1. Theory

Definition: An atomic force microscope scans a sharp tip on a flexible cantilever over a surface and records the cantilever’s deflection or oscillation to map topography with sub-nanometre vertical and nanometre lateral resolution (DTU).
Forces: Van der Waals, contact, adhesion and electrostatic forces act between tip and sample; the cantilever converts them into measurable deflections (Giessibl).
Modes: Contact mode holds a constant deflection, tapping (amplitude modulation) a constant reduced amplitude, and non-contact (frequency modulation) a constant frequency shift (DTU; Giessibl).
Calibration: The spring constant and optical-lever sensitivity are calibrated from thermal noise, dimensions or reference cantilevers (Sumbul et al.).

2. Working Principle

Optical lever: A laser reflected from the cantilever onto a split photodiode detects its deflection, the beam-bounce method (Giessibl).
Feedback: A piezo scanner adjusts height to keep the chosen signal constant; the height signal forms the image (Giessibl; DTU).
Probes: Silicon or silicon nitride tips with radii of 2 to over 20 $nm$ and a range of stiffness and frequency (DTU).
Environment: Air, liquid or vacuum, with lower quality factors in liquid (Sumbul et al.).

3. Applications

Roughness: Wafer, film and post-CMP roughness (DTU).
Step heights and trenches: Etch depths and line profiles (DTU).
Mechanical mapping: Modulus, adhesion and deformation in PeakForce modes (DTU).
Platform for electrical modes: Conductive AFM, KPFM, SCM and SSRM in this folder (those pages).

Cantilever and Calibration

Name	Formula	Related Formulae	Variables/Constants
Cantilever spring constant and resonance	$k = \frac{E w t ^{3}}{4 L ^{3}}$	$f_{0} = \frac{1}{2 π} \frac{k}{m ^{*}} \approx 0.162 \frac{t}{L ^{2}} \frac{E}{ρ}$	$k : s p r in g co n s t an t$ $E : Y o u n g^{'} s m o d u l u s$ $w, t, L : w i d t h, t hi c k n ess an d l e n g t h o f a r ec t an g u l a r b e am$ $f_{0} : f u n d am e n t a l r eso nan ce, m^{*} : e f f ec t i v e ma ss$ $ρ : d e n s i t y; G i ess ib l, E q s . 14 an d 15 : t h ese b e am r e l a t i o n s; D T U : T a p 150 A l - G, T a p 300 A l - G, S c an A sy s t, s u p er - s ha r p an d hi g h - a s p ec t - r a t i o p r o b es; S u mb u l e t a l . : A C 10 (8.15 b y 1.94 μ m, 1425 kHz, Q 48, 0.164 N/m) an d A C 40 (38.2 b y 17.0 μ m, 108 kHz, Q 40, 0.089 N/m) in ai r$ $(a ss u m e d : s i l i co n E = 169 GPa, ρ = 2330 kg/ m^{3}; co m p u t e d : 125 b y 40 b y 4 μ m g i v es 55 N/m an d 353 kHz, a t a pp in g p r o b e; 450 b y 50 b y 2 μ m g i v es 0.19 N/m an d 13.6 kHz, a co n t a c t p r o b e)$
Thermal-noise calibration	$\frac{1}{2} k_{B} T = \frac{1}{2} k ⟨ d^{2} ⟩$	$k = β \frac{k _{B} T}{χ ^{2} ⟨ d ^{2} ⟩}, ⟨ d^{2} ⟩_{S H O} = \frac{π B ^{2} f _{R}}{2 Q}$	$⟨ d^{2} ⟩ : m e an s q u a r e d t h er ma l d e f l ec t i o n$ $β : m o d e cor r ec t i o n (0.971 f or r ec t an g u l a r b e am s)$ $χ : f r ee - t o - co n t a c t I n v O L S cor r ec t i o n (1.09)$ $B : t h er ma l am pl i t u d e f r o m t h e P S D f i t$ $f_{R} : r eso nan ce$ $Q : q u a l i t y f a c t or; S u mb u l e t a l . : e q u i p a r t i t i o n, β an d χ, S H O an d P i r z er - H ug e l f i t s d i f f er in g b y u p t o ab o u t 40 p er ce n t a t Q n e a r 1, f i t w in d o w f_{R} /2 t o 2 f_{R}, l o w - p a ss a t 3 MHz$ $(co m p u t e d : k_{B} T / k = 0.216 nm f or 0.089 N/m an d 0.010 nm f or 40 N/m a t 300 K)$
Sader and global calibration	$k = 0.1906 ρ_{f} b^{2} L Q Γ_{i} (ω_{R}) ω_{R}^{2}$	$k = A Q f_{R}^{1.3}$	$ρ_{f} : f l u i d d e n s i t y$ $b : w i d t h$ $Γ_{i} : ima g ina r y h y d r o d y nami c f u n c t i o n$ $ω_{R} : an g u l a r r eso nan ce$ $A : p r o b e - t y p e coe f f i c i e n t; S u mb u l e t a l . : S a d er f or m (p a r tl y g a r b l e d) an d t h e g l o ba l c a l ib r a t i o n ini t ia t i v e, A_{ai r} = 0.0344 (A C 10) an d 0.634 (A C 40) n N s^{1.3} / m, k w i t hin ab o u t 5 p er ce n t o f ai r v a l u es an d I n v O L S w i t hin ab o u t 1 p er ce n t, s p r e a d o f 20 t o 40 p er ce n t in l i q u i d; S a d er m e t h o d n ee d in g o n l y pl an - v i e w d im e n s i o n s (C a l t ec h p a g e, n o t c i t e d)$ $(co m p u t e d : A Q f_{R}^{1.3} r e t u r n s 0.089 N/m f or A C 40 an d 0.165 N/m f or A C 10, ma t c hin g t h e t ab u l a t e d k)$

Modes and Forces

Name	Formula	Related Formulae	Variables/Constants
Contact mode	$F = k d$	$d = \frac{Δ V _{P S D}}{S} (InvOLS 1/ S)$	$F : t i p f or ce$ $d : c an t i l e v er d e f l ec t i o n$ $Δ V_{P S D} : p h o t o d e t ec t or s i g na l$ $S : o pt i c a l - l e v er se n s i t i v i t y; G i ess ib l : s t a t i c d e f l ec t i o n F / k an d b e am - b o u n ce d e t ec t i o n; D T U : co n t a c t, t a pp in g an d P e ak F or ce t a pp in g m o d es, t o p o g r a p h y, r o ug hn ess an d s t e p h e i g h t s$ $(co m p u t e d : 0.19 N/m d e f l ec t e d 5 nm p r esses w i t h ab o u t 1 nN)$
Amplitude and frequency modulation	$Δ f \approx - \frac{f _{0}}{2 k} \frac{\partial F}{\partial z}$	$A_{s p} < A_{0}, τ_{A M} \approx \frac{2 Q}{f _{0}}$	$Δ f : r eso nan ce s hi f t f r o m t h e t i p - s am pl e f or ce g r a d i e n t$ $A_{s p}, A_{0} : se tp o in t an d f r ee am pl i t u d es$ $τ_{A M} : am pl i t u d e r es p o n se t im e; G i ess ib l : Δ f f or m an d τ_{A M} \approx 2 Q / f_{0}; S u mb u l e t a l . : Q f a l l in g f r o m ab o u t 40 t o 48 in ai r t o 1.3 t o 1.9 in l i q u i d$ $(co m p u t e d : f_{0} = 300 kHz, k = 40 N/m an d a 0.1 N/m g r a d i e n t g i v e Δ f = - 375 Hz; Q = 400 a t 300 kHz g i v es τ_{A M} = 2.7 m/s)$
Tip-sample forces	$F_{v d W} = - \frac{H R}{6 D ^{2}}$	$V_{v d W} = - \frac{H R}{6 D}$	$F_{v d W} : v an d er W aa l s f or ce b e tw ee n a s p h er e o f r a d i u s R an d a f l a t a t d i s t an ce D$ $H : H amak er co n s t an t$ $V_{v d W} : v an d er W aa l s e n er g y; G i ess ib l : t h e e n er g y f or m, d i f f er e n t ia t e d h er e f or t h e f or ce (co m p u t e d); m e ni sc u s f or ces f r o m w a t er l a y er s in ambi e n t co n d i t i o n s$ $(a ss u m e d : H = 1 \times 1 0^{- 19} J, R = 10 nm, D = 0.5 nm; co m p u t e d : 0.67 nN)$

Image Quality

Name	Formula	Related Formulae	Variables/Constants
Tip convolution	$w_{a pp} \approx w + 2 2 R h$	$image = tip \oplus sample (dilation)$	$w_{a pp} : a pp a r e n t w i d t h o f a f e a t u r e o f t r u e w i d t h w an d h e i g h t h ima g e d w i t h a t i p o f r a d i u s R; g eo m e t r i c r e l a t i o n (co m p u t e d); D T U : ima g e a s co n v o l u t i o n o f t i p an d s am pl e, t i p r a d i u s 2 t o o v er 20 nm (s t an d a r d b e l o w 12 nm), ha l f - co n e 3 t o o v er 25 d e g r ees, w or n t i p s b l u r r in g f e a t u r es an d l o w er in g r o ug hn ess, t i p c h ec k w i t h a w or n p r o b e (R_{q} 2.49, R_{a} 1.97, R_{ma x} 18.8 nm) a g ain s t a n e w p r o b e (10.5, 8.45, 78.3 nm), t r e n c h d e pt h - t o - w i d t h r a t i o ab o u t 1, ab o u t 10 w i t h t h e r i g h t p r o b e$ $(co m p u t e d : a 10 nm t a l l f e a t u r e ima g e d w i t h a 10 nm t i p b r o a d e n s b y 28 nm; t h e w or n t i p u n d er - r e p or t s R_{q} b y a f a c t or o f 4.2)$
Roughness parameters	$R_{a} = \frac{1}{N} \sum ∣ z_{i} - \overset{z}{ˉ} ∣, R_{q} = \frac{1}{N} \sum (z_{i} - \overset{z}{ˉ})^{2}$	$R_{q} \approx 1.25 R_{a} (Gaussian surface)$	$R_{a}, R_{q} : a r i t hm e t i c an d r oo t - m e an - s q u a r e r o ug hn ess$ $R_{ma x} : p e ak - t o - v a l l ey; d e f ini t i o n s an d G a u ss ian r a t i o π /2 (co m p u t e d); D T U : r o ug hn ess an d s t e p h e i g h t s, Gw y dd i o n, N an o S co p e A na l y s i s an d S P I P so f tw a r e$ $(co m p u t e d : D T U^{'} s n e w - t i p v a l u es g i v e R_{q} / R_{a} = 1.24 an d w or n - t i p v a l u es 1.26, c l ose t o t h e G a u ss ian 1.25)$
Noise and height accuracy	$z_{n o i se} < 30 pm RMS, z_{se n sor} = 35 pm RMS$	$u_{c} = u_{c a l}^{2} + u_{QC}^{2}$	$z_{n o i se} : v er t i c a l n o i se f l oor$ $u_{c} : co mbin e d h e i g h t u n cer t ain t y$ $u_{c a l} : c a l ib r a t i o n - s t an d a r d u n cer t ain t y$ $u_{QC} : a cce pt e d QC o f f se t; D T U : n o i se b e l o w 30 pm, c l ose d - l oo p X Y n o i se b e l o w 0.15 nm, Z se n sor 35 pm, n o n l in e a r i t y b e l o w 0.5 p er ce n t, r an g es 90 μ m X Y an d 13 μ m Z, h e i g h t a cc u r a cy b e tt er t han 2 p er ce n t a t 200 nm (t y p i c a l l y 0.75), c a l ib r a t i o n w i t h 1.5 nm u n cer t ain t y an d 4 nm QC o f f se t g i v in g 4.3 nm (2.1 p er ce n t), l a t er a l r eso l u t i o n ab o u t 1 nm an d v er t i c a l b e l o w 1 an g s t r o m, s t a g e 180 b y 150 mm$ $(co m p u t e d : 1. 5^{2} + 4^{2} = 4.27 nm, ma t c hin g D T U^{'} s 4.3 nm)$

Important Questions

Ques: How does an AFM measure topography?

By keeping a tip-sample interaction constant with feedback while scanning, and recording the height correction. A laser reflected off the cantilever onto a split photodiode detects its deflection or oscillation, and a piezo moves the sample or tip to hold that signal at a setpoint, so the height map reproduces the surface (Giessibl: beam-bounce detection with deflection, amplitude or frequency as the feedback signal). DTU’s Bruker Dimension Icon systems give about 1 $nm$ lateral and below 1 angstrom vertical resolution, a noise floor below 30 $pm$ RMS, and ranges of 90 $μ m$ laterally and 13 $μ m$ vertically. A thermally excited soft cantilever of 0.089 $N/m$ fluctuates by about 0.22 $nm$ at room temperature (computed), so stiff cantilevers are needed for the quietest imaging. Reference: DTU Nanolab LabAdviser, ‘AFM: Atomic Force Microscopy’ (Bruker Dimension Icon-Pt) (fetch-tool summary); F. Sumbul, N. Hassanpour, J. Rodriguez-Ramos and F. Rico, ‘One-Step Calibration of AFM in Liquid’, Front. Phys. 8, 301 (2020) (fetch-tool summary).

Ques: What distinguishes contact, tapping and non-contact modes?

Which cantilever signal the feedback holds. Contact mode keeps a constant deflection, so force is $k d$ : a 0.19 $N/m$ cantilever deflected 5 $nm$ presses with about 1 $nN$ (computed), and lateral forces can damage soft samples. Tapping mode drives a stiffer cantilever near resonance and keeps the amplitude below its free value, touching only briefly each cycle (Giessibl: amplitude and phase as feedback). Non-contact or frequency-modulation mode tracks the resonance shift $Δ f \approx - (f_{0} /2 k) \partial F / \partial z$ ; a 300 $kHz$ , 40 $N/m$ cantilever shifts 375 $Hz$ for a 0.1 $N/m$ gradient (computed). High Q slows amplitude response, $τ_{A M} \approx 2 Q / f_{0}$ (Giessibl), about 2.7 $m/s$ at Q = 400 (computed), which frequency modulation avoids by responding within about one cycle. DTU also offers PeakForce tapping. Reference: DTU Nanolab LabAdviser, ‘AFM: Atomic Force Microscopy’ (Bruker Dimension Icon-Pt) (fetch-tool summary); F. J. Giessibl, ‘Advances in atomic force microscopy’, arXiv:cond-mat/0305119 (Rev. Mod. Phys. 75, 949, 2003): k = Y w t^3/(4L^3) (Eq. 14), f0 = 0.162 (t/L^2) sqrt(Y/rho) (Eq. 15), Delta f = -(f0/2k) dF/dz, V_vdW = -A_H R/(6z) for a sphere, tau_AM about 2Q/f0, beam-bounce detection, static deflection F/k, AM and FM feedback, jump-to-contact and meniscus forces in ambient conditions (fetch-tool summary).

Ques: How is the spring constant calibrated?

Most often from thermal noise. By equipartition, $\frac{1}{2} k_{B} T = \frac{1}{2} k ⟨ d^{2} ⟩$ , so measuring the thermal fluctuation gives k, with corrections of 0.971 for the mode shape and 1.09 for the free-to-contact sensitivity of rectangular beams (Sumbul et al.). The Sader method uses only plan-view dimensions, resonance and Q with the fluid’s hydrodynamic function. Sumbul et al. also apply the global calibration $k = A Q f_{R}^{1.3}$ , with A fitted for each probe type; using their coefficients, it returns 0.089 $N/m$ for AC40 and 0.165 $N/m$ for AC10, matching their tabulated values (computed). They found k within about 5 per cent of air values and sensitivity within about 1 per cent, though the spread in liquid reached 20 to 40 per cent. Reference: F. Sumbul, N. Hassanpour, J. Rodriguez-Ramos and F. Rico, ‘One-Step Calibration of AFM in Liquid’, Front. Phys. 8, 301 (2020) (fetch-tool summary).

Ques: Are the AC10 and AC40 resonances in liquid about 40 and 500 kHz, or 571 and 29 kHz?

Sumbul et al.’s methods section gives about 40 $kHz$ for AC10 and about 500 $kHz$ for AC40 in liquid, while their measured table gives 571 $kHz$ for AC10 and 29 $kHz$ for AC40. The table is the more likely correct: the shorter AC10 cantilever has the higher air resonance, 1425 $kHz$ against 108 $kHz$ , and liquid lowers both by similar factors, 2.5 and 3.7 (computed), so the methods text appears to have the two probe names swapped (my reading). The beam relation confirms the ordering: resonance scales as $t / L^{2}$ , and AC10 is about 4.7 times shorter (computed). Their quality factors fall from 40 to 48 in air to about 1.3 to 1.9 in liquid, which is why calibration in liquid is harder. Reference: F. Sumbul, N. Hassanpour, J. Rodriguez-Ramos and F. Rico, ‘One-Step Calibration of AFM in Liquid’, Front. Phys. 8, 301 (2020) (fetch-tool summary); F. J. Giessibl, ‘Advances in atomic force microscopy’, arXiv:cond-mat/0305119 (Rev. Mod. Phys. 75, 949, 2003): k = Y w t^3/(4L^3) (Eq. 14), f0 = 0.162 (t/L^2) sqrt(Y/rho) (Eq. 15), Delta f = -(f0/2k) dF/dz, V_vdW = -A_H R/(6z) for a sphere, tau_AM about 2Q/f0, beam-bounce detection, static deflection F/k, AM and FM feedback, jump-to-contact and meniscus forces in ambient conditions (fetch-tool summary).

Ques: Why do AFM images depend on the tip, and how much?

Because the image is a dilation of the sample by the tip shape. DTU notes that tip radii range from 2 to over 20 $nm$ and half-cone angles from under 3 to over 25 degrees, so features narrower or steeper than the tip are widened: a 10 $nm$ high feature imaged with a 10 $nm$ tip broadens by about $2 2 R h = 28$ $nm$ (computed). Broad tips cannot reach trench bottoms, which limits standard probes to depth-to-width ratios of about 1, or about 10 with high-aspect-ratio probes (DTU). DTU’s tip check is striking: a worn probe gave an $R_{q}$ of 2.49 $nm$ against 10.5 $nm$ for a new one on the same sample, 4.2 times lower (computed). Tips must therefore be checked on reference samples. Reference: DTU Nanolab LabAdviser, ‘AFM: Atomic Force Microscopy’ (Bruker Dimension Icon-Pt) (fetch-tool summary).

Ques: Which forces act between tip and sample?

Long-range attraction, short-range repulsion and adhesion. A sphere of radius R at distance D from a flat feels van der Waals attraction $H R / (6 D^{2})$ ; with a Hamaker constant of $1 \times 1 0^{- 19}$ $J$ , a 10 $nm$ tip at 0.5 $nm$ feels about 0.67 $nN$ (Giessibl’s energy form differentiated; computed). Giessibl notes that soft cantilevers can jump to contact and that in air meniscus forces from water or hydrocarbon layers add adhesion. Force-distance curves record deflection while the tip approaches and retracts, showing snap-in, contact stiffness and pull-off, and DTU’s PeakForce modes use them to map modulus, adhesion and deformation. Keeping forces near a nanonewton or below protects both tip and sample. Reference: F. J. Giessibl, ‘Advances in atomic force microscopy’, arXiv:cond-mat/0305119 (Rev. Mod. Phys. 75, 949, 2003): k = Y w t^3/(4L^3) (Eq. 14), f0 = 0.162 (t/L^2) sqrt(Y/rho) (Eq. 15), Delta f = -(f0/2k) dF/dz, V_vdW = -A_H R/(6z) for a sphere, tau_AM about 2Q/f0, beam-bounce detection, static deflection F/k, AM and FM feedback, jump-to-contact and meniscus forces in ambient conditions (fetch-tool summary); DTU Nanolab LabAdviser, ‘AFM: Atomic Force Microscopy’ (Bruker Dimension Icon-Pt) (fetch-tool summary).

Ques: How accurate are AFM height measurements?

About 2 per cent on calibrated steps, worse for very small heights. DTU specifies height accuracy better than 2 per cent at 200 $nm$ , typically better than 0.75 per cent, calibrating the Z piezo on a certified step of about 200 $nm$ with uncertainty of 1.5 $nm$ or less and accepting a 2 per cent, 4 $nm$ , QC offset; combined, $1. 5^{2} + 4^{2} = 4.27$ $nm$ (computed), matching its 4.3 $nm$ or 2.1 per cent. Integral nonlinearity is below 0.5 per cent and closed-loop sensor noise 35 $pm$ RMS. For sub-nanometre to few-nanometre features, DTU notes that tip properties, sample stiffness and scan force matter more, so roughness of very smooth wafers should be measured with sharp tips and low forces. Reference: DTU Nanolab LabAdviser, ‘AFM: Atomic Force Microscopy’ (Bruker Dimension Icon-Pt) (fetch-tool summary).

Ques: How do Ra and Rq compare, and what does their ratio reveal?

Rq weights large deviations more than Ra. For a Gaussian height distribution, $R_{q} = π /2 R_{a} \approx 1.25 R_{a}$ (computed). DTU’s tip-check data follow this: the new tip gave 10.5 and 8.45 $nm$ , a ratio of 1.24, and the worn tip 2.49 and 1.97 $nm$ , 1.26 (computed), so wear changed the magnitude but not the shape of the height distribution. Ratios well above 1.25 indicate spikes or particles, and below it, plateau-like surfaces. Peak-to-valley values such as $R_{ma x}$ , 78.3 against 18.8 $nm$ in DTU’s example, are most sensitive to tip condition and outliers. Roughness also depends on scan size and sampling, so the scan length, pixel count and filtering should always be reported, for example DTU’s 1 $μ m$ scans at 256 samples per line. Reference: DTU Nanolab LabAdviser, ‘AFM: Atomic Force Microscopy’ (Bruker Dimension Icon-Pt) (fetch-tool summary).

References

DTU Nanolab, “AFM: Atomic Force Microscopy,” LabAdviser (Bruker Dimension Icon-Pt). Available: https://labadviser.nanolab.dtu.dk/index.php/Specific_Process_Knowledge/Characterization/AFM:_Atomic_Force_Microscopy
F. Sumbul, N. Hassanpour, J. Rodriguez-Ramos and F. Rico, “One-Step Calibration of AFM in Liquid,” Frontiers in Physics, vol. 8, 301, 2020. Available: https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00301/pdf
F. J. Giessibl, “Advances in atomic force microscopy,” Reviews of Modern Physics, vol. 75, pp. 949-983, 2003; arXiv:cond-mat/0305119. Available: https://arxiv.org/pdf/cond-mat/0305119v1

Quartz 5

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Atomic Force Microscope (AFM)