Deformation potential extraction and computationally efficient mobility calculations in silicon from first principles

We present a first-principles framework to extract deformation potentials in silicon based on density-functional theory (DFT) and density-functional perturbation theory (DFPT). We compute the electronic band structures, phonon dispersion relations, and electron-phonon matrix elements to extract deformation potentials for acoustic and optical phonons for all possible processes. The matrix elements clearly show the separation between intra- and intervalley scattering in the conduction band, and quantify the strength of the scattering events in the degenerate bands of the valence band. We then use an advanced numerical Boltzmann transport equation (BTE) simulator that couples DFT electronic structures and energy/momentum-dependent scattering rates to compute the transport properties for electrons and holes. By incorporating ionized impurity scattering as well, we calculate the $n$-type and $p$-type mobility versus carrier density and make comparisons to experiments, indicating excellent agreement. The fact that the method we present uses well-established theoretical tools and requires the extraction of only a limited number of matrix elements, makes it generally computationally very attractive, especially for semiconductors with a large unit cell and lower symmetry.

Figure 1

Schematics of (a) acoustic and (b) optical vibration modes in the long-wavelength limit for Si visualized by forces on atoms.

Figure 2

(a) The electronic structure and (b) phononic spectrum of Si. In (a) we depict with large red dots for the initial electronic state at the VBM for holes and at the CBM for electrons, and red line segments for the final electronic states. In (b) the large red dots and red line segments indicate the corresponding phonons that are involved in the transitions for intravalley and $g$-type intervalley scatterings. The phonons for $f$-type scattering are not shown here as they are not located at the L-$\mathrm{\Gamma }$-X path.

Figure 3

Matrix elements [(a)–(d)] and $M_{mn}^{\nu }(\mathbf{k},\mathbf{q})$ [(e), (f)] for the valence band of Si vs the phonon wave vector $\mathbf{q}$. The initial electronic state $\mathbf{k}$ is at the VBM for holes ($\mathrm{\Gamma }$ point), whereas the corresponding final states are ${\mathbf{k}}^{\prime }=\mathbf{k}+\mathbf{q}$. (a) Our calculated matrix elements for the TO phonon mode (solid line) compared to Luis's work (dotted line) [62]. (b) Matrix elements for the LA phonon mode (solid line), compared to that using Wannier interpolations (dotted line). (c) Calculated matrix elements for the LA phonon with a separate combination of transitions from the heavy hole (HH) or light hole (LH) bands as initial/final states. The blue regions near the $\mathrm{\Gamma }$ point indicate the most relevant matrix elements for the scattering processes. The corresponding four combinations of transitions between HH and LH are indicated. (d) Same as (c) but for the LO phonon mode. (e) $M_{mn}^{\nu }(\mathbf{k},\mathbf{q})$ of the VBM for the LA and (f) LO phonon modes. The $\mathbf{q}$ vector range is the zoom around the blue regions of (c) and (d).

Figure 4

3D view of $M_{mn}^{\nu }(\mathbf{k},\mathbf{q})$ for LA phonons for scattering into the HH along different $\mathbf{q}$ vectors.

Figure 5

Calculated intravalley electron-phonon coupling matrix elements for Si as a function of the phonon momentum $\left|\mathbf{q}\right|$, for transitions via LA (red solid lines) or TA (blue solid lines) phonons. For $\mathbf{q}\to$ 0, the relation between the $M_{mn}^{\nu }(\mathbf{k},\mathbf{q})$ and the dilatation and uniaxial deformation potentials for LA phonons and the sum of the two TA phonons (see Table 2) are shown with the dashed lines. (a) The first Brillouin zone of $\mathbf{q}$ vectors, where the center is located at the CBM of the electronic Brillouin zone, at $\mathbf{k}$ = (0, 0, 0.8375). [(b)–(i)] Electron-phonon $M_{mn}^{\nu }(\mathbf{k},\mathbf{q})$ (or the zero-order deformation potential) in the direction of $\mathbf{q}$ as shown in the insets.

Figure 6

[(a)–(c)] 2D cross-section view of the first Brillouin zone for electrons (red) and phonons (blue), with the 6 CBM depicted by the green ellipsoids. The corresponding matrix elements for electrons vs the phonon wave vector $\mathbf{q}$ along (d) $\mathrm{\Gamma }$-X direction, also indicating the $g$-type intervalley process, (e) $\mathrm{\Gamma }$-K direction, also indicating the $f$-type process, and (f) $\mathrm{\Gamma }$-L direction, with initial electronic state $\mathbf{k}$ at the CBM, where the blue zone within 1/10 of half Brillouin zone is considered for the derivation of the coupling matrix M for acoustic phonons.

Figure 7

Scattering rate of Si at 300 K vs energy for (a) holes under ADP and ODP scattering, (b) electrons under ADP and IVS scattering. The total scattering rates for (c) holes and (d) electrons in comparison with other ab initio results from Refs. [64, 65, 90] and empirical results from Refs. [91, 92, 93, 94].

Figure 8

(a) Mobility for holes in Si at 300 K calculated in this paper with solid (phonon-limited) and circled (phonon plus IIS) lines, compared to that using EPW for e-ph scattering from Ref. [32], in which IIS is described using a phenomenological ad hoc equation in the effective mass approximation. Comparison is also made with experimental mobility values from Refs. [101, 102, 103, 104]. (b) Mobility for electrons in Si at 300 K calculated in this paper with solid (phonon-limited) and circled (phonon plus IIS) lines, compared to that using EPW for e-ph scattering from Ref. [32] and experiments from Refs. [25, 103, 104]. For the case of the blue circled lines (phonon plus IIS), we use the Brooks-Herring model to compute IIS. Notice that one blue circle for the case of phonon plus IIS at low carrier concentrations is slightly higher compared to the phonon-limited mobility due to the usual large amount of numerical noise associated with the IIS scattering rates [65].

Figure 9

The coupling matrix elements for LO, TO, LA, and TA modes near the $\mathrm{\Gamma }$ point of holes for scattering into the HH of Si.

Figure 10

The (a) $g$ and (b) $f$ processes shown in 3D Brillouin zones to identify the wave vectors of the phonons involved. The wave vectors to the same and different Brillouin zones are illustrated with the dashed and solid lines, respectively.