1. zyxwvutsrqp Phys. F: Met. Phys. 14(1984) 1205-1215. Printed in Great Britain zyxwv Quick iterative scheme for the calculation of transfer matrices: application to zyxw MO( 100) M P Lopez Sancho, J M Lopez Sancho and J Rubio Instituto de Fisica de Materiales, CSIC, Serrano 144, Madrid 6, Spain Received 14 September 1983 Abstract. The transfer matrix of a solid described by the stacking of principal layers is obtained by an iterative procedure which takes into account 2" layers after n iterations, in contrast to usual schemes where each iteration includes just one more layer. The Green function and density of states at the surface of the corresponding semi-infinite crystal are then given by well known formulae in terms of the transfer matrix. This method, especially convenient near singularities, is applied to the calculation of the spectral as well as the total densities of states for the (100) face of molybdenum. The Slater-Koster algorithm for the calculation of tight-binding parameters is used with a basis of nine orbitals per atom (4d, 5s, 5p). Surface states and resonances are first identified and then analysed into orbital components to find their dominant symmetry. Their evolution along the main symmetry lines of the two-dimensional Brillouin zone is given explicitly. The surface- state peak just below the Fermi level (Swanson hump) is not obtained. This is traced to the difficulty in placing an appropriate boundary condition at the surface with the tight-binding parametrisation scheme. 1. Introduction Iterative methods for the calculation of the Green function at the surface layer of a solid (Haydock zyxwvuts et a1 1972, Ainshchik et a1 1976, Foo et zyxwv a1 1976, Mele and Joannopoulos 1978) have sometimes been criticised on the basis of convergence arguments (Dy et a1 1979, Lee and Joannopoulos 198 la, b). Thus the 'effective field' or transfer-matrix approach gives the surface (zeroth-layer) Green function by the equation GW(0) =(U zyxwvu - zyx HW - HoI T(W))-] (1) where H, and zyxwvutsr H,, are matrix elements of the Hamiltonian between layer Bloch states (see below) and the transfer matrix Tis given by T(0) = zyxw (0 - H, - H,, T(w))- Hi, (2) which must be calculated by iterating until self-consistency is achieved. This usually involves many iterations (an average of -SO), especially in the neighbourhood of the singularities of G(o) where several hundred may be needed to get an accurate result. In this paper we propose a new iterative scheme for the calculation of the transfer matrix which converges very quickly. After n iterations 2" layers are taken into account instead of the n layers one would have included with the usual method based on iterating equation (2). Away from singularities, five or six iterations usually suffice to get a convergent result (2' = 32, 26 = 64 with equation (2)). Close to singularities, the number of 0305-4608/84/05 1205 + 11 $02.25 zyxwv 0 1984 The Institute of Physics 1205