D3Q

What D3Q is

D3Q extends Quantum ESPRESSO with the third derivative of the energy from density-functional perturbation theory, giving anharmonic force constants without finite displacements. Its thermal transport code solves the Boltzmann equation variationally and computes phonon linewidths. On the map it serves the third-order force constants, the linewidths and the conductivities.

WebsiteRepositoryOn the mapAll codesLicense: GPL-2.0 (thermal2: CeCILL-2.1 OR GPL-2.0-or-later; some files GPL-3.0; bundled minpack: Minpack)

In more detail

Third-order force constants without supercells

The three-phonon scattering coefficients are third derivatives of the total energy with respect to three phonon displacements with wave vectors q, q' and q'' summing to a reciprocal lattice vector. The 2n+1 theorem of density functional perturbation theory expresses them through the first-order change of the density and wave functions, so no supercell is needed. Paulatto 2013 extended the D3 code of Quantum ESPRESSO from the (0, -q, q) case to three arbitrary wave vectors and to metals and zero-gap systems (Paulatto 2013). The coefficients are computed on a coarse grid, 4x4x4 for diamond, and Fourier interpolated to the dense mesh of the transport calculation (Fugallo 2013).

The exact solution of the Boltzmann equation

The Omini-Sparavigna iteration from the single-mode approximation converges only when resistive processes dominate. Fugallo 2013 recast the problem as the minimization of a quadratic functional whose minimum is the conductivity and solved it with a preconditioned conjugate gradient. For diamond at 100 K on a 30x30x30 mesh with 20 cm-1 smearing the iteration becomes unstable after 77 steps at a precision of about 300 W/m/K, while the conjugate gradient converges in four. Without borders the iteration fails after 91 steps with an error near 3000 W/m/K (Fugallo 2013). The transport step cost two orders of magnitude more CPU time than the DFPT force constants.

Linewidths and the single-mode approximation

The anharmonic broadening of a mode is the imaginary part of its self energy, summed over a fine q grid with the two energy-conserving deltas smeared. In graphene the broadening of the optical branches shows sudden steps and spikes, and the TA and LA branches keep a finite width at small q at finite temperature. Within the single-mode approximation the out-of-plane conductivity of graphite agrees with experiment while the in-plane value is about a factor two too low (Paulatto 2013). The exact solution closes this gap. The in-plane conductivity of graphite is an order of magnitude above the SMA and matches the 2000 W/m/K measured on large perfect crystals (Fugallo 2014).

Validation

For diamond the exact BTE follows the measured temperature dependence for samples with 98.9%, 99.9% and 99.93% C12, and a Casimir length of 0.3 cm with shape factor 0.5 describes the finite size (Fugallo 2013). For graphene the exact solution reaches the diffusive limit at lengths two orders of magnitude above the SMA, between 100 um and 1 mm, so an intrinsic kappa needs samples of order 1 mm. Hydrogenation halves the conductivity and fluorination cuts it by an order of magnitude (Fugallo 2014). In graphene, boron nitride and graphane second sound survives to room temperature, and in graphene it propagates over distances of order 1 mm (Cepellotti 2015).

What it computes on the map

5 quantities, drawn by tier with the data flow between them. Dashed boxes are intermediate quantities. Select one for its formula and the other codes that compute it.

Each quantity with the interface the representation records.

• ForceConstants[order=2]d3_q2r.x mat2R (thermal2 format, not flfrc; centered by default, nfar=2), Ry/bohr^2Ry_per_bohr2
• ForceConstants[order=3]d3_qq2rr.x mat3R (d3_asr3.x, d3_sparse.x refine it), Ry/bohr^3Ry_per_bohr3
• Linewidth[channel=anharmonic_3ph]d3_lw.x calculation='lw imag' ($prefix_T$T_s$sigma.out), cm^-1inverse_cm
• ThermalConductivity[bte_solver=direct_inverse]d3_tk.x calculation='cgp' (alias 'exact'; last row of $prefix.$grid_T$T_s$sigma.out), W/(m K)W_per_m_per_K
• ThermalConductivity[bte_solver=rta]d3_tk.x calculation='sma' ($prefix.$grid_P_sma.out, K_P), W/(m K)W_per_m_per_K

Releases on record

Versions on record for this code, with the license read from the release itself.

References

The code

The papers that describe the code; cite them when you use it.

1. G. Fugallo, M. Lazzeri, L. Paulatto et al., Ab initio variational approach for evaluating lattice thermal conductivity, Physical Review B, 2013. doi.org/10.1103/PhysRevB.88.045430
2. L. Paulatto, F. Mauri, M. Lazzeri, Anharmonic properties from a generalized third-order ab initio approach: Theory and applications to graphite and graphene, Physical Review B, 2013. doi.org/10.1103/PhysRevB.87.214303

Applications

Highly cited studies that cite the code, with a few picked by hand.

1. G. Fugallo, A. Cepellotti, L. Paulatto et al., Thermal Conductivity of Graphene and Graphite: Collective Excitations and Mean Free Paths, Nano Letters, 2014. doi.org/10.1021/nl502059f
2. A. Cepellotti, G. Fugallo, L. Paulatto et al., Phonon hydrodynamics in two-dimensional materials, Nature Communications, 2015. doi.org/10.1038/ncomms7400

How to cite

L. Paulatto, F. Mauri, M. Lazzeri, Anharmonic properties from a generalized third-order ab initio approach: Theory and applications to graphite and graphene, Phys. Rev. B 87, 214303 (2013); exact BTE: G. Fugallo, M. Lazzeri, L. Paulatto, F. Mauri, Ab initio variational approach for evaluating lattice thermal conductivity, Phys. Rev. B 88, 045430 (2013), doi:10.1103/PhysRevB.88.045430

doi.org/10.1103/PhysRevB.87.214303