Fermi Energy and Fermi Level in Semiconductors


by physio
Tags: energy, fermi, semiconductors
physio
physio is offline
#1
Nov12-12, 09:33 AM
P: 55
I have searched a lot on the internet for a simple explanation of these two terms. As I have understood these things (quantum physics), after reading from a lot of sources, is that in an atom there are discrete energy levels and electrons occupy the levels according to the least energy of the levels and no two electrons have the same quantum number i.e. electrons have different states in a particular energy level i.e. they have a unique quantum number, also the Fermi level is the highest energy occupied by electrons in a particular state at 0K. Now when many atoms come together the discrete energy levels form a band of energies such as the valence band. Now, in semiconductors we have the valence band, the conduction band and the forbidden band in the band diagram, so for an intrinsic semiconductor the Fermi level lies in the forbidden gap. How is this possible? What actually is Fermi Energy? What I understood about the Fermi-Dirac statistics is that it gives us the probability of finding an electron in a given band according to the no. of states present in the band and the no. of electrons etc. Am I correct?

Can anyone please explain Fermi Energy and Fermi Level in layman terms?

Thanks!!!
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BackEMF
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#2
Nov13-12, 03:27 AM
P: 53
The Fermi-Dirac distribution gives us the probabilty that, should there exist a quantum state at a particular energy level, we find an electron occupying that state. It is the "probabilty of occupation" of an available energy level. To get the total number of electrons per unit volume, in a given energy range (between [itex] E_{1} [/itex] and [itex] E_{2} [/itex] say), we must integrate the product of the Fermi-Dirac function [itex] f(E) [/itex] with the density of states function [itex] s(E) [/itex], between those energy levels, that is

[tex] n = \int_{E_1}^{E_2} f(E)s(E) .[/tex]

In the forbidden gap, there are no states available, so s(E) is zero, and hence there are no electrons to be found.

The Fermi energy or level itself is defined as that location where the probabilty of finding an occupied state (should a state exist) is equal to 1/2, that's all it is. At 0 K all states below the Fermi level are filled, and all above are empty. At any other temperature, you find some electrons just below the Fermi level have gained enough thermal energy to rise above it by a small amount, but always, at the Fermi level, the probability is 1/2.
physio
physio is offline
#3
Nov13-12, 04:34 AM
P: 55
I didn't quite understand your second point which says that
The Fermi energy or level itself is defined as that location where the probabilty of finding an occupied state (should a state exist) is equal to 1/2, that's all it is.
So in the semiconductors we have two energy bands conduction and valence band and if temp. increases the Fermi Level should increase, is that correct?

Is the Fermi Energy the average energy of an ensemble of electrons? What I understood is that the electrons don't have enough energy to jump to a particular state and hence they remain in a particular state and if they are provided with energy corresponding to (Ec-Ef) then they can occupy the conduction band and the Fermi Level will then again shift upwards towards the conduction band. Is this correct?

Thanks for your reply!!

BackEMF
BackEMF is offline
#4
Nov22-12, 08:03 AM
P: 53

Fermi Energy and Fermi Level in Semiconductors


Quote Quote by physio View Post
I didn't quite understand your second point which says that

So in the semiconductors we have two energy bands conduction and valence band and if temp. increases the Fermi Level should increase, is that correct?
Yep, that's it - as the energy of the electrons increases the probability of finding some at higher energy also increases, and in particular the energy level at which the probabilty of find an electron is 0.5 i.e. the Fermi level, will rise up from somewhere near the middle of the bandgap in an intrinsic (undoped) semiconductor, to somewhere nearer the conduction band.

Quote Quote by physio View Post
Is the Fermi Energy the average energy of an ensemble of electrons? What I understood is that the electrons don't have enough energy to jump to a particular state and hence they remain in a particular state and if they are provided with energy corresponding to (Ec-Ef) then they can occupy the conduction band and the Fermi Level will then again shift upwards towards the conduction band. Is this correct?
The Fermi energy is a distinct concept from the Fermi level. I suggested otherwise in my previous post because they are often used interchangably by semiconductor physisists.

The Fermi energy is the energy of the highest occupied electron when the system is in the ground state. So it won't change with temperature (since as T increases the systems will leave the ground state). The Fermi energy is not the average energy, but it is related to the average energy.

Electrons that occupy the valence band can jump up to the conduction band provided they are supplied with the energy difference, which is Ec - Ev.
kpw1
kpw1 is offline
#5
Nov24-12, 02:23 PM
P: 5
I'm struggling with the same problem of distinguishing Fermi level from Fermi energy.

Quote Quote by BackEMF View Post
At 0 K all states below the Fermi level are filled, and all above are empty.
So when the temperature is at 0 K, where is the Fermi level in relation to the valence band and conduction band edges? Is the Fermi level between them? Or is the Fermi level at or below the valence band edge?

Thanks in advance!
imroze99
imroze99 is offline
#6
Dec23-12, 10:37 AM
P: 10
We generally say that the Fermi energy level is the highest occupied energy level at 0K. Fermi level is compared to be surface of the electron sea in a material. Thus it is considered as the top most occupied energy level by the electrons.
This is one definition and if we apply concept of Fermi level in semiconductors we say it is an energy level between the forbidden band gap(where no electron exists as per the definition) where only the probability is 50%. Thus accordingly Fermi level shouldn't contain any electron.But according to the above para Fermi Level should be the top most occupied energy Level at 0K like Ev(Valence band edge). And obviously room temp is above 0K. But this contradicts the statement in the above para.
Darwin123
Darwin123 is offline
#7
Dec24-12, 04:15 PM
P: 741
Quote Quote by physio View Post
I have searched a lot on the internet for a simple explanation of these two terms. As I have understood these things (quantum physics), after reading from a lot of sources, is that in an atom there are discrete energy levels and electrons occupy the levels according to the least energy of the levels and no two electrons have the same quantum number i.e. electrons have different states in a particular energy level i.e. they have a unique quantum number, also the Fermi level is the highest energy occupied by electrons in a particular state at 0K. Now when many atoms come together the discrete energy levels form a band of energies such as the valence band. Now, in semiconductors we have the valence band, the conduction band and the forbidden band in the band diagram, so for an intrinsic semiconductor the Fermi level lies in the forbidden gap. How is this possible? What actually is Fermi Energy? What I understood about the Fermi-Dirac statistics is that it gives us the probability of finding an electron in a given band according to the no. of states present in the band and the no. of electrons etc. Am I correct?

Can anyone please explain Fermi Energy and Fermi Level in layman terms?
The probability of finding an electron at a given energy level in thermal equilibrium is proportional to the Fermi Dirac function times the density of states at that energy level.

The density of states of an extrinsic semiconductor in the forbidden gap is zero. No matter how large the value of the Fermi Dirac function, the probability of finding an electron in the forbidden gap is zero for an extrinsic crystal in complete thermal equilibrium.

I think the closest analog to Fermi level in everyday life is the sea level. The sea level describes a state where water on the surface of the earth flows. If a region is not at sea level, then the water in that region is not in "marine" equilibrium.

Consider water above and below sea level. The water above sea level is in a type of "forbidden zone." The water on land above sea level will tend to flow toward the sea. Most rivers on continents are above sea level, so they tend to flow toward the sea. However, there are land locked regions that are below sea level. Water will tend to flow into these regions unless blocked by something. The region around the Dead Sea is below sea level. The only reason it is a desert is because land blocks the flow of water. Much of current Holland is currently below sea level. The area used to be marsh land. However, dams were built to keep the water out. So now these marshes are dry land where people live.

The water in rivers and lakes are usually above sea level. You can consider the marshes and lakes to be in a "forbidden zone" for marine water. However, there are very large lakes below sea level such as the Dead Sea.

There is no water at sea level in the area of the Dead Sea. All the water of the Dead Sea is below sea level. Water is pouring into the Dead Sea from the Mediterranean Sea. Water that pours in from the Mediterranean Sea evaporates leaving the salt.

The density of water molecules in liquid water can be considered an analog to the density of states of electrons. Liquid water has a fixed density when it is present. Liquid water has almost the same density in the deepest mines then on a mountain. Liquid water has the same density even at sea level. However, there are regions of the world where there is no liquid water at sea level.

The "sea level" is still an important concept in hydrology. However, there isn't always water at "sea level".

The Fermi level is defined in terms of the Fermi Dirac function, not the density of states. The Fermi level can be considered the energy where the electrons would tend to flow if there was a positive density of states at that energy. However, where there is a zero density of states there are no electrons independent of Fermi level.
imroze99
imroze99 is offline
#8
Dec27-12, 04:35 AM
P: 10
I am very much satisfied with the definition of Fermi Level But again unable to apply this analogy. If we have considered Fermi Level as electron sea level but not containing electron. Then while describing carrier flow across a p-n junction we say that due to Fermi level difference the electrons flow from n to p region until the both Fermi levels align themselves. If Fermi Level doesn't contain any electron then shouldn't we align the Valence Band of the two regions. How would we explain this scenario considering the water level analogy?
Darwin123
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#9
Dec27-12, 11:43 AM
P: 741
Quote Quote by imroze99 View Post
I am very much satisfied with the definition of Fermi Level But again unable to apply this analogy. If we have considered Fermi Level as electron sea level but not containing electron. Then while describing carrier flow across a p-n junction we say that due to Fermi level difference the electrons flow from n to p region until the both Fermi levels align themselves. If Fermi Level doesn't contain any electron then shouldn't we align the Valence Band of the two regions. How would we explain this scenario considering the water level analogy?
The Fermi level of a specific crystal in thermal equilibrium would be like the sea level of a specific sea. The individual crystal would be like the region around a land locked sea. There could one sea level for the Atlantic, one sea level for the Pacific, one sea level for the Mediterranean, one sea level for the dead sea, and one sea level for each of the Great Lakes in the United States.

Some of the sea levels would be the same and some not. Currently, the Dead Sea has a "sea level" far below the other sea levels. The Great Lakes have a sea level which are above the other sea levels.

A p-n junction would be like a channel connecting two seas. There is a connection between the Atlantic and Pacific, so I suppose they have the same "sea level". I think the sea level of the Atlantic and Pacific is by default the sea level that geographers talk about. Water flows from the Atlantic flows freely to the Pacific and vica versa. The "channels" include the southern tips of Africa and South America. Other channels include the Arctic seas. Their sea levels tend to align due to gravity. This is a type of equilibrium, much like the equilibrium in a single crystal.

Note that the solid interface of the earth does not align even if there are "channels". The top of a continent does not align with the ocean bottom no matter how much time has passed. Rock, like the valence band, does not align easily.


Currently, the Mediterranean has a higher "sea level" then the "Dead Sea." water from the Mediterranean is slowly pouring into the Dead Sea. If there was no evaporation, the two sea levels would settle down to the same level. However, rapid evaporation makes the "Dead Sea level" to be lower then the Mediterranean "sea level". That is one reason that the Dead Sea has gotten so salty. This would be analogous to a crystal which is not in thermal equilibrium.


The valence band of a solid at thermal equilibrium would be analogous to the geodesic solid-fluid interface of the earth. On dry land continent, the dry land surface would be the solid phase surface of the earth. In mid ocean, the ocean bottom would be the dry land surface of the earth.

If a pn-junction connects two regions of crystal, the electrons from the n-doped region pour into the p-doped regions. If the biasing voltage doesn't change, eventually the Fermi levels of the two regions align.

Cautiously carrying the analogy a bit further:

Suppose the two seas in contact have different salinities. Again take the Dead Sea, which is much saltier then the Mediterranean. There is a salinity gradient across the channels that connect the Dead Sea to the Mediterraneon. The salinity gradient sould be similar to the electric field that goes across a pn junction.

In a transistor, there is an electric field across the junction due to the difference in dopant. Once the two Fermi levels are aligned, there is still that electric field.
Darwin123
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#10
Dec27-12, 01:10 PM
P: 741
Quote Quote by imroze99 View Post
We generally say that the Fermi energy level is the highest occupied energy level at 0K. Fermi level is compared to be surface of the electron sea in a material. Thus it is considered as the top most occupied energy level by the electrons.
This is one definition and if we apply concept of Fermi level in semiconductors we say it is an energy level between the forbidden band gap(where no electron exists as per the definition) where only the probability is 50%. Thus accordingly Fermi level shouldn't contain any electron.But according to the above para Fermi Level should be the top most occupied energy Level at 0K like Ev(Valence band edge). And obviously room temp is above 0K. But this contradicts the statement in the above para.
You keep on asking how there can be no electrons at the Fermi level. I tried to offer an analogy. You know that one must be careful with analogies. However, analogies can be useful within limits.

I presented the analogy of the Fermi level in solid state physics to sea level in geography. However, the sea level is defined in terms of the geodesic for the water surface of a sea. I just thought of an extension to this analogy.

Think of water droplet as conduction electrons and air bubble in water. As before, the Fermi level of a sea is the boundary between the liquid water in a sea and the gases in the atmosphere.

There is sea spray in the atmosphere and air bubbles beneath the surface. The presence of sea spray and the presence of air bubbles does not affect the sea level. The sea level consists of an unbroken continuum of water.

Gravity is analogous to an electric field. The effective mass is analogous to electric charge. The effective mass of an air bubble is negative due to buoyancy.

There may be water droplets above the surface of the ocean. They are pulled downward. There may be air bubbles under the surface of a sea. They are subject to buoyant forces and gravity. The total effect is that the air bubble are pushed upward by the buoyant forces. The bubbles effectively have a negative charge!

An intrinsic semiconductor crystal is like a sea with no air bubbles below the surface and no water droplets above the surface. A metal with conduction electrons is like a sea with no air bubbles but lots of sea spray far above the surface of the sea.

The forbidden gap is like a hot dry boundary layer of air that evaporates water droplets that evaporates water droplets that are not high enough. The hot layer of air can also prevent the formation of air bubbles on the surface. The visible part of clouds is made of water droplets. Therefore, one can think of clouds like conduction band electrons.

A peaceful day at sea is like an n-type metal (e.g., silver). If you are out on the sea, and observe clouds in the sky, then you can think of the clouds as conduction electrons. The droplets are falling toward the earth, but evaporate before they reach the ground. New droplets form. The convection cell between sea level and the tropopause (where clouds are visible) is the forbidden gap. Because the sea is peaceful, there are no air bubbles below or on the surface.

You can make up your own analogy concerning p-type metals (e.g., aluminum). It has to have lots of air bubbles below the surface and no sea spray or clouds above the surface.

Much semiconductor physics can be mapped onto this geographical analogy. Sea spray can then be analogous to n-type impurities. Bubbles near the surface can be considered like p-type impurities. Clouds are like free conduction band electrons. Bubbles trapped on the sea bottom (by vegetation, etc.) can be considered holes. A channel between on sea and another is like a p-n junction.

Much of this was discussed in another post. However, let me repeat with additons based on these new analogies.


The Mediterranean can be considered like an n-doped crystal. The Dead sea is like a p-doped crystal. Because of channels between them, water is flowing from the Mediterranean to the Dead Sea. However, salt is being left in the Dead Sea by evaporation. The salinity gradient is like an electric field. Some salt is being forced by the gradient to move opposite the flow of water into the Mediterranean.

I repeat the caution about analogies. This analogy is not meant to be a replacement for quantum mechanics.
imroze99
imroze99 is offline
#11
Dec31-12, 03:15 AM
P: 10
The top of the sea level is analogous to valence band or Fermi Level. Let me clear the matter if we say that Fermi level is just an energy level at which probability is 0.5 but having zero electron density being in the forbidden gap. But it is only used as a reference energy level and actually the top of sea is the Valence band and the electrons rise to the conduction band like clouds. While talking about p-n junction we generally say due to Fermi Level difference electrons flow from n to p like the sea level concentration or salinity gradient and the two quasi Fermi Level align themselves to a single Fermi Level of the device. But to be more precise we should say that the electrons flow due to difference in valence band energy level of the two regions and to make this simpler we use concept Fermi Level alignment. Am I Right? Does explaining something based on analogy is correct ? Can we know the very actual mechanism?
Darwin123
Darwin123 is offline
#12
Dec31-12, 04:21 PM
P: 741
[/QUOTE]The top of the sea level is analogous to valence band or Fermi Level. Let me clear the matter if we say that Fermi level is just an energy level at which probability is 0.5 but having zero electron density being in the forbidden gap. But it is only used as a reference energy level and actually the top of sea is the Valence band and the electrons rise to the conduction band like clouds. While talking about p-n junction we generally say due to Fermi Level difference electrons flow from n to p like the sea level concentration or salinity gradient and the two quasi Fermi Level align themselves to a single Fermi Level of the device. But to be more precise we should say that the electrons flow due to difference in valence band energy level of the two regions and to make this simpler we use concept Fermi Level alignment. Am I Right?[/QUOTE]
Not really.

The main problem is that the electrons will stop flowing when the Fermi levels are aligned. Thus, the alignment of the valence bands doesn't determine whether or not the electrons flow. Regardless of the alignment of the valence bands, the time will come when the electrons stop flowing.

The fact that you are at the edge of a cliff doesn't ensure that a wind is blowing. The fact that the rock has a discontinuity in height does not tell you whether or not the air will flow from the low to the high side of the cliff, or vica versa.

When there is a mountain, and there is a wind, clouds will stay the same height as they are blown along. In fact, the mountain may intercept the cloud. A driver going over the mountain will observe that the top of the mountain has fog. However, the fact there is a mountain is not sufficient to predict the presence of wind.

[/QUOTE]Does explaining something based on analogy is correct ? Can we know the very actual mechanism?[/QUOTE]
I did not propose the analogy as an explanation. I was presenting the analogy as an aid to understanding the physics of materials.

Explaining by analogy between systems is not correct. Analogy can only help understanding if theories describing the two systems are similar in certain ways.

Analogy can help a person understand an explanation that is already there. Analogy can also serve as a short cut to doing math, providing certain mathematical similarities in the two theories.

I thought that you wanted an analogy. In fact, you presented your own analogy. I was just tuning your analogy up a bit for accuracy.

With me, the analogy worked the other way. I am a solid state physicist who had to work with meteorologists for a while. I had trouble with such concepts as "what keeps a cloud from falling." Although I could read the explanations and even the mathematics, I had difficulty visualizing what happens in the atmosphere.

It helped me through some of the mathematics to say to myself, "the troposphere is like the forbidden gap in a semiconductor crystal". While this isn't actually an explanation of what keeps the cloud from falling, it helped me access some of the expertise that I acquired in solid state for application to meteorological problems. Very often, I remembered the solution to a certain equation only because I had worked on an analogous semiconductor problem.

Although I really like analogies, I am also very aware of the pitfalls in analogy. So it is only fair to warn you. Analogy is like any tool. The most important thing to know about any tool is safety.

At some point, you have to look at the problem straight on without analogy. However, analogy can help you get to that point.

The actual dynamics of a semiconductor has to do with many body quantum mechanics and Schroedingers equation. The force that makes electrons flow across a pn junction are similar to the forces that cause chemical bonding.

Now there is another analogy! If you want me to explain what happens in a semiconductor using the chemical bonding analogy, then we are in business. I know chemistry very well. However, I know even this analogy has limits.
imroze99
imroze99 is offline
#13
Dec31-12, 08:34 PM
P: 10
With our discussion I have come across the following points regarding Fermi level
*Fermi level is the just an energy level where the value of Fermi Dirac function comes to be .5
*Fermi level is in forbidden gap where density of states is zero therefore no electron will exist at FL.
*When analysed by sea level analogy the top of the sea level is valence band and the electron flow should take place due to concentration gradient of Valence electrons.
Are these points correct?
Considering the above points I wanted to know that what is so special about the Fermi Level that in p-n junction the electron movement flow takes place till the alignment of their respective quasi Fermi level.
imroze99
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#14
Dec31-12, 08:36 PM
P: 10
Of course I want to study this concept with more precise analogy.
Darwin123
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#15
Jan1-13, 04:38 AM
P: 741
Quote Quote by imroze99 View Post
With our discussion I have come across the following points regarding Fermi level
*Fermi level is the just an energy level where the value of Fermi Dirac function comes to be .5
*Fermi level is in forbidden gap where density of states is zero therefore no electron will exist at FL.
This last statement is wrong. Or rather, it isn't always true. It is true in the case of an insulator, but false in the case of a metal.

The Fermi level is not always in a forbidden gap. The Fermi level is only in a forbidden gap in the case of an insulator. In the case of a metal, the Fermi level is never in a forbidden gap. One special case of insulator is high resistivity semiconductors.

In the case of an n-type metal (e.g., silver, copper) , the Fermi level is above the bottom of the conduction band. In that case, there are electrons in the conduction band. In the case of a p-type metal (e.g., aluminum), the Fermi level is below the bottom of the conduction band. In that case, there are holes in the valence band. In the case of an insulator (e.g., sulfur), the Fermi level is right between the two bands. There are no free electrons or holes in an insulator.

Semiconductors are not "semi" in having an in-between conductivity. Semiconductors are "semi" because their conductivity is easier to change then true metals or true insulators. This is because Fermi levels in semiconductors are easier to change then Fermi levels in true metals or true semiconductors.

Quote Quote by imroze99 View Post
*When analysed by sea level analogy the top of the sea level is valence band and the electron flow should take place due to concentration gradient of Valence electrons.
This analogy is wronger than other analogies. All analogies by definition are wrong, but many of the facilitate analysis of more precise theories. Your analogy does not facilitate the analysis.


I don't think this analogy as stated facilitates accurate analysis because the valence bands don't align. The Fermi levels align, not the valence bands. So your analogy, as stated, leads to incorrect predictions.

I modified your analogy in a previous post. I said that the valence band is more closely analogous to the sea bottom. Perhaps you can address this point.

In my sea level analogy, the surface of the sea is the Fermi level. The bottom of the sea is the valence band. The tropopause is the conduction band.

The conduction band and the valence band are defined in terms of electron density of states, not electron occupancy. The conduction band can be empty of electrons and still exist. The valence band can be filled of electrons and still exist. After one dopes the crystal, the conduction band can have electrons and the valence band can have holes. Obviously, the still exist.

The forbidden gap still exists whether or not the bands are full or empty. The energy gap varies only a little with doping. Adding electrons to the conduction band can lower the energy gap by a small fractional amount, but it doesn't change it. The energy gap is mostly determined by the lattice and types of atom that make up the lattice. Occupancy doesn't affect the energy gap very much.

Because the conduction band is defined in terms of density of states, the conduction band doesn't align with other conduction bands. Because the valence band is defined in terms of density of states, the valence band doesn't align with other valence bands. The Fermi level aligns with other Fermi levels.

My suggested analogy was that the Fermi level of a region of crystal is like the surface level of a sea, lake or river. The valence band is like the bottom of a sea, lake, or river.

Water flows from a high lake level to a low sea level when the seas are connected. If the system reaches equilibrium, then the lake and sea levels are aligned.

The bottom of the sea, lake, or river doesn't move no matter how much water is being moved. Well, there is a little erosion. However, the bottom of the sea, lake or river doesn't move.

If because of rain and evaporation there is no equilibrium, then water can pour through the river lake indefinitely. The lake and sea levels can remain constant. If the lake runs out of water, then the lake will be dry.

Equilibrium comes about when the sea has the same level as the lake. There may or may not be water in the lake when equilibrium is reached. However, the new level of both sea and lake are the same when there is equilibrium.

When the lake has dried up, the lake level will be the same as the sea level. However, the sea level is below the bottom of the lake. If enough water has flowed in the sea, there may not even be an aquifer below the bottom of the lake. Even though there is no water in the lake, the lake level is well defined. It is the same as the sea level.

The point is that the local sea level is not defined by the surface of water at that location. It is defined as the level to which the water would go if it were allowed to move.

Quote Quote by imroze99 View Post
Are these points correct?
Considering the above points I wanted to know that what is so special about the Fermi Level that in p-n junction the electron movement flow takes place till the alignment of their respective quasi Fermi level.
I said that these points were wrong in a previous post. I repeat my argument in this post. Perhaps you could tell me how I am incorrect.


What makes the Fermi level special is the Pauli exclusion principle and thermodynamics. Two electrons can not have the same quantum numbers by the Pauli exclusion principle. The crystal has a minimum energy according to thermodynamics. The electrons lose energy until they reach a minimum energy allowed by the Pauli exclusion principle and thermodynamics.
imroze99
imroze99 is offline
#16
Jan4-13, 07:59 PM
P: 10
How can we relate the surface of the sea level with the Fermi Level knowing that surface of sea Level contains water but the Fermi Level lies in the forbidden gap at which electron density is zero?
imroze99
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#17
Jan13-13, 10:47 PM
P: 10
Sir you haven't replied as the analogies pertaining to Fermi Level are bit confusing so I want to crystal clear my concept or you can give me some reference.
Kholdstare
Kholdstare is offline
#18
Jan15-13, 08:22 AM
P: 390
Quote Quote by imroze99 View Post
How can we relate the surface of the sea level with the Fermi Level knowing that surface of sea Level contains water but the Fermi Level lies in the forbidden gap at which electron density is zero?
You're misunderstanding one thing. Fermi level is not an energy level, its just a reference line. There's no special electron orbital at Fermi level ready to hold two electrons. At 0K all electrons must occupy orbitals below this line. It does not mean they can occupy everywhere starting from Fermi level.

The actual occupation of electrons at a specific energy is determined by whether there is an electron orbital present in which electron assumes that energy. It is determined by solving Schrodinger's equation, not by looking at Fermi energy.


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