Method of producing thermoelectric material

Abstract

A process for manufacturing a thermoelectric material having a plurality of grains and grain boundaries. The process includes determining a material composition to be investigated for the thermoelectric material and then determining a range of values of grain size and/or grain boundary barrier height obtainable for the material composition using current state of the art manufacturing techniques. Thereafter, a range of figure of merit values for the material composition is determined as a function of the range of values of grain size and/or grain boundary barrier height. And finally, a thermoelectric material having the determined material composition and an average grain size and grain boundary barrier height corresponding to the maximum range of figure of merit values is manufactured.

Claims

1. A thermoelectric material comprising: a first matrix phase; an inter-grain phonon scattering second phase comprising a plurality of oxide nanoparticles; and a plurality of third phase grain boundaries, wherein said plurality of third phase grain boundaries are zinc antimony modified grain boundaries consisting of zinc and antimony with an average thickness within a range of 2-75 nm and wherein said plurality of third phase grain boundaries have a grain boundary barrier height within a range of 10-300 meV.

2. The thermoelectric material of claim 1, wherein said first matrix phase has an average grain size within a range of 5-100 nm.

3. The thermoelectric material of claim 2, wherein said average grain size is within a range of 5-50 nm.

4. The thermoelectric material of claim 3, wherein said average grain size is within a range of 5-25 nm.

5. The thermoelectric material of claim 1, wherein said first matrix phase is a bismuth antimony telluride phase.

6. The thermoelectric material of claim 1, wherein said plurality of oxide nanoparticles have an average diameter within a range of 2-100 nm.

7. The thermoelectric material of claim 6, wherein said average diameter is within a range of 4-50 nm.

8. The thermoelectric material of claim 7, wherein said average diameter is within a range of 6-14 nm.

9. The thermoelectric material of claim 6, wherein said plurality of oxide nanoparticles are zinc oxide nanoparticles.

10. The thermoelectric material of claim 1, wherein said average thickness is within a range of 5-70 nm.

11. The thermoelectric material of claim 10, wherein said average thickness is within a range of 10-70 nm.

12. The thermoelectric material of claim 11, wherein said average thickness is within a range of 15-65 nm.

13. The thermoelectric material of claim 12, wherein said average thickness is within a range of 27-61 nm.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1a-1d are schematic illustrations of: (a) the grain structure of a prior art thermoelectric material; (b) a schematic illustration of a thermoelectric material having altered grain boundaries according to an embodiment of the present invention; (c) a schematic illustration of a prior art nanocomposite thermoelectric material; and (d) a schematic illustration of a nanocomposite thermoelectric material having modified grain boundaries according to an embodiment of the present invention;

(2) FIG. 2 is a graph illustrating calculated electron mean free path, electron wavelength, and carrier percentage occupation as a function of dimensionless electron energy;

(3) FIG. 3 is a schematic illustration of a model for treatment of grain boundary barrier height according to an embodiment of the present invention;

(4) FIG. 4 is a graph illustrating effect of grain boundary properties on electrical conductivity;

(5) FIG. 5 is a graph illustrating effect of grain size on lattice (Kl) and electronic (Ke) thermal conductivity;

(6) FIG. 6 is a graph illustrating effect of grain boundary properties on total thermal conductivity;

(7) FIG. 7 is a graph illustrating effect of grain boundary properties on Seebeck coefficient;

(8) FIG. 8 is a graph illustrating calculated normalized Seebeck coefficient as a function of electron energy;

(9) FIG. 9 is a graph illustrating effect of grain boundary properties on dimensionless figure of merit (ZT);

(10) FIG. 10 is a graph illustrating Seebeck coefficient for a bulk (Bulk) and nanocomposite (NC) thermoelectric material as a function of grain boundary barrier height;

(11) FIG. 11 is a graph illustrating electrical conductivity and electronic thermal conductivity for a bulk (Bulk) and nanocomposite (NC) thermoelectric material as a function of grain boundary barrier height;

(12) FIG. 12 is a graph illustrating ZT as a function of grain boundary barrier height;

(13) FIG. 13 is a flowchart for a process according to an embodiment of the present invention;

(14) FIG. 14 is a flowchart for a process according to an embodiment of the present invention;

(15) FIG. 15 is a flowchart for a process according to an embodiment of the present invention;

(16) FIG. 16 is a flowchart for a process according to an embodiment of the present invention;

(17) FIG. 17 is a flowchart for a process according to an embodiment of the present invention;

(18) FIG. 18 is a schematic illustration of a computer with a processor according to an embodiment of the present invention;

(19) FIG. 19 is a series of graphs illustrating: (A) figure of merit ZT versus temperature for a bismuth antimony telluride matrix with zinc oxide nanoparticles and zinc antimony modified grain boundaries (BAZTfilled square data points) and a bismuth antimony telluride matrix with no zinc oxide nanoparticles or modified grain boundaries (BATopen square data points); (B) electrical conductivity versus temperature for the BAZT and BAT materials; (C) Seebeck coefficient versus temperature for the BAZT and BAT materials; (D) power factor versus temperature for the BAZT and BAT materials; and (E) thermal conductivity versus temperature for the BAZT and BAT materials;

(20) FIG. 20 is a transmission electron microscopy (TEM) image illustrating: (A) a bismuth antinomy telluride matrix with zinc antinomy modified grain boundaries; and (B) zinc oxide nanoparticles within the bismuth antinomy telluride matrix; and

(21) FIG. 21 is a graph illustrating the x-ray diffraction (XRD) results of the BATZ material.

DETAILED DESCRIPTION OF THE INVENTION

(22) The present invention discloses a process for determining an optimum range of compositions for a thermoelectric material system, within which the material system may exhibit generally high figure of merit values. As such, the process has utility for improving the efficiency of experimental design and production of thermoelectric materials.

(23) The process for determining an optimum range of compositions for a thermoelectric material system considers a variety of relevant factors, parameters and the like in order to determine which material systems should be considered and/or which range of compositions should be studied in more detail. A thermoelectric material exhibiting a dimensionless high figure of merit (ZT) needs to possess a high Seebeck coefficient (S) for high voltage generation, a low electrical resistivity () to minimize Ohmic losses and a low thermal conductivity () to minimize heat conduction.

(24) The relationship between ZT, S, , and can be expressed as:
ZT=S.sup.2T/Eqn 1
and/or as:

(25) $\begin{array}{cc}\mathrm{ZT}=\frac{S^{2}T}{\left(k_{\mathrm{el}}+k_{\mathrm{ph}}\right)}& \mathrm{Eqn}2\end{array}$
where .sub.el and .sub.ph are the electronic and phonon contribution to the overall thermal conductivity k.

(26) Typically, S, , and are interdependent with an increase of the Seebeck coefficient resulting in an increase in electrical resistivity, whereas a decrease in the electrical resistivity results in an increase of the thermal conductivity. At least one approach for obtaining high figure of merit values has investigated the insertion of nanoparticles within a thermoelectric material (U.S. Pat. No. 7,309,830). Materials using this approach can result in phonons being scattered by the nanoparticles, thereby reducing the lattice thermal conductivity while leaving the electrical resistivity and Seebeck coefficient for the thermoelectric host matrix unchanged.

(27) Elemental substitutions, also known as atomic substitutions, in potential thermoelectric materials have imperfections on the order of 1 angstrom (). Thus alloying additions can result in the scattering of short-wavelength phonons much more effectively than mid- and long-wavelength phonons. Therefore, mid- and long-wavelength phonons dominate the heat conduction in alloys and thermoelectric materials that have been doped with other elements not originally within the starting material. In the alternative, the inclusion of additions such as nanoparticles in the size range of phonon wavelengths introduces another scattering mechanism that affects mid- and/or long-wavelength phonons, thereby providing an opportunity to reduce the thermal conductivity of such materials below the alloy limit. However, which nanoparticles with respect to their composition, size and size distribution, and which host matrix the nanoparticles should be added to has heretofore been a difficult task to predict. In response to the difficulty in predicting successful thermoelectric material systems, a process to perform just this task has been developed by Banerjee et al. (U.S. Pat. No. 7,734,428).

(28) An embodiment of the current process includes determining a material composition to be investigated for the thermoelectric material and determining a range of values for a grain related property that is obtainable for the material composition using state of the art manufacturing techniques. Once the material composition and the range of values for the grain related property have been determined, a plurality of Seebeck coefficients for the material composition as a function of the range of values can be calculated. In addition, a plurality of electrical resistivity values and a plurality of thermal conductivity values for the material composition as a function of the range of values for the grain related property can also be calculated.

(29) It is appreciated that once the plurality of Seebeck coefficients, electrical resistivity values, and thermal conductivity values have been determined, a range of figure of merit values as a function thereof can be calculated and a generally maximum range of figure of merit values can be determined, such values being a function of the range of values of the grain related property. Naturally, once the maximum range of figure of merit values has been determined, a thermoelectric material having the determined material composition and the grain related property(ies) corresponding to the maximum range of figure of merit values is manufactured.

(30) In the alternative to the above outlined embodiment, a plurality of material compositions can be investigated in this manner and a single material composition or a limited range of compositions having a potential and desired ZT are manufactured.

(31) The grain related property can include any grain related property known to those skilled in the art, illustratively including grain size, grain boundary barrier height, and the like. For the purposes of the present invention, the term grain size is defined as the average mean diameter of grains within a thermoelectric material obtained through any method and/or technique known to those skilled in the art. For example and for illustrative purposes only, a grain size can be determined by taking a statistical average of a plurality of grain diameters from a metallographic cross-section of the material with a single grain diameter obtained by averaging at least two linear and orthogonal measurements across a given grain.

(32) Also for the purposes of the present invention, the term grain boundary barrier height is defined as the energy potential of a grain boundary that will scatter an electron having less energy than the potential and allow an electron having more energy than the potential to pass therethrough.

(33) The material composition to be investigated can be a bulk thermoelectric material composition, or in the alternative, a nanocomposite thermoelectric material composition. It is appreciated that the term bulk thermoelectric material refers to a polycrystalline material without the presence of second phase particles such as nanoparticles of an insulating type material. In the alternative, the term nanocomposite thermoelectric material refers to a bulk thermoelectric material having second phase particles such as nanoparticle insulating material inclusions, e.g. nanoparticle inclusions such as silicon oxide, zinc oxide, and the like.

(34) The range of values for grain size of the material composition to be investigated can be between 5 and 100 nanometers (nm) while the range of values of grain boundary barrier height can be between 10 and 300 milli-electron volts (meV). In addition, the grain size of the manufactured thermoelectric material can be obtained by consolidating a plurality of nanoparticles having a mean diameter that is less than or generally equal to the final grain size of the material. The grain boundary barrier height of the manufactured thermoelectric material can be altered and/or obtained by doping of the material and/or altering a surface of the plurality of nanoparticles used to manufacture the thermoelectric material. In some instances, the surface of the plurality of nanoparticles is altered by applying a coating thereon before the nanoparticles are consolidated to produce the thermoelectric material.

(35) Not being bound by theory, it is appreciated that a grain boundary is a result of and/or forms from a crystallographic misalignment between adjacent grains. In addition, the misalignment results in a residual electric charge across the grain boundary which can produce an electrostatic potential commonly referred to as an interfacial barrier and/or grain boundary barrier height which can be measured using AC impedance. As a first approximation, the magnitude of this interfacial barrier, also known as the grain boundary barrier height, can be calculated from the expression:

(36) $\begin{array}{cc}{E}_b=\frac{{\mathrm{eN}}_t^2}{8\mathrm{.Math.}{N}_D}& \mathrm{Eqn}3\end{array}$
where N.sub.t is the number density of traps, is the permittivity and N.sub.D is the doping concentration. The trap density is generally unknown and can vary widely, however assuming a generally high doping level and reported values for N.sub.t in the range of 10.sup.11-10.sup.13 cm.sup.3, an E.sub.b of around 25 meV can be calculated.

(37) The process can provide a thermoelectric material as schematically illustrated in FIGS. 1a and 1d. In particular, FIG. 1b illustrates a bulk thermoelectric material 100 having altered and/or engineered grain boundaries 110 when compared to the grain boundaries 110 of the material 100 illustrated in FIG. 1a. In addition, FIG. 1d illustrates a nanocomposite thermoelectric material 200 having altered and/or engineered grain boundaries 210 in addition to nanoparticle inclusions 205 when compared to the material 200 having the nanoparticles 205 and grain boundaries 210.

(38) The grain size of the thermoelectric material 100 and/or 200 can be altered and/or engineered, e.g. by using nanoparticles with a desired average size to consolidate and manufacture the material. In addition, local electronic and thermal properties of the grain boundaries can be altered and/or engineered by controlling the interfacial composition between the grains, that is the interfacial composition of the grain boundaries. For example and for illustrative purposes only, a second phase can be engineered to be present at the interface between the grains such as Pb.sub.0.75Sn.sub.0.25Se coatings on Pb.sub.0.75Sn.sub.0.25Te; CoSb.sub.3 coatings on La.sub.0.9CoFe.sub.3Sb.sub.12; and alkali-metal salt coatings on (Bi.sub.0.2Sb.sub.0.8).sub.2Te.sub.3. In fact, results from CoSb.sub.3/La.sub.0.9CoFe.sub.3Sb.sub.12 and coated (Bi.sub.0.2Sb.sub.0.8).sub.2Te.sub.3 materials have shown moderate improvements in the figure of merit ranging from 15-30%.

(39) In order to incorporate a grain related property into a modeling and/or manufacturing process, the scattering behavior of electrons, holes and/or phonons within a material can be useful. Not being bound by theory, a theoretical simulation can be based on the Boltzmann equation with relaxation time approximation. For example, a modified Callaway model with respect to the lattice of a thermoelectric material can be incorporated with scattering of phonons through grain boundaries, defects, nanoparticles, and the like provided by Equation 4 below:
.sub.c.sup.1=.sub.B.sup.1+.sub.U.sup.1+.sub.N.sup.1+.sub.A.sup.1+.sub.D.sup.Eqn 4
where corresponds to scattering time and the subscripts B, U, N, A and D correspond to boundary, Umpklamp, normal, alloy, and nanoparticle, respectively, related scattering.

(40) With respect to carriers, that is electrons and holes, Equation 5 can be used where Op, DOp, and DAp represent optical phonon, deformation potential of optical phonon, and deformation potential of acoustic phonon related scattering.
.sub..sup.1=.sub.Op.sup.1+.sub.DOp.sup.1+.sub.DAp.sup.1Eqn 5

(41) In addition to scattering time, the total electrical conductivity can be expressed as a summation of the contributions from both electron and hole bands, while the overall Seebeck coefficient can be obtained through weighting each band's contribution using a normalized electrical conductivity. In order to obtain the electronic thermal conductivity, the electronic thermal conductivity from the Lorentz number (L) can be obtained using Equations 6-8 below. In particular, Equation 6 is an expression of the total electrical conductivity (), Equation 7 is an expression of the overall Seebeck coefficient, and Equation 8 is an expression for the electronic thermal conductivity. It is appreciated that the bipolar thermal conductivity contribution to the electronic thermal conductivity must also be considered and that this type of conduction occurs when carriers moving between different bands carry heat via the Peltier effect and as such can still transport heat even if the net electric current is zero.

(42) $\begin{array}{cc}=\underset{i}{\overset{e,h}{.Math.}}{}_i& \mathrm{Eqn}6\\ S=\underset{i}{\overset{e,h}{.Math.}}\frac{S_i{}_i}{}& \mathrm{Eqn}7\\ k_e={\left(\frac{k_B}{e}\right)}^2\left(\underset{i}{\overset{e,h}{.Math.}}{L}_i+L_b\right)T& \mathrm{Eqn}8\end{array}$

(43) In addition to the above, the nature of grain boundary scattering exhibited by carriers can be estimated from the electron wavelength and electron mean free path (MFP) and the cumulative distribution function of the electron occupation number versus electron energy can provide the percentage of electrons that have energy less than a certain value. In particular, Equations 9-13 afford for the electron MFP, electron wavelength, and carrier percentage occupation as a function of dimensionless electron energy shown in FIG. 2 where the electron MFP can be calculated using the expression 1= where and are provided by Equations 11 and 4, respectively, g is the density of state function and a is equal to the inverse of the bandgap for the material (1/E.sub.g), in some instances referred to as the parabolicity factor.

(44) $\begin{array}{cc}g=\frac{{}_0^{E/k_{b}T}g\left(E\right)dE}{{}_0^{}g\left(E\right)dE}& \mathrm{Eqn}9\\ g\left(E\right)=v^{2}D\left(E\right)\left(-\frac{f}{E}\right)& \mathrm{Eqn}10\\ v=\frac{{\left(2E\left(1+E\right)/m_c^{*}\right)}^{0.5}}{\left(1+2E\right)}& \mathrm{Eqn}11\\ D\left(E\right)=\frac{\sqrt{2}{\left(m_d^{*}\right)}^{1.5}}{{}^2{}^3}E\left(1+2E\right)\left(1+E\right)& \mathrm{Eqn}12\\ =\frac{2}{m_c^{*}v}=\frac{2\left(1+2E\right)}{\sqrt{2m_c^{*}E\left(1+E\right)}}& \mathrm{Eqn}13\end{array}$

(45) As shown in FIG. 2, the majority of electrons have a MFP less than 30 nanometers which is comparable and/or of the same order of magnitude of a grain size of between 20 to a few hundred nanometers. As such, FIG. 2 confirms that a majority of carriers will experience multiple scattering due to the grain boundaries and memory of a previous collision by a carrier will be retained by the carrier when it reaches another portion of the grain interface, i.e. each scattering point on a grain boundary is not independent from others. In addition, since the dominant electron wavelength is approximately 28 nm and an assumed grain boundary width of 1 to 2 nm is much smaller, there is no possibility of diffuse scattering of electrons. Finally, the electron MFP of less than 30 nanometers is comparable to a hole wavelength of 28 nm which implies that the Boltzmann equation is at the edge of its validity using this process. As such, it is appreciated that other expressions with respect to relaxation time approximation can be used for the basis of a theoretical simulation that incorporates grain boundary effect in determining various properties of a thermoelectric material and still fall within the scope of the present invention.

(46) Turning now to the actual effect of grain boundary properties on thermoelectric characteristics, FIG. 3 provides a model of a grain having a grain boundary with a width w and a grain boundary potential barrier of E.sub.b. In addition, the grain size has a dimension of L which naturally separates the grain boundary on opposing sides of the grain. As illustrated in the figure, if an electron has an energy of E, the electron will pass through the grain boundary barrier height if E>E.sub.b and will scatter if E<E.sub.b.

(47) Not being bound by theory, assuming T(E) is a transmission probability of an electron passing through a grain boundary barrier height and there are N grain boundaries, the MFP of the electron due to scattering by the grain boundary can be expressed as Equation 14 when N is assumed to be infinity.

(48) $\begin{array}{cc}{}_{\mathrm{grainboundary}}=\underset{n=1}{\overset{N.fwdarw.}{.Math.}}{T\left(E\right)}^n\left(1-T\left(E\right)\right)\mathrm{nL}=\frac{T\left(E\right)L}{1-T\left(E\right)}& \mathrm{Eqn}14\end{array}$
which further provides a relaxation time of:
.sub.B=.sub.grainboundary/Eqn 15
where is given by:

(49) $\begin{array}{cc}{}_B=\frac{L}{v}\left(1+\frac{4\frac{E}{E_b}.Math.1-\frac{E}{E_B}.Math.}{{\sinh }^2.Math.\sqrt{\frac{2m_c^{*}{E}_B{w}^2}{{}^2}.Math.1-\frac{E}{E_B}}.Math.}\right)& \mathrm{Eqn}16\end{array}$

(50) In order to better understand the effect of grain related properties on the thermoelectric material behavior, and based on the model shown in FIG. 3, the effect of grain size on electrical conductivity was examined with results shown in FIG. 4. The width w of the grain boundaries was assumed to be constant at 2 nm while the grain boundary barrier height was varied from 20 meV to 300 meV. In addition, the electrical conductivity of a bulk thermoelectric material and a nanocomposite thermoelectric material was investigated and is shown in the graph. In the case of the nanocomposite thermoelectric material, SiO.sub.2 nanoparticles of 3 nm diameter were used for the calculations. It is appreciated that FIG. 4 illustrates that with increasing grain size, the electrical conductivity increases, which can be explained due to decreasing probability of scattering events. In addition, with the inclusion of ceramic nanoparticles within the material, significantly lower electrical conductivities were observed. Finally, varying the grain boundary barrier height significantly affects the conductivities of both the bulk thermoelectric material and the nanocomposite thermoelectric material. It is appreciated that this effect is stronger at smaller grain sizes simply due to the fact that smaller grains increase the number of scattering events and thus reduce a carrier MFP.

(51) Turning now to FIG. 5, a graph illustrating the lattice and electronic thermal conductivity as a function of grain size is shown. Similar to FIG. 4, grain boundary scattering clearly affects both lattice and electronic thermal conductivity with the most significant effect occurring for grain sizes below 25 nm. In addition, FIG. 6 illustrates that total thermal conductivity illustrates a similar behavior to electrical conductivity which provides evidence that reduction in grain size for a thermoelectric material can be an effective way of reducing the material's thermal conductivity.

(52) Regarding the Seebeck coefficient for a thermoelectric material, FIG. 7 illustrates a complicated relationship between the Seebeck coefficient, grain size, and grain boundary barrier height. In particular, and for bulk thermoelectric material, the highest Seebeck coefficient occurred for a grain boundary barrier height of 60 meV while for a nanocomposite thermoelectric material, the highest Seebeck coefficient was observed for a grain boundary barrier height of 20 meV. It is appreciated that the difference between the two materials and the associated Seebeck coefficient can be the result of filtering of low energy electrons within the grains of the nanocomposite thermoelectric material. In addition, FIG. 8 provides a typical normalized Seebeck coefficient distribution as a function of electron energy. As shown by this figure, a maximum value or maximum range of values for the Seebeck coefficient does not result from electron energies that are too low or too high. Stated differently, there is an intermediate value or range of values for electron energy that provides a desired Seebeck coefficient. In addition, low energy electrons pose a negative impact to the Seebeck coefficient.

(53) Based on these figures and their teachings, it is clear that small grains with high grain boundary barrier potentials, for example E.sub.b=300 meV, have the least effect on the Seebeck coefficient since such high potential barriers can filter even high energy electrons. On the other hand, FIG. 7 illustrates that the Seebeck coefficient behavior flipped or was inverted for the nanocomposite material versus the bulk material when the grain boundary barrier height was 20 meV and 60 meV, respectively. Not being bound by theory, this is postulated to be due to the Seebeck coefficient distribution being different for the two materials, and depending on the location of the peak of normalized Seebeck coefficient as shown in FIG. 8, either 20 meV or 60 meV can be more effective in increasing the Seebeck effect.

(54) Regarding the dimensionless figure of merit ZT, FIG. 9 provides a comparison of ZT as a function of grain size, bulk thermoelectric material, nanocomposite thermoelectric material, and grain boundary barrier height. From this figure, it is appreciated that grain sizes below 25 nanometers can provide a dramatic improvement in the performance of bulk thermoelectric material and nanocomposite thermoelectric material. In addition, the grain boundary barrier height can significantly alter the ZT for a particular material. For example, at lower grain sizes, the ZT for the bulk and nanocomposite materials overlap, thereby suggesting that the benefit of adding second phase nanoparticles to a thermoelectric material can be diminished in cases where the grain boundary barrier potential is high due to impurities, doping, and the like.

(55) The effect of ceramic nanoparticle inclusions within a bulk thermoelectric material on grain boundary barrier height can also be of interest with FIG. 10 illustrating a graphical representation of Seebeck coefficient as a function of grain boundary barrier height for a bulk thermoelectric material (Bulk) and a nanocomposite thermoelectric material (NC). As shown in FIG. 10, and for which a grain size of 30 nm was assumed, smaller grain boundary barrier heights are preferred with potentials over 100 meV virtually having no effect on the Seebeck coefficient of the material. In addition, FIG. 11 shows or illustrates the same behavior with respect to electrical conductivity and electronic thermal conductivity and FIG. 12 provides a graph illustrating the effect of grain boundary barrier height on ZT for bulk thermoelectric material and nanocomposite thermoelectric material.

(56) It is appreciated that FIG. 12 could lead to the conclusion that lower grain boundary barrier heights are desired in all cases in order to achieve an increase in ZT for any thermoelectric material. However, such a conclusion can be false, for example when the grain size is also considered as discussed above in relation to FIG. 9.

(57) Turning now to FIG. 13, a process for manufacturing a thermoelectric material is shown generally at reference numeral 10. The process 10 includes determining a thermoelectric material composition to be investigated at step 100, followed by determining a range of ZT values for the thermoelectric material composition at step 110. Given the range of determined ZT values determined at step 110, step 120 determines a range of maximum ZT values and a thermoelectric material is manufactured at step 130.

(58) FIG. 14 illustrates a flowchart for a process according to another embodiment of the invention at reference numeral 20. The process 20 includes determining a thermoelectric material composition (C.sub.i) to be investigated at step 200, followed by determining a range of matrix grain sizes that are obtainable for C.sub.i at step 210 and determining a range of grain boundary barrier energy heights obtainable for C.sub.i using current state of the art manufacturing techniques at step 220. The determined thermoelectric material composition may or may not include inter-grain phonon scattering particles. In addition, step 210 or step 220 may or may not include determining a range of particle sizes obtainable for inter-grain phonon scattering particles using current state of the art manufacturing techniques.

(59) At step 230, ZT values are calculated for C.sub.i as a function of the obtainable range of matrix grain sizes and obtainable range of grain boundary heights that were determined in steps 210 and 220. Thereafter, a matrix grain size and grain boundary height is selected as a function of the calculated ZT values for C.sub.i at step 240. In some instances, step 240 can include selecting a desired average grain size for the matrix phase and a desired particle size for inter-grain phonon scattering particles as a function of the calculated ZT values. Finally, a thermoelectric material is manufactured at step 250, the thermoelectric material having the composition C.sub.i from step 200, the selected matrix grain size, grain boundary height and/or inter-grain phonon scattering particles with an average particle size that is equivalent to the desired particle size for the inter-grain phonon scattering second phase from step 240. For the purposes of the present invention, the term equivalent is defined to be within 10%, i.e. the average particle size of the inter-grain phonon scattering particles is within +/10% of the desired particle size.

(60) Another flowchart illustrating a process according to another embodiment of the present invention is shown generally at reference numeral 30 in FIG. 15. The process 30 includes determining a thermoelectric material composition (C.sub.i) to be investigated at step 300 followed by determining a range of matrix grain sizes and a range of grain boundary heights that are obtainable for a thermoelectric material having composition C.sub.i at steps 302 and 304, respectively. It is appreciated that the composition C.sub.i can include a solid solution material, i.e. a material having a relatively uniform composition without the addition of second phase particles, second phase precipitates, and the like. In the alternative, the composition C.sub.i can be a precipitation type material such as a precipitation type alloy in which second phase precipitates/particles are formed within the alloy/material during processing. In another alternative, the composition C.sub.i can be a composite type material in which second phase particles are present and/or added to a matrix material. The matrix material can be a metal, alloy, semiconductor, ceramic, e.g. an oxide, nitride, etc., and the like. Also, the second phase particles can be a metal, alloy, semiconductor, ceramic, e.g. an oxide, nitride, etc., and the like.

(61) The process 30 also includes calculating Seebeck coefficients for the C.sub.i composition as a function of the obtainable range of matrix grain sizes and obtainable grain boundary heights at step 306. At step 308, the process 30 includes calculating electrical resistivity values for C.sub.i as a function of the obtainable range of matrix grain sizes and grain boundary heights. At step 310, the calculation of thermal conductivity values for C.sub.i as a function of the obtainable range of matrix grain sizes and grain boundary heights is performed. Next, the calculation of ZT values for C.sub.i as a function of the calculated Seebeck coefficients, electrical resistivity values, and thermal conductivity values is performed at step 312.

(62) Once the ZT values have been calculated, a maximum range of the calculated ZT values for C.sub.i is determined at step 314 and step 316 includes determining a matrix grain size and a grain boundary height that is within the maximum determined range from step 314. At step 318, a thermoelectric material having the composition C.sub.i is manufactured, the material having the determined matrix grain size and grain boundary height from step 316. It is appreciated that in order to manufacture the thermoelectric material at step 318, material corresponding to the composition C.sub.i is provided and processed to produce the material.

(63) Referring to FIG. 16, another process according to an embodiment of the present invention is shown generally at reference numeral 40. The process 40 includes determining a thermoelectric material composition C.sub.i to be investigated at step 400, followed by determining a range of matrix grain sizes and a range of grain boundary heights for the composition C.sub.i to be investigated at steps 410 and 420, respectively. At step 430, ZT values for C.sub.i are calculated as a function of the determined ranges of matrix grain sizes and grain boundary heights from steps 410 and 420. It is appreciated that the ZT values can be calculated as a function of Seebeck coefficients, electrical resistivity values, and thermal conductivity values for C.sub.i that are also a function of the range of matrix grain sizes and grain boundary heights determined in steps 410 and 420, respectively.

(64) A desired range of the calculated ZT values for C.sub.i is determined at step 440, e.g. a desired maximum range of the calculated ZT values can be determined. Based on the determined desired range of the calculated ZT values, the range of matrix grain sizes and grain boundary heights that fall within this range are determined at step 450. Then, matrix grain sizes and grain boundary heights that are obtainable using current state of the art production techniques and that fall within the determined desired range of calculated ZT values for C.sub.i are determined at step 460.

(65) A powder that has the C.sub.i composition and an average particle diameter that is equivalent to an obtainable matrix grain size that falls within the grain sizes of step 460 is provided. It is appreciated that the powder can be a homogeneous powder, i.e. a powder made of individual powder particles that have a generally uniform composition with each other or, in the alternative, a non-homogeneous composition, i.e. the powders being made from first component powder particles and second component powder particles. As stated above, the first component powder particles can be in the form of be a metal, alloy, semiconductor, ceramic, e.g. an oxide, nitride, etc., and the like. Also, the second component particles can be in the form be a metal, alloy, semiconductor, ceramic, e.g. an oxide, nitride, etc., and the like. In addition, third component particles, fourth component particles, etc. can be included in the powder having the C.sub.i composition at step 470, the third, fourth, etc. particles being of the form of be a metal, alloy, semiconductor, ceramic, e.g. an oxide, nitride, etc., and the like. Also, the powders, particles, etc., may or may not be in the form of nanoparticles such a nano-spheres, nano-rods, nano-discs nano-ellipsoids, and the like.

(66) The powder having the C.sub.i composition is processed to produce a component. For example, a process for processing the powders is shown in FIG. 17 at reference numeral 60. The process 60 includes taking or obtaining the powders of the thermoelectric material at step 600, the powders having desired chemical and/or physical properties/characteristics. For example, the powders can include a mixture of first component particles P.sub.1, second component particles P.sub.2, etc. as shown at 602. In addition, the powders can have a desired particle diameter or desired particle size as shown at step 604 and/or a desired surface modification as shown at step 606. It is appreciated that the desired particle diameter is determined from a process disclosed herein, e.g. the processes illustrated in FIGS. 13-16.

(67) It is also appreciated that the particle diameter of the first component particles P.sub.1 can be the same or different than the particle diameter of the second component particles P.sub.2. Finally, it is appreciated that the terms particle diameter and matrix grain size refer to an average particle diameter and average matrix grain size, respectively, as is known to those skilled in the art. The variation in particle diameters and matrix grain sizes can have a half width at half maximum height (HWHM) to modal diameter ratio between 0.4-0.6 for a differential distribution as is known to those skilled in the art. In other instances, the HWHM/modal diameter ratio is between 0.3-0.4 or, in the alternative, between 0.2-0.3. In the alternative, the HWHM/modal diameter ratio is between 0.6-0.7 or, in the alternative, between 0.7-0.8.

(68) The process 60 further includes compaction of powders to produce a thermoelectric component at step 610. The step 610 can be a compaction by any means or method known to those skilled in the art, illustratively including sintering, hot isostatic pressing (HIP), cold isostatic pressing (CIP), die pressing, continuous particle or powder processing (CPP), etc. as shown at 612. In some instances, the component is further treated at step 620 in order to obtain a desired matrix grain size and/or desired grain boundary barrier height. For example and for illustrative purposes only, the treatment at step 620 can include a thermal treatment, a mechanical treatment, and/or a thermal-mechanical treatment. Other treatments such as exposure to electromagnetic radiation, nuclear radiation, and the like can be included.

(69) Regarding the calculation of the various Seebeck coefficients, electrical resistivity values, thermal conductivity values, ZT values, and the like, FIG. 18 provides a schematic illustration of a computer at reference numeral 50. The computer 50 can include a CPU 500 with a plurality of components 502, 504, 506, . . . 516. For example and for illustrative purposes only, the CPU 500 can include a processor 502, memory 504, database 506, and the like. It is appreciated that the various components 502-516 are in communication with each other and with the CPU 500 as is known to those skilled in the art.

(70) The various calculations can include algorithms with respect to Equations 1-16, assumptions, electric constants and/or physical constants known to those skilled in the art. For example, the temperature T referenced in Equation 1 can be assumed to be room temperature. In the alternative, the calculations can assume a plurality of temperatures, e.g. temperatures ranging from 0 C. to 200 C. at increments of 1 C., increments of 5 C., increments of 10 C., and the like.

(71) In order to illustrate a sample calculation, but not limit the scope of the instant disclosure in any way, an example of a ZT calculation is provided below.

(72) The figure of merit (ZT) for a given grain size and grain barrier height combination j, is given by:
ZT=S.sup.2.Math.T.Math.C/(K.sub.e+K.sub.l)Eqn 17
where S is the Seebeck coefficient for the given grain size and grain boundary barrier height, C is the electrical conductivity, K.sub.e is the electron contribution to the overall thermal conductivity and K.sub.l is the phonon contribution to the overall thermal conductivityall for the given grain size and grain boundary barrier height. It is appreciated that the index j for the given grain size and grain boundary energy barrier height is not shown for convenience, unless needed for clarity. Stated differently, each of the expressions discussed are for the given grain size and grain boundary energy barrier height j.

(73) In order to properly define S, C, K.sub.ph and K.sub.e, a series of constants known to those skilled in the art are required and provided below. It should be appreciated that the values for the constants listed below are presented without units, as used in computer code for the calculation of ZT as a function of grain size and grain boundary energy height. However, it should also be appreciated that the units for the constants provided below would be known to one skilled in the art and be in accordance with units that afford calculation of the Seebeck coefficient in microvolts per meter (V/m), electrical conductivity in siemens per meter (S/m) and thermal conductivity in watts per meter kelvin (W/mK).

(74) General constants used in an example calculation include:
=3.14
.sub.B=1.3810.sup.23=Boltzmann constant
h.sub.c=1.05410.sup.34=Planck's constant
e.sub.v=1.610.sup.19=1 electron volt
e=1.610.sup.19=electron charge
m.sub.e=9.110.sup.31=electron effective mass
.sub.0=8.8510.sup.12=permittivity of air
e.sub.1=69.8.Math..sub.o.Math.4=high frequency dielectric constantEqn 18
e.sub.0=400.Math..sub.o.Math.4=static dielectric constantEqn 19
Also, density of states constants and expressions related to effective mass include:
N.sub.v=12=number of valleys in the electron bandstructure
E.sub.g=0.13.Math.e.sub.v1.0810.sup.4.Math.T.Math.e.sub.v=electron bandgapEqn 20
.sub.L=E.sub.g/(k.sub.BT)=bandgap in k.sub.BTEqn 21
=k.sub.BT/E.sub.g=inverse of bandgap in k.sub.BTEqn 22
e.sub.g=E.sub.g/e.sub.v=bandgap in electron voltEqn 23
m.sub.h1=0.0308.Math.m.sub.e=hole effective massEqn 24
m.sub.h2=0.441.Math.m.sub.e=hole effective massEqn 25
m.sub.h3=0.0862.Math.m.sub.e=hole effective massEqn 26
m.sub.e1=0.0213.Math.m.sub.e=electron effective massEqn 27
m.sub.e2=0.319.Math.m.sub.e=electron effective massEqn 28
m.sub.e3=0.0813.Math.m.sub.e=electron effective massEqn 29
md.sub.e=N.sub.v.sup.2/3.Math.(m.sub.e1.Math.m.sub.e2.Math.m.sub.e3).sup.1/3=density of state electron effective massEqn 30
md.sub.e1=(m.sub.e1.Math.m.sub.e2.Math.m.sub.e3).sup.1/3=density of state electron effective massEqn 31
md.sub.h=N.sub.v.sup.2/3.Math.(m.sub.e1.Math.me.sub.e2.Math.m.sub.e3).sup.1/3=density of state hole effective massEqn 32
md.sub.h1=(m.sub.e1.Math.m.sub.e2.Math.m.sub.e3).sup.1/3=density of state hole effective massEqn 33
Mc.sub.e=3/(1/m.sub.e1+1/m.sub.e2+1/m.sub.e3)=Total effective mass of electronEqn 34
Mc.sub.h=3/(1/m.sub.h1+1/m.sub.h2+1/m.sub.h3)=Total effective mass of holeEqn 35

(75) Fermi energy expressions include:

(76) $\begin{array}{cc}\mathrm{zF}=1.68000000000002000000{10}^{-9}.Math.T^3-2.32525000000003000000{10}^{-6}.Math.T^2+1.18722500000002000000{10}^{-3}.Math.T-2.36675820000002000000{10}^{-1}& \mathrm{Eqn}36\\ E_f=\mathrm{zF}.Math.e_v=\mathrm{Fermi}\mathrm{energy}\mathrm{in}{e}_v& \mathrm{Eqn}37\\ x_f=E_f/\left(k_B.Math.T\right)=\mathrm{Fermi}\mathrm{energy}\mathrm{in}{k}_B.Math.T& \mathrm{Eqn}38\\ F_e=\left(E_f-E_g/2\right)/\left(k_B.Math.T\right)=\mathrm{energy}\mathrm{of}\mathrm{electron}& \mathrm{Eqn}39\\ F_h=-\left(E_g/2+E_f\right)/\left(k_B.Math.T\right)=\mathrm{energy}\mathrm{of}\mathrm{hole}& \mathrm{Eqn}40\end{array}$
with Fermi functions:

(77) $\begin{array}{cc}{f}_e=1/\left(1+\exp \left(z-F_e\right)\right)& \mathrm{Eqn}41\\ d{1}_e=\frac{d{f}_e}{dz}=\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}& \mathrm{Eqn}42\\ f_h=1/\left(1+\exp \left(z-F_h\right)\right)& \mathrm{Eqn}43\\ d{1}_h=\frac{d{f}_h}{dz}=\frac{d(1/\left(1+\exp \left(z-F_h\right)\right)}{dz}& \mathrm{Eqn}44\end{array}$
and conductivity effective mass expressions:

(78) 0$\begin{array}{cc}L{1}_e={}_{-F_e}^{}{\left(z+z^2\right)}^{1.5}\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}dz& \mathrm{Eqn}45\\ L{2}_e={}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}}{\left(1+2z\right)}dz& \mathrm{Eqn}46\\ L{1}_h={}_{-F_h}^{}{\left(z+z^2\right)}^{1.5}.Math.\frac{d(1/\left(1+\exp \left(z-F_h\right)\right)}{dz}dz& \mathrm{Eqn}47\\ L{2}_h={}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.\frac{d(1/\left(1+\exp \left(z-F_h\right)\right)}{dz}}{\left(1+2z\right)}dz& \mathrm{Eqn}48\\ \begin{array}{c}m{c}_e=M{c}_e.Math.L{1}_e/L{2}_e\\ =\mathrm{conductivity}\mathrm{effective}\mathrm{mass}\mathrm{of}\mathrm{electron}\end{array}& \mathrm{Eqn}49\\ \begin{array}{c}m{c}_h=M{c}_h.Math.L{1}_h/L{2}_h\\ =\mathrm{conductivity}\mathrm{effective}\mathrm{mass}\mathrm{of}\mathrm{hole}\end{array}& \mathrm{Eqn}50\end{array}$

(79) Regarding scattering terms, the polar optical phonon scattering of electrons (.sub.poe) and holes (.sub.poh) can be determined from the following constants and expressions:
.sub.o=8.8510.sup.12=permittivity of air
e.sub.1=69.8=high frequency dielectric constant
e.sub.0=400=static dielectric constant
K.sub.0=0.1e.sub.v=optical phono energy
N.sub.0=1/(exp(K.sub.0/(B.Math.T))1)=phonon Plank functionEqn 51
cop.sub.e=(4.Math..sub.o.Math.h.sub.c.sup.2/(3e.sup.2.Math.(1/e.sub.11/e.sub.0).Math.N.sub.0)).Math.(2/(md.sub.c.Math.K.sub.0.Math.(1+K.sub.0/E.sub.g))).sup.0.5Eqn 52
cpo.sub.h=(4.Math..sub.o.Math.h.sub.c.sup.2/(3e.sup.2.Math.(1/e.sub.11/e.sub.0).Math.N.sub.0)).Math.(2/(md.sub.h.Math.K.sub.0.Math.(1+K.sub.0/E.sub.g))).sup.0.5Eqn 53
b.sub.e=(1+2z)/(z+z.sup.2)Eqn 54
.sub.z3=1/b.sub.eEqn 55
with:
.sub.poe=cpo.sub.e.Math..sub.z3=lifetime for optical phonon-electron scatteringEqn 56
and:
.sub.poh=cpo.sub.h.Math..sub.z3==lifetime for optical phonon-hole scatteringEqn 57

(80) Deformation potential scattering of electrons (.sub.dae) and holes (.sub.dah) by acoustic phonons can be determined using the following constants and expressions:
K.sub.a=1.1=fitting constant
C.sub.1=7.110.sup.10=combination electric constant
E.sub.ac=3.5e.sub.v=deformation potential for acoustic phononEqn 58
B.sub.a=8z(1+z).Math.K.sub.a/(3(1+2z).sup.2)Eqn 59
A.sub.a=z(1K.sub.a)/(1+2z)Eqn 60
Cda.sub.e=2h.sub.c.sup.4.Math.C.sub.1/(E.sub.ac.sup.2.Math.(2.Math.md.sub.c.Math..sub.B.Math.T).sup.1.5)Eqn 61
Cda.sub.h=2h.sub.c.sup.4.Math.C.sub.1/(E.sub.ac.sup.2.Math.(2.Math.md.sub.h.Math..sub.B.Math.T).sup.1.5)Eqn 62
.sub.z1=1/(((z+z.sup.2).sup.0.5).Math.(1+2z).Math.((1A.sub.a).sup.2B.sub.a))Eqn 63
with:
.sub.dae=Cda.sub.e.Math..sub.z1=lifetime for acoustic phonon-electron scatteringEqn 64
and:
.sub.dah=Cda.sub.h.Math..sub.z1=lifetime for acoustic phonon-hole scatteringEqn 65

(81) Deformation potential scattering of electrons (.sub.doe) and holes (.sub.doh) by optical phonons can be determined using the following constants and expressions:
r.sub.ho=7.8610.sup.3=density
a=10.4510.sup.10=lattice constant
K.sub.0=1.1=fitting constant
E.sub.oc=60e.sub.v=deformation potential for optical phononEqn 66
B.sub.o=8z(1+z).Math.K.sub.o/(3(1+2z).sup.2)Eqn 67
A.sub.o=z(1K.sub.o)/(1+2z)Eqn 68
Cdo.sub.e=2h.sub.c.sup.2.Math.a.sup.2.Math.K.sub.0.sup.2.Math.r.sub.ho/(E.sub.oc.sup.2.Math.(2.Math.md.sub.e.Math..sub..Math.T).sup.1.5)Eqn 69
Cdo.sub.h=2h.sub.c.sup.2.Math.a.sup.2.Math.K.sub.0.sup.2.Math.r.sub.ho/(E.sub.oc.sup.2.Math.(2.Math.md.sub.h.Math..sub.B.Math.T).sup.1.5)Eqn 70
.sub.z2=1/(((z+z.sup.2).sup.0.5).Math.(1+2z).Math.((1A.sub.o).sup.2B.sub.o))Eqn 71
with:
.sub.doe=Cdo.sub.e.Math..sub.z2=lifetime for deformation potential-electron scatteringEqn 72
and:
.sub.doh=Cdo.sub.h.Math..sub.z2=lifetime for deformation potential-hole scatteringEqn 73

(82) In the event that the thermoelectric material includes nanoparticles, the scattering of elections (.sub.ie) and holes (.sub.ih) by the nanoparticles can be determined using the following constants and expressions:

(83) $\begin{array}{cc}{}_e=\mathrm{Real}\mathrm{Portion}\mathrm{of}\text{:}(\left(\left(\left(2e^2.Math.{\left(2.Math.{\mathrm{md}}_e\right)}^{1.5}/\left(e_0.Math.h_c^3\right)\right).Math.{\left(k_{B}T\right)}^{0.5}.Math.{\left(F_e+F_e^2\right)}^{0.5}.Math.{\left(1+2F_e\right))}^{-0.5}\right)/{10}^{-9}\right)& \mathrm{Eqn}74\\ {}_h=\mathrm{Real}\mathrm{Portion}\mathrm{of}\text{:}(\left(\left(\left(2e^2.Math.{\left(2.Math.{\mathrm{md}}_h\right)}^{1.5}/\left(e_0.Math.h_c^3\right)\right).Math.{\left(k_{B}T\right)}^{0.5}.Math.{\left(F_h+F_h^2\right)}^{0.5}.Math.{\left(1+2F_h\right))}^{-0.5}\right)/{10}^{-9}\right)& \mathrm{Eqn}75\\ \mathrm{WF}=300=\mathrm{work}\mathrm{function}\mathrm{of}\mathrm{an}\mathrm{inclusion},\text{.Math.}e.g.a\mathrm{ceramic}\mathrm{inclusion}& \\ \mathrm{EE}=4.76=\mathrm{electron}\mathrm{affinity}\mathrm{of}\mathrm{host}\mathrm{material}& \\ {\mathrm{Vb}}_e=\mathrm{WF}-\mathrm{EE}+F_e.Math.\left(k_{B}T\right)/e_v& \mathrm{Eqn}76\\ {\mathrm{Vb}}_h=\mathrm{WF}-\mathrm{EE}+F_h.Math.\left(k_{B}T\right)/e_v& \mathrm{Eqn}77\\ W_e={\mathrm{Vb}}_e.Math.R/\exp \left(-R/{}_e\right)& \mathrm{Eqn}78\\ W_h={\mathrm{Vb}}_h.Math.R/\exp \left(-R/{}_h\right)& \mathrm{Eqn}79\\ W=10=\mathrm{fitting}\mathrm{constant}& \\ U=0.3=\mathrm{volume}\%\mathrm{of}\mathrm{nanoparticles}& \\ R=1.5=\mathrm{radius}\mathrm{of}\mathrm{nanoparticles}\mathrm{in}\mathrm{nm}& \\ {\mathrm{Vr}}_e=W_e.Math.\left(1/1\right).Math.\exp \left(-1/{}_e\right).Math.e_v\text{:}{V}_e={\mathrm{Vr}}_e/e_v& \mathrm{Eqn}80\\ {\mathrm{Vr}}_h=W_h.Math.\left(1/1\right).Math.\exp \left(-1/{}_h\right).Math.e_v\text{:}{V}_h={\mathrm{Vr}}_h/e_v& \mathrm{Eqn}81\\ a_e={}_{R,0}^{,W}\left(q.Math.\sin \left(\frac{1t}{R}\right)v_{\mathrm{re}}\right)d\mathrm{ld}t& \mathrm{Eqn}82\\ {}_e=\left(1/R^4\right).Math.a_e& \mathrm{Eqn}83\\ a_h={}_{R,0}^{,W}\left(1.Math.\sin \left(\frac{1t}{R}\right)v_{\mathrm{rh}}\right)d\mathrm{ld}t& \mathrm{Eqn}84\\ {}_h=\left(1/R^4\right).Math.a_h& \mathrm{Eqn}85\end{array}$
with:
.sub.ie=(.sub.BT).sup.1.5.Math.(z.sup.1.5.Math.(1+z).sup.1.5.Math.4R.Math.(2md.sub.e1).sup.0.5)/((1+2z).Math.U.Math.3.sub.e)=lifetime for nanoparticles-electron scatteringEqn 86
and:
.sub.ih=(.sub.BT).sup.1.5.Math.(z.sup.1.5.Math.(1+z).sup.1.5.Math.4R.Math.(2md.sub.h1).sup.0.5)/((1+2z).Math.U.Math.3.sub.h)=lifetime for nanoparticles-electron scatteringEqn 87

(84) Grain boundaries can naturally be a source of scattering, and the scattering of electrons (.sub.be2) and holes (.sub.bh2) can be determined using the following constants and expressions:
e.sub.b=0.3(values range from 0.003-0.3 with 0.3 corresponding to a very strong electrical conductivity)=Barrier height
d.sub.1=30=grain boundary constant
E.sub.b=e.sub.b.Math.e.sub.v=grain boundary barrier energy height=E.sub.b in FIG. 3Eqn 88
DL=d.sub.1.Math.10.sup.9=grain size=L in FIG. 3Eqn 89
Gw=510.sup.9=grain boundary width=W=FIG. 3
E=.sub.BTzEqn 90
GN=4(E/E.sub.b).Math.(1E/E.sub.b)Eqn 91
GD.sub.e=((2md.sub.e.Math.E.sub.b.Math.Gw.sup.2/h.sub.c.sup.2).Math.(1E/E.sub.b)).sup.0.5Eqn 92
Z.sub.e=(exp(GD.sub.e)exp(GD.sub.e))/2Eqn 93
Z.sub.e1=GD.sub.eEqn 94
with:
.sub.be2=DL.Math.(md.sub.e/(2.Math.E)).sup.0.5.Math.(1+GN/(Z.sub.e1).sup.2)=lifetime for grain boundary-electron scatteringEqn 95
and:
GN=4.Math.(E/E.sub.b).Math.(1E/E.sub.b)Eqn 96
GD.sub.h=((2.Math.md.sub.h.Math.E.sub.b.Math.Gw.sup.2/h.sub.c.sup.2).Math.(1E/E.sub.b)).sup.0.5Eqn 97
Z.sub.h=(exp(GD.sub.h)exp(GD.sub.h))/2Eqn 98
Z.sub.h1=GD.sub.hEqn 99
with:
.sub.bh2=DL.Math.(md.sub.h/(2E)).sup.0.5.Math.(1+GN/(Z.sub.h1).sup.2)=lifetime for grain boundary-hole scatteringEqn 100

(85) Though not required, interfacial surface roughness scattering of an inclusion particle can be considered with such scattering of electrons (.sub.ifre) and holes (.sub.ifrh) determined by the following constants and expressions:
d=1=roughness height in nm
c.sub.1=2.4=correlation length in nm
=c.sub.110.sup.9Eqn 101
=d10.sup.9Eqn 102
.sub.0=8.8510.sup.12
For holes:

(86) $\begin{array}{cc}{h}_{\mathrm{WL}}=11.4{10}^{-9}=\mathrm{hole}\mathrm{constant}& \\ {\mathrm{Ns}}_h={10}^{18}=\mathrm{carrier}\mathrm{concentration}& \\ k_h=2/h_{\mathrm{WL}}=\mathrm{counting}\mathrm{constant}\mathrm{for}\mathrm{phonon}\mathrm{energy}& \mathrm{Eqn}103\\ K_h=2k_h& \mathrm{Eqn}104\\ {\mathrm{.Math.}}_s=400=\mathrm{high}\mathrm{frequency}\mathrm{dielectric}\mathrm{constant}& \\ b_h={\left(6*2{\mathrm{md}}_h.Math.e^2.Math.N_{\mathrm{Sh}}/\left(h_c^2{\mathrm{.Math.}}_s.Math.{\mathrm{.Math.}}_o\right)\right)}^{0.333}& \mathrm{Eqn}105\\ {\mathrm{fz}}_h={\left(0.5.Math.b_h^3.Math.z^2\right)}^{0.5}.Math.\exp \left(-0.5b_{h}z\right)& \mathrm{Eqn}106\\ {\mathrm{fzq}}_h={\left(0.5.Math.b_h^3.Math.z_q^2\right)}^{0.5}.Math.\exp \left(-0.5b_h{z}_q\right)& \mathrm{Eqn}107\\ {\mathrm{Fq}}_h={}_{0,0}^{{10}^9,{10}^9}(({\mathrm{fz}}_h^2.Math.\exp \left(-z.Math.q\right).Math.{\mathrm{fzq}}_h^2.Math.\exp \left(z_{q}q\right)dz.Math.d{z}_q& \mathrm{Eqn}108\\ {\mathrm{qs}}_h=\left(e^2.Math.{\mathrm{md}}_h/\left(h_c^2.Math.2.Math.{\mathrm{.Math.}}_s.Math.{\mathrm{.Math.}}_o\right)\right).Math.{\mathrm{Fq}}_h& \mathrm{Eqn}109\\ J_h={}_0^{K_h}\frac{\left(-q^2{}^4/4\right).Math.q^4}{2{k_h^3\left(q+{\mathrm{qs}}_h\right)}^2\sqrt{{\left(1-\left(\frac{q}{2k_h}\right)\right)}^2}}dq& \mathrm{Eqn}110\\ w_{\mathrm{ifrh}}=\left({\left(e^2.Math.{\mathrm{Ns}}_h.Math..Math./\left(2{\mathrm{.Math.}}_s.Math.{\mathrm{.Math.}}_o\right)\right)}^2.Math.{\mathrm{md}}_h.Math.J_h/h_c^3\right)& \mathrm{Eqn}111\\ {}_{\mathrm{ifrh}}=1/{\mathrm{wifr}}_h=\mathrm{lifetime}\mathrm{for}\mathrm{roughness}\text{-}\mathrm{hole}\mathrm{scattering}& \mathrm{Eqn}112\end{array}$
and for electrons:

(87) $\begin{array}{cc}{e}_{\mathrm{WL}}=11.4{10}^{-9}=\mathrm{electron}\mathrm{constant}& \\ {\mathrm{Ns}}_e={10}^{18}=\mathrm{carrier}\mathrm{concentration}& \\ k_e=2/e_{\mathrm{WL}}& \mathrm{Eqn}113\\ K_e=2k_e& \mathrm{Eqn}114\\ {\mathrm{.Math.}}_s=400=\mathrm{high}\mathrm{frequency}\mathrm{dielectric}\mathrm{constant}& \\ b_e={\left(6.Math.2{\mathrm{md}}_e.Math.e^2.Math.{\mathrm{Ns}}_e/\left(h_c^2.Math.{\mathrm{.Math.}}_s.Math.{\mathrm{.Math.}}_o\right)\right)}^{0.333}& \mathrm{Eqn}115\\ {\mathrm{fz}}_e={\left(0.5.Math.b_e^3.Math.z^2\right)}^{0.5}.Math.\exp \left(-0.5b_{e}z\right)& \mathrm{Eqn}116\\ {\mathrm{fzq}}_e={\left(0.5.Math.b_e^3.Math.z_q^2\right)}^{0.5}.Math.\exp \left(-0.5b_e{z}_q\right)& \mathrm{Eqn}117\\ {\mathrm{Fq}}_e={}_{0,0}^{{10}^9,{10}^9}(({\mathrm{fz}}_e^2.Math.\exp \left(-z.Math.q\right).Math.{\mathrm{fzq}}_e^2.Math.\exp \left(z_{q}q\right)dz.Math.d{z}_q& \mathrm{Eqn}118\\ {\mathrm{qs}}_e=\left(e^2.Math.{\mathrm{md}}_e/\left(h_c^2.Math.2.Math.{\mathrm{.Math.}}_s.Math.{\mathrm{.Math.}}_o\right)\right).Math.{\mathrm{Fq}}_e& \mathrm{Eqn}119\\ J_e={}_0^{K_e}\frac{\exp \left(-q^2{}^4/4\right).Math.q^4}{2{k_e^3\left(q+{\mathrm{qs}}_e\right)}^2\sqrt{{\left(1-\left(\frac{q}{2k_e}\right)\right)}^2}}dq& \mathrm{Eqn}120\\ w_{\mathrm{ifre}}=\left({\left(e^2.Math.{\mathrm{Ns}}_e.Math..Math./\left(2{\mathrm{.Math.}}_s.Math.{\mathrm{.Math.}}_o\right)\right)}^2.Math.{\mathrm{md}}_e.Math.J_e/h_c^3\right)& \mathrm{Eqn}121\\ {}_{\mathrm{ifre}}=1/w_{\mathrm{ifre}}=\mathrm{lifetime}\mathrm{for}\mathrm{roughness}\text{-}\mathrm{electron}\mathrm{scattering}& \mathrm{Eqn}122\end{array}$

(88) Thus the total scattering for electrons (.sub.ze) and holes (.sub.zh) can be obtained from:
.sub.ze=1/(1/.sub.doe+1/.sub.dae+1/.sub.poe+1/.sub.be2+1/.sub.ie+1/.sub.ifre)=total lifetime for electron scattering Eqn 123
.sub.zh=1/(1/.sub.doh+1/.sub.dah+1/.sub.poh+1/.sub.bh2+1/.sub.ie+1/.sub.ifrh)=total lifetime for hole scattering Eqn 124

(89) Once the scattering terms have been determined, electrical conductivity (C.sub.e, C.sub.h), Seebeck coefficient (S.sub.e, S.sub.h) can be determined for the given grain size and grain boundary energy barrier height using the following expressions. In particular, C.sub.e and C.sub.h can be determined from:

(90) $\begin{array}{cc}{c}_e=\left(e^2/\left(3{}^2.Math.h_c^3.Math.m_{\mathrm{ce}}\right)\right).Math.{\left(2.Math.m_{\mathrm{de}}{k}_{B}T\right)}^{1.5}& \mathrm{Eqn}125\\ C_e=c_e.Math.{}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{ze}}.Math.\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}}{\left(1+2z^2\right)}dz& \mathrm{Eqn}126\\ c_h=\left(e^2/\left(3{}^2.Math.h_c^3.Math.m_{\mathrm{ch}}\right)\right).Math.{\left(2.Math.m_{\mathrm{dh}}{k}_{B}T\right)}^{1.5}& \mathrm{Eqn}127\\ C_h=c_h.Math.{}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{zh}}.Math.\frac{d(1/\left(1+\exp \left(z-F_h\right)\right)}{dz}}{\left(1+2z\right)}dz& \mathrm{Eqn}128\end{array}$
and the total electrical conductivity (C) is simply:
C=(C.sub.e+C.sub.h)Eqn 129
Also, S.sub.e and S.sub.h can be determined by:

(91) $\begin{array}{cc}\mathrm{Se}=-\left(k_B/e\right).Math.\frac{{}_{F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{ze}}.Math.\frac{d\left(\frac{1}{1+\exp \left(z-F_e\right)}\right)}{z}.Math.\left(z-F_e\right)}{\left(1+2z\right)}dz}{{}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{ze}}.Math.\frac{d\left(\frac{1}{1+\exp \left(z-F_e\right)}\right)}{dz}}{\left(1+2z\right)}dz}& \mathrm{Eqn}130\\ S_h=-\left(k_B/e\right).Math.\frac{{}_{F_h}^{}\frac{\begin{array}{c}{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{zh}}.Math.\\ \frac{d(1/\left(1+\exp \left(z-F_h\right)\right)}{dz}.Math.\left(F_h-z\right)\end{array}}{\left(1+2z\right)}dz}{{}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{zh}}.Math.\frac{d(1/\left(1+\exp \left(z-F_h\right)\right)}{dz}}{\left(1+2z\right)}dz}& \mathrm{Eqn}131\end{array}$
and the total Seebeck coefficient (S) is simply:
S=(S.sub.e.C.sub.e+S.sub.h.Math.C.sub.h)/CEqn 132

(92) It is appreciated that the power factor (P) is given by:
P=S.sup.2.Math.CEqn 133
Also, it should be appreciated that the preceding constants and expressions afford for the calculation of electrical conductivity and Seebeck coefficient as a function of grain size, grain boundary width and/or grain boundary barrier height. Naturally, certain constants would be assumed and varied as needed. For example, the temperature (T) for the calculations used to produce FIGS. 3 and 7 was 300 C., however, the temperature could be varied to determine the effect of temperature on C and S. Also, the size/radius, volume percent, fitting constants, etc., of any inclusions, nanoparticles, etc., can be varied to determine their effect on calculated results.

(93) Regarding thermal conductivity of the material, it is appreciate that additional expressions and scattering terms are needed. For example, Lorentz numbers for electrons (L.sub.e) and holes (L.sub.h) are useful and can be obtained from the following expressions:

(94) $\begin{array}{cc}{L}_e={\left(\begin{array}{c}\left(\frac{\frac{C_e}{C}{}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{ze}}.Math.z^2.Math.\frac{d\left(\frac{1}{1+\exp \left(z-F_e\right)}\right)}{dz}}{\left(1+2z\right)}dz}{{}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{ze}}.Math.\frac{d\left(\frac{1}{1+\exp \left(z-F_e\right)}\right)}{dz}}{\left(1+2z\right)}dz}\right)-\\ \left(\frac{{}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.t_{\mathrm{ze}}.Math.z.Math.\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}}{\left(1+2z\right)}dz}{{}_{-F_e}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{ze}}.Math.\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}}{\left(1+2z\right)}dz}\right)\end{array}\right)}^2& \mathrm{Eqn}134\\ \mathrm{and}& \\ L_h={\left(\begin{array}{c}\left(\frac{\frac{C_h}{C}{}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{zh}}.Math.z^2.Math.\frac{d\left(\frac{1}{1+\exp \left(z-F_e\right)}\right)}{dz}}{\left(1+2z\right)}dz}{{}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{zh}}.Math.\frac{d\left(\frac{1}{1+\exp \left(z-F_e\right)}\right)}{dz}}{\left(1+2z\right)}dz}\right)-\\ \left(\frac{{}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.t_{\mathrm{zh}}.Math.z.Math.\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}}{\left(1+2z\right)}dz}{{}_{-F_h}^{}\frac{{\left(z+z^2\right)}^{1.5}.Math.{}_{\mathrm{zh}}.Math.\frac{d(1/\left(1+\exp \left(z-F_e\right)\right)}{dz}}{\left(1+2z\right)}dz}\right)\end{array}\right)}^2& \mathrm{Eqn}135\\ \mathrm{and}& \\ \begin{array}{c}{L}_b={\left(e/k_B\right)}^2.Math.C_e.Math.C_h.Math.{\left(S_e-S_h\right)}^2/C^2\\ =\mathrm{bipolar}\mathrm{Lorentz}\mathrm{number}\end{array}& \mathrm{Eqn}136\end{array}$
which afford for the electronic thermal conductivity (K.sub.e) to be calculated from:
K.sub.e=T.Math.C.Math.(k.sub.B.sup.2/e.sup.2).Math.(L.sub.e+L.sub.h+L.sub.b)Eqn 137
As such, the electronic thermal conductivity as a function of grain size can be calculated and plotted as shown in FIG. 5.

(95) Regarding lattice thermal conductivity, the following constants and expressions are useful:

(96) $\begin{array}{cc}{K}_0=0.1e_v=\mathrm{phonon}\mathrm{energy}& \mathrm{Eqn}138\\ L_{\mathrm{EO}}=1{10}^{-3}=\mathrm{grain}\mathrm{size}\mathrm{parameter}\mathrm{of}\mathrm{sample}{B}_u=10.9{10}^{-18}=\mathrm{Umpklamp}\mathrm{scattering}\mathrm{constant}n=5.2=\mathrm{fitting}\mathrm{parameter}\text{.Math.}{B}_n=42{10}^{-24}=\mathrm{normal}\mathrm{scattering}\mathrm{constant}\text{.Math.}A=1.9{10}^{-42}=\mathrm{alloy}\mathrm{scattering}\mathrm{constant}& \\ E_d=25e_{v}v=3000=\mathrm{sound}\mathrm{velocity}{}_d=164.9=\mathrm{Debye}\mathrm{temperature}& \mathrm{Eqn}139\\ Q={}_d/TU=0.3=\mathrm{volume}\%\mathrm{of}\mathrm{particles}r=1.5=\mathrm{radius}\mathrm{of}\mathrm{particles}\mathrm{in}\mathrm{nm}& \mathrm{Eqn}140\\ A_1=r^2/1^2& \mathrm{Eqn}141\\ B_1=1^2/rm=-0.925=\mathrm{mass}\mathrm{constant}k=0.3=\mathrm{spring}\mathrm{constant}q=80=\mathrm{fitting}\mathrm{parament}& \mathrm{Eqn}142\\ r_1=\left({}^2/4\right).Math.m^2+3.Math.\left({}^8\right).Math.\left(k^2\right).Math.{\left(\sin \left(.Math.\mathrm{q\_}/2\right)\right)}^4/\left({\left(.Math.\mathrm{q\_}/2\right)}^4\right)& \mathrm{Eqn}143\\ r_2=3.14.Math.\left(\cos \left(4r.Math.q\right)-1+\left(4r.Math.q\right).Math.\sin \left(4r.Math.q\right)+32{\left(r.Math.q\right)}^4-8{\left(r.Math.q\right)}^2\right)/\left(16{\left(r.Math.q\right)}^6\right)& \mathrm{Eqn}144\\ {}_{\mathrm{\_ray}}={\left(r.Math.q\right)}^4.Math.r_1.Math.r_2& \mathrm{Eqn}145\\ =1/2=\mathrm{trigonometric}\mathrm{ratio}\mathrm{q\_}=0.1=\mathrm{wave}\mathrm{vector}\mathrm{parameter}& \\ X=\left(\sqrt{1+m)}.Math.\sqrt{1+k)}\right)-1& \mathrm{Eqn}146\\ {}_{\mathrm{\_ng}}=2.Math.\left(1-\sin \left(2.Math.r.Math.q.Math.X\right)/\left(r.Math.q.Math.X\right)+\sin \left(r.Math.q.Math.X\right).Math.\sin \left(r.Math.q.Math.X\right)/\left({\left(r.Math.q.Math.X\right)}^2\right)\right)& \mathrm{Eqn}147\\ {}_{\mathrm{\_total}}=1/\left(\left(1/{}_{\mathrm{\_ray}}\right)+\left(1/{}_{\mathrm{\_ng}}\right)\right)& \mathrm{Eqn}148\\ {}_1=\frac{U\left(3{10}^{27}\right)}{4{}_0^{}\frac{r^3{r}^{\left(A1-1\right)}\exp \left(\frac{-r}{B_1}\right)}{B_1^{A1}\left(A_1\right)}dr}& \mathrm{Eqn}149\end{array}$
which afford for grain boundary scattering (t_B), Umpklamp scattering (t_u), normal scattering (t_N), alloy scattering (t_A) and nanoparticle scattering (t_D) to be determined from the following expressions:

(97) $\begin{array}{cc}\begin{array}{c}{t}_{\_B}=v/L_{\mathrm{EO}}\\ =\mathrm{total}\mathrm{lifetime}\mathrm{for}\mathrm{phonon}\mathrm{and}\mathrm{boundary}\mathrm{scattering}\end{array}& \mathrm{Eqn}150\\ \begin{array}{c}{t}_{\mathrm{\_U}}=B_u.Math.s^2.Math.k_B^2.Math.T^3.Math.\exp \left(-{}_d/\left(n.Math.T\right)\right)/h_c^2\\ =\mathrm{total}\mathrm{lifetime}\mathrm{for}\mathrm{phonon}\text{-}\mathrm{and}\mathrm{UmKlapp}\\ \mathrm{scattering}\end{array}& \mathrm{Eqn}151\\ \begin{array}{c}{t}_{\mathrm{\_N}}=B_n.Math.s^2.Math.k_B^2.Math.T^5/h_c^2\\ =\mathrm{total}\mathrm{lifetime}\mathrm{for}\mathrm{phonon}\text{-}\mathrm{and}\mathrm{Normal}\\ \mathrm{scattering}\end{array}& \mathrm{Eqn}152\\ \begin{array}{c}{t}_{\mathrm{\_A}}=A.Math.s^4.Math.k_B^4.Math.T^4/h_c^4\\ =\mathrm{total}\mathrm{lifetime}\mathrm{for}\mathrm{phonon}\text{-}\mathrm{and}\mathrm{alloy}\mathrm{scattering}\end{array}& \mathrm{Eqn}153\\ t_{\mathrm{\_D}}=v{}_1\left(1/\left({\left(A_1.Math.B_1\right)}^2\right)\right).Math.\left(1{10}^{-18}\right).Math.{}_0^{}{}^2{r}^4{}_{\_\mathrm{total}}.Math.r^{\left(A1\text{-}1\right)}.Math.\exp \frac{\left(-r/B_1\right)}{B_1^{A1}\left(A_1\right)}dr=\mathrm{total}\mathrm{lifetime}\mathrm{for}\mathrm{phonon}\text{-}\mathrm{and}\mathrm{nanoparticle}\mathrm{scattering}& \mathrm{Eqn}154\end{array}$