Topological superconductivity induced by ferromagnetic metal chains

Recent experiments have provided evidence that one-dimensional (1D) topological superconductivity can be realized experimentally by placing transition-metal atoms that form a ferromagnetic chain on a superconducting substrate. We address some properties of this type of system by using a Slater-Koster tight-binding model to account for important features of the electronic structure of the transition-metal chains on the superconducting substrate. We predict that topological superconductivity is nearly universal when ferromagnetic transition-metal chains form straight lines on superconducting substrates and that it is possible for more complex chain structures. When the chain is weakly coupled to the substrate and is longer than superconducting coherence lengths, its proximity-induced superconducting gap is $\sim \Delta E_{\text{SO}}/J$ where $\Delta$ is the $s$-wave pair potential on the chain, $E_{\text{SO}}$ is the spin-orbit splitting energy induced in the normal chain state bands by hybridization with the superconducting substrate, and $J$ is the exchange splitting of the ferromagnetic chain $d$ bands. Because of the topological character of the 1D superconducting state, Majorana end modes appear within the gaps of finite length chains. We find, in agreement with the experiment, that when the chain and substrate orbitals are strongly hybridized, Majorana end modes are substantially reduced in amplitude when separated from the chain end by less than the coherence length defined by the $p$-wave superconducting gap. We conclude that Pb is a particularly favorable substrate material for ferromagnetic chain topological superconductivity because it provides both strong $s$-wave pairing and strong Rashba spin-orbit coupling, but that there is an opportunity to optimize properties by varying the atomic composition and structure of the chain. Finally, we note that in the absence of disorder, a new chain magnetic symmetry, one that is also present in the crystalline topological insulators, can stabilize multiple Majorana modes at the end of a single chain.

Figure 1

Bogoliubov quasiparticle bands of a system with strong exchange splitting strength $J$, and pair potential $\Delta$ and spin-orbit coupling strengths $E_{\text{SO}}$ that are by comparison weaker. This illustration assumes that the majority-spin $d$ bands (red) are full and the minority-spin $d$ bands (blue) are partially filled, the usual case for transition-metal ferromagnets. The minority-spin electron (solid) and hole (dashed) bands which cross at the Fermi level are coupled via a virtual process in which the pair potential $\Delta$ couples minority-spin electrons (holes) to majority-spin holes (electrons) and $E_{\text{SO}}$ couples minority-spin holes (electrons) to majority-spin holes (electrons). It follows that the quasiparticle gap at the Fermi energy indicated in the inset is $\sim \Delta E_{\text{SO}}/J$.

Figure 2

Model band structures for straight Fe chain. (a) ${\lambda }_{\text{SO}}=0$. (b) ${\lambda }_{\text{SO}}=0.2$ eV. (c) BdG spectrum with ${\lambda }_{\text{SO}}=0.2$ eV and $\Delta =0.2$ eV. The pair-potential value used in this illustration is unrealistically large and has been chosen for easy visualization. The inset highlights the quasiparticle bands which cross at the Fermi energy. (d) Same as (c) but with an orbital-independent Rashba spin-orbit term with coupling constant $t_R=0.05$ eV (see text).

Figure 3

Heuristic example of Rashba spin-orbit coupling induced by hopping on the chain via substrate sites with strong spin-orbit coupling. In this illustration, the two blue atoms can be associated with neighboring atoms on the chain and the green atom with a Pb atom in the substrate. Inversion symmetry is broken because the substrate atom is below the chain atoms. The Rashba coupling strength is proportional to the difference between left to right and right to left, spin-$\uparrow$ to spin-$\downarrow$ virtual hopping between the chain atoms. The Rashba spin-orbit coupling for this simple toy model is plotted here as a function of the chain-substrate-chain bond angle.

Figure 4

Topological phase diagram for the $3d$ straight ferromagnetic chain model. Blue regions in chemical potential vs exchange coupling strength phase diagram have Majorana number $\mathcal{M}=-1$ while white regions have $\mathcal{M}=1$. This figure was constructed using the hopping parameters listed in Table 1. When the exchange splitting is larger than the bandwidth, the realistic case for transition-metal chains, the gapped superconducting state is almost always topological.

Figure 5

Majorana phase diagram of a zigzag chain with nearest-neighbor hopping vs chemical potential and bond angle (cf. Fig. 3) at fixed exchange coupling $J=2.65$ eV. The blue and the white regions correspond to $\mathcal{M}=-1$ and 1, respectively.

Figure 6

Spatial distribution of one of the two Majorana states ($E\approx \pm 2\times {10}^{-6}$ eV) for a finite chain with 200 atoms. $\Delta =0.9$ eV, $t_R=0.1$ eV, so that the zero-energy gap is $\sim 0.1$ eV in the infinite chain.

Figure 7

(a) Local density of states at the end of a semi-infinite chain (blue) and in the middle of an infinite chain (red). Both calculations were performed for chains with $\mathcal{M}=-1$. $\Delta =1.5$ meV, $t_R=0.1$ eV, $J=2.65$ eV, and $\mu =1.3$ eV. (b), (c) Nonmonotonic dependence of the height of the zero-energy peak with the parameters (b) $t_R$ and (c) $\Delta$. The left panel in each figure is the spectral function of the end Green's function in an energy window around zero energy, and the right panel is the zero-energy value of the spectral function. In (b), $\Delta$ is fixed at 0.1 eV and in (c), $t_R$ is fixed at 0.1 eV. Values of the other parameters in (b) and (c) are the same as those in (a).

Figure 8

Geometry of the hybrid system: an Fe (orange) chain is embedded into the (110) surface of a bulk Pb (gray) superconducting substrate.

Figure 9

Typical low-energy spectra of a finite-size hybrid system (left axis) with a linear Fe chain along with the corresponding values of both topological invariants (right axis), the Majorana number $\nu$, and the winding number $w$, computed from the bulk Hamiltonian. In these calculations, the Fe chain is 120 unit cells long, $\Delta =0.1$ eV, and the finite size of the Pb substrate is 21 unit cells in the $y$ direction, 1 in the $z$ direction, and 140 in the $x$ direction. The parameter ${\mu }_{\text{Fe}}$ which is varied specifies the band line up between the Fe and Pb states as described in the text. Results are presented for two different strengths of the Pb/Fe hybridization parameter $V_{pd\pi }$.

Figure 10

Phase diagrams of a hybrid structure with a linear Fe chain and a 2D superconducting Pb substrate. Panel (a) shows the band structure (for the minority band only) of the Fe chain when it is suspended. Panels (b), (c), and (d) are all plotted as a function of ${\mu }_{\text{Fe}}$ and $V_{pd\pi }$, showing the gaps of the hybrid structure, the Majorana number $\nu$, and the topological invariant (winding number) $w$, respectively. The parameter $V_{pd\pi }$ signifies the strength of the hybridization. The calculations of $w$ suffer significantly from numerical errors when the system gap is small. In order to reduce these errors, we set $\Delta =0.1$ eV in the present calculations. When the errors of $w$ are insignificant, we find $\nu =wmod2$, as expected. The size of the Pb substrate used here is 21 unit cells in the $y$ direction and 1 in the $z$ direction (infinite in the $x$ direction).

Figure 11

Phase diagrams of a hybrid structure with a triple Fe chain coupled to a 2D superconducting Pb substrate. The upper panel shows the phase diagram in terms of the Majorana number $\nu$, while the lower panel shows the phase diagram in terms of the the winding number $w$. In these calculations, $\Delta =0.1$ eV, and the size of the Pb substrate is 15 unit cells in the $y$ direction and 1 in the $z$ direction (infinite with periodic boundary conditions in the $x$ direction).

Figure 12

A representative eigenstate wave function of a finite-size hybrid structure composed of a 60-unit-cell-long triple Fe chain and a Pb substrate of size $90\times 21\times 2$ unit cells. The upper panel shows the wave-function amplitude on both the Fe chain and the Pb substrate (amplitudes for overlap points in the $xy$ plane are summed up); the lower panel shows details of the wave function on the Fe chain separately for each sublattice (see text). The energy corresponding to this eigenstate is 0.09 meV (see the highlighted point in Fig. 13, upper panel); the ratio of the total weight of this wave function on the Pb substrate to that on the Fe chain is approximately 9.3 because of the strong hybridization and the significantly larger size of the substrate. Periodical boundary conditions for the substrate are adopted in both the $x$ and the $y$ directions in order to stabilize our numerical calculations, which leads to relatively strong coupling between the end states across the boundaries (see the enhanced wave-function amplitudes in the substrate across the boundaries). The pairing parameter in Pb has a realistic value $\Delta =1.3$ meV; the coupling parameter in this specific example is $V_{pd\pi }=0.65$ eV.

Figure 13

Comparison between the low-energy spectrum in the hybrid system (upper panel) and that in a suspended triple Fe chain with artificially added Rashba spin-orbit coupling and spin-singlet pairing terms (lower panel). In the upper panel, the spectrum is plotted as a function of the coupling between Fe and Pb atoms; in the lower panel, the spectrum is plotted as a function of the Rashba spin-orbit coupling strength ($\alpha$) in a realistic range. The two models have Fe chains of the same length (about 210 Å), and the same pairing parameter ($\Delta =1.3$ meV). The small circle in the upper panel highlights the eigenstate that is plotted in Fig. 12.

Figure 14

Effective hopping parameters $t_{\Sigma }$ (upper panel) and inverse-lifetime parameters ${\tau }_{\Sigma }$ (lower panel; see text for the definitions of these parameters), extracted from the self-energy due to the superconducting substrate (assuming zero energy and $V_{pd\pi }=1$ eV), as a function of distance $x$. Only two of the largest parameters are shown in each plot. The insets show the same curves scaled by appropriate power-law prefactors in terms of $x$.

Figure 15

Spatial dependence of the zero-energy spectral functions in a 60-unit-cell-long triple Fe chain, with different values of coupling $V_{pd\pi }$ between Fe and Pb atoms. Data for different $V_{pd\pi }$ are shifted by an increment of 0.01 for clarity. The Pb substrate is infinite in the $xy$ plane and semi-infinite in the $z$ direction, and has a spin-singlet pairing gap $\Delta =1.3$ meV.