Using High Energy Angle Resolved Photoelectron Spectroscopy to Reveal the Charge Density in Solids M. Ma ˚nsson, 1 T. Claesson, 1 M. Finazzi, 2 C. Dallera, 2 N. B. Brookes, 3 and O. Tjernberg 1, * 1 Materials Physics, Royal Institute of Technology KTH, Electrum 229, S-164 40 Kista, Sweden 2 INFM-Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy 3 European Synchrotron Radiation Facility, ESRF, BP220, 38043 Grenoble cedex, France (Received 21 August 2008; published 25 November 2008) The charge density in solids is a fundamental parameter. Here we demonstrate that the charge density can be determined by the use of angle resolved photoelectron spectroscopy. The method, which involves a Fourier-like transform from momentum space to real space, is demonstrated by utilizing soft x-ray angle resolved photoelectron spectroscopy to sample the complete three-dimensional Brillouin zone of copper. It is also shown that this can be done in an energy resolved way as to extract the charge density contribution from states of a particular energy. DOI: 10.1103/PhysRevLett.101.226404 PACS numbers: 71.20.b Ever since the advent of density functional theory [1] in the 1960s the charge density has been an omnipresent and fundamental parameter in condensed matter physics. The local density approximation (LDA) [2] and its extensions have permitted the calculation of charge densities and related physical parameters in a wide variety of systems. On the experimental side, the advances in x-ray scattering notably with the aid of synchrotron radiation sources has permitted the charge density to be determined experimen- tally [3]. Thus far the experimental possibilities have, in contrast to theory, been limited to determining the total charge density, and the possibility of determining energy resolved charge densities is very limited. Angle resolved photoelectron spectroscopy (ARPES) is a powerful tool to study the electronic structure in solids. Until recently, the technique has to a large extent been limited to the exploration of two-dimensional systems. The reason for this is to be found in the extreme surface sensitivity and large emission angles that follow from the use of photon energies in the 20–100 eV range, the energy range where most experiments are performed. During the past few years, the technique has, however, been extended into the soft x-ray range. The resulting increase in probing depth, accompanying improvement in momentum space resolution [4], and smaller emission angles has permitted the investigation of fully three-dimensional systems [5–7]. Here we present a new application of soft x-ray ARPES to three-dimensional systems by demonstrating a method for extracting charge densities in crystalline solids. Furthermore, we demonstrate that this can be done in an energy resolved way. ARPES measurements using soft x rays involve only small angular ranges, which makes it straightforward to collect spectra along an approximately straight line across the Brillouin zone. An example of this is given in Fig. 1(a) where the valence band dispersion in Cu is shown. The corresponding k-space line along which the data were collected is indicated by a black line in the Brillouin zone of Fig. 1(b). Overlayed in Fig. 1(a) is the result of a LDA calculation for the same k-space line. The high intensity of the Cu 3d related bands can be seen in the 2– 6 eV binding energy range of Fig. 1(a), accompanied by the weaker Cu 4s related band, which crosses the Fermi level. The experiments were carried out at the ID08 beam line of the European Synchrotron Radiation Facility (ESRF) using a Scienta SES-2002 electron energy analyzer. The Cu(001) single crystal was cleaned by ion bombardment and annealing. Surface cleanliness and orientation was verified by core level spectroscopy and low energy electron diffraction. Energy distribution curves (EDC) were col- lected for almost 4400 k points at a sample temperature of 20 K. In order to minimize any geometrical variation in the optical matrix elements [8], the incidence angle was kept fixed at 30  and circularly polarized light was used. Further, by retaining only the k points within the first Brillouin zone we used the smallest set of emission angles possible, resulting in 2400 inequivalent k points [dark gray (red) points of Fig. 1(b)]. The total energy resolution employed varied from 80 to 140 meV over the 430– 580 eV photon energy range and the momentum space resolution was 0:04  0:1  0:1  A 3 based on a probing depth of 10 A ˚ [9]. The LDA calculations were performed with the WIEN2K code [10]. The recorded intensity in an ARPES measurement is proportional to the spectral function Aðk; !Þ convoluted with the optical matrix elements [11]. Since the spectral function is the imaginary part of the Green function for the system, it gives a direct handle on many of the fundamental properties of the system. The energy dependence is related to cross section and final state effects but these can be normalized out if the change is smooth as a function of energy. In the soft x-ray range this seems to be the case as evidenced by final state measurements on Al [12] and by the fact that we observe only a slow overall change in valence band intensity as the photon energy is varied. As a consequence, it is possible to directly extract the spectral PRL 101, 226404 (2008) PHYSICAL REVIEW LETTERS week ending 28 NOVEMBER 2008 0031-9007= 08=101(22)=226404(4) 226404-1 Ó 2008 The American Physical Society