and their possible experimental signatures. We also find that, within the
nominally topological regime, the system can host well separated Majorana
modes even in the presence of significant disorder levels, but typically the
presence of these modes is not associated with a significant edge-to-edge
correlation. A key finding in this context is that disorder may often push
Majorana zero modes away from the wire ends, thus making them invisible to
local (end-of-wire) tunnel spectroscopy. Thus, it is entirely possible (and
likely) to miss the presence of Majorana zero modes in a disordered nanowire
when using tunneling spectroscopy simply because this is a local probe
sensitive only to states localized at the wire ends. Hence, in the presence of
disorder, long segments within the bulk of the wire may be topologically
nontrivial, with Majorana modes emerging at their ends, but the wire ends
themselves may contain no Majorana modes, which dramatically reduces the
probability of observing edge-to-edge correlations.

We establish that detailed two-dimensional maps of the zero-bias conductance
as a function of Zeeman splitting (i.e. magnetic field in the laboratory) and
chemical potential (i.e. gate voltage in the laboratory) may be the most
effective operational way to search for the “hidden” topological
superconductivity and the associated Majorana modes. The current experimental
focus on looking for large zero bias peaks with conductance $\sim 2e^{2}/h$ by
fine-tuning the control parameters is unlikely to solve the outstanding
questions regarding the nature of the low-energy states responsible for these
peaks. First, a large zero bias peak obtained through careful fine-tuning and
post selection may have nothing to do with topological Majorana modes, and
second, this procedure is likely to lead to strong confirmation bias in the
experiment. Instead, creating zero bias conductance maps in the extensive
parameter space of gate voltage and magnetic field using the cleanest possible
samples and comparing these maps to our theoretical results may be a much more
systematic way of searching for Majorana physics, without suffering from any
confirmation bias. In addition, this would provide much needed estimates of
the disorder strength characterizing actual superconductor-semiconductor
hybrid devices and an effective way of testing future materials improvements
that aim at reducing disorder.

We find that for reasonably realistic (but still somewhat optimistic)
parameter choices, genuine, well-separated topological Majorana modes should
exist in nanowires for impurity densities up to $5\cdot 10^{15}~{}$cm-3, which
corresponds to around 15 impurities per micron. This would mean that a $2-4$
micron long nanowire can contain up to $30-60$ charge impurities, but cleaner
samples, with charge impurity density below $10^{15}~{}$cm-3, may be necessary
in practice, since we ignored any disorder arising from possible interface
defects or imperfections. Such a low intrinsic doping of less than
$10^{15}~{}$cm-3 is a challenge, but is by no means out of reach in
semiconductor materials growth, as impurity contents below $10^{13}~{}$cm-3
have been achieved in MBE-grown GaAs structures [59].

Our final conclusion is that charge impurities cause serious problems, but by
no means destroy the topology in hybrid nanowires, as long as their
concentration is maintained below a certain threshold. Future experiments
should provide estimates of the disorder levels that characterize existing
hybrid systems, while a systematic effort should be dedicated to the
production of much cleaner wires, with significantly lower impurity content,
where Majorana zero modes could emerge easily and manifest the full range of
their expected phenomenology.

Implications of our work for the realization of Majorana zero modes and
topological qubits are obvious and far-reaching. Since semiconductor nanowire-
superconductor hybrid platforms are by far the leading TQC candidates, by
virtue of the tunability of the system through electrical gating and varying
magnetic field, and because semiconductor growth enables the realization of
very pure materials, our detailed macroscopic, quantitative analysis of all
relevant aspects of Majorana physics in the SM-SC platform in the presence of
charge impurity disorder provides the community with clear and quantitative
guidelines on how to make progress: Obtain samples with $10^{15}$ per cm3 or
less impurity content, produce two-dimensional parameter maps of the zero bias
differential conductance over extended parameter regions, beware of the
possibility that impurities may push the Majorana bound states away from the
edges, so that topology may be hidden in tunnel spectroscopy at the ends, do
not focus on trying to find Majorana quantization (which are often spurious),
instead focus on the totality of the parameter space rather than fine-tuning,
carry out conductance correlations the way proposed in the current work as a
function of magnetic field and gate voltage by doing tunneling from both ends,
and try to obtain nonlocal correlations not just from the two ends, but along
the wire. Our work establishes the existence of topological Majorana modes in
the system in the presence of some amount of charge impurities even when the
disorder potential is significantly larger than the SC gap, provided that the
impurity concentration is not too high. This is a highly encouraging result
which should inspire new efforts toward creating Majorana qubits.

###### Acknowledgements.

This work is supported by NSF Grant No. 10026278 and by the Laboratory for
Physical Sciences.

Figure A1: Examples of effective potential (black solid lines) and the fitted
effective potential (red dashed lines) from impurities of charge (a) $Q=-e$
and (b) $Q=e$ using the fitting Eqs. (50-52). The transverse positions of the
impurities are indicated in the panels. The average absolute error between the
exact and fitted potentials for $|z|\leq 100~{}\text{nm}$ is (a)
$0.03~{}\text{meV}$ and (b) $0.07~{}\text{meV}$, respectively.

##  Appendix A Fitting effective impurity potentials to an analytic function

Figure A2: Effective redistribution amplitude $B^{\alpha}_{red}$ vs effective
impurity amplitude $B^{\alpha}_{imp}$ when the $\alpha=2\text{ (black dots),
}3\text{ (red), or }4\text{ (green)}$ subband is tuned to the Fermi level.
Amplitudes are extracted from fitting the effective potential of $19$ evenly
spaced impurity locations within the transverse profile of the nanowire.
Dashed lines are linear regression fits to the matching color data. See Table
1 for fitting parameters.

As alluded to in the main text [see also Eq. (29)], the effective potential of
a single charge impurity located at $z=0$ can be well captured by fitting both
the effective impurity and redistribution potentials to exponential functions,

$\displaystyle
V_{\alpha,\alpha}\left(z\right)=V_{\alpha,\alpha}^{imp}\left(z\right)+V_{\alpha,\alpha}^{red}\left(z\right),$
(50) $\displaystyle
V_{\alpha,\alpha}^{imp}\left(z\right)=B^{\alpha}_{imp}e^{-|z|/\lambda^{\alpha}_{imp}},$
(51) $\displaystyle
V_{\alpha,\alpha}^{red}\left(z\right)=-B^{\alpha}_{red}e^{-|z|/\lambda^{\alpha}_{red}},$
(52)

where $B^{\alpha}_{imp}$ and $B^{\alpha}_{red}$ are the amplitudes of the
effective impurity and redistribution potentials, respectively,
$\lambda^{\alpha}_{imp}$ and $\lambda^{\alpha}_{red}$ are the corresponding
decay lengths, and $\alpha$ is the subband index. We place a minus sign in
front of $B^{\alpha}_{red}$ to emphasize that the redistribution potential
(partially) suppresses the impurity charge potential. Two examples of this
fitting are shown in Fig. A1, for a negative and positive elementary impurity
charge with the $\alpha=2$ subband tuned to the Fermi level. We find excellent
agreement between the actual and fitted potential in both cases. Indeed, the
average absolute error, $|V_{2,2}^{exact}-V_{2,2}^{fit}|$, within
$100~{}\text{nm}$ of the impurity is only (a) $0.03~{}\text{meV}$ and (b)
$0.07~{}\text{meV}$, respectively. Note that we found the average absolute
error to be of this order for all impurity locations sampled. Generically, we
find that cases with a negative charge impurity fit slightly better to Eqs.
(50-52) than positive charge impurity cases. This is due to a more prominent
“hump” feature after first crossing $V_{\alpha,\alpha}=0$ for positively
charged impurities compared to negatively charged impurities. Nevertheless,
the fitting is exceptional for both impurity charge signs.

While Eqs. (50-52) represent an excellent approximation for the effective
potential, it requires $4$ fitting parameters, which may be cumbersome if one
wants to construct a phenomenological model of charge impurity disorder
without explicitly performing numerically expensive Schrödinger-Poisson
calculations. This motivates us to investigate whether the various fitting
parameters display correlations to reduce the number of necessary input
parameters. We indeed find this to be the case. The resulting correlations are
shown in Figs. A2, A3, and A4 and discussed below.

Figure A3: Comparison of impurity and redistribution (inverse) decay lengths,
$(\lambda^{\alpha}_{imp})^{-1}$ and $(\lambda^{\alpha}_{red})^{-1}$, when the
$\alpha=2\text{ (black dots), }3\text{ (red), or }4\text{ (green)}$ subband is
tuned to the Fermi level. Decay lengths are extracted from fitting the
effective potential of $19$ evenly spaced impurity locations within the
transverse profile of the nanowire. (a) Top and (b) bottom panels correspond
to $Q=+e$ and $Q=-e$, respectively. Dashed lines are linear regression fits to
the matching color data. See Table 2 for fitting parameters.

The effective redistribution amplitude $B^{\alpha}_{red}$ as a function of the
effective impurity amplitude $B^{\alpha}_{imp}$ is shown in Fig. A2 for
several different subbands tuned to the Fermi level. The amplitudes come from
fitting the effective impurity and redistribution potentials to Eqs. (51, 52)
with each data point corresponding to a different impurity location in the
transverse profile of the wire. For this data set, 19 evenly spaced positions
in the nanowire’s cross section were sampled. We observe a general linear
trend between the two amplitudes for all three subbands in which the magnitude
of the redistribution amplitude increases with increasing magnitude of the
impurity amplitude, as seen in the linear fit lines (dashed lines). The
positive sign of the slope makes physical sense since increasing the magnitude
of the impurity amplitude should increase the redistribution of free charge
around the impurity to (partially) counteract the perturbation of the
electrostatic environment. What’s not obvious, however, is that a linear
relationship should capture the dependency rather well. After all, the
Schrödinger-Poisson equations should be expected to behave non-linearly due to
the interplay between the various occupied subbands. To quantify how well the
linear fit captures the relationship, we have gathered the fitting parameters
into Table 1. In particular, we wish to bring attention to the coefficient of
determination, $r^{2}$, which indicates how much of the variance of the data
is explained by the linear model. For all except the ($\alpha=4,Q=-e$) case,
the linear fit explains over half of the variance ($r^{2}>0.5$). Moreover,
$Q=e$ cases display particularly high $r^{2}$ value. We also notice that the
$r^{2}$ coefficient diminishes on average with increasing subband index,
$\alpha$, suggesting the Schrödinger-Poisson equations are behaving with
increasing non-linearity as occupation is increased.

Similar to the effective potential amplitudes, we compare the (inverse) decay
lengths of the effective redistribution and impurity potentials in Fig. 3 with
the subband $\alpha=2,3,\text{ or }4$ tuned to the Fermi level. Again we
observe general linear trends and fit the data from each subband to a line.
The fitting parameters are gathered in Table 2. The $r^{2}$ coefficients are
of similar size to what was found in studying the relationship between the
potential amplitudes (Table 1), but are slightly smaller, indicating the decay
lengths behave in a slightly more non-linear manner.

Figure A4: Comparison of impurity amplitude $B^{\alpha}_{imp}$ and (inverse)
decay length $(\lambda^{\alpha}_{imp})^{-1}$ when the $\alpha=2\text{ (black
dots), }3\text{ (red), or }4\text{ (green)}$ subband is tuned to the Fermi
level. Amplitude and decay lengths are extracted from fitting the effective
potential of $19$ evenly spaced impurity locations within the transverse
profile of the nanowire. Impurity charge $Q=-e$. Dashed lines are linear
regression fits to the matching color data. See Table 3 for fitting
parameters.

Finally, we study the correlation between the effective impurity potential’s
amplitude $B^{\alpha}_{imp}$ and (inverse) decay length
$(\lambda_{imp}^{\alpha})^{-1}$ in Fig. A4. In contrast to Figs. A2 and A3, we
only consider $Q=-e$, since flipping the sign to $Q=e$ only changes the sign
of the amplitude, $B^{\alpha}_{imp}$. The fitting parameters are gathered in
Table 3. On the one hand, we observe large coefficients of determination,
$r^{2}=0.87,0.80$, for $\alpha=2,3$, respectively. On the other hand,
$r^{2}=0.07$ for $\alpha=3$. These indicate that the relationship between the
effective impurity amplitude and inverse decay length is well captured by a
linear fit for $\alpha=2,3$, but not $\alpha=4$. Evidently, as the
wavefunction moves away from the SM-SC interface with increasing $\alpha$ the
electrostatics become more subtle and the relationship between the amplitude
and (inverse) decay length becomes more complicated.

We’re now in a strong position to create realistic phenomenological models of
charge impurity disorder in SM-SC hybrids nanowires using only 1 or 2
parameters from which we need to sample. We accomplish this by leveraging the
information we’ve just laid out regarding the linear relationships between the
various fitting parameters. In the case of low-occupancy ($\alpha\leq 3$ in
this case) the relationships between the 4 fitting parameters,
$B^{\alpha}_{imp}$, $B^{\alpha}_{red}$, $(\lambda^{\alpha}_{imp})^{-1}$, and
$(\lambda^{\alpha}_{red})^{-1}$, are well described by all three linear
relationships studied in this appendix. Therefore, one only needs to sample
the $B^{\alpha}_{imp}$ distribution to create a realistic model of disorder.
Given a $B^{\alpha}_{imp}$ value, we only have to plug it into the linear
equations given in Tables 1-3 and the corresponding parameters, $m$ and $b$
(also in the tables), to obtain the other three fitting parameters. In the
case of higher occupancy ($\alpha\geq 4$ in this case) the relationship
between $B^{\alpha}_{imp}$ and $(\lambda^{\alpha}_{imp})^{-1}$ is not
represented well by a linear fit. Therefore, we need to sample from both
$B^{\alpha}_{imp}$ and $(\lambda^{\alpha}_{imp})^{-1}$ to create a realistic
disorder model. The other two fitting parameters, $B^{\alpha}_{red}$ and
$(\lambda^{\alpha}_{red})^{-1}$, however, can still be obtained using the
linear equations and corresponding parameters in Tables 1 and 2. We then have
a convenient and accurate way of producing realistic disorder potential
profiles due to charge impurities in Majorana SM-SC hybrid nanowires.

Table 1: Fitting parameters of dashed lines in Fig. A2 corresponding to the fitting equation, $B^{\alpha}_{red}=mB^{\alpha}_{imp}+b$. Coefficient of determination $r^{2}$ for each linear fit is given in the final column, where $r^{2}=1$ indicates a perfect fit. $(\alpha,Q)$ | $m$ | $b~{}\text{(meV)}$ | $r^{2}$
---|---|---|---
$(2,+e)$ | 0.30 | -0.46 | 0.91
$(2,-e)$ | 0.07 | 0.41 | 0.53
$(3,+e)$ | 0.29 | -0.64 | 0.75
$(3,-e)$ | 0.13 | 0.30 | 0.52
$(4,+e)$ | 0.43 | -0.36 | 0.79
$(4,-e)$ | 0.12 | 0.81 | 0.34
Table 2: Fitting parameters of dashed lines in Fig. A3 corresponding to the fitting equation, $(\lambda^{\alpha}_{red})^{-1}=m(\lambda^{\alpha}_{imp})^{-1}+b$. Coefficient of determination $r^{2}$ for each linear fit is given in the final column, where $r^{2}=1$ indicates a perfect fit. $(\alpha,Q)$ | $m$ | $b~{}(\text{nm}^{-1})$ | $r^{2}$
---|---|---|---
$(2,+e)$ | 0.18 | 0.016 | 0.83
$(2,-e)$ | 0.11 | 0.013 | 0.46
$(3,+e)$ | 0.13 | 0.017 | 0.63
$(3,-e)$ | 0.22 | 0.009 | 0.59
$(4,+e)$ | 0.12 | 0.023 | 0.45
$(4,-e)$ | 0.19 | 0.014 | 0.32
Table 3: Fitting parameters of dashed lines in Fig. A4 corresponding to the fitting equation, $(\lambda^{\alpha}_{imp})^{-1}=mB^{\alpha}_{imp}+b$. Coefficient of determination $r^{2}$ for each linear fit is given in the final column, where $r^{2}=1$ indicates a perfect fit. $(\alpha,Q)$ | $m~{}(\text{meV}^{-1}\text{ nm}^{-1})$ | $b~{}(\text{nm}^{-1})$ | $r^{2}$
---|---|---|---
$(2,-e)$ | 0.012 | 0.018 | 0.87
$(3,-e)$ | 0.014 | 0.005 | 0.80
$(4,-e)$ | 0.003 | 0.042 | 0.07

a

a

a

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