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Caption : fig.s1 schematic illustrations of aa, ab, and ac stacking configurations of ti₃c₂o₂. (a) aa stacking; (b) ab stacking; and (c) ac stacking.

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Caption : fig.s2 total energy evolution profiles obtained from 10 ps first-principles molecular dynamics (fpmd) simulations at 300 k with a time step of 1 fs for the twisted bilayer ti3c2o2 moiré superlattices with aa stacking: (a) 0°; (b) 21.78°; (c) 27.8°; and (d) 38.22°.

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Caption : fig.s3 total energy evolution profiles obtained from 10 ps first-principles molecular dynamics (fpmd) simulations at 300 k with a time step of 1 fs for the twisted bilayer ti3c2o2 moiré superlattices with ab stacking: (a) 0°; (b) 21.78°; (c) 27.8°; and (d) 38.22°.

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Caption : fig.s4 total energy evolution profiles obtained from 10 ps first-principles molecular dynamics (fpmd) simulations at 300 k with a time step of 1 fs for the twisted bilayer ti3c2o2 moiré superlattices with ac stacking: (a) 0°; (b) 21.78°; (c) 27.8°; and (d) 38.22°.

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Caption : fig.s5 final atomic structures of twisted bilayer ti3c2o2 moiré superlattices after 10 ps fpmd simulations at 300 k. panels (a-c) correspond to different stacking configurations (aa, ab, and ac), while different twist angles are presented within each panel.

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Caption : fig.s6 calculated phonon spectra of twisted bilayer ti₃c₂o₂ moiré superlattices with different stacking configurations and twisting angles: (a) 0° aa, (b) 21.78° aa, (c) 27.8° aa, (d) 38.22° aa; (e) 0° ab, (f) 21.78° ab, (g) 27.8° ab, (h) 38.22° ab; (i) 0° ac, (j) 21.78° ac, (k) 27.8° ac, and (l) 38.22° ac.

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Caption : fig.s7 3d contour plots of charge density difference distributions of ti3c2o2 with different stacking configurations at 0°. (a) aa stacking; (b) ab stacking; and (c) ac stacking. the isovalue contours are set to ±0.0001 e·å-3.

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Caption : fig.s8 3d contour plots of charge density difference for ti3c2o2 moiré superlattices with different stacking configurations at 21.78°. (a) aa stacking; (b) ab stacking; and (c) ac stacking. the isovalue contours are set to ±0.0001 e·å-3.

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Caption : fig.s9 3d contour plots of charge density difference for ti3c2o2 moiré superlattices with different stacking configurations at 27.8°. (a) aa stacking; (b) ab stacking; and (c) ac stacking. the isovalue contours are set to ±0.0001 e·å-3.

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Caption : fig.s10 3d contour plots of charge density difference for ti3c2o2 moiré superlattices with different stacking configurations at 38.22°. (a) aa stacking; (b) ab stacking; and (c) ac stacking. the isovalue contours are set to ±0.0001 e·å-3. the red box indicates the minimal supercell.

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Caption : fig.s11 relaxed structures after interlayer adsorption of a single li atom at different adsorption sites and twist angles in aa-stacked ti₃c₂o₂: (a) 0°; (b) 21.78°; (c) 27.8°; (d) 38.22°.

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Caption : fig.s12 relaxed structures after interlayer adsorption of a single li atom at different adsorption sites and twist angles in ab-stacked ti₃c₂o₂: (a) 0°; (b) 21.78°; (c) 27.8°; (d) 38.22°.

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Caption : fig.s13 relaxed structures after interlayer adsorption of a single li atom at different adsorption sites and twist angles in ac-stacked ti₃c₂o₂: (a) 0°; (b) 21.78°; (c) 27.8°; (d) 38.22°.

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Caption : fig.s14 representative surface diffusion pathways of li ions on ti₃c₂o₂ moiré superlattices. since the surface diffusion paths are nearly identical across different stacking configurations and twist angles, only a representative structure is displayed.

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Caption : fig.s15 spatial distribution of the normalized z-coordinate percentage of interlayer li adsorption in aa-stacked twisted bilayer ti₃c₂o₂ moiré superlattices, reflecting the adsorption preference of li ions toward the upper or lower ti₃c₂o₂ layer: (a) 0°; (b) 21.78°; (c) 27.8°; and (d) 38.22°.

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Caption : fig.s16 spatial distribution of the normalized z-coordinate percentage of interlayer li adsorption in ab-stacked twisted bilayer ti₃c₂o₂ moiré superlattices, reflecting the adsorption preference of li ions toward the upper or lower ti₃c₂o₂ layer: (a) 0°; (b) 21.78°; (c) 27.8°; and (d) 38.22°.

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Caption : fig.s17 spatial distribution of the normalized z-coordinate percentage of interlayer li adsorption in ac-stacked twisted bilayer ti₃c₂o₂ moiré superlattices, reflecting the adsorption preference of li ions toward the upper or lower ti₃c₂o₂ layer: (a) 0°; (b) 21.78°; (c) 27.8°; and (d) 38.22°.

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Caption : fig.s18 interlayer diffusion pathway of li ions in the 0° aa-stacked ti₃c₂o₂ moiré superlattice, showing the evolution of the li-o coordination environment from sixfold prismatic coordination to a fourfold planar coordination at the saddle point and back to sixfold prismatic coordination.

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Caption : fig.s19 interlayer diffusion pathway of li ions in the 21.78° aa-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s20 interlayer diffusion pathway of li ions in the 27.8° aa-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s21 interlayer diffusion pathway of li ions in the 38.22° aa-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s22 interlayer diffusion pathway of li ions in the 0° ab-stacked ti₃c₂o₂ moiré superlattice, showing the evolution of the li-o coordination environment from octahedral coordination to tetrahedral coordination at the saddle point and back to octahedral coordination.

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Caption : fig.s23 interlayer diffusion pathway of li ions in the 21.78° ab-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s24 interlayer diffusion pathway of li ions in the 27.8° ab-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s25 interlayer diffusion pathway of li ions in the 38.22° ab-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s26 interlayer diffusion pathway of li ions in the 0° ac-stacked ti₃c₂o₂ moiré superlattice, showing a tetrahedral-octahedral-tetrahedral migration mechanism. the initial and final states correspond to fourfold tetrahedral li-o coordination, while the saddle point exhibits sixfold octahedral coordination.

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Caption : fig.s27 interlayer diffusion pathway of li ions in the 21.78° ac-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s28 interlayer diffusion pathway of li ions in the 27.8° ac-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s29 interlayer diffusion pathway of li ions in the 38.22° ac-stacked ti₃c₂o₂ moiré superlattice, illustrating the migration path together with the local coordination environments at the minimum- and maximum-energy positions.

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Caption : fig.s30 atomic structures of li-saturated ti3c2o2 moiré superlattices with different stacking configurations and twist angles. panels (a-c) correspond to different stacking configurations, and different twist angles are shown within each panel.

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Caption : fig.s31 representative atomic structures in which surface-adsorbed li atoms exhibit aggregation during first-principles molecular dynamics simulations of ti₃c₂o₂ moiré superlattices: (a) 21.78°aa; (b) 38.22°ab; (c) 21.78°ac.

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Caption : fig.s32 electron localization function (elf) maps of saturated li adsorption structures for different stacking configurations and twist angles. panels (a-c) correspond to different stacking configurations, and different twist angles are shown within each panel.

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Caption : fig.s33 schematic illustration of the definitions of interlayer distance and interlayer sliding in ti₃c₂o₂ moiré superlattices.

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Caption : fig.s34 electronic band structures of li-intercalated ti3c2o2 with different stacking configurations and their moiré superlattices.
(a) 0°aa ti39c26o26∥ti39c26o26@li12,
(b) 21.78°aa ti63c42o42∥ti63c42o42@li22,
(c) 27.8°aa ti39c26o26∥ti39c26o26@li14,
(d) 38.22°aa ti63c42o42∥ti63c42o42@li18;
(e) 0°ab ti39c26o26∥ti39c26o26@li13,
(f) 21.78°ab ti63c42o42∥ti63c42o42@li23,
(g) 27.8°ab ti39c26o26∥ti39c26o26@li13,
(h) 38.22°ab ti63c42o42∥ti63c42o42@li24;
(i) 0°ac ti39c26o26∥ti39c26o26@li13,
(j) 21.78°ac ti63c42o42∥ti63c42o42@li24,
(k) 27.8°ac ti39c26o26∥ti39c26o26@li13,
(l) 38.22°ac ti63c42o42∥ti63c42o42@li21.

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Caption : fig.s35 ordered li interlayer structure of ti₃c₂o₂ formed after li insertion at a 0° twist angle: (a) 0°aa; (b) 0°ab; (c) 0°ac.

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Caption : fig.s36 comparison of electronic band structures and density of states (dos) of ti₃c₂o₂ moiré superlattices with different stacking configurations and twist angles before and after li intercalation. (a-d) aa stacking with twist angles of 0°, 21.78°, 27.8°, and 38.22°; (e-h) ab stacking with the corresponding twist angles; (i-l) ac stacking with the corresponding twist angles.