Thermoelectric properties of topological chains coupled to a quantum dot

Abstract

Topological one-dimensional superconductors can sustain zero energy modes protected by different kinds of symmetries in their extremities. Observing these excitations in the form of Majorana fermions is one of the most intensive quests in condensed matter physics. We are interested in another class of one-dimensional topological systems in this work, namely topological insulators. Which present symmetry-protected end modes with robust properties and do not require the low temperatures necessary for topological superconductivity. We consider a device in the form of a single electron transistor coupled to the simplest kind of topological insulators, namely chains of atoms with hybridized sp orbitals. We study the thermoelectric properties of the device in the trivial, non-trivial topological phases and at the quantum topological transition of the chains. We show that the device’s electrical conductance and the Wiedemann–Franz ratio at the topological transition have universal values at very low temperatures. The conductance and thermopower of the device with diatomic sp-chains, at their topological transition, give direct evidence of fractional charges in the system. The former has an anomalous low-temperature behavior, attaining a universal value that is a consequence of the double degeneracy of the system due to the presence of zero energy modes. On the other hand, the system can be tuned to exhibit high values of the thermoelectric figure of merit and the power factor at high temperatures.

Introduction

The origin of thermoelectricity can be traced back to the discovery of the Seebeck effect in the 19th century. It consists in the production of electrical energy directly from heat, and its inverse, the Peltier effect, that transforms electrical energy into thermal energy. After the development of the first thermoelectric generators (TEGs) with applications in industry¹, these lost the competition with the dynamoelectric machines due to the high costs of their electrical energy generation. Their technological development was interrupted for several decades. Only in the middle of the last century, due to the needs of the aerospacial and military industries, did the interest in developing new TEGs reappear. The thermoelectricity acquires some practical applications in those strategic areas after the discovery that the doped semiconductor $Bi_{2}Te_{3}$ and its alloys $Sb_{2}Te_{3}$, and $Bi_{2}Se_{3}$^2,3,4, present high electric conductivities $\sigma$ and low thermal conductivities $\kappa$. In consequence, those thermoelectric materials (TEM) exhibit at ambient temperatures a higher dimensionless thermoelectric figure of merit (zT)^1,2 and a high power factor (PF) and, until now, dominate the commercial industry of TEGs⁵.

Recently, it was shown that the usual thermoelectric materials (TEM), like $Bi_{2}Te_{3}$, $Bi_{2}Se_{3}$, $Sb_{2}Te_{3}$, and $FeSb_{2}$^6,7,8 are also three-dimensional topological insulators exhibiting surface states with a single Dirac cone and some of their striking properties are due to their strong spin–orbit coupling^5,9,10 and their conducting surface states^11,12. A promising route to explore the effects of the topologically non-trivial surface states (TNSS) on the TE properties was followed in Ref.¹³, which studied thin films of $Bi_{2}Te_{3}$. The authors used first-principles calculations and Boltzmann theory to obtain zT for different film thicknesses. They defined a unit (QL) of quintuple layers of the real material Te–Bi–Te–Bi–Te and observed a $p-$type and $n-$type $zT \simeq 2$ peak in $QL=3$ when the system enters the topologically non-trivial regime from the trivial one. The results show a relevant enhancement of zT due to the contribution of TNSS compared to the pristine form of bulk $Bi_{2}Te_{3}$, $zT=0.4$. Another step in the direction of the use of TNSS states in real systems was obtained after the recent advances in the synthesis of $Bi_{2}Te_{3}$ thin films, which allows separating the bulk from the TNSS states in order to design quantum devices with improved thermoelectric properties¹⁴.

The study of topological systems is now one of the most active areas of research in condensed matter physics^15,16,17,18. The theoretical efforts to understand the properties of these systems has lead to the predictions of emergent excitations with unexpected properties that make them potentially useful for different types of applications. Among these works, Kitaev model for a p-wave superconducting chain¹⁹ has played a fundamental role and many suggestions have appeared of how to realize this model in actual physical systems. In the topological phase, the finite one-dimensional Kitaev superconducting chain presents Majorana, zero energy modes, at its ends. The physical implementation of the p-wave Kitaev model and the detection^{20,21,22,23,24,25,26,27,28} of the zero energy Majorana modes is a modern Graal in materials research. In this pursuit an initial major difficulty is to obtain a p-wave superconductor, since this is far from being common in nature²⁹. Several proposals have been put forward to generate this type of pairing in a chain, mostly using proximity effects and magnetic fields¹⁵.

Besides one-dimensional p-wave superconductors there is a class of topological insulating^30,31 chains that is much simpler and also presents protected zero energy modes at their ends. Representative members of this class are sp-chains consisting of atoms with hybridized s and p orbitals³². The mixing between s and p orbitals in neighboring ions is antisymmetric and this gives rise to non-trivial topological properties³³, in close analogy with the antisymmetric p-wave paring of the Kitaev chain. Notice that the asymmetry of the mixing holds for any pair of orbitals that have angular momentum quantum numbers differing by an odd number. In spite of their symmetry protection, the edge modes in topological sp-chains have distinct features from the Majoranas in the Kitaev chain. The former are quasi-particles with a hybrid sp-character that are formed of two different types of Majoranas³².

The sp-chains may be easier to realize in practice then p-wave superconductors. Also, they do not require the low temperatures necessary for superconductivity, to manifest their topological properties. A possible realization of the sp-chain is carbyne, the one-dimensional allotropic form of carbon^{34,35,36,37,38}. In this system the 2s orbital hybridizes with a single 2p orbital favoring a linear atomic alignment³⁹. A significant effort has been made in the synthesis of these materials that in principal can exist in a metallic state (cumulene) and in an insulating, broken symmetry state, with alternating single and triple bonds³⁵.

As we show in the appendix monoatomic and diatomic sp-chains can be mapped in two very well known topological chains, the Su–Schrieffer–Heeger (SSH)^40,41 and the Rice–Mele (RM)^42,43 chains, respectively. These chains have been intensively studied and their topological properties are well known. For this reason we study here the latter two models since they yield results for the thermoelectric properties similar to those of the sp-chains.

This paper studies the thermoelectric properties of two semi-infinite Rice–Mele chains connected to a quantum dot. We investigate the device’s electrical and thermal transport properties as a function of temperature, in the topologically non-trivial and trivial phases and at the topological transition. According to Refs.^13,14 we expect an increase of zT due to topological states at the edges of these chains.

This work has the following structure: in “he Rice–Mele model”, we introduce the Rice-Mele model and present its topological properties. In “The surface density of states”, we employ a method developed in Ref.⁴⁴ to obtain the local Green’s function at the edge of the chain. This yields the surface density of states for the Rice-Mele chain. In “Thermoelectric properties of two semi-infinite chains coupled to a quantum dot”, we present the device consisting of two identical semi-infinite topological chains connected to a singly occupied quantum dot⁴⁵, without correlations effects. We use linear response theory to define the thermoelectric coefficients. In “Results for SSH chains or monoatomic sp-chains” and “Results of the diatomic sp or Rice–Mele chains”, we calculate, electrical and thermal conductances, thermopower, Wiedemann–Franz ratio, power factor, and the dimensionless thermoelectric figure of merit of our device when the quantum dot is connected to monoatomic sp or SSH chains, and to diatomic sp or Rice–Mele chains, respectively. Notice that the figure of merit measures the usefulness of the device to produce electrical power. In “High temperature results”, we present the high temperature results and finally, we conclude with a discussion of our results and the perspectives of our approach.

The Rice–Mele model

The Rice–Mele model has been used to describe polymeric chains with alternating bonds⁴². It is generally associated with fractional charges that arise due to their topological properties and it is used here to model diatomic sp-chains. Its Hamiltonian is given by

$$\begin{aligned} {\mathcal {H}}_{RM}&= - V_1 \sum _n c_{A,n}^\dagger c_{B,n} - V_2 \sum _n c_{A,n+1}^\dagger c_{B,n} \nonumber \\&\quad +(\epsilon _A-\mu ) \sum _n c_{A,n}^\dagger c_{A,n} + (\epsilon _B - \mu ) \sum _n c_{B,n}^\dagger c_{B,n} + H.c., \end{aligned}$$

(1)

where $c_{(A,B),n}^\dagger$ and $c_{(A,B),n}$ create and annihilate electrons on site n of sub-lattice (A,B), respectively. The hopping $V_1$ connect electrons in the same unit cell n, and $V_2$ those in different unit cells. The site energies $\epsilon _{(A,B)}$ are different in sub-lattices A and B and $\mu$ is the chemical potential. For a semi-infinite chain the sum extends from $n=0$ to $n=\infty$. The SSH model is obtained from the RM model, Eq. (1), when the site energies are taken equal zero, i.e., $\epsilon _A=\epsilon _B=0$.

The energy of the bands of the infinite, translation invariant RM chain can be obtained transforming to momentum space and diagonalizing the Hamiltonian³⁰. They are given by

$$\begin{aligned} \tilde{\omega }_1(k)=-\tilde{\mu } +\sqrt{2 \tilde{V} \cos (k)+\tilde{V}^2+\tilde{\epsilon }^2+1} \end{aligned}$$

(2)

$$\begin{aligned} \tilde{\omega }_2(k)=-\tilde{\mu } -\sqrt{2 \tilde{V} \cos (k)+\tilde{V}^2+\tilde{\epsilon }^2+1}. \end{aligned}$$

(3)

The extrema of the bands occur for $k=\pi$. Notice that there is always a gap between the bands, which is given by

$$\begin{aligned} \tilde{\Delta }=|\tilde{\omega }_1(\pi )-\tilde{\omega }_2(\pi )|=2 \sqrt{(1-\tilde{V})^2+\tilde{\epsilon }^2}. \end{aligned}$$

(4)

The tilde quantities are dimensionless, normalized by the hopping $V_2$ and $\tilde{V}=V_1/V_2$. We took $\epsilon _A=-\epsilon _B=\epsilon$. In the case of the SSH model, with $\epsilon =0$, the band gap closes for $\tilde{V}=1$, at the topological transition.

The topological properties of the RM and SSH chains are well known^18,30. For the latter there is a non-trivial topological phase for $\tilde{V}<1$ characterized by a non-trivial winding number. For $\tilde{V}=1$ there is a topological transition for a topologically trivial phase with $\tilde{V}>1$. In the non-trivial topological phase there are edge modes at the ends of a finite chain. These edge states decay into the bulk with a characteristic length that depends on the distance to the topological transition, $\xi =(1-\tilde{V})^{-\nu }$. At the topological transition $\xi$ diverges and the surface state spreads into the bulk³². For the SSH model the critical exponent $\nu =1$.

The topological properties of the RM model are more complex, but also well known^18,30. The topological phases can be characterized by Chern numbers^{18,30,43,46,47}, $n_C=-{\textrm{sgn}}[\epsilon (V_2-V_1)]$, such that for $V_1=V_2$ or $\epsilon =0$ there are topological quantum phase transitions^30,46,47. The phase with $\tilde{V} < 1$ is topologically non-trivial.

The surface density of states

In order to obtain the thermoelectric properties of our device, we need to calculate the surface density of states of the semi-infinite RM and SSH chains. Here, we use a method developed in Ref.⁴⁴ that yields the local Green’s functions at the edge of this chain. This Green’s function is obtained from the self-consistent equation,

$$\begin{aligned} G_{00}(\omega ) = \begin{pmatrix} \omega +\mu -\epsilon &{} V_{1} &{} 0 \\ V_{1} &{} \omega +\mu +\epsilon &{} V_{2} \\ 0 &{} V_{2} &{} G_{00}^{-1} \end{pmatrix}_{(00)}^{-1}, \end{aligned}$$

(5)

from which, we can get the surface density of states,

$$\begin{aligned} \rho (\omega )=\frac{-1}{\pi } \textrm{Im} G_{00}(\omega ). \end{aligned}$$

(6)

SSH chains

Let us start with the simpler case of the semi-infinite SSH chain, which corresponds to the RM model with $\epsilon =0$. From Eq. (5), with $\epsilon =0$, we obtain a self-consistent problem involving a second degree algebraic equation for the local Green’s function,

$$\begin{aligned}{}[V_{2}G_{00}]^{2}-2\alpha [V_{2}G_{00}] + 1=0, \end{aligned}$$

(7)

with

$$\begin{aligned} \alpha (\omega )=\frac{\omega ^{2}+V_{2}^{2}-V_{1}^{2}}{2V_{2}\omega }. \end{aligned}$$

(8)

We consider the case of half-filled band and take $\mu =0$. The surface Green’s function can be directly obtained from Eq. (7). It is given by,

$$\begin{aligned} G_{00}(\omega )=\frac{1}{2} \frac{1}{ \omega }\left[ \tilde{\omega }^2-\tilde{V}^2+1\pm \sqrt{(\tilde{\omega }^2-\tilde{V}^2+1)^2-4 \tilde{\omega }^2}\right] \end{aligned}$$

(9)

where $\tilde{\omega }=\omega /V_{2}$, and $\tilde{V}=V_1/V_2$.

The surface density of states is obtained from Eqs. (6) and (9) and is given by

$$\begin{aligned} \rho (\omega )= \frac{1}{2} \left\{ D \delta (\tilde{\omega }) + \frac{1}{\pi }{\textrm{Im}} [\frac{ \sqrt{(\tilde{\omega }^2+D)^2-4 \tilde{\omega }^2}}{\tilde{\omega }} ]\right\} . \end{aligned}$$

(10)

where $D=1-\tilde{V}^2$ and $\tilde{\omega } \rightarrow \tilde{\omega }+i \epsilon$. The sign of the root is chosen so that the density of states is positive and from now on we take $V_2=1$. There is an additional contribution to the zero energy mode due to the second, square root term. Considering this explicitly, we can rewrite Eq. (10) as

$$\begin{aligned} \rho (\omega )= \frac{1}{2} \left\{ (D+|D|) \delta (\tilde{\omega }) + \frac{1}{\pi }\frac{{\textrm{Im}}\sqrt{(\tilde{\omega }^2+D)^2-4 \tilde{\omega }^2}}{\tilde{\omega }}\right\} , \end{aligned}$$

(11)

where one sees that the zero energy mode only appears for $D>0$, or $\tilde{V}<1$, i.e., in the topological phase of the chain (see Fig. 1). In the trivial phase there is a cancellation and the zero energy surface mode disappears. Notice that the zero energy mode is a true surface state since its energy does not coincide with any of the bulk states.

Figure 1

(a) Density of states at the surface of the semi-infinite SSH chain and (b) real part of the surface Green’s function. In the topological phase (red, dashed), the trivial phase (black, continuous) and at the topological transition (blue, dotted). We took $V_2=1$, such that $\tilde{\omega }=\omega$.

For completeness and since it will be used further on, we also obtain the real part of the surface Green’s function (see Fig. 1). This is given by

$$\begin{aligned} {\textrm{Re }} G_{00}(\tilde{\omega })=\frac{1}{2}\left\{ D\frac{1}{\tilde{\omega }} + \tilde{\omega } +\frac{{\textrm{Re}}\sqrt{(\tilde{\omega }^2+D)^2-4 \tilde{\omega }^2}}{\tilde{\omega }} \right\} . \end{aligned}$$

(12)

Notice that

$$\begin{aligned} \lim _{\tilde{\omega } \rightarrow 0}{\textrm{Re }} G_{00}(\tilde{\omega })=\frac{1}{2} \frac{(D+|D|)}{\tilde{\omega }}. \end{aligned}$$

Then, we find that the surface Green’s function gives direct information on the topological state of the chain. Furthermore, the weight of the zero energy mode vanishes linearly with the distance to the topological transition ($D \propto (1-\tilde{V})$).

Rice–Mele chains

The surface density of states of the semi-infinite RM chain, obtained from Eqs. (5) and (6) is given by

$$\begin{aligned} \rho (\omega )\!=\!\left( D+\left| D \right| \right) \delta (\omega + \mu -\epsilon )+ \frac{\text {sgn}(\omega + \mu )\Im m [R(\omega )]}{2 (\omega + \mu -\epsilon )} \end{aligned}$$

(13)

with

$$\begin{aligned} R(\omega ){} & {} =\left( -(\mu +\omega )^2+(1-\tilde{V})^2+\epsilon ^2\right) ^{\frac{1}{2}} \nonumber \\{} & {} \quad \times \left( -(\mu +\omega )^2+(1+\tilde{V})^2+\epsilon ^2\right) ^{\frac{1}{2}} \end{aligned}$$

(14)

Notice the presence of a surface mode at a finite energy $\omega _S=\epsilon -\mu$, for $\tilde{V}<1$ (since $V_2=1$ we keep the tilde only in $\tilde{V}$). Differently from the SSH model, as seen in  “SSH chains”, the RM system is always gapped even at $\tilde{V}=1$. However, the phases with $\tilde{V} > 1$ and $\tilde{V} < 1$ can still be distinguished by the absence or presence, respectively of the surface mode (besides their Chern numbers)^{18,30,43,46,47}. The phase with $\tilde{V} < 1$ is the topologically non-trivial.

Figure 2

Surface density of states of the semi-infinite diatomic sp-chain (RM chain), Eq. (13), at the topological transition ($\tilde{V}=1$), for (a) $\mu =\epsilon =- 0.3$; (b) $\mu =\epsilon =0.3$. Real part of the surface Green’s function for (c) $\mu =\epsilon =- 0.3$ and (d) $\mu =\epsilon =0.3$.

Figure 2 shows the surface density of states and the real part of the surface Green’s function for RM-chains at the topological transition $\tilde{V}=1$. The figures are for two values of the energy of the local surface mode, $\epsilon =\pm 0.1$. The chemical potential is located on the energies of these modes ($\mu =\epsilon$).

Thermoelectric properties of two semi-infinite chains coupled to a quantum dot

In this section we study the transport properties of a device consisting of two identical semi-infinite chains connected to a quantum dot^45,48, as shown in Fig. 3. Since we are dealing with spinless fermions, the dot can either be empty, or singly occupied. The non-interacting quantum dot has a single state with energy $E_0$ and is coupled to the chains by a hopping term $t_d$ that transfers quasi-particles in and out of the dot. Then, the dot provides a connection between the semi-infinite chains and allows to probe the nature of the edge states through their contribution to the thermal and electrical conductances of the device, as we discuss below. The coupling Hamiltonian between the dot and the semi-infinite chains is given by, $H_c=-\sum _{\alpha }t_{d,\alpha } c^{+}_{\alpha ,0} d+H.c.$, where the second quantization operators c and d refer to the chains and dot and $\alpha =r,l$ to the right and left chains, respectively. The dot couples to the first site of each chain (site 0)⁴⁹. For simplicity, we take here $t_{d,r}=t_{d,l}=t_{d}$.

The full local Green’s function of the dot connected to the two semi-infinite chains is given by⁴⁹,

$$\begin{aligned} G_d(\omega )=\frac{g_d}{1-2|t_d|^2 g_d G_{00}} \end{aligned}$$

(15)

where,

$$\begin{aligned} g_d=\frac{1}{\omega -E_0} \end{aligned}$$

(16)

is the Green’s function of the non-interacting dot. The Green’s function $G_{00}$ is that of the edge of the chains and is given by the self-consistent solution of Eq. (5). Notice that Eq. (15) can be rewritten as,

$$\begin{aligned} G_d(\omega )=\frac{1}{\omega -E_0-2|t_d|^2{\textrm{Re}}G_{00}-i2|t_d|^2{\textrm{Im}}G_{00}}. \end{aligned}$$

(17)

Figure 3

Two semi-infinite sp-chains connected to a quantum dot (QD). A very small potential difference $V_c$ is applied in the chains. Notice that $t_d$ is the coupling between the dot and the chains.

The dimensionless electrical conductance of the device, chain-dot-chain can be obtained as in Ref.⁵⁰. It is given by,

$$\begin{aligned} G/G_0=\int d\omega (-\frac{\partial f}{\partial \omega }) {\mathcal {T}}(\omega ) \end{aligned}$$

(18)

where $f(\omega )$ is the Fermi–Dirac distribution and

$$\begin{aligned} {\mathcal {T}}(\omega ) =-\Gamma {\textrm{Im}} G_d(\omega ) \end{aligned}$$

with $\Gamma = 2 \pi |t_d|^2 {\textrm{Im}} G_{00}$, the Anderson broadening⁵⁰. In Eq. (18), $G_0=e^2/h$ is the quantum of conductance.

More generally, we define the quantities,

$$\begin{aligned} {\mathcal {L}}_n=\frac{1}{h} \int d\omega (-\frac{\partial f}{\partial \omega }) \omega ^n {\mathcal {T}}(\omega ), \end{aligned}$$

(19)

in terms of which we can obtain the thermoelectric coefficients. The conductance can be rewritten as $G=e^2\mathcal {L}_0$. The thermal conductance K and the thermopower S are given, respectively, by

$$\begin{aligned} K= & {} \frac{1}{T} \left( {\mathcal {L}}_2 - \frac{{\mathcal {L}}_1^2}{{\mathcal {L}}_0} \right) , \end{aligned}$$

(20)

$$\begin{aligned} S= & {} -\left( \frac{1}{e T}\right) \frac{\mathcal {L}_1}{\mathcal {L}_0}. \end{aligned}$$

(21)

These in turn define the Wiedemann–Franz ratio (WF) and the dimensionless figure of merit ZT that are given, respectively, by

$$\begin{aligned} WF= & {} \frac{1}{T} \left( \frac{K}{G}\right) , \end{aligned}$$

(22)

$$\begin{aligned} ZT= & {} \frac{S^2GT}{K}, \end{aligned}$$

(23)

where the former ratio WF is given in units of the Lorenz number $L_0=(\pi ^2/3)(k_B/e)^2$.

The Mahan–Sofo parameter $\zeta$⁵¹ is defined in terms of the thermoelectric coefficients

$$\begin{aligned} \zeta = \frac{L_{1}^{2}(T)}{L_{0}(T)L_{2}(T)} , \end{aligned}$$

(24)

and using this parameter, the dimensionless thermoelectric figure of merit, defined in Eq. (23), can be written as

$$\begin{aligned} ZT=\frac{\zeta }{1-\zeta }. \end{aligned}$$

(25)

The best ZT occurs at the limit $\zeta \rightarrow 1$.

Results for SSH chains or monoatomic sp-chains

We start obtaining the thermoelectric properties of the device in the case the dot is coupled to SSH chains. We calculate, using the equations above, the thermoelectric properties of the coupled system, dot-chains, in the different topological phases of the SSH chains and at the topological transition. When the chains are in either the trivial or topological phases, i.e., for $\tilde{V}>1$ and $\tilde{V}<1$, respectively, the conductances are zero at zero temperature, since the bulk of the chains are insulators. At finite temperatures these conductances become finite due to thermal activation of quasi-particles above the band gap. The results presented are obtained for the chemical potential of the chains $\mu =0$, i.e., for a full lower band (half-filling). The dot energy is $E_0=0$, and the coupling between the dot and the chains is taken as, $t_d/V_2=0.15$.

Figure 4

(a) Dimensionless electrical conductance, (b) thermal conductivity divided by temperature in units of $G_0L_0$ and (c) Wiedemann–Franz ratio ($WF=(K/T)/(G/G_0)$) in units of $G_0L_0$ as functions of temperature for the device with SSH chains. In the trivial phase ($\tilde{V}=1.03$) black continuous, topological phase ($\tilde{V}=0.97$) red dashed and at the topological transition ($\tilde{V}=1$) blue continuous.

Figure 4a shows the conductance of the device in the trivial and topological phases. As expected the conductances vanish at $T=0$ in both phases and become finite at finite temperatures. The finite temperature conductance is larger in the topological phase. Notice that in both cases shown, $\tilde{V}=1.03$ and $\tilde{V}=0.97$, the system is at the same distance of the topological transition at $\tilde{V}=1$. The increment of the conductance in the topological phase can be attributed to the presence of the edge mode.

Figure 4b shows the thermal conductivity divided by temperature, in units of $G_0L_0$. They also vanish for $T=0$, in both the trivial and topological phases, as expected, since the bulk of the chains is insulating.

Figure 4c shows the Wiedemann–Franz ratio, defined as $WF=(K/T)/(G/G_0)$ and in units of $G_0L_0$, as a function of temperature. Away from the topological transition in both trivial and topological phases the Wiedemann–Franz law is violated. This can occur in topological systems^21,52, as for the monoatomic chains and in general for diatomic chains, as we discuss below and show in Fig. 11.

Figure 4a–c, also show the conductance, thermal conductivity and Wiedemann–Franz ratio at the topological transition, i.e., at $\tilde{V}=1$. The zero temperature dimensionless electrical conductance in this case is unity showing that a quantum of charge flows through the system. Then, at the transition the surface modes recombine to form a quasi-particle that transports electric current through the dot. The current can flow through the device since, at $V_1=1$, the chains are in a semi-metallic state (Dirac semi-metal). We point out that the zero temperature electrical conductance at the transition does not depend on the coupling $t_d$ between the dot and the chains. The thermal conductance K, differently from the electrical conductance vanishes at zero temperature, even at the topological transition. However, the temperature normalized thermal conductance (K/T) at the topological transition goes in this limit to 1, in units of $G_0L_0$ , as shown in Fig. 4b. The Wiedemann–Franz ratio, at the topological transition of the monoatomic chain, starts as unity at $T=0$ and remains constant at very low temperatures showing that the Wiedemann–Franz law is obeyed in this case.

Finally, we remark that the thermopower, Eq. (21), vanishes at the trivial and topological phases and also at the topological transition. This occurs since the quantity $\mathcal {L}_1$ in this equation cancels out due to equal but opposite contributions of electrons and holes to this quantity in this particle-hole symmetric case.

Results of the diatomic sp or Rice–Mele chains

In this section, we obtain the thermoelectric properties of the device when Rice–Mele chains are attached to the quantum dot. Notice that in this case the chiral symmetry of the SSH chain is broken for RM chains. We consider the situation where the chemical potential coincides with the local energy of one of the sub-lattices, i.e., we take $\mu =\pm \epsilon$. Furthermore we consider that the quantum dot is in resonance with the energy of the edge mode, which for the condition $\mu =\pm \epsilon$ corresponds to take $E_0=0$. Since $\epsilon \ne 0$, the topological transition of the model occurs for $\tilde{V}=1$. We start showing the normalized temperature dependent conductances of the Rice-Mele model at the topological transition. As can be seen in Fig. 5, the normalized conductances at zero temperature now attains a value of 1/2, expected when fractional charges e/2 are responsible for the electronic transport in the system. This result is universal in the sense that it is independent of the coupling $t_d$ between the dot and the chains and the value of $\epsilon$, for the conditions specified above ($\mu =\pm \epsilon$, $E_0=0$). Whenever we use this term here we refer to this type of universality. The figure shows the normalized conductance for two values of $\epsilon /V_2$. Notice that for $\epsilon /V_2 \ll 1$ the finite temperature conductance reaches a maximum value close to one, as if there is a recombination of the fractional charges in the system due to thermal effects.

Figure 5

Normalized conductances as a function of temperature for a system consisting of two semi-infinite Rice–Mele chains attached to a quantum dot at the topological phase transition of the chains ($\tilde{V}=1$). In blue continuous $\epsilon /V_2=0.1$, and in red dashed $\epsilon /V_2=5 \times \, 10^{-5}$. In both cases, the low temperature saturation value $G/G_0=1/2$ gives evidence of fractional charges flowing in the system. The curves for $G/G_0$ are independent of the coupling to the quantum dot and of the sign of $\epsilon$, for $\mu =\epsilon$ and the dot in resonance with the edge mode. Notice that for small values of $\epsilon /V_2$ as temperatures increases there is a kind of recombination of the fractional charges.

We point out that the fractional charge as evidenced by the zero temperature conductance is a direct consequence of the breaking of chiral symmetry of the original SSH model, due to the finite and distinct energies of the sub-lattices of the Rice–Mele model.

Thermopower

The thermopower is an interesting and unique physical property that contains fundamental information on both, transport and thermodynamic properties of the system. The temperature dependence of the thermopower of the device consisting of two Rice–Mele chains coupled to the quantum dot can be obtained using Eq. (21). At the topological transition ($\tilde{V}=1$), this is shown in Fig. 6 for $\mu = \epsilon$ and the cases of $\epsilon$ positive and negative. The corresponding surface density of states for these two cases is shown in the upper panels of Fig. 2. The thermopower is positive or negative depending whether the charge carriers are holes or electrons, respectively. It is constant at low temperatures and its absolute value decreases with increasing temperature. It is remarkable that it does not vanish for $T \rightarrow 0$, as expected from the third law of thermodynamics. Mathematically, this arises since the function $\mathcal {T}(\omega )$ in Eq. (19) has a jump discontinuity and is non-differentiable at $\omega =0$, which precludes a low temperature Sommerfeld expansion.

The constant low temperature values for the thermopower, $S(T \rightarrow 0) \approx \pm 1.386$ can be rationalized in terms of the properties of the quantum dot and of the chains at the topological transition. Since we took $\mu =\epsilon$, the doubly degenerate zero energy surface mode⁵³ becomes delocalized at the transition and every site in the system including the dot has a double degenerescence. For a system of charged particles, the thermopower represents the entropy per carrier divided by the charge of the carrier⁵⁴,

$$\begin{aligned} S_0=\frac{\text {entropy per carrier}}{q^*}. \end{aligned}$$

(26)

This is also known as the Kelvin formula for the Seebeck coefficient⁵⁵. The entropy per site is $S= \ln 2$ and remains finite at $T=0$ due to the double degeneracy of the states, whether a site is occupied by a particle or by a hole. If the carriers have a fractional charge, $q^*= \pm 1/2$ (in units of electric charge) as evidenced by the zero temperature conductance, we get

$$\begin{aligned} S_0=\frac{\ln 2}{(\pm 1/2)}=\pm 2 \ln 2 \approx \pm 1.386 (k_B/e), \end{aligned}$$

(27)

which are exactly the low temperature saturation values, obtained numerically for the thermopower using Eq. (21), as shown in Fig. 6. These values are universal in the same sense we used for the conductance, i.e., they are independent of $\epsilon$ and $t_d$ (for $\mu =\epsilon$, $E_0=0$).

Figure 6

Thermopower of the device as a function of temperature in units of $(k_B/e)$ at the topological transition of the RM chains. $S>0$ correspond to $\mu =\epsilon =+0.1$, and $\mu =\epsilon =+3.3 \times \, 10^{-4}$ (light curve). Negative thermopower ($S<0$) corresponds to $\mu =\epsilon =-0.1$ and $\mu =\epsilon =-3.3 \times \, 10^{-4}$ (light curve). The light color curves show the trend to the results the SSH chain with $\epsilon =0$. The energy scale for the low temperature saturation of the thermopower is given by the difference in site energies, $2 \epsilon$. The numerical results for the saturation values, $S(T=0) \approx \pm 1.38634$ are in close agreement with $S_0=\pm 2 \ln 2 \approx \pm 1.38634$, as discussed in the text.

Figure 7

Thermopower of the device as a function of temperature in units of $(k_B/e)$ away and at the topological transition of the RM chains. Red dashed corresponds to $\tilde{V}=0.95$, such that the chains are in the topological phase. Black continuous shows the thermopower in the trivial phase, with $\tilde{V}=1.05$ and blue at the topological transition. The gray dashed line shows the classical result for a semiconductor with activation energy $\Delta$.

Then this result for the thermopower, together with that for the electrical conductance, corroborate the existence of carriers with fractional charges, $q=\pm e/2$, flowing in the device with RM chains at the topological transition. This transport charge does not necessarily coincide with the notion of a boundary charge in topological chains⁵⁶. This concept is useful when the system is in the non-trivial topological phase but away from the topological transition⁵⁶. The wave function of the edge modes is finite inside the bulk of the material, and the boundary charge is obtained by integrating over a finite characteristic length that depends on the model’s parameters. The fractional charges we get from the transport properties appear at the topological transition. In this case, the relevant length scale is the penetration depth of the edge modes that diverge at this transition^32,57,58. It implies that the edge mode charge is spread all over the system; consequently, the concept of a boundary charge becomes meaningless.

For completeness we show in Fig. 7 the temperature dependent thermopower away from the topological transition in both trivial and non-trivial topological phases.

Thermal conductance and Wiedemann–Franz ratio

The thermal conductance divided by temperature (K/T) at the topological transition of the diatomic sp-chain is shown in Fig. 9. From Eqs. (19) to (22), we can write

$$\begin{aligned} \frac{K}{T} = \frac{{\mathcal {L}}_2}{T^2}- \left( \frac{{\mathcal {L}}_1}{e T {\mathcal {L}}_0}\right) ^2 e^2 {\mathcal {L}}_0, \end{aligned}$$

(28)

and using the expressions for the thermopower and conductance we get,

$$\begin{aligned} \frac{K}{T} = \frac{{\mathcal {L}}_2}{T^2}- S^2 G. \end{aligned}$$

(29)

Figure 8

Thermal conductance divided by temperature of the device at the topological transition of the RM chains in units of $G_0L_0$, where $L_0$ is the Lorenz number for $\epsilon =01$ blue continuous, and $\epsilon =0.3$ red dashed. The zero temperature limiting value $(K/T)_0 \approx 0.20792$ (see text). is independent of the values of $t_d$ and $\epsilon$ as long as, $\mu =\epsilon$ and $E_0=0$.

We can obtain the limit of zero temperature analytically, $(K/T)_0 = ( K/T)_{T \rightarrow 0}$, using the results for the thermopower, Eq. (27), and for the conductance. We find,

$$\begin{aligned} (K/T)_0= \frac{1}{2}(1 - \frac{3}{\pi ^2}(2 \ln 2)^2) \approx 0.20792, \end{aligned}$$

(30)

in units of $G_0L_0$. This is in agreement with the numerical result shown in Fig. 8 and it is independent of $\epsilon$ and $t_d$. The dimensionless Wiedemann–Franz ratio attains at zero temperature the value, $\mathcal {W}= (WF/L_0)= 1/2$. This value of $\mathcal {W}$ is different from that for metallic chains where $\mathcal {W}=1$. Violation of the Wiedemann–Franz law has been found in interacting systems^21,59 and in devices with interacting quantum dots^60,61.

For completeness, we point out that away from the topological transition, both in the trivial and non-trivial topological phases we obtain that the conductance and thermal conductivities are thermally activated as in a semi-conductor.

Figure of merit and power factor

Figure 9 shows the dimensionless power factors⁶² and figures of merit $ZT=(S^2GT)/K$ of the device, as functions of temperature, at the topological transition, $\tilde{V}=1$, and in the trivial $\tilde{V}=1.05$ and topological $\tilde{V}=0.95$ phases of the RM chains. The power factor is defined as $PF=(\widetilde{PF}/S_0^2 G_0)$, where $S_0$ is the zero temperature thermopower and $G_0$ the unit of conductance. The quantity $\widetilde{PF}=S^2G$ is the full dimensional power factor⁶³. These quantities ZT and PF do not depend on the sign of $\epsilon$, only on its absolute value.

Figure 9

(a) Dimensionless electrical conductance, (b) thermal conductivity divided by temperature in units of $G_0L_0$ and (c) Wiedemann–Franz ratio ($WF=(K/T)/(G/G_0)$) in units of $G_0L_0$ as functions of temperature for the device with SSH chains. In the trivial phase ($\tilde{V}=1.03$) black continuous, topological phase ($\tilde{V}=0.97$) red dashed and at the topological transition ($\tilde{V}=1$) blue continuous.

Notice that the figures of merit ZT at the trivial and topological phases assume large values, for the parameters used in Fig. 9 at a temperature of$T/V_2 \approx 0.01$, where the power factor is close to a maximum. In order to translate this in physical temperature notice that the energy scale $V_2$ is of the order of a bandwidth ($\sim 1$ eV or $\sim 10^4$ K). In the trivial semiconductor phase although the figure merit increases at lower temperatures, the power factor drops to very small values, while it continues significant at the topological transition. The significance of this quantity (PF) is that, in a time reversible system at steady state, the maximum power for conversion of heat into work is given by $P_{max}= (1/4) \widetilde{PF}$ for two heat reservoirs with a difference in temperature $\Delta T=1$ K. The efficiency of a device at this maximum power is given by⁶³,

$$\begin{aligned} \eta (P_{max})=\frac{\eta _{ca}}{2} \frac{ZT}{ZT+2} \end{aligned}$$

(31)

where $\eta _{ca}$ is the efficiency of a Carnot engine working between the same reservoirs.

It is worth emphasizing that the relevant characteristic temperatures we obtain, for example, for the saturation of the thermopower at low temperatures, maxima of PF, saturation of WF using reasonable values for the parameters of the dot-chains system are much larger than the actual Kondo temperature of realistic quantum dots^64,65.

High temperature results

In this section, we present the results for the thermally activated thermoelectric properties of the device coupled to RM chains.

Figure 10

Density of states corresponding to different values of $V_{1}/V_{2}=1.2;1.0;0.8;0.5$. The legends represent the values of $\tilde{V}=V_1/V_2$.

In Fig. 10, we present the density of states for different values of the ratio $V_{1}/V_{2}=1.2;1.0;0.8;0.5$. Two points should be noticed here: First, at the topological phase transition, $V_{1}/V_{2}=1.0$, the density of states (red curve) presents a sharp behavior at $\mu =0$ that gives rise to an electrical conductance $G/G_{0}=0.5$ at low temperatures. On the contrary, inside the topological region, the density of states at the chemical potential presents a delta function, as indicated in the curves with $V_{1}/V_{2}=0.8,0.5$. On the other hand, the curve $V_{1}/V_{2}=1.2$, outside the topological region, exhibits a full gap. The second point, and the most important for our purposes, is that inside the topological region, as $V_{1}/V_{2}$ decreases, the electrons migrate from the valence band to the peak located at the chemical potential, increasing its weight and the gap, allowing for tuning the thermoelectric properties to the room temperatures region.

Figure 11

Temperature activated thermoelectric properties for different values of the hybridization $V_{1}/V_{2}: ({\textbf {a}}) 1.2; ({\textbf {b}}) 1.0; ({\textbf {c}}) 0.8; ({\textbf {d}}) 0.5$.

In Fig. 11a–d, we plot the transport thermoelectric properties for different values of the ratio $V_{1}/V_{2}$: (a) 1.2; (b) 1.0; (c) 0.8; (d) 0.5. Figure 11a shows a high ZT value, but the Power factor is very low, which limits the usefulness of this region. At the topological transition $V_{1}/V_{2}=1.0$ (Fig. 11b), ZT, $\xi$ and the PF attain robust values. As $V_{1}/V_{2}$ decreases, the gap increases as indicated in Fig. 10; the ZT, $\xi$ and the PF attain high values for $V_{1}/V_{2}=0.95$ (Fig. 11c). For $V_{1}/V_{2}=0.80$, the peak of ZT, $\xi$ and the PF occur at around $T/V_{2}=0.1$ (Fig. 11d). However, the thermoelectric properties value tends to decrease for low values of $V_{1}/V_{2}$.

Conclusions and perspectives

Topological insulating chains have many exciting properties. These chains can be realized in materials with hybridized sp-states where the anti-symmetric nature of the hybridization between orbitals of different parities guarantees their topological properties. We consider in this work monoatomic and diatomic sp-chains that map directly in the SSH and Rice–Mele problems, respectively. We obtain the density of states at the edge of a semi-infinite chain, which varies according to the topological phase of the chain. We show that the weight of the zero energy modes in the non-trivial topological phase vanishes continuously with the distance to the topological transition. In order to study the transport properties of the chains, we considered a simple device consisting of a quantum dot connected to two identical semi-infinite sp or Rice–Mele chains. Away from the topological transition and at $T=0$, the current through the device vanishes since the chains are insulators in their bulk, whether they are in the topologically trivial or non-trivial phases. However, at finite temperatures there is activated transport that is different in the trivial and topological phases. This work focuses on the linear response regime or zero voltage limit. It would be interesting to study this same problem at finite voltage. It will be the subject of future research, as it requires an entirely different approach, namely the Keldish formalism⁶⁶.

At the topological transition of the monoatomic, or SSH chains, and zero temperature, the conductance in the device has a finite universal value $G/G_0 = 1$, independent of the parameters of the model like the coupling between the chains and the dot, as long as the energy of the dot $E_0=0$. Since, at the transition, the surface modes penetrate into the bulk, the system carries current even at $T=0$. The normalized Wiedemann–Franz ratio turns out to be equal unity in terms of the Lorenz number. The thermal conductivity vanishes at $T=0$ even at the topological transition and the thermopower of the monoatomic chains always vanishes due to particle-hole symmetry.

A different behavior arises when we consider diatomic sp-chains with different sub-lattices local energies. In this case the finite local energies break the chiral symmetry of the SSH Hamiltonian and the chain is now modeled by the Rice–Mele Hamiltonian. This system still presents non-trivial topological phases that are now characterized by Chern numbers. Interestingly, the zero temperature dimensionless conductance at the topological transition assumes the value $G/G_0=1/2$, as would be expected for carriers with a fractional charge and is a consequence of the breakdown of chiral symmetry of the SSH model. The thermopower of the device in this case has an anomalous behavior and does not vanish at low temperatures. It attains a universal value at $T=0$ consistent with the result for the conductance that implies fractional charges $q^*=1/2$ flowing in the system. This is due to the double degeneracy of the system associated with the presence of zero energy modes. it is interesting to compare the physical properties of the fractional charge carriers in the topological insulators with those of Majorana modes in p-wave superconductors⁶⁷.

Notice that the antisymmetric hybridization responsible for the non-trivial topological properties of chains does not mix the spins of the carriers. This is quite distinct from the case of spin–orbit interactions that mixes the spins. The consequence is that it is much easier to produce a singly polarized material in the former case.

The sp-chains, with edge modes in their topological phases are easier to realize in practice than p-wave superconductors. Carbyne, the one-dimensional allotropic form of carbon with hybridized sp orbitals provides a realization of these chains. They are potentially useful systems exhibiting properties that can be explored in a large temperature range. In particular, we show that varying $V_{1}/V_{2}$, the figure of merit and power factors can attain high values at high temperatures, making the system very attractive to be explored in technological applications.