Q.In an intrinsic semiconductor the energy gap Eg is 1.2eV. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at 600 K and that at 300 K? Assume that the temperature dependence of intrinsic carrier concentration ni is given by
Intrinsic carrier concentration is a foundational idea in semiconductor physics — let’s build it from the ground up, with no prior knowledge of semiconductors needed.
1. Intuition: What does "intrinsic" mean?
Imagine a pure, perfect crystal of silicon — no impurities, no defects. At absolute zero temperature (0 K), all electrons are tightly bound in the crystal lattice. No current flows.
Now, heat it up. Thermal energy shakes the atoms. Some electrons gain enough energy to break free from their bonds. When an electron leaves, it leaves behind a hole — a missing electron that behaves like a positive charge.
In this pure crystal, every free electron comes from a broken bond, and every broken bond creates one hole. So:
Number of free electrons = Number of holes
This balance is the hallmark of an intrinsic semiconductor.
2. The precise definition
Intrinsic carrier concentration (ni) is the number of free electrons (or holes) per unit volume in a pure, undoped semiconductor at thermal equilibrium.
It is denoted by ni and has units of cm−3 or m−3.
Key points:
It depends only on the material and temperature — not on doping.
For silicon at room temperature (300 K):
ni≈1.5×1010 cm−3
For germanium: ni≈2.5×1013 cm−3
For gallium arsenide: ni≈1.8×106 cm−3
3. The formula (for exams)
The precise expression is:
ni=NcNv⋅e−Eg/(2kT)
Where:
Nc = effective density of states in the conduction band
Nv = effective density of states in the valence band
Eg = bandgap energy (eV)
k = Boltzmann constant (8.617×10−5 eV/K)
T = absolute temperature (K)
Important: The exponential term e−Eg/(2kT) dominates — a small change in Eg or T causes a huge change in ni.
4. Why does it matter?
It sets the baseline for all semiconductor devices. Doping increases one carrier type, but the product n⋅p=ni2 always holds at equilibrium.
Temperature sensitivity: ni roughly doubles for every 10∘C rise in silicon. This is why circuits fail in heat. …
Why this formula?
Why Intrinsic Carrier Concentration Has That Formula
The intrinsic carrier concentration ni is the number of electrons (or holes) per unit volume in a pure, undoped semiconductor at thermal equilibrium. The formula you see in every textbook is:
ni=NcNve−Eg/2kT
where Nc and Nv are the effective density of states in the conduction and valence bands, Eg is the bandgap energy, k is Boltzmann's constant, and T is absolute temperature.
This isn't pulled from thin air. It comes from a simple physical balance: in an intrinsic semiconductor, every electron in the conduction band leaves behind a hole in the valence band. So the electron concentration n must equal the hole concentration p, and both equal ni.
Step 1: The electron and hole concentrations individually
Electrons in the conduction band follow Fermi-Dirac statistics. For non-degenerate semiconductors (which intrinsic ones are, since the Fermi level lies near midgap), the distribution approximates the Maxwell-Boltzmann tail:
n=Nce−(Ec−EF)/kT
Similarly, holes in the valence band:
p=Nve−(EF−Ev)/kT
Here Ec is the conduction band edge, Ev is the valence band edge, and EF is the Fermi level. The effective densities Nc and Nv come from integrating the density of states times the Boltzmann factor — they depend on the effective masses of electrons and holes and on temperature.
Step 2: The intrinsic condition
In an intrinsic semiconductor, there are no dopants. Every electron that jumps to the conduction band creates exactly one hole. So:
n=p
Set the two expressions equal:
Nce−(Ec−EF)/kT=Nve−(EF−Ev)/kT
Take natural logs and solve for EF:
−(Ec−EF)+lnNc=−(EF−Ev)+lnNv
EF=2Ec+Ev+2kTlnNcNv
The Fermi level in an intrinsic semiconductor sits very close to the middle of the bandgap, shifted slightly by the ratio Nv/Nc. For most practical purposes, it's at midgap.
Step 3: Multiply to eliminate EF
Now here's the clever part. Instead of solving for EF directly, multiply n and p:
np=NcNve−(Ec−EF)/kTe−(EF−Ev)/kT
The EF terms cancel:
np=NcNve−(Ec−Ev)/kT=NcNve−Eg/kT
This product np is a constant for a given material at a given temperature — it does not depend on the Fermi level. This is the law of mass action for semiconductors.
Since conductivity of an intrinsic semiconductor is proportional to the intrinsic carrier concentration ni (hole mobility is negligible and electron mobility is roughly constant here), the ratio of conductivities at two temperatures reduces to the ratio of ni at those temperatures, computed from the given exponential temperature …
σ∝ni here (hole mobility negligible, electron mobility ~constant), so form the ratio ni(600K)/ni(300K) using the given exponential law; the conductivity increases by a factor of roughly 1.1×105.
Step 1 — Why σ∝ni
Conductivity of an intrinsic semiconductor is:
σ=nie(μe+μh)
Since μh≪μe and μe is treated as essentially independent of temperature here, σ∝ni to good approximation.
Step 2 — Temperature dependence of ni
ni=n0exp(−2kBTEg)
So the ratio of conductivities at T2=600K and T1=300K is:
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
AHSEC Higher Secondary (HS) Final Examination 2026Set ANNUAL2 marks
Q.Write True or False:
(i) An intrinsic semiconductor behaves as a conductor at 0 K.
(ii) In intrinsic semiconductor, n_e = n_h = n_i.
(iii) For p-type semiconductor, n_e > n_h.
(iv) In a semiconductor in thermal equilibrium, n_h n_e = n_i^2.
›Reveal solutionSolution
(i) False, (ii) True, (iii) False, (iv) True.
(i) 'An intrinsic semiconductor behaves as a conductor at 0 K' — FALSE. At 0 K there is no thermal energy to free electrons, so the valence band is full and the conduction band empty; the intrinsic semiconductor behaves as a perfect insulator.
(ii) 'n_e = n_h = n_i' — TRUE. In a pure (intrinsic) semiconductor, each broken bond produces one free electron and one hole, so the electron and hole densities are equal, both equal to the intrinsic carrier concentration n_i.
AHSEC Higher Secondary (HS) Final Examination 2023Set ANNUAL2 marks
Q.What do you mean by thermal equilibrium? How does the situation is mathematically expressed in terms of electron and hole concentrations and intrinsic carrier concentration?
›Reveal solutionSolution
Thermal equilibrium means generation and recombination of carriers balance out, expressed by the constant relation np = ni².
In a semiconductor at a given temperature, electron-hole pairs are continuously generated (thermal energy breaking covalent bonds) and simultaneously recombining (free electrons falling back into holes). Thermal equilibrium is the steady condition — at constant temperature, with no external disturbance such as applied voltage or illumination — in which the rate of generation of new electron-hole pairs exactly equals the rate of recombination, so the carrier concentrations n (electrons) and p (holes) each remain constant with time.
Mathematically, this balance is expressed by the law of mass action: