Finite-size effects in a nanowire strongly coupled to a thin superconducting shell
Christopher Reeg Daniel Loss Jelena Klinovaja
cond-mat.mes-hall cond-mat.supr-con






Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland


We study the proximity effect in a one-dimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finite-size effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any fine-tuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the “hard gap" regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.

74.45.+c,71.10.Pm,73.21.Hb,74.78.Na

Finite-size effects in a nanowire strongly coupled to a thin superconducting shell
    Jelena Klinovaja
    December 30, 2023
==================================================================================





§ INTRODUCTION
 Topological superconductivity has been a subject of intense study in recent years <cit.> both theoretically  and experimentally because the localized Majorana excitations of such systems obey non-Abelian statistics and can potentially be utilized for applications in quantum computing <cit.>. The most promising proposal to date for engineering Majorana bound states in nanowires combines Rashba spin-orbit coupling, proximity-induced s-wave superconductivity, and an external magnetic field applied parallel to the nanowire <cit.>. An alternative proposal which has also received a great deal of attention involves coupling a ferromagnetic atomic chain to an s-wave superconductor with strong intrinsic spin-orbit coupling  <cit.>. Since the first generation of nanowire experiments <cit.>, there has been significant progress made in both the fabrication of cleaner devices as well as in the quality of the proximity-induced superconductivity <cit.>. The most significant advance in this respect has been the development of thin shells (with thickness d∼10 nm) of superconducting Al grown epitaxially on either InAs <cit.> or InSb <cit.> nanowires, thus ensuring a very strong proximity contact which has led to very hard induced gaps in the nanowires in the absence of a magnetic field.

Despite the recent experimental development of these thin superconducting shells, the most comprehensive theories describing proximity-induced superconductivity in a nanowire treat the superconductor as infinitely large <cit.>. Such an assumption implies that there is a continuum of states in the superconductor, and therefore there are always states available to couple to the nanowire and open a gap. However, in reality, the superconductor has a finite level spacing δ E_s∼ħ v_F/d due to its finite size. For the thin Al shells studied experimentally (v_F∼10^6 m/s and d∼10 nm), the level spacing of the shell δ E_s∼10 meV exceeds the Fermi energy of the nanowire (∼0.1-1 meV for typical semiconducting nanowires). Thus, for the experimental system, the limit of a bulk superconductor is not the relevant one and finite-size effects are expected to play an important role in determining the strength of proximity-induced superconductivity.


In this paper, we show that the finite size of the shell can be very detrimental to inducing superconductivity in the nanowire. In order to induce a sizable superconducting gap without finely tuning the thickness of the shell on the scale of the Fermi wavelength (of the superconductor), the energy scale describing tunneling between the nanowire and superconductor (γ) must be made larger than the level spacing of the shell (γ≳δ E_s). However, such strong tunneling induces a shift in the effective chemical potential of the nanowire which greatly exceeds the semiconducting energy scale. As a result, it is possible for the system to exhibit a hard gap even if the nanowire is effectively depleted; in this case, the gap is determined by the lowest subband in the superconductor rather than the nanowire. Additionally, in order to reach the topological phase, the Zeeman energy induced by an external magnetic field must counteract the large chemical potential shift. As a result, the field strength needed to reach the topological phase greatly exceeds the critical field of Al.

The remainder of this paper is organized as follows. In Sec. <ref>, we describe a simple theoretical model which can be applied to the experimental geometry of a thin superconducting shell strongly coupled to a semiconducting nanowire. In Sec. <ref>, we analyze the spectrum of our model, showing that a large tunneling strength is needed to overcome the large level spacing of the superconductor and open a sizable gap in the nanowire. We consider the case when the nanowire is located near the edge of the superconductor in Sec. <ref>, while we consider the case when the nanowire is located in the middle of the superconductor in Sec. <ref>. We present a numerical tight-binding calculation to back up the theoretical analysis of our model in Sec. <ref>. In Sec. <ref>, we determine how the finite size of the superconductor affects the critical field strength needed to reach the topological phase in the nanowire. In Sec. <ref>, we relate the results of our simple model directly to the experimental setup and provide estimates for the level spacing of the superconducting shell, the tunneling strength needed to induce a sizable gap in the nanowire, and the critical field strength needed to reach the topological phase. Our conclusions are given in Sec. <ref>.



§ MODEL
 
The system we consider is displayed in Fig. <ref>(a). We consider a nanowire which is an infinitely long one-dimensional channel oriented along the y-direction (with zero width). The nanowire is tunnel coupled at a position x=x_w to a superconductor which is infinitely long in the y-direction and has finite extent d in the x-direction (the need for a finite x_w will be explained below).



We consider a Hamiltonian of the form

    H=H_w+H_s+H_t.

For now, we take a simple model for the Hamiltonian of the nanowire,

    H_w=∑_σ∫dk_y/2π ψ^†_σ(k_y)ξ_kψ_σ(k_y),

where k_y is a conserved momentum in the direction parallel to the nanowire, ψ_σ^†(k_y) is the creation operator in the nanowire, and ξ_k=k_y^2/2m_w-μ_w (m_w and μ_w are the effective mass and chemical potential of the nanowire, respectively). The superconductor is described by a BCS Hamiltonian,

    H_s=1/2∫dk_y/2π∫_0^d dx η^†(k_y,x)ℋ_BCSη(k_y,x),

where η(k_y,x)=[η_↑(k_y,x),η_↓^†(-k_y,x)]^T, η_σ^†(k_y,x) is the creation operator in the superconductor, and ℋ_BCS=(-∂_x^2/2m_s+k_y^2/2m_s-μ_s)τ_z+Δτ_x, with Δ the (constant in space) superconducting pairing potential and τ_x,y,z Pauli matrices acting in Nambu space. The two systems are coupled at a position x=x_w by a tunneling term which we assume preserves spin and momentum,

    H_t=-t∑_σ∫dk_y/2π[ψ_σ^†(k_y)η_σ(k_y,x_w)+H.c.],

where t is the (spin-independent) tunneling amplitude. Such a model corresponds to local tunneling along the superconductor/semiconductor interface.

In the absence of tunneling, the spectrum of the finite-sized superconductor is given by (n∈ℤ^+)

    E_n(k_y)=√((μ_s-k_y^2/2m_s-π^2 n^2/2m_sd^2)^2+Δ^2).

When the quantization scale exceeds the gap, 1/m_sd^2≫Δ, there are very few subbands available to couple to the low-energy modes of the nanowire, as shown in Fig. <ref>(b). In this case, the relevant subbands follow a linearized form

    E_n(k_y)=√([(k_Fd-π n)δ E_s-k_y^2/2m_s]^2+Δ^2),

where we define the level spacing δ E_s=v_F/d (v_F=k_F/m_s is the Fermi velocity of the superconductor and k_F=√(2m_sμ_s) is the Fermi momentum). As we will show explicitly, when δ E_s≫Δ, it is the level spacing which is the relevant scale (rather than Δ) in determining the strength of the proximity effect.

The Hamiltonian in Eq. (<ref>) can be diagonalized by means of a Bogoliubov transformation <cit.>. The resulting Bogoliubov-de Gennes (BdG) equation is given by

    [ℋ_BCS+t^2δ(x-x_w)G_0^R(E,k_y)]ψ_s(x)=Eψ_s(x),

where ψ_s(x) is the (Nambu spinor) wave function of the superconductor and G_0^R(E,k_y)=(E-ξ_kτ_z+i0^+)^-1 is the bare retarded Green's function of the nanowire (in the absence of tunneling). The nanowire itself enters only through the boundary condition at x=x_w [corresponding to the δ-function term in Eq. (<ref>)]. Inside the superconductor, we solve Eq. (<ref>) on both the left (x<x_w) and right (x>x_w) sides of the nanowire to give
 

    ψ_L(x)=c_1([ u_0; v_0 ])sin(k_+d)+c_2([ v_0; u_0 ])sin(k_-d), 
    ψ_R(x)=c_3([ u_0; v_0 ])sin[k_+(d-x)]+c_4([ v_0; u_0 ])sin[k_-(d-x)],


where k_±=(k_F^2-k_y^2±2im_sΩ)^1/2, u_0(v_0)=√((1± iΩ/E)/2), and Ω=√(Δ^2-E^2). The vanishing boundary conditions which we impose at the free ends of the superconductor (x=0 and x=d) are accounted for already in Eqs. (<ref>); the boundary conditions at x=x_w due to tunneling are given by <cit.>
 

    ψ_L(x_w)=ψ_R(x_w), 
    1/k_F[∂_xψ_R(x_w)-∂_xψ_L(x_w)]=2γτ_zG_0^R(E,k_y)ψ_s(x_w),


where γ=t^2/v_F is a tunneling energy scale.

Imposing boundary conditions at x=x_w, the solvability condition of the resulting system of equations determines the excitation spectrum E(k_y). Assuming that μ_s≫|Ω|, we make a semiclassical expansion

    k_±=k_Fφ± iΩ/(v_Fφ)≡ζ± iχ,

where φ=(1-k_y^2/k_F^2)^1/2 parametrizes the quasiparticle trajectory inside the superconductor (0<φ≤1). We note that the semiclassical approximation breaks down for grazing trajectories k_y≈ k_F within the superconductor, and we do not consider such trajectories. After some algebra (see Appendix <ref> for details), the solvability condition can be expressed as

    E^2/Γ^2(E,k_y)-Δ^2(1/Γ(E,k_y)-1)^2-[ξ_k-δμ(E,k_y)]^2=0,

where we define the effective parameters


    Γ=(1+γ/Ωφ[cosh(2χ d)-cos(2ζ d)]{sinh(2χ d)-cos(2ζ x_w)sinh[2χ(d-x_w)]-cos[2ζ(d-x_w)]sinh(2χ x_w)})^-1, 
    δμ=-γ/φ[cosh(2χ d)-cos(2ζ d)]{sin(2ζ d)-sin(2ζ x_w)cosh[2χ(d-x_w)]-sin[2ζ(d-x_w)]cosh(2χ x_w)}.


The quantity Γ(E,k_y), which takes values 0<Γ<1 for E<Δ, renormalizes the energy and is responsible for inducing superconductivity in the nanowire, while the quantity δμ(E,k_y) corresponds to a tunneling-induced shift in the effective chemical potential of the nanowire.

Note that both tunneling-induced terms vanish (δμ=0 and Γ=1) if the nanowire is taken to be strictly at the edge of the superconductor, x_w=0 or x_w=d. As a result, the system behaves as though there is no tunnel coupling [i.e., Eq. (<ref>) reduces to simply E^2=ξ_k^2]. This is a direct consequence of the fact that the nanowire was taken to have zero width. The tunneling term in Eq. (<ref>) relates the wave functions in the nanowire and superconductor at x=x_w, and the superconducting wave function vanishes at the boundaries. We choose to keep the approximation of a zero-width nanowire, as it is more consistent with previous related theories and is easier to treat analytically, and therefore the nanowire must be chosen to be located at some position x=x_w≠0 in order to have a non-vanishing tunnel coupling. We will show in Sec. <ref> that such an approximation is consistent with a numerical tight-binding calculation in which the wire can be placed strictly at the edge of the superconductor. Alternatively, if the nanowire is located strictly at the edge of the superconductor, it must be given a finite width so that the superconducting wave function does not vanish at the interface. Related calculations were carried out in Refs. <cit.>, where proximity-induced superconductivity was studied in a quasi-two-dimensional layer of finite width coupled to a semi-infinite three-dimensional superconductor.

We also note that we can equivalently express the solvability condition in Eq. (<ref>) in the language of Green's functions. We can rewrite Eq. (<ref>) in the form 
(G^R_w)^-1=0, where 
 G^R_w=[(G_0^R)^-1-Σ^R]^-1
is the retarded Green's function of the nanowire with a self-energy induced by the superconductor. From Eq. (<ref>), we can identify the retarded self-energy as (see also Appendix <ref>)

    Σ^R(E,k_y)=(1/Γ-1)(Δτ_x-E)-δμ τ_z.

with Γ and δμ as defined in Eq. (<ref>).

Before moving on, we pause to compare our result for the self-energy of a nanowire coupled to a finite-sized superconductor to the self-energy that has appeared extensively in the literature to describe proximitized nanowires beyond the weak coupling limit <cit.>. In these works, all based on the approach of integrating out the superconducting degrees of freedom, the superconductor is implicitly assumed to be infinitely large, with a nanowire coupled to the middle of the superconductor. In this geometry, one obtains the same self-energy as given in Eq. (<ref>), but with the vastly simplified effective parameters δμ=0 and Γ=(1+γ/Ω)^-1. We find that we recover this form for the self-energy by setting x_w=d/2 and taking the limit d→∞ in Eq. (<ref>) (the momentum dependence must also be neglected by setting φ=1). For maximum transparency in relating the current work to the previous ones, we show in Appendix <ref> how Eqs. (<ref>) and (<ref>) can be equivalently derived by integrating out the superconductor.





§ EXCITATION SPECTRUM
 
In this section, we analyze the excitation spectrum of our model in two simplified limiting cases. First, in Sec. <ref>, we consider the case when the nanowire is placed very close to the boundary of the superconductor, such that k_Fx_w≪1 (i.e., the distance between the nanowire and the edge of the system is much smaller than the Fermi wavelength of the superconductor λ_F). In Sec. <ref>, we consider the case when the nanowire is placed in the middle of the superconductor, x_w=d/2. Throughout this section, we assume that the width of the superconductor is much smaller than its coherence length, d≪ξ_s; equivalently, its level spacing is much larger than the gap, δ E_s≫Δ.




 §.§ Wire near edge of superconductor
 


  §.§.§ Analytical calculation of excitation gap
 
We first look to analytically determine the excitation gap of the semiconductor/superconductor system when the wire is placed near the edge of the superconductor (k_Fx_w≪1). In this limit, and taking E<Δ (Δ is the upper bound on the size of the excitation gap), the effective parameters of Eq. (<ref>) can be simplified to

    Γ=(1+2γ(k_Fx_w)^2/δ E_ssin^2(k_Fd))^-1, 
    δμ=2γ(k_Fx_w)[1-(k_Fx_w)(k_Fd)].

In Eq. (<ref>), we have additionally assumed that |sin(k_Fd)|≫Δ/δ E_s (recall that we are assuming Δ/δ E_s≪1). Therefore, Eq. (<ref>) breaks down when the thickness of the shell approaches k_Fd→π n (n∈ℤ^+). We have also neglected the momentum dependence of φ(k_y) by setting φ=1; this assumption is justified provided that k_y/k_F≪1/√(k_Fd).

Because the effective parameters of Eq. (<ref>) are not functions of energy or momentum, it is particularly simple to solve for the spectrum,

    E^2=Γ^2(k_y^2/2m_w-μ_eff)^2+Δ^2(1-Γ)^2,

where we define μ_eff=μ_w+δμ. [Remember, Eq. (<ref>) should be taken to describe the spectrum only for E<Δ.] If μ_eff>0, the spectrum of Eq. (<ref>) describes a superconductor with an induced gap of size

    E_g/Δ=1-(1+2γ(k_Fx_w)^2/δ E_ssin^2(k_Fd))^-1

which is opened around the effective Fermi momentum k_F,eff=√(2m_wμ_eff). We note that Eq. (<ref>) cannot be applied if the tunneling energy is made too large, such that k_F,eff/k_F≳1/√(k_Fd). In terms of energy scales, we find that Eq. (<ref>) breaks down when √(γ/δ E_s)≳√((m_s/m_w)/(k_Fx_w))≫1. If μ_eff<0, then Eq. (<ref>) describes the spectrum of an insulator with gap |μ_eff|. In this case, one must take into account the full momentum dependence φ(k_y) in order to calculate the gap and Eq. (<ref>) does not apply.

In the limit when the induced gap is small E_g≪Δ, it is necessarily given by

    E_g/Δ=2γ(k_Fx_w)^2/δ E_ssin^2(k_Fd)≪1.

This result has several important implications. First, assuming that the thickness of the shell is not finely tuned on the scale of the Fermi wavelength of the superconductor [i.e., sin^2(k_Fd)∼1], we see that the induced gap can be small even if the tunneling energy greatly exceeds the gap of the superconductor (γ≫Δ). This result is purely a finite-size effect and is due to the suppression of the gap by a factor Δ/δ E_s≪1. Second, we see that the gap is additionally suppressed by a factor (k_Fx_w)^2≪1, which is a direct consequence of the smallness of the superconducting wave function in the vicinity of the edge. Finally, we note that it is still possible to induce a sizable gap if the thickness of the shell is fine-tuned to the limit sin^2(k_Fd)≪(γ/δ E_s)(k_Fx_w)^2≪1; in this limit, we find from Eq. (<ref>) that E_g≫Δ and our original expansion breaks down. This leads to a resonance behavior, with sharp peaks in the induced gap when the resonance condition sin(k_Fd)=0 is satisfied. The width of the resonance peaks is estimated as x_w√(γ/δ E_s)≪λ_F.

Rearranging Eq. (<ref>), we can express the tunneling energy γ in terms of the experimentally observable quantities E_g and Δ (similarly to what is done in Refs. <cit.> for the case of a bulk superconductor). However, due to the presence of the quantities k_Fx_w and k_Fd, which would be impossible to determine experimentally, we can obtain only an order of magnitude estimate for γ for the case of a finite superconductor. Assuming that sin^2(k_Fd)∼1, we find

    γ_edge∼E_g/Δ-E_gδ E_s/(k_Fx_w)^2.

From Eq. (<ref>), it is clear that the lower bound on the tunneling strength needed to induce a sizable gap in the system (such that E_g∼Δ) is given by the level spacing δ E_s.











  §.§.§ Numerical calculation of spectrum
 

While we were able to solve for the excitation spectrum at energies E<Δ to determine the gap in certain limits [see Eq. (<ref>)], it is much more difficult to solve for the full spectrum. Because the full spectrum E(k_y) obeys a transcendental equation that cannot be solved analytically in general, we must resort to solving Eq. (<ref>) numerically.

In Fig. <ref>, we plot the excitation gap E_g as a function of superconductor width d. We calculate the gap numerically by computing the spectrum and finding the minimum of the lowest subband, allowing us to treat values of k_Fd for which Eq. (<ref>) breaks down [namely, for sin(k_Fd)→0 and μ_eff<0]. Overall, we find very good agreement between the numerical solution for the gap and the analytical form given in Eq. (<ref>). For weak tunneling [Fig. <ref>(a)], the gap is in general very small with very sharp resonance peaks around k_Fd=π n. As the tunneling is increased, the resonance peak is broadened and the size of the gap is generally shifted to larger values [Fig. <ref>(b)]. When γ∼δ E_s/(k_Fx_w)^2, the gap is always of the same order as that of the superconductor, E_g∼Δ [Fig. <ref>(c)].

To better understand the behavior of the gap as a function of γ, we plot the spectrum for various γ and fixed superconductor width (chosen to be off resonance) in Fig. <ref>. In the absence of tunneling [Fig. <ref>(a)], there is a large separation in energy between the band of the nanowire and the lowest subband of the superconductor (a consequence of the fact that δ E_s≫μ_w). As the tunneling strength is increased [Figs. <ref>(b) and <ref>(c)], the effective chemical potential of the nanowire μ_eff increases and the two lowest subbands move closer in energy; as a result, the nanowire can more efficiently couple to the superconductor and the proximity-induced gap increases. When γ∼δ E_s/(k_Fx_w)^2 [Fig. <ref>(d)], the tunneling is strong enough to overcome the large subband spacing of the superconductor. This creates significant overlap between the two lowest subbands of the system and a sizable excitation gap E_g∼Δ.














  §.§.§ Simple two-band model
 

In this section, we present a simple two-band model which can be used to better understand the “weak tunneling" limit γ≪δ E_s/(k_Fx_w)^2. In this limit, we can safely assume that the nanowire couples only to the lowest subband of the superconductor. Taking into account only the lowest superconducting subband, we can write down a simple tunneling Hamiltonian

    H=1/2∫dk_y/2π Ψ^†[  ξ_k    0   -t    0;    0 -ξ_k    0    t;   -t    0  ξ_n    Δ;    0    t    Δ -ξ_n;      ]Ψ,

where Ψ=(Ψ_w,Ψ_n)^T (Ψ_w describes states in the nanowire, while Ψ_n describes states in subband n of the superconductor) and t is a coupling between the two bands with dimensions of energy [note that this is not the same t which was introduced in Eq. (<ref>)]. Quantization of the superconducting bands is accounted for through ξ_n=k_y^2/2m_s-μ_n, with μ_n=δ E_s(k_Fd-π n) for n∈ℤ^+ [see Eq. (<ref>)].



The corresponding spectrum is given by

    2 E^2    = Δ^2 + ξ_k^2+  ξ_n ^2 +2t^2 
       ±√((Δ^2 - ξ_k^2+ξ_n^2)^2+4t^2[Δ^2+(ξ_k+ξ_n)^2]).

In general, |μ_n|≫μ_w,Δ,t and we can expand Eq. (<ref>) to give

    E=±√(ξ_k^2+t^4/μ_n^2).

In this case, the lower subband takes on a superconducting dispersion with a small induced gap E_g=t^2/μ_n≪Δ. The gap can only be enhanced when |μ_n|≲μ_w,Δ,t, which occurs only when a new superconducting subband becomes occupied, k_Fd≈π n. While we cannot solve analytically for the gap in this limit, we find by plotting the spectrum that the gap is approximately Δ.

We plot the spectrum Eq. (<ref>) for different superconductor widths d in Fig. <ref>. Away from resonance [Fig. <ref>(a)], the lowest subband has a superconducting dispersion with very small induced gap [Eq. (<ref>)]. As the resonance is approached [Fig. <ref>(b)], the lowest superconducting subband becomes available to more strongly couple to the nanowire band and the gap is enhanced. On resonance [Fig. <ref>(c)], overlap between the two subbands is maximal and the full gap Δ is opened. As a new subband in the superconductor becomes occupied and moves away from the nanowire band [Fig. <ref>(d)], the gap is again suppressed by the large subband spacing in the superconductor. By plotting the spectrum, we indeed see that the excitation gap is sharply peaked as a function of d around k_Fd=π n, consistent with the resonance behavior discussed in Sec. <ref><ref> and shown in Fig. <ref>.












 §.§ Wire in middle of superconductor
 

If the wire is placed in the middle of the superconductor, x_w=d/2, the effective parameters of Eq. (<ref>) for energies E<Δ can be simplified to

    Γ=(1+γ/δ E_s1/2cos^2(k_Fd/2))^-1, 
    δμ=γtan(k_Fd/2).

Again, we neglect the momentum dependence of the effective parameters by setting φ=1 and assume |sin(k_Fd)|≫Δ/δ E_s.

Since the parameters in Eq. (<ref>) are independent of energy and momentum, the spectrum is again given by Eq. (<ref>). Assuming that μ_eff>0, we find an excitation gap

    E_g/Δ=1-(1+γ/δ E_s1/2cos^2(k_Fd/2))^-1.

While the gap can still be small for γ≫Δ, there are two important differences when comparing to the case when the nanowire is at the edge of the superconductor [Eq. (<ref>)]. First, the gap is no longer suppressed by the factor (k_Fx_w)^2≪1 which originated from the smallness of the superconducting wave function near the edge. Instead, without any fine-tuning of the superconductor width [cos^2(k_Fd/2)∼1], a sizable gap can be induced for γ≳δ E_s. Second, the periodicity of the gap as a function of k_Fd is twice as large. In the weak tunneling limit γ≪δ E_s, resonance peaks are half as frequent and occur near k_Fd=π(2n+1) (n∈ℤ^+). The width of each resonance peak is larger than in the case of the nanowire at the edge of the superconductor, but it is still much smaller than the Fermi wavelength, λ_F√(γ/δ E_s)≪λ_F.







The excitation gap E_g is plotted as a function of d for several different tunneling strengths γ in Fig. <ref>. Similarly to Sec. <ref><ref>, we solve for the gap numerically by computing the full spectrum E(k_y) and finding the minimum. Again, all parameters are chosen the same as in Fig. <ref> (except for x_w=d/2). As previously discussed, the periodicity of the gap as a function of k_Fd is increased by a factor of two compared with the case when the nanowire is located at the edge of the superconductor. We see that the gap is maximized near k_Fd/π=2n+1 and is minimized near k_Fd/π=2n. This result can be inferred from the structure of the wave function corresponding to the lowest quantized subband of the superconductor in the absence of the nanowire. For k_Fd/π=2n+1, there is a superconducting subband available at low energies with which the nanowire can couple; because the wave function of this subband is extremal at x=d/2, the nanowire efficiently couples and a large gap is induced. For k_Fd/π=2n there is again a superconducting subband at low energies; however, the wave function of this subband has a node at x=d/2 and does not couple efficiently to the nanowire. In this case, a sufficiently large gap is opened only when the tunneling is very strong.


We also find that as the tunneling strength is increased, extended plateaus emerge as a function of k_Fd. To better understand this behavior, in Figs. <ref>(a)–<ref>(c) we plot the spectrum for different γ choosing k_Fd/π=35.75. As the tunneling strength is increased from γ=0 [Fig. <ref>(a)], a superconducting gap is induced on the band which originates in the nanowire. However, at the same time, the nanowire band gets depleted. For some critical tunneling strength, the nanowire band becomes depleted completely and enters an insulating phase. The minimum excitation gap is then given by the insulating gap in the nanowire at k_y=0 [Fig. <ref>(b)]. As the tunneling strength is increased further and the insulating gap on the nanowire band exceeds Δ, the minimum excitation gap is determined by the lowest occupied subband of the superconductor [Fig. <ref>(c)]. This behavior can be understood as follows. The depletion of the nanowire band can be inferred from δμ given in Eq. (<ref>). When the wire is in the middle of the superconductor, δμ<0 for precisely half of a period, including for k_Fd/π=35.75. In the strong tunneling limit, |δμ|≫μ_n in general and μ_eff<0 (i.e., the nanowire becomes insulating) also for half of a period (and, as shown in Fig. <ref>, the plateau extends over half of a period in the strong tunneling limit). We also find that the lowest subband of the superconductor (corresponding to n=36) remains almost completely unaffected by tunneling. This is consistent with our previous analysis and results from the fact that this subband cannot efficiently couple to the nanowire because the corresponding wave function has a node at x=d/2. Rather, the nanowire band couples most efficiently to the second-lowest subband (corresponding to n=35). While the presence of the nanowire modifies this subband somewhat in the vicinity of k_y=0, it has no effect on the gap of this subband at finite k_y. These findings are summarized in Fig. <ref>(d), where we plot the excitation gap as a function of tunneling strength for k_Fd/π=35.75.



Using our result for the gap [Eq. (<ref>)], we provide an order of magnitude estimate for the tunneling strength γ assuming that the superconductor width d is tuned outside of the plateau region and satisfies cos^2(k_Fd/2)∼1. In this case, we estimate

    γ_middle∼E_g/Δ-E_gδ E_s.

Compared with Eq. (<ref>), we find that a sizable gap (E_g∼Δ) can be induced with a much smaller tunneling strength when the wire is placed in the middle of the superconductor.

Finally, we compare three different cases by plotting the induced gap in the nanowire as a function of γ in Fig. <ref>. In Fig. <ref>(a), we show a direct comparison between the case of a nanowire located at the edge of the superconductor and the case of a nanowire located in the middle. We also contrast our result for a finite superconductor against that for a bulk superconductor by plotting the induced gap as a function of γ for the latter case in Fig. <ref>(b). For a bulk system, the induced gap obeys the equation <cit.>

    γ_bulk=E_g√(Δ+E_g/Δ-E_g).

We see that to open a sizable gap in a finite superconductor, a tunneling strength which is several orders of magnitude larger than in the bulk case is needed.
















§ TIGHT-BINDING MODEL
 

In this section, we check our analytical results by comparing them with a numerical tight-binding model for proximity-induced superconductivity <cit.> in the geometry shown in Fig. <ref>. The system is again assumed infinite in the y-direction, so that the Hamiltonian takes a block-diagonal form in momentum k_y, H=∑_k_y H_k_y.  The size of the superconductor in the x direction is N a_x (a_x,y are lattice constants), while the size of the nanowire is taken to be a single site. The Hamiltonian of the superconductor is given by

    H_s,k_y   = ∑_σ∑_i=1^N{ [μ_s- 2 t_0 cos (k_y a_y)] c_k_y,i,σ^† c_k_y,i,σ
       -(t_0c_k_y,i+1,σ^† c_k_y,i,σ -Δ c_k_y ,i ,↑^† c_-k_y ,i ,↓^† + H.c.)},

where c_k_y,i,σ destroys a state of momentum k_y and spin σ in the superconductor at site i, t_0 is the hopping amplitude, μ_s is the chemical potential (calculated from the bottom of the band), and Δ is the pairing potential. The Hamiltonian of the nanowire is given by

    H_w,k_y=∑_σ[μ_w-2t_wcos(k_ya_y)]b^†_k_y,σb_k_y,σ,

where b_k_y,σ destroys a state of momentum k_y and spin σ in the nanowire, μ_w is the chemical potential, and t_w is the hopping amplitude. The nanowire is coupled to the superconductor at site j,

    H_t,k_y=-t∑_σ(c^†_k_y,j,σb_k_y,σ+H.c.),

where t is the tunneling amplitude between the nanowire and superconductor. We assume that tunneling preserves both spin and momentum.

We consider two separate cases. First, the nanowire is placed at the end of the superconducting chain (j=1). Whereas analytically we were unable to place the nanowire strictly at the edge of the superconductor due to its vanishing width, we can do so in the tight-binding formulation (the nanowire has a finite width of one site). Second, the nanowire is placed in the middle of the superconductor (j=N/2). The results of our tight-binding calculation are shown in Fig. <ref>. We plot the excitation gap as a function of superconductor width choosing μ_s=0.1t_0, μ_w=10^-4t_0, Δ=10^-5t_0, and t_w=50t_0 (all parameters are chosen to coincide with those used previously in the analytical calculation). For Fig. <ref>(a), which corresponds to j=1, we find very good agreement with the analytics of Sec. <ref> and Fig. <ref>. We note that the resonance peaks in the curve corresponding to weak tunneling [black curve in Fig. <ref>(a)] are narrower than a single site and therefore do not appear in the numerics. For Fig. <ref>(b), which corresponds to j=N/2, we find very good agreement with the analytics of Sec. <ref> and Fig. <ref>. Notably, a sizable gap E_g∼Δ is seen only when the tunneling strength t exceeds the chemical potential of the superconductor.








§ TOPOLOGICAL CRITERION
 
To access the topological phase, we now assume that the nanowire has Rashba spin-orbit coupling and a Zeeman splitting that results from the application of an external magnetic field B parallel to the nanowire, corresponding to the typical setup for realizing topological superconductivity <cit.>. The Hamiltonian of the nanowire in this case is given by 

    H_w=1/2∫dk_y/2π Ψ^†(k_y)ℋ_w(k_y)Ψ(k_y),

where Ψ(k_y)=[ψ_↑(k_y),ψ_↓(k_y),ψ_↑^†(-k_y),ψ_↓^†(-k_y)]^T. The Hamiltonian density, which is a 4×4 matrix in Nambu ⊗ spin space, is given by

    ℋ_w(k_y)=ξ_kτ_z-α k_yσ_z-Δ_Zτ_zσ_x,

where α is the Rashba constant, Δ_Z=gμ_BB/2 is the Zeeman splitting (g is the g-factor of the nanowire and μ_B is the Bohr magneton), and σ_x,y,z are Pauli matrices acting in spin space. Following convention, we neglect the effect of the external magnetic field on the superconductor. Generalization of the solution of the BdG equation given in Eq. (<ref>) to the case where one additionally has to account for the spin degree of freedom is straightforward. The addition of spin-orbit coupling and Zeeman splitting simply modifies the retarded Green's function which enters the boundary conditions [Eqs. (<ref>)], G_0^R=(E-ℋ_w+i0^+)^-1. Solving the boundary conditions, we find that the self-energy given in Eq. (<ref>) still holds, with the simple replacement Δτ_x→-Δτ_yσ_y to account for the spin-singlet nature of the induced pairing <cit.>.


Given the self-energy, we find that the excitation spectrum is determined from the implicit equation

    E^2/Γ^2=Δ_Z^2+Δ^2(1/Γ-1)^2+(ξ_k-δμ)^2+α^2k_y^2 
       ±2√(Δ_Z^2Δ^2(1/Γ-1)^2+(ξ_k-δμ)^2(Δ_Z^2+α^2k_y^2)).

The critical Zeeman splitting needed to close the excitation gap at k_y=0, which we find by setting k_y=E=0 in Eq. (<ref>), is found to be

    ΓΔ_Z^c   =√(Γ^2(μ_w+δμ)^2+E_g^2),

where we replace Δ(1-Γ)=E_g, noting that this replacement is strictly valid only when δ E_s≫Δ. [The explicit forms of δμ(0,0) and Γ(0,0) are given in Eq. (<ref>).]
If the chemical potential shift δμ can be compensated by tuning the chemical potential of the wire (such that μ_w+δμ=0), then the critical field strength is Δ_Z^c∼ E_g/Γ (this is a similar criterion used, for example, to analyze the data of Ref. <cit.>; note that the critical Zeeman splitting needed is larger than the induced gap E_g). If, however, the chemical potential shift is made too large (|δμ|≫μ_w), the critical Zeeman splitting is determined solely by this shift, Δ_Z^c∼|δμ|, and the finite size of the superconductor pushes the topological threshold to significantly higher magnetic field strength. We will provide numerical estimates in the next section to argue that the latter case is more relevant to thin superconducting shells that have a large level spacing δ E_s≫Δ.







§ RELATION TO EXPERIMENTS WITH EPITAXIAL SUPERCONDUCTING SHELLS
 

In this section, we argue that the theoretical model that we have considered to this point is applicable to recent experiments studying InAs or InSb nanowires strongly coupled to thin superconducting Al shells (see Fig. <ref>) <cit.>. We also provide an order of magnitude estimate for the level spacing of the shell, which determines the critical field strength needed to reach the topological phase in such a setup.



First, in order to treat the nanowire as one-dimensional, we assume that the wave function is uniform across the entire cross-section of the nanowire (which spans roughly ∼100 nm). Second, because we have replaced the nanowire cross-section by a single point, we must also neglect the width of the superconducting shell, W∼100 nm (see Fig. <ref>). To justify this assumption, let us define a phenomenological tunneling strength that originates from the coupling between the nanowire and superconductor along this dimension [call it t(W)]. Such a tunneling should be given by the product of the nanowire and superconducting wave functions, integrated over the width,

    t(W)∼∫_0^Wdz ψ_w^*(z)ψ_s(z).

Here, we denote the dimension along the width of the shell by z, imagining for simplicity that the shell cross-section is rectangular rather than kinked. The superconducting wave function is quantized as ψ_s∼sin(k_sz)/√(W), where k_s=nπ/W and 1/√(W) is a normalization factor. The wave function of the nanowire is uniform, but it also must be normalized, ψ_w∼1/√(W_w). Because the normalization factor of the nanowire wave function should scale with the width of the shell, W_w∼ W, the effective tunneling strength is given by

    t(W)∼1/W∫_0^Wdz sin(k_sz)∼1.

Because the tunneling does not scale with W, this additional dimension is unimportant and can be neglected.

We now turn our attention toward applying our results to make qualitative predictions about the experimental setup. The relevant geometrical parameter (which corresponds to d) is the thickness of the superconducting shell, d∼10 nm. For shells of this thickness, we estimate a level spacing δ E_s=ħ v_F/d∼10 meV, where we take v_F∼10^6 m/s for Al.

Given the level spacing of the shell, we can also estimate the tunneling strength needed to induce a gap E_g∼Δ. Because the experimental systems universally exhibit sizable induced gaps (and therefore it is safe to assume that the shell thicknesses are not fine-tuned to a resonance point), we use Eq. (<ref>) to estimate

    γ_edge≳100 meV,

where we take (k_Fx_w)^2∼0.1.

Therefore, the chemical potential shift induced by tunneling is estimated from Eq. (<ref>) as

    |δμ(0,0)|∼γ(k_Fx_w)≳30 meV.

Given that the characteristic energy scale of the nanowire is the spin-orbit energy (E_so=m_wα^2/2ħ^2), which takes typical values E_so∼0.1-1 meV in semiconducting nanowires <cit.>, the topological phase transition is controlled entirely by tunneling, Δ_Z^c∼|δμ(0,0)|≳30 meV. Such a large Zeeman energy corresponds to a critical field strength of B_c≳60 T for Al/InAs (g_InAs∼20) and B_c≳30 T for Al/InSb (g_InSb∼40). In either case, the magnetic field threshold needed to reach the topological phase greatly exceeds the field strength at which superconductivity in Al is destroyed (which occurs for B∼2 T).

As discussed previously, in order to negate the effect of the tunneling-induced shift in the chemical potential, one must tune to μ_w=-δμ(0,0). However, this is difficult in practice because |δμ(0,0)|≫ E_so. If δμ(0,0)>0, it is not possible to properly tune the chemical potential without completely depleting the semiconducting band; if δμ(0,0)<0, the nanowire would have to be gated outside of the regime for which it is semiconducting.

We note that there are several aspects of the experimental setup which are not accounted for by our simple model. For example, our model does not include an additional renormalization of the nanowire g-factor by the superconductor beyond that which is contained in Eq. (<ref>). This is because the g-factor in the superconductor was taken to be g_s=0 [and therefore no Zeeman term enters the self-energy of Eq. (<ref>)]. However, the additional renormalization should be small, as we estimate the effective g-factor of the wire as Γ g+(1-Γ)g_s≈Γ g given that g≫ g_s and Γ∼(1-Γ)∼1 <cit.>. We also do not account for any orbital effects in the nanowire, as the nanowire was taken to have zero width. In the experimental setup, however, the nanowire has a diameter W_w∼100 nm; compared with a typical cyclotron radius for electrons in the wire r_c=m_wv_Fw/eB∼10 nm (evaluated for a field strength B∼1 T and Fermi velocity v_Fw∼10^5 m/s), orbital effects can be non-negligible <cit.>. Additionally, we do not consider the superconductor to be disordered. It has been shown that disorder in a bulk superconductor can be detrimental to the proximity effect <cit.>; however, this does not seem to be an issue experimentally, as sizable hard proximity-induced gaps are universally observed <cit.> along with ballistic transport in the absence of a magnetic field <cit.>. Relatedly, given the fact that applied magnetic fields are small, we do not determine Δ(x) self-consistently as done, for example, for Shiba states, where the exchange interaction (effective Zeeman field) is comparable to the Fermi energy <cit.>. Finally, we do not account for the fact that there may be multiple subbands in the nanowire which contribute to transport. While this possibility seems to be excluded by the fact that typical nanowires exhibit a quantized conductance of 2e^2/h in the ballistic transport limit over a wide range of gate voltages <cit.>, it could be that with the introduction of the large tunneling energy scale, higher subbands can become important. The intersubband spacing in an InSb nanowire was estimated in the absence of a magnetic field and a superconducting shell to be ∼20 meV in Ref. <cit.>, so we cannot rule out the possibility that a |δμ(0,0)|∼30 meV simply places the chemical potential into a higher subband. However, this possibility requires a more in-depth theoretical treatment that cannot be captured by a single-band model.




§ CONCLUSIONS
 
We have studied the proximity effect in a semiconducting nanowire strongly coupled to a thin superconducting shell with thickness d. We have shown that finite-size effects become detrimental to the induced superconductivity when the level spacing of the shell δ E_s∼ħ v_F/d exceeds its gap, δ E_s≫Δ. In this limit, a large tunneling strength γ≳δ E_s is needed to overcome the level spacing to induce a gap in the nanowire (we estimate γ≳100 meV for typical experimental setups with d∼10 nm). In turn, this large tunneling energy induces a very large shift in the effective chemical potential of the nanowire (δμ≳30 meV) which would be difficult to compensate by gating.

In order to overcome the detrimental finite-size effects, the thickness of the superconducting shell should be made larger than its coherence length, d≫ξ_s. In this limit, the level spacing and chemical potential shift become negligible (δ E_s,δμ≪Δ), and a sizable proximity gap can be induced with a much smaller tunneling energy (γ∼Δ). However, this requirement may prove problematic when using Al, which has a very long coherence length ξ_s∼1 μm, to induce superconductivity. It would therefore be beneficial to choose a superconductor with shorter coherence length (for example, Nb has ξ_s∼10 nm).















This work was supported by the Swiss National Science Foundation and the NCCR QSIT. 







§ SOLVABILITY CONDITION AND SELF-ENERGY
 
In this appendix, we show how to arrive at the solvability condition given in Eq. (<ref>) of the main text. The boundary conditions that must be imposed on the wave function of Eq. (<ref>) are given in Eq. (<ref>). These boundary conditions can be significantly simplified by assuming that μ_s≫Δ,E. In this limit, we can make a semiclassical approximation k_±=(k_F^2-k_y^2±2m_siΩ)^1/2≈ k_Fφ± iΩ/(v_Fφ), where φ=(1-k_y^2/k_F^2)^1/2 (0<φ≤1). The boundary conditions in Eq. (<ref>) can be written in matrix form as Mc=0, where c=(c_1,c_2,c_3,c_4)^T and


    M   =([ u_0ϕcos(k_+x_w) v_0ϕcos(k_-x_w); v_0ϕcos(k_+x_w) u_0ϕcos(k_-x_w); -u_0sin(k_+x_w) -v_0sin(k_-x_w); -v_0sin(k_+x_w) -u_0sin(k_-x_w) ]. ⋯
       ⋯.[ u_0(φcos[k_+(d-x_w)]+2γ/E-ξ_ksin[k_+(d-x_w)]) v_0(φcos[k_-(d-x_w)]+2γ/E-ξ_ksin[k_-(d-x_w)]); v_0(φcos[k_+(d-x_w)]-2γ/E+ξ_ksin[k_+(d-x_w)])  u_0(φcos[k_-(d-x_w]-2γ/E+ξ_ksin[k_-(d-x_w)]);                            u_0sin[k_+(d-x_w)]                            v_0sin[k_-(d-x_w)];                            v_0sin[k_+(d-x_w)]                            u_0sin[k_-(d-x_w)] ])


In Eq. (<ref>), we have approximated k_±=k_Fφ outside of the trigonometric functions while keeping k_±=k_Fφ± iΩ/(v_Fφ) inside. Taking the determinant of the matrix in Eq. (<ref>), we find a solvability condition given by

    0   =Ωφ^2sin(k_+d)sin(k_-d){E^2-ξ_k^2-γ^2/φ^2β_+β_- 
       +γ/φ[ξ_k(β_++β_-)+E^2/iΩ(β_+-β_-)]},

where we define β_±={cos[k_±(d-2x_w)]-cos(k_± d)}/sin(k_± d). Dividing Eq. (<ref>) through by the common factor Ωφ^2sin(k_+d)sin(k_-d) and rearranging, we obtain

    0   =E^2(1+γ/Ωφ Im(β_+))^2-Δ^2γ^2/Ω^2φ^2[Im(β_+)]^2 
       -(ξ_k-γ/φ Re(β_+))^2

Defining the quantities Γ={1+γ Im(β_+)/(Ωφ)}^-1 and δμ=γ Re(β_+)/φ, we arrive at the solvability condition presented in Eq. (<ref>). Substituting k_±=k_Fφ± iΩ/(v_Fφ) into the expressions for Γ and δμ, we arrive at the definitions presented in Eq. (<ref>).

We also note that the solvability condition of Eq. (<ref>) can be expressed as [G_w^R(E,k_y)]^-1=0, where

    (G_w^R)^-1=([ E/Γ-ξ_k+δμ  -Δ(1/Γ-1);  -Δ(1/Γ-1) E/Γ+ξ_k-δμ ])

Noting that the bare retarded Green's function of the nanowire is given by (G_0^R)^-1=E-ξ_kτ_z+i0^+, the full Green's function can be written as (G_w^R)^-1=(G_0^R)^-1-Σ^R with self-energy

    Σ^R=([ E(1-1/Γ)-δμ    Δ(1/Γ-1);    Δ(1/Γ-1) E(1-1/Γ)+δμ ]),

as given in Eq. (<ref>).




§ INTEGRATING OUT SUPERCONDUCTOR
 
In this section, we show how to alternatively derive Eqs. (<ref>) and (<ref>) by integrating out the superconductor. For completeness, we first go through the steps of performing the integration. We start with the same model as considered in the main text, expressed in terms of the Euclidean action. The action of the nanowire is given by

    S_NW=1/2∫dω/2π∫dk_y/2π Ψ_w^†(G_w^0)^-1Ψ_w,

where (G_w^0)^-1=iω-ξ_kτ_z is the inverse Matsubara Green's function of the nanowire in the absence of tunneling (ω is a Matsubara frequency) and Ψ_w=[ψ_↑(k_y,ω),ψ_↓^†(-k_y,-ω)]^T is a Heisenberg spinor field describing states in the nanowire. The action of the superconductor is given by

    S_BCS=1/2∫dω/2π∫dk_y/2π∫_0^ddx Ψ_s^†(iω-ℋ_BCS)Ψ_s,

where ℋ_BCS is as defined below Eq. (<ref>) and Ψ_s=[η_↑(x,k_y,ω),η^†_↓(x,-k_y,-ω)] is a spinor field describing states in the superconductor. The tunneling action, which couples the nanowire to the superconductor at x=x_w, is taken to be

    S_t=-t/2∫dω/2π∫dk_y/2π∫_0^ddx[Ψ_w^†τ_zΨ_sδ(x-x_w)+H.c.].

The path integral representation of the partition function is then given by

    𝒵=∫ D[Ψ_w^†,Ψ_w]∫ D[Ψ_s^†,Ψ_s]e^-S_w-S_BCS-S_t.

In the exponential, we rewrite


    S_BCS+S_t   =1/2∫dω/2π∫dk_y/2π{∫_0^ddx[Ψ_s^†-tΨ_w^†τ_zG_s^0(x_w,x)](iω-ℋ_BCS)[Ψ_s-tG_s^0(x,x_w)τ_zΨ_w]-t^2Ψ_w^†τ_z G_s^0(x_w,x_w)τ_zΨ_w}.


In Eq. (<ref>), we introduce a function G_s^0(x,x') that must satisfy (iω-ℋ_BCS)G_s^0(x,x')=G_s^0(x',x)(iω-ℋ_BCS)=δ(x-x'); i.e. G_s^0(x,x') corresponds to the Green's function of the superconductor in the absence of tunneling. Evaluating the Gaussian path integral over superconducting fermions, we obtain an effective action describing the nanowire given by

    S_eff=1/2∫dω/2π∫dk_y/2πΨ^†_w[(G_w^0)^-1-Σ]Ψ_w,

with the self-energy given by

    Σ=t^2τ_zG_s^0(x_w,x_w)τ_z.


To explicitly evaluate the self-energy, we must choose the appropriate bare Green's function G_s^0(x,x') for the geometry under consideration. For our purposes, we evaluate the self-energy using the Green's function of a finite-sized superconductor satisfying vanishing boundary conditions at x=0 and x=d. The bare Green's function must satisfy the equation

    [iω+(∂_x^2/2m_s-k_y^2/2m_s+μ_s)τ_z-Δτ_x]G_s^0(x,x')=δ(x-x').

The solution to Eq. (<ref>) can be written as the sum of a homogeneous solution G_h(x,x') and a particular solution G_p(x-x'),

    G_s^0(x,x')=G_h(x,x')+G_p(x-x'),

with the particular solution corresponding to the bulk superconducting Green's function. We determine the bulk Green's function in real space by Fourier transformation. Defining ξ_ks=(k_x^2+k_y^2)/2m_s-μ_s, we have

    G_p(x-x')   =-∫dk_x/2πiω+ξ_ksτ_z+Δτ_x/Δ^2+ξ_ks^2+ω^2e^ik_x(x-x')
       =-1/v_FΩφ[(iω+Δτ_x)cos(ζ|x-x'|) 
       -Ωτ_zsin(ζ|x-x'|)]e^-χ|x-x'|,

where, as we have done throughout, we replace k_±=ζ=k_Fφ outside of the exponentials while keeping k_±=ζ± iχ=k_Fφ± iΩ/(v_Fφ) inside of the exponentials (in Matsubara frequency space, Ω=√(Δ^2+ω^2)). The homogeneous solution is given by

    G_h(x,x')   =(iω+Δτ_x+iΩτ_z)[c_1e^ik_+x+c_2e^-ik_+x] 
       +(iω+Δτ_x-iΩτ_z)[c_3e^ik_-x+c_4e^-ik_-x].

The unknown coefficients, which are functions of the coordinate x', are determined by imposing the boundary conditions G_s^0(0,x')=G_s^0(d,x')=0. Solving the boundary conditions gives

    c_1(x')   =sin[k_+(d-x')]/2v_FΩφsin(k_+d), 
    
    c_2(x')   =1/2v_FΩφ[i+(k_+d)]sin(k_+x'), 
    
    c_3(x')   =1/2v_FΩφ[-i+(k_-d)]sin(k_-x'), 
    
    c_4(x')   =sin[k_-(d-x')]/2v_FΩφsin(k_-d).

Substituting Eqs. (<ref>)-(<ref>) into Eq. (<ref>) and setting x=x'=x_w, we find the bare Green's function


    G_s^0(x_w,x_w)   =-1/v_FΩφ[cosh(2χ d)-cos(2ζ d)]{(iω+Δτ_x){sinh(2χ d)-cos(2kζ x_w)sinh[2χ(d-x_w)] 
       -cos[2ζ(d-x_w)]sinh(2χ x_w)}-Ωτ_z{sin(2ζ d)-sin(2ζ x_w)cosh[2χ(d-x_w)]-sin[2ζ(d-x_w)]cosh(2χ x_w)}}.


Substituting Eq. (<ref>) into Eq. (<ref>) and defining Γ and δμ as in Eq. (<ref>), we obtain a self-energy given by (recall γ=t^2/v_F)

    Σ=(Δτ_x-iω)(1/Γ-1)-δμ τ_z.

After analytic continuation, we reproduce the retarded self-energy given in Eq. (<ref>).

We note that choosing the bare Green's function to be translationally invariant and equal to the bulk superconducting Green's function, as is typically done to describe proximitized nanowires, corresponds to the geometry of a nanowire coupled to the middle of an infinitely large superconductor. In this case, the bare Green's function is simply G_s^0(x_w,x_w)=G_p(0), yielding a self-energy

    Σ_bulk=-γ(iω-Δτ_x)/φ√(Δ^2+ω^2).















