Thermoelectric Efficiency of Single-Molecule Junctions: Phase

Article pubs.acs.org/JPCC

Thermoelectric Efficiency of Single-Molecule Junctions: Phase Diagram Constructed from First-Principles Calculations Ilias Amanatidis,† Jing-Yao Kao,† Li-Yang Du,† Chun-Wei Pao,‡ and Yu-Chang Chen*,† †

Department of Electrophysics, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan Research Center for Applied Sciences, Academia Sinica, 128 Section 2, Academia Road, Nankang, Taipei 11529, Taiwan

‡

ABSTRACT: To understand the behavior of thermoelectric efficiency of single-molecule junctions from 0 K to room temperature, we investigated the thermoelectric properties of a dibenzenedithiol (DBDT) single-molecule junction. We investigated its Seebeck coefficient (S), electric conductance (σ), and electron’s thermal conductance (κel) in the framework of parameter-free density functional theory combined with the Lippmann−Schwinger formalism in scattering approach. We observe that the nanojunction is p-type and the value of the Seebeck coefficient at room temperature is around 40 μV/A, in agreement with the results of the experiment. In addition, we investigate the phonon’s thermal conductance (κph) using (i) the weak-link model suitable for ballistic phonon transport mechanism in the low-temperature quantum regime and (ii) the nonequilibrium molecular dynamics (NEMD) simulation in the high-temperature classical regime. We finally construct the phase diagram for ZT, where the value of ZT reveals the power law behavior that falls into four phases because of the competition between κel and κph and the crossover from the quantum to classical phonon transport mechanism for κph. Our theory shows the following ZT ∝ Tx where x = 2, 0, 2.26, and 3 in different temperature regimes labeled by I, II, III, and IV, respectively.

■

chemical potentials μL, μR and temperatures TL and TR, respectively. An important quantity that describes that aforementioned phenomena is the Seebeck coefficient S,34,43 defined as S = −dV/dT, where dV is the voltage difference generated by the temperature difference between the two electrodes of the nanojunction. Its value is closely related to the slope of the density of states (DOS) at the chemical potentials. In terms of semiconductor terminologies, the sign is related to the carrier type. When the Seebeck coefficient is positive, the nanojunction is p-type (holes), with the Fermi energy Ef of the electrodes close to the highest occupied molecular orbital (HOMO) level. By contrast, when the Seebeck coefficient is negative, the nanojunction is an n-type (electrons) with the Fermi energy Ef of the electrodes close to the lowest unoccupied molecular orbital (LUMO) level. The Seebeck coefficient is sensitive to the slope of the DOS. As a result, it can provide more information than conductance measurement in exploring the electronic structures.

INTRODUCTION One of the modern topics in the field of mesoscopic and nanoscopic physics that is rapidly developing in the past decade is that of molecular electronics.1−4 The quantum transport properties enhanced by the extreme miniaturization of molecular junctions offer the possibility of a new form of electronic components at the atomic level. We have been able to investigate their novel fundamental quantum transport properties under nonequilibrium conditions. More specific, a large number of theoretical and experimental studies have examined the quantum transport properties of molecular/ atomic junctions such as the current−voltage characteristics,5−13 shot noise,14−17 counting statistics,18 local heating,19−22 and gate-controlled effects.23−27 As a result of the evolution of the ongoing research on molecular electronics, a new field has recently been emerged; this new field examines the thermoelectric properties of molecular nanojunctions (mM-m).28−48 Thermoelectric effects describe the phenomena involved in the conversion of thermal energy to electric energy and vice versa. Thermoelectric single-molecule junction is defined as a molecule sandwiched by left (L) and right electrodes (R) with © 2015 American Chemical Society

Received: September 21, 2015 Revised: November 4, 2015 Published: December 7, 2015 28728

DOI: 10.1021/acs.jpcc.5b09221 J. Phys. Chem. C 2015, 119, 28728−28736

Article

The Journal of Physical Chemistry C

The dibenzenedithiol (DBDT) single-molecule junction is an ideal testbed to compare theories and experiments. In this paper, we investigated the fundamental properties of ZT of a dibenzenedithiol (DBDT) single-molecule junction, as depicted in Figure 1a. Examination is conducted in a two terminal

Over the past few years, a growing number of studies have focused on the thermoelectric nanojunctions. The Seebeck coefficient has been experimentally measured34,38,39 in singlemolecule junctions by scanning tunneling microscope (STM) techniques, showing its importance in understanding the underlying physics of thermoelectric nanojunctions. Motivated by these experiments, researchers have developed theoretical approaches to determine the thermoelectric properties of nanoscale systems.30−33,35−37,40−50 The Seebeck coefficients in nanoscale scale systems show unprecedented unusual behaviors related to detailed electronic structures in contrast to the bulk materials. For example, the Seebeck coefficients of carbon atom chains oscillate with the length and can be negative (n-type) or positive (p-type) due to the full- and half-filled π*-orbital.51 Surface reconstruction occurs in C60 STM junction around 300−400 K and may also lead to phase transition in the Seebeck coefficient.52 Nanojunctions, employing the Seebeck effect, are proposed theoretically to work as thermoelectric nanodevices, such as a transistor powered by heat50 and a thermoelectric (TE) nanorefrigerator,49,53 paving the way to be demonstrated by the experimentalists. The Seebeck coefficient has been experimentally examined in benzenedithiol-type molecules in STM, which consist of one, two, and three benzene type rings sandwiched by Au electrodes.34 They show that the Seebeck coefficient is positive, thereby suggesting that the benzene ring-type molecular junctions are p-type and their Fermi level EF is close to HOMO. Although conductance in these type of molecules has been studied extensively both in theory and experimentally, the position of the Fermi level EF relative to the HOMO and the LUMO is ambiguous. Some researchers suggest that EF is close to the HOMO, while others suggest that is close to the LUMO.34 Thermoelectric junctions convert thermal energy into electric energy through the Seebeck effect provided that there is a temperature difference between the electrodes of the nanodevice. Apart from the underlying physics of the Seebeck coefficient, researchers revealed through the thermoelectric figure of merit ZT how efficient the thermoelectric junctions could serve as power generators. ZT is defined as, ZT = ((S2σ)/ (κel + κph))T, where S, σ, κel, κph are the Seebeck coefficient, electric conductance, electronic thermal conductance, and phonon’s thermal conductance, respectively. As it is well-known in textbooks,54 the degrees of freedom of vibrational motion classically contribute to the heat capacity, such that CV = 3NkB because all phonon modes are excited in high temperature limit (T ≫ TDebye), as shown by Dulong− Petit law. By contrast, CV ∝ T3, known as the Debye T3 law, is attributed to the decreased number of phonon modes in lowtemperature quantum regime (T ≪ TDebye). Similarly, the phonon’s thermal conductance can reflect the quantum behavior in the low temperature regime and the classical behavior in the high-temperature regime. Moreover, thermal conductance is marked by the competition between the contributions of phonon and electron. Although the fundamental properties of ZT have been widely studied, full analysis of the effect of the aforementioned questions on the value of ZT seems to be lacking. We aim at answering the above-mentioned fundamental questions: (i) the difference in the phonon’s thermal conductance in low-temperature quantum regime and the high-temperature classical regime, and (ii) the effect of the competition between the contributions of phonon and electron on the thermal conductance and ZT.

Figure 1. (a) Schematic of the DBDT single-molecule junction where a dibenzenedithiol bridges the Au electrodes via Au−S covalent bonds. The structure is relaxed using the VASP code. In the NEMD simulation, the thermal current carried by phonon is evaluted within the Nose−Hoover thermostat, where both Au electrodes have a total of 256 atoms. In the framework of ”DFT+Lippmann−Schwinger equation” calculations, the electrodes are described by the jellium model. In the weak link model, the molecule is taken as a weak elastic spring with the stiffness calculated from total energy calculations using VASP. (b) Density of states (DOS) of the continuum electrons calculated self-consistently in the framework of DFT in scattering approach as a function of energy, where zero energy corresponds to the chemical potential. (c) Phonon density of states (PDOS) calculated in NEMD simulation as a function of frequency ω in units of THz. 28729


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Inclusion of the nanostructured object is considered in the scattering approach. The wave functions of the total system (bimetallic junction +atom/molecule) are calculated by solving the Lippmann−Schwinger equation iteratively until selfconsistency is reached, as follows:

device, where the left and right metal Au electrodes were served as temperature and electron reservoirs, respectively. We investigated the Seebeck coefficient, electric conductance, and electric thermal conductance as a function of temperature based on the density functional theory combined with the Lippmann−Schwinger formalism in scattering approach. Calculation values show that they are comparable with the experimental results,34 thereby suggesting that our approach can be used to determine the underlying physics of these thermoelectric junctions. We further calculate the phonon’s thermal conductance (κph) in a low-temperature quantum regime using the weak-link model and in a high-temperature classical regime using nonequilibrium molecular dynamics (NEMD) simulations. We observe that the value of ZT reveals a power law with different exponents in different temperature regimes because of the distinct classical and quantum behaviors of the phonon’s thermal conductance and the competition between κel and κph.

0,L(R) ΨL(R) EK (r) = Ψ EK (r) +

ΨL(R) EK (r2)

where stands for the effective single-particle wave functions which carry the current, thereby corresponding to propagating electron incident from the left (right) electrode. The quantity G0E is Green’s function for the bimetallic junction. The potential V(r1,r2) that the electrons experience when they scatter through the nanojunction is expressed as follows: ⎧ V (r1, r2) = Vps(r1, r2) + ⎨(Vxc[n(r1)] − Vxc[n0(r1)]) ⎩

THEORY a. Density Functional Theory combined with Lippmann−Schwinger Equation. In the following section, we describe how to calculate the electric current in the framework of the DFT combined with the Lippmann−Schwinger equation in scattering approach. We consider a two-terminal nanojunction where a scattering object (single molecule) is attached to left and right electrodes with a certain applied source-drain bias drain VB = (μR − μL)/e; μR and μL are the chemical potentials for right and left electrodes, respectively. The full Hamiltonian of the open system is H = H0 + V where H0 represents the Hamiltonian of the electrodes and V represents the effective potential of the scattering object. In this study, the scattering object is the dibenzenedithiol single molecule. The optimized structures of the dibenzenedithiol single-molecule junction is relaxed using the Vienna Ab-Initio Simulation Package (VASP), with a geometry depicted in Figure 1a. First, we calculate the unperturbed wave functions of the bare electrodes described by the Hamiltonian H0 when a bias voltage VB is applied across of them. The unperturbed wave functions L(R) have the form, Ψ0,L(R) (r) = (2π)−1eiK·R·uL(R) EK EK (z), where uEK (z) describes the electrons incident from the left (right) electrode before the inclusion of the nanostructured object. The wave function uL(R) EK (z) is calculated by solving the Schrödinger equation and Poisson equations iteratively until self-consistency is obtained. Deep inside the electrodes (z→ ± ∞), the rightand left-moving waves satisfy the scattering boundary conditions, as follows:

+

I(μL , TL ; μR , TR ) =

1 2 k 2 L

=E−

1 2

1 2

K|2 − veff (∞)

2e h

∫ dEτ(E)[fE (μR , TR )

− fE (μL , TL)]

(4) −1

where = [exp[(E − μL(R))/(kBTL(R))] + 1] represents the electronic population for the left(right) electrode described by the Fermi−Dirac distribution with μL(R) and TL(R) as the chemical potential and the temperature for the left(right) electrode, respectively. kB is the Boltzmann constant. Also, τ(E) = τL(R)(E) because we assume that the electrodes from both sides are made by the same material. τL(R)(E) is the transmission of electron as a function of energy E incident from the left (right) electrode and is defined as follows, f L(R) E

⎧ 1 (e−ikRz + R eikRz), z → ∞ ⎪ ⎪ kR ×⎨ ⎪ 1 T e−ikLz , z → −∞ ⎪ k ⎩ L

=E−

(3)

3

where Vps(r1,r2) is the electron−ion interaction potential represented with pseudopotential, Vxc[n(r)] is the exchangecorrelation potential calculated at the level of the local-density approximation, n0(r) is the electron density for the pair of biased bare electrodes, n(r) is the electron density for the total system, and δn(r) is their difference. The quantity G0E is the Green’s function for the bare electrodes. The wave functions that achieve self-consistency in the DFT framework in plane wave basis are applied to calculate the electric current and the electron’s thermal current. A basis of round 5000 plane waves is chosen for self-consistent calculations. Given than the wave functions written in a plane-wave basis achieve self-consistency in the DFT framework, we are able to calculate the electric current. We consider left and right Au electrodes, shown in Figure 1a, as two semi-infinite electron jellium metal surface (rs ≈ 3) separated by a distance. The current calculated from the probability current density can be casted into a Landauer formula, as follows:

where K is the electron momentum in the plane parallel to the electrode surfaces and z is the coordinate parallel to the direction of the current. The condition of energy conservation 1 2 k 2 R

δn(r ) ⎫

∫ dr3 |r − 3r | ⎬⎭ δ(r1 − r2) 1

(1)

gives

(2)

ΨL(R) EK (r)

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uERK (z) = (2π )−1/2

∫ d3r1 ∫ d3r2GE0(r, r1)V (r1, r2)

τ L(R)(E) =

π ℏ2 mi

LL(RR) ∫ dR ∫ dKIEE, K (r)

and

K|2 − veff ( −∞), where veff(z) is the effective

potential comprising the electrostatic and exchange-correlation potentials. 28730

[ΨL(R) E,K ]

*∇ΨL(R) E,K

(5)

with IEE,K defined as = − L(R) *ΨL(R) E,K . ΨE (r,K) represents the single particle wave function when an electron is injected from the left(right) electrode with energy E and momentum K parallel in the transport direction of the electrode’s surface. dR is an element of the electrode surface. The electronic wave function ΨL(R) E (r, K) is calculated ILL(RR) EE,K

∇[ΨL(R) E,K ]


Article

The Journal of Physical Chemistry C Table 1. Classical Interatomic Potential Parameters Employed in the Present Studya interaction

potential model

Chemical Bond Potential C−C C−H C−S S−Au Au−Au Valence Angle Potential C−C−C Dihegral Angle Potential C−C−C−C, C−C−C−H Inversion Angle Potential S−C−C−C, H−C−C−C Lennard-Jones Potential C−C C−H C−S H−H H−S S−S a

parameters (in atomic unit)

0.5kr(r − r0)2 0.5kr(r − r0)2 0.5kr(r − r0)2 0.5kr(r − r0)2 EAM Potential

kr kr kr kr

0.5kθ(cos θ − cos θ0)2

kθ = 0.213, θ0 = 120°

kϕ[1 + cos(mϕ−δ)]

kϕ = 0.002, δ = 180°, m = 2

kφ(1 − cos φ)

kφ = 0.064

Ar−12−Br−6 Ar−12−Br−6 Ar−12−Br−6 Ar−12−Br−6 Ar−12−Br−6 Ar−12−Br−6

A A A A A A

= = = =

= = = = = =

0.470, 0.314, 0.314, 0.043,

0.904, 0.116, 2.101, 0.013, 0.276, 0.487,

r0 r0 r0 r0

B B B B B B

= = = =

= = = = = =

2.627 1.928 3.269 4.516

0.024 0.005 0.049 0.001 0.011 0.104

Interaction parameters from the Dreiding potential (ref 64).

Thus far, we have ignored the phonon’s contribution to the thermal current, which is driven by the vibrations of atoms. In the low-temperature quantum regime, phonons are ballistic and can carry heat with no internal scattering when the size of the nanojunction is smaller than the phonon mean-free path. We applied the weak link model suitable for describing ballistic phonons to estimate the phonon’s thermal conductance in the low-temperature quantum transport regime.59 The weak link model considers the macroscopic electrodes as ideal thermal conductors with distinct temperatures. To the leading order in the strength of the weak elastic link, the nanostructure connecting the electrodes is modeled by a harmonic spring with a given stiffness K. The phonon’s thermal conductance is estimated by an equation equivalent to the Landauer formula with ballistic phonons, as follows:

iteratively until self-consistency is achieved through the Lippmann−Schwinger scattering approach.55−57 The localdensity approximation is used for the exchange-correlation potential.58 Figure 1b shows the density of states (DOS) of the continuum electrons for the DBDT single-molecule junction in this calculations. b. Thermoelectricity Theory. In this section, we briefly present the theory on how to calculate the Seebeck coefficient, the electric conductance, and the electron’s thermal conductance based on the effective single-particle current-carrying wave functions obtained self-consistently in the DFT framework in scattering approach, as presented above. For more details, please see refs 43, 46, 47, 49, 50. Moreover, we calculate the phonon’s thermal conductance κph by using the weak link model and NEMD simulations. The Seebeck coefficient S (defined by S = −dV/dT) is calculated by the following formula: 1 K1(μ , T ) S(μ , T ) = − eT K 0(μ , T )

κ ph =

∫ dEτ(E)(E − μ)n

∂fE (μ , T ) ∂E

(7)

The electron’s thermal conductance, kel, is calculated by the following formula: κel(μ , T ) =

K 2(μ , T ) ⎤ 2⎡ ⎢eK1(μ , T )S(μ , T ) + ⎥ ⎦ h⎣ T

(8)

The electric conductance in the linear response regime is as follows: σ (T ) =

2e 2 h

⎛ ∂f ⎞

∫ dE⎜⎝− ∂E ⎟⎠τ(E)

∫0

∞

dEE3

∂n(E , T ) ∂T

(10)

where n(E,T) = 1/[eE/(kBT) − 1] is the Bose−Einstein phonon statistics; the stiffness of the DBDT single-molecule junction is estimated by total energy calculations using VASP with K = 6.7 × 104 erg/cm2. The phonon density of states, N(E) = CE, is evaluated using the longitudinal and transverse sound velocities of Au: vl = 3.2 × 105 cm/s and vt = 1.2 × 105 cm/s, respectively,59 which gives C = 1.07 × 109 cm2/erg2. In the high-temperature classical regime, we evaluate the phonon’s thermal conductance as a function of temperature, TMD, using the NEMD simulation method developed by Müller-Plathe.60 These NEMD simulations of thermal conductivity across the DBDT single-molecule junction are carried out by the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package.61,62 The DBDT molecule is bridging Au electrodes, where both electrodes have in total 256 Au atoms, as shown in Figure 1a. The NEMD simulations are performed in a temperature ranging from 50 to 300 K. The temperature of the hot reservoir is set 10% higher, whereas the temperature of the cold reservoir is set 10% lower than the average temperature of the system to generate the thermal current flowing from the hot to cold temperature reservoirs.

(6)

where K n(μ , T ) = −

2πK 2C 2 ℏ

(9)

2

defining G0 = 2e /h. The electric conductance is insensitive to temperature because the main transport mechanism is tunneling. 28731


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The Journal of Physical Chemistry C The temperatures in the hot and cold reservoir are controlled by using the Nose−Hoover thermostat method. The thermal current driven by the temperature difference due to vibrations of atoms is evaluated by employing LAMMPS. The NEMD simulations are performed with a time step of 0.5 fs for a time period of 10 ns. In Table 1, we present the classical interatomic force field for two-, three-, and four-body potentials which are used in the NEMD simulations. The phonon’s thermal conductance κph is then calculated by computing the heat flux and the temperature gradient.61,62 The temperature TMD is calculated in NEMD simulation according to the Boltzmann distributions and is valid only for temperatures higher than the Debye temperature (TDebye ≈ 100 K in our case). To identify the value of the TDebye, we employ the quantum correction to NEMD simulations. The TDebye is defined when the quantum correction to NEMD simulations is negligibly small when T > TDebye. The quantum correction uses an equality of the system energies written in the equipartition law and mechanical pictures. The quantum correction of the phonon’s thermal conductance κph and the temperature T are given by63 κ ph = κ ph,MD

dTMD dT

(11)

and kBTMD =

∫ dωDph(ω)ℏω(n(ω , T ) + 1/2)

(12)

where Dph(ω) is the phonon density of states (PDOS) calculated from the Fourier transform of the atoms’ velocity autocorrelation functions. ω is the phonon frequency. n(ω,T) = 1/(eℏω/kBT−1) is the Bose−Einstein phonon distribution, and ℏω/2 the zero-point energy. A typical phonon density of states is shown in Figure 1c.

Figure 2. (a) The Seebeck coefficient S and in the inset the electric conductance σ as a function of the temperature for the DBDT singlemolecule junction. Its positive value suggests that it is a p-type thermoelectric nanojunction. (b) The electron’s thermal conductance κel versus the temperature obtained from first-principles calculations (black solid line) and from the Wiedemann−Franz law (green dotted line). The inset shows the characteristic temperature T1 ≈ 1.7 K, determined by the competition between κel and κph. κph is evaluated by the weak-link model valid in low-temperature quantum transport regime. Note that κel > κph for T < T1 and κel < κph for T > T1. κph well exceeds κel in all temperature range except for T < T1 due to the insulating behavior of the DBDT single-molecule junction.

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RESULTS AND DISCUSSION The Seebeck coefficient S, electric conductance σ, and electron’s thermal conductance κel are calculated using eqs 6, 8, and 9, respectively. Moreover, the phonon’s thermal conductance κph is calculated (i) using the weak link model [eq 10] which is valid in low-temperature (T ≪ TDebye) ballistic phonon transport regime and (ii) using the NEMD simulation which is valid in high-temperature (T ≫ TDebye) classical phonon transport regime. The quantum correction to NEMD simulation is also attempted to correct κph in the crossover from classical to quantum regime. Numerical results are presented in Figures 2 and 3. To gain insight into the properties of the thermoelectric nanojunction on the dependence of the temperatures, we expand S, κel, and κph from weak link model to the lowest order in temperature, as follows: S ≈ αT

(13)

κel ≈ βT

(14)

kelWF(T ) = L WFσT

where, the Lorenz number LWF is L WF =

2 π 2 ⎛ kB ⎞ ⎜ ⎟ 3⎝e ⎠

(17)

Figures 2a shows the Seebeck coefficient as a function of temperature. The Seebeck coefficient is related to the slope of the transmission function and shows a linear dependence up to 150 K at low temperatures, as described by eq 13. The sign of the Seebeck coefficient is positive, thereby indicating that the DBDT single-molecule junction is p-type, suggesting that the Fermi energy Ef is possibly closer to the highest occupied molecular orbital (HOMO). The slope of DOS (roughly speaking τ(E) ∝ DOS) at the Fermi energy is then negative. We note that the value of the Seebeck coefficient S ≈ 42 μ V/K at room temperature calculated from first-principles is in agreement with the experiment measurement S = 12.9 μV/K.34 The inset of Figure 2a shows the DBDT molecule is an insulator with a small electric conductance σ ≈ 4.3 × 10−3 G0, where G0 = 2 e2/h is the quantum unit conductance. We note that the

and κ ph ≈ γT 3

(16)

(15)

where α = −π2k2B[δτ(μ)/δE]/[3eτ(μ)], β = 2π2k2Bτ (μ)/(3h), and γ = 8π 5 k B4 C 2 K 2 /(15ℏ). Here we show that the Wiedemann−Franz law can be derived from the above theory. With the electric conductance σ ≈ (2e2/h)τ(μ) and eq 14, we obtain the electron’s thermal conductance, 28732


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+Lippimann−Schwinger equation” based on the “electrode states” in scattering approaches. Figure 2b shows the electron’s thermal conductance calculated from first-principles calculations (black dotted line) and from the Wiedemann−Franz law (green dotted line) as a function of the temperature. The electron’s thermal conductance κel varies linearly for T < 100 K, in good agreement with the Wiedemann−Franz law (eq 16), while it starts to deviate for T > 100 K but still at the same order with the κel obtained from first-principles calculations. The inset of Figures 2b shows the competition between the electron’s thermal conductance κel and the phonon’s thermal conductance κph. In the quantum phonon regime (T < 10 K),59 the κph is evaluated by the weak link model which can suitably describe the quantum transport of the ballistic phonon, where phonons can carry thermal energy with no internal scattering. The phonon transport dominates the contribution to the thermal conductance except at very low temperatures because the junction is electrically insulating. We observe a characteristic temperature, T1 ≈ 1.7 K, which is defined as the temperature where κph = κel. When T ≪ T1, the electron’s contribution dominates the thermal conductance (κel ≫ κph). Similarly, when T ≫ T1, the phonon’s contribution dominates the thermal conductance (κph ≫ κel). Figure 3 compares the phonon’s thermal conductance κph as a function of temperature calculated from (i) the weak link model (solid line with purple triangles), (ii) the NEMD simulation without the quantum correction (solid line with open circles), and (iii) the NEMD simulation with the quantum correction (dashed line with solid squares). The κph calculated from the weak link model show linear relation in log κph−log T plot, thereby indicating that κph ∝ γT3. The cubic exponent is attributed to the decreased number of excited phonons at low temperatures similar to the Debye T3 law for heat capacitance. The slope in log κph−log T plot corresponds to the exponent of the power law. The ballistic phonon quantum transport regime may exist only below T = 10 K.59 Thus, we define as T2 ≈ 10 K as the characteristic temperature, below which the ballistic quantum transport of phonon is valid. The comparison between the values of κph calculated from NEMD simulation with and without the quantum correction shows that the quantum correction to the NEMD simulation is insignificant for T > 100 K. This allows us to define the characteristic temperature (Debye temperature) T3 = 100 K. For T > T3, the transport mechanism of phonon is classical, i.e., the quantum correction to the NEMD simulation is insignificant. The applicability of the weal link model and the NEMD simulation is restrictive in the low-temperature phonon quantum transport regime and high-temperature classical regime, respectively. The enormously large value of κph described by the weak link model implies that the model fails in the high temperature regime, where the preassumption of ballistic phonon is no longer valid. The quantum correction to the classical NEMD simulation is important for T < T3, but it fails at around 50 K. Before the breakdown of quantum correction (80 K < T < 100 K), we notice that κph after the quantum correction is roughly linear in log−log plot, and thus it has power law behavior in this reliable regime. Moreover, κph reveals power law behaviors in regimes I, II, and IV. Therefore, we choose to present κph in power law by inter- and extrapolation in regime III, as shown in Figure 3. In this crossover regime from classical to quantum mechanics, κph obviously needs more sophisticated theoretical approach beyond the scope of the current study.

Figure 3. Phonon’s thermal conductance κph as a function of temperature calculated from (i) the weak link model (solid line with purple triangles), (ii) the NEMD simulations without quantum correction (solid line with open circles), and (iii) the NEMD simulation with quantum correction (dashed line with solid squares). The effect of quantum correction on the NEMD simulation is insignificant in high-temperature classical regime (T > T3 ≈ TDebye ≈ 100 K). We note that the κph calculated from the weak link model is valid only in ballistic−phonon quantum transport regime at very low temperatures (T < T2, where T2 ≈ 10 K). We also assume that the κph follows a power law behavior and is drawn as a red dashed line in the quantum-classical crossover regime(T2 κel). The transport mechanism of phonons in this regime experiences a crossover from the quantum to classical limits. Both weak link model and NEMD simulations fail to describe the phonon’s thermal conductance in full range. However, we assume κph ≈ δ·Tc where the values of c and δ are estimated by interpolation, as shown in Figure 3. For T > T3 (regime IV), ZT ≈ α2σ/κph·T3. In this regime, the classical phonon’s transport dominates the thermal conductance (i.e., κph > κel). In this case, κph is roughly a constant similar to the physics of Dulong−Petit law for heat capacitance.

In the crossover area from quantum mechanics to classical regime (T2 < T < T3), the thermal conductance κph is calculated assuming a power law behavior (see Figure 3) because its behavior below T2 and above T3 is also power law. This is done using the interpolation method, having in mind the value of κph at T2 and κph at T3 from the NEMD simulation. Figure 4 shows ZT as a function of temperature. The linear slope in the log κph−log T plot implies the power law behavior

■

CONCLUSIONS In summary, we investigate the Seebeck coefficient (S), electric conductance (σ), and electronic thermal conductance (κel) in the framework of density functional theory combined with the Lippmann−Schwinger formalism in scattering approach. Our result for the Seebeck coefficient is in quantitative agreement with the experiment, thereby suggesting that our approach can be used to determine the underlying physics of these thermoelectric junctions. The positive value of the Seebeck coefficient suggests that the main type of carriers is p-type (holes), whereas the Fermi energy of the DBDT singlemolecule junction is close to the HOMO. In addition, the phonon’s thermal conductance (κph) is investigated using (i) the weak link model in the low-temperature ballistic phonon transport regime and (ii) using the nonequilibrium molecular dynamics simulation in the high-temperature classical regime. We finally construct the phase diagram for ZT which shows the power law behaviors, falling into four phases according to exponents. The values of exponents can be explained by the competition between κel and κph and the transport mechanism of phonons which cross from the quantum mechanical to classical regime. We observe that ZT is falling into four regimes separated by the characteristic temperatures T1 ≈ 1.7 K, T2 ≈ 10 K, and T3 ≈ 100 K. In regime I, the ZT value is proportional to T2 where the electron’s quantum transport dominates the thermal conductance. In regime II, the ZT value approach to a saturation value where the ballistic phonon transport dominates the thermal conductance. In regime III, the ZT value is proportional to T3−c where the phonon experiences crossing over quantum mechanical into the classical transport regime. In this temperature regime, the thermal conductance is also dominated by the phonon’s contribution. Although the dependence of κph on temperature remains unknown, we could roughly estimate κph ≈ δ·Tc by interpolation. In regime IV, the value of ZT is proportional to T3, where the transport mechanism of phonon is classical and dominates the thermal conductance.

Figure 4. Thermoelectric figure of merit ZT as a function of the temperature. The value of ZT reveals power law with different exponents in four regimes determined by T1, T2, and T3. In the plot, κel and κph along with its corresponding color denote the main carrier mechanism in each four regimes with the indigo color being the quantum regime, red color the quantum crossover to classical regime and the black color the classical regime.

of ZT. The power law behavior of ZT can be differentiated into four regimes according to exponents in T. For the explanation of the exponents, we expand S ≈ αT and κel ≈ βT to the leading order in temperature. Moreover, we approximate (i) κph ≈ γT3 in regime I (T < T1) and regime II (T1 < T < T2); (ii) κph ≈ δTc in regime III (T2 < T < T3); (iii) κph ≈ const in regime IV (T > T3). Consequently, we obtain the expression of ZT to the leading order in temperature, as follows: ZT =

S2σ T→ κel + k ph

⎧(α 2σ /β)T 2 for T < T 1 ⎪ ⎪ (regime I, κel dominates) ⎪ ⎪ α 2σ /γ for T1 < T < T2 ⎪ (regime II, QM, κ dominates) ph ⎪ ⎨ ⎪(α 2σ /δ)T 3 − c for T2 < T < T3 ⎪ (regime III, κ dominates) ph ⎪ ⎪ 2 3 ⎪(α σ /k ph)T for T > T3 ⎪ (regime IV, classical, κ dominates) ph ⎩

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. (18)

Notes

The authors declare no competing financial interest.

where c ≈ 0.74. The above equation can explain the exponents of the power law. For T < T1 (regime I), ZT ≈ (α2σ/β)T2. In this regime, the electron quantum transport mechanism dominates the thermal conductance (i.e., κel ≈ β T > κph). For T1 κel).

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ACKNOWLEDGMENTS We acknowledge the support of the National Science Council (Taiwan) under grants MOST-103-2112-M-009-014-MY3, 104-2627-M-009-007-MY3, 104-2622-8-002-003-MY3, and NSC-102-2628-M-001-004-MY3, the Ministry of Education 28734


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through the Aiming for Top University Plan (MOE ATU), and the National Center for Theoretical Sciences. Also, we thank National Center for High-Performance Computing for computing time and facilities.

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Thermoelectric Efficiency of Single-Molecule Junctions: Phase

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