We study scaling laws characterizing the interdiffusive zone between two miscible fluids flowing side by side in a Y-shape laminar micromixer using the lattice Boltzmann method. The lattice Boltzmann method solves the coupled three-dimensional (3D) hydrodynamics and mass transfer equations and incorporates intrinsic features of 3D flows related to this problem. We observe the different power-law regimes occurring at the center of the channel and close to the top/bottom wall. The extent of the interdiffusive zone scales as the square root of the axial distance at the center of the channel. At the top/bottom wall, we find an exponent 1/3 at early stages of mixing as observed in the experiments of Ismagilov [Appl. Phys. Lett. 76, 2376 (2000)]. At a larger distance from the entrance, the scaling exponent close to the walls changes to 1/2 [J.-B. Salmon and A. Adjari, J. Appl. Phys. 101, 074902 (2007)]. Here, we focus on the effect of the finite aspect ratio on diffusive broadening. Interestingly, we find the same scaling laws regardless of the channel's aspect ratio. However, the point at which the exponent 1/3 characterizing the broadening at the top/bottom wall reverts to the normal diffusive behavior downstream strongly depends on the aspect ratio. We propose an interpretation of this observation in terms of the shear rate at the side walls. A criterion for the range of aspect ratios with non-negligible effect on diffusive broadening is also provided.