Memristive Systems Analysis of 3-Terminal Devices

Blaise MouttetICECS 2010

December 12-15Athens, Greece

OverviewReview of Memristive, Mem-capacitive, and

Mem-inductive SystemsIntroduction to Mem-Transistor SystemsSmall signal analysis of Mem-Transistor

SystemsExamples

Widrow-Hoff memistorSynaptic floating gate transistorNano-ionic MOSFET

Memristive SystemsResistive dynamic

system defined by a state vector x v= R(x,i,t)idx/dt = f(x,i,t)

Degenerates to linear resistor at high frequency (property 6)

Examples of memristive behavior was originally noted from thermistor, neural models and discharge tubes.L.O. Chua, S.M. Kang. “Memristive Devices and Systems,” Proceedings of the IEEE,

Vol. 64, iss.2 (1976)

In 1967 Argall demonstrated zero-crossing resistance hysteresis and frequency dependence for thin film TiO2.

F.Argall “Switching Phenomena in Titanium Oxide Thin Films,” Solid-State Electronics, Vol. 11, pp.535-541 (1968).

Mem-capacitive SystemsCapacitive dynamic

system defined by a state vector x q= C(x,v,t)vdx/dt = f(x,v,t)

Degenerates to linear capacitor at high frequency

Examples of mem-capacitive behavior are found in nanocrystal and perovskite thin filmsM. DiVentra, Y. V. Pershin, L.O. Chua, “Putting Memory into Circuit Elements:

Memristors, Memcapacitors, and Meminductors,” Proceedings of the IEEE, vol 97, iss.8, (2009)

Mem-inductive SystemsInductive dynamic

system defined by a state vector x f= L(x,i,t)idx/dt = f(x,i,t)

Degenerates to linear inductor at high frequency

Examples of mem-inductive behavior are found in MEMS inductors

Y. V. Pershin, M. DiVentra, “Memory effects in complex materials and nanoscale systems,” arXiv:submit/0144853 (2010)

Transistor = Transfer Resistor (can amplify signals but is memory-

less)

Memristor = Memory Resistor (has memory but dissipates signal

energy)

Is it possible to build a singular non-linear circuit element having features of both a memristor and a transistor?

2-Port Model of Voltage-Controlled Transistor

I1 = g(V1,V2)I2= h(V1,V2)

I1 = Y11V1+Y12V2

I2= Y21V1+Y22V2

Small signallinearization

2-Port Model of Voltage-Controlled Mem-Transistor

I1 = g(V1,V2,x)I2= h(V1,V2,x)dx/dt =f(V1,V2,x)

Small Signal LinearizationFor a 1st order system in which one of the input voltages is held constant and the other input voltage is denoted as a gate voltage (Vg) a small variation around a fixed state x0 and voltage Vg0 is expressible as:

δxx

)x,f(VδV

V

)x,f(V

dt

dxδ

δxx

)x,h(VδV

V

)x,h(V)δ(I

δxx

)x,g(VδV

V

)x,g(V)δ(I

0g0g

g

0g0

0g0g

g

0g02

0g0g

1

0g01

TransconductanceIn the LaPlace domain the transconductance is determined from the small signal linearization. System stability is determined

by the sign of

g

0g0

0g0

g

0g00g0

g

dm v

)x,h(V

x

)x,f(Vs

v

)x,f(V

x

)x,h(V

(s)ΔV

(s)ΔIg

x

)x,f(V 0g0

Transconductance

g

0g0

0g0

g

0g00g0

g

dm v

)x,h(V

x

)x,f(Vjω

v

)x,f(V

x

)x,h(V

(s)ΔV

(s)ΔIg

For periodic excitation frequencies (s=jw) the transconductance of a mem-transistor is generally a frequency dependent complex number and represents both gain and a phase shift between the input and output signals.

Transconductance

g

0g0

0g0

g

0g00g0

g

dm v

)x,h(V

x

)x,f(Vj

v

)x,f(V

x

)x,h(V

(s)ΔV

(s)ΔIg

At high excitation frequencies (w∞) the first term reduces to zero and the transconductance reduces to that of an ordinary transistor.

Examples

Example #1:Widrow-Hoff Memistor

B. Widrow, “An Adaptive ADALINE Neuron Using Chemical Memistors,” Stanford Electronics Laboratories Technical Report 1553-2, October 1960.

In 1960 a 3-terminal electrochemical “memistor” was developed by Bernard Widrow and Marcian Hoff.

The memistor formed a central component to an early ANN and the development of the LMS algorithm.

Example #1:Widrow-Hoff Memistor

B. Widrow, “An Adaptive ADALINE Neuron Using Chemical Memistors,” Stanford Electronics Laboratories Technical Report 1553-2, October 1960.

The memistor was experimentally shown to demonstrate a charge- dependent conductance in a similar fashion to the later predicted memristor of Chua.

Example #2: Synaptic Transistor

C.Diorio,P.Hasler,B.A.Minch,C.A.Mead, “A single-transistor silicon synapse,” IEEE Transactions on Electron Devices, Vol. 43, No. 11, Nov. 1996.

Since the 1990’s analog floating gate MOSFET transistors have been designed to act as synapses for neuromorphic hardware.

Example #2: Synaptic Transistor

C.Diorio,P.Hasler,B.A.Minch,C.A.Mead, “A single-transistor silicon synapse,” IEEE Transactions on Electron Devices, Vol. 43, No. 11, Nov. 1996.

VVdc

Vm

d

VVdc

V

Vgc

V

VbiVdg

V

bidgg

VVdc

V

Vgc

V

Tcorner

ewi

weeVVi

ewQww

ww

dt

dw

1

)(

2

2

0

2

21

1max

The sub-threshold modeling equations developed by Diorio et al. represent a 1st order, voltage-controlled mem-transistor with the state variable equal to the source current and Vgc as the control voltage.

Example #3: Nano-ionic FET

D.B. Strukov, G.S. Snider, D.R. Stewart, R.S. Williams “The missing memristor found,” Nature, Vol. 453, May 2008.

Nano-ionic motion of oxygen vacancies in TiO2/TiO2-x has been used by Strukov et al. to explain memristive effects.

If minimum TiO2 thickness > tunneling gap then memcapacitive rather than memristive effects would be expected as a result of ionic drift.

Example #3: Nano-ionic FET

2kT

w)a,qE(Vsinhfae

dt

dw

(w)V

V1(w)VV

2L

WC(w)μI

0.5VV(w)VVL

WC(w)μI

n(w)kT

qV(w)expII

GCU/kT

A

DS2TGSnd

2DSDSTGSnd

GS0d sub-threshold region

saturation region

triode region

Ionic drift equation:

Long channel MOSFET equations:

N-port Memory Resistive SystemsI1 = g1(V1,V2,..,Vn,x)I2= g2(V1,V2,..,Vn,x) . .In= gn(V1,V2,..,Vn,x)

dx/dt =f(V1,V2,..,Vn,x)

B.Mouttet “Memristive Transfer Matrices,” arXiv:1004:0041 (2010)B.Mouttet “Programmable Crossbar Signal Processor,” U.S. Patent 7 302 513, (2007)

SummaryMem-transistor systems analysis has been proposed

based on non-linear, dynamic 2-port systems.A generalized transconductance of mem-transistors

includes both gain and phase shift and may be useful to determining mem-transistor stability.

Some 3-terminal electronic devices have been shown to exhibit memory effects over the past 50 years but the transistor models have rarely included non-linear dynamic systems analysis.

In addition to memristors, memcapacitors, and meminductors, mem-transistors will likely play an increasingly important role in 21st century electronics to achieve neuromorphic and bio-inspired computing.