The PDF file includes:

• Section S1. Details of simulation methods, models, and data analysis
• Section S2. Details of analytical models
• Fig. S1. Characterizations of the prepared 2D material of GO.
• Fig. S2. Cryo-TEM images of the blank liposomes, the formation process of the sandwiched GO superstructure, and the sandwiched structure at different detection angles.
• Fig. S3. Tomography views of the 3D map for the GO-membrane superstructure and the blank liposome vesicles.
• Fig. S4. A series of TEM images of the GO–cell membrane interaction and the cells after exposure to different dimensional materials.
• Fig. S5. The interaction between GO and the cells.
• Fig. S6. Molecular models for the individual entities used in the simulations.
• Fig. S7. Translocation pathways of GO across the lipid membrane toward the sandwiched GO structure.
• Fig. S8. Translocation pathways of GO, with the model representing outcomes from standard oxidization, across the lipid membrane.
• Fig. S9. The displacement probability distributions and the translational diffusion coefficients of the GO sandwiched inside the membrane.
• Fig. S10. A schematic diagram illustrating the definition of the turning angle between the neighboring persistent segments.
• Fig. S11. Diffusive properties of GO with χ_GT = 7.15.
• Fig. S12. Transition of diffusion patterns of the sandwiched GO from Brownian to Lévy and even directional dynamics with a membrane size of 40×40 rc2.
• Fig. S13. Diffusive dynamics and membrane-pore states of a circular GO.
• Fig. S14. Simulation results demonstrate various membrane-pore states and the mechanism of pore formation.
• Fig. S15. Representative snapshots from simulations feature the sandwiched GO–induced pores in the single leaflet of cell membranes.
• Fig. S16. The energy of the sandwiched GO–induced pore as a function of the radius of the pore R at K_a ≥ K_a0 (K_a0 ≈ 25 k_BT/nm²).
• Fig. S17. Correlation between the analytical model and simulation results.
• Fig. S18. Diffusive dynamics of lipids varies from Fickian to superdiffusive.
• Fig. S19. Sandwiched GO–induced pores in the single leaflet of the cell membrane for the GO model representing outcomes from standard oxidization processes.
• Fig. S20. The efficacy of the GO-sandwiched structure on drug delivery.
• Fig. S21. Diffusive dynamics of a representative drug bead captured by the transmembrane receptor.
• Fig. S22. Probability distribution of the capturing time for the drug beads released from the sandwiched GO and the center of the intracellular region.
• References (55–59)

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Other Supplementary Material for this manuscript includes the following:

• Movie S1 (.mov format). Detailed translocation pathway of GO across the lipid membrane toward the sandwiched GO structure at χ_GT = 15.73.
• Movie S2 (.mov format). Detailed translocation pathway of the GO model, representing outcomes from standard oxidization processes with the oxidation degree ρ_O = 0.3, across the lipid membrane toward the sandwiched GO structure.
• Movie S3 (.mov format). Detailed diffusive dynamics of a sandwiched GO exhibiting Brownian motion at χ_GT = 1.43.
• Movie S4 (.mov format). Detailed diffusive dynamics of a sandwiched GO exhibiting Lévy walk at χ_GT = 10.01.
• Movie S5 (.mov format). Detailed diffusive dynamics of a sandwiched GO exhibiting directional motion at χ_GT = 14.3.