At which electric field or voltage does field emission occur? Imagine our environment is a vacuum. At which value of electric field or voltage, does field emission occur? I just want to know what is the maximum electric field we are allowed, to place between two electrodes, without field emission phenomenon. So, you can choose any metal that you wish in order to mention this maximum value.
 A: Field emission is a quantum tunneling process, and as such can occur at any voltage, although for low enough fields it will be exponentially suppressed. Intuitively one should expect the emission rate to go as something like:
$$dN/dt \propto \exp(-eEd/W)\equiv \exp(-E/E_c),\, E_c= W/ed$$
where $e$, $E$ and $W$ are the electron charge, applied electric field and the work function of the emitting material respectively. $d$ is the distance determining how much the bulk electrons orditals leak out into vacuum. You can estimate if as $d\approx \hbar v_F/W$, $v_F$ being fermi velocity. The work function $W$ is typically in the range of several electron-volts, i.e. comparable to typical Fermi energies, therefore $d$ is roughly equal lattice constant. So now that we have an idea of the order of magnitude of all involved quantities, we can estimate the characteristic field $E_c$ at which field emission becomes noticeable: $E_c$~$5eV/(e*1nm)=5$ V/nm$=5e7$ V/cm.
You can achieve such fields by either increasing your voltage or by reducing the distance between electrodes. If you work with an STM you can get filed emission already with several-to-ten volts of bias voltages. Also, note that the actual local field can be enhanced near surface inhomogeneities, that will facilitate field emisssion.
A: Well, I am a new master circuit student and I do know so much about physics.
The definition of work function is here:

In solid-state physics, the work function (sometimes spelled workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface.

And also, according to field electron emission:

Field electron emission, also known as field emission (FE) and electron field emission, is emission of electrons induced by an electrostatic field. The most common context is field emission from a solid surface into a vacuum.

So, we can say that field emission occurs when the electron can absorb enough energy to overcome the work function the metal.
So according to my research, I can say that based on Work functions of elements the maximum work functions of materials is 5.93eV (and I do not know a material with a higher work function is discovered or not). So, the voltage required to remove an electron from metal to a point in vacuum will be equal to:
5.93/(1.6e-19)=3.7e19 volts
