Semiconductor depletion and charge collection

Consider an abrupt one-sided planar junction with uniform effective ionized dopant density in the lightly doped bulk. The highly doped side contributes little depletion width. Solving Poisson’s equation gives a width proportional to the square root of built-in plus applied reverse-bias voltage. Use the dopant-density magnitude in that width formula, and use a consistent permittivity rather than a dimensionless relative dielectric constant alone. Assume the built-in field alone does not fully deplete the sensor. The approximation neglects complicated space-charge profiles and edge fields.

The width cannot exceed the physical sensor thickness. At full depletion, increasing reverse bias changes the field and collection speed rather than indefinitely increasing sensitive thickness. A parallel-plate capacitance estimate decreases as depleted width increases until that geometric limit. Breakdown, leakage, surface conditions and radiation damage impose further constraints. In particular, a fully depleted sensor can still lose carriers through trapping or recombination, so full depletion should not be equated with unit charge-collection efficiency.

Deposited energy produces electron-hole pairs with a mean creation energy larger than the bandgap, because energy also goes into lattice excitations. Use a specified material and temperature for this mean energy. A fully collected pair supplies one elementary charge magnitude to a readout electrode after both carrier contributions are included, not two elementary charges merely because two carriers exist. The carrier paths and readout geometry determine how the signal develops in time.

The weighting potential is calculated with the readout electrode at unit potential, other electrodes at zero and no space charge. It differs from the actual bias potential driving carrier motion. The induced electrode-charge change from a signed moving carrier depends on its weighting-potential endpoints. Electrons and holes moving to opposite electrodes contribute complementary pieces for a simple fully collecting planar detector. Trapping, partial depletion, finite shaping time and segmented weighting fields can all alter the measured integral or pulse shape.

At a fixed fully deposited energy, the charge-creation variance is often smaller than the Poisson value because electronic and lattice energy channels are constrained by energy conservation. A Fano factor describes that intrinsic variance in the stated material model. Independent electronic noise can be expressed as an equivalent input number of electrons and added in quadrature. This convention must be input referred; using an output noise without undoing gain gives the wrong energy width.

The resulting expression is a useful ideal peak-width budget, not a complete model for every spectrum. Incomplete collection can produce asymmetric low-energy tails rather than another Gaussian variance. Electronic noise also depends on detector capacitance, leakage current, temperature and shaping time. A Gaussian FWHM conversion applies to the approximately Gaussian core only. Measuring a broader peak therefore does not by itself disprove the Fano description; collection, calibration, electronic and source widths must be investigated separately.