The Majorana spin in magnetic atomic chain systems
In this paper, we establish that Majorana zero modes emerging from a topological band structure of a chain of magnetic atoms embedded in a superconductor can be distinguished from trivial localized zero energy states that may accidentally form in this system using spin resolved measurements. To demonstrate this key Majorana diagnostics, we study the spin composition of magnetic impurity induced in-gap Shiba states in a superconductor using a quantum impurity model (at the mean-field level). By examining the spin and spectral densities in the context of the Bogoliubov-de Gennes (BdG) particle-hole symmetry, we derive a sum rule that relates the spin densities of localized Shiba states with those in the normal state without superconductivity. Extending our investigations to ferromagnetic chain of magnetic impurities, we identify key features of the spin properties of the extended Shiba state bands, as well as those associated with a localized Majorana end mode when the effect of spin-orbit interaction is included. We then formulate a phenomenological theory for the measurement of the local spin densities with spin-polarized scanning tunneling microscopy (STM) techniques. By combining the calculated spin densities and the measurement theory, we show that spin-polarized STM measurements can reveal a sharp contrast in spin polarization between an accidentally-zero-energy trivial Shiba state and a Majorana zero mode in a topological superconducting phase in atomic chains. We further confirm our results with numerical simulations that address generic parameter settings.
Experimental breakthroughs Mourik et al. (2012); Das et al. (2012); Rokhinson et al. (2012); Deng et al. (2012); Nadj-Perge et al. (2014); Sun et al. (2016) have advanced the research on Majorana zero modes (MZMs) Wilczek (2009); Alicea (2012); Beenakker (2013); Elliott and Franz (2015); Beenakker and Kouwenhoven (2016); Lutchyn et al. (2017) from appealing theoretical ideas Kitaev (2001); Fu and Kane (2008); Nilsson et al. (2008); Lutchyn et al. (2010); Oreg et al. (2010); Choy et al. (2011); Martin and Morpurgo (2012); Nadj-Perge et al. (2013); Klinovaja et al. (2013); Braunecker and Simon (2013); Vazifeh and Franz (2013); Pientka et al. (2013); Li et al. (2014) to an exciting new stage. Intensive efforts are being made in laboratories Ruby et al. (2015); Qu et al. (2015); Albrecht et al. (2016); Deng et al. (2016); Pawlak et al. (2016); Feldman et al. (2017); Ruby et al. (2017) towards realizing the full potential of MZMs, for example, to demonstrate non-abelian braiding statistics Read and Green (2000); Ivanov (2001); Alicea et al. (2011); Aasen et al. (2016); Li et al. (2016), and to ultimately perform topological quantum computation Kitaev (2003); Sarma et al. (2015); Nayak et al. (2008). Experimental evidence of MZMs is not only necessary to consolidate existing observations but also valuable for a deeper understanding of the current platforms. In the semiconductor nanowire Majorana platform Lutchyn et al. (2010); Oreg et al. (2010); Mourik et al. (2012); Das et al. (2012); Rokhinson et al. (2012); Deng et al. (2012), for instance, observations of the exponential decay of MZMs, albeit by indirect means, have been recently reported Albrecht et al. (2016). In the atomic chain Majorana platform Nadj-Perge et al. (2014); Ruby et al. (2015); Choy et al. (2011); Martin and Morpurgo (2012); Nadj-Perge et al. (2013); Klinovaja et al. (2013); Braunecker and Simon (2013); Vazifeh and Franz (2013); Pientka et al. (2013); Li et al. (2014); KirÅ¡anskas et al. (2015); Schecter et al. (2016); Christensen et al. (2016); Andolina and Simon (2017) a series of recent high resolution measurements have placed more stringent bounds on the MZM splitting at low temperatures, revealed new spatial structures of MZMs, and established their equal electron-hole weights using spectroscopy with a superconducting scanning tunneling microscopy (STM) tip Feldman et al. (2017). In this latter platform, a strong localization of the MZMs has been observed Nadj-Perge et al. (2014); Ruby et al. (2015) with spatially resolved STM spectroscopy measurements, and is now theoretically well understood Peng et al. (2015); Li et al. (2014). In all these Majorana platforms, the however small possibility that trivial end states can be accidentally tuned to zero energy and hence incorrectly identified as MZMs still exists Liu et al. (2017); Danon et al. (2017). For example, in the atomic chain platform, it is possible that conventional localized Shiba states Yu (1965); Shiba (1968); Rusinov (1969) are accidentally tuned to nearly zero energy by a local potential at the end of a magnetic chain. Such a possibility, however improbable, cannot be excluded by energy resolved spectroscopic measurements alone but requires other types of diagnostics Rosdahl et al. (2017). In this paper, we show that spin polarization Sticlet et al. (2012); BjÃ¶rnson et al. (2015) can distinguish MZMs and trivial Shiba states and show how this is revealed by spin-polarized STM measurements Wiesendanger (2009). Experimental demonstration that establishes this distinction has recently been accomplished for magnetic chains of Fe atoms on the surface of Pb Jeon et al. (2017).
The paper is organized as follows. In Section II, we illustrate the key idea of this paper, using a one-body toy model for one proximitized magnetic impurity. In Section III, we examine the spin properties of Shiba states induced by a single magnetic impurity using a fully quantum mechanical model of such a impurity hybridized with a generic conventional s-wave superconductor. A key result discussed in this section is the relation between the spin densities induced locally by the impurity in the normal state and those associated with the in-gap Shiba states in the superconducting state. These results show how such localized states differ in their spin properties from a MZM that emerges in the topological phase of the atomic chains, and set the stage to understand the spin properties of Shiba bands induced by a ferromagnetic chain of magnetic atoms in a superconductor discussed in Section IV. In Section IV, we derive in detail the spin properties of a MZM when spin-orbit coupling drives the system into a topological superconducting phase. To relate our theory to a recent experiment Jeon et al. (2017), we develop a phenomenological theory in Section V that captures how spin-polarized STM probes the properties of the in-gap states in a superconductor. An important result established in this section is a diagnostic test: unlike a MZM, a trivial zero mode with wavefunction support on the end of the chain would show no contrast in spin-polarized STM measurements performed under typical constant current conditions. In Section VI, we complement our analytical results with numerical simulations, using generic but realistic parameters. Aside from confirming our analytical results, our simulations allow us to demonstrate the utility of our Majorana spin diagnostic test.
Ii Toy model
In order to demonstrate the main idea of this paper, we start with a toy model comprising only a single site and two spins. The Hamiltonian is given by
where , and are real, non-negative parameters representing the (magnetization) exchange energy, the chemical potential and the (induced) pairing potential, respectively; and represent the Nambu particle (without bar) and hole (with bar) creation operators, respectively; () stands for the Pauli matrices for spin, and () stands for the Pauli matrices for the Nambu particle-hole spinors. From now on we will drop the identity matrices and where no ambiguity will arise.
The above Hamiltonian is easily solved and the two low energy eigenstates are given by (see Fig. 1)
where, by definition, and . When both low energy states are exactly at zero energy, which requires , we can artificially construct Majorana states by superposing and as (in the , basis)
where stands for the gauge freedom in constructing the Majorana states.
In this paper, we are particularly interested in the particle components of the Nambu spinors because of their relevance to the STM measurement in the single-electron sequential tunneling regime. To this end we define spin densities for an eigenstate to be
is the projector to the spin-/ particle component, respectively. From Eq. (4), it is clear that () only contains nonvanishing spin- (spin-) component:
In other words, the two low energy states are both fully spin-polarized with opposite spin polarization (see Fig. 1). Therefore we may formally write down () as a function of energy () that is associated with (), hence we have
By assuming , which is always true if , we obtain and . This implies that the spin densities are dominated by the spin polarization of the original state that is closer to the chemical potential. In the case of Fig. 1, this is the electronic spin- state, whether occupied or unoccupied.
Moreover, the difference between the two spin densities satisfies
Namely, reaches its maximum at . On the other hand, for the artificial Majorana states in Eq. (5), we straightforwardly find
Apart from a factor of owing to the fact that the spin densities at are split equally to two Majorana states, we see that for the Majorana states corresponds to the maximum of , at , associated with generic eigenstates . This behavior, as we see in later sections, where Shiba and Majorana states are considered, is key for a MZM to be distinguished from trivial Shiba states in spin-polarized STM measurement.
Incidentally, if we choose a different orientation for the spin basis, which, for example, amounts to replacing in Eq. (7) by or and denoting the corresponding by or , it is easy to verify that if , whereas , and . This indicates that Majorana states, albeit artificial in the toy model, may acquire finite in-plane spin polarization in contrast to trivial quasiparticle states represented by Sticlet et al. (2012). Such in-plane spin polarization, however, is very weak by realistic measure as ; therefore we will focus on only the spin polarization along the magnetization () in the following of this paper.
Iii Single magnetic impurity model
We now build up towards our Shiba chain model by considering the model that consists of a 2D or 3D bulk superconductor coupled to a single quantum magnetic impurity. We assume the superconductor to be infinite (without any surface) in all its dimensions and neglect spin-orbit coupling for simplicity. This hybrid system can be described by a Bogoliubov-de Gennes (BdG) Hamiltonian
Here, , and stand for the band-width parameter, the chemical potential and the (real) pairing potential for the superconductor, respectively; and represent the Nambu particle and hole creation operators for the superconductor, respectively; and are the Fourier transforms of and , respectively; , as well as and , is similarly defined as in Eqs. (1) and (2). With the tunneling Hamiltonian (19), we have assumed the single magnetic impurity to be sitting at .
Throughout this paper we will focus on the spin densities on the -orbitals – namely, the spin densities evaluated with and operators, . The reason for this is twofold: first, the experimental technique considered in this paper is STM, which measures locally with atomic resolution; second, although the major weight of a Shiba state is distributed in the superconductor, as we will see at the end of this section, the length scale of the distribution is given by the superconducting coherence length which is generally large ( 80 nm in Pb, for example), such that the local weight of a Shiba state is small on a superconductor atom compared with that on the magnetic adatom (such as Fe) -orbitals.
The retarded Green’s function for the -orbital degrees of freedom is given by (see Appendix A)
where with a positive infinitesimal, and is the normal DOS of the superconductor at its Fermi energy ( if the superconductor is 3D, and if the superconductor is 2D). Throughout this paper we will assume . Note that Eq. (21) is valid for the full energy range (below or above the superconducting gap) as long as we adopt the convention of taking the square root such that . Peng et al. Peng et al. (2015) have previously explored the limit of when which leads to a deep-Shiba-limit effective Hamiltonian of the single impurity problem (see Appendix B).
The -orbital spin densities, defined as
In Fig. 2, we show the spin densities obtained directly from Eq. (23) with a small finite , which accounts for the finite width of the delta functions; in the upper panel of Fig. 3, we show the energy of a Shiba state, , as a function of , from the exact solution of the poles in Eq. (21), as well as from the approximate expression Eq. (27) in the limit .
are the normal-state spin densities in the full energy range ( is irrelevant in the normal state). This convergence is certainly expected and can be seen straightforwardly from Eq. (21) by noticing that implies with vanishing pairing terms. In Fig. 2, we plot in broken lines the normal-state spin densities in the same energy range as the superconducting-state spin densities, where the convergence is clearly seen. The energy range shown in Fig. 2 is small compared with or ; therefore are roughly constants. The superconducting-state spin densities approach these constants under the condition .
In the more interesting subgap regime, Eq. (25) shows that the Shiba state occurring at has only nonvanishing spin-() electronic components. More importantly, the total spin densities inside the gap are given by (see also the lower panel of Fig. 3)
where the second equation stands for a shorthand notation and are defined in Eq. (29). This represents a sum rule regarding the redistribution of -orbital spin densities, as well as its spectral density, upon the opening of a superconducting gap in the host. Moreover, in a larger energy range where , the sum rule resumes a more transparent form (see Appendix B)
which leads to a deep-Shiba-limit effective Hamiltonian
This effective Hamiltonian has the same form of the toy model Eq. (1), except that has replaced , and there is an additional scaling factor which represents the portion of the weight of a Shiba state actually on the -orbitals Peng et al. (2015). Here, we emphasize that for self-consistency the low-energy eigenstates of the Hamiltonian (34) correspond to Shiba states only if the eigenvalues associated with these eigenstates are small compared to , which amounts to the condition . Under such a condition, and to the leading order in , it is straightforward to show that Eqs. (27) and (25) reduce to Eqs. (3) and (8), respectively (see Appendix B).
Given that Shiba states are the only way to produce a localized in-gap state in a conventional superconductor, the calculations of this section will provide an important spin signature of such states if they were to form accidentally at zero energy. As we describe below, the analytical results of this section can be combined with an STM measurement theory to provide a key difference between the measured spin contrast from such states and those from a MZM that emerge in a topological magnetic chain.
Iv Magnetic impurity chain model
Now we are in the position to discuss a 3D model composed of a bulk superconductor and a magnetic impurity chain:
where is given by Eq. (20), and
with a real symmetric function of and a real anti-symmetric function of , representing the spin-independent and the spin-orbit-coupling energies, respectively; , , and . For simplicity, we have assumed that: first, the chain is embedded in a 3D bulk superconductor with no surface; second, spin-orbit coupling, although mainly induced from the host material in reality, is added only to the Hamiltonian for the chain. Since is a good quantum number in the above Hamiltonian, we can solve the model for each fixed separately such that the problem is reduced to 2D with a single magnetic impurity, as we have solved in the previous section. With this dimensional reduction, we define an effective -dependent chemical potential , and modify Eq. (22) to be with the normal DOS of the substrate at its Fermi energy with a fixed .
Before we proceed, let us first examine the behavior of . By assuming the normal-state Hamiltonian of the host to be [cf. Eq. (20)], we obtain to be a rectangular function
where is the Fermi wave-vector for the bulk superconductor. Therefore also appears to be a rectangular function that is given by a constant inside the cutoff momentum range and 0 otherwise. In the rest of this paper, we shall assume the range to be sufficiently large such that the low energy (smaller than or comparable to ) states of the pristine -orbital bands always fall into this momentum range – outside this momentum range, because of the vanishing and hence the vanishing , the self-energy term in in Eq. (21) is also vanishing, thus the -orbital states outside become irrelevant at low energy by our assumption.
iv.1 Shiba bands in the absence of spin-orbit coupling
We start with the limit of vanishing spin-orbit coupling, namely, . The Shiba band dispersion relation is given by replacing with in Eq. (27):
which is an integration of normal-state spin densities over the momentum range where the magnetic chain is strongly hybridized with the superconductor. In other words, the spin densities in the normal-state limit are contributed by an extended momentum range of the pristine -orbital bands with broadening . In the subgap regime , by using Eq. (25) we have
where are the solutions of the equations , respectively for the two signs, in the range of . By definition, we have for all , which is a manifestation of particle-hole symmetry. In contrast to the normal-state case, Eq. (44) shows that the subgap spin densities at any specific energy () are only contributed by a small set of momenta () from the pristine -orbital bands, and hence can vary strongly with energy.
Here, are the DOS of the Shiba bands; are the spin densities inherited from the sum rule Eq.(30). In essence, Eq. (46) is the same as Eq. (25) with the -function replaced by the DOS. Note that the -dependence in Eq. (48) corresponds to the -dependence in Eq.(30).
Owing to the particle-hole symmetry , satisfy
Moreover, contain Van Hove singularities of the Shiba bands that are dominant in the -dependence of . In Fig. 4 (a) and (b), we show an example of the Shiba bands from Eq. (41) with , and its corresponding with a specific . On the other hand, do not apparently exhibit any particle-hole symmetry [see Fig. 4 (c)] as they both originate from the broadening of the pristine -orbital bands. In particular, and can differ significantly in magnitude when the chemical potential is much closer to one of the spin bands (the minority spin, ), therefore the spectral density defined as a summation of and will be mostly dominated by the former and hence exhibit a strong asymmetry inside the gap. This is indeed the observation of spin-independent STM measurements Nadj-Perge et al. (2014); Feldman et al. (2017). We will further see, in Sec. V, how spin-polarized STM measurements can effectively amplify the spin density by normalizing with their associated normal-state background [see Fig. 4 (d)].
which relates directly the Shiba state energy with the normal-state spin densities. In the context of Shiba bands, this relation becomes
which immediately leads to the symmetry presented in Eq. (50).
More importantly, we find the maximum of the ratio at , with
This is also the ratio of at zero energy as from Eq. (49). We will see that Majorana zero modes acquire precisely this maximum ratio.
iv.2 Majorana zero modes with perturbative spin-orbit coupling
We now investigate the spin densities associated with the Majorana zero modes by including spin-orbit coupling perturbatively. To this end we assume well-behaved -orbital bands such that there exist and only exist two solutions, , to the equation in the limit of vanishing spin-orbit coupling. Namely,
Here we have used the fact that in Eq. (39) is an even function of . By solving the effective Hamiltonian in the vicinity of (see Appendix C), we obtain the -orbital components of the Majorana zero modes to be (up to a normalization factor; note that the Majorana wavefunctions have support both in the magnetic chain and in the superconductor, but here we focus on the chain part only)
and correspond to two Majorana zero modes at two ends of the chain. The spinor parts of these wavefunctions, which are position-independent, are precisely given by the artificial Majorana solutions Eq. (5) in the toy model with replacing and . The ratio of Majorana spin densities , for both and , is given by
iv.3 Effects of finite spin-orbit coupling
The case of finite spin-orbit coupling can be solved directly from Eqs. (21), (23) and (39), although the analytical expressions in general become lengthier and less transparent compared with the vanishing spin-orbit coupling case. We will focus on the spin densities associated with the Majorana zero modes in this case, and discuss briefly the spin densities associated with the Shiba bands at the end of this section.
where we have dropped the dependence of , and to shorten the expression. This equation is to be compared with Eq. (41). Clearly, . The solutions of exist only if and , the former generically requiring or , and the latter imposing in addition a condition for the values of (and hence ) at these special momenta. When fully gapped (see Fig. 6 for an example), the Shiba bands are topologically nontrivial if (see Appendix D)
If this condition is true, the induced -wave gap estimated at , where by definition
is given by
If , then , which recovers the estimation given by Ref. Li et al., 2014 (see also Eq. (140) in Appendix C). Furthermore, the Majorana zero mode solutions are given by (up to a normalization factor; see Appendix D)