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Conductance Spectroscopy of Majorana Zero Modes in Superconductor-Magnetic Insulating Nanowire Hybrid Systems

There has been a recent interest in superconductor-magnetic insulator hybrid Rashba nanowire setups for potentially hosting Majorana zero modes at smaller external Zeeman fields. Using the Keldysh non-equilibrium Green's function technique, we develop a detailed quantum transport approach that accounts for the hybrid structure comprising the Rashba nanowire coupled to the bilayer structure which includes the proximity inducing superconductor and the magnetic insulator. We provide a detailed analysis of three terminal setups to probe the local and non-local conductance spectra in both the pristine as well as the disordered nanowire setups. We uncover the conductance quantization scaling with the bilayer coupling and the signatures of the gap closing followed by the emergence of near-zero energy states, which can be attributed to topological zero modes in the clean limit. However, in the presence of a smoothly varying disorder potential, trivial Andreev bound states may form with signatures reminiscent of topological zero modes in the form of a premature gap closure in the non-local conductance spectra. Our results therefore provide transport-based analysis of the operating regimes that support the formation of Majorana modes in these hybrid systems of current interest, while investigating the effect of disorder on experimentally relevant device structures.

Recent proposals have considered measuring all the elements of the conductance matrix, particularly focusing on non-local transport measurements using three-terminal normal-topological superconductor-normal (N-TS-N) setups to identify the bulk-gap closing and reopening, which separates the trivial and topological regimes [31][32][33][34][35][36]. Non-local conductance signatures could supplement the local conductance measurements in identifying MZM signatures by detecting non-trivial correlations particularly in the presence of disorder.
With the aforementioned on one hand, the basic Rashba wire setup has further drawbacks which includes the requirement of large magnetic fields that could potentially destroy superconductivity [37][38][39][40] apart from the practicalities of precise magnetic field alignment [41]. Recently, efforts are being made towards realizing * bm@ee.iitb.ac.in topological superconductivity with zero external magnetic fields by using proximity effects from magnetic insulators (MI) [42][43][44][45][46][47][48][49][50][51][52]. Recent experimental [43] and theoretical works [44][45][46][47][48]51] featuring this setup indicate that at very low external magnetic fields, or even zero external magnetic fields, a topological MZM phase can emerge. The object of this paper is hence to provide an in-depth analysis of the transport signatures of MZMs in these structures, particularly focusing on the local and non-local conductance spectra in both pristine and disordered nanowires.
Using the Keldysh non-equilibrium Green's function (NEGF) technique, we develop a detailed quantum transport approach that accounts for the hybrid structure comprising the Rashba nanowire coupled to the bilayer structure which includes the proximity inducing superconductor and the magnetic insulator. We provide a detailed analysis of three terminal setups to probe the local and non-local conductance spectra in both the pristine as well as the disordered cases. We uncover the conductance quantization scaling with the bilayer coupling and the signatures of the gap closing followed by the emergence of near-zero energy states, which can be attributed to the zero modes in the clean nanowire. However, in the presence of a smoothly varying disorder potential, trivial Andreev bound states may form with signatures reminiscent of topological zero modes.

II. RESULTS AND DISCUSSIONS
We consider semiconductor nanowires (SM) with Rashba-spin-orbit coupling with epitaxial layers of superconductors (SC) (usually Al/Pb) and magnetic insulators (MI) (usually EuS), as depicted in Fig. 1(a). We then consider the device geometry where the MI and SC are in contact with the SM individually and overlap with each other, and connected to metallic leads as depicted in Fig. 1(b). The isolated Rashba nanowire is described by the following Hamiltonian: Where V SM Z is the Zeeman Hamiltonian in the SM, µ is the electrochemical potential, α R is the strength of the Rashba spin-orbit coupling, m * is the effective mass of the electron andσ i ,τ i , are the Pauli matrices in the spin and the particle-hole space, respectively.
The effects of the SC-MI bilayer are accounted for as a self-energy term in the Green's function for the nanowire and the effect of the direct coupling of the MI to the nanowire is taken to be an effective Zeeman field in the wire. We also use the self-consistent value of the superconducting gap, ∆, calculated from the bare superconducting gap ∆ 0 of the parent superconductor in the presence of the Zeeman field and scattering processes. The process of obtaining this is involved and has been described in the supplementary material. We use ∆ 0 = 0.23 meV, m * = 0.015m e , where m e is the electron rest mass, for all our simulations. In order to model the system to simulate transport measurements, we reduce the hexagonal nanowire to a quasi one-dimensional system [48] as shown in Fig. 1 (c) and (d). An external Zeeman field is applied which is anti-parallel to the magnetization in the MI. It reduces the Zeeman term in the Hamiltonian of the SC, but increases the Zeeman field in the normal metal. We parameterize the Zeeman fields in the SC and the SM in terms of the field directly induced in the SC due to the MI (V SC 0 ), and the coupling strengths of the SC and SM nanowire to the external magnetic field, (g SC , g SM ) as follows: We use g SC = 2, g SM = −15 for our calculations, closely following the setup in [48], used for equilibrium calculations.
We first present the numerical results for the pristine nanowire, in the absence of an inhomogeneous potential. The low energy density of states (DOS) shown in Figs. 2 (a) and (b) illustrates the lowest ABS which form a nearzero energy oscillating mode after the topological transition, which is marked by the bulk gap closing and reopening. The bulk gap closing and reopening is more prominent in the longer nanowire than the shorter one, since there are more sub-gap states. As the length of the nanowire increases, the oscillations around zero-energy are exponentially suppressed.
The local density of states (LDOS) corresponding to The splitting of the zero bias peak can be seen for the shorter nanowire, consistent with Fig. 2.(c) Shows that as the coupling to the normal contacts, γ, is increased, so that it becomes much larger than the coupling, γSC , between the nanowire and the SC-MI bilayer, the peak asymptotically reaches the expected quantized value.
these nanowires shows that the zero energy state is well localised at both ends, and more prominently so in the longer nanowire. The LDOS plots also show a greater  Fig. 1(c). Both the local and the non-local conductances show the bulk gap closing and reopening, which signals a topological transition, followed by the emergence of a near-zero energy state with a splitting that oscillates as a function of the externally applied magnetic field. splitting in energy around the zero energy for the shorter nanowire than the longer nanowire. While on closer inspection, the longer nanowire also shows some splitting, in reasonable experimental measurements we expect to see a single broadened peak. These observations are due to the hybridization of the MZMs when they overlap in finite nanowires, resulting in a splitting of the zero mode [53]. The hybridization of the MZMs through the nanowire is suppressed with increasing length [54][55][56], which is consistent with our observations. In Fig. 3, we plot the differential conductance for a clean nanowire with chemical potential µ = 0.125meV and find a clear zero bias conductance peak in the topological regime, and no peak near zero energy in the topologically trivial regime. The peak is close to the quantized value of e 2 h expected from an MZM in an N-S-N setup under symmetric biasing [54,55], but is smaller. We attribute this observation to the level broadening in the presence of the proximitising SC-MI bilayer which effectively acts as an extra contact [57] and induces further broadening compared to a two-terminal N-TS-N device. This is also borne out by the fact that as the coupling to the metallic contacts becomes much stronger than the coupling to the bilayer, the ZBCP asymptotically reaches the quantized value, since the broadening due to the effective MI-SC bilayer becomes negligible in comparison to the broadening induced by the metallic contacts. It should also be noted that the exactness of the quantization would also depend on the external magnetic field because the Majorana overlap energy oscillates with the externally applied magnetic field. As the overlap energy of the MZMs at the ends of the nanowire varies, the peak value of the ZBCP also changes [58]. This figure clearly elucidates the effect of introducing the bilayer on the actual conductance quantization of MZMs in the setup.
Both the local and non-local conductances spectra for the pristine nanowire are shown in Fig. 4, which exhibit similar features, that is, the bulk gap closing and reopening followed by the emergence of Majorana oscillations around zero energy. Since the zero modes appear after the closure of the bulk gap, we can conclude that they are not quasi-MZMs, but may indeed be topological MZMs [59]. We also note that a finite low-bias non-local conductance only emerges after the topological transition. The low bias non-local conductance is rectifying in nature [31] and switches sign as the the voltage polarity is reversed. At the turning points, the non-local conductance vanishes.
For a single ABS level in the sub-gap region, there is a correspondence between the non-local conductance and the BCS charges of the bound state at the leads [60]. We can see that the BCS charge and the energy splitting oscillate out of phase. The vanishing of the non-local conductance at the turning points is a signature expected from hybridized MZMs, which should be chargeless at turning points. The local conductance is almost quantized at 2e 2 h where the Majorana splitting goes to zero. The deviation from the precise quantization value may be attributed to contact broadening, and due to the fact that we have three contacts in our system, as elaborated previously. At the points where the Majorana overlap energy becomes significant, the value of the local con-ductance drops further.
In presence of a smoothly-varying potential barrier, the formation of quasi-MZMs is expected [61][62][63][64][65][66][67]. They characteristically appear in the topologically trivial regime before the closure of the bulk gap, mimicking many signatures originally considered as 'smoking gun signatures' for MZMs including oscillations with the externally applied magnetic field, and with the associated local conductance quantized at values close to 2e 2 h . It is expected that systems which show quasi-MZMs, will also exhibit true MZMs in the topological phase on increasing the external Zeeman field. For such a disordered case, for the longer nanowire, as shown in Fig. 5(a),(b) and (c), in the DOS, we find signatures characteristic of a quasi-MZM state, followed by a gap reopening signature and the emergence of a potential topological MZM. The local conductance in this case is quite deceptive since we see a premature gap closing and the bulk-gap reopening signature is extremely faint.
The quasi-MZM and the true MZM regions are quite difficult to distinguish. In the non-local conductance plot seen in Fig. 5(b), the bulk-gap reopening is seen more prominently. The non-local conductance shows signatures of both the quasi-MZM and the topological MZM states, which can be distinguished by their position with respect to the reopening of the bulk gap [61]. For the shorter nanowire, as seen in Figs. 5 (d),(e) and (f), the local and the non-local conductance spectra both show the gap closing and reopening followed by the emergence of MZMs, which oscillate in energy as the Zeeman field is increased. At the points where the splitting is zero, the local conductance is quantized at values very close to 2e 2 h . We do not find any signatures of quasi-MZM states in this device. An interesting point to note is that the local conductance exhibits signs of negative differential conductance. It is also worth noting that especially for longer nanowires, neither the local nor the non-local conductance alone has the entire information regarding the channel DOS, that arises from its eigenspectrum.
The MZMs are protected by a clear topological gap both for the pristine nanowire and for the nanowire with a smoothly varying background potential. The local conductance fails to probe the bulk states for sufficiently long nanowires. Before one lays claim to having observed topological MZMs, it is necessary to measure the entire conductance matrix to probe a device and investigate whether the zero bias peaks at both the contacts or on both the sides are correlated and emerging after the bulk gap closing and reopening, To conclude, we used the NEGF technique to develop a detailed quantum transport approach that accounts for the hybrid structure comprising the Rashba nanowire coupled to the bilayer structure which includes the proximity inducing SC and the MI. We provided a detailed analysis of three terminal setups to probe the local and non-local conductance spectra in both the pristine as well as the disordered limits. We uncovered the conductance quantization scaling with the bilayer coupling and the signatures of the gap closing followed by the emergence of near-zero energy states, which can be attributed to the topological zero modes in the clean nanowire limit. However, in the presence of a smoothly varying potential, trivial Andreev bound states may form with signatures reminiscent of topological zero modes in the form of a premature gap closure in the non-local conductance spectra. Our results therefore provide transport-based analysis of the operating regimes that support the formation of Majorana modes in these hybrid systems of current interest, while investigating the effect of disorder on experimentally relevant device structures. The analysis in this paper sets the stage for detailed investigations that extend this technique for realistic geometries [68,69] accounting also for the quantum dot formation [70] with the inclusion of scattering effects [54,71].

III. METHODS
We discretize the Hamiltonian of the system (2) on a 1D lattice with N sites, and write the Green's func-tion in the Nambu spinor basis [69] (ψ ↑ , ψ ↓ , −ψ † ↑ , ψ † ↓ ) T . The Hamiltonian for the Rashba nanowire and the selfenergies corresponding to the metallic contacts and the SC-MI bilayer, are then used obtain the retarded Green's function for the hybrid device which is used for our transport calculations, where η is an infinitesimal positive damping parameter introduced for numerical stability, and I is the identity matrix of the dimension of the Hamiltonian matrix in Nambu space. In the wide-band approximation [54][55][56], the self energies for the metallic contacts, Σ L(R) , are written in their eigenbasis and are hence diagonal, as detailed in the supplementary material. We use the Usadel equation, which is derived from a quasi-classical approximation to the Gorkov equations, to find the Green's function, and hence, the self-energy, Σ for the SC-MI bilayer [48]. The effect of the proximity of the MI on the SC can be taken into account in the boundary conditions of the Usadel equation. The MI layer induces a uniform Zeeman field V SC Z in the diffusive superconductor [72]. The Usadel equation is then solved with the self-consistent value of ∆ to obtain the quasi-classical Green's function for the bilayer,ǧ. The gap is induced in the bare Rashba nanowire by considering the proximity effect of the bilayer, which is taken into account using the self energy, Σ which can be obtained from the semi-classical Green's function,ǧ. The imaginary part of Σ connects the electron and hole subspaces, thus, inducing a gap in the system.
We also take into account spin-orbit and spin-flip scattering in the SC, by adding a scattering self-energy term in the Usadel equation. The energy scales for the spinorbit and spin-flip relaxation processes are characterised by Γ so , Γ sf respectively. We take Γ so = Γ sf = 0.4∆ 0 for our simulations unless stated otherwise. We also use the Usadel Equation to calculate the self-consistent value of the superconducting gap in the presence of an external magnetic field. For this, we solve the Usadel equation self-consistently with the superconducting gap equation S.7 and a thermodynamic constraint as outlined in the supplementary material.
We use the retarded Green's function defined above to calculate the conductance matrix for this setup [54][55][56]69]. As shown in Fig. 1, we apply voltages V L(R) to the left and right contacts and measure terminal currents I L(R) . We use the Keldysh non-equilibrium Green's function formalism to evaluate the terminal currents [54][55][56]69]. The terminal electronic current at the left contact [73] can be derived in the Landauer Büttiker form as: CAR (E) represent the energy resolved transmission probabilities for the direct, Andreev and crossed-Andreev processes involving the left and right contacts for the electronic sector of the Nambu space and I' is the extra current due to the SC-MI bilayer acting as an effective contact, derived in the supplementary material.
Using the expressions for the terminal currents from above, the conductance matrix [G] can be defined as: The diagonal matrix elements represent the local conductance at the left and right contacts, and the off-diagonal components represent the non-local conductance.
The local conductance at the left contact can be derived by taking a partial derivative of the left terminal current (I L ), as given in (4), with the left contact voltage (V L ), and the right contact voltage (V R ) set to zero, and is given by . Using this, we derive the following expression for the local conductance using the Landauer Büttiker form The term G LL (V ) is due to currents flowing into the SC-MI bilayer [56].
The non-local conductance formula can similarly be derived by taking a partial derivative of the left terminal current (I L ), as given in (4), over the right terminal voltage (V R ), with the left terminal voltage (V L ) set to zero, such that, (7) Using the above, we have analysed the local and nonlocal conductances of the device in both the pristine and disordered setups.

A. Usadel Equation
The Usadel equation is the standard way to describe superconductors in the diffusive limit, yielding a quasiclassical Green's function which we use to compute the self-energy of the SC-MI bilayer in our setup. It is valid when the mean free path is less than the superconducting coherence length but still much greater than the metallic Fermi velocity. This approximation is reasonable for superconductors such as Al, which are commonly used in such experiments. We also consider a 'dirty' superconductor, in which the scattering with non-magnetic impurities is also much smaller than the superconducting coherence length.
The Usadel equation comes from a quasiclassical approximation to the Gorkov equations. The Gorkov equations form a closed set of equations that describe the equations of motion for the Green's function of a device and incorporate all the results of microscopic BCS theory. This is done by introducing an anomalous Green's function to account for electron pairing in addition to the normal electronic Green's function. However, this is often difficult to work with, and the Usadel equation, which is derived from it is often used to describe 1D superconducting systems. Both the Gorkov equation, and by extension, the Usadel equation can easily account for dirty systems with impurities.
We take into account spin-orbit and spin-flip scattering in the SC which may be intrinsic or arise from scattering off magnetic impurities. The MI layer also induces a uniform Zeeman field V SC Z in the superconductor. In a dirty superconductor, the effect of the MI on the SC is microscopically equivalent to applying a magnetic field [72]. The Usadel equation in the Nambu spinor basis is: where, the quasi-classical Green's function for the bilayer isǧ. Here we have ignored the orbital effects, so the vector potential, i.e., A = 0, D is the diffusion constant, V SC Z is the Zeeman field induced in the SC due to the MI including any additional external magnetic field, ∆ is the superconducting gap, andΣ represents the self energy due to scattering. The self energy due to scattering incorporates the spin-orbit (Σ so ) scattering and the spin-flip scattering (Σ sf ), which are characterised by the timescales τ so , τ sf respectively.

(S.2)
We define the energy scale for the relaxation processes as: Γ so(sf ) = 3 2τ so(sf ) . Parameterizing the quasi-classical green's function in terms of θ and φ, we can write: This yields the following equations: In our case, the effect of the MI and SM on the superconductor are uniform, and thereforeǧ(ω, r), and thus, θ(ω, r) and φ(ω, r) are only functions of ω and not r. This means that the diffusive term drops out of the Usadel equation, and we can simply solve the nonlinear algebraic equations for θ(ω, r) and φ(ω, r). Since this bilayer is in contact with the entire Rashba nanowire, we consider the self energy to be Σ (ω) = I N ⊗Σ(ω) . For the Green's function calculations, we solve the Usadel equations in the real time domain. However, before we solve for the Green's function for the device, we use the Usadel equation to find the self-consistent value of the superconducting gap in the presence of external magnetic fields and the magnetic field due to the magnetic insulator. We solve the Usadel equation self-consistently with the superconducting gap equation S.7, along with a thermodynamic constraint S.8 that the free energy of the superconducting phase should be greater than that of the normal metal phase, such that, and f sn = πT ν 0 ωn>0 4ω n − 2 cosh φ (2ω n cos θ + ∆ sin θ) + 4V SC Z sinh φ sin θ + D ∇ 2 θ − ∇ 2 φ + We start with the initial guess of ∆ being ∆ 0 , calculate the Green's function for each ω n , find ∆, and then repeat until it converges. The entire procedure for this calculation is schematized in Fig. S.3. Figure S.2 depicts the results of the self-consistent calculation for various values of the magnetic field and the scattering times. These results agree with [48]. The Chandrasekhar-Clogston limit places a stringent restriction on the maximum value of a magnetic field that can co-exist with superconductivity. The superconducting pairing potential undergoes a first-order transition into the normal state. This limit is clearly seen in our computational results. The Clogston limit increases with increase in spin-orbit coupling and decreases with decrease in spin-orbit coupling.

B.
The Keldysh NEGF approach

Current calculation
In the NEGF approach, the device is partitioned into a central channel, which comprises the Rashba nanowire and the leads as depicted in Fig. S.3, which are incorporated using the self-energies. Here, the interface with the SC-MI bilayer, is also incorporated as a self-energy, and effectively acts as a third contact. This enables us to calculate the retarded Green's function for the channel as: where E is the free variable energy, and I is the identity matrix of the dimension of the channel Hamiltonian, η is a small positive damping parameter, and Σ r L and Σ r R represent the retarded self energies for the semi-infinite contacts and Σ represents the self energy of the SC-MI bilayer, whose calculation was detailed in the previous section.  S.3. Schematic for the numerical calculations. The typical NEGF scheme involves the discretization of the channel Hamiltonian and expressing the coupling to various contacts using self-energies. In our setup, the voltages are applied at the left and right contacts, and the Usadel equations are employed to calculate the self energy due to the bilayer that is in contact with the Rashba nanowire channel.
In the wide-band approximation, the self-energy of the contacts can be represented in their own eigenbasis, and are hence diagonal. They are characterised by a parameter γ L/R , which represents the escape rate into the contacts. The current operator for an N-S-N device, derived from first principles in various references [55,56], to discern the various components of the current: where the term (i) represents the direct transmission process of either the electron or the hole, (ii) represents the direct Andreev transmission and (iii) represents the crossed Andreev transmission. At this point it is worth noting that f ee α = f (E − µ α ), f hh α = f (E + µ α ), and that Γ α is the imaginary part of the respective self energy. In our device, we need to consider the extra contribution to the current arising due to the presence of the effective third-contact [56].
C. Local and non-local conductance

Floating superconductor configuration
For a floating SC, we consider the only effect of the SC is proximity induced superconductivty, and we do not consider as a terminal for current flow. We derive the expression for the current in a generic bias situation, (V L ,V R ) for the left and right contacts with µ L = eV L and µ R = eV R Both the electron and hole flows contribute to the current at any contact. The net current at the left contact is given by: The individual electron and hole components can then be derived using: L in order to calculate the net current from a contact. We also consider the conductance matrix, which explicitly evaluates the local and non-local components of the conductance.
Henceforth, we consider only the electron current since I L = I (e) L . The local conductance is then given as: (S. 16) At zero temperature, the Fermi distribution function, f (x) = Θ (−x) where Θ is the Heaviside step function. This implies f (E − eV ) = Θ (eV − E). Utilising the fact that the derivative of the Heaviside function is the Dirac delta function, we get: The expression for non-local conductance can be derived similarly.

Grounded superconductor configuration
For our device setup, current can flow into the SC-MI bilayer terminal as well. This leads to an additional self-energy term (Σ ) in the Green's function. The Green's function for the grounded setup is thus given by : where Σ can be derived as outlined in S.6. where γ represents the coupling strength between the bilayer and the Rashba nanowire.
In the presence of the superconducting contact, an additional term gets added to the previously derived formula for local conductance (S.17), and no additional term gets added to the non-local conductance formula (S.19).
We once again start with the general current operator for the left contact, and obtain the electron current for the left contact as: