Electron-multiplying CCD ($g$)

An electron-multiplying CCD adds a high-field multiplication register before the output amplifier. Impact ionization generates a large mean gain, reducing input-referred read noise but adding multiplication excess noise.

Random multiplication increases the noise relative to deterministic gain. If the output from one input electron has mean $g$ and variance $g^2$, a Poisson distribution of input electrons with mean $\mu$ produces output variance $2g^2\mu$, by the law of total variance. The high-gain analog-mode excess-noise factor is $\sqrt{2}$.

An energetic charge carrier creates an additional electron–hole pair. In a multiplication register this is stochastic, so the output charge fluctuates even when the input electron number is fixed.

For a direct-gap semiconductor with equal parabolic conduction-electron effective masses, restrict the final valence hole to zero crystal momentum and consider a normal process. Momentum conservation gives ${\mathbf{k}}_1={\mathbf{k}}_2+{\mathbf{k}}_3$, while energy conservation gives $E_1=E_g+\alpha (k_2^2+k_3^2)$ with $\alpha ={\hslash }^2/(2m^{\ast })$. The Cauchy-Schwarz inequality gives $k_2^2+k_3^2\ge k_1^2/2$, so $E_1\ge E_g+E_1/2$ and the threshold is $2E_g$. Equal final wavevectors attain the bound; opposite spins allow the two electrons to occupy that wavevector. This is the threshold under the stated hole constraint, not the general threshold when the final hole can carry momentum.