Appearance
2.1 — Semiconductor Physics: Silicon, Doping and the PN Junction
Volume I, Chapter 1.1 treated the transistor as a black box: a switch you turn on with a voltage. It said explicitly that the physics was being deferred. This is where the debt is paid.
The question to answer is this. Copper conducts. Glass does not. Silicon sits in between, and by adding roughly one foreign atom per hundred million you can move it a million times towards the conducting end — and, crucially, you can do it to one region of a crystal and not the region next door. That last sentence is the entire semiconductor industry.
1. Why some materials conduct and others do not
An atom's electrons occupy shells at fixed energies. In a single isolated atom those energies are sharp lines. Bring 10^{23} atoms together into a crystal and every line splits into an enormous number of very slightly different levels, so close together that they form a continuous band of allowed energies. Between the bands lie ranges of energy that no electron can have, called band gaps.
Two bands matter:
- The valence band holds the electrons that are bound into chemical bonds. They are stuck; they hold the crystal together and cannot move around.
- The conduction band is the next band up. An electron that gets there is free to roam the whole crystal, and a roaming electron is a current.
The energy gap between them, written E_g, decides everything.
| Material class | Gap E_g | Result at room temperature |
|---|---|---|
| Metal | none — bands overlap | electrons already free, conducts |
| Semiconductor | ~0.7 to 1.5 eV | a few electrons make it across |
| Insulator | 5 eV and up | essentially none make it |
The unit is the electron-volt (eV), the energy an electron gains falling through one volt: 1\ \text{eV} = 1.602\times10^{-19} J. Silicon's gap is 1.12 eV, germanium's is 0.66 eV, gallium arsenide's 1.42 eV, and diamond's 5.5 eV — which is why diamond is a superb insulator despite being made of the same column of the periodic table.
What lifts an electron across the gap? Heat, mostly. At any temperature above absolute zero the atoms vibrate and a few electrons get lucky and pick up enough energy. The number that succeed depends on temperature exponentially:
n_i \propto e^{-E_g/2kT}
where k is Boltzmann's constant and T is absolute temperature. Exponential means violently sensitive. In silicon, the number of free carriers roughly doubles for every 10 °C rise. That single fact explains why semiconductor devices drift with temperature, why they need heatsinks, and why thermal runaway (Chapter 2.3) is a real failure mode rather than a textbook curiosity.
2. Pure silicon, and why it is nearly useless
Silicon is in group IV of the periodic table, meaning each atom has four valence electrons — the outer electrons available for bonding. In a silicon crystal, each atom sits at the centre of a tetrahedron and shares one electron with each of its four neighbours, forming four covalent bonds. Every atom ends up surrounded by eight shared electrons, which is the stable arrangement, and the whole crystal locks together.
At absolute zero, every electron is in a bond, nothing is free, and the crystal is a perfect insulator.
At room temperature, thermal energy breaks a small fraction of bonds. In pure (intrinsic) silicon at 300 K, the free electron concentration is
n_i \approx 1.5\times10^{10}\ \text{per cm}^3
Compare that with silicon's 5\times10^{22} atoms per cm³. Roughly one atom in three trillion has given up an electron. That is why intrinsic silicon has a resistivity of about 2300 Ω·m — nearly an insulator, and useless as it stands.
The hole is a real thing, not a bookkeeping trick
When an electron leaves a bond, it leaves behind an empty bonding position. A neighbouring bound electron can hop into that empty position, which fills it but creates a new empty position where that electron came from. Repeat, and the emptiness moves through the crystal in the opposite direction to the electrons.
We call the emptiness a hole, treat it as a particle with positive charge +e, and give it its own mobility. This is not a fiction to make the sums easier — it is genuinely how the physics behaves, and holes have measurable properties. The Hall effect can tell you experimentally whether the moving charges in a piece of material are negative or positive, and in some semiconductors the answer really is positive.
Holes move more slowly than electrons, because moving a hole requires a chain of bound electrons to shuffle rather than one free electron to fly. In silicon, electron mobility is about 1350 cm²/V·s and hole mobility about 480 cm²/V·s. Electrons are roughly three times more mobile than holes. Remember that number; it is why NMOS transistors are faster than PMOS ones, why n-channel devices dominate power switching, and why CMOS logic (Volume I, 1.1) has to make its PMOS transistors physically wider than its NMOS ones to balance the two.
In intrinsic silicon, every free electron came from a broken bond, so electrons and holes are created in pairs and their concentrations are equal:
n = p = n_i
3. Doping — deliberately spoiling the crystal
Now the trick that made electronics possible. Replace a few silicon atoms with atoms from the neighbouring columns of the periodic table.
N-type: add group V
Phosphorus, arsenic and antimony sit in group V and have five valence electrons. Substitute one for a silicon atom in the lattice: four of its electrons form the normal bonds with the four silicon neighbours, and the fifth has nothing to bond to.
That fifth electron is only loosely held — it takes about 0.045 eV to free it in silicon, compared with 1.12 eV to break a real bond. Room temperature supplies that easily, so essentially every dopant atom donates one free electron. Group V dopants are therefore called donors.
The result is n-type silicon: n for negative, because the majority of moving carriers are negatively charged electrons.
The picture below is worth more than the paragraph. It shows one phosphorus atom sitting in a silicon lattice, and the whole of n-type doping is visible in it.
Walk the drawing. Count the lines leaving each silicon atom: four, one to each neighbour, every one a shared pair. Now look at the phosphorus. Four of its electrons are doing the same job as any silicon atom's, and the fifth has no bond to join — that is the loose one in the picture, and it is the free carrier. There is one of those for every phosphorus atom you put in, which is why the conductivity is set by how much you dope rather than by the silicon itself.
And notice what the drawing does not show: any net charge. The phosphorus nucleus still has the extra proton that came with the extra electron. Nothing has been added or removed from the crystal's total, which is the neutrality point made below.
A vital point that confuses everybody once. N-type silicon is electrically neutral. The phosphorus atom donated an electron but kept its extra proton, so the crystal has exactly as much positive charge as negative. What changed is not the total charge but the mobility of the charge: one carrier per dopant atom is now free to move, and the compensating positive charge is a fixed ion locked into the lattice, unable to go anywhere. That distinction between mobile carriers and fixed ions is the key to understanding the junction in section 5, so hold on to it.
P-type: add group III
Boron, gallium and indium sit in group III with three valence electrons. Substitute one in: it forms three bonds and the fourth bond position is left with a missing electron — a hole.
A nearby bound electron can hop in to complete that bond, and then the hole is somewhere else. Group III dopants are called acceptors, because they accept an electron from the lattice.
The result is p-type silicon, with holes as the majority carriers. It too is electrically neutral overall; the boron atom that accepted an electron becomes a fixed negative ion, balancing the mobile positive hole.
How much doping
Typical doping is 10^{15} to 10^{17} atoms per cm³, against silicon's own 5\times10^{22}. That is one dopant atom per hundred thousand silicon atoms at the heavy end, and one per fifty million at the light end.
The effect is out of all proportion to the quantity. At 10^{16} donors per cm³ you have about a million times more free electrons than pure silicon offered, so the conductivity rises by roughly a factor of a million. A contamination too small to detect by weight changes the material's electrical character completely, which is also why semiconductor fabrication plants are as obsessive about cleanliness as they are.
Heavy doping, around 10^{19} and up, is written n+ or p+. It makes the material behave almost like a metal and is used for the contact regions where a metal wire touches silicon, because a lightly doped contact would form an unwanted diode of its own.
The law that ties it together
Doping does not simply add carriers; it also suppresses the other kind. More electrons around means a hole is more likely to meet one and recombine. The equilibrium result is remarkably clean:
\boxed{np = n_i^2}
The product of electron and hole concentrations is fixed at any given temperature, regardless of doping. This is the mass action law, and it is the semiconductor version of the same equilibrium mathematics that governs chemical reactions.
Worked example. Silicon doped with N_D = 10^{16} donors per cm³ at 300 K, where n_i = 1.5\times10^{10}.
Essentially every donor contributes an electron, so n \approx 10^{16} per cm³. Then
p = \frac{n_i^2}{n} = \frac{(1.5\times10^{10})^2}{10^{16}} = \frac{2.25\times10^{20}}{10^{16}} = 2.25\times10^4\ \text{per cm}^3
Electrons outnumber holes by twelve orders of magnitude. Electrons are the majority carriers here and holes the minority carriers — and the minority carriers, despite being so rare, turn out to control the entire behaviour of the transistor in Chapter 2.3.
4. How carriers move: drift and diffusion
There are exactly two ways charge moves in a semiconductor, and every device in this Part is built from combinations of them.
Drift is motion caused by an electric field, which is what happens in an ordinary resistor. The current density is
J_{drift} = q(n\mu_n + p\mu_p)E
where \mu (mu) is mobility and E is the field. This is Ohm's law in disguise.
Diffusion is motion caused by a concentration difference, with no field involved at all. Carriers wander randomly because of thermal energy, and if there are more of them on the left than the right, more will wander rightwards than leftwards purely by counting. Drop ink in still water and it spreads without anything pushing it; this is the same process.
J_{diff} = qD_n\frac{dn}{dx} - qD_p\frac{dp}{dx}
where D is the diffusion coefficient and dn/dx is how steeply the concentration changes with position.
Diffusion has no analogue in ordinary wire circuits, and it is the mechanism that makes the junction work. Einstein showed the two coefficients are not independent:
\frac{D}{\mu} = \frac{kT}{q} = V_T
V_T is the thermal voltage, about 25.9 mV at 300 K and usually rounded to 26 mV. This number appears in every semiconductor equation from here to the end of Part 2, so it is worth knowing on sight. It is simply the energy scale of thermal jiggling expressed in volts.
5. The PN junction — the device that starts everything
Take a single crystal and dope one region p-type and the neighbouring region n-type. What happens at the boundary?
Step 1: diffusion starts
On the n side there is a huge concentration of electrons; on the p side, almost none. That is a concentration gradient, so electrons diffuse from n to p. Symmetrically, holes diffuse from p to n.
Step 2: fixed ions are left behind
Here is where section 3's point about mobile carriers versus fixed ions earns its keep. When an electron leaves the n side, it leaves behind the donor ion — a phosphorus atom with a spare proton, locked in the lattice. The n side near the boundary is now positively charged. Likewise the p side near the boundary is left with negative acceptor ions.
So a thin region either side of the junction is stripped of mobile carriers and full of fixed, charged ions. That is the depletion region — depleted of carriers, not of atoms. It is typically 0.1 to 1 µm wide.
Step 3: a field appears and stops the process
Those exposed ions create an electric field pointing from the n side (positive ions) to the p side (negative ions). That field pushes electrons back towards the n side — exactly opposing the diffusion that created it.
Equilibrium arrives when the field's push exactly cancels diffusion's push. At that point net current is zero, the depletion region stops growing, and a permanent voltage exists across it.
The built-in potential
The voltage across the depletion region at equilibrium is the built-in potential:
V_{bi} = V_T\ln\!\left(\frac{N_AN_D}{n_i^2}\right)
with V_T = 26 mV. Worked example for N_A = N_D = 10^{16} per cm³:
V_{bi} = 0.026\times\ln\!\left(\frac{10^{32}}{2.25\times10^{20}}\right) = 0.026\times\ln(4.44\times10^{11}) = 0.026\times26.8 = 0.70\ \text{V}
There is the famous 0.7 V, and it was not a measured constant handed down from a table. It came out of the doping levels and the thermal voltage. Germanium, with its much larger n_i (because of the smaller band gap), gives about 0.3 V by the same calculation, which is exactly the number quoted for germanium diodes.
The trap to avoid: you cannot measure this 0.7 V with a voltmeter across an unconnected diode. The reading will be zero. The metal-to-silicon contacts at each end form their own small barriers that cancel it exactly, which they must — otherwise you could build a battery from a diode and violate energy conservation.
6. Biasing the junction
Now apply an external voltage and watch the barrier move.
Forward bias: positive on p, negative on n
The external voltage opposes the built-in field. The barrier gets lower, so more carriers have enough thermal energy to climb it. Since the number with a given energy falls exponentially (that Boltzmann factor from section 1), lowering the barrier a little raises the current enormously.
The depletion region narrows. Once the applied voltage approaches V_{bi}, the barrier is nearly flat and current flows freely. This is why a silicon diode "turns on" at about 0.7 V.
Reverse bias: negative on p, positive on n
The external voltage adds to the built-in field. The barrier gets higher, the depletion region gets wider, and essentially no majority carriers can cross.
A tiny current still flows — the reverse saturation current I_S, typically nanoamps or less — carried by minority carriers, the rare holes on the n side and electrons on the p side. They are swept across gladly, since the field pushes them the right way. There are just very few of them.
I_S doubles roughly every 10 °C, because it depends on n_i^2, which is the exponential from section 1. That is the origin of nearly every temperature problem in analog electronics.
The diode equation
Combining the barrier physics with the Boltzmann distribution gives the Shockley diode equation:
\boxed{I = I_S\left(e^{V/nV_T}-1\right)}
where n is the ideality factor, between 1 and 2 depending on how the device is made.
Read it in two halves. Forward bias with V more than about 0.1 V: the exponential dwarfs the 1, so I \approx I_Se^{V/nV_T} — current rises exponentially with voltage. Reverse bias: the exponential collapses towards zero, leaving I \approx -I_S, a tiny constant leak.
The exponential is startlingly steep. At n=1 and V_T = 26 mV, every 60 mV of extra forward voltage multiplies the current by ten, because \ln(10)\times26\ \text{mV} = 60 mV. That is the number that makes a diode look like a fixed 0.7 V drop: to change the current by a factor of a thousand, the voltage only needs to move 180 mV.
7. Junction capacitance, and why it limits speed
The depletion region is an insulating layer with conductive material either side. That is a capacitor.
Its capacitance changes with reverse voltage, because more reverse bias means a wider depletion region, which means a bigger plate separation:
C_j = \frac{C_{j0}}{\left(1+\frac{V_R}{V_{bi}}\right)^m}
with m around 0.5 for an abrupt junction. A diode is therefore a voltage-controlled capacitor, and this is deliberately exploited in the varactor diode, which is how a radio can be tuned electronically instead of with a mechanical variable capacitor. Chapter 7.2 uses one.
In every other diode and transistor it is a nuisance, because it must be charged and discharged whenever the device switches, and that takes time and current. Junction capacitance is one of the two things that limit how fast a transistor can switch, and the reason a modern chip's speed improved as its transistors got physically smaller.
There is a second capacitance in forward bias, the diffusion capacitance, coming from the charge stored as carriers in transit across the junction. It is larger than the junction capacitance and it is why an ordinary diode cannot switch off instantly — the stored charge must be swept out first, giving a reverse recovery time of tens or hundreds of nanoseconds. Chapter 9.5 shows what that costs in a switching power supply, and why Schottky diodes, which have no stored minority charge at all, are used where speed matters.
8. Breakdown
Push reverse bias high enough and the current suddenly rises steeply. Two distinct mechanisms cause it, and they behave differently.
Avalanche breakdown happens in lightly doped junctions with wide depletion regions. A minority carrier picks up enough energy from the strong field to knock an electron out of a bond when it collides, creating a new pair; those carriers do the same; the process multiplies. It dominates above about 6 V, and its breakdown voltage rises with temperature, because hotter lattice vibrations make collisions happen sooner, before carriers can gather enough energy.
Zener breakdown happens in heavily doped junctions with very narrow depletion regions, where the field is so intense that electrons tunnel straight through the barrier — a quantum mechanical effect with no classical analogue. It dominates below about 5 V, and its breakdown voltage falls with temperature.
Between roughly 5 and 6 V both act, and their opposite temperature drifts cancel. This is why 5.6 V Zener diodes are the most temperature-stable voltage reference you can make from a single junction, a fact used in precision instruments for decades.
Breakdown is not damage. A junction in breakdown is fine as long as the power dissipated stays within its rating; the current must simply be limited externally. Chapter 2.2 builds a voltage regulator out of exactly that behaviour.
Everything in this chapter has been about one junction sitting still. Chapter 2.2 puts it to work: rectifying AC into DC, clamping and clipping signals, emitting light, and regulating a supply.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Semiconductor basics
n_i \propto T^{3/2}e^{-E_g/2kT}
The exponential is a Boltzmann factor: the fraction of electrons with enough thermal energy to cross a gap of E_g. The factor of 2 in the denominator appears because the pair creation shares the gap between an electron state and a hole state. Practical consequence: n_i roughly doubles per 10 °C in silicon.
Silicon at 300 K: n_i = 1.5\times10^{10}\ \text{cm}^{-3}, E_g = 1.12 eV.
\boxed{np = n_i^2}
The mass action law. At equilibrium the rate of pair generation (fixed by temperature) equals the rate of recombination (proportional to np), so the product is pinned.
n \approx N_D, \qquad p = \frac{n_i^2}{N_D} \qquad\text{(n-type)}
p \approx N_A, \qquad n = \frac{n_i^2}{N_A} \qquad\text{(p-type)}
The thermal voltage
V_T = \frac{kT}{q}
At 300 K: V_T = \dfrac{1.381\times10^{-23}\times300}{1.602\times10^{-19}} = 0.0259 V, universally rounded to 26 mV. It appears in every junction equation in the Part.
\frac{D}{\mu} = V_T \qquad\text{(the Einstein relation)}
Conductivity
\sigma = q(n\mu_n + p\mu_p), \qquad \rho = \frac1\sigma
Silicon mobilities at 300 K: \mu_n \approx 1350, \mu_p \approx 480\ \text{cm}^2/\text{V}\cdot\text{s}. Electrons are about 2.8 times more mobile than holes, which is why NMOS beats PMOS and NPN beats PNP.
The PN junction
V_{bi} = V_T\ln\!\left(\frac{N_AN_D}{n_i^2}\right)
Where it comes from: at equilibrium the drift current exactly cancels the diffusion current. Writing both and setting them equal, then integrating across the depletion region, produces exactly this logarithm. With N_A = N_D = 10^{16} it evaluates to 0.70 V — the famous silicon 0.7 V is a derived number, not a measured constant.
Depletion width:
W = \sqrt{\frac{2\varepsilon}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)(V_{bi}-V)}
The width grows with reverse bias (making V negative increases the bracket), which is why junction capacitance falls with reverse bias.
C_j = \frac{\varepsilon A}{W} = \frac{C_{j0}}{\left(1+V_R/V_{bi}\right)^m}, \quad m \approx 0.5
The Shockley diode equation
I = I_S\left(e^{V/nV_T}-1\right)
Forward, with V \gg V_T: I \approx I_Se^{V/nV_T}. Reverse: I \approx -I_S.
The decade rule. For the current to change by a factor of 10:
\Delta V = nV_T\ln 10 = 1\times0.026\times2.303 = 60\ \text{mV}
Sixty millivolts per decade at n=1. This is why the diode looks like a constant 0.7 V drop over a huge current range.
Temperature: V_F falls about 2 mV per °C at constant current, and I_S roughly doubles per 10 °C.
Dynamic resistance of a forward-biased diode:
r_d = \frac{dV}{dI} = \frac{nV_T}{I}
At 1 mA with n=1: 26 Ω.
What the next chapter fixes
The junction derived here is a one-way valve, and that alone is enough to build something useful. Chapter 2.2 turns it into the diode: its real current–voltage curve, the rectifier circuits that turn mains AC into the DC every device runs on, and the special cases — Zener, LED, photodiode — that are the same junction operated in different regions.