The Effect of Electrode Size on Memristor Properties: An Experimental and Theoretical Study

1. The Effect of Electrode Size on Memristor Properties: An Experimental and Theoretical Study Ella Gale, Ben de Lacy Costello and Andrew Adamatzky

2. We Want To Know… • Which Model of Memristance Works Best • What Effect Electrode Size has on Memristor Properties

3. Theories of Memristance

4. ChUa’sPhenomenalogical Definition • M = memristance • q= charge • φ = magnetic flux

5. There Are Three Theoretical Memristor Models • Strukov et al’sPhenomenalogical Model • Georgiou et al’s Bernoulli Equations • Mem-Con Model 1. Strukov et al 2. Georgiou et al 3. Gale,

6. 1. Phenomenological Model = ionic mobility of the O+ vacancies Roff = resistance of TiO2 Ron = resistance of TiO(2-x) This is a 1-D model Strukov et al, The Missing Memristor Found, Nature, 2008

7. 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

8. 2. Georgiou et Al’s Model The Model Predictions • = ‘Dimensionless Lumped Parameter’ • Contains: • ‘all’ physical dimensions of device • all parameters of experiment • is related to • is related to This is a 1-D model

9. 3. Memristance, as Derived from Ion Flow • Universal constants: • , Experimental constants: product of surface area () and electric field (), • , Material variable, =, where This is a 3-D model Gale, The Missing Magnetic Flux in the HP Memristor Found, 2011

10. Mem-Con Model Conservation Function Memory Function

11. Our Premise • Goal: To Investigate Which Theoretical Model Works Best • Method: • Spatial Dimension Effects (Strukov and Mem-Con) • Test Hysteresis Predicitons (Georgiou)

12. Size Predictions • Strukov et al’ssuggests 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

13. 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

14. Two Different Types of Memristor Behaviour Seen in Our Lab Curved (BPS-like) Memristors Triangular (UPS-like) Memristors • Pictures

15. Test 1 • The Effect of • Varying Electrode Size

16. Curved Switching Memristors

18. As , • , • Fit Memory Function to as a function of • Fit Conservation Function to as a function of F

19. Memory function Describes ’s variation with F Only 1 fitting parameter needed: (, ~ )

20. Conservation function Describes ’s variation with F One Fitting Parameter, , = Ωm-1 (Bulk value: 1012Ωm-1)

21. 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

22. Test 2 • Is the Hysteresis Related to the ‘dimensionless lumped parameter’, ?

23. 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) Simulated Result Ref…

24. Measured Hysteresis Versus Experimental Values of

25. Does GeorGiouet Al’s Predicted relate to measured ?

26. Hysteresis Size Depends on F and Ron

27. 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)

28. Filamentary Extension of the Mem-Con Theory of Memristance and its Application to Titanium Dioxide Sol-Gel Memristors Ella Gale, Ben de Lacy Costello and Andrew Adamatzky

29. Two Different Types of Memristor Behaviour Seen in Our Lab Curved (BPS-like) Memristors Triangular (UPS-like) Memristors • Pictures

30. Memristor Structure and Function

31. Shape of the Filament

32. We Want To… • Extend the Mem-Con Model to Describe Filamentary (Triangular) Memristors

33. The Mem-Con Theory

34. Definition of the Memristor Inductor Memristor Capacitor Resistor

35. What the Flux? But, where is the magnetic flux? Strukov et al, 2008 Chua, 1971

36. 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

37. Mem-Con Theory Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011

38. ExTending the Mem-Con Theory to Filaments

39. Shape of the Filament

40. Equivalent Circuit Diagram To The Device Chemistry e

41. M: Time-DepedendAnt Expression for the Volumes • Memristance based on • Due to the shape, varies with

42. Vacancy Magnetic Field • Vacancy Magnetic Field  • G can be solved by • where we use • and Cuboid of

43. Vacancy Magnetic Field

44. Memory FuNction • is the surface normal for area infinitesimal • Wb • For Strukov’s device: • b [1] • As • [2] • , • and Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011 Chua, Memristor: The Missing !!!

45. Resistance of a Conical Resistor • Not as easy as it looks.

46. Filament Resistance Ref?

47. Equivalent Circuit Diagram To The Device Chemistry e

48. Comparison to Experiment Theoretical Model Experiment

50. Starting From The Ions… • Memristance is a phenomenon associated with ionic current flow • Therefore  calculate the magnetic flux of the IONS • Vacancy Volume Current  , L = eLectric field • Vacancy Magnetic Field  • Vacancy Magnetic Flux  Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011