ABSTRACT - DRUM - University of Maryland

Similar to the previous analysis, we first look for non-degenerate Majorana
zero-energy state. The Majorana condition ΞΨ ∝ Ψ fixes the value of m to be l−1
2
for odd l. For even l, there is no integer m satisfying Majorana condition so no
Majorana zero mode exists. The radial part of BdG equation then becomes
⎛
⎞
H r f(r)
⎜
⎟
⎝
⎠ ˜Ψ 0 (r) = 0 (3.23)
f(r) −σ y H r σ y
⎛
⎞
−µ v ( )
∂ r + m+1
r
H r = ⎜
⎝
−v ( )
∂ r − m −µ
r
⎟
⎠ . (3.24)
Here ˜Ψ 0 is assumed real. Since we are interested in non-degenerate solution, Ψ 0 must
be simultaneously an eigenstates of σ y τ y (particle-hole symmetry). This condition
implies that η ↑ = −λχ ↑ , η ↓ = λχ ↓ where λ = ±1. Taking into account above
constraints 4 × 4 BdG equation reduces to
⎛
⎜
⎝
−v ( ∂ r − m r
−µ v ( ⎞
)
∂ r + m+1
r + λf
⎟
) ⎠
− λf −µ
⎛ ⎞
χ ↑
⎜ ⎟
⎝ ⎠ = 0. (3.25)
χ ↓
The solution of the above equation can be easily obtained for µ ≠ 0:
⎛ ⎞ ⎛ ⎞
χ ↑
⎜ ⎟
⎝ ⎠ = N J m ( µ
⎜
r) v ⎟
∫ r
3 ⎝ ⎠ e−λ 0 dr′ f(r ′) . (3.26)
χ ↓ J m+1 ( µ r) v
Obviously, we should take λ = 1 to make radial wave functions normalizable. Here
N 3 is the normalization constant, which is determined by
4πN 2 3
∫ ∞
0
rdr
[J 2 m( µ v r) + J 2 m+1( µ v r) ]
e −2r/ξ = 1, (3.27)
yielding
N 2 3 =
8
πξ 2[ 8 2 F 1 ( 1 2 , 3 2 ; 1; −λ2 )+3λ 2 2F 1 ( 3 2 , 5 2 ; 3; −λ2 ) ] (3.28)
56