How would you explain the photoelectric effect using quantum theory?

Photoelectric effect can only be explained by the quantum concept of radiation.

The observations of photoelectric effect experimentally provides with the following conclusions -

1) The photocurrent is proportional to the intensity of incident radiation.

2) The magnitude of stopping potential and hence the maximum kinetic energy of emitted photoelectrons is proportional to the frequency of emitted radiation.

3) There exists a minimum threshold frequency so that if radiation of frequency lesser than this threshold frequency is incident on the metal surface, there is no photoemission irrespective of intensity of radiation.

Einstein's explanation -

Einstein used Max Planck's concept of quantized radiation packets each of an energy
#E = hnu#
where
#h# is Planck's constant and
#nu# is frequency of radiation.

According to Einstein, a single photon upon being incident on the metal surface interacts with one electron and transfers it's entire energy to the electron.
Then it is emitted with a maximum kinetic energy,
#K_(max) = hnu - phi_0#
Where, #phi_0# is the work function of the metal.

This explains the fact that maximum kinetic energy increases linearly with frequency of incident radiation.

Now for a frequency, #nu_0# so that #hnu_0 = phi_0# implies that for radiation of frequency less than #nu_0# the maximum kinetic energy of the photoelectron turns out to be negative and hence, no photoemission is possible.

This explains the concept of threshold frequency #nu_0# below which photoemission is not possible even with an extremely intense beam of light.

Since #phi_0# depends on the metal and the nature of the surface, the work function and hence the threshold frequency varies from substance to substance.

Also intensity of radiation from the quantum concept is associated with number of photons incident on a unit area in unit time.
That is, a greater intensity implies that a greater number of photons are incident on the metal surface per unit area in unit time.
Now since a single photon interacts with a single electron causing the emission of the electron, a higher number of incident photons implies that a greater number of photoelectrons are emitted in unit time. Consequently, the current increases with the number of photons incident and hence with the intensity of radiation used.