The Thermopower of a Quantum Dot Coupled to Luttinger Liquid System

With the progress in nanofabrication technique and nanometer scale materials, research on electron transport properties of mesoscopic systems has become a very active field in condensed matter physics. Considerable researches have been mainly focused on charge transport in nanostructures and nanodevices. Besides the charge transport, a detailed understanding of heat transport through mesoscopic systems is of equally importance (Afonin, 1995; Small, 2003) because they can provide additional information on the kinetics of carriers not available in the measurement of current voltage characteristics (Heremans, 2004). For instance, thermoelectric properties are very sensitive to dimensionality, the electronic spectrum near the Fermi level, scattering processes (Koch, 2004), electron-phonon coupling strength (Yang1, 2010) and electron-hole symmetry (Small, 2003). There have been several theoretical studies on the thermopower S, which mainly focused on quantum dot (QD) coupled to the normal Fermi liquid (FL) leads (Boese, 2001; Dong, 2002; Kim, 2003; Krawiec, 2007; Yang1, 2010), denoted hereafter as FL-QD-FL. As for systems containing a quantum dot coupled to one-dimensional (1D) interacting electron leads, although their charge transport phenomena have been investigated (Yang2, 2001; Yang3, 2010), yet there have been much less efforts devoted to the thermoelectric properties of them (Kane, 1996; Krive, 2001; Romanovsky, 2002). It is well known that the 1D interacting electron systems can be described by the Luttinger liquied (LL) theory (Luttinger, 1963), which holds some unique features such as spin-charge separation, suppression of the electron tunneling density of states, power-law dependence of the electrical conductance on temperature and bias voltage, etc.. The LL behaviour has been experimentally reported in singleand multi-wall carbon nanotubes (Bockrath, 1999; Kanda, 2004; Yao, 1999) and fractional quantum Hall edge states (Chang, 1996). Recently, the use of carbon nanotubes as a thermoelectric material has gained great interest due to their 1D structure. The thermopower of single-walled carbon nanotubes have beenmeasured in experiments (Bradley, 2000; Choi, 1999; Collins, 2000; Hone, 1998; Kong, 2005; Small, 2003). For example, Kong et al. have shown a linear temperature dependence of the thermopower at low temperature (Kong, 2005). Small et al. have observed strong modulations of thermopower as the function of gate voltage Vg in individual Carbon nanotubes (Small, 2003). Dresselhaus et al. have found that the low-dimensional The Thermopower of a Quantum Dot Coupled to Luttinger Liquid System


Introduction
With the progress in nanofabrication technique and nanometer scale materials, research on electron transport properties of mesoscopic systems has become a very active field in condensed matter physics. Considerable researches have been mainly focused on charge transport in nanostructures and nanodevices. Besides the charge transport, a detailed understanding of heat transport through mesoscopic systems is of equally importance (Afonin, 1995; because they can provide additional information on the kinetics of carriers not available in the measurement of current voltage characteristics (Heremans, 2004). For instance, thermoelectric properties are very sensitive to dimensionality, the electronic spectrum near the Fermi level, scattering processes (Koch, 2004), electron-phonon coupling strength (Yang1, 2010) and electron-hole symmetry . There have been several theoretical studies on the thermopower S, which mainly focused on quantum dot (QD) coupled to the normal Fermi liquid (FL) leads (Boese, 2001;Dong, 2002;Krawiec, 2007;Yang1, 2010), denoted hereafter as FL-QD-FL. As for systems containing a quantum dot coupled to one-dimensional (1D) interacting electron leads, although their charge transport phenomena have been investigated (Yang2, 2001;Yang3, 2010), yet there have been much less efforts devoted to the thermoelectric properties of them (Kane, 1996;Romanovsky, 2002). It is well known that the 1D interacting electron systems can be described by the Luttinger liquied (LL) theory (Luttinger, 1963), which holds some unique features such as spin-charge separation, suppression of the electron tunneling density of states, power-law dependence of the electrical conductance on temperature and bias voltage, etc.. The LL behaviour has been experimentally reported in single-and multi-wall carbon nanotubes (Bockrath, 1999;Kanda, 2004;Yao, 1999) and fractional quantum Hall edge states (Chang, 1996). Recently, the use of carbon nanotubes as a thermoelectric material has gained great interest due to their 1D structure. The thermopower of single-walled carbon nanotubes have been measured in experiments Choi, 1999;Hone, 1998;Kong, 2005;. For example, Kong et al. have shown a linear temperature dependence of the thermopower at low temperature (Kong, 2005). Small et al. have observed strong modulations of thermopower as the function of gate voltage Vg in individual Carbon nanotubes . Dresselhaus et al. have found that the low-dimensional thermoelectric materials performed better than bulk ones (Dresselhaus, 1999). Several theoretical works have been developed to predict the enhancement of thermopower in nanoscaled materials by the intralead electron interaction (Kane, 1996;Krive2, 2001;Romanovsky, 2002). Krive et al. (Kane, 1996;) used a phenomenological approach to investigate the thermopower of a LL wire containing an impurity. In spite of the above work, an explicit thermopower formula in the LL leads was not given out. In the following, we use the notation S to denote the thermopower S of systems comprosing LL and S 0 to those comprising noninteracting FL. Theoretically, the thermopower S of a LL with an impurity can be represented by the thermopower S 0 multiplied by an interaction-dependent renormalization factor. Alternatively, one may intentionally introduce a QD into the LL, denoted as LL-QD-LL. Thus, he may attach a QD, instead of an impurity atom, to the end of a carbon nanotube. A quantum dot is experimentally more controllable than an impurity. For instance, the energy of a quantum dot can be tuned by the gate voltage V g .
In this chapter, we will first give the stationary thermopower formula of a QD coupled to LL leads through tunneling junctions, a system of LL-QD-LL (see Fig.1) by applying the nonequilibrium Green function technique (Haug, 1996) instead of phenomenal theories. And then we later turn our attention to the time-dependent phenomena. The generalized thermopower formula is obtained under time-dependent gate voltage. Although there are many studies on time-dependent nonequilibrium transport, the research on the time-dependent thermopower and the formula under the ac field are still lack. Here we will fill the blanks for the low dimension system. Fig. 1. The two-terminal electron transport through a single-level quantum dot weakly coupled to the Luttinger liquid leads with the chemical potentials µ L and µ R . Electrons tunnel from one lead to another by hopping on and off the dot level with the energy ε. The position of the dot levels with respect to the Fermi energy can be uniformly shifted by applying a voltage V g to the gate electrode.

The Model
In the considered LL-QD-LL system, the QD is weakly connected with semi-infinite LL electrodes. The Hamiltonian of this system includes three parts: which represents the Hamiltonians of the left and right LL leads (H leads = H L + H R ), the central dot and the tunneling interactions between them H T , respectively. Firstly, we present a detailed discussion of bosonization for a continuum model of length L with open boundary conditions (Eggert0, 1992;Fabrizio, 1995;Furusaki0, 1994) and electron-electron interaction.

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Fingerprints in the Optical and Transport Properties of Quantum Dots www.intechopen.com The specify Hamiltonian of the LL leads can be easily written where the first term represents the kinetic energy, and the second one describes the electron-electron interaction, ε k is the dispersion law of the 1D band, and ψ σ (x) is the spin σ electron annihilation operator subject to the open boundary conditions: We apply the boundary conditions Eq.(5) to expand electron annihilation operator ψ which takes the form with k = πn/L, n being a positive integer. Usually a 1D system with periodic boundary conditions has two Fermi points ±k F . Here we only have single Fermi point given by k = k F . The 1D fermion field ψ σ can be expanded about the Fermi point k F in terms of the left moving and right moving fields as In the case of periodic boundary conditions (Haldane, 1981), these left moving and right moving fields are not independent and satisfy Then the fermion fields automatically satisfy the boundary conditions whereas the condition ψ σ (L)=0 implies that the operator ψ σR (x) should obey Therefore, we can actually work with the right moving field only, the left moving one is then defined by the above relation. Thus the field ψ σR (x) can be defined for all x, and obeying the 449 The Thermopower of a Quantum Dot Coupled to Luttinger Liquid System www.intechopen.com periodicity condition with the period 2L: In terms of the right moving field, the kinetic energy terms in the Hamiltonian Eq.(1) takes the form where we have linearized the electron spectrum. The single fermion operators for right-moving electrons with spin σ on lead α can be bosonized in the position representation by applying the periodic boundary condition Eq.(12) as The operator η σ is real fermion and satisfies the anti-commutation relations with δ is Delta function. Eq.(15) assure the correct anti-commutation rules for electron operators with different σ. In order to calculate the correlation function, a method of dealing with this was suggested by Luther and Perchel (Luther, 1974). It used a limiting process, where the wave function contained s parameter α ′ , and the limit α ′ → 0 is taken at the end of the calculation of the correlation function. Using the parameter α ′ , we can represent the electron operator ψ σR (x) as Where α ′ is a short-distance cutoff of the order of the reciprocal of the Fermi wave number k F . The phase field φ σ (x) satisfies periodic boundary condition: and can be expressed as follows here, a † q and a q are the creation and annihilation operators of bosons. These operators satisfy the canonical bosonic commutation relations [a q , a † q ′ ]=δ(q − q ′ ). q = πn/L , n is an integer. The density of right moving electrons is given by Applying the boundary conditions Eq.(8), we have 450 Fingerprints in the Optical and Transport Properties of Quantum Dots

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The bosonized form of the kinetic energy is where colon represents the normal order form of the operators.
In order to deal with electron-electron interacting terms in Hamiltonian Eq.
(1), we continue to make use of the above bosonization procedure expressing the electron interaction Hamiltonian in terms of the right moving Fermi field ψ σR only.
Before we turn to the interaction effects, we introduce the bosonic variables corresponding to charge and spin excitations: and The interaction part of the Hamiltonian contains several terms classified as: the diagonal terms in the electron densities and the mixing left and right densities term. Consequently, the Hamiltonian becomes where where v F is the Fermi velocity, g 4 and g 2 represent forward scatterings; in our work, we will not consider the backscattering interaction. In the absence of backscattering, the Luttinger Hamiltonian, H LL , is exactly soluble using the technique of bosonization. In order to express the Hamiltonian in diagonal form, we introduce the canonically conjugate Boson operators, in a standard way by the Bogolubov rotation, The Hamiltonian can be achieved by the canonical transformation in terms of b νq and b † νq where

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The Thermopower of a Quantum Dot Coupled to Luttinger Liquid System www.intechopen.com The unitary operator U is defined by In the next step we find how the Fermi operators transform by applying U. Employing the method of Mattis and Lieb (Mattis, 1964), after lengthy but straightforward calculations, we arrive at the expression for the electron annihilation operator in terms of free bosons for the case of the interacting Fermi system with open boundaries: where ε ν σ is +1 unless σ =↓ and ν = σ when its value is −1. We have defined In the continuum limit, the Hamiltonian can be expressed This Hamiltonian describes the propagation of the charge density fluctuations in the leads with renormalized velocity v c . From Krönig's relation (Krönig, 1935), the kinetic term has been written in a quadratic form of the density operators, because the bosons are defined as excitations above an N particle ground state, Hamiltonian must include terms that include the energy of the different bosonic ground states. These terms are not required for the calculations in this chapter, and are hence omitted.
The Hamiltonian of the single-level QD takes the form of where ε is the energy of the electron on the dot, and d † and d are fermionic creation and annihilation operators satisfying canonical commutation relation {d, d † } = 1.
The tunneling Hamiltonian is given by the standard expression: where t α is the electron tunneling constant and ψ † α , ψ α (α = L/R) are the Fermi field operators at the end points of the left/right lead. The operator ψ α could be written in a "bosonized" form (Furusaki, 1998) where and K ρα = e 2ϕ ν is the interaction parameter in the "fermionic" form of the LL Hamiltonian (33), which restricts the LL parameter g to vary between 0 and 1. The noninteracting case corresponds to v c = v F and K ρα = 1. For repulsive interactions, K ρα < 1. Because of the SU(2) spin symmetry under no magnetic filed, K σ = 1. Thus the correlation functions the end point of the left LL lead without the coupling to the quantum dot ψ † σ (0, t)ψ σ (0, 0) can be obtained after long calculation Where c A is a dimensionless constant of order 1, Λ is a high-energy cutoff or a band width, δ is positive infinitesimal, and g −1 Similarly, the correlation function at the end point of the the right lead is obtained as the above method. The electron-electron interaction parameters of the left and right LL leads are assumed equal g L = g R = g for convenience.

The thermopower formula under no ac field
The charge current J L flowing from the left lead L into the quantum dot can be evaluated as follows: We introduce the time-diagonal parts of the correlation functions: With the help of the Langreth analytic continuation rules (Haug, 1996). By means of them, it is easy to express the current as ). After applying Langreth theorem of analytic continuation, the average current can then be expressed as In terms of a long derivation, we can easily establish an expression for the expectation value of the electric current where G r(a) d is retarded (advanced) Green function of the quantum dot and Γ L/R , proportional to |t L/R | 2 , describes the effective level broadening of the dot. g (ω) is the Fourier transform of the lesser (greater) Green function at the end point of the left (right) LL lead without the coupling to the QD, which has been obtained by (Furusaki, 1998): Now we define the Luttinger liquid distribution functions F <,> L/R as

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The Thermopower of a Quantum Dot Coupled to Luttinger Liquid System www.intechopen.com The function F < (ω) is the electron occupation number for interacting electrons which is analogous to the Fermi distribution function f (ω) of noninteracting electrons and F > (ω) is analogous to 1 − f (ω) of FL leads. T L/R is temperature and µ L/R the chemical potential of the left or right lead where µ L = µ + ηV and µ R = µ +(η − 1). Γ(z) is the Gamma function. Following the derivation in Ref. (Yang3, 2010), the current can be obtained as is the transmission probability. If g = 1, Eq. (43) will degrade to the usual well-known current expression for a FL-QD-FL system.
Our goal is to find the general thermopower formula of the model described by the Hamiltonian Eq. (62). The thermopower S is defined in terms of the voltage V generated across the quantum dot when temperature gradient ΔT = T L − T R is much less than T L and T R and when current J is zero (Cutler, 1969): where L 12 and L 11 are linear response coefficients when the current J L is presented by small bias voltages and small temperature gradients ΔT: where . Comparing both sides of the Eqs.(78) (let e = 1,h = 1), we obtain The formulas Eqs.(46) and Eqs. (47) are independent of the approximation adopted in deriving the retarded (advanced) Green function. However, the partial derivatives ∂F ∂µ and ∂F ∂T are not yet expressed evidently. In the following we will show the explicit expression for L 11 and L 12 . The linear expansion of the Luttinger liqiud distribution function becomes In order to achieve a compact expression, we define and expand them to the first order derivatives: and

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Fingerprints in the Optical and Transport Properties of Quantum Dots www.intechopen.com where F 1 (µ, T) and F 2 (µ, T) are the equilibrium LL distribution functions, and F 1 (µ, T)= F 2 (µ, T). Then Substituting of Eq.(51) into Eq.(43)enables one to obtain the expressions of ∂F ∂V and ∂F ∂(ΔT) required in Eqs. (46) and (47). We arrive at that and In derivation we have used the relation |Γ( With the same deriving process, we obtain the partial derivation with respect to temperature as ∂F It follows from Eqs. (54), (55), (46) and (47) that and with T(ω)| δV=0,δ(ΔT)=0 . And they become functions related to the QD density of states and LL distribution function. We stress that Eqs. (56) and (57) are the linear response coeffcients in a LL-QD-LL system. These equations will naturally degrade to those of a FL-QD-FL system if g = 1. The thermopower can be obtained by the equation Eq. (44) in which the current equals to zero. Substituting Eqs. (56) and (57) into Eq. (44), we have

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The Thermopower of a Quantum Dot Coupled to Luttinger Liquid System www.intechopen.com As shown in Eq. (78), when temperature difference between the leads is zero, conductance is then given by G = e 2 L 11 /T. Comparison of the explicit expressions of the conductance G and thermopower S exhibits that the latter contains information different from the former.
In calculation, the Green functions of the QD are required as shown in Eq. (40). The retarded Green function is defined by G r (t)=−iθ(t) {d(t), d † (0)} and can be derived by means of the equation of motion method. Its analytical expression is where the retarded self-energy is originated from the tunneling into the leads and is given by: In the next section we will give our numerical results and discuss the thermoelectric properties.

Numerical results
The expressions (56) and (57)enable us to calculate numerically the conductance and thermopower as functions of the applied voltage and temperature. It is assumed that the system is of structural symmetric: Γ L = Γ R = Γ. In calculation we take the coupling strength Γ as the energy unit and set the Fermi level of the lead to be zero. Then the energy level ε of the QD represents the gate voltage Vg. No other bias is applied, i.e., we always consider the zero bias case. Figures 1(a) and (b) show the gate voltage dependence of the conductance and thermopower, respectively. The conductance varies smoothly, which is in agreement with the previous scanning gate microscopy experiments (Small2, 2003;Woodside, 2002). The thermopower S varies rapidly with the variation of the gate voltage and can reach a large absolute value at low temperature. Obviously, the gate voltage violates electron-hole asymmetry and its value tunes the thermopower. Experiments did show the features Small3, 2004).
From Fig. 1 it is seen that the conductance is an even function of ε , while the thermopower is an odd function: S(ε)=−S(−ε) , which is coincide to experiments (Staring, 1993). It is easily understood that the Hamiltonian in this paper possesses electron-hole symmetry: as V g is changed to −V g , the form of the Hamiltonian remains unchanged if the electrons are simultaneously converted to holes. This is the foundation of discussing the symmetry relations for the dependence of the G and S on V g . When both bias voltage V and the current J L change their signs, the sign of the conductance G = dJ L /dV remains unchanged. Subsequently, G(−ε)=G(ε). On the other hand, the temperature difference is irrelevant to the change of the current carriers, i.e., the kinetic coefficient L 12 = dJ L /d(δT) , changes sign: L 12 (ε)=−L 12 (−ε) when V g changes sign. Thus we conclude that G(−ε)=G(ε). Note that G is proportional to L 11 . Therefore, the thermopower is also an odd function S = − 1 e 2 T L 12 L 11 of the gate voltage: S(ε)=−S(−ε). The result is qualitative agreement with the experiments (Dzurak, 1997;Egger, 1997;Moller, 1998) where there was no evidence of other significant contributions of transport mechanisms, such as phonon drag, to the observed thermopower.

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Comparing the curves G in Fig. 1(a) and S in Fig. 1(b), we find that the sign of S coincides with dG/dV g . The reason of the lowering of the conductance with the energy level increasing of the quantum dot is ascribed to the change of the electron tunneling from the resonant tunneling to sequential behavior. Now we turn to the effect of temperature on the conductance and thermopower. Figure 2 plots their curves. Figure 2(a) shows that at low temperature the conductance scales as power laws with respect to temperature, G(T) ∝ T α where α = 2/g − 2(g < 1). This functional form and the power index are in good agreement with experimental results (Bockrath, 1999;Kong, 2005;Yao, 1999). Some theoretical works with respect to impurity-contained systems gained the same conclusion (Dresselhaus, 1999;Krive2, 2001;Romanovsky, 2002). The temperature-dependent power-law scalings of conductance is associated with the suppression of tunneling to a LL in which the density of states vanishes as a power law in the energy of the tunneling electron, and the suppression becomes stronger with the decrease of g, which manifests a signature for electron-electron correlations (Harman, 2002;Kane, 1996). With increasing temperature, the mechanism of electron transport gradually turns from a resonant tunneling-like process to a sequential process. At higher temperature, the conductance is inverse to the temperature. This reflects that the effect of the electron-electron interaction on transport mechanism decays. In the temperature range between, there appears a conductance peak. Fig. 2. The schematic picture of the two-terminal electron transport through a single-level quantum dot weakly coupled to the Luttinger liquid leads with the chemical potentials µ L and µ R . The position of the dot levels with respect to the Fermi energy can be uniformly shifted by applying a voltage V g to the gate electrode.
In Fig. 2(b), the thermopower S shows linear behavior as temperature rises from zero. This is because in the low temperature regime electron tunneling transport mechanism is dominant. This behavior is the same as that of a LL containing an impurity (Krive2, 2001) and in agreement with experiments (Hone, 1998;Kong, 2005). With the electron-electron interaction enhancement, the thermopower is also increased, which has been proved in experiment (Lerner, 2008). We fit numerically the thermopower relation between the thermopower S of the LL and S 0 of Fermi liquid at low temperature with S =( 3 2g − 1 2 )S 0 (T) where S 0 (T) ∝ T. The electron-electron interaction in LL systems enhances and renormalizes the thermopower. In the limit of strong interaction g << 1, this thermopower S can be expressed as S(T) ∝ S 0 (T)/g. In this case, the thermopower of the LL is enhanced by a factor of order of magnitude of 3/(2g). Figure 2(b) reveals that at a fix temperature, a smaller g results in a lager S value. Hence, a larger slope of the S − T curve at low temperature means a stronger interaction in LL leads. It is worth to note that the thermopower S of LL is much greater than the value S 0 of FL (g = 1), which reflects that the intralead electron interactions in the LL enhance the electron-hole asymmetry. With further increasing temperature a peak-like structure emerges. This is due to the mechanism at low temperature of electron transport switching from a tunnelling process to a diffusive process at high temperature. It is worth mentioning that the result is qualitatively agreement with the works (Romanovsky, 2002) with respect to an impurity in the LL lead connected to noninteracting electrons or a FL. At low temperature, a small potential barrier can strongly influence the transport properties of a LL system, so that the thermopower induced by electron backscattering dominates. This behavior is similar the case of an impurity (Romanovsky, 2002), where the impurity backscattering is considered to be a main origin of the thermopower. The impurity can also be modeled as tunnelling junctions between two decoupled semi-infinite LLs , and the tunneling junction between impurity and the LL is described by the tunneling Hamiltonian (Barnabe, 2005;Goldstein, 2010)[45-47].
At high temperature, the thermomotion of electrons become predominant and the interaction between them will be less important. Thus discrepancy between the LL and FL systems will disappear. As a result, in the high temperature limit, S becomes identical to S, as shown in Fig. 2(b). We recall that in a weak interaction system, the thermopower S M is related to conductance G by Mott's formula (Kane, 1996): which was originally derived for bulk systems. Note that this approximation is independent of the specific form of the transmission probability T(ω) . The quantity S M is different from S 0 of a noninteraction FL. Dependence of the zero bias conductance G on the chemical potential can be in practice measured under the variation of the gate voltage V g . Since the gate voltage shifts the energy levels of the QD, one may assume that ∂lnG ∂µ , ∂lnG ∂V g . Then Eq. (26) becomes S M = − π 2 3 k 2 B T e dlnG dV g | E F . Figure 3 shows the variation of the thermopower S obtained from Eq. (23) and S M from Mott relation Eq. (26) at T = 1.0 for four electron-electron interactions. It is seen that the relation between and G holds qualitatively for weak electron-electron interaction (Appleyard, 2000;Kane, 1996;. However, even in the noninteraction case g = 1, there is some quantitative difference between S and S M . The difference is obviously enhanced by the strong electron-electron interaction. Experiments (Bockrath2, 2001) evidenced the deviation from the Mott formula Eq. (26). We interpret it as a manifestation of many-body effects in the 1D electron gas. The intralead electron interactions affect the thermopower through the dependence of the transmission probability on electron-electron interaction.
From the above numerical results, we can observe that both the thermopower and conductance manifest linear temperature-dependent power-law scaling, a behavior the same as that of an impurity-contained LL system (Kane, 1996) at low temperature. The electron-electron interaction in the leads brings a significant improvement of the thermopower, a conclusion similar to that of a LL with an impurity Krive2, 2001). As is well known, in a perfectly electron-hole symmetric system, the thermopower S = 0. The strong suppression of thermopower arises from the exact counteraction of the currents of electrons and holes induced by temperature gradient, which results in a zero net electric current. Only when the electron-hole symmetry is broken, the nonzero thermopower emerges. Our numerical fittings show that at low temperature the thermopower S can be expressed by the thermopower S 0 of noninteracting electrons multiplied by an 458 Fingerprints in the Optical and Transport Properties of Quantum Dots www.intechopen.com interaction factor as: S =( 3/2g − 1/2)S 0 (T) . In the limit of the strong intralead electron interaction, we have S(T) ≈ 3S 0 (T)/2g which has an additional 3/2 factor compared to S(T) ≈ S 0 (T)/g of the impurity-induced thermopower in 1D systems (Romanovsky, 2002). A slight deviation from the electron-hole symmetry will cause a considerable thermopower. In low-dimensional materials, the electron-hole asymmetry is usually strong and can be modulated experimentally by tuning external parameters, such as gate voltage and magnetic field. Our results reveal that how the thermopower of a LL system containing a QD is modulated by tuning the gate voltage.

The thermopower formula with a time-dependent gate voltage
The thermopower formula has been derived at the stationary system. Below we will derive an expression for the time-dependent thermopower for the Luttinger liquid leads connected to the central region. It is well known that more rich physics could be exploited if the QD device is subject to a microwave irradiation field. The perturbations of ac fields can give rise to some very interesting phenomena, such as photon-electron pumping effect, the turnstile effect, the sideband effect, and photon-assisted tunneling (Blick, 1995;Kouwenhoven et al., 1997;Tien, 1963). It has been reported that the microwave spectroscopy is a possible tool to probe the energy spectrum of small quantum systems (Wiel, 2002). So the photon-assisted tunneling could provide a new way of understanding the electron-electron influence on the transport properties of the dot. Indeed, the influence of the ac field on the current-voltage characteristics in the strongly correlated interaction model was discussed by some authors. The essential effect of photon-assisted tunneling on transport properties is that the electrons tunneling through the system can exchange energy with microwave fields, opening new inelastic tunneling channels and introducing many effects. The measurement of the thermopower at ac field frequency in the order of GHz regime may offer more value information on the electron interaction. However, the explicit thermopower formula under the time-dependent gate has been still lacking. Here we will fill the blanks. We start out by introducing a model for a QD coupled to the LL leads under a time-dependent gate voltage. The Hamiltonian of the system (see Fig.2) can be described as follows where H leads = H L + H R represents the Hamiltonians of the left and right LL leads and its standard form is given as above, H D = ε(t)d † d is the Hamiltonian of the QD, with {d † , d} the creation/annihilation operators of the electron in the QD, ε(t)=ε + ΔcosΩt, ε is the time-independent single electron energy of the QD without microwave fields, Δ and Ω are the amplitude and frequency of the ac gate voltage, respectively. It causes an alternating current through the dot. H T is the tunneling Hamiltonian and can be written as by applying a time-dependent canonical transformation (Bruder, 1994) Under this transformation, we obtain H ′ is now time-dependent. The current operator which describes tunneling from the L lead into the QD at time t is found to be: (in units ofh = 1)) Using nonequilibrium-Green-function technique and Langreth theorem of analytic continuation, the current can then be expressed as: where G r (t, t 1 ) and G < (t, t 1 ) are the Green's function of the QD. The retarded Green function G r (t, t 1 ) and lesser Green function G < (t, t 1 ) can be calculated from the following Dyson equation: and the Keldysh equation where g r (t, t 1 ) is the free retarded Green function of isolated dot which depends only on the time difference t − t 1 . Σ r/a,< (τ, τ ′ )=∑ α=L,R t * α (τ)g r/a,< α (τ, τ ′ )t α (τ ′ ) is the self-energy.

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We now make Fourier transformation over the two times t and t ′ which switches from the time-domain into energy representation through a double-time Fourier-transform defined as (Wang, 1999;Xing, 2004) F(ω, ω 1 )= dtdt 1 F(t, t 1 )e iωt e −iω 1 t 1 and And with the help of the above equation with respect to τ =(t + t 1 )/2 and let t ′ = t − t 1 .
we finally obtain following expression for Dyson equation and the following expression for the lesser Green function from Eq. (69) Where the time-average greater (lesser)self-energy Σ > (ω)(Σ < (ω)) is which can be obtained by the time-average double-time self energy After using Langreth theorem of analytic continuation, and taking Fourier transformation over the current equation Eq. (67) and the time-averaged tunneling current can be expressed as, with g Lm/Rm (ω)=g L/R (ω + mΩ).
As the above step, we also introduce the electron occupation number for interacting electrons F(ω), then we finally obtains the photon-assisted tunneling current where J m (z) is the mth-order Bessel function and F Lm/Rm (ω)=F L/R (ω + mΩ). The more detail derivation process of the time-dependent current formula Eq. (77) can be found in the work (Yang3, 2010).
In the next we only give the time-dependent thermopower formula using the above procedure. Under the small bias voltages and small temperature gradients and with the help of the linear expansion, we have . omparing both sides of the Eqs.(78) (let e = 1,h = 1), we obtain where the partial derivatives ∂F mn ∂µ and ∂F mn ∂T are not yet expressed evidently. In the following we will show the explicit expression for L 11 and L 12 . In order to obtain L 11 and L 12 ,w e must arrive at and where ). Substituting of Eq.(81) and Eq.(82) into Eq.(77) enables one to obtain the expressions of ∂F mn ∂V and ∂F mn ∂(ΔT) required in Eqs. (79) and (80). After a long calculation using the same steps above we finally obtain With the same deriving process, we obtain the partial derivation with respect to temperature as

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Fingerprints in the Optical and Transport Properties of Quantum Dots www.intechopen.com with ω m = ω + mΩ and F m = F(ω + mΩ). In terms of the linear expansion of the time-dependent current formula, The coefficients L 11/12 of the linear response theory can be determined from the corresponding correlation functions. and From the expression of the coefficients L 11/12 , we can see the coefficients L 11/12 containing a additional term caused by the time-dependent gate voltage.
The time-dependent zero bias conductance is then given by G(0)= e 2 T L 11 , and the time-dependent thermopower can be obtained from the ratio between voltage gradient ΔV and and temperature gradient ΔT between the two reservoirs, when both left and right time-dependent electric currents cancel Thus the conductance and thermopower take the form and where ψ mn = Ψ −m − Ψ +m − Ψ −n + Ψ +n , ψ ±m = Ψ +m − Ψ −m and ψ ±n = Ψ +n − Ψ −n . When no ac filed, the above formula return to the equation (58). This formula describes the time-averaged thermopower through the LL-QD-LL system in the presence of ac fields which contains more information than the equation (58). The numerical results of the time-dependent thermopower will be published in the future. Fig. 4. The conductance (a) and thermopower S (b) as a function of temperature with ε = 2.0 for g=0.2, 0.5, 0.8 and 1.0. At low temperature, the conductance exhibits a power-law dependence of the temperature and the thermopower manifests the linear and positive temperature dependence, respectively. The interaction factor g can be inferred from the slopes of thermopower. With the enhancement of the electron-electron, the thermopower is increased.  The book "Fingerprints in the optical and transport properties of quantum dots" provides novel and efficient methods for the calculation and investigating of the optical and transport properties of quantum dot systems. This book is divided into two sections. In section 1 includes ten chapters where novel optical properties are discussed. In section 2 involve eight chapters that investigate and model the most important effects of transport and electronics properties of quantum dot systems This is a collaborative book sharing and providing fundamental research such as the one conducted in Physics, Chemistry, Material Science, with a base text that could serve as a reference in research by presenting up-to-date research work on the field of quantum dot systems.