!> Author: Jabir Ali Ouassou
!> Category: Materials
!>
!> This submodule is included by conductor.f, and contains equations which model
!> spin-flip scattering, spin-orbit scattering, and magnetic orbital depairing.
module spinscattering_m
use :: condmat_m
use :: material_m
private
! Public interface
public spinscattering, spinscattering_construct
! Type declarations
type :: spinscattering
! Metadata
class(material), pointer :: material => null() !! Pointer to the material modelled by this instance
type(nambu), dimension(0:7) :: nambuv !! Pauli matrices spanning the 4×4 Spin-Nambu space
! Physical fields
real(wp) :: depairing = 0.0_wp !! Orbital depairing coefficient
real(wp) :: spinflip = 0.0_wp !! Spin-flip scattering coefficient (1/8τΔ)
real(wp) :: spinorbit = 0.0_wp !! Spin-orbit scattering coefficient (1/8τΔ)
contains
procedure :: diffusion_equation => spinscattering_diffusion_equation !! Diffusion equation
procedure :: kinetic_equation => spinscattering_kinetic_equation !! Kinetic equation
end type
! Type constructors
interface spinscattering
module procedure spinscattering_construct
end interface
contains
function spinscattering_construct(parent) result(this)
!! Constructs a spinscattering object with a given parent material.
type(spinscattering) :: this
class(material), target :: parent
integer :: i
! Save a pointer to the parent object
this % material => parent
! Memoize the most used basis matrices
do i=0,7
this % nambuv(i) = nambuv(i)
end do
end function
pure subroutine spinscattering_diffusion_equation(this, p)
!! Calculate the spin-flip and spin-orbit scattering terms in the diffusion
!! equation, and update the second derivatives of the Riccati parameters.
class(spinscattering), intent(in) :: this
type(propagator), intent(inout) :: p
type(nambu) :: G, R
type(nambu) :: Gsf, Gso, Gdp
real(wp) :: Csf, Cso, Cdp
! Construct the propagator matrix
G = p % retarded()
! Construct the self-energy matrices
associate(N => this % nambuv)
Gdp = N(4)*G*N(4)
Gsf = N(1)*G*N(1) + N(2)*G*N(2) + N(3)*G*N(3)
Gso = N(4)*Gsf*N(4)
end associate
! Calculate the self-energy prefactors
! (including 1/2i from the Riccati-parametrized Usadel equation)
Cdp = (this % depairing) / (2 * 4 * this % material % thouless)
Csf = (this % spinflip ) / (2 * 1 * this % material % thouless)
Cso = (this % spinorbit) / (2 * 1 * this % material % thouless)
! Calculate the self-energy commutators
R = Cdp*Gdp + Csf*Gsf + Cso*Gso
R = R*G - G*R
! Update the second derivatives of the Riccati parameters
associate( U => R % matrix, &
g => p % g, gt => p % gt, &
dg => p % dg, dgt => p % dgt, &
d2g => p % d2g, d2gt => p % d2gt )
d2g = d2g + (pauli0 - g*gt) * (U(1:2,3:4) - U(1:2,1:2)*g )
d2gt = d2gt + (pauli0 - gt*g) * (U(3:4,1:2) - U(3:4,3:4)*gt)
end associate
end subroutine
pure subroutine spinscattering_kinetic_equation(this, Gp, R)
!! Calculate the self-energies in the kinetic equation.
class(spinscattering), intent(in) :: this
type(propagator), intent(in) :: Gp
complex(wp), dimension(0:7,0:7), intent(inout) :: R
complex(wp) :: Csf, Cso, Cdp
! Calculate the self-energy prefactors
Cdp = (0,1) * (this % depairing) / (4 * this % material % thouless)
Csf = (0,1) * (this % spinflip ) / (1 * this % material % thouless)
Cso = (0,1) * (this % spinorbit) / (1 * this % material % thouless)
! Calculate the self-energy contributions
associate(N => this % nambuv)
R = R &
+ Csf * Gp % selfenergy2(N(1)) &
+ Csf * Gp % selfenergy2(N(2)) &
+ Csf * Gp % selfenergy2(N(3)) &
+ Cdp * Gp % selfenergy2(N(4)) &
+ Cso * Gp % selfenergy2(N(5)) &
+ Cso * Gp % selfenergy2(N(6)) &
+ Cso * Gp % selfenergy2(N(7))
end associate
end subroutine
end module
