The Seebeck coefficient $S$ (a.k.a. thermopower) tells you how much voltage a material develops when one side is made hotter than the other:

$$S \;=\; -\frac{\Delta V}{\Delta T}\quad(\text{units: µV K}^{-1})$$

It is essentially the average energy carried per charge carrier, measured relative to the Fermi level $E_F$.

A positive $S$ indicates hole-like (p-type) carriers, while a negative $S$ signals electron-like (n-type) transport. In thermoelectric devices, a large magnitude of $S$ is prized because the efficiency figure of merit $ZT = S^{2}\sigma T/(\kappa_e+\kappa_L)$

2. A handy formula

For metals and degenerate semiconductors the Mott expression links $S$ to how sharply the transport quantities change with energy right at $E_F$:

$$S \;\approx\; \frac{\pi^{2}k_B^{2}T}{3e}\; \left.\frac{d\ln\sigma(E)}{dE}\right|_{E=E_F}.$$

Here $\sigma(E)=e^{2}N(E)\,v^{2}(E)\,\tau(E)$ is the energy-resolved electrical conductivity, built from

• $N(E)$ – electronic density of states (DOS),

• $v(E)$ – carrier velocity,

• $\tau(E)$ – scattering/relaxation time.

Because the logarithmic derivative distributes, the term $\tfrac{d\ln N(E)}{dE}\big|_{E_F}$

3. Intuitive picture

• Heat makes carriers diffuse from hot → cold.

• A voltage appears only if carriers above $E_F$ contribute differently than those below it—i.e. if the distribution of states is asymmetric.

• A steep DOS slope means many more states just above $E_F$

If the DOS is flat, carriers above and below $E_F$ cancel and $S$ stays small.

4. Practical knobs to steepen the DOS

5. The trade-off

A large $S$ alone is not enough—you also need high electrical conductivity $\sigma$ and low lattice thermal conductivity $\kappa_L$ to maximise

$$ZT \;=\; \frac{S^{2}\sigma T}{\kappa_e+\kappa_L}.$$

A sky-high DOS slope often comes from heavy effective masses or flat bands, which can reduce carrier mobility (and hence $\sigma$). Much of modern thermoelectric design is finding band structures that keep mobility reasonably high while still providing a steep enough DOS slope.

Key takeaway

“Thermoelectrics like a high DOS slope at $E_F$” because the Seebeck coefficient is proportional to the energy derivative of the carrier population at the Fermi level. A rapidly rising DOS around $E_F$ creates the asymmetry in carrier energies that drives a big thermoelectric voltage. Balancing that steep slope with good mobility and low thermal conductivity is what turns a material into a high-$ZT$

How to calculate

Below is a pragmatic “menu” of first-principles routes you can take, from the quick-and-approximate to the fully ab-initio scattering treatment. Pick the level of rigor that matches your time budget and how sensitive $S$ is to scattering in your material.

Thanks to the GNN, we may be able to skip the costly DFT portion and use the neural network to output the material's Hamiltonian.

1. Semiclassical Boltzmann Transport with a Constant Relaxation Time (τ)

“DFT + BoltzTraP2” — 1 day turnaround

Assumptions & caveats

• τ cancels out of $S$, so the magnitude you get is often reasonable even if you don’t know scattering.

• Works best for lightly doped semiconductors or semimetals where energy dependence of τ is mild.

• Misses phonon-drag and any strong energy or momentum dependence of scattering.

2. Energy-Dependent τ(E) from empirical or ML scattering models

“DFT + AMSET/BoltzTraP-τ(E)” — several days

1. Run DFT + dense bands (as above).

2. Feed the bands into AMSET (or similar) which builds τ(E) from:

   • Deformation potentials → acoustic and polar optical phonon scattering

   • Ionized impurity scattering (need carrier density & dielectric constant)

3. Evaluate $S(T,\mu)$ with the energy-dependent τ(E) plugged into the Boltzmann integrals.

Gives much better doping- and temperature-trends without a full electron-phonon calculation.

3. First-Principles Electron-Phonon (EP) Scattering

“DFPT/EPW → full BTE” — 1–2 weeks

Strengths: no adjustable parameters; captures energy & momentum dependence of τ, multiband effects, and phonon-drag (if included).
Weakness: heavy compute cost; convergence with k/q grids can be painful.