[5] 반도체 기초 8-２

So now again the second is a band structure, second topic. So we just talking about the hydrogen. Hydrogen is just one electrons in the ball ring. That's why I do one very simple one electron atom. which means hydrogen. So we just got the conclusion for the potential energy and the radius equation.

How we get to this equation? So here first say what is the energy and what is the radius? N-dependent. n dependent which is PIM is potential energy. So from these numbers, how we got this one? This is from here. How we got the radius from this equation? The simple calculation we got the number here.

So how? The mass is given, charge is given, pi, epsilon is given, and the bank constant is given, only, and dependent. That is, all the thing is given, given, given, and then, and dependent. So, these numbers are purely dependent on the n. This number was purely dependent which means discrete which is discrete number we got.

Okay, so very simple that's why here nucleus here center point here and the n1 whatever here, inner circle one, the second circle and third circle, that radius is changeable purely depends on the n number. Right. The radius. Also energy level. This energy level, this tell you this number, this number. this number exactly here and here and here.

So these all the equation is dependent on Schrodinger equation. So this is a nanoelectron. I'm not going to cover in detail, but only one electron potential that is designed by something, Psi function, which is a quantum number n L m which is a quantum number. So low energy state which is a n number 1 L m. I'm

gonna tell you later what is a quantum number in here. The wave function is given by equals to Bose radius something like that which is discrete number. So next, the token is hydrogen, not a case. This actual case is we name it like x, y, z, the real atomic

We are dealing with something spherical coordinate. Why? You can see not just linear motion, the electron circulating. like a circular motion. So that's why we real case is spherical coordinate. That is why R, Sena, V in electron motion.

But you don't have to know about detail because in nanoelectroning there is dealing with something we do. Schrodinger equation. Schrodinger equation in nanoelectronics. So this is the subject. Maybe later you can access this topic. So in here something that depends on the radius.

Here is a Vr means here is potential energy. Yeah, okay, potential energy. So this is V delimited previously. So electron and hydrogen atom is attracted by the central force, okay, which is a plus in here, minus circulating. It's already directed toward the positive nucleus. So this direction. Therefore, spherical coordinates enter the nucleus to describe a portion of electron.

So potential, energy, electron depends on the unit, R. So R depends on everything. So here, so hydrogen is atom. That is another atom, which is two hydrogen atom is a molecule. Okay. So this separate thing is put together with a molecule. This is a molecule. So this energy level, this energy, something lower energy level, lower potential potential

potential means more stable. This is why make molecules lower energy, lower potential is more stable. So this is a echondra bonding. The one separate atom put together is a molecule which is a lower potential, which is a more stabilized condition.

This is the atomic structure. So we look at the hydrogen atom. The hydrogen atom is the first. of one electron circulating. Periodic elbow means something, which is something, whatever atomic structure, electronic spin, intrinsic angular momentum spin, all kinds of notation here, which is N. quantum number here. Share number based on the n number is KLMN.

Okay, so this is the share notation. Sub share is something. We notating something. L number. Zero means something and one two three four. Depends on the n number. just notating which is also SPD okay as you know like a 2 6 10 14 this is atomic number something 1s is hydrogen is something 1s case is a number 1 0 0

electron spin number something like this Okay, so halium is 2 means 1, right? Lithium is 3, that's why 2 in here. So 2, 1, 2, whatever. So all the things in notation is dependent on the phi and Lm has four number. It's basic simple. What I'm talking about is chemistry.

Chemistry is called activity element, something to determine, something balance or outermost electromagnetic. So this is I just want to mention about this number. L, M and what is this number? Depends on the atom. So we can figure out the already. So as you know, like a periodic table here is a limit 2, 3, 4, whatever.

That is it. Depends 1 here, 2 here. 3, 4, 5, 6, 7, 8, 9, 10, that's near here Whatever. Okay, so all this is a periodic table Something how is involved. So depends on the SPD. L number 1, 2, 3 here located and then here and then 2P located in here because the six electrons one two three four five six so three state okay so this is two here because

s right this is something like that things going on so so on so on so one is here two s which is a energy level Atomic number is increased, energy level, this kind of a shape. So kind of the balance selector. Balance means something of a physical conductor, which is the balance selector. So fill out the force here.

something moving up layer. So this is carrier number. So this is balance. We mainly focusing starting point is balance band. So which is atomic structure as you know it's mentioned about the quantum number and 1, 2, 3, 4 increase and then these numbers we learn about it. What is the energy level? Depends on the quantum number, which is kinetic energy level.

So from the ground state something increase level. As you know which is here something 1s, 2s, 2p, 3s, 3p, whatever that I previously mentioned. So all the electronic structures. This is the atom structure and atomic number. So now back to the something

band structure and the one atomic structure and number one here and two here three four five so radius depends on the purely a n square which means n dependent okay this constant some constant is something we call the bold radius this is your times some quantum number squared is radius. As you know what is the energy level? This is fixed depends on the n numbers and 1 here sorry 1 here 2 3 3 4 whatever

just put the number here we can get energy level. So this is How we got this number, this number, this number. This is the energy level. Pretty much 4, 1, and 2, but getting to the answer. This is the electron energy. So electrons in atoms. which is electrons in solid. There is something stating atom. We're just looking

up to one hydrogen atom. Okay. Which is electrons attractive potential atom. Which is E1, E2. You know the number we got here. Do you know how we got to this number? The vacuum state is the bound state which allows the energy and is separated by a gap. So this is the gap. This energy level is many many states composed of a state.

Energy number, whatever, we got everything here. something constant. So we figure out this number 13.6 and divide by n-square. So n is 1, 2, 3, 4 increase. This energy level number is lower lower. So in here something electronic. One electron is it, which is what?

Move around. Okay. Move around here. So which is a crystallized structure, electrons move around. Okay. Periodic crystal. Which is a periodic crystal, electron moving around. Whatever. which is the I mentioned before, Psi function. Big Psi function, NL, M is a quantum number. Well, principal quantum number, L is the orbital quantum number. The other one is the angular momentum of electron, which is NL.

M number. So whatever. Big Psi NLM. So which is NL. Big Psi number. That is a wave function. Okay. This is a shredding equation that I told you before. So now I'm talking about the silicon. Silicon is atomic number is 14. so 14 electrons okay so very uh very simple very simple uh look up to very carefully okay

now i just separate the two part here so one is same thing same thing one is atom number one for the silicon okay so atom number two to the silicon So two atoms put together like hydrogen. Two atoms are separated by getting closer, getting closer and then bonding each other. One silicon. How is that going to happen? I'm going to tell you right now. So here, so 14. Here.

So which is 1s2, 2 electrons, 2s2, 2s2, 2p6, total 8. So this is 8, sorry, this is 8, that is 2. 2 plus 8 is 10 so here is 10 electron but outside the four left over right so same thing here is a

10 electron to inside okay here is a 8 electron and a 2 electron plus 10 electrons inside then outside the eight elect now four electrons in here Okay, which is the inner one is S. Sorry. Inner one is S, right? Our word is P. So this is what? 3, S, 2, 3, P, 2, right? The same thing. 3, S, 2, 3, P, 2.

So which means these two electrons and these two electrons is a different state. Not same state. This is here and here. One is lower, two is lower, two is higher. So not same state. So our ratio has four electrons. 3s, 2, 2p, one is equal to 3t and separate condition which is like this which is more than 10 atoms. So outer shell has four electrons unequally, this is important, unequally discrete and separate condition.

Separate condition means what? Different state. This is mentioned one state, different state. So once combined the bond, the four electrons equally dissipated. So S, P, L, dissipated by equally. So this is very important. I want to see you next page. Okay. Actually, initially, separate and different location but must combine getting closer getting closer and

these things these things in same state means equally disparate first unequally but later equally disparate. So here, so examining electrons as solid, examining electrons stated, let us examine hydrogen. So back to the hydrogen status.

So one electron halo. So radiate this direction of the probability density function. the probability finding the electron at particular distance from the nucleus. So, nucleus here, where is located the electron? That is the probability function. So, electron is not the clearly expected location of electron, which is

Too many electrons, too small space, people make try to understand, introduced by probability function, which means not clear. So, depending on the time location where the electron can find which location it is. So that's why here something 1s2 state this probability function. You find the most highest point here and something like that.

What is 2s2 which means this is here. n number is 2 here. L number is 0. This is also L number is 0. So whatever is the location. This is the highest probability. Find out the electron, whatever. So proportional to the product of blah, blah, blah. So you don't have to know about the side function. I don't know. The differential volume of the shell around the nucleus.

The probability density function of the lost energy probably is something right here. So 1s state, 2s state is more like this. So this is the most probable distance from the nucleus, which is the same as the Bohr's theory, Bohr's radius. So considering that it is spherically symmetric, because it's all spherically.

So this is all spherically. This is half of the line. So half of the line is like this. Gas-symmetric probability function can conceive the concept of the electron cloud. the electron energy shell, which is on the nucleus then discrete particle orbiting around the nucleus, which is in radical probability density function for next higher spherical symmetric wave function.

So this is the all-wave function and then this is symmetric. okay so it depends on the n number and number and number whatever this is what it is so next which is it energy difference extremely small just so for practical purpose we have something very small so this distance small very extremely small so this look So it looks like a quasi-continuous because it's too close. That's why it's quasi-continuous distribution.

Allowed energy band. This is allowed, not allowed, not allowed. It's allowed, allowed. So the X-axis is more different between two energy states. Okay, we change something like this. So now what is quantum of N1, N2, N3? which is located like this and then also is now here something diagram which is infinity status something you want it to cinema whatever okay so this is

so hydrogen is in diagram So next, which is the formation of energy band? I mentioned this is a separate one, but this one is left and right is the getting closer, getting closer overlap. Okay, which is here something that A, this one shows the vertical property density function

follows the electron energy status of this single Non-interacting because only one single atom. Non-interacting hydrogen atom. But figure B shows something, then probably call for two atoms, which is two atoms getting together, closer, closer, proximity to each other. The wave function of two electron overlap, which is two electrons. will inner act, which means inner act something. This one and this one is getting closer.

What's here? Two electrons, no, no, two statuses here and here, which is colliding, right? Not colliding. Something is getting closer, closer. One is up, one is shifted down. So a little split. That's why the this data sees something split each other okay so which is atom one atom two is getting closer closer and then this state is going to be splitting like this which is more denser or more electron quantum number n3 is more denser and then

closer to the other atoms. So this n number is pretty much 4. If you're getting closer, getting closer, which is colliding first. n3 is first, n2 is next, n1 is third. So which is our most layer is colliding first, then like this. later on. So this is it, which is formation, something energy band. Here. This is the whole band structure, which is a combination of the state. All the states

is getting closer, closer. make band structure. This is what it is. So it's solid in silicon then atom is to compose the band structure of silicon. So if you take a look the first atom and the second atom the silicon was 14 right so 10 of the electron in here 10 of

electrons in here four of electrons in here four of electrons in here But for electrons, two of them is outside, two of them is an inner circle inside. Two of them is an outer side, two of them is an inner side. So S and P, different state. Actually, this is the entry. N3, this is N1, N2, N1, N2.

So those four electrons located in different states here and here and here and here, which is before make molecule. So atom state is different. Different state. One is S and one is P. Here is the atom state S and P different.

This thing is here. So this figure This computational figure makes this band structure. So look it up here. So how many states, how many electrons in here? So here, this is 3s2, right? This is 3s2. This is 3 p2 this is 3 p2 right so four electron to four electron so here so this is in a circle

this is a outer circle this is a inner circle this is a outer circle so how many state and electron here. So this is 3s 3p. This is 3s is 2n state to 2n electron. So 3p is what? 6 states And then how many electrons? Two electrons.

Right? So that was here. Two electrons and two electrons, which is the inner one is this one. outer one is this one. So how many states? This is the sixth state. two states. This is six states and two states. So here and here is clearly different state.

But this is the atom state. This is the atom state. This is a molecule. Okay, so which is two states, different states, when you do, based on the reference point A0 means, which is a combination of a molecule which is

bonding distance. So this is two atoms make a bond here. After passed this point, these different states are going to be combined. So 6n, 2n is 4n, 4n which is same path, same location. This is clearly different state but this is the same. It is 2n state and 2n state. How many electrons?

Tab is zero and tab is 4. So same electron number. Here is 2n and this is 2n. So this is four-end. So electrons come to the down side, then no electrons upper side, which is conduction band here, the balance band here. So, we have a very complicated band structure to make it simplified like this. Okay, balance band and then conduction band. So this area makes simplify this band structure.

So here is how many electron balance? 1, 2, 3, 4. Conduction is nothing in there. So four electrons in here nothing in there. Okay that is it some combination of the Vendor structure. This is the x-direction. This is E means energy. Right so molecular status. So separate atom status make the unequalized status.

When you do after bonding, equalize state, which is this. This is silicon band structure. So this is how combine atom state to molecular state. Atom state, no band structure, but the bonding atom by atom that is making some band structure like this so this is typical band structure of this silicon this is all about it

so now we are not talking about it so one atom is in here one atom, atom number one and atom number two the same atom so which is the same structure same electronic structure so E1 here E2 here E3 here This is an outer layer. Same thing. And even the radius is R1, R2, R2, R3.

The same. So exactly the same state is going to be getting closer. This separate atom getting together which is a make of molecule like a bonding. Bonding means here something about we call crystal structure. So which is a crystal structure. crystal structure. So which is totally separate this electronic system but combined like this.

Okay, this is crystal. So atomic spacing one, two. to Armstrong. Whatever. So this state is overlap here. This state, this state is overlapped here and here, here, then here. Whatever. So whatever. So this is something. Atoms brought up closer to each other which make a form crystal. which is discrete atomic level.

This is discrete atomic level. discrete atomic level start broadening. Broadening means make band here. So all these things separate the state is going to be make broadening which is band. So this state is a state. Change it to bend. Okay. And one more thing is a low line core level.

So let it be. So low line means this side. This is the higher side. Right. The lower side is it. In affected, unaffected. Unaffected means what? Stable. So getting this way is stable. This way is unstable. unstable stable lower is going to be stable higher is unstable so higher

level broaden into this segment to the form depend why the significantly burn in higher level because higher is more state more state here is a that's why state more state means broader than here. So this is filled up here. All the field state is gonna be we call it allowed band. allowed band is state exist. Okay the band gap is no state

state exists. State exists means electron exists. Here is no electron exists. Okay, so which is we just make sure the which is what is different band gap and they allowed band gap. Okay so there is one important thing.

So this is a free state means more than this much energy again getting electron free. Okay that is a free state. This free state in same thing in atomic structure or so more than this energy gonna be free. So this is about some case of the bonding which is energy bending.

So I want to just make sure it's sentence by sentence. I just follow the things we just went through. Silicon is the most important silicon semiconductor material. Okay, it's the most popular material and the atom number is 14. Okay. Two electrons in the first shell. Which is 4. 2 electrons is what? 1 has 2. 8 in second. What is that? 2 has 2. 2p is 6. That's why

2 and 6 is the total A. Here 2 is 2 electrons. It's 4. And the 4 electrons have 4. Don't show what it is. Four electrons. Four electrons is what? 3s2, 3p2. Right? What is the atomic bond? Atomic bond is this one. Right? When you do make the combine, which is a cobalent bonding,

this different state s state and the p state getting to closer and then what same state which means something here is all the things of four electrons here is four electrons getting to the what balance band. So this thing four electron is into the balance band. Since interatomic spacing is small, so order of this is a

quattam nanometer, pretty much small space, electron is influenced by the oil-near-bine nucleus. So except near the surface, the potential energy E_p is periodic function. As you know, the potential is periodic, which is periodicity of the crystal lattice. And discrete state is associated with the isolated atom. Now it's floating into the energy band. So discrete state is floating into the what?

the band, the transition state to the band which is crystallized. Okay, that is a we call the band. Okay, in crystal containing billions of zillions of the atoms and each state has infinitely close each other. Each band is often considered to be continuous. Considered to be continuous which means not exactly continuous.

Looks like continuous. Too much state which looks like continuous. So each shell of the discrete silicon atoms are combined into the band of the state of the silicon crystal. While the lower energy state is corresponding to the inner shell. Lower state is inner shock, highly localized, near the nuclei.

Lower state means what? Lower state energy, which means getting this side. Lower, which means inner state. That's why I correspond to inner shock. okay which is highly localized near the atomic nuclei. So this is a lot of things going on. So this lower state, lower state means closer to nuclei.

only slightly influenced by the presence of a neighboring atom. It does spread slightly on higher energy compared to the higher. So which means higher state. The lower state is slightly influenced. Okay so higher state is severely, highly influenced. Highly inference state. Okay. Higher level is also outer shell. Outer layer.

Okay. Lower state means what? Inner state or inner state. Same. So electrons of two lower energy show are tightly bounded in respect to atoms and the plays a no role in operation in the device. So lower state it means not that much player

Okay, that's why. Inner is very much stable. Okay, that's why not much play. Which means unstable, unstable case is more actively play. The third shell of our silicon is split into the two band which is crystallized silicon.

The lower of this band which is balanced band has a told you 3s2 3p2 getting to the lower means getting to the balance band. Top is a conduction band. So this is a separation. So these four electrons into the lower okay so that's why here the content of four electrons these balance

electron responsible for cobalent bonding so these four electron into the balance band the other one the four electrons so here is four electrons in here but top layer which is a conduction band here is 4 empty. So which is 4 vacant. Empty means vacant. Stead is in the third shell. Third shell means quantum number N3. Silicon formed band crystal. Silicon is called conduction band.

The conduction band is split from the balance band by So this match 1.12 electron volt band gap in room temperature. So the electrons since they are moving slowly spend much of their time in three regions between the atom and the electron cloud.

So attraction between the negative cloud and the positive nuclei, which is something closer. Attraction, right? What holds the crystal together. This is holding energy. So holding energy. Crystal put together. Okay. So now this is the summarized figure.

So what we've been through the old silicon. bonding, band structure. Okay. So this is some kind of a changing point here. Here and here is different. This side, what? Separated atoms, right? This side one born-did-crystal.

Okay? Structure. Okay? So things are gonna be changed here. This side and this side. So look at the first separated. state and electron here first. First here is 3s in the lower, 3p is upper. How many states available? p means 6, okay, s means 2. So it's always 6 states available here, 2 states

available here. So here is this p. region. And this region is unequal. Why? Because here is S and P. Six states available. Two states available. How many electrons? Here is two electrons. here. But here is two electrons here. Y3P2, 3S2. Okay. This is this one. This is this one.

Stage is available but two electrons. two states available, two electrons. Okay, that is unequal. So these two atoms are getting closer, closer, make a bonding here. The certain point is a bond. This much distance. So after that, these two, how many? Two electrons in here. P, two electrons here, S. This electron moving to the bottom.

This is moving electrons to the bottom. So two electrons, two electrons, so four electrons. Okay. So here is zero electrons. This electron? No. This electron moving? No. Yeah, all the things go lower level. So this is zero electron and four electron. And the state is 6 and 2, right? The 6 and 2 total 8. But coming, getting to do bonding each other.

So that is something. 6 and 2, that is 4 and 4, which means equal. Equally. Here is unequally. Okay. So this region is unequal state. This is equal state. Okay. So state number is same. Here is a total of 8. Here is a total of 8. 4 and 4 = 8. Here is a 6 and 2 = 8.

So unequal state changes to equal state. And upper level is zero electron. Here is a 4. Here is a full. So that's why Balance band is full. So this state is full. This state is empty. So we call it up the balance band here. And here is the conduction band. It's zero electron and four electron. The full and partially no no nothing there.

Okay. So we call it up. This is EC means EV means balance band. EG means what? Energy band gain. There is no electron but state is available. Here is a state is available. Okay, so this is the state available means band. This is band. Here is no state. No state means no electrons exist in this region between here and here.

That is the energy band gap. So now I'm going to tell you about this. Now I'm gonna tell you about the metal and semiconductor. So band structure. So first the metal first. So things are gonna be the same thing.

structure means not atom state this is a crystallized structure metal crystal rise structure okay so things this is very much figure is very much familiar okay we've been through all that so first look it up here first and then have a quantum number n equal one n=2, n=3. Okay? Fully fail, fully fail, fully fail. Okay? The outside layer, quantum

number n4 is half-fuel. Half-fuel means very much unstable. Unstable means Electron is hang around, going around very easily. Freedom is very much free. Okay, that's why free means active band. Okay, but here is fully filled, means no empty space, which means very much inner.

means very much stable. So here is something about the state is available. The state is available but half fill to fully fill between those things is what? No state. It means no electronic jesting. Now, this thing is I'm gonna tell you first. I just remind you what is conductivity. is what? N mu q. What is it? Conductivity decided by number of carriers which is electron

mobility and the charge. The charge and mobility is fixed. This is a changeable. So we, I told you, as I told you, like a conductivity control by number of carriers. Okay, so carrier control means conductivity control. Then how we control this side by carrier based on the sum. energy variable number of k are decided by density of a state times Fermi function.

So this is very much tell you about the density of something state here times the Fermi function. The Fermi function is probability function. Probability function is maximum is 1, minimum is 0. So in electronic there is so many billions, zillions of electrons available. So given time, given location, how many electrons are in there? We cannot decide how many electrons

it is. Why? too much electron and too much unstable which means unstable means good conductivity. So people introduce some probability function. How many possibility electrons exist at given time, given location. That's why I introduced some probability function. And then state. I told you what this state means. The only different thing is given. space how many states available. More states means more carriers.

So sooner or later, maybe next week, I'm gonna talk more detail about this. Okay, so just I'm gonna remind you number of carriers is important. This is the state is available fully filled fully filled here one two three so band five here and the five here and one is here so this band structure rotating clockwise like this way okay so this thing like this

rotating this way okay exactly same thing. Then here is something empty spot. This is empty spot too. F is filled here. F is filled here. But something drawing little bit make slightly lean So we choose something smeared out and smeared in here. So this means empty spot is more carrier.

Looks like. So here it is not exactly half-fiel. Something is more empty and more filled up here. This means exactly the same. your textbook something didn't make it confused. I just just make it some collection. Here is the metal. Okay, next one is a semiconductor. Same thing. The band structure as you guys know here, fill it out. Electron quantum number one,

2, 3, 4, whatever. Fully fuel 5. And this is empty. You choose some. Active means when some empty state is present. Not fully fuel. Empty means something. Make empty here. and then feed it up here. So some electrons excited into here to do here is going to be active. Okay, active means unstable. Okay, so the thing is going. So it's a fully

bottom state we call it a balanced band in the conductive band here. So the same thing here exactly this figure rotating like a clockwise. Same thing here. So 6 here, 6 here, 1 here, 1 here. The fully filled up to here. The fully filled here. This is what? Between is forbidden band gap.

This is forbidden band gap too. Okay. So fully filled. Then here is the same thing. This thing, not even a straight line which is smeared out and in something here. So this much space change into here. So there is going to be something make empty here. Make empty means more unstable. More unstable means active.

Okay, so this figure and this figure is the most important. part. But I'm gonna here. So see here is a perfectly filled, right? Perfectly empty. Here is what? Not perfectly. So this part is not matching. Okay. This is empty. White, this is gray. And this is full.

This electron moving to the here. So make empty, partially filled. We should draw it like this. Make empty and partially filled here. So then this figure, this figure is matching up. Your text makes some people confused. That's wrong. So, after that, this figure, this figure here. So this electron how moving this way? Get something supply energy. Energy means electrons excited.

Something moving. Get over value here. So that is gonna be I'm gonna talk about soon later. Okay, more details about the density of state or forming function. So that's it. So this is for semiconductor. You just compare. The third case is F4, semiconductor means fully filled balance band. Empty is a conduction band.

When you try some energy, something excited into top level, partially filled, partially empty here. So then it's something like this that's gonna be conducting. This thing is going on. So now I'm gonna summarize all we went through. So this figure, so many figures. Metal case is a semiconductor case. Everything is filled up.

However, it's not that much. unstable, which is a stable portion here, up to here. And then semiconductor is, which can separate some band gap, conduction band, balanced band. But metal case is no band gap between here and the partial refuel. So all the time, this is free. moving around. This is not free. But after partially empty and partially filled, then active.

That stage is called on. And this is down, it's gonna be off. On and off. Like switching. The main thing is the highest torque to the band partially filled. This is the partial filled. Okay, highest occupied band energy, completely filled. This is the complete refill. Okay, this thing is here. This thing is regarding this one. Okay, we just compare about this matter.

This is the insular semiconductor, highest occupied fluid. completely filled and low temperature. Okay, low temperature means what? No thermal energy. It means no supplying energy from the outside. Okay, usually the semiconductor is very much the low conductivity. That's why it is fully filled. Low conductivity means inactive. Inactive means stable. Metal has very good conductivity.

Conductivity means active and unstable. This is all about it. metal, insulator, and semiconductor portion. So this is a band structure which is comparison metal, insulator, semiconductor.