VOLUME 81, NUMBER 14 PHYSICAL REVIEW LETTERS 5OCTOBER 1998 Theoretical Aspects of the Charge Density Wave in Uranium Lars Fast, 1 Olle Eriksson, 1,2 Börje Johansson, 1 J. M. Wills, 2 G. Straub, 2 H. Roeder, 2 and Lars Nordström 1 1 Condensed Matter Theory Group, Department of Physics, University of Uppsala, Box 530 75121, Uppsala, Sweden 2 Center of Materials Science and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544 (Received 17 February 1998) Using a first principles total energy method, we have reproduced the observed charge density wave (CDW) state of a-uranium (called a 1 ). This CDW is found to be a result of a Peierls-like transition, i.e., by opening of partial gaps at the Fermi level. The part of the Fermi surface affected by the distortion shows a strong nesting of fairly narrow f bands. In addition we suggest that the slightly modified a 1 CDW state, which is called a 2 and is observed by cooling the a 1 phase, is caused by a closely related mechanism, namely, by a nesting of the Fermi surface in the b direction. This is consistent with the observed CDW ordering. [S0031-9007(98)07255-X] PACS numbers: 71.45.Lr, 71.15.Mb, 71.18. + y, 71.20. – b Today the charge density wave (CDW) [1–3] state has almost exclusively been observed in quasi-one- dimensional systems, as, for example, in NbSe 3 [4]. However, there is one important exception to this experi- mental fact, namely, uranium metal [5]. Indeed, uranium is also the only element in the periodic table which exhibits such a behavior. Thus it becomes particularly important to verify this unique property theoretically. In fact, it was only after several decades of thorough experimental work that it was experimentally established that uranium metal exhibits a sequence of low tempera- ture states, which have been identified as charge density waves (the different phases are called a 1 , a 2 , and a 3 ). The first transition takes place at 43 K (a 1 ) and the last one stabilizes below 23 K (a 3 ). After the completion of the last CDW transition, uranium has in fact transformed to an element where the primitive cell has a volume of 6000 Å 3 [5]. On the theoretical side there has not been a corresponding refinement of the theoretical treatment to cope with these fine details of the low temperature crystal structure of uranium. Despite the fact that the conceptual possibility of CDW states was suggested many years ago for simple metals [6], uranium has remained a unique exception among the elements showing such behavior (the spin-density wave of chromium is accompanied by a weak CDW [7,8], but the latter is simply induced by the former). One may wonder why a CDW state has not been observed in other elements since in compounds (especially compounds with “one- dimensional” character) the CDW state is more frequently observed [6,3]. An “ideal” one-dimensional CDW system has a periodic charge density given by [3] r  r 0 1 Dr cos2k F r 1f , (1) where r 0 is the density of the normal state and Dr is the magnitude of the charge density wave, whereas k F is the Fermi wave vector of the undistorted lattice and f is a phase factor. As a consequence of this added periodicity in the system, a so called “Peierls gap” opens up in the energy level distribution at the Fermi level, E F . This modified electron density is then normally accompanied by a movement of atomic positions (dimerization) and the CDW may be identified from this structural distortion. The origin of the CDW in one-dimensional compounds has been discussed in terms of Peierls distortions [2] and Kohn anomalies [9]. The superconducting properties of the CDW condensate, suggested by Fröhlich [1], has until this date not been discovered, presumably due to the pinning of the CDW [3]. Instead, normally the resistivity behavior is characteristic of a gaped system, and in addition there is the complication of the sometimes observed nonlinear current-voltage (I -V ) relationship [3]. The most characteristic identification of a CDW in uranium has been found from the observation of the structural transition [5]. Neutron experiments indicate that the CDW at 43 K is associated with a significant phonon softening, a fact which may help understanding this martensitic transition [10]. From a materials science point of view, the CDW state in uranium manifests itself by a small but still drastic change in many physical properties: lattice parameter, resistivity, elastic response, and thermal expansion [5]. The transition at 43 K (a 1 ) is much simpler than the other two transitions at lower temperatures. It involves only a doubling of the conventional unit cell along the a direction. The corresponding atomic displacements are larger, by an order of magnitude, than the displacements occurring at the other two transitions. Furthermore the survival of this state to higher temperatures (compared to the others) signals that this transition to the a 1 phase is energetically also the most important one. For these reasons we will here focus our attention on the a 1 CDW state. The structural arrangement of this distortion is shown in Fig. 1. Notice that the a 1 CDW state is characterized by one parameter, labeled u in Fig. 1. The doubled unit cell volume associated with a 1 -U is built up from two atomic layers, distinguished by open and closed circles in the figure. A possible primitive cell is indicated by the box in the figure (thin line). When 2978 0031-9007 98  81(14)  2978(4)$15.00 © 1998 The American Physical Society