THEORY

When the miniaturization of thermoelectric junctions reaches the atomic scale, it indeed opens new field for the thermoelectricity due to the quantum transport for electrons and phonons in atomistic system. Thus, the studies of thermoelectricity in molecular junctions require the development of new theories.

Motivated by Majumdar’s experiments, we develop theories allied to first-principles calculations to investigate the Seebeck coefficients and the thermoelectric figure of merit in the molecular tunneling junction.

2-1 Current of atomic wires

We started by a brief introduction of the DFT calculations for a molecule sandwiched between two bulk electrodes with external source-drain

bias. The effective single-particle wave functions of the whole system in the continuum states were calculated in scattering approach by solving the Lippmann-Schwinger equation with exchange and correlation energy included within the local density approximation. Two planar metallic electrodes, represented as a uniform-background (jellium) model.

The calculations proceeded in the following way: First, within the framework of the density-functional formalism, the single-particle wave functions and self-consistent density distribution were obtained by solving the coupled Poisson equation and Shrödinger equation for the pair of bare metallic electrodes in the presence of the bias voltage. Next, corresponding to each of these wave functions, a Lippmann-Schwinger equation involving a Green’s function for the biased bimetallic junction was solved to obtain an effective single-particle wave function for the total system, consisting of the two electrodes plus a group of atoms. From these wave functions, the charge density for the total system was obtained, and the problem was solved self-consistently using a modified iterative procedure. Atomic units were used here, with 0'0=m=1=1.

2-1-1 Bimetal junction

Fig. 5. The schematic of bimetal junction

Two bulk electrodes are modeled as two semi

by Jellium model. The wavefunction of the bare electrodes can be obta solving the Shrödinger equation and Poisson equation.

the left hand side can be partial transmitted and partially reflected.

biased electrode (which we will take henceforth has the form of a linear combination of left with wave vector

described by ^

7₈^

µ

Bimetal junction

The schematic of bimetal junction with external source

Two bulk electrodes are modeled as two semi-infinite bulk metals described by Jellium model. The wavefunction of the bare electrodes can be obta

solving the Shrödinger equation and Poisson equation. Electrons incident from the left hand side can be partial transmitted and partially reflected.

We first note that Ψ³ :, the superscripts M refer to the pair of bare biased has the form '^;25<=4_,25 ?, where R is the coordinate parallel to the surfaces and z the coordinate normal to them. Deep in the positively biased electrode (which we will take henceforth to be the left

has the form of a linear combination of left-moving and right-moving plane waves with wave vector 7₈. Here, the energies of electrons are

  ! #^_J2₅J^# υ_KK³ #∞, where E is the energy eigenvalue

µ

with external source-drain bias.

infinite bulk metals described by Jellium model. The wavefunction of the bare electrodes can be obtained by Electrons incident from the left hand side can be partial transmitted and partially reflected. The of the electrons satisfy the boundary

 L

∞( M

superscripts M refer to the pair of bare biased , where R is the coordinate parallel Deep in the positively left electrode), 4_,25 ?

moving plane waves are conservative as , where E is the energy eigenvalue

in the single-particle equations for the pair of biased electrodes and the total effective potential (electrostatic plus exchange correlation).

Next we will specify the character of

We investigate the electron transport, phonon transport, and thermoelectricity of molecules wire and the total effective potential (electrostatic plus exchange correlation).

Next we will specify the character of 4_,25 ? by an additional . For propagating states, we will replaceα either by “

correspond to a wave incident from the left (together with its reflected and transmitted parts) or by “#,” which will correspond to a wave incident from the

is defined as ^

investigate the electron transport, phonon transport, and thermoelectricity of molecules wire and atomic wires¹ sandwiched between two bulk electrodes with finite external source-drain bias and

shown in Fig. 6, where the bias is given by _&

schematic of aluminum atomic junction: the atomic size conductors where both chemical potential and temperature gradients are present.

µ

T R

particle equations for the pair of biased electrodes and υ_KK³ ? is the total effective potential (electrostatic plus exchange correlation).

by an additional subscript α:

investigate the electron transport, phonon transport, and sandwiched between two drain bias and finite temperature

^µ.E6µ_ ^.I.

: the atomic size conductors gradients are present.

The continuum wave functions Ψ^3W, the superscripts MA refer to the complete system consisting of the metal electrodes and the group of atoms between them, are solved in scattering approaches. The continuum wave functions Ψ^3W are solved by the Lippmann-Schwinger equation, where Ψ³ will have the same labeling (! 2₅ α), even though 2₅ no longer refers to a conserved quantity. These solutions will also have the same normalization as the Ψ³, a fact that facilitates the calculation of the electron density distribution and the current. For !_X8 Y ! Y Z_XA, where !_X8 is the Fermi level in the left electrode and !_XA  !_X8 _& is the Fermi level in the right electrode (bias _& taken positive), we occupy only states corresponding to a wave incident from the right, i.e., only Ψ_,2^3W₅₆ and not Ψ_,2^3W_5[.

The Hamilton of complete system can be put into Lippmann-Schwinger form :

Ψ^3W :  Ψ³ :  N ^@:^U^@:^UU\³ : :^Uδ :^U :^UUΨ^3W :^UU ]

this equation embodies the motion that electrons in states of the electrodes impinge on and are scattered elastically by the potential δ : :^U , which describes the difference in potential between the complete system and the bare electrodes. It can be written

δ : :^U  υ ^ : :^U atomic pseudopotentials introduced by Hamann¹⁷.

2-1-3 Current in nanojunction

First, we calculate the current with left/right electrode temperature at 0K, Fermi-Dirac distribution equal to 1. The electron number density is given by the sum of squares of the occupied states Ψ_,2^3W_5α, with a factor 2 included for spin degeneracy (we take the system to be unpolarized) the electric current density in the full system is given by group of atoms. Then the quantity of interest to us is

  N ^G?n < oe^3W# e³p 

which is independent of z since our system has no current sources or sinks.

Here ?n is the unit vector point to the right and perpendicular to the surfaces of the electrodes. If we write Ψ^3W  Ψ³ δΨ, then

different, Fermi-Dirac distribution not equal to 1 or 0.

  #r N ! N ^= N ^2₅Ss_,⁸_,,⁸⁸  s_,^A_,,^AAT 

where

_,,;t ′  hΨ_,^; : 2₅i^RjΨ_,^t′ : 2₅ # jhΨ_,^; : 2₅i^RΨ_,t′ : 2₅ /

where r e  u G. Ψ_,^{8 A} : 2₅ is the single-particle wave function incident from the left (right) electrode with energy E and component of the momentum 2₅ parallel to the electrode surface^18,19, and d²R represents an element of the electrode surface. We assume that the left/right electrode serves as the electron and thermal reservoir with the electron population described by the Fermi-Dirac distribution function

s_,^{8 A}bµ_{8 A} _{8 A}c   v'^{,6µDwI EI E} x

 L

where µ_{8 A} and _{8 A} are the chemical potential and the temperature in the left (right) electrode, respectively, and 7_y is the Boltzmann constant. For simplicity, we define the transmission probability of electron with energy E incident from the left (right) electrode as

8 A !  #r$ N ^= N ^2₅_,,^{88 AA} : 2₅ M

By using the relation !  ^A !  ⁸ ! , a direct consequence of the time-reversal symmetry, the current in Eq. (14) can be rewritten as

Sµ₈ ₈ µ_A _AT 

$ N !hs^,ASµ_A _AT # s_,⁸Sµ₈ ₈Ti ^A ! V

where the left and right electrode have different chemical potentials given by the bias (the source drain bias is _z{ ^µE6µ_ ^I ) . We also assume that the left/right electrode can be connected to its own heat bath such that ₈ can be different from _A.

2-2 Thermoelectric figure of merit

Molecular tunneling junctions consist of source-drain electrodes as independent electron and heat reservoirs with distinct temperatures TL(R) and chemical potentials µL(R). The efficiency of energy conversion depends on several factors: the electrical conductance (σ), the Seebeck coefficient (S), the electron thermal conductance (κ_) and the phonon thermal conductance (κ ).

The efficiency can be described by the dimensionless thermoelectric figure of merit:

  ^σ

κ_ κ  ]

where T = (TL +TR)/2 is the average temperature in the source-drain electrodes.

The ideal thermoelectric molecular junction would have a large S, a large σ and a small combined thermal conductance (κ_ κ ). Thermoelectric materials with a large σ are usually accompanied by a large κ_, which makes the enhancement of the thermoelectric figure of merit a challenging task.

We reported theoretical calculations of the thermoelectric figure of merit in nanojunctions. It aimed to obtain a qualitative and quantitative descriptions of ZT for temperatures and lengths of the nanojunctions. The self-consistent density functional theory (DFT) was performed together with the derivation of an analytical expression for ZT to investigate its dependence on the temperatures and lengths of nanojunctions. The thermal current carried by electron transport and phonon transport were considered.

In the following subsections, we briefly introduce the Seebeck coefficient, the electrical conductance, the electron thermal conductance, and the phonon thermal conductance, respectively.

2-2-1 Seebeck coefficient^7,8

Consider a tunnel junction (as mention above) that may have different temperatures in the source and drain electrodes, a small thermoelectric voltage (∆V) in the junctions can be induced by an additional temperature difference (∆T) applied in the electrodes. The ratio of the thermoelectric voltage to the temperature difference is defined as the Seebeck coefficient.

 

 "

From Eq. (18) and (20) when the system comes to equilibrium, the electric current generated by an additional infinitesimal temperature ∆T across the electrodes is compensated by an induced small voltage ∆V across the junction.

For simplicity, we assume that the additional temperature and the induced voltage are distributed symmetrically in the left/right electrodes, that is,

 |µ₈ ₈ Fermi-Dirac distribution function to the first order of ∆T and ∆V

s ! µ  '     s ! µ   

µ s !µ '  

 s !µ  

and then obtain the Seebeck coefficient in the linear response regime.

Sµ₈ µ_A ₈ _AT  # '

Г_⁸

₈ Г_A^A

Г_~⁸ Г_~^A 

where

Г_^{8 A}  # N ! b! # µ_{8 A}c^s_,^{8 A}

! ! 

We can rewrite the above equation in a specific form

Sµ₈ µ_A ₈ _AT

In the low temperature regime, where we neglect the higher order terms in the temperature, the above equation can be simplified using Sommerfeld expansion^8,12,20,

which implies that the Seebeck coefficient is closely related to the slope of the transmission probability at Fermi level. When S>0, the carriers are p type. In this case the direction of electric current is the same as the direction of thermal current. When S<0, the carriers are n type. In this case, the direction of electric current is opposite to the direction of thermal current.

2-2-2 Electrical conductance

The electrical conductance, typically insensitive to temperature in cases that the direct tunneling is the major transport mechanism, may be expressed as

σ _„†^„. Using Eq. (18), add e and 1 because we use atomic units before, and system. So the conductance with finite temperature is given as

σ ' by Eq. (29) and (22), in the linear response regime and TLT^R approximation.

κ_  

‡ |Г^' Г_

 } "

where Г_  # € ! ! # µ^{ ˆK}_ˆ,^‰ !.

2-2-4 Phonon thermal conductance

So far, the physical quantities that have been discussed have been related to the propagation of electrons. However, in most cases, the thermal current is dominated by the contributions from phonon transport. In the absence of the phonon thermal conductance, the research on ZT is incomplete. To consider the phonon contribution to ZT, it is assumed that the nanojunction is a weak elastic link, with a given stiffness that may be evaluated from total energy calculations, attached to the electrodes modeled as phonon reservoirs. We estimate the contribution of the thermal current from phonon scattering (‹ ), following the approach of Patthon and Geller²¹.

Fig. 7. Weak link model. Two macroscopic dielectrics, at temperatures TL and TR, are joined by a harmonic spring of stiffness K. [21]

The model we consider is shown in Fig.7 [21]: Two macroscopic solids L and R are held at fixed temperature TL and TR, each in thermodynamic equilibrium, and joined by a weak mechanically elastic link: a harmonic spring of stiffness K.

The two bodies are assumed to be electrically insulating, so that thermal energy is carried only by phonons. In Patthon and Geller’s paper we can get the formulation of the thermal current from phonon scattering (‹ ).

‹  $Г^ respectively. After expanding the Bose-Einstein distribution function to the first order of ∆T in the expression of phonon thermal current, the phonon thermal bias and low temperature regimes. The results of this study may be of interest to experimentalists attempting to develop thermoelectric nanoscale devices.

The Seebeck coefficient and the electron (phonon) thermal conductance can be characterized by the power low expansions: S αT, κel ” βhT  ηT³i  βT and κ_ph  ™ ℓT³ in the common range of temperatures(T_L  TR  T and in

the linear response regime µ_L  µ_R µ, where α  #^Ÿ2_3e^{KB2 ¡ µ}_{¢ µ} ^E ; β ^2Ÿ2K_3h^{B2¢ µ};

η ^{bŸKB ¡ µ E c}

2

3¢ µ² and ™ ℓ  ^8Ÿ5K₁₅^B4C_1ℓ²2^A2Y2. Consequently, ZT in the nanojunctions have a simple form

ZT α²σT³

βT ™ ℓT³ 

which is valid in small bias and low temperature regimes.