††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Bulk transport paths through defects in floating zone and Al flux grown SmB6

Yun Suk Eo<EMAIL_ADDRESS>Department of Physics, University of Michigan,
Ann Arbor, MI 48109 USA  Maryland Quantum Materials Center and Department of
Physics, University of Maryland, College Park, Maryland, USA  Alexa Rakoski
<EMAIL_ADDRESS>Department of Physics, University of Michigan, Ann Arbor, MI
48109 USA  Shriya Sinha  Department of Physics, University of Michigan, Ann
Arbor, MI 48109 USA  Dmitri Mihaliov  Department of Physics, University of
Michigan, Ann Arbor, MI 48109 USA  Wesley T. Fuhrman  Maryland Quantum
Materials Center and Department of Physics, University of Maryland, College
Park, Maryland, USA  Institute for Quantum Matter and Department of Physics
and Astronomy, The Johns Hopkins University, Baltimore, MD 21218 USA  Shanta
R. Saha  Maryland Quantum Materials Center and Department of Physics,
University of Maryland, College Park, Maryland, USA  Priscila F. S. Rosa  Los
Alamos National Laboratory, Los Alamos, NM 87545 USA  Zachary Fisk  Department
of Physics and Astronomy, University of California Irvine, Irvine, CA 92697
USA  Monica Ciomaga Hatnean  Department of Physics, University of Warwick,
Coventry, CV4 7AL, UK  Geetha Balakrishnan  Department of Physics, University
of Warwick, Coventry, CV4 7AL, UK  Juan R. Chamorro  Institute for Quantum
Matter and Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, MD 21218 USA  Department of Chemistry, The Johns Hopkins
University, Baltimore, MD 21218 USA  Seyed M. Koohpayeh  Institute for Quantum
Matter and Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, MD 21218 USA  Department of Materials Science and Engineering, The
Johns Hopkins University, Baltimore, MD 21218 USA  Tyrel M. McQueen  Institute
for Quantum Matter and Department of Physics and Astronomy, The Johns Hopkins
University, Baltimore, MD 21218 USA  Department of Chemistry, The Johns
Hopkins University, Baltimore, MD 21218 USA  Department of Materials Science
and Engineering, The Johns Hopkins University, Baltimore, MD 21218 USA  Boyoun
Kang  Department of Materials Science and Engineering, Gwangju Institute of
Science and Technology, Gwangju 61005 Korea  Myung-suk Song  Department of
Materials Science and Engineering, Gwangju Institute of Science and
Technology, Gwangju 61005 Korea  Beongki Cho  Department of Materials Science
and Engineering, Gwangju Institute of Science and Technology, Gwangju 61005
Korea  Michael S. Fuhrer  Monash University, Melbourne, Victoria 3800,
Australia  ARC Centre of Excellence in Future Low-Energy Electronics
Technologies, Monash University, Victoria 3800 Australia  Johnpierre Paglione
Maryland Quantum Materials Center and Department of Physics, University of
Maryland, College Park, Maryland, USA  Canadian Institute for Advanced
Research, Toronto, Ontario M5G 1Z8, Canada  Çağlıyan Kurdak  Department of
Physics, University of Michigan, Ann Arbor, MI 48109 USA

###### Abstract

We investigate the roles of disorder on low-temperature transport in SmB6
crystals grown by both the Al flux and floating zone methods. We used the
inverted resistance method with Corbino geometry to investigate whether low-
temperature variations in the standard resistance plateau arises from a
surface or a bulk channel in floating zone samples. The results show
significant sample-dependent residual bulk conduction, in contrast to smaller
amounts of residual bulk conduction previously observed in Al flux grown
samples with Sm vacancies. We consider hopping in an activated impurity band
as a possible source for the observed bulk conduction, but it is unlikely that
the large residual bulk conduction seen in floating zone samples is solely due
to Sm vacancies. We therefore propose that one-dimensional defects, or
dislocations, contribute as well. Using chemical etching, we find evidence for
dislocations in both flux and floating zone samples, with higher dislocation
density in floating zone samples than in Al flux grown samples. In addition to
the possibility of transport through one-dimensional dislocations, we also
discuss our results in the context of recent theoretical models of SmB6.

##  I Introduction

SmB6 is the oldest known Kondo insulator (KI), G. Aeppli and Z. Fisk (1992);
Riseborough (2000) in which strong correlations between the $f$ and $d$
electrons lead to the opening of a small hybridization gap at the Fermi energy
below about 100 K. The initial narrow-gap picture resulting from hybridization
of $f$\- and $d$\- bands was proposed by N. Mott in 1974. Mott (1974) Since
then, many reports have elaborated on this picture, but lingering problems
persisted. Experimentally, one problem was the resistivity saturation below
about 4 K which could not be explained with impurities or a minimum
conductivity model. Menth  _et al._ (1969); Allen  _et al._ (1979)
Theoretically, the Kondo hybridization that opened the gap had parity
violation at the high symmetry points. Martin and Allen (1979)

After nearly 40 years, shortly after the advent of three-dimensional
topological insulators (TI) in bismuth-based materials, these two lingering
mysteries of SmB6 were revisited and arguably solved by introducing a
topological band inversion. When the hybridization gap forms from two bands of
opposite parities, it leaves a surface conduction channel that is
topologically protected. Dzero  _et al._ (2010); Takimoto (2011); Dzero  _et
al._ (2012) Transport experiments confirmed that the surface is conductive
while the bulk is insulating. Wolgast  _et al._ (2013); Kim  _et al._ (2013,
2014) Experimental evidence supporting the TI proposal was also obtained via
methods including angle resolved photoemission spectroscopy (ARPES), Neupane
_et al._ (2013); Jiang  _et al._ (2013); Xu  _et al._ (2014); Denlinger  _et
al._ (2014) point contact spectroscopy, Zhang  _et al._ (2013) scanning
tunneling microscopy, Rößler  _et al._ (2016); Pirie  _et al._ (2020) and
inelastic neutron scattering. Fuhrman  _et al._ (2015) Conversely, some
reports favor a non-topological explanation for surface conduction in SmB6.
Hlawenka  _et al._ (2018); Hermann  _et al._ (2020); Frantzeskakis  _et al._
(2013)

Samples used in modern studies are grown either by the aluminum flux method or
the optical floating zone method. Single crystals can be grown below their
melting point by the Al flux method, which may enhance the stoichiometry of
the target sample by preventing vaporization of Sm at high temperatures.
Phelan  _et al._ (2016) Flux grown samples are small (a few mm in each
direction) and can contain inclusions of the flux. P. C. Canfield and Z. Fisk
(1992) In contrast, floating zone samples are grown at or above the melting
point, and the high temperatures used can introduce defects due to thermal
stresses Koohpayeh  _et al._ (2008) or through vaporization of Sm. Phelan  _et
al._ (2016) Floating zone samples are quite large (a few cm long) and are
uncontaminated by flux. In general, characterization methods like powder x-ray
diffraction show no obvious difference between samples grown by the two
methods. Hatnean  _et al._ (2013); Phelan  _et al._ (2016) However, there
appears to be a clear difference in the experimental results, especially at
low temperatures, when comparing Al flux and floating zone grown samples.
ARPES results that find evidence for a trivial surface in SmB6 were performed
on floating zone samples, Hlawenka  _et al._ (2018) while some of the most
compelling evidence for a topological surface comes from flux grown samples.
Xu  _et al._ (2014)

De Haas van Alphen (dHvA) quantum oscillations were also used to search for TI
states, Li  _et al._ (2014); Tan  _et al._ (2015); Xiang  _et al._ (2017);
Hartstein  _et al._ (2017) but instead they revealed deeper mysteries about
the bulk of SmB6 and the origin of the low temperature behavior. Reports
finding evidence for a 2D surface Li  _et al._ (2014); Xiang  _et al._ (2017)
used flux grown samples, but these results have also been attributed to
aluminum inclusions in the samples. Thomas  _et al._ (2019) Tan, $et~{}al.$
Tan  _et al._ (2015) observe bulk quantum oscillations in floating zone grown
SmB6 at low temperatures when the bulk gap is opened. In addition, heat
capacity and thermal conductivity on some floating zone grown samples show a
large residual density of states in the $T\rightarrow 0$ limit, which could
imply that charge-neutral fermions exist in the bulk. Hartstein  _et al._
(2017) However, flux-grown samples have never shown evidence for charge-
neutral quasiparticles in the bulk. Xu  _et al._ (2016); Boulanger  _et al._
(2018) These and other subsequent experimental and theoretical studies
attempting to resolve the bulk dHvA result take two opposite approaches. Bulk
quantum oscillations could be intrinsic to SmB6, for example due to charge-
neutral quasiparticles, or they could have an extrinsic origin, for example,
pockets of an unknown metallic phase.

Much of the theoretical work on quantum oscillations has focused on a possible
intrinsic origin. Some of these scenarios have included oscillations by
excitonic states J. Knolle and N. R. Cooper (2017); Chowdhury  _et al._ (2018)
or a Majorana fermion band that breaks gauge symmetry. Baskaran (2015); Erten
_et al._ (2016a) Others have proposed breakdown of the gap under magnetic
field, Erten  _et al._ (2016b); P. S. Riseborough and Z. Fisk (2017) or ways
for oscillations to occur in gapped systems based on the unhybridized band
structure or as an effect of the band edges. J. Knolle and N. R. Cooper
(2015); Zhang  _et al._ (2016)

The other possibility is that the quantum oscillations have an extrinsic
origin from disorder or impurities. In the presence of generic short-range
disorder, states from the conduction and valence band could spill into the
gap, which could be responsible for the oscillations. H. Shen and L. Fu
(2018); Harrison (2018) Alternatively, natural magnetic impurities could be
responsible for the excess heat capacity at low temperatures. Fuhrman  _et
al._ (2018) These local moments in the lattice would be screened, and the
amount of screening, and thus the magnetization, would oscillate in magnetic
field. W. T. Fuhrman and P. Nikolić (2018) Still another report focused on
nonmagnetic impurities, which were found to form a deep impurity band as in a
metal as well as an in-gap band, Abele  _et al._ (2020) and another proposal
revisited the idea of in-gap impurity states. Skinner (2019) Historically,
hydrogenic in-gap impurity states as are found in doped semiconductors were
proposed in SmB6. This model is unjustified in SmB6, Rakoski  _et al._ (2017),
one reason being that the standard hydrogenic impurity model relies on a
parabolic band structure. Instead, Ref. Skinner (2019) shows that the hybrid
band structure of SmB6 has its own model of hydrogen-like in-gap impurity
states. Interestingly, the density of defects required for an insulator-to-
metal transition is orders of magnitude higher than the required density for
the same in parabolic semiconductors.

These proposals for extrinsic sources of quantum oscillations have
experimental consequences well beyond bulk quantum oscillations. For example,
in transport, the nodal semimetal scenario would imply linear-in-T behavior in
the bulk resistivity at low temperatures. Harrison (2018) In the presence of
hydrogenic-like impurities, low-temperature bulk resistivity would be
dominated by an activated term corresponding to hopping in the impurity band;
the activation energy would be different from the one arising from the Kondo
gap. Skinner (2019) In our previous work, we used the inverted resistance
method to find the bulk resistivity even when the surface channel dominates
below about 4 K. We found that SmB6 grown by the aluminum flux method shows a
continuous exponential rise in resistivity of nearly 10 orders of magnitude
from 40 K to 2 K. Eo  _et al._ (2018) Samples grown with Sm deficient off-
stoichiometry still showed an exponential rise of 7-8 orders of magnitude, but
at about 2 K they reveal a bulk saturation distinct from the surface channel.
Eo  _et al._ (2019)

We previously argued that the resistivity values of this newly discovered bulk
channel at low temperatures are extremely high. In fact, such resistivity
saturation after a high magnitude increase is only seen in ultra clean
semiconductors. Debye and Conwell (1954); Chapman  _et al._ (1963) This would
correspond to a tiny conduction channel, which does not help resolve the
question of quantum oscillations. Nevertheless, understanding the origin of
this low-temperature bulk conduction is important for understanding the unique
role of disorder in SmB6. Previously, we speculated that the mysterious third
channel could be conduction through one-dimensional defects, or dislocations,
that are topologically protected. Eo  _et al._ (2019)

Dislocations have been studied extensively in semiconductor thin films such as
GaN, where they are a significant source of scattering in electronic devices
and provide recombination sites in optoelectronic devices. Ourmazd (1984) In
thin films, dislocations form during growth, especially at the interface
between a substrate and a film with different lattice constants. This lattice
mismatch between the two materials strains the layer, leading to the formation
of dislocations. Hall and Bean (1992) The density of dislocations present in
the film is related to the difference in lattice constants between the
substrate and the film, with lower dislocation density corresponding to more
closely matched lattice constants. Dislocations are also present in crystals.
They can form from internal stresses in the growth, especially stresses due to
thermal fluctuations, local impurities in the growth, or even vibrations in
the environment. Hall and Bean (1992) Impurities in the growth can provide
nucleation sites where dislocations start to form, and high temperatures used
in the growth can compound the effect of internal stresses as well. D. Hull
and D. J. Bacon (2011) Additionally, dislocations can extend from a seed
crystal containing dislocations to new growth based on that seed. D. Hull and
D. J. Bacon (2011) However, not much is known about dislocations in
topological materials. Previously, dislocations in Bi-based topological thin
films have been shown to create unwanted bulk current paths. Hamasaki  _et
al._ (2017); Ran  _et al._ (2009) Dislocations in SmB6 would be especially
interesting to study in light of the previous report of a truly insulating
bulk in the dc limit.

The level of disorder can generally be measured in transport via the mobility.
At low temperatures, measurements of mobility in SmB6 are not straightforward.
Experimental reports - ARPES, dHvA oscillations, and transport - disagree on
the order of magnitude of the mobility, which ranges from about 10
cm2/V$\cdot$s in transport Luo  _et al._ (2015) to about 1000 cm2/V$\cdot$s in
quantum oscillations; Li  _et al._ (2014) surface preparation can even affect
the extracted mobility. Eo  _et al._ (2020) A recent study also shows that the
two proposed topological surface channels would have very different
mobilities, Eo  _et al._ (2020) and accounting for a disorder-based channel
could be an additional challenge.

Detailed transport results have also shown that the resistivity saturation at
low temperature is non-universal. Phelan  _et al._ (2016) Al-flux grown
samples generally yield resistivity with temperature-independent plateaus.
Cooley  _et al._ (1995); Sluchanko  _et al._ (1999); Luo  _et al._ (2015)
Floating zone samples are generally less consistent and can behave similarly
to flux-grown samples, show temperature-dependent behavior, or even a step-
like behavior. Flachbart  _et al._ (2001); Gabáni  _et al._ (2015); Phelan
_et al._ (2016) An open question is whether these differences in behavior are
due to different surface characteristics or bulk characteristics. If the
differences are due to surface characteristics, the low-temperature bulk
behavior should be similar between both types of samples. But, if bulk
characteristics differ at low temperatures, this would be reflected in
inverted resistance measurements.

In this work, we perform inverted resistance measurements on a Corbino disk
geometry on floating zone grown samples. We find that these samples all
demonstrate bulk conduction at lowest temperatures, but with significant
sample-to-sample variation. In combination with previous results identifying a
low-temperature bulk conduction channel on flux-grown samples of different
defect levels, Eo  _et al._ (2019) we discuss the possible origins of this new
bulk channel in the context of recent impurity models and dislocations. To
expand experimental understanding of the role of disorder, we also perform
chemical etching to verify the presence of dislocations in our flux and
floating zone grown samples. The wide variation in our low-temperature results
depending on the sample used suggests that many discrepancies in experimental
reports on SmB6 may have an extrinsic origin.

##  II Inverted resistance on a Corbino disk

To investigate the origin of different low-temperature behavior in SmB6, we
used a recently developed method called inverted resistance. This method can
distinguish whether the resistance originates from the bulk or the surface and
allows us to find the bulk resistivity even if the surface conduction
overwhelms the bulk. Eo  _et al._ (2018) We briefly illustrate the method.
Consider the simple case where the resistance only depends on the bulk
resistivity. This is when the bulk conduction overwhelms the surface
conduction or the surface conduction does not exist at all. In this case, the
electric current flows only through an isotropic bulk, and the resistance is
proportional to the bulk resistivity, $\rho_{b}$:

$R=C_{b}\rho_{b},$ (1)

where $C_{b}$ is a prefactor that is determined by the geometry of the sample
and the position of the electrodes. The resistance measurements from a
different selection of electrodes will only change the value $C_{b}$. Those
different resistance measurements will have the same temperature dependence,
originating from $\rho_{b}$, and therefore the $R$ vs. temperature and
$\rho_{b}$ vs. temperature curves will have the same shape and be parallel.

If the two resistances are not parallel to each other as a function of
temperature, Eq. (1) cannot be used. One possible reason for not being
parallel is when disorder in the crystal creates a large inhomogeneity and
therefore the temperature dependence of the bulk resistivity is not global. A
more dramatic case is when an extra conduction channel is present, for example
a surface conduction channel. This is indeed the case for SmB6 below 4 K, and
the resistance measurement can be explained simply by the following. The bulk
resistivity increases exponentially with inverse temperature while the surface
sheet resistance changes only very moderately. At low enough temperatures, the
bulk conduction is so low that it becomes overwhelmed by the surface
conduction. In this temperature regime, the measured resistance is
proportional to the sheet resistance, $R_{s}$:

$R=C_{s}R_{s},$ (2)

where $C_{s}$ is a prefactor that is determined by the geometry of the surface
and the position of the electrodes.

In the case of both surface and bulk conduction, a different type of
resistance measurement can be used. Here, the current flows inside a Corbino
disk geometry and the voltage is measured exterior to that disk. This method
is known as inverted resistance, and the details can be found in Ref. Eo  _et
al._ (2018). The inverted resistance, $R_{\textrm{Inv}}$, now depends on both
$R_{s}$ and $\rho_{b}$:

$R_{\textrm{Inv}}=C_{\textrm{Inv}}\frac{R_{s}^{2}}{\rho_{b}},$ (3)

We use Eqs. 1, 2, and 3 to analyze our resistance measurement.
$R_{\textrm{Inv}}$ is particularly useful because it allows us to access
$\rho_{b}$ at temperatures where surface conduction dominates. In the case
where the bulk resistivity exhibits ideal activated behavior,
$\rho(T)\propto\exp{(E_{a}/T)}$, $R_{Inv}$ will also follow the inverse
temperature dependence ($\propto\exp{(-E_{a}/T)}$). This ideal relation
surprisingly holds true for the case of stoichiometric flux-grown SmB6 case Eo
_et al._ (2019) However, this temperature dependence in the resistance can be
interrupted if a second bulk conduction channel, exists.

Figure 1:  Inverted resistance measurement of four floating-zone grown SmB6
samples.

##  III Results

Four floating zone samples were prepared for the inverted resistance
measurement. Details of the samples’ origins and growth methods can be found
in Table 1. All samples were polished with grits down to 0.3 $\mu$m. The
Corbino disks were patterned by photolithography. Ti/Au was deposited on the
samples using e-beam evaporation and later a lift-off process with acetone was
used to define the pattern and electrodes. We used gold wires to make
electrical connection from the electronics to the sample, and attached them
using either silver paste or wire bonding.

Sample | Growth details | Starting powder origin | Reference
---|---|---|---
Warwick 1 | Standard growth | Alfa Aesar | Hatnean  _et al._ (2013)
Warwick 2 | Standard growth | American Elements | Hatnean  _et al._ (2013)
JHU 1 | Standard growth | Testbourne Ltd. | Phelan  _et al._ (2016)
JHU 2 | Doubly-isotope enriched, Sm deficient | Alfa Aesar | Phelan  _et al._ (2016)
Table 1: Details of the four floating zone samples measured.

Fig. 1 shows the resistance vs. temperature of all four samples. The blue
curves are the standard resistance measurements and the red curves are the
inverted resistance measurements. In the standard measurements (blue curves),
all four samples show a change in slope around 4 K that would conventionally
be regarded as a surface plateau. Fig. 1 (a) and (d) show little to no
temperature dependence in the standard measurement below about 4 K, especially
compared to Fig. 1 (b) and (c). Using the inverted curves (red), we can
determine whether these “plateaus” arise from surface or bulk conduction
channels. We see dramatic differences in the inverted resistance results for
each sample. Fig. 1 (a) shows a resistance that drops and saturates, similar
to the non-stoichiometric flux growths observed previously. Eo  _et al._
(2019) The inverted resistance measurement shown in Fig. 1 (d) has a feature
similar to Fig. 1 (a) but also a moderate drop at lower temperatures. In Fig.
1 (b) and (c), we see a temperature dependence that is close to parallel to
the standard measurement.

Figure 2:  Bulk resistivity conversion from Fig. 1 measurements.

We convert the measured resistances to bulk resistivity in Fig. 2, except for
the sample Warwick 1 (Fig. 1(a)) which was complicated to convert due to the
Corbino disks being placed on different crystal planes. We compare the results
from this study to the previously reported result on a stoichiometric flux-
grown sample (black trace). Eo  _et al._ (2019) We find that the remaining
three floating zone-grown samples have a significant slope change in bulk
resistivity which indicates that the intrinsic exponential temperature
dependence in bulk resistivity is interrupted. That is, another bulk
conduction mechanism is present in these samples in addition to the standard
mechanism responsible for activated behavior.

##  IV Discussion

Figs. 1 and 2 show that the four floating zone samples presented here have
non-negligible bulk conduction, but the characteristics of the bulk conduction
differ by sample. In Fig. 1 (b) and (c) the standard and inverted resistance
curves are parallel to one another, demonstrating that these samples are bulk
conductors and can be described by Eq. 1. In terms of transport, this means
that either the surface conduction is nonexistent or that the bulk conduction
channel dominates.

Many researchers have proposed impurities as a possible origin for residual
bulk conduction in SmB6. Most point defects come from the starting material,
as rare earth elements are notoriously difficult to purify. In our samples,
the starting materials were sourced from different companies, so the purity of
the Sm used in the growth may differ among samples. One way to reduce rare
earth impurities is via isotopic purification to Sm-154. The only remaining
rare-earth impurity is Gd-154, which is magnetic. Fuhrman  _et al._ (2018) Gd
impurities have been studied previously, beginning with the observation that
the substitution of even 1% Gd could dramatically change the electrical
properties of SmB6. Geballe  _et al._ (1970) Later, a Gd doped sample was used
to test the TI hypothesis by searching for time reversal symmetry breaking
below 4 K. Kim  _et al._ (2014) Gd impurities have also been shown to increase
the residual heat capacity at low temperatures Fuhrman  _et al._ (2018) and
have been suggested as an avenue for screening of the Kondo effect at low
temperatures. Fuhrman  _et al._ (2019) Debate is ongoing about the role of Gd
impurities; recent reports have suggested that it is not responsible for bulk
dHvA oscillations. Hartstein  _et al._ (2020) However, the local environment
of Gd impurities is metallic even at very low concentrations, and at higher
concentrations this could lead to percolation through the sample in transport
measurements. Souza  _et al._ (2020)

Results from the isotopically purified sample shown in Fig. 1 (d) have
different features in the inverted resistance curve compared to the non-
purified samples (Figs. 1 (b) and (c)). Since the standard and inverted curves
are not parallel, this sample does not have dominant bulk conduction, but it
may have parallel surface and bulk channels. Since the sample is isotopically
pure, the bulk channel could come from the remaining Gd impurities, Sm
vacancies introduced during growth, or both.

A general model of impurities used in semiconductors and other materials is
the effective mass approximation, where the impurity is treated
hydrogenically, with an effective Bohr radius and binding energy. Kohn (1957)
In SmB6, the conditions for standard hydrogenic impurites are not satisfied
when the model for semiconductors is used. Rakoski  _et al._ (2017) However,
the model introduced by B. Skinner in Ref. Skinner (2019) demonstrated that
the effective mass approximation can be modified for SmB6. In the Skinner
model, the quadratic potential used in the original treatment of the effective
mass approximation Kohn (1957) is swapped for the Mexican hat type potential
seen in SmB6. New conditions for the effective radius and binding energy of
the impurity state are determined. The total dc conductivity in the presence
of these new impurity states is also derived and found to be a combination of
the standard activated behavior with activation energy $E_{1}$ and an
activated hopping term with activation energy $E_{3}$, Skinner (2019)

$\sigma(T)=\sigma_{1}\exp{\bigg{(}\frac{E_{1}}{k_{B}T}\bigg{)}}+\sigma_{3}\exp{\bigg{(}\frac{E_{3}}{k_{B}T}\bigg{)}}.$
(4)

This type of impurity band could be present in all samples and could describe
both magnetic and nonmagnetic point impurities, including Sm vacancies. Since
the addition of Sm vacancies to flux grown samples has been shown to induce
bulk conductivity, Eo  _et al._ (2019) some portion of the residual bulk
conductivity seen in this work in floating zone samples could also be due to
Sm vacancies as described above. However, the magnitude of the bulk conduction
seen in the inverted measurements is much greater in all the floating zone
samples, including the isotopically purified sample, so it is unlikely that
vacancies or impurities alone could be the origin. The possibility of hopping
conduction and even insulator-to-metal transition by heavily doped foreign
magnetic impurities will be discussed elsewhere in our future work.

Another possibility for the source of the residual bulk conduction is one-
dimensional defects, or dislocations. As discussed earlier, a mismatch in
lattice constant between a substrate and a semiconductor thin film can lead to
the formation of dislocations which terminate on the surface of the film. Hall
and Bean (1992) The density of dislocations in a film can be estimated by the
change in lattice parameter, $n_{\textrm{dis}}=|1/a_{1}^{2}-1/a_{2}^{2}|$,
where $a_{1}$ and $a_{2}$ are the lattice parameters on the two surfaces of
the film.

In SmB6, one study reports a change in lattice constant over the length of a
floating zone sample. Phelan  _et al._ (2016) Unlike the case of thin films,
to the best of our knowledge there is no literature describing how to estimate
dislocation density in bulk crystals where there is a variation in lattice
constant. Here, we introduce a new method to understand the formation of
dislocations in bulk materials. Later, we use our model to estimate the
dislocation density in SmB6 samples. We consider a floating zone sample as its
size allows for more variation of lattice parameter in the crystal compared to
a flux grown sample, but dislocations are still expected to be present in flux
grown crystals.

Generally, dislocations that form in films terminate on the surface of the
film. In thin films, the dislocations form in the growth direction and
terminate on the top surface of the film. In crystals, however, dislocations
do not have to form and terminate only along the direction of growth;
dislocations could also terminate on the side surfaces of the crystal, as
shown in Fig. 3. Thus, the dislocation density in crystals is expected to
depend both on the change in lattice constant in the direction of growth and
on the size (radius) of the crystal.

Figure 3:  Sketch of dislocations in a floating zone crystal. Some
dislocations (green) initiate within a slice of thickness $\epsilon$, and
others (blue) terminate within that slice. The remaining dislocations (gray)
form and terminate elsewhere in the sample.

To estimate the dislocation density, we model the floating zone rod as forming
from a series of thin slices of thickness $\epsilon$ as the molten zone passes
through the furnace. In analogy with the equation above for semiconductor thin
films, the total number of dislocations that nucleate within the slice is

$\left|\frac{1}{(a(x))^{2}}-\frac{1}{(a(x+\epsilon))^{2}}\right|(\pi r^{2})$
(5)

where $a(x)$ is the lattice parameter at a location $x$ along the growth
direction and $r$ is the radius of the crystal. A sketch of these is shown in
green in Fig. 3. All the dislocations that form in this slice must terminate
somewhere on the surface of the sample, whether on the sides or the ends. To
account for the dislocation terminating on the sides, we introduce an angle
$\theta_{\textrm{d}}$ which the dislocation makes with respect to the growth
direction. Then, the number of dislocations that terminate within a slice
(shown in blue in Fig. 3) is related to the surface area of the slice, the
dislocation density ($n_{\textrm{dis}}$), and $\theta_{\textrm{d}}$ by

$(2\pi r\epsilon)n_{\textrm{dis}}\cos\theta_{\textrm{d}}.$ (6)

Here, $\cos\theta_{\textrm{d}}=1$ would correspond to all dislocations
oriented along the growth direction. We expect $cos\theta_{\textrm{d}}<1$ in
an actual sample, since dislocations are expected to terminate randomly on the
surface but form with the growth of the rod. Since all the dislocations that
formed must terminate, Eqs. 5 and 6 are equal. Expanding Eq. 5 as a Taylor
series, we calculate that the estimated dislocation density is

$n_{\textrm{dis}}=\frac{r}{\cos\theta_{\textrm{d}}}\frac{|\nabla
a(x)|}{(a(x))^{3}}$ (7)

for the dislocation density.

We can estimate the dislocation density in the sample with reported change of
lattice constant from Ref. Phelan  _et al._ (2016). The lattice parameter in
that sample was $a_{1}=4.134309$ Å on one end of the crystal and
$a_{2}=4.133343$ Å on the other end, 8 cm away. The radius of the crystal was
3 mm. Using these values with Eq. 7, we estimate that the dislocation density
in this floating zone sample is $\sim 10^{10}$ cm-2.

Dislocations are commonly imaged by preparing samples as for transmission
electron microscopy (TEM). However, the estimated density of dislocations we
calculated is too small to use this method. Instead, we used chemical etching
to reveal points where dislocations terminate on the surface. During etching,
material is removed from the area near a crystal defect at a different rate
than from the lattice. The etching method allows defects, including
dislocations, to be imaged optically. Ourmazd (1984) The “etch pit” that forms
also mirrors the crystal structure of the sample; for SmB6 we expect to see
square etch pits. With longer etching time, the size of the etch pits
increases and more etch pits start to form, so that the etch pit density
observed provides a lower bound on the actual dislocation density.

Figure 4:  Examples of etch pits in (a) a floating zone sample etched for 340
seconds and (b) a flux-grown sample etched for 600 seconds.

We used equal parts nitric and sulfuric acid diluted to 10% to etch flux grown
and floating zone SmB6 crystals. After etching, we observed etch pits using a
scanning electron microscope in both types of samples. Examples of etch pits
are shown in Fig. 4. The floating zone sample shown was etched for 340 seconds
and had an etch pit density of $10^{5}$ cm-2. The flux grown sample shown was
etched for 600 seconds and had an etch pit density of $2\times 10^{3}$ cm-2.
Even though the flux grown sample was etched longer than the floating zone
sample, it has a lower density of etch pits observed, suggesting that the
floating zone sample hosts more dislocations than the flux grown sample. In
both samples, the locations of the etch pits is nonuniform, which suggests
that local inhomogeneities in temperature or stoichiometry, for example,
during sample growth are important to the formation of dislocations. In both
cases, the observed densities are much lower than the calculated estimate of
1010 cm-2, but since we imaged immediately after identifying that etch pits
were present, our values are lower bounds on the number of dislocations
actually present in the samples.

In addition to impurity hopping and dislocations, we briefly consider other
theories proposed to explain some of the novel results of SmB6. First, the
proposal that SmB6 is a nodal semimetal H. Shen and L. Fu (2018); Harrison
(2018) is inconsistent with the low-temperature bulk conduction that we
observe, and it does not explain the difference between flux- and floating
zone-grown samples. Our inverted resistance curves show two regions of
activated behavior (above and below about 4 K) rather than activated behavior
above 4 K and linear-in-T behavior below 4 K. Next, in heat transport, excess
thermal conduction at low temperature was not found in flux-grown samples, and
reports have disagreed about whether thermal conduction is present universally
in floating zone samples. Hartstein  _et al._ (2017); Xu  _et al._ (2016);
Boulanger  _et al._ (2018). Theories attempting to reconcile these conflicting
results have focused on the possibility that the floating zone samples contain
charge-neutral excitations and primarily explored their relevance to dHvA
oscillations rather than transport. Our data do not provide evidence for
charge-neutral excitations, but the low-temperature bulk channel we observe
could conduct heat and contribute to dHvA. Even in samples with very few rare
earth impurities, Hartstein  _et al._ (2020) dislocations could still
contribute to these effects. A better understanding of the role of
dislocations, or more generally, the conduction channel we observe here, will
be an intriguing area of further study.

##  V Conclusion

In this work, we performed transport measurements on floating zone grown SmB6
using the inverted resistance method. Standard four point and Hall bar
geometry resistance results show nonuniversal temmperature dependence below
about 4 K, but the origin of this behavior is difficult to pinpoint as both
bulk and surface channels are present. The inverted resistance method we used
allowed us to characterize the bulk behavior at temperatures at which surface
conduction dominates. We found that the four floating zone samples show bulk
conduction with characteristics differing by sample. On the other hand, a
stoichiometric flux-grown sample (along with other results from Ref. Eo  _et
al._ (2019)) has a truly insulating bulk, and the introduction of Sm vacancies
in flux grown samples was previously shown to induce bulk conduction.

We discussed various possibilities for the origin of the new conducting
channel observed here, as well as the differences between the floating zone
results presented here and the flux-grown samples presented in the previous
work. Eo  _et al._ (2019) We especially considered impurities, which could be
magnetic, like Gd, or non-magnetic, including defects like Sm vacancies. Our
experimental results are consistent with the Skinner model Skinner (2019) for
impurity hopping conduction at low temperatures with an activated transport
behavior. In addition, we considered one-dimensional defects, or dislocations,
extending throughout the sample. We observed a small dislocation density in
both flux and floating zone samples via chemical etching, with a larger
dislocation density observed in floating zone samples compared to flux grown
samples. While this is consistent with the relative amounts of bulk conduction
observed in samples grown by each technique, further work to explore the
characteristics of the dislocations is needed to verify that they contribute
to bulk conduction with a magnitude agreeing with our inverted resistance
data. Future work could include characterizing the mobility of the channel or
thermal studies of the role of dislocations in SmB6.

###### Acknowledgements.

A.R. would like to acknowledge funding support from NSF Grant #DGE-1256260.
W.T.F. is grateful to the Schmidt Science Fellows program, in partnership with
the Rhodes Trust, for support of this work. M.S.F. and Y.S.E. were supported
by the Australian Research Council grant CE170100039. Work at the University
of Maryland was supported by NSF award no. DMR-1905891 and the Gordon and
Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF9071. The work
at Los Alamos National Laboratory was performed under the auspices of the U.S.
Department of Energy, Office of Basic Energy Sciences, Division of Materials
Sciences and Engineering. The work at the University of Warwick was supported
by EPSRC, UK through Grant EP/T005963/1. The work at the Johns Hopkins
University was supported as part of the Institute for Quantum Matter, an
Energy Frontier Research Center funded by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences under Award No. DE-SC0019331. Device
fabrication was performed in part at the University of Michigan Lurie
Nanofabrication Facility. The authors acknowledge the financial support of the
University of Michigan College of Engineering and NSF grant #DMR-0320740, and
technical support from the Michigan Center for Materials Characterization.

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