Stripe order and vibrational properties of ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4+\delta }$ for $\delta =2∕15$: Measurements and ab initio calculations

The optical properties of the static charge- and stripe-ordered material ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4+\delta }$ for $\delta =\frac{2}{15}$ have been measured over a wide frequency and temperature range for light polarized within the $a\text{−}b$ planes and along the $c$ axis. Below the charge-ordering temperature, $T_{\mathrm{co}}\simeq 319\mathrm{K}$, a charge gap opens and the electronic background, upon which four strong infrared-active phonons are superimposed, drops towards zero. As the temperature decreases, many new spectral features are observed in response to the ordering of interstitial oxygen as well as the formation of a superlattice due to the charge order in the $\mathrm{Ni}{\mathrm{O}}_2$ planes. In particular, the prominent mode at $354{\mathrm{cm}}^{-1}$ splits into three components; while the frequencies do not shift below the magnetic-ordering transition at $T_{\mathrm{m}}=110\mathrm{K}$, there is a transfer of oscillator strength in response to the change in registry of the charge stripes with respect to the underlying lattice. Ab initio calculations have been performed using density-functional theory, and the phonon dispersion curves were obtained using the direct method. Likely assignments of the new modes activated by stripe order are discussed. In some crystals, two antiresonances are observed in the conductivity for $T\simeq T_{\mathrm{co}}$, which change to a resonant character for $T\lesssim T_{\mathrm{m}}$; these modes are shown to be due to longitudinal optic $c$-axis modes which appear as a result of surface misorientation.

Figure 1

(Color online) The unit cell of ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_4$ in the tetragonal $I4∕mmm$ setting with $a=b=3.888\mathrm{\AA }$ and $c=12.555\mathrm{\AA }$. The $\mathrm{Ni}{\mathrm{O}}_2$ layers are separated by LaO bilayers. There are two different oxygen environments; the oxygen atoms in the $\mathrm{Ni}{\mathrm{O}}_2$ planes are referred to in the text as O(1), while the apical oxygen atoms in the LaO layers are referred to as O(2).

Figure 2

(Color online) The reflectance for light polarized in the $a\text{−}b$ planes of ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4.133}$ shown from $\simeq 20{\mathrm{cm}}^{-1}$ to over $12000{\mathrm{cm}}^{-1}$ at a variety of temperatures above and below $T_{\mathrm{m}}=110\mathrm{K}$. The reflectance at low frequency drops quickly with decreasing temperature in response to the formation of a charge gap. At low temperature the reflectance is typical of an insulator and is dominated by the infrared-active modes; below $T_{\mathrm{m}}$ a great deal of vibrational fine structure develops. Inset: The temperature-dependent reflectance of ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4.133}$ for light polarized along the $c$ axis.

Figure 3

(Color online) The temperature dependence of the optical conductivity in the region of the infrared-active lattice vibrations for ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4.133}$ for $E\Vert ab$. The weak antiresonances attributed to the $c$-axis LO modes are denoted by asterisks. The reduction of the electronic background with decreasing temperature signals the formation of a charge gap. In addition, there is a great deal of fine structure below $T_{\mathrm{m}}$.

Figure 4

(Color online) (a) The real part of the optical conductivity in the region of the vibration at $354{\mathrm{cm}}^{-1}$. At room temperature, the vibration appears as a single mode. At $\simeq 180\mathrm{K}$ the mode has a pronounced asymmetry and some additional weak features in the conductivity are also discernible. Below the magnetic transition at $T_{\mathrm{m}}=110\mathrm{K}$ the asymmetry is resolved as the vibration splits into a three modes at 332, 354, and $369{\mathrm{cm}}^{-1}$ (arrows). (b) The detailed temperature dependence of the modes close to $T_{\mathrm{m}}$. (c) The temperature dependence of the mode at about $660{\mathrm{cm}}^{-1}$, which develops a shoulder at about $630{\mathrm{cm}}^{-1}$ below $T_{\mathrm{m}}$, as well as additional fine structure at 478 and $511{\mathrm{cm}}^{-1}$.

Figure 5

(Color online) Temperature dependence of the (a) frequencies and (b) relative intensities of the three peaks identified in Figs. 4a, 4b. The dashed line denotes $T_{\mathrm{m}}=110\mathrm{K}$; the dotted lines are guides to the eye. (c) Temperature dependence of the charge stripe incommensurability, $ϵ$, and the fractions of O-centered, $f_{\mathrm{O}}$, and Ni-centered, $f_{\mathrm{Ni}}$, stripes.

Figure 6

Calculated total energy for the formula unit ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_4$ in the $I4∕mmm$ space group as a function of volume with constant $c∕a$ ratio. The calculated equilibrium volume of $88.3{\mathrm{\AA }}^3$ per formula unit is in good agreement with experimental value of $94.8{\mathrm{\AA }}^3$ (Ref. 34). The dotted line is a fit to the Murnaghan equation of state (Ref. 53).

Figure 7

The calculated phonon dispersion curves in ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_4$ for the tetragonal $I4∕mmm$ space group along the high-symmetry $\Gamma \to M$ direction (zone diagonal), shown in the region of the two infrared-active $E_u$ modes (solid lines). The transverse optic (TO) and longitudinal optic (LO) branches of the $E_u$ modes are labeled. The $A_{2u}$ modes (active along the $c$ axis) are denoted by short-dashed line, while the remaining modes that are active in the $a\text{−}b$ planes are denoted by long-dashed lines. New zone-center modes are expected to be folded back from the ${\mathbf{q}}_{\mathrm{co}}$ branch point (dotted line); the range of variation of $ϵ$ for $T<T_{\mathrm{m}}$ is indicated by the shaded region [see Fig. 5c].

Figure 8

(Color online) The reflectance of ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4.133}$ at room temperature in the region of the two low-frequency antiresonances for light polarized within the $a\text{−}b$ planes. A polarization angle ${\theta }_0$ (dashed line) has been identified where the antiresonance features are totally absent from the reflectance spectra. These features are most pronounced at ${\theta }_0+\frac{\pi }{2}$ (dotted line); by ${\theta }_0+\pi$ (dashed line) they have once again been totally removed from the reflectance.

Figure 9

(Color online) Model calculations. (a) The normal reflectance of the bulk (solid line) based on the experimentally determined oscillator parameters at room temperature for the $ab$-plane and $c$-axis (Table ) lattice modes, as well as the $s$ and $p$ polarized reflectance assuming a $\theta ={30}^{\circ }$ angle of incidence (dotted and dashed lines, respectively). The $c$-axis loss function (short-dashed line) has two prominent peaks which correspond to the frequencies of the LO modes; these peaks correspond exactly with the antiresonance dips in the reflectance spectra for $p$-polarized light. (b) The experimentally determined room-temperature reflectance for ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_{4.133}$ for $E\Vert {\theta }_0$ (solid line) and $E\Vert {\theta }_0+\frac{\pi }{2}$ (dashed line), compared with the $c$-axis loss function. The antiresonant features for $E\Vert {\theta }_0+\frac{\pi }{2}$ occur at the same frequency as the peaks in the $c$-axis loss function, calculated from the experimentally determined $c$-axis dielectric constant, in agreement with the model results in the upper panel.