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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 

as well, indicating constructive superposition of the os- cillations of the individual T P $_{maj}$( k$_{|}$$_{|}$ ). 

Although the cloverleaf-like structures within the mi- 

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