CHARGE TRANSPORT (Katharina Broch, 27.04.2017) References The following is based on these references: β’ K. Seeger βSemiconductor Physicsβ, Springer Verlag, 9 th edition 2004 β’ D. Jena βCharge Transport in Semiconductorsβ β’ C.R. King Jr. βIntroduction to Microelectronics Theoryβ, Georgia Instiute of Technology β’ A. Hohlleitner βNanosystems I, Chapter 03: Electronic transport in 3D, 2D and 1Dβ Technische UniversitΓ€t MΓΌnchen β’ P.-A. Blance (ed.) Photorefractive Organic Materials and Applications, Springer Series in Materials Science (2016), Chapter 2 β’ V. Coropceanu, et al., Chem. Rev. (2007) β’ 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 β β( / ) 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 βSemiconductor Physicsβ 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 , is much smaller than for the C-atoms. At elevated temperatures, all donors are thermally ionized and the concentration of conducting electrons is the same as the concentration of donor atoms . =
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CHARGE TRANSPORT - soft-matter.uni-tuebingen.de the Drude model, π£ is determined from ( π£ )+ π£ γπ Γ γ = β where γπ Γ γis an average relaxation time (to equilibrium).The
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CHARGE TRANSPORT (Katharina Broch, 27.04.2017)
References
The following is based on these references:
β’ K. Seeger βSemiconductor Physicsβ, Springer Verlag, 9th edition 2004
β’ D. Jena βCharge Transport in Semiconductorsβ
β’ C.R. King Jr. βIntroduction to Microelectronics Theoryβ, Georgia Instiute of
Technology
β’ A. Hohlleitner βNanosystems I, Chapter 03: Electronic transport in 3D, 2D and 1Dβ
Technische UniversitΓ€t MΓΌnchen
β’ P.-A. Blance (ed.) Photorefractive Organic Materials and Applications, Springer Series
in Materials Science (2016), Chapter 2
β’ V. Coropceanu, et al., Chem. Rev. (2007)
β’ 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
π β πβ(ππΊ/ππ΅π)
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 βSemiconductor Physicsβ 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
νπΊ,π· is much smaller than for the C-atoms. At elevated temperatures, all donors are thermally
ionized and the concentration of conducting electrons π is the same as the concentration of
donor atoms ππ· . π = ππ·
In this range, the carrier concentration is independent on temperature.
At even higher temperatures, π > ππ· 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 νπΊ,π· is smaller than νπΊ 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 π£π‘β, which is given
by: 1
2πβπ£π‘β
2 = 3
2ππ΅π
π£π‘β 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.
β 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 π£π· 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 π£π· is proportional to the
applied field and the proportionality constant is the mobility Β΅ (units: cmΒ²/Vs):
π£π· = Β΅πΈ
If there were no scattering events, the velocity would go to infinity (since πΈβπ
πβ = const), resulting
in an infinite mobility.
β 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 βhot electronβ 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 βCharge Transport in Semiconductorsβ, Fig. 6a.
4.1.1.4 Drude model In the range of extrinsic conductivity (i.e. n is independent on temperature), the current density
π, caused by applying an external field is:
π = π β π β π£π·
with π being the charge carrier concentration and π£π· being the beforementioned drift velocity.
In the Drude model, π£π· is determined from π
β’ First term: change in distribution function due to the total field force πΉ π‘ = οΏ½βοΏ½ + π£ ΓοΏ½βοΏ½ β’ Second term: change in distribution function due to concentration gradients
β’ 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
from which we get the momentum relaxation time ππ via 1
ππβ = ππππ£
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:
π(π) = [πΎ/2
π ππΒ²(π/2)] Β²
with K being the distance for which the potential energy equals twice the kinetic energy (πΎ =ππ2/(4ππ π 0ππ£Β²). Here, we assume that the Coulomb potential of the impurity extends to
infinity.
The ππ derived from this scattering cross section via 1
ππ= ππ£ππΌ has no finite value for π = 0.
This problem is solved by the fact that the Coulomb potential is a screened potential
π(π) = β(π|π|
4πνν0π) π
βππΏπ·
We introduce the screening length πΏπ· 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
νππ 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
ππβ = π£
πππβ
with the mean free path
πππ =πΔ§ππ
πΒ²νππ2 ππ΅π
(ππ longitudinal elastic constant)
From this we obtain the mobility (for details see Seegerβs 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 kΒ²dk.
β At higher temperatures, the mobility in bulk semiconductors is limited by scattering at
phonons.
π β πβ3/2
Relaxation of hot electrons.
From A. Hohlleitner βNanosystems I, Chapter 03:
Electronic transport in 3D, 2D and 1Dβ TUM.
For temperatures below 100K, acoustic phonon scattering dominates, since optical phonons
have energies of 30 β 60 meV and are therefore inefficient scatterers at low temperatures.
However, they are important for the relaxation of highly energetic charge carriers (βhot
electronsβ). These βhot electronsβ relax by emitting a cascade of optical phonons until πΈ < Δ§ππΏπ/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 (π β πβ1.67).
Temperature dependence of mobility in a
semiconductor exhibiting band transport.
In summary, in a βrealβ 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 |ππ > and |ππ > is described
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:
π‘ = πΈ(πΏ+1)π» β πΈπΏ(π»β1)
2
where We are the energies of the LUMO +1 (L+1), LUMO (L), HOMO (H) and HOMO-1 (H
β 1) levels in the neutral states of the dimer A β 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 BΓ€ssler 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 ππ·ππ. 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.
β’ Energies of defect electrons or transport states exhibit Gaussian distribution of DOS
G ( E ) ~ exp ( - E2 / 2 2 )
β’ 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:
β’ Transfer integral: Looking at electron (charge) transfer processes, the transfer integral
describes the βinteractionβ of wavefunctions of the two molecules involved in the charge
transfer process, i.e. the electronic coupling.
β’ Reorganization energy: Tends to localize charge and has two contributions, an internal
(Energy necessary to βaccommodateβ 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 Δ§Ο 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 Ο the momentum relaxation time.
with t the transfer integral, g the electron-phonon coupling parameter and Δ§Ο0 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 π β exp (ββ/ππ) with β 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 doesnβt 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