Probing charge carrier movement in organic semiconductor thin films

Article Cite This: ACS Appl. Electron. Mater. XXXX, XXX, XXX−XXX

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Probing Charge Carrier Movement in Organic Semiconductor Thin Films via Nanowire Conductance Spectroscopy Mykhailo V. Klymenko,*,†,‡ Jesse A. Vaitkus,† and Jared H. Cole†,‡ †

Chemical and Quantum Physics, School of Science, and ‡ARC Centre of Excellence in Exciton Science, RMIT University, Melbourne, Victoria 3001, Australia

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ABSTRACT: Understanding movement of charge carriers within organic semiconductor films is crucial for applications in photovoltaics and flexible electronics. This study theoretically investigates the use silicon nanowires for probing spatial and temporal distribution of charge carriers in organic thin films and estimates the sensitivity of electrical conductance of silicon nanowires to changes of charge states within an organic semiconductor physisorbed on the surface of the nanowire. Elastic scattering caused by motion of charge carriers near the nanowire modifies the mean-free path for backscattering of electrons propagating within it, which we have mathematically expressed in terms of the causal Green’s functions. The scattering potential has been computed by using a combination of a polarizable continuum model and density functional theory with a range-separated exchange-correlation functional for organic molecules and the semiempirical tight-binding model for silicon. For a single charge carrier in crystalline tetracene, ultrathin silicon nanowires with characteristic sizes of the cross section below 2 nm produce a detectable conductance change at room temperature. For larger nanowires the sensitivity is reduced; however, the conductance change grows with the number of charged molecules, with sub-4 nm nanowires being sensitive enough to detect several tens of charge carriers. We propose using noise spectroscopy to access the temporal evolution of the charge states. Information regarding the spatial distribution of charge carries in organic thin films can be obtained by using a grid of nanowire resistors and electric impedance tomography. KEYWORDS: charge sensing, nanowire, electron transport, organic thin film, charge carrier, electrical impedance tomography, noise spectroscopy

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INTRODUCTION

morphology near the interface as well as the distribution of trap states and impurities. In this work we consider an array of silicon nanowires (NWs) as a measurement setup to access spatial and temporal information about the free charge carriers distribution in organic semiconductor thin films, illustrated in Figure 1. The goal is to estimate the sensitivity of the electron transport in silicon NWs to changes of the charge states of molecules in organic semiconductors. Previously, NWs have established themselves as highly sensitive chemical sensors.6 For instance, they can detect extremely small amounts of charge transferred from ammonia molecules physisorbed at their surfaces in both gas and liquid environments.7,8 The possibility of noninvasive charge sensing by electrical current placed in close proximity to a confined charge has been also proven with quantum-point contact charge sensors.9,10 The problem of conductance change caused by motion of elastic scatterers, fluctuation of chemical potential or magnetic fields, and their implications for 1/f noise had been intensively studied in the 1980s and early 1990s for metallic wires both

A general property of semiconductors that makes them so widely used in electronics and optoelectronics is their strong dependence of the concentration of free charge carriers on various parameters such as temperature, concentration of dopants, stress and strain, and applied electromagnetic fields. Organic semiconductors are no exception.1 Understanding the distribution and transport of charge carriers under nonequilibrium conditions in organic semiconductor films is crucial for designing new organic field-effect transistors2 and hybrid organic−inorganic photovoltaic devices.3 Several different types of charge and excitonic excitations can be generated in such films via electrical or photoexcitation. Each type of excitation will undergo different dynamics within the film and will induce both spatial and temporal changes in the local electrostatic environment. A local noninvasive charge sensing probe of free carrier density induced by a spatially localized photoexcitation is of particular interest since it may give information regarding the anisotropic diffusion, quantum efficiency, mobility, and lifetimes of charge carriers, similarly to the spatially resolved optical pump−probe experiments.4,5 Measuring a spatially resolved net stationary charge distribution in an organic thin films reveals details of the crystal © XXXX American Chemical Society

Received: June 4, 2019 Accepted: July 31, 2019 Published: July 31, 2019 A

DOI: 10.1021/acsaelm.9b00354 ACS Appl. Electron. Mater. XXXX, XXX, XXX−XXX

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ACS Applied Electronic Materials

sites,26 and scanning tunneling microscope directed selfassembly of dopants in silicon.27 On the basis of the obtained results, we propose a concept for the 2D spatially and temporarily resolved charge sensing via noise spectroscopy and electric impedance tomography using a 2D grid of silicon NWs.

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RESULTS AND DISCUSSION We restrict our treatment to the case where an organic semiconductor is in contact with one side of a rectangular silicon NW as is shown in Figure 1. Moreover, the organic semiconductor is physisorbed at the hydrogen passivated (100) silicon surface to ensure minimal deformations caused by the silicon crystal lattice. Although our treatment is applicable for any organic semiconductor held at the surface of NWs by van der Waals forces, here we consider the thin film of tetracene crystal as an example. To reduce the number of defects in the crystalline structure of the ultrathin NW, the channel must be an intrinsic semiconductor. The population of charge carriers in the NW can be controlled by so-called electrostatic doping applying an external electrostatic field,28 by temperature that controls the concentration of thermally generated charge carriers, or by properly chosen electrodes facilitating an injection of charge carriers into the NW. The conductance sensitivity to a single scattering event is estimated using a model system represented by a single silicon NW and single molecule possessing different charge states (see Figure 2). If the interference between electrons scattered on

Figure 1. Linear array of silicon nanowires for probing charge states of organic molecules physisorbed on their surfaces. The charge carriers within the organic semiconductor (denoted by red shaded molecules) modify the conductance of the nanowires. The movement and position of the charges can be probed by fluctuations of currents Ij=1−4 in the linear transport regime.

numerically and analytically.11−14 The theoretical models developed during that time concerned mostly the interference and localization effects in systems with relatively simple electronic structure (a single parabolic energy band) and lacked details on the scattering potential required for a quantitative treatment.12 Nevertheless, those works have produced upper estimates for the conductance change and useful relationships between conductance sensitivity and dimensionality of the systems consistent with the results obtained in this work. In this paper we address the limit when the mean distance between scattering centers is large, and the scattering potential is weak, so that interference between scattering events may be neglected. At the same time the system itself is characterized via a realistic band structure with a scattering potential known in detail. In this work, the conductance change caused by scattering from the potential induced by physisorbed charged molecules has been computed using the relationship between the conductance and mean-free path for backscattering15 which may be used in conjunction with Matthiessen’s rule16 for various types of scatterer. We have expressed the mean free path in terms of causal Green’s functions using the Caroli formula for the transmission probability. The Green’s functions are computed by using the semiempirical tight-binding Hamiltonian for silicon NW with the reduced mode space transformation.17,18 To obtain the scattering potential generated by a physisorbed molecule with an excess charge in a crystalline environment, we use a recently developed technique representing a combination of the polarizable continuum model and density functional theory with the range-separated exchange-correlation functional ωB97XD with an adjustable parameter ω.19 We apply our model to a range of NWs with widths from 2 to 6 times the silicon lattice constant (1.38−3.49 nm including hydrogen passivation layers). Sub-3 nm NWs have been fabricated by several authors. A particularly successful technology for fabricating ultrathin NWs is chemical vapor deposition with gold nanoclusters as catalysts.20−22 The conventional CMOS technology also has perspectives for manufacturing such NWs according to recent advances (see for example the Omega-gate NW field-effect transistor23). Other perspectives in this context technologies are the self-limiting oxidation,24,25 synthesis of vertically assembled nanocompo-

Figure 2. A simplified model of the system used in our conductance change computations. The model consists of a single hydrogen passivated rectangular silicon NW with a single physisorbed tetracene molecule. The molecule is surrounded by other neutral tetracene molecules in the crystal lattice (not shown in the figure). The effect of the tetracene crystal lattice is taken into account through the effective polarizable continuum media model.

neighboring charged molecules is negligibly small, the information about single molecule scattering can be easily generalized to the case of multiple scatterers as is shown in the following section. Finite Temperature Linear Transport Regime Model for Dilute Elastic Scattering Centers. For the linear transport regime close to equilibrium, the conductance of a NW can be computed by using the Landauer formula for a B


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quantum interference between electrons scattered at different molecules is negligibly small. As a result of these approximations, the mean free path can be computed independently for each kind of scattering and then summed up by using Matthiessen’s rule:16

two-terminal device (see Figure 2) modified to take into account interactions in the device region:29 G=−

2e 2 ℏ

∫ dε

d f (ε ) dε

tr(GaΓ R Gr Γ LΣ0−1Σ) (1)

εF

λ−1(ε) = λ+−1(ε) + λ−−1(ε) + λph−1(ε)

where Gr and Ga are retarded and advanced Green’s functions expressed in the matrix form ΓL,R = i(ΣrL,R − ΣaL,R), ΣrL,R, and ΣaL,R are the retarded and advanced self-energies describing coupling to semi-infinite left and right leads respectively, Σ0 = −i(ΓL + ΓR), and Σ is the total self-energy describing all elastic and inelastic scattering in the system. In the case where only elastic scattering is present, Σ = Σ0 (see ref 29 for more details), and we obtain the Caroli formula for transmission T(ε) = tr(GaΓRGrΓL) in eq 1. Alternatively, scattering in the device region can be phenomenologically expressed as15 G=−

2e 2 ℏ

∫ dε

d f (ε ) dε

The mean free path determined by the phonon scattering, λph(ε), exceeds 500 nm in ultrathin NWs,16,31 so the ballistic transport can be observed even at room temperature. Therefore, we neglect the electron−phonon contribution in this work. Each term in eq 6 is computed independently by using a formula derived from eq 3. For instance, in the case of elastic scattering, the mean free path reads λelastic−1(ε) = λ+−1(ε) + λ−−1(ε) ÄÅ ÉÑ ÅÅ tr(Ga Γ R Gr Γ L) ÑÑ Å ÑÑ 0 0 ÑÑ = ∑ ⟨l⟩j−1ÅÅÅÅ − 1 a R r L ÑÑ ÅÅ tr(G j Γ G jΓ ) ÑÑÖ j =+, − ÅÇ

T (ε ) M ( ε) εF

(2)

where M(ε) = tr(Ga0ΓRGr0ΓL) is the number of modes defined in terms of the Green’s function of the infinite uniform NW, Gr0, and T(ε) is the transmission probability given by the formula15 T (ε) =

λ (ε) λ (ε) + L

(7)

In the case when a single type of elastic scattering dominates, the transmission probability in eq 3 reads −1 ÅÄÅ ÑÉÑo l o ÅÅ tr(Ga0Γ R Gr0Γ L) ÑÑ| o o o o Å Ñ T ( ε) = m 1 + nLÅÅ − 1ÑÑ} o ÅÅ tr(Ga Γ R Gr Γ L) ÑÑo o o o +, − +, − ÑÖo (8) ÅÇ n ~

(3)

where n = ⟨l⟩+,−−1 is the linear density of either positively or negatively charged scattering centers. At zero temperature, the conductance change ΔG = |G0 − G+,−| is proportional to the transmission change ΔT = 1 − T(ε). When nL ≪ 1, according to eq 8, ΔT ∝ nL. In the limit when nL → ∞, ΔT → 1. The observed dependence of the conductance on the concentration of scattering centers and length of the device region is consistent with the results published in ref 11. In this work we do not consider charge and energy transfer effects across the interface, since we are interested in how sensitive a NW is to movement of charges in the covering layer, not to charge transfer processes. As a consequence of this approximation, the conductance change has been computed assuming that the transport processes in tetracene do not modify the Fermi level for electrons in silicon. In reality, Dexter and Foster exciton transfer as well as charge transfer processes across interface may be possible for certain configurations of the interface between organic and inorganic semiconductors. However, it is possible to purposely suppress those effects by using hydrogen passivated surfaces with thin lithium fluoride interlayers.32 Because of the small concentration of scattering centers, a good approximation is to neglect their effect on the charge density in NWs. That allows us to avoid a self-consistent solution of Poisson and transport equations. Including such higher order corrections will be necessary when designing a particular device with large variations of electron density, but this is beyond the scope of this proposal. Scattering Potential. For solving a scattering problem it is crucial to determine the scattering potential that is, in our case, the electrostatic fields of a tetracene molecule physisorbed at the NW surface. For metallic substrates, the most energetically favorable geometrical configuration of the interface is such that the molecular axis c of tetracene (see Figure 2) is parallel to the

where λ(ε) is the energy-dependent mean free path for backscattering and L is the length of the device region. Comparing eq 1 and eq 2, we can establish the relationship between the mean free path and the causal Green’s functions: tr(GaΓ R Gr Γ LΣ0−1Σ) λ (ε) = λ (ε) + L tr(Ga0Γ R Gr0Γ L)

(6)

(4)

If the system is represented by a few sets of indistinguishable scatterers and scattering events are independent, it is reasonable to derive the mean free path for a scattering center of a particular type j and then sum up contributions from each scattering center. In this case, the length of device region L in eq 4 has to be chosen such that the device region contains only a single scattering center. Because of the random distribution of the scattering centers, this length varies along the NW. Averaging over all device regions of different lengths gives the expression for the mean-free path for backscattering: ÄÅ ÉÑ−1 ÅÅ tr(Ga Γ R Gr Γ L) ÑÑ ÅÅ Ñ 0 0 λj(ε) = ⟨l⟩j ÅÅÅ − 1ÑÑÑÑ a R r L ÅÅ tr(G j Γ G jΓ ) ÑÑ (5) ÅÇ ÑÖ where ⟨l⟩j=+,− is the average distance between positively (index +) or negatively (index −) charged scattering centers. According to eq 3, the resistance depends linearly on the length of the device L. This implies that we work in the Ohmic regime neglecting localization effects.30 For our system this is a good approximation since the volume of the device affected by a single molecule has a characteristic size less than the total mean free path, and at the same time, the average distance between molecules with excess charge is much larger than this number. This is usually the case for organic films as they contain relatively low free carrier concentration at all temperatures. In this case, the electron scattering by phonons and by each charged molecule are independent events, and the C


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The model has two parameters: a static dielectric constant, ϵ, and a range-separation parameter, ω. For the example of the crystalline pentacene, it has been shown in ref 19 that a proper choice of those parameters can reproduce energies of frontier orbitals with qualitative accuracy. For the case of tetracene, we take the static dielectric constant of 3.7 from the literature.38 The range-separation parameter omega ω has been optimized to reproduce the ionization potential and electron affinity for bulk crystalline tetracene, following the procedure described in ref 19. This results in the optimal value of ω = 0.032 bohr−1. The results of the electrostatic field computations are shown in Figure 3 for the case of the tetracene molecular anion. In the first-order approximation, the charge density and electrostatic potential are obtained neglecting an inhomogeneity of the environment, i.e., assuming that tetracene is surrounded by other tetracene molecules only. The environment of a molecule physisorbed at the silicon surface includes not only crystalline tetracene but also silicon NWs and SiO2 substrate as is illustrated in Figure 1. A rough estimate of the static dielectric constant for the silicon NW can be computed by using a simple analytic expression based on Penn’s model39 adapted to quantum-confined systems:40,41 ϵ = 1 + (ϵB − 1)/ (1 + ΔE2/Eg2), where ϵB = 11.3 is the static dielectric constant for bulk silicon, Eg = 4 eV is the direct band gap for the singleoscillator model, ΔE = π ℏ E F / 2m a, m is the electron effective mass, and a is the wire radius. All material parameters are taken from ref 41. As a result, we have an effective dielectric constant in the range 8.11−10.62 for the NWs we consider here. While the dielectric constant of 3.9 for SiO242 is close to the corresponding value of tetracene, the dielectric mismatch between a silicon NW and tetracene is rather large. To obtain corrections to the electrostatic potential caused by the dielectric mismatch, we propose to solve the Poisson equation with the tetracene charge density, ρ0, obtained with neglecting the dielectric mismatch and with a spatially dependent dielectric permittivity, ϵ(r): ∇[ϵ(r)∇ϕ(r)] = −4πρ0. The Poisson equation has been solved numerically using the induced charge computation method43 with a self-consistent iteration loop. Conductance Analysis: Static Analysis. We start our analysis by considering scattering at the single molecule limit for which nL = 1 in eq 8. The NW operates as a linear conductor above the threshold. The dependence of the conductance on the Fermi energy, shown in Figure 4, demonstrates that the elastic scattering by the electrostatic fields of tetracene molecules with excess charge suppresses the ballistic electron transport around the sub-band edges. The

interface, while the axis b forms some angle with the surface due to the herringbone structure of the organic crystal. For insulators and semiconductors with passivated surfaces, the angle α between the axis c and the surface is rather large, and unlike for metal substrates, the herringbone structure is formed in the direction parallel to the interface. Detailed information about the crystalline tetracene orientation relative to the silicon (100) surface is reported in ref 33 where the authors have measured the averaged angle α = 65 ± 3° for the ultrathin tetracene films using the near-edge X-ray absorption fine structure. We use that average value in our calculations (see Figure 3). To the best of our knowledge, the spacing d

Figure 3. Contribution to on-site energies of the NW tight-binding model from the electrostatic field of the tetracene radical anion physisorbed at the (100) surface. The plots show the magnitude of the electrostatic field across the NW in the (a) xy plane, (b) zy plane, and (c) xz plane. The orientations of those planes relative to the NW are illustrated in (d).

between physisorbed tetracene molecules and silicon surface is not known so we perform simulations for a set of values of the parameter d. Electron transport in organic semiconductors is characterized by an interplay between coherent transport described by band theory and incoherent polaronic transport.34 At low temperatures, the coherent transport may dominate in crystalline organic semiconductors, resulting in delocalization of charge carriers over large length scales. As the temperature increases, the electron−phonon scattering becomes more intense, leading to so-called polaron bandwidth narrowing.35 At room temperature the incoherent transport dominates, and the electron transport becomes hopping between first nearest neighbors, one at a time, with a charge localized mostly on a single molecule. In the case of organic semiconductor films with an amorphous structure, typically the charge carriers are localized within a single molecule even at low temperatures.36 To compute the electrostatic fields of the tetracene molecules in their crystalline environment, we employ a method proposed in refs 19 and 37. The method treats the effect of environment on the electronic structure of a single molecule using the polarizable continuum model (PCM) and tuned range-separated hybrid exchange-correlation potentials.

Figure 4. Conductance of a silicon NW having rectangular cross section with a side length of 1.38 nm. D


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Figure 5. (a) Fermi energy dependence of the pristine nanowire conductance at T = 5 K and (b) the conductance change determined by a scattering on the electrostatic field of the tetracene cation physisorbed at the surface at T = 5, 100, and 300 K. The spacing between silicon surface and molecules is taken to be 5 Å. The series of rectangular NWs are characterized by the length of the side of 1.38, 1.86, 2.40, 2.95, and 3.49 nm.

electron wave packets with smaller group velocity have larger probability to reflect at the molecular potential. The only detectable difference between positively and negatively charged molecules is the overall suppression of the conductance change in the case of radical cation comparing to the radical anion. For both charge signs, the shapes of the conductance change energy spectra are similar. To estimate the conductance sensitivity as a function of NW widths and temperature, we have computed the conductance change for the case of the molecular cation for a series of rectangular NWs of various sizes at different temperatures (see Figure 5). Increasing the width of NWs leads to the overall shift of the conductance spectra and to a reduction of spacing between propagating modes implying enhancement of the density of states. At zero temperature, for each propagating mode the conductance change caused by the flip of the charge state of a molecule decreases with larger widths of NWs. However, at higher temperatures, the Fermi window, defined by the

expression df(ε)/dε|εF in eq 1, becomes wider, and the conductance is determined not by a single mode but by a contribution from several modes within the Fermi window. Therefore, the conductance change computed at a certain Fermi energy is defined by both the energy distribution of modes and conductance change per mode. As a result, the interplay between the density of states (modes) and sensitivity per mode determines a complicated temperature dependence of the peak conductance change (see Figure 6). We define the peak conductance change as the conductance change at the Fermi energy corresponding to the first maximum at T = 5 K. The Fermi energies corresponding to those peaks are denoted in Figure 5b by vertical thin lines. In Figure 6, because of the above-mentioned interplay, we see lots of intersections of the curves at low temperatures for wider NWs and at relatively high temperatures for the smallest NW of 1.38 nm. The nonmonotonic dependence of conductance on the NW size can be explained by the fact that the density of states in the smallest NW is so small that despite its high sensitivity per E


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monolayers (see Figure 7). For larger NWs, charges in the first monolayer are only detectable. The sensitivity significantly degrades at higher temperatures resulting in the maximal conductance change of several percent from 2e2/ℏ in the NW with the side length of 3.49 nm (see Figure 5b). However, such a conductance change is caused by elastic scattering on a single molecule only. Going beyond the single molecule limit by increasing the concentration of scattering centers and the length of the device region enhances the conductance change as is illustrated in Figure 8. The conductance change grows with the product nL and saturates at the conductance of the pristine NW. This effect has been first discussed in ref 11, showing results similar to ours. Extremely small sizes of NWs considered in this work are required to detect motion of a single molecule. As have been shown in Figure 8, the conductance change increases with number of charged molecules in the device region. For significant larger concentrations of uniformly distributed charges at the surface, another study has been performed in ref 44. They have studied the effect of surface charges on the threshold voltage and also observed a dependence of the sensitivity on the surface-to-volume ratio. According to their results, in the limit of large surface charge density the NW remains sensitive for widths up to 150 nm. Noise Spectroscopy: Dynamic Analysis. Besides the static charge distribution sensitivity, information about the dynamic evolution of charge carriers can also be obtained. The time evolution of molecular charge states leads to fluctuations of the NW conductance which, in turn, can be probed by noise spectroscopy of the electric current in the NW.46 The noise in ultrathin wires consists of shot noise and thermal noise that can be expressed in terms of the transmission probabilities:

Figure 6. Temperature dependence of the peak conductance change for a series of rectangular NWs with a side length of 1.38, 1.86, 2.40, 2.95, and 3.49 nm.

mode, the maximal conductance change at low temperatures is observed in a slightly larger NW with larger density of propagating modes. At temperatures above 200 K, the peak sensitivity of NWs depends monotonically on their widths. The dependence of the peak conductance change on the spacing between organic molecules and silicon surface is shown in Figure 7 for two NWs of different sizes, computed for

ij qV yz 2q2 zz S = 2qV cothjjj j 2kBT zz ℏ k {

∑ Tj(1 − Tj) + 4kBT j

2q2 ℏ

∑ Tj j

(9)

We assume that charge state fluctuations of tetracene molecules represent a discrete Markov process like telegraph noise or generation−recombination noise. This kind of noise is described by the Lorentzian power spectrum:47,48 S(ω) =

4V 2ΔGs 2P+, −P0τ 1 + ω 2τ 2

(10)

where V is the applied electrical bias, ΔGs is the conductance change caused by charge state switching of a single molecule, P+,− and P0 are the probabilities of finding the molecule in one of positive or negative charge states and in the neutral state, respectively. These probabilities can be expressed as P+,− = nL/ N and P0 = 1 − nL/N, where n is the mean linear density of molecules with excess charge, L is the length of NW, and N is the overall number of tetracene molecules covering NW. If we assume that the number of charged molecules is much less than the overall number of molecules, the resulting expression for the noise power spectral density reads

Figure 7. Dependence of the peak conductance change on the spacing between tetracene molecules and NW surface. The green dashed line denotes the width of one monolayer of the crystalline tetracene (13.53 Å according to ref 45).

both cation and anions placed at the center of NW as well as displaced to its edge. The obtained results show that the conductance change is very sensitive to the width of NWs while it is less sensitive to changes of the lateral coordinates of molecules within the NW width and the sign of their charge. Therefore, we may average computed conductance over lateral displacement of molecules and charge signs (see gray thick curves in Figure 7). In the smallest NW at T = 5 K, the conductance is sensitive to the charge states in the first and second molecular

S(ω) =

2 2 nL 4V ΔGs τ N 1 + ω 2τ 2

(11)

Equation 11 together with eq 9 can be used to fit the measured noise power density. As a result of the fitting procedure, one obtains the value of the characteristic time τ. The values of n F


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Figure 8. Fermi energy dependence of the conductance change caused by scattering on the electrostatic field of the tetracene cation for different values of nL, effective number of molecules, and for (a) T = 5 K and (b) T = 300 K.

and ΔGs can be obtained from measurements of the mean conductance change discussed in the previous section. It has been suggested in ref 49 that to reduce the contributions of thermal noise and shot noise, the measurements of the noise power density can be performed in the frequency range up to 100 GHz where telegraph noise dominates. The proposed method can be used to measure fluctuations of the intrinsic charge carriers as well as the photoinduced charge carriers. In the latter case ultrafast pulses combined with phase locked detection allow the charge movement within the film to be distinguished from the inherent photoconductance response of the NWs. In addition, comparison of neighboring NWs in an array (as visualized in Figure 1) provides a method of common mode rejection as neighboring NWs will see a different electrostatic environment but the same optical intensity. Spatially Resolved Measurements: Electrical Impedance Tomography. As well as temporal resolution, the spatial layout of NWs allows the position of the mobile charges to be determined. The 1D array of silicon NWs shown in Figure 1 provides information about a linear distribution of charge carriers in the direction along the NW array. To access information about the 2D distribution of charge carriers in the organic semiconductor thin film, we propose a setup representing a grid of NW resistors shown in Figure 9a. In principle, the resistance of each NW resistor can be read out with having contacts only at the edge nodes by means of electrical impedance tomography.50 Electrical impedance tomography is a standard tool in modern medical imaging and has also recently found several applications for charge sensing of nanoengineered thin films.51,52 In the equivalent circuit of the proposed device shown in Figure 9b, all resistances are unknown while one has access to set up and/or to measure the voltages of the edge nodes and/or the currents injected in the edge nodes. It has been proven that the problem of finding resistances for such an equivalent circuit has a unique solution.50 The technology relies on how fast this problem can be solved on the device level. The proposed setup requires additional analysis on the trade-offs between accuracy, resolution, and bandwidth, which can be optimized for the parameters of the given film to be studied. The NW resistors can be implemented in several ways with different types of contacts. One possible implementation is a

Figure 9. (a) Setup for the nanoscale electrical impedance tomography of organic thin films and (b) its equivalent circuit. We show a 4 × 4 array of NW resistors only for illustrating purposes. In reality, the number of the resistors determining the resolution of the measurement setup could be much larger.

stand-alone NW with metallic contacts, similar to one fabricated in refs 20 and 21. In this case, the work function of the metal electrodes has to be chosen such that its Fermi level is aligned with the conduction band of the NW. It is desirable to make Ohmic contacts between channel and metal electrodes. If not, a gate electrode can to be introduced to reduce the Schottky barrier electrostatically. Another implementation relies on a conventional CMOS technology reduced in size to the nanoscale.23 For our purpose the device has to be fabricated without spacers and the gate stack has to be etched out down to the channel. In this case, the contacts are degenerate n-type semiconductors. Another alternative for the room-temperature operation is the junctionless NW transistor.53 In this case, the contacts are intrinsic semiconductors, and conduction is provided by thermally generated charge carriers. For CMOS devices, an additional electrostatic gate is required to move the operating point above the threshold.

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CONCLUSIONS We have considered silicon NWs as a probe for charge states in tetracene molecules physisorbed on its surface as an example. We used a linear transport model and expressed the mean-free path determined by elastic scattering in terms of causal Green’s functions. The Green’s functions have been computed by using the recursive Green’s function algorithm and a semiempirical tight-binding model with the reduced mode space transformation. The electrostatic potentials of molecules with excess G


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the unit cell to the left and right neighbors, respectively, ε is the energy, and λn = e−ikn is an eigenvalue expressed as an analytic function of the wavenumber k. Following the procedure described in ref 60, the eigenvectors un and eigenvalues λn can be divided in two classes: the left-propagating modes, u< and Λ and Λ>. The selfenergies describing couplings to the semi-infinite leads can be expressed in terms of those modes:60

charge have been computed by using a combination of the polarizable continuum model and density functional theory with the range-separated exchange-correlation functional. The conductance change caused by elastic scattering on the electrostatic field of charged molecules decreases with increasing temperature and transverse sizes of NWs. For a single molecule, ultrathin silicon NWs with characteristic cross sections below 2 nm are characterized by the conductance change of about 0.1·2e2/ℏ at room temperature. The conductance change grows with the number of charged molecules. Thus, sub-4 nm NWs are sensitive enough to detect several tens of charge carriers. At room temperature, only charge carriers in the first molecular monolayer are detectable. Our results show that silicon NWs can be used to measure the concentration of charge carriers in organic thin films assuming that their charge is localized within a single molecule. In addition, the time evolution of molecular charge states leads to fluctuations of the NW conductance which, in turn, can be probed by noise spectroscopy. We propose using a grid of NW resistors to access the information about 2D spatial distribution of charge carriers in organic thin films via electrical impedance tomography.

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ΣrL = h1u >Λ>u >−1,

ΣrR = h−1uu15 Å from the molecular center of mass. At the shorter distances, electrostatic fields are characterized by a nonuniform angular dependence. The dielectric permittivities of Si NWs are collected in Table 1. To take into account the dielectric mismatch between tetracene and the Si NW, we solve the Poisson equation

(13)

where h0 is the tight-binding Hamiltonian of the NW unit cell shown in Figure 2, h−1 and h1 are the Hamiltonians describing couplings of H


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ACS Applied Electronic Materials Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.K. thanks Barbara Fresch and Francoise Remacle for fruitful discussions on DFT computations. M.K. and J.C. thank Pegah Maasoumi for discussions on relevant technological advances. The authors acknowledge support of the Australian Research Council through Grant CE170100026. This research was undertaken with the assistance of resources and services from the National Computational Infrastructure, which is supported by the Australian Government.

Figure 11. Electrostatic potential of tetracene (a) anion and (b) cation computed in the homogeneous media approximation.

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Table 1. Relative Dielectric Permittivity of Silicon NWs Obtained from Penn’s Model NW width (nm) ϵNW

1.38 8.11

1.86 9.26

2.40 9.97

∇[ϵ(r)∇ϕ(r)] = − 4πρ0

2.95 10.38

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3.49 10.62

(16)

for the charge density ρ0 obtained in the homogeneous media approximation. The Poisson equation (16) has been solved for the electron density of the tetracene molecule to take into account the effect of the nonuniform dielectric environment on the electrostatic potential. The charge density can be extracted from the electrostatic potential, ϕ0, shown in Figure 11 by using the relationship ρ0 = −(ϵTc/4π)∇2ϕ0. Equation 16 can be rewritten in the form

∇2 ϕ(r) = −

1 [4πρ0 + ∇ϵ(r)∇ϕ(r)] ϵ(r)

(17)

The right-hand side of the equation can be thought as an effective charge, ρ′, dependent on the electrostatic field. This is the essence of the so-called the induced charge computation method.43 The equation can be solved iteratively assuming ρ′ = 4πρ0/ϵ(r) as a first guess and updating charge ρ′ for each next step. The advantage of the iteration procedure is that at each iteration step we solve the standard Poisson equation with constant coefficients, for which a variety of standard Poisson solvers exists. In this work we use the so-called fast Poisson solver from GPAW package.63 The solver uses a combination of Fourier and Fourier sine transforms in combination with parallel array transposes. The discussion on the effect of dielectric mismatch and results for the neutral tetracene molecule are given in the Supporting Information, section S2.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsaelm.9b00354. Brief description of the reduced mode space method (section S1), electrostatic potential of tetracene molecule (section S2), and listing of the input file for Gaussian90 needed to reproduce results of DFT computations (section S3) (PDF)

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Phone +613 (0) 99259555. ORCID

Mykhailo V. Klymenko: 0000-0002-4641-8977 I


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