Quantum Tunneling Currents in a Nanoengineered Electrochemical

Article pubs.acs.org/JPCC

Quantum Tunneling Currents in a Nanoengineered Electrochemical System Chaitanya Gupta,*,† Ross M. Walker,§ Shuai Chang,∥ Sean R. Fischer,‡ Matthew Seal,⊥ Boris Murmann,‡ and Roger T. Howe‡ †

ProbiusDx, Inc., El Cerrito, California 94530, United States Department of Electrical Engineering, Stanford University, Stanford, California 94305, United States § Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, Utah 84112, United States ∥ College of Materials and Metallurgy, Wuhan University of Science and Technology, Wuhan, Hubei 430081, China ⊥ OpenGov, Inc., Redwood City, California 94063, United States ‡

ABSTRACT: We develop an equivalent circuit model for charge transfer across a nanoscale electrochemical interface and apply it to tune the interface parameters so that tunneling electrons transduce information about the vibronic structure of the interface. Model predictions are broadly consistent with cyclic voltammograms acquired using a custom, low-noise potentiostat on a 50 nm diameter Pt(80%)−Ir(20%) electrode functionalized with 10 nm diameter gold nanoparticles and immersed in a phosphate buffer with a redox couple. Conductance−voltage sweeps for 1 μM 2-Dleucine have shifted vibronic peaks from those for 1 μM leucine, indicating promise for label-free sensing. The model is based on two interdependent lengths that describe the interaction strengths between the participant electronic states in the electrolyte and the participant reaction coordinates, and between the latter and the surrounding bath modes. These lengths translate into capacitive elements which are positioned in parallel to the classical capacitance defined by the interface geometry. We identify an optimal charge-transfer regime, defined by a specific interface geometry, in which the energy transferred between the transitioning electron and a specific reaction coordinate mode is dissipated exactly by the interaction of the recipient mode with the surrounding bath. The perturbative effect of coupling external potentiostatic instrumentation to the nanoelectrochemical interface for measuring charge transfer is defined by an equivalent interface “temperature”.

1. INTRODUCTION Electron transfer is ubiquitous in physical and biochemical systems and is associated with the transduction of energy between a donor and an acceptor state. This exchange of energy is often mediated by the dielectric environment surrounding the donor and acceptor species to an extent depending on the nature of the electron transfer mechanism. In the case of adiabatic charge transfer, the electron transitions from donor to acceptor state along an isoenergetic surface. Temporal fluctuations in the intervening dielectric medium equalize the energy of the electron in the donor and acceptor states at the time of transition. The dynamics of the transfer process are determined solely by the relaxation rate of the participating nuclear vibrational modes in the medium, which follow the rapid oscillation of the electronic wave function between the donor and acceptor states along the reaction coordinates. The time scale for the dynamics of the transitioning electron are set by the slower, participant nuclear vibrational modes.1−5 Nonadiabatic charge transfer, on the other hand, is characterized by the transition of the electron across multiple © XXXX American Chemical Society

energy states, with several quanta of energy being exchanged between the electron and the environment via a scattering process that is accompanied by rapid environmental moderelaxation. The bath vibrational modes involved in the dissipation of energy are indistinguishable from one another, but for their differing frequencies, unlike in the case of the adiabatic reaction where a specific, limited number of modes, designated together as a reaction coordinate vector, collectively map the evolution of the state of the system due to their direct participation in the rate-limiting dynamics. The electronic and nuclear time scales are nonseparable in the nonadiabatic limit, so the measured charge-transfer rate reflects the coupled dynamics.6−9 A theoretical framework that interpolates between the adiabatic and nonadiabatic charge -transfer limits was recently presented,10,11 wherein polarity-based interactions and friction effects influenced the static reaction energetics and dynamics, Received: May 7, 2017 Revised: June 23, 2017

A

DOI: 10.1021/acs.jpcc.7b04350 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C respectively.12 A modification in the frictional coupling between the reaction coordinate modes and the surrounding bath that affects the longitudinal relaxation of the participant solvent reaction coordinates13 or a change in the electronic coupling matrix element was shown to induce the crossover between the two regimes, where the relaxation time for the coordinate modes is determined to be a function of the difference in dielectric polarization energy between the donor and acceptor states.14,15 The parameters that determine the nature of the interaction between the environment and the transitioning electronic wave function, namely the electronic coupling and the relaxation time constant, are usually treated independently and are prescribed empirically, depending on the material properties of the solid and liquid phases of the electrochemical interface.16 In this paper, we also define two parameters that characterize the nature of the charge-transfer reaction, which determine the strength of the electrostatic interaction between the tunneling electron and the reaction coordinate, and between the coordinate modes and their surrounding environment. We show that the two variables have an inverse dependence on the donor−acceptor polarization energy difference and hence the intrinsic charge at the interface and are therefore interdependent. Furthermore, the two parameters may be expressed as equivalent length scales that are also amenable to representation as equivalent circuit elements in a modified Randles circuit model of the interface.17 The characterization of the charge-transfer process as current flow through a collection of model circuit elements allows the elucidation of quantitative rules for the sizing of the interface, in order to specify the charge-transfer mechanism. When the postulated length scales describing the electronic state−reaction coordinate interactions and the reaction coordinate−bath interactions are comparable to one another, the charge-transfer mechanism becomes a resonant exchange of energy between the tunneling charge and a specific coordinate mode. This mechanism is of interest for transducing vibrational mode information within the measured electron-transfer rate. The electrochemical interface geometry requisite for this resonant exchange is described with the use of the length scales, as well as an equivalent circuit model. In addition, the perturbative effect on the properties of an optimal interface structure due to the coupling with an external measurement apparatus is characterized by an interface “temperature” figureof-merit that relates the geometry of the electrochemical system to the voltage noise power spectral density characteristics of the instrument. Nanoscale electrochemical interfaces are fabricated based on design guidelines derived from the circuit model. We characterize these interfaces through cyclic voltammograms, measured using an ultralow noise, three-terminal potentiostat. For an optimal interface, the cyclic voltammograms show characteristics of polaronic charging and vibronic-mode transduction. We also demonstrate an ability to differentiate between single-atom isotopic substitutions on the amino-acid leucine as well as to differentiate between two structural isomers of leucine using the cyclic voltammogram.

conventional nonadiabatic or adiabatic process. We will also explore a third possibility, where the reaction coordinate modes are only weakly perturbed by the neighboring bath. Relevant physical parameters that determine the nature of the transition are summarized in the context of this elementary theoretical description, and these parameters are then applied in a “toy” model of a molecular redox-active species dissolved in the liquid electrolyte and an ideal metal electrode. Length scales and related equivalent capacitances are derived from this basic model of the electrochemical interface, and a lumped circuit parameter representation of the charge-transfer phase space is presented both for the quantum and classical regimes. The charge-transfer system comprises the charge carrier that tunnel transitions between the donor and acceptor states, the harmonic reaction-coordinate modes accompanying the transition process that are associated with the donor and acceptor states, and the bath vibrational modes coupled linearly to the reaction coordinates, as described in the Garg−Onuchic− Ambegaokar (GOA) model18 H = Hd(1 − σz)|d⟩⟨d| + Haσz|a⟩⟨a| + Vad|a⟩⟨d| + Vda|d⟩⟨a| Ṗ 2 ⎞ 1⎛ 1 1 ⎞⎛ ⎟⎜(PΩ + Pd + (Pa − Pd)σz)2 + Ω2 ⎟dV + ∑ ⎜ 2 − εΩ ⎠⎝ Ω ⎠ Ω 2 ⎝ nΩ εo

∫

+

∫∑ k

⎛⎛ ⎜ 1⎛ 1 1 ⎞⎜⎜ ⎜ 2 − ⎟ ⎜Pk + 2 ⎝ nk εo εk ⎠⎜⎜⎜ ⎜⎝ ⎝

⎞ ⎟ Pk̇ 2 ⎟ dV + ωk 2 ⎟ ⎟ ⎠

PΩ

(1 − )ε ( εo εk

Ω

1 nk 2εo

−

1 εk

)

⎞2 ⎟ ⎟ ⎟⎟ ⎠

(1.1)

The first four terms on the right-hand side (rhs) of eq 1.1 represent the state of the charge carrier when it is occupying the donor state (Hd), the acceptor state (Ha), and the energy of coupling between the participant electronic states (Vad,Vda). The parameter σz is the charge-transfer equivalent of the spin, which can asymptotically attain values between 0 and 1, thereby indicating occupancy of the charge carrier in the donor or acceptor electronic states. These terms define the energies of the two participant manifolds and their interactions, which εσ ℏΔ replace the spin Hamiltonian terms 2 o σz + 2z in the GOA model. The states of the reaction coordinate modes Ω and the bath modes ωk are each represented by harmonic vibrations of a vector of specific polarization density, PΩ and Pk, respectively, in the limit that the displacements of the modes are small. These polarization vibrational modes of the reaction coordinates and the bath are assumed to be normal to each other respectively in accordance with the Kubo description of the energetics of dielectric media.19 The potential energy of the polarization modes associated with the reaction coordinates reduces to the harmonic manifold corresponding to the donor states for σz = 0 and for σz = 1 reduces to the energy of the states associated with the harmonic acceptor manifold. The reaction coordinates in eq 1.1 are also linearly coupled to the charge state through the parameter σz and also to the bath modes, as in the GOA model. The coupling to the bath mode k 1 is determined by the coupling constant cΩ−k = ,

2. THEORY 2.1. Unified Model for the Description of Two-State Charge Transfer. In this section, we develop the dynamical characterization of charge transfer from a donor to an acceptor state, with emphasis on those conditions under which the probabilistic transition of the system can be reduced to a

(1 − εo / εk)εΩ

estimated from the electrostatic coupling energy between the polarization densities PΩ and Pk in the volume of the electrolyte B


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The Journal of Physical Chemistry C phase that is bounded by a solid metal electrode surface on one side:20 EΩ‐bath =

λΩ =

P

∫ εΩ ∑ Dk dV Ω

is the polarization density of the harmonic reaction coordinate mode Ω in its ground state and associated with configuration i, and VΩ is the spatial volume required to sustain the modal oscillations. Therefore, the square of the displacement parameter, λΩ2, is a nondimensional measure of the difference in energies of the donor and acceptor species in their respective ground states. The proposed transform enables estimates for the transition probability under the Born− Oppenheimer approximation by a separation of the dynamics of the electronic and reaction coordinate degrees of freedom.9 The vibrating polarization modes associated with the reaction coordinate couple linearly to the bath polarization modes. The weak interaction between the coordinate and the significantly more numerous bath states is characterized by lumped, macroscopic parameters like the damping viscosity and thermal noise for separable reaction coordinate and bath states. Specifically, the damped oscillatory dynamics of a reaction mode coordinate of frequency Ω are expressed in terms of the evolution of the creation operator as22

where Dk is the electrical displacement field associated with mode k. The sixth term on the rhs of eq 1.1 includes this linear coupling energy in the bath Hamiltonian. In addition, ni2 and εi in eq 1.1 are the high-frequency refractive index and lowfrequency dielectric permittivity associated with mode i. The probability amplitude A for the collective states of the charge-transfer system to transition from an initial donor configuration |d⟩ to a final acceptor configuration |a⟩ is estimated to be t Aad (t ) = ⟨a|exp( −j /ℏ ∫0 V̂ (t ′)dt ′)|d⟩,

in the strong electronic coupling limit

(2.1)

The configuration of the states includes all participating degrees of freedom, i.e., both nuclear and electronic. The transfer rate, in the Fourier space, is directly estimated from the probability amplitude as21 kd → a

1 = lim t →∞ t

∫0

t

⎛ (Δε − E + E ) ⎞ d d a dt exp⎜ −j t ′⎟ ⎠ dt ′ ℏ ⎝

⟨Aad (t ′)Ada⟩E , T

aΩ†(t ) = aΩ†(0)e (2.2)

⎛ ω ∑ ⎜⎜ 1 i i ⎝ n 2ε − i o

⟨ ⟩E,T in eq 2.2 denotes an ensemble average of the transition probability over the electronic and nuclear degrees of freedom, where the term “nuclear” includes the reaction coordinate and bath modes. On the other hand, the transfer rate, for the same transition from donor to acceptor, in the weak electronic coupling limit is given by the Fermi golden rule approximation as9,21 kd → a =

1 ℏ2

∞

⎛

j

⎜

(

Ω̃ = Ω −

̃

εo εi

)

e jΩt − γ /2t − e jωit † bi (0) Ω̃ − ωi + jγ /2

∫ dω g(ω)

(2.3)

Ωω

where Ed and Ea are the electronic energy eigenvalues associated with the donor and acceptor Hamiltonians, respectively, and V̂ is the electronic coupling operator defined in eq 1.1, with V̂ (t) being the interaction operator for V̂ in the Heisenberg representation. The expression in eq 2.3 is derived from a perturbative expansion of the probability amplitude in eq 2.1, to first order in V̂ for the weak electronic coupling limit. The notation ⟨ ⟩T in eq 2.3 is an ensemble thermal average of the quantity within the brackets over the nuclear degrees of freedom. A unitary polaron transform of the electronic coupling operator decouples the reaction coordinate modes from the electronic states and transforms the relevant transfer operator ̂ as9,19 f(t)

(

1 nΩ 2εo

−

1 εΩ

)(

1 n2(ω)εo

−

1 ε(ω)

2

)(1 − ) ε εo ε(ω)

2

⎛ 1 ⎞ ⎜ ⎟ ⎝Ω − ω⎠

Ω

(4.2)

and γ=

2πg (Ω)Ω2 (εΩ − εo)2

(

1 2

nΩ εo

−

1 εΩ

2

)

(4.3)

represent the mode shift and effective viscosity acting on the coordinate mode as a result of interaction with the bath, both of which are functions of the bath density of states g(ω). The second term on the rhs of eq 4.1 is representative of an effective “noise” from the thermalized environment that affects the dynamics of oscillator mode Ω. In the weak electronic coupling limit, it is understood that the coordinate and bath degrees of freedom are indistinguishable from one another, implying that the coupling term responsible for the damping and noise and/ or the bath density of states reduce to zero in this asymptotic case. The time-dependent dynamics of the coordinate oscillator factor in the unitary transformation of the transfer operator in eq 3.1 and render bath and reaction coordinate states separable, which in turn influences the estimation of the transition rates in

(3.1)

where f(t) is given by Vad|a⟩⟨d|e−jωadt + Vda|d⟩⟨a|e−jωdat and j

⎞1/2 1 ⎟ 1 ⎟ 1− εi ⎠

⎞1/2 ⎟ ⎛1 ⎞ ⎜ ⎟ 1 ⎟ ⎝ εΩ ⎠ εΩ ⎠

with the corresponding conjugate expression for the evolution of the annihilation operator. In eq 4.1

⎟

f ̂ (t ) = f (t )exp(−j ∑Ω λΩ(aΩ†(t ) − aΩ(t )))

⎛ Ω ⎜ − j⎜ 1 ⎝ nΩ2εo −

j Ω̃t − γ /2t

(4.1)

⎞

∫−∞ dt exp⎝− ℏ (Ed − Ea)t ⎠⟨V̂ (t )V̂ ⟩T

(3.2)

PΩi

(1.2)

k

1/2 (Pa Ω − Pd Ω)VΩ ⎛ 1 1 ⎞ ⎟ ⎜ − 2 εΩ ⎠ (ℏΩ)1/2 ⎝ nΩ εo

t

exp( − ℏ ∫0 (Vad|a⟩⟨d|e−jωadt ′ + Vda|d⟩⟨a|e−jωdat ′)dt ′) in the weak and strong electronic coupling scenarios. In eq 3.2, ωad is (Ea − Ed)/ℏ, and λΩ is a nondimensional measure of the shift between the harmonic potential energy surfaces of the reaction coordinate modes in the donor and acceptor configurations, given by C


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The Journal of Physical Chemistry C

the mode frequencies. As described previously by the GOA model, for the strong electronic coupling case, the transition rate is independent of the electronic coupling energy and proportional to the root of the reorganization energy and to the modified mode frequency Ω̃, whereas the transition rate in the weak electronic coupling regime is characterized by a proportional dependence on the electronic coupling energy and inverse dependence on the root of the reorganization energy and the real mode frequency Ω.18,24 Having demonstrated the validity of our model for transition rate estimation by comparing its asymptotic results with those yielded by an oft-cited and previously published model, a specific scenario for determining the charge-transfer rate is considered when (i) λΩ(2nΩ̃ + 1) → 0 for all Ω and (ii) when damping effects on the motion of the reaction coordinates are vanishingly small. With these assumptions, the transition rate reduces to

eqs 2.2 and 2.3). For the specific case when there is large damping on the motion of the reaction coordinate mode Ω, subject to Ω̃ > γ, we have

⟨Aad (t )Ada⟩E , T

⎛ ⎜ = exp⎜∑ −λΩ 2(2nΩ̃ + 1)(1 − e−γt ) ⎜Ω ⎜ ⎝

⎛ ⎛ ⎜ ⎜ 1 + λΩ 2e−γt ⎜ −j Ω̃t ⎜1 − 1 ⎜⎜ ⎜ 2(ε Ω̃ − εo) 2 − nΩ̃ εo ⎝ ⎝

(

1 ε Ω̃

)

⎞ ⎟ ⎟ ⎟ ⎠

⎞⎞ ⎟⎟ − Ω t (2nΩ̃ + 1)⎟⎟ ⎟⎟⎟⎟ ⎠⎠ ̃2 2

(5.1)

A coherent exchange of energy between a resonant bath mode and the frequency-shifted reaction coordinate mode with frequency Ω̃ results in a modification to the oscillatory part of the function on the rhs of eq 5.1. All other interactions between the bath and the reaction coordinates are captured by the effective frequency shift Ω − Ω̃ and the damping term γ. The thermalized occupancy of the reaction coordinate oscillator energy levels is described by nΩ̃ in eq 5.1. The transition rate is shown to be kd → a ∝

∑ λΩ 2(2nΩ̃ + 1)Ω̃

kd → a

⎛ ⎜ ⎜ ⎜ γ ⎜ ⎜⎛ ⎛ ⎜⎜ ⎜ ⎜ ⎜Δε − Ed + Ea − ∑Ω Ω⎜1 − ⎜ ⎜ ⎜ ⎝ ⎝⎝

2

Ω

⎛ ⎜ exp⎜ − ⎜ ⎝

(

Δε − Ed + Ea ℏ

+ ∑Ω γλΩ 2(2nΩ̃ + 1)

2

∑Ω λΩ 2(2nΩ + 1)Ω2 Ed − Ea ℏ

∑Ω λΩ 2(2nΩ + 1)Ω2

2

⎞

)⎟

+ ∑Ω λΩ 2(2nΩ + 1)Ω

⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ 2 ⎟ ⎞⎞ ⎟⎟ 1 2⎟ +γ ⎟ ⎛ 1 1 ⎞ ⎟⎟ ⎟ (εΩ − εo)⎜ 2 − ⎟ ⎟⎟ ⎝ nΩ εo εΩ ⎠ ⎠⎠ ⎠

where the rate peaks at the resonance between the energies of the electronic states and the collective of harmonic reaction mode oscillators. This resonance is both shifted and broadened by the interaction of the oscillators with the bath. The result in eq 5.4 indicates that a reaction system with constrained degrees of freedom for the motion of the reaction coordinate oscillators may be interrogated for the vibrational frequencies of the oscillators by sweeping the energy of the electronic states and tracking the appearance of resonance features in the measured transition rate as a function of the electronic energy of the reaction system. The remaining portion of this section will identify design rules for an electrochemical interface that enables the existence of the conditions necessary for the transduction of the reaction coordinate vibrational frequency information, namely (i) λΩ,nΩ → 0, (ii) vanishing interaction of coordinate subsystem with the external bath, and (iii) restricted degrees of freedom for the reaction coordinate in the electronic transition from donor to acceptor. These design rules identify specific parameters associated with the materials and geometry of the interface that isolate the electrochemical charge-transfer process from classical effects, which normally dominate the measured electronic current. 2.2. Molecular Model for Electrolyte-Specific Redox Species. In this section, we develop an elementary onedimensional (1D) model for the vibronic structure of an electrolyte-dissolved molecular redox-active species and derive two length scales that describe the interactions between the electronic states of the redox-active molecule and the harmonic states of the reaction coordinate, as well as the interactions between the bath and reaction coordinate degrees of freedom,

|Vad|2

(

)

⎟ ⎟ ⎠

by a straightforward substitution of eq 5.1 in eq 2.2, for the strong electronic coupling scenario. In a similar vein, the transition rate for the weak electron coupling limit is given by

⎛ ⎜ exp⎜ − ⎜ ⎝

1 εΩ

(5.4)

(5.2)

kd → a ∝

(

⎞ ⎟ ⎟ ⎟ ⎠

⎞

)⎟

2 ∑Ω (Ω̃ − γ 2)(2nΩ̃ + 1)λΩ 2

⎛ ⎜ 1 ∝ ∑ λΩ 2γ Ω⎜1 − 1 ⎜ Ω (εΩ − εo) 2 − nΩ εo ⎝

(5.3)

by a substitution of the time-dependent and dissipation-free dynamics of the reaction coordinate oscillator modes in eq 2.3. The rate expressions derived above assume single donor and acceptor energy levels in the participant species, and when more than one energy level is involved in the transfer, the rates are ensemble averaged over the occupancy statistics of the multiple energy states.23 The barrier inhibiting the transition, referred to as the reorganization energy, is a measure of the energy expended, either to the ambient bath as in eq 5.2 or to the electronic degrees of freedom in eq 5.3, in the process of altering the polarization density from a configuration at equilibrium with a donor to a set of states at equilibrium with an acceptor species. Equations 5.2 and 5.3, as outlined above, capture the salient dependencies of the transition rate on the electronic coupling energy, the reorganization energy, and D


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The Journal of Physical Chemistry C respectively. These length scales are shown to characterize the nature of the charge-transfer process, based on the relative magnitude of the interaction modalities they describe. The potential energy of interaction between the electronic states of the redox-active species and the surrounding electrostatic environment comprising solvent polarization modes and other charged species is described by a conventional, continuum electrostatics formulation, in 1D,20 as V (x ) =

( −e ∫ dx″|ψ (x″)|2 ) 4πεo +

with UN−N defining the “shifted” ground state for the redox system arising from coupling with the polarization modes. From a dimensional analysis of the expression of the electronic state energy in eq 6.4, a length scale can be identified for the interaction between the electronic states of the redox moiety and its external electrostatic environment, which includes the polarization density harmonic oscillator states:

∫ dx′P(x′)·∇ |x −1 x′|

λe −Ω =

where ρ and P are the free charge and reaction coordinate polarization densities associated with the surrounding electrostatic environment. The existence of the wave function terms in the rhs of eq 6.1 couples the potential energy to the solution of the Schrödinger’s wave equation (SWE), requiring an iterative procedure for the estimation of the wave function. Our objective is not an exact analytical calculation of the functional dependence of the wave function on the spatial coordinates of the reaction system, but rather to find a form of the solution that yields the empirical length scale parameters. In this context, we set ∫ dx″|ψ(x″)|2 = α, where the integral is over the spatial extent of the redox molecule. α is an initial guess input to the iterative algorithm for the solution of the SWE and is a parameter between 0 and 1. The application of Gauss’ law and boundary conditions at the metal−electrolyte interface yields a simplification of eq 6.1 ⎛1 ( − αe) ⎛ Q int 1⎞ − αe ⎜ Q int⎜ − ⎟ + + 4πx 4πε ⎜⎝ x ε⎠ ⎝ εo

∑ i

4πεoℏ2 meeQ int

(7.1)

A large λe−Ω corresponds to a strong coupling between the reaction coordinates and electronic states and implies enhanced responsivity of the redox electronic states to perturbing potentials from sources external to the subsystem of electronic states. Another length scale can be identified in eq 6.4 that characterizes the self-energy of reaction coordinate modes: λΩ− n

⎞ Q int 2 ⎛ 1 1 ⎟ ⎜ = − ⎜ 2 ℏΩm ⎝ nΩm εo εΩm ⎟⎠

(7.2)

where Ωm is the largest mode frequency of the participant reaction coordinate modes. The parameter λΩ−n measures the energy content of a collection of oscillatory degrees of freedom comprising of relevant reaction coordinates. A large λΩ−n is indicative of increased energy transferred via dissipative forces to the coordinate modes during the charge-transfer process. The largest mode frequency is representative for the coordinate mode frequencies and results in a conservative estimate for λΩ−n. A comparison of λΩ−n in eq 7.2 with the nondimensional shift parameter λΩ in eq 3.2 indicates that both parameters are representative of the same physical quantity, namely the energy of the nonelectronic modes of the system. 2.3. Lumped Equivalent Circuit Representations of Charge-Transfer System. The interaction between electronic and reaction coordinate degrees of freedom as well as between the coordinate degrees of freedom of the reaction system can also be represented as equivalent capacitances by utilizing the phenomenological length scales described in the previous section. We define

⎞ ⎟ x − xi ⎟⎠ qi

(6.2)

where Qint is the interface charge in the tunneling path between the metal and the redox-active species dissolved in the electrolyte and qi represents the collection of charges in the immediate environment of the redox species outside the transfer path of the tunneling charge. The first term on the rhs represents the contribution to the potential energy of the electronic states from the coupling to neighboring reaction coordinate polarization densities. The second term represents the contribution from Coulombic coupling to the neighboring charges in the electrostatic environment around the redox molecule and, in the classical case, determines the fraction of the applied potential bias driving the electron transfer.25 In eq 6.2, the interface is set at x = 0. A solution for the SWE with the simplified potential energy function from eq 6.2 ⎡ −ℏ2 2 ⎤ ∇ + V (x) + UN − N ([Pi ])⎥ψ (x) = Eψ (x) ⎢ ⎣ 2me ⎦

⎞2 ∑i qi)⎟⎟ + UN − N ⎠ (6.4)

(6.1)

V (x ) =

me 2k 2ℏ2

⎛ − αe ⎛1 ( − αe) 1⎞ ⎜⎜ Q int⎜ − ⎟ + (Q int + ε⎠ 4πε ⎝ εo ⎝ 4π

∫ dx′ |xρ−(x′x)′|

( −e ∫ dx″|ψ (x″)|2 ) 4πεo

Ek = −

cq =

meeQ int 4π ℏ2

(8.1)

and ⎞−1 εΩmℏΩm ⎛ 1 1 ⎜ ⎟ − cn = εΩm ⎟⎠ Q int 2 ⎜⎝ nΩm 2εo

(6.3)

can be found using the same approach used to determine the energy states of the hydrogen atom.21 UN−N([Pi]) represents the self-energy of all the polarization states Pi of the reaction system, and this self-energy term varies on a significantly slower time-scale than does V(x). The polarization and electronic degrees of freedom of the redox system are thus separable under the Born−Oppenheimer approximation, and with this consideration, the energies of polarization-dressed electronic states are estimated as

(8.2)

as the “quantum” and “nuclear” capacitances per unit area, respectively. The quantum capacitance cq, as defined in eq 8.1, is the electrochemical analog to the Luryi quantum capacitance for a two-dimensional conductor26 and is a measure of the screening of the redox electronic states by the reaction coordinate polarization modes. The nuclear capacitance cn, as defined in eq 8.2, is an estimate of the self-screening by the reaction coordinate polarization modes and is a physical entity E


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dependent field in the charge-transfer pathway and from the geometry of the interface outside of the charge tunneling path, respectively. In addition, a nonlinear dissipative element Zdiss characterizes the bath coupled dynamics of a collection of harmonic reaction-coordinate oscillators, and a Warburg-type impedance element ZW models the diffusional and field-driven electro-migration transport of reactants to the interface. For subsequent analysis, we shall ignore transport-related limitations to the measured charge-transfer rate and focus on the transition dynamics accompanying the charge-transfer process. The equivalent circuit description of the electrochemical interface in Figure 1B reduces to the well-known, often-used Randles equivalent circuit in the classical limit that ℏ → 0 (Figure 1C). In a similar vein, the classical Randles circuit description is also recovered for large and small Qint, indicating that the magnitude of Qint serves as a metric for the degree of quantum behavior at an electrochemical interface, like the Planck constant or temperature for other physical systems. However, unlike these parameters the dependence of the nature of the system on interface charge is nonmonotonic, i.e., there is an optimal magnitude of Qint for which the electrochemical interface manifests quantum behavior, which we define as Qintquant in eq 8.3b. In this context, a “quantum” electrochemical interface is gated by Qint and is characterized by interactions of comparable strength between the molecular electronic states and the harmonic polarization states associated with the reaction coordinates, as well as between the reaction coordinate polarization states themselves.

that is very similar to a classical capacitance but is specific to the field due to Qint alone. In addition, the interface geometry contributes to another charge distribution that is distinct from Qint. These charges are localized on surfaces that are either outside the physical charge-transfer path or arise from a classical field that is superposed directly on the transfer path due to the geometry, as shown in Figure 1A. The latter charge distribution,

Figure 1. (A) Pictorial representation of Qint as charge in the path of the electron’s tunneling transition from the electrode to the redox couple, as distinct from Qclass which originates from classical fields at the interface (Qclass = ∑qi). (B) Equivalent circuit representation of an electrochemical interface operating in the quantum regime, where cq is the quantum capacitance, cn the nuclear capacitance, Cgeom the classical capacitance along the charge-transfer path, Cpar the classical capacitance outside the charge-transfer path, Zdiss a dissipative element modeling the bath-coupled dynamics, and ZW a Warburg-type impedance modeling diffusional and field-driven transport. (C) Reduction of equivalent circuit to the Randles classical equivalent circuit model in the limit of ℏ → 0.

λe −Ω = λΩ− n or cq = cn

Q int quant

collectively represented as ∑qi in eq 6.2, usually determines the total potential drop at the interface that drives the electrontransfer reaction in the classical case.17,25 However, in the general case when quantum effects are also deemed important, the potential energy from the applied bias would be split between the field due to Qint and the field from ∑qi. Accordingly, the equivalent circuit representation for the charge-transfer interface is depicted in Figure 1B, where Cgeom and Cpar are classical capacitances arising from the geometry-

(8.3a)

⎛ ⎞1/3 ⎜ 4π ℏ3ε Ω ⎟ Ωm m ⎟ ≈⎜ ⎜ ⎛ 1 1 ⎞⎟ ⎜ mee⎜ n 2ε − ε ⎟ ⎟ Ωm ⎠ ⎠ ⎝ ⎝ Ωm o

(8.3b)

A physical interpretation of eqs 8.3a and 8.3b is that there is a coherent exchange of energy between the electronic and polarization degrees of freedom upon the transition from the donor to acceptor configuration. 2.4. Geometric Sizing Rules for a Quantum Electrochemical Interface. The representation of molecular interactions as lumped circuit elements enables a quantitative estimation of the three-dimensional geometry of the charge-

Figure 2. (A) Schematic of nanoscale electrochemical interface of area AS coated with a self-assembled monolayer of thickness δ. The electrode outside of the nanoscale interface is coated with an insulator of thickness tins and has area Apar. (B) Schematic of a modified interface, in which a functionalized metallic nanoparticle is used to modify the tunneling energy barrier at the interface. F


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The Journal of Physical Chemistry C transfer interface. The interaction capacitances cq and cn are compared with geometry-dependent capacitances that arise from the micro- and nanoscale structures at the electrochemical interface to ensure that classical shielding effects do not limit the interactions between the electronic and reaction-coordinate polarization states. The proposed geometry of the interface comprises an area AS where the electrode is in direct contact with the electrolyte and across which active charge transfer occurs and an insulating film of thickness tins and dielectric permittivity εins that covers and insulates an area Apar of the conductive metal surface from the liquid electrolyte. In addition, the area AS may be functionalized by a molecular film of thickness δ and permittivity εm to impart specific electronic properties and surface chemistries to the chargetransfer interface (Figure 2A). For the molecular thin film, the consideration that interaction effects, as described by cq or cn in the quantum limit (eq 8.3), are dominant over the classical screening of bath polarizations and charges across the thin film yields δ 4π ℏ2 < εm meeQ int quant

i.e., transition probabilities for contributions from coordinate modes with higher frequencies tend to be thermalized for AS as estimated from eq 10.2. The total energy for reorganization of the coordinate modes in the electronic transition is usually determined from the equivalent electronic energy required or expended for transition from the donor to acceptor state for an interface in a static equilibrium nuclear configuration.17,29 In this context, the reaction reorganization energy, from our model, is 2 quant ⎡ me ⎛ eQ int 1 1 ⎤⎞ ⎜ ⎟ Er ∼ ⎢ − ⎥⎟ ⎜ ε ⎦⎠ 2ℏ2 ⎝ 4π ⎣ εo

which follows from eq 6.4, where only the contribution of the reaction-coordinate polarization is considered in the determination of the equivalent energy required for the electronic transition. The appropriate redox chemistry for the electrolyte phase is selected using the criterion that the reorganization energy, Δr, of the redox couple is closest to but smaller than Er in magnitude. Such a rationale for selection implies that with the chosen redox chemistry, a smaller magnitude of the interface charge exists in the tunneling path, i.e. QintΔr < Qintquant. The difference in the tunneling path charge from the optimum value is accommodated by selecting the chemistry and thickness of the molecular thin film functionalizing the sensor interface. The tunneling barrier to overcome for the electronic transition to occur across the thin film depends on the coupling between the thin film and the donor (hD−f), between the film and the acceptor species (hf−A), as well as the coupling between the molecular units comprising the thin film (hf−f) which are assumed to be equal.30

(9.1)

A similar consideration for the capacitance due the insulator between the electrolyte and the underlying metal electrode describes a lower limit for the ratio AS/Apar AS 4π ℏ2εins > A par t insmeeQ int quant

(9.2)

The lateral dimension for AS is estimated from the volume required to contain the equivalent thermal spread in the harmonic polarization density, when the spread is of sufficient magnitude (∼Qintquant) to facilitate a transition event between the relevant polarization states solely via thermal excitation. The area AS introduces a spatial limit on the reaction coordinate states that can sustain the polarization density variations necessary for thermalized charge transfer, which in turn stipulates an upper limit on the number of reaction modes that can contribute to the measured transition dynamics in eq 5.4. For coherent polarization states, the uncertainty in the polarization state is given by22,27 ⎛ ⎜ ⟨ΔPΩ 2⟩ = ⎜ ⎜ 2V ⎝ Ω

(

ℏΩ 1 nΩ 2εo

−

1 εΩ

)

Δ Er = r + βf − f δf − f + βf − A δf − A kBT kBT

(11.3)

where βf−f and βf−A are the electron hopping matrix elements of the film and the functional end group and δf−f and δf−A are the thicknesses associated with the film and functionalization chemistry, respectively. From eq 11.2, the electron hopping matrix elements30 are

where the time dependence in eq 10.1 occurs over the slower time scales associated with the ensemble-averaged parameters like γ. The rationale for assuming coherence in the polarization states presupposes a classical trajectory for the heavier nuclear modes under the influence of an external biasing field.28 VΩ is the effective volume of the electrolyte (AStins as defined by the geometry of the interface in Figure 2A) that “contains” the polarization mode in eq 10.1. Therefore, the required lateral dimensions for gating the contributions from modes with frequencies smaller than Ωm is given by

βf − f =

⎡ Δε ⎤ ln⎢ ⎥ ζf − f ⎣ hf − f ⎦

(11.4a)

⎡ Δε ⎤ ln⎢ ⎥ ζf − A ⎣ hf − A ⎦

(11.4b)

2

and

1 ⎤⎞ εΩm ⎥ ⎦⎟

⎟ ⎟ ⎟ ⎠

(11.2)

In eq 11.2, Δε is the tunneling energy gap, which is the energy required to liberate a hole from the acceptor species and locate it on the film valence band and ζ and ζf−A are the thicknesses of the individual molecular units associated with thin film and the end-group functionalization chemistry that is in contact with the electrolyte, respectively. The total charge-transfer barrier, expressed nondimensionally in units of thermal energy, for specific redox chemistry, molecular film chemistry, and functionalization end-group chemistry is

⎞ ⎟ −γt ⎟(e + (1 − e−γt /2)2 ) ⎟ ⎠

⎛ ⎡ 1 2t ins⎢ 2 − ⎜ (Q int ) ⎣ nΩm εo ⎜ AS = 2 ℏΩm ⎛ εo ⎞ ⎜ ⎜1 − ⎟ ⎜ ε ⎝ Ωm ⎠ ⎝

2 2 ⎡ Δε ⎤⎞ ℏ2 ⎛ 2 ⎡ Δε ⎤ ⎜⎜ ln⎢ ln⎢ ⎥⎟⎟ ⎥+ 8me ⎝ ζ ⎣ hf − f ⎦ ζf − A ⎣ hf − A ⎦⎠

ΔEtunn =

(10.1)

quant 2

(11.1)

βf − A =

2

which are dependent on the geometry and the tunneling energy gap normalized by the respective coupling energies. Equations

(10.2) G


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The Journal of Physical Chemistry C 10.2 and 11.1 in conjunction with eq 11.3 model electrochemical interfaces suited for operating in the quantum regime, subject to the limitations prescribed by eqs 9.1 and 9.2. Additional structures may be introduced at the interface for the reduction of the tunneling energy gap, which enable the use of thicker functionalization films. Thicker films are attractive because off-the-shelf functionalization chemistries can be used to self-assemble a stable film onto the sensing area AS. In addition, if the functionalization monolayers are prepared by self-assembly, these energy gap-reducing structures provide tolerance against multilayer thin films that may form because of inhomogeneity in the self-assembly process. The functionalization of metallic nanoparticles onto AS with a molecular linker (Figure 2B) results in a built-in electric field when there is a work function difference with the electrode metal. This field reduces the effective tunneling energy gap associated with the rate-limiting tunneling path in the thicker functionalization film. The molecular linker used to tether the nanoparticle bridge should be thinner than the functionalization film thickness to ensure that the macroscopic electrode−linker−nanoparticle assembly does not become rate-limiting in the tunneling path. Also, the nanoparticle should be larger than 2 nm in diameter for a Fermi level associated with a significant density of states to exist for the cluster of metal atoms.31 The reduced tunneling energy barrier due to the built-in field allows a thicker film to satisfy the energy balance in eq 11.3 for the rate-limiting pathway. These thicker films, however, continue to be subject to the equivalent thickness constraint outlined in eq 9.1. The functionalization of the nanoparticle with the increased density of states at a higher energy may be thought of as equivalent to the existence of a surface state at the metal electrode surface with concomitant band-bending of the Fermi energy at the interface. The distortion of the Fermi level at the interface reduces the effective tunneling barrier for transport across the molecular film, resulting in a larger β value for the film. The incorporation of these additional degrees of freedom in the interface design allows for the use of a larger range of material chemistries in the fabrication of the electrochemical interface and also provides for design tolerances that guard against the formation of a nonideal thin film at the nanoscale interface. A representative nanoscale electrochemical interface is sketched in Figure 3, consisting of (a) an organic alkanethiol monolayer as the functionalization thin film on gold nanoparticles, with diameter larger than 2 nm, linked to a Pt(80%)− Ir(20%) macro electrode by ethylene dithiol linker molecules; (b) silicon dioxide (SiO2) as the insulating film; and (c) a 100 mM phosphate buffer aqueous electrolyte. The cutoff maximum frequency Ωm is estimated to be 1.5 × 1014 rad/s based on eqs 9.2, 10.2, and 11.3. For the above choice of materials and mode frequencies characteristic of the interface, the upper limit of the functionalization film thickness δ is 0.5 × 10−9 m and the reorganization energy of the interface is Δreorg ∼ 0.5 eV; therefore, suitable redox chemistry for the interface is the ferro-/ferricyanide redox couple.32−34 2.5. Potentiostatic Feedback Control of Electrochemical Interface with Quantum Behavior Using a Low Voltage Noise Three-Terminal Potentiostat Apparatus. The nanoscale electrochemical system is interfaced with an electronic, three-terminal potentiostatic apparatus for the application of a bias voltage and the measurement of the resulting current. The instrumentation acts as an external bath that couples into the dynamics of the quantum electrochemical interface, which in turn perturbs the transition dynamics from

Figure 3. Schematic depiction of a representative nanoscale electrochemical interface operating in the quantum regime (not to scale), consisting of gold nanoparticles (diameter larger than 2 nm) functionalized with an alkanethiol monolayer, linked to a Pt(80%)− Ir(20%) macro electrode by ethane dithiol linker molecules, an insulating film of silicon dioxide (SiO2) with thickness of 1 μm. The electrolyte is a 100 mM phosphate buffer solution, and ferro-/ ferricyanide is a suitable redox chemistry with a reorganization energy of 0.5 eV.

the donor to acceptor state in the event that a voltage stimulus is applied to the system. The influence of the accompanying electronics on the evolution of the transition event is described in this section, including the mitigation of the resulting disturbances that arise from a coupling with an external measurement system. In this context, the use of potentiostatic feedback for the application of a low voltage-noise excitation bias and for the measurement of the resulting current between the “source” and “drain” of a molecular-scale charge-transfer system has been characterized in Gupta et al.28 The source and drain in this context refer to the entities participating in the energy exchange interaction that accompanies the physical transition of the system from donor to acceptor state, with the energy flow occurring from source to drain. For the chargetransfer interface, the source and drain elements of the system are, interchangeably, the vibration-dressed electronic (vibronic) energy levels associated with the redox species and the vibrational modes of the participant reaction coordinates. A reference electrode in the electrolyte phase is used as an imperfect measurement proxy for the vibronic energy level of the molecular redox-active species, with dissipation in the measurement of the voltage in the electrolyte phase occurring because of coupling of the measurement apparatus with the redox electronic and vibrational degrees of freedom. The physical nanoscale electrochemical interface is connected to a grounded working node, and the counter electrode node supplies the current necessary to maintain a desired bias between the redox species in the electrolyte and the Fermi level in the metal electrode (Figure 4). The reaction coordinate harmonic modes at the interface between the nanoscale electrode structure and the electrolyte phase constitute the complementary source/drain to the redox dressed electronic energy levels. The different nondissipative coupling modalities between the source and drain elements are modeled by the circuit elements described in section 2.3. In addition, a dissipative element between the source and drain elements of the charge-transfer system characterizes the coupling of the participant resonant modes, namely, the vibronic states of the redox species and the reaction coordinate modes at the interface to the thermodynamic bath. For the weak linear coupling approximation, this element is modeled by a resistive H


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Figure 4. Nanoscale electrochemical interface. (A) Fabrication sequence: electrochemical sharpening of the tip of the Pt−Ir wire, coating with an insulating HfO2 film using atomic layer deposition (ALD), etching a small hole in the film using a focused ion beam (FIB), functionalization of the exposed Pt−Ir in the hole, and mounting in hole in PDMS membrane. (B) SEM image of tip of coated Pt−Ir wire showing 50 nm diameter hole in the HfO2 film after FIB etching (red circle in panel A). (C) Photograph of electrochemical cell with Ag-AgCl reference electrode and gold wire for counter electrode. Nanoscale electrode on tip of wire mounted at the surface of the PDMS gasket at the bottom of the sample well (see panel A).

for kBT/ℏΩm ≫ 1. The inverse relationship between the two characteristic length scales follows from their respective dependencies on the tunneling path charge Qint as seen in eqs 7.1 and 7.2. The attenuation of damping and the thermalized disturbances acting on the charge-transfer reaction resulting from the application of high gain feedback action is expected to result in a smaller λΩ−n and a concomitantly larger λe−Ω, which in turn arise from a smaller Qint. We next develop a mechanistic characterization of damping effects and thermal noise at the electrochemical interface and its relation to Qint, which uses a Hamiltonian description of the energy of a single reaction coordinate mode, linearly coupled to a large collection of harmonic degrees of freedom, to estimate the dynamic evolution of the polarization density associated with the reaction coordinate mode. The reaction coordinate polarization mode is subject to bath-induced damping and random thermalized forces as a consequence of the linear coupling.

element. The lumped circuit elements dictate the relaxation of the electrochemical interface structure in response to a timevarying exciting bias, which is usually separable from the much faster dynamics of the nuclear and electronic modes of the system. In Gupta et al.28 it was demonstrated that for a large gain feedback action, driven by the difference between the measured and desired electrochemical bias at the reference electrode probe, when the dynamics of the feedback response are comparable to the relaxation time scale of the electrochemical interface, the dissipative coupling of the participant modes with the environment is attenuated, thereby rendering the source-to-drain energy transfer diabatic in nature. This effect is also predicted for the charge-transfer mechanism at the nanoelectrochemical interface when subject to the biasing action of the high-gain feedback loop. The damping forces inhibiting the oscillatory dynamics of the reaction-coordinate oscillators will be attenuated, and the charge-transfer process will assume an increasingly nonadiabatic character. The length scales λe−Ω and λΩ−n described in section 2.2 that measure the interaction strength between the dressed electronic states of the redox species and the reaction coordinate modes as well as in between the reaction coordinate modes themselves, characterize the nature of the transport regime. A small λe−Ω and large λΩ−n imply independent electronic and reaction coordinate degrees of freedom and dissipative charge transfer from donor to acceptor state, indicating an adiabatic transition event. Conversely, a large λe−Ω and small λΩ−n interface characterizes an electronic transition with strong coupling to the nuclear modes, which describes a nonadiabatic transition

2 1⎛ 1 1 ⎞⎡ 2 PΩ̇ ⎤ ⎟⎢PΩ + 2 ⎥ ⎜ 2 − 2 ⎝ nΩ εo εΩ ⎠⎣ Ω ⎦

HΩ = +

∑ k

2 εo ⎞ ⎡ 2 σ̇ 2 ⎤ P 1⎛ 1 1 ⎞⎛ ⎜ 2 − ⎟⎜1 − ⎟ ⎢σk + k 2 ⎥ + Ω σk 2 ⎝ nk εo εk ⎠⎝ εk ⎠ ⎣ ε ωk ⎦ Ω

(12.1)

where a unimodal, one-dimensional reaction coordinate polarization density is assumed for the charge-transfer system and bath degrees of freedom are multimodal though also oneI


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where G(ω) is the frequency-dependent closed loop gain of the feedback. Equation 12.5 presupposes an ideal noiseless feedback action applied on the physical interface by the accompanying electronics, which is modified by additional additive noise sources in the case there are added fluctuation sources from the feedback instrumentation.28 The gain also attenuates the damping forces acting on the reaction coordinate polarization density as

dimensional in nature. A linear coupling between the reactioncoordinate and bath modes is considered in the evaluation of polarization dynamics, with bath polarization density modes Pk in eq 1.1 replaced by an equivalent modal tunneling path charge density σk where Q int(t ) AS

=

∑ σk(t )

(12.2a)

k

The reaction-coordinate polarization modes are treated semiclassically here and not with a rigorous quantummechanical approach as was done in section 2.1 for the sake of demonstrating the influence of feedback with a simpler, intuitive description and without loss of core physics.35,36 The equivalent damping coefficient and the amplitude of the thermal fields acting on the reaction coordinate polarization density are given by37,38 ⎡ ⎢ g (Ω)Ω ⎢∑ 1 1 2⎢ − ε εΩ ⎢ k nΩ 2εo Ω ⎣ 2

γ (t ) =

(

)

cos ωkt

(

1 2

nk εo

−

1 εk

)(

1−

εo εk

γeff (ω) ≈

γ(ω) Re(G(ω))

(12.6a)

where

γ(ω) =

⎤ ⎥ ⎥ 2 ⎥ ⎥⎦

)

∫ ∫

d t e − jω t

d k g (k )

(12.3a)

E (t ) = − ∑

σkocos ωkt +

σk̇ o ωk 2

(12.3b)

Q int(t ) ASεΩ

(12.3c)

Accordingly, the power spectral density of the field fluctuations at the electrochemical interface is frequency independent, up to a cutoff frequency determined by the maximum frequency of the bath modes (ωkmax), with a magnitude that is determined by contributions from the individual mode components of the tunneling path charge density ⟨ΔE2⟩ =

Teff =

2εΩ 2[1 + (ω/ωkmax )2 ] 1 εΩ (1 + (ω/ωkmax )2 ) 2

∫ dk g(k) σo2(k) (12.4)

σo2

where is the mode-dependent surface charge power spectral density and the sum over all bath degrees of freedom in eq 12.3 is replaced by an integral of the contributions from the individual modal charge densities weighted by the density of states in the space of available modes. The fluctuations in the bath field, when referred to the input of the biasing feedback network, describe the equivalent thermal noise source for the electrochemical system as demonstrated in Gupta et al.28 ⟨ΔEmeas 2⟩ ≈

⟨ΔE2⟩ |1 + G(ω)|2

−

1 εΩ

)ε

2

Ω

cos[ω(k)t ]

(

1 n2(k)εo

−

1 ε(k)

2

)(1 − ) εo ε(k)

Ω2

(

1 nΩ 2εo

−

1 εΩ

⟨ΔEmeas 2⟩ γeff kB

)

(12.7)

in the semiclassical limit. The bath temperature is attenuated by the closed loop feedback bandwidth and gain-dependent factor Re(G(ω))/|1 + G(ω)| 2 in the ideal case when the accompanying biasing network is noiseless. In the nonideal situation, with contributions from noise sources in the instrumentation electronics, the bath temperature is, accordingly, higher. The underlying dependence of the field fluctuations on Qint makes the tunneling path charge a surrogate for the effective bath temperature. It is important to note that in the scenario when there are multiple, independent, and mutually normal coordinate polarization states participating in the transition process and not just a single mode as described above

∑k σko 2[δ(ω − ωk) + δ(ω + ωk)]

∼

1 nΩ 2εo

and Re[G(ω)] is the real part of G(ω). The application of gain and the subsequent attenuation of the field fluctuations applied to the quantum electrochemical interface results in a reduction of Qint as seen from eqs 12.3 and 12.5. The extent of the reduction is determined by the gain-bandwidth characteristic of the feedback system and the fluctuations imparted by the external amplifier to the electrochemical system at the input terminals of the feedback loop, where additional noise sources increase Qint. In addition, by the fluctuation−dissipation relationship, the effective bath temperature as “seen” by the polarization mode associated with the reaction-coordinate is described by

respectively, where σko and σ̇ko are the randomly determined values of tunneling path interface charge density and tunneling path charge flux, respectively. The thermal fluctuation in bath field acting on the reaction coordinate polarization is directly related to fluctuations in the tunneling path interface charge as E (t ) = −

(

(12.6b)

sin ωkt

εΩ

k

g (Ω)Ω2

Q int(t ) As

∼

∑ σk + ∑ σΩ k

Ω

(12.8)

The feedback affects the contribution to Qint(t) from the linearly coupled bath modes, k, that contribute to the thermal noise and the damping experienced by the transitioning system. Consequently, as the number of coordinate polarization states contributing to the observed transition rate increases, the feedback is less effective in modulating the dynamics of the

(12.5) J


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wires to be coated in a single ALD run. A reference wafer is also loaded into the ALD chamber with the tips for convenient thickness measurement using a Woollam M2000 Spectroscopic Ellipsometer. Prior to deposition of the primary oxide material, a 0.5 nm thick seed layer of Al2O3 is deposited on the Pt−Ir metal to ensure initial nucleation of the HfO2 and SiO2 precursors on the surface. After deposition of the oxide, the sharp tips are annealed in forming gas (20% H2, 80% N2 by volumetric flow rates) at 300 °C for 30 min to reduce surface states in the oxide film that could contribute to parasitic charge transfer at the interface. The nanoscale electrode area is defined near the apex of each tip by etching holes in the insulating film with a gallium focused ion beam (FIB) produced by a Hitachi Dual Beam 325 SEM/ FIB system. The etch was timed carefully to avoid overetching the hole and doping the Al2O3 sidewalls with gallium, and the insulator film quality was kept consistent by using a specific etch recipe. Figure 4B is a scanning electron microscopy (SEM) image showing the end of a sharp tip, after the FIB mill, of a 50 nm diameter hole in the HfO2 film. Custom holders are used for mounting the annealed wires into the load lock of the FIB tool in batches of 10. The sharpened wires are manually and sequentially aligned to the electron beam optics for imaging and the ion beam optics for milling. After ion milling, the sharp wires are rinsed in ethanol and then immersed in (a) a 1 mM ethanolic solution of 1,2-ethanedithiol (Sigma-Aldrich, St. Louis, MO) for 24 h, (b) an aqueous solution of gold nanoparticles (Sigma-Aldrich, St. Louis, MO) of varying diameters for 24 h, and (c) a 1 mM ethanolic solution of an alkanethiol monolayer (Sigma-Aldrich, St. Louis, MO) with specific functionalization chemistry for 48 h. The nanoscale electrochemical interface was functionalized with different chemistries as part of the experimental plan. The oxide-coated and ion-milled tips are immersed in the ethanolic or aqueous deposition medium within sealed vials. After immersion, the sharp wires are rinsed with ethanol and blow dried with dry N2 in preparation for loading into the electrochemical cell. The long immersion times and the increased likelihood of oxidation in the thiol functional groups may lead to the formation of a heterogeneous monolayer/multilayer thin-film structure. Moreover, the interaction of the monolayer functionalization chemistry with specific analytes in the electrolytic sample may also result in a film structure that is different from the ideal thin film depicted in Figure 3. The incorporation of a nanoparticle as well as the selection of functionalization chemistry and film thickness guards against deterioration in signal transduction due to the formation of these nonideal films. The sharpened wires are threaded through a poly di-methyl siloxane (PDMS) gasket, using a precision z-stage to manually pull the tips, blunt end first, through an existing hole in the gasket. The sharpened tip is carefully positioned 1 μm or less above the upper surface of the PDMS gasket (Figure 4A). The gasket is made by mixing 100:10 parts by volume of monomer and curing agent (Sylgard Dow Corning, Midland, MI), degassing the mixture in vacuum, and curing at 50 °C. The gasket is cut out of a cured block of the PDMS with approximate dimensions 3 cm × 3 cm × 1 cm. A blunted 30 gauge (inner diameter) syringe needle is used to create a hole in the middle of the gasket for the sharpened tip. Once loaded in the PDMS gasket, the functionalized and ion-milled tip is connected to a flexible wire for future connection to a custom potentiostat. The gasket/tip/wire structure is mounted on the bottom half of a Teflon electrochemical cell which is secured to

transition, eventually culminating in the nonadiabatic chargetransfer limit when the feedback action is likely to prove ineffective. The description of the effect of high gain, low noise feedback action on a physical nanoelectrochemical interface provided above assumes an interface operating in a quantum regime as described in sections 2.2−2.4 and not limited by classical phenomena, such as charge transport through the interface geometric capacitance or transport-related effects. The attenuation of the damping forces and the thermal fluctuations acting on the reaction coordinate polarization dynamics are essential functions of the feedback system to retain the quantum nature of the interface, and the geometry of the interface may require adjustment as per the rules described in sections 2.2−2.4 to accommodate the influence of the electronics on Qint.

3. EXPERIMENTAL SECTION 3.1. Sample Preparation. The design parameters and guidelines developed in the previous section are tested experimentally by fabricating and testing the nanoscale electrochemical interface. A 250 μm diameter Pt (80%) − Ir (20%) wire (Nanoscience Instruments Inc., Alexandria, VA) is cut into 2 cm long segments and then cleaned in a standard SC1 solution (100 mL deionized (DI) water:10 mL H2O2:1 mL NH4OH). The cleaned wire segment is mounted in a custom holder on a z-axis positioning stage, as illustrated schematically in Figure 4A. The stage lowers the wire segment into an electrochemical etch bath, at the center of a copper-ring counter electrode. The etch bath consists of 3 M NaCl (SigmaAldrich, St. Louis, MO) and 0.25 M HClO4 (Sigma-Aldrich, St. Louis, MO) in 200 mL of DI water. The wire is dipped into the bath, and a 60 Vpp (peak-to-peak amplitude) square wave with frequency of 1 kHz is applied between the wire segment and the counter electrode. The etching process is terminated manually when a spark is observed at the etched tip and hydrogen bubbles in the etchant bath disappear. The meniscus at the wire−etchant−air interface and the radial geometry of the etching setup results in a sharp tiplike profile at the breakpoint of the short wire segment.39−41 After the etch process is terminated, the holder is retracted and the etched wire segment is removed for cleaning. A sharp tip is desirable for minimizing the parasitic capacitance after mounting in the polymer gasket, as will be explained later in this section. The etched wire is rinsed in DI water and dipped in SC-1 solution for 1 min, prior to a repeated rinse with DI water followed by blow-drying with dry N2. A 40 nm thick insulating oxide film is deposited on the electrochemically etched wire using an atomic layer deposition (ALD) tool. Sharpened wires stored for more than 24 h prior to deposition are sonicated in ethanol and blow-dried with dry N2. ALD ensures uniformity of the deposited oxide film over the sharp tip and minimizes pinhole defects in the film, which would produce undesired electrochemical leakage currents. After the initial deposition of a seed layer, the ALD process becomes self-limiting and produces monolayer-by-monolayer growth of the oxide film on the Pt−Ir surface. Two types of insulating oxide films were investigated experimentally in this paper: hafnium dioxide (HfO2) and silicon dioxide (SiO2). The Savannah thermal deposition tool (Ultratech Inc., Waltham, MA) and Fiji plasma deposition system (Ultratech, Waltham, MA) were used to deposit 40 nm of HfO2 and SiO2, respectively. A custom tip-holder allows up to 20 etched K


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Figure 5. Schematic of potentiostatic feedback loop for (a) application of low voltage noise signal bias between the functionalized nanoscale electrode and an electrochemical reference and (b) measurement of resulting tunneling interface current.

digitized by analog-to-digital converters (ADC) for measurement recording. The measurement is performed in a Faraday cage to minimize environmental signals. The output signals Vmeas and VTIA are heavily averaged as the excitation voltage is swept, to reduce noise introduced by the electronics and residual environmental signals; details are provided in the following section. A MATLAB script on a dedicated PC workstation automates the voltage sweep and the measurement of the resulting interface current. The custom potentiostat was benchmarked against the Gamry 600, a commercial low-noise potentiostat, which was also used for some of the I−V measurements on nanoscale electrochemical interfaces.

the top half by screws. An O-ring forms a compression seal between the gasket and top half of the Teflon cell to prevent leaking. Figure 4C is a photograph of the electrochemical cell. The liquid electrolyte is an aqueous buffer consisting of a mixture of potassium mono- and dihydrogen phosphate (Sigma-Aldrich, St. Louis, MO) in DI water at a total concentration of 100 mM, in a ratio such that the solution is buffered at pH 7.5. In addition, the aqueous electrolyte contains one of two redox couples at 1 mM concentration. Additional analyte species at requisite concentrations may be added to the electrolyte. A volume of 2.5 mL of the electrolyte solution is pipetted into the well of the electrochemical cell. A 2 mm diameter gold wire (Alfa Aesar, Ward Hill, MA) and a miniature Ag/AgCl reference electrode (BASi, West Lafayette, IN) are secured using alligator clips to wires connected to the counter and reference electrode nodes, respectively. 3.2. Electrochemical Instrumentation. Figure 5 shows a simplified schematic of an ultralow-noise, custom potentiostat implemented with off-the-shelf components.28 The circuit implements a high-gain negative feedback loop which forces Vmeas ∼ Vexcitation by applying a corrective signal to the counter electrode that is proportional to the difference between the Vexcitation and the measured reference electrode voltage Vmeas. The potential of the reference electrode Vref is measured using a high-input impedance buffer (>100 GΩ). Mismatch between the junction field effect transistor (JFET) devices and the biasing 33 Ω resistors introduces a 0 (Figure 7C), where the bias ϕr is applied to the redox state via the reference electrode and is referenced to a grounded set of metal electronic states. The path to transition from redox state to metal electrode via the nanoparticle linker is without barrier for ϕr < 0, as can be seen in Figure 7A. Specifically, the manifold of available electronic states in the nanoparticle bridge allow for a direct transition from the redox ground state to a vibronic state on the nanoparticle element, as indicated by path a. Subsequently, a barrierless charge-transfer event occurs between the ground state of the nanoparticle linker and the underlying metal electrode (path b). The applied bias must be less than a threshold value for the first channel to open, which enables the transfer from the ground redox state to the lowest vibronic nanoparticle state. For near-resonance conditions between the metal Fermi level and the vibronic ground state of the redox molecule, the transition between the redox species and the nanoparticle bridge is thermally activated (path a′ in Figure 7B), and for off-resonance case, with ϕr > 0, the transition from the highest-energy vibronic state of the nanoparticle element to the ground state of the redox moiety in the electrolyte (path a″ in Figure 7C) is without barrier beyond a threshold positive bias voltage. Thus, the availability of the additional channels for transport occurs outside a bias window centered on the electronic resonance between the metal electrode and the redox-active species, as can be observed in Figure 6B,C. The nature of the hysteresis and the appearance of the barrier-free channels in the I−V traces can be modulated by modifying the geometry and the redox chemistry at the nanoelectrochemical interface, as is shown in Figure 8. For all

traces demonstrated in Figure 8, the underlying metal electrode is platinum (80%)−iridium (20%); the metal to gold nanoparticle linker is 1,2-ethanedithiol; and the monolayer on the nanoparticle is 1-mercapto-3-propionic acid. The gold nanoparticle is 10 nm in diameter, and the redox chemistry used is the ferro-ferri-cyanide couple, unless indicated otherwise. The default insulator coating the etched tip is, in most cases, 40 nm thick hafnium oxide, and the FIB-etched electrode is 50 nm in diameter. Reducing the nanoparticle size to 5 nm (Figure 8B) yields a current−voltage characteristic with larger hysteresis, with more pronounced turn-on and turnoff voltages and without the presence of the vibronic conduction channels. The functionalization of the interface with smaller nanoparticles increases the effective charge-transfer flux area and simultaneously reduces the effective tunneling energy gap Δε due to increased Fermi level distortion at the metal-1,2 ethanedithiol interface. The reduced tunneling energy gap results in higher transition currents and compensates for the increased Qint due to increased area. Increasing the interface area to 1 μm in diameter and reducing the nanoparticle size to 5 nm yields a further increase in Qint, as observed in Figure 8D, resulting in smaller overall transition rates and less pronounced electronic resonance features. The large interface area results in increased thermal contributions from high-frequency modes as described in the section 2.4, yielding a reduced transition rate as well (eq 5.4), because barrierless resonant transfer channels are not available for charge transfer. The use of hexaamine ruthenium (II and III) chloride as the interface redox couple instead of the ferro-/ferri-cyanide redox chemistry in Figure 8C results in a complete collapse of the hysteresis and the appearance of a classical I−V characteristic for the electroO


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Figure 9. (A) Input referred voltage noise power spectral density for the reference electrode node compared between the Gamry Ref. 600 potentiostat and our custom built low voltage noise potentiostatic instrument. I−V traces for (B) 40 nm thick SiO2 and (C) 40 nm thick HfO2 coated interface structures as acquired on the Gamry and the custom board. Comparison between panels B and C also indicates that film with lower dielectric permittivity exhibits quasihysteretic behavior even when scanned with the Gamry instrument. In either case, the interface comprises a 50 nm ion milled hole in the insulator with the underlying Pt(80%)−Ir(20%) metal functionalized by a self-assembled monolayer film comprising 1,2ethanedithiol, a 5 nm gold nanoparticle, and 1 propanethiol. Electrolyte is composed of 100 mM potassium mono- and dihydrogen phosphate buffered at pH 7.5 and 1 mM ferro-/ferricyanide redox couple.

which computes a discrete Fourier transform on sampled time records of varying length to return the frequency domain representation of the measured signal. The noise of the custompotentiostat is amplified above the noise floor of the HP89410a (∼5 × 10−17 V2/Hz) using a custom amplifier built into the board. The low-frequency spectrum of each potentiostat is dominated by 1/f noise from traps in active transistors. The high -requency spectrum of each potentiostat is dominated by thermal noise from active transistors and circuit resistances. The spectrum of the custom-potentiostat has tones at 60, 120, and 180 Hz from coupling to the mains voltage. These tones do not appear in the spectrum of the Gamry Ref. 600, possibly because the mains tones are below the 1/f + thermal noise floor of the instrument. The spectrum of the Gamry Ref. 600 instead has tones at 80 Hz, 320 Hz, and 1 kHz, which are attributed to the internal circuitry of the instrument. Increased voltage noise at the reference node results in quasi-classical I−V traces for nanoelectrochemical interfaces that are subject to biasing by the feedback network on the Gamry instrument, wherein electronic resonances and the hysteresis due to discrete charging effects are both suppressed, and the I−V trace, particularly in the case of the hafnia-coated interface (Figure 9C), is classically adiabatic in nature as would be expected for a high Qint interface with no damping attenuation. The difference in the I−V characteristic between the silicon dioxide and the hafnium dioxide interfaces in Figure 9B,C, specifically when the Gamry

chemical interface. The ruthenium-based redox chemistry has a reorganization energy that is nearly twice as large as the reorganization energy for the ferro-/ferri-cyanide couple.45−47 Consequently, cq for the ruthenium chemistry is large enough that the system is limited by the classical components of the impedance, rendering the charge-transfer process largely nonadiabatic in nature. The effect of the application of the low voltage-noise signal at the interface, as well as the application of the high-gain feedback for attenuation of damping effects in charge transfer at the nanoscale interface, is demonstrated in Figure 9 by contrasting measurements made on the same interface type by two different potentiostats: the first instrument is our custom, low-noise board and the second, a commercial instrument, the Gamry Ref. 600. The nanoscale interfaces considered here consist of a 50 nm diameter, ion milled pore in a 40 nm insulator film that coats the sharp platinum (80%)− iridium (20%) tip. The interface is functionalized by a heterogeneous self-assembled film comprising 1,2-ethanedithiol, a 5 nm gold nanoparticle, and 1-propanethiol, unless otherwise indicated. The measured voltage noise spectral density between the reference electrode and the working electrode for the Gamry Ref. 600 potentiostat and the custom potentiostat are shown in Figure 9A, demonstrating the superior noise performance of the custom board. The spectra are measured using an HP89410a Vector Signal Analyzer, P


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Figure 10. I−V traces demonstrating the effects of systematic variation in interface parameters. In each case, the electrolyte is 100 mM potassium mono- and dihydrogen phosphate and the metallic substrate is Pt(80%)−Ir(20%) etched into a sharp tip. (A) Interface was 50 nm milled hole in a hafnia coated insulator, functionalized by a monolayer assembly comprising 1,2-ethanedithiol, 10 nm gold nanoparticles, and 3-amino-1-propanethiol with variable redox chemistry (reorganization energy). (B) I−V and (C) conductance−voltage traces for interfaces composed of 50 nm milled hole in a hafnia coated insulator, functionalized by monolayer assembly comprising 1,2-ethanedithiol, 10 nm gold nanoparticles, and a three-carbon alkanethiol monolayer with variable end group chemistry. The redox species used is the ferro-/ferricyanide couple in each experiment. Effect of functionalizing monolayer thickness on electronic transition as seen in (D) I−V trace and (E) conductance−voltage characteristic for interface that is 50 nm hole in 40 nm thick hafnia dielectric, where underlying metal is functionalized by 1,2-ethanedithiol, 10 nm nanoparticles, and films of differing thickness. Inset in panel D shows a close-up of I−V traces for 1-hexanethiol and 1-propanethiol. Redox species used for all experiments is ferro-/ ferricyanide couple. Effect of deuterating functionalizing monolayer on transition current as seen from (F) I−V and (G) conductance−voltage trace. Interface in each case 50 nm hole in 40 nm thick hafnia dielectric, where underlying metal is functionalized by 1,2-ethanedithiol, 10 nm nanoparticles, and deuterated and hydrogenated alkanethiol monolayer films that are three carbon atoms thick. Redox species used for all experiments is ferro-/ferricyanide couple.

ent of the feedback action, wherein the hafnia dielectric coated interface exhibits lower overall current and more diffuse turn-on and turn-off transitions. The effect of redox chemistry on the transfer kinetics is further elaborated in Figure 10A. The redox couples Fe[CN]63−,4−, Ru[NH3]62+,3+, the ferrocene redox system, and a redox-free electrolyte were used in these measurements, wherein the interface was a 50 nm diameter ion-milled hole in the hafnia insulator, which was functionalized by a monolayer assembly comprising 1,2-ethanedithiol, 10 nm diameter gold nanoparticles, and 3-amino-1-propanethiol. The electrolytedissolved redox moieties display I−V behaviors ranging from the purely quantized to the purely classical, in increasing order

is used for the cyclic voltammetry measurement, suggests that the geometric capacitance from the coating oxide determines if the interface charge transfer is in the quantum or classical regime, which is consistent with the model proposed earlier. For the smaller parasitic capacitance arising from the silicon dioxide insulator, the charge-transfer process appears to have quantum characteristics in the resonance features and the presence of the hysteresis, which are absent in the I−V trace for the hafnia-coated insulator. The differences between I−V characteristics of the two oxides persist when the interfaces are subject to low voltage noise, high gain feedback, suggesting that the oxide material chemistry continues to play an important role in determining classical, parasitic contributions independQ


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Figure 11. Interface is a 50 nm ion-milled pore in 40 nm thick hafnium oxide insulator and exposed metal is functionalized by a film comprising1,2ethanedithiol, 10 nm gold nanoparticles, and 1-mercapto-3-propionic acid. Redox chemistry used in each instance is the ferro-/ferricyanide couple at 1 mM concentration, and background electrolyte is phosphate buffer at 100 mM concentration. Analyte comparison, at 1 μM electrolyte concentration, is done between (A) leucine and 2-D-leucine in and (C) between leucine and isoleucine as seen from the I−V trace. Panel B demonstrates a shift in channel energies for the deuterated analyte as seen from conductance−voltage trace.

of reorganization energy associated with the charge-transfer system (Fe[CN]63−,4− < ferrocene < Ru[NH3]62+,3+ < redoxfree electrolyte).48 Figure 10B,C illustrates the effect of terminal end chemistry at the functionalization monolayer on the nanoparticle. The redox chemistry consistently used in this set of experiments was the Fe[CN]63−,4− couple, and the monolayer end-chemistry was varied between −NH 2 , −COOH, and −CH3. The interaction between the charge on the end group and the negatively charged redox entity influences the tunneling energy gap, which in turn determines both the Qint value, as seen from eq11.4, and the efficiency of the tunneling transition in transducing information about vibrational channels available for the charge-transfer process. Figure 10C graphs the forward-scan conductances for the different monolayer end-groups, showing the improved resolution in the discrete channel signatures in the order −CH3 < COOH < −NH2. The effect of functionalization monolayer thickness is demonstrated in Figure 10D,E. The interface consists of a 50 nm diameter ion-milled hole in a 40 nm thick hafnia film, and the underlying Pt−Ir metal electrode is functionalized by the 1,2-ethanedithiol, 10 nm diameter gold nanoparticle heterogeneous film. The nanoparticle is then subsequently either kept bare or functionalized by 1-propanethiol or by 1-hexanethiol to generate the I−V traces recorded in Figure 10D. The bare nanoparticle interface is characterized by a near classical cyclic voltammogram, whereas the monolayer-functionalized nanoparticle interfaces are characterized by the discrete, quantizedcharge I−V traces with well-defined hysteresis and turn-on and turn-off voltages, as seen in the inset. The conductance from the forward scan (Figure 10E) indicates a decreasing resolution

in the transduction of vibrational channels for the thicker functionalization film, because a thicker film would result in increased interactions between the larger numbers of reaction polarization modes present in the tunneling path and the redox vibronic states. The increased number of mode channels presented to the tunneling electron at a given bias results in loss of resonance between the polarization modes and the electronic states, rendering the polarization modes indistinct from the bath degrees of freedom. However, the electronic resonances are more pronounced and the magnitude of the measured current is also higher for the thicker monolayer. A thicker film between the nanoparticle electrode and the electrolytedissolved redox-active species uncouples the electronic degrees of freedom between the donor and acceptor states, qualitatively rendering the transition more nonadiabatic in nature. Consequently, Qint for a thicker film is lower than that for a thinner film, subject to the film thickness being below the threshold value prescribed by eq 9.1. In the quantum regime, a lower Qint would yield larger transition rates and reduced damping associated with transition dynamics as seen in eqs 11.1 and 12.3, respectively. Qualitatively, the increased numbers of vibrational channels available for the transition result in larger transfer rates, as long as classical fields due to the interface geometry do not become rate-limiting. The appropriate choice of monolayer chemistry and film thickness for a transducing electrochemical interface would be determined experimentally from the nature of the heterogeneous film structure at the interface that forms because of the self-assembly process as well as because of the interaction between the monolayer functionalization and the electrolytic sample being characterized. R


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5. CONCLUSIONS We have presented a framework to ascertain the degree of quantum behavior exhibited by a nanoscale electrochemical interface, based on specific criteria that address (a) the lateral dimension of the interface, parallel to the charge-transfer flux; (b) the thickness and chemistry of the functionalization film on the interface; (c) the thickness and dielectric permittivity of insulator on the macro-lead that connects to the interface; (d) the area of macro-lead; (e) the redox chemistry in electrolyte phase; (f) voltage noise at the electrochemical interface; and (g) feedback loop gain for the potentiostat used to apply a voltage bias across the interface. These criteria are expressed as macroscopic lumped impedance elements that form an equivalent circuit model for the interface; the model reduces to the Randles equivalent circuit in the classical limit. In addition, the perturbation introduced by electrochemical instrumentation on the quantum electrochemical system has been described and methods have been developed to minimize these disturbances. The transition flux for an interface operating in the quantum regime, as measured by the charge-transfer current at the nanoscale interface, contains signatures that indicate discrete interface charging, as well as the presence of discrete vibrational channels for the tunneling transition. Therefore, there is an opportunity to leverage these nanoscale electrochemical interfaces, controlled by low-noise instrumentation, for the probe- and label-free sensing of analyte molecules dissolved in the electrolyte. A recent body of work has described a related concept of recognition tunneling in electrochemical tunneling junctions between functionalized macroscopic metal electrodes and a scanning microscopy tip electrode, also functionalized by “recognition” reagents.49,50 Stochastic fluctuations in bond formation between the electrolyte-dissolved analytes and the recognition elements within the tunneling gap result in telegraph noise in the tunneling current chronoamperogram.51,52 Because the characteristics of the noise waveforms are related to the nature of the temporary bond between the analyte and functionalization chemistry, the noise signatures provide a means of identifying the analyte molecules from the transient current measurement.53,54 The sensing paradigm based on the nanoscale electrochemical systems described in this paper would transduce information about the participant polarization modes, as well as the tunneling and total interface charge, by means of a heterogeneous signal consisting of multiple higher-dimensional features. A mechanistic analysis of the underlying features in the signal would require development of a complicated, multiscale spatial and temporal physicochemical model for the chargetransfer process and the accompanying surface interactions between multiple potential analytes and the sensing interface. An alternative methodology for signal analysis would utilize reference data sets of signatures from “pure” analytes for pattern-matching and feature identification, as is usually done for spectroscopy-like platforms. We will demonstrate this new electronic, label- and probe-free sensing paradigm in a subsequent publication, with an application targeted toward the multiplexed identification of specific target analytes in human serum.

Mass effects on the charge-transfer reaction in the quantum regime are explored in Figure 10F,G. In these experiments, as in the previous measurements, the interface is a 50 nm diameter ion-milled hole in a 40 nm thick hafnia insulator functionalized by the 1,2-ethanedithiol linker and 10 nm diameter gold nanoparticles. The functionalization monolayer on the nanoparticle is varied between 1-propanethiol and 1-propane-d7thiol, in which all hydrogen atoms are replaced by the heavier isotope deuterium. A reduced reaction polarization mode frequency, due to substitution with the heavier isotope, results in a smaller Qintquant, as seen from eq 8.3b. Therefore, with the existing choice of redox chemistry (ferro-/ferri-cyanide couple) and functionalization monolayer thickness and chemistry, the actual Qint value is likely to be larger in magnitude than the optimum Qintquant, making the transfer process more adiabatic in nature. The I−V trace for this nonoptimal interface has reduced hysteresis and less pronounced turn-on and turn-off transitions at the electronic resonances, as seen in Figure 10F. However, the presence of resonant vibrational channels that participate in the transition process can still be seen in the forward scan conductance (Figure 10G). The sensitivity of the nanoelectrochemical interface to electrolyte-dissolved analytes is demonstrated in Figure 11. In each example, the sensor is a 50 nm diameter, ion-milled hole in 40 nm thick HfO2 insulating film that coats a sharpened platinum (80%)−iridium (20%) wire. The exposed metal at the bottom of the hole is functionalized by a heterogeneous film comprising 1,2-ethanedithiol as a linker, a 10 nm diameter gold nanoparticle, and 1-mercapto-3-propionic acid. The redox chemistry used in each case is the ferro-/ferri-cyanide couple at 1 mM concentration, and the background electrolyte is a phosphate buffer at 100 mM concentration, as with previous experiments. The analyte is dissolved at 1 μM concentration for each experiment and is an isotope or isomer of the amino acid leucine. Figure 11A compares the measured I−V traces for the nanoscale electrochemical interfaces, for the cases when the electrolyte-dissolved analyte species is leucine or 2-D-leucine, in which the hydrogen atom on the carbon adjacent to the carboxyl group of the leucine molecule is substituted by deuterium. Figure 11C compares the I−V characteristics for the case when the dissolved analyte is leucine versus its structural isomer, isoleucine. The I−V trace for the deuterated analyte exhibits a larger hysteresis, as seen in Figure 11A, while isoleucine has a quasi-classical response to a low voltage-noise scanning bias, as observed in Figure 11C. The diversity in response of the nanoelectrochemical interface to the mass as well as structure of the analyte suggests that the transition electron flux is affected by the intrinsic vibrational modes of the analyte through the presentation of these modes to the monolayer functionalization chemistry. Although a mechanistic description of the analyte’s orientation with respect to the monolayer’s end-group chemistry is beyond the scope of this paper, the I−V characteristics indicate that the charge-transfer behavior is affected. Analyte polarization states could also contribute additional channels to facilitate transition events. Figure 11B contrasts the conductance data for two cases, where the analytes are 1 μM leucine and 1 μM 2-D-leucine. The vibrational channels for the 2-D-leucine analyte all shift to lower bias values by 20 mV on average with a standard deviation of 3 mV. The nearly uniform shift in the channel energy to a reduced bias is consistent with the reduced vibrational mode frequency associated with the heavier deuterated analyte.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. S


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The Journal of Physical Chemistry C ORCID

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Chaitanya Gupta: 0000-0002-7570-0867 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Defense and Advanced Research Projects Agency Grant N66001-11-1-4111, sponsored by the Mesodynamic Architectures program; Program Manager Dr. Jeffrey Rogers. Device fabrication utilized the Stanford Nanofabrication Facility (SNF) for atomic layer deposition and the Stanford Nano Shared Facilities (SNSF) for focused-ion beam etching. We thank Dr. Michelle Rincon of SNF for her help with the ALD deposition.

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