Optical phonon production by upconversion: Heterojunction-transmitted versus native phonons

Seungha Shin and Massoud Kaviany

Phys. Rev. B 91, 165310 – Published 27 April 2015

Abstract

High-energy optical phonons are preferred in phonon-absorbing transitions, and regarding their production we analyze the phonon upconversion processes under nonequilibrium created by heterojunction transmission. For heterojunctions, steady phonon flux from a low-cutoff-frequency layer (e.g., Ge) is transmitted to a high cutoff layer (e.g., Si), creating a nonequilibrium population of low-energy phonons for upconversion. Using quantum spectral phonon transmission and first-principles calculations of the phonon interaction kinetics, we identify the high-conversion efficiency channels, i.e., modes and wave vectors. Junction-transmitted phonons, despite suffering from the interface reflection and from spreading interactions with equilibrium native phonons, have a high upconversion rate to Brillouin zone-boundary optical phonons, while nonequilibrium native phonons are efficiently upconverted over most of the zone. So, depending on the harvested optical phonon, one of these nonequilibrium phonons can be selected for an efficient upconversion rate.

Received 23 June 2014
Revised 11 April 2015

DOI:https://doi.org/10.1103/PhysRevB.91.165310

Authors & Affiliations

Seungha Shin^*

Department of Mechanical, Aerospace and Biomedical Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA

Massoud Kaviany

Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

^*sshin@utk.edu

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Vol. 91, Iss. 16 — 15 April 2015

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Figure 1
Three-phonon processes involving a phonon of wave vector $κ$ and mode $α$ interacting with $κ^{'} α^{'}$ and $κ^{''} α^{''}$ , where $κ α$ is created (a and b) or annihilated (c and d), with energy and momentum (g is reciprocal lattice vector) conservations.
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Figure 2
Si optical-phonon creation rates in interaction with an equilibrium phonon of energy $E_{p}$ , at 300 K (for created phonons $hp = LO$ , TO1, and TO2). Rates (per unit time per unit energy in a primitive cell) are normalized using the equilibrium population distribution $d n_{p}^{o} / d E_{p}$ .
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Figure 3
Schematics of phonon harvesting at the Ge/Si heterojunction (HPUC). (a) Phonon dispersion, density of states $(D_{p})$ , and population distribution for Ge and Si. Under flux, due to the larger $D_{p}$ at Ge acoustic and optical modes, low-energy phonons in Si are overpopulated and then upconverted to Si optical phonons. (b) Anticipated spatial distributions of the optical and acoustic phonon temperatures $(T_{p, O}$ and $T_{p, A}$ with an interfacial temperature drop due to boundary resistance). The phonon source is in Ge and the phonon flux in the $z$ direction creates the temperature variations in Si. The Si optical mode is less populated, because none is transmitted from Ge. The targeted phonon mode (hp) is harvested (absorbed by the other system) in $[z_{o}, z_{o} + Δ δ_{hp}]$ , and this leads to the underpopulation. The harvested phonon flux $q_{hp}$ is due to transport of harvested $(q_{tr, hp})$ –mode and conversion of nonharvested (nhp)–mode phonons $(q_{up, hp})$ .
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Figure 4
(a) Phonon population (per primitive cell) distributions adjacent to junctions, based on the NEGF transmission spectrum [40] for Ge/Si phonons (right axis), at 300 K. Shaded regions represent larger overpopulation by transmitted Ge phonons. Si and Ge equilibrium distributions are also shown for contrast. (b) Nonequilibrium phonon distribution created by Ge-transmitted and Si-native phonons under the same phonon flux $(3.8 MW / {cm}^{2})$ . Si phonon modes matching the Ge phonons are more pronouncedly overpopulated by the Ge-transmitted phonon flux.
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Figure 5
(a) Variations of dimensionless, local net creation (upconversion) rates for LO phonons $(d {\dot{s}}_{κ α}^{*} / d κ^{*}$ ; $α = LO)$ from Ge-transmitted and Si-native overpopulated phonons, along high-symmetry axes. The created optical phonon energy $(E_{p, LO})$ is also shown (upper), and it is evident that overpopulation by Ge-transmitted phonons results in a higher LO phonon creation rate in the regime [49, 56 meV] compared to NE Si-native phonons. (b) Difference between $d {\dot{s}}_{κ α, Si}^{*} / d κ^{*}$ and $d {\dot{s}}_{κ α, Ge / Si}^{*} / d κ^{*}$ in an irreducible BZ wedge. Spheres are centered at the sampling points of a $27 \times 27 \times 27$ grid, showing the wave vector of the upconverted (harvested) phonon. Red represents a larger $d {\dot{s}}_{κ α, Ge / Si}^{*} / d κ^{*}$ , blue a larger $d {\dot{s}}_{κ α, Si}^{*} / d κ^{*}$ , and the diameter indicates the magnitude. Results are for 300 K.
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Figure 6
Spatial variations of the local harvesting efficiency $(d η_{hp = κ α} / d κ^{*})$ for two wave vectors, $κ^{*} = (0.81, 0.52, 0.07)$ near the zone boundary (solid line) and $(0.15, 0.15, 0.15)$ near the zone center (dotted line). For the first, the optimal location for efficiency (filled square) exists before relaxation, while for the second the efficiency increases monotonically. $d η_{hp = κ α} / d κ^{*}$ in bulk Si (dashed line) is larger with no loss by heterojunction transmission $(τ_{p, Ge \to Si})$ .
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