Calculated magnetic exchange interactions in high-temperature superconductors

Using a first-principles linear-response approach, we study the magnetic exchange interactions $J$ for a series of superconducting cuprates. We reproduce the observed spin-wave dispersions together with other experimental trends, and show that different cuprates have similar $J^{\prime }$s regardless of their $T_c$. The nearest-neighbor $J$ is not sensitive to the hole doping, which agrees with recent experiments. For the undoped cuprates, the second-nearest-neighbor $J$ is ferromagnetic, but changes its sign with hole doping. We also find that, in contrast to the hopping integral, the exchange interaction is not sensitive to the position of apical oxygen. To see the effect of the long-range nature of the exchange on the superconducting $T_c$, we study the dynamical spin susceptibility $\chi \left(q,\omega \right)$ within the $t\text{−}J$ model using a dynamical cluster approximation.

Figure 1

(Color online) Definitions of nearest-neighbor, $J_1$, next-nearest-neighbor, $J_2$, and third-nearest-neighbor, $J_3$, exchange interactions for Cu spins, which are the parameters of a Heisenberg model $H={\sum }_{⟨ij⟩}{J}_{ij}{S}_i{S}_j$.

Figure 2

(Color online) Comparison between calculated (solid lines) and experimental (symbols) spin-wave dispersions for ${\text{La}}_2{\text{CuO}}_4$. The triangles and squares are the experimental results at $T=10$ and 295 K, respectively (Ref. 11).

Figure 3

(Color online) Calculated spin-wave dispersions for ${\text{HgBa}}_2{\text{CaCu}}_2{\text{O}}_6$, ${\text{Sr}}_2{\text{CuO}}_2{\text{Cl}}_2$, and ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_6$.

Figure 4

(Color online) Spin susceptibility ${\chi }^{″}\left(q,\omega \right)$ for $q=\left(\pi ,\pi \right)$ versus $\omega$ for different exchange interactions $J_2$ and $J_3$. Here, $J_1=0.3t$ is fixed and a filling $⟨n⟩=0.8$ and temperature $T=0.08t$ have been used.

Figure 5

Superconducting transition temperature for different exchange interactions $J_2$ and $J_3$. Here, we have used the same values for $J_1$, $J_2$, and $J_3$ and the filling as in Fig. 4