Charge Carrier Transfer: A Neglected Process in Chemical Engineering

Ind. Eng. Chem. Res. 2010, 49, 9565–9579

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Charge Carrier Transfer: A Neglected Process in Chemical Engineering Jacek Tyczkowski* DiVision of Molecular Engineering, Faculty of Process and EnVironmental Engineering, Technical UniVersity of Lodz, 90-924 Lodz, Wolczanska 213, Poland

This paper presents the discussion that charge carrier transfer should be treated by modern chemical engineering on equal terms with other transfer processes such as mass, heat, and momentum transfer. In an age when chemical engineering is entering the field of electronics, electrostatics, and electrochemistry, more and more often involving examination down to the molecular level, it is impossible to describe much phenomena without taking into consideration the charge carrier transfer. The discussion, the main goal of which is to familiarize chemical engineers with the fundamentals of charge transfer processes, is focused on covalent amorphous materials that are particularly interesting for advanced technologies. The paper presents a review of the generation mechanisms (band-to-band generation, Poole-Frenkel mechanism, charge carrier injection from contacts, photogeneration) and transport mechanisms (band transport, hopping transport) governing the charge carrier transfer on the molecular level. The problems with determining the dominant mechanisms for a given material are also discussed. Finally, specific examples of the use of the charge carrier transfer in chemical engineering are presented. The main attention is focused on electrochemical devices, electrocatalysis, photocatalysis, charging phenomena, and single-molecule engineering. 1. Introduction Recently, we have observed a rapid transformation of the chemical engineering from a discipline that is largely focused on macroscopic phenomena to a branch of science that is more interdisciplinary and molecularly based. This development catalyzes the creation of new frontiers at the interface between engineering and molecular sciences. It becomes clear that, presently, a new (third) paradigm of chemical engineering is undoubtedly emerging. This new approach may be best characterized as molecular engineering of products and processes.1-5 It is evident that molecular engineering will be playing a critical role in advanced product technologies in the foreseeable future. The design of products and control of processes on the molecular level opens new fascinating opportunities in such fields as nanocatalysis, fuel and solar cells, nanocomposites with special optical, electrical, magnetic and mechanical properties, nanoporous materials, and many others.6,7 In all of these cases, the “bottom-up” chemical method of building systems, atom by atom, is very different from the traditional engineering approach. The classical chemical engineering, starting from a model of unit operations, presently is strongly based on transport phenomena being the result of efforts to bring physics into the quantitative description of these operations. Generally, the transport phenomena have been classified into three fundamental transfer processes: momentum transfer, heat transfer, and mass transfer (see Table 1). Since the 1960s, these three basic transfer processes have been, in principle, enough to solve all problems of chemical engineering.8,9 Very recently, one more fundamental transfer process, namely, charge carrier transfer, so far neglected by chemical engineering, has started to attract more and more attention. It is no wonder that in an age when chemical engineering is entering the fields of electrochemistry, electrostatics, and even electronics, we cannot imagine describing the processes that occur without taking into consideration just the charge carrier transfer. The growing interest in this field concerns, for example, * E-mail: [email protected].

such serious chemical engineering problems as the behavior of charged particles and droplets,10-12 processes in electrochemical devices (e.g., fuel cells),13-16 electronic properties of porous media,17,18 catalytic and photocatalytic systems,19-21 and electronic materials,22-24 as well as the modeling and application of various plasma processes.25-27 On the other hand, the developing molecular engineering requires a completely new look at the processes. Of course, momentum transfer, heat transfer, and mass transfer are still the governing laws here, but now their description needs a quantum approach to the problem. We have to take into account, for instance, the quantum states of molecules, interactions between single molecules and phonon and photon transports. It has also swiftly become clear that we cannot solve many problems at the molecular level if we do not consider the charge carrier transfer processes.28 As one can see, the charge carrier transfer deserves a permanent place in the field of chemical engineering and all the transfer phenomena should be treated equally. Accordingly, in this paper, I set two goals for myself: the first is to convince chemical and process engineers to treat charge carrier transfer on equal terms with the other transfer processes, and the second is to present, as an example of the problem, the charge carrier transfer processes in amorphous solid materials, which are a huge group of materials, being specially interesting for advanced technologies. Covalent glasses (e.g., hydrogenated amorphous silicon, silicon carbide, etc.), carbon films (e.g., diamond-like films), ceramics, films and nanoparticles of metal oxides, and even, to some extent, conventional polymers are included among these materials.29 Can we construct, by adopting such materials and optimizing the operation, for example, an electrochemical reactor (such as a fuel cell) without taking the charge carrier transfer into consideration? This is practically unfeasible.30 2. Basic Approach In transfer processes, which take place in nonequilibrium thermodynamic systems, we are concerned with the transfer of a given property through a medium that can be a fluid (gas or liquid) or a solid. This property, which is being transferred, can be mass, thermal energy (heat), or momentum. All these transfer

10.1021/ie101069w  2010 American Chemical Society Published on Web 09/09/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010

Table 1. Three Fundamental Transfer Processes in Classical Chemical Engineering equation

description of the equation

name of the equation

areas of application

Mass Transfer dc jx ) -D dx

(1)

jx: the flux of molecules of a given substance in the x-direction D: the diffusion coefficient dc/dx: the concentration gradient of this substance in the x-direction

(2)

qx: the heat flux in the x-direction κ: the thermal conductivity dT/dx: the temperature gradient in the x-direction

Fick’s law

distillation, absorption, drying, liquidsliquid extraction, adsorption, ion exchange, crystallization, membrane processes, etc.

Fourier’s law

evaporation, distillation, drying, etc.

Newton’s law

fluid flow, mixing, sedimentation, filtration, etc.

Heat Transfer dT qx ) -κ dx

Momentum Transfer dυx τzx ) -η dz

τzx: the flux of x-directed momentum in the z-direction η: the viscosity dυx/dz: the gradient of the x-directed velocity in the z-direction

(3)

processes are characterized in the elementary sense by the same general type of transfer equation:31 rate of transfer process )

driving force resistance

(4)

where the rate of transfer process is defined as the flux of a property, expressed as the amount of property being transferred per unit time and per unit cross-sectional area perpendicular to the direction of flow. We can formalize eq 4 for the fundamental transfer processessmass, heat and momentum transfersby writing appropriate equations that are collected in Table 1. By analogy, the transfer of charge carriers can also be included in this category. In this case, the charge carrier flux (in other words, the density of electric current) is proportional to the voltage drop (driving force), which can be expressed by Ohm’s law: Jx ) -σ

dUx dx

(5)

where Jx is the electric current density in the x-direction and σ is the electrical conductivity. The term dUx/dx is the voltage gradient in the x-direction and -

dUx ) Fx dx

(6)

where Fx is the electric field strength. If the electric current density is expressed in units of A/m2 and the electric field strength is given in units of V/m, the electrical conductivity is presented in units of S/m. Generally, σ is a function of Fx and temperature. Also note that eq 5 is valid for both ionic and electronic (electrons and holes) charge carriers. However, for a large majority of covalent glasses that are being chosen in this paper to illustrate the charge carrier transfer in solids (except for solid electrolytes), electrons and holes are the only carriers responsible for this process.29 Thus, in our further discussions, we will focus mainly on these carriers. The coefficient σ in eq 5, similar to the terms D, κ, and η in eqs 1, 2, and 3, respectively, includes information concerning transfer mechanisms. We can interpret it from the macroscopic point of view, and we can delve deeper into the problem, going

down to the molecular level. For example, for the case of heat transfer, we can use the common macroscopic description based on simple theories employing severe approximations to achieve a closed-form equation.32 However, more and more often, such an approach is not enough to predict heat-transfer processes in new complex materials. A molecular approach with molecular dynamic simulations, the scattering of phonons and free electrons, quantum size effects (very important in nanostructures) etc., has recently emerged as a powerful new tool for chemical engineering.28,33-37 An analogous situation takes place in the case of the electrical conductivity (σ). Nevertheless, we can look at the charge carrier transfer processes macroscopically and interpret the coefficient σ as the reciprocal of a resistivitysa typical parameter used for simple calculations of the electric current flowing through a given system. On the other hand, if we want to understand the charge carrier transfer phenomenon more thoroughly, we must go down to the molecular level and we should discuss the generation and transport processes of charge carriers, which constitute this phenomenon, taking into account the electronic structure of the system.38 3. How Should We Understand the Coefficient σ? As it has been already mentioned, the phenomenon of charge carrier transfer (electric conduction) in all materials consists of two basic processes: charge carrier generation and transport of generated carriers. It can be expressed by a simple fundamental relation: σ ) enµ

(7)

where e is the elementary charge, n the density of carriers, and µ the carrier mobility. The number of carriers is connected to generation processes, whereas the mobility is related to transport processes. When the carriers can be transported through the bulk of material faster than they are generated, generation processes play the dominant role in the conduction phenomenon. In the opposite case, transport processes decide, with regard to the electrical conductivity. Both the generation and transport processes are usually very complicated and, in many cases, the knowledge of the theoretical details of these processes is still relatively poor. Nevertheless, the basic mechanisms governing the generation and transport


Figure 1. Fundamental band models of solid state: (A) crystal, (B) amorphous semiconductor (Mott-CFO model), and (C) amorphous insulator. The regions of localized states are shaded.

of charge carriers have already been established, and the most important of them are briefly presented in the next sections. Before we move on to the details of these mechanisms, we should, however, familiarize ourselves with typical band models of the solid state. 3.1. Band Models. Taking covalent amorphous materials into account, the application of Mott’s theory to describe their electronic structure is an entirely justified approach.39 To understand the nature of the electronic structure of the amorphous solid state, it is simplest to start from the electron structure of a crystalline material. It is widely recognized that such a structure is described by a band model based on the periodicity of the crystal lattice. In this case, the band model (the energy distribution of the density of states), in its simplest version, can be described as follows: two bands of extended states exist, and they are the conduction and valence bands; they are separated by the forbidden band (bandgap) Egap (see Figure 1A). The term EF denotes the Fermi levelsby analogy to chemical thermodynamicssalso called the chemical potential. Electrons and holes can be transported through the crystal if their energies are in a range of the conduction or valence bands, respectively.40 If we provoke a disorder of periodicity, we transform the crystal structure to the amorphous state. The basic idea of the theory describing the electron structure of amorphous solids is grounded on the notion of a localized state. A criterion for electron localization in amorphous materials was first introduced by Anderson,41 who showed that, under specific conditions of disorder, the probability of diffusion of an electron to a certain distance decreases to zero with the increase of this distance. Mott,39 who analyzed the localization of charge carriers in both the conduction and the valence bands in amorphous materials, proved that the Anderson’s localization may occur only in certain fragments of both bands, with the remaining part of each band consisting of extended states. This implies, in a manner identical to that for the case of crystalline state, that a charge carrier excited to an energy level belonging to the range of extended states can freely move along the entire system, and its diffusion ability is limited only in the energy range of localized states. Based on these considerations, a band model was proposed for the covalent amorphous solids.39,42 This model, which is known as the Mott-CFO model, is schematically presented in Figure 1B. The energies denoted by EV and EC constitute the borders between localized and extended states. The distance between these two values is called the mobility gap (Eg) and is equivalent to the bandgap for crystalline materials. In this case, the charge carrier (electron and hole) transport can proceed via the extended states (as in a crystal) or via a hopping mechanism, using the localized states.43,44 All materials of which the electronic structure is described by such a model are called amorphous semiconductors (a-S). Based on the theory of electron state localization, one can predict that, with increasing disorder, the ranges of extended

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states in the conduction and valence bands become more and more narrow and above some critical strength of disorder, all states are localized.42,44 It leads to a new category of materials, of which the electronic structure is qualitatively different from that for a-S. To distinguish between both types of amorphous materials, the materials without the extended states have been called amorphous insulators (a-I).29 In Figure 1C, a typical band model of a-I is illustrated. All states are localized. It means that electrons and holes can be transported through the material only by hopping. This process proceeds above EC and below EV, respectively, where the densities of states are sufficiently high. It has been shown that a step change in carrier mobility in the vicinity of EC and EV justified the concept of the mobility gap but its meaning is different from that for a-S.45 Generally, the distance ECEV for both a-S and a-I is often called the transport gap (EG).29 Summarizing, it should be emphasized that the evident differences between electronic structures of crystals, amorphous semiconductors, and amorphous insulators must result in drastic differences in mechanisms of the charge carrier generation and transport. 3.2. Generation Processes. Two fundamental types of generation processes can be distinguished: that is, the bulk generation and the charge carrier injection from other materials. In the former case, the charge carriers are permanently present in the bulk of material and a part of them is excited to the transport states; in the latter case, in turn, the excess electrons or holes are injected to the bulk from electrical contacts. For amorphous semiconductors, where EG is narrow (e3 eV), the generation process consists of the thermal creation of electron-hole pairs; electrons are excited from the valence band to the conduction band of extended states (Figure 2A). In this case, the total number of electrons or holes per unit volume in the conduction (C) or valence (V) band, respectively, is given by46

[

] )

(E - EF) dE kT |E - EF | ) N(EC,V)kT exp - C,V kT ∆EF ) N(EC,V)kT exp kT

ne,h )

∫

∞

EC,V

N(EC,V) exp -

( (

)

(8)

where N(EC,V) is the density of states at EC or EV (expressed in units of eV-1 m-3) (see Figure 1). If we substitute eq 8 into eq 7, we obtain

(

σ ) N(EC,V)kTeµ exp -

(

∆E ) σ01 exp - F kT

)

∆EF kT

)

(9)

where σ01 is the pre-exponential factor. Using eqs 5, 6, and 9, and making an assumption that the electric field distribution is uniform, the electric current density can be expressed as

(

J ) σ01F exp -

∆EF kT

)

(10)

where F)

U d

(11)

U is the applied voltage and d is the thickness of the sample. As one can see, when the band-to-band generation is the dominant process of the charge carrier transfer, the electric

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Figure 2. Bulk generation processes: (A) band-to-band generation and (B) Poole-Frenkel generation for donor and acceptor levels.

we obtain, depending on r, βPF values in the range of 2 × 10-5-4 × 10-5 eV m1/2/V1/2. With the reduction of the barrier, the higher density of generated carriers and then the higher density of the electric current occur. Thus, the final form of the equation that describes the current density when the Poole-Frenkel generation dominates is the following:

(

J ) σ02F exp Figure 3. Reduction of the barrier height by the electric field in the Poole-Frenkel generation. The effect is shown for donor centers.

current density should be directly proportional to the applied voltage (J ∝ U) and inversely proportional to the sample thickness (J ∝ d-1). Also note that, for undoped amorphous semiconductors, the Fermi level usually lies in the vicinity of the transport gap center (∆EF ≈ 1/2EG).29,47,48 In the case of amorphous insulators, the problem is more complicated, because of a wide EG. If the Fermi level EF is in the middle of the gap, the distances between EF and the transport edges (∆EF) amount to a few electron volts; thus, any thermal generation is not observed. However, if a band of acceptors or donors lying in the vicinity of one of the transport edges, with the density of states sufficient to shift EF to the position inside the band or between the band and the transport edge, occurs in the insulator, we can observe the bulk generation (Figure 2B). This is the result of the thermal ionization of donor or acceptor states (generation centers). This process is called the Poole-Frenkel (P-F) generation.38 The density of generated carriers in the P-F process is dependent, at a given temperature, on the electric field applied to the sample. A hole (h) or electron (e) released from the center must overcome a Coulombic potential barrier Ed0 that is created between the carrier and the oppositely charged center. If the external electric field F is applied, the barrier height is reduced, according to the relation: Ed ) Ed0 - βPFF1/2

(12)

(see Figure 3). Ed is the effective barrier height and βPF is the Poole-Frenkel coefficient, which is given by βPF )

( ) 3

e πεε0

1/2

1 r

(13)

where εε0 is the permittivity of the material and r is a parameter that is associated with the compensation process (1 e r e 2).29 For ε ) 3.5, which is a typical value for amorphous insulators,

Ed0 - βPFF1/2 rkT

)

(14)

where σ02 is the pre-exponential factor and F is given by eq 11. For the simplest case with a monoenergetic level of donors or acceptors, the factor σ02 is σ02 ) (N(EC,V)kTNd)1/2eµ

(15)

where Nd is the density of generation centers (donors or acceptors) (expressed in units of m-3). However, note that the factor βPFF1/2 in eq 14 has an influence on the current only at very high electric fields (F > 106 V/m). The bulk generation can be also created by the absorption of light. For perfect crystalline material completely free of states in the transport gap (free of charge carrier traps) (Figure 1A), it is easily shown that the steady-state carrier concentration is38

( Gγ )

1/2

ne,h )

(16)

where G is the density of carriers generated per unit time by the light excitation and γ is the recombination coefficient. Consequently, the steady-state photoconductivity (σph), based on eq 7, is given by σph ) e

( Gγ )

1/2

(µe + µh)

(17)

In real solid materials that contain various states in the transport gap (see Figures 1B and 1C), the trapping and the recombination through the states become dominant and the calculation of ne,h becomes quite mathematically involved. For an arbitrary distribution of states in the transport gap N(E), the following rate equation for electrons can be written (an analogous equation also can be written for holes):38 ∂ne )G+ ∂t

∫

EC

EV

[

]

(EC - E) N(E)fr(E) dE kT EC CnneN(E)[1 - fr(E)] dE (18) E

CnN(EC) exp -

∫

V

where Cn is the capture rate coefficient for electrons and fr is the probability for a recombination center to be occupied. The first term on the right-hand side of the equation represents the density


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estimation of the current density for this process requires the solution of very complex quantum equations (in many cases by numerical methods).38,49 However, it has been suggested that this current density can be correctly described by a simple empirical relation of the form50

( ) [(

J ) A exp -

Figure 4. Charge carrier injection. Three main processes are presented: (1) thermionic emission (Schottky generation), (2) thermionic field emission, and (3) field emission.

of carriers generated per unit time by the light excitation, the second term represents the rate of re-excitation of trapped carriers, and the third term represents the carrier loss due to trapping in the shallow traps and in the recombination centers. Equation 18 becomes even more complicated when the external electric field is applied and the diffusion effects are taken into consideration. The solution of such equations is generally formidable, even with the aid of numerical techniques. However, making certain assumptions, we can describe the photogeneration processes quite well. Another way to generate charge carriers in the material is the injection of excess carriers from electrode contacts. Three main processes of the injection can be distinguished: thermionic emission, thermionic field (thermally assisted field) emission, and field emission (see Figure 4). Generally, these processes are strictly dependent on the type and shape of contact barriers and can proceed both in semiconductors and in insulators.38,49 For intermediate electric field and temperature, most often, the thermionic emission is the dominant process of the charge carrier injection from contacts (process 1 in Figure 4). This is the Schottky generation. In this process, charge carriers must overcome the contact barrier Φ0. This barrier can be modified, as for the P-F mechanism, by the external electric field F. For the standard neutral contact, the field lowering of its barrier is expressed by Φ ) Φ0 - βSchF1/2

(19)

where Φ is the effective barrier height and the Schottky coefficient (βSch) is given by βSch )

( ) e3 4πεε0

1/2

(20)

Assuming that ε ) 3.5, similar for the Poole-Frenkel process, we obtain βSch ) 2 × 10-5 eV m1/2/V1/2. If the injected carriers are transported through the bulk of material faster than they are injected, the final form of the equation describing such a current under steady-state conditions is dependent only on the generation process and is the following:

(

J ) ART2 exp -

Φ0 - βSchF1/2 kT

)

(21)

where AR is the Richardson constant (theoretically, AR ≈ 1.2 × 106 A m-2 K-2). For a relatively narrow barrier (W e 10-20 nm), when a carrier tunneling through the barrier becomes possible, the thermionic field emission occurs (process (2) in Figure 4). The

Φ0 B + C F1/2 exp kT kT

) ]

(22)

where A, B, and C are constants. By comparison of this relation to the model numerical calculations (for a triangular-shaped barrier), it has been found that A ) 6.8 × 1011 A m-2, B ) 1.65 × 10-24 J m1/2/V1/2], and C ) 1.2 × 10-4 m1/2/V1/2. For the sake of formality, we should otherwise refer to the field emission process (process (3) in Figure 4), which generally occurs in covalent glasses relatively rarely. Similarly, as for the thermionic field emission, numerical approaches are the only way to properly calculate the tunneling current in that case. However, in a first approximation, we can express the field emission current density as38

(

)

2RΦ03/2 3eF J) 1/2 RΦ0 kT πRΦ01/2kT sin eF eF

(

ART2π exp -

) (

)

(23)

where R)

4π(2m)1/2 h

(24)

Here, m is the charge carrier mass and h is the Planck’s constant. For low temperatures, the field emission process should give, according to eq 23, a linear relationship between (ln J)/F2 and F-1. Just the same as for the bulk generation, the photoexcitation of charge carriers in the electrode contacts can be used instead of the thermal excitation. This is a photoinjection (internal photoemission) process that generally can be described, according to the Fowler theory, by the following relation:29,38 Y ∝ (hν - Φ)2

(25)

where Y is the photocurrent quantum yield, hν the exciting photon energy, and Φ the contact barrier height (for electrons or holes), given by eq 19. Usually, an exponent of 2 is used in this relation; however, in some cases, other values, such as 2.5 or 3, have been proposed.29 The generation by injection processes (thermally excited or photoexcited) is associated with electrodes that control the electric current. However, if an electrode emits more electrons or holes than the bulk can acceptsin other words, the injected carriers are transported through the bulk of material slower than they are injectedsa space charge is formed with a distribution being dependent on the distribution of density-of-states (see Figure 1). This charge creates a field that reduces the rate of carrier emission from the electrode. Thus, the current is now controlled not by the charge-injection electrode, but by the bulk of the material. In such a case, the electric conduction is described by the theory of space charge limited current (SCLC).38,51 The most frequently used form of the equation describing the current density in the case of SCLC is the following: J ) ΘUn′d-m′

(26)

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where n′ g 2 and m′ ) 2n′ - 1. The parameter Θ is a complicated function of the density-of-states distribution. 3.3. Transport Processes. From the point of view of the band model, we can distinguish two fundamental mechanisms of charge carrier transport through amorphous materials: transport via extended states (band transport) and transport via localized states (hopping transport). As one can see in Figures 1B and 1C, both of these mechanisms occur in amorphous semiconductors; however, for amorphous insulators, only the hopping transport is possible.29,46 The main parameter that describes the transport mechanisms is the carrier mobility (µ). For the band transport, in the extended states just above EC or below EV, the carrier mean free path is of the order of the interatomic spacing a. Transport through these states, by analogy to the mass transfer on the molecular level, may be likened to Brownian motion.46,52 In this regime, the mobility can be obtained with the help of Einstein’s relation: µ)

eD kT

where νph is the phonon frequency (∼1013 Hz), Rf the rate of falloff of the wave function at the state, R the distance covered in one hop, and ∆Ehop the activation energy of the hopping transport. The factor µ0hop is 102s103 times lower than the corresponding factor for the band transport (recall eq 29). As the temperature is reduced, the number and energy of phonons decrease, and the more-energetic phonon-assisted hops will progressively become less favorable. Carriers will tend to hop to larger distances to find states lying energetically closer than the nearest neighbors, and, at the same time, the hopping will be constrained only to the narrow band of states around the Fermi level. This mechanism is the so-called variable range hopping (VRH). In this case, the factor exp{-2RfR - [∆Ehop/ (kT)]} from eq 31 will not have its maximum value for the nearest neighbors. To determine the most probable hopping distance, Mott55 used an optimization procedure and he obtained, for the average energy spacing between states near the Fermi level,

(27)

∆Ehop )

The diffusion coefficient (D) may be written as

3 4πR N(EF)

(32)

3

and for the most probable jump distance, 1 D ) νela2 6

(28)

where νel is a typical electronic frequency (∼1015 Hz). The mobility in the Brownian-motion regime is then given by µ)

( )

1 ea2 ν ) µ0b 6 kT el

(29)

The aforementioned equation predicts a band transport mobility of µ0b ≈ 5 × 10-4 m2 V-1 s-1 at room temperature. In all amorphous semiconductors, apart from the extended states, the localized states also exist (see Figure 1B). In the band transport, they can play the role of traps for charge carriers. The trapped electron (or hole) is released over time, depending on the depth of the trap and the temperature. Such a carrier can be trapped many times during the band transport through the material. The mobility, which is controlled by multiple trapping, can be expressed by the following equation, which can be applied to a typical distribution of localized states (for instance, that shown in Figure 1B):39

( )

µ ) µ0bβ exp -

∆Et kT

(30)

where β is a constant, depending on the distribution of localized states and ∆Et is the activation energy of the band transport that is associated with trapping. The second fundamental mechanism of charge carrier transport through covalent amorphous materials is the hopping transport via localized states.44,53 Charge carriers are hopping from state to state via a phonon-assisted tunneling process. According to Miller and Abrahams,54 who gave the first analytical-based report of hopping transport, the hops proceed between the nearest states and every such hop results in the energy exchange between the carrier and a phonon. Thus, it may be expected that the mobility will have a thermally activated nature: µ)

( )

(

)

(

∆Ehop ∆Ehop 1 eR2 V exp(-2RfR) exp ) µ0hop exp 6 kT ph kT kT

)

(31)

R)

(

9 8πRfN(EF)kT

)

1/4

(33)

where N(EF) is the density of states at the Fermi level. Hence, the final form of the equation that describes the electrical conductivity, when the variable range hopping mechanism governs the charge carrier transfer process (using eqs 7 and 31, and assuming that n ) N(EF)kT), is the following:

( )

σ ) σ03 exp -

T0 T

1/4

(34)

where σ03 )

(

e2VphN(EF) 9 6 8πRfN(EF)kT

)

1/2

(35)

and T0 ) Λ

( ) Rf3 kN(EF)

(36)

Λ is a numerical factor that, for the above derivation, is ∼18. There have been several more derivations of the conductivity formula for variable range hopping. Generally, in all of these cases, the T -1/4 relationship remains unchanged, but different values were found for Λ, which changed from 10 to 37.8.44,46,53 The problem of charge carrier transport mechanisms is still open. Profound understanding of how the charge carriers are transported through covalent amorphous solids, descending to the molecular level, is a serious challenge. Every now and then, new ideas occur; for example, a multiphonon hopping mechanism has been proposed.56-58 The electrical conductivity, in this case, is expressed as σ ) n′′Tm′′

(37)

where n″ and m″ are parameters. For example, the multiphonon hopping model has turned out to be very useful for describing the charge carrier transfer through amorphous carbon films.56 Considering the hopping transport via localized states, it should be added that the analogous concept is successfully


applied to the description of charge carrier transport processes in organic disorder solids, despite the fact that their electronic structure drastically differs from that for covalent amorphous solids discussed in this paper (see Section 3.1). In covalent amorphous solids, all atoms are incorporated into the covalent network. In contrast, organic disorder solids, such as molecularly doped polymers, low-molecular-weight glasses, and conjugated polymers, are composed of individual molecules (or macromolecules) bound by van der Waals interactions. Although, in this case, one cannot talk about the Mott-CFO model, the concept of localized states and the charge carrier transport via these states is fully justified. A major step forward in the description of charge transport in amorphous organic films was the introduction of an uncorrelated Gaussian disorder model by Ba¨ssler in 1993.59,60 The model assumes that charge carriers move via variable range hopping (VRH) between states strongly localized on molecules or molecular segments and randomly distributed in space with some concentration N. The energies of charge carriers on these states are subject to a Gaussian distribution; therefore, the density of states takes the form N(E) )

N

√2πσ′

( )

exp

E2 2σ′2

(38)

The energy E is measured relative to the center of the distribution, with a distribution width of σ′, which is a measure of the energetic disorder within the material. Such a distribution of localized states is fundamentally different from that in covalent amorphous materials, in which the tails of localized states in the transport gap (see Figures 1B and 1C) are believed to have an exponential shape. Because the distribution of localized states evidently determines the hopping transport, the electric conduction phenomena for organic disorder solids and covalent amorphous solids usually are essentially different. The interested reader can find details of the charge carrier transport processes in organic disorder solids, for example, in the referenced literature.44,60 4. Investigations of the Charge Carrier Transfer in Amorphous Solids An enormous number of papers and books on electric conduction of amorphous materials have been published over the last 30 years. Generally, two main goals have motivated these studies. On one hand, they enable us to characterize the generation and transport processes taking place in investigated materials; on the other hand, they are an important source of information about the electronic structure of these materials, such as distributions of density-of-states, position of the Fermi level, height and type of contact barriers, etc.29,38 For most of these works, the investigations come down to measurements of the electric current as a function of U, d, T, and, if need be, the light intensity and its wavelength. Unfortunately, although the various mechanisms of electric conduction processes differ essentially (see Sections 3.2 and 3.3), it is still an open problem to establish explicitly which process is dominant in the studied material. Especially controversial are attempts to describe charge carrier transport processes by the SCLC mechanism, as well as attempts to distinguish between the generation mechanisms solely on the basis of currentsvoltage dependences.61 Let us discuss, for example, the case of differentiation between the SCLC and generation processes. According to the SCLC theory (eq 26), a bilogarithmic plot of the currentsvoltage dependence (ln J vs ln U) should give a straight line with a slope of b ) n g 2. Thus, the experimentally obtained rectilinear

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Figure 5. Current-voltage characteristic simulated for a generation process (Poole-Frenkel or Schottky), presented according to eq 39 in the ln JsU1/2 coordinate system. (Parameters ai ) βi/(kTd1/2) and C are assumed to be equal 1 and 0, respectively.)

branch in the ln Jsln U characteristic with a slope b g 2 can suggest the presence of SCLC. On the other hand, if one of the generation processes determines the electric conduction through the material (the Schottky generation (eq 21) or the Poole-Frenkel generation (eq 14)), the current density for both these cases may be generally expressed as ln J )

βi kTd1/2

U1/2 + C

(39)

where βi is βSch or βPF and C is a constant (in the case of the Poole-Frenkel mechanism, C is a very weak function of the electric field; however, for our discussion, it may be neglected). The plot of eq 39 in the ln J vs ln U coordinate system is an exponential curve described by the transformed form of eq 39: ln J )

βi kTd1/2

( 21 ln U) + C

exp

(40)

Thus, in cases in which generation processes dominate, the experimental results should not give rectilinear J-U characteristics in the lnJ-lnU coordinate system. However, the measurement errors, as well as the limited range of the measurements, make it impossible to distinguish between the real rectilinear branch and the rectilinear approximation of a segment of the exponential curve. In Figure 5, a simulated generation process, described by eq 39, in which the factor ai ) βi/(kTd1/2) has been assumed to be equal to 1, is presented in the ln J-U1/2 coordinate system. The same data are plotted in the ln Jsln U coordinate system (see Figure 6). Here, we can distinguish three excellent rectilinear branches approximated by the method of least squares (the same procedure is used for real experimental results). Both the second and third branches with slopes of b g 2, de facto describing the generation process, could erroneously be taken to indicate the presence of SCLC. As one can see from this simple example, the problem of distinguishing between charge carrier transfer processes is very complicated and requires, apart from JsU characteristics, other sophisticated measuring techniques, e.g., time of flight, photoinjection, dielectric spectroscopy, etc.29,61 Measurements of the electric current as a function of temperature are sometimes also useful for determining the character of charge carrier transfer process. Examining the relationships that describe the generation and transport processes (see Sections 3.2 and 3.3), we draw the conclusion that, in the

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Figure 6. Currentsvoltage characteristic from Figure 5 presented in the ln Jsln U coordinate system. Rectilinear branches characteristic of the SCLC process are visible.

majority of cases, the electric current density may be expressed by the following phenomenological form:

( )

J ) J0 exp -

EA kT

(41)

where J0 is the pre-exponential factor and EA is the activation energy of the electric conduction. Although the physical interpretation of J0 and EA is very often difficult and ambiguous, the analysis of these parameters obtained from experiments can give, in some cases, valuable information on charge carrier transfer mechanisms.62,63 5. Charge Carrier Transfer Processes and Chemical Engineering As it has been already mentioned in Section 1, more and more often, the need arises to involve charge carrier transfer in a detailed description of the chemical processes. Generally, such a description is limited to a phenomenological approach in which only the macroscopic parameter σ intrinsically (see eq 5), but not its nature and origin, is taken into consideration. However, more and more of these problems today address the molecular engineering area, which definitely requires a deeper analysis of the generation and transport processes of the charge carrier transfer phenomenon. A few examples presented below show where and how such a molecular approach to the charge carrier transfer is (or should be) applied to chemical engineering problems. 5.1. Electrochemical Devices. High-temperature fuel cells, ion pumps, separation membranes, and sensors are examples of the electrochemical devices. For all of them, solution of the basic charge carrier transfer equations for solids is the first step in assessing their functional performance characteristics. In many cases, however, such equations must consider not only electronic, but also ionic, transfer, because the solid electrolytes used in the devices are characterized by both types of conductivity. Thus, the basic equations of the transient, according to eq 5, are the following:64 dUx Jel ) -σel dx

(42)

and Jion ) -σion

(

dUx 1 dµchem + zFa dx dx

)

(43)

where the subscripts “el” and “ion” refer to electronic (electrons and holes) and ionic (cations and anions) carriers, respectively. For electronic carriers, only the voltage gradient is the driving force, whereas, for ionic carrier transfer, the chemical potential gradient (the first term in eq 43) also must be taken into account. In this case, the charge transient is accompanied by mass transport of the ionic species. Solution of these transport equations allows the calculation of potential profile inside the solid electrolyte and other performance characteristics such as currentsvoltage and (efficiency of electrochemical processes)svoltage curves. Determination of the potential at the interface and inside electrolyte allows, in turn, tailoring of the thickness of each element of multielectrolyte devices, whereas performance characteristic curves help in deciding the optimum operating conditions.64 In the overwhelming majority of cases, however, the charge carrier transfer models that are applied to solid electrolytes assume constant values of ionic and electronic conductivities, or, at the most, take σ as an empirical function of a selected parameter.65,66 In reality, these conductivities (σel and σion) are dependent on the charge carrier mobility and the charge carrier concentration (see eq 7). The mobility and concentration of charge carriers, in turn, are dependent on the chemical structure of the solid electrolyte and its contact with a gas or liquid phase, as well as being dependent on conditions under which the device works (temperature, pressure etc.). If we are looking for rigorous and accurate descriptions of transfer phenomena in solid electrolytes, knowledge of the generation and transport processes of charge carriers is absolutely necessary. The first attempts have already been undertaken;14,67 however, this is only the beginning of the road. The analysis of σ on the molecular level becomes especially important for sensing devices, when we want to model sensor responses to a given stimulus. In cases in which electric signals are the response, knowledge of the mechanism of charge carrier generation and the carrier mobility is necessary to solve this problem. Both ionic68 and electronic carriers69 can be the source of such signals. For example, let me briefly mention the humidity sensors based on nitrogen-containing organosilicones. One of these materials includes plasma-deposited thin silazane films. To describe the relationship between relative humidity (RH) and the electrical current flowing in these films, JsF characteristics for various RH were measured.70 The results plotted according to eq 14 are presented in Figure 7. The analysis of these dependences, as well as other investigations (e.g., photoconductivity), showed that holes are responsible for the electrical conductivity and the process is governed by the Poole-Frenkel mechanism. A substantial increase in the current with increasing relative humidity for a given value of F1/2 is indicative of a growth in the concentration of the acceptor centers (Nd in eq 15), which are created by water molecules. On the other hand, the different slope of the characteristics in Figure 7 proves that changes in Ed0 (eq 14) must happen. Deeper analysis of the nature of the acceptor center and its cooperation with water molecules, which leads to an initial charge separation in the center, gave the following relation:70 Ed0 )

(

E0d0

4πεε0FE0d0 1+ eRr

)

(44)

0 whereEd0 is the potential barrier Ed0 for RH ) 0, and Rr is the average value of the restoring force. Substituting eq 44 into eq 0 ) 0.65 eV, r ) 2, and 14 and taking from experiments that Ed0


Figure 7. JsF characteristics for a sample of plasma-deposited silazane, at various relative humidity (RH) values.70

ε0 ) 2.25, one obtains the following, at room temperature (295 K):

[

J ) σ02F exp

25.59

()

F 1 + 1.02 × 10 Rr 9

- 7.86 × 10-4F1/2

]

(45)

where σ02, according to eq 15, can be expressed as σ02 ) σ02 ′ Nd1/2

(46)

The concentration of the acceptor centers Nd and the restoring force Rr are functions of RH, which can be determined experimentally: Nd can be obtained from measurements of anomalous transient photocurrents,71 and Rr can be obtained from the slopes of the JsF characteristics (see Figure 7). As one can see, studies of the interaction between water molecules and the sensor on the molecular level lead not only to a better understanding of its sensing mechanisms but also provide a much more complete description of the relationship between signal (electrical current) and relative humidity (RH), voltage applied to the sensor, temperature, etc., which enables better design and application of such devices. Many systems that use various types of sensors (e.g., for O2, CO, CO2, NO2, hydrocarbons, explosive materials) require a similar approach. 5.2. Electrocatalysis. Membrane reactors are a particular form of the integration of chemical conversion and separation. Many combinations can be realized in such reactors from the large variety of different reactions, as well as membrane separation processes. However, on a large scale, only electrocatalytic membrane reactors have succeeded in gaining industrial relevance today.72 An effective electrocatalyst must satisfy many requirements, such as high activity, high electrical conductivity, and longterm stability, which may be in conflict with each other. One possible way to solve these conflicts is the use of composite materials, where the matrix and the dispersed phase are independently selected. A successful approach, for instance, is that of associating a highly conducting (though catalytically inert) matrix with an active (though less conducting) dispersed phase. When the matrix is permeable, one has a threedimensional (3D) catalyst: all catalytic particles are active, irrespective of their position in the composite.73 Figure 8

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Figure 8. Model of a three-dimensional (3D) electrocatalytic membrane.

schematically represents the operation of such a system. Two types of processes must be considered to describe it, namely, charge carrier transport and mass transport. However, if one wants to have a comprehensive model of the electrocatalytic membrane, the value of the matrix conductivity itself is insufficient and the electron and hole transport processes must be examined in detail. Electronically conductive polymers are the most widely used as the matrixes.74-76 Let us take thin acrylonitrile films fabricated by plasma deposition as an example of such a matrix. These films doped with Co atoms were tested as 3D electrodes for the reduction of molecular oxygen.77 As it turns out, the electrical conductivity of the acrylonitrile films is strongly dependent on the plasma deposition parameters. For example, a change in deposition temperature (Td) from 295 K to >570 K causes a drastic increase in the conductivity from ∼10-16 S/m to 10-8 S/m. Such a change in σ has a dramatic influence on electrochemical processes taking place in the membrane. Thorough investigations of the charge carrier transfer processes in the acrylonitrile films were performed in an effort to search for a source of this effect.78 Especially interesting results concern the electron and hole mobility in the films. In Figure 9A, these mobilities are presented as a function of σ (that was controlled by Td). Two regions can be distinguished in this relationship: for σ < 10-9 S/m (Td < 570 K), where the mobility values are initially constant and then slowly increase; and for σ > 10-9 S/m, where a rapid increase in µ is observed. The first region has been connected with the hopping transport (eq 31), whereas for the second one, the band transport controlled by multiple trapping has been deduced (eq 30). More-detailed investigations of these processes and the fact that the Poole-Frenkel mechanism governs the charge carrier generation have allowed one to explain changes in the activation energy of the conductivity estimated from JsT measurements, according to eq 41 (see Figure 9B). Taking eqs 14 and 30 or 31, the activation energy can be expressed as78 EA )

Ed0 - βPFF1/2 + ∆Eµ r

(47)

where ∆Eµ ) ∆Ehop or ∆Et and is determined from measurements of µsT dependences. Since ∆Eµ is very small over the entire range of σ (0.02-0.05 eV), the changes in EA have been connected with a decrease in Ed0, being the result of the

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Figure 10. Primary steps in the photocatalytic process: (1) generation of electronshole pairs by a photon; (2) bulk recombination; (3) surface recombination; (4) trapping; (5) initiation of a reductive pathway; (6) initiation of an oxidative pathway.79,81,83

Figure 9. (A) Drift mobility µ of holes (b) and electrons (4) and (B) activation energy of conductivity EA, as a function of the conductivity σ in plasma-deposited acrylonitrile films.78

formation of conjugated sequences (-CdC-C)) in the film structure with growing Td. If the charge carrier mobility and the activation energy of the conductivity, and, as a consequence, the density of generation and transport states in the matrix, are known, the electrocatalytic process can be described precisely as a function of the process parameters (e.g., temperature, applied voltage) and the parameters of the matrix deposition. 5.3. Photocatalysis. Another prime example of the need for the use of charge carrier transfer in chemical engineering are photocatalytic processes. Photocatalysis has emerged to be one of the most promising pollution remediation technologies in recent decades.79,80 This technology can make use of the energy from the sun, which advocates energy sustainability. Semiconductor photocatalysts generate electron and hole pairs upon irradiation by light energy above the transport gap (Figure 2A). The holes and electrons generated in the bulk of the semiconductor particle are moved to its surface and can be utilized in initiating oxidation and reduction reactions, respectively. Numerous studied have shown that photocatalytic technology is capable of degrading and mineralizing a variety of harmful organic pollutants into innocuous products.81-85 This technology has also been applied for heavy-metal-ion reduction, whereby the toxic heavy-metal ions are reduced to their insoluble states for subsequent recovery or removal from an industrial effluent.86,87 Apart from the pollution remediation technologies, starting from the pioneering work of Fujishima and Honda,88 photocatalysis also opens wide the prospects for the process of water splitting to hydrogen and oxygen. It has been known already for quite a long time that the understanding of photocatalysis mechanisms and the presentation of its comprehensive kinetic model is impossible without profound analysis of the charge carrier generation and transport processes. Numerous researchers have claimed that the rate of degradation of chemical compounds by photocatalysis follows the classical Langmuir-Hinshelwood (LH) model. However, the LH model turned out to be highly ambiguous in this case;

for instance the model is not able to define the relationship between the photodegradation rate and the illumination intensity. Besides, there are many examples of discrepancies between the LH model and experimental results.89 A unified quantitative model of photocatalysis must consider all competitive processes involved in such systems together. Apart from mass transport processes, generation of the electronshole pairs by the light excitation, charge trapping and recombination pathways, transport mechanisms of both electron and hole through the semiconductor (see: eqs 17 and 18), and transfer of the charges from the semiconductor surface to reactants are the fundamental processes of interest in this respect.90 Primary steps in the photocatalytic process are schematically shown in Figure 10.79,81,83 Recently, comprehensive kinetic models have been developed, successfully gathering the above-addressed phenomena together to describe the photoelectrochemical behavior of nanostructured TiO2.89-91 The basic reaction equations used in these models, illustrating the necessity to take into consideration the charge carrier transient processes, are summarized in Table 2.81,92 However, it should be emphasized that all charge transfer mechanisms in photocatalytic systems (thin films or nanoparticles) are not yet fully understood. Even less is known about the new, more-sophisticated, photocatalytic systems that contain nanotubes or nanowires.93,94 These systems exhibit peculiar optical and electronic properties, as a result of quantum confinement effects. The problem becomes much more complicated when the photocatalytic system is permeated with an electrolyte95 or an insulating gas,96 doping with other atoms,92-95 and contacted with other metal oxides97 or nanoparticles of a noble metal.19 In all of these cases, the electronic structure of the photocatalytic system is modified. The doping creates new donor and acceptor states (and also trapping states), the contact with metal nanoparticles is the source of all consequences arising from the contact barrier (see Figure 4), etc. There is no doubt that the understanding of all these effects, and their appropriate utilization in models that would be useful for chemical engineering, require a thorough analysis of the charge carrier transfersthe process whose elements have been presented in the previous chapters of this paper. 5.4. Charging Phenomena. In practice, powders and other particulate solids are widely used as raw materials, intermediates, or final products. The electrostatic charges in such particles often


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Table 2. Basic Processes Connected with Charge Carrier Generation and Transport in TiO2 Photocatalysts Used for Degradation of Organic Pollutantsa reaction -

TiO2 + hV f e + h

+

type of process

e- + h+ f energy (phonons)

photogeneration (then the transport through the semiconductor with µe and µh, respectively) direct recombination

Ti4+ + e- f Ti3+

examples of bulk trapping

O2- + h+ f OTi3+ + h+ f Ti4+ -

-

O +e fO

examples of bulk recombination Figure 11. Schematic view of the surface molecular structure of polymer particle in charging experiments (according to Yoshida et al.102).

2-

Ti4+(OH) + h+ f Ti4+(OH•+)

examples of surface trapping

Ti4+(OH) + e- f Ti3+(OH) Ti4+(OH•+) + e- f Ti4+(OH)

examples of surface recombination

Ti3+(OH) + h+ f Ti4+(OH) Ti4+(OH•+) + organic pollutantf fTi4+(OH) + oxidized pollutant

examples of interfacial charge transfer

Ti3+(OH) + O2 f Ti4+(OH) + O•2 a The solid circle symbol (b) denotes a radical center. Data taken from refs 81 and 92.

play an important role in the process involved. On the one hand, this charging is considered as impediments and disruptions of the process. For example, the charged particles in pneumatic transport or a fluidized bed experience electrostatic forces and, thus, tend to adhere to the walls or cohere to each other. If the particles are excessively charged, an electric discharge will occur, which can give rise to fire and explosion hazards. On the other hand, electrostatic forces can control the movement of the charged particles. It can be used for electrostatic precipitation, powder coating, laser printer and copying machines in electrophotography, etc.10,98-100 In many of the aforementioned cases, the electrostatic charging, charge decay and neutralization, and charge distribution on the surface and in the bulk of particles (and, thus, the problems closely connected with the charge carrier transfer processes) still remain unexplained. For instance, although it seems to be well-known that particles are generally charged by ions arising from corona discharge or by contact with another material, there are still many unknowns and problems to solve. It also became obvious that the control of electrostatic phenomena is impossible, if the charge and its distribution on individual particles are not determined.100 Hence, the mechanisms of generation and transport of charge carriers for various particles (insulating, semiconducting, metallic, with different shapes) must be clarified and studied experimentally as well as theoretically.11,101 Recently, a molecular approach to the particle charging has been developedsspecifically for chemical engineeringsfor better understanding of the process and more advanced modeling

of electrophotographic systems.102 The charging in polymersmetal contact (between a single polymer particle and a metal plate) was investigated on the molecular level, employing quantum chemistry methods. Very interesting results were obtained. It was found, for instance, that the charge transfer process drastically depends on the polymer surface structure: if the polymer chains are arranged there perpendicular to the metal surface, the charge transfer is much more efficient than that for the parallel arrangement (see Figure 11). It was also shown that the dangling bonds of polymers play a large role in the charge transfer process (as charge trapping states). The above example is just enough to show that investigations of the charge carrier transfer on the molecular level lead to valuable results, which can be very useful for modeling and optimizing processes involving charged particles. Another interesting phenomenon that is associated with the charging processes is the electrokinetic transport and separations in fluidic nanochannels. The recent interest in such systems is constantly increasing, because of the wide range of applications that they offer. Nanochannels have been used for chemical and biomolecular sensing, as well as for the separation of charged analytes,103,104 and even for the construction of nanofluidic batteries,105 diodes,106 and transistors.107 The transport of fluid and electric current in very small channels of nanometer size presents not only practical but also fundamental interest. At these dimensions, the electric double layer that usually forms on the channel walls becomes comparable in size to the channel width. The origin of the charges at the surface is due to tribocharging processes. It is obvious that the density and distribution of the charges, their mobility, immobilization by trapping at the surface, recombination, and transport through the nanochannel material are closely connected with the molecular and electronic structures of this material. The molecular structure of the fluid is also not without significance. Thus, although in many cases, the consideration of nanofluidic systems is limited to continuum approaches, where the molecular level does not need to be explicitly taken into account,103,108 more nanofluidic research on the charge carrier transfer will be needed if we want to demonstrate the real potential of nanofluidic devices (for example, that of energy conversion systems).109 5.5. Single Molecule Engineering. The rapid progress in molecular manipulation with a scanning tunneling microscope (STM) tip opens up entirely new opportunities in nanoscience and technology. With these advances, the elementary chemical reaction steps such as dissociation, diffusion, adsorption, readsorption and bond formation processes become possible to be

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Table 3. General Characterization of the Electronic Structure and the Electrical Conductivity for a-SiXCY:H Filmsa

transport gap, EG Fermi level, EF charge carriers electrical conductivity, σ pre-exponential factor, σ0 generation mechanism transport mechanism a

Insulator (a-I)

Semiconductor (a-S)

6.8 eV 0.7 eV holes 10-16 S/m 10-4-10-5 S/m Poole-Frenkel generation hopping via the localized states

3.7 eV 0.3 eV holes 10-8 S/m 10-2-10-3 S/m band-to-band generation and thermionic field emission with SCLC band transport controlled by multiple trapping

Data taken from ref 114.

performed by using the STM tip at the single molecule level with an atomic scale precision. Using a variety of manipulation techniques in a systematic and step-by-step manner, a complete chemical reaction sequence has been induced with the STM tip leading to the synthesis of molecules on an individual basis. It should be emphasized that, under the influence of the tip, reactions that otherwise may not occur in nature can be forced to proceed.110 Both the molecular bond-breaking processes and the formation of new chemical bonds are mainly induced by tunneling electrons from the STM tip. When electron energies higher than the contact barrier between the tip and a molecule are used, the processes are in the Schottky emission regime (see process (1) in Figure 4). Otherwise, when the energies are lower, the processes are involved in the tunneling regime (see process (2) in Figure 4). In the latter case, the processes can be precisely controlled.110 There is no doubt that an understanding of singlemolecule charging by the tunneling electrons, electron transport through the molecule, altering the reaction pathway of the molecule through redox energetics, etc., is the key to successfully developing the single-molecule engineering.111 Of course, thinking about scaling up of such processes, a system that consists of billions of “molecular tips” should be constructed. It is suggested that, for example, oriented carbon nanotubes or plasma-deposited a-Ssa-I films could play such a role. The latter system, in which an insulating matrix is filled by semiconducting filaments forming on the film surface closepacked nanoislands separated by the insulator, seems to be especially interesting. The creation of such a nanocomposite was proved by AFM measurements of reversible charge storage on the surface.112 From the chemical structure point of view, the films are hydrogenated covalent alloys of carbon and one of the other elements of the carbon family (e.g., a-GeXCY:H, a-SiXCY:H). It has been found that these films exist in two forms with totally different electronic structures, namely, the semiconducting (a-S) and insulating (a-I) form (see Figures 1B and 1C).29 Both forms can be fabricated by plasma deposition from a single-source precursor (e.g., tetramethylgermane or tetramethylsilane) in the same plasma process. Transformation in the structure from a-I to a-S (called the a-Isa-S transition) is caused only by small changes in the impact energy of ions bombarding the growing film.113 The intermediate state between a-S and a-I forms, where the a-Ssa-I nanocomposite is created, has been proven to be very fascinating. The surface of these films resembles the system of a huge number of tip nanoreactors. The electric potential of the semiconducting tips can be controlled by voltage applied to the metal electrode upon which the film was deposited. It is obvious that electrical properties of the a-S and a-I forms determine the mechanism of chemical processes that would proceed on the a-S tips. However, if we think about the design and preparation of a multitip reactor, first we need detailed information on the electronic structure, as well as the charge carrier generation and transport processes that characterize the

a-Ssa-I nanocomposite. Ever since the a-Isa-S transition was found, extensive studies have been performed on these films. The studies have provided many results concerning the molecular structure and electrical properties of the a-I and a-S forms of various hydrogenated covalent glasses fabricated on the carbon basis.29 As an example, the most recent results for a-SiXCY:H films are presented in Table 3.114 However, it is only the first step in the design of a-Isa-S nanocomposite multitip reactors. A detailed understanding of the charge carrier transfer in the a-Isa-S nanocomposite, where contacts between the a-I matrix and the a-S filaments play the crucial role in the process, then is necessary. Investigations in this direction have already been undertaken. 6. Conclusions Now, when chemical engineering is undergoing a rapid transformation from a discipline focused on macroscopic phenomena to one being more interdisciplinary and molecularly based, the role of the charge carrier transfer in description of many processes becomes indisputable. By analogy to the other transfer processes, such as the transfer of mass, heat, and momentum, the charge carrier transfer on the macroscopic level is also characterized by the general phenomenological equation (eq 4), which, in this particular case, brings the well-known Ohm’s law (eq 5) into play. However, the chemical engineering problems that are currently discussed more and more often need a quantum approach to understand them. If we want to go down to the molecular level of the charge carrier transfer, we have to study in depth the coefficient σ that is the electrical conductivity. Such investigations as JsU, JsT, and Jsd characteristics, photoinjection of electrons and holes, time-of-flight of charge carriers, dielectric spectroscopy, optical absorption, etc. enable us to extract information on the generation and transport processes of charge carriers: these are the processes that constitute the charge carrier transfer phenomenon. In covalent amorphous solids, which make materials especially interesting for modern technology, we can distinguish two fundamental types of generation mechanisms: the bulk generation and the injection from contacts. The generated carriers (electrons and holes) can be then transported through the material in two fundamental ways: via extended states (band transport) and via localized states (hopping transport). Without extensive knowledge of the charge carrier transfer processes, as well as the generation and transport mechanisms governing these processes, it is impossible to make any further progress in advanced technologies. Such areas as electrochemical devices (fuel cells, separation membranes, sensors, etc.), electrocatalytic and photocatalytic reactors, systems that used charging phenomena, nanoreactors with electrical control of chemical reactions on the molecular level, and many others require, for their development, a sound knowledge of the charge carrier transfer processes.


Nomenclature Notation a ) interatomic spacing (m) ai ) βi/kTd1/2 (V-1/2) AR ) Richardson constant (A/(m2 K2)) Cn ) capture rate coefficient (trapping coefficient) for electrons (m3/s) c ) concentration (mol/m3) D ) diffusion coefficient (m2/s) d ) sample thickness (m) ∆E ) activation energy of a process (eV) e ) elementary charge; e ) 1.6022 × 10-19 C E ) energy, energy level (eV) EA ) activation energy of the electric conduction (eV) Eg ) mobility gap (eV) EG ) transport gap (eV) F ) electric field (V/m) Fa ) Faraday constant; Fa ) 9.64853 × 104 C/mol fr ) probability for a recombination center to be occupied G ) photogeneration rate of electron-hole pairs (m-3 s-1) h ) Planck’s constant; h ) 4.1357 × 10-15 eV s hν ) photon energy (eV) J ) electric current density (A/m2) j ) mass flux (mol/(m2 s)) k ) Boltzmann’s constant; k ) 8.6173 × 10-5 eV/K m ) mass of charge carrier (kg) n ) density of carriers (m-3) m′, n′ ) indexes in the SCLC theory m″, n″ ) parameters in the multiphonon hopping model N ) concentration of states (m-3) N(E) ) density of states (eV-1 m-3) Nd ) density of donors or acceptors (m-3) q ) heat flux (J/(m2 s)) r ) compensation parameter R ) hop distance (m) T ) temperature (K) Td ) deposition temperature (temperature of substrate) (K) U ) applied voltage (V) W ) contact barrier thickness (m) Y ) photocurrent quantum yield (C) z ) valence of ion Greek Letters Rf ) rate of falloff of the wave function (m-1) Rr ) restoring force (eV/m2) β ) constant depending on the distribution of localized states βi ) βPF or βSch (eV m1/2/V1/2) βPF ) Poole-Frenkel coefficient (eV m1/2/V1/2) βSch ) Schottky coefficient (eV m1/2/V1/2) γ ) recombination coefficient (m3/s) ε ) relative permittivity ε0 ) permittivity of a vacuum; ε0 ) 8.8542 × 10-12 C2/(N m2) Φ0, Φ ) contact barrier height (eV) η ) viscosity (kg/(m s)) κ ) thermal conductivity (J/(m s K)) Λ ) numerical factor in the theory of variable range hopping νel ) electronic frequency (Hz) νph ) phonon frequency (Hz) Θ ) SCLC parameter τ ) shear stress (N/m2) µ ) carrier mobility (m2/(V s)) µchem ) chemical potential (J/mol) σ ) electrical conductivity (S/m) σ0 ) pre-exponential factor (S/m) σ′ ) Gaussian distribution width (eV)

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υ ) velocity (m/s) Subscripts b ) band transport C ) conduction band d ) donor or acceptor level e ) electrons h ) holes el ) electronic carriers F ) Fermi level hop ) hopping transport ion ) ionic carriers ph ) photoelectronic processes t ) trap level V ) valence band x, z ) direction µ ) carrier mobility

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ReceiVed for reView May 12, 2010 ReVised manuscript receiVed August 11, 2010 Accepted August 16, 2010 IE101069W

Charge Carrier Transfer: A Neglected Process in Chemical Engineering

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