Intrinsic Semiconductors

The Crystal and Its Bonds

Si and Ge crystallise in the diamond-like structure: every atom is surrounded by four nearest neighbours (lattice spacings: C 3.56, Si 5.43, Ge 5.66 angstrom). Each atom has four valence electrons and shares one with each neighbour, receiving a share back — the shared pairs forming covalent bonds that shuttle between atoms and hold the crystal strongly together.

At low temperatures all bonds are intact — the idealised picture of NCERT Fig. 14.4 — and there are no free carriers.

Thermal generation: electrons AND holes

As temperature rises, thermal energy breaks a few bonds. Each break releases a free electron (charge -q) into the crystal — and the vacancy left behind acts as an effective positive charge +q: a hole.

Key Point: The hole is not a particle in its own right — it is a missing bound electron. But it behaves exactly like an apparent free particle of charge +q, and treating it so is the master trick of semiconductor physics.

In an intrinsic (pure) semiconductor:

$$n_e = n_h = n_i$$

where $n_i$ is the intrinsic carrier concentration.

How Holes Move, and the Two Currents

Suppose a hole sits at site 1. A bound electron from a neighbouring bond (site 2) can jump into the vacancy — now the hole is at site 2. The vacancy has effectively moved — without the originally freed electron being involved at all. That free electron wanders independently as a conduction electron.

Under an electric field:

• Free electrons drift (against the field) → electron current $I_e$.
• Holes drift toward negative potential (along the field) → hole current $I_h$.

$$\boxed{I = I_e + I_h}$$

(Remember: hole motion is 'only a convenient way of describing the actual motion of bound electrons' — but the convenience is total.)

Generation and recombination

Alongside generation, recombination runs continuously: a free electron collides with a hole and the pair vanishes (the bond reforms). At equilibrium, rate of generation = rate of recombination, fixing $n_i$ at each temperature.

The band picture

• T = 0 K: valence band full, conduction band empty — the intrinsic semiconductor behaves like an insulator.
• T > 0 K: thermal energy lifts some electrons VB → CB; the CB is partially occupied, the VB carries an equal number of holes (NCERT Fig. 14.6).

[NEET Important] Three instant-recall facts: intrinsic means $n_e = n_h$; total current has TWO components ($I_e + I_h$); at 0 K the material is an insulator. [JEE Tip] $n_i$ grows steeply (exponentially) with temperature — so intrinsic conductivity RISES with T, opposite to metals, where lattice vibrations increase resistivity. 'Semiconductor resistance falls on heating' is the classic discriminator.

Solved Examples

Example 1: Counting bonds and carriers [Board Conceptual]

In pure silicon at low temperature, how many covalent bonds does each atom form, and what carriers exist?

Solution:

1. Each Si atom shares its 4 valence electrons with 4 neighbours: 4 covalent bonds per atom (each bond shared between two atoms).
2. At low temperature all bonds are intact: no free electrons, no holes — the crystal is effectively an insulator.
3. Takeaway: carriers in intrinsic material exist only by thermal bond-breaking; cold crystals have none.

Example 2: The hole's charge and origin [NEET Conceptual]

What exactly is a hole, and why does it carry +q?

Solution:

1. A broken bond leaves a vacancy where an electron (-q) used to be.
2. The neighbourhood, previously neutral, is now short one electron: net effective charge +q localised at the vacancy.
3. Bound electrons hopping into the vacancy shift it — the vacancy moves like a free positive particle.
4. Takeaway: hole = electron vacancy with effective charge +q; its 'motion' is bookkeeping for bound-electron hops.

Example 3: Electron-hole pair arithmetic [Board Numerical]

In a pure Ge sample at 300 K, $n_i = 2.4 \times 10^{19}$ m⁻³. State $n_e$ and $n_h$, and the effect of doubling the temperature qualitatively.

Solution:

1. Intrinsic: $n_e = n_h = n_i = 2.4 \times 10^{19}$ m⁻³.
2. Higher temperature breaks exponentially more bonds → $n_i$ rises steeply (not linearly).
3. Takeaway: in intrinsic material the two carrier types are always equal — every broken bond makes exactly one of each.

Example 4: Two currents, one total [JEE Conceptual]

In an intrinsic semiconductor with an applied field, electrons carry 60% of the current. What carries the rest, and in which direction do the two carrier types move?

Solution:

1. The remaining 40% is hole current $I_h$: I = $I_e + I_h$.
2. Electrons (negative) drift opposite to the field; holes (positive) drift along the field, toward the negative terminal.
3. Both currents ADD in the conventional sense (opposite charges moving opposite ways = same conventional current direction).
4. Takeaway: the two contributions never cancel — they cooperate.

Example 5: Why intrinsic conductivity is feeble [Conceptual Numerical]

Silicon has ~$5 \times 10^{28}$ atoms/m³ but $n_i \approx 1.5 \times 10^{16}$ m⁻³ at 300 K. What fraction of atoms is ionised?

Solution:

1. Fraction: $\dfrac{1.5 \times 10^{16}}{5 \times 10^{28}} = 3 \times 10^{-13}$.
2. Roughly one atom in three trillion has a broken bond — conduction is a rare-event phenomenon.
3. Takeaway: this feebleness is exactly why doping (next section) is needed to make useful devices.

Example 6: Recombination equilibrium [Board Conceptual]

Why doesn't the number of carriers in an intrinsic semiconductor grow without limit at fixed temperature?

Solution:

1. Generation (bond-breaking) creates pairs at a temperature-dependent rate.
2. Recombination — an electron colliding with a hole and refilling the bond — destroys pairs at a rate that grows with carrier numbers.
3. Steady state: generation rate = recombination rate, pinning $n_i$(T).
4. Takeaway: $n_i$ is a dynamic equilibrium value, not a fixed stock.

Example 7: Semiconductor vs metal on heating [NEET Discriminator]

A metal wire and a pure Si rod are both heated. Compare the changes in their resistance and explain.

Solution:

1. Metal: carrier number ~fixed; lattice vibrations scatter electrons more → resistance increases.
2. Silicon: thermal generation multiplies carriers exponentially — this swamps extra scattering → resistance decreases.
3. Takeaway: opposite temperature coefficients — the single most-tested intrinsic-semiconductor fact (and the basis of thermistors).

Example 8: Which electron moves in hole conduction? [Conceptual]

When a hole 'moves' from site 1 to site 2, trace the actual electron motion.

Solution:

1. A bound electron in the covalent bond at site 2 jumps to the vacancy at site 1.
2. Site 1 is now complete; site 2 carries the vacancy: the hole has apparently moved 1 → 2, opposite to the electron's hop.
3. The originally freed conduction electron is elsewhere entirely, contributing to $I_e$ independently.
4. Takeaway: hole current = choreography of bound electrons; conduction-electron current = free-particle motion. Two distinct channels.

Example 9: Band diagram at two temperatures [Board Sketch]

Describe the intrinsic band diagram at T = 0 K and T > 0 K.

Solution:

1. T = 0 K: valence band completely full, conduction band completely empty, gap $E_g$ between — an insulator (NCERT Fig. 14.6a).
2. T > 0 K: a few electrons occupy the conduction band (drawn as filled circles), leaving an equal number of holes (empty circles) at the top of the valence band (Fig. 14.6b).
3. Takeaway: always draw equal electron and hole counts for intrinsic material — unequal counts signal doping.

Example 10: Photogeneration preview [JEE Link]

Light of wavelength 600 nm falls on Ge ($E_g$ = 0.7 eV) and on C ($E_g$ = 5.4 eV). In which does it create electron-hole pairs?

Solution:

1. Photon energy: $\dfrac{1240}{600} \approx 2.07$ eV.
2. Ge: 2.07 > 0.7 eV ✔ — pairs are generated (photoconductivity).
3. C: 2.07 < 5.4 eV ✘ — diamond stays transparent and insulating.
4. Takeaway: any energy source beating $E_g$ — heat, light, voltage — creates pairs; the comparison photon-energy-vs-gap runs through LEDs and photodiodes later.