Extinction of quasiparticle scattering interference in cuprate superconductors arXiv:1001.3232v2 [cond-mat.supr-con] 26 May 2010 Zhi Wang, Bin Liu, and Shiping Feng ∗ Department of Physics, Beijing Normal University, Beijing 100875, China The quasiparticle scattering interference phenomenon characterized by the peaks in the local density of states is studied within the kinetic energy driven superconducting mechanism in the presence of a single impurity. By calculation of the Fourier transformed ratio of the local density of states at opposite energy, it is shown that the quasiparticle scattering interference phenomenon can be described qualitatively by a single impurity in the kinetic energy driven homogeneous d- wave superconducting state. The amplitude of the peak increases with increasing energy at the low energy, and reaches a maximum at the intermediate energy, then diminishes to zero at the high energy. The theory also predicts that with increasing doping, the position of the peak along the nodal direction moves towards to the center of the Brillouin zone, while the position of the peak along the antinodal direction is shifted to large momentum region. PACS numbers: 74.20.-z, 74.25.Jb, 74.50.+r, 74.72.-h Keywords: Quasiparticle scattering interference; Local density of states; Cuprate superconductors; Kinetic energy driven superconducting mechanism In the study of the physical properties of cuprate super- conductors, one of powerful techniques is scanning tun- neling microscopy (STM) $^{2}$, since it is the only method to probe the real-space inhomogeneous electronic struc- ture of cuprate superconductors. Application of STM to cuprate superconductors would allow one to explore some important issues $^{2–5}$, including (1) the physical pro- cesses dominating quasiparticle scattering, (2) the degree to which the quasiparticles are well defined and coherent, and (3) the relation between the commensurate and in- commensurate magnetic signatures and the quasiparticle scattering processes. In particular, STM has been used to infer the momentum-space information of quasiparticles from the Fourier transform of the position ( r ) and energy ( ω )-dependent local density of states (LDOS), ρ ( r , ω ), then both real-space and momentum-space modulations for LDOS are explored simultaneously $^{4}$. The typical fea- ture observed by the Fourier transformed LDOS is dom- inated by the peaks at well-defined wavevectors q$_{i}$ obey- ing the octet model $^{3–5}$, since the quasiparticle disper- sion in cuprate superconductors in the superconducting (SC) state has closed constant-energy contours surround- ing the d-wave nodes. Theoretically, there is a general consensus that the peak emerges due to the quasiparticle scattering interference (QSI) $^{6–8,10}$. This QSI is mani- fested itself as spatial modulation in the Fourier trans- formed LDOS ρ ( q , ω ) whose wavevectors change with en- ergy. Several attempts have been made to make this ar- gument more precise. In particular, some calculations based on the phenomenological d-wave Bardeen-Cooper- Schrieffer (BCS) formalism have been performed by con- sidering the effect of impurity scattering $^{6–11}$, where it has been shown that a single or few impurities in a homoge- neous d-wave SC state leads to a result that is in quali- tative agreement with the STM experimental data $^{3–5}$. Recently, the improvements in the resolution of STM experiments 12–14 allowed one to resolve additional fea- tures in QSI. Among these new achievements is the mea- surement of the ratio of differential conductances at op- posite bias, where V is the bias voltage and g ( r , V ) the differential conductance. The advantage of this procedure is that it cancels the severe systematic errors in the measure- ment of g ( r , + V ) due to tip elevation errors $^{12–14}$, yet re- tains all the quasiparticle information in the differential conductance. In particular, an remarkable phenomenon observed from these new STM experiments is that QSI always disappears at a scale energy (high energy) that is indistinguishable from the energy at which electronic ho- mogeneity is lost. In other words, at energies below the high energy, the STM response is similar to the dispers- ing Bogoliubov quasiparticle, however, at energies above the high energy, the response becomes highly spatially in- homogeneous. In corresponding to the constant-energy contour of the high energy, an extinction line exists 14 in momentum space [near the diagonal line between ( ± π, 0) and (0 , ± π )], beyond which most of the dispersing peaks disappear, to be replaced by a reduced set of roughly non- dispersive peaks. These new STM experimental results lead naturally to speculation about the appearance of the QSI extinction in cuprate superconductors in the SC state near the boundary of the Brillouin zone (antinodal regime). Within the phenomenological BCS formalism, the QSI phenomenon in a d-wave SC state with coexist- ing short-range antiferromagnetic (AF) order has been studied 15,16 recently by considering the scattering aris- ing from a single point-like impurity, and the result shows that dispersing peaks are then extinguished at the high energy $^{15}$. To the best of our knowledge, the dispers- ing peaks below the high energy in cuprate superconduc- tors and its extinction at the high energy has not been treated starting from a microscopic SC theory, and no explicit calculations of the doping dependence of the dis- persing peaks has been made so far. In this paper we show very clearly if the effect of a single point-like impu- rity scattering potential is considered within the kinetic energy driven SC mechanism $^{17}$, the calculated ratio of LDOS at opposite energy in momentum-space can re- produce some main features observed experimentally on ∗ Corresponding author. Tel.: +86 10 58806408; Fax: +86 10 58801764; E-mail address: spfeng@bnu.edu.cn Z ( r , V ) ≡ g ( r , + V ) /g ( r , − V ) , (1)