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Sentaurus Device
11. Special Focus: Device Electrical Breakdown Simulation

11.1 Overview
11.2 Comparison of Breakdown Methods
11.3 Approximate Breakdown Analysis
11.4 Ionization Integrals With Carriers Analysis
11.5 External Resistor Method
11.6 Voltage-to-Current Boundary Condition Switching Method
11.7 Continuation Method
11.8 Transient Method
11.9 Tracking Additional Data in the Current File
11.10 Driving Force for Avalanche Generation and High-Field Saturation Models

Objectives

To demonstrate how to perform device electrical breakdown simulations in Sentaurus Device for a 2D NMOSFET device, operating in the off-state regime.

11.1 Overview

In general, device breakdown simulations and off-state device breakdown simulations, in particular, are often difficult perform. In the prebreakdown regime, a current is small and almost constant or, at least, relatively flat. In this case, a voltage-controlled bias sweep works best. In the postbreakdown regime, however, even a small voltage change can lead to a huge current increase, so that sometimes even a snapback can occur due to the dominance of the avalanche generation term in that regime.

Different breakdown methods are available in Sentaurus Device, each having certain advantages for a particular device operation regime.

11.2 Comparison of Breakdown Methods

Table 1 lists the methods suitable for device electrical breakdown simulations and briefly compares their advantages and disadvantages in terms of simulator performance, accuracy, and applicability.

Table 1. Comparison of different breakdown methods.
Method	Advantages	Limitations
Approximate breakdown analysis	Fast; it does not consume much memory since it requires only the Poisson analysis. Qualitatively produces reasonable results.	Can be applied only to off-state breakdown problematics, where currents are still negligibly small. Quantitatively not very precise; usually underestimates a breakdown.
Ionization integrals with carriers analysis	Similar to approximate breakdown analysis in terms of ionization integrals computation. This method solves the entire carrier transport task, having the avalanche term deactivated in the right-hand side of the continuity equation, but computed in a postprocessing step.	Requires educated guess from user to decide when to stop a simulation.
External resistor method	Simulation is performed in one effort with the bias applied through an external resistor connected to a device electrode in series.	Requires user intervention or iterations to establish a load resistor. Simulation might not converge if the device goes into snapback.
Voltage-to-current boundary condition (BC) switching method	Flexible method for most complicated curve shapes, including curves with negative differential resistance behavior.	Sensitive to a chosen time step at the voltage-to-current BC switching point.
Continuation method	Automatically defines the external load resistance by monitoring the curve slope.	Sensitive to grid quality and linear solver settings. Does not like noisy currents. Can be more computationally expensive than other methods.
Transient method	Similar to the external resistor method, but uses the transient bias ramp instead of quasistationary. Useful in cases where traps are considered or a simulated structure includes floating semiconductor regions.	Same as the external resistor method. Since a bias ramp time can potentially affect the results, the transient method should be used with care.

The complete project that demonstrates these breakdown methods and focuses on a MOSFET off-state breakdown simulation can be investigated from within Sentaurus Workbench in the directory Applications_Library/GettingStarted/sdevice/BVmethods.

The BVMethods project parameter value decides which method to use for a breakdown simulation. After project preprocessing in Sentaurus Workbench, the corresponding command inputs are generated.

To visualize characteristic curves, select the Inspect nodes of interest, and click the Visualize Selected Nodes Together toolbar button (see Section 6.5 Visualize Selected Nodes Together).

Click to view the master file BV_des.cmd.

11.3 Approximate Breakdown Analysis

Approximate breakdown analysis (ABA) is an inexpensive alternative to the breakdown simulations performed with the drift-diffusion or hydrodynamic transport models. Its main advantages are:

Is fast and less memory and CPU expensive, since it requires only the Poisson equation to be solved.
Predicts breakdown trends in the case of doping and geometry variations.
Has no problem with floating semiconductor regions.
Tolerates coarse grids.
Can be used in combination with the drift-diffusion or hydrodynamic carrier transport models.

On the other hand, its applicability is limited by the following factors:

This method considers constant quasi-Fermi potentials. Therefore, it can be applied only to (quasi-)equilibrium conditions assuming no current flow. This limits ABA to off-state and low-current condition device simulations only.
This method usually underestimates a breakdown because phenomena such as carrier recombination, tunneling, surface scattering, and high-field carrier velocity saturation are not accounted for.

11.3.1 Math Section

Math {
  Extrapolate
  Iterations=20
  Notdamped =100
  RelErrControl

  ComputeIonizationIntegrals
  BreakAtIonIntegral(3 1.)
}

To activate ABA, you must specify ComputeIonizationIntegrals to switch on the computation of the ionization integrals for ionization paths crossing the local field maxima in the semiconductor. The keyword BreakAtIonIntegral terminates the quasistationary simulation when a specified number of integrals (3, in example) is greater than the specified value (1, in example).

Having an integral value higher than unit indicates that the avalanche condition is reached (at least one extra electron–hole pair is generated).

11.3.2 Physics Section

Physics {
  Recombination(
    Avalanche(ElectricField)
  )

  Fermi
}

To enable carrier generation under ABA, the Avalanche model must be switched on within the Recombination statement, with the ElectricField specification for the driving force.

11.3.3 Plot Section

Plot {
  eIonIntegral hIonIntegral MeanIonIntegral eAlphaAvalanche hAlphaAvalanche
}

To plot the electron- or hole-associated and mean ionization integrals, the keywords eIonIntegral, hIonIntegral, and MeanIonIntegral are specified in the Plot section. To plot the integral quantities, they are saved automatically in the .plt file under the names PhiElectron and PhiHole.

11.3.4 Solve Section

Solve {
  *- Build-up of initial solution:
  Coupled(Iterations=100){ Poisson }
  Coupled { Poisson Electron Hole }
  
  Quasistationary(
    InitialStep=@<1e-2/100>@ Increment=1.41 
    MinStep=@<1e-5/100>@ MaxStep=0.025
    Goal{ Name="drain" Voltage=_Vdd_ }
  ) { Coupled { _EQUATIONSET_ } }
}

To calculate the ionization integrals in ABA, it is sufficient to solve only the Poisson equation along the drain bias sweep. Therefore, the _EQUATIONSET_ macro in the project tool input for this method is defined as:

#define _EQUATIONSET_   Poisson

Figure 1 shows the results of the ionization integrals computation as a function of the applied bias. The arrow indicates the breakdown bias condition, where either of the ionization integrals exceeds the unit value.

Figure 1. MOSFET off-state breakdown simulations performed with ABA. Electron and hole ionization integrals are shown as a function of the drain bias for V_gs = 0. The breakdown is estimated as a point where either of the ionization integrals exceeds the unit value as indicated by the arrow. (Click image for full-size view.)

11.4 Ionization Integrals With Carriers Analysis

Ionization integrals with carriers (IIC) analysis follows the concept of ABA by computing the ionization integrals along a bias sweep, but quantitatively it is more precise than ABA because it solves the entire system of carrier transport equations and computes the ionization integrals in a postprocessing mode based on a given driving force specification for the avalanche model.

At the same time, the corresponding avalanche generation rates are excluded from the carrier continuity equations, even if the model is activated in the Physics section, and the decision when to stop a simulation is defined by either the given Goal value (see Section 11.4.4 Solve Section) or the BreakAtIonIntegral criterion (see Section 11.4.3 Math Section).

Such an approach can be applied to a large number of electrical breakdown simulation tasks, including 3D device simulations.

The same Electrode section description is used for IIC analysis as for ABA.

11.4.1 Physics Section

Physics {
  EffectiveIntrinsicDensity( OldSlotboom )     
  Mobility(
    DopingDep
    HighFieldsaturation( GradQuasiFermi )
    Enormal
  )
  Recombination(
    SRH( DopingDep )
    Band2Band(E2)
    Avalanche( Eparallel )
  )

  Fermi
}

The typical physical model description is applied with IIC analysis. Even if the avalanche model is switched on, it will be ignored by activating the AvalPostProcessing keyword in the Math section (see Section 11.4.3 Math Section). However, for the avalanche coefficient computation, the driving force, as specified as a parameter of the Avalanche model, will be used (Eparallel is used in this example).

11.4.2 CurrentPlot Section

CurrentPlot {
  AvalancheGeneration (Integrate(Semiconductor))
}

This section provides instructions to integrate the computed avalanche generation rate over the entire silicon domain and to save the computed quantity into the .plt file for each bias ramp step. This can be used further to extract the breakdown voltage (see Section 11.4.5 Breakdown Voltage Extraction).

11.4.3 Math Section

Math {
  ComputeIonizationIntegrals
  BreakAtIonIntegral(3 1.)

  AvalPostProcessing
}

In addition to ComputeIonizationIntegrals, which activates the impact ionization computation in a postprocessing mode, the keyword AvalPostProcessing instructs Sentaurus Device to exclude the avalanche generation term from the self-consistently solved carrier transport equations.

However, the carrier generation due to the carrier impact ionization is still computed in a postprocessing step and, together with ionization integrals, can be plotted in Sentaurus Visual for further analysis.

11.4.4 Solve Section

Solve {
  *- Build-up of initial solution:
  Coupled(Iterations=100){ Poisson }
  Coupled { Poisson Electron Hole }
  
  Quasistationary(
    InitialStep=@<1e-2/100>@ Increment=1.41 
    MinStep=@<1e-5/100>@ MaxStep=0.025
    Goal{ Name="drain" Voltage=_Vdd_ }
  ) { Coupled { _EQUATIONSET_ } }
}

To compute the ionization integrals with IIC analysis, the entire set of carrier transport equations must be solved. Therefore, the _EQUATIONSET_ macro in the project tool input for this method is defined as:

#define _EQUATIONSET_   Poisson Electron Hole

The target drain bias is given by the _Vdd_ macro definition:

#define _Vdd_ 100.

whose value should be higher than the actual breakdown voltage. The decision as to when to stop a simulation is based on the BreakAtIonIntegral criterion specified in the Math section.

11.4.5 Breakdown Voltage Extraction

The breakdown voltage is estimated by looking at the avalanche current component, which is computed from the integral value of the avalanche generation rate, plotted according to the statement provided in the CurrentPlot section (see Section 11.4.2 CurrentPlot Section). For each bias condition, the avalanche generation current is computed. The point where its value exceeds the value of the conduction current is recognized as a breakdown point (see Figure 2).

Figure 2. Currents computed with IIC analysis. The blue curve shows the conduction current, directly computed by Sentaurus Device. The red curve indicates the portion of the current due to the avalanche generation only, computed in a postprocessing step. The breakdown is estimated as a point where these two currents meet. (Click image for full-size view.)

11.5 External Resistor Method

A simple but effective way to switch from a prebreakdown voltage-controlled sweep to a postbreakdown current-controlled sweep is to attach an appropriate resistor to an electrode. The resistor value should be large enough such that, in the postbreakdown regime, most of the applied voltage drops across the resistor. The combination of the resistor + voltage source then acts effectively as a current source.

A simple way to establish the value of the resistor is to make a test run without an external resistor. In general, the simulation fails to converge near the onset of breakdown or shortly after. Divide the applied voltage by the obtained electrode current and use a resistor value of the order of V/I.

For most breakdown simulation tasks, the external resistor method provides the most reliable solution and can be recommended, despite its tendency to be the slowest of all the methods discussed in this section.

11.5.1 Electrode Section

Electrode {
  { Name="drain"     Voltage=0.0 Resistor= 1e7  }

  { Name="source"    Voltage=0.0 }
  { Name="gate"      Voltage=0.0 Barrier= -0.55 }
  { Name="substrate" Voltage=0.0 }
}

In the Electrode section, the series resistor is attached to the drain electrode. The value of 10⁷ Ω is determined by the test run as previously mentioned.

11.5.2 Math Section

Math {
  Extrapolate
  Iterations=20
  Notdamped =100
  RelErrControl

  AvalDerivatives
  ErrRef(Electron)=1.e10
  ErrRef(Hole)=1.e10
  BreakCriteria{ Current(Contact="drain" AbsVal=_Idmax_) } 
}

The keyword AvalDerivatives improves robustness for breakdown simulations. It includes derivatives with respect to the avalanche generation rates into the Jacobian matrix. Do not use this option if the impact ionization model is not activated since, in that case, the option does not help convergence, but requires a larger memory use and a longer runtime.

The simulation stops when a specified drain current level is reached. In this example, the BreakCriteria statement terminates the bias sweep when the drain current exceeds a value of _Idmax_ = 1.443x10^-3 A/μm. _Idmax_ is defined as a local Sentaurus Workbench variable.

11.5.3 Solve Section

Solve {
  *- Build-up of initial solution:
  Coupled(Iterations=100){ Poisson }
  Coupled { Poisson Electron Hole }
  
  Quasistationary(
    InitialStep=@<1e-2/100000>@ Increment=1.41 
    MinStep=@<1e-5/100000>@ MaxStep=0.025
    Goal{ Name="drain" Voltage=_Vdd_ }
  ) { Coupled { _EQUATIONSET_ } }
}

The drain is ramped in the Goal statement up to a specified value, which is much higher than the expected breakdown voltage because, in the postbreakdown regime, most of the voltage drops across the resistor.

When plotting the resulting I–V curve from the external resistor method, use the InnerVoltage value as an argument to obtain the pure voltage drop on a simulated device under investigation.

11.6 Voltage-to-Current Boundary Condition Switching Method

As an alternative to the external resistor method to simulate a device electrical breakdown, you can use the voltage-to-current method, which applies the boundary condition (BC) switch from the voltage BC (Dirichlet) to the current BC (Neumann).

When specifying the switching point, you must ensure that beyond this point the current increases monotonically as shown in Figure 3. Otherwise, the method fails. Note, however, that the voltage can decrease again (snapback).

Figure 3. For monotonic curve (red), the voltage-to-current BC switching method can be applied to any bias point. For nonmonotonic curve (black), showing negative differential resistance behavior, the voltage-to-current BC switching method can be applied only to the indicated bias, where the current starts to grow monotonically. (Click image for full-size view.)

11.6.1 Electrode Section

Electrode {
  { Name="drain"     Voltage=0.0 }

  { Name="source"    Voltage=0.0 }
  { Name="gate"      Voltage=0.0 Barrier= -0.55 }
  { Name="substrate" Voltage=0.0 }
}

The voltage-to-current BC switching method does not require a specific resistor to be attached to an electrode of interest. If not specified, the default resistor value of 1 mΩ is used.

11.6.2 Solve Section

Solve {
  *- Build-up of initial solution:
  Coupled(Iterations=100){ Poisson }
  Coupled { Poisson Electron Hole }
  
  Quasistationary(
    InitialStep=@<1e-2/5.0>@ Increment=1.41 
    MinStep=@<1e-5/5.0>@ MaxStep=0.025
    Goal{ Name="drain" Voltage=6. }
  ) { Coupled { _EQUATIONSET_ } }
  
  Set ("drain" mode current)
  QuasiStationary ( 
    InitialStep=1e-4 Maxstep=0.05 MinStep=1e-9 Increment=1.41 
    Goal { name=drain current=_Idmax_ }
  ) { Coupled { _EQUATIONSET_ } }
}

For the MOSFET off-state breakdown regime with the first Quasistationary statement, the drain electrode is first biased up to 6 V. Next, the Set command instructs the simulator to switch from the finally achieved voltage BC to the current BC at the drain electrode.

The Set("drain" mode current) command is used here for demonstration purposes only and can be omitted.

In the next Quasistationary statement, the drain current is ramped up to a value (_Idmax_ = 1.443x10^-3 A/μm) specified inside the Goal statement.

Figure 4. MOSFET off-state breakdown I–V curve as a result of the voltage-to-current BC switching method. Inset: Magnification around the switching point (6 V) shows the adaptive step refinement as specified in the consequent Quasistationary statement. (Click image for full-size view.)

Sometimes, it is not known in advance where to specify a switching point. In this case, it is recommended to ramp up a device bias to as high a value as possible and switch to the current BC only after the voltage BC no longer converges. For example, when a device starts snapping back (that is, inner dV becomes negative).

11.7 Continuation Method

The continuation method enables automated tracing of arbitrarily shaped curves such as breakdown or latch-up. Its implementation is based on a dynamic load-line technique adapting the boundary conditions along the traced curve to ensure convergence.

Because the curve derivative computation is involved in the continuation method, sometimes it can show less stable convergence behavior, compared to the external resistor method or the voltage-to-current BC switching method.

Without user intervention, an external load resistor is connected to a device electrode at which the curve is traced and, therefore, the device is indirectly biased through the load resistance. By monitoring the slope of a traced curve, an optimal BC is determined by adjusting the load line so that it is orthogonal to the local tangent of the curve. The continuation method automatically decides on a BC without prior knowledge of the curve characteristics.

11.7.1 Solve Section

Solve {
  *- Build-up of initial solution:
  Coupled(Iterations=100){ Poisson }
  Coupled { Poisson Electron Hole }
  
  Continuation(
    Name="drain"                     * Curve is traced at this electrode
    InitialVStep=0.1                 * Initial voltage step
    Increment=1.41 Decrement=2
    MaxVstep=0.5                     * limits voltage step along curve trace
    MinVoltage=0 MaxVoltage=100      * Define the I-V window where
    MinCurrent=0 MaxCurrent=_Idmax_  * IV curve is traced
    Iadapt=1e-8                      * up to this current a Quasistationary
                                     * with voltage ramping is used instead
                                     * of continuation
  ) { Coupled { _EQUATIONSET_ } }
}

The continuation method traces the drain I–V curve automatically according to the specified I–V curve window. MinVoltage, MaxVoltage, MinCurrent, and MaxCurrent define the window where the curve is traced. The value of MaxVstep limits the maximum voltage step along the bias sweep, thereby allowing the curve to look smoother.

The Iadapt keyword indicates the threshold current value that must be reached to initiate the continuation method. Below this value, a regular voltage bias ramp is applied.

Figure 5. I–V curve obtained with the continuation method. The arrow indicates the position where the continuation method starts. (Click image for full-size view.)

11.8 Transient Method

In some situations, for example, with a floating semiconductor region present inside a simulated structure or when the dynamic trap occupation by electrons or holes is considered, it is required that a device simulation is performed in transient.

The following sections show how to perform a transient device breakdown simulation for a 2D NMOSFET device.

11.8.1 Electrode Section

In certain cases, having a device bias ramped in the time domain helps to achieve faster and more reliable results for a device breakdown simulation. To do this, time-dependent boundary conditions are used for a specific electrode (drain electrode in this case), with a given list of voltage–time pairs. The initial boundary condition in this case is still defined by a separate "static" voltage whose value is used to compute the initial guess solution.

Electrode {
  { Name="drain" Voltage=0.0 Voltage=(0. at 0., _Vdd_ at _tend_) Resistor= 1e7 }

  { Name="source"    Voltage=0.0 }
  { Name="gate"      Voltage=0.0 Barrier= -0.55 }
  { Name="substrate" Voltage=0.0 }
}

Here, the first voltage–time pair defines the initial condition from where the transient starts. The second pair, _Vdd_ at _tend_, defines the final time and the goal bias to be reached. Between these, as many voltage–time pairs as needed can be specified, thereby allowing a complex piecewise bias transient ramp to be simulated.

11.8.2 Math Section

Simulations in transient require discretization of the transient equations. Two transient discretization schemes are supported in Sentaurus Device: simple backward Euler (BE) method and composite trapezoidal rule/backward differentiation formula (TRBDF) method.

The composite TRBDF method, a second-order approximation, is used by default in Sentaurus Device. On the other hand, in most cases, the BE method, despite its lower accuracy, is suggested as the method of choice to solve most standard simulation tasks in transient, due to its better performance compared to the composite TRBDF method.

To activate the BE method, use:

Math {
...
  Transient= BE
}

The same break criterion, as used in the external resistor method, is used here to terminate the simulation when the given current level is reached.

For all of these methods, linear solution extrapolation is used, which helps to speed up the convergence on a single Newton iteration. However, this is not always the case for transient simulations. The decision to apply or not apply the solution extrapolation can be made individually for each particular case, by switching on or off the Extrapolate keyword in the Math section of the command file.

11.8.3 Solve Section

Solve {
  *- Build-up of initial solution:
  Coupled(Iterations=100){ Poisson }
  Coupled { Poisson Electron Hole }
  
  Transient (
    InitialTime= 0. FinalTime= _tend_
    InitialStep= @<_tend_*0.1/_Vdd_>@ Increment= 1.41
    Minstep= @<_tend_*1e-5/_Vdd_>@  MaxStep= @<_tend_/100.>@ 
  ) { Coupled { _EQUATIONSET_ } }
}

Remembering that a large external lumped resistor is attached to the drain electrode, the _Vdd_ bias is defined as:

#define _Vdd_ 100000.

The _tend_ value defines how fast the drain bias is ramped in time and is defined in seconds:

#define _tend_ 1.

The Transient keyword indicates that the simulation should be performed in transient. All time-related parameter values are given in seconds. Computations performed for InitialStep, MinStep, and MaxStep values reflect the fact that a large lumped resistor is attached to the drain electrode, and a large bias is applied to the drain electrode accordingly.

When plotting the resulting I–V curve from the transient method, use the drain InnerVoltage value as an argument to obtain the pure voltage drop on a simulated device under investigation.

The additional displacement current component J_D=ε₀·δE/δt is computed in transient, which contributes to the integral current value. Therefore, with the faster ramp times, you might expect different results obtained in transient, compared to a steady-state solution. In general, you should double-check whether a certain ramp time value satisfies the task requirement.

However, in this particular example, the breakdown value is not affected by the ramp speed, as shown in Figure 6.

Figure 6. Resulting I(t) (solid lines) and V(t) (dots) curves obtained with different bias ramp times (from left to right, 1 s, 10 s, 100 s, and 1e5 s). (Click image for full-size view.)

Figure 7 shows the Applications_Library/GettingStarted/sdevice/BVmethods project, with the extracted breakdown voltage values obtained with different solution methods.

Figure 7. Sentaurus Workbench project showing the extracted breakdown voltage values for each breakdown method. (Click image for full-size view.)

Table 2 compares the CPU times for the MOSFET off-state breakdown simulation performed using the different breakdown methods.

Table 2. CPU times for different breakdown methods (average data is shown).
Method	ABA	IIC	External resistor	Voltage-to-current	Continuation	Transient
CPU time [s]	19	37	121	170	120	88

11.9 Tracking Additional Data in the Current File

As mentioned in Section 11.4.2 CurrentPlot Section, the top-level CurrentPlot statement provides a way to write additional scalar data to the current file. Data depending on coordinates can be written.

When writing coordinate-dependent data, the values are interpolated as required. For example, the maxima of the electric field and the impact ionization, and their respective coordinates, can be written to the .plt file with:

CurrentPlot {
  ImpactIonization(Maximum(Semiconductor Coordinates))
  ElectricField   (Maximum(Semiconductor Coordinates))
}

This capability is used in the electrical breakdown simulation in Section 11.8 Transient Method. Figure 8 shows the maximum of the electric field and the maximum of the impact ionization, depending on time.

Figure 8. Maximum of the electric field (left axis) and the impact ionization (right axis) versus time. (Click image for full-size view.)

Furthermore, it is possible to output averages, integrals, maximum, and minimum of quantities over specified domains. To do this, specify the keyword Average, Integrate, Maximum, or Minimum, respectively, followed by the specification of the domain in parentheses.

A domain can be a region, a material, region and material interfaces, a window (given by coordinates), a doping well, or any of Semiconductor, Insulator, or Everywhere. For example, avalanche generation can be integrated over the silicon material with:

CurrentPlot {
  AvalancheGeneration (Integrate(Semiconductor))
}

This capability is used in the electrical breakdown simulation example with the approximate breakdown analysis provided in Section 11.4 Ionization Integrals With Carriers Analysis. Figure 9 plots the impact ionization rate, integrated over the silicon region, versus drain voltage.

Figure 9. Integrated impact ionization rate versus drain voltage. (Click image for full-size view.)

11.10 Driving Force for Avalanche Generation and High-Field Saturation Models

Since the process of avalanche generation depends exponentially on the driving force that causes it, Sentaurus Device provides many options for choosing the appropriate driving force model for avalanche generation and also for high-field saturation (HFS) models. The driving force for avalanche generation affects breakdown behavior directly; whereas, the driving force choice for HFS models affects the behavior after breakdown, since the current density in power devices is limited by velocity saturation.

11.10.1 Driving Force for Avalanche Generation

For avalanche generation, Sentaurus Device offers the following driving forces:

GradQuasiFermi is the default and uses the magnitude of the gradient of eQuasiFermi or hQuasiFermi as the driving force.
EParallel uses the electric field in the direction of the electron or hole current as the driving force.
ElectricField is used only for approximate breakdown analysis (see Section 11.3 Approximate Breakdown Analysis).

The driving force for the avalanche model is chosen as following:

Physics {
  Recombination(
    Avalanche( <drivingforce> )
  )
}

When the carrier density is small, the direction of the current vector cannot be well established numerically, leading to large fluctuations in the driving force field. This can result in sudden changes in avalanche generation and, therefore, leads to convergence problems.

The possibility to interpolate a given avalanche driving force to the electric field at low carrier concentrations, similar to what is described in Section 6.3.2 Nonlinear Solver Controls, is available in Sentaurus Device. It can sometimes improve the simulator convergence behavior.

For the GradQuasiFermi driving force, the following syntax should be used to interpolate it to the ElectricField within regions of small carrier densities:

Math {
  RefDens_eGradQuasiFermi_ElectricField = <refdens>
  RefDens_hGradQuasiFermi_ElectricField = <refdens>
}

where <refdens> is the electron or hole concentration below which this interpolation becomes active.

Note that these keywords will affect the computation of driving forces for both high-field saturation mobility and avalanche generation.

Alternatively, if the EParallel driving force is used as a driving force, it can be also interpolated to the ElectricField using the following syntax:

Math {
  RefDens_eEParallel_ElectricField_Aval = <refdens>
  RefDens_hEParallel_ElectricField_Aval = <refdens>
}

Using the above specification, the driving force interpolation is applied to the avalanche term only.

11.10.2 Driving Force for Velocity High-Field Saturation Models

For velocity high-field saturation (HFS), Sentaurus Device offers the following driving forces:

GradQuasiFermi is the default and uses the magnitude of the gradient of eQuasiFermi or hQuasiFermi as the driving force.
EParallel uses the electric field in the direction of the electron or hole current as the driving force.
ElectricField uses the magnitude of the electric field as the driving force.
EParallelToInterface uses the electric field component parallel to the nearest interface as the driving force.
SqrtEfieldProdGQF uses the square root of the electric field multiplied by the gradient of the quasi-Fermi level as the driving force.

The driving force is chosen in the Physics section:

Physics {
  Mobility(
    HighFieldSaturation( <drivingforce> )
  )
}

As with avalanche generation, EParallel is the most physically accurate model, but it suffers from the same stability problems. ElectricField overestimates the driving force in the channel of a MOS device, since the perpendicular electric field is also taken into account. For MOS channels, EParallelToInterface is a better model, because the current flows primarily parallel to the metal–insulator interface. However, at rounded trench corners where breakdown can occur, this model underestimates the driving force, because there the current flow becomes primarily perpendicular to the interface. As such, GradQuasiFermi offers the best balance between accuracy and robustness in terms of the choice of driving force model.

Note that the choice of the driving force for velocity HFS will have little or no impact on the breakdown voltage (the point where breakdown starts), but it can significantly affect snapdown behavior. Therefore, a careful driving force choice is needed only when snapdown behavior is of interest.

11.10.3 Driving Force Interpolation for Velocity High-Field Saturation Models

Even though GradQuasiFermi offers the best choice for the driving force for HFS models, due to its mathematical nature, it can show instabilities in areas of low-carrier concentration. To overcome these instabilities, Sentaurus Device can be instructed to interpolate from GradQuasiFermi to a different driving force model.

The first option is to interpolate from GradQuasiFermi to ElectricField:

Math {
  RefDens_eGradQuasiFermi_ElectricField_HFS = <refdens>
  RefDens_hGradQuasiFermi_ElectricField_HFS = <refdens>
}

where refdens is the electron or hole concentration below which this interpolation becomes active.

The second option is to interpolate from GradQuasiFermi to EParallelToInterface:

Math {
  RefDens_eGradQuasiFermi_EParallelToInterface_HFS = <refdens>
  RefDens_hGradQuasiFermi_EParallelToInterface_HFS = <refdens>
}

Lastly, there is an option to interpolate from EParallel to ElectricField, just like for avalanche generation:

Math {
  RefDens_eEParallel_ElectricField_HFS = <refdens>
  RefDens_hEParallel_ElectricField_HFS = <refdens>
}

Which of these interpolation options is appropriate for the given structure depends on whether the breakdown behavior occurs primarily in the channel or at a trench corner. In the first case, interpolation to EParallelToInterface is more accurate; whereas in the latter case, interpolation to ElectricField is more accurate.

If no prior information is available about the breakdown location, the following setup is recommended for a typical power device:

Physics {
  Recombination(
    Avalanche( GradQuasiFermi )
  )
  Mobility(
    HighFieldSaturation( GradQuasiFermi )
  )
}
Math {
  RefDens_eGradQuasiFermi_ElectricField_HFS = 1e12
  RefDens_hGradQuasiFermi_ElectricField_HFS = 1e12
}