Spin Dynamics of the Electron-Doped High-$T_c$ Superconducting Cuprates

We show that a basic difference between the electron- and hole-doped cuprates is their proximity to two different quantum critical points in a 2D free fermion system on a square lattice and that the spin dynamics observed recently for the first time in the electron-doped ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$, very different from that in the hole-doped cuprates, can be understood as a consequence of this effect.

Figure 1

Static electron-hole response function $t{\chi }^0(\mathbf{Q})$ as a function of electron concentration $\delta =1-n$.

Figure 2

Wave vector dependence of the static susceptibility. (a) $t{\chi }^0(\mathbf{q},0)$ as a function of $\mathbf{q}$ [direction $(1,0)$] for different doping from QCP1 to QCP2 ($Z_1>0$, $Z_2>0$) and outside ($Z_1<0$ or $Z_2<0$). (b) Characteristic wave vectors as functions of chemical potential [dashed area indicates the region of the plateau in the $q$ dependence of ${\chi }^0(\mathbf{q},0)$]. [The calculations are done for $t^{\prime }/t=-0.3$, so that $(Z_1+Z_2)/t=1.2$.]

Figure 3

$t\mathrm{Im}{\chi }^0(\mathbf{q},\omega )$ and $t\mathrm{Re}{\chi }^0(\mathbf{q},\omega )$ as functions of $\omega$ for different wave vectors in the $(1,0)$ direction. Calculations are done for $Z_2/t=0.1$ (electron doping) and $t^{\prime }/t=-0.3$. The singularity at ${\omega }_{c1}$ is not seen being out of the energy scale.

Figure 4

Electron-hole continuum for the case of electron doping ($\mu /t=-0.1$ as deduced from ARPES data for ${\mathrm{Nd}}_{1.75}{\mathrm{Ce}}_{0.15}{\mathrm{CuO}}_4$ [1]). The blue line in (a) and (b) is a pseudogap line, ${\omega }^{ps}(q)={\omega }_{c2}(q/q_{c2}-1)$. Its continuation in (b) towards low $\omega$ shown by the red line [described by the same analytical expression, $q_{c2}(\omega )=q_{c2}(1+\omega /{\omega }_{c2})$] is a true gap line. Two other gap lines forming the total true gap line in (b) are discussed in the text. [In (b), $\Delta /t=0.1$.]

Figure 5

Momentum dependences [$(1,0)$ direction] of the collective fluctuations in the normal and SC states in the case of electron doping ($Z_2/t=0.1$, $\Delta /t=0.04$, $t/J=1.6$).