PSCAN DIMENSIONLESS UNITS

SUNY RSFQ Cell Library

PSCAN UNITS

Introduction and Definition

The definition of dimensionless variables and units used in PSCAN and many other Josephson junction circuit simulators stems from the basic equations of Josephson junction dynamics - see, e.g., Ref. 1. In dimensional units the equations describing the time evolution of Josephson phases even in simplest circuits are rather bulky. For example, for a single Josephson junction (described within the RSJ model [1]) closed with a superconductor loop we have:

Here C is the junction capacitance, R its normal resistance (including external shunt, if any), I_c critical current, while L is the loop inductance. Introducing the dimensionless current, voltage, time, resistance, inductance, and capacitance as follows:

we can reduce Eq. (1a) to a much simpler form:

PSCAN uses these dimensionless variables (i, v, tau, r, l, and c). Notice that according to Eq. (2a) the choice of the critical current unit (I_u) remains arbitrary. (For a single junction, I_u = I_c is an evident choice [1], but a typical RSFQ circuit uses many junctions with a variety of areas and hence I_cvalues.) PSCAN accepts a value of 0.125 mA which is equal to the typical critical current of the smallest junctions in RSFQ circuits operating at liquid helium temperatures.

Typical Values of Units

Variable	Notation of the Unit	Unit Values for Various Levels of Nb-trilayer Technology
Variable	Notation of the Unit	3.5 um		0.8 um		0.4 um
Voltage	V_u	0.3	mV	1.1	mV	2.0	mV
Current	I_u	0.125	mA	0.125	mA	0.125	mA
Time	t_u	1.1	ps	0.30	ps	0.165	ps
Resistance	R_u	2.4	Ohm	8.8	Ohm	16	Ohm
Inductance	L_u	2.64	pH	2.64	pH	2.64	pH
Capacitance	C_u	455	fF	37	fF	10	fF

Comments:

At proper shunting (or no shunting) the I_cR product (and hence V_u) does not depend on the junction area.
For RSFQ applications, larger (>0.5 um) Josephson junctions should be externally shunted to avoid hysteretic behavior [1]. The standard choice of the shunt resistance is that resulting in critical damping: c = 1, i.e. C_u = A_minC₀, where A_min is the smallest junction area and C₀ is its specific capacitance. For Nb-trilayer junctions, C₀ slowly increases at logarithm of the critical current density j_c: from 5 uF/cm² at j_c = 1 kA/cm² to approximately 8 uF/cm² at j_c= 100 kA/cm² [2, 3].
For deep-submicron, self-shunted Nb-based junctions, the theoretical maximum of V_u = I_cR is 2.2 mV; the first experimental data [4] are surprisingly close (~2.0 mV). Theoretically, V_u should not depend on the junction area, but may depend on the details of the particular fabrication technology, so that the table above lists only approximate values of this parameter.
The unit of current (I_u) depends solely on convention (see above).
As soon as V_u has been determined (generally, from experiment) and I_u chosen, all other units are defined exactly by Eqs. (2).
The time unit defined by Eq. (2c) is natural, but a little bit small for Josephson junction dynamics description. For example, typical FWHM duration of the SFQ pulse is about 3 tau_u. This is why the canonical RSFQ text [5] uses the units of tau₀ = (pi)x(t_u). (tau₀ in picoseconds is conveniently close to the linear size of a 0.125 mA junction in microns.) When comparing various results, it is necessary to mind this duality.

References:

K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach, New York, 1986), Chapters 1 and 2.
M. Maezawa, M. Aoyagi, H. Nakagawa, and S. Takada, Appl. Phys. Lett., vol. 66, pp. 2134-2136, Apr. 1995.
V. Patel and J. Lukens, to be published in IEEE Trans. on Appl. Supercond., vol. 9, June 1999.
W. Chen, A. Rylyakov, V. Patel, J. Lukens, and K. Likharev, Appl. Phys. Lett., vol. 73, pp. 2817-2819, Nov. 1998.
K. Likharev and V. Semenov, IEEE Trans. on Appl. Supercond., vol. 1, pp. 3-28, March 1991.

K. Likharev