Quantum Efficiency (QE)

Quantum efficiency is a unitless number that measures the sensitivity of the photocathode to light. It is the ratio of the number of electrons emitted to the number of incident photons.^[1] This property depends on the wavelength of light being used to illuminate the photocathode. For many applications, QE is the most important property as the photocathodes are used solely for converting photons into an electrical signal.

Quantum efficiency may be calculated from photocurrent (${\displaystyle I}$), laser power (${\displaystyle P_{\text{laser}}}$), and either the photon energy (${\displaystyle E_{\text{photon}}}$) or laser wavelength (${\displaystyle \lambda _{\text{laser}}}$) using the following equation.^[1][2]

${\displaystyle {\text{QE}}={\frac {N_{\text{electron}}}{N_{\text{photon}}}}={\frac {I\cdot E_{\text{photon}}}{P_{\text{laser}}\cdot e}}\approx {\frac {I[{\text{A}}]\cdot 1240}{P_{\text{laser}}[{\text{W}}]\lambda _{\text{laser}}[{\text{nm}}]}}}$

Mean Transverse Energy (MTE) and Thermal Emittance

For some applications, the initial momentum distribution of emitted electrons is important and the mean transverse energy (MTE) and thermal emittance are popular metrics for this. The MTE is the mean of the squared momentum in a direction along the photocathode's surface and is most commonly reported in units of milli-electron volts.^[3]

${\displaystyle {\text{MTE}}={\frac {p_{\perp }^{2}}{2m_{e}}}}$

In high brightness photoinjectors, the MTE helps to determine the initial emittance of the beam which is the area in phase space occupied by the electrons.^[4] The emittance (${\displaystyle \varepsilon }$) can be calculated from MTE and the laser spot size on the photocathode (${\displaystyle \sigma _{x}}$) using the following equation.

${\displaystyle \varepsilon =\sigma _{x}{\sqrt {\frac {\text{MTE}}{m_{e}c^{2}}}}}$

where ${\displaystyle m_{e}c^{2}}$ is the rest mass of an electron. In commonly used units, this is as follows.

${\displaystyle \varepsilon [{\text{um}}]\approx \sigma _{x}[{\text{um}}]{\sqrt {\frac {{\text{MTE }}[{\text{meV}}]}{511\times 10^{6}}}}}$

Because of the scaling of transverse emittance with MTE, it is sometimes useful to write the equation in terms of a new quantity called the thermal emittance.^[5] The thermal emittance is derived from MTE using the following equation.

${\displaystyle \varepsilon _{\text{th}}={\sqrt {\frac {\text{MTE}}{m_{e}c^{2}}}}}$

It is most often expressed in the ratio um/mm to express the growth of emittance in units of um as the laser spot grows (measured in units of mm).

An equivalent definition of MTE is the temperature of electrons emitted in vacuum.^[6] The MTE of electrons emitted from commonly used photocathodes, such as polycrystalline metals, is limited by the excess energy (the difference between the energy of the incident photons and the photocathode's work function) provided to the electrons. To limit MTE, photocathodes are often operated near the photoemission threshold, where the excess energy tends to zero. In this limit, the majority of photoemission comes from the tail of the Fermi distribution. Therefore, MTE is thermally limited to ${\displaystyle k_{B}T}$, where ${\displaystyle k_{B}}$ is the Boltzmann constant and ${\displaystyle T}$ is the temperature of electrons in the solid.^[7]

Due to conservation of transverse momentum and energy in the photoemission process, the MTE of a clean, atomically-ordered, single crystalline photocathode is determined by the material's band structure. An ideal band structure for low MTEs is one that does not allow photoemission from large transverse momentum states. ^[8]

Outside of accelerator physics, MTE and thermal emittance play a role in the resolution of proximity-focused imaging devices that use photocathodes.^[9] This is important for applications such as image intensifiers, wavelength converters, and the now obsolete image tubes.

Lifetime

Many photocathodes require excellent vacuum conditions to function and will become "poisoned" when exposed to contaminates. Additionally, using the photocathodes in high current applications will slowly damage the compounds as they are exposed to ion back-bombardment. These effects are quantified by the lifetime of the photocathode. Cathode death is modeled as a decaying exponential as a function of either time or emitted charge. Lifetime is then the time constant of the exponential.^[10][11]