This page describes various properties and models available in the Semiconductor material model. You may also find the Creating a New Semiconductor Material page to be useful.

## Electronic

- DC PERMITTIVITY: The relative permittivity (or dielectric constant) of the material is equal to the square of the refractive index, and is assumed to be the DC (zero frequency) value.

- WORK FUNCTION: In a semiconductor, the work function φs describes the energy cost of removing an electron from the intrinsic energy level (the Fermi energy of the undoped semiconductor) and placing it at "infinity." A related value is the electron affinitiy χs, which is the energy cost of removing an electron from the conduction band edge.

$$ \chi_s = \phi_{s,i} - \frac{E_G}{2} +\frac{k_BT}{2}\text{ln}\frac{N_C}{N_V} $$

where EG is the band gap and NC and NV are the effective density of states in the conduction band and valence band, respectively.

- EC VALLEY: The conduction band of semiconductors can have several valleys and by default the lowest valley is enabled for each semiconductor in the default list of materials in the material database. For each valley, the different semiconductor properties can be specified and by default only those from the lowest valley are used. The user can choose to change this by picking between the L, X or Gamma valleys.

### Fundamental

- EFFECTIVE MASS: To account for the influence of the crystal lattice potential of the semiconductor, electrons and holes can be approximated as free charges with an effective mass (relative to the electron rest mass) that depends on the electronic band-structure of the material. In CHARGE and HEAT, the effective mass is treated as a parameter of the material model.The temperature variation in the effective mass can be accounted for with a quadratic model

$$ m^*_{n, p} (T) = + m^*_{n, p} (0) \alpha T + \beta T^2$$

where coefficients α and β, and the effective mass at T=0K are inputs to the model.

Related to the effective mass is the effective density of states in the conduction and valence bands

$$ N_C = 2\Big( \frac{2\pi m^*_n k_BT}{h^2}\Big)^{3/2}$$

$$ N_V = 2\Big( \frac{2\pi m^*_p k_BT}{h^2}\Big)^{3/2}$$

where h is Planck’s constant.

- BAND GAP: A key physical property of the material is the band gap, which, like the effective mass, is derived from the electronic band-structure of the material. In CHARGE and HEAT, the band gap energy is treated as a parameter of the material model.The temperature variation in the band gap can be accounted for with a "universal" empirical model

$$E_{G}(T)=E_{G, 0}-\frac{\alpha T^{2}}{\beta+T}$$

where coefficients α and β, and the band gap energy at T=0K are inputs to the model.

- BAND GAP NARROWING: When impurities are added to the intrinsic (pure) semiconductor, localized allowed energy states may be introduced at energies that lie within the band-gap. In the case of dopants, these impurity states will exist with energies near the conduction or valence band edges (such that the dopants readily ionize at moderate temperatures). When the concentration of dopants is large, these discrete states will begin to merge and form a thin "band" of allowed states within the band gap, effectively narrowing the band gap. This can be viewed as a narrowing of the band gap or an increase in the effective density of states.

The Slotboom model [1] for band gap narrowing is provided in CHARGE and HEAT to account for this effect,

$$ \Delta E_G = -V_1 \Bigg[ \text{ln}\bigg(\frac{N^+_D+N^-_A}{N_0}\bigg) + \sqrt{\text{ln}^2\bigg(\frac{N^+_D+N^-_A}{N_0}\bigg)+C} \Bigg]$$

where the coefficients V1, N0, and C are inputs to the model, and the effect can be specified independently for electrons and holes. Note that the sign implies a narrowing effect for positive coefficients.

- INTRINSIC CARRIER CONCENTRATION: The intrinsic carrier concentration is calculated from the effective mass and band gap, and is only displayed in the Material Database for reference. It is calculated as

$$ n_i = \sqrt{N_CN_V}\text{ exp}(-E_G/2 k_BT) $$

where T=300K is assumed, and the effective density of states and band gap are treated are treated as intrinsic quantities (before band gap narrowing).

### Mobility

The mobility parameter in the drift-diffusion equations is the physical link between the motion of carriers (electrons and holes) and the semiconductor material. The mobility can be viewed as a measure of how readily electrons and holes can move through the crystal lattice of the semiconductor. In the absence of any interactions with the lattice, impurities, or other carriers, electrons and holes would move freely in the periodic potential of the lattice; interactions that change the momentum of the carriers are termed scattering events. Different types of scattering contribute to the mobility of the electrons and holes, including lattice scattering, ionized/neutral impurity scattering, and carrier-carrier scattering.

In addition, the velocity of the carriers is observed to saturate at high-fields. Each of these scattering mechanisms can be addressed in CHARGE by applying the appropriate models, which are detailed in the following sections.

- LATTICE SCATTERING: The fundamental process that impedes the free motion of the carriers in the lattice is thermal scattering off of the lattice itself. The mobility due to lattice scattering is treated as a basic input into the CHARGE semiconductor model, and may be entered as a constant value or with a temperature dependence described by the "universal" temperature model,

$$ A(T) = A(300)\Big(\frac{T}{300}\Big)^{\eta} $$

where A(300) is the value of the parameter at T=300K, and η is a temperature exponent. In the case of the lattice scattering mobility μL, the temperature dependence reads

$$ \mu^L_{n,p}(T) = \mu^L_{n,p}(300)\Big(\frac{T}{300}\Big)^{\eta} $$

where subscripts n and p refer to electrons and holes, respectively. - IMPURITY AND FREE-CARRIER SCATTERING: Many models exist to account for the influence of impurities on the carrier mobility. CHARGE provides support for three common models with wide-ranging applicability: the Caughey-Thomas model [2], the Masetti model [3], and the Klaassen model [4]. Each model requires a variety of coefficients; default values are provided with CHARGE for common semiconductors.

For general modeling purposes, the Caughey-Thomas model or Masetti models are often sufficient, and coefficients are available for multiple semiconductor materials. The Klaassen model is primarily tuned for silicon at T=300K, and coefficients for other materials are not available. At moderate doping densities, the mobility predicted by all models reduces to that of the Caughey-Thomas model.- Caughey-Thomas model: This is the most basic model.

$$ \mu^{LI}_{n,p} = \mu^{min} _{n,p} + \frac{ \mu^L _{n,p}-\mu^{min} _{n,p}}{1+(N/N _{ref})^{\alpha}}$$

where N is the total doping concentration (N = NA + ND), μL is the lattice scattering mobility (as determined from the model chosen in the previous section), and μmin, Nref, and α are temperature-dependent coefficients described the universal temperature model. - Masetti model: To account for extremely large doping concentrations, choose this model, which adds a correction to the Caughey-Thomas model for large N:

$$ \mu^{LI}_{n,p} = \mu^{min} _{n,p} + \frac{ \mu^L _{n,p}-\mu^{min} _{n,p}}{1+(N/C _{r})^{\alpha}} - \frac{\mu^{(2)} _{n,p}}{1 + (C_s/N)^{\beta}}$$

Again, N is the total doping concentration (N = NA + ND) and μL is the lattice scattering mobility. Parameters μmin, μ(2), Cr (replacing Nref of the Caughey-Thomas model), Cs, α, and β are each temperature-dependent coefficients described the universal temperature model. - Klaassen model: This model can be used to account for the aforementioned doping effects (at moderate and high impurity concentrations), as well as the influence of carrier-carrier scattering. The Klaassen model combines the basic lattice scattering with the impurity and carrier-carrier scattering using Matthisen’s rule

$$ \frac{1}{\mu^{LIC} _{n,p} } = \frac{1}{\mu^{L} _{n,p} } + \frac{1}{\mu^{IC} _{n,p} }$$

where μL is the lattice scattering mobility and μIC is Klaassen’s impurity and carrier-carrier (IC) scattering mobility. The formulation of the IC scattering mobility is complex and involves multiple levels of coefficients and models accounting for

- majority carrier scattering by dopants,

- minority carrier scattering by dopants, and

- electron-hole scattering.

- Caughey-Thomas model: This is the most basic model.

To begin, the IC mobility is defined as a function of the dopant and carrier concentrations,

$$ \mu^{IX} _{ \nu, \pi } \big( N _{\Delta} , \nu \nu \big) = \frac{(\mu ^{\Lambda} _{\nu,\pi})^2}{\mu ^{\Lambda} _{\nu,\pi}-\mu ^{\mu \nu} _{\nu,\pi}}\Bigg( \frac{N^{\sigma \chi} _{\nu,\pi}}{N ^{\sigma \chi,\varepsilon \phi \phi} _{\nu,\pi}} \Bigg) \Bigg( \frac{N _{\rho, \varepsilon \phi}}{N ^{\sigma \chi} _{\nu,\pi}} \Bigg) ^{\alpha} + \frac{\mu ^{\Lambda} _{\nu,\pi} \mu ^{\mu \nu} _{\nu,\pi}}{\mu ^{\Lambda} _{\nu,\pi}-\mu ^{\mu \nu} _{\nu,\pi}} \Bigg( \frac{\nu +\pi}{N ^{\sigma \chi,\varepsilon \phi \phi} _{\nu,\pi}} \Bigg) $$

where μL is the lattice scattering mobility, and coefficients μmin, Nref1 (equivalent to Nref or Cr), and α are defined as for the Caughey-Thomas or Masetti models. Note that the Klaassen model accounts for temperature dependence separately, therefore a constant value should be used for the lattice scattering mobility.

In the preceding equation, the "scattering densities" are

$$ N_n^{SC} = N_D + N_A + p$$

$$ N_p^{SC} = N_D + N_A + n$$

where the donor and acceptor densities, ND and NA respectively, are corrected according to the clustering function:

$$ N_D = N_D^+ + \frac{N_D^+}{C_D + (N_{ref,D}/N_D^+)^2 }$$

$$ N_A = N_D^+ + \frac{N_A^+}{C_A + (N_{ref,A}/N_A^-)^2 }$$

Here, CD, Nref,D, CA, and Nref,A are coefficients of the model.

The "effective scattering densities" are defined as (using the same clustering-corrected acceptor and donor concentrations)

$$ N_n ^{sc\text{, }eff} = N_D + G(P_n)N_A + \frac{p}{F(P_n)}$$

$$ N_p ^{sc\text{, }eff} = N_A+ G(P_p)N_D + \frac{n}{F(P_p)}$$

The function G describes the ratio of scattering cross-sections between repulsive and attractive screened Coulomb potentials as a function of the factor P (itself a function of carrier density and majority dopant density). The factor P accounts for the screening effect, and is calculated as the weighted harmonic mean of two parameters accounting for the free-carrier and ionized impurity screening,

$$ P_n(N_D.n)=\Big(\frac{f_{CW}}{P_{CW,n}(N_D)}+\frac{f_{BH}}{P_{BH,n}(n)} \Big)^{-1} $$

$$ P_p(N_A.p)=\Big(\frac{f_{CW}}{P_{CW,p}(N_A)}+\frac{f_{BH}}{P_{BH,p}( p)} \Big)^{-1} $$

Weights fCW and fBH are coefficients of the model.

The same factor P is used in the calculation of the function F, which describes the mobility ratio between stationary, infinite-mass secondary scatters (e.g. ionized impurities) and mobile, finite-mass secondary scatters (e.g. free carriers). Both functions F and G are parameterized fitting functions to physical processes, and the coefficients of those functions (r1 to r6 for function F and s1 to s7 for function G) are also coefficients of the model.

- HIGH-FIELD MOBILITY: As the electric field within the semiconductor increases, the drift-velocity of the carriers is commonly observed to saturate, reducing the mobility accordingly. To account for this effect, CHARGE includes high-field mobility models that describe the monotonic (silicon-like) or overshoot (GaAs-like) velocity saturation behaviour.
- Monotonic model:

$$ \mu ^{LICE} _{n,p} = \frac{\mu ^{LIC} _{n,p}}{\Big(1 +\big( \frac{\mu ^{LIC} _{n,p} F _{n,p}}{\nu^{sat} _{n,p}}\big) ^{\beta}\Big)^{1/\beta}} $$

where μLIC is the mobility accounting for lattice, impurity, and carrier-carrier scattering (as calculated using the active models for those processes) and vsat is the model coefficient that determines the saturation velocity. F is the driving field, which may be defined as the magnitude of the quasi-Fermi level gradient or the component of the electric field in the direction of the current density. - Overshoot model:

$$ \mu ^{LICE} _{n,p} = \frac{\mu ^{LIC} _{n,p} + \frac{\nu ^{sat} _{n,p}}{F _{n,p}} (\frac{F _{n,p}}{F_0} ) ^{\beta}}{1+(\frac{F _{n,p}}{F_0})^{\beta}} $$

where again F is the driving field, F0 is the critical field, and vsat is the saturation velocity.This model is typically applied to the electrons in GaAs.

- Monotonic model:

## Recombination / Generation

Recombination describes the processes by which an electron from the conduction band makes an energetic transition and neutralizes a hole in the valence band. Generation describes the complementary behavior, where an electron is excited from the valence band to the conduction band, creating a hole in the process (often, the term electron-hole-pair is used when referring to generation). The models for bulk recombination and generation processes relate to the physical mechanisms by which the carriers make the energetic transition. CHARGE provides models describing

- trap-assisted (Shockley-Read-Hall) recombination,
- Auger recombination,
- radiative recombination,
- impact ionization,
- band to band tunneling, and
- stimulated recombination

These models and their parameterizations are the subject of the following sections.

### Trap-Assisted (Shockley-Read-Hall) Recombination

The recombination process in the trap-assisted model assumes that there are unoccupied "trap" states (also referred to deep-level defect states) within the band gap. Typically, these states result from impurities (either intentional or unintentional), and the most active have energy levels near the middle of the band gap. Recombination occurs when an electron relaxes (transfers energy to the lattice or emits a photon) to the trap state from the conduction band, and sequentially, a hole from the valence band relaxes to the same trap state. This process is modeled using the Shockley-Read-Hall (SRH) equation,

$$R_{SRH} = \frac{np-n_i^2}{\tau _p(n+n_1) + \tau_n(p+p_1)} $$

where τn and τp are the electron and hole lifetimes, respectively, and n1 and p1 are the effective densities of carriers in the trap states. The trap states are characterized by their densities Nt, capture cross-section σt, and energy level Et - Ei (commonly abbreviated as Et and referenced to the intrinsic energy level). The constants n1 and p1 are calculated as

$$ n_1 = n_i \text{exp}(E_t/k_bT)$$

$$ p_1 = p_i \text{exp}(-E_t/k_bT)$$

The carrier lifetime can be determined from the capture cross-section and trap density as

$$ \tau_{n,p} = \Big(\sigma_{n,p} N_t \sqrt{\frac{3k_B T}{m^* _{n,p}}} \Big)^{-1} $$

but is commonly taken as an input to the model.

CHARGE provides a temperature dependent model for the SRH carrier lifetime, as well as models that include corrections for doping density and field effects. The general form of the carrier lifetime can be expressed as

$$\tau _{n,p} (T,N,F) = \frac{\tau _{n,p} ^0 (T)f(N)}{1+g(T,F)} $$

where f is a function of the total dopant concentration N and g is a function of the magnitude of the applied field F. The basic temperature-dependent model for the carrier lifetime follows the usual power-law relation

$$\tau ^{srh,0} _{n,p} (T) = \tau ^{srh,0} _{n,p} (300) (\frac{T}{300}) ^{\eta _{n,p}} $$

Alternately, a constant value can be supplied for both electrons and holes.

To account for doping concentration effects, CHARGE provides two correction models that use the previous expression for the SRH carrier lifetime as an input. First, a modified model in the form proposed by Fossum is described by

$$\tau ^{srh,0} _{n,p} (T) = \frac{ \tau ^{srh,0} _{n,p}}{\alpha _{n,p}+\beta _{n,p} N _{n,p}+ \gamma _{n,p} N _{n,p} ^{\sigma _{n,p}}} \text{, where } N _{n,p} = \frac{N_A +N_D}{ N _{n,p} ^{ref}}$$

The original model of Fossum can be obtained by setting coefficients α, β, and σ to one (1) and setting γ to zero (0), and this is the default model used in CHARGE.

Alternately, a formulation proposed by Klaassen can be selected, where the SRH carrier lifetime correction is given by the equation

$$\tau ^{srh,0} _{n,p} (T) = \frac{ \tau ^{srh,0} _{n,p}}{1 + \alpha _{n,p} \tau ^{srh,0} _{n,p} N _{n,p} } \Big(\frac{T}{300}\Big) ^{\eta _{n,p}} \text{, where }N _{n,p} = \frac{N_A +N_D}{ N _{n,p} ^{ref}}$$

Note: that this model explicitly includes the temperature dependence, and should only be used in concert with a constant value for the baseline SRH carrier lifetime. |

To account for field effects, either the Hurkx [5] or Schenk [6] model may be selected. These models represent the corrections for trap-assisted tunneling, where carriers can transition to a deep-level trap state by tunneling through an electrostatic barrier.

**Hurkx Model **

The correction term g(F) for the Hurkx model (referred to as Γ in the reference) is

$$ g_{v}=\Delta \widetilde{E} _{v} \int _{0}^{1} \exp \big( \Delta \widetilde{E} _{v}u - K_{v} u^{3 / 2} d u \big) $$

where ν is the carrier index (n or p),

$$ \Delta \tilde{E}_{v}=\Delta E_{v} / k T $$

$$\Delta E_{n}=\left\{\begin{array}{cc}{0} & {E_{F n}>E_{C}} \\ {E_{C}-E_{F n}} & {E_{T}<E_{F n} \leq E_{C}} \\ {E_{C}-E_{T}} & {E_{F n} \leq E_{T}}\end{array}\right.$$

(and a similar expression for holes), and

$$K_{v}=\frac{4}{3} \frac{\sqrt{2 m^{*} \Delta E_{v}^{3}}}{q \hbar F}$$

**Schenk Model**

The correction term g(F) for the Schenk model is

$$ \begin{aligned} g_{v}=&\left(1+\frac{(\hbar \Theta)^{3 / 2} \sqrt{E_{t}-E_{0}}}{E_{0}^{\hbar} \hbar_{0}}\right)^{-1 / 2} \frac{(\hbar \Theta)^{3 / 4}\left(E_{t}-E_{0}\right)^{1 / 4}}{2 \sqrt{E_{0} E_{t}}}\left(\frac{\hbar \Theta}{k T}\right)^{3 / 2} \\ & \times \exp \left[-\frac{E_{t}-E_{0}}{\hbar \omega_{0}}+\frac{\hbar \omega_{0}-k T}{2 \hbar \omega_{0}}+\frac{E_{t}+k T / 2}{\hbar \omega_{0}} \ln \left(\frac{E_{t}}{\varepsilon_{R}}\right)-\frac{E_{0}}{\hbar \omega_{0}} \ln \left(\frac{E_{0}}{\varepsilon_{R}}\right)\right] \\ & \times \exp \left(\frac{E_{t}-E_{0}}{k T}\right) \exp \left[-\frac{4}{3}\left(\frac{E_{t}-E_{0}}{\hbar \Theta}\right)^{3 / 2}\right] \end{aligned} $$

where the electro-optic frequency is \( \Theta=\left(q^{2} F^{2} / 2 \hbar m_{t}\right)^{1 / 3} \) with effective tunneling mass \( m_t \), and the transition energy is

$$ E_{0}=2 \sqrt{\varepsilon_{F}}\left[\sqrt{\varepsilon_{F}+E_{t}+\varepsilon_{R}}-\sqrt{\varepsilon_{F}}\right]-\varepsilon_{R} $$

with

$$\varepsilon_{F}=\frac{\left(2 \varepsilon_{R} k T\right)^{2}}{(\hbar \Theta)^{3}}$$ and $$\varepsilon_{R}=S \hbar \omega_{0}$$

which is the product of the optical phonon energy ħω0 and the Huang-Rhys (coupling) factor S. The trap energy Et is referenced to the valence band edge, and is computed from the mid-gap offset specified for the trap-assisted tunneling model.

### Auger Recombination

Auger transitions are three-particle transitions (two carriers scatter and transfer energy and/or momentum to a third carrier) that describe four related processes, which are illustrated in the figures below. Each process has an associated rate coefficient. According to the principle of detailed balance, the net rate for each type of carrier must be zero at equilibrium, such that

$$C_{c n}^{A U} n_{i}^{2}=C_{e n}^{A U} \text { and } C_{c p}^{A U} n_{i}^{2}=C_{e p}^{A U}$$

Assuming that the value of the rate coefficients does not change as the system moves from equilibrium, the net Auger recombination rate is

$$R_{A U}=\left(C_{c n}^{A U} n+C_{c p}^{A U} p\right)\left(n p-n_{i}^{2}\right)$$

Note: that Auger transitions depend only on carrier density, differentiating them from other recombination processes. |

CHARGE supports three models for the capture rate coefficients, including

- the universal temperature model proposed by Klaassen,
- an empirical model by White accounting for a reduction in the recombination rate at high carrier concentrations, and
- a model by Clugston and Basore accounting for both high and low injection conditions.

The universal temperature model proposed by Klaassen takes the usual power-law form,

$$ C_{n, p}^{0}=C_{n, p}(300)\left(\frac{T}{300}\right)^{\eta_{n, p}} $$

and is suitable for devices where Auger recombination is moderate (low injection conditions). The Auger rate coefficients are only weakly dependent on temperature, and constant values may be used as well.

An alternate empirical model proposed by White can be used as a correction to the previous model, taking that coefficient as an input. The White model accounts for the reduction in the Auger recombination rate observed at high carrier densities (due to strong screening effects), and is expressed as

$$ C_{n}=\frac{C_{n}^{0}}{1+\alpha n}, \quad C_{p}=\frac{C_{p}^{0}}{1+\alpha p} $$

where the coefficient α determines the transition density.

A related model proposed by Clugston and Basore is designed to account for the two regimes related to minority carrier injection:

$$ \begin{array}{l}{C_{n}=C_{n}^{0}\left(\frac{N_{D}}{N_{D}+p}\right)+\frac{C_{n}^{H I}}{2}\left(\frac{p}{N_{D}+p}\right)} \\ {C_{p}=C_{p}^{0}\left(\frac{N_{A}}{N_{A}+n}\right)+\frac{C_{p}^{H I}}{2}\left(\frac{n}{N_{A}+n}\right)}\end{array} $$

CHARGE will use the Auger capture rate coefficient defined in the Klaassen model (or a constant value) for the low injection conditions, and apply a second coefficient when strong minority carrier injection dominates according to the preceding formulations.

### Radiative Recombination

In a radiative transition, a conduction band electron will relax directly, emitting a photon whose energy approximately equals that of the band gap, and then recombine with a hole in the valence band. The opposite process occurs when a photon is absorbed by an electron in the valence band, promoting it to the conduction band and leaving a hole in its place. Radiative recombination transitions are typically significant only in materials with a narrow bandgap, or a bandstructure that permits direct transitions in momentum (e.g. GaAs). Radiative recombination is typically negligible in bulk silicon.

The recombination rate is determined from the product of a capture rate coefficient and the carrier density product,

$$R_{OPT}=C_{c} ^{OPT} n p$$

and the corresponding generation rate is simply the emission rate constant,

$$G_{OPT}=C_{e} ^{OPT}$$

Once again, the coefficients are related by the principle of detailed balance at thermal equilibrium, such that

$$ R_{OPT}=C_{c} ^{OPT} ( np-n_{i}^{2} ) $$

The optical capture rate coefficient can be modeled in CHARGE either as a constant or using the universal temperature power-law,

$$C_{c}^{O P T}(T)=C_{c}^{O P T}(300)\left(\frac{T}{300}\right)^{\eta}$$

### Impact Ionization

Impact ionization is a carrier generation process where an electron or hole, accelerated by a high field, will relax by transferring energy to the lattice. When energy exceeding the band gap is transferred to the lattice, an electron-hole pair is excited (and separated by the strong local field), generating additional free carriers. Above a critical threshold, this process leads to avalanche breakdown.

Impact ionization described by the Selberherr model [7]:

$$R_{I I}=-\alpha_{n} \frac{J_{n}}{q}-\alpha_{p} \frac{J_{p}}{q}$$

where Jn,p is the magnitude of the current density for the indicated carrier, and

$$\alpha_{v}=\alpha_{v}^{\infty} \exp \left[-\left(\frac{E_{v}^{c r i t }}{F_{v}}\right)^{\beta_{\nu}}\right]$$

The impact ionization process is exponentially dependent on the driving field F (either the quasi Fermi level gradient or electric field component in the direction of the current density) and moderated by the critical field Ecrit.

Note: Impact ionization process is exponentially dependent on the electric field and the local variations in the quasi-Fermi levels (through the current density). Consequently, it is a highly non-linear process, and it's inclusion in the physical model for the semiconductor can cause divergences in the simulation. By default, the impact ionization process is not enabled. When simulating avalanche breakdown, ensure that adequate steps are taken to ensure simulation convergence, including reducing step size, increasing iteration limits, and enabling gradient mixing. |

### Band to Band Tunneling

The band-to-band tunneling processes are a local approximations to a non-local (tunneling) process. The models are derived from the band-bending physics at high-fields: when there exists a sufficient gradient, the band edges present a narrow tunneling barrier to the carriers. Two models for this process are included in CHARGE: the Hurkx [5] and Schenk [8] formulations.

**Hurkx Model**

The Hurkx model for band to band tunneling takes the form

$$ R_{b b t}=-B F^{\sigma} D \exp \left(-\frac{F_{0}}{F}\right) $$

where B is a scaling parameter, F the magnitude of the electric field, F0 the critical field,

$$\sigma=\left\{\begin{array}{ll}{2} & {\text { direct }} \\ {5 / 2} & {\text { indirect }}\end{array}\right.$$

and

$$ D=\frac{n p-n_{i}^{2}}{\left(n+n_{i}\right)\left(p+n_{i}\right)} $$

**Schenk Model**

The Schenk model is similar in form to the Hurkx model, but with explicit reference to phonon-assisted tunneling.

$$ R_{b b t}=A F^{\sigma} D\left[\frac{\left(F_{c}^{\mp}\right)^{-3 / 2} \exp \left(-\frac{F_{c}^{\mp}}{F}\right)}{\exp \left(\frac{\hbar \omega_{T A}}{k_B T}\right)-1}+\frac{\left(F_{c}^{ \pm}\right)^{-3 / 2} \exp \left(-\frac{F_{c}^{ \pm}}{F}\right)}{1-\exp \left(-\frac{\hbar \omega_{T A}}{k_B T}\right)}\right] $$

where F the magnitude of the electric field, A is a scaling factor, σ = 7/2 for indirect transitions,

$$ D=\frac{n p-n_{i}^{2}}{\left(n+n_{i}\right)\left(p+n_{i}\right)} $$

and the critical field

$$ F_{c}^{ \pm}=B\left(E_{g} \pm \hbar \omega_{T A}\right)^{3 / 2} $$

with the acoustic phonon energy ħωTA.

### Stimulated recombination

Stimulated recombination model can include the effect of the optical stimulated emission in lasers, even though CHARGE solver does not include a laser simulator. This effect is a combination of material gain and laser cavity properties and can be calculated using a laser simulator, such as TWLM in INTERCONNECT. Material properties can be defined per domain, so the stimulated recombination property can be turned on only for those domains inside a cavity.

The stimulated recombination model is based on the following equation,

$$R_{stim}=\left\{\begin{array}{ll}{c\left(\frac{I_{recomb}}{qV}-\frac{I^{th}_{recomb}}{qV}\right)} & {\textrm{if}\;\;I_{recomb} \gt I^{th}_{recomb},} \\ {0} & {\textrm{if}\;\;I_{recomb} \leq I^{th}_{recomb},}\end{array}\right.$$

where \(c\) is a unitless coefficient in the range 0-1 modeling how much of the total recombination current above threshold undergoes stimulated recombination, \(I_{recomb}\) is the total recombination current, \(I^{th}_{recomb}\) is the total recombination current at threshold, and \(V\) is the volume of the gain region, which should correspond to the volume of the domain where \(R_{stim}\) model is turned on. This equation can be rewritten as

$$R_{stim}=\left\{\begin{array}{ll}{c\left(\frac{J_{recomb}}{qt}-\frac{J^{th}_{recomb}}{qt}\right)} & {\textrm{if}\;\;J_{recomb} \gt J^{th}_{recomb},} \\ {0} & {\textrm{if}\;\;J_{recomb} \leq J^{th}_{recomb},}\end{array}\right.$$

where \(J_{recomb}\) is the total recombination current density (current per unit area), \(J^{th}_{recomb}\) is the total recombination current density at threshold, and \(t\) is the thickness of the gain region, which should correspond to the thickness of the domain where \(R_{stim}\) model is turned on. This latter form of the equation is directly related to how are input parameters defined in the material editor.

The parameters for this model can be found, for example, by running the corresponding laser simulation using MQW and TWLM solvers. The LI curve generated by TWLM can be used to find the total recombination current at lasing threshold and to extract the coefficient c, which is related to the slope.

### References

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**35**, 953 (1992); Klaassen, D. B. M., Solid State Electronics,**35**, 961 (1992) - Hurkx, G. et al., IEEE Trans. Electron Devices,
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**35**, 1585 (1992) - Selberherr, S. Analysis and Simulation of Semiconductor Devices, Springer Verlag, Vienna (1984)
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