Impact of dynamical screening on the phonon dynamics of metallic ${\text{La}}_2{\text{CuO}}_4$

It is shown within a quantitative calculation that poor dynamical screening of the long-ranged Coulomb interaction due to the slow charge dynamics around the $c$ axis leads in metallic ${\text{La}}_2{\text{CuO}}_4$ to low-energy electronic collective excitations in a small region around this axis, strongly mixing with certain interlayer phonons. The manifestation of such a phonon-plasmon scenario in layered systems based on a nonadiabatic charge response recently proposed on a qualitative level is quantitatively investigated by a realistic calculation of the frequency and wave-vector-dependent irreducible polarization part of the density response function. Our calculation corrects for a generic deficit of LDA calculations which as a rule predict a too large electronic $c$-axis dispersion insufficient to describe the $c$-axis charge response in general and in particular the phonons along this axis in the cuprates. As shown in this work, the latter are highly sensitive with respect to the interlayer coupling and thus a very accurate electronic dispersion along the $c$ axis is crucial. Linear response theory is used to calculate the coupled mode dispersion in the main symmetry directions of the Brillouin zone and the charge-density redistributions excited by certain strongly coupling phononlike and plasmonlike modes. The corresponding mode induced orbital averaged changes in the self-consistent potential felt by the electrons are assessed. Our analysis should be representative for the optimally to overdoped state of the cuprates where experimental evidence of a coherent three-dimensional Fermi surface and a coherent $c$-axis charge transport is given. It is demonstrated that modes from the outside of a small region around the $c$ axis can reliably be calculated within the adiabatic limit. On the other hand, modes inside this sector have to be determined nonadiabatically. In particular, the relevance of a strongly coupling nonadiabatic phononlike apex oxygen $Z$-point breathing mode ${\text{O}}_z^Z$ at about 40 meV is emphasized whose mode energy decreases with less doping.

Figure 1

Electronic band structure $E_n\left(\mathbf{k}\right)$ of ${\text{La}}_2{\text{CuO}}_4$ in the 31BM obtained from an accurate tight-binding representation of the first-principles linearized augmented-plane-wave band structure (Ref. 28). (a) The Fermi energy $E_F$ corresponding to the undoped material is taken as the zero of the energy. (b) Density of states $Z\left(E\right)$ of the 31BM.

Figure 2

Calculated phonon dispersion for metallic ${\text{La}}_2{\text{CuO}}_4$ in the adiabatic approximation for the $c$-axis polarized ${\Lambda }_1$ modes $\left[\sim \left(0,0,1\right)\right]$ based on the band structure within the 31BM as input for the proper polarization part ${\Pi }_{\kappa {\kappa }^{\prime }}$ (a). (b)–(d) calculated ${\Lambda }_1$ modes as in (a) but using the modified 31BM (M31BM) as discussed in the text as input for ${\Pi }_{\kappa {\kappa }^{\prime }}$. The panels (b)–(d) represent the phonon dispersion for three different doping levels $x=0.156\left(E_F=-0.08\right)$, $0.224\left(E_F=-0.10\right)$, and $0.297\left(E_F=-0.135\right)$, by applying a rigid-band approximation of the M31BM to represent the optimally doped to overdoped state of ${\text{La}}_2{\text{CuO}}_4$. The experimental results (open squares ◻) are taken from Refs. 12, 13. The full dot (●) denotes the ${\text{O}}_z^Z$ mode. $E_F$ is in units of eV.

Figure 3

(a) Electronic band structure $E_n\left(\mathbf{k}\right)$ of ${\text{La}}_2{\text{CuO}}_4$ in the M31BM taking into account the enhanced anisotropy of the real material as compared with the 31BM typical for LDA. (b) Density of states $Z\left(E\right)$ in the M31BM. The vertical lines indicate the different doping levels from optimal to overdoping as in Fig. 2.

Figure 4

Electronic $k_z$ dispersion for ${\text{La}}_2{\text{CuO}}_4$ along the cut of the Fermi surface highlighted in Fig. 6a by the black bar. (a) 31BM and (b) M31BM. The corners of these surfaces are $\text{A}=\left(0.4\frac{2\pi }{a},0.05\frac{2\pi }{a},0\right)$, $\text{B}=\left(0.5\frac{2\pi }{a},0.05\frac{2\pi }{a},0\right)$, $\text{C}=\left(0.4\frac{2\pi }{a},0.05\frac{2\pi }{a},\frac{2\pi }{c}\right)$, and $\text{D}=\left(0.5\frac{2\pi }{a},0.05\frac{2\pi }{a},\frac{2\pi }{c}\right)$.

Figure 5

Comparison of the doping evolution of the measured Fermi surface of ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ from Refs. 42, 43 for optimal doping $\left(x=0.15\right)$ to overdoping $\left(x=0.22;0.30\right)$ with the corresponding calculated Fermi surface within the rigid-band approximation of the M31BM, respectively. The dots indicate the experimental results.

Figure 6

(Color online) Fermi-surface maps for different $k_z$ values ($k_z\equiv z=0$ and $k_z\equiv z=\frac{2\pi }{c}$) as a function of ${\mathbf{k}}_{\parallel }=\left(k_x,k_y\right)$ for different doping levels in rigid-band approximation of the M31BM. (a) Model OP $\left(x=0.156\right)$, (b) model OD1 $\left(x=0.224\right)$, and (c) model OD2 $\left(x=0.297\right)$. For the explanation of the small black bar in (a) see Fig. 4.

Figure 7

Calculated coupled phonon-plasmon dispersion of the ${\Delta }_1^{\prime }$, $\left({\text{O}}_z^{Z^{\prime }}\right)$ modes (—) from the $Z$ point $\left(\epsilon =0\right)$ along ${\Delta }^{\prime }=\left(\epsilon \frac{2\pi }{a},0,\frac{2\pi }{c}\right)$ to $\epsilon =0.02\left(Z^{\prime }\right)$ for the models (a) OP, (b) OD1, and (c) OD2, respectively. The coupling ${\text{O}}_z^Z$ modes at $Z$ are represented by ●; – –: free plasmon branch; $\cdots$ borderline for damping due to electron-hole decay.

Figure 8

Nonadiabatic coupled phonon-plasmon dispersion as calculated with model OP for the proper polarization part, representing the optimally doped state of ${\text{La}}_2{\text{CuO}}_4$ from ${\Gamma }^{\prime }=\left(\epsilon \frac{2\pi }{a},0,0\right)$ to $Z^{\prime }$ along the ${\Lambda }^{″}=\left(\epsilon \frac{2\pi }{a},0,\zeta \frac{2\pi }{c}\right)$ direction. In the different panels of the figure from left to right the following values of $\epsilon$ have been used: $\epsilon =0.003,0.004,0.006,0.01,0.025,0.05,0.100$. The rightmost panel shows the results of a calculation in adiabatic approximation for $\epsilon =0.100$. Only the ${\Lambda }^{″}$ branches (—) coupling to the charge fluctuations are shown. – –: free-plasmon branch; $\cdots$: borderline for damping. Dots at $Z^{\prime }$: ● ${\text{O}}_z^{Z^{\prime }}\left(\text{na}\right)$; ○ ${\text{O}}_z^{Z^{\prime }}\left(\text{ad}\right)$. na: nonadiabatic; ad: adiabatic.

Figure 9

Nonadiabatic coupled phonon-plasmon dispersion as calculated with model OP for the optimally doped state of ${\text{La}}_2{\text{CuO}}_4$ from $\Gamma$ to $Z^{\prime }=\left(\epsilon \frac{2\pi }{a},0,\frac{2\pi }{c}\right)$ along the ${\Lambda }^{\prime }=\zeta \left(\epsilon \frac{2\pi }{a},0,\frac{2\pi }{c}\right)$ direction. In the different panels from left to right the following values for $\epsilon$ have been used: $\epsilon =0.003,0.004,0.006,0.010,0.025,0.050,0.100$. The rightmost panel shows the result of a calculation within adiabatic approximation for $\epsilon =0.100$. Only the ${\Lambda }^{\prime }$ branches (—) coupling to the charge fluctuations are shown. – –: free plasmon branch; $\cdots$: borderline for damping. Dots at $Z^{\prime }$: ● ${\text{O}}_z^{Z^{\prime }}\left(\text{na}\right)$, ○ ${\text{O}}_z^{Z^{\prime }}\left(\text{ad}\right)$. Dots at $\Gamma$: ● $A_{2u}^{\text{LO}}$ (ferro, na); ○ $A_{2u}^{\text{TO}}$ (ferro, ad). na: nonadiabatic; ad: adiabatic.

Figure 10

Calculated nonadiabatic coupled phonon-plasmon dispersion of the ${\Lambda }_1$ modes of model OP for the optimally doped state of ${\text{La}}_2{\text{CuO}}_4$ along the $\Lambda \sim \left(0,0,1\right)$ direction. The coupling modes at $\Gamma$ [$A_{2u}^{\text{LO}}$ (ferro, na)] and at $Z\left({\text{O}}_z^Z\right)$ are shown as black dots (●). The open dot (○) at $\Gamma$ indicates $A_{2u}^{\text{TO}}$ (ferro, ad).—: ${\Lambda }_1$ modes; – –: free plasmon branch; $\cdots$: borderline for damping; – ⋅ –: steep ${\Lambda }_1$ branch in the adiabatic approximation.

Figure 11

Contour plot in the CuO plane of the nonlocal part of the displacement-induced charge-density redistribution for the ${\text{O}}_z^Z$ mode in metallic ${\text{La}}_2{\text{CuO}}_4$ calculated according to Eq. (22) in adiabatic approximation for the M31BM (model OP) for the proper polarization part (a). (b) Same as in (a) but for the nonadiabatic phononlike ${\text{O}}_z^Z$ mode. (c) Same as (a) but for the nonadiabatic plasmonlike ${\text{O}}_z^Z$ mode. The units are in ${10}^{-4}{e}^2/a_B^3$. The phase of ${\text{O}}_z^Z$ is as shown in Fig. 12. Full lines (—) mean that electrons are accumulated in the corresponding region of space and the broken dotted lines $\left(–\cdots –\right)$ indicate regions where the electrons are pushed away.

Figure 12

Contour plot perpendicular to the CuO plane of the nonlocal part of the displacement-induced charge-density redistribution for the ${\text{O}}_z^Z$ mode in ${\text{La}}_2{\text{CuO}}_4$ calculated from Eq. (22) in adiabatic approximation for the M31BM (model OP) (a). (b) Same as (a) but for the nonadiabatic phononlike ${\text{O}}_z^Z$ mode. (c) Same as (a) but for the nonadiabatic plasmonlike ${\text{O}}_z^Z$ mode. The units and the meaning of the line types are as in Fig. 11. The arrows denote the displacements of the apex oxygen ions.

Figure 13

(a) Calculated displacement-induced charge fluctuations $\delta {\zeta }_{\kappa }\left(\mathbf{q},\sigma ,\omega \right)$ [Eq. (11)] and (b) corresponding orbital averaged changes in the crystal potential $\delta V_{\kappa }\left(\mathbf{q},\sigma ,\omega \right)$ [Eq. (14)] for the phononlike ${\text{O}}_z^{Z^{\prime }}$ mode along the ${\Delta }^{\prime }$ direction $\left(\epsilon \frac{2\pi }{a},0,\frac{2\pi }{c}\right)$, $\epsilon ∊\left(0.00,0.02\right)$. $\text{Cu}3d$ (—) and ${\text{O}}_{xy}2p$ $(\cdots )$ orbital degrees of freedom are considered. The calculations have been performed for model OP.