Memristors ISL1- 1 the Perfect Storm in Nonlinear Circuit Theory ! ISL1- 2 Esaki Diode

10 Things You Didn’t Know About

Memristors ISL1- 1 The Perfect Storm in Nonlinear Circuit Theory ! ISL1- 2 Esaki Diode

Leo Esaki Nobel Prize in Physics, 1973 L. Esaki New Phenomenon in Narrow Germanium p-n junctions

Physics review 109(2):603, 1958 ISL1- 3 Simplest Tunnel Diode Circuit I P2 (I2, V2) + I I2

I1 P1 (I1, V1) L = 1 V

- P* Impasse I* point dI V  0 V V* V V dt 1 2 Solution does NOT exist beyond P* ! ISL1- 4 Two Points of View ! For mathematician, no solution is a Perfectly valid solution For everybody else, no solution means nonsence. ISL1- 5 Crisis in Circuit Theory Pre-1970 Definitions of the 3 Basic Circuit Elements Capacitors, Resistors, and Inductors give wrong circuit solutions when the elements are

time-varying or nonlinear ISL1- 6 3 Basic Circuit Elements dv(t) 1745 i(t)  C dt

1827 v(t)  R i(t)

di(t) 1831 v(t)  L dt ISL1- 7 To Recover from the perfect storm Capacitors, Resistors, Inductors must be redefined via an

AXIOMATIC APPROACH ISL1- 8 All Results Derived from An Axiomatic Approach are

Timeless ! ISL1- 9 Four Basic Circuit Variables

i(t) + v(t) 

voltage current v(t) i(t)

t q( tid )( )  t ≜   (tvd )(≜ )  

charge flux q(t) (t)

ISL1- 10 GEDANKEN i + PROBING B _v CIRCUITS -

ISL1-11 Linear resistor: v = Ri or i = Gv Nonlinear Resistor R R = Resistance, G  = Conductance1 R

slope = R2

slope = G1 Q1

R1 1 R3 1 Q2

slope = G2 0

G3 1 Current-controlled Resistor: v vˆ() i Voltage-controlled Resistor: i iˆ() v

Ri = small-signal resistance at Qi Gi = small-signal conductance at Qi ISL1- 12 current, Ampere A voltage, Volt V RESISTOR v i R(v,i)=0

q φ charge, Coulomb C flux, Weber Wb ISL1- 13 current, Ampere A voltage, Volt V RESISTOR v i R(v,i)=0

)= i

L ,



(

L INDUCTOR

q φ charge, Coulomb C flux, Weber Wb ISL1- 14 current, Ampere A voltage, Volt V RESISTOR v i R(v,i)=0

q,v

i )= C L ,



(

INDUCTOR CAPACITOR

q φ charge, Coulomb C flux, Weber Wb ISL1- 15 4 Basic Circuit Elements current, Ampere A voltage, Volt V RESISTOR v i R(v,i)=0

q,v

i )= C L ,



(

L INDUCTOR CAPACITOR ? Missing Link q φ charge, Coulomb C flux, Weber Wb ISL1- 16 4 Basic Circuit Elements current, Ampere A voltage, Volt V RESISTOR v i R(v,i)=0

q,v

i )= C L ,



(

L INDUCTOR CAPACITOR M

M ( , q)= 0 q φ charge, Coulomb C MEMRISTOR flux, Weber Wb ISL1- 17 Memristor

tangent ddfqdq () v   = f(q) slope = M(q) dtdqdt q i 0 M(q)  = f(q) v = M(q) i M(q) is called the Memristance. ISL1- 18 A Fourth Basic Element Called the Memristor was postulated in 1971 Leon O. Chua Memristor : The missing circuit element IEEE Transactions on Circuit Theory, vol.18, no.5, p.507-519, 1971. and found in 2008 D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams The Missing Memristor Found Nature, vol.453, p.80-83,2008. ISL1- 19 HP Memristor D

Pt -1 TiO2 TiO2-x Pt - 2 i  v +

v = M (q) i Memristance

vOR N M(qq ) ROFF  1 - D2 where D is the device thickness (can be scaled to less than 2 nano meters) ROFF, RON, v are device parameters ISL1-20 Memristor is defined by a State - Dependent

Ohm’s Law ISL1- 21 1973 Discovers Nobel Super- Prize conducting Josephson in tunneling Physics junctions

B. D. Josephson ISL1- 22 Brian Josephson 1973 Nobel Prize in Physics: JOSEPHSON JUNCTION CIRCUIT MODEL

i33 As i n iBv444[cos] i + i1 i2 i3 i4 v ? _

2-terminal element to model the Josephson Pair-tunneling current

2-teminal element to model the Quasi-Particle Pair interference current ISL1- 23 qB sin iA33 sin 44

ISL1- 24 Why is the Memristor Non-Volatile ? q v (t) + i s  P qP E +_ v  t _ 0  = E  0 t0 t0 +  P 

Assume t   (t ) (tv )( )  d0 0 0 E  t0 = 0 P = t 0 t0 t0 +  ISL1- 25 Example: A two-state Charge-Controlled Memristor Charge - controlled  i = q , Weber  - q curve :  =  (q) 1 + 10 10 low-resistance Charge- (high-conductance) state controlled high-resistance  (low-conductance) state v =  memristor q, 0 1  =  (q) Coulomb _ M(q), Ω 10

Memristance 10 Ω d 1 Mq()≜ q, dq 1 0.1 Ω 0 Coulomb ISL1- 26 Non-volatile memories are estimated to be a 400 billion dollar Industry by 2020 ! Imagine a PC which turns on instantly ! ISL1- 27 Why not Flash ? • Can not be economically scaled below 10 nanometers • Poor Retention time: Fails after switching between 10,000 and 100,000 times • Low Speed • Power Hungry • They lose about 20 percent of information for decade. ISL1- 28 Non-Volatile Nano Memristors will eventually replace the following conventional computer memories •Flash Memories •DRAMs •Hard Drives

ISL1- 29 I = ? q

+ V = 6 volts - φ 0 q = φ3

What happen when you connect a Memristor across a battery ?

ISL1- 30 ISL1- 31 q to infinity i = ? i for E > 0

+v = E volts- 3E(φ(0))2 φ t 0 0 q = φ3 3E(φ(0)+6t)2 t (td ) (0)v ( ) 0 Shocking Truth ! i (0) Et q (t ) (0) E t3 The DC V-I + curve consists dtq () 2 I E i (t )  3 (0)  E t ( E ) of only one v dt point V 3E (0) E t 2 -   (V, I) = (0, 0). 0

 as t   ISL1- 32 The Ideal Memristor does not have a DC V-I Curve ! ISL1- 33 Standing Assumption

All state variables xi in the state equation x = f ()x , v (Voltage-Controlled or Memristor) (Current-Controlled x = f ()x , i Memristor) have infinite range: -∞ < x < ∞ i ISL1- 34 An 8 nm Memristor

From: S. Pi, P. Lin, Q. Xia,“Cross point arrays of 8 nm × 8 nm memristive devices fabricated with nano imprint lithography”, J. Vac. Sci. Technol. B 31, 06FA02-1 - 06FA02-6, 2013 ISL1- 35 Memristor made from

A single Layer of the Molecule MoS2

100

A) μ I( 0 From: iv i V. K. Sangwan, D. Jariwala, I. S. Kim, K. S. Chen, T. J. Marks, L. J. Lauhon, iii M. C. Hersam, “Gate-tunable memristive phenomena mediated by -100 grain boundaries in single-layer MoS2”, Nature Nanotechnology 10, p. 403-406, -20 0 20

2015 V(V) ISL1- 36 How Do You Know Your Device is a Memristor ? Since hp’s 2008 publication in Nature of a nano-scale memristor, numerous other memristors have been published. Less than 5 such publications have an equation describing their device ! How then can they claim their device are memristors ? ISL1- 37 Experimental Definition of the i Memristor + v ̶

If it’s Pinched, i, (mA) 0.15

0.075 it’s a v, (V) 0

-0.075

Memristor -0.15 ISL1- 38 Genealogy of Memristors

Extended Generic Ideal Ideal Memristor Memristor Generic Memristor Memristor

Voltage – Controlled Voltage – Controlled Voltage – Controlled Voltage – Controlled i  G(,)x vv i  G()x v i  G()φ v G(,x 0)  dx dx ˆ dφ  g(,)x v  g()x v  v dt dt dt

ISL1- 39 The Memristor Universe EXTENDED MEMRISTOR v  R(,)x ii dx  f (,)x i R(,0)x  dt GENERIC MEMRISTOR v  R()x i IDEAL GENERIC MEMRISTOR dx  fˆ ()x i dt IDEAL MEMRISTOR dq v  R()q i  i dt ISL1- 40 Every Ideal Memristor spawns an Infinite family of Equivalent Generic Memristor

Siblings ISL1-41 Ideal Memristor Cousins All Generic and Extended JournalMemristors of Applied Physics 68, 6535 (1990); doi: 10.1063are/1.346832 Qualitatively Identical to Ideal memristors ISL1- 42 All Non-volatile Memories based on Resistance Switchings are

Memristors ISL1- 43 Following non-volatile memory devices are memristors • Re RAMS • Phase Change Memories • MRAMS • Ferro-Electric Non-volatile Memories • Atomic Switch ISL1- 44 Examples of Non-Volatile Memristors • RRAM Memristors (metal oxides Tio2, TaOx, etc.) • Polymeric Memristors (conducting polymers) • Ferroelectric Memristors (Ferroelectric films) • Manganite Memristors (Perovskite manganite) • Spintronic Memristors (spin-transfer torque magnetic layers)

ISL1- 45 Cat' s Whisker from the First Radios are Memristors

Bistable memristive behavior Input current versus voltage across device. Philmore cat’s whisker in contact with a Galena crystal. One clearly observes pinched hysteresis loop for cat’s whisker. ISL1- 46 Material Views Advanced Functional Materials www.MaterialViews.com 2012, 22, 4493-4499 www.afm-journal.de A Natural Silk Fibroin Protein-Based Transparent Bio-Memristor Mrinal K. Hota, Milan K. Bera, Banani Kundu, Subhas C. Kundu, and Chinmay K. Maiti

Cocoons of Cut pieces of Degumming: removal Degummed fibre mulberry silkworm cocoons of protein sericin of protein fibroin

Fibroin solution Thorough dialysis of fibroin Dissolved fibre Device Structure (conc. 2% w/v) solution to remove excess LiBr in 9.3 M LiBr ISL1- 47 Pinched hysteresis loop in the i – v plane resembles a seagull-wing in the log | i | - v plane

ISL1- 48 ISL1- 49 Scientific Reports 5 Article Number: 10022 www.nature.com/scientificreports Published: 7 May 2015 Nonvolatile Bio-Memristor Fabricated with Egg Albumen Film Ying-Chih Chen, Hsin-Chieh Yu, Chun-Yuan Huang, Wen-Lin Chung, San-Lein Wu & Yan-Kuin Su

Egg White Solid

I–V characteristics of the thermal-baked and dry-cured albumen devices ISL1-50 The Quest for building nano-scale solid state non-volatile memories dates back from 1939 ISL1- 51 Trans. Faraday Society, Vol. 35, pp. 1436-1439, 1939 ELECTRICAL CONDUCTION OF COMMERCIAL BORON CRYSTALS By J. H. BRUCE and A. HICKLING

ISL1- 52 Extracted from page 301 circuits protection make high conductor normal very certain transforms with The large the voltages poor . voltages conditions it temperature drop for conductor of valuable boron is of into abnormally certain resistance electrical from a which under good for for ISL to a 1 - 53 JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 9 SEPTEMBER 1962 Low-Frequency Negative Resistance in Thin Anodic Oxide Films T. W. HICKMOTT i, mA 70 60 50 40 30 20 -12 -10 -8 -6 -4 -2 10 v, V 0 2 4 6 8 12 -70 10 -60 -50 -40 -30 Preparation of metal-anodic oxide-metal sandwiches. -20 -10 Circuit for measuring electrical characteristics. t Negative resistance is not found at 60 Hz !

v(t) Small-signal conductance at ΔT zero DC bias voltage can be A varied continuously over a wide range by applying t voltage pulses, and tuning Conductance the pulse amplitude A, or Change of conductivity of aluminium oxide film by tuning voltage pulse 10-µsec and 100-µsec pulses of varying voltage. the pulse width ΔT . ISL1- 54 Journal of Applied Physics, Vol. 34, pp. 711-712, 1963 Negative Resistance in Thin Niobium Oxide Films By S. PAKSWER and K. PRATINIDHI

I-V characteristics of Al-Al2O3-Hg sandwich ISL1- 55 Proc. of the IEEE, Vol. 51, pp. 941-942, 1963 Current-Controlled Negative Resistance in Thin Niobium Oxide Film By K. L. CHOPRA

I-V characteristics with multiple negative resistance regions ISL1- 56 Some Excerpts of Confused and Ambiguous statements on Non-Volatile Memories • A “memory” effect where no negative resistance would normally occur….. [Hickmott, 1962; page 2673] • In all cases when a new memory state is induced, hysteresis is manifest in the V-I characteristic and the V-I loop is generated…. • Furthermore, a memory state never accompanies a V-I characteristic that does not exhibit hysteresis… • Hysteresis is also exhibited in the V-I characteristic when the memory is erased.. • In no circumstances is erasure observed when there is no hysteresis… [Simmons and Verderber, 1969; page 91] ISL1- 57 First Hint of Pinched Hysteresis Loop: A Device Dubbed LETTER 8 MEMORY was published in 1971 POLARIZED (LETETR ‘8’) MEMORY IN CdSe POINT CONTACT DIODES M. Kikuchi, M. Saito, H. Okushi Electrotechnical Laboratory, Mukodai, Tanashi, Tokyo, Japan. and A Matsuda Nippon Columbia Co. Ltd., Kawasaki, Kanagawa, Japan. (Received 10 March 1971 by T. Muto) Solid State Communications

vol. 9, p. 705-707, 1971. ISL1- 58 Researchers were mystified by hysteresis loops which pass through the origin ! Magnetic Hysteresis The lag or delay of a magnetic material known commonly as Magnetic Hysteresis, relates to the magnetization properties of a material by which it firstly becomes magnetized and then demagnetized.

ISL1- 59 Then Research was Frozen for the next 30 years ! Less than 10 papers on Solid state non-volatile memory devices were published between 1970 and 2000 ! ISL1- 60 55 Years of confused and Nanoelectronic Programmable Synapses Based on Pha Change Materialsmisunderstood for Brain-Inspired Computing non-volatile memory

Devices ISL1- 61 APPLIED PHYSICS LETTERS VOLUME 77, NUMBER 1 3 JULY 2000

Pinched Hysteresis loop of Switching performance of a capacitor-like structure based on Cr-doped

Cr-doped SrZrO3 memristor SrZrO3. (a) Applied voltage vs time and (b) readout current vs time. A negative voltage pulse of 2 ms switches the system into the low-impedance state. After applying a positive voltage pulse of 2 ms, the ‘‘information’’ written to the device is erased and the high impedance state is recovered.

Faster switching speeds, i.e., shorter write and erase pulses as used here, are also possible but require

higher voltage amplitudes. ISL1- 62 1987 Discovers Nobel high- Prize temperature in Super- conductors Physics

Johannes Georg Bednorz ISL1- 63 v i Minimum Pulse-Area Switching w H Theorem v There is a minimum pulse area A(x1 , x2) 0 t iv G()x w × H = A(x1 , x2) to switch between any 2 states x1 and x2 xx f (), v of any non-volatile memristor. f ()x , 0  0 H G A H= w 0 x x x 1 2 0 w ISL1- 64 2012, Yang, et al., 2011, Szot, et al., (a) 2009, Kim, et al., (b) (c) Nano Letters Applied Physics Letters Nanotechnology

2009, Jo, et al., 2012, Pickett , et al., (d) 2011, Hino, et al., (e) (f) Sci. Technol. Adv. Mater. Nano Letters Nature Materials

5 500 100 v, v, v, 0 0 0 0 0 mV V 0 V -500 -5 -100 -1000 -100 0 100 -6 0 3 6 -3 -2 -1 0 1 2 2003, Sakamoto, et al., (d) 2013, Nardi et al., (e) 2000, Beck, et al., (f) IEEE Trans. Electron Devices Applied Physics Letters Applied Physics Letters i, mA i, μA i, mA 3 Reset 40 2 1 Set 20 v, 1 0 v, v, 0 0 0 V 0 V 0 V -1 Set -20 -1 -2 Reset -40 -2 -3 -2 -1 -0.5 0 0.5 1 2 -0.5 0 0.5 -0.2 0 0.2 ISL1- 66 Not all memristors are Non-Volatile ! ISL1- 67 Non-Volatile Memristor Theorem : Two implies Infinite ! If a passive memristor exhibits 2 stable memory states, then it has a continuum of stable memory states. ISL1- 68 Fundamental memristor memory Theorem All passive non-volatile memristor memories are continuum (analog) memories.

ISL1- 69 Aplysia with a Nobel Prize Medal ISL1- 70 2000 Discovers Nobel the Prize molecular in basis of Physiology memory in Aplysia

ISL1- 71 ISL1- 72 (a)

Sensory Neuron Stimulus

(b) i (c) i(t), Amps 1 + 20 2 3 4 5 +_ 15 6 7 v(t) v 8 9 10 10 11 12 _ 13 14 5 15

q 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1415 1213 t, seconds 10 11 v(t), Volts 8 9 1 area = 50 7 5 seconds 6 Slope = 1 5 8 10 2 3 5 6 7 8 9 10 11 12 13 14 15 4 (G = 1 Siemens) 3 8 2 Slope = 2 (G = 2 Siemens) 1  0 750 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 t, seconds ISL1- 73 Synapses are Memristors

axon of neuron j  dq(t) pre-synaptic i(t)= terminal dt  Q d (t) v(t)= neuro- dt transmitters 0 qQ q

Slope post-synaptic  terminal M (Q) = memristance at q=qQ dendrite of neuron k Synaptic strength  memristance ISL1- 74 1961 Nobel Prize in Physiology

Hodgkin- Huxley Nerve Membrane Model

+ + INa + IK V V _Na _K GNa GK GL C V

E E E _ Na K L

Time-varying Sodium Time-varying Potassium conductance conductance Sir A. L. Hodgkin Sir A. F. Huxley From A.L. Hodgkin and A. F. Huxley A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. Journal of Physiology, Vol. 117, pp.500-544, 1952 ISL1- 75 The suggestion of an inductive reactance anywhere in the system was shocking to the point of being unbelievable. Kenneth Cole

ISL1- 76 ISL1- 77 Hodgkin’s Blunder

Hodgkin had struggled in vain searching for a physical interpretation of the squid axon inductance. He failed because he had mistaken the axon for a time-varying conductance, when in fact it has a simple explanation if the Potassium and Sodium ion channels are identified as memristors. A. L. Hodgkin, “The ionic basis of electrical activity in nerve and muscle,”

Biological Review, Vol. 26, pp. 339-409, 1951 ISL1- 78 (a) (c) I Outside Hodgkin-Huxley Cell +

V CM E

_ Inside (b) (d) I Outside Cell Na Channel K Channel + I INa IK IL C M + + + G G (b) V CM VNa Na VK K VL GL _ Na _ K _

ENa EK E Inside Cell _ L

ISL1-79 Axons are made of Memristors !

ISL1- 80 1904 Discovers Nobel Associative Prize memory in and Physiology learning phenomenon Ivan P. Pavlov ISL1- 81 ISL1-82 ISL1-83 Emulating Pavlov’s Dog Associative Learning Phenomenon Memristor Circuit for Emulating Pavlov’s Dog

Ag/AgInSbTe/Ta Memristor

Bell sound stimulus Meat sight R1=500Ω VCS(t) VUS(t) stimulus + + - - + R2=2400Ω Vout - From: Y. Li, L. Xu, Y. P. Zhong, Y. X. Zhou, S. J. Zhong, Y. Z. Hu, L. O. Chua, X. S. Miao Associative Learning with Temporal Contiguity in a Memristive Circuit for Large-Scale Neuromorphic Networks Adv. Electronic Materials, 1500125, p.1-8,2015. ISL1- 84 Waveforms Measured from Memristor Circuit Emulating Pavlov’s Experiment

) 0.6

CS 0.4 V ( 0.2 0.0 Bell -0.2

) 1.5 US

V 1.0 ( 0.5 0.0 Food Food -0.5

) 1.5

OUT 1.0 V ( 0.5 0.0 -0.5

0 10 20 30 40 50 60 Response Response

ISL1- 85 ISL1- 86 A Medieval Catapult K. Liu, C. C. Cheng, J. Suh, R. T.Kong. D. Fu, S. Lee, J. Zhou, L. O. Chua and J. Wu Advanced Materials Vol.26,no.11,pp.1746-1750 March 2014

ISL1- 87 Dr. Junqiao Wu with President Obama at the award ceremony for Presidential Early Career Awards for Scientists and Engineers Whitehouse, April 14, 2014.

ISL1-88 A Micro-Catapult Memristor i(mA)

-4 -3 -2 -1 0 1 2 3 4 v(V) i + v -

Micro-Catapult Memristor Pinched Hysteresis Loop Micro-Catapult Memristor ISL1- 89 A Vanadium Dioxide Micro Catapult Memristor

From: Advanced Materials, 2013

50 µm Vanadium Dioxide Memristor • 1000 times more powerful than a human muscle • Can Catapult objects 50 times heavier than itself • Can catapult objects over a distance 5 times its length • Faster than the blink of an eye. ISL1- 90 GE90 Turbofan for Boeing 777

Performance of Turbofan for Boeing 777: Power Density: 9 kW/kg Rotation Speed: 10,000 rpm Performance of Micro Catapult Memristor : Power Density: 39 kW/kg Rotation Speed: 200,000 rpm ISL1- 91 Some Memristor Circuits have Hamiltonians !

ISL1- 92 φ + i State Equations dx  i L = 1 v dt 0 x di 2 - 1 3 (1xi )  xx dt Hamiltonian Equations 3 dx H dx dx   1 ddt i 1 d i d ≜ i dt

di H di 2 di    (1) x  (1 x2 ) d x d i dt From : 1 3 M. Itoh and L. Chua Hamiltonian: H (x , ixxi ) ()  Memristor Hamiltonian Circuits 3 Int. J. of Bifurcation and Chaos Pseudo Potential Pseudo Kinetic Energy H (i) vol. 21, pp.2395-2425, 2011 Energy Hx (x) i ISL1- 93 H(x, i) = 0 ii The Hamiltonian has a constant value

H(x, i) = H0 along each curve. x Hamiltonian 1 H (,)x i x x3  i 3

ISL1- 94 x(t) p(t) = v(t) i(t)

x(0) = -0.8 (a) i(t) i(0) = 4 (d)

t E(t)=dt p(t′)  0 v(t) (b)

(e) (c) ISL1- 95 H (t) Pseudo Potential Energy x 1 H =+x + x 3 i x 3

H (t) Pseudo Kinetic Energy i

H i = i

Hamiltonian H(t)

H = Hxi+ H Pseudo Pseudo Potential Kinetic Energy Energy ISL1- 96 i (-0.8, 4)

H0 ≈ 3.03

(0.8, -4)

Two typical trajectories which tend to the x-axis along on the contour H ≈ 3.03. 0 ISL1- 97 Shocking Revelation ! Not all Hamiltonian Systems are

Conservative ISL1- 98 Question : What physical quantity is conserved in the Hamiltonian H(x , i) ? ISL1- 99 Answer : The Hamiltonian H(x , i) conserved the total flux φ(t) = φ(t) + φ(t) Total Inductor Memristor

ISL1-100 Mendeleev’s First Published Periodic Table, 1869

Ga Gallium

Sc Ge Scandium Germanium

ISL1-101 The First 25 Circuit Elements dt v()   , 1, 2 ,... if    dt  v (tif ) , 0   v() ()t  t  vdif , 1    t   ( ),  2 ,2 v 3 ,...dd  dif        112 

dt i()   , if   1, 2 ,...  dt  i (t ) , if   0 i() ()t  t  i d, if   1  t     2 i ( ) d  d  d  , if    2 ,  3 ,...        1 1 2  ISL1-102 Universal Pinched Hysteresis Loop Theorem The ((α + 1) , (β + 1)) element associated with any (α , β) element must exhibit a pinched hysteresis loop. i i i(α + 1) i(β) + + α α + 1 v β v β + 1 ̶ ̶ 0 v(β + 1) 0 v(α) i i i i i i + + + + + + -1 0 -1 0 -2 -1 v -1 v 0 v -2 v -1 v -1 v 0 ̶ ̶ ̶ ̶ ̶ ̶ Memristor Resistor Memcapacitor Capacitor Meminductor Inductor

Example 1 Example 2 Example 3 ISL1-103 ()α , β - Elements

i i(β)

v v(α)

(a) (b) (a) Symbol for a v(α) – i(β) element. (b) The constitutive relation of a curve or subset of points in the v(α) – i(β) plane. Leon Chua Device Modelling Via Basic Nonlinear Circuit Elements IEEE Transactions on Circuits and Systems vol. CAS-27, No. 11, pp. 1014-1044, Circuit-element array: Each dot with coordinates 1980 (α) (β) (α, β) denotes a v – i element. ISL1-104 ()α , β - Elements

i i(β) Memristor (α, β) = (-1 , -1) v v(α)

(a) (b) (a) Symbol for a v(α) – i(β) element. (b) The constitutive relation of a curve or subset of Meminductor (α) (β) points in the v – i plane. (α, β) = (-2 , -1) Leon Chua Device Modelling Via Basic Nonlinear Circuit Elements IEEE Transactions on Circuits and Systems vol. CAS-27, No. 11, pp. 1014-1044, Memcapacitor (α, β) = (-1 , -2) 1980 ISL1-105 The Golden Strip i i(β)

v v(α)

(a) (b) (a) Symbol for a v(α) – i(β) element. (b) The constitutive relation of a curve or subset of points in the v(α) – i(β) plane. Leon Chua Device Modelling Via Basic Nonlinear Circuit Elements IEEE Transactions on Circuits and Golden Strip : α ≤ 0, β ≤ 0 Systems |α - β| ≤ 2 (-1 , -2) vol. CAS-27, No. 11, pp. 1014-1044, Theorem: Only elements belonging to the Golden Strip are passive and physically realizable. 1980 ISL1-106 ISL1-107 ISL1-108 Inspiration from Chemistry The latest Periodic Table contains 118 elements : • 92 elements exist in nature. • 26 elements are artificially synthesized. Most synthesized elements are unstable --- some exist for only a few milliseconds, then decomposed into two lighter elements.

ISL1-109 Periodic Table of the Elements

The periodic table lists all the known elements of the universe. Here, the human-made ones are called out. Scientists are working to find out if the table ever ends.

ISL1- 110 Nonlinear Circuit Theory predicts the existence of infinitely many passive (α , β) - elements So far, only 4 passive circuit elements have been built without power supply: • Resistor • Inductor • Capacitor • Memristor ISL1- 111 A Cap for Number of Chemical Elements No Element beyond 137 can exist ; because it would violate the laws of physics. ISL1- 112 The Mysterious 137

Fy137 Feynmanium

ISL1- 113 Fractional - Order mem – (α , β) Circuit elements 0 ≤ α ≤ 1 0 ≤ β ≤ 1

ISL1- 114 Fractional Derivative Definition : Fractional Derivative Laplace Transform of Riemann (Riemann – Liouville definition) - Liouville differential Operator 1 d n t RnD f( ttfd )()( )  1 The Laplace Transform of the α-order at n  (n)  dt a Riemann-Liouville differential n operator is: d n n  atjf tt( a ) ,, R  dt L 0 Dt fs(t)(t) L  f   nn1,  n1 where Γ is the gamma function and  sDfk Rk  1 (t).   0 t t  0 atj is the Riemann-Liouville k0 integral operator defined by: For zero initial conditions we have 1 t j()(). t1 f  d   at  L 0 Dt f(t)  s L f (t) () a

ISL1- 115 ISL1- 116 ISL1- 117 Interpolated characteristics of memfractor between a memcapacitor, a memristor,

a meminductor and a second-order memristor. ISL1- 118 Graphical representation of the coefficient Graphical representation of the coefficient ()α + α b=αα 12. c=αα(1 ). ()α1 ,α2 1 2 (α1 ,α2 ) 2 1 2 ISL1- 119 ISL1-120 The Curse of Impasse Points

1 iR ivv=-+ 3 i i Q RRR R iC L iR 1 2 3 3 v R = 1Ω v (a) v R C L L = 1H vR R 0 -1 1 vR

2 1 - qC qvv=-+ 3 3 Q Q CCC 2 1 2 3 (a) (b) 3 (b) iR -1 0 1 vC iR Q 2 iL 1 2  3 Q 3 2 vR v C vC i L iR C i3 -1 0 1 vR

2 vR 1Ω v (c) - C v3 3 Q2 (c) (d) Circuit for demonstrating the necessity of Circuit for demonstrating the necessity of connecting a capacitor across a connecting a v (0) – i (-2) element across a nonmonotonic voltage-controlled Capacitor. nonmonotonic voltage-controlled Resistor. ISL1-121 Memristors are Ubiquitous !

ISL1-122 Sweat Ducts are Memristors

(a) (c) Pinched Hysteresis Loop Measured from the Sweat duct v, V

(b) - + i,

+ 0 μA +

+ + + + +

+ + +

+ + + + ISL1-123 Memristive Properties of Skin

Measured by: Olivier Pabst, PhD Student

Supervisor: Professor Orjan Martinsen

Department of Physics, University of Oslo September 21, 2015

Voltage over voltage plot over 2 periods. Vpp =10:5V, f =20mHz. Left Hand in saline solution, other electrode on left forearm

ISL1-124 Our body is covered with Memristors ! ISL1- 125 (a)

i + -

(b)

(a) Ag/AgCl (a)+ PinchedElectrodes Hysteresis Loop Measured (a) from- the Venus FlytrapVenus Flytrap (a)(a) (c) (c) i, µA i i 2014, Volkov+ , e+t al., i i J. of Plant+ Signaling- - and Behavior- + - v, V (b)(b) (b) 0 Ag/AgCl + v, V Ag/ElectrodesAgCl ++ Ag/AgCl + Ag/AgCl (b) Venus Electrodes- Electrodes Venus Flytrap Electrodes Flytrap - Ag/AgCl + (c) - Electrodes- VenusVenus Flytrap Flytrap i, µA Pinched Hysteresis Loop 2014, Volkov, et al., - ISL1-126 J. of Plant Signaling Venus Flytrap (c)and Behavior (c) i, µA i, µA 0 v, V 2014, Volkov, et al., (c)2014, Volkov, et al., J. of Plant Signaling J. of Plant Signaling i, µA and Behavior 2014and Behavior, Volkov, et al., J. of Plant Signaling and Behavior 0 0 v, V v, V 0 v, V Venus Flytrap

ISL1-127 i, µA

Ag/AgCl Electrodes

0 v, V

Venus Flytrap

ISL1-128 Action Potential Generated by calcium Ion-channel Memristors in the Venus Flytrap

From : A special pair of phytohormones controls excitability, slow closure, and external stomach formation in the Venus flytrap From : María Escalante-Pérez, Elzbieta Krol, Annette Stange, Dietmar Geiger, Khaled A. S. Al-Rasheid, Bettina Hause, Erwin Neher, and Rainer Hedrich Proc. of National Academy of Science, vol. 108, September 13, 2011. ISL1-129 Discovers 1991 the function Nobel of single Prize ion in channels in Physiology cells

Erwin Neher ISL1-130 Voltage-Gated Ion Channels are Memristor ISL1-131 Where there is smoke there is fire ISL1- 132 Where there is ion there is memristor ISL1- 133 ISL1-134 A SunPatiens Flower

Platinum wire i v

Platinum wire

ISL1-135 2

Pt µA 0

, , i

i /

-1 Pt -2

-4 -2 0 2 4 vv ,/ VV

ISL1-136 ISL1-137 Alessandro Volta

Volta explains the principle of the “electrode column” to Napoleon

Voltaic Pile In honor of his invention Volta was (Invented by made a count by Napoleon in 1801. Volta 1800) ISL1-138 Setup of the carbon rod electrodes

ISL1-139 Carbon arc discharge in operation ISL1-140 40 40 Voltage(V) 30 Current(A) 30 20 20

10 10

0 0 Voltage (V) Voltage -10 -10 (A) Current

-20 -20

-30 -30

-40 -40 Davy’s Carbon0.00E+00 -1.00E-03Rod Electrodes2.00E-03 3.00E-03 Exhibits4.00E-03 the5.00E-03 Memristor Time (s) Pinched-Hysteresis Loop(a) Fingerprints

700Hz

Current (A) Current -10

-20

-30

-40 -40 -30 -20 -10 0 10 20 30 40 Voltage (V) (b) Input Testing Signal: 3 KHz Sinusoidal Voltage ISL1-141 (a) Experimental voltage and current waveforms and (b) V-I hysteresis loop of carbon rod discharge at 700 Hz Video

ISL1-142 Bonaparte Napoleon Sir Humphry Davy

In 1808, when France was at war with England, Bonaparte Napoleon has decided to award Sir Humphry Davy

The Prix Napoleon de Institut ! ISL1-143 Faraday Invented the Inductor in 1831

Michael Faraday ISL1-144 4 Basic Circuit Elements dv(t) 1745 i(t)  C dt Memristor v(t)=R(x )i(t) dx 1801 M =f(x ,i ) Sir Humphry Davy dt

1827 v(t)  R i(t)

di(t) 1831 v(t)  L dt ISL1-145 The Most General Memristors are called Extended Memristors i State and Input-Dependent Ohm’s Law Nanoelectronic Programmable Synap Based on Phase Change iMaterialss(t) for Brainv-Inspired Computing vi R(,)x i where R(,)x 0  State Equation Current-Controlled dx Memristor  f (,)x i dt ISL1-146 Why is the Condition R( ,0)x  necessary in the definition of the state and Current-Dependent Ohm’s Law v  R(,)x ii

ISL1-147 An Extended Memristor with Unbounded Memristance i(t) i(t) = sin t + dx ==i sint sin t v(t) dt Assuming x(0) = -1 - t xs(t)dt =1-11  + it n -cos t-c os t State-Dependent Ohm’s Law 0   x x(t) v =i i v (t)=(t)=(t)i x  i(t) R(x, i) State Equation v(t)  -cos t dx Hence, = i dt 2 2 2 2 f ()x, i v (t)i (t)=( -cos t ) +(sint) 1 ISL1-148 An Extended Memristor t = π v i = sint q t = 3π/2 t = π/2 v=-cos t q = v i v 2 (t)+(t)=i 2 1 v Slope = 1 t = 0 The v vs. i loci is NOT Pinched ! Observations : i dx x (2). (1).  = i + v ==i x dt i f ()x, i v C = 1F  x = q (Charge) R()x, i - q = v (3). (1) and (2) q = v ISL1-149 If It’s Not Pinched It’s Not A MEMRISTOR

ISL1- 150 Example of An Erroneous Memcapacitor

Hysteresis loop in q - v plane is not pinched 0 not a memcapacitor General scheme of a solid-state memcapacitor. A metamaterial medium consisting of N metal Charge-voltage plot at different applied voltage frequencies layers embedded into an insulator is inserted f. The decrease in the hysteresis at higher frequencies is a between the plates of a “regular” capacitor. signature of memcapacitors. ISL1-151