---
abstract: 'The description of the proximity effect in superconducting/ferromagnetic heterostructures requires to use spin-dependent boundary conditions. Such boundary conditions must take into account the spin dependence of the phase shifts acquired by electrons upon scattering on the boundaries of ferromagnets. The present article shows that this property can strongly affect the critical temperature and the energy dependence of the density of states of diffusive heterostructures. These effects should allow a better caracterisation of diffusive superconductor/ferromagnet interfaces.'
author:
- Audrey Cottet
title: 'Spectroscopy and critical temperature of diffusive superconducting/ferromagnetic hybrid structures with spin-active interfaces'
---

Introduction
============

When a ferromagnetic metal ($F$) with uniform magnetization is connected to a BCS superconductor ($S$), the singlet electronic correlations characteristic of the $S$ phase can propagate into $F$ because electrons and holes with opposite spins and excitation energies are coupled coherently by Andreev reflections occurring at the $S/F$ interface. Remarkably, the ferromagnetic exchange field induces an energy shift between the coupled electrons and holes, which leads to spatial oscillations of the superconducting order parameter in $F$ [@Buzdin1982; @Golubov]. This effect has been observed experimentally through oscillations of the density of states (DOS) in $F$ with the thickness of $F$ [@TakisN], or oscillations of the critical current $I_{0}$ through $S/F/S$ structures [@Ryazanov; @TakisI; @SellierPRB; @Blum], with the thickness of $F$ or the temperature. The oscillations of $I_{0}$ have allowed to obtain $\pi$-junctions[@Guichard], i.e. Josephson junctions with $I_{0}<0$, which could be useful in the field of superconducting circuits [@Ioffe; @Taro]. A reentrant behavior of the superconducting critical temperature of $S/F$ bilayers with the thickness of $F$ has also been observed [@TcSF]. At last, some $F/S/F$ trilayers have shown a lower critical temperature for an antiparallel alignment of the magnetizations in the two $F$ layers as compared with the parallel alignment[@SSspinswitch], which should offer the possibility of realizing a superconducting spin-switch[@DeGennes; @TcFSFth].

![a. Diffusive $F/S/F$[ ]{}trilayer consisting of a BCS superconductor $S$ with thickness $d_{S}$ placed between two ferromagnetic electrodes $F_{1}$ and $F_{2}$ with thickness $d_{F}$. In this picture, the directions of the magnetic polarizations in $F_{1}$ and $F_{2}$  are parallel \[antiparallel\], which corresponds to the configuration $\mathcal{C}=P$ $[AP]$. b. $S/F$[ ]{}bilayer consisting of a BCS superconductor $S$ with thickness $d_{S}/2$ contacted to a ferromagnetic electrode $F$ with thickness $d_{F}$.](structure.eps){width="0.7\linewidth"}

For a theoretical understanding of the behavior of $S/F$ hybrid circuits, a proper description of the interfaces between the different materials is crucial. For a long time, the only boundary conditions available in the diffusive case were spin-independent boundary conditions derived for $S/$normal metal interfaces[@Kuprianov]. Recently, spin-dependent boundary conditions have been introduced for describing hybrid diffusive circuits combining BCS superconductors, normal metals and ferromagnetic insulators [@condmatHuertas]. These boundary conditions take into account the spin-polarization of the electronic transmission probabilities through the interface considered, but also the spin-dependence of the phase shifts acquired by electrons upon transmission or reflection by the interface. The first property generates widely known magnetoresistance effects[@magn]. The second property is less commonly taken into account. However, the Spin-Dependence of Interfacial Phase Shifts (SDIPS) can modify the behavior of many different types of mesoscopic circuits with ferromagnetic elements, like those including a diffusive normal metal island [@FNF], a resonant system[@CottetEurophys; @SST], a Coulomb blockade system[@CB; @Cottet06; @SST], or a Luttinger liquid[@Luttinger]. It has also been shown that the SDIPS has physical consequences in $S/F$ hybrid systems[@Tokuyasu; @otherBC; @mixing; @condmatHuertas]. One can note that, in some references, the SDIPS is called ”spin-mixing angle” or ”spin-rotation angle” (see e.g. Refs. ).** **In the diffusive $S/F$ case, the spin-dependent boundary conditions of Ref.  have been applied to different circuit geometries[@demoBC; @applications; @Morten; @cottet05; @Braude] but the only comparison to experimental data has been performed in Ref. . The authors of this Ref. have generalized the boundary conditions of Ref.  to the case of metallic $S/F$ interfaces with a superconducting proximity effect in $F$. They have showed that the SDIPS can induce a shift in the oscillations of the critical current of a $S/F/S$ Josephson junction or of the DOS of a $S/F$ bilayer with the thickness of $F$. Signatures of this effect have been identified in the hybrid structures of Refs. . Nevertheless, the problem of characterizing the SDIPS of diffusive $S/F$ interfaces has raised little attention so far, in spite of the numerous experiments performed.

A good characterization of the properties of diffusive $S/F$ interfaces would be necessary for a better control of the superconducting proximity effect in diffusive heterostructures. The present article presents other consequences of the SDIPS than that studied in Ref. , which could be useful in this context. In particular, the SDIPS can generate an effective magnetic field in a diffusive $S$ in contact with a diffusive $F $, like found for a ballistic $S$ in contact with a ferromagnetic insulator [@Tokuyasu]. This effective field can be detected, in particular, through the DOS of the diffusive $F$ layer, with a visibility which depends on the thickness of $F$. A strong modification of the variations of the critical temperature of diffusive $S/F$ structures with the thickness of $F$ is also found. These effects should allow to characterize the SDIPS of diffusive $S/F$ interfaces through DOS and critical temperature measurements, by using the heterostructures currently fabricated. The calculations reported in this paper are also appropriate to the case of a diffusive $S$ layer contacted to a ferromagnetic insulator ($FI $).

This paper is organized as follows: Section II presents the initial set of equations used to describe the heterostructures considered. The case of $F/S/F$ trilayers is mainly addressed, but the case of $S/F$ (or $S/FI$) bilayers follows straightforwardly. Section III specializes to the case of a weak proximity effect in $F$ and a superconducting layer with a relatively low thickness $d_{S}\leq\xi_{S}$, with $\xi_{S}$ the superconducting coherence length in $S$. The spatial evolution of the electronic correlations in the $S$ and $F$ layers is studied in Section III.A. The energy-dependent DOS of $S/F$ heterostructures is calculated in Section III.B. Section III.C considers briefly the limit of $S/FI$ bilayers. Section III.D discusses SDIPS-induced effective field effects in other types of systems. Section III.E compares the present work to other DOS calculations for data interpretation in $S/F$ heterostructures. Critical temperatures of $S/F$ circuits are calculated and discussed in Section III.F. Conclusions are presented in Section IV. Throughout the paper, I consider conventional BCS superconductors with a s-wave symmetry.

Initial description of the problem
==================================

This article mainly considers a diffusive $F/S/F$[ ]{}trilayer consisting of a BCS superconductor $S$ for $-d_{S}/2<x<d_{S}/2$, and ferromagnetic electrodes $F_{1}$ for $x\in\{-d_{S}/2-d_{F},-d_{S}/2\}$ and $F_{2}$ for $x\in\{d_{S}/2,d_{S}/2+d_{F}\}$ (see Figure 1.a). The magnetic polarization of the two ferromagnets can be parallel (configuration $\mathcal{C}=P$) or anti-parallel (configuration $\mathcal{C}=AP$), but the modulus $\left|  E_{ex}\right|  $ of the ferromagnetic exchange field is assumed to be the same in $F_{1}$ and $F_{2}$. Throughout the structure, the normal quasiparticle excitations and the superconducting condensate of pairs can be characterized with Usadel normal and anomalous Green’s functions $G_{n,\sigma}=\mathrm{sgn}(\omega_{n})\cos(\theta_{n,\sigma})$ and $F_{n,\sigma}=\sin(\theta_{n,\sigma})$, with $\theta_{n,\sigma}(x)$ the superconducting pairing angle, which depends on the spin direction $\sigma
\in\{\uparrow,\downarrow\}$, the Matsubara frequency $\omega_{n}(T)=(2n+1)\pi
k_{B}T$, and the coordinate $x$ (see e.g. Ref. ). The Usadel equation describing the spatial evolution of $\theta_{n,\sigma}$ writes $$\frac{\hbar D_{S}}{2}\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}%
}=\left|  \omega_{n}\right|  \sin(\theta_{n,\sigma})-\Delta(x)\cos
(\theta_{n,\sigma}) \label{UsadelS}%$$ in $S$ and $$\frac{\hbar D_{F}}{2}\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}%
}=\left(  \left|  \omega_{n}\right|  +iE_{ex}\sigma\mathrm{sgn}(\omega
_{n})\right)  \sin(\theta_{n,\sigma}) \label{UsadelF2}%$$ in $F_{1}$ and $F_{2}$, with $D_{F}$ the diffusion constant of the ferromagnets and $D_{S}$ the diffusion constant of $S$. The self-consistent superconducting gap $\Delta(x)$ occurring in (\[UsadelS\]) can be expressed as $$\Delta(x)=\frac{\pi k_{B}T\lambda}{2}\sum\limits_{\substack{\sigma
\in\{\uparrow,\downarrow\}\\\omega_{n}(T)\in\{-\Omega_{D},\Omega_{D}\}}%
}\sin(\theta_{n,\sigma}) \label{Delta}%$$ with $\Omega_{D}$ the Debye frequency of $S$, $\lambda^{-1}=2\pi k_{B}T_{_{c}%
}^{BCS}\sum\nolimits_{\omega_{n}(T_{_{c}}^{BCS})\in\{0,\Omega_{D}\}}\omega
_{n}^{-1}$ the BCS coupling constant and $T_{_{c}}^{BCS}$ the bulk transition temperature of $S$. I assume $\Delta=0$ in $F_{1}$ and $F_{2}$. The above equations must be supplemented with boundary conditions describing the interfaces between the different materials. First, one can use $$\left.  \frac{\partial\theta_{n,\sigma}}{\partial x}\right|  _{x=\pm
(d_{S}/2+d_{F})}=0 \label{derZero}%$$ for the external sides of the structure. Secondly, the boundary conditions at the $S/F$ interfaces can be calculated by assuming that the interface potential locally dominates the Hamiltonian, i.e. at a short distance it causes only ordinary scattering (with no particle-hole mixing) (see e.g. Ref. ). This ordinary scattering can be described with transmission and reflection amplitudes $t_{n,\sigma}^{S(F)}$ and $r_{n,\sigma
}^{S(F)}$ for electrons coming from the $S$($F$) side of the interface in channel $n$ with a spin direction $\sigma$. The phases of $t_{n,\sigma}%
^{S(F)}$ and $r_{n,\sigma}^{S(F)}$ can be spin-dependent due to the exchange field $E_{ex}$ in $F_{1(2)}$ and a possible spin-dependence of the barrier potential between $S$ and $F_{1(2)}$. Boundary conditions taking into account this so-called Spin-Dependence of Interfacial Phase Shifts (SDIPS) have been derived for $|t_{n,\uparrow}^{S}|^{2},|t_{n,\downarrow}^{S}|^{2}\ll1$ and a weakly polarized $F$[@condmatHuertas; @cottet05]. When there is no SDIPS, the boundary conditions involve the tunnel conductance $G_{T}=G_{Q}%
\sum\nolimits_{n}T_{n}$ and the magnetoconductance $G_{MR}=G_{Q}%
\sum\nolimits_{n}(|t_{n,\uparrow}^{S}|^{2}-|t_{n,\downarrow}^{S}|^{2})$, with $\uparrow(\downarrow)$ the majority(minority) spin direction in the $F$ electrode considered, $G_{Q}=e^{2}/h$, and $T_{n}=|t_{n,\uparrow}^{S}%
|^{2}+|t_{n,\downarrow}^{S}|^{2}$. In the case of a finite SDIPS, one must also use the conductances $G_{\phi}^{F(S)}=2G_{Q}\sum\nolimits_{n}(\rho
_{n}^{F(S)}-4[\tau_{n}^{S(F)}/T_{n}])$, $G_{\xi}^{F(S)}=-G_{Q}\sum
\nolimits_{n}\tau_{n}^{S(F)}$ and $G_{\chi}^{F(S)}=G_{Q}\sum\nolimits_{n}%
T_{n}(\rho_{n}^{F(S)}+\tau_{n}^{S(F)})/4$, with $\rho_{n}^{m}%
=\operatorname{Im}[r_{n,\uparrow}^{m}r_{n,\downarrow}^{m~\ast}]$ and $\tau
_{n}^{m}=\operatorname{Im}[t_{n,\uparrow}^{m}t_{n,\downarrow}^{m~\ast}]$ for $m\in\{S,F\}$. In the following, I will focus on the effects of $G_{\phi}^{F}$ and $G_{\phi}^{S}$, and I will assume $G_{MR}$, $G_{\xi}^{F(S)}$ and $G_{\chi
}^{F(S)}$ to be negligible, like found with a simple barrier model in the limit $T_{n}\ll1$ and $E_{ex}\ll E_{F}$[@cottet05]. In this case, one finds that the boundary conditions for the $S/F$ interface located at $x=x_{j}=(-1)^{j}d_{s}/2$, with$\ j\in\{1,2\}$, write $$\xi_{F}\left.  \frac{\partial\theta_{n,\sigma}^{F}}{\partial x}\right|
_{x_{j}}=(-1)^{j}\gamma_{T}\sin[\theta_{n,\sigma}^{F}-\theta_{n,\sigma}%
^{S}]+i\gamma_{\phi}^{F}\epsilon_{n,\sigma}^{\mathcal{C},j}\sin[\theta
_{n,\sigma}^{F}] \label{BCright}%$$ and $$\xi_{F}\left.  \frac{\partial\theta_{n,\sigma}^{F}}{\partial x}\right|
_{x_{j}}-\frac{\xi_{S}}{\gamma}\left.  \frac{\partial\theta_{n,\sigma}^{S}%
}{\partial x}\right|  _{x_{j}}=\sum\limits_{m\in\{F,S\}}i\gamma_{\phi}%
^{m}\epsilon_{n,\sigma}^{\mathcal{C},j}\sin[\theta_{n,\sigma}^{m}]
\label{BCleft}%$$ where the indices $S$ and $F$ indicate whether $\theta_{n,\sigma}$ and its derivative are taken at the $S$ or $F$ side of the interface. These equations involve the reduced conductances $\gamma_{T}=G_{T}\xi_{F}/A\sigma_{F}$ and $\gamma_{\phi}^{F(S)}=G_{\phi}^{F(S)}\xi_{F}/A\sigma_{F}$, the barrier asymmetry coefficient $\gamma=\xi_{S}\sigma_{F}/\xi_{F}\sigma_{S}$, the superconducting coherence lengthscale $\xi_{S}=(\hbar D_{S}/2\Delta
_{BCS})^{1/2}$, the magnetic coherence lengthscale $\xi_{F}=(\hbar
D_{F}/\left|  E_{ex}\right|  )^{1/2}$, the gap $\Delta_{BCS}$ for a bulk $S$, the normal state conductivity $\sigma_{F(S)}$ of the $F(S)$ material, and the junction area $A$. The coefficient $\epsilon_{n,j}^{\mathcal{C}}$ takes into account the direction of the ferromagnetic polarization of electrode $F_{j}$ in configuration $\mathcal{C}\in\{P,AP\}$. One can use the convention $\epsilon_{n,\sigma}^{P,j}=(-1)^{j}\sigma\mathrm{sgn}(\omega_{n})$ and $\epsilon_{n,\sigma}^{AP,j}=\sigma\mathrm{sgn}(\omega_{n})$, in which the factor $\mathrm{sgn}(\omega_{n})$ arising from the definition of $\theta_{n,\sigma}$ and the terms $(-1)^{j}$ and $\sigma$ arising from the boundary conditions have been included for compactness of the expressions. Note that in the presence of a finite SDIPS i.e. $\gamma_{\phi}^{F(S)}\neq0$, the right hand side of equation (\[BCleft\]) is not zero contrarily to what found in the spin-degenerate case [@Kuprianov]. In the general case, $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$ are different (see e.g. Appendix A). This implies that, with the present approximations, each interface is characterized by three parameters: $\gamma_{T}$, $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. For the sake of simplicity, symmetric $F/S/F$ trilayers are considered, so that $\gamma_{T}$, $\gamma_{\phi}^{F}$ and $\gamma_{\phi
}^{S}$ are the same for the two $S/F$ interfaces.

Before working out the above system of equations, it is interesting to note that the angle $\theta_{n,\sigma}$ calculated in the parallel configuration $\mathcal{C}=P$ for $x>0$ also corresponds to the angle $\theta_{n,\sigma}$ expected for a $S/F$ bilayer consisting of a superconductor $S$ for $0<x<d_{S}/2$, and a ferromagnetic electrode $F$ for $x\in\{d_{S}%
/2,d_{S}/2+d_{F}\}$ (Figure 1.b). In practice, using a $F/S/F$ geometry can allow one to obtain more information on spin effects, as shown below.

Case of a thin superconductor and a weak proximity effect in $F$
================================================================

Spatial variations of the pairing angle
---------------------------------------

I will assume that the amplitude of the superconducting correlations in $F_{1(2)}$ is weak, i.e. $\left|  \theta_{n,\sigma}\right|  \ll1$ for $x\in\{-d_{S}/2-d_{F},-d_{S}/2\}$ and $x\in\{d_{S}/2,d_{S}/2+d_{F}\}$ (*hypothesis 1)* so that one can develop the Usadel equation (\[UsadelF2\]) at first order in $\theta_{n,\sigma}$. This leads to $$\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}}-\left(  \frac
{k_{n,\sigma}^{\mathcal{C},j}}{\xi_{F}}\right)  ^{2}\theta_{n,\sigma}=0
\label{UsadelF}%$$ in the ferromagnet $F_{j}$, with $j\in\{1,2\}$, $k_{n,\sigma}^{\mathcal{C}%
,j}=(2[i\eta_{n,\sigma}^{\mathcal{C},j}+\left|  \omega_{n}/E_{ex}\right|
)])^{1/2}$ and $\eta_{n,\sigma}^{\mathcal{C},j}=(-1)^{j}\epsilon_{n,\sigma
}^{\mathcal{C},j}$. Combining Eqs. (\[derZero\]) and (\[UsadelF\]), one finds in $F_{j}$ $$\theta_{n,\sigma}(x)=\theta_{n,\sigma}^{F}(x_{j})\frac{\cosh\left(  \left[
x-(-1)^{j}\left(  d_{F}+\frac{d_{S}}{2}\right)  \right]  \frac{k_{n,\sigma
}^{\mathcal{C},j}}{\xi_{F}}\right)  }{\cosh\left(  d_{F}\frac{k_{n,\sigma
}^{\mathcal{C},j}}{\xi_{F}}\right)  } \label{theta}%$$ This result together with boundary condition (\[BCright\]) leads to $$\theta_{n,\sigma}^{F}(x_{j})=\frac{\gamma_{T}\sin(\theta_{n,\sigma}^{S}%
(x_{j}))}{\gamma_{T}\cos(\theta_{n,\sigma}^{S}(x_{j}))+i\gamma_{\phi}^{F}%
\eta_{n,\sigma}^{\mathcal{C},j}+B_{n,\sigma}^{\mathcal{C},j}} \label{thetaF}%$$ with $B_{n,\sigma}^{\mathcal{C},j}=k_{n,\sigma}^{\mathcal{C},j}\tanh
[d_{F}k_{n,\sigma}^{\mathcal{C},j}/\xi_{F}]$. This allows to rewrite the boundary condition (\[BCleft\]) in closed form with respect to $\theta_{n,\sigma}^{S}$, i.e. $$\xi_{S}\left.  \frac{\partial\theta_{n,\sigma}^{S}}{\partial x}\right|
_{x_{j}}=(-1)^{j+1}\mathcal{L}_{j,n,\sigma}^{\mathcal{C}}\sin[\theta
_{n,\sigma}^{S}(x_{j})]$$ for the $S/F$ interface located at $x=x_{j}$, with $$\frac{\mathcal{L}_{j,n,\sigma}^{\mathcal{C}}}{\gamma}=\frac{\gamma
_{T}(B_{n,\sigma}^{\mathcal{C},j}+i\gamma_{\phi}^{F}\eta_{n,\sigma
}^{\mathcal{C},j})}{\gamma_{T}\cos(\theta_{n,\sigma}^{S})+B_{n,\sigma
}^{\mathcal{C},j}+i\gamma_{\phi}^{F}\eta_{n,\sigma}^{\mathcal{C},j}}%
+i\gamma_{\phi}^{S}\eta_{n,\sigma}^{\mathcal{C},j}%$$ In the following, I will assume $\left|  E_{ex}\right|  \gg\Delta_{BCS}$ like in most experiments, so that $k_{n,\sigma}^{\mathcal{C},j}=1+i\eta_{n,\sigma
}^{\mathcal{C},j}$. I will also assume $d_{S}/\xi_{S}\leq1$, so that one can use, for the $\mathcal{C}$ configuration and $-d_{S}/2<x<d_{S}/2$, $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}%
-\alpha_{n,\sigma}^{\mathcal{C}}(x/\xi_{S})-\beta_{n,\sigma}^{\mathcal{C}%
}(x/\xi_{S})^{2} \label{thetaQuad}%$$ with $\left|  \theta_{n,\sigma}(x)-\widetilde{\theta}_{n,\sigma}^{\mathcal{C}%
}\right|  \ll1$ (*hypothesis 2*). Note that although experiments are often performed in the limit of thick superconducting layers $d_{S}>\xi_{S}$, assuming $d_{S}\leq\xi_{S}$ is not unrealistic since one can obtain diffusive superconducting layers with a thickness $d_{S}\sim\xi_{S}$ (see e.g. Ref. ). Furthermore, using relatively low values of $d_{S}$ is more favorable for obtaining efficient superconducting spin-switches[@Baladie]. Hypothesis 2 allows one to develop $\sin
(\theta_{n,\sigma})$ and $\cos(\theta_{n,\sigma})$ at first order with respect to $\theta_{n,\sigma}-\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$ in $S$. Accordingly, I will neglect the space-dependence of $\Delta(x)$ and assign to it the value $\Delta^{\mathcal{C}}$ in configuration $\mathcal{C}$. Note that I do not make any assumption on the value of the angle $\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$, which is not necessarily close to the bulk BCS value. The coefficient $\mathcal{L}_{j,n,\sigma}%
^{\mathcal{C}}$ is transformed into its conjugate when the magnetic polarization of electrode $F_{j}$ is reversed. Therefore, I will note $\mathcal{L}_{1,n,\sigma}^{P}=\mathcal{L}_{2,n,\sigma}^{P(AP)}=\mathcal{L}%
_{n}^{\sigma}$ and $\mathcal{L}_{1,n,\sigma}^{AP}=(\mathcal{L}_{n}^{\sigma
})^{\ast}$. The above assumptions lead to $\alpha_{n,\sigma}^{P}=0$, $$\beta_{n,\sigma}^{P}=\frac{4\mathcal{L}_{n}^{\sigma}\sin(\widetilde{\theta
}_{n,\sigma}^{\mathcal{C}})}{4\delta_{S}+\cos(\widetilde{\theta}_{n,\sigma
}^{\mathcal{C}})\delta_{S}^{2}\mathcal{L}_{n}^{\sigma}} \label{bP}%$$$$\alpha_{n,\sigma}^{AP}=i\operatorname{Im}[\mathcal{L}_{n}^{\sigma}]\frac
{4\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})-\cos(\widetilde{\theta
}_{n,\sigma}^{\mathcal{C}})\delta_{S}^{2}\beta_{AP}}{4+2\operatorname{Re}%
[\mathcal{L}_{n}^{\sigma}]\cos(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}%
})\delta_{S}}%$$ and $$\beta_{n,\sigma}^{AP}=\frac{\left(  8\operatorname{Re}[\mathcal{L}_{n}%
^{\sigma}]+4\left|  \mathcal{L}_{n}^{\sigma}\right|  ^{2}\cos(\widetilde
{\theta}_{n,\sigma}^{\mathcal{C}})\delta_{S}\right)  \sin(\widetilde{\theta
}_{n,\sigma}^{\mathcal{C}})}{8\delta_{S}+6\operatorname{Re}[\mathcal{L}%
_{n}^{\sigma}]\cos(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})\delta_{S}%
^{2}+\left|  \mathcal{L}_{n}^{\sigma}\right|  ^{2}\cos^{2}(\widetilde{\theta
}_{n,\sigma}^{\mathcal{C}})\delta_{S}^{3}} \label{bAP}%$$ with $\delta_{S}=d_{S}/\xi_{S}$. On the other hand, from (\[UsadelS\]), one finds $$\beta_{n,\sigma}^{\mathcal{C}}=\frac{\Delta^{\mathcal{C}}\cos(\widetilde
{\theta}_{n,\sigma}^{\mathcal{C}})-\left|  \omega_{n}\right|  \sin
(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})}{2\Delta_{BCS}} \label{bc}%$$ The comparison between equations (\[bP\]), (\[bAP\]) and (\[bc\]) allows one to find $\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$ as a function of $\Delta^{\mathcal{C}}$. Then, one has to calculate $\Delta^{\mathcal{C}}$ by using the self-consistency relation (\[Delta\]). I will study below the DOS and the critical temperature following from these Eqs., in a limit which leads to simple analytical expressions.

Low-temperature density of states of $S/F$ heterostructures
-----------------------------------------------------------

The DOS of the ferromagnets $F_{1}$ and $F_{2}$ of Figure 1.a can be probed at $x=\pm(d_{F}+d_{S}/2)$ by performing tunnelling spectroscopy through an insulating layer. So far, this quantity has been less measured[@TakisN; @Kontos04; @Courtois; @Reymond] than critical temperatures or supercurrents. However, this way of probing the superconducting proximity effect is very interesting because it allows one to obtain spectroscopic information. It has been shown that the zero-energy DOS of a $F$ layer in contact with a superconductor oscillates with the thickness of $F$. For certain thicknesses, this zero-energy DOS can even become higher than its normal state value $N_{0}$ [@theoDOS; @Zareyan; @Bergeret; @yokohama], as shown experimentally in Ref. . Remarkably, the SDIPS can shift these oscillations[@cottet05]. Although the energy dependence of the DOS of diffusive $S/F$ structures has raised some theoretical and experimental interest, the effect of the SDIPS on this energy dependence has not been investigated so far.

For calculating analytically the low-temperature DOS of the structure of Fig. 1.a., one can assume $\gamma_{T}^{2}\cos(\theta_{n,\sigma}^{S}(x_{j}))/\left|
(\right.  \gamma_{T}\cos(\theta_{n,\sigma}^{S}(x_{j}))+i\gamma_{\phi}^{F}%
\eta_{n,\sigma}^{P,2}+B_{n,\sigma}^{P,2})(\gamma_{T}+i\gamma_{\phi}^{S}%
\eta_{n,\sigma}^{P,2}\left.  )\right|  \ll1$ (*hypothesis 3*), which leads to $\mathcal{L}_{n}^{\sigma}=\gamma\left(  \gamma_{T}+i\gamma_{\phi}%
^{S}\sigma\mathrm{sgn}(\omega_{n})\right)  $. This hypothesis is e.g. valid for $d_{F}\geq\xi_{F}$ and any value of $\gamma_{\phi}^{S(F)}$ if $\gamma_{T}$ is relatively small (see e.g. Fig. \[DOS1\]). I will also assume that the lowest order terms in $\delta_{S}$ prevail in the numerators and denominators of expressions (\[bP\]) and (\[bAP\]), i.e. $\beta_{n,\sigma}^{P}%
\sim\mathcal{L}_{n}^{\sigma}\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}%
})\delta_{S}^{-1}$ and $\beta_{n,\sigma}^{AP}\sim\operatorname{Re}%
[\mathcal{L}_{n}^{\sigma}]\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}%
})\delta_{S}^{-1}$ (*hypothesis 4*). Taking into account hypothesis 3 and $\gamma\sim1$, hypothesis 4 is valid provided $\gamma_{T}$ and $\left|
\gamma_{\phi}^{S}\right|  $ are relatively small compared to $1$. Importantly, hypotheses 3 and 4 are less restrictive regarding the value of $\gamma_{\phi
}^{F}$. Accordingly, I will often use $\gamma_{T},\left|  \gamma_{\phi}%
^{S}\right|  \ll\left|  \gamma_{\phi}^{F}\right|  $ in the Figs. of this paper. Hypotheses 3 and 4 lead to $$\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}=\mathrm{\arctan}\left(
\frac{\Delta^{\mathcal{C}}}{\left|  \omega_{n}\right|  +\Omega_{n,\sigma
}^{\mathcal{C}}}\right)  \label{thetaS}%$$ with $$\Omega_{n,\sigma}^{P}=2\Delta_{BCS}\gamma(\gamma_{T}+i\gamma_{\phi}^{S}%
\sigma\mathrm{sgn}(\omega_{n}))\delta_{S}^{-1} \label{omegaa}%$$ and $$\Omega_{n,\sigma}^{AP}=2\Delta_{BCS}\gamma\gamma_{T}\delta_{S}^{-1}
\label{Om2}%$$

![Zero energy density of states $N^{F}(\varepsilon=0)$ at $x=d_{F}+d_{S}/2$ as a function of $d_{F}$, for the $F/S/F$ structure of Figure 1.a in the $P$ configuration, with $\gamma=1$, $\gamma_{T}=0.12$, and $d_{S}/\xi_{S}=1$. The different curves correspond to different SDIPS parameters:$\ \gamma_{\phi}^{F}=\gamma_{\phi
}^{S}=0$ (black dotted curve), $\{\gamma_{\phi}^{F}=-2.1$, $\gamma_{\phi}%
^{S}=0\}$ (black full curve), and $\{\gamma_{\phi}^{F}=-2.1$, $\gamma_{\phi
}^{S}=0.12\}$ (red full curve). The low-temperature self-consistent gap $\Delta^{P}$ found for these three different cases is indicated at the bottom right of the Figure. Using a strong value for $\gamma_{\phi}^{F}$ allows one to change significantly the phase of the oscillations of $N_{\sigma}^{F}(0)$ with $d_{F}$ (effect similar to Ref. ). One can check that the hypotheses 1 to 4 are valid for the parameters used in this Figure. Note that for the different cases considered here, the critical temperature $T_{c}^{P}$ of the structure (see Sec. III.F) is such that $0.615T_{c}^{BCS}<T_{c}^{P}<0.635T_{c}^{BCS}$. The four red points correspond to the red full curves shown in Fig. \[DOS2\].[]{data-label="DOS1"}](DOS1.eps){width="0.8\linewidth"}

Equations (\[thetaS\]) and (\[omegaa\]) show that an effective magnetic field $H_{eff}$ appears in the $S$ layer in the $P$ configuration, due to $\gamma_{\phi}^{S}\neq0$. From Eqs. (\[thetaS\]) and (\[omegaa\]), $H_{eff}$ can be expressed as $$g\mu_{B}H_{eff}=\frac{\hbar v_{F}^{S}}{d_{s}}\frac{2G_{\phi}^{S}\ell_{e}^{S}%
}{3\sigma_{S}A}=2E_{TH}^{S}\frac{G_{\phi}^{S}}{G_{S}} \label{heff}%$$ Here, $v_{F}^{S}$ and $\ell_{e}^{S}$ denote the Fermi velocity and mean free path in $S$, and $G_{S}=\sigma_{S}A/d_{S}$ and $E_{TH}^{S}=\hbar D_{S}%
/d_{S}^{2}$ denote the normal state conductance and the Thouless energy of the $S$ layer of Fig. 1.a. From Equations (\[thetaS\]) and (\[Om2\]), the effective field effect disappears in the $AP$ configuration because the two contacts are assumed to be symmetric, and therefore, their contributions to $H_{eff}$ compensate each other in the $AP$ case. Note that, in principle, the $\gamma_{\phi}^{F}$ term can induce an effective field analogue to $H_{eff}$ in the $F$ layer, but this effect is not relevant in the regime studied in this paper (see Appendix B). The effects of $H_{eff}$ on the DOS of the structure will be investigated in next paragraphs. In order to calculate $\Delta^{\mathcal{C}}$, one has to combine the self-consistency relation (\[Delta\]) with Eq. (\[thetaS\]), which gives, at low temperatures, $$\operatorname{Re}[\log(\frac{\Omega_{n,\sigma}^{\mathcal{C}}+\sqrt{\left(
\Omega_{n,\sigma}^{\mathcal{C}}\right)  ^{2}+\left(  \Delta^{\mathcal{C}%
}\right)  ^{2}}}{\Delta_{BCS}})]=0 \label{DeltaEq}%$$ This equation can be solved numerically. The resulting $\Delta^{\mathcal{C}}$ is independent from the values of $n$ and $\sigma$ used in Eq. (\[DeltaEq\]). Then, the value of the pairing angle $\theta_{n,\sigma}$ in the ferromagnets can be found by using Eqs. (\[theta\]), (\[thetaF\]), (\[thetaQuad\]) and (\[thetaS\]). Note that for $\gamma\ll1$, the above Eqs. are in agreement with formula (5) of Ref. , obtained with rigid boundary conditions, i.e. $\theta_{n,\sigma}$ equal to its bulk value at the $S$ side. The energy dependence of $\theta_{n,\sigma}$ can be found by performing the analytic continuation $\omega_{n}=-i\varepsilon
+\Gamma$ and $\mathrm{sgn}(\omega_{n})=1$ in the above equations. The rate $\Gamma=0.05$ is used to account for inelastic processes [@theseW]. At last, the density of states $N_{\sigma}(x,\varepsilon)$ at position $x$ for the spin direction $\sigma\in\{\uparrow,\downarrow\}$ can be calculated by using $N_{\sigma}(x,\varepsilon)=\left(  N_{0}/2\right)  \operatorname{Re}%
[\cos[\theta_{n,\sigma}(x)]]$, where $N_{0}/2$ is the normal density of states per spin direction.

![Density of states $N^{F}(\varepsilon)$ at $x=d_{F}+d_{S}/2$ as a function of the energy $\varepsilon$, for the $F/S/F$ structure of Figure 1.a with $\gamma=1$, $\gamma_{T}=0.12$, $d_{S}/\xi_{S}=1$, $\gamma_{\phi}^{F}=-2.1$ and $\gamma_{\phi}^{S}=0.12$. The different panels correspond to different values of $d_{F}/\xi_{F}$. The red full curves correspond to the $P$ configuration and the blue dashed curves to the $AP$ configuration. For finite values of $\gamma_{\phi}^{S}$, some characteristic ”double-gap” structures appear in $N_{\sigma}^{F}(\varepsilon)$ in the $P$ case, due to a SDIPS-induced effective field appearing in $S$. The visibility of this effect strongly depends on the thickness $d_{F}$ of the $F$ layers. For the different cases considered here, one can check that hypotheses 1 to 4 are valid and that the critical temperature $T_{c}^{\mathcal{C}}$ of the structure (see Sec. III.F) is such that $0.62T_{c}^{BCS}<T_{c}^{\mathcal{C}}<0.67T_{c}^{BCS}$. Note that for the above parameters and $\gamma_{\phi}^{S}=0$, the curves obtained in the $P$ and $AP$ case would be identical and very close to the $AP$ curves shown here. []{data-label="DOS2"}](DOS2.eps){width="0.85\linewidth"}

In the following, I will mainly focus on $N^{F}(\varepsilon)=\sum
\nolimits_{\sigma\in\{\uparrow,\downarrow\}}N_{\sigma}(x=d_{F},\varepsilon)$. Figure \[DOS1\] shows the variations of the zero energy density of states $N^{F}(\varepsilon=0)$ as a function of $d_{F}$, for interface parameters $\gamma_{T}=\gamma_{\phi}^{S}=0.12$ and $\gamma_{\phi}^{F}=-2.1$. Importantly, the value $\gamma_{T}=0.12$ seems realistic, at least for the weakly polarized $_{0.9}$$_{0.1}$ bilayers used in Ref. , for which one finds $\gamma_{T}\sim0.15$ (see Ref. ). In addition, a simple barrier model suggests that the situation $\left|  G_{\phi}^{F}\right|  \gg G_{T}$, with $G_{\phi}^{F}<0$ can happen (see Appendix A). The value $\gamma_{\phi}^{F}=-2.1$ used in Fig. \[DOS1\] thus seems possible**.** One can see that $\gamma_{\phi}^{F}$ can change significantly the phase of the oscillations of $N^{F}(\varepsilon=0)$ with $d_{F}$ (this effect has already been studied in Ref.  in the case of rigid boundary conditions but I recall it here for the sake of completeness). In Fig. \[DOS1\], using for $\gamma_{\phi}^{F}$ a strong negative value allows to get $N^{F}(0)<N_{0}$ for the lowest values of $d_{F}%
$, like often found in experiments. Note that from Fig. \[DOS1\], $\gamma_{\phi}^{S}$ can also shift the oscillations of $N^{F}(\varepsilon=0)$ with $d_{F}$. Here, this effect is much weaker than that of $\gamma_{\phi}%
^{F}$, but one has to keep in mind that the limit $\gamma_{\phi}^{S}\ll
\gamma_{\phi}^{F}$ is considered.

Equation (\[heff\]) shows that a measurement of $H_{eff}$ should allow to determine the conductance $G_{\phi}^{S}$ of a diffusive $S/F$ interface. In this context, studying the energy dependence of $N^{F}(\varepsilon)$ is very interesting, because it can allow to see clear signatures of $H_{eff}$, as shown below. Figure \[DOS2\] shows the energy dependence of $N^{F}%
(\varepsilon)$ in the $P$ and $AP$ configurations, for a finite value of $\gamma_{\phi}^{S}$ and different values of $d_{F}$. For $\mathcal{C}=P$, $N^{F}(\varepsilon)$ shows some ”double-gap” structures which disappear if the device is switched to the $AP$ configuration. These double structures are an indirect manifestation of the effective magnetic field $H_{eff}$ occurring in $S$ in the $P$ configuration, due to $\gamma_{\phi}^{S}\neq0$. Although $H_{eff}$ is localized in the $S$ layer, the double-gap structure that this field produces in the DOS of $S$ is transmitted to the DOS of $F$ due to Andreev reflections occurring at the $S/F$ interfaces, as shown by Eq. (\[thetaF\]). Interestingly, Rowell and McMillan have already observed that an internal property of a $S$ layer can be seen through the superconducting proximity effect occurring in a nearby normal layer. More precisely, these authors have found that the DOS of an layer can reveal the phonon spectrum of an adjacent superconducting layer[@Rowell]. Remarkably, the visibility of $H_{eff}$ in $N^{F}(\varepsilon)$ is modulated by quantum interferences occurring in $F$. Indeed, $H_{eff}$ is more visible for certain values of $d_{F}$ (e.g. $d_{F}/\xi_{F}=1.0$ or $1.2$ in Fig. \[DOS2\]) than others (e.g. $d_{F}/\xi_{F}=2.1$ in Fig. \[DOS2\]), due to the $d_{F}$-dependence of Eq.(\[thetaF\]).

It is useful to note that the SDIPS-induced effective field $H_{eff}$ should also occur in the $S/F$ bilayer of Figure 1.b. In this case, the Thouless energy and normal state conductance of the $S$ layer correspond to $\widetilde{E}_{TH}^{S}=4E_{TH}^{S}$ and $\widetilde{G}_{S}=2G_{S}$ respectively, so that one finds $g\mu_{B}H_{eff}=\widetilde{E}_{TH}^{S}%
G_{\phi}^{S}/\widetilde{G}_{S}$. Double gap structures strikingly similar to those shown in Figure \[DOS2\] were indeed measured very recently by P. SanGiorgio et al., at the ferromagnetic side of diffusive bilayers, in the absence of any external magnetic field[@SanGiorgio]. Remarkably, the visibility of the observed double structures varies with $d_{F}$, as predicted above. Note that in $S/F$ bilayers, the field $H_{eff}$ should also be observable directly at the $S$ side by measuring $N^{S}%
(\varepsilon)=\sum\nolimits_{\sigma\in\{\uparrow,\downarrow\}}N_{\sigma
}(x=0,\varepsilon)$. However, for parameters comparable to those of Figure \[DOS2\], this should not enhance the resolution on $H_{eff}$ (see figure \[DOS3\], left). For certain values of $d_{F}$, $H_{eff}$ is even more visible in $N^{F}(\varepsilon)$ than in $N^{S}(\varepsilon)$(see Figure \[DOS2\]).

Before concluding this section, I would like to emphasize that from Eqs. (\[thetaS\]) and (\[Om2\]), in the $AP$ configuration, the SDIPS-induced effective field $H_{eff}$ disappears for the $F/S/F$ structure considered in this paper because the two contacts are assumed to be symmetric and have thus opposite contributions to $H_{eff}$ in the $AP$ case. In the case of a dissymmetric structure, this should not be true anymore, but the SDIPS-induced effective field should nevertheless vary from the $P$ to the $AP$ case. This is one practical advantage of working with $F/S/F$ trilayers instead of $S/F$ bilayers.

![Energy dependent density of states $N^{S}(\varepsilon)$ measurable at the S side ($x=0$) of the $S/F$ structure of Figure 1.b. The left panel corresponds to the case of a metallic $F$ contact with parameters corresponding to that of Figs. \[DOS1\] and \[DOS2\]. In this case, the DOS measured at the $S$ side does not allow to resolve the SDIPS-induced effective field $H_{eff}$ better than a DOS measurement at the $F$ side of the structure, as can be seen from a comparison with Fig. \[DOS2\]. For certain values of $d_{F}$, $H_{eff}$ is even more visible in $N^{F}(\varepsilon)$ than $N^{S}(\varepsilon)$. The right panel corresponds to the case in which $F$ is not a metal but an insulating ferromagnet, i.e. $\gamma_{T}=0$ . In this case, one can use the reduced SDIPS parameter $\lambda_{\phi}^{S}=G_{\phi}^{S}\xi_{S}/A\sigma_{S}$. For $\lambda_{\phi}^{S}\neq0$, $N^{S}(\varepsilon)$ shows strong signatures of the effective field $H_{eff}$ $\ $induced by the $FI$ layer in $S$. One can check that the hypotheses 1 to 4 are valid for the parameters used in this Figure.[]{data-label="DOS3"}](DOS3.eps){width="1\linewidth"}

Low-temperature density of states of $S/FI$ bilayers
----------------------------------------------------

Twenty years ago, internal Zeeman fields were observed in superconducting Al layers contacted to different types of ferromagnetic insulators ($FI$) (see Refs. ). Using a *ballistic* $S/FI$ bilayer model, Ref.  suggested that the observed internal fields could be induced by the SDIPS[@Tokuyasu]. However, the inadequacy of this ballistic approach for modeling the actual experiments was pointed out in Ref. . Most of the experiments on Al$/FI$ interfaces were interpreted by their authors in terms of a diffusive approach with no SDIPS, and an internal Zeeman field added arbitrarily in the Al layer (see Refs. ). The calculations of Section III.B. provide a microscopic justification for the use of such an internal field in the diffusive model. Indeed, using $\gamma_{T}=0$ in the above calculations allows one to address the case of diffusive $S/FI$ bilayers. One finds that the SDIPS-induced effective field $H_{eff}$ of Eq. (\[heff\]) can occur in a thin diffusive $S$ layer contacted to a $FI$ layer. This effective field effect can be seen e.g. in the density of states $N^{S}(\varepsilon)$ of the $S$ layer at $x=0$ (see Figure \[DOS3\], right). Remarkably, it was found experimentally[@Hao] that $H_{eff}$ scales with $d_{s}^{-1}$, in agreement with Eq. (\[heff\])[@NoteDG].

SDIPS-induced effective fields in other types of system
-------------------------------------------------------

Interestingly, the SDIPS can induce effective field effects in other types of systems. First, the case of $S/N/FI$ trilayers with a thickness $d_{N}$ of normal metal $N$ has been studied theoretically[@demoBC; @applications]. In this case, a conductance $G_{\phi}^{N}$ similar to $G_{\phi}^{S}$ can be introduced to take into account the SDIPS for electrons reflected by the $FI$ layer. The $N$ layer is subject to an effective field $g\mu_{B}H_{eff}%
^{\prime}=E_{TH}G_{\phi}^{N}/G_{N}$ with $G_{N}$ the conductance of $N$. The expression of $H_{eff}^{\prime}$ is analogue to that of $H_{eff}$ (see Eq. \[heff\]), up to a factor 2 which accounts for the symmetry of the $F/S/F$ structure with respect to $x=0$ in the $P$ configuration. Secondly, an effective field $H_{eff}^{^{\prime\prime}}$ defined by $g\mu_{B}%
H_{eff}^{^{\prime\prime}}=(\hbar v_{F}^{W}/2L)(\varphi_{L}^{\uparrow}%
+\varphi_{R}^{\uparrow}-\varphi_{L}^{\downarrow}-\varphi_{R}^{\downarrow})$ has been predicted for a resonant single-channel ballistic wire with length $L$ placed between two ferromagnetic contacts[@CottetEurophys], with $\varphi_{L(R)}^{\sigma}$ the reflection phase of electrons with spin $\sigma$ incident from the wire onto the left(right) contact, and $v_{F}^{W}$ is the Fermi velocity in the wire. The expression of $H_{eff}^{^{\prime\prime}}$ also shows strong similarities with that of $H_{eff}$ (see Eq. \[heff\], middle term). In practice, signatures of the field $H_{eff}^{^{\prime\prime}}$ could be identified in a carbon nanotube contacted with two ferromagnetic contacts [@Cottet06; @Sahoo]. The fields $H_{eff}$, $H_{eff}^{^{\prime}}$ and $H_{eff}^{^{\prime\prime}}$ have the same physical origin: the energies of the states localized in the central conductor of the structure depend on spin due to the spin-dependent phase shifts acquired by electrons at the boundaries of this conductor. In all cases, the DOS of the central conductor reveals the existence of the SDIPS-induced effective field only if it already presents a strong energy dependence near the Fermi energy in the absence of a SDIPS. In the $F/S/F$ case, this energy dependence is provided by the existence of the superconducting gap in $S$[@cond]. In the $S/N/FI $ case, it is provided by the existence of a superconducting minigap in $N$. At last, in the case of the ballistic wire, it is provided by the existence of resonant states in the wire.

Comparison between the present work and other models for data interpretation in $S/F$ heterostructures
------------------------------------------------------------------------------------------------------

For characterizing the properties of $S/F$ interfaces, one has to interpret the experimental data showing the oscillations of the density of states $N^{F}(\varepsilon)$ in $F$ with the thickness $d_{F}$ of $F$ (or the oscillations of the critical current $I_{0}$ of a $S/F/S$ Josephson with $d_{F}$). However, if one uses a simple description with spin-degenerate boundary conditions, the amplitude and the phase of these signals are not independent, which makes the agreement with experimental results impossible in most cases. The SDIPS concept can solve this problem since it produces a shift of the signals oscillations with respect to the $G_{\phi}^{S(F)}=0$ case. However, in many cases, the observed shifts were attributed to the existence of a magnetically dead layer (MDL) at the $F$ side of the $S/F$ interface (see e.g. Refs.  ). In other cases, the discrepancy between the theory and the data was solved by taking into account spin-scattering processes in the $F$ layer (see e.g. Ref. ). In order to have a better insight into superconducting proximity effect experiments, one must stress the importance of estimating experimentally the MDL thickness and the spin-scattering rate. In principle, spin-scattering rates can be estimated experimentally, as was done for instance for the alloy[@Pratt] which is frequently used in proximity effect measurements (see e.g. ). An experimental determination of the MDL thickness has also been performed in a few structures used to measure $T_{c}$ or $I_{0}%
$[@Muhge; @Aarts; @Piano2; @Piano; @Bell], but, so far, this parameter has not been used for a real quantitative analysis of the data. In some situations, a model combining the SDIPS with spin-scattering and/or a MDL may be necessary. In any case, it is important to point out that descriptions based on spin-degenerate boundary conditions are, in principle, incomplete since they do not account for the effective field effect described in section III.B.

Before concluding this section, it is interesting to note that the effective field effect produced by $G_{\phi}^{S}$ in $S$ or the phase shift of the spatial oscillations of the DOS provided by $G_{\phi}^{F}$ in $F$ will remain qualitatively similar when the signs of $G_{\phi}^{S}$ and $G_{\phi}^{F}$ are changed (not shown). The sign of $G_{\phi}^{F}$ in the experiments of Refs.  could be determined from a quantitative comparison between the theory and the data[@cottet05]. Below, we present a study of the critical temperature of $S/F$ structures which can give information on the signs of $G_{\phi}^{S}$ and $G_{\phi}^{F}$ through qualitative signatures.

Critical temperature of $S/F$ heterostructures
----------------------------------------------

The critical temperature of $S/F$ hybrid structures has already been the topic of many theoretical (see e.g. Refs. ) and experimental (see e.g. Refs. ) studies, but the effects of the SDIPS on this quantity have raised little attention so far. I show below that the SDIPS can significantly modify the critical temperatures of $S/F$ diffusive structures. Calculating the critical temperature $T_{c}^{\mathcal{C}}$ of the structure of Figure 1.a in configuration $\mathcal{C}$ requires to consider the limit in which superconducting correlations are weak in $S$ as well as in $F$ (hypotheses 1 and 2 are then automatically satisfied). Equations (\[bP\]), (\[bAP\]) and (\[bc\]), then lead to

![Critical temperature $T_{c}^{P}$ for the $F/S/F$ structure of Figure 1.a in the $P$ configuration (top panels) and difference $\Delta T_{c}=T_{c}^{AP}-T_{c}^{P}$ between the critical temperatures in the $P$ and $AP$ configurations (bottom panels), as a function of the thickness $d_{F}$ of the $F$ layers. The left panels show the effect of a finite $\gamma_{\phi}^{S}$ and the right panels the effect of a finite $\gamma_{\phi}^{F}$. The four panels show the case $\gamma_{\phi}%
^{S}=\gamma_{\phi}^{F}=0$ with full red lines, for comparison. The other parameters used in the Figure are $\gamma=1$, $\gamma_{T}=0.12$ and $d_{S}%
/\xi_{S}=1$. The SDIPS modifies $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ in a quantitative or qualitative way, depending on the case considered. The type of effects produced by the SDIPS on $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ strongly depend on the signs of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$.[]{data-label="FSFTc"}](FSFTc.eps){width="1\linewidth"}

$$\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}=\frac{\Delta^{\mathcal{C}}%
}{\left|  \omega_{n}\right|  +2\Delta_{BCS}b_{n,\sigma}^{\mathcal{C}}%
\delta_{S}^{-1}}%$$

with $$b_{n,\sigma}^{P}=\frac{4\mathcal{L}_{n,\sigma}^{0}}{4+\delta_{S}%
\mathcal{L}_{n,\sigma}^{0}} \label{bp}%$$$$b_{n,\sigma}^{AP}=\frac{8\operatorname{Re}[\mathcal{L}_{n,\sigma}%
^{0}]+4\left|  \mathcal{L}_{n,\sigma}^{0}\right|  ^{2}\delta_{S}%
}{8+6\operatorname{Re}[\mathcal{L}_{n,\sigma}^{0}]\delta_{S}+\left|
\mathcal{L}_{n,\sigma}^{0}\right|  ^{2}\delta_{S}^{2}} \label{bap}%$$ and $$\frac{\mathcal{L}_{n,\sigma}^{0}}{\gamma}=\frac{\gamma_{T}(B_{n,\sigma}%
^{P,2}+i\gamma_{\phi}^{F}\sigma\mathrm{sgn}(\omega_{n}))}{\gamma
_{T}+B_{n,\sigma}^{P,2}+i\gamma_{\phi}^{F}\sigma\mathrm{sgn}(\omega_{n}%
)}+i\gamma_{\phi}^{S}\sigma\mathrm{sgn}(\omega_{n})$$ These Eqs. together with (\[Delta\]) lead to $$\log\left(  \frac{T_{c}^{BCS}}{T_{c}^{\mathcal{C}}}\right)  =\operatorname{Re}%
\left[  \Psi\left(  \frac{1}{2}+\frac{b_{n,\sigma}^{\mathcal{C}}}{\delta
_{S}\exp(\Gamma)}\frac{T_{c}^{BCS}}{T_{c}^{\mathcal{C}}}\right)  \right]
-\Psi\left(  \frac{1}{2}\right)  \label{Tc}%$$ where $\Gamma$ denotes Euler’s constant. The resulting $T_{c}^{\mathcal{C}}$ is independent from the values of $n$ and $\sigma$ used in Eq. (\[Tc\]). Note that in the case $\gamma_{\phi}^{S}=\gamma_{\phi}^{F}=0$ and $\delta
_{S}\rightarrow0$, this equation is in agreement with Eqs. (18) and (19) of Ref. .

Performing a numerical resolution of Eq. (\[Tc\]) together with (\[bp\]) and (\[bap\]), one obtains the results of Fig. \[FSFTc\], which shows the critical temperature $T_{c}^{P}$ of the structure in the $P$ configuration (top panels) and the difference $\Delta T_{c}=T_{c}^{AP}-T_{c}^{P}$ (bottom panels) as a function of $d_{F}$, for different interface parameters[@approx]. Here, for simplicity, I consider cases where $T_{c}^{\mathcal{C}}$ does not show a strongly reentrant behavior, i.e. a cancellation of $T_{c}^{\mathcal{C}}$ in a certain interval of $d_{F}$. Such a behavior can happen for instance for larger values of $\gamma_{T}$ (see e.g. Ref. ) or $\gamma_{\phi}^{(S)F}$, but this corresponds to a minority[@Tccoupe] of the experimental observations made so far. The four panels of Fig. \[FSFTc\] show the case $\gamma_{\phi}^{S}=\gamma_{\phi
}^{F}=0$ with full red lines, for comparison with the other cases, where the SDIPS is finite. I first comment the results obtained for the $T_{c}^{P}%
(d_{F})$ curves (top panels of Fig. \[FSFTc\]). In some cases, the SDIPS can modify quantitatively the $T_{c}^{P}(d_{F})$ curves, for instance by amplifying the dip expected in $T_{c}^{P}(d_{F})$ in the absence of a SDIPS (see e.g. dotted curve in the upper left panel, corresponding to $\gamma
_{\phi}^{S}>0$). In some other cases the SDIPS can modify qualitatively the $T_{c}^{P}(d_{F})$ curves, for instance by transforming the minimum expected in $T_{c}^{P}(d_{F})$ into a maximum (see e.g. curve with circles in the upper right panel, corresponding to $\gamma_{\phi}^{F}<0$). I now comment the results obtained for the $\Delta T_{c}(d_{F})$ curves (bottom panels of Fig. \[FSFTc\]). In the range of parameters studied in this work, one always finds $\Delta T_{c}>0$ because the effects of the two $F$ layers on $S$ are partially compensated in the $AP$ case. In the absence of a SDIPS and for a low $\gamma_{T}$, the $\Delta T_{c}(d_{F})$ curve presents a maximum at a finite value of $d_{F}$ (see red full lines in bottom panels). In some cases, the SDIPS can modify quantitatively the $\Delta T_{c}(d_{F})$ curves, for instance by increasing the value of this maximum (see dotted curve in the bottom left panel, corresponding to $\gamma_{\phi}^{S}>0$). In other cases, the SDIPS can modify qualitatively the $\Delta T_{c}(d_{F})$ curves, for instance by turning the maximum expected in $\Delta T_{c}(d_{F})$ into a minimum (see dashed curve in the bottom left panel, corresponding to $\gamma_{\phi}^{S}<0$), or by increasing the complexity of the variations of $\Delta T_{c}$ with $d_{F}$ (see curve with circles in the bottom right panel, corresponding to $\gamma_{\phi}^{F}<0$), or by transforming $\Delta
T_{c}(d_{F})$ into a monotonically decreasing curve with a reduced amplitude (see dot-dashed curve in bottom right panel, corresponding to $\gamma_{\phi
}^{F}>0$). Remarkably, the type of effects produced by the SDIPS on the $T_{c}^{\mathcal{C}}(d_{F})$ and $\Delta T_{c}(d_{F})$ curves depend on the sign of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. Critical temperature measurements can thus be an interesting way to determine the values of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. For simplicity, I have shown here results for $\gamma_{\phi}^{F}\neq0$ and $\gamma_{\phi}^{S}\neq0$ separately. Nevertheless, the behavior predicted for $\gamma_{\phi}^{F}\neq0$ and $\gamma_{\phi}^{S}\neq0$ simultaneously remain highly informative on the values of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. Note that if $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$ are increased compared to the values used in Fig. \[FSFTc\], the $T_{c}^{\mathcal{C}}(d_{F})$ curve gets some cancellation points like in Ref. , but the behaviors of $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ remain, in many cases, qualitatively dependent on the signs of $\gamma_{\phi}^{F}$ and $\gamma_{\phi
}^{S}$ (not shown).

Before concluding, I note that with the approximations used in the previous section (III.B), the electronic correlations inside $S$ were affected by the presence of the $F$ electrodes through the parameter $\gamma_{\phi}^{S}$ only. Consequently, the self-consistent gap $\Delta^{\mathcal{C}}$ of the $S$ layer was independent from $d_{F}$, and it was furthermore identical for the $P$ and $AP$ configurations for $\gamma_{\phi}^{S}=0$. In the present section, I have not neglected the dependence of $T_{c}^{P}$ and $T_{c}^{AP}$ on $d_{F}$ and $\gamma_{\phi}^{F}$ because I have considered parameters for which hypothesis 3 is not acceptable anymore. In particular, I have used lower values for $d_{F}$[@regimes].

Conclusion
==========

This article shows that the Spin-Dependence of Interfacial Phase Shifts (SDIPS) can have a large variety of signatures in diffusive superconducting/ferromagnetic ($S/F$) heterostructures. Ref. had already predicted that the SDIPS produces a phase shifting of the oscillations of the superconducting correlations with the thickness of $F
$ layers. This article shows that this is not the only consequence of the SDIPS in $S/F$ circuits. In particular, the SDIPS can produce an effective magnetic field in a diffusive $S$ layer contacted to a diffusive $F$ layer. This effective field can be seen e.g. through the DOS of the diffusive $F$ layer, with a visibility which oscillates with the thickness of $F$. The SDIPS can also modify significantly the variations of the critical temperature of a $S/F$ bilayer or a $F/S/F$ trilayer with the thickness of $F $, either in a quantitative or in a qualitative way, depending on the regime of parameters considered and in particular the sign of the conductances $G_{\phi}^{S}$ and $G_{\phi}^{F}$ used to account for the SDIPS of the $S/F$ interfaces. In the case of a $F/S/F$ spin valve, this last result also holds for the thickness-dependence of the difference between the critical temperatures in the parallel and antiparallel lead configurations. These effects should help to determine the parameters $G_{\phi}^{S}$ and $G_{\phi}^{F}$ of diffusive $S/F$ interfaces. The calculations shown in this paper are also appropriate to the case of thin diffusive $S$ layers contacted to ferromagnetic insulators.

*I thank P. SanGiorgio for showing me his experimental data prior to publication, which stimulated part III.B of this work. I acknowledge discussions with T. Kontos and W. Belzig. This work was supported by grants from the Swiss National Science Foundation and Région Ile-de-France.*

Appendix A: Parameters of a $S/F$ interface from a Dirac barrier model
======================================================================

![Conductances $G_{T}$ (full line), $\left|  G_{\phi}^{S}\right|  $ (dashed lines), and $\left|
G_{\phi}^{F}\right|  $ (dash-dotted lines) reduced by the number of channels $n$, as a function of the spin-averaged barrier strength $Z$ of a $S/F$ interface modeled with a Dirac barrier (see text). The curves were obtained with typical Fermi energies $E_{f}^{F}=0.8~\mathrm{eV}$ and $E_{f}%
^{S}=12~\mathrm{eV}$ in $F$ and $S$ respectively, a spin-polarization $P=0.06$ of the density of states in $F$, and a spin asymmetry $\alpha=0.06$ for the barrier. Note that for the set of parameters used this Figure, one finds $G_{\phi}^{F}<0$ and $G_{\phi}^{S}<0$. However, for other parameters (e.g. by using $\alpha<0$), one can reverse the signs of $G_{\phi}^{F}$ and $G_{\phi
}^{S}$, or obtain opposite signs for $G_{\phi}^{F}$ and $G_{\phi}^{S}$ (not shown).[]{data-label="Dirac"}](Dirac.eps){width="0.85\linewidth"}

The exact values of the conductances $G_{T}$, $G_{\phi}^{F}$ and $G_{\phi}%
^{S}$ of a $S/F$ interface depend on the details of this interface and on the microscopic structure of the contacted materials. Nevertheless, it is already interesting to study a simplified Dirac barrier model which shows that the parameters regime assumed in this paper is, in principle, possible. Neglecting the transverse part of the electrons motion, one finds $r_{n,\sigma}%
^{S(F)}=(k_{S(F)}^{\sigma}-k_{F(S)}^{\sigma}-iZ^{\sigma})/D^{\sigma}$ and $t_{n,\sigma}^{S(F)}=2(k_{S}^{\sigma}k_{F}^{\sigma})^{1/2}/D^{\sigma}$ with $D^{\sigma}=k_{S}^{\sigma}+k_{F}^{\sigma}+iZ^{\sigma}$. Here, $k_{S(F)}%
^{\sigma}$ is the Fermi electronic wavevector in $S(F)$ and $Z^{\sigma}$ is the strength of the Dirac barrier for electrons with spin $\sigma$. I use $\hbar k_{S}^{\sigma}=(2m_{e}E_{f}^{S})^{1/2}$, with $m_{e}$ the free electron mass and $E_{f}^{S}$ the Fermi level in $S$. For $F$, I use an s-band Stoner model, in which $\hbar k_{F}^{\sigma}=(2m_{e}E_{f}^{F}[1\pm2P/(1+P^{2}%
)])^{1/2}$, with $P$ the spin-polarization of the density of states in $F$ and $E_{f}^{F}$ the Fermi level in $F$. I assume that the barrier can have a spin dependence $\alpha=(Z^{\downarrow}-Z^{\uparrow})/(Z^{\uparrow}+Z^{\downarrow
})$ due to the ferromagnetic contact material used to form the interface. The results given by this approach are shown in Fig. \[Dirac\]. The conductances $G_{T}$, $\left|  G_{\phi}^{F}\right|  $ and $\left|  G_{\phi}^{S}\right|  $, reduced by the number of channels $n$, are shown as a function of the spin-averaged barrier strength $Z=(Z^{\uparrow}+Z^{\downarrow})/2$. The three types of conductances go to zero when $Z$ goes to infinity with $\alpha$ constant[@lim]. One can see that in the limit $T_{n}\ll1$ (i.e. here $G_{T}h/ne^{2}\ll1$) and $P\ll1$ in which the boundary conditions (\[BCright\],\[BCleft\]) have been derived, $\left|  G_{\phi}^{F}\right|
$ can be significantly stronger than $G_{T}$, as assumed in Figs. \[DOS1\] and \[DOS2\]. Note that for the set of parameters used in Fig. \[Dirac\], one finds $G_{\phi}^{F}<0$ and $G_{\phi}^{S}<0$. However, for other parameters (e.g. by using $\alpha<0$), one can reverse the signs of $G_{\phi}^{F}$ and $G_{\phi}^{S}$, or obtain opposite signs for $G_{\phi}^{F}$ and $G_{\phi}^{S}$ (not shown). This suggests that there is no fundamental constraint on the signs of $G_{\phi}^{F}$ and $G_{\phi}^{S}$ in the general case.

Appendix B: Effective field effect in a $F$ layer
=================================================

This appendix reconsiders the case of the $S/F$ bilayer of Fig.1.b, in the limit $d_{F}\ll\xi_{F}$ where $\theta_{n,\sigma}(x)$ can be approximated with a quadratic form in the $F$ layer. From Eqs. (\[UsadelF2\]) and (\[BCright\]), one finds, for $x\in\lbrack d_{S}/2,d_{S}/2+d_{F}]$, $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{F}+b_{n,\sigma}^{F}\left[
\frac{d_{S}-2x}{d_{F}}+\left(  \frac{d_{S}-2x}{2d_{F}}\right)  ^{2}\right]$$ with $$\widetilde{\theta}_{n,\sigma}^{F}=\arctan\left(  \frac{\gamma_{T}%
\sin(\widetilde{\theta}_{n,\sigma}^{S})}{\gamma_{T}\cos(\widetilde{\theta
}_{n,\sigma}^{S})+\frac{2\xi_{F}}{d_{F}}\frac{\Omega_{n}}{E_{TH}^{F}}%
+i\gamma_{\phi}^{F}\eta_{n,\sigma}^{P,2}}\right)  \label{ThetaF}%$$$$b_{n,\sigma}^{F}=\frac{\Omega_{n}\sin(\widetilde{\theta}_{n,\sigma}^{F}%
)}{E_{TH}^{F}}%$$ $\Omega_{n}=\left|  \omega_{n}\right|  +iE_{ex}\eta_{n,\sigma}^{P,2}$ and $\widetilde{\theta}_{n,\sigma}^{S}=\theta_{n,\sigma}^{S}(x=d_{S}/2)$. Here, $G_{F}=\sigma_{F}A/d_{F}$ and $E_{TH}^{F}=\hbar D_{F}/d_{F}^{2}$ denote the conductance and Thouless energy of the $F$ layer (note that $d_{F}\ll\xi
_{F}\Longleftrightarrow E_{ex}\ll E_{TH}^{F}$). From completeness, I also give the analogous equations for $x\in\lbrack0,d_{S}/2]$. Assuming that $\theta_{n,\sigma}(x)$ can be approximated with a quadratic form in the $S$ layer, one finds, from Eqs. (\[UsadelS\]) and (\[BCleft\]), $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{S}+b_{n,\sigma}^{S}\left[
\frac{2(2x-d_{S})}{d_{S}}+\left(  \frac{2x-d_{S}}{d_{S}}\right)  ^{2}\right]$$ with $$\widetilde{\theta}_{n,\sigma}^{S}=\mathrm{\arctan}\left(  \frac{\gamma
\gamma_{T}\sin(\widetilde{\theta}_{n,\sigma}^{F})+\frac{4\xi_{S}}{d_{S}}%
\frac{\Delta}{\widetilde{E}_{TH}^{S}}}{\gamma\gamma_{T}\cos(\widetilde{\theta
}_{n,\sigma}^{F})+\frac{4\xi_{S}}{d_{S}}\frac{\left|  \omega_{n}\right|
}{\widetilde{E}_{TH}^{S}}+i\gamma\gamma_{\phi}^{S}\eta_{n,\sigma}^{P,2}%
}\right)  \label{ThetaS}%$$ and $$b_{n,\sigma}^{S}=\frac{\left|  \omega_{n}\right|  \sin(\widetilde{\theta
}_{n,\sigma}^{S})-\Delta\cos(\widetilde{\theta}_{n,\sigma}^{S})}{\widetilde
{E}_{TH}^{S}}%$$ The notations used in the above equations are the same as in Section III. In particular, the Thouless energy and the normal state conductance of the $S$ layer with thickness $d_{S}/2$ are denoted $\widetilde{G}_{S}=2\sigma
_{S}A/d_{S}$ and $\widetilde{E}_{TH}^{S}=4\hbar D_{S}/d_{S}^{2}$, and one has $\eta_{n,\sigma}^{P,2}=\sigma\mathrm{sgn}(\omega_{n})$. In the limit $\widetilde{\theta}_{n,\sigma}^{F}\ll1$, Eq. (\[ThetaS\]) is in agreement with Eqs. (\[thetaS\]) and (\[omegaa\]). The analytic continuation of Eq. (\[ThetaS\]) shows that the $S$ layer is subject to the effective field $H_{eff}=\widetilde{E}_{TH}^{S}G_{\phi}^{S}/\widetilde{G}_{S}$, in agreement with Eq. (\[heff\]). The analytic continuation of Eq. (\[ThetaF\]) shows that the $F$ layer is subject to an analogue effective field $H_{eff}^{F}$, defined by $$g\mu_{B}H_{eff}^{F}=E_{TH}^{F}\frac{G_{\phi}^{F}}{G_{F}}%$$ It is interesting to compare Eqs. (\[ThetaF\]) and (\[thetaF\]). First, note that in the regime $\left|  \widetilde{\theta}_{n,\sigma}^{F}\right|
\ll1$, Eq. (\[ThetaF\]) can be recovered from the low $d_{F}$-limit of Eq. (\[thetaF\]). The interest of Eq. (\[ThetaF\]) is that it is valid beyond the regime of a weak superconducting proximity effect studied in part III. In particular, Eq. (\[ThetaF\]) indicates that, in principle, the field $E_{TH}^{F}$ can occur in a thin $F$ layer for any value of the tunneling conductance $G_{T}$ of the $S/F$ interface. Interestingly, in the case of a thick $F$ layer $d_{F}\geq\xi_{F}/2$, from Eq. (\[thetaF\]), the contribution of $G_{\phi}^{F}$ to the pairing angle $\theta_{n,\sigma}^{F}(x)$ cannot be put anymore under the form of an effective exchange field, due to the non-linearity of the $B_{n,\sigma}^{\mathcal{C},j}$ term. Thus, strictly speaking, the concept of a SDIPS-induced effective field is valid only in the limit of thin metallic layers. However, it might be possible, in principle, to observe reminiscences of the double-gap structure appearing in $N^{F}%
(\varepsilon)$ in the regime of intermediate layer thicknesses $d_{F}\sim
\xi_{F}$ at least, due to the continuity of the equations. Then, one can wonder why this effect does not appear in section III. This is due to the regime of parameters chosen: section III assumes $E_{ex}\gg\Delta_{BCS}$ like in most experiments, and it furthermore focuses on the typical energy range probed in superconducting proximity effect measurements, i.e. $\left|
\varepsilon\right|  \lesssim2\Delta_{BCS}$. In such a regime, one can neglect the term $\left|  \omega_{n}\right|  $ compared to $iE_{ex}\sigma
\mathrm{sgn}(\omega_{n})$ in Eqs. (\[thetaF\]) and (\[ThetaF\]), so that $H_{eff}^{F}$ does not emerge in the model. Accordingly, signatures of the SDIPS-induced effective field $H_{eff}^{F}$ are not likely to be observed in practice.

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With the parameters of Fig. \[FSFTc\], the approximations $b_{n,\sigma}^{P}=\mathcal{L}_{n,\sigma}^{0}$ and $b_{n,\sigma}^{AP}%
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Note that both regimes $d_{F}\leq\xi_{F}$ (see e.g. Ref ) and $d_{F}\geq\xi_{F}$ (see e.g. Ref. ) seem to be accessible in practice.

For $Z\rightarrow+\infty$, $G_{\phi}^{F(S)}$ vanishes because $\arg(r_{n,\sigma}^{F(S)})\rightarrow\pi$ and $\arg(t_{n,\sigma}%
^{F(S)})\rightarrow-\pi/2$ for $\sigma\in\{\uparrow,\downarrow\}$.
