When an electric field is applied across a semiconductor (a) electrons move from lower energy level to higher

NCERT Exemplar · Q8

Q. When an electric field is applied across a semiconductor

(a) electrons move from lower energy level to higher energy level in the conduction band.

(b) electrons move from higher energy level to lower energy level in the conduction band.

(d) holes in the valence band move from lower energy level to higher energy level.

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Concept understanding — P-N Junction Formation

P-N Junction Formation: From Intuition to Precision

Imagine two rooms connected by a door. One room is filled with people who have extra energy (they want to give it away), and the other room is filled with people who are missing energy (they want to take it). The moment you open the door, what happens? People rush from the high-energy room to the low-energy room until both rooms reach a balance. That rush, and the final balanced state, is the essence of a P-N junction.

In a semiconductor, the "people" are charge carriers: electrons (negative charge) and holes (the absence of an electron, which behaves like a positive charge). A P-type semiconductor has an excess of holes (positive carriers), and an N-type semiconductor has an excess of electrons (negative carriers). When you bring them together, they don't just sit still — they interact.

The Intuitive Picture

Take a P-type crystal and an N-type crystal. At the instant they touch, there is a huge concentration difference: lots of holes on the P-side, lots of electrons on the N-side. Nature hates steep gradients, so carriers begin to diffuse — they move from where they are abundant to where they are scarce.

Electrons from the N-side cross into the P-side.
Holes from the P-side cross into the N-side.

But here's the catch: when an electron from the N-side meets a hole on the P-side, they recombine — the electron fills the hole, and both disappear as free carriers. This recombination doesn't happen everywhere; it happens in a narrow region near the interface, called the depletion region (or space-charge region).

Why "depletion"? Because in that region, free electrons and free holes have been used up. All that remains are the fixed, immovable ions: positive donor ions on the N-side (which lost their electron) and negative acceptor ions on the P-side (which gained an electron). These fixed charges create an electric field that points from the N-side (positive ions) to the P-side (negative ions).

This electric field is crucial. It acts like a bouncer: it pushes electrons back toward the N-side and holes back toward the P-side. This drift motion opposes the initial diffusion. Eventually, the diffusion current (driven by concentration difference) exactly balances the drift current (driven by the electric field). The system reaches thermal equilibrium — no net current flows.

The Precise Statement

\text{A P-N junction is formed by bringing P-type and N-type semiconductors into intimate contact. At equilibrium, a depletion region of fixed ions creates a built-in electric field that prevents further net diffusion of carriers.}

More formally:

Diffusion: Majority carriers (electrons from N-side, holes from P-side) diffuse across the junction due to the concentration gradient.
Recombination: These carriers recombine near the interface, leaving behind fixed ionized impurities (donors on N-side, acceptors on P-side).
Depletion region: A region devoid of free carriers, containing only fixed charges, forms at the junction.
Built-in electric field: The fixed charges create an electric field ( $\vec{E}$ ) pointing from N to P.
Equilibrium: The drift current due to $\vec{E}$ exactly cancels the diffusion current. The net current is zero.

The width of the depletion region ( $W$ ) depends on the doping concentrations. For a one-sided junction (heavily doped on one side), the depletion region extends mostly into the lightly doped side. …

Why this formula?

Why the P-N Junction Forms: The Physics Behind the Barrier

A p-n junction isn't just two pieces of semiconductor stuck together. The key to understanding it is this: nature hates sharp gradients in carrier concentration. When you bring p-type (excess holes) and n-type (excess electrons) material into contact, carriers immediately begin to diffuse across the junction — holes from p to n, electrons from n to p.

This diffusion is the engine that drives everything else.

Step 1: Diffusion Creates a Depletion Region

As holes leave the p-side, they leave behind fixed, negatively charged acceptor ions ( $A^-$ ). As electrons leave the n-side, they leave behind fixed, positively charged donor ions ( $D^+$ ). These ions are immobile — they're locked in the crystal lattice.

The region near the junction that gets stripped of mobile carriers is called the depletion region (or space-charge region). It contains only fixed ions, creating an electric field that points from the n-side (positive ions) toward the p-side (negative ions).

Watch out

Do not confuse "depletion" with "no charge." The depletion region is highly charged — it's just that the charge is from fixed ions, not mobile carriers.

Step 2: The Electric Field Opposes Diffusion

The built-in electric field $E$ exerts a force on any mobile carrier that tries to cross:

Holes (positive) feel a force pushing them back toward the p-side.
Electrons (negative) feel a force pushing them back toward the n-side.

This field grows stronger as more carriers diffuse and more ions are uncovered. Eventually, the field becomes strong enough that the drift current (carriers swept by the field) exactly balances the diffusion current (carriers moving due to concentration gradient). At this point, the net current is zero — thermal equilibrium is reached.

Step 3: The Built-in Potential Barrier

Because the electric field exists over a distance, there is a potential difference across the depletion region. This is the built-in potential $V_0$ (also called $V_{bi}$ ). It represents the energy barrier that a majority carrier must overcome to cross to the other side.

V_0 = \frac{kT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right)

Where:

$k$ = Boltzmann constant
$T$ = absolute temperature
$q$ = electron charge magnitude
$N_A$ = acceptor doping concentration (p-side)
$N_D$ = donor doping concentration (n-side)
$n_i$ = intrinsic carrier concentration

Why This Formula Holds: The Derivation

The derivation comes from equating the Fermi levels on both sides. In equilibrium, the Fermi level must be constant throughout the entire structure.

On the p-side, the Fermi level $E_F$ lies close to the valence band. The position relative to the intrinsic Fermi level $E_i$ is:

E_F - E_i = kT \ln\left(\frac{N_A}{n_i}\right) \quad \text{(for p-type)}

On the n-side, the Fermi level lies close to the conduction band:

E_F - E_i = -kT \ln\left(\frac{N_D}{n_i}\right) \quad \text{(for n-type)}

The difference in $E_i$ between the two sides (which is the same as the difference in $E_F$ between the two sides before contact) must be accommodated by the built-in potential. The total band bending $qV_0$ equals this difference:

qV_0 = \left[ kT \ln\left(\frac{N_A}{n_i}\right) \right] - \left[ -kT \ln\left(\frac{N_D}{n_i}\right) \right]

$qV_0 = kT \left[ \ln\left(\frac{N_A}{n_i}\right) + \ln\left(\frac{N_D}{n_i}\right) \right]$ …

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The field tilts the energy bands. Conduction-band electrons drift opposite to the field and roll toward lower energy levels (B), while valence-band holes drift along the field and rise toward higher energy levels (D). Correct options: (B) and (D).

Principle

When an electric field $\vec{E}$ is applied across a semiconductor, it tilts the energy bands -- the conduction and valence band edges are no longer flat but slope in the direction of the field. Both electrons and holes drift in response to the field, but because their charges are opposite, they drift in opposite spatial directions -- and, plotted on the same band-energy diagram, they move in opposite directions along the energy axis too.

Electrons in the conduction band

An electron carries charge $-e$ , so it accelerates opposite to $\vec{E}$ . On the tilted band diagram, moving opposite to the field means moving toward the lower side of the tilt. An electron therefore "rolls downhill," settling toward a lower energy level in the conduction band as it drifts.

\text{electron drifts opposite to } \vec{E} \;\Rightarrow\; \text{moves to a lower level in the conduction band.}

So statement (B) is correct and (A) is wrong.

Holes in the valence band …

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Q.When an electric field is applied across a semiconductor

Concept understanding — P-N Junction Formation

P-N Junction Formation: From Intuition to Precision

The Intuitive Picture

The Precise Statement

Why this formula?

Why the P-N Junction Forms: The Physics Behind the Barrier

Step 1: Diffusion Creates a Depletion Region

Step 2: The Electric Field Opposes Diffusion

Step 3: The Built-in Potential Barrier

Why This Formula Holds: The Derivation

Principle

Electrons in the conduction band

Holes in the valence band …

Method: Deriving Carrier Drift Direction from Band Bending, Not Kinetic Energy

Q.When an electric field is applied across a semiconductor

Principle

Electrons in the conduction band

Holes in the valence band …

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