1. Introduction
Since the Majorana fermion (a particle that is its own antiparticle) was first proposed by Majorana in 1937 [1], the hunt for it has become one of the paramount research tasks and received much attention from the physics, chemistry and biology communities [2–14]. In condensed matter physics, Kitaev proposed a seminal model in which Majorana fermions could emerge as quasiparticle excitations of Majorana zero-energy modes (MZMs) at the ends of a one-dimensional spinless p-wave superconducting chain [2]. This model is now known as the Kitaev model or Kitaev chain. He also emphasized that Majorana fermions [15–19] obey non-Abelian statistics [20, 21] and are ideal candidates for topological quantum computation [22–27]. However, p-wave superconductivity is extremely rare in nature. Fu and Kane proposed an original approach that topological insulators proximitzed by s-wave superconductors could lead to MZMs at vortices [3]. Das Sarma's group used semiconductor nanowires with strong spin–orbit coupling to couple s-wave superconductors [6, 28], leading to the appearance of MZMs at both ends of the nanowires under magnetic fields along the nanowire direction. Vazifeh et al constructed one-dimensional magnetic impurity chains [29] by regularly arranging magnetic impurity atoms on the surface of s-wave superconductors. Due to the proximity effect of superconductivity, the chain of magnetic atoms appears topologically and spontaneously in a topologically non-trivial state.
In general, the observation of a quantized zero-bias conductance peak (ZBP) in units of 2e2/h in differential conductance measurements is considered one of the experimental features confirming the presence of MZMs [4–6]. Kouwenhoven's team reported the observation of a ZBP in the system of InSb semiconductor nanowires covered by NbTiN superconductors in 2012 [30]. This observation was later reproduced in other experiments [9–12]. Furthermore, ZBPs have also been detected in atomic chains [31] and topological insulators proximitized by s-wave superconductors [13]. Although great progress has been made in the detection of Majorana fermions, there are still many controversies and significant discrepancies with theory. The reason is that the Andreev reflection process induced by the MZMs has electron-hole symmetry, but no quantized ZBP in units of 2e2/h has been observed. On the other hand, the cause of the ZBP is not unique, such as disorder [16, 32–36], weak antilocalization [37], nonuniform parameters [38–45], and coupling a quantum dot to a nanowire can also cause the ZBP [46, 47]. Remarkably, a quantized conductance plateau at 2e2/h at zero-bias was recently observed at the center of the vortex in an iron-based superconductor, confirming the existence of MZMs [48].
The new perspectives on antiferromagnetic (AFM) systems have been brought about by recent advances in experimental techniques that allow the preparation of atomic chains [49]. In this way, the self-organized spin helix order [29, 50, 51] can be stabilized by the Ruderman–Kittel–Kasuya–Yosida mechanism [52–54]. Additionally, ideal monoatomic chains can be built atom by atom [55], allowing various types of magnetic order to exist in the chain [54, 56, 57]. For example, AFM order has been experimentally observed in chains of Fe [58, 59]. Moreover, the proximity effect can transfer the AFM order to the nanowire by bringing it into contact with a strong AFM system. Strong antiferromagnets such as V5S8 [60], NiPS3 [61], and Mn2C [62], can be good candidates for the substrate in the investigated system. In such cases, the topological phase can emerge in the presence of an external magnetic field while the AFM order of the substrate remains intact.
As far as we know, most of the present work on the one-dimensional topological superconductor with (quasi) periodic potential is concerned only with the nearest hopping and pairing terms And they usually focus on the transitions from the topological to the topologically trivial phases induced by the (quasi) periodic potential. However, the study of the interplay between the periodic potential and the AFM chain is still lacking, and the periodic potential-induced topological phases in superconductors are also uncovered, which motivates us to investigate whether the periodic potential can induce topological phases in the one-dimensional topological superconductor with AFM order.
In this paper, we study the topological superconductivity of an AFM chain with an on-site periodic potential. The energy spectra and the conductance of this AFM chain-superconductor system are calculated by considering several factors, including the Zeeman field and the periodic potential. Our conclusions indicate that a topologically nontrivial superconductor can occur in the proposed device under suitable parameters, and the AFM chain in the topologically nontrivial phase can host Majorana zero modes (MZMs) at both ends. In a typical situation, the phase transition from a topologically trivial to a nontrivial state is induced by the on-site periodic potential.
The rest of the paper is organized as follows. In section 2 , we present the model Hamiltonian of the AFM chain proximity-coupled by a superconductor and the nonequilibrium Green's function method. In section 3 , we investigate the energy spectrum of the 1D AFM chain in the external Zeeman magnetic field and calculate the wave-function ∣$Psi$n∣2 to prove the existence of Majorana states. Using the nonequilibrium Green's function, we also calculate the transmission coefficients of electron tunneling through the proposed AFM chain. Finally, the main conclusions are summarized in section 4 .
2. Model and method
The system consists of a one-dimensional Rashba nanowire with superconducting and AFM order both induced by proximity effects, as shown in figure 1, in the presence of an external magnetic field directed along the nanowire. The Hamiltonian of this system can be written as [63]:
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal H }}_{0} & = & \displaystyle \sum _{i,s,\sigma }-(\mu +\sigma h){c}_{{is}\sigma }^{\dagger }{c}_{{is}\sigma }-{\rm{i}}\lambda \\ & & \times \,\displaystyle \sum _{i,\sigma {\sigma }^{{\prime} }}\left[{c}_{{iA}\sigma }^{\dagger }{\sigma }_{\sigma {\sigma }^{{\prime} }}^{y}{c}_{{iB}{\sigma }^{{\prime} }}+{c}_{{iB}\sigma }^{\dagger }{\sigma }_{\sigma {\sigma }^{{\prime} }}^{y}{c}_{i+1,A{\sigma }^{{\prime} }}\right]+{\rm{H}}.{\rm{c}}.\\ & & -\displaystyle \sum _{{ij}\,,{{ss}}^{{\prime} },\sigma }{t}_{{ij}}^{{{ss}}^{{\prime} }}{c}_{{is}\sigma }^{\dagger }{c}_{{{js}}^{{\prime} }\sigma }+{\rm{\Delta }}\displaystyle \sum _{{is}}\left({c}_{{is}\uparrow }^{\dagger }{c}_{{is}\downarrow }^{\dagger }+{c}_{{is}\downarrow }{c}_{{is}\uparrow }\right)\\ & & -{m}_{0}\displaystyle \sum _{i\sigma }\sigma \left({c}_{{iA}\sigma }^{\dagger }{c}_{{iA}\sigma }-{c}_{{iB}\sigma }^{\dagger }{c}_{{iB}\sigma }\right).\end{array}\end{eqnarray}$
Here, ${c}_{{is}\sigma }^{\dagger }\left({c}_{{is}\sigma }\right)$ describes the creation (annihilation) of an electron with spin σ ∈ { ↑ , ↓ } in sublattice s ∈ {A, B} of the ith unit cell. μ is the chemical potential and h is the external Zeeman magnetic field. In our work, we assume that the magnetic field is parallel to the nanowire and ignore orbital effects. λ represents the spin–orbit coupling strength, where σy is a Pauli matrix. We consider equal hopping between the nearest-neighbor sites (when ${t}_{{ij}}^{{{ss}}^{{\prime} }}=t=1$ in appropriate energy units) and zero otherwise. The last two terms describe the proximity effects induced by the AFM order and superconductor, where m0 is the amplitude of the AFM order and Δ is the superconducting order parameter.
Figure 1. Schematics of an AFM chain deposited on the surface of an s-wave superconductor (green rectangle) and contacted by left (L) and right (R) normal-metal electrodes at the two ends. The nearest-neighbor sites due to the proximity effect are two non-equivalent sites with opposite magnetic moments (blue and orange). In the presence of a Zeeman field parallel to the nanowire, MZMs could emerge at both ends of the AFM chain. |
To control the MZM, we consider the following periodic potential [64]:
$\begin{eqnarray}{{ \mathcal H }}_{1}=\displaystyle \sum _{n=1}^{N}{{\rm{\Omega }}}_{n}{c}_{n}^{\dagger }{c}_{n},\end{eqnarray}$
where ${c}_{n}^{\dagger }{c}_{n}$ is the creation (annihilation) operator on the site n of the 1D AFM chain. ${{\rm{\Omega }}}_{n}=V\cos [2\pi \beta (n-q)+\phi ]$ is the on-site potential, V is the modulation amplitude of the on-site potential, β controls the periodicity of the modulation and an offset q is introduced in the cosine modulation. φ is the modulation phase. The chemical potential μ can be replaced by the periodic potential Ωn.To verify the existence of MZM in a 1D AFM chain in more detail, we employ the nonequilibrium Green's function method [65] to investigate the transport properties through the 1D AFM chain connected by two normal-metal electrodes. There are three kinds of transport processes in this two-electrode device, such as quantum tunneling, the Andreev reflection, and crossed Andreev reflection. The transmission coefficients can be described as:
$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{QT}}(E) & = & \mathrm{Tr}\left[{{\rm{\Gamma }}}_{{ee}}^{L}{G}_{{ee}}^{r}{{\rm{\Gamma }}}_{{ee}}^{R}{G}_{{ee}}^{a}\right],\\ {T}_{\mathrm{AR}}(E) & = & \mathrm{Tr}\left[{{\rm{\Gamma }}}_{{ee}}^{L}{G}_{{eh}}^{r}{{\rm{\Gamma }}}_{{hh}}^{L}{G}_{{he}}^{a}\right],\\ {T}_{\mathrm{CAR}}(E) & = & \mathrm{Tr}\left[{{\rm{\Gamma }}}_{{ee}}^{L}{G}_{{eh}}^{r}{{\rm{\Gamma }}}_{{hh}}^{R}{G}_{{he}}^{a}\right],\end{array}\end{eqnarray}$
where e and h represent electron and hole, respectively. L and R represent the left and right electrodes of the device. ${G}^{r}(E)={\left[{G}^{a}(E)\right]}^{\dagger }={\left[E-H-{{\rm{\Sigma }}}_{L}^{r}-{{\rm{\Sigma }}}_{R}^{r}\right]}^{-1}$ is the retarded Green's function, and ${{\rm{\Sigma }}}_{\beta }^{r/a}$ is the retarded/advanced self-energy due to the coupling to electrode β. Here, we consider the wide-band limit and the self-energy is taken as ${{\rm{\Sigma }}}_{\beta }^{r}={\left({{\rm{\Sigma }}}_{\beta }^{a}\right)}^{\dagger }=-{\rm{i}}{\rm{\Gamma }}/2$, where Γ is the coupling strength between the electrode and the AFM chain.At zero temperature, the Fermi level E dependent conductance G(E) can be obtained by Green's function method and Landau-Büttiker formula as:
$\begin{eqnarray}G(E)=\displaystyle \frac{1}{2}\left({T}_{{\rm{Q}}{\rm{T}}}+2{T}_{{\rm{A}}{\rm{R}}}+{T}_{{\rm{C}}{\rm{A}}{\rm{R}}}\right){G}_{0},\end{eqnarray}$
where G0 = 2e2/h.3. Results and discussion
In the numerical calculations, the hopping value of the nearest sites is taken as the energy unit, t = 1, and the chemical potential μ = 0 which could be replaced by the on-site potential Ωn. Other periodic potential parameters are then taken as β = 1/3, q = 2, φ = 5π/3, and N = 200. The coupling strength between the electrodes and the AFM chain is Γ = 0.05t [66], the superconducting order parameter Δ = 0.2t, the spin–orbit coupling strength λ = 0.15t, and the AFM amplitude m0 = 0.3t [63]. These parameters will be used throughout the paper, unless stated otherwise.
3.1. Signatures of MZMs in 1D topological superconducting AFM chain devices
We consider a one-dimensional Rashba nanowire with superconducting and antiferromagnetic order. Figures 2(a)–(f) show the energy spectrum of the 1D AFM chain as a function of external Zeeman magnetic field with different periodic potential strengths, respectively. By inspecting figure 2(a), it clearly shows that the midgap states can cross the Fermi level E = 0 and the energy spectrum is symmetric with respect to E = 0, owing to the electron-hole symmetry in the superconductor. However, a topologically nontrivial phase does not exist for any value of the Zeeman magnetic field.
Figure 2. The energy spectrum of the 1D AFM chain as a function of external Zeeman magnetic field with different periodic potential strengths V = 0 (a), V = 0.5 (b), V = 1 (c), V = 1.5 (d), V = 2 (e), and V = 2.5 (f). The colorful circles correspond to the electronic states at the Fermi level E = 0. |
When the periodic potentials are increased sequentially, it can be seen from figure 2(b)–(f) that the energy spectrum remains symmetrical with respect to E = 0, similar to the case when the periodic potential strength equals 0. Interestingly, different phenomena can be observed in the AFM chain-superconductor system in a relatively large Zeeman field regime when the strengths of the periodic potential are sequentially increased. When the Zeeman magnetic field is increased beyond h ∼0.55t (see figure 2(c)), a topological gap emerges in the energy spectrum, demonstrating a topological phase transition from a topological trivial phase to a topological nontrivial one. It can be seen that the band gap closes or reopens while considering the periodic potential, as shown in figures 2(c)–(f). Here, a zero energy level emerges in the energy spectrum in the AFM chain, i.e. the MZM. The realizations of MZMs demonstrate that an AFM chain proximity-coupled by an s-wave superconductor can transit from trivial superconductor phase to topological nontrivial superconductor phase under the effect of periodic potentials.
3.2. Confirmations of MZMs in 1D topological superconducting AFM chain devices
However, it is not sufficient for identifying the topological phase transition only considering the closing and reopening of the energy spectrum of 1D AFM chain. For further confirmation, figure 3. shows the probability distribution ∣$Psi$n∣2 of energy level at V = 1t and h = 0.6t (see the colorful dots in figure 2), which are obtained by diagonalizing the Hamiltonian of AFM chain-superconductor system (see equation (1 )). The probability distributions are localized at the ends of the AFM chain, indicating that the AFM chain-superconductor system is a topological nontrivial phase.
Figure 3. Spatial distributions of the wave-functions ∣$Psi$n∣2 of the degenerate electronic states marked by the colorful circles in figure 2. Here, the length of AFM chain is N = 200. |
To get a better view of the topological of the AFM chain-superconductor system, we calculate the conductance G as a function of h with different periodic potential strengths V in figure 4. For the left and right normal-metal electrodes, the symmetric condition is taken into account and the linewidth function is fixed to ΓL = ΓR = 0.05t. One can see that no plateau is present when the periodic potential strength V = 0 (see figure 4(a)). With the increase of the periodic potential strength V, the plateau of conductance G gradually appears. Here, a conductance plateau always corresponds to a new phase or the emergence of a novel phenomenon. As compared with the energy spectrum (see figure 2), the conductance plateau corresponds to the topological nontrivial phase hosting MZMs. Thus the production of MZMs can be easily controlled and adjusted by this parameter, e.g. the strength of periodic potential.
Figure 4. The conductance G as a function of h with different periodic potential strengths V = 0 (a), V = 0.5 (b), V = 1 (c), V = 1.5 (d), V = 2 (e), and V = 2.5 (f). E = 0, and the remaining parameters are the same as in figure 2. The blue solid line denotes G. |
In figure 5, we present the transmission coefficients TQT, TAR, and TCAR as a function of h with different periodic potential strengths V. It can be seen that the conductance is mainly contributed by the Andreev reflection and quantized TAR is observed. The transmission coefficient TAR shows two plateaus of conductance with the given external Zeeman magnetic field ranges, and the location of two plateaus corresponds to the position of zero energy level of the energy spectrum in figures 2(d) and (f). This indicates a topological phase transition from a topological trivial phase to a topological nontrivial phase in the AFM chain-superconductor system. The quantum tunneling transmission TQT is strictly equal to the crossed Andreev reflection transmission TCAR, and both of them are shown a tiny peak when the quantum phase transition happens. Such features also demonstrate the existence of MZMs.
Figure 5. Transmission coefficients TQT, TAR, and TCAR as a function of h with different periodic potential strengths V = 1.5 for (a) and V = 2.5 for (b). E = 0, and the remaining parameters are the same as in figure 4. The blue solid line, the green solid line, and the red dot-dashed line represent TAR, TCAR, and TQT, respectively. |
4. Summary
In summary, we have investigated the topological phases induced by the periodic potential in the one-dimensional topological superconducting AFM chain, where the intrinsic spin–orbit coupling and a Zeeman energy along with the AFM chain are considered. It is shown that the MZMs can be formed at both ends of the AFM chain in a topologically nontrivial phase. Using the nonequilibrium Green's function method, we also investigate the transport properties of the AFM chain connected by two normal-metal electrodes. It is found that the MZMs can induce a conductance, which shows a stable plateau-like structure as a function of the Zeeman field. Moreover, the resonant peaks of transmission coefficients TQT and TCAR correspond to the critical points of topological phase transition.