Efficient Anharmonic Lattice Dynamics Calculations of Thermal Transport in Crystalline and Disordered Solids using kALDo

kALDo is a versatile and scalable open-source software to compute phonon transport in crystalline and amorphous solids. It features real space QHGK calculations and three different solvers of the linearized BTE: direct inversion, self-consistent cycle, and RTA. The algorithms are implemented using linear algebra operations on tensors, to take advantage of multithreading on GPU and CPU using Numpy, Tensorflow, and optimized tensor libraries. Using the Atomic Simulation Environment, κALDo can calculate the IFCs using several ab initio and molecular dynamics codes, thus enabling the use of first-principles DFT, empirical forcefields, and semi-empirical tight- binding. A native LAMMPS interface is also available in the USER-PHONON package. Finally, through seamless integration with the hiPhive package, the IFC calculation can take advantage of compressing-sensing machine learning algorithms. The code is released open-source for the community to use and contribute with edits and suggestions. It is designed on modern software best practices, and we hope to provide a development platform to implement new theory and methods.

The code is available on GitHub, while kALDO website can be found here.

Introduction

Understanding heat transport in semiconductors and insulators is of fundamental importance because of its technological impact in electronics and renewable energy harvesting and conversion. Anharmonic Lattice Dynamics provides a powerful framework for the description of heat transport at the nanoscale. One of the advantages of this method is that it naturally includes quantum effects due to atoms vibrations, which are needed to compute thermal properties of semiconductors widely use in nanotechnology, like Silicon and Carbon, even at room temperature. While heat transport in amorphous and crystalline semiconductors has a different microscopic origin, a unified approach to simulate both crystals and glasses has been devised. Here we introduce a unified workflow, which implements both the Boltzmann Transport equation (BTE) and the Quasi Harmonic Green-Kubo (QHGK) methods. We discuss how the theory can be optimized to exploit modern parallel architectures, and how it is implemented in kALDo: a versatile and scalable open-source software to compute phonon transport in solids.

Theory

In semiconductors, electronic and vibrational dynamics often occur over different time scales, and can thus be decoupled using the Born Oppenheimer approximation. Under this assumption, the potential \(\phi\) of a system made of \(N_{atoms}\) atoms, is a function of all the \(x_{i\alpha}\) atomic positions, where \(i\) and \(\alpha\) refer to the atomic and Cartesian indices, respectively. Near thermal equilibrium, the potential energy can be Taylor expanded in the atomic displacements, \(\mathbf{u}=\mathbf x-\mathbf{x}_{\rm equilibrium}\),

\[\phi(\{x_{i\alpha}\})=\phi_0 + \sum_{i\alpha}\phi^{\prime}_{i\alpha }u_{i\alpha} +\frac{1}{2} \sum_{i\alpha i'\alpha'} \phi^{\prime\prime}_{i\alpha i'\alpha '}u_{i\alpha} u_{i'\alpha'}+\] \[+ \frac{1}{3!}\sum_{i\alpha i'\alpha 'i''\alpha ''} \phi^{\prime\prime\prime}_{i\alpha i'\alpha 'i''\alpha ''} u_{i\alpha }u_{i'\alpha '} u_{i''\alpha ''} + \dots,\]

where

\[\phi^{\prime\prime}_{i\alpha i'\alpha '}=\frac{\partial^{2} \phi}{\partial u_{i\alpha } \partial u_{i'\alpha '} },\qquad \phi^{\prime\prime\prime}_{i\alpha i'\alpha 'i''\alpha ''}=\frac{\partial^{3} \phi}{\partial u_{i\alpha } \partial u_{i'\alpha '} \partial u_{i''\alpha ''}},\]

are the second and third order interatomic force constants (IFC). The term \(\phi_0\) can be discarded, and the forces \(F = - \phi^{\prime}\) are zero at equilibrium.

The IFCs can be evaluated by finite difference, which consists in calculating the difference between the forces acting on the system when one of the atoms is displaced by a small finite shift along a Cartesian direction. The second and third order IFCs need respectively, \(2N_{atoms}\), and \(4N_{atoms}^2\) forces calculations. In crystals, this amount can be reduced exploiting the spatial symmetries of the system, or adopting a compressed sensing approach. In the framework of DFT, it is also possible and often convenient to compute IFCs using perturbation theory.

The dynamical matrix is the second order IFC rescaled by the masses, \(D_{i\alpha i'\alpha}=\phi^{\prime\prime}_{i\alpha i'\alpha'}/\sqrt{m_im_{i'}}\). It is diagonal in the phonons basis

\[\sum_{i'\alpha'} D_{i\alpha i'\alpha'}\eta_{i'\alpha'\mu} =\eta_{i\alpha\mu} \omega_\mu^2\]

and \(\omega_\mu/(2\pi)\) are the frequencies of the normal modes of the system.

For crystals, where there is long range order due to the periodicity, the dimensionality of the problem can be reduced. The Fourier transfom maps the large direct space onto a compact volume in the reciprocal space: the Brillouin zone. More precisely we adopt a supercell approach, where we calculate the dynamical matrix on \(N_{\rm replicas}\) replicas of a unit cell of \(N_{\rm unit}\) atoms, at positions \(\mathbf R_l\), and calculate

\[D_{i \alpha k i' \alpha'}=\sum_l \chi_{kl} D_{i \alpha l i' \alpha'},\quad \chi_{kl} = \mathrm{e}^{-i \mathbf{q_k}\cdot \mathbf{R}_{l} },\]

where \(\mathbf q_k\) is a grid of size \(N_k\) indexed by \(k\) and the eigenvalue equation becomes

\[\sum_{i'\alpha'} D_{i \alpha k i' \alpha'} \eta_{i' \alpha'k s}=\omega_{k m}^{2} \eta_{i \alpha k s }.\]

which now depends on the quasi-momentum index, \(k\), and the phonons mode \(s\).

Boltzman Transport Equation

At finite temperature \(T\), the Bose Einstein statistic is the quantum distribution for atomic vibrations

\[n_{\mu}=n(\omega_{\mu})=\frac{1}{e^{\frac{\hbar\omega_{\mu}}{k_B T}}-1}\]

where \(k_B\) is the Boltzmann constant and we use \(\mu =(k,s)\).

We consider a small temperature gradient applied along the \(\alpha\)-axis of a crystalline material. If the phonons population depends on the position only through the temperature, \(\frac{\partial n_{\mu\alpha}}{\partial x_\alpha} = \frac{\partial n_{\mu\alpha}}{\partial T}\nabla_\alpha T\), we can Taylor expand it

\[\tilde n_{\mu\alpha} \simeq n_\mu + \lambda_{\mu\alpha} \frac{\partial n_\mu}{\partial x_\alpha} \simeq n_\mu + \psi_{\mu\alpha}\nabla_\alpha T\]

with \(\psi_{\mu\alpha}=\lambda_{\mu\alpha} \frac{\partial n_\mu}{\partial T}\), where \(\lambda_{\mu\alpha}\) is the phonons mean free path. Being quantum quasi-particles, phonons have a well-defined group velocity, which, for the acoustic modes in the long wavelength limit, corresponds to the speed of sound in the material,

\[v_{ ks\alpha}=\frac{\partial \omega_{k s}}{\partial {q_{k\alpha}}} = \frac{1}{2\omega_{ks}}\sum_{i\beta l i'\beta'} i R_{l \alpha} D_{i\beta li'\beta'}\chi_{kl} \eta_{ks i\beta}\eta_{ksi'\beta}\]

and the last equality is obtained by applying the derivative with respect to \(\mathbf{q}_k\) directly to the eigenvectors Equation

The heat current per mode is written in terms of the phonon energy \(\hbar \omega\), velocity \(v\), and out-of-equilibrium phonons population, \(\tilde n\):

\[j_{\mu\alpha'} =\sum_\alpha \hbar \omega_\mu v_{\mu\alpha'} (\tilde n_{\mu\alpha} - n_{\mu})\simeq- \sum_\alpha c_\mu v_{\mu\alpha'} \mathbf{\lambda}_{\mu\alpha} \nabla_\alpha T .\]

As we deal with extended systems, we can assume heat transport in the diffusive regime, and we can use Fourier’s law

\[J_{\alpha}=-\sum_{\alpha'}\kappa_{\alpha\alpha'} \nabla_{\alpha'} T,\]

where the heat current is the sum of the contribution from each phonon mode: \(J_\alpha = 1/(N_k V)\sum_\mu j_{\mu\alpha}\). The thermal conductivity then results:

\[\kappa_{\alpha \alpha'}=\frac{1}{ V N_k} \sum_{\mu} c_\mu v_{\mu\alpha} \lambda_{\mu\alpha'},\]

where we defined the heat capacity per mode

\[c_\mu=\hbar \omega_\mu \frac{\partial n_\mu}{\partial T},\]

which is connected to total heat capacity through \(C = \sum_\mu c_\mu /NV\).

We can now introduce the BTE, which combines the kinetic theory of gases with collective phonons vibrations: \({\mathbf{v}}\_{\mu} \cdot {\boldsymbol{\nabla}} T \frac{\partial n\_{\mu}}{\partial T}=\left.\frac{\partial n\_{\mu}}{\partial t}\right|\_{\text {scatt}},\) where the scattering term, in the linearized form is

\[\left.\frac{\partial n\_{\mu}}{\partial t}\right|\_{\text {scatt}}=\] \[\frac{\nabla_\alpha T}{\omega_\mu N_k}\sum_{\mu^{\prime} \mu^{\prime \prime}}^{+} \Gamma_{\mu \mu^{\prime} \mu^{\prime \prime}}^{+} \left(\omega_\mu\mathbf{\psi}_{\mu\alpha} +\omega_{\mu^{\prime}}\mathbf{\psi}_{\mu^{\prime}\alpha} -\omega_{\mu^{\prime \prime}} \mathbf{\psi}_{\mu^{\prime \prime}\alpha}\right) +\] \[+\frac{\nabla_\alpha T}{\omega_\mu N_k}\sum_{\mu^{\prime} \mu^{\prime \prime}}^{-} \frac{1}{2} \Gamma_{\mu \mu^{\prime} \mu^{\prime \prime}}^{-} \left(\omega_\mu\mathbf{\psi}_{\mu\alpha} -\omega_{\mu^{\prime}} \mathbf{\psi}_{\mu^{\prime}\alpha} -\omega_{\mu^{\prime \prime}} \mathbf{\psi}_{\mu^{\prime \prime}\alpha}\right) .\]

\(\Gamma^{+}_{\mu\mu'\mu''}\) and \(\Gamma^{-}_{\mu\mu'\mu''}\) are the scattering rates for three-phonon scattering processes, and they correspond to the events of phonons annihilation \(\mu, \mu'\rightarrow\mu''\) and phonons creation \(\mu \rightarrow\mu',\mu''\)

\[\Gamma\_{\mu \mu^{\prime} \mu^{\prime \prime}}^{\pm} =\frac{\hbar \pi}{8} \frac{g\_{\mu\mu'\mu''}^{\pm}}{\omega\_{\mu} \omega\_{\mu'} \omega\_{\mu''}}\left|\phi\_{\mu \mu^{\prime} \mu^{\prime \prime}}^{\pm}\right|^{2},\]

and the projection of the potentials on the phonon modes are given by

\[\phi^\pm _{ksk's'k'' s''}= \sum_{il'i'l''i''} \frac{ \phi_{il'i'l''i''}} {\sqrt{m_{i}m_{i'}m_{i''}}} \eta_{i ks}\eta^{\pm}_{i'k' s'} \eta^*_{i''k''s''}\chi^\pm_{k'l'}\chi^*_{k''l''}\]

with \(\eta^+ =\eta\), \(\chi^+=\chi\) and \(\eta^- =\eta^*\), \(\chi^-=\chi^*\). The phase space volume \(g^\pm_{\mu\mu^\prime\mu^{\prime\prime}}\) in the previous equation are defined as

\[g^+_{\mu\mu^\prime\mu^{\prime\prime}} = (n_{\mu'}-n_{\mu''}) \delta^+_{\mu\mu^\prime\mu^{\prime\prime}}\] \[g^-_{\mu\mu^\prime\mu^{\prime\prime}} = (1 + n_{\mu'}+n_{\mu''}) \delta^-_{\mu\mu^\prime\mu^{\prime\prime}},\]

and include the \(\delta\) for the conservation of the energy and momentum in three-phonons scattering processes,

\[\delta_{ks k's' k''s''}^{\pm}= \delta_{\mathbf q_{k}\pm\mathbf q_{k'}-\mathbf q_{k''}, \mathbf Q} \delta\left(\omega_{ks}\pm\omega_{k's'}-\omega_{k''s''}\right),\]

with \(Q\) the lattice vectors. Finally, the normalized phase-space per mode \(g_\mu=\frac{1}{N}\sum_{\mu'\mu''}g_{\mu\mu'\mu''}\), provides useful information about the weight of a specific mode in the anharmonic scattering processes.

In order to calculate the conductivity, we express the mean free path in terms of the 3-phonon scattering rates \(v_{\mu\alpha} = \tilde \Gamma_{\mu\mu' }\lambda_\mu = (\delta_{\mu\mu'}\Gamma^0_\mu + \Gamma^{1}_{\mu\mu'})\lambda_{\mu\alpha},\)

where we introduced

\[\Gamma^{0}_\mu=\sum_{\mu'\mu''}(\Gamma^+_{\mu\mu'\mu''} + \Gamma^-_{\mu\mu'\mu''} ),\]

and

\[\Gamma^{1}_{\mu\mu'}= \frac{\omega_{\mu'}}{\omega_\mu} \sum_{\mu''}(\Gamma^+_{\mu\mu'\mu''} -\Gamma^+_{\mu\mu''\mu'} -\Gamma^-_{\mu\mu'\mu''} -\Gamma^-_{\mu\mu''\mu'} ).\]

In RTA, the off-diagonal terms are ignored, \(\Gamma^{1}_{\mu\mu'}=0\), and the conductivity is

\[\kappa_{\alpha\alpha'} =\frac{1}{N_kV} \sum_{\mu}c_\mu v_{\mu\alpha}{\lambda_{\mu\alpha'}} =\frac{1}{N_kV} \sum_\mu c_\mu v_{\mu\alpha} {\tau_\mu}{v_{\mu\alpha'}},\]

where \(\tau_\mu=1/2\Gamma_{\mu}^0\) corresponds the phonons lifetime calculated using the Fermi Golden Rule.

It has been shown that, to correctly capture the physics of phonon transport, especially in highly conductive materials, the off diagonal terms of the scattering rates cannot be disregarded. More generally, the mean free path is calculated inverting the scattering tensor

\[\lambda_{\mu\alpha} = \sum_{\mu'}(\tilde \Gamma_{\mu\mu' })^{-1}v_{\mu'\alpha}.\] \[\kappa_{\alpha\alpha'} =\frac{1}{N_kV} \sum_{\mu\mu'} c_\mu v_{\mu\alpha}(\tilde \Gamma_{\mu\mu' })^{-1}v_{\mu'\alpha'}.\]

This inversion operation is computationally expensive; however, when the off-diagonal elements of the scattering rate matrix are much smaller than the diagonal, we can rewrite the mean free path obtained from the BTE as a series:

\[\lambda_{\mu\alpha} = \sum_{\mu'}\left(\delta_{\mu\mu'} + \frac{1}{\Gamma^0_\mu}\Gamma^{1}_{\mu\mu'}\right)^{-1}\frac{1}{\Gamma^0_{\mu'}}v_{\mu'\alpha} =\] \[= \sum_{\mu'} \left[ \sum^{\infty}_{n=0}\left(- \frac{1}{\Gamma^0_\mu}\Gamma^{1}_{\mu\mu'}\right)^n \right]\frac{1}{\Gamma^0_{\mu'}}v_{\mu'\alpha} ,\]

where in the last step we used the identity \(\sum_0 q^n = (1 - q)^{-1}\), true when \(|q|=|\Gamma^1/\Gamma^0|<1\). This equation can then be written in an iterative form

\[\lambda^0_{\mu\alpha} = \frac{1}{\Gamma^0_\mu}v_\mu \qquad \lambda^{n+1}_{\mu\alpha} = - \frac{1}{\Gamma^0_\mu}\sum_{\mu'}\Gamma^{1}_{\mu\mu'} \lambda^{n}_{\mu'\alpha}.\]

Hence, the inversion in of the scattering tensor is obtained by a recursive expression. Once the mean free path is calculated, the conductivity is straightforwardly computed.

Quasi-Harmonic Green Kubo

In non-crystalline solids with no long range order, such as glasses, alloys, nano-crystalline, and partially disordered systems, the phonon picture is formally not well-defined. While vibrational modes are still the heat carriers, their mean-free-paths may be so short that the quasi-particle picture of heat carriers breaks down and the BTE is no longer applicable. In glasses heat transport is dominated by a diffusive processes in which delocalized modes with similar frequency transfer energy from one to another. Whereas this mechanism is intrinsically distinct from the underlying hypothesis of the BTE approach, the two transport pictures have been recently reconciled in a unified theory, in which the thermal conductivity is written as:

\[\kappa_{\alpha \alpha'}=\frac{1}{V} \sum_{\mu \mu'} c_{\mu \mu'} v_{\mu \mu' \alpha} v_{\mu \mu' \alpha'} \tau_{\mu \mu'}.\]

This expression is analogous to the RTA one, where modal heat capacity, phonon group velocity and lifetimes are replaced by the generalized heat capacity,

\[c_{\mu \mu'}=\frac{\hbar \omega_{\mu} \omega_{\mu'}}{T} \frac{n_{\mu}-n_{\mu'}}{\omega_{\mu}-\omega_{\mu'}},\]

the generalized velocities,

\[v_{\mu\mu'\alpha}=\frac{1}{2\sqrt{\omega_\mu\omega_{\mu'}}} \sum_{ii'\beta'\beta''}(x_{i\alpha}-x_{i'\alpha })D_{i\beta i'\beta'}\eta_{\mu i\beta}\eta_{\mu'i'\beta'},\]

and the generalized lifetime \(\tau_{\mu\mu'}\). The latter is expressed as a Lorentzian, which weighs diffusive processes between phonons with nearly-resonant frequencies:

\[\tau_{\mu\mu'} = \frac{\gamma_{\mu}+\gamma_{\mu'}}{\left(\omega_{\mu}-\omega_{\mu'}\right)^{2}+\left(\gamma_{\mu}+\gamma_{\mu'}\right)^{2}}\]

where \(\gamma_\mu\) is the line width of mode \(\mu\) that can be computed using Fermi Golden rule. These equations have been derived from the Green-Kubo theory of linear response applied to thermal conductivity, by taking a quasi-harmonic approximation of the heat current, from which this approach is named quasi-harmonic Green-Kubo (QHGK). It has been proven that for crystalline materials QHGK is formally equivalent to the BTE in the relaxation time approximation, and that its classical limit reproduces correctly molecular dynamics simulations for amorphous silicon up to relatively high temperature (600 K). Finally, we provide a microscopic definition of the mode diffusivity,

\[D_{\mu} =\frac{1}{N_k V} \sum_{\mu'}v_{\mu\mu'} \tau_{\mu\mu'}v_{\mu\mu'},\]

which conveniently provide a measure of the temperature-independent contribution of each mode to thermal transport.

Scalable and general implementation: the kALDo package

The theory outlined in the previous section is implemented in kALDo, an efficient and scalable software package which applies the ALD framework to compute the thermal transport in crystalline and non-crystalline solids at various levels of accuracy. kALDo features real space QHGK calculations and three different solvers of the linearized BTE: direct inversion, self-consistent cycle and RTA.

Finally, we want to show the kALDo code architecture

Benchmarks applications

The workflow for ALD calculations is illustrated below

Here, we present two example simulations of both a periodic and an amorphous structure.

Ab initio silicon diamond

In this example we calculate the second order IFC using Density Functional Perturbation Theory as implemented in the Quantum-Espresso package. The phonon lifetimes and thermal conductivity calculations are performed using a (19, 19, 19) q-point grid.

The kALDo minimal input file, looks like

# IFCs object creation using ase.build.bulk
fc = ForceConstants(atoms=bulk('Si', 'diamond', a=2.699),
                    supercell=(5, 5, 5)))

# input is the ASE input for QE
finite_difference.calculate_second(calculator=Espresso(**input))
finite_difference.calculate_third(calculator=Espresso(**input))

# Phonons object creation
phonons = Phonons(force_constants=fc,
               kpts=[19, 19, 19],
               temperature=300)


# Conductivity calculations
cond = Conductivity(phonons=phonons))

print('Thermal conductivity matrix, in (W/m/K):')
print(cond.conductivity(method='inverse').sum(axis=0))

We performed the simulation using the local density approximation for the exchange and correlation functional and a Bachelet-Hamann-Schluter norm-conserving pseudoptential. Kohn-Sham orbitals are represented on a plane-waves basis set with a cutoff of 20 Ry and (8, 8, 8) k-points mesh. The minimized lattice parameter is 5.398A. The third-order IFC is calculated using finite difference displacement on (5, 5, 5) replicas of the irreducible fcc-unit cell, including up to the 5th nearest neighbor. We obtained the following thermal properties

The silicon diamond modes analysis is shown above. Quantum (red) and classical (blue) results are compared. a) Normalized density of states, b) Normalized phase-space per mode \(g\), c) lifetime per mode \(\tau\), d) mean free path \(\lambda\), and e) cumulative conductivity \(\kappa_{cum}\).

Amorphous silicon

Here we study a-Si generated by LAMMPS molecular dynamics simulations of quenching from the melt a 4096 atom crystal silicon structure, with 1989 Tersoff interatomic potential and the QHGK method. The minimal input file looks like the following

# IFCs object creation
fc = ForceConstants.from_folder(atoms,
                                folder='./input_data'))

# Phonons object creation
phon = Phonons(force_constants=fc,
               temperature=300)

# Conductivity calculations
cond = Conductivity(phonons=phonons))

print('Thermal conductivity matrix, in (W/m/K):')
print(cond.conductivity(method='qhgk').sum(axis=0))

In a simliar treatment to the silicon crystal, a full battery of modal analysis can be calculated with both quantum and classical statistics on the amorphous systems returning the phonon DoS as well as the associated lifetimes, generalized diffusivities, normalized phase space and cumulative conductivity

Classical and quantum properties for 4096 atom amorphous silicon system are shown above. a) density of states, b) lifetimes, c) diffusivities, and e) cumulative thermal conductivity. In spite of the increased quantum lifetimes, a decrease of 0.17W/m/K is seen in the quantum conductivity. The difference in conductivity is primarily a result of the overestimation of classical high frequency heat capacities.

Conclusions

We introduced a new implementation of ALD approaches to compute the thermal conductivity of solids. The widely used BTE approach is supplemented by a recently developed unified theory, QHGK, that extends the application of ALD to compute heat transport in disordered materials.

kALDo is a modern Python-based software that can seamlessly link to different molecular simulation packages to get IFC either from DFT or semiempirical forcefields. kALDo runs on GPUs and CPUs, and is released open-source, for the scientific community to use and develop further.

Acknowledgements

Giuseppe Barbalinardo was supported by a fellowship from The Molecular Sciences Software Institute under NSF grant OAC-1547580

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