Report

The Effect of Electrode Size on Memristor Properties: An Experimental and Theoretical Study Ella Gale, Ben de Lacy Costello and Andrew Adamatzky We Want To Know… A. Which Model of Memristance Works Best B. What Effect Electrode Size has on Memristor Properties THEORIES OF MEMRISTANCE CHUA’S PHENOMENALOGICAL DEFINITION = M = q = φ = memristance charge magnetic flux There Are Three Theoretical Memristor Models 1. Strukov et al’s Phenomenalogical Model 2. Georgiou et al’s Bernoulli Equations 3. Mem-Con Model 1. Strukov et al 2. Georgiou et al 3. Gale, 1. Phenomenological Model = off − 2 off on () This is a 1-D model = ionic mobility of the O+ vacancies Roff = resistance of TiO2 Ron = resistance of TiO(2-x) Strukov et al, The Missing Memristor Found, Nature, 2008 2. GEORGIOU ET AL’S MODEL • Rewrote Strukov et al’s model as Bernoulli Equations • Gained Some Analytical Solutions • Predicts the Size of the Hysteresis, , in Memristor I-V curves 2. GEORGIOU ET AL’S MODEL The Model Predictions = ‘Dimensionless Lumped Parameter’ is related to is related to Contains: • ‘all’ physical dimensions , , , of device • all parameters ( , 0 , 0 ) of experiment = 2 = This is a 1-D model 2 0 0 2 2 −1 3. Memristance, as Derived from Ion Flow • Universal constants: 0 4 0 = (()) 4 This is a 3-D model • = ∙ ∙ , Experimental constants: product of surface area () and electric field (), • , Material variable, = ( ), where ( , , ) Gale, The Missing Magnetic Flux in the HP Memristor Found, 2011 MEM-CON MODEL Memor y Function = ∙ + 2 C o n s e r va t i o n F u n c t i o n − = = + OUR PREMISE Goal: To Investigate Which Theoretical Model Works Best Method: A. Spatial Dimension Effects (Strukov and Mem-Con) B. Test Hysteresis Predicitons (Georgiou) SIZE PREDICTIONS • Strukov et al’s suggests no effect of size of E or F • Georgiou et al suggest no effect of E or F • Mem-Con model suggests that changing E or F will affect memristance • ∴ Test whether there is an effect of altering E or F Our Memristors • Crossed Aluminium electrodes • Thin-film (40nm) TiO2 sol-gel layer • E = 4mm • F = 1, 2, 3, 4 or 5mm 1. Gergel-Hackett et al, A Flexible Solution Processed Memristor, IEEE Elec. Dev. Lett., 2009 2. Gale et al, Aluminium Electrodes Effect the Operation of Titanium Dioxide Sol-Gel Memristors, Submitted 2012 Two Different Types of Memristor Behaviour Seen in Our Lab C ur ved (BP S - li ke) M em ri s to r s Pictures Tri a n gu la r ( U P S - li ke) M em ri s to r s TEST 1 The Effect of Varying Electrode Size CURVED SWITCHING MEMRISTORS As → 0 , → , → Fit Memory Function to as a function of = = ∙ () + 2 Fit Conservation Function to as a function of F = () Memory function Describes ’s variation with F Only 1 fitting parameter needed: = 2.09 × 1025 (2 = 1.25 × 10−20 , ~ 1 × 10−23 %) CONSERVATION FUNCTION DESCRIBES ’S VARIATION WITH F One Fitting Parameter, , = 6.82 × 1010 Ωm-1 (Bulk value: 1012 Ωm-1) SO, • Measured and vary with electrode size • This relationship is well described by the Mem-Con theory • Hysteresis is effected by Electrode Size • The Mem-Con Theory Correctly Predicts that Memristance Should be a Function of the Three Spatial Dimensions • The Strukov Theory Incorrectly Asserts that it is Only a Function of 1 Spatial Dimenion TEST 2 Is the Hysteresis Related to the ‘dimensionless lumped parameter’, ? THE EXAMPLE GIVEN IN GEORGIOU ET AL’S PAPER Voltage Source Waveforms: • Green: Bipolar Piece-Wise Linear (analytically calculable) • Red: Sinusoidal (not analytically calculable) • Blue: Triangular (analytically calculable) Ref… Simulated Result MEASURED HYSTERESIS VERSUS EXPERIMENTAL VALUES OF DOES GEORGIOU ET AL’S PREDICTED RELATE TO MEASURED ? HYSTERESIS SIZE DEPENDS ON F AND RON SUMMARY • Georgiou et al’s Bernoulli Equation Formulation does not work at predicting hysteresis* • Electrode Size can be changed to control hysteresis size* • The Mem-Con Model can be used to predict which electrode sizes will give a certain max or min resistance value (at the same omega)* • All three spatial dimensions of the memristor are important in describing memristance • The Mem-Con Model is a good model for real world memrstors * For Curved Type Devices (see next talk for an explanation) FILAMENTARY EXTENSION OF THE MEMCON THEORY OF MEMRISTANCE AND ITS APPLICATION TO TITANIUM DIOXIDE SOLGEL MEMRISTORS Ella Gale, Ben de Lacy Costello and Andrew Adamatzky Two Different Types of Memristor Behaviour Seen in Our Lab C ur ved (BP S - li ke) M em ri s to r s Pictures Tri a n gu la r ( U P S - li ke) M em ri s to r s Memristor Structure and Function SHAPE OF THE FILAMENT We Want To… Extend the Mem-Con Model to Describe Filamentary (Triangular) Memristors THE MEM-CON THEORY Definition of the Memristor Inductor Memristor Resistor Capacitor What the Flux? But, where is the magnetic flux? Strukov et al, 2008 Chua, 1971 = − 2 () = = CALCULATING THE CHUA MEMRISTANCE The Mem-Con model is based on calculating the MAGNETIC FLUX of the IONS for several reasons: • The IONS are the memory property, i.e. they hold the state of the memristor • The IONS move slower than the electrons and it is this that causes both the lag (hysteresis) and frequency response • The ION mobility, , is the physical quantity that controls the dynamics of the system Therefore, using magnetostatics to calculate the relationships between the ionic magnetic flux and charge we will arrive at a formula for memristance that satisfies Chua’s definition Mem-Con Theory • = + • = • = (− )TiO2 ↔ () ↔ Ionic ↑ ↔ () ↔ Electronic Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 EXTENDING THE MEM-CON THEORY TO FILAMENTS SHAPE OF THE FILAMENT EQUIVALENT CIRCUIT DIAGRAM TO THE DEVICE CHEMISTRY e () = 1 1 + + 1 + 2 − M: TIME-DEPEDENDANT EXPRESSION FOR THE VOLUMES • • Memristance based on Due to the shape, () varies with () = 1 3 (1 2 + 21 + ()2 ) = 1 3 (()2 + 2 2 + 2 2 ) 1 = 2 −21 − 22 + 22 + 2 + 4 1 2 − 1 2 + 1 2 + 2 2 = 21 − 22 + 22 = 2 + 2 Vacancy Magnetic Field Vacancy Magnetic Field 0 × = = 4 || G can be solved by = where we use =1 2 + 2 + 2 = × 3 2 and Cuboid of ∙ 2 ∙ 2 ≡ { , , } Vacancy Magnetic Field = 0, − , ∙=0 MEMORY FUNCTION + + = ∙ − − is the surface normal for area infinitesimal = = 6.52 × 10−30 Wb For Strukov’s device: = ‼! Wb As = [2] = () , and = [1] 1. Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 2. Chua, Memristor: The Missing !!! RESISTANCE OF A CONICAL RESISTOR Not as easy as it looks. = 2()2 − 2 − 2 2 + − 2 2 + ( − ) FILAMENT RESISTANCE = 1 − +1 Ref ? EQUIVALENT CIRCUIT DIAGRAM TO THE DEVICE CHEMISTRY e () = 1 1 + + 1 + 2 − COMPARISON TO EXPERIMENT Experiment Theoretical Model Starting From The Ions… • Memristance is a phenomenon associated with ionic current flow • Therefore calculate the magnetic flux of the IONS Vacancy Volume Current = (2−) Vacancy Magnetic Field = Vacancy Magnetic Flux = , L = eLectric field × || 0 || ( 4 ) Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 Calculate the Magnetic B field Associated with the ions Vacancy Volume Current = , ( , ) L = eLectric field Starting From The Ions… • Memristance is a phenomenon associated with ionic current flow • Therefore calculate the magnetic flux of the IONS Vacancy Volume Current = (2−) Vacancy Magnetic Field = Vacancy Magnetic Flux = , L = eLectric field × || 0 || ( 4 ) Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 CONCLUSIONS & FURTHER WORK Conclusions F u r t h e r Wo r k • Filamentary addition to the MemCon model gives good qualitative agreement to experiment Work out the quantitative values Re-do derrivation allowing a background bulk memristance With Thanks to • Ben de Lacy Costello • Andrew Adamatzky • David Howard • Larry Bull • Steve Kitson (HP UK) • David Pearson (HP UK) • Bristol Robotics Laboratory FURTHER WORK • A larger study to test Georgiou et al’s model has been undertaken • Repetition of size experiments with a different memristor at a different lab How does a Neuron Compute? Influx of Ionic I Voltage Spike Axon: Transmission along neuron Synapse: Transmission between neurons Memristive Systems to Describe Nerve Axon Membranes Synapse Long-Term Potentiation Starting From The Ions… • Memristance is a phenomenon associated with ionic current flow • Therefore calculate the magnetic flux of the IONS Vacancy Volume Current = (2−) Vacancy Magnetic Field = Vacancy Magnetic Flux = , L = eLectric field × || 0 || ( 4 ) Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 A HUGE PROBLEM OF TERMINOLOGY ReRAM • • • • Definition based on behaviour UPS – Voltage polarity irrelevant BPS –Voltage polarity relevant Pinched hysteresis loop in I-V space • Different behaviour based on forming process, complience current Memristor • Satisfy Chua’s definition: = V= • Pinched hysteresis loop in I-V space • -- The Memristor as a Synapse Before learning Before learning After learning After learning During learning Spike-Time Dependent Plasticity, STDP • • • • Process by which synapses are potentiated Related to Hebb’s Rule Possibly a cause of memory and learning Relative timing of spike inputs to a synapse important Bi and Poo, Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength and Postsynaptic Cell Type, J. Neurosci., 1998 Memristor Structure and Function Chua’s Definitions of Types of Memristors Charge-Controlled Memristor = () () ≡ F lux-Controlled Memristor = () () () ≡ L. Chua, Memristor – The Missing Circuit Element, IEEE Trans. Circuit Theory, 1971 Definition of the Memristor Inductor Memristor Resistor Capacitor What the Flux? But, where is the magnetic flux? Strukov et al, 2008 Chua, 1971 = − 2 () = = Starting From The Ions… • Memristance is a phenomenon associated with ionic current flow • Therefore calculate the magnetic flux of the IONS Vacancy Volume Current = (2−) Vacancy Magnetic Field = Vacancy Magnetic Flux = , L = eLectric field × || 0 || ( 4 ) Mem-Con Theory • = + • = • = (− )TiO2 ↔ () ↔ Ionic ↑ ↔ () ↔ Electronic Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 Two Different Types of Memristor Behaviour Seen in Our Lab C ur ved (BP S - li ke) M em ri s to r s Pictures Tri a n gu la r ( U P S - li ke) M em ri s to r s Memristor I-V Behaviour Our Intent: To make a memristor brain & thus a machine intelligence Connecting Memristors with Spiking Neurons to Implement STDP Simulation Results 1. Zamarreno-Ramos et al, On Spike Time Dependent Plasticity, Memristive Devices and Building a Self-Learning Visual Cortex, Frontiers in Neuroscience, 2011 0. Linares-Barranco and Serrano-Gotarredona, Memristance can explain Spike-TimeDependent-Plasticity in Neural Synapses, Nature Preceedings, 2009 But, Memristors Spike Naturally! Current Spikes Seen in I-t Plots Spikes are Reproducible Volta ge Square Wave Cur rent Spike Response Spikes are Repeatable Voltage Ramp Current Response Memristor Behaviour Looks Similar to Neurons Memristor Neuron Bal and McCormick, Synchronized Oscilliations in the Inferior Olive are controlled by the Hyperpolarisation-Activated Cation Current Ih, J. Neurophysiol, 77, 3145-3156, 1997 SPIKES SEEN IN THE LITERATURE Spintronic Memristor Current Spikes Pershin and Di Ventra, Spin Memristive Systems: Spin Memory Effects in Semi-conductor Spintronics, Phys. Rev. B, 2008 Properties of Spikes Direction of Spikes is related to ∆ not V The switch to 0V has a associated current spike Spikes are repeatable Spikes are reproducable Spikes are seen in bipolar switching memristors/ReRAM • Spikes are not seen in unipolar switching, UPS ReRAM type memristors • • • • • Two Different Types of Memristor Behaviour Seen in Our Lab Cur ved (BPS-like) Memristor s Triangular (UPS -like) Memristor s Where do the Spikes Come From? Does Current Theory Predict Their Existence? Mem-Con Model Applied to Memristor Spikes Neurons Memristors q φ q φ I V V I In Chua’s Model Neuron Volta ge Spikes ∆ = (())∆ • Dynamics related to min. response time, τ, related to speed of ion diffusion across membrane • Memory property = ??? • Neuron operated in a current-controlled way Memristor Cur rent Spikes ∆ = ∆ • Dynamics related to τ, which is related to • Memory property = qv • Memristor operated in voltage controlled way What is the Memory Property of Neurons? • More complex system than a single memristor • Short-term memory associated with membrane potential • Long term memory associated with the number of synaptic buds Memristor Models Fit the Data Sol-Gel Memristor Negative V Sol-Gel Memristor Positive V Memristor Model Fits the PEO-PANI Memristor Al-TiO2-Al Sol-Gel Memristor Time & Frequency Dependence of Hysteresis for Al-TiO2-Al Au-TiO2-Au WORMS Memory Au-TiO2-Au WORMS Memory I-t Response to Stepped Voltage Time Dependent I-V Al-TiO2-Al Current Response to Voltage Ramp Voltage Ramp Cur rent Response Further Work Neurology: • Modelling Neurons with the Mem-Con Theory to prove that they are Memristive • Investigate the Memory Property for neurons Unconventional Computing: • Further Investigation of memristor and ReRAM properties • Attempt to build a neuromorphic control system for a navigation robot Summary • • • • Neurons May Be Biological Memristors Neurons Operate via Voltage Spikes Memristors can Operative via Current Spikes Thus, Memristors are Good Candidates for Neuromorphic Computation • A Memristor-based Neuromorphic Computer will be Voltage Controlled and transmit data via Current Spikes