Effects of the electrostatic environment on the Majorana nanowire devices

We discuss electrostatic effects in a device design as used by Mourik et al  [8], however our methods are straightforward to adapt to similar layouts (see appendix B for a calculation using a different geometry). Since we are interested in the bulk properties, we require that the potential and the Hamiltonian terms are translationally invariant along the wire axis and we consider a 2D cross section, shown in figure 1. The device consists of a nanowire with a hexagonal cross section of diameter $W=100\;\mathrm{nm}$ on a dielectric layer with thickness ${d}_{\mathrm{dielectric}}=30\;\mathrm{nm}$. A superconductor with thickness ${d}_{\mathrm{SC}}=187\;\mathrm{nm}$ covers half of the wire. The nanowire has a dielectric constant ${\epsilon }_{{\rm{r}}}=17.7$ (InSb), the dielectric layer has a dielectric constant ${\epsilon }_{{\rm{r}}}=8$ (Si₃N₄). The device has two electrostatic boundary conditions: a fixed gate potential ${V}_{{\rm{G}}}$ set by the gate electrode along the lower edge of the dielectric layer and a fixed potential ${V}_{\mathrm{SC}}$ in the superconductor, which we model as a grounded metallic gate. We set this potential to either ${V}_{\mathrm{SC}}=0\;{\rm{V}}$, disregarding a work function difference between the NbTiN superconductor and the nanowire, or we assume a small work function difference [22, 23] resulting in ${V}_{\mathrm{SC}}=0.2\;{\rm{V}}$.

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We model the electrostatics of this setup using the Schrödinger–Poisson equation. We split the Hamiltonian into transverse and longitudinal parts. The transverse Hamiltonian ${{ \mathcal H }}_{{\rm{T}}}$ reads

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{T}}}=-\displaystyle \frac{{{\hslash }}^{2}}{2{m}^{*}}\left(\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}}{\partial {y}^{2}}\right)-e\phi (x,y)+\displaystyle \frac{{E}_{\mathrm{gap}}}{2},\end{eqnarray} \tag{ 1 }$$

with $x,y$ the transverse directions, ${m}^{*}=0.014{m}_{{\rm{e}}}$ the effective electron mass in InSb (with m_e the electron mass), $-e$ the electron charge, and ϕ the electrostatic potential. We assume that in the absence of electric field the Fermi level ${E}_{{\rm{F}}}$ in the nanowire is in the middle of the semiconducting gap ${E}_{\mathrm{gap}}$, with ${E}_{\mathrm{gap}}=0.2\;\mathrm{eV}$ for InSb (see figure 2(a). We choose the Fermi level ${E}_{{\rm{F}}}$ as the reference energy such that ${E}_{{\rm{F}}}\equiv 0$.

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The longitudinal Hamiltonian ${{ \mathcal H }}_{{\rm{L}}}$ reads

$$\begin{eqnarray}&&{{ \mathcal H }}_{{\rm{L}}}=-\displaystyle \frac{{{\hslash }}^{2}}{2{m}^{*}}\displaystyle \frac{{\partial }^{2}}{\partial {z}^{2}}-{\rm{i}}\alpha \displaystyle \frac{\partial }{\partial z}{\sigma }_{y}+{E}_{{\rm{Z}}}{\sigma }_{z},\end{eqnarray} \tag{ 2 }$$

with z the direction along the wire axis, α the spin–orbit coupling strength, ${E}_{{\rm{Z}}}$ the Zeeman energy and ${\boldsymbol{\sigma }}$ the Pauli matrices. The orientation of the magnetic field is along the wire in the z direction. In this separation, we have assumed that the spin–orbit length ${l}_{\mathrm{SO}}={{\hslash }}^{2}/({m}^{*}\alpha )$ is larger or comparable to the wire diameter, ${l}_{\mathrm{SO}}\gtrsim W$ [24, 25]. Furthermore, we neglect the explicit dependence of the spin–orbit strength α on the electric field. We ignore orbital effects of the magnetic field [26], since the effective area of the transverse wave functions is much smaller than the wire cross section due to screening by the superconductor, as we show in section 3.

Since the Hamiltonian is separable in the limit we are using, the charge density in the transverse direction $\rho (x,y)$ is:

$$\begin{eqnarray}&&\rho (x,y)=-e\displaystyle \sum _{i}{| {\psi }_{i}(x,y)| }^{2}\ n({E}_{i},{E}_{{\rm{Z}}},\alpha ),\end{eqnarray} \tag{ 3 }$$

with ${\psi }_{i}$ the transverse wave function and E_i the subband energy of the ith electron mode defined by ${{ \mathcal H }}_{{\rm{T}}}{\psi }_{i}={E}_{i}{\psi }_{i}$. Further, $n({E}_{i},{E}_{{\rm{Z}}},\alpha )$ is the 1D electron density, which we calculate in closed form from the Fermi momenta of different bands in appendix C. The subband energies E_i depend on the electrostatic potential $\phi (x,y)$, and individual subbands are occupied by 'lowering' subbands below ${E}_{{\rm{F}}}$ (shown schematically⁴ in figure 2(b)).

The Poisson equation that determines the electrostatic potential $\phi (x,y)$ has the general form:

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}\phi (x,y)=-\displaystyle \frac{\rho (x,y)}{\epsilon },\end{eqnarray} \tag{ 4 }$$

with the dielectric permittivity. Since the charge density of equation (3) depends on the eigenstates of equation (1), the Schrödinger and the Poisson equations have a nonlinear coupling.

We calculate the eigenstates and eigenenergies of the Hamiltonian of equation (1) in tight-binding approximation on a rectangular grid using the Kwant package [27]. We then discretize the geometry of figure 1 using a finite element mesh, and solve equation (4) numerically using the FEniCS package [28].

Equations (1) and (3) together define a functional $\bar{\rho }[\phi ]$, yielding a charge density from a given electrostatic potential ϕ. Additionally, equation (4) defines a functional $\bar{\phi }[\rho ]$, giving the electrostatic potential produced by a charge density ρ. The Schrödinger–Poisson equation is self-consistent when

$$\begin{eqnarray}&&\bar{\phi }[\bar{\rho }[\phi ]]-\phi =0.\end{eqnarray} \tag{ 5 }$$

We solve equation (5) using an iterative nonlinear Anderson mixing method [29]. We find that this method prevents the iteration process from oscillations and leads to a significant speedup in computation times compared to other nonlinear solver methods (see appendix E). We search for the root of equation (5) rather than for the root of

$$\begin{eqnarray}&&\bar{\rho }[\bar{\phi }[\rho ]]-\rho =0,\end{eqnarray} \tag{ 6 }$$

since we found equation (5) to be better conditioned than equation (6). The scripts with the source code as well as resulting data are available online as ancillary files for this manuscript.

Having solved the electrostatic problem for the normal system, i.e. taking into account only the electrostatic effects of the superconductor, we then use the electrostatic potential $\phi (x,y)$ in the superconducting problem. To this end, we obtain the Bogoliubov-de Gennes Hamiltonian ${{ \mathcal H }}_{\mathrm{BdG}}$ by summing ${{ \mathcal H }}_{{\rm{T}}}$ and ${{ \mathcal H }}_{{\rm{L}}}$ and adding an induced superconducting pairing term:

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{BdG}}=\left[\left(-\displaystyle \frac{{{\hslash }}^{2}}{2{m}^{*}}{{\rm{\nabla }}}^{2}-e\phi (x,y)+\displaystyle \frac{{E}_{\mathrm{gap}}}{2}\right){\sigma }_{0}-{\rm{i}}\alpha \displaystyle \frac{\partial }{\partial z}{\sigma }_{y}\right]\otimes {\tau }_{z} \\ &&+{E}_{{\rm{Z}}}{\sigma }_{z}\otimes {\tau }_{0}+{\rm{\Delta }}{\sigma }_{0}\otimes {\tau }_{x},\end{eqnarray} \tag{ 7 }$$

with ${\boldsymbol{\tau }}$ the Pauli matrices in electron–hole space and Δ the superconducting gap.

The three-dimensional BdG equation (7) is still separable and reduces for every subband with transverse wave function ${\psi }_{i}$ to an effective one-dimensional BdG Hamiltonian:

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{BdG},i}=\left[\left(\displaystyle \frac{{p}^{2}}{2{m}^{*}}-{\mu }_{i}\right){\sigma }_{0}+\displaystyle \frac{\alpha }{{\hslash }}p{\sigma }_{y}\right]\otimes {\tau }_{z}+{E}_{{\rm{Z}}}{\sigma }_{z}\otimes {\tau }_{0}+{\rm{\Delta }}{\sigma }_{0}\otimes {\tau }_{x},\end{eqnarray} \tag{ 8 }$$

where $p=-{\rm{i}}{\hslash }\partial /\partial z$ and we defined ${\mu }_{i}=-{E}_{i}$ (see figure 2(b). Since the different subbands are independent, ${\mu }_{i}$ can be interpreted as the chemical potential determining the occupation of the ith subband.

While the Fermi level is kept constant by the metallic contacts, the chemical potential ${\mu }_{i}$ of each subband does depend on the system parameters: ${\mu }_{i}={\mu }_{i}({V}_{{\rm{G}}},{E}_{{\rm{Z}}})$. Most of the model Hamiltonians for Majorana nanowires used in the literature are of the form of equation (8) (or a two-dimensional generalization) using one chemical potential μ. To make the connection to our work, μ should be identified with ${\mu }_{i}$, and not be confused with the constant Fermi level ${E}_{{\rm{F}}}$. For example, the constant chemical potential limit of [20] refers to the special case that ${\mu }_{i}$ is independent of ${E}_{{\rm{Z}}}$, and it is not related to ${E}_{{\rm{F}}}$ being always constant⁵ .

Properties of Majorana modes formed in the ith subband only depend on the value of ${\mu }_{i}$ (or equivalently E_i). In the following we thus determine the effect of the electrostatics on ${\mu }_{i}$ before we finally turn to Majorana bound states.

The full self-consistent solution of the Schrödinger–Poisson equation is computationally expensive and also hard to interpret due to a high dimensionality of the space of unknown variables. We find a simpler form of the solution at a finite Zeeman field relying on the large level spacing ∼10 meV in typical nanowires. It ensures that the transverse wave functions stay approximately constant, i.e. $| \langle \psi ({E}_{{\rm{Z}}})| \psi (0)\rangle | \;\approx \;1$ up to magnetic fields of $\sim 7\;{\rm{T}}$. In this limit we may apply perturbation theory to compute corrections to the chemical potential for varying ${E}_{{\rm{Z}}}$.

We write the potential distribution for a given ${E}_{{\rm{z}}}$ in the form

$$\begin{eqnarray}&&\phi (x,y,{E}_{{\rm{Z}}})={\phi }_{{\rm{b}}.{\rm{c}}.}(x,y)+\displaystyle \sum _{i=0}^{N}{\phi }_{i}(x,y,{E}_{{\rm{z}}}),\end{eqnarray} \tag{ 9 }$$

where ${\phi }_{{\rm{b}}.{\rm{c}}.}$ is the potential obeying the boundary conditions set by the gate and the superconducting lead, and solves the Laplace equation

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{\phi }_{{\rm{b}}.{\rm{c}}.}(x,y)=0.\end{eqnarray} \tag{ 10 }$$

The corrections ${\phi }_{i}$ to this potential due to the charge contributed by the ith mode out of the N modes below the Fermi level then obeys a Poisson equation with Dirichlet boundary conditions (zero voltage on the gates):

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{\phi }_{i}(x,y,{E}_{{\rm{Z}}})=\displaystyle \frac{e}{\epsilon }{| {\psi }_{i}(x,y)| }^{2}n(-{\mu }_{i}-\delta {\mu }_{i},{E}_{{\rm{Z}}},\alpha )\end{eqnarray} \tag{ 11 }$$

where we write the chemical potential at a finite value of ${E}_{{\rm{Z}}}$ as ${\mu }_{i}({E}_{{\rm{Z}}})={\mu }_{i}+\delta {\mu }_{i}$ where ${\mu }_{i}$ is the chemical potential in the absence of a field.

We now define a magnetic field-independent reciprocal capacitance as

$$\begin{eqnarray}&&{P}_{i}(x,y)=\displaystyle \frac{{\phi }_{i}(x,y,{E}_{{\rm{Z}}})}{-e\;n(-{\mu }_{i}-\delta {\mu }_{i},{E}_{{\rm{Z}}},\alpha )}\end{eqnarray} \tag{ 12 }$$

which solves the Poisson equation

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{P}_{i}(x,y)=-\displaystyle \frac{1}{\epsilon }{| {\psi }_{i}(x,y)| }^{2}.\end{eqnarray} \tag{ 13 }$$

Having solved the Schrödinger–Poisson problem numerically for ${E}_{{\rm{Z}}}=0$, we define $\delta {\phi }_{i}={\phi }_{i}(x,y,{E}_{{\rm{Z}}})\;$ $-\;{\phi }_{i}(x,y,0)$ and $\delta n=n(-{\mu }_{i}-\delta {\mu }_{i},{E}_{{\rm{Z}}},\alpha )-n(-{\mu }_{i},0,\alpha )$. The correction $\delta {E}_{i}$ to the subband energy E_i is then given in first order perturbation as

$$\begin{eqnarray}&&\delta {E}_{i}=-e\langle {\psi }_{i}| \displaystyle \sum _{j=0}^{N}\delta {\phi }_{j}| {\psi }_{i}\rangle .\end{eqnarray} \tag{ 14 }$$

Using equations (12), (14) and $\delta {\mu }_{i}=-\delta {E}_{i}$ we then arrive at:

$$\begin{eqnarray}&&\delta {\mu }_{i}=-{e}^{2}\displaystyle \sum _{j=0}^{N}{P}_{{ij}}\delta {n}_{j},\end{eqnarray} \tag{ 15 }$$

with the elements of the reciprocal capacitance matrix P given by

$$\begin{eqnarray}&&{P}_{{ij}}=\langle {\psi }_{i}| {P}_{j}| {\psi }_{i}\rangle .\end{eqnarray} \tag{ 16 }$$

Solving the equation (15) self-consistently, we compute corrections to the initial chemical potentials ${\mu }_{i}$. The equation (15) has a much lower dimensionality than equation (5) and is much cheaper to solve numerically. Further, all the electrostatic phenomena enter equation (15) only through the reciprocal capacitance matrix equation (16).

We start by computing the electrostatic response to changes in the magnetic field when the Fermi level is close to the band bottom for a single band (N = 1, and we write the index ${\mu }_{1}\equiv \mu$ for brevity). We study the influence of the electrostatic environment and assess whether the device is closer to a constant charge density or constant chemical potential situation (using the nomenclature of [20] explained in appendix A).

The top panel of figure 5 shows the chemical potential response to Zeeman field. Without a superconducting contact, the electron-electron interactions in the nanowire are screened the least, and the Coulomb effects are the strongest, counteracting density changes in the wire. In agreement with this observation, we find the change in chemical potential μ comparable to the change in ${E}_{{\rm{Z}}}$. Hence, in this case the system is close to a constant-density regime.

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A superconducting contact close to the nanowire screens the electron-electron interaction in the wire due to image charges. The chemical potential is then less sensitive to changes in magnetic field. We find that this effect is most pronounced for a positive work function difference with the superconductor ${V}_{\mathrm{SC}}=0.2\;{\rm{V}}$, when most of the electrons are pulled close to the superconducting contact. Then, the image charges are close to the electrons and strongly reduce the Coulomb interactions. In this case the system is close to a constant chemical potential regime. For ${V}_{\mathrm{SC}}=0\;{\rm{V}}$ screening from the superconducting contact is less effective, since electric charges are further away from the interface with the superconductor. Therefore in this case, we find a behavior intermediate between constant density and constant chemical potential.

Besides the dependence on the electrostatic surrounding, the magnetic field response of the chemical potential depends on the spin–orbit strength. Specifically, the chemical potential stays constant over a longer field range when the spin–orbit interaction is stronger⁶ . The bottom panel of figure 5 explains this: when the spin–orbit energy ${E}_{\mathrm{SO}}\gg {E}_{{\rm{Z}}}$, the lower band has a W-shape (bottom left). A magnetic-field increase initially transforms the lower band back from a W-shape to a parabolic band, as indicated by the dashed red lines. During this transition, the Fermi wavelength is almost constant. Since the electron density is proportional to the Fermi wavelength, this means that both the density and the chemical potential change very little in this regime. We thus identify the spin–orbit interaction as another phenomenon driving the system closer to the constant chemical potential regime, similar to the screening of the Coulomb interaction by the superconductor.

At large Zeeman energies ${E}_{{\rm{Z}}}\gtrsim {E}_{\mathrm{SO}}$, the spin-down band becomes parabolic (bottom right of figure 5). This results in the slope of $\mu ({E}_{{\rm{Z}}})$ becoming independent of the spin–orbit coupling strength, as seen in the top panel of figure 5 at large values of ${E}_{{\rm{Z}}}$.

Close to the band bottom and when spin–orbit interaction is negligible, we study the asymptotic behavior of μ and n by combining the appropriate density expression equation (C.6) with the corrections in the chemical potential equation (15). In that case, the chemical potential becomes

$$\begin{eqnarray}&&\mu =-\displaystyle \frac{{e}^{2}P}{\pi {\hslash }}\sqrt{2{m}^{*}(\mu +{E}_{{\rm{Z}}})}.\end{eqnarray} \tag{ 17 }$$

We associate an energy scale E_P with the reciprocal capacitance P, given by

$$\begin{eqnarray}&&{E}_{{\rm{P}}}=\displaystyle \frac{2{m}^{*}{e}^{4}{P}^{2}}{{\pi }^{2}{{\hslash }}^{2}}\end{eqnarray} \tag{ 18 }$$

and study the two limits ${E}_{{\rm{P}}}\gg {E}_{{\rm{Z}}}$ and ${E}_{{\rm{P}}}\ll {E}_{{\rm{Z}}}$. In the strong screening limit ${E}_{{\rm{P}}}\gg {E}_{{\rm{Z}}}$ we find the asymptotic behavior $\mu \approx -{E}_{{\rm{Z}}}$, corresponding to a constant-density regime. The opposite limit ${E}_{{\rm{P}}}\ll {E}_{{\rm{Z}}}$ yields $\mu \approx -\sqrt{{E}_{{\rm{P}}}{E}_{{\rm{Z}}}}$, close to a constant chemical potential regime. We computed E_P explicitly for the chemical potential variations as shown in the top panel of figure 5. For a nanowire without a superconducting lead, we find an energy ${E}_{{\rm{P}}}\approx 42\;\mathrm{meV}\gg {E}_{{\rm{Z}}}$, indicating a constant-density regime. Using the classical approximation of a metallic cylinder above a metallic plate, we find an energy of the same order of magnitude. For a nanowire with an attached superconducting lead at ${V}_{\mathrm{SC}}=0\;{\rm{V}}$, we get ${E}_{{\rm{P}}}\approx 7\;\mathrm{meV}\sim {E}_{{\rm{Z}}}$, intermediate between constant density and constant chemical potential. Finally, a superconducting lead at ${V}_{\mathrm{SC}}=0.2\;{\rm{V}}$ yields ${E}_{{\rm{P}}}\approx 0.5\;\mathrm{meV}\ll {E}_{{\rm{Z}}}$, indicating a system close to the constant chemical potential regime.

Since integrating over density-of-states measurements yields $\delta {n}_{0}$, the inverse self-capacitance $-e\langle {\psi }_{0}| {P}_{0}| {\psi }_{0}\rangle$ can be inferred from experimental data by fitting the density variation curves to the theoretical dependence $\mu ({E}_{{\rm{Z}}})$. This allows to experimentally measure the effect of the electrostatic environment, when knowing the remaining Hamiltonian parameters.

We compare the response to Zeeman field in the multi-band case for N  =  3 and N  =  10 to the single band behavior in figure 6. We observe that presence of extra charges further reduces the sensitivity of the chemical potential to the magnetic field. We interpret the non-monotonous behavior of the chemical potential (most pronounced for N = 10 in figure 6, but in principle present for all N) as being due to a combination of the Van Hove singularities in the density of states and screening by charges. For a fixed chemical potential, the upper band, moving up in energy due to the magnetic field, loses more states than the lower band acquires, since it approaches the Van Hove singularity in its density of states. To keep the overall density fixed, the chemical potential increases. Once the density in the lower band equals the initial density, the upper band is empty and the chemical potential starts dropping again. In the limit of constant density and a single mode the magnetic field dependence of the chemical potential can be solved analytically, reproducing the non-monotonicity and kinks (see appendix D).

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Relating the variation in ${\mu }_{i}$ to density measurements is experimentally inaccessible for $N\gt 1$, since corrections to ${\mu }_{i}$ depend on the density changes of each individual mode, as expressed in equation (15).

The nanowire enters the topological phase when the bulk energy gap closes at a Zeeman energy of ${E}_{{\rm{Z}}}=\sqrt{{\mu }^{2}+{{\rm{\Delta }}}^{2}}$. The electrostatic effects affect the shape of the topological phase boundary through the dependence of μ on ${E}_{{\rm{Z}}}$. To find the topological phase boundary as a function of both experimentally controllable parameters ${V}_{{\rm{G}}}$ and ${E}_{{\rm{Z}}}$, we perform a full self-consistent simulation at ${E}_{{\rm{Z}}}=0$. We then compute corrections to the resulting chemical potential at arbitrary ${E}_{{\rm{Z}}}$ using equation (15), and find topological phase boundary ${E}_{{\rm{Z}}}=\sqrt{{\mu }^{2}+{{\rm{\Delta }}}^{2}}$ by recursive bisection.

Figure 7 shows the resulting phase boundary corresponding to ${\rm{\Delta }}=0.5\;\mathrm{meV}$. The phase boundary has a non-universal shape due to the interplay between electrostatics and magnetic field. In agreement with our previous conclusions, the electrostatic effects are the strongest with absent work function difference ${V}_{\mathrm{SC}}=0\;{\rm{V}}$ (top panel of figure 7) when the nanowire is intermediate between constant density and constant chemical potential⁷ . Close to the band bottom, the charge screening reduces changes in density, and thus lowers the chemical potential by an amount that is similar to ${E}_{{\rm{Z}}}$. Hence, the lower phase boundary (at smaller ${V}_{{\rm{G}}}$) has a weaker slope than the upper phase boundary (at larger ${V}_{{\rm{G}}}$). Note that in the limit of constant density, the lower phase boundary would be a constant independent of ${E}_{{\rm{Z}}}$ (see appendix D).

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For a work function difference ${V}_{\mathrm{SC}}=0.2\;{\rm{V}}$, the system is closer to the constant chemical potential regime. In this regime, μ changes linearly with ${V}_{{\rm{G}}}$, yielding a hyperbolic phase boundary with symmetric upper and lower arms and its vertex at ${E}_{{\rm{Z}}}={\rm{\Delta }}$. When spin–orbit interaction is strong, a transition in the lower arm of the phase boundary from constant chemical potential (hyperbolic phase boundary) to constant density (more horizontal lower arm) occurs, resulting in a 'wiggle' which is most pronounced for ${V}_{\mathrm{SC}}=0\;{\rm{V}}$ and ${l}_{\mathrm{SO}}=60\;\mathrm{nm}$. This feature is less pronounced for ${V}_{\mathrm{SC}}=0.2\;{\rm{V}}$ due to the screening by the superconductor suppressing the Coulomb interactions.

The wave functions of the two Majorana modes at the endpoints of a finite-length nanowire have a finite overlap that results in a finite nonzero energy splitting ${\rm{\Delta }}E$ of the lowest Hamiltonian eigenstates [17–21]. This splitting oscillates as a function of the effective Fermi wave vector ${k}_{{\rm{F}},\mathrm{eff}}$ as ${\rm{cos}}({k}_{{\rm{F}},\mathrm{eff}}L)$ [20]. We investigate the dependency of the oscillation frequency, or the oscillation peak spacing on magnetic field and the electrostatic environment.

A peak in the Majorana splitting energy occurs when Majorana wave functions constructively interfere, or when the Fermi momentum equals $q\pi /L$, with q the peak number and L the nanowire length. The momentum difference between two peaks is

$$\begin{eqnarray}&&{k}_{{\rm{F}},\mathrm{eff}}({E}_{{\rm{Z}},q+1})-{k}_{{\rm{F}},\mathrm{eff}}({E}_{{\rm{Z}},q})=\displaystyle \frac{\pi }{L},\end{eqnarray} \tag{ 19 }$$

where ${E}_{{\rm{Z}},q}$ is the Zeeman energy corresponding to the qth oscillation peak. In the limit of small peak spacing, we expand ${k}_{{\rm{F}},\mathrm{eff}}({E}_{{\rm{Z}},q+1})-{k}_{{\rm{F}},\mathrm{eff}}({E}_{{\rm{Z}},q})$ to first order in ${E}_{{\rm{Z}}}$:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{k}_{{\rm{F}}}}{{\rm{d}}{E}_{{\rm{Z}}}}{\rm{\Delta }}{E}_{{\rm{Z}}}=\displaystyle \frac{\pi }{L},\end{eqnarray} \tag{ 20 }$$

yielding the peak spacing

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{{\rm{Z}},\mathrm{peak}}=\displaystyle \frac{\pi }{L}{\left(\displaystyle \frac{{\rm{d}}{k}_{{\rm{F}}}}{{\rm{d}}{E}_{{\rm{Z}}}}\right)}^{-1}.\end{eqnarray} \tag{ 21 }$$

Since ${k}_{{\rm{F}},\mathrm{eff}}={k}_{{\rm{F}},\mathrm{eff}}({E}_{{\rm{Z}}},\mu ({E}_{{\rm{Z}}}))$, we substitute

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{k}_{{\rm{F}}}}{{\rm{d}}{E}_{{\rm{Z}}}}=\displaystyle \frac{\partial {k}_{{\rm{F}}}}{\partial {E}_{{\rm{Z}}}}+\displaystyle \frac{\partial {k}_{{\rm{F}}}}{\partial \mu }\displaystyle \frac{{\rm{d}}\mu }{{\rm{d}}{E}_{{\rm{Z}}}}.\end{eqnarray} \tag{ 22 }$$

We obtain the values of $\partial {k}_{{\rm{F}}}/\partial {E}_{{\rm{Z}}}$ and $\partial {k}_{{\rm{F}}}/\partial \mu$ from the analytic expression for ${k}_{{\rm{F}}}$, presented in appendix C. The value ${\rm{d}}\mu /{\rm{d}}{E}_{{\rm{Z}}}$ results from the dependence $\mu ({E}_{{\rm{Z}}})$ shown in figure 5.

Figure 8 shows the peak spacing as a function of ${E}_{{\rm{Z}}}$ for a nanowire of length $L=2\;\mu {\rm{m}}$. Stronger screening reduces the peak spacing (i.e. increases the oscillation frequency) by reducing the sensitivity of the chemical potential to the magnetic field, as discussed in section 4. In addition, spin–orbit strength has a strong influence on the peak spacing, since for ${E}_{{\rm{Z}}}\ll {E}_{\mathrm{SO}}$ the density, and thus ${k}_{{\rm{F}},\mathrm{eff}}$, stays constant. This results in a lower oscillation frequency and hence a larger peak spacing. Correspondingly, we find that the peak spacing may increase, decrease, or roughly stay constant as a function of the magnetic field.

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Similarly to the shape of the Majorana transition boundary, figure 8 shows that the peak spacing does not follow a universal law, in contrast to earlier predictions [21]. In particular, our findings may explain the zero-bias oscillations measured in  [11], exhibiting a roughly constant peak spacing.

Figure 9 shows Majorana energy oscillations as a function of both gate voltage and magnetic field strength for ${V}_{\mathrm{SC}}=0.2\;{\rm{V}}$, with $L=1000\;\mathrm{nm}$ to increase the Majorana coupling. The diagonal ridges are lines of constant chemical potential. The difference in slope between the ridges of both plots indicates a difference in the equilibrium situation: closer to constant density for weak spin–orbit coupling, closer to constant chemical potential for strong spin–orbit coupling. The bending of the constant chemical potential lines in the lower panel indicates a transition from the latter mechanism to the former mechanism, due to the increase of magnetic field, as explained in section 4.

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