Measuring the Silicon Bandgap

What I set out to do

Measure two physical constants, the silicon bandgap and Boltzmann's constant, from a single transistor, using a 9 V battery, three NPN transistors, a switch, a few resistors and a multimeter. Then combine the two results into a voltage that barely changes with temperature. That's a bandgap reference, and almost every integrated circuit has one inside it.

Why it works

A forward-biased junction follows V_BE = (kT/q)·ln(I/I_S). The saturation current I_S rises steeply with temperature because it carries the bandgap inside it as exp(−E_g/kT), and that growth wins out, so V_BE falls almost linearly as the transistor heats up, by roughly 2 mV per kelvin in silicon. If you extend that line down to 0 K, the thermal terms go to zero and the intercept equals E_g/q. So a temperature sweep gives you the bandgap.

For Boltzmann's constant, run the same transistor at two different currents and subtract. I_S is different for every transistor, but it's the same in both readings, so it cancels and leaves ΔV_BE = (kT/q)·ln(I₁/I₂). That difference grows linearly with absolute temperature, and since the current ratio is known, you can solve for k.

Schematic: an all-NPN current mirror acting as a current sink, a current selector switch, and the diode-connected test transistor — Current mirror, current selector, and the device under test.

A beaker of colored water with a thermometer and the test transistor's leads submerged — Only the test transistor goes in the bath. The mirror stays on the bench.

The measurement

The transistor under test is a diode-connected NPN, with the collector tied to the base. Its current is set by an all-NPN current mirror running as a sink, so the bias holds steady as it heats up. A switch selects either a 10 k or 100 k reference resistor, giving two currents about 10:1 apart for the ΔV_BE half of the experiment.

The mirror stays out of the bath, at room temperature. If the current source heated up along with the transistor, the current would drift with temperature and I'd be measuring two things changing at once. I used eight baths from about −1 °C to 75 °C and gave each one three minutes to settle.

Data I threw out

I threw out every 1 k reading. At that current the 9 V battery sagged badly and the readings drifted while I watched. I also dropped three 10 k points as outliers because they showed a higher voltage at a higher temperature, which can't happen, since V_BE only falls as the transistor warms up. That left five clean points for the bandgap fit and four paired points for Boltzmann.

I didn't want to drop data just because it didn't fit. Both cuts had a physical reason: one set had a known supply problem, and the other broke the rule that V_BE can only fall with temperature.

Results

Plot of base-emitter voltage against temperature with a linear fit extrapolated back to 0 K, intercepting at 1.31 eV — V_BE against temperature, extrapolated to 0 K. Excluded outliers marked separately.

Plot of delta V-BE against temperature with a linear fit through the origin giving k = 1.56e-23 J/K — ΔV_BE against temperature, fitted through the origin.

Quantity	Measured	Accepted	Error
V_BE slope	−2.16 mV/K (R² = 0.97)	≈ −2 mV/K	—
Silicon bandgap	1.31 eV	1.12 eV at room temperature, ≈1.17 eV at 0 K	+17 % / +12 %
Boltzmann's constant	1.56 × 10⁻²³ J/K	1.38 × 10⁻²³ J/K	+13 %

Building the reference

The last step adds the falling V_BE to the rising ΔV_BE, multiplied by about 9.8 so the two slopes cancel. The sum is flat at roughly 1.31 V across the whole temperature range. That's a bandgap reference, and it comes out at the same 1.31 V as the extrapolated intercept, which is a good check that the two measurements agree.

Plot showing the falling V-BE and rising weighted delta V-BE summing to a flat 1.31 volt line across temperature — The two temperature slopes cancel, leaving a flat 1.31 V.

Where the error comes from

Both constants came out high, and I think there are two reasons. The real V_BE curve bends slightly rather than being perfectly straight, so fitting a line and pushing it all the way to 0 K overshoots the true intercept. Also, my bath temperatures weren't perfectly settled. One cold point even read below a warmer one, and you can see the scatter in the plot.

That scatter hurts Boltzmann more than the bandgap, because ΔV_BE is a difference of about 60 mV between two much larger readings, so a small error in either one gets magnified. Tighter temperature control and a wider sweep would pull both closer.

For a 9 V battery and a handheld multimeter, getting within 13–17% of the textbook values seems reasonable to me.

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