1. Memristive Systems Analysis of 3-Terminal Devices Blaise Mouttet ICECS 2010 December 12-15 Athens, Greece

2. Overview Review of Memristive, Mem-capacitive, and Mem- inductive Systems Introduction to Mem-Transistor Systems Small signal analysis of Mem-Transistor Systems Examples Widrow-Hoff memistor Synaptic floating gate transistor Nano-ionic MOSFET

3. Memristive Systems Resistive dynamic system defined by a state vector x v= R(x,i,t)i dx/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)

4. In 1967 Argall demonstrated zero-crossing resistance hysteresis and frequency dependence for thin film TiO 2. F.Argall Switching Phenomena in Titanium Oxide Thin Films, Solid-State Electronics, Vol. 11, pp.535-541 (1968).

5. Mem-capacitive Systems Capacitive dynamic system defined by a state vector x q= C(x,v,t)v dx/dt = f(x,v,t) Degenerates to linear capacitor at high frequency Examples of mem- capacitive behavior are found in nanocrystal and perovskite thin films M. 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)

6. Mem-inductive Systems Inductive dynamic system defined by a state vector x = L(x,i,t)i dx/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)

7. 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?

8. 2-Port Model of Voltage-Controlled Transistor I 1 = g(V 1,V 2 ) I 2 = h(V 1,V 2 ) I 1 = Y 11 V 1 +Y 12 V 2 I 2 = Y 21 V 1 +Y 22 V 2 Small signal linearization

9. 2-Port Model of Voltage-Controlled Mem-Transistor I 1 = g(V 1,V 2,x) I 2 = h(V 1,V 2,x) dx/dt =f(V 1,V 2,x)

10. Small Signal Linearization For a 1 st order system in which one of the input voltages is held constant and the other input voltage is denoted as a gate voltage (V g ) a small variation around a fixed state x 0 and voltage V g0 is expressible as:

11. Transconductance In the LaPlace domain the transconductance is determined from the small signal linearization. System stability is determined by the sign of

12. Transconductance For periodic excitation frequencies (s=j ) 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.

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

14. Examples

15. 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.

16. 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.

17. 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 1990s analog floating gate MOSFET transistors have been designed to act as synapses for neuromorphic hardware.

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

19. 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 TiO 2 /TiO 2-x has been used by Strukov et al. to explain memristive effects. If minimum TiO 2 thickness > tunneling gap then memcapacitive rather than memristive effects would be expected as a result of ionic drift.

20. Example #3: Nano-ionic FET sub-threshold region saturation region triode region Ionic drift equation: Long channel MOSFET equations:

21. N-port Memory Resistive Systems I 1 = g 1 (V 1,V 2,..,V n,x) I 2 = g 2 (V 1,V 2,..,V n,x). I n = g n (V 1,V 2,..,V n,x) dx/dt =f(V 1,V 2,..,V n,x) B.Mouttet Memristive Transfer Matrices, arXiv:1004:0041 (2010) B.Mouttet Programmable Crossbar Signal Processor, U.S. Patent 7 302 513, (2007)

22. Summary Mem-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 21 st century electronics to achieve neuromorphic and bio-inspired computing.