Intrinsic semiconductors

Hold a wire of pure silicon and a wire of pure copper in your hand and warm both gently. The copper wire conducts a little worse as it heats up — its resistance rises, as it does for every ordinary metal. The silicon wire does the opposite: it conducts better as it warms. That single reversed sign is the whole personality of a semiconductor, and it comes from something happening deep inside the crystal lattice — covalent bonds between silicon atoms occasionally shaking themselves apart under thermal agitation, each broken bond setting one electron free and leaving behind a strange, mobile "hole" where the electron used to sit. A chemically pure crystal of silicon or germanium, with no impurity atoms deliberately added, is called an intrinsic semiconductor, and it is the starting point for the entire semiconductor-electronics story that follows in this chapter — doping, p-n junctions, diodes, rectifiers. Before any of those devices can make sense, you need to understand how a pure crystal, at absolute zero a perfect insulator, wakes up at room temperature into a feeble but genuine conductor, and why it conducts using two kinds of moving charge rather than one. This article builds that picture from the covalent bond upward: how a broken bond creates an electron and a hole in one stroke, why the two populations are forced to stay exactly equal in a pure crystal, how a hole's motion is really a relay race of electrons hopping between bonds, and how those two carrier types combine into a single conductivity formula you can actually calculate with — turning "pure silicon barely conducts" into a number you can check against a real textbook result.

Key takeaways

An intrinsic semiconductor is a chemically pure crystal (silicon or germanium) with no added impurities — group-14 elements with four valence electrons each, bonded into a tetrahedral lattice by covalent bonds.
At $0\ \text{K}$ every valence electron is locked into a bond: the valence band is completely full, the conduction band completely empty, and the crystal is a perfect insulator.
Thermal energy occasionally breaks a bond, promoting an electron to the conduction band and leaving a hole (a missing bond electron) behind — this is electron-hole pair generation, and it always produces the two carriers in equal numbers.
A hole is not a real moving particle; it is the apparent motion of a vacancy as neighbouring bond electrons hop into it one after another, and it behaves exactly like a mobile charge of $+e$ .
Because $n_e = n_h = n_i$ always holds in a pure crystal, both electrons and holes carry current, and the total conductivity is $\sigma = e\,n_i(\mu_e + \mu_h)$ — silicon at room temperature has a resistivity of a few thousand ohm-metres, far worse than a metal but far better than an insulator.
Generation (bond breaking, needs energy) is balanced by recombination (an electron falling back into a hole, releasing energy); the steady $n_i$ at a given temperature is where these two rates match.
Key idea: thermal energy breaks covalent bonds in a pure Si/Ge crystal to create equal numbers of free electrons and holes ( $n_e=n_h=n_i$ ), and both carriers drift under an applied field to give a conductivity $\sigma = e\,n_i(\mu_e+\mu_h)$ that rises with temperature — the opposite of a metal.

The core insight

In a pure silicon or germanium crystal, thermal energy breaks covalent bonds to create free electrons and holes in exactly equal numbers ( $n_e = n_h = n_i$ ), and current flows as the sum of both carriers drifting under a field, $\sigma = e\,n_i(\mu_e + \mu_h)$ , a conductivity that grows with temperature instead of shrinking as it does in a metal.

Where it fits in the syllabus

This topic opens Unit IX, "Semiconductor Electronics: Materials, Devices and Simple Circuits," right after the qualitative classification of solids by energy bands into conductors, insulators, and semiconductors. The board syllabus treats energy bands and this classification qualitatively only — no band-theory derivation is required — and the numerical anchor this article leans on, the conductivity formula $\sigma = e\,n_i(\mu_e+\mu_h)$ , is standard NCERT-level algebra rather than anything derived from first principles. What is explicitly kept out for 2025-26, and therefore absent here, is the exponential temperature-dependence of $n_i$ (the Arrhenius-type law relating $n_i$ to $E_g$ and $T$ ) — that belongs to university solid-state physics, not this board syllabus, so temperature effects are described qualitatively while the worked numbers below use a fixed, quoted room-temperature value of $n_i$ .

The physical picture

Silicon and germanium sit in group 14 of the periodic table, each atom carrying four valence electrons. In the solid, every atom shares one electron with each of its four nearest neighbours, forming four covalent bonds and completing a stable outer shell of eight shared electrons around each atom — a tetrahedral crystal lattice. At absolute zero every one of those bonding electrons is exactly where it should be: the valence band (the collection of these bonding energy states) is completely filled, and the conduction band (the higher-energy states in which an electron is free to roam and carry current) is completely empty. With no electrons available in the conduction band and none of the filled valence band able to move (a completely full band carries no net current, just as a completely full car park has no room for any car to shift into), the crystal behaves as a perfect insulator at $0\ \text{K}$ . Raise the temperature and thermal vibrations of the lattice occasionally supply enough energy to snap a covalent bond. The freed electron jumps across the energy gap $E_g$ into the conduction band, and the vacancy it leaves behind in the bond is called a hole:

\text{bond electron} \xrightarrow{\;\text{thermal energy} \geq E_g\;} e^-_{\text{(conduction band)}} + h^+_{\text{(valence band)}}

where: $E_g$ is the energy gap between the top of the valence band and the bottom of the conduction band — about $1.1\ \text{eV}$ for silicon and $0.7\ \text{eV}$ for germanium — and the process is called electron-hole pair generation. Every such event creates exactly one free electron and one hole, so in a pure crystal the two populations must always match:

n_e = n_h = n_i

where: $n_e$ is the free-electron concentration in the conduction band, $n_h$ is the hole concentration in the valence band, and $n_i$ is the intrinsic carrier concentration, the shared value both populations settle at for a given temperature. The physical reading is simple: because carriers are born in pairs and nothing else creates or destroys them separately in a pure crystal, you can never have more electrons than holes, or the reverse — this equality is the single defining fact of an intrinsic semiconductor, and it stops holding the moment impurities are added (the subject of the next article). The hole deserves special care because it is easy to misread as a fictional bookkeeping trick with no physical consequence — it is not. Once a vacancy opens in a bond, an electron from a neighbouring intact bond can hop across and fill it. That hop does not create a new carrier; it simply relocates the vacancy to the neighbour's old bond. Repeat this many times along a chain of bonds and the vacancy appears to travel through the crystal, in the direction opposite to the actual electron hops — exactly as a bubble in a queue of people "moves" backward through the queue while every person only steps forward once. Under an applied electric field, valence electrons preferentially hop against the field (toward lower potential energy), so the hole drifts with the field, precisely like a real particle carrying charge $+e$ .

Figure

Energy band diagram comparing an insulator with a wide gap, a semiconductor with a narrow gap of about one electron volt, and a conductor with overlapping bands and no gap

Derivation AConductivity of an intrinsic semiconductor from its two carriers

The goal of this derivation is to combine the separate drift contributions of free electrons in the conduction band and holes in the valence band into one usable expression for the conductivity of a pure semiconductor crystal, so that the abstract statement "both carriers conduct" becomes an actual formula you can plug numbers into and check against a textbook value, the way you already do for a metal's $\sigma = n e \mu$ but now with two carrier species that must be added rather than just one.

Step 1. Recall how a single carrier type conducts. For any charge carrier of number density $n$ , charge magnitude $q$ , and drift mobility $\mu$ (drift speed per unit field, $v_d = \mu E$ ), the current density it contributes under a field $E$ is $J = nq\mu E$ , so that carrier type alone gives a conductivity $\sigma = nq\mu$ . This is exactly the free-electron picture used for a metal.

Step 2. Identify the two carrier types in an intrinsic semiconductor. Free electrons in the conduction band, of density $n_e$ and mobility $\mu_e$ , drift opposite to the field (their charge is $-e$ ) and carry a conventional current in the direction of the field. Holes in the valence band, of density $n_h$ and mobility $\mu_h$ , drift along the field (their effective charge is $+e$ ) and also carry conventional current in the direction of the field. Crucially, both contributions add — they do not cancel — because both currents point the same way even though the two carriers physically move in opposite directions.

Step 3. Write the total current density as the sum of the two contributions. $J = J_e + J_h = n_e e \mu_e E + n_h e \mu_h E = e(n_e \mu_e + n_h \mu_h)E$ . Reading off the conductivity from $J = \sigma E$ gives the general two-carrier result $\sigma = e(n_e\mu_e + n_h\mu_h)$ , valid for any semiconductor, doped or pure.

Step 4. Specialise to an intrinsic (pure) crystal. Since pair generation forces $n_e = n_h = n_i$ , the expression collapses to a single-quantity formula:

\sigma = e\,n_i(\mu_e + \mu_h)

where: $e = 1.6\times10^{-19}\ \text{C}$ is the electronic charge magnitude, $n_i$ is the intrinsic carrier concentration at the temperature of interest, and $\mu_e,\mu_h$ are the electron and hole mobilities of the material (in $\text{m}^2\text{V}^{-1}\text{s}^{-1}$ ), with $\mu_e > \mu_h$ always, because electrons move more freely through the lattice than the bond-hopping relay of holes. This is the formula the worked examples below use directly; its only temperature dependence, left unstated here per the board syllabus, is that $n_i$ itself grows rapidly as $T$ rises, which is why intrinsic conductivity rises with heating rather than falling as a metal's does.

Worked examples

Board level - resistivity of intrinsic silicon at room temperature

For pure silicon at $300\ \text{K}$ , take $n_i = 1.5\times10^{16}\ \text{m}^{-3}$ , $\mu_e = 0.135\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$ ( $1350\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$ ), and $\mu_h = 0.0475\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$ ( $475\ \text{cm}^2\text{V}^{-1}\text{s}^{-1}$ ). Find the conductivity and resistivity of the crystal.

Step 1. Add the two mobilities, since both carriers contribute to conduction: $\mu_e + \mu_h = 0.135 + 0.0475 = 0.1825\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$ .

Step 2. Apply the conductivity formula derived above: $\sigma = e\,n_i(\mu_e+\mu_h) = (1.6\times10^{-19})(1.5\times10^{16})(0.1825)$ .

Step 3. Multiply in stages to avoid a slip: $(1.6\times10^{-19})(1.5\times10^{16}) = 2.4\times10^{-3}$ ; then $(2.4\times10^{-3})(0.1825) = 4.38\times10^{-4}\ \text{S/m}$ .

Step 4. Invert for resistivity: $\rho = 1/\sigma = 1/(4.38\times10^{-4}) \approx 2.28\times10^{3}\ \Omega\text{m}$ . Answer: $\sigma \approx 4.38\times10^{-4}\ \text{S/m}$ and $\rho \approx 2.28\times10^{3}\ \Omega\text{m}$ — thousands of times more resistive than a metal such as copper ( $\rho \sim 10^{-8}\ \Omega\text{m}$ ), which is exactly why a pure semiconductor is called only a feeble conductor.

Competitive level - current through a silicon bar

A bar of the same intrinsic silicon (using $\rho = 2.28\times10^{3}\ \Omega\text{m}$ from the board example) has length $L = 1\ \text{cm}$ and uniform cross-section $A = 1\ \text{mm}^2$ . A potential difference of $5\ \text{V}$ is applied across its length. Find the resistance and the current that flows.

Step 1. Convert every length to SI before touching the resistance formula: $L = 1\ \text{cm} = 1\times10^{-2}\ \text{m}$ , $A = 1\ \text{mm}^2 = 1\times10^{-6}\ \text{m}^2$ . This is the step most students skip, and it is the one that wrecks the answer if skipped.

Step 2. Apply $R = \rho L/A = (2.28\times10^{3})(1\times10^{-2})/(1\times10^{-6}) = 2.28\times10^{7}\ \Omega$ , about $22.8\ \text{M}\Omega$ — a huge resistance for a centimetre-scale bar, again underlining how poorly a pure semiconductor conducts compared with a metal wire of the same size.

Step 3. Apply Ohm's law to find the current: $I = V/R = 5/(2.28\times10^{7}) \approx 2.19\times10^{-7}\ \text{A}$ , about $0.22\ \mu\text{A}$ . Answer: $R \approx 22.8\ \text{M}\Omega$ , $I \approx 0.22\ \mu\text{A}$ — a current so small it would be swamped by noise in most real circuits, which is precisely why devices are built from doped semiconductors rather than intrinsic ones. Timing note: this should take under ninety seconds once $\rho$ is known. The whole chain is unit conversion to SI, then $R=\rho L/A$ , then $I=V/R$ — do the conversions first and write them down explicitly, because mixing centimetres, millimetres and metres in the same line is the single most common way marks are dropped on this kind of problem, far more often than an error in the physics itself.

Common traps

Treating the hole as a real particle physically travelling through the crystal — it is the apparent motion of a vacancy as many different valence electrons each hop one step; only electrons actually move bodily.
Writing $n_e = n_h$ for any semiconductor — this equality is the defining signature of an intrinsic (pure) crystal only; it fails the moment donor or acceptor impurities are added, which is the very next topic in this chapter.
Assuming a semiconductor's resistance behaves like a metal's, rising with temperature — an intrinsic semiconductor's resistance falls as temperature rises, because more bonds break and $n_i$ grows.
Forgetting that both carriers add to the current rather than partially cancelling — even though electrons and holes drift in opposite physical directions, both conventional currents point the same way, so $\sigma = e(n_e\mu_e+n_h\mu_h)$ has a plus sign, not a minus sign.
Mixing up generation and recombination in a definition-based question — generation breaks a bond and needs energy input; recombination is an electron falling back into a hole and releases energy; at a fixed temperature the two rates balance, keeping $n_i$ steady.

Common trapA hole is a bookkeeping device for many real electron hops, not a physical particle drifting through the crystal

It is tempting, and almost always wrong on an exam, to picture a hole as a tiny positive ball that physically travels from one end of the crystal to the other, the way a free electron does in the conduction band. What is actually happening is a relay: electron A, still bound in its own bond, hops into a neighbouring vacancy; that hop empties A's original bond, creating a new vacancy one step away from where the first one was. Chain enough of these hops together and the vacancy appears to glide across the crystal, in the direction opposite to the individual electron hops, purely because each hop relocates where the "missing spot" is. The reason this bookkeeping is not just a convenient fiction is that it produces exactly the right physics: under an applied field, valence electrons preferentially hop against the field, so the chain of vacancies drifts with the field, delivering a current contribution indistinguishable from a real particle of charge $+e$ moving that way. Describe a hole as "a vacancy whose position moves through a chain of electron hops," not as "a positive particle" — the first is correct physics, the second loses marks for imprecision even though the electrical behaviour is genuinely that of a $+e$ carrier.

You should be able to…

Explain, using covalent bonds in the Si/Ge lattice, how thermal energy generates an electron-hole pair, and state why the two populations are always equal in a pure crystal.
Describe hole motion correctly as a chain of valence-electron hops between neighbouring bonds, and explain why it behaves electrically like a $+e$ charge carrier.
Derive and use $\sigma = e\,n_i(\mu_e+\mu_h)$ starting from the single-carrier result $\sigma = nq\mu$ , and compute resistivity, resistance or current from it given $n_i$ , $\mu_e$ , $\mu_h$ and the geometry of a sample.
Explain qualitatively why intrinsic conductivity rises with temperature, in contrast with the falling conductivity of a metal.
Distinguish generation (bond breaking, energy absorbed) from recombination (electron-hole pair annihilation, energy released), and describe their balance at a fixed temperature.
State why intrinsic semiconductors, despite genuinely conducting, are far closer to insulators than to metals in resistivity, which motivates deliberately adding impurities.

Ask the tutor: if the idea of a hole "moving" while no actual positive particle exists still feels strange, ask your AI tutor to walk through several consecutive electron hops along a row of bonds, one at a time, and point out exactly where the vacancy sits after each hop.

Why does heating make a semiconductor conduct better when it makes a metal conduct worse?

In a metal, the number of free electrons is already fixed and huge — heating just makes the lattice vibrate more, so electrons scatter more often and mobility drops, raising resistance. In an intrinsic semiconductor almost no electrons are free to begin with; heating's main effect is breaking more covalent bonds, which creates more electron-hole pairs and shoots up $n_i$ faster than any drop in mobility can offset. Since $\sigma = e\,n_i(\mu_e+\mu_h)$ , a rapidly rising $n_i$ wins, and conductivity increases with temperature — the opposite trend from a metal, and one of the clearest markers that a material is a semiconductor rather than a conductor.

Is a hole really "nothing," or does it have properties like mass and charge?

A hole is not literally a particle sitting in the periodic table, but within the crystal it behaves in every measurable electrical sense like a real particle of charge $+e$ : it drifts under a field, it has a mobility $\mu_h$ you can measure, and it even behaves as if it carries an effective mass (heavier than a free electron's, which is why $\mu_h < \mu_e$ and why holes move more sluggishly). Physically it is the collective effect of many valence electrons taking turns to fill a vacancy, but treating it as a mobile positive charge carrier gives correct predictions for current, so physicists and this syllabus both use it as a genuine second carrier type, not merely a figure of speech.

Why does silicon have such a huge resistivity compared to a metal, if it does conduct?

The numbers in the worked example above make the gap concrete: silicon's resistivity works out to about $2.3\times10^{3}\ \Omega\text{m}$ , while copper's is around $1.7\times10^{-8}\ \Omega\text{m}$ — silicon is roughly a hundred billion times more resistive. The reason is carrier density: a metal has on the order of $10^{28}$ free electrons per cubic metre already available with no thermal activation needed, while pure silicon at room temperature has only about $10^{16}$ carriers per cubic metre, all of them thermally generated one broken bond at a time. Fewer carriers means less current for the same field, so an intrinsic semiconductor, despite genuinely conducting, sits many orders of magnitude closer to an insulator than to a metal — which is exactly why it is called a semiconductor.

Does germanium behave the same way as silicon, or is it different?

The mechanism is identical — covalent bonds breaking, electron-hole pairs generated in equal numbers, both carriers contributing to $\sigma = e\,n_i(\mu_e+\mu_h)$ — but the numbers differ because germanium's energy gap, about $0.7\ \text{eV}$ , is narrower than silicon's $1.1\ \text{eV}$ . A smaller gap means bonds break more easily at a given temperature, so germanium has a larger intrinsic carrier concentration and conducts noticeably better in its pure state than silicon does at the same temperature. This is also why silicon, not germanium, dominates modern electronics: silicon's larger gap keeps its intrinsic (unwanted, uncontrolled) conductivity lower, so devices behave more predictably once impurities are deliberately added to engineer the wanted conductivity.