Thermoelectric Figure of Merit in a One-Dimensional Model with k⁴-Dispersion: An Extension of the Theory by Hicks and Dresselhaus

Abstract

Motivated by the strategy developed by Hicks and Dresselhaus in a quantum wire corresponding to a single-chain model with $k^2$-dispersion, we study a one-dimensional double-chain model with two carriers of electrons and holes, characterized by $k^4$-dispersion. To understand the role of the enhancement of the density of state derived from $k^4$-dispersion, we calculate an optimized dimensionless thermoelectric figure of merit ($ZT$) depending on the side length of the cross section, a, in the same way as discussed by Hicks and Dresselhaus. We find that $ZT$ enhances as a decreases similarly to the results obtained in the single-chain model, while the enhancement of $ZT$ is smaller than that of single-chain model. We discuss the reason in connection with the difference of electronic state between the single- and double-chain models.

1. Introduction

The search for materials with a high thermoelectric figure of merit, $ZT$, is one of the most important tasks in thermoelectric research. In 1993, Hicks and Dresselhaus discussed the promising possibility of high $ZT$ in low-dimensional systems [1,2]. Guided by their proposal, many experimental and theoretical studies have been carried out to look for the appropriate materials [3,4,5]. In particular, in one-dimensional systems such as quantum wires and carbon nanotubes, it was discussed that phonon scattering from the surfaces of one-dimensional wires reduces lattice thermal conductivity, which enhances $ZT$. Furthermore, in one-dimensional electron systems, the density of state at the band edge has a divergence, which also gives a chance to enhance the Seebeck coefficient that is related to the energy derivative of the density of state. In the present paper, we extend this strategy of Hicks and Dresselhaus to electron systems with $k^4$-dispersion, which leads to increased divergence of the density of state at the band edge. We investigate theoretically $ZT$ in comparison with the results in the $k^2$-dispersion.

2. Inorganic Materials as a Candidate of $k^4$-Dispersion

As discussed in the introduction, there are several candidate materials of the Hicks–Dresselhaus theory based on the $k^2$ dispersion. Firstly, Hicks and Dresselhaus predicted the enhancement of ZT in the Bi₃Te₂ nanowire. After that, the theory was extended to not only the organic materials such as carbon nanotube, but also the inorganic materials of silicon nanowire [6], bismuth nanowire [7], etc. For example, the bismuth nanowire has been studied from both the experimental and theoretical side, because bismuth has peculiar properties of long mean free path and small effective mass [7,8,9,10,11]. Extending the Hicks–Dresselhaus theory to the bismuth nanowire, it was found that ZT improves in the realistic parameters as the diameter of nanowire decreases [8]. In contrast to the theoretical expectation, the experimental result obtained in the bismuth nanowire has a peak of $ZT\simeq 0.3$ at 100 nm diameter, and decreases drastically as the diameter decreases [9]. This is due to the classical and quantum size effect. Although accurate measurements of electrical and thermoelectric transports and the size (boundary) effect in the bismuth nanowire were studied [10,11], the size of the nanowire is several hundred nm and a thinner nanowire is needed to compare the theory in detail. In addition, the optimization of the carrier concentration and size effect are suggested to play an important role in realizing the prediction of Hicks–Dresselhaus.

On the other hand, there are candidate materials with $k^4$-dispersion. Here, we introduce the excitonic insulator with a one-dimensional chain structure. The excitonic insulator is a condensed state of electrons and holes by the Coulomb interaction [12]. As discussed in the inorganic materials Ta₂NiSe₅, the excitonic insulator is suggested to appear at room temperature [13,14,15]. In Ta₂NiSe₅, there are three one-dimensional chains of electron and hole bands with coulomb interaction between chains [13,14]. In the excitonic insulator, the bottom of the conduction band and the top of the hole band become flat, similar to the $k^4$-dispersion shown in Figure 1.

Due to the analogy with the excitonic insulator, it is possible to have the $k^4$-dispersion in one-dimensional materials with the hybridization between two chains. Therefore, in this paper, we introduce the two-chain model as a minimum model with $k^4$-dispersion and study the thermoelectric properties of this model using the same method as the Hicks–Dresselhaus model.

3. Model Hamiltonian

Figure 2 shows schematic pictures of (a) the single-chain model corresponding to the Hicks–Dresselhaus theory and (b) the double-chain model, consisting of two carriers of electrons and holes depending on the chain. We consider that the conductor is square in cross section with a side of length a and the one-dimensional chains are packed in the cross section of $a^2$. In the following, we investigate the double-chain model shown in Figure 2b. The effective Hamiltonian is given by

$$H=\sum _i\left(-t{c}_{A,i+1}^{†}{c}_{A,i}+t{c}_{B,i+1}^{†}{c}_{B,i}+\Delta c_{A,i}^{†}{c}_{B,i}\right)+\mathrm{H}.\mathrm{C}.+\sum _i{\epsilon }_0{c}_{A,i}^{†}{c}_{A,i},$$

(1)

where $c_{A,i}$ ($c_{B,i}$) represents the electron annihilation operator on the i-th site of the one-dimensional chain A (B), t is the hopping integral along the chains, $\Delta$ is the hopping between A and B chains, ${\epsilon }_0$ represents the energy difference between the fermions on the chains A and B, and H.C. means the hermite conjugate terms. After the Fourier transform, this Hamiltonian is represented by a matrix form

$$H=\left(\begin{array}{cc}{\epsilon }_0-2tcoskb& \Delta \\ \Delta & 2tcoskb\end{array}\right),$$

(2)

where b is the lattice constant along the chain. Here, the A (B) chain plays the role of the electron (hole) band. If we choose ${\epsilon }_0=4t$, then the two bands (without $\Delta$) touch at $k=0$, leading to the $k^4$-dispersion when $\Delta$ is nonzero. In this case, the energy dispersion in the vicinity of $k=0$ becomes

$$E=2t\pm \sqrt{{\Delta }^2+t^2{\left(kb\right)}^4}\sim 2t\pm \Delta \left\{1+{\displaystyle \frac{t^2}{2{\Delta }^2}}{\left(kb\right)}^4\right\},$$

(3)

which gives $k^4$-dispersion near the bottom of the conduction band and at the top of the valence band. In the following, we consider this specific case near the bottom of the conduction band, which has the dispersion, ${\epsilon }_k=t_{\mathrm{eff}}{\left(kb\right)}^4$, with $t_{\mathrm{eff}}=t^2/2\Delta$.

The density of state in the present model diverges as ${\epsilon }^{-3/4}$ near the bottom of the band, which is stronger than ${\epsilon }^{-1/2}$ realized in the one-dimensional $k^2$-dispersion. We expect that this strong divergence of the density of state can enhance the thermoelectric properties in the present model.

4. Formula of Thermoelectric Figure of Merit (ZT)

In the standard linear-response theory, the electric conductivity in one-dimensional electron system is given by

$$\sigma ={\displaystyle \frac{2e^2}{L{a}^2}}\sum _k{v}_k^2\tau \left(-f^{\prime }\left({\epsilon }_k\right)\right),$$

(4)

where $v_k$ is the velocity defined as $v_k=\partial {\epsilon }_k/\hslash \partial k$ and $f\left(\epsilon \right)$ is the Fermi distribution function $f\left(\epsilon \right)=1/[e^{(\epsilon -\mu )/k_{\mathrm{B}}T}+1]$ with $\mu$ and $k_{\mathrm{B}}$ being the chemical potential and the Boltzmann constant, respectively. We have used the assumption of a constant relaxation time $\tau$ and the prefactor 2 comes from the spin degrees of freedom.

In the present case of ${\epsilon }_k=t_{\mathrm{eff}}{\left(kb\right)}^4$, it is straightforward to obtain

$$\sigma =A{F}_{-1/4},$$

(5)

where

$$A={\displaystyle \frac{6e^2\tau t_{\mathrm{eff}}b}{\pi a^2{\hslash }^2}}{\left({\displaystyle \frac{k_{\mathrm{B}}T}{t_{\mathrm{eff}}}}\right)}^{3/4},$$

(6)

and $F_i$ is defined as [2]

$$F_i=F_i\left(\eta \right)={\int }_0^{\infty }{\displaystyle \frac{x^{i}dx}{e^{x-\eta }+1}},$$

(7)

with $\eta =\mu /k_{\mathrm{B}}T$. Here, we have changed the integral variable from k to $x={\epsilon }_k/k_{\mathrm{B}}T$. Note that in the $k^2$-dispersion case, we have $\sigma =A^{\prime }{F}_{-1/2}$.

The other linear response coefficients, $L_{ij}$ ($i,j=1,2$), are defined by [16]

$$\begin{array}{cc}\hfill \mathit{j}& =L_{11}\mathit{E}+L_{12}\left(-{\displaystyle \frac{\mathsf{\nabla }T}{T}}\right),\hfill \\ \hfill {\mathit{j}}_Q& =L_{21}\mathit{E}+L_{22}\left(-{\displaystyle \frac{\mathsf{\nabla }T}{T}}\right),\hfill \end{array}$$

(8)

where $\mathit{j}$ and ${\mathit{j}}_Q$ are electric and heat currents, respectively, and $L_{11}$ is the electric conductivity $\sigma$. When we express the conductivity as

$$\sigma ={\int }_{-\infty }^{\infty }d\epsilon \sigma (\epsilon ,T)\left(-f^{\prime }\left(\epsilon \right)\right),$$

(9)

we can see that the following relations hold in the present case:

$$\begin{array}{cc}\hfill L_{12}=L_{21}& ={\int }_{-\infty }^{\infty }d\epsilon (\epsilon -\mu )\sigma (\epsilon ,T)\left(-f^{\prime }\left(\epsilon \right)\right),\hfill \\ \hfill L_{22}& ={\int }_{-\infty }^{\infty }d\epsilon {(\epsilon -\mu )}^2\sigma (\epsilon ,T)\left(-f^{\prime }\left(\epsilon \right)\right),\hfill \end{array}$$

(10)

which we call the Sommerfeld–Bethe (SB) relation [17]. The SB relation is derived from the Kubo–Luttinger theory [18,19]. The validity of SB relation was studied in detail [17] and it was found that the origin of the thermoelectric effect is classified into inside of SB relation and out of SB relation. Phonon drag effect, which is an origin of huge Seebeck effects, is out of SB relation [17,20]. Recently, based on this classification, the microscopic theories of thermoelectrics have been extensively studied [21].

Using this SB relation, it is straightforward to obtain

$$\begin{array}{cc}\hfill L_{12}=L_{21}& =A{\displaystyle \frac{k_{\mathrm{B}}T}{e}}\left({\displaystyle \frac{7}{3}}{F}_{3/4}-\eta F_{-1/4}\right),\hfill \\ \hfill L_{22}& =A{\displaystyle \frac{{\left(k_{\mathrm{B}}T\right)}^2}{e^2}}\left({\displaystyle \frac{11}{3}}{F}_{7/4}-{\displaystyle \frac{14}{3}}\eta F_{3/4}+{\eta }^2{F}_{-1/4}\right).\hfill \end{array}$$

(11)

Then, the Seebeck coefficient S and the electronic thermal conductivity ${\kappa }_e$ are given by

$$\begin{array}{cc}\hfill S& ={\displaystyle \frac{L_{12}}{T{L}_{11}}}={\displaystyle \frac{k_{\mathrm{B}}}{e}}\left({\displaystyle \frac{7}{3}}{\displaystyle \frac{F_{3/4}}{F_{-1/4}}}-\eta \right),\hfill \\ \hfill {\kappa }_e& ={\displaystyle \frac{1}{T}}\left(L_{22}-{\displaystyle \frac{L_{12}{L}_{21}}{L_{11}}}\right)=A{\displaystyle \frac{k_{\mathrm{B}}^{2}T}{e^2}}\left({\displaystyle \frac{11}{3}}{F}_{7/4}-{\displaystyle \frac{49}{9}}{\displaystyle \frac{{\left(F_{3/4}\right)}^2}{F_{-1/4}}}\right).\hfill \end{array}$$

(12)

Finally, introducing the lattice thermal conductivity ${\kappa }_L$, we find the thermoelectric figure of merit

$$ZT={\displaystyle \frac{S^2\sigma T}{{\kappa }_e+{\kappa }_L}}={\displaystyle \frac{{\left(\frac{7}{3}\frac{F_{3/4}}{F_{-1/4}}-\eta \right)}^2{F}_{-1/4}}{\frac{1}{B}+\frac{11}{3}{F}_{7/4}-\frac{49}{9}\frac{{\left(F_{3/4}\right)}^2}{F_{-1/4}}}},$$

(13)

with

$$B=A{\displaystyle \frac{k_{\mathrm{B}}^{2}T}{e^2{\kappa }_L}}.$$

(14)

5. Numerical Results

The result in Equation (13) shows that the parameter dependence of $ZT$ comes only from the values of $\eta =\mu /k_{\mathrm{B}}T$ and B. Microscopic parameters such as $\tau$, $t_{\mathrm{eff}}$, and the lattice parameters $a,b$ are included in B. Therefore, when we fix the temperature T and the value of B, we can find an optimal value of $\eta$ that maximizes $ZT$. Experimentally, $\eta$ can be adjusted by, for example, doping, as discussed by Hicks and Dresselhaus [2].

To evaluate the parameter B, Hicks and Dresselhaus assumed that the mean free path of phonons are limited by surface scattering. Then, the lattice thermal conductivity was estimated as

$${\kappa }_{\mathrm{L}}={\displaystyle \frac{1}{3}}{C}_{v}v\ell ,$$

(15)

with ℓ being the length of the cross section ($\ell =a$). We use the same parameters as used by Hicks and Dresselhaus: $C_v=1.2\times {10}^6 \mathrm{J}$${\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, $v=3\times {10}^3 \mathrm{m}$${\mathrm{s}}^{-1}$, $\ell =10$ Å. In this case, the lattice thermal conductivity becomes ${\kappa }_{\mathrm{L}}=1.2 \mathrm{W}$${\mathrm{K}}^{-1}$${\mathrm{m}}^{-1}$. For the prefactor of the conductivity, A, we use $\tau =1\times {10}^{-14}$ s, $\Delta =1$ eV and $b=3$ Å.

Figure 3 shows $\eta$-dependence of ZT for the Hicks–Dresselhaus theory (black lines) and the double-chain model (red lines). Here we fixed the parameters as $m^{*}=0.01m_0$ with $m_0$ being the electron mass for Hicks-Dresselhaus, and $t=0.5$ eV ($t_{\mathrm{eff}}\simeq 0.13$ eV) for the double-chain model, $T=300$ K and the length of the side of the cross section, a is $a=5$ Å and 10 Å. The optimal value of $\eta$ ($\eta ={\eta }^{*}$) is determined as the value that ZT is maximum.

Figure 4 shows the optimal value of $\eta$ ($\eta ={\eta }^{*}$) as a function of the length of the side of the cross section, a. The black and gray lines indicate the results of Hicks–Dresselhaus for $m^{*}/m_0=0.01$ and 1, respectively. In this paper, we use $m^{*}/m_0=0.01$~1 to compare the dependence of the mass in Hicks–Dresselhaus with the dependence of hopping (t) in the present model. The pink, red, and blue lines show the numerical results of double-chain model for $t=1$ eV ($t_{\mathrm{eff}}\simeq 0.5$ eV), $0.5$ eV ($t_{\mathrm{eff}}\simeq 0.13$ eV) and $0.1$ eV ($t_{\mathrm{eff}}\simeq 0.005$ eV), respectively. The dependence of the double-chain model on a is the same as that of the single-chain model, in the view that ${\eta }^{*}$ decreases as a decreases.

In our actual numerical calculations, we use another universal function $T_i\left(\eta \right)$ instead of $F_i\left(\eta \right)$, because the linear response coefficients are expressed by the energy derivative of the Fermi distribution function $f^{\prime }\left(\epsilon \right)$. $T_i\left(\eta \right)$ is related to $f^{\prime }\left(\epsilon \right)$. Some details are shown in Appendix A.

Using ${\eta }^{*}$, the obtained maximum of $ZT$ is shown in Figure 5. The black and gray lines are the result of the single-chain model (Hicks–Dresselhaus [2]), and the pink, red, and blue lines indicate the results of the double-chain model. As the length of the side, a, decreases, ZT increases drastically in both models, while we find that the enhancement of ZT in the double-chain model is smaller than that of the single-chain model in the realistic parameter region. This difference between the single- and double-chain models is mainly due to the difference of B. We can rewrite the prefactor of the conductivity obtained by Hicks and Dresselhaus, $\sigma =A^{\prime }{F}_{-1/2}$, as follows:

$$A^{\prime }={\displaystyle \frac{1}{\pi a^2}}{\left({\displaystyle \frac{2k_{\mathrm{B}}T}{{\hslash }^2}}\right)}^{1/2}{\left(m^{*}\right)}^{1/2}e{\mu }_e={\displaystyle \frac{2e^2\tau t_{\mathrm{eff}}^{\prime }b}{\pi a^2{\hslash }^2}}{\left({\displaystyle \frac{k_{\mathrm{B}}T}{t_{\mathrm{eff}}^{\prime }}}\right)}^{1/2},$$

(16)

where ${\mu }_e=e\tau /m^{*}$ is the electron mobility and $t_{\mathrm{eff}}^{\prime }$ in the single-band model is defined by

$$t_{\mathrm{eff}}^{\prime }={\displaystyle \frac{{\hslash }^2}{2m^{*}{b}^2}},$$

(17)

because the energy dispersion is written as ${\epsilon }_{\mathbf{k}}={\hslash }^2{k}^2/2m^{*}=t_{\mathrm{eff}}^{\prime }{\left(kb\right)}^2$. This prefactor $A^{\prime }$ has the same parameter dependence as that in the present model in Equation (6). However, $t_{\mathrm{eff}}^{\prime }$ in the single-chain model (Hicks–Dresselhaus) is very large: $t_{\mathrm{eff}}^{\prime }=0.6$ eV for $m^{*}=m_e$ and $t_{\mathrm{eff}}^{\prime }=60$ eV for $m^{*}=0.01m_e$. This large $A^{\prime }$ causes small $1/B$ in the expression of $ZT$, or the relative amplitude of the lattice thermal conductivity compared to ${\kappa }_e$ is very small. As a result, the figure of merit $ZT$ becomes large for the single-chain model. In fact, when we compare the result of $ZT$ in Figure 5 for the single-chain model with $m^{*}=m_e$ (corresponding to $t_{\mathrm{eff}}^{\prime }=0.6$ eV) and that for the double-chain model with $t=0.5$ eV (corresponding to $t_{\mathrm{eff}}=0.13$ eV), we find that the values of $ZT$ for both cases are comparable, but $ZT$ for the latter is slightly larger than the former.

Finally, we show the temperature (T) dependence of ZT for the single- and double-chain models in Figure 6. Here, we fixed the parameters as shown in Figure 3. We find that ZT gradually increases as the temperature increases in both cases. This will be because the relative weight of the lattice thermal conductivity decreases as the temperature increases. Note that the difference between the magnitudes of $ZT$ for the double-chain model with $t=0.5$ and that in the single-chain model with $m^{*}=m_0$ slightly enhanced as the temperature increases.

6. Conclusions

In conclusion, the figure of merit of the present model (double-chain model) is not so large compared with the results obtained by Hicks and Dresselhaus. Nevertheless, the obtained value of $ZT$ is promising to be realized in actual materials. The reason why $ZT$ is not so large is that the effective hopping energy $t_{\mathrm{eff}}$ is not so large compared to that for the single-chain model. As shown in the preceding section, if the effective mass is $m^{*}=0.01m_e$, $t_{\mathrm{eff}}^{\prime }$ in the single-chain model becomes as large as 60 eV. Thus, if we can choose a material which has a small effective mass as $m^{*}=0.01m_e$ in the double-chain model, we can have a large value of $ZT$ comparable to that in the single-chain model.