Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

ESWC ’23: Trusting Decentralised Knowledge Graphs and Web Data Workshop –
TrusDeKW, May 28–June 1, 2023, Hersonissos, Greece

[1]

[orcid=0000-0001-8390-6171<EMAIL_ADDRESS>url=https://patrickhochstenbach.net/ ] [1]

[orcid=0000-0001-8862-0666<EMAIL_ADDRESS>]

[orcid=0000-0002-8596-222X<EMAIL_ADDRESS>url=https://ruben.verborgh.org/ ]

[1]Corresponding author.

# RDF Surfaces: Computer Says No

Patrick Hochstenbach  Ghent University Library, Rozier 9, Ghent, 9000, Belgium
Ghent University IDLab - ELIS, Technologiepark-Zwijnaarde 122, Zwijnaarde,
9052, Belgium  Jos De Roo  Ruben Verborgh

(2023)

###### Abstract

Logic can define how agents are provided or denied access to resources, how to
interlink resources using mining processes and provide users with choices for
possible next steps in a workflow. These decisions are for the most part
hidden, internal to machines processing data. In order to exchange this
internal logic a portable Web logic is required which the Semantic Web could
provide. Combining logic and data provides insights into the reasoning process
and creates a new level of trust on the Semantic Web. Current Web logics
carries only a fragment of first-order logic (FOL) to keep exchange languages
decidable or easily processable. But, this is at a cost: the portability of
logic. Machines require implicit agreements to know which fragment of logic is
being exchanged and need a strategy for how to cope with the different
fragments. These choices could obscure insights into the reasoning process. We
created RDF Surfaces in order to express the full expressivity of FOL
including saying explicitly ‘no’. This vision paper provides basic principles
and compares existing work. Even though support for FOL is semi-decidable, we
argue these problems are surmountable. RDF Surfaces span many use cases,
including describing misuse of information, adding explainability and trust to
reasoning, and providing scope for reasoning over streams of data and queries.
RDF Surfaces provide the direct translation of FOL for the Semantic Web. We
hope this vision paper attracts new implementers and opens the discussion to
its formal specification.

###### keywords:

Semantic WebFirst-order Logic Logical Reasoning

##  1 Introduction

RDF [1] is the standard for modeling data on the Semantic Web. From a
syntactic viewpoint, RDF is a simple data model for expressing relations
between triples where each triple is a first-class web object. From a semantic
viewpoint, a triple is an assertion expressing what is believed to be _true_.
Combinations of triples form a logical conjunction ( and ). Any combination of
resources on the Semantic Web creates a universal and statement [2]. This is
not unusual as the majority of database systems assert truth similarly.

Human and software agents interpret data and use internal logic to turn these
assertions into decisions regarding what data to _trust_ or not (negation),
provide _options_ to select data to ingest or compute (disjunction) and make
_inferences_ from this data (implication). Current Web logics in the form of
rule and ontology languages provides insights into the reasoning process but
only carries a fragment of first-order logic (FOL). These fragments of FOL are
tuned to make the processing of Web logic decidable or easily processable, at
the cost of portability. _Portability_ is the ability to represent data and
logic independently irrespective of the processing environment. RDF is
portable, any RDF triple expresses the same data and meaning irrespective of
the source. Adding Web logic, this becomes much harder: machines must first
agree on an entailment regime before they know what each triple represents. In
this position paper, we present our case for a portable Web logic language
extending RDF semantics called _RDF Surfaces_ , which is as powerful as FOL.
RDF Surfaces is able to group RDF data within surfaces and provide semantics
to say ‘no’: expressing an explicit scope and classical negation.

##  2 Background

We started our research into portable logic in relation to our involvement in
the SolidLab Flanders [3] and Mellon-funded project “Scholarly Communications
in the Decentralized Web” [4]. In both projects, a decentralized network of
_data pods_ exist on which individuals store (personal) data. In the case of
SolidLab, data are documents Flemish citizens share with the government or
businesses. In the case of the Mellon project, data are research artifacts
such as scholarly articles and research data that are shared within a research
community. Both projects are investigating how automated systems can assist in
managing these types of data, interlink data with existing (external)
resources, and provide enforcement of data policies using logic-based rules.
In each use case, automated systems can’t assume that only one actor creates
and manages these rules. In a decentralized world, many agents can set their
own requirements of what should or shouldn’t happen with data. For instance,
in the case of data policies, different actors on a personal, institutional,
and governmental level can define the permissions and prohibitions (re)using
data. These policies can overlap and possibly contradict themselves. Such
clashes can benefit from being detected at an early stage and not at run time.
Another challenge is the single language problem. Machines should be capable
to implement potentially heterogeneous collection policy languages with many
possible dialects. The strength of RDF is that the data model for these
variations is portable, there is a common format that can express the overlap
between the different policy languages. But, it is not the case that the logic
– what all RDF triples entail – is portable. Without knowing which kind of
inferences are possible (using entailment regimes such as RDFS, OWL-DL, OWL2,
Notation3,..), it is possible that designers of policies can reach other
conclusions than the consumer (the policy enforcer on the pod). For these use
cases, we need a common portable logic, with a common syntax and semantics.

##  3 RDF Surfaces

Only two extensions are required to the semantics of RDF to make it
expressable as FOL: a notion of a (possible nested) _surface_ to group zero or
more RDF graphs with a truth-functional type, and a notion of _graffiti_ as
marks on such as a surface with the function of existentially quantified
variables.

Surfaces can be regarded as nestable virtual sheets of paper on which RDF
graphs are written. A _positive surface_ asserts all triples written on it; as
with RDF, multiple triples form a logical conjunction ( and ). The default
surface is positive. All existing RDF graphs are regarded to be on the default
positive surface. A _negative surface_ negates all triples written on it, and
multiple triples form a negation (not) of a conjunction ( and ).111Individual
triples are not necessarily negated; only their conjunctions. With logical
connectives and and not, any truth-functional statement can be created by
composition, similar to how logical gates on a computer chip can be created by
combining nand gates.

Graffiti are marks on a surface representing quantified logical variables.
Graffiti marks on a positive surface are interpreted as _existentially_
quantified variables, this is how blank nodes are currently interpreted in
RDF. The difference between blank nodes and graffiti marks is that blank nodes
act as _coreferences_ to graffiti marks, and that every surface contains its
own unique set of graffiti marks. Transporting marks to a new surface requires
creating new graffiti marks (not relabeling or merging graffiti marks, as is
the case with blank nodes). Using De Morgan’s duality222$\lnot\exists x\colon
P(X)\equiv\forall x\colon\lnot P(x)$ , graffiti marks on a negative surface
are interpreted as _universally_ quantified variables.

Given the notion of a surface and graffiti, we define an H-graph as the
combination of a (typed) surface $S$, graffiti $Gr$, and a graph $H$ which is
again an H-graph. Each RDF graph is an H-graph on the default positive typed
surface. The empty H-graph is regarded as a tautology on a positive-typed
surface and a contradiction on a negative-typed surface. By combining and
nesting H-graphs any truth-functional statement can be created. H-graphs have
similar semantics as the alpha and beta Existential graphs of
Pierce333https://plato.stanford.edu/entries/peirce-logic/#AlphSyst which are
as expressive as FOL[5][6]. Our position is that these two added primitives
(surfaces and graffiti marks) are a much more economical way to express FOL
than in OWL Full, plus they are an immediate natural extension to RDF
semantics.

We need a notation to transport the H-graphs over the Web and require an RDF
syntax for that. This syntax should provide a way to scope RDF graphs and
codify the graffiti marks. Our first choice was to use Notation3 [7], because
it provides syntactic support for nested quoted graphs (surfaces), built-ins
(to reason with surface types), and lists (codifying the graffiti marks).
Using Notation3, an H-graph is expressed using graffiti marks as the subject
list, a typed surface as a predicate, and an H-graph as the object. In our
implementations, the surface predicate is implemented as a built-in to allow
for reasoning with H-graphs.

Listing 1 provides an example of an H-graph with semantics $\forall
s\colon\mathit{learns}(s,Physics)\Rightarrow\mathit{reads}(s,Newton)\lor\mathit{reads}(s,Einstein)$
by stating it as $\forall
s\colon\lnot(\mathit{learns}(s,Physics)\land\lnot\mathit{reads}(s,Newton)\land\lnot\mathit{reads}(s,Einstein))$
which means “Anyone that learns physics reads Newton or Einstein (or both)”.

Listing 1: The RDF Surfaces semantics of a material implication with a
disjunction using the Notation3 syntax.

⬇

(_:S) log:onNegativeSurface {

_:S :learns :Physics .

() log:onNegativeSurface { _:S :reads :Newton } .

() log:onNegativeSurface { _:S :reads :Einstein } .

} .

For a detailed introduction to RDF Surfaces we refer to our RDF Surface Primer
[8].

##  4 Related Work

The case for portable Web logic is not new. Already in the 2000s Berners-Lee
[9] called for the development of a language on top of RDF in his Semantic Web
Logic Language (SWeLL) proposal. SWeLL was imagined as a unifying language
acting as a logical bus to “allow any web software to read and manipulate data
published by any other web software”. For all logical relations to be
expressed, SWeLL had to extend RDF by including negation and explicit
quantification.

A similar portability argument was given in 2009 by Hayes in his ISWC invited
talk [10]. According to his argumentation, the idea of Web logic portability
has not been achieved due to the layering of logic in the Semantic Web stack.
Different layers of the stack cannot guess which entailment regimes are used
when receiving data. Without this information, two independent machines cannot
arrive at the same conclusions given a model. As a first step towards a
solution for this conundrum, Hayes presented RDF Redux which provides a syntax
for expressing Web logic with FOL semantics. His ideas were very influential
for our work and led to our current research into RDF Surfaces.

There are two main reasons why classical negation has thus far been avoided.
The first reason is negation opens the door to create _paradoxes_ on the
Semantic Web. One of the main motivations of Berners-Lee et al. for N3Logic
[11] was to introduce the ability to compare Web documents and make inferences
about assertions in each document. For this reason, a quoting mechanism was
introduced. Quoting in combination with classical negation can lead to
paradoxes when assertions in resources are self-referential. This self-
referential problem resembles the liar paradox in the philosophy of language:
“This sentence is false”. To avoid paradoxes, Scoped Negation As Failure
(SNAF) was introduced to N3Logic and acts as a monotonic version of Negation
As Failure (NAF). SNAF disallows self-referential negative statements about
resources and defines a scope for negation by the absence of information
(Scope + NAF = SNAF). However, SNAF cannot avoid semantic inconsistencies nor
has a syntactic mechanism for expressing them. With RDF Surfaces we introduce
a formalism requiring negative facts to be explicitly expressed as negative
surfaces, without using (S)NAF. Inconsistencies can be expressed as an
assertion plus a negative surface thereof. These inconsistencies still need to
be detected which leads to our second reason.

Classical negation is also avoided due to combining the expressive power of
logics with negation and _decidability_ problems. _Satisfiability_ is a
decidability problem to find inconsistencies. _Completeness_ is a decidability
problem, related to satisfiability, to find all valid statements from a
knowledge base. _Halting_ is a decidability problem requiring a machine to
stop in a finite time. Logics containing logical variables and negation can be
as powerful as first-order logic (FOL) for which it is proven [12] solving any
of these problems is undecidable. Three options are available for dealing with
undecidability for which, alas, only two can be chosen: (P) portable, allow
any ontology input, (C) be complete, (H) always halt [13]. OWL 2 Full is an
example of a Web logic that can express any ontology but is undecidable. OWL 2
Description Logics drops option (P) and creates decidable fragments of FOL and
reasoners can always return all answers and always halt, i.e. (C+H). This
choice raises the question of what fragment of FOL to choose. This is at the
cost of Web portability. There are at least as many choices as there are OWL
Description Logics. Reasoners that cannot guess which fragment to choose to
have to implement them all to provide portability. Reasoners should be as
expressive as FOL in order to do so. In general, if we want portable Web
logic, option (P) seems to be unavoidable. We argue: any of the (P+C), (P+H),
and (C+H) axis are valuable depending on the use case. A reasoner as powerful
as (P+C) or (P+H) is required to implement a portable Web logic, but the logic
could be ’tunable’ for specific use cases. A ‘tuned down’ (C+H) reasoner could
implement OWL Description Logics as syntactic sugar. (P+C) and (P+H) can
provide a portable logic for reasoning over decentralized sources.
Undecidability is a hard problem, but unavoidable in the latter case. The
logic provided by RDF Surfaces, we argue, can be created with fewer primitives
than OWL 2 Full and is in line with a large body of research on FOL.
Undecidability itself does not stop the Web from evolving as can be seen from
the popularity of Web programming languages such as JavaScript.

##  5 Conclusion and future work

We believe that RDF Surfaces provides an expressivity comparable to Peirce’s
alpha and beta graphs and FOL. We still need proof that a complete mapping
from RDF Surfaces to FOL (or Peirce’s graphs) exists. This is part of our
current research. Existential rules have shown already to be sufficient to
implement the inference rules of RDFS, OWL-RL, and OWL-EL [14, 15, 16]. These
existential rules were implemented in the Notation3 language, but we are
porting them to RDF Surfaces. We started to port the Notation3 language itself
in RDF Surfaces as ongoing
work.444https://github.com/eyereasoner/Notation3-By-
Example/tree/main/examples/n3s

The expressivity of RDF Surfaces goes beyond what is provided by OWL
Description Logics. It could be impractical to express every inference rule
solely in the language of first-order logic. The Notation3 language provides
built-in functions and relations for arbitrary operations on RDF graphs of
practical assistance to the programmer. A pure subset of these functions
(deterministic and without side effects) could form the basis for an assembly
language of the web. In addition to Web portability, use cases for RDF
Surfaces can be found in implementing policy languages where one would
explicitly want to express what is regarded as misuse of information and
reason over that. RDF Surfaces can provide the semantics to limit the scope of
reasoning over RDF data streams or limit the scope for queries and provide
alternatives to query whatever is in reach [17].

The authors are currently experimenting with reasoners implementing RDF
Surfaces along the (P+H) [18] and (C+H) [19] axis and formed a W3C Community
Group 555https://www.w3.org/community/rdfsurfaces/ to define the semantics and
find new implementers.

###### Acknowledgements.

This work is funded by the Andrew W. Mellon Foundation (grant number:
1903-06675) and supported by SolidLab Vlaanderen (Flemish Government, EWI, and
RRF project VV023/10). The authors thank Dörthe Arndt and Ruben De Decker for
several insightful discussions about logic and applications of RDF Surfaces.

## References

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  * Hayes and Patel-Schneider [2014] P. Hayes, P. F. Patel-Schneider, RDF 1.1 semantics, 2014. URL: https://www.w3.org/TR/rdf11-mt/.
  * Sol [2023] SolidLab flanders – SolidLab flanders, 2023\. URL: https://solidlab.be/.
  * Foundation [2019] M. Foundation, Scholarly communications in the decentralized web : Ghent university, 2019. URL: https://mellon.org/grants/grants-database/grants/ghent-university/1903-06675/.
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  * Roberts [1973] D. D. Roberts, The Existential Graphs of Charles S. Peirce, Approaches to Semiotics, 1973.
  * Arndt et al. [2023] D. Arndt, W. Van Woensel, D. Tomaszuk, G. Kellogg, Notation3, 2023. URL: https://w3c.github.io/N3/spec/.
  * Hochstenbach and De Roo [2023] P. Hochstenbach, J. De Roo, RDF surfaces primer, 2023. URL: https://w3c-cg.github.io/rdfsurfaces/.
  * Berners-Lee [2000] T. Berners-Lee, SWELL - semantic web logic language, 2000. URL: https://www.w3.org/2000/01/sw/DevelopmentProposal.
  * Hayes [2009] P. Hayes, BLOGIC. (ISWC 2009 invited talk), 2009. URL: https://www.slideshare.net/PatHayes/blogic-iswc-2009-invited-talk.
  * Berners-Lee et al. [2007] T. Berners-Lee, D. Connolly, L. Kagal, Y. Scharf, J. Hendler,  N3logic: A logical framework for the world wide web (2007). URL: http://arxiv.org/abs/0711.1533. arXiv:0711.1533.
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  * Tomaszuk [2016] D. Tomaszuk, Inference rules for RDF(s) and OWL in n3logic, 2016. URL: http://arxiv.org/abs/1601.02650. arXiv:1601.02650 [cs].
  * Arndt et al. [2015] D. Arndt, B. De Meester, P. Bonte, J. Schaballie, J. Bhatti, W. Dereuddre, R. Verborgh, F. Ongenae, F. De Turck, R. Van de Walle, E. Mannens,  Ontology reasoning using rules in an eHealth context,  in: N. Bassiliades, G. Gottlob, F. Sadri, A. Paschke, D. Roman (Eds.), Rule Technologies: Foundations, Tools, and Applications, Lecture Notes in Computer Science, Springer International Publishing, 2015, pp. 465–472. doi:10.1007/978-3-319-21542-6_31.
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  * De Roo [2023] J. De Roo, Euler yet another proof engine, 2023\. URL: https://github.com/eyereasoner/eye.
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††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Transformation of hexagonal Lu to cubic LuH2+x single-crystalline films

Peiyi Li  Ningbo Institute of Materials Technology and Engineering, Chinese
Academy of Sciences, Ningbo 315201, China  Jiachang Bi  Ningbo Institute of
Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo
315201, China  Shunda Zhang  Ningbo Institute of Materials Technology and
Engineering, Chinese Academy of Sciences, Ningbo 315201, China  Rui Cai
Ningbo Institute of Materials Technology and Engineering, Chinese Academy of
Sciences, Ningbo 315201, China  Guanhua Su  Ningbo Institute of Materials
Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China
Fugang Qi  Ningbo Institute of Materials Technology and Engineering, Chinese
Academy of Sciences, Ningbo 315201, China  Ruyi Zhang  Ningbo Institute of
Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo
315201, China  Zhiyang Wei  Ningbo Institute of Materials Technology and
Engineering, Chinese Academy of Sciences, Ningbo 315201, China  Yanwei Cao
<EMAIL_ADDRESS>Ningbo Institute of Materials Technology and Engineering,
Chinese Academy of Sciences, Ningbo 315201, China  Center of Materials Science
and Optoelectronics Engineering, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

With the recent report of near ambient superconductivity at room temperature
in the N-doped lutetium hydride (Lu-H-N) system, the understanding of cubic
Lu-H compounds has attracted worldwide attention. Generally, compared to
polycrystal structures with non-negligible impurities, the single-crystalline
form of materials with high purity can provide an opportunity to show their
hidden properties. However, the experimental synthesis of single-crystalline
cubic Lu-H compounds has not been reported thus far. Here, we developed an
easy way to synthesize highly pure LuH2+x single-crystalline films by the
post-annealing of Lu single-crystalline films (purity of 99.99%) in the H2
atmosphere. The crystal and electronic structures of films were characterized
by X-ray diffraction, Raman spectroscopy, and electrical transport.
Interestingly, Lu films are silver-white and metallic, whereas their
transformed LuH2+x films become purple-red and insulating, indicating the
formation of an unreported electronic state of Lu-H compounds possibly. Our
work provides a novel route to synthesize and explore more single-crystalline
Lu-H compounds.

##  I Introduction

A very recent report of near ambient superconductivity at room temperature in
the Lu-H-N system immediately ignited the worldwide interest of studying Lu-H
compounds Nature-2023-Dias ; arXiv-2023-Jin ; arXiv-2023-LiuP ; CPL-2023-Cheng
; arXiv-2023-Wen ; arXiv-2023-Sun ; arXiv-2023-Liu ; arXiv-2023-Cheng ;
arXiv-2023-Wen2 ; SB-2023-Cheng ; arXiv-2023-Boeri ; arXiv-2023-Cao ;
arXiv-2023-MLiu ; arXiv-2023-Ho ; arXiv-2023-Cui ; arXiv-2023-Zurek ;
arXiv-2023-MM ; arXiv-2023-Heil ; ArXiv-2020-Duan ; Research-2022-Cui ;
IC-2021-Cui ; M-2021-Wang . Regarding this subject, one fundamental
identification is about the composition and structure of the red matter
showing near ambient superconductivity at room temperature. In the initial
report, the authors assigned the emergent superconductivity to cubic
$Fm\bar{3}m$-LuH3-δNε (92.25% purity) with both N-substitution and H-vacancy
defects Nature-2023-Dias . However, identifying the composition and structure
of $Fm\bar{3}m$-LuH3-δNε is still a big challenge.

Generally, the stable phase of LuH3 has a rhombohedra structure with space
group $P\bar{3}c1$, not the cubic structure with space group $Fm\bar{3}m$
arXiv-2023-Ho ; arXiv-2023-Boeri . Therefore, based on a combination of
pressure-dependent color, the X-ray diffraction data, and the Raman data, the
composition of experimental $Fm\bar{3}m$-LuH3-δNε was considered to be
LuH2±xNy by several experimental works arXiv-2023-Wen2 ; arXiv-2023-Liu ;
CPL-2023-Cheng . Moreover, high-throughput first-principles calculations
pointed out that there are no thermodynamic stable Lu-H-N ternary structures
and the experimental $Fm\bar{3}m$-LuH3-δNε may be a mixture of LuH2 and LuN
arXiv-2023-MLiu . However, a very recent experimental work shows that N-doped
$Fm\bar{3}m$-LuH3 can be synthesized by applying a 2 GPa pressure
arXiv-2023-MM . Furthermore, it is found that both N-doping and pressure can
stabilize $Fm\bar{3}m$-LuH3, and Lu8H21N is a stable Lu-H-N ternary at 1 GPa
arXiv-2023-Ho . It is noteworthy that both $Fm\bar{3}m$-LuH2 and
$Fm\bar{3}m$-LuH3 are cubic with a lattice parameter $\sim$ 5 Å arXiv-2023-Ho
; arXiv-2023-Boeri . Therefore, it is not easy work to distinguish them
experimentally. Synthesizing single-crystalline Lu-H compounds is very helpful
to understand their intrinsic properties, but it has not been reported thus
far.

Figure 1:  The photographs and crystal structures of hexagonal Lu (top panel)
and cubic LuH2+x (bottom panel) films.

To address the above, we developed an easy method of synthesizing single-
crystalline cubic LuH2+x films by the post-annealing of single-crystalline Lu
films in the H2 atmosphere. Unexpectedly, in sharp contrast to metallic LuH2
powders, the LuH2+x films become purple-red and insulating, indicating the
formation of an unreported electronic state of Lu-H compounds. Our work
provides a novel route to synthesize more single-crystalline Lu-H compounds.

Figure 2:  Temperature-dependent sheet resistance of the Lu film from 300 to 2
K.

##  II Experiments

Single-crystalline Lu thin films (thicknesses $\sim$ 80 nm) were grown on
(111)-MgO single crystal substrates (5$\times$5$\times$ 0.5 mm3) by a home-
made high-pressure radio frequency (RF) magnetron sputtering, the setup of
which is analogous to our previous reports APLM-2021-Zhang ; PRM-2021-Bi ;
ACSP-2021-Zhang ; ACSAMI-2021-Zhang . For this sputtering system, the base
vacuum pressure is $\sim$ 1 $\times$10-7 Torr, and the purities of the 2-inch
Lu target and the Ar reactive gas are 99.99% and 99.999%, respectively. During
growth, the Ar pressure was kept at 0.02 torr with a gas flow of 5.6 sccm, and
the substrate temperature was held at 300 ∘C. The power of the RF generator
was 80 W. Furthermore, to ensure the uniformity of the films, the heating
stage was rotated during growth. After deposition, the samples were cooled
down to room temperature in the same Ar atmosphere. Then, the Lu films on MgO
substrates were annealed at 280 ∘C for 4 hours in a flow of H2/Ar mixture gas
(the volume fraction of H2 is 8%, purity of 99.99%). During the post-
annealing, the pressure of Ar/H2 was kept at 0.14 MPa.

The crystal structures of Lu and LuH2+x films were characterized by a high-
resolution X-ray diffraction (XRD, Bruker D8 Discovery) with the Cu Kα source
($\lambda$ = 1.5406 Å) and confocal Raman spectroscopy (Renishaw inVia Reflex)
using a 532 nm laser. The electrical properties were measured from 300 to 2 K
by Physical Property Measurement System (PPMS) in a van der Pauw geometry
(DynaCool, Quantum Design).

Figure 3:  Wide-range 2$\theta$-$\omega$ scans of LuH2+x (red) and Lu (green)
films on (111)-MgO substrates. Figure 4:  Raman spectra of LuH2+x films (red
curves) and LuH2 powders (grey curves).

##  III Results and Discussion

Figure 1 shows the schematic transformation of hexagonal Lu (top panel) to
cubic LuH2+x (bottom panel). Generally, single-crystalline Lu has a hexagonal
close-packed (hcp) crystal structure with space group $P6_{3}/mmc$. Its
lattice parameters are a = b = 3.516 Åand c = 5.573 Å PRB-1998-Vohra .
Experimentally, the color of Lu films is silver-white before post-annealing
(see the top panel in Fig. 1). Unexpectedly, arfter post-annealing the color
of the films change to purple-red (see the bottom panel in Fig. 1), indicating
the formation of Lu-H compounds. However, it has been known that bulk
$Fm\bar{3}m$-LuH and $Fm\bar{3}m$-LuH3 are unstable at ambient conditions
arXiv-2023-Boeri . Therefore, the composition and structure of our purple-red
samples are likely to be $Fm\bar{3}m$-LuH2+x. Also, it is noted that both
$Fm\bar{3}m$-LuH2 and $Fm\bar{3}m$-LuH3 are cubic with a lattice parameter
$\sim$ 5 Å arXiv-2023-Ho ; arXiv-2023-Boeri . Moreover, N-doped
$Fm\bar{3}m$-LuH3 has been synthesized at 2 GPa pressure arXiv-2023-MM .
Considering the existence of strain effect and defects in our purple-red
films, the value of x may be positive.

First, to understand the conductivity of Lu and LuH2+x films, we measured
electrical transport from 300 to 2 K. As seen in Fig. 2, the Lu films are
highly conducting, the sheet resistance of which is $\sim$ 11 $\Omega$ at room
temperature. With decreasing the temperature from 300 to 2 K, the resistance
of Lu films has a linear dependence of the temperature down to 20 K, then
becomes a constant below 20 K. This behavior is consistent with our previous
observation of polycrystal Lu bulk arXiv-2023-Cao . Unexpectedly, in sharp
contrast to metalic LuH2 powders and polycrystals arXiv-2023-Cao ;
CPL-2023-Cheng ; arXiv-2023-Wen ; arXiv-2023-Sun ; arXiv-2023-Liu ;
arXiv-2023-Cheng ; arXiv-2023-Wen2 ; SB-2023-Cheng , our LuH2+x films is
highly insulating, indicating the formation of an unreported electronic state
of Lu-H compounds.

Next, to investigate the crystal structures of Lu and LuH2+x films, we carried
out XRD characterization. As seen in Fig. 3, the wide-range 2$\theta$-$\omega$
scan on Lu films (see the green curve) only shows the (002) and (004)
diffraction peaks without detectable secondary phases, indicating the
epitaxial growth of single-crystalline Lu films. The peak position of (002)
peak is around 31.998∘, corresponding to the lattice parameter $d_{002}$ =
2.495 Å, which is close to the value of bulk Lu ($d_{002}$ = 2.491 Å),
indicating the epitaxy of Lu films on MgO substrates. Interestingly, after the
post-annealing, the two diffraction peaks have significant shifts to low
diffraction angles (see the red curve in Fig. 3). The peak positions of LuH2+x
are around 30.478∘ and 63.398∘, respectively, which agree well with the (111)
and (222) diffractions peak positions of cubic LuH2 or cubic LuH3. The
extracted out-of-plane lattice parameter of our LuH2+x films is around 5.07 Å,
which is consistent to the value of bulk LuH2 or LuH3 ($\sim$ 5 Å)
arXiv-2023-Ho ; arXiv-2023-Boeri .

To further understand the crystal structure of LuH2+x films, we measured Raman
spectroscopy on both LuH2+x films and LuH2 powders. It is noted that the
contribution of MgO single-crystalline is as small as being ignored.
Therefore, the Raman signals of LuH2+x films are strong. As seen in Fig. 4,
the two characteristic peaks ($\sim$ 250 and 1200 cm-1) can be observed in
both LuH2+x films and LuH2 powders. In contrast to the reported peak positions
(248 and 1200 cm-1) of LuH2 powders and polycrystals Nature-2023-Dias ;
CPL-2023-Cheng ; arXiv-2023-Liu ; arXiv-2023-MM , these two Raman
characteristic peaks have a blue shift to $\sim$ 260 and 1226 cm-1,
respectively. It has been pointed out that the strong mode 1200 cm-1 can be
associated with the T2g Raman-active mode of the cubic structure arXiv-2023-MM
, further demonstrating the formation of cubic LuH2+x. Based on the purple-red
color, the insulating state, and the cubic structure of our LuH2+x films, it
indicates a formation of an unreported electronic state in LuH2+x films.

##  IV Conclusion

In this work, we developed an easy way to synthesize single-crystalline cubic
LuH2+x films by the post-annealing of single-crystalline Lu films in the H2
atmosphere. The crystal and electronic structures of films were characterized
by XRD, Raman, and electrical transport. Interestingly, Lu films are silver-
white and metallic, whereas their transformed LuH2+x films become purple-red
and insulating. Based on the purple-red color, the insulating state, and the
cubic structure of LuH2+x films, it indicates the formation of an unreported
electronic state in our LuH2+x films. Our work provides a novel route to
synthesize and explore more single-crystalline Lu-H compounds.

##  V Acknowledgments

We acknowledge insightful discussions with Rui Peng. This work was supported
by the National Key R&D Program of China (Grant No. 2022YFA1403000), the
National Natural Science Foundation of China (Grant Nos. U2032126 and
11874058), the Pioneer Hundred Talents Program of the Chinese Academy of
Sciences, the Zhejiang Provincial Natural Science Foundation of China under
Grant No. LXR22E020001, the Beijing National Laboratory for Condensed Matter
Physics, the Ningbo Natural Science Foundation (Grant No. 20221JCGY010338),
and the Ningbo Science and Technology Bureau (Grant No. 2022Z086).

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