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I am beginner of material modelling that's why my question may be basic and I have no direction. If we want to see whether two crystal layer(semiconductor) can be put together, can the DFT calculation let us to find the answer?

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Speaking as someone who does both computational modeling and experiment, there is not a trivial answer! I'm not sure if you are referring to 2D materials (which Jack addressed in his answer) but I am not experienced with that, so I will try to answer in the context of general thin films.

In experiment, you can of course deposit films on any substrate you wish, but what you really get at the end of it isn't always straightforward. Let's assume that you can grow the films epitaxially with good lattice matching, and that you have a good understanding of the steps needed to ensure a single, known surface termination on the substrate. There's still no guarantee that you will get a clean interface, since different materials can still chemically react with each other. For example, growth of non-silicon oxides (MOx) on silicon can result in a Si/SiO2/MOx structure rather than Si/MOx. A nice paper on why SrTiO3 is not a good gate dielectric material is Kolpak and Ismail-Beigi, Phys. Rev. B 83, 165318.

On the computational side, predicting what the interface will be is also not easy. There is a large space of possible terminations and arrangements to explore. The paper I referenced shows a small exploration of the possibilities with SrTiO3 and Si. There is no way of knowing for sure what your real-life termination will be without experiment. Usually you have some known idea of termination and structure (from high resolution cross-sectional transmission electron microscopy, for example) which you can then model to get some insights from electronic structure. It's challenging to make predictions purely from modeling in these cases.

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  • In the experiment, you can always put two semiconducting crystal layers together.

  • For DFT simulation, you can also stack two semiconducting crystal layers together, but you must consider the lattice match. For example, monolayer MoS$_2$ and monolayer WS$_2$ can be stacked directly because these two monolayers have almost the same lattice constant. But if you want to stack graphene and MoS$_2$, you must build supercells (with almost the same lattice constants) of both monolayers and then stack together.

  • These ideas to build a reasonable 2D heterostructure model for DFT simulation can be applied to the bulk materials.

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Since we focusing on matter modeling I would try to provide a method a general method of actually creating these models. As others have mentioned, you will need to deal with the problem of mismatching lattices. This can be deal with by either directly stretching the lattice of one material to match the other and simply stacking them or we can make a model which allows for the material to be closer to its original lattice constant.

One way to do this for surfaces that share the same cell shape is to find cell transformations which preserve the cell shape but result in repetitions of the cell. This can easily be done by generating 2x2 or 3x3 cell repetitions for example, but we can also find other valid cell transformations that will work. For example, in a FCC surface we can find cells which repeat the cell the following number of times within the same shape.

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49

These cannot be identified visually very easily, but these cells are generally referred to as root cells in literature with various symbols. Finding them has been automated in ASE within the build module and it has been generalized to all surface shapes. This cell can then be minimally strained by stretching or compressing it to match two surfaces together.

The final result is that you can match an oxide such as TiO2 to WO3 for example if both surfaces share the same cell shape. One surface will need to be slightly strained (typically the overlayer) but this can be minimized. It may be that some strain is favorable as well, and this can be modeled in the same way. This avoids a strain that is only applied in one direction which may be unphysical.

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