AFM principles

An atomic force microscope converts tip–sample interaction into a calibrated height or force signal. The image is not a direct photograph: it is the output of a sensor, feedback controller, scanner and reconstruction convention.

The optical lever amplifies cantilever angle. Feedback turns the detector error into a scanner Z command that is recorded as topography in constant-interaction modes.

Tip, cantilever and interaction

A sharp tip sits at the end of a compliant cantilever. When the interaction force \(F\) is small enough for a linear cantilever model,

\[ F=-k\,d, \]

where \(k\) is spring constant in N m\(^{-1}\) and \(d\) is cantilever deflection in metres. The sign depends on the detector and coordinate convention. Van der Waals, capillary, electrostatic, chemical and repulsive contact forces can all contribute; “height” is therefore inferred through an operating-mode-specific control law, not read by a camera.

Optical lever and detector

A laser reflected from the back of the cantilever moves across a quadrant photodiode. A vertical difference signal such as \((A+B)-(C+D)\) is converted from volts to deflection with a deflection-sensitivity calibration. Multiplying by \(k\) converts deflection to force. Detector saturation, laser drift, cross-talk and an incorrect sensitivity propagate into force results.

Feedback and scanner

The controller compares the measured interaction signal with a setpoint and commands the Z piezo. The reported topography can correspond to this correction, the residual/error signal, or another vendor-defined channel. Feedback gains, line speed and bandwidth determine whether steep features are tracked or distorted.

Coordinates and physical scale

SPM arrays have at least four coordinate choices:

• array row/column indexing;
• physical X/Y coordinates derived from field of view and pixel count;
• fast/slow scan axes and forward/backward direction;
• Z/force sign and unit conventions.

A transpose or vertical flip can preserve roughness statistics while changing spatial interpretation. A unit error can change a modulus by orders of magnitude. SPM-Kit domain objects therefore keep array shape, X/Y range, unit, direction and source metadata where the reader can recover them.

Raw signal versus calibrated quantity

Raw/near-raw signal	Required context	Calibrated/derived quantity
detector voltage	deflection sensitivity	cantilever deflection (m)
cantilever deflection	spring constant	force (N)
scanner command/counts	scanner calibration and axis convention	topography (m)
force and ramp coordinate	tip/sample geometry, contact point, Poisson ratio	indentation and modulus
electrostatic null voltage	sign convention and calibrated tip work function	CPD/work function

Calibration metadata are inputs to a result, not decorative labels.

SPM-Kit implementation and evidence

Concept	Current path	Evidence	Limitation
image/force inspection	spmkit.core.io.load_any, reader contracts	format-specific Level 1/2	not every variant is demonstrated
image domain model	spmkit.core.models.SPMData / SPMChannel	Level 1	metadata depend on source availability
force calibration functions	spmkit.core.analysis.calibration	Level 1/2 paths	calibration constants must be externally defensible
file inspection CLI	spmkit info	Level 1	inspection does not validate an analysis

Fathom routes image data to Imagen and force data to Curva de fuerza or Mapa. It uses the same Core reader/domain objects.

Theory overview · Operating modes