# Perspective: Molecular beam epitaxy of antiperovskite oxides

H. Nakamura  Department of Physics, University of Arkansas, Fayetteville, AR
72701, USA  D. Huang  Max Planck Institute for Solid State Research, 70569
Stuttgart, Germany  H. Takagi  Max Planck Institute for Solid State Research,
70569 Stuttgart, Germany  Department of Physics, University of Tokyo, 113-0033
Tokyo, Japan  Institute for Functional Matter and Quantum Technologies,
University of Stuttgart, 70569 Stuttgart, Germany

###### Abstract

Antiperovskites, or inverse perovskites, have recently emerged as a material
class with a plethora of promising electronic properties. This perspective
describes the molecular beam epitaxy (MBE) growth of oxide antiperovskites
Sr3PbO and Sr3SnO. We show that MBE offers great potential not only in growing
antiperovskites with high structural quality, but also in providing a means to
seamlessly connect with advanced characterization tools, including x-ray
photoelectron spectroscopy (XPS), low-energy electron diffraction (LEED),
reflection high-energy electron diffraction (RHEED), and scanning tunneling
microscopy (STM), to facilitate the analyses of their intrinsic properties.
The initial results point toward the feasibility of atomically controlled
antiperovskite growth, which could open doors to study topological and
correlated electronic states in an electronic environment quite distinct from
what is available in conventional complex oxides.

##  I Introduction

Antiperovskites with $A_{3}B$O structure ($A$ = Ca, Sr, Ba, Yb, Eu; $B$ = Sn,
Pb) have recently acquired considerable interest in the condensed matter
physics community, in part due to the prediction of their topological
electronic properties. From a chemistry perspective, the $A_{3}B$O
antiperovskites belong to the Zintl compounds, which are characterized by a
nearly complete electron transfer from the more electropositive $A$ atom to
the more electronegative $B$ atom Widera and Schäfer (1980); Velden and Jansen
(2004); Nuss  _et al._ (2015). In this case, the alkaline earth cation (A2+)
and the electronegative ion (B4-) both achieve a closed-shell valence state,
resulting in a narrow-gap semiconductor or semimetal that may host topological
phases. In recent years, the hunt for new topological materials has opened
researchers’ eyes to wider materials within the Zintl phase. The Dirac
semimetal Na3Bi is a well known example of Zintl phase Liu  _et al._ (2014).
Sr2Pb and Sr2Sn, which were proposed to be topological insulators under strain
Sun  _et al._ (2011), are also Zintl compounds closely related to the
$A_{3}B$O antiperovskites. Unusual electronegative Sn and Pb ions in
antiperovskites have been experimentally verified in recent works Huang  _et
al._ (2019); Ikeda  _et al._ (2019).

Three-dimensional (3D) Dirac states with a small mass gap $E_{g}$ were
identified theoretically in antiperovskites by Kariyado et al Kariyado and
Ogata (2011, 2012, 2017). Later, Hsieh et al. gave a rigorous topological
classification, showing that some of the $A_{3}B$O antiperovskites are
topological crystalline insulators (TCIs) Hsieh  _et al._ (2014). Many
theoretical studies followed, clarifying the nature of surface states Chiu
_et al._ (2017), superconductivity Kawakami  _et al._ (2018), hinge states
Fang and Cano (2020), and surface magnetism Arras  _et al._ (2021). One of the
most appealing features is that there is a large energy window in the $E(k)$
diagram ($\sim$0.4 eV in Sr3PbO) in which topologically interesting states,
such as the bulk 3D Dirac bands, do not overlap in energy with topologically
trivial bands. The existence of such an energy window offers a unique
opportunity to study topological electronic states. Furthermore, the Dirac
nodes are located at the Fermi level ($E_{F}$) for stoichiometric compounds,
which is beneficial to the study of those states by electronic transport.

Transport and thermoelectric properties of $A_{3}B$O antiperovskites have also
been extensively studied experimentally. The observations of superconductivity
in highly Sr-deficient Sr3-xSnO Oudah  _et al._ (2016), large thermoelectric
power Okamoto  _et al._ (2016), quantum oscillations Suetsugu  _et al._
(2018), and large diamagnetism associated with Dirac electrons Suetsugu  _et
al._ (2021), whose dispersion was also confirmed by ARPES measurements Obata
_et al._ (2017), were some of the early data obtained using bulk crystals.
Molecular beam epitaxy (MBE), first developed for Sr3PbO Samal  _et al._
(2016), enabled tunable carrier densities ($n$) in Sr3SnO and facilitated
detailed analyses of three-dimensional weak antilocalization Nakamura  _et
al._ (2020). Alternative MBE growth approaches for Sr3SnO were also reported
Ma  _et al._ (2020); Wu  _et al._ (2021). All the experimental evidence,
including ARPES, points to the correctness of the bulk band structure
predicted by theories, although critical information such as the value of
$E_{g}$ in $A_{3}B$O antiperovskites is still missing. The experimental
evidence of TCI surface states is also thus far lacking.

Despite these experimental and theoretical progress, the understanding of
antiperovskites is still at its infancy. For example, we do not know what kind
of impurity phases could exist in this system, especially when the
stoichiometry is deliberately changed from pure compositions. This complicates
the interpretation of superconductivity observed in Sr3-xSnO, where secondary
phases were also found Oudah  _et al._ (2016). Another question is associated
with the interface or surface of antiperovskites. Along the [001] direction of
antiperovskites is a sequence of positively ($+2$) and negatively ($-2$)
charged layers ($BA$ and O$A_{2}$, respectively), which should trigger
electronic or atomic reconstructions at the surface or interface, as per polar
discontinuity Nakagawa  _et al._ (2006). The physics of reconstructions is
currently poorly understood for antiperovskites. On the other hand, because of
the close match in lattice constants among antiperovskites, fabricating
perfect interfaces of topological materials seems feasible. Emerging phases at
interfaces is another agenda that deserves continued effort. A recent
discovery of interface superconductivity at the KTaO3(111) surface, at a much
higher $T_{c}$ than that observed on the (001) surface, reaffirms that even
the interfaces of conventional oxides are still rich in unexpected electronic
phenomena Liu  _et al._ (2021). Antiperovskites hold strong promise for the
discovery of novel electronic phenomena at the interface, especially once
aspects of material growth are better understood.

Figure 1: Band gap vs. lattice constant of selected $A_{3}B$O antiperovskite
materials ($A$ = Ca, Sr, Ba, Eu, Yb; $B$ = Pb, Sn). All the lattice constants
come from X-ray diffraction data Velden and Jansen (2004); Nuss  _et al._
(2015), while the band gaps are computed from first-principles DFT
calculations Chiu  _et al._ (2017); Pertsova  _et al._ (2019). For Ba
compounds, lattice constants for the room temperature cubic phase are shown.

The structure-property relationship of different antiperovskites within the
$A_{3}B$O family may be understood by focusing on two key parameters, bandgap
($E_{g}$) and lattice constant. As shown in Fig. 1, the lattice constants of
antiperovskites are largely determined by the element $A$. For instance,
$A_{3}B$O antiperovskites with $A$ = Ca and Yb show similar lattice constants,
so do the ones with $A$ = Sr and Eu, largely independent of whether the $B$
element is Pb or Sn. Next, antiperovskites with Pb show universally larger
band gaps compared to their Sn counterparts. This is because the opening of
the small band gap ($\sim$10 meV) is controlled by the strength of spin-orbit
coupling. An exception to this simple rule is the antiperovskites with Eu,
where $E_{g}$ disappears both for the Pb and Sn compounds, suggesting the
influence of $f$ electron states Pertsova  _et al._ (2019). The valence and
conduction bands consist of the $p$ orbitals of Sn/Pb and the $d$ orbitals of
alkaline earth elements, respectively, and the formation of Dirac bands is
driven by a band inversion at the $\Gamma$ point. This generates six identical
Dirac nodes along equivalent $\Gamma$–$X$ lines. Since the Dirac nodes are
located at $E_{F}$, the small gaps shown in Fig. 1 are the ones at Dirac
crossings, making the Dirac electrons massive.

In this Perspective, we provide insights into the MBE growth of Sr3PbO and
Sr3SnO antiperovskites. We show that MBE is one of the ideal techniques to
study the intrinsic properties of as-grown antiperovskites, which are air-
sensitive, as is the case with many other Zintl compounds. The study of the
two antiperovskites, grown and analyzed using identical methods, sheds light
on finer aspects of structure-property relationships. This paper is organized
as follows. In Subsection II.1, we discuss a global phase diagram for Sr3PbO
that is mapped out by co-depositing elements under vacuum with varying flux
ratios. In Subsection II.2, we discuss a sequenced shutter approach, inspired
by many earlier MBE works, that we believe could improve the quality of Sr3PbO
films, and also enabled us to quickly stabilize the Sr3SnO phase. In
Subsection II.3, we describe an alternative approach to the MBE of Sr3SnO by
using SnO2 as a solid source. In Subsection II.4, we discuss ex-situ transport
measurements of the films. In Subsection II.5, we discuss various
characterizations of the surface of antiperovskite films in ultra-high vacuum
(UHV). We note some distinctions between Sr3PbO and Sr3SnO that highlight
their complex chemistries, including signatures of surface reconstructions in
the latter, but not in the former.

##  II MBE growth of $A_{3}B$O antiperovskites

###  II.1 Identification of Growth Window: Co-deposition

The lattice constants of $A_{3}B$O antiperovskites are $a\sim
5\,\mathrm{\AA}$, significantly larger than the standard perovskite lattice
constant of around 4 $\mathrm{\AA}$ Schlom  _et al._ (2008). As noted earlier,
Sr3PbO ($a=5.15\,\mathrm{\AA}$) and Sr3SnO ($a=5.14\,\mathrm{\AA}$) have
almost identical lattice constants due to a common $A$ element, and can be
grown on the same substrate of choice. Convenient substrates include yttria-
doped ZrO2 (YSZ) with $a$ = 5.14 Å ($-$0.2%), LiAlO2 with
$a=5.17\,\mathrm{\AA}~{}(+0.35\%)$, LaAlO3(LAO) with
$a=3.79\times\sqrt{2}=5.36\,\mathrm{\AA}~{}(+4.1\%)$, and silicon with
$a=5.43\,\mathrm{\AA}~{}(+5.4\%)$. SrO, which is an easily grown binary oxide,
also functions as a perfect buffer layer with
$a=5.16\,\mathrm{\AA}~{}(+0.2\%)$. Here, we consider the (001) facet of these
substrates for growth, the percentage denotes the lattice mismatch with
respect to Sr3PbO, and a $\sqrt{2}$ factor for LAO reflects an in-plane 45∘
rotation of the film unit cell with respect to the substrate. An initial
report on the MBE growth of $A_{3}B$O was for Sr3PbO on LAO Samal  _et al._
(2016). Despite a moderate lattice mismatch of 4.1$\%$, this choice of
substrate was important in the initial stages of film growth, since it allowed
an easy identification of all the antiperovskite ($00l$) x-ray diffraction
(XRD) peaks without overlap with the substrate peaks.

Figure 2: Antiperovskite MBE system as developed at the Max Planck Institute
for Solid State Research in Stuttgart, Germany. A room-temperature STM is
integrated to enable direct transfer of grown films from the MBE. Two UHV
suitcases could be mounted to the load lock (LL) chamber of the MBE, allowing
transfer to a neighboring glove box (inset), as well as external facilities.
Figure 3: The “phase diagram” of MBE films obtained by XRD as a function of
flux ratios (co-deposition). The substrate temperature was 450∘C. Pb and Sr
fluxes were determined by a QCM, while the oxygen flux is reported as a flow
rate (sccm); thus, the ratios involving oxygen are not calibrated to atomic
ratios. The first two successful films of Sr3PbO are shown as stars. The
lesson is that the Sr flux had been too small during the initial phase of the
growth efforts. Small numbers inside the circles indicate sample numbers.

We performed growth using a customized oxide MBE system (Eiko, Japan; Fig. 2).
We first review Sr3PbO growth obtained by co-deposition of elemental sources
on LAO substrates at a substrate temperature of 450∘C. No ex-situ pre-
treatment of the substrates was performed; the as-bought substrates were
loaded into the growth chamber and degassed at the growth temperature of the
film. As MBE sources, we used pure Sr and Pb from standard effusion cells, and
oxygen gas from a mass flow controller. The oxygen gas was later changed to an
Ar-O2 mixture. The early attempts of growth at various flux ratios are shown
in Fig. 3. Adjusting the flux or determining the appropriate substrate
temperature posed a challenge due to the lack of a prior report on MBE growth.
This is especially the case when three elements need to be controlled
independently, and we do note that other recent MBE works have utilized
SnO/SnO2 as a source to circumvent this issue Ma  _et al._ (2020); Wu  _et
al._ (2021). To streamline the growth, we used a diagram like the one in Fig.
3 to avoid making duplicate films at similar conditions. By this approach, we
found that (1) at lower Sr flux, we obtain a SrO phase, largely independent of
the Pb flux; (2) upon increasing the Sr flux at a constant O2 and Pb flux, we
destroy the SrO phase and obtain an amorphous phase (determined by the lack of
any peaks in XRD) with low electrical resistance ($<$M$\Omega$). The phase
space for this “amorphous” region was large, hampering an immediate success.
However, there was a relatively clear boundary where the SrO phase was
destroyed by increasing the Sr/O2 ratio. By increasing the Sr flux to reach
three times the Sr/O2 value needed to destroy the SrO phase, we obtained the
first few films of Sr3PbO. This occurred at impractically large Sr flux of 6
to 7 $\rm{\AA/s}$, as measured by a quartz crystal monitor (QCM), which
depleted the Sr cell in 1–2 weeks. Ar-diluted O2 (10% O2, and later 2% O2)
enabled the use of a lower Sr flux, making the MBE growth more sustainable.

Further optimization with the co-deposition technique was possible by
utilizing XRD (00$l$) peaks with odd $l$. In fact, since the (00$l$) peaks
with even $l$ originate also from SrO impurity, they do not serve as evidence
of an antiperovskite phase. To structurally confirm the antiperovskite phase,
(00$l$) peaks with odd $l$ need to be observed. This, in turn, provided an
approach to further optimize the growth: it was found that the ratio of the
amplitudes between odd and even (00$l$) peaks was maximized at optimized
growth conditions Samal  _et al._ (2016). This could be explained by taking
into account long-range structural order. Because the odd (00$l$) peaks in XRD
are only allowed due to the distinct layers on the (001) plane, namely, OSr2
and PbSr layers for Sr3PbO, any structural disorder that mixes those layers
revives the extinction rule for rock-salt structures that suppresses odd
diffraction peaks (as in SrO). For example, an insertion of a SrO layer
between two antiperovskite layers could easily impact the long-range
structural coherence of the film.

Although experiments using LAO substrates enabled the identification of the
growth window, the grown antiperovskite films had limited quality. For Sr3PbO,
the best rocking curve of the (001) peak directly grown on LAO showed a full-
width half-maximum (FWHM) of 0.88∘. By using a SrO buffer layer on LAO, we
obtained a FWHM = 0.65∘. By using YSZ as the substrate, the FWHM was further
reduced down to 0.43∘. These results obtained for co-deposition clearly
demonstrate the impact of substrates. Substantial improvements in film quality
were seen by using lattice-matched YSZ substrates. Next, we show that further
improvements in the antiperovskites films could be achieved by utilizing
sequential (shuttered) deposition.

###  II.2 Sequential Epitaxy

After we identified the antiperovskite phase, we tested several strategies to
further improve the quality of antiperovskite films grown by MBE. One issue
that seemed to impact the quality was the presence of SrO as a dominant
impurity phase. Since it is perfectly lattice-matched to antiperovskites, it
seemed hard to completely eliminate its formation during standard co-
deposition conditions. Therefore, we decided to separate the flux of oxygen
from other fluxes by the use of shutters. Also, to better control the O2 at a
low flux, a regulated UHV leak valve was installed (instead of a mass flow
controller). Both the shutters and the UHV leak valve were controlled by
computer via Labview (National Instruments). In most of the sequential epitaxy
growths, the Sr flux at the open shutter condition, as monitored by a QCM, was
$\sim$0.28–0.30 Å/s, while the Pb (and later Sn) flux was tuned to achieve a
Sr/Pb (or Sr/Sn) ratio of 10–20, which corresponded to 0.015–0.029 Å/s.

Figure 4: (a) Sr flux variation during the sequential (element-shuttered)
growth of Sr3PbO as monitored by AAS. The absorption signal carries
contributions from Sr atoms both impinging the substrate and being re-
evaporated from the substrate. The shutter sequences for Sr, Pb, and O2 are
also depicted, with the high state corresponding to an open shutter, and the
low state corresponding to a closed shutter. (b) XRD rocking curve of the
(001) reflection for Sr3PbO (film thickness: 200 nm) obtained using this
method. (c) and (d) Corresponding Sr absorption curve, shutter sequences and
(001) rocking curve for a Sr3SnO film (film thickness: 300 nm). Both Sr3PbO
and Sr3SnO were grown on YSZ substrates at 450∘C.

The best pattern for sequential growth was identified by trial and error. When
all the fluxes were separated, that is, when we deposited Sr, Pb, and O2 one
after the other without overlap, the antiperovskite phase did not form. The
best result was obtained by opening the Pb shutter in the middle of Sr
deposition, while introducing O2 flux at a separate timing [Fig. 4(a)]. Atomic
absorption spectroscopy (AAS; Accuflux from SVTA) provided insights into the
growth mode. In our chamber, AAS was installed via a pair of counter-facing
CF16 viewports positioned right below the substrate position. In this
position, a large portion of the measured Sr flux was re-evaporated flux from
the substrate heating stage, which we confirmed by comparing the Sr flux
measured at different substrate temperatures. (Apparently, the Sr flux
measured by AAS dropped substantially with decreasing substrate temperature,
although the flux independently monitored by a QCM remained constant). This
means that the MBE growth is at an adsorption-controlled condition, wherein
excess amounts of Sr atoms re-evaporate from the substrate. As shown in Fig.
4(a), the Sr flux as seen by AAS exhibited a noticeable decrease when the Pb
shutter was opened. This is interpreted as the result of the formation of a
PbSr layer on the substrate, preventing the re-evaporation of a portion of Sr
that combined with Pb. How the subsequent OSr2 layer is formed upon
introducing O2 is less clear, but we speculate that either (i) O2 sticks on
the PbSr surface, then is later converted to OSr2 by excess Sr flux, or (ii)
the completed PbSr layer flooded with excess Sr atoms (as our sequence does)
prepares adsorbed Sr atoms to act as reaction sites with O2. The combination
of both processes (i) and (ii) is also possible. The shutter sequence was
fine-tuned such that one full cycle corresponded to one unit cell (one PbSr
and one OSr2 layer). We confirmed that the growth rate obtained by post-growth
thickness measurements (stylus profilometer; Dektak) from several Sr3SnO films
(described later) was 0.46 nm/cycle, which is close to the lattice constant of
5.14$\rm{\AA}$ for Sr3SnO.

After the adoption of this sequential growth, we observed an improvement in
the film quality, as demonstrated by XRD rocking curves of the (001) peaks,
reaching 0.27∘ for Sr3PbO [Fig. 4(b)]. (A further comparison of Sr3PbO films
grown via co-deposition and shuttered epitaxy, including carrier density and
mobility, is given the Supplemental). With this sequenced approach, we were
also able to successfully grow Sr3PbO at substrate temperatures of 370 and
500∘C, in addition to 450∘C. At even higher temperatures ($T$ $>$ 500∘C), the
Sr flux needed to be increased substantially to compensate for the increased
re-evaporation from the substrate, and thus was judged to be impractical.
Growth at temperatures lower than 370∘C should work as well, although they
remain untested.

Having established the sequential growth of Sr3PbO, we began to grow Sr3SnO
thin films as well. A key electronic difference between the two arises from
the smaller spin-orbit coupling of Sn, which results in a second Dirac point,
protected by symmetry against gapping, within 100 meV below $E_{F}$ Kariyado
and Ogata (2017). We were able stabilize the target antiperovskite phase much
faster by adapting the sequential deposition technique, instead of exploring
the entire Sr-Sn-O phase diagram with co-deposition. We note some differences
in the growths of Sr3SnO and Sr3PbO: First, higher Sr flux was needed in the
former. Not only did we increase the Sr/Sn flux ratio ($\sim$12-15 compared to
Sr/Pb$\sim$10-12), we also increased the amount of time in each cycle during
which the Sr shutter was open [note different sequence in Fig. 4(c)]. Second,
during the optimization of Sr3SnO films, we uncovered recognizable impurity
peaks in XRD in the vicinity of the antiperovskite phase, in contrast to the
large “amorphous” region near the Sr3PbO antiperovskite phase (Fig. 3). We
could achieve a FWHM of 0.61∘ for the (001) reflection of Sr3SnO in a film
with no visible impurity peaks in XRD [Fig. 4(d)]. Further improvement of the
crystallinity and reduction of the FWHM down to 0.35∘ was possible; however,
such Sr3SnO films showed an impurity phase in XRD, possibly SrSn (see
Supplemental).

###  II.3 Solid-source SnO2 for Sr3SnO growth

In this section, we describe an alternative MBE growth approach for Sr3SnO
films. Instead of controlling Sr, Sn, and O separately, both Ma et al. and Wu
et al. used Sr and the binary oxide SnO2 as solid sources Ma  _et al._ (2020);
Wu  _et al._ (2021). Upon heating, SnO2 gives rise to SnO and O2 fluxes, which
provide Sn and O species presumably close to an ideal atomic ratio of 1:1.
Meanwhile, excess amounts of Sr flux are used to achieve adsorption-controlled
growth. Confirming this scenario, high Sr/SnO flux ratios of 12.5 Wu  _et al._
(2021) or 13-17 Ma  _et al._ (2020) were needed to produce high quality Sr3SnO
films. The growth temperatures were 400-600∘C Ma  _et al._ (2020) and 650∘C Wu
_et al._ (2021), showcasing that at least for Sr3SnO, this adsorption-
controlled growth can be realized in a wide range of substrate temperatures.
This is also reasonable because SnO is much less volatile compared to Sr Ma
_et al._ (2020). We note that a much higher Sr flux (1.9–40 Å/s) was used Ma
_et al._ (2020) compared to that for the sequential MBE ($\sim$0.3 Å/s), which
likely helped the growth at higher substrate temperatures ($>$500∘C) by
overcoming the rapid re-evaporation of Sr from the substrate.

With the MBE-grown Sr3SnO films, Ma et al. successfully performed in-situ
ARPES of the valence bands of Sr3SnO, which showed good agreement with DFT
calculations Ma  _et al._ (2020). Wu et al. used scanning electron microscopy
(SEM) to investigate their MBE-grown Sr3SnO filmsWu  _et al._ (2021). In Sn-
rich films that exhibited similar transport properties to that of pure
stoichiometric films and showed no impurity peaks in XRD, SEM nevertheless
revealed the presence of impurity phases as dark columnar regions. Hence, the
use of microscopy and other advanced characterization tools may provide a
fuller picture of antiperovskite films.

###  II.4 Transport

The antiperovskite films suffer from extreme air sensitivity and degrade
within seconds of exposure to ambient environment, as evidenced by a
pronounced change in color from metallic silver to opaque brown. To
characterize the films after MBE growth, two pathways are possible. (1) For
ex-situ measurements, we transported the films in a vacuum suitcase from the
load lock of the MBE to a glove box filled with inert Ar gas (inset of Fig.
2). Inside the glove box, we evaporated a Au capping layer, which then enabled
the antiperovskite films to survive in air for several hours during XRD
measurements or thickness measurements. For transport measurements of the
films, we deposited Au electrodes inside the glove box, covered the remaining
areas of the film with a grease droplet (Apiezon N), and performed wiring onto
a puck either with Ag epoxy (EPOTEK E4110; room-temperature cure) inside the
glove box, or bonding outside the glove box. These steps allowed the films to
survive when transferring the wired device to a PPMS. (2) For measurements in
UHV, these were performed either in the MBE growth chamber for RHEED, or in an
adjoining chamber for STM (Fig. 2), or transported via a vacuum suitcase to a
separate chamber for XPS and LEED.

Both Sr3PbO and Sr3SnO revealed metallic behavior with similar residual
resistivity ratios around 1.5–2 [Figs. 5(a) and 5(b)]. The resistance curve of
Sr3SnO is qualitatively similar to that of Refs. Ma  _et al._ (2020); Wu  _et
al._ (2021) [overlaid in Fig. 5(b)], showing good consistency among the MBE-
grown Sr3SnO films of different groups, measured both with an in-situ four-
probe setup Ma  _et al._ (2020) and ex-situ with a grease protection layer
(here and Wu  _et al._ (2021)). Hall measurements [Figs. 5(c) and 5(d)] point
to hole carriers, presumably arising from Sr vacancies. While the hole carrier
concentration could be tuned by varying the Sr/Pb and Sr/Sn ratios, the lowest
concentrations we could attain were around 1$\times$1020 cm-3 for Sr3PbO and
1$\times$1019 cm-3 for Sr3SnO. For Sr3SnO, the corresponding Fermi energy is
within the range of the Dirac dispersion, making the films ideal for
magnetotransport measurements Nakamura  _et al._ (2020); Huang  _et al._
(2021); Wu  _et al._ (2021).

Figure 5: Transport characterization of antiperovskite films. (a) and (b)
Temperature-dependent sheet resistances of a Sr3PbO (120 nm thick) and Sr3SnO
(100 nm thick) film, respectively. In (b), similar resistance curves of Sr3SnO
films from Refs. Ma  _et al._ (2020); Wu  _et al._ (2021) are scaled and
overlaid, showing qualitative consistency among films measured in situ, or ex
situ with a protection layer of Apiezon grease. (c) and (d) Transverse
resistances in a perpendicular magnetic field. The carrier densities and
mobilities are 1.9 $\times$ 1020 holes/cm3 and 121 cm2/(Vs)-1 at 10 K for the
Sr3PbO film and 2.6 $\times$ 1019 holes/cm3 and 162 cm2/(Vs)-1 at 2 K for the
Sr3SnO film.

###  II.5 Surface characterization

We now direct our focus to the characterization of antiperovskite surfaces. It
is well established that the perovskite oxides exhibit rich surface chemistry.
For example, the workhorse substrate SrTiO3 hosts a plethora of surface
reconstructions, including (2$\times$1), (2$\times$2), $c$(4$\times$2),
$c$(4$\times$4), $c$(4$\times$6), (6$\times$2),
($\sqrt{5}$$\times$$\sqrt{5}$)$R$26.6∘, and
$\sqrt{13}$$\times$$\sqrt{13}$)$R$33.7∘ superstructures for the nonpolar (001)
surface alone Erdman  _et al._ (2003). Many of these reconstructions are
oxygen deficient and are formed simply by annealing at high temperatures in
UHV. Different reconstructions also appear during the MBE growth of SrTiO3
films by making fine changes in flux ratios Kajdos and Stemmer (2014). In
contrast, there is scant information regarding surface reconstructions in
antiperovskite oxides. As previously described, the Sr3PbO(001) and
Sr3SnO(001) surfaces are polar (type III according to Tasker’s classification
Tasker (1979)), and without an atomic or electronic reconstruction, the
electrostatic potential would diverge at the surface. It is also natural to
ask whether Sr vacancies in Sr3PbO and Sr3SnO can play an analogous role to O
vacancies in perovskites in stabilizing surface reconstructions. Understanding
the surface atomic structure of antiperovskites is crucial, since topological
Chiu  _et al._ (2017) and/or spin-polarized Arras  _et al._ (2021) surface
states have been predicted, and their properties would be affected by
reconstructions. Edge states arising from higher-order topology are also
predicted Fang and Cano (2020), so investigating terrace structures at
surfaces would also be useful. Many systematic studies are needed in the
antiperovskites; here, we provide a brief overview of preliminary
investigations in this direction.

XPS is a surface-sensitive technique that probes the core levels of elements
and reveals their valence state and chemical environment Moulder and Chastain
(1992). By measuring at normal and grazing emission angles, the probing depth
can be varied. In the case of Sr3PbO and Sr3SnO, it is instructive to check
whether anionic Pb and Sn can be stabilized in thin films, given that the
massive Dirac cones are formed partly by the nearly filled $p$ shells of Pb
and Sn. (Although the formal oxidation states of Pb and Sn are $-$4, the Bader
charges computed by DFT is closer to $-$2, indicating a degree of covalency
Huang  _et al._ (2019)). We found that indeed in our films, the Pb 4$f$ and Sn
3$d$ core levels showed a component with lower binding energy than metallic Pb
and Sn, consistent with anionic Pb and Sn Huang  _et al._ (2019). However,
traces of cationic and/or neutral Pb and Sn were also observed within a
surface layer, estimated to be on the order of 1 nm thick. This surface layer
appeared to be stronger in Sr3SnO. Furthermore, we observed both cationic and
neutral Sn on the surface of Sr3SnO, but only cationic Pb on the surface of
Sr3PbO.

Figure 6: RHEED images of various (a) and (b) Sr3PbO(001) and (c)–(f)
Sr3SnO(001) films taken along the [100] direction. Electron energy: 15 keV.
Corresponding line cuts intersecting the (01) and (02) streaks are shown.
Sample numbers correspond to those in Fig. 7. Superstructure streaks in
Sr3SnO(001) films were obtained by tuning growth conditions, such as (d)
increasing the substrate temperature by 50∘C, (e) increasing the Sn flux, or
(f) annealing the film in Sr flux for several hours after growth.

Further insights into the surfaces of Sr3PbO and Sr3PbO could be derived from
LEED and RHEED measurements. Our LEED measurements revealed 1$\times$1
diffraction spots consistent with the expected (001) surfaces of Sr3PbO and
Sr3SnO Huang  _et al._ (2019). RHEED measurements similarly exhibited streaks
indicative of a flat, 2D film [Fig. 6(a)-(c)]. We note that measurements of
RHEED during the film deposition were rather useful, because we could
differentiate between rock-salt SrO and antiperovskite by the appearance of
(0$l$) streaks with odd $l$ that are extinct in the former, but not in the
latter (see supplemental). In Sr3SnO, we found that it was possible to induce
superstructure streaks in the RHEED pattern by increasing the substrate
temperature or tuning the Sr/Sn flux ratio from optimized values. Three
examples are shown in Figs. 6(d)–(f). Despite differences in their patterns,
the superstructure streaks appear to be multiples of $l$ = 1/8, which suggests
that the underlying real-space pattern has a periodicity of 8$a$. In Fig.
6(d), we observe $l$ = 1/8, 2/8, and 3/8 superstructure streaks; in Fig. 6(e),
$l$ = 3/8, 4/8, and 5/8 streaks; in Fig. 6(f), $l$ = 4/8 streaks. These
patterns are reminiscent of the $4a$, $5a$, and $6a$ superstructure streaks
detected in La2-xSrxCuO4 thin films and associated with ordering of oxygen
vacancies Suyolcu  _et al._ (2022); Wu  _et al._ (2017). In the case of
Sr3SnO, future systematic studies are needed to map out the full phase diagram
of surface reconstructions. In the case of Sr3PbO, despite tuning the
substrate temperature and Sr/Pb flux ratio, we consistently observed only a
1$\times$1 surface with no superstructure.

Figure 7: STM topographic images of (a) and (b) Sr3PbO and (c) Sr3SnO thin
films, revealing step edges with preferential orientation along [110] and
[1$\bar{1}$0]. Setpoints: (a) $-$1 V, 10 pA, (b) $-$2 V, 5 pA, (c) $-$1.75 V,
10 pA. (d) Averaged line cut along the horizontal red line in (b). The steps
mostly correspond to one unit cell (UC). (e)–(g) Scans over small areas on a
single terrace, revealing different kinds of globular surface morphologies.
The root-mean-square (RMS) roughness are 3.5, 0.8, and 3.6 Å, respectively.
Setpoints: (e) $-$1 V, 10 pA, (f) $-$1 V, 5 pA, (g) $-$1.5 V, 10 pA.

We performed room-temperature STM (Unisoku, Japan) on several antiperovskite
films immediately after MBE growth. Generally, we observed terraces on the
order of tens of nanometers in lateral size [Fig. 7(a)–(c)]. The step edges
appear to be preferentially oriented along the [110] and [1$\bar{1}$0]
directions, especially in Fig. 7(a), but steps aligned along [100] and [010]
were also observed in Fig. 7(c), as well as rounder steps that are more
densely bunched [Fig. 7(b)]. A sample line cut in Fig. 7(d) shows that the
step heights are consistent with integer multiples of the antiperovskite unit
cell.

Our STM measurements failed to resolve any atomic lattices. Instead, we
observed different surface morphologies, consisting of globules of varying
sizes [Figs. 7(e)–7(g)], which sometimes showed quasi-ordering into stripes
[Fig. 7(f)]. The STM images showed little correspondence with RHEED patterns
taken on the same films. For example, the Sr3PbO films AP424 and AP426 have
the same RHEED pattern with 1$\times$1 streaks [Figs. 6(a) and 6(b)], but STM
did not resolve a 1$\times$1 atomic lattice in either case [Figs. 7(e) and
7(f)], and the terracing behavior is quite distinct. In the case of Sr3SnO, we
were similarly unable to resolve a 1$\times$1 atomic lattice [sample SS121;
Fig. 7(g)], even when RHEED patterns indicated such a structure [Fig. 6(c)].

This discrepancy between RHEED and STM has previously been reported in the
perovskite SrTiO3(001). Although simultaneous RHEED measurements showed sharp
streaks associated with a 1$\times$1 surface, STM revealed a disordered,
globular surface Iwaya  _et al._ (2018). One hypothesis is that an amorphous
layer that is invisible to electron diffraction covers the 1$\times$1 surface.
In our case, this could be the $\sim$1-nm-thick surface layer detected in XPS.
In La1-xCaxMnO3 thin films, a similar phenomenology has been attributed to the
result of an unstable growth front Tselev  _et al._ (2015), which may be more
significant for polar surfaces. As the constituent atoms land on the surface
of the film, they cluster in highly disordered, 3D globules, for which STM
cannot resolve an atomic lattice. As more atoms are deposited and these
globules become buried beneath the surface, their crystallinity improves,
although defects and vacancies get incorporated. Hence, the surface layer is
always disordered, whereas the layers beneath are crystalline. This mechanism
may explain the appearance of clear step edges and terrace structure in Sr3PbO
and Sr3SnO, indicating epitaxial growth, yet the appearance of globules on
each terrace. In the case of SrTiO3(001), a 1$\times$1 surface could be
visualized not with STM, but with non-contact AFM Sokolović  _et al._ (2019).
Furthermore, once the SrTiO3(001) surface reconstructs and deviates from
1$\times$1, STM can resolve the different and diverse reconstructions. In our
case, we could not obtain stable STM imaging on Sr3SnO(001) films showing
superstructure streaks in RHEED, nor resolve the superstructures.

In the characterization of MBE-grown antiperovskite films, RHEED is more
sensitive than XRD, but STM is even more sensitive than RHEED. RHEED can
resolve the appearance of surface reconstructions in Sr3SnO(001) films that
otherwise show the same XRD patterns. However, STM can resolve different
terrace structures and surface morphologies in Sr3PbO(001) films that
otherwise show the same RHEED patterns. Our preliminary measurements indicate
that antiperovskites show a similar richness in surface chemistry that is well
explored in the perovskite oxides, but many more systematic studies are
needed.

##  III Conclusion

We have demonstrated pathways for the MBE growth of the antiperovskites
Sr3PbO(001) and Sr3SnO(001). By using sequential (element-shuttered) MBE that
separates oxygen flux from other fluxes (Sr, Pb, or Sn), high quality films
with sharp XRD rocking curves were obtained at relatively low substrate
temperatures of 450∘C. Complex surface reconstructions that indicate a long-
range periodicity of eight times the lattice constant were observed in
Sr3SnO(001) by deviating from an optimal growth condition, whereas signs of
reconstruction were absent in Sr3PbO(001). Such long-range patterns are
reminiscent of the ordering of oxygen vacancies in La2-xSrxCuO4 films, but the
microscopic origin in the case of the antiperovskites needs to be clarified in
subsequent studies. Looking towards the future, there is still much to be
improved in terms of film quality, especially in the light of the ultrahigh
crystallinity and carrier mobilities that have been recently achieved in the
MBE of conventional oxides Schlom  _et al._ (2008); Son  _et al._ (2010);
Falson  _et al._ (2016). Further advancement in the control of material
quality will most certainly open a novel research arena at the intersection of
correlated electrons and topological electronic states. We also anticipate
that novel electronic and magnetic phases at surfaces or heterointerfaces of
antiperovskites are within experimental reach by using MBE. The exploration of
such avenues will likely involve the seamless integration of advanced
experimental apparatuses with the MBE system, even beyond what is outlined in
this Perspective.

##  IV Acknowledgement

We thank D. Samal and J. Merz for contributions in the development of
antiperovskite MBE. We are also grateful to K. Pflaum, M. Dueller, U.
Engelhardt, and C. Mühle for technical support. D.H. acknowledges support from
a Humboldt Research Fellowship for Postdoctoral Researchers.

##  V Supplementary Material

See supplementary material for additional XRD, RHEED, and transport data of
antiperovskites grown by MBE.

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$\mu_{ij}:\mathbbm{1}_{S}(1)[2]\stackrel{{\scriptstyle\iota_{i}}}{{\hookrightarrow}}\Pi_{S}(\partial_{i}X)\xrightarrow{\nu_{i*}}\Pi_{S}(\bar{X})\xrightarrow{\nu_{j}^{!}}\Pi_{S}(\partial_{j}X,\langle
N_{j}\rangle)\stackrel{{\scriptstyle\pi_{j}}}{{\twoheadrightarrow}}\operatorname{Th}(N_{j}^{0})$

Here, $\iota_{i}$ (resp. $\pi_{j}$) is the (canonical) split monomorphism
(resp. epimorphism) in the isomorphism $\epsilon_{1}$ (resp. $\epsilon_{2}$)
of 6.2.2 (resp. 6.2.5). Moreover, if one chooses a trivialization of
$N_{j}^{0}$ over $S$, $\mu_{ij}$ becomes an element of
$\operatorname{GW}(\mathcal{O})$ such that

$\mu_{ij}=(\partial_{j}X,\partial_{i}X)^{or}_{quad}$

Here, we use the natural orientation of the tangent space of
$\partial_{i}X\stackrel{{\scriptstyle\epsilon_{i}}}{{\simeq}}\mathbf{P}^{1}_{k}$
(that is, ${\mathcal{O}}_{\mathbf{P}^{1}_{k}}(2)$) and the chosen orientation
of $N_{j}$ to define the oriented quadratic intersection degree on the right
hand-side (see Definition 6.1.7). Our identification of $\mu_{ij}$ follows
from the definitions: $\nu_{j}^{!}$ is essentially determined by the
fundamental class $\eta_{\bar{X}}(\partial_{j}X)$, and the composition with
$\nu_{i*}$ is to be understood as the pullback by $\nu_{i}$ in the homotopy
category.

###  6.3. Triangulated and abelian mixed motives

######  6.3.1.

The motive of a Thom space depends only on the rank of the vector bundle,
i.e., for $\Lambda$-linear motives we do not need to assume normal bundles are
relatively oriented. Thus the motive at infinity $M^{\infty}(X)$ – or
Wildeshaus’ boundary motive – is the homotopy fiber of the canonical map

$M(\partial X)\rightarrow M(\partial X)^{\vee}(2)[4]$

associated with the fundamental class of the diagonal of $\partial X/S$ (see
5.2.5). Let $\mathcal{D}_{X}^{M}$ be the smooth Artin motive over $S$ defined
as the homotopy cofiber of (6.2.2.a) seen as a rigid object in
$\operatorname{DM}(S,\Lambda)$. 5.3.2 implies the next result.

######  Theorem 6.3.2.

In the setting of 6.0.1, the motive at infinity $M^{\infty}_{S}(X)$ of the
relative surface $X/S$ is the homotopy fiber of the map

$\beta:\mathcal{D}^{M}_{X}\oplus\bigoplus_{i\in
I}\mathbbm{1}_{S}(1)[2]\xrightarrow{\begin{pmatrix}a&b^{\vee}(2)[4]\\\
b&\mu\end{pmatrix}}(\mathcal{D}^{M}_{X})^{\vee}(2)[4]\oplus\bigoplus_{j\in
I}\mathbbm{1}_{S}(1)[2]$

Moreover, $\mu$ is given by the matrix $(\mu_{ij})_{i,j\in I}$ where

$\mu_{ij}=\deg(\nu_{i}^{*}(\eta_{\bar{X}}(\partial_{j}X))\in
H^{00}_{M}(S,\Lambda)$

Here, the quadratic intersection degree is an integer,
$\mu_{ij}\in\mathbb{Z}$, in the following cases:

  * •

$S$ is regular connected and $\Lambda=\mathbb{Q}$ (use [CD19])

  * •

$S$ is smooth connected over a field or a Dedekind ring and
$\Lambda=\mathbb{Z}$ (use [Spi18])

Note that $M^{\infty}_{S}(X)$ is a triangulated mixed and smooth Artin-Tate
motive. Next we relate the above to the motivic $t$-structure, which exists
for rational coefficients $\Lambda=\mathbb{Q}$ when:

  * •

$S=\operatorname{Spec}(K)$, $K$ is a finite field, or a global field (see
[Lev93])

  * •

$S=\operatorname{Spec}(O_{K})$, $\mathcal{O}_{K}$ is a number ring (see
[Sch11])

The homology motive at infinity of $X/S$ in the abelian category of Artin-Tate
mixed motives $\operatorname{MM^{AT}}(S,\mathbb{Q})$ is defined by

$\operatorname{\underline{H}}_{i}^{\infty}(X):=\operatorname{\underline{H}}_{i}^{\mu}(M_{S}^{\infty}(X))$

where $\operatorname{\underline{H}}_{i}^{\mu}$ is the $i$-th homology functor
for the motivic t-structure (with standard homological conventions).

######  Corollary 6.3.3.

In the notation of 6.3.2, the homology motive at infinity
$\operatorname{\underline{H}}_{i}^{\infty}(X)$ vanishes for $i\not\in[0,3]$
and there is an exact sequence in the abelian category
$\operatorname{MM^{AT}}(S,\mathbb{Q})$

$\displaystyle 0\rightarrow$
$\displaystyle\operatorname{\underline{H}}_{3}^{\infty}(X)\rightarrow\bigoplus_{i\in
I}\mathbbm{1}_{S}(2)\xrightarrow{\sum_{i<j}p_{ij}^{i!}-p_{ij}^{j!}}\bigoplus_{i<j}M_{S}(\partial_{ij}X)(2)$
$\displaystyle\rightarrow\operatorname{\underline{H}}_{2}^{\infty}(X)\rightarrow\bigoplus_{i\in
I}\mathbbm{1}_{S}(1)\xrightarrow{\ \mu\ }\bigoplus_{j\in I}\mathbbm{1}_{S}(1)$
$\displaystyle\rightarrow\operatorname{\underline{H}}_{1}^{\infty}(X)\rightarrow\bigoplus_{i<j}M_{S}(\partial_{ij}X)\xrightarrow{\sum_{i<j}p^{i}_{ij*}-p^{j}_{ij*}}\bigoplus_{i\in
I}\mathbbm{1}_{S}\rightarrow\operatorname{\underline{H}}_{0}^{\infty}(X)\rightarrow
0$

where $\mu$ is the Mumford matrix, and
$M_{S}(\partial_{ij}X)=\operatorname{\underline{H}}^{\mu}_{0}(M_{S}(\partial_{ij}X))$
is seen as an abelian Artin-Tate motive.

Note in particular that $\operatorname{\underline{H}}_{0}^{\infty}(X)$ and
$\operatorname{\underline{H}}_{3}^{\infty}(X)$ are pure of respective weights
$0$ and $-4$, while $\operatorname{\underline{H}}_{1}^{\infty}(X)$ and
$\operatorname{\underline{H}}_{2}^{\infty}(X)$ are in general mixed of weights
$\\{0,-2\\}$ and $\\{-2,-4\\}$, respectively.

###  6.4. Three general lines in projective plane

Over a field $k$, let $L_{1},L_{2},L_{3}\cong\mathbb{P}^{1}_{k}$ be three
general lines in $\mathbb{P}^{2}_{k}$. Let
$\sigma:\bar{X}\to\mathbb{P}^{2}_{k}$ be the surface obtained by blowing-up
one $k$-rational point on each of the lines $L_{i}$ distinct from the
$k$-rational points $p_{12}=L_{1}\cap L_{2}$, $p_{13}=L_{1}\cap L_{3}$,
$p_{2,3}=L_{2}\cap L_{3}$. Then $\bar{X}$ is a del Pezzo surface of degree $6$
and the proper transform $\partial X$ of $L_{1}\cup L_{2}\cup L_{3}$ on
$\bar{X}$ a strict normal crossing divisor; it is the support of an anti-
canonical divisor on $\bar{X}$ and thus an ample divisor. We now compute the
homology motive at infinity of the affine surface $X=\bar{X}-\partial X$. In
the exact sequence of Corollary 6.3.3 the homomorphisms
$\bigoplus_{i<j}M_{k}(\partial_{ij}(X))\cong\bigoplus_{i=1}^{3}\mathbbm{1}_{k}\rightarrow\bigoplus_{i\in
I}\mathbbm{1}_{k}\cong\bigoplus_{i=1}^{3}\mathbbm{1}_{k}$ and $\bigoplus_{i\in
I}\mathbbm{1}_{k}(2)\cong\bigoplus_{i=1}^{3}\mathbbm{1}_{k}(2)\rightarrow\bigoplus_{i<j}M_{k}(\partial_{ij}(X))(2)\cong\bigoplus_{i=1}^{3}\mathbbm{1}_{k}(2)$
are given respectively by the matrix

$N=\left(\begin{array}[]{ccc}1&1&0\\\ -1&0&1\\\ 0&-1&-1\end{array}\right)$

and its transpose. Moreover, since $L_{i}$ has trivial normal bundle in
$\bar{X}$ for $i=1,2,3$, the quadratic Mumford intersection matrix equals

$\mu=\left(\begin{array}[]{ccc}0&\langle 1\rangle&\langle 1\rangle\\\ \langle
1\rangle&0&\langle 1\rangle\\\ \langle 1\rangle&\langle
1\rangle&0\end{array}\right)$

The matrix $N\in\mathcal{M}_{3,3}(\mathbb{Z})$ is equivalent to the diagonal
matrix $\mathrm{diag}(1,1,0)$ and the matrix $\mu$ in
$\mathcal{M}_{3,3}(\mathrm{GW}(k))$ is equivalent to the diagonal matrix
$\mathrm{diag}(1,1,2)$. In summary, the above yields

$\underline{H}_{i}^{\infty}(X)=\left\\{\begin{array}[]{ll}\mathbbm{1}_{k}&i=0\\\
(\mathbbm{1}_{k}/2)(1)\oplus\mathbbm{1}_{k}&i=1\\\ \mathbbm{1}_{k}(2)&i=2\\\
\mathbbm{1}_{k}(2)&i=3\end{array}\right.$

###  6.5. Danielewski hypersurfaces

For a field $k$ and $n\geq 1$, the Danielewski hypersurface $D_{n}$ is the
smooth affine surface $D_{n}$ in $\mathbf{A}_{k}^{3}$ cut out by the equation
$x^{n}z=y(y-1)$. Owing to [Dan88], $D_{n}$ becomes a Zariski locally trivial
$\mathbf{G}_{a}$-bundle over the affine line with two origins
$\breve{\mathbf{A}}_{k}^{1}$ (using the factorization of the surjective
projection $\pi_{n}=\mathrm{pr}_{x}:D_{n}\to\mathbf{A}_{k}^{1}$). Thus $D_{n}$
is $\mathbf{A}^{1}$-equivalent to $\breve{\mathbf{A}}_{k}^{1}$ and
$\mathbf{P}_{k}^{1}$. The threefolds $D_{n}\times\mathbb{A}^{1}_{k}$ are
isomorphic, but the surfaces $D_{n}$ are pairwise non-isomorphic. Over
$\mathbb{C}$, Danielewski [Dan88], Fieseler [Fie94] established this by
showing the underlying complex analytic manifolds have non-isomorphic first
singular homology groups at infinity. Our methods provide a base field
independent argument which allows to distinguish between the $D_{n}$’s via
homology motives at infinity.

We begin by constructing explicit smooth projective completions $\bar{D}_{n}$
of the surfaces $D_{n}$, whose boundaries are strict normal crossing divisors.
The morphism $\varphi_{n}=\mathrm{pr}_{x,y}:D_{n}\to\mathbf{A}_{k}^{2}$
expresses $D_{n}$ as the affine modification of $\mathbf{A}_{k}^{2}$ with
center at the closed subscheme $Z_{n}$ with ideal $(x^{n},y(y-1))$ and divisor
$D_{n}=\mathrm{div}(x^{n})$, cf. [DPØ19]. Furthermore, $\varphi_{n}$
decomposes into a sequence of affine modifications

(6.5.0.a) $\varphi_{n}=\varphi_{1}\circ\psi_{2}\cdots\circ\psi_{n}\colon
D_{n}\to D_{n-1}\to\cdots D_{2}\to D_{1}\to{\mathbf{A}}_{k}^{2}$

given by $\psi_{\ell}:D_{\ell}\to D_{\ell-1}$; $(x,y,z)\mapsto(x,y,xz)$, with
center at the closed subscheme $Y_{\ell-1}=(x,z)$ and divisor
$H_{\ell}=\mathrm{div}(x)$. That is, $\varphi_{1}:D_{1}\to\mathbf{A}_{k}^{2}$
is the birational morphism obtained by blowing-up the points $(0,0)$, $(0,1)$
in $\mathbf{A}_{k}^{2}$ and removing the proper transform of
$\\{0\\}\times\mathbf{A}_{k}^{1}$, and $\psi_{\ell}:D_{\ell}\to D_{\ell-1}$ is
the birational morphism obtained by blowing-up the points $(0,0,0)$, $(0,0,1)$
in $\pi_{\ell}^{-1}(0)$ and removing the proper transform of
$\pi_{\ell-1}^{-1}(0)$.

Now consider the open embedding
$\mathbf{A}_{k}^{2}\hookrightarrow\mathbf{P}_{k}^{1}\times\mathbf{P}_{k}^{1}$;
$(x,y)\mapsto([x:1],[y:1])$. Then $C_{\infty}=\mathbf{P}_{k}^{1}\times[1:0]$
and $F_{\infty}=[1:0]\times\mathbf{P}_{k}^{1}$ are irreducible components of
$\mathbf{P}_{k}^{1}\times\mathbf{P}_{k}^{1}$ and we set
$F_{0}=[0:1]\times\mathbf{P}_{k}^{1}$. Let
$\bar{\varphi}_{1}:\bar{D}_{1}\to\mathbf{P}_{k}^{1}\times\mathbf{P}_{k}^{1}$
be the blow-up of the points $([0:1],[0:1])$, $([0:1],[1:1])$ in $F_{0}$, with
respective exceptional divisors $E_{1,0}$, $E_{1,1}$. From now on the proper
transform of $F_{0}$ in $\bar{D}_{1}$ is also denoted by $F_{0}$. With these
definitions, there is a commutative diagram

$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\textstyle{\bar{D}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\varphi}_{1}}$$\textstyle{\mathbf{A}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{P}^{1}\times\mathbf{P}^{1}}$

Here, $D_{1}\hookrightarrow\bar{D}_{1}$ is the open immersion given by the
complement of the support of the strict normal crossing divisor $\partial
D_{1}=C_{\infty}\cup F_{\infty}\cup F_{0}$. The closures in $\bar{D}_{1}$ of
the two irreducible components $\\{x=y=0\\}$ and $\\{x=y-1=0\\}$ of
$\pi_{1}^{-1}(0)$ equal the exceptional divisors $E_{1,0}$ and $E_{1,1}$,
respectively. We calculate the self-intersection numbers
$C_{\infty}^{2}=F_{\infty}^{2}=0$, $F_{0}^{2}=-2$ in $\bar{D}_{1}$; that is,
the usual degrees of the respective normal line bundles of these curves in
$\bar{D}_{1}$, see e.g., [GH94, Chapter 5.6], [Sha13, Chapter IV].

To construct $\bar{D}_{n}$, $n\geq 2$, we start with $\bar{D}_{1}$ and proceed
inductively by performing the same sequence of blow-ups as for the affine
modifications $\psi_{\ell}:D_{l}\to D_{\ell-1}$ in (6.5.0.a). This yields
birational morphisms $\bar{\psi}_{\ell}:\bar{D}_{\ell}\to\bar{D}_{\ell-1}$
consisting of the blow-up of one point on $E_{\ell,0}-E_{\ell-1,0}$ and
another point on $E_{\ell,1}-E_{\ell-1,1}$ with respective exceptional
divisors $E_{\ell+1,0}$ and $E_{\ell+1,1}$ (by convention
$E_{0,0}=E_{0,1}=F_{0})$. Moreover, $D_{\ell}$ embeds into $\bar{D}_{\ell}$ as
the complement of the support of the strict normal crossing divisor $\partial
D_{\ell}=C_{\infty}\cup F_{\infty}\cup
F_{0}\cup\bigcup_{i=1}^{\ell-1}(E_{i,0}\cup E_{i,1})$ in such a way that the
closures of the two irreducible components $\\{x=y=0\\}$ and $\\{x=y-1=0\\}$
of $\pi_{\ell}^{-1}(0)$ coincide with the divisors $E_{\ell+1,0}$ and
$E_{\ell+1,1}$, respectively. By construction, there is a commutative diagram

$\textstyle{\bar{D}_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi_{\ell}}}$$\textstyle{\bar{D}_{\ell-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{D}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi_{2}}}$$\textstyle{\bar{D}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\varphi}_{1}}$$\textstyle{\mathbf{P}^{1}\times\mathbf{P}^{1}}$$\textstyle{D_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\ell}}$$\textstyle{D_{\ell-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{2}}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\textstyle{\mathbf{A}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

For every $n\geq 2$, we may visualize the boundary divisor $\partial D_{n}$ as
a fork of $\mathbf{P}^{1}$’s

|  |
---|---|---
$\textstyle{(E_{1,0},-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E_{n-1,0},-2)}$$\textstyle{(F_{\infty},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C_{\infty},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(F_{0},-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E_{1,1},-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(E_{n-1,1},-2)}$

with the indicated self-intersection numbers for each irreducible component.
We may order the irreducible components of $\partial D_{n}$ as follows

$F_{\infty}<C_{\infty}<F_{0}<E_{1,0}<\ldots<E_{n-1,0}<E_{1,1}<\ldots<E_{n-1,1}$

The Euler class of $\mathcal{O}_{\mathbf{P}^{1}}(2)$ equals the hyperbolic
plane $\mathbb{H}=\langle 1,-1\rangle\in\operatorname{GW}(k)$. Thus the Euler
class of $\mathcal{O}_{\mathbf{P}^{1}}(-2)$ is $-\mathbb{H}$. Moreover, by
Proposition 6.1.4, the transversal intersection of two copies of
$\mathbf{P}^{1}$ in a unique $k$-rational point yields the class $\langle
1\rangle\in\operatorname{GW}(k)$. The quadratic Mumford intersection matrix
thus takes the form (with zero entries mostly left out of the notation)

$\mu_{n}=\left(\begin{array}[]{ccccccccc}0&\langle 1\rangle\\\ \langle
1\rangle&0&\langle 1\rangle\\\ &\langle 1\rangle&-\mathbb{H}&\langle
1\rangle&0&&\langle 1\rangle\\\ &&\langle 1\rangle&-\mathbb{H}&\langle
1\rangle\\\ &&0&\langle 1\rangle&\ddots&\langle 1\rangle\\\ &&&&\langle
1\rangle&-\mathbb{H}&0\\\ &&\langle 1\rangle&&&0&-\mathbb{H}&\langle
1\rangle\\\ &&&&&&\langle 1\rangle&\ddots&\langle 1\rangle\\\ &&&&&&&\langle
1\rangle&-\mathbb{H}\end{array}\right)$

######  Proposition 6.5.1.

Over a field $k$ and $n\geq 1$, the homology motives at infinity of the
Danielewski surfaces $D_{n}$ are given by

$\underline{H}_{i}^{\infty}(D_{n})=\left\\{\begin{array}[]{ll}\mathbbm{1}_{k}&i=0\\\
(\mathbbm{1}_{k}/2n)(1)&i=1\\\ 0&i=2\\\
\mathbbm{1}_{k}(2)&i=3\end{array}\right.$

###### Proof.

The boundary $\partial D_{n}$ has $2n+1$ irreducible components which
intersect in $2n$ $k$-rational points. As in Example 6.4, one infers from the
description of $\partial D_{n}$ that the homomorphism

$\bigoplus_{i<j}M_{k}(\partial_{ij}(D_{n}))\cong\bigoplus_{i=1}^{2n}\mathbbm{1}_{k}\rightarrow\bigoplus_{i\in
I}\mathbbm{1}_{k}\cong\bigoplus_{i=1}^{2n+1}\mathbbm{1}_{k}$

is given by an element $N$ of $\mathcal{M}_{2n,2n+1}(\mathbb{Z})$ equivalent
to $\mathrm{I}_{2n}$. This implies that
$\underline{H}_{0}^{\infty}(D_{n})=\mathbbm{1}_{k}$ and
$\underline{H}_{1}^{\infty}(D_{n})=\mathrm{Coker}(\mu_{n})$. Since the
homomorphism

$\bigoplus_{i\in
I}\mathbbm{1}_{k}(2)\cong\bigoplus_{i=1}^{2n+1}\mathbbm{1}_{k}(2)\rightarrow\bigoplus_{i<j}M_{k}(\partial_{ij}(D_{n}))(2)\cong\bigoplus_{i=1}^{2n}\mathbbm{1}_{k}(2)$

is given by the transpose of $N$, it follows that
$\underline{H}_{3}^{\infty}(D_{n})=\mathbbm{1}_{k}(2)$ and
$\underline{H}_{2}^{\infty}(D_{n})=\mathrm{Ker}(\mu_{n})$. Finally, one can
check by elementary transformations on rows and columns that as an element of
$\mathcal{M}_{2n+1,2n+1}(\mathrm{GW}(k))$, the quadratic Mumford matrix
$\mu_{n}$ is equivalent to the diagonal matrix
$\mathrm{diag}(1,\ldots,1,n\mathbb{H})$. This implies in turn that
$\underline{H}_{2}^{\infty}(D_{n})=0$ and
$\underline{H}_{3}^{\infty}(D_{n})=(\mathbbm{1}_{k}/2n)(1)$. ∎

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