We have investigated the electron transport properties and the thermoelectric efficiency in the amino-substituted (-NH2) and nitro-substituted (-NO) 1,4-benzenedithiolates molecular junctions. Our calculations are based on ab initio self-consistent density functional theory. We have computed the I-V characteristics by using the N. D. Lang’s methods24. I will briefly introduce the density functional theory in section 2-1 and the method of appllying DFT to compute the electric current in section 2-2 and the Seebeck coefficient S in section 2-3.
2-1 Density Functional Theory
The fundamental physical quantities in the ground state can be uniquely described from the electron density n r in many-particle system. All ground state properties of ( )
the many electron system are functional of n r . In 1964 Hohenberg and Kohn prove ( ) that the ground state electron density uniquely determines the external potential. Kohn and Sham extended the theorem by separating the total energy into the kinetic energy of electron, the potential energy of attraction between electrons and nuclei, the coulomb potential energy of repulsion, and the exchange-correlation energy between electrons.
2-1-1 Hohenberg and Kohn theorem
The external potential is uniquely determined by the ground state electron density.
The above theorem can be proved as follows： We assume that two different potential V1 and V2 have the same n r . Suppose ( ) V1V2constant and 1 2 where
1 is the ground state wave function. The Schrodinger equation can be expressed as
1 1 1 1
H E
2 2 2 2,
H E
where E1 and E2 are eigen-energies of H1 and H2, respectively. With different external potentials, the Hamiltonian can be expressed as
1 2 1 2.
Combine with Eq.(2.1) and Eq.(2.2), we obtain
3
1 2 ( 1 2) ( ) 1 2,
E E
d r V V n r E E (2.3)which leads to a contradiction and means that the assumptions are wrong. Thus, two different external potentials cannot correspond to the same non-degenerate ground state density. The total energy can be expressed as a functional of ground state charge density
( )
2-1-2 Kohn-Sham equation
From the Hohenberg and Kohn theorem, it is known that the ground state properties of many-particle system can be determined by the electron density n r . The ( ) charge density in the ground state can be solved iteratively until the self-consistent is achieved.
The ground state energy of a homogeneous interacting electron gas can be written as the fourth term is the exchange-correlation energy functional of an interacting system with density n r . By the variational principle with the total electron ( ) N
n r d r( ) 3 single-electron wave equation which is called Kohn-Sham Equation.2
The early calculations of first-principles pseudopotential are made within the scheme of orthogonalized-plane-wave (OPW) atomic calculation. The wave functions in this way exhibit the correct shape outside the core region; however, they differ from the real wave functions by a normalization factor. Hamann, Schluter and Chiang (HSC) propose a model pseudopotential to solve the problems that have four properties：(1) real and pseudo valence eigenvalues agree for a chosen atomic configuration; (2) real and pseudo wave functions agree beyond a chosen core radius rc; (3) the integrals from 0 to r of the real and pseudo charge densities agree for rrc for each valence state, this is norm conservation condition; (4) The logarithmic derivatives of the real and pseudo wave function and their first energy derivates agree for rrc.
Because the lattice has the periodic characteristic, the wave functions must satisfy the Bloch theorem. It can be written as expansion of the following form:
( ) pseudo-Hamiltonian of the valence electrons can be expressed as
2
2 ps H xc,
H p V V V
m (2.8)
where
( ). angular momentums of the electron in the s, p and d orbitals are 0, 1 and 2, respectively.
The potential can be expressed as
satisfies the Poisson equation and it can be written as
2VH( )r 8n r( ). local potential. The Hamiltonian can be rewritten as
loc nl
' '
( , ).
nl nl
kG V kG V kG kG (2.15)
(B) Pseudopotential
Pseudopotentials are introduced to simplify electronic structure calculations by eliminating the need to atomic core states and the strong potentials responsible for binding them.
To construct atomic pseudopotential lm at a given energy which are identical to atomic eigenfunctions. The lm are continued inside rc with the condition that
The pseudopotentials are obtained by inverting the Schrodinger equation
2
The complete pseudopotential is then written as
( ) ( ) ( ) , coordinate) of the Hamman-Schluter-Chiang (HSC) pseudopotential which used in an expansion of N plane waves requires the evaluation of ( 1)
2
l lm l lm .
which is not Hermitian and the generalized norm-conservation requirement,
3 * * 3
which is substituted into the Eq.(2.24) and one can obtain the pseudopotential
,
ps loc i ij j .
i j
V V
B (2.22)In general, it is difficult to apply Eq.(2.20), results in lm whose plane-wave expansions are extremely slowly converging. To avoid applying Eq.(2.20), Chou constructed norm-conserving nlm at two energies En and inverted the Schrodinger
The Anl is Hermitian and the nlm( )r are solutions of the pseudo Schrodinger equation at En with nl replaced by l.
(C) Local Density Functional Approximation
The exchange-correlation energy is relation to the electronic distribution in the system. It is difficult to give an exact expression for Exc because of its complexity. In order to simplify this complexity, Kohn and Sham suggested using the homogeneous electron gas system to approximate the energy contribution from Exc[ ]n in 1965. If the electronic density varies slowly, the exchange-correlation functional can be written as
[ ] [ ] ( ) 3 ,
xc xc
E n
n n r d r (2.24)where the exchange-correlation potential can be expressed as
( ) xc[ ] { ( )},
where xc[ ]n is the exchange-correlation energy density of the homogeneous electron gas. Vxc( )n is the exchange and correlation contribution to the chemical potential of a homogeneous gas of density n.
The exchange-correlation energy density can be separated into x[( )]n and [( )]
c n
. x[( )]n is the exchange energy of a homogeneous electron gas and c[( )]n is the correlation energy of a homogenous electron gas.
Within Hartree-Fock approximation the exchange energy density can be obtained by solving the Schrodinger equation of the non-interacting homogenous electron gas.
0.458
x( )
s
r r
rs is Wigner-Seitz radius,
where calculations by Ceperley and Alder. The wave function for electrons in a finite volume subject to periodic boundary conditions and extrapolated the energy per electron to infinite volume. The Ceperley’s parameterization of the correlation energy for rs 1 is
1 2
Substituting Eq.(2.26) into Eq.(2.25), the relation between exchange-correlation potential and electronic density can be expressed as
[1 ] . difference in value between the new electronic density and the initial electronic density is too big, they will be mixed to generate another electronic density, and repeat the above procedures until the difference in value between the new density and last density is very small. The above procedures are called self-consistent procedure.
The convergence of the flow chart：
2-2 Current of atomic wires24
We start with a brief introduction of how to calculate the electric current carried by the electron transport in the DFT framework. We picture a nanoscale junction as formed by two semi-infinite electrodes held a fixed distance apart, with a nano-structured object bridging the gap between them. The full Hamiltonian of the system is H = H0+V, wherein H0 is the Hamiltonian due to the bare electrodes, and V is the scattering potential of the nano-structured object. The nano-structured object could be a single atom, a chain of atoms, a molecule, or any system with nanoscale dimension. The effective single-particle wave functions of the whole system in the continuum states are 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 applied bias is given by VB FR FL e
, where L(R)is the chemical potential deep in the left (right) electrode. The single-particle wave functions and self-consistent density distribution are 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 is 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 is obtained, and the problem is solved self-consistently using a modified iterative procedure. Atomic units are used here, withe m 1.
2-2-1 Bimetal junction
Fig. 11. The schematic of bimetal junction with external source-drain bias.
Two bulk electrodes are modeled as two semi-infinite bulk metals described by Jellium model. The wavefunction of the bare electrodes can be obtained by solving the Shrödinger equation and Poisson equation until the self consistency is achieved.
Electrons incident from the left hand side can be partial transmitted and partially reflected. The unperturbed wavefunctions EKM ( )r of the electrons satisfy the boundary condition as shown in Eq. (2.30), i.e.,
( ) i K R ( ), and z the coordinate normal to them. Deep in the positively biased electrode (which we
will take henceforth to be the left electrode), uEK ( )z has the form of a linear
combination of left-moving and right-moving plane waves with wave vector kL. Here, the energies of electrons are conservative as described by
u z . For propagating states, we will replace either by “+,” which will 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 right and thus kR is as defined continuum normalization which we impose on the wave functions M, specified by
3 *
We investigate the electron transport and thermoelectricity of molecules wire and atomic wires sandwiched between two bulk electrodes with finite external source-drain bias and finite temperature difference as shown in Fig. 11, where the bias is given by
system consisting of the metal electrodes and the group of atoms between them, are solved in scattering approaches. The continuum wave functions MA are solved by the Lippmann-Schwinger equation, where M will have the same labeling ( ,E K , ) ,
even though K no longer refers to a conserved quantity. These solutions will also have the same normalization as the M, a fact that facilitates the calculation of the electron density distribution and the current. For EFL E EFR, where EFL is the
Fermi level in the left electrode and EFR EFLeVB is the Fermi level in the right electrode (bias VB taken positive), we occupy only states corresponding to a wave incident from the right, i.e., only EKMA and not EKMA.
Fig. 12. The installed system on the source-drain bias and gate voltage in the -NH2 and –NO 1,4-benzenedithiolates molecular junction: the atomic size conductors where both chemical potential and temperature gradients are present.
The Hamilton of complete system can be put into Lippmann-Schwinger form :
this equation embodies the motion that electrons in states of the electrodes impinge on and are scattered elastically by the potential V r r( , '), which describes the difference in potential between the complete system and the bare electrodes. It can be written
3 ( ") Fermi-Dirac distribution equal to 1. The electron number density is given by the sum of squares of the occupied states EKMA, 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
the current density for the pair of biased electrodes in the absence of the group of atoms.Then the quantity of interest to us is
2 MA M ,
I
d R zj j (2.35)which is independent of z since our system has no current sources or sinks. Here z is the unit vector point to the right and perpendicular to the surfaces of the electrodes. If we write MAM , then
Next, we calculate the current trough the tunnel junction where the temperatures and chemical potentials in source and drain electrodes can be different, Fermi-Dirac distribution not equal to 1.
where ,i jL R, . EL R( )( ,r K ) is the single-particle wave function incident from the
left (right) electrode with energy E and component of the momentum K parallel to the electrode surface11,26, and dR 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
( ) electrode, respectively, and kB is the Boltzmann constant. For simplicity, we define the transmission probability of electron with energy E incident from the left (right) symmetry, the current in Eq. (2.37) can be rewritten as
( L, L, R, R) 1 ER( R, R) EL( L, L) R( ),
2-3 Seebeck coefficient
Consider a tunnel junction 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. using the second-quantization field-operator method with the effective single-particle wave functions, we presented the form of current.
1 [ ER( R, R) R( ) EL( L, L) L( )].
I dE f T E f T E
(2.43) The current is described by the Fermi-Dirac distribution function Eq. (2.44) and the transmission function Eq. (2.45). Moreover, the parameters, μL(R) and TL(R), are the chemical potential and the temperature in the left (right) electrode, respectively.( ) single-particle wave function incident from the left(right) electrode with energy E and component of the momentum K∥ parallel to the electrode surface and dR represents an element of the electrode surface. The stationary wave function EL R( )( ,r K) can be calculated by solving the Lippmann-Schwinger equation iteratively to self-consistency.
The exchange-correlation potential is included in density-functional formalism by using
the local-density approximation. Once the single-particle wave functions are calculated self-consistently, the transmission function of electron with energy E can be calculated using Eq. (2.45). We had completely explained at the section 2-2.
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,
We make the differences of extra current from the changed temperatures and voltages be zero to control the relation between T and V . The next step, we expand the Fermi-Dirac distribution function to the first order in Tand V . So we can obtain the Eq. (2.47)
We can simplify the Eq.(2.47) by using the Sommerfeld expansion27.
2 2
From Eq. (2.49), the Seebeck coefficient relates closely to the transmission function in the vicinity of the left and right Fermi levels.
In the first step for our analysis, we study the Seebeck coefficient in a
When we consider the case of low-temperature regime, the Seebeck coefficient can be also further simplified by using the Sommerfeld expansion as
2 2 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. The final Eq. (2.51) has been applied to several atomic and molecular systems.27 The investigation explores the dependence of the Seebeck coefficient on the gate voltages, temperatures of the electrodes, and the source-drain biases in both the linear and nonlinear response regimes.