Chapter 4. Results and Discussion
4.5 ω-F ESC-tip from the Single GaN Island
In this section, we would discuss the tip-sample distance dependence of the ω-force acting on the tip from the surface charge on the fixed GaN island. The sample is AlGaN epilayer with GaN islands and its morphology is shown in Fig.
4.4.3. When the tip is above the fixed island during the scanning, we believed that the ω-force acting on the tip, Fω, consists of two parts. One is the ω-force between the tip and the island, Ftip-is,ω, and the other is the ω-force between the tip and the entire plane area except of island itself, Ftip-p,ω. And they both consist of capacitive and Coulombic components, respectively. Thus, the ω-force acting on the tip can be expressed as the sum of the two kinds of ω-forces: (see Fig.
4.5.1)
tip p, tip is,
tip p
dc CPD,p ac p,ESC tip,
tip is
dc CPD,is ac is,ESC tip,
F (Z) F (Z) F (Z)
where Ftip-p,ω is the ω-force between the tip and the plane, Ftip-is,ω is the ω-force between tip and the specific island, Fp,ESC-tip,ω is the ω-term Coulombic force between the tip and the surface ESC on the plane, Fis,ESC-tip,ω is the ω-term Coulombic force between tip and the surface ESC on the specific island, and VCPD,p , VCPD,is , Ctip-p , Ctip-is denote the VCPD on the plane around the specific island, the VCPD on the specific island, the capacitance between the tip and the
plane around the specific island, and the capacitance between the tip and the specific island, respectively. In the experiment, when the applied dc voltage Vdc= VCPD,p, the capacitive term of the island can be neglected. Thus, we can have an equation with a much simple form, and obtain the Coulombic force between tip as well as the surface ESC on the specific island, Fis,ESC-tip,ω, by the following equation.
tip p
dc CPD,p ac p,ESC tip,
tip is
dc CPD,is ac is,ESC tip,
tip p
dc CPD,p ac p,ESC tip, is,ESC tip,
C (Z) We conducted the experiments on the GaN islands, for example, the specific GaN island circled in Fig. 4.4.3. Its height and the diameter are about 9nm and 250nm, respectively. The value of Fω used here represents the averaged ω-force value over the 70nm
×
70nm area of the island on the ω-force mapping (Fig. 4.4.3 (b)). The value of Ftip-p,ω represents the averaged ω-force value over the 0.2µm×
0.2µm region measured at a distance of ~1µm away from islands in Fig. 4.4.3 (b), assuming that the value of Ftip-p,ω does not change no matter at region with or without GaN islands. Since the obtained value of Ftip-p,ω is measured at a featureless plane area away from islands, where exhibits uniformcharge distribution on surface, as shown in Fig. 4.5.2 (c), which is different from the area region around the island, where we assume no ESC in the interface between AlGaN epilayer and the GaN island, as shown in Fig. 4.5.2 (a). Using charge superposition principle, the charge distribution on the island and surrounding area (Fig. 4.5.2 (a)) can be resembled by the equivalent charge distribution described as follows. In equivalent charge distribution, we assume there exhibits charge distributed on the AlGaN under the GaN/AlGaN interface with exactly the same charge density as that of plane area. In order to balance the charge above, we also assume the same interface charge density, however, with opposite polarity, also exists on the GaN island side (signed as solid circles), as shown in Fig. 4.5.2 (d). Consequently, by this arrangement we can subtract the force Ftip-p,ω resulted from the uniform charge distribution signed as the open circles in Fig. 4.5.2 (c) from Fω by using Eq. (4-7) to obtain the value of the ω-term Coulombic force Fis,ESC-tip,ω due to the interaction of charges between tip and the specific island, as shown in Fig. 4.5.2 (b). From Fig. 4.5.2 (d), we learn that Fis,ESC-tip,ω is not due to the surface charge (signed as “x“) of the island itself, instead, it represents ω-term Coulombic force due to the net surface charge effect owing to the surface charge of island (signed as “x“) and imaginary interface charge (signed as solid circles) on the island side because the thickness of island (~10nm) is relatively thin as compared with the distance between tip and sample (100~500nm).
Figure 4.5.3 shows the ω-force and 2ω-force intensities vs. tip-sample
distance, which the forces consist of the interactions between tip and both the specific island as well as the plane area around the specific island. Fig. 4.5.4 shows the ω-force and 2ω-force intensities vs. tip-sample distance, which the forces are both between the tip and the plane around the specific island. By referring Eq. (4-7), we can deduce the ω-term Coulombic force intensity vs.
tip-sample distance, which the force is between the tip and the surface ESC on the specific GaN island, as shown in Fig. 4.5.5.
Figure 4.5.6 shows the fitting result of the data in Fig. 4.5.4 (a) by cone model, which represents that the tip-sample distance dependence of the ω-force can be interpreted by cone model. The fitting parameters L, R, and θ are 14µm, 11nm, and 12°, respectively. We would use those parameters to fit the following data.
In our fittings, we viewed the single GaN island as a thin disk because of its disk-like shape and the height of the island being much smaller than the tip to sample distance. And in the range of the tip to sample distance in our experiments, the charge distribution of the tip was viewed as uniformly charged line, point charge, and cone, respectively, which were proposed in the published literature. [40-41] Therefore, we used uniformly charged line-disk model, charged point-disk model, and cone model to fit the data in Fig. 4.5.5, respectively. The calculation of the models is discussed as follows.
(I) Uniformly charged line-disk model
Figure 4.5.7 shows the schematic of uniformly charged line-disk model.
The force between uniformly charged line and the uniform ESC density on the GaN island whose radius is 125nm as mentioned above. We use a well-known example in the textbook about electromagnetics to calculate the force between the uniformly charged line and the disk, as shown in Fig. 4.5.7. The formula [42]
for the electric field of the well-known example is
2 2 0.5 charge density of the disk in Fig. 4.5.8. In Fig. 4.5.7, when integrating the Eq.
(4-8) with respect to the charged line length element dZ’, we can obtain the total electrostatic force F
According to the report of S. Belaidi et al. [40], they analyzed the electrostatic forces acting on the tip in AFM and the result is in Table 1 [40]. It is noted that the results in Table 1 are in the assumption of infinite plane. And the λ(θ) is obtained as follows when an infinite cone with a half angle θ [40]:
0 0 (II) Charged point-disk model
The schematic diagram is shown in Fig. 4.5.9. If we take point discharge into account and ignore the interaction between side wall of the tip and the single disk-like GaN island, the interaction between the sphere on the pointed
end of the tip and the island is the main contribution of the force. [43-44] We took the sphere as a point-charge located in the center because of the smaller size (~20nm in diameter) relative to the diameter of the disk-like island (~250nm) and convenience. By Eq. (4-8), this model is easily calculated as
2 2 0.5
where Q is the point-charge on the pointed end of the tip.
(III) Cone model
Eq.(4-6) in section 4.3 has represented the result.
The fitted results of those models for Fig. 4.5.5 are shown in Fig. 4.5.10.
We found that the charged point-disk model is the better model to interpret the interaction between the tip and the surface ESC on the specific GaN island, which is different from the model of the interaction between the tip and the infinite Si wafer resulted from the section 4.3.
Above the infinitely charged plane, the electric field is uniform. Thus, the electric field from the sample surface acts on all part of the tip above the plane.
In the experiment on Si wafer, the dimension of Si wafer is so large that the Si wafer is infinite compared with the tip. In this condition, we must take all part of the tip into account when computing the ω-term Coulombic force acting on the tip. Therefore, we must use the cone mode to interpret the ω-term Coulombic force between the tip and the surface ESC on the Si wafer. In addition, in the
single disk-like GaN island whose dimension is finite compared with the tip, the electric field is not uniform and decreases rapidly with increasing height (see Appendix.). So, the more the position approaches to the sample and the more the quantity of charge is, the more the electrical force is. On the probe, the sphere on the pointed end of the tip is the most close to the island and more induced charge assembles there (phenomenon of point discharge). The ω-term Coulombic force acting on the sphere of the tip from the surface ESC on the single disk-like GaN island is the main origin. As a result, we used the charged point-disk model to interpret the force Fis,ESC-tip,ω(Z).
Table 1. Analytic expressions for the electrostatic force. R is the radius of the tip. Z is the tip to sample distance. L is the length of the tip. θ is half angle of the cone. s(θ)=aθ+b, a=0.13 and b=0.72 can be used in the 5<θ<50. [40]
Very small tip-
Model Sphere Uniformly
charged line
Fp
- - - -Fig. 4.5.1. Schematic diagram of the interactions between the tip and the sample surface with GaN islands. The “+” and “-“ represent the induced charge by biasing, and the solid circles are surface ESC.
It is noted that the twin-arrows represent the force. Fis is the force between the tip and the specific island. The Fp is the force between the tip and the plane around the specific island. The size of the arrows does not express the intensity of the force.
Plane (AlGaN)
Fig. 4.5.2. Schematic diagram of the structure (a) which represents the realistic distribution of surface ESC on the AlGaN epilayer with GaN island being combined by (b) and (c). (b) and (c) represent the ESC distribution on the GaN island and AlGaN epilayer, respectively. The open circles and “x“ are expressed as the surface ESC on plane and island, respectively. And the solid circles are expressed as the charge on the lowerside of the GaN island, which is opposite pole to the charge expressed by the open circles. (d) is equivalent to (a) and the circles in the dotted ellipse in (d) express the counter-balanced charge.
(a)
tip to sam ple distance (nm )
2ω-force
tip to sam ple distance (nm ) ω-force
(b)
Fig. 4.5.3. The intensities of (a) ω-force and (b) 2ω-force acting on the tip vs. tip-sample distance which the forces consist of Fp and Fis indicated in Fig. 4.5.1
.
(a)
Fig. 4.5.4. The intensities of (a) ω-force and (b) 2ω-force acting on the tip vs. tip-sample distance which the forces are between the tip and the plane around the island.
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0
0 100 200 300 400 500 600 0.00
0.02 0.04 0.06 0.08 0.10
electric force intensity
tip to sample distance (nm)
Fig. 4.5.5. The calculated ω-term Coulombic force intensity between the tip and the surface ESC on the specific island vs. tip-sample distance.
0 100 200 300 400 500 600 2.5
3.0 3.5 4.0 4.5 5.0
0 100 200 300 400 500 600
2.5 3.0 3.5 4.0 4.5 5.0
force intensity
tip to sample distance (nm)
F
ω,plane-tip
cone model
L=14
µm R=11nm
θ=12
0Fig. 4.5.6. The ω-force intensity acting on the tip vs. tip-sample distance on the plane region of the sample and their fitting result by the cone model.
Z’
- - - -Fig. 4.5.7. Schematic of uniformly charged line-disk model. The “+” and
“-“ represent the induced charge by biasing, and the solid circles are surface ESC whose uniform charge density is ρs in our assumption.
Fig. 4.5.8. Schematic diagram of Eq. (4-8). [42]
V
dc+V
acsinωt
Fig. 4.5.9. Schematic of charged point-disk model.
0 100 200 300 400 500 600
tip to sample distance (nm)
Fω between ESC on island and tip charged point-disk model
uniformly charged line-disk model Cone model
R=11nm L=14µm θ=120
Fig. 4.5.10. The ω-term Fis,ESC-tip(Z) experimental data and the three model fitted results.