# Continuity vacuum level at interface

## Main Question or Discussion Point

Hi,

I am studying the Metal-Semiconductor junction. I cannot find any clear explanation of the continuity of the vacuum level at the interface. Can someone help me?

My reference book provides me with the following:

When two different materials are brought into contact, they must share the same free electron level at the interface, i.e. the free electron level is continuous from one material to the next. This is because at the interface of two materials, an electron that is free from the crystal field of one material is also free from the crystal field of the other material.

This explanation doesn't convince me because at the interface the bands are not continuos. So speaking of " the energy of an electron at the interface" doesn't make any sense for me...

Thank you

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mfb
Mentor
Think of the metal/semiconductor/vacuum border in equilibrium: if you move an electron from the metal to the vacuum, then to the semiconductor and back to the metal the net energy needed should be zero, otherwise you violate energy conservation. In equilibrium there is no current flow between metal and semiconductor so there the electron doesn't need energy for the transition. The other two transitions have to have the same energy.

The equality of the free energy of electrons at both sides of the junction is a consequence of the equilibrium condition. An analog for this is the equality of temperature under equilibrium when bringing two bodies into contact.

I cannot find any clear explanation of the continuity of the vacuum level at the interface. Can someone help me?
What is not clear to me is that you are asking about electron free energy but you formulated the title of your thread and started your post by mentioning the continuity of "vacuum level".

What is not clear to me is that you are asking about electron free energy but you formulated the title of your thread and started your post by mentioning the continuity of "vacuum level".
The discontinuity of the bands at the interface is a consequence of the continuity of the vacuum level. In fact, for each of the materials electron affinity (e.a.) is inherently constant; since, in general, the metal and the semiconductor have different electron affinities, if the vacuum level is continuos then bands must be discontinuous.

Think of the metal/semiconductor/vacuum border in equilibrium: if you move an electron from the metal to the vacuum, then to the semiconductor and back to the metal the net energy needed should be zero, otherwise you violate energy conservation. 1) In equilibrium there is no current flow between metal and semiconductor so there the electron doesn't need energy for the transition. 2)The other two transitions have to have the same energy.
This one seems a wonderful explanation to me, but I do not understand the statements in bold and I am going to explain why:
1) Consider a transition semiconductor-metal. Are you saying that, since no external forces are applied to the device, and no dissipative force come into play, the total energy of the electron before the transistion equals the one after it? Are you?
2) The transitions metal-vacuum and semiconductor-vacuum imply different amounts of energy, ruled by the electron affinity of the material. Could you try to clarify this point to me?

In order to make the discussion easier I post the band diagram.

Thank you both of you.

ZapperZ
Staff Emeritus
There is something puzzling in this whole thread, and it could be simply a matter of being sloppy with the terminology. Let's start from the very beginning with the quote from the book:

When two different materials are brought into contact, they must share the same free electron level at the interface, i.e. the free electron level is continuous from one material to the next. This is because at the interface of two materials, an electron that is free from the crystal field of one material is also free from the crystal field of the other material.
First of all, there is continuity everywhere! There may not be continuity in the first derivative of a particular band, but that band is, nevertheless, continuous!

Secondly, the quote mentions nothing about "vacuum level". What it is referring to is that when two different materials are brought in contact, the Fermi level, or Fermi energy will be the same for both metals, due to thermal equilibrium (the same as bringing two objects of different temperature together - at thermal equilibrium, they will have the same temperature).

So now that we have gotten those straighten out, what exactly is the problem here?

Zz.

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mfb
Mentor
1) Consider a transition semiconductor-metal. Are you saying that, since no external forces are applied to the device, and no dissipative force come into play, the total energy of the electron before the transistion equals the one after it? Are you?
Electrons move from one material to the other if this is energetically favorable. As a result, the Fermi levels get adjusted.
2) The transitions metal-vacuum and semiconductor-vacuum imply different amounts of energy, ruled by the electron affinity of the material. Could you try to clarify this point to me?
Well, the Fermi energy can be empty (so you need more energy to emit an actual electron), but if there would be electrons, the energy would be the same.

mfb: I need some time to think about it. I hope we can continue the conversation tomorrow. Thank you.

First of all, there is continuity everywhere! There may not be continuity in the first derivative of a particular band, but that band is, nevertheless, continuous!

Zz.
The conduction band seem to be discontinuous at the interface (please, refer to the image I have posted). Actually the bands of the metal are not showed, so I guess the conduction band is somewhere below the Fermi level. Where is the mistake?

Secondly, the quote mentions nothing about "vacuum level". What it is referring to is that when two different materials are brought in contact, the Fermi level, or Fermi energy will be the same for both metals, due to thermal equilibrium (the same as bringing two objects of different temperature together - at thermal equilibrium, they will have the same temperature).
My quote mentions :... they must share the same free electron level at the interface, i.e. the free electron level is continuous from one material to the next...
So it clearly refers to the vacuum level. I do not get your point here.

Ps: English is not my mother toungue. If there is any problem because something is no clear, please tell and I will try to express myself differently.

Thank you

ZapperZ
Staff Emeritus
Free electron level IS NOT the vacuum level. You need to go back and figure out the definition of Fermi level and vacuum level. The top-most occupied level of a metal is the Fermi level, and this resides within the conduction band of the metal. What you don't see is the valence band of the metal, because this is usually at a deeper energy.

The vacuum level is the energy above which the electrons have sufficient energy to escape the metal. This is NOT the free election level as drawn in that figure!

Zz.

Vacuum level is the energy of a still electron in the vacuum, and it is taken as reference. Free electron energy is the energy required to set an electron free from the lattice, to free the electron in the vacuum.
In the image I have posted, vacuum level is taken as reference and is drawn in the top part of the figure. Do we agree about what I wrote till now?

mfb: Maybe it is better for me not to address two conversations at the same time because I am getting a bit confused. I am going to clarify these point to myself and then, if you are still available, I will answer to you in detail.

Thank you

ZapperZ
Staff Emeritus
Vacuum level is the energy of a still electron in the vacuum, and it is taken as reference. Free electron energy is the energy required to set an electron free from the lattice, to free the electron in the vacuum.
In the image I have posted, vacuum level is taken as reference and is drawn in the top part of the figure. Do we agree about what I wrote till now?
No, because the "vacuum level" changes! You can't use something as a "reference" when it changes when going across the metal and semiconductor.

What doesn't change is the Fermi level, and this is what we use most often as the reference! In fact, thermodynamics require that the Fermi level of both material line up at equilibrium.

I've never heard of this "free electron energy" the way you have defined it. I've heard of photoemission threshold, work function, etc. And I specialize in photoemission phenomenon and photocathodes.

I still do not understand the problem that you are having with the quote you cited.

Zz.

Take a look at the image. The highest level in the figure, called "vacuum level", is indeed the vacuum level according to the definition you wrote in post #8?

ZapperZ
Staff Emeritus