Current Electricity | Class 12 CBSE Physics | Concept Notes
The question this note answers
We already know R = ρL/A: resistance depends on geometry (L, A) and on resistivity ρ, a material property. But ρ itself isn’t a true constant — it depends on temperature. This note asks: why does heating a conductor change its resistivity, and why do metals and semiconductors respond in opposite directions?
Metals: resistance increases with temperature
The microscopic picture
In a metal, electrical conduction is mainly due to a large number of free electrons moving through a lattice of positively charged ions.
In the drift-velocity model, the resistivity is related to the electron relaxation time τ by:
ρ = m / (n e² τ)
and therefore the resistance of a uniform conductor is:
R = ρL/A = mL / (nAe²τ)
where:
- ρ = resistivity of the material
- R = resistance of the conductor
- m = electron mass
- n = number density of free electrons
- e = magnitude of electronic charge
- τ = average relaxation time between successive scattering events
- L = length of the conductor
- A = cross-sectional area
What happens when temperature increases?
As the temperature of a metal rises, the ions in its crystal lattice vibrate more strongly.
These increased lattice vibrations produce stronger electron–phonon scattering. As a result, conduction electrons are scattered more frequently and their average relaxation time τ decreases.
Since:
ρ ∝ 1/τ
a decrease in τ causes the resistivity to increase.
Therefore:
Temperature ↑ → lattice vibrations ↑ → electron scattering ↑ → τ ↓ → resistivity ↑ → resistance ↑
What happens to the number of free electrons?
For an ordinary metal, the number density of conduction electrons, n, remains approximately constant over the usual temperature range considered in elementary electrical-conduction problems.
Therefore, the dominant temperature dependence of metallic resistivity comes from the change in electron scattering and relaxation time τ, rather than a significant change in the number of free electrons.
This is an important difference between metals and semiconductors, where carrier concentration can change strongly with temperature.
Temperature dependence of resistance
Over a moderate temperature range, the resistance of a metal can be approximated by:
Rₜ = R₀ [1 + α(T − T₀)]
where:
- Rₜ = resistance at temperature T
- R₀ = resistance at the reference temperature T₀
- α = temperature coefficient of resistance
- T and T₀ must be expressed in the same temperature unit
For a typical metal:
α > 0
Therefore, resistance increases as temperature increases.
The same relationship can be written for resistivity:
ρₜ = ρ₀ [1 + α(T − T₀)]
provided the temperature coefficient is being treated as approximately constant over the temperature range considered.
What does the temperature coefficient mean?
The temperature coefficient of resistance tells us how strongly the resistance changes with temperature.
For a positive temperature coefficient:
α > 0 → resistance increases with temperature
For a negative temperature coefficient:
α < 0 → resistance decreases with temperature
Most ordinary metallic conductors have a positive temperature coefficient near room temperature.
Important limitation of the formula
The equation
Rₜ = R₀ [1 + α(T − T₀)]
is an approximation, not a universal law valid over every temperature range.
It works reasonably well over a moderate temperature interval where α can be treated as approximately constant.
Over a wider temperature range, the temperature dependence can become non-linear and α itself may change with temperature.
Connection with the V–I graph
This temperature dependence explains why a metallic filament bulb is non-ohmic.
For an ordinary fixed resistor maintained at approximately constant temperature:
V ∝ I → straight-line V–I graph
For a tungsten filament bulb:
Current ↑ → heating ↑ → temperature ↑ → resistance ↑
Therefore the V–I relationship becomes non-linear and the graph curves.
Semiconductors: resistance decreases with temperature
The microscopic picture — a different mechanism entirely
In a semiconductor, the most important temperature-dependent quantity is usually the number of mobile charge carriers, rather than the relaxation time τ.
The electrical conductivity of a semiconductor can be written as:
σ = neμ
for a simple single-carrier picture, where:
- σ = electrical conductivity
- n = number density of mobile charge carriers
- e = magnitude of electronic charge
- μ = mobility of the charge carriers
Since:
ρ = 1/σ
an increase in the number of mobile charge carriers causes the resistivity to decrease, provided that the increase in carrier concentration dominates any decrease in mobility.
What happens at low temperature?
At relatively low temperatures, fewer electrons have enough thermal energy to be excited from the valence band to the conduction band.
Therefore:
- The number of free electrons in the conduction band is relatively small.
- The number of holes in the valence band is also relatively small.
- The concentration of mobile charge carriers is low.
- Conductivity is relatively low.
- Resistivity, and therefore resistance, is relatively high.
What happens when temperature increases?
As temperature rises, more electrons acquire sufficient thermal energy to cross the energy gap between the valence band and conduction band.
This produces:
More electrons in the conduction band + more holes in the valence band
Therefore, the number of mobile charge carriers increases significantly.
For an intrinsic semiconductor:
n = p = nᵢ
where nᵢ is the intrinsic carrier concentration.
The intrinsic carrier concentration increases strongly with temperature, approximately according to an exponential relationship.
Consequently, semiconductor conductivity increases rapidly with temperature and resistivity decreases.
But what about carrier mobility?
Temperature also affects the mobility of charge carriers.
As temperature increases, lattice vibrations become stronger, leading to increased carrier–phonon scattering. This tends to reduce mobility.
Therefore, two competing effects occur:
Temperature ↑ → carrier concentration ↑ → conductivity ↑
but also:
Temperature ↑ → lattice vibrations ↑ → scattering ↑ → mobility ↓ → conductivity ↓
For intrinsic semiconductors over the usual temperature range considered in basic physics, the large increase in carrier concentration generally dominates the reduction in mobility.
Hence:
Temperature ↑ → carrier concentration increases strongly → conductivity increases → resistivity decreases
Why is the behaviour opposite to that of metals?
The key difference is which temperature-dependent effect dominates.
| Property | Metal | Semiconductor |
| Main temperature effect | Relaxation time / scattering changes | Carrier concentration changes strongly |
| Carrier concentration | Approximately constant | Can increase strongly with temperature |
| Mobility | Generally decreases as temperature rises | Generally decreases because of increased scattering |
| Net conductivity | Generally decreases | Generally increases |
| Resistivity | Generally increases | Generally decreases |
| Temperature coefficient | Usually positive | Usually negative over the relevant intrinsic/semiconducting range |
The key contrast
For a metal:
Temperature ↑ → lattice vibrations ↑ → scattering ↑ → τ ↓ → resistivity ↑
The number of conduction electrons remains approximately constant.
For an intrinsic semiconductor:
Temperature ↑ → more electron–hole pairs generated → carrier concentration ↑↑ → conductivity ↑ → resistivity ↓
Although increased lattice vibrations can reduce carrier mobility, the strong increase in the number of charge carriers generally dominates.
Important qualification: doped semiconductors
The simple statement “semiconductor resistance always decreases with temperature” is an important approximation, not an absolute rule for every semiconductor under every condition.
In a doped semiconductor, different temperature regions can occur:
- Low-temperature / freeze-out region: many dopant atoms are not ionized, so carrier concentration is low.
- Intermediate / extrinsic region: dopants provide most of the charge carriers, so carrier concentration can remain approximately constant over a range of temperatures.
- High-temperature / intrinsic region: thermally generated electron–hole pairs become dominant, and carrier concentration rises rapidly.
Therefore, the temperature dependence of resistance can be more complicated than a single straight-line relationship.
Temperature coefficient
Because semiconductor resistance generally decreases as temperature rises over the relevant operating range, semiconductors commonly exhibit a:
Negative temperature coefficient of resistance (NTC)
That means:
α < 0
for the effective temperature coefficient over that range.
This behaviour is used in devices such as thermistors, particularly NTC thermistors used for temperature sensing and measurement.
Exam-ready summary
For an intrinsic semiconductor:
Temperature ↑ → electron–hole pair generation ↑ → carrier concentration ↑↑ → conductivity ↑ → resistivity ↓ → resistance ↓
The dominant reason for the negative temperature coefficient of resistance in semiconductors is the strong increase in the number of mobile charge carriers with temperature.
One important comparison to remember
Metal:
Carrier concentration ≈ constant
→ scattering increases with temperature
→ resistance increases
Semiconductor:
Carrier concentration increases strongly with temperature
→ this usually dominates the mobility decrease
→ resistance decreases
Memory aid:
Metal: temperature mainly changes scattering.
Semiconductor: temperature strongly changes carrier availability.
Worked Example 1 — metal
A resistor has resistance 100 Ω at 20°C. Its temperature coefficient of resistance is α = 0.004 /°C. Find its resistance at 70°C.
Worked Example 2 — the trap
A student assumes a thermistor (a semiconductor-based temperature sensor) behaves like a metal resistor and expects its resistance to rise as it’s heated. What actually happens, and why is the student’s assumption wrong?
Answer: A thermistor’s resistance typically decreases sharply as it’s heated (this is precisely why thermistors are useful as temperature sensors — resistance is a sensitive, monotonic function of temperature). The student incorrectly generalized the metal behavior (α positive) to a semiconductor device, where the carrier-density effect (n increasing rapidly with temperature) dominates and produces the opposite sign. This is a case where “resistance changes with temperature” is true for both material types, but the direction of that change depends entirely on which microscopic mechanism dominates.
Case study — why this matters for real instruments
A copper wire used inside a precision measuring instrument (e.g., a galvanometer coil) will have its resistance drift measurably if the instrument’s temperature changes during use — this is an actual source of experimental error in careful lab work, not just a textbook abstraction. By contrast, special alloys like manganin are deliberately engineered to have an extremely small α (close to zero) specifically so that resistance boxes and standard resistors built from them stay accurate across a range of room temperatures. This is a direct, practical consequence of the physics in this note: if you need a resistor whose value you can trust regardless of small temperature fluctuations, you actively want a material with small |α|, not just “some metal.”
Quick Practice
Q1. A platinum resistance thermometer has resistance 50 Ω at 0°C and α = 0.0039/°C. Find its resistance at 100°C.
Answer: R = 50×[1+0.0039×100] = 50×1.39 = 69.5 Ω
Q2. Explain, using the concept of relaxation time, why a metal’s resistivity increases with temperature.
Answer: Higher temperature means larger-amplitude lattice vibrations, which increases the collision frequency experienced by drifting electrons. This decreases the average time between collisions (relaxation time, τ). Since ρ = m/(ne²τ) and τ decreases while n stays essentially constant, ρ increases.
Q3. Two materials — one a metal, one a semiconductor — both start at the same resistance at room temperature. If both are heated by the same amount, will they still have the same resistance afterward? Explain your reasoning.
Answer: No. The metal’s resistance will increase (positive α, dominated by decreasing τ) while the semiconductor’s resistance will decrease (negative effective α, dominated by increasing n) — the two materials’ resistances will diverge in opposite directions even though they started equal.
Summary
| Property | Metals | Semiconductors |
| Temperature coefficient (α) | Positive (α > 0) | Generally negative over the useful operating range (α < 0) |
| Dominant microscopic effect | Lattice vibrations ↑ → electron–phonon scattering ↑ → relaxation time τ ↓ | Thermally generated charge carriers ↑↑ → carrier concentration ↑↑ |
| Carrier concentration with temperature | Approximately constant | Increases strongly with temperature, especially in the intrinsic region |
| Mobility with temperature | Generally decreases because of increased lattice scattering | Generally decreases due to increased phonon scattering, but this is usually outweighed by the strong increase in carrier concentration |
| Conductivity with temperature | Generally decreases because mobility decreases | Generally increases because the increase in carrier concentration dominates |
| Resistivity with temperature | Increases | Generally decreases |
| Practical consequence | Resistance changes with temperature, which can introduce errors in precision measurements. Low-temperature-coefficient alloys such as manganin are therefore used when resistance stability is important. | The strong temperature dependence of resistance is deliberately used in thermistors and temperature-sensing circuits. |
| Key physical idea | Temperature mainly changes the scattering of existing charge carriers. | Temperature strongly changes the number of available charge carriers. |
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