Hall Voltage, Single-Carrier Density & Seebeck Thermoelectric ZT

Two independent illustrative solid-state models

The Hall example assumes a uniform slab with current and magnetic field perpendicular. With the chosen lead and field orientation, VH=RH*I*B/t. Signed RH, current and field determine the voltage sign; reversing one reverses the voltage. Thickness must be positive.

For a simple single-carrier model with charge magnitude e, n=1/(ABS(RH)*e). This is an effective single-carrier estimate, not a universal inversion for multi-carrier materials or a full transport model. A zero Hall coefficient cannot be converted into a finite density by this expression.

The thermoelectric example uses illustrative coefficients, not a certified material specification. ZT=S²*sigma*T/kappa uses properties at the same temperature and total thermal conductivity, including electronic and lattice contributions. ZT=0.8 is dimensionless; it does not mean 80% conversion efficiency.

The open-circuit relation VS=S*deltaT assumes a constant or suitable effective coefficient and a chosen voltage polarity. For a thermocouple loop the effective coefficient is the difference between its two legs. Reversing deltaT reverses this voltage. Open-circuit voltage alone does not specify deliverable power or efficiency.

B9 is the representative temperature in kelvin, treated as the arithmetic mean for the voltage example. The endpoints B9 plus/minus B10/2 must be above zero kelvin. Strongly temperature-dependent coefficients require integration rather than this constant-coefficient approximation.

Hall voltage in volts, D2

B2 is RH in m³/C, B3 current in A, B4 magnetic flux density in tesla and B5 slab thickness in metres.

=IF(COUNT(B2:B5)<>4,"Enter numeric Hall inputs",IF(B5<=0,"Thickness must be positive",B2*B3*B4/B5))

Single-carrier number density in m⁻³, D3

Uses the SI elementary charge magnitude 1.602176634E-19 C. The absolute Hall coefficient gives positive number density; its sign remains relevant to carrier type in this model.

=IF(COUNT(B2)<>1,"Enter numeric Hall coefficient",IF(B2=0,"Carrier density undefined for zero Hall coefficient",1/(ABS(B2)*1.602176634E-19)))

Open-circuit Seebeck voltage in volts, D4

B6 is S in V/K and B10 is signed temperature difference in K. This sign convention is VS=S*deltaT.

=IF(COUNT(B6,B9:B10)<>3,"Enter numeric Seebeck and temperature inputs",IF(B9-ABS(B10)/2<=0,"Both endpoint temperatures must be positive Kelvin",B6*B10))

Material figure of merit ZT, D5

B7 is electrical conductivity in S/m, B8 total thermal conductivity in W/(m*K), and B9 temperature in K. Negative S is allowed because it is squared; conductivities and temperature must be positive.

=IF(COUNT(B6:B9)<>4,"Enter numeric thermoelectric inputs",IF(MIN(B7:B9)<=0,"Conductivities and Kelvin temperature must be positive",B6^2*B7*B9/B8))

Further reading

• Microsoft Support — ROUND function
• NIST — Resistivity and Hall measurements
• NIST researchers — Applications of high-throughput screening tools for thermoelectric materials