FullShell

Package to build full-shell hybrid semiconductor-superconductor hamiltonians.

Uses Quantica.jl.

How to install

This package is not registered in Julia. In order to use it, run the following code in a Julia REPL:

julia> using Pkg
julia> Pkg.activate()
julia> Pkg.add("https://github.com/CarlosP24/FullShell.jl.git")
julia> Pkg.instantiate()

Examples

Building a full-shell Hamiltonian

Default parameters are in meV and nm.

julia> using FullShell

julia> model = (;
        R = 70,
        w = 0,
        d = 0,
        preα = 0,
        Δ0 = 0.23,
        ξd = 70,
        τΓ = 1,
        g = 0,
        a0 = 5,
        μ = 0.75,
        α = 0,
        )
(R = 70, w = 0, d = 0, preα = 0, Δ0 = 0.23, ξd = 70, τΓ = 1, g = 0, a0 = 5, μ = 0.75, α = 0)

julia> hSM, hSC, parameters = build_cyl(; model...)
(ParametricHamiltonian{Float64,2,1}: Parametric Hamiltonian on a 1D Lattice in 2D space
  Bloch harmonics  : 3
  Harmonic size    : 1 × 1
  Orbitals         : [4]
  Element type     : 4 × 4 blocks (ComplexF64)
  Onsites          : 1
  Hoppings         : 2
  Coordination     : 2.0
  Parameters       : [:Vmax, :Vmin, :Z, :preα, :Φ, :α, :μ], ParametricHamiltonian{Float64,2,1}: Parametric Hamiltonian on a 1D Lattice in 2D space
  Bloch harmonics  : 3
  Harmonic size    : 1 × 1
  Orbitals         : [4]
  Element type     : 4 × 4 blocks (ComplexF64)
  Onsites          : 1
  Hoppings         : 2
  Coordination     : 2.0
  Parameters       : [:Vmax, :Vmin, :Z, :preα, :Φ, :α, :μ, :τΓ, :ω], FullShell.Params
  ħ2ome: Float64 76.1996
  μBΦ0: Float64 119.6941183
  m0: Float64 0.023
  g: Float64 0.0
  P: Float64 919.7
  Δg: Float64 417.0
  Δs: Float64 390.0
  preα: Float64 0.0
  a0: Float64 5.0
  t: Float64 66.26052173913044
  echarge: Float64 1.0
  R: Float64 70.0
  w: Float64 0.0
  d: Float64 0.0
  Vmax: Float64 0.0
  Vmin: Float64 0.0
  Vexponent: Float64 2.0
  Δ0: ComplexF64
  ξd: Float64 70.0
  α: Float64 0.0
  μ: Float64 0.75
  τΓ: Float64 1.0
  Φ: Float64 1.0
)

Calculating the LDOS at the edge of the nanowire

First, we need to obtain the Greens function at the edge of the nanowire. As we work with a semi-infinite system, we need to use a Schur Quantica.GreenSolver(see Quantica documentation).

julia> using Quantica
julia> g = hSC |> greenfunction(GS.Schur(boundary = 0))
GreenFunction{Float64,2,1}: Green function of a Hamiltonian{Float64,2,1}
  Solver          : AppliedSchurGreenSolver
  Contacts        : 0
  Contact solvers : ()
  Contact sizes   : ()
  ParametricHamiltonian{Float64,2,1}: Parametric Hamiltonian on a 1D Lattice in 2D space
    Bloch harmonics  : 3
    Harmonic size    : 1 × 1
    Orbitals         : [4]
    Element type     : 4 × 4 blocks (ComplexF64)
    Onsites          : 1
    Hoppings         : 2
    Coordination     : 2.0  
    Parameters       : [:Vmax, :Vmin, :Z, :preα, :Φ, :α, :μ, :τΓ, :ω]

Second, we build the LDOS function at the edge of the nanowire (cells -1or 1as the Shur boundary is at cell 0), which depends on the energy and all parameters of our Greens function (and Hamiltonian):

julia> ρ = ldos(g[cells = (-1)])
LocalSpectralDensitySlice{Float64} : local density of states at a fixed location and arbitrary energy
  kernel   : LinearAlgebra.UniformScaling{Bool}(true)

We are now able to calculate the LDOS at the edge of the semi-infinite for whatever set of parameters that we want, having set all defaults when defining the modelobject. If not specified there, defaults are those defined by FullShell.jl:

julia> ω = 0.0 + 1e-4im
0.0 + 0.0001im
julia> ρ(ω; ω = ω, Φ = 1, Z = 0)
1-element OrbitalSliceVector{Vector{Float64}}:
 1.6677498905272993e-6