# An NLG pipeline for a legal expert system: a work in progress

Inari Listenmaa
Jason Morris
Alfred Ang
Maryam Hanafiah
Regina Cheong
{ilistenmaa, jmorris, alfredang, maryammh<EMAIL_ADDRESS>
Singapore Management University

###### Abstract

We present the NLG component for L4, a prototype domain-specific language
(DSL) for drafting laws and contracts. As a concrete use case, we describe a
pipeline for a legal expert system created from L4 code. The NLG component is
used in two steps. The first step is to create an interview, whose answers are
processed into a query for an automated reasoner. The second step is to render
the answers of the reasoner in natural language.

##  1 Introduction

We introduce L4, a prototype111L4 is a work in progress, and this article
presents a snapshot of the project as of June 2021. Any concrete examples of
L4 code may change in a few months. domain-specific language (DSL) for
drafting laws and contracts. L4’s applied focus places it within the “Rules as
Code” movement (e.g. OpenFisca , Catala Merigoux et al. (2021)) that itself
draws on early computational law thinking Sergot et al. (1986); Love and
Genesereth (2005). But rather than focusing on encoding laws into existing
programming languages, we devise an external DSL designed for legal
specification.

Drafting in a high-level, declarative DSL makes it possible to separate the
legal layer and the application layer. We encode the rules once, and that
encoding is a source for further applications, such as legal expert systems,
consistency checking, visualisation and natural language generation. If the
rules change, they need to be changed only once at the source, and the
applications can be automatically updated.

In this paper, we describe an NLG pipeline from a set of rules drafted in L4
into natural language. The generated natural language must adapt to different
communicative needs: a piece of code may become, for example, a declaration, a
prohibition, a condition, an interview question or an answer to a user query.
Because we require flexibility in output, it is important to have a deep
understanding of any source material. Section 3 describes the first step, of
acquiring a base for NLG from user’s descriptions of classes and
predicates—this step is a general prerequisite to all kinds of NLG tasks, not
limited to the one we describe here. Section 4 describes a concrete use case
of creating a legal expert system from a L4 encoding. Finally, Section 5
outlines future work.

##  2 L4 example

As an introduction to L4, we will encode the rules of Rock-Paper-Scissors
(RPS) for two players.

###  2.1 L4 basics

#### Types and values

First, we declare data types for the entities needed to represent the rules. A
round of RPS is a Game with two Players, each throwing Sign. Then we introduce
the three signs: rock, paper and scissors.

class # types

Player

Game

Sign

decl # values

Rock : Sign

Paper : Sign

Scissors : Sign

#### Predicates

Next, we define four predicates, used later in our rules. The current syntax
of L4 is adopted from functional programming languages. For example, the type
signature for Win : Player $\rightarrow$ Game $\rightarrow$ Bool means “the
function Win takes a Player and a Game, and returns a Boolean”.

decl # predicates

Participate

: Player → Game → Bool

Throw : Player → Sign → Bool

Win : Player → Game → Bool

Beat : Sign → Sign → Bool

For the purposes of constructing an interview, it is useful to have
information on how to group the different predicates. L4 offers an alternative
syntax for the predicates to be fields in the data types, as follows.

# equivalent to the standalone

# versions Participate, Throw, Win

class

Player {

participate : Game → Bool

throw : Sign → Bool

win : Game → Bool

}

The class definition with the fields throw, win and participate is equivalent
to the stand-alone predicates in capital letters. Note that capital letters
have no significance in L4, we just use them in this example to distinguish
between standalone predicates and those in a field.

As an alternative, participate and win can be defined as fields of the class
Game. With this grouping, the type signature becomes Game $\rightarrow$ Player
$\rightarrow$ Bool. We will use this grouping in the examples in Section 4.1,
because that makes the best order for asking the questions.

class # different grouping

Game {

participate : Player → Bool

win : Player → Bool

}

#### Rules

We formulate the rules for winning a game: both players must throw a sign, and
the winning player’s sign must beat the other player’s sign. The syntax for
function application follows functional programming conventions: to say that
“x wins rps”, we write Win x rps. The relations between the signs are also
encoded as rules—implementation omitted for brevity.

rule <winner> # rules

for a : Player, g : Game,

r : Sign, s : Sign

if exists b : Player .

Participate a g &&

Participate b g &&

Throw a r && Throw b s &&

Beat r s

then Win a g

###  2.2 Natural language descriptions

For NLG purposes, the L4 file may include optional descriptions in controlled
natural language (CNL). If no natural language description is given, the
predicate name is used instead. For example, we can record that Participate
takes its object with the preposition “in”.

lexicon

Participate @ "participate in"

The current default is to include only the predicate, and assume the argument
order subject–object–indirect object. Subject agreement doesn’t matter:
“participate in” and “participates in” are equivalent. For higher than ternary
predicates, alternative argument order or several prepositions, the types may
used as placeholder arguments. This is how to avoid NLG like “game wins
player”, when the predicates are grouped as fields in the classes.

class

Game {win : Player → Bool}

# normalised into predicate

# win : Game → Player → Bool

lexicon

win @ "[Player] wins [Game]

##  3 CNL to NLG

Listenmaa et al. (2021) describes in detail the CNL and the process of parsing
the descriptions, including ambiguity resolution, so we include only a short
version here.

#### CNL

As seen in Section 2.2, the L4 classes and predicates can be enriched with a
natural language description. These descriptions are written in a controlled
natural language (CNL) which is implemented in Grammatical Framework (GF,
Ranta 2004), a programming language for multilingual grammar applications.

Our CNL is based on the GF Resource Grammar Library (RGL) Ranta et al. (2009),
which provides a library of syntactic structures and morphology for over 30
languages, and a large, multilingual morphological lexicon Angelov (2020)
based on Princeton WordNet Fellbaum (1998), enriched with syntactic features
like valency.

On top of the RGL and the GF-WordNet lexicon, we have added a set of
constructions common to legal text. In addition, we have relaxed the rules for
forming sentences, so that we can parse predicates without arguments,
(participates in), or arguments in brackets ([Player] wins [Game]).

So far we have only used English, but the method is scalable to any of the
languages that are in the RGL and have a large lexicon.

#### Goal of the CNL

The purpose of our CNL is to be able to parse the user input correctly, and
thus use it correctly in the NLG. Unlike many other CNLs out there (see Kuhn
(2014) for a survey), ours is not concerned with well-defined semantics—L4,
being a programming language, is much better suited for that. We only care
about well understood syntactic structure, so that we can have flexible NLG
for different communicative needs.

##  4 NLG pipeline for an expert system

We construct an interview in order to find out who wins a game of RPS. The
main puzzle, who wins RPS, is not one of the questions we pose to the
user—instead, we ask about all the prerequisite information, and give the
user’s answers to an automated reasoner, which will announce the winner.

NLG-wise, this involves two steps. In the first step, we generate interview
questions from the L4 code (Section 4.1). In the second step, we generate
natural language from the output of the reasoner (Section 4.2).

###  4.1 Interview questions

The RPS interview will ask about the players and which signs they threw.

  1. 1.

Is there a game?

     * •

Answer options: yes/no

     * •

Answered: yes

  2. 2.

Who participates in the game?

     * •

Answer options:

       * –

slot for free text

       * –

loop ”Are there more players?”

     * •

Answered:

       * –

Alice, yes for more players

       * –

Bob, no for more players

  3. 3.

Which sign does Alice throw?

     * •

Answer options: rock/paper/scissors

  4. 4.

Which sign does Bob throw?

     * •

Answer options: rock/paper/scissors

#### Docassemble

The questions are embedded in a Docassemble interview. Docassemble is an open-
source legal expert system, where users answer a set of questions through a
browser interface. The responses are then compiled together into a document or
processed further in other applications—in our case, an automated reasoner.

In order to let the user customise the interview—for instance, the NLG,
question ordering, additional links to source material— our system outputs a
LExSIS (Legal Expert System Interface Schema) file, which is short and
declarative. The final interview, a long and imperative piece of Python code,
is then constructed from the LExSIS file.

#### Types of questions

The interview includes three types of questions: a yes–no question (1), an
open-ended wh-question (2) and two enumeration questions (3,4). These are
determined by the L4 code.

Game is a class, so the first question we ask is whether a game exists. When
we know that a game exist, we ask wh-questions about the predicates related to
the game: who participates and which sign do they throw. The L4 code lists
three signs, but no pre-existing players: that’s why the answer options for
(2) is a slot for free text, and for (3,4) an enumerated list.

The choice of who in (2) is because player is a human noun. The GF-WordNet
lexicon Angelov (2020) contains this information for about 6500 words, so we
just make it as a rule in the grammar. If some human nouns are missing the
animacy information, they will just be verbalised with the general strategy,
which players.

#### Order of questions

The order of the questions is determined by Docassemble and the L4 encoding.

Docassemble has its internal logic: the goal of the interview is this case to
find out the winner of a game of RPS, and thus it starts out from the first
prerequisite, is there a game?

L4 too has its own logic: if we used the encoding where participate is a field
of the class Game, then it will group the questions accordingly. Participation
is a feature of a game, so after ensuring that a game exists, the user is
asked about the game’s participants—game is now old information, so in English
it gets a definite article and is placed later in the question. With a
different grouping of predicates in L4, the questions will also be grouped and
phrased differently: for instance, Are there any players, followed by Which
game does Alice participate in.

#### Constructing the questions

Table 2 shows the GF expressions needed to construct different types of
sentences. All of the functions in normal typeface come from the language-
independent API of the GF RGL222
http://www.grammaticalframework.org/lib/doc/synopsis/. The variables in
italics are generic: any noun phrase could be in place of np, any transitive
verb for v2. For a verb like participate in, the preposition is part of the
verb’s lexical entry—that’s why the expression to construct a VP is identical
for participates in RPS and throws rock.

None of the expression templates which includes a noun phrase specifies a
determiner. The NP can be constructed in different phases: it can be new or
old information, or it may be a proper noun. Unfortunately, the lexicon
doesn’t contain properties like mass noun vs. count noun, nor idiomatic
expressions, like throw rock instead of throw a rock. Currently, we just
accept that sometimes the determiner choice is unnatural, and leave it to the
user to postprocess the output.

###  4.2 Reasoner output in natural language

The second output target is a verbalisation of the output from an automated
reasoner. Using the user answers to the interview from Section 4.1, a solution
satisfying all the constraints is generated. The solution is then restructured
in natural language, becoming the conclusion the user receives.

Alice wins RPS, because

  * •

Alice throws paper and Bob throws rock, and

  * •

paper beats rock.

#### s(CASP) query

The reasoner we use is s(CASP) Arias et al. (2018): a logic programming
language for Constraint Answer Set Programming.

The L4 code is translated into s(CASP) rules and statements. The query “who
wins a game of RPS” is phrased as ?- win(Game,Player), where Player and Game
are s(CASP) variables. The order of the arguments comes from the L4 encoding
where win is a field of the class Game.

The L4 encoding doesn’t list any players or games, so s(CASP) needs more
information before it can meaningfully run the query. As we have seen in 4.1,
this information comes from the interview.

#### s(CASP) answers

The answer consists of one or more sets of s(CASP) statements. These
statements, just like their source in L4, are connected to the CNL
descriptions, which are stored as GF abstract syntax trees. This makes it
possible to aggregate them, such as grouping them by subject or predicate. In
case of multiple sets for the same query, we group repeating statements in one
block, grouped by and, and the rest under another, grouped by or. An example
of this strategy is shown in Table 2, which shows an answer to a slightly
different query: “list all ways that Alice can win a game”.

Currently, our order of aggregating predicates is intransitive before
transitive, and nouns/adjectives before verbs. This results in sentences like
are players and participants in RPS, are players and participate in RPS, play
and participate in RPS, or less fluently, are participants in RPS and play.

##  5 Future work

As of June 2021, the whole L4 system, including the NLG component, is just a
prototype. Future work for the CNL and user descriptions is listed in
Listenmaa et al. (2021). The quality of NLG depends on the quality of user
input—no amount of sophisticated information structure will cover for badly
parsed description, so improving the CNL is our first priority.

We have demonstrated in this paper the system’s capability to answer questions
of type What Is True (Alice wins RPS), When Is This True (when Alice throws
paper and Bob throws rock) and Why (because paper beats rock). As we scale up
to larger examples, we will need more fine-grained heuristics in aggregation,
information structure and ordering, as well as adding variation to the
structures. We also plan to add more languages.

In the longer term, we plan to support more types of queries, with
hypotheticals and goal-seeking, such as What Would Make This True and What
Else Is True, and new types of questions and answers require new NLG
functions. As another long-term goal, we would like to incorporate more
semantic knowledge in our NLG.

Table 1: GF expressions to construct different questions

Is there a game |  mkQS (mkCl noun)
---|---
Does Bob participate in RPS |  mkQS (mkCl np (mkVP v2 np))
Who participates in RPS |  mkQS (mkQCl who_IP (mkVP v2 np))
Which player throws rock |  mkQS (mkQCl (mkIP which_IDet noun) (mkVP v2 np))
Which sign does Alice throw |  mkQS (mkQCl (mkIP which_IDet noun) (mkClSlash np (mkVPSlash v2)))

Table 2: Aggregation of multiple models: answer to all ways Alice can win

Alice wins RPS, if all of the following hold:
---
|  RPS is a game, and
|  Alice and Bob are players and participate in RPS
and one of the following holds:
|  rock beats scissors, Alice throws rock and Bob throws scissors,
|  scissors beats paper, Alice throws scissors and Bob throws paper, or
|  paper beats rock, Alice throws paper and Bob throws rock.

## References

  * Angelov (2020) Krasimir Angelov. 2020.  A Parallel WordNet for English, Swedish and Bulgarian.  In _Proceedings of The 12th Language Resources and Evaluation Conference_ , pages 3008–3015.
  * Arias et al. (2018) Joaquin Arias, Manuel Carro, Elmer Salazar, Kyle Marple, and Gopal Gupta. 2018.  Constraint answer set programming without grounding.  _Theory and Practice of Logic Programming_ , 18(3-4):337–354.
  * (3) Docassemble.  https:/docassemble.org.
  * Fellbaum (1998) Christiane Fellbaum. 1998.  _WordNet: An electronic lexical database_.  MIT press.
  * Kuhn (2014) Tobias Kuhn. 2014.  A survey and classification of controlled natural languages.  _Computational linguistics_ , 40(1):121–170.
  * Listenmaa et al. (2021) Inari Listenmaa, Maryam Hanafiah, Regina Cheong, and Andreas Källberg. 2021\.  Towards CNL-based verbalization of computational contracts.  In _Proceedings of the 7th International Workshop on Controlled Natural Language (CNL 2020/21)_.
  * Love and Genesereth (2005) Nathaniel Love and Michael Genesereth. 2005.  Computational Law.  In _Proceedings of the 10th international conference on Artificial intelligence and law_ , ICAIL ’05, pages 205–209, New York, NY, USA. Association for Computing Machinery.
  * Merigoux et al. (2021) Denis Merigoux, Nicolas Chataing, and Jonathan Protzenko. 2021.  Catala: A programming language for the law.
  * (9) OpenFisca.  https://openfisca.org.
  * Ranta (2004) Aarne Ranta. 2004.  Grammatical Framework.  _Journal of Functional Programming_ , 14(2):145–189.  Publisher: Cambridge University Press.
  * Ranta et al. (2009) Aarne Ranta, Ali El Dada, and Janna Khegai. 2009.  The GF Resource Grammar Library.  _Linguistic Issues in Language Technology_ , 2(2):1–63.
  * Sergot et al. (1986) M. J. Sergot, F. Sadri, R. A. Kowalski, F. Kriwaczek, P. Hammond, and H. T. Cory. 1986.  The British Nationality Act as a logic program.  _Communications of the ACM_ , 29(5):370–386.


# Superconductivity and magnetic and transport properties of single-
crystalline CaK(Fe1-xCrx)4As4

M. Xu  J. Schmidt  Ames National Laboratory, Iowa State University, Ames, Iowa
50011, USA  Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA  M. A. Tanatar  Ames National Laboratory, Iowa State
University, Ames, Iowa 50011, USA  Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA  R. Prozorov  Ames National
Laboratory, Iowa State University, Ames, Iowa 50011, USA  Department of
Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  S. L.
Bud’ko  Ames National Laboratory, Iowa State University, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011,
USA  P. C. Canfield  Ames National Laboratory, Iowa State University, Ames,
Iowa 50011, USA  Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA<EMAIL_ADDRESS>
(September 3, 2024)

###### Abstract

Members of the CaK(Fe1-xCrx)4As4 series have been synthesized by high-
temperature solution growth in single crystalline form and characterized by
X-ray diffraction, elemental analysis, magnetic and transport measurements.
The effects of Cr substitution on the superconducting and magnetic ground
states of CaKFe4As4 ($T_{c}$ = 35 K) have been studied. These measurements
show that the superconducting transition temperature decreases monotonically
and is finally suppressed below 1.8 K as $x$ is increased from 0 to 0.038. For
$x$-values greater than 0.012, signatures of a magnetic transition can be
detected in magnetic measurements with the associated features in the
transport measurements becoming detectable for $x$ $\geq$ 0.038. The magnetic
transition temperature increases in a roughly linear manner as Cr substitution
increases. A temperature-composition (T-x) phase diagram is constructed,
revealing a half-dome of superconductivity with the magnetic transition
temperature, $T^{*}$, appearing near 22 K for $x$ $\sim$ 0.017 and rising
slowly up to 60 K for $x$ $\sim$ 0.077. The $T$-$x$ phase diagrams for
CaK(Fe1-x$T$x)4As4 for $T$ = Cr and Mn are essentially the same despite the
nominally different band filling; this is in marked contrast to $T$ = Co and
Ni series for which the $T$-$x$ diagrams scale by a factor of two, consistent
with the different changes in band filling Co and Ni would produce when
replacing Fe. Superconductivity of CaK(Fe1-xCrx)4As4 is also studied as a
function of magnetic field. A clear change in $H^{\prime}_{c2}$($T$)/$T_{c}$,
where $H^{\prime}_{c2}$($T$) is d$H_{c2}$($T$)/d$T$, at $x$ $\sim$ 0.012 is
observed and probably is related to change of the Fermi surface due to
magnetic order. Coherence length and the London penetration depths are also
calculated based on $H_{c1}$ and $H_{c2}$ data. Coherence lengths as the
function of $x$ also shows changes near $x$ = 0.012, again consistent with
Fermi surfaces changes associated with the magnetic ordering seen for higher
$x$-values.

##  I introduction

The study of Fe-based superconductors has lead to extensive experimental
interest and their variety offers the opportunity of understanding
unconventional superconductivity in a broader sense. Fe-based superconductors
families share similar crystal structures [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and
phase diagrams [11, 12, 13, 14, 15, 16, 17, 18, 19], which suggests a
relationship between, or proximity of, superconducting and magnetic and/or
nematic ordering or fluctuations. These relationships between
superconductivity and magnetic as well as structural transitions and
fluctuations are believed to be key to understand unconventional
superconductivity [20].

Though Fe-based superconductors have a moderate degree of structural
diversity, three, main structural classes: $Ae$Fe2As2 ($Ae$=Alkaline Earth)
(122) families [11, 12, 13, 2, 3, 14, 15, 4], $AeA$Fe4As4 (A=Alkaline Metal)
(1144) family [5, 6, 7, 16, 9, 10] and FeSe [8, 17] provide a microcosm of
many key questions at hand. Compared with the interplay between stripe-like
antiferromagnetic order, nematicity, and superconductivity in the doped
122-systems at ambient pressure [14, 3], the electron-doped
CaK(Fe${}_{1-x}T_{x}$)4As4, $T$ = Ni and Co system has hedgehog-spin-vortex-
crystal (h-SVC) type antiferromagnetic (AFM) order and superconductivity
interacting with each other without any structural phase transition[16, 21].

On one hand, since the phase diagrams of Co and Ni substitutions of CaKFe4As4
scaled almost exactly as a function of band filling change, the comparison
between CaKFe4As4 and Ba0.5K0.5Fe2As2 based on their similar, nominal electron
counts, seems justified. One the other hand, given that
CaK(Fe${}_{1-x}T_{x}$)4As4 allows for the study of how nominal hole-doping
with Mn and Cr can affect the superconducting and magnetic properties of this
system, it is very important to see how their $T$-$x$ phase diagrams compare
with each other as well as those for $T$ = Co and Ni.

We have recently found that for CaK(Fe${}_{1-x}T_{x}$)4As4, $T$ = Mn, Mn is a
far more local-moment-like impurity than $T$ = Co or Ni are. We also found
that the substitution level of CaK(Fe1-xMnx)4As4 can only go to up to $x$ =
0.036. Beyond that level, 1144 phase is not stabilized with the similar
synthesis condition. This limited the exploration of hole -doped 1144 phase
diagram and the evolution of h-SVC type antiferromagnetic transition. Cr
offers twice the amount of nominal hole-doping per $x$ and, like Mn, can
sometimes manifest local-moment-like properties in intermetallic samples. As
such, Cr substitution offers a great of opportunity to further our
understanding of the behavior of h-SVC type antiferromagnetism in the 1144
system.

In this paper, we detail the synthesis and characterization of
CaK(Fe1-xCrx)4As4 single crystals. A temperature-composition ($T$-$x$) phase
diagram is constructed by elemental analysis, magnetic and transport
measurements. In addition to creating the $T$-$x$ phase diagram, coherence
lengths and the London penetration depths are also calculated based on
$H_{c1}$ and $H_{c2}$ data obtained from measurements. The data for Cr-
substituted 1144 are added on the $\lambda^{-2}$ versus $\sigma$ $T_{c}^{2}$
plot and compared with the Mn substitution. Finally, temperature vs change of
electron count, |$\Delta e^{-}$|, phase diagram for CaK(Fe${}_{1-x}T_{x}$)4As4
single crystals, $T$ = Cr, Mn, Ni and Co, is also presented and discussed. By
comparing all four $T$ = Cr, Mn, Ni and Co substitutions we find that whereas
for $T$ = Ni and Co CaK(Fe${}_{1-x}T_{x}$)4As4 the temperature -substitution
phase diagrams scale with additional electrons (in much the same way that the
Ba(Fe${}_{1-x}TM_{x}$)2As2 phase diagrams do for $T$ = Ni and Co), for $T$ =
Cr and Mn the temperature substitution phase diagrams are essentially
identical when plotted more simply as $T$-$x$ diagrams, suggesting that for Cr
and Mn there may be other variables or mechanisms at play.

##  II Crystal Growth and Experimental Method

Single crystalline CaK(Fe1-xCrx)4As4 samples were grown by high-temperature
solution growth [22] out of FeAs flux in the manner similar to
CaK(Fe1-xMnx)4As4 [19]. Lumps of potassium metal (Alfa Aesar 99.95%),
distilled calcium metal pieces (Ames Laboratory, Materials Preparation Center
(MPC 99.9%) and Fe0.512As0.488 and Cr0.512As0.488 precursor powders were
loaded into a 1.7 ml fritted alumina Canfield Crucible Set [23] (LSP
Industrial Ceramics, Inc.) in an argon filled glove-box. The ratio of
K:Ca:Fe0.512As0.488 and Cr0.512As0.488 was 1.2:0.8:20. A 1.3 cm outer diameter
and 6.4 cm long tantalum tube which was used to protect the silica ampoule
from reactive vapors was welded with the crucible set in partial argon
atmosphere inside. The sealed Ta tube was then itself sealed into a silica
ampoule and the ampoule was placed inside a box furnace. The furnace was held
for 2 hours at 650 ℃ before increasing to 1180 ℃ and held there for 5 hours to
make sure the precursor was fully melted. The furnace was then fast cooled
from 1180 ℃ to 980 ℃ in 1.5 hours. Crystals were grown during a slow cool-down
from 980 ℃ to 915 ℃ over 100-150 hours dependent on substitution level. After
1-2 hours at 915 ℃ the ampoule was inverted into a centrifuge and spun to
separate the remaining liquid from the grown crystals. Metallic, plate-like,
crystals were obtained. The average size and thickness decreased by factor 2-4
as $x$ is increased. The largest crystal is about centimeter size as shown in
figure 1

Single crystals of CaK(Fe1-xCrx)4As4 are soft and malleable as CaKFe4As4 and
are difficult to grind for powder X-ray diffraction measurements. Diffraction
measurements were carried out on single crystal samples, which were cleaved
along the (001) plane, using a Rigaku MiniFlex II powder diffactometer in
Bragg-Brentano geometry with Cu K$\alpha$ radiation ($\lambda$ = 1.5406 Å)
[24].

The Cr substitution levels ($x$) of the CaK(Fe1-xCrx)4As4 crystals were
determined by energy dispersive spectroscopy (EDS) quantitative chemical
analysis using an EDS detector (Thermo NORAN Microanalysis System, model
C10001) attached to a JEOL scanning-electron microscope. The compositions of
platelike crystals were measured at three separate positions on each crystal’s
face (parallel to the crystallographic ab-plane) after cleaving them. An
acceleration voltage of 16 kV, working distance of 10 mm and take off angle of
35∘ were used for measuring all standards and crystals with unknown
composition. Pure CaKFe4As4 was used as a standard for Ca, K, Fe and As
quantification. LaCrGe3 and YCr6Ge6 were used as standards for Cr, both
leading to consistent results without significant difference within the
experimental error ($\sim$ 0.001). The spectra were fitted using NIST-DTSA II
Microscopium 2020-06-26 software[25]. Different measurements on the same
sample reveal good homogeneity in each crystal and the average compositions
and error bars were obtained from these data, accounting for both
inhomogeneity and goodness of fit of each spectra.

Temperature- and magnetic-field-dependent magnetization and resistance
measurements were carried out by using Quantum Design (QD), Magnetic Property
Measurement Systems (MPMS and MPMS3) and Physical Property Measurement Systems
(PPMS). Temperature- and magnetic-field-dependent magnetization measurements
were taken for H$\parallel$ab by placing the plate-like sample between two
collapsed plastic straws with the third, uncollapsed, straw providing support
as a sheath on the outside or by using of a quartz sample holder. The single
crystal samples of CaK(Fe1-xCrx)4As4 measured in the MPMS and MPMS3 have
plate-like morphology with length and width from 3 mm to 10 mm and thickness
(c axis) 50 - 200 $\mu$m. The approximate effective demagnetizing factor $N$
ranges from 0.007 to 0.077 with magnetic field applied parallel to the
crystallographic ab plane[26]. AC electrical resistance measurements were
performed in a standard four-contact geometry using the ACT option of the
PPMS, with a 3 mA excitation and a frequency of 17 Hz. 50$\mu$m diameter Pt
wires were bonded to the samples with silver paint (DuPont 4929N) with contact
resistance values of about 2-3 Ohms. The magnetic field, up to 90 kOe, was
applied along c or ab directions, perpendicular to the current, with the
current flowing in the ab plane in both cases.

Contacts for inter-plane resistivity measurements were soldered using tin. The
top and bottom surfaces of the samples were covered with Sn solder [27, 28]
and 50 $\mu$m silver wires were attached to enable measurements in a pseudo-
four-probe configuration. Soldering produced contacts with resistance
typically in the 10 $\mu\Omega$ range. Inter-plane resistivity was measured
using a two-probe technique with currents in 1 to 10 mA range (depending on
sample resistance which is typically 1 m$\Omega$). A four-probe scheme was
used down to the sample to measure series connected sample, $R_{s}$, and
contact, $R_{c}$ resistance. Taking into account that $R_{s}\gg R_{c}$,
contact resistance represents a minor correction of the order of 1 to 5%. The
details of the measurement procedure can be found in Refs. [29, 30, 31]. The
results of the measurements are in good agreement with similar measurements on
pure CaKFe4As4 [6].

Measurements with current along the $c-$axis suffer strongly from inter-layer
connectivity due to the micacious nature of single crystals. To ascertain
reproducibility, we performed measurements of $\rho_{c}$ on two to five
samples and obtained qualitatively similar temperature dependencies of the
electrical resistivity, as represented by the ratio of resistivities at room
and low temperatures, $\rho_{c}(0)/\rho_{c}(300)$. The resistivity
$\rho_{c}(300K)$ was in the range 1- 2 m$\Omega cm$, corresponding to an
anisotropy ratio $\rho_{c}/\rho_{a}\approx$ 3 to 6 at 300 K.

##  III CaK(Fe1-xCrx)4As4 Structure and Composition

Figure 1:  X-ray diffraction data of CaK(Fe0.923Cr0.077)4As4 showing ($00l$)
diffraction peaks from in-lab diffraction measurements on a single-crystalline
plate plotted on a semi-log scale. The value of $l$ is shown in red under each
peak. The inset shows the picture of CaK(Fe0.988Cr0.012)4As4 single crystal.
Note that $l$ = odd ($00l$) lines are evidence of the ordered CaKFe4As4
structure [5, 6].

Figure 2: (a)EDS measured Cr concentration vs nominal Cr concentration for the
CaK(Fe1-xCrx)4As4 series. The line is the linear fit of the data with fixed
intercept to 0. (b) Plot of $c$ lattice parameters as a function of
substitution level $x$. $c$ lattice parameters are calculated by the single-
crystalline plate X-ray diffraction plot [24].

Figure 1 presents single crystal diffraction data of CaK(Fe1-xCrx)4As4 with
$x_{EDS}=0.077$ which is the largest substitution level obtained. Attempts to
grow crystals with $x_{EDS}$ > 0.077 failed to yield mm-sized or larger
samples that could be identified as Cr-doped 1144. From the figure, we can see
that all ($00l$), $l\leq 12$, are detected. The $h+k+l$ = odd peaks which are
forbidden for the I4/mmm structure [5] can be clearly found. This indicates
that sample has the anticipated P4/mmm structure associated with the CaKFe4As4
structure [5, 16, 6].

The Cr substitution, $x$EDS, determined by EDS is shown in figure 2a for
different crystals as a function of the nominal Cr fraction, $x$nominal, that
was originally used for the growth. Error bars account for both possible
inhomogeneity of substitution and goodness of fit of each EDS spectra. A clear
correlation can be seen between the nominal and the measured substitution
levels, with a proportionality factor of 0.47 $\pm$ 0.02. For comparison, the
ratio of measured to nominal Mn, Ni and Co fraction in the corresponding
CaK(Fe1-xTx)4As4 are 0.60, 0.64 and 0.79 respectively [16, 19]. From this
point onward, when substitution level $x$ is referred to, it will be the EDS
value of $x$. Figure 2b presents $c$ lattice parameters as the function of
$x$. $c$ lattice parameter monotonically increases as Cr substitution level
increase, which is consist with the larger radius of Cr than Fe. $c$ lattice
parameter values are calculated by the single-crystalline plate X-ray
diffraction plot [24]. In Mn-1144 (CaK(Fe1-xMnx)4As4), the evolution of the
$c$ lattice parameter is difficult to determine due to the small difference of
radius between Fe and Mn and low substitution levels. The highest $x$ is 0.036
in Mn-1144 and it’s smaller than, 0.039, lowest substitution of Co-1144.

##  IV Data analysis and Phase Diagram

Figure 3: Zero-field-cooled-warming (ZFCW) low temperature magnetization as a
function of temperature for CaK(Fe1-xCrx)4As4 single crystals with a field of
50 Oe applied parallel to the crystallographic ab plane. $M$ is the volumetric
magnetic moment with cgs unit emu cm-3 or Oe.  Figure 4: Low temperature
magnetization divided by applied field as a function of temperature for
CaK(Fe1-xCrx)4As4 single crystals with a field of 10 kOe applied parallel to
the crystallographic ab plane. The inset shows the CaK(Fe0.983Cr0.017)4As4
single crystal magnetization for 5 K < $T$ < 300 K. Small vertical arrows
indicate the location of $T^{*}$, see Appendix for the criterion.  Figure 5:
Temperature dependence of normalized resistance,
$\textit{R}(\textit{T})/\textit{R}$(300 K), of CaK(Fe1-xCrx)4As4 single
crystals showing the suppression of the superconducting transition $T_{c}$ and
the appearance and evolution of a kink-like feature, marked with arrows. The
criterion used to determine $T^{*}$ from this kink-feature is outlined and
discussed in the appendix.

Figure 3 shows the low temperature (1.8 K - 45 K), zero-field-cooled-warming
(ZFCW) magnetization for CaK(Fe1-xCrx)4As4 single crystals for $H_{||ab}$ = 50
Oe (ZFCW magnetization and field-cooled (FC) data for an $x$ = 0.017 sample
can be found in figure 19 in the Appendix). $M$ is the volumetric
magnetization in this figure and is calculated by using the density of
CaKFe4As4, which is determined to be 5.22 g/cm3 from the lattice parameters at
room temperature [5]. A magnetic field of 50 Oe was applied parallel to ab
plane (i.e. parallel to the surface of the plate-like crystal). The
superconducting transitions ($T_{c}$) are clearly seen in this graph except
for the substitution value x = 0.038. As the value of the Cr substitution,
$x$, increases, the superconducting transition temperature decreases. For
$x=0.025$, a full magnetic shielding is not reached by 1.8 K.

Figure 4 shows the low temperature (5 K - 150 K) $M$($T$)/$H$ data for
CaK(Fe1-xCrx)4As4 single crystals with 10 kOe field applied parallel to the
crystallographic ab plane. The appearance of first a Curie-Weiss tail and
later a kink-like feature after adding Cr is similar to Mn substituted 1144.
Kink-like features are found above 20 K for $x>0.012$, which very likely
indicate antiferromagnetic transition. Similar kink-like features were
correlated with AFM order in CaK(Fe1-xMnx)4As4 [19, 32]. The inset shows
$M$($T$)/$H$ of a CaK(Fe0.983Cr0.017)4As4 single crystal over a wider
temperature range. As Cr is added the Curie-tail-like feature grows. The
$M$($T$) data above transitions can be fitted by a C/($T$+$\theta$) +
$\chi_{0}$ function as long as Cr doping levels is larger than 0.005 ($x$ >
0.005). The effective moment versus $x$ data is shown in figure 18 in the
Appendix; $\mu_{eff}$ calculated per Cr, is found to be $\sim$ 4 $\mu$B. For
$x$ > 0.012, a kink-like feature can be seen at a temperature $T^{*}$. As $x$
increases from 0.017 to 0.077, the temperature $T^{*}$ increases from $\sim$
20 K to $\sim$ 60 K. The criterion for determining $T^{*}$ and more discussion
about the Curie-tail are shown in the Appendix.

Figure 5 presents the temperature dependent, normalized, electrical resistance
of CaK(Fe1-xCrx)4As4 single crystals. RRR (the ratio of 300 K and low
temperature resistance just above $T_{c}$) decreases as Cr substitution
increases, which is consistent with the disorder increasing. The
superconducting transition temperatures decrease as Cr is added to the system.
When x = 0.038, there is no signature of a superconducting transition
detectable above 1.8 K. With increasing Cr content, a kink appears for $x$ >
0.025 and rises to about 60 K for $x$ = 0.077 and features become more clearly
resolved with increasing substitution. A similar feature also appeared in Mn,
Ni and Co-substituted CaKFe4As4 electrical resistance measurements [16, 19].
The criterion for determining the transition temperature, $T^{*}$, associated
with this kink is shown in the Appendix in figure 21c where $R(T)$ and
d$R(T)$/d$T$ are both shown.

Figure 6: Temperature dependence of normalized resistance,
$\textit{R}(\textit{T})/\textit{R}$(300 K), of CaK(Fe1-xCrx)4As4 single
crystals with $x=$0.017 for electrical currents along $a-$axis (black) and
along $c$-axis (red). Inset shows resistivity derivatives plotted against
d($MT$/$H$)/d$T$ (blue). The onset of the feature in the d($MT$/$H$)/d$T$
curve at $\sim$25 K is accompanied by flattening in the $c$-axis resistivity
derivative, but without a clear anomaly. No features are observed in
$d(R/R(300K))/dT$ curve for in-plane current (black line in the inset.)
Figure 7: Temperature dependence of normalized resistance,
$\textit{R}(\textit{T})/\textit{R}$(300 K), of CaK(Fe1-xCrx)4As4 single
crystals with $x$ = 0.038 for electrical currents along $a$-axis (black) and
along $c$-axis (red). Inset shows resistivity derivatives plotted against
d($MT$/$H$)/d$T$ (blue, right axis). The peak in the d($MT$/$H$)/d$T$ curve at
$\sim$ 42 K is accompanied by a clear feature in the $c$-axis resistivity
derivative, but only with a very broad anomaly for the in-plane transport.

Figures 6 and 7 compare the normalized electrical resistivity of
CaK(Fe1-xCrx)4As4 single crystals for electrical currents along $a$-axis in
the tetragonal plane (black lines in the main panels) with those along the
tetragonal $c$-axis (red curves in the main panels). Samples with $x$ = 0.017,
figure 6, are in the range of SC and AFM coexistence; samples with $x$ =
0.038, figure 7, are in the range where superconductivity is suppressed (see
figure 8, below). The inter-plane resistivity of the samples with $x$ = 0.017
shows a broad cross-over close to room temperature, much milder feature is
found in in-plane transport. This is very similar to the results on the parent
compound $x$ = 0 [6]. For samples with $x$ = 0.038, figure 7, the inter-plane
resistivity (red curve, main panel) reveals clearly non-monotonic dependence.
The cross-over transforms into a clear maximum above 200 K, followed by a
second maximum centered at about 35 K, close to the temperature of long-range
magnetic ordering.

Insets in the figures compare the derivatives of the normalised resistivities
for two current directions with d($MT$/$H$)/d$T$ (blue lines, right scale)
[33]. For sample $x$ = 0.017 in figure 6, no clear features are observed in
the resistivity derivatives, however some flattening is observed for $c$-axis
resistivity. For samples with $x$ = 0.038, the peak in d($MT$/$H$)/d$T$ at
$\sim$ 42 K is in good agreement with the feature in the derivative of inter-
plane resistivity. A feature in the derivative of in-plane resistivity is
notably less pronounced (black curve in inset in figure 7).

In some cases the features associated with magnetic ordering in the FeAs-based
superconductors are clearer for current flow along the $c$-axis as opposed to
current flow in the basal $ab$-plane [31]. This is believed to be due to an
anternating arrangements of the magnetic moments along the $c$-axis direction,
providing partial gapping of the Fermi surface affecting more strongly inter-
plane transport. The clarity of features increases with $x$ in
CaK(Fe1-xCrx)4As4, making then clearly visible in the raw resistivity data for
$x=$ 0.077 (see Appendix figure 23).

Figure 8 summarizes the transition temperature results inferred from
magnetization and resistance measurements, plots the superconducting and
magnetic transitions as a function of substitution and constructs the $T$-$x$
phase diagram for the CaK(Fe1-xCrx)4As4 system. As depicted in this phase
diagram, increasing Cr substitution (i) suppresses $T_{c}$ monotonically with
it extrapolating to 0 K by $x$ $\sim$ 0.03 and (ii) stabilizes a new
transition, presumably an antiferromagnetic one, for $x$ $\gtrsim$ 0.017 with
the transition temperature rising from $\sim$ 20 K for $x$ = 0.017 to $\sim$
60 K for $x$ = 0.077. Each phase line is made out of data points inferred from
R(T) and M(T) measurements, illustrating the good agreement between our
criteria for inferring $T_{c}$ and $T^{*}$ from magnetization and resistivity
data. The CaK(Fe1-x$T$x)4As4 series, $T$ = Mn, Co and Ni, have qualitatively
similar phase diagrams, with the quantitative differences being associated
with the substitution levels necessary to induce the magnetic phase and to
suppress superconductivity. We were not able to infer the behavior of $T^{*}$
once it drops below $T_{c}$, but if it is similar to other $T$
substitution[34, 32], $T^{*}$ should be suppressed very fast in the
superconductivity state. Further comparison of the CaK(Fe1-xCrx)4As4 phase
diagram to the phase diagrams of the CaK(Fe1-x$T$x)4As4 series will be made in
the discussion section below.

Given that the $R$($T$) data were taken in zero applied field whereas the
$M/H$($T$) data shown in figure 4 were taken in 10 kOe, it is prudent to
examine the field dependence of transition associated with $T^{*}$. In figure
9 we show the d($MT/H$)/d$T$ data [35] for the $x$ = 0.077 sample for $H$ ||
ab = 10, 30 and 50 kOe. As is commonly seen for antiferromagnetic phase
transition, increasing a magnetic field leads to a monotonic suppression of
$T^{*}$. The inset to figure 9 shows that the extrapolated, $H$ = 0, $T^{*}$
value would be 57.4 K as compared to the value of 57.2 K for 10 kOe. This
further confirms that there should be (and is) good agreement between the
$T^{*}$ values inferred from 10 kOe magnetization data and the $T^{*}$ values
inferred from and resistance data in figure 8. In addition these data suggest
that magnetic field could be used to fine tune the value of $T^{*}$, if
needed.

Figure 8: Temperature - composition phase diagram of CaK(Fe1-xCrx)4As4 single
crystals as determined from resistance ($R$($T$)) and magnetization ($M$($T$))
measurements. The circular symbols denote the $T_{c}$ phase line and the
cross-like symbols denote the $T^{*}$ phase line, most likely associated with
antiferromagnetic order. Superconducting (SC), magnetically ordered (MO) and
paramagnetic (PM) regions are denoted. Details of how the MO line extends into
the SC state are not addressed in this phase diagram.  Figure 9:
d($MT/H$)/d$T$ vs. $T$ of CaK(Fe0.923Cr0.077)4As4 single crystal with 10 kOe,
30 kOe and 50 kOe applied parallel to the crystallographic ab plane. $T^{*}$
at different field are shown as points on the figure. Inset shows transition
temperature, $T^{*}$, inferred for different applied field values using the
same criterion shown in appendix. The solid green line is linear fit to the
data points, extrapolating to 54.7 K for $H$ = 0.

##  V Superconducting critical fields and anisotropy

Figure 10: Magnetization of a single crystal of CaK(Fe0.983Cr0.017)4As4 as a
function of magnetic field applied parallel to the crystallographic ab plane
at 2 K. In zero applied field cooling (ZFC) to 2 K so as to demagnetization is
done at 60 K before cooling to minimize the remnant magnetic field.

Superconductivity can be studied as a function of field (in addition to
temperature and doping). Before we present our $H$c2($T$) results, based on
$R$($T,H$) data, it is useful to check the $M$($H$) data. We start with M(H)
data for $x$ = 0.012, $T_{c}$ = 21.3 K, taken over a wide field range. The 2 K
M(H) data for shown in figure 10 is classically non-linear, showing a local
minimum near $H$ $\sim$ 2.5 kOe. For $T$ = 2 K < $T_{c}$ the $H_{c2}$ value is
clearly higher than the 65 kOe maximum field we applied (see discussion and
figures below). $H_{c1}$ can be inferred from the lower field M(H) data.

In order to better estimate $H_{c1}$ values we performed low field, $M(H)$
sweeps at base temperature. In figure 11a we show the M(H) data for 0
$\leqslant$ $x$ $\leqslant$ 0.025 for $H$ $\leqslant$ 100 Oe. As $x$ increases
the deviation from the fully shielded, linear behavior, that occurs at
$H_{c1}$, appears at lower and lower fields. As shown in the inset of figure
11a, $\Delta$$M$ is determined by subtracting the linear, lowest field
behavior of $4\pi M$ from $H$. Given the finite thickness of samples and field
direction applied in $ab$ plane, there is a small demagnetizing factor ($N$ <
0.077), therefore, $H_{c1}$ is taken as the vortices start to enter the sample
and determined as the point when the $M$($H$) data deviates from the linear,
lowest field behavior. The non-zero value is due to remnant field of MPMS. The
standard error of $H_{c1}$ comes from at least 4 different samples’
measurements. Figure 11b shows $H_{c1}$ at 2 K with different substitution.
$x$ = 0.025 is not shown in this plot since $T_{c}$ $\sim$ 7 K is close to 2
K. The data shown in figure 11b are roughly linear in $x$, but there may be a
hint of a change in behavior near $x$ $\sim$ 0.01, where $T_{c}$ drops below
$T^{*}$ in figure 8. This will be discussed further when examine the London
penetration depth below.

In order to further study the effects of Cr substitution on the
superconducting state, anisotropic $H_{c2}(T)$ data for temperatures near
$T_{c}$ were determined for the substitution levels that have
superconductivity. Figure 12a shows a representative set of $R$($T$) data
taken for fixed applied magnetic fields, $H$ $\parallel$ $c$ axis and $ab$
plane $\leq$ 90 kOe for $x$ = 0.017. Figure 12a also shows an example of the
onset and offset criteria used for the evaluation of $T_{c}$. Figure 12b
presents the anisotropic $H_{c2}(T)$ curves for CaK(Fe0.983Cr0.017)4As4 single
crystals with both $H$ || $c$ and $H$ || $ab$, showing both the onset (Tonset)
and offset (Toffset) temperatures. For $x$ = 0.005, 0.009, 0.012 and 0.025,
$H_{c2}(T)$ curves are shown in Appendix figures 24 \- 27. From $H_{c2}(T)$
plots, we can see that $T_{c}$ is only suppressed by about 4 K when 90 kOe
magnetic field is applied, so the complete $H_{c2}(T)$ plots of the
CaK(Fe1-xCrx)4As4 compounds cannot be fully determined, however we still can
observe several trends in these data.

Figure 13 shows the temperature dependent anisotropy ratio,
$\gamma=H_{c2}^{\parallel ab}(T)/H_{c2}^{\parallel c}(T)$, is around 2.5 for
these samples over the 0.6 < $T$/$T_{c}$ < 1.0 range. This value is similar to
other 122 and 1144 materials [13, 6, 19, 31]. $\gamma$ $\sim$ 2.5 is also
qualitatively consistent with the estimated resistivity anisotropy ratio
$\gamma_{\rho}=\rho_{c}/\rho_{a}\approx$$3-6$ at 300 K, increasing to $6-10$
at $T$ = 0 K, with $\gamma_{H}\sim\sqrt{\gamma_{\rho}}$ [36]. Black circles
and dashed line present the data for pure CaKFe4As4 [6]. The anisotropies of
coherence length and penetration depth are expected to be the same close to
$T_{c}$, but can have opposite temperature dependence upon cooling below that
[37]. However, almost no temperature dependence of $\gamma$ is seen in the
temperature range measured. Based on that, the average values of gamma can be
taken as a good estimate of both anisotropies at low temperature as well.

Figure 11: (a) Magnetization as a function of magnetic field applied parallel
to the crystallographic ab axis at 2 K for different substitution levels.
Arrows mark the value of the magnetic field ($H_{c1}$) where $M$(H) deviates
from linear behavior. Inset shows the criterion we use to determine the
$H_{c1}$ values. Remnant field of measurements are smaller than 1 Oe, which is
consistent with $M$($H$) plots shown in figure. Temperature is ZFC to 2 K and
demagnetization is done at 60 K before cooling to minimize the remnant
magnetic field (b) $H_{c1}$ value versus $x$.

Figure 12: (a) Temperature-dependent electrical resistance of
CaK(Fe0.983Cr0.017)4As4 single crystal for magnetic field parallel to the
crystallographic c axis (upper panel) and $ab$ plane (lower panel) for
representative fields $\textit{H}\leq 90$ kOe. Onset and offset criteria for
$T_{c}$ are shown by the red solid lines. (b) Anisotropic $H_{c2}(T)$ data
determined for two single crystalline samples of CaK(Fe0.983Cr0.017)4As4 using
onset criterion (solid) and offset criterion (hollow) inferred from the data
shown in (a).

Given that we have determined $H_{c2}$($T$) for temperatures close to $T_{c}$,
we can evaluate the $H^{\prime}_{c2}$($T$)/$T_{c}$ close to $T_{c}$, where
$H^{\prime}_{c2}$($T$) is d$H_{c2}$($T$)/d$T$, specifically seeing how it
changes as $T_{c}$ drops below $T^{*}$ with increasing x. Error of
$H^{\prime}_{c2}$($T$)/$T_{c}$ comes from linear fit of $H_{c2}$($T$) near the
$T_{c}$. In the case of other Fe-based systems [38, 39, 40, 41, 19] clear
changes in $H^{\prime}_{c2}$($T$)/$T_{c}$ were associated with changes in the
magnetic sublattice coexisting with superconductivity (i.e. ordered or
disordered). In figure 14 we can see that there is a change in the
x-dependence of $H^{\prime}_{c2}$($T$)/$T_{c}$ for $x$ $>$ 0.012, beyond which
substitution level suppresses $T_{c}$ below $T^{*}$. Comparison with the slope
change of $H_{c2}$ in the pressure-temperature phase diagram of
CaK(Fe1-xNix)4As4 [38], further suggest that this is probably related to
changes in the Fermi surface, caused by the onset of the new periodicity
associated with the AFM order.

Figure 13: Anisotropy of the upper critical field, $\gamma=H_{c2}^{\parallel
ab}(T)/H_{c2}^{\parallel c}(T)$, as a function of effective temperature,
$T$/$T_{c}$, for CaK(Fe1-xCrx)4As4 single crystals, using onset criterion
(upper panel) and offset criterion (lower panel), inferred from the
temperature-dependent electrical resistance data. The $T_{c}$ value used to
calculate the effective temperature (T/$T_{c}$) is the zero-field
superconductivity transition temperature for each Cr-substitution levels (see
figure 8 above). Data CaKFe4As4 are shown in black circles with dashed line.
Figure 14: Substitution dependence of slope of $H_{c2}$($T$) at $T_{c}$,
-$H^{\prime}_{c2}$($T$)/$T_{c}$, where $H^{\prime}_{c2}$($T$) is
d$H_{c2}$($T$)/d$T$ at $T_{c}$. $T_{c}$ is determined by $T_{offset}$ from
criteria shown in figure 12. Qualitatively similar, albeit somewhat weaker,
results can be seen using $T_{onset}$ criterion data. Dash lines show the
linear fit for $x$ < 0.01 and $x$ > 0.01, which indicate the change of trend
of $H_{c2}$($T$) at $T_{c}$ versus $x$ as substitution increase.

##  VI discussion and summary

The $T$-$x$ phase diagram for CaK(Fe1-xCrx)4As4 (Figure 8) is qualitatively
similar to those found for Co-, Ni- and Mn- substituted CaKFe4As4, there is a
clear suppression of $T_{c}$ with increasing Cr substitution as well as an
onset of what is likely to be a AFM ordering for $x$ > 0.012.

In figures 11 and 14 we presented measurements and analysis of $H_{c1}$ and
$H_{c2}$ data. Whereas we see only subtle, if any effect of the onset of AFM
ordering on $H_{c1}$ (figure 11b), there is a clear effect on $H_{c2}$ (figure
14). Using our $H_{c1}$ and $H_{c2}$ data we can extract information about the
superconducting coherence length and London penetration depth as well.

Figure 15: (a) Coherence length versus $x$ plot of CaK(Fe1-xCrx)4As4 single
crystals with applied field in $c$ direction and $ab$ plane. (b) shows the
London penetration depth as a function of $x$ obtained by magnetic field
applied parallel to the crystallographic ab axis with different substitution
levels. Dash lines show the linear fit for $x$ < 0.01 and $x$ > 0.01, which
show the change of coherence lengths and London penetration depths at $T_{c}$
versus $x$ as substitution increase.

Figure 15a shows coherence length, $\xi$, of CaK(Fe1-xCrx)4As4 as a function
of $x$. $\xi$ is estimated by using the anisotropic scaling relations
0.69|d$H_{c2}^{||c}$/d$T$| = $\phi_{0}$/2$\pi$$\xi^{2}_{ab}$$T_{c}$ and
0.69|d$H_{c2}^{\perp c}$/d$T$| = $\phi_{0}$/2$\pi$$\xi_{c}\xi_{ab}$$T_{c}$
[6]. We estimate $H_{c2}(0)$ with $H_{c2}(0)/(T_{c}|$d$H_{c2}(T_{c})/$d$T|)$ =
0.69 [42]. Figure 15b shows the London penetration depth, $\lambda_{ab}$, as a
function of $x$. Since, according to figure 13, $\gamma$ of $H_{c2}$ does not
change much as the temperature decreases below Tc, the anisotropy of the
penetration depth is estimated as the average of the anisotropy of $H_{c2}$ at
low temperature. $\lambda_{ab}$ is obtained by using,

$\displaystyle H_{c1}^{||ab}$ $\displaystyle=\leavevmode\nobreak\
\phi_{0}/(4\pi\lambda_{ab}\lambda_{c})(ln\kappa_{ab}+0.5)$ (1)
$\displaystyle=\phi_{0}/(4\pi\lambda_{ab}^{2}\gamma)(ln(\lambda_{ab}/\xi_{ab})+ln\gamma+0.5)$
(2)

$\lambda_{c}/\lambda_{ab}$ = $\xi_{ab}/\xi_{c}$ = $H_{c2}^{\parallel
ab}(T)/H_{c2}^{\parallel c}(T)$ = $\gamma$ = $1/\varepsilon$ with
$\varepsilon$ being the angle-dependent anisotropy parameter and $\kappa_{ab}$
= $\sqrt{\lambda_{ab}\lambda_{c}/\xi_{ab}\xi_{c}}$ [43, 44, 45]. Coherence
lengths and penetration depths increase as substitution levels increase, and,
given that $\xi$ depends on d$H_{c2}$/d$T$ and penetration depths depends on
$H_{c1}$ and $\xi$. Both figure 15a and b show breaks in behavior near $x$
$\sim$ 0.01, the substitution level at which $T^{*}$ emerges from below
$T_{c}$.

Figure 16: (a) shows $\lambda^{-2}$ versus $\sigma$ $T_{c}^{2}$ (Homes
scaling) of Cr (black) and Mn (red) substitution of CaKFe4As4. (b) shows
Homes’scaling with other superconductors [46]. Except CaK(Fe1-xCrx)4As4 and
CaK(Fe1-xMnx)4As4, Homes’ scaling is given by 1/$\lambda_{s}^{2}$ $\propto$
$T_{c}\sigma_{dc}$. $\sigma_{dc}$ is DC conductivity and data points are
obtained from optical spectroscopes. Red dash-dot ellipse marks the Mn and Cr
substituted 1144 data points.

Figure 16a shows $\lambda^{-2}$ versus $\sigma$ $T_{c}^{2}$ in Cr and Mn
substitution of CaKFe4As4, where $\sigma$ is normal state conductivity which
was measured just before $T_{c}$ (Resistivity plot is shown in appendix figure
20). Whereas the CaK(Fe1-xMnx)4As4 data roughly follow the behavior associated
with the Homes type scaling in the presence of pair breaking [46, 47, 48]. The
CaK(Fe1-xCrx)4As4 data also follows the Homes scaling with slightly different
slope. Figure 16b shows the Homes’ scaling of superconductors on a log-log
scale [46]. Other pnictides are: Ba(Fe0.92Co0.08)4As4 and
Ba(Fe0.95Ni0.05)4As4. Cuprates are YBa2Cu3O6+y. On this log-log scale, both
the CaK(Fe1-xCrx)4As4 and CaK(Fe1-xCrx)4As4 data sets agree rather well with
other data, although being shifted up somewhat. It should be noted that the
other data were determined from optical measurements [46] and differences in
criteria as well as measurement techniques may be responsible for the offset.

The $T$-$x$ phase diagrams of $T$ = Co and Ni substitutions in
CaK(Fe1-x$T$x)4As4 scaled almost exactly as a function of band filling change
(i.e. when each Ni atom brings two extra electrons and each Co atom brings
only one extra electron) [16, 21]. This led to the conclusion that for
electron doping of CaKFe4As4 the number of electrons added was the control
parameter for both the stabilization of magnetic ordering as well as for the
suppression of superconductivity. This scaling did not seem to work for the
case of Mn substitution [19], but with only one "hole-like" transition metal
substitution, it was hard to make clear conclusions.

When Cr is substituted into CaKFe4As4, though, there is a qualitatively
similar suppression of superconductivity as well as the stabilization of
magnetic order, as was found for $T$ = Mn, Co and Ni. However there is a clear
and important difference on a quantitative level, as shown in figure 17a. The
$T$-$x$ phase diagram of CaK(Fe1-xCrx)4As4 is essentially identical to that of
CaK(Fe1-xMnx)4As4. This is very different behavior from, figure 17b, that seen
for electron doped 1144 where electron count seemed to be the key variable.

In figure 17c the CaK(Fe${}_{1-x}T_{x}$)4As4 phase diagrams for $T$ = Cr, Mn,
Co and Ni are plotted on the same $T$ and $\Delta$e- axes. Comparison of
figures 17a, b and c reveal a clear and striking difference between the hole
and electron doped CaK(Fe${}_{1-x}T_{x}$)4As4 systems. Whereas for the
electron doped ($T$ = Co, Ni) there is very clear scaling with number of added
electrons, for hole doped ($T$ = Mn, Cr) there is clear scaling with the
number of substituted atoms, $x$. These striking differences in the phase
diagrams beg the question of what is different between the two types of
substitution. When there was only data on Mn substitution to compare with the
Co and Ni substitutions, one possible explanation could be based on an
asymmetric density of electronic states on either side of $E$F. Given that the
Cr- and Mn- substituted phase diagrams scale with $x$ rather than $e^{-}$,
this is no longer a possibility.

Figure 17: (a)(b) Temperature versus substitution level x phase diagram of
CaK(Fe${}_{1-x}T_{x}$)4As4 single crystals, $T$ = Cr, Mn, Ni and Co [16]. (c)
Temperature versus $\Delta e^{-}$ change of electrons phase diagram of
CaK(Fe${}_{1-x}T_{x}$)4As4 single crystals, $T$ = Cr, Mn, Ni and Co The
circular symbols denote the $T_{c}$ phase transitions and the cross-like
symbols denote the $T^{*}$ and $T_{N}$ phase transitions, which are obtained
from resistance and magnetic moment measurements, denoted as "$R$($T$)" and
"$M$($T$)".

A different approach to these data is to note that there is one other clear
difference between the Mn and Cr substitutions as compared to the Co and Ni
ones. Mn and Cr clearly bring local-moment-like behavior as manifest by their
conspicuous, high temperature, Curie tails that grow with increasing $x$. This
is absent for the Co and Ni substitutions. The effective moments coming from
the Mn and Cr Curie tails ($\sim$ 5 $\mu_{B}$ [19] and $\sim$ 4 $\mu_{B}$
respectively) are consistent with Mn3+ and Cr3+ valencies. The Cr and Mn
appear to behave like local moment impurities. As such it is not surprising
that they lead to a stronger suppression of $T_{c}$ (via Abrikosov - Gor’kov
pair breaking [49]). In a similar manner, it is not surprising that the
addition of relatively large, local moment impurities to an itinerant,
relatively small moment system helps to stabilize magnetic order. Given that
the size of the local moments are similar, it is not surprising that we find
that the $T$-$x$ phase diagrams scale well. The fact that CaKFe4As4 manifests
rather bi-modal responses to $T$ substitution for $T$ = Co and Ni versus $T$ =
Mn and Cr is consistent with the growing understanding of many of the Fe-based
superconductors[13, 50] families as manifesting properties in between those of
a wide band metal (which would support rigid band shifting) and a more ionic-
or Zintl-like compound (that would support valence-counting-like behavior).

In summary, we have been able to grow and study the CaK(Fe1-xCrx)4As4 system.
Based on magnetic and transport measurements, we assemble a $T$-$x$ phase
diagram that clearly shows the suppression of the superconducting $T_{c}$ with
the addition of Cr, with $T_{c}$ dropping from 35 K for $x$ = 0 to zero for
$x$ $\sim$ 0.03, as well as the stabilization of magnetic order for $x$ >
0.012, with 22 K $\leq$ $T^{*}$ $\leq$ 60 K. As $x$ becomes greater than 0.012
and $T_{c}$ becomes less than $T^{*}$, a clear change in the behavior of
$H^{\prime}_{c2}$($T$)/$T_{c}$ and the associated superconducting coherence
length, $\xi$, can be seen. These are associated with the probable changes in
the Fermi surface that accompany the AFM ordering at $T^{*}$. Comparable
features in $H_{c1}$ or the London penetration depth are not clearly
resolvable.

The $T$-$x$ phase diagram for CaK(Fe1-xCrx)4As4 is qualitatively identical to
CaK(Fe1-xMnx)4As4 phase diagram due to the effect of local moment of Cr and
Mn. Similarity of phase diagrams also indicate importance of the local moment
impurity on hole doped 1144, which is different from the electron doped 1144.
On the other hand, both hole and electron doping in 1144 stabilize
antiferromagnetic ordering with increasing substitution level and the
suppression of $T_{c}$ in CaK(Fe1-xCrx)4As4 is faster than Ni- and Co-1144.

###### Acknowledgements.

We thank B. Kuthanazhi for useful discussions. Work at the Ames National
Laboratory was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division. The Ames
National Laboratory is operated for the U.S. Department of Energy by Iowa
State University under Contract No. DE-AC02-07CH11358.

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##  VII Appendix

Figure 18: (a) shows Curie-Weiss temperature, $\theta$, effective moment,
$\mu_{eff}$, obtained from Curie-Weiss fit to the difference magnetization
($\Delta M^{\prime}$) between CaKFe4As4 and CaK(Fe1-xCrx)4As4 single crystals
as a function of temperature for with a field of 10 kOe applied parallel to
the crystallographic ab plane. (b) shows temperature-independent
susceptibility , $\chi_{0}$.

The magnetization plots shown in figure 4 have the appearance of Curie-Weiss
tails potentially associated with the Cr-substitution. We fit the
magnetization difference ($\Delta M^{\prime}$) between CaK(Fe1-xCrx)4As4 and
CaKFe4As4 single crystals as a function of temperature from 20 K above $Tc$ to
250 K with a field of 10 kOe applied parallel to the crystallographic ab plane
by a C/($T$+$\theta$) + $\chi_{0}$ function assuming that tail behavior is
only due to Cr. Figure 18 shows the result of fitting. The value of
$\mu_{eff}$ is around 4 $\mu_{B}$. The fitting results comes from at least 4
different samples $M$($T$) measurements. The values of $\theta$, $\mu_{eff}$
and $\chi_{0}$ become more stable after $x$ = 0.17. Instability for small
substitution levels may come from difficulty of fitting small tail-like
contribution and when $x$ = 0.005, there is no clear tail shown.

Figure 19: Zero-field-cooled-warming (ZFCW) and Field-cooled (FC) low
temperature magnetization as a function of temperature for
CaK(Fe0.983Cr0.017)4As4 single crystals with a field of 50 Oe applied parallel
to the crystallographic ab plane. $M$ is the volumetric magnetic moment with
cgs unit emu cm-3 or Oe.

Figure 19 shows Zero-field-cooled-warming (ZFCW) and Field-cooled-warming (FC)
low temperature magnetization as a function of temperature for
CaK(Fe0.983Cr0.017)4As4 single crystals with a field of 50 Oe applied parallel
to ab plane. The large difference between ZFCW and FCW is consistent with the
large pinning found even in pure CaKFe4As4[51].

Figure 20: Temperature dependence of Resistivity, $\rho$, of CaK(Fe1-xCrx)4As4
single crystals showing the suppression of the superconducting transition
$T_{c}$.

Figure 20 shows temperature dependence of resistivity, $\rho$, of
CaK(Fe1-xCrx)4As4 single crystals with $x$ < 0.38. Thickness is estimated by
the density of pure CaKFe4As4, the mass and area of plate-like samples. The
superconducting transition temperature is suppressed and resistivity before
$T_{c}$ is increased by increasing substitution. Given the inevitably large
geometric errors associated with the precised determination of length between
voltage contacts as well as the sample thickness we consider the uncertainty
in our resistivity values to be on the order of 20$\%$ and, as such we plot
the data as R(T)/R(300 K) in the main text.

Figure 21: Onset and offset criteria for $T_{c}$ and $T^{*}$ based on
magnetization and resistance measurement.

Criteria for inferring $T_{c}$ and $T^{*}$ are shown in figure 21. For $T_{c}$
(figure 21a) we use an onset criterion for M(T) data and an offset criterion
for R(T) data. As is often the case, these criteria agree well, especially in
the low field limit. The error bar of $T_{c}$ is determined by the half of
difference between onset and offset. Since according to the [33],
d($\chi$$T$)/d$T$, d($\rho$)/dT behave like $C_{p}$ which gives transition
temperature between onset and offset points, we use average of onset and
offset value of d($\chi$$T$)/d$T$ and d($\rho$)/dT as $T^{*}$. For $T^{*}$,
although the feature is much clearer for Cr substitution than it was for Mn,
Ni or Co substitutions [16, 19], the features in M(T) and R(T) are still
somewhat subtle in low substitution level. We infer $T^{*}$ as the average of
onset and offset value and use the half of the difference between onset and
offset as the error.

Figure 22: The $T^{*}$ anomaly appears clearly as a step in both plot
d($MT/H$)/d$T$ and the derivative of resistance, dR/dT. Only the data above
$T_{c}$ are plotted. Rhombuses symbols show the transition temperature of
$T^{*}$ and error bars come from the criteria introduced above.

dR(T)/dT and dM(T)/dT data for several different x-values are shown in figure
22, showing good agreement between the position of the $T^{*}$ features.

Figure 23: Temperature dependence of normalized resistance,
$\textit{R}(\textit{T})/\textit{R}$(300 K), of CaK(Fe1-xCrx)4As4 single
crystals with $x$ = 0.077 for electrical currents along $a$-axis (black) and
along $c$-axis (red). Magnetic susceptibility shows clearly distinguished
anomaly, similar to the features observed in raw resistivity data for both
current directions. Inset shows resistivity derivatives plotted against
d($MT$/$H$)/d$T$ (blue). The onset of the feature in the d($MT$/$H$)/d$T$
curve at $\sim$50 K is accompanied by the clear feature in both $c$\- and
$a$-axis resistivity derivative.  Figure 24: Anisotropic $H_{c2}(T)$ data
determined for two single crystalline samples of $x$EDS=0.005
CaK(Fe1-xCrx)4As4 using onset criterion (solid) and offset criterion (hollow)
inferred from the temperature-dependent electrical resistance data.  Figure
25: Anisotropic $H_{c2}(T)$ data determined for two single crystalline samples
of $x$EDS=0.009 CaK(Fe1-xCrx)4As4 using onset criterion (solid) and offset
criterion (hollow) inferred from the temperature-dependent electrical
resistance data.  Figure 26: Anisotropic $H_{c2}(T)$ data determined for two
single crystalline samples of $x$EDS=0.012 CaK(Fe1-xCrx)4As4 using onset
criterion (solid) and offset criterion (hollow) inferred from the temperature-
dependent electrical resistance data.  Figure 27: Anisotropic $H_{c2}(T)$ data
determined for two single crystalline samples of $x$EDS=0.025
CaK(Fe1-xCrx)4As4 using onset criterion (solid) and offset criterion (hollow)
inferred from the temperature-dependent electrical resistance data.

Figures 24 \- 27 present $H_{c2}(T)$ curves which is obtained from $R$($T$)
data for fixed applied fields and the criteria shown in Figs 12a of
CaK(Fe1-xCrx)4As4 single crystals for $x$ = 0.005, 0.009, 0.012, 0.017 and
0.025.