Superexchanges and Charge Transfer in the La$_3$Ni$_2$O$_7$ Thin Films

Abstract

The recent discovery of ambient-pressure superconductivity with $T_c$ above 40 K in La$_3$Ni$_2$O$_7$ thin films represents a significant advance in the field of nickelate superconductor. Motivated by the experimental reports, here we study an 11-band $d-p$ Hubbard model with tight-binding parameters derived from \textit{ab initio} calculations, using large scale determinant quantum Monte Carlo and cellular dynamical mean-field theory. Our results reveal that the major superexchange couplings in La$_3$Ni$_2$O$_7$ thin films can be substantially weaker than in the bulk material at 29.5 Gpa. Specifically, the out-of-plane antiferromagnetic correlation between Ni$-d_{3z^2-r^2}$ orbitals is reduced by about 27\% in film, while the in-plane magnetic correlations remain largely unaffected. We evaluate the corresponding antiferromagnetic coupling constants, $J_{\perp}$ and $J_{\parallel}$ using perturbation theory. With regard to charge transfer properties, we find that the biaxial compression in films reduces charge transfer gap. We also resolve the orbital distribution of doped holes and electrons among the in-plane (Ni$-d_{x^2-y^2}$ and O$-p_x/p_y$) and the out-of-plane (Ni$-d_{3z^2-r^2}$ and O$-p_z$) orbitals, uncovering a pronounced particle-hole asymmetry. Theses findings lay a groundwork for the study of low-energy $t-J$ model of La$_3$Ni$_2$O$_7$ films and provide key insights into the understanding of physical distinctions between the film and bulk bilayer nickelates.

Figures (4)

• Figure 1: Schematic of the 11-band bilayer LNO thin film lattice model with the seven major hopping parameters and the illustration of the two-dimensional NiO$_2$ lattice for DMQC or CDMFT simulations. (a). The orange, blue and green electron clouds denote Ni-3$d_{3z^2-r^2}$, Ni-3$d_{x^2-y^2}$ and O-2$p$ orbitals. For brevity, some of the orbitals on plane $b$ is not displayed. Four hopping processes between Ni-$d$ and O-$p$ orbitals predominantly contribute to superexchange interactions between Ni-$d$ orbitals are: $t_1=-1.4795,t_2=0.6605,t_3=-1.4565,t_6=1.2045$. The site energies are: $\varepsilon_{d_{x^2-y^2}}=-0.962,\varepsilon_{d_{3z^2-r^2}}=-1.070,\varepsilon_{p_{x}/p_{y}}=-4.650,\varepsilon_{p_{z}}=-4.078,\varepsilon_{p_{z}^{'}}=\varepsilon_{p_{z}^{"}}=-2.977$hu2025electronic. The superexchanges between the intra-layer $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals (not shown here) vanishes due to symmetry. Here $t_4=0.5675,t_5=0.4315,t_7=0.4115$ denote electrons hopping processes between O-$p$ orbitals. (b). Dark balls denote nickel positions while light balls show oxygen positions in the NiO$_2$ plane. The 6 $\times$ 6 unit cells used in DQMC simulations is shown here directly, while the 2 $\times$ 2 CDMFT effective cluster is outlined by a blue dashed rectangle. For clarity, only one NiO$_2$ layer is shown here.
• Figure 2: The spin-spin correlation function $\langle S_{ia\alpha}\cdot S_{jb\beta}\rangle$ for four neighboring $d$-orbitals is shown in numbers to demonstrate the relative strength of the antiferromagnetic superexchange couplings in the system. The orange and blue symbols represent $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ orbitals respectively. We have used the specific values $U = 7$ and $J = 0.15U$, very similar to those obtained from constrained RPA christiansson2023correlated. (a) are from DQMC at half-filling ($n_h=1$, $\mu=1.5$) at $T = 0.25$, and (b) are from CDMFT at $\mu = 0$ ($n_h \approx 1.234$). Note that the magnetic correlations between the on-site inter-layer $d_{x^2-y^2}$ and $d_{3z^2-r^2}$ orbitals, which are due to Hund’s coupling, are not shown here.
• Figure 3: $\langle S_{ia\alpha}\cdot S_{jb\beta}\rangle$ between pairs of nearest-neighboring (NN) d-orbitals as a function of temperature $T$ at pristine configuration, as obtained from DQMC simulations. The squares indicate results for the intra-layer (IR) $d_{x^2-y^2}-d_{x^2-y^2}$ correlations, while the circles show the results for the inter-layer (IT) $d_{3z^2-r^2}-d_{3z^2-r^2}$ correlations.
• Figure 4: Charge transfer properties and charge carrier evolution of the La$_3$Ni$_2$O$_7$ thin film. (a). DQMC and CDMFT result of hole concentration $n_h$ as a function of hole chemical potential $\mu_h$. An inflection point suggests the opening of a charge transfer gap at half-filling when the hole chemical potential $\mu_h$ approaches $\mu_h \sim -1.55$. Here $T=0.3$. (b). Carrier concentration as a function of Hund's coupling $J_H$ for each orbital in the LNO thin film under half-filling (dashed line) and pristine configuration ($\mu=0$, solid line) conditions by CDMFT at $T=0.125$. The left axis represents the in-plane and out-of-plane hole concentrations, while the right axis shows the hole concentration of oxygen orbitals. (c) and (d) show electron(hole) concentration variation $\Delta n_{e(h)}^\alpha$ as functions of chemical potential $\mu_{e(h)}$ and $\text{Sr}^{2+}(\text{Ce}^{4+})$ nominal doping levels $x$, respectively. Here, $\Delta n_h^{\mathrm{IP}} = \Delta n_h^{d_{x^2-y^2}} + \Delta n_h^{p_x} + \Delta n_h^{p_y}$ counts the number of holes in the in-plane $d_{x^2-y^2}$ orbital and $p_x/p_y$ orbitals combined, while $\Delta n_h^{\mathrm{OP}} = \Delta n_h^{d_{3z^2-r^2}} + 0.5(\Delta n_h^{p_z} + \Delta n_h^{p_z'})$ counts the number of holes in the out-of-plane. Note that the $\Delta n_h^\alpha$ here represents the difference in hole concentration relative to the pristine LNO thin film (Ni-$3d^{7.5}$), i.e. Hole concentration variation $\Delta n_{h}^\alpha \equiv n_{h}(\mu) - n_{h}(\mu_h = 0)$. When Sr doping level $x$ = 0.203, the nominal hole content in La$_{3-x}$Sr$_x$Ni$_2$O$_7$ corresponds to $\mu_h=0.2$, while at $\mu_h$ of 0.4, the equivalent Sr doping level is $x$ = 0.406. At this doping level, superconductivity is no longer observed according to Ref. hao2025superconductivityphasediagramsrdoped. (e). The electron concentration variation $\Delta n_e^\alpha$ as a function of the electron chemical potential $\mu_e$ in the regime where $\mu_e > 1.55$ (corresponding to electron doping with $n_e > 1$) and hole concentration variation $\Delta n_h^\alpha$ in the regime where hole doping with $n_h = 2-n_e > 1$($n_{e/h} = \sum_{a,\alpha}n^{a\alpha}_{e/h}$), respectively. Here, $\alpha$ denotes distinct $\alpha$-orbitals and charge concentration variation $\Delta n_{e/h}^\alpha \equiv n_{e/h}(\mu) - n_{e/h}(\mu_e = 1.55)$. Results in (c), (d) and (e) above are from DQMC simulations at $T = 0.3$.