What if kinetic energy of ejected electron = 0

Question

When hf ≥ work function, Then the electron still comes out. So, if I say kinetic energy of ejected electron = 0, it should still come out. Right ?

Then, how does the electron even move out or gets ejected if its $v=0$?

Answer 1

Rather than debating whether an electron with no kinetic energy is ejected consider how an experiment to measure the photoelectron effect might be conducted. A photosensitive electron emitter is irradiated and the kinetic energy of ejected electrons is measured. One way to do this experiment is described in MIT Physics Department lab course notes. The setup is described in the following schematic:

A photodiode is irradiated with the filtered output of a mercury lamp. Only one of the available narrow bands of the lamp's spectrum is selected by the filter. The light strikes the K coated surface of the cathode of the photodiode. Emitted electrons are collected at a ring-shaped anode separated from the cathode by a gap, setting up a small current. An adjustable retarding voltage can be applied between the cathode and the anode to suppress the current, and the minimal cutoff voltage V which completely suppresses the current is measured. This voltage is theoretically related to the work function $\phi$ of the material and to the energy $h\nu$ of the incident light as $$V = \frac{h}{e}\nu - \frac{1}{e} \phi$$

Now note perhaps the most important point with regard to your question: only a limited number of light frequencies are available in the experiment, corresponding to the bands generated by the mercury lamp. In addition the motion of a collection of electrons (a current) is measured, not the velocity of a single emitted electron. While it is possible to perform single particle experiments, at no point in the MIT experiment is the kinetic energy of individual electrons directly measured.

The lab notes also provide a short history of the discovery of the photoelectric effect:

Crude though the early data were, the qualitative fact of the dependence of the critical cutoff voltage on the wavelength of light emerged with sufficient clarity to induce the young Albert Einstein, working as a patent examiner in the Swiss Patent Office in 1905, to link the effect with the recent idea, introduced by Planck in 1900,that matter radiates its energy in quanta of energy hν.

and

It was not until 1912 that the technical problems of making precision measurements of the photoelectric effect were overcome [...]

In other words, 7 years passed between the time that Einstein postulated his theoretical description and a confirmation based on "clean" data.

Note also from the lab notes how inherently "messy" such an experiment can be:

It is not possible to precisely determine the work function for removing an electron, because the cathode surface interacts with the remaining gases in the photocell as a getter, so that the surface characteristics change a little from the ideal case. It is important to note that the electronic work function φ is a material constant which incorporates the different emission potentials of the cathode and anode. The former is a difficult quantity to estimate due to the manufacturing process which makes the cathode surface inhomogeneous. It is composed of a mixture of potassium, potassium oxide and oxidized silver. For this reason, you need to take care that the same area is always illuminated. The emission work function of the photoelectrons can vary locally!

Answer 2

There is the direct analogy to the Solar system.

Imagine an asteroid ejected from the Solar system with zero kinetic energy. It makes sense only if the asteroid is at infinite distance from Sun, otherwise it is not ejected yet. It would start long fall on the Sun.

Or, if other stellar systems are considered, than gravity of the Sun must be counteracted by gravity of the other stars, so there is zero potential gradient.

Now, replace stars by atoms and the asteroid by an electron. An electron with zero kinetic energy would be truly ejected only if there is mutually cancelled attraction toward neighbour atoms and ions, or if it is at infinite distance to them.

So the zero kinetic energy of ejected electron has sense and meaning rather merely mathematical as the limit point on the chart where electron energy as function of wavelength crosses the zero point.

Zero kinetic energy of electron has also quantum obstacles. Zero kinetic energy means zero speed and that would mean certainty of the electron position. Following Heisenberg uncertainty principle , momentum of electron could have any value.

Electron wave function, determining electron energy in an atom, is not function of time, so precision of predicted energy is not limited.