Intrinsic Semiconductors

Every silicon chip on Earth begins as intrinsic material. Raw silicon is refined to better than one foreign atom in a billion — "nine-nines" electronic-grade silicon — and grown into single crystals the width of a dinner plate, because only a near-perfect intrinsic starting crystal lets engineers add impurities exactly where they want them. That near-empty canvas is then patterned into the billions of transistors on a modern processor. The steep temperature dependence you met here is used directly in the thermistor, a semiconductor whose resistance drops predictably as it warms — the sensor behind digital thermometers and battery-management systems. The absorption edge set by the gap is why silicon photodiodes and solar cells respond to visible light while germanium reaches into the infrared for fibre-optic receivers. And the equal-carrier balance that defines intrinsic material is the reference point from which the next article departs: dope the crystal, tilt the balance toward electrons or holes, and you have the raw ingredient of the p–n junction, the diode, and the transistor — the whole of modern electronics.

Take a single atom of silicon. It has four electrons in its outermost shell — it sits in group IV of the periodic table, like germanium below it and carbon (diamond) above it. In a crystal each silicon atom shares one of those four electrons with each of its four nearest neighbours, forming four covalent bonds. Every valence electron is thereby spoken for, glued into a shared bond. This is the same picture as the filled valence band of the previous article — "valence band full" and "every bond occupied" say the same thing. At absolute zero this crystal carries no current at all. Not one electron is free to wander; the conduction band above the $1.1\ \text{eV}$ gap is completely empty. A perfect intrinsic crystal at $T = 0$ is a perfect insulator, indistinguishable electrically from diamond — the only difference is the size of the gap it would take to liberate an electron. Now warm the crystal to room temperature. Its atoms jiggle with thermal energy, and the typical thermal quantum is $kT \approx 0.026\ \text{eV}$ at $300\ \text{K}$. That is forty times smaller than silicon's $1.1\ \text{eV}$ gap, so almost every bond survives — but "almost" is the crucial word. In the lottery of $10^{28}$ jiggling atoms, a rare few bonds absorb enough of a thermal fluctuation to break, and the freed electron is kicked across the gap into the conduction band, where it roams the whole crystal. That is thermal generation of an electron–hole pair.

The electron is only half the story. The bond it left behind now misses an electron — a little pocket of net positive charge. This vacancy is called a hole, and it is the concept that trips up most beginners. The hole does not sit still: an electron in a neighbouring intact bond can slip sideways into the vacancy, which costs almost nothing since it is a hop between bonds, not a jump across the gap. Now the bond it came from is the one missing an electron — the vacancy has moved. Repeat this and the hole shuffles through the crystal like a bubble rising through water, or the empty seat that travels down a row as each person shifts over. A hole is genuinely mobile, and being a region of missing negative charge, it behaves in every measurable way as a positive charge carrier of magnitude $+e$. So a pure semiconductor conducts by two carriers at once. Apply a field and the free electrons drift one way, the holes the other; because their charges are opposite, both push current in the same direction. This two-carrier conduction — electrons in the conduction band, holes in the valence band — is the whole secret of the semiconductor, and why it is far more versatile than a metal, which conducts by electrons alone.

Here is the most important consequence of demanding a pure crystal. Every conduction electron got there by breaking a bond, and breaking a bond makes exactly one hole. There is no other source of either carrier — no impurity atoms donating spare electrons or grabbing them. Electrons and holes are therefore created strictly in pairs and destroyed in pairs (when a wandering electron falls back into a hole, both vanish — that is recombination). At any temperature the crystal settles into a balance where generation and recombination run at equal rates, and in that steady state the carrier counts are equal:

where: $n_e$ is the free-electron concentration, $n_h$ the hole concentration, and $n_i$ their common value, the intrinsic carrier concentration. The physical reading is bookkeeping made rigorous: you cannot make a free electron in a pure crystal without simultaneously making a hole, so you always have exactly as many of one as the other. It is also charge neutrality — the crystal is uncharged, the negative electrons exactly balanced by the positive holes. The value of $n_i$ depends only on material and temperature. At $300\ \text{K}$ the measured figures are $n_i \approx 1.5\times10^{16}\ \text{m}^{-3}$ for silicon and $2.4\times10^{19}\ \text{m}^{-3}$ for germanium — germanium has about sixteen hundred times more carriers, purely because its smaller gap ($0.7$ vs $1.1\ \text{eV}$) breaks bonds more readily. Diamond's $5.5\ \text{eV}$ gap gives an $n_i$ so vanishingly small that it is an insulator at any temperature you can survive. These numbers sound large until you compare them with the number of atoms: a cubic metre of silicon holds about $5\times10^{28}$. Dividing, only about one atom in $3\times10^{12}$ has given up an electron at room temperature. A pure semiconductor is a spectacularly poor conductor — but that scarcity is not a defect, it is the blank canvas. Because pure silicon has so few carriers of its own, adding a trace of impurity (the doping of the next article) can multiply the carrier count a millionfold and put the conductivity entirely under our control.

Now to the fact that feels wrong the first time you meet it. Cool a copper wire and it conducts better; cool a silicon crystal and it conducts worse, until near absolute zero it stops altogether. Semiconductors have a negative temperature coefficient of resistance — resistance falls as temperature rises — the cleanest fingerprint distinguishing a semiconductor from a metal. The reason is already in the physical picture. In a metal the carriers are always there; heating cannot add electrons, it only makes the lattice vibrate more and jostle them, so resistance rises. In a semiconductor the carriers must first be created by breaking bonds, and heat is exactly what breaks bonds. Raise the temperature and you make more electron–hole pairs, flooding the crystal with carriers. This carrier-multiplication swamps the modest rise in scattering, so conductivity climbs — steeply — with temperature. How steeply? The number of pairs follows a law borrowed from statistical physics, which we quote rather than derive (the full result comes from Fermi–Dirac statistics and a density-of-states integral over the bands — both beyond this article's algebra, so we take the formula on trust and check that it reproduces the measured $n_i$):

where: $T$ is the absolute temperature, $E_g$ the band gap, $k = 8.617\times10^{-5}\ \text{eV K}^{-1}$ Boltzmann's constant, and $A$ a material constant. The exponential factor dominates everything: $E_g/2kT$ for silicon at $300\ \text{K}$ is about $21$, so $\exp(-21) \approx 6\times10^{-10}$, only a minuscule fraction of bonds broken at any instant. But because the gap sits divided by temperature inside an exponential, a small rise in $T$ shrinks the exponent and multiplies $n_i$ dramatically — warming silicon from $300$ to $400\ \text{K}$ multiplies its carrier concentration roughly three-hundred-fold. The smaller the gap, the milder the exponent and the more carriers, which is exactly why germanium out-carries silicon.

We can now write down how well an intrinsic crystal conducts. Under a field $E$ each carrier drifts at a speed proportional to the field, the constant of proportionality being its mobility $\mu$. Both carriers contribute, so conductivity is the sum of an electron and a hole term; for the intrinsic case $n_e = n_h = n_i$ it collapses to a single clean expression:

where: $\sigma$ is the conductivity (siemens per metre), $e = 1.6\times10^{-19}\ \text{C}$, $\mu_e$ and $\mu_h$ are the electron and hole mobilities, and $n_i$ the intrinsic concentration. Conductivity is carrier density times charge times mobility, summed over both carrier types. Electrons are always the more mobile (they roam a real band, while holes advance by the slower relay of bond-hopping), so $\mu_e > \mu_h$ in every semiconductor — for silicon $\mu_e \approx 0.135$ and $\mu_h \approx 0.048\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$. The derivation below turns these into a resistivity of about $2.3\times10^{3}\ \Omega\,\text{m}$ for pure silicon — vastly more resistive than copper's $1.7\times10^{-8}\ \Omega\,\text{m}$, yet vastly less than a true insulator. A photon can do the same job as heat: if its energy $h\nu$ exceeds the gap it can break a bond and make a pair. The longest wavelength that still carries enough energy sets an absorption edge:

which for silicon ($E_g = 1.1\ \text{eV}$, $hc = 1240\ \text{eV nm}$) gives $\lambda_{\max} = 1240/1.1 \approx 1127\ \text{nm}$, in the near-infrared. Shorter wavelengths — all the visible spectrum — are absorbed, which is why a silicon wafer looks opaque grey; longer wavelengths pass through, which is why silicon is a favourite infrared window and detector. Germanium's smaller gap pushes its edge out to $1240/0.7 \approx 1771\ \text{nm}$.

Board level — the absorption edge of silicon and germanium (CBSE and ICSE pattern)

A photon can create an electron–hole pair only if its energy is at least the band gap. Using $hc = 1240\ \text{eV nm}$, find the longest wavelength of light that a pure silicon crystal ($E_g = 1.1\ \text{eV}$) can absorb by this mechanism, and repeat for germanium ($E_g = 0.7\ \text{eV}$). Which of the two can detect ordinary visible light, and which reaches further into the infrared?

Step 1. The threshold condition is $h\nu = hc/\lambda \ge E_g$, so the longest usable wavelength is $\lambda_{\max} = hc/E_g$.

Step 2. For silicon: $\lambda_{\max} = 1240/1.1 = \boxed{1127\ \text{nm}}$, in the near-infrared. Any wavelength shorter than this — all visible light, $400$–$700\ \text{nm}$ — carries more than $1.1\ \text{eV}$ and is absorbed, which is why silicon photodetectors and solar cells work across the whole visible band.

Step 3. For germanium: $\lambda_{\max} = 1240/0.7 = \boxed{1771\ \text{nm}}$. Its smaller gap lets it respond to longer infrared wavelengths that silicon cannot see — which is exactly why germanium and other narrow-gap semiconductors are chosen for infrared sensing.

Step 4. Both absorb visible light (visible photons exceed both gaps), but germanium reaches further into the infrared. The rule to carry away: a smaller gap means a longer cut-off wavelength, since a less energetic, longer-wavelength photon suffices to bridge it.

Competitive level — intrinsic conductivity and resistivity of germanium (JEE and NEET pattern)

Pure germanium at $300\ \text{K}$ has $n_i = 2.4\times10^{19}\ \text{m}^{-3}$, electron mobility $\mu_e = 0.39\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$ and hole mobility $\mu_h = 0.19\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$. Take $e = 1.6\times10^{-19}\ \text{C}$. Find its conductivity and resistivity, and compare with pure silicon's resistivity of about $2.3\times10^{3}\ \Omega\,\text{m}$.

Step 1. Both carriers contribute, and for an intrinsic crystal $n_e = n_h = n_i$, so $\sigma = n_i e(\mu_e + \mu_h)$. Add the mobilities first: $\mu_e + \mu_h = 0.39 + 0.19 = 0.58\ \text{m}^2\text{V}^{-1}\text{s}^{-1}$.

Step 2. Substitute: $\sigma = (2.4\times10^{19})(1.6\times10^{-19})(0.58)$. Group the powers of ten: $(2.4)(1.6) = 3.84$ and $10^{19}\times10^{-19} = 1$, so $\sigma = 3.84 \times 0.58 = \boxed{2.2\ \text{S m}^{-1}}$.

Step 3. Resistivity is the reciprocal: $\rho = 1/\sigma = 1/2.23 = \boxed{0.45\ \Omega\,\text{m}}$.

Step 4. Compare germanium's $0.45\ \Omega\,\text{m}$ against silicon's $2.3\times10^{3}\ \Omega\,\text{m}$ — germanium conducts roughly five thousand times better, a direct consequence of its $1600\times$ larger carrier density. The gap is doing the work: germanium's narrower $0.7\ \text{eV}$ gap breaks far more bonds at $300\ \text{K}$. Timing note: about ninety seconds — one substitution into $\sigma = n_i e(\mu_e+\mu_h)$ and one reciprocal. The exam-saving trick is to add the mobilities first and pair each $10^{19}$ with a $10^{-19}$ so the powers of ten annihilate, leaving the single multiplication $3.84\times0.58$ to do in your head.

Competitive level — how many times more carriers on heating (JEE Advanced pattern)

Using $n_i = A\,T^{3/2}\exp(-E_g/2kT)$ for silicon ($E_g = 1.1\ \text{eV}$, $k = 8.617\times10^{-5}\ \text{eV K}^{-1}$), estimate the factor by which the intrinsic carrier concentration increases when the crystal is heated from $300\ \text{K}$ to $400\ \text{K}$.

Step 1. Take the ratio at the two temperatures so the unknown constant $A$ cancels: $\dfrac{n_i(400)}{n_i(300)} = \left(\dfrac{400}{300}\right)^{3/2}\exp\!\left[-\dfrac{E_g}{2k}\left(\dfrac{1}{400}-\dfrac{1}{300}\right)\right]$.

Step 2. The exponential group: $E_g/2k = 1.1/(2\times8.617\times10^{-5}) = 6.38\times10^{3}\ \text{K}$, and $\left(\tfrac{1}{300}-\tfrac{1}{400}\right) = 8.33\times10^{-4}\ \text{K}^{-1}$, so the exponent is $+5.32$ and $\exp(5.32) = \boxed{204}$.

Step 3. The prefactor: $(400/300)^{3/2} = (1.333)^{1.5} = 1.54$.

Step 4. Multiply: $\dfrac{n_i(400)}{n_i(300)} = 1.54\times204 \approx \boxed{314}$. A $100\ \text{K}$ warming multiplies the carriers — and roughly the conductivity — by about three hundred, almost all from the exponential. That sensitivity is why a bare intrinsic semiconductor makes a superb thermistor but a treacherous circuit element unless temperature-controlled. Timing note: two minutes, and it lives or dies on the exponent. Compute $E_g/2k = 6.4\times10^{3}\ \text{K}$ once, note the bracket $1/300-1/400 = 100/120000 = 8.3\times10^{-4}$, and the rest is one $\exp$ and one $1.5$-power. Do not evaluate $A$ or the absolute $n_i$ — the ratio was engineered to make them vanish.

Saying resistance rises with temperature. This is metal reasoning misapplied. A semiconductor's resistance falls as it is heated, because heat manufactures new electron–hole pairs. Writing "the atoms vibrate more so the semiconductor conducts less" is the most common way to throw away marks here in CBSE, NEET and JEE alike.

Treating a hole as mere empty space that cannot carry current. A hole is a genuinely mobile positive carrier: bond electrons hop into the vacancy, so it drifts and carries real current, at roughly half the mobility of a free electron. In intrinsic material electrons and holes carry the current together, in equal numbers.

Forgetting that $n_e = n_h$ holds only for the pure crystal. The instant you dope it — next article — the equality breaks: one carrier becomes majority, the other minority, though the product $n_e n_h = n_i^2$ stays fixed. Quoting $n_e = n_h$ for a doped sample is a classic slip.

Confusing $kT$ with $E_g$. The thermal quantum $kT \approx 0.026\ \text{eV}$ at $300\ \text{K}$ is about forty times smaller than silicon's $1.1\ \text{eV}$ gap. Room heat crosses the gap only in the rare tail of the thermal distribution, which is why $n_i$ is so tiny — do not write "$kT$ excites electrons across the gap" as though it were easy.

Getting the gap–carrier direction backwards. A smaller gap gives more carriers and a longer absorption edge, because less energy breaks a bond. Germanium ($0.7\ \text{eV}$) beats silicon ($1.1\ \text{eV}$) on both. Students routinely reverse this.

Board scope. The quantitative intrinsic-carrier physics — $n_i$, the mobilities, $\sigma = n_i e(\mu_e+\mu_h)$ and the temperature law — is core on CBSE, ICSE (ISC), JEE Main, JEE Advanced and NEET; the qualitative electron–hole and band picture gives useful background to the semiconductor-device topics of Cambridge A-Level, but its syllabus does not itself examine intrinsic-carrier concentration or the conductivity formula, so treat those numbers as enrichment for an A-Level answer. AP Physics 2 does not carry semiconductor electronics at all, and current IB Physics has moved electronics out of the core — do not assume this material is on either. IGCSE 0625 meets diodes and logic gates only qualitatively and does not require the intrinsic-carrier or conductivity numbers; give an IGCSE-style answer, if asked, in words rather than in mobilities and exponentials.

Texts. NCERT Physics Class 12, Chapter 14 ("Semiconductor Electronics"), the intrinsic-semiconductor sections · H. C. Verma, Concepts of Physics Vol. 2, "Semiconductors and Semiconductor Devices" · Halliday, Resnick & Walker, Fundamentals of Physics, "Conduction of Electricity in Solids" · Kittel, Introduction to Solid State Physics, for the carrier-statistics treatment behind equation (2).

Provenance. LANE_SPEC anchor values and standard textbook mobilities. Recompute: $kT = 8.617\times10^{-5}\times300 = 0.0259 \approx 0.026\ \text{eV}$; $E_g/kT(\text{Si}) = 42.6$; $\exp(-E_g/2kT) = \exp(-21.3) = 5.8\times10^{-10}$; $n_i$: Si $1.5\times10^{16}$, Ge $2.4\times10^{19}\ \text{m}^{-3}$, ratio $1600$; Si atom density $5.0\times10^{28}$, atoms per electron $3.3\times10^{12}$; edges $\lambda_{\max} = 1240/E_g$ give $1127$ and $1771\ \text{nm}$; $\sigma = n_i e(\mu_e+\mu_h)$ gives Si $4.4\times10^{-4}\ \text{S m}^{-1}$ ($\rho = 2.3\times10^{3}\ \Omega\,\text{m}$) and Ge $2.2\ \text{S m}^{-1}$ ($\rho = 0.45\ \Omega\,\text{m}$); temperature ratio $(400/300)^{1.5}\exp[(E_g/2k)(1/300-1/400)] = 1.54\times204 = 314$.

Prerequisites. Energy Bands in Solids — the article that established the valence band, the conduction band and the forbidden gap, and showed how the gap's width sorts conductor from semiconductor from insulator. Everything here rests on that gap being small (Si $1.1\ \text{eV}$, Ge $0.7\ \text{eV}$). Leads to. Extrinsic Semiconductors and Doping, which breaks the $n_e = n_h$ symmetry on purpose to make n-type and p-type material, and the mass-action law $n_e n_h = n_i^2$. For the statistics behind equation (2) — where the $T^{3/2}$ prefactor and the mid-gap Fermi level come from — the university seam on the chemical potential of a Fermi gas at finite temperature treats it properly (go-deeper only, never an exam dependency here).