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In Semiconductor Physics and Devices by Donald A. Naemen (pg. 176) the following figure appears:

Energy-band diagram for nonuniform doping

The donor concentration is a decreasing function of x.

My question: why are the energies of the conduction and valence band ($E_c$ and $E_v$) a function of the concentration? Why is the Fermi-level the only value in the figure that is not affected by the non-uniform distribution of donors?

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  • $\begingroup$ $E_{fi}$ is the intrinsic Fermi energy at mid gap. $E_{f}$ is the result of varying the doping. To be in equilibrium the Fermi energy is constant across the material. So, what is your confusion more specifically? $\endgroup$
    – Jon Custer
    Jun 4 '17 at 21:01
  • $\begingroup$ I understand why $E_f$ is constant, and why $E_{fi}$ varies. I don't understand why $E_c$ and $E_v$ vary with donor concentrations. $\endgroup$
    – YoA
    Jun 5 '17 at 11:44
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    $\begingroup$ Because in equilibrium the Fermi energy has to be constant across the material. What does the band diagram look like across a $pn$ junction? The Fermi energy is constant, the conduction and valence bands bend. $\endgroup$
    – Jon Custer
    Jun 5 '17 at 12:41
  • $\begingroup$ How do we know that the material is in equilibrium? $\endgroup$
    – Kashmiri
    Dec 28 '20 at 8:09
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Due to concentration difference there is a flow of mobile charge Carrier from region of higher concentration to lower concentration this results into diffusion current in the s/c. Thus there is a potential generated which energises the charge carriers according to Energy=-q.V (q=e) Due to this energy we see the Ec Ei and Ev are bent Now coming to the Fermi energy level : as the semiconductor is in equilibrium the net current is zero therefore no electron flow happens in equilibrium and the Fermi energy level stays flat. In conclusion the Ec/Ev/Ei bend due to diffusion owing to concentration difference while Fermi level remains flat due to equilibrium condition.

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  • $\begingroup$ Thank you , so $E_i$ bends because its defined to be at the middle of $Ec$ and $Ev$? $\endgroup$
    – Kashmiri
    Dec 28 '20 at 7:52
  • $\begingroup$ How do we know that the material is in equilibrium? $\endgroup$
    – Kashmiri
    Dec 28 '20 at 8:09

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