Seebeck Effect

The intuition: heat drives carriers downhill

Put one end of a metal bar on a hot plate and leave the other end cool. Inside the metal, the free electrons behave like a gas. At the hot end they jiggle faster and spread out more; at the cold end they are slower and denser. Just like air rushing from a high-pressure region to a low-pressure one, electrons diffuse from the hot end toward the cold end faster than they trickle back.

That net flow is self-limiting. As electrons accumulate at the cold end, it becomes negatively charged and the hot end becomes positively charged. The growing electric field pushes electrons back uphill. Equilibrium is reached when the diffusion current exactly cancels the field-driven drift current — and at that point a steady voltage sits across the bar even though no current flows in the open circuit. That voltage is the Seebeck effect.

The voltage is proportional to the temperature difference:

V = −S · ΔT       (open circuit, single material)

where S is the Seebeck coefficient (also called thermopower), measured in volts per kelvin, and ΔT = T_hot − T_cold. The minus sign is a convention: it makes S positive when the cold end ends up at higher potential (the p-type / hole case).

Why it happens microscopically

The differential form is more honest, because S generally varies with temperature:

S = −dV/dT

The size and sign of S come from how charge carriers respond to a temperature gradient. Two pictures combine:

• Carrier diffusion. Hot carriers have more kinetic energy and longer mean free paths, so the hot-to-cold flux beats the cold-to-hot flux. This is the dominant term in semiconductors and is captured by the Mott formula, which ties S to how steeply the electrical conductivity depends on energy at the Fermi level.
• Phonon drag. At lower temperatures, the lattice vibrations (phonons) streaming from hot to cold collide with carriers and sweep them along, adding an extra contribution that can dominate near 50–200 K in some materials.

The Mott relation for metals makes the energy-dependence explicit:

S = −(π² k_B² T) / (3 e) · [ d(ln σ(E)) / dE ]_(E = E_F)

where k_B is Boltzmann's constant, e the elementary charge, σ(E) the energy-resolved conductivity, and E_F the Fermi energy. Because S scales with k_B/e ≈ 86 µV/K and is reduced by the small factor (k_B T / E_F), metals — with their large Fermi energies (a few eV) — end up with tiny coefficients of order microvolts per kelvin. Semiconductors have E_F near the band edge, so the suppression factor is far weaker and S climbs to hundreds of microvolts per kelvin.

The sign tells you the carrier type

The Seebeck coefficient is one of the cleanest ways to identify the majority charge carrier:

Material type	Majority carrier	Cold end charge	Sign of S
n-type semiconductor	Electrons (−)	Negative	S < 0
p-type semiconductor	Holes (+)	Positive	S > 0
Most metals	Electrons	Usually negative	Small, sign varies

Copper is a notable exception among metals: its measured S is about +1.8 µV/K, positive, because of the detailed shape of its band structure near the Fermi surface — a reminder that the simple "electrons make S negative" rule is only a first approximation.

Why thermocouples need two metals

You cannot read the Seebeck voltage of a single uniform wire by attaching a voltmeter, because the leads of the meter are themselves conductors with their own Seebeck coefficient sitting in the same temperature gradient. Any homogeneous loop gives zero. The trick is to use two different materials joined at a junction. The net voltage around the loop then depends on the difference of their Seebeck coefficients:

V = (S_A − S_B) · (T_hot − T_cold)

This is exactly a thermocouple. One junction sits at the temperature you want to measure (the hot junction), the other is held at a known reference temperature (the cold junction, historically an ice bath at 0 °C, now compensated electronically). The voltage maps to the temperature difference, and standardized alloy pairs give well-tabulated, reproducible curves.

Thermocouple type	Materials	Sensitivity (near 25 °C)	Typical range
Type K	Chromel / Alumel	≈ 41 µV/K	−200 to 1260 °C
Type J	Iron / Constantan	≈ 52 µV/K	−40 to 750 °C
Type T	Copper / Constantan	≈ 43 µV/K	−200 to 350 °C
Type E	Chromel / Constantan	≈ 68 µV/K	−200 to 900 °C
Type S	Pt-10%Rh / Pt	≈ 6 µV/K	0 to 1600 °C

A type-K thermocouple reading a 100 °C difference produces only about 4.1 mV — small, but a stable, repeatable signal that instrumentation amplifiers handle easily. Thermocouples are rugged, cheap, self-powered, and span enormous temperature ranges, which is why they remain the workhorse of industrial temperature sensing.

Real numbers

Scenario	Result
Copper bar, S ≈ 1.8 µV/K, ΔT = 100 K	V ≈ 0.18 mV
Bi₂Te₃ leg, S ≈ 200 µV/K, ΔT = 100 K	V ≈ 20 mV per leg
Type-K thermocouple, ΔT = 100 °C	V ≈ 4.1 mV
TEG module, ~127 thermocouple pairs, ΔT = 70 K	≈ a few volts open circuit
Voyager RTG (Si-Ge legs, Pu-238 heat)	≈ 110 W electrical at launch
Commercial Bi₂Te₃ module efficiency	≈ 5–8 % (ZT ≈ 1)

From sensing to power: thermoelectric generators

A single leg makes only millivolts, so a thermoelectric generator (TEG) stacks many of them. The key trick is to alternate n-type and p-type legs, connect them electrically in series but thermally in parallel. Because n-type legs push current one way and p-type legs the other for the same heat flow, wiring them head-to-tail makes their voltages add instead of cancel. A typical module packs roughly 127 such couples between two ceramic plates.

How good a material is for this job is summarized by the dimensionless figure of merit:

ZT = S²σT / κ

where σ is electrical conductivity and κ is thermal conductivity. You want a large Seebeck coefficient, good electrical conduction (to extract current without resistive loss), and poor thermal conduction (so the temperature difference is not short-circuited by heat leaking through). These requirements fight each other — in metals σ and κ rise together via the Wiedemann–Franz law — which is why good thermoelectrics are heavily doped semiconductors like Bi₂Te₃, PbTe, and Si-Ge alloys engineered to scatter phonons while letting electrons through. Commercial materials sit near ZT ≈ 1; the maximum Carnot-limited efficiency rises with ZT.

The same module run in reverse — pushing current through it — pumps heat and gets cold on one side. That is the Peltier effect, the Seebeck effect's thermodynamic twin, used in portable coolers, CPU spot-cooling, and laboratory temperature control.

JavaScript — Seebeck and thermocouple calculations

// Open-circuit Seebeck voltage for a single material
// S in volts per kelvin, deltaT in kelvin
function seebeckVoltage(S, deltaT) {
  return -S * deltaT; // volts
}

// Copper bar, S = 1.8 uV/K, 100 K difference
const Vcu = seebeckVoltage(1.8e-6, 100);
console.log(`Copper: ${(Vcu * 1e3).toFixed(3)} mV`); // -0.180 mV

// Bismuth telluride n-type leg, S = -200 uV/K
const Vbi = seebeckVoltage(-200e-6, 100);
console.log(`Bi2Te3 leg: ${(Vbi * 1e3).toFixed(1)} mV`); // 20.0 mV

// Thermocouple: voltage depends on the DIFFERENCE of coefficients
function thermocoupleVoltage(S_A, S_B, deltaT) {
  return (S_A - S_B) * deltaT; // volts
}

// Type-K (~41 uV/K net), 100 C difference
const Vk = thermocoupleVoltage(41e-6, 0, 100);
console.log(`Type-K, dT=100C: ${(Vk * 1e3).toFixed(2)} mV`); // 4.10 mV

// Recover temperature difference from a measured voltage
function deltaTfromVoltage(voltage, netSeebeck) {
  return voltage / netSeebeck; // kelvin
}
console.log(`4.1 mV -> dT = ${deltaTfromVoltage(4.1e-3, 41e-6).toFixed(0)} K`); // 100

// Thermoelectric figure of merit ZT = S^2 * sigma * T / kappa
function figureOfMerit(S, sigma, kappa, T) {
  return (S * S * sigma * T) / kappa;
}
// Bi2Te3-like: S=200 uV/K, sigma=1e5 S/m, kappa=1.5 W/mK, T=300 K
const ZT = figureOfMerit(200e-6, 1e5, 1.5, 300);
console.log(`ZT ~= ${ZT.toFixed(2)}`); // ~0.80

// TEG module open-circuit voltage: N couples each with two legs
function tegVoltage(S_p, S_n, deltaT, couples) {
  // each couple contributes (S_p - S_n) * deltaT
  return (S_p - S_n) * deltaT * couples; // volts
}
// 127 couples, S_p=+200, S_n=-200 uV/K, dT=70 K
const Vmod = tegVoltage(200e-6, -200e-6, 70, 127);
console.log(`Module: ${Vmod.toFixed(2)} V`); // ~3.56 V

Where the Seebeck effect shows up

• Temperature measurement. Thermocouples in furnaces, engines, kilns, and the gas-valve flame sensors in older water heaters and stoves.
• Spacecraft power. Radioisotope thermoelectric generators (RTGs) power Voyager 1 and 2, Cassini, Curiosity, and Perseverance — decades of reliable watts with zero moving parts.
• Waste-heat recovery. Harvesting energy from car exhaust, industrial flue gas, and wood stoves to trickle-charge electronics.
• Material characterization. The sign and magnitude of S identify carrier type and probe band structure, used heavily in studying semiconductors and superconductors.
• Cooling (Peltier twin). The reverse effect drives solid-state coolers for picnic boxes, laser diodes, and infrared detectors.
• Body-heat harvesters. Wearable TEGs that run a small sensor from the few-degree gradient between skin and air.

Common mistakes

• Expecting voltage from a single uniform wire. A homogeneous loop in a temperature gradient gives zero net voltage — you always need two materials with different Seebeck coefficients, which is why thermocouples are bimetallic.
• Confusing Seebeck and Peltier. Seebeck turns ΔT into voltage; Peltier turns current into ΔT. They are reciprocal (Peltier coefficient Π = S·T), but they are not the same measurement.
• Ignoring the sign convention. Sign depends on carrier type (electrons vs holes) and a choice of definition; always state which end is referenced before quoting a sign.
• Assuming bigger ΔT alone means more power. Output power also depends on ZT and on the load matching; a high-conductivity material with low S can deliver less power than a moderate one with high S.
• Treating S as constant. The Seebeck coefficient varies with temperature, so thermocouple tables are nonlinear and you should integrate dV = −S(T) dT rather than multiply by a single slope over a wide range.
• Neglecting cold-junction compensation. A thermocouple reads only a temperature difference; forgetting to measure or compensate the reference junction is the classic source of reading errors.