Induced triplet pairing in clean $s$-wave superconductor/ferromagnet layered structures

We study induced triplet pairing correlations in clean ferromagnet/superconductor/ferromagnet heterostructures. The pairing state in the superconductor is the conventional singlet $s$ wave, and the angle $\alpha$ between the magnetizations of the two ferromagnetic layers is arbitrary. We use a numerical fully self-consistent solution of the microscopic equations and obtain the time-dependent triplet correlations via the Heisenberg equations of motion. We find that in addition to the usual singlet correlations, triplet correlations, which are odd in time as required by the Pauli principle, are induced in both the ferromagnets and the superconductor. These time-dependent correlations are largest at times of order of the inverse of the Debye cutoff frequency ${\omega }_D$, and we find that within that time scale, they are often spatially very long ranged. We discuss the behavior of the characteristic penetration lengths that describe these triplet correlations. We also find that the ferromagnets can locally magnetize the superconductor near the interface and that the local magnetization then undergoes strongly damped oscillations. The local density of states exhibits a variety of energy signatures, which we discuss, as a function of ferromagnetic strength and $\alpha$.

Figure 1

(Color online) Schematic of the FSF junction. The $y$ axis is normal to the interfaces. The left ferromagnet layer denoted $F_1$ has a magnetization oriented at an angle $-\alpha /2$ in the $x\text{−}z$ plane, while the other magnet $F_2$ has a magnetization orientation at an angle $\alpha /2$ in the $x\text{−}z$ plane. All layer widths are labeled.

Figure 2

(Color online) The real parts, $f_0$ and $f_1$, of the triplet pair amplitudes ${\mathbf{f}}_{\mathbf{0}}$ and ${\mathbf{f}}_{\mathbf{1}}$ [Eqs. (2.18)], plotted as a function of position (in terms of $Y\equiv k_{F}y$) for three values of $I$ at different times $\tau \equiv {\omega }_{D}t$ indicated in the legends of the top panels. These quantities are normalized to the value of the singlet pairing amplitude in a bulk $S$ material. The main plots show half of the $S$ region (right side of the vertical dashed line) and part of the $F_1$ region. The insets are blow ups of the region near the interface. The angle $\alpha$ is zero in the left panel and $\pi /2$ in the right panel.

Figure 3

(Color online) Penetration depths for the triplet amplitudes [see Eq. (3.1)], which are plotted as a function of $\tau$, as calculated from $f_0$ and $f_1$ in both the $S$ and $F$ regions, for the values of $I$ indicated and the same angles as in Fig. 2.

Figure 4

(Color online) Maximum absolute values of $f_0$ and $f_1$ (see text) as a function of dimensionless time $\tau$ at $I=1$. In the top panel, we consider $f_0$ at both $\alpha =0$ and $\alpha =\pi /2$, while in the bottom panel, we consider $f_1$ at $\alpha =\pi /2$.

Figure 5

(Color online) Imaginary parts, ${\tilde{f}}_0\left(y,t\right)$ and ${\tilde{f}}_1\left(y,t\right)$, of the complex triplet amplitudes ${\mathbf{f}}_{\mathbf{0}}\left(y,t\right)$ and ${\mathbf{f}}_{\mathbf{1}}\left(y,t\right)$. These quantities are normalized and exactly plotted as their corresponding real parts are in Fig. 2, except that here we consider only the $I=1$ case, and both plots are for $\alpha =\pi /2$. As in Fig. 2, the main plot shows the behavior over an extended region. There are now two insets to each main plot, each showing the detailed behavior near the interface itself, on either the $S$ or $F$ side.

Figure 6

(Color online) Magnetic moment component $m_x$ [see Eq. (2.21)], which is normalized as explained in the text, plotted vs position (at $I=1$) for several values of $\alpha$. The main plot shows the behavior near the interface (vertical dashed line), while the inset covers the whole sample.

Figure 7

(Color online) Normalized (see text) magnetic moment component $m_z$ [Eq. (2.20)] plotted vs dimensionless position for three values of $I$. Again, the main plot is the behavior near the interface, while the insets cover the whole sample. It is evident that $m_z$ points in the same direction for both magnets.

Figure 8

(Color online) Normalized local DOS from Eq. (2.22) integrated over (see text) over either the $F$ region (left panels) or the $S$ region (right panels) for three values of $I$ and several relative magnetization orientations. The temperature is at $T=0.01T_c$.

Figure 9

(Color online) Temperature and position dependence of $f_0$ and $f_1$ at $I=0.5$ and $\tau =4$. In the top panel $\alpha =0$ and in the bottom panel $\alpha =\pi /2$. The values of $T/T_c$, where $T_c$ is the transition temperature of bulk $S$, are indicated. The main plots show a rather wide region near the interface (vertical dashed line) and the insets focus on the region very near the interface.

Figure 10

(Color online) The maximum values of the real parts of the triplet amplitudes at $I=0.5$ and $\tau =4$ as a function of temperature and the same values of $\alpha$ as in Fig. 9. The inset shows the corresponding peak values of the ordinary singlet amplitude, $f_3\equiv \Delta \left(y\right)/g$ [Eq. (2.9)].

Figure 11

(Color online) Temperature and position dependence of the ordinary singlet pair amplitude (or $\Delta \left(y\right)/g$), which is normalized to its value in bulk $S$ material, for moderate magnetic strength, $I=0.5$. Top panel: the magnetizations are parallel $\left(\alpha =0\right)$. Bottom panel: the magnetizations are perpendicular $\left(\alpha =\pi /2\right)$.