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Slow processes in disordered solids M. Pollak and A. View via Publisher. Alternate Sources. Save to Library. Create Alert. Share This Paper. Citations Publications citing this paper. Magneto-transport and localization in disordered systems with local superconductive attraction Thi Thuong Huyen Nguyen. Theoretical studies of the interplay between superconductivity and disorder Anirban Gangopadhyay. Synthesis, structural and electrical characterizations of LaSrCu0. Magnetic and electronic properties of iron-based superconducting systems Sebastian Sambale.

Field theory of disordered systems -- Avalanches of an elastic interface in a random medium Alexander Dobrinevski. Because the hopping rates depend exponentially both on the energy difference and on the distance between pairs of sites, analytical calculation of such an average is usually very difficult, but it is partially simplified in a system with a steep distribution of localized states, where the hopping process is well described with the concept of transport energy.

The most probable upward jump corresponds to an optimized combination of the distance and energy difference. The average distance for states below any energy E 1 is. The result 12,16 is that, independent of the energy of the starting site, the fastest hops occur toward sites in the vicinity of a specific level, the transport energy E tr , as indicated in the scheme of Figure 1, that is given by where.

The average jump distance is therefore evaluated by eq The jump frequency from the energy E to the transport energy is For the calculation of the jump diffusion coefficient, we average eq 19 as follows The main contributions arise from carriers between E F and E tr ; therefore, we can write. Calculating the integral in eq 21, we arrive at the result. Therefore, the chemical diffusion coefficient is.

Using eqs 9 and 4 we obtain the conductivity. Let us remark on some features of these results. First of all, eqs 24 and 25 show the same Fermi-level dependence as that in MT, which is understandable because the transport has been assumed to be governed by activated hops to E tr. But there are important differences between the models. These parameters are independent of traps, and are needed separately to describe steady-state transport. In contrast, the hopping model lacks such parameters. The results of the numerical calculation eq 20 and the analytical formula eq 24 are plotted in Figure 1b.

We remark that both lines in Figure 1b i have exactly the same slope and ii differ by a small multiplicative factor due to the approximations involved in eq 24 , of about 3, that becomes larger as T 0 increases. Therefore, eq 24 is a good practical approximation.

### Hopping transport in solids

In addition, the absolute values of D n are close to those obtained in the MT model, and hence, in agreement with the literature results. The exact match between all of the lines in Figure 1b is coincidental because in MT the line can be shifted by changing the bulk value D 0 of TiO 2. This is a first indication that indeed the hopping model is a good candidate to describe the D n E F typically measured in DSC. Further features of the hopping model concern the temperature dependence of transport quantities and are illustrated in Figure 3.

According to eq 17, the transport level changes with the temperature, and for the assumed parameters the change is about 60 meV over K, see Figure 3a. Thus, the intercept of D n E F lines at different temperatures, as shown in Figure 3c, does not indicate the transport level exactly, in contrast to MT.

Still, in a first approximation the intercept shows roughly the position of E tr , which is substantially below the conduction band level E 0. Thus, we can appreciate that the model results in Figure 3b show good agreement with the experimental results of Figure 2c. The hopping model seems to describe this feature much better than the MT model. We are therefore encouraged to fit the results in Figure 2c with the model of eq 24 in order to obtain the electronic parameters included in the hopping model. However, one must be careful when treating the D n E F data because, as mentioned above in the MT analysis, the slopes in the experimental data of Figure 2c only give the parameter T 0 which can already be inferred from the capacitance.

The two values differ by a small amount of about 30 meV, in agreement with the stationary values of the capacitance. These values are furthermore reasonably close to those presented in the literature. In comparison with previous literature calculations, we note that the Fermi-level dependence of the conductivity obtained in eq 25 is in agreement with the result of Vissenberg and Matters, 27 which was obtained using the critical path analysis based on percolation theory.

Such analysis is more reliable than our simple approach that assumes from the start the activated hops to the transport energy. Therefore, concerning the absolute values of the hopping diffusion coefficient, our analytical results based on averaging the hopping rates must be regarded as an estimation, and for further check of the validity of eqs 24 and 25 one should compare them with Monte Carlo simulations as discussed by Baranovskii et al.

In addition, it should be emphasized that the transport energy approach does not work well when the Fermi level approaches the level E tr. In fact, when the Fermi energy rises, the average distance between sites decreases rapidly and the tunneling becomes favorable over thermal activation to the higher energies. Therefore, diffusion in the high carrier density range requires a different approach. Another work worth commenting on is that of Nelson et al. This is not in agreement with the core, and almost universal, assumption of transition rates in hopping conductivity, indicated in eq 11, above, namely, that the energy dependence is only on the second term of the exponent of upward jumps, while the tunneling factor is independent of energy.

Therefore the results in ref 29 are not claims about standard hopping theory and are not in conflict with ours. It is well agreed that recombination in DSC is influenced by trapping effects. A model based on trapping and detrapping followed by interfacial charge transfer at the conduction band levels 31,32 seems to be so far the best approach for the interpretation of electron lifetimes. Therefore, it is obvious that the clarification of the interpretation of the transport level for electrons in the nanostructured metal oxides used in DSC can have a substantial impact on the understanding of recombination processes.

This important question requires further work. In conclusion, we have derived a useful working formula for the interpretation of the chemical diffusion coefficient of electrons in DSC using the hopping model. This approach seems to describe well some features of the data that are not captured correctly in the multiple trapping model, especially concerning the temperature dependence and energy levels governing electron transport. However, it must be recognized that the linear region of log D n E F contains very little information on the transport mechanism.

We hope that the present considerations will encourage further experimental work in the temperature dependence and high carrier density regime, where the transport mechanism should be unambiguously identified. If the hopping model is confirmed by such measurements, then it can provide interesting new insight into the operation of dye-sensitized solar cells.

Nature , , A photoelectrochem. The overall light-to-elec. The c. B , , American Chemical Society. The properties and phys. The chem. The thermodn. From these considerations, the chem. This last model is shown to correspond to the mean-field approxn. A general expression contg. C , , A review. Recent progress toward understanding the processes taking place in dye-sensitized nanocryst. These methods include direct measurement of the quasi-Fermi level using an indicator electrode and charge extn. The influence of electron trapping on dynamic measurements of electron transfer and transport is discussed within the framework of the quasistatic assumption, and a new assessment of the electron diffusion length in the dye-sensitized nanocryst.

Transient photovoltage and photocurrent measurements were employed to det. Photocurrent transients were taken at the open circuit potential, as opposed to the std. Kinetic results were used to calc. In the calcn. This new method gives activation energies that decrease linearly as the Fermi level position moves toward the conduction band edge, as expected, but not found in previous studies.

The results are consistent with the presence of a distribution of traps below the TiO2 conduction band, the detrapping from which limits both the transport and the recombination of electrons.

Kopidakis, Nikos; Benkstein, Kurt D. Titanium dioxide films were doped electrochem. Photocurrent transients of doped nonsensitized TiO2 films indicate that lithium doping decreases the diffusion coeff. Photocurrent and photovoltage transients of sensitized TiO2 films provide the 1st evidence that electron transport limits recombination with the redox electrolyte in working cells. As the Li d. The electron diffusion coeff.

With increasing doping, the dependence of the electron diffusion coeff. The photovoltaic characteristics of dye-sensitized solar cells are largely unaffected by lithium intercalation, implying that intercalation has only a small effect on the charge collection efficiency and the rate of recombination.

A simple model is presented that explains the obsd. Probably increasing the electron transport rate will not significantly improve the solar cell performance. Google Scholar There is no corresponding record for this reference. Status Solidi B , 94 , A variable-range hopping model is examd. For an exponential d. This model can be applied to activated transport in n-type amorphous Si. Monroe, D. Hopping of photoexcited carriers between localized band-tail states gives rise to a new regime of energy relaxation which is manifested at low temps.

Even in the multiple-trapping regime, however, the current is carried at a transport energy within the band tail. A simple description is given which includes both of these processes in a single self-consistent model and est. Solids , 74 , A model is presented in which equil. This results in carrier mobilities that are apparently thermally activated, due to the temp.

The model resolves the problem of the obsd. Solids , , A particular energy level in the band tail of a disordered semiconductor, called the transport energy, plays a role in hopping transport of carriers in both equil. The transport energy is re-derived in a way that explains the universality of this energy level for different phys. The transport energy dets. The concept of the transport energy is illustrated by comparing 2 recently suggested approaches to the description of the steady-state photocond.

Electron transport in dye-sensitized nanocryst. Values of the apparent electron diffusion coeff. The slow transport of electrons was attributed to multiple trapping MT at energy levels distributed exponentially in the band gap of the nanocryst. In the MT model, release of immobile electrons from occupied traps to the conduction band is a thermally-activated process, and it can be expected that the apparent electron diffusion coeff.

In fact, rather small activation energies 0. The MT model can give rise to such anomalously-low apparent activation energies as a consequence of the boundary conditions imposed by the short-circuit condition and the quasi-static relation between changes in the densities of free and trapped electrons. This conclusion is confirmed by exact numerical solns. Matter , 9 , Institute of Physics Publishing.

In disordered semiconductors with purely exponential energy distribution of localized band-tail states, as in amorphous semiconductors, all transport phenomena at low temps. The authors analyze whether such a transport level exists also in materials with densities of localized states DOSs different from the purely exponential one. In both cases the transport energy exists, implying that it also exists for all intermediate forms of the DOS. Special attention is paid to the dependences of the transport level and of its width on the DOS parameters and temp.

Arkhipov, V. American Institute of Physics. An analytic model of the equil. Doping is shown to create addnl. In the case of weak intersite coupling relevant for org. More detailed data anal. Acta , doi The spatial dependence of the electron quasi-Fermi level QFL in dye-sensitized nanocryst. The calcns. The predicted QFL profiles depend on assumptions made about energy positions, electron mobility, and the conduction band d. The position of the QFL at the electrolyte side of the dye sensitized TiO2 film in a DSC was measured using a thin passivated Ti contact deposited on top of the nanocryst.

TiO2 by evapn. The method allows changes in the electron QFL at all points on the IV characteristic of the cell to be monitored under dark and photo-stationary conditions. Cells incorporating the Ti electrode can give information about the behavior of the QFL under dynamic conditions. O'Regan, Brian C. Charge d. Together, these changes increase the quantum efficiency of charge sepn. The decrease of the recombination rate const. Photocurrent transients and charge extn. Further improvements in transport cannot improve the beneficial effect of TiCl4 treatment. Verification of the CCTPV technique is undertaken by comparison to transient absorption and by a model of the technique.

Charge sepn. Transient optical expts. Bisquert, J. A review of the understanding of dye-sensitized solar cells DSSCs with ref. Solar cells are discussed in terms of the chem. DSSCs are photovoltaic energy converters and the common basis of all photovoltaic systems as well as their most important differences are also presented. The onset wavelengths of the surface photovoltage SPV in dye-sensitized solar cells DSSCs with different mesoporous, wide-band gap electron conductor anode materials, viz.

We find a clear dependence of these onset wavelengths on the conduction band edge energies ECB of these oxides.

The ECB levels of Nb2O5 and SrTiO3 are known to be some meV closer to the vacuum level than that of our anatase films, while there is no significant difference between the optical absorption spectra of the dye on the various films. Such injection comes about because, in contrast to what is the case for anatase, the LUMO of the adsorbed dye in the soln. While for Nb2O5 hot electron injection has been proposed earlier, on the basis of flash photolysis expts.

Nb2O5 can be understood in terms of hole injection from the dye into the oxide via intraband gap surface states. Solid-state dye-sensitized photovoltaic cells were fabricated with TiO2 as the electron conductor and CuSCN as the hole conductor. These cells involve the nanoscale mixing of cryst. Charge transport and field distribution in this kind of material are as yet unexplored. Photocurrent and photovoltage transients were combined with variation in the layer thickness to examine the limiting factors in charge transport and recombination.

In solid-state cells, the similarity of the charge transport and recombination rates results in a low fill factor, and photocurrent losses, both important limiting factors of the efficiency. A simple model was given, and suggestions were made for improvements in efficiency. The cond. T and low impurity concn. The model is that of phonon-induced electron hopping from donor site to donor site where a fraction K of the sites is vacant owing to compensation. To 1st order in the elec. The theory of diffusion and the principal methods of detg.

A summary of major exptl.

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## Postdoc thermoelectric

Extended networks of nanosized semiconductor particles permeated with an electrolyte display unique electrochem. The authors report on a general study of the electrochem. Models were developed accounting for the fundamental characteristics of these electrodes: charge accumulation, charge transport, and interfacial charge transfer. These characteristics can be translated into simple elec.

A key point for describing the exptl. The phys. The authors describe in detail the numerical simulation methods, and despite the simplicity of the approach, these methods allow for quant. B , 57 , American Physical Society. The field-effect mobility in an org. From a percolation model of hopping between localized states and a transistor model an analytic expression for the field-effect mobility was obtained. The theory is applied to describe the expts. Good agreement was obtained, with respect to both the gate voltage and the temp.

Thin Solid Films , , 2. Elsevier B. Incoherent hopping conduction has been revealed as the electronic transport mechanism in lightly doped microcryst. Although microcryst. We review several theor. In order to verify the validity of anal. The simulation results support the anal. B , 63 , We use transient and steady-state optical spectroscopies to study the recombination reaction between electrons and dye cations in a dye-sensitized nanocryst.

TiO2 electrode in several different chem. Kinetic decay curves are approx. We have developed a model of electron transport in the presence of an energetic distribution of trap states and consider two regimes. In the first, the continuous-time random-walk CTRW electrons are free to diffuse through the lattice, by means of multiple trapping events mediated by the conduction band. In the second, the hopping regime, trapped electrons are allowed to tunnel to other, vacant trap sites, or to the dye cation, according to a Miller-Abrahams model for the transition rate.

We carry out Monte Carlo simulations of the recombination kinetics as a function of electron d. The hopping model is ruled out by subnanosecond measurements. We conclude that multiple trapping with a broad energetic distribution of electron traps is responsible for the slow recombination kinetics. When applied to recombination in a nanocryst. Anta, J. To be submitted for publication. The processes of charge sepn. TiO2 solar cells are characterized by certain time consts.

These are measured by small perturbation kinetic techniques, such as intensity modulated photocurrent spectroscopy IMPS , intensity modulated photovoltage spectroscopy IMVS , and electrochem. The authors study the phys.

The authors describe in detail a simple kinetic model for diffusion, trapping, and interfacial charge transfer of electrons, and the authors demonstrate the compensation of trap-dependent factors when forming steady-state quantities such as the diffusion length, Ln, or the electron cond. Electron lifetime measurements in nanoparticles as a function of the Fermi level position at high resoln.

This model predicts a behavior divided in three domains for the electron lifetime dependence on open-circuit voltage that is in excellent agreement with the exptl. Cited By.

## Inverse Temperature Dependence of Charge Carrier Hopping in Quantum Dot Solids

This article is cited by 75 publications. ACS Omega , 4 6 , DOI: Mahyar Taherpour, Yaser Abdi. The Journal of Physical Chemistry C , 3 ,