4 Figure 4: (Color online) Majority (P, bottom row), minority (P, middle row) and AP (top row) transmission T ( k$_{|}$$_{|}$ , E$_{F}$ ) within two-dimensional Brillouin zones for Fe(001)/ x (Cr)/6(MgO)/Fe(001) MTJs with Cr layer thicknesses of x = 0 , . . . , 7 ML (from left to right). The two-dimensional Brillouin zones cover the range between − π/a and π/a . as well, indicating constructive superposition of the os- cillations of the individual T P $_{maj}$( k$_{|}$$_{|}$ ). Although the cloverleaf-like structures within the mi- tions of the point group 4 mm (Δ$_{1}$, Δ$_{5}$, Δ$_{2}$ and Δ$_{2}$ ′ ) of the associated Bloch states (Fig. 3). A complex band structure of a periodic system is the conventional band structure extended to Bloch vectors ( k$_{|}$$_{|}$ , k$_{⊥}$ ) with complex wave numbers k$_{⊥}$ . The associated bands can be cast into four categories $^{39}$: (i) real bands which correspond to the conventional band structure and have Im k$_{⊥}$ = 0; (ii) imaginary bands of the first kind have Re k$_{⊥}$ = 0 and Im k$_{⊥}$ = 0; (iii) imaginary bands of the second kind with Re k$_{⊥}$ = π / a and Im k$_{⊥}$ = 0; and ( iv ) complex bands with Re k$_{⊥}$ = 0, Re k$_{⊥}$ = π / a and Im k$_{⊥}$ = 0. The imaginary part of k$_{⊥}$ is denoted as κ and repre- sents a measure for the decay rate of evanescent Bloch states $^{43}$. At the Fermi energy, a complex band of the second kind shows up at k$_{⊥}$ = (0 . 922 , 0 . 503) π / a (circles in Fig. 3). The corresponding decay rate of κ = 0 . 78 / ML matches well the estimated exponential decay of the ma- jority transmission ( κ = 0 . 75 / ML, Fig. 1a). Due to the nonvanishing real part the exponential decay exhibits an oscillatory envelope with a wavelength λ = π / Re[ k$_{⊥}$ ] ≈ 2 . 17 ML which agrees well with the fit in Fig. 1b. We conclude therefore that the thickness dependence of the transmission for the majority states at the ¯ Γ point is very likely governed by this Δ$_{1}$ state, provided the electronic structure of ultra-thin Cr films is well described by that of bulk Cr. Transmission maps display the spin-resolved transmis- sion for P and AP versus k$_{|}$$_{|}$ for each Cr thickness x (Fig. 4). As observed for T P $_{maj}$( k$_{|}$$_{|}$ = 0) one finds within the majority transmission maps a clearly visible modula- tion of T P $_{maj}$( k$_{|}$$_{|}$ ) for k$_{|}$$_{|}$ -points in the center regions with maxima (minima) at even (odd) multiples of x . It is reasonable to assume that the oscillations of these trans- missions are caused by the same bandstructure effect as it was discussed for T P maj at ¯ Γ. Furthermore, the major- ity RA products in Fig. 5 exhibit even-odd oscillations Figure 5: (Color online) Cr thickness dependence of (a) optimistic and (b) normalized TMR ratios in Fe(001)/ x (Cr)/6(MgO)/Fe(001) MTJs. Since variations of the Cr layer thickness cannot be ruled out in experiment, a model with resistors in parallel connection is assumed to mimic Cr-thickness fluctuations (line styles indicate the weight w ; see text). (c) Resistance area product RA for par- allel magnetic (P: majority, minority) and anti-parallel mag- netic configurations (AP), shown on a logarithmic scale.