Renormalization scheme for the transfer-matrix method and the surfaces of wurtzite ZnO

Abstract

The recently introduced transfer-matrix method for surface-band calculations relies solely on the eigensolutions of this matrix. Potential problems may arise, however, if the matrix is too large or ill defined. The latter occurs if any interlayer-coupling Hamiltonian matrix has zero determinant. These difficulties arise in the study of the wurtzite surfaces of ZnO when one uses a tight-binding Hamiltonian that includes the Zn $3d$ orbitals and only nearest-neighbor interactions. To overcome these problems, a renormalization scheme is introduced that systematically removes irrelevant degrees of freedom, reducing the size of matrices and eliminating singularities. This scheme is used, in conjunction with the transfer-matrix method, to study the ($10\overline{1}0$) and ($11\overline{2}0$) surfaces of ZnO. The results show that (1) there are no surface states in the thermal gap, (2) the electronic structure of the two surfaces is very similar, (3) $d$ character is present in all the surface states, and (4) surface states exist in the $d$ bands as well.