Bicontinuous oxide heteroepitaxy with enhanced photoconductivity

Self-assembled systems have recently attracted extensive attention because they can display a wide range of phase morphologies in nanocomposites, providing a new arena to explore novel phenomena. Among these morphologies, a bicontinuous structure is highly desirable based on its high interface-to-volume ratio and 3D interconnectivity. A bicontinuous nickel oxide (NiO) and tin dioxide (SnO2) heteroepitaxial nanocomposite is revealed here. By controlling their concentration, we fabricated tuneable self-assembled nanostructures from pillars to bicontinuous structures, as evidenced by TEM-energy-dispersive X-ray spectroscopy with a tortuous compositional distribution. The experimentally observed growth modes are consistent with predictions by first-principles calculations. Phase-field simulations are performed to understand 3D microstructure formation and extract key thermodynamic parameters for predicting microstructure morphologies in SnO2:NiO nanocomposites of other concentrations. Furthermore, we demonstrate significantly enhanced photovoltaic properties in a bicontinuous SnO2:NiO nanocomposite macroscopically and microscopically. This research shows a pathway to developing innovative solar cell and photodetector devices based on self-assembled oxides.

Phase-field modeling has been routinely used to simulate microstructure evolution during phase separation via spinodal decomposition or nucleation and growth 1, 2, 3 . Here, we perform phase-field modeling to reconstruct the 3D equilibrium microstructure of the 1-xSnO2-xNiO (x=0.33, 0.5, 0.67) nanocomposite thin films based on their plane-view ( Supplementary Fig.  10 Fig. 2) TEM images.

8) and cross-sectional (Supplementary
In the present phase-field model, the equilibrium microstructure is described by the spatial distribution of the local concentration of the NiO, cNiO (denoted as "c" hereafter), in the SnO2-NiO two-phase mixture. The temporal evolution of the c is governed by the Cahn-5 Hilliard equation, where is the total Helmholtz free energy of the two-phase system 4 ; is the mobility which should be a function of the mobilities of all the diffusing species as well as their local concentration 3 . For simplicity, we assume following refs. 1, 5 , where a=0 10 and a=1 correspond to bulk-diffusion-controlled and interface-diffusion-controlled dynamics, respectively. In the simulations, we set M=1 and a=0.9 to model interface-diffusiondominated dynamics considering the moderate bulk interdiffusion across the SnO2-NiO interface. Yet, it is noteworthy that the equilibrium morphology is mainly determined by the competition between the local chemical free energy and the energy of the SnO2-NiO interface, 15 both of which contribute to the total Helmholtz free energy F. In a diffuse-interface description, F is written as, Here (c) is the local chemical free energy density (J/m 3 ), which typically takes the form of a double-well potential 2, 3 , 20 where is the energy density barrier (J/m 3  The second term in the integrand of Eq. (2) is the gradient energy density describing the short-range interaction between the diffusing species, where is the gradient energy coefficient (J/m). The gradient energy density is part of the total interface energy density.
Since (c) is isotropic, the shape anisotropy of the equilibrium microstructure is determined 10 by anisotropy of the specific SnO2-NiO interface energy (J/m 2 ), i.e., the , , , as discussed in the main text.
The specific interface energy is proportional to the gradient energy coefficient. In the limit of a 1D system, one can have the analytical solution of 3 , where is the interface width. 15 Thus, the anisotropy in the specific interface energy can be considered by parametrizing the diagonal components of the gradient energy coefficient matrix . If assuming λ~6nm (a rough estimate from Fig. 3d) and considering a specific interface energy γ~0.3 J/m 2 (a typical value for semi-coherent interfaces), one has κ=κ0~2.710 -9 J/m (estimated using the formula mentioned above). Since the actual length scale of the phases depends on the both 20 the gradient energy and the local chemical free energy (related to w), we performed highthroughput 3D phase-field simulations 3 to identify the values of κ and w that can lead to microstructures patterns that are both visually and statistically similar to the TEM images at equilibrium. To this end, we numerically solve the Cahn-Hilliard equation (Eq. (1)) in a discrete 3D periodic system using a semi-implicit Fourier spectral algorithm 6 with 25 dimensions of 128 ∆x×128∆y×128∆z. The grid sizes ∆x=∆y=∆z=l0=2 nm, and note that for nondimensionalization. For spinodal decomposition, the initial microstructure is a uniform concentration (c=0.5 or 0.67) superimposed with a sinusoidal concentration wave with a peak amplitude of ~0.02. In the case of nucleation and growth, the initial microstructure was generated based on classical nucleation theory as mentioned above 30 (details of implementation are available in ref. 3 ).
To evaluate the structural statistics of the microstructure images, we performed a 2D fast Fourier transform (FFT) to extract the feature length l * for every 2D slice of the simulated 3D microstructures and the 2D TEM images. As an example, Supplementary Fig. 9 below shows the diffraction pattern of the TEM image for the 0.5SnO2-0.5NiO sample ( Supplementary Fig. 9f). From the 2D diffraction pattern, we can calculate the structure factor via the following equation 7  peaks. This approach has been commonly used to determine the feature length of Labyrinthine patterns 7,8  According to the band diagram in Figure 4b, the built-in potential is 1.08 eV. To investigate the length of the depletion region, Hall effect measurement is applied to NiO and SnO2. Major carrier concentration is for electrons in SnO2, and for holes in NiO. Based on the potential-related depletion length equation below, the depletion length for NiO and SnO2 are summarized in Supplementary Table 3. xp 5 and xn are the negative and positive depletion layer widths. At the same time, ND and NA are the carrier concentration of donors and acceptors. q is the electron charge, and Vd is the builtin potential.
Supplementary Table 3 Carrier concentration and depletion region analyzed by Hall effect 10 measurement.

Carrier concentration
Depletion region width P-type N-type -