Chapter 2: Carrier Modeling

The big question of this chapter. A silicon crystal contains about $5 \times 1 0^{22}$ atoms in every cubic centimetre, each with 14 electrons. Yet a piece of silicon conducts electricity only through a tiny, controllable handful of them (roughly $1 0^{10}$ per cm $^{3}$ in pure Si). Who are those few special electrons, what is the strange “anti-particle” called a hole, how many of each are there, and how can we dial that number up or down on purpose? By the end of the chapter you will be able to answer all of that with a single, tidy set of equations.

Carriers are the entities that carry charge from place to place inside a material and hence give rise to electrical currents. In daily life the carrier you meet is the electron in a copper wire. In semiconductors you also meet a second, equally important carrier, the hole.

Equilibrium, the “rest” condition. Everything in this chapter assumes equilibrium: no external voltages, magnetic fields, stresses, light, or other perturbations, and every observable is independent of time. Think of a perfectly still pond. If you understand the still pond, you can predict what happens when you throw a stone in. Equilibrium is the still pond of semiconductor physics. Whenever a formula in this chapter is only valid at equilibrium, remember that it will not hold once we apply a voltage or shine light (later chapters).

The author warns that some results are quoted without proof; the philosophy is that knowing how to interpret and use a result matters more than being able to derive it. (In this study version I have nevertheless filled in every algebraic step that the book leaves out.) Appendix A of the book gives a deeper account of quantization, and Section 2.1 may be replaced by it without loss of continuity.

2.1 The Quantization Concept

Instead of diving straight into crystalline silicon (14 electrons per atom, $5 \times 1 0^{22}$ atoms/cm $^{3}$ ), we first learn the ground rules on the simplest possible atom, hydrogen.

The puzzle that started it all

In the early 1900s, scientists knew a hydrogen atom was a light negative particle orbiting a much heavier positive nucleus. What they could not explain: when hydrogen gas is heated it glows only at certain discrete wavelengths (sharp colored lines), whereas classical physics predicted a smooth rainbow (continuum).

Analogy. A violin string clamped at both ends can only sustain certain notes (standing waves); you cannot get a pitch “in between” the allowed harmonics. The hydrogen electron, it turns out, is also a wave trapped around a nucleus, so only certain “notes” (energies) are allowed.

Bohr’s solution (1913)

Niels Bohr postulated that the electron can only occupy certain well-defined orbits; equivalently, its angular momentum can only take the values $n ℏ$ . That restriction is quantization. Combining it with Coulomb attraction and circular-orbit mechanics gives the binding energy of the electron:

E_{H} = - \frac{m _{0} q ^{4}}{2 ( 4 π ε _{0} ℏ n ) ^{2}} = - \frac{13.6}{n ^{2}} eV, n = 1, 2, 3, \dots (2.1)

where

$E_{H}$ is the electron energy (binding energy; negative means “bound”),
$m_{0}$ is the free-electron rest mass,
$q$ is the magnitude of the electron charge,
$ε_{0}$ is the permittivity of free space,
$ℏ = h /2 π$ ( $h$ = Planck’s constant),
$n$ is the energy quantum number (orbit identifier).

The electron-volt is the energy gained by one electron falling through 1 V: $1 eV = 1.6 \times 1 0^{- 19} J .$

Energy ladder (Fig. 2.1). The figure shows the first allowed orbits and their energies:

$n$	$E_{H}$
1	$- 13.6$ eV
2	$- 3.4$ eV
3	$- 1.51$ eV

(Check: $13.6/4 = 3.4$ , $13.6/9 = 1.51$ .)

Why spectral lines are discrete. When the electron drops from a higher $n$ to a lower $n$ , the atom releases the energy difference as a photon. Because only certain energies exist, only certain differences (and therefore only certain wavelengths) exist. For instance $n = 2 \to 1$ releases $Δ E = - 3.4 - (- 13.6) = 10.2 eV .$

Ionization energy. To free a ground-state ( $n = 1$ ) electron entirely you must supply $∣ E_{H} (n = 1) ∣ = 13.6$ eV.

From hydrogen to silicon (Fig. 2.2)

The one big lesson: the energy of an electron in an atomic system is restricted to a limited set of values. A multi-electron atom is more complicated, but its energy-level structure is still easy to describe for silicon.

Of silicon’s 14 electrons, 10 occupy very deep-lying, tightly bound levels. They stay essentially undisturbed in chemical reactions or normal atom-atom interactions. The nucleus plus these ten electrons is called the core.
The remaining 4 electrons are weakly bound and take part strongly in chemistry; they are the valence electrons.
Figure 2.2: the four valence electrons, if unperturbed, occupy four of the eight allowed states of the next-higher shell ( $n = 3$ ): two states at the “s” energy and six states at the (slightly higher) “p” energy. Deeper, the $n = 1$ shell holds 2 electrons and the $n = 2$ shell holds 8, totalling 10 core electrons.
Germanium (32 electrons) has essentially the same valence structure with a 28-electron core.

(Footnote of the book: not only energy, but many observables of atomic-size particles are quantized; Quantum Mechanics is the field developed to describe them.)

2.2 Semiconductor Models

This section introduces two visualization aids used constantly in device analysis. It is appropriate in a chapter on carrier modeling because we are modeling the carrier “container,” the crystal.

2.2.1 Bonding Model

An isolated Si atom has 4 valence electrons. In the diamond lattice (Fig. 1.4c) each atom has 4 nearest neighbors and shares one valence electron with each. This equal sharing is covalent bonding.

Fig. 2.3 (the bonding model). A flat grid in which

each circle is the core of a Si atom,
each line is a shared valence electron. Eight lines touch each atom because an atom both contributes four shared electrons and accepts four from neighbors (two lines per bond, one from each partner).

The 2-D picture is an idealization to help mental visualization.

Sample uses (Fig. 2.4):

(a) A missing atom (point defect) is drawn by erasing one circle with its eight line-ends.
(b) A broken bond is drawn by deleting one line segment (one shared electron) and showing that electron wandering free.

Strictly, the perfect picture of Fig. 2.3 is valid only at $T \approx 0$ K with no defects and no impurities, since bond breaking (at $T > 0$ ) and defects occur naturally.

What the bonding model cannot do: it says nothing about electron energies. For that we need the energy band model.

2.2.2 Energy Band Model

Building the band picture step by step (Fig. 2.5).

Start with $N$ isolated Si atoms very far apart. Each has the same discrete levels. Ten electrons per atom are core, so ignore them. Four valence electrons per atom occupy 4 of the 8 states of the outer shell. In total: $2 N$ s-states and $6 N$ p-states (that is $8 N$ states) and $4 N$ valence electrons.
Now bring the atoms closer together. Their wavefunctions overlap and interatomic forces shift the energy levels. Because the Pauli exclusion principle forbids two electrons occupying the same state, each of the $N$ nearly identical levels must split into slightly different levels. With $N \sim 1 0^{22}$ the levels are so closely spaced that they form quasi-continuous energy bands.

Analogy: $N$ identical tuning forks placed on a shared table begin to couple and ring at slightly different frequencies; the single pure note becomes a smear (a band) of tones.
At the actual Si lattice spacing, the allowed states divide into two bands separated by an energy gap:
- upper band: the conduction band,
- lower band: the valence band,
- gap in between: the forbidden gap or band gap $E_{G}$ . No allowed electron states exist there.
Each band contains $4 N$ states. Since there are exactly $4 N$ valence electrons and electrons fall to the lowest available energies (single occupancy per state, Pauli), the valence band is (almost completely) filled and the conduction band is (almost completely) empty. At $T \to 0$ K the valence band is completely full and the conduction band completely empty.
A further crucial fact: unlike the valence electrons of an isolated atom, band electrons are not tied to any one atom. On average four electrons are shared between an atom and its neighbors, but which electrons they are changes in time; the electrons roam. The allowed states belong to the crystal as a whole. In a perfect crystal one sees the same allowed-state configuration at every point.

Therefore, a plot of allowed electron energy versus position $x$ along a crystal direction gives horizontal bands (Fig. 2.5, right). We name:

$E_{c}$ : the lowest possible conduction band energy (conduction band edge),
$E_{v}$ : the highest possible valence band energy (valence band edge),
$E_{G} = E_{c} - E_{v}$ : the band gap energy.

Fig. 2.6, the energy band diagram. In practice we draw only two horizontal lines $E_{c}$ and $E_{v}$ with a gap between them. The line showing the top of the conduction band, the line showing the bottom of the valence band, the filling pattern, the energy axis label (vertical) and the position axis label (horizontal) are all understood but not drawn. This is the “shorthand” version used everywhere afterwards.

2.2.3 Carriers

No broken bonds, no carriers (Fig. 2.7a). In the bonding model a perfect lattice has no mobile charges. Equivalent energy-band statement: valence band completely full, conduction band completely empty means no carriers and no current.

Subtle puzzle. In the band model valence electrons do move around the crystal. Why do they not carry current? Answer: momentum is quantized too. For every state with momentum $+ p$ there is a state with momentum $- p$ of equal magnitude. If a band is completely filled, all momenta cancel in pairs and the net momentum is exactly zero: $\sum_{filled band} p = 0 \Rightarrow no net current .$ Think of a completely full auditorium: everyone is in motion in their heads, but nobody can go anywhere, so nothing is transported.

The electron (Fig. 2.7b).

Bonding model: a Si-Si bond breaks and the released electron wanders through the lattice. That wandering electron is a carrier.
Band model: an electron is excited from the valence band to the conduction band; electrons in the conduction band are carriers.

The energy needed to break a bond is the band gap energy $E_{G}$ . “Free bonding-model electron” and “conduction band electron” are two names for the same thing; from now on “electron” with no modifier means a conduction band electron.

The hole (Fig. 2.7c). Breaking a bond leaves a missing bond (a void).

Bonding model: a neighboring bound electron can hop into the void, which moves the void to where that electron came from. The void migrates.
Band model: removing an electron from the valence band leaves an empty state in an otherwise filled sea of states. Like a bubble in a liquid, the empty state moves rather freely because of the cooperative motion of valence electrons.

That mobile missing bond or empty valence-band state is the hole, the second kind of carrier.

Intuition. Picture a parking garage with every space full except one, and cars shift into the vacancy one by one. Watching the cars, you track dozens of moving objects. Watching the empty space, you track a single object moving the opposite way. It is much simpler to follow the hole than the sea of electrons. That is why a hole behaves as a positive particle.

A valence-band hole is on an equal footing with a conduction-band electron. Both participate in most devices, and holes are the main carrier in some devices. With familiarity one thinks of the hole as a subatomic particle.

2.2.4 Band Gap and Material Classification

The band picture of Fig. 2.6 applies, with slight modification, to all materials. What differs is the size of the gap (Fig. 2.8).

Class	Band gap	Result at room temperature
Insulator	very wide, e.g. diamond $\approx 5$ eV, SiO $_{2}$ $\approx 8$ eV	thermal energy excites almost no electron across the gap, so few carriers, poor conductor
Semiconductor	moderate: GaAs 1.42 eV, Si 1.12 eV, Ge 0.66 eV (at 300 K)	thermal excitation makes a moderate number of carriers, intermediate conductivity
Metal	very narrow or zero (valence and conduction bands overlap)	abundant carriers always, excellent conductor

Analogy. The gap is a wall between the ground floor (valence band) and the upper floor (conduction band). In an insulator the wall is a skyscraper’s height, in a metal there is no wall, and in a semiconductor it is a wall that thermal “jumps” can occasionally clear.

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