Quantum Boltzmann equation

Application to semiconductor physics

A typical model of a semiconductor may be built on the assumptions that:

1. The electron distribution is spatially homogeneous to a reasonable approximation (so all x-dependence may be suppressed)
2. The external potential is a function only of position and isotropic in p-space, and so ${\displaystyle \mathbf {F} }$ may be set to zero without losing any further generality
3. The gas is sufficiently dilute that three-body interactions between electrons may be ignored.

Considering the exchange of momentum ${\displaystyle \mathbf {q} }$ between electrons with initial momenta ${\displaystyle \mathbf {k} }$ and ${\displaystyle \mathbf {k_{1}} }$, it is possible to derive the expression $${\displaystyle {\begin{aligned}{\mathcal {Q}}[f](\mathbf {k} )&={\frac {-2}{\hbar (2\pi )^{5}}}\int d\mathbf {q} \int d\mathbf {k_{1}} |{\hat {v}}(\mathbf {q} )|^{2}\\&\quad \times \delta \!\left({\frac {\hbar ^{2}}{2m}}(|\mathbf {k-q} |^{2}+|\mathbf {k_{1}+q} |^{2}-\mathbf {k} _{1}^{2}-\mathbf {k} ^{2})\right)\\&\quad \times \left[f_{\mathbf {k} }f_{\mathbf {k_{1}} }(1-f_{\mathbf {k-q} })(1-f_{\mathbf {k_{1}+q} })-f_{\mathbf {k-q} }f_{\mathbf {k_{1}+q} }(1-f_{\mathbf {k} })(1-f_{\mathbf {k_{1}} })\right]\end{aligned}}}$$

Phonon–phonon scattering

Phonon–phonon scattering, governed by anharmonic lattice interactions, thermalizes the phonon population on timescales of tens to hundreds of picoseconds and is the principal mechanism of thermal resistance in insulators and semiconductors ^[9][10]. In this context, the time-depended quantum Boltzmann equation is also known as the Peierls-Boltzmann equation. The underlying Hamiltonian is written in second quantization as follows:

${\displaystyle {\hat {\mathcal {H}}}_{\text{anh}}={\frac {1}{3!}}\sum _{\mathbf {q} \nu ,\,\mathbf {q} '\nu ',\,\mathbf {q} ''\nu ''}\Psi _{\mathbf {q} \nu ,\mathbf {q} '\nu ',\mathbf {q} ''\nu ''}\left({\hat {a}}_{\mathbf {q} \nu }^{\dagger }-{\hat {a}}_{-\mathbf {q} \nu }\right)\left({\hat {a}}_{\mathbf {q} '\nu '}^{\dagger }-{\hat {a}}_{-\mathbf {q} '\nu '}\right)\left({\hat {a}}_{\mathbf {q} ''\nu ''}^{\dagger }-{\hat {a}}_{-\mathbf {q} ''\nu ''}\right)}$

where ${\displaystyle {\hat {a}}_{\mathbf {q} \nu }^{\dagger }}$ and ${\displaystyle {\hat {a}}_{\mathbf {q} \nu }}$ are phonon creation and annihilation operators. The coupling coefficient ${\displaystyle \Psi _{\mathbf {q} \nu ,\mathbf {q} '\nu ',\mathbf {q} ''\nu ''}}$ is constructed from the third-order interatomic force constants (IFCs) ${\displaystyle \Phi _{0b,\,nb',\,mb''}^{\alpha \beta \gamma }}$, which give the change in force on atom ${\displaystyle b}$ in the unit cell along direction ${\displaystyle \alpha }$ due to simultaneous displacements of atoms ${\displaystyle b'}$ and ${\displaystyle b''}$ in unit cells ${\displaystyle n}$ and ${\displaystyle m}$ along directions ${\displaystyle \beta }$ and ${\displaystyle \gamma }$.^[11]

After Fourier transformation and projection onto phonon polarization vectors ${\displaystyle e_{b,\mathbf {q} \nu }^{\alpha }}$, one obtains

${\displaystyle \Psi _{\mathbf {q} \nu ,\mathbf {q} '\nu ',\mathbf {q} ''\nu ''}={\frac {i}{\sqrt {V}}}\,\delta _{\mathbf {G} ,\,\mathbf {q} +\mathbf {q} '+\mathbf {q} ''}\sum _{b,b',b''}{\sqrt {\frac {\hbar ^{3}}{m_{b}m_{b'}m_{b''}\,\omega _{\mathbf {q} \nu }\,\omega _{\mathbf {q} '\nu '}\,\omega _{\mathbf {q} ''\nu ''}}}}\;e_{b,\mathbf {q} \nu }^{\alpha }\,e_{b',\mathbf {q} '\nu '}^{\beta }\,e_{b'',\mathbf {q} ''\nu ''}^{\gamma }\;\Phi _{\mathbf {q} b,\mathbf {q} 'b',\mathbf {q} ''b''}^{\alpha \beta \gamma }}$

where ${\displaystyle V}$ is the crystal volume, ${\displaystyle m_{b}}$ is the mass of atom ${\displaystyle b}$, and the Kronecker delta ${\displaystyle \delta _{\mathbf {G} ,\,\mathbf {q} +\mathbf {q} '+\mathbf {q} ''}}$ enforces crystal momentum conservation modulo a reciprocal lattice vector ${\displaystyle \mathbf {G} }$.

Applying Fermi's golden rule to ${\displaystyle {\hat {\mathcal {H}}}_{\text{anh}}}$ yields the phonon–phonon collision term. For a phonon mode ${\displaystyle (\mathbf {q} ,\nu )}$, three distinct three-phonon processes govern the time-evolution of the phonon occupation number^[12]:

${\displaystyle {\dot {n}}_{\mathbf {q} \nu }{\big |}_{\text{ph-ph}}=-{\frac {2\pi }{\hbar ^{2}}}\sum _{\mathbf {k} \lambda ,\,\mathbf {k} '\lambda '}\left|\Psi _{\mathbf {q} \nu ,\mathbf {k} \lambda ,\mathbf {k} '\lambda '}\right|^{2}{\Big [}\,{\mathcal {D}}^{-}(\mathbf {q} \nu$ ;\mathbf {k} \lambda ,\mathbf {k} '\lambda ')+{\mathcal {D}}^{+}(\mathbf {q} \nu ;\mathbf {k} \lambda ,\mathbf {k} '\lambda ')+{\mathcal {D}}^{+}(\mathbf {q} \nu ;\mathbf {k} '\lambda ',\mathbf {k} \lambda )\,{\Big ]}}

where the three terms correspond to distinct scattering channels. The process, in which one phonon decays into two,

${\displaystyle {\mathcal {D}}^{-}=\left[n_{\mathbf {q} \nu }(n_{\mathbf {k} \lambda }+1)(n_{\mathbf {k} '\lambda '}+1)-(n_{\mathbf {q} \nu }+1)\,n_{\mathbf {k} \lambda }\,n_{\mathbf {k} '\lambda '}\right]\delta (\omega _{\mathbf {q} \nu }-\omega _{\mathbf {k} \lambda }-\omega _{\mathbf {k} '\lambda '})}$

and the coalescence process, in which two phonons merge into one,

${\displaystyle {\mathcal {D}}^{+}=\left[n_{\mathbf {q} \nu }\,n_{\mathbf {k} \lambda }(n_{\mathbf {k} '\lambda '}+1)-(n_{\mathbf {q} \nu }+1)(n_{\mathbf {k} \lambda }+1)\,n_{\mathbf {k} '\lambda '}\right]\delta (\omega _{\mathbf {q} \nu }+\omega _{\mathbf {k} \lambda }-\omega _{\mathbf {k} '\lambda '})}$

In each case the delta function enforces energy conservation, and the occupation-number prefactors respect Bose-Einstein statistics. The third term in is the time-reverse of the coalescence process.

Electron-phonon dynamics

Ultrafast dynamics simulations based on the time-dependend Boltzmann equation, parameterized from ab initio, have been widely used to describe non-equilibrium processes involving electrons and phonons in condensed matter physics ^[13]. Applications include the investigation of the thermalization of electrons and phonons following photoirradiation ^[14][15], and the framework has since been extended to ultrafast magnetization dynamics in photo-excited ferromagnets ^[16][17].