The Corbino disk is a specialized geometry used for measuring resistivity and Hall effect in high-conductivity materials, offering distinct advantages over conventional Hall bar structures. Unlike the rectangular Hall bar, the Corbino disk consists of a thin, annular-shaped semiconductor sample with concentric inner and outer electrical contacts. This configuration eliminates the need for transverse Hall voltage contacts, making it particularly suitable for materials with very high conductivity or low carrier mobility, where traditional Hall measurements may suffer from poor signal-to-noise ratios.

In a Corbino disk, current flows radially between the inner and outer contacts when a voltage is applied. Under an external magnetic field perpendicular to the plane of the disk, the Lorentz force deflects charge carriers azimuthally, creating a circular Hall current. Since the disk lacks lateral boundaries, the Hall voltage cannot develop as in a Hall bar. Instead, the magnetic field modifies the effective resistance between the inner and outer contacts, allowing the determination of carrier mobility and concentration through resistivity measurements alone.

The resistivity in the Corbino geometry is derived from the potential difference between the inner and outer contacts. For a disk with inner radius \( r_i \), outer radius \( r_o \), thickness \( t \), and an applied voltage \( V \), the current density \( J \) is radial and given by \( J = I / (2 \pi r t) \), where \( I \) is the total current. The electric field \( E \) relates to the current density via \( E = \rho J \), where \( \rho \) is the resistivity. Integrating the electric field from \( r_i \) to \( r_o \) yields the total voltage drop:
\[ V = \int_{r_i}^{r_o} E \, dr = \frac{\rho I}{2 \pi t} \ln \left( \frac{r_o}{r_i} \right). \]
Thus, the resistivity is obtained as:
\[ \rho = \frac{2 \pi t V}{I \ln (r_o / r_i)}. \]

When a magnetic field \( B \) is applied perpendicular to the disk, the Lorentz force induces an azimuthal current, altering the effective resistivity. The Hall angle \( \theta_H \) is defined by \( \tan \theta_H = \mu B \), where \( \mu \) is the carrier mobility. The resulting current density acquires an azimuthal component, modifying the radial resistivity to an effective value \( \rho_{\text{eff}} \):
\[ \rho_{\text{eff}} = \rho (1 + \mu^2 B^2). \]
The measured resistance \( R \) under the magnetic field becomes:
\[ R(B) = \frac{\rho (1 + \mu^2 B^2)}{2 \pi t} \ln \left( \frac{r_o}{r_i} \right). \]
By comparing the resistances with and without the magnetic field, the mobility can be extracted:
\[ \mu = \frac{1}{B} \sqrt{ \frac{R(B)}{R(0)} - 1 }. \]
The carrier concentration \( n \) is then calculated using \( \sigma = n e \mu \), where \( \sigma = 1 / \rho \) is the conductivity and \( e \) is the electron charge.

Compared to standard Hall bars, the Corbino disk offers several advantages for high-conductivity materials. First, it avoids the difficulty of measuring small Hall voltages in highly conductive samples, where the Hall signal may be obscured by noise. Second, the absence of lateral edges eliminates geometric magnetoresistance effects that complicate Hall bar measurements. Third, the circular symmetry ensures uniform current distribution, reducing errors from contact misalignment or inhomogeneities.

However, the Corbino geometry also has limitations. It cannot directly measure the Hall voltage, which is often desirable for determining the carrier type (electrons or holes). Additionally, the analysis assumes ideal radial current flow, which may not hold if the contacts are imperfect or the material is highly inhomogeneous. Despite these drawbacks, the Corbino disk remains a powerful tool for characterizing high-mobility or high-conductivity semiconductors, such as graphene, highly doped silicon, or certain III-V compounds.

In summary, the Corbino disk provides a robust method for resistivity and Hall measurements in materials where conventional Hall bars are impractical. By leveraging its unique geometry, researchers can accurately determine key transport parameters without the challenges posed by high conductivity or low mobility. The derived equations enable straightforward extraction of carrier mobility and concentration, making the Corbino disk an essential technique in semiconductor characterization.

A comparison between Corbino disk and Hall bar geometries highlights their complementary roles:

| Feature | Corbino Disk | Hall Bar |
|-----------------------|-------------------------------|-------------------------------|
| Geometry | Annular, radial current flow | Rectangular, transverse Hall voltage |
| Magnetic Field Effect | Modifies effective resistance | Generates transverse voltage |
| Best For | High-conductivity materials | Moderate-conductivity materials |
| Carrier Type Detection| Indirect (requires analysis) | Direct (via Hall voltage sign) |
| Edge Effects | Absent | Present (may cause artifacts) |

The choice between the two depends on the material properties and the specific parameters of interest. For high-conductivity samples, the Corbino disk’s insensitivity to edge effects and its reliance on resistance measurements alone make it the preferred option. In contrast, Hall bars remain indispensable for direct Hall voltage measurements and detailed studies of carrier dynamics in lower-mobility systems.

Practical considerations for Corbino measurements include ensuring ohmic contacts at both inner and outer electrodes and minimizing contact resistance. The disk thickness should be uniform to avoid variations in current density, and the magnetic field must be applied precisely perpendicular to the plane to prevent spurious effects. Advanced lithographic techniques enable the fabrication of microscale Corbino disks for nanomaterial characterization, further expanding their utility in modern semiconductor research.

In conclusion, the Corbino disk geometry is a vital tool for characterizing high-conductivity semiconductors, offering a reliable alternative to Hall bars when traditional methods fall short. Its mathematical framework allows precise determination of mobility and carrier concentration, while its design circumvents common pitfalls associated with high-mobility materials. As semiconductor technology advances toward ever-higher performance materials, the Corbino disk will continue to play a critical role in understanding and optimizing their electronic properties.