Carrier scattering
Encyclopedia
Defect types include atom vacancies, adatoms
, steps, and kinks which occur most frequently at surfaces due to finite material size causing crystal discontinuity. What all types of defects have in common, whether they be surface or bulk, is that they produce dangling bonds which have specific electron energy levels not similar to those of the bulk. This is because these states cannot be described with periodic Bloch waves due to the change in electron potential energy caused by the missing ion cores just outside of the surface. Hence, these are localized states which one must solve the Schrödinger equation for separately such that electron energies can be properly described. The break in periodicity results in a decrease in conductivity due to defect scattering.
A simpler and more qualitative way of determining dangling bond energy levels is with Harrisson diagrams. Metals have non-directional bonding and a small Debye length
which, due to their charged nature, makes dangling bonds inconsequential if they can even be considered to exist. Semiconductors are dielectrics so electrons can feel and become trapped at defect energy states. The energy levels of these states are determined by the atoms that make up the solid. Figure 1 shows the Harisson diagram for the elemental semiconductor Si. From left to right, s-orbital and p-orbital hybridization promotes sp3 bonding which, when multiple sp3 Si-Si dimers are combined to form a solid, defines the conduction and valence bands. If a vacancy were to exist, such as those on each atom at the solid/vacuum interface, it would result in at least one broken sp3 bond which has an energy equal to that of single self hybridized Si atoms as shown in Figure 1. This energy corresponds to roughly the middle of the bandgap of Si, ~0.55eV above the valence band. Certainly this is the most ideal case whereas the situation would be different if bond passivation (see below) and surface reconstruction
, for example, were to occur. Experimentally, the energies of these states can be determined using absorption spectroscopy
or X-ray photoelectron spectroscopy
, for example, if instrument sensitivity and/or defect density are high enough.
Compound semiconductors, such as GaAs, have dangling bond states that are nearer to the band edges (see Figure 2). As bonding becomes increasingly more ionic, these states can even act as dopants. This is the cause of the well known difficulty of GaN p-type doping where N vacancies are abundant due to its high vapor pressure resulting in high Ga dangling bond density. These states are close to the conduction band edge and therefore act as donors. When p-type acceptor dopants are introduced, they are immediately compensated for by the N vacancies. With these shallow states, their treatment is often considered as an analogue to the hydrogen atom as follows for the case of either anion or cation vacancies (hole effective mass, m*, for cation and electron m* for anion vacancies). The binding energy, Ec-Edb, is
where U=-q2/(4πεεrr) is the electrostatic potential between an electron occupying the dangling bond and its ion core with ε, the free space permittivity constant, εr, the relative permittivity, and r the electron-ion core separation. The simplification that the electron translational energy, KE=-U/2, is due to the virial theorem for centrosymmetric potentials. As described by the Bohr model
, r is subject to quantization
.
The electron momentum is p=mv=h/λ such that
resulting in
and
.
This treatment loses accuracy as the defects tend away from either band edge.
with H' being the interaction parameter and the delta function, δ(Ef-Ei), indicating elastic scattering
. The simple relation 1/τ= Σk',k Sk'k makes this a useful equation for characterizing material transport properties when used in conjunction with σ = ne2τ /m* and Matthiessen's rule to incorporate other scattering processes.
The value of Sk'k is primarily determined by the interaction parameter, H'. This term is different depending on whether shallow or deep states are considered. For shallow states, H' is the perturbation term of the redefined Hamiltonian H=Ho+H', with Ho having an eigenvalue energy of Ei. The matrix for this case is
where k' is the final state wavevector of which there is only one value since the defect density is small enough to not form bands (~<1010/cm2). Using the Poisson equation for Fourier periodic point charges,
,
gives the Fourier coefficient of the potential from a dangling bond Vq=e/(q2εεrV) where V is volume. This results in
where qs is the Debye length
wavevector correction due to charge screening. Then, the scattering frequency is
where n is the volumetric defect density. Performing the integration, utilizing |k|=|k'|, gives
.
The above treatment falters when the defects are not periodic since dangling bond potentials are represented with a Fourier series. Simplifying the sum by the factor of n in Eq (10) was only possible due to low defect density. If every atom (or possibly every other) were to have one dangling bond, which is quite reasonable for a non-reconstructed surface, the integral on k' must also be performed. Due to the use of perturbation theory in defining the interaction matrix, the above assumes small values of H' or, shallow defect states close to band edges. Fortunately, Fermi's Golden Rule itself is quite general and can be used for deep state defects if the interaction between conduction electron and defect is understood well enough to model their interaction into an operator that replaces H'.
In a similar experiment, one can just lower the temperature of the conductor (Figure 3) so that phonon density decreases to negligible allowing defect dominant resistivity. With this case, σ = ne2τ /m* can be used to directly calculate τ for defect scattering.
Surface defects can always be “passivated” with atoms to purposefully occupy the corresponding energy levels so that conduction electrons cannot scatter into these states (effectively decreasing n in Eq (10)). For example, Si passivation at the channel/oxide interface of a MOSFET
with hydrogen (Figure 4) is a typical procedure to help reduce the ~1010 cm−2 defect density by up to a factor of 12 thereby improving mobility and, hence, switching speeds. Removal of intermediary states which would otherwise reduce tunneling barriers also decreases gate leakage current and increases gate capacitance as well as transient response. The effect is that the Si sp3 bonding becomes fully satisfied. The obvious requirement here is the ability for the semiconductor to oxidize the passivating atom or, Ec-Edb + χ > EI, with the semiconductor electron affinity
χ and atom ionization energy
EI.
, which consequently results in a time-dependent strain, 
where a simple plane wave is used to describe the phonon propagation, 
. Displacement of atoms away from their equilibrium positions generally causes a change in the electronic band structure
(Figure 5) where, for scattering, we are concerned with electrons in the conduction band with energy ~ECB,
.
The empirical parameter, ZDP, is termed the deformation potential and describes electron-phonon coupling strength. Multiplying by the phonon population (Bose-Einstein distribution, Nq) gives the total deformation potential,
(the reason for the root will be apparent below). Here, the + corresponds to phonon emission and – for phonon absorption during the scattering event. A note, because 
for transverse phonons, only interactions with longitudinal phonons are non-zero. Therefore, the complete interaction matrix is
where the Kronecker delta enforces momentum conservation and arises from assuming the electronic wavefunctions (final state,) are plane waves as well.
with the phonon radial frequency ωq=cq, volume V, solid density ρ, and phonon group velocity c. Plugging this into Eq. 6 gives
.
With the assumptions that Nq>>1, ħω<van Hove singularity
) gives the scattering rate:
where g(E) is the electronic density of states
for which the 3-dimensional solution with parabolic dispersion was used to obtain the final answer.
q>>1 cannot be made. Yet, a reasonable route that still provides a detour from dealing with complex phonon dispersions is using the Einstein model which states that only one phonon mode exists in solids. For optical phonons, this approximation turns out to be sufficient due to very little slope variation in ω(q) and, thus, we can claim ħω(q) ≅ ħω, a constant. Consequently, Nq is also a constant (only T dependent). The last approximation, g(E')=g(E±ħω) ~ g(E), cannot be made since ħω ~ E and there is no workaround for it, but the added complexity to the sum for τ is minimal.
.
The sum turns to the density of states at E' and the Bose-Einstein distribution can be taken out of the sum due to ħω(q) ≅ ħω.
Adatoms
An adatom is an atom that lies on a crystal surface, and can be thought of as the opposite of a surface vacancy. This term is used in surface chemistry, when describing single atoms lying on surfaces and surface roughness. The word is a contraction of "adsorbed atom"...
, steps, and kinks which occur most frequently at surfaces due to finite material size causing crystal discontinuity. What all types of defects have in common, whether they be surface or bulk, is that they produce dangling bonds which have specific electron energy levels not similar to those of the bulk. This is because these states cannot be described with periodic Bloch waves due to the change in electron potential energy caused by the missing ion cores just outside of the surface. Hence, these are localized states which one must solve the Schrödinger equation for separately such that electron energies can be properly described. The break in periodicity results in a decrease in conductivity due to defect scattering.
Electronic Energy Levels of Semiconductor Dangling Bonds
Debye length
In plasma physics, the Debye length , named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge...
which, due to their charged nature, makes dangling bonds inconsequential if they can even be considered to exist. Semiconductors are dielectrics so electrons can feel and become trapped at defect energy states. The energy levels of these states are determined by the atoms that make up the solid. Figure 1 shows the Harisson diagram for the elemental semiconductor Si. From left to right, s-orbital and p-orbital hybridization promotes sp3 bonding which, when multiple sp3 Si-Si dimers are combined to form a solid, defines the conduction and valence bands. If a vacancy were to exist, such as those on each atom at the solid/vacuum interface, it would result in at least one broken sp3 bond which has an energy equal to that of single self hybridized Si atoms as shown in Figure 1. This energy corresponds to roughly the middle of the bandgap of Si, ~0.55eV above the valence band. Certainly this is the most ideal case whereas the situation would be different if bond passivation (see below) and surface reconstruction
Surface reconstruction
Surface reconstruction refers to the process by which atoms at the surface of a crystal assume a different structure than that of the bulk. Surface reconstructions are important in that they help in the understanding of surface chemistry for various materials, especially in the case where another...
, for example, were to occur. Experimentally, the energies of these states can be determined using absorption spectroscopy
Absorption spectroscopy
Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons, from the radiating field. The intensity of the absorption varies as a...
or X-ray photoelectron spectroscopy
X-ray photoelectron spectroscopy
X-ray photoelectron spectroscopy is a quantitative spectroscopic technique that measures the elemental composition, empirical formula, chemical state and electronic state of the elements that exist within a material...
, for example, if instrument sensitivity and/or defect density are high enough.
where U=-q2/(4πεεrr) is the electrostatic potential between an electron occupying the dangling bond and its ion core with ε, the free space permittivity constant, εr, the relative permittivity, and r the electron-ion core separation. The simplification that the electron translational energy, KE=-U/2, is due to the virial theorem for centrosymmetric potentials. As described by the Bohr model
Bohr model
In atomic physics, the Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with electrostatic forces providing attraction,...
, r is subject to quantization
.The electron momentum is p=mv=h/λ such that
resulting in
and
.This treatment loses accuracy as the defects tend away from either band edge.
Defect Scattering
The dangling bond energy levels are eigenvalues of wavefunctions that describe electrons in the vicinity of the defects. In the typical consideration of carrier scattering, this corresponds to the final state in Fermi's Golden Rule of scattering frequency:with H' being the interaction parameter and the delta function, δ(Ef-Ei), indicating elastic scattering
Elastic scattering
In scattering theory and in particular in particle physics, elastic scattering is one of the specific forms of scattering. In this process, the kinetic energy of the incident particles is conserved, only their direction of propagation is modified .-Electron elastic scattering:When an alpha particle...
. The simple relation 1/τ= Σk',k Sk'k makes this a useful equation for characterizing material transport properties when used in conjunction with σ = ne2τ /m* and Matthiessen's rule to incorporate other scattering processes.
The value of Sk'k is primarily determined by the interaction parameter, H'. This term is different depending on whether shallow or deep states are considered. For shallow states, H' is the perturbation term of the redefined Hamiltonian H=Ho+H', with Ho having an eigenvalue energy of Ei. The matrix for this case is
where k' is the final state wavevector of which there is only one value since the defect density is small enough to not form bands (~<1010/cm2). Using the Poisson equation for Fourier periodic point charges,
,gives the Fourier coefficient of the potential from a dangling bond Vq=e/(q2εεrV) where V is volume. This results in
where qs is the Debye length
Debye length
In plasma physics, the Debye length , named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge...
wavevector correction due to charge screening. Then, the scattering frequency is
where n is the volumetric defect density. Performing the integration, utilizing |k|=|k'|, gives
.The above treatment falters when the defects are not periodic since dangling bond potentials are represented with a Fourier series. Simplifying the sum by the factor of n in Eq (10) was only possible due to low defect density. If every atom (or possibly every other) were to have one dangling bond, which is quite reasonable for a non-reconstructed surface, the integral on k' must also be performed. Due to the use of perturbation theory in defining the interaction matrix, the above assumes small values of H' or, shallow defect states close to band edges. Fortunately, Fermi's Golden Rule itself is quite general and can be used for deep state defects if the interaction between conduction electron and defect is understood well enough to model their interaction into an operator that replaces H'.
Experimental Measurements
Determination of the extent these dangling bonds have on electrical transport can be experimentally observed fairly readily. By sweeping the voltage across a conductor (Figure 3), the resistance, and with a defined geometry, the conductivity of the sample can be determined. As mentioned before, σ = ne2τ /m*, where τ can be determined knowing n and m* from the Fermi level position and material band structure. Unfortunately, this value contains effects from other scattering mechanisms such as due to phonons. This gains usefulness when the measurement is used alongside Eq (11) where the slope of a plot of 1/τ versus n makes Ec-Edb calculable and the intercept determines 1/τ from all but defect scattering processes. This requires the assumption that phonon scattering (among other, possibly negligible processes) is independent of defect concentration.In a similar experiment, one can just lower the temperature of the conductor (Figure 3) so that phonon density decreases to negligible allowing defect dominant resistivity. With this case, σ = ne2τ /m* can be used to directly calculate τ for defect scattering.
Passivation
MOSFET
The metal–oxide–semiconductor field-effect transistor is a transistor used for amplifying or switching electronic signals. The basic principle of this kind of transistor was first patented by Julius Edgar Lilienfeld in 1925...
with hydrogen (Figure 4) is a typical procedure to help reduce the ~1010 cm−2 defect density by up to a factor of 12 thereby improving mobility and, hence, switching speeds. Removal of intermediary states which would otherwise reduce tunneling barriers also decreases gate leakage current and increases gate capacitance as well as transient response. The effect is that the Si sp3 bonding becomes fully satisfied. The obvious requirement here is the ability for the semiconductor to oxidize the passivating atom or, Ec-Edb + χ > EI, with the semiconductor electron affinity
Electron affinity
The Electron affinity of an atom or molecule is defined as the amount of energy released when an electron is added to a neutral atom or molecule to form a negative ion....
χ and atom ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...
EI.
Phonon Scattering
We now consider carrier scattering with lattice deformations termed phonons. Consider the volumetric displacement such a propagating wave produces, , which consequently results in a time-dependent strain, where a simple plane wave is used to describe the phonon propagation, . Displacement of atoms away from their equilibrium positions generally causes a change in the electronic band structureElectronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
(Figure 5) where, for scattering, we are concerned with electrons in the conduction band with energy ~ECB,
. The empirical parameter, ZDP, is termed the deformation potential and describes electron-phonon coupling strength. Multiplying by the phonon population (Bose-Einstein distribution, Nq) gives the total deformation potential,
for transverse phonons, only interactions with longitudinal phonons are non-zero. Therefore, the complete interaction matrix is where the Kronecker delta enforces momentum conservation and arises from assuming the electronic wavefunctions (final state,
Acoustic Phonons
Using Fermi's Golden Rule, the scattering rate for low energy acoustic phonons can be approximated. The interaction matrix for these phonons iswith the phonon radial frequency ωq=cq, volume V, solid density ρ, and phonon group velocity c. Plugging this into Eq. 6 gives
.With the assumptions that Nq>>1, ħω<
Van Hove singularity
A Van Hove singularity is a kink in the density of states of a solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone...
) gives the scattering rate:
where g(E) is the electronic density of states
Density of states
In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...
for which the 3-dimensional solution with parabolic dispersion was used to obtain the final answer.
Optical Phonons
Typically, phonons in the optical branches of vibrational dispersion relationships have energies on the order of or greater than kT and, therefore, the approximations ħω< .The sum turns to the density of states at E' and the Bose-Einstein distribution can be taken out of the sum due to ħω(q) ≅ ħω.