Magnetic fluctuations in $n$-type high-$T_c$ superconductors reveal breakdown of fermiology: Experiments and Fermi-liquid/RPA calculations

By combining experimental measurements of the quasiparticle and dynamical magnetic properties of optimally electron-doped ${\mathrm{Pr}}_{0.88}\mathrm{La}{\mathrm{Ce}}_{0.12}\mathrm{Cu}{\mathrm{O}}_4$ with theoretical calculations, we demonstrate that the conventional fermiology approach cannot possibly account for the magnetic fluctuations in these materials. In particular, we perform tunneling experiments on the very same sample for which a dynamical magnetic resonance has been reported recently and use photoemission data by others on a similar sample to characterize the fermionic quasiparticle excitations in great detail. We subsequently use this information to calculate the magnetic response within the conventional fermiology framework as applied in a large body of work for the hole-doped superconductors to find a profound disagreement between the theoretical expectations and the measurements: this approach predicts a steplike feature rather than a sharp resonance peak, it underestimates the intensity of the resonance by an order of magnitude, it suggests an unreasonable temperature dependence of the resonance, and most severely, it predicts that most of the spectral weight resides in incommensurate wings which are a key feature of the hole-doped cuprates but have never been observed in the electron-doped counterparts. Our findings strongly suggest that the magnetic fluctuations reflect the quantum-mechanical competition between antiferromagnetic and superconducting orders.

Figure 1

The nonmonotonic $d$-wave gap $\Delta$ of PLCCO along the Fermi surface as a function of the Fermi surface angle $\varphi$ (see the inset) calculated with the set of parameters listed in Table . The inset shows the relation between the Fermi surface and the magnetic Brillouin zone. The hot spots relevant for the magnetic response at ${\mathbf{q}}_{\mathrm{AF}}$ are shown as open circles. The position of the gap maximum close to the hot spots and the ratio ${\Delta }_{\max }∕{\Delta }_0$ of the maximum gap value and the antinodal gap are in good agreement with ARPES measurements (Ref. 19). The absolute gap values are extracted from our tunneling experiment (see Sec. 3).

Figure 2

(Color online) Geometry and results of direct point-contact tunneling measurements on single crystals of PLCCO. (a) The schematic diagram of the experimental setup, where a Au tip is pointed toward the $a∕b$ axis direction determined by neutron diffraction. (b) The relationship between standard $d$-wave gap and tunneling direction. (c) Calculated quasiparticle density of states using gap values at different temperatures showing Van Hove singularities at the antinodal gap ${\Delta }_0$ and the maximum gap ${\Delta }_{\max }$. (d) Temperature dependence of the $dI∕dV$ spectra from $2\text{to}20\mathrm{K}$ with increments of $2\mathrm{K}$. The spectra were obtained by normalizing the corresponding backgrounds at temperatures well above $T_c$. (e) Temperature dependence of the gap value ${\Delta }_0$; the solid line denotes the BCS prediction. (f) Magnetic field dependence of the $dI∕dV$ spectra for a $c$-axis aligned magnetic field. The theoretical calculations are indicated by red lines in (d) and (f). All the spectra and fitting lines except for the lowest ones are shifted upward for clarity. (g) Superconducting gap as a function of increasing magnetic field; the solid line is a guide for the eyes. The ${\Delta }_0$ values in (e) and (g) are determined by fitting the normalized spectra to the extended Blonder-Tinkham-Klapwijk model (Ref. 21) with a $d$-wave-type gap function along the $a∕b$ axes.

Figure 3

(Color online) Magnetic response ${\chi }^{″}(\mathbf{q},\omega )$ calculated within the FL/RPA approach using a set of parameters optimized for optimally doped YBCO (Ref. 14) plotted along the $(H,0.5)$ and $(H,H)$ (inset) directions close to ${\mathbf{q}}_{\mathrm{AF}}$. White and black points show neutron scattering data from Refs. 3, 8, respectively.

Figure 4

(Color online) Evaluation of ${\chi }^{″}({\mathbf{q}}_{\mathrm{AF}},\omega )$ in the superconducting state for different values of $U$ showing the narrow energy window $515\mathrm{meV}<U<528\mathrm{meV}$ for which a bound state in the gap $2\Delta =10\mathrm{meV}$ of the particle-hole is formed. For $U=500\mathrm{meV}$, we find an intensity enhancement at the experimentally observed resonance energy ${\omega }_{\mathrm{res}}=11\mathrm{meV}$. Since the feature is located slightly above the gap, the intensity is significantly reduced compared to the bound-state situation and the line shape is very asymmetric.

Figure 5

(Color online) (a) Evolution of the resonance feature calculated for PLCCO within the FL/RPA approach with temperature. With increasing temperature, the resonance feature shifts continuously to lower energy and decreases in intensity. (b) Calculated intensity at ${\omega }_{\mathrm{res}}$ as a function of temperature showing a strong decrease at temperatures well below $T_c$.

Figure 6

(Color online) Comparison of the resonance in absolute units with FL/RPA calculations for optimally hole-doped YBCO and electron-doped PLCCO. (a) Local susceptibility in absolute units for optimally doped YBCO at $10\mathrm{K}$ from Ref. 8. The solid blue line is the calculation based on RPA model scaled to match the experimental data. (b) Local susceptibility in both the normal and superconducting states for PLCCO obtained from converting the raw data of Ref. 16 to absolute units. Solid lines are guide for the eyes. The dashed lines represent the results of the FL/RPA calculations with the same scale factor as used for YBCO. Note that the theoretical values are about six times smaller than the experimental results.

Figure 7

(Color online) Comparison of the magnetic excitation spectrum ${\chi }^{″}(\mathbf{q},\omega )$ along the $[H,H]$ direction in the vicinity of ${\mathbf{q}}_{\mathrm{AF}}=(1∕2,1∕2)\mathrm{r.l.u}$. resulting from the FL/RPA calculations with neutron scattering data (Ref. 16) on optimally doped PLCCO $(T_c=24\mathrm{K})$ measured at $T=2\mathrm{K}$. The very strong incommensurate wings predicted by the calculations highlight the failure of the FL/RPA approach.

Figure 8

(Color online) Bare Lindhard functions ${\chi }_0^{″}(\mathbf{q},\omega )$ of YBCO and PLCCO in the superconducting phases calculated with band structure and $d$-wave gap parameters listed in Table . Whereas the momentum dependence of the gap of the particle-hole continuum looks very similar for the $p$-type and $n$-type materials, the distribution of spectral weight is completely different. For the $p$-type material, spectral weight is accumulated at ${\mathbf{q}}_{\mathrm{AF}}$, whereas for the $n$-type materials, a lot of intensity has shifted from ${\mathbf{q}}_{\mathrm{AF}}$ to incommensurate wave vectors.

Figure 9

(Color online) Comparison of the normal-state dispersions, Fermi surfaces (left panel), and Fermi velocities (right panel) of YBCO and PLCCO.