<EMAIL_ADDRESS>
# Band Structure and Fermi Surface Nesting in LaSb2

Evan O’Leary1,2, Lin-Lin Wang1,2, Yevhen Kushnirenko1,2, Ben Schrunk1,2,
Andrew Eaton1,2, Paula Herrera-Siklody1, Paul C. Canfield1,2 [  Adam
Kaminski1,2<EMAIL_ADDRESS>1Iowa State University, Department of Physics
and Astronomy, Ames, IA, 50014  2Ames Laboratory US Department of Energy,
Ames, Iowa 50011, USA

###### Abstract

We use high-resolution angle resolved photoemission spectroscopy (ARPES) and
density functional theory (DFT) to investigate the electronic structure of the
charge density wave (CDW) system ${\rm LaSb_{2}}$. This compound is among an
interesting group of materials that manifests both a CDW transition and lower
temperature superconductivity. We find the DFT calculations to be in good
agreement with our ARPES data. The Fermi surface of ${\rm LaSb_{2}}$ consists
of two small hole pockets close to $\Gamma$ and four larger pockets near the
Brillouin zone (BZ) boundary. The overall features of the Fermi surface do not
vary with temperature. A saddle point is present at $-0.19$ $eV$ below the
Fermi level at $\Gamma$. Critical points in band structure have more
pronounced effects on a materials properties when they are located closer to
the Fermi level, making doped ${\rm LaSb_{2}}$ compounds a potential
interesting subject of future research. Multiple peaks are present in the
generalized, electronic susceptibility calculations indicating the presence of
possible nesting vectors. We were not able to detect any signatures of the CDW
transition at 355 K, pointing to the subtle nature of this transition. This is
unusual, given that such a high transition temperature is expected to be
associated with the presence of a large CDW gap. This is confirmed through
investigation of the Fermi surface and through analysis of momentum
distribution curves (MDC). It is possible that changes are subtle and occur
below current sensitivity of our measurements.

##  I Introduction

The rare earth diantiminides have been known crystallographically since 1969,
but until the systematic growth of single crystals from excess Sb-solution
little was known about their physical properties [1, 2, 3]. The RSb2 (R = La -
Sm) series was first studied in single crystal form in with subsequent papers
studying effects of doping, pressure and high magnetic field [4, 5]. The
crystal structure of ${\rm LaSb_{2}}$ is orthorhombic and highly layered, with
a minor difference between the in-plane lattice constants [6]. ${\rm
LaSb_{2}}$ can be described as metallic-micaceous in that it readily
exfoliates into highly malleable 2-D like sheets[4]. As such it is very well
suited for ARPES experiments. ${\rm LaSb_{2}}$ is part of a special group of
materials that exhibit superconductivity and charge density wave (CDW) order
[6, 7, 8]. The coexistence of superconductivity and CDW order have been
studied previously in the transition metal dichalcogenide systems ${\rm
NbSe_{2}}$, ${\rm 1T-TaS_{2}}$, and ${\rm TaSe_{2}}$ [9, 10, 11]. Similarly to
${\rm LaSb_{2}}$ they are also highly layered. The superconducting transition
in ${\rm LaSb_{2}}$ occurs at Tc = 1.2 K, while two separate CDW transitions
are proposed at 355 K and 13 K [6, 7, 8]. Another interesting feature of ${\rm
LaSb_{2}}$ is the large, linear magneto resistance that does not saturate at
high fields [6].

The transport properties of ${\rm LaSb_{2}}$ have been extensively studied.
The resistivity of ${\rm LaSb_{2}}$ is expected to be anisotropic, based on
the crystal structure. This was confirmed by single crystal four-probe
resistivity measurements [4]. A more recent study claims that the single
crystal ${\rm LaSb_{2}}$ has an isotropic resistivity tensor [12]. The
unexpected isotropic resistivity tensor is explained by crystal imperfections.
Further, scanning tunneling microscopy measurements show that ${\rm LaSb_{2}}$
enters a superconducting state at Tc $\approx$ 1.2 K, while more recent
resistivity measurements place the transition at Tc $\approx$ 0.4 K [6, 5].
Unfortunately, this temperature is so far unattainable for ARPES measurements.

Several experiments have attempted to verify the CDW state in ${\rm LaSb_{2}}$
below 355 K, with evidence remaining fairly elusive [6, 8, 12, 5]. Resistivity
data indicates CDW order in ${\rm LaSb_{2}}$ through a dip and a hysteresis of
2 K at around 350 K [8, 5]. The CDW transition is more apparent and shifts to
roughly 150 K upon substitution of La with Ce to around 50% [8]. The
sharpening of the hysteresis is suspected to be caused by the introduced
disorder in the system. Single crystal X-ray diffraction on pure ${\rm
LaSb_{2}}$ showed peaks where systematic absences were expected, indicating a
mixing of the a- and b- directions potentially as a result of twinning or
stacking faults [12]. This is potential evidence that innate disorder in pure
${\rm LaSb_{2}}$ may be the origin of the CDW state. The resistivity
hysteresis may also be enhanced by applied pressure, increasing to a width of
50 K, and shifting to lower temperature until it is completely suppressed at
6-7 kbar [5, 13].

Another interesting property of ${\rm LaSb_{2}}$ is the large, linear
magnetoresistance that does not saturate up to a field of 45 T [6, 13, 8]. It
is speculated that scattering rate variations with applied magnetic field are
the cause of the linear magnetoresistance [14]. This is based on the minor
changes seen in the optical reflectivity when the magnetic field is varied.
The large linear magnetoresistance makes ${\rm LaSb_{2}}$ a good candidate for
magnetic field sensors [15].

The electronic band structure of ${\rm LaSb_{2}}$ has been previously
investigated using de Haas-van Alphen (dHvA) technique, ARPES , and density
functional theory calculations [16, 17]. The dHvA measurements showed a quasi-
two-dimensional Fermi surface with two nearly cylindrical extremal orbits, and
one small ellipsoidal orbit [16]. These measurements were in rough agreement
with the calculated Fermi surface. The DFT calculations seemed to agree well
with the observed Fermi surface measured above the proposed CDW transition.
The latest ARPES measurements showed that ${\rm LaSb_{2}}$ has multiple
topological nodal points, lines and surfaces, including eightfold degenerate
nodal points [18]. The multiple topological nodal structure is cited as
another potential explanation of the large linear magnetoresistance in this
material. We are not aware of any previous measurements of the band structure
near the CDW transition at 355 K, motivating this study.

##  II Methods

Large single crystals of ${\rm LaSb_{2}}$ were grown out of antimony flux [4,
2, 5]. Elemental La +(99.99+ Ames Laboratory) and Sb (99.999+ Alfa Aesar) were
combined in the ratio of ${\rm La_{5}Sb_{95}}$ and placed into the bottom part
of a 2 ml Canfield Crucible set (CCS) [19, 20]. The CCS was sealed into an
amorphous silica tube with silica wool on top of the CCS to act as a cushion
during the decanting process. The sealed ampule was heated over 3 hours to
1100°C, held at 1100°C for 5 hours and then cooled to 1000°C over 1 hour.
After sitting at 1000°C for 5 hours, the ampoule was cooled to 675°C over 99
hours. Upon reaching 675°C, the ampule was removed from the furnace and
decanted in a lab centrifuge [21]. After cooling to room temperature, the
ampule was opened, revealing large, sometimes crucible limited crystals of
${\rm LaSb_{2}}$.

The ARPES measurements were performed by cleaving ${\rm LaSb_{2}}$ in situ at
base pressure lower than 5 $\times$ 10-11 Torr. The 330 K and 360 K data were
measured from the same sample which was cleaved at 330 K. The 7 K data was
measured and cleaved at 7 K. A Scienta DA30 analyzer was used, in conjunction
with a tunable, picosecond Ti:Sapphire laser and fourth harmonic generator
operating at 6.7 $eV$ [22]. A second, fixed wavelength laser was also used in
conjunction with the fourth harmonic generator and operated at 7 $eV$. The
chemical potential was determined by measuring a polycrystalline Cu reference
sample that was in electrical contact with the ${\rm LaSb_{2}}$, and fitting a
Fermi function to the data. One data set, shown in Fig. 1 (c) was taken while
applying a -7 V bias potential to the sample. A copper disk of radius 1.2 cm
was electrically isolated from the rest of the ARPES system and the potential
was supplied by a DC Power Supply 9116 from BK Precision directly to the disk.
This technique has been tested and theoretically analyzed, where for a
comparable setup there were little to no distortions when the applied bias is
greater than -10 V [23].

Band structures of ${\rm LaSb_{2}}$ have been calculated in density functional
theory [24, 25] (DFT) using PBE [26] exchange-correlation functional with
spin-orbit coupling (SOC) and experimental lattice parameters. DFT
calculations were performed in VASP [27, 28] with a plane-wave basis set and
projector augmented wave [29] method. We used the primitive unit cell of 12
atoms with a $\Gamma$-centered Monkhorst-Pack [30] (11x11x3) $k$-point mesh
and a Gaussian smearing of $0.05$ $eV$ to converge charge density. A tight-
binding model based on maximally localized Wannier functions [31, 32, 33] was
constructed to reproduce closely the bulk band structure including SOC in the
range of $E_{F}\pm 1eV$ with La $sdf$ and Sb $p$ orbitals. Then the spectral
functions and Fermi surface of a semi-infinite ${\rm LaSb_{2}}$ (001) surface
were calculated with the surface Green’s function methods [34, 35, 36, 37] as
implemented in WannierTools [38]. The real part of susceptibility function has
been calculated on a dense k-point mesh of (160x160x40) to study Fermi surface
nesting.

##  III Results and discussion

Density functional theory calculations are presented in Fig. 1. The crystal
structure and the proposed cleaving plane, indicated by the arrow, is shown in
Fig. 1 (a). The termination occurs at the ${\rm(LaSb)_{2}}$ layer. Comparing
Fig. 1 (d),(e) with Fig. 2 (e),(h) respectively, shows this termination gives
a calculated band structure that has good agreement with our ARPES data. Both
show a saddle like dispersion centered on $\Gamma$. Comparison of Fig. 1 (b)
and (c), reveals that two small round pockets appear in both the DFT
calculations and ARPES data and have very similar shape. There are multiple
outer pockets predicted by the calculations, which are also seen in the ARPES
data. They are similar in shape and location, but ARPES intensity fades out in
the upper corner of Fig. 1 (c). The ARPES data appear to have good agreement
with the DFT calculations.

Critical points in the energy dispersion of 3-dimensional materials produces a
divergence in the density of states (DOS), and causes the derivative of the
DOS to be non-continuous, known as a van Hove singularity. When critical
points in electronic structure are near or coincide with the Fermi energy,
materials may present with interesting properties such as superconductivity,
magnetism, or charge and spin density waves [39, 40]. It is general practice
to try and tune the location of the van Hove singularity to a lower binding
energy. This can be accomplished either via doping or other mechanical methods
as in twisted graphene [40, 39]. In Fig. 2 we present ARPES data that reveals
the presence of a saddle point located at $\Gamma$. The portion of the pockets
discussed previously that are closest to $\Gamma$ make up the upwards
dispersion of the saddle point. The dispersions across the pockets for several
cuts are shown in Fig. 2 (c)-(g), showing an upwards dispersion. Fig. 2 (h)
shows a cut perpendicular through $\Gamma$ and shows a downwards dispersion.
In Fig. 2 (c)-(g), the bottom of the band is exactly the dispersion shown in
Fig. 2 (h). The energy distribution curve (EDC) shown in Fig. 2 (b), indicates
the location of the saddle point is $-0.19$ $eV$. Given the saddle point
occurs at a rather high binding energy, it is unlikely to be tuned to the
Fermi level. It is worth recalling that the CDW was enhanced as ${\rm
LaSb_{2}}$ was doped with Ce [8]. This enhancement could potentially be an
effect of the saddle point shifting lower in binding energy.

Charge density wave states occur when electrons and phonons become coupled,
causing the electron density to be periodically displaced. The period of the
CDW can be either commensurate or incommensurate with the period of the
lattice. A CDW that is commensurate with the lattice can cause small
displacements of the ions that compose the lattice. This disrupts the
periodicity of the lattice, leading to changes in the size and shape of the BZ
and potentially causing band folding. A CDW that is incommensurate with the
lattice is generally much weaker in its affects. Incommensurate CDWs may
interact with impurities in the lattice, leading to a pinning of the phase of
the CDW. Strong pinning can cause more pronounced affects such as lattice
distortions and large frequency dependent responses in the conductivity [41].
A CDW transition often leads to subtle or less subtle changes in the
electronic structure, such as band back folding caused by new periodicity, and
or development of new energy gaps due to Fermi surface nesting like what is
seen in the CDW material ${\rm 2H-NbSe_{2}}$ [42, 43]. We looked carefully for
these effects in the portion of the BZ accessible in our experiments.

Proposed nesting vectors predicted by DFT calculations are shown in Fig. 3
(a). The lengths of the nesting vectors were determined through investigation
of the electronic susceptibility, shown in Fig. 3 (b) and (c). A peak in the
electronic susceptibility may help to satisfy the Chan-Heine criterion for
strong electron-phonon coupling, increasing the probability of CDW formation
[44]. The nesting vectors connect parallel contours on the calculated Fermi
surface. The calculations provide insights on where potential effects due to
the CDW might be observed in ARPES data.

In Fig. 4 (a)-(c), the temperature evolution of the Fermi Surface (FS) is
shown. The only noticeable difference in the ARPES data through the transition
is a slight shift in intensity of the spectra. This can be seen in the cuts
along the X-$\Gamma$-X direction shown in Fig. 4 (d),(e). The band appears
sharper and more intense at 7 K, with high intensity at $-0.19$ $eV$. The
spectrum at 330 K shows a band with the same shape to the band in the 7 K
spectra, but with a much lower intensity. At 330 K the band has slightly
shifted to lower binding energy, relative to its position at 7 K. The shift of
the band is an effect of changing temperature and not caused by the CDW
transition. In Fig. 4 (h),(i) we examine one of the locations that DFT
calculations predict possible nesting vectors may form. From these data, we
see no changes indicative of Fermi surface nesting. An EDC taken at the band
crossing is shown in Fig. 5 (c), there is no indication of an energy gap
forming when entering the CDW state. Given the high transition temperature,
one would expect an energy gap to be easily detectable. We show MDCs at the
Fermi level in Fig. 5 (d), again no observable changes occur when going
through the CDW transition. To more precisely investigate any potential
changes due to the CDW, we studied changes to the quasi-particle lifetime
across the CDW transition. This was done by fitting the MDC, like the ones
shown in Fig. 5 (d), with Lorentzian functions for a range of binding
energies. As shown in Fig. 5 (e), there are no quantitative changes in the
width of the electron bands when going through the CDW transition,
demonstrating that the CDW did not cause any changes in the quasi-particle
lifetime in this portion of the BZ.

##  IV Conclusions

In summary, we preformed detailed ARPES measurements and DFT calculations to
investigate the electronic properties of ${\rm LaSb_{2}}$ that hosts high and
low temperature CDW states. The ARPES data agrees relatively well with DFT
calculations and shows a Fermi surface with two small pockets located near
$\Gamma$, and four larger pockets around the edge of the BZ. These results
agree with previous dHvA data, ARPES data and DFT calculations [16, 17]. No
visible changes in the band structure occur when going through the CDW
transitions. The most interesting feature was a saddle point predicted by DFT
calculations and confirmed by our ARPES measurements. The saddle point was
centered at $\Gamma$ and $-0.19$ $eV$ below the Fermi level. This is
relatively high in binding energy, but worth investigating further through
doped ${\rm LaSb_{2}}$ compounds.

Figure 1: Density functional theory band structure calculation, with predicted
surface termination for ${\rm LaSb_{2}}$. (a) Crystal structure of ${\rm
LaSb_{2}}$, arrow indicates termination plane. (b) Predicted Fermi surface for
(LaSb)2 termination, showing BZ labeling. (c) ARPES intensity at Fermi level
taken at 7 K integrated within 10 $meV$ of the Fermi cutoff. ARPES data was
taken with a bias potential of -7 V applied to the sample. (d),(e) Energy
dispersion cuts, compare with Fig. 2 (e),(h) respectively. Figure 2: ARPES
data showing saddle point. (a) Map shown at 7 K integrated within 10 $meV$
around the Fermi cutoff. (b) EDC taken at cut shown in panel (e) with peak
indicating the location of the saddle point. (c)-(h) Energy dispersion cuts
through lines in panel (a) showing a saddle point in the band structure.
Figure 3: Density functional theory electronic susceptibility calculation. (a)
Calculated band structure at the Fermi level. (b),(c) Electronic
susceptibility shown in momentum space, (b) shows largest peak in electronic
susceptibility, (c) shows second largest peak in electronic susceptibility.
Predicted nesting vectors are shown by arrows. Figure 4: ARPES intensity at
the Fermi level, at several temperatures across the CDW transition. (a)-(c)
Measured at 7 K, 330 K, and 360 K respectively. (d)-(i) Band dispersion at
cuts shown by red lines in panel (a). Figure 5: Comparison of bandwidth when
crossing through the CDW transition. (a),(b) Cuts taken of electron band
centered at $\Gamma$, at 335 K and 370 K. (c),(d) Energy distribution curve
and momentum distribution curve, respectively, through lines in panel (a). (e)
Full width at half maximum (FWHM) of the Lorentzian fit of MDCs, solid line is
FWHM of the left branch of the band, the dashed line is the right branch.
Color indicates temperature.

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