Near Doping-Independent Pocket Area from an Antinodal Fermi Surface Instability in Underdoped High Temperature Superconductors

N. Harrison

Phys. Rev. Lett. 107, 186408 – Published 28 October 2011

Abstract

Fermi surface models applied to the underdoped cuprates predict the small pocket area to be strongly dependent on doping whereas quantum oscillations in ${YBa}_{2} {Cu}_{3} O_{6 + x}$ find precisely the opposite to be true—seemingly at odds with the Luttinger volume. We show that such behavior can be explained by an incommensurate antinodal Fermi surface nesting-type instability—further explaining the doping-dependent superstructures seen in cuprates using scanning tunneling microscopy. We develop a Fermi surface reconstruction scheme involving orthogonal density waves in two dimensions and show that their incommensurate behavior requires momentum-dependent coupling. A cooperative modulation of the charge and bond strength is therefore suggested.

Received 8 August 2011

DOI:https://doi.org/10.1103/PhysRevLett.107.186408

Authors & Affiliations

N. Harrison

Los Alamos National Laboratory, MS E536, Los Alamos, New Mexico 87545, USA

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Vol. 107, Iss. 18 — 28 October 2011

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Images

Figure 1
(a) Charge modulation periods seen using x-rays or NMR in ${YBa}_{2} {Cu}_{3} O_{6 + x}$ [3, 6] (large triangles), ${La}_{1.875} {Ba}_{0.125} {CuO}_{4}$ and ${La}_{1.48} {Nd}_{0.4} {Sr}_{0.12} {CuO}_{4}$ [4] (pentagon) and STM in ${Bi}_{2 - y} {Pb}_{y} {Sr}_{2 - z} {La}_{z} {CuO}_{6 + x}$ (diamonds), ${Bi}_{2} {Sr}_{2} {CaCu}_{2} O_{8 + δ}$ (squares) and ${Ca}_{2 - x} {Na}_{x} {CuO}_{2} {Cl}_{2}$ (small triangle) taken from Ref. 8. In comparing different materials, we neglect possible differences in $ε (k)$ [23]. The line and circles show the doping $p$ for each $λ$ extracted from the model density-of-states minimum [e.g., Fig. 3b]. (b) Measured leading ${YBa}_{2} {Cu}_{3} O_{6 + x}$ quantum oscillation frequency [16, 17, 18, 19, 20] (circles) compared to its strong $p$ dependence expected in the 4 hole pocket [1] (dotted line), Millis and Norman stripe [13] (dot-dash line) and fixed $λ = 4$ bidirectional charge [15] (dashed line) models, where $F = (ℏ / 2 π e) A_{e}$ . The present model (solid line) uniquely yields a weakly $p$ -dependent $F$ [20].Reuse & Permissions
Figure 2
(a) The unreconstructed Fermi surface at $p = 8.5 %$ (black) [23] together with itself translated by multiples of $Q_{x}$ (cyan) and multiples of $Q_{x}$ and $Q_{y}$ (grey) for $λ = 4$ . (b) Reconstructed Fermi surface (black) resulting from a unidirectional charge modulation (i.e., $n^{'} = 1$ ) in which $V_{x, 0} = 0.3 t$ and $r = 0$ in Eq. (2), shown for a quadrant of the extended Brillouin zone. (c) Same as (b) but with a momentum-dependent $V_{x}$ in which we choose $r = 1$ [29]. (d) The calculated electronic density-of-states (DOS) for the unreconstructed band [23] exhibiting a van Hove singularity near $ε \approx - 2 t$ . (e) The calculated DOS (black line) for $r = 0$ in (b). (f) The calculated DOS (black line) for $r = 1$ in (c). Red lines in (e) and (f) are the corresponding DOS calculated for concurrent charge modulations along $a$ and $b$ (i.e., such that $n^{'} = n$ ) in which we assume $V_{x, 0} = V_{y, 0}$ (by no means a required constraint).Reuse & Permissions
Figure 3
(a) Electronic density-of-states (DOS) calculated for bidirectional charge modulations with periodicities corresponding to different multiples $λ$ of the $a$ and $b$ lattice vectors as indicated, assuming uniform couplings $V_{x, 0} = V_{y, 0} = 0.3 t$ in which $r = 0$ . b Same as (b) but assuming momentum-dependent couplings in which $r = 1$ in Eq. (2). Curves have been offset for clarity. The dashed line in (b) indicates the minimum in the DOS near to which the Fermi energy is likely to be located.Reuse & Permissions
Figure 4
(a), (b), (c) and (d) Reconstructed Fermi surface for selected $λ$ ’s in Fig. 3b when the Fermi energy is situated at the minimum in the density-of-states, with the corresponding hole doping given. Solid red lines indicate the $k$ -space area of the $λ a \times λ b$ superstructure, while dashed red lines indicate the $n^{2}$ -fold reduced Brillouin zone [which coincides with the superstructure in (a) and (d)]. In (b) and (c), a magenta line is used to trace the path of the electronlike orbit that occurs in strong magnetic fields. $t_{10} \sim 100 meV$ [23] produces an effective mass and gap consistent with experiments.Reuse & Permissions

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