Quantum Noise of Current in a 1,4-Benzenedithiol Single-Molecule

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C: Physical Processes in Nanomaterials and Nanostructures

Quantum Noise of Current in a 1,4-Benzenedithiol Single-Molecule Junction: First Principle Calculations Bin Ouyang, Bailey C. Hsu, and Yu-Chang Chen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b06710 • Publication Date (Web): 01 Aug 2018 Downloaded from http://pubs.acs.org on August 7, 2018

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The Journal of Physical Chemistry

Quantum Noise of Current in a 1,4-Benzenedithiol Single-Molecule Junction: First Principle Calculations. Bin OuYang, Bailey C. Hsu, and Yu-Chang Chen∗ Department of Electrophysics, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan E-mail: [email protected]

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Shot noise refers to the autocorrelation of current related to the quantum statistics of discrete electrons. The study of shot noise can provide a deep understanding of the quantum fluctuation of current, owing to the interference between incident and reflected electrons. Electrons experience scattering when in the mesoscopic quantum point contact with materials of intermediate lengths. Thus, electron transport is typically incoherent and quantum channels are reformed in leads through adiabatic constriction. The starting point of the shot noise theory is based on a field operator constructed from incoherent wave functions in leads. However, electron transport is coherent in single-molecule junctions with length scales smaller than the dephasing length. In investigations regarding shot noise in systems with coherent electron transport, using a field operator constructed from coherent wave functions is natural. In this study, we investigated shot noise in a 1,4-benzenedithiol single-molecule junction on the basis of coherent wave functions obtained self-consistently from first principles approaches. The theoretical value of shot noise in our calculations is S ≈ 4.03 × 10−26 A2 /Hz at VBias = 0.01 V. This value is in good agreement with that obtained from a recent experimental measurement (S ≈ 4.37 × 10−26 A2 /Hz).

;ĂͿ

TOC Entry: (a) Schematic of the 1,4-benzenedithiol single-molecule junction at high conductance state σ = 0.23 G0 at VBias = 0.01 V; (b) shot noise as a function of bias.

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Introduction In recent years, nanoscale electronics has elicited considerable attention owing to endeavors in miniaturized devices and the desire to understand novel quantum phenomena caused by miniaturization. 1 Single-molecule junctions have provided new opportunities and challenges in this field. 1–12 Single-molecule junctions of nanometer scales are signalled by a coherent wave nature provided by an atomic-level quantum control. They are ideal test-beds for verifying the consistency between experimental measurement and the novel quantum theory. Counter-intuitive phenomena beyond classical expectation may be observed. For instance, the Pauli exclusion principle may cause local cooling instead of heating in asymmetric nanojunction because of inelastic electron-vibration interactions when passing a current via finite voltages. 13 Unlike the Johnson-Nyquist noise caused by classical thermal fluctuations of the current, 14,15 shot noise is due to the quantum fluctuation of current at zero temperature. Meanwhile, shot noise is attributed to the quantum statistical correlation of electrons. It can provide a deep insight into quantum transport, particularly the statistics of quantized charge, the role of the Pauli exclusion principle, dephasing, and the quantum mechanical wave nature of interference between incident and reflected electrons. Shot noise is of particular interest in the study of electron transport because of its rich quantum physics. 16,17 Shot noise S reaches the classical limit 2eI when electrons are completely uncorrelated in statistics, as described by a Poissonian distribution. That is, the Fano factor defined by F = S/(2eI) reaches 1, where I and e are the average current and the electron charge, respectively. In mesoscopic systems with materials of intermediate lengths, electrons encounter inelastic scattering and loose coherent phases in wave functions. Consequently, the starting point for the shot noise theory is based on a field operator constructed from incoherent wave

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functions, where electrons in the wave guides are quantized: 16,18,19

ˆ L (r, t) = Ψ

Z

NL (E)

X

iEt/~

dEe

n=1

χLn (r⊥ ) [ˆ aLn eikLn z + ˆbLn e−ikLn z ], [2πvLn (E)]1/2

(1)

where z is the coordinates along the leads, and r⊥ is the transverse coordinates, a ˆ and ˆb are annihilation operators for incoming and outgoing electrons in the wave guides,respectively. χLn (r⊥ ) is the transverse wave function of electrons that are quantized in the left lead, and n is the quantum channels that are reformed through adiabatic constriction. The above field operator is the starting point of the shot noise theory. The field operator is formed by incoherent wave functions that describe incoming and outgoing electrons, which are quantized in the left and right leads. Electrons are scattered from one channel in the left lead into another in the right lead, as described by a scattering matrix s,                

ˆbL1 .. . ˆbLN L ˆbR1 .. . ˆbRN R





ˆL1  a   .   ..         a  ˆ  = s  LNL    a   ˆR1   .   ..      a ˆRNR

              



(2)

where a ˆL(R)n and ˆbL(R)n are annihilation operators of the incoming and outgoing electrons of the n-th channel in the L (R) lead. The transverse component of the wave functions χLn (r⊥ ) is quantized. The transverse wave functions are finally integrated by orthonormal relations. Owing to the quantum statistical expectation value of the product of the creation and annihilation operators, shot noise is reduced to a simple form expressed by the eigenvalues of P the scattering matrix. 16 Consequently, shot noise can be expressed as S ∝ n Tn (1 − Tn ), where Tn represent the eigenchannels of the scattering matrix. The dimensionless shot noise described as the Fano factor is F =

P

Tn (1−Tn ) nP n Tn

4

. 16 The preceding expression requires a scat-


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ܽො௅

ܽොோ

ܾ෠௅ i r

ܾ෠ோ t

୐ ሺ‫ݎ‬Ԧ ) ߰୉୏ ோ ሺ‫ݎ‬Ԧ ) ߰୉୏

t

i r

Figure 1: (a) Schematics of a mesoscopic quantum point contact with a length scale larger than the dephasing length. The system is constructed by a junction connected to the electrodes via well-defined leads, which act as wave guides. The field operator is formed by incoherent wave functions. a ˆL(R) and ˆbL(R) are annihilation operators of electrons quantized in the transverse direction in the left and right leads, respectively. Electrons scattered from one channel to another is described by a ˆL(R) and ˆbL(R) connected via a scattering matrix; (b) a nanoscale junction with a length scale shorter than the dephasing length. The system may lack the wave guides as the junction rapidly opens up into the electrodes. The L(R) wave functions ΨEK (r) for electrons are coherent. The field operator is constructed from coherent wave functions obtained from DFT+LS self-consistent transport calculations. The transverse component of the momentum K is continuum. tering matrix connecting incoherent wave functions in leads, where the transverse quantum channels are reformed, as schematically described in Figure 1(a). Shot noise in the mesoscopic quantum point contact is essentially 1D because the transverse components of the incoherent wave functions are separable and can be integrated out. One can expect that the interference effect between channels may become less significant. 19 By contrast, a short single-molecule junction may have a length scale shorter than the dephasing length. Therefore, the wave functions that describe the tunneling electrons are coherent, as schematically depicted in Figure 1(b). The streamline expectation values of the current density flowing through the molecule region show 3D configurations. 20 Moreover, 5



the decomposition of 3D wave function into the product of quantized transverse components and longitudinal component may not be possible because single-molecule junctions have abrupt constrictions and lack smooth matching regions to reform wave guides. In this study, we investigated the shot noise of a 1,4-benzenedithiol single-molecule junction that is based on a field operator constructed from coherent wave functions, where the effective single-particle wave functions are calculated self-consistently in scattering approaches using density functional theory (DFT) combined with the Lippmann-Schwinger equation (LS), referred to as DFT-LS hereafter. The theory based on a combination of DFT and the nonequilibrium Greens function (DFT-NEGF) provides another alternative to study electron transport in atomistic junctions. 21 It has been shown that the Landauer+DFT approach is flaw and needs an additional orbital for nonequilibrium effects. 22 First-principles claculations based on DFT-NEGF approaches have been applied to investigate shot noise in junctions in elastic, 23–25 and inelastic scattering. 26 In the strongly correlated transport regime, the nonequilibrium system can be mapped to an effective equilibrium system. 27 In addition to this, vortex correction contributes significantly to inelastic noise. 28 Steady-state current for open quantum systems based on scattering states has been investigated through numerical renormalized approaches. 29 A considerable number of studies have been conducted on shot noise. 16–19,21–42 However, very few attempts have been made on the exploration of shot noise from the coherent wave function perspective. 43–45 Previous attempts are limited to toy models formed by monatomic chains. 43–45 Our calculations show that shot noise based on coherent wave functions obtained from parameter-free first-principles calculations is in good agreement with the value measured in a recent experiment. 42 Our theory and approach provide a useful tool for the exploration of shot noise in coherent transport regime from first principles calculations without any fitting parameters. Our shot noise theory is based on the effective single-particle wave functions obtained from DFT-LS, where shot noise is owing to the quantum statistics of non-interacting electrons and interference between wave functions of incident and reflected electrons. The present study can also provide extra opportunity

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(a) X Z

L Jellium surface

(b)

Jellium surface

(c)

X Z Figure 2: (a) The optimized Au-BDT-Au junction at high conduction state. (b) Static charge density of the Au-BDT-Au nano-structure obtained from VASP. The charge density is used as the initial guess density in the self-consistent DFT+LS transport calculations. (c) Optimized geometry depicted in (a) is determined by the total energy calculated using VASP as a function of L, where L is the distance between the surfaces of the electron jelliums. for the exploration of unknown interference effects due to x− and y−components from the perspective of 3D coherent wave functions.

Theoretical Methods Calculations of wave functions of the Metal-Molecule-Metal (M-Molecule-M) junction are separated into two steps. First, we consider the metal-vacuum-metal (M-V-M) junction, where the molecule bridging the electrodes is removed. The bimetal junction are modeled as two electron reservoirs connected to a battery. Two electrodes are modeled as electron jellium in a positive background with two semi-infinite planar surfaces with a vacuum region in between. Wave functions of the M-V-M junction are obtained by iteratively solving the Schr¨odinger equation coupled to Poisson equation within the framework of density functional

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theory until self-consistency is achieved. Second, wave functions of the M-Molecule-M junction are calculated using the Lippmann-Schwinger equation, where the Green’s function is calculated using wave functions of the M-V-M junction. The calculations is iteratively performed in plane wave basis in the framework of DFT-LS until self-consistency is achieved.

(a) The M-V-M junction The M-V-M junction is modeled as two terminal electrodes connected to a battery with an external bias, VB = (µR −µL )/e, where µL(R) is the chemical potential deep in the left (right) electrodes. The electrode is described as a semi-infinite metal modeled as electron jellium r3 )−1 , where rs is the Wigner-Seitz with a uniform positive-charge background n+ = ( 4π 3 s radius. Quantum tunnelling through the vacuum enables the charge transfer from the metal region to vacuum region, which forms dipolar layers on the metal surfaces. The characteristic length scales of the dipolar layers are comparable to the Fermi wavelength λF . Moreover, when electrons are scattered on the interfaces between the metal region and vacuum region, a Friedel oscillation of charge density in metal region is induced. The voltage drop between electrodes due to the battery causes an additional transfer of charge like a capacitor, which generates an electric field in the vacuum region. The wave functions are calculated by solving the Schr¨odinger coupled to Poisson equations in the framework of density functional theory until self-consistency is reached. that is, ~2 d2 {− + vef f [z, n(z)] − E}uαE (z) = 0, 2 2m dz

(3)

∂z2 Ves (z) = 4π[n+ − n(z)],

(4)

and

L(R)

where uE

(z) describes electrons with the energy (E) traveling along the z-direction incident

from the left (or right) to right (or left) electrode, respectively; n(z) and n+ are the electron density and positive background, respectively; vef f [z, n(z)] is the effective potential which 8


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comprises the exchange-correlation potential Vxc in local density approximation and the electrostatic potential Ves owing to the charge distribution, that is,

vef f [z, n(z)] = Ves (z) + Vxc [n(z)].

(5)

Note that the electrostatic potential Ves is recalculated using the Poisson equation according to the distribution of charges, where the electron density n(z), that is, Z 1 X kF α 2 n(z) = 2 (kF α − kα2 )uαE (z)dkα , π α=L,R 0 L(R)

which is recalculated using uE

(6)

(z) obtained from Eq. (3).

Electrons away from the scattering region are free to move, i.e., wave functions uαE (z) are asymptotically equivalent to plane waves, as z → ±∞. Consequently, wave functions are solved using Eq. (3) with the following boundary conditions,

uLE (z) =

r

and uR E (z) =

r

   eikL z + RL e−ikL z , z < −∞ me × 2π~2 kL   TL eikR z , z>∞

(7)

   TR e−ikL z , z < −∞ me ×  e−ikR z + R eikR z , z > ∞. 2π~2 kR  R

(8)

Note that conservation of total charge must be maintained in the iterative calculations to warranty that the Vef f (z) has a flat band deep inside in the electrodes, as z → ±∞.

(b) The M-Molecule-M junction When the molecule is included, wave functions that describe electrons in the M-V-M junction are, in further, scattered by the bridging molecule. We consider the molecule as a scattering center. Wave functions of the M-Molecule-M junction are calculated iteratively using the

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Lippmann-Schwinger equation in the framework of DFT-LS until self-consistency is achieved, L(R) ΨEK (r)

=

0,L(R) ΨEK (r)

0,L(R)

Z +

3

Z

d r1

L(R)

where ΨEK (r) = (2π)−1 eiK·R × uE

L(R)

d3 r2 G0E (r, r1 )V (r1 , r2 )ΨEK (r2 ),

(9)

(z) stands for the effective single-particle wave func-

tions of the M-V-M junction. Note that electrons are free to move in the x- and y-direction in deep electrodes, and thus are described as (2π)−1 eiK·R , where K and R are the momentum L(R)

of electron and coordinates in the x-y plane, respectively; ΨEK (r) are the effective singleparticle wave functions of the entire M-Molecule-M junction solved from Eq. (9); V (r1 , r2 ) represents the potential of the scattering center, that is,

V (r1 , r2 ) = Vps (r1 , r2 )+ Z δn (r3 ) δ(r1 − r2 ), (Vxc [n (r1 )] − Vxc [n0 (r1 )]) + dr3 |r1 − r3 |

(10)

where Vps (r1 , r2 ) is the pseudopotential; Vxc [n (r)] is the exchange-correlation potential in local-density approximation; n0 (r) and n (r) represents the charge density of the M-V-M and M-Molecule-M junctions, respectively, and δn(r) =n (r) − n0 (r) is their difference. Electrons may form bound states which are the eigenvalues of the Hamiltonian. Consequently, electron density of the M-Molecule-M junction has a form of

n(r) = 2

X

2

|Ψi | + 2

i

X Z

Z dE

d2 K |ΨαEK (r)|2 .

(11)

α=L,R

The quantity G0E in Eq. (9) is the Greens function of the M-V-M junction, which has a form of G0E (r1 , r2 )

1 = 2 2π

Z

d2 KeiK·(R1 −R2) ×

10

uLE (z< )uR E (z> ) , WE


(12)

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L(R)

where uE

(z) are wave functions of the M-V-M junction; z< (z> ) is the lesser (greater) of

z1 and z2 ; WE is the Wronskian, WE = uLE (z)

d L d R uE (z) − uR u (z). E (z) dz dz E

(13)

A plane waves basis, {eikn ·r }, has be selected for numerical calculations. We consider a plane wave basis in a box of size (2Lx ) × (2Ly ) × (2Lz ), where kn = ( Lπx nx , Lπy ny , Lπz nz ) and n = (nx , ny , nz ) is an integer vector. In plane wave basis, wave functions of the M-V-M and X L(R) R(L) 0,R(L) M-Molecule-M junctions can be expressed as ΨEK (r) = ΨEKn eikn ·r and ΨEK (r) = n X 0,L(R) ΨEKn eikn ·r , respectively; the scattering potential, V (r1 , r2 ), can also be expressed as n X eikn1 ·r1 Vn1 ,n2 eikn2 ·r2 ; the Green’s function, G0E (r1 , r2 ), can also be expressed V (r1 , r2 ) = n1 ,n2

as

G0E (r1 , r2 )

=

X

eikn1 ·r1 G0E,n1 ,n2 eikn2 ·r2 . After integrating out the position coordinates,

n1 ,n2

Eq. (9 can be cast into a system of linear equations, that is, X

L(R)

0,L(R)

CEnn0 ΨEKn0 = ΨEKn ,

(14)

n0

L(R)

where ΨEKn0 can be solved from the system of linear equations, and CEnn0 has a form of CEnn0 = δnn0 − 8Lx Ly Lz

X

G0E,n,n00 Vn000 n0 .

(15)

n00

The LippmannSchwinger equation is one of the most commonly used equations for the characterization of scattering in quantum mechanics. It has been widely applied in the study of collision in atomic physics, molecular physics, optical physics, nuclear physics, and particle physics. Our approaches embed the LippmannSchwinger equation in the density functional theory, where the scattered wave functions are solved in plane wave basis. Instead of solving the the Lippmann-Schwinger equation in the integral equation form [Eq. (9)], the LippmannSchwinger equation is transformed into a problem of linear algebra [Eq. (14)] that is solved

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non-perturbatively. 47

(c) Theory of Shot Noise Our starting point for the shot noise theory is based on a field operator constructed from coherent wave functions obtained from DFT-LS calculations:

ˆ = Ψ

X

aαEK (t) ΨαEK (r) .

(16)

α,E,K

where wave functions ΨαEK (r) are iteratively calculated within DFT-LS until self-consistency is obtained; subindex K in the wave function describes the momentum of the electron in L(R)

x-y plane and it is continuum because of the lack of wave guide; 47 α = L or R; aEK (t) = L(R)

exp(−iωt)aEK , and the annihilation operator for electrons incident from the left (right) L(R)

electrodes is aEK which satisfies anti-commutation relations {aαE1 K1 , aβ† E2 K2 } = δαβ δ (E1 − E2 ) δ(K1 − K2 ),

(17)

where β = L or R. The expectation value of the product of the electron creation and annihilation operators at thermal equilibrium describes the quantum statistical distribution of electron β α < aα† E1 K1 aE2 K2 >= δαβ δ (E1 − E2 ) δ(K1 − K2 )fE ,

(18)

where the statistics of electrons coming from the left (right) electrodes are assumed to be L(R)

determined by the equilibrium Fermi-Dirac distribution function fE

= 1/{1 + exp[(E −

µL(R) )/(kB T )]} in the left (right) electron reservoir. We also assume that the scattering of electrons are elastic such that

2 }2 kR 2me

2

2

= E− }2mKe −vef f (∞) and

2 }2 kL 2me

2

2

= E− }2mKe −vef f (−∞). 46,47

The quantum statistical expectation values of the products of the four operators are

12


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carried out and are given by D

E E D ED i† j k† j k† l l = a a a a ai† a a a E1 K1 E2 K2 E3 K3 E4 K4 E1 K1 E2 K2 E3 K3 E4 K4 E ED D k† j i† l − aE1 K1 aE4 K4 aE2 K2 aE3 K3 .

(19)

Following the same procedure used to derive the conventional theory of shot noise, 16 we obtain the expression for shot noise based on Eq. (16) and is given by e~ S(z1 , z2 ) = 2π~( )2 mi

Z

EF R

Z dE

Z dr1⊥

Z dr2⊥

Z dK1

RL LR dK2 I˜EK (r1 )I˜EK (r2 ). 1 ,EK2 2 ,EK1

EF L

(20) where I˜Eαβ1 K1 ,E2 K2 (r) = ΨαE1 K1

∗

∇ΨβE2 K2 − ∇ ΨαE1 K1

∗

ΨβE2 K2 .

The value of shot noise is obtained by averaging S(z1 , z2 ) in space. Equation (20) indicates that shot noise results from the interference between the right-going and left-going waves. In our theory, the channels are described by the component of momentum K perpendicular to the direction of the current. The momentum K are not quantized in discrete channels in the transverse direction because our system lacks the wave guide. Thus, it may be more difficult to find the eigenchannels because the channels K are continuum.

Results and discussion The conductances of the Au-BDT-Au junctions method are widespread in the mechanically controllable break junction (MCBJ) experiments. 42,48–53 The distribution of conductances was attributed to the diversified contact geometries of the junctions. To compare our theoretical shot noise calculations with that measured in the experiment, 42 we select a contact geometry of the Au-BDT-Au junctions at a high conductance state, as shown in Figure 2(a). First, the geometry of the Au-BDT-Au cluster is relaxed when the Vienna Ab-Initio Simu-

13



lation Package (VASP) is used. This charge density of the Au-BDT-Au cluster, depicted in Figure 2(b), was adopted as the initial guess density for self-consistent DFT + LS transport calculations. The distance between the Au-BDT-Au cluster and the electrodes was determined by total energy calculations using VASP, as shown in Figure 2(c). The most stable junction had a distance L = 19.0 a.u. between the two surfaces of electron jelliums, which was determined by using the minimum value in the total energy as a function of L. (a)

(b) 0.5 V

168

1.0 V

360

112

240

56 120

0

Density of states (arb.unit)

Density of states (arb.unit)

0.4 V

840 560 280 0

0.3 V

600 400 200 0

0.2 V

330 220 110 0

0.1 V

510

0

0.9 V

279 186 93 0

0.8 V

450 300 150 0

0.7 V

246 164 82

340 170

0

0

0.6 V

510

960

0.01 V

340

640 170

320

0

0

-6

-4

-2

-6

0

-4

-2

0

2

Energy (eV)

Energy (eV)

(c) S wave (S)

PDOS (arb. unit)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Pz wave (S) Px-y wave (S)

100

S wave (Au) Pz wave (Au) Px-y wave (Au) Total

0 -5

-4

-3

-2

-1

0

Energy (eV)

Figure 3: DOS of the AuBDTAu single-molecule junction in the continuum region at various biases: (a) VBias =0.01, 0.1, 0.2, 0.3, 0.4, 0.5 V and (b) VBias = 0.6, 0.7, 0.8, 0.9, 1.0 V. The left Fermi level EF L (red-dashed lines) is set as the zero value of energy. The DOS that lies above EF L represents the current-carrying windows formed between EF L and EF R . The bias is defined by VBias =(EF R -EF L )/e. (c) The partial DOS at VBias = 0.5 V contributed by the S-wave, Pz -wave, and the Px−y -wave centered at the Au and S atoms in a radius of 3 a. u., respectively, where the Au-S bond forms the bottleneck for the current. The Px−y orbital of the sulfer and the Px−y orbital of the gold atom form a π bond, which dominates the states in the current-carrying window between EF L and EF R .

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Figures 3(a) and 3(b) show the density of states (DOS). These states gradually developed from a partial peak to a full peak located between EF L and EF R , which formed an energy window for the current. The sulfur atom and the neighboring gold atom formed a covalent bond. The Au-S bond in the Au-BDT-Au junction became the bottleneck for the current. The majority of the streamlines of the current density were directed to flow through the Au-S bonds. To understand how the electric current flows through the junction, we showed the projections of the orbitals localized on the sulfur and gold atoms and compared them with the total DOS in Figure 3(c). It shows that Px−y waves localized on the sulfur and gold atoms dominated the DOS in the energy windows, where the current can flow. The result confirms that between the Au and S atoms a -bond, which is the major channel for the tunneling current, was formed. Wave functions obtained through DFT+LS calculations were applied to calculate the current and shot noise in the Au-BDT-Au junction under various bias voltages. Figure 4(a) shows the current I (left axis) and conductance σ (=

I ; VBias

right axis) as functions of

applied voltage. Nonlinear current-voltage characteristics were observed. A marked increase in conductance was recorded from σ ≈ 0.23 G0 at VBias = 0.01 V to σ ≈ 0.57 G0 at VBias = 0.3 V, where G0 =

2e2 h

is the quantum unit of conductance. The increase in conductance was

attributed to the fact that more states were included in the current-carrying energy windows in DOS. These states gradually developed from a partial peak to a full peak located between EF L and EF R , as shown in Figure 3(a) and 3(b). Shot noise, expressed in Eq. (20), may be interpreted as the interference between transmitted and reflected wave functions. Shot noise can be interpreted as quantum fluctuation of the current due to the interference between the incident and reflected wave functions. Figure 4(b) shows the shot noise as a function of applied voltage. The theoretical value of shot noise is S ≈ 4.03×10−26 A2 /Hz at VBias = 0.01. This value is in good agreement with that measured in the experiment at high conductance state (S ≈ 4.37 × 10−26 A2 /Hz, where G = 0.23 G0 at VBias = 0.01). 42 Figure 5 shows the correlation of conductance (left axis) and Fano factor (right axis)

15



(a)

60

Current (

A)

0

Conductance (G )

0.6 40

0.5

20

0 0.4 -20 0.3 -40

0.2

-60 -1

0

V

(b)

Bias

1

(V)

500

-26

2

A /Hz)

600

Shot noise (10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 23

400

300

200

100

0 0.0

0.2

0.4

0.6

V

BIAS

0.8

1.0

(V)

Figure 4: (a) Current [(black) circle; left vertical axis] and conductance [(blue) square; right vertical axis] and (b) shot noise as a function of bias up to VBias = 1 V.

16


0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2 0.0

Fano factor

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Conductance (G )

Page 17 of 23

0.2 0.2

0.4

0.6

V

BIAS

0.8

1.0

(V)

Figure 5: Fano factor [(black) circle; left vertical axis] and conductance [(blue) square; right vertical axis] as functions of bias. as a function of bias. We observed that conductance and Fano factor exhibited a high correlation, that is, the Fano factor decreased as conductance increased. Such relationship greatly resembles that in a single-particle picture (or one-channel system). The results of our calculations indicate that the Au-BDT-Au junction is roughly a single-channel case, as G/G0 = T and F = R = 1 − T . This finding can be attributed to the fact that the sulfur and the neighboring Au atom form a π bond, which is the bottleneck in the current, where the px−y waves of the sulfur and Au atoms contributes a dominant channel in the conduction window of energy. It may be the reason why the theoretical values of the Fano factor imply that the Au-BDT-Au junction is roughly a one-channel system consistent with the results of recent experiments 42

Conclusions The shot noise of the Au-1,4-benzenedithiol-Au (Au-BDT-Au) single-molecule junction was investigated as a function of bias from the wave function perspective, obtained from the DFT-

17



LS calculations . The theoretical value of shot noise is approximately 4.03 × 10−26 A2 /Hz at a high-conductance state with σ ≈ 0.23 G0 at V= 0.01 V. The results are in good agreement with a recent experimental measurement where S = 4.37 × 10−26 A2 /Hz and G = 0.23 G0 at VB = 0.01 V. Our calculations show conductance and Fano factor as a function of bias are highly correlated. That is, the Fano factor decreases as conductance increases. Moreover, G ≈ G0 ·T and F ≈ (1−T ) indicate that the Au-BDT-Au junction is similar to a one-channel system, and this finding is consistent with the results of recent experiments.

Acknowledgement The authors acknowledge the support of the MOST (Taiwan) under Grants MOST-1062112-M-009-010-MY3, 106-2627-M-009-001-, and 106-2112-M-009-015-MY3. This work was financially supported by the Center for Semiconductor Technology Research from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. Also supported in part by the Ministry of Science and Technology, Taiwan, under Grant MOST-107-3017-F-009-002. We thank National Center for High-performance Computing for computing time and facilities. We also acknowledge the support of National Center for Theoretical Sciences.

References (1) Di Ventra, M 2008, Electrical Transport in Nanoscale Systems, Cambridge University Press, Cambridge. (2) Aviram. A.; Ratner, M. A. Molecular Rectifiers. Chem. Phys. Lett. 1974, 29, 277-283. (3) Cui, X. D.; Primak, A.; Zarate, X.; Tomfohr, J.; Sankey, O. F.; Moore, A. L.; Moore, T. A.; Gust, D.; Harris, G.; Lindsay, S. M. Reproducible Measurement of Single-Molecule Conductivity. Science 2003, 294, 571-574. 18


Page 18 of 23

Page 19 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(4) Xu, B.; Tao, N. J. Measurement of Single-Molecule Resistance by Repeated Formation of Molecular Junctions. Science 2003, 301, 1221-1223. (5) Donhauser, Z. J.; Mantooth, B. A.; Kelly, K. F.; Bumm, L. A.; Monnell, J. D.; Stapleton, J. J.; Price, Jr D. W.; Rawlett, A. M.; Allara, D. L.; Tour, J. M. et al. Conductance Switching in Single Molecules Through Conformational Changes. Science 2001, 292, 2303-2307. (6) Datta, S.; Tian, W.; Hong, S.; Reifenberger, R.; Henderson, J. I.; Kubiak, C. P. Current-Voltage Characteristics of Self-Assembled Monolayers by Scanning Tunneling Microscopy. Phys. Rev. Lett., 1997, 79, 2530-2533. (7) Kushmerick, J. G.; Holt, D. B.; Pollack, S. K.; Ratner, M. A.; Yang, J. C.; Schull, T. L.; Naciri, J.; Moore, M. H.; Shashidhar, R. Effect of Bond-Length Alternation in Molecular Wires. J. Am. Chem. Soc. 2002, 124, 10654-10655. (8) Wold, D. J.; Frisbie, C. D. Fabrication and Characterization of Metal-Molecule-Metal Junctions by Conducting Probe Atomic Force Microscopy. J. Am. Chem. Soc. 2001, 123, 5549-5556. (9) Chen, J.; Reed, M. A.; Rawlett, A. M.; Tour, J. M. Large On-Off Ratios and Negative Differential Resistance in a Molecular Electronic Device. Science 1999, 286, 1550-1552. (10) Nitzan, A.; Ratner, M. A. Electron Transport in Molecular Wire Junctions. Science 2003, 300, 1384-1389. (11) Amanatidis, I.; Kao, J. K. ; Du, L. Y.; Pao, C. W.; Chen, Y. C. Thermoelectric Efficiency of Single-Molecule Junctions: Phase Diagram Constructed from First-Principles Calculations. J. Phys.Chem. C 2015, 119, 28728-28736. (12) Kamenetska, M.; Quek, S. Y.; Whalley, A. C.; Steigerwald, M. L.; Choi, H. J.; Louie, S. G.; Nuckolls, C.; Hybertsen, M. S.; Neaton, J. B.; Venkataraman, L. Conductance 19



and Geometry of Pyridine-Linked Pingle-Molecule Junctions. J. Am. Chem. Soc. 2010, 132, 6817-6821. (13) Hsu, B. C.; Chen, Y. C. Band-Engineered Local Cooling in Nanoscale Junctions. Sci. Rep. 2017, 7, 42647. (14) Johnson, J. Thermal Agitation of Electricity in Conductors. Phys. Rev. 1928, 32, 97109. (15) Nyquist, H. Thermal Agitation of Electric Charge in Conductors. Phys. Rev. 1928, 32, 110-113. (16) Blanter, Y. M.; B¨ uttiker, M. Shot Noise in Mesoscopic Conductor. Phys. Rep. 1998, 336, 1-166. (17) Song, Y.; Lee, T. Electronic Noise Analyses on Organic Electronic Devices. J. Mater. Chem. C 2017, 5, 7123-7141. (18) Beenakker, C. W. J.; B¨ uttiker, M. Suppression of Shot Noise in Metallic Diffusive Conductors. Phys. Rev. B 1992, 46, 1889-1892 (19) B¨ uttiker, M.; Imry, Y.; Landauer, R.; Pinhas, S. Generalized Many-Channel Conductance Formula with Application to Small rings. Phys. Rev. B 1985, 31, 6207-6215. (20) Hsu, B. C.; Chiang, C. W.; Chen, Y. C. Effect of Electron-Vibration Interactions on the Thermoelectric Efficiency of Molecular Junctions. Nanotech. 2012, 23, 275401. (21) Taylor, J.; Guo, H.; Wang, J. Ab Initio Modeling of Quantum Transport Properties of Molecular Electronic Devices. Phys. Rev. B 2001, 63, 245407. (22) Baratz, A.; Galperin, M.; Baer, R. Gate-Induced Intramolecular Charge Transfer in a Tunnel Junction: A Nonequilibrium Analysis. J. Phys. Chem. C 2013, 117, 1025710263. 20


Page 20 of 23

Page 21 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(23) Wang, B.; Wang, J. Spin Polarized I-V Characteristics and Shot Noise of Pt Atomic Wires. Phys. Rev. B 2011, 84, 165401. (24) Li, D.; Zhang, L.; Xu, F.; Wang, J. Enhancement of Shot Noise due to the Fluctuation of Coulomb Interaction. Phys. Rev. B 2012, 85, 165402. (25) Yu, Z.; Wang, J. Transport Properties of WSe2 Nanotube Heterojunctions: A FirstPrinciples Study. Phys. Rev. B 2015, 91, 205431 (26) Avriller, R.; Frederiksen, T. Inelastic Shot Noise Characteristics of Nanoscale Junctions from First Principles. Phys. Rev. B 2012, 86, 155411. (27) Han, J. E. Mapping of Strongly Correlated Steady-State Nonequilibrium System to an Effective Equilibrium. Phys. Rev. B 2007, 75, 125122. (28) Haupt, F.; Novotn´ y , T.; Belzig, W. Current Noise in Molecular Junctions: Effects of the Electron-Phonon Interactions. emphPhys. Rev. B 2010, 82, 165441. (29) Anders, F. B. Steady-State Currents through Nanodevices: A Scattering-States Numerical Renormalization-Group Approach to Open Quantum Systems. Phys. Rev. Lett. 2008, 101, 066804. (30) Bulashenko, O. M.; Rub´ı, J. M. Shot Noise as a Tool to Probe an Electron Energy Distribution. Physica E 2002, 12, 857-860. (31) Landauer, R. The Noise is the Signal. Nature 1998, 392, 658-659. (32) Liu, R. C.; Odom, B.; Yamamoto, Y.; Tarucha, S. Quantum Interference in Electron Collision. Nature 1998, 391, 263-265. (33) Levinson, Y.; Imry, Y. 2000 Dephasing and Shot-Noise in Mesoscopic Systems. In: Kulik I.O., Ellialtio˘ g lu R. (eds) Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics. NATO Science Series (Series C: Mathematical and Physical Sciences), vol. 559. Springer, Dordrecht. 21



(34) Iannaccone, G.; Macucci, M.; Pellegrini, B. Theory of Conductance and Noise Additivity in Parallel Mesoscopic Conductors. Phys. Rev. B 1997, 56, 12104-12107. (35) Shimizu, A.; Ueda, M. Effects of Dephasing and Dissipation on Quantum Noise in Conductors. Phys. Rev. Lett. 1992, 69, 1403-1406. (36) Nagav, K. E. On the Shot Noise in Dirty Metal Contacts. Phys. Lett. A 1992, 169, 103-107. (37) Reznikov, M.; Heiblum, M.; Shtrikman, H.; Mahalu, D. Temporal Correlation of Electrons: Suppression of Shot Noise in a Ballistic Quantum Point Contact. Phys. Rev. Lett. 1995, 75, 3340-3343. (38) Agra¨ıt, N.; Yeyati, A. L.; van Ruitenbeek, J. M. Quantum Properties of Atomic-Sized Conductors. Phys. Rep. 2003, 377, 81-273. (39) Djukic, D.; van Ruitenbeek, J. M. Shot Noise Measurements on a Single Molecule. Nano Lett. 2006, 6, 789-793. (40) Chen, R.; Wheeler, P. J.; Di Ventra, M.; Natelson, D. Enhanced Noise at High Bias in Atomic-Scale Au Break Junctions. Sci. Rep. 2014, 4, 4221. (41) Kiguchi, M.; Tal, O.; Wohlthat, S.; Pauly, F.; Krieger, M.; Djukic, D.; Cuevas, J. C.; van Ruitenbeek, J. M. Highly Conductive Molecular Junctions Based on Direct Binding of Benzene to Platinum Electrodes. Phys. Rev. Lett. 2008, 101, 046801. (42) Karimi, M. A.; Bahoosh, S. G.; Herz, M.; Hayakawa, R.; Pauly, F.; Scheer, F. Shot Noise of 1,4-Benzenedithiol Single-Molecule Junctions. Nano Lett. 2016, 16, 1803-1807. (43) Chen, Y. C.; Di Ventra, M. Shot Noise in Nanoscale Conductors from First Principles. Phys. Rev. B 2003, 67, 153304. (44) Lagerqvist, J.; Chen, Y. C.; Di Ventra, M. Shot Noise in Parallel Wires. Nanotechnol. 2004, 15, S459-464 22


Page 22 of 23

Page 23 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(45) Liu, Y. S.; Chen, Y. C. Counting Statistics in Nanoscale Junctions. Phys. Rev. B 2011, 83, 035401. (46) Di Ventra M.; Lang, N. D. Transport in Nanoscale Conductors from First Principles. Phys. Rev. B 2001, 65, 045402. (47) Lang, N. D. Resistance of Atomic Wires. Phys. Rev. B 1995, 52, 5335. (48) Tsutsui, M.; Teramae, Y.; Kurokawa, S.; Sakai, A. High-Conductance States of Single Benzenedithiol Molecules. Appl. Phys. Lett. 2006, 89, 163111. (49) Sergueev, N.; Tsetseris, L.; Varga, K.; Pantelides, S. Configuration and Conductance Evolution of Benzene-Dithiol Molecular Junctions under Elongation. Phys. Rev. B 2010, 82, 073106. (50) Pontes, R. B.; Rocha, A. R.; Sanvito, S.; Fazzio, A.; da Silva, A. J. R. Ab Initio Calculations of Structural Evolution and Conductance of Benzene-1,4-dithiol on Gold Leads. ACS Nano 2011, 5, 795804. (51) Martin, C. A.; Ding, D.; van der Zant, H. S. J.; van Ruitenbeek, J. M. Lithographic Mechanical Break Junctions for Single Molecule Measurements in Vacuum: Possibilities and Limitations. New J. Phys. 2008, 10, 065008. (52) Taniguchi, M.; Tsutsui, M.; Yokota, K.; Kawai, T. Inelastic Electron Tunneling Spectroscopy of Single Molecule Junctions using a Mechanically Controllable Break Junction. Nanotechnology 2009, 20, 434008. (53) Haiss, W.; Wang, C. S.; Jitchati, R.; Grace, I.; Martin, S.; Batsanov, A. S.; Higgins, S. J.; Bryce, M. R.; Lambert, C. J.; Jensen, P. S.; Nichols, R. J. Variable Contact Gap Single-Molecule Conductance Determination for a Series of Conjugated Molecular Bridges. J. Phys.: Condens. Matter 2008, 20, 374119.

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Quantum Noise of Current in a 1,4-Benzenedithiol Single-Molecule

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