# Magnetotransport Properties and Fermi Surface Topology of Nodal line
Semimetal InBi

Sambhab Dana, Kuldeep Kargetib, R. C. Sahooc, Shovan Dand, Debarati Pala,
Sunil Vermaa, Sujay Chakravartye, S. K. Pandab, S Patila
<EMAIL_ADDRESS>aDepartment of Physics, Indian Institute of Technology
(BHU), Varanasi 221005, India  bDepartment of Physics, Bennett University,
Greater Noida 201310, Uttar Pradesh, India  cDepartment of Chemical Science
and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo
152-8552, Japan  dDepartment of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400005, India  eUGC-DAE Consortium for Scientific Research, Kalpakkam
Node, Kokilamedu 603104, India

###### Abstract

In the present study, we have discussed the up-turn behavior in the
resistivity pattern of the topological nodal line semimetal InBi. We argued
that such nature could be generalized with a mathematical model, that can be
applied to any compounds exhibiting similar behavior. The extremely high
magnetoresistance (XMR) has also been explained by the carrier compensation in
the compound, estimated from the Hall conductivity. Moreover, from the study
of Subhnikov-de Haas (SdH) oscillation and density functional theory (DFT), we
obtained the complete three-dimensional (3D) Fermi surface topology of the
compound InBi. A detailed understanding of carriers’ behavior has been
discussed using those studies. We have also unfurled the topology of each
electron and hole pocket and its possible modulation with electron and hole
doping.

##  I Introduction

Dealing with exotic bandstructure of the topological nodal line semimetals
(TNLS) is flourishing field in modern condensed matter physics. The electronic
structure originating from the band theory describes the topological
classification of the materials. For example, in the case of a topological
insulator (TI), the surface band is topologically non-trivial, whereas the
bulk band remains in an insulating phase[1]. Unlike TI, a lot of metals and
semimetals also poses topologically non-trivial bulk band. Nevertheless, the
recently discovered Dirac and Weyl semimetals significantly divert the
attention toward topological semimetals[2]. Unlike the TI, Dirac and Weyl
semimetals show the Dirac cone in the bulk band, and conduction occurs through
its bulk state. In such a semimetallic phase, the conduction and valence band
touches at the Dirac Point (DP), which can be called a zero-dimensional nodal
point. But, in the case of TNLS, this nodal point becomes one dimension nodal
line extended either in a line or a loop in the momentum space. The compounds
having such features in the electronic bandstructure are called either nodal
line[3], nodal chain[4], nodal knot[5], nodal-link[6, 7, 8] or nodal-net[9,
10].

Recently discovered a few promising nodal line materials are ZrSiX (X= S, Se,
Te )[11, 12], Ag2S[13], AX2 (A = Ca, Sr, Ba; X = Si, Ge, Sn).[14], Ca3P2[15],
CaP3[16], CaTe[17], CaAuAs[18], CaAgX (X = P, As)[19], IrF4[4], Mg3Bi2[20],
XB2 (X = Ti, Zr)[21], SrAs3[22], Co2TiX (Si, Ge or Sn)[23], Ta3X (X = Al, Ga,
Sn, Pb)[24], X2Y (X = Ca, Sr, Ba; Y = As, Sb, Bi)[25], YCoC2[26], MnN[27] and
YH3[28] etc. As one dimensional band-crossing contains a larger density of
states compering to the nodal points; such materials show novel properties
like large spin Hall effect[29], long-range Coulomb interaction[30], and flat
Landau level[31]. etc. Interestingly, when spin-orbit coupling comes into the
scenario, nodal line of the material is either protected by inversion,
reflection, and nonsymmorphic symmetry or even gaped due to the lack of such
symmetries[3, 32, 33, 9, 34]. For example, reflection of the Ta atomic plane
protects the nodal line in XTaSe2 (X = Pb, Tl)[35, 36]. In the case of Cu3PdN,
nodal lines are protected by inversion symmetry[10]. For ZrSiS[11] and
InBi[37], nodal lines are protected by non-symmorphic symmetry. Although a
vast studies is done on ZrSiS, InBi remains the least studied compound till
the date. InBi has two distinct nodal lines protected by non-symmorphic
symmetry[37] and also having topologically non-trivial type-II Dirac cone
protected by four-fold cyclic symmetry[37]. The compound InBi exhibits unique
transport properties viz. extremely high magnetoresistance (XMR)[38], highly
anisotropic MR[38], and pressure-induced superconductivity[39]. The compound
also shows an upturn nature of the temperature (T) dependent resistivity
($\rho$)[38] in the presence of magnetic field (B), but a detailed discussion
on that topic is not available to the date. The quantum oscillation study of
the compound is also reported earlier[40], but a deep understanding of Fermi
surface topology and its role in transport is still lacking.

In this rapid communication, we report structural, transport, electronic
bandstructure and Fermi surface topology of InBi. The report is organized in
the following way. We discussed structural properties of the compound in Sec.
III.1. In Sec. III.2, we have studied the magnetotransport properties of InBi.
In this section, we have focused on the interesting result, viz. up-turn
nature in $\rho(T)$. We have given a mathematical model based on up-turn
behaviour. Our experimental result well justifies the model. Another aspect of
our study is establishing the role of electron and hole carriers in the
transport phenomena. Although the attentive studies have been performed
earlier[38], the important role of carrier concentration in the transport
studies was still missing. We bring some exciting results to address the
issue. The results are compelling and could unravel the origin of XMR in InBi.
In Sec. III.3, We have discussed the Fermi surface topology by Shubnikov-de
Haas (SdH) oscillation. We verified our experimental result by theoretical
analysis based on density functional theory (DFT). Our DFT result
satisfactorily matches the experimental data. Accumulating the empirical and
theoretical tools, we have successfully mapped the compound’s Fermi surface.
Such work gives a complete three dimensional (3D) visualization of the Fermi
surface. We provide a detailed discussion of each Fermi sheet (FS). The
modulation of each FS due to electron and hole doping and the FS’s role in
transport phenomena are also discussed rigorously. At the end of this section,
we have discussed about some important physical parameters derived from our
SdH oscillation.

##  II Experimental details and theoretical methods

We prepared single crystal InBi using the modified Bridgman method. Elemental
In (Alfa Aesar) and Te (Alfa Aesar), with at least 99.9$\%$ purity, have been
used as the starting materials. The stoichiometric amount of starting
materials was taken in an evacuated (10-6 mBar) quartz ampule. The ingots were
melted at 200∘ C in the quartz ampule and kept for 24 hr at that temperature.
The liquid of the ingots was subsequently cooled to 150∘ C with a cooling rate
of 20∘.hr-1. Then we cooled the molten material from 150∘ to 30∘ C in slow
cooling rate 3∘.hr-1. As the bottom edge of the quartz tube is a conical
shape, the nucleation process starts from the bottom point of the quartz tube,
and the crystallization process has been completed at this stage. The shiny
conical shape single crystal is obtained from the quartz tube. The small parts
of the crystal have been used for further characterization.

X-ray diffraction (XRD) of powder compound was performed for structural
analysis. We also performed XRD on compound’s cleaved surface to check the
single crystallinity. The cleaved surface’s XRD study identifies the
particular planes’ Miller indices. The Hall study of the compound was
performed on a rectangle-shaped sample with the four-point probe in the van
der Pauw configuration. The Hall study was completed using a 9T Physical
Property Measurement System(PPMS), and the Hall voltages were detected along
the direction perpendicular to the current flow in the sample plane. The MR
measurement was performed in the four probe puck in 16 T PPMS. During MR and
Hall measurements, the external magnetic field was applied perpendicular to
the ab-plane.

To interpret our experimental results, we have carried out density functional
theory (DFT) calculations using full potential linearized augmented plane-wave
based method implemented in WIEN2k code[41]. All the calculations are
performed in a non-spin polarized set-up with generalized gradient
approximation (GGA)[42] approach for the exchange-correlation functional. We
include spin-orbit coupling (SOC) into the system, and all result is obtained
from GGA+SOC calculation. In order to obtain a very accurate Fermi energy,
$R_{mt}K_{max}$ value is considered to be 7 and Brillouin zone integration is
done using tetrahedron method[43] with a sufficiently dense k-mesh ($14\times
14\times 14$). We have generated the 3D Fermi surface using wannier90[44] and
used xcrysden[45] for plotting.

##  III Results and discussion

###  III.1 Structural characterization

Figure 1: (a) The unit cell of InBi compound. (b) The BZ structure and the
high symmetric points along which DFT calculation is performed. (c) The
bandstructure along high symmeyric points. The band-1 (red) and band-2 (blue)
are only two bands that contribute at $E_{F}$. (d) The ab-planes of InBi. (e)
XRD that performed on c-axis. The XRD peaks are coming from the ab plane.
(Inset) The photograph of conical shape InBi single crystal. (f) The XPS
spectra of InBi

We performed a powder XRD to obtain the crystal structure of the compound. Our
study suggests that the compound crystallizes in p4/nmm (Space group no. 129)
crystal structure. The refinement of the powder XRD pattern reveals a =
5.01(2) $\AA$, b = 5.012(2) $\AA$ and c = 4.77(9) $\AA$. The unit cell of the
crystal indicating all atoms is shown in FIG. 1 (a). The shaded area in the
unit cell indicates a plane where all In atoms lie. The Brillouin zone (BZ)
structure indicating high symmetric points and corresponding electronic
bandstructure is shown in FIG. 1 (b) and (c). The bandstructure indicates that
only two bands crossed the Fermi level. We assigned these two bands as Band-1
and band-2. Band-1 and band-2 are highlighted in red and blue colour in FIG. 1
(c). The layered crystal structure of the compound is shown in FIG. 1 (d). The
shaded planes indicate the ab-plane that contain In and Bi atoms. FIG. 1 (e)
shows the XRD pattern taking from the ab-plane of the crystal. As the
diffraction occur only among the ab-planes, the diffractogram only contains
$(00l)$ peaks. The very sharp peaks of the XRD pattern indicate the good
crystallinity of the compound. The $(00l)$ peaks are also consistent with the
space group and unit cell parameters derived from the powder XRD. We shown the
photograph of our conical-shaped single crystal along with a centimeter scale
in the inset of FIG. 1 (e). The X-ray photoelectron spectroscopy (XPS) of InBi
is shown FIG. 1 (f). The spectra from $4f$ and $3d$ orbitals of Bi and In
atoms well verified the compound’s elemental composition. The XPS spectra also
agree with the experimental stoichiometric ratio of the elemental composition.

###  III.2 Magnetotransport properties

####  III.2.1 Magnetic field dependent $\rho(T)$

Figure 2: (a) $\rho(T)$ measured at 3–100 K with the application of 0–9 T
magnetic field. (b)After performing $\frac{\partial\rho(T)}{\partial T}$ (to
calculate turn-on temperature). The $T_{m}$ and $T_{i}$ are indicated in the
inset figure. (c) The variation of $T_{m}$ and $T_{i}$ with the magnetic
field. Triangle formed by $T_{m}$ and $T_{i}$ are shown by a pink shaded area.

Figure 3: (a) Calculated $MR\%$ with the application of 3 T, 6 T and 9 T
magnetic field. (b) Experimental $\rho(T)$ without any magnetic field and
differnt type of fitted functions. Data upto 20 K is shown in the inset
figure. (c) Experimental and simulated $\rho(T)$ with the application of the 6
T magnetic field. (d) Simulated $\rho(T)$ under the application of 0–9 T
magnetic field. Experimental $\rho(T)$ (shown as black dashed lines) are
plotted over the theoretically generated one. The $\rho_{turn-on}$ as a
function of $T_{m}$ is indicated by a solid magenta line. The
$\frac{\partial\rho(T)}{\partial T}$ of the theoretically generated patterns
are shown in the inset figure. (e) Variation of turn-on temperature as a
function of B and the fitted pattern. (f) Simulated $\rho(6,T)$ with the
variation of $\alpha$

.

Figure 4: 3D contour plot of simulated data. (a) $\rho(B,T)$ (b)
$\frac{\partial[\rho(B,T)]}{\partial T}$

The temperature dependent resistivity [$\rho(T)$] measured at 3–100 K is shown
in FIG.2(a). The resistivity of the compound at 3 K and 100 K are
0.5$\times$10-2 and 14.7 $\times$10-2 $\mu$$\Omega$m, respectively, which is
very close to the earlier reported result[38]. To calculate the RRR of our
compound, we performed our resistivity study upto 300 K. The RRR of the
compound based on the 3 and 300 K resistivity data is estimated to be 93. The
RRR of the compound in the earlier reported data lies between 91-271[38]. Our
study suggests that the RRR of our compound lies within the range of earlier
reported data. From FIG.2(a), we have observed that the resistivity of the
compound sharply increases with the increase of magnetic field, especially in
the low-temperature regime. Whene B = 0, resistivity of the compound have been
increased monotonically with temperature. On applying the high magnetic field,
the $\rho(T)$ rapidly decreases first, and then it increases with the increase
of temperature. Such kind of up-turn behaviour is also observed in other non-
trivial topological systems[46, 47, 48, 49, 50], where applying certain
magnetic field the $\rho(T)$ shows minima at a particular temperature
($T_{m}$).

We have shown the 1st order derivative $\frac{\partial[\rho(T)]}{\partial T}$
in FIG.2(b). Turn-on temperature is indicated in the figure as Tm, where the
slope of the $\rho(T)$ changes its sign from negative to positive. Apart from
turn-on behaviour, we have also identified the minima of the
$\frac{\partial[\rho(T)]}{\partial T}$, indicated as $T_{i}$ in FIG.2(b). The
$T_{i}$ of the compound is approximately estimated as 12 K for our compound,
and it changes negligibly with the change of applied magnetic field. We have
plotted the variation of $T_{m}$ and $T_{i}$ with the applied magnetic field
in FIG.2(c). The triangle formed by the $T_{m}$ and $T_{i}$ is one of the
vital features observed in the compensated semimetals[51]. The value of the
slope inside the shaded area of the triangle is negative, whereas the upper
outside region of the triangle shows the positive slope. For the case of
graphite and bismuth, the triangular phase diagram can be explained by the
inequality $\hslash/\tau\lesssim\hslash\omega_{c}\lesssim k_{\rm B}T$, where
$\tau$ is the electron-phonon scattering time and $\omega_{c}$ is the
cyclotron frequency[52]. In clean semimetals with low carrier density
$\tau^{-1}\ll k_{B}T/\hslash$, and hence, there exists a wide temperature-
field range where XMR and turn-on like behaviour in resistivity appear. On the
other hand, large carrier density and strong impurity scattering limit the MR
in conventional metals.

We have calculate the T-dependent MR from the $\rho(T)$ data. The MR of the
compound defined as

$MR=[\rho(B,T)-\rho(0,T)]/\rho(0,T)$ (1)

Here, $\rho(B,T)$ denoted the resistivity of the compound as a function of
magnetic field (B) and temperature (T). The compound’s $MR\%$ at 3-100 K is
shown in FIG.3(a). The $MR\%$ of the compound is very high at low
temperatures. It goes up to 2500% at 9 T. We have observed that $MR\%$ of the
compound reduces drastically from 0 to 20 K and it almost becomes negligible
above 40 K.

To observe the temperature dependency of $\rho(0,T)$, we have fitted
$\rho(0,T)$ with several functions [FIG.3(b)]. First, to observe the
temperature dependency at low temperature, we have fitted $\rho(0,T)$ as
$A+BT^{n}$ (where $A$ and $B$ are the constant) up to 20 K. The result from
the best-fitted parameter gives $n$ = 3. In the second step, we fixed $n$ = 2,
and fitted the $\rho(0,T)$ up to 20 K. However, the fitted data using $n$ = 2
is not well-matched compared to the previous case. Lastly, we have fitted
$\rho(0,T)$ in the whole temperature range, and the best-fitted parameter for
that scenario gives $n$ = 1.44. In the last case, the fitted data is well-
matched in high temperature, but it does not fit well in low-temperature
region. The observation of $n$ = 3 for our case indicates the departure of
pure electron dominated scattering. Generally, $n$ = 2 is observed for pure
electron-correlated dominated scattering[53]. Similar kinds of behaviours (n =
3) are also observed for other compensated semimetals viz., LaSbTe[54] and
LaBi[55]. Semimetal LaSb ($n$ = 4)[48], elemental yttrium[56] and transition-
metal carbide[57] also shown interband electron-phonon scattering.

####  III.2.2 origin of turn-on behaviour

The MR of the compound based on the Kohler’s law[58, 59] is defined as

$MR=\alpha[B/\rho(0,T)]^{m}$ (2)

Putting the Eq. 1 in to the Eq. 2, we get

$\rho(B,T)=(\alpha B^{m}/[\rho(0,T)]^{m-1})+\rho(0,T)$ (3)

Using Eq. 3, we have simulated the $\rho(T)$ under the 6 T magnetic field.
During the simulation, we have fixed $\alpha$ = 1.2$\times$10-13
($\Omega$mT-1)1.6 and m = 1.6. Ideally, m = 2 for the perfectly compenseted
system but, our result m = 1.6 suggests that our compound slightly departs
from the perfectly compnsated case. The simulated patterns of $\rho(6,T)$
along with the experimental data are shown in FIG. 3 (c). Our study suggests
that simulated $\rho(6,T)$, derived from the experimental $\rho(0,T)$
reasonably matches the experimental $\rho(6,T)$. we have also predicted
$\rho(6,T)$ for all the corresponding cases of $\rho(0,T)$. Our simulated
$\rho(6,T)$ in FIG. 3 (c) follows the same colour code for the corresponding
cases of $\rho(0,T)$ in FIG. 3 (b). The result suggests that for $n$ = 3 and
2, the theoretically generated $\rho(6,T)$ matches well at low temperature but
deviates at high temperature. For the case of $n$ = 1.44, simulated
$\rho(6,T)$ fairly matches with experimental one at high temperature.

Using Eq. 3, we have simulated $\rho(B,T)$ for 0.5–9 T. We have plotted the
theoretically generated $\rho(B,T)$ in FIG. 3 (d). The result matches the
experimental data excellently. The experimental data are plotted as a black
dashed line over the corresponding simulated pattern. Such an analysis
strongly validates our model that finely replicate our unique $\rho(B,T)$. We
also shown the first order derivative of $\rho(B,T)$ in the inset of FIG. 3
(d) to calculate the turn-on temperature ($T_{m}$). Derived $T_{m}$ with the
corresponding magnetic field are potted in FIG. 3 (e). The variation of turn-
on temperature with respect to the magnetic field is fitted with the equation

$T_{m}=\zeta(B-B_{c})^{\nu}$ (4)

Such kind of formulation is theoretically predicted by D. V. Khveshchenko[60],
and validated for graphene and bismuth. For the graphite and bismuth the $\nu$
= 1/2. The best fitted parameters in our case give $\zeta$ = 12.75 K.T-0.5,
$B_{c}$ = 1.36 T and $\nu$ = 1/2. $\nu$ = 1/2 indicates that the turn-on
nature of our compound follows Khveshchenko’s rule.

In FIG. 3 (f), we simulate $\rho(B,T)$ with the variation of $\alpha$. In the
simulation, we fixed B = 6 T, and observe how the variation of $\alpha$
affects the turn-on behaviour. Our analysis suggests that the lower the
$\alpha$, the higher the critical magnetic field needed to produce turn-on
behaviour. Our analysis indicates that if $\alpha$ $<$ 0.48$\times$10-13
($\Omega$mT-1)1.6, we could not observe any turn-on behaviour even at 6 T. The
analysis suggests that there is an enormous role of $\alpha$ in producing
turn-on phenomena.

For further studies , we give some mathematical analysis regarding to the
turn-on behaviour. First, We define the resistivity minima at $T_{m}$ as
$\rho_{turn-on}$. To calculate the minima in $\rho(B,T)$, we take
$\frac{\partial[\rho(T,B)]}{\partial T}$ and make it zero at $T=T_{m}$. The
mathematics is followed as

$\frac{\partial[\rho(B,T)]}{\partial T}\arrowvert_{T=T_{m}}=0$ (5)

Solving the Eq. 5, we get

$\rho(0,T_{m})=\alpha^{1/m}(m-1)^{1/m}B$ (6)

To calculate the resistivity at turn-on temperature, we put $T=T_{m}$ in Eq.
3. The equation follows as:

$\rho(B,T_{m})=\alpha^{1/m}(m-1)^{1/m}\left(\frac{1}{m-1}+1\right)B$ (7)

From Eq. 4, we already know that the parameter $T_{m}$ is a explicit function
of B. Hence, putting the value of B from Eq. 4 to Eq. 7, we get $\rho_{turn-
on}$ as a function of $T_{m}$. After rewriting Eq. 7, we get

$\rho_{turn-
on}(T_{m})=\alpha^{1/m}(m-1)^{1/m}\left(\frac{1}{m-1}+1\right)\left(\frac{T_{m}^{2}}{\zeta^{2}}+B_{c}\right)$
(8)

Eq. 8 is the general formula we developed which valid for particular compound
that shows turn-on nature in $\rho(T)$ on the application of magnetic field.
Putting the value of $\alpha$ = 1.2$\times$10-13 ($\Omega$mT-1)1.6, $m$ = 1.6,
$\zeta$ = 12.75 K.T-0.5 and Bc = 1.36 T, Eq. 8 in our case becomes

$\rho_{turn-on}(T_{m})=\left[\frac{T^{2}_{m}}{163}+1.36\right]\times 10^{-8}$
(9)

We plot the Eq. 9 in Fig. 3 (d). The magenta line in the figure indicates the
$\rho_{turn-on}$. The simulated $\rho_{turn-on}$ based on Eq. 9 excellently
matches with the experimental results.

From the whole transport study, we want to give some conclusive remarks.
First, without the application of the magnetic field resistivity of the
compound follows as A +B$T^{n}$. At low temperature n = 3. But at higher
temperature n becomes 1.44. Secondly, based on Eq. 3, we have generated a
complete two dimensional data set (contains B and T ) that fully capable of
replicating the experimental data. We shown the simulated resistivity data and
its 1st derivative in a 3D contour plot in FIG. 4 (a)-(b). Thirdly, the
parameter $\alpha$ has a great role in producing the turn-on behaviour in
$\rho(T)$. Turn-on phenomena can be completely vanished if $\alpha$ becomes
lower than critical value. Finally, to calculate the resistivity at turn-on
temperature ($T_{m}$), we developed a generalized formulation that finely
simulate the experimental result. The mathematical formulation based on Eq. 8
is solely developed by ourselves.

####  III.2.3 Temperature dependent $\rho(B)$

Figure 5: (a) The $\rho(B)$ measured at 3–100 K with the application of 0–9 T
magnetic field. (b) The $MR\%$ as a function of B that is calculated from the
experimental $\rho(B)$. (c) The $\log-\log$ plot of $MR\%$ as a function of
$B/\rho(0,T)$. The fitted line based on the Kohler’s law is shown in a solid
magenta line. The fitted line for 3 K is separately shown in the inset figure.

The variation of $\rho(B)$ at several temperatures is shown in FIG. 5 (a). We
have shown that the change of resistivity with temperature is very high when
applied magnetic field is very low and $\rho(B)$ of all temperatures trend to
converge in a singular region at higher magnetic field. It is observed that
with the increase of B, $\rho(B,3)$ is growing firstly, and it almost catches
up the value of $\rho(B,15)$, $\rho(B,25)$ an $\rho(B,50)$ at 3 T, 4 T and 6 T
respectively. The MR% as a function of B is shown in FIG. 5 (b). The  MR% of
the compound is very high below 10 K and reaches up to 2500% at 3 K. On the
other hand, MR% at higher temperature reduces drastically and almost becomes
negligible above 25 K. We have also shown MR% vs $B/\rho(T,0)$ for all the
temperatures in FIG. 5 (c). The $\log-\log$ plot of all the data points falls
in a single straight line. The fitted line based on the Eq. 2 is shown by a
solid magenta line. During the fitting, we have put corresponding value of
$\alpha$ and $m$ as stated earlier. The fitted data at 3 K before the
$\log-\log$ plotting is shown in the inset of FIG. 5 (c). The falling of all
data points in a singular straight line also verified our earlier assumption
that the parameter $\alpha$ and $m$ are almost temperature-insensitive for our
compound.

####  III.2.4 Hall resistivity study

Figure 6: The $\rho_{xy}(B)$ from +9 T to -9 T (four-quadrant plot) at (a) 3
K, (b) 10 K, (c) 25 K and (d) 50 K. (e) The $\rho_{xy}(B)$ from 0–9 T with the
temperature range 3–50 K Figure 7: (a) $\sigma_{xx}(B)$ and its corresponding
fittings, (inset) logarithmic plot of $\sigma_{xx}(B)$ as a function of
magnetic field and temperature in a 3D contour plot. (b) $\sigma_{xy}(B)$ and
its corresponding fittings, (inset) logarithmic plot of $\sigma_{xy}(B)$ as a
function of magnetic field and temperature in a 3D contour plot. (c) The
temperature-dependent electron and hole density that is estimated from the two
carrier model. (d) The temperature-dependent electron and hole mobility that
is estimated from the two carrier model.

The Hall effect study in 0–9 T magnetic field at different temperature are
shown in FIG. 6. From the figure, $\rho_{xy}(B)$ shows a nonlinear nature at
all temperatures. Such kind of behaviour suggests that both carriers (electron
and hole) are present in the transport phenomena. We have also observed that
at T $<$ 25 K, $\rho_{xy}(B)$ increases rapidly. It is shown that on
application of 9 T magnetic field, $\rho_{xy}$ at 3 K is almost twice
comparing to the $\rho_{xy}$ at 25 K. Interestingly, the change in  MR%
calculated from $\rho_{xx}$ also twice in the same temperature and magnetic
field.

Using experimental result of $\rho_{xx}(B)$ and $\rho_{xy}(B)$, we have
calculated the $\sigma_{xx}(B)$ and $\sigma_{xy}(B)$. After the inverting of
the resistivity matrix, $\sigma_{xx}(B)$ and $\sigma_{xy}(B)$ can be written
as

$\sigma_{xx}(B)=\frac{\rho_{xx}(B)}{[\rho_{xx}(B)]^{2}+[\rho_{xy}(B)]^{2}}$
(10)

$\sigma_{xy}(B)=\frac{\rho_{xy}(B)}{[\rho_{xx}(B)]^{2}+[\rho_{xy}(B)]^{2}}$
(11)

Based on the Eq. 10 and 11, calculated $\sigma_{xx}(B)$ and $\sigma_{xy}(B)$
are plotted in FIG. 7 (a) and 7 (b).

We have fitted $\sigma_{xx}(B)$ and $\sigma_{xy}(B)$ with two band two carrier
model[61] stated as

$\sigma_{xx}(B)=e\left[n_{h}\mu_{h}\frac{1}{1+(\mu_{h}B)^{2}}+n_{e}\mu_{e}\frac{1}{1+(\mu_{e}B)^{2}}\right]$
(12)

$\sigma_{xy}(B)=eB\left[n_{h}\mu_{h}^{2}\frac{1}{1+(\mu_{h}B)^{2}}-n_{e}\mu_{e}^{2}\frac{1}{1+(\mu_{e}B)^{2}}\right]$
(13)

As we stated earlier that both the carrier types contribute to our transport
phenomena, we have performed the global fitting based on the two carrier model
where simultaneous presence of electron and hole carrier is considered. Here
electron density ($n_{e}$), hole density ($n_{h}$), electron mobility
($\mu_{e}$) and hole mobility ($\mu_{h}$) are termed as a fitting parameters.
The fitted lines are shown as a solid red line in FIG. 7 (a) and 7 (b).
Extracted parameters from the Eq. 12 and 13 give almost same result. The
fitting parameters viz., $n_{e}(T)$, $n_{h}(T)$, $\mu_{e}(T)$ and $\mu_{h}(T)$
are shown in FIG. 7 (c) and 7 (d).

From FIG. 7 (c), it is observed that $n_{e}(T)$ is slightly higher at 3 K but,
as a broad view the $n_{e}(T)$ and $n_{h}(T)$ are approximately same ($\sim$
1.5$\times$10-26 m-3) below 25 K. After 25 K, $n_{e}(T)$ almost remains the
same and $n_{h}(T)$ increases to $\sim$ 7$\times$10-26 m-3. The results
suggest that the number of electrons and holes contributes equally at lower
temperature whereas, at higher temperatures, the transport phenomena are
slightly dominated by holes. Observing the MR data from FIG. 5 (b), we
observed that MR% also very high bellow 25 K. Such kind of correlation
indicates that the phenomena XMR is accompanied by the electron-hole
compensation of the compound. Regarding to the carrier mobility, we have
observed that the electron mobility of the compound at 3 and 15 K are 3.98 and
0.29 m2V-1s-1 respectively. The hole mobility at 3 and 15 K are 25 and 15
times lower than the corresponding electron mobility at that temperature. The
effective mobility ($\mu_{eff}$ $\sim$$\sqrt{\mu_{e}\mu_{h}}$) at 3 K is 0.79
m2V-1s-1, which is consistent with the earlier reported data[38].

###  III.3 Shubnikov-de Haas oscillation and Fermi surface topology study

Figure 8: (a) The electron pockets come from band-1. The $\gamma$ and $\alpha$ sheets are lying at high symmetric Z and R points respectively. (b) The hole pockets come from band-2. The $\beta$ and $\delta$ sheets are lying at high symmetric $\Gamma$ and M points respectively. Figure 9: The bandstructure of the band-1 is shown in red. The allowed states marked by red shaded area denoted the electron-like pocket. The $\gamma$ and $\alpha$ pockets are generated from the band-1. The bandstructure of the band-2 is shown in blue. The allowed states marked by blue shaded area denoted the hole-like pockets. The $\beta$ and $\delta$ pockets are generated from the band-1. Figure 10: Fermi sheets in kx-ky plane at several kz cut. (a) Several kx-ky plane in 3D BZ where the 2D contours are plotted. (b)-(d) The contours in blue coming from band-2. the areas of $\beta$ and $\delta$ sheets are continuously decreasing from CUT-1 to CUT-4. The arrow mark in CUT-1 indicates the extrimum area of the corresponding pockets. (f)-(i) The contours in red coming from band-1. the areas of $\alpha$ and $\gamma$ sheets are continuously increasing from CUT-5 to CUT-8. The arrow mark in CUT-8 indicates the extrimum area of the corresponding pockets. Figure 11: The FFT spectra and SdH oscillation. (a) The FFT spectra calculated at 7-15 T field. The red arrow indicates the extrimum orbits of the corresponding pockets from where frequency peaks are coming. (b) The actual MR data from where FFT is performed. (c)-(e) The deconvoluted spectra coming from corresponding pocket at 3 K.  Figure 12: Data taken from Meyer et al.[40]. Angle dependent variation of three major frequencies $F{{}_{\beta}}$, $F{{}_{\gamma}}$ and $F{{}_{\delta}}$ Figure 13: The veriation of Fermi sheets from hole doping to electron doping. (a)-(e) The evolution of 3D BZ from hole doping to electron doping. (f) The shape of the $\gamma$ pocket at E = EF. (g)-(i) The evolution of $\alpha$ pocket from hole doped to electron doped. The extrimum orbit parallel to kx-ky plane is shown by the red arrow. The substantial hole doping split the $\alpha$ pocket by two parts. (j) The shape of the $\beta$ pocket at E = EF. (k)-(m) The evolution of $\beta$ pocket from hole doped to electron doped. The extrimum orbit parallel to kx-ky plane is shown by red dotted line. Table 1: Fermi sheets properties FS pocket | Origin | Type |  | Experimental SdHa
---
Frequency (T)
| Experimental
---
areab (nm-2)
| Theoretical
---
area(c) (nm-2)
$\beta$ | Band-2 | Hole | 330 | 3.1 | 3.4
$\delta$ | Band-2 | Hole | 543 | 5.1 | 5.3
$\alpha$ | Band-1 | Electron | Merged with $\beta$ | – | 3.3
$\gamma$ | Band-1 | Eectron | 792 | 7.5 | 6.7
| (a) At 3 K and B along the c-axis, (b) Area derived from our SdH frequency
---
(c) Extremum area (from our DFT calculation) in kx-ky plane

####  III.3.1 The FS topology of InBi

Before introducing the experimental part, we discussed the basic structure of
Fermi surface of InBi. We have shown the 3D Fermi surface in FIG. 8. Our study
suggest that only two bands contribute at the Fermi level. The FS coming out
from band-1 creates two distinct pockets. In FIG. 8 (a), we marked these
pockets as $\gamma$ and $\alpha$. The $\gamma$ and $\alpha$ pockets are placed
at high symmetric points Z and R respectively. Similarly, the FS coming out
from band-2 also construct two pockets. We have termed these two pockets
$\beta$ and $\delta$ as indicated in FIG. 8 (b). The $\beta$ and $\delta$
pockets lie at high symmetric points $\Gamma$ and M respectively. Energy
dispersion of the band-1 and band-2 are shown in FIG. 9 where allowed electron
and hole-like states are shown in red and blue shaded areas respectively. FIG.
9 indicates that $\gamma$ and $\alpha$-pockets coming from band-1 are electron
types. Similarly, $\beta$ and $\delta$-pockets coming from band-2 are hole
types.

After identifying all the electron and hole pockets, we have calculated the
extremum area of these pockets. To draw the extrema, we take several cuts in
$k_{x}-k_{y}$ plane in 3D BZ. The contours from several cut perpendicular to
the $k_{z}$ axis are shown in FIG. 10. The red and blue contours come from
band-1 and band-2 respectively. FIG. 10 (b)-(e) suggests that the extremum
area of $\beta$ and $\delta$ sheets lie at the $\Gamma$ point. The cut
corresponding to the $\Gamma$ point is shown in FIG. 10 (b) where maxima of
the Fermi surface are indicated by a arrow. It is visible that the area of the
$\beta$ and $\delta$ sheets continuously decreases from CUT-1 to CUT-4.
Similarly, from CUT-5 to CUT-8, the area of the $\gamma$ and $\alpha$ sheets
continuously increase. The maximum area of $\gamma$ and $\alpha$ sheets lies
at Z-point (CUT-8). The extremum area of $\gamma$ and $\alpha$-sheets are
indicated by the arrow in FIG. 10 (i). The extremum area of all four Fermi
sheets is listed in TABLE-1.

####  III.3.2 Experimental verification

The quantum oscillation work of InBi was performed by Y. Saito[62] in 1964. In
this same work, the author performed de Haas-van Alphen (dHvA) oscillation for
InBi. After Saito’s work, many authors performed the quantum oscillation
study, but none of them successfully resolved the nature of electron and hole
pocket from the oscillatory data. In 1973 Meyer et al.[40] performed dHvA
study of InBi in all crystallographic directions. We have shown a part of
Meyer’s result in FIG. 12 as a reference. The author predicts that $F_{\beta}$
and $F_{\gamma}$ are originating from the InBi and $F_{\delta}$ is originating
from the impurity band. Observing the angular variation of dHvA oscillation,
Meyer et al. first predicted that the 3D topology of the FS originated from
$F_{\beta}$ and $F_{\gamma}$ should be distorted ellipsoid. From the angle-
dependent data, they cannot reveal the origin of each FS from the BZ
structure. We brought the Meyer’s result as it plays a crucial role in the
context of our current study. We started our experiment with magneto-transport
studies viz. MR at low temperature and high magnetic field. To get the SdH
signal, we measured MR at 3 K, 5 K, 10 K, and 15 K in the B-field 7–15 T. The
MR data and corresponding FFT spectra are shown in FIG. 11 (b) and (a)
respectively. All FFT peaks named as $F_{\beta}$, $F_{\delta}$ and
$F_{\gamma}$ are highlighted in FIG. 11 (a). The time scale spectra for
corresponding frequencies are shown in FIG. 11 (c)-(e). We observed that there
is a considerable amount of beat formation in time scale spectra. The all
beats originate from the finite broadening of the FFT peaks. The formation of
beats in the oscillation is an acceptable phenomena for any compound[63]. The
detail discussion of each frequency peaks is given in the subsequent section.

####  III.3.3 Frequency from $\beta$-sheets ($F_{\beta}$)

The $F_{\beta}$ of FFT spectra is observed at 330 T as shown in FIG. 11 (a).
The extremum area of the corresponding FS is close to our theoretical
calculation (see the TABLE-1). As shown in FIG. 13 (j), the 3D shape of
$\beta$-pocket looks like a distorted ellipsoid structure. The angle dependent
study of $F_{\beta}$ by Meyer et al. is shown by a red dashed line in FIG. 12
It indicates that if we change the crystal orientation from [001] to [100],
$F_{\beta}$ changes from 330 T to 1052 T. The angular dependency of
$F_{\beta}$ suggests that the semi-major axis of the pocket is 3.3 times
longer than its semi-minor axis. Such kind of experimental result is very
consistent with our theoretically generated FS structure. Our theoretical
calculation also suggests that if we move the Fermi level slightly upward
(electron dope), the pocket size becomes lower. Our system might behave as an
electron-doped compound, as it shows slightly lower FS area than the
theoretical one.

####  III.3.4 Frequency from $\delta$-sheets ($F_{\delta}$)

The $F_{\delta}$ of the FFT spectra is observed at 543 T as shown in FIG. 11
(a). Origin of the particular frequency was not established earlier. Our study
suggests that the $F_{\delta}$ comes from the distorted triangular shape FS
lying at the M point. This FS has a four-fold symmetry as shown in FIG. 13
(k)-(m). The FIG. 13 (l) indicates that at E = $E_{\rm F}$ the four petals are
connected and electrons can move through the FS to make a complete loop. The
extremum orbit of the electron path is marked by a red dashed line. There is
an additional possibility that electrons can circulate within a single petal
of FS as the linkage between two petals is infinitesimally small. Our study
suggests that any finite upward movement of $E_{\rm F}$ can disconnect the
linkage among the petals. We show an example in FIG. 13 (m) where a 10 meV
upward shift of $E_{\rm F}$ (electron-doped) breaks the connection among
petals. As our system behaves as an electron-doped (as discussed earlier), we
can easily assume that the four petals are disconnected from each other and
electron move through the extremum orbit within a single petal as shown in
FIG. 13 (m). The theoretically calculated extremum area of a single petal is
very close to the experimental one which unambiguously supports the claim
stated earlier. The DFT calculation slightly overestimates the FS area,
indicating the electron-doped system.

The structure of the $\delta$ pockets in different angle are shown in FIG. 13
(c). Observing the 3D nature of the pockets we can say, area of the pocket in
$k_{x}-k_{y}$ plane is much larger than the area along $k_{x}-k_{y}$ plane.
Interestingly, $F_{\delta}$ from angle dependent dHvA changes according to
this FS topology. The angular variation of $F_{\delta}$ is shown by the dashed
magenta line in FIG. 12. The data suggest that when crystal lies in [001],
$F_{\delta}$ has its highest value. But with the tilting towards [100],
$F_{\delta}$ started to decrease. As the area of the pocket in $k_{x}-k_{y}$
is very small, no signal was observed along [100] direction. Such an angle
dependent study again strongly verify the nature of the FS.

####  III.3.5 Frequency from $\alpha$-sheets ($F_{\alpha}$)

The frequency corresponding to the $\alpha$ sheet is not observed by us
because the extremum area of the $\alpha$ and $\beta$ sheets are very close to
each other (see TABLE-1) and hence, both frequencies are superimposed to each
other. We show how the FS evolves from hole-doping to electron-doping in FIG.
13 (g)-(i) for the completeness of our study. We observed that a 100 meV
downward shift of $E_{\rm F}$ (hole doping) can break the $\alpha$ sheet into
two equal parts. On the other hand 100 meV upward shift of $E_{\rm F}$
(electron doping) enhance the volume of the FS. The extremum orbit in the
$k_{x}-k_{y}$ plane is indicated by the red arrow in FIG. 13 (h).

####  III.3.6 Frequency from $\gamma$-sheets ($F_{\gamma}$)

$F_{\gamma}$ is observed at 792 T in the FFT spectra as shown in FIG. 11 (a).
The FS area corresponding to $\gamma$-sheets is the largest and most intense
compared to all other pockets. The particular frequency comes from ellipsoid-
shaped FS lying at the Z-point of the BZ. The 3D shape of the $\gamma$-pocket
is shown in FIG. 13 (f) The angle dependent study of $F_{\gamma}$ is shown by
the blue dashed line in FIG. 12. It indicates that if we change the crystal
orientation from [001] to [100], $F_{\gamma}$ changes from 790 T to 1124 T.
The angular dependency of $F_{\gamma}$ suggests that the semi-major axis of
the pocket is 1.5 times longer than its semi-minor axis. Such kind of
experimental result is very consistent with our theoretically generated FS
structure. Unlike the $\beta$ and $\delta$, experimental FS area of
$\gamma$-pocket is slightly lower than the theoretically predicted one (see
TABLE-1). Lowering the FS area of any electron pocket is justifiable for an
electron-doped system. The amplitude of $F_{\gamma}$ shows the adequate
temperature dependency that help us to calculate the ‘effective mass’. All
parameters viz. ‘effective mass’ ‘Dingle temperature’ and ‘Berry phase’
derived from the $F_{\gamma}$ are discussed in details in the subsequent
section.

###  III.4 The study of effective mass, Dingle Temperature and Berry phase

Table 2: Berry phase ($\phi_{B}$) calculated from $F_{\beta}$, $F_{\delta}$ and $F_{\gamma}$. $\phi_{B}$ of the corresponding frequencies are listed for various $\Delta$. List of frequencies |  | Berry phase ($\phi_{B}$) |
---|---|---|---
$\Delta$ = -1/8 |  $\Delta$ = 0 |  $\Delta$ = 1/8
$F_{\beta}$ | -0.5$\times\pi$ | -0.76$\times\pi$ | 1.01$\times\pi$
$F_{\delta}$ | 0.13$\times\pi$ | -0.13$\times\pi$ | -0.38$\times\pi$
$F_{\gamma}$ | 0.41$\times\pi$ | 0.16$\times\pi$ | -0.06$\times\pi$
Table 3: Parameters derived from the $F_{\gamma}$ oscillation. $m^{\ast}$, effective mass; $T_{D}$, Dingle temperature; $\tau$, relaxation time and $\mu_{q}$, quantum mobility. | $m^{\ast}$
---
(Units of $m_{0}$)
| $T_{D}$
---
(K)
| $\tau$
---
(ps)
| $\mu_{q}$
---
($m^{2}V^{-1}s{-1}$)
0.40 | 1.1 | 1.2 | 0.53
Figure 14: (a) Calculation of the effective mass from T-dependent amplitude of
SdH oscillation. (b) The amplitude-simulation with varying $T_{D}$. (c) The
phase part of the L-K formula is fitted for Fβ. Simulated oscillations with
varying $\phi_{B}$ are indicated by the red and blue line.

The oscillation pattern of the resistivity is described by the Lifshitz-
Kosevich (LK) formula

$\Delta R\propto-B^{\lambda}R_{T}R_{\rm
D}R_{S}\sin\left[2\pi\left(\frac{F}{B}-\Gamma-\Delta\right)\right]$ (14)

where $R_{T}=\alpha T\mu/B\sinh(\alpha T\mu)$, $R_{D}=\exp(-\alpha
T_{D}\mu/B)$ and $R_{S}=\cos(\pi g\mu/2)$. $\mu$ is the ratio of the effective
cyclotron mass $m^{\ast}$ to free electron mass $m_{0}$. $T_{\rm D}$ is the
Dingle temperature, and $\alpha$ = $(2\pi^{2}k_{B}m_{0})/(\hbar e)$. The
oscillation phase is described as a sine term with an additional phase factor
$\Gamma-\Delta$, in which $\Gamma$ = $\frac{1}{2}$ \- $\phi_{B}/2\pi$ and
$\phi_{B}$ is the Berry phase. The dimensionality of FS is determined by the
phase shift $\Delta$. For 2D FS, $\Delta$ = 0. For 3D FS, $\Delta$ can take
value $\pm$ 1/8 according to the minima and maxima of the FS area. The term
$\lambda$ is also determined by the FS dimensionality. $\lambda$ can takes
value $\frac{1}{2}$ and 0 for 3D and 2D FS respectively. From the L-K formula,
effective mass $m^{\ast}$ can be calculated from the fit of the thermal
damping factor $R_{T}$ with the temperature dependence amplitude of the
oscillation. From the SdH oscillation, we have observed that an adequate
T-dependent signal comes only from Fγ. From the T-dependent amplitude-damping,
we have calculated the effective mass ($m^{\star}$) of the electron. The
fitting of RT for the Fγ is shown in FIG. 14 (a) The derived m⋆ from the
calculation is approximately 0.40$\times m_{0}$. Putting the value of
$m^{\star}$, we have theoretically estimated the Dingle temperature ($T_{\rm
D}$). The amplitude of the oscillation with varying $T_{\rm D}$ is shown in
FIG. 14 (b). We have also marked the beat formation phenomena (discussed
earlier) in the figure. The extraction of $T_{\rm D}$ from a beat forming data
was also done earlier[63]. Our estimated $T_{\rm D}$ from the fitted data is
approximately 1.1 K.

The Dingle temperature of the compound has a microscopic origin closely
related to the carrier’s scattering. Our low $T_{\rm D}$ indicates that sample
has low imperfections viz. dislocations and mosaic structure[63]. We have
fitted the phase part of the oscillation from the L-K formula for calculating
the Berry phase. As all Fermi sheets of InBi have the maximum area, $\Delta$
of our case is equal to -1/8. For the completeness of our study, we have
derived $\phi_{B}$ for $\Delta$ = 1/8, 0, and -1/8 for all the frequencies.
Derived Berry phase for Fβ, Fδ and Fγ are listed in TABLE 2. The estimated
$\phi_{B}$ from Fβ is very close to $\pi$ whereas $\phi_{B}$ coming from Fδ
and Fγ are close to zero. The $\pi$-Berry phase of the band indicates the non-
triviality of the FS. If the $\phi_{B}$ departure from $\pi$ to 0, the FS
becomes non-trivial to trivial. Our study suggests except $\beta$-sheet, all
other FS shows trivial nature. Using Dingle temperature $T_{\rm D}$ ($\sim$
1.1 K), we have calculated the quantum relaxation time $\tau_{q}=\hbar/2\pi
k_{\rm B}T_{\rm D}=1.20\times 10^{-12}s$ and quantum mobility
$\mu_{q}=e\tau_{q}/m^{\star}=0.53m^{2}V^{-1}s^{-1}$. All estimated parameters
from the Fγ are listed in TABLE 3. The quantum mobility ($\mu_{q}$) at 3 K is
very close to the compound’s effective mobility ($\mu_{eff}$) estimated from
Hall data at the same temperature. The consistent result suggests that our
experimental outcome from SdH oscillation and the Hall measurement is very
reliable. Our estimated quantum mobility is slightly lower than the classical
mobility. Such a phenomenon is expected because quantum mobility is sensitive
to both small-angle and large-angle scattering. On the other hand, classical
Drude’s mobility is only influenced by large-angle scattering[64].

##  IV Conclusion

We presented a detailed discussion of transport phenomena and the Fermi
surface topology of InBi. The turn-on phenomena of resistivity are intensely
studied. We gave an in-depth analysis of the origin of turn-on behavior that
justified our mathematical model. We have calculated the temperature-dependent
carrier density and mobility for both carriers. We observed electron-hole
carrier compensation follows up to 25 K. The high MR below 25 K indicates
carrier compensation is one of the primary origins of the XMR. We successfully
resolved the 3D structure of every electron and hole pocket with the help of
Shubnikov-de Haas oscillation. Our study reveals that two bands, each electron
and hole-like, majorly contribute to transport phenomena. The holes contribute
from one pocket lying at the central and four pockets lying at the corner of
the BZ. Similarly, the electrons come from one pocket lying at the body center
and four pockets at the edge of the BZ. The collective participation of each
electron and hole pockets makes the compound nearly perfect compensated semi-
metal at low temperature. The modulation of electron and hole pockets occurred
as if the compound behaved slightly electron doped at 3 K. The structure of
each Fermi sheets excellently matches with the theoretical result. The unique
Fermi surface topology governing the carrier compensation is closely
correlated to the magneto-transport phenomena of the compound.

###### Acknowledgements.

SP acknowledges the financial support from DST-SERB, India under the project
no. ECR/2017/001243. One of the authors Shovan Dan would like to thank SERB,
India (Grant No. PDF/2021/004624) for the financial assistance. We thank Dr.
Arkadeb Pal for helping in a scientific collaboration.

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