Intrinsic & Extrinsic Semiconductors

PHYSemiconductors & Energy Bands

An intrinsic semiconductor is a pure crystal (Si, Ge) in which thermally generated electron-hole pairs give equal carrier concentrations, while an extrinsic semiconductor is doped to make one carrier type dominate. This subtopic is examined through carrier-concentration calculations using the mass-action law and conductivity/resistivity calculations using carrier mobilities, so you must handle SI units of $n$ ( $\text{m}^{-3}$ ) and $\mu$ ( $\text{m}^{2}\,\text{V}^{-1}\,\text{s}^{-1}$ ) carefully.

Mass-action law

n_e \, n_h = n_i^{2}

n_i

= intrinsic carrier concentration. After doping, the minority concentration is found from

n_h = n_i^{2}/n_e

(n-type) or

n_e = n_i^{2}/n_h

(p-type).

Conductivity of a semiconductor

\sigma = e\left(n_e \mu_e + n_h \mu_h\right)

\sigma

= conductivity (

\text{S\,m}^{-1}

\Omega^{-1}\text{m}^{-1}

e = 1.6 \times 10^{-19}\,\text{C}

\mu_e,\mu_h

= electron and hole mobilities.

Intrinsic conductivity

\sigma_i = e \, n_i \left(\mu_e + \mu_h\right)

Special case with

n_e = n_h = n_i

. Used for pure Ge/Si.

Resistivity

\rho = \dfrac{1}{\sigma}

\rho

= resistivity (

\Omega\,\text{m}

). Reciprocal of conductivity.

In intrinsic material $n_e = n_h = n_i$ ; doping raises one carrier hugely and (by mass action) suppresses the other.
Donor (pentavalent) doping gives n-type with $n_e \gg n_h$ ; acceptor (trivalent) doping gives p-type with $n_h \gg n_e$ .
Even in heavily doped material the sample stays electrically neutral; the dopant supplies fixed ionised cores.
$n_i$ rises sharply with temperature (more pairs), so semiconductor conductivity increases with $T$ — opposite to metals.
Electron mobility $\mu_e$ exceeds hole mobility $\mu_h$ because electrons move in the conduction band more freely.
In an extrinsic sample the majority-carrier term dominates $\sigma$ , but include both terms unless told otherwise.
Keep all $n$ in $\text{m}^{-3}$ and $\mu$ in $\text{m}^{2}\,\text{V}^{-1}\,\text{s}^{-1}$ for SI consistency.

Where the marks go

Forgetting to use mass action: students set the minority carrier to zero instead of $n_i^{2}/n_{majority}$ .
Dropping the factor $e$ in $\sigma = e(n_e\mu_e + n_h\mu_h)$ or using $e$ in the wrong power of ten.
Mixing CGS-style $\text{cm}^{-3}$ with SI $\text{m}^{-3}$ ( $1\,\text{cm}^{-3} = 10^{6}\,\text{m}^{-3}$ ).
Reporting resistivity without inverting $\sigma$ , or quoting $\rho$ in $\Omega$ instead of $\Omega\,\text{m}$ .

How the board asks it

Numericalconductivity from carrier concentration and mobility
A specimen of silicon has electron and hole concentrations $n_e = 5 \times 10^{19}\,\text{m}^{-3}$ and $n_h = 5 \times 10^{19}\,\text{m}^{-3}$ with mobilities $\mu_e = 0.13\,\text{m}^2\,\text{V}^{-1}\,\text{s}^{-1}$ and $\mu_h = 0.05\,\text{m}^2\,\text{V}^{-1}\,\text{s}^{-1}$ . Calculate the conductivity and resistivity of the sample. (Take $e = 1.6 \times 10^{-19}\,\text{C}$ .)
Numericalmass-action law
A pure germanium crystal has intrinsic carrier concentration $n_i = 2.4 \times 10^{19}\,\text{m}^{-3}$ . It is doped with donor atoms so that the electron concentration becomes $n_e = 4.8 \times 10^{22}\,\text{m}^{-3}$ . Calculate the hole concentration $n_h$ and state whether the sample is n-type or p-type.
Give reasons $n_i$ rises with temperature
Account for the fact that the conductivity of a pure semiconductor increases with rise in temperature, whereas that of a metallic conductor decreases.
Distinguishdonor vs acceptor doping
Distinguish between an n-type and a p-type extrinsic semiconductor with respect to the type of dopant used and the majority charge carriers.
Define / stateintrinsic semiconductor and $n_e = n_h = n_i$
Define an intrinsic semiconductor and state the relation between its electron, hole and intrinsic carrier concentrations at a given temperature.
Give reasonselectrical neutrality on doping
An n-type semiconductor has a large number of free electrons as majority carriers. Explain, with reason, why the crystal as a whole remains electrically neutral.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.