# Monte Carlo simulation of thermoelectric properties in silicon nanofilms: diffusive seebeck coefficient analysis

Mohammad Ghanem, Philippe Dollfus, Jérôme Saint-Martin

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Mohammad Ghanem, Philippe Dollfus, Jérôme Saint-Martin. Monte Carlo simulation of thermoelec- tric properties in silicon nanofilms: diffusive seebeck coefficient analysis. Solid-State Electronics, 2025, 229, pp.109157. 10.1016/j.sse.2025.109157. hal-05091543

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https://hal.science/hal-05091543v1

Submitted on 5 Jun 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
particle Monte Carlo code [8] used here is self-consistently coupled to the Poisson's equation. It is based on an analytical conduction band structure of bulk silicon which consists of six ellipsoidal non parabolic Δ valleys along [100] directions. Their effective masses and non- parabolicity coefficient are $m_l = 0.916*m_0$, $m_t = 0.19*m_0$, and $\alpha = 0.5$, respectively. This conduction band description is sufficient since the studies are only conducted close to equilibrium and in low energy scales. The parameters used for intravalley and intervalley electron-phonon scatterings mechanisms are given in [9], and the dependence of the scattering rates on temperature is detailed in [10]. The surface rough- ness scattering is considered specular to allow comparison with the bulk case.

### 3. Results

In Fig. 2 (a), non-uniform ($T_{phonon} = [280 \text{ K-320 K}]$) and uniform ($T_{phonon} = [300 \text{ K-300 K}]$) temperature profiles applied along the device are shown in addition to the temperature of electrons ($T_{electron}$) after moving and scattering inside the device without any voltage bias applied ($V_{applied} = 0 \text{ V}$). When a non-uniform phonon temperature is applied to the device, the electron temperature is not constant and very close to the phonon one. This occurs because the temperature difference between electrons and phonons tends to be limited by the inelastic electron-phonon scattering. In Fig. 2 (b), the obtained I-V curve for the device is presented in the case of the non-uniform temperature (black dots) and in the case of a uniform temperature of 300 K (red dots). A linear regression (depicted by the red and black solid lines) was applied to the simulated current data points. The total device's Seebeck coefficient, under non-uniform temperature profile, was obtained by dividing the open voltage (extracted as the voltage corresponding to zero current from the linear fit) $V_{oc}$, by the temperature difference $\Delta T$. The mobility of the total device was obtained from the slope of the linear fit. The obtained total Seebeck coefficient (S) was determined to be $451 \pm 12 \mu V/ \text{K}$. The mobility of the device was determined to be $612 \pm 8 \frac{\text{cm}^2}{\text{V}\text{s}}$. The difference in S between $10^{17} \text{ cm}^{-3}$ and $10^{19} \text{ cm}^{-3}\text{N-doped bulk silicon}$ without the phonon-drag effect, i.e. only diffusion part, was calculated by [11] to be $378 \mu V/\text{K}$ at 300 K (where in their work the phonon drag part was shown to take a strong and crucial role).

In Fig. 3, at a bias voltage of 0 V, 0.05 V, and 0.1 V the kinetic energy of electrons, the electrostatic potential, and the electric field profiles along the length of the simulated device are plotted. As previously mentioned, the increase in electron energy along the device (Fig. 3 (a)) primarily stems from the initial imbalance between electron and phonon energies. The effect of the applied voltage on the kinetic energy is barely noticeable due to the small applied voltages.

Fig. 3 (b) shows the electrostatic potential difference between the contacts and the device's middle due to doping variation. The non-uniform electron temperature drives electrons from hot to cool regions, altering their distribution. This causes a linearly decreasing electrostatic potential even when no voltage bias is applied. As $V_{app}$ increases, the potential flattens at $V_{app} = V_{oc}$ where the applied field counteracts this movement and forces uniform distribution. For $V_{app} > V_{oc}$, the field dominates, reversing the imbalance and creating a linear increase in the potential. The electric field which is the negative derivative of the potential along the device, is presented in the inset of Fig. 3 (b). Understanding the impact of non-uniform temperature profiles and applied voltage on electron kinetics not only enriches our understanding of thermoelectric transport mechanisms but also offers valuable insights for the designing better thermoelectric devices.

In Fig. 4, the total diffusive Seebeck coefficient (which is the difference between the one of the low doping material - $10^{17} \text{ cm}^{-3}$ - and the reference material, i.e. contacts at $10^{19} \text{ cm}^{-3}$) was calculated for different average temperatures and for different temperature gradients at 300 K. The Seebeck coefficient decreases as the temperature increases, primarily due to higher scattering rates, which hinder electron movement toward the colder side. On the other hand, the Seebeck coefficient seems to be constant and barely changes for various temperature differences in Fig. 4(b). The slight variation can be attributed to simulation uncertainty. This was also observed in [12], where in their TE device the Open Circuit Voltage $V_{oc}$ seems to increase linearly with the increase of temperature gradient which ensures that the obtained results are consistent, and the Seebeck coefficient is not affected.

This difference in the diffusive thermopower was also investigated with respect to different device's length and doping at 300 K and plotted in Fig. 5. The Seebeck coefficient slightly increases with device length because longer devices experience more scattering, enhancing energy distribution asymmetry. In shorter, ballistic devices, the energy distribution remains more uniform, resulting in a lower Seebeck coefficient. In contrast, Fig. 5(b) shows that the diffusive Seebeck coefficient for bulk silicon decreases with increasing carrier concentration. This decline is attributed to the reduced energy difference between carriers and the Fermi level. Our results, when compared to those in [11], are found to be close, where the coupled Boltzmann transport equations are solved.

### 4. Conclusion

The device's Seebeck coefficient and thermoelectric transport properties were investigated for a silicon-based nanofilm by using an ensemble Monte Carlo solver for electrons considering a phonon bath

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic view of the TE device studied.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) Non-uniform (black solid line) and uniform (red solid line) applied temperatures in addition to the calculated temperature of electrons (dashed lines) along the device length. (b) Simulated I-V curves (dots) and their linear regression (solid lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Electrons kinetic energy (a) and electrostatic potential created (b) as a function of the position. Inset: corresponding electric field.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Seebeck coefficient of the simulated nanofilm with respect to different (a) temperatures and (b) temperature gradients at 300 K.</figcaption></figure>

with a non-uniform temperature. The obtained Seebeck coefficient in this work seems to be comparable and consistent with previous experimental and simulation works, and the computed microscopic thermoelectric transport results are expected to help designing better thermoelectric devices. In future work, different polytype and dual doping nanowires will be investigated, and phonon drag effects will also be implemented in our simulator, which will provide a more accurate

tool to optimize and test the TE device performance in these kinds of nanostructures.

CRediT authorship contribution statement

Mohammad Ghanem: Writing – original draft. Philippe Dollfus: Supervision. Jerome Saint-Martin: Supervision.
<figure><img src="image_6.png" /><figcaption>Fig. 5. Seebeck coefficient of the simulated nanofilm with respect to different (a) device length and (b) carrier concentration at 300 K.</figcaption></figure>

Declaration of competing interest

This work was financially supported in part by the French ANR under
PLACHO Project (ANR-21-CE50-0008).

Data availability

Data will be made available on request.

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