Three-terminal semiconductor junction thermoelectric devices: improving performance^*

In the following discussions we specifically consider a p–i–n junction made of 'extremely narrow-gap semiconductors'. The structure is depicted in figure 1(a), for the case of converting thermal to electrical energy. It can be viewed as an analogue of the p–i–n photodiode, where photons are replaced by phonons, i.e. the phonon-assisted inter-band transitions lead to current generation in the junction. The device can also be used to e.g. cool the thermal terminal via the electric current between the electronic terminals. We focus on the situation where single-phonon-assisted inter-band transitions are allowed. (The electronic band gap is hence required to be smaller than the phonon energy.) Such processes may play a significant role in the transport across the junction, whereas the transition rates of multi-phonon processes are much smaller. In semiconductors, the optical phonon energy is usually in the range of 20–100 meV. There are several candidates with such small band gaps: (i) Gapless semiconductors resulting from accidental band degeneracy in solid solutions, such as Pb_1−xSn_xTe and Pb_1−xSn_xSe [21]. At a certain mole fraction x the band gap closes, around which it can be very small. (ii) Gapless semiconductors originating from band inversion, such as HgTe and HgSe [21]. The band gap can be tuned via the quantum-well effect in the superlattices composed of a gapless semiconductor and a normal semiconductor without band inversion [22]. An example is the HgTe/CdTe superlattices with a tunable band gap [23]. (iii) Multilayers (superlattices) of 'topological' insulators with ordinary insulator layers sandwiched between the topological ones [24]. For example, in Bi₂Te₃/Si superlattices the energy gap can be tuned by varying the thickness of the topological and ordinary layers [24]. A significant merit of superlattices is that the lattice thermal conductivity along the growth direction can be much smaller than both of the two bulk constituents. For example, the Si/Ge superlattices have a thermal conductivity about two orders of magnitude smaller than the bulk values [25]. We also point out that the same idea can be applied to devices where the role of optical phonons is played by other bosons. If the energy of such bosons is higher, the requirement for a small band gap can be softened.

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For simplicity, we consider a linear junction where the conduction band edge varies linearly in the intrinsic region with the coordinate along the junction z from z = −L/2 to L/2 as

$$\begin{equation} E_{\mathrm{c}}(z) = E_{\mathrm{g}}(1/2-a z/L), \end{equation} \tag{ 5 }$$

with E_g being the band gap [26]. We set 0 < a < 1 so that the p-doped (n-doped) region is at the left (right) side of the junction (see figure 1(b)). These electronic terminals can have temperatures different from that of the thermal terminal. It is favorable to bend the two electronic terminals away from the thermal one so that they will be better thermally isolated from the thermal terminal and from each other (see figure 1(a)). Another possibility is a 'thermal finger' for the boson bath (figure 1(c)), which can be well isolated from the electronic leads.

We start by considering only phonon-assisted transport, ignoring phononic thermal conduction and normal diode transport. The phonon-assisted inter-band transitions generate current flow in the junction. In the linear-response regime, the thermoelectric transport equations are written as [16]

$$\begin{equation} \left( \begin{array}{c} I_{\rm e}\\ I_{Q}^{\mathrm{e}}\\ I_{Q}^{\mathrm{pe}}\end{array}\right) = \left( \begin{array}{cccc} G_{\rm in} & L_1 & L_2 \\ L_1 & K_{\rm e}^0 & L_3 \\ L_2 & L_3 & K_{\mathrm{pe}} \\ \end{array}\right) \left(\begin{array}{c} \delta\mu/e \\ \delta T/T\\ \Delta T/T \end{array}\right) . \end{equation} \tag{ 6 }$$

Here I and I^e_Q are the electronic charge and heat currents flowing between the two electronic terminals, e < 0 is the electronic charge, I^pe_Q is the heat current from the thermal terminal to the two electronic ones, G_in is the conductance in the inelastic channels, K⁰_e is related to the electronic heat conductance between the electronic terminals and K_pe is that between the thermal terminal and the electronic ones. L₃ is the off-diagonal heat conductance. L₁ and L₂ are related to the currents induced by the temperature differences (thermopower effect) and the current-induced temperature differences (refrigerator and heater effects). δμ = μ_L − μ_R (δT = T_L − T_R) is the chemical potential (temperature) difference between the two electronic terminals, and $\Delta T=T_{\mathrm { p}}-\frac {1}{2}(T_{\mathrm { L}}+T_{\mathrm { R}})$ is the difference between the temperature of the thermal terminal and the average temperature of the two electric ones, with T_L, T_R and T_p being the temperatures of the left and right electronic terminals and the phonon terminal, respectively. With these definitions of ΔT, δT and I^e_Q, I^pe_Q, the Onsager reciprocal relationships are satisfied. In such a setup, as found in [16], the three-terminal Seebeck coefficient and figure of merit are, when G_el and K_pp are neglected,

$$\begin{eqnarray} &&S_{\rm p} = \frac{L_2}{TG_{\rm in}} , \quad \tilde{Z}T = \frac{ L_2^2}{G_{\rm in}K_{\mathrm{pe}} - L_2^2}, \end{eqnarray} \tag{ 7 }$$

respectively. To obtain the transport coefficients and the figure of merit, we need to calculate the phonon-assisted currents through the system.

The Hamiltonian of the inter-band electron–phonon coupling is [27]⁷

$$\begin{equation} H_{\rm e-ph} = \frac{1}{\sqrt{V}}\sum_{{\bf k}{\bf q}\lambda \nu \rho} M_{{\bf q}\lambda \nu \rho}c^\dagger_{{\bf k+q}\rho}c_{{\bf k}\nu}(a_{{\bf q}\lambda} + a^\dagger_{-{\bf q}\lambda}) + {\rm H.c.}, \end{equation} \tag{ 8 }$$

where V is the volume of the system, λ is the phonon branch index, ν (ρ) runs through the valence (conduction) band indices, M _qλνρ is the matrix element of the electron–phonon coupling and c^† (a^†) is the electron (phonon) creation operator. Due to momentum (when valid) and energy conservations, phonons involved in such processes will be in a small energy range. For indirect-band semiconductors and with momentum conservation, these phonons can be acoustic as well as optical ones. For simplicity and definiteness, we consider a direct-band semiconductor system and assume that the contribution from the optical phonons is the dominant one. From equation (8), the net electron–hole generation rate per unit volume, g_p, is given for single-phonon transitions by the Fermi golden rule as

$$\begin{eqnarray} &&\fl g_{\rm p} = \frac{2\pi}{\hbar} \sum_{{\bf k}{\bf q}\lambda \nu \rho}|M_{{\bf q}\lambda \nu \rho}|^2 [(1-f_{{\bf k+q}\rho})f_{{\bf k}\nu}N_{{\bf q}\lambda}- f_{{\bf k+q}\rho} (1-f_{{\bf k}\nu})(N_{{\bf q}\lambda} +1) ]\delta(E_{{\bf k+q}\rho}-E_{{\bf k}\nu}-\omega_{{\bf q}\lambda})\nonumber\\ &&= \int \,{\rm d}E_{ i}\,{\rm d}E_j \Gamma_{\rm p}(E_{ i},E_j) [(1-f(E_j))f(E_{i})N(\omega_{ji})\nonumber\\ &&-f(E_j)(1-f(E_{\rm i}))(N(\omega_{ji})+1)] , \end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray} \fl \Gamma_{\rm p}(E_{i},E_j) &=& \frac{2\pi}{\hbar} \sum_{{\bf k}{\bf q}\lambda \nu \rho} |M_{{\bf q}\lambda \nu \rho}|^2 \delta(E_j-E_{i}-\omega_{{\bf q}\lambda}) \delta(E_{{\bf k+q}\rho}-E_j) \delta(E_{{\bf k}\nu}-E_{i}) . \end{eqnarray} \tag{ 10 }$$

Here, E _k+qρ, E _kν and ω _qλ are the electron and phonon energies when the two systems are uncoupled, and f and N are the non-equilibrium distributions of electrons and phonons in the intrinsic region, with ω_ji ≡ E_j − E_i > 0 due to energy conservation. According to [28], when the length of the intrinsic region L is sufficiently smaller than the carrier diffusion length, the electronic distribution in the conduction (valence) band in the intrinsic region can be well approximated as the distribution in the n-doped (p-doped) electronic terminal [28]. Similarly the phonon distribution is almost the same as that in the thermal terminal when the contact between the intrinsic region and the thermal terminal is good. Finally, a small amount of disorder that always exists in real systems and relaxes the momentum conservation can enhance the phonon-assisted inter-band transitions.

The transport coefficients are determined by studying the currents at a given bias and/or a temperature difference. The key relation is the continuity equation [28, 29]

$$\begin{equation} \partial_t n_\alpha(z,t) = \frac{n_\alpha^{eq}(z)-n_\alpha(z,t)}{\tau_\alpha} - \frac{1}{q_\alpha}\partial_z I_\alpha(z,t) + g_{\rm p} , \end{equation} \tag{ 11 }$$

where n_α (α = e,h) are the electron and the hole densities in the conduction and valence bands, and n^eq_α are the equilibrium values of those densities. I_α are the charge currents, q_α are the charges of the electron and the hole, τ_α are the carrier lifetimes limited by the recombination processes other than the phonon-assisted ones that have already been taken into account in g_p, and g_p is the net carrier density generation rate given in equation (9). The currents I_α, which consist of diffusion and drift parts, are

$$\begin{equation} I_\alpha(z) = -{e} \chi_\alpha n_\alpha(z) {\cal E} - q_\alpha D_\alpha\partial_z n_\alpha(z), \end{equation} \tag{ 12 }$$

where χ_α and D_α are the mobilities and the diffusion constants, respectively. They are related by the Einstein relation, D_α = −(k_BT/e)χ_α. ${\cal E}= aE_{\mathrm {g}}/(eL)$ is the built-in electric field in the intrinsic region.

If Boltzmann statistics for the electrons can be assumed everywhere, the net generation rate g_p will depend on z very weakly such that its spatial dependence can be ignored. In this situation, the total carrier densities can be divided into two parts, n_α = n_α,g + n_α,n where n_α,g = g_pτ_α are the spatially independent carrier densities generated by the phonon-assisted inter-band transitions and n_α,n are the 'normal' densities in the junction, determined by the continuity equation with g_p = 0. Similarly, the current is divided into two parts, I_α = I^α_n + I^α_g. The current in the normal diode channel can be obtained from equations (11) and (12) with proper boundary conditions, yielding the celebrated rectification current–voltage relation

$$\begin{equation} I_n^\alpha = I_{ns}^{\alpha}({\rm e}^{-\delta \mu/k_{\rm B}T}-1), \end{equation} \tag{ 13 }$$

with I^α_ns = −eD_αL⁻¹_αn^m_α being the saturated currents. Here n^m_α is the density of the minority carrier and L_α is its diffusion length [28].

Inserting n_α,g into equation (12), one obtains the currents in the phonon-assisted channel⁸

$$\begin{equation} I_{\mathrm{g}}^\alpha = -{e} \chi_\alpha g_{\rm p}\tau_\alpha {\cal E} = - \chi_\alpha \tau_\alpha E_{\mathrm{g}}aL^{-1} g_{\rm p} \ . \end{equation} \tag{ 14 }$$

In the linear-response regime,

$$\begin{eqnarray} &&\fl g_{\rm p} = \int\,{{\rm d}E_{\rm i}}\,{\rm d}E_j \Gamma_{\rm p}(E_{i},E_j) f^0(E_{i})(1-f^0(E_j)) N^0(\omega_{ji}) \left[\frac{\delta \mu}{k_{\rm B}T}+ \frac{\overline{{E}}_{ij} - \mu}{k_{\rm B}T}\frac{\delta T}{T} + \frac{\omega_{ji}}{k_{\rm B}T}\frac{\Delta T}{T} \right] \nonumber\\ &&\equiv g_{\rm p}^0 \left[\frac{\delta \mu}{k_{\rm B}T}+ \frac{\langle \overline{{E}}_{ij} - \mu\rangle}{k_{\rm B}T}\frac{\delta T}{T} + \frac{\langle \omega_{ji}\rangle}{k_{\rm B}T}\frac{\Delta T}{T} \right], \end{eqnarray} \tag{ 15 }$$

where f⁰ and N⁰ are the equilibrium distribution functions of the electrons and the phonons, respectively, and $\overline {{E}}_{ij}\equiv (E_{\mathrm { i}}+E_j)/2$. Consequently

$$\begin{eqnarray} & G_{\rm in} = -\frac{eE_{\mathrm{g}} g_{\rm p}^0 a}{k_{\rm B}TL}\sum_\alpha \chi_\alpha \tau_\alpha,\quad L_1 = \frac{G_{\rm in}}{e} \langle \overline{{E}}_{ij}-\mu\rangle,\nonumber \\ & L_2 = \frac{G_{\rm in}}{e} \langle \omega_{ji}\rangle ,\quad K_{\rm e}^0 = \frac{G_{\rm in}}{e^2} \langle (\overline{{E}}_{ij}-\mu)^2\rangle ,\nonumber \\ & L_3 = \frac{G_{\rm in}}{e^2} \langle (\overline{{E}}_{ij}-\mu)\omega_{ji}\rangle ,\quad K_{\rm pe} = \frac{G_{\rm in}}{e^2} \langle \omega_{ji}^2\rangle , \end{eqnarray} \tag{ 16 }$$

where g⁰_p is the equilibrium transition rate (defined in equation (15)) and the average is defined in equation (4) with

$$\begin{eqnarray} \fl g_{\rm in}(E_{i},E_j)&=&-\frac{eE_{\mathrm{g}} a}{k_{\rm B}TL} \Gamma_{\rm p}(E_{i},E_j) f^0(E_{i})(1-f^0(E_j)) N^0(\omega_{ji})\sum_\alpha \chi_\alpha \tau_\alpha . \end{eqnarray} \tag{ 17 }$$

The above results are very similar to those obtained in [16]: due to the inelastic nature of the transport, the carrier energies at the p and n terminals are different; the heat transferred between the two terminals is the average one $\overline {{E}}_{ij} - \mu$, whereas the energy difference ω_ji is transferred from the thermal terminal to the two electronic ones. This is also manifested in the way the temperature differences are coupled to the heat flows [16] in equation (15), ensuring the Onsager relations. An important feature is that the three-terminal Seebeck coefficient, L₂/(TG_in), is negative definite because ω_ji > 0. In contrast, the two-terminal Seebeck coefficient, L₁/(TG_in), does not possess this property. It can be positive or negative due to the partial cancelation of the contributions from electrons and holes, whereas there is no such cancelation for L₂ (see equation (16)). Equation (16) is a generalization of the results in [16] where there was only a single microscopic energy channel. When many inelastic processes coexist, the contribution of each process is weighed by its conductance. From equations (7) and (16), we find that the 'ideal' three-terminal figure of merit is

$$\begin{equation} \left. \tilde{Z}T\right|_{\rm ideal} = \frac{\langle \omega_{ji}\rangle^2}{\langle \omega_{ji}^2\rangle - \langle \omega_{ji}\rangle^2 } . \end{equation} \tag{ 18 }$$

For a single microscopic energy channel system where ω_ji is fixed, this figure of merit goes to infinity [16]. When there are several such energy channels, it becomes finite due to the non-zero variance of ω_ji. We estimate the figure of merit when 〈ω〉 − E_g ≃ k_BT and γ,k_BT ≪ 〈ω〉. Here 〈ω〉 is the average phonon energy and γ is the effective bandwidth (i.e. the variance of ω_ji due to spectral dispersion) of the involved phonons. From equations (16) and (17) one finds that the variance of ω_ji is limited by γ² or (k_BT)², whichever is smaller. For example, when the effective phonon bandwidth γ is much smaller than k_BT, the variance is rather limited by γ². In the situations where $\gamma , k_{\mathrm { B}}T\ll \langle \omega _{ji} \rangle$, the numerator in equation (18) is much larger than the denominator. The 'ideal' figure of merit can be very high thanks to the electronic band gap when E_g ≫ k_BT or the narrow bandwidth of the optical phonons 〈ω〉 ≫ γ. However, in realistic situations, as often happens, the parasitic heat conduction is another major obstacle to a high figure of merit. This will be analyzed in the next subsection.

Besides the phonon-assisted transport channel, there is the normal diode channel that is dominated by the (elastic) barrier transmission and the diffusion of minority carriers. It contributes to G, L₁ and K⁰_e as well. In addition there are the 'parasitic' heat currents carried by phonons flowing between the two electronic terminals and those from the thermal terminal to the electronic ones. Taking into account all these, the thermoelectric transport equations are written as

$$\begin{equation} \left( \begin{array}{c} I_{\rm e}\\ I_{Q}\\ I_{Q}^{\mathrm{T}}\end{array}\right) = \left( \begin{array}{cccc} G_{\rm in} + G_{\rm el} & L_1 + L_{1,{\rm el}} & L_2 \\ L_1 + L_{1,{\rm el}} & K_{\rm e}^0 + K_{{\rm e},{\rm el}}^0 + K_{\rm p} & L_3 \\ L_2 & L_3 & K_{\mathrm{pe}} + K_{\mathrm{pp}} \\ \end{array}\right) \left(\begin{array}{c} \delta\mu/e \\ \delta T/T\\ \Delta T/T \end{array}\right). \end{equation} \tag{ 19 }$$

Here, I_Q = I^e_Q + I^p_Q is the total heat current between the two electronic terminals which consists of the electronic I^e_Q and the phononic I^p_Q contributions; I^T_Q = I^pe_Q + I^pp_Q is the total heat current flowing out of the thermal terminal to the two electronic ones with I^pp_Q being the purely phononic part. Finally, G_el, L_1,el and K⁰_e,el are the contributions to the transport coefficients from the normal diode (elastic) channel, and K_p and K_pp are the heat conductances of phonons flowing between the two electronic terminals and those flowing from the thermal terminal to the two electronic ones, respectively. Note that the elastic channel does not contribute to L₂ and L₃ which are solely related to the inelastic processes. The three-terminal figure of merit can be obtained by optimizing the efficiency of a refrigerator working at δT = 0, but with finite δμ and ΔT. This figure of merit is [16]

$$\begin{equation} \tilde{Z}T = \frac{L_2^2}{( G_{\rm in} + G_{\rm el} )( K_{pe} + K_{pp} ) - L_2^2 }, \end{equation} \tag{ 20 }$$

which is equivalent to equation (3). The diode conductance is $G_{\mathrm { el}} = -(e/k_{\mathrm { B}}T)\sum _{\alpha } I_{ns}^{\alpha }$. A high figure of merit requires G_in ≫ G_el, which is not difficult to achieve according to the analysis in the next subsection (2.4). In such a situation and when γ ≪ 〈ω〉 or k_BT ≪ 〈ω〉, i.e. when the energy width due to k_BT or γ gives a much weaker limitation to $\tilde {Z}T$ than K_pp, one finds from equation (20)⁹

$$\begin{equation} \tilde{Z}T \simeq \frac{K_{\mathrm{pe}}}{K_{\mathrm{pp}}} . \end{equation} \tag{ 21 }$$

Estimations carried out in section 2.4 indicate that there are parameter regimes where the figure of merit equation (20) can be greater than the usual two-terminal ones in the same material. We repeat that, unlike the two-terminal case, the thermal conductance K_p between the electronic terminals does not affect the three-terminal figure of merit.

Often K_p and K_pp are of the same order of magnitude. We note that K_p and K_pp are small in several gapless semiconductors such as PbSnTe, PbSnSe, BiSb, HgTe and HgCdTe. In addition, superlattice structures (and other planar composite structures) usually have much lower K_p and K_pp along the growth direction than the bulk materials [30]. The geometry where the electric current flows along that direction is promising for high figures of merit.

We defined τ_α (α = e,h) as the carrier lifetimes due to recombination processes other than the phonon-assisted ones that have already been taken into account in g_p (see equation (9)). The carrier lifetimes due to the phonon-assisted processes, τ_e,p for the electrons and τ_h,p for the holes, satisfy the detailed balance relations g⁰_pτ_e,p = g⁰_pτ_h,p = n_i in the intrinsic region. Here n_i is the electron (hole) density in that region and g⁰_p is the equilibrium transition rate per unit volume defined in equation (15). The transition between minibands in semiconductor superlattices is usually dominated by the phonon-assisted processes in the dark limit when the miniband gap is smaller than the phonon energy [31]. We introduce the parameter ζ to write g⁰_pτ_e ≃ g⁰_pτ_h = ζn_i. This parameter is governed by the electron–phonon interaction strength, being of order unity when such a coupling is strong as is the case for some III–V (and other) semiconductors, or smaller (it will be taken as 1/4 below) [31]. Inserting this into equation (16), one finds

$$\begin{equation} G_{\rm in} \simeq e^2 (k_{\rm B}T)^{-2} E_{\rm g} \zeta n_{\rm i} \sum_\alpha D_\alpha L^{-1} \ , \end{equation} \tag{ 22 }$$

where L stands for the length of the intrinsic region, which will be taken to be much smaller than the diffusion length L_α [28]. In equation (22) we have chosen a (defined in equation (5)) to be close to 1, which amounts to high doping density $N_{\mathrm { d}}\lesssim N_0\equiv n_{\mathrm { i}}\exp [E_{\mathrm {g}}/(2k_{\mathrm { B}}T)]$ in the p-and n-doped regions. The elastic conductance is the slope of the I–V rectification characteristics, equation (13), at V =0,

$$\begin{equation} G_{\rm el} \simeq e^2 (k_{\rm B}T)^{-1} n_{\rm i}^{2} N_{\rm d}^{-1} \sum_\alpha D_\alpha L_\alpha^{-1}, \end{equation} \tag{ 23 }$$

with L_α denoting the carrier diffusion length. The majority carrier densities (doping densities) in both the n and p regions are taken to be N_d. It follows then that

$$\begin{equation} \frac{G_{\rm el}}{G_{\rm in}} \simeq \frac{k_{\rm B}T}{E_{\mathrm{g}}}\frac{n_{\rm i}}{\zeta N_{\rm d}}\frac{\sum_\alpha D_\alpha L_\alpha^{-1}}{\sum_\alpha D_\alpha L^{-1} }. \end{equation} \tag{ 24 }$$

A small ratio can be easily achieved because L < L_α,k_BT < E_g and n_i ≪ N_d. Therefore, the contribution of G_el is not the main obstacle for a high figure of merit in this device.

For G_el ≪ G_in with γ (the effective width of the relevant energy band) and k_BT being much smaller than 〈ω〉 (the typical phonon energy), the figure of merit $\tilde {Z}T$ is given by equation (21). The phononic heat conductance K_pp is, as usual, the least known and an extremely important obstacle for increasing $\tilde {Z}T$. K_pp can be reduced, in principle, by engineering the interfaces. Even without such improvement, according to the geometry K_pp ≃ 4K_p where K_p is the bulk phonon thermal conductance across the junction. In the relevant regime, K_pe ≃ e⁻²G_in〈ω〉² ≳ e⁻²G_inE²_g. From equation (22) we then obtain

$$\begin{equation} K_{\mathrm{pe}} \simeq \left(\frac{E_g}{k_{\rm B}T}\right)^{2} \mathrm{e}^{-\frac{E_{\mathrm{g}}}{2k_{\rm B}T}} E_{\mathrm{g }}\zeta N_{\rm d}\sum_\alpha D_\alpha L^{-1} . \end{equation} \tag{ 25 }$$

We shall introduce yet another parameter to characterize the ratio K_e/K_p with K_e being the electron thermal conductance in a doped sample of the same geometry and size as the junction, with doping density N_d. The temperature dependence of the ratio K_e/K_p varies with the doping, structure and temperature. The temperature dependence is assumed to be of the form K_e/K_p∝T^β with β being a constant. A temperature-independent pre-factor parameter ξ = K_eE^β_g/(K_pk^β_BT^β) is then introduced. We further assume that the transport properties in the n and p regions are similar (up to signs). The Wiedemann–Franz law implies that K_e = η(k_BT)²e⁻²G_D, where G_D denotes the electrical conductance in the doped sample and η ≃ 2 for Boltzmann statistics. According to [28], $G_{\mathrm { D}}\simeq 0.25G_{\mathrm { el}}\exp [E_{\mathrm {g}}/(k_{\mathrm { B}}T)]$ when N_d ≃ N₀ and L/L_α is small. Using equation (23) one can write $K_{\mathrm {pp}}\simeq 4K_{\mathrm { p}}\simeq 4\xi ^{-1}K_{\mathrm { e}} \simeq E_{\mathrm {g}}^\beta (k_{\mathrm { B}}T)^{1-\beta }\xi ^{-1}\eta N_0\sum _\alpha D_\alpha L _\alpha ^{-1}$. The figure of merit is then estimated as

$$\begin{eqnarray} \tilde{Z}T &\simeq& \frac{K_{\mathrm{pe}}}{K_{\mathrm{pp}}}\simeq \frac{ \left(\frac{E\mathrm{_g}}{k_{\rm B}T}\right)^{2} \,{\rm e}^{-\frac{E\mathrm{_g}}{2k_{\rm B}\mathrm{T}}} E\mathrm{_g }\zeta N_0\sum_\alpha D_\alpha L^{-1}}{ E_{\mathrm{g}}^\beta (k_{\rm B}\mathrm{T})^{1-\beta}\xi^{-1} \eta N_0\sum_\alpha D_\alpha L_\alpha^{-1}}\nonumber\\ &\simeq & \left(\frac{E_{\mathrm{g}}}{k_{\rm B}T}\right)^{3-\beta} \,{\rm e}^{-\frac{E_{\mathrm{g}}}{2k_{\rm B}\mathrm{T}}} \frac{\zeta\xi}{\eta}\frac{L_\alpha}{L} . \end{eqnarray} \tag{ 26 }$$

If the temperature dependence comes mainly from the first two factors, then a high figure of merit can be obtained for E_g/(2k_BT) = 3 − β. For large L_α/L, the three-terminal figure of merit can be larger than the two-terminal one. The power factor of the device, P = G_inS²_p, can be estimated similarly.

An especially appealing setup exploiting the topological insulator–ordinary insulator–superlattices can be built as follows (taking the superlattice Bi₂Te₃/Si as an example). On the front and back surfaces of each thin Bi₂Te₃ layer, there are protected surface states with a gapless Dirac-cone like spectrum [32]. Tunneling between the two surfaces opens a gap in the spectrum of each surface band [24]. In superlattices these states form a pair of conduction and valence minibands where the band gap can be controlled via the thickness of the two types of layers (for details see [24] and footnote¹⁰). As the energy of the optical phonon in Si is much larger than that in Bi₂Te₃, the optical phonons in Si layers are well localized in these layers. So are the optical phonons in the Bi₂Te₃ layers. Similarly, due to the significant mismatch of the mechanical properties of the two types of layers, the acoustic phonons also have difficulty in being transmitted across the interfaces. The phononic thermal conductivity along the growth direction is then considerably reduced. It should be smaller than the values for both materials in the bulk [25]. On the other hand, the phononic heat conductivity within each thin layer of Si is large. At T = 300 K, the former is 1.5 Wm⁻¹ K⁻¹ [33], whereas the latter is about 10² Wm⁻¹ K⁻¹ [34]. When the electric current is along the growth direction, the small phononic heat conductance along this direction greatly reduces the parasitic heat conduction K_pp and benefits the figure of merit. On the other hand, within each Si layer phonons are transferred efficiently from the thermal terminal to the system, which is good for enhancing the output power of the device.

We now estimate the figure of merit of the device using the example of the semiconductor made of the Bi₂Te₃/Si superlattice. We shall use the transport parameters of bulk Bi₂Te₃ to do the estimation, although the superlattice should have better thermoelectric performance [35]. First tune the superlattice structure to make E_g ≲ 〈ω〉 with 〈ω〉 being the energy of the optical phonon in Si, which is about 60 meV (see [19]) (equivalent to about 700 K). We will choose E_g = 600 K. In Bi₂Te₃ with N_d ≃ 10²⁰ cm⁻³, one finds from  [33] that κ_e ≃ κ_p at 300 K. This determines the parameter ξ to be 2^β. Using these parameters we calculate and plot the figure of merit as a function of temperature in figure 2 for a practically achievable value L_α/L = 4 and a modest value of ζ = 1/4. In the calculation we ignored the temperature dependence of ζ and L_α. The results are computed for three situations with β = 1,2 and 2.5. It is seen that the figure of merit with the several underestimations made can be larger than 1 around room temperature for all the three situations, indicating potential usefulness. Finally, we note that the same strategy and analysis can also be applied to the superlattice made of an inverted-band gapless semiconductor and a normal semiconductor, where the band gap can be tuned via the quantum-well effect [22] and the normal semiconductor can be chosen to have the proper optical phonon frequency and the high thermal conductivity to enable better thermoelectric performance.

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