Investigation of Timing Parameters in Single-Flux-Quantum Circuits Using Low Critical-Current Junctions and Low Bias Voltages

Abstract

In this study, the trade-off among power consumptions, operating frequencies, and error rates in low-power single-flux-quantum (SFQ) circuit is investigated. The power consumption of SFQ circuits can be reduced by the lowering critical currents ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I_{\mathrm{c}}$</tex-math></inline-formula> values) of the Josephson junctions (JJs), or the bias voltage ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V_{\mathrm{b}}$</tex-math></inline-formula> ). The shifts and spreads of timing parameter fluctuations when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I_{\mathrm{c}}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V_{\mathrm{b}}$</tex-math></inline-formula> were lowered were numerically analyzed, with consideration of thermal noise at 4.2 K. The timing behaviors after <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I_{\mathrm{c}}$</tex-math></inline-formula> was lowered were relatively simple, being based on the signal-to-noise ratio, whereas those for lowered <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V_{\mathrm{b}}$</tex-math></inline-formula> values were complicated because of the dynamic behaviors of JJs driven at reduced voltage. The simulation results imply that the operating frequencies decreased because the timing windows narrowed, but the highest energy efficiency was obtained when we lowered <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I_{\mathrm{c}}$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V_{\mathrm{b}}$</tex-math></inline-formula> to 6.25% of their conventional values, and the energy reduction after lowering of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$V_{\mathrm{b}}$</tex-math></inline-formula> was 26% better than that after lowering of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I_{\mathrm{c}}$</tex-math></inline-formula> . Moreover, if a small error rate (for example <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-6}$</tex-math></inline-formula> ) is acceptable, further improvement in energy efficiency would be expected at significantly higher clock frequencies than would be the case if the SFQ circuit must exhibit error-free operation.