# How does surface potential depend on doping level in semiconductors

d.arbitman
I am reading Analysis and Design of Analog ICs by Gray and Meyer. In Ch. 2, they describe using MOS technology for fabricating on-chip capacitors.

First and foremost, what exactly does surface potential mean in the context of semiconductors?

Quoted from Gray and Meyer (5e), Page 149:
Because the plates of the capacitor are a heavily doped semiconductor rather than an ideal conductor, some variation in surface potential relative to the bulk material of the plate occurs as voltage is applied to the capacitor. This effect is analogous to the variation in surface potential that occurs in an MOS transistor when a voltage is applied to the gate. However, since the impurity concentration in the plate is usually relatively high, the variations in surface potential are small. The result of these surface potential variations is a slight variation in capacitance with applied voltage.

So, how does surface potential depend on doping? I just can't picture what's happening in the semiconductor itself.

P.S. Please no equations (unless absolutely necessary) because I want to develop intuition.

## Answers and Replies

Staff Emeritus
A Kelvin probe, or a Kelvin Probe Force Microscopy, measures the "surface potential" of a material. For a semiconductor, this is the energy between the Fermi level and the vacuum level.

I'm not sure if, in the context of your question, there is a different meaning of surface potential.

Zz.

d.arbitman
A Kelvin probe, or a Kelvin Probe Force Microscopy, measures the "surface potential" of a material. For a semiconductor, this is the energy between the Fermi level and the vacuum level.

I'm not sure if, in the context of your question, there is a different meaning of surface potential.

Zz.

What does the Fermi level represent? I don't have any intuition on that. I looked through my physics textbooks but they don't do a good job of explaining it besides giving an equation and a graph or two. Wikipedia doesn't do a good job of it either. What is vacuum level?

My main question is: How does the surface potential depend on the doping level?

Staff Emeritus
What does the Fermi level represent? I don't have any intuition on that. I looked through my physics textbooks but they don't do a good job of explaining it besides giving an equation and a graph or two. Wikipedia doesn't do a good job of it either. What is vacuum level?

My main question is: How does the surface potential depend on the doping level?

The first part of your question requires that you learn about band diagram of material. That will take quite a lot of time (and a lot of sketching) to be done here.

The surface potential is affected by doping. When you dope a semiconductor, the Fermi level (or more accurately, the Chemical potential if one doesn't want to be sloppy about it) changes. A dope with donors, causing it to be n-type, will move the Chemical potential upwards (i.e. making the difference between the vacuum and chemical potential smaller), while doping with acceptors (p-type) will cause it to go the other way.

I think you need to first learn about the band structure of solids, especially semiconductors, before you can try to understand what you were reading earlier.

Zz.

Staff Emeritus
That only tells me about how one kind of doping affects the Fermi level as opposed to the other. I was curious as to how the LEVEL of doping affects surface potential (i.e. lightly doped vs. heavily doped)

The more you dope a semiconductor, the MORE the Fermi level will change, and thus, changing the surface potential (energy between the Fermi level and the vacuum level).

{I hate to say "this is assuming rigid band model", because I'm sure I'll get asked "what is a rigid band model?"}

Zz.

d.arbitman
The more you dope a semiconductor, the MORE the Fermi level will change, and thus, changing the surface potential (energy between the Fermi level and the vacuum level).

{I hate to say "this is assuming rigid band model", because I'm sure I'll get asked "what is a rigid band model?"}

Zz.

That makes sense. (the relationship, not the concept)

I have never heard of the rigid band model. Any suggestions for a textbook where I can learn about semiconductor physics? At the moment I am using the following two books for reference:
Device Electronics for Integrated Circuits; Muller & Kamins, 2e
Semiconductor Devices, Physics and Technology, Sze & Lee, 3e

My initial model of how doping affected the surface potential involved an analogy with an ideal conductor. In other words, the higher the doping level in a semiconductor, the more charge carriers it has and the closer it mimics the behavior of an ideal conductor. So as the Grey and Meyer textbook stated, the higher the impurity concentration, the less variation in surface potential. So, I thought that the potential difference between one edge of the semiconductor and the opposing edge would approach 0V as the doping level increased.

d.arbitman
The surface potential is affected by doping. When you dope a semiconductor, the Fermi level (or more accurately, the Chemical potential if one doesn't want to be sloppy about it) changes. A dope with donors, causing it to be n-type, will move the Chemical potential upwards (i.e. making the difference between the vacuum and chemical potential smaller), while doping with acceptors (p-type) will cause it to go the other way.

I was reading Physics of Semiconductor Devices by Sze, and I came across the following band diagrams. If we look at Fig 11b, that represents n-type doping, and the energy level difference between the Fermi level and the energy of the conduction band is less than the energy level difference of Fig 11a, in which there is no doping. Is that what you meant when you said:
A dope with donors, causing it to be n-type, will move the Chemical potential upwards (i.e. making the difference between the vacuum and chemical potential smaller)

Is it this relationship that explains why a large n-type doping level causes less variation in the surface potential?

Staff Emeritus
Is it this relationship that explains why a large n-type doping level causes less variation in the surface potential?

No, it doesn't. I need to see the context of that description.

If the entire surface has the same doping, then how can there be a "variation", large or small, of the surface potential?

Zz.

d.arbitman
No, it doesn't. I need to see the context of that description.
Zz.

Which description, the figure or the quote from my first post?

In any case, I attached pictures of the section from which I extracted the quote for my first post.

#### Attachments

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Staff Emeritus
Read carefully. Notice that it says variation in surface potential with APPLIED field! This means that you can vary such potential as you vary the applied field! The applied field effectively changes the vacuum level, and thus, even with a fixed Fermi level, you can vary the surface potential!

Zz.

d.arbitman
Read carefully. Notice that it says variation in surface potential with APPLIED field! This means that you can vary such potential as you vary the applied field! The applied field effectively changes the vacuum level, and thus, even with a fixed Fermi level, you can vary the surface potential!

Zz.

EDIT:
The applied field increases the vacuum level, which in turn increases the difference between the vacuum level and the Fermi level. In this particular case, does the surface potential represent the same thing that the work function would?

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Staff Emeritus
EDIT:
The applied field increases the vacuum level, which in turn increases the difference between the vacuum level and the Fermi level. In this particular case, does the surface potential represent the same thing that the work function would?

Careful. You can increase or decrease the difference between those two energy levels by changing the polarity of the applied field.

The surface potential becomes the work function for a metal, assuming you know the work function of the "probe metal", i.e. you know the work function of one of the electrode. In a semiconductor, this difference between the vacuum level and the fermi level may also be called the work function, but this is not the same as the photoemission threshold as in metals. This is because the photoemission threshold for a semiconductor is the electron affinity plus the energy gap.

Zz.

d.arbitman
The surface potential becomes the work function for a metal, assuming you know the work function of the "probe metal", i.e. you know the work function of one of the electrode. In a semiconductor, this difference between the vacuum level and the fermi level may also be called the work function, but this is not the same as the photoemission threshold as in metals. This is because the photoemission threshold for a semiconductor is the electron affinity plus the energy gap.

The electron affinity is the difference between the vacuum level and the energy of the conduction band at absolute zero, correct? If so, then I understand what you said above. Thank you.

Careful. You can increase or decrease the difference between those two energy levels by changing the polarity of the applied field.

I hesitated to ask that question because I thought the magnitude of the surface potential would be the same regardless of the polarity.

So back to the section on Poly-Poly Capacitors. If we apply a potential difference, positive for the top plate and grounded on the bottom plate (i.e. the E field is pointing down from top to bottom), would the surface potential of the top plate increase, while the surface potential of the bottom plate remain the same?