# Role of Dimensionality and Size in Controlling the Drag Seebeck Coefficient
of Doped Silicon Nanostructures: A Fundamental Understanding

Raja Sen<EMAIL_ADDRESS>Nathalie Vast  Jelena Sjakste
<EMAIL_ADDRESS>Laboratoire des Solides Irradies,
CEA/DRF/IRAMIS, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris,
91120 Palaiseau, France

###### Abstract

In this theoretical study, we examine the influence of dimensionality, size
reduction, and heat-transport direction on the phonon-drag contribution to the
Seebeck coefficient of silicon nanostructures. Phonon-drag contribution, which
arises from the momentum transfer between out-of-equilibrium phonon
populations and charge carriers, significantly enhances the thermoelectric
coefficient. Our implementation of the phonon drag term accounts for the
anisotropy of nanostructures such as thin films and nanowires through the
boundary- and momentum-resolved phonon lifetime. Our approach also takes into
account the spin-orbit coupling which turns out to be crucial for hole
transport. We reliably quantify the phonon drag contribution at various doping
levels, temperatures, and nanostructure geometries for both electrons and
holes in silicon nanostructures. Our results support the recent experimental
findings, showing that a part of phonon drag contribution survives in 100 nm
silicon nanostructures.

The nanostructuring of semiconductors provides a viable route to enhance the
thermoelectric efficiency as compared to that of the bulk by tuning the
transport properties (He and Tritt, 2017; Dangić  _et al._ , 2020; Yang  _et
al._ , 2021). Together with the electrical and thermal conductivities, the
Seebeck coefficient - which links the electrical current to the temperature
gradient - is a key physical quantity characterizing the performance of
thermoelectric materials. For a nondegenerate semiconductor, there are two
contributions to the total Seebeck coefficient ($\mathrm{\mathit{S}^{tot}}$):
the diffusive ($\mathrm{\mathit{S}^{diff}}$) and the phonon drag
($\mathrm{\mathit{S}^{drag}}$) contributions. While the former comes from the
diffusion of charge carriers under a temperature gradient, the latter arises
from the momentum transfer between the out-of-equilibrium phonon populations
and the charge carriers (Gurevich, 1945; Herring, 1954; Cantrell and Butcher,
1987a, b; Gurevich and Mashkevich, 1989). Despite the impossibility to
separately measure the $\mathrm{\mathit{S}^{diff}}$ and
$\mathrm{\mathit{S}^{drag}}$ contributions, the important role played by the
phonon drag has been recognized experimentally by the strong increase of
$\mathrm{\mathit{S}^{tot}}$ at low temperatures in semiconductors, where
anharmonicity is reduced and out-of-equilibrium phonon populations are very
large (Frederikse, 1953; Geballe and Hull, 1954, 1955; Bentien  _et al._ ,
2007; Takahashi  _et al._ , 2016; Jaoui  _et al._ , 2020).

The individual contributions to $\mathrm{\mathit{S}^{tot}}$ can, however, be
quantified theoretically by means of models with effective parameters
(Herring, 1954; Mahan  _et al._ , 2014; Battiato  _et al._ , 2015) or by ab
initio calculations (Zhou  _et al._ , 2015; Fiorentini and Bonini, 2016;
Protik and Kozinsky, 2020; Xu  _et al._ , 2021; Protik  _et al._ , 2022). For
example, it has been recently shown by density functional theory (DFT) that at
$300$ K and at low electron doping ($10^{14}\leavevmode\nobreak\
\mathrm{cm^{-3}}$), more than $30\%$ of $\mathrm{\mathit{S}^{tot}}$ in silicon
comes from $\mathrm{\mathit{S}^{drag}}$. Moreover, the relative contribution
of $\mathrm{\mathit{S}^{drag}}$ with respect to $\mathrm{\mathit{S}^{tot}}$
increases even further at higher doping (and fixed temperature) or at (fixed
doping and) temperatures lower than $300$ K (Zhou  _et al._ , 2015; Fiorentini
and Bonini, 2016; Protik and Kozinsky, 2020; Xu  _et al._ , 2021; Protik  _et
al._ , 2022). Nevertheless, downsizing a semiconductor to the sub-micron scale
is expected to drastically reduce the mean free path (MFP) of phonons and in
consequence, the drag contribution to $\mathrm{\mathit{S}^{tot}}$. Therefore,
the Seebeck coefficient is expected to decrease monotonically with decreasing
the size of the nanostructure (Frederikse, 1953; Geballe and Hull, 1954, 1955;
Takahashi  _et al._ , 2016).

In spite of the great effort invested in studying the effect of
nanostructuring on $\mathrm{\mathit{S}^{tot}}$ (Pokharel  _et al._ , 2013;
Wang  _et al._ , 2013; Kockert  _et al._ , 2019; Nadtochiy  _et al._ , 2019;
Cabero  _et al._ , 2021), no consensus has been reached about the role of
$\mathrm{\mathit{S}^{drag}}$ at the nanoscale, even in the case of silicon
(Zhou  _et al._ , 2015; Sadhu  _et al._ , 2015; Salleh  _et al._ , 2014;
Boukai  _et al._ , 2008; Fauziah  _et al._ , 2020; Ryu  _et al._ , 2010).
Indeed, from the theoretical side, Zhou et al. (Zhou  _et al._ , 2015) have
pointed out that phonons contributing to $\mathrm{\mathit{S}^{drag}}$ have
longer MFPs than those contributing to the lattice thermal conductivity of Si.
A consequence is that $\mathrm{\mathit{S}^{drag}}$ is expected to be strongly
suppressed at $1$ $\mu$m by the effect of size reduction (Zhou  _et al._ ,
2015). In addition, in the experimental study of $\mathrm{\mathit{S}^{tot}}$,
Sadhu et al. (Sadhu  _et al._ , 2015) have concluded that the
$\mathrm{\mathit{S}^{drag}}$ component vanishes completely in Si nanowires
having a characteristic length smaller than $100$ nm. However, these studies
contradict the findings of other experimental works which have suggested that
the drag contribution in Si ultrathin films (Salleh  _et al._ , 2014),
nanowires (Boukai  _et al._ , 2008; Fauziah  _et al._ , 2020), and nanoribbons
(Ryu  _et al._ , 2010) does not vanish. The situation is further complicated
by the possible presence of various competing effects at the nanoscale, such
as the energy filtering effect induced by defects, which can lead to an
increase of the total Seebeck coefficient Zide  _et al._ (2006); Bennett  _et
al._ (2015); Liang  _et al._ (2017). Thus, to disentangle the intricate
effects that govern the magnitude of $\mathrm{\mathit{S}^{tot}}$ at the
nanoscale, a detailed understanding of the dependence of
$\mathrm{\mathit{S}^{drag}}$ on the dimensionality and size of nanostructures
is necessary and can only be achieved through theory.

In this paper, we report the results of the systematic investigation of the
influence of both size reduction and dimensionality on the phonon drag Seebeck
coefficient of electron- and hole-doped silicon, by solving the coupled
linearized Boltzmann transport equation (BTE) for charged carriers and for
phonons, in combination with a fully ab initio description of the carrier-
phonon interaction (Murphy-Armando and Fahy, 2006; Calandra  _et al._ , 2010;
Bernardi, 2016; Giustino, 2017; Sjakste  _et al._ , 2018; Poncé  _et al._ ,
2020; Brunin  _et al._ , 2020). The coupling of the BTEs enables us to include
in particular the effect of the out-of-equilibrium phonon populations which
arise in presence of the temperature gradient. At variance with previous ones,
the computational approach implemented in this work allows to account for the
anisotropy of phonon scattering by nanostructure boundaries in the
calculations of phonon drag Seebeck coefficient. In our work, the role of
anisotropy and dimensionality of boundary scattering has been studied by
considering two-dimensional (2D) nanofilms and one-dimensional (1D) nanowires,
as well as isotropic boundaries (Fig. 1, panel a). The direction-resolved out-
of-equilibrium phonon populations have been determined as a function of the
nanostructure size, with the aim of quantifying the effect of the transport-
direction-dependent phonon-boundary scattering in the phonon drag
contribution. Moreover we have included the effect of spin-orbit coupling on
the $\mathrm{\mathit{S}^{tot}}$ for holes, an effect which has been neglected
so far for the Seebeck coefficient.

Figure 1: Panel a (b): Schema of the in-plane (out-of-plane) direction of heat
and current flows as a function of the nanostructure dimensionality: 2D-Film,
1D-wire, and the isotropic boundary. Panel c (e): Variation of hole (electron)
phonon drag Seebeck coefficient as a function of the Casimir length
($L^{\mathrm{Cas}}$) in the in-plane direction at $300$ K and
$10^{14}\leavevmode\nobreak\ \mathrm{cm^{-3}}$ doping concentrations. Panel d
(f): Same as panel c (e) in the out-of-plane direction.

The electrical and heat currents produced by a temperature gradient experience
a mutual drag via the interaction between charge carriers and phonons. This
means that, in principle, the carrier-phonon scattering terms that govern the
BTE for charge carriers and sometimes play a role in the BTE for phonons,
depend both on the charge-carrier out-of-equilibrium distribution functions,
$f_{\mathrm{n}\mathbf{k}}$, and on the out-of-equilibrium phonon populations
$n_{\mathbf{q}\nu}$, where $\mathrm{n},\leavevmode\nobreak\
\mathbf{k},\leavevmode\nobreak\ \nu,\leavevmode\nobreak\ \mathbf{q}$ are
respectively the electronic band index, wave vector, phonon mode index, and
wave vector. However, the electron-phonon scattering terms in the phonon BTE
has proven necessary only close to the degenerate semiconductor limit, e.g.,
for carrier concentrations larger than $10^{19}\leavevmode\nobreak\
\mathrm{cm^{-3}}$ at 300 K in silicon Zhou  _et al._ (2015). In that case, a
partial decoupling scheme Cantrell and Butcher (1987a, b) can be used, in
which the electron-phonon scattering terms in the phonon BTE are made
dependent on the (equilibrium) Fermi-Dirac distribution function
$f_{\mathrm{n}\mathbf{k}}^{0}$ Zhou  _et al._ (2015); Protik and Kozinsky
(2020); Xu  _et al._ (2021); Protik  _et al._ (2022). For low to moderate
doping concentrations which do not exceed $10^{19}\leavevmode\nobreak\
\mathrm{cm^{-3}}$, the effect of electron-phonon scattering on the phonon
populations can be safely neglected Mahan  _et al._ (2014); Fiorentini and
Bonini (2016); Macheda and Bonini (2018). In the present study, we follow the
latter approximation and obtain the Seebeck coefficient of silicon
nanostructures, including the phonon drag mechanism, by solving the charge-
carrier BTE in the relaxation time approximation:

$\displaystyle-\frac{\partial
f_{\mathrm{n}\mathbf{k}}^{0}}{\partial\varepsilon_{\mathrm{n}\mathbf{k}}}\mathrm{\mathbf{v}_{n\mathbf{k}}}\cdot\left[\mathbf{E}e+\frac{\nabla_{\mathbf{r}}T}{T}\left(\varepsilon_{\mathrm{n}\mathbf{k}}-\mu\right)\right]$
$\displaystyle-D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}\left(g,\delta
n\right)=$ $\displaystyle-\left(\frac{\partial
f_{\mathrm{n}\mathbf{k}}}{\partial t}\right)_{coll}$ (1)

where $\mathbf{E}$ and $\nabla_{\mathbf{r}}T$ denote a small electric field
and the temperature gradient, $\mu$ is the chemical potential for holes
(electrons), $\mathbf{v}_{\mathrm{n}\mathbf{k}}$ and
$\varepsilon_{\mathrm{n}\mathbf{k}}$ are respectively the charge-carrier group
velocity and energy. The term $\left(\frac{\partial
f_{\mathrm{n}\mathbf{k}}}{\partial t}\right)_{coll}$ includes all of the
collisions associated with carrier-phonon and carrier-impurity scatterings.

The term $D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}\left(g,\delta n\right)$ is
the key quantity that describes the phonon drag mechanism, and can be written
as (Fiorentini and Bonini, 2016; Macheda and Bonini, 2018):

$\displaystyle D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}\left(g,\delta
n\right)=$
$\displaystyle\frac{2\pi}{\hbar}\sum_{\mathrm{m}\nu}\int\frac{d\mathbf{q}}{\Omega_{\mathrm{BZ}}}\left|g_{\mathrm{mn}\nu}\left(\mathbf{k},\mathbf{q}\right)\right|^{2}$
$\displaystyle\times\left\\{\delta
n_{\mathbf{q\nu}}\delta\left(\varepsilon_{\mathrm{n}\mathbf{k}}-\varepsilon_{\mathrm{m}\mathbf{k}+\mathbf{q}}+\hbar\omega_{\mathbf{q}\nu}\right)\right.$
$\displaystyle+\left.\delta
n_{\mathbf{-q\nu}}\delta\left(\varepsilon_{\mathrm{n}\mathbf{k}}-\varepsilon_{\mathrm{m}\mathbf{k}-\mathbf{q}}-\hbar\omega_{-\mathbf{q}\nu}\right)\right\\}$
$\displaystyle\times\left(f_{\mathrm{m}\mathbf{k}+\mathbf{q}}^{0}-f_{\mathrm{n}\mathbf{k}}^{0}\right)$
(2)

where $g_{\mathrm{mn}\nu}\left(\mathbf{k},\mathbf{q}\right)$ is the carrier-
phonon interaction matrix element and $\Omega_{\mathrm{BZ}}$ is the volume of
the first Brillouin zone (BZ). The (linearized) out-of-equilibrium phonon
populations, $\delta
n_{\mathbf{q}\nu}=n_{\mathbf{q}\nu}-n_{\mathbf{q}\nu}^{0}$, are expressed as:

$\delta
n_{\mathbf{q\nu}}=-\tau_{\mathbf{q\nu}}\frac{\nabla_{\mathbf{r}}T\cdot\mathrm{\mathbf{c}_{\mathbf{q\nu}}}}{k_{B}T^{2}}\hbar\omega_{\mathbf{q\nu}}n_{\mathbf{q\nu}}^{0}\left(1+n_{\mathbf{q\nu}}^{0}\right)$
(3)

and have been obtained by solving the phonon BTE in the single mode
approximation with the D3Q code (Paulatto  _et al._ , 2013; Fugallo  _et al._
, 2013). Here, $\mathbf{c}_{\mathbf{q}\nu}$, $\tau_{\mathbf{q}\nu}$,
$\omega_{\mathbf{q}\nu}$, and $n_{\mathbf{q}\nu}^{0}$ are respectively the
group velocity, lifetime, frequency, and the (equilibrium) Bose-Einstein
phonon populations. We have used our in-house modified version of the
Electron-Phonon Wannier (EPW) code (Ponce  _et al._ , 2016) to solve Eq. 1
(see Supplemental Material (SM) for implementation description and
computational details (Sen, )).

Eqs. 1 and 2 give rise to an electrical current in the same direction as the
heat flow (Fig. 1, panels a and b). Indeed, although expressed in a form
similar to standard electron-phonon scattering processes, the term
$D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}\left(g,\delta n\right)$ is not a
resistive process and contributes to the enhancement of the electrical current
in the direction of the heat flow. We obtain the diffusion contribution when
$\delta n_{\mathbf{q\nu}}=0$, and thereby calculate
$\mathrm{\mathit{S}^{drag}}$ as $\mathrm{\mathit{S}^{tot}}|_{\delta
n_{\mathbf{q\nu}}\neq 0}-\mathrm{\mathit{S}^{diff}}$.

In addition to the phonon-phonon and isotope scattering rates (Paulatto  _et
al._ , 2013; Fugallo  _et al._ , 2013), the inverse of the phonon lifetime
$(\tau_{\mathbf{q\nu}})^{-1}$ is also determined by the phonon-boundary
scattering rate which, for nanostructures, is the crucial quantity that
controls the magnitude of $D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}$. We have
used the phonon momentum-resolved Casimir model to determine the transport-
direction-dependent phonon-boundary scattering in nanostructures. The Casimir
scattering rate for a phonon is given by (Markov  _et al._ , 2016; Fauziah
_et al._ , 2020):

$(\tau_{\mathbf{q\nu}}^{\mathrm{bound}})^{-1}=\left(\frac{1-p}{1+p}\right)\frac{|\mathbf{c}_{\mathbf{q}\nu}^{\mathrm{proj}}|}{L^{\mathrm{Cas}}}$
(4)

where the Casimir scattering length, $L^{\mathrm{Cas}}$, represents the
nanostructure size (Fig. 1, panels a and b). The specularity, $p$, ranges from
0 to 1 for completely diffusive to completely specular scattering,
respectively. The velocity, $|\mathbf{c}_{\mathbf{q}\nu}^{\mathrm{proj}}|$, is
the phonon group velocity $\mathbf{\mathbf{c}_{\mathbf{q}\nu}}$ projected on
the direction(s) in which the phonon transport is limited by the boundaries
(see SM). It should be mentioned that while the anisotropy of the boundary
scattering has been taken into account in several theoretical studies of
lattice thermal conductivity (McGaughey  _et al._ , 2011; Ma, 2012; Jeong  _et
al._ , 2012; Markov  _et al._ , 2016), only the isotropic boundary (Fig. 1,
panel a) has been considered so far in the studies of the phonon drag effect
(Fauziah  _et al._ , 2020; Li  _et al._ , 2022). The case of anisotropic
phonon-boundary scattering presents a computational challenge, due to the
reduced symmetry of the term
$D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}\left(g,\delta n\right)$ of Eq. 2 in
presence of the anisotropic phonon lifetime $\tau_{\textbf{q}\nu}$. This
challenge can be overcome by performing the calculation of
$D_{\mathrm{n}\mathbf{k}}^{\mathrm{drag}}\left(g,\delta n\right)$ without
making use of crystal symmetry considerations and employing an additional
$\mathbf{q}$-points filtering scheme (see SM).

Figure 2: Calculated Seebeck coefficient of hole-doped bulk silicon (a) with
and (b) without SOC interaction as a function of the carrier concentration at
300 K. Solid line: $S^{\mathrm{tot}}$, Dashed line: $S^{\mathrm{diff}}$, and
Dot-dashed line: $S^{\mathrm{drag}}$. The circle, diamond, and triangle
respectively denote the theoretical value of $S^{\mathrm{tot}}$,
$S^{\mathrm{diff}}$, and $S^{\mathrm{drag}}$ taken from Ref. (Zhou  _et al._ ,
2015). Squares: Experimental data (Geballe and Hull, 1955).

We start by examining the effect of spin-orbit coupling (SOC) on the Seebeck
coefficient for holes, as no such report is available in the literature. Our
results for $\mathrm{\mathit{S}^{tot}}$ in hole-doped bulk silicon with and
without SOC, together with the corresponding contributions from the diffusion
and drag parts, are shown in Fig. 2 (panels a and b) as a function of carrier
concentration at 300 K. Taking SOC into account leads to a decrease of
$\mathrm{\mathit{S}^{tot}}$ at all concentrations, significantly improving the
agreement with the available experimental data (Geballe and Hull, 1955).
Turning now to the analysis of $\mathrm{\mathit{S}^{diff}}$ and
$\mathrm{\mathit{S}^{drag}}$ contributions, the absolute value of
$\mathrm{\mathit{S}^{diff}}$ (dashed line), as expected, decreases linearly
with the increase of carrier doping (Zhou  _et al._ , 2015; Fiorentini and
Bonini, 2016; Protik and Kozinsky, 2020; Xu  _et al._ , 2021; Protik  _et al._
, 2022), and is not affected by the presence/absence of SOC. In wide contrast,
$\mathrm{\mathit{S}^{drag}}$ (dot-dashed line) remains nearly independent of
the carrier concentration and is found to be strongly affected by SOC. Indeed,
we find that the $\mathrm{\mathit{S}^{drag}}$ contribution is reduced by
$30\%$ when SOC interaction is accounted for. As the effect of SOC on the
electron-phonon matrix elements in Si is found to be weak for phonons
contributing to phonon drag (see SM), this reduction is explained by the
change of the band structure around the top of the valence band (VB) induced
by SOC (Poncé  _et al._ , 2018), which, in turn, affects the number of allowed
electron-phonon interactions contributing to the phonon drag (the Dirac
distribution in Eq. 2). Indeed, Poncé et al. (Poncé  _et al._ , 2018) pointed
out the improvement of the calculated hole effective masses in the valence
band of silicon with SOC. We show that this is crucial also for the drag
Seebeck coefficient for holes.

We now turn to the role of dimensionality, size, and direction in governing
the drag Seebeck coefficient of silicon nanostructures. Our theoretical
results (Fig. 1) show the effect of size reduction on
$\mathrm{\mathit{S}^{drag}}$ for monocrystalline intrinsic samples of
different geometries at 300 K. These results have been obtained for $p=0$ in
Eq. 4 (completely diffusive boundary) and thus, should be regarded as the
lowest threshold value of $\mathrm{\mathit{S}^{drag}}$. Our calculations show
that $\mathrm{\mathit{S}^{drag}}$ is almost size-independent down to
$L^{\mathrm{Cas}}\sim 100$ $\mu\mathrm{m}$, and then decreases monotonically
with the decrease in $L^{\mathrm{Cas}}$ for all dimensionalities and heat
transport directions.

However, one can observe a different rate of decrease of
$\mathrm{\mathit{S}^{drag}}$ for in- and out-of-plane directions. For both
hole- and electron-phonon drag effects, $\mathrm{\mathit{S}^{drag}}$ along the
out-of-plane direction of thin films and nanowires behaves as in the case of
the isotropic boundary, and is quenched almost completely as
$L^{\mathrm{Cas}}$ approaches $\sim 100$ nm. At the same time,
$\mathrm{\mathit{S}^{drag}}$ decreases at a slower rate along the in-plane
direction of thin films and nanowires than in the out-of-plane one, reflecting
the fact that the phonons are scattered less frequently by boundaries when
traveling in the in-plane direction. Our study shows that for both low hole
and electron doping, a silicon thin film (nanowire) of thickness (diameter)
$100$ nm can still preserve more than $20\%$ ($10\%$) of the bulk
$\mathrm{\mathit{S}^{drag}}$ when measured along the in-plane direction.

It must be noted that boundary scattering strongly affects the phonons with
long mean free paths which contribute mostly to phonon drag. In that respect,
the effect of boundary scattering on the phonon drag differs from that of
alloying, which was recently studied in Ref. (Xu  _et al._ , 2021) for Si-Ge
alloys. Indeed, it was shown in Ref. (Xu  _et al._ , 2021) that mass disorder
in alloys scatters mostly high-frequency phonons, affecting the phonons with
long mean free paths in lesser extent, in contrast to boundary scattering.

Figure 3: Variation of hole-phonon drag Seebeck coefficient of silicon thin
film (with $L^{\mathrm{Cas}}=1$ $\mu$m and doping $10^{14}\leavevmode\nobreak\
\mathrm{cm^{-3}}$) as a function of temperature. The solid and dashed lines
represent $\mathrm{\mathit{S}^{drag}}$ along the in-plane and out-of-plane
directions of the thin film, respectively. Figure 4: Variation of the total
Seebeck coefficient ($\mathrm{\mathit{S}^{tot}}$) of (a) hole-doped
($3.6\times 10^{17}\leavevmode\nobreak\ \mathrm{cm^{-3}}$) and (b) electron-
doped ($1.5\times 10^{16}\leavevmode\nobreak\ \mathrm{cm^{-3}}$) silicon
nanowires as a function of the diameter ($L^{\mathrm{Cas}}$) at 300 K.
Squares: Experimental data from Ref. (Fauziah  _et al._ , 2020) for holes and
Ref. (Bennett  _et al._ , 2015) for electrons.

Turning to the comparison with previous experimental works, our results do not
entirely confirm the conclusions of Ref. (Sadhu  _et al._ , 2015), in which
the phonon drag Seebeck coefficient is reported to be quenched completely in
silicon nanowires of diameter smaller that $100$ nm. Rather, our results are
found to be compatible with Refs. (Fauziah  _et al._ , 2020; Bennett  _et al._
, 2015), in which it has been found that a part of
$\mathrm{\mathit{S}^{drag}}$ survives in nanostructures. One must note here
that the remaining drag contribution which we predict for silicon
nanostructures with $L^{\mathrm{Cas}}=100$ nm would be within the error bar of
the experimental measurements of Sadhu et al. (Sadhu  _et al._ , 2015) at
moderate doping ($3\times 10^{18}\leavevmode\nobreak\ \mathrm{cm^{-3}}$).

We next predict the temperature dependence of $\mathrm{\mathit{S}^{drag}}$ in
silicon nanostructures (Fig. 3). For hole-doped silicon thin film
($10^{14}\leavevmode\nobreak\ \mathrm{cm^{-3}}$) with $L^{\mathrm{Cas}}=1$
$\mu$m, the $\mathrm{\mathit{S}^{drag}}$ contribution is found to grow with
decreasing temperature. The in-plane temperature dependence is found similar
to the out-of-plane one. This behaviour is consistent with the increase of
$\mathrm{\mathit{S}^{tot}}$ of bulk silicon (Geballe and Hull, 1955; Zhou  _et
al._ , 2015) and is predicted here for the first time for nanostructures.

Finally, in Fig. 4, we compare our theoretical results with recent experiments
available in literature: Ref. (Fauziah  _et al._ , 2020) for hole doped (panel
a) and Ref. (Bennett  _et al._ , 2015) for electron doped (panel b) silicon
nanowires. These experiments were performed at room temperature and moderate
doping concentrations ($3.6\times 10^{17}\leavevmode\nobreak\
\mathrm{cm^{-3}}$ for hole doped and $1.5\times 10^{16}\leavevmode\nobreak\
\mathrm{cm^{-3}}$ for electron doped samples). As mentioned in the
introduction, nanostructuring can induce several competing effects on the
experimentally measured Seebeck coefficient (Hu  _et al._ , 2020). Apart from
the phonon drag reduction, other effects discussed in literature are: energy
filtering (creation of energy barriers due to extended defects), changes in
the band structure due to confinement, changes in band structure due to
fabrication and/or measurement procedures (Hu  _et al._ , 2020) as well as
additional 1D-like phonon transport in case of nanowires (Boukai  _et al._ ,
2008; Zhou  _et al._ , 2015). The latter phenomena may lead to the increase of
the Seebeck coefficient. Although the study of those effects is beyond the
scope of our formalism, the comparison between our calculated results and
experiments allows to gain insight into the relative role of phonon drag with
respect to other effects. Indeed, one can see in panel a of Fig. 4 that in
agreement with our calculations, the experimentally measured hole Seebeck
coefficient in Ref. (Fauziah  _et al._ , 2020) was found to decrease with the
decreasing nanostructure size, which is a clear indication that phonon drag
contribution is still present for the nanostructure sizes under study. At the
same time, some other effect is clearly playing a role for the Seebeck
coefficient measured in Ref. (Fauziah  _et al._ , 2020), which is demonstrated
by the fact that the experimental Seebeck value at 10 $\mu$m exceeds the bulk
value at the same doping concentration.

For the case of electrons (Bennett  _et al._ , 2015), the Seebeck coefficients
which were measured for defect-free nanowires with diameters 0.6-1 $\mu$m are
found in good agreement with our calculated data, demonstrating the
effectiveness of our theoretical scheme (Fig. 4, panel b). We note that the
isotropic boundary model underestimates $\mathrm{\mathit{S}^{tot}}$ for both
cases of electrons and holes.

In conclusion, in this work we have provided a detailed ab initio study of the
effect of the dimensionality, size, and heat-transport direction on the phonon
drag Seebeck coefficient in silicon nanostructures, accounting both for the
anisotropy of the boundary scattering and for the spin-orbit coupling.
Inclusion of the latter is shown to be mandatory to obtain a predictive
description of the hole-phonon scattering. The implementation of the phonon
drag term in combination with anisotropic scattering of phonons by
nanostructure boundaries turns out to be crucial to evaluate the transport-
direction-dependent out-of-equilibrium phonon populations in silicon
nanostructures, and predict the phonon-drag contribution to the Seebeck
coefficient. In particular, we have shown that even if phonon drag
contribution is strongly reduced by nanostructuring, a silicon thin film
(nanowire) of thickness (diameter) $100$ nm can still preserve, at 300 K, more
than $20\%$ ($10\%$) of the bulk $\mathrm{\mathit{S}^{drag}}$ when measured
along the in-plane direction, for both electrons and holes. Our findings for
silicon nanowires support the conclusion of the recent experimental work of
Ref. (Fauziah  _et al._ , 2020) concerning the impact of phonon-boundary
scattering on the hole Seebeck coefficient of silicon nanowires and show an
excellent agreement with the electron Seebeck coefficient measured in Ref.
(Bennett  _et al._ , 2015). At the same time, the remaining drag contribution
which we predict for silicon nanostructures with $L^{\mathrm{Cas}}=100$ nm
would be within the error bar of the experimental measurements of Ref. (Sadhu
_et al._ , 2015) at moderate doping. Therefore, our results allow to resolve
the apparent contradiction in previous literature. Furthermore, we also
predict that even if the remaining contribution of
$\mathrm{\mathit{S}^{drag}}$ at 300 K is relatively small in silicon
nanostructures, a remarkable increase of the phonon drag contribution is to be
expected at low temperatures.

###### Acknowledgements.

Calculations have been performed with the Quantum ESPRESSO computational
package Giannozzi  _et al._ (2017), the EPW code Ponce  _et al._ (2016), the
D3Q code Paulatto  _et al._ (2013); Fugallo  _et al._ (2013) and the Wannier90
code Pizzi  _et al._ (2020). This work has been granted access to HPC
resources by he French HPC centers GENCI-IDRIS, GENCI-CINES and GENCI-TGCC
(Project 2210) and by the Ecole Polytechnique through the 3L-HPC project.
Financial supports from the ANR (PLACHO project ANR-21-CE50-0008, Macacqu
flagship Labex Nanosaclay ANR-10-LABX-0035), from the DIM SIRTEQ, from the
CNRS-CEA program "Basic reseach for energy" are gratefully acknowledged. We
acknowlege useful discussions with Dr. Natalio Mingo and Dr. Samuel Poncé, as
well as the contribution of Dr. Gaston Kane on the preliminary stage of the
project.

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