As mentioned earlier, physicists thought it was light waves that liberate the electrons from the metal lattice. But when they looked at the experimental results more carefully, they began to have doubts.
Below are the I-V graphs for experiments conducted using different light intensities (but the same light frequency and emitter). They show that increasing the light intensity results in higher saturation currents. Is this a problem? Nope. At higher light intensity, more energy per unit time (per unit area) is being delivered to the emitter, which causes more photoelectrons to be liberated per unit time, resulting in a higher saturation current. This observation is totally in agreement with the wave model of light.
What raises the eye brows is that these I-V graphs show the same stopping potential. This indicates that while a higher light intensity does increase the rate of emission, it does not increase the maximum KE of emitted photoelectrons. But if a light wave is flooding the emitter with more energy per unit time, the electrons should on average be more energetic. Not only should more of them should break free per unit time, those which break free should escape at higher speed and KE. The mystery deepens when we look at the I-V graphs for experiments conducted using different light frequencies (but still using the same emitter). They show that higher light frequency results in higher stopping potential. This shows that the maximum KE of emitted photoelectrons increases with the light frequency.
This is very odd because it is well established that the rate of transmission of energy by an EM wave is proportional to amplitude-square, and independent of the frequency. So the wave model predicts that the maximum KE of photoelectrons should increase with the amplitude, not the frequency, of the light wave. Yet the results indicate that the maximum KE of photoelectrons increases with the frequency, not the amplitude, of the light. This observation flies in the face of the wave model of light!
In fact, it was discovered that if the frequency of the light used is too low, no photoelectric current can be measured, no matter how strong the light is. This shows that there is a particular threshold frequency f0 (for the particular type of metal used as the emitter) below which no photoelectric effect can occur, regardless of the intensity of light. This is extremely odd because as an EM wave, light of any frequency should be able to deliver energy to the emitter at a high enough rate, provided the amplitude is high enough. If anything, there should be a threshold amplitude, not a threshold frequency, before photoelectrons can be knocked out of the emitter. Once again, the experimental results contradict the wave model.
Last but not least, there seems to be zero time lag between the shining of the light and the registration of a steady photoelectric current on the micro-ammeter. As a wave, light can only deliver energy in a distributed (over space) and continuous (over time) manner. In the context of the photoelectric effect, this means that the light energy must be shared by all the trillions of electrons hit by the wave front. It will take time for each individual electron to accumulate energy from the light wave. If we keep reducing the light intensity, we should be able to prolong the length of time electrons take to accumulate sufficient energy before they can be liberated. Yet, experimental results show that photoelectrons are emitted without any delay the instant the light arrives at the emitter, regardless of how low the light intensity is. The transfer of energy seems to be instantaneous.
 Mechanical waves would transfer energy at a rate that is proportional to amplitude-square and frequency-square. EM waves are not mechanical waves.
 Light diffracts. It is impossible to focus the light beam on one single electron.