@@ -633,7 +633,7 @@ Perfectly ordered crystalline solids are characterized by a faultlessly regular
Real solid-state systems are not perfectly smooth and have impurities: they are disordered.
Disorder is always present and it has a strong impact on the physical properties of proximitized nanowires.
Research has shown that both disorder in the semiconductor and at the semiconductor-superconductor interface is detrimental to the creation of Majoranas in proximitized nanowires~\cite{Lobos2012,Lutchyn2012,Sau2012a,Sau2013,Hui2015,Cole2016,Liu2018}.
However, semiconductor nanowire with epitaxially grown aluminium~\cite{Lutchyn2018,Krogstrup2015} minimize the disorder effects and have high-quality semiconductors and semiconductor-superconductor interfaces; however, its outer surface is oxidized and therefore strongly disordered.
However, semiconductor nanowire with epitaxially grown aluminum~\cite{Lutchyn2018,Krogstrup2015} minimize the disorder effects and have high-quality semiconductors and semiconductor-superconductor interfaces; however, its outer surface is oxidized and therefore strongly disordered.
Scattering of the disordered superconducting boundary randomizes the quasiparticle's motion inside the superconductor.
Fortunately, because of the large difference in the effective masses and Fermi energies between the semiconductor and the superconductor, as argued by the authors of~\cite{Sticlet2017,Lutchyn2012,Liu2018}, disorder in the superconductor is not detrimental to the manipulation and observation of Majoranas, however, a more systematic numerical study is required.
This problem is complex to study numerically because of the aforementioned high density differences, which means that the discretization in the superconductor has to be smaller than the Fermi wavelength of Al, $\approx\SI{0.1}{\AA}$.
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@@ -912,7 +912,7 @@ The Fermi wavelength $\lambda_\textrm{F}$ sets a lower limit on the lattice cons
This means that systems easily become larger than $1e6$ sites, and including the spin and electron-hole degrees orbitals results in matrices of $4e^6\times4e^6$.
To calculate a phase diagram $E_\textrm{gap}(B_x,\mu)$ using this system, we need to diagonalize this large matrix---which is an expensive operation by itself---many times.
In the case of strongly coupled superconductors, it becomes more problematic because the wavefunction has most of its weight in the superconductor, which means we have to simulate it on the level of its Fermi wavelength.
Aluminium is typically used as the superconductor and has a Fermi energy of $E_\textrm{F}\approx\SI{11.7}{eV}$, which translates into $\lambda_\textrm{F}\approx\SI{0.36}{nm}$.
Aluminum is typically used as the superconductor and has a Fermi energy of $E_\textrm{F}\approx\SI{11.7}{eV}$, which translates into $\lambda_\textrm{F}\approx\SI{0.36}{nm}$.
So, to properly simulate even the thin layer of superconducting material on top of a nanowire, the lattice constant has to be smaller then an ångström, resulting in matrix sizes that go far beyond the current capabilities of even the largest computers.
@@ -19,7 +19,7 @@ The main experimental effort in the last few years has focused on detecting thes
However, SOI, the mechanism providing the topological protection, has been challenging to detect directly in Majorana nanowires.
The electric field that gives rise to SOI in our system mainly results from structural inversion asymmetry of the confinement potential (Rashba SOI), which depends on the work function difference at the interface between the nanowire and the superconductor and on voltages applied to nearby electrostatic gates~\cite{Vuik2016,Antipov2018,Woods2018,Mikkelsen2018}.
The Rashba SOI in nanowires has been investigated extensively by measuring spin-orbit related quantum effects: level repulsion of quantum dot levels~\cite{Fasth2007,Nadj-Perge2012}, and of Andreev states~\cite{Moor2018,Deng2016}, weak antilocalization in long \mbox{diffusive} wires~\cite{Hansen2005,Weperen2015}, and a helical liquid signature in short quasiballistic wire~\cite{Kammhuber2017}.
The Rashba SOI in nanowires has been investigated extensively by measuring spin-orbit related quantum effects: level repulsion of quantum dot levels~\cite{Fasth2007,Nadj-Perge2012}, and of Andreev states~\cite{Moor2018,Deng2016}, weak anti-localization in long \mbox{diffusive} wires~\cite{Hansen2005,Weperen2015}, and a helical liquid signature in short quasiballistic wire~\cite{Kammhuber2017}.
However, the SOI strength relevant to the topological protection is affected by the \mbox{presence} of the superconductor, necessitating direct observation of SOI in Majorana nanowires.
Here, we reveal SOI in an InSb nanowire coupled to a NbTiN superconductor through the dependence of the superconducting gap on the magnetic field, both strength and orientation.
We find that the geometry of the superconductor on the nanowire strongly modifies the direction of the spin-orbit field, which is further tunable by electrostatic gating, in line with the expected modifications of the electric field due to work function difference and electrostatic screening at the nanowire-superconductor interface.
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@@ -68,8 +68,8 @@ This pronounced anisotropy of the gap closing with respect to different $B$ dire
\section{Interpretation of the anisotropy}
Before we discuss the SOI in more detail, we rule out alternative mechanisms for the anisotropy which can originate in the bulk superconductor, or the InSb nanowire.
First, an anisotropic magnetic field-induced closing of the bulk superconducting gap is excluded for the fields we apply, which are far below the critical field of NbTiN ($>$9 T~\cite{VanWoerkom2015}.
We note that this is different from aluminium films~\cite{Chang2015,Deng2016,Gazibegovic2017,Zhang2018}, where a small magnetic field ($<$0.3 T) perpendicular to the film completely suppresses superconductivity, making them unsuitable to reveal SOI from an anisotropic gap closing.
First, an anisotropic magnetic field-induced closing of the bulk superconducting gap is excluded for the fields we apply, which are far below the critical field of NbTiN ($>$9 T~\cite{VanWoerkom2015}.
We note that this is different from aluminum films~\cite{Chang2015,Deng2016,Gazibegovic2017,Zhang2018}, where a small magnetic field ($<$0.3 T) perpendicular to the film completely suppresses superconductivity, making them unsuitable to reveal SOI from an anisotropic gap closing.
Next, we consider Meissner screening currents in NbTiN that can cause deviations in the magnetic field in the nanowire.
Our Ginzburg-Landau simulations show that the field corrections due to Meissner screening are negligible (see Fig.~\ref{fig:GL}), since the dimensions of the NbTiN film ($<$1 $\mu$m) are comparable to the penetration depth ($\sim$290 nm).
The simulations also show that vortex formation is most favorable along the $z$ axis, which implies that the observed anisotropic gap closing is not caused by gap suppression due to vortices near the nanowire~\cite{Takei2013}, since we do not observe the fastest gap closing along $z$ [Fig.~\ref{fig:fig1}(f)].
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@@ -113,7 +113,7 @@ We note that adding the orbital effect in Fig.~\ref{fig:fig2}(c) shifts the gap
By fitting the curvature of the gap closing~\cite{Heck2017,Pan2019} along $x$ [white dashed line in Fig.~\ref{fig:fig1}(e)] we estimate a range of the SOI strength $\alpha$ of 0.15 -- 0.35 eV\AA\ from devices $A$-$D$ (for fitting details and fits to additional devices, see Sec.~\ref{sec:SOI_extraction}).
This SOI strength is in \mbox{agreement} with the values extracted from level repulsion of Andreev states~\cite{Stanescu2013,Moor2018} in an additional device $E$, see Sec.~\ref{sec:level_rep} for more details.
\mbox{Since}$\alpha$ depends on the electric field in the wire, we expect the observed variation in the SOI strength of devices to be caused by differences in the applied gate voltages and wire diameter.
Recently, the level repulsion of Andreev states in InSb nanowires covered with epitaxial aluminium has shown a SOI strength of approximately 0.1 eV\AA~\cite{Moor2018}, slightly lower than we find for NbTiN covered nanowires, most likely due to strong coupling to the aluminium superconductor, leading to stronger renormalization of the InSb material parameters~\cite{Stanescu2011,Cole2015,Antipov2018,Woods2018,Mikkelsen2018,Reeg2018}.
Recently, the level repulsion of Andreev states in InSb nanowires covered with epitaxial aluminum has shown a SOI strength of approximately 0.1 eV\AA~\cite{Moor2018}, slightly lower than we find for NbTiN covered nanowires, most likely due to strong coupling to the aluminum superconductor, leading to stronger renormalization of the InSb material parameters~\cite{Stanescu2011,Cole2015,Antipov2018,Woods2018,Mikkelsen2018,Reeg2018}.
We fit the eigenenergies of $H$ to our experimental data [Fig.~\ref{fig:SOIlevelrep}a] to extract $\delta_{\mathrm{SO}}$.
The precise value of the coupling parameter $\delta_{\mathrm{SO}}$ depends not only on $\alpha$, but also on the details of the confinement and on the coupling strength to the superconducto~\cite{Moor2018}.
The precise value of the coupling parameter $\delta_{\mathrm{SO}}$ depends not only on $\alpha$, but also on the details of the confinement and on the coupling strength to the superconducto~\cite{Moor2018}.
A rough estimate, with reasonable agreement to numerical simulations, was proposed to be: $2\delta_{\mathrm{SO}}$ = $\alpha\pi/ L$, where $L$ is the length of the wire.
The extracted $\delta_{\mathrm{SO}}$ is shown in Fig.~\ref{fig:SOIlevelrep}(b) for various values of the super gate voltage $V_{\mathrm{SG}}$.
As $V_{\mathrm{SG}}$ becomes more negative, we see an increase in $\delta_{\mathrm{SO}}$, consistent with an increasing electric field in the nanowire.
