Algorithm for Random Walk Simulation of Modulated Surface

Article pubs.acs.org/JPCC

Algorithm for Random Walk Simulation of Modulated Surface Photovoltage Signals in Nanostructured Systems with Localized States Steffen Fengler‡ and Thomas Dittrich* Institut für Silizium-Photovoltaik, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Kekuléstrasse 5, 12489 Berlin, Germany ABSTRACT: The simulation of modulated surface photovoltage (SPV) signals has been demonstrated for nanostructured systems with localized electronic states by using a random walk approach. The proposed algorithm was adapted to a real measurement time regime and allows for the investigation of in-phase and phase-shifted by 90° SPV signals and of frequency, phase angle−amplitude, and temperature dependencies of modulated SPV signals. The simulation of modulated SPV signals can give a strong impact for the interpretation of SPV spectra concerning localized electronic states and related phenomena in nanostructured systems.

1. INTRODUCTION In nanostructured systems, localization of charge carriers due to disorder in space and due to electronic defect states is a major limitation for charge separation and charge transport. Localized states play a decisive role in nanostructured photoactive materials for photovoltaic and photocatalytic solar energy conversion. The investigation of the influence of localized states on charge separation in nanostructured materials is important for better understanding of limiting factors in related devices. Therefore, the further development of sensitive methods and analysis tools for the investigation of localized states in nanostructured photoactive materials is of broader interest. Surface photovoltage (SPV)1 measurements can provide useful information about electronic states and mechanisms of modulated charge separation in nanostructured systems containing, for example, organic molecules,2 semiconductor quantum dots (QDs),3 or carbon nanotubes in contact with a metal halide perovskite.4 Modulated SPV measurements are highly sensitive with respect to charge separation in extremely thin photoactive layers5 or even monolayers of molecules, such as oriented donor−acceptor dyads,6 or colloids, such as CdSe QDs.7 Regarding the parallel plate capacitor, for example, a density of charge carriers of only 108 cm−2 separated in space by 1 nm would result in a SPV signal of about 1 μV (see also ref 8), which is above the noise level for measurements with lockin amplifiers. Further, modulated SPV measurements are sensitive to only electronic states and processes that can follow the modulation period (T) in charge separation and relaxation, that is, modulated SPV signals are practically independent of photoinduced changes of permanent dipoles or photoinduced surface chemical reactions. This favors modulated SPV © XXXX American Chemical Society

measurements for the investigation of electronically active electronic states and transport processes. Recently, random walk simulations have been applied for the analysis of SPV transients in TiO2 layers with localized states9 or in single layers of surface-treated CdSe QDs.7,10 However, an appropriate approach allowing for a detailed analysis of modulated SPV signals is still missing. In this work, an algorithm for random walk simulations of modulated SPV signals in nanostructured systems with localized states has been developed. For this purpose, basic building blocks of hard spheres were defined and distinguished by the presence or absence of localized and/or delocalized states. In a next step, hard spheres were combined with stacks, layers, or layer systems. A random variation of distances between localized states, which is a characteristic property for disorder, was introduced by arranging hard spheres in a softened lattice, which led to a reduction of memory resources required for random walk simulations in comparison with static state ensembles. The random walk simulations were based on Miller−Abrahams hopping11 between localized electronic states. The in-phase (fast response in relation to T) and phase-shifted by 90° (slow response in relation to T) SPV signals were obtained from four averaged integrated values of the distance of separated charge defined at full, quarter, half, and three-quarters modulation periods. As an example, a model system of metal nanospheres with varied distances in a softened fcc lattice and with Gaussian distributions of electron and hole states was applied to demonstrate the simulation of in-phase Received: May 27, 2016 Revised: July 15, 2016

A

DOI: 10.1021/acs.jpcc.6b05341 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

probability of subsequent trapping of the charge at a localized state.10 Hard spheres with localized and/or delocalized states can be placed close to a surface of a substrate serving as a chargeselective contact. In our simplified model, a sharp chargeselective contact is defined by a charge transfer from a hard sphere into a geometric plane at which the transferred charge will be delocalized. This case has been considered, for example, for the simulation of SPV transients within the so-called single QD approximation.10 Usually, SPV spectra of more complex systems with charge transfer or transport across localized states shall be analyzed. Related model systems can be created from hard spheres described above as the elementary building blocks. Localized states arise, for example, due to disorder. The variation of distances between localized states is the decisive property for charge transfer in a disordered system with localized states. Random walk simulations in systems with random distributions of localized states in space demand large resources in the memory of the computer used for simulation. However, the definition of a random or amorphous distribution of localized states in space is not necessary for describing a random distribution of distances between nearest neighbors. In our model, we implemented a so-called softened lattice, which defines the positions of the centers of the hard spheres. A softened lattice (see, for example, Figure 1d for a softened fcc lattice) means that the distance between the centers of two neighbored hard spheres (dij) is obtained from a random distribution for each individual step of charge transfer. In the case that transferred charge is localized within metal spheres, the distance for charge transfer is then the difference between dij and D. In a similar way, layers with localized states in insulating sintered nanoparticles or with semiconductor QDs can be built, whereas the average distances between the surfaces of two neighboring hard spheres or colloids are close to zero or to the doubled thickness of the surfactant layer, respectively. 2.2. Processes Considered in the Simulation of Modulated SPV Signals. The processes of light absorption and photogeneration, charge separation, charge transport, and recombination have to be taken into account in the simulation of modulated SPV signals. A typical light intensity for real spectral-dependent modulated SPV measurements with a halogen lamp and a monochromator is on the order of μW/ cm2 depending on the wavelength and spectral resolution, resulting in a photon flux on the order of 1011 cm−2 s−1. The generation rate is given by the product considering the photon flux and the absorption law. The maximum generation rate for a monolayer of dye molecules, for example, would be on the order of 108−109 cm−2 s−1, corresponding to about 5 × 106 to 5 × 107 photogeneration events of electron−hole pairs in onehalf period (light on) for a modulation frequency of 10 Hz. In the following, it will be assumed that each photogeneration event of an electron−hole pair leads to one elementary charge separated in space (event), whereas the number of events per half period (light on) is proportional to the generation rate and the events are distributed randomly in time (Figure 2a). For simplicity, it will be further assumed that charge separation takes place at a sharp charge-selective contact, which is the case, for example, for the dissociation of excitons at an interface (for example, layers of conjugated organic molecules) or for injection (for example, layer of dye molecules at the surface of an insulator). In this sense, the initial distance between the planar charge-selective contact and the charge

and phase-shifted by 90° modulated SPV spectra, to show the influence of modulation frequency and temperature, and to present typical dependencies of the phase angle on the amplitude. The proposed algorithm allows simulation of modulated SPV signals in nanostructured systems and can be extended, in principle, as well to the simulation of other modulated signals related, for example, to modulated photoluminescence or photocurrent measurements.

2. METHODS 2.1. Localized and Delocalized States in the HardSphere Approach. For the simulation of modulated SPV signals, the distribution of localized states in space shall be defined. The extension of a localized state is on the order of a bond length in a chemical compound or of a unit cell in a crystal. In the simplest case, a localized defect can be described by an S orbital and can be therefore expressed by a hard metal sphere with a diameter (D) between about 0.2 and 1 nm (see also Figure 1a). As an example, an electron captured by a C60

Figure 1. Hard spheres with diameter D with delocalized and without localized (a), without delocalized and with localized (b), and with localized and delocalized (c) electronic states. The extension of localized states (δ) is much smaller than D. Hard spheres distributed within a softened fcc lattice and a sharp charge-selective contact (d). The indexes i and j and the parameter dij mark two neighboring spheres and the distance between the centers of both.

molecule can be considered as an electron localized at a metal sphere with a diameter of about 0.7 nm.12 As remark, an electron is delocalized in a hard metal sphere in the sense of the parameter D but localized in the sense of the isolated metal sphere. Localized states appear often at the surface or in the bulk of insulating nanoparticles, for example, defect states in TiO2 nanoparticles.13 In such case, the value of D is much larger than 1 nm, localization occurs randomly within the sphere with diameter D, and the extension of localized states (δ) is much smaller than D (Figure 1 b). Localized and delocalized states can be combined with each other (Figure 1c), for example, in chemically treated semiconductor QDs.7 The delocalized state is extended over the diameter of the QD, and localized states are distributed at the surface of the QD. A coupling between the localized and delocalized states can be defined by a probability of charge transfer from the localized into the delocalized state and by a B


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distribution of localized states in space, the energy levels, and a matrix of tunneling rates between localized states were defined. For the simulations presented, the localized states were defined by metal spheres distributed within a softened fcc lattice with a random distortion of the lattice position by maximal 40% of the shortest distance between the surfaces of neighboring hard spheres in the undisturbed lattice. The time dependence of dcc is decisive for the simulation of SPV signals. Figure 2b shows schematically the time dependence of dcc for the simulation of several events. The time dependence of dcc can last over several periods. The time dependence of a SPV signal (see also the schematic in Figure 3b) can be calculated by the following equation: N

Figure 2. Schematic of the modulated light intensity resulting in single events of charge separation (a) and of the time-dependent projected distance of a given charge separated in space for some events (b).

(2)

The parameter N denotes the number of events that are large and amounts usually between 105 and 106.

separated in space is equal to the distance between the planar charge-selective contact and the localized state at which the charge separated in space appeared. The initial distance of charge separation depends on the nature of the process of charge separation and the considered time scale. For example, the initial distance of charge separation can range from values significantly below 1 nm for dissociation of excitons at donor− acceptor heterojunctions14 to hundreds of nanometers for charge separation in a space charge region within the transit time, which is on the order of picoseconds for crystalline semiconductors. In the following, an initial distance of charge separation of the distance of the first monolayer of localized states to the charge-selective contact will be taken. A random walk based on tunneling between localized states was calculated for each event, whereas the tunneling rates were taken for Miller−Abrahams tunneling11 under consideration of the electric field

Figure 3. Schematic of the modulated light intensity (a), of the sum of all projected distances of numerous events (b), and of the projected distances integrated in time for integration intervals of a quarter modulation period (c). See the text for details.

⎛ (E − E f ) + | E i − E f | ⎞ 1 1 = ·exp(2·α ·R if ) ·exp⎜ i ⎟· τ τK 2 · k B· T ⎝ ⎠ ⎛ e 1 ⎞ · ⎟ exp⎜ ⎝ 4π ·ε ·ε0 dcc ⎠

e · ∑ dcck ε ·ε0 k = 1

SPV(t ) =

(1)

In-phase and phase-shifted by 90° SPV signals have to be obtained from the parameters S1, S2, S3, and S4, which are calculated from the time dependence of the SPV signal by taking into account the integrated over quarter periods values (Figure 3) and by averaging over the number of all periods within the time of measurement at a given generation rate (M)

where τK, α, Rif, Ei, Ef, kB, T, e, ε, ε0, and dcc denote the hopping time of an elementary step, the inverse tunneling length, the distance between the initial and final states for tunneling, the energies in the initial and final states, the Boltzmann constant, the temperature, the elementary charge, the relative dielectric constant, the dielectric constant of vacuum, and the distance of the charge carrier to the plane of the charge-selective contact. For the presented simulations, the value of Rif was given as the difference between dif and D (Figure 1d). As remark, a selfconsistent calculation of the electric field or effective contributions of individual charge carriers shall be considered in case of generation of more than one electron−hole pair in the simulation box. Here, separation of only one electron−hole pair was taken into account. The random walk of one event was finished after recombination described by tunneling of the charge carrier separated in space from a localized state into the plane of the charge-selective contact. 2.3. Simulation of Modulated In-Phase and PhaseShifted by 90° SPV Signals. Before starting the simulation of a three-dimensional random walk for a separated single charge carrier, an average distance between localized states, the

M

Sj =

i=1

t = (j /4)·T + i·T

∫t =(j−1)·T+i·T

∑{

N

(∑ [dcck (t )]) dt }

j = 1, ..., 4

k=1

(3)

The in-phase and phase-shifted by 90° signals (X and Y, respectively) are calculated by using the parameters S1, S2, S3, and S4 and the definition of the X and Y signals X = S1 + S2 − S3 − S4

(4a)

Y = +S1 − S2 − S3 + S4

(4b)

Often, the amplitude (R) and the phase angle (φ) are analyzed instead of the X and Y signals. The values of R and φ are obtained from the values of X and Y by using the following equations: R= C

X2 + Y 2

(5a) DOI: 10.1021/acs.jpcc.6b05341 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C φ=

⎛Y ⎞ 180 ·arctan⎜ ⎟ ⎝X⎠ π

(5b)

3. RESULTS AND DISCUSSION 3.1. Adaptation of a Real Measurement Regime with Simulations. The qualitative behavior of the simulation algorithm for modulated SPV signals in systems with localized states will be demonstrated on an example with localized states expressed by metal spheres distributed within a softened fcc lattice. This model system reflects the properties of a disordered transport system with fluctuating distances between nearest neighbors of localized states (Figure 4a). Further,

Figure 5. Example for the adaptation of a real time-dependent measurement regime of modulated SPV signals for photon energies increasing in time (filled circles) and corresponding light intensity of the light source (open circles) and absorption coefficient of the model system (triangles) (a), resulting in the dependencies for the simulated in-phase signals (filled circles), phase-shifted by 90° signals (open circles), and PV amplitude (open triangles) (b) and for the phase angle (c).

Figure 4. Model system consisting of a transport layer for holes with localized states and an electron-selective contact (a). The densities of localized states for electrons and holes are distributed by a Gaussian (b).

were much smaller than those of the in-phase SPV signals, and the amplitude followed therefore mainly the in-phase SPV signal. In the following, the time dependence of the modulated SPV signals with changing photon energy will be presented as spectra. The phase angle (Figure 5c) changed from about −20° at low generation rate to values close to −5° at high generation rates, that is, the average time holes spend in deeper defect states decreased with increasing generation rate. This is not surprising because more of the deeper localized states are occupied due to partial charging of the layer. 3.2. Frequency-Dependent In-Phase and PhaseShifted by 90° Signals. In real modulated measurements of SPV spectra with a lamp and a monochromator, the modulation frequency can be easily varied with a mechanical chopper from about 1 Hz to several kHz. Related modulated SPV spectra have been simulated for the example model system at different modulation frequencies. Figure 6 presents in-phase and phaseshifted by 90° SPV spectra simulated for a temperature of 30 °C and for modulation frequencies of 8, 400, and 8000 Hz. The in-phase SPV signals decreased with increasing modulation frequency. This is not surprising because more and more deep localized states are getting permanently charged with increasing modulation frequency and do not follow the modulation period anymore, that is, they do not contribute to modulated SPV signals at higher frequencies. In contrast, phase-shifted by 90° SPV signals first increase with increasing frequency and decrease at high frequencies. For the given example, the spectrum of the phase-shifted SPV signals simulated at 8 Hz showed two maxima at photon energies closer to the lower generation rates. For the analysis of SPV spectra, a so-called onset energy (Eon) is often used, which is determined from the intersection point of the slope in the inflection point of the spectrum and

electronic states for electrons and holes are distributed by a Gaussian (Figure 4b) defining (i) the behavior of the absorption spectrum as well as (ii) the behavior of the transport of the charge remaining in the transport layer. As such, the width of the Gaussian has strong influence on trapping and detrapping of charge carriers and therefore on charge transport. A typical real measurement of a modulated SPV spectrum lasts for about 10 min, that is, the photon energy is increased step by step within this time (Figure 5a). A modulated SPV signal is measured with a lock-in amplifier for each corresponding generation rate in a time interval starting after 1 s after reaching the required photon energy (after so-called stabilization of the signal) until the time when the monochromator starts to move to the next photon energy. The light intensity and the absorption coefficient change in time with respect to the change of the photon energy. Figure 5a gives a real spectrum of the intensity of a quartz prism monochromator and an absorption spectrum obtained for the Gaussian distribution of electron and hole states if assuming constant matrix elements for all transitions. Figure 5b shows the simulated spectra of the in-phase and phase-shifted by 90° SPV signals for the modulation frequency of 80 Hz and a temperature of 230 °C. The sign of the in-phase signals was positive because the electron was separated into the charge-selective plane at the substrate side. The scaling of the values of the SPV signals depends on the generation rate and was set to about 1 mV in the maximum of the given spectrum of the in-phase SPV signals. As usual for one mechanism of charge separation with electrons separated toward the internal interface, the phase-shifted by 90° SPV signals were negative. Further, absolute values of the phase-shifted by 90° SPV signals D


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range between 2 and 200 Hz and decreased further by up to 10 times at 8 kHz. The frequency dependencies of Xmax and Ymax behave like a system with a characteristic time constant of 5 ms. The values of EonX and EonY increased with increasing frequency and saturated at values close to 1.1 eV between 200 and 400 Hz for EonX and between 1 and 2 kHz for EonY, whereas the difference between EonX and EonY decreased from about 0.08 eV at 2 Hz to less than 0.02 eV at 2 kHz. The absolute values of the phase-shifted by 90° SPV signals can be plotted versus the in-phase SPV signals (Figure 8). At

Figure 6. In-phase (black lines) and phase-shifted by 90° (red lines) SPV spectra for the given model system simulated for a temperature of 30 °C and modulation frequencies of 8, 400, and 8000 Hz (thin, medium, and thick lines, respectively). The determination of the empirical values of the so-called onset energies is given.

the axis of the photon energy. The onset energy is close to the band gap for SPV measurement on bulk semiconductors.15 In systems with localized states, the onset energies are usually different for in-phase (EonX) and phase-shifted by 90° (EonY) SPV spectra, where EonY is lower than EonX. For the given example, the values of EonX and EonY amounted to 0.989 ± 0.002 and 0.919 ± 0.003 eV, respectively, at 8 Hz and to 1.110 ± 0.007 and 1.095 ± 0.005 eV, respectively, at 4 kHz. For comparison, the onset energy of the absorption spectrum was about 1.12 eV. The values of the maximum in-phase (Xmax) and phaseshifted by 90° (Ymax) SPV signals as well as of EonX and EonY are plotted as a function of the modulation frequency in Figure 7a,b, respectively. The value of Xmax decreased by about 100 times in the frequency range between 4 Hz and 4 kHz. The value of Ymax increased by about 30 times in the frequency

Figure 8. Plot of the maximum negative absolute values of the phaseshifted by 90° SPV signals versus the maximum in-phase SPV signals simulated for a temperature of 30 °C.

first glance, this plot is reminiscent of a depressed semiarc in a Cole−Cole plot with a constant phase element.16 This analogy is not surprising because the in-phase and phase-shifted by 90° SPV signals can be treated, for the given example, like the real and imaginary parts of an impedance leading to a reduction or retardation of the SPV signal, respectively. It is interesting to note the asymmetry with higher phase-shifted SPV signals toward the lower values of the in-phase SPV signals following, obviously, from the disorder near the charge-selective contact. Related asymmetries have been observed also by impedance spectroscopy but could not be, however, explained with really satisfaction so far by macroscopic models such as thin films with an exponential decay of resistivity.17 In this sense, the consideration of disorder at electrodes by modulated random walk simulations can become very useful for the interpretation not only of modulated SPV measurements but also of modulated photoelectric or modulated electric measurements such as intensity modulated photovoltage or photocurrent spectroscopies18 or impedance spectroscopy,17 respectively. 3.3. Analysis of Phase−Amplitude Diagrams. The dependence of the phase angle on the amplitude of SPV signals contains information about mechanisms in photogeneration, transport, or relaxation. For example, the phase angle was practically independent of R for charge separation within an ordered monolayer of dyad molecules but depended strongly on charge separation due to electron injection from the same molecules into TiO2 containing a large number of localized states.6 Figure 9 shows the dependence of the phase angle on R simulated for the model system at 30 °C and modulation frequencies of 8 and 80 Hz. A phase angle closer to 0° means a faster response of the SPV signal in relation to the modulation period. At low amplitudes, the phase angles were about −40 and −60° for modulation frequencies of 8 and 80 Hz, respectively. With increasing R, the phase angles reduced toward −2 and −15° for modulation frequencies of 8 and 80 Hz, respectively. Such behavior is typical for modulated SPV signals limited by charge transport in a system with localized

Figure 7. Frequency dependencies of the maximum values of the inphase (squares) and phase-shifted by 90° (circles) SPV signals (a) and of the onset energies of the in-phase (squares) and phase-shifted by 90° (circles) SPV signals (b) simulated for a temperature of 30 °C. E


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The inset of Figure 10 gives Arrhenius plots of the maxima of the PV amplitudes simulated for 8, 80, and 800 Hz. The saturation of the PV amplitudes at lower temperatures and the influence of thermally activated emission of charge carriers from localized states at higher temperature can be clearly distinguished.

4. CONCLUSIONS Finally, an algorithm has been developed for the random walk simulation of modulated SPV signals in systems with localized electronic states. In our model, the basic spatial units are defined by hard spheres with localized and/or delocalized states. The hard spheres can be combined to simple and more complex layer systems. The approach of a softened lattice with varying distances between metal spheres has been proposed for implementing transport properties of disordered systems with localized states. The procedure for getting the in-phase and phase-shifted by 90° SPV signals has been given. The qualitative behavior of typical modulated SPV signals depending on generation rate, modulation frequency, and temperature as well as a typical dependence of the phase angle on the PV amplitude was demonstrated for a simple model system containing localized electron and hole states with a Gaussian distribution in energy. The proposed simulation algorithm is relatively fast and can therefore be (i) implemented as well into fitting procedures for the analysis of real modulated SPV measurements, (ii) extended to more complex systems, and (iii) applied to the simulation of other than SPV modulated photoelectric measurements. Our approach is suitable as well for extension to the simulation of photoluminescence and photoconductivity signals in systems with localized states. In these cases, radiative recombination of an electron and a hole in a state or of extraction and recombination of charge carriers at contacts, respectively, shall be analyzed instead of the analysis of the distance of charge separation. In this sense, the development of robust and reliable simulation tools will be of interest for a broader community in areas of photovoltaics, photocatalysis, and materials science.

Figure 9. Dependence of the phase angle on the PV amplitude simulated for a temperature of 30 °C and frequencies of 8 and 80 Hz (squares and circles, respectively).

states. As such, modulated SPV signals can depend on more than only one distribution of localized states and on different conditions for initial charge separation. In such a case, the dependence of the phase angle on R becomes more complex and can include, for example, hysteresis and/or local extrema (see, for example, ref 3). In this sense, the dependence of the phase angle on R derived from SPV spectra can serve as an additional fingerprint for a given sample. 3.4. Temperature-Dependent In-Phase and PhaseShifted by 90° Signals. The emission probability of charge carriers from localized electronic states increases with increasing temperature. This has a tremendous influence on temperature-dependent modulated SPV signals. Figure 10

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel +49 30 8062-41313. Fax +49 30 8062-41333. Notes

The authors declare no competing financial interest. ‡ S.F.: on leave.

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Figure 10. In-phase (black lines) and phase-shifted by 90° (red lines) SPV spectra for the given model system simulated for a frequency of 80 Hz and temperatures of 180, 130, and 80 °C (thin, medium, and thick lines, respectively). The inset represents Arrhenius plots of the maxima of the PV amplitudes simulated for 8, 80, and 800 Hz (squares, circles, and triangles, respectively).

ACKNOWLEDGMENTS S.F. is grateful to the Helmholtz-Gemeinschaft Deutscher Forschungszentren e.V. (HGF) for financial support within the Hybrid-PV project.

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ABBREVIATIONS SPV, surface photovoltage

shows in-phase and phase-shifted by 90° SPV spectra for the given model system simulated at temperatures of 180, 130, and 80 °C for a frequency of 80 Hz. The modulated SPV signals decreased with increasing temperature due to faster transport of separated holes toward the charge-selective contact where the holes can recombine. The phase-shifted by 90° SPV signals decreased stronger with increasing temperature than the inphase SPV signals, that is, the response of the modulated SPV signals became faster with respect to the modulation period.

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Algorithm for Random Walk Simulation of Modulated Surface

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