Chapter 8 The effect of surface roughness on the alignment of grown 2D materials
8.1 Modeling and discussions
Figure 8.2(a) shows the atomic structure of an ideal Cu(111)-based high-index low symmetric Cu surface with parallel step edges along a constant direction. The heights of the atoms are coded by different colors and the tilted angle of this surface with respect to its high-symmetric terrace plane (that
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is Cu(111) in this example) is denoted by α. Obviously, the step density increases with the increase of α. As discussed above, a realistic substrate surface must have a surface roughness. Figure 8.2(b) shows an example of the possible structure of a realistic surface, where we use θ to define the angle between the substrate surface and the tangent line of a nanoprotrusion (or nanoindent) on the surface, to describe the sharpness of the nanoprotrusion (or nanodent). As shown in Figure 8.2(b), the step edges are no longer aligned along a constant direction but become meandering because of the surface roughness.
Figure 8.2(e) shows the top views of three high-index Cu surfaces with a θ of 3.5º. From these top views, we can easily find the direction change of step edges and we use ∆𝜑 to describe the maximum change of the step edge direction from its original direction, ∆𝜑 = 𝜑𝑚𝑎𝑥− 𝜑𝑚𝑖𝑛.
Figure 8.2 The effects of roughness on the surface topography of a Cu foil. (a) Atomic models of an ideal high-index Cu surface which has a tilted angle of 𝛼 with its low-index terrace. (b) Different views of a fluctuated Cu foil surfaces with 𝜃 denoting the fluctuation degree. (c) Model of constructing a fluctuated Cu surface. (d) Relationship between Cu foil fluctuation degree and variation of Cu step edge direction for various high-index Cu foils. (e) Top views of various high-index Cu surfaces under a same surface fluctuation with Height profile colored. (e) Relationship between 𝜃 and 𝛼 regard to different step variation.
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To quantitatively calculate ∆𝜑, a simplified cone model is used to descibe the nanoprotrusion on the Cu surface, as shown in Figure 8.2(c). Suppose the Cu surface is in the horizon plane, the surface of the cone can be described by:
𝑧 = −√(tan 𝜃)2∙ (𝑥2+ 𝑦2) (8.1)
We further assume that the angle between the high-index surface and its nearest low-index terrace is 𝛼 and planes parallel to the low-index terrace satisfies that:
𝑧 = tan 𝛼 ∙ 𝑥 − 𝐶 (8.2)
The step edges on the cone are thus the interception lines of the cone surface and the planes of the terraces, and they are curved lines. Moreover, the direction the projections of the tangential directions of the interception lines are thus the directions of the step edges. Mathematically the projection of the interception curves to the high-index surface (z = 0) satisfies:
−√(tan 𝜃)2∙ (𝑥2+ 𝑦2) = tan 𝛼 ∙ 𝑥 − 𝐶 (8.3) From Eq. (8.3), we can get:
𝑦 = ±√((tan 𝛼)2
(tan 𝜃)2− 1) ∙ 𝑥2− 2𝐶 tan 𝛼
(tan 𝜃)2∙ 𝑥 + 𝐶2
(tan 𝜃)2 (8.4)
𝑑𝑦
𝑑𝑥 = ± 1 tan 𝜃
((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥 − 𝐶 tan 𝛼
√((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥2− 2𝐶 tan 𝛼 ∙ 𝑥 + 𝐶2 (8.5) The highest and lowest boundaries of the step edge direction (𝜑𝑚𝑎𝑥 and 𝜑𝑚𝑖𝑛)can be calculated by:
𝑑2𝑦
𝑑𝑥2= ∓ tan 𝜃 ∙ 𝐶2
[((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥2− 2𝐶 tan 𝛼 ∙ 𝑥 + 𝐶2]3/2= 0 (8.6) From Eq. (8.6), it can be seen that 𝑑2𝑦
𝑑𝑥2= 0 at 𝑥 → ±∞, suggesting that 𝑑𝑦
𝑑𝑥 reaches its maximum and minimum values at 𝑥 → ±∞, respectively, which are:
𝑥→+∞lim 𝑑𝑦
𝑑𝑥 = − 1 tan 𝜃
((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥
√((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥2= √(tan 𝛼 tan 𝜃)
2
− 1 (8.7)
𝑥→−∞lim 𝑑𝑦 𝑑𝑥 = 1
tan 𝜃
((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥
√((tan 𝛼)2− (tan 𝜃)2) ∙ 𝑥2= −√(tan 𝛼 tan 𝜃)
2
− 1 (8.8)
From Eqs. (8.7-8.8), we finally obtain:
∆𝜑 = 𝜑𝑚𝑎𝑥− 𝜑𝑚𝑖𝑛= 𝑎𝑡𝑎𝑛 ( lim
𝑥→+∞
𝑑𝑥
𝑑𝑦) − 𝑎𝑡𝑎𝑛 ( lim
𝑥→−∞
𝑑𝑥
𝑑𝑦) (8.9)
and equivalently we can finally have:
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∆𝜑 = {
2 × atan [√ (tan 𝜃)2
(tan 𝛼)2− (tan 𝜃)2] , 𝑖𝑓 𝜃 < 𝛼 2𝜋 , 𝑖𝑓 𝜃 ≥ 𝛼
(8.10)
It should be noted when 𝜃 ≥ 𝛼, ∆𝜑 will abruptly become 2𝜋 which means the appearance of step edges along all directions because two step edges will eventually intersect with each other once their direction difference is larger than 𝜋. Take Cu(111) surface as an example, its 𝛼 is 0° and thus any surface roughness can lead to the appearance of step edges along all possible directions. Figure 8.2(d) shows the ∆𝜑 curves as a function of 𝜃 for four different Cu surfaces. It can be seen that ∆𝜑 increases significantly with 𝜃, suggesting that surface roughness has a significant effect on the step edge direction and further the alignment of 2D materials on the substrate. In addition, from Figure 8.2(e) we can also qualitatively reveal the effect of the density of step edges on the change of their directions under the same 𝜃. Obviously, the variation in the direction of step edges decreases with the increase of step edge density. Therefore, a high step edge density can effectively weaken the effect of surface roughness on the change of the directions of step edges.
Figure 8.3(a) shows a map of largest variation of the step edge direction as a function of both 𝜃 and 𝛼. From Eq. (8.10), we can know that for high-index substrates with small 𝜃 and 𝛼, the largest variation of the step edge direction can be estimated by:
∆𝜑 = 2 × atan [
√ (𝜃 𝛼⁄ )2 1 − (𝜃 𝛼⁄ )2
]
(8.11)
Figure 8.3(b) shows the ∆𝜑 curve as a function of 𝜃⁄𝛼 when both 𝜃 and 𝛼 are small. We can see that for ∆𝜑 < 5º, 10º, 20º and 30º, 𝜃⁄ <0.044, 0.088, 0.174 and 0.259 is required, respectively. 𝛼 On a typical annealed Cu foil, 𝜃 is ~1º,67,84 and thus the critical 𝛼 of the Cu surface that satisfies the above requirements can be calculated to be 22.72º, 11.36º, 5.75º and 3.86º, respectively. If a substrate is carefully polished and its 𝜃 is as low as 0.01º,366,367 unidirectionally aligned step edges on such a substrate can exist if 𝛼 is greater than 1º.
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Figure 8.3 (a) Variation of the step edge direction regarding to (𝜃, 𝛼). (b) Variation of n of the step edge direction regarding to 𝜃/𝛼 with small 𝜃 and 𝛼.
Above analysis clearly shows that high-index surfaces with large 𝛼 (or equivalently large step edge density) is superior to those with small 𝛼 in templating the synthesis of unidirectionally aligned 2D materials, because of the smaller change in the step edge direction. Take hBN grown on high-index Cu substrates as an example, Figure 8.4 shows the most stable alignment of hBN islands on Cu(5 5 6) and the Cu(10 10 17) surfaces, which is obtained based on the results shown in Chapter 7. Obviously, although these two surfaces have the same 𝜃 = 3.5º, there are two orientations of hBN islands grown on the Cu(5 5 6) surface, which has a low step edge density, and only one orientation for the hBN islands on the Cu(10 10 17) surface with a high step edge density.
Figure 8.4 Alignments of hBN islands on Cu(5 5 6) and Cu(10 10 17) surfaces under same surface roughness of 𝜃 = 3.5º.
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