300$ K is shown in Fig. 2(a). At $T=27.5$ K a sharp drop in the
$\chi$ confirms the antiferromagnetic ordering. At high temperature $\chi$
shows a clear Curie-Weiss behaviour and the $\chi(T)$ data follows the Curie-
Weiss law: $\chi(T)=C/(T-\theta_{p})$, where $C$ is the Curie constant and
$\theta_{\rm p}$ is the paramagnetic Weiss temperature. The effective magnetic
moment $\mu_{\rm eff}$ of Gd3+ ions can simply be obtained by the relation
$\mu_{\rm eff}=\sqrt{8C}$. The inverse $\chi(T)$ plot is shown in the inset of
Fig. 2(a). The solid line shows the fit to the Curie-Weiss law. From the
fitting we obtained $\theta_{\rm p}=-48.6$ K and the Curie constant $C=8.02$
emu/mol. The estimated $\mu_{\rm eff}$ from the fitting is $8.1~{}\mu_{\rm
B}$/Gd which is nearly equal to the theoretical value of $7.9~{}\mu_{\rm B}$
of a free Gd3+ ion. The negative value of $\theta_{\rm p}$ confirms the
antiferromagnetic correlations.

Figure 2: (a) Magnetic susceptibility $(\chi)$ as function of Temperature in
$0.1$ T magnetic field, (inset) $\chi^{-1}$ fitted in Curie-Weiss law in
paramagnetic region. (b) Field dependence of Magnetization at different
temperature for field parallel to [100] (or equivalent) direction.

The isothermal magnetization $(M)$ measured at various fixed temperatures is
shown in Fig. 2(b). The $M(B)$ curves for $T<T_{\rm N}$ shows a small change
of slope at around $1$ T signalling a subtle spin re-orientation followed by
steady increase without any sign of saturation up to a magnetic field of $7$
T. An estimation of the critical field at which the magnetization attains the
saturation value for an antiferromagnet at $T=0$ K can be estimated using the
mean field model. The expression for the critical field $H_{\rm c}$ is given
by [24]:

$H_{\rm c}=\frac{M_{\rm s}}{\chi(T_{\rm N})},$ (1)

where $M_{\rm s}=7~{}\mu_{\rm B}$/Gd, the saturation magnetization of Gd3+ ion
and $\chi(T=T_{\rm N})$ is obtained from Fig. 2(a). Substituting the value of
$\chi(T_{\rm N})=0.09454$ emu/mol $=1.6927~{}\times~{}10^{-5}~{}\mu_{\rm B}$
Oe-1 Gd-1, the critical field $H_{\rm c}$ is estimated as $41$ T. This
estimated value of $H_{\rm c}$ is in good agreement with the high field
magnetization data, where the $M_{\rm sat}$ is attained at a critical field of
$H_{\rm c}=42$ T [25].

Figure 3: (a) Temperature dependence of electrical resistivity of GdBi at
different applied magnetic fields, (top inset) fitting of resistivity at $14$
T magnetic field with Eq.(2), (bottom inset) power law fitting of low
temperature resistivity. (b) Temperature dependence of MR normalized with its
value at $T2$ K at different magnetic fields. (inset) MR as function of
temperature. (c) Field dependence of MR at various temperatures, (inset) power
law fitting of MR at $2$ K . (d) Kohler’s scaling of MR at different
temperature.

###  III.3 Electrical Resistivity and Magnetoresistance

The temperature dependence of electrical resistivity $\rho_{\rm xx}(T)$
measured in zero field is shown in the main panel of Fig. 3(a). In the absence
of magnetic field the $\rho_{xx}(T)$ decreases linearly as the temperature is
decreased and at $T_{\rm N}=27.5$ K a sharp drop in $\rho_{xx}(T)$ is observed
due to the reduction in the spin disorder scattering. Below $T_{\rm N}$, the
resistivity drops rapidly and attains a value of about $390$ n$\Omega$cm at
$2$ K. The residual resistivity ratio (RRR) of the sample is estimated as 82
suggesting a high quality of crystal. The zero field $\rho_{xx}(T)$ data,
below $T_{\rm N}$ was fitted to the power law $\rho_{xx}(T)=\rho_{0}+aT^{n}$
as shown in the inset of Fig. 3(a). The best fit to the data was obtained for
$n=1.5$, typically for most of the rare-earth compounds the resistivity show a
power law behavior, due to the $e-e$ scattering, however in the present case
the low $T$ resistivity data shows a $T^{1.5}$ behaviour. According to the
spin fluctuation theory by Moriya et al. [26], the $T^{1.5}$ dependence is
observed in antiferromagnetic materials near a quantum critical point [27,
26]. The $T^{1.5}$ behaviour in a localized $f$ electron system GdBi warrants
further investigation.

The temperature dependence of $\rho(T)$ at various applied magnetic fields is
also shown in Fig. 3(a). The overall behaviour of $\rho_{xx}(T)$ remains the
same in the paramagnetic state in applied magnetic fields. The
antiferromagnetic transition at $T_{\rm N}=27.5$ K remains robust for fields
as high as $14$ T without any shift which is substantiated with the magnetic
susceptibility and heat capacity data as well. However, $\rho_{xx}(T)$ shows
an upturn well below $T_{\rm N}$ for fields greater than $1$ T. The upturn
increases more rapidly with higher magnetic fields and resembles a metal-to-
insulator like transition (MIT). Similar behaviour has been observed in other
$RBi$ (R = Pr, Ho, Er) compounds [17, 20, 19]. It is interesting to note that
the $\rho_{xx}(T)$ goes through a minimum before the upturn and this minimum
shifts to higher temperature as the magnetic field is increased. The magnetic
field driven MIT has been observed in systems like WTe2, NbP [10, 13] and
semimetallic compounds like MoSi2, WSi2, WP2 [11, 12, 14] . Different
mechanisms have been put forward for this type of field induced MIT and large
MR in topological materials. For example, in WTe2 the extremely large MR is
attributed to the perfect electron-hole resonance, while in the case of LaBi
and LaSb the large MR is attributed to the orbital texture [15]. All these
mechanisms are for non-magnetic systems. GdBi exhibits an antiferromagnetic
transition and typically, in such kind of materials the positive
magnetoresistance is attributed to the suppression of the $T_{\rm N}$ due to
applied magnetic fields. However, $T_{\rm N}$ of GdBi is robust and no change
in $T_{\rm N}$ has been observed for fields as high as $14$ T, furthermore the
$M(B)$ data by Li $et~{}al.$ [25], have revealed that the magnetization
increases linearly and attains the field induced ferromagnetic state at around
$42$ T, which suggests the gradual spin re-orientation. The scattering due to
this staggered moment together with near compensation of charge carriers and a
very low residual resistivity (vide infra) results in such a large MR.

To understand the field dependence of the electrical resistivity further, we
plotted the normalized temperature dependence of resistivity MR($T$)/MR($2$K),
measured in different fields from $4$ to $14$ T in steps of $2$ T, as shown in
Fig. 3(b). It is interesting to see that the normalized curves fall on to a
single curve suggesting the $T$-dependent MR remains almost the same for all
magnetic fields. Hence, it can be said that the low temperature behavior of
$\rho_{xx}(T)$ is metallic rather insulating in high magnetic fields [28]. A
similar behavior is observed in the magnetically ordered ErBi [19]. The upturn
in $\rho_{xx}(T)$ at low temperature can be well described by the Kohler’s
scaling rule [29, 30, 31], according to which, the field dependent MR at
different temperature will follow the same functional form MR $\propto
f(B\tau)$ where $\tau$ is the relaxation time that is inversely proportional
to $\rho_{0}$ as long as the scattering mechanisms at different temperatures
remain same. Following the Kohler’s rule, the resistivity $\rho(B,T)$ of GdBi
can be written as:

$\rho(B,T)=\rho_{0}(0,T)\left[1+\gamma\left(\frac{B}{\rho_{0}(0,T)}\right)^{m}\right]$
(2)

where $\rho_{0}(0,T)$ is the measured resistivity of GdBi at zero applied
magnetic field. A fit of Eqn. 2, keeping $\gamma$ as the only adjustable
parameter, with data at $14$ T is shown in the top inset of Fig.3(a). Value of
$m$ is kept fixed at 1.7, obtained from power law fitting of
magnetoresistance. Deviation becomes large when the temperature is above the
transition temperature $T_{\rm N}$, which maybe attributed to the change of
scattering mechanism, which limits the use of Kohler’s law. The above equation
also shines light on the presence of a minimum in resistivity at magnetic
field due to the coexistence of $\rho(0,T)$ and inverse of that term in
$\rho(H,T)$.

Magnetoresistance of GdBi at different temperature is shown in Fig. 3(c) as a
function of magnetic field applied in transverse direction of current. At $2$
K, MR reaches $5.1\times 10^{3}\%$ in field $10$ T and $10.9\times 10^{3}\%$
in $14$ T (not shown here) which is extremely large. Best fit of the MR data
at $2$ K with power law is shown in the inset of 3(c) where the obtained
exponent value is $1.7$. For a perfectly compensated semimetal one would
expect, this value to be 2. In this present case the exponent value reveals
that the charge carriers are nearly compensated and this has been confirmed
from the Hall data (to be discussed below). Value of the exponent decreases as
temperature is increased from $2$ K up to $20$ K after that it again increases
till $100$ K with value 1.8, probably due to change of carrier concentration.
To verify the applicability of Eqn.2 to the resistivity data, we performed the
Kohler’s scaling rule to the field dependent magnetoresistance plot, shown in
Fig. 3(d), where MR is plotted against $(B/\rho_{0})$ at different
temperatures, and as can be seen, all MR are collapsing onto a single curve as
predicted by the scaling rule. Deviation from a straight line behaviour is
mainly attributed to the different scattering mechanism in GdBi. This also
describes why the fitting shown in the top inset of Fig. 3(a) using Eqn.2 is
deviating at temperature above $T_{N}$.

We have used Hall resistivity $(\rho_{xy})$ and linear resistivity
$(\rho_{xx})$ data to estimate the carrier concentration of GdBi. Hall
measurements were performed in five-probe geometry followed by
antisymmetrization of the data to minimize the contribution of linear
resistivity $(\rho_{xx})$. The curved nature of Hall data implies the
existence of multiple charge carriers. Two-band model is used here to fit the
linear conductivity $(\sigma_{xx})$ and hall conductivity $(\sigma_{xy})$. In
semi-classical two band model, complex conductivity can be written as:

$\sigma=e\left[\frac{n_{e}\mu_{e}}{1+i\mu_{e}B}+\frac{n_{h}\mu_{h}}{1-i\mu_{h}B}\right]$
(3)

where $e$ is the magnitude of elementary charge and $\sigma_{xx}$ and
$\sigma_{xy}$ are obtained from real part and imaginary part of $\sigma$
respectively. $\sigma_{xx}$ and $\sigma_{xy}$ are calculated from experimental
$\rho_{xx}$ and $\rho_{xy}$ using the following relations:

$\sigma_{xx}=\frac{\rho_{xx}}{\rho_{xx}^{2}+\rho_{xy}^{2}}$ (4)

$\sigma_{xy}=\frac{\rho_{xy}}{\rho_{xx}^{2}+\rho_{xy}^{2}}$ (5)

Figure 4: (a) $\sigma_{xx}$ and $\sigma_{xy}$ of GdBi as a function of
magnetic field. Solid lines show the fit with two band model. (b) (top)
Electron and hole density, (bottom) electron and hole mobility at different
temperature, obtained from two band model fitting.

We simultaneously fitted the $\sigma_{xx}$ and $\sigma_{xy}$ data at $2$ K to
the real and imaginary part of Eqn.3 and the best fit is shown in Fig.4(a).
Calculated values of carrier density and mobility at different temperatures is
shown in Fig.4(b). At $2$ K the electron and hole density are estimated as
$7.08\times 10^{26}$ m-3 and $7.30\times 10^{26}$m-3 which explains the
deviation of MR from quadratic behaviour in field. As temperature is
increased, we see a crossover in electron and hole density, which is
consistent with the observation that the exponent value ($n$) in power law
fitting of MR is increasing towards the value 2 as temperature is increased.
The estimated mobility at $2$ K is also quite large, reaching $0.914$ m2V-1s-1
for electrons and $0.994$ m2V-1s-1 for holes. These values of carrier
concentration and mobility are nearly the same as obtained for ErBi and HoBi
compounds [20, 19]. The nearly compensated nature of the charge carriers
together with relatively high mobility are the reasons for the observed large
MR.

###  III.4 Shubnikov-de Hass (SdH) Oscillations

Table 1: Parameters estimated from SdH oscillation analysis: $m^{*}$, effective mass; $T_{D}$, Dingle temperature; $A$, extremal area of the Fermi surface; $k_{\rm F}$, Fermi wave vector; $v_{\rm F}$, Fermi Velocity; $\tau$, quantum relaxation time, and $\mu_{\rm Q}$, quantum mobility. |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
Fermi pocket |  Frequency |  $m^{*}$ |  $T_{D}$ |  $A$ |  $k_{F}$ |  $v_{F}$ |  $\tau$ |  $\mu_{Q}$
|  (T) |  $(m_{e})$ |  (K) |  (nm-2) |  (107 cm-1) |  (107 cm/s) |  (10-13 s) |  (cm2/V s)
|  |  |  |  |  |  |  |
$\alpha_{1}$ |  462 |  0.36 |  6.13 |  4.40 |  1.18 |  3.75 |  1.98 |  954.5
$\alpha_{2}$ |  872 |  0.35 |  14.66 |  8.30 |  1.62 |  5.38 |  0.83 |  416.3
$\alpha_{3}$ |  1641 |  0.72 |  5.22 |  15.63 |  2.23 |  3.56 |  2.33 |  564.1
$\alpha_{4}$ |  2205 |  0.72 |  5.24 |  21.01 |  2.58 |  4.13 |  2.32 |  562.4
$\alpha_{5}$ |  2564 |  0.78 |  $\dots$ |  24.43 |  2.79 |  4.11 |  $\dots$ |  $\dots$

We have observed oscillation in MR when the field is ramped from $11-14$ T, up
to $8$ K temperature beyond which it is not discernible. The background
subtracted SdH oscillation ($\Delta R_{xx}$) is shown in Fig. 5(a). It is
quite obvious from the shape of the oscillation that it possess multiple
frequencies. A fast Fourier transform (FFT) of the oscillation revealed as
many as $5$ fundamental frequencies at 461 T ($F_{\alpha_{1}}$), 871 T
($F_{\alpha_{2}}$), 1640 T ($F_{\alpha_{3}}$), 2203 T ($F_{\alpha_{4}}$) and
2562 T ($F_{\alpha_{5}}$), as shown in the inset of Fig. 5(a). The frequencies
may correspond to five pockets in the Fermi surface [32, 33], two hole pockets
at $\Gamma$ point and three electron pockets at $X$ points in the Brillouin
zone.

The extremal cross section area ($A_{\alpha_{i}}$) of these Fermi pockets are
calculated from the Onsager relation: $F_{\alpha_{i}}=(\hbar
A_{\alpha_{i}}/2\pi e)$, see Table 1. The oscillatory part $\Delta R_{xx}$ can
be described by Lifshitz-Kosevitch (LK) expression [34]. The temperature
dependent amplitude, obtained from FFT of $\Delta R_{xx}$, is shown in Fig.
5(b), that shows the oscillation amplitude of individual frequency decreases
with increase in temperature, and it follows thermal damping factor $X/sinhX$,
where $X=(2\pi^{2}m^{*}_{\alpha_{i}}k_{B}T)/(e\hbar B)$. Here, index $i$
represents i-th frequency, $m^{*}_{\alpha_{i}}$ is the effective mass
corresponding to the frequency $F_{\alpha_{i}}$. Fig. 5(c) shows mass plot for
different frequencies, and the extracted effective masses are given in Table
1. Estimated effective mass of carriers are very similar to other rare-earth
monopnictides [30, 33, 17]. Also, the field induced oscillation amplitude
damping follows
$exp[-(2\pi^{2}m^{*}_{\alpha_{i}}k_{B}T_{D_{\alpha_{i}}})/(e\hbar B)]$, where
$T_{D_{\alpha_{i}}}$ is the Dingle temperature of $F_{\alpha_{i}}$, and
$\frac{1}{B}=\frac{1}{2}(\frac{1}{B_{1}}+\frac{1}{B_{2}})$ where $B_{1}$ and
$B_{2}$ represents the range of applied magnetic field used. As multiple
frequencies results to such $\Delta R_{xx}$ oscillation, FFT amplitude
$(A_{\Delta R})$ of individual frequency in different field segments can give
the field dependent amplitude variation. From the $ln[A_{\Delta R}sinh(X)/X]$
vs. $1/B$ fit, see Fig. 5(d), obtained Dingle temperatures are used to
calculate the quantum relaxation time $\tau_{\alpha_{i}}(=\hbar/2\pi
k_{B}T_{D_{\alpha_{i}}})$ and quantum mobility
$\mu_{Q_{\alpha_{i}}}(=e\hbar/2\pi k_{B}T_{D_{\alpha_{i}}}m^{*}_{\alpha_{i}})$
of carriers for different pockets, listed in Table 1. Here $F_{\alpha_{5}}$ is
relatively large to extract the field dependent amplitude variation within
applied field range. A much lower temperature and high magnetic field are
necessary to investigate further.

Figure 5: (a) Oscillating part of electrical resistivity of GdBi as a function
of inverted magnetic field, (inset) Fast Fourier transform of oscillating part
of electrical resistivity. (b) FFT amplitude at different temperature. (c)
Mass plot of different frequencies. The solid lines are the fits to the
thermal damping factor of the Lifshitz-Kosevich expression. (d) Dingle plots
of different frequencies.

###  III.5 Specific heat

The heat capacity $(C_{p})$ of GdBi single crystal in zero magnetic field in
the temperature range $2$ to $200$ K is shown in the main panel of Fig.6(a).
The sharp $\lambda$ shaped peak at $27.5$ K is observed confirming the
antiferromagnetic ordering in this compound. The heat capacity attains the
value of 49.89 J K-1mol-1 at $200$ K which is the expected Dulong-Petit
limiting value of $3nR$. The field dependence of heat capacity is shown in the
inset of Fig. 6(a). It is evident that the magnetic field does not have any
effect on the magnetic ordering. Typically, in antiferromagnetic materials the
long range interaction competes with the applied magnetic field, resulting in
the lowering of the ordering temperature. As we have already seen in the
$M(H)$ data, the $4f$ moments align to the applied field direction in an
extremely slow rate and the field induced ferromagnetic state is achieved at
high magnetic field of $42$ T [25]. Hence fields up to $8$ T do not have any
effect to the heat capacity peak.

Figure 6: (a) Specific heat $(C_{p})$ of GdBi and LuBi as function of
temperature, (inset) $C_{p}$ vs. Temperature at different magnetic field. (b)
(top) $C_{4f}/T$ versus temperature and (bottom) magnetic entropy as a
function of temperature.

Next, we estimate the magnetic contribution $C_{4f}(T)$ to the heat capacity
of GdBi, to accomplish this we have grown the single crystal of LuBi which
possesses the same rock-salt type cubic crystal structure and a completely
filled $4f$-shell. Due to the difference in atomic mass of Gd and Lu, obtained
$C_{p}$ of LuBi cannot completely mimic the lattice contribution of GdBi. A
correction to the $C_{p}$ is required [35] to obtain the correct value of
specific heat. The correction factor in this case is calculated to be 1.02,
that is very small and here neglected. After subtracting the lattice and
electronic part, obtained magnetic specific heat of GdBi is shown in
Fig.6(b)(bottom) as $C_{4f}/T$. A broad hump is observed in the $C_{4f}/T$
plot at low temperature. Such type of hump arises for the systems with
$(2S+1)$-fold degenerate ground state with large $S$-values [36]. For large
$S$ the entropy is large, in order to accommodate the increased entropy a hump
appears in the heat capacity. For $S=7/2$ systems according the mean field
theory the hump $C_{4f}(T)$ data appears at $T~{}\leq~{}T_{\rm N}/3$ [36]. The
hump appears at $9.2$ K, which agrees well with the MFT for $T_{\rm N}=27.5$ K
in GdBi. The magnetic entropy $S_{4f}$ attains a value of 13.02 J K-1mol-1 at
$T_{\rm N}$ which is $75\%$ of $R$ln$(2S+1)=17.3$ J K-1mol-1 for $S=7/2$. The
reduction in the entropy may be attributed to the inaccurate estimate of the
lattice contribution to the heat capacity.

##  IV Conclusion

High quality single crystal of GdBi has been grown by high temperature
solution growth. We have performed a systematic study on the magnetic and
electrical transport properties. From the SdH quantum oscillations we have
analysed the Fermi surface properties. The magnetic measurements revealed that
GdBi undergoes an antiferromagnetic transition at $27.5$ K. The isothermal
magnetization $M(H)$ did not show any sign of saturation and reached value of
about $1.25$ $\mu_{\rm B}$/Gd at $7$ T, where as the saturation moment is $7$
$\mu_{\rm B}$/Gd. The electrical resistivity confirmed the antiferromagnetic
ordering by displaying a sharp drop in the resistivity at $27.5$ K and the
overall resistivity behaviour was typical metallic like. With application of
magnetic field the electrical resistivity displayed a huge upturn in the
magnetically ordered state at low temperature as observed in most of the
compensated semimetallic systems. The MR also displayed a large value of the
order 104% without any sign of saturation. Hall effect studies revealed
multiple type of charge carriers and a near compensation of the charge
carriers. From the SdH oscillation studies we have estimated the effective
masses of the observed five different frequencies which are almost of the same
order as observed in other rare-earth monopnictides. A high field MR
measurements at low temperatures and fields greater than $14$ T will throw
more light on the observed non-saturating magnetoresistance.

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