Spin stiffness and domain walls in Dirac-electron mediated magnets

Spin interactions of magnetic impurities mediated by conduction electrons is one of the most interesting and potentially useful routes to ferromagnetism in condensed matter. In recent years, such systems have received renewed attention due to the advent of materials in which Dirac electrons are the mediating particles, with prominent examples being graphene and topological insulator surfaces. In this paper, we demonstrate that such systems can host a remarkable variety of behaviors, in many cases controlled only by the density of electrons in the system. Uniquely characteristic of these systems is an emergent long-range form of the spin stiffness when the Fermi energy $\mu$ resides at a Dirac point, becoming truly long-range as the magnetization density becomes very small. It is demonstrated that this leads to screened Coulomb-like interactions among domain walls, via a subtle mechanism in which the topology of the Dirac electrons plays a key role: the combination of attraction due to bound in-gap states that the topology necessitates and repulsion due to scattering phase shifts yields logarithmic interactions over a range of length scales. We present detailed results for the bound states in a particularly rich system, a topological crystalline insulator surface with three degenerate Dirac points and one energetically split off. This system allows for distinct magnetic ground states, which are either twofold or sixfold degenerate, with either short-range or emergent long-range interactions among the spins in both cases. Each of these regimes is accessible, in principle, by tuning the surface electron density via a gate potential. A study of the Chern number associated with different magnetic ground states leads to predictions for the number of in-gap states that different domain walls should host, which we demonstrate using numerical modeling are precisely borne out. The nonanalytic behavior of the stiffness on magnetization density is shown to have a strong impact on the phase boundary of the system and opens a pseudogap regime within the magnetically ordered region. We thus find that the topological nature of these systems, through its impact on domain wall excitations, leads to unique behaviors distinguishing them markedly from their nontopological analogs.

Figure 1

(a) Illustration of length scale $\xi$ over which impurity spins are coupled in the two-dimensional systems we consider, which grows faster than the distance between impurities as the impurity density $n_{\mathrm{imp}}$ becomes small. (b) Illustration of a pair of domain walls separated by a distance $d$.

Figure 2

Summary of different magnetization behaviors expected for a (111) TCI surface. In (a) and (b), magnetization aligns in such a way that the Dirac cone at the $\overline{\mathrm{\Gamma }}$ point develops a large gap. Filling by electrons (red regions) in (a) yields emergent long-range interactions, while in (b), they are short-range. This situation arises when the Fermi energy resides near the Dirac point energy of the $\overline{\mathrm{\Gamma }}$ point. (c) and (d) represent the analogous situations when the magnetization aligns to create a large gap at one of the $\overline{M}$ points, which is expected when the Fermi energy is near these Dirac point energies. The expected of dependence on $T_c$ on the magnetic impurity concentration $n_{\mathrm{imp}}$ will vary linearly or quadratically depending on whether there are long- or short-range spin-gradient interactions in the system. (Inset) Locations of Dirac points in the surface Brillouin zone.

Figure 3

Numerical result for $G\left(u\right)$ [Eq. (20)] as a function of $u\equiv b_0/g$. This function determines how the energy of the system increases when a spin gradient of wave vector $g$ is introduced, which for $g\ll b_0$ is quadratic in $g$ but only linear in $b_0$, while for $g\gg b_0$, one obtains behavior linear in $g$.

Figure 4

Numerical calculation of energy per unit length required for single overturn of the staggered magnetization $\mathrm{\Delta }$ in a graphene ribbon of width $L_w$, relative to that with a uniform staggered magnetization, for various values of $L_w$ and $\mathrm{\Delta }$. When $L_w$ is held fixed, this energy grows linearly with $\mathrm{\Delta }$. For fixed $\mathrm{\Delta }$, the energy approaches a constant as $1/L_w$ grows.

Figure 5

Piecewise constant domain wall configuration with two domain walls of width $w$ each separated by a center-to-center distance $d$. (Top) The parameter $b_x$ for which we allow the possibility of values that are equal or opposite, allowing for magnetizations that rotate with the same or with opposite senses. Piecewise constant regions I–V are labeled. (Bottom) $\mathrm{\Delta }\left(x\right)$, illustrating that the magnetization rotates from up to down and back again. Arrows indicate the orientation of the magnetization vector in each region.

Figure 6

DW pair energy (adding the contributions from both the bound state and phase shift) as a function of the distance between the domain walls $d$ when $\mu$ is in the gap. Different lines indicate different values of ${\mathrm{\Delta }}_0$, as indicated. Energies expressed in units of $e_0\equiv \hslash v_F/a_0$.

Figure 7

DW pair energy (adding the contributions from both the bound state and phase shift) as a function of the distance between the domain walls $d$ when ${\mathrm{\Delta }}_0=0.005$ for various values of $\mu$, as indicated. Energies expressed in units of $e_0\equiv \hslash v_F/a_0$.

Figure 8

Contributions to DW pair energy from bound states and phase shifts shown separately, for ${\mathrm{\Delta }}_0=0.005e_0$ and for various values of chemical potential as indicated. Energies expressed in units of $e_0\equiv \hslash v_F/a_0$. The near cancellation of the two contributions is apparent.

Figure 9

Energy of DW superlattice as a function of separation $d$ for graphene, for different values of $\mu$ and ${\mathrm{\Delta }}_0$ as indicated, in energy units of $e_0\equiv \hslash v_F/a_0$. Domain walls are one unit cell wide. Note, $x$ axis is on a logarithmic scale in (a), but on a linear scale in the other figures.

Figure 10

(a) The fcc Brillouin zone containing $L_i(i=0,1,2,3)$ points and their projections onto the (111) surface, which yield the $\overline{\mathrm{\Gamma }}$ and ${\overline{M}}_i(i=1,2,3)$ points. (b) Extended real space unit cells with two atoms per unit cell, used in constructing a domain wall. (c) Surface Brillouin zone for the two surface atom unit cell, which folds the original hexagonal Brillouin zone for the single atom (real space) unit cell into a rectangular one.

Figure 11

The band structure around $\overline{\mathrm{\Gamma }}$ and $\overline{M}$ with magnetic moment $\left|J\mathbf{S}\right|=0.05$. A small potential gradient has been introduced to lift the surface degeneracy.

Figure 12

The Berry's curvature for top surface state around $\overline{\mathrm{\Gamma }}$ point with the magnetic moment $\left|J\mathbf{S}\right|$ as indicated.

Figure 13

The Berry's curvature for top surface state around $\overline{M}$ point with the magnetic moment $\left|J\mathbf{S}\right|$ as indicated.

Figure 14

The extrapolation of Chern number with magnetic moment $\left|J\mathbf{S}\right|$ for top surface state. The bottom surface state has opposite behavior, i.e., the values are opposite in sign.

Figure 15

(a) Energy bands $E_n$ near $\overline{\mathrm{\Gamma }}$ when a Neel domain wall of width $d=0$ runs along $k_2$. The electron density $n\left(i_1\right)$ for the representative DW states (indicated by blue cross points and numbered $1,2,\cdots )$ along the unit cell direction $a_1$ are shown in (b) and (c) for top and bottom surfaces, respectively, of the slab with (111) surfaces. The components of magnetic moments $b_z$ along $\mathrm{\Gamma }\text{−}{L}_0$ direction are shown by red arrows between (b) and (c).

Figure 16

Energy bands $E_n$ (a) and electron densities [(b) and (c)] of bound DW states near the $\overline{Y}$ point for the DW configuration as described in Fig. 15.

Figure 17

(a) Energy bands $E_n$ near $\overline{\mathrm{\Gamma }}$ when a Neel domain wall of width $d=10$ runs along $k_2$. The electron density $n\left(i_1\right)$ for DW states (indicated by blue cross points and numbered $1,2,\cdots )$ along the unit cell direction $a_1$ are shown in (b) and (c) for the top and bottom surfaces, respectively, of the slab with (111) surfaces. The component of magnetic moments $b_z$ along $\mathrm{\Gamma }\text{−}{L}_0$ and $b_x$ along $a_1$ directions are shown by red arrows between (b) and (c).

Figure 18

Energy bands $E_n$ (a) and electron densities [(b) and (c)] of bound DW states near the $\overline{Y}$ point for the DW configuration as described in Fig. 17.

Figure 19

(a) DW configurations connecting (i) $\mathrm{\Gamma }\text{−}{L}_1$ to $\mathrm{\Gamma }\text{−}{L}_2$ directions (magenta) and (ii) $\mathrm{\Gamma }\text{−}{L}_1$ to $\mathrm{\Gamma }\text{−}{L}_3$ directions (blue). (b) The energy difference between these configurations $\mathrm{\Delta }E=E_{ii}-E_i$ as a function of chemical potential $\mu$ for $d=0,10$ and $\left|J\mathbf{S}\right|=0.01$ showing minimum when $\mu$ is close to $\overline{M}$ Dirac point energy $E_{\overline{M}}$. (c) $\mathrm{\Delta }E$ for larger $\left|J\mathbf{S}\right|=0.10$.

Figure 20

(a) Energy bands $E_n$ at $\overline{\mathrm{\Gamma }}$ for DW configuration corresponding to (ii) in Fig. 19, with $d=0$. The density of electron $n\left(i_1\right)$ for the representative DW states (indicated by blue cross points and numbered $1,2,\cdots )$ along the unit cell direction $a_1$ are shown in (b) and (c) for top and bottom surfaces respectively of the slab with (111) surfaces.

Figure 21

(a) Energy bands $E_n$ near $\overline{Y}$ for DW configuration corresponding to (ii) in Fig. 19, with $d=0$, for momentum along $k_2$. The electron density $n\left(i_1\right)$ for the representative DW states (indicated by blue cross points and numbered $1,2,\cdots )$ along the unit cell direction $a_1$ are shown in (b) and (c) for top and bottom surfaces respectively of the slab with (111) surfaces.

Figure 22

Schematic phase diagram for classical magnetic impurities coupled by surface Dirac electrons, contrasting behavior of the critical temperature $T_c$ vs impurity density $n_{\mathrm{imp}}$ when the Fermi energy $\mu$ passes through a Dirac point (red) with when it does not (blue). Dashed illustrates illustrates a trajectory in which the in-gap states associated with domain walls will produce pseudogap behavior in the electronic spectrum, and where the interactions among the DW's become increasingly long-range moving down the trajectory, enhancing the density of states within the mean-field gap.