AFM Tip Measurements: Cantilever Motion to Surface Data

An Atomic Force Microscope (AFM) maps a surface by scanning a sharp tip attached to a flexible cantilever across it; the cantilever bends or changes its vibration as it feels atomic-scale forces, and a sensor measures that motion to build a topographical image. You enter the raw cantilever deflection or amplitude signal, and the AFM software converts it into height data using the cantilever’s known spring constant and sensitivity. The result is a three-dimensional map of the surface with nanometer or even sub-nanometer vertical resolution.

Formula

The core relationship in AFM is that the cantilever acts as a linear spring for small deflections, following Hooke’s law: F = k × d, where F is the force (in newtons), k is the cantilever spring constant (in N/m), and d is the deflection (in meters). The surface height is simply the deflection value when the AFM operates in constant-force (contact) mode, because the feedback loop keeps the force (and thus deflection) constant by moving the z-piezo. The raw electrical signal from the position-sensitive photodetector is converted to deflection using the sensitivity s (in nm/V): d = V × s. In dynamic modes (tapping or non-contact), the measured quantity is the amplitude or phase of the cantilever oscillation, which is related to the tip–sample interaction through more complex equations, but the output is still a height map.

Worked Example

Using the default values from the calculator: cantilever deflection d = 25 nm, sensitivity s = 50 nm/V, and spring constant k = 0.5 N/m.

• Output voltage: V = d / s = 25 nm / 50 nm/V = 0.5 V
• Force on tip: F = k × d = 0.5 N/m × 25×10â�»â�¹ m = 12.5×10â�»â�¹ N = 12.5 nN
• Surface height: In constant-force mode, the height equals the deflection, so h = 25 nm (but in practice the z-piezo movement is recorded, not the deflection, after feedback).

The calculator shows h = 25 nm, V = 0.5 V, and F = 12.5 nN, matching the manual calculation.

How the Cantilever Motion Is Measured

The most common method is the optical lever: a laser beam reflects off the top of the cantilever onto a position-sensitive photodetector (usually a four-quadrant photodiode). As the cantilever bends, the reflected beam moves, changing the relative intensity on the quadrants. The difference signal (e.g., (A+B) – (C+D) for vertical deflection) is proportional to the cantilever slope, not the absolute deflection. Calibration converts this voltage to nanometers. Other detection methods include piezoelectric cantilevers (which generate a voltage when bent) and interferometry (which measures displacement directly using light interference).

From Deflection to a Topographic Image

The AFM raster-scans the tip across the sample line by line. At each pixel, the system records the z-piezo position that maintains a setpoint (constant deflection, constant amplitude, or constant frequency shift). This z-position is the height value for that pixel. The feedback loop is critical: it must respond quickly enough to track surface features without overshooting. The raw data is a matrix of heights, which is then processed to remove tilt, bow, and noise, and rendered as a color-scale image or a 3D surface plot.

Choosing the Right Mode

Different AFM modes suit different samples:

• Contact mode: The tip touches the surface; good for hard, flat samples, but can damage soft or delicate samples.
• Tapping mode (AC mode): The cantilever oscillates near its resonance frequency; the tip lightly taps the surface. Ideal for soft, fragile, or loosely attached samples (e.g., DNA, polymers).
• Non-contact mode: The tip oscillates above the surface without touching; uses van der Waals forces. Best for very soft or liquid samples, but lower resolution and slower.
• Force spectroscopy: The tip is pushed into the surface and retracted while measuring force vs. distance, giving mechanical properties like stiffness and adhesion.

Common Problems and Calibration

The most frequent issues are tip artifacts (the image is a convolution of the tip shape with the sample), feedback instability (causing streaks or overshoot), and thermal drift (causing image distortion over time). Calibration requires measuring the cantilever’s spring constant (often by thermal noise method) and the optical lever sensitivity (by pressing the tip against a hard surface and recording the deflection vs. z-piezo movement). Without proper calibration, height measurements are only relative, not absolute.

Limits of the Method

AFM measures the surface, not the bulk. The vertical resolution is sub-nanometer, but lateral resolution is limited by tip radius (typically 5–20 nm). The scan speed is slow (minutes per image), and the maximum scan area is usually around 100 µm × 100 µm. The sample must be reasonably flat (roughness less than the z-range of the scanner, which is typically a few micrometers). In liquid, the cantilever’s resonance frequency and quality factor change, complicating dynamic mode operation. Also, the measured height is a convolution of the true topography and the tip–sample force interaction, so on soft samples the height may be underestimated due to indentation.

Alternatives and Complementary Techniques

For larger areas or faster scans, scanning electron microscopy (SEM) and profilometry are alternatives, but they lack the vertical resolution and force sensitivity of AFM. For non-topographic information, scanning tunneling microscopy (STM) gives electronic structure but requires conductive samples. AFM can be combined with Raman spectroscopy or infrared spectroscopy to correlate topography with chemical composition. For high-speed imaging, high-speed AFM (HS-AFM) uses smaller cantilevers and faster electronics to capture dynamic processes in real time.

Frequently Asked Questions

What is the difference between deflection and height in AFM?

Deflection is the physical bending of the cantilever due to tip–sample forces. Height is the z-position of the scanner that maintains a constant deflection (or setpoint). In constant-force mode, the height is the actual surface topography; the deflection is kept nearly zero by the feedback loop.

Why do I need to calibrate the cantilever spring constant?

Because the force calculation (F = k × d) depends on k. If k is wrong, all force measurements are wrong. The spring constant varies from cantilever to cantilever even within the same batch, so it must be measured individually for quantitative work.

Can AFM measure soft samples like cells or polymers?

Yes, but you must use tapping or non-contact mode to minimize lateral forces. Contact mode can deform or damage soft samples. In force spectroscopy, you can actually measure the sample’s elasticity by analyzing the force–distance curves.

What is the optical lever sensitivity and why does it matter?

It is the conversion factor between the photodetector voltage and the cantilever deflection (in nm/V). It depends on the laser spot position, the cantilever length, and the detector distance. Without it, you cannot convert voltage to actual deflection, so height and force values are meaningless.

Why does my AFM image show streaks or stripes?

Streaks are usually caused by feedback loop oscillation (the gain is too high), by a dirty or damaged tip, or by large surface features that the feedback cannot track quickly. Reduce the scan speed, lower the feedback gains, or replace the tip.

Can AFM measure in liquid or vacuum?

Yes. AFM operates in air, liquid, and vacuum. In liquid, it is often used for biological samples in their native environment. In vacuum, the quality factor of the cantilever resonance is higher, improving sensitivity in dynamic modes.

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in d | Cantilever deflection (nm) | nm | 25 | 0
in s | Cantilever sensitivity (nm/V) | nm/V | 50 | 0
in k | Spring constant (N/m) | N/m | 0.5 | 0
out h | Surface height | nm | d | 2
out v | Output voltage | V | d / s | 3
out f | Force on tip | nN | k * d | 2