Why the DOS-slope at the Fermi level matters for thermoelectrics

Came across this idea when I was doing some research on what we can do with the Hamiltonian of a material. Turns out there's signal for determining what a good thermoelectric material is in its density of states.

Previously, I had avoided Hamiltonians as they are fundamental outputs of DFT but I don't have the resources to be able to run them. Turns out there are some GNNs that can predict them very cheaply. If that is the case, we could quickly screen a large number of materials for good candidates.

Check out this nice intro into thermoelectrics from some folks at my alma mater! Figures in this post come from there. https://thermoelectrics.matsci.northwestern.edu/thermoelectrics/index.html

What are we looking for

The Seebeck coefficient $S$.

$S$ quantifies how strongly a material converts a temperature difference into an electrical voltage. This property

1. What the Seebeck coefficient measures

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 $ZT = S^{2}\sigma T/(\kappa_e+\kappa_L)$ scales with $S^{2}$. Thus, materials design often focuses on electronic structures that maximize the asymmetry of carrier energies around the Fermi level—steepening the density-of-states slope near $E_F$—while maintaining good electrical conductivity and low thermal conductivity.

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 $\tfrac{d\ln N(E)}{dE}\big|_{E_F}$ is often the largest contributor—especially when scattering is weakly energy-dependent.

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 $E_F$ (or just below, for $p$-type) than the opposite side, so the average energy of carriers that flow is farther from $E_F$.
  → larger energy per carrier → larger Seebeck.

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-$ZT$ thermoelectric.

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.