M EMRISTORS & C O.

An introduction to the physics and electronics

of passive circuit elements with memory

Sebastiano Peotta

Quantum Transport and Information Group

Scuola Normale Superiore di Pisa

Internal Seminar
20 January 2010
S UMMARY

Introduction
The fourth element
Motivations

The first memristor

Linear and non linear drift models
Memcapacitors and meminductors

Physical realizations
Overview
Semiconductor/Ferromagnet junction
Phase change memristors
Spintronic devices
Memcapacitor model
Higher order memristive systems
RRAM memory
S UMMARY

Introduction
The fourth element
Motivations

The first memristor

Linear and non linear drift models
Memcapacitors and meminductors

Physical realizations
Overview
Semiconductor/Ferromagnet junction
Phase change memristors
Spintronic devices
Memcapacitor model
Higher order memristive systems
RRAM memory
S UMMARY

Introduction
The fourth element
Motivations

The first memristor

Linear and non linear drift models
Memcapacitors and meminductors

Physical realizations
Overview
Semiconductor/Ferromagnet junction
Phase change memristors
Spintronic devices
Memcapacitor model
Higher order memristive systems
RRAM memory
The fourth element

Current i
Voltage v
R
Charge q = dt i
Flux φ = dt v
R

[Chua 1971, Strukov et al. 2008]

Linear and non-linear memristors
Memristor defining relation

dϕ = M dq (1)
A linear (M = cost) memristor is trivial, it’s just a resistor!

dϕ dq
=M ⇒ v=Mi (2)
dt dt
But more generally memristance can depend on charge, so

dϕ dq
µZ ¶
= M(q) ⇒ v=M dt i i (3)
dt dt
Memristance is a resistance that depends on the past history.

⇒ M EMORY
Why memristors?

Fundamental interest:
Ï Standard non-linear passive circuit elements cannot simulate
memristors [Chua 1971]
Ï Memristance generally increases by reducing the scale
[Strukov et al. 2008]
Ï Many systems have memristance [Chua and Kang 1976]
Practical interest:
Ï RRAM (Resistive Random Access Memory), simplest and
densest RAM ever [Jo 2009]
Ï Simple programmable analog and digital electronics
[Pershin and DiVentra 2009a, Mouttet 2008]
Ï Neuromorphic devices to simulate biological systems
behaviour
[Pershin and Di Ventra 2009b, Pershin and DiVentra 2008a]
Experimental realization of a memristor
The linear drift model
At the nanometer scale small voltage imply large electric field

Memristance depends on width of doped region w

w ³ w´
M(w) = R ON + 1 − R OFF (4)
D D
Equation of motion of mobile dopants (oxygen vacancies)

dw R ON i(t) R ON
= µV ⇒ w(t) − w0 = µV q(t) (5)
dt D }
| {z D
Electric field
The linear drift model
Finally
µV R ON
M(q(t)) = R OFF + (R ON − R OFF ) q(t) (6)
D2
The memristive power is measured by the quantity

µV R ON
Q0−1 = (7)
D2

Typical values Q0 ≈ 10−2 C

Memristance goes as D−2

Memristance becomes more important

for understanding the electronic char-
acteristics of any device as the critical
dimensions shrink to the nanometer scale.
[Strukov et al. 2008]
Signs of memristance
Frequency dependent Lissajous figures
or
pinched hysteresis loops

[Strukov et al. 2008]

Non-linear drift models
The linear drift model does not model the drift near the boundaries
of real devices.
Introduce a window function, example: F(w/D) = w(D − w)/D2

dw R ON i(t) ³ w ´
= µV F (8)
dt D D
Typical hysteresis curve [Strukov et al. 2008]
Some relevant points

Ï Memristors have a polarity

Ï Memristors are purely dissipative elements, no energy storage
(the hysteresis loops is pinched)
Ï Introducing a window functiong means that we have no pure
memristive behaviour, but we are dealing with more general
memristive systems [Chua and Kang 1976]

v = R(w, i)i (9)

dw
= f (w, i) (10)
dt
Ï The resistence can be controlled by an n-component vector w,
we then have a n−th order memristive system. (Interesting
example later with memcapacitance)
Variations: memcapacitors and meminductors
Definition of a generalized memory device
[Pershin, Di Ventra and Chua 2009]

y(t) = g(x, u, t)u(t) (11)

ẇ = f (w, u, t) (12)

Different choices of the couple of conjugate variables (y, u) leads to

different types of components.
Voltage controlled memcapacitor
µZ ¶
(y, u) = (q, v) ⇒ q = C dt v v (13)

Current controlled meminductor

µZ ¶
(y, u) = (φ, i) ⇒ φ=L dt i i (14)
Possible implementations

Ï TiO2−x /TiO2 thin films, seen previously [Strukov et al. 2008]

Ï Other metal-oxides thin film [Krzystenczo et al 2009]
Ï Ferromagnet-semiconductor junctions, spin density as the
state variable [Pershin and Di Ventra 2008b]
Ï Hysteretic insulator-metal transition in correlated electron
systems (VO2 ) [T. Driscoll et al. 2009a, T. Driscoll et al. 2009b]
Ï Spin-torque-induced magnetization motion in Magnetic
Tunnel Junctions and Spin Valves [Wang et al 2009]
Ï Chalcogenides, perovskites, organic films, multistable
molecules and many more... (see [Strukov et al. 2008])
Half-metal/semiconductor junction

Spin blockade Drift-Diffusion model for spin

[Pershin and Di Ventra 2008c] density

∂n↑(↓) e
e = div j↑(↓) + (n↑(↓) − n↓(↑) )
∂t 2τsf
j↑(↓) = σE + eD∇n↑(↓)

Voltage drop

0 N0
µ ¶
V = ρsL + ρc I
2n↑ (0)
Spin memory effect
Z L
Integrated Spin density N↑ = dx n↑ (x) (15)
0
Assuming n↑ = g(N↑ ) (16)
∂N↑ 1
Neglecting diffusion e =− i (17)
∂t 2
The junction behaves as a perfect memristor at high frequency
N0
µ ¶
V = ρ s L + ρ 0c ¡ ¢ I (18)
2 N↑ − q(t)/2e

[Pershin and Di Ventra 2008b]

Insulator-Metal Transition (IMT)
in Vanadium Dioxide VO2 [T. Driscoll et al. 2009a]
Proof of principle of phase transition-based memristive systems

Local heating + Hysteretic transition ⇒ Memristive system

Resistance always increases with current 6= normal memristor

V = R(x, I, t)I (19)

2
ẋ = f (x, I, t) ∝ I (20)
Frequency agile metamaterial on VO2 thin film.
[T. Driscoll et al. 2009b]

Microresonators (Split Ring Resonator, SRR) modify the

electromagnetic response of a VO2 .
IMT affects permittivity εeff (ω) ⇒ shift in resonant frequency
Memory effect in the electromagnetic response:
memcapacitive model

Hysteretic behaviour of permittivity allows voltage controlled

persistent tuning of metamaterial frequency.

[T. Driscoll et al. 2009b]

Convenient circuit model using memristors and memcapacitors.
More on spintronic devices: Magnetic tunnel junctions
State variable: orientation of magnetization (θ) in MJT free layer

1 GHIGH − GLOW
R(θ) = TMR
TMR = (21)
G0 1 + TMR+2 cos θ
¡ ¢
GLOW

Spin-torque-induced magnetization alignment

θ̇ = − |α sin{z
θ cos θ} + βI sin θ θ̇ = f (θ, I) (22)
| {z }
damping term torque

[Wang et al 2009]
More on spintronic devices: Spin valves
Long spin valve with domain wall on free layer [Wang et al 2009]

State variable: domain wall position x

x ³ x´
R(x) = RLOW + RHIGH 1 − (23)
D D
Current driven domain wall velocity ẋ = ΓI (24)
Spintronic analog of TiO/TiO2−x memristor

Γq(t)
· ¸
Memristance M(q) = RHIGH − (RHIGH − RLOW ) (25)
D
Simple model of a charge controlled memcapacitor
Solid state memcapacitor Circuit model

Idea: use a multi-layered metamaterial as a dielectric medium

in a capacitor.

C0 £ ¤
C(Q1 , q) = state variable: Q1 (26)
δ Q1
1+ d q
dQ1 £ ¤
= −IR tunneling current (27)
dt
Flash memory cells are very similar [Lee et al 2009]
Solid state memcapacitor behaviour
Ï Hysteretic charge-voltage curves (non-pinched however)
(compare to [Pershin and Di Ventra 2008b])
Ï Negative and diverging values of capacitance (some real
examples in
[Krems, Pershin and Di Ventra 2010, Gommans et al. 2005])
Ï Dissipative behaviour due to tunneling
Coexistence of resistive switching and tunnel
magnetoresistence [Krzystenczo et al 2009]
MJT with MgO insulator layer


v(t) = R(w1 , w2 , i, t)i(t)

Second order memristive system ẇ1 = f1 (w1 , i, t)i(t)


ẇ2 = f2 (w2 , i, t)i(t)
magnetization direction: w1 oxide insulator state: w2
Crossbar RRAM with amorphous silicon active region
[Jo 2009]
Memcapacitor and meminductor emulators
Once you have a memristor is easy to simulate other memory
passive components

Memcapacitor
emulator

Meminductor
emulator
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