!> Author: Jabir Ali Ouassou !> Category: Materials !> !> This submodule is included by conductor.f, and contains the equations which model spin-active interfaces. !> !> @TODO !> Reimplement shortcut-evaluation of the current for nonmagnetic interfaces. module spinactive_m use :: propagator_m use :: material_m use :: condmat_m private ! Public interface public spinactive ! Type declarations type :: spinactive ! Physical parameters of the interface real(wp) :: conductance = 1.0 !! Interfacial conductance real(wp) :: polarization = 0.0 !! Interfacial spin-polarization real(wp) :: spinmixing = 0.0 !! Interfacial 1st-order spin-mixing real(wp) :: secondorder = 0.0 !! Interfacial 2nd-order spin-mixing real(wp), dimension(1:3) :: magnetization = [0,0,1] !! Interfacial magnetization direction real(wp), dimension(1:3) :: misalignment0 = [0,0,0] !! Interfacial magnetization misalignment (this side) real(wp), dimension(1:3) :: misalignment1 = [0,0,0] !! Interfacial magnetization misalignment (other side) ! Fields used internally by the object type(nambu), private :: M !! Magnetization matrix (transmission) type(nambu), private :: M0 !! Magnetization matrix (reflection, this side) type(nambu), private :: M1 !! Magnetization matrix (reflection, other side) contains procedure :: diffusion_current => spinactive_diffusion_current procedure :: kinetic_current => spinactive_kinetic_current procedure :: update_prehook => spinactive_update_prehook end type contains impure subroutine spinactive_update_prehook(this) !! Updates the internal variables associated with spin-active interfaces. class(spinactive), intent(inout) :: this ! Transmission magnetization this % M = nambuv(this % magnetization) ! Reflection magnetization (this side) this % M0 = this % M if (nonzero(this % misalignment0)) then this % M0 = nambuv(this % misalignment0) end if ! Reflection magnetization (other side) this % M1 = this % M if (nonzero(this % misalignment1)) then this % M1 = nambuv(this % misalignment1) end if end subroutine pure function spinactive_diffusion_current(this, G0, G1) result(I) !! Calculate the matrix current at an interface with spin-active properties. The equations !! implemented here should be valid for an arbitrary interface polarization, and up to 2nd !! order in the transmission probabilities and spin-mixing angles of the interface. class(spinactive), intent(in) :: this type(nambu), intent(in) :: G0, G1 !! Propagator matrices type(nambu) :: S0, S1 !! Matrix expressions type(nambu) :: I !! Matrix current ! Evaluate the 1st-order matrix functions S0 = spinactive_current1_transmission(G1) S1 = spinactive_current1_reflection() ! Evaluate the 1st-order matrix current associate(S => S0 + S1) I = (S*G0 - G0*S) end associate ! Calculate the 2nd-order contributions to the matrix current. Note that we make a ! number of simplifications in this implementation. In particular, we assume that ! all interface parameters except the magnetization directions are equal on both ! sides of the interface. We also assume that the spin-mixing angles and tunneling ! probabilities of different channels have standard deviations that are much smaller ! than their mean values, which reduces the number of new fitting parameters to one. if (abs(this % secondorder) > 0) then ! Evaluate the 1st-order matrix functions associate(M1 => this % M1) S1 = spinactive_current1_transmission(G1*M1*G1 - M1) end associate ! Evaluate the 2nd-order matrix current I = I & + spinactive_current2_transmission() & + spinactive_current2_crossterms() & + spinactive_current2_reflection() end if ! Scale the final result based on conductance I = (this % conductance/2) * I contains pure function spinactive_current1_transmission(G) result(F) !! Calculate the 1st-order transmission terms in the matrix current commutator. type(nambu), intent(in) :: G type(nambu) :: F real(wp) :: Pr, Pp, Pm associate(P => this % polarization, M => this % M) Pr = sqrt(1 - P**2) Pp = 1 + Pr Pm = 1 - Pr F = G + (P/Pp)*(M*G+G*M) + (Pm/Pp)*(M*G*M) end associate end function pure function spinactive_current1_reflection() result(F) !! Calculate the 1st-order spin-mixing terms in the matrix current commutator. type(nambu) :: F associate(Q => this % spinmixing, M0 => this % M0) F = ((0,-1)*Q) * M0 end associate end function pure function spinactive_current2_transmission() result(I) !! Calculate the 2nd-order transmission terms in the matrix current. type(nambu) :: I associate(R => this % secondorder, Q => this % spinmixing) I = (-0.50*R/Q) * (S0*G0*S0) end associate end function pure function spinactive_current2_reflection() result(I) !! Calculate the 2nd-order spin-mixing terms in the matrix current. type(nambu) :: I, U associate(R => this % secondorder, Q => this % spinmixing, M0 => this % M0) U = M0*G0*M0 I = (0.25*R*Q) * (U*G0 - G0*U) end associate end function pure function spinactive_current2_crossterms() result(I) !! Calculate the 2nd-order cross-terms in the matrix current. type(nambu) :: I, U associate(R => this % secondorder, M0 => this % M0) U = S0*G0*M0 + M0*G0*S0 + S1 I = ((0.00,0.25)*R) * (U*G0 - G0*U) end associate end function end function pure subroutine spinactive_kinetic_current(this, G0, G1, C0, C1) !! Calculate the kinetic boundary coefficients at an interface with spin-active properties. !! These can be used to calculate the generalized current according to J = C₀H₀ - C₁H₁. class(spinactive), intent(in) :: this type(propagator), intent(in) :: G0 !! Propagator (this side) type(propagator), intent(in) :: G1 !! Propagator (other side) complex(wp), dimension(0:7,0:7), intent(out) :: C0 !! Boundary coefficient (this side) complex(wp), dimension(0:7,0:7), intent(out) :: C1 !! Boundary coefficient (other side) type(nambu), dimension(0:7) :: N type(nambu) :: GR0, GA0 type(nambu) :: GR1, GA1 integer :: i, j ! Construct the basis matrices do i=0,7 N(i) = nambuv(i) end do ! Construct the propagator matrices (this side) GR0 = G0 % retarded() GA0 = G0 % advanced() ! Construct the propagator matrices (other side) GR1 = G1 % retarded() GA1 = G1 % advanced() ! Construct the boundary coefficients do j=0,7 do i=0,7 ! Calculate the boundary matrix coefficients C0(i,j) = (this % conductance/8) * trace( ( R(GA1)*N(i) - N(i)*R(GR1) ) * ( (GR0*N(j) - N(j)*GA0) ) ) C1(i,j) = (this % conductance/8) * trace( ( GA0 *N(i) - N(i)* GR0 ) * ( T(GR1*N(j) - N(j)*GA1) ) ) end do end do contains pure function T(U) ! Calculates the contents of the spin-active boundary condition commutators: ! I ~ [F(G₁), G₀] ! This is used for the calculation of the boundary coefficient matrices. ! Note that this version of the function only includes transmission terms, ! and used to evaluate contributions from the other side of the interface. type(nambu), intent(in) :: U type(nambu) :: T real(wp) :: GMR real(wp) :: GT1 associate(M => this % M, P => this % polarization) ! Calculate the normalized interface conductances GMR = P/(1 + sqrt(1-P**2)) GT1 = (1 - sqrt(1-P**2))/(1 + sqrt(1-P**2)) ! Calculate the transmission function T = U + GMR * (M*U+U*M) + GT1 * M*U*M end associate end function pure function R(U) ! Calculates the contents of the spin-active boundary condition commutators: ! I ~ [F(G₁), G₀] ! This is used for the calculation of the boundary coefficient matrices. ! Note that this version of the function also includes reflection terms, ! and is used to evaluate contributions from this side of the interface. type(nambu), intent(in) :: U type(nambu) :: R complex(wp) :: Gphi associate(M => this % M0, Q => this % spinmixing) ! Calculate the normalized interface conductance Gphi = (0,-1) * Q ! Calculate the reflection function R = T(U) + Gphi * M end associate end function end subroutine end module