# Superconductivity with and without glue and the role of the double-occupancy
forbidding constraint in the $t$-$J$-$V$ model

Luciano Zinni, Matías Bejas, and Andrés Greco  Facultad de Ciencias Exactas,
Ingeniería y Agrimensura and Instituto de Física Rosario (UNR-CONICET).
Avenida Pellegrini 250-2000 Rosario-Argentina.

###### Abstract

The occurrence of retarded (with glue) and unretarded (without glue) pairing
is thoroughly discussed in cuprates. We analyze some aspects of this problem
in the context of the $t$-$J$-$V$ model in a large-$N$ approximation. When
$1/N$ renormalizations are neglected the mean-field result is recovered, where
the unretarded $d$-wave superconducting pairing triggered by the spin-exchange
interaction $J$ is obtained. However, the presence of a non-negligible
nearest-neighbors Coulomb interaction $V({\bf q})$ kills superconductivity. If
the non-double-occupancy constraint and its fluctuations are considered, the
situation changes drastically. In this case, $V({\bf q})$ is screened making
$d$-wave superconductivity very robust. In addition, we show that the early
proposal for the presence of an unretarded pairing contribution triggered by
the spin-exchange interaction $J$ can be discussed in this context.

##  I Introduction

The origin of superconductivity in high-$T_{c}$ cuprates is under an intense
debate since its discovery in 1986. Not only the high value of the
superconducting critical temperature $T_{c}$ is surprising, but these
materials are also anisotropic and the metallic and superconducting properties
show a two-dimensional character. The phase diagram in the temperature and
doping plane of these materials shows unconventional characteristics, as the
dome-shaped behavior of $T_{c}$ against doping in the proximity to the
antiferromagnetic insulator and the pseudogap phase at low doping (see Ref.
[Keimer et al., 2015] for a review). In addition to these features, the
superconducting gap has $d$-wave symmetryKeimer et al. (2015); Timusk and
Statt (1999). All members of the cuprate family share similar characteristics,
suggesting the existence of a universal physics.

Phenomenological theories where pairing is due to antiferromagnetic
fluctuationsMonthoux et al. (1991); Monthoux and Pines (1993) were proposed
for explaining the $d$-wave symmetry of the superconducting gap in the
proximity to antiferromagnetism. In this scenario, the two-dimensional
antiferromagnetic fluctuations play the role of a retarded glue, as phonons in
conventional low-temperature superconductors.

In the early times, the Hubbard and the $t$-$J$ models were recognizedAnderson
(1987) as minimal microscopic models for cuprates. The Hubbard model treated
in the framework of a weak coupling random phase approximation shows $d$-wave
superconductivityBulut and Scalapino (1996), where the effective pairing
interaction is mediated by the dynamical spin susceptibility which acts as a
pairing glue. In this approach, the nearest-neighbors Coulomb interaction,
$V({\bf q})=2V[\cos(q_{x})+\cos(q_{y})]$, which is expected to be non-
negligible in cupratesFeiner et al. (1996), affects superconductivity because
it has a $d$-wave repulsive projection. It is therefore important to
understand why $T_{c}$ remains large even if a nearest-neighbors Coulomb
interaction is present. The same effect of $V({\bf q})$ is expected in the
antiferromagnetic phenomenological theoriesMonthoux et al. (1991); Monthoux
and Pines (1993).

The study of $d$-wave superconductivity and the role of the nearest-neighbors
Coulomb interaction in the Hubbard model is hugeSénéchal et al. (2013);
Esirgen et al. (1999); Husemann and Metzner (2012); Jiang et al. (2018). In
Ref. [Sénéchal et al., 2013] it was shown that $d$-wave superconductivity in
the Hubbard model is almost unaffected by $V({\bf q})$ if the strong coupling
limit is properly treated. In addition, since the two-dimensional Hubbard
model reduces to the $t$-$J$ model in the large-$U$ limit Chao et al. (1978),
the question about the role of $V({\bf q})$ on superconductivity is also of
interest in the $t$-$J$ model. On the other hand, retarded (with glue) and
unretarded (without glue) paringAnderson (2007); Maier et al. (2008) is under
discussion in the $t$-$J$ model. While in Ref. [Maier et al., 2008] pairing
was discussed as composed by a retarded and an unretarded contributions, in
Ref. [Anderson, 2007] only an unretarded pairing was considered as the
relevant one.

At the mean-field level, the $t$-$J$ model shows $d$-wave
superconductivityBaskaran et al. (1993) arising from the unretarded exchange
interaction $J[\cos(q_{x})+\cos(q_{y})]$, where $J$ is the spin-exchange
coupling. Since the exchange interaction has the same form of the nearest-
neighbors Coulomb interaction, the last one is, in principle, detrimental to
superconductivity. Using a large-$N$ approach on the $t$-$J$-$V$ model, in
this paper we discuss superconductivity and the role of the nearest-neighbors
Coulomb interaction $V({\bf q})$. When the local constraint that prohibits
double occupancy is not included superconductivity is strongly affected by
$V({\bf q})$, even for $V$ of the order of $J$. Including the constraint
$d$-wave superconductivity is robust against $V({\bf q})$, even for $V\gg J$.
We also found that the leading contribution to superconductivity is mainly
provided by the unretarded exchange, however, this contribution is efficient
only if the constraint is properly included. In Sec. II we present a summary
of the formalism, in Sec. III the results and discussions, and in Sec. IV the
conclusions.

##  II Model and summary of the formalism

As a minimal model, we study the $t$-$J$-$V$ model on a square lattice,

$H=-\sum_{\langle
i,j\rangle,\sigma}t_{ij}\tilde{c}^{\dagger}_{i\sigma}\tilde{c}_{j\sigma}+\sum_{\langle
i,j\rangle}J_{ij}\left(\vec{S}_{i}\cdot\vec{S}_{j}-\frac{1}{4}n_{i}n_{j}\right)+\sum_{{\langle
i,j\rangle}}V_{ij}n_{i}n_{j}\,,$ (1)

where $\tilde{c}^{\dagger}_{i\sigma}$ ($\tilde{c}_{i\sigma}$) is the creation
(annihilation) operator of electrons with spin $\sigma(=\uparrow,\downarrow)$
in the Fock space without double occupancy at any site,
$n_{i}=\sum_{\sigma}\tilde{c}^{\dagger}_{i\sigma}\tilde{c}_{i\sigma}$ is the
electron density operator, $\vec{S}_{i}$ is the spin operator. The hopping
(spin exchange) $t_{ij}$ ($J_{ij}$) takes the value $t$ ($J$) between the
first nearest-neighbors sites. $V_{ij}$ is a nearest-neighbors Coulomb
interaction with strength $V$.

It is non-trivial to study the $t$-$J$ model because of the local constraint
that prohibits the double occupancy at any site. In addition, the operators
involved in the $t$-$J$ model are Hubbard operatorsHubbard (1963) which
satisfy non-standard commutation rules. We employ here a large-$N$ technique
based on a path integral representation in terms of the Hubbard operators (see
Refs. [Foussats and Greco, 2002, 2004] and references therein). In the
large-$N$ scheme, the number of spin components is extended from 2 to $N$ and
the physical quantities are computed in powers of $1/N$. In what follows the
spin index $\sigma$ is called $p$.

In the framework of the large-$N$ path integral approach, the $t$-$J$ model is
mapped to an effective theory described in terms of fermions, bosons, and
their mutual interactionsFoussats and Greco (2004).

a) Fermions: We obtain a fermionic propagator [solid line in Fig. 1(a)],

$G^{(0)}_{pp^{\prime}}({\bf k},{\rm
i}\omega_{n})=\frac{\delta_{pp^{\prime}}}{{\rm i}\omega_{n}-\varepsilon_{{\bf
k}}}\,,$ (2)

with the electronic dispersion

$\varepsilon_{{\bf
k}}=-2\left(t\frac{\delta}{2}+\Delta\right)\left[\cos\left(k_{x}\right)+\cos\left(k_{y}\right)\right]-\mu\,.\\\
$ (3)

For a given doping $\delta$, the chemical potential $\mu$ and $\Delta$ are
determined self-consistently by solving

$1-\delta=\frac{2}{N_{s}}\sum_{{\bf k}}n_{F}\left(\varepsilon_{\bf
k}\right)\;,$ (4)

and

$\displaystyle{}\Delta=\frac{J}{4N_{s}}\sum_{{\bf
k}}\left[\cos\left(k_{x}\right)+\cos\left(k_{y}\right)\right]n_{F}\left(\varepsilon_{\bf
k}\right)\;,$ (5)

where $n_{F}$ is the Fermi function and $N_{s}$ is the total number of lattice
sites. The momentum ${\bf k}$ is measured in units of the inverse of the
lattice constant. In Eq. (2) $\omega_{n}$ is a fermionic Matsubara frequency.
The Green’s function $G^{(0)}_{pp^{\prime}}({\bf k},{\rm i}\omega_{n})$ is
$O(1)$ in the context of the $1/N$ expansion.

In the present formalism, the spin-exchange term or $J$-term of the
$t$-$J$-$V$ model [Eq. (1)] is treated by introducing a bond-field variable
that describes charge fluctuations on the bond connecting nearest-neighbors
sites along the $x$\- and $y$-directions. $\Delta$ is the static mean-field
value of this bond field. Although the electronic dispersion [Eq. (3)] looks
like that in a free electron system, the hopping integral $t$ is renormalized
by doping $\delta$ because of electron-correlation effects. In addition, there
is a contribution $\Delta$ which depends on $J$.

b) Bosons: We define a six-component bosonic field

$\delta X^{a}=(\delta
R\;,\;\delta{\lambda},\;r^{x},\;r^{y},\;A^{x},\;A^{y})\,,$ (6)

where $\delta R$ describes the fluctuations of the number of holes at a given
site, thus it is related to on-site charge fluctuations, $\delta\lambda$ is
the fluctuation of the Lagrange multiplier introduced to enforce the
constraint that prohibits the double occupancy at a given site, and $r^{x}$
and $r^{y}$ ($A^{x}$ and $A^{y}$) describe fluctuations of the real
(imaginary) part of the bond field coming from the $J$-term.

The $6\times 6$ bare bosonic propagator associated with $\delta X^{a}$ [dashed
line in Fig. 1(a)], connecting two generic components $a$ and $b$ is

$\displaystyle\left[D^{(0)}_{ab}({\bf
q},\mathrm{i}\nu_{n})\right]^{-1}=N\left(\begin{array}[]{cccccc}\frac{\delta^{2}}{2}\left(V-\frac{J}{2}\right)\left[\cos\left(q_{x}\right)+\cos\left(q_{y}\right)\right]&\frac{\delta}{2}&0&0&0&0\\\
\frac{\delta}{2}&0&0&0&0&0\\\ 0&0&\frac{4}{J}\Delta^{2}&0&0&0\\\
0&0&0&\frac{4}{J}\Delta^{2}&0&0\\\ 0&0&0&0&\frac{4}{J}\Delta^{2}&0\\\
0&0&0&0&0&\frac{4}{J}\Delta^{2}\end{array}\right)\;,$ (13)

where ${\bf q}$ and $\nu_{n}$ are the momentum and bosonic Matsubara
frequencies, respectively. The factor $N$ shows that the $6\times 6$ bosonic
propagator $D^{(0)}_{ab}$ is $O(1/N)$, and it is frequency independent. The
element $(1,1)$ in Eq. (13) carries the information of
$\frac{1}{4}J_{ij}n_{i}n_{j}$ and $V_{ij}n_{i}n_{j}$ of Eq. (1), while the
information of $J_{ij}\vec{S}_{i}\cdot\vec{S}_{j}$ is contained in the
elements $(a,a)$ with $a=3$-$6$.

c) Interaction vertices: For computing quantities up to $O(1/N)$ the present
large-$N$ scheme leads to three-legs and four-legs vertices [Fig. 1(a)].

The three-legs vertex

$\displaystyle\Lambda^{pp^{\prime}}_{a}=$ $\displaystyle(-1)\left[\frac{{\rm
i}}{2}(\omega_{n}+{\omega^{\prime}}_{n})+\mu+2\Delta\sum_{\eta}\cos\left(k_{\eta}-\frac{q_{\eta}}{2}\right)\cos\frac{q_{\eta}}{2};\;1\;;-2\Delta\cos\left(k_{x}-\frac{q_{x}}{2}\right)\;;\right.$
(14)
$\displaystyle\left.-2\Delta\cos\left(k_{y}-\frac{q_{y}}{2}\right);\;2\Delta\sin\left(k_{x}-\frac{q_{x}}{2}\right);\;2\Delta\sin\left(k_{y}-\frac{q_{y}}{2}\right)\right]\delta^{pp^{\prime}}\,,$

where $\eta=x$, $y$, represents the interaction between two fermions and one
boson.

The four-legs vertex $\Lambda^{pp^{\prime}}_{ab}$ represents the interaction
between two fermions and two bosons. $\Lambda^{pp^{\prime}}_{ab}$ fulfills the
symmetry of $\Lambda^{pp^{\prime}}_{ab}=\Lambda^{pp^{\prime}}_{ba}$, and the
only elements different from zero are:

$\displaystyle\Lambda^{pp^{\prime}}_{\delta R\delta R}=$
$\displaystyle\left[\frac{{\rm
i}}{2}(\omega_{n}+{\omega^{\prime}}_{n})+\mu\right.$ (15)
$\displaystyle\left.+\Delta\sum_{\eta}\cos\left(k_{\eta}-\frac{q_{\eta}+q^{\prime}_{\eta}}{2}\right)\;\left(\cos\frac{q_{\eta}}{2}\;\cos\frac{q^{\prime}_{\eta}}{2}\;+\;\cos\frac{q_{\eta}+q^{\prime}_{\eta}}{2}\right)\right]\delta^{pp^{\prime}}\,,$
$\Lambda^{pp^{\prime}}_{\delta
R\delta\lambda}=\frac{1}{2}\delta^{pp^{\prime}}\,,$ (16)
$\Lambda^{pp^{\prime}}_{\delta
R\,r^{\eta}}=-\Delta\cos\left(k_{\eta}-\frac{q_{\eta}+q^{\prime}_{\eta}}{2}\right)\cos\frac{q^{\prime}_{\eta}}{2}\,\delta^{pp^{\prime}}\,,$
(17)

and

$\Lambda^{pp^{\prime}}_{\delta
R\,A^{\eta}}=\Delta\sin\left(k_{\eta}-\frac{q_{\eta}+q^{\prime}_{\eta}}{2}\right)\cos\frac{q^{\prime}_{\eta}}{2}\,\delta^{pp^{\prime}}\,.$
(18)

Each vertex conserves momentum and energy and it is $O(1)$. For readability
reasons we drop the frequencies and momenta in the left hand side of the
definitions of the three- and four-legs vertices $\Lambda^{pp^{\prime}}_{a}$
and $\Lambda^{pp^{\prime}}_{ab}$ [see Fig. 1(a) for the frequency and momentum
dependence].

By using the propagators and vertices summarized in Fig. 1(a) we can draw
Feynman diagrams as usual.

Figure 1: (a) Summary of the Feynman rules. Solid line represents the
fermionic propagator $G_{pp^{\prime}}^{(0)}$. Dashed line represents the
$6\times 6$ bosonic propagator $D_{ab}^{(0)}$. $\Lambda^{pp^{\prime}}_{a}$ and
$\Lambda^{pp^{\prime}}_{ab}$ represent the interaction between two fermions
and one and two bosons, respectively. (b) Diagrammatic representation of the
Dyson equation. (c) The two different contributions to the irreducible boson
self-energy. (d) Effective interaction between fermions. Looking at the order
of the propagators and vertices we see that $V_{{\rm eff}}$ is $O(1/N)$, thus
superconductivity arises at $O(1/N)$ in this large-$N$ scheme.

From the Dyson equation [Fig. 1(b)], the bosonic bare propagator
$D^{(0)}_{ab}$ is renormalized at $1/N$ order,

$[D_{ab}({\bf q},\mathrm{i}\nu_{n})]^{-1}=[D^{(0)}_{ab}({\bf
q},\mathrm{i}\nu_{n})]^{-1}-\Pi_{ab}({\bf q},\mathrm{i}\nu_{n})\,,$ (19)

where the $6\times 6$ boson self-energy matrix $\Pi_{ab}$ [Fig. 1(c)] is:

$\displaystyle\Pi_{ab}({\bf q},\mathrm{i}\nu_{n})=-\frac{N}{N_{s}}\sum_{{\bf
k}}h_{a}({\bf k},{\bf q},\varepsilon_{\bf k}-\varepsilon_{{\bf k}-{\bf
q}})\frac{n_{F}(\varepsilon_{{\bf k}-{\bf q}})-n_{F}(\varepsilon_{\bf
k})}{\mathrm{i}\nu_{n}-\varepsilon_{\bf k}+\varepsilon_{{\bf k}-{\bf
q}}}h_{b}({\bf k},{\bf q},\varepsilon_{\bf k}-\varepsilon_{{\bf k}-{\bf q}})$
$\displaystyle\hskip
71.13188pt-\delta_{a\,1}\delta_{b\,1}\frac{N}{N_{s}}\sum_{\bf
k}\frac{\varepsilon_{\bf k}-\varepsilon_{{\bf k}-{\bf
q}}}{2}n_{F}(\varepsilon_{\bf k})\;,$ (20)

with $h_{a}$ given by

$\displaystyle h_{a}({\bf k},{\bf q},\nu)=$
$\displaystyle\left\\{\frac{2\varepsilon_{{\bf k}-{\bf
q}}+\nu+2\mu}{2}+2\Delta\left[\cos\left(k_{x}-\frac{q_{x}}{2}\right)\cos\left(\frac{q_{x}}{2}\right)+\cos\left(k_{y}-\frac{q_{y}}{2}\right)\cos\left(\frac{q_{y}}{2}\right)\right];1;\right.$
$\displaystyle\left.-2\Delta\cos\left(k_{x}-\frac{q_{x}}{2}\right);-2\Delta\cos\left(k_{y}-\frac{q_{y}}{2}\right);2\Delta\sin\left(k_{x}-\frac{q_{x}}{2}\right);2\Delta\sin\left(k_{y}-\frac{q_{y}}{2}\right)\right\\}\,.$
(21)

The vertices $\Lambda^{pp^{\prime}}_{a}$ and $\Lambda^{pp^{\prime}}_{ab}$ not
only represent interactions from the Hamiltonian Eq. (1) but, as they come
from the path integral, they contain also contributions from the algebra of
the Hubbard operators and the non-double-occupancy constraint, which introduce
a frequency dependence. Due to this frequency dependence, the computation of
the first and second diagrams in Fig. 1(c) leads to finite and infinite
contributions. However, the ghost fields from the Jacobian in the path
integral give rise to terms that cancel exactly these infinitiesFoussats and
Greco (2002, 2004).

The $6\times 6$ dressed bosonic propagator $D_{ab}$ contains all possible
charge fluctuations of the $t$-$J$ model on the square lattice, and all are
treated on equal footingBejas et al. (2012). The large-$N$ approach weakens
the effective spin interaction compared with the one associated with the
charge degrees of freedom. $D_{ab}$ with $a,b=1,2$ describes on-site charge
fluctuations associated to $\delta R$ and $\delta{\lambda}$. The presence of
$\delta{\lambda}$ indicates that the non-double-occupancy constraint and its
fluctuations are taken into account in the calculation. The element $(1,1)$ of
$D_{ab}$ is related to the usual charge-charge correlation function Foussats
and Greco (2002). $D_{22}$ and $D_{12}$ correspond to fluctuations associated
with the non-double-occupancy condition and correlations between non-double-
occupancy condition and charge-density fluctuations, respectively. We call
this case as on-site charge sector or the $2\times 2$ sector. If $a,b=3$-$6$,
$D_{ab}$ describes bond-charge fluctuations associated to $r^{x}$, $r^{y}$,
$A^{x}$, and $A^{y}$. We call this case as the bond-charge sector or the
$4\times 4$ sector. $D_{ab}$ also contains the mixing of both sectors, however
it was shown that the coupling between on-site and bond-charge fluctuations is
negligibleBejas et al. (2017). If $J=0$ the $6\times 6$ $D_{ab}$ reduces to
the $2\times 2$ sector, and only on-site charge fluctuations are involved.

The superconducting effective interaction between fermions, $V_{{\rm
eff}}({\bf k},{\bf k^{\prime}};\omega_{n},\omega_{n}^{\prime})$, can be
calculated using the diagram in Fig. 1(d), which shows that in the present
theory pairing is mediated by charge fluctuations contained in $D_{ab}$. Note
that we can also draw a diagram containing two vertices
$\Lambda^{pp^{\prime}}_{ab}$ and two bosonic propagators $D_{ab}$, however,
this contribution is omitted because it is $O(1/N^{2})$. The analytical
expression for the effective interaction is

$V_{{\rm eff}}({\bf k},{\bf
k^{\prime}};\omega_{n},\omega_{n}^{\prime})=\Lambda_{a}D_{ab}({\bf k}-{\bf
k^{\prime}},\omega_{n}-\omega_{n}^{\prime})\Lambda_{b}\,,$ (22)

where $\Lambda_{a}$ and $\Lambda_{b}$ are the three-legs vertices from Eq.
(14) with $p=p^{\prime}$.

We use a weak coupling approximation to compute the effective couplings
$\lambda_{i}$ in the different pairing channels or irreducible representations
of the order parameter on the square lattice, $i$
$[i=(d_{x^{2}-y^{2}},d_{xy},s^{\prime},p)]$,

$\displaystyle\lambda_{i}=\frac{1}{(2\pi)^{2}}\frac{\int(d{\bf k}/|v_{\bf
k}|)\int(d{\bf k^{\prime}}/|v_{\bf k^{\prime}}|)g_{i}({\bf k^{\prime}})V_{{\rm
eff}}({\bf k^{\prime}},{\bf k})g_{i}({\bf k})}{\int(d{\bf k}/|v_{\bf
k}|)g_{i}({\bf k})^{2}}\,,$ (23)

where the functions $g_{i}({\bf k})$ encode the different pairing symmetries,
$g_{d_{x^{2}-y^{2}}}({\bf k})=\cos(k_{x})-\cos(k_{y})$, $g_{d_{xy}}({\bf
k})=\cos(k_{x})\cos(k_{y})$, $g_{s^{\prime}}({\bf
k})=\cos(k_{x})+\cos(k_{y})$, and $g_{p}({\bf k})=\sin(k_{x})$. $v_{\bf k}$ is
the quasiparticle velocity at momentum ${\bf k}$. The integrations are
restricted to the Fermi surface, i.e., ${\bf k}$ and ${\bf k^{\prime}}$ run
over Fermi surface momenta and ${\rm i}\omega_{n}={\rm
i}\omega^{\prime}_{n}=0$. $\lambda_{i}$ measures the strength of the
interaction between electrons at the Fermi surface in a given symmetry channel
$i$. If $\lambda_{i}>0$, electrons are repelled hence, superconductivity is
only possible when $\lambda_{i}<0$. The critical temperatures, $T_{c}$, can
then be estimated using a BCS expression:
$T_{ci}=1.13\omega_{0}\exp(-{1/|\lambda_{i}|})$, where $\omega_{0}$ is a
suitable cutoff frequency which encodes retardation effects. If $\lambda_{i}$
is negligible, superconductivity is unexpected, no matter the value of
$\omega_{0}$. Although it is an approximation, the weak coupling scheme gives
a way to select, in principle, the dominant pairing channels from all
different contributions independently of their retarded or unretarded nature.
It was introduced in retarded (with glue) cases as the electron-phonon
oneRickayzen (1980), where $\lambda$ is the dimensionless coupling strength
due to the electron-phonon interaction. This approach was also used for spin-
fluctuation interaction in the context of cupratesMonthoux et al. (1991). The
fact that we calculate on the Fermi surface in Eq. (23) does not invalidate
the study of retarded interactions. Obtaining an accurate value of $T_{c}$
requires considering retardation effects in more detail, but that is not our
aim. We study the main tendencies to superconductivity and from where they
arise by computing the coupling strength $\lambda$ of each contribution.

##  III Results and discussions

We chose $J=0.3$, $T=0$, and $0\leq V\ll V_{c}$, where $V_{c}$ is the onset of
the instability to a checkerboard charge density waveFoussats and Greco
(2004); Hoang and Thalmeier (2002). Energy is given in units of $t$. There is
no tendency to superconductivity, i.e., $\lambda_{i}>0$, for any pairing
channel except for $d_{x^{2}-y^{2}}$ for $\delta<0.5$. Thus, in the following
we focus only on the $d_{x^{2}-y^{2}}$ channel. For simplicity we call
$\lambda_{d_{x^{2}-y^{2}}}$ ($d_{x^{2}-y^{2}}$-wave) as $\lambda$ ($d$-wave)
in what follows.

Figure 2: (Color online) The superconducting coupling $\lambda$ and
$\lambda^{(0)}$ versus doping $\delta$ for $V=0$ and $V=0.3$. Inset: The
superconducting coupling $\lambda$ and $\lambda^{(0)}$ versus $V$ for
$\delta=0.25$. For this $\delta$, $V_{c}\sim 1.9$.

Using the $6\times 6$ $D_{ab}$ in Eq. (22) we compute $\lambda$ as a function
of $\delta$ for $V=0$ and $V=0.3$. Figure 2 shows that, although $V({\bf q})$
has a repulsive $d$-wave projection, $\lambda$ is almost unaffected by $V$. In
addition, $d$-wave superconductivity enhances with decreasing doping. This
result is contrary to other results that suggest that superconductivity is
killed by $V$ already for values of the order of $J$ (Refs. [Plakida and
Oudovenko, 2014; R. Zeyher and A. Greco, 1998; Zhou and Wang, 2004]).

Using the $6\times 6$ $D_{ab}^{(0)}$ instead of $D_{ab}$ in Eq. (22) $V_{{\rm
eff}}$ is given byneg

$\displaystyle V_{{\rm eff}}^{(0)}({\bf k},{\bf
k^{\prime}};\omega_{n},\omega_{n}^{\prime})=$
$\displaystyle\left(\frac{J}{2}-V\right)\left[\cos\left(k_{x}-k^{\prime}_{x}\right)+\cos\left(k_{y}-k^{\prime}_{y}\right)\right]+$
$\displaystyle\frac{J}{2}\left[\cos\left(k_{x}-k^{\prime}_{x}\right)+\cos\left(k_{y}-k^{\prime}_{y}\right)\right].$
(24)

Note that $V_{{\rm eff}}^{(0)}$ is frequency independent. The first term in
$V_{{\rm eff}}^{(0)}$ containing $J$ and $V$ comes from the $2\times 2$ sector
of the $t$-$J$ model. The second term originates from the $4\times 4$ sector
which is proportional to $J$. We call attention that considering $\tilde{c}$
as usual fermions in Eq. (1) $V_{{\rm eff}}^{(0)}$ can be recovered as a mean-
field approximation of the $t$-$J$ model.

Using $V_{{\rm eff}}^{(0)}$ the corresponding $\lambda^{(0)}$ can be computed.
In contrast to $\lambda$, while superconductivity is robust for $V=0$,
$\lambda^{(0)}$ vanishes for $V=0.3$ (see Fig. 2). These results show that,
among other effects discussed later, the renormalization of $D^{(0)}_{ab}$ by
the $6\times 6$ boson self-energy $\Pi_{ab}$ [Eq. (19)] screens out the effect
of $V$. The inset in Fig. 2 shows $\lambda$ and $\lambda^{(0)}$ versus $V$ for
$\delta=0.25$. These results for $\lambda$ indicate that superconductivity is
mostly unaffected by the Coulomb interaction even for $V\gg J$ when the full
dressed $D_{ab}$ bosonic propagator is considered. On the other hand, for the
case of the bare propagator $D^{(0)}_{ab}$ no superconductivity occurs for
$V>0.3$, as can be expected from Eq. (24).

One difference between $\lambda$ and $\lambda^{(0)}$ for $V=0$ is that while
$\lambda^{(0)}$ smoothly decreases with decreasing doping, $\lambda$ tends to
large negative values at $\delta\sim 0.13$. This behavior for $\lambda$ can be
explained in the context of the flux phase instability, which occurs at a
critical doping $\delta_{c}\sim 0.13$ for present parametersBejas et al.
(2012). See the Appendix for details about the flux phase. Since $\lambda$ is
calculated on the Fermi surface, i.e., $\omega_{n}=\omega^{\prime}_{n}=0$,
when approaching $\delta_{c}$ the effective superconducting coupling $\lambda$
tunes the instability and diverges.

Figure 3: (Color online) (a) $\lambda^{{\rm Ch-FP}}$ versus doping for $V=0$
and $V=0.3$. $\lambda$ from Fig. 2 is included for comparison. (b)
$\lambda^{{\rm Ch}}$ for $V=0$ and $V=0.3$, and $\lambda^{\rm J}$ versus
doping. (c) $\lambda^{{\rm FP}}$ and $\lambda^{\rm J}$ versus $\delta$. Inset:
$\lambda^{{\rm FP}}$ and $\lambda^{\rm J}$ versus $\delta$ for $\Gamma=0.05$.

It was shown that $\lambda$, which includes the bosonic self-energy
$\Pi_{ab}$, is robust against $V$, but such robustness is not present in the
case of $\lambda^{(0)}$. Next, we discuss which are the relevant components of
$\Pi_{ab}$ that lead to the different behavior between $\lambda$ and
$\lambda^{(0)}$. Since the flux phase belong to the $5$-$6$ sector of $D_{ab}$
(see the Appendix), we calculate $\lambda$ including only the $2\times 2$
sector $\Pi_{11}$, $\Pi_{12}$, $\Pi_{22}$, and the flux sector $\Pi_{55}$,
$\Pi_{56}$ and $\Pi_{66}$ in the Dyson equation [Eq. (19)], i.e., leaving the
other components of $\Pi_{ab}$ as zero. We call this $\lambda^{{\rm Ch-FP}}$.
Figure 3(a) shows $\lambda^{{\rm Ch-FP}}$ for $V=0$ and $V=0.3$. For
completeness, in the figure we included the results of $\lambda$ for the full
$6\times 6$ case of Fig. 2. These results show that $\Pi_{11}$, $\Pi_{12}$,
$\Pi_{22}$, $\Pi_{55}$, $\Pi_{56}$, and $\Pi_{66}$ are the most important
components of the bosonic self-energy $\Pi_{ab}$ since they capture the same
$\lambda$ behavior as using the full $6\times 6$ $\Pi_{ab}$. It is important
to note that the inclusion of $\Pi_{ab}$ in the Dyson equation introduces a
frequency dependence in the dressed bosonic propagator $D_{ab}$, i.e., the
effective interactions are retarded in contrast to the unretarded interactions
from the undressed $D^{(0)}_{ab}$. This point is important for later
discussions.

Next we analyze the influence of the $2\times 2$ on-site-charge and FP sectors
separately.

Considering only $\Pi_{11}$, $\Pi_{12}$, and $\Pi_{22}$ in the Dyson equation
the effective paring interactions can be written as

$\displaystyle V_{eff}^{({\rm Ch})}({\bf k},{\bf
k^{\prime}};\omega_{n},\omega_{n}^{\prime})=$
$\displaystyle\frac{-2\Lambda_{1}(\delta-\Pi_{12})+\Lambda_{1}^{2}\Pi_{22}-\\{\frac{\delta^{2}}{2}(2V-J)F_{{\bf
k},{\bf
k}^{\prime}}-\Pi_{11}\\}}{(\delta-\Pi_{12})^{2}+\Pi_{22}\\{\frac{\delta^{2}}{2}(2V-J)F_{{\bf
k},{\bf k}^{\prime}}-\Pi_{11}\\}}+$
$\displaystyle\frac{J}{2}\left[\cos\left(k_{x}-k^{\prime}_{x}\right)+\cos\left(k_{y}-k^{\prime}_{y}\right)\right],$
(25)

where $F_{{\bf k},{\bf
k}^{\prime}}=\cos\left(k_{x}-k^{\prime}_{x}\right)+\cos\left(k_{y}-k^{\prime}_{y}\right)$.
Using Eq. (25), we compute $\lambda^{{\rm Ch}}$. Figure 3(b) shows results for
$\lambda^{{\rm Ch}}$ versus $\delta$ for $V=0$ and $V=0.3$. It can be seen
that $\lambda^{{\rm Ch}}$ is almost unaffected by $V$ showing that the Coulomb
repulsion is indeed screened by the $\Pi_{ab}$ components that belong to the
$2\times 2$ on-site charge sector. The second term on the right hand side of
Eq. (25) is the same as in Eq. (24). We call $\lambda^{J}$ the contribution
from this term and its behavior is shown in Fig. 3(b). The fact that the three
curves are nearly coincident give us the clue that the components of the
$2\times 2$ on-site charge sector of $\Pi_{ab}$ screen the first term of Eq.
(24) and consequently, only the effective
$(J/2)\,\left[\cos\left(k_{x}-k^{\prime}_{x}\right)+\cos\left(k_{y}-k^{\prime}_{y}\right)\right]$
interaction from the $4\times 4$ sector survives.

When the $t$-$J$ model is treated at mean-field level, superconductivity is
expected to be triggered by the exchange term $J({\bf k}-{\bf k^{\prime}})$.
However, superconductivity is killed by a small nearest-neighbors Coulomb
interaction. When the non-double-occupancy constraint is treated properly, the
effect of the Coulomb interaction is screened. Thus, present results show a
clear difference between a treatment of superconductivity at the mean-field
level and a treatment in strong coupling. We think that our results support
the early point of viewAnderson (2007) that superconductivity in cuprates has
a contribution from the unretarded $J({\bf k}-{\bf k^{\prime}})$ term, but we
claim that for such a pairing to be realized the non-double-occupancy
constraint should be treated beyond mean-field.

The screening effect from the $2\times 2$ sector can be understood as follows.
The second contribution of the first term in Eq. (25) is mainly $s$-wave and
gives a negligible contribution in the $d$-wave channel, i.e., this term is
not relevant for our analysis. The third term has the form of the screening of
the Coulomb interaction from the usual RPA, because $\Pi_{22}$ is just a
simple bubble. It is important to remember that $\Pi_{22}$ arises here from
fluctuations of the Lagrange multiplier introduced to impose the constraint.
Then, this contribution screens the $J$ and $V$ terms (first term of $V_{{\rm
eff}}^{(0)}$) from the $2\times 2$ sector, while the $J$-term from the
$4\times 4$ sector (second term in $V_{{\rm eff}}^{(0)})$ remains. In
addition, the first contribution $-2\Lambda_{1}(\delta-\Pi_{12})$, which is
independent of $J$ and $V$ has a small repulsive $d$-wave projection. Then, if
$V=J=0$, i.e., only the $2\times 2$ sector is present, superconductivity is
not expected to be mediated by charge fluctuations.

It is important to mention that the doping dependence of $\lambda^{{\rm Ch}}$
and $\lambda^{\rm J}$ does not show the steep behavior near $\delta_{c}$ seen
in Fig. 2 for $\lambda$. This is due to the fact that we did not include
$\Pi_{55}$, $\Pi_{56}$ and $\Pi_{66}$ from the FP sector (see the Appendix).
To understand the influence of only these components on $\lambda$ we take the
dressed bosonic propagator $D_{ab}$ and compute $V_{{\rm eff}}$ by projecting
$D_{ab}$ onto the FP eigenvector $(0,0,0,0,1/\sqrt{2},-1/\sqrt{2})$ [Ref.Bejas
et al., 2012]. We obtain

$V_{{\rm eff}}^{({\rm FP})}\left({\bf k},{\bf
k^{\prime}};\omega_{n},\omega_{n}^{\prime}\right)=-(\Lambda_{5}-\Lambda_{6})^{2}{\rm
Re}\chi_{{\rm FP}}({\bf k}-{\bf k^{\prime}},{\rm i}\omega_{n}-{\rm
i}\omega^{\prime}_{n})\,,$ (26)

where $\Lambda_{5}$ and $\Lambda_{6}$ are the fifth and sixth component of the
vertices in Eq. (14), and

$\displaystyle\chi_{{\rm FP}}({\bf q},{\rm
i}\nu_{n})=[(8/J)\Delta^{2}-\Pi_{{\rm FP}}({\bf q},{\rm i}\nu_{n})]^{-1}\,,$
(27)

which is the flux phase susceptibilityBejas et al. (2012) and $\Pi_{{\rm
FP}}({\bf q},{\rm i}\nu_{n})$ the electronic polarizability given by

$\displaystyle\Pi_{{\rm FP}}({\bf q},{\rm
i}\nu_{n})=-\frac{1}{N_{s}}\;\sum_{{\bf k}}\;\gamma_{{\rm FP}}^{2}({\bf
q},{\bf k})\frac{n_{F}(\epsilon_{{\bf k}+{\bf q}})-n_{F}(\epsilon_{{\bf
k}})}{\epsilon_{{\bf k}+{\bf q}}-\epsilon_{{\bf k}}-{\rm i}\nu_{n}}\,,$ (28)

with the form factor $\gamma_{{\rm FP}}({\bf q},{\bf
k})=2\Delta[\sin(k_{x}+q_{x}/2)-\sin(k_{y}+q_{y}/2)]$. For ${\bf
q}=(\pi,\,\pi)$ the form factor $\gamma_{{\rm FP}}({\bf q},{\bf k})$
transforms as $[\cos(k_{x})-\cos(k_{y})]$, i.e., the flux instability has
$d$-wave symmetry. $\chi_{{\rm FP}}({\bf q},{\rm i}\nu_{n})$ plays the role of
a bosonic glue, as phonons in usual superconductors.

This projection isolates the FP sector and allows us to check its effect on
$\lambda$. $\lambda^{{\rm FP}}$ versus $\delta$, where $\lambda^{{\rm FP}}$ is
calculated using $V_{{\rm eff}}^{({\rm FP})}$, is shown in Fig. 3(c). While at
large doping $\lambda^{{\rm FP}}$ goes to zero, the curve shows the steep
behavior approaching $\delta_{c}$. In Fig. 3(c), we also plot $\lambda^{\rm
J}$ versus $\delta$. Comparing $\lambda^{{\rm FP}}$ with $\lambda^{\rm J}$ we
conclude that the flux phase enhances superconductivity only near the quantum
critical point at $\delta_{c}$ associated with the flux instability. This
tendency to enhance superconductivity can be seen as triggered by quantum
critical fluctuationsAbanov et al. (2019).

Figure 3 shows that the total coupling strength $\lambda$ can be computed in a
good approximation as the sum of $\lambda^{{\rm FP}}$ and $\lambda^{\rm J}$,
i.e., as coming from an effective pairing interaction $V_{{\rm eff}}\sim
V_{{\rm eff}}^{({\rm FP})}+J({\bf k}-{\bf k}^{\prime})$. While $V_{{\rm
eff}}^{({\rm FP})}$ is retarded, $J({\bf k}-{\bf k}^{\prime})$ is unretarded.
It is well known that when introducing a finite broadening $\Gamma$ in the
analytical continuation ${\rm i}\nu_{n}=\nu+{\rm i}\Gamma$ in Eq. (28), the
flux phase is pushed to lower dopingsYamase et al. (2019). In the inset of
Fig. 3(c) we show results for $\lambda^{{\rm FP}}$ and $\lambda^{\rm J}$ for
$\Gamma=0.05$. For this $\Gamma$ the flux-phase does not set down at a finite
doping, and $J({\bf k}-{\bf k}^{\prime})$ is a good approximation for
computing the total coupling strength for all dopings.

The authors of Ref.[Maier et al., 2008] showed that the pairing strength is
composed by a retarded spin fluctuation contribution and an unretarded term
$J({\bf k}-{\bf k}^{\prime})$, and that the retarded pairing dominates. In
agreement with this work we also found an unretarded $J({\bf k}-{\bf
k}^{\prime})$ contribution. As discussed in our paper the large-$N$
approximation weakens spin fluctuations over charge fluctuations, then we
cannot rule out the presence of a retarded spin-fluctuation pairing. In Ref.
[Maier et al., 2008] the Coulomb potential $V({\bf q})$ was not included,
which can kill the superconductivity from the spin fluctuation term. However,
we showed that the constraint in the $t$-$J$ model, when included, screens
$V({\bf q})$. Then, we think that our paper and that of Ref. [Maier et al.,
2008] are complementary. If pairing in cuprates is mainly retarded or mainly
unretarded is an open discussion. Although one can expect a retarded pairing
as in conventional superconductors, some experiments suggest that pairing may
certainly be unretardedLorenzana et al. (2013); Park et al. (2013).

##  IV Conclusions

Using a large-$N$ approach on the microscopic $t$-$J$-$V$ model we studied
$d$-wave superconductivity and the role of a nearest-neighbors Coulomb
repulsion on it. In this approach, pairing is mediated by a bosonic propagator
which contains on-site charge and bond-charge fluctuations, both treated at
the same footing in present formalism. When the bare bosonic propagator is
considered, superconductivity arises from the unretarded exchange term $J({\bf
k}-{\bf k^{\prime}})$. However, the presence of the nearest-neighbors Coulomb
repulsion $V({\bf q})$ is detrimental to superconductivity and cancels pairing
for values of $V\sim J$, suggesting a fragile $d$-wave superconductivity. The
situation changes drastically when the bosonic propagator is dressed by
interactions. In this case, superconductivity becomes almost unaffected by $V$
and remains robust even for $V\gg J$. The inclusion of the non-double-
occupancy constraint and its fluctuations screens the effect of $V({\bf q})$,
while a pairing contribution from the $J$-term remains. In other words, the
scenario for a possible unretarded (without glue) pairing contribution
triggered by $J$ emerges in strong coupling, i.e., only if the local
constraint is considered properly.

Our results may be useful for the comparison with similar calculations in the
Hubbard and $t$-$J$ models. A robust $d$-wave superconductivity against a
nearest-neighbors Coulomb repulsion $V({\bf q})$ requires the non-double
occupancy to be considered, and at this level an unretarded pairing
contribution is obtained. In the large-$U$ limit, the Hubbard model is mapped
to the $t$-$J$ model. Then it would be interesting to check the role of
$V({\bf q})$ on superconductivity and, in addition, to disentangle retarded
and unretarded interactions from the obtained pairing in the Hubbard model.

###### Acknowledgements.

The authors thank P. Bonetti, W. Metzner, D. Vilardi, and H. Yamase for
fruitful discussions. A.G. thanks the Max-Planck-Institute for Solid State
Research in Stuttgart for hospitality and financial support.

##  Appendix A Some characteristics and discussions on the flux phase

In this Appendix we briefly discuss the main characteristics of the flux phase
(FP) and its possible connection with the physics of the pseudogap. As
discussed in Ref. [Bejas et al., 2012], for present parameters ($J=0.3$ and
$T=0$) the flux phaseAffleck and Marston (1988); Marston and Affleck (1989);
Morse and Lubensky (1991); Cappelluti and Zeyher (1999); Foussats and Greco
(2004) occurs at $\delta=\delta_{c}\sim 0.13$, with a modulation vector ${\bf
Q}$ close to $(\pi,\pi)$, i.e., the FP breaks the translational symmetry. In
present large-$N$ approximation the FP occurs when one eigenvalue of
$D^{-1}_{ab}$ is zero, and since the associated eigenvector is of the form
$(0,0,0,0,1/\sqrt{2},-1/\sqrt{2})$, the flux instability is located in the
sector $5$-$6$ of the $6\times 6$ matrix $D_{ab}$ (Ref.[Bejas et al., 2012]).
For $\delta<\delta_{c}$ the imaginary components $A^{x}$ and $A^{y}$ of the
bond field become finite. The commensurate FP is characterized by the
modulation vector ${\bf q}=(\pi,\,\pi)$ and describes staggered circulating
currents. In the FP state a $d$-wave gap, similar to the pseudogap in
cuprates, opens, and Fermi pockets with low intensity in the outer part are
developedChakravarty et al. (2003) instead of a large Fermi surface. The FP is
equivalent to the $d$CDW which was proposed phenomenologically for describing
the pseudogapChakravarty et al. (2001).

The FP is a bond-charge instability. As discussed in Ref. [Bejas et al.,
2012], besides the FP there are several kinds of bond-charge fluctuations, and
in principle all of them can lead to an instability depending on the model
parameters. However, in the context of the present large-$N$ method, for hole-
doped cuprates it was found that the flux instability is robust in a
realistic-parameters regime. In contrast, for electron-doped cuprates the
leading bond-charge instability can occurs for the real components $r^{x}$ and
$r^{y}$ of the fluctuations of the bond field Bejas et al. (2014).

Although the flux phase or $d$CDW is a candidate for describing the pseudogap,
its existence in the $t$-$J$ and Hubbard models is controversial. While some
reports show the presence of the flux instability or its fluctuationsLeung
(2000); Dong and Gull (2020), others do notMacridin et al. (2004). The FP is
also controversial from the experimental point of view. While the authors of
Refs. [Chakravarty et al., 2001; Honerkamp and Lee, 2004; Lee et al., 2006]
show that a series of experiments in the pseudogap phase can be described in
the context of the FP, angle-resolved photoemission spectroscopy (ARPES)
experiments do not show pockets but Fermi arcsNorman et al. (1998); Damascelli
et al. (2003) which are considered as an indication that translational
symmetry is not broken in the pseudogap. In Refs. [Greco, 2009; Greco and
Bejas, 2011, 2014] the interaction between the flux-phase fluctuations and
carriers in the proximity to the flux-phase instability leads to a reasonable
description of the Fermi arcs and Raman scattering without the necessity of
the translational-symmetry breaking. RecentlyGourgout et al. (2020), it was
proposed that the FP is a good candidate for describing the pseudogap.

The connection between the FP and the antiferromagnetism and its fluctuations,
which lead to $d$-wave superconductivitySénéchal et al. (2013); Vilardi et al.
(2019), is an interesting point. The FP occurs at much larger doping
($\delta=0.13$ in the present calculation) than the onset of
antiferromagnetism. Then, at the onset of the FP both phases may interact
weakly while, with decreasing doping approaching the antiferromagnetic-
insulating phase, antiferromagnetism and its fluctuations may lead against the
FP. On the other hand, the FP develops staggered magnetic moments much weaker
than those in the antiferromagnetic phaseHsu et al. (1991) which, in
principle, indicates that the FP and antiferromagnetism are distinct phases.
In spite of that, it was claimed that antiferromagnetism can be also
understood in the framework of the flux phaseHo et al. (2001).

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Magnetization Reversal Mechanism in Exchange-Biased Spring-like Thin-Film
Composite

Marcin Perzanowski, Jakub Gregor-Pawlowski, Arkadiusz Zarzycki, Marta
Marszalek
Institute of Nuclear Physics Polish Academy of Sciences, Deparment of
Materials Science, Radzikowskiego 152, 31-342 Krakow, Poland

email<EMAIL_ADDRESS>
Abstract: Development of modern spintronic devices requires materials
exhibiting specific magnetic effects. In this paper, we investigate a
magnetization reversal mechanism in a [Co/Pdx]7/CoO/[Co/Pdy]7 thin-film
composite where an antiferromagnet is sandwiched between a hard and a soft
ferromagnets with different coercivities. The antiferromagnet/ferromagnet
interfaces give rise to the exchange bias effect. The application of soft and
hard ferromagnetic films causes exchange-spring-like behavior while the choice
of the Co/Pd multilayers provides large out-of-plane magnetic anisotropy. We
observed that the magnitude and the sign of the exchange bias anisotropy field
are related to the arrangement of the magnetic moments in the
antiferromagnetic layer. This ordering is induced by the spin orientation
present in neighboring ferromagnetic films which is, in turn, dependent on the
orientation and strength of the external magnetic field.

DOI: 10.1021/acsami.0c14115 (OPEN ACCESS)

Keywords: exchange bias, spring magnet, perpendicular magnetic anisotropy,
magnetization reversal, FORC, thin films, multilayers, antiferromagnet

##  1 Introduction

Exchange bias effect is a phenomenon occurring at the interface between a
ferromagnet (FM) and an antiferromagnet (AFM). The exchange coupling occurs
after cooling such a FM/AFM system in the external magnetic field below the
Néel temperature of the AFM, giving rise to the magnetic hysteresis loop shift
along the field axis.1, 2 The shift is called the exchange bias field
$H_{\mathrm{ex}}$ and its magnitude is inversely proportional to the thickness
of the FM material revealing the interfacial nature of the effect. In most
cases the bias field decreases monotonically with increasing temperature to
the field $H_{\mathrm{ex}}=0$ for the blocking temperature for exchange bias.
However, there are also systems where the bias field first increases, and then
drops down as temperature rises.3 Usually, the blocking temperature is lower
than the Néel temperature of the AFM due to the structural imperfections
present in FM and AFM materials, as well as the condition of the interface. To
describe the exchange bias phenomenon various models have been applied
considering the magnetic domains in the antiferromagnet,4 the role of
uncompensated spins 5 as well as the roughness of the FM/AFM interface.6

The possible technological application of the exchange bias effect has been
studied in the context of its implementation in sensors, 7 biomedicine, 8, 9
and magnetic read heads and spintronic devices.10, 11, 12 Most of the research
on the exchange bias effect has been done on flat multilayers, however, there
are also studies for materials in the form of magnetic antidots and dots,13,
14, 15 core-shell structures,16, 17, 18 or rings and disks.19, 20 One of the
key features especially important for application in flat and patterned
spintronic devices is a perpendicular magnetic anisotropy present in the
magnetic material. For this reason, we focused on system including Co/Pd
ferromagnetic multilayers with easy axis of magnetization perpendicular to a
film plane.21, 22 As an antiferromagnetic material for the exchange bias
effect studies we have chosen cobalt oxide CoO since its properties are well
known and the Co/CoO interface is considered to be a model system for such
investigations.23, 24, 25, 26

Here, we present studies on the cooling field influence on the magnetization
reversal mechanism for the exchange-biased system where the CoO
antiferromagnetic layer is sandwiched between two [Co/Pd] ferromagnetic
multilayers with different coercivities. The issue of the cooling field impact
on the magnitude and sign of the exchange bias field has been recently raised
in a few research papers. However, these works focused on AFM/FM bilayer,27
spin glass/FM interface,28 or system where AFM and FM layers are separated by
a paramagnetic material.29 This paper develops this field of research further
by combining two magnetic effects in one study — exchange bias and magnetic
exchange spring, to find how they affect the reversal process. Such type of
FM/AFM/FM composite that we study here is similar to the hard-soft exchange
spring materials which can be applied in high density magnetic recording
devices.30, 31, 32 We find that the magnetization switching process and the
magnitude and the sign of the exchange bias field are different depending on
the magnetic state of both the ferromagnetic and the antiferromagnetic films
induced by cooling the system in various external magnetic fields. The studies
of the magnetic hysteresis loops obtained under different conditions are
supported by the First Order Reversal Curve (FORC) measurements. The FORC
investigations were carried out for both ascending and descending branches of
the hysteresis loop which is especially significant for the exchange-biased
systems with bias loop shift from zero position.

##  2 Experimental

The samples were fabricated by thermal evaporation at room temperature in
ultrahigh vacuum under a pressure of $10^{-7}$ Pa. The systems were deposited
on single-crystal Si(100) substrates with a 2 nm thick Pd buffer layer.
Ferromagnetic multilayers consisted of [Co/Pd]7 stacks with a Co thickness of
0.3 nm. The Pd layers have a thickness of 0.6 or 1.2 nm. The antiferromagnetic
CoO layer (AFM) was obtained by oxidizing 1-nm-thick Co layer in the
atmosphere of pure oxygen under a pressure of $3\times 10^{2}$ Pa for 10
minutes. In this paper four systems were studied — the [Co/Pd${}_{\mathrm{0.6\
nm}}$]7 and [Co/Pd${}_{\mathrm{1.2\ nm}}$]7 multilayers, the
[Co/Pd${}_{\mathrm{0.6\ nm}}$]7/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 and the
[Co/Pd${}_{\mathrm{0.6\ nm}}$]7/CoO/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7
composites. During the deposition the substrates were kept at 300 K.

The multilayer structure of the samples was investigated by X-ray reflectivity
(XRR) measurements carried out using an X’Pert Pro PANalytical diffractometer
equipped with a Cu X-ray tube operated at 40 kV and 30 mA. For systems without
AFM layer the XRR method was used only for validation of the assumed layered
structure, while for AFM-based composite it was also applied to determine the
CoO thickness and density.

Magnetic studies were done using Quantum Design MPMS XL SQUID magnetometer.
The zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves were
measured as follows. First, the sample was demagnetized at 300 K by
application of an osciltextitg external magnetic field, inducing zero
magnetization. Then, the sample was cooled down to 10 K without the presence
of the external field. At 10 K, the field of 500 Oe was applied in the
direction perpendicular to the sample surface, and the ZFC magnetization curve
was measured during heating to 300 K. After reaching the final temperature,
the FC curve was recorded during cooling down to 10 K with the same external
field. Both ZFC and FC curves were measured at a temperature change rate of 3
K/min. The hysteresis loops were measured in out-of-plane geometry at 10 K, 50
K, and 100 K, with the external magnetic field perpendicular to the sample
plane. During the cooling down to low a temperature various external magnetic
cooling fields were applied. For more details see further in the text.

First-order reversal curve (FORC) measurements were done at 10 K in out-of-
plane geometry. During an FORC measurement along a single hysteresis branch,
first, a $\pm$10 kOe field was set to magnetically saturate the ferromagnetic
components of the system. Then, the field was changed to the reversal field
$H_{\mathrm{R}}$ and the magnetization was measured as a function of the
external field $H$ toward the initial $\pm$10 kOe point. Next, the succeeding
$H_{\mathrm{R}}$ field was set prior to the subsequent magnetization
measurement. The set of $M(H,H_{\mathrm{R}})$ curves measured for different
starting $H_{\mathrm{R}}$ fields was transformed into an FORC distribution.

##  3 Results and discussion

First, we studied the magnetic properties of two ferromagnetic components of a
complex composite system. The hysteresis loops together with the d$M$/d$H$
switching field distributions, measured at 10 K after field cooling in +50 kOe
for [Co/Pd${}_{\mathrm{0.6\ nm}}$]7 and [Co/Pd${}_{\mathrm{1.2\ nm}}$]7
multilayers, and [Co/Pd${}_{\mathrm{0.6\ nm}}$]7/[Co/Pd${}_{\mathrm{1.2\
nm}}$]7 system, are presented in Figure 1. To obtain quantitative information
on the magnetic properties of the systems both upper and lower magnetization
branches of the hysteresis loops for the [Co/Pd${}_{\mathrm{0.6\ nm}}$]7 and
[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 multilayers were fitted using the following
expression13, 33, 34

$M(H)=\frac{2}{\pi}M_{\mathrm{s}}\arctan\left(g\left[\frac{H-H_{\mathrm{c}}}{H_{\mathrm{c}}}\right]\right)\
,$ (1)

where $M_{\mathrm{s}}$ is the saturation magnetization of the system,
$H_{\mathrm{c}}$ is its coercitivy, and $g$ represents the slope of the
magnetization curve.

The [Co/Pd${}_{\mathrm{0.6\ nm}}$]7 multilayer (Figure 1a) shows remanence
magnetization equal to 0.98 of saturation magnetization $M_{\mathrm{s}}$,
indicating that the easy axis of magnetization is perpendicular to the sample
plane.

Figure 1: Magnetic hysteresis loops measured at 10 K in out-of-plane geometry
for (a) [Co/Pd${}_{\mathrm{0.6\ nm}}$]7, (b) [Co/Pd${}_{\mathrm{1.2\ nm}}$]7,
and (c) [Co/Pd${}_{\mathrm{0.6\ nm}}$]7/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7
systems after cooling in a +50 kOe external magnetic field. The upper panels
present d$M$/d$H$ switching field distributions. The points represent
experimental data, and solid lines are fits (see text). In Figure (c) the hard
magnet (HM) and soft magnet (SM) components are marked with corresponding
color fields.

The switching field distributions d$M$/d$H$ for both upper and lower
magnetization branches are symmetrical and sharp denoting an abrupt
magnetization reversal process. The coercivity of the sample obtained from the
fit is 1 kOe. The [Co/Pd${}_{\mathrm{1.2\ nm}}$]7 multilayer (Figure 1b) has a
similar remanence demonstrating large out-of-plane magnetic anisotropy. In
comparison to the [Co/Pd${}_{\mathrm{0.6\ nm}}$]7 multilayer, the switching
field distributions d$M$/d$H$ of this sample are broader showing that the
rotation of the magnetic moments induced by the magnetic field sweep takes
place in a more gradual way. The coercivity of this multilayer is 5.4 kOe, and
therefore, it will be denoted further in the text as a hard magnet (HM), while
the [Co/Pd${}_{\mathrm{0.6\ nm}}$]7 multilayer will be labeled as a soft
magnet (SM). In both cases the same total Co thickness was deposited, and the
saturation magnetizations normalized to the surface area are equal. Due to the
symmetrical single-step magnetization reversal both SM and HM systems can be
described as rigid magnets.

The hysteresis loop and the d$M$/d$H$ distributions for the
[Co/Pd${}_{\mathrm{0.6\ nm}}$]7/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 system, being
a superposition of the hard and soft ferromagnets and further labeled as
SM/HM, are shown in Figure 1c. The d$M$/d$H$ curves demonstrate asymmetrical
switching field distributions related to the presence of different magnetic
phases SM and HM reversing at distinct external magnetic fields. To acquire
quantitive information on coercive fields and saturation magnetizations of the
components each magnetization branch was fitted using a sum of two $M(H)$
functions expressed by Eq. (1)

$\begin{split}M(H)=\frac{2}{\pi}M_{\mathrm{s}}^{\mathrm{SM}}\arctan\left(g^{\mathrm{SM}}\left[\frac{H-H_{\mathrm{c}}^{\mathrm{SM}}}{H_{\mathrm{c}}^{\mathrm{SM}}}\right]\right)\\\
+\frac{2}{\pi}M_{\mathrm{s}}^{\mathrm{HM}}\arctan\left(g^{\mathrm{HM}}\left[\frac{H-H_{\mathrm{c}}^{\mathrm{HM}}}{H_{\mathrm{c}}^{\mathrm{HM}}}\right]\right)\
,\end{split}$ (2)

where the SM and HM superscripts denote the quantities associated with the
corresponding constituent multilayers.

The sharper maxima, indicating swift magnetization reversal, occur for the
magnetic field of $\pm 1.7$ kOe and they are associated with the reversal of
the SM component. Accordingly, the broader d$M$/d$H$ maxima, suggesting slower
process of magnetization rotation, giving a coercivity of 2.2 kOe are related
to the HM multilayer. The saturation magnetizations of both SM and HM
components are similar to those observed for the single [Co/Pd] multilayers
(Figures 1a and 1b). As a consequence, the SM/HM system has the saturation
magnetization twice as large as the individual [Co/Pd] stack. The two-step
reversal process demonstrates that the SM/HM composite acts like an exchange-
spring system rather than like a single rigid magnet.35, 36 The increase of
the SM and the decrease of the HM coercivities, in comparison to the single
[Co/Pd] stacks, result from the exchange coupling between magnetic materials.
Similar changes observed for exchange-spring films with out-of-plane
anisotropy were reported by Casoli et al.37

In order to study the magnetization reversal mechanism of a spring-like
exchange-biased system, an antiferromagnetic (AFM) CoO layer was sandwiched
between SM and HM multilayers. The XRR measurement (Figure 2) showed that the
density of the cobalt oxide layer is 6.38 g/cm3 which is close to the bulk CoO
value of 6.44 g/cm3.

Figure 2: XRR measurement (red points) and fitted curve (blue line) for the
[Co/Pd${}_{\mathrm{0.6\ nm}}$]7/CoO/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 composite.

Therefore, we can expect that the Co layer was oxidized to CoO and it has
appropriate magnetic properties to create magnetic exchange coupling with the
ferromagnetic layers. The thickness of the CoO film determined from the XRR
equals to 1.8 nm which is a larger value than the deposited Co thickness due
to the incorporation of atoms in the material. According to the work by van
der Zaag et al.,38 for such a CoO thickness the blocking temperature for the
exchange bias is approximately 160 K. Accordingly, the temperature of 10 K, at
which magnetization reversal studies were carried out, is sufficiently low to
register meaningful exchange bias field. The fit shows that the roughness of
the Co layers within the [Co/Pd] stacks is comparable to their thickness equal
to 0.3 nm, indicating large jaggedness of the Co-Pd interfaces. Therefore, it
is highly unlikely that the deposited Co forms continuous layers. Taking into
account the calculated roughness of each Pd layer being approximately 0.4 nm,
the [Co/Pd${}_{\mathrm{0.6\ nm}}$]7 stack should instead be considered as an
intermixed Co-Pd film. In the case of the [Co/Pd${}_{\mathrm{1.2\ nm}}$]7
multilayer, due to the larger thickness of the Pd layers, the intermixed Co-Pd
interface regions are separated from each other by the layers of pure Pd.

The zero-field-cooled (ZFC) and field-cooled (FC) curves for the SM/AFM/HM
system are shown in Figure 3. The system was demagnetized at 300 K and, due to
the cooling in zero magnetic field, that state was preserved at 10 K.
Application of the 500 Oe external field induced nonzero magnetization caused
by the partial orientation of the magnetic moments along the field direction.
Heating of the system led to the increase of thermal fluctuations of the
magnetic moments leading to more moments being unblocked and able to align
with the field. This process can be observed in the ZFC curve as a progressive
rise of the magnetization up to 300 K. Cooling the system from 300 K (the FC
curve) resulted in a gradual decrease of thermal fluctuation energy, resulting
in the blocking of the magnetic moments. On the other hand, the presence of
the external magnetic field caused stepwise reorientation of the spins along
the field direction. These two factors are responsible for the progressive
increase of the magnetization seen in the FC measurement.

Figure 3: ZFC/FC curves for the [Co/Pd${}_{\mathrm{0.6\
nm}}$]7/CoO/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 composite measured with +500 Oe
external magnetic field.

The shape of the ZFC and FC curves is typical for ferromagnets and indicates
that the composite reveals ferromagnetic behavior in the whole temperature
range from 10 K to 300 K. Since both curves are monotonic without any maxima,
no superparamagnetic or superferromagnetic effects have to be taken into
account during further analysis of the magnetization switching process.24
Additionally, in both FC and ZFC curves there is a small increase of the
signal at low temperatures. Such behavior suggests that a small paramagnetic
contribution is present in the system.

The magnetization reversal mechanism of the exchange-biased SM/AFM/HM
composite was studied by a series of hysteresis loops measured at 10 K in the
out-of-plane geometry. Prior to the measurement the external perpendicular
magnetic field of +50 kOe was set at 300 K to align all ferromagnetic moments
in the SM and HM layer in the positive out-of-plane direction. Then, the field
was changed to $H_{\mathrm{cool}}$ in which the sample was cooled down to 10 K
and the loops were measured. Representative hysteresis loops for various
$H_{\mathrm{cool}}$ values are shown in Figure 4a.

Figure 4: (a) Representative hysteresis loops measured for the
[Co/Pd${}_{\mathrm{0.6\ nm}}$]7/CoO/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 composite
at 10 K for different cooling fields. (b) Hysteresis loops measured at 10 K,
50 K, and 100 K after field cooling in +50 kOe. The upper panels in both
figures show the d$M$/d$H$ switching field distributions. The points represent
experimental data, and the solid lines are fits (see text). The hard (HM) and
soft (SM) magnetization components are marked with corresponding color fields.

In all loops two pairs of switching fields per magnetization branch are
present, similar to those observed for SM/HM system, indicating a comparable
spring-like behaviour of the composite film with SM and HM multilayers
reversing at different external fields, lower and higher, respectively.
Moreover, in the $M(H)$ loops are observed small signal steps around $H=0$,
reflected in the d$M$/d$H$ curves as small maxima centered around zero
external field.

Similarly to the previous cases, the upper and lower magnetization branches of
the loops were fitted by the function of Eq. (1). However, here Eq. (2) used
for the SM/HM system was complemented by another component labeled as C to
reflect the presence of the signal observed for $H\\!\approx\\!0$ Oe, giving
the following expression

$\begin{split}M(H)=\frac{2}{\pi}M_{\mathrm{s}}^{\mathrm{SM}}\arctan\left(g^{\mathrm{SM}}\left[\frac{H-H_{\mathrm{c}}^{\mathrm{SM}}}{H_{\mathrm{c}}^{\mathrm{SM}}}\right]\right)\\\
+\frac{2}{\pi}M_{\mathrm{s}}^{\mathrm{HM}}\arctan\left(g^{\mathrm{HM}}\left[\frac{H-H_{\mathrm{c}}^{\mathrm{HM}}}{H_{\mathrm{c}}^{\mathrm{HM}}}\right]\right)\\\
+\frac{2}{\pi}M_{\mathrm{s}}^{\mathrm{C}}\arctan\left(g^{\mathrm{C}}\left[\frac{H-H_{\mathrm{c}}^{\mathrm{C}}}{H_{\mathrm{c}}^{\mathrm{C}}}\right]\right)\
.\end{split}$ (3)

The fits showed that the C component reveals no coercivity and its saturation
magnetization $M_{\mathrm{s}}^{\mathrm{C}}$ accounts in average for 5% of the
total magnetization exhibited by the SM/AFM/HM system. Figure 4b demonstrates
the evolution of the hysteresis loop for increased temperature indicating a
gradual decrease of the saturation magnetization as well as the reduction of
the coercive field, accompanied by the reduction of the exchange bias field.
Moreover, after heating the SM/SFM/HM systems to 50 K or 100 K the signal for
$H=0$ does not appear. Taking into account the shape of the ZFC/FC curves
(Figure 3) and the results obtained from the hysteresis loops it can be
concluded that a small fraction of a paramagnetic phase is present in the
SM/AFM/HM system. This is due to the $M\\!\propto\\!T^{-1}$ Curie law
according to which a paramagnetic contribution to the magnetic signal becomes
more prominent as temperature approaches zero. The origin of such phase can be
related to the oxidation procedure applied to obtain antiferromagnetic CoO.
Most of the Co volume deposited between the SM and HM stacks transformed into
antiferromagnet. Therefore, the antiparallel arrangement of the magnetic
moments gives zero net magnetization and does not contribute to the
magnetization recorded in the measurements. However, there are still some Co
atoms which were not oxidized and provide paramagnetic contribution to the
system. Due to their magnetic nature they are not magnetically coupled to the
other magnetic regions. Thereby, this phase does not have influence on the
magnetic properties revealed by the soft, hard, and antiferromagnetic
components of the system, and does not affect the reversal mechanism present
in the composite.

For each value of the cooling field $H_{\mathrm{cool}}$ the saturation
magnetizations $M_{\mathrm{s}}^{\mathrm{SM}}$ and
$M_{\mathrm{s}}^{\mathrm{HM}}$ of the soft and hard ferromagnetic components
are approximately equal and similar to those observed for the SM and HM stacks
(Figure 1a and b), and to the magnetization components obtained from fitting
of the SM/HM hysteresis loop (Figure 1c). Thereby, the overall magnetization
of the SM/AFM/HM system, neglecting the paramagnetic contribution described
above, is similar to that recorded for the SM/HM system, testifying that the
magnetic signal comes only from the Co atoms within the [Co/Pd] stacks. The
oxidized Co layer becomes antiferromagnetic below the Néel temperature, and
preserves this property regardless of the cooling procedure, which is observed
as a lack of the overall magnetization change upon the $H_{\mathrm{cool}}$
alteration.

Changes of the exchange bias fields and coercivities on the cooling field
$H_{\mathrm{cool}}$, obtained from the fits, are presented in Figures 5a and
b.

Figure 5: (a) Dependencies of the full loop exchange bias field
$H_{\mathrm{ex}}$ (black squares) and coercivity $H_{\mathrm{c}}$ (green
squares) on the cooling field $H_{\mathrm{cool}}$ for the SM/AFM/HM composite.
(b) Dependencies of the exchange bias fields $H_{\mathrm{ex}}^{\mathrm{SM}}$
and $H_{\mathrm{ex}}^{\mathrm{HM}}$ (full squares) and coercivities
$H_{\mathrm{c}}^{\mathrm{SM}}$ and $H_{\mathrm{c}}^{\mathrm{HM}}$ (full
triangles) on the cooling field $H_{\mathrm{cool}}$ for the soft (SM) and hard
(HM) magnetization components of the SM/AFM/HM composite. (c) Schematic
representation of different magnetization reversal mechanisms for the
different regions marked in subfigures (a) and (b) by Roman numbers.

The dependencies can be divided into four regions in which the magnetization
reversal process takes place in a different manner.

In the first region (I), covering the $H_{\mathrm{cool}}$ range from high
positive fields down to -0.2 kOe, there is no significant change of the full
loop bias field $H_{\mathrm{ex}}$, equal to -0.5 kOe. The corresponding
$H_{\mathrm{ex}}^{\mathrm{SM}}$ and $H_{\mathrm{ex}}^{\mathrm{HM}}$ values
also do not alter with the $H_{\mathrm{cool}}$, the
$H_{\mathrm{ex}}^{\mathrm{SM}}$ is -0.4 kOe, and the
$H_{\mathrm{ex}}^{\mathrm{HM}}$ is slightly larger than the $H_{\mathrm{ex}}$.
The initial field of +50 kOe forces all magnetic moments in both SM and HM
multilayers to align and to point in a positive out-of-plane direction. The
absence of the $H_{\mathrm{ex}}$ changes suggests that in this region the
$H_{\mathrm{cool}}$ field in this range is too weak to alter the original
arrangement of the spins. Therefore, the system is cooled down with all
ferromagnetic moments pointing in the direction perpendicular to the sample
plane (see corresponding path I in Figure 5c). This lack of a spin
reorientation upon external field change, even for slightly negative
$H_{\mathrm{cool}}$ values, is a consequence of a large out-of-plane
anisotropy accompanied by the high magnetic remanence present in both SM and
HM layers. Below the Néel temperature the CoO orders antiferromagnetically,
and below the exchange bias blocking temperature its magnetic moments start to
couple with the SM and HM ferromagnetic spins. Due to this exchange coupling
the orientation of the ferromagnetic spins imposes the out-of-plane alignment
of the AFM moments and freezes them, making the AFM spin structure insensitive
to further changes of the external magnetic field. Such highly-ordered
perpendicular arrangement of the SM, HM, and AFM magnetic moments favors a
strong out-of-plane exchange bias coupling on both SM/AFM and AFM/HM
interfaces which is reflected as a largest value of $H_{\mathrm{ex}}$, seen
also for both SM and HM components.

The second region (II) covers the $H_{\mathrm{cool}}$ range from -0.2 kOe to
-0.5 kOe. Here, the $H_{\mathrm{ex}}$ value changes rapidly and alters its
sign from negative to positive. As previously, the external field was set to
+50 kOe and then changed to $H_{\mathrm{cool}}$, prior to the cooling and loop
measurement. Contrary to the previous case, the cooling field becomes strong
enough to drive the changes in ferromagnetic spin orientation and to start
magnetization reversal process within the SM and HM layers. Since the SM has a
lower coercivity we can expect that arrangement of its magnetic moments is
more sensitive to the external field change. Additionally, we can anticipate
that the spin arrangement in the SM layer diverges from the ideal out-of-plane
orientation to a greater extent than for the HM with larger coercivity (see
path II in Figure 5c). Therefore, at the AFM/HM interface the CoO spins are
coupled antiferrmagnetically to each other and aligned mostly perpendicularly
as imposed by the exchange coupling to the HM magnetic moments whose out-of-
plane orientation is driven by the external field. On the other hand, the
random orientation of the SM magnetic moments induced by the cooling field
caused formation of the variously oriented regions in the neighboring CoO
layer. Within these regions the AFM spins are coupled antiparallel and the
spatial direction along which they are arranged is determined by the
orientation of the nearby ferromagnetic moments due to the exchange
interaction induced during the cooling procedure. Since the cooling field
caused a high degree of spin disorder in the SM layer the number of the
perpendicularly oriented AFM magnetic moments is smaller than in the region
(I). This results in the decrease of the out-of-plane exchange bias field
$H_{\mathrm{ex}}$ and is also reflected in a faster reduction of the
$H_{\mathrm{ex}}^{\mathrm{SM}}$ value upon $H_{\mathrm{cool}}$ change than for
the $H_{\mathrm{ex}}^{\mathrm{HM}}$ field (see Figure 5b).

Further change of the $H_{\mathrm{cool}}$ to the value of -0.45 kOe (see path
in Figure 5c) leads to the bias field $H_{\mathrm{ex}}=0$. In this situation
only a small fraction of the HM spins is aligned in the initial positive out-
of-plane direction. Thereby, this restricts the amount of the CoO material
being frozen in the same orientation and causes a meaningful reduction of the
$H_{\mathrm{ex}}^{\mathrm{HM}}$ field measured in the out-of-plane direction.
In the case of the SM layer, since it has lower coercivity than the HM, the
cooling field was sufficiently large to rotate a limited number of the
magnetic moments and to point them in the negative out-of-plane direction. As
a consequence, the amount of the AFM spins coupled in the same manner is
restricted, which leads to the appearance of poor positive out-of-plane
$H_{\mathrm{ex}}^{\mathrm{SM}}$ bias field of the opposite sign in comparison
to the $H_{\mathrm{ex}}^{\mathrm{HM}}$. The contributions from these two
effects, taking place in the opposite spatial directions, compensate each
other and give the overall exchange bias field $H_{\mathrm{ex}}$ equal to
zero.

In the third region of the $H_{\mathrm{ex}}$ dependence, for the
$H_{\mathrm{cool}}$ from -0.5 kOe to -2 kOe, the positive exchange bias field
rises constantly but in a less rapid way than in the second region. In this
regime, it can be expected that the $H_{\mathrm{cool}}$ field is sufficiently
large to reverse most of the spins in the SM layer and point them in the
direction opposite to the initial state set by +50 kOe. However, the field is
still not strong enough to fully rotate magnetic moments in the HM. Therefore,
a good out-of-plane arrangement of AFM spins extorted by the ordered
ferromagnet is present mainly at the SM/AFM interface where the spins point in
a negative perpendicular direction. This leads to the perpendicular exchange
coupling and gives rise to the positive bias loop shift (see accompanying path
in Figure 5c). A gradual change of the $H_{\mathrm{cool}}$ extorts an out-of-
plane spin alignment also in the HM. This results in an increase of the
exchange coupling strength at the AFM/HM interface which, together with the
coupling at the SM/AFM interface, is responsible for a constant rise of the
exchange bias field. The situation is similar to that observed in the second
region with a considerable difference in the $H_{\mathrm{ex}}$ slope. The
magnetization reversal process in the SM is more abrupt than in the HM as it
is seen in Figure 1 for the [Co/Pd] multilayers. Therefore, when
$H_{\mathrm{cool}}$ drives the magnetization reversal, a small modification in
the external field value can cause a larger spin rotation in the SM than in
the HM. Since the arrangement of the ferromagnetic moments is a key parameter
for the magnitude of the exchange bias field, the sharp reversal of the SM
spins leads to a substantial change of $H_{\mathrm{ex}}$. The reversal of the
HM requires larger $H_{\mathrm{cool}}$ fields and is more protracted, and for
these reasons it provides less dynamic change of $H_{\mathrm{ex}}$ in the
third region.

The fourth region extends from -2 kOe up to -3 kOe $H_{\mathrm{cool}}$ fields.
The $H_{\mathrm{ex}}$ values are maximal and the same as in region I, but with
the opposite sign, which indicates the same magnetization reversal mechanism.
In the first region the $H_{\mathrm{cool}}$ field was too weak to change the
orientation of the spins ordered by the initial +50 kOe field, so the spins
were pointing in the positive out-of-plane direction. Here, the
$H_{\mathrm{cool}}$ field is large enough to magnetically saturate the SM and
HM layers in the opposite, negative, out-of-plane direction (see corresponding
path in Figure 5c). Such orientation of the SM and HM moments maximizes their
exchange coupling strength to the AFM producing the largest hysteresis loop
shift.

We observed that in region (I) the coercive field of the full loop has a value
of approximately 3.3 kOe. In region (II) it starts to increase, reaches its
maximum for $H_{\mathrm{cool}}\\!=\\!-0.45$ kOe, and then decreases in region
(III), reaching in region (IV) the same value as observed in region (I).
Similar behavior is observed for the $H_{\mathrm{c}}^{\mathrm{HM}}$ and
$H_{\mathrm{c}}^{\mathrm{SM}}$ coercive fields. In region (I), the
magnetization reversal taking place along the upper hysteresis branch is more
energetically demanding than the reversal along the lower branch due to the
unidirectional anisotropy induced by the ferromagnetic-antiferromagnetic
coupling. In region (IV) this situation is mirrored since the cooling field
strength was sufficient to magnetically saturate both SM and HM components.
Considering region (II), a stepwise change of the cooling field leads to
progressive spin orientation disorder induced in the ferromagnetic layers. The
SM layer is more susceptible to this disarranging process due to its softer
ferromagnetic properties, contrary to the HM component revealing larger
coercitivy. As a result, the exchange coupling process causes formation of the
variously spatially oriented regions within the antiferromagnetic layer below
the blocking temperature during the cooling procedure. These regions,
maintaining their antiferromagnetic nature, couple the ferromagnetic layers in
diverse spatial directions and constrict the reversal process along both upper
and lower magnetization branches. Since that, the reversal process becomes
more energetically demanding as the spatial orientations of the pinning AFM
regions get more diverse, leading to the coercivity rise. For the cooling
field of -0.45 kOe, at which the coercivity is maximal, some of the regions in
the antiferromagnet at the SM/AFM interface extort already the appearance of
small positive exchange bias field, while the bias field associated with the
AFM/HM interface still remains negative. This means that some of the AFM
regions counteract against the reversal along the upper magnetization branch,
due to the pinning effects driven by the exchange coupling, while the others
constrict the magnetization switching in the opposite field sweep direction.
Coexistence of these two processes causes the reversal is energetically
demanding along both branches, leading to the increase of coercivity. In
region (III) both $H_{\mathrm{ex}}^{\mathrm{HM}}$ and
$H_{\mathrm{ex}}^{\mathrm{SM}}$ bias fields changed their sign to positive. In
this situation, a stepwise change of the external cooling field extorts
gradual spin ordering along the negative perpendicular direction. Thereby, the
magnetization reversal along the upper hysteresis branch becomes less
energetically demanding than in region (II) due to the fact that the exchange
anisotropy at the SM/AFM and AFM/HM interfaces tends to couple the
ferromagnetic layers along the same direction. At the same time, since the
cooling field orientation induced the unidirectional anisotropy in the
opposite direction than in region (I), the reversal along the lower branch
becomes less difficult. This results in the progressive decrease of the
coercivity to the minimal value observed in regions (I) and (IV).

To support the information on the magnetization reversal mechanism provided by
the analysis of the $H_{\mathrm{ex}}$ and $H_{\mathrm{c}}$ dependencies on the
cooling field $H_{\mathrm{cool}}$, a series of FORC measurements were carried
out. The experiments were done at 10 K for the SM/AFM/HM system cooled in the
external fields of $+50$ kOe and $-0.45$ kOe to obtain FORC distributions for
the cases of maximal exchange bias field value, and for no biased loop.
Materials with the exchange bias effect show training effect which is observed
as a gradual decrease of the exchange bias field magnitude through consecutive
hysteresis loop cycling.39 The $|H_{\mathrm{ex}}|$ falloff is mainly present
during the first few cycles, and after that the changes of the exchange field
become less prominent. During the FORC measurement the external field is
switched many times between the reversal $H_{\mathrm{R}}$ and saturation $\pm
10$ kOe fields. Therefore, starting the measurement from the untrained state
would be associated with the presence of the training effect affecting the
resultant distribution, especially for a few initial $M(H_{\mathrm{R}},H)$
curves. To minimize the influence of this effect on the final FORC, prior to
the measurements at low temperature, the external magnetic field was switched
from +50 kOe to -50 kOe 15 times.

Figure 6: FORC measurements for the [Co/Pd${}_{\mathrm{0.6\
nm}}$]7/CoO/[Co/Pd${}_{\mathrm{1.2\ nm}}$]7 composite obtained at 10 K for
different $H_{\mathrm{cool}}$ cooling fields: (a) +50 kOe, and (b) -0.45 kOe.
Left: $M(H,H_{\mathrm{R}})$ FORC loop families for ascending and descending
magnetization branches, plotted in canvas of the hysteresis loops. The
characteristic features are marked with color arrows (see text).

The initial magnetization data dependent on the reversal field
$H_{\mathrm{R}}$ and the external field $H$, measured for gradual ascending
and descending $H$ and $H_{\mathrm{R}}$, were transformed into FORC
distributions $\rho(H_{\mathrm{R}},H)$ by calcutextitg mixed second-order
derivatives of the magnetization $M(H_{\mathrm{R}},H)$:40

$\rho(H_{\mathrm{R}},H)=-\frac{1}{2}\frac{\partial^{2}M(H_{\mathrm{R}},H)}{\partial
H\partial H_{\mathrm{R}}}\ .$ (4)

Such a definition of the FORC $\rho(H_{\mathrm{R}},H)$ distribution eliminates
reversible components of the magnetization reversal process. Therefore, any
nonzero $\rho(H_{\mathrm{R}},H)$ signal represents an irreversible
magnetization switching. The FORC $\rho$ distribution can be plotted in
coordinate system rotated by 45∘ with the coercive field $h_{\mathrm{c}}$ and
interaction field $h_{\mathrm{u}}$ on the horizontal and vertical axes,
respectively. These coordinates are defined as follows:

$h_{\mathrm{c}}=\frac{H-H_{\mathrm{R}}}{2}\ ,\
h_{\mathrm{u}}=\frac{H+H_{\mathrm{R}}}{2}\ .$ (5)

In a such coordination system, $h_{\mathrm{c}}$ represents the local coercive
field of each hysteron, and $h_{\mathrm{u}}$ corresponds to the local
interaction field shifting the hysteron’s center along the field axis.41 The
obtained FORC distributions $\rho(h_{\mathrm{c}},h_{\mathrm{u}})$ for both
ascending and descending magnetization branches, together with FORC curve
families $M(H_{\mathrm{R}},H)$ plotted in canvas of the hysteresis loop, are
shown in Figure 6. In each FORC distribution there are two main prominent
features related to the irreversible magnetization switching processes. These
features are characteristic of the reversal process taking place by the
nucleation of the magnetic domains and subsequent domain-wall motion.42

First, the ridges seen in all FORCs and marked by the green arrows in Figure 6
are directly related to the nucleation and subsequent rapid propagation of the
magnetic domains with reversed magnetization orientation within the system. It
can be seen in both FORCs measured for descending magnetization branches that
the ridge is centered on the point with no shift along the vertical
$h_{\mathrm{u}}$ axis. On the other hand, the ridge centre observed for
ascending fields after cooling the sample in +50 kOe is moved along
$h_{\mathrm{u}}$ for approximately -0.3 kOe (blue arrow in Figure 6a). This
feature is connected to the presence of the exchange interaction in the system
which is, in our case, the ferromagnetic-antiferromagnetic coupling present at
the SM/AFM and AFM/HM interfaces in the composite. The value of the
$h_{\mathrm{u}}$ shift is lower than the exchange bias field -0.5 kOe observed
in the hysteresis loop. However, it has to be taken into account that the
FORCs were measured on trained system where the value of the $H_{\mathrm{ex}}$
is expected to be lower than for untrained composite. Therefore, the
$h_{\mathrm{u}}$ shift is the evidence for the presence of ferromagnetic-
antiferromagnetic coupling in the system which drives the exchange bias
effect. For the case of $H_{\mathrm{cool}}\\!=\\!-0.45$ kOe when no hysteresis
loop shift was recorded (see Figure 5) the corresponding ascending ridge also
reveals no shift along the interaction axis $h_{\mathrm{u}}$.

The second feature observed in each FORC is a positive-negative pair of
irreversible regions along the lines marked by the purple arrows in Figures 6a
and b. Such paired signals are distinctive for switching of the orientation of
the magnetization by domain nucleation and domain-wall motion, and are related
to the annihilation of magnetic domains at the end of the magnetization
reversal process as the external field induces magnetic saturation of the
system. It is also seen in the FORCs for the system cooled in
$H_{\mathrm{cool}}\\!=\\!-0.45$ kOe (Figure 6b) that the irreversible regions
related to the domain annihilation are better pronounced than the
corresponding features observed after field cooling in +50 kOe (black arrows
in Figure 6b).

All FORCs show the lack of irreversible signal around
$h_{\mathrm{c}}\\!=\\!0$. This confirms that the magnetization drop observed
in the hysteresis loops for external magnetic field around 0 (see Figure 4a)
has a paramagnetic nature. The paramagnetic contribution is also reflected in
the FC and ZFC curves (Figure 3) and observed as a small magnetization rise at
a low temperature. The origin of the effect is the presence of unoxidized Co
atoms in the AFM. They do not exhibit ferromagnetic behavior and are not
magnetically coupled to the other magnetic regions. Due to their nature they
do not influence the magnetization reversal process of the composite.

##  4 Conclusions

In this paper, we describe the magnetization reversal mechanism of the
exchange-biased [Co/Pd${}_{\mathrm{0.6\ nm}}$]7/CoO/[Co/Pd${}_{\mathrm{1.2\
nm}}$]7 composite where the antiferromagnetic film is sandwiched between the
soft and hard ferromagnetic layers. Upon cooling such an exchange-spring-like
system in the external magnetic field $H_{\mathrm{cool}}$ the initial
alignment of the ferromagnetic spins forces the antiferromagnet to freeze in a
certain state. This state, induced by the cooling field, is then crucial for
the magnetization switching process, which has been reflected as the change of
the magnitude and the sign of the exchange bias field $H_{\mathrm{ex}}$ with
the $H_{\mathrm{cool}}$. Since we have two different ferromagnetic films from
both sides of the AFM, the shape of the exchange bias field $H_{\mathrm{ex}}$
of the cooling field $H_{\mathrm{cool}}$ dependence is conditioned by the
reversal mechanisms manifested by these two materials. In case where the
external field is large enough to affect the spin orientation in the soft
magnet, simultaneously being too weak to rotate moments of the hard magnet, we
observed rapid change of the exchange bias field. On the other hand, the
rotation of the magnetic moments of the hard magnetic film induced by the
$H_{\mathrm{cool}}$ resulted in less prominent slope of the $H_{\mathrm{ex}}$
curve. Therefore, the dependence is not symmetrical around the point
$H_{\mathrm{ex}}\\!=\\!0$. We also observed that cooling the system in zero
field does not result in the lack of the exchange bias loop shift, which is
caused by large out-of-plane anisotropy, leading to high magnetic remanence
displayed by the system. The FORC studies show that the magnetization reversal
is driven by the nucleation of magnetic domains with opposite orientation
followed by the domain-wall motion. Additionally, we record the shift of the
FORC irreversible region caused by the presence of the exchange bias field in
the system.

$\star$ $\star$ $\star$

##

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