Alignment of graphene islands on twinned Cu substrates

Chapter 6 Single-crystal graphene grown on twinned Cu substrates

6.3 Theoretical modeling and discussions

6.3.2 Alignment of graphene islands on twinned Cu substrates

To investigate the orientations of graphene islands on Cu surface, the grain texture of the Cu surfaces should be figure out first. As mentioned above, the orientation of any grain can be confirmed by three rotational operations of Euler angles. A rotational operation with an angle of 𝜃 around a rotational axis U= (x, y, z) where 𝑥²+ 𝑦²+ 𝑧²= 1 can be implemented by multiplying a rotate matrix G on the coordinates of all Cu atoms:

G=[

cos 𝜃 + 𝑥²(1 − cos 𝜃) 𝑥𝑦(1 − cos 𝜃) − 𝑧 sin 𝜃 𝑥𝑧(1 − cos 𝜃) + 𝑦 sin 𝜃 𝑥𝑦(1 − cos 𝜃) + 𝑧 sin 𝜃 cos 𝜃 + 𝑦²(1 − cos 𝜃) 𝑦𝑧(1 − cos 𝜃) − 𝑥 sin 𝜃 𝑥𝑧(1 − cos 𝜃) − 𝑦 sin 𝜃 𝑦𝑧(1 − cos 𝜃) + 𝑥 sin 𝜃 cos 𝜃 + 𝑧²(1 − cos 𝜃)

] (6.1)

After knowing the orientation of the Cu grain, the surface index {hkl} of this grain can be obtained by calculating the normal direction of the surface plane. Here, we chose three arbitrary points of the surface plane, namely A (X1, Y1, Z1), B (X2, Y2, Z2), and C (X3, Y3, Z3). The normal direction

<hkl> can be obtained by 𝐴𝐵⃑⃑⃑⃑⃑⃑ × 𝐵𝐶⃑⃑⃑⃑⃑⃑, then we get:

ℎ =(𝑌₂− 𝑌₁)(𝑍₃− 𝑍₂)

(𝑍₂− 𝑍₁)(𝑌₃− 𝑌₂) (6.2)

𝑘 =(𝑍₂− 𝑍1)(𝑋₃− 𝑋2)

(𝑋₂− 𝑋₁)(𝑍₃− 𝑍₂) (6.3)

𝑙 =(𝑋₂− 𝑋₁)(𝑌₃− 𝑌₂)

(𝑌₂− 𝑌₁)(𝑋₃− 𝑋₂) (6.4)

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Because Cu steps play a critical role in determining the orientation of graphene grown on the surface, as proved in Chapter 4. The step direction of an arbitrary Cu surface should also be explored.

Due to that the close-packed <110> steps or segments are energetically favored and appears more frequently, here in this Chapter to make it sample, only <110> dominated steps are considered.

FCC Cu crystal has totally six <110> close-packed direction, from the two examples of Cu surfaces in Figure 6.6 (a-b), we can see the lengths of the projections of the six <110> directions on a surface are dependent on the angle between their direction and the surface plane, and the <110>

direction which has the smallest angle with respect to the surface plane acts as step segment on the surface, as marked by blue arrows. The angles between six <110> directions with the two surfaces in Figure 6.6 (a-b) are listed in Table 6.1, when the angle is 0º, the <110> direction is exactly the step edge direction of surface, e.g., the step edges of Cu(611) is along Cu[01̅1]. Figure 6.6 (c) illustrates a projection of a <110> axis (denoted by 𝐵𝐴⃑⃑⃑⃑⃑⃑) on a Cu plane, which can be obtained by calculating the projected points of A (a1, b1, c1) and B (a2, b2, c2).

The plane equation of the Cu foil surface is:

hx + ky + lz + d = 0 (6.5)

where 𝑑 = −(ℎ ∙ 𝑋₄+ 𝑘 ∙ 𝑌₄+ 𝑙 ∙ 𝑍₄), and (X4, Y4, Z4) is a known point of the surface plan. The projected point (Pa, Pb, Pc) can be calculated as followed:

𝑃_𝑎= 𝑎 + 𝑡 ∙ ℎ (6.6)

𝑃_𝑏= 𝑏 + 𝑡 ∙ 𝑘 (6.7)

𝑃_𝑐 = 𝑐 + 𝑡 ∙ 𝑙 (6.8)

𝑡 = −ℎ ∙ 𝑎 + 𝑘 ∙ 𝑏 + 𝑙 ∙ 𝑐 + 𝑑

ℎ²+ 𝑘²+ 𝑙² (6.9)

Therefore, the projected direction of 𝐵𝐴⃑⃑⃑⃑⃑⃑ can be obtained by:

𝐵^′𝐴^′

⃑⃑⃑⃑⃑⃑⃑⃑⃑ = 𝐴^′(𝑃_𝑎1, 𝑃_𝑏1, 𝑃_𝑐1) − 𝐵^′(𝑃_𝑎2, 𝑃_𝑏2, 𝑃_𝑐2) (6.10) And the angle (α) between a <110> axis and the Cu foil surface can be obtained by:

cos 𝛼 = 𝐵𝐴⃑⃑⃑⃑⃑⃑ ∙ 𝐵⃑⃑⃑⃑⃑⃑⃑⃑⃑^′𝐴^′

|𝐵𝐴⃑⃑⃑⃑⃑⃑||𝐵⃑⃑⃑⃑⃑⃑⃑⃑⃑|^′𝐴^′ (6.11)

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Figure 6.6 (a-b) Illustrations of the relationship between 6 <110> directions and Cu steps on Cu(542) and Cu(611) surfaces. (c) Sketch of a <110> direction projected on the surface of Cu foil.

Table 6.1 The angles (°) between <110> direction and Cu surface.

[110] [1̅10] [101] [101̅] [011] [01̅1]

(5 4 2) 39.2 12.2 47.5 18.4 71.5 6.05

(6 1 1) 53.4 35.00 53.4 35.00 13.3 0.00

After the direction of <110> segment of the step is confirmed, graphene orientation on both sides of a twined Cu substrate can be roughly known by applying the rule that graphene ZZ segment aligning along <110> segment of the step, which means that the misalignment angle between graphene islands on two sides of a grain boundary can be obtained by calculating the angle between <110>

segments of the surfaces of grain A and B.

As introduced above in the modeling part, we can use two degrees of freedom to decide the orientation of a <111>/60^o Cu twin, 𝜃 and 𝜓. Because of the 6-fold symmetric of (111) plane of FCC, 𝜓 can be restricted into 60^o. Therefore, by changing 𝜓 from -30º to 30º and 𝜃 from 0º to 90º, all possible orientations of the <111>/60^o Cu twin can be reached. The map in Figure 6.7 shows the misorientation angles of graphene islands on the two sides of the <111>/60^o Cu twin along all orientations, and the misorientation angle values are denoted by colors. It can be seen that there is a large possibility (56.92% for misorientation angle < 0.01 and 56.66% for misorientation angle < 0.001) to get unidirectionally aligned graphene islands on <111>/60^o Cu twins.

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Figure 6.7 Map of the misangles of graphene islands on two sides of <111>/60^o twined Cu foils. θ is the angle between the normal direction of the (111) boundary plane with that of the Cu foil surface, 𝜓 is the rotate angle of the twin around their <111> coaxis.

Table 6.2 lists the experimental and theoretical results of 14 samples, where ∆_𝐸𝑥𝑝 is the graphene misorientation angles obtained by experimental measurements, and ∆_𝑀𝑎𝑝 is that obtained by theoretical calculations and marked in the map. The perfect agreement between experimental measurements and theoretical calculations strongly supports our theory.

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Table 6.2 Experimentally obtained misalignment angles (∆_𝐸𝑥𝑝) and theoretically predicted ones (∆_𝑀𝑎𝑝) of graphene islands on 14 twinned Cu foils.

No. Measured Euler Angles Assessment of <1 1 1>/60˚ twin boundary Misalignment Analysis 𝜑₁ (˚) 𝜙 (˚) 𝜑₂ (˚) θ (˚) 𝜓 (˚) 𝛥θ (˚) 𝛥𝜓 (˚) ∆_𝐸𝑥𝑝(˚) ∆_𝑀𝑎𝑝(˚) 1-A 221.00 12.20 44.70 66.94 -0.07

0.02 59.79 0±0.5 0.00

1-B 283.30 52.90 41.20 66.96 -59.86 2-A 227.90 12.80 37.70 57.32 15.14

1.00 59.66 27.2 ± 0.5 27.21 2-B 15.50 37.00 37.50 58.32 -44.52

3-A 136.10 39.10 19.20 47.77 -80.08

0.51 59.17 0±0.5 0.00

3-B 284.20 9.60 90.00 48.28 -20.91 4-A 233.10 46.70 46.80 67.83 -81.76

0.36 59.71 0±0.5 0.08

4-B 144.40 14.50 75.70 67.47 -22.05 5-A 139.10 41.20 14.50 80.71 5.11

0.73 60.37 11.0±0.5 11.00 5-B 222.00 35.30 88.10 81.44 -55.26

6-A 141.10 39.30 12.50 80.28 2.82

0.13 59.72 11.0±0.5 11.28 6-B 224.90 35.50 84.80 80.41 -56.90

7-A 262.1 28.4 58.8 27.75 -14.10

0.62 61.23 1±0.5 0.00

7-B 354.2 39.9 13.6 27.13 47.13 8-A 64.5 36.6 53.7 67.05 -39.79

0.68 61.00 25±0.5 26.42

8-B 312.8 22.6 15.6 67.73 21.21 9-A 123.1 41.4 46.2 63.62 47.57

0.01 60.19 27±0.5 27.47

9-B 236.3 14 81 63.63 -12.62 10-A 71.3 39.8 80.8 82.08 -31.61

0.35 60.24 10±0.5 11.03

10-B 349.3 37.6 6 81.73 28.63 11-A 126.9 31.4 27.4 84.99 50.90

0.21 60.19 3±0.5 2.16

11-B 281 41 59.2 85.21 -9.29 12-A 51.9 44.3 87.1 33.25 -58.66

0.61 60.13 0±0.5 0.00

12-B 142 20.9 42.7 33.86 1.47 13-A 236.4 44.9 10.7 42.8 -59.12

0.15 59.86 0±0.5 0.00

13-B 336.3 12.1 42.6 42.65 0.74 14-A 261.8 36.6 45.4 18.14 -0.77

0.1 59.02 0±0.5 0.00

14-B 335.5 47 23.9 18.04 58.25

Note: i-A and i-B (i=1, 2, …, 14) represent the Cu surface on both sides of the twin boundary, measured Euler angles are obtained from EBSD measurements.

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Besides, it is worth noting that the misorientation angles are exactly 0º along the dashed line of the map, indicating perfect alignment of graphene islands on the Cu foils with those twin grain boundaries. Figure 6.8(a) shows three atomic configurations with <111>/60^o Cu twins along the dashed line, we can see that step edges of the surfaces are all along the same Cu<110> direction. Figure 6.8(b) shows the SEM images and atomic models of graphene on Cu twin surfaces of sample #12-14 that are located on the dashed line. Similarly to those grown on the (116)/(111) Cu twin, graphene islands on the two sides of the twin boundaries are highly orientated with one pair of their ZZ edges parallel with the grain boundary, even though their shapes are either stranded hexagons or stretched hexagons along the step edge direction. The consistency between the four samples with misaligned graphene islands and our theoretical predictions shown in Figure 6.9 also validates our theory.

Figure 6.8 (a) Three atomic configurations of <111>/60^o twin Cu foils along the point line of the map.

(b) SEM images and atomic models of well-aligned graphene islands on surface of <111>/60^o Cu twin located in the point line of the map.

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Figure 6.9 SEM images and atomic models of misaligned graphene islands on surface of <111>/60^o Cu twin.