# Coherent and non-coherent conductance signatures of Majorana zero modes

Chaitrali Duse  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai-400076, India  Praveen Sriram  Department of Applied Physics,
Stanford University, 348 Via Pueblo, Stanford, CA 94305, United States of
America  Kaveh Gharavi  Institute for Quantum Computing, University of
Waterloo, Waterloo, Ontario, Canada N2L 3G1  Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1  Jonathan
Baugh  Institute for Quantum Computing, University of Waterloo, Waterloo,
Ontario, Canada N2L 3G1  Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, Canada N2L 3G1  Department of Chemistry,
University of Waterloo, Waterloo, Ontario, Canada N2L 3G1  Bhaskaran
Muralidharan  Department of Electrical Engineering, Indian Institute of
Technology Bombay, Powai, Mumbai-400076, India<EMAIL_ADDRESS>
###### Abstract

Conductance signatures that signal the presence of Majorana zero modes in a
three terminal nanowire-topological superconductor hybrid system are analyzed
in detail, in both the clean nanowire limit and in the presence of non-
coherent dephasing interactions with fluctuating impurities. In the coherent
transport regime for a clean wire, we point out contributions of the local
Andreev reflection and the non-local transmissions toward the total
conductance lineshape clarifying the role of contact broadening on the
Majorana conductance lineshapes at the magnetic field parity crossings.
Interestingly, at larger $B$-field parity crossings, the contribution of the
Andreev reflection process decreases which is compensated by the non-local
processes in order to maintain the conductance quantum regardless of contact
coupling strength. In the non-coherent transport regime, we include dephasing
that is introduced by momentum randomization processes, that allows one to
smoothly transition to the diffusive limit. Here, as expected, we note that
while the Majorana character of the zero modes is unchanged, there is a
reduction in the conductance peak magnitude that scales with the strength of
the impurity scattering potentials. Important distinctions between the effect
of non-coherent dephasing processes and contact-induced tunnel broadenings in
the coherent regime on the conductance lineshapes are elucidated. Most
importantly our results reveal that the addition of dephasing in the set up
the does not lead to any notable length dependence to the conductance of the
zero modes, contrary to what one would expect in a gradual transition to the
diffusive limit. This work thus paves the way for systematic introduction of
scattering processes into the realistic modeling of Majorana nanowire hybrid
devices and assessing topological signatures in such systems in the presence
of scattering processes.

##  I Introduction

Figure 1:  Device schematics: (a) The N-TS-N system under consideration in
this work. $V_{G}$ controls the chemical potential in the nanowire, which we
assume to be spatially uniform. $V_{G1}$ and $V_{G12}$ control the coupling to
the normal contacts on either side. Voltages $V_{L}$ and $V_{R}$ are applied
to the respective contacts. Throughout this work, we will be focusing on
symmetric biasing ($V_{L}=-V_{R}=V/2$). (b) After discretizing the nanowire as
a N-site tight-binding lattice (with site index m), the self-energy matrices
$\Sigma_{L}$ and $\Sigma_{R}$ in the NEGF formalism take into account the
coupling of the channel to the contacts. Non-coherent scattering processes
within the nanowire are taken into account by the scattering self-energy
$\Sigma_{R}$.

Quasi one-dimensional Rashba nanowire-superconductor hybrids Alicea (2010);
Lutchyn  _et al._ (2010) are currently leading candidates for the detection
and manipulation of Majorana zero modes (MZMs) Kitaev (2001), with the
ultimate goal of realizing topological quantum computing Sarma  _et al._
(2015); Aasen  _et al._ (2016); O’Brien  _et al._ (2018). The past few years
have witnessed tremendous experimental advancements Mourik  _et al._ (2012);
Deng  _et al._ (2016); Nichele  _et al._ (2017); Gül  _et al._ (2018); Zhang
_et al._ (2021) aimed at the ascertained detection of MZMs via conductance
measurements, with the zero bias conductance quantization as a possible tell-
tale signature. The zero bias conductance peak (ZBCP) of measure $2e^{2}/h$
arising out of a coherent and perfect Andreev reflection Prada  _et al._
(2020) is regarded as a strong signature of the MZM. This signature is
expected from a local conductance measurement in a normal - topological
superconductor (N-TS) link Zhang  _et al._ (2019); Prada  _et al._ (2020);
Zhang  _et al._ (2018, 2021).
In recent times, there has also been a keen interest in pursuing transport
measurements across a three terminal normal-topological superconductor-normal
(N-TS-N) configuration Puglia  _et al._ (2020), where specific non-local
conductance signatures Rosdahl  _et al._ (2018) could augment the certainty of
MZM detection. Before graduating toward the actual manipulation or harnessing
of MZMs Zhang  _et al._ (2019); Prada  _et al._ (2020), many unresolved
aspects exist, that include the possibility of a non-topological origin of the
zero bias conductance as well as understanding the role of interactions within
the topological superconductor channel Stanescu and Tewari (2013); Liu  _et
al._ (2017a); Vuik  _et al._ (2019); Liu  _et al._ (2018); Moore  _et al._
(2018a); Deng  _et al._ (2018); Moore  _et al._ (2018b); Chiu and Das Sarma
(2019); Pan and Sarma (2020).
All this has led to a side-by-side strong theoretical push Stanescu and Tewari
(2013); Liu  _et al._ (2017a); Vuik  _et al._ (2019); Liu  _et al._ (2018);
Moore  _et al._ (2018a); Deng  _et al._ (2018); Moore  _et al._ (2018b); Chiu
and Das Sarma (2019); Pan and Sarma (2020), where a lot of focus has been on
resolving the issue about the origins of the observed ZBCP. In parallel, there
is also a strong need to push for theoretical works that go beyond minimal 1D
models in order to include multi-mode effects Prada  _et al._ (2020); Lahiri
_et al._ (2018); Sriram  _et al._ (2019), realistic potentials Mikkelsen  _et
al._ (2018); Antipov  _et al._ (2018); Domínguez  _et al._ (2017); Vuik  _et
al._ (2016); Pan  _et al._ (2019); Winkler  _et al._ (2019), the inclusion of
disorder Deng  _et al._ (2018); Moore  _et al._ (2018b); Chiu and Das Sarma
(2019); Pan and Sarma (2020) and also scattering and relaxation effects Liu
_et al._ (2017b). While the importance of ascertaining the topological origin
of the observed experimental conductance signatures is of great importance, in
this paper, we take the route of analyzing the nature of a topological MZM in
a typical three terminal N-TS-N configuration and the corresponding ZBCP,
first in the coherent transport regime, and then in the non-coherent regime
where dephasing effects due to relaxation mechanisms are present.
Theoretical works have been based on minimal 1D models used to analyze the
N-TS links Stanescu and Tewari (2013); Liu  _et al._ (2017a); Vuik  _et al._
(2019); Liu  _et al._ (2018); Moore  _et al._ (2018a); Deng  _et al._ (2018);
Moore  _et al._ (2018b); Chiu and Das Sarma (2019); Pan and Sarma (2020) and
have typically been based on a scattering formulation of quantum transport
Groth  _et al._ (2014) in the coherent regime. In this work, we use the
Keldysh non-equilibrium Green’s function (NEGF) technique San-Jose  _et al._
(2013); Cayao  _et al._ (2018, 2015); Doornenbal R. J.  _et al._ (2015);
Zazunov  _et al._ (2016); Sriram  _et al._ (2019); Bartolo  _et al._ (2020);
Leumer  _et al._ (2020) to analyze the transport across the N-TS-N junction,
with the inclusion of channel interactions in the non-coherent regime Datta
(1997); Golizadeh-Mojarad and Datta (2007); Sharma  _et al._ (2016, 2017);
Singha and Muralidharan (2018).
A three terminal geometry depicted schematically in Fig. 1, comprises a
topological superconducting region flanked by two normal contacts and tunnel
barriers. Such a geometry permits local and non-local transport measurements,
with the local Andreev reflection, the crossed Andreev and direct
transmissions respectively Rosdahl  _et al._ (2018); Leumer  _et al._ (2020);
Bartolo  _et al._ (2020). In the coherent regime, we point out contributions
of the local Andreev reflection and the non-local transmissions toward the
conductance signatures and that a half of the conductance quantum $(e^{2}/h)$
is indeed maintained. The half fraction, as pointed out in earlier works
Leumer  _et al._ (2020) is indeed due to a voltage divider formed at the two
N-TS junctions, under a symmetric bias situation. The effect of contact
broadening clearly shows the maintenance of the ZBCP at zero temperatures
Nichele  _et al._ (2017); Zazunov  _et al._ (2016). In addition, we clarify on
the roles of the contact broadenings on the local and the non-local components
of the transmission around the parity crossings, and the conductance
lineshapes of the MZM. Interestingly, we note that at larger $B$-field parity
crossings, the contribution of the Andreev reflection process decreases which
is compensated by the non-local processes in order to maintain the conductance
quantum regardless of contact coupling strength.
We then analyze the non-coherent transport regime by including dephasing due
to fluctuating impurities and the resulting momentum randomization processes
which allows one to transition smoothly from the coherent ballistic limit all
the way to the diffusive limit Camsari  _et al._ (2020). It is pointed out
that the Majorana character of the zero mode is unchanged, as seen in the
spatial variation of the wavefunction, and the zero-bias conductance peak is
retained. However, we observe a reduction in the peak magnitude that scales
with the strength of the impurity potential fluctuations. Important
distinctions between dephasing processes in the non-coherent regime and the
contact-induced tunnel broadenings on the conductance lineshapes in the
coherent regime Nichele  _et al._ (2017) are clearly elucidated. Most
importantly our results reveal that the addition of dephasing in the set up
the does not lead to any notable length dependence to the conductance of the
zero modes, contrary to what one would expect in a gradual transition to the
diffusive limit. We believe that this work paves the way for systematic
introduction of scattering processes into the realistic modeling of Majorana
nanowire hybrid devices.
This paper is organized as follows. In Sec. II, we introduce the 1-D Rashba
nanowire channel and the NEGF formalism in both the coherent and non-coherent
regime. Following this, we describe the main results of coherent transport in
Sec. III.2, and that of non-coherent transport in Sec. III.3. We also depict
the results of the longer wire in Sec. III.4. Many of the results presented
deserve a longer discussion that is dealt with in Sec. IV and we conclude in
Sec. V.

##  II Set up and formalism

The transport set up as depicted in Fig. 1 comprises the topological
superconducting channel, connected to two normal metallic contacts. The
schematic also consists of two tunnel gates labeled $V_{G1}$ and $V_{G2}$ for
modulating the tunnel barriers and a central gate $V_{G}$. We consider a one-
dimensional semiconducting nanowire with Rashba spin-orbit coupling
$\alpha_{R}$, which is placed in an external axial-Zeeman field $B$ coupled to
an s-wave superconductor in order to induce a proximity effect Zhang  _et al._
(2019). Within the Keldysh NEGF formalism, we can translate the set up as
depicted in Fig. 1, where the channel is represented via the Hamiltonian and
the connection to contacts as well as dephasing via self energies. We begin by
detailing the computational approach we employ to pursue the calculations in
this paper.

###  II.1 The Rashba nanowire Hamiltonian

For our computational purposes, we consider the 1-D Rashba Hamiltonian Cayao
(2017) in the tight-binding representation in a discrete lattice San-Jose  _et
al._ (2013); Cayao (2017) with $N$ sites and effective lattice constant $a$,
written as

$\mathcal{\hat{H}_{0}}=\sum_{i}c_{i}^{\dagger}\alpha_{S}c_{i}+\sum_{<i,i+1>}c_{i}^{\dagger}\beta
c_{i+1}+\sum_{<i,i+1>}c_{i+1}^{\dagger}\beta^{\dagger}c_{i},$ (1)

where $c_{i}={\begin{bmatrix}d_{i,\uparrow}&d_{i\downarrow}\end{bmatrix}}^{T}$
represents the annihilation operator of the spinor, with individual spin
components $d_{i\sigma}$ of spin $\sigma=\uparrow,\downarrow$ on a site $i$.
We then have $\mathcal{\hat{H}_{0}}=\sum_{i,j}c_{i}^{\dagger}H_{0}c_{j}$,
where $H_{0}$ is the first quantized Rashba nanowire Hamiltonian such that

$\displaystyle\alpha_{S}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}2t-\mu&V_{Z}\\\
V_{Z}&2t-\mu\end{array}\right)$ (4) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-t&t_{SO}\\\
-t_{SO}&-t\end{array}\right),$ (7)

where Zeeman energy $V_{Z}=g\mu_{B}B/2$, where $g$ is the electronic g-factor,
$\mu_{B}$ is the Bohr magneton, and $B$ is the applied axial magnetic field.
The tight-binding hopping parameter $t=\frac{\hbar^{2}}{2m^{*}a^{2}}$ with
$m^{*}$ being the effective mass and $\hbar$ being the reduced Planck’s
constant, and $t_{SO}=\frac{\alpha_{R}}{2a}$, where $\alpha_{R}$ is the Rashba
spin-orbit coupling term. $\mu$ is the chemical potential in the nanowire,
which is taken to be spatially uniform. Here $\alpha_{S}$ represents the on-
site $2\times 2$ matrix and $\beta$ represents the $2\times 2$ hopping matrix
in the discretized tight binding representation.
In the Bogoliubov-de Gennes (BdG) representation for the superconducting
state, we can write the Hamiltonian of the Rashba nanowire in its 4-component
site Nambu spinor
$\Psi_{i}={\begin{bmatrix}d_{i,\uparrow}&d_{i\downarrow}&d_{i\uparrow}^{\dagger}&d_{i\downarrow}^{\dagger}\end{bmatrix}}^{T}$
representation as
$\mathcal{H}=\frac{1}{2}\sum_{i,j}\Psi^{\dagger}_{i}H_{BdG}\Psi_{j}$, where

$H_{BdG}=\left(\begin{array}[]{cc}H_{0}&\Delta_{m}\\\
\Delta^{\dagger}_{m}&-H_{0}^{*}\end{array}\right)$ (8)

with the pairing potential given as
$\Delta_{m}=i\Delta(\mathbbm{1}_{N}\otimes\sigma_{y})$, where $\Delta$ is the
superconducting order parameter induced in the nanowire by proximity effect
and $\sigma_{y}$ is the $y$ Pauli matrix. Increasing the magnetic field $B$
(and thereby $V_{Z}$) in the wire drives the system into the topological phase
beyond the critical Zeeman energy $V_{Z}^{c}=\sqrt{\Delta^{2}+\mu^{2}}$, which
corresponds to a critical magnetic field $B_{c}$.Alicea (2010); Lutchyn  _et
al._ (2010)
We take the contacts to be in the eigenbasis with their Hamiltonians given by

$\hat{H_{C}}=\sum_{k\sigma,\alpha\in L,R}\epsilon_{\alpha k\sigma}c_{\alpha
k\sigma}^{\dagger}c_{\alpha k\sigma},$ (9)

where $c_{\alpha k\sigma}^{(\dagger)}$ represents the annihilation (creation)
operator of an electronic momentum eigenstate of energy $\epsilon_{\alpha
k\sigma}$ indexed with $k$ of spin $\sigma=\uparrow,\downarrow$ in contact
labeled $\alpha=L,R$. Since the contacts are normal metal, the energy levels
are spin degenerate such that $\epsilon_{\alpha k\uparrow}=\epsilon_{\alpha
k\downarrow}=\epsilon_{\alpha k}$
The coupling between the contacts and the topological superconductor is
represented by the spin-conserving tunneling Hamiltonian given by

$\hat{H_{T}}=\sum_{k\sigma m\alpha\in L,R}\left[\tau_{\alpha
km\sigma}c_{\alpha k\sigma}^{\dagger}d_{m\sigma}+{\tau}^{*}_{\alpha k\sigma
m}d_{m\sigma}^{\dagger}c_{\alpha k\sigma}\right],$ (10)

where $\tau_{\alpha km\sigma}$ is an element of the tunnel coupling matrix
between a state $k$ in contact $\alpha=L,R$ with spin
$\sigma=\uparrow,\downarrow$ and a site $m$ with the same spin
$\sigma=\uparrow,\downarrow$ in the channel. In this paper, and of relevance
to the Majorana nanowire, we take a spin-independent tunnel coupling
$\tau_{\alpha km\uparrow}=\tau_{\alpha km\downarrow}=\tau_{\alpha km}$. We now
proceed to describe the transport formulation that involves the proximitized
Rashba nanowire system as the central system coupled to two normal contact as
leads $\alpha=L,R$. Furthermore, referring to Fig. 1, we note that $m=1(m=N)$
when $\alpha=L(\alpha=R)$, implying that $\tau_{Lk}=\tau_{Lk1}$ and
$\tau_{Rk}=\tau_{RkN}$.

###  II.2 The Keldysh NEGF formalism

Though the foundations of scattering theory Groth  _et al._ (2014) allow us to
evaluate various quantities and currents in various structures, a systematic
computational framework for arbitrary geometries including unconventional
pairing Samanta and Datta (1998) may be set up using the Keldysh non-
equilibrium Green’s function (NEGF) technique Haug and Jauho (1996); Datta
(1997). The formalism also allows one to capture intra-channel interactions in
a systematic way, in principle facilitating a smooth transition from a purely
quantum ballistic regime all the way toward the diffusive regime Datta (1997);
Golizadeh-Mojarad and Datta (2007); Camsari  _et al._ (2020). This aspect of
the inclusion of dephasing will also be dealt with in this paper. First, we
will cover the coherent ballistic regime, in which the Landauer-Büttiker type
expressions for the conductance signatures of the MZMs will be derived. Based
on the Hamiltonian from (8), for steady state transport calculations, we
define the retarded Green’s function in the matrix representation as

$[G^{r}(E)]=\left[(E+i\eta)I-H_{BdG}-\Sigma^{r}_{L}-\Sigma^{r}_{R}-\Sigma^{r}_{s}\right]^{-1},$
(11)

where $E$ is the energy free variable and $I$ is the identity matrix of the
dimension of the Hamiltonian. The quantities
$\Sigma^{r}_{\alpha},\Sigma_{s}^{r}$ represent the retarded self energies of
the contacts $\alpha$, and the scattering processes respectively. The latter
will be considered in the non-coherent regime. To evaluate transport
properties, one also requires the lesser Green’s function, which is given by a
matrix product

$[G^{<}(E)]=[G^{r}][\left(\Sigma_{L}^{<}+\Sigma_{R}^{<}+\Sigma_{s}^{<}\right)][G^{a}],$
(12)

where $\Sigma_{\alpha}^{<}$ is the lesser self energy of the contacts are
defined in Appendix in (29).
The calculation of retarded contact self energies $[\Sigma^{r}_{L(R)}]$ can be
done in any basis of the contacts depicted in Fig. 1.
$\Gamma_{L(R)}=i[\Sigma^{r}_{L(R)}]-[\Sigma^{r}_{L(R)}]^{\dagger}$ denotes the
broadening matrix of either lead. Here, we consider the contacts in the
eigenbasis of the Hamiltonian in (9), where the self energy matrix is
diagonal. In the wide band approximation described in Appendix A, the contact
broadening matrix is real and independent of energy, so that the self energy
matrix can be written as $[\Sigma^{r}_{L(R)}]=-i[\Gamma_{L(R)}]/2$. Further,
$\Sigma^{r}_{L}(i,i)=-i\gamma_{L}/2$ for $i=1$ to $i=4$ and
$\Sigma^{r}_{R}(i,i)=-i\gamma_{R}/2$ for $i=4N-3$ to $i=4N$, and all other
diagonal elements are zero. These represent the up (down)-spin electron (hole)
submatrices at each site $i$ in the 4-component site Nambu representation. The
energy-independent self energy is given by $\gamma_{L(R)}=2\pi\int
d\epsilon_{k\alpha}\mid\tau_{\alpha,k}\mid^{2}\delta(E-\epsilon_{k\alpha})$,
which we treat as a parameter here. In this paper we take
$\gamma_{L}=\gamma_{R}=\gamma$.
With the evaluation of the retarded Green’s function, one can calculate the
spectral function and the density of states as follows:

$[A(E)]=i[(G^{r}-G^{a})]=[G^{r}][\Gamma][G^{a}],$ (13)

where, the diagonal elements of the spectral function $D(E)=A(E)/2\pi$ are
related to the local density of states (LDOS), and
$[G^{a}(E)]=\left[G^{r}(E)\right]^{\dagger}$ represents the advanced Green’s
function.
Coherent transport: The current operator Datta (1997) from the Keldysh NEGF
formalism can be derived from the lesser Green’s function in Nambu space, as
detailed in Appendix A as follows:

$I^{op}_{L}(E)=\frac{1}{2}\left[\frac{e}{h}Tr\left(\tau_{z}\left[[G^{r}][\Sigma_{L}^{<}]-[\Sigma_{L}^{<}][G^{a}]+[G^{<}][\Sigma_{L}^{a}]-[\Sigma_{L}^{r}][]G^{<}]\right]\right)\right],$
(14)

where $\tau_{z}=\sigma_{z}\otimes I$, with $\sigma_{z}$ being the Pauli-z
matrix, representing the fact that the net current is the difference between
the currents of the composite electron and hole blocks. The factor of half
comes from the BdG representation of the original Hamiltonian. Following the
detailed derivation in Appendix A, we can obtain the electron (hole) current
across the device as a sum of three components:

$\displaystyle
I^{e(h)}_{L}(E)=\frac{e}{h}\left(Tr\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{ee(hh)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{R}^{ee(hh)}(E)\right]\right)$
(15)
$\displaystyle\qquad+\frac{e}{h}\left(Tr\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{L}^{hh(ee)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{L}^{hh(ee)}(E)\right]\right)$
(16)
$\displaystyle\qquad+\frac{e}{h}\left(Tr\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{hh(ee)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{R}^{hh(ee)}(E)\right]\right),$
(17)

where the superscripts $ee~{}(hh)$ represent the block diagonal sector of the
corresponding matrices in the electron (hole) Nambu space, and (15) represents
the direct transmission process of either the electron or the hole, (16)
represents the direct Andreev transmission and (17) represents the crossed
Andreev transmission. The three components $T_{D}(E)$, $T_{A}(E)$ and
$T_{CA}(E)$ corresponding to direct transmission, Andreev reflection and and
crossed Andreev transmission processes shown above are

$\displaystyle
T_{D}(E)=Tr\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{ee(hh)}G^{a}\right)$ (18)
$\displaystyle
T_{A}(E)=Tr\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{L}^{hh(ee)}G^{a}\right)$ (19)
$\displaystyle
T_{CA}(E)=Tr\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{hh(ee)}G^{a}\right).$
(20)

At this point it is worth noting that
$f_{\alpha}^{ee}(E)=f(E-\mu_{\alpha}),f_{\alpha}^{hh}(E)=f(E+\mu_{\alpha})$,
representing the Fermi-Dirac distributions of the electronic and hole sector
respectively. In these calculations, we must recall that the up(down)-spin
degrees of freedom in the electron and hole sectors respectively are included
in (15), (16) and (17).

Figure 2: Equivalent Landuer-Büttiker network for our device. Each contact is
further divided into “two contacts”, the electron part and the hole part,
namely $Le$, $Lh$, $Re$ and $Rh$ respectively.

Figure 3:  Energy spectrum of a short (a), $l=1.6\mu m$) and a long (b),
$l=8\mu m$) Majorana nanowire with varying normalized Zeeman field $B/B_{c}$.
The gap closing and re-opening at $B=B_{c}$ is seen clearly confirming the
topological regime for $B\geq B_{c}$. The oscillations in energy are due to
hybridization of the zero-energy Majorana modes at either ends in a finite
nanowire. While these oscillations are seen prominently for the short wire,
their amplitude is suppressed for longer wires where the Majorana localization
length $l_{M}\ll l$ (but are still present, as seen in the inset). The points
marked $B_{1}$ and $B_{2}$ in (a) represent the first two distinct zero energy
parity crossings for the shorter wire.

A somewhat rudimentary understanding of the above results can also be sought
from the Landauer-Büttiker formalism. Even though our device has two contacts,
we conceptually have a four terminal device as sketched in Fig. 2. The current
across such a multi-terminal device Datta (1997) as seen from a terminal $p$
can be written as

$I_{p}(E)=\frac{1}{q}\left[\sum_{q}G_{qp}(E)f_{p}(E)-G_{pq}(E)f_{q}(E)\right],$
(21)

where $G_{pq}$ represents the conductance of the segment $q\rightarrow p$, and
$f_{p(q)}$ refers to the Fermi-Dirac distribution in terminal $p(q)$, with
chemical potential $\mu_{p(q)}$. In the 1D channel that we consider, the
conductances obey Onsager symmetry such that $G_{qp}=G_{pq}$, and hence
$I_{p}=\sum_{q}G_{pq}\left(f_{p}-f_{q}\right)$. Each contact has its
electronic and hole component labeled as $Le(1)$, $Lh(2)$, $Re(3)$ and $Rh(4)$
respectively. Now we can write the terminal current for (say) the $Le(1)$
contact as

$I_{1}=G_{12}(f_{1}-f_{2})+G_{13}(f_{1}-f_{3})+G_{14}(f_{1}-f_{4}),$ (22)

where the energy argument is understood as described in (21). Now comparing
the above equation with (15), (16) and (17), we note at zero temperature that
$G_{12}=\frac{e^{2}}{h}T_{A}(E=0)$, $G_{13}=\frac{e^{2}}{h}T_{D}(E=0)$ and
$G_{14}=\frac{e^{2}}{h}T_{CA}(E=0)$, representing the Andreev transmission
representing the local conductance, the direct and the crossed Andreev
transmissions that comprise non-local conductance respectively. Throughout
this work, we work in the zero temperature limit with the thermal energy much
less than the other energy scales involved in the system ($k_{B}T\ll\Delta$).
For reasons involving current conservation in systems featuring
superconducting regions, well noted in previous works Cuevas  _et al._ (1996);
Lim  _et al._ (2012); Doornenbal R. J.  _et al._ (2015); Leumer  _et al._
(2020), we resort to a symmetric voltage bias, i.e., $\mu_{L}=qV/2$ and
$\mu_{R}=-qV/2$. However, the issue of current conservation can only be via a
self consistent calculation of the order parameter as noted in Cuevas  _et
al._ (1996), or other number conserving methods beyond the mean-field
description Lapa and Levin (2020). We leave such a calculation for future
work.
Non-coherent transport: In the case of non-coherent transport, intra-channel
interactions such as electron-electron interactions, dephasing and inelastic
scattering can be included via the scattering self energy $\Sigma^{r}_{s}$. In
this paper, we include intra-channel interactions in the form of channel
fluctuating impurities with localized potentials $U(r_{i})$ at the sites, such
that their correlator $\bar{D}(i,j)=\left\langle
U(r_{i})U^{*}(r_{i})\right\rangle$ can be used to find the scattering self
energies within the self consistent Born approximation Danielewicz (1984);
Datta (1997); Golizadeh-Mojarad and Datta (2007); Sharma  _et al._ (2018);
Lahiri  _et al._ (2018); Sriram  _et al._ (2019)

$\displaystyle\Sigma_{s}^{r}(i,j)=\bar{D}(i,j)G^{r}(i,j)$ (23)
$\displaystyle\Sigma_{s}^{<}(i,j)=\bar{D}(i,j)G^{<}(i,j),$ (24)

where, the superscripts $r(<)$ stand for the usual retarded (lesser) Green
functions. In this paper, we adopt a homogeneous model with uniform, elastic,
and spatially uncorrelated interactions, resulting in a diagonal form of
$\bar{D}(i,j)=d_{0}\delta_{ij}$. This model discards the off-diagonal elements
of the Green’s function, thus relaxing both the phase and momentum of
quasiparticles in the nanowire. Danielewicz (1984); Datta (1997). The quantity
$d_{0}$ is the dephasing parameter which represents the magnitude squared of
the fluctuating scattering potentials. This can also be extended to include
non-local fluctuations via the spatial correlations of the impurity potentials
Danielewicz (1984); Datta (1997). Using (24) solved self consistently with the
equations for the retarded (lesser) Green’s functions given in (11) and (12)
we next obtain the currents given by (14). We must importantly note that here
one cannot write the current operator in a compact Landauer-Büttiker form and
hence cannot separate the local and non-local contributions.

Table 1: These parameters Mourik  _et al._ (2012) are used in all further analysis, unless otherwise stated. Parameter |  | Value
---|---|---
Effective mass | $m^{\ast}$ | 0.015 $m_{e}$
Induced order parameter | $\Delta$ | 0.25 meV
Tight-binding hopping parameter | t | 10 meV
Rashba spin-orbit coupling strength | $\alpha_{R}$ | 20 meV nm
Chemical potential | $\mu$ | 0.5 meV
Contact coupling energy | $\gamma$ | 0.025 meV
NEGF infinitesimal parameter | $\eta$ | 10-10 eV

##  III Results

###  III.1 Energy Spectra

We begin by reiterating some well known results Cayao (2017) in connection
with the equilibrium energy spectra of the wires that we consider in this
paper. The energy spectrum of the system is evaluated by numerically
diagonalizing the discretized tight-binding Hamiltonian (8) using the
parameter set described in Tab. I . The energy eigenvalues with varying
magnetic field $B$ are shown in Fig. 3 for two different lengths of the
nanowire: $l\approx 1.6\mu m$ (Fig. 3) and $l\approx 8\mu m$ (Fig. 3),
corresponding to $100$ and $500$ sites respectively in the tight-binding
model. The two eigenvalues closest to zero energy are highlighted. The energy
gap at decreases as the Zeeman field is increased, finally closing at the
topological transition point corresponding to $B=B_{c}$. Beyond $B=B_{c}$,
topological Majorana bound states are formed at each end of the nanowire.

Figure 4:  Local density of states map with varying magnetic field $B$ at a
particular lattice site close to the edge on site 5 (a) and at the center of
the nanowire at site 50 (b). We clearly see that the zero-mode contribution to
the density of states is mainly from the edge sites as compared to the center,
where the zero-mode local density of states contribution is much smaller.

These zero energy states decay exponentially into the bulk of the wire, with
their spatial spread characterized by the Majorana localization length
$l_{M}$. The Majorana localization length has a non-trivial dependence on the
parameters chosen Cayao (2017); Klinovaja and Loss (2012), and may increase or
decrease with the applied magnetic field depending on the spin-orbit length
scale. For the range of realistic parameters chosen, $l_{M}$ indeed increases
with increasing Zeeman field, following a minima close to the critical field
$B_{c}$ Cayao (2017).
In a finite nanowire, there is non-zero overlap between the wavefunctions of
the the states, and this hybridization leads to oscillations in the
corresponding energy levels. The length of the nanowire $l$ relative to the
localization length $l_{M}$ determines the amplitude of these oscillations;
however the oscillations are always present, as depicted in the inset of Fig.
3. For long enough nanowires, the broadening in the energy levels due
temperature or coupling to contacts would be larger than the amplitude of
these oscillations at Zeeman energies close to the topological transition, and
hence the states are effectively zero modes.

Figure 5:  Local density of states at the Zeeman energies $B_{1}$ (Fig. 5) and
$B_{2}$ (Fig. 5) where the oscillations meet to give exactly zero-energy
modes, as marked in Fig. 3. These modes are clearly seen to be localized at
the ends of the nanowire at both values of $B$. The localization is greater at
$B_{1}$ compared to $B_{2}$ due to a shorter Majorana localization length
$l_{M}$.

Figure 6: Local and non-local transmission at zero energy with varying Zeeman
field. Plots are resolved into the direct, Andreev and crossed Andreev process
components, with (a) and (b) evaluated for $\gamma=0.1\Delta$ around the
points $B_{1}$ and $B_{2}$ (Fig. 3) respectively, (c) and (d) are evaluated
for $\gamma=\Delta$ at the points $B_{1}$ and $B_{2}$. In all cases, at the
Zeeman energies with exact zero-modes, the direct and crossed Andreev
contributions (non-local processes) drop and only the local Andreev process
makes the maximum contribution to the transmission peak with a total magnitude
of unity. For larger $\gamma$, the peaks are broadened out in energy, and
therefore the peak is more robust to small changes in B around $B_{1}$ or
$B_{2}$, however, retaining the unity maximum. It is also worth noting by
comparing (a) and (c) with (b) and (d) that the individual contributions of
the local and non-local processes change at the two unique parity crossings,
while to total magnitude remains quantized at unity at the magnetic field
corresponding to either crossing.

Figure 7: Coherent transport through a clean Majorana nanowire, exhibiting the
quantized zero-bias conductance peak signature of zero-modes. (a) Differential
conductance as a function of voltage, at magnetic field $B_{1}$ corresponding
to the first zero-mode shows the zero-bias conductance peak quantized at
$e^{2}/h$. (b) Zero-bias conductance with varying magnetic field shows
quantized peaks at the values of $B$ ($B_{1}$ and $B_{2}$) where the
oscillating states become MZMs.

###  III.2 Coherent Transport

Next, we employ the NEGF formalism in the clean wire limit in the coherent
transport regime through the nanowire. Using the spectral function defined in
(13), we calculate the local density of states (LDOS) at a specific site in
the nanowire with varying magnetic field as shown in Fig. 4. The total density
of states follows the energy spectrum in Fig. 3, but the contributions from
each site vary across the eigenstates. Figure 4 shows the LDOS at the fifth
site in the $100$-site nanowire, which is a location near the left end of the
wire. We see that this site mainly contributes to the oscillating mode close
to zero energy, and its contribution to the other states is small. The LDOS at
the center of the nanowire (site $50$) in Fig. 4 on the other hand shows a
negligible contribution to the zero-mode. In contrast, the higher energy
states have nearly equal LDOS at either site, which indicates that they are
not localized. To examine more closely the localization of states, in Fig. 5
we focus on the spatial variation of the LDOS particular Zeeman energies
corresponding to the point $B_{1}$ (Fig. 5) and $B_{2}$ (Fig. 5) as marked in
Fig. 3, in which the oscillating low-energy modes in the energy spectrum cross
to give to give exact zero-energy modes. We see clear end-localization of the
zero-energy Fermionic wavefunctions, with maximum density at site 5 near the
edge. We also see the finite spatial spread into the bulk of the nanowire and
the finite Majorana localization length. Since $l_{M}$ increases with
increasing Zeeman field in our sysstem, the zero-modes are seen to be more
localized at the point $B_{1}$ than at the point $B_{2}$ as expected. We also
note that the parity crossings have a node at the center of the nanowire at
$B_{2}$ but a non-zero value at $B_{1}$, which is also seen in Fig. 4 in the
density of the oscillating modes.

Figure 8: Local density of states at $B_{1}$ with momentum-relaxing dephasing
interactions, characterized by $d_{0}=10^{-8}t_{0}^{2}$ (a) and
$d_{0}=10^{-6}t_{0}^{2}$ (b). Non-coherent effects cause the states to be
broadened out in energy. However, the end-localized nature remains preserved,
confirming that these modes are still topologically significant.

Figure 9: Probability density $\mid\Psi_{i}\mid^{2}$ at each site in the
nanowire at Zeeman energies $B_{1}$ (a) and $B_{2}$ (b), for a clean nanowire
and for two different magnitudes of the dephasing interactions $d_{0}$. In all
the cases, the density has exactly the same spatial variation, indicating that
the localized nature of the MZMs is preserved even in the presence of
fluctuating impurities.

Figure 10: Differential conductance at magnetic field $B_{1}$, in the presence
of non-coherent interactions characterized by $d_{0}=10^{-8}t_{0}^{2}$ (a) and
$d_{0}=10^{-6}t_{0}^{2}$ (b). The ZBCP is present but no longer quantized at
$e^{2}/h$, and its magnitude decreases with increasing $d_{0}$, along with a
significantly more broadened lineshape.

Figure 11: Zero-bias conductance near $B_{1}$ (a) and $B_{2}$ (b) for two
different strengths of non-coherent interactions. The peaks are slightly
higher at $B_{2}$ compared to $B_{1}$. The lineshapes are more broadened in B
for stronger interactions.

Having verified the presence of edge-localized zero-energy MZMs at specific
Zeeman energies, we now turn our attention to the transmission through the
nanowire. We focus on the the zero-energy transmission $T(E=0)$ near the
crossing point Zeeman energies corresponding to $B_{1}$ and $B_{2}$ and for
two different values of the contact coupling $\gamma$, in Fig. 6. The total
transmission has a peak of magnitude unity at $B_{1}$ and $B_{2}$, and drops
to zero at all other intermediate values of magnetic field. The total zero-
energy transmission is directly related to the zero-bias conductance as noted
in Sec. II.2.

Figure 12: Probability density $\mid\Psi_{i}\mid^{2}$ at each site in the
nanowire at a magnetic field $B>B_{C}$ in the topological regime, for a clean
nanowire and for two different magnitudes of the dephasing interactions
$d_{0}$. The end-localization is better in this case, with zero probability
density for most region in the bulk of the wire. As in Fig. 9, the spatial
variation is the same in the coherent and non-coherent regimes. Figure 13:
Local and non-local transmission at zero energy with varying magnetic field
for a long wire ($l=8\mu m$), resolved into the direct, Andreev and crossed
Andreev process components. The total transmission shows a plateau of
magnitude 1 in the topological regime, dominated by Andreev reflection.

Figure 14: Zero-bias conductance for a longer nanowire with varying magnetic
field for two different strengths of non-coherent interactions,
$d_{0}=10^{-8}t_{0}^{2}$ in Fig. (a) and $d_{0}=10^{-6}t_{0}^{2}$ in Fig. (b).
The corresponding conductance for the short wire are shown as green dashed
lines. The coherent conductance is also shown for reference in (a) as a dotted
grey line.

We note that the dominant process when the modes have exactly zero energy is
the local Andreev reflection process. At Zeeman energies close to the parity
crossings corresponding to $B_{1}$ or $B_{2}$, the non-local direct and
crossed Andreev components have a significant non-zero contribution to
transmission, but it drops to a minimum at precisely $B_{1}$ and $B_{2}$,
leading to their split-peaks feature in Fig. 6. We also note the significant
robustness of the peak height to varying contact coupling. Increasing the
contact coupling causes the states to be more broadened out in energy, while
maintaining the unity transmisison peak. In Figs. 6 and 6, $\gamma$ is ten
times that in Figs. 6 and 6, which manifests as broadening of the transmission
peak. The relative contributions of the three processes may change but the
peak height of the total transmission is preserved at unity. It is also worth
noting by comparing Fig. 6 and Fig. 6 with Fig. 6 and Fig. 6 that the
individual contributions of the local and non-local processes change at the
two unique parity crossings, while to total magnitude remains quantized at
unity at the magnetic field corresponding to either crossing. This point will
be taken up in detail in the discussion section and is intimately related to
the localization length and the competition between the direct and crossed
Andreev processes. In fact, the broadening with respect to magnetic field
leads to a range of $B$ at which the transmission is nearly unity. This is one
possible mechanism that would lead to robustness to varying Zeeman field, as
is observed in experiments.
We now analyze quantum transport through the N-TS-N system, as shown in Fig.
1, by applying a symmetric bias. The current in the NEGF formalism is given by
Eqs. 15, 16 and 17. It is worth noting that, due to the symmetric biasing,
$f_{L}^{ee}=f_{R}^{hh}$, and hence, the crossed Andreev transmission processes
does not contribute to the current Leumer  _et al._ (2020), and the non-local
current only consists of direct transmission contribution. Also, for the
N-TS-N three terminal set up, the conductance is quantized at $e^{2}/h$
instead of $2e^{2}/h$ due to the presence of two N-TS junctions in series and
the resulting voltage divider Leumer  _et al._ (2020); Li and Xu (2020), as
opposed to only one junction in a typical two terminal N-TS set up.
We first focus on the conductance at the magnetic field corresponding to
$B_{1}$ in Fig. 7. As expected, we observe a zero-bias conductance peak of
magnitude $e^{2}/h$. The peak height remains robust to varying contact
coupling $\gamma$ and the peak is seen to be broadened in the voltage axis. We
reiterate that the peak is expected to be quantized at $e^{2}/h$ for symmetric
biasing, unlike the experimentally measured value of $2e^{2}/h$, which
corresponds to asymmetric biasing $V_{L}=V$, $V_{R}=0$ Leumer  _et al._
(2020); Li and Xu (2020).
Next, we vary the magnetic field $B$ and look at the ZBCP by applying a small
voltage ($V\approx 10^{-5}\Delta$) where $G\approx I/V$. In Fig. 7, we observe
quantized conductance peaks at $B_{1}$ and $B_{2}$, with the quantization
robust to changing contact coupling. As seen in the transmission components,
the ZBCP is dominated by the local Andreev reflection process.

###  III.3 Effect of dephasing

We now turn the attention to the effect of dephasing to demonstrate its effect
using two values of the non-coherent interactions $d_{0}=10^{-8}t_{0}^{2}$ and
$d_{0}=10^{-6}t_{0}^{2}$. We calculate the LDOS at each site as a function of
energy, at the magnetic field corresponding to $B_{1}$ at the two values of
interaction strength in Fig. 8. We see that the states are broadened out in
energy and this broadening increases with the strength of interactions
$d_{0}$. However, the zero-energy states remain localized at the edges with
maximum density at the fifth site, as before. This points to the preservation
of how local the MZM is even in the presence of non-coherent interactions,
that is, there is no spatial broadening.
To reinforce the nature of localization and the changes due to non-coherent
processes, we compute the probability density of these states in the nanowire,
at site $i$ as

$\mid\Psi_{i}\mid^{2}=\int-iG^{<}(i,i)dE.$ (25)

Here we note that the integral must run over the entire energy range over
which the MZM is broadened. Once that is ensured, we surprisingly see no
change in the spatial variation of the probability density
$\mid\Psi_{i}\mid^{2}$ as we go from a clean nanowire to stronger non-coherent
interactions in Fig. 9. This is a strong indicator that the edge-local nature
of the zero-modes remains completely unchanged. Further, the wavefunctions
match the localization seen in the LDOS, with the fifth and the ninety fifth
sites having the highest density of states. The oscillations in density due to
hybridization of the end-localized states are also seen clearly here.
Finally, we calculate the transport through the nanowire with non-coherent
interactions using the current formula in (14). We reiterate that there is no
simple relation between current and transmission as in (15), (16) and (17),
and hence cannot express the currents in terms of the local and non-local
contributions like how we could in the coherent case.
As seen in Fig. 10, the differential conductance thus calculated still
exhibits a ZBCP at the parity crossings, even in the presence of dephasing and
momentum-relaxation. However, we observe a marked decrease in the height of
this ZBCP - and that it is no longer quantized, unlike in the case of
increasing contact broadening, which preserved the quantization. The peak is
further suppressed for stronger non-coherent interactions, dropping by almost
one order of magnitude in Fig. 10. As expected, the ZBCP shows significant
broadening in voltage, corresponding to the energy-broadening of the states.
Next, we vary the magnetic field $B$ and examine the peak value of the
conductance in Fig. 11. We now note that the conductance is significantly
broadened along the $B$ axis as well, for higher values of the interaction
strength $d_{0}$. Notably, the peak height is higher at $B_{2}$ (Fig. 11) as
compared to $B_{1}$ (Fig. 11), which points to a possible variation in the
robustness of the ZBCP at different parity crossings.

###  III.4 Long Nanowire

Having shown some key features related to the Majorana ZBCP in shorter wires,
in the coherent and non-coherent cases, specifically at the parity crossings,
we now take a quick look at the case of the long wire by considering the case
depicted in Fig. 3 where parity crossings cannot be discerned as easily. The
inclusion of dephasing and momentum-relaxation as done earlier in Sec. III.3
preserves the end-localization as seen in the energy-integrated probability
densities as seen in Fig. 12. The densities when integrated over the energy
range over which the MZMs are broadened, remain the same as in the clean
nanowire limit in the presence of non-coherent interactions of any strength.
Since the wire length is much greater than the Majorana localization length in
this case, we see a strong localization of the Majorana modes to the ends of
the nanowire, with the probability densities going to zero for a large part of
the bulk region in the middle.
In the long wire, the amplitude of Majorana oscillations in the energy
spectrum is smaller than the thermal and contact broadening of the energy
states. Therefore, we see unity transmission at magnetic fields above the
critical field, i.e., $B>B_{C}$, clearly seen in Fig. 13 arising from perfect
local Andreev reflection. Further, in Fig. 13, we also note that the total
transmission follows the local Andreev transmission with a minor non-local
contribution right before the onset of the zero bias plateau, which is related
to the small contribution the non-local processes make right before the
critical field is reached. This leads to the ZBCP quantized at $e^{2}/h$ for
all $B>B_{C}$ for a clean wire being a plateau instead of discrete peaks, as
seen in the grey dotted line in Fig. 14.
Finally, we look at the conductance in the non-coherent regime. A ZBCP is
present as for the short wire, but the peak height gets suppressed as the
strength of non-coherent interactions increases. This effect is best seen in
Fig. 14 in the zero-bias conductance with varying magnetic field $B$. As
expected, for the long wire, we have a conductance plateau instead of discrete
peaks at parity crossings as was seen in the case of the short wire. However
the value of the zero-bias conductance (ie., the peak height) remains almost
the same as that at the crossing Zeeman energies corresponding to $B_{1}$ and
$B_{2}$ of the short wire. Thus, we note that the zero-bias peak is no more
robust to dephasing in a longer nanowire than in a short one, which is a very
interesting result indicating a lack of length dependence of the MBS
conductance even as we transition to the diffusive limit. Besides this, we
also note that the magnitude of the conductance increases as the B-field
increases, indicating to some extent the greater degree of robustness to
dephasing.

##  IV Discussion

There are several aspects in this paper that merit further consideration.
Figure 6 requires the first discussion that summarizes some important aspects
of coherent transport across an MZM and the related ZBCP signatures in a three
terminal set up. We note that for the second parity crossing, comparing Fig. 6
and Fig. 6 with Fig. 6 and Fig. 6, that the individual contributions of the
local and non-local processes change at the two unique parity crossings, while
to total magnitude remains quantized at unity at the magnetic field
corresponding to either crossing. Most importantly, the direct transmission
process also contributes to the total conductance with a corresponding
reduction in the local Andreev reflection. This can be attributed to the fact
that for the realistic parameters used in our paper, the Majorana localization
length actually increases between the two parity crossings Cayao (2017). For
the short wire, longer localization lengths lead to overlaps due to which the
non-local transmissions will increase. Furthermore, the direct transmission
dominates over the crossed Andreev process due to the fact that the hopping
parameter is much larger than the superconducting pairing parameter. Note that
there is an effective p-wave pairing that results in a competition between the
direct transmission process and the crossed Andreev transmission. In the case
of the long wire, it is clear from Fig. 12, that the length of the wire is
much greater than the localization lengths, leading one to note that the only
dominant process above the critical field is the local Andreev reflection
process.
In the non-coherent limit, the values of dephasing introduced can be mapped to
the elastic momentum relaxation lengths, that are in turn dependent on the
magnetic field. The first important aspect that arises out of dephasing is the
broadenend line shape of the ZBCP as witnessed in Fig. 10. Specific comparison
between Fig. 10 (a) and Fig. 10(b) clearly depict a drastic reduction in the
peak height along with significant broadening as the strength of the dephasing
is increased, corresponding to reducing the elastic momentum relaxation
length. The most important aspect here is the gradual transition from the
coherent ballistic limit to the diffusive regime as the parameter $d_{0}$ is
increased. This helps us uncover aspects of transport in such an intermediate
regime where scattering events compete with coherent reflections. This aspect
is well noted and summarized in Fig. 14 (a) and (b), for both the long and the
short wire. Firstly, we do note the interesting aspect, as mentioned before
that the long wire is more or less as robust as the shorter wire when it comes
to the conductance peak heights at the parity crossings. The absence of any
such strong length dependence at the zero mode parity crossings indicates that
the local Andreev reflection process is a dominant contributor to the total
zero mode conductance, whose reduction in magnitude can be attributed to the
energy broadening of the mode depicted in the LDOS plots in Fig. 8.
Interestingly, as pointed out before, the momentum relaxation processes
contribute only to the energy broadening of the zero modes and not to the
spatial broadening.

##  V Conclusions

To summarize, the nature of the conductance signatures that signals the
presence of Majorana zero modes in a three terminal nanowire-topological
superconductor hybrid system was analyzed in detail, in both the coherent and
the non-coherent transport regimes. In the coherent regime, we pointed out
contributions of the local Andreev reflection and the non-local transmissions
toward the conductance signatures. In the non-coherent regime, via the
inclusion of dephasing due to the fluctuating impurities and the resulting
momentum randomization processes, it was pointed out that while the Majorana
character of the mode is unchanged, a reduction in the conductance magnitude
that scales with the strength of the impurity potentials. Important
distinctions between dephasing processes in the non-coherent regime and the
contact induced tunnel broadenings on the conductance lineshapes were clearly
elucidated. Most importantly our results revealed that the addition of
dephasing in the set up the does not lead to any notable length dependence to
the conductance of the zero modes, contrary to what one would expect in a
gradual transition to the diffusive limit as a result of dephasing. This work
paves the way for systematic introduction of scattering processes into the
realistic modeling of Majorana nanowire hybrid devices.

Acknowledgements: We wish to acknowledge Supriyo Datta, Milena Grifoni,
Magdalena Marganska, Nico Leumer, Sumanta Tewari, Abhishek Sharma and Atri
Dutta for stimulating discussions. The research and development work
undertaken in the project under the Visvesvaraya Ph.D Scheme of the Ministry
of Electronics and Information Technology (MEITY), Government of India, is
implemented by Digital India Corporation (formerly Media Lab Asia). This work
is also supported by the Science and Engineering Research Board (SERB),
Government of India, Grant No EMR/2017/002853 and Grant No. STR/2019/000030,
the Ministry of Human Resource Development (MHRD), Government of India, Grant
No. STARS/APR2019/NS/226/FS under the STARS scheme. The authors BM and JB
acknowledge the Shastri Indo-Canadian Insitute for the Shastri Mobility Grant
2019.

##  Appendix A Derivation of current components in the coherent limit

Since the leads effectively form an infinite dimensional matrix, the NEGF
approach begins by partitioning the channel and the leads Datta (1997); Haug
and Jauho (1996) and working with the channel Green’s function and
incorporating the leads via self-energies. The retarded channel Green’s
function can now be written as:

$G^{r}(E)=\left[(E+i\eta)I-H_{BdG}-\Sigma^{r}_{L}-\Sigma^{r}_{R}\right]^{-1},$
(26)

where $E$ is the free variable energy, and $I$ is the identity matrix of the
dimension of the Hamiltonian. The energy is indeed a Fourier transform
variable of the difference time of the two time Keldysh Green’s function
defined on the Schwinger-Keldysh contour Haug and Jauho (1996). For brevity,
we will use the energy variable on the left hand side, with the energy
dependence of the elements of the matrix product on the right hand side
implicit.

Here a straightforward calculation of the lead self energies matrix elements
$\Sigma^{r}_{\alpha}(i,i)=\sigma_{\alpha\in L/R}$ read Cuevas  _et al._
(1996); Haug and Jauho (1996); Datta (1997)

$\sigma_{\alpha\in
L/R}(E)=\sum_{k}\frac{\mid\tau_{\alpha,k}\mid^{2}}{\left(E-\epsilon_{k\alpha}+i\eta\right)}={P}\int
d\epsilon_{k\alpha}\frac{\mid\tau_{\alpha,k}\mid^{2}}{\left(E-\epsilon_{k\alpha}\right)}-i\pi\int
d\epsilon_{k\alpha}\mid\tau_{\alpha,k}\mid^{2}\delta(E-\epsilon_{k\alpha}),$
(27)

where $P$ stands for the Cauchy principal value. This integral can be recast
as $\sigma_{\alpha}=\epsilon_{\alpha}(E)-i\gamma_{\alpha}/2$, where the real
part $\epsilon_{\alpha}$ is calculated from the principal value integral and
the imaginary part
$\gamma_{\alpha}=i\left(\sigma^{r}_{\alpha}-\sigma^{a}_{\alpha}\right)$
represents the level broadening or decay constant.
In this work, we will focus on including the contacts described by the
eigenbasis representation or the wide band limit. Here with the assumption of
a broad-band contact dispersion ($V_{k\alpha,m}$ being independent of energy),
the principal value integral representing the real part of the self energy
vanishes and we are just left with the broadening matrix $\Gamma_{\alpha}(E)$
such that the self energies can be represented as
$\Sigma^{r}_{\alpha}=-i\Gamma_{\alpha}/2$, which can be treated as an input
parameter. Once we have the self energies, we need to evaluate the “lesser”
Green’s function or equivalently, the correlation function given by:

$G^{<}(E)=G^{r}(E)\left(\Sigma_{L}^{<}(E)+\Sigma_{R}^{<}(E)\right)G^{a}(E),$
(28)

where $G^{a}$ is the advanced Green’s function, which is the Hermitian
conjugate of the retarded Green’s function calculated from (26). The
quantities $\Sigma^{<}_{L,R}(E)$ represent the in-scattering functions from
leads $L$ and $R$ respectively evaluated as

$\displaystyle\Sigma_{\alpha}^{<}(E)$ $\displaystyle=$
$\displaystyle-\left[\Sigma^{r}_{\alpha}(E)-\Sigma^{a}_{\alpha}(E)\right]f_{\alpha}(E)$
(29) $\displaystyle=$ $\displaystyle i\Gamma_{\alpha}(E)f_{\alpha}(E),$

where $f_{\alpha}=f(E-\mu_{\alpha})$, the Fermi-Dirac distribution in either
lead $\alpha=L(R)$ with its own chemical potential $\mu_{\alpha}$. In this
formulation, all different components of the currents can then be deduced from
the current operator formula, which in turn, can be derived from fundamental
considerations Haug and Jauho (1996) and the current operator at the left
contact now reads:

$\displaystyle I^{op}_{L}(E)$ $\displaystyle=$
$\displaystyle\frac{e}{h}[G^{r}(E)\Sigma_{L}^{<}-\Sigma_{L}^{<}G^{a}(E)$ (30)
$\displaystyle+G^{<}(E)\Sigma_{L}^{a}(E)-\Sigma_{L}^{r}(E)G^{<}(E)]$

and making some basic manipulations and taking the trace of the above equation
to find the net charge current, we arrive at the celebrated current formula
Meir and Wingreen (1992):

$I_{L}(E)=\frac{ie}{h}Trace\left[\Gamma_{L}(E)f_{L}(E)\left(G^{r}(E)-G^{a}(E)\right)+\Gamma_{L}(E)G^{<}(E)\right].$
(31)

The above formula can be further simplified by the notation introduced in
Datta (1997) with the spectral function $A(E)=i(G^{r}-G^{a})$ and the electron
correlation matrix $G^{n}(E)=-iG^{<}(E)$ as

$I_{L}(E)=\frac{e}{h}Trace\left[\Gamma_{L}(E)f_{L}(E)A(E)-\Gamma_{L}(E)G^{n}(E)\right].$
(32)

Furthermore, it can be easily shown that

$\displaystyle A(E)$ $\displaystyle=$ $\displaystyle i(G^{r}-G^{a})$ (33)
$\displaystyle=$ $\displaystyle
G^{r}i\left[\left({G^{a}}\right)^{-1}-\left({G^{r}}\right)^{-1}\right]G^{a}$
$\displaystyle=$ $\displaystyle G^{r}i\left[\Sigma^{r}-\Sigma^{a}\right]G^{a}$
$\displaystyle=$ $\displaystyle G^{r}\Gamma G^{a},$

where $\Gamma(E)=\Gamma_{L}(E)+\Gamma_{R}(E)$. Using the above in conjunction
with (28) and (29), along with the properties of the trace operation, yields
the Landauer transmission formula for the current as

$I_{L}=\int
dETrace\left[\Gamma_{L}G^{r}\Gamma_{R}G^{a}\right]\left(f_{L}(E)-f_{R}(E)\right),$

where $f_{L(R)}(E)$ represents the Fermi-Dirac distribution in either lead.
However, in the case involving superconductors, either functioning as a
contact or as the central system or both, the net current through the contact
$\alpha$ is the difference between electron and hole currents in Nambu space.
We can now use the full fledged current operator in (30) to evaluate the
current from first principles as

$I^{op}_{L}(E)=\frac{e}{h}\tau_{z}\left[G^{r}(E)\Sigma_{L}^{<}(E)-\Sigma_{L}^{<}(E)G^{a}(E)+G^{<}(E)\Sigma_{L}^{a}(E)-\Sigma_{L}^{r}(E)G^{<}(E)\right],$
(34)

where $\tau_{z}=\sigma_{z}\otimes I_{N\times N}$, where $\sigma_{z}$ is the
Pauli-z matrix and $I_{N\times N}$ is the identity matrix of dimension
$N\times N$. The next step is to take a trace of the above equation, which in
general need not take the form of (31), typically when the contacts are
superconducting due to the non-diagonal structure of $\Sigma_{L(R)}$, and may
lead to erroneous results, specifically when evaluating the Josephson
currents. However, in our case, as the contacts are normal, and hence
$\Sigma_{L(R)}$ diagonal, (34) indeed takes the form of (31) with the net
current becoming

$I_{\alpha}=\int
dE\frac{\left(I_{\alpha}^{e}(E)-I_{\alpha}^{h}(E)\right)}{2},$

where each current $I_{\alpha}^{e(h)}$ can be evaluated separately using the
form in (31) or (32). The factor of a half comes since the original
Hamiltonian been doubled in order to write it consistently in the Nambu space.
It is then instructive in our context to note that various matrices defined
within the approach will have a matrix structure due to the electron-hole
Nambu space. In particular, the contact broadening matrices etc., can be
written with a general diagonal structure (due to the diagonal structure of
normal contacts in Nambu space) as
$\Gamma_{\alpha}=\Gamma^{ee}_{\alpha}+\Gamma^{hh}_{\alpha}$ and
$\Sigma_{\alpha}^{(r,a,<)}=\Sigma_{\alpha}^{(r,a,<,ee)}+\Sigma_{\alpha}^{(r,a,<,hh)}$,
where the superscripts $ee(hh)$ represent the electron (hole) diagonal part of
the self energy or broadening matrix. Following this and using the above
observations on the current operator formula in (32), we obtain the electron
(hole) current across the TS as a sum of three components:

$\displaystyle
I^{e(h)}_{L}(E)=\frac{e}{h}\left(Trace\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{ee(hh)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{R}^{ee(hh)}(E)\right]\right)$
(35)
$\displaystyle\qquad+\frac{e}{h}\left(Trace\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{L}^{hh(ee)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{L}^{hh(ee)}(E)\right]\right)$
(36)
$\displaystyle\qquad+\frac{e}{h}\left(Trace\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{hh(ee)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{R}^{hh(ee)}(E)\right]\right),$
(37)

where (36) represents the direct transmission process of either the electron
or the hole, (37) represents the direct Andreev transmission and (37)
represents the crossed Andreev transmission. At this point it is worth noting
that
$f_{\alpha}^{ee}(E)=f(E-\mu_{\alpha}),f_{\alpha}^{hh}(E)=f(E+\mu_{\alpha})$.
The matrix structure of the broadening matrix $\Gamma$ itself is diagonal such
that $\Gamma_{L}^{ee}=\Gamma_{L}(1,1)=\gamma$,
$\Gamma_{L}^{hh}=\Gamma_{L}(2,2)=\gamma$,
$\Gamma_{R}^{ee}=\Gamma_{R}(2N-1,2N-1)=\gamma$,
$\Gamma_{R}^{hh}=\Gamma_{L}(2N,2N)=\gamma$ and zeros otherwise. The Andreev
transmission can then be written as $T_{A}(E)=\gamma^{2}\mid
G^{r,eh}_{11}(E)\mid^{2}$ , the crossed Andreev transmission as
$T_{CA}(E)=\gamma^{2}\mid G^{r,eh}_{1N}(E)\mid^{2}$ and the direct
transmission can be written as $T_{D}(E)=\gamma^{2}\mid
G^{r,ee}_{1N}(E)\mid^{2}$. Here $G^{r,ee(hh)}_{ij}$ represents the $i,j$
element of the electron (hole) diagonal block of the retarded Green function
in Nambu space and $G^{r,eh(he)}_{ij}$ represents that of the off-diagonal
block of the retarded Green function in Nambu space.

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