Nature | Letter
Intra-unit-cell electronic nematicity of the high- T copper-oxide pseudogap states c
(15 July 2010)
05 March 2010
04 May 2010
In the high-transition-temperature (high-
T ) superconductors the pseudogap phase becomes predominant when the density of doped holes is reduced c . Within this phase it has been unclear which electronic symmetries (if any) are broken, what the identity of any associated order parameter might be, and which microscopic electronic degrees of freedom are active. Here we report the determination of a quantitative order parameter representing intra-unit-cell nematicity: the breaking of rotational symmetry by the electronic structure within each CuO 1 unit cell. We analyse spectroscopic-imaging scanning tunnelling microscope images of the intra-unit-cell states in underdoped Bi 2 Sr 2 CaCu 2 O 2 8 + and, using two independent evaluation techniques, find evidence for electronic nematicity of the states close to the pseudogap energy. Moreover, we demonstrate directly that these phenomena arise from electronic differences at the two oxygen sites within each unit cell. If the characteristics of the pseudogap seen here and by other techniques all have the same microscopic origin, this phase involves weak magnetic states at the O sites that break 90°-rotational symmetry within every CuO δ unit cell. 2
View full text
Figures at a glance
Figure 1: CuO 2 electronic structure and ω ≈ Δ 1 pseudogap states.
a, Schematic of the spatial arrangements of CuO 2 electronic structure with Cu sites and orbitals indicated in blue and O sites and 2 p orbitals in yellow. σ E F, Fermi energy. The inset shows the approximate energetics of the band structure when such a charge-transfer insulator is doped by removing electrons from the O atoms (the respective bands are indicated by the same colours as in the CuO 2 schematic). The ‘real’ part of any q-space electronic structure at the Bragg wavevector ( Q, ) is defined throughout this paper as being in phase with the Cu lattice (and therefore even about each Cu site). This definition is shown schematically as modulations in ω which would contribute to ( Q, ) at the two Bragg wavevectors ω Q and x Q . DOS, density of states. y b, Schematic copper-oxide phase diagram. Here T c is the critical temperature circumscribing a ‘dome’ of superconductivity, T is the maximum temperature at which phase fluctuations are detectable within the pseudogap phase, and φ T* is the approximate temperature at which the pseudogap phenomenology first appears. c, The two distinct classes of excitations as identified by multiple spectroscopies in underdoped copper-oxides (reproduced with permission from ref. 3) as a function of hole-density p. The excitations to energies Δ 1( p) are referred to as the ‘pseudogap states’ both because Δ 1( p) tracks T*( p) and because they exist unchanged in both the pseudogap and superconducting phases. Those excitations to energies ω < Δ 0( p) can be associated with the Bogoliubov quasiparticles . 6, 7, 8 d, Evolution of the spatially averaged tunnelling spectra of Bi 2Sr 2CaCu 2O 8 with diminishing + δ p, here characterized by T c( p). The energies Δ 1( p) (blue dashed line) are easily detected as the pseudogap edge while the energies Δ 0( p) (red dashed line) are more subtle but can be identified by the correspondence of the “kink” energy with the extinction energy of Bogoliubov quasiparticles, following the procedure in ref. 4 8.
Figure 2: Imaging the spatial symmetries of the ω ≈ Δ 1 pseudogap states.
a, Spatial image ( R-map ) of the Bi 5 2Sr 2CaCu 2O 8 + pseudogap states δ ω ≈ Δ 1 at T ≈ 4.3 K for an underdoped sample with T c = 35 K. The inset shows the Fourier transform upon which the inequivalent Bragg vectors Q = (1, 0)2 x π/ a 0 and Q y = (0, 1)2 π/ a 0 are identified by red arrows and circles. The inequivalent wavevectors S = (~3/4, x 0)2 π/ a 0 and S = (0, y ~3/4)2 π/ a 0 are identified by blue arrows and circles. b, Spatial image ( R-map ) of the Bi 5 2Sr 2CaCu 2O 8 pseudogap states + δ ω ≈ Δ 1 at T ≈ 55 K for the same sample with T c = 35 K. Again, the inset shows the Fourier transform with the inequivalent Bragg vectors Q = (1, 0)2 x π/ a 0 and Q = (0, 1)2 y π/ a 0 identified by red arrows and S = (~3/4, 0)2 x π/ a 0 and S = (0, ~3/4)2 y π/ a 0 identified by blue arrows and circles. The phenomenology of the ω ≈ Δ 1 pseudogap states, especially their broken spatial symmetries, appear indistinguishable whether in the superconducting phase ( a) or in the pseudogap phase ( b). c, A schematic representation of how electronic contributions from multiple sites within the CuO 2 unit cell could lead to global electronic nematicity in the copper oxides. Here the two O sites are labelled using different colours to represent the inequivalent electronic structure at those locations within each unit cell.
Figure 3: Nematic ordering and O-site specificity of ω ≈ Δ 1 pseudogap states.
a, Topographic image T( r) of the Bi 2Sr 2CaCu 2O 8 surface. The inset shows that the real part of its Fourier transform Re + δ T( q) does not break C 4 symmetry at its Bragg points because plots of T( q) show its values to be indistinguishable at Q = (1, 0)2 x π/ a 0 and Q = (0, 1)2 y π/ a 0. Importantly, this means that neither the crystal nor the tip used to image it (and its Z( r, ) simultaneously) exhibits C ω 2 symmetry ( Supplementary Information section V). The bulk incommensurate crystal supermodulation is seen clearly here; as always, it is at 45° to, and therefore is the mirror plane between, the x and y axes. For this symmetry reason it has no influence on the electronic nematicity discussed in this paper. b, The Z( r, e = 1) image measured simultaneously with T( r) in a. The inset shows that the Fourier transform Z( q, e = 1) does break C 4 symmetry at its Bragg points because . This means that, on average throughout the FOV of a and b, the modulations of Z( r, ω ≈ Δ 1) that are periodic with the lattice have different intensities along the x axis and along the y axis. This is a priori evidence for electronic nematicity in the pseudogap states ω ≈ Δ 1. c, The value of defined in equation (3) computed from Z( r, e) data measured in the same FOV as a and b. Its magnitude is low for all ω < Δ 0 and then rises rapidly to become well established near e ≈ 1 or ω ≈ Δ 1. Thus the quantitative measure of intra-unit-cell electronic nematicity established in equations (1) and (3) reveals that the pseudogap states in this FOV of a strongly underdoped Bi 2Sr 2CaCu 2O 8 sample are nematic. d, Topographic image T( r) from the region identified by a small black box in a. It is labelled with the locations of the Cu atom plus both the O atoms within each CuO 2 unit cell (labels shown in the inset). Overlaid is the location and orientation of a Cu and four surrounding O atoms using a representation similar to that of Fig. 2c. e, The simultaneous Z( r, e = 1) image in the same FOV as d (the region identified by small white box in b) showing the same Cu and O site labels within each unit cell (see inset). Thus the physical locations at which the nematic measure of equation (4) is evaluated are labelled by the dashes. Overlaid is the location and orientation of a Cu atom and four surrounding O atoms using a representation similar to that of Fig. 2c. f, The value of in equation (4) computed from Z( r, e) data measured in the same FOV as a and b. As in c, its magnitude is low for all ω < Δ 0 and then rises rapidly to become well established at e ≈ 1 or ω ≈ Δ 1. If the function in equation (4) is evaluated using the Cu sites only, the nematicity is about zero (black diamonds), as it must be. This independent quantitative measure of intra-unit-cell electronic nematicity again shows that the pseudogap states are strongly nematic and, moreover, that the nematicity is due primarily to electronic inequivalence of the two O sites within each unit cell.
Figure 4: Rapid increase of correlation length of nematicity at ω ≈ Δ 1.
a, A large FOV T( r) image which preserves C 4 symmetry. b, Correlation lengths for the nematic ordering (red solid diamonds), and for the possibly smectic ordering (blue solid squares). (See Supplementary Information section VI for an evaluation of and .) The coarsening length scales 1/ Λ n and 1/ Λ s are one-third that of typical spatial variations, as determined from the 3 radius of the respective peaks. The correlation lengths are determined from the full-width at half-maximum of the spatial auto-correlation functions. Comparison of the nematic order parameter σ evaluated from equation (3) (open diamonds) and smectic order parameter ) from equation (6) (open squares) shows that has low magnitude and is energy independent, whereas rises rapidly to become well established at the pseudogap energy. c, Image of ( Supplementary Information section VI) from the same FOV showing that has diverged to the size of the image. Thus, if there are Ising nematic domains (as there should be), they must be larger than 0.025 micrometres square. We note that the spatial resolution is limited by the cut-off scales shown in the figure. d, Images of ( Supplementary Information section VI) from the same FOV as a showing that is spatially disordered with very short correlation length; this is equally true at all energies. The evolution of and as a function of e are available in Supplementary movies 1 and 2, respectively.