Fermi surface reconstruction and quantum oscillations in underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7\text{−}x}$ modeled in a single bilayer with mirror symmetry broken by charge density waves

Hole-doped high-temperature cuprate superconductors below optimum doping have electronlike Fermi surfaces occupying a small fraction of the Brillouin zone. There is strong evidence that this is linked to charge density wave (CDW) order, which reconstructs the large holelike Fermi surfaces predicted by band structure calculations. Recent experiments have revealed the structure of the two CDW components in the benchmark bilayer material ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7\text{−}x}$ in high field where quantum oscillation (QO) measurements are performed. We have combined these results with a tight-binding description of the bands in a single bilayer to give a minimal model revealing the essential physics of the situation. Here we show that this approach, combined with the effects of spin-orbit interactions and the pseudogap, gives a good qualitative description of the multiple frequencies seen in the QO observations in this material. Magnetic breakdown through weak CDW splitting of the bands will lead to a field dependence of the QO spectrum and to the observed fourfold symmetry of the results in tilted fields.

Figure 1

Schematic illustration of CDW hybridization of $\mathrm{Cu}{\mathrm{O}}_2$ bilayer electron states. In (a) are represented the FS sheets arising from the antibonding $\left(A\right)$ and bonding $(B$, dashed line) bands of the $\mathrm{Cu}{\mathrm{O}}_2$ bilayers in a $k_x-k_y$ cross section of the BZ for YBCO. These bands form large hole surfaces of approximately cylindrical shape around the corner of the BZ. The basal-plane parts of the CDW wave vectors ${\mathbf{\delta }}_a$ and ${\mathbf{\delta }}_b$ are shown approximately to scale. The shaded region in (a), which is centered on the point $\frac{1}{2}({\mathbf{\delta }}_a,{\mathbf{\delta }}_b)$, is shown in the other five panels. The effects of the CDW components may be represented by translation of the bands in the other corners of the BZ by ${\mathbf{\delta }}_a$ and ${\mathbf{\delta }}_b$, giving crossings between states connected by one of the CDW wave vectors. The positions where the $A$ and $B$ states are degenerate and $A\text{−}B$ hybridization may occur are marked with open black circles. At the red dots, there is $A\text{−}A$ and $B\text{−}B$ degeneracy, and weaker hybridization occurs. The resulting small electronlike FS pockets are shown in red and black. If the hybridization at the $A\text{−}A$ crossings is weak, it can be crossed at high magnetic fields, and new FS areas, including those represented in (e) and (f), will result.

Figure 2

Reconstructed Fermi surfaces in zero magnetic field created by the CDWs. In (a) are shown the bands in the absence of hybridization by the CDWs. The antiferromagnetic zone boundary is marked by the diagonal red dashed line. In (b) are shown the Fermi surfaces as reconstructed from the Fermi arcs by the CDWs. For both (a) and (b) the SO interaction is nonzero. We assume that the paler parts are removed by the PG. (c) and (d) are calculated with the SO interaction set to zero. By comparison of (b) with (d), one can see which crossings are turned into avoided crossings by the SO interaction. Careful comparison of (a) with (c) shows that the SO interaction also leads to a greater $A\text{−}B$ separation, as expected from Eq. (6).

Figure 3

Reconstructed Fermi surfaces in a magnetic field and details of crossings. (a) The spin-split bands in the absence of hybridization by the CDWs; (b) a close-up of one of the hybridization regions for comparison with (c) and (d). The spin directions in the various bands are marked by arrows. (c),(d) The same regions with the avoided crossings and spin flips caused by the CDW.

Figure 4

Effects of tilted magnetic field and SO interaction on Fermi surface reconstruction. (a) The same magnitude of field as in Fig. 3 is applied at an angle of ${45}^{\circ }$ to c, tilted towards the $y$ direction. (b) Detail of the removal of spin flips and their replacement by small gaps for the crossings near the $k_x$ direction. (c),(d) are for the field tilted towards x and confirm that the SO-induced spin flips are not removed for the crossings close to the field-tilt direction. If the field is tilted towards the $xy$ direction, smaller gaps are created at both crossing points, as indicated in (e) and (f).

Figure 5

Fermi surface orbits, magnetic breakdown, and spin-flip. The straight lines in k space represent schematically the $A$ and $B$ (dashed) bands that intersect to form the small electron pocket Fermi surfaces. Strong hybridization (which would round the corners) and reflection is expected at the black points marking $A\text{−}B$ crossings. Weak hybridization, spin flip on reflection, and a larger probability of transmission is expected at the red $A\text{−}A$ crossing points. The shaded areas represent the different kinds of possible k-space closed orbits of carriers in a magnetic field which would be detected by QO measurements. The change in color of the orbit from black to red and back represents a flip of the spin of an electron by the SO interaction.