Intrinsic Semiconductor

Fig. 14.3 is a three-dimensional ball-and-stick model of the diamond-cubic crystal structure — the arrangement of atoms in carbon (diamond), silicon, and germanium. The figure shows a cube with atoms placed at every corner, at the centre of each face, and at four interior positions that sit in a tetrahedral pattern. Each atom is connected by a stick (representing a covalent bond) to its four nearest neighbours, giving the entire lattice its characteristic tetrahedral bonding. A faint cube outline frames the structure, and along one edge a dimension arrow marks the lattice constant $a$ — the side length of the cube. The caption notes that for carbon $a = 3.56\ \text{Å}$, for silicon $a = 5.43\ \text{Å}$, and for germanium $a = 5.66\ \text{Å}$.

The physical idea this figure teaches is that each atom in the crystal is surrounded by exactly four nearest neighbours, and each atom shares one of its four valence electrons with each neighbour. These shared electron pairs form covalent bonds that hold the crystal together. At low temperatures, all bonds are intact and no free charge carriers exist — the material behaves like an insulator. As temperature rises, thermal energy can break some covalent bonds, freeing an electron and leaving behind a vacancy called a hole. The free electron and the hole both contribute to electrical conduction.

The textbook uses this figure to introduce the concept of intrinsic carrier concentration. In an intrinsic semiconductor, the number of free electrons $n_e$ equals the number of holes $n_h$, and this common value is denoted $n_i$:

Here $n_i$ is the intrinsic carrier concentration — the density of electron-hole pairs thermally generated in a pure semiconductor at a given temperature. The total current $I$ in such a material is the sum of the electron current $I_e$ and the hole current $I_h$:

$I = I_e + I_h$ …

What Fig. 14.4 Shows

The figure is a simplified two-dimensional sketch of the actual three-dimensional diamond-like crystal structure of silicon or germanium. Each atom is drawn as a circle labelled +4, which represents the inner core of the atom — the nucleus plus the ten tightly bound inner electrons (for Si, the $1s^2 2s^2 2p^6$ configuration; for Ge, the $1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10}$ core). The +4 charge indicates that after the four valence electrons are shared, the core effectively has a net charge of $+4e$.

The atoms are arranged in a regular square grid. Every atom is connected to its four nearest neighbours (up, down, left, right) by a covalent bond, drawn on the real figure as a dashed/scalloped curved line with small dots placed along it — the dashed line represents the bond itself and each dot represents one of the two shared bonding electrons (the figure's own legend lists these as two separate symbols: the bond line, and the bonding electron dot). This is the textbook's "overemphasised" two-dimensional picture: in reality, the bonds are not all in one plane, but the essential idea of four bonds per atom is correct.

The key point of the figure is that every bond is intact. No bonds are broken, no electrons have been knocked loose. This is the situation at very low temperatures (near 0 K), where thermal energy is insufficient to break any covalent bonds. The crystal is a perfect insulator at this stage — there are no free charge carriers.

The Physical Idea

The figure establishes the reference state for understanding how semiconductors conduct. In this idealised low-temperature picture, all four valence electrons of each Si or Ge atom are locked into covalent bonds. These electrons are not free to move through the crystal; they are localised between pairs of atoms. Consequently, there are no conduction electrons and no holes — the material behaves like an insulator.

As temperature rises, thermal energy becomes available. Some electrons gain enough energy to break free from their covalent bonds. When an electron leaves a bond, it becomes a free electron (charge $-q$) that can move through the crystal and contribute to conduction. The vacancy it leaves behind — a missing electron in what was a complete bond — behaves as a hole with an effective positive charge $+q$. This is shown in the subsequent Fig. 14.5(a), which builds directly on the intact-bond picture of Fig. 14.4.

The Key Formula

The textbook uses this figure to introduce the fundamental relation for intrinsic semiconductors:

Here:

• $n_e$ is the number density of free electrons (conduction electrons) per unit volume
• $n_h$ is the number density of holes per unit volume
• $n_i$ is the intrinsic carrier concentration — the common value for a pure, undoped semiconductor at a given temperature

The equality $n_e = n_h$ follows directly from the generation process shown in Fig. 14.5(a): every time a covalent bond breaks, it creates exactly one free electron and one hole. No other source of charge carriers exists in an intrinsic semiconductor. The total current is the sum of electron and hole contributions:

where $I_e$ is the electron current and $I_h$ is the hole current.

Why This Matters for the Energy Band Picture …

Fig. 14.5 is the key picture that makes the idea of a “hole” physically concrete. It shows two snapshots of the same small piece of a silicon or germanium crystal, drawn as a 2‑D grid of +4 cores connected by covalent bonds (each bond is a pair of shared electrons).

Panel (a) shows what happens when thermal energy breaks one covalent bond. At the site labelled site 1, the bond is missing one electron — that electron has been knocked free and is now a conduction electron, shown as a separate particle moving away from the lattice with the text label "(Thermally generated free electron)." The empty bond position left behind is the hole, marked with the text label "Hole at site 1 (electron vacancy)." (Note: the real figure labels these two events with plain text only — it does not print "$-q$"/"$+q$" charge symbols directly on the diagram; the charge notation is discussed only in the surrounding body text.) So in (a) you see the generation of an electron–hole pair: one free electron and one hole, created together.

Panel (b) shows how the hole can appear to move. An electron from a neighbouring intact bond at site 2 (lower‑left in the figure) jumps along an arrow into the hole at site 1. After that jump, site 1 now has a complete bond (the hole is filled), but site 2 now has a missing electron — a new hole. The hole has apparently shifted from site 1 to site 2. The original free electron from panel (a) is not involved in this motion; it moves independently through the crystal.

The physical idea is that in an intrinsic semiconductor at moderate temperatures, thermal energy creates a small number of electron–hole pairs. Both the free electrons and the holes can carry current. The textbook uses this figure to introduce the intrinsic carrier concentration:

where $n_e$ is the number density of free electrons, $n_h$ is the number density of holes, and $n_i$ is the intrinsic carrier concentration (a material‑ and temperature‑dependent constant). Because each broken bond creates one electron and one hole, the two densities are equal in a pure (intrinsic) semiconductor.

The total current under an applied electric field is the sum of the electron current and the hole current:

Here $I_e$ is due to the drift of free electrons (negative charges moving toward the positive terminal), and $I_h$ is due to the apparent drift of holes (positive‑effective charges moving toward the negative terminal). The figure makes clear that these two contributions come from the same fundamental process — bond breaking — but the carriers move independently once created. …

Fig. 14.6 is an energy-band diagram, not a plot with axes. It shows two vertical panels side by side, labelled (a) and (b). In each panel, you see two horizontal bands: the upper one is the conduction band and the lower one is the valence band. The vertical gap between them is the forbidden energy gap $E_g$. The left panel, (a), is labelled “$T = 0$ K”. Here the conduction band is completely empty — no electrons — and the valence band is completely filled. The right panel, (b), is labelled “$T > 0$ K”. In this panel, four filled circles (electrons) appear in the conduction band, and four empty circles (holes) appear in the valence band. The labels $E_C$ and $E_V$ mark the bottom of the conduction band and the top of the valence band, respectively.

The physical idea is straightforward. At absolute zero, every electron sits in the lowest possible energy state. In an intrinsic semiconductor, that means the valence band is full and the conduction band is empty. With no electrons in the conduction band, there is no way for current to flow — the material behaves exactly like an insulator. That is panel (a).

As temperature rises, thermal energy becomes available. Some electrons in the valence band gain enough energy to jump across the gap $E_g$ into the conduction band. Each such jump leaves behind a vacancy in the valence band — a hole. The figure shows four such events: four electrons have moved up, and four holes remain below. Because the semiconductor is intrinsic (no impurities), every electron in the conduction band comes from the valence band, so the number of electrons $n_e$ equals the number of holes $n_h$. This is the defining relation:

Here $n_i$ is the intrinsic carrier concentration — the number of free electrons (or holes) per unit volume in a pure semiconductor at a given temperature. The value of $n_i$ depends strongly on $E_g$ and on temperature $T$ through the relation

where $N_C$ and $N_V$ are the effective densities of states in the conduction and valence bands, and $k_B$ is Boltzmann’s constant. The figure itself does not show this formula, but it is the quantitative consequence of the picture: the larger $E_g$, the fewer electron-hole pairs are generated at a given temperature. …