Tunnel current through a redox molecule coupled to classical phonon ...

PHYSICAL REVIEW B 76, 125312 共2007兲

Tunnel current through a redox molecule coupled to classical phonon modes in the strong tunneling limit I. G. Medvedev A. N. Frumkin Institute of Physical Chemistry and Electrochemistry of Russian Academy of Sciences, Leninsky Prospect 31, 119991, Moscow, Russia 共Received 17 April 2007; revised manuscript received 21 June 2007; published 18 September 2007兲 The tunnel current through a redox molecule coupled to classical phonon modes is calculated in the strong tunneling limit in the Born-Oppenhiemer approximation. The effective “potential energy” of the phonon modes is introduced in the steady state of the in situ scanning tunneling microscope system. The expression for the tunnel current is obtained by using thermal averaging over this potential energy calculated within the spinless model in the wideband approximation. The dependence of the tunnel current on the width of the valence level of the redox molecule, the bias voltage, and the overvoltage is studied and discussed. The results of calculations of the tunnel current and the spectral density are used for interpretation of the experimental data obtained by Tao 关Phys. Rev. Lett. 76, 4066 共1996兲兴 and Visoly-Fisher et al. 关Proc. Natt. Acad. Sci. U.S.A. 103, 8686 共2006兲兴. DOI: 10.1103/PhysRevB.76.125312

PACS number共s兲: 73.23.⫺b, 71.38.⫺k, 81.07.Nb, 85.65.⫹h

I. INTRODUCTION

The study of electron tunneling through a redox molecule placed into the polar solvent between an electrode and a scanning tunneling microscope 共STM兲 tip 共in situ STM of redox molecules兲 is of great importance both for the development of molecular devices1–4 and for the understanding of the photosynthesis process in biology.5 One of the important features of this system is that in the oxidized and reduced states the valence level of the molecule is located above and below the energy window of the STM 共the energy gap between the electrode and tip Fermi levels兲, respectively. Therefore, as in the usual mesoscopic systems with strong electron-phonon coupling,6–8 the electron transport through the redox molecule is assisted by phonon modes associated with the motion of the solvent nuclei in this case. The coupling of the valence electron to the thermally activated nuclear motion 共the fluctuations of the solvent polarization兲 brings the molecular valence level into the energy window of the STM where electron transport through the molecule becomes possible.9 This results in the appearance of the FrankCondon factor in the expression for the tunnel current. This process, analogous to the Frank-Condon process encountered in electron transfer reactions, was revealed first for the case of STM of redox molecules10,11 and then rediscovered for usual mesoscopic systems6,12–15 as the Frank-Condon blocade. Another important feature of the in situ STM of redox molecules is the possibility of controlling the tunnel current using two independent parameters: the bias voltage Vb 共the difference of potentials between the electrode and the tip of the STM兲 as the source-drain voltage and the overvoltage ␩ 共the electrode potential vs a reference electrode inserted in the polar solvent兲 as the gate voltage.1,9–11 There is a large amount of work in the theory of electron tunneling through a single level with coupling to phonon modes. The first works16,17 in this field present the exact solution for the model when the presence of the Fermi seas of the leads is ignored.18–20 Two limits are the most suitable for study of the tunneling problem in the general case. They 1098-0121/2007/76共12兲/125312共10兲

are the weak tunneling limit 共the limit of weak coupling of the molecule to the leads兲 and the strong tunneling one. The kinetic equations can be used for the weak tunneling case.9,10,12–14,20,21 The study of the strong tunneling limit is based mainly on an approximate equations of motion.19–22 In the present paper, we study the strong tunneling limit using the Born-Oppenheimer approximation 共BOA兲 which permits the electron and phonon degrees of freedom to be separated and is valid in the considered case under the conditions obtained below. The BOA was applied in Refs. 23 and 24 to equilibrium systems for study of the image force effect on chemisorption and polaronic effects in mixedvalence compounds. We apply it to a nonequilibrium system and show that an effective “potential energy” can be introduced for the phonon subsystem in the steady state. The condition for applicability of the BOA is presented in the next section. The standard BOA considered in the present paper differs from its variant used in Ref. 25 as discussed in the next section. The expression for the effective potential energy for the classical phonon subsystem at zero temperature was obtained earlier in Ref. 26 using adiabatic switching on of the coupling of the redox molecule with the leads and equations of motion for the electron creation and annihilation operators in the Heisenberg representation. In Ref. 27, the Keldysh nonequilibrium Green function method was used for direct calculation of the nonequilibrium steady-state energy of the electronic subsystem which represents part of the zerotemperature effective potential energy. The zero-temperature effective actions, which take into account the quantum effects in the phonon subsystem, were obtained in Refs. 28 and 29. These actions yield the same effective potential energy for the classical phonon subsystem as that obtained in Refs. 29 and 27. In the next section a simpler and more direct method of calculation of the effective potential energy is presented, which is valid for arbitrary temperatures. For most polar solvents the frequencies of the phonon modes ប␻ corresponding to the Debye region of dielectric losses are much smaller than kBT ⬇ 0.025 eV at room temperatures 共e.g., ប␻ ⬃ 10−4 eV for water兲. Here kB is the Bolt-

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©2007 The American Physical Society


I. G. MEDVEDEV

zmann constant and T is the temperature. It should be noted that some intramolecular vibrations also have energies of the order of 0.001– 0.01 eV. Therefore, the fluctuations of the solvent polarization and these intramolecular vibrations can be treated in the classical limit at room temperatures so that an expression for the tunnel current can be suggested based on thermal averaging of the tunnel current calculated at fixed values of phonon coordinates over the effective potential energy. The same region of physical parameters of the STM system has been considered in a number of works.9,11,25,30–32 However, the theory of electron tunneling through a redox molecule coupled to classical phonon modes in the strong tunneling limit cannot be considered as complete. For example, the effective potential energy of the phonon subsystem for the nonequilibrium case was not obtained in Ref. 25 so that the problem of calculation of the tunnel current using thermal averaging over the effective potential energy was not considered. The method of Refs. 11 and 32 neglects the presence of the Fermi seas of the leads and, therefore, takes into account only the oxidized 共empty兲 state of the redox molecule, as is discussed in detail in the next section. Although the strong tunneling limit is considered in Ref. 9, the dependence of the tunnel current on the total strength ⌬ of the coupling of the redox molecule with the leads is neglected in the small-bias-voltage limit. In the large-biasvoltage limit30 the coupling strength ⌬ is taken into account only during the estimation of a steady-state average valence level population na共兵qk其兲 where qk are the coordinates of the normal modes describing the slow fluctuations of the solvent polarization. However, the exact dependence of na共兵qk其兲 on qk that can be obtained in the strong tunneling limit is ignored in Ref. 30 and na共兵qk其兲 is taken to be a constant in some characteristic regions. A simplified effective potential energy of the phonon subsystem depending on this stepwise function na共兵qk其兲 is used for calculation of the tunnel current. An interpolation formula for the current incorporating the interpolation parameter is used for description of the intermediate overvoltage and bias voltage ranges. The full dependence of the average occupation na共兵qk其兲 on qk in the strong tunneling limit is taken into account in Ref. 31. However, the effective potential energy of the classical phonon subsystem for the nonequilibrium case was calculated in Ref. 31 using the integration of na over ␧a without any justification of this procedure. The tunnel current was calculated in Ref. 31 through simulation of the stochastic motion of the classical phonon subsystem on the effective potential energy surface. Unlike Ref. 31, the present paper obtains an explicit expression for the tunnel current as the statistical average over the equilibrium distribution of the phonon coordinates on the effective potential energy. The obvious advantage of this approach is the possibility of analytical study of the dependence of the tunnel current on the parameters of the system and the derivation of the equations for other physical characteristics of the system related to the tunnel current. The use of an explicit expression for the tunnel current also permits us to obtain approximate expressions for different particular cases. So, in Sec. II the dependence of the tunnel current j on

the overvoltage ␩ is studied for all values of the parameters within the parameter region under consideration. In this section the expression for the spectral function in the strong tunneling limit is derived. In Sec. III the approximate expressions for the tunnel current, the spectral density, and the width of the curve j共␩兲 are obtained in the small- and largebias-voltage limits. The results of the numerical calculations of the tunnel current and the spectral density are presented in Sec. IV and used for interpretation of the experimental data obtained in Refs. 1 and 5. II. MODEL AND EXPRESSIONS FOR THE CURRENT AND THE SPECTRAL DENSITY

For the sake of simplicity, we neglect the spin degrees of freedom and use the wideband approximation for the leads. As a result, the Hamiltonian of the system has the usual form16,17 H = Hel +

1 兺 ប␻k共p2k + q2k 兲 2 k

共1兲

where Hel is the effective electronic Hamiltonian in the BOA: * † Hel = 兺 ␧mnm + ␧a共qk兲n + 兺共Vamc†acm + Vam cmca兲, 共2兲 m

m

the subscript m in the right-hand side 共RHS兲 of Eq. 共2兲 is either k or p, and ␧k and ␧ p are the electronic energies of the electrode and the tip quasiparticle states 兩k典 and 兩p典, respec† cm is the occupation number operator for these tively. nm = cm states. ␧a共qk兲 and n = c†aca are the energy and the occupation number operator of the valence orbital 兩a典 of the redox molecule. The third term in the RHS of Eq. 共2兲 describes the coupling between the electronic states of the electrode and the tip of the STM with the redox orbital with Vam as the coupling constants. The second term in the RHS of Eq. 共1兲 is the Hamiltonian of the phonon subsystem where pk and qk are the dimensionless momenta and coordinates of the slow solvent modes, and ␻k are the effective frequencies corresponding to the normal modes qk. The well-known parameter ␭ = 0.5兺k␥2k / ប␻k is called here the reorganization energy.1,9–11 The electronic energy of the valence orbital in the BOA has the form9 ␧a共qk兲 = ␧0a − 兺 ␥kqk + e␸ + ␥eVb .

共3兲

k

Here ␧0a is the bare electron energy of the valence orbital,␥k are the coupling constants describing the interaction of the valence orbital with the solvent polarization, e is the absolute value of the electron charge, and ␸ is the difference of the potentials in the bulk of the electrode and at the redox molecule site. The last term in the RHS of Eq. 共3兲 is the shift of the redox level caused by the bias voltage where the parameter ␥, 0 ⬍ ␥ ⬍ 1, describes the fraction that bias voltage contributes to this shift. For example, ␥ ⬇ 1 / 2 for the symmetric potential distribution in the tunnel gap in the case when the redox molecule is placed in the middle between the electrode

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TUNNEL CURRENT THROUGH A REDOX MOLECULE…

and the tip of the STM and ␥ ⫽ 1 / 2 for the spatially nonsymmetric case. As in Ref. 17, we use the wideband approximation for the electronic structures of the electrode and the tip of STM, so that the coupling of the electronic levels of the electrode and the tip with the redox orbital can be characterized by only and ⌬tip two parameters ⌬el = ␲兺k兩Vak兩2␦共␧ − ␧k兲 2 = ␲兺 p兩Vap兩 ␦共␧ − ␧ p兲, which are independent of energy. The total coupling strength ⌬ equals ⌬el + ⌬tip. The parameter ⌬ has also the physical meaning of the half-width of the redox level appearing due to the coupling of this level with the electrode and the tip. The electrode and the tip of the STM are in thermal equilibrium at a given temperature T and have their own Fermi distribution functions f i共␧兲, which are characterized by the nonperturbed chemical potentials ␮i, so that eVb = ␮tip − ␮el 共i = el or tip兲. The characteristic time of the electron subsystem is ␶e = ប / ⌬ in the case under consideration. Apart from its usual interpretation, this time has also a physical meaning of the time required for the transition to the new steady state in the case when the molecular valence level is shifted from a given fixed position to another one. It is assumed below that ␭ and 兩eVb兩 Ⰷ kBT. In order that the BOA can be used, the time ␶e should be shorter than ␶v, where ␶v is the time of the shift of the valence level by kBT due to the thermal fluctuations of the phonon coordinates. It is due to the fact that the energy 兩␧a兩 is of the order of ␭ or 兩eVb兩 for almost all values of qk, so that the electron subsystem does not “feel” the shift of ␧a by We obtain that 兩d␧a共␶兲 / d␶兩 = 兺␥k兩dqk / d␶兩 kBT. ⬃ 兺共␥2k ␻kkBT / ប兲1/2 = ␻av共kBT兲1/2 兺共␥2k / ប␻k兲1/2, where ␻av is the average value of the phonon frequencies and the last sum over k is of the order of 共2␭兲1/2. Then ␶v = 共kBT / 2␭兲1/2 / ␻av, and the condition of applicability of the BOA is ⌬ 艌 ប␻av共2␭ / kBT兲1/2, where ␻av Ⰶ kBT. The opposite inequality ⌬ Ⰶ ប␻av共2␭ / kBT兲1/2 can be considered as the condition for the weak tunneling limit. The value of ␭ for the case of the STM of a redox molecule can be estimated as 0.2– 0.5 eV.5 If we adopt a very rigid estimation ប␻av ⬇ kBT / 5 for the classical phonon modes in order to take into account the intramolecular vibrations, we obtain that ⌬ ⬎ 0.03 eV⬃ kBT. As a result, under this condition, the electronic properties of the system can be calculated at fixed values of qk. Using the equations of motion for the operators pk共␶兲 and qk共␶兲 in the Heisenberg representation, it can be shown that d2qk共␶兲/d␶2 = − ␻2k qk共␶兲 + ␻k␥kn共␶兲/ប,

共4兲

where n共␶兲 is the occupation number operator of the valence orbital in the Heisenberg representation. It follows from Eqs. 共1兲 and 共4兲 that the operator Fk共␶兲 = ␥kn共␶兲 is the operator of the “force” acting on the phonon mode k due to its interaction with the valence electron. In the spirit of the BOA, let us perform quantum-statistical averaging of both sides of Eq. 共4兲 over the electronic states of the steady state of the STM system considering at fixed values of qk. An average timeindependent number of electrons na(␧a共qk兲 , T) = 具n共␶兲典 in the valence orbital can be calculated for the steady state using the Fourier transform of the corresponding lesser Green ⬍ 共␧兲. Therefore, a well-defined time-independent function Gaa

value of the average force Fk共qk , T兲 = 具Fk共␶兲典 = ␥kna(␧a共qk兲 , T) = −⳵⌽e共qk , T兲 / ⳵qk also exists, where ⌽e共qk , T兲 is the part of the effective potential energy of the phonon modes due to their interaction with the valence electron. It can be easily shown that the total effective potential energy ⌽共qk , T兲 of the phonon modes is given by the expression 1 兺 ប␻kq2k + 2 k

⌽共qk,T兲 =

冕

␧a共qk兲

na共␧a⬘,T兲d␧a⬘ .

共5兲

0

In equilibrium, the second term in the RHS of Eq. 共5兲 becomes the qk-dependent free energy of the electronic subsystem. Our derivation of the effective potential ⌽共qk , T兲 seems to be simple and straightforward, and differs from that of Refs. 28 and 29. The latter is based on the nonequilibrium Keldysh path integral technique. The effective action for the phonon subsystem obtained in Ref. 28 and the more general one of Ref. 29 also take into account quantum effects in the phonon subsystem. The effective classical potential of the phonon modes at T = 0 is obtained from these effective actions in the limit when the quantum effects in the phonon subsystem are ignored. We mention also that, in the single-phonon mode limit, Eq. 共5兲 differs from Eq. 共7兲 of Ref. 33. We assume below that in the steady state the phonon subsystem is in equilibrium at a given T in the external static field defined by the effective potential ⌽共qk , T兲. Then the tunnel steady-state current j can be obtained as the thermal average of the steady-state current j共qk , T兲, calculated at fixed values of qk, over the effective potential energy ⌽共qk , T兲:

冕兿冕兿

dqk j共qk,T兲exp关− ⌽共qk,T兲/kBT兴

j=

k

.

共6兲

dqk exp关− ⌽共qk,T兲/kBT兴

k

Equations 共5兲 and 共6兲 are valid in the BOA for arbitrary forms of the effective electronic Hamiltonian Hel including the case when the Coulomb interaction between electrons in the valence orbital of the bridge molecule is taken into account. However, the condition of applicability of the BOA can differ from that obtained above. In particular, for the spinless model, we have 共see, e.g., Ref. 18兲 tip na共␧a,T兲 = nel a 共␧a,T兲 + na 共␧a,T兲,

=

j共qk,T兲 =

⌬i ␲

冕

⬁

−⬁

nia共␧a,T兲

f i共␧兲d␧ , 关␧ − ␧a共qk兲兴2 + ⌬2

2e⌬el⌬tip ␲ប

冕

⬁

−⬁

关f tip共␧兲 − f el共␧兲兴d␧ . 关␧ − ␧a共qk兲兴2 + ⌬2

共7兲

共8兲

The calculations are simplified if we replace the Fermi distribution functions f i by the step functions in Eqs. 共7兲 and 共8兲. It can be shown that the errors are of the order of 共␲ / 3兲⌬i共kBT兲2共␧a − ␮i兲 / 关共␧a − ␮i兲2 + ⌬2兴2 and

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I. G. MEDVEDEV

共␲ / 6兲共kBT兲2兺i⌬i / 关共␧a − ␮i兲2 + ⌬2兴 for nia and the absolute value of the qk-dependent part of ⌽e共qk , T兲, respectively. For the tunnel current the temperature correction is of the order of eVb共kBT兲2⌬ / 关共eVb / 2兲2 + ⌬2兴2 in the case when the valence level lies in the middle of the energy window. Therefore, it is sufficient to take ⌬ 艌 0.1 eV or eVb 艌 0.1 eV in order to neglect the temperature corrections. Then nel a 共␧a兲 =

ntip a 共␧a兲 =

冉

冊

⌬el 1 1 ␧a共qk兲 , − arctan ⌬ 2 ␲ ⌬

冉

j=

2e⌬el⌬tip ប⌬

冉

共10兲

is the dimensionless effective coordinate. Performing the integration over ␰ j in Eqs. 共6兲 and 共13兲, one obtains that

冊

j=

The last term in the RHS of Eq. 共5兲 takes the form

k

⌬el + ln共␧2a + ⌬2兲 + 共␧a − eVb兲ntip a 共␧a兲 2␲

A共␧兲 =

共12兲

dqk兵关␧ − ␧a共qk兲兴2 + ⌬2其−1 exp关− ⌽共qk兲/kBT兴

冕兿

共16兲

⬁

dq exp关− U共q兲/kBT兴

⌬ ␲

冕

⬁

−⬁

dq兵关␧ − ␧a共q兲兴2 + ⌬2其−1 exp关− U共q兲/kBT兴

冕

.

⬁

dq exp关− U共q兲/kBT兴

−⬁

共17兲 Here U共q兲 = ␭q2 + ⌽e共q兲

共13兲

共18兲

and ⌽e共q兲 and j共q兲 are expressed in terms of ␧a共q兲, where ␧a共q兲 = ␧0a − 2␭q + e␸ + ␥eVb. Using Eqs. 共8兲 and 共18兲, it can be shown that the tunnel current j tends to the maximum value jmax = 2e⌬el⌬tip / ប⌬ in the case when Vb → ⬁ and 0 ⬍ ␥ ⬍ 1 i.e., for eVb Ⰷ ␭. It is well known that the potential energy U共q兲 can have two potential wells at Vb = 0.24,34 Moreover, U共q兲 can also have three potential wells in some regions of the parameter space in the case27,28 when Vb ⫽ 0 共see, e.g., Fig. 2 of Ref. 27兲. If for Vb = 0 and the spinless model the potential energy U共q兲 has two potential wells where na ⬇ 0 and na ⬇ 1, then the depths of these wells are the same at e␸ = e␸0 = ␭ − ␧0a. Therefore, the overvoltage ␩ = ␸ − ␸0 can be defined in the case when the potential at the redox molecule site coincides with that in the bulk of the solvent. The sum ␧0a + e␸ in the RHS of Eq. 共3兲 equals ␭ + e␩ in this case. However, the potential at the redox molecule site does not coincide with that in the bulk of the solvent in general, so that only the part ␰␩, 0 ⬍ ␰ Ⰶ 1, of the overvoltage leads to the shift of the redox level.9 As a result, Eq. 共3兲 can be rewritten in terms of the overvoltage in the following form: ␧a共q兲 = ␭共1 − 2q兲 + e␩V + eVb/2,

dqk exp关− ⌽共qk兲/kBT兴

k

so that36

dq j共q兲exp关− U共q兲/kBT兴

−⬁

−⬁

and

It was shown in Ref. 27 using the Keldysh nonequilibrium Green function method that the potential energy given by Eq. 共12兲 coincides up to the qk-independent term with the lowest time-independent electronic energy of the system in the steady state at fixed qk. Let us compare the standard BOA considered in the present paper and its variant used in Ref. 25. Unlike the standard BOA, only the part 兺k␥kqkna 共in our notation兲 of the full effective potential is used in Eq. 共2兲 of Ref. 25 for the description of the coupling of phonons with electrons. The expression 共11兲 of Ref. 25 for ␧a includes the term −2␭na instead of −兺k␥kqk entering the RHS of Eq. 共3兲 of the present paper. However, it is obvious that the value of the last term differs considerably from na共qk兲 and coincides with it only at stable points 共the positions of the potential wells and the transition points of the effective potential energy兲. The expression for the electronic energy obtained in Ref. 25 differs from Eq. 共12兲 due to a number of approximations made in Ref. 25. We also use below the spectral function35,36 A共␧兲 = 共r兲共r兲共␧兲 / ␲ 关where Gaa 共␧兲 is the retarded Green function兴 −ImGaa which is normalized to unity and for the spinless model has the form

冕兿

冕冕 ⬁

共11兲

⌬ A共␧兲 = ␲

共15兲

k

冊

⌬tip + ln关共␧a − eVb兲2 + ⌬2兴. 2␲

共14兲

that the energy of the valence through the linear combination use the new orthogonal set of instead of the original set 兵qk其,

q = 兺 ␥kqk/2␭

2e⌬el⌬tip eVb − ␧a共qk兲 ␧a共qk兲 + arctan arctan . j共qk兲 = ␲ប⌬ ⌬ ⌬

⌽e共qk兲 =

关f tip共␧兲 − f el共␧兲兴A共␧兲d␧.

−⬁

It follows from Eq. 共3兲 orbital depends on qk only 兺k␥kqk. Therefore, we can classical coordinates 兵q , ␰ j其 where

共9兲

␧a共qk兲 − eVb ⌬tip 1 1 , − arctan ⌬ 2 ␲ ⌬

␧anel a 共␧a兲

冕

⬁

共19兲

where ␩V = ␰␩ + 共␥ − 1 / 2兲Vb. The tunnel current and the spectral function given by Eqs. 共16兲 and 共17兲 are functions of ⌬i, Vb, and ␩V. The dependence of j on ␩ at fixed Vb is often studied in experimental

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work. Using Eq. 共16兲 it can be shown that the current j共␩V兲 has a maximum at ␩V = 0 in the case when ⌬el = ⌬tip. This result, obtained first in Ref. 9 for the case of small values of eVb, e␩, and ⌬ Ⰶ ␭ is generalized here to arbitrary values of these parameters with a single restriction on ⌬, which should satisfy the condition of the applicability of BOA. It is instructive to present here the expression for the spectral function in the weak tunneling limit. In this case A共␧兲 has a simple form and is a linear combination of the spectral functions of the oxidized and reduced states of the redox molecule with coefficients which can be found using, e.g., kinetic equations. The spectral function for the oxidized state

A共␧⬘兲 ⬇

冉

1 1 2 ␲␭kBT

冊冉 1/2

is presented, e.g., in Ref. 35. The spectral function for the reduced state can be easily found using the same arguments as in Ref. 35. We can also calculate A共␧兲 for ⌬el = ⌬tip using Eq. 共17兲 in the limit ⌬ → 0 because, as in the BOA, the weak tunneling limit implies that the electron tunneling is not taken into account during the calculation of the spectral function 共the latter, however, can depend on ⌬i / ⌬兲. The phonon subsystem “sees” only the fully empty or fully filled valence levels in this case. Substituting ␲␦(␧ − ␧a共q兲) / ⌬ for 兵⌬2 + 关␧ − ␧a共q兲兴2其−1 and exp关−Uox共q兲 / kBT兴 + exp关−Ured共q兲 / kBT兴 for exp关−U共q兲 / kBT兴, where Uox共q兲 = ␭q2 and Ured共q兲 = ␭共1 − q兲2 + e␩V, one obtains

冊

exp关− 共␧⬘ − e␩V − ␭兲2/共4␭kBT兲兴 + exp关− 共␧⬘ − e␩V + ␭兲2/共4␭kBT兲兴exp共− e␩V/kBT兲 , 1 + exp共− e␩V/kBT兲

where ␧⬘ = ␧ − eVb / 2. The method of calculation of the tunnel current and the spectral density presented by Eqs. 共6兲 and 共13兲, respectively, differs from that of Refs. 11 and 32. The latter uses the exact solution for the tunnel current obtained in Refs. 16 and 17 within the model of the leads having no Fermi seas of electrons 共see, for example, the discussion of this model in Refs. 18–20兲. Since the exact solution is not possible in the general case,18 the results of Refs. 16 and 17 are of importance because they consist of a complete analytical description of the electron tunneling for the model system. However, for the problem of the calculation of the tunnel current through the redox molecule, the presence of the Fermi seas cannot be ignored since the effect of the reduced form of the bridge molecule, which is neglected in Refs. 11, 16, 17, and 32, should be taken into account. Indeed, the spectral function A共␧兲 entering Eq. 共14兲 for the tunnel current characterizes the background where an electron described by the lesser Green ⬍ 共␧兲 propagates. Within the model of Refs. 16 function Gaa ⬍ and 17, the lesser Green function Gaa 共␧兲 describes the propagation of a single electron in the system consisting of the leads and the redox molecule. It implies that, in this case, the spectral function A共␧兲 characterizes the empty system having no electrons at all, so that, as in the weak tunneling limit discussed above, the electron tunneling has no influence on A共␧兲. Therefore, there is no essential difference between the weak tunneling, intermediate tunneling, and strong tunneling regimes within the theory of Refs. 16 and 17. Thus, as in the strong tunneling limit but for all values of ⌬, the spectral function A共␧兲 of Refs. 16 and 17 for the case of the classical phonon modes is given by Eq. 共13兲, where, however, the thermal averaging should be performed only over the free phonon modes since the phonon subsystem “sees” the empty valence level. This means that the method of Refs. 11, 16, 17, and 32 can take into account only the oxidized state of the redox molecule so that, e.g., only the first term in the numerator in

共20兲

the large parentheses in the RHS of Eq. 共20兲 is used for calculations of the tunnel current in the weak tunneling limit. However, it is obvious that both terms give a contribution of the same order to the tunnel current in the general case. In particular, substituting step functions for the Fermi distribution functions in Eq. 共14兲 and using Eq. 共20兲, one obtains the simple expression j⬇

冉冊

4e⌬el⌬tip kBT ប⌬ ␲␭

1/2

exp关− ␭/共4kBT兲兴

sinh关eVb/共4kBT兲兴 cosh关e␩V/共2kBT兲兴共21兲

for the tunnel current in the weak tunneling limit for 共eVb ± 2e␩V兲2 Ⰶ 16␭kBT, which takes into account both oxidized and reduced states of the redox molecule and does not follow directly from the equations of Ref. 11. In the strong tunneling limit considered in the present paper, the current j共qk兲 is averaged in Ref. 11 only over the free phonon modes, i.e., only the first term in the RHS of Eq. 共5兲 is taken into account in Eq. 共6兲 because the second term equals zero for a system with no electrons. III. APPROXIMATE EXPRESSIONS FOR THE CURRENT IN THE STRONG TUNNELING LIMIT FOR DIFFERENT PARTICULAR CASES

To understand the behavior of the current in the strong tunneling limit, it is worthwhile to consider a number of different particular cases. For ␭ 艋 0.5 eV, the main contribution to the total tunnel current given by Eq. 共16兲 arises from values of q which correspond to the location of the valence level within or near the energy window of STM 共resonance electron tunneling兲. Let us consider the small-⌬ region for the strong tunneling limt 共i.e., ␭ and 兩eVb兩 Ⰷ ⌬兲 for ␭ 艋 0.5 eV in the case when the BOA is still valid. Then one obtains from Eqs. 共17兲 and 共19兲 that

125312-5


I. G. MEDVEDEV

A共␧兲 ⬇

1 2共␲kBT␭兲1/2 ⫻

A共␧兲 ⬇

exp兵− U关− 0.5共␧ − ␭ − e␩V − eVb/2兲/␭兴/共kBT兲其 . 1 + exp共− e␩V/kBT兲

Here, during the calculation of the denominator, we take into account that U共q兲 = Uox共q兲 for q ⬍ q* and U共q兲 = Ured共q兲 for q ⬎ q*, respectively, in the small-⌬ limit, where q* = 0.5共␭ + e␩V兲 / ␭ is the position of the top of the potential barrier. The conditions ␭ Ⰷ 兩eVb兩 , 兩e␩V兩 , kBT were also used. In contrast to the weak tunneling limit, the potential energy U共q兲 is a smooth function of q for Vb ⫽ 0 near the top of the potential barrier even for small values of ⌬ ⬃ kBT. It can be shown that for ⌬el = ⌬tip, 兩eVb兩 Ⰷ ⌬ ⬃ kBT, 兩eVb兩 ⬎ 2兩e␩V兩, and 兩eVb ± 2e␩V兩 Ⰶ ␭ the top of the potential barrier lies at q* ⬇ 1 / 2 and

and

j = jmax

near the top of the potential barrier, where ␹c = −⳵na / ⳵␧a is the charge susceptibility and 2␭␹c共q*兲 − 1 ⬎ 0 at the top of the potential barrier. Using Eqs. 共14兲, 共22兲, and 共23兲, it can be shown that, similar to Eq. 共21兲, there is a simple expression for the tunnel current also in the strong tunneling limit for 共eVb ± 2e␩V兲2 Ⰶ 16␭kBT 共the small-bias-voltage region兲 and under the conditions presented above: 1/2

eVb exp关− 共␭ − 兩eVb兩兲/共4kBT兲兴 . cosh关e␩V/共2kBT兲兴 k BT 共24兲

It follows from Eq. 共24兲 that the tunnel current is proportional to ⌬ and Vb in this case.9,37 Equation 共24兲 simulates rather well the results of numerical calculations of the current obtained using Eq. 共16兲 for, e.g., ␭ = 0.2 eV, ⌬ ⬃ kBT, and Vb ⬃ 0.1 V, and describes the behavior of the current qualitatively for ␭ = 0.5 eV and small values of ⌬. It is obvious from Eq. 共24兲 as well as from Eq. 共21兲 that the tunnel current has a maximum at ␩V = 0, as it should. It can also be readily shown from these equations that the width W at half maximum of the curve j共␩V兲 equals 4kBT ln共2 + 31/2兲 / e ⬇ 5.2kBT / e = 0.13 V, and is independent of ␭ and Vb in the small-bias-voltage region. The absolute value of the exponent in Eq. 共24兲 describes the FrankCondon barrier which decreases with the increase of 兩Vb兩. If 兩Vb兩 increases and becomes larger than 2␭关1 − 2共⌬ / ␲␭兲1/2兴 for small values of ⌬ and 兩e␩V兩, the FrankCondon barrier disappears, so that the potential energy U共q兲 has now a single potential well which corresponds to the adsorbed molecule having a certain average number ns = qs ⬇ 1 / 2 of electrons at the valence orbital. Since ␭␹c共q*兲 Ⰶ 1 in this case, Eq. 共23兲 becomes U共q兲 ⬇ U0 + ␭共q − ns兲2 in the adsorption regime so that

冕

共25兲

共1−␥兲eVb

−␥eVb

exp兵− 关␧ − ␭共1 − 2ns兲共27兲

Equation 共27兲 shows explicitly that j tends to jmax in the case when Vb / ␭ → ⬁ and 0 ⬍ ␥ ⬍ 1. It follows from Eq. 共27兲 that j = jmax sgn共Vb兲关1 − f„eVb/2 + ␭共1 − 2ns兲 + e␩V… − f„eVb/2 − ␭共1 − 2ns兲 − e␩V…兴

共23兲

冉冊

1 1 2 共␲kBT␭兲1/2

− e␰␩兴2/共4␭kBT兲其d␧.

U共q兲 ⬇ 共␭ + 2e␩V − 兩eVb兩兲/4 − ␭关2␭␹c共q*兲 − 1兴共q − 1/2兲2

e⌬el⌬tip kBT 2ប⌬ ␲␭

共26兲

− eVb/2兴2/共4␭kBT兲其

共22兲

j⬇

1 exp兵− 关␧ − ␭共1 − 2ns兲 − e␩V 2共␲kBT␭兲1/2

共28兲

for 兩eVb / 2 ± 关␭共1 − 2ns兲 + e␩V兴兩 Ⰶ 2共␭kBT兲1/2 where f共x兲 = 共␭kBT兲1/2 exp关−x2 / 共4␭kBT兲兴 / x. The current j共␩V兲 given by Eq. 共28兲 takes its maximum value j共0兲 = jmax兵1 − 4共␭kBT兲1/2 exp关−共eVb兲2 / 共16␭kBT兲兴 / eVb其 at ␩V = 0. The small-⌬ limit in the case ␭ 艋 0.5 eV means that the current j共q兲 equals approximately jmax for 0.5共␭ + e␩V − eVb / 2兲 / ␭ 艋 q 艋 0.5共␭ + e␩V + eVb / 2兲 / ␭ 关i.e., for 0 艋 ␧a共q兲艋 eVb兴 and zero otherwise. However, when the intramolecular classical modes are also coupled to the valence level, the reorganization energy ␭ becomes larger than 0.5 eV. The Frank-Condon barrier is too large in this case so that the major contribution to the tunnel current in Eq. 共16兲 in the small-⌬ limit for eVb Ⰶ ␭ is given by the neighborhoods of the coordinates of the potential wells of the potential energy U共q兲 for ␩V ⬇ 0. The corresponding boundary value of ␭ can be estimated from the relation j共qox兲exp关−U共qox兲 / kBT兴 = j共q*兲exp关−U共q*兲 / kBT兴, where qox ⬇ ⌬共1 − e␩V / ␭兲 / 共␲␭兲 is the coordinate of the potential well corresponding to the oxidized state of the redox molecule, U共qox兲 ⬇ ⌬ / ␲, and U共q*兲 can be calculated using Eq. 共23兲. For example, it can be shown that the contribution to the total tunnel current produced by the values of q corresponding to the location of the valence level within or near the energy window of STM is very small for ␭ = 1 eV, Vb ⬃ 0.1 V, and small values of ⌬. For large positive values of ␩V the major contribution to the tunnel current in Eq. 共16兲 is given only by the neighborhood of the coordinate qox. For ␩V ⬇ 0 or for the opposite case of large positive values of ␩V, the tunnel current can be calculated approximately using the unified expression j⬇

冉冊冕 ␭ ␲ k BT

1/2

⬁

j共q兲exp关− ␭共q − qox兲2/kBT兴dq. 共29兲

−⬁

Substituting into the RHS of Eq. 共29兲 the expansion of j共q兲 in series in q − qox and e␩V for e␩V, ⌬ Ⰶ ␭, up to the second order in q − qox and the first order in e␩V, one obtains that

125312-6



j⬇

冋

冉

冊

册

4⌬ 2e␩V 8⌬ 6kBT jmax eVb⌬ 1+ − 1+ + . 2 ␲ ␭ ␭␲ ␭ ␭␲ ␭ 共30兲

1 0.10

It should be noted that the current given by Eq. 共30兲 is proportional to ⌬el⌬tip, so that off-resonance electron tunneling takes place in this case. It follows from Eq. 共30兲 that W⬇

冉

4⌬ 6kBT ␭ 1− + 2e ␭␲ ␭

冊

共31兲

so that the width of the curve j共␩V兲 equals approximately half of the reorganization energy in the large-␭ region. Equation 共31兲 shows that the width decreases with the increase of ⌬ due to the decrease of the Frank-Condon barrier and, therefore, the increase of the contribution to the current from the values of q corresponding to the location of the valence level within or near the energy window of STM. Although the RHS of Eq. 共31兲 is independent of Vb, the width calculated with the use of Eq. 共16兲 decreases with increase of Vb for the same reason. That is, if we estimate the tunnel current for the case when ␭ = 1 eV and Vb ⬎ 0.1 V as the sum of the contributions from the resonance and off-resonance tunneling presented by Eqs. 共24兲 and 共30兲, respectively, then it can be shown that the width decreases with increase of Vb as ␭ / 2e − ␺共Vb兲exp关eVb / 共4kBT兲兴, where ␺ ⬎ 0 is a certain function of Vb. Finally, let us consider the case when eVb Ⰷ ␭. We can neglect the dependence of the position of the valence level on q in this case so that j=

冋冉

冊

冉

eVb/2 + e␩V eVb/2 − e␩V jmax arctan + arctan ␲ ⌬ ⌬

J/Jmax

冊册

共32兲 for all values of ⌬. Using Eq. 共32兲 it can be shown that W = 关共eVb兲2 + 4⌬2兴1/2. / e.

2

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

∆ (eV)

FIG. 1. Tunnel current j in units of jmax as a function of ⌬ for ⌬el = ⌬tip = ⌬ / 2, Vb = 0.1 V, ␩ = 0, kBT = 0.025 eV, and ␥ = 1 / 2. Curve 1, ␭ = 0.25 eV; curve 2, ␭ = 0.5 eV.

rier. This barrier disappears finally at some value of ⌬共␭兲 so that in the large-⌬ region the potential energy U共q兲 has a single potential well which corresponds to the adsorbed molecule having a certain average number of electrons in the valence orbital. As a result, the tunnel current reaches its maximum value in the intermediate-⌬ region and then decreases in the adsorption region due to increase of the total width of the valence level 共i.e., the decrease of the number of resonance electronic sates of the valence level lying within the energy window of the STM兲. Figure 2 presents the tunnel current as a function of the bias voltage for three different values of ⌬. The curves 1, 2, and 3 show the crossover from the small-bias-voltage region 1.00

IV. THE RESULTS OF CALCULATIONS AND DISCUSSION

1

J/Jmax

The results of calculations of the tunnel current and the spectral density presented below are based on Eqs. 共9兲–共12兲 and 共16兲–共19兲. For the sake of simplicity only the case ⌬el = ⌬tip = ⌬ / 2 is considered in what follows. At small values of 兩eVb兩 and ⌬, the potential energy U共q兲 has two potential wells 共bistability property兲 due to the existence of two valence states of the redox molecule. These wells are separated by the Frank-Condon barrier with the height which is roughly proportional to ␭ / 4 关see, e.g., Eqs. 共21兲 and 共24兲 as the simplest particular cases兴. Figure 1 presents the dependence of the tunnel current on ⌬ for Vb = 0.1 V, ␩V = 0, and two values of ␭ corresponding to the smallest and highest estimations of this parameter for the in situ STM system.5 In the small-⌬ region, the values of the current are determined by the height of the Frank-Condon barrier and, at given ⌬, are larger in the case of smaller values of ␭. Figure 1 shows that the tunnel current increases with increase of ⌬ in the small-⌬ region due to the decrease of the Frank-Condon bar-

0.75

2

0.50

3

0.25

0.00

0.0

0.2

0.4

0.6

0.8

1.0

Vb (V)

FIG. 2. Tunnel current j in units of jmax as a function of the bias voltage Vb for ␭ = 0.5 eV, ⌬el = ⌬tip = ⌬ / 2, ␩ = 0, kBT = 0.025 eV, and ␥ = 1 / 2. Curve 1, ⌬ = 0.05 eV; curve 2, ⌬ = 0.1 eV; curve 3, ⌬ = 0.2 eV.

125312-7


I. G. MEDVEDEV 0.075

3

1

0.5

A(ε)

J/Jmax 2

2

0.4

4

0.050 0.3

3

0.2

1 0.025

0.1

-1.0

-0.5

0.0

0.5

1.0

ε (eV) -0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

ηV (V)

FIG. 3. Tunnel current j in units of jmax as a function of the overvoltage ␩ for ␭ = 0.5 eV, ⌬el = ⌬tip = ⌬ / 2, Vb = 0.1 V, kBT = 0.025 eV, ␰ = 1, and ␥ = 1 / 2. Curve 1, ⌬ = 0.1 eV; curve 2, ⌬ = 0.2 eV; curve 3, ⌬ = 0.3 eV.

where the tunnel current is determined by the value of the Frank-Condon barrier to the large-bias-voltage region where the Frank-Condon barrier disappears 关see, e.g., Eqs. 共21兲 and 共24兲, which show that the height of the barrier decreases with increase of 兩Vb兩 as 共␭ − 兩eVb兩兲 / 4兴. Therefore the large-biasvoltage region is also the adsorption region. As a result, Fig. 2 shows that the tunnel current increases and decreases with increase of ⌬ in the small- and in the large-bias-voltage regions, respectively. Figure 3 displays the results of calculations of the dependence of the tunnel current on the overvoltage at fixed value of the bias voltage. As was discussed above, the function j共␩V兲 has a maximum at ␩V = 0 关or at ␩ = 共1 / 2 − ␥兲Vb / ␰兴 so that the position ␸max of the maximum of the function j共␸兲 is close to the equilibrium potential ␸0 for ␰ ⬇ 1 and ␥ ⬇ 1 / 2. The experimental data of Tao1 confirm the theoretical prediction that ␸max ⬇ ␸0. Figure 3 also shows that the width of the curve j共␸兲 observed in Ref. 1 共⬃0.25 V兲 can be reproduced by the widths of the curves 1 or 2 corresponding to ⌬ = 0.1 or 0.2 eV 共the widths W at half maximum equal 0.22, 0.29, and 0.38 V for curves 1, 2, and 3, respectively兲. These results differs from those of Ref. 32 where the position of the maximum of j共␸兲 was shifted by 0.3 V from the experimental value and the observed width of the curve was obtained for very small values of ␭ and ⌬. It is due to the fact that the form of the curve j共␸兲 is related in Ref. 32 to the form of the part of the spectral function A共␧兲 corresponding only to the oxidized state of the redox molecule. The width of this part indeed depends essentially on ␭ and ⌬. However, it was shown above 共see Fig. 3兲 and already follows from the simple equations 共21兲 and 共24兲 that, in the case when the full spectral function is taken into account, the width of the curve j共␸兲 for small values of ⌬ and ␭ 艋 0.5 eV is entirely independent of ␭. For the values of ⌬ and Vb under consider-

FIG. 4. Spectral function 共in 1/eV兲 for ␭ = 0.5 eV, ⌬el = ⌬tip = ⌬ / 2, Vb = 0.1 V, ␩ = 0, kBT = 0.025 eV, ␥ = 1 / 2, and different values of ⌬: Curve 1, ⌬ = 0.05 eV; curve 2, ⌬ = 0.1 eV; curve 3, ⌬ = 0.2 eV; curve 4, ⌬ = 0.3 eV.

ation, the width equals approximately 共5.2kBT + ⌬兲 / e but not W ⬇ 4共␭kBT ln 2兲1/2 / e. The spectral functions for fixed Vb = 0.1 V, ␩ = 0, ␥ = 1 / 2, ␭ = 0.5 eV, and different values of ⌬ are compared in Fig. 4. For small ⌬ Ⰶ ␭ the curves A共␧兲 consist of two smeared spectral functions corresponding to the oxidized and reduced states of the bridge molecule. At ⌬ = 0.3 eV the spectral function describes already the adsorbed state of the bridge molecule having a certain average number of electrons at valence orbital. The potential energy U共q兲 has a single potential well in this case. Figures 3 and 4 show that, in accordance with Eq. 共14兲, values of the tunneling current are determined by the values that the spectral function takes in the energy window of the STM 共0 艋 ␧ 艋 0.1 eV兲. Figures 3 and 4 also show that the width of j共␸兲 does not depend directly on ␭ and ⌬ through the widths of the parts of A共␧兲 corresponding to the oxidized and reduced states of the bridge molecule. In contrast to Ref. 32, here it depends on ␭ and ⌬ through the form of A共␧兲 in the energy window. The closer the bridge molecule is to the adsorption region, the weaker is the dependence of j共␸兲 on ␸ and, therefore, the larger is the width of the curve j共␸兲. This result clearly shows that for ␭ = 0.5 eV and values of ⌬ relevant to localized d states of the bridge molecule1 and having an order of a few tenths of 1 eV, the curve j共␸兲共see Fig. 3兲 also has a width of the order of a few tenths of 1 V. The next example concerned with the experimental data of Ref. 5 is the dependence of the width W on the bias voltage at fixed ⌬. It can be shown that W increases with increase of Vb for ⌬ ⬎ 0.1 eV due to the approach of the system to the state where U共q兲 has a single potential well. However, as was discussed above, for small values of ⌬, the dependence of W on Vb for ␭ = 0.5 eV is weak and not monotonic. In particular, for ⌬ = 0.1 eV, W is almost independent of Vb for 0.1艋 Vb 艋 0.5 V and equals 0.22, 0.21, and 0.23 V for Vb = 0.1, 0.3, and 0.5 V, respectively. To imitate the ex-

125312-8


TUNNEL CURRENT THROUGH A REDOX MOLECULE… 0.4

3

0.04

3

J/Jmax

j/jmax 0.3

0.03

0.2

0.02

2

2

0.1

0.01

1

1 0.0 0.0

0.4

0.8

1.2

1.6

-0.3

ϕ (V)

FIG. 5. Tunnel current j in units of jmax as a function of the potential ␸ for ␭ = 0.5 eV, ␸0 = 1.2 V, ⌬el = ⌬tip = ⌬ / 2, ⌬ = 0.1 eV, kBT = 0.025 eV, ␰ = 0.4, and ␥ = 0.9. Curve 1, Vb = 0.1 V; curve 2, Vb = 0.3 V; curve 3, Vb = 0.5 V.

perimental data presented in Fig. 4共a兲 of Ref. 5, we put ␭ = 0.5 eV, ␸0 = 1.2 V, ␥ = 0.9, ␰ = 0.4, and ⌬ = 0.1 eV. The results of calculations of j共␸兲 are shown in Fig. 5. Although we could not reproduce the decrease of the width with increasing bias voltage observed in Ref. 5, our calculations show almost constant width. The decrease of the observed width may be attributed to change of the other parameters of the system 共e.g., ⌬, the position of the molecule, etc.兲 with increase of Vb. Tacking account of the Coulomb repulsion between the electrons with different spin projections on the valence orbital may also be of importance. However, decrease of the width with increasing bias voltage can be obtained when possible participation of the intramolecular modes in the tunneling process is taken into account, as was discussed in the previous section. The curves of j共␩兲 for ␭ = 1 eV and ⌬ = 0.1 eV are presented in Fig. 6. Calculations show that the widths W equal 0.51, 0.45, and 0.29 V for curves 1, 2, and 3, respectively. V. CONCLUSION

-0.2

-0.1

0.0

0.1

0.2

0.3

η (V)

FIG. 6. Tunnel current j in units of jmax as a function of the overvoltage ␩ for ␭ = 1 eV, ⌬el = ⌬tip = ⌬ / 2, ⌬ = 0.1 eV, kBT = 0.025 eV, ␰ = 1, and ␥ = 1 / 2. Curve 1, Vb = 0.1 V; curve 2, Vb = 0.3 V; curve 3, Vb = 0.5 V.

modes in the strong tunneling limit using thermal averaging over the potential energy of the phonon modes introduced in the steady state of the in situ STM system. In contrast to Refs. 11 and 32, our method takes into account not only the oxidized state of the redox molecule but also the full spectral density. As a result, the current j共qk兲 is averaged in Eqs. 共6兲 and 共16兲 not only over the free phonon modes but over the total effective potential energy. It allows one to describe the thermally activated regime, the adsorption regime and the crossover between them. The results of calculations of the tunneling current as a function of the parameters of the system are presented and discussed. In particular, the j共␸兲 curves calculated for a physically reasonable parameter set exhibit the main features presented in experimental data obtained in Refs. 1 and 5. ACKNOWLEDGMENTS

We presented a method of calculation of the tunnel current through a redox molecule coupled to classical phonon

The author is grateful to A. M. Kuznetsov for helpful discussions. This work was partly supported by the Russian Foundation for Basic Research 共Grant No. 06-03-32193a兲.

N. J. Tao, Phys. Rev. Lett. 76, 4066 共1996兲. D. L. Gittins, D. Bethell, D. J. Schiffrin, and R. J. Nichols, Nature 共London兲 408, 67 共2000兲. 3 B. Xu and N. G. Tao, Science 301, 1221 共2003兲. 4 F. Chen, J. He, C. Nuckolls, T. Roberts, J. E. Klare, and S. Lindsay, Nano Lett. 5, 503 共2005兲. 5 I. Visoly-Fisher, K. Daie, Y. Terazono, C. Herrero, F. Fungo, L.

Otero, E. Durantini, J. J. Silber, L. Sereno, D. Gust, T. A. Moore, A. L. Moore, and S. M. Lindsay, Proc. Natl. Acad. Sci. U.S.A. 103, 8686 共2006兲. 6 H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P. Alivisatos, and P. L. McEuen, Nature 共London兲 407, 57 共2000兲. 7 J. Park, A. N. Pasupathy, J. I. Goldsmith, C. Chang, Y. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D. Abruna, P. L. McEuen,

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