[5] 반도체 기초 8-1

자, 지금까지 carrier, electron, 호홀로 되어 있다. so current가 어떻게 흘리나 So we'll learn the band. Band is important.

"Modern" is a word"finger print" 모든 material은 band structure 트래요. So all kinds of material have different band structure. all kinds of different material. Why is it different? Because atomic structure is different.

atom as a different band structure. So now I'm going to talk about how band structure created. How band structure created. This structure is the most fundamental theory for electronic structure.

characteristic or behavior depends on the band. That's why the band is important. And then what is a band? Band is a whole state. Band structure. is composed of state. So then what is state means? Stays place electron

sit well located. Okay, so electron sit well located place is state. So state is a component of the band structure. The state, you should make sure what is the band, what is the state means. So now I'm going to talk about the state and the band structure. What is the relation?

So what we want to be? How composed of the state? So first I want to take a look at the silicon. It's the most popular material. Okay, most popular material. Atomic number is 14. Okay, so as I told you before, so first, the inner circle is 1, quantum number n = 1, n = 2,

third is n = 3. n = 1 means 1s2. 2 means 2s2, 2p6. total a here. S2 3 P2. That is a silicon atomic structure. So this is one atom and an atom put together.

Okay, so this is all composition of silicon crystal structure. So one of which The two atoms are here. Most outer layer is 1,2,3,4 in here. electron is here. Red. One, two, three, four. That's why total eight electrons located.

How about the inner one? Inside here and here is it. You don't need that. You don't need to even think about it. Why? Because all structure is the combination. What structuralized and contribution of the bonding is only outside electron not inside. I'm going to talk about more detail later. Okay how we compose. How about here is the copper. Copper is what?

quantum number N = 1, 2, 3, 4. So 1 is what? 2 electrons. 2 is 2 plus 8 is 10. How about the N3? number is one. You figure out what it is something so most our electron is one is

one. Very this is electron is stable or unstable very much unstable. Okay so very easy to going around so which means free electron like a free electron. So that's why the copper is a very good conductor. Very good conductor. So this electron is contributing transfer energy by electron.

which means a column. So this is a column. So I just briefly introduce some of the atom. This bonding is a silicon case. So this is a crystallized structure. So this case is what? all bonding together. So fully occupied. So free electron?

No. Why? Because all the things are bonding each other. Okay. So this is it. So this structure is not conducting. So next, I'm gonna introduce electrons, same as vehicles. a nucleus here is circulating by something the path, specific path.

So this vehicle is circulating specific line. Okay, not just hang around any other line. Okay. specific location. Same as electron. Electron is circling around specific location. So that I'm gonna talk about with you some theory I want to introduce. Okay so most fundamental is Bohr's model. Bohr's model which is electron

revolving around the nucleus circular path which means specific circular path. Like this. Electron and vehicle. So, what is the model? Then mass of the nucleus is something like this much weight, okay? Which is this much times that of electrons.

So nucleus is much much heavier than electrons. So nucleus is just sit there, then electrons hang around, circling around. the nucleus is considered fixed space. Okay? This nucleus is always fixed. So now the coulombic force. What is the coulombic force? coulombic force is 2 charge. Q1, Q2. Q1, Q2 times pi

4 pi epsilon r square. So that is false for two charge. There is dispersed this much radius something. So this is the notation by same charge q. One is plus one is minus. So that's why it's minus here. So r is the distance between two charges. Epsilon is this one. Permanent possibility. It's free space because there is only free space between nucleus and electron.

So q1 is equal to something like that. One charge plus q. So this is the force. Coulombic force. This is what about it? The expression for the attractive. Centrifugal force between two particles. So this is the force again. That is something. something del, some kind of derivative which is this one, same as the fifth sum, based on the radius.

So gradient is taken in the r direction. So this is the potential energy of electron at the position of r, which is the potential energy. D is potential equal. That is energy derivative of the same as the force dr. This is something about D. So energy is a force something related.

Okay. So dr and dr is the same. You have to invest that. So now, the autonomic force is the infinity distance is zero. Okay. That's why infinity distance means something r is infinity. That is what? Something 2,4 then gonna be 4,6,0. Right? Reference point.

If we 2,4 is gonna be almost zero. Getting to the zero. Define potential energy at infinity is a vacuum level. so this is the vacuum level. E vacuum. right? so this is energy required free electron from the influencer of the nucleus. so the nucleus here

electron circling around based on the nucleus. So moving the electron infinitely far away from it. So electron effect from the nucleus, right? If R is too far, there's gonna be no effect. No force on it. So this is gonna be free electron. How do we? Okay? So an electron infinity for from the nucleus it is now free electron.

We call it a free electron and we're back. Okay, so no force interaction from the nucleus. R is, case is infinity. So R is infinity is almost vacuum level. So E_p is an electron potential energy and some distance R from the nucleus. So which is integrating both sides rearranged. So now something potential. We just integrate something potential into something

make state of energy minus this one. So there is potential whatever integration. This one to this one. So the force is something centrifugal force. mv square over r. The electron is revolving in a circular radius around the nucleus. You know the Newtonian mechanism. So mechanically, it is a centrifugal force that is noted by this one.

okay so revolving force something this one and then sorry this is minus here so charge one nucleus and then electron is moving so you take it a force is something we mentioned before right so this is centrifugal force and then this is Sorry. So 4 pi epsilon is gone. So this is the total zero. Sum is zero.

So centrifugal force and some potential energy. Sum is zero. Right. Which is our own is stable. So net force is something electron must be zero. This is very simple. One is potential energy, one is kinetic energy, something sum is zero. So now I introduced here. Same with the Bohr's model. R1, R2, R3, R4 here.

So what it is? This nucleus here. So this is a quantum number n = 1, this n = 2, n = 3, n = 4. So I want to prove here. So it depends on the radius. This radius from here or here or here orLAUGHTER fixed by N number which is a quantum number. So I'm going to prove it now, right now.

So this is the same as here. So this figure and this figure what is that? The same thing, same thing here. What is the ground status? This is the unit here. And this is the R4 is what? Here. R3 is here. R2 is here. What it is? So actual case is something electron is circulating. But as you are convenient to understand, start to make a straight line.

here. The actual case is something ground level to the E2, E3, something like that. But this is something make convenient to understanding, something make the same distance here and here. But actual case is like this. I want to prove it. How this radius, this is R1, this is R2, this is R3. So whatever. So I want to prove it right now.

So first, something to charge it first is this one. So this is something like that. So we just integrate for the energy level. So that is potential infinity E vacuum. So now energy, that is something this one. So this one is integrated to the energy. So that is R^2, but this is only R^1.

So, this is be careful. Okay, because it integrates. So, electron has the energy greater than E vacuum. The coulombic force is not enough to keep electrons bound to the atom. Which means what? R is 2,4 not bounding, which is make free from the center point. sent out. So now this is what. Demand the quantum number E1, R1, whatever.

En number, Rn number. So this is fixed. So E1, E2, E3, E4 is something different. So I want to explain how we got this number. So R1 is 0.053 nanometer. Why this number come up? Why this electron volt something come up depends on the level. This level, this level, the fixed calculation.

So very simple. So I'm gonna explain how we got this number. And then this number. So now, next one is the kinetic energy and the potential energy in the energy. some atomic structure. Previously we mentioned hydrogen, right? Hydrogen. Why? Because hydrogen is only one nucleus and then one electron. Atom number is one. So one electron, that's why I simplify. That's why for example I presented one

model that is hydrogen. So now I'm integral of the angular momentum and around one complete orbit with the integer. There is n number right so n number is something quantum number. Here is integer multiplied over a frank constant h. That is the definition with the speed. So h over 2 pi is h bar. The Planck constant is the same thing.

One is the Planck constant over 2 pi. That is h bar. So now this is circulation, right? So this motion is not linear motion, it's circular motion. So that's why integrating not dx, d theta. Something momentum. M is momentum is mv. So here is mv. So r theta.

circulation theta. So that is the total momentum integration over all over circle. So that is something notation by N. Quantum number N and the Planck constant H. So NH. So this thing, momentum is integrating 0 to the 2pi. So we got this one. m, v and r, theta, 2pi, minus 0 is 2pi.

So 2pi times m. here R, V is N, Planck constant. Okay, so this is the definition. So this is also, I just make here M, velocity, R, N, and H. h = nh, Planck constant here. Same thing.

Because why 2 pi here over something 2 pi. This is the addition. 2 pi is Planck constant bar. So m, velocity V, r is n h bar. Now. Okay, this is done on one occasion. And subscript n indicates something particular, orbit, radius or speed. Here is a n-dependent. Radius r, n-dependent, velocity n.

So let's take a look what it is. So specific quantum number. quantum number n. n depends on radius and velocity. Here, velocity and radius. So this is force. Centric power force. This is force. This is force I mentioned before. And this is something potential energy zero. This is what I previously done.

This is newly done. So this equation and this equation put in solve this equation. How? Because this is with n, v, n over n h bar over m no no Rv. Rv. So Vm. Alright so this Vm is here. So Sol

This equation put together we got this one. Okay. So this is radius, right? Radius. The ball's radius of n state is something properly associated with a particular value of r. Sorry, n. So I figured out what radius definition. So this Rn is put into the easy equation. Then what? I know Rn here. So putting the Rn here, I can figure out velocity n.

This is velocity n. What I figured out. So we got to radius velocity. About what? Something molecule and then circulating electron. Now I got radius and what velocity in? V and or radius Rn. So what it is here? So atomic structure, atom structure of the radius and the velocity depends on the what?

Purely one single value. What it is? Because what? Let's figure out. First, radius here. Frank constant is given, right? Constant, fixed. charge is fixed. Okay, so mass of electron whatever fixed. Okay, so 4 and pi is given and then this permissibility is given. So what is left over? Only quantum number.

n value decides radius. Pretty simple. And what it is? The velocity. Take a look to the velocity. Charge is given. Constant. Planck constant is given. Epsilon is given. Only n. So n depends. Only n change. Then velocity change. So radius changes. So very interesting is that only the quantum number depends on that quantum number decides radius and then velocity.

Right? So find energy associated with this state. So the state number is dependent and dependent. right, and decide which velocity and radius. So total energy of the system is equal to kinetic energy plus the potential energy. The kinetic answer is over. So what is the kinetic energy? That's kn. So n, quantum number,

whatever that kinetic energy m the v square is it we got here v we got here so 1 over mv equal so this one is square put in here we got here so kinetic energy so mass is given charge is given If the Lund is given, the Planck constant is given. So kinetic energy only depends on the N. So N, energy level, something is potential energy, something.

Same thing, kinetic energy, blah, blah. Vacuum minus kinetic energy, there is potential energy. So say energy is quantized. quantized, right? Why? Because it depends on the n number, which is quantized. It can have only discrete value associated with quantum number n. So, in atom structure, atom structure, velocity, then what? Radius.

radius. velocity V, radius R, energy. E, all three values are decided by quantum number N here. Okay, so what does this mean? Take a look here. previously here. This radius is decided by N. So radius is fixed N1, N2, N3, N4.

How about the kinetic energy? Velocity in here V1, V2, V3, V4. Fixed. Depends on the N number. So energy, E1, E2, E3, E4, whatever, these are by n number. So these values, these values are decided by only n. Whatever, 1, 2, 3, 4. That's why the number is here, here, here is fixed. So N decides. N number is no radius in between here, no energy between here, no velocity between here.

So atomic structure depends on the atom number. So all the material or atom number in the Earth, this radius is the same exactly. This value is exactly the same. It depends on the atom number, but what is different? The number of electrons is different. So structure-wise, the thing is the structure of all the material is R1 is the same, R2 is the same, V1 is the same, V2 is the same.

E1 is the same, E2 is the same, whatever. Different things are only number of electrons. Well located. Structure-wise, everything is the same.

So now, next thing is the molecule. Previously, we talked about atom. Atom is one nucleus and one electron for the hydrolon. H, right? So now, H2. So molecule case which is what? Two atoms put together like this. Right? So this is one Atom number one here. Atom number two here. Okay, so it's separate

This is a separate atom number one and then two Something about the energy of It's a potential energy. So the E p means potential energy here. E1 here, E2 here, E3 here. We know what it is, right? So this is what? Something potential energy for the R is going to be infinity. Something going further, further, and then what? Force is going to be zero here. and this is gonna be 0,2. So R1 here, R2 here, R3 here. So this energy R1, R2, R3

decides by what? Only n number. n2 the quantum number. So this is quantum number decides or energy level, okay? Where is it going to be energy level? Nothing but quantum number, right? So this is a potential energy in here and here. Something here and here. So same atoms, so same structure in here and here.

So R1 R2 is the distance between the electron each of the nucleus Okay, so nucleus to the electron Nucleus so that's why two electron are two nucleus and then two electron whatever Electrons in here and here and the nucleus in here. So what is going to be? The radius is going further and further, we got the potential energy. So this is to show something about the electron energy state.

So what is the state here? State is here, and here, and here. The state means the well-located electron. Right? The state here. which is to quantize. State is always not continuous which is quantized here. State in here. So nothing stayed here. Nothing stayed here. So nothing stayed here. Nothing stayed here. So we change the electron. Can exist not here. Not here. Only state here can exist electron. to test what state it is.

So energy-main diagram which is showing up here. So first, this is each atom. these two atoms but make these one separate atoms put together make the molecule so molecule means what? atoms getting closer closer what's going to be happening? then this structure this kind of shape structure

getting closure so this one is here and here this one is here and here so these things again closure makes like this way so over like this so this is what H2 molecule okay and then what State is discretized. Depends on the N number. Quantum number N = here, N = here, N = 3 in here. So it is a discretized allowed energy state.

So electrons can sit here. Whatever. So atom number 4 may be sitting here. Something like that. So getting closer, so this is something range of a movement of electron one state. This match. What? State one is what? And number one here. All right. So this E1, right? E2, E3, E4. So E1 state is here. So range of movement of electron only where electron and hydrogen case is N1.

So 1 has 2. So 2 electron can be initiated. So these electrons only move in this location. So, E2 is this much. Getting bigger and bigger. So, this is where separation is small enough to potential energy maximum between nucleus.

potential energy maximum between nucleus. This is the potential energy. This is the molecular. So we are going through what I explained here, everything here. Electron, the ground state. It goes to energy state, would be one of the nucleus, but one electron excited state could

be traveled back and forth between nucleus. And in fact, they shall buy two atoms. Since the electron tends to seek their lowest possible energy. So here ground, more and more higher energy. So electron find out where is stable condition. Most stable condition is ground, whatever, lowest possible.

So this is the first one of the electrons being the one of the upper level would be not less strong. Which means upper level not less strong, so down. Losing energy is going to be down. More stabilized conditions. So electron would quickly be able to the ground state. Ground state E1 is something more stable state. So electron, the ground state would be shared by the two nucleus.

And oscillating, sorry. oscillating between two positions, which is something like that. Since each nucleus has the ground state associated with it, tunnel out of two electrons can occupy this ground state. for neutral, so hydrogen molecule. So region between the nucleus, which is kinetic energy, is something that does this, which is velocity is small.

So electrons will travel more slowly in this region. Then average electrons feed most of their time between two nuclear. So electrons therefore create the negative charge of the electron crowd, the region that tending to attract the two nucleus together. Inter-nuclear spacing is too small. However, the potential energy E_p is decreased which is increased kinetic energy.

Since total energy conserved as kinetic energy and the non-electron speed increase, the electron crowd effectively reduces the lessen the attractive force. Now I mention about the quantum number. Quantum number is something n number. So quantum mechanics is energy, its atom is quantized. Quantized means not continuous.

a certain discrete value. What is that? Value means n. Something number. So this is what? Value of the state. So state. Like I mentioned something band structure. Band structure is a composite of the state.

State is a mole means more electron can exist. The quantum number N is called the principle of quantum number. It describes the energy of electron and allowed state. Then physical meaning of these quantum numbers that are not essential to this transistor. But Polar X principle is essential. So I'm going to talk about moreLAUGHTER things. The lost energy orbit of the atom is n1.

This is the lost energy level. The state is both two electrons. So one state is two electrons. These electrons can must have different spin quantum number. One is plus one and half, one is minus one and half, like a spin number. So N2 state means there are two possible orbit shades. One orbit is spherically symmetric, all the two electrons are the spin.

So N2 state is a composition of S and P. There are three elliptical orbital shapes but different orientations. Each of these holds two electron of the opposite space, bringing maximum number of the electron and sequential to the A. This is 2+6. That's why A. This is S and P. It's very capable of building in the quantum number.

Definitely. This is the quantum number, polyexclusive principle. So now, previously I just talked about atomic structure. right? Atomable H. Then molecule. Molecule H2. So two atoms put together like

one nucleus one. One nucleus one electron getting closer getting closer it just interact with each other. There is a bonding, there is a molecule. That's why we make a structure like this and then like this previously. So now what it is? Something, this is two atoms put together. But here is more, more than two atoms.

Like six atoms put together, there's a potential energy between like this, and then this is the most out surface layer, and here and here. All right, this is the surface of crystal. So more than two electrons is a crystallized structure. we call it up that's why crystallized structure covalent bonding so each atom makes the bonding with covalent bonding to make the molecule so y axis is energy here is energy right so this energy level

because the most lower one is N1 quantum number. N2 here maybe. So N2 depends on the atom number. N3 and N4 more and more. The upper layer is higher energy. So we just make a curve dot here something like this. This is the actual case of the band structure. But people make more convenient to understand it, to make a straight line here.

That's why all the band structure is a straight line. Actual case is not straight line. make it easier to understand the band. So this is E1 and E3. So look up here. E1 is thinner than E2. Why? 1s2. So two electrons can sit. So two electrons can sit. The state is going to be thinner.

How about E2? E2 is 2s2, 2p6. Eight electrons can sit here. So more thicker. So outer, outer is more thicker and thicker. Right? So And outer structure. We are interested only outer layer of the band structure. We are interested. Not inside. We are not interested in the inner one. Okay, so I'm gonna talk about later more like how you bonding approach the theory.

So now I'm gonna tell you here, so all electronic structure outside layer, outer layer, lower in the balance band, top is a conduction band here. So this is, we just, this portion makes simplify this one. destruction. Okay, so because we not interested in inner Sun because this is too much stable nothing is changeable. Energy in and out, electron changing only

outside not inner side. Why? Because inner side is more stable. More stable then outer side. Okay, so here is a very unstable vision. This is stable. Not much change inside here. That's why we only drawing here every material. You know, interesting enough. Okay only interested outside band structure. So this is outside

In here you more and more but we not interest because nothing changed. Okay only changing outside layer So thing is here So conduction band the balance band the forbidden band is something easy means well band gap Band gear means electrons can exist inside here. Only here or here. Something, electron, something, up, get energy up, down, energy loose. If you get

What is located in the balance band? Something electrons are here. The crystallized structure. Here is empty. Then, perfectly full and perfectly empty. That case is not conducting. Electrons can move around here. okay so this is the case not conducting but here something is this is a fully occupied case but this is not fully occupied so empty here one two three

four five six seven Electrons empty. Empty means what? This is a hole. So these make a void which is a hole. 1, 2, 3, 4, 5, 6, 7. Electrons where is going to be? Moving up. So how many electrons? 1, 2, 3, 4, 5, 6, 7 electrons. Exactly 7 holes down here.

7 electrons a hole. Exactly matching up. In this case, we called it an intrinsic semiconductor. I'm going to tell you later. Intrinsic semiconductor. Okay, so number of electrons, number of holes is exactly the same. So if these seven electrons sit here, perfectly full and perfectly empty here, no conducting. But electrons moving up is one, two, three, four, five, six, seven electrons.

And seven holes in down here. Okay, so then this is the how many carrier? How many? Seven Electron and seven holes. So total 14 carrier existing here. So more and more carrier means more better conductor. Okay this is a better conductor so this is what it is

Next, so like this is one important parameter. Three other parameters I'm going to tell you here. So every this is a band structure, right? Most outside layer. So key parameter is band gate, electron affinity in here. and ionization electron affinity, ionization potential, and the band gap here. So first look it up.

Ghost energy in a conduction band is designed for the EC. What is EC? Conduction band. Which is the conduction band and the highest energy in the balance band. highest energy in the balance band, highest band is here. This is E-V. Key parameter is E-C, E-V. Where is located? That is the key point. So, like a fingerprint, every material right here.

Silicon, gallium, arcylidin, germanium, all kinds of materials are different. band structure is three values here. The band gap first here, electron affinity here, electron affinity, then electron affinity. So once Electron affinity band gap is known. We can find out. Ionization energy 2. This is defined as minimum energy required to excite one electron from the top of the balance band to the vacuum level.

So here to do here. This is the ionization energy. So actually, if every material has something, band gap and electron affinity, we know everything. So electron most likely ionization, this requires the eldest amount of energy to leave the atom. These are top of the balance bands. Electron affinity is defined by energy difference between vacuum level to the vacant state of

the lost energy E_c. Electron affinity. Energy band gives minimum energy quality excited electron from the energy. balance band to the conduction band here to the here. So that is the energy band gap and the electron affinity. The two important parameters are the key factor for the every material.

So now, all band structure of a semiconductor. So how many carriers is important here and here. So electron affinity. Before we don't band gap, we normally refer to it simply band gap. fundamental property of the semiconductor. This is the key point: band gap and electron affinity. Band gap is determined by energy requires

some excited electron from balance band to conduction band. So, major electron affinity, however, the excited electron must be passing through the semiconductor surface to the the collector vacuum which means what? Here is the electron affinity something more than vacuum level something out of the structure. So this value and this value is adding up there is a

ion is an energy. The same thing. So we say the two parameters are a key factor for the material property. The band property of the surface can differ from the bulk because the surface contamination is a mechanical strain. So occupancy of these material bands is all things tending to seek their lowest energy possible. Okay, lowest energy. Why? Because it's stable. Lowest energy stable is stable.

So find the sick point of the lowest energy level that is most stable. That's why all the, that's natural law. So all the things is going to be stabilized. Trying to be. What's the electron? Baker's balance band was lost at a given time. The effect of the absolute zero and every electron is the lowest possible state, implying that of this lowest to allow the band is full of 2D including entire

Plus beta. So, absolute zero, which is a Kelvin, which is a no thermal energy. Every electron is the lowest possible energy state. The perfect semiconductor, every stage's balance band is occupied by an electron. Every stage's conduction band is empty. This is natural. So this case is what? Non-conducting. Means what? No carrier.

Means no electron and hole. Balance band is failed. No state is forbidden. the minimum energy can be absorbed by balanced electron EEG, enough to excite the electron from balanced band to conduction band. At room temperature, because of thermal agitation, the few electrons excite into the conduction

band. One eventually fall back in a vacant state with a balance band, re-emitting excess energy, something as heat-like. The average time electrons bend in conduction band is called electron lifetime. The balance band excites, then coming back to the balance band. How long stay in conduction band?

That is the lifetime of a lecture. Usually 10^-10^-3 seconds. Depends on the material. Lecture and conduction bands are free to move around the equator. constant energy between collision data of some and empty state same energy into the electron can move. This band is called conduction band.