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For frictional experiments the lateral force sensitivity also needs to be calibrated. As with the normal spring constant, there are many ways to achieve this, such as pushing against a known sample [157] or calibrated force sensor [158], changing oscillation frequency with the addition of mass [159], and the torsional Sader method [159,160]. Here, the improved wedge calibration method is used, in which a hard calibration sample with sloped and flat sections is scanned [161]. This gives a calibration value for converting from the raw deflection (𝑉𝑙𝑎𝑡, in volts) to the lateral force between

the tip and sample (𝐹𝑙𝑎𝑡, in N), though other techniques (such as the torsional

Sader method) use a two step method like the normal force calibration, with an intermediary term corresponding to either the rotation of the tip (𝜙, in radians) or

the displacement of the tip apex (𝑑𝑙𝑎𝑡, in m). The relation between these terms and

the torque of the cantilever due to torsion,𝜏, is given by

𝐹𝑙𝑎𝑡 =𝛼𝑉𝑙𝑎𝑡 =

𝜏

ℎ+𝑇₂ =𝛽𝜙=

𝛽𝑑𝑙𝑎𝑡

ℎ+𝑇₂ , (2.6)

whereℎ is the tip height, 𝑇 is the cantilever thickness and 𝛼 and 𝛽 are calibration

factors, of which𝛼 is the one of interest here, measured in NV−1.

In the wedge method the tip is scanned up and down a sloped sample, re- sulting in a balance of forces illustrated previously in figure 1.11 of section 1.1.4.3. With the tip travelling up the slope the measured lateral signal will have one value,

𝑉𝑢, while going down the slope it will have a different value, 𝑉𝑑. These two values

can be combined to give two new values, the half-width,

𝑉𝑊 =

𝑉𝑢−𝑉𝑑

2 , (2.7)

and the offset,

𝑉Δ=

𝑉𝑢_+𝑉𝑑

2 . (2.8)

These values can also be calculated for the flat regions, where the up and down directions are replaced with trace and retrace directions. The width of the flat section,𝑉_𝑊𝑓 𝑙𝑎𝑡, is determined by the friction force (i.e.𝛼𝑉_𝑊𝑓 𝑙𝑎𝑡=𝐹𝜇), while the offset,

𝑉_Δ𝑓 𝑙𝑎𝑡, represents any errors with the alignment of the system, such as a non-zero

free lateral deflection or crosstalk from the deflection signal. As these errors affect the value on the slope equally, the slope offset is corrected to give𝑉_Δ′ =𝑉Δ−𝑉Δ𝑓 𝑙𝑎𝑡.

to terms of𝑉𝑊 and𝑉Δ′, giving

𝛼𝑉𝑊 =𝜇(𝐹 +𝐹𝑎𝑑cos𝜃)

cos2_𝜃−𝜇2_sin2_𝜃 and (2.9)

𝛼𝑉_Δ′ =𝜇

2_{sin𝜃(𝐹cos𝜃+𝐹𝑎𝑑) +𝐹sin𝜃cos𝜃}

cos2_𝜃−𝜇2_sin2_𝜃 , (2.10)

a full derivation of which can be seen in [161]. Equation 2.10 can then be divided by equation 2.9 to give

sin𝜃(𝐹cos𝜃+𝐹𝑎𝑑)𝜇2− 𝑉

′ Δ

𝑉𝑊

(𝐹+𝐹𝑎𝑑cos𝜃)𝜇+𝐹sin𝜃cos𝜃= 0, (2.11)

which is a quadratic equation for the unknown 𝜇, consisting of measured (𝑉𝑊 and 𝑉_Δ′) and known values — 𝐹 is the set point force, a user chosen value, 𝐹𝑎𝑑 is the

adhesion, which is measured prior to the scan, and𝜃is the slope angle, taken from the

calibration grid’s manufacturer specifications or measured using the height image. Solving equation 2.11 then gives two values of𝜇𝑖, which in turn can be substituted

into equation 2.9 or 2.10 to give two values of𝛼𝑖. At this point one of the𝛼𝑖 values

may be negative, in which case it can be discarded and the positive𝛼𝑖is taken as the

calibration coefficient,𝛼(this can be done before substitution — a value of𝜇𝑖 < _tan1_𝜃

will always yield a negative𝛼𝑖). If both the values of𝛼𝑖 are positive then one must

be selected over the other, which can be done by considering the friction on the flat regions. New friction coefficients corresponding to the flat regions are given by

𝜇𝑓 𝑙𝑎𝑡_𝑖 = 𝛼𝑖𝑉

𝑓 𝑙𝑎𝑡 𝑊

𝐹+𝐹𝑎𝑑

. (2.12)

While it is unlikely that 𝜇𝑖 =𝜇𝑓 𝑙𝑎𝑡_𝑖 , as the different crystallographic planes being

scanned will have different true physical values of 𝜇, the values should at least be

similar. Thus the value of 𝛼 that minimises |𝜇𝑖−𝜇𝑓 𝑙𝑎𝑡𝑖 |is chosen as the calibration

coefficient,𝛼.

Here, the wedge method was implemented by scanning a Mikromasch TGF11 trapezoidal calibration grid (with𝜃= 54.74∘) in contact mode with the fast scan axis

perpendicular to the length of the trapezoids. The scan size was selected such that both sides of the trapezoid were included within the scan range and both the lateral trace and retrace signals were recorded, with example images shown in figures 2.5(a– c). The two lateral images were then processed according to equations 2.7 and 2.8 and histograms of these results were produced, shown in figures 2.5(d) and (e), respectively. On the histogram of 𝑉𝑊 the large peak at low values corresponds to

a

b

c

1 µm

Figure 2.5: a–c) Contact mode AFM image of a TGF11 calibration grid taken with a load force of 324 nN. The channels shown and their corre- sponding full data scales are a) height, 2 𝜇m, b) lateral trace, 200 mV, and

c) lateral retrace, 200 mV and offset -150 mV relative to (b). d,e) Histograms of the lateral width and offset, respectively, with Gaussian fits using Igor Pro’s multi-peak fitting tool, showing the peak centre and standard devi- ation. With an adhesion of 22 nN (measured by force curve) these values give a lateral calibration of 27±7𝜇mV−1.

𝑉_𝑊𝑓 𝑙𝑎𝑡 and the smaller peak corresponds to𝑉𝑊 on the slopes. On the 𝑉Δ histogram

the large central peak corresponds to𝑉_Δ𝑓 𝑙𝑎𝑡while the smaller side peaks correspond

to 𝑉Δ on the slopes — there are two such peaks because the force behaviour is

mirrored for the second slope, effectively meaning that the direction (and thus the measured signal) is reversed relative to𝑉_Δ𝑓 𝑙𝑎𝑡. There is also a double main peak effect

on the offset histogram, which is due to differences between the plateau and valley regions of the calibration grid, possibly caused by the manufacturing process.

The histogram peaks were then fit using Gaussian curves, the central value for which is used as the appropriate value for fitting. The value of𝑉_Δ𝑓 𝑙𝑎𝑡is the average

of the two associated peaks and𝑉_Δ′ is a weighted average of the two measured peaks

given by 𝑉_Δ′ = 1 𝜎𝑎|𝑉_Δ𝑎−𝑉 𝑓 𝑙𝑎𝑡 Δ |+ 1 𝜎𝑏|𝑉_Δ𝑏 −𝑉 𝑓 𝑙𝑎𝑡 Δ | 1 𝜎𝑎 +_𝜎1𝑏 , (2.13)

where𝜎 is the standard deviation of each separate peak, denoted by superscript 𝑎

and𝑏. These values were then used according to the calculations previously described

to give a value for𝛼.

There are some considerations that must be made for an accurate calibration using the wedge method. Firstly, the orientation of the sample is important — if it is misaligned relative to the scan direction this will cause the effective value of

𝜃 to vary, though this can be accounted for by measuring𝜃 in the height image as

opposed to using the manufacturer specification. The sample can also be placed on a slant, which will affect both the value of 𝜃 and the measured values for the

flat regions — this can be hard to accurately correct in most AFM systems, though can be monitored through the height image. Scanning over the slope causes the deflection to change, which is then corrected through the feedback loop adjusting the height position. If this feedback is not correctly tuned there can be a significant lag in the movement resulting in a different deflection, and thus load force, on the slopes. This can be corrected through careful tuning of the feedback parameters while monitoring the recorded deflection values.