Characteristics of STM Processes as a Probe Detecting Vibronic I

J. Phys. Chem. B 1998, 102, 1833-1844

1833

V-I Characteristics of STM Processes as a Probe Detecting Vibronic Interactions at a Redox State in Large Molecular Adsorbates Such as Electron-Transfer Metalloproteins Hitoshi Sumi† Institute of Materials Science, UniVersity of Tsukuba, Tsukuba, 305-8573 Japan ReceiVed: September 26, 1997

Large redox molecules such as electron-transfer metalloproteins have a special relevance to scanning tunneling microscopy (STM) processes, since a redox state (RS) immersed therein provides a station of electron-hopping mediation for STM currents when they are adsorbed on the substrate. The currents flow when the tip is adjusted at the location of the RS on the substrate. Flowing into the RS, an electron induces both the innersphere reorganization (in atomic arrangements in a prosthetic group that provides the RS) and the outersphere one (in amino-acid-residue arrangements in the protein matrix around the RS). The mediation is, therefore, accompanied by simultaneous emission of large-energy-quantum intramolecular phonons participating in the former and by relaxation along the coordinate of the latter. The intramolecular-phonon-assisted mediation is opened when the bias voltage exceeds a series of threshold values determined in conjunction with the latter. It is predicted, therefore, that the V-I characteristics should have a phonon-progressional staircase structure. It enables us to derive fundamental physical parameters for the RS due to the prosthetic group, such as its location with respect to both energy and height from the substrate, energies of inner-sphere and outer-sphere reorganization, and the energy quantum of intramolecular phonons participating in the former.

1. Introduction Scanning tunneling microscopy (STM) is very powerful in providing an atomic-scale surface image of conducting materials. As an extension of its wide applicabilities, it has been pursued to obtain an atomic-scale image of large biomolecules, such as DNA1 and proteins,2 adsorbed on the substrate, although they are not conducting. It can be regarded as the first stage of these investigations that STM was successful in the case of large molecular adsorbates, such as phthalocyanine3 and porphyrin,4 which are closely related to the activities of biomolecules. It has been observed in these studies that the image changes drastically depending on the bias voltage between the tip and the substrate: The image of molecular adsorbates is obtained at a certain bias, while at another bias it disappears but the image of the substrate appears. It seems, therefore, that an electronic state due to molecular adsorbates becomes resonant for mediation of electron tunneling at a certain bias. A clear-cut example of these phenomena was recently reported for a coadsorbed monolayer of porphyrin (P) and iron porphyrin (FeP),5 where only the image of FeP could be obtained at an appropriate bias, but no difference between P and FeP was obtained at another bias. In this case, the LUMO (the lowest unoccupied molecular orbital) of FeP becomes resonant for mediation of STM currents.6 FeP, in general, metalloporphyrin, is immersed as a prosthetic group in large redox proteins such as cytochrome c and plays an essential role in mediation of biological electrontransfer chains, providing a redox couple.7 Observation of an atomic-scale image of proteins adsorbed on the substrate might be difficult, however, since they have in general a size of the order of several tens of angstroms: STM currents flowing through a protein body between the tip and the substrate must be intervened by several tens of atoms †

E-mail: [email protected]. Fax: 81(Japan)-298-55-7440.

comprising the protein matrix,8,9 and the atomic-scale difference in the position of the tip is blurred after averaging over many possible paths for the currents.10,11 In this situation, redox proteins seem to have a special relevance to STM processes: In these proteins, prosthetic groups such as metalloporphyrin are immersed as mediation stations for electron tunneling through a protein body.7 To facilitate the mediation, either the LUMO or the HOMO (the highest occupied molecular orbital) due to a prosthetic group has been adjusted so as to be much lower than the LUMO but much higher than the HOMO due to the protein itself. This adjustment has been accomplished in order for biological electron-transfer chains to be mediated most efficiently at the protein, as a result of evolution for a long time. When a redox protein adsorbed on the substrate is seen by STM, therefore, tunneling currents through the protein body should be mediated by the LUMO or HOMO due to a prosthetic group immersed therein, at an appropriate bias. Presence of specific paths for STM currents through redox proteins has in fact been suggested from the observed bias-dependent contrast of the STM image.2,12,13 From this contrast, we can detect the horizontal position of the prosthetic group on the substrate, as the position of FeP was discriminated in the coadsorbed monolayer of FeP and P.5 Since a redox protein has a size of the order of several tens of angstroms, the prosthetic group immersed in it is in general more than 10 Å high from the substrate on which the protein is adsorbed. When the tip in STM is adjusted to the neighborhood of the horizontal position of the prosthetic group, it is vertically positioned midway between the tip and the substrate. In STM, therefore, the bias potential at the prosthetic group relative to the substrate is a fraction of the total bias between the tip and the substrate. When an electron flows into the redox state in the prosthetic group, it induces two kinds of reorganization, in atomic arrangements inside the prosthetic group and in molecular

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1834 J. Phys. Chem. B, Vol. 102, No. 10, 1998 (amino-acid-residue) arrangements in the protein matrix outside it. In other words, the electron-tunneling mediation at the redox state is accompanied by simultaneous emission of large-energyquantum intramolecular phonons participating in the former and by relaxation along the coordinate of the latter, which can be treated semiclassically. STM processes accompanying the latter were theoretically investigated by Kuznetsov and his collaborators.14-16 In electron-transfer reactions in solution, the former has been called the inner-sphere reorganization and the latter the outer-sphere one.17 These two kinds of reorganization should play important roles when STM currents are mediated by a redox protein, although refs 14-16 take into account only the latter. Especially, the bias condition for the rise of STM currents through a redox protein is determined not only by the energy of the redox couple due to the prosthetic group but also by the interaction of an electron with vibrational motions around and inside the prosthetic group. The present author18 pointed out, in this situation, that the intramolecular-phonon-assisted mediation associated with the inner-sphere reorganization is opened when the bias voltage exceeds a series of threshold values determined in conjunction with the outer-sphere reorganization. It was predicted, thus, that the V-I characteristics of the STM processes should have an intramolecular-phononprogressional staircase structure. In refs 14-16, however, no staircase-structured V-I characteristic has ever been pointed out even qualitatively,19 although it should manifest itself even if the inner-sphere reorganization is neglected as in refs 1416. This characteristic is very important and will be disclosed in more detail in the present work. Its observation would enable us to derive fundamental physical parameters for the redox state due to the prosthetic group, such as its energy-level position, energies of inner-sphere and outer-sphere reorganization, and the energy quantum of the intramolecular phonons in the former. The pattern of this staircase-structured V-I characteristic depends strongly on the free energy Gm for the redox state relative to the Fermi level in the substrate that is assumed to be made of a metal. The Gm value can be changed by changing the metal of the substrate or by the electrochemical control adopted in ref 5. The aim of the present work is to describe theoretically the Gm variation in the staircase-structured V-I characteristic, and to show how we can derive the fundamental physical parameters for the redox state from the data of its observation. Thus, STM processes mediated by a redox protein can provide a powerful experimental tool for probing the vibronic (i.e., interacting electron-vibration) state at the prosthetic group immersed in it. The plan of the present work is as follows. Electron mediation by a redox protein in STM processes is described qualitatively by introducing an adiabatic-potential diagram in Section 2. Its theoretical formulation is given in Section 3. These two sections include an extended description of the content of ref 18 which corresponds to a brief communication of the present work. It is indispensable for understanding Section 4, which gives examples of the Gm variation in the staircase-structured V-I characteristic calculated by this formulation. General discussions are given in Section 5. 2. Adiabatic-Potential Diagram for STM Currents via Redox Proteins Let us consider that a redox protein is adsorbed on the substrate and the tip in STM is adjusted horizontally in the neighborhood of the position of a prosthetic group immersed in it. Both the tip and the substrate are assumed to be made of metals. Without an applied bias between them, Fermi levels for the electron sea in the tip and the substrate are balanced

Sumi

Figure 1. Transitions a and b of an electron from the tip to the substrate mediated by the redox state (RS) in a large molecular adsorbate.

with each other, while they deviate from each other under an applied bias where STM currents flow. We consider, moreover, that either the LUMO or HOMO due to the prosthetic group, working as a redox state in the protein, has an energy level in the same energy region as the Fermi levels in the tip and the substrate. In this case, other electronic states in the protein should have energy levels much higher or lower than the Fermi levels, and they are irrelevant as mediation stations for electron translocation through the protein. The prosthetic group with the redox state (abbreviated as RS hereafter) is, in general, at least more than 10 Å apart from both the tip and the substrate, since the protein has a size of the order of several tens of angstroms. Then, it is reasonable to consider that STM currents are carried by incoherent electron hopping from the tip to the RS, shown as a in Figure 1, and that from the RS to the substrate, shown as b therein. When STM currents flow steadily, the RS should be partially occupied (i.e., the RS is completely occupied at a time, but completely emptied at another time). When the tip bias potential is positive, both a and b in Figure 1 carry the forward current. We are trying here to construct a formula for the STM current bridging the positive and the negative bias regions. Then, electron hopping reverse to a and b must also be taken into account even in the positive bias region, although they are completely neglected in refs 1416. When the RS is provided by the LUMO due to the prosthetic group, electron occupation at the RS means single-electron occupancy at the LUMO. When the RS is provided by the HOMO, on the other hand, electron occupation at the RS means double-electron occupancy at the HOMO. Fermi levels for the electron sea in the tip and the substrate are written respectively as Ft and Fs. Without applied bias, Ft equals Fs. Their nonzero difference under an applied bias

Ft - Fs ) φ

(2.1)

is regarded, for simplicity, as the bias potential at the tip relative to the substrate, given by the bias voltage at the tip multiplied by the electron charge. Since the RS is located midway between the tip and the substrate, the electric potential at the RS is smaller than φ. It is written as Rφ with 0 < R < 1, where R is determined by the relative spatial position of the RS between the tip and the substrate and also by the distribution of the dielectric constant inside the protein in which the RS is immersed. Let Gm represent the free energy relative to Fs when the RS is occupied without applied bias. Under the applied bias, therefore, the free energy for the RS occupied by an electron shifts, from its zero-bias value Gm + Fs, to

Em ) Gm + R φ + Fs

(0 < R < 1)

(2.2)

Electronic states comprising the Fermi sea are delocalized in both the tip and the substrate. Then, electrons in these states are little influenced by molecular (i.e. amino-acid residue)arrangement distortions in the protein matrix surrounding the

V-I Characteristics of STM Processes

J. Phys. Chem. B, Vol. 102, No. 10, 1998 1835

RS and also by atomic-arrangement distortions inside the prosthetic group. When an electron hops to the RS from either the tip or the substrate, on the other hand, its energy begins to be influenced by the distortions, since its wave function shrinks at the prosthetic group. Induced as its reaction to the hopping are both the outer-sphere reorganization outside the prosthetic group and the inner-sphere one inside it, as mentioned earlier. Participating in the outer-sphere reorganization are intermolecular vibrations in the protein matrix, and they have energy quanta at most of the order of several tens of wave numbers.20,21 They are usually much smaller than the energy dissipated by this reorganization, written as λm, which is usually of the order of 1000 cm-1 as seen typically in the reaction center of photosynthesis.22 Then, we can treat the outer-sphere reorganization in terms of a classical, or at most semiclassical, coordinate Q. Participating in the inner-sphere reorganization are, on the other hand, intramolecular vibrations inside the prosthetic group, and they have energy quanta, in general, of the order of 1000 cm-1. Since the energy dissipated by this reorganization is, in general, also of the order of 1000 cm-1,17 the inner-sphere reorganization must be treated quantum mechanically. In this case, important roles are played by large energy quanta themselves of these vibrations. Adopting a single-mode approximation, hereafter, we take into account only a single mode of intramolecular vibrations as participating in the inner-sphere reorganization, and their energy quantum is written as pωa. In this approximation, it is convenient to express the energy dissipated by this reorganization as Sapωa, where Sa represents a dimensionless coupling constant called the HuangRhys factor.17 The origin of the coordinate Q introduced above is put at the molecular arrangement in the protein matrix equilibrated when the RS is unoccupied. Then, Q describes a distortion from the equilibrated arrangement. Since Q does not necessarily coincide with a normal mode of intermolecular vibrations in the protein matrix, it is described by a linear combination of normal modes of these vibrations. The energy of the distortion is quadratic in Q as long as its amplitude is small. It is written as Q2/2 by multiplying all the coefficients in the linear combination by an appropriate common number. In the classical, or semiclassical, approximation for Q, it is convenient to introduce an adiabatic potential that is defined by the sum of the energy of an electron interacting with the distortion, and the energy for producing the distortion which is given by Q2/2. Energies of electronic states in the tip are written as Ei with i ) 1, 2, 3, ..., while those in the substrate are as Ef with f ) 1, 2, 3, ... They closely pile up with infinitesimal separation in each group. Let us first consider a case that the RS is unoccupied, In this case, an electron that is to make a transition to the RS is staying at an electronic state in the tip or in the substrate. Regarding the electron as not interacting with the distortion described by the coordinate Q, we can get the adiabatic potentials of

Ei + Q2/2 and Ef + Q2/2, with i and f ) 1, 2, 3, ... (2.3) for the electron respectively in the ith electronic state in the tip and in the fth electronic state in the substrate. Occupancy at these states is described by the Fermi distribution functions p(Ei - Ft) and p(Ef - Fs), respectively, for the Fermi energies Ft and Fs, with

p(E) ) 1/(eβE + 1), where β ) 1/(kBT)

(2.4)

Figure 2. Configuration among the adiabatic potential Vm(Q) for the RS and those for electronic states around Fermi levels Ft and Fs, respectively, in the tip and the substrate at a small bias potential φ, shown as a function of the coordinate Q of molecular-arrangement distortions around the RS.

After the hopping of the electron to the RS, on the other hand, it begins to interact with the molecular-arrangement distortion described by the coordinate Q. As a result of the interaction, molecular arrangements in the protein matrix distort automatically along the coordinate Q. The amount of energy dissipated in this relaxation corresponds to the outer-sphere reorganization energy λm. After the relaxation, the total energy of the electrondistortion system should come to the free energy, Em of (2.2), for an RS occupied under the applied bias. This means that the adiabatic potential for the RS is given by

Vm(Q) ) Em + (Q - x2λm )2/2

(2.5)

since it satisfies correctly that Vm(0) is larger by λm than the lowest value Em of Vm(Q). Figure 2 shows an example of the Q dependence of Vm(Q) of (2.5) relative to those of the adiabatic potentials in (2.3). Shown explicitly by multiple parabolas in Figure 2, however, are the adiabatic potentials in (2.3) for delocalized states only with energies below the Fermi energies Ft and Fs, respectively, in the tip and the substrate, since it is until below about Ft and Fs, respectively, that these states are occupied. These multiple parabolas should be separated only infinitesimally, but they are drawn with nonvanishing spacing for visualization by the eyes. Figure 2 shows an example that Ft is larger than Fs for φ > 0 in (2.1), and Em is located between them. Before electron hopping from the tip to the RS by a step a in Figure 1, molecular arrangements in the protein matrix are in the neighborhood of Q ) 0 since adiabatic potentials become minimum there in Figure 2 irrespective of whatever state the electron occupies in the tip. At the molecular arrangement of Q ) 0, energy Vm(Q) at the RS is higher than the highest value Ft of the adiabatic potentials for states occupied in the tip, in Figure 2. This means that on electron hopping a from the tip to the RS there exists a nonvanishing barrier that must be surmounted by the help of thermal distortions along the coordinate Q, although the minimum value Em of Vm(Q) is lower than Ft. To be more exact, the electron hopping a in Figure 1 takes place dominantly by the trajectory a in Figure 2, since

1836 J. Phys. Chem. B, Vol. 102, No. 10, 1998 the thermal activation energy is lowest from an electronic state with energy Ft in the tip in those with energies not higher than Ft. Before electron hopping from the RS to the substrate by a step b in Figure 1, molecular arrangements in the protein matrix are in the neighborhood of the Q value where Vm(Q) becomes minimum with energy Em in Figure 2. At this Q value, adiabatic potentials for electronic states unoccupied with energy above about Fs in the substrate are higher than Em. This means that also on electron hopping b from the RS to the substrate there exists a nonvanishing barrier that must be surmounted by the help of thermal distortions along the coordinate Q, although the minimum value Fs of these adiabatic potentials is lower than Em. To be more exact, electron hopping b from the RS to the substrate in Figure 1 takes place dominantly by the trajectory b in Figure 2, since the thermal activation energy is lowest to an electronic state with energy Fs in the substrate in those with energies not lower than Fs. We see, therefore, that even if the energy Em at the RS is lower than the Fermi level Ft in the tip and simultaneously higher than that Fs in the substrate, it occurs that both the electron hoppings from the tip to the RS and from the RS to the substrate must be thermally activated over a distortion barrier. This situation is realized for a small φ. The resonant electron tunneling cannot occur in the neighborhood of the initial equilibrated molecular arrangement in the protein matrix as long as its coordinate Q is kept unchanged semiclassically, as was first noted by Kuznetsov and his collaborators.14-16 It is not practical to describe this situation purely quantum mechanically in terms of phonon-assisted electron tunneling, not in terms of the semiclassical coordinate Q. Such a description is appropriate for participation of at most two or three phonons.23-25 It would require a limit of multiphonon participation in the present problem since energy quanta of molecular-arrangement distortions in the protein matrix are much smaller than the reorganization energy λm by them as mentioned earlier. The semiclassical coordinate Q was introduced also in ref 26 to describe STM processes mediated by the RS. It was assumed there, however, that a resonant electron tunneling took place at any value of Q, and the total tunneling rate was obtained by thermally averaging a tunneling rate at each Q over its Gaussian distribution with a variance (2λmkBT)1/2 around Q ) 0. This assumption is inconsistent with the present picture, widely accepted in electron-transfer reactions,17 that electron tunneling is possible only when thermal fluctuations in molecular-arrangement distortions enable the coordinate Q to reach a special value at the barrier top (corresponding to the transition state) along the trajectories a and b in Figure 2. Since Ft - Fs ) φ while Em - Fs ) Gm + R φ with 0 < R < 1 under a bias potential φ at the tip from the substrate, increasing φ cast the configuration in Figure 2 into that in Figure 3. It is realized there that at the molecular arrangement of Q ) 0, Vm(Q) ()Fm there) is lower than Ft, and simultaneously at the Q value where Vm(Q) becomes minimum, the adiabatic potential passing through Fs is lower than Vm(Q) ()Em there). In this configuration, electron hopping a from the tip to the RS in Figure 1 can take place without thermal activation, from the bottom of the adiabatic potential for an electronic state with energy Fm below the Fermi level Ft in the tip to Vm(Q). Simultaneously, electron hopping b from the RS to the substrate in Figure 1 can take place also without thermal activation, from the bottom of Vm(Q) to the adiabatic potential (shown by a dashed line in Figure 3) for an electronic state with energy Em - λm above the Fermi level Fs in the substrate. At this bias potential φ, therefore, STM currents should flow barrierlessly

Sumi

Figure 3. Configuration for the stepwise rise in STM currents, among the adiabatic potential Vm(Q) for the RS and those for electronic states in the tip and the substrate at a tip bias potential φ.

and rise almost stepwise at least at low temperatures. This stepwise rise in STM currents was first pointed out in ref 18, but not mentioned in refs 14-16 even qualitatively. Although the configuration among adiabatic potentials in Figure 2 was first noted in refs 14-16, it was not pointed out therein that the configuration in Figure 2 transforms itself into that in Figure 3 with a change in the bias potential φ, giving rise to barrierless flow of STM currents.27 The condition for the stepwise rise in STM currents mentioned above is Ft > Fm () Em + λm) and Em > Fs + λm for φ > 0, with suffixes t and s exchanged with each other for φ < 0. Introducing (2.2) into these inequalities, the condition can be expressed in terms of the tip bias potential φ and the free energy Gm for the RS occupied without applied bias, as

φ>

Gm + λm Gm - λm and φ > , for φ < 0 1-R R

(2.6)

Gm - λm Gm + λm and φ < , for φ < 0 R 1-R

(2.7)

or

φ 0, while for electron hopping reverse to b from the substrate



(n) Figure 4. Region for the (φ, G(n) m ) pair realizing the stepwise rise in STM currents (in Figure 3 for n ) 0), with Gm t Gm ( npωa for an integral number n, where Gm represents the free-energy height of the RS above the Fermi energy Fs in the substrate without applied bias.

to the RS they become opened every time Fm + n pωa becomes lower than Fs for n ) 1, 2, 3, ... for φ < 0. Under these conditions, electron inflow to the RS can take place, accompanied by emission of n intramolecular phonons with energy quantum pωa, without a thermal-activation barrier due to molecular-arrangement distortions along the coordinate Q. Introducing G(n) m t Gm + npωa, we note that these conditions are the same as the first inequality in (2.6) or (2.7) as long as Gm therein is replaced by G(n) m . This means in Figure 4 that STM currents should flow barrierlessly every time a point on horizontal lines of G(n) m ) Gm + npωa for n ) 1, 2, 3, ... written therein crosses the upper boundaries of the hatched regions, marked as Tip for φ > 0 and as Substrate for φ < 0, with an increase in |φ|. Concerning electron outflow from the RS, they become opened for electron hopping b to the substrate every time Em exceeds Fs + λm + n pωa for n ) 1, 2, 3, ... in Figure 3 for φ > 0, while for electron hopping reverse to a to the tip they become opened every time Ft + λm + n pωa becomes lower than Em for n ) 1, 2, 3, ... for φ < 0. Under these conditions, electron outflow from the RS can take place also without thermal activation, accompanied by emission of n intramolecular phonons with energy quantum pωa. Introducing G(n) m t Gm npωa here, we note that these conditions are the same as the second inequality in (2.6) or (2.7) as long as Gm therein is replaced by G(n) m . This means in Figure 4 that STM currents should flow barrierlessly every time a point on horizontal lines of G(n) m ) Gm - npωa for n ) 1, 2, 3, ... written therein crosses the lower boundaries of the hatched regions, marked as Substrate for φ > 0 and as Tip for φ < 0, with an increase in |φ|. In order for these higher-order channels to be operative, the fundamental channel with condition (2.6) or (2.7) must have already been opened. It is opened at the φ values (one for φ > 0 and one for φ < 0) where the horizontal line with height Gm crosses the hatched region in Figure 4. In order to determine where the stepwise rise in STM currents takes place, therefore, each of the two hatched regions in Figure 4 must be cut by a vertical line at these φ values, and a triangle made must be removed from each hatched region. These two φ values give the fundamental threshold for the stepwise rise. Then, we can write horizontal lines of G(n) m ) Gm ( npωa for n ) 1, 2, 3, ... in Figure 4. At each of the φ values where these horizontal lines cross the upper or lower boundaries of the triangle-removed hatched regions, STM currents should rise stepwise successively

with an increase in |φ|, increasing the order of channels, and construct a staircase structure. 3. Phonon-Progressional Staircase-Structured V-I Characteristics Under steady flow of STM currents, we can define an average occupancy at the RS. Since the RS is temporally occupied and then emptied in this case, the occupancy written as Pm should be neither zero nor unity but has a value between them, as long as the currents do not vanish. It is determined by the steadystate condition that currents flowing into the RS are balanced with currents flowing out of it. To describe these currents, let us express, by ktmPm, the rate of electron hopping to the tip from the RS. The rate of its reverse process should be written as (1 - Pm)kmt, since electron hopping to the RS is prohibited if it has already been occupied. Similarly, let us express, by ksmPm, the rate of electron hopping to the substrate from the RS, and the rate of its reverse process by (1 - Pm)kms. Then, the balance of currents mentioned above means

(1 - Pm)[kmt + kms] ) [ktm + ksm]Pm

(3.1)

Pm ) (kmt + kms)/(kmt + kms + ktm + ksm)

(3.2)

giving

When electron hopping a occurs first and b follows in Figure 1, it is appropriate to interpret a two-step process composed of a and b as carrying electron currents. In this case, the forward current from the tip to the substrate can be obtained from a scheme ksmPm

(1 - Pm)kmt

] tip substrate 79 RS[\ k P

(3.3)

tm m

The standard formula for a rate in a two-step process gives the forward current as

A ) (1 - Pm) ksmkmt/(ktm + ksm) ) ksmkmt/(kmt + kms + ktm + ksm)

(3.4) (3.5)

where we should note that Pm in ksmPm and ktmPm in (3.3) cancels between the numerator and the denominator in (3.4), and (3.5) was obtained by substituting (3.2). The backward current B is given by replacing ksmkmt in the numerator in (3.5) with ktmkms. From the law of mass action, however, B should equal A e-βφ, since φ given by (2.1) equals the free energy at

1838 J. Phys. Chem. B, Vol. 102, No. 10, 1998

Sumi

the tip relative to the substrate. Thus, the total STM current is given by

I ) A - B ) A (1 - e-βφ) , with β ) 1/(kBT)

(3.6)

When the free energy, Gm in (2.2), for the RS relative to the substrate has a negative value much lower than -kBT, it is more appropriate to interpret the two-step process composed of electron hopping a and b in Figure 1 as carrying currents by positive holes. Since the RS has initially been occupied, in this case, electron hopping b should occur first from the RS to the substrate, with a following from the tip to the RS emptied by b. It is more intelligible to interpret these processes such that hole hopping from the substrate to the RS occurs first, and that from the RS to the tip follows. It was mentioned in refs 1416 as if there existed two channels of the electron and the hole currents in which the former dominated the latter when Gm is positive and much higher than kBT, while the latter dominated the former when Gm is negative and much lower than -kBT. It will be shown in the Appendix, however, that the expression of (3.6) with (3.5) originally derived from the standpoint of electron currents agrees completely with that derived from the standpoint of hole currents as long as the condition (3.1) for steady flow of STM currents is satisfied. There exists, therefore, not the two channels by electron or hole flows, but only a single one through which the total STM current can be expressed by (3.6) with (3.5). Electronic states for the Fermi sea in both the tip and the substrate are continuously distributed with energy Ei for i ) 1, 2, 3, ... in the tip and with energy Ef for f ) 1, 2, 3, ... in the substrate. Then, we can introduce the rate constant kmi of an elementary electron hopping to the RS from the ith electronic state in the tip and that kim of its reverse process. Similarly, let us introduce also the rate constant kmf of an elementary electron hopping to the RS from the fth electronic state in the substrate, and that kfm of its reverse process. With these rate constants, kmt and ksm in (3.3) are given by

kmt )

∑i kmi p(Ei - Ft)

and ksm )

∑f [1 - p(Ef - Fs)] kfm

and

t Em ( npωa ) (Gm ( npωa) + Rφ + Fs E((n) m

(3.10)

with

and

W(E) t

{

Dm-1

x2π Dm-1 x2π

[ [

exp -

(E - λm)2

exp -

]

2Dm2 (E + λm)2 2Dm2

, +

]

E , kBT

for E > 0 for E < 0 (3.11)

where Jmi and Jmf represent, respectively, the transfer integral for electron hopping between the RS and the ith electronic state in the tip and between the RS and the fth electronic state in the substrate. Separation of W(E) into two cases in (3.11) was introduced to ensure the relation of detailed balance, W(-E) ) W(E) e-βE. Width Dm of W(E) is determined by

Dm2 ) λmpωb coth(βpωb/2)

(3.12)

with the use of the average energy quantum pωb of intermolecular phonons participating in the outer-sphere reorganization in the protein matrix. As mentioned earlier, pωb is at most several tens of wave numbers. Physically, Dm represents the root-mean-square average of the energy, -(2λm)1/2 Q in (2.5), of interaction of an electron at the RS with moleculararrangement distortions. In (3.8) and (3.9), e-Sa San/n! represents the Franck-Condon factor associated with emission of n intramolecular phonons of energy quantum pωa. We can get kmi and kfm by exchanging suffixes i and f in (3.8) and (3.9). Since electronic states in both the tip and the substrate are delocalized, their amplitude at a point therein is infinitesimal, and both Jmi and Jmf are infinitesimal quantities.14-16 In this case, it is convenient to introduce functions, with finite magnitudes, of energy E as

gt(E) t

2π p

∑i Jmi2 δ(E - Ei)

(3.13)

∑f Jmf2 δ(E - Ef)

(3.14)

and

gs(E) t

2π p

They represent spectral functions, with a dimension of inverse time, for an interaction-weighted density of electronic states respectively in the tip and the substrate. These functions, together with kmi and kfm in (3.8) and (3.9), enable us to express kmt and ksm of (3.7) in the form of -Sa

kmt ) e

∞

San

∑ n! ∫ gt(E) p(E - Ft) W(E - E(n)m ) dE

(3.15)

n)0

and

n

∑

(3.9)

∑

(3.7)

where p(Ei - Ft) and p(Ef - Fs) with p(E) defined by (2.4) represent, respectively, the occupancy at the ith electronic state in the tip and at the fth one in the substrate. We can get kms and ktm in (3.5) by exchanging i and f and simultaneously t and s in (3.7). Intramolecular vibrations, of energy quantum pωa, participating in the inner-sphere reorganization must be treated quantummechanically. Then, we can calculate kmi, kfm, kmf, and kim as giving an electron transition accompanied by emission or absorption of phonons of energy quantum pωa at the transitionstate value of the coordinate Q of molecular-arrangement distortions determined in the semiclassical approximation.28 In the present problem, we can neglect the electron transition with phonon absorption, since pωa is of the order of 1000 cm-1, much larger than kBT. Then, kmi and kfm are given by ∞ S a 2π W(Ei - E(n) kmi ) Jmi2 e-Sa m) p n)0 n!

n

2π 2 -Sa ∞ Sa kfm ) Jmf e W(E(-n) - Ef) m p n)0 n!

(3.8) -Sa

ksm ) e

∞

San

- E) dE ∫ gs(E)[1 - p(E - Fs)]W(E(-n) ∑ m n! n)0

(3.16)



Both kms and ktm can be obtained by exchanging suffixes t and s in (3.15) and (3.16). With these ktm, ksm, kms, and ktm, we can calculate the total STM current I of (3.6) by (3.5). With an increase in the bias potential φ, it should rise stepwise every time φ satisfies the conditions for opening of the fundamental and then high-order channels mentioned in Section 2. Then, the φ dependence of I posseses a staircase structure. Each step height of the staircase can be obtained by regarding each stepwise rise as a vertical one, which can be obtained by regarding the Fermi-distribution function p(E) of (2.4) and the outer-sphere-interaction broadening function W(E) of (3.11), respectively, as a step function that equals unity for E < 0 and zero for E > 0, and a δ function δ(E - λm). In this approximation, in order for the nth component in the summation on the right-hand side of (3.15) or (3.16) to be nonvanishing, Ft > E(n) m + λm for (3.15) or E(-n) > Fs + λm for (3.16) must be satisfied. Since E((n) is m m defined by (3.10), they correspond to (1 - R) φ > (Gm + npωa) + λm for (3.15) or R φ > -(Gm - npωa) + λm for (3.16). Two conditions obtained by n ) 0 are just the same as (2.6) for φ > 0 since only kmt and ksm are relevant to the stepwise rise in STM currents there. They must first be satisfied simultaneously as the start of the stepwise rise in the positive φ region, to which we confine ourselves for simplicity. For n g 1, the former and the latter are, respectively, just the same as the condition for opening of the nth higher-order channel for electron inflow to and outflow from the RS in the positive φ region. Then, on the basis of the consideration in Section 2, we can understand them by Figure 4 as follows. First, we write a horizontal line with height Gm in Figure 4, and determine a φ value by its crossing with the hatched region in the positive φ region. It gives the threshold value for opening of the fundamental channel. For the stepwise rise in STM currents to start, the applied bias potential φ must exceed this fundamental threshold. Under this condition, we write in Figure 4 a vertical line at the value of φ and cut a triangle from the hatched region. Then, we write a series of horizontal lines of G(n) m ) Gm + npωa for n ) 0, (1, (2, (3, ..., and select horizontal lines that penetrate the triangle. Let us express by M (g0) the highest number of n in these horizontal lines, and by -L (e0) its lowest number. Then, (3.15) and (3.16) can be approximated as -Sa

kmt ≈ e

M

San

gt(E(n) ∑ m + λm) n! n)0

(3.17)

and -Sa

ksm ≈ e

L

San

- λm) ∑ n! gs(E(-n) m

(3.18)

n)0

When both kmt and ksm become activationless as given by (3.17) and (3.18), both ktm and kms become negligibly small since λm . kBT. Moreover, (3.6) reduces to I ≈ A, since |φ| . kBT is satisfied when we enter the hatched region in Figure 4 for the stepwise rise in I. With these kmt and ksm, we can get A (≈I) in (3.5) and the occupancy Pm at the RS in (3.2) by

I ≈ 1/(ksm-1 + kmt-1) and Pm ≈ kmt/(ksm + kmt) (3.19) This I with kmt and ksm of (3.17) and (3.18) gives the step height in the staircase structure of the STM current when the tip bias potential φ is positive. For the height of the first step, especially, at least either M in (3.17) or L in (3.18) is equal to zero.

When φ < 0, and φ exceeds the threshold value for the fundamental channel determined by (2.7), on the other hand, both kmt and ksm become negligibly small with both kms and ktm becoming activationless. Determining L and M values similarly in Figure 4 also in this case, we can obtain approximate values for both kms and ktm by exchanging t and s in (3.17) and (3.18). Both I and Pm can be obtained by the same exchange in (3.19). 4. Gm Variation in the V-I Characteristics First, we note in Figure 4 that when the tip bias potential φ is as small as |φ| < 2λm, neither (2.6) nor (2.7) can be satisfied by any Gm. In this small φ region, therefore, the STM current I should be kept very small at low temperatures and the occupancy Pm at the RS should be nearly zero or unity. In the calculation, shown below, of I of (3.6) by (3.5), (3.15), and (3.16), a simplification was adopted that both gt(E) and gs(E) were approximated as constant, given respectively by gt and gs. Their magnitudes depend on the distance respectively between the tip and the RS and between the RS and the substrate through the dependence of Jmi and Jmf. To investigate the Gm variation in the φ dependence of I, let us start with the most intelligible case. It is when Gm is positive and sufficiently high such that L in (3.18) becomes much larger than unity, to be more exact, much larger than Sa. In this case, we can approximate (3.18) as ksm ≈ gs for φ > 0 (or ktm ≈ gt for φ < 0) under the approximation for gt(E) and gs(E) mentioned above. For the height of the first step, especially, M must be equal to zero with a very large L, and for φ > 0 the height is given, together with Pm, by

I ≈ 1/(gs-1 + gt-1 eSa) and Pm ≈ 1/[1 + (gs/gt) eSa] (4.1) Those for φ < 0 can be obtained by exchanging t and s in (4.1). They are brought about by the opening of the fundamental channel. The higher-order channels appear with a step height, given by (3.19) for φ > 0, where M increases one by one from zero as |φ| increases. It is due to opening of electron inflow to the RS from the tip for φ > 0, or from the substrate for φ < 0, accompanied by emission of increasing numbers of the intramolecular phonon. The saturation value of the step can be obtained by setting M in (3.17) to infinity, as

I ≈ 1/(gs-1 + gt-1) and Pm ≈ 1/[1 + (gs/gt)] (4.2) In the calculation shown hereafter, a dimensionless quantity (gt-1 + gs-1)I measuring the STM current was calculated together with Pm. They depend on gt and gs only through gt/gs, which depends on R in (2.2). The ratio gt/gs was taken at unity, while R was taken at 2/3. Moreover, λm was taken at 1000 cm-1, pωa at 1500 cm-1, Sa at 0.7, and pωb at 30 cm-1 (as a typical energy quantum of phonons in the protein matrix20,21). The φ dependence of I expected above are realized by Gm ) 3500 cm-1 as seen in Figure 5, which shows the φ/(pωa) dependence of (gt-1 + gs-1)I and Pm calculated at T ) 10 K. In Figure 5 we should note, first of all, a staircase-structured V-I characteristic which has been anticipated so far. In more detail, the current is practically vanishing and Pm is zero in the small φ region, in response to a situation that the RS is located at an energy Gm ()3500 cm-1) much higher than kBT (≈7 cm-1) from the Fermi levels when φ ) 0. When φ/(pωa) exceeds about 20, both (gt-1 + gs-1) I and Pm become saturated respectively at unity and 1/2, in agreement with (4.2) for gt/gs ) 1. In the negative φ region, on the other hand, they are saturated respectively at -1 and 1/2 when |φ|/pωa exceeds about


Figure 5. Tip-bias-potential dependence of the STM current and the occupancy at the RS at 10 K when the free-energy height Gm of the RS above the Fermi energy Fs in the substrate is 3500 cm-1 without applied bias.

Figure 6. Tip-bias-potential dependence of the differential conductance of the STM current and the occupancy at the RS at 10 K when Gm is 3500 cm-1 without applied bias.

10. Approaching these saturation values, both of them rise or drop stepwise several times in the intermediate φ region. The threshold for the first stepwise rise is determined by the crossing of a horizontal line of height Gm with the hatched regions in Figure 4, and is located at φ/(pωa) ) 9 for φ > 0 or -4.5 for φ < 0. When φ exceeds this threshold, both (gt-1 + gs-1) I and Pm reach the first step with height given respectively by 0.66 and 0.33 for φ > 0 in agreement with (4.1), while by -0.66 and 0.33 for φ < 0. The subsequent stepwise rises in both (gt-1 + gs-1) I and Pm are located at φ/(pωa) ) 9 + 3n for φ > 0, and at φ/(pωa) ) -4.5 - 1.5n for φ < 0 with n ) 1, 2, 3, ..., derived from the crossing of horizontal lines of G(n) m ) Gm + npωa with the hatched regions in Figure 4. Their separation is, in general, given by 1/(1 - R) for φ > 0, and by 1/R for φ < 0. The height of these steps is given by (3.19) with (3.17) and (3.18) for L ) ∞ and M ) 1, 2, 3, ... All of them are brought about by the opening of electron inflow to the RS from the tip for φ > 0, or from the substrate for φ < 0, accompanied by emission of M intramolecular phonons. In fact, they are concomitant with an increase in Pm in Figure 5. The staircase structured V-I characteristic in Figure 5 becomes more pronounced when differentiated as in Figure 6 where the differential conductance (gt-1 + gs-1)pωa dI/dφ is plotted, together with Pm, as a function of φ/(pωa). A stepwise rise in Figure 5 is converted in Figure 6 into a sharp peak, which is more easily discernible. The amount of the stepwise rise is converted into the integration intensity of the peak. Next, let us change the value of Gm to 1100 cm-1. The φ/(pωa) dependence of the differential conductance (gt-1 + gs-1)pωa dI/dφ and Pm is shown in Figure 7. In this case, successive peaks in the positive φ region are similar to those in

Sumi


Figure 6, located at φ/(pωa) ) 4.2 + 3n for n ) 0, 1, 2, 3, ... in Figure 7, derived from the crossing in Figure 4. In the negative φ region, however, main progressions of peaks at φ/(pωa) ) -2.1 - 1.5n for n ) 0, 1, 2, 3, ... are supplemented by subsidiary progressions at φ/(pωa) ) 0.2 - 3n for n ) 1, 2, 3, ... without n ) 0, derived also from Figure 4. The former is similar to those in Figure 6, but the latter newly appears, showing the opening of electron outflow from the RS (to the tip for φ < 0) accompanied by emission of n intramolecular phonons. In fact, they are brought about by a decrease in Pm in Figure 7. The opening of electron outflow from the RS to the tip, expected at φ/(pωa) ) 0.2 for n ) 0, does not occur as derived from Figure 4, since Pm is kept zero there in Figure 7. The height of the first step in the φ dependence of the STM current I is represented by the total area of the first peak in the differential conductance in Figure 7, as mentioned earlier. The L and M values associated with the first step are, respectively, 3 and 0 for φ > 0 since the opening of electron outflow from the RS to the substrate is expected at φ/(pωa) ) -0.1 + 1.5n for n ) 0, 1, 2, 3, ..., not sufficiently exceeding 4.2 for n e 3, while both of them are 0 for φ < 0. Since L ) 3 has the same effect as L ) ∞ for Sa ) 0.7, the height of the first step calculated from (3.17) and (3.18) is given by

1/(gs-1 + gt-1 eSa) for φ > 0 and

e-Sa/(gs-1 + gt-1) for φ < 0

(4.3)

with a ratio of about 1.34 in agreement with the intensity ratio between the first peaks in Figure 7. Next, let us change the value of Gm to |1 - 2R|λm ≈ 333 cm-1. The φ/(pωa) dependence of the differential conductance (gt-1 + gs-1)pωa dI/dφ and Pm is shown in Figure 8. In this case, a horizontal line of height Gm drawn in Figure 4 crosses with the hatched region for φ < 0 just at the apex at φ/(pωa) ) -1.33. Then, for φ < 0, we have two series of progressional peaks at φ/(pωa) ) -1.33 - 1.5n and -1.33 - 3 n for n ) 0, 1, 2, 3, ..., as seen in Figure 8. The former is brought about by the opening of electron inflow to the RS from the substrate, with an increase in Pm in Figure 8, while the latter by that of electron outflow from the RS to the tip with a decrease in Pm. Since both series appear from n ) 0, neither is main nor subsidiary. For φ > 0, on the other hand, the horizontal line of height Gm drawn in Figure 4 crosses with the upper boundary of the hatched region at φ/(pωa) ) 2.67 where the first peak is located. Starting from this peak, main progressions of peaks





are located at φ/(pωa) ) 2.67 + 3n for n ) 0, 1, 2, 3, ..., as seen in Figure 8. They are brought about by the opening of electron inflow to the RS from the tip, with an increase in Pm in Figure 8. Subsidiary progressions of peaks for φ > 0 can be derived from Figure 4 by the crossing of horizontal lines of G(n) m ) Gm - npωa with the lower boundary of the hatched region, located at φ/(pωa) ) 0.67 + 1.5 n for n ) 2, 3, ... without n ) 0 and 1. In Figure 8, in fact, we can see a small peak located at φ/(pωa) ) 3.67 for n ) 2. It is brought about by the opening of electron outflow from the RS to the substrate, with a decrease in Pm in Figure 8. The L and M values associated with the first step in the φ dependence of the STM current I are respectively 1 and 0 for φ > 0, while both of them are 0 for φ < 0. Calculated from (3.17) and (3.18), the height of the first step is given by

e-Sa/[gs-1(1 + Sa)-1 + gt-1] for φ > 0 and

e-Sa/(gs-1 + gt-1) for φ < 0

(4.4)

with ratio about 1.26 in agreement with the intensity ratio between the first peaks in Figure 8. Finally, let us put the value of Gm at 0 cm-1. The φ/(pωa) dependence of (gt-1 + gs-1) pωa dI/dφ and Pm calculated in this case is shown in Figure 9. The former is symmetric with respect to φ ) 0, while Pm - 1/2 is antisymmetric. Moreover, Pm changes very quickly from unity to zero as φ passes zero increasingly although the STM current I, which is not shown in Figure 9, is kept very small (practically zero) around φ ) 0. Main progressions of peaks are located at φ/(pωa) ) ((2 +

3n) for n ) 0, 1, 2, 3, ..., while subsidiary progressions are located at φ/(pωa) ) ((1 + 1.5n) for n ) 1, 2, 3, ... without n ) 0, derived from the crossing in Figure 4. The former is brought about by the opening of electron inflow from the tip to the RS for φ > 0, with an increase in Pm in Figure 9, and by that of electron outflow from the RS to the tip for φ < 0 with a decreases in Pm. The latter is brought about by the opening of electron outflow from the RS to the substrate for φ > 0, and by that of electron inflow from the substrate to the RS for φ < 0. When we regard the n ) 1 peak in the subsidiary progressions as inseparable from the n ) 0 peak in the main progressions in Figure 9, the L and M values for the first step in the φ dependence of the STM current I are, respectively 1 and 0. Then

e-Sa/[gs-1(1 + Sa)-1 + gt-1], irrespective of the sign of φ (4.5) is obtained as the height of the first step by (3.19) with (3.17) and (3.18). Equation 4.5 is applicable without the simplification of gs ) gt made in the calculation of Figure 9. The mirror symmetry of the STM current I with respect to φ ) 0 for Gm ) 0 in Figure 9 is not related to this simplification. It originates in the electron-hole symmetry, since it is considered that when φ > 0, currents are carried by electrons from the tip to the substrate mediated by the RS, and when φ < 0, they are carried by positive holes also from the tip to the substrate. Because of this electron-hole symmetry, the φ/(pωa) dependence of the differential conductance (gt-1 + gs-1)pωa dI/dφ for Gm of a certain negative value, say -∆, and that for Gm ) ∆ should be a mirror image of one another with respect to φ ) 0, irrespective of the value of gt/gs. The φ/(pωa) dependence of Pm - 1/2 for Gm ) -∆ and that for Gm ) ∆ should be an antimirror image of one another with respect to φ ) 0. These relations can be checked easily by obtaining the crossing of horizontal lines of G(n) m for n ) 0, 1, 2, 3, ... with the hatched regions in Figure 4 and calculating the height of the steps in the φ dependence of I and Pm by (3.17)-(3.19) for φ > 0 or their counterparts for φ < 0. Because of this electron-hole symmetry, it is not necessary to explicitly calculate them for Gm of a negative value as long as those for Gm of a positive value have been calculated as in Figures 6-8. Widths of the peaks in Figures 6-9 are mainly brought about by the width of function W(E) of (3.11) by which kmt and ksm are determined as (3.15) and (3.16). The width Dm is determined by (3.12) and increases with temperature T when kBT exceeds about pωb, where pωb was put at 30 cm-1 in the calculation of Figures 6-9. The φ/(pωa) dependence of the differential conductance (gt-1 + gs-1)pωa dI/dφ and the occupancy Pm at the RS for Gm ) 1100 cm-1 were calculated at T ) 300 K in Figure 10, although they were calculated at T ) 10 K in Figure 7. As expected, intramolecular-phononprogressional structures apparent at T ) 10 K cannot be distinguished at room temperature because of the broadening of each peak. They were calculated at T ) 80 K in Figure 11. We see therein that essential features in the phonon-progressional structures are maintained until about liquid-nitrogen temperature. In the expressions (3.15) and (3.16) for kmt and ksm, the width Dm of (3.12) of W(E) as a function of E is much larger than kBT, while the Fermi distribution function p(E) of (2.4) changes stepwise with width of the order of kBT. In (3.15) and (3.16), therefore, p(E) can be approximated by a step function that equals unity for E < 0 and vanishes for E > 0. Adopting further the same simplification as adopted in Section 4 that both gt(E)


Sumi



and gs(E) in (3.15) and (3.16) are approximated as constant, given respectively by gt and gs, (3.15) and (3.16) can be approximated respectively by elementary functions of

kmt )

gt 2

-Sa

e

∞

San

and

ksm )

gs 2

e

-Sa

(

∑ n! erfc -

n)0

∞

San

(

∑ n! erfc -

n)0

)

Ft - λm - E(n) m

x2Dm

)

(4.6)

E(-n) - λm - Fs m

x2Dm

(4.7)

where erfc(x) [≡(2/xπ)∫∞x exp(-t2) dt] represents the complementary error function of variable x. In deriving these equations, W(E) of (3.11) was approximated by its expression for E > 0 in the entire E region, since W(E) for E < 0 is negligibly small as long as λm . kBT and it can be replaced by the first expression in (3.11), which becomes also very small for E < 0. Approximate elementary-function expressions for both kms and ktm can be obtained by exchanging t and s in (4.6) and (4.7). We should note here that erfc(x) changes from 2 to 0, with a transition region of width of the order of unity, as x passes 0 increasingly. Therefore, kmt of (4.6) with E(n) m of (3.10) changes stepwise, as a function of φ () Ft - Fs), with a transition region of width of the order of Dm/(1 - R), while ksm of (4.7) does with a transition region of width of the order of Dm/R. When these stepwise changes are approximated by vertical ones without transition width, (4.6) and (4.7) reduce,

respectively, to (3.17) and (3.18), from which step heights in the total STM current I were calculated. With these kmt, ksm, kms, and ktm, approximate elementaryfunction expressions for both I and Pm can be obtained respectively by (3.6) with (3.5) and by (3.2). The φ/(pωa) dependence of the differential conductance (gt-1 + gs-1)pωa dI/dφ and Pm calculated with these expressions are practically the same as those calculated in the previous section with the original expressions (3.15) and (3.16) both at 10 and 80 K. At 300 K, the former shows about 10% deviations from the latter at the peaks in the differential conductance. Therefore, these approximate elementary-function expressions are practically useful at low temperatures. 5. Discussion It has already been observed, as noted in Section 1, that the STM image of large molecular adsorbates changes drastically depending on the applied bias. This means that if the tip in STM is fixed at an appropriate position, the STM current would change drastically depending on the applied bias. The present work has predicted a staircase structure of this dependence together with its interpretation and presented a formula to describe it. In the present work, we considered only the STM current mediated by the RS, but there might exist parasitic channels. The STM current obtained above must be supplemented by that through these channels. The parasitic component in general does not show such a staircase-structured bias dependence as seen in the former component, constituting a structureless background, and it can easily be separated in the differtial conductance. When we can observe a staircase-structured bias dependence in the STM current or a sequence of separated peaks in the differential conductance both in the positive and the negative bias region as in Figures 6-9, analysis of them enables us to derive fundamental physical parameters characterizing the RS, as shown below. First, we can set the free-energy difference Gm at the RS from the substrate so as to be sufficiently large, as in Figures 5 and 6, without applied bias. For Gm > 0 in this case, separation between adjacent peaks is given by pωa/(1 - R) for φ > 0, and by pωa/R for φ < 0. Then, we can get pωa and R, the latter of which enables us to obtain an estimate of the relative height of the RS between the tip and the substrate. Height of the first and the second steps in the STM current is given, respectively, by 1/(gs-1 + gt-1 eSa) and 1/[gs-1 + gt-1 eSa (1 + Sa)-1] for φ > 0, and by 1/(gt-1 + gs-1 eSa) and 1/[gt-1 + gs-1 eSa (1 + Sa)-1] for φ < 0. Data of these quantities enable us to derive Sa and gt/gs. Position of the first peak in the differential conductance is given by (Gm + λm)/(1 - R) for φ > 0, and by -(Gm + λm)/R for φ < 0. Then we can get the value of Gm + λm, but we cannot get Gm and λm separately from these data alone. When Gm is set to be negative with |Gm| sufficiently large, we similarly cannot separate Gm and λm from the value of λm - Gm derived from the position of the first peak, although other parameters can be obtained separately. Even in these cases, an approximate value of λm can independently be estimated from the width of the first peak, as understood from (4.6) and (4.7): The width is about Dm/(1 - R) in the positive φ region and about Dm/R in the negative φ region in the former case for a large positive Gm, where Dm is determined by λm in (3.12). In the latter case for a large negative Gm, on the other hand, R and 1 - R are exchanged with each other in the statement mentioned above.

V-I Characteristics of STM Processes When |Gm| is not sufficiently large, as in Figures 7-9, the differential conductance is composed of two series of separated peaks, with separation of pωa/(1 - R) and pωa/R, both in the positive and the negative bias region. Detecting these two series, we can get pωa and R. With this value of R obtained, we can draw the hatched regions of Figure 4 by assuming an appropriate value for λm. Then, assuming also an appropriate value for Gm, we can draw horizontal lines of G(n) m ) Gm ( npωa for n ) 0, 1, 2, 3, ... in Figure 4 obtained above and examine crossing points of these lines with the hatched regions therein. We can determine both Gm and λm by adjusting them in such a way that φ values at these crossing points reproduce those at the peaks observed in the differential conductance. Then, equating to theoretical values the observed heights of the first and the second, or other, peaks in the STM current both for φ > 0 and for φ < 0, we can derive the values of Sa and gt/gs. The φ value at the first peak in the differential conductance is determined by the fundamental threshold with the conditions in (2.6) or (2.7). In the positive φ region, therefore, it is given by (Gm + λm)/(1 - R) when Gm > -(2R - 1)λm, and by (λm - Gm)/R when Gm < -(2R - 1)λm. In the negative φ region, on the other hand, it is given by -(Gm + λm)/R when Gm > (2R - 1)λm, and by -(λm - Gm)/(1 - R) when Gm < (2R 1) λm. They provide the fundamental condition for deriving λm and Gm values. The calculation in Section 4 was performed in a simplified situation that the interaction-weighted density of electronic states, gt(E) and gs(E) of (3.13) and (3.14), in the tip and the substrate are nearly independent of the energy variable E. This situation simplifies the analysis of experimental data, too. It is desirable, therefore, that metals used for the tip and the substrate satisfy this requirement. For φ > 0, kmt in (3.17) is obtained when the Fermi energy Ft in the tip exceeds all of E(n) m + λm ()Em + λm + npωa) for n ) 0, 1, 2, ..., M, which extend over a width of Mpωa. In (3.18), all of E(-n) - λm () Em - λm m npωa) for n ) 0, 1, 2, ..., L, extending over a width of Lpωa, must exceed the Fermi energy Fs in the substrate. For φ < 0, these statements are true with the exchange of suffixes t and s. Here, both L and M can take an integral number of several (to be more exact, much larger than Sa), and pωa is of the order of 1000 cm-1. In order for both functions gt(E) and gs(E) in (3.17) and (3.18) to be regarded as nearly E independent; therefore, they must be so over a width of about 1 eV respectively around the Fermi energies Ft and Fs in the tip and the substrate. The E dependence of gt(E) and gs(E) originates mainly from the energy dependence of the density of electronic states in metals used for the tip or the substrate. Then, these metals must have a nearly flat density-of-states function around their Fermi energy. In the present work, it has implicitly been assumed that electron hopping from the RS to the tip or the substrate takes place from the bottom of the adiabatic potential Vm(Q) in Figures 2 and 3. This assumption can be justified if we are in the limit of weak electronic coupling between the RS and the tip or the substrate since electron hopping between them becomes the slowest process. In the actual situation, it requires that the time for both inner-sphere and outer-sphere reorganization, occurring when an electron flows into the RS from the tip or the substrate, is much shorter than the time for subsequent electron outflow from the RS to the substrate or the tip. The time for the latter can be considered to be larger than about 1 ps, inferred from the primary process in photosynthesis29,30 whose electrontransfer system has been adjusted so as to have a rate as fast as possible in the course of evolution for a long time. In bacterial

J. Phys. Chem. B, Vol. 102, No. 10, 1998 1843 photosynthesis, for example, an electron is transferred from a bacteriochlorophyll dimer to a bacteriochlorophyll monomer, and subsequently from the monomer to a bacteriopheophytin.31 In each pair of pigments with a separation about 10 A,32 the electron-hopping time is of the order of 1 ps. Concerning the time for inner-sphere reorganization, it has been observed to be completed in subpicoseconds in typical dye molecules33,34 and others35,36 in solution. The time for outer-sphere reorganization can be estimated by the inverse of the width of the frequency distribution of vibrations participating in the reorganization.37,38 The width for intermolecular vibrations in the protein matrix has been observed to be of the order of several tens of wavenumbers in units of energy,20,21 corresponding to several terahertz. This means that the outer-sphere reorganization is completed also in subpicoseconds. In these situations, it seems reasonable to accept the assumption mentioned above. Short-time contributions of electron hopping in the course of the outer-sphere reorganization was taken into account by Kuznetsov and his collaborators14-16 for STM currents mediated by redox molecular adsorbates. Not only these contributions but also those in the course of the inner-sphere reorganization become unnegligible with an increase in the transfer integral for electron hopping between the RS and the tip or the substrate. In usual cases, however, it seems that they can be neglected, for the reasons mentioned in the previous paragraph. In the present calculation of the STM current mediated by an RS, it was treated as providing a real intermediate state for real transitions of an electron to or from the tip and the substrate. In the configuration shown in Figure 2, these transitions require thermal activation by molecular-arrangement fluctuations, taking place on trajectories a and b. It is possible in Figure 2, however, that the STM current is carried by virtual mediation at the RS around Q ) 0 without participation of these fluctuations. A real transition in this case is only from an electronic state in the tip around Ft to that in the substrate with the same energy in Figure 2. It can be decomposed into a virtual transition from the former to the RS with energy around Fm at Q ) 0 and a subsequent virtual transition from there to the latter. This process, called the superexchange one, is fourth order in electronic coupling, Jmi and Jmf in (3.8) and (3.9), and is higher order than the sequential process, composed of two subsequent real transitions, taken into account in the present work. Then, the superexchange process can be neglected in the limit of weak electronic coupling. However, it takes place without thermal activation by molecular-arrangement fluctuations, and in some cases, it can dominate the sequential process. When the configuration in Figure 2 transforms itself into that in Figure 3 with a change in the bias, the real transitions comprising the sequential process can take place also without thermal activation, and the superexchange process becomes completely negligible. The staircase structure in STM currents treated in the present work is brought about by the configuration in Figure 3. The superexchange process, therefore, can be neglected here. It is possible to develop a unified description of STM processes, in which both the superexchange and the sequential process are incorporated as realized in mutually opposite limits, by extending a similar description30,39 for long-range electron transfer in the initial process of photosynthesis.

Acknowledgment. The author thanks Professor Jens Ulstrup of the Technical University of Denmark for drawing the author’s attention to the present subject.


Sumi

Appendix. STM Currents Carried by Holes When STM currents are described as carried by holes, the forward current A is due to translocation of holes from the substrate to the tip. The two-step process for obtaining A is described with partial currents ksmPm by hole hopping from the substrate to the RS, (1 - Pm)kms by its reverse process and (1 - Pm)kmt by hole hopping from the RS to the tip, as (1 - Pm)kmt

ksmPm

tip 79 RS [\ ] substrate (1 - P )k m

(A1)

ms

In this case, the standard formula for a rate in a two-step process gives A as

A ) Pmksmkmt/(kmt + kms)

(A2)

where we should note that 1 - Pm in (A1) has been canceled between the numerator and the denominator in (A2). This expression of A looks different from that of (3.4), but we see that (A2) is exactly the same as (3.5) under the steady-state condition (3.1). References and Notes (1) Driscoll, R. J.; Youngquist, M. G.; Baldeschwieler, J. D. Nature 1990, 346, 294. (2) Nawaz, Z.; Cataldi, T. R. I.; Knall, J.; Somekh, R.; Pethica, J. B. Surf. Sci. 1992, 265, 139. (3) Strohmaier, R.; Ludwig, C.; Petersen, J.; Gompt, B.; Eisenmenger, W. J. Vac. Sci. Technol. B 1996, 14, 1079. (4) Kunitake, M.; Batina, N.; Itaya, K. Langmuir 1995, 11, 2337. (5) Tao, N. J. Phys. ReV. Lett. 1996, 76, 4066. (6) Lever, A. B. P., Gray, H.B., Eds. Iron Porphyrin; AddisonWesley: Reading, MA, 1983; Part 1. (7) Salemme, F. R. In Tunneling in Biological Systems; Chance, B., DeVault, D. C., Frauenfelder, H., Marcus, R. A., Schrieffer, J. R., Sutin, N., Eds.; Academic: New York, 1979; p 523. Bowler, B. E.; Raphael, A. L.; Gray, H. B. Prog. Inorg. Chem. 1990, 38, 259. (8) Beratan, D. N.; Betts, J. N.; Onuchic, J. N. Science 1991, 252, 1285. (9) Therien, M. J.; Chang, J.; Raphael, A. L.; Bowler, B. E.; Gray, H. B. In Long-Range Electron Transfer in Biology; Palmer, G. A., Ed.; Structure and Bonding 75; Springer-Verlag: Berlin, 1991; p 109. (10) Kuki, A.; Wolyness, P.G. Science 1987, 236, 1647.

(11) Kuki, A. In Long-Range Electron Transfer in Biology; Palmer, G. A., Ed.; Structure and Bonding 75; Springer-Verlag: Berlin, 1991; p 49. (12) Alekperov, S. D.; Vasiljev, S. I.; Kononenko, A. A.; Lukashev, E. P.; Panov, V. I.; Semenov, A. E. Chem. Phys. Lett. 1989, 164, 151. (13) Andersen, J. E. T.; Møller, P.; Pedersen, M. V.; Ulstrup, J. Surf. Sci. 1995, 325, 193. (14) Kuznetsov, A. M.; Sommer-Larsen, P.; Ulstrup, J. Surf. Sci. 1992, 275, 52. (15) Kuznetsov, A. M.; Ulstrup, J. Mol. Phys. 1996, 87, 1189. (16) Andersen, J. E. T.; Kornyshev, A. A.; Kuznetsov, A. M.; Madsen, L. L.; Møller, P.; Ulstrup, J. Electrochim. Acta 1997, 42, 819. (17) Kuznetsov, A. M. Charge Transfer in Physics, Chemistry and Biology; Gordon and Breach: Reading, MA, 1995. (18) Sumi, H. Chem. Phys. 1997, 222, 269. (19) Instead, it was pointed out in ref. 14 (p 59, the fourth paragraph) that the V-I characteristic is approximately Gaussian in the formulation developed there. (20) Lyle, P. A.; Kolaczkowski, S. V.; Small, G. J. J. Phys. Chem. 1993, 97, 6924. (21) Ahn, J. S.; Kanematsu, Y.; Enomoto, M.; Kushida, T. Chem. Phys. Lett. 1995, 215, 336. (22) Bixon, M.; Jortner, J.; Michel Beyerle, M. E. Chem. Phys. 1995, 197, 389. (23) Sumetskii, M. Yu.; Fel’shtyn, M. L. SoV. Phys. JETP 1988, 67, 1610. (24) Baratoff, A.; Persson, B. N. J. J. Vac. Sci. Technol. A 1988, 6, 331. (25) Glazman, L. I.; Shekhter, R. I. Solid State Commun. 1988, 66, 65. (26) Schmickler, W. Surf. Sci. 1993, 295, 43. (27) Instead, a phenomenon was pursued that even in the configuration in Figure 2, STM currents can flow barrierlessly during relaxation along the coordinate Q only in a short time immediately after electron inflow to or outflow from the RS. It is not probable, however, that this phenomenon dominates the currents, as discussed in Section 5 (the ninth paragraph). (28) Jortner, J. J. Chem. Phys. 1976, 64, 4860. (29) As a review, Friesner, R.A.; Won, Y. Biochim. Biophys. Acta 1989, 977, 99. (30) See also, Sumi, H. J. Electroanal. Chem. 1997, 438, 11. (31) Arlt, T.; Schmidt, S.; Kaiser, W.; Lauterwasser, C.; Meyer, M.; Scheer, H.; Zinth, W. Proc. Natl. Acad. Sci. U.S.A. 1993, 90, 11757. (32) Moser, C. C.; Keske, J. M.; Warncke, K.; Farid, R. S.; Dutton, P. L. Nature 1992, 355, 796. (33) Weiner, A. M.; Ippen, E. P. Chem. Phys. Lett. 1985, 114, 456. (34) Mokhtari, A.; Chesnoy, J.; Laubereau, A. Chem. Phys. Lett. 1989, 155, 593. (35) Yu, J.; Berg, M. J. Chem. Phys. 1992, 96, 8741, 8750; J. Phys. Chem. 1993, 97, 1758. (36) Fourkas, J. T.; Berg, M. J. Chem. Phys. 1993, 98, 7773; 99, 8552. (37) Toyozawa, Y. J. Phys. Soc. Jpn. 1976, 41, 400. (38) Toyozawa, T.; Kotani, A.; Sumi, A. J. Phys. Soc. Jpn. 1977, 42, 1495. (39) Sumi, H.; Kakitani, T. Chem. Phys. Lett. 1996, 252, 85.

Characteristics of STM Processes as a Probe Detecting Vibronic I

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