Thermoelectric performance of granular semiconductors

We study the effects of doping and confinement on the thermoelectric properties of nanocrystalline semiconductors. We calculate the thermopower and figure of merit for temperatures less than the charging energy. For weakly coupled semiconducting grains it is shown that the figure of merit is optimized for grain sizes of order 5 nm for typical materials, and that its value can be larger than one. Using the similarities between granular semiconductors and electron or Coulomb glasses allows for a quantitative description of inhomogeneous semiconducting thermoelectrics.

Figure 1

(Color online) Sketch of a nanogranular material showing typical electron (e) and hole (h) transport. In the upper row of grains an inelastic electron tunneling process is shown and in the lower row a co-tunneling loop responsible for the electronic part of the heat transport is presented. The energy $\left(\epsilon \right)$ transport goes from the “hot” $\left(H\right)$ to the ”cold” $\left(C\right)$ side.

Figure 2

(Color online) Simulation results for the density of state $n\left(\epsilon \right)$ vs energy for different filling factors $\nu$, indicated by the arrow across the curves from $\nu =0.5$ to $\nu =0.8$ in steps of $\Delta \nu =0.05$ ($\nu >1/2$ corresponds to $n$-type doped semiconductors). The simulations were done for a 2D Coulomb glass model, simulating the whole weakly coupled granular sample[16, 17] for a system size of ${500}^2$. The inset shows the dependence of the shift of the chemical potential $\Delta \mu$ and the asymmetry of the density of states $\Delta n$ on $\nu$ (see text). Using these data we extract the numerical coefficients $a_1\approx 2.9$ and $a_2\approx 0.4$ in Eq. (7) by linear fitting.

Figure 3

(Color online) Diagrams describing lowest (a) and higher (b) order co-tunneling processes (multiple tunneling events are indicated by the dotted lines). The solid lines denote the propagator of electrons. The tunneling vertices are described by the circles. The wavy lines indicate the external coupling to the heat vertices.

Figure 4

(Color online) Plots of the figure of merit $ZT$ vs temperature $T$ (a) and grain size $a$ (b) for two (2D) and three (3D) dimensional samples and two different filling factors $\nu =0.6,0.65$ ($n$-type doped granular semiconductors). Typical material parameters are given in the text. In a), the grain size is chosen to be $a=5\text{nm}$ and in b), the temperature is fixed at 200 K. One sees that $ZT$ can be optimized by choosing appropriate grain sizes: For the used parameters, these sizes are about 4 nm in 3D and 6 nm in 2D.