Non-collinear magnetism in WIEN2k

wien workshop 2003 – p.2/40

wien workshop 2003 – p.2/40

Outlinetheoretical introduction;wien workshop 2003 – p.3/40

Outlinetheoretical introduction;try to place implemented equations in a sequance ofapproximations begining from Dirac equationwien workshop 2003 – p.3/40

Outlinetheoretical introduction;try to place implemented equations in a sequance ofapproximations begining from Dirac equationimportant details of implementations;wien workshop 2003 – p.3/40

Outlinetheoretical introduction;try to place implemented equations in a sequance ofapproximations begining from Dirac equationimportant details of implementations;examples of calculations;wien workshop 2003 – p.3/40

Outlinetheoretical introduction;try to place implemented equations in a sequance ofapproximations begining from Dirac equationimportant details of implementations;examples of calculations;how to run the code?;wien workshop 2003 – p.3/40

Dirac HamiltonianSpin as a dynamical variable is of great importance inmagnetism. Thus we start with Dirac Hamiltonian.H D = c⃗α · ⃗p + βmc 2 + Vwien workshop 2003 – p.4/40

¡¢ £¡¡¢ £¢ £Dirac HamiltonianH D = c ⃗α · ⃗p + βmc 2 + V⃗α is 3-component vectorof 4x4 matrixes:α k =⎛⎝ 0 σ kσ k 0⎞⎠Pauli matrixes:σ 1 =0 11 0σ 2 =0 −ii 0σ 3 =1 00 −1wien workshop 2003 – p.5/40

Dirac HamiltonianH D = c⃗α · ⃗p +β mc 2 + Vβ is 4x4 matrix: β =⎛⎝ 1 00 −1⎞⎠ 1 =⎛⎝ 1 00 1⎞⎠wien workshop 2003 – p.6/40

Dirac HamiltonianH D = c⃗α · ⃗p + βmc 2 + Vβ is 4x4 matrix: β =⎛⎝ 1 00 −1⎞⎠ 1 =⎛⎝ 1 00 1⎞⎠m - is electron mass, c - speed of the lightp - momentum operatorV - efective potential, which is related to density (DFT)wien workshop 2003 – p.6/40

Dirac equationH D = c⃗α · ⃗p + βmc 2 + V⎛H D ⎜⎝ψ 1ψ 2ψ 3⎞ ⎛= ε⎟ ⎜⎠ ⎝ψ 1ψ 2ψ 3⎞⎟⎠ψ 4ψ 4Dirac equationwien workshop 2003 – p.7/40

Dirac equationH D = c⃗α · ⃗p + βmc 2 + V⎛H D ⎜⎝ψ 1ψ 2ψ 3⎞ ⎛= ε⎟ ⎜⎠ ⎝ψ 1ψ 2ψ 3⎞⎟⎠ψ 4ψ 4Hamiltonian is 4x4 matrix operatorwave function is 4 component vectorwien workshop 2003 – p.7/40

Dirac equationH D = c⃗α · ⃗p + βmc 2 + Vlargecomponent⎛H D ⎜⎝ψ 1ψ 2ψ 3ψ 4⎞⎟⎠⎛= ε⎜⎝ψ 1ψ 2ψ 3ψ 4⎞⎟⎠smallcomponentHamiltonian is 4x4 matrix operatorwave function is 4 component vectorwien workshop 2003 – p.7/40

Dirac equationH D = c⃗α · ⃗p + βmc 2 + Vlargecomponent⎛H D ⎜⎝ψ 1ψ 2ψ 3ψ 4⎞⎟⎠⎛= ε⎜⎝ψ 1ψ 2ψ 3ψ 4⎞⎟⎠smallcomponentneglecting contribution of small component to the charge density, onecan derive 2x2 Pauli like Hamiltonianwien workshop 2003 – p.7/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .wien workshop 2003 – p.8/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .Hamiltonian is 2x2 matrix operatorwave function is 2 component vectorwien workshop 2003 – p.8/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .Hamiltonian is 2x2 matrix operatorwave function is 2 component vector⎛ ⎞ ⎛ ⎞H P⎝ ψ 1⎠ = ε ⎝ ψ 1⎠ψ 2 ψ 2wien workshop 2003 – p.8/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .Hamiltonian is 2x2 matrix operatorwave function is 2 component vectorspin upcomponentH P⎛⎝ ψ 1ψ 2⎞⎠ = ε⎛⎝ ψ 1ψ 2⎞⎠spin downcomponentwien workshop 2003 – p.8/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗ Beff + ζ ·(⃗σ · ⃗l)+ . . .effective electrostaticpotentialeffective magneticfieldspin-(orbit)interaction termwien workshop 2003 – p.9/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗ Beff + ζ ·(⃗σ · ⃗l)+ . . .effective electrostaticpotentialeffective magneticfieldspin-(orbit)interaction termV eff = V ext + V H + V xc⃗ Beff = ⃗ B ext + ⃗ B xcwien workshop 2003 – p.9/40

Pauli HamiltonianH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗ Beff + ζ ·(⃗σ · ⃗l)+ . . .effective electrostaticpotentialeffective magneticfieldspin-(orbit)interaction termV eff = V ext + V H + V xc⃗ Beff = ⃗ B ext + ⃗ B xcquite important and difficult to define, describemany body effectswien workshop 2003 – p.9/40

Exchange potential and fieldIn DFT V xc and ⃗ B xc are defined by:V xc = ∂E xc (n, ⃗m), Bxc ⃗ = ∂E xc (n, ⃗m)∂n∂ ⃗mIn local density approximation (LDA):E xc (n, ⃗m) =∫nɛ xc (n, m) dr 3what results in:V xc = ɛ xc (n, m)+n ∂ɛ xc (n, m)∂n, ⃗ Bxc = n ∂ɛ xc (n, m)∂mˆmwien workshop 2003 – p.10/40

Exchange potential and fieldIn DFT V xc and ⃗ B xc are defined by:V xc = ∂E xc (n, ⃗m)∂nIn local density approximation (LDA):, ⃗ Bxc = ∂E xc (n, ⃗m)∂ ⃗mE xc (n, ⃗m) = nɛ xc (n, m) dr 3what results in:V xc = ɛ xc (n, m) + n ∂ɛ xc (n, m)∂n, ⃗ Bxc = n ∂ɛ xc(n,m)∂mˆmexchange field is parallel to local magnetisationdensity vectorwien workshop 2003 – p.10/40

¡¢ £Non-collinear caseH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .If we take everything, and also allow magnetisation to vary its directionfrom point to point, we will end up with 2x2 Hamiltonian:− 22m ∇2 + V eff + µ B ⃗σ · ⃗B z + . . . µ B (B x − iB y ) + . . .µ B (B x + iB y ) + . . . − 22m ∇2 + V eff − µ B ⃗σ · ⃗B z + . . .ϕ = εϕwien workshop 2003 – p.11/40

¡¢ £Non-collinear caseH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .If we take everything, and also allow magnetisation to vary its directionfrom point to point, we will end up with 2x2 Hamiltonian:− 22m ∇2 + V eff + µ B ⃗σ · ⃗B z + . . . µ B (B x − iB y ) + . . .µ B (B x + iB y ) + . . . − 22m ∇2 + V eff − µ B ⃗σ · ⃗B z + . . .ϕ = εϕsolutions are non-pure spinors⎛ ⎞ϕ =⎝ ψ ↑ψ ↓⎠ , ψ ↑ , ψ ↓ ≠ 0this is non-collinearitywien workshop 2003 – p.11/40

¡¢ £Collinear caseH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .If we ignore only spin-orbit term, and allow magnetisation to point in onedirection (z), resulting Hamiltonian will be diagonal in spin-space.− 22m ∇2 + V eff + µ B ⃗σ · ⃗B z 00 − 22m ∇2 + V eff − µ B ⃗σ · ⃗B zϕ = εϕϕ ↑ =⎛⎝ ψ ↑0⎞⎠ , ϕ ↓ =⎛⎝ 0 ψ ↓⎞⎠solutions are pure spinorswith non-degenerateenergiesε ↑ ≠ ε ↓collinear magnetismwien workshop 2003 – p.12/40

¡¢ £Non-magnetic caseH P = − 22m ∇2 + V eff + µ B ⃗σ · ⃗B eff + ζ ·(⃗σ · ⃗l)+ . . .If we ignore spin-orbit term and magnetic exchange field, resultingHamiltonian will be diagonal in spin-space.− 22m ∇2 + V eff 00 − 22m ∇2 + V effϕ = εϕϕ ↑ =⎛⎝ ψ 0⎞⎠ , ϕ ↓ =⎛⎝ 0 ψ⎞⎠solutions are pure spinorswith degenerate energiesnon-magnetic solutionε ↑ = ε ↓wien workshop 2003 – p.13/40

ConsequencesImplementation of non-collinearity means that we allow bothpart of the eigen-spinor (ψ ↑ , ψ ↓ ) to be non zero at the sametime. As a result:wien workshop 2003 – p.14/40

ConsequencesImplementation of non-collinearity means that we allow bothpart of the eigen-spinor (ψ ↑ , ψ ↓ ) to be non zero at the sametime. As a result:we have to deal with 2x2 Hamiltonian, instead of 1x1 asin collinear caseswien workshop 2003 – p.14/40

ConsequencesImplementation of non-collinearity means that we allow bothpart of the eigen-spinor (ψ ↑ , ψ ↓ ) to be non zero at the sametime. As a result:we have to deal with 2x2 Hamiltonian, instead of 1x1 asin collinear casesthis means that diagonalisation is 4 times more expensivewien workshop 2003 – p.14/40

ConsequencesImplementation of non-collinearity means that we allow bothpart of the eigen-spinor (ψ ↑ , ψ ↓ ) to be non zero at the sametime. As a result:we have to deal with 2x2 Hamiltonian, instead of 1x1 asin collinear casesthis means that diagonalisation is 4 times more expensivethis is usually much more because of losing symmetryoperationswien workshop 2003 – p.14/40

Implementations of NCM in LAPWψ ↑ , ψ ↓ can be non-zero only when basis allows for that.wien workshop 2003 – p.15/40

Implementations of NCM in LAPWψ ↑ , ψ ↓ can be non-zero only when basis allows for that.L. Nordström and D. J. Singh Phys. Rev. Lett. 76, 4420 (1996).pure spinor non polarised basis, given in one global spincoordinate frame,basis must be supplemented by additional local orbitals,magnetisation is a continuous field.wien workshop 2003 – p.15/40

Implementations of NCM in LAPWPh. Kurz at al. Phys. Rev. B 63, 96401 (2001),interstitial region and each atomic sphere have their ownquantisation axis,quantisation axis of a sphere is supposed to point in adirection of average magnetisation,basis functions are pure spinor in interstitial region,inside spheres non-pure spinors but polarised,atomic moment approximation (AMA).This is implemented in WIEN2kwien workshop 2003 – p.16/40

Spin coordinate setsα 1α2wien workshop 2003 – p.17/40

Spin coordinate setsα 1α2⎛⎞⎛⎞interstices: ϕ ⃗ Gσ = ei ( ⃗ G+ ⃗ k)·⃗r χ σ , where χ σ =⎝ 1 0⎠, or⎝ 0 1⎠wien workshop 2003 – p.17/40

Basis functionsspheres - combination of pure spinors in a local coordinate frame:the direction of the quantisation axis is along an averagemagnetisation inside the sphere,α 1α2wien workshop 2003 – p.18/40

Basis functionsspheres - combination of pure spinors in a local coordinate frame:the direction of the quantisation axis is along an averagemagnetisation inside the sphere,ϕ AP W⃗Gσ(⃗k)ϕ LO⃗Gσ α == ∑ σ α ∑LGσ(A ⃗ αL(A ⃗ Gσσ αu σαlLu σαl+ B ⃗ Gσσ αL+ B ⃗ Gσσ αL˙u σαl+ C ⃗ Gσ αL)˙u σαl Y L χ σ α,)u σα2,l Y L χ σ αY L is a spherical harmonic, L stands for (l, m),χ σ α is a spinor given in a local coordinate frame,u, ˙u are radial function and its energy derivative.wien workshop 2003 – p.18/40

Augmentation (...)ϕ AP W⃗Gσ(⃗k)ϕ LO⃗Gσ α == ∑ σ α ∑(LA ⃗ Gσ αL(A ⃗ Gσσ αu σαlLu σαl+ B ⃗ Gσσ αL+ B ⃗ Gσσ αL˙u σαl+ C ⃗ Gσ αL)˙u σαl Y L χ σ α,)u σα2,l Y L χ σ αA L , B L , C L are calculated from sphere boundary conditions.wien workshop 2003 – p.19/40

Augmentation (...)ϕ AP W⃗Gσ(⃗k)ϕ LO⃗Gσ α == ∑ σ α ∑(LA ⃗ Gσ αL(A ⃗ Gσσ αu σαlLu σαl+ B ⃗ Gσσ αL+ B ⃗ Gσσ αL˙u σαl+ C ⃗ Gσ αL)˙u σαl Y L χ σ α,)u σα2,l Y L χ σ αA L , B L , C L are calculated from sphere boundary conditions.for APW/LAPW matching is done to ↑ and ↓ plane waves in aglobal spin coordinate frame. Thus A L , B L depend on global σand local σ α spin indexes, and on k-point.wien workshop 2003 – p.19/40

Augmentation (...)ϕ AP W⃗Gσ(⃗k)ϕ LO⃗Gσ α == ∑ σ α ∑(LA ⃗ Gσ αL(A ⃗ Gσσ αu σαlLu σαl+ B ⃗ Gσσ αL+ B ⃗ Gσσ αL˙u σαl+ C ⃗ Gσ αL)˙u σαl Y L χ σ α,)u σα2,l Y L χ σ αA L , B L , C L are calculated from sphere boundary conditions.for APW/LAPW matching is done to ↑ and ↓ plane waves in aglobal spin coordinate frame. Thus A L , B L depend on global σand local σ α spin indexes, and on k-point.LO has to vanish at a sphere boundary. It is a pure spinor in alocal coordinate frame (A L , B L , C L depend only on σ α ).wien workshop 2003 – p.19/40

Augmentation (...)ϕ AP W⃗Gσ(⃗k)ϕ LO⃗Gσ α == ∑ σ α ∑(LA ⃗ Gσ αL(A ⃗ Gσσ αu σαlLu σαl+ B ⃗ Gσσ αL+ B ⃗ Gσσ αL˙u σαl+ C ⃗ Gσ αL)˙u σαl Y L χ σ α,)u σα2,l Y L χ σ αA L , B L , C L are calculated from sphere boundary conditions.for APW/LAPW matching is done to ↑ and ↓ plane waves in aglobal spin coordinate frame. Thus A L , B L depend on global σand local σ α spin indexes, and on k-point.LO has to vanish at a sphere boundary. It is a pure spinor in alocal coordinate frame (A L , B L , C L depend only on σ α ).for APW type of basis B = 0 in PW and LO, and additional localorbital “lo” is introduced with C = 0, and B ≠ 0.wien workshop 2003 – p.19/40

Hamiltonian (...)In the interstitial region we don’t have spin-orbit,Hamiltonian is a sum of only kinetic energy and effective potentials:Ĥ = − 22m ∇2 + ˆV ,where ˆV combines V eff and B eff in a form of 2x2 potential matrix.wien workshop 2003 – p.20/40

Hamiltonian (...)In the interstitial region we don’t have spin-orbit,Hamiltonian is a sum of only kinetic energy and effective potentials:Ĥ = − 22m ∇2 + ˆV ,where ˆV combines V eff and B eff in a form of 2x2 potential matrix.In the spheres w can have everythingThus, Hamiltonian is more complicated:Ĥ = − 22m ∇2 + ˆV + Ĥ so + Ĥ orb + Ĥ cAs a result, there are same choices:wien workshop 2003 – p.20/40

Hamiltonian - spheres (...)inside sphere potential matrix: ˆV = ˆV d + ˆV off .wien workshop 2003 – p.21/40

Hamiltonian - spheres (...)inside sphere potential matrix: ˆV = ˆV d + ˆV off .AMA mode (atomic moment approximation) ˆV off is ignored.⎛ ⎞ˆV = ⎝ V ↑↑ 0⎠0 V ↓↓wien workshop 2003 – p.21/40

Hamiltonian - spheres (...)inside sphere potential matrix: ˆV = ˆV d + ˆV off .AMA mode (atomic moment approximation) ˆV off is ignored.⎛ ⎞ˆV = ⎝ V ↑↑ 0⎠0 V ↓↓FULL mode (non-collinearity inside spheres)⎛ ⎞ ⎛ˆV = ⎝ V ↑↑ 0⎠ + ⎝ 0 V ↑↓0 V ↓↓ V ↓↑ 0⎞⎠wien workshop 2003 – p.21/40

¡¢ £Hamiltonian - spheres (...)SO (spin-orbit coupling) -Ĥ so = ξ⃗σ · ⃗l = ξˆlzˆlx + iˆl yσ x − iˆl y−ˆl z,wien workshop 2003 – p.22/40

¡¡¢ £¢ £Hamiltonian - spheres (...)SO (spin-orbit coupling) -Ĥ so = ξ⃗σ · ⃗l = ξˆlzˆlx + iˆl yσ x − iˆl y−ˆl z,ORB (lda+U) -Ĥ orb =m,m ′|m〉 v ↑ mm ′ 〈m ′ | 00 |m〉 v ↓ mm ′ 〈m ′ |,wien workshop 2003 – p.22/40

¡¡¡¢ £¢ £¢ £Hamiltonian - spheres (...)SO (spin-orbit coupling) -Ĥ so = ξ⃗σ · ⃗l = ξˆlzˆlx + iˆl yσ x − iˆl y−ˆl z,ORB (lda+U) -Ĥ orb =m,m ′|m〉 v ↑ mm ′ 〈m ′ | 00 |m〉 v ↓ mm ′ 〈m ′ |,constrain field ( ⃗ B c ⊥ ẑ α ) -Ĥ c = µ B⃗ Bc · ⃗σ =0 µ B (B c,x − iB c,y )µ B (B c,x + iB c,y ) 0.wien workshop 2003 – p.22/40

¡¡¡¢ £¢ £¢ £Hamiltonian - spheres (...)SO (spin-orbit coupling) -Ĥ so = ξ⃗σ · ⃗l = ξˆlzˆlx + iˆl yσ x − iˆl y−ˆl z,ORB (lda+U) -Ĥ orb =m,m ′|m〉 v ↑ mm ′ 〈m ′ | 00 |m〉 v ↓ mm ′ 〈m ′ |,constrain field ( ⃗ B c ⊥ ẑ α ) -Ĥ c = µ B⃗ Bc · ⃗σ =0 µ B (B c,x − iB c,y )µ B (B c,x + iB c,y ) 0.wien workshop 2003 – p.22/40

Allocation of the Hamiltonian and overlap matrixes (...)APW upLO upMatrix elements(integrals):H ⃗G, ⃗ G′ =〈 ∣ 〉 ∣∣ ψ ⃗G Ĥ∣ φ ⃗G ′ ,S ⃗G, ⃗ G′ = 〈 ψ ⃗G | φ ⃗G ′〉,APW downwhere ψ and φ can beAPW or LO, ↑ or ↓LO downwien workshop 2003 – p.23/40

Allocation of the Hamiltonian and overlap matrixes (...)APW upcollinear setupLO upAPW downLO downwien workshop 2003 – p.23/40

Allocation of the Hamiltonian and overlap matrixes (...)collinear setupAPW upLO upAPW downLO downwien workshop 2003 – p.23/40

Allocation of the Hamiltonian and overlap matrixes (...)collinear setupAPW upLO upAPW downLO downwien workshop 2003 – p.23/40

Spin-spiralsspin-spiral is defined by a vector ⃗q given in reciprocal space and anangle υ between magnetic moment and rotation axis,spin rotation (α)lattice translation (R)⃗m n = ⃗mα = ⃗R · ⃗q( (cos ⃗q · ⃗R n) (sin υ, sin ⃗q · ⃗R n) )sin υ, cos υthe direction of the rotation axis is arbitrary.wien workshop 2003 – p.24/40

Spin-spiralspure translations are obviously not symmetry operations of H,H(⃗r + ⃗R n) ≠ H (⃗r) ,wien workshop 2003 – p.25/40

Spin-spiralspure translations are obviously not symmetry operations of H,H(⃗r + ⃗R n) ≠ H (⃗r) ,translations coupled with spin space rotations T n =are symmetry operations of H;T n † H (⃗r) T n = U † ( −⃗q · ⃗R n) H(⃗r + ⃗ R n) U{−⃗q · ⃗R n |ɛ| ⃗ R n }(−⃗q · ⃗R n) ,wien workshop 2003 – p.25/40

Spin-spiralspure translations are obviously not symmetry operations of H,H(⃗r + ⃗R n) ≠ H (⃗r) ,translations coupled with spin space rotations T n =are symmetry operations of H;T n † H (⃗r) T n = U † ( −⃗q · ⃗R n) H(⃗r + ⃗ R n) U{−⃗q · ⃗R n |ɛ| ⃗ R n }(−⃗q · ⃗R n) ,Group of T n is Abelian, thus it has one dimensional representations,T n ψ k (r) = U (−q · R n ) ψ k (r + R n ) = e ik·Rn ψ k (r) .wien workshop 2003 – p.25/40

Spin-spiralspure translations are obviously not symmetry operations of H,H(⃗r + ⃗R n) ≠ H (⃗r) ,translations coupled with spin space rotations T n =are symmetry operations of H;T n † H (⃗r) T n = U † ( −⃗q · ⃗R n) H(⃗r + ⃗ R n) U{−⃗q · ⃗R n |ɛ| ⃗ R n }(−⃗q · ⃗R n) ,Group of T n is Abelian, thus it has one dimensional representations,T n ψ k (r) = U (−q · R n ) ψ k (r + R n ) = e ik·Rn ψ k (r) .Bloch theorem!!!wien workshop 2003 – p.25/40

Spin-spiralsWave function of a spiral structure is of the form:⎛⎞ψ k (r) = e ik·r ⎝ e −iq·r2 u ↑ k (r) ⎠e iq·r2 u ↓ k (r) ,where u σ (r) has translational periodicity.wien workshop 2003 – p.26/40

Spin-spiralsWave function of a spiral structure is of the form:⎛⎞ψ k (r) = e ik·r ⎝ e −iq·r2 u ↑ k (r) ⎠e iq·r2 u ↓ k (r) ,where u σ (r) has translational periodicity.↑ part of a spinor transform with k − q 2, ↓ part of a spinor transformwith k + q 2. The basis has to be composed using condition:∣∣G + k ± q ∣ ≤ G max ,2with “−” for ↑, and “+” for ↓. This can generate different numbers ofbasis functions for ↑ and ↓ spins.wien workshop 2003 – p.26/40

Intra-atomic NCM, fcc P u(a) plane x = 0 (b) plane z = 1/10Spin density maps of fcc Pu. Calculation in FULL mode with SO. Average momenta point to〈001〉wien workshop 2003 – p.27/40

Intra-atomic NCM, bcc U(c) plane x = 1/2 (d) plane z = 6/10Spin density maps of bcc U (unit cell size 9 a.u.). Calculation in FULL mode with SO.Average momenta point to 〈001〉wien workshop 2003 – p.28/40

γ Fe, spin spiral0total energy [Ry]-0,001-0,002-0,003AMAFULLlocal spin moment [µ B]1,81,71,61,51,4AMAFULL-0,004ΓqX1,3ΓqXLocal spin moment and total energy versus spin-spiral ⃗q vector.wien workshop 2003 – p.29/40

γ Fe, spin spiralAMAFULLSpin density maps for q = 0.6 (0-Γ, 1-X)wien workshop 2003 – p.30/40

Magnetic structure of Mn 3 SnChemical structure of Mn 3 Snwien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snfmwien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snfmafmwien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snfm afm ncm 1wien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snfm afm ncm 1ncm 2wien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snfm afm ncm 1ncm 2 ncm 3wien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snfm afm ncm 1ncm 2 ncm 3 ncm 4wien workshop 2003 – p.31/40

Magnetic structure of Mn 3 Snso fm afm ncm 1 ncm 2 ncm 3 ncm 4E fm − E [Ry] - 0.0 0.0131 0.0444 0.0444 0.0444 0.0444+ 0.0 0.0133 0.0441 0.0439 0.0444 0.0445M s [µ B ] - 3.012 2.684 3.037 3.037 3.037 3.037+ 3.008 2.679 3.034 3.034 3.038 3.037efg on Mn - -1.657 -2.111 -0.894 -0.894 -0.894 -0.894[10 21 V/m 2 ] + -1.661 -2.119 -0.892 -0.899 -0.891 -0.894-0.898 -0.881hff on Mn - -309.9 -153.1 31.2 31.2 31.2 31.2[kGauss] + -309.6 -152.9 31.1 31.5 31.5 30.9in non-so case all ncm structures are symmetry equivalent32.2 32.1with so ncm structures become inequivalent, additionally for ncm3 and ncm4 Mn 2 andMn 5 are no longer equivalent to the rest of Mn’swien workshop 2003 – p.32/40

NCM input filecase.inncm:FULL0 0 060.00000 90.00000 0180.00000 90.00000 0300.00000 90.00000 060.00000 90.00000 0180.00000 90.00000 0300.00000 90.00000 00.00000 0.00000 00.00000 180.00000 00.50000wien workshop 2003 – p.33/40

NCM input filecase.inncm:FULL0 0 060.00000 90.00000 0180.00000 90.00000 0300.00000 90.00000 060.00000 90.00000 0180.00000 90.00000 0300.00000 90.00000 00.00000 0.00000 00.00000 180.00000 00.50000first record defines mode of calculation(FULL/AMA)second record defines spin-spiralvector;next records contain polar angles (φ, θ)defining directions of magnetisation ineach sphere and optimisation switch;last number is a mixing parameter usedduring calculation of the constrain field;wien workshop 2003 – p.33/40

NCM input filecase.inncm:FULL0 0 060.00000 90.00000 0180.00000 90.00000 0300.00000 90.00000 060.00000 90.00000 0180.00000 90.00000 0300.00000 90.00000 00.00000 0.00000 00.00000 180.00000 00.50000first record defines mode of calculation(FULL/AMA)second record defines spin-spiralvector;next records contain polar angles (φ, θ)defining directions of magnetisation ineach sphere and optimisation switch;last number is a mixing parameter usedduring calculation of the constrain field;case.inncm is used in initialisation stage, to lower chemical symmetrydue to magnetic moments;lapw1, lapw2, lapwdm use it to define quantisation axes for atomicspheres;wien workshop 2003 – p.33/40

How to run it?initialisation – looks like in collinear code, but symmetry must be lowered because ofmagnetic momenta;x ncmsymmetrySCF cycle looks like:x lapw0 -ncm [...][ x orb -up, x orb -du ]x lapw1 -ncm [-p -orb -so ...]x lapw2 -ncm [-p ....][ x lapwdm -ncm [-p ...] ]x lcore -upx lcore -dnx mixer -ncmSCF job can be run with runncm script which is modified version of WIEN2k run scriptrunncm [-p -so -orb -cc 0.0001 ...]wien workshop 2003 – p.34/40

AcknowledgementsMany thanks to:Georg MadsenPeter BlahaKarlheinz SchwarzThe Endwien workshop 2003 – p.35/40

¢£¢¢££¡¡Appendix - augmentation go backExpansion coefficients A ⃗ Gσσ αLand B ⃗ Gσσ αLare calculated from matching condition:e i ( ⃗ G+ ⃗ k)⃗r χ g σ =σ α LGσσA ⃗ αL u σαl+ B ⃗ Gσσ αL˙u σαlY L χ g σ α ,where the both spinors are represented in a global coordinate frame. Multiplying both sides∗by , integrating over spin variable, and comparing to the collinear expression:χ g σ αe i ( ⃗ G+ ⃗ k)·⃗r =LGσA ⃗ αL u σαl+ B ⃗ Gσ αL˙u σαlY L .A ⃗ Gσσ αLand BGσσ ⃗ αLare given by:A ⃗ Gσσ αL =B ⃗ Gσσ αL =χ g σ αχ g σ α∗ χgσ A ⃗ Gσ αL ,∗ χgσ B ⃗ Gσ αL .wien workshop 2003 – p.36/40

§¦¡¡¤¤¦¦¡¥¡¡¤Appendix - Hamiltonian setupgo backinterstices - basis functions are plane waves: ϕ ⃗Gσ = e i ( ⃗ G+ ⃗ k)⃗·r χ σ , and matrixelements:ϕ ⃗Gσ¡¢¡£¡Ĥ¡¢¡£¡ϕ ⃗G ′ σ′= (V σσ ′Θ) ( ⃗ G− ⃗ G′ ) + δ σσ ′2m⃗G ′ + ⃗ k2Θ( ⃗ G− ⃗ G′ )where Θ is a step function, V σσ ′ are components of the potential matrix.spheres - ϕ P ⃗Gσ W = σ α φG ⃗σσ αχ α =Hamiltonian looks like:σ αL A ⃗ Gσσ αLu σαlY L χ α , setup of theϕ P W⃗Gσ |H | ϕ P W⃗G ′ σ ′=σ αLA σαL⃗G, σB σαL⃗G ′ , σ ′,A σαL⃗G, σ= A ⃗ Gσσ αL ,B σαL⃗G ′ , σ ′=GA ⃗ ′ σ ′ σ ′αL ′σ ′α L ′u σαl Y L | H σ α σ ′α | uσ′αl ′ Y L ′.wien workshop 2003 – p.37/40

¦¡¡¦¢ £Appendix - density matrixn + m zm x − i m yρ σσ ′ = 1 2 nI 2 + σ · −→ m = 1 2m x + i m yn − m zinterstices (ρ ⃗ Gσσ ′)ρ σσ ′ (⃗r) =〈σ | ψ υ 〉 ψ υ | σ ′=ρ ⃗ Gσσ ′e i ⃗ G⃗·rυ⃗Gwave function is generated in real space: ψυ σ (⃗r) =indexes band and k-point.ρ σσ ′ (⃗r) =υ (ψσ υ (⃗r)) ∗ ψ σ′υ (⃗r) ,ρ σσ ′ (⃗r) is Fourier transformed into ρ ⃗ Gσσ ′.⃗G c ⃗ Gυ e i ( ⃗ G+⃗r)⃗·r χ σ , υwien workshop 2003 – p.38/40

¡¢£Appendix - density matrixspheres (ρ L σ α σ ′α )ρ σ α σ ′α (⃗r) =υ〈σ α | ψ υ 〉 ψ υ | σ ′α=Lρ L σ α σ ′α (r) Y s L (ˆr)ρ L σ α σ ′α are calculated with the following expression:ρ L σ α σ ′α =ij l ′ l ′′ υ m ′ m ′′ D iσαυL ′ D jσ′αυL ′′ GL, L ′ , L ′′u σαl ′ u σ′αl ′′ ,where D iσαυL(i stands for A, B, C) are given by:P W :D A,σαυL=σGc ⃗ Gυ A ⃗ Gσσ αLLO, lo :D A,σαυL=Gc ⃗ Gυ A ⃗ Gσ αLwien workshop 2003 – p.39/40

¡¢ £¡¢ £Appendix - potential matrix1. local real space diagonalisation:ρ L σ α σ ′αρ ⃗ Gσσ ′→ρ σ α σ ′α (⃗r)ρ σσ ′ (⃗r)→ρ ↑ (r) 00 ρ ↑ (r)→ρL ↑ , ρL ↓ρ ⃗ G↑, ρ ⃗ G↓2. generation of collinear potentials:ρ L ↑ , ρL ↓ρG ⃗↑, ρG ⃗ → V ↑ L,V ↓L↓VG ⃗↑, VG ⃗↓3. local de-diagonalisation of collinear potentials in real space:V L↑ , V L ↓V ⃗ G↑, V ⃗ G↓→V ↑ (r) 00 V ↑ (r)→V σ α σ ′α (⃗r)V σσ ′ (⃗r)→ V L σ α σ ′αV ⃗ Gσσ ′wien workshop 2003 – p.40/40