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If in an electric field, greater potential energy and voltage are toward the positive charge and away from negative charge, then why do PN energy band diagrams indicate lower energies toward the positive charge across the depletion region?

Skateboarder

Depletion Region

Band Diagram

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3 Answers 3

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Think of it like this, if you let go of a ball, it will fall until the next stable point it finds right? because of gravity and the gravitational field, and one obvious thing to note is, that the ball is going from a higher energy level to a lower one.

If you look at a normal battery, the electrons are moving from the negative end to the postive end, and this is the same story as it was with the ball.

So in the depletion region, after stability has been reached, you have an electric field from your N-side to your P-side (as shown in your pictures). Now think about the battery again, the electric field is from the positive end to the negative end, but the voltage (as its scientific term) is in the opposite direction.

So this was an intuitive understanding of fields and energies, but if you are interested enough in maths like me, you would see that in physics, when you are talking about work being done on sth or by sth, there is usually a negative sign invovled.

To expand, lets imagine you have a fixed charge that doesn't move (Q). The field generated by this electrostatic charge is: $$E= Q/(4πεr^2 )$$ So to calculate potential, you will need to integrate the field, but then a problem pops up! Integration is done over a range (definite): $$∫_O E(r).dr =∫_O Q/(4πεr^2 ).dr =[-Q/(4πεr )]_O$$

The point O is our reference, and you need a reference to measure your potential against. The way physicists go around this is by putting infinity the reference point, so the answer to the above integration would be: $$potential = (-Q/(4πεr )) - (-Q/(4πεR ) $$ $$=$$ $$\lim_{R\rightarrow ∞}(-Q/(4πεr )) - (-Q/(4πεR )$$

where

$$R = distance\ from \reference$$

So, 1/∞ is really close to 0! and we will only be left with: $$potential = (-Q/(4πεr ))$$

And thus, the negative sign! That is why in your depletion region, where you have an electric field from your N-side to P-side, the energy (In the way that I explained above) and thus the voltage, is in the opposite direction. it is worth noting that energy is a scalar value and does not have a direction, but I am trying to give an intuitive answer.

I hope this will help undestand the relationship better. For your understanding of P-N junctions, start looking into fermi level, energy band diagrams, band gap enegy and Recombination/generation in a semiconductor.

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  • \$\begingroup\$ Lovely work. For that reason alone, I'll accept yours as the answer and give it a like. I believe you're telling me that due to the junction at equilibrium, an electron (our skateboarder) and the N-type's neutral region have lost potential energy relative to the P-type and in order to cross the depletion region (the ramp), requires some kinetic energy to do so. Note that I have changed that graphic in my original question - putting the skateboarder into agreement with the voltage/E-field and reversing both (directionwise) to be in agreement with the Wiki graphic (both ruthlessly stolen). \$\endgroup\$ Commented Aug 2, 2020 at 0:25
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    \$\begingroup\$ As I understand it, a (mobile) positive charge gains potential energy in moving towards positive charge and away from negative charge (towards the right) and loses kinetic energy, whereas an electron loses potential energy and gains kinetic energy in being propelled by the E-field towards the positive charge. And that agrees with the electrons want to fall (drops of water) and holes want to rise (bubbles in water) analogy. \$\endgroup\$ Commented Aug 2, 2020 at 0:35
  • \$\begingroup\$ @TedJackson That is right, if I may refer back to the falling ball towards the earth example, the ball is going parallel to the direction of the gravitational field and as you mentioned, gains kinetic energy and loses potential. The same analogy applies here, if a POSITIVE test charge (i.e. a hole) is travelling in the direction of the E-field (towards our N-type semi-con) it will lose potential. but please do note that this is not the only reason governing the energy bands diagram during thermal equilibrium. \$\endgroup\$
    – NeuroEng
    Commented Aug 3, 2020 at 7:14
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in an electric field, greater potential energy and voltage are toward the positive charge and away from negative charge

This is correct from the view point of very definition of Electric field.

then why do PN energy band diagrams indicate lower energies toward the positive charge across the depletion region

this is because, in energy band diagram, on y-axis energy is not the energy of a positive test charge (remember, to define electric field, we take positive unit test charge) but the energy of the electron in particular state in the semiconductor.

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E=-qv in the band diagram The energy is defined minius the potential times the charge (in elcectron volte units)

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