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
The Seebeck coefficient .
quantifies how strongly a material converts a temperature difference into an electrical voltage. This property
The Seebeck coefficient (a.k.a. thermopower) tells you how much voltage a material develops when one side is made hotter than the other:
It is essentially the average energy carried per charge carrier, measured relative to the Fermi level .
A positive indicates hole-like (p-type) carriers, while a negative signals electron-like (n-type) transport. In thermoelectric devices, a large magnitude of is prized because the efficiency figure of merit scales with . 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 —while maintaining good electrical conductivity and low thermal conductivity.
For metals and degenerate semiconductors the Mott expression links to how sharply the transport quantities change with energy right at :
Here is the energy-resolved electrical conductivity, built from
– electronic density of states (DOS),
– carrier velocity,
– scattering/relaxation time.
Because the logarithmic derivative distributes, the term is often the largest contributor—especially when scattering is weakly energy-dependent.
Heat makes carriers diffuse from hot → cold.
A voltage appears only if carriers above 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 (or just below, for -type) than the opposite side, so the average energy of carriers that flow is farther from .
→ larger energy per carrier → larger Seebeck.
If the DOS is flat, carriers above and below cancel and stays small.
Strategy | What it does | Examples |
---|---|---|
Band-edge “energy filtering” | Place slightly inside the valence or conduction band so carriers come from the band edge where DOS rises sharply | Classic PbTe, BiTe |
Valley/band convergence | Merge several near-degenerate pockets ⇒ higher DOS effective mass without killing mobility | Half-Heuslers, SnSe, MgSb |
Narrow or flat bands | Flat bands ⇢ high DOS; works if mobility is not destroyed | CuSe super-ionic phase |
Resonant impurities | Localized impurity level inside a band produces a sharp DOS spike | Tl-doped PbTe |
A large alone is not enough—you also need high electrical conductivity and low lattice thermal conductivity to maximise
A sky-high DOS slope often comes from heavy effective masses or flat bands, which can reduce carrier mobility (and hence ). 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 ” because the Seebeck coefficient is proportional to the energy derivative of the carrier population at the Fermi level. A rapidly rising DOS around 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- thermoelectric.
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 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.
“DFT + BoltzTraP2” — 1 day turnaround
Step | What you do | Typical tools |
---|---|---|
a. Ground-state DFT | Relax structure → compute bands on a dense k-mesh (≥ 20 × 20 × 20). | VASP, QE, WIEN2k, FP-LO |
b. Interpolate bands & velocities | Feed the eigenvalues (and optional k-derivatives) to a Fourier/Wannier interpolator. | BoltzTraP2, BoltzWann |
c. Solve BTE (constant τ) | Specify a (temperature-independent) τ or leave it symbolic; code integrates the transport tensors and returns . | BoltzTraP2 CLI or Python API |
Assumptions & caveats
τ cancels out of , 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.
“DFT + AMSET/BoltzTraP-τ(E)” — several days
Run DFT + dense bands (as above).
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)
Evaluate with the energy-dependent τ(E) plugged into the Boltzmann integrals.
Gives much better doping- and temperature-trends without a full electron-phonon calculation.
“DFPT/EPW → full BTE” — 1–2 weeks
Step | Notes |
---|---|
a. DFPT: compute dynamical matrix & EP matrix elements on coarse q-mesh. | QE, ABINIT |
b. Wannier-interpolate matrix elements to dense meshes (EPW). | |
c. Solve linearized BTE for the nonequilibrium distribution → obtain σ(Τ, μ) and . | EPW 6.x, ShengBTE-e, Gollum-EP |
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.