Large enhancement in thermoelectric efficiency of quantum dot junction due to increase of level degeneracy

TL;DR

This paper theoretically shows that increasing the degeneracy of energy levels in quantum dot junctions significantly boosts thermoelectric efficiency, mainly through enhanced electrical conductance, with potential applications in energy conversion.

Contribution

It introduces a Green-function theoretical model demonstrating how increased level degeneracy enhances the thermoelectric figure of merit ($ZT$) in quantum dot systems.

Findings

Higher degeneracy increases electrical conductance ($G_e$) and $ZT$.

Seebeck coefficient ($S$) remains unaffected by degeneracy.

Double quantum dots also show $ZT$ enhancement with increased degeneracy.

Abstract

It is theoretically demonstrated that the figure of merit ($ZT$) of quantum dot (QD) junctions can be significantly enhanced when the degree of degeneracy of the energy levels involved in electron transport is increased. The theory is based on the the Green-function approach in the Coulomb blockade regime by including all correlation functions resulting from electron-electron interactions associated with the degenerate levels ($L$). We found that electrical conductance ($G_e$) as well as electron thermal conductance ($\kappa_e$) are highly dependent on the level degeneracy ($L$), whereas the Seebeck coefficient ($S$) is not. Therefore, the large enhancement of $ZT$ is mainly attributed to the increase of $G_e$ when the phonon thermal conductance ($\kappa_{ph}$) dominates the heat transport of QD junction system. In the serially coupled double-QD case, we also obtain a large enhancement of $ZT$ arising from higher $L$. Unlike $G_e$ and $\kappa_e$, $S$ is found almost independent on electron inter-dot hopping strength.

Equations

$$H_0$$

$$H_{QD}$$

$$J_{\alpha }^n$$

$$G_e$$

$$S$$

$${\kappa }_e$$

$$(\frac{\delta J_{\alpha }^0}{\delta \Delta V}{)}_{\Delta T=0}=\frac{ie}{h}\sum _{j\sigma }\int dϵ{\Gamma }_j^{\alpha }(ϵ)\times$$

$$[\frac{\delta G_{j\sigma }^{<}(ϵ)}{\delta f_{\alpha }(ϵ)}+(G_{j\sigma }^r(ϵ)-G_{j\sigma }^{(a}(ϵ))]\frac{\delta f_{\alpha }(ϵ)}{\delta \Delta V},$$

$$(\frac{\delta J_{\alpha }^0}{\delta \Delta T}{)}_{\Delta V=0}=\frac{ie}{h}\sum _{j\sigma }\int dϵ{\Gamma }_j^{\alpha }(ϵ)\times$$

$$[\frac{\delta G_{j\sigma }^{<}(ϵ)}{\delta f_{\alpha }(ϵ)}+(G_{j\sigma }^r(ϵ)-G_{j\sigma }^{(a}(ϵ))]\frac{\delta f_{\alpha }(ϵ)}{\delta \Delta T},$$

$$(\frac{\delta J_{\alpha }^1}{\delta \Delta T}{)}_{\Delta V=0}=\frac{i}{h}\sum _{j\sigma }\int dϵ{\Gamma }_j^{\alpha }(ϵ)(ϵ-E_F)\times$$

$$[\frac{\delta G_{j\sigma }^{<}(ϵ)}{\delta f_{\alpha }(ϵ)}+(G_{j\sigma }^r(ϵ)-G_{j\sigma }^a(ϵ))]\frac{\delta f_{\alpha }(ϵ)}{\delta \Delta T},$$

$$(\frac{\delta J_{\alpha }^1}{\delta \Delta V}{)}_{\Delta T=0}=\frac{i}{h}\sum _{j\sigma }\int dϵ{\Gamma }_j^{\alpha }(ϵ)(ϵ-E_F)\times$$

$$[\frac{\delta G_{j\sigma }^{<}(ϵ)}{\delta f_{\alpha }(ϵ)}+(G_{j\sigma }^r(ϵ)-G_{j\sigma }^a(ϵ))]\frac{\delta f_{\alpha }(ϵ)}{\delta \Delta V}.$$

$${\kappa }_{ph}=\frac{1}{h}\int d\omega \mathcal{T}(\omega {)}_{ph}\frac{{\hslash }^3{\omega }^2}{k_B{T}^2}\frac{e^{\hslash \omega /k_{B}T}}{(e^{\hslash \omega /k_{B}T}-1{)}^2},$$

$${\mathcal{T}}_{ph}(\omega )=\frac{N_{ph,1}(\omega )}{1+\tilde{L}/{\ell }_0(\omega )}+\frac{N_{ph,2}(\omega )}{1+\tilde{L}/D}$$

$$\frac{1}{{\ell }_0(\omega )}=B\frac{{\delta }^2}{D^3}(\frac{\omega }{{\omega }_D}{)}^2{N}_{ph}(\omega ),$$

$${\kappa }_{ph}=F_s{g}_0(T),$$

$$\frac{{\mathcal{T}}_{LR}(ϵ)}{4{\Gamma }_L{\Gamma }_R}=\frac{1-N_{-\sigma }}{(ϵ-E_0{)}^2+{\bar{\Gamma }}^2}+\frac{N_{-\sigma }}{(ϵ-E_0-U_0{)}^2+{\bar{\Gamma }}^2},$$

$${\mathcal{T}}_{LR}(ϵ)\approx \frac{4{\Gamma }_L{\Gamma }_{R}L{P}_{L,1}}{(ϵ-E_0{)}^2+{\bar{\Gamma }}^2},$$

$${\mathcal{L}}_n=\frac{2}{h}\int dϵ{\mathcal{T}}_{LR}(ϵ)(ϵ-E_F{)}^n\frac{\partial f(ϵ)}{\partial E_F},$$

$$ZT=\frac{(\Delta /eT{)}^2{G}_0\Gamma \pi L{P}_{L,1}T}{k_{B}Tcos{h}^2(\frac{\Delta }{2k_{B}T})k_{ph}},$$

$${\mathcal{T}}_{LR}(ϵ)/(4t_{LR}^2{\Gamma }_L{\Gamma }_R)=\frac{P_{1,1}}{|{\mu }_L{\mu }_R-t_{LR}^2{|}^2}$$

$${\mathcal{T}}_{LR}(ϵ)\approx \frac{4{\Gamma }_L{\Gamma }_R{t}_{LR}^{2}L{P}_{L,1}}{|(ϵ-E_0+i{\Gamma }_L)(ϵ-E_0+i{\Gamma }_R)-t_{LR}^2{|}^2},$$

$${\mathcal{L}}_0=\frac{2}{h{k}_{B}T}\frac{\pi \Gamma t_{LR}^{2}L{P}_{L,1}}{(4t_{LR}^2+{\Gamma }^2)}\frac{1}{cos{h}^2(\frac{\Delta }{2k_{B}T})}$$

$${\mathcal{L}}_1=\frac{2}{h{k}_{B}T}\frac{\pi \Gamma t_{LR}^{2}L{P}_{L,1}}{(4t_{LR}^2+{\Gamma }^2)}\frac{\Delta }{cos{h}^2(\frac{\Delta }{2k_{B}T})}.$$

Full text

I Introduction

Recently, many efforts have been devoted to the search of high-efficiency thermoelectric (TE) materials, because of the high demand of energy-saving solid state coolers and power generators.1,2 TE devices have very good potential for green energy applications due to their desirable features, including low air pollution, low noise, and long operation time. However, there exists certain barrier for TE devices to replace conventional refrigerators and power generators since TE materials with figure of merit ($ZT$) larger than three are not yet found.1,2 The figure of merit, $ZT=S^{2}G_{e}T/\kappa$,defined in the linear response regime is composed of the Seebeck coefficient ($S$), electrical conductance ($G_{e}$), thermal conductance ($\kappa$) and equilibrium temperature ($T$). $\kappa$ is the sum of the electron thermal conductance ($\kappa_{e}$) and phonon thermal conductance ($\kappa_{ph}$). It has been shown that low-dimensional systems including quantum wells3, quantum wires4 and quantum dots (QDs)5 have very impressive $ZT$ values when compared with bulk materials.3-9 In particular, $ZT$ of PbSeTe QD array (QDA) can reach two,5 which is mainly attributed to the reduction of $\kappa_{ph}$ in QDA.2 However, QD junctions with $ZT\geq 3$ are not yet reported experimentally. There are some technical difficulties in using the QD junction to achieve $ZT\geq 3$ via the reduction of phonon thermal conductivity.1,2

More than two decades ago, Hicks and Dresselhauss theoretically predicted that $ZT$ values of BiTe quantum wells and quantum wires can be larger than one at room temperature.10,11 In particular, $ZT$ values of nanowires (with diameter smaller than 1 nm) may reach 10 based on the assumption of very low lattice thermal conductivity ($\kappa_{L}=1.5Wm^{-1}K^{-1}$ for Bi2Te3). Recently, there are considerable interest on $ZT$ values of nanowires filled with QDs,1,2 because it is expected that $\kappa_{ph}$ can be reduced significantly due to the introduction of QDs. Such a reduction of $\kappa_{ph}$ due to phonon scattering with QDs in SiGe nanowire filled with QDs was verified theoretically in Ref. 12. However, the behaviors of $G_{e}$, $S$ and $\kappa_{e}$ in nanowires filled with QDs remain unclear because of the complicated many-body problem involved. The full many-body effect on the behaviors of electron thermoelectric coefficients may be analyzed by considering a single QD or double QDs (DQD) embedded in a single nanowire to reveal the importance of the electron Coulomb interaction.

Theoretical studies have indicated that a TE device made of molecular QD junction13-14 can reach the Carnot efficiency if one can neglect $\kappa_{ph}$. Such a divergence of $ZT$ for QDs is related to the divergence of $G_{e}/\kappa_{e}$, which violates the Wiedeman-Franz law (WFL).15 The violation of WFL is a typical feature for QDs with discrete energy levels.16 It is hard to realize thermal devices with Carnot efficiency as considered in Refs. 13 and 14, because it is impossible to blockade acoustic phonon heat flow completely in the implementation of solid state TE devices1,2. Therefore, finding a way to enhance $ZT$ of QD junctions under an achievable $\kappa_{ph}$ value is crucial. Here, we demonstrate that by increasing the level degeneracy in QDs, it is possible to enhance the thermoelectric efficiency significantly given the condition $\kappa_{ph}/\kappa_{e}\gg 1$. The level degeneracy in a QD can be determined by its point-group symmetry. For spherical QDs made of semiconductors with zincblende (e.g. III-V compounds) or diamond crystal structure (e.g. Si or Ge), the point group is Td. Thus, the orbital degeneracy $L$ can be described by singlet ($A_{1}$), doublet ($E_{2}$) or triplet ($T_{2}$). If the QD energy levels are well described by the effective-mass model (neglecting the crystal-field effect), then the orbital degeneracy is determined by the associated orbital angular momentum quantum number $\ell$, and the level degeneracy becomes $L=2\ell+1$. For example, the $p$-like states in a spherical QD are 3-fold degenerate with $L=3$ (not including spin degeneracy). In an QD junction, one can tune the gate voltage to access the level with desired degeneracy. The high level degeneracy ($L$) is also feasible in QDs made of multi-valley semiconductors such as Si or Ge. Our theoretical results may serve as useful guideline for optimizing $ZT$ of semiconductor QD1,2 or molecular QD systems13, in which a dominating phonon thermal conductivity cannot be avoided.

II Formalism

Here we consider nanoscale semiconductor QDs embedded in a nanowire connected with metallic electrodes. An extended Anderson model is employed to simulate a QD junction with degenerate levels.17-19 The Hamiltonian of the QD junction system considered is given by $H=H_{0}+H_{QD}$, where

[TABLE]

The first two terms of Eq. (1) describe the free electron gas in the left and right electrodes. $a^{\dagger}_{k,\sigma}$ ($b^{\dagger}_{k,\sigma}$) creates an electron of momentum $k$ and spin $\sigma$ with energy $\epsilon_{k}$ in the left (right) electrode. $V^{L}_{k,\ell}$ ($V^{R}_{k,\ell}$) describes the coupling between the the $\ell$-th energy level of the QD system and left (right) electrode. $d^{\dagger}_{\ell,\sigma}$ ($d_{\ell,\sigma}$) creates (destroys) an electron in the $\ell$-th energy level of the QD.

[TABLE]

where $E_{\ell}$ is the spin-independent QD energy level, and $n_{\ell,\sigma}=d^{\dagger}_{\ell,\sigma}d_{\ell,\sigma}$, $U_{\ell}$ and $U_{\ell,j}$ describe the intralevel and interlevel Coulomb interactions, respectively. For nanoscale semiconductor QDs, the interlevel Coulomb interactions as well as intralevel Coulomb interactions play a significant role on the electron transport in semiconductor junctions. It is worth noting that $H_{QD}$ possesses the particle-hole symmetry. One can prove it with a simple swap of electron and hole operators ($d_{\ell,\sigma}\rightarrow c^{\dagger}_{\ell,\sigma}$). The form of $H_{QD}$ is changed only by constant terms when QD energy levels are degenerate. This indicates that dynamic physical quantity is unchanged in the hole picture.

To reveal the transport properties of a QD junction connected with metallic electrodes, it is convenient to use the Green-function technique. The electron and heat currents from reservoir $\alpha$ to the QD are calculated according to the Meir-Wingreen formula19

[TABLE]

where $n=0$ is for the electrical current and $n=1$ for the heat current. $\Gamma^{\alpha}_{j}(\epsilon)=\sum_{k}|V_{k,j}|^{2}\delta(\epsilon-\epsilon_{k})$ is the tunneling rate for electrons from the $\alpha$-th reservoir and entering the $j$-th energy level of the QD. $f_{\alpha}(\epsilon)=1/\{\exp[(\epsilon-\mu_{\alpha})/k_{B}T_{\alpha}]+1\}$ denotes the Fermi distribution function for the $\alpha$-th electrode, where $\mu_{\alpha}$ and $T_{\alpha}$ are the chemical potential and the temperature of the $\alpha$ electrode. $e$, $h$, and $k_{B}$ denote the electron charge, the Planck’s constant, and the Boltzmann constant, respectively. $G^{<}_{j\sigma}(\epsilon)$, $G^{r}_{j\sigma}(\epsilon)$, and $G^{a}_{j\sigma}(\epsilon)$ denote the frequency-domain representations of the one-particle lessor, retarded, and advanced Green’s functions, respectively.

II.1 Thermoelectric coefficients

Thermoelectric coefficients including $G_{e}$, $S$ and $\kappa_{e}$ in the linear response regime can be evaluated by using Eq. (3) with small $\Delta V=(\mu_{L}-\mu_{R})/e$ and $\Delta T=T_{L}-T_{R}$. We obtain the following expressions of thermoelectric coefficients:

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

$\frac{\delta G^{<}_{j\sigma}(\epsilon)}{\delta f_{\alpha}(\epsilon)}$ is obtained by taking the derivative of the equation of motion with respect to the change in Fermi-Dirac distribution, $f_{\alpha}(\epsilon)$. Here we have assumed the variation of the correlation functions with respect to $\delta f_{\alpha}(\epsilon)$ is of the second order. Note that we have to take the limit $\Delta V\rightarrow 0$ for the calculation of $(\frac{\delta J^{0}_{\alpha}}{\delta\Delta V})_{\Delta T=0}$ and $(\frac{\delta J^{1}_{\alpha}}{\delta\Delta V})_{\Delta T=0}$. $E_{F}$ is the Fermi energy of electrodes. The one-particle Green’s functions in Eqs. (7)-(10) are related recursively to high-order Green’s functions and correlation functions via a hierarchy of equations of motion (EOM)20. This hierarchy self terminates at the $2N$-particle Green function, where $N$ is the number of levels considered in the QD or coupled QDs.

To reveal the effect of degenerate levels on the thermoelectric efficiency of QD junction system, all needed Green’s functions and correlation functions arising from electron-electron interactions in the QDs considered are computed self-consistently following the procedures described in our previous work.20,21 Our procedure is beyond the mean-field theory, which is widely used in solving the equation of motion in the Green function calculation.14 For $L=4$, our calculation involves solving one-, two-, $\cdots$, up to eight-particle Green functions.

II.2 Phonon thermal conductance

The thermoelectric efficiency of a QD junction embedded in a nanowire is determined by the figure of merit, $ZT=S^{2}G_{e}T/(\kappa_{e}+\kappa_{ph})$, which involves the $\kappa_{ph}$ of the QD junction system. The optimization of molecular QD junctions under the condition of $\kappa_{e}/\kappa_{ph}\gg 1$ has been theoretically investigated in references[13,14]. However, the condition of $\kappa_{e}/\kappa_{ph}\gg 1$ is very difficult to realize in practice. The main goal of this study is to investigate the effect of energy level degeneracy on thermoelectric efficiency under the realistic condition with $\kappa_{ph}/\kappa_{e}>1$. The phonon thermal conductance of nanowires have been extensively studied experimentally and theoretically.22-32 In Refs. 22-24 it has been shown experimentally that $\kappa_{ph}$ displays a linear $T$ behavior from $20K$ to $300K$ for silicon nanowires with diameter $22~{}nm$. The linear $T$ behavior of $\kappa_{ph}$ also holds for $T$ between $100K$ and $400K$ for germanium nanowires with diameter $19~{}nm$.25 Due to the reduction of $\kappa_{ph}$, $ZT$ of silicon nanowires increases significantly (with $ZT=1$ at $200K$) in comparison with $ZT=0.01$ for bulk silicon at room temperature.4,26 The linear $T$ behavior of nanowires is an interesting topic. Many theoretical efforts have been devoted to clarifying why nanowires with diameters near $20~{}nm$ exhibit the linear $T$ behavior.27-32 For a true one-dimensional system, the linear $T$ behavior of $\kappa_{ph}$ is expected.33 To include $\kappa_{ph}$, we have adopted the Landauer formula given in Refs. 28 and 30.

[TABLE]

where $\omega$ and ${\cal T}_{ph}(\omega)$ are the phonon frequency and throughput function, respectively. In a perfect wire throughput is unity for each open channel, then $\kappa_{ph}$ in such a perfect case is given by $\kappa_{ph,0}=\frac{k^{2}_{B}\pi^{2}TN_{ph}}{3h}=g_{0}(T)N_{ph}$, where $N_{ph}$ is the total number of open modes and $g_{0}(T)=\frac{k^{2}_{B}\pi^{2}}{3h}T=9.456\times 10^{-4}T(nW/K^{2})$ is called the quantum conductance.[34] Experimentally, it was found that the linear-$T$ behavior in ballistic regime only holds for temperature below $0.8K$ for wires of size  200nm with $N_{ph}=4$ (which includes one longitudinal, one torsional, and two flexural modes)34. Beyond $0.8K$, the nonlienar-$T$ behavior was observed due the contribution of high-energy phonon modes being thermally populated. Low-temperature thermoelectric properties of Kondo insulator nanowire was also studied in Ref. 35 by using the Callaway model to describe $\kappa_{ph}$.

If there exists phonon elastic scattering from the disorder effects of nanowire surface27-33 or interface boundary of QDs embedded in the nanowire,36,37 the throughput function becomes more complicated.27-33 In general, the ${\cal T}_{ph}(\omega)$ depends on the length ($\tilde{L}$) and diameter ($D$) of the nanowire, phonon mean free path $\ell_{0}(\omega)$, and Debye frequency ($\omega_{D}$). Realistic calculation of ${\cal T}_{ph}$ requires heavy numerical work for treating the detailed phonon dispersion curves,29-33,36-37 which is beyond the scope of this article. However, an empirical expression which works well in general for semiconductor nanowires at a wide range of temperature can be found in Ref. 24, which reads

[TABLE]

with the frequency-dependent mean free path $\ell_{0}(\omega)$ given by

[TABLE]

where $N_{ph}(\omega)=4+A(\frac{D}{a})^{2}(\frac{\omega}{\omega_{D}})^{2}$ (for $\omega<\omega_{D}$) denotes the number of phonon modes. The dimensionless parameters are chosen to be $A=2.17$ and $B=1.2$. Notation $a$ denotes the lattice constant of nanowire. $N_{ph,1}(\omega)=N_{ph}(min(\omega,v_{s}/\delta))$ and $N_{ph,2}(\omega)=N_{ph}(\omega)-N_{ph,1}(\omega)$. $v_{s}$ is the sound velocity of nanowire and $\delta$ describes the thickness of the rough surface of nanowire24,28 In Eq. (12), one essentially replaces the frequency-dependent mean free path $\ell_{0}(\omega)$ by a constant $D$ for the high-frequency modes ($\omega>v_{s}/\delta$).

For a certain range of temperatures, a simple expression of $\kappa_{ph}(T)$ for molecular QD junction system may be used. One can approximately write13

[TABLE]

where $F_{s}$ is a dimensionless correction factor used to describe the effect of non-ballistic phonon transport due to surface roughness and phonon scattering from QDs, which replaces the throughput function ${\cal T}_{ph}(\omega)$ used in Eq. (11). The simple expression of Eq. (14) with $F_{s}=0.1$ will be used to describe $\kappa_{ph}$ throughout this article except in Fig. 4. In Fig. 4, we compare $ZT$ as a function of temperature obtained by using both Eq. (12) and Eq. (14). It is found that with the simple scaling factor $F_{s}$ we can describe the behavior of $\kappa_{ph}$ reasonably well for thin nanowires in the temperature range of interest. With this simple scaling we can clarify the effect of level degeneracy ($L$) on $ZT$ for different magnitudes of $\kappa_{ph}$.

III Results and discussion

Based on Eqs. (4)-(6), we numerically calculate thermoelectric coefficients including all correlation functions arising from electron Coulomb interactions in the QDs. Fig. 1 shows $ZT$ of a single QD junction as a function of the QD level $E_{0}$, which is tuned by gate voltage $V_{g}$ according to $E_{0}=E_{F}+50\Gamma_{0}-eV_{g}$ for the case of non degeneracy ($L=1$) and 3-fold degeneracy ($L=3$). Note that the role of gate voltage introduced here allow us to tune the difference between the QD level energy and Fermi energy. Throughout this article, we adopt a symmetrical tunneling rate with $\Gamma_{L}=\Gamma_{R}=\Gamma=\Gamma_{0}$ and all energy scales are in terms of $\Gamma_{0}$. $\Gamma_{0}\approx 1meV$ in typically QD junctions; thus, reasonable values for $U_{0}$ and $U_{I}$ in realistic semiconductor QDs are in the range of 20-100 $\Gamma_{0}$. Fig. 1(a), (b) and (c) are for $k_{B}T=1\Gamma_{0}$, $k_{B}T=5\Gamma_{0}$ and $k_{B}T=10\Gamma_{0}$, respectively. It is seen that the maximum $ZT$ for the 3-fold case is significant higher than the corresponding value for the non-degenerate case when the temperature is high. For example, the maximum $ZT$ (labeled by $(ZT)_{max}$) is enhanced by near two-times for $k_{B}T=5\Gamma_{0}$ and more than two-times for $k_{B}T=10\Gamma_{0}$, although the enhancement of $ZT$ for $L=3$ is small at $k_{B}T=1\Gamma_{0}$. We observe several new spectral features with similar $ZT$ values at $E_{0}$ values spaced apart approximately by the charging energy $U_{0}$ or $U_{I}$, which is caused by the intralevel and interlevel Coulomb interactions. For the non-degenerate case ($L=1$), thermoelectric coefficients can be calculated in terms of the transmission coefficient ${\cal T}_{LR}(\epsilon)$, which can be expressed as

[TABLE]

where $\bar{\Gamma}=(\Gamma_{L}+\Gamma_{R})$, and $N_{-\sigma}$ denotes the single-particle occupation number. Eq. (15) illustrates two resonant peaks at $\epsilon=E_{0}$ and $\epsilon=E_{0}+U_{0}$ with the probability weights of ($1-N_{-\sigma}$) and $N_{-\sigma}$, respectively, which are related to the two $M$-shaped spectral features in $ZT$ (labeled by $\epsilon_{1,1}$ and $\epsilon_{1,2}$) with the dip position corresponding to the resonance energies. (Here the intralevel Coulomb interaction used is $U_{0}=60\Gamma_{0}$) Similarly, for $L=3$ at $k_{B}T=1\Gamma_{0}$ we label the six $M$-shaped spectral features by $\epsilon_{3,n}$ ($n=1,\cdots 6)$, which result from the resonant channels at $E_{0}$, $E_{0}+U_{I}$, $E_{0}+2U_{I}$, $E_{0}+U_{0}+2U_{I}$, $E_{0}+U_{0}+3U_{I}$, and $E_{0}+U_{0}+4U_{I}$, respectively. (Here, we have adopted $U_{0}=U_{I}=20\Gamma_{0}$. $U_{I}$ denotes the interlevel Coulomb interactions for the $L=3$ case.) These channels correspond to physical processes of filling the QD with one to six electrons. At higher temperatures ($k_{B}T=5\Gamma_{0}$ and $k_{B}T=10\Gamma_{0}$), the 1st $M$-shaped spectral feature for $ZT$ is broadened and enlarged. (The last M-shaped $ZT$ feature for $L=3$ is not shown in Fig. 1) For $L=3$, the other spectral features of $ZT$ (at $\epsilon_{3,n}$; $n=2,3,4,5$) are suppressed. This is attributed to the significant reduction of maximum $S^{2}$ for those channels. Because of electron-hole symmetry in the system Hamiltonian, it is expected that the spectrum of $ZT$ is symmetrical about the middle point of the Coulomb gap (MPCG). For $L=3$, MPCG occurs at $eV_{g}=100\Gamma_{0}$. Therefore, we only need to focus on the analysis of $ZT$ optimization near the first spectral feature in the level-depletion regime, which is defined as the regime when the average occupation number of the QD summed over spin ($N_{t}$) is less than one. In general, it occurs at $E_{0}>E_{F}$.

To gain better understanding of the enhancement mechanism for $(ZT)_{max}$ resulting from increased degeneracy, we calculate the $G_{e}$, $S$, $\kappa_{e}$ and $ZT$ of the QD junction as functions of the level energy ($\Delta=E_{0}-E_{F}$) at $k_{B}T=5\Gamma_{0}$ for $L=1$ and $L=3$ and the results are shown in Fig. 2. In Fig. 2(a) the maximum $G_{e}$ value is enhanced with increasing of degeneracy, although its dependence of $L$ is not linear.This is mainly attributed to complicated correlation functions arising from the electron Coulomb interactions in QD. $G_{e}$ is much smaller than $G_{0}=\frac{2e^{2}}{h}$ (the electron quantum conductance) even for L=3, which is mainly attributed to strong electron Coulomb interactions. We note that the Seebeck coefficient is almost independent of $L$, whereas $G_{e}$ and $\kappa_{e}$ are enhanced with increasing $L$. However, since $\kappa_{ph}/\kappa_{e}\gg 1$, the $L$ dependence of $\kappa_{e}$ won’t affect $ZT$ appreciably. Thus, the enhancement of $ZT$ shown in Fig. 1(b) mainly comes from the increase of $G_{e}$, not $S$. In the Coulomb blockade regime, $\kappa_{e}$ and $G_{e}$ are highly suppressed. Thus, if one can introduce a mechanism to reduce $\kappa_{ph}$ (with $F_{s}=0.1$ for example), then the maximum value of $ZT$ can reach 1 for $L=1$ and around 2 for $L=3$ as illustrated in Fig. 2 (d).

The calculation of thermoelectric coefficients for the $L=3$ case including all correlation functions arising from electron Coulomb interactions is quite complicated. To reveal $L$-dependent $(ZT)_{max}$, we consider the transmission coefficient ${\cal T}_{LR}(\epsilon)$ including only the contribution from the resonant channel $\epsilon-E_{0}$, which is approximately given by

[TABLE]

where $P_{L,1}$ is the $L$-dependent probability weight for the resonant channel at $\epsilon=E_{0}$. We have $P_{3,1}=(1-N_{-\sigma})(1-(N_{-\sigma}+N_{\sigma})+c)(1-(N_{-\sigma}+N_{\sigma})+c)$, where $c=\langle n_{\ell,\sigma}n_{\ell,-\sigma}\rangle$ denotes the intralevel two particle correlation function.38

Thermoelectric coefficients determined by the ${\cal T}_{LR}(\epsilon)$ of Eqs. (15) and (16) can be calculated by $G_{e}=e^{2}{\cal L}_{0}$, $S=-{\cal L}_{1}/(eT{\cal L}_{0})$ and $\kappa_{e}=\frac{1}{T}({\cal L}_{2}-{\cal L}^{2}_{1}/{\cal L}_{0})$. ${\cal L}_{n}$ is given by

[TABLE]

where $f(\epsilon)=1/(exp^{(\epsilon-E_{F})/k_{B}T}+1)$ is the Fermi distribution function of electrodes.

Because Eq. (16) does not take into account the interlevel correlation functions arising from $U_{I}$, Eq. (16) is not adequate for illustrating thermoelectric coefficients for the situation $\Delta/k_{B}T\leq 1$. Nevertheless, we see that $(ZT)_{max}$ of Fig. 2 does not occur in the $\Delta/k_{B}T\leq 1$ regime. Therefore, we consider the limit of weak coupling between QD and electrodes ($\Gamma_{L}=\Gamma_{R}=\Gamma\rightarrow 0$) in Eqs. (15) and (16) and obtain $G_{e}=\frac{G_{0}\Gamma\pi LP_{L,1}}{k_{B}Tcosh^{2}(\frac{\Delta}{2k_{B}T})}$, $S=-\Delta/(eT)$, and $\kappa_{e}=0$. The $L$-dependent behavior of $(ZT)_{max}$ is then determined by the simple expression

[TABLE]

which explains that the $L$-dependent $ZT$ is determined by $G_{e}$ rather than $S$ and why $ZT$ approaches 0 as $E_{0}\rightarrow E_{F}$ for $L=1$. Note that $ZT$ for $L=3$ does not approach zero as $\Delta\rightarrow 0$, because $S$ has a finite value at $\Delta=0$ (see the dashed line of Fig. 2(b)). Such a result also indicates that the correlation arising from $U_{I}$ can not be neglected for QD when $E_{0}$ is close to $E_{F}$. Some novel nanoscale TE devices resulting from the inclusion of $U_{I}$ were theoretically discussed for designing electron heat rectifiers38 and current diodes.39

Figure 3 shows $G_{e}$, $S$, $\kappa_{e}$ and $ZT$ as functions of temperature with $\Delta=15\Gamma_{0}$ for $L=1,3$, and 4. From the application point of view, the temperature dependence of $ZT$ is an important consideration for developing room temperature power generators used in consumer electronics.2 $G_{e}$ is highly enhanced for $L=3$ and 4 in the whole temperature regime, but the difference of $L=3$ and $L=4$ is small, indicating a saturation behavior as $L$ exceeds 3, mainly because of the factor $P_{L,1}$ in Eq. (16). As seen in Fig. 3(b), $S$ is nearly independent of $L$ for $k_{B}T<7\Gamma_{0}$, but becomes weakly dependent on $L$ at higher temperature. This implies that the effect of resonances at $\epsilon_{L,2}$ and $\epsilon_{L,3}$ can not be ignored for $L>1$ in the high temperature regime ($k_{B}T\geq 7\Gamma_{0}$). Although $\kappa_{e}$ is enhanced with increasing $L$ as shown in Fig. 3(c), its effect is insignificant since $\kappa_{e}$ is much smaller than $\kappa_{ph}$. Therefore, the behavior of $ZT$ with respect to $k_{B}T$ is determined by the power factor ($PF=S^{2}G_{e}T$). We found impressive enhancement of $ZT$ for $L=3$ and 4 for $k_{B}T\geq 4\Gamma_{0}$. Comparing Figs. 2(d) and 3(d), we see that the maximum values of $ZT$ occur near $k_{B}T=\Delta/2.4$ for the tunneling rate considered ($\Gamma=1\Gamma_{0}$). Based on such a condition, we can infer that the maximum $ZT$ at room temperature $k_{B}T=25~{}meV$ will occur near $\Delta=60~{}meV$ for $\Gamma_{0}=~{}1meV$.

In Figures (1)-(3), $\kappa_{ph}$ is assumed to obey the simple expression give in Eq. (14). Here we examine whether the large enhancement of $ZT$ due to level degeneracy will be destroyed when a more realistic throughput function given by Eq. (12) is considered. Fig. 4 shows the comparison of $ZT$ and $\kappa_{ph}$ calculated by both Eq. (12) and Eq. (14). In Fig. 4(a), $\kappa_{ph}$ used to obtain the solid ($L=3$) and dashed curves ($L=1$) for ZT are calculated by using Eqs. (11) and (12) with $D=4~{}nm$, $\delta=2~{}nm$ and $\tilde{L}=2\mu m$. Other parameters are given by physical properties of silicon semiconductors.24,28 The triangle ($L=3$) and square marks ($L=1$) for ZT are calculated by using $\kappa_{ph}$ based on Eq. (14). The maximum $ZT$ values of solid and dashed lines are near 1.8 and 0.9, respectively. The results of Fig. 4(a) indicate that the large enhancement of $ZT$ resulting from $L$ is unchanged even when a more realistic expression for $\kappa_{ph}$ (which is nonlinear in temperature) is used. When we compare the spectra of $ZT$ given by the solid curve and the curve with triangle marks for the case of $L=3$, the curves with triangle marks have better $ZT$ value at low temperature due to the lower value of $\kappa_{ph}$ in the linear-$T$ expression. Fig. 4(b) shows $\kappa_{ph}$ for nanowires with diameter of $10,15$, and $20~{}nm$, which agree well with the experimental results of Ref. 24, lending support for the validity of this model. The comparison of behaviors of $\kappa_{ph}$ for a 4 nm nanowire obtained by both Eq. (12) (solid curve) and Eq. (14) (triangles) is shown in Fig. 4(c). It is found that the results obtained by the simple linear-$T$ expression of Eq. (14) are fairly close to that obtained by the realistic expression of Eq. (12) for temperatures between 50K and 200K. In Fig. 4(c), $\kappa_{ph}$ shows a nonlinear-$T$ behavior between $1K$ and $50$, which is mainly attributed to frequency-dependent mean free path. The dashed and doted lines show the behavior of electronic thermal conductance ($\kappa_{e}$) with respect to temperature. Note that the $G_{e}$, $S$ and $\kappa_{e}$ in Fig. 4 are calculated according to the simplified method described in Ref. 38, where we only considered single-particle occupation numbers and intralevel two-particle correlation functions. The curves with triangle and square marks shown in Fig. 4(a) are almost identical to the black solid line and red dashed line of Fig. 3(d) obtained by the full calculation.

Next we examine whether the large enhancement of $ZT$ due to increase of level degeneracy still exists in the case of coupled double QDs (DQDs). The Hamiltonian of a DQD is given by $H_{DQD}=H_{QD,L}+H_{QD,R}+U_{LR}\sum_{\ell,j}n_{L,\ell,\sigma}n_{R,j,\sigma^{\prime}}+t_{LR}\sum_{\ell,j}(d^{\dagger}_{L,\ell,\sigma}d_{R,j,\sigma}+h.c)$.40-42 $H_{QD,L}$ ($H_{QD,R}$) denotes the Hamiltonian of the left (right) QD as defined in Eq. (2). For simplicity, the interdot electron hopping strengths $(t_{LR})$ and electron Coulomb interactions ($U_{LR}$) are assumed uniform. Although electron tunneling currents through DQDs have been extensively studied by several authors,40-42 the optimization of $ZT$ including the effect of all correlation functions arising from electron Coulomb interactions has not been reported. Here, we assume one nondegenerate energy level for each QD ($L=1$). The energy levels of left QD and right QD are the same (denoted $E_{0}$). Based on Eqs. (4)-(6), $G_{e}$, $S$, $\kappa_{e}$ and $ZT$ as functions of the QD energy level (which is related to the gate voltage by $E_{0}=E_{F}+50\Gamma_{0}-eV_{g}$) for $K_{B}T=3$, 5, and 7 $\Gamma_{0}$ are plotted in Fig. 5. There are four peaks labeled by $\epsilon_{1,n}$ ($n=1,2,3,4$) in the spectrum of electrical conductance ($G_{e}$). The Seebeck coefficient ($S$) behaves like the derivative of $-G_{e}$ and is vanishingly small at the MPCG due to electron-hole symmetry. The maximum $S$ occurs near the onset of the first peak in $G_{e}$ or the ending of the last peak. Both $\kappa_{e}$ and $G_{e}$ are symmetrical with respect to MPCG. Similar to the single QD $L=1$ case in Fig. 1(d) the maximum $ZT$ values occur at either the level-depletion regime or the full-charging regime as seen in Fig. 5(d). Here, the maximum $ZT$ is close to 1.7 at $k_{B}T=3\Gamma_{0}$ with $F_{s}=0.1$.

To further understand the relationship between physical parameters and thermoelectric coefficients, we consider some approximations which include only the dominant correlation functions to derive ${\cal T}_{LR}(\epsilon)$ for DQD with L=1. Simple analytic expressions of $G^{<}_{j,\sigma}(\epsilon)$,$G^{r}_{j,\sigma}(\epsilon)$ and $G^{a}_{j,\sigma}(\epsilon)$ can be found in our previous work42 when we only include intradot two-particle correlation functions in the probability weights. Here, we include all two-particle and three-particle correlation functions. Then, the following expression of ${\cal T}_{LR}(\epsilon)$ is obtained by solving the hierarchy of equations of motion (which terminates at the 4-particle Green function) via a similar procedure as described in Ref. 42.

[TABLE]

where $\mu_{L}=\epsilon-E_{L}+i\Gamma_{L}$ and $\mu_{R}=\epsilon-E_{R}+i\Gamma_{R}$. The probability weights are given by $P_{1,1}=1-N_{L,\bar{\sigma}}-N_{R,\bar{\sigma}}-N_{R,\sigma}+\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}\rangle+\langle n_{L,\bar{\sigma}}n_{R,\sigma}\rangle+\langle n_{R,\bar{\sigma}}n_{R,\sigma}\rangle-\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, $P_{1,2}=N_{R,\bar{\sigma}}-\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}\rangle-\langle n_{R,\bar{\sigma}}n_{R,\sigma}\rangle+\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, $P_{1,3}=N_{R,\sigma}-\langle n_{L,\bar{\sigma}}n_{R,\sigma}\rangle-\langle n_{R,\bar{\sigma}}n_{R,\sigma}\rangle+\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, $P_{1,4}=\langle n_{R,\bar{\sigma}}n_{R,\sigma}\rangle-\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, $P_{1,5}=N_{L,\bar{\sigma}}-\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}\rangle-\langle n_{L,\bar{\sigma}}n_{R,\sigma}\rangle+\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, $P_{1,6}=\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}\rangle-\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, $P_{1,7}=\langle n_{L,\bar{\sigma}}n_{R,\sigma}\rangle-\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, and $p_{1,8}=\langle n_{L,\bar{\sigma}}n_{R,\bar{\sigma}}n_{R,\sigma}\rangle$, where $\langle n_{\ell,\bar{\sigma}}n_{j,\sigma}\rangle$ denote the two particle correlation functions and $\langle n_{\ell,\bar{\sigma}}n_{j,\sigma}n_{j,\bar{\sigma}}\rangle$ the three-particle correlation functions (including both intradot and interdot terms). Note that the probability weights satisfy the conservation law $\sum_{m}P_{1,m}=1$. $U_{L(R)}$ and $U_{L,R}$ denote the intradot and interdot Coulomb interactions. When all correlation functions of DQDs are included as in Refs. 21 and 40, it is difficult to find an analytical expression for ${\cal T}_{LR}(\epsilon)$. The thermoelectric coefficients, $G_{e}$, $S$, $\kappa_{e}$, and $ZT$ obtained by using Eq. (III) are plotted in Fig. 6. The results are in very good agreement with those shown in Fig. 5, which are obtained by the full calculation, including all correlation functions. It should be noted that Eq.(III) works well only in the limit $t_{LR}/k_{B}T\ll 1$. If one would like to study the spin-dependent thermoelectric coefficients of DQD in the low temperature regime ($k_{B}T<t_{LR}$), all correlations functions should be included.40

In Figs. 5 and 6, the peak positions $\epsilon_{1,n}$ ($n=1,2,3,4$) correspond to the channels with probability weights $P_{1,1}$, $P_{1,3}$, $P_{1,6}$ and $P_{1,8}$, respectively. There exists an interesting behavior for $S$ at low temperature ($k_{B}T=3\Gamma_{0}$). We found that $S$ is a linear function of $eV_{g}$ near the maximum of $G_{e}$. From the $\kappa_{e}$ behavior shown in Figs. 5(c) and 6(c), the electron heat flow is maximized near the mid point between the first (or last) two resonant channels, and it increases with increasing temperature. Eq. (III) allows us to obtain an analytic form of thermoelectric coefficients, which is very useful for clarifying how thermoelectric coefficients are influenced by tunneling rates, inter-dot hopping strength, and electron Coulomb interactions. Our analysis shows that $S$ is independent of $t_{LR}$ and it has a linear dependence of $\Delta=E_{0}-E_{F}$ (see Eqs. (21) and (22)) near maximum $ZT$. As a consequence, the trend of $ZT$ with respect to $t_{LR}$ is determined by that of $G_{e}$ for $\kappa_{ph}/\kappa_{e}\gg 1$.

Next we consider the case with two-fold degenerate levels ($L=2$) for each QD in the DQD junction. Such two-fold degeneracy can be realized in QDs with suitable symmetry. For example, the $x$- and $y$-like states in a disk-shaped QD are degenerate. Due to symmetry, the intradot electron hopping process is prohibited, whereas the interdot electron hopping strength is nonzero. We assume $t_{LR}=1\Gamma_{0}$, the same as that of the $L=1$ case. The intradot Coulomb interactions are taken to be $U_{L,i,j}=U_{R,i,j}=U_{I}=50\Gamma_{0}$, where $i,j=1,2$ denote the two degenerate levels within the same QD. The interdot Coulomb interaction is taken as $U_{L,R}=40\Gamma_{0}$. The calculation of thermoelectric coefficients of DQD with $L=2$ involves solving one-, two-, $\cdots$, up to eight-particle Green functions. Due to the presence of $t_{LR}$ term, the numerical procedure is much more complicated than that of a single QD with $L=4$. Based on Eqs. (4)-(6), we calculate the thermoelectrical coefficients of DOD for the $L=2$ as functions of the QD energy level as shown in Fig. 7. The first four resonant channels of $G_{e}$ (on the left hand side of MDCG) are approximately given by $\epsilon_{2,1}=E_{0}$, $\epsilon_{2,2}=E_{0}+U_{LR}$, $\epsilon_{2,3}=E_{0}+U_{LR}+U_{I}$, and $\epsilon_{2,4}=E_{0}+2U_{LR}+U_{I}$, in which the small $t_{LR}$ is neglected since $t_{LR}\ll U_{L,R}$. The oscillatory behavior of $G_{e}$ displayed in Fig. 7(a) is similar to the $G_{e}$ spectra observed experimentally in tunneling current measurements of PbSe QD (which has a six-fold degenerate excited state) and carbon nanotube QD (which has an eight-fold state).43,44 Although the $S$ spectrum exhibits more bipolar oscillatory structures, the maximum $S$ value does not increase with increasing $L$. This feature is the same as that of a single QD case. Fig. 7(d) shows an large enhancement of maximum $ZT$ arising from the degeneracy effect. In the current case, $ZT_{max}$ reaches around 2.7. Comparing Fig. 7(d) with Fig. 5(d), we see an enhancement of maximum $ZT$ from around 1.7 to 2.7 when $L$ increases from 1 to 2 for DQD junction when $F_{s}=0.1$. We expect even larger enhancement to occur for higher level degeneracy. Unfortunately, the computation effort for $L>2$ for a DQD junction is prohibitively large if all Green functions and correlations functions are to be included.

To clarify the behavior of $ZT_{max}$ in the level-depletion regime (with $N_{t}<1$), we can approximately write (by keeping only the dominant channel)

[TABLE]

where $P_{L,1}$ is the probability weight for DQD in the level-depletion regime. Under the assumption of $\Gamma_{L}=\Gamma_{R}\rightarrow 0$ and $t_{LR}/k_{B}T\ll 1$, we have

[TABLE]

and

[TABLE]

From Eqs. (21) and (22), we have $G_{e}=e^{2}{\cal L}_{0}$ and $S=-\Delta/(eT)$. This reveals the behavior of $S$ around the maximum of $G_{e}$ at $k_{B}T=3\Gamma_{0}$ in Fig. 6(b) and L-dependent $ZT_{max}$ determined by $G_{e}$ in Fig. 7(d). Note that if we artificially set $F_{s}=0$ (i.e. neglecting $\kappa_{ph}$), one can prove that the enhancement of $ZT_{max}$ arising from $L$ will disappear due to the $L$-independence of the ratio $G_{e}/\kappa_{e}$ and $L$-independence of $S$. If we choose a much higher value of $F_{s}$ (e.g. $F_{s}=1$) in Eq. (14), we will have the condition $\kappa_{ph}/\kappa_{e}\gg 1$. In this situation, $L$ dependence of $ZT_{max}$ is fully determined by $G_{e}$ and we have $ZT$ linearly proportional to $L$, since $P_{L,1}$ in Eq. (18) is close to 1 under the condition $(E_{0}-E_{F})/k_{B}T\gg 1$, and the $ZT$ values will be approximately proportional to $1/F_{s}$. Finally, we would like to point out that QD junctions embedded in a silicon nanowire can be realized by the advanced technique reported in Refs. 45 and 46.

IV Conclusion

We have theoretically investigated the effects of level degeneracy on thermoelectric properties of QDs embedded in a thin nanowire junction in the Coulomb blockade regime. All the correlation functions arising from electron Coulomb interactions for electrons in the degenerate levels are included in our calculation. We found that the maximum values of $ZT$ can be highly enhanced with level degeneracy under the typical condition with $\kappa_{ph}$ much larger than $\kappa_{e}$. When $(E_{0}-E_{F})/k_{B}T\gg 1$, $S$ is independent on $L$. Therefore, the enhancement of $ZT_{max}$ in the level-depletion regime is mainly attributed to the increase of $G_{e}$. Large enhancement of $ZT$ due to the increase of level degeneracy is also found in the presence of finite electron hopping in coupled QD system. In our studies, we assumed a simple expression $\kappa_{ph}=F_{s}g_{0}(T)$ for the phonon thermal conductance. However, it is worth pointing out that our conclusion on the effect of level degeneracy on $ZT$ is not limited to the linear $T$-behavior of $\kappa_{ph}$ (as illustrated in Fig. 4). The enhancement due to level degeneracy holds as long as $\kappa_{ph}>\kappa_{e}$, regardless of the temperature dependance of $\kappa_{ph}$. This implies that the design principle based increasing level degeneracy is applicable for a composite materials involving arrays of coupled QDs1,2 and molecular QDs.13

**Acknowledgments

**This work was supported under Contract Nos. MOST 103-2112-M-008-009-MY3 and MOST 104-2112-M-001-009-MY2.

E-mail address: mtkuo@ee.ncu.edu.tw

E-mail address: yiachang@gate.sinica.edu.tw