# Thermoelectric Properties of InA Nanowires from Full-Band Atomistic Simulations

^{*}

## Abstract

**:**

## 1. Introduction

_{e}, and the phonon/lattice part of the thermal conductivity, κ

_{l}. The quantity σS

^{2}is the power factor (PF). Over the last several years, a myriad of materials and concepts for high ZT have evolved [1], including GeTe [2], PbTe [3], half-Heuslers [4], skutterudites [5], etc. Low-dimensional materials such as nanowires (NWs) are one of these concepts, as they can achieve extremely low thermal conductivities due to strong phonon-interface scattering. Significant increases in TE performance and ZT in NWs and their networks have been reported [6,7,8,9,10,11,12,13]. ZT values up to 1 for NWs based on several materials (Si, SiGe, InAs, InSb, Bi, PbTe, ZnO, SnSe, NiFe, and many more) have been investigated [11,14,15,16]. Since the pioneering work by Hicks and Dresselhaus, efforts have also been focused on utilizing the sharp features in the low-dimensional density-of-states to improve the power factor as well [17,18]. Theoretical studies on the thermoelectric power factor of NWs showed that one-dimensional (1D) modes could provide power factor improvements even up to 30% [19,20,21,22]. Experimentally, however, this has not yet been achieved, because to observe the true 1D nature, one needs to consider NW diameters down to a few nanometers (as in the case of Si) [23]. At those dimensions, however, surface roughness scattering (SRS) drastically reduces σ [23], but also distorts the sharp features in the density-of-states [24].

^{*}~0.2 m

_{0},) at diameters below ~10 nm [23]. In the case of InAs, however, with effective mass m

^{*}~0.02 m

_{0}, we expect such effects to appear at a larger length scale, as it was observed in the case of Bi nanowires as well [25,26]. The Seebeck coefficient, in particular, as we have previously shown, begins to increase in an almost linear fashion with diameter reduction from the point where quantum confinement splits the NW subbands at such degree, which leaves only a few subbands (ideally one) in the vicinity of the Fermi level [19,23]. Thus, low-effective mass materials, which reach the ‘few subband’ condition at larger diameters, could provide a larger Seebeck coefficient increase with further diameter reduction compared to channels with larger effective masses. Nanowires with larger diameters are practically more feasible and controllable as well. Power factor benefits would then be more easily realized. The subband quantization, a signature of low dimensionality, has been observed at lower temperatures in InAs NWs, where the effect of individual subband features was observed in all three coefficients, the electrical conductivity, the Seebeck coefficient, and the power factor [27,28]. In another low temperature work, InAs/InP NW superlattices were fabricated, and quantum dots were formed, exhibiting promising thermoelectric energy power extraction and conversion efficiency [29]. Doping and planar defects are also investigated in order to optimize the PF and decrease the NW thermal conductivity [30]. References [31,32] have also measured promising TE performance for InAs NWs with diameters as low as 20 nm.

_{l}is reduced and ZT can be improved [6,7,33]. We believe that our results will further add to the understanding of the effects of low-dimensionality on the PF and the conditions under which improvements can be observed. We also believe that the method we employ can prove useful in other TE material investigations which require capturing accurately the energy dependence of the scattering times, especially when extended to two-dimensional (2D) and three-dimensional (3D) materials.

## 2. Theoretical and Computational Method

#### 2.1. Bandstructure Features under Confinement

^{3}d

^{5}s

_{*}tight-binding model with the parametrization of Ref. [34]. The model is validated to capture all of the relevant features of the bandstructure of semiconductors that appear at the nanoscale. Previous works showed that tight-binding methods could capture essential bandstructure features beyond band quantization, such as band splitting, non-parabolicity, band warping, effective mass variation, etc. [35,36,37,38,39,40,41]. Importantly for this work, tight-binding is robust enough to calculate the bandstructure for NWs up to 40 nm in diameter (structures of up to 30,000 atoms in the unit cell). We consider [100] n-type InAs nanowires, and we ignore spin-orbit coupling. The bandstructures of InAs NWs of diameters d = 3 nm, 10 nm, 20 nm, and 40 nm are shown in Figure 1a–d. The position of the Fermi level for the carrier density of n = 10

^{18}/cm

^{3}(approximate concentration where the thermoelectric power factor peaks for the d = 10 nm NW) is indicated by the red lines in each sub-figure. The two important things to notice here as the diameter is scaled are the following: (i) the number of subbands is reduced to very few, even to a single subband, and (ii) the position of the Fermi level, which directly determines the Seebeck coefficient (S), shifts lower compared to the band edge (comparing here at the same carrier density). Indeed, the distance of the Fermi level from the band edge η

_{F}= E

_{C}− E

_{F}increases substantially with diameter reduction as shown in the inset of Figure 1d (in units of k

_{B}T). The Seebeck coefficient is proportional to the average energy of the current flow as S ~ <E − E

_{F}>, which depends linearly on η

_{F}. This increase in η

_{F}originates from the fact that the number of subbands decreases slower compared to the NW area (and cannot be reduced to zero subbands at ultra-narrow diameters). The only way then to retain a constant carrier (3D) density is to increase η

_{F}(to lower E

_{F}), and this is the reason that Seebeck coefficient improvements are expected at low dimensions, as we show below [19,23].

^{*(−3/2)}, whereas the differential of the band edges determines the strength of the SRS rate as r

_{SRS}~(ΔE/Δd)

^{2}[42]. Results for the two nanowire orientations [100] (blue lines) and [110] (orange lines) are shown. Both quantities begin to increase when the diameter is scaled below d ~ 20 nm, whereas significant increases are observed for diameters below d ~ 10 nm. The increase in the effective mass originates from the behavior of non-parabolic bands under confinement and is well explained in previous works and was observed in Si [35,36] and Bi nanowires as well [43]. For diameters down to d = 3 nm, an increase of ~ 3× is observed. In contrast, in the case of Si for example, a less non-parabolic material, the corresponding increase in the effective masses is somewhat less than half up to 1.4× (inset of Figure 2a). In a similar way, the differential of the shifts in the band edges increases largely with diameter reduction (Figure 2b), in a more sensitive fashion compared to Si channels (inset of Figure 2b). Both the effective masses and band edges are more sensitive to quantization in InAs compared to Si due to the much smaller effective mass and larger non-parabolicity. This sensitivity, however, would have a negative impact on its transport properties, as we show below.

#### 2.2. Transport Theory–Linearized BTE Formalism

_{x}is the wavevector index, ${\tau}_{{k}_{x},n}\left(E\right)$ is the carrier relaxation time, ${g}_{{k}_{x},n}^{1\mathrm{D}}\left(E\right)$ is the one-dimensional density of states, and ${\nu}_{k}$ is the band velocity computed as:

_{ω}is the number of phonons given by the Bose-Einstein distribution, and Ω is the unit cell volume. For acoustic deformation potential scattering (ADP), optical deformation potential scattering (ODP), and polar optical phonon scattering (POP), respectively, for the strength of the scattering event, it holds that:

_{ADP}and D

_{O}are the scattering deformation potential amplitudes, ${\epsilon}_{\infty}$ is the high frequency dielectric constant, and ${\epsilon}_{s}$ is the static dielectric constant (for InAs we use ${\epsilon}_{\infty}=12.3$, and ${\epsilon}_{\mathrm{S}}=15.15$) [49,50].

_{x}is the length of the unit cell, ${F}_{n/m}\left(\overrightarrow{R}\right)$ is the cross-sectional part of the wave function of the initial/final state, and the integral is performed over the cross section of the nanowire.

_{X}, we obtain the transition rates and relaxation times as:

_{X}, we obtain the transition rates and relaxation times as:

_{X}, we obtain the transition rates and relaxation times as:

_{x}still remains within the summation.

^{6}behavior. The scattering rate is then evaluated as previously by:

#### 2.3. Calibration to Bulk Mobility

^{2}/V-s, with a slight downward trend with an increasing diameter. The bulk phonon-limited low-field mobility value is ~40,000 cm

^{2}/V-s [49], and our quantitative overestimation could show that, indeed, larger NW diameters are needed to reach the bulk mobility, or that the deformation potentials chosen, which are bulk values, are not that accurate for NWs. Nevertheless, we still use bulk values, although it is observed that phonon confinement can lead to larger deformation potential values. Our goal is not to accurately map the bulk mobility, but to quantitatively present the trend of the TE coefficients with the diameter.

## 3. Results and Discussion

#### Thermoelectric Performance of InAs Nanowires

^{2}versus the carrier density for [100] nanowires of diameters from d = 40 nm down to d = 3 nm are shown in Figure 4a–c, respectively, at T = 300 K. Following the mobility trend, the electrical conductivity for the narrower nanowires is significantly lower compared to that of the larger nanowire diameters (Figure 4a), with the exception of the d = 10 nm NW (red line), which overpasses all others from densities n > 10

^{18}/cm

^{3}and above. This is a consequence of the reduction in the POP scattering rates as the average exchange vector decreases with reduced diameter and reduced number of bands. On the other hand, as the diameter is decreased, a significant increase is observed in the Seebeck coefficient across all carrier concentrations as shown in Figure 4b. This is a consequence of the increase in the η

_{F}as indicated in the inset of Figure 1d, which essentially increases the average energy of the current flow and consequently the Seebeck coefficient. As a consequence of these trends, the power factor in Figure 4c exhibits a somewhat erratic behavior, where the narrower nanowires (d < 5 nm) indicate a clear advantage only at higher carrier concentrations, beyond n = 10

^{18}/cm

^{3}. The power factor is maximized for the d = 10 nm NW at n = 10

^{18}/cm

^{3}and for the d = 5 nm NW at n = 10

^{19}/cm

^{3}. For these wires, the PF reaches large values of > 5 mW/mK

^{2}, which signals promising TE performance. The larger diameter NWs (d = 20 and 40 nm) lack significantly in performance, and their peak appears at lower densities.

^{2}.

^{18}/cm

^{3}, the density at which the d = 10 nm NW PF peaks. The TE coefficients σ, S, and σS

^{2}are plotted versus the nanowire diameter, d. In this case, we plot the phonon-limited TE coefficients in blue lines, and we then include SRS in addition in red lines. In the phonon-limited transport case, the electrical conductivity in Figure 6a increases by ~ 40% as the diameter is scaled down to d ~ 12 nm compared to the larger diameter value, but further diameter scaling results in its sharp drop. This is a consequence of the increase in electron-phonon scattering (form factors) and effective mass increase. As shown in Figure 2b, for diameters below d = 10 nm and carrier densities n = 10

^{18}/cm

^{3}, only one subband participates in transport and the Fermi level is pushed below the band edge, in which case carriers with lower velocities participate in transport, and the conductivity is reduced. On the other hand, the shift in E

_{F}increases the Seebeck coefficient significantly (Figure 6b).

_{F}much more with confinement, resulting in a much larger increase in S. If this effect begins at larger diameters as in InAs, then there is more room for scaling and larger Seebeck coefficient increases can be achieved.

_{F}upon confinement, which in turn increases the energy of the current flow, and (ii) quantization, which reduces the strength of POP scattering around NW diameters of d = 10 nm. In light mass materials, this effect begins at larger diameters, which allow for design flexibility by scaling. On the other hand, this same light effective mass, which causes strong confinement, also causes a similarly large SRS as a result of larger sensitivity in the band edges of the electronic structure. Thus, the same effect that provides the benefits also takes most of them away. In comparison, for heavier effective mass materials, such as Si, moderate improvements in the power factor are observed upon confinement, but at smaller NW diameters, of d ~ 5 nm. At such narrow diameters, SRS is also strong, and benefits are also suppressed, even eliminated [23]. Thus, the benefits in polar, light mass materials are expected to be larger compared to non-polar materials with larger effective masses.

_{l}, originating from enhanced phonon-boundary scattering [6,7]. The fact that SRS also drastically affects phonons even at a larger degree compared to electrons, makes it so that rough boundaries are actually favorable. However, the knowledge at which length scales and for which materials the power factor is less affected, or even increased, can provide opportunities for improving the ZT figure of the merit of low-dimensional TE materials.

^{2}. From Figure 4, the simulated phonon-limited (upper limit) PF at those Seebeck coefficient values is ~ 3 mW/mK

^{2}(green line). When SRS is introduced in Figure 5, the PF drops to ~ 1 mW/mK

^{2}for the roughness amplitude of Δ

_{rms}= 1 nm we used, suggesting that the experimental Δ

_{rms}might have been somewhat smaller. In the second work, Ref. [32], the authors measured the TE PF of a d ~ 23 nm InAs NW, again using gating techniques, and found it to be ~ 0.05 mW/mK

^{2}at densities of 10

^{18}/cm

^{3}, which is, however, significantly lower compared to what we compute, possibly due to numerous other scattering mechanisms present and not accounted for in the simulation.

_{e}+ κ

_{l}. The κ

_{e}is given by κ

_{e}= LσT, where L is the Lorenz number. Under the simple acoustic phonon scattering conditions and parabolic bands, the Lorenz number resides mostly between L = 2.45 × 10

^{−8}W Ω K

^{−2}in the degenerate limit and L = 1.49 × 10

^{−8}W Ω K

^{−2}in the non-degenerate limit. These values are routinely used to estimate κ

_{e}when limited knowledge about thermal transport details exists. However, we have shown that the Lorenz number can be reduced significantly from the degenerate limit in the presence of multi-band effects, and inter-band scattering [61]. The Lorenz number of the InAs NWs we consider is shown in Figure 7 for the case of phonon-limited transport (blue line) and phonon plus SRS limited transport (red line) for NWs with a carrier density of n = 10

^{18}/cm

^{3}. Indeed, the Lorenz number at large diameters resides at values around the degenerate limit, as expected since E

_{F}resides well into the bands (Figure 1d). The Lorenz number takes a sudden drop to the non-degenerate limit (and even below) at d ~ 12 nm in the presence of SRS, which lowers κ

_{e}. This is a consequence of the E

_{F}shifting lower, towards non-degenerate conditions, still at the same density. The important thing here, however, is that the power factor can increase (at least at the best case around the d ~ 10 nm NW), and the ZT would also benefit from reduction in both κ

_{l}and κ

_{e}. For example, the thermal conductivity of such narrow NWs is reported to be around ~ 2 mW/mK

^{2}, in which case a ZT of ~ 0.15 can be reached, which is a significant value for room temperature operation.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Derivation of Equation (19)

#### Derivation of Equation (20) for ADP Scattering:

#### Derivation of Equation (24) for ODP Scattering:

#### Derivation of Equation (28) for POP Scattering:

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**Figure 1.**Electronic bandstructures for [100] InAs nanowires of diameters (

**a**) d = 3 nm, (

**b**) d = 10 nm, (

**c**) d = 20 nm, and (

**d**) d = 40 nm. The position of the Fermi level E

_{F}for carrier density n = 10

^{18}/cm

^{3}at room temperature is indicated. The difference of the Fermi level from the band edges η

_{F}= (E

_{C}− E

_{F})/k

_{B}T, which determines the Seebeck coefficient, is indicated as well. The inset of (

**d**) shows η

_{F}versus the NW diameter.

**Figure 2.**(

**a**) The effective mass of the first subband of InAs nanowires as a function of the nanowire diameter. An increase in the mass is observed as the diameter is reduced. (

**b**) The differential of the band edge of the nanowires versus their diameter. Nanowire orientations in [100] (blue—circle lines) and in [110] (orange—triangle lines) are shown. The insets show the corresponding mass variation and band edge differential changes for Si nanowires, as shown in Refs [42,44], which indicate less variation for both quantities.

**Figure 3.**Low-field electron mobility vs. nanowire diameter for [100] InAs nanowires at room temperature. Different scattering cases are shown: (i) the blue line shows phonon-limited transport (including acoustic and optical deformation potentials scattering and polar optical phonon scattering), (ii) the red line shows the case of when surface roughness scattering (SRS) is added in addition to phonon scattering.

**Figure 4.**Thermoelectric coefficients under phonon scattering-limited transport conditions at room temperature for [100] InAs with different diameters, as indicated in the figure. (

**a**) Electrical conductivity, (

**b**) Seebeck coefficient, and (

**c**) power factor versus carrier concentration. As the diameter is reduced, the Seebeck coefficient is increased. The power factor is increased for the smaller NW diameters around d ~ 3–10 nm.

**Figure 5.**Thermoelectric coefficients under phonon plus surface roughness scattering (SRS) transport conditions at room temperature for [100] InAs with different diameters, as indicated in the figure. (

**a**) Electrical conductivity, (

**b**) Seebeck coefficient, and (

**c**) power factor versus carrier concentration. As the diameter is reduced, the Seebeck coefficient is increased. The power factor is increased for the larger diameters around d ~ 10–40 nm.

**Figure 6.**Thermoelectric coefficients under phonon (blue lines) and phonon plus surface roughness scattering (red lines) transport conditions at room temperature for [100] InAs NWs versus diameter at a fixed carrier concentration of n = 10

^{18}/cm

^{3}. (

**a**) Electrical conductivity, (

**b**) Seebeck coefficient, and (

**c**) power factor. As the diameter is reduced, the Seebeck coefficient is increased. The power factor is increased for diameters around d ~ 10 nm.

**Figure 7.**The Lorenz number versus nanowire diameter under phonon scattering conditions (blue line) and under phonon plus surface roughness scattering (red line) transport conditions at room temperature at a fixed carrier concentration of n = 10

^{18}/cm

^{3}.

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## Share and Cite

**MDPI and ACS Style**

Archetti, D.; Neophytou, N.
Thermoelectric Properties of InA Nanowires from Full-Band Atomistic Simulations. *Molecules* **2020**, *25*, 5350.
https://doi.org/10.3390/molecules25225350

**AMA Style**

Archetti D, Neophytou N.
Thermoelectric Properties of InA Nanowires from Full-Band Atomistic Simulations. *Molecules*. 2020; 25(22):5350.
https://doi.org/10.3390/molecules25225350

**Chicago/Turabian Style**

Archetti, Damiano, and Neophytos Neophytou.
2020. "Thermoelectric Properties of InA Nanowires from Full-Band Atomistic Simulations" *Molecules* 25, no. 22: 5350.
https://doi.org/10.3390/molecules25225350