Q.Suppose a pure Si crystal has $5 \times 10^{28}$ atoms $\text{m}^{-3}$. It is doped by $1\ \text{ppm}$ concentration of pentavalent As. Calculate the number of electrons and holes. Given that $n_i = 1.5 \times 10^{16}\ \text{m}^{-3}$.

Net doping (the sample is n-type)

Donor (phosphorus) $N_D = 3\times10^{22}\,\text{m}^{-3}$ exceeds acceptor (boron) $N_A = 2.8\times10^{22}\,\text{m}^{-3}$, so under complete ionization the majority electron concentration is:

Minority (hole) concentration via the mass-action law

$n\,p = n_i^{2} \;\Rightarrow\; p = \frac{n_i^{2}}{n}$ …

Concept and Intuition

In an intrinsic (pure) semiconductor, every electron excited to the conduction band leaves a hole in the valence band, so the electron concentration $n$ equals the hole concentration $p$, and both equal the intrinsic carrier concentration $n_i$. When we dope the semiconductor to make it extrinsic, we add impurities that greatly increase one type of carrier. However, the material remains in thermal equilibrium, and a fundamental principle—the law of mass action—holds: the product $n \times p$ is constant at a given temperature, equal to $n_i^2$. So if doping raises the hole concentration, the electron concentration must drop to keep the product constant. This is not a guess; it's a direct consequence of the balance between generation and recombination of electron-hole pairs.

Step-by-Step Solution

1. Identify the intrinsic carrier concentration. The problem states that in the intrinsic semiconductor, the concentrations of holes and electrons are equal and given as $4 \times 10^8 \, \mathrm{m}^{-3}$. Therefore, the intrinsic carrier concentration is

2. Recall the law of mass action. For any non-degenerate semiconductor in thermal equilibrium, the product of the electron concentration $n$ and the hole concentration $p$ is constant:

This holds for both intrinsic and extrinsic semiconductors at the same temperature.

3. Apply the law to the extrinsic case. After doping, the hole concentration becomes

Let the new electron concentration be $n$. Then:

[!TLDR]

Reverse current depends on thermally generated minority carriers, which increase as the band gap decreases; Ge's smaller gap gives it the larger reverse current.

Concept

In CBSE Class 12 Semiconductor Electronics, the reverse saturation current of a diode is due to minority carriers produced by thermal generation across the band gap. The intrinsic carrier concentration follows $n_i^2\propto e^{-E_g/kT}$, so a larger band gap means far fewer carriers and a smaller reverse current.

Solution

• For identical dimensions and the same temperature and reverse bias, the only difference is the band gap $E_g$.
• Silicon has a larger $E_g$ than germanium, so $n_i(\text{Si}) < n_i(\text{Ge})$; fewer thermally generated minority carriers means a smaller reverse current in Si.
• Germanium, with the smaller band gap, generates more minority carriers, so its reverse current is larger. …

Concept & Intuition

An intrinsic semiconductor is a pure crystal with no intentional impurities. At absolute zero, all electrons are tightly bound in the valence band, so there are no free charge carriers. As temperature rises, thermal energy can break some covalent bonds, promoting an electron from the valence band to the conduction band. This process creates two mobile carriers: the electron (now free to move in the conduction band) and the hole (the empty state left behind in the valence band, which behaves like a positive charge carrier). Crucially, each promotion creates one electron and one hole together — they come in pairs. No other mechanism generates carriers in a pure material. Therefore, the number of electrons ($n$) must always equal the number of holes ($p$) in an intrinsic semiconductor, at any temperature.

Step-by-step reasoning

1. Definition of intrinsic semiconductor

   An intrinsic semiconductor has no dopant atoms. The only source of charge carriers is thermal excitation across the band gap. There are no impurities to donate extra electrons or accept extra holes.

2. Carrier generation mechanism

   When an electron gains enough energy (≥ band gap energy $E_g$), it jumps from the valence band to the conduction band. This single event produces exactly one free electron and one hole. The process is called electron-hole pair generation.

3. Charge neutrality condition

   The crystal as a whole must remain electrically neutral. In an intrinsic semiconductor, the total negative charge from conduction electrons equals the total positive charge from holes. Since each electron carries charge $-e$ and each hole carries $+e$, this neutrality condition is:

where $n$ is the electron concentration and $p$ is the hole concentration.

4. No other sources of carriers …

In a semiconductor $\sigma=ne\mu$ with $\mu\propto\tau$. Heating:

• promotes many electrons across the gap, so $n$ rises steeply (dominant effect);
• increases lattice vibration/scattering, so relaxation time $\tau$ (and hence mobility) falls slightly. …

Step 1 — The resistivity ladder.

Resistivity $\rho = 1/\sigma$ is what separates the three classes of solid, and the ranges are enormous (spanning ~20 orders of magnitude):

Why: in a metal the conduction band is partly filled, so carriers are freely available. In a semiconductor a band gap $E_g \approx 1$ eV must be crossed thermally, so only a small fraction of electrons ($\propto e^{-E_g/2k_BT}$) contribute — far fewer carriers, far higher $\rho$. In an insulator $E_g \gtrsim 3$ eV and essentially none cross.

Step 2 — Pin it down with a real number.

Intrinsic (pure) silicon at $300$ K:

That value lies squarely inside $10^{3}$–$10^{6}\ \Omega$cm. …

Step 1 — The condition for floating.

The iceberg floats in equilibrium, so its weight is exactly balanced by the buoyant force:

Step 2 — Write each side.

• Weight of the whole iceberg (volume $V$, density $\rho_i$):

• Archimedes' principle: the buoyant force equals the weight of the displaced water. Only the submerged volume $V_{\text{sub}}$ displaces water:

Step 3 — Equate and solve for the fraction.

The $g$ cancels (the answer is the same on any planet), leaving

Step 4 — Substitute the numbers.

Water: $\rho_w = 1.00\ \text{g cm}^{-3}$. Ice: $\rho_i = 0.917\ \text{g cm}^{-3}$ (both in the same units, so no conversion is needed).

Step 5 — Physical sense-check. …

The concept of a "hole" in semiconductor physics is one of the most subtle ideas you'll encounter. Many students try to think of it as a real particle, like an electron, but with positive charge — that's a trap. The hole is a conceptual convenience, not a fundamental particle.

In a pure semiconductor crystal, atoms are bonded together by covalent bonds — each atom shares electrons with its neighbours to complete its valence shell. At absolute zero, every electron is locked in a bond, and no current flows. But when you add energy (heat, light), an electron can break free from its covalent bond and become a "free electron" that can move through the crystal.

What's left behind? That broken bond is missing an electron. The neighbouring electrons can now "jump" into this empty spot, effectively moving the vacancy around. This moving vacancy behaves exactly like a positive charge carrier — it's called a hole.

Now let's examine each option carefully.

1. Option (A): "An anti-particle of electron."

   This sounds scientific, but it's wrong in this context. An anti-particle (like a positron) is a real fundamental particle with the same mass as an electron but opposite charge. A hole is not a fundamental particle — it's an absence in a crystal lattice. The hole's effective mass is different from an electron's, and it only exists inside the semiconductor. So (A) is incorrect.

2. Option (B): "A vacancy created when an electron leaves a covalent bond."

   This is exactly right. When an electron absorbs enough energy to break its covalent bond, it becomes a free electron. The empty bond site is the hole. This hole can move as neighbouring electrons fill it, creating a new vacancy elsewhere. This is the standard definition used in every textbook and exam.

3. Option (C): "Absence of free electrons." …

Step 1 — The concept.

In a pure (intrinsic) semiconductor electrons and holes are created in pairs, so $n_e = n_h = n_i$. When the crystal is doped, the total carrier densities change, but at thermal equilibrium the rates of thermal generation and recombination still balance, which gives the mass-action law

This is why doping to raise one carrier density automatically suppresses the other (extra holes recombine with the electrons).

Step 2 — Identify the given quantities.

"The density of an electron–hole pair in pure germanium" is the intrinsic density:

Aluminium is a trivalent (Group 13) dopant in Ge, hence an acceptor: it makes the crystal p-type, so holes become the majority carriers:

Step 3 — Compute the minority (electron) density. …