Vertical electric field is using to tune the electrical properties of channel in FETs. In new 2D material this is referred to the shifting of electrochemical potential. In contrast in simple Nonequilibrium Green function formulism applying voltage to the gate or gates (DG/double gate channels) does not influenced Fermi level position in numerical calculation. Instead, it will change the shape of conduction band.
Is there any mathematical relation or physics formula that can explain the displacement or shifting of the Fermi level with respect to Fermi level on the time that no external voltage applied to the gate or gates, (considering martial properties)?
Anybody can explain this paradox? Is there any good literatures on the shifting of Fermi level due to applying electric field?
Dear Vahid, there is no paradox. The Fermi level shift is a consequence of applying the bias. Example: if you apply a negative bias to the gate, the channel region will become less attractive to electrons (the conduction band shifts upwards). This will change the carrier balance in the channel (less electrons compared to holes), and the Fermi level will shift towards the valence band.
Hi!
There are two kinds of vertical electric fields. First it is perpendicular to the plane of 2D materials e.g., in MoS2 structure it is perpendicular to Mo-plane. Since one S-atom is placed above and one S-atom is placed below the Mo-plane, a potential differences V is seen between them. In numerical calculation, you should add +V/2 scalar potential to on-sites energy of one S-atom and -V/2 scalar potential to other S-atom.
Second it is parallel to the Mo-plane but it is perpendicular to the direction of current. If current direction is along x-axis, the field is along y-axis and S-atoms are placed along z-axis above and below the Mo-plane. Now Mo and both S-atoms fill a scalar potential V which is proportional to their y-coordinates. It should be also added to on-site energy of each atoms.
Both fields may/may not split the spin and you may/may not see spin splitting. It depends on many things, of course.
Now if you apply a scalar potential in the current direction i.e., x-direction, you should not only solve the self consistent field equation but also you should solve the Poisson equation. In this case it is necessary you find the new Fermi energy level since you disturb the equilibrium charge distribution along your transport channel.
Best Regards
Thank you Filip, your description is quite acceptable but why E_f=0 for simulation of conventional semiconductor. you can check it in the MATLAB codes in chapter 7 of Prof Datta's book (quantum transport).
Thank you Tillmann, I am familiar with imerf but I think in Nano scale maybe it is not so convenient to use quasi Fermi level because the channel is very thin, plus temperature is considered to be very low (4K) and using approximation of Maxwell distribution is not quite true.
Thank you very Hamidreza , I have seen several articles about "screening" or etc and how the external potential should be add to the Hamiltonian. among them I picked an article entitled "Electrostatic Doping of Graphene through Ultrathin Hexagonal Boron Nitride Films". it is appear form last formula in article. in simple way another's related change of Fermi level to the important parameters such as backgate voltage, relative permittivity of isolator, thinness of isolator, ... . what do you think? is it possible to use such formula for more thick isolator such as 100nm Sio2 and different channel like Mos2?
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