The reasons of Quantum Mechanics
In 1888 a Swedish researcher called Rydberg was observing the emission lines of several materials. He noticed that coloredlines are emitted with some regularity, and he though there should be a way to predict the position of each line at wavelength
λ with a single formula. After several trials using hidrogen gas, he finally found this formula:
1
1
1
= R(
− )
λ
nf
ni

(1)

This is called it Rydberg formula and predict the exact position of hydrogen emission lines, the very important fact is that
there are these strange numbers ni and nf , that now we call radial quantum number that were consecutive integer number ofa then called series of lines.
This fact discovered in 1888, was again put forward later by the experiments of the British Rutherford. Rutherford in 1911
irradiated with alpha particles (two protons and two neutrons) a gold foil. With his surprise he found that almost all particles
went straight through the gold, like he was inconsistent, but in some rare case, the particle was rejected back, like she collided
with something very heavy. He made several calculations that pointed that the atom of gold were almost completely empty,
and all the mass was concentrated in a heavy positive charged atomic nucleus. Then he developed his own simple model ofatom, a sort of planetary system, were negatively charged electrons have almost no mass and they were orbiting a central heavy
nucleus.
Later Bohr and Rutherford developed the so called Rutherford-Bohr model of the atom. This model is very similar tothe planetary model firstly introduced by Rutherford, but electrons were kept at fixed orbits in such a way that the Rydberg
formula can be satisfied.
Why electrostatic forces kept negative electrons in this fixed orbits, without radiating energy and without being attractedend collapsing to the positive nucleus was still a mystery.
Another phenomena was suggesting the presence of a new physics. If we illuminate a metal, like Zinc for example, it waspossible to observe the emission of electron. This was firstly observed by Hertz, and was actually called Hertz effect. The
strange thing was that there was a sort of threshold of wavelength under which no emission was possible. If the wavelength of
the radiation was bigger than this value, there was no electron emission from the metal even increasing the light intensity. Thisphenomena was impossible to explain with classical physics, because the energy of an oscillator (Maxwell showed in 1861 thatthe electromagnetic field is mathematically equivalent to an oscillator) should be proportional to its amplitude. If I increase
the amplitude, I should get electron out sooner o later (!) However, no electron was emitted, whatever intensity, if frequency(the inverse of the wavelength) was over the threshold. Moreover, even if light had the same intensity, changing increasing thefrequency was increasing the electron emission ! What was happening here ?
Einstein explained that the energy of light is divided in tiny quanta of energy, and introduced for the first time the basic
quantum mechanics concepts in the effort to explain the photoelectric effect. However, his theory was not widely accepted andremained controversial for many years.

The Black Body radiation, the first Quantum Mechanics FormulaSuppose we want to study the radiation emission of a hot body. In a hot body, all the molecules vibrates. We can approximate
these vibrations as elastic oscillations of a mass particle. If there is radiation emission, it means this particle is charged. It
is known to physicists that a charged particle oscillating radiate away an amount of energy proportional to the energy of theparticle, accordingly to this formula:
dW
I=
∝ γW
(2)
dt
This means that the energy radiated away per unit of time is proportional to the energy of the charge with a coefficient γ. On
average, at a certain temperature T the energy of the particle is known to be KB T , so we have:dW∝ γKB Tdt

(3)

We do not derive here the detail of the calculations, however γ is ω/Q where Q is the oscillation quality factor. This quality
factor is proportional to 1/ω. If soγ ∝ ω2
Finally we find in the relation for the intensity I(ω) a direct dependence with ω 2 and the molecules kinetic energy: kT ,I(ω) ∝ ω 2 KB T

(4)

this equation is called the Rayleigh’s law for the black body radiation. Interestingly enough, this equation fits very well theexperimental data for low energies, however, it fail completely to predict a diminished emission at higher energies. Accordingly
to eq. 4, because of the ω 2 , we should observe a lot of X-Ray and other unhealthy emission from a hot body, much more
UV and X-rays than other radiation! Instead, of course, in the experiments, we never observe emission over UV or X-rays,even if we heat-up a body at very high temperatures. Moreover, a low without w2 will be in conflict with the conservationof energy, that would be very bad indeed! There must be something wrong in this equation, what is going on here ?!? Many
researchers at that time studied the problem, and nobody was able to find an answer. The problem was called by physicists of10ruggero.sci.yokohama-cu.ac.jp — —Author: Ruggero Micheletto, Aki Tosaka

the time the UV catastrophe. Max Planck also studied the problem and was getting mad at it. Inspired by Rutherford-Bohrmodel of the atom, where electrons stay in a fixed orbit regulated by quantum numbers, one day he made the very fancy
hypothesis that the kinetic energy cannot be a continuous value, but must be quantized in multiple steps of hω. In this way
he calculated the average energy that is not any more kT as in the equation 4, but a different value that drops down very fastat higher frequencies. This supposition that Energy cannot be continuous, was defined by Planck and his colleagues as an act
of desperations ! How it was done ?
Planck thought that for some reason, the oscillation of electrons must be quantized, exactly as the strings of a guitar areforced to vibrate only to multiple frequencies. This idea was not unfamiliar to physicists, in fact standing waves in musicalinstruments behave exactly this way: frequency is quantized in multiples of a base tone (see fig. 1)

Fig. 1. Examples of standing waves of a string at different frequencies. The arrows represent the position of the string at the minimum and maximum of the oscillation.
Planck thought then that the vibration of the molecules in a hot body, must behave exactly this way. So he simply calculated
Etot
where Etot is the total energy of the system, andthe average energy of such systems. The average energy is : < E >= N
tot
Ntot is the total number of available states.
Let’s calculate first Ntot : lets choose one frequency ω. For this single ω, if we have many possible levels of energies E0 , E1 ,E2 etcetera. Each of this has energy multiple of the minimum E0 , in this succession: E1 = hω, E2 = 2hω, E3 = 3hω etcetera.
We will have N0 , N1 , N2 molecules on each of these energy levels, right? The problem now is: ”what will be the distributionof these energies?”, in other words, how many molecules will be in N0 , N1 , N2 and so on? Of course, Planck wanted to be
simple and he thought at the well known Boltzmann distribution that we know very well. So he said, the number of moleculesin each energy states E0 , E1 , E2 . . ., will be distributed as a Boltzmann distribution, so like this :−E1 /kTN1 = N0 e

N0−hω/kT= N0 eN2 = N0 e−E2 /kT = N0 e−2hω/kT
N3 = N0 e−E3 /kT = N0 e−3hω/kT

(5)

...
where h is a constant. If we have an infinite number of these states, what is the total Ntot that we are looking for ? Well, let’s
simplify the equation by calling x = e−hω/kT , then our eq. 6 becomes:
N0
N1 = N0 x
N2 = N0 x2
N3 = N0 x3

(6)

...
and so on. Then the total Ntot is
Ntot = N0 (1 + x + x2 + x3 + . . .)For the theory of series it is very easy to demonstrate (simply multiply the series by x and compare it with itself) that
Ntot = N0

1
1−x

(7)

At this point we only need to calculate the total energy of the system. Planck simply added up the energy for each level. The
total Energy at the lowest ground level was for simplicity set to zero, the total energy at the fist level was N1 ∗ E1 , for thesecond N2 ∗ E2 and so on... since Planck assumed that energy proceed in multiple of a base (ground) value E0 = hω, thenE1 = 2hω, E3 = 3hω. If we proceed this way, using again x = e−hω/kT the total energy is thenEtot = hω(x + 2x2 + 3x3 + . . .)(8)

Again, if this series is infinite, the theory says that (use the fact that x + x2 + x3 + . . . ≈ x/(1 − x))
Author: Ruggero Micheletto, Aki Tosaka

Etot = hωx(1 − x)2

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so the average system energy that we are looking for is:< Ek >=

Etot
x
= hω
Ntot
1−x

(10)

hω
ehω/kT − 1(11)

or substituting back x< Ek >=(note that the exponent of the exponential now is positive and not negative as usual!). This formula is what we have to
substitute to < Ek >= kT in eq. 4, it goes down very fast for higher frequency. We obtain a curve that fits perfectly
experimental data:ω 2 < Ek >
hω 3
(12)
I(ω) == 2 2 hω/kT22π c
π c (e− 1)This equation grows initially with ω as before, but at higher frequencies the exponential wins and everything goes to zero as
it should be, the UV catastrophe was finally solved! This was the first quantum equation ever, it became very famous as the
Planck radiation law, or Planck’s blackbody equation. The problem of the ultraviolet emission of hot body of equation 4 wassolved for ever and quantum mechanics was born.

The electron double-slit interference ExperimentThis experiment was at first conceived partially as a thought experiment, or from a German word a Gedanken experiment.
However, everything we will say about it is now a confirmend experimental fact.
Suppose we have an electron cannon. This is a machine that shoot electron, exactly like an old CRT screen, or like theelectron source of an electron microscope. These electron, at first, they were considered to be particles of matter, so theymust behave like that. What happens in the experiment is that if we make the electron pass through a doble slit and on theother side we place a fotographic plate, instead to have a localize pattern of the same shape of the slits, we observe a figure ofinterference... (!)[FIG]
The conclusion appears to be absurd! How is possible that electrons behave like waves? The researchers then tried to shootone electron by one, at big distance in time. When only one electron was shot on the double-slit, one little spot was visualizedon the photografic plate. If few electron where shot, few little dots were observed. If many electrons were shot, one by one,many little dots appeared, but incredibly enough, the dots formed the shape of an inferference pattern again!
This made researchers think hard of what can be the cause of this. It was like the electron was interfering with himself,
somehow. When the researchers tried to visualize which of the two slits it passed through, putting a detector near the slits, thensomething even more strange happened: the electron behaved again as a particle. No interference pattern, just dots aligned
perfectly with the slit.
In conclusion, if the electron have the option to pass through slit one or slit two, it behave like somehow a wave, and we
see an interference pattern on the screen. If we measure where the particle has passed through using a detector, we do not see
any interference, simply the electron go straight on the screen, and we see the shape of the slit.
So electrons have a dualistic behaviour. This idea was firstly established by professor De Broglie. He said that there is adualism on the behaviour of matter, that depending on conditions it behave like a particle or like a wave.
To formalize mathematically what was happening, we have to say that our experimental result is an interference patternon our photografic plate. The intensity of the pattern is proportional to the number of particles that hit the plate. In otherwords it is proportional to the probability that a particle hit there in a given time. So our measurable parameter is not anymore
a position of a particle, like in Newtonian physics, now we only have a probability P . Because the pattern formed by thisprobability is an interference pattern, we can associate to the particle a probability wave, or wave function to the particle. This
wave can be written this way:
Φ = e−i(ωt−kx)(13)
like any other wave. We have no idea at this stage of the physical meaning of ω or k, or the time t, we just know that x is thespace. If we do this, we can say that my particle can pass either through the slit one or slit two s1 or s2 , and the final wave isthe sum of the two waves:
Φ12 = Φ1 + Φ2
Of course, the sum of two waves, produce an interference, so Φ12 has the shape of an interference pattern, please do not forgetthat this is a complex function that has the shape of a wave. Our experimental result is insteat a real value, the interference
pattern we observe, that is the module of this interference wave, so we can write:
P12 = |Φ12 |2 = |Φ1 + Φ2 |2
The observation and the study on this experimental facts brought the researcher to astonishing conclusions that can besummarized in the so called three principles of Quantum Mechanics:

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0) A particle has within itself a wave that is called wave function and generally expressed as the complex wave
Φ = e−i(ωt−kx)
1) The probability to find the particle is expressed as the module of φ :P = |φ|22) If a particle has two alternatives ways to reach a point in space x, the probability to find it has the shape of an interference
pattern:
P12 = |Φ12 |2 = |Φ1 + Φ2 |23) If the particle has no alternatives (it has been measured where it has passed), then the probability to find the particle is not
an interference, but the bare sum of two probabilities
P12 = P1 + P2 = |Φ1 |2 + |Φ2 |2
Please note that since Φ is defined as the complex wave in eq.(13), its module square is of course just a real number, and in
ideal case it is just 1 !

De Broglie Hypothesis and Heisember principle
We have the experimental fact that electrons diffract exactly like waves, like light or like water waves. This is a fact, so
researchers had to think a way to represent particles as wave. The idea is that particles have an inner probability wave calledwave function Φ and that their probability to be found in a place is the square of this function. The De Broglie Hypothesis ishhthat the energy of such a particle is given by E = 2π
ω and that is momentum by 2πk, where ω = 2π
and k = 2πτλNow if a interference pattern is created, what is the relation between position X and momentum of a particle? We can calculate
this directly from the diffraction equation. We use simply the diffraction triangle in figure [FIG] and find that, for small angles,∆P is equal to P0 ∗ α. From De Broglie Hypothesis, P0 = h̄k, so∆P = h̄kα. If this is just a diffraction problem, we can find immediately α = λd where d is the aperture. But the aperture is actually how
well we know the position of our particle, so we can call d as ∆x. If this is so we haveλ∆x
λ∆P = h̄k
∆X
α=

(14)
(15)

Knowing that k = 2π
we finally deduce the fundamental relation of Heisemberg principle
λ∆P ∗ ∆X = h
This simply states that the knowledge of a particle position limits the knowledge of particle speed (momentum), and viceversa.
Remember that this definition is not related to particle physics. It is a relation that can be derived by wave physics only. The
Heisemberg principle is valid for audio waves, sea waves or any other real wave too !

The Bound States, stationary waves!
We have the experimental fact that electrons diffract exactly like waves, like light or like water waves. If we assume that therelation of the wave function ψ = eωt−kx and the physical propertiesof a particle is E = h̄w and p = h̄k, then, as we foundabove, the particles interfere and produce a behaviour in which ∆x∆p > h, that is called also Heinsenberg principle.
How we can be sure that the De Broglie Hypothesis E = h̄w and p = h̄k is really correct ? After all, it looks really strange.
A particle of a definite energy E, would have a probability to be found in a certain place as
P = |ψ|2that is simply P = const., a value the same everwhere. In other words, if the particle can be found anywhere we do not know
where it is; this sounds very very strange !
Now we consider an atom, where particles are not free, but confined (bound) around the nucleus of it. Electrons are to
be found only in a certain place. If is true that there is a wave function associated with it, it must be a stationary wave, likethe one of a guitar or a violin. If this is so, a mechanical analogous implies that the violine wave must resonate ONLY with
fixed wavelength, the fundamental and all its harmonics (the harmonics are the frequencies multiples of a certain fundamental
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frequency fo , that is 2fo , the second harmonic, 3fo the third harmonics and so on). Then also in our atom, the probability tofind the particle will be confined to fixed energies
En = eωn twere wn are multiples of a fundamental wo : wn = {wo , 2wo , 3wo . . .}. This is a characteristics of bound states, where the particle
is confined somewhere, so there is no x dependence, and the wave function depends only on time.
Is there a way to localize a particle still using the De Broglie Hypothesis? What happens if we superimpose different
frequencies on a wave? If we remember wave mathematics, we should realize that the sum of slightly different frequencies
produce beats. Beats are waves modulated in amplitude. Beats are very well known to musicians. If we play two guitar strings
tuned almost the same f1 ≈ f2 , easily we can hear the beats that have frequency centered about the difference between thetwo frequency df = f2 − f1 .
More interestingly, if we add up more nearby frenquency, not just two as above, but many in a almost continuous way, we
obtain a particular set of interference, generally called a wave packet. It is a wave distributed along many frequencies with abump of amplitude localized around center frequency. To localize a particle, we have to consider that the wave function
ψ = ei (wt − kx)(16)contains both a space and time dependence. If we fix the time t and consider only k, our wave packet will be composed by
the sum of many waves of different wave number, instead of frequency, but the mathematics will be exactly the same. So,
because the wave function represent the probability to find the particle, the particle is localized at the center of the wave packet!
This means that a particle is represented by the superposition of many wave functions (16) with similar, but different wave
numbers. Moreover, if we shorten the difference in wave number, we obtain a wider localization and viceversa. Because thewave number represent the momentum of the particle we have again the situation described in the Heisemberg principle, a sortof ∆P ∆X > const...! If we use the definitions of energy and wave packet as before, we will prove this fact easily again. See
figure 2.

Fig. 2. The superimposition of several waves with similar wave number k. If we increase the width of the wave number span (we increase ∆P in Quantum Mechanicalanalogous) we have a better localization, as Heisemberg principle should indicate. Also, if we increase time (lower panel on on the right), the wave packet moves right, theanalogous of a particle moving in space toward the right.

The Dirac representation
Now we know that what we thoght it was a particle, an electron or a photon, are not corpuscole, but they behave like waves,so they contain wave functions within themselves. We can write this wave function like this: ψ = e−i(ωt−kx) . Also we learned
from Planck black-body equation, the Bohr atom, Einstein photoelectric effect and other experiments that in this equation theparameter ω has the physical meaning of Energy and the parameter k is the momentum. The De Broglie hypotesis is E = hωand p = hk. Also, we have learned that the probability to find a particle is simply given by
P = |ψ|2
This implies that a moving particle is represented by a wave packet where many waves of different energies (momentum andkinetic energy are related) overlap together, instead a steady particle is bound in a place and must have only fixed falues ofenergy E = hωn like a resonanting violin string. Now that we established these things we come back to the original double
slit problem we discussed above. Everything is expressed mathematically in terms of probability, not enymore in position as
in classical mechanics. How we can develop a simpler mathematical framework to calculate these probabilities? One way was
introduced by prof. Dirac and still used today.
Suppose we indicate the position of a particle in one place, in our formulation we can call it like this e−i(ωt−kx) , to makeit a little bit smaller we use this indication: |ψ >= e−i(ωt−kx) . What happens if two of these wave functions interfere? The
result depend in how many dimensions we consider the problem. In one single dimension x, we will have beats, the probability
to find the particle |ψ|2 will be a figure of interference, as already discussed above. However, to reproduce the infamous two
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slits electron experiment we must proceed in two dimensions, we need two space variables x and y and two wave numbers kx
and ky . How we can write mathematically the interference of two two dimensional waves? We do not go in details, but takein account that for each point in x, we need to integrate (sum up) the effects of all waves coming from differents y. So themathematics is still conceptually the same, but heavier to write and calculate. Dirac representation help us to simplify. Letsconsider that our waves are complex vectors, so the result of the sum of two of them is something like this:
~ ~

eA·B
If we use the De Broglie hypothesis this will be of the form:e−i(~p~r12 )where r12 is the vectorial difference between position x1 and x2 . Dirac introduced this notation: < ψ1 |ψ2 >= e−i(~p~r12 ) we
intentianolly ignored the presence of an integral here. In this way we can express very compactly the probability to find theparticle if has moved from one place to another. Equivalence- a method to represent the physics of particles in QM without writing all the waves - the example of slit problem. - anexample with multiple levels slits

The equivalence with vector methematics- the similarity with decomposition in base states to the projection of a vector in its base coordinates. - the physical meaningof base states: they represent alternative, mutually exclusive states

The Hamiltonian Matrix- the amplitude of transition from r¯1 to r¯2 is< r1 |r2 >= ep̄r̄12- The definition of operator: hχ|a|si - how the operators changes in time:ψ(t + dt)i = U (t + dt, t)ψ(t)i(17)

hj|ψ(t + dt)i = hj|U (t + dt, t)|ψ(t)i(18)

hj|ψ(t + dt)i = Σi hj|U (t + dt, t)|iihi|ψ(t)i(19)

Uij (t + dt, t) = hj|U (t + dt, t)|ii(20)

Cij (t) = hi|ψ(t)i(21)

Cj (t + dt) = Σi Uij (t + dt, t)Ci (t)(22)

Uij = δij + aij ∆t(23)

Cj (t + dt) = Σi (δij + aij ∆t)Ci (t)(24)

Cj (t + dt) − Cj (t) = ∆tΣi aij Ci (t)Cj (t + ∆t) − Cj (t)= Σi aij Ci (t)∆t
Cj (t)= Σi aij Ci (t)dt
Cj (t)ih̄= Σi Hij Ci (t)
dt

(25)
(26)
(27)
(28)

Propagation in a crystal lattice- explain that it is normally difficoult to the electron to circulate in a solid crystal.
- cross section of electron 1 Amstrong- cross section calculation:- calculate the volume ”shadow” behind the target ballNint =

nbeam V

Nint =

nbeam πR2 vt

NintT ime =
σ=

(29)
(30)

2

nbeam πR v

(31)

NintT ime /Jbeam(32)

where J is the flux defined as Jbeam = nbeam v and nbeam is the beam density, or the number of particle per unity of volume.
We can define another important parameter λ, that is called usually mean free path. It is the average length that theparticle can run free. This parameter is as important as τ (the average time a particle runs free without collisions) and has
almost the same meaning. The mean distance the gas particle can go -on average- before it experience a collision. Now let’s
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try to relate some gas parameters that we know. Let’s ask ourselves, what is the probability of collision after the particle has
moved a distance dx ? We know the answer, this probability (for only one particle) is:
dx
λ

(33)

Why? Because if I am located in the middle of tables separated one meter each other and I move of one meter... I am sure -on
average- that I will bump on a table. However, if I move only half a meter, on average, I have 50% probability to bump on atable. That’s the simple linear relation expressed by 33. dx is how much I move among tables and λ is the distance between
tables in our example. Now, lets consider it from another point of view. If we have a box of volume dV , with particle density
n, what is the probability of collision in a small section of it of length dx ? Well, as we know the number of particles in thissection is nAdx, where A is the area of this section (see figure 3).

Fig. 3. A scheme for the modeling of the collision probability. The cross section is represented by the average area of a single gas molecule.
If the every single particle has an average area σ, then the total area occupied by the particle is σnAdx. If we divide thisarea with the total area available (which of course is A), we obtain the probability of collision within this section of length dx:
σndx. But this value must be equal to eq.33, so finally we have:dx= σndxλ(34)

simplifying this yields this very important relation:
1
(35)
σn
this equation is very important because it relate the particle density, their cross section σ and their mean free path in a particle
system each other with a very simple and compact formula.
λ=A semiconductor device: PN junctions
We studied how the energy that a particle has, in quantum mechanics formalism, is represented by the parameter E in theequation:pE
ψ = e−ı(wt−kx) = e−ı( h̄ t− h̄ x)(36)
because we define the energy E = h̄ω and the momentum p = h̄k. In the previous chapters we came to the conclusion thatthe probability amplitude of transition between states |ii and |ψi that is Ci = hi|ψi changes accordingly to the Hamiltonian
equations studied above. As a consequence, if the parameter E represents the Energy of a particle that is adsorbed by an atom,
this energy E must be contained between a range (a band) of values within
E = E0 ± 2A cos(bk)where E0 is an arbitrary base energy reference, A is the energy necessary for the particle to leak from one atom to the adjacent,and b is the distance between them. As we have seen, the very same conclusions can be reached if we represent a particle thatleaves an atom, not one that is absorbed by one. So we have two energy bands, one for the particles absorbed by an atom, one
for those that leave it. These two allowed bands are usually called the electron band and the hole band because in the mostcommon situation the particle studied is an electron, and a missing electron condition is called a hole.
The meaning of the bands is that any particle that does not have enough energy to be within the band, cannot exist. So,
we can say that there is an Energy gap between the hole band and the electron band. Where do particles gets their energy? In
absence of anything else of course by thermal energy (!)
So, because of thermal Energy, we expect to have a certain number of electrons and holes in their band in a certain distribution

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in function of temperature. The most natural distribution is the Boltzmann distribution
Ne = Noe e−Ee /kT(37)

Nh = Noh e+Eh /kT

(38)

Eg/kT

(39)

Ne Nh = const ∗ e(40)
where we have multiplied the first two to obtain the third equation, naturally Eg = ∆E = (Ee − Ek ). From this last equation,
we understand that if the temperature increases, we put more electrons or holes in the band, so we have more carriers forpossible conduction of electricity.
Suppose now that in a crystal of a certain atomic species, we introduce a different atom with different valence. If we do so,
we are able to introduce a possible hole or a possible electron for the conduction bands. Then a material with many impurities
that introduce a hole, will have an energy band lower than a meterial with impurities that introduce electrons. The question
is: what will happen I put in contact these two materials? We form a so called junction where the potential will change along
the crystal axis x that runs along the direction orthogonal to it.
We have to consider that Energy potential is related to electric potential by the simple relation E = qV , where V is the voltage(difference of potential).
Near the interface between the two material, the juction, we will have a slope of potential, so a diffusion process will begin.
Lets consider only ONE type of carriers here, for example the ”p” carriers. The more ”p” carriers in one the ”P” side, want
to diffuse and migrate to the other side of the crystal. However, the ratio of carriers that will actually reach the ”N” side will
depend on the energy barrier in the junction accordingly to the usual Boltzmann distribution. So at the thermal equilibrium:
Np (N − side)= e−qV /kT
Np (P − side)(41)

where with P and N we indicate the two doping and majority carriers of the material. Notice also that qV is the energy required
to carry a charge q through a potential difference V . What is the physical meaning of this equation? This means that carriers
will diffuse from P-side to N-side because of their inherent thermal energy, but only the fraction that has enough energy qV
will reach the other side. Please notice very well, that also the opposite phenomena will occur: P carriers in the N-side, will try
to migrate to the P-side... this is a current in the opposite direction. In thermal equilibrium, we expect that these two equal
to zero! The absolute value can be called I0 and is equal to:I0 ≈ Np (N − side) = Np (P − side)e−qV /kT(42)

So I0 is a diffusion p-type carrier current running from left to right and from right to left within the Junction, just for reason ofthermal carrier diffusion. If we apply an external potential difference −∆V to the junction, what will happen? The potential
will be no longer V , but become V − ∆V . So the current of positive carriers running in the junction from P-side to N-side willbe:I1 ≈ Np (N − side) = Np (P − side)e−q(V −∆V )/kT
(43)
This current is from P-side to N-side and it is bigger than I0 by a factor ∆V , so the ratio between the two is:
I1= e+q∆V /kT
I0

(44)

I1 = I0 e+q∆V /kT

(45)

or
it increases exponentially by external variation of potential ∆V . What about the current due to Np carriers in the opposite
side, the N-side? The potential ∆V cannot change exponentially these carriers that are residing on the N-side! Why ? Because
a positive carrier on the N-side sees a big potential slope V and goes straight on the other side, almost indipendently of ∆V(∆V is of course considered to be much smaller than the junction potential V ).
So we have the sum of two Np currents. One is due to carrier originated in the p-side. These are majority carriers and as
we said increase exponentially with ∆V . The others are Np carriers originated in the N-side, these are minority carriers andeven if the cross to P side, they encounter a positive slope, so they all reach P side without change. If ∆V is small comparedto the junction V potential, the value of this current is always I0 . Summing up these two contributions and taking in accountthat they run in opposite directions:
I = I0 (eq∆V /kT − 1)
if we plot this current I in function of ∆V we obtain the typical curve of a rectifier. The first electronic device was born: theDiode !

Author: Ruggero Micheletto, Aki Tosaka

PNAS

Data di uscita

VolumeNumber17