  Micro-magnets are key components for quantum information processing with
individual spins, enabling arbitrary rotations and addressability. In this
work, characterization of sub-micrometer sized CoFe ferromagnets is performed
with Hall bars electrostatically defined in a two-dimensional electron gas. Due
to the ballistic nature of electron transport in the cross junction of the Hall
bar, anomalies such as the quenched Hall effect appear near zero external
magnetic field, thus hindering the sensitivity of the magnetometer to small
magnetic fields. However, it is shown that the sensitivity of the diffusive
limit can be almost completely restored at low temperatures using a large
current density in the Hall bar of about 10 A/m. Overcoming the size limitation
of conventional etched Hall bars with electrostatic gating enables the
measurement of magnetization curves of 440 nm wide micro-magnets with a
signal-to-noise ratio above 10^3. Furthermore, the inhomogeneity of the stray
magnetic field created by the micro-magnets is directly measured using the
gate-voltage-dependent width of the sensitive area of the Hall bar.
