Semiconductors formulas

Energy bands and carrier concentration. 17 formulas for GATE EC Electronic Devices, each with when to use it. Each is explained in the topic notes.

All 17 Semiconductors formulas

FERMI–DIRAC DISTRIBUTION

f (E) = \frac{1}{1 + exp ( \frac{E - E _{F}}{k T} )}

Probability that an allowed state at energy $E$ is occupied by an electron, in thermal equilibrium. $k = 8.62 \times 1 0^{- 5} eV/K$ . For any $T > 0$ , $f (E_{F}) = \frac{1}{2}$ . At $T = 0 K$ , $f = 1$ for $E < E_{F}$ and $f = 0$ for $E > E_{F}$ .

PROBABILITY THAT A STATE IS EMPTY (HOLE OCCUPANCY)

1 - f (E) = \frac{1}{1 + exp ( \frac{E _{F} - E}{k T} )}

Used for holes in the valence band; follows directly from $f (E)$ .

EQUILIBRIUM ELECTRON AND HOLE CONCENTRATIONS

n_{0} = N_{c} exp (- \frac{E _{c} - E _{F}}{k T}), p_{0} = N_{v} exp (- \frac{E _{F} - E _{v}}{k T})

Valid in thermal equilibrium for a non-degenerate semiconductor ( $E_{F}$ at least about $3 k T$ from $E_{c}$ and from $E_{v}$ ), whatever the doping. $N_{c}$ and $N_{v}$ are the effective densities of states of the conduction and valence bands.

EFFECTIVE DENSITIES OF STATES

N_{c} = 2 (\frac{2 π m _{n}^{*} k T}{h ^{2}})^{3/2}, N_{v} = 2 (\frac{2 π m _{p}^{*} k T}{h ^{2}})^{3/2}

In SI units these come out in m $^{- 3}$ ; divide by $1 0^{6}$ for cm $^{- 3}$ . Both scale as $(m^{*})^{3/2} T^{3/2}$ , so they depend on the carrier effective masses and rise with temperature. They are material properties, not constants: in a numerical problem, use the values given.

INTRINSIC CARRIER CONCENTRATION

n_{i} = N_{c} N_{v} exp (- \frac{E _{g}}{2 k T})

Holds in thermal equilibrium, non-degenerate; $E_{g} = E_{c} - E_{v}$ . For silicon at $T = 300 K$ the commonly used value is $n_{i} \approx 1.5 \times 1 0^{10} cm^{- 3}$ (some texts use $1.0 \times 1 0^{10} cm^{- 3}$ ). $n_{i}$ depends on the material and the temperature, so always use the value the problem states.

TEMPERATURE DEPENDENCE OF $N_I$

n_{i}^{2} = N_{c} N_{v} exp (- \frac{E _{g}}{k T}) \propto T^{3} exp (- \frac{E _{g}}{k T})

This follows from $N_{c} N_{v} \propto T^{3}$ , and the exponential factor dominates. Taking $E_{g}$ as constant, two temperatures are related by $\frac{n _{i} ( T _{2} )}{n _{i} ( T _{1} )} = (\frac{T _{2}}{T _{1}})^{3/2} exp [\frac{E _{g}}{2 k} (\frac{1}{T _{1}} - \frac{1}{T _{2}})]$ .

INTRINSIC FERMI LEVEL

E_{i} = \frac{E _{c} + E _{v}}{2} + \frac{k T}{2} ln (\frac{N _{v}}{N _{c}}) = \frac{E _{c} + E _{v}}{2} + \frac{3}{4} k T ln (\frac{m _{p}^{*}}{m _{n}^{*}})

Obtained by setting $n_{0} = p_{0}$ in the Boltzmann expressions. If $m_{p}^{*} > m_{n}^{*}$ , $E_{i}$ lies slightly above mid-gap. The shift is of order $k T$ , small compared with $E_{g}$ .

CARRIER CONCENTRATIONS REFERRED TO $E_I$

n_{0} = n_{i} exp (\frac{E _{F} - E _{i}}{k T}), p_{0} = n_{i} exp (\frac{E _{i} - E _{F}}{k T})

Equivalent to the $N_{c}$ , $N_{v}$ forms and valid under the same conditions (equilibrium, non-degenerate, any doping). Multiplying the two gives $n_{0} p_{0} = n_{i}^{2}$ directly.

FERMI LEVEL POSITION, N-TYPE

E_{F} - E_{i} = k T ln (\frac{n _{0}}{n _{i}}), E_{c} - E_{F} = k T ln (\frac{N _{c}}{n _{0}})

From the Boltzmann expressions; non-degenerate, equilibrium. $E_{F}$ lies above $E_{i}$ when $n_{0} > n_{i}$ .

FERMI LEVEL POSITION, P-TYPE

E_{i} - E_{F} = k T ln (\frac{p _{0}}{n _{i}}), E_{F} - E_{v} = k T ln (\frac{N _{v}}{p _{0}})

From the Boltzmann expressions; non-degenerate, equilibrium. $E_{F}$ lies below $E_{i}$ when $p_{0} > n_{i}$ .

MASS-ACTION LAW

n_{0} p_{0} = n_{i}^{2}

Holds in thermal equilibrium in a non-degenerate semiconductor, doped or undoped, with $n_{i}$ evaluated at the same temperature. It does not hold under illumination or carrier injection (non-equilibrium).

CHARGE NEUTRALITY (GENERAL)

n_{0} + N_{a}^{-} = p_{0} + N_{d}^{+}

Total negative charge equals total positive charge in the bulk, where $N_{a}^{-}$ and $N_{d}^{+}$ are the ionised acceptor and donor concentrations. With full ionisation, $N_{a}^{-} = N_{a}$ and $N_{d}^{+} = N_{d}$ .

COMPENSATED DOPING (FULL IONISATION)

n_{0} - p_{0} = N_{d} - N_{a}

Combine with $n_{0} p_{0} = n_{i}^{2}$ to solve for $n_{0}$ and $p_{0}$ . If $∣ N_{d} - N_{a} ∣ ≫ n_{i}$ , the material is n-type when $N_{d} > N_{a}$ (then $n_{0} \approx N_{d} - N_{a}$ ) and p-type when $N_{a} > N_{d}$ (then $p_{0} \approx N_{a} - N_{d}$ ); the minority carrier follows from mass action.

EXACT EQUILIBRIUM SOLUTION ($N_D > N_A$)

n_{0} = \frac{N _{d} - N _{a}}{2} + (\frac{N _{d} - N _{a}}{2})^{2} + n_{i}^{2}, p_{0} = \frac{n _{i}^{2}}{n _{0}}

This is the positive root of $n_{0}^{2} - (N_{d} - N_{a}) n_{0} - n_{i}^{2} = 0$ (full ionisation, non-degenerate). For $N_{a} > N_{d}$ swap the roles: $p_{0} = \frac{N _{a} - N _{d}}{2} + (\frac{N _{a} - N _{d}}{2})^{2} + n_{i}^{2}$ . Needed whenever $∣ N_{d} - N_{a} ∣$ is not much larger than $n_{i}$ . With $N_{d} = N_{a}$ it gives $n_{0} = p_{0} = n_{i}$ .

THERMAL VOLTAGE

V_{T} = \frac{k T}{q}

$V_{T} \approx 25.9 mV$ at $T = 300 K$ , with $k = 1.38 \times 1 0^{- 23} J/K$ and $q = 1.6 \times 1 0^{- 19} C$ . Equivalently $k T \approx 0.0259 eV$ at 300 K, so an energy difference in eV divided by $k T$ in eV gives the dimensionless exponent in the Boltzmann expressions directly.

EFFECTIVE MASS FROM THE $E$–$K$ DIAGRAM

m^{*} = \frac{ℏ ^{2}}{( d ^{2} E / d k ^{2} )}

Evaluated at the band edge, with $ℏ = h /2 π$ . A sharply curved band gives a small effective mass. For a parabolic conduction band, $E = E_{c} + \frac{ℏ ^{2} k ^{2}}{2 m _{n}^{*}}$ .

BAND-GAP WAVELENGTH

λ = \frac{h c}{E _{g}} \Rightarrow λ (μ m) \approx \frac{1.24}{E _{g} ( eV )}

The wavelength of a photon of energy $E_{g}$ . It gives the approximate emission wavelength of band-to-band recombination and the longest wavelength (cut-off) the material absorbs band-to-band. Uses $h c \approx 1.24 eV μ m$ .

Where this was asked: 2018 (1), 2020 (1), 2022 (1), 2023 (3), 2024 (2), 2025 (1). See the 9 previous-year questions on Semiconductors. This shows what was asked, not what will be asked.

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Semiconductors: the notes and 9 solved previous-year questions