Nanotechnology/AFM/Overview of properties of various cantilevers

=Examples of AFM cantilevers=

For a BS-75kHz
L=225 w=28 t=3 h=17+3/2 (is actually trapezoidal)}

$$k_{N}=\frac{1}{4}Yw\left( \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left(160000\right)  28\left(  \frac{3}{225}\right)  ^{3}=2.\,654\,8$$

$$f[Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}} =\frac{\left( 3\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left( 225\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  } {3\ast2330}}=79318.Hz$$

$$2\left( \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{28\ast18.5}{3\ast 225}\right)  ^{2}=1.\,177\,8$$ so $$k_{lat}~k_{tor}.$$

$$k_{tor}=k_{N}\frac{1}{2}\left( \frac{L}{h}\right)  ^{2}=2.65\frac{1} {2}\left( \frac{225}{18.5}\right)  ^{2}=195.\,99$$

$$k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left( 160000\right) 3\left( \frac{28}{225}\right)  ^{3}=231.\,26$$

For a MPP311
Specs: 13 kHz, 0.45 N/m : L=440 w=30 t=4 h=17.5+2 (is it actually trapezoidal??)

$$k_{N}=\frac{1}{4}Yw\left( \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left( 160000\right)  30\left(  \frac{4}{440}\right)  ^{3}=0.901\,58$$

$$f$$ $$[Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}} =\frac{\left( 4\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left( 440\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  } {3\ast2330}}=27655.Hz$$

$$2\left( \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{30\ast19.5}{4\ast 440}\right)  ^{2}=0.220\,96$$ so $$k_{lat}<k_{tor}.$$

$$k_{tor}=k_{N}\frac{1}{2}\left( \frac{L}{h}\right)  ^{2}=0.9\frac{1} {2}\left( \frac{440}{19.5}\right)  ^{2}=229.\,11$$

$$k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left( 160000\right) 4\left( \frac{30}{440}\right)  ^{3}=50.\,714$$

For a MPP211
Specs: 50 kHz, 1.5 N/m : L=215 w=30 t=4 h=17.5+2 (is it actually trapezoidal??)

$$k_{N}=\frac{1}{4}Yw\left( \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left( 160000\right)  30\left(  \frac{4}{215}\right)  ^{3}=7.\, 727\,6$$

$$f$$ $$[Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}} =\frac{\left( 4\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left( 215\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  } {3\ast2330}}=1.\,158\,2\times10^{5}Hz$$

$$2\left( \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{30\ast19.5}{4\ast 215}\right)  ^{2}=0.925\,43$$

so $$k_{lat}~k_{tor}.$$

$$k_{tor}=k_{N}\frac{1}{2}\left( \frac{L}{h}\right)  ^{2}=7.7\frac{1} {2}\left( \frac{215}{19.5}\right)  ^{2}=468.\,02$$

$$k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left( 160000\right) 4\left( \frac{30}{215}\right)  ^{3}=434.\,68$$

For a MPP111
Specs: 200 kHz, 20 N/m : L=115 w=30 t=4 h=17.5+2 (is it actually trapezoidal??)

$$k_{N}=\frac{1}{4}Yw\left( \frac{t}{L}\right)  ^{3}=\frac{1}{4}\left( 160000\right)  30\left(  \frac{4}{115}\right)  ^{3}= 50.\,497$$

$$f$$ $$[Hz]=\frac{t\beta_{i}^{2}}{4\pi L^{2}}\sqrt{\frac{Y}{3\rho}} =\frac{\left( 4\ast10^{-6}\right)  \left(  1.875\right)  ^{2}}{4\pi\left( 115\ast10^{-6}\right)  ^{2}}\sqrt{\frac{\left(  160\ast10^{9}\right)  } {3\ast2330}}=4.\,048\,4\times10^{5}Hz$$

$$2\left( \frac{wh}{tL}\right)  ^{2}=2\left(  \frac{30\ast19.5}{4\ast 115}\right)  ^{2}=3.\,234\,6$$ so $$k_{lat}>k_{tor}.$$

$$k_{tor}=k_{N}\frac{1}{2}\left( \frac{L}{h}\right)  ^{2}=50.4\frac{1} {2}\left( \frac{115}{19.5}\right)  ^{2}=876.\,45$$

$$k_{lat}=\frac{1}{4}Y\frac{tw^{3}}{L^{3}}=\frac{1}{4}\left( 160000\right) 4\left( \frac{30}{115}\right)  ^{3}=2840.\,5$$