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Related: How will crystal orientation affect the mobility? (I separate one question out. I wrote my understanding of the concept there, but I am not sure if it is accurate.)

We know that crystal orientation will affect the mobility. What is crystal orientation, in particular for 2d crystals? Is there a basic textbook/lecture explaining this concept?

hkl (and hk for 2d crystals). are Miller Indices Geometry of Crystals for crystal planes, right? Is it the same as, or sufficient to describe, crystal orientation?

Charlie Chang
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  • What direction is the electron moving in? Not all directions are equivalent, so mobility depends on direction. – Jon Custer May 30 '22 at 11:57
  • I have more details in the linked post. Yes mobility depends on orientation. So my question is what is exactly crystal orientation. Is it the relative orientation of a crystal w.r.t. the electric field, or to the substrate or to something else? And how the orientation is geometrically defined, for 2d crystals (in 3d crystals it seems we can use crystal plane; plz see what is quoted in the linked post)? – Charlie Chang May 30 '22 at 12:02
  • It is orientation to the crystal. In a 2d material you still have directions in the plane. – Jon Custer May 30 '22 at 12:04
  • To make it clear, it is orientation of the electrical field (or another layer of crystal, e.g. a SiO2 substrate) to the crystal? And in 2d crystals the direction can be described by $(m,n)$, similar to $(i,j,k)$ in 3d?\It seems we don't need the electrical field/substrate in defining the crystal orientation, e.g. we can say a crystal has x mobility in (1,1), y mobility in (1,3), (if we simply say (m,n) is the orientation, then geometrically the definition is complete) but even so we possibly have assumed a field/substrate? So the definition is still a bit confusing... – Charlie Chang May 30 '22 at 12:24
  • .. In other words, I might be wondering in what cases and how we use the geometrical crystal orientation. – Charlie Chang May 30 '22 at 12:27
  • hkl etc. are Miller Indices, right? Is it the same as crystal orientation? – Charlie Chang May 30 '22 at 14:41

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The orientation in this context is the direction of motion of the electron with respect to the crystal structure. This is typically discussed in terms of a state in reciprocal (“k”) space. An applied electric field is usually parallel to the electron motion and could be used as a stand-in directional reference (no so for any version of the Hall effect). The substrate is irrelevant to the question of orientation.

Gilbert
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  • The electron moves in the crystal (periodic potential); the two correspond to momentum $k'$ of the electron and to the $k$ space, respectively. (It is easy to describe $k'$ in the lattice. How to describe $k'$ in $k$ space?) Then we will have something like Miller Indices to describe the direction of $k'$ in $k$-space? Your answer is concise, so is there a related textbook, paper or note? – Charlie Chang May 31 '22 at 01:03
  • Find one here 'The families of lattice planes are in one-to-one correspon- dence with the possible directions of reciprocal lattice vectors'. So reciprocal lattice vectors, crystal planes, and Miller indices ('for describing lattice planes (or reciprocal lattice vectors)') can all be used to describe crystal orientation, since there are 1-1 correspondences between the three? – Charlie Chang May 31 '22 at 01:25
  • It is like each crystal plane corresponds to a point (instead of a (hyper)plane in $k$-space). (I am not familiar with relevant math so the following discussion may be inaccurate.)\ We may understand this from the perspective of Fourier transform FT (which the transform from lattice to reciprocal lattice is). .. – Charlie Chang May 31 '22 at 01:45
  • .. In $1d$ crystal (or FT), possible periods of the lattice is $nT$, and so the frequencies $f$ (gotten by FT) = $\frac 1 {nT}$. In $2d$ or higher dim crystal (or $2d/3d$/complex FT...), possible periods of the lattice could be represented by the crystal planes/Miller indices (this is intuitively clear), and so it seems natural that the 'frequencies' $f$ (gotten by the FT of the continuous periodic potential function)= points in the reciprocal lattice (FT of the discrete lattice). – Charlie Chang May 31 '22 at 01:45
  • ..Thus (mathematically) part of the significance of the concept of crystal orientation is that it provides an intuitive description of--and connects--reciprocal lattice (in $k$-space) and normal lattice, as well as the discrete/continuous Fourier transform involved. – Charlie Chang May 31 '22 at 01:59
  • ============================================================= But then why it is the direction of motion of the electron instead of that of an electric field or something else (w.r.t. the crystal structure, which could be described by points in the reciprocal lattice, by crystal planes, by Miller indices)? – Charlie Chang May 31 '22 at 02:05