II. PORE SIZE DISTRIBUTION AND NET POROSITY

Consider a porous medium in the presence of different groups of nonoverlapping spherical hollow pores with various size scales, e.g., nanoscale pores with randomly distributed pore diameters d ₁ < 100 nm around average diameter $\overline {{d_1}}$, microscale ones with diameters d ₃ ≥ 1 μm around average diameter $\overline {{d_3}}$, and pores with intermediate size diameters d ₂ around average diameter $\overline {{d_2}}$ (Fig. 1). As in Refs. Reference Tarkhanyan and Niarchos15 and Reference Tarkhanyan, Ioannidou and Niarchos16, we assume that the pore size distribution in each group is described by probability density function

(1)$$f\left( {{\rho _i}} \right) = {{N\left( {{\rho _i}} \right)} \over {{N_i}}} = {{{\rho _i}^{{\alpha _i} - 1}\exp \left( {{{ - {\rho _i}} \mathord{\left/ {\vphantom {{ - {\rho _i}} {{\beta _i}}}} \right. \kern-\nulldelimiterspace} {{\beta _i}}}} \right)} \over {\beta _i^{{\alpha _i}}\Gamma \left( {{\alpha _i}} \right)}},\quad i = 1,\,2,\,3\quad ,$$

where $\rho _i = {{d_i } \mathord{\left/ {\vphantom {{d_i } {\overline {d_i } }}} \right. \kern-\nulldelimiterspace} {\overline {d_i } }}$ are the nondimensional diameters, N(ρ_i)dρ_i is the number of pores with given size scale in the interval, ρ_i,ρ_i + dρ_i,N _i is the total number of pores in the group, α_i > 1 and β_i are the “shape” and “scale” parameters of the gamma distribution, respectively,^{Reference Jambunathan18} and Γ(α_i) is the gamma function (the Euler integral of the second kind). The mean (first moment) and variance of the distribution are given by α_iβ_i = 1 and ${\alpha _i}\beta _i^2 = \alpha _i^{ - 1}$, respectively, using which for the second and third moments of the distribution, one can obtain^{Reference Tarkhanyan and Niarchos15,Reference Tarkhanyan, Ioannidou and Niarchos16}

(2)$$\overline {\rho _i^2} = 1 + \alpha _i^{ - 1}\quad ,$$

(3)$$\overline {\rho _i^3} = {{\beta _i^3\Gamma \left( {{\alpha _i} + 3} \right)} \over {\Gamma \left( {{\alpha _i}} \right)}} = {{\left( {{\alpha _i} + 1} \right)\left( {{\alpha _i} + 2} \right)} \over {\alpha _i^2}}\quad .$$

FIG. 1.

Schematic illustration of the medium with randomly distributed nano, intermediate, and microscale pores.

Note that if the standard deviation $\alpha _i^{ - 1/2}\overline {{\rho _i}}$ is much less than the mean, i.e., when $\alpha _i^{1/2} \gg 1$, the distribution function $f\left( {{\rho _i}} \right) \sim \delta \left( {{\rho _i} - \overline {{\rho _i}} } \right)$. The porosity φ_i of each group is given by

(4)$${\varphi _i} = {{{V_i}} \mathord{\left/ {\vphantom {{{V_i}} V}} \right. \kern-\nulldelimiterspace} V} = {{\pi {c_i}{{\overline {{d_i}} }^3}\overline {\rho _i^3} } \mathord{\left/ {\vphantom {{\pi {c_i}{{\overline {{d_i}} }^3}\overline {\rho _i^3} } 6}} \right. \kern-\nulldelimiterspace} 6}\quad ,$$

where V is the total volume of the sample.

(5)$${V_i} = {{\pi {{\overline {{d_i}} }^3}{N_i}\overline {\rho _i^3} } \mathord{\left/ {\vphantom {{\pi {{\overline {{d_i}} }^3}{N_i}\overline {\rho _i^3} } 6}} \right. \kern-\nulldelimiterspace} 6}\quad ,$$

is the net volume of the pores with given size scale, and c _i = N _i/V is the number density (concentration) of the pores. It is evident that the net porosity of the material is given by

(6)$$\varphi = \mathop\sum\limits_i {{\varphi _i}} \quad .$$

III. EFFECTIVE POROSITY AND FERMI ENERGY IN THE PRESENCE OF INTERFACE TRAP CENTERS

In porous materials under consideration, free charge carriers are absent not only inside the pores but also in a small spherical region around the pores due to the presence of deep surface states, which act as electron traps. These trap centers are situated at pore/medium interfaces and are characterized by localized energy E _tr near or above the middle of the band gap. Total number of traps per pore is then $\pi d_i^2{N_{\rm{s}}}$, where N _s is the areal density of trap centers. Consequently, the volume density of traps localized at interfaces of the pores with given size scale is equal to

(7)$${N_{{\rm{tr}},i}} = \pi d_i^2{N_{\rm{s}}}{c_i} = {{6{\varphi _i}{N_{\rm{s}}}\rho _i^2} \mathord{\left/ {\vphantom {{6{\varphi _i}{N_{\rm{s}}}\rho _i^2} {\overline {{d_i}} \overline {\rho _i^3} }}} \right. \kern-\nulldelimiterspace} {\overline {{d_i}} \overline {\rho _i^3} }}\quad ,$$

where c _i is eliminated with the use of Eq. (4). Usually N _tr,i « N _D, where N _D is the net density of shallow donors with energy of ionization ε_D. In the temperature range ε_D < k _BT < ε_G/2, where ε_G is the width of the band gap, all the donors are assumed to be ionized and then the density of electrons in bulk material ${n_0} \cong {N_{\rm{D}}}$. In porous material, as electrons are trapped, spherical regions depleted of free electrons are forming around the pores. Assuming that all trap states are filled by frozen-in electrons and there is no thermal emission from these states, the condition of electrical neutrality can be written as

(8)$${4 \over 3}\pi \left[ {R_i^3 - {{\left( {{{{d_i}} \over 2}} \right)}^3}} \right]{N_{\rm{D}}}q = \pi d_i^2{N_{\rm{s}}}q\quad ,$$

using which for the radius of the depleted region, one can obtain

(9)$${R_i} = {{{d_i}} \over 2}{\left( {1 + {{6{N_{\rm{s}}}} \over {{d_i}{N_{\rm{D}}}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}\quad .$$

Thus, the volume per pore with no mobile electron is equal to V _p,i(1 + 6N _s/N _Dd _i), where $V_{{\rm{p}},i} = {{\pi d_i^3 } \mathord{\left/ {\vphantom {{\pi d_i^3 } 6}} \right. \kern-\nulldelimiterspace} 6}$ is a pore volume. Introducing now “effective porosity” with respect to mobile electrons for each group of the pores

(10)$${\phi _i} = {\varphi _i}\left[ {1 + {{6{N_{\rm{s}}}\overline {\rho _i^2} } \over {{N_{\rm{D}}}\overline {{d_i}} \overline {\rho _i^3} }}} \right]\quad ,$$

for the net effective porosity and the effective density of free electrons in porous material we obtain, respectively,

(11a)$$\phi \equiv \sum\limits_i {{\phi _i}} = \varphi + {\varphi _{{\rm{tr}}}}\quad ,$$

and

(11b)$$n = {n_0}\left( {1 - \phi } \right)\quad ,$$

where

(12)$${\varphi _{{\rm{tr}}}} = {{6{N_{\rm{s}}}} \over {{N_{\rm{D}}}}}g,\quad g \equiv \mathop\sum\limits_{i = 1}^3 {{{{\alpha _i}{\varphi _i}} \over {\left( {{\alpha _i} + 2} \right)\overline {{d_i}} }}} \quad .$$

Thus, the presence of pores and traps leads to a reduction of free charge carriers' density with respect to that in bulk material with zero porosity and consequently to the shifting of the Fermi energy and modification of all transport coefficients. Consider, for example, n-type isotropic material with a single parabolic conduction band. The density of electrons at the arbitrary level of the Fermi energy is given by

(13)$$n = {{{{\left( {2{m^*}{k_{\rm{B}}}T} \right)}^{3/2}}} \over {2{\pi ^2}{\hbar ^3}}}{F_{1/2}}\left( {{\zeta ^*}} \right)\quad ,$$

where m ^∗ is the effective mass, k _B is the Boltzmann constant, T is the absolute temperature, h is the Planck constant, ζ is the chemical potential relative to the conduction band edge, ζ^∗ ≡ ζ/k _BT is the reduced chemical potential, and

(14)$${F_n}\left( {{\zeta ^*}} \right) = \int\limits_0^\infty {{{{x^n}{\rm{d}}x} \over {\exp \left( {x - {\zeta ^*}} \right) + 1}}} \quad ,$$

is the Fermi integral. Using Eq. (11b) in the case of nondegenerate semiconductors, where exp(−ζ^∗) » 1, for the reduced chemical potential, we obtain

(15a)$${\zeta ^*} = \zeta _0^* + \ln \left( {1 - \phi } \right)\quad ,$$

where

(15b)$$\zeta _0^* = - \ln \left( {{{{N_{\rm{c}}}} \mathord{\left/ {\vphantom {{{N_{\rm{c}}}} {{n_0}}}} \right. \kern-\nulldelimiterspace} {{n_0}}}} \right)\quad ,$$

is the reduced chemical potential in bulk material with zero porosity, and

(16)$${N_{\rm{c}}} = {{2{{\left( {2\pi {m^*}{k_{\rm{B}}}T} \right)}^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}} \mathord{\left/ {\vphantom {{2{{\left( {2\pi {m^*}{k_{\rm{B}}}T} \right)}^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}} {{h^3}}}} \right. \kern-\nulldelimiterspace} {{h^3}}}\quad ,$$

is the effective density of states in the conduction band. Since ζ₀ is negative, from Eq. (15a), one can conclude that the presence of porosity leads to the enhancement in the absolute value of the Fermi level. Unlike this, in the case of degenerately doped materials, where exp(ζ^∗) » 1, the Fermi energy decreases:

(17)$$\zeta = {\zeta _0}{\left( {1 - \phi } \right)^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}}\quad ,$$

Here

(18a)$${\zeta _0} = \varepsilon _{\rm{F}}^0\left[ {1 - {{{\pi ^2}} \over {12}}{{\left( {{{{k_{\rm{B}}}T} \over {\varepsilon _{\rm{F}}^0}}} \right)}^2}} \right]\quad ,$$

where

(18b)$$\varepsilon _{\rm{F}}^0 = {{{\hbar ^2}} \over {2{m^*}}}{\left( {3{\pi ^2}{n_0}} \right)^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}}\quad ,$$

is the Fermi energy at T = 0 K.

IV. POWER FACTOR IN THE PRESENCE OF DIFFERENT GROUPS OF PORES WITH VARIOUS SIZE SCALES

Consider now a composite with all-scale hierarchical disorder in pore sizes, e.g., a material consisting of different phases: host porous medium with volume fraction 1 − φ and different groups of nanoscale and microscale insulating particles (or hollow pores) with volume fractions φ_i. We will use classical expressions for the Seebeck coefficient and electrical conductivity based on the solution of the Boltzmann transport equation under relaxation time approximation. They can be written in the form

(19)$$S = {1 \over {qT}}\left[ {{{{G_{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}}}\left( {{\zeta ^*}} \right)} \over {{G_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}\left( {{\zeta ^*}} \right)}} - \zeta } \right]\quad ,$$

(20)$$\sigma = {{2{q^2}{{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {3{\pi ^2}{\hbar ^3}}}{G_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}\left( {{\zeta ^*}} \right)\quad ,$$

using which for the power factor, one can obtain

(21)$$P = {{2{{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {3{\pi ^2}{\hbar ^3}{T^2}}}{{{{\left( {{G_{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}}} - \zeta {G_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right)}^2}} \over {{G_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad ,$$

where

(22)$${G_n} \equiv {G_n}\left( {{\zeta ^*}} \right) = \int\limits_0^\infty {{\varepsilon ^n}\left( {{{ - \partial {f_0}} \mathord{\left/ {\vphantom {{ - \partial {f_0}} {\partial \varepsilon }}} \right. \kern-\nulldelimiterspace} {\partial \varepsilon }}} \right)\tau \left( \varepsilon \right){\rm{d}}\varepsilon \quad .}$$

$f_0 \left( {\varepsilon ,T} \right) = \left( {{\rm{exp}}\left[ {{{\left( {\varepsilon - \zeta } \right)} \mathord{\left/ {\vphantom {{\left( {\varepsilon - \zeta } \right)} {k_{\rm{B}} T}}} \right. \kern-\nulldelimiterspace} {k_{\rm{B}} T}}} \right] + 1} \right)^{ - 1}$ is the Fermi distribution function, and τ(ε) is the combined relaxation time. Using Matthiessen's rule for scattering events, which are independent from each other, τ(ε) can be expressed as^{Reference Ziman19–Reference Lundstrom22}

(23)$${\tau ^{ - 1}} = \tau _{\rm{b}}^{ - 1} + \tau _1^{ - 1} + \tau _2^{ - 1} + \tau _3^{ - 1}\quad ,$$

where τ_b is the electronic momentum relaxation time in the bulk matrix, and τ_i (i = 1, 2, 3) is that due to scattering at pore/medium interfaces. To find the relaxation time, let us estimate electron mean free path (MFP). When a charge carrier travels a distance L, the average number of its collisions with pores of given size scale is equal to

(24)$${\gamma _i} = L{A_i}{c_i}\quad ,$$

where

(25)$${A_i} = {{\pi \overline {d_i^2} } \mathord{\left/ {\vphantom {{\pi \overline {d_i^2} } 4}} \right. \kern-\nulldelimiterspace} 4} = \pi {\overline {{d_i}} ^2}{{\left( {1 + {\alpha _i}} \right)} \mathord{\left/ {\vphantom {{\left( {1 + {\alpha _i}} \right)} {4{\alpha _i}\quad ,}}} \right. \kern-\nulldelimiterspace} {4{\alpha _i}\quad ,}}$$

is the average classical cross-sectional area. Therefore,

(26)$${\Lambda _i} = {L \mathord{\left/ {\vphantom {L {{\gamma _i}}}} \right. \kern-\nulldelimiterspace} {{\gamma _i}}} = {\left( {{A_i}{c_i}} \right)^{ - 1}}\quad ,$$

and then τ_i can be found as

(27)$${\tau _i} = {{{\Lambda _i}} \over v} = {{\left( {{\alpha _i} + 2} \right)\overline {{d_i}} {{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {3{\alpha _i}{\varphi _i}}}{\varepsilon ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}\quad ,$$

where v = v(ε) is the electronic velocity. Thus, from Eq. (23) for the combined relaxation time, we obtain

(28)$$\tau \left( \varepsilon \right) = {{{\tau _{\rm{b}}}} \over {1 + 3g{\tau _{\rm{b}}}{{\left( {{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {2{m^*}}}} \right. \kern-\nulldelimiterspace} {2{m^*}}}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad ,$$

where g is given in Eq. (12). For definiteness, we will further consider the case of relatively low impurity concentrations, so that the main contribution in the carrier relaxation time at zero porosity is from the scattering on deformation potential of acoustic phonons. Then, setting^{Reference Ziman19,Reference Ansel'm20}

(29)$${\tau _{\rm{b}}} = a{\varepsilon ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}\quad ,$$

where

(30)$$a = {{2\pi {\hbar ^4}\rho {u^2}} \over {{k_{\rm{B}}}T{{\left( {2{m^*}} \right)}^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}}{D^2}}}\quad ,$$

is the constant for acoustic phonons, ρ is the mass density, u is the longitudinal velocity of the sound, and D is the deformation potential constant, Eq. (28) can be transformed to the form

(31)$$\tau \left( \varepsilon \right) = {{a{\varepsilon ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad .$$

Substituting Eq. (31) into Eqs. (19)–(21), we obtain the following expressions:

(32)$$S = {{{k_{\rm{B}}}} \over q}\left[ {{{2{F_1}\left( {{\zeta ^*}} \right)} \over {{F_0}\left( {{\zeta ^*}} \right)}} - {\zeta ^*}} \right]\quad ,$$

(33)$$\sigma = {{n{q^2}} \over {{m^*}}}{{b{F_0}\left( {{\zeta ^*}} \right)} \over {{F_{1/2}}\left( {{\zeta ^*}} \right)}}\quad ,$$

(34)$$b \equiv {{2a} \over {3\sqrt {{k_{\rm{B}}}T} \left[ {1 + 3ag{{(2{m^*})}^{ - 1/2}}} \right]}}\quad ,$$

and

(35)$$P = {{nbk_{\rm{B}}^2{{\left[ {2{F_1}\left( {{\zeta ^*}} \right) - {\zeta ^*}{F_0}\left( {{\zeta ^*}} \right)} \right]}^2}} \over {{m^*}{F_0}\left( {{\zeta ^*}} \right){F_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\left( {{\zeta ^*}} \right)}}\quad ,$$

where F _n(ζ^∗) is given in Eq. (14). Eqs. (32)–(35) for the transport coefficients are valid for an arbitrary level of the Fermi energy.

V. ENHANCEMENT OF THE THERMOPOWER AND POWER FACTOR IN NONDEGENERATE AND DEGENERATELY DOPED POROUS COMPOSITES WITH ALL-SCALE HIERARCHICAL DISORDER

In this section, we will examine peculiarities of the power factor in two limiting cases.

A. Nondegenerate limit

Let us begin from the case of nondegenerate bulk materials with zero porosity. For such a material from Eqs. (32)–(35), one can find well-known classical results^{Reference Ziman19–Reference Yu and Cardona24}

(36)$${S_0} = {{{k_{\rm{B}}}} \over q}\left( {2 - \zeta _0^*} \right)\quad ,$$

(37)$${\sigma _0} = {{{n_0}{q^2}\left\langle {{\tau _{\rm{b}}}} \right\rangle } \over {{m^*}}}\quad ,$$

(38)$${P_0} = {{{n_0}k_{\rm{b}}^2{{\left( {2 - \zeta _0^*} \right)}^2}\left\langle {{\tau _{\rm{b}}}} \right\rangle } \over {{m^*}}}\quad ,$$

where $\zeta _0^*$ is given by Eq. (15b) and

(39)$$\left\langle {{\tau _{\rm{b}}}} \right\rangle = {{4a} \over {3\sqrt {\pi {k_{\rm{B}}}T} }}\quad .$$

For porous materials under consideration, Eqs. (32)–(35) with the use of Eqs. (13) and (15a) are transformed to the form

(40)$$S = {S_0}\left[ {1 + {{\left| {\ln \left( {1 - \phi } \right)} \right|} \over {2 - \zeta _0^*}}} \right]\quad ,$$

(41)$$\sigma = {{{\sigma _0}\left( {1 - \phi } \right)} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad ,$$

(42)$$P = {{{P_0}\left( {1 - \phi } \right)} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}}{\left[ {1 + {{\left| {\ln \left( {1 - \phi } \right)} \right|} \over {2 + \ln \left( {{{{N_{\rm{c}}}} \mathord{\left/ {\vphantom {{{N_{\rm{c}}}} {{n_0}}}} \right. \kern-\nulldelimiterspace} {{n_0}}}} \right)}}} \right]^2}\quad ,$$

where ϕ and g are given by Eqs. (11a) and (12).

For relative growth of S and relative change in power factor in this nondegenerate approximation, we obtain

(43)$${\gamma _{{\rm{s}}1}} \equiv {{S - {S_0}} \over {{S_0}}} = {{|\ln \left( {1 - \phi } \right)|} \over {2 - \zeta _0^*}}\quad ,$$

and

(44)$${\gamma _{{\rm{p}}1}} \equiv {{P - {P_0}} \over {{P_0}}} = {{\left( {1 - \phi } \right){{\left( {{\gamma _{s1}} + 1} \right)}^2}} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}} - 1\quad ,$$

respectively. Before that to examine physical consequences of these equations, let us consider opposite limiting case too.

B. Strongly degenerate limit

For degenerately doped materials from Eqs. (32)–(35), we obtain

(45)$$S = {S_0}{\left( {1 - \phi } \right)^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}}}\quad ,$$

(46)$$\sigma = {{{\sigma _0}{{\left( {1 - \phi } \right)}^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}}} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad ,$$

(47)$$P = {{{P_0}{{\left( {1 - \phi } \right)}^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}}}} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad ,$$

where S ₀, σ₀, and P ₀ are the corresponding quantities for strongly degenerate bulk material of zero porosity:

(48)$${S_0} = {{{\pi ^2}k_{\rm{B}}^2T} \over {3q\varepsilon _{\rm{F}}^0}}\quad ,$$

(49)$${\sigma _0} = {{{n_0}{q^2}a} \over {{m^*}\sqrt {\varepsilon _F^0} }}\quad ,$$

(50)$${P_0} = {{2{\pi ^2}{{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}ak_{\rm{B}}^4{T^2}} \over {27{\hbar ^3}\varepsilon _{\rm{F}}^0}}\quad .$$

Relative growth in S and relative change in power factor in this case are given by

(51)$${\gamma _{{\rm{s}}2}} = {\left( {1 - \phi } \right)^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}}} - 1\quad ,$$

and

(52)$${\gamma _{{\rm{p}}2}} = {{{\gamma _{{\rm{s}}2}} - 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad .$$

According to both Eqs. (40) and (45), the presence of hollow pores of different size scales leads to the enhancement in the thermopower with respect to its value in the bulk material with zero porosity. The enhancement in the absolute value of the Seebeck coefficient and the reduction in electrical conductivity depend on the net effective porosity ϕ. Despite the quite different form of expressions in the right sides of Eqs. (43) and (51), the relative growth of S in both these cases increases monotonically with increasing of the porosity and also (at the given value of the ratio N _s/N _D) with decreasing of characteristic pore sizes $\overline {{d_i}}$ [see Eq. (12) for φ_tr]. Except that, γ_s depends also on the shape parameters of the gamma distribution and increases slightly with increasing of all three parameters α_i.

It is evident that because of the reduction in conductivity, the enhancement in S is not sufficient for the enhancement of the power factor. Indeed, from Eqs. (44) and (52), we conclude that an enhancement in power factor is only possible if the relative growth in S exceeds a definite critical value:

(53)$${\gamma _{{\rm{s}}1}} > \sqrt {{{1 + 3ag{{\left( {2{m^*}} \right)}^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \over {1 - \phi }}} - 1\quad ,$$

in nondegenerate limit and

(54)$${\gamma _{{\rm{s}}2}} > 3ag{\left( {2{m^*}} \right)^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}\quad ,$$

in degenerate limit. In the case when the porosity is governed by one length-scale, the conditions (53) and (54) can be rewritten in the form

(53a)$$|\ln \left( {1 - \phi } \right)| > \left( {2 - \zeta _0^*} \right)\left[ {\sqrt {{{\left( {1 + \Gamma \phi } \right)} \mathord{\left/ {\vphantom {{\left( {1 + \Gamma \phi } \right)} {1 - \phi }}} \right. \kern-\nulldelimiterspace} ({1 - \phi) }}} - 1} \right]\quad ,$$

(54a)$$\phi > 1 - {\left( {1 + \Gamma \phi } \right)^{{{ - 3} \mathord{\left/ {\vphantom {{ - 3} 2}} \right. \kern-\nulldelimiterspace} 2}}}\quad ,$$

where

(54b)$$\Gamma = 3a{g_0}{\left( {2{m^*}} \right)^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}{\left( {1 + {{6{g_0}{N_{\rm{s}}}} \mathord{\left/ {\vphantom {{6{g_0}{N_{\rm{s}}}} {{N_{\rm{D}}}}}} \right. \kern-\nulldelimiterspace} {{N_{\rm{D}}}}}} \right)^{ - 1}}\quad ,$$

a is a nondimensional positive parameter and

(54c)$${g_0} = {\alpha \mathord{\left/ {\vphantom {\alpha {\left( {\alpha + 2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\alpha + 2} \right)}}\bar d\quad .$$

We have to recollect that the modulus of the reduced chemical potential in n-type nondegenerate semiconductors satisfies condition $\exp |\zeta _0^*| \gg 1$. Taking these fact into account and carrying out a careful numerical examination, it is not difficult to be convinced that in nondegenerate porous semiconductors, both conditions (53) and (53a) are practically unrealizable, i.e., the increase of the Seebeck coefficient does not compensate the decrease in electrical conductivity. As to the case of a material with strongly degenerated electronic gas, the fulfillment of condition (54a) depends on the value of parameter Γ. At a given value of the effective porosity $\phi$, the enhancement of the power factor occurs only if

(54d)$$\Gamma \lt {\phi ^{ - 1}}\left[ {{{\left( {1 - \phi } \right)}^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}}} - 1} \right]\quad .$$

In Fig. 2 are presented the graphs of the function $f\left( \phi \right) = 1 - {\left( {1 + \Gamma \phi } \right)^{ - 3/2}}$ in the right-hand side of Eq. (54a) for four different values of Γ. A numerical analysis shows that at Γ ≤ 0.666 the condition (54a) is satisfied for all possible values of ϕ. At Γ = 0.67, 0.7, and 0.8, such a situation occurs only if ϕ > 0.006, 0.054, and 0.167, respectively. Thus, at Γ > 2/3, there is a minimum value

(54e)$${\phi _{{\rm{min}}}} = {4 \over {5\Gamma }}\left( {1 - {2 \over {3\Gamma }}} \right)\quad ,$$

so as the enhancement of the power factor occurs only if the porosity ϕ > ϕ_min.

FIG. 2.

Graphs of the function $f\left( \phi \right) = 1 - {\left( {1 + \Gamma \phi } \right)^{ - 3/2}}$ for four different values of Γ.

Note that at small enough porosity values for which Γϕ « 1, Eq. (54a) gives Γ < 2/3. This inequality means that in the case when the ratio N _s/N _D ≥ 3a/4(2m ^∗)^1/2, the enhancement of the power factor becomes possible at an arbitrary value of $\overline d$. In the opposite case when

(54f)$${{{N_{\rm{s}}}} \mathord{\left/ {\vphantom {{{N_{\rm{s}}}} {{N_{\rm{D}}}}}} \right. \kern-\nulldelimiterspace} {{N_{\rm{D}}}}} \lt {{{3a} \mathord{\left/ {\vphantom {{3a} {4\left( {2{m^*}} \right)}}} \right. \kern-\nulldelimiterspace} {4\left( {2{m^*}} \right)}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\quad ,$$

the enhancement of the power factor in degenerate semiconductors with small porosity occurs only if the average pore diameter satisfies the condition $\overline d > {d_{{\rm{min}}}}$ where

(54g)$${d_{{\rm{min}}}} = {{6\alpha } \over {\left( {\alpha + 2} \right)}}\left( {{{3a} \over {4{{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}}} - {{{N_{\rm{s}}}} \over {{N_{\rm{D}}}}}} \right)\quad .$$

Let us take, e.g., N _s = 10⁹ cm⁻², N _D = 2.5 × 10¹⁶ cm⁻³, and m ^∗ = 0.26m ₀ = 2.366 × 10⁻³¹ kg (Si). For the constant a of acoustic phonons [see Eq. (30)] in Si at T = 200 K, one can find a = 1.16 × 10⁻²² kg^1/2 m. Then, Eq. (54g) at “shape” parameter of the distribution, α = 5 gives $\overline d > 0.54\mu {\rm{m}}$. Setting $\overline d = 0.6\mu {\rm{m}}$, we obtain g ₀ = 1.19 × 10⁶m ⁻¹ and Γ = 0.6, i.e., in this case, the enhancement of the power factor can occur at any porosity in the range ϕ ≤ 15%.

VI. EFFECTIVE THERMOPOWER OF A MULTILAYER COMPOSITE WITH PORES OF VARIOUS SIZE SCALES IN DIFFERENT LAYERS

In the previous section, we have assumed that the pores of different size scales are distributed in the same layer of the thermoelectric material. It is extremely interesting to compare the enhancement of the Seebeck coefficient in such a material with that in multilayer composite consisting of alternating layers with various porosities and pores of different size scales when the pores in each layer have the same size scale. As before, the pores are assumed to be randomly distributed in the layers and described by probability density functions f(ρ_i) given by Eq. (1). For definiteness, we will consider again 3-layer composite: let one of the layers contains small (d < 100 nm) nanoscale pores with volume fraction φ₁, the second one pores with intermediate size scale and volume fraction φ₂, and the third layer contains microscale size pores with volume fraction φ₃. The layers are assumed to be perpendicular to the direction of the temperature gradient and having the same volume and thickness. The latter is assumed significantly larger than the maximum value of electron MFP in the layers Λ_e,i. For example, in the case of classical electronic gas, one can find

(55a)$${\Lambda _{{\rm{e}},i}} = {{\left\langle {{\tau _{\rm{b}}}} \right\rangle \sqrt {{{{{3{k_{\rm{B}}}T} \mathord{\left/ {\vphantom {{3{k_{\rm{B}}}T} m}} \right. \kern-\nulldelimiterspace} m}}^*}} } \over {1 + {{3a{\alpha _i}{\varphi _i}} \mathord{\left/ {\vphantom {{3a{\alpha _i}{\varphi _i}} {\left( {{\alpha _i} + 2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\alpha _i} + 2} \right)}}\overline {{d_i}} {{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}}}\quad ,$$

where a and $\left\langle {{\tau _{\rm{b}}}} \right\rangle$ are given by Eqs. (30) and (39), respectively. Since in main thermoelectric materials Λ_e ∼ 10–100 nm at room temperatures,^{Reference Chen, Dresselhaus, Dresselhaus, Fleurial and Caillat25} it is evident that the layers with thickness ∼1 mm can be very suitable for experiments.

The electrical conductivity of the composite is given by

(55b)$${\sigma _{\rm{c}}} = {\sigma _0}{\left( {\mathop\sum\limits_i {\left[ {{{1 + {{3a{\alpha _i}} \mathord{\left/ {\vphantom {{3a{\alpha _i}} {\left( {{\alpha _i} + 2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\alpha _i} + 2} \right)}}{{\left( {2{m^*}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\overline {{d_i}} } \over {3{{\left( {1 - {\phi _i}} \right)}^{\rm{\nu }}}}}} \right]} } \right)^{ - 1}}\quad ,$$

where ν = 1 in nondegenerate case and ν = 2/3 in strongly degenerate limit.

Using the fact^{Reference Callen26,Reference de Groot and Mazur27} that the thermopower is equivalent to entropy per carrier per charge, and the entropy of a system in local equilibrium conditions is an additive quantity, in Ref. Reference Tarkhanyan and Niarchos17, it has been shown that the effective Seebeck coefficient of a multilayer composite can be expressed as

(56)$${S_{\rm{c}}} = {{\mathop\sum\limits_i {{n_i}{S_i}} } \mathord{\left/ {\vphantom {{\sum\limits_i {{n_i}{S_i}} } {\sum\limits_i {{n_i}} }}} \right. \kern-\nulldelimiterspace} {\mathop\sum\limits_i {{n_i}} }}\quad ,\quad i = 1,\,2,\,3\quad ,$$

where n _i is the density of free charge carriers in the ith layer, and ${S_i}$ is the thermopower of the corresponding single layer. Note that there is no discrepancy between different arrangements of the layers with distinct pore size scales. Assuming that the bulk density of electrons n ₀ and the ratio N _s/N _D are constants, from Eq. (56) in nondegenerate case, we obtain

(57)$${S_{\rm{c}}} = {S_0}\left[ {1 + {{\mathop\sum\limits_i {\left[ {\left( {1 - {\phi _i}} \right)|\ln \left( {1 - {\phi _i}} \right)} \right]} } \over {\left( {2 - \zeta _0^*} \right)\mathop\sum\limits_i {\left( {1 - {\phi _i}} \right)} }}} \right]\quad ,$$

and relative growth

(58)$${\gamma _{{\rm{s}}3}} = {{\mathop\sum\limits_i {\left[ {\left( {1 - {\phi _i}} \right)|\ln \left( {1 - {\phi _i}} \right)} \right]} } \over {\left( {2 - \zeta _0^*} \right)\mathop\sum\limits_i {\left( {1 - {\phi _i}} \right)} }}\quad ,$$

where

(59)$${\phi _i} = {\varphi _i}\left[ {1 + {{6{N_{\rm{s}}}{\alpha _i}} \over {{N_{\rm{D}}}\left( {{\alpha _i} + 2} \right)\overline {{d_i}} }}} \right]\quad .$$

In strongly degenerate case, Eqs. (57) and (58) should be replaced by

(60)$${S_{\rm{c}}} = {S_0}{{\mathop\sum\limits_i {{{\left( {1 - {\phi _i}} \right)}^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}} } \over {\mathop\sum\limits_i {\left( {1 - {\phi _i}} \right)} }}\quad ,$$

(61)$${\gamma _{{\rm{s}}4}} = {{\mathop\sum\limits_i {{{\left( {1 - {\phi _i}} \right)}^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}} } \over {\mathop\sum\limits_i {\left( {1 - {\phi _i}} \right)} }} - 1\quad .$$

Using Eqs. (57) or (60) for the thermopower in common with Eq. (55b) for the electrical conductivity, it is not difficult to make analogical calculations for the power factor and its relative change. Furthermore, we will restrict ourselves to the consideration of the thermopower only.

Note that total porosity for ν-layer composite material containing layers of various volume V _i and porosity φ_i is given by average porosity

(62a)$$\varphi = {{\mathop\sum\limits_{i = 1}^{\rm{\nu }} {{\varphi _i}{V_i}} } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^{\rm{\nu }} {{\varphi _i}{V_i}} } {\sum\limits_{i = 1}^{\rm{\nu }} {{V_i}} }}} \right. \kern-\nulldelimiterspace} {\mathop\sum\limits_{i = 1}^{\rm{\nu }} {{V_i}} }}\quad ,$$

where φ_iV _i is the total volume of the pores in a layer with number i. If the layers have the same volume (V _i = V ₀), Eq. (62a) gives $\varphi = {{\left( {\mathop\sum\limits_{i = 1}^\nu {\varphi _i } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 1}^\nu {\varphi _i } } \right)} \nu }} \right. \kern-\nulldelimiterspace} \nu }$ or for 3-layer composite

(62b)$$\varphi = {{\left( {{\varphi _1} + {\varphi _2} + {\varphi _3}} \right)} \mathord{\left/ {\vphantom {{\left( {{\varphi _1} + {\varphi _2} + {\varphi _3}} \right)} 3}} \right. \kern-\nulldelimiterspace} 3}\quad .$$

A simple comparison of Eq. (58) with Eq. (43) and Eq. (61) with Eq. (51) shows that the enhancement in the absolute value of S in the multilayer composite is considerably less than that in single layer composite with multiscale size pores:

(63)$${{{\gamma _{{\rm{s}}3}}} \over {{\gamma _{{\rm{s}}1}}}} = {{\mathop\sum\limits_i {\left[ {\left( {1 - {\phi _i}} \right)|\ln \left( {1 - {\phi _i}} \right)|} \right]} } \over {|\ln \left( {1 - \phi } \right)|\mathop\sum\limits_i {\left( {1 - {\phi _i}} \right)} }}\quad ,$$

in nondegenerate case and

(64)$${{{\gamma _{{\rm{s}}4}}} \over {{\gamma _{{\rm{s}}2}}}} = {{{{\left[ {\mathop\sum\limits_i {\left( {1 - {\phi _i}} \right)} } \right]}^{ - 1}}\mathop\sum\limits_i {{{\left( {1 - {\phi _i}} \right)}^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} - 1} } \over {{{\left( {1 - \phi } \right)}^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}}} - 1}}\quad ,$$

in the case of strongly degenerate electronic gas. In Eqs. (63) and (64), ϕ_i and ϕ can be expressed by the same relations, which are given in Eqs. (10) and (11a), respectively. It is evident that to compare the enhancement in the thermopower of the 3-layer material and homogeneous material with the same volume fraction of the pores in each one, for the total porosity φ in Eq. (11a) must be used by the value given by Eq. (62b) so that the materials differ only in physical arrangement of the pores.

Setting, for example, $\overline {{d_1}} = 10\;{\rm{nm}}$, $\overline {{d_2}} = 0.2\;\mu {\rm{m}}$, $\overline {{d_3}} = 1\;\mu {\rm{m}}$, α₁ = 8, α₂ = 6, α₃ = 3, and φ₁ = φ₂ = φ₃ = 0.05, from Eq. (63) at N _s = 10⁹ cm⁻² and N _D = 10¹⁶ cm⁻³, we obtain γ_s3/γ_s1 = 0.767. However, for heavily doped samples at N _D = 2.5 × 10¹⁶ cm⁻³ and the same values of the other parameters, Eq. (64) gives γ_s4/γ_s2 = 0.886.

Thus, in both these limiting cases, the medium with multiscale size pores in the same host material provides stronger enhancement in the thermopower compared with that in layered composite in which each layer contains pores of single length-scale: more than 30% stronger enhancement in the case of nondegenerate electronic gas and ∼10% in strongly degenerate case.

It is not difficult to see that the enhancement in the material with multilength-scale pores exceeds that also in the case of an identical material with single-scale pores when they have the same total porosity. Really, the relative growth of the thermopower in a material with nondegenerate electronic gas and single-scale pores is given by

(65a)$${\gamma _{\rm{s}}} = {{|\ln \left( {1 - {\phi _{\rm{s}}}} \right)|} \over {2 - \zeta _0^*}}\quad ,$$

where

(65b)$${\phi _{\rm{s}}} = \varphi \left[ {1 + {{6{N_{\rm{s}}}\alpha } \over {{N_{\rm{D}}}\left( {\alpha + 2} \right)\overline d }}} \right]\quad .$$

A comparison of Eqs. (65a) and (43) shows immediately that the ratio γ_s/γ_s1 is always less than 1. For example, consider a material with monoscale pores described by $\overline d = 0.2\;\mu {\rm{m}}$, α = 6, and φ = 0.05. Then for the same values N _s = 10⁹ cm⁻² and N _D = 10¹⁶ cm⁻³, one can find ϕ_s = 0.051 and γ_s/γ_s1 = 0.669. In the case of microscale pores with $\overline d = 1\;\mu {\rm{m}}$, α = 3, and the same porosity φ = 0.05, the ratio is γ_s/γ_s1 = 0.655. Thus, the medium with multiscale size pores provides ∼1.5 times stronger enhancement in the thermopower than a material with monoscale porosity.

It is evident that by combining the significant reduction of lattice thermal conductivity predicted in Refs. Reference Tarkhanyan and Niarchos15 and Reference Tarkhanyan, Ioannidou and Niarchos16 and possible enhancement in the power factor due to the increasing value of the thermopower, high thermoelectric performance of porous materials can be achieved.