CHARGE TRANSPORT (Katharina Broch, )

Transcription

1 CHARGE TRANSPORT (Katharina Broch, ) References The following is based on these references: K. Seeger Semiconductor Physics, Springer Verlag, 9 th edition 2004 D. Jena Charge Transport in Semiconductors C.R. King Jr. Introduction to Microelectronics Theory, Georgia Instiute of Technology A. Hohlleitner Nanosystems I, Chapter 03: Electronic transport in 3D, 2D and 1D Technische Universität München P.-A. Blance (ed.) Photorefractive Organic Materials and Applications, Springer Series in Materials Science (2016), Chapter 2 V. Coropceanu, et al., Chem. Rev. (2007) P.M. Zimmermann et al., Nat. Chem. 2 (2010) General remarks about charge transport Intrinsic vs. extrinsic semiconductors Intrinsic semiconductor: In a pure semiconductor, an electron in the valence band can be thermally excited into the conduction band and the concentration of conduction electrons is n e (ε G/k B T) For every conduction electron, there is a hole produced in the valence band and it is always n = p. This is the condition for intrinsic conduction. The three regimes of charge carrier concentration in an extrinsic semiconductor. K. Seeger Semiconductor Physics Fig. 1.3 Extrinsic semiconductor: In a doped semiconductor, atoms are replaced by impurity atoms with more valence electrons than the atoms of the semiconductor (e.g. a 4-valent C atom by a 5-valent phosphorous atom). The 5 th electron is bound loosely and therefore, its binding energy ε G,D is much smaller than for the C-atoms. At elevated temperatures, all donors are thermally ionized and the concentration of conducting electrons n is the same as the concentration of donor atoms N D. n = N D

2 In this range, the carrier concentration is independent on temperature. At even higher temperatures, n > N D and the semiconductor will become intrinsic. At lower temperatures, there is a so-called carrier freeze out at the donor molecules (which act as traps for charge carriers). The charge carrier concentration in this regime again depends exponentially on the temperature, but since ε G,D is smaller than ε G the slope is smaller than in the intrinsic region Movement of charges without applied field Movement of charge without applied field Charge carriers in a material are constantly moving with a thermal velocity v th, which is given by: 1 2 m v 2 th = 3 2 k BT v th at room temperature is approx cm/s. Importantly, the charge carriers do not continue indefinitely along their path, which would otherwise result in indefinite conductivity or zero resistance. In reality, with the important exception of superconductors, the charge carriers collide with lattice vibrations (phonons), impurities (neutral or ionized) or other charge carriers resulting in scattering processes. Each scattering event randomizes the direction of the thermal velocity and thus, the net velocity averaged over time is zero. The thermal velocity does not contribute to charge transport!

3 Movement of charges under applied field, definition of mobility Movement of charge with applied field In order to observe the movement of charges (i.e. a net current), an electric field has to be applied. Under the influence of this electric field, electrons and holes move with a net velocity, the drift velocity v D which is superimposed on the random motion of the thermal velocity, resulting in a drift current. In the so-called low field regime (or ohmic regime) the drift mobility v D is proportional to the applied field and the proportionality constant is the mobility µ (units: cm²/vs): v D = µe If there were no scattering events, the velocity would go to infinity (since E q = const), resulting m in an infinite mobility. Scattering is the origin of resistance/finite conductivity! For sake of completeness, we note that for increasing field strength, the velocity will reach a saturation value, which is the high field regime (or hot electron regime). We will focus in the following only on the low field regime for which the mobility is defined by the beforementioned equation. The two regimes of transport depending on the field strength. D. Jena Charge Transport in Semiconductors, Fig. 6a Drude model In the range of extrinsic conductivity (i.e. n is independent on temperature), the current density j, caused by applying an external field is: j = n e v D with n being the charge carrier concentration and v D being the beforementioned drift velocity.

4 In the Drude model, v D is determined from d dt (mv D) + mv D τ m = ee where τ m is an average relaxation time (to equilibrium). The second term describes friction the charge carriers experience as they move due to the thermal vibration of atoms, which will depend on the temperature of the crystal. We note here, that in general τ m is energy dependent. In the steady state ( d dt (mv D) = 0) and for low fields, v D will be proportional to the field, as we discussed before. We can now define the proportionality constant µ (the charge carrier mobility) as: µ = e m τ m Inserting this into the equation for the current density j we get j = σe with the conductivity σ: σ = neµ = n e² m τ m σ(t) describes the conductivity which can have different origins: 1) Intrinsic, if carriers are generated by thermal excitation 2) Extrinsic, if carriers are generated by introducing dopants 3) Injection controlled, if carriers are injected using electrodes 4) Photoconductivity, if carriers are generated by absorption of light Movement of charges in periodic potential (perfect crystal) So far, we discussed the general motion of a charged particle in a material, without considering the crystal structure. We turn now to a discussion of the motion of an electron in the valence band of a perfect semiconductor under the influence of an applied electric field. The discussion of the electron movement in the conduction band can be done accordingly. The general equation of motion of the electron in k-space can be written as: F = ( e) [E + v B ] = ħ dk dt As the electron reaches the end of the 1. Brioullin Zone (BZ) an Umklapp-process takes place, meaning that the electron re-enters the 1. BZ. These Umklapp-processes occur as long as the field is too small to cause the transition of an electron to the next nearest empty band across the energy gap. This transition, which happens at higher fields, is called Zener tunneling, but will not be discussed in more detail here. The force the electron experiences by the applied electric field mainly affects the electron momentum, the velocity and the acceleration of the electron is determined by the bandstructure. A typical way of including the bandstructure in the discussion of charge transport is by introducing an effective mass m*, as discussed in the introductory chapter. We can then assume that the electron still moves according to E = ħ²k² 2m

5 when we replace m by m, with 1 m = 1 ħ² d²e dk² We can now re-do the above calculations of the Drude model, to take into account the band structure and obtain: e µ = m τ m If m* is larger than m (i.e., if for small bands), the mobility µ is correspondingly smaller. This is one of the origins of the smaller mobility of organic semiconductors compared to inorganic semiconductors. Although the physics of charge transport in inorganic and organic semiconductors can be discussed using similar models, disorder effects and the lower dielectric constants, effects which will be discussed later in more detail, lead to narrower bands in organic semiconductors Theoretical description of charge transport Theoretical description of band transport Transport in materials with delocalized electronic wave functions occurs within the electronic bands and mean-field approximations are used for a theoretical description. One of the most common semi-classical models to describe band transport is the drift-diffusion model: 1) The electrons are treated as classical particles 2) Charge carriers are in local equilibrium (thermalized by electron-electron and electronlattice interactions) 3) Electrical current is the result of two processes: a drift under an applied electric field and diffusion from regions of high charge carrier concentration to low charge carrier concentration First, we introduce the quasi-fermi levels ε f,n and ε f,p for electrons (n) and holes (p), respectively. These quasi-fermi levels originate from the thermalization of excess free charge carriers (introduced by dopands or by injection) by electron-electron and electron-lattice scattering processes and lead to the formation of thermal disctributions of electrons and hole in the conduction and valence band, respectively. They are defined as: n c (r, T) = N c (T)e ( E c (r ) ε f,n (r ) kt ) and p v (r, T) = N v (T)e ( E v (r ) ε f,p (r ) ) kt Here, n c and p v are the concentrations of electrons and holes, N c (T) and N v (T) are the DOS in the conduction and valence band and E c (r) and E v (r) is the conduction and valence band edge. The non-equilibrium conditions which lead to the formation of these quasi-fermi levels are also the origin of the two processes leading to the electric current. The change in the quasi-fermi levels due to thermalization effects can then be written as: ε f,n (r, T) = E c (r) kt n c(r, T) n c (r, T) ε f,p (r, T) = E v (r) kt p v(r, T) p v (r, T) The first term corresponds to an electric force qf = E c,v (r) and describes the drift current, the second term is a force driven by a concentration gradient, i.e. a diffusion force leading to a diffusion current. In total, we have

6 j = j drift + j diff = [σ n (T) + σ p (T)]F + kt q [σ n n c n c σ p p v p v ] This leads us back to the Drude model, in which the scattering force is taken into account by a scattering term. A second model is the Boltzmann model, which follows a similar idea and uses distribution functions to describe the impact of an external perturbation on the distribution of charge carriers. We briefly summarize the main results and refer the reader to the excellent review by D. Jena Charge Transport in Semiconductors. A distribution function f(r, k, t) describes the probability of finding an electron at time t at a position r with a wavevector k ± dk. At equilibrium and in the absence of external fields the distribution function is given by Fermi-Dirac statistics. Any external perturbation drives the distribution function away from equilibrium. This is described by the Boltzmann transport equation (BTE), which is formally written as df dt = F t ħ kf(k ) + v r f(k ) + f t First term: change in distribution function due to the total field force F t = E + v B Second term: change in distribution function due to concentration gradients Last term: local change in distribution function Since the number of charge carriers (if no charge carriers are injected via electrodes) is constant, the total rate of change of the distribution is zero. This means that we can write the local change as: f t = f t coll F t ħ kf(k ) + v r f(k ) After sufficient time, the system will have reached an equilibrium distribution f 0 and in this case the relaxation time approximation can be made: f t = f t coll = f f 0 τ Here, τ is the time scale of relaxation of the system. In the steady state, the BTE can be written as: f t = f f 0 F t τ ħ kf(k ) + v r f(k ) = 0 And has, in the absence of concentration gradients the solution f(k ) = f 0 (k ) τf t v f 0(k ) ε (We have used the definition of the velocity: v = 1/ħ( ε k / k)) From this, we can now determine the total number of charge carriers per unit volume in a d- dimensional space n = dd k f(k) (2π) d and the current in response to the electric field (j = nev)

7 J = 2e dd k (2π) d vf(k ) From this, the mobility can be derived based on J i = enμ d F i and reduces in the degenerate case (i.e. the distribution function is a Heavyside step function around the Fermi energy) to with n>0. Real semiconductors: Temperature dependence of mobility In general, low energy sites, which can lead to localisation of the charge carrier wavefunction or lead to scattering of the charge carriers, are called trap states. In inorganic as well as in organic semiconductors, such trap states are for example impurities (neutral or ionized) or structural imperfections of the crystal lattice, see Fig. xx. Depending on the charge transport mechanism, these trap states can strongly influence the absolute value and the temperature dependence of the charge carrier mobility especially at low temperatures. At higher temperatures, another scattering mechanism dominates the behaviour of the material, which we will discuss later. Scattering at ionized impurities (screened Coulomb potential): The dimensionality of charge transport In this lecture series, effects which are taking place if the size of the system (its dimensionality) is reduced are of special importance. One very interesting effect is the temperature dependence of the mobility, which is shown here for a bulk (3D) sample and a 2D electron gas (2DEG).

8 Temperature dependence of charge carrier mobility for a bulk sample and a 2DEG. D. Jena Charge Transport in Semiconductors Fig. 6b We will see that the origin of the temperature dependence at low temperatures is mainly scattering from ionized impurities which has a different temperature dependence for 3D and 2D systems. We will discuss scattering of charge carriers at ionized impurities by summarizing the main results from the discussion in Seeger s book. The interested reader is referred to this book for details. First, we will define two important time scales: 1) The quantume lifetime (or inter-collision time), which is the mean free time between collisions. The probability per unit time for a collision is given by 1 = Nσ τ Q v Q with the differential cross section: π σ c = 2π σ(θ)sinθdθ 0 D. Jena Charge Transport in Semiconductors Fig. 8 2) The momentum relaxation time, which is related to the change of momentum

9 p p cosθ = 1 cosθ p The momentum-transfer cross section σ m is given by π σ m = 2π σ(θ)(1 cosθ)sinθdθ from which we get the momentum relaxation time τ m via 1 τm = Nσ m v 0 Coulomb scattering of an electron and a hole by a positive ion. From Seeger, Fig. 6.3 We consider now a singly ionized impurity atom of charge Ze fixed inside the crystal. The scattering cross section obtained from classical mechanics is: K/2 σ(θ) = [ sin²(θ/2) ] ² with K being the distance for which the potential energy equals twice the kinetic energy (K = Ze 2 /(4πκκ 0 mv²). Here, we assume that the Coulomb potential of the impurity extends to infinity. The τ m derived from this scattering cross section via 1 = σvn τ I has no finite value for θ = 0. m This problem is solved by the fact that the Coulomb potential is a screened potential V(r) = ( Z e r 4πεε 0 r ) e L D We introduce the screening length L D to take into account that the electrostatic field of the ionized impurity is screened by the surrounding carriers. Therefore, in the vicinity of an ionized impurity, the density of charge carriers will be: n(r) = ne ev(r)/kt n(1 e V(r)/kT) For example, in n-type Germanium κ = 16 and m*/m0=0.12 with n=2.5x10 17 cm -3 at 100K. This gives a screening length of 5.5nm, while the averaged nearest distance between two ionized impurities is 15.9nm. Interestingly, the screening is weaker in 2DEG and independent on the charge carrier density! In a quantum mechanical picture, impurities can be treated as perturbations of the exact Hamiltonian H0 of the perfect semiconductor, which is determined by the crystal potential and which has been discussed in detail in BM KoMa as well as in the introductory chapters. The perturbation is described by introducing a second part H1 of the Hamiltonian. The general Schrödinger-equation has the form: iħ t ψ(r, t) = (H 0 + H 1 )ψ(r, t) Based on this, the probability of finding an electron with initial state k> in final state k > after scattering from an ionized impurity can be derived based on Fermi s Golden Rule.

10 The mean free time between two scattering events is: 1 τ Q = σvn I = (N I )S(k, k k ) with V the scattering potential and sigma the scattering cross section This is the rate k S(k, k ) at which an electron on momentum k is scattered by a single impurity, multiplied by by the toal number of impurities and gives us the inter-collision time. However, the mobility is determined by the averaged momentum scattering time τ m : µ = e τ m m 1 with = 1 (1 cos(θ)), which is τ τ m τ Q corrected for the fact that the change in k depends on Q the scattering angle. Applying this general relation to scattering from ionized impurities (again the reader is refered to textbooks for details) and considering that < τ m > is the total momentum scattering time averaged for a Fermi-Dirac distributed carrier population, we obtain the following dependence of the mobility µ on temperature T and impurity density NI: µ T3/2 N I This is the first important result: At low temperatures, the mobility in bulk semiconductors is limited by scattering at ionized impurities. Density of states for different dimensions. From C.R.King Jr. Introduction to Microelectronics Theory, Georgia Instiute of Technology, Fig Using this quantum mechanical considerations, we can understand the differences in the lowtemperature behaviour of a bulk semiconductor and the 2DEG, which we have seen before.

11 First of all, due to the quantization effects and the degeneracy of conducting electrons (as we have seen earlier, all conducting electrons in a 2D system have k k F = 2πn 2D ) the averaging procedure is different. Secondly, and more importantly, there is no temperature activation necessary, since the electron density is independent of temperature for a 2DEG. This results in the observed temperature-independence of the mobility at low temperatures in a 2DEG. Charge transport in 1D will be discussed later, but from the discussion above it becomes already clear: The density of states has strong impact on the charge transport physics of semiconductors. Scattering at lattice vibrations (acoustic phonons) In the previous chapter, we discussed the temperature behaviour of the mobility at low temperatures and we will now discuss scattering at lattice vibrations as the origin of the decrease in mobility at higher temperatures. We will summarize only the results here, for a detailed derivation, the interested reader is referred to Seeger (p ). We can describe an acoustic wave which propagates in a crystal lattice by: δr = A l e ±i(q lr) ω l t q_l wavevector and w_l angular frequency, sound velocity v_l=w_l/q_l The difference in displacement between two neighbouring atoms can be approximated by: with r δr = ±i(q l δr) δr(a) δr(0) = ( r δr)a Due to the lattice deformation, the edge of valence and conduction band will vary linearly for small deformations (as it is the case for acoustic phonons) and the change in energy for a hole/an electron at atom A or atom B is then δε h = ε ac,v ( r δr) δε e = ε ac,c ( r δr) ε ac are here proportionality constants, the so-called deformation potential constants, named after the deformation potential introduced 1950 by Bardeen and Shockley. The mean relaxation time is 1 τm = v l ac with the mean free path l ac = πħc l m²ε 2 ac k B T (c l longitudinal elastic constant) From this we obtain the mobility (for details see Seeger s book)

12 The mean free path decreases with increasing temperature since at higher temperatures more phonons are excited and therefore, more scattering centers exist. The dependence on the effective mass arises from the DOS k²dk. At higher temperatures, the mobility in bulk semiconductors is limited by scattering at phonons. μ T 3/2 Relaxation of hot electrons. From A. Hohlleitner Nanosystems I, Chapter 03: Electronic transport in 3D, 2D and 1D TUM. For temperatures below 100K, acoustic phonon scattering dominates, since optical phonons have energies of mev and are therefore inefficient scatterers at low temperatures. However, they are important for the relaxation of highly energetic charge carriers ( hot electrons ). These hot electrons relax by emitting a cascade of optical phonons until E < ħω LO /2π and thereafter, loose their energy by emission of acoustic phonons. At temperatures above 100K, scattering at optical phonons has to be taken into account and modifies the temperature dependence slightly (μ T 1.67 ). Temperature dependence of mobility in a semiconductor exhibiting band transport. In summary, in a real semiconductor for which charge transport can be described by band theory, but which has impurities, we expect the temperature dependence of the mobility which is shown in the figure Density of states distribution for differences in the long-range order We have already seen before that the DOS will depend on the dimensionality of the system. In organic semiconductors, the extent of electronic delocalization depends on the strength of the electronic coupling between molecules as well as the energetic and positional disorder, which we will discuss later and which leads to electronic localization.

13 Organic semiconductors display a wide range of long-range order, from highly crystalline samples such as single crystals of pentacene or rubrene to polycrystalline or even amorphous thin films. Depending on the sample preparation procedure, polycrystalline and amorphous domains can coexist in one sample. In combination with energetic disorder due to chemical impurities, grain boundaries or crystal defects, electron-phonon interactions and the low dielectric constant in organic molecules results in weak Coulomb screening and high electronic localization. This results in narrow bandwidth of <500meV. In addition, the weak Coulomb screening leads to strong binding energies between electrons and holes and to the formation of exciton states as discuss in previous chapters. This has important implications for charge transport and the functioning of optoelectronic devices such as solar cells or light emitting diodes, which will be discussed in following chapters. In contrast, due to the large dielectric constant of inorganic semiconductors and the high Coulomb screening, the binding energy of excitons is in the order of the thermal energy and therefore, they can easily be dissociated even at room temperature. Density of states distribution in crystalline, polycrystalline and amorphous solids. The mobility edge describes the transition from localized to extended tail states. From P.-A. Blance (ed.) Fig In the following, we will discuss briefly how long-range order affects the density of states and accordingly, the theoretical models which can be used to describe charge transport. In crystalline solids, electronic wave functions overlap significantly, and electronic bands are formed with energy gaps in between due to the symmetry of the unit cells. The present of defects or dopants leads to the occurrence of localized states within the band gap which can act as traps or as doping sites creating free charge carriers. The electronic and optical properties of such crystalline solids can be well described with band theory. Band theory is a one-electron independent particle theory that assume the existence of a set of stationary extended one-electron states distributed according to the Fermi-Dirac distributions. Polycrystalline solids consist of crystalline grains which are separated by grain boundaries. These grain boundaries and other defects such as impurities or lattice dislocations introduce localized electronic states in the energy gap for which the wave function of the electron only extends over a few nearest neighbouring atoms. If the material has a lot of grain boundaries or other defects, a continuum of localized tail states can form, which is often observed in organic semiconductors. These tail states are obviously important for devices, in particular for field

14 effect transistors with low gate voltages, since by varying the gate voltage the Fermi level is moved through the DOS. Finally, in amorphous solids we only have short range order and accordingly, the electronic wave function only extends over a few nearest neighbours. This results in narrow bandwidths and there is also a continuum of localized and extended tail states observed due to disorder induced localised states. These materials cannot longer be described by band theory and other models have to be used Transport mechanisms Schematics of band and hopping transport. From D. Jena, Fig Theoretical description of hopping transport In materials with significant degrees of disorder, charge transport cannot be described by band transport, but involves incoherent electron transfer reactions, which strongly depend on the electronic coupling. The coupling between two localized states ψ a > and ψ b > is described by the transfer matrix element: t ab = ψ a H ψ b where H is the Hamiltonian of the system. In a disordered system, calculations of the transfer matrix elements are challenging and rely on simplifications. In the most commonly used approximation, the transfer integral of an electron (hole) transfer form molecule A to B is given by: t = E (L+1)H E L(H 1) 2 where We are the energies of the LUMO +1 (L+1), LUMO (L), HOMO (H) and HOMO-1 (H 1) levels in the neutral states of the dimer A B. Disorder models There are several, very successful models which incorporate energetic disorder to describe charge transport. We will briefly introduce the most common model, the Gaussian Disorder Model developed by Bässler and co-workers. As for many other models (such as e.g. the famous Marcus model), it is assumed that the hopping sites are distributed following a Gaussian distribution of the DOS with a standard deviation σ DOS. Hopping occurs between non-equivalent sites ei and ej within this Gaussian DOS, which are randomly selected.

15 The model is based on a few reasonable assumptions about the disorder. Energies of defect electrons or transport states exhibit Gaussian distribution of DOS G ( E ) ~ exp ( - E 2 / 2 2 ) The hopping rate is given by the product of the Boltzmann factor, a pre-exponential factor and a factor that takes into account the overlap of the wavefunctions (which is (also) not sharp, but subject to a distribution) The central result is that for small electrical fields the mobility behaves as = 0 exp [ - ( T / T0 ) 2 ] with T0 = 2 / ( 3 kb ) Importantly, this model disregards polaronic effects which we will discuss in the following. Reorganisation energy, polarons Change in nuclear coordinate of potential minimum in ground and excited state of a molecule. On the right the difference in the electronic wavefunction of HOMO and LUMO in pentacene are shown. From V. Coropceanu, Chem. Rev. (2007) and P.M. Zimmermann et al, Nat. Chem. 2 (2010).

16 Schematic of polaron formation by adding charge to molecule. Adding a free charge carrier in an organic solid leads to the deformation of the molecular geometry. This can be understood in the Frank-Condon picture in which the minimum of the electronic ground state and the electronic excited state are not at the same configurational coordinate. Therefore, the molecule has to deform slightly to find the energetically lowest configuration when a charge is placed on the molecule. The energy necessary for this deformation is called the reorganisation energy. In addition to changes in the molecular configuration, the whole lattice, i.e. the surrounding molecules will react to the charge on the molecule. When the charge moves to the next molecule, the lattice deformation will follow this movement, resulting in a coupling of the charge carrier to the lattice vibration (=phonon). The quasiparticle corresponding to this coupled state of charge carrier and phonon is called polaron and dominates the charge transport in organic semiconductors. The reorganization energy describes the polaron binding energy. A polaron describes a charge carrier surrounded by a polarized medium and therefore, the effective mass of this quasiparticle will be larger than the effective mass of a free charge carrier in an inorganic solid. The larger the effective mass of the polaron, the more localized is its wavefunction. In organic semiconductors, polarons are usually highly localized and referred to as small polarons. The polaron binding energy is and results from the deformation in molecular geometry and lattice geometry as the carrier localizes on a given site. Such polarons can move between equivalent localized polaron states by tunnelling or by hopping between non-equivalent sites, which requires the absorption or emission of a phonon. Tunnelling of polarons results in band-like behaviour and usually dominates at low substrate temperatures. Hopping is a thermally activated process and thus, dominates at higher temperatures. In summary, these are the quantities, which determine charge transport in highly disordered systems: Transfer integral: Looking at electron (charge) transfer processes, the transfer integral describes the interaction of wavefunctions of the two molecules involved in the charge transfer process, i.e. the electronic coupling. Reorganization energy: Tends to localize charge and has two contributions, an internal (Energy necessary to accommodate charge on molecule) and an external (Energy necessary for local environment to react to change in polarisation)

17 In the following we will briefly introduce models which describe charge transport via hopping in organic semiconductors. Holstein model (small polarons) The Holstein model uses perturbation theory to calculate an electron transfer rate in the hopping regime, which will depend on the temperature T and the energy of the phonon ħω to which the electron couples. Importantly, this theory excludes the existence of disorder! An important quantity is the electron-phonon coupling term g which is related to the polaron binding energy First, we focus on the strong electron-phonon coupling regime with g>>1 In this case, the mobility can be split into two parts, a part describing coherent electron transfer processes (tunneling) and another part which describes incoherent electron transfer processes (hopping) In a semiconductor with narrow bands with a being the spacing between neighbouring molecules and τ the momentum relaxation time. with t the transfer integral, g the electron-phonon coupling parameter and ħω0 the phonon energy. This means that g strongly affects the transfer integral, which is a very important result and we will come back to it later From these two equations we see that hopping transport is described by three regimes: 1) Tunnelling regime: Coherent electron transfer processes. The mobility exhibits a bandlike temperature dependence. This dominates at low temperatures 2) Hopping regime: Incoherent electron transfer processes dominate above a certain temperature T1. These electron transfer processes are field-assisted and thermally activated and only involve nearest neighbours. The mobility shows an activation behaviour μ exp ( /kt) with being half the polaron binding energy 3) Electron scattering regime, the polarons are dissociated and the charge carriers are scattered at thermal phonons, leading to a decrease in mobility with increasing temperature.

18 From P.-A. Blance (ed.) Fig. 2.3 As an attempt for a description of hopping transport including polaronic and disorder effects we mention the effective medium model, which unfortunately, exceeds the scope of this lecture Boundary between band and hopping transport Finally, we will discuss the crossover from band to hopping transport. So far, we have only discussed the regime of strong electron phonon coupling. The strength of the electron phonon coupling is described by the coupling term g which is related to the polaron binding energy We have seen that the coupling term g reduces the transfer integral

19 V. Coropceanu et al., Chem. Rev. (2007) In case of weak electron-phonon coupling, the mobility will be dominated by coherent electron transfer processes and display a band-like transport behaviour in which the mobility increases with decreasing temperature. In case of medium electron-phonon coupling, bandlike transport dominates a low temperatures, but due to the hopping contribution it will exhibit a weaker temperature dependence at higher temperatures. In case of strong electron-phonon coupling we are back at the three regimes discussed before. As a final note: Although the Holstein model is very useful in the description of charge transport, it explicitly doesn t include disorder and therefore, cannot be a complete model of charge transport. There are more sophisticated attempts for a description of charge transport, for example the effective medium model or the multiple trapping and detrapping model, but a discussion of those models exceeds the scope of the lecture.