Renato Mariz de Moraes, Marshal Miller, Alex Glasser, Anand Banerjee, Ed Ott, Tom Antonsen,

and Steven M. AnlageCSR, Department of Physics

MURI Review

14 November, 2003

Inducing Chaos in the p/n Junction

Funded by STIC/STEP and Air Force MURI

Motivation

• Identify the origins of chaos in the driven resistor – inductor - varactor diode series circuit

• Establish a “universal” picture of chaos in circuits containing p/n junctions

• Identify new opportunities to induce chaos exploiting the p/n junction nonlinearity⇒ John Rodgers’ talk

Driven Resistor-Inductor-Diode CircuitStudied since the 1980’s

Why is the driven RLD circuit so important?Simplest passive circuit that displays period doubling and chaosIt is a good model of the ubiquitous p-n junction and its nonlinearities

L D

RVDRIVEN

L D

RVDRIVEN

Chaos in the Driven RLD Circuit

Voltage acrossResistor R ~ I

V0 sin(ωt)

1

2

4

chaos

I

Driving Amplitude V0 (V)

Bifurcation diagramMaximum

Voltage acrossResistor R R = 25 Ω

L = 50 µHD = NTE610f = 2.5 MHz

12

4 chaos

R L D

Period-doubling phase diagrams

(b)

f (MHz)

RLD

+ o

p-am

p ci

rcui

t

HF signal appliedNo HF signal

Low

Fre

q. D

rivin

g V

olta

ge (V

)

f (MHz)

Low

Fre

q. D

rivin

g V

olta

ge V

0(V

)

RLD

VLF

Diode = NTE610L = 390 nHR = 25 Ωf0 ≈ 70 MHz

VLF = V0 cos(2π f t)

Period-1Period-1

Period-2 or morecomplicated behavior

)0(1

0jLC

f ≈

L D

R

resonance

Nonlinearity of the p-n Junction

The diffusive dynamics of majority and minority charge carriersin the p-n junction is complex and nonlinear

pn(x,t)np(x,t)

x

t = 0 (forward biased and conducting)

t = 0

np0pn0

t = τRR (reverse biased and off)

t = τs

Charge distribution profiles fora forward-biased diode that issuddenly switched off at t = 0

V D

I(V D)

V D

C (V D)

B attery

Cor H unt

N L C RV D

I(V D)

V D

C (V D)

B attery

Cor H unt

N L C R

V D

I(V D)

V D

C (V D)

Cor H unt

N L C RV D

I(V D)

V D

C (V D)

Cor H unt

N L C R

All models of chaotic dynamics in p-n junctions approximate thecharge dynamics using nonlinear lumped-elements

≈RealDiode

ApproximateNonlinearLumped-element

-1 -0.5 0.5 1

2´ 10 -1 0

4´ 10 -1 0

6´ 10 -1 0

8´ 10 -1 0

1´ 10 - 9

Reverse Voltage (-Vv )

There are 3 competing forms of nonlinearity in this problem:1. Nonlinear I-V curve Irv(Q). Traditional focus ⇒ rectification

2. Nonlinear C(V) => Vv(Q). C increases by x4 ⇒ period doublingVan Buskirk + Jeffries, Chua, Crevier, etc.

3. Finite minority carrier lifetime or reverse recovery time. Delayed feedbackThe p/n junction retains memory of previous fwd-bias current swings

Rollins + Hunt

Resistor-Inductor-Diode Circuit

C(V) (F)

)(21

0 VLCf

π=

=> No consensus on the origin of chaos

What is the cause of chaos?

Time (µs)

I DIO

DE(

A)Diode 1N4007

R = 25 Ω

Driv

e V

olta

ge (V

)

Piecewise linear capacitor

RV

IDIODE

Reverse Recovery Time τRR

τs τt

τRR

A forward-biased diode that is shutoff will continue to conduct for thereverse recovery time, τRR

τs = storage timeτt = transition time

τRR = τs + τt

τrr vs. Diode

t (s)

Vol

tage

dro

p at

re

sist

or (V

)R=25 ΩD

A=5VVDC=0

f=70

0 kH

zf=

700

kHz

f=70

0 kH

zf=

100

kHz

1N5475B,varactor

NTE610,varactor

1N4148,fast recovery

1N4007,rectifier

1.2µs

120ns

60ns

≈0

Diode τRR(ns)

Cj(pF)

Results with f0 ~ 1/τRR

Results with f0 ~ 10/τRR

Results withf0 ~ 100/τRR

1N5400 7000 81 Period-doublingandchaos for f/f0 ~ 0.11 – 1.64

Period-doubling andchaos f/f0 ~ 0.16 –1.76

No chaos, nor period-doubling

1N4007 700 19 Period-doublingandchaos for f/f0 ~ 0.13 – 2

Period-doubling andchaos for f/f0 ~ 0.23 – 1.3

No period doubling or chaos

1N5475B 160 82 Period-doublingandchaos for f/f0 ~ 0.66 – 2.2

No chaos, nor period-doubling

No chaos, nor period-doubling

NTE610 45 16 Period-doublingandchaos for f/f0 ~ 0.14 – 3.84

Period-doubling only for f/f0 ~ 1.17 –3.25

No chaos, nor period-doubling

Search for Period Doubling and Chaos in Driven RLD Circuit

RLD

VLF jLCf

π21

0 =

V0 sin(2π f t)

L D

R

Circuit Chaos: Rule of ThumbDriven Nonlinear Diode Resonator

L D

RVDRIVEN

L D

RVDRIVEN

sin(ωt)

For lumped element nonlinear diode resonator circuits, Period Doubling and Chaos are observed for sufficiently large driving amplitudes when;

ω ~ ω0/10 ↔ 4 ω0

andω0 < 10/τRR to 100/τRR

wherejLC

10 =ω and Cj is the diode junction capacitance

τRR is the “reverse recovery” time of the diode

Circuit ChaosDriven Nonlinear Diode Resonator

Several kinds of models of the driven RLD circuit show behavior consistent with experiment

V D

I(V D)

V D

C (V D)

B attery

Cor H unt

N L C RV D

I(V D)

V D

C (V D)

B attery

Cor H unt

N L C R

Universal Feature of All Models:All models display a “reverse-recovery-like” phenomenon, associated with a charge storage mechanism.

When ω >> 1/τRR period doubling and chaos are strongly suppressed

(technical detail)

Time (µs)

I DIO

DE(

A)

Diode 1N4007 R = 25 Ω

Driv

e V

olta

ge (V

)

Piecewise linear capacitor

RV

IDIODE

Piecewise Linear Capacitor has a “Reverse Recovery Time τRR”

τs τt

τRR

V

QE0 slope 1/C1

slope1/C2

PLC Modelfor the diode

Circuit ChaosDriven Nonlinear Diode Resonator

The reverse recovery time of the diode is itself a nonlinear function of many parameters, including;

History of current transients in the diodePulse amplitudePulse frequencyPulse duty cycleDC bias on the junction

These nonlinearities can expand the range of driving parametersover which period doubling and chaos are observed.

InfluenceτRR

Stor

age

Tim

e (n

s)

------ PSpice____ NTE610

DC Offset VDC (V)

------ PSpice____ 1N4007

Stor

age

Tim

e (n

s)DC Bias Dependence of Reverse Recovery Time

RNTE610

A=5Vf=100kHz

VDC

Forward bias: broadercharge distribution inp/n junction, and a longer recovery time

Factor of10 change in τs

-1 -0.5 0.5 1

2 ´ 10 - 1 0

4 ´ 10 - 1 0

6 ´ 10 - 1 0

8 ´ 10 - 1 0

1 ´ 10 - 9

Reverse Voltage (-Vv

Incident Power (dBm)

Max

. of C

ircui

t Cur

rent

(Arb

. Uni

ts)

0.0 V

+0.2 V (forward bias)

-0.4 V (reverse bias)

-0.6 V (reverse bias)

DC Bias Dependence of Period Doubling

Reverse bias enhancesperiod doubling in thiscase

But reverse bias shouldreduce the C(V) nonlinearity!

CD(V) (F)

RLDNTE 610 diodeR = 25 Ω, L = 10 µHf =29 MHz >1/τRR=22 MHz> f0 = 12.3 MHz

Drop in τRR and CD with VDC makes f ~ f0 ~1/τRR

Circuit ChaosMore Complicated Circuits

RLD-TIATrans-ImpedanceAmplifier

Our conclusion:The combination of rectification and nonlinear dynamics in thiscircuit produces qualitatively new ways to influence circuit behavior by means of rf injection.

2-tone injection experiments: frequencyω0

~ MHz~ GHz

ωLF ωHF

The ωHF signal is rectified, introducing a DC bias on the p/njunction and increasing the circuit nonlinearity at ωLF.

(similar to Vavriv)

Two-Tone Injection of Nonlinear Circuits

Driven RLD Circuit Driven RLD/TIA Circuit

No change in period doublingbehavior with or without RF

RF injection causes significantdrop in driving amplituderequired to produce period-doubling!

Driving Frequency (MHz)(Low Frequency)

Low

Fre

q. D

rivi

ng

Vol

tage

(V

)

Period-2

Period-1

Region not explored

Period Doubling (With RF)

Period Doubling (No RF)

Driving Frequency (MHz)(Low Frequency)

Low

Fre

q. D

rivi

ng

Vol

tage

(V

)

0 90

8

Period-Doubling

Max

. of

Op-

amp

AC

V

olta

ge O

utpu

t

No

Inci

dent

Po

wer

P HF=

+20d

Bm

P HF=

+40d

Bm

P HF=

+30d

Bm

Low Frequency Driving Voltage VLF (V)

RF Injection Lowers the Threshold for Chaos in Driven RLD/TIA

LF = 5.5 MHz+ HF = 800 MHz

Max

. of

Op-

amp

AC

V

olta

ge O

utpu

t

No

DC

O

ffse

tD

C

Off

set=

+40

mv

DC

O

ffse

t=+4

40m

v

DC

O

ffse

t=+3

00m

v

Low Frequency Driving Voltage VLF (V)

LF = 5.5 MHz+ VDC OffsetVLF + VHF

VDC

Period 1

Two-Tone Injection of Nonlinear Circuits

In general …To understand the p/n junction embedded in more complicated circuits:

Nonlinear capacitanceRectificationNonlinearities of τRR

All play a role!

In this case …The combination of rectification, nonlinear capacitance, andthe DC-bias dependence of τRR produce complex dynamics

⇒ More surprises are in store …

Rg

Vref

VincVg(t)

+V(t)-

Transmission Line

Chaos in the Driven Diode Distributed Circuit

Delay differential equations for the diode voltage

Z0

delay T

A simple model of the ESD circuit on an IC

Vg = .5 V Period 1

Vg = 2.25 V Period 2

Vg = 3.5 V Period 4

Vg = 5.25 V Chaos

V(t)

(Vol

ts)

V(t)

(Vol

ts)

V(t)

(Vol

ts)

V(t)

(Vol

ts)

Time (s)

Time (s)

Time (s)

Time (s)

Chaos in the Driven Diode Distributed Circuit

Simulation results

Vg (Volts)

Stro

be P

o in t

s (V

o lts

)

Period 2

Period 1

Period 4

Chaos

Chaos in the Driven Diode Distributed Circuit

f = 700 MHzT = 87.5 psRg = 1 ΩZ0 = 70 ΩPLC, Cr = Cf/1000

Challenges for the Future

• Ten parameters to explore:Cf, Cr, g, Zo, Rg, Vg, ω, T, Vf, Vgap

• Experimental verification of numerical results

2D projectionof chaotic orbit

Conclusions about Chaosin the Driven p/n Junction

• A history-dependent recovery/discharge time scale is the key physics needed to understand chaos in the driven RLD circuit

• Nonlinear Capacitor (NLC) models have a τRR-like time scale

• Both the Hunt and NLC models have a history-dependentrecovery time scale due to charge storage mechanisms

• Real diodes have strong nonlinearities of the reverse recovery timethat are not captured in current models

• The addition of a TIA to the RLD circuit introduces a new wayto influence nonlinear circuit behavior through rectification

• Embedding a diode in a distributed circuit offers new opportunities to induce chaos. See John Rodgers’ talk

Recent Papers on the Nonlinear Diode Resonator andRelated Circuits:

Renato Mariz de Moraes and Steven M. Anlage, "Unified Model, and Novel Reverse Recovery Nonlinearities, of the Driven Diode Resonator,"Phys. Rev. E 68, 026201 (2003).

Renato Mariz de Moraes and Steven M. Anlage, "Effects of RF Stimulus and Negative Feedback on Nonlinear Circuits,"IEEE Trans. Circuits Systems I (in press).

http://arxiv.org/abs/nlin.CD/0208039

T. L. Carroll and L. M. Pecora, “Parameter ranges for the onset of period doubling in the diode resonator,” Phys. Rev. E 66, 046219 (2002)

DURIP 2004 proposal: “Nonlinear and Chaotic Pulsed Microwave Effects on Electronics”Anlage, Granatstein and Rodgers