CHARGE TRANSPORT (Katharina Broch, 27.04.2017)

References

The following is based on these references:

• K. Seeger “Semiconductor Physics”, Springer Verlag, 9th 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).

4.1.1 General remarks about charge transport

4.1.1.1 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−(𝜀𝐺/𝑘𝐵𝑇)

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 5th electron is bound loosely and therefore, its binding energy

휀𝐺,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 𝑁𝐷 . 𝑛 = 𝑁𝐷

In this range, the carrier concentration is independent on temperature.

At even higher temperatures, 𝑛 > 𝑁𝐷 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 휀𝐺,𝐷 is smaller than 휀𝐺 the slope is smaller than in

the intrinsic region.

4.1.1.2 Movement of charges without applied field

Movement of charge without applied field

Charge carriers in a material are constantly moving with a thermal velocity 𝑣𝑡ℎ, which is given

by: 1

2𝑚∗𝑣𝑡ℎ

2 = 3

2𝑘𝐵𝑇

𝑣𝑡ℎ at room temperature is approx. 107 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!

4.1.1.3 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 𝑣𝐷 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 𝑣𝐷 is proportional to the

applied field and the proportionality constant is the mobility µ (units: cm²/Vs):

𝑣𝐷 = µ𝐸

If there were no scattering events, the velocity would go to infinity (since 𝐸∗𝑞

𝑚∗ = const), resulting

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.

4.1.1.4 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 ∙ 𝑣𝐷

with 𝑛 being the charge carrier concentration and 𝑣𝐷 being the beforementioned drift velocity.

In the Drude model, 𝑣𝐷 is determined from 𝑑

𝑑𝑡(𝑚𝑣𝐷) +

𝑚𝑣𝐷

⟨𝜏𝑚⟩= 𝑒�⃗�

where ⟨𝜏𝑚⟩ 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 ⟨𝜏𝑚⟩ is energy dependent.

In the steady state (𝑑

𝑑𝑡(𝑚𝑣𝐷) = 0) and for low fields, 𝑣𝐷 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 ⟨𝜏𝑚⟩

Inserting this into the equation for the current density 𝑗 we get

𝑗 = 𝜎�⃗�

with the conductivity 𝜎:

𝜎 = 𝑛𝑒µ = 𝑛𝑒²

𝑚⟨𝜏𝑚⟩

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

4.1.1.5 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) ∙ [�⃗� + 𝑣 × �⃗� ] = ħ𝑑�⃗�

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

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:

µ = 𝑒

𝑚∗ ⟨𝜏𝑚⟩

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.

4.1.2. 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 electron-

lattice 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” 휀𝑓,𝑛 and 휀𝑓,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:

𝑛𝑐(𝑟 , 𝑇) = 𝑁𝑐(𝑇)𝑒(−𝐸𝑐(�⃗⃗� )− 𝜀𝑓,𝑛(�⃗⃗� )

𝑘𝑇) and 𝑝𝑣(𝑟 , 𝑇) = 𝑁𝑣(𝑇)𝑒(−

𝐸𝑣(�⃗⃗� )− 𝜀𝑓,𝑝(�⃗⃗� )

𝑘𝑇)

Here, 𝑛𝑐 and 𝑝𝑣 are the concentrations of electrons and holes, 𝑁𝑐(𝑇) and 𝑁𝑣(𝑇) are the DOS in

the conduction and valence band and 𝐸𝑐(𝑟 ) and 𝐸𝑣(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:

∇휀𝑓,𝑛(𝑟 , 𝑇) = ∇𝐸𝑐(𝑟 ) − 𝑘𝑇∇𝑛𝑐(𝑟 , 𝑇)

𝑛𝑐(𝑟 , 𝑇)

∇휀𝑓,𝑝(𝑟 , 𝑇) = ∇𝐸𝑣(𝑟 ) − 𝑘𝑇∇𝑝𝑣(𝑟 , 𝑇)

𝑝𝑣(𝑟 , 𝑇)

The first term corresponds to an electric force 𝑞𝐹 = ∇𝐸𝑐,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

𝑗 = 𝑗 𝑑𝑟𝑖𝑓𝑡 + 𝑗 𝑑𝑖𝑓𝑓 = [𝜎𝑛(𝑇) + 𝜎𝑝(𝑇)]𝐹 + 𝑘𝑇

𝑞[𝜎𝑛

𝑛𝑐∇𝑛𝑐 −

𝜎𝑝

𝑝𝑣∇𝑝𝑣]

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 , �⃗� , 𝑡) describes the probability of finding an electron at time t at a

position 𝑟 with a wavevector �⃗� ± 𝑑�⃗� . 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 𝑡ħ

∇𝑘𝑓(�⃗� ) + 𝑣 ∇𝑟𝑓(�⃗� ) +𝜕𝑓

𝜕𝑡

• First term: change in distribution function due to the total field force 𝐹 𝑡 = �⃗� + 𝑣 ×�⃗� • 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 𝑡ħ

∇𝑘𝑓(�⃗� ) + 𝑣 ∇𝑟𝑓(�⃗� )

After sufficient time, the system will have reached an equilibrium distribution 𝑓0 and in this

case the relaxation time approximation can be made: 𝜕𝑓

𝜕𝑡=

𝜕𝑓

𝜕𝑡|𝑐𝑜𝑙𝑙 = −

𝑓 − 𝑓0𝜏

Here, 𝜏 is the time scale of relaxation of the system.

In the steady state, the BTE can be written as:

𝜕𝑓

𝜕𝑡= −

𝑓 − 𝑓0𝜏

− 𝐹 𝑡ħ

∇𝑘𝑓(�⃗� ) + 𝑣 ∇𝑟𝑓(�⃗� ) = 0

And has, in the absence of concentration gradients the solution

𝑓(�⃗� ) = 𝑓0(�⃗� ) − 𝜏𝐹𝑡⃗⃗ ⃗𝑣

𝜕𝑓0(�⃗� )

𝜕휀

(We have used the definition of the velocity: 𝑣 = 1/ħ(𝜕휀𝑘/𝜕𝑘))

From this, we can now determine the total number of charge carriers per unit volume in a d-

dimensional space

𝑛 = ∫𝑑𝑑𝑘

(2𝜋)𝑑𝑓(𝑘)

and the current in response to the electric field (𝑗 = 𝑛𝑒𝑣)

𝐽 = 2𝑒 ∫𝑑𝑑𝑘

(2𝜋)𝑑𝑣 𝑓(�⃗� )

From this, the mobility can be derived based on

𝐽𝑖 = 𝑒𝑛𝜇𝑑𝐹𝑖 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).

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

𝜏𝑄= 𝑁𝜎𝑄𝑣

with the differential cross section:

𝜎𝑐 = 2𝜋 ∫ 𝜎(𝜃)𝑠𝑖𝑛𝜃𝑑𝜃

𝜋

0

D. Jena “Charge Transport in Semiconductors Fig. 8

2) The momentum relaxation time, which is related to the change of momentum

𝑝 − 𝑝 𝑐𝑜𝑠𝜃

𝑝= 1 − 𝑐𝑜𝑠𝜃

The momentum-transfer cross section 𝜎𝑚 is given by

𝜎𝑚 = 2𝜋 ∫𝜎(𝜃)(1 − 𝑐𝑜𝑠𝜃)𝑠𝑖𝑛𝜃𝑑𝜃

𝜋

0

from which we get the momentum relaxation time 𝜏𝑚 via 1

𝜏𝑚⁄ = 𝑁𝜎𝑚𝑣

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:

𝜎(𝜃) = [𝐾/2

𝑠𝑖𝑛²(𝜃/2)] ²

with K being the distance for which the potential energy equals twice the kinetic energy (𝐾 =𝑍𝑒2/(4𝜋𝜅𝜅0𝑚𝑣²). Here, we assume that the Coulomb potential of the impurity extends to

infinity.

The 𝜏𝑚 derived from this scattering cross section via 1

𝜏𝑚= 𝜎𝑣𝑁𝐼 has no finite value for 𝜃 = 0.

This problem is solved by the fact that the Coulomb potential is a screened potential

𝑉(𝑟) = −(𝑍|𝑒|

4𝜋휀휀0𝑟) 𝑒

−𝑟𝐿𝐷

We introduce the screening length 𝐿𝐷 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=2.5x1017 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𝜕

𝜕𝑡𝜓(𝑟 , 𝑡) = (𝐻0 + 𝐻1)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.

The mean free time between two scattering events is: 1

𝜏𝑄= 𝜎𝑣𝑁𝐼 = ∑(𝑁𝐼)𝑆(�⃗�

′, �⃗�

𝑘′

)

with V the scattering potential and sigma the scattering cross section

This is the rate ∑ 𝑆(�⃗� ′, �⃗� 𝑘′ ) at which an electron on momentum �⃗� 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 𝜏𝑚:

µ =𝑒⟨𝜏𝑚⟩

𝑚∗

with 1

𝜏𝑚=

1

𝜏𝑄(1 − cos(𝜃)), which is 𝜏𝑄 corrected for the fact that the change in �⃗� depends on

the scattering angle.

Applying this general relation to scattering from ionized impurities (again the reader is refered

to textbooks for details) and considering that < 𝜏𝑚 > 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

𝑁𝐼

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. 12.7.

Using this quantum mechanical considerations, we can understand the differences in the low-

temperature behaviour of a bulk semiconductor and the 2DEG, which we have seen before.

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 ≅ 𝑘𝐹 = √2𝜋𝑛2𝐷 ) 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. 173-175).

We can describe an acoustic wave which propagates in a crystal lattice by:

𝛿𝑟 = 𝐴𝑙𝑒±𝑖(�⃗� 𝑙𝑟 )−𝜔𝑙𝑡

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:

|𝛿𝑟 (𝑎) − 𝛿𝑟 (0)| = (∇𝑟⃗⃗⃗⃗ ∙ 𝛿𝑟 )𝑎

with ∇𝑟⃗⃗⃗⃗ ∙ 𝛿𝑟 = ±𝑖(𝑞𝑙⃗⃗ ⃗𝛿𝑟 )

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

𝛿휀ℎ = 휀𝑎𝑐,𝑣(∇𝑟⃗⃗⃗⃗ ∙ 𝛿𝑟 )

𝛿휀𝑒 = 휀𝑎𝑐,𝑐(∇𝑟⃗⃗⃗⃗ ∙ 𝛿𝑟 )

휀𝑎𝑐 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

𝑙𝑎𝑐⁄

with the mean free path

𝑙𝑎𝑐 =𝜋ħ𝑐𝑙

𝑚²휀𝑎𝑐2 𝑘𝐵𝑇

(𝑐𝑙 longitudinal elastic constant)

From this we obtain the mobility (for details see Seeger’s book)

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 30 – 60 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 < ħ𝜔𝐿𝑂/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.

4.1.2.2 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.

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. 2.2.

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

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.

4.1.2.3 Transport mechanisms

Schematics of band and hopping transport. From

D. Jena, Fig. 10.

4.1.2.4 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 |𝜓𝑎 > and |𝜓𝑏 > is described

by the transfer matrix element:

𝑡𝑎𝑏 = ⟨𝜓𝑎|𝐻|𝜓𝑏⟩

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 − 𝐸𝐿(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 𝜎𝐷𝑂𝑆. Hopping occurs between non-equivalent sites ei and ej within this Gaussian DOS, which are

randomly selected.

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 ( - E2 / 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).

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)

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 band-

like 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.

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.

4.1.2.5 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

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.