Photoelectric effect (work function)

The photoelectric effect is the release of an electron from a material by the absorption of a single photon. An electron leaves the surface only if the photon energy $h\nu$ is at least the work function $W$, the energy needed to remove an electron from the Fermi level to the vacuum outside the surface. Work functions of metals lie between about 2 and 6 eV: cesium is about 2.14 eV, so it responds to light shorter than 579 nm, while zinc at about 4.3 eV needs ultraviolet below 288 nm. Hertz observed the effect in 1887; Lenard showed in 1902 that the electron energy is independent of the light's intensity; Einstein explained it in 1905 by treating light as quanta, and Millikan's measurements of 1916 confirmed the linear relation and gave an accurate value of Planck's constant.

Einstein's equation

Energy conservation for one photon absorbed by one electron gives

$$E_\mathrm{kin,max} = h\nu - W.$$

Electrons from deeper states leave with less energy, so this is the maximum. The threshold, or cutoff, wavelength is where the kinetic energy reaches zero:

$$\lambda_c = \frac{hc}{W} = \frac{1239.84\ \mathrm{eV\,nm}}{W}.$$

Worked example. Light at 405 nm carries $1239.84/405$ = 3.06 eV per photon. On a cesium surface ($W$ = 2.14 eV) the fastest electrons leave with 0.92 eV, about 570 km/s, and a retarding potential of 0.92 V between the surface and a collector stops the photocurrent. Red light at 650 nm (1.91 eV) releases no electrons from cesium at any intensity, while a weak violet beam releases them immediately.

How it is observed

The classic measurement places the emitting surface (the photocathode) and a collecting anode in vacuum and measures the photocurrent as a function of the voltage between them. With the anode positive, the current saturates at a value proportional to the incident power; with the anode made negative, the current falls to zero at the stopping potential $V_s = E_\mathrm{kin,max}/e$. Plotting $V_s$ against frequency for several wavelengths gives a straight line of slope $h/e$ whose intercept on the frequency axis is the threshold frequency $W/h$ (the voltage-axis intercept is $-W/e$). Three observations distinguish the effect from any classical wave picture: no emission below the threshold frequency at any intensity, maximum electron energy independent of intensity, and no measurable delay between illumination and emission even at very low power.

External and internal photoeffect

The process above, with electrons leaving the material, is the external photoelectric effect. It is the basis of vacuum photodetectors: the photomultiplier tube, image intensifiers, and the photocathode of a streak camera. Practical photocathodes use alkali compounds such as bialkali (Sb–K–Cs) or multialkali layers, and semiconductor cathodes such as GaAs treated with cesium to lower the effective barrier, because clean metals emit very few electrons per photon. Quantum efficiency peaks at roughly 25 to 35% for bialkali near 400 nm; the corresponding responsivity at 25% and 400 nm is $0.25 \times 400/1239.84$ ≈ 0.081 A/W.

In the internal photoelectric effect the absorbed photon lifts an electron from the valence band into the conduction band of a semiconductor, creating an electron-hole pair that stays inside the material. The threshold is then the bandgap instead of the work function: 1.12 eV in silicon, a cutoff of 1107 nm. This is the mechanism of every photodiode, CCD and CMOS sensor, solar cell and photoconductive detector. Because bandgaps are smaller than work functions, internal-effect detectors reach far into the infrared, where photocathodes become inefficient.

Pitfalls

• Surface condition. Work functions depend on crystal face, contamination and adsorbed layers by several tenths of an electronvolt, so tabulated values vary between sources; alkali photocathodes degrade on exposure to air.
• Threshold vs efficiency. A photon above threshold is not guaranteed to produce an electron; the quantum efficiency near threshold is very small and rises gradually with photon energy.
• Dark emission. Thermionic emission from low-work-function cathodes produces dark counts that rise steeply with temperature, which is why infrared-sensitive photocathodes are cooled.

Common questions

What is the work function?

The minimum energy needed to remove an electron from the Fermi level of a solid to rest just outside its surface. It is a property of the surface as well as the bulk material, typically 2–6 eV for metals and lower for cesiated surfaces.

Does brighter light give faster electrons?

No. Higher intensity means more photons per second, which increases the number of emitted electrons and hence the current, but each electron still receives the energy of one photon. Only a shorter wavelength raises the maximum kinetic energy.

Is a photodiode based on the photoelectric effect?

Yes, on the internal photoelectric effect: photons create electron-hole pairs inside the semiconductor, and the junction field separates them. No electron leaves the material, and the threshold is the bandgap.

References: A. Einstein, Ann. Phys. 17, 132 (1905); R. A. Millikan, Phys. Rev. 7, 355 (1916); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); E. Hecht, Optics, 5th ed. (Pearson, 2017).