Type Name: jj
The Josephson junction model is an extended version of the RSJ model as used by Jewett.
The parameters marked with an asterisk in the area column scale with the ics parameter given in the device line, not necessarily linearly. The present model paradigm assumes that the model parameters apply to a ``reference'' junction, which is a typical mid-critical current device as produced by the fouhdry. Instantiations derive from the reference device for a desired critical current. Appropriate scaling, not necessarily linear, will be applied when formulating instance capacitance and conductances.
JJ Model Parameters name area parameter units default level model type - 1 pijj pi junction - 0 rtype Quasiparticle branch model - 1 cct Critical current model - 1 icon Critical current first zero A 1.0e-2 icrit * Reference junction critical current A 1.0e-3 vg or vgap Gap voltage V 2.6e-3 delv Gap voltage spread V 80.0e-6 cap * Reference junction capacitance F 0.7e-12 cpic Reference junction capacitance per critical current F/A 0.7e-9 cmu Capacitance area/edge scaling 0.0 vm Reference icrit*rsub V 16.5e-3 rsub or r0 * Reference subgap resistance vm/icrit icrn Reference icrit*rnorm V 1.65e-3 rnorm or rn * Reference normal state resistance icrn/icrit gmu Conductance area/edge scaling 0.0 icfct or icfact Ratio of critical to gap currents - /4 force Don't check conductances 0 vshunt Voltage to specify fixed shunt resistance V 0.0 lsh0 Shunt resistor series inductance, constant part H 0.0 lsh1 Shunt resistor series inductance, proportional part H/ 0.0 tsfactor Time step phase change limit /2 tsaccel time step accelerator 1.0 tc superconducting transition temperature K 9.26 tnom model reference temperature K 4.2 temp operating temperature K tnom tcfct temperature dependence fitting parameter 1.74 vdbpak dropback voltage (read only)
Detailed information about these parameters is presented below. Unless stated otherwise, this information also applies to the Verilog-A Josephson junction model provided with WRspice in the Verilog-A examples, and the microscopic model.
0 The junction is completely unshunted, all shunt conductances set to zero. 1 Standard piecewise-linear model. 2 Analytic exponentially-derived approximation. 3 Fifth order polynomial expansion model. 4 ``Temperature'' variation, allow modulation of the gap parameter.
Values for rtype larger than 1 are not currently supported in the Verilog-A model supplied with WRspice in the Verilog-A examples.
The default is rtype=1. Setting rtype=0 will disable modeling of the quasiparticle current, effectively setting the RSJ shunt resistance to infinity. Conditions with rtype=1 and 2 are as described by Jewett, however it is not assumed that the normal resistance projects through the origin. The icfact parameter can be set to a value lower than the default BCS theoretical value to reflect the behavior of most real junctions. The quasiparticle resistance is approximated with a fifth order polynomial if rtype=3, which seems to give good results for the modeling of some NbN junctions (which tend to have gently sloping quasiparticle curves).
Rtype=4 uses a piecewise-linear quasiparticle characteristic identical to rtype=1, however the gap voltage and critical current are now proportional to the absolute value of the control current set with a control=src_name entry in the device line. This is to facilitate modeling of temperature changes or nonequilibrium effects. For control current of 1 (Amp) or greater, the full gap and critical current are used, otherwise they decrease linearly to zero. If no device control source is specified, the algorithm reverts to rtype=1. It is expected that a nonlinear transfer function will be implemented with a controlled source, which will in turn provide the controlling current to the junction in this mode. For example, the controlling current can be translated from a circuit voltage representing temperature with an external nonlinear source. The functional dependence is in general a complicated function, but a reasonable approximation is 1 - (T/Tc)4 . See the examples (A.3) for an example input file (ex10.cir) which illustrates rtype=4.
It is currently not possible to use other than the piecewise linear model with temperature variation. If rtype=4, then legal values for the critical current parameter are cct=0 (no critical current) and cct=1 (fixed critical current). If another value is specified for cct, cct reverts to 0. Thus, magnetic coupling and quasiparticle injection are not simultaneously available.
0 No critical current. 1 Fixed critical current. 2 Sin(x)/x modulated supercurrent. 3 Symmetric linear reduction modulation. 4 Asymmetric linear reduction modulation.
Values for cct larger than 1 are not currently supported in the Verilog-A model supplied with WRspice in the Verilog-A examples.
The control instance parameter should be used with devices using cct 2,3, or 4. With cct=2, the first zero is equal to the value of the model parameter icon. For cct=3, the maximum critical current is at control current zero, and it reduces linearly to zero at control current = icon . Junctions with cct=4 have maximum critical current at control current = - icon, and linear reduction to zero at control current = + icon. If cct is specified as 2, 3, or 4, the area parameter, if given, is set to unity. Otherwise, the model parameters are scaled appropriately by the area before use.
The parameter is not currently recognized by the Verilog-A Josephson junction model provided with WRspice, as that model does not currently support values of cct larger than 1.
I = Icsin()where Ic is the critical current. and the junction ``phase'' is
= (2/)V(t)dt .The V(t) is the junction voltage, and is the magnetic flux quantum.
The icrit parameter should not be confused with the ics instance parameter. The latter is actually a scale factor which specifies the instantiated device critical current as well as appropriately scaling conductances and capacitance, from the model reference current which is icrit.
C = cap(A(1 - cmu) + cmu)Here, A is the ``area'' scaling factor, which is the ratio of the junction critical current to the reference critical current.
Gx = Gx0(A(1 - gmu) + gmu)Here, Gx refers to either the subgap or normal conductance, Gx0 is the same parameter for the reference junction. The A is the scaling parameter, that is, the ratio of instance to reference critical currents. The default value is 0, meaning that scaling is assumed purely linear, which will be the case until a number is provided through additional data analysis. It may prove necessary to have separate scaling parameters for subgap and above gap condutance, at which time a new model parameter may be added.
When simulating SFQ circuits, between SFQ pulses there is often significant time where signals are quiescent and one could probably take larger time steps, speeding simulation. This appears true to an extent, however one can see signs of instability if steps are too large.
The tsaccel parameter is the ratio of the longest time step allowed to that allowed at the dropback voltage. In computing the time step, the low voltage threshold is reduced to the dropback voltage divided by tsaccel, so time steps will be inversely proportional to voltages above this value.
Experimentation suggests that a value of 2.5 is a good choice for RSFQ circuits, your results may vary.
= tanh(tcfcttc/T - 1))Reference: https://physics.stackexchange.com/questions/192416/
Here, is the gap parameter, with the subscript indicating the value at temperature T=0. The tcfct parameter is semi-empirical, commonly cited as 1.74, which is the default value. It is taken that the junction gap voltage and critical current are directly proportional to the gap parameter. The quasiparticle resistance is assumed invariant with temperature, which should be true for the low temperatures of interest.
In order to apply temperature correction one takes the assumption that parameters for the reference junction are measured at nominal temperature tnom, and we therefor have
= tanh(tcfcttc/tnom - 1))
The correction factor, which will multiply device critical current and gap voltage, is the ratio / . This is applied, in addition to area scaling, when extrapolating device parameters from reference junction parameters.