Magnetic impurities in a superconductor: Effect of domain walls and interference

We consider the effect of magnetic impurities, modeled by classical spins, in a conventional superconductor. We study their effect on the quasiparticles, specifically on the spin density and local density of states. As previously emphasized, the impurities induce multiple scatterings of the quasiparticle wave functions, leading to complex interference phenomena. Also, the impurities induce quantum phase transitions in the many-body system. Previous authors studied the effect of either a small number of impurities (from 1 to 3) or a finite concentration of impurities, typically in a disordered distribution. In this work, we assume a regular set of spins distributed inside the superconductor in such a way that the spins are oriented, forming different types of domain walls, assumed stable. This situation may be particularly interesting in the context of spin transfer due to polarized currents traversing the material.

Figure 1

Various Néel-type domain walls considered in this paper.

Figure 2

(Color online) (a) Energy levels for one impurity. There is one pair of bound states originated by the impurity. As $J$ grows, the level appears to “cross” zero energy. The crossing is between $J=1$ and $J=2$ (see below). (b) Energy levels for DW1 domain wall.

Figure 3

(Color online) Lowest energy level as a function of $J$ for the various domain walls. Note the various level crossings (near zeros) for the various cases. The crossings are concentrated for $1<J<2$. Note that for the antiferromagnetic chain, there is a single level crossing. Domain walls DW1 and DW2 have the same lowest state energy and the same happens with domain walls DW3 and DW4.

Figure 4

Comparison of the spatial dependence of the gap function for the case of a single impurity, for the cases (a) $15\times 15$ and (b) $25\times 25$ for open boundary conditions and (c) periodic boundary conditions, for $J=1$.

Figure 5

$\Delta$ for $J=2$ for one impurity. $\Delta$ decreases at the impurity site (this is the same as for a nonmagnetic impurity). Note that there is a $\pi$ shift in $\Delta$ (it becomes negative) at the impurity site in contrast to the case for $J=1$, shown in Fig. 4b.

Figure 6

$\Delta$ for DW1 for $J=1,2$. Note that for $J=2$, there is again a $\pi$ shift.

Figure 7

$\Delta$ for DW4 for $J=1,2$. Once again there is a $\pi$ shift.

Figure 8

[(a) and (b)] Plots of $s_z$, the local $z$ component of the electronic spin density for the case of two impurities for $J=1,2$. [(c) and (d)] Plot of $s_z$ for $J=1,2$ for DW1. Note that along the line of spins constituting the domain wall, the spin density is large and positive and that for $J=1$ in the neighbors it is negative, and for the other case $J=2$ it is positive. As for the cases of one and two impurities, this transition is associated with a transition from $s_z^T=0$ to an $s_z^T\ne 0$. [(e) and (f)] Plot of $s_z$ for $J=1,2$ for DW4. Note that now the total magnetizations are zero but if we divide the system in two-halves, similar considerations apply.

Figure 9

(Color online) Various quantities for DW1 as a function of $J$. We plot $s_z^T$, $s_x^T$, $s_z^l$, $s_x^l$, $s_z^r$, and $s_x^r$. Here, $l$ and $r$ stand for left half of the system and right half of the system. Due to the symmetry of the domain wall chosen, $s_x^T=0$. Note, however, that if we consider the magnetization of the left or right halves, there are interesting phase transitions.

Figure 10

(Color online) Various quantities for DW2 as a function of $J$. We plot $s_z^T$, $s_x^T$, $s_z^l$, $s_x^l$, $s_z^r$, and $s_x^r$.

Figure 11

(Color online) Various quantities for DW3 as a function of $J$. We plot $s_z^T$, $s_x^T$, $s_z^l$, $s_x^l$, $s_z^r$, and $s_x^r$.

Figure 12

(Color online) Various quantities for DW4 as a function of $J$. We plot $s_z^T$, $s_x^T$, $s_z^l$, $s_x^l$, $s_z^r$, and $s_x^r$.

Figure 13

(Color online) Various quantities for the AF spin configuration (DW5) as a function of $J$. We plot $s_z^T$, $s_x^T$, $s_z^l$, $s_x^l$, $s_z^r$, and $s_x^r$.

Figure 14

LDOS for the lowest energy state (with positive energy) and the second lowest state (also with positive energy) for $J=1$ in the case of a single impurity.

Figure 15

LDOS for the lowest energy state (with positive energy) for $J=2,6$ in the case of a single impurity.

Figure 16

LDOS for DW1 for the first seven levels for $J=1$ and a state in the continuum.

Figure 17

LDOS for DW1 for the first seven levels for $J=2$ and a state in the continuum.

Figure 18

LDOS for a single impurity. (a) ${\rho }_{+}({ϵ}_1,i,\uparrow )$ for $J=1$, (b) ${\rho }_{-}({ϵ}_1,i,\downarrow )$ for $J=1$, (c) ${\rho }_{+}({ϵ}_1,i,\downarrow )$ for $J=2$, and (d) ${\rho }_{-}({ϵ}_1,i,\uparrow )$ for $J=2$. (Note that in the vertical axis, labels $u=\uparrow$ and $d=\downarrow$.)

Figure 19

LDOS for the domain wall DW1. (a) ${\rho }_{+}({ϵ}_1,i,\uparrow )$ for $J=1$, (b) ${\rho }_{+}({ϵ}_1,i,\downarrow )$ for $J=1$, (c) ${\rho }_{-}({ϵ}_1,i,\uparrow )$ for $J=1$, and (d) ${\rho }_{-}({ϵ}_1,i,\downarrow )$ for $J=1$. (Note that in the vertical axis, label $u=\uparrow$ and $d=\downarrow$.)

Figure 20

LDOS for the domain wall DW1. (a) ${\rho }_{+}({ϵ}_3,i,\uparrow )$ for $J=1$, (b) ${\rho }_{+}({ϵ}_3,i,\downarrow )$ for $J=1$, (c) ${\rho }_{-}({ϵ}_3,i,\uparrow )$ for $J=1$, and (d) ${\rho }_{-}({ϵ}_3,i,\downarrow )$ for $J=1$. (Note that in the vertical axis, label $u=\uparrow$ and $d=\downarrow$.)

Figure 21

LDOS for the domain wall DW1. (a) ${\rho }_{+}({ϵ}_1,i,\uparrow )$ for $J=4$, (b) ${\rho }_{+}({ϵ}_1,i,\downarrow )$ for $J=4$, (c) ${\rho }_{-}({ϵ}_1,i,\uparrow )$ for $J=4$, and (d) ${\rho }_{-}({ϵ}_1,i,\downarrow )$ for $J=4$. (Note that in the vertical axis, label $u=\uparrow$ and $d=\downarrow$.)

Figure 22

(Color online) Hopping matrix elements for up spin electrons from a bulk point and the center point, in the horizontal (along $x$) and vertical (along $z$) directions.

Figure 23

(Color online) Hopping matrix elements for down spin electrons from a bulk point and the center point, in the horizontal (along $x$) and vertical (along $z$) directions.

Figure 24

(Color online) Profile of $S_x$ of the impurity spins for DW1 and for for $J_f=0.1,1,4$ as a function of $J$.

Figure 25

(Color online) $⟨\Delta ⟩$ and $s_z^T$ for one impurity and DW1 as a function of temperature for various values of $J$.

Figure 26

(Color online) Effect of temperature on the results for the various magnetizations (total and left or right) for DW1.