Significant Reduction of Lattice Thermal Conductivity by the Electron-Phonon Interaction in Silicon with High Carrier Concentrations: A First-Principles Study

Abstract

The electron-phonon interaction is well known to create major resistance to electron transport in metals and semiconductors, whereas fewer studies are directed to its effect on phonon transport, especially in semiconductors. We calculate the phonon lifetimes due to scattering with electrons (or holes), combine them with the intrinsic lifetimes due to the anharmonic phonon-phonon interaction, all from first principles, and evaluate the effect of the electron-phonon interaction on the lattice thermal conductivity of silicon. Unexpectedly, we find a significant reduction of the lattice thermal conductivity at room temperature as the carrier concentration goes above 10 19 cm −3 (the reduction reaches up to 45% in p-type silicon at around 10 21 cm −3 ), a range of great technological relevance to thermoelectric materials. The coordinates of electrons and atomic nuclei represent the most common degrees of freedom in a solid. The full quantum mechanical treatment of the excitations in a solid thus requires the solution of the Schrödinger equation involving the coordinates of all electrons and atomic nuclei, which appears intractable in most cases. A widely applied simplification, the Born-Oppenheimer approximation (BOA) [1], makes use of the fact that the electrons' mass is much smaller than that of the nuclei, and the electrons respond to the motions of the nuclei so quickly that the nuclei can be treated as static at each instant. Under the BOA, the coordinates of the nuclei enter the electronic Schrödinger equation as external parameters, and in turn the electronic ground-state energy acts as part of the interaction energy between the nuclei given a specific configuration, with which the quantized excitations of the atomic nuclei, namely phonons, can be investigated separately from the electrons [2]. It is important to note, however, that the BOA does not separate the electronic and atomic degrees of freedom completely, and a remaining coupling term can cause transitions between the eigenstates of the electron and phonon systems [3]. This electronphonon interaction (EPI) problem was first studied by Bloch [4], and later understood as the main source of resistance to electrical conduction in metals and semiconductors at higher temperatures [3,5,6], and played the key role in the microscopic theory of superconductivity [7].While the effect of the EPI on electron transport has been widely studied in great detail and has become standard content in textbooks [3,5,6], its effect on phonon transport has received much less attention. In our opinion the reason is twofold. First of all, the carrier concentration in semiconductors for conventional microelectronic and optoelectronic applications is typically below 10 19 cm −3 [8], and as we shall show later, the impact of the EPI on phonon transport in this concentration range turns out to be too small to invoke any practical interest. On the other hand, in metals with typical carrier concentrations greater than 10 22 cm −3 , the thermal conduction is dominated by electrons, an...

On the tuning of electrical and thermal transport in thermoelectrics: an integrated theory–experiment perspective

“…In Si, large amount of heat will be scattered if the carrier concentration is high due to a much enhanced electron-phonon interaction, as reported by Liao et al 157 Considering that many TE materials have fairly high carrier concentrations (10 20 -10 21 cm − 3 ), the electronphonon interaction is likely to be a player in reducing the κ L . Experimentally, the noticeable influence of electron-phonon interaction has been pointed out in some heavily doped TE materials, such as SiGe 158 and Mo 3 Se 7 .…”

First-principles calculations of thermal, electrical, and thermoelectric transport properties of semiconductors

“…The effect of electron scattering of phonons can then be evaluated at different temperatures and doping levels. Surprisingly, it has been found in Si that the thermal conductivity starts to decrease as the carrier concentration goes beyond 19 3 10 cm  and the reduction reaches 45% in p-type silicon at around 21 3 10 cm  [111], as shown by Fig. 7a.…”

“…7b. In fact, based on the deformation potential model, it can be found that the phonon scattering rates due to electrons linearly scales with phonon frequency [111] while as we have discussed above the intrinsic phonon-phonon scattering is dominated by the normal process for low frequency phonons, with a frequency dependence as frequency to the second power.…”

Fröhlich Electron-Phonon Vertex from First Principles

“…This quantity has the meaning of probability amplitude for the scattering between the initial electronic state |ψ nk and the final state |ψ mk+q via the perturbation ∆ qν V due to a phonon with crystal momentum q, branch ν and frequency ω qν . The matrix elements g mnν (k, q) can be calculated starting from density functional perturbation arXiv:1510.06373v1 [cond-mat.mtrl-sci] 21 Oct 2015 theory [21], and have been employed to investigate many properties involving EPIs, for example the electron velocity renormalization [22] and lifetimes [23,24], phonon softening [3] and lifetimes [25,26], phonon-assisted absorption [27,28], critical temperature in conventional superconductors [10,29,30], and resistivity [13,14]. The key ingredient of all these calculations is the evaluation of g mnν (k, q) on extremely dense Brillouin zone grids, which is computationally prohibitive.…”