Generating Input Data within DFT. The adapted GGA+DMFT scheme to calcu-late electronic structures exhibiting commensurate antiferromagnetic ordering needs input from a converged DFT calculation and again the FLEUR code was used to generate this input. Chromium with commensurate antiferromagnetic structure can be modeled by a unit cell of simple cubic structure with two atoms, one at the corner and one at the center of the cell, with opposite spin-magnetic moments. The exper-imental lattice constant was used in all calculations. Furthermore the parameters Kmax and lmax were set to 3.4 and 8 in all calculations.
The choice of the number of k points has to be done carefully, since the magnetic properties of chromium are sensitive to the Brillouin-zone sampling which reflects
10.2 Calculating Commensurate AFM within GGA+DMFT 131 the proximity to the magnetic instability close to the experimental lattice constant as mentioned above. Therefore a large number of 969kpoints in the irreducible part of the 1st BZ was used in the iterations to solve the Kohn-Sham equations. With this sampling the Cr moment is within 0.002µB of the converged value calculated for the experimental lattice constant. The densities of states and the eigenvalues to construct the DFT lattice Green function were determined with an even larger number of 1140k points.
Finally a parametrization of the exchange-correlation potential has to be chosen.
As pointed out in the introduction of this chapter, one may chose for Cr between a GGA yielding accurate lattice constants but too high values for the spin-magnetic moment, or a LDA parametrization which usually gives a value for the spin-magnetic moment close to experiment but underestimates the lattice constant substantially.
Furthermore, it was shown by Singh and Ashkenazi [SA92] that the LDA actually predicts the ground state of chromium to be paramagnetic if the equilibrium lattice constant determined within LDA is used instead of the experimental lattice constant.
As in previous calculations for Fe, Co and Ni I therefore use the PBE parametrization within the GGA.
Construction of the Lattice Green Functions. The two most important steps in the derivation of the new LDA+DMFT scheme are the construction of the lat-tice Green function within DFT for a crystal symmetry with two atoms per unit cell and the incorporation of this lattice Green function into the GGA+DMFT self-consistency cycle. The construction of the Green function can be done straightfor-wardly as described in chapter 4 and I only recall the most important steps here.
The general form of the TB-FLAPW basis functions is used
χµσL (rµ) = uσl(rµ)YL(ˆrµ), (see 3.29) where L = (l, m) is the combined orbital index, µdenotes the muffin tin inside the unit cell thus corresponding to a label of the different atoms per unit cell andrµ gives the position inside the muffin tin µ. If the Kohn-Sham eigenfunctions are expanded in terms of the general form of the TB-FLAPW basis functions and the lattice Green function is constructed using this expansion of the Kohn-Sham eigenstates the Green function is obtained as a matrix with elements
G0LLµµ00σ(k;) = 1 N
X
ν
AµσL,ν(k)
AµL00σ,ν(k)∗
−σk,ν±iη . (see 4.34) The numberN gives the number of atoms in the crystal andσk,ν are the Kohn-Sham eigenvalues. The deviation between this Green function and the Green function for a crystal structure with one atom per unit cell stems form the restriction ofkto the first Brillouin zone which differs in both cases.
For chromium with two different atomsµ∈ {A, B}in the unit cell, the DFT lattice Green function in k space can be written in the following matrix form
G0σ(k;) =
G0AAσ(k;) G0ABσ(k;) G0BAσ(k;) G0BBσ(k;)
. (10.1)
From the Green function in k space a Green function in real space is obtained using the lattice Fourier transformation introduced in chapter 4. The Fourier transforma-tion can be applied to each matrix element separately. Thus, we can make use of the fact that within DMFT only the local elements of the Green functions are needed, i.e. only the sub-matrices G0AAσ(k;) and G0BBσ(k;) need to be transformed into real space. The lattice transformation for these matrix elements however is of the simple form
G0µµσ() = X
k
G0µµσ(k;). (10.2)
The new GGA+DMFT Iteration Cycle. As discussed by Georgeet al. [GKKR96]
within DMFT the same mean-field equations as derived for crystal structures with a single atom per unit cell can be derived for each atom A and B from the two sub-lattices of Cr. This is due to the single-site approximation in DMFT. If the mean-field equations presented in chapter 6 now hold for each atom separately the sub-matrices G0AAσ() and G0BBσ() can be used directly as input in the first iteration step for the FLEX solver to calculate a self-energy contribution ΣAAσ() and ΣBBσ() for each atomic site. The resulting matrix
Σσ() =
ΣAAσ() 0 0 ΣBBσ()
. (10.3)
is used together with the DFT lattice Green function inkspace from equation (10.1) to solve a Dyson equation yielding the interacting lattice Green function G in k space as a matrix of the same form as the DFT lattice Green function in (10.1).
The sub-matrices GAAσ() andGBBσ() of the interacting lattice Green function can now be obtained by applying the lattice Fourier transformation to the newly derived Green function G. The bath Green function for each atom is obtained from these sub-matrices using the same formulas as in the case of a single atom per unit cell since the mean-field equations hold. The bath Green functions are then used to calculate new self-energy matrix elements ΣAAσ() and ΣBBσ() and this scheme is iterated until convergence is reached.
Due to the specific structure of chromium, i.e. all local properties of an atom Ain the unit cell in real space are the same as the other atom B with reversed spin, the self-energy contribution has to be calculated only once e.g. for atom A. To obtain the self-energy for atom B the relation
ΣBBσ() = ΣAA−σ() (10.4)