Mobility Models
In The Semiconductor Equations section, a closed-form expression was computed for the electron and hole mobilities by making the relaxation-time approximation and by assuming a particular form for the scattering time as a function of energy. Any mechanism that disturbs the perfect periodicity of the lattice can produce scattering of the carriers. Such mechanisms include:
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Phonons (L): Thermally generated acoustic waves traveling through the crystal (scattering by phonons is frequently referred to as lattice scattering, which is somewhat confusing as the lattice itself never scatters the carriers unless perturbed from perfect periodicity).
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Ionized impurities (I): These begin to have an effect at doping levels above approximately 1015 cm-3 at room temperature.
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Carrier-carrier scattering (C).
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Neutral impurity scattering (N): This is important at low temperatures only, typically below 77K.
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High field velocity saturation (E).
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Surface scattering (S): This includes effects such as interface charges, scattering by surface phonons, and so on. It is important in field effect devices, such as field effect transistors.
In the Semiconductor interface, the letters in brackets in the list above are used to label the effects included in a specific mobility model (the convention used in Ref. 6 is adopted).
In practice the mobility is typically computed using analytic functions with or without a rigorous physical motivation and designed to fit the experimental data to a good degree of accuracy. Often, mobility models are designed to address one particular effect (for example high field velocity saturation) and require other mobility models as a basis (for example a model incorporating scattering due to phonons and impurities). As an example, high field effects are often incorporated into a model by defining the high field mobility as a function of the mobility due to phonon and impurity scattering. In other cases the scattering due to different mechanisms is combined using Matthiessen’s rule — an approximate method for combining mobilities that, while not rigorously correct, is frequently employed to produce empirical fits to experimental data in the context of mobility models. Matthiessen’s rule combines mobilities in the following manner:
An example of the manner in which mobility models are often combined is:
(3-75)
where μE, the mobility including high field effects, is a function of μs (the surface mobility), which in turn is a function of the mobility model for phonon (or lattice) and impurity scattering (μLI).
COMSOL Multiphysics uses a general mechanism to combine both user-defined and predefined mobility models that accommodate combinations of the form given in Equation 3-75. Mobility models are added as subnodes to the Semiconductor Material Model node. If the mobility model requires an input mobility, this is selected from the available mobilities appropriate for this model (individual selections are required for both the electron and hole mobility inputs). Multiple user-defined mobility models can be added and these can be used as inputs to predefined or other user-defined mobility models. The user-defined mobility models are always available as inputs in predefined mobility models (if the model requires an input). Within this system it is possible to mix user-defined mobility models with models predefined by COMSOL Multiphysics in a very general manner. The model used within the simulation is selected for electrons and holes by changing the electron mobility and the hole mobility settings in the Mobility Model section of the Semiconductor Material Model node, which by default uses a constant mobility obtained from the material properties.
It is important to understand that each type of mobility model subnode only defines mobility variables for electrons and holes that can be used by other models, or by the Semiconductor Material Model node. The mobility models for electrons and holes actually used in the simulation are determined by the selections or settings in the Semiconductor Material Model node (the parent node) which do not change when additional mobility models are added. In order to add a mobility model to a simulation it is therefore necessary to both add the sequence of mobility model subnodes to the model tree and then to select the desired final mobility for both electrons and holes in the Semiconductor Material Model node.
The following sections describe the predefined mobility models currently available in the Semiconductor interface.
Theory for the Power Law Mobility Model (L)
The Power Law Mobility Model (L) is a simple mobility model and requires no input as it includes scattering due to phonons. The electron (μn,pl) and hole (μp,pl) mobilities are determined by the equations:
where T is the lattice temperature and μp,pl, μp,pl, αn, αp, and Tref are material properties. For silicon the values of the material properties are taken from Ref. 16.
Theory for the Arora Mobility Model (LI)
The empirical Arora Mobility Model (LI) includes both phonon and impurity scattering. The electron (μn,ar) and hole (μp,ar) mobilities are determined by the equations:
where T is the lattice temperature, Na− is the ionized acceptor concentration, and Nd+ is the ionized donor concentration. All the other parameters are material properties. For silicon the values of the material properties are taken from Ref. 16.
Theory for the Fletcher Mobility Model (C)
The Fletcher Mobility Model (C) adds carrier-carrier scattering to an existing mobility model (or to a constant input mobility). It accepts input mobilities of type L or LI, as well as user-defined input mobilities.
The model uses Matthiessen’s rule to combine the input mobility with a carrier-carrier scattering mobility term that is identical for electrons and holes. The model is based on Ref. 17. The electron (μn,ar) and hole (μp,ar) mobilities are determined by the equations:
where T is the lattice temperature, μin,n and μin,p are the electron and hole input mobilities, n is the electron concentration, and p is the hole concentration. F1 (SI unit: s2A/(m3kg)), F2 (SI unit: 1/m2), and Tref are material properties. For silicon the values of the material properties are taken from Ref. 18.
Theory for the Lombardi Surface Mobility Model (S)
The Lombardi Surface Mobility Model (S) adds surface scattering resulting from surface acoustic phonons and from surface roughness. Mobility contributions corresponding to these effects are combined with the input mobility using Matthiessen’s rule. The model accepts input mobilities of type L, LI, or C as well as user defined input mobilities. The model is based on Ref. 19. The electron (μn,lo) and hole (μp,lo) mobilities are determined by the following equations:
where T is the lattice temperature, μin,n and μin,p are the electron and hole input mobilities, Na- is the ionized acceptor concentration, Nd+ is the ionized donor concentration, is the component of the electric field perpendicular to the electron current and is the component of the electric field perpendicular to the hole current. All other parameters in the model are material properties (note that δn and δp have units of V/s). The material properties for silicon are also obtained from this reference Ref. 19.
Theory for the Caughey-Thomas Mobility Model (E)
The Caughey-Thomas Mobility Model (E) adds high field velocity scattering to an existing mobility model (or to a constant input mobility). It accepts input mobilities of type L, LI, C, or S as well as user defined input mobilities. The model is based on Ref. 20. The electron (μn,lo) and hole (μp,lo) mobilities are determined by the following equations:
where T is the lattice temperature, μin,n and μin,p are the electron and hole input mobilities and Fn and Fp are the driving forces for electrons and holes (currently Fn=E||,n and Fp=E||,p where E||,n is the component of the electric field parallel to the electron current and E||,p is the component of the electric field parallel to the electron current). All other parameters in the model are material properties (note that vn,0 and vp,0 are the saturation velocities for electrons and holes and have units of m/s). The material properties for silicon are also obtained from Ref. 20.
Theory for the User-defined Mobility Model
The User-Defined Mobility Model can be used to create electron and hole mobilities with user-defined expressions for the electron and hole mobilities. These mobility models can be combined with other user-defined or predefined mobility models in arbitrary combinations. These mobility models can take other defined mobilities as inputs. By default the output mobility is set to the input mobility for both electrons and holes. However, it is possible to change the expression for the output mobility so that it is any function of the input mobility (or indeed a function that does not depend on the input mobility). The default value is set in this way to make it straightforward to access the variable in which the input mobility is stored.
In the finite volume method, the dependent variables are constant within each mesh element and the gradients cannot be defined using the differentiation operator in COMSOL. A detailed understanding of the finite volume method is required in order to set up mobility models that depend on the gradients of the dependent variables in COMSOL (∇n, ∇p, and ∇V). These variables include the currents jn and jp, the electric field E, and the electric displacement field D, as well as the gradients of the quasi-Fermi levels ∇Efn and ∇Efp. In the finite element method these limitations do not apply and the gradients of the dependent variables (and quantities which depend on them) can be used in expressions.
Defining the Carrier Mobility