Thermoelectric power factor of nanocomposite materials from two-dimensional quantum transport simulations

Nanocomposites are promising candidates for the next generation of thermoelectric materials since they exhibit extremely low thermal conductivities as a result of phonon scattering on the boundaries of the various material phases. The nanoinclusions, however, should not degrade the thermoelectric power factor, and ideally should increase it, so that benefits to the ZT figure of merit can be achieved. In this work we employ the nonequilibrium Green's function quantum transport method to calculate the electronic and thermoelectric coefficients of materials embedded with nanoinclusions. For computational effectiveness we consider two-dimensional nanoribbon geometries, however, the method includes the details of geometry, electron-phonon interactions, quantization, tunneling, and the ballistic to diffusive nature of transport, all combined in a unified approach. This makes it a convenient and accurate way to understand electronic and thermoelectric transport in nanomaterials, beyond semiclassical approximations, and beyond approximations that deal with the complexities of the geometry. We show that the presence of nanoinclusions within a matrix material offers opportunities for only weak energy filtering, significantly lower in comparison to superlattices, and thus only moderate power factor improvements. However, we describe how such nanocomposites can be optimized to limit degradation in the thermoelectric power factor and elaborate on the conditions that achieve the aforementioned mild improvements. Importantly, we show that under certain conditions, the power factor is independent of the density of nanoinclusions, meaning that materials with large nanoinclusion densities which provide very low thermal conductivities can also retain large power factors and result in large ZT figures of merit.

Figure 1

(a) Calibration of the simulations’ scattering parameters. The scattering strength is increased in an $L=15-\mathrm{nm}$ channel until the conductance falls to half of the ballistic value (dashed-black line), thereby setting the mean-free path of the electrons to 15 nm. (b) The power factor (defined as $G{S}^2$) of a pristine (without nanoinclusions) channel as the Fermi level is scanned across the bands. (c) A schematic of a typical geometry we consider. $V_{\mathrm{B}}$ is the barrier height, $d$ is the nanoinclusion diameter, and $E_{\mathrm{F}}$ is the Fermi level. (d) A comparison of the transmissions for an empty channel under ballistic coherent conditions (blue line), a channel with nanoinclusions under coherent transport (light-blue line), an empty channel under phonon scattering transport conditions (red line), and a channel with nanoinclusions under phonon-scattering transport conditions (light-red line).

Figure 2

The thermoelectric coefficients of an $L=60$-nm channel with an $8\times 4$ hexagonal arrangement of nanoinclusions [inset of (c)] and acoustic-phonon-scattering transport conditions vs nanoinclusion barrier height $V_{\mathrm{B}}$. (a) The conductance. (b) The Seebeck coefficient. (c) The power factor defined as $G{S}^2$. Five different Fermi levels are considered: $E_{\mathrm{F}}=-0.025\mathrm{eV}$ (purple-diamond lines), $E_{\mathrm{F}}=0\mathrm{eV}$ (green-star lines), $E_{\mathrm{F}}=0.025\mathrm{eV}$ (black-cross lines), $E_{\mathrm{F}}=0.05\mathrm{eV}$ (red-square lines), and $E_{\mathrm{F}}=0.075\mathrm{eV}$ (blue-circle lines).

Figure 3

The thermoelectric coefficients of an $L=60$-nm channel with $E_{\mathrm{F}}=0.05\mathrm{eV}$ (dashed red line) and acoustic-phonon-scattering transport conditions vs nanoinclusion barrier height $V_{\mathrm{B}}$. (a) The conductance. (b) The Seebeck coefficient. (c) The power factor defined as $G{S}^2$. Hexagonal arrays of four different nanoinclusion densities are considered as shown in the inset of (c): $2\times 4$ array (green lines), $4\times 4$ array (black lines), $6\times 4$ array (blue lines), and $8\times 4$ array (red lines).

Figure 4

The distribution of the energy of the current flow for an $L=60$-nm channel with an $8\times 4$ array of nanoinclusions and $E_{\mathrm{F}}=0.05\mathrm{eV}$. The stars denote the average energy of the current flow. A zoomed-in version of these is shown in the inset. Six different nanoinclusion barrier heights are shown: $V_{\mathrm{B}}=0\mathrm{eV}$ (black), $V_{\mathrm{B}}=0.02\mathrm{eV}$ (red), $V_{\mathrm{B}}=0.04\mathrm{eV}$ (blue), $V_{\mathrm{B}}=0.06\mathrm{eV}$ (green), $V_{\mathrm{B}}=0.08\mathrm{eV}$ (purple), and $V_{\mathrm{B}}=0.1\mathrm{eV}$ (brown). The dotted line in the inset indicates from right to left the trend of increase in $V_{\mathrm{B}}$.

Figure 5

The thermoelectric coefficients of $L=60$-nm channels [insets of (c)] with $8\times 4$ array of nanoinclusions vs nanoinclusion barrier height $V_{\mathrm{B}}$, for two different nanoinclusion diameters: $d=1.5\mathrm{nm}$ (red lines) and $d=3\mathrm{nm}$ (black lines), and a $15\times 7$ array with $d=1.5\mathrm{nm}$ (blue lines) whose density is equivalent to the $8\times 4$ array with $d=3\mathrm{nm}$. (a) The conductance. (b) The Seebeck coefficient. (c) The power factor defined as $G{S}^2$.

Figure 6

The transmission vs energy for the channels shown in the inset of Fig. 5: $8\times 4$ arrays of $d=1.5\mathrm{nm}$ (red lines) and $d=3\mathrm{nm}$ (black lines), and a $15\times 7$ array with $d=1.5\mathrm{nm}$ whose density is equivalent to the $8\times 4$ array with $d=3\mathrm{nm}$ (blue lines). (a) The case for nanoinclusions with $V_{\mathrm{B}}=0.02\mathrm{eV}$ barrier height, i.e., before the $E_{\mathrm{F}}$, where the power factor reaches the maximum point in Fig. 5. (b) The case for nanoinclusions with $V_{\mathrm{B}}=0.07\mathrm{eV}$ barrier height, i.e., after the $E_{\mathrm{F}}$, where the power factor starts to saturate in Fig. 5.

Figure 7

(a) Color map of the current flow directed along the length of the channel (L directed) through an $8\times 4$ hexagonal array of nanoinclusions ($d=3\mathrm{nm}$, $V_{\mathrm{B}}=0.02\mathrm{eV}$). Nanoinclusions can be seen as the blue areas and the matrix material as the yellow and green areas. (b) The channel length directed current along the dashed black line shown in (a) at four different energies, below the barrier at $E=0.01\mathrm{eV}$ (green line), at the barrier $E=0.02\mathrm{eV}$ (black line), at the Fermi level $E=0.05\mathrm{eV}$ (blue line), and above the barrier and Fermi level at $E=0.075\mathrm{eV}$ (purple line). The location of the first nanoinclusion (NI), which extends for 3 nm, is denoted. (c) The current flow at two points in the structure: at the center of a nanoinclusion (blue line, position shown by dotted blue line in (b) at $L\sim 40\mathrm{nm}$) and in the pristine matrix [black line, position shown by dotted black line in (b) at $L\sim 46\mathrm{nm}]$. The barrier height is shown by the dashed black line and Fermi level by the dashed red line.

Figure 8

The transmission vs energy for an $L=60\mathrm{nm}$ ballistic coherent channel (no phonon scattering) for the following cases as shown in the insets: (i) pristine material without nanoinclusions (red line), (ii) material with an $8\times 4$ hexagonal array of nanoinclusions (blue line) with $V_{\mathrm{B}}=0.1\mathrm{eV}$, and (iii) a superlattice material (black line) with $V_{\mathrm{B}}=0.1\mathrm{eV}$. The barrier height $V_{\mathrm{B}}$ is marked by a dashed black line. It can be seen that the superlattice is effective at cutting out the contribution of low-energy electrons (achieving an increase in the Seebeck coefficient) whereas the nanoinclusions act to reduce the transmission uniformly in the entire energy region.

Figure 9

The thermoelectric coefficients of $L=60\mathrm{nm}$ channels with an $8\times 4$ array of nanoinclusions vs nanoinclusion barrier height $V_{\mathrm{B}}$, for four different simulation conditions: Ballistic transport (black-cross lines), mean-free path $\mathrm{mfp}=30\mathrm{nm}$ and $m^{*}=m_0$ (blue-diamond lines), mean-free path $\mathrm{mfp}=15\mathrm{nm}$ and $m^{*}=0.5m_0$ (green-star lines), and mean-free path $\mathrm{mfp}=15\mathrm{nm}$ and $m^{*}=m_0$ (red-square lines—same as in Figs. 2 and 3). (a) The conductance. (b) The Seebeck coefficient. (c) The power factor defined as $G{S}^2$. Inset of (b): The transmission probability vs energy in the four cases for $V_{\mathrm{B}}=0\mathrm{eV}$.

Figure 10

The shape of the barrier around the nanoinclusion for different cases using 1D self-consistent calculations. (a) The perfectly square barrier as used in the simulations. (b) The barrier shape when uniform doping is applied in all domains—Schottky barriers are formed around the nanoinclusion. (c) The barrier shape around the nanoinclusion when only the matrix material is doped, whereas the nanoinclusion remains undoped. (d) A case with variable doping where the doping in the nanoinclusion is reduced to 30% of that in the matrix material. In the latter case the barrier profile looks very similar to the one simulated.