Quasiparticle Nernst effect in stripe-ordered cuprates

Experiments on underdoped cuprate superconductors suggest an intricate relation between the normal-state Nernst effect and stripe order. The Nernst signal appears enhanced near 1/8 hole doping and its onset temperature scales with the stripe-ordering temperature over some range of doping. Here, we employ a phenomenological quasiparticle model to calculate the normal-state Nernst signal in the presence of stripe order. We find that Fermi pockets caused by translational symmetry breaking lead to a strongly enhanced Nernst signal, with a sign depending on the modulation period of the ordered state and other details of the Fermi surface. This implies differences between antiferromagnetic and charge-only stripes. We also analyze the anisotropy of the Nernst signal and compare our findings with recent data from ${\text{La}}_{1.6-x}{\text{Nd}}_{0.4}{\text{Sr}}_x{\text{CuO}}_4$ and ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_y$.

Figure 1

Real-space structure of (a) site-centered and [(b) and (c)] bond-centered stripes with period-4 (period-8) order in the charge (spin) sector. Shown are spin and charge distributions, with the circle radii corresponding to on-site hole densities. In panel (c), showing “valence-bond” stripes (Refs. 49, 54), the structure of spin-singlet bond modulations is shown as well which has a dominant $d$-wave form factor.

Figure 2

Fermi surfaces for the bond-centered period-8 stripe states with (a) pure bond modulation, $\delta t=0.05\text{eV}$ and (b) pure spin modulation, $V_s=0.09\text{eV}$, plotted in the first quadrant of the Brillouin zone of the underlying square lattice. The Fermi surfaces are qualitatively equivalent to those obtained from site-centered spin or charge potentials. Without spin order (case a), besides open orbits only small holelike closed orbits with a large aspect ratio are present. Spin order (case b) induces both holelike and electronlike closed orbits.

Figure 3

Fermi surfaces for the bond-centered period-8 stripe states with combined spin and charge modulation, plotted in the first quadrant of the Brillouin zone of the underlying square lattice. (a) $V_s=0.09\text{eV}$ and $\delta t=0.02\text{eV}$. (b) $V_s=0.09\text{eV}$ and $\delta t=0.055\text{eV}$. With increasing bond modulation, the small holelike pockets shrink (case a) and disappear (case b).

Figure 4

As in Fig. 3 but for site-centered period eight stripe order. (a) $V_s=0.1\text{eV}$ and $V_c=0.1\text{eV}$. (b) $V_s=0.1\text{eV}$ and $V_c=0.15\text{eV}$. As above, with increasing charge modulation the Fermi pockets disappear in favor of open one-dimensional orbits.

Figure 5

Nernst effect for period-8 antiferromagnetic stripes at doping $x=1/8$ as a function of the spin modulation; the results are identical for the site-centered and bond-centered cases. The Nernst coefficient becomes negative at $V_s\simeq 0.1\text{eV}$, corresponding to maximal local moments of $2{\mu }_B⟨S_z⟩\simeq 0.3{\mu }_B$. Here and in the following, ${\nu }_{yx}$ is the Nernst signal for $\vec{\nabla }T\parallel \widehat{x}$. The stripes have a modulation wave vector $\parallel \widehat{x}$, i.e., run along $\widehat{y}$, such that ${\nu }_{xy}\left({\nu }_{yx}\right)$ is defined with $\vec{\nabla }T$ parallel (perpendicular) to the stripes.

Figure 6

Doping dependence of the Nernst coefficient for period-8 antiferromagnetic stripes, assuming a doping dependence of the stripe order described by Eq. (18) and $V_0=0.15$ eV. It can be seen that the Nernst coefficient is similarly enhanced near $x=1/8$ for both types of stripe order.

Figure 7

Temperature dependence of the Nernst coefficient for period-8 antiferromagnetic stripes. Upon increasing temperature, the Nernst coefficient increases strongly to a large positive value which becomes maximal at around 20 K. Slightly below the ordering temperature $T_{\text{sp}}\simeq 60\text{K}$, the coefficient becomes negative, as observed in experiment. The different scattering rates have been parametrized with $a={\tau }_0^{-1}$, $b=a/70\text{K}$, and $c=a/800{\text{K}}^2$, and we set $V_0=0.1\text{eV}$.

Figure 8

Nernst effect for period-4 charge-only stripes at doping $x=1/8$ as a function of (a) site-centered chemical-potential modulation and (b) bond-centered bond modulation. The direction-averaged Nernst coefficient is clearly either negative or much less enhanced than for spin stripe order for site-centered stripe order. In addition, it is small everywhere where modulation in the charge channel does not exceed 30%, corresponding to $\delta t\lesssim 0.06\text{eV}$ and $V_c\lesssim 0.1\text{eV}$. The large anisotropy in panel (b) is due to the presence of extremely elongated hole pockets.

Figure 9

Nernst effect for site-centered period-8 stripes with combined spin and charge orders. (a) Fixed $V_c=0.03\text{eV}$ as a function of $V_s$. (b) Fixed $V_c=0.1\text{eV}$ as a function of $V_s$. (c) Fixed $V_s=0.1\text{eV}$ as a function of $V_c$. For a spin-stripe potential of $V_s=0.1\text{eV}$, charge potentials above the moderate value $V_c=0.05\text{eV}$ lead to a negative or small Nernst coefficient, see panel (c).

Figure 10

As in Fig. 9 but for bond-centered period-8 stripes. (a) Fixed $\delta t=0.055\text{eV}$ as a function of $V_s$. (b) Fixed $V_s=0.09\text{eV}$ as a function of $\delta t$. As is depicted in panel (a), for a wide range of spin-stripe potentials below $V_s\simeq 0.09\text{eV}$ the Nernst coefficient is positively enhanced. ($V_s=0.1\text{eV}$ corresponds to an ordered moment of $\simeq 0.3{\mu }_B$). For bond modulations $\delta t\lesssim 0.05\text{eV}$, the Nernst coefficient can remain positive, see panel (b).

Figure 11

Nernst coefficient ${\nu }_{yx}$ for a period-16 SDW order as a function of $V_s$ with $x=0.1$. For $V_s\gtrsim 0.07\text{eV}$ (corresponding to a maximal local moment of $m\gtrsim 0.20{\mu }_B$) the Nernst coefficient turns negative with an enhanced amplitude in comparison to the nonordered state. Note that in a stripe (or SDW) picture for ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_y$, the stripes run along the $b$ axis [as inferred from neutron scattering (Refs. 21, 40)] and our ${\nu }_{yx}$ corresponds to the Nernst signal with $\vec{\nabla }T$ along the $a$ axis.

Figure 12

Nernst effect for site-centered period-10 stripe order. Spin only stripe order (a) leads to a negative Nernst coefficient for spin potentials above $V_s=0.09\text{eV}$. (b) Adding additional charge order to a spin potential of $V_s=0.1\text{eV}$ does not change the sign of the Nernst coefficient for charge potentials $V_c\le 0.1\text{eV}$, which correspond to realistic charge modulations of up to 30%.

Figure 13

Nernst effect for period-10 stripe order at finite temperatures. Upon decreasing temperature to below about 25 K, the Nernst coefficient changes sign and becomes negative. Upon increasing temperature above about 25 K, the coefficient becomes positive and significantly enhanced. Slightly below the ordering temperature $T_{\text{sp}}\simeq 50\text{K}$, the coefficient becomes negative again. The different scattering rates have been parameterized with $a={\tau }_0^{-1}$, $b=a/70\text{K}$, and $c=a/800{\text{K}}^2$, and we set $V_0=0.12\text{eV}$.

Figure 14

Fermi surfaces as resulting from period-10 stripe order, plotted in the first quadrant of the Brillouin zone of the underlying square lattice. Pure spin stripe order with $V_s=0.08\text{eV}$ produces both electronlike and holelike closed orbits, see (a). Adding additional charge stripe order with $V_c=0.07\text{eV}$ eliminates the electron like orbits and the remaining closed orbits are all holelike, see (b).