1542 LABEL:LastPageJul. 10, 2018Oct. 14, 2019

*Extended version of a paper with the same title presented at CONCUR 2017.

# Rule Formats for Nominal Process Calculi*

Luca Acetoa,b  ,  Ignacio Fábregasc  ,  Álvaro García-Pérezc  ,  Anna
Ingólfsdóttirb  and  Yolanda Ortega-Mallénd  aGran Sasso Science Institute,
L’Aquila, Italy<EMAIL_ADDRESS>bICE-TCS, School of Computer Science,
Reykjavik University, Iceland<EMAIL_ADDRESS>cIMDEA Software Institute,
Madrid, Spain<EMAIL_ADDRESS>dDepartamento
de Sistemas Informáticos y Computación, Universidad Complutense de Madrid,
Spain<EMAIL_ADDRESS>
###### Abstract.

The nominal transition systems (NTSs) of Parrow et al. describe the
operational semantics of nominal process calculi. We study NTSs in terms of
the nominal residual transition systems (NRTSs) that we introduce. We provide
rule formats for the specifications of NRTSs that ensure that the associated
NRTS is an NTS and apply them to the operational specifications of the early
and late pi-calculus. We also explore alternative specifications of the NTSs
in which we allow residuals of abstraction sort, and introduce translations
between the systems with and without residuals of abstraction sort. Our study
stems from the Nominal SOS of Cimini et al. and from earlier works in nominal
sets and nominal logic by Gabbay, Pitts and their collaborators.

###### Key words and phrases:

nominal sets, nominal structural operational semantics, process algebra,
nominal transition systems, scope opening, rule formats

Research partially supported by the project Nominal SOS (nr. 141558-051) of
the Icelandic Research Fund, the project 001-ABEL-CM-2013 within the NILS
Science and Sustainability Programme, the Spanish Projects TRACES
(TIN2015-67522-C3-3-R) and Bosco (PGC2018-102210-B-I00), and by Comunidad de
Madrid as part of the program S2018/TCS-4339 (BLOQUES-CM) co-funded by EIE
Funds of the European Union, and the projects RACCOON (H2020-EU 714729) and
MATHADOR (COGS 724.464) of the European Research Council, and the Spanish
addition to MATHADOR (TIN2016-81699-ERC)

##  1\. Introduction

The goal of this paper is to develop the foundations of a framework for
studying the meta-theory of structural operational semantics (SOS) [Plo04] for
process calculi with names and name-binding operations, such as the
$\pi$-calculi [MPW92, SW01]. To this end, we build on the large body of work
on rule formats for SOS, as surveyed in [AFV01, MRG07], and on the nominal
techniques of Gabbay, Pitts and their co-workers [UPG04, CP07, GM09, Pit13].

Rule formats provide syntactic templates guaranteeing that the models of the
calculi, whose semantics they specify, enjoy some desirable properties. A
first design decision that has to be taken in developing a theory of rule
formats for a class of languages is therefore the choice of the semantic
objects specified by the rules. The target semantic model we adopt in our
study is that of _nominal transition systems_ (NTSs), which have been
introduced by Parrow et al. in [PBEGW15, PWBE17] as a uniform model to
describe the operational semantics of a variety of calculi with names and
name-binding operations. Based on this choice, a basic sanity criterion for a
collection of rules describing the operational semantics of a nominal calculus
is that they specify an NTS, and we present a rule format guaranteeing this
property (Thm. LABEL:the:alpha-conversion).

As a first stepping stone in our study, we introduce _nominal residual
transition systems_ (NRTSs), and study NTSs in terms of NRTSs (Section 2).
More specifically, the only requirement of an NRTS is that its transition
relation is equivariant, which means that it treats names uniformly. This is a
desirable property of models of nominal calculi, such as NTSs. Moreover, NTSs
are NRTSs that, in addition to having an equivariant transition relation,
satisfy a property Parrow et al. call _alpha-conversion of residuals_ (see
Def. 2 for the details). The latter property formalises a key aspect of
calculi in which names can be scoped to represent local resources. To wit, one
crucial feature of the $\pi$-calculus is scope opening [MPW92]. Consider a
transition $p\mathrel{\raisebox{-2.84526pt}{$\buildrel\overline{a}(\nu
b)\over{\longrightarrow}$}}p^{\prime}$ in which a process $p$ exports a
private/local channel name $b$ along channel $a$. Since the name $b$ is local,
it ‘can be subject to alpha-conversion’ [PBEGW15] and the transitions
$p\mathrel{\raisebox{-2.84526pt}{$\buildrel\overline{a}(\nu
c)\over{\longrightarrow}$}}p\\{c/b\\}$ should also be present for each ‘fresh
name’ $c$.

In contrast to related work [CMRG12, FG07], our approach uses _nominal terms_
[Pit13] to connect the specification system with the semantic model. This has
the advantage of capturing the requirement that transitions be ‘up to alpha-
equivalence’ (typical in nominal calculi) without instrumenting alpha-
conversion explicitly in the specification system.

We specify an NRTS by means of a nominal residual transition system
specification (NRTSS), which describes the syntax of a nominal calculus in
terms of a nominal signature (Section 3) and its semantics by means of a set
of inference rules (Section LABEL:sec-specification-nrts). We develop the
basic theory of the NRTS/NRTSS framework, building on the nominal algebraic
datatypes of Pitts [Pit13] and the nominal rewriting framework of Fernández
and Gabbay [FG07]. Based on this framework, we provide rule formats [AFV01,
MRG07] for NRTSSs (Section LABEL:sec-rule-format-nrts) that ensure that the
induced transition relation is equivariant (Thm. LABEL:thm:rule-format-
equivariance) and enjoys alpha-conversion of residuals (Thm. LABEL:the:alpha-
conversion), and is therefore an NTS. Section LABEL:sec:example-nts presents
an example of application of these rule formats to the setting of the
$\pi$-calculus. Section LABEL:sec:nts-abstraction-sort explores alternative
specifications of the NTSs in which we allow a residual to be an atom
abstraction (hereafter referred to as _residual with abstraction sort_). We
introduce translations between the systems with and without residuals of
abstraction sort (Defs. LABEL:def:nts-to-nrts and LABEL:def:nrts-to-nts). We
develop a rule format that guarantees that these translations are the inverse
of each other (Thms. LABEL:thm-composition-identity and LABEL:thm-composition-
identity-abstraction). Section LABEL:sec:application-BA-format presents an
example of application of this rule format to the early $\pi$-calculus and to
a slightly modified version of the late $\pi$-calculus. We also show that both
the specification with and without residuals of abstraction sort induce the
same model of computation. Finally, Section LABEL:sec:conclusions discusses
related work, as well as avenues for future work, and concludes.

The appendix accompanying the paper collects some proofs that are omitted in
the main text.

This paper is an extended version of a paper with the same title presented at
CONCUR 2017 [AFGIO17]. The novel content in this extended version is
summarised below:

  * •

In Section 2 we recall the notion of finite renamings, which play a prominent
role throughout this paper since they replace the permutations in the
moderated terms of the CONCUR 2017 paper.

  * •

In Section LABEL:sec:early-pi-calculus we introduce an NRTSS that faithfully
captures the original semantics of the early $\pi$-calculus [San96]. The NRTSS
of the CONCUR 2017 paper induced a semantics that failed to capture some
transitions in the original early $\pi$-calculus.

  * •

In Section LABEL:sec:late-pi-calculus we introduce an NRTSS whose induced
semantics differs minimally from the original semantics of the late
$\pi$-calculus [San96] (see Remark LABEL:rem:prevent-capture-in for further
discussion). We also apply the rule format for alpha-conversion of residuals
to this version of the late $\pi$-calculus. This section is entirely novel.

  * •

Section LABEL:sec:nts-abstraction-sort, where we study alternative
formulations of the NTSs in which we allow residuals of abstraction sorts, is
entirely novel.

  * •

In Section LABEL:sec:application-BA-format we apply the rule formats from
Section LABEL:sec:nts-abstraction-sort to the early $\pi$-calculus and to our
version of the late $\pi$-calculus. This section is entirely novel too.

  * •

We have included the detailed proofs of all lemmas and theorems in the paper,
some of which were missing in the conference version.

##  2\. Preliminaries

This section collects some earlier foundational work by Gabbay and Pitts on
nominal sets and finitary renamings [GH08, GP02, Pit13, Pit16] on which our
work builds, and recalls the nominal transition systems of Parrow et al.
[PBEGW15].

### Nominal Sets

We assume a countably infinite set $\mathbb{A}$ of _atoms_ and consider
$\mathrm{Perm}\;\mathbb{A}$ as the group of _finite permutations of atoms_
(hereafter _permutations_) ranged over by $\pi$, where we write $\iota$ for
the _identity_ , $\circ$ for _composition_ and $\pi^{-1}$ for the _inverse_ of
permutation $\pi$. We are particularly interested in _transpositions_ of two
atoms: $(a\,b)$ stands for the permutation that swaps $a$ with $b$ and leaves
all other atoms fixed. Every permutation $\pi$ is equal to the composition of
a finite number of transpositions, i.e.
$\pi=(a_{1}\,b_{1})\circ\ldots\circ(a_{n}\,b_{n})$ with $n\geq 0$.

An _action_ of the group $\mathrm{Perm}\;\mathbb{A}$ on a set $S$ is a binary
operation mapping each $\pi\in\mathrm{Perm}\;\mathbb{A}$ and $s\in S$ to an
element $\pi\cdot s\in S$, and satisfying the identity law $\iota\cdot s=s$
and the composition law $(\pi_{1}\circ\pi_{2})\cdot
s=\pi_{1}\cdot(\pi_{2}\cdot s)$. A _$\mathrm{Perm}\;\mathbb{A}$ -set_ is a set
equipped with an action of $\mathrm{Perm}\;\mathbb{A}$.

We say that a set of atoms $A$ _supports_ an object $s$ iff $\pi\cdot s=s$ for
every permutation $\pi$ that leaves each element $a\in A$ invariant. In
particular, we are interested in sets all of whose elements have finite
support (Def. 2.2 of [Pit13]).

[Nominal sets] A _nominal set_ is a $\mathrm{Perm}\;\mathbb{A}$-set all of
whose elements are finitely supported.

For each element $s$ of a nominal set, we write $\mathrm{supp}(s)$ for the
least set that supports $s$, called the _support_ of $s$. (Intuitively, the
action of permutations on a set $S$ determines that a finitely supported $s\in
S$ only depends on atoms in $\mathrm{supp}(s)$, and no others.) The set
$\mathbb{A}$ of atoms is a nominal set when $\pi\cdot a=\pi\,a$ since
$\mathrm{supp}(a)=\\{a\\}$ for each atom $a\in\mathbb{A}$. The set
$\mathrm{Perm}\;\mathbb{A}$ of finite permutations is also a nominal set where
the permutation action on permutations is given by conjugation, i.e.
$\pi\cdot\pi^{\prime}=\pi\circ\pi^{\prime}\circ\pi^{-1}$, and the support of a
permutation $\pi$ is $\mathrm{supp}(\pi)=\\{a\mid\pi a\not=a\\}$.

Given two $\mathrm{Perm}\;\mathbb{A}$-sets $S$ and $T$ and a function $f:S\to
T$, the action of permutation $\pi$ on function $f$ is given by conjugation,
i.e. $(\pi\cdot f)(s)=\pi\cdot f(\pi^{-1}\cdot s)$ for each $s\in S$. We say
that a function $f:S\to T$ is _equivariant_ iff $\pi\cdot f(s)=f(\pi\cdot s)$
for every $\pi\in\mathrm{Perm}\;\mathbb{A}$ and every $s\in S$. The intuition
is that an equivariant function $f$ is atom-blind, in that $f$ does not treat
any atom preferentially. It turns out that a function $f$ is equivariant iff
$\mathrm{supp}(f)=\emptyset$ (Rem. 2.13 of [Pit13]). The function
$\mathrm{supp}$ is equivariant (Prop. 2.11 of [Pit13]).

Let $S$ be a $\mathrm{Perm}\;\mathbb{A}$-set, we write $S_{\mathrm{fs}}$ for
the nominal set that contains the elements in $S$ that are finitely supported.
Let $S_{1}$ and $S_{2}$ be nominal sets. The product $S_{1}\times S_{2}$ is a
nominal set (Prop. 2.14 of [Pit13]). The permutation action for products is
given componentwise (Eq (1.12) of [Pit13]).

Conjugation yields that, for every $\mathrm{Perm}\;\mathbb{A}$-set $S$, the
action of $\pi$ on $s\in S$ is equivariant. Indeed,

$\pi\cdot(\pi_{1}\cdot s)=(\pi\circ\pi_{1})\cdot
s=(\pi\circ\pi_{1}\circ\pi^{-1}\circ\pi)\cdot
s=((\pi\cdot\pi_{1})\circ\pi)\cdot s=(\pi\cdot\pi_{1})\cdot(\pi\cdot s).$

It is also straightforward to show that composition of permutations is
equivariant. In fact,

$\pi\cdot(\pi_{1}\circ\pi_{2})=\pi\circ(\pi_{1}\circ\pi_{2})\circ\pi^{-1}=(\pi\circ\pi_{1}\circ\pi^{-1})\circ(\pi\circ\pi_{2}\circ\pi^{-1})=(\pi\cdot\pi_{1})\circ(\pi\cdot\pi_{2}).$

An element $s_{1}\in S_{1}$ _is fresh in_ $s_{2}\in S_{2}$, written
$s_{1}\\#s_{2}$, iff $\mathrm{supp}(s_{1})\cap\mathrm{supp}(s_{2})=\emptyset$.
The freshness relation is equivariant (Eq. (3.2) of [Pit13]).

We consider _atom abstractions_ (Sec. 4 of [Pit13]), which represent alpha-
equivalence classes of elements.

[Atom abstraction] Given a nominal set $S$, the _atom abstraction_ of atom $a$
in element $s\in S$, written $\langle a\rangle s$, is the
$\mathrm{Perm}\;\mathbb{A}$-set $\langle a\rangle s=\\{(b,(b\,a)\cdot s)\mid
b=a\lor b\\#s\\}$, whose permutation action is $\pi\cdot\langle a\rangle
s=\\{(\pi\cdot b,\pi\cdot((b\,a)\cdot s))\mid\pi\cdot b=\pi\cdot a\lor\pi\cdot
b\\#\pi\cdot s\\}$.

We write $[\mathbb{A}]S$ for the set of _atom abstractions_ in elements of
$S$, which is a nominal set (Def. 4.4 of [Pit13]), since
$\mathrm{supp}(\langle a\rangle s)=\mathrm{supp}(s)\setminus\\{a\\}$ for each
atom $a$ and element $s\in S$.

######  Remark 1.

Notice that, by Lemma 4.3 in [Pit11], $s=s^{\prime}$ whenever $\langle
a\rangle s=\langle a\rangle s^{\prime}$. $\blacksquare$

Nominal sets are the objects of a category $\mathbf{Nom}$ whose morphisms are
the equivariant functions. The category $\mathbf{Nom}$ is closed under finite
products and both finite and infinite coproducts.111In $\mathbf{Nom}$,
coproducts correspond to disjoint unions. We write
$s=\mathrm{inj}_{i}s^{\prime}$ with $i\in I$ and $s^{\prime}\in S_{i}$ for an
element $s$ in a coproduct $\sum_{i\in I}(S_{i})$. (For a finite coproduct
${S_{1}+\cdots+S_{n}}$ we let $I=\\{1,\ldots,n\\}$.) For other set-theoretical
operations (i.e. infinite products, functions, partial functions, power sets)
the following caveat applies. The category of nominal sets is closed under the
variant of each operation that restricts any universal quantification that is
involved in the operation to quantify only over finitely supported elements
(see Sections 2.2 to 2.5 of [Pit13]).

The _nominal function set_ between nominal sets $S$ and $T$ (Definition 2.18
of [Pit13]) is the nominal set ${(T^{S})}_{\mathrm{fs}}$ of finitely supported
functions from $S$ to $T$—be they equivariant or not; recall that an
equivariant function has empty support. (We may write $S\to_{\mathrm{fs}}T$ in
lieu of ${(T^{S})}_{\mathrm{fs}}$.) The application and currying functions can
be respectively restricted to equivariant functions
${\mathrm{app}:(X\to_{\mathrm{fs}}Y)\times X\to Y}$ and
$\mathrm{curry}:(Z\times X\to_{\mathrm{fs}}Y)\to Z\to(X\to_{\mathrm{fs}}Y)$
such that the nominal function set coincides with the _exponential object_ in
$\mathbf{Nom}$, i.e. there is a bijection between hom-sets
$\mathbf{Nom}(Z\times X,Y)$ and $\mathbf{Nom}(Z,X\to_{\mathrm{fs}}Y)$ given by
sending $f\in\mathbf{Nom}(Z\times X,Y)$ to
$\mathrm{curry}(f)\in\mathbf{Nom}(Z,X\to_{\mathrm{fs}}Y)$. (Section 2.4 in
[Pit13] spells out all the details on this isomorphism.)

Finally, the category $\mathbf{Nom}$ is Cartesian closed (Theorem 2.19 in
[Pit13]), i.e., $\mathbf{Nom}$ admits all the finite products (including the
empty product 1 which is the terminal object) and all the exponentials.

### Renamings

We consider the _finitely supported renamings_ (hereafter _renamings_) ranged
over by $\rho$, which are finitely supported functions
$\rho:\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}$, that is, functions that act like
the identity on all but finitely many atoms. We write $\iota$ for the
_identity function_ and ‘;’ for _diagrammatical composition_ , that is, $f;g$
denotes the function $g\circ f$. We are particularly interested in
_replacements_ of an atom by another: $b/a$ stands for the replacement that
substitutes $a$ with $b$ and leaves all other atoms fixed. Every renaming
$\rho$ is equal to the composition of a finite number of replacements [GH08],
i.e. $\rho=b_{1}/a_{1};\ldots;b_{n}/a_{n}$ with $n\geq 0$. Notice that
$\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}$ with ‘;’ as composition operator and
$\iota$ as identity element is a monoid [GH08].

An _action_ of the monoid $\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}$ on a set $S$
is a binary operation mapping each
$\rho\in\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}$ and $s\in S$ to an element
$s\\{\rho\\}\in S$, and satisfying the identity law $s\\{\iota\\}=s$ and the
composition law $(s\\{\rho_{1}\\})\\{\rho_{2}\\}=s\\{\rho_{1};\rho_{2}\\}$. We
will provide an action of renaming for the raw terms to be defined in Section
3. An action of renaming could be defined for every object in $\mathbf{Nom}$,
which ultimately gives rise to the category $\mathbf{Ren}$ of renamings as
described in [GH08], which is a generalisation of $\mathbf{Nom}$. We are
interested in interpreting our terms as the nominal algebraic datatypes of
[Pit13], which live in $\mathbf{Nom}$, and therefore we refrain ourselves from
interpreting our terms in the category $\mathbf{Ren}$, and we treat renamings
as the exponential objects $\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}$ in the
former category.

Notice that every permutation is an instance of a renaming. For every
permutation $\pi$, we may write $s\\{\pi\\}=\pi\cdot s$ for the action of
renaming $\pi$ on $s$, and for every renaming $\rho$, we may write $\pi;\rho$
for the diagrammatical composition of $\rho$ after $\pi$. As we have mentioned
above, the renamings are the exponential object
$\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}$ in the category $\mathbf{Nom}$, and
therefore they are equipped with a permutation action given by
$\pi\cdot\rho=\pi^{-1};\rho;\pi$. As for any other element of an object in
Nom, the support of a renaming $\rho$ is the least set $A$ such that
$\pi\cdot\rho=\rho$ for every permutation $\pi$ that leaves each element of
$A$ invariant.

Consider the replacement $b/a$. Its support is $\mathrm{supp}(b/a)=\\{a,b\\}$,
as we show next. Let $\pi$ be a permutation such that $\pi\cdot a=a$ and
$\pi\cdot b=b$. We show that $\pi^{-1};b/a;\pi=b/a$. For atom $a$,

$a\\{\pi^{-1};b/a;\pi\\}=(\pi^{-1}\cdot
a)\\{b/a;\pi\\}=a\\{b/a;\pi\\}=b\\{\pi\\}=\pi\cdot b=b=a\\{b/a\\}.$

For any other atom $c\\#a$,

$c\\{\pi^{-1};b/a;\pi\\}=(\pi^{-1}\cdot c)\\{b/a;\pi\\}=(\pi^{-1}\cdot
c)\\{\pi\\}=\pi\cdot(\pi^{-1}\cdot c)=c=c\\{b/a\\},$

since $\pi^{-1}\cdot c\not=a$ by the assumptions on $\pi$. Therefore
$\\{a,b\\}$ supports $b/a$, while it is not hard to see that no subset of
$\\{a,b\\}$ does so. $\blacksquare$

Since every renaming $\rho$ is finitary, its support can be defined
alternatively as in the proposition below.

######  Proposition 2.

Let $\rho$ be a renaming. The support $\mathrm{supp}(\rho)=\\{a,(\rho\
a)\mid\rho\ a\not=a\\}$.

The proof of Proposition 2 is in Appendix LABEL:ap:preliminaries.

### Nominal Transition Systems

Nominal transition systems adopt the state/residual presentation for
transitions of [BP09], where a residual is a pair consisting of an action and
a state. In [PBEGW15], Parrow et al. develop modal logics à la Hennessy-Milner
for process nominal calculi. Here we are mainly interested in the transition
relation and we adapt Definition 1 in [PBEGW15] by removing the predicates. We
write $\mathcal{P}_{\omega}(\mathbb{A})$ for the _finite power set_ of
$\mathbb{A}$.

[Nominal transition system] A _nominal transition system_ (NTS) is a quadruple
$(S,\mathit{Act},\mathrm{bn},\buildrel\over{\longrightarrow})$ where $S$ and
$\mathit{Act}$ are nominal sets of _states_ and _actions_ respectively,
$\mathrm{bn}:\mathit{Act}\to\mathcal{P}_{\omega}(\mathbb{A})$ is an
equivariant function that delivers the _binding names_ in an action, and
${\buildrel\over{\longrightarrow}}\subseteq S\times(\mathit{Act}\times S)$ is
an equivariant binary transition relation from states to _residuals_ (we let
$\mathit{Act}\times S$ be the set of residuals). The function $\mathrm{bn}$ is
such that $\mathrm{bn}(\ell)\subseteq\mathrm{supp}(\ell)$ for each
$\ell\in\mathit{Act}$. We often write
$p\buildrel\over{\longrightarrow}(\ell,p^{\prime})$ in lieu of
$(p,(\ell,p^{\prime}))\in{\buildrel\over{\longrightarrow}}$.

Finally, the transition relation $\buildrel\over{\longrightarrow}$ must
satisfy _alpha-conversion of residuals_ , that is, if
${a\in\mathrm{bn}(\ell)}$, $b\\#(\ell,p^{\prime})$ and
$p\buildrel\over{\longrightarrow}(\ell,p^{\prime})$ then also
$p\buildrel\over{\longrightarrow}((a\,b)\cdot\ell,(a\,b)\cdot p^{\prime})$, or
equivalently $p\buildrel\over{\longrightarrow}(a\,b)\cdot(\ell,p^{\prime})$.

We will consider an NTS (without its associated binding-names function
$\mathrm{bn}$) as a particular case of a nominal residual transition system,
which we introduce next. [Nominal residual transition system] A _nominal
residual transition system_ (NRTS) is a triple
$(S,R,\buildrel\over{\longrightarrow})$ where $S$ and $R$ are nominal sets,
and where ${\buildrel\over{\longrightarrow}}\subseteq S\times R$ is an
equivariant binary transition relation. We say $S$ is the set of _states_ and
$R$ is the set of _residuals_. The connection between NTSs and NRTSs will be
studied in more detail in Section LABEL:sec-rule-format-nrts.

##  3\. Nominal Terms

This section is devoted to the notion of nominal terms, which are syntactic
objects that make use of the atom abstractions of Definition 2 and represent
terms up to alpha-equivalence. As a first step, we introduce raw terms, devoid
of any notion of alpha-equivalence. Our raw terms resemble those from the
literature, mainly from [UPG04, FG07, CP07, Pit13], but with some important
differences. In particular, our terms include both variables (i.e. unknowns)
and moderated terms (i.e. explicit renamings over raw terms), and we consider
atom and abstraction sorts. (The raw terms of [Pit13] do not include moderated
terms, and the ones in [UPG04, FG07] only consider moderated variables where
the delayed renaming is a permutation. In [CP07] the authors consider neither
atom nor abstraction sorts.) We also adopt the classic presentation of free
algebras and term algebras in [GTWW77, BS00] in a different way from that in
[CP07, Pit13]. The raw terms correspond to the standard notion of free algebra
over a signature generated by a set of variables. We then adapt the
$\Sigma$-structures of [CP07] to our sorting schema. Finally, the nominal
terms are the interpretations of the ground terms in the initial
$\Sigma$-structure; we show that they coincide with the nominal algebraic
terms of [Pit13].

[Nominal signature and nominal sort] A _nominal signature_ (or simply a
_signature_) $\Sigma$ is a triple $(\Delta,A,F)$ where
$\Delta=\\{\delta_{1},\ldots,\delta_{n}\\}$ is a finite set of _base sorts_ ,
$A$ is a countable set of _atom sorts_ , and $F$ is a finite set of _function
symbols_. The _nominal sorts_ over $\Delta$ and $A$ are given by the grammar

$\sigma~{}::=~{}\delta\mid\alpha\mid[\alpha]\sigma\mid\sigma_{1}\times\cdots\times\sigma_{k},$

with $k\geq 0$, $\delta\in\Delta$ and $\alpha\in A$. The sort $[\alpha]\sigma$
is the _abstraction sort_. Symbol $\times$ denotes the _product sort_ , which
is associative; $\sigma_{1}\times\cdots\times\sigma_{k}$ stands for the sort
of the empty product when $k=0$, which we may write as 1. We write
$\mathsf{S}$ for the set of nominal sorts. We arrange the function symbols in
$F$ based on the sort of the data (base sort) that they produce. We write
$f_{ij}\in F$ with $1\leq i\leq n$ and $1\leq j\leq m_{i}$ such that $f_{ij}$
has arity $\sigma_{ij}\to\delta_{i}$, where $\delta_{i}$ is a base sort.

The theory of nominal sets extends to the case of many-sorted atoms (see Sec.
4.7 in [Pit13]). We assume that $\mathbb{A}$ contains a countably infinite
collection of atoms $a_{\alpha}$, $b_{\alpha}$, $c_{\alpha}$, … for each atom
sort $\alpha$ such that the sets of atoms $\mathbb{A}_{\alpha}$ of each sort
are mutually disjoint. We write
$\mathrm{Perm}_{s}\,\mathbb{A}=\\{\pi\in\mathrm{Perm}\;\mathbb{A}\mid\forall\alpha\in
A.\,\forall a\in\mathbb{A}_{\alpha}.\;\pi\,a\in\mathbb{A}_{\alpha}\\}$ for the
subgroup of finite permutations that respect the sorting. The sorted nominal
sets are the $\mathrm{Perm}_{s}\,\mathbb{A}$-sets whose elements are finitely
supported. We also consider renamings that respect the sorting, which we write
${(\mathbb{A}\to_{\mathrm{fs}}\mathbb{A})}_{s}=\\{\rho\in\mathbb{A}\to_{\mathrm{fs}}\mathbb{A}\mid\forall\alpha\in
A.~{}\forall a\in\mathbb{A}_{\alpha}.~{}\rho\ a\in\mathbb{A}_{\alpha}\\}$.
(Notice that every permutation in $\mathrm{Perm}_{s}\,\mathbb{A}$ is a
renaming that respects the sorting.) In the sequel we may drop the $s$
subscript in $\mathrm{Perm}_{s}\,\mathbb{A}$ and in
${(\mathbb{A}\to_{\mathrm{fs}}\mathbb{A})}_{s}$, and omit the ‘sorted’ epithet
from ‘sorted nominal sets’.

We let $\mathcal{V}$ be a set that contains a countably infinite collection of
_variable names_ (variables for short) $x_{\sigma}$, $y_{\sigma}$,
$z_{\sigma}$, … for each sort $\sigma$, such that the sets of variables
$\mathcal{V}_{\sigma}$ of each sort are mutually disjoint. We also assume that
$\mathcal{V}$ is disjoint from $\mathbb{A}$.

[Raw terms] Let $\Sigma=(\Delta,A,F)$ be a signature. The set of _raw terms
over signature $\Sigma$ and set of variables $\mathcal{V}$_ (_raw terms_ for
short) is given by the grammar

$\begin{array}[]{rcl}t_{\sigma}::=&x_{\sigma}\mid
a_{\alpha}\mid{(t_{\sigma}\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}})}_{\sigma}\mid{([a_{\alpha}]t_{\sigma})}_{[\alpha]\sigma}\mid{(t_{\sigma_{1}},\ldots,t_{\sigma_{k}})}_{\sigma_{1}\times\cdots\times\sigma_{k}}\mid{(f_{ij}(t_{\sigma_{ij}}))}_{\delta_{i}},\end{array}$

where term $x_{\sigma}$ is a _variable_ of sort $\sigma$, term $a_{\alpha}$ is
an _atom_ of sort $\alpha$, term
${(t_{\sigma}\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}})}_{\sigma}$
is a _moderated term_ (i.e. the explicit, or delayed, renaming $\rho$ over
term $t_{\sigma}$), term ${([a_{\alpha}]t_{\sigma})}_{[\alpha]\sigma}$ is the
_abstraction of atom $a_{\alpha}$ in term $t_{\sigma}$_, term
${(t_{\sigma_{1}},\ldots,t_{\sigma_{k}})}_{\sigma_{1}\times\cdots\times\sigma_{k}}$
is the _product of terms_ $t_{\sigma_{1}}$, …, $t_{\sigma_{k}}$, and term
${(f_{ij}(t_{\sigma_{ij}}))}_{\delta_{i}}$ is the _datum of base sort
$\delta_{i}$ constructed from term $t_{\sigma_{ij}}$ and function symbol
$f_{ij}:\sigma_{ij}\to\delta_{i}$_. When they are clear from the context or
immaterial, we leave the arities and sorts implicit and write $x$, $a$,
$t\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}}$,
$[a]t$, $(t_{1},\ldots,t_{k})$, $f(t)$, etc. Given a raw term $t$, the _size
of $t$_ is the number of nodes of $t$’s abstract syntax tree.

The raw terms are the inhabitants of the carrier of the free algebra over the
set of variables $\mathcal{V}$ and over the $\mathsf{S}$-sorted conventional
signature that consists of the function symbols in $F$, together with a
constant symbol for each atom $a_{\alpha}$, a unary symbol that produces
moderated terms for each renaming $\rho$ and each sort $\sigma$, a unary
symbol that produces abstractions for each atom $a_{\alpha}$ and sort
$\sigma$, and a $k$-ary symbol that produces a product of sort
$\sigma_{1}\times\cdots\times\sigma_{k}$ for each sequence of sorts
$\sigma_{1}$, …, $\sigma_{k}$. (See [GTWW77] for a classic presentation of
term algebras, initial algebra semantics and free algebras.)

We write $\mathbb{T}(\Sigma,\mathcal{V})_{\sigma}$ for the set of raw terms of
sort $\sigma$. A raw term $t$ is _ground_ iff no variables occur in $t$. We
write $\mathbb{T}(\Sigma)_{\sigma}$ for the set of ground terms of sort
$\sigma$. The sets of raw terms (resp. ground terms) of each sort are mutually
disjoint as terms carry sort information. Therefore we sometimes identify the
family ${(\mathbb{T}(\Sigma,\mathcal{V})_{\sigma})}_{\sigma\in\mathsf{S}}$ of
$\mathsf{S}$-indexed raw terms and the family
${(\mathbb{T}(\Sigma)_{\sigma})}_{\sigma\in\mathsf{S}}$ of
$\mathsf{S}$-indexed ground terms with their respective ranges
$\bigcup_{\sigma\in\mathsf{S}}\mathbb{T}(\Sigma,\mathcal{V})_{\sigma}$ and
$\bigcup_{\sigma\in\mathsf{S}}\mathbb{T}(\Sigma)_{\sigma}$, which we
abbreviate as $\mathbb{T}(\Sigma,\mathcal{V})$ and $\mathbb{T}(\Sigma)$
respectively.

The set $\mathbb{T}(\Sigma,\mathcal{V})$ of raw terms is a nominal set, with
the $\mathrm{Perm}\;\mathbb{A}$-action and the support of a raw term given by:

$\begin{array}[]{cc}\begin{array}[t]{rcl}\pi\cdot x&=&x\\\ \pi\cdot
a&=&\pi\,a\\\
\pi\cdot(t\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}})&=&(\pi\cdot
t)\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\pi\cdot\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}}\\\
\pi\cdot[a]t&=&[\pi\,a](\pi\cdot t)\\\
\pi\cdot(t_{1},\ldots,t_{k})&=&(\pi\cdot t_{1},\ldots,\pi\cdot t_{k})\\\
\pi\cdot(f(t))&=&f(\pi\cdot t),\\\\[8.0pt]
\end{array}&\begin{array}[t]{rcl}\mathrm{supp}(x)&=&\emptyset\\\
\mathrm{supp}(a)&=&\\{a\\}\\\
\mathrm{supp}(t\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}})&=&\mathrm{supp}(t)\cup\mathrm{supp}(\rho)\\\
\mathrm{supp}([a](t))&=&\\{a\\}\cup\mathrm{supp}(t)\\\
\mathrm{supp}((t_{1},\ldots,t_{k}))&=&\mathrm{supp}(t_{1})\cup\ldots\cup\mathrm{supp}(t_{k})\\\
\mathrm{supp}(f(t))&=&\mathrm{supp}(t).\end{array}\end{array}$

It is straightforward to check that the permutation action for raw terms is
sort-preserving (remember that permutations are also sort-preserving). The set
$\mathbb{T}(\Sigma)$ of ground terms is also a nominal set since it is closed
with respect to the $\mathrm{Perm}\;\mathbb{A}$-action given above.

Below on the left we introduce the action of renaming for a raw term $t$,
which replaces each occurrence of a free atom $a$ in $t$ by $a\\{\rho\\}$. On
the right we present the function
$\mathrm{fa}:\mathbb{T}(\Sigma,\mathcal{V})\to\mathcal{P}_{\omega}(\mathbb{A})$,
which delivers the set of free atoms in a raw term:

$\begin{array}[]{cc}\begin{array}[t]{rcl}x\\{\rho\\}&=&x\\\
a\\{\rho\\}&=&\rho\,a\\\
(t\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho_{1}\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}})\\{\rho_{2}\\}&=&t\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho_{1};\rho_{2}\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}}\\\
([a]t)\\{\rho\\}&=&[\rho\,a](t\\{\rho\\})\\\
(t_{1},\ldots,t_{k})\\{\rho\\}&=&(t_{1}\\{\rho\\},\ldots,t_{k}\\{\rho\\})\\\
(f(t))\\{\rho\\}&=&f(t\\{\rho\\}),\\\\[8.0pt]
\end{array}&\begin{array}[t]{rcl}\mathrm{fa}(x)&=&\emptyset\\\
\mathrm{fa}(a)&=&\\{a\\}\\\
\mathrm{fa}(t\boldsymbol{\\{}\\!\\!\\!\boldsymbol{\\{}\rho\boldsymbol{\\}}\\!\\!\\!\boldsymbol{\\}})&=&\mathrm{fa}(t\\{\rho\\})\\\
\mathrm{fa}([a]t)&=&\mathrm{fa}(t)\setminus\\{a\\}\\\
\mathrm{fa}(t_{1},\ldots,t_{k})&=&\mathrm{fa}(t_{1})\cup\ldots\cup\mathrm{fa}(t_{k})\\\
\mathrm{fa}(f(t))&=&\mathrm{fa}(t).\end{array}\end{array}$

Notice that the set of free atoms in a raw term differs from the support of
the term. For instance, $\mathrm{fa}([a](a,b))=\\{b\\}$, but
$\mathrm{supp}([a](a,b))=\\{a,b\\}$.

######  Remark 3.

Let $t$ be a raw term and $\rho$ a renaming. Then the size of $t\\{\rho\\}$
equals the size of $t$, which can be checked in a straightforward way by the
definition above. $\blacksquare$

Observe that the action of renaming is equivariant.

######  Lemma 4.

Let $t$ be a term, $\rho$ be a renaming and $\pi$ be a permutation. Then,
$\pi\cdot(t\\{\rho\\})=(\pi\cdot t)\\{\pi\cdot\rho\\}$.

As expected, the free atoms of a raw term are contained in its support.

######  Lemma 5.

Let $t$ be a raw term. Then $\mathrm{fa}(t)\subseteq\mathrm{supp}(t)$.

The proof of Lemmas 4 and 5 are in Appendix LABEL:ap:terms.

[$\pi$-calculus] Consider a signature $\Sigma$ for the $\pi$-calculus [SW01,
CMRG12] given by a single atom sort $\mathsf{ch}$ of channel names, and base
sorts $\mathsf{pr}$ and $\mathsf{ac}$ for processes and actions respectively.
The function symbols (adapted from [SW01]) are the following:


# Tuneable vertical hysteresis loop shift in Ni80Fe${}_{20}/$SrRuO3
heterostructures

Manisha Bansal1 Samir Kumar Giri2 Weiwei Li2 Judith L. MacManus-Driscoll2
Tuhin Maity1,2<EMAIL_ADDRESS>1School of Physics, Indian Institute of
Science Education and Research Thiruvananthapuram, Thiruvananthapuram, Kerala
695551, India
2Department of Materials Science and Metallurgy, University of Cambridge, 27
Charles Babbage Road, Cambridge CB3 0FS, United Kingdom

###### Abstract

A novel vertical hysteresis loop shift is observed for a bilayer thin film of
Ni80Fe20/SrRuO3 (1:2 thickness ratio) on SrTiO3 substrate, after field cooled
through the Curie temperature $\sim$125 K of SrRuO3, whereas no shift is found
for zero field cooled measurement. The vertical shift is noticed below
$\sim$125 K which increases with decreasing temperature and reaches a maximum
of 33% at 2 K. The vertical shift is modelled closely using micromagnetic
simulations. It is shown that the amount of vertical shift depends on the
competition between the relative thicknesses and the anisotropies of the two
films governed by Spring Thickness law and Imperial Law, respectively. We
propose a generalized model of vertical shift which is applicable for other
material systems, and hence could be informative for use of vertical shift in
future spintronic devices.

††preprint: APS/123-QED

##  I Introduction

Exchange bias (EB) has garnered huge amount of interest due to its tremendous
success in several breakthrough applications in spintronics[1, 2]. Decades of
research have been carried out in search for fascinating EB related phenomena
in several material systems. In conventional EB system, the magnetic
hysteresis loop ($MH$) shifts in the horizontal or field ($H$) axis[3, 4] due
to the interfacial coupling at ferromagnetic (FM)/antiferromagnetic (AFM)
interface. Researchers have been curious to find the $MH$ shift along the
magnetization ($M$) axis, i.e., a vertical bias (VB) in continuation of the
conventional EB for years. This shift allows an extra degree of freedom to be
harnessed, potentially giving additional functionality in future extraordinary
spintronic devices, e.g., multi-state magnetic recording technology, and also
perhaps greater robustness[5, 6, 7]. The vertical shift like behavior has been
observed in few systems including magnetic heterostructures, multiferroic
ceramics, core-shell magnetic nanoparticles, manganite based superlattices,
etc[8, 9, 10, 11, 12, 13]. Some of these reports have indicated that the
interfacial spin glass state arising from the uncompensated spins may account
for such shift[12, 13]. On the other hand, other researches have indicated
that vertical shift could result from variations in the domain structure
leading to uncompensated spins or pinned magnetic moments at the interface as
found in some FM/AFM systems[8, 10, 14, 15, 16]. However, the vertical shift
has been reported even with a compensated G-type AFM[17]. So the actual reason
for the elusive VB is debatable due to the discrepancies so far. No research
has come up with the concrete results with a quantitative model of tuneable VB
and its explanation. Indeed, most of the research to date has reported only an
insignificant and non-tuneable shift, which is not so promising for
applications. Although observations of VB like behavior in some systems have
intrigued the scientific community, a quantitative modelling and an in-depth
understanding for tuneable VB is still outstanding.

In this report, we show a clear VB ($\sim$33%) in a Ni80Fe20/SrRuO3 (SRO) bi-
layer thin film at 2 K after cooling under ±1 T bias field and $MH$ loop
measured with ±0.5 T field range. The zero field cooled (ZFC) measurement does
not show any VB similar to the conventional horizontal EB behavior.
Temperature ($T$) dependent VB measurements are done to show the evolution of
VB. Micromagnetic simulations are also undertaken which are found in good
agreement with experimental results. We show that the VB strongly depends on
the thickness, anisotropy, temperature, etc. By using both experimental and
simulation results, we derive a generalized model of VB to understand its
origin and to explore VB in different material systems for its potential
applications in spintronic devices.

##  II Experimental methods

13-unit cells of epitaxial SRO film ($\sim$5 nm) were deposited on a SrTiO3
(STO) substrate by advanced pulsed laser deposition (PLD) technique. The
thickness was monitored by Reflection high-energy electron diffraction (RHEED)
in PLD. On top of that, a 2 nm Ni80Fe20 layer was deposited by DC/RF
sputtering. The detailed magnetic measurements were carried out in a Quantum
Design MPMS3 magnetometer. When it was required, the sample was demagnetized
at RT by a proper demagnetization protocol and the magnet was reset before the
measurement to ensure that there was no trapped flux present in the sample or
the superconducting coils of the magnetometer[18]. ZFC and positive/negative
field cooled (FC) $MH$ measurements were carried out similar to conventional
EB measurements with both bias fields of ±1 T and ±7 T to observe VB, if any.
For ZFC $MH$ measurement, the sample was cooled down from 350 K to 2 K under
no bias field and then the loop was swept in the loop tracing field range of 0
T→ +0.5 T→ -0.5 T→ +0.5 T while for +FC and -FC $MH$ measurement, the sample
was cooled from 350 K to 2 K under bias field and the loop was swept in the
loop tracing field range of +0.5 T→ -0.5 T→ +0.5 T (under +1 T bias field) and
-0.5 T→ +0.5 T→ -0.5 T (under -1 T bias field), respectively. Moreover, the FC
$MH$ was measured at various temperatures to investigate $T$ dependent VB
study. The amount of the vertical shift of $MH$, $M_{\textrm{VB}}$ was
calculated as ($M_{\textrm{S+}}$ \+ $M_{\textrm{S-}}$)/2, where
$M_{\textrm{S+}}$ ($M_{\textrm{S-}}$) corresponds to the positive (negative)
saturation magnetization ($M_{\textrm{S}}$) value (signs are included). The
magnetization _vs._ temperature ($MT$) measurements for ZFC, FC and remanence
(REM) were carried out to identify the phase transition of the film. For ZFC
$MT$ measurement, the sample was first demagnetized at room temperature,
cooled down from 350 K to 2 K with zero bias field. A field of 1 mT was
applied and the $M$ was measured as a function of $T$ while ramping from 2 K
up to 350 K. Then, similar measurement was done immediately for FC $MT$
measurement while cooling from 350 K to 2 K under 1 mT bias field. Then the
applied field was removed and magnetization was measured while heating from 2
K to 350 K to get REM $MT$.

##  III Simulation methods

In order to verify the experimental results, Object Oriented Micromagnetic
Framework (OOMMF) simulations were performed[19]. A bilayer system of 20 nm
SRO film and 10 nm Ni80Fe20 was simulated with the thickness ratio
($t_{\textrm{R}}$) 2:1 almost tallying with the $t_{\textrm{R}}$ of the two
films taken experimentally. The total area of the atlas was taken as 320 nm x
320 nm. Only in plane 75% of the middle portion of the atlas (Fig. S2) was
considered magnetic to avoid edge effects[20]. Rest of the area of the atlas
was non-magnetic. The cell dimensions were taken as 10 nm x 10 nm x 5 nm along
x, y and z directions, respectively to include exchange interaction effect for
Ni80Fe20 and SRO. These cell dimensions were kept constant all through the
simulation process to maintain the consistency of the results. The typical
simulation parameters were $M_{\textrm{S}}$ of 600 kA/m (Ni80Fe20) and 350
kA/m (SRO), anisotropy constant ($K_{\textrm{1}}$) of 0 kJ/m3 (Ni80Fe20)[21]
and 500 kJ/m3 (SRO)[22] and exchange stiffness constant ($A$) of 13 pJ/m
(Ni80Fe20)[21] and 1.8 pJ/m (SRO)[22]. In our case, a pulse of a very high
field was applied initially for a short time to replicate the FC biasing-like
experiment. Then, $MH$ were simulated in the range of ±0.9 T and ±7 T in a
sequence to understand the field dependence on VB. $M_{\textrm{VB}}$ was
calculated with the remanence magnetization ($M_{\textrm{R}}$) values, i.e.,
($M_{\textrm{R+}}$ \+ $M_{\textrm{R-}}$)/2, where $M_{\textrm{R+}}$
($M_{\textrm{R-}}$) is the positive (negative) $M_{\textrm{R}}$ (signs are
included) correspond to the moment in descending (ascending) branch of the
$MH$ at zero magnetic field. Further, due to the linear change of
$H_{\textrm{C}}$ with changing $t_{\textrm{R}}$ and $K_{\textrm{1}}$ (Fig.
S3), $M_{\textrm{VB}}$ was divided by $H_{\textrm{C}}$ to normalize its
effect. Thickness ($t$) and $K_{\textrm{1}}$ of the films were also varied to
estimate their effect on $M_{\textrm{VB}}$ manifesting a generalized ground.
Since, the total $M_{\textrm{S}}$ can be manipulated by the change of
thickness, we have not considered its effect as both of them are strongly
correlated. Also, the varying $A$ had no noteworthy effect over the vertical
shift of the $MH$ so we kept it constant throughout. We also changed the
anisotropy direction, and found $MH$ for (113) direction to be most comparable
with the experimental results ($MH$ plots for different anisotropy directions
(Fig. S7) are provided in SI).

##  IV Experimental Results

Fig. 1 shows the $MH$ curves for ZFC (grey) and ±FC (±1 T) (blue and red)
measurements for a bilayer system of Ni80Fe20/SRO films on STO (110) substrate
at 2 K. The field range of $MH$ were taken to be ±0.5 T to confirm the
saturation of Ni80Fe20. The grey curve clearly shows no VB for ZFC measurement
whereas a positive (negative) VB is observed for positive (negative) bias
field of ±1 T FC. The ±FC loops shift $\sim$±33% at 2 K compared to the ZFC
loop. The significant decrease of the coercivity ($H_{\textrm{C}}$) from
$\sim$4.7 mT in ZFC to $\sim$2.8 mT in ±FC is also observed (Fig. 1(b)).
Besides, a horizontal EB of magnitude $\sim$1.7 mT is observed for VB curves
from $(H_{\textrm{C+}}+H_{\textrm{C-}})/2$, where $H_{\textrm{C+}}$ and
$H_{\textrm{C-}}$ are the coercivities (signs are included) in the ascending
and descending branches, respectively. Further, similar $MH$ measurements were
done using a high field range of ±7 T for FC (7 T bias) and ZFC (olive green
curve in Fig. 2(a)) clearly depicting no VB whereas shifts are observed for
±0.5 T field range (blue and red curve), also shown in Fig. 1.

Figure 1: (a) ZFC (grey) - ±FC (blue and red) measurements for a bilayer
system of Ni80Fe20/SRO films on STO substrate at 2 K. (b) Zoomed-in version of
the main figure illustrates the $H_{\textrm{C}}$ change. Figure 2: (a)
Experimental result of VB for Ni80Fe20/SRO heterostructure measured at 2 K
with 7 T bias field and zero bias field/±7 T field range $MH$ (green) and ±1 T
bias field/±0.5 T field range $MH$ (blue and red). (b) Simulation result of VB
similar to the experimental results. Green curve for ±7 T loop tracing field
range and blue and red shifted curves for high pulse and ±0.9 T field range.
(Insets) Zoomed positive and negative shifts are explicitly shown. Figure 3:
(a) Trend of asymmetricity of the hysteresis loops at different $T$. Above
$\sim$125 K, no shift is seen. (b) Variation of $M_{\textrm{VB}}$ (%) with $T$
in black circles. FC (red), ZFC (green) and REM (blue) $MT$ measurements show
the phase transition point at the $T_{\textrm{C}}$ of 125 K. The blue curve
refers to the REM _vs._ $T$ plot extracted from the experiments similar to the
$M_{\textrm{VB}}$ (%) _vs._ $T$ curve.

We carried out temperature-dependent measurements to explore the evolution of
VB since the competition among the Zeeman energy ($E_{\textrm{Z}}$),
anisotropy energy ($E_{\textrm{A}}$) and thermal energy ($E_{\textrm{T}}$)
should be considered in accordance with the Stoner-Wohlfarth model[9]. From
the $MH$ curves recorded at different $T$, a decreasing trend of
$M_{\textrm{VB}}$ from low $T$ (2 K) to $\sim$125 K is seen and it becomes
zero at temperatures higher than $\sim$125 K (Fig. 3(a)). The
$M_{\textrm{VB}}$ with different $T$ has been plotted (black circles) in Fig.
3(b). The $MT$ (ZFC-FC-REM) measurement in Fig. 3(b) shows the Curie
temperature ($T_{\textrm{C}}$) of SRO to be $T_{\textrm{C}}$ $\sim$ 125 K[23,
24], the exact temperature below which VB is observed (explicitly illustrated
in Fig. 3(a)). Essentially, SRO becomes FM from paramagnetic (PM) below
$T_{\textrm{C}}$ $\sim$ 125 K while cooling down from 350 K to 2 K whereas
Ni80Fe20 is FM throughout (because of its high $T_{\textrm{C}}$ : 500-800
K)[25, 26] confirmed from its steady positive REM above $T_{\textrm{C}}$
$\sim$ 125 K of SRO. Note that the trend followed by the the REM _vs._ $T$
curve (blue) is similar to the calculated $M_{\textrm{VB}}$ _vs._ $T$ (black
circles) implying the close relation of REM of SRO with the $M_{\textrm{VB}}$.
Both the curves show a steep rise once they reach the $T_{\textrm{C}}$ of SRO
($T_{\textrm{C}}$ $\sim$ 125 K).

##  V Simulation Results

From OOMMF simulations, we found that the loop swept in the range of ±0.9 T
field, achieves VB (blue and red $MH$ in Fig. 2(b)). The positive (negative)
shift of the $MH$ loop was observed for positive (negative) bias field,
similar to the experimental shifts obtained (Fig. 2(a)). On the contrary, the
hysteresis loop swept in the field range ±7 T shows no VB (olive green $MH$
Fig. 2(b)) exactly homogenous to the experimental curves (Fig. 2(a)) achieved.

From the simple Meiklejohn-Bean model for the EB system which states the
reliance of horizontal shift over the $t$ and $M$ of the films in the form of
an energy expression[27], we can intuitively extend the dependence of these
parameters along with the anisotropy over $M_{\textrm{VB}}$. A variation of
these parameters is believed to achieve a tuneable VB, consequently giving a
valuable insight into the mechanism. We have studied the effect of anisotropy
and thickness on $M_{\textrm{VB}}$. The effect of $K_{\textrm{1}}$ on
$M_{\textrm{VB}}$ is shown in Fig. 4(a) where $K_{\textrm{1}}$ is varied from
very low value of $200\;kJ/m^{3}$ to very high value of $2000\;kJ/m^{3}$. This
includes well-known hard materials (e.g., Co, MnAl, FePd, etc.) with their
$K_{\textrm{1}}$ marked in Fig. 4(a) to compare their $M_{\textrm{VB}}$ shift.
$MH$ plots for different $K_{\textrm{1}}$ is given in SI Fig. S5. It is found
that large values of $K_{\textrm{1}}$, i.e., $K_{\textrm{1}}>600\;kJ/m^{3}$ do
not lead to further increase in $M_{\textrm{VB}}$. At a certain point, the
proportion of the two layers reaches the maximum compatibility ratio after
which VB becomes almost constant. Hence, the data points up to
$K_{\textrm{1}}\sim 600\;kJ/m^{3}$ are considered for fitting to estimate the
shift change with respect to anisotropy. These data points have been
successfully fitted (solid green curve) by the Imperial Law (Eq. (1)) which
states an even power dependence of $M_{\textrm{VB}}$ on $K_{\textrm{1}}$[28].

$M_{\textrm{VB}}=A+B{K_{\textrm{1}}}^{2}+CK_{\textrm{1}}^{4}$ (1)

where A, B, and C are the fitting parameters (values are given in SI Table
S1).

In addition, we varied $t_{\textrm{R}}$ (hard/soft) of the two layers to
attain the thickness dependence on $M_{\textrm{VB}}$ by simulations (the $MH$
plots are given in SI Fig. S4). The trend is depicted in the inset of Fig.
4(b). In the same figure, we also did a best fit (solid blue curve) of the
thickness ratio data points (blue spheres) with the Spring Thickness Law (Eq.
(2)) which is analogous to the $t$ dependence of the horizontal EB
version[27].

$M_{\textrm{VB}}=D+E/t_{\textrm{R}}+F/({t_{\textrm{R}}}^{2})$ (2)

where D, E and F are the involved fitting parameters (values are given in SI
Table S1). The curve implicates that the maximum $M_{\textrm{VB}}$ is obtained
for the 2:1 ratio for the bilayer system of SRO and Ni80Fe20 (close to
experimental ratio). The shift increases rapidly to this $t_{\textrm{R}}$
while it goes down and then saturates.

Figure 4: (a) Dependence of $M_{\textrm{VB}}$ over $K_{\textrm{1}}$ fitted by
the solid green curve (Imperial Law fit) from Eq. (1) for the data points
(green spheres) obtained by simulations. Different hard materials are marked
on the curve for their $K_{\textrm{1}}$ accordingly. (b) Dependence of
$M_{\textrm{VB}}$ over $t_{\textrm{R}}$ is shown in inset. The solid blue line
shows the best fitted curve by Eq. (2) (Spring Thickness fit) for the data
points (blue spheres) achieved computationally. Correspondence of the
reciprocal of $T$ fit data with the reciprocal of $t$ fit (Spring Thickness
fit) and $K_{\textrm{1}}$ fit (Imperial Law fit) independently and in combined
form.

Finally, we attempted to understand the relationship between temperature,
anisotropy and thickness of the films which governs the VB. Here, using curve
fittings and analysis of the experimental as well as theoretical results, we
show a coherent dependence of the three parameters on each other. As we can
clearly see from the Fig. 4(b), the reciprocal of the normalized experimental
$M_{\textrm{VB}}$ curve for different temperatures exactly fits with the
curves obtained from the reciprocal of the Spring Thickness fit from the
thickness dependent simulated data, as well as independently with the Imperial
Law fit of the simulated anisotropy data. Surprisingly, it has been seen that
they not only fit the experimental ${M_{\textrm{VB}}}^{-1}$ _vs._ $T$ data
individually but in the combined form as well. This implies that the function
governing the reciprocal of the shift gained from the temperature data is
directly proportional to the product of the function of $M_{\textrm{VB}}$ for
anisotropy (Eq. (1)) and the inverse function of $M_{\textrm{VB}}$ for
thickness (Eq. (2)). We could express this perfectly fitted data in combined
form by Eq. (3).

$M_{\textrm{VB}}^{-1}=G^{\prime}\frac{A^{\prime}+B^{\prime}{T^{2}}+C^{\prime}{T^{4}}}{D^{\prime}+E^{\prime}/T+F^{\prime}/T^{2}}$
(3)

where $A^{\prime}-G^{\prime}$ are the required fitting parameters (provided in
SI Table S1). Hence, it can be inferred that experimental and theoretical
results are absolutely fitting and complementing each other. Therefore, there
is strong correlation between the parameters such as temperature, anisotropy
and the thickness.

##  VI Discussion

To gain an insight into the mechanism of obtaining VB, we explain how a simple
combination of hard and soft magnetic layers results into the novel VB without
any complex phenomena being involved. The loop tracing field is of critical
importance. SRO saturates at a very high field of $\sim$2.5 T[23], whereas
Ni80Fe20 saturates at $\sim$20 mT[29] (Fig. 5(a)). Initially, an +FC (-FC)
bias field aligns both the hard and the soft layer resulting into a net $M$ in
the positive (negative) direction. Further, during field reversal for positive
(negative) VB measurement, due to an insufficient amount of loop tracing field
applied ($H<H_{\textrm{S}}$ (saturation field) of SRO), the effect of moment
imprinted from the hard uncompensated spins of SRO in the non-collinear state
cancels out from the moment of Ni80Fe20 because of the favourable anisotropy
of the hard spins in the direction of FC bias. This decreases the negative
(positive) moment and causes asymmetry in the loop.

Figure 5: (a) Step-like hysteresis for the Ni80Fe20/SRO heterostructure for 7
T bias and no bias field. (b) VB subtracted $M^{\prime}H$ curves illustrating
a change in $H_{\textrm{C}}$ for ZFC-FC hysteresis loops but no horizontal EB.

One another important observation is that the descending (ascending) branch of
the positive (negative) biased $MH$ is exactly aligned over the descending
(ascending) branch of the symmetric ZFC $MH$ near the coercive field region
(where $M$ $\sim$ 0), clearly seen from Fig. 1 (experimental) and Fig. 2(b)
(simulation). However, the corresponding opposite branches are not the parts
of the symmetric loop. Hence, a significant decrease of $H_{\textrm{C}}$ (Fig.
1(b)) from $\sim$4.7 mT to $\sim$2.8 mT is measured experimentally for the
vertically shifted FC hysteresis loops as compared to the non-shifted ZFC loop
and observed in simulations as well (Fig. 2(b)). This observed decrease in
$H_{\textrm{C}}$ can be understood as the pinning of the hard spins
originating from FC in one direction (due to high anisotropy) under low field
loop tracing range (±0.5 T). The frozen hard spins from FC favour the
direction of external field when the loop is swept in the direction of FC
bias. This now requires less field to achieve the required $M$, leading to low
$H_{\textrm{C}}$ as compared to the case when loop tracing field direction
opposes FC bias. Hence, the ascending (descending) branch for positive
(negative) VB loop pulls away from the symmetric hysteresis loop.

Moreover, the observed horizontal shift ($\sim$1.7 mT) in Fig. 1 could be
misunderstood as conventional EB at first instance. But interestingly, after
removing the VB from the ±FC curves and plotting the $MH$, it is observed that
there is no horizontal EB (Fig. 5(b)). Although, the decrease of
$H_{\textrm{C}}$ for FC with respect to ZFC remains. This is similar to the
remanence magnetization shift often observed in EB system due to curvature of
$MH$ at $H=0$. Hence, the VB does not originate from interfacial spin coupling
like conventional EB rather due to the competition between the magnetic
parameters.

Figure 6: $MH$ showing VB for different combination of soft and hard magnetic
materials.

Another interesting observation is that the positive (negative)
$M_{\textrm{S}}$ of positive (negative) shifted curves coincides with the
$+M_{\textrm{S}}$ ($-M_{\textrm{S}}$) of the fully saturated loop (refer Fig.
2) even at the low loop field tracing range. This is because the maximum
saturation ($M_{\textrm{S}}$) for SRO is achievable for $T<T_{\textrm{C}}$
after applying a field (FC) at $T>T_{\textrm{C}}$ as SRO is PM at
$T>T_{\textrm{C}}$ importantly. One can further use this result to exploit the
applications where the maximum saturation needs to be achieved for VB without
using very high fields similar to recent heat-assisted magnetic recording
(HAMR) technology in modern hard disk drives[30].

Nevertheless, volume fraction or $t_{\textrm{R}}$ plays an essential role in
the whole picture. It was already shown, 2:1 is the $t_{\textrm{R}}$ for the
maximum $M_{\textrm{VB}}$ in our case for the corresponding anisotropies of
the two materials. The optimum thickness ratio could vary for different
materials for different anisotropies. As we can see, for our case,
$K_{\textrm{1}}\text{(hard layer)}>600\;kJ/m^{3}$ leads to an ineffective
change in $M_{\textrm{VB}}$ possibly due to the maximum compatibility ratio
attained for the two layers at a certain point. A large increase of the hard
anisotropy with the same $t_{\textrm{R}}$ may imbalance the ratio of the
compensated spins to the uncompensated spins which may not change
$M_{\textrm{VB}}$ much eventually.

Table 1:  Different materials combination for observing VB phenomenon.
No. | Material | Type | $M_{\textrm{S}}$ | $K_{\textrm{1}}$ | $A$ | Shift
---|---|---|---|---|---|---
|  |  | (kA/m) | (kJ/m3) | (pJ/m) | %
1 | SRO | Hard | 350 | 500 | 1.8 | 54.7
| Ni-amorphous | Soft | 493 | 0 | 8.5 |
| alloy |  |  |  |  |
2 | SRO | Hard | 350 | 500 | 1.8 | 17.8
| FeNiB | Soft | 836 | 0 | 8.07 |
| (40/40/20)- |  |  |  |  |
| amorphous alloy |  |  |  |  |
3 | Co | Hard | 1420 | 700 | 30.2 | 1.04
| Ni80Fe20 | Soft | 600 | 0 | 13 |

To reproduce the proof of concept for different materials from our proposed
model, we have also simulated for different combination of materials (e.g.,
Co/Ni80Fe20, SRO/Ni-amorphous alloy, etc.) with their individual
characteristic parameters like $K_{\textrm{1}}$, $M_{\textrm{S}}$ and $A$ and
found similar VB behaviour (refer to Table 1 and Fig. 6). It is clearly
visible that Ni80Fe20/Co has come up with the minimum shift. This will allow
us to choose the right material combination in different cases according to
the application point of view.

##  VII Conclusion

We have demonstrated and modelled a simple bilayer thin film heterostructure
of a soft and a hard layer to obtain a large VB shift. By experiment and
detailed micromagnetic simulations, we have observed that the important
parameters for achieving the large VB are the respective anisotropies and
thickness ratio of the two layers. An optimum amount of loop tracing field
range below $T_{\textrm{C}}$ of the hard material is also essential to achieve
the vertical shift. We have developed a generalized model by fitting the data
incorporating both experimental and simulation results, for the first time to
demonstrate achievable VB by different bilayer material combinations. This VB
system is much simpler, easy to reproduce, tuneable and controllable compared
to alternative systems, including complex FM/AFM spin glasses, or doped
magnetic systems, or systems containing vacancies or topological defects. The
proposed model can also be applicable to the other VB systems with quantified
spin glass, pinned/uncompensated spins or coupled interfaces. The work is an
important step forward to a comprehensive understanding of tuneable VB and
therefore has important implications in future spintronic devices.

## Acknowledgement

MB and TM acknowledge IISER Thiruvananthapuram, India for funding and support
for HPC time at the Padmanabha cluster, IISER Thiruvananthapuram, India. We
are also funded from the Leverhulme Trust grant $\\#$ RPG-2015-017, the EPSRC
grants EP/N004272/1, and EP/M000524/1, the Royal Academy of Engineering Chair
in Emerging Technologies Grant CiET1819$\backslash$24, the EU grant
H2020-MSCA-IF-2016 (745886)-MuStMAM, and the Isaac Newton Trust (RG96474).

Additional Information

This additional Information contains the details of the micromagnetic
simulations performed to effectuate the phenomena of novel vertical bias.

A. Simulation protocols: Initially, a 10 nm layer of Ni80Fe20 film and 20 nm
of SRO film were simulated individually to estimate the correctness of the
$H_{\textrm{C}}$ and the $M_{\textrm{S}}$ of the two films (shown in Fig. 7).
$H_{\textrm{C}}$ for Ni80Fe20 and SRO is noted to be 8 mT and 1.48 T,
respectively which are close to the experimental values[31, 29, 32]. Further,
a bilayer film of 10 nm Ni80Fe20 (top) and 20 nm SRO (bottom) was simulated
for obtaining the VB. Later on, various parameters including $t_{\textrm{R}}$
of the two films, $K_{\textrm{1}}$, etc., were varied to estimate their effect
on $M_{\textrm{VB}}$. Fig. 9 shows the trend of change of $H_{\textrm{C}}$
with respect to $t_{\textrm{R}}$ and $K_{\textrm{1}}$.

Figure 7: Magnetic hysteresis loop for Ni80Fe20 (10 nm) : $H_{\textrm{C}}$ = 8
mT. Magnetic hysteresis loop for SRO (20 nm) : $H_{\textrm{C}}$ = 1.48 T.
Figure 8: Bitmaps used in simulations for the two films: Ni80Fe20 (320 nm x
320 nm x 10 nm) and SRO (320 nm x 320 nm x 20 nm). Figure 9: Variation of
$H_{\textrm{C}}$ with $t_{\textrm{R}}$ and $K_{\textrm{1}}$ of hard layer.

B. Thickness variation: Fig. 10 shows few plots representing varied
$t_{\textrm{R}}$ (other parameters are constant) of the two magnetic layers,
i.e., SRO and Ni80Fe20 to achieve the maximum $M_{\textrm{VB}}$. The black
loop is the fully saturated loop achieved with loop field range ±7 T whereas
red (blue) corresponds to the loops with field sweep range ±0.9 T whose
unsaturated spins have gained the VB. It has been found that the maximum has
been obtained for the $t_{\textrm{R}}$ 2:1 for SRO (hard) and Ni80Fe20 (soft)
layer respectively (highlighted in red text) which is similar to the ratio
taken experimentally (2 nm for Ni80Fe20 and 5 nm for SRO) too. As we go for
very high ratios, the overall shift is decreasing. The increasing effect of
the hard layer for higher $t$ of SRO decreases the effectiveness of the soft
layer. Such an imbalance reduces the $M_{\textrm{VB}}$.

Figure 10: $MH$ showing VB for the bilayer system of Ni80Fe20/SRO for
$t_{\textrm{R}}$ ranging from 1:1 to 6:1. Ratio 2:1 (text highlighted in red)
is approximately same as that of experimental ratio of the two layers.

C. Anisotropy constant variation: $K_{\textrm{1}}$ of the hard layer was
varied while keeping the other parameters constant by the help of simulations.
Some examples are shown in Fig. 11. Almost negligible vertical shift is seen
for the anisotropy of hard layer as 200 kJ/m3 (shown in the first graph of
Fig. 11). Anisotropy constant $K_{\textrm{1}}$=500 kJ/m3 (highlighted in red)
matches most with the experimental results. Increasing $K_{\textrm{1}}$
further doesn’t change $M_{\textrm{VB}}$ much, and hence is not considered.

Figure 11: $MH$ showing VB for the bilayer system of Ni80Fe20/SRO for
different anisotropy constants of the hard layer. Anisotropy of the soft layer
was kept constant as 0 J/m3. Anisotropy constant $K_{\textrm{1}}$ = 500 kJ/m3
(text highlighted in red) is the anisotropy of SRO serving as our main result.

D. Anisotropy direction variation: According to the Stoner- Wohlfarth
model[33], the overall energy of a system is expressed by Eq. (4) as:

$E=K_{\textrm{U}}sin(\phi-\theta)^{2}-\mu MHcos(\phi)$ (4)

where $K_{\textrm{U}}$ is the uniaxial anisotropy constant, $M$ is the
magnetization, $H$ is the external magnetic field, $\theta$ and $\phi$ are the
angles of magnetic field with anisotropy and magnetization, respectively
(refer to Fig. 12.).

Figure 12: Magnetization and applied external magnetic field is in the same
direction implying $\phi=0$. Anisotropy easy axis is at an angle $\theta$ with
the magnetization/magnetic field.

The first and second term in Eq. (4) refer to the anisotropy energy
($E_{\textrm{A}}$) and Zeeman energy ($E_{\textrm{Z}}$) and both the energies
compete with each other. Anisotropy energy is crucially defined by the
direction of the easy axis. If the anisotropy easy axis is in the direction of
applied magnetic field, then it is easy for the spin moments to align in the
common direction which in effect will give a constant $M$ on applying a very
small amount of $H$, giving a square-like hysteresis loop. This is because
both $E_{\textrm{Z}}$ and $E_{\textrm{A}}$ are favouring each other
parallelly. However, when the magnetic field is applied along the hard axis,
both the energies start competing with each other. It can be further
understood from the Eq. S1 depicting an angular dependence between the two
energies.

Figure 13: $MH$ showing $M_{\textrm{VB}}$ for the bilayer system of
Ni80Fe20/SRO for different directions of anisotropy of the hard layer i.e.,
(100), (110) and (113). Anisotropy direction (113) (text highlighted in red)
is the anisotropy direction of SRO taken for the main result.

For our case, the initial $M$ axis was kept same as that of the external $H$
direction, i.e., ((100) ; X axis). We have tried changing the anisotropy
direction such as (100), (110) and (113) (depicted in Fig. 13). The VB shift
is observed for all three directions. However, direction (100) shows the
constant square-like $MH$ whereas direction (113) gives the most comparable
result to the experimental outcomes, therefore we have chosen (113) for
anisotropy direction as our most significant result.

Table 2:  Fitting parameters of modelling for Eq. 1, 2 and 3 respectively in the main text.  Fitting parameters | Equation 1 | Equation 2 | Equation 3
---|---|---|---
$A/A^{\prime}$ | 0 | - | -0.00642
$B/B^{\prime}$ | 2.10736 x 10-4 | - | 229.36768
$C/C^{\prime}$ | 2.85514 x 10-9 | - | -0.00131
$D/D^{\prime}$ | - | 96.8043 | 1.96123
$E/E^{\prime}$ | - | 560.341 | 2.23799 x 107
$F/F^{\prime}$ | - | -593.348 | 57011
$G^{\prime}$ | - | - | 0.05486

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