Uncovering the important role of transverse acoustic phonons in the carrier-phonon scattering in silicon

Carrier mobility is an essential parameter of semiconductors, characterizing how quickly carriers can move in a material when driven by an external electric field. Because electron-phonon (e-ph) scattering limits the room-temperature carrier mobility in high-quality semiconductors, understanding the mechanisms of interaction between carriers and phonons at the microscopic level is vital to investigate the transport properties, especially in nanoelectronic devices. Here, we reproduce the experimentally measured electron and hole mobility in silicon (Si) over a wide temperature range without relying on adjustable parameters by performing the first-principles calculations. By decomposition of the first-principles calculation-predicted e-ph scattering into the contributions from different phonon modes and electronic valleys, we show that the transverse acoustic (TA) phonon mode has a comparable contribution to the longitudinal acoustic (LA) phonon mode in scattering of both electrons and holes on limiting the carrier mobilities in Si. This is in striking contrast with the common sense that the TA mode is negligible based on the classical e-ph interaction modes. We unravel that the neglect of TA scattering is due to the substantial underestimation of the shear deformation potential (associated with the TA mode) with respect to the dilation deformation potential (associated with the LA mode). We also find that the transverse optical (TO) phonon mode, rather than the conventionally presumed longitudinal optical (LO) and LA modes, provides the leading scattering channel (accounting for 58%) in $f$-type intervalley scattering and the LO mode is dominant over the LA mode in $g$-type intervalley scattering for electrons in Si. These findings illustrate why the technology computer-aided design device simulation loses the predictive capability, although it is possible to obtain reasonable results using adjustable parameters based on the incorrect physics models. It calls for a revisit of the mechanisms underlying the carrier mobility in semiconductors.

Figure 1

Phonon-limited electron and hole mobilities as a function of temperature predicted using the first-principles Boltzmann formalism compared with experimental data. The discrete points represent experimental values adopted from the publications cited in Refs. [31, 32, 33, 34, 35]. The solid lines are computed by the first-principles method as implemented in the epw package [28].

Figure 2

(a) Intravalley (green dots) and intervalley (blue and orange dots for $f$ and $g$ types, respectively) scattering and total scattering rates (black dots) for electrons in Si at 300 K. The dashed red line represents the total scattering rate based on the semiempirical model including acoustic-deformation-potential and intervalley optical-deformation-potential scattering, which has been shown in earlier work [8]. (b)–(d) Decomposed phonon-limited intravalley, $g$-type, and $f$-type scattering into TA, LA, LO, and TO modes, respectively. (e) Pie chart representation of the percentages of intravalley scattering (green region), $g$-type intervalley scattering (orange region), and $f$-type intervalley (blue region) electron-phonon (e-ph) scattering at room temperature. (f) The contributions of TA, LA, LO, and TO modes to the total e-ph scattering rate. Note that TA is a sum of two transverse acoustic modes (TA1 and TA2); LA represents the longitudinal acoustic mode; TO is a sum of two transverse optical modes (TO1 and TO2); and LO represents the longitudinal optical mode.

Figure 3

(a)–(i) Intraband electron-phonon (e-ph) scattering matrix elements $g_{mn\nu }(\mathbf{k},\mathbf{k}+\mathbf{q})$ for the conduction band of GaAs vs the phonon wave vector q along several directions with $\theta =0,0.25,0.306,0.5,0.694,\mathrm{and}0.75\pi$. The initial electronic state $|n,\mathbf{k}\rangle$ is located at the conduction band minimum (CBM; $\mathrm{\Gamma }$) for electrons, whereas the corresponding final states $|n,\mathbf{k}+\mathbf{q}\rangle$ are in the same band.

Figure 4

(a) Electronic band structure of Si. The conduction band minima occur along the $\mathrm{\Gamma }\text{−}X$ at $\sim 0.82$ of the Brillioun zone (BZ) edge with ${\mathrm{\Delta }}_1$ representation. The valence band maximum is located at the center of the BZ with ${\mathrm{\Gamma }}_5^{+}$ representation. (b) Phonon dispersion curves of Si. The lowest three branches are acoustic phonons and high-lying three branches are optical phonons, which are both threefold degenerate at $\mathrm{\Gamma }$ (acoustic phonons belong to ${\mathrm{\Gamma }}_4^{-}$ and optical phonons belong to ${\mathrm{\Gamma }}_5^{+}$ of diamond structure $O_h$ point group) and split into nondegenerate longitudinal acoustic (LA) and longitudinal optic (LO) and twofold degenerate transverse acoustic (TA) and transverse optic (TO), respectively, along the $\mathrm{\Gamma }-X$ direction (phonons along this direction are so-called $\mathrm{\Delta }$ phonons). Along the $\langle 110\rangle$ direction from $\mathrm{\Gamma }$ to $K$, the threefold degenerate acoustic (optical) phonons (called the $\mathrm{\Sigma }$ phonons) split into three modes belong to irreducible representations ${\mathrm{\Sigma }}_1, {\mathrm{\Sigma }}_3$, and ${\mathrm{\Sigma }}_4$ (${\mathrm{\Sigma }}_1, {\mathrm{\Sigma }}_2$, and ${\mathrm{\Sigma }}_3$) of the wave vector group $C_{2v}$.

Figure 5

The phonon structure of Si with a color map of the electron-phonon (e-ph) scattering matrix elements $|g_{mn\nu }(\mathbf{k},\mathbf{k}+\mathbf{q})|$ at 300 K for band indexes $m$ and $n$ located at the lowest conduction band $\mathbf{k}=(0.82,0,0)(2\pi /a)$ and phonon wave vector $\mathbf{q}$ along the high-symmetry path: (a) ${\mathrm{\Gamma }}_1\text{−}{X}_1\text{−}\mathrm{\Gamma }\text{−}X$ along the $\left[100\right]$ direction entering into the adjacent Brillouin zone (BZ; here, the high-symmetry path $X_1\text{−}\mathrm{\Gamma }\text{−}X$ is in the first BZ, while $X_1\text{−}{\mathrm{\Gamma }}_2$ are inside the adjacent BZ), and (b) $\mathrm{\Gamma }\text{−}{K}_2\text{−}{K}_3\text{−}{\mathrm{\Gamma }}_3$ along the $\left[\overline{1}10\right]$ direction entering into the second and third BZ (here, $\mathrm{\Gamma }$ and $K$ points are in the first BZ and ${\mathrm{\Gamma }}_3$ and $K_3$ points are their counterparts in the other BZ). An electron with ${\mathrm{\Delta }}_1$ symmetry at $\mathbf{k}=(0.82,0,0)(2\pi /a)$ can be scattered into the opposite valley ${\mathbf{k}}^{\prime }=(-0.82,0,0)(2\pi /a)$ along the same axis or equivalent nearby valley ${\mathbf{k}}^{\prime }=(1.18,0,0)(2\pi /a)$ in the adjacent BZ via phonons with finite wave vector $\mathbf{q}$, labeled as $g_N$ and $g_U$, respectively. The former and the latter represent normal and Umklapp $g$-type intervalley scattering, respectively. Similarly, an electron with ${\mathrm{\Delta }}_1$ symmetry at $\mathbf{k}=(0.82,0,0)(2\pi /a)$ can be scattered into the equivalent perpendicular valley ${\mathbf{k}}^{\prime }=(0,0.82,0)(2\pi /a)$ in the same BZ along the $\mathrm{\Sigma }$ axis via $\mathrm{\Sigma }$ phonons with a wave vector labeled as $f$. (c) Schematic diagram of the first BZ in Si. (d) Schematic diagram of the $g$- and $f$-type intervalley scattering for electrons in the conduction band minimum (CBM) of Si, showing the involved phonon wave vectors.

Figure 6

Intravalley scattering matrix elements $|g_{mn\nu }(\mathbf{k},\mathbf{k}+\mathbf{q})|$ for electrons in the conduction band of Si against the phonon wave vector $\mathbf{q}$ along (a) [100], (b) $\left[1\overline{1}0\right]$, (c) [110], (d) $\left[1\overline{1}1\right]$, (e) [111], (f) [$0\overline{1}\overline{1}]$, (g) [010], (h) $[\overline{1}11$], and (i) $[\overline{1}\overline{1}0$] directions with $\theta =0$, 0.25, 0.36, 0.5, 0.694, or 0.75 $\pi$. The initial electron state $|n,\mathbf{k}\rangle$ is located at one $\mathrm{\Delta }$ valley, whereas the corresponding final states are $|n,\mathbf{k}+\mathbf{q}\rangle$ within the same $\mathrm{\Delta }$ valley.

Figure 7

Comparison of the deformation potentials for intravalley scattering of electrons in Si as a function of angle $\theta$ between the phonon wave vector $\mathbf{q}$ and the principal axis of the valley. The $\theta$ dependence of effective transverse acoustic (TA) and longitudinal acoustic (LA) deformation potentials obtained from the theory of Herring and Vogt [4, 6, 9, 12] according to Eqs. (12a) and (12b) by setting constants ${\mathrm{\Xi }}_u$ and ${\mathrm{\Xi }}_d$ of the hydrostatic deformation potential and uniaxial shear deformation potential to (a) $({\mathrm{\Xi }}_u,{\mathrm{\Xi }}_d)$ = (10.0, $-11.5$) eV and $({\mathrm{\Xi }}_u,{\mathrm{\Xi }}_d)$ = (8.3, 5.0) eV, which are two sets frequently quoted in the literature [6, 57, 58], and (b) our extracted $({\mathrm{\Xi }}_u,{\mathrm{\Xi }}_d)$ = (0.8, 9.0) eV from the first-principles calculation-predicted TA (marked by red dots) and LA deformation potentials (marked by black dots). Note that the TA deformation potential is a sum of two TA modes.

Figure 8

(a) Heavy hole (hh) scattering rate decomposed into different phonon branches with respect to different electron energies from the band extrema at 300 K. (b) Schematic diagram of intraband and interband scattering induced by acoustic (AC) and optical (OP) phonons for hh. Pie chart representation of the proportion of average electron scattering rate induced by different phonon-limit scattering channels (c) to hh intraband scattering and (d) interband scattering from hh to light hole (lh) band. Here, the transverse modes consider both branches.

Figure 9

Hole-phonon scattering matrix elements $g_{mn\nu }(\mathbf{k},\mathbf{k}+\mathbf{q})$ in Si for a hole initially located at the valence band maximum (VBM; heavy hole [hh] band) transition to the hh band (intraband scattering) as a function of the phonon momentum $\mathbf{q}$ along (a) [100], (b) [110], (c) $\left[1\overline{1}0\right]$, (d) [111], (e) [$11\overline{1}]$, and (f) $[\overline{1}11$] directions. Since optical phonons are forbidden by symmetry in the hole-phonon scattering, here, we only show the acoustic phonon scattering matrix elements for two transverse acoustic (TA) modes (denoted as TA1 and TA2) and one longitudinal acoustic (LA) mode.

Figure 10

(a)–(i) Hole-phonon scattering matrix elements $g_{mn\nu }\left(\mathbf{k},\mathbf{k}+\mathbf{q}\right)$ for the transition of holes in Si from the heavy hole (hh) band to the light hole (lh) band as a function of the phonon momentum q.

Figure 11

(a)–(i) Hole-phonon scattering matrix elements $g_{mn\nu }(\mathbf{k},\mathbf{k}+\mathbf{q})$ for the transition of holes in GaAs from the heavy hole (hh) band to the hh band as a function of the phonon momentum $\mathbf{q}$.

Figure 12

(a)–(i) Hole-phonon scattering matrix elements $g_{mn\nu }\left(\mathbf{k},\mathbf{k}+\mathbf{q}\right)$ for the transition of holes in GaAs from the heavy hole (hh) band to the light hole (lh) band as a function of the phonon momentum q.

Figure 13

Comparison of angular-dependent hole-phonon scattering matrix elements $\left|g\right|$ obtained from the classic deformation model [42] (solid lines) and from the first-principles calculations (dots with error bar) for intraband transitions by longitudinal acoustic (LA) and transverse acoustic (TA) modes.

Figure 14

Comparison of phonon-limited hole mobility in Si as a function of temperature. The black solid line and dots represent hole mobilities predicted by first-principles methods and measured by experiment, respectively, as shown in Fig. (1). Dashed lines represent the predicted hole mobilities by the classical model (see details in Ref. [73]) with deformation potentials ($a, b, d$) of (2.36, $-1.87, -3.60$) eV [orange, fitted from heavy hole (hh) matrix elements], (3.21, $-1.76, -3.37$) eV (blue, fitted from hh intraband matrix elements), and (2.1, $-2.33, -4.75$) eV (green, obtained by Fischetti and Laux [14]), respectively. The solid green line is the data adopted directly from Ref. [14].