1. Introduction

Majorana fermions are fermions that are identical to their antiparticles^[1]. In condensed matter physics, they usually appear as quasiparticle excitations and play an important role in both topological quantum computation and condensed matter theory^[2-4]. In recent years, topological superconductors and superfluid systems have attracted great interest because of the possible existence of Majorana bound states (MBSs)^[5]. Research on MBSs has long been centered on electronic systems, such as iron-based superconductors^[6], topological insulators^[7], and superconductor heterostructures^[8]. Owing to their material diversity, relatively high critical temperatures, and versatile experimental techniques, these systems suggest the possibility of observing and partially controlling MBSs.

The phenomenon of topological physics is not exclusive to electronic systems; it is also realized in systems like superfluid ³He-B. A common feature of unconventional superconductors and superfluids is the presence of low-energy midgap states near surfaces or interfaces^[9-11]. Superfluid ³He, the first experimentally established p-wave superfluid^[12], exhibits multiple distinct superfluid phases stabilized under different pressure, temperature, and magnetic fields. This makes ³He an ideal candidate system for exploring the intrinsic properties of topological bound states. Theoretically, in the B phase of superfluid ³He, the p-wave order parameter allows Andreev reflection at the surface to induce topologically protected zero energy bound states^[13]. The wave functions of these states are localized within a nanometer scale region near the surface, and these subgap surface states are predicted to be MBSs^[14,15]. Although studies of MBSs in ³He-B are relatively limited, its unique physical properties provide irreplaceable advantages for investigating Majorana fermions. Recent experiments by Autti et al.^[16] have directly observed a two-dimensional superfluid layer of ³He-B at the container boundaries, distinct from the bulk phase. In this confined geometry, quasiparticles are trapped in a quantum well due to the suppression of the energy gap. Theoretical predictions indicate that such a two-dimensional system hosts Majorana zero modes.

Several experimental studies have reported efforts to detect Majorana fermions in ³He-B. Experimentally, the surface acoustic impedance can be measured in ³He-B^[17]. The measured surface impedance can be compared with theoretical results to support the presence of MBSs. Jiang et al.^[18] measured the effect of the surface specular reflection rate on the surface-state gap and quasiparticle transport in superfluid ³He-B, confirming the sensitivity of surface-bound-state transport to the specular reflectivity. Okuda et al.^[19] used acoustic spectroscopy to measure the surface acoustic impedance of ³He-B. Their results showed that the surface bound states have features consistent with Majorana excitations. They also reported detailed information on the dispersion relation of these states; a smoother surface produced a dispersion relation that more closely approached the ideal Majorana cone. At low temperatures, ³He-B exhibits enhanced heat capacity^[20], which has been attributed to contributions from MBSs. Murakawa et al.^[21] further confirmed that Majorana excitations are more likely to form on smooth surfaces through impedance measurements, while Ikegami et al.^[22] demonstrated that the free surface of ³He-B behaves as a specular reflecting surface by studying the mobility of Wigner crystals formed on it. These results motivate the use of smooth ³He-B surfaces as ideal platforms for studying MBSs. Thus, most previous studies have focused on the energy gap of the bound states on smooth surfaces^[23].

Despite these advances, studies of surface bound states in ³He remain incomplete. A clear experimental probe to verify their Majorana nature is still missing. Compared with other probes of Majorana surface states, such as acoustic impedance and thermodynamic measurements, the electron-bubble probe provides several unique advantages. It acts as a mobile and tunable local scatterer, allows depth-selective coupling to surface bound states via an external electric field, and yields direct transport observables such as mobility. This makes it a sensitive and minimally invasive probe for resolving surface-state scattering properties. Recently, Carvalho et al.^[24] studied quasiparticle scattering in ³He-A, focusing on topological defects such as radial disgyrations and symmetric vortices. Ikegami et al.^[25] employed impurities as local probes to investigate the Majorana characteristics of surface bound states on the free surface of ³He-B, but the detailed mechanisms of quasiparticle–ion interactions were not thoroughly explored.

To address these issues, we focus on impurity scattering at the surface of ³He-B, aiming to determine whether the surface bound states exhibit the features of Majorana zero modes. The study is based on the quasiclassical Green’s function method combined with quantum scattering theory, which describes the collective behavior of low-energy quasiparticles and the spatial dependence of the surface bound states. Specifically, we construct a scattering potential model for impurities near the ³He surface and solve the Bogoliubov–de Gennes (BdG) equations for p-wave superfluidity to obtain the spectral characteristics. The local density of states (LDOS) is then calculated using the quasiclassical Green’s function method. In addition, we examine how impurity scattering affects the transport properties of the surface bound states.

This study provides a unified description for understanding the dynamical properties of surface bound states in superfluid ³He-B, and demonstrates that external electric and magnetic fields can effectively manipulate the energy spectrum and scattering characteristics of surface Majorana fermions.

2. Theoretical Model

2.1 Majorana fermions formed at the surface of ³He-B

We study the motion of charged particles in superfluid ³He-B and their scattering with Majorana fermions. This approach allows us to probe properties of the superfluid and clarify the mechanisms of these scattering processes. Both negative and positive ions can serve as effective probes. A negative ion, or electron bubble, is a small spherical cavity created by injecting an electron into liquid helium. The cavity is formed due to the strong repulsive interaction between the injected electron and surrounding ³He atoms, which arises from the Pauli exclusion principle. The resulting electron bubble has a radius of approximately 1-2 nm^[26], depending on the externally applied pressure^[27]. The positive ion, often referred to as a snowball, is a cluster of helium atoms surrounding a positively charged center^[28] The snowball ion has a radius of approximately 6A under zero pressure conditions^[29,30], and its effective mass is roughly one-tenth of that of an electron bubble under low-pressure conditions. The electron bubble serves as an effective probe because of its large effective mass, which is about 100-300 times the mass of a ³He atom at zero pressure, and its dynamics are dominated by elastic scattering. Therefore, in this study, we choose the electron bubble as the probe to investigate quasiparticle dynamics in ³He-B.

An electron bubble is injected into ³He-B by applying an external electric field perpendicular to the surface of ³He-B, and its penetration depth is denoted by z. Near the surface, the coherence length ξ of the surface bound states is approximately 80 nm, while the size of the electron bubble is much smaller than the coherence length. At this stage, the electron bubble acts merely as an impurity, with no significant distortion of either the superfluid gap or the superflow. The drag force is dominated by ordinary scattering^[31]. When an additional electric field parallel to the surface is applied, the electron bubble begins to move laterally, as shown in Figure 1a,b. The bubble collides with Majorana fermions excited near the surface. In the ideal case of a specular surface, these Majorana fermions are reflected specularly and scattered at the same angle as their angle of incidence, as shown in Figure 1c.

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Figure 1. (a) Schematic of the scattering model, the white sphere represents an electron bubble, which moves in the horizontal direction. The coherence length of the surface bound states is 80 nm; (b) The electron bubble moves laterally along the surface and collides with surface Majorana fermions, p and p′ denote the momenta of the Majorana fermions before and after the collision, respectively. During the collision, momentum p - p′ is transferred to the electron bubble; (c) Elastic scattering between Majorana fermions and a moving electron bubble, leading to specular reflection; (d) The trapping depth of the electron bubble in the potential well under applied electric fields of 2 kv/m (red line), 5 kV/m (blue line), and 10 kV/m (green line), the minima of the potential energy are indicated by arrows.

In typical mK ³He experiments, electron bubble transport measurements are performed in ultra-low-vibration cryogenic platforms, such as dilution refrigerators, where mechanical noise and heat leakage are carefully minimized. The present theoretical model assumes an ideal, well-stabilized environment and sufficiently low temperatures. In practice, residual mechanical vibration, surface fluctuations, and small heat flux may slightly modify the local quasiparticle distribution and surface profile, which could introduce higher-order corrections to the bubble dynamics. However, these effects mainly influence quantitative details and do not change the leading-order scattering mechanisms considered here. Therefore, they are neglected within the present quasiclassical transport framework.

The distance z between the electron bubble and the free surface can be continuously and reproducibly tuned via the vertical electric field $E_{\perp }$. Following the resonance method established by Poitrenaud et al.^[32], the electron bubble is trapped in an anharmonic potential well formed by the image force and an applied vertical electrostatic field $E_{\perp }$. The combined force creates a unique potential minimum below the surface, where the electron bubble stably resides. The potential is given by:

(1)$U\left(z\right)=\frac{A}{z}+e{E}_{\perp }z$

Here A = e²(ε - 1)/4ε(ε + 1), ε is the relative dielectric constant of liquid ³He, and z represents the distance from the electron bubble to the free surface. As indicated by Eq. (1), its depth z can be tuned by varying $E_{\perp }$ through the equilibrium position. Thus, varying the vertical electric field precisely controls the position of the minimum, enabling tunable control of the distance. Figure 1d illustrates the changes in the depth z of bubble. At ultralow temperatures, the thermal motion of the electron bubble is strongly suppressed, and it oscillates harmonically about the potential minimum with angular frequency ω. Precise calibration of z is achieved by measuring the harmonic oscillation frequency ω of the electron bubble at the bottom of the well. With a fixed radio frequency ω, the vertical electric field $E_{\perp }$ is swept. When the frequency matches the eigenfrequency ω/2π of the electron bubble, resonant absorption occurs, producing a resonance peak. The corresponding field satisfies ${\omega }^2=2{\left(e{E}_{\perp }\right)}^{3/2}/m^{\ast }{A}^{1/2}$, from which z can be extracted using the known values of the effective mass of the electron bubble m^*. Moreover, the anharmonicity of the potential well induces a shift in the resonance frequency. Experimentally, this can be corrected in situ by monitoring the resonance lineshape and harmonic signals, further improving the calibration accuracy of z.

Under ultralow temperatures and appropriate electric fields, the trapping lifetime of the electron bubble is much longer than the experimental scan time, and $\omega \tau \gg 1$. This ensures the reproducibility of z, with no position drift due to ion escape or thermal diffusion.

In ³He-B, the fermionic excitations are described by the BdG equation:

(2)$\begin{array}{c}\hat{H}\Psi \left(r\right)=\left(\begin{array}{cc}{\hat{p}}^2/2m-E_F& \left(\Delta /k_F\right)\sigma \cdot \hat{p}\\ \left(\Delta /k_F\right)\sigma \cdot \hat{p}& -{\hat{p}}^2/2m+E_F\end{array}\right)\Psi \left(r\right)=E\Psi \left(r\right)\\ \hat{H}\Psi \left(r\right)=\left(\begin{array}{cc}{\hat{p}}^2/2m-E_F& \left(\Delta /k_F\right)\sigma \cdot \hat{p}\\ \left(\Delta /k_F\right)\sigma \cdot \hat{p}& -{\hat{p}}^2/2m+E_F\end{array}\right)\Psi \left(r\right)=E\Psi \left(r\right)\end{array}$

where E_F denotes the Fermi energy, and k_F is the Fermi wave vector. The momentum operator $\hat{p}=-iħ\nabla$, and ħ is reduced Planck constant. The four-component Bogoliubov–Nambu spinor wave function ψ(r) is defined as a coherent superposition of particle and hole components, ψ(r) = (u_↑(r), u_↓(r), v_↑(r), v_↓(r))^T, where r = (x,y,z).

The Hamiltonian $\hat{H}$ can be divided into components parallel and perpendicular to the surface:

(3)$\hat{H}\left(k_{\Vert }\right)=\left(\begin{array}{cc}0& \left(\Delta /k_F\right)k_{\Vert }\cdot \sigma \\ \left(\Delta /k_F\right)k_{\Vert }\cdot \sigma & 0\end{array}\right)$

(4)$\hat{H}\left(k_{\perp }\right)=k_{\perp }\left(\begin{array}{cc}\left(1/m\right)(-i{\partial }_z)& \left(\Delta /k_F\right){\sigma }_z\\ \left(\Delta /k_F\right){\sigma }_z& -\left(1/m\right)(-i{\partial }_z)\end{array}\right)$

where k = k_F(cosϕsinθ,sinϕsinθ,cosθ), $k_{\Vert }=(k_x,k_y)$, $k_{\perp }=\sqrt{k_F^2-{\left|k_{\Vert }\right|}^2}$. The polar angle θ is defined as the angle between the incident Majorana fermions’ momentum vector k and the z-axis, while the azimuthal angle ϕ represents the angle between the projection of k onto the x - y plane and the reference axis for the azimuthal direction. The free surface is located at z = 0, and the region z > 0 corresponds to the superfluid ³He-B. A specular reflection boundary condition is imposed at the surface. When a Majorana fermion moves along the z-axis, the surface reverses its momentum component k_z to -k_z. Since the B phase is a p-wave superfluid, the component of the pair potential along the z-axis is suppressed at the surface, while the parallel momentum components (k_x,k_y) remain unchanged under specular reflection. Surface scattering suppresses the perpendicular component of the order parameter^[33]; consequently, the parallel component ${\Delta }_{\Vert }$ can be preserved as its bulk value ∆, whereas the perpendicular component ${\Delta }_{\perp }$ must be reduced or even vanish near the surface. Under the Andreev approximation, the order parameter can be simplified as: ${\Delta }_{\perp }\left(z\right)=\Delta \tanh \left(z/\xi \right),{\Delta }_{\Vert }=\Delta$. Here, the coherence length ξ = ħv_F/2∆.

We impose the boundary condition that the surface wave function ψ(r) = 0 at z = 0 and that it decays exponentially inside the ³He-B region below the surface. The application of the boundary conditions yields the wave function and its corresponding energy spectrum. The resulting wave function, which exhibits spatial localization and decays exponentially as sech(z/2ξ) along the surface normal, satisfies the Majorana relation u = -iv. This eigenfunction is given by:

(5)${\Psi }_{\pm }\left(r\right)=N\mathrm{sech}\left(\frac{z}{2\xi }\right)e^{i{k}_{\Vert }\cdot r}(e^{-i\varphi /2}{\Phi }_{\uparrow }\mp e^{i\varphi /2}{\Phi }_{\downarrow })$

where

(6)${\Phi }_{\uparrow }\equiv \frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 0\\ 0\\ i\end{array}\right),{\Phi }_{\downarrow }\equiv \frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ i\\ -1\\ 0\end{array}\right)$

Here, N is the normalization constant. Near the surface, sech(z/2ξ) can be approximated as 1. The normalized wave function for each spin component takes the form as follows.

(7)$\begin{array}{cc}& |{\Psi }_{k,\uparrow }^{\pm }⟩=e^{-i\varphi /2}|{\Phi }_{\uparrow k}⟩=\frac{1}{\sqrt{2}}{e}^{-i\varphi /2}(1,0,0,i{)}^T|k⟩\\ & |{\Psi }_{k,\downarrow }^{\pm }⟩=\mp e^{i\varphi /2}|{\Phi }_{\downarrow k}⟩=\mp \frac{1}{\sqrt{2}}{e}^{i\varphi /2}(0,i,-1,0{)}^T|k⟩\end{array}$

$|{\Psi }_k^{\pm }⟩$ corresponds to positive and negative eigenvalues, with spin σ = ↑↓. The Hamiltonian has a conelike linear dispersion relation as follows.

(8)$E=\left(\Delta /k_F\right)\left|k_{\Vert }\right|$

As shown in Figure 2, the linear dispersion relation of the surface bound states constitutes a characteristic signature of Majorana fermions. In the topological superfluid ³He-B, the surface Majorana fermions are topologically protected and exhibit a massless Dirac-like linear dispersion relation, analogous to the Dirac cone observed in graphene, but originating from the intrinsic order of the superfluid. The dispersion relation of these Majorana modes preserves time reversal symmetry, leading to a linear energy dependence of the density of states at low energies. The topological order of the superfluid in ³He-B is maintained by the topology of the bulk pairing potential, while the surface bound states, protected by a nonzero topological invariant, retain their gapless linear dispersion relation.

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Figure 2. The dispersion relation of surface states in superfluid ³He-B shows symmetric quasi-electron and quasi-hole branches, the colorbar indicates the energy magnitude.

The Green’s function enables the calculation of the density of states and the formulation of the scattering T-matrix. Applying analytic continuation to all components and combining contributions from both continuum states and bound-state poles yields the 4 × 4 matrix in Nambu spin space. The final analytical expression of the retarded Green’s function can thus be written in matrix form as^[34]:

(9)$g^R(\theta ,E,z)=-i\pi \left(\begin{array}{cccc}{g}_0& -g_{\Vert }{e}^{-i\varphi }& -f_{\Vert }{e}^{-i\varphi }& f_z\\ g_{\Vert }{e}^{i\varphi }& g_0& f_z& f_{\Vert }{e}^{i\varphi }\\ -f_{\Vert }{e}^{i\varphi }& f_z& g_0& g_{\Vert }{e}^{i\varphi }\\ f_z& f_{\Vert }{e}^{-i\varphi }& -g_{\Vert }{e}^{-i\varphi }& g_0\end{array}\right)$

where

(10)$\begin{array}{cc}& g_0=-i\frac{E}{\sqrt{{\Delta }^2-E^2}}[1-\frac{1}{2}P(\theta ,E\left){\mathrm{sech}}^2\frac{z}{2\xi }\right]\\ & g_{\Vert }=\frac{\Delta \sin \theta }{2\sqrt{{\Delta }^2-E^2}}P(\theta ,E){\mathrm{sech}}^2\frac{z}{2\xi }\\ & f_{\Vert }=i\frac{\sin \theta }{\sqrt{{\Delta }^2-E^2}}[1-\frac{1}{2}P(\theta ,E\left){\mathrm{sech}}^2\frac{z}{2\xi }\right]\\ & f_z=i\frac{\Delta \cos \theta }{\sqrt{{\Delta }^2-E^2}}\tanh \frac{z}{2\xi }+\frac{iE}{2\sqrt{{\Delta }^2-E^2}}P(\theta ,E){\mathrm{sech}}^2\frac{z}{2\xi }\\ & P(\theta ,E)=\frac{{\Delta }^2{\cos }^2\theta }{{(E+i{0}^{+})}^2-{\Delta }^2{\sin }^2\theta }\end{array}$

The Majorana fermions’ momentum resolved LDOS is given by the imaginary part of the quasi-classical Green’s function^[35] as follows:

(11)$N(\hat{k};E,z)=-\frac{1}{\pi }\mathrm{Im}{g}^R(\hat{k},E,z)$

LDOS can be divided into two components: one is the contribution of the bound state.

(12)$N_B(p,z,E)=\left(\frac{\pi }{2}\frac{{\Delta }_{\perp }}{\cos \theta }{e}^{-2{\Delta }_{\perp }z/\left(ħv_F\right)}\right)[\delta (E-E_{B+}\left(p_{\Vert }\right))+\delta (E-E_{B-}\left(p_{\Vert }\right))]$

Here, $E_{B\pm }\left(p_{\Vert }\right)=\pm \left(\Delta /p_F\right)\left|p_{\Vert }\right|$, $p_{\Vert }$ denotes the in-plane momentum of the fermions confined to the surface, p_F is the fermi momentum. The spectral weight $\exp (-2{\Delta }_{\perp }z/ħv_F)$ decays with the increase in z, and is therefore significant only near the surface. The other contribution comes from the continuum spectrum.

(13)$N_C(p,z,E)=\frac{\left|E\right|}{\sqrt{E^2-|\Delta {|}^2}}[1-\frac{{\Delta }_{\perp }^2{\cos }^2\theta }{E^2-{\Delta }_{\Vert }^2{\sin }^2\theta }\cos \left(\frac{2z\sqrt{E^2-|\Delta {|}^2}}{ħv_F\cos \theta }\right)]$

Therefore, the total LDOS is as follows.

(14)$N(p,z,E)=N_B(p,z,E)+N_C(p,z,E)$

The spatial distribution of the LDOS is shown in Figure 3. In Figure 3a, bright stripes approximately parallel to the z axis appear within the energy gap at E = ∆. The LDOS increases linearly with energy, as shown in Figure 3c, which is particularly pronounced at the surface. The bulk density of states would normally diverge at E = ∆, but this divergence is suppressed at the surface. The spectral weight removed from the bulk is redistributed to the bound states, confirming their spatial confinement near the surface, with a bound-state wave function whose spectral weight decays along the z-direction. In Figure 3b, the bound state energy is proportional to the parallel momentum, with symmetric positive and negative branches.

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Figure 3. (a) The LDOS as a function of the normalized energy and normalized depth, the colorbar indicates the magnitude of the LDOS; (b) The variation of the LDOS with normalized energy and momentum; (c) The energy dependence of the LDOS at z = 0; (d) The depth dependence of the LDOS at |E/∆| = 1. LDOS: local density of states.

A zero-energy mode is present at zero momentum, resulting in a Dirac cone. The surface excitations exhibit Majorana character, and their linear dispersion within the bulk gap reflects the topologically nontrivial nature of the bulk state. The real-space distribution of a quantum state at E = ∆ is shown in Figure 3d, which clearly illustrates how the wave function decays as $\exp (-2{\Delta }_{\perp }z/ħv_F)$. In experiments, Arrayas et al.^[36] used a levitating sphere as a probe to detect the surface excitation spectrum of superfluid ³He-B, enabling quantitative analysis of the quasiparticle energy spectrum, lifetime, and transport behavior.

2.2 Dynamics of electrons bubble and surface-bound states in superfluid ³He-B

Based on the equation of motion for an electron bubble, its momentum change rate dP/dt is governed by the total external drag force, as indicated by Eq. (15). For a slow-moving electron bubble in ³He-B driven by an electric field, the momentum variation stems from the scattering of all Majorana fermions.

(15)$\frac{dP}{dt}=-\sum _{k,k^{\prime },\sigma ,{\sigma }^{\prime }}ħ(k^{\prime }-k)(1-f_{k^{\prime }})f_k\Gamma (k,\sigma \to k^{\prime },{\sigma }^{\prime })$

Here, k denotes the momentum of a Majorana fermion before scattering. A Majorana fermion with initial momentum k is scattered by the electron bubble into a final state with momentum k′, transferring momentum ħ(k′ - k) to the electron bubble. The velocity of the electron bubble is sufficiently low that the motion of the electron bubble does not appreciably perturb the initial and final Majorana fermion distribution functions, which remain well described by the Fermi–Dirac distribution. Owing to the large mass of the electron bubble compared to that of a ³He atom, its recoil energy can be neglected in scattering events. Consequently, the scattering can be treated as elastic, and the energy of Majorana fermions is conserved. The scattering rate $\Gamma (k,\sigma \to k^{\prime },{\sigma }^{\prime })$ depends on the velocity of the impurity.

According to Fermi’s golden rule, the $\Gamma (k,\sigma \to k^{\prime },{\sigma }^{\prime })$ can be calculated using perturbation theory in quantum mechanics. For elastic scattering, it is proportional to the squared modulus of the scattering amplitude and the density of final states.

(16)$\Gamma (k,\sigma \to k^{\prime },{\sigma }^{\prime })=\frac{2\pi }{ħ}\delta (E_{k^{\prime }}-E_k){\left|t(k,\sigma \to k^{\prime },{\sigma }^{\prime })\right|}^2$

Considering only the first order of velocity yields, and from Eq. (15) and Eq. (16), we obtain Eq. (17).

(17)$\frac{dP}{dt}=\frac{-ħ\pi }{k_{B}T}\sum _{k,k^{\prime },\sigma ,{\sigma }^{\prime }}[v\cdot (k^{\prime }-k)](k^{\prime }-k)(1-f_k)f_k\delta (E_{k^{\prime }}-E_k){\left|t(k,\sigma \to k^{\prime },{\sigma }^{\prime })\right|}^2$

This result shows that the drag force is proportional to the velocity in the low-velocity limit, allowing the introduction of a drag coefficient η defined via dP/dt = -ηv. Considering only the component parallel to the surface, the corresponding parallel component is given by:

(18)${\eta }_{\Vert }=\frac{\pi ħ}{2}\sum _{k,k^{\prime }}{(k_{\Vert }^{\prime }-k_{\Vert })}^2(-\frac{\partial f_k}{\partial E_k})\delta (E_{k^{\prime }}-E_k)\sum _{\sigma ,{\sigma }^{\prime }}{\left|t(k,\sigma \to k^{\prime },{\sigma }^{\prime })\right|}^2$

where |t(k,σ → k′,σ′)|² is the square of the scattering T-matrix^[37].

(19)$\sum _{\sigma ,{\sigma }^{\prime }}{\left|t(k,\sigma \to k^{\prime },{\sigma }^{\prime })\right|}^2=\sum _{\sigma ,{\sigma }^{\prime }=\uparrow ,\downarrow }\left|⟨{\Psi }_{k^{\prime },{\sigma }^{\prime }}|T_S\right|{\Psi }_{k,\sigma }⟩{|}^2$

The T-matrix for Majorana fermions scattering in the superfluid state is governed by the Lippmann–Schwinger equation, which is formulated using the normal state T-matrix. By incorporating the quasiclassical Green’s function into the Lippmann–Schwinger framework, the corresponding T-matrix elements in the superfluid can be determined as follows.

(20)$T_S({\hat{k}}^{\prime },\hat{k},E,z)=T_N({\hat{k}}^{\prime },\hat{k})+N_F\int \frac{d{\Omega }_{k^{\prime \prime }}}{4\pi }{T}_N({\hat{k}}^{\prime },{\hat{k}}^{\prime \prime })\left|g^R\right(\theta ,E,z)-g_N|T_S({\hat{k}}^{\prime \prime },\hat{k},E,z)$

The intermediate momentum ${\hat{k}}^{\prime \prime }$ in the Lippmann–Schwinger equation serves as the integration variable, accounting for all virtual scattering pathways. The integration spans the entire Fermi surface. In this formulation, N_F is the normal state density of states, g^R denotes the quasiclassical Green’s function for surface bound states, and g_N = -iπτ₀ (with τ₀ being the unit matrix) represents the normal-state Green’s function. The T-matrix in the normal phase, denoted as T_N, enters the self-consistent equation as follows:

(21)$T_N({\hat{k}}^{\prime },\hat{k})=\left(\begin{array}{cc}{t}_N({\hat{k}}^{\prime },\hat{k}){\sigma }_0& 0\\ 0& -t_N{(-{\hat{k}}^{\prime },-\hat{k})}^{\ast }{\sigma }_0\end{array}\right)$

where $t_N({\hat{k}}^{\prime },\hat{k})=-\frac{1}{\pi N_F}\sum _{l=0}^{\infty }(2l+1)e^{i{\delta }_l}\sin {\delta }_l{P}_l(\hat{k}\cdot {\hat{k}}^{\prime })$ is the Legendre polynomial, l is the angular momentum quantum number. Considering the potential energy between Majorana fermions and electron bubbles, which features a finite short-range repulsion, the hard sphere potential model is most suitable.

(22)$U\left(r\right)=\{\begin{array}{cc}\infty & r\le R\\ 0& r>R\end{array}$

Within the hard sphere scattering model, the phase shifts of scattering for the T-matrix are defined by tanδ_l = j_l(k_FR)/n_l(k_FR), where R = 1.42 nm is the radius of the electron bubble determined from normal-state mobility experiments^[38]. The dimensionless parameter k_FR = 2πR/λ_F, with the Fermi wavelength λ_F = 2π/k_F ≈ 0.127 nm, characterizes the relative size of the bubble compared to the λ_f. In this expression, j_l and n_l represent the spherical Bessel functions of order l for the first kind and second kind, respectively. At low angular momenta (l < 12), as illustrated in Figure 4, the scattering phase shifts δ_l exhibit pronounced oscillations. The short-range repulsion of the hard-sphere potential causes low-l Majorana fermions to undergo head-on collisions with the electron bubble, resulting in strong wave function reflection and substantial phase shifts. In the high-l regime (l > 12), the phase shifts rapidly approach zero, and scattering effects become negligible. Consequently, the large phase shifts at low angular momenta dominate the scattering cross section, and only phase shifts with l < k_FR are retained in the computation of the scattering matrix.

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Figure 4. Phase shifts as a function of the angular momentum quantum number l for the hard-sphere potential model.

The polar angle averaged differential cross section, which gives the probability distribution per unit solid angle after a Majorana fermion is scattered by an electron bubble, is determined by the squared scattering amplitude. After averaging over the azimuthal angle difference φ = ϕ′ - ϕ between the incident and outgoing Majorana fermions, the polar angle averaged differential cross section is obtained as follows:

(23)$\frac{\overline{d\sigma }}{d\Omega }(\phi ,E,z)={\left(\frac{\pi N_F}{k_F}\right)}^2{\left(\frac{E}{\Delta }\right)}^4\frac{1}{4}\sum _{S,S^{\prime }}\sum _{\sigma ,{\sigma }^{\prime }}{\left|t(k_{\Vert },s{k}_{\perp },\sigma \to k_{\Vert }^{\prime },s^{\prime }{k}_{\perp },{\sigma }^{\prime })\right|}^2$

The in-plane component $k_{\Vert }$ represents the projection of the wave vector onto the surface. The presence of surface bound states introduces a sign index s = ±1 for the perpendicular momentum component $k_{\perp }$. This index distinguishes between incident and reflected waves. Specifically, s = ±1 corresponds to positive $k_{\perp }$, indicating propagation away from the surface, while s = -1 denotes negative $k_{\perp }$, representing propagation toward the surface. The wave function of the surface-bound state must satisfy the boundary condition ψ = 0 at z = 0. During scattering, the combinations of s and s′ values correspond to different scattering channels.

3. Results and Discussions

3.1 Electron transport cross section and mobility in electric field

The total cross section σ_tot(E,z) and transport cross section σ_tr(E,z) are defined as follows:

(24)${\sigma }_{\mathrm{tot}}(E,z)\equiv \frac{3}{2}{\int }_0^{2\pi }d\phi \frac{d\sigma }{d\Omega }(\phi ,E,z)$

(25)${\sigma }_{\mathrm{tr}}(E,z)=\frac{3}{2}{\int }_0^{2\pi }d\phi (1-\cos \phi )\frac{d\sigma }{d\Omega }(\phi ,E,z)$

Figure 5a shows the σ_tot(E,z) and σ_tr(E,z) versus energy at depth z = 0. The σ_tr(E,z) is consistently smaller than the σ_tot(E,z). At low energy (E < 0.2∆), both exhibit small, narrow peaks, with negligible contribution from surface bound states. In the intermediate range 0.6∆ < E < 0.8∆, near the bulk excitation threshold, the rise and oscillations remain weak. At higher energy, a strong peak emerges near the bulk gap. This resonance originates from the increased density of states of surface bound states and quasibound states around the electron bubble. Momentum transfer is enhanced due to the concentration of states, while the opening of combined bulk-surface scattering channels causes a sudden phase change, triggering resonant enhancement. Furthermore, the enhanced quasiparticle scattering near the gap edge amplifies the response, leading to a pronounced peak in the transport cross section. Figure 5b shows the energy dependence of the transport cross section σ_tr(E,z) at various depths. Near the surface, it varies smoothly with energy, whereas at larger depths, pronounced oscillations with sharp resonance peaks emerge.

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Figure 5. (a) Total cross section σ_tot(E,z) (red line) and transport cross section σ_tr(E,z) (blue line) as a function of energy at z = 0.9ξ; (b) The energy dependence of the transport cross section at different depths: z = 2ξ (red line), z = ξ (blue line), and z = 0.2ξ (green line).

Figure 6 presents the depth dependence of the σ_tot(E,z) at different energies. At low energies, it remains nearly constant, while near E～∆, it increases markedly with depth. This behavior reflects the suppression of Majorana fermion scattering by low-energy surface bound states. Owing to particle–hole symmetry and the topological protection of the p-wave pairing, the scattering cross section is strongly reduced near the surface. As the depth z increases, the bound-state contribution decays, while the contribution from the bulk quasiparticle continuum becomes increasingly dominant, and the σ_tot(E,z) rises until it saturates once the depth exceeds the coherence length ξ.

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Figure 6. Total cross section σ_tot(E,z) as a function of depth at (a) E = 0.7∆ (red line), and (b) E = 0.9∆ (blue line).

To further elucidate the transport mechanism and clarify why the σ_tr(E,z) remains smaller than the σ_tot(E,z), the differential scattering cross section is plotted in polar coordinates. As shown in Figure 7a, when E = 0.7∆, scattering is predominantly concentrated in small-angle forward directions. Backscattering is strongly suppressed, and momentum transfer is dominated by forward and near-forward processes, leading to a distinct forward enhancement. Comparing Figure 7a,b reveals that as the energy of Majorana fermions increases, the forward peak becomes sharper and more intense with a narrower angular width. In addition, secondary peaks emerge, reflecting a strong forward focusing tendency in the differential cross section. A comparison between Figure 7b,c further indicates that the forward peak becomes more pronounced at larger depths. As z increases, the effective strength of the T-matrix, T(E,z), becomes enhanced, leading to a larger scattering amplitude $f(E,z)\propto ⟨\Psi (E,z)|T(E,z)|\Psi (E,z)⟩$. Consequently, the peak value of the differential scattering cross section increases with z. When the quasiparticle energy approaches the gap edge, as shown in Figure 7d, the scattering exhibits strong anisotropy, and the angle-averaged cross section typically reaches a pronounced maximum. The drag coefficient can be simplified to:

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Figure 7. Normalized polar angle averaged differential cross sections for electron bubble at various energies and depths. The normalization is performed with respect to πR², corresponding to the dashed reference in the figure. The specific parameters are (a) z = 0.5ξ, E = 0.7∆ and (b) z = 0.5ξ, E = 0.8∆, (c) z = ξ, E = 0.8∆, (d) z = 1.5ξ, E = 0.9∆.

(26)${\eta }_{\Vert }=\frac{{\pi }^2}{16}{\mathrm{sech}}^4\left(\frac{z}{2\xi }\right)n_3{p}_F{\int }_{-\Delta }^{\Delta }dE(-\frac{\partial f}{\partial E}){\sigma }_{\mathrm{tr}}(E,z)$

where n₃ = k_F³/3π²) is the particle density of ³He. The drag coefficient consists of two parts: the contribution from the surface bound state ${\eta }_{\Vert }$, and the contribution from the continuum Majorana fermions η_B^[39].

(27)${\eta }_B=2n_3{p}_F{\int }_{\Delta }^{\infty }dE(-\frac{\partial f}{\partial E}){\sigma }_{tr}^B\left(E\right)$

where ${\sigma }_{tr}^B$ is the transport cross section in the bulk ³He-B. The mobility is obtained as follows:

(28)$\mu =\frac{e}{{\eta }_{\Vert }+{\eta }_B}$

As shown in Figure 8, the mobility µ rises as the temperature T decreases. At low temperatures, scattering is dominated by surface bound states with linear dispersion. The Fermi–Dirac distribution restricts Majorana fermion occupancy near the Fermi energy, enhancing the electron bubble–Majorana state scattering rate and increasing the drag coefficient η, thereby suppressing µ. At higher temperatures, the superfluid gap closes and bulk Majorana fermions dominate, causing µ to approach the bulk value. Above T/T_c > 0.4, mobility becomes independent of depth z, as seen in the overlapping curves for z = 28, 48, 68, and 500 nm. At lower temperatures, surface effects become significant, and µ increases more rapidly with decreasing temperature for larger z.

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Figure 8. The mobility of electron bubble as a function of the normalized temperature is shown for different depths, z = 28 nm (black line), 48 nm (red line), 68 nm (blue line), and 500 nm (green line), T_c denotes the critical temperature of the superfluid ³He-B phase. The inset shows the mobility as a function of the inverse reduced temperature, with data digitized from Refs^[25,34].

To enable a quantitative comparison between theory and experiment, we digitized the mobility data of Ikegami et al.^[25,34]. The extracted data points are plotted in the inset of Figure 8. As shown in the figure, for electron bubble depths smaller than the coherence length (z = 21, 40, and 58 nm), the curves nearly coincide, indicating weak sensitivity of the mobility to depth. This z-independent mobility arises from a compensation mechanism: the surface state density decays as sech⁴(z/2ξ), while the scattering cross section σ_tr(E,z) grows with z due to the prolonged lifetime of quasibound states. The resulting balance between decaying density and enhanced resonant scattering yields a depth-insensitive transport response.

3.2 Electron transport mobility in magnetic field

The surface bound states are sensitive only to the normal component of the magnetic field. When the field is applied perpendicular to the surface, even a weak magnetic field opens a Zeeman gap in the dispersion relation. This anisotropy originates from the Ising-type magnetic character of the surface-bound states^[37]. In contrast, for an in-plane magnetic field, chiral symmetry protects the gapless spectrum of the surface Majorana fermions, and an energy gap develops only when the field strength exceeds a critical value of about 3 mT. For a magnetic field applied along the z-direction, the characteristic equation of the surface bound states takes the form:

(29)$\left(\begin{array}{cc}\Delta \sin \theta & -\frac{\gamma H}{2}{e}^{-i\varphi }\\ -\frac{\gamma H}{2}{e}^{i\varphi }& -\Delta \sin \theta \end{array}\right)\Psi \left(r\right)=E\Psi \left(r\right)$

$E=\sqrt{{\Delta }^2{\sin }^2\theta +(ħ\gamma H/2{)}^2}$. The magnetic field opens an energy gap and alters the energy spectrum of the bound states, where γ is the gyromagnetic ratio of the ³He nucleus, and the Zeeman energy gap is ħλH/2. The application of a magnetic field breaks time-reversal symmetry in superfluid ³He-B. This leads to the opening of a Zeeman gap in the surface Majorana fermions, shifting their spectral weight away from zero energy. As a consequence, these states no longer fulfill the condition for Majorana zero modes, and their dispersion relation becomes gapped. Figure 9a shows the resulting gapped spectrum, where a finite energy gap appears at $k_{\left|\right|}=0$ while the linear dispersion relation is recovered in the large-momentum limit $(k_{\left|\right|}\to \infty )$. A direct comparison of the energy gaps with and without the magnetic field is presented in Figure 9b.

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Figure 9. (a) Edge state dispersion relation on the superfluid ³He-B surface under magnetic field; (b) Comparison of dispersion relation with and without magnetic field, the red dashed line represents the dispersion relation under a magnetic field, while the blue line represents the dispersion relation in the absence of a magnetic field.

Figure 10 displays the energy dependence of the transport cross section σ_tr(E) for electron bubbles interacting with surface Majorana fermions near a free surface, under magnetic fields of H = 0.1∆ and H = 0.2∆. At low energy, the surface bound states are strongly suppressed and redistributed toward the bulk gap edge. The absence of available surface scattering channels leads to a sharp reduction in the scattering cross section, and the cross section approaches zero. For H = 0.1∆, the Zeeman gap remains relatively narrow, suppressing only a portion of the low-energy scattering channels. This results in a decrease of σ_tr(E,z) at low energies, accompanied by an enhancement near E ≈ 0.8∆. When the magnetic field increases to H = 0.2∆, the Zeeman gap widens to 0.2∆, further suppressing the σ_tr(E,z) in the low-energy region until it nearly vanishes. Simultaneously, a more pronounced enhancement emerging of σ_tr(E,z) emerges around E ≈ 0.8∆. Clearly, a stronger magnetic field enhances the contrast between low-energy suppression and high-energy enhancement, flattening the profile at low energies while sharpening the peak at higher energies. This behavior reflects the magnetic field induced reconstruction of the energy spectrum and redistribution of scattering channels, consistent with the evolution of the density of states under a magnetic field. As the field strength increases, the opening of the Zeeman gap reduces scattering from low-energy quasiparticles, thereby diminishing the drag on electron bubbles in the low-energy regime and enhancing their mobility.

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Figure 10. Energy dependence of the transport cross section under a magnetic field for external field strengths of (a) H = 0.1∆ and (b) H = 0.2∆.

Figure 11 shows the normalized mobility of electron bubbles in the ³He-B phase as a function of the reduced temperature under three different magnetic field conditions (0 T, 0.25 T, 0.35 T). In the medium to high temperature range, the magnetic field exhibits a suppressive effect on mobility, while in the low-temperature region it shows an enhancement trend. The mobility decreases with increasing magnetic field. At a field of 0.25 T, ³He remains in the A-phase at higher temperatures. When the temperature decreases below the transition temperature T_c, a first-order phase transition to the B-phase occurs. Due to the completely different energy gap structures and surface states between the A-phase and B-phase, electron bubbles experience significantly different scattering efficiencies in these two phases, resulting in a distinct jump in mobility at the phase transition point.

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Figure 11. Mobility as a function of temperature at 0 T (black line), 0.25 T (blue line), 0.35 T (red line).

We note that the current magnetic-field-dependent analysis assumes that an ultra-low-temperature environment can be maintained under moderate magnetic fields. Experimentally, the combination of nuclear demagnetization refrigeration and a superconducting magnet system can provide steady magnetic fields of several tesla while maintaining temperatures in the mK range, though additional technical thermal-stability constraints may arise. In our theoretical treatment, we assume an ideal low-temperature environment in which the system remains in the ³He-B phase, and the magnetic field does not cause significant heating. The theoretical study focuses on the scattering effects of surface states, and therefore is not limited by any specific cooling scheme.

4. Conclusion

Based on a transport analysis of electron bubbles confined near the free surface of superfluid ³He-B, this work has elucidated the key dynamical properties of surface bound states. These states display a linear Dirac-cone dispersion $E\propto k_{\Vert }$ and a LDOS that increases linearly with energy within the bulk gap. The scattering cross section between electron bubbles and Majorana fermions is primarily mediated by these surface modes, with small-angle forward scattering dominating at higher energies, and backscattering suppressed by particle–hole symmetry. The nearly depth-independent mobility stems from a compensation between the exponential decay of the surface-state spectral weight and the progressive enhancement of the quasibound-state scattering amplitude in the integrated transport response. Under a perpendicular magnetic field, Zeeman splitting opens a gap in the linear spectrum, whereas in zero field, the well-defined gapless dispersion is consistent with the Majorana nature of the surface excitations.

Acknowledgments

The author thanks Dr. Qingxuan Wang for helpful discussions and for suggestions on the manuscript.

Authors contribution

Liu M: Conceptualization, data curation, investigation, writing-original draft.

Zhang K: Methodology, validation, writing-review & editing.

Zhou J: Supervision, project administration, funding acquisition, writing-review & editing.

Conflicts of interest

The authors declare no conflicts of interest.

Ethical approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Funding

This work is supported by the Key-Area Research and Development Program of Guangdong Province (Grant No. 2020B0303060001).

Copyright

© The Author(s) 2026.