We present an experimental method to determine the electronic E(k) band structure in crystalline solids absolutely, i.e., with complete control of the three-dimensional wave vector k. Angle-dependent very-low-energy electron diffraction is first applied to determine the unoccupied states whose k is located on a high-symmetry line parallel to the surface. Photoemission via these states, employing the constant-final-state mode, is then utilized to map the valence bands along this line. We demonstrate the method by application to Cu, and find significant deviation from free-electron-like behavior in the unoccupied states, and from density-functional theory in the occupied states.

V. N. Strocov, R. Claessen, G. Nicolay, S. Hüfner, A. Kimura, A. Harasawa, S. Shin, A. Kakizaki, P. O. Nilsson, H. I. Starnberg, P. Blaha
Phys. Rev. Lett. 81 (22), pp. 4943ff.

The present ray-tracing scheme employs the electrostatic potential calculation by finite-difference solution of the Laplace equation in three dimensions. The run time reduction, achieved by using successively refined grids and optimizing the iteration cycle, allows one to perform calculations on a personal computer. The code is applied to a standard four-grid LEED optics, operated in the retarding field mode, to calculate electron trajectories in the case of off-symmetry incidence. In particular, dependences of incidence on the sample rotation and primary energy are calculated. A simple empirical formula is derived, which allows accurate experimental determination of without extensive calculations. Ray-tracing calculations are further applied to analyse the influence of residual asymmetries of the electron optics.

V.N. Strocov
Meas. Sci. Technol. 7, pp. 1636-1642

A method of band mapping providing full control of the three-dimensional k is described in detail. Angle-dependent very-low-energy electron diffraction is applied to determine the photoemission final states along a Brillouin zone symmetry line parallel to the surface; photoemission out of these states is then utilized to map the valence bands in the constant-final-state mode. The method naturally incorporates the non-free-electron and excited-state self-energy effects in the unoccupied band, resulting in an accuracy superior over conventional techniques. Moreover, its intrinsic accuracy is less limited by lifetime broadening. As a practical advantage, the method provides access to a variety of lines in the Brillouin zone using only one crystal surface. We extensively tested the method on Cu. Several new aspects of the electronic structure of this metal are determined, including non-free-electron behavior of unoccupied bands and missing pieces of the valence band.

V.N. Strocov, R. Claessen, G. Nicolay, S. Hüfner, A. Kimura, A. Harasawa, S. Shin, A. Kakizaki, P. O. Nilsson, H. I. Starnberg, P. Blaha
Phys. Rev. B. 63, 205108 ff.