---
title: "Hall resistance and its quantization"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:hall-resistance-quantization"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/hall-resistance-quantization
site: BCT Engineering
---

# Hall resistance and its quantization

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2071 Shrawan, 5 marks
- Video: [Quantum Hall Effect Full Explanation | Hall Effect, Landau Levels, 2DEG, Hall Resistance](https://www.youtube.com/watch?v=IdPNGaQYbIA) — Plume Verse
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/hall-resistance-quantization
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Explain Hall Effect. Derive a relation for Hall resistance. From this relation explain the meaning of quantization of Hall resistance.

## Short answer

Hall effect produces transverse voltage V_H = IB/(net). Hall resistance is R_H = B/(e n_s). In quantized Hall effect, 2D systems exhibit plateaus R_H = h/(ν e²) (ν integer) from Landau level filling, used as a universal resistance standard (~25.8 kΩ for ν=1).

## Answer

#### Hall effect
When a current-carrying conductor is placed in a magnetic field perpendicular to the current, a voltage appears across the conductor, perpendicular to both the current and the field. This is the **Hall effect**.

```text
        B (into page)
   +-----------------------+
   |  ->  ->  ->  I  ->    |  top edge:    + (or -)
   |                       |
   +-----------------------+  bottom edge: - (or +)
        V_H across width
```

Moving carriers feel the force F = qv_d × B, so they pile up on one edge. This builds a transverse field E_H until it cancels the magnetic force.

#### Hall resistance
1. Balance of forces: eE_H = e v_d B, so E_H = v_d B.
2. Hall voltage across width w: V_H = E_H w = v_d B w.
3. Drift speed from I = n e v_d (w t): v_d = I/(n e w t), where t is the thickness.
4. So V_H = IB/(n e t).
5. Hall resistance: **R_H(res) = V_H/I = B/(n e t) = B/(e n_s)**, where n_s = nt is the number of electrons per unit area (sheet density).

It rises linearly with B in an ordinary conductor.

#### Quantization of Hall resistance
In a 2D electron gas at very low temperature (~1 K) and strong field (~10 T), the electron energies form Landau levels. Each level holds n_s = eB/h electrons per unit area. If ν levels are completely filled, n_s = ν eB/h. Putting this in step 5:

**V_H/I = h/(ν e²)**, with ν = 1, 2, 3, …

So the Hall resistance does not vary smoothly with B. It stays on flat **plateaus** at h/e², h/2e², h/3e², … with h/e² = 25 812.8 Ω. The values depend only on the fundamental constants h and e, not on the sample, and this is used as the resistance standard (integer quantum Hall effect).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Hall voltage appears:
   A. along the conductor, parallel to the current
   B. across the conductor, perpendicular to both the current and the magnetic field
   C. parallel to the magnetic field
   D. only when the field is parallel to the current
   Answer: B. Moving carriers are pushed sideways by qv×B and pile up on one edge, creating a transverse voltage. (Key point: Hall effect)

2. The transverse Hall field E_H builds up until:
   A. the electric force eE_H balances the magnetic force e v_d B
   B. the current drops to zero
   C. the magnetic force vanishes
   D. the carriers stop moving along the wire
   Answer: A. At balance eE_H = e v_d B, so E_H = v_d B. (Key point: Force balance)

3. The Hall resistance V_H/I of a conductor is:
   A. n e t/B
   B. B t/(n e)
   C. e n_s/B
   D. B/(n e t) = B/(e n_s)
   Answer: D. From V_H = IB/(net), with n_s = nt the sheet density. (Key point: Hall resistance)

4. With ν Landau levels completely filled (n_s = ν eB/h), the Hall resistance becomes:
   A. ν h/e²
   B. h/(ν e)
   C. h/(ν e²)
   D. B/(ν e²)
   Answer: C. Substituting n_s into B/(e n_s) gives h/(νe²); B cancels. (Key point: Landau levels)

## Related in Engineering Physics

- [Hall effect, coefficient and mobility](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ahall-effect-coefficient-mobility-conductivity) — Engineering Physics, chapter 8
- [Hall voltage and type of carriers](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ahall-effect-voltage-carrier-type) — Engineering Physics, chapter 8, asked 2×
- [Drift velocity from Hall experiment](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ahall-numerical-conductor-drift-velocity) — Engineering Physics, chapter 8

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