Subsections

• 2.3.1 Photon Transition
• 2.3.2 Phonon Transition
  • 2.3.2.1 Dynamic Case
  • 2.3.2.2 Stationary Case
  • 2.3.2.3 Surface Generation/Recombination
  • 2.3.2.4 Distributed Traps
  
  
• 2.3.3 Auger Generation/Recombination
• 2.3.4 Impact Ionization

2.3 Carrier Generation and Recombination

Carrier generation is a process where electron-hole pairs are created by exciting an electron from the valence band of the semiconductor to the conduction band, thereby creating a hole in the valence band. Recombination is the reverse process where electrons and holes from the conduction respectively valence band recombine and are annihilated. In semiconductors several different processes exist which lead to generation or recombination, the most important ones are:

• photon transition or optical generation/recombination,
• phonon transition or Shockley-Read-Hall generation/recombination,
• Auger generation/recombination or three particle transitions, and
• impact ionization.

In thermal equilibrium the generation and recombination processes are in dynamic equilibrium. When the system is supplied with additional energy, for example through the absorption of photons or the influence of temperature, additional carriers are generated. The most important generation/recombination processes for the simulation of semiconductor devices are summarized in the following.

2.3.1 Photon Transition

Figure 2.1: Direct generation/recombination process. During photon assisted recombination an electron from the conduction band re-combines with a hole in the valence band. The excess energy is transferred to a photon. The reverse process obtains its energy from radiation and generates an electron hole pair.

The photon transition is a direct, band-to-band, generation/recombination process. An electron from the conduction band falls back to the valence-band and releases its energy in the form of a photon (light). The reverse process, the generation of an electron-hole pair, is triggered by a sufficiently energetic photon which transfers its energy to a valence band electron which is excited to the conduction band leaving a hole behind. The photon energy for this process has to be at least of the magnitude of the band-gap energy . Figure 2.1 gives an overview of this process. The initial electron/hole constellation is found in (a) while the constellation after the generation/recombination process is found in (b).

For these state changes in the semiconductor the energy and the momentum have to be conserved. The energy is emitted or absorbed via a photon with the energy

$$E = h \nu ,$$ (2.44)

where $h$ is Planck's constant and $\nu$ is the frequency of the emitted or absorbed photon. However, as the momentum of a photon is very small no momentum transfer is possible in the transition process. Therefore, only direct band-to-band transitions are possible, where no change in momentum is necessary. As silicon and germanium are indirect semiconductors and have their valence band maximum and their conduction band minimum on different positions in momentum space, direct transitions are very unlikely to occur and can in most cases be neglected for those materials. In direct semiconductors like GaAs, this effect is very important.

The process of carrier recombination is directly proportional to the amount of available electrons and holes. By assuming the capture and emission rates $C_\mathrm{c}^\mathrm{OPT}$ and $C_\mathrm{e}^\mathrm{OPT}$ the recombination and generation ($G=-R$) rates can be written as

$$R_{np}^\mathrm{OPT} = C_\mathrm{c}^\mathrm{OPT} n p ,$$ (2.45)

$$G_{np}^\mathrm{OPT} = C_\mathrm{e}^\mathrm{OPT} .$$ (2.46)

As the recombination and generation are balanced in thermal equilibrium, where $n p = n_\mathrm{i}^2$

$$R_{np}^\mathrm{OPT} = G_{np}^\mathrm{OPT} ,$$ (2.47)

$$C_\mathrm{c}^\mathrm{OPT} n_\mathrm{i}^2 = C_\mathrm{e}^\mathrm{OPT} ,$$ (2.48)

the total band-to-band recombination is calculated as the deviation from the thermal equilibrium

$$R^\mathrm{OPT} = C_\mathrm{c}^\mathrm{OPT} (n p - n_\mathrm{i}^2) .$$ (2.49)

This process always strives for thermal equilibrium. For an excess concentration of carriers $n p - n_\mathrm{i}^2 >0$ and carrier recombination dominates, while for low carrier densities $n p - n_\mathrm{i}^2 <0$ and carrier generation prevails.

2.3.2 Phonon Transition

Figure 2.2: Four sub-processes in the Shockley-Read-Hall generation/recombination process. 1. electron capture, 2. hole capture, 3. hole emission, and 4. electron emission.

Another process is the generation/recombination by phonon emission. This process is trap-assisted utilizing a lattice defect at the energy level $\ensuremath{E_\textrm{t}}$ within the semiconductor band-gap. The excess energy during recombination and the necessary energy for generation is transferred to and from the crystal lattice (phonon). A theory describing this effect has been established by Shockley, Read, and Hall [7,8]. Therefore, the effect is throughout the literature referenced as Shockley-Read-Hall (SRH) generation/recombination. Four sub-processes are possible:

1. Electron capture. An electron from the conduction band is captured by an empty trap in the band-gap of the semiconductor. The excess energy of $\ensuremath {E_\textrm{c}}-\ensuremath{E_\textrm{t}}$ is transferred to the crystal lattice (phonon emission).
2. Hole capture. The trapped electron moves to the valence band and neutralizes a hole (the hole is captured by the occupied trap). A phonon with the energy $\ensuremath{E_\textrm{t}}- \ensuremath {E_\textrm{v}}$ is generated.
3. Hole emission. An electron from the valence band is trapped leaving a hole in the valence band (the hole is emitted from the empty trap to the valence band). The energy necessary for this process is $\ensuremath{E_\textrm{t}}- \ensuremath {E_\textrm{v}}$.
4. Electron emission. A trapped electron moves from the trap energy level to the conduction band. For this process additional energy of the magnitude $\ensuremath {E_\textrm{c}}-\ensuremath{E_\textrm{t}}$ has to be supplied.

These sub-processes are illustrated in Figure 2.2. (a) gives the initial electron/hole constellation, the arrows schematically depict the transition, and (b) gives the constellation after the sub-process.

Both, the recombination and the generation processes are two-step processes. The sequential occurrence of sub-processes 1 and 2 leads to recombination of an electron-hole pair. The excess energy of approximately the band-gap energy is transferred to the crystal lattice via lattice vibrations, phonons. For the SRH generation of an electron-hole pair sub-processes 3 and 4 are responsible. Here external energy has to be supplied from the lattice.

To derive an expression for the total recombination rate $\ensuremath{R^\textrm{SRH}}$, rates for every sub-process are introduced. Here, acceptor like traps are assumed, which are neutral when empty and negatively charged when occupied by an electron. The derivation for donor traps, which are neutral when occupied by an electron and positively charged when empty, is similar and delivers the same result.

The electron capture rate $v_\mathrm{ec}$ is proportional to the electron concentration in the conduction band $n$, the concentration of empty traps $\ensuremath{N_\textrm{t}}^0$, and a proportionality constant $k_\mathrm{ec}$. As the available electrons are spread in energy in the conduction band, we must consider the electron capture rate for different energies $\ensuremath{E}$. With the energy dependent distribution function for electrons $\ensuremath{f_\textrm{e}}(\ensuremath{E})$ and the density-of-states $\ensuremath{g_\textrm{e}}(\ensuremath{E})$ we get

$$\ensuremath{\textrm{d}v_\textrm{ec}}= k_\mathrm{ec}(\ensure... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.50)

This is the differential electron capture rate at energy $\ensuremath{E}$. The total amount of conduction band electrons is

$$n = \int_{\ensuremath {E_\textrm{c}}}^{\infty} \ensuremath{... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.51)

The hole capture rate $v_\mathrm{hc}$ is proportional to the hole concentration in the valence band $p$, the concentration of filled traps $\ensuremath{N_\textrm{t}}^-$, and a proportionality constant $k_\mathrm{hc}$. Again, we consider the spread of the holes in energy,

$$\mathrm{d}v_\mathrm{hc} = k_\mathrm{hc}(\ensuremath{E}) ... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.52)

Here, $\ensuremath{f_\textrm{h}}(\ensuremath{E})$ is the distribution function for holes and $\ensuremath{g_\textrm{h}}(\ensuremath{E})$ the density-of-states. The total amount of holes in the valence band is

$$p = \int_{\ensuremath {E_\textrm{v}}}^{\infty} \ensuremath{... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.53)

The hole emission rate $v_\mathrm{he}$ is proportional to the concentration of empty traps, the proportionality constant $k_\mathrm{he}$,

$$\mathrm{d}v_\mathrm{he} = k_\mathrm{he}(\ensuremath{E}) ... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.54)

And finally the electron emission rate $v_\mathrm{ee}$ is proportional to the concentration of filled traps and the proportionality constant $k_\mathrm{ee}$

$$\mathrm{d}v_\mathrm{ee} = k_\mathrm{ee}(\ensuremath{E}) ... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.55)

The total trap concentration $\ensuremath{N_\textrm{t}}$ is

$$\ensuremath{N_\textrm{t}}= \ensuremath{N_\textrm{t}}^0 + \ensuremath{N_\textrm{t}}^- ,$$ (2.56)

and the fraction of occupied traps $\ensuremath{f_\textrm{t}}$ is given by

$$\ensuremath{f_\textrm{t}}= \frac{\ensuremath{N_\textrm{t}}^... ...c{\ensuremath{N_\textrm{t}}^0}{\ensuremath{N_\textrm{t}}} .$$ (2.57)

With these definitions the net recombination rate for electrons becomes

$$\mathrm{d}\ensuremath{R^\textrm{SRH}_\textrm{e}}= \ensurema... ...}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.58)

In thermal equilibrium ( $n p = n_0 p_0 = \ensuremath {n_\textrm{i}}^2$) the net generation equals zero, which means that the respective capture and emission rates for electrons and holes must be equal

$$\ensuremath{v_\textrm{ec}}= \ensuremath{v_\textrm{ee}} , \quad \ensuremath{v_\textrm{hc}}= \ensuremath{v_\textrm{he}} .$$ (2.59)

From (2.58) we obtain using (2.59)

$$\frac{\ensuremath{k_\textrm{ee}}(\ensuremath{E})}{\ensurema... ...uremath{E})}{1-\ensuremath{f_\textrm{e}}(\ensuremath{E})} .$$ (2.60)

In thermal equilibrium is given by Fermi-Dirac statistics

$$f(\ensuremath{E}) = \frac{1}{1+\exp \left(\displaystyle \fr... ...nsuremath{\textrm{k$_\textrm{B}$}} T_\textrm{L}}\right)} ,$$ (2.61)

with the property

$$\frac{f}{1-f} = \exp \left(-\frac{\ensuremath{E}-\ensuremat... ...{\ensuremath{\textrm{k$_\textrm{B}$}}T_\textrm{L}}\right) .$$ (2.62)

The ratio (2.60) then calculates as

$$\frac{\ensuremath{k_\textrm{ee}}(\ensuremath{E})}{\ensurema... ...ensuremath{\textrm{k$_\textrm{B}$}} T_\textrm{L}}\right) .$$ (2.63)

Using (2.63) we can further develop (2.58)

$$\mathrm{d}\ensuremath{R^\textrm{SRH}_\textrm{e}}=\left[ \en... ...rm{e}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}}$$ (2.64)

$$=\left[ (1-\ensuremath{f_\textrm{t}}) \ensuremath{f_\tex... ...rm{e}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}}$$ (2.65)

$$=\left[ 1 - \frac{\ensuremath{k_\textrm{ee}}(\ensuremath{E})... ... \ensuremath{N_\textrm{t}}\ensuremath{\textrm{d}\ensuremath{E}}$$ (2.66)

$$=\left[ 1 - \exp \left( \frac{\ensuremath{E_\textrm{t}}- \en... ...emath{\textrm{k$_\textrm{B}$}}T_\textrm{L}} \right) \right]$$ (2.67)

$$(1-\ensuremath{f_\textrm{t}})\ensuremath{f_\textrm{e}}(\ensur... ... \ensuremath{N_\textrm{t}}\ensuremath{\textrm{d}\ensuremath{E}}$$ (2.68)

$$=\left[ 1 - \exp \left( \frac{\ensuremath {E_\textrm{Ft}}- \... ...emath{N_\textrm{t}} \ensuremath{\textrm{d}\ensuremath{E}} ,$$ (2.69)

with the trap's quasi Fermi energy .

Integrating over all possible electron energies gives the total electron recombination rate

$$\ensuremath{R^\textrm{SRH}_\textrm{e}}= \left[ 1 - \exp \le... ...{e}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.70)

Typically a capture cross section $\ensuremath{\sigma_\textrm{e}}(\ensuremath{E})$ is introduced to rewrite as

$$\ensuremath{k_\textrm{ec}}(\ensuremath{E}) = \ensuremath{\s... ...}(\ensuremath{E}) \ensuremath{v_\textrm{th}^\textrm{e}} ,$$ (2.71)

with the thermal velocity for electrons

$$\ensuremath{v_\textrm{th}^\textrm{e}}= \sqrt{\frac{3\ensuremath{\textrm{k$_\textrm{B}$}}T_\textrm{L}}{m}} ,$$ (2.72)

resulting in

$$\ensuremath{R^\textrm{SRH}_\textrm{e}}= \left[ 1 - \exp \le... ...{e}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.73)

For non-degenerate semiconductors near equilibrium a Maxwell-Boltzmann distribution can be assumed

$$f(\ensuremath{E}) = \exp \left( - \frac{\ensuremath{E}- \en... ...\ensuremath{\textrm{k$_\textrm{B}$}}T_\textrm{L}} \right) ,$$ (2.74)

and one obtains for the integral in (2.73)

$$n \ensuremath{\langle \! \langle \ensuremath{\sigma_\textrm... ...{e}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} ,$$ (2.75)

with the properties

$$n = \ensuremath {N_\textrm{c}}\exp \left(-\frac{\ensuremath... ...langle \ensuremath{\sigma_\textrm{e}} \rangle \! \rangle} ,$$ (2.76)

where is the effective density-of-states for electrons we have

$$\ensuremath{R^\textrm{SRH}_\textrm{e}}= \left[ n - \ensurem... ..., (1-\ensuremath{f_\textrm{t}}) \ensuremath{K_\textrm{ec}} ,$$ (2.77)

and

$$\ensuremath{R^\textrm{SRH}_\textrm{e}}= \left[ n (1-\ensure... ...uremath{f_\textrm{t}}\right] \ensuremath{K_\textrm{ec}} .$$ (2.78)

Introducing the auxiliary quantity

$$n_1 = \ensuremath {N_\textrm{c}}\exp \left(-\frac{\ensurema... ...{\ensuremath{\textrm{k$_\textrm{B}$}}T_\textrm{L}}\right) ,$$ (2.79)

we finally get for the electron recombination rate

$$\ensuremath{R^\textrm{SRH}_\textrm{e}}= \left( n (1-\ensure... ...ensuremath{f_\textrm{t}}\right) \ensuremath{K_\textrm{ec}} .$$ (2.80)

Analogously the hole recombination rate can be obtained as

$$\ensuremath{R^\textrm{SRH}_\textrm{h}}= \left( p \ensuremat... ...suremath{f_\textrm{t}}) \right) \ensuremath{K_\textrm{hc}} ,$$ (2.81)

by introducing

$$p_1 = \ensuremath {N_\textrm{v}}\exp \left(\frac{\ensuremat... ...{\ensuremath{\textrm{k$_\textrm{B}$}}T_\textrm{L}}\right) .$$ (2.82)

2.3.2.1 Dynamic Case

In transient simulations the capture and emission rates are not equal. Therefore, no further simplifications are possible and an additional equation has to be solved for each trap

$$\frac{{\textrm{d}} n_\mathrm{t}}{{\textrm{d}} t} = \ensurem... ...m{SRH}_\textrm{e}}- \ensuremath{R^\textrm{SRH}_\textrm{h}} .$$ (2.83)

This increases the computational effort significantly but is necessary for example to simulate the charge pumping effect (Section 4.1).

2.3.2.2 Stationary Case

In the stationary case electrons and holes always act in pairs thus the recombination rates for electrons and holes are equal,

$$\ensuremath{R^\textrm{SRH}}_n = \ensuremath{R^\textrm{SRH}}_p =\ensuremath{R^\textrm{SRH}} .$$ (2.84)

Therefore, we can calculate $\ensuremath{f_\textrm{t}}$ from (2.80) and (2.81) as

$$\ensuremath{f_\textrm{t}}= \frac{k_\mathrm{ec} n + k_\mathrm{hc} p_1}{k_\mathrm{ec}(n+n_1) + k_\mathrm{hc}(p+p1)} .$$ (2.85)

Using this expression for the total recombination rate we get

$$\ensuremath{R^\textrm{SRH}}= \ensuremath{R^\textrm{SRH}}_n ... ...suremath{f_\textrm{t}}) - n_1 \ensuremath{f_\textrm{t}}\right)$$ (2.86)

$$= k_\mathrm{ec} \ensuremath{N_\textrm{t}}\left( n (1-\frac... ...m{hc} p_1}{k_\mathrm{ec}(n+n_1) + k_\mathrm{hc}(p+p1)} \right)$$ (2.87)

$$= k_\mathrm{ec} k_\mathrm{hc} \ensuremath{N_\textrm{t}}\fra... ...n_1 p_1}{k_\mathrm{ec}(n+n_1) + k_\mathrm{hc} (p + p_1)} .$$ (2.88)

It is very common to introduce carrier lifetimes for electrons and holes $\ensuremath{\tau_n}$ and $\ensuremath{\tau_p}$

$$\ensuremath{\tau_n}= \frac{1}{k_\mathrm{ec}\ensuremath{N_\t... ...tau_p}= \frac{1}{k_\mathrm{hc}\ensuremath{N_\textrm{t}}} .$$ (2.89)

By using the capture cross sections for electrons and holes, $\ensuremath{\sigma_\textrm{e}}$ and $\ensuremath{\sigma_\textrm{h}}$, and the thermal velocities $\ensuremath{v_\textrm{th}^\textrm{e}}$ and $\ensuremath{v_\textrm{th}^\textrm{h}}$

$$\ensuremath{\tau_n}= \frac{1}{\ensuremath{\sigma_\textrm{e}... ...ath{v_\textrm{th}^\textrm{h}} \ensuremath{N_\textrm{t}}} ,$$ (2.90)

we come to the final formulation of the Shockley-Read-Hall model for carrier generationrecombination

$$\ensuremath{R^\textrm{SRH}}= \frac{n p - \ensuremath {n_\te... ...{\ensuremath{\tau_p}(n+n_1) + \ensuremath{\tau_n}(p+p_1)} .$$ (2.91)

Regarding the efficiency of trap centers it can be seen that the energy transfer necessary for generationrecombination is always approximately the band-gap energy, no matter where the trap energy level is. The reason is that carriers are transferred from one energy band-edge to the trap level and further to the other band-edge, giving in total the band-gap energy. But when the two sub processes capture and emission are considered, it can be seen that the further the trap energy is away from the mid-gap energy, the higher is the necessary energy for either capture or emission and the lower for the respective other process. The highest energy in this two-step process is always limiting the total generation/recombination. When the trap is located in the middle of the band-gap, the resulting energy barrier height is half the band-gap energy. As the trap is moved away from the mid-gap energy, the limiting energy barrier is increased and the probability of generation/recombination is reduced.

Impurities used for doping semiconductors are usually energetically situated very close to either the valence or the conduction band in order to be effective doping centers. They are therefore not very effective for carrier generation/recombination and are called ``shallow'' traps. ``Deep'' traps on the other hand are located close to the mid-gap which can be used to artificially increase the carrier generation or recombination.

2.3.2.3 Surface Generation/Recombination

For the investigation of NBTI the generation and recombination mechanisms at the silicondielectric interface are of major importance (Chapter 3). The Shockley-Read-Hall generation/recombination mechanism can also be applied to traps at the interface, which is for example obligatory for the simulation of the charge pumping effect (Section 4.1).

The derivation for recombination at surface traps is similar to the derivation for bulk traps. The major difference is the different unit for interface traps $[\ensuremath{N_\textrm{it}}]=1/\mathrm{cm}^2$ and the resulting unit for the surface recombination velocity $[\ensuremath{R^\textrm{SRH}_\textrm{it}}]=1/\mathrm{cm}^2 \mathrm{s}$.

2.3.2.4 Distributed Traps

As described in detail in Section 3.1.1, interface traps are not located on discrete energy levels but distributed in the band-gap instead. When accounting for the trap density-of-states $\ensuremath{D_\textrm{it}}(\ensuremath{E})$, we get for the interface trap concentration

$$\ensuremath{N_\textrm{it}}= \int_{\ensuremath {E_\textrm{v}... ...}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.92)

The interface trap recombination rate is then obtained as

$$\ensuremath{R^\textrm{SRH}_\textrm{it}}= \int_{\ensuremath ... ...it}}(\ensuremath{E}) \ensuremath{\textrm{d}\ensuremath{E}} .$$ (2.93)

2.3.3 Auger Generation/Recombination

Figure 2.3: Four sub-processes in the Auger generation/recombination mechanism. 1. electron capture, 2. hole capture, 3. electron emission, and 4. hole emission.

In the direct band-to-band Auger mechanism three particles are involved. During generation an electron hole pair is generated consuming the energy of a highly energetic particle. In the opposite process, when an electron hole pair recombines, the excess energy is transferred to a third particle. In detail the four possible processes are as follows:

1. Electron capture. An electron from the conduction band moves to the valence band neutralizing a hole in the valence band. The excess energy is transferred to an electron in the conduction band.
2. Hole capture. Again, an electron from the conduction band moves to the valence band and recombines with a valence hole. The excess energy is, in contrast to Process 1, transferred to another hole in the valence band.
3. Electron emission. A highly energetic electron from the conduction band transfers its energy to an electron in the valence band. The valence electron moves to the conduction band generating an electron hole pair.
4. Hole emission. A highly energetic hole from the valence band transfers its energy to an electron in the valence band which is then excited to the conduction band generating an electron hole pair.

These sub-processes are illustrated in Figure 2.3. (a) gives the initial electron/hole constellation, the arrows schematically depict the transition, and (b) gives the constellation after the sub-process.

As for the Shockley-Read-Hall effect a model can be derived by setting up rates for the four processes. For electron capture two electrons in the conduction band and one hole in the valence band are necessary. Using $k_\mathrm{ec}$ as the rate constant, the electron capture rate $v_\mathrm{ec}$ becomes

$$v_\mathrm{ec} = k_\mathrm{ec} n^2 p .$$ (2.94)

Analogical for hole capture where two holes and one electron are involved $v_\mathrm{hc}$ evaluates as

$$v_\mathrm{hc} = k_\mathrm{hc} n p^2 .$$ (2.95)

For electron and hole emission only one respective carrier is necessary

$$v_\mathrm{ee} = k_\mathrm{ee} n ,$$ (2.96)

$$v_\mathrm{he} = k_\mathrm{he} p .$$ (2.97)

In thermal equilibrium the respective capture and emission rates are in equilibrium, and therefore

$$v_\mathrm{ec,0} = v_\mathrm{ee,0} , \quad k_\mathrm{ec} \ensuremath {n_\textrm{i}}^2 = k_\mathrm{ee} ,$$ (2.98)

$$v_\mathrm{hc,0} = v_\mathrm{he,0} , \quad k_\mathrm{hc} \ensuremath {n_\textrm{i}}^2 = k_\mathrm{he} .$$ (2.99)

This leads us to the final model for the Auger recombination rate $\ensuremath{R^\textrm{AUG}}$

$$\ensuremath{R^\textrm{AUG}}= v_\mathrm{ec} + v_\mathrm{hc} ... ... n + k_\mathrm{hc} p)(n p - \ensuremath {n_\textrm{i}}^2) .$$ (2.100)

Although the Auger mechanism is microscopically exactly the same as the mechanism during impact ionization described in the next section, the energy source is completely different. Whereas impact ionization relies on high current density, only a very large carrier density is of importance for Auger generation/recombination, as can be seen in the final formulation of (2.100).

2.3.4 Impact Ionization

Figure 2.4: Impact ionization and avalanche multiplication. An energetic electron donates its energy to the generation of an electron hole pair. The newly generated electron can, due to the high electric field, obtain high energy and generate further carriers, leading to avalanche multiplication.

Impact ionization is a pure generation process. Microscopically it is exactly the same mechanism as the generation part of the Auger process: a highly energetic carrier moves to the conduction or valence band, depending on the carrier type, and the excess energy is used to excite an electron from the valence band to the conduction band generating another electron hole pair. The major difference is the cause of the effect. While it is purely the carrier concentration in the Auger mechanism, for impact ionization it is the current density.

Two partial processes can be distinguished:

1. Electron emission. A highly energetic electron from the conduction band transfers its energy to an electron in the valence band. The valence electron moves to the conduction band generating an electron hole pair.
2. Hole emission. A highly energetic hole from the valence band transfers its energy to an electron in the valence band which is then excited to the conduction band generating an electron hole pair.

Figure 2.4 depicts the effect of impact ionization and avalanche multiplication. The leftmost, highly energetic, electron excites a new electron/hole pair which gains energy and generates further carriers. The result is an avalanche multiplication of carrier generation.

As already mentioned, the generation rates are modeled proportional to the current densities $\ensuremath{{\vec{J}}_n}$ and $\ensuremath{{\vec{J}}_p}$ and can be written as:

$$v_\mathrm{e} = \alpha_\mathrm{e} \frac{\vert\ensuremath{{\vec{J}}_n}\vert}{\mathrm{q}} ,$$ (2.101)

$$v_\mathrm{h} = \alpha_\mathrm{h} \frac{\vert\ensuremath{{\vec{J}}_p}\vert}{\mathrm{q}} ,$$ (2.102)

with the ionization rates for electrons and holes, $\alpha_\mathrm{e}$ and $\alpha_\mathrm{h}$. These rates are typically described with an exponential dependence upon the electric field component in the direction of the current flow $E$. With the critical electric fields for electrons and holes, $E_\mathrm{e}^\mathrm{crit}$ and $E_\mathrm{h}^\mathrm{crit}$, and the ionization rates at infinite field, $\alpha_\mathrm{e}^\infty$ and $\alpha_\mathrm{h}^\infty$, the ionization rates evaluate as

$$\alpha_\mathrm{e} = \alpha_\mathrm{e}^\infty \exp\left(-\l... ...{e}^\mathrm{crit}}{E} \right)^{\beta_\mathrm{e}} \right) ,$$ (2.103)

$$\alpha_\mathrm{h} = \alpha_\mathrm{h}^\infty \exp\left(-\l... ...{h}^\mathrm{crit}}{E} \right)^{\beta_\mathrm{h}} \right) .$$ (2.104)

Here, $\beta_\mathrm{e}$ and $\beta_\mathrm{h}$ are additional model parameters, which are in the range of 1-2. The total impact ionization rate is now found as

$$R^\mathrm{II} = -v_\mathrm{e} -v_\mathrm{h} = -\alpha_\math... ...{h} \frac{\vert\ensuremath{{\vec{J}}_p}\vert}{\mathrm{q}} .$$ (2.105)

The impact ionization rate does not actually depend on the local electric field but on the carrier temperature and, thus, on the high-energy tail of the distribution function. Therefore, the model is not very accurate, especially in small devices. Carriers need to travel a certain distance in the high electric field in order to gain energy. For the exact modeling semiconductor device equations of higher order are necessary.