Effect of Nanoparticles on Electron and Thermoelectric Transport

0 1.Department of Electrical Engineering, University of California , Santa Cruz, CA 95064 , USA . 2.Department of Physics , University of California , Santa Cruz, CA 95064 , USA . 3.Electrical and Computer Engineering Department , University of California , Santa Barbara, CA 93106 , USA . 4.Materials Department , University of California , Santa Barbara, CA 93106, USA. 5. 1 Zebarjadi, Esfarjani, Shakouri, Bian, Bahk, Zeng, Bowers, Lu, Zide, and Gossard Recent experimental results have shown that adding nanoparticles inside a bulk material can enhance the thermoelectric performance by reducing the thermal conductivity and increasing the Seebeck coefficient. In this paper we investigate electron scattering from nanoparticles using different models. We compare the results of the Born approximation to that of the partial-wave method for a single nanoparticle scattering. The partial-wave method is more accurate for particle sizes in the 1 nm to 5 nm range where the point scattering approximation is not valid. The two methods can have different predictions for the thermoelectric properties such as the electrical conductivity and the Seebeck coefficient. To include a random distribution of nanoparticles, we consider an effective medium for the electron scattering using the coherent potential approximation. We compare various theoretical results with the experimental data obtained with ErAs nanoparticles in an InGaAlAs matrix. Reasonably good agreement is found between the measured and theoretical electrical conductivity and Seebeck data in the 300 K to 850 K temperature range. - In recent years, along with advancement in materials synthesis, it has been possible to embed nanoparticles with controlled size in bulk materials. Such structures can be used for various applications such as thermoelectric power generators1 and solar cells.2 The advantage of incorporating nanoparticles inside thermoelectric materials is to reduce the lattice thermal conductivity3 and enhance the Seebeck coefficient due to electron filtering.4 Adding nanoparticles also reduces the electrical conductivity. Therefore the size and the material forming the nanoparticles should be chosen in such a way that they scatter phonons more effectively than electrons. (Received July 8, 2008; accepted December 31, 2008; published online January 28, 2009) Scattering of electrons by localized defect potentials,5 by space-charge regions,6 and by neutral impurities7 have been widely studied. Busch and Soukoulis8 investigated the transport of a classical wave in a random medium composed of dielectric spheres. They obtained the mean free path of the transport and the energy transport velocity, which were in agreement with the experimental results. Sheng9 calculated the electronic transport in granular metal films. His theory is extendable to the case of metallic spherical nanoparticles inside a host matrix. Kim and Majumdar10 calculated the phonon scattering cross section of spherical nanoparticles within the Born approximation, and found an oscillatory behavior in the scattering cross section, presumably due to acoustic mismatch. Recently, Faleev and Leonard11 showed enhancement of the power factor of semiconductors with metallic nanoinclusions at high doping concentrations. They have used the partial-wave technique to calculate the scattering rates from nanoparticles; however, the results are based on Eq. 35 in their paper, which is not correct (see Eq. 6 of this paper). In contrast to their prediction, we find that there is a big difference between a well potential and a barrier potential. In this paper, we consider finite-size nanoparticles inside a host semiconductor and investigate the effect of adding nanoparticles on the thermoelectric transport using three different methods. We use the Born approximation and the partial-wave technique to calculate the scattering cross section and transport properties in the dilute limit (low concentration of nanoparticles). For higher concentrations we use the coherent potential approximation (CPA),12,13 which is known as the best single-site approximation for a disordered system. The predictions of these models of thermoelectric transport are studied in this work. MODEL AND THEORY Single Nanoparticle Scattering Here we develop the theory to calculate the scattering cross section of a single nanoparticle. The theory is applicable to the case of any arbitrary spherically symmetric potential. Born Approximation The Born approximation is based on the perturbation theory and therefore has the intrinsic assumption that the potential of the scatterer is weak. This approximation works well for energies which are several times the barrier height. For a square barrier/well potential, the Born approximation yields the following results for the differential scattering cross section (r(h)), the total scattering cross section (r), and the momentum cross section (rm), respectively.14 g 2ka sin rh sin hdh cos h sin hdh where, a is the radius of the nanoparticle, V0 is the barrier/well height, k is the wavevector of the incident electron, and m* is its effective mass in the host material. As can be seen from these formulas, the sign of the potential (barrier or well) does not matter, and the strength of the nanoparticle potential (and its size, in the high-energy limit) comes as a multiplicative factor. The Partial-Waves Method The partial-waves method is a way to calculate the scattering from a spherically symmetric potential exactly. The details of the method can be found in standard textbooks.14 The result for the total cross section is well known. We find for the momentum cross section the expression shown in Eq. 6, which is in contrast to the (incorrect) results of Faleev and Leonard. rh k12 X1 2l 1eidl sin dlPlcos h 2; where dl is called the phase shift of the lth partial wave (l being the angular momentum quantum number) and / is the wavefunction of the electron inside the nanoparticle. For a known potential shape and a given energy, the solution of the Schrodinger equation can be found numerically (for example, by the Numerov method) inside the nanoparticle. Then yl is calculated by matching the slope of the wavefunction at the boundary between the nanoparticle and the host material. We use the shooting method to find the wavefunction within the cutoff radius of the nanoparticle. The solution is exact and can be applied to any arbitrary spherically symmetric potential of finite range. Scattering from a Random Medium In the dilute limit, when the nanoparticles are far apart, each nanoparticle can be treated independently; however, when the concentration of nanoparticles increases, this approximation ceases to be valid. We need to add the effect of the nanoparticles volume, meaning that the electron wave sees a random medium instead of individual scatterers. One approach to solve such a problem is to use an effective medium theory. Instead of a random potential due to na (...truncated)