Phase separation and proximity effects in itinerant ferromagnet/superconductor heterostructures

Heterostructures made of itinerant ferromagnets and superconductors are studied. In contrast to most previous models, ferromagnetism is not enforced by an effective Zeeman field but induced in a correlated single-band model (CSBM) that displays itinerant ferromagnetism as a mean-field ground state. In this model superconductivity and magnetism are both calculated self-consistently. We calculate the magnitude of the magnetization, the superconducting correlations, and variations of the charge density self-consistently for a superconducting-magnetic bilayer by solving the Bogoliubov–de Gennes equations on a two-dimensional lattice. We determine all three quantities as a function of the Coulomb repulsion $U$ and the ferromagnetic exchange interaction $J$. The CSBM displays a variety of features not present in the Zeeman exchange model—for example, the occurrence of electronic phase separation and the competition of magnetic and superconducting orders far away from the interface.

Figure 1

Sketch of the SF bilayer proximity system with periodic boundary conditions along the $x$ and $y$ directions. $L_{x,y}=N_{x,y}a$ where $a$ denotes the lattice constant which in the following is set to $a\equiv 1$.

Figure 2

Ground state energy vs spin-spiral modulation vector $\mathbf{q}\left[S_i=S_{0}exp\left(i\mathbf{q}·{\mathbf{R}}_i\right)\right]$ along the diagonal direction of the Brillouin zone at half filling (a) and density $n=0.9$ (b). Note that for other $U_F/t$ the results are qualitatively similar but energy variations decrease with decreasing $U_F/t$. (c) Phase diagram in the $(U_F/t,J/t)$ plane for density $n=0.2$. (d) Chemical potential vs density. The horizontal dotted lines are determined by the Maxwell construction. The compressibility diverges at the local extrema.

Figure 3

Charge density (a), magnetization (b), and singlet pair correlations (c) as a function of $x$ and various values of the exchange coupling $J/t$ in the ferromagnetic region. For all cases the local potential $V^{\mathrm{loc}}$ is adjusted in such a way as to obtain a similar charge density deep in the S layer. The thin dashed green line reports the result within the EFM for $h_{i,z}/t=3$ (see text). Parameters: $U_F/t=2$, $U_S/t=-2$, $n=0.625$; $f_B\approx 0.1$ is the bulk value of $f_0$.

Figure 4

Close look near the interface of the charge density (a), magnetization [(b), (c)], and singlet pair correlations (d) shown in Fig. 3 for various values of the exchange coupling $J/t$ in the F. For all cases the local potential $V^{\mathrm{loc}}$ is adjusted as described in Fig. 3. Shown for comparison as a thin dashed green line is also the pair correlation in the EFM for $h_{i,z}/t=3$. The S correlations inside the F in panel (d) have been fitted with Eq. (11) (light red and black lines; see text for fitting parameters). Parameters: $U_F/t=2$, $U_S/t=-2$, $n=0.625$, $f_B\approx 0.1$.

Figure 5

Charge density (a), magnetization [(b), (c)], and singlet pair correlations (d) close to the interface as in Figs. 3 and 4 but for a larger value of the Coulomb potential in the F, $U_F/t=5$. The phase separation regime is substantially extended when compared to Figs. 3 and 4. Both pair correlations in the F and the charge and magnetic configurations in S are rapidly suppressed away from the interface. Parameters: $U_F/t=5$, $U_S/t=-2$, $n=0.625$, $f_B\approx 0.1$.

Figure 6

Local density of states (a) inside the S (black; $i_x=60$) and the F (red; $i_x=110$) for $J/t=0.5$. Other parameters: $U_F/t=2$, $U_S/t=-2$, $n=0.625$.

Figure 7

Position- and frequency-dependent singlet correlations, Eq. (13), for (a) $J/t=0.35$ and (b) $J/t=0.7$. Pair correlations oscillate both in space and frequency and extend deeper into the F for small $J/t$. $i_x$ indexes the site as in Fig. 3. A horizontal cut at $\omega =0$ for these two values of $J/t$ is shown in Fig. 3. Vertical cuts at fixed values of $i_x$ are shown in Fig. 8. Parameters: $U_S/t=-2$, $U_F/t=2,n=0.625$, $f_B\approx 0.1$.

Figure 8

Imaginary part of the singlet Gor'kov pair correlation $F(i_x,\omega )$ dependence on frequency at specific points in the heterostructure and for various exchange couplings as indicated in the panels. The correlations at $x=79$ and $x=80$ are in the S while those for $x\ge 81$ are in the F. Note that the singlet order parameter for $J/t=0.5$ and $x=81$ has opposite sign as compared to the other couplings [cf. panel (c) of Fig. 3]. For better comparison we have therefore multiplied the $J/t=0.5$ in panel (c) by $-1$, symbolized by the dashed line. Parameters: $U_F/t=2$, $U_S/t=-2$, $n=0.625$, $f_B\approx 0.1$.