Tunneling chirality Hall effect in type-I Weyl semimetals
W. Zeng




[E-mail: ]zeng@ujs.edu.cn
Department of physics, Jiangsu University, Zhenjiang 212013, China






We propose a tilt-assisted chirality Hall effect in the normal metal-superconductor (NS) junctions based on the time-reversal broken type-I Weyl semimetals. It is found that the chirality-contrasting skew reflection occurs at the NS interface due to the tilt of the Weyl cones, which is responsible for the nonzero transverse chirality Hall currents. Distinct from the Hall effect induced by the Berry curvature, we further illustrate that the transverse chirality current here is determined by the symmetry of the tilt. Specifically, both the transverse chirality Hall current and the transverse charge Hall current may occur when the tilt breaks the mirror symmetry (ℳ). However, a pure transverse chirality Hall current with zero net charge is present when the tilt breaks ℳ symmetry but preserves the combined ℳ𝒞 symmetry, where 𝒞 represents the ℤ_2 exchange symmetry.






 




Tunneling chirality Hall effect in type-I Weyl semimetals
    W. Zeng
    April 29, 2024
=========================================================





§ INTRODUCTION




Weyl semimetals are three-dimensional topological materials with the conduction and valence bands touching at two or more crossing points in the bulk, which are known as the Weyl nodes <cit.>. The emergence of the nontrivial and stable Weyl
nodes requires the breaking of either time-reversal symmetry (𝒯) or spatial inversion symmetry (ℐ). As a consequence of the Nielsen-Ninomiya theorem <cit.>, the minimal model of the 𝒯 symmetry broken Weyl semimetal contains a single pair of Weyl nodes, whereas the ℐ symmetry broken one contains four Weyl nodes <cit.>. Each of the paired Weyl nodes acts like a topological charge with the charge sign corresponding to its chirality <cit.>. The manipulation of the chirality is one of the hot topics in Weyl physics. Up to now, several works have been devoted to such chirality-dependent physics of Weyl semimetals, such as the chiral anomaly <cit.>, chirality-dependent Hall effect <cit.> and chirality Josephson effect <cit.>.












A finite tilt of the Weyl cones can be generated because the Lorentz symmetry is not necessarily the symmetry group in condensed matter systems <cit.>. Distinct from the graphene-like materials where the tilting is usually weak <cit.>, the tilting can be strong in Weyl semimetals. Depending on whether the Weyl cone is overtilted or not, Weyl semimetals can be classified into two subgroups, i.e., type-I and type-II Weyl semimetals <cit.>. The type-I Weyl semimetals possess a closed Fermi surface enclosing either an electron or a hole pocket, with a vanishing density of states at the Weyl nodes. The type-II Weyl semimetals host overtilted Weyl cones and the Fermi surface near the Weyl nodes is hyperboloidal with a large density of states, leading to the electron and hole pockets near the Weyl nodes <cit.>. Many tilt-induced intriguing transport properties have been reported in Weyl semimetals. Such as the anomalous Nernst and thermal Hall effects <cit.>, double Andreev reflection <cit.>, tilt-assisted π-phase Josephson current <cit.>, and linear magnetochiral transport <cit.>.






 

A very recent interesting work <cit.> reported that the tilt mechanism can lead to the tunneling valley Hall effect in Dirac systems, where a strong tilt-dependent transverse valley Hall current can be generated by the momentum filtering of the tunneling Dirac fermions. Subsequently, the nonlinear valley Hall effect in tilted massless Dirac fermions in strained graphene and organic semiconductors was predicted <cit.>, where a valley Hall current occurs with both inversion and time-reversal symmetry. Inspired by this, here we propose a tilt-induced chirality Hall effect in
Weyl semimetals. We theoretically investigate the transverse charge and chirality transport in the normal metal-superconductor (NS) junctions based on the type-I Weyl semimetals, which breaks the time-reversal symmetry but preserves the inversion symmetry. It is found that the scattering at the NS interface is chirality-dependent. For the electrons with a given chirality, the incident angle resolved reflection probability is asymmetric, which is responsible for the transverse chirality Hall current. The chirality Hall conductance arsing from the skew scattering can be obtained within the Landauer formalism <cit.>. It is found that the transverse chirality current is determined by the tilt parameter c and the intersection angle α between the line connecting the two opposite chiral Weyl nodes and the normal of the NS interface. For the tilt breaking the mirror symmetry ℳ, which requires α≠π/2, both the transverse chirality Hall current and charge Hall current may occur. However, for the tilt breaking the ℳ symmetry but preserving the combined ℳ𝒞 symmetry with 𝒞 being the ℤ_2 exchange symmetry, requiring α=0 or π, a pure transverse chirality Hall current with zero net charge appears.

The remainder of the paper is organized as follows. The model Hamiltonian and the scattering approach are explained in detail in Sec. <ref>. The numerical results and discussions are presented in Sec. <ref>. Finally, we conclude in Sec. <ref>.



 



















§ MODEL

We consider the NS junction along the z axis, where the normal and superconduting regions are located at z<0 and z>0, respectively, as shown in Fig. <ref>. The Weyl nodes ±𝐊_0 are in the q_x-q_z plane and the line connecting ±𝐊_0 can make an angle of α with q_z axis. In the crystal coordinates, the minimal model for the tilted Weyl semimetal is described by the effective two-band Hamiltonian <cit.>

    H=∑_χ,𝐪Ψ^†_χ,𝐪ℋ_χ(𝐪)Ψ_χ,𝐪,

where Ψ_χ,𝐪=(ψ_χ,𝐪↑,ψ_χ,𝐪↓)^T is the spinor basis with χ=± being the chirality of the Weyl nodes and 𝐪=(q_1,q_2,q_3) is the momentum measured from χ𝐊_0. Around the Weyl nodes, the low-energy Hamiltonian reads <cit.>

    ℋ_χ(𝐪)=   ħ v_χ q_1σ_0+ħ v_F(q_1σ_1+q_2σ_2-χ q_3σ_3),
 
where v_F is the Fermi velocity, σ_0 is the identity matrix and σ_i (i=1,2,3) are Pauli matrices acting on the spin space. The tilt of the Weyl cones is along the q_1 direction with the parameter v_χ. Here we focus on the tilting effect in the type-I Weyl semimetals, i.e., |v_χ|<v_F. The inversion symmetry of the Hamiltonian in Eq. (<ref>) requires σ_3ℋ_+(𝐪)σ_3=ℋ_-(-𝐪), leading to v_+=-v_-, which implies that the opposite chiral Weyl cones have tilts in opposite directions. It is convenient to work with the junction coordinates, where the transport direction is assumed to be along the z axis, as shown in Fig. <ref>. The two different coordinate systems are related by the rotation transformation

    q_1=   q_xcosα-q_zsinα,   q_2=q_y,
    
    q_3=   q_zcosα+q_xsinα,

and, similarly, σ_1=σ_xcosα-σ_zsinα, σ_2=σ_y, σ_3=σ_zcosα+σ_xsinα, where α is the angle between the line connecting two Weyl nodes and the q_z axis.







In the superconducting region, the zero-momentum BCS pairing is preferred for inversion-symmetric Weyl semimetals <cit.>, for which the paired electrons are from two Weyl nodes with opposite chirality. This BCS superconductivity can be induced by the conventional superconductor via the proximity effect in the Weyl semimetal based junction <cit.>. The pairing Hamiltonian reads

    ℋ_Δ=∑_χ,s∫ d𝐫 Δψ^†_χ,s(𝐫)ψ^†_-χ,-s(𝐫)+h.c.,

where s={↑,↓} is the spin index and Δ is the pairing potential. In the Nambu basis (ψ_χ↑,ψ_χ↓,ψ_-χ↓^†,-ψ_-χ↑^†)^T, the NS junction is described by the Bogoliubov-de Gennes (BdG) Hamiltonian <cit.>
 

    ℋ_BdG=   [          ℋ_χ(-i∇-χ𝐊_0)-μ(z)                        Δ(z);                      Δ^*(z) μ(z)-σ_yℋ^*_-χ(-i∇+χ𝐊_0)σ_y ],

where the chemical potential μ(z)=μ for z<0 and μ(z)=μ_s for z>0, the pairing term Δ(z)=0 for z<0 and Δ(z)=Δ_0 for z>0. By performing the translation transformation D=exp(iχ𝐊_0·𝐫) and the spin rotation transformation U=σ_xexp(iασ_y) <cit.>, Eq. (<ref>) can be written as

    ℋ_BdG=   [ ħ v_F(χ c q_tσ_0+(q_xσ_x-χ q_yσ_y+ q_zσ_z))-μ(z)                                           Δ_B(z);                                           Δ_B(z) ħ v_F(-χ c q_tσ_0-(q_xσ_x+χ q_yσ_y+q_zσ_z))+μ(z) ],

where c=|v_χ/v_F| is a dimensionless parameter employed here to characterize the tilt (0<c<1), q_t=q_xcosα-q_zsinα and Δ_B(z)=-Δ(z)sinασ_z+Δ(z)cosασ_x.


In the normal segment of the junction (z<0), the scattering states propagating along the +z axis are given by

    |φ_e^>⟩=[      Γ_e^>; |𝐪_∥|e^-iϕ;          0;          0 ]e^iq_e^> z,  
    |φ_h^>⟩=[         0;         0;     Γ_h^>; |𝐪_∥|e^iϕ ]e^iq_h^> z,

where the subscripts `e/h' of the wave functions denote the electron/hole states, respectively, 𝐪_∥=(q_x,q_y) is the conserved transverse wave vector, ϕ=arctan(χ q_y/q_x), and Γ_e(h)^>=(1+χ csinα)q_e(h)^>+(-)p_e(h) with p_e(h)=(E+(-)μ)/ħ v_F-(+)χ q_x c cosα. We note that the factor e^i𝐪_∥·𝐫_∥ with 𝐫=(x,y) is omitted in Eq. (<ref>) for simplicity.
The longitudinal wave vectors are given by   

    q_e(h)^>=+(-)χ p_e(h)c sinα+ζ_e(h)√(p_e(h)^2-κ|𝐪_∥|^2)/κ,

where

    ζ_e(h)=sgn(p_e(h)+√(κ)|𝐪_∥|),

with κ=1-(csinα)^2.
Similarly, the scattering states propagating along the -z axis, i.e., |φ^<⟩, can be obtained by the replacement ζ_e(h)→-ζ_e(h) in Eqs. (<ref>-<ref>).

In the heavily doping limit |μ_s|≫|μ| <cit.>, only the excitations quasiperpendicular transmitting to the superconducting region need to be considered, resulting in the effective excitation gap Δ_B=Δ_0|sinα| <cit.>. The transmitted states in the superconducting region are given by

    |φ_s,1^>⟩=[    𝒫;    0; -Δ_B;    0 ]e^iq_s,1^>z,   |φ_s,2^>⟩=[   0; Δ_B;   0;   𝒫 ]e^iq_s,2^>z,

where 𝒫=√(E^2-Δ_B^2)+E and the subscript `1(2)' denotes the electron-like (hole-like) quasiparticle state. The longitudinal wave vectors in the superconducting region are given by

    q_s,1(2)^>=(ħ v_F)^-1+(-)μ_s+√(E^2-Δ_B^2)/1-(+)χ csinα.





The total wave function describing the scattering process reads

    ψ(z)=
    |φ_e^>⟩+r|φ_e^<⟩+r_A|φ_h^<⟩,    z<0, 
     
    t_1|φ_s,1^>⟩+t_2|φ_s,2^>⟩,    z>0.

Here t_1(2), r and r_A are the transmission amplitude, normal reflection amplitude and Andreev reflection amplitude, respectively, which can be obtained by matching the wave function at z=0. 

The longitudinal conductance for the χ chirality Weyl node is given by the Blonder-Tinkham-Klapwijk approach <cit.>

    σ_z,z^χ=e^2/h∑_𝐪_∥(1+R_A-R),

where

    R_A=|v_h,z^</v_e,z^>||r_A|^2,   R=|v_e,z^</v_e,z^>||r|^2,

are the Andreev reflection probability and normal reflection probability, respectively. We note that both r and r_A are the function of χ. v_ς,i^ϱ (ς=e,h, ϱ=>,< and i=x,y,z) in Eq. (<ref>) is the group velocity along the i axis for the excitation state |φ^ϱ_ς⟩, which can be obtained by the Hellmann-Feynman theorem <cit.>:

    v_ς,i^ϱ=   ∂ E_ς/ħ∂ q_i^ϱ=∂/ħ∂ q_i⟨φ^ϱ_ς|ℋ_BdG|φ^ϱ_ς⟩
    
    =   ħ^-1⟨φ^ϱ_ς|∂/∂ q_iℋ_BdG|φ^ϱ_ς⟩
    
    =   ħ^-1⟨φ^ϱ_ς|ĵ_i|φ^ϱ_ς⟩,

with ĵ_x=τ_z(σ_x+χ ccosα), ĵ_y=-χτ_zσ_y and ĵ_z=τ_z(σ_z-χ csinα).



The transverse conductance can be calculated within the Landauer formalism <cit.> (see Appendix for details)

    σ_η,z^χ=e^2/h∑_𝐪_∥(v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2+v_h,η^</v_e,z^>|r_A|^2),

where η=x,y. The charge Hall angle (ϑ) and the chirality Hall angle (ϑ_chi) are given by

    tan(ϑ)=∑_χσ_η,z^χ/∑_χσ_z,z^χ,  tan(ϑ_chi)=∑_χχσ_η,z^χ/∑_χσ_z,z^χ.














§ RESULTS




 §.§ Skew reflection and transverse chirality and charge currents
 


We first consider the situation where the effective superconducting gap Δ_B=Δ_0|sinα| is nonzero, i.e., α≠0,π. Both the Andreev process and the normal reflection process contribute to the transverse Hall currents. It is sufficient to consider the normal reflection probability R in the subgap regime (|E|<Δ_0|sinα|) on account of R+R_A=1. 

The normal reflection probability R versus the incident angle θ_i is shown in Fig. (<ref>)(a) for α=π/3, where the effective superconducting gap is Δ_B=0.86Δ_0. Due to the Weyl cones tilt in the q_x-q_z plane in our model, we focus on the reflection in the x-z plane, where the incident angle is given by

    θ_i=arctan(v^>_e,x/v^>_e,z).

It is shown that the electrons from the χ=+1 chiral node have large reflection probabilities for -90^∘<θ_i<0^∘ [gray line in Fig. (<ref>)(a)], which is responsible for the nonzero transverse chirality Hall current. For the electrons from the χ=-1 chiral node, the skew reflection also exists [green line in Fig. (<ref>)(a)]. However, the scattering is asymmetric for the electrons from different chiral nodes, i.e., R_χ,α=π/3(θ_i)≠ R_-χ,α=π/3(-θ_i), indicating the presence of a nonzero transverse charge Hall current.


The other two scenarios with α=-π/3 and α=2π/3 are also considered, where the effective superconducting gaps are both 0.86Δ_0 as well. The normal reflection probabilities for α=-π/3 and α=2π/3 are shown in Figs. (<ref>)(b) and (<ref>)(c), respectively. The chirality-contrasting reflection remains present. However, compared with the α=π/3 case, the electrons with the opposite (same) chirality are skew reflected to the opposite direction for α=-π/3 (2π/3). Consequently, the scattering patterns in Figs. (<ref>)(a) and (<ref>)(b) are symmetric for different chiralities whereas the scattering patterns in Figs. (<ref>)(a) and (<ref>)(c) are symmetric for the same chirality. The above symmetric relations can be expressed in terms of R, namely 

    R_χ,α=π/3(θ_i)=R_-χ,α=-π/3(-θ_i),
    
    R_χ,α=π/3(θ_i)=R_χ,α=2π/3(-θ_i).

For α=π/2, the skew reflection is absent, and the scattering pattern for χ=+1 and χ=-1 are identical to each other , as shown in Fig. (<ref>)(d). 




This chirality-contrasting skew reflection mentioned above may result in a transverse chirality Hall current as well as a transverse charge Hall current, which can be characterized by the Hall angle. The charge and chirality Hall angles versus the incident energy at different α are shown in Figs. (<ref>)(a) and (<ref>)(b), respectively. For α=π/3, the charge Hall angle ϑ is positive and increases with the increasing of E in the subgap energy regime (|E|<0.86Δ_0), as the shown in Fig. (<ref>)(a) (red solid line). However, the chirality Hall angel is not monotonically dependent on E in the subgap regime, as the shown in Fig. (<ref>)(b) (red solid line). For α=-π/3 and 2π/3, the charge Hall angles are equal to each other but negative, as the gray and green lines shown in Fig. (<ref>)(a), respectively. The chirality Hall angle remains unchanged for α=-π/3 [gray line in Fig. (<ref>)(b)], but reverses its sign for α=2π/3 [green line in Fig. (<ref>)(b)]. For α=π/2, both the charge and chirality Hall angle are zero due to the absence of the skew reflection, as shown in Figs. (<ref>)(a) and (<ref>)(b) (blue dashed lines).


The α-dependence of the Hall angle is shown in Figs. (<ref>)(c) and (<ref>)(d). The charge Hall angle ϑ(α) is odd parity, as shown in Figs. (<ref>)(c), whereas the chirality Hall angle ϑ_chi(α) is even parity, as shown in Figs. (<ref>)(d). It is shown that both ϑ and ϑ_chi are absent at α=±π/2. However, for α=0 and π, ϑ_chi is finite but ϑ=0.

 



The different α-dependence of ϑ and ϑ_chi can be understood by analyzing the reflection probability R. In fact, the symmetric scattering behaviors expressed in Eqs. (<ref>-<ref>) are valid for arbitrary parameter set {α,-α,π-α}, which are given by

    R_χ,α(θ_i)=R_-χ,-α(-θ_i),
    
    R_χ,α(θ_i)=R_χ,π-α(-θ_i).

For the Andreev reflection probability R_A, the similar identities can be obtained by using the current conservation relation R_A=1-R. With the help of Eqs. (<ref>-<ref>), the α-dependent charge and chirality Hall angles can be expressed in terms of R, which are given by

    ϑ(α)=𝒬∑_χ∫_-π/2^π/2dθ_isinθ_iR_χ(θ_i,α),
    ϑ_chi(α)=𝒬∑_χ∫_-π/2^π/2dθ_iχsinθ_iR_χ(θ_i,α),

with 𝒬 being a parameter independent of θ_i and 𝒬(α)=𝒬(-α). By substituting Eqs. (<ref>-<ref>) into Eqs. (<ref>-<ref>) and changing the integration variable θ_i→-θ_i, one finds

    ϑ(α)=-ϑ(-α)=-ϑ(π-α),
    ϑ_chi(α)=ϑ_chi(-α)=-ϑ_chi(π-α),

which indicates that the odd-parity charge Hall angle ϑ(α) disappears at α={π/2,-π/2,0,π} and the even-parity chirality Hall angle ϑ_chi(α) disappears at α={π/2,-π/2}.



 §.§ Pure transverse chirality currents
 


It is noted that the charge Hall angle ϑ vanishes whereas the chirality Hall angle ϑ_chi remains finite at α={0,π} [see Figs. (<ref>)(c) and (<ref>)(d)], implying a pure chirality current. 

In this scenario the effective pairing potential is zero. Under the heavily doping condition, the incident electron is local reflected with probability R in the normal region and quasiperpendicular transmitted (𝐪_∥=0) as an electron-like quasiparticle with probability T_1=|v_s1,z^>/v_e,z^>||t_1|^2 in the superconducting region. The current conservation requires R+T_1=1. In the heavily doping limit |μ_s|≫|μ|, the normal reflection probability is only determined by the tilting parameter c. For α=0, the normal reflection probability is obtained analytically as

    R=|√(^2θ_i-c^2)-1/tanθ_i-χ c|^2,

where θ_i=arctan(v^>_e,x/v^>_e,z) is the incident angle. For the Weyl node with a given chirality χ, the θ_i-resolved reflection is asymmetric, as shown in Fig. <ref>(a), where the electrons with positive chirality (χ=+1) have large reflection probabilities for 0^∘<θ_i<90^∘ (gray lines), whereas the electrons with negative chirality (χ=-1) have large reflection probabilities for -90^∘<θ_i<0^∘ (green lines). The carries with opposite chiralities turn into different transverse directions, leading to a transverse chirality current. With the help of Eqs. (<ref>) and (<ref>), the chirality Hall angle at α=0 is obtained by

    tan(ϑ_chi)=2sinℓ×pcos 2ℓ+cosℓ(πsinℓ-2)/πcos2ℓ-2psin 2ℓ+4sinℓ,

where p=arctanh(cosℓ) with ℓ=arccos(c). The chirality Hall angle is only determined by the tilting parameter c and approaches its maximum value at c≃0.82 with ϑ_chi≃0.22, as shown in Fig. <ref>(c) (black solid line). For the non-tilting energy dispersion (c=0), the transverse chirality current is absent.




Furthermore, it is found that the scattering pattern is mirror symmetric between two different chiral nodes, i.e.,

    R_χ,α=0(θ_i)=R_-χ,α=0(-θ_i),

as shown in Fig. <ref>(a), leading to a zero net transverse charge current with

    tan(ϑ)=0.

Consequently, the pure chirality current with zero net charge is predicted for α=0.





For α=π, the line connecting two opposite chiral Weyl nodes makes a 180^∘ rotation, implying the interchange of the chiralities between the nodes, which leads to the θ_i-resolved R_α=π being a copy of R_α=0 with the substitution χ→-χ, as shown in Fig. <ref>(b). Consequently, the pure transverse chirality Hall current is reversed, while its absolute value remains unchanged, as shown in Fig. <ref>(c) (black dashed line).










 §.§ Symmetry analysis
 









Distinct from the Berry curvature induced Hall effect <cit.>, the physical origin of the transverse Hall current in our model is attributed to the symmetry breaking caused by the the tilt. The low-energy Hamiltonian for the opposite chiral Weyl nodes at ±𝐊_0 is given by ℋ_±=ħ v_F(± c q_tσ_0+q_xσ_x∓ q_yσ_y+ q_zσ_z). For the Weyl cones tilted in q_x-q_z plane, the transverse Hall chirality current and charge current may occur when the angle-resolved reflection is asymmetric for a given chiral node, which requires the tilt breaking the mirror symmetry in the q_y=0 plane, i.e.,

    ℳℋ_±(q_x,q_z)ℳ^-1≠ℋ_±(-q_x,q_z),

where ℳ=σ_z is the mirror reflection operator about the y-z plane. For q_t=q_xcosα-q_zsinα, Eq. (<ref>) leads to cosα≠0, i.e., α≠±π/2, which is in agreement with Eqs. (<ref>-<ref>).

Furthermore, in order to generate a pure transverse chirality Hall current, the reflection between two different chiral nodes should be symmetric to cancel out the net transverse charge current. This requires an additional ℤ_2 exchange symmetry 𝒞 <cit.>, which swaps the opposite chiral sector: 𝒞ℋ_+(q_x,q_z)𝒞^-1=ℋ_-(q_x,q_z). Consequently, the transverse charge Hall current vanishes when the tilt preserves the combined ℳ𝒞 symmetry

    (ℳ𝒞)ℋ_+(q_x,q_z)(ℳ𝒞)^-1=ℋ_-(-q_x,q_z),

which results in α=0,π.


















§ CONCLUSIONS
 

To conclude, we study the transverse transport in the NS junctions based on the time-reversal symmetry broken type-I Weyl semimetals. We focus on the inversion symmetric tilt, where the two Weyl cones with opposite chiralities have tilts in opposite directions. Our investigation reveals that a chirality-contrasting skew reflection occurs at the NS interface due to the tilt of the Weyl cones, resulting in nonzero transverse chirality Hall currents. We further illustrate that the transverse chirality current here is determined by the symmetry of the tilt. Specifically, both transverse chirality current and charge current may occur when the tilt breaks the mirror symmetry (ℳ). However, a pure transverse chirality current with zero net charge is present when the tilt breaks ℳ symmetry but preserves the combined ℳ𝒞 symmetry with 𝒞 being the ℤ_2 exchange symmetry.



















§ APPENDIX



In this appendix we provide the details of the derivation of the transverse conductance and the Hall angle in Eqs. (<ref>) and (<ref>) of the main text.


The transverse current along the η axis (η=x,y) is given by I_η=I_η^>+I_η^< with I^>_η (I^<_η) being the net current flowing from left to right (right to left) <cit.>. In order to get a balanced current in the barrier region, the state propagating towards (outwards) the barrier has a positive (negative) contribution to the transverse current. I^>_η is carried by the state |φ^>⟩=|φ^>_e⟩+r|φ^<_e⟩+r_A|φ^<_h⟩ with energy E and transverse wave vector 𝐪_∥, which is given by


    I^>_η=   e/L∑_𝐪_∥,q_z(v_e,η^>-v_e,η^<|r|^2)f(E-eV)[1-f(E)]
       -(-e/L)∑_𝐪_∥,q_zv_h,η^<|r_A|^2[1-f(E+eV)]f(E)
    
    =   e∑_𝐪_∥∫dq_z/2π[(v_e,η^>-v_e,η^<|r|^2)f(E-eV)[1-f(E)]
       +v_h,η^<|r_A|^2[1-f(E+eV)]f(E)]
    
    =   e/h∑_𝐪_∥dE [(v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2)f(E-eV)[1-f(E)]
       +v_h,η^</v_e,z^>|r_A|^2[1-f(E+eV)]f(E)].

Here V is the longitudinal voltage drop along the junction and f(E)=1/(exp(E/k_BT)+1) is the Fermi-Dirac distribution function with k_B and T being the Boltzmann constant and temperature, respectively.



The net current from right to left (I^<_η) can be obtained by considering the incoming states in the right superconducting region.  Instead of dealing with quasiparticles in the superconducting region, it is equivalent to suppose the incident hole state with energy -E and transverse wave vector -𝐪_∥ in the left normal region, i.e., |ψ^<⟩=|ψ^>_h⟩+r̅|ψ^<_h⟩+r̅_A|ψ^<_e⟩, where r̅ (r̅_A) is the normal (Andreev) reflection amplitude for the hole state. Consequently, the net current carried by |ψ^<⟩ reads

    I^<_η=   -e/L∑_𝐪_∥,q_z(v_h,η^>-v_e,η^<|r̅|^2)f(-E+eV)[1-f(-E)]
       -e/L∑_𝐪_∥,q_zv_e,η^<|r̅_A|^2[1-f(-E-eV)]f(-E)
    
    =   -e/h∑_𝐪_∥∫ dE [(v_h,η^>/v_h,z^>-v_h,η^</v_h,z^>|r̅|^2)f(-E+eV)[1-f(-E)]
       +v_e,η^</v_h,z^>|r̅_A|^2[1-f(-E-eV)]f(-E)].

The particle-hole symmetry leads to the follow identities

    v_e,η^>/v_e,z^>=   v_h,η^>/v_h,z^>,  v_e,η^</v_e,z^>|r|^2=v_h,η^</v_h,z^>|r̅|^2,  v_h,η^</v_e,z^>|r_A|^2=v_e,η^</v_h,z^>|r̅_A|^2,



which simplify the expression for the total current 

    I_η=   I^>_η+I^<_η
    
    =   e/h∑_𝐪_∥∫_-∞^∞dE ([v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2][f(E-eV)-f(E)]+v_h,η^</v_e,z^>|r_A|^2[f(E)-f(E+eV))
    
    =   e/h∑_𝐪_∥∫_-∞^∞dE ([v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2]_(E)+[v_h,η^</v_e,z^>|r_A|^2]_(-E))[f(E-eV)-f(E)].

Consequently, the transverse conductance is given by

    σ_η,z=   ∂ I_η/∂ V
    
    =   e^2/h∑_𝐪_∥∫_-∞^∞dE([v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2]_(E)+[v_h,η^</v_e,z^>|r_A|^2]_(-E))[-∂(f(E-eV)-f(E))/∂ (E-eV)]
    
    =   e^2/h∑_𝐪_∥∫_-∞^∞dE([v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2]_(E)+[v_h,η^</v_e,z^>|r_A|^2]_(-E))δ(E-eV)
    
    =   e^2/h∑_𝐪_∥([v_e,η^>/v_e,z^>-v_e,η^</v_e,z^>|r|^2]_(eV)+[v_h,η^</v_e,z^>|r_A|^2]_(-eV)).

The longitudinal current can be calculated by the similar method, which reads

    I_z=   I_z^>+I_z^<
    
    =   e/h∑_𝐪_∥∫_-∞^∞ dE ([1+v_e,z^</v_e,z^>|r|^2][f(E-eV)-f(E)]-v_h,z^</v_e,z^>|r_A|^2[f(E)-f(E+eV)])
    
    =   e/h∑_𝐪_∥∫_-∞^∞ dE ([1+v_e,z^</v_e,z^>|r|^2]_(E)[f(E-eV)-f(E)]-[v_h,z^</v_e,z^>|r_A|^2]_(-E)[f(E-eV)-f(E)]).

Consequently, the longitudinal conductance is given by

    σ_z,z=   ∂ I_z/∂ V
    =   e^2/h∑_𝐪_∥∫_-∞^∞ dE([1+v_e,z^</v_e,z^>|r|^2]_(E)-[v_h,z^</v_e,z^>|r_A|^2]_(-E))[-∂(f(E-eV)-f(E))/∂ (E-eV)]
    
    =   e^2/h∑_𝐪_∥∫_-∞^∞ dE([1+v_e,z^</v_e,z^>|r|^2]_(E)-[v_h,z^</v_e,z^>|r_A|^2]_(-E))δ(E-eV)
    
    =   e^2/h∑_𝐪_∥(1+[v_e,z^</v_e,z^>|r|^2]_(eV)-[v_h,z^</v_e,z^>|r_A|^2]_(-eV)).


It is noted that, for the incident (reflected) states, the longitudinal group velocities are always positive (negative), i.e., v_e,z^><0 and v_e/h,z^<<0. However, the sign of the transverse group velocity is uncertain. Consequently, for the longitudinal conductance, the reflection coefficients can be written as

    v_e,z^</v_e,z^>|r|^2=-|v_e,z^</v_e,z^>||r|^2=-R,
    v_h,z^</v_e,z^>|r_A|^2=-|v_h,z^</v_e,z^>||r_A|^2=-R_A.

where R_A (R) is positive and denotes the reflection probability for the Andreev reflection (normal reflection). In the subgap regime, A+B=1 due to the current conservation. The longitudinal conductance [Eq. (<ref>)] can be expressed in terms of R and R_A

    σ_z,z=e^2/h∑_𝐪_∥(1-R+R_A),

which is the Blonder-Tinkham-Klapwijk formalism. The charge Hall angle (ϑ) and the chirality Hall angle (ϑ_chi) are given by

    tan(ϑ)=∑_χσ_η,z^χ/∑_χσ_z,z^χ,  tan(ϑ_chi)=∑_χχσ_η,z^χ/∑_χσ_z,z^χ.






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