Semiconductor Material Model Properties

Reference：Semiconductor Material Model Properties – Ansys Optics

Electronic

DC PERMITTIVITY（直流介电常数）

DC PERMITTIVITY（直流介电常数）: 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.
$$n=\sqrt{{\epsilon }_r}$$

WORK FUNCTION（功函数）

WORK FUNCTION（功函数）: 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.
$${\chi }_s={\varphi }_{s,i}-\frac{E_G}{2}+\frac{k_{B}T}{2}\ln \frac{N_C}{N_V}$$
where EG is the band gap and NC and NV are the effective density of states in the conduction band and valence band, respectively.

${\chi }_s$：电子亲和能，即将一个电子从导带边缘移除所花费的能量；

${\varphi }_{s,i}$：半导体的功函数，即从半导体的本征能级（即未掺杂半导体的费米能级）移除一个电子并将其放置在“无穷远”处所需的能量；

$E_G$：半导体的能带间隙，即导带和价带之间的能量差；

$k_B$：玻尔兹曼常数，用于将温度从开尔文转换为能量单位；

$T$：绝对温度；

$N_C$：导带中的有效态密度；

$N_V$：价带中的有效态密度

EC VALLEY

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.

半导体的导带可以有多个能谷。

默认情况下，使用半导体在材料数据库默认材料列表中最低的能谷。

对于每个能谷，可以指定不同的半导体属性。默认情况下只使用最低能谷的属性。

如果想选择其它能谷，用户可以通过选择L、X或Gamma能谷来改变这一点。

• L、X、Gamma 能谷是什么？

Fundamental

EFFECTIVE MASS（有效质量）

EFFECTIVE MASS（有效质量）: To account for the influence of the crystal lattice potential（晶格势能） 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, MQW, 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

可以使用有效质量来引入晶格势能对电子或空穴的影响，有效质量取决于材料的能带结构。

在CHARGE、MQW和HEAT模型中，有效质量被视为材料模型的一个参数。有效质量随温度的变化可以通过二次模型来表示，下面的公式即为该二次模型。

$$m_{n,p}^{\ast }(T)=+m_{n,p}^{\ast }(0)\alpha T+\beta T^2$$

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

系数 $\alpha$、$\beta$ 和 $T=0K$ 时的有效质量为模型的输入参数

$m^{\ast }$ 表示有效质量，$n,p$ 分布代表电子和空穴

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

导带和价带中的有效态密度跟有效质量相关，公式如下。其中的 $h$ 是普朗克常数。

$$\begin{gathered} N_C=2{\left(\frac{2\pi m_n^{\ast }{k}_{B}T}{h^2}\right)}^{\frac{3}{2}} \\ {N}_V=2{\left(\frac{2\pi m_p^{\ast }{k}_{B}T}{h^2}\right)}^{\frac{3}{2}} \end{gathered}$$

BAND GAP（带隙）

BAND GAP（带隙）: 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, MQW, 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

材料的一个关键物理属性是能带间隙，它像有效质量一样，是从材料的电子能带结构中得出的。在CHARGE、MQW和HEAT模型中，能带间隙能量被视为材料模型的一个参数。能带间隙的温度变化可以通过一个“通用”的经验模型来表示。

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

系数 $\alpha$、$\beta$ 和 $T=0K$ 时的带隙能量为模型的输入参数

BAND GAP NARROWING（禁带收缩）

BAND GAP NARROWING（禁带收缩）: 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 in Silicon and other semiconductor materials,

当给本征（纯）半导体掺杂时，可能会在带隙内的局部区域引入新的能级。当掺杂浓度加大时，这些在带隙内的离散能级将开始合并，并在带隙内形成一个薄薄的允带，即缩小了带隙的宽度。这可以被视为能带间隙的缩小或有效态密度的增加。

Slotboom模型¹用于描述能带间隙的缩小，在CHARGE和HEAT中提供，以考虑硅和其他半导体材料中的这种效应：

$$\Delta E_G=-V_1\left[\ln \left(\frac{N_D^{+}+N_A^{-}}{N_0}\right)+\sqrt{{\ln }^2\left(\frac{N_D^{+}+N_A^{-}}{N_0}+C\right)}\right]$$

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.

其中系数 $V_1$、$N_0$ 和 $C$ 是模型的输入参数，并且可以分别为电子和空穴指定这种效应。符号为正意味着带隙收缩。

$N_D^{+}$：表示施主载流子（donor）掺杂浓度，施主掺杂剂是能够提供额外电子的杂质，它们在半导体中引入额外的导电电子，通常用于n型半导体。

$N_A^{-}$：表示受主载流子（acceptor）掺杂浓度，受主掺杂剂是能够捕获电子的杂质，它们在半导体中产生额外的空穴，通常用于p型半导体。

INTRINSIC CARRIER CONCENTRATION（本征载流子浓度）

INTRINSIC CARRIER CONCENTRATION（本征载流子浓度）: 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

本征载流子浓度是根据有效质量和能带间隙计算得出的，仅在材料数据库中显示以供参考。其计算公式为

$$n_i=\sqrt{N_C{N}_V}exp(-\frac{E_G}{2k_{B}T})$$

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

其中假设温度T=300K，在禁带收缩前，有效态密度（$N_C,N_V$）和带隙（$E_G$）被视为固有的数值（这几个数值已经在前面的材料属性中得到过了）。

Mobility（迁移率）

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.

漂移-扩散方程中的迁移率参数是载流子（电子和空穴）运动与半导体材料之间的物理联系。迁移率可以被视为衡量电子和空穴在半导体的晶体晶格中移动的容易程度。在没有任何与晶格、杂质或其他载流子的相互作用的情况下，电子和空穴将在晶格的周期性势场中自由移动；改变载流子动量的相互作用被称为散射事件。不同类型的散射对电子和空穴的迁移率都有贡献，包括晶格散射、电离/中性杂质散射和载流子-载流子散射。

此外，观察到在高电场下载流子的速度会达到饱和。这些散射机制中的每一个都可以通过在CHARGE中应用适当的模型来解决，这些模型在以下各节中详细说明。

LATTICE SCATTERING（晶格散射）

LATTICE SCATTERING（晶格散射）: 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,

晶格散射：阻碍载流子在晶格中自由运动的基本过程是与晶格本身的热散射。由于晶格散射导致的迁移率在CHARGE的半导体模型中被视为一个基本输入，可以输入为一个常数或者用一个“通用”的温度模型来描述晶格散射随温度的变化。

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

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 [2, Ch. 3.4.1] ²

以上这个“通用”温度模型里，A表示变量A在 T=300K 时的数值，用一个与温度相关的指数来近似表示温度对变量A的影响，$\eta$ 是其需要输入的指数。

$${\mu }_{n,p}^L(T)={\mu }_{n,p}^L(300){\left(\frac{T}{300}\right)}^{\eta }$$

where subscripts n and p refer to electrons and holes, respectively.

${\mu }^L$ 表示电子迁移率，下标n和p分别指的是电子和空穴

IMPURITY AND FREE-CARRIER SCATTERING（杂质和自由载流子散射）

IMPURITY AND FREE-CARRIER SCATTERING（杂质和自由载流子散射）: 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, Ch. 3.4.2.1] ²³, the Masetti model [4] ⁴, and the Klaassen model [5] ⁵. 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.

有许多模型可以解释杂质对载流子迁移率的影响。CHARGE支持三种常见的模型，这些模型具有广泛的适用性：Caughey-Thomas模型[2, Ch. 3.4.2.1] ²³、Masetti模型[4] ⁴和Klaassen模型[5] ⁵。每个模型都需要一系列的系数；对于常见的半导体材料，CHARGE提供了默认值。

Caughey-Thomas model

Caughey-Thomas model: This is the most basic model.
$${\mu }_{n,p}^{LI}={\mu }_{n,p}^{min}+\frac{{\mu }_{n,p}^L-{\mu }_{n,p}^{min}}{1+(N/N_{ref}{)}^{\alpha }}$$
where N is the total doping concentration (N = NA + ND), μL is the lattice scattering mobility (as described by the model chosen in the previous section), and μmin, Nref, and α are temperature-dependent coefficients also described by the universal temperature model, with A(T) replaced by the coefficients listed above. The overall form of this equation follows the form described in [2, Ch. 3.4.2.1] ², which is a temperature-dependent adaptation of the basic Caughey-Thomas model [3] ³.

Caughey-Thomas模型是最基本的模型。

其中N是总掺杂浓度（$N=NA+ND$），${\mu }^L$是晶格散射迁移率（如前一节中所选模型所述），${\mu }^{min}$、$N_{ref}$ 和 $\alpha$ 是温度依赖系数，也由通用温度模型描述，其中 $A(T)$ 被上述系数替换。这个方程的整体形式遵循[2, Ch. 3.4.2.1] ²中描述的形式，这是基本Caughey-Thomas模型[3] ³的温度依赖性改编。

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

掺杂浓度非常大时，请选择此模型，它为大 $N$ 添加了对Caughey-Thomas模型的修正。

同样，N是总掺杂浓度（$N=NA+ND$），${\mu }^L$是晶格散射迁移率。参数 ${\mu }^{min}$、${\mu }^{(2)}$、$C_r$（替换Caughey-Thomas模型中的 $N_{ref}$）、$C_s$、$\alpha$ 和 $\beta$ 都是每个温度依赖系数，由通用温度模型描述。

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

此模型考虑了在中、高掺杂浓度下的杂质和自由载流子散射，以及载流子-载流子散射的影响。Klaassen模型使用Matthisen规则将基本晶格散射与杂质和载流子-载流子散射结合起来。

其中 ${\mu }^L$ 是晶格散射迁移率，${\mu }^{IC}$ 是Klaassen的杂质和载流子-载流子（IC）散射迁移率。IC散射迁移率的公式是复杂的，涉及多个级别的系数和模型，以解释：

• 多数载流子与掺杂剂的散射
• 少数载流子与掺杂剂的散射
• 电子-空穴散射

杂质和载流子-载流子（IC）散射迁移率

• 其它参数是什么含义？

To begin, the IC mobility is defined as a function of the dopant and carrier concentrations,
$${\mu }_{\nu ,\pi }^{IX}(N_{\Delta },\nu \nu )=\frac{({\mu }_{\nu ,\pi }^{\Lambda }{)}^2}{{\mu }_{\nu ,\pi }^{\Lambda }-{\mu }_{\nu ,\pi }^{\mu \nu }}\left(\frac{N_{\nu ,\pi }^{\sigma \chi }}{N_{\nu ,\pi }^{\sigma \chi ,\epsilon \varphi \varphi }}\right){\left(\frac{N_{\rho ,\epsilon \varphi }}{N_{\nu ,\pi }^{\sigma \chi }}\right)}^{\alpha }+\frac{{\mu }_{\nu ,\pi }^{\Lambda }{\mu }_{\nu ,\pi }^{\mu \nu }}{{\mu }_{\nu ,\pi }^{\Lambda }-{\mu }_{\nu ,\pi }^{\mu \nu }}\left(\frac{\nu +\pi }{N_{\nu ,\pi }^{\sigma \chi ,\epsilon \varphi \varphi }}\right)$$
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.

杂质和载流子-载流子（IC）散射迁移率被定义为掺杂和载流子浓度的函数。

其中 ${\mu }^L$ 是晶格散射迁移率，系数 ${\mu }^{min}$、$N_{ref1}$（等同于 $N_{ref}$ 或 $C_r$），以及 $\alpha$ 的定义与Caughey-Thomas或Masetti模型中相同。请注意，Klaassen模型单独考虑了温度依赖性，因此应使用晶格散射迁移率的恒定值。

In the preceding equation, the “scattering densities” are

在前面的方程中，“散射密度”是

$$\begin{array}{r}{N}_n^{SC}=N_D+N_A+p\\ N_p^{SC}=N_D+N_A+n\end{array}$$

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

其中施主和受主载流子浓度 $N_D$ 和 $N_A$ 分别根据 clustering 函数进行校正：

$$\begin{array}{r}{N}_D=N_D^{+}+\frac{N_D^{+}}{C_D+(N_{ref,D}/N_D^{+}{)}^2}\\ N_A=N_D^{+}+\frac{N_A^{+}}{C_A+(N_{ref,A}/N_A^{-}{)}^2}\end{array}$$

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

这里，$C_D$、$N_{ref}$、$D$、$C_A$ 和 $N_{ref}$、$A$ 是模型的系数。

The “effective scattering densities” are defined as (using the same clustering-corrected acceptor and donor concentrations)
$$\begin{array}{r}{N}_n^{sc,eff}=N_D+G(P_n)N_A+\frac{p}{F(P_n)}\\ N_p^{sc,eff}=N_A+G(P_p)N_D+\frac{n}{F(P_p)}\end{array}$$
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).

“有效散射密度”定义为 $N^{sc,eff}$（其模型中使用的是用 clustering 函数校正过的受主和施主载流子浓度），其中函数 $G$ 描述了散射截面上排斥性库仑势和吸引性库仑势之间的比率。

函数 $G$ 是因子 $P$ 的函数（ $P$ 是载流子密度和多数掺杂密度的函数）。

**注：**屏蔽库仑势（Screened Coulomb Potential）是一种考虑了介质或环境对库仑势的屏蔽效应的势能。在实际应用中，库仑势通常会因为介质的存在而被部分屏蔽，从而导致库仑势的强度减弱。屏蔽库仑势可以分为吸引性和排斥性两种类型。吸引性库仑势是由于原子核与电子之间的库仑吸引作用产生的，排斥性库仑势是由于带相同电荷的粒子之间的库仑斥力产生的。在分子动力学中，库仑势被认为是长程相互作用的一部分，而排斥力则是短程相互作用的一部分。

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,
$$\begin{array}{r}{P}_n(N_{D\cdot }n)={\left(\frac{f_{CW}}{P_{CW,n}(N_D)}+\frac{f_{BH}}{P_{BH,n}(n)}\right)}^{-1}\\ P_p(N_{A\cdot }p)={\left(\frac{f_{CW}}{P_{CW,p}(N_A)}+\frac{f_{BH}}{P_{BH,p}(p)}\right)}^{-1}\end{array}$$
Weights fCW and fBH are coefficients of the model.

$P$ 考虑了屏蔽效应，并且是自由载流子和杂质离子屏蔽效应的加权调和平均值，权重 $f_{CW}$ 和 $f_{BH}$ 是模型的系数。

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.

同样的，$P$ 还用于计算函数 $F$，函数 $F$ 描述了静止的、无限质量的次级散射体（例如掺杂离子）和移动的、有限质量的次级散射体（例如自由载流子）之间的迁移率比率。函数 $F$ 和 $G$ 都是物理过程的参数化拟合函数，这些函数的系数（函数 $F$ 的r1到r6和函数 $G $的s1到s7）也是模型的系数。

HIGH-FIELD MOBILITY（强电场迁移率）

• HIGH-FIELD MOBILITY（强电场迁移率）: 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.

强电场迁移率：在强电场作用下，半导体材料中电子或空穴的输运特性。当电场强度超过某一阈值时，电子或空穴的迁移率会受到限制，达到一个饱和值，这种现象称为高场效应。为了考虑这种效应，CHARGE包含了强电场迁移率模型，这些模型描述了单调的（硅类）或过冲的（砷化镓类）速度饱和行为。

• Monotonic model:

$${\mu }_{n,p}^{LICE}=\frac{{\mu }_{n,p}^{LIC}}{{\left(1+{\left(\frac{{\mu }_{n,p}^{LIC}{F}_{n,p}}{v_{n,p}^{sat}}\right)}^{\beta }\right)}^{1/\beta }}$$

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.

其中 ${\mu }^{LIC}$ 是考虑了晶格(Lattice)、杂质(Impurity)和载流子-载流子(Carrier)散射的迁移率，$v^{sat}$ 是决定饱和速度的模型系数。$F$ 是驱动场，可以定义为准费米能级梯度的大小或电流密度方向上的电场分量。

• Overshoot model:

$${\mu }_{n,p}^{LICE}=\frac{{\mu }_{n,p}^{LIC}+\frac{v_{n,p}^{sat}}{F_{n,p}}{\left(\frac{F_{n,p}}{F_0}\right)}^{\beta }}{1+{\left(\frac{F_{n,p}}{F_0}\right)}^{\beta }}$$

where again F is the driving field (unit is V/cm), F0 is the critical field, and vsat is the saturation velocity.This model is typically applied to the electrons in GaAs.

其中 $F$ 是驱动场（单位是V/cm）与 Monotonic model 的 $F$ 一致，$F_0$ 是临界场，$v^{sat}$ 是饱和速度。这个模型通常应用于砷化镓材料中的电子。

Recombination / Generation（复合 / 激发）

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.

复合描述了一个来自导带的电子通过能量转换并与价带中的空穴中和的过程。产生则描述了相反的行为，即电子从价带激发到导带，在此过程中产生一个空穴（通常在提到产生时，会使用电子-空穴对这一术语）。体复合和产生过程的模型与载流子进行能量转换的物理机制有关。CHARGE提供了描述以下过程的模型：

• trap-assisted（Shockley-Read-Hall）复合
• Auger复合
• 辐射复合
• 影响电离
• 带间隧穿
• 受激发复合

这些模型及其参数化是以下各节的主题。

Trap-Assisted (Shockley-Read-Hall) Recombination（SRH复合）

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,

在Shockley-Read-Hall模型中，复合过程假设在能带间隙内存在未占据的“陷阱”态（也称为深能级缺陷态）。通常，这些状态是由杂质（无论是有意还是无意的）引起的且其最活跃的能级在带隙的中间附近。当一个电子从导带释放能量（将能量传递给晶格或发射光子）到陷阱态，并且相应的，一个空穴从价带放松到同一个陷阱态时，就会发生复合。这个过程是使用Shockley-Read-Hall（SRH）方程来建模的。

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

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
$$\begin{array}{rl}{n}_1& =n_{i}exp(E_t/k_{b}T)\\ p_1& =p_{i}exp(-E_t/k_{b}T)\end{array}$$
The carrier lifetime can be determined from the capture cross-section and trap density as
$${\tau }_{n,p}={\left({\sigma }_{n,p}{N}_t\sqrt{(}\frac{3k_{B}T}{m_{n,p}^{\ast }})\right)}^{-1}$$
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
$${\tau }_{n,p}^{srh,0}(T)={\tau }_{n,p}^{srh,0}(300)(\frac{T}{300}{)}^{{\eta }_{n,p}}$$
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
$${\tau }_{n,p}^{srh,0}(T)=\frac{{\tau }_{n,p}^{srh,0}}{{\alpha }_{n,p}+{\beta }_{n,p}{N}_{n,p}+{\gamma }_{n,p}{N}_{n,p}^{{\sigma }_{n,p}}},where{N}_{n,p}=\frac{N_A+N_D}{N_{n,p}^{ref}}$$
The original model of Fossum can be obtained by setting coefficients α, β, and σ to one (1) and setting γ to zero (0).

Alternately, a formulation proposed by Klaassen can be selected, where the SRH carrier lifetime correction is given by the equation
$${\tau }_{n,p}^{srh,0}(T)=\frac{{\tau }_{n,p}^{srh,0}}{1+{\alpha }_{n,p}{\tau }_{n,p}^{srh,0}{N}_{n,p}}{\left(\frac{T}{300}\right)}^{{\eta }_{n,p}},where{N}_{n,p}=\frac{N_A+N_D}{N_{n,p}^{ref}}$$
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.

……（未完待续）

KdotP

k · p model is used in MQW to calculate the electronic band structure of quantum wells. By default, the model is enabled only for semiconductor materials supported by MQW. See buildmqwmaterial for a list of supported materials.

k·p模型在多量子阱（MQW）中用于计算量子阱的电子能带结构。默认情况下，该模型仅对MQW支持的半导体材料启用。支持的材料列表可以在 buildmqwmaterial 中找到。

Crystal Structure（晶体结构）

• 闪锌矿结构（zincblende）
• 纤锌矿结构（wurtzite）

Lattice Constant（晶格常数）

• lc a (m): in-plane lattice constant. A cubic lattice is assumed for zincblende, and hexagonal is assumed for wurtzite materials.

lc a (m): 平面内晶格常数。对于闪锌矿结构，假设为立方晶格；对于纤锌矿结构，假设为六角晶格。

• lc c (m): hexagonal lattice constant along the c-axis of a wurtzite crystal

lc c (m): 纤锌矿晶体沿c轴的六角晶格常数。

• c轴？晶格的基本理论？

The temperature variation in the lattice constant can be accounted for with a quadratic model

晶格常数随温度的变化可以通过二次模型来表示：
$$lc(T)=lc(300)+\alpha (T-300)+\beta (T-300{)}^2$$
where coefficients α and β, and the lattice constant at T=0K, are inputs to the model.

其中系数 $\alpha$ 和 $\beta$，以及 T=0K 时的晶格常数是模型的输入参数。

Lattice constant enables calculating biaxial strain from the known difference in lattice constants between the base material and the QW material. The base material must be defined in the simulation involving MQW solver, or strain can be defined as an input by the user, in which case lc is not used. More details on how the strain is calculated from lattice constants can be found in section Strain in the MQW solver reference manual.

晶格常数可以用来计算基底材料和量子阱材料之间已知的晶格常数差异所产生的双轴应变。在涉及MQW求解器的模拟中，必须定义基底材料，或者用户可以输入应变，此时不使用lc。关于如何从晶格常数计算应变的更多细节，可以在 MQW求解器参考手册 的应变部分找到。

Stiffness Tensor（刚度张量）

• c11 (N/m2): Elastic stiffness coefficient.
• c12 (N/m2): Elastic stiffness coefficient.
• c44 (N/m2): Elastic stiffness coefficient.
• c13 (N/m2): Elastic stiffness coefficient, only relevant to wurtzite materials.
• c33 (N/m2): Elastic stiffness coefficient, only relevant to wurtzite materials.

弹性刚度系数，仅与纤锌矿材料相关

Coefficients of the elastic stiffness tensor are used for the purpose of calculating all required material strain tensor components from the input biaxial strain tensor components. More details can be found in section Strain in the MQW solver reference manual.

弹性刚度张量的系数用于从输入的双轴应变张量分量计算所有所需的材料应变张量分量。更多信息可以在 MQW求解器参考手册 的应变部分找到。

Deformation Potentials（形变势能）

• ac (eV): zincblende conduction band deformation potential.（闪锌矿导带形变势）
• av (eV): zincblende valence band deformation potential.（闪锌矿价带形变势）
• b (eV): zincblende valence band deformation potential.（闪锌矿价带形变势）
• a1, a2 (eV): wurtzite conduction band deformation potential.（纤锌矿导带形变势）
• D1, D2, D3, D4, D5, D6 (eV): wurtzite valence band deformation potential.（纤锌矿价带形变势）

Deformation potentials determine how the strain affects the conduction and valence band structure. For more details see the definition of conduction and valence band Hamiltonian in the MQW solver introduction.

形变势决定了应变如何影响导带和价带结构。更多细节可见 MQW求解器介绍 中的导带和价带哈密顿量的的定义。

Effective mass parameters

• GAMMA1, GAMMA2, GAMMA3: zincblende Luttinger parameter for the valence band k · p Hamiltonian.（闪锌矿价带k·p哈密顿量的Luttinger参数）
• A1, A2, A3, A4, A5, A6: wurtzite effective mass parameter for the valence band k · p Hamiltonian.（纤锌矿价带k·p哈密顿量的有效质量参数）
• me in-plane: wurtzite effective mass for conduction band, parallel to the quantum well plane.（纤锌矿导带有效质量，平行于量子阱平面）
• me out-of-plane: wurtzite effective mass for conduction band, along the quantum well growth direction, in other words, the c-axis of hexagonal lattice.（纤锌矿导带有效质量，沿量子阱生长方向，换句话说，即六角晶格的c轴）

For more details see the definition of the valence band Hamiltonian in the MQW solver introduction.

更多细节可见 MQW求解器介绍 中的导带和价带哈密顿量的的定义。

Band splitting parameters

• delta spin-orbit (eV): band splitting due to spin-orbit coupling.（由自旋轨道耦合引起的能带分裂）
• delta crystal (eV): crystal field splitting, only relevant to wurtzite materials.（晶体场分裂，仅与纤锌矿材料相关）

Energy Parameter for the Optical Matrix Element

• EP (eV): Energy parameter for the optical matrix element.（光学矩阵元素的能量参数）

Energy parameter is used for the purpose of calculating the optical transition matrix element needed to calculate gain and spontaneous emission. This parameter is usually given in tables for different semiconductors.

能量参数用于计算光学跃迁矩阵元素，这是计算增益和自发发射所必需的。这个参数通常在不同半导体的表格中给出。

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1. Slotboom, J.W., Solid-State Electron., 20, 279 (1979) ↩︎ ↩︎

2. Palankovski, V., Simulation of Heterojunction Bipolar Transistors [Doctoral dissertation, University of Vienna] (2000) https://www.iue.tuwien.ac.at/phd/palankovski/diss.html ↩︎ ↩︎ ↩︎ ↩︎ ↩︎

3. Caughey, D. M. and Thomas, R. E., Proc. IEEE, 52, 2192 (1967) ↩︎ ↩︎ ↩︎ ↩︎

4. Masetti, G., et al., IEEE Trans. Electron Devices, ED-30, 764 (1983) ↩︎ ↩︎

5. Klaassen, D. B. M., Solid State Electronics, 35, 953 (1992); Klaassen, D. B. M., Solid State Electronics, 35, 961 (1992) ↩︎ ↩︎