Proximity effects in superconductor-ferromagnet heterostructures

The proximity effect at superconductor-ferromagnet interfaces produces damped oscillatory behavior of the Cooper pair wave function within the ferromagnetic medium. This is analogous to the inhomogeneous superconductivity, predicted long ago by Fulde and Ferrell (P. Fulde and R. A. Ferrell, 1964, “Superconductivity in a strong spin-exchange field,” Phys. Rev. 135, A550–A563), and by Larkin and Ovchinnikov (A. I. Larkin and Y. N. Ovchinnikov, 1964, “Inhomogeneous state of superconductors,” Zh. Eksp. Teor. Fiz. 47, 1136–1146 [Sov. Phys. JETP 20, 762–769 (1965)]), and sought by condensed-matter experimentalists ever since. This article offers a qualitative analysis of the proximity effect in the presence of an exchange field and then provides a description of the properties of superconductor-ferromagnet heterostructures. Special attention is paid to the striking nonmonotonic dependence of the critical temperature of multilayers and bilayers on the ferromagnetic layer thickness as well as to the conditions under which “$\pi$” Josephson junctions are realized. Recent progress in the preparation of high-quality hybrid systems has permitted the observation of many interesting experimental effects, which are also discussed. Finally, the author analyzes the phenomenon of domain-wall superconductivity and the influence of superconductivity on the magnetic structure in superconductor-ferromagnet bilayers.

Figure 1

The $(T,H)$ phase diagram for the 3D superconductor. At temperatures below $T^{*}=0.56T_c$ the second-order transition occurs from the normal to the nonuniform superconducting FFLO phase. The bold line corresponds to the first-order transition into the uniform superconducting state, and the dotted line represents the second-order transition into the nonuniform superconducting state.

Figure 2

Energy band of the 1D superconductor near the Fermi energy. Due to the Zeeman splitting the energy of the electrons with spin orientation along the magnetic field (↑) decreases, dotted line, while the energy of the electrons with the opposite spin orientation (↓) increases, dotted line. The splitting of the Fermi momenta is $\pm \delta k_F$, where $\delta k_F={\mu }_{B}H∕v_F$. The Cooper pair comprises one electron with spin (↑) and momentum $k_F+\delta k_F$, and another electron with spin (↓) and momentum $-k_F+\delta k_F$. The resulting momentum of the Cooper pair is nonzero: $k_F+\delta k_F+(-k_F+\delta k_F)=2\delta k_F\ne 0$.

Figure 3

Schematic behavior of the superconducting order parameter near the (a) superconductor-normal metal and (b) superconductor-ferromagnet interfaces. The continuity of the order parameter at the interface implies the absence of the potential barrier. In the general case at the interface the jump of the superconducting order parameter occurs.

Figure 4

Measurements of the differential conductance by 125 for two $\mathrm{Al}∕{\mathrm{Al}}_2{\mathrm{O}}_3∕\mathrm{Pd}\mathrm{Ni}∕\mathrm{Nb}$ junctions with two different thicknesses (50 and $75\mathrm{\AA }$) of the ferromagnetic PdNi layer. A $1500\text{−}\mathrm{\AA }$-thick aluminum layer was evaporated on SiO and then quickly oxidized to produce a ${\mathrm{Al}}_2{\mathrm{O}}_3$ tunnel barrier. Tunnel junction areas were defined by evaporating $500\mathrm{\AA }$ of SiO through masks. A PdNi thin layer was deposited and then backed by a Nb layer.

Figure 5

Experimental data of 119 on the oscillation of the critical temperature of $\mathrm{Nb}∕\mathrm{Gd}$ multilayers vs thickness of Gd layer $d_G$ for two different thicknesses of Nb layers: (a) $d_{\mathrm{Nb}}=600\mathrm{\AA }$ and (b) $d_{\mathrm{Nb}}=500\mathrm{\AA }$. Dashed line in (a) is a fit by the theory of 172.

Figure 6

S/F multilayer. The $x$ axis is chosen perpendicular to the planes of the S and F layers with the thicknesses $2d_s$ and $2d_f$, respectively. (a) The curve $\Psi \left(x\right)$ represents schematically the behavior of the Cooper pair wave function in the 0 phase. Due to symmetry the derivative of $\Psi$ (and $F$) is zero at the centers of the S and F layers. The case of the 0 phase is equivalent to the S/F bilayer with the S and F layer thicknesses $d_s$ and $d_f$, respectively. (b) The Cooper pair wave function in the $\pi$ phase vanishes at the center of the F layers and $\Psi \left(x\right)$ is antisymmetric toward the center of the F layer.

Figure 7

The dependence of the critical temperature on the thickness of the F layer for the 0 phase (solid line) and the $\pi$ phase (dotted line) for the transparent S/F interface. Note that the highest transition temperature $T_c^{*}$ corresponds to the lowest point. The dimensionless thickness of the F layer $2y=2d_f∕{\xi }_f$ and the first transition from the 0 to the $\pi$ phase occurs at $2d_f=2.36{\xi }_f$. The parameter is ${\tau }_0=(2d_s{\xi }_f∕D_s){\sigma }_s∕{\sigma }_f$.

Figure 8

The critical temperature of the 0 phase (solid line) and the $\pi$ phase (dashed line) as a function of the dimensionless thickness of the F layer $2y=2d_f∕{\xi }_f$ for different S/F interface barriers $\tilde{\gamma }={\gamma }_B({\xi }_n∕{\xi }_f)$. (a) The dimensionless pair-breaking parameter ${\tilde{\tau }}_0=4\pi T_c(2d_s{\xi }_f∕D_s){\sigma }_s∕{\sigma }_f=21$. (b) The dimensionless pair-breaking parameter ${\tilde{\tau }}_0=20.05$.

Figure 9

Variation of the critical temperature of the $\mathrm{Nb}∕{\mathrm{Cu}}_{0.43}{\mathrm{Ni}}_{0.57}$ bilayer with the F layer thickness (176). The theoretical fit (84) gives an exchange field value $h\sim 130\mathrm{K}$ and an interface transparency parameter ${\gamma }_B\sim 0.3$.

Figure 10

Geometry of the $\mathrm{S}∕\mathrm{F}∕\mathrm{S}$ junction. The thickness of the ferromagnetic layers is $2d_f$ and both S/F interfaces have the same transparencies, given by the coefficient ${\gamma }_B$.

Figure 11

Critical current of the $\mathrm{S}∕\mathrm{F}∕\mathrm{S}$ Josephson junction near $T_c$ as a function of the dimensionless thickness of the F layer $2y=2d_f∕{\xi }_f$. There are no barriers at the S/F interfaces $({\gamma }_B=0)$, $R_n$ is the resistance of the junction, and $V_0=\pi {\Delta }^2∕2e{T}_c$.

Figure 12

Temperature dependences of the critical thickness $2d_f^c$ of the F layer, corresponding to the crossover from the 0 to the $\pi$ phase in the limit of very small boundary transparency for different values of the exchange field.

Figure 13

Nonmonotonic temperature dependences of the normalized critical current for the low transparency limit. Curve 1, $h∕T_c=10$ and $2d_f∕{\xi }_f=0.84$; curve 2, $h∕T_c=40$ and $2d_f∕{\xi }_f=0.5$; curve 3, $h∕T_c=100$ and $2d_f∕{\xi }_f=0.43$.

Figure 14

Critical current $I_c$ as a function of temperature for ${\mathrm{Cu}}_{0.48}{\mathrm{Ni}}_{0.52}$ junctions with different F layer thicknesses $2d_F$. At the thickness of the F layer of $27\mathrm{nm}$ the temperature mediated transition between the 0 and $\pi$ phase occurs. Adapted from 177.

Figure 15

Critical current $I_c$ at $T=4.2\mathrm{K}$ of ${\mathrm{Cu}}_{0.47}{\mathrm{Ni}}_{0.53}$ junctions as a function of the F layer thickness (174). Two $0\text{−}\pi$ transitions are revealed. The theoretical fit corresponds to Eq. (A9) in the Appendix, Sec. 2, taking into account the presence of magnetic scattering with $\alpha =1∕{\tau }_{s}h=1.33$ and ${\xi }_f=2.4\mathrm{nm}$. The inset shows the temperature mediated $0\text{−}\pi$ transition for an F layer thickness of $11\mathrm{nm}$.

Figure 16

Experimental points correspond to critical current measurements, by 124, vs the PdNi layer thickness. The theoretical curve is the fit of 47. The fitting parameters are ${\xi }_f\sim 30\mathrm{\AA }$ and $\pi {\Delta }^2∕e{T}_c\sim 110\mu \mathrm{V}$.

Figure 17

Experiments of 100 on the diffraction pattern of SQUID with 0 and $\pi$ junctions. There is no shift of the pattern between 0-0 and $\pi \text{−}\pi$ SQUIDs. The ${\Phi }_0∕2$ shift is observed between $0\text{−}\pi$ and 0-0 or $\pi \text{−}\pi$ SQUIDs. The 0 and $\pi$ junctions were obtained by varying the PdNi layer thickness.

Figure 18

Earlier observation by 77 of the spin-valve effect on In film between oxidized FeNi and Ni layers. The resistive measurements of the critical temperature are presented in zero field: dashed line, after application of the $1\text{−}\mathrm{T}$ field parallel to the ferromagnetic layers; solid line, after application of the $-1\text{−}\mathrm{T}$ field and subsequently $+0.03\text{−}\mathrm{T}$ field to return the magnetization of the FeNi layer.

Figure 19

Geometry of the $\mathrm{F}∕\mathrm{S}∕\mathrm{F}$ sandwich. The thickness of the S layer is $2d_s$ and two F layers have identical thicknesses $d_f$.

Figure 20

Influence of the S/F interface transparency [parameter $\tilde{\gamma }={\gamma }_B({\xi }_n∕{\xi }_f)$] on the $T_c^{*}$ vs $d_f$ dependence (17). The thickness of the F layer is normalized by ${\xi }_f$. The dimensionless pair-breaking parameter ${\tilde{\tau }}_0=4\pi T_c(2d_s{\xi }_f∕D_s){\sigma }_s∕{\sigma }_f$ is chosen constant and equal to 4. The solid line corresponds to the antiparallel case, and the dashed line to the parallel case. One can distinguish four characteristic types of behavior: (a) weakly nonmonotonic decay to a finite value $T_c^{*}$, (b) reentrant behavior for the parallel orientation, and (c) and (d) monotonic decay to $T_c^{*}=0$ with (d) or without (c) switching to a first-order transition in the parallel case. In (d), the dotted line represents schematically the first-order transition line.

Figure 21

The calculated dependence of the superconducting transition temperature vs inverse reduced half thickness $d^{*}∕d_s$ of the superconducting layer for parallel and antiparallel alignments for the transparent interface $({\gamma }_B=0)$ and thick ferromagnetic layer $(d_f⪢{\xi }_f)$. The effective length is $d^{*}=({\sigma }_f∕{\sigma }_s)(D_s∕4\pi T_c){(h∕D_f)}^{1∕2}$.

Figure 22

The $(T,h)$ phase diagram of the atomic S/F multilayer in the limit of the small transfer integral $t⪡T_c$.

Figure 23

S/F bilayer with domain structure in the ferromagnetic layer. The period $D$ of the domain structure $(D=2\pi ∕Q)$ is smaller than the superconducting coherence length ${\xi }_s$.

Figure 24

The ferromagnetic film with perpendicular anisotropy on a superconducting substrate.