Why silicon dominates over germanium: Visual walkthrough

We will link back to the parent parent topic and to the vault notes on Bandgap and intrinsic carrier concentration, Fermi-Dirac distribution and thermal excitation, and Intrinsic vs extrinsic semiconductors as each idea comes up.

Step 1 — What is a "band", really? (drawing the energy ladder)

WHAT: we draw energy on a vertical axis — higher up means more energy — and mark the two allowed shelves plus the empty gap between them.

WHY: every later symbol ( $E_{g}$ , "carrier", "leakage") is just a statement about this picture. If we don't fix the picture first, the letters mean nothing.

PICTURE: the low grey shelf is packed with electrons (dots). The high shelf is empty. The white stripe in the middle is the forbidden zone.

Figure — Why silicon dominates over germanium

Step 2 — The bandgap $E_{g}$ : the exact height of the jump

WHAT: we label the gap height. Silicon's gap is $E_{g}^{S i} \approx 1.12 eV$ ; germanium's is $E_{g}^{G e} \approx 0.66 eV$ — a shorter gap.

WHY this quantity and not another? Because the only thing that decides whether an electron can become free is whether it can pay the entry fee $E_{g}$ . Everything downstream is "how often does nature pay this fee?"

PICTURE: two ladders side by side — silicon's rung is tall, germanium's is short. Same picture, different jump heights.

Step 3 — Heat as a coin-flip: where $e^{- E / k_{B} T}$ comes from

WHAT: we introduce the Boltzmann factor $e^{- E / k_{B} T}$ . Read it as: "the odds of borrowing energy $E$ from the heat bath."

WHY the exponential and not, say, $1/ E$ ? Because independent lucky pushes multiply, and multiplying many small probabilities gives an exponential decay. This is the same rule behind the Fermi-Dirac distribution and thermal excitation tail — deep in the gap, Fermi-Dirac collapses to exactly this Boltzmann shape.

Term by term: $decay e^{- E_{g} fee / k_{B} T typical jiggle}$

$E_{g}$ in the top — the fee is big, the exponent is very negative, the odds are tiny.
$k_{B} T$ on the bottom — hotter ( $T$ up) makes the exponent less negative, so more jumps. Colder freezes the jumps out.

PICTURE: a curve of "chance of a jump" versus energy needed. It plummets. We mark where $E_{g}^{S i}$ and $E_{g}^{G e}$ land on it — germanium sits far higher up the curve (easier).

Step 4 — Why the exponent gets a factor of 2 (the pair rule)

WHAT: we define the intrinsic carrier concentration $n_{i}$ — the number of these thermally-freed carriers per cubic centimetre in a pure crystal (see Bandgap and intrinsic carrier concentration).

WHY the square root? The rate at which pairs are born — equivalently the product $n \cdot p$ of electron count $n$ times hole count $p$ — scales as $e^{- E_{g} / k_{B} T}$ (one whole fee per pair). But in a pure crystal every electron has a matching hole, so $n = p = n_{i}$ , giving $n \cdot p = n_{i}^{2}$ . Therefore: $n_{i} = n p = C e^{- E_{g} / k_{B} T} = C e^{- E_{g} /2 k_{B} T}$

$n p$ — the pair product, costs the full $e^{- E_{g} / k_{B} T}$ .
— because $n_{i}$ is the count of one species, not the pair product.
the square root halves the exponent → the $2$ in $e^{- E_{g} /2 k_{B} T}$ .

PICTURE: one jump, two travellers — a filled dot leaving on the high shelf, an open circle (hole) left behind on the low shelf, arrows showing both drifting.

Step 5 — The full shape of $n_{i}$ (folding in the prefactor)

WHAT: we now have the complete room-temperature law.

WHY carry $T^{3/2}$ at all if it is weak? Honesty: it is there. But for a ratio of Si to Ge at the same temperature, the $T^{3/2}$ is identical on top and bottom and cancels — so it will vanish in the next step. We keep it visible so the cancellation is a choice, not a hidden cheat.

PICTURE: $n_{i}$ versus temperature on a log axis — a nearly straight rising line dominated by the exponential, with the gentle $T^{3/2}$ curvature barely visible. Silicon's line sits far below germanium's at every temperature.

Step 6 — The ratio: watch everything cancel

WHAT: we divide germanium's $n_{i}$ by silicon's, both at $T = 300 K$ .

$\frac{n _{i} ( Ge )}{n _{i} ( Si )} = \frac{B T ^{3/2} e ^{- E_{g}^{G e} /2 k_{B} T}}{B T ^{3/2} e ^{- E_{g}^{S i} /2 k_{B} T}}$

WHY: the messy prefactors are the same physics for two group-IV crystals at the same $T$ , so $B$ cancels and $T^{3/2}$ cancels. What survives is only the difference in the jump fee:

$= e^{(E_{g}^{S i} - E_{g}^{G e}) /2 k_{B} T}$

Term by term, watch each symbol earn its place:

$E_{g}^{S i} - E_{g}^{G e} = 0.46 eV$ — germanium's shorter gap is the whole story; a smaller fee means far more jumps.
$2 k_{B} T = 2 \times 0.02585 = 0.0517 eV$ — the halved thermal budget from Step 4.
the sign is positive (Si gap is bigger), so the exponent is positive and the ratio is greater than 1: germanium has more carriers, exactly as intuition (short gate → more sneak-throughs) demanded.

$\frac{0.46}{0.0517} = 8.9 \Rightarrow e^{8.9} \approx 7.3 \times 1 0^{3}$

PICTURE: the two exponential curves from Step 3 with a vertical bracket showing the $0.46 eV$ horizontal shift, and the resulting $\approx 7300 \times$ vertical gap between the two carrier levels.

Step 7 — The degenerate cases (never leave a gap in the argument)

Case $T \to 0$ (absolute zero): the exponent $- E_{g} /2 k_{B} T \to - \infty$ , so $e^{- \infty} = 0$ . Both crystals have $n_{i} \to 0$ : no heat, no free carriers, perfect insulators. Silicon and germanium become identical (both zero) — the ratio is undefined ( $0/0$ ) because the shape only differs when heat is present.

Case $T \to \infty$ (very hot): the exponent $\to 0$ , so $e^{0} = 1$ . The gap stops mattering; both fill up with carriers and the ratio collapses toward 1. This is the physical meaning of thermal runaway — the semiconductor stops behaving like a switch and acts like a plain conductor. Germanium reaches this ruin at a lower temperature because it starts thousands of times closer to it.

Case $E_{g}^{S i} = E_{g}^{G e}$ (imaginary equal gaps): the difference in the exponent is $0$ , ratio $= e^{0} = 1$ . Confirms the exponent is driven purely by the gap difference — exactly what we claimed in Step 6.

PICTURE: the ratio $n_{i} (Ge) / n_{i} (Si)$ plotted against temperature: it starts undefined near $0 K$ , peaks huge at low-but-nonzero $T$ , and decays toward $1$ as $T$ climbs. Room temperature ( $300 K$ ) is marked at the $\approx 7300$ level.

The one-picture summary

Everything above compresses into a single frame: two energy ladders, the shared Boltzmann curve, the $0.46 eV$ horizontal shift, and the $\approx 7300 \times$ vertical carrier gap that decided which element runs the world.

Recall Feynman retelling — the whole walkthrough in plain words

Picture two electric fences with electrons grazing behind them. Silicon's fence is tall ( $1.12 eV$ ), germanium's is short ( $0.66 eV$ ). Heat is a crowd of electrons randomly trying to hop over — and the chance of clearing a fence drops exponentially with its height ( $e^{- E_{g} / k_{B} T}$ ). Because each successful hop leaves behind an empty seat (a hole) that also wanders, we count pairs, and taking a square root to get one species halves the exponent — that's the mysterious 2 in $e^{- E_{g} /2 k_{B} T}$ . When we ask "how many more hoppers does the short fence let through?", every messy factor cancels and only the fence-height difference survives: $0.46 eV$ divided by the halved thermal budget $0.0517 eV$ gives $8.9$ , and $e^{8.9} \approx 7300$ . So germanium leaks thousands of times more even when it should be OFF — it overheats, misbehaves, and loses the market. At absolute zero both fences are perfect (nobody hops). Crank the heat high enough and both fences drown — germanium drowns first. Same tall silicon fence that keeps things quiet also grows a free glassy raincoat (SiO₂) we build switches on. Tall fence, quiet OFF state, free raincoat: silicon wins.

Connections

Bandgap and intrinsic carrier concentration
Intrinsic vs extrinsic semiconductors
Fermi-Dirac distribution and thermal excitation
Carrier mobility and drift velocity
MOSFET operation and the gate oxide
SiO2 and the planar process
Doping n-type and p-type