Verilog-A Large-Signal RF MEMS Model Library

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This SourceForge project hosts an open-source Verilog-A [1,2] library of large-signal RF MEMS capacitor, resonator and switch models [3,4]. The electrostatically-actuated RF MEMS capacitor and switch models are based on a nonlinear damped mass spring system and a nonlinear capacitive transducer, whereas the electrostatically-actuated RF MEMS resonator model is based on a nonlinear damped mass-spring system and two nonlinear capacitive transducers. The reasons for nonlinear behavior are:

The contact force consists of the attractive van der Waals force, written from a Hamaker constant, and the repulsive short-range force at contacting asperities. Both are nonlinear with beam displacement. The repulsive force is Born (Pauli) repulsion arising from electron-cloud overlap; it is represented by a Hertzian wall standing off at an effective surface roughness height, which is the parameter that sets the down-state capacitance of a real switch.
The damping coefficient is the sum of two terms that behave differently. Anchor loss is elastic-wave radiation into the substrate: it is set by geometry and material and does not vary with beam displacement. Squeeze-film damping does vary, rising as the inverse cube of the gap. Squeeze-film damping involves gas dynamics, and both the gas mean free path and the viscosity are scaled with ambient pressure and temperature, so the models are not restricted to atmospheric pressure.
The electrostatic force is nonlinear time-varying with beam displacement and drive voltage. Because it rises as the inverse square of the gap while the restoring force is very nearly linear, the stable branch ends part-way through the travel and the beam collapses. The same asymmetry means the beam leaves the down state from a gap of tens of nanometres rather than from the full gap, so RF MEMS switches and switched capacitors can be held down at a voltage well below their pull-in voltage. In vibrating RF MEMS resonators the same mechanism appears as electrostatic spring softening, a negative electrical stiffness that grows with bias. Because the transducer is periodically time-varying under large-signal drive, noise is folded around the carrier and its harmonics.
The spring force is nonlinear with beam displacement, although it is often linearized (Hooke's law of elasticity). Taking into account the cubic spring force coefficient, k₃ [N/m³], is necessary to model spring-hardening and spring-softening of Duffing resonators. For a fixed-fixed beam the mechanical cubic term is mid-plane stretching, which hardens the spring; softening comes from the electrostatic transducer and is produced by the electrostatic force rather than folded into k₃. The net behavior is therefore bias-dependent.

The Verilog-A models include electromechanical gas dynamics effects, such as anchor loss, hold-down, hysteresis, self-actuation, squeeze-film damping, spring-hardening, spring-softening, asperity contact and van der Waals interaction. In addition, noise sources, such as Brownian noise sources and thermal noise sources, are also modeled, the Brownian force source being tied to the instantaneous total damping. Dielectric charging, which is the dominant cause of long-term capacitance-voltage drift and eventual stiction failure in capacitive switches, is available as an optional trapped-charge state. Other multiphysics effects, such as acoustoelectric, electrothermal, piezoelectric, pyroelectric and thermomechanical effects, are not included yet.

The 1-DOF models were revised in version 2.0.0. That release corrects the van der Waals coefficient, the Lennard-Jones repulsive exponent, the Veijola rarefaction constant [7], the gas mean free path, the separation of anchor loss from squeeze-film damping, the sign of the resonator cubic spring coefficient, and the solver tolerances. It is not parameter-compatible with the 1.x series; the changelog shipped with the archive lists every change that moves a number and why.

The Verilog-A models can be used with SPICE solvers for DC, small-signal (AC, noise, S-parameters (SP)) and large-signal (harmonic balance (HB), periodic steady-state (PSS), quasi-periodic steady-state (QPSS), transient) simulation of analog/RF circuits based on RF MEMS components. The mechanical state is carried on kinematic nodes, so it is state the simulator can see rather than hidden state [2], and HB and PSS converge on it. Some remarks on simulation:

Large-signal simulation (1 dB compression point, third order intercept) of high-Q resonator based analog/RF circuits: While transient analysis is feasible, it requires a large number of time steps to reach the steady-state regime. It is therefore recommended to use the HB method, which directly converges to the steady-state solution, with a limited number of harmonics.
Solver tolerances: disciplines.vams defaults position, velocity and force to an absolute tolerance of 1e-6. A MEMS beam travels about 1e-6 m in total, so the default tolerance is the size of the entire mechanical signal and the solver will report convergence before the mechanical state has converged. The file mems_disciplines.vams corrects this through the override macros the standard already provides, and must be included before any other disciplines.vams.
Stiffness through contact: the asperity wall raises the damping coefficient by more than four orders of magnitude over the last few nanometres of travel. Allow the engine a small minimum timestep through contact rather than a large one, and tighten reltol.
Noise simulation: Supply-voltage noise should be generated by the test bench. Note that white_noise contributes in small-signal noise and pnoise analyses; it appears in transient only on simulators with transient noise enabled.
Power consumption simulation: At rest, electrostatically-actuated RF MEMS capacitors and switches exhibit no static (DC) power consumption. While switching, electrostatically-actuated RF MEMS capacitors and switches do exhibit dynamic (AC) power consumption (10-100 nJ per switching cycle). The dynamic power consumption is due to current transients, which (dis)charge the capacitive transducer. It is therefore recommended to perform a transient simulation.
Switching time simulation of switch based analog/RF circuits: RF MEMS switches are slow compared to III-V compound semiconductor and silicon semiconductor switches. It is therefore recommended to perform an envelope simulation. Squeeze-film damping is the dominant lever on switching time: a switch that closes in 14 us at 1 atm closes in under 4 us at 10 mbar, so package pressure is the first thing to check when measured switching times disagree with simulation.

The Verilog-A models allow for specification of beam dimensions and material properties (Poisson ratio, Young's modulus, etc...). Examples of schematics based on gallium nitride (GaN), polysilicon, silicon carbide (SiC) fixed-fixed beam RF MEMS resonators and oscillators, as well as golden capacitive fixed-fixed beam and ohmic cantilever RF MEMS switches, are available upon request. The archive ships a README, a changelog, worked example netlists, and two test scripts: an independent Python implementation of the same equations that serves as the executable specification, and a set of regression checks that also compiles every model.

Coming soon

Revision of the multiple DOF models, which still carry the 1.x contact and damping formulations and need the same correctness pass as the 1-DOF models.
A reduced-order beam model. A 1-DOF model holds the whole plate at a single gap and therefore cannot represent the beam peeling off progressively from its edges, which is what sets the release voltage of a real capacitive switch. The release voltage is consequently under-predicted.
Inclusion of non-electromechanical multiphysics effects, such as acoustoelectric, electrothermal, piezoelectric, pyroelectric and thermomechanical effects. I will probably start with implementation of electrothermal and thermomechanical effects, because it allows analysis of heat handling, which is of importance in the heterogeneous integration with GaN high electron mobility transistor (HEMT) technology. It also allows analysis of infrared (IR) detection properties.
Verilog-A models of comb-drive, disc, free-free beam and ring RF MEMS resonators
Verilog-AMS models of analog mixed-signal circuits based on RF MEMS components, such as a digital phase locked loop (DPLL) based on a phase frequency detector (PFD) with charge pump (CP) output, and Δ/Σ fractional-N frequency synthesizers

Examples

Examples of schematics based on the OHMIC_CANTILEVER_RF_MEMS_SWITCH and the RF_MEMS_CAPACITOR Verilog-A models can be downloaded. Note that as of version 2.0.0 the ohmic switch has a separate actuation electrode pair, so its port list is (rf_a, rf_b, act_p, act_n).

Requirements

A SPICE solver supporting Verilog-A/Verilog-AMS, such as Keysight ADS, Cadence SpectreRF, or Synopsys HSPICE, is required. An open-source flow built on OpenVAF, such as ngspice 39 or later with OSDI, can also be used; all three 1-DOF models compile clean under OpenVAF. Note that HSPICE supports Verilog-A but not full Verilog-AMS, and that QPSS is Cadence nomenclature. Some remarks on installing the Verilog-A models:

1-DOF Models:
- Keysight ADS (see [5])
  - Copy the files with ael extension into the networks subdirectory of the ADS project.
  - If necessary, make a veriloga subdirectory in the ADS project.
  - Copy the files with va extension, and the file mems_disciplines.vams, into the veriloga subdirectory of the ADS project.
  - Close and reopen the ADS project.
  - Create a new design.
  - Insert a random component and swap it with OHMIC_CANTILEVER_RF_MEMS_SWITCH or RF_MEMS_CAPACITOR.
- Cadence (see [6])
  - CIW: File → New → Cellview...
  - Create New File: Cell Name: OHMIC_CANTILEVER_RF_MEMS_SWITCH or RF_MEMS_CAPACITOR, View Name: veriloga, Tool: VerilogA-Editor. Click "OK".
  - The editor will appear. Copy/paste the Verilog-A code into the editor. Save it and exit the editor.
  - A dialog box will appear and ask you if you want to create a new symbol. Click "Yes".
  - The Symbol Generation Options window will appear. Make appropriate changes and click "OK".
  - Create a new schematic and insert an instance of the OHMIC_CANTILEVER_RF_MEMS_SWITCH component or the RF_MEMS_CAPACITOR component.
  - Make sure mems_disciplines.vams is on the include path, and that it is read before any other disciplines.vams.
- OpenVAF and ngspice
  - Run openvaf RF_MEMS_CAPACITOR.va to produce RF_MEMS_CAPACITOR.osdi.
  - Load it from the netlist with pre_osdi RF_MEMS_CAPACITOR.osdi inside a .control block.
Multiple DOF Models (by Dr. J. Iannacci):
- Cadence (see [6])
  - The RF MEMS compact model library is treated by Cadence as a standard design kit. This means it has to be copied in a proper location of the file system which we suppose to be /users/OPUS.
  - If we now name the two libraries as MEMS_MODELER_CONTAINER_VerilogA and MEMS_MODELER_CONTAINER_SpectreHDL, the components within each of them will be seen as subdirectories of the two following paths: /users/OPUS/MEMS_MODELER_CONTAINER_VerilogA and /users/OPUS/MEMS_MODELER_CONTAINER_SpectreHDL.
  - The previous absolute paths must then be added to the Library Path Editor within Cadence in order to make the two libraries visible and usable within Cadence together with the other pre-existing libraries and design kits.
  - The new configuration must be saved in the cds.lib file.
  - The RF MEMS compact models deal with mixed domain magnitudes, i.e. electrical and mechanical. Proper definitions for the electrical and mechanical magnitudes are declared within the libraries. Moreover, for each of them two parameters (abstol and blowup) used by the Spectre simulator to define convergence and non-convergence cases are defined for all the magnitudes.
  - In the VerilogA library the implementation definitions are included in the file /users/OPUS/MEMS_MODELER_CONTAINER_VerilogA/discipline/discipline.h and no special settings are required to properly link such a file.
  - In the SpectreHDL the implementation information are defined within the file /users/OPUS/MEMS_MODELER_CONTAINER_SpectreHDL/quantity.spectre. This file must be linked to make it visible during simulations. To do this a line containing the information on where the file is located must be added to the cds.lib file (absolute path: /users/OPUS/ cds.lib) before the lines referring to the actual libraries and design kits. In this case the line has to be as follows: DEFINE MEMS_MODELER_CONTAINER_SpectreHDL /users/OPUS/MEMS_MODELER_CONTAINER_SpectreHDL.

License

Links

[1] Verilog-AMS Language Reference Manual, Version 2.4, Accellera Systems Initiative.

[2] Hidden State in SpectreRF, Version 2, May 16, 2019, Ken Kundert, Designer's Guide Consulting, Inc.

[3] Modeling RF MEMS Devices, Koen Van Caekenberghe, IEEE Microwave Magazine, Volume 13, Issue 1, January - February 2012 Page(s): 83 - 110.

[4] Mixed-Domain Fast Simulation of RF and Microwave MEMS-based Complex Networks within Standard IC Development Frameworks, Jacopo Iannacci, Fondazione Bruno Kessler - FBK, MemSRaD Research Unit, Italy

[5] Using Verilog-A and Verilog-AMS in Advanced Design System, Keysight Technologies

[6] Verilog-AMS resources and example models, Designer's Guide Community

[7] Equivalent-circuit model of the squeezed gas film in a silicon accelerometer, T. Veijola, H. Kuisma, J. Lahdenperä and T. Ryhänen, Sensors and Actuators A, Volume 48, 1995 Page(s): 239 - 248.

[8] Intermolecular and Surface Forces, 3rd edition, Jacob N. Israelachvili, Academic Press, 2011. Source of the flat/flat Lennard-Jones pressure used for the contact model.