Theoretical investigation of electron-phonon interaction in one-dimensional silicon quantum dot array interconnected with silicon oxide layers

Electronic and phononic states and their interactions in one-dimensional arrays of Si quantum dots interconnected with thin oxide layers is theoretically investigated. Electronic states under low electric field condition are obtained in the Kronig-Penny potential. Approximate expression for phonon wave functions is developed and numerically calculated using the linear atomic chain model. Simulated dispersion relation shows acoustic phonon modes, phonon band gaps, and confined optical phonon modes. Electron-phonon scattering rate is written using a one-dimensional expression. Intraminiband scattering rates and energy relaxation rates are simulated both for absorption and emission processes. The scattering rate varies from $\sim {10}^{12}$ to $\sim {10}^{14}$, depending on the initial electron energy. The scattering rate for absorption/emission processes rapidly decreases at near the top/bottom of minibands due to limited number of phonon branches that can mediate the scattering processes. Negative energy relaxation rate is observed near the bottom of minibands, which is due to larger scattering rate for absorption process and smaller phonon energy mediating the scatterings for emission process. The scattering rate for absorption decreases rapidly with decreasing temperature. Once the temperature drops down to 100 K, the energy relaxation rate for emission process dominates the absorption process.

Figure 1

Model of the idealized one-dimensional Si quantum dot array interconnected with thin oxide layers (1DSiQDA).

Figure 2

Linear atomic chain model used for phonon calculation.

Figure 3

One-dimensional electron energy dispersion in (a) 1DSiQDA and (b) one-dimensional coupled quantum-box structures of $\mathrm{GaAs}∕\mathrm{AlAs}$ (1DGaAsQDA). The horizontal axis is normalized to the edge value of the first Brillouin zone, $\pi ∕d$. Broken line shows the barrier height. Parameters used in the calculation for the 1DSiQDA is shown in Table . For the 1DGaAsQDA, the barrier height is 0.75 eV, and the effective masses in GaAs and AlAs are set 0.067 and 0.150, respectively.

Figure 4

One-dimensional phonon energy dispersions for (a) 1DSiQDA (b) Si quantum wire, and (c) oxide quantum wire. The horizontal axis is normalized to the edge value of the first Brillouin zone, $\pi ∕d$.

Figure 5

(Color online) One-dimensional phonon wave functions in the 1DSiQDA, $S_{q_x}\left(x\right)$. (a) Real part of $S_{q_x}\left(x\right)$ calculated for an acoustic phonon having phonon energy of 0.003 eV. (b) Same plot for an optical phonon. (c) Squared amplitude of the acoustic phonon wave shown in (a). (d) Squared amplitude of the optical phonon wave shown in (b).

Figure 6

Scattering rates as a function of initial electron energy calculated for absorption (broken lines) and emission (solid lines) processes. Each group of curves was obtained from the second to sixth electron minibands. The calculation was performed at $T=300\mathrm{K}$. The inset is the blow-up of the results calculated for the fifth miniband.

Figure 7

Scattering rate for absorption process calculated for miniband 5 mediated by the phonons on branches (a) 1–5, (b) 6–10, (c) 10–15, and (d) 16–20. Gray curve shows total scattering rate for absorption, which is given by the sum of the results for each phonon branch.

Figure 8

Same plot as Fig. 7 calculated for emission process.

Figure 9

Energy relaxation rate as a function of initial electron energy. Solid lines: emission processes; broken lines: absorption processes; and gray lines: total energy relaxation rate. The calculation was performed at $T=300\mathrm{K}$.

Figure 10

Blow-up of the results for the fifth miniband shown in Fig. 9.

Figure 11

Scattering rate for (a) absorption and (b) emission processes calculated for $T=300$, 100, 77, 33, 10, and 4.2 K.

Figure 12

Energy relaxation rate for absorption (broken line) and emission process (solid line) calculated for $T=300$, 100, 77, 33, 10, and 4.2 K.

Figure 13

Minibands considering transverse electron energy excitation. Excitation index indicates different transverse excitation energies, and “Total” denotes result obtained by superposing the minibands obtained for each excitation index.

Figure 14

Minibandgap energies as a function of minibandgap number $n$ calculated for different Si dot sizes. (a) 4.0 nm, (b) 3.0 and 7.0 nm. The $n\text{th}$ minibandgap is defined as energy gaps between $n\text{th}$ and $n+1\text{st}$ minibands. The broken lines indicate the maximum phonon energy in the 1DSiQDA, 0.062 eV.