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Presented by:•Sabikeena Sadeque•Pranjal Rahman
American International Uni versity Bangladesh – AIUB
Modeling Memristive Behavior Using Drude Model
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
1
Introduction to Drude Model
Paul Drude proposed the simple classical Drude Model of electrical conduction in 1900.
To model the behavior of a memristor through mathematical formulae, HP’s TiO2 memristor was taken into consideration.
It consists of two platinum electrodes with a thin film of titanium dioxide sandwiched in the middle.
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
2
Introduction to Drude Model (contd.)
Figure 1: A simple circuit used as a basis of the Drude Model.
Vdx
Ex
V
I
vxi = velocity of the ith electron in x direction at time t.
ti = lost collision time.
(t – ti) = time for which electron accelerated free of collisions.
uxi = velocity of electron i in x-direction just after the collision (initial velocity).
me = mass of electron.
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
3
Introduction to Drude Model (contd.)
Since as electric field strength E is the force F acting per unit charge q, we can say that for an electron,
(1)
We know that from Newton’s Second Law of Motion, (2)
From the above two equations we obtain an equality which can be arranged to obtain,
Now, and from this,
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
eEF
maF
e
x
meEa
atuv
e
ixxx
mtteEuv ii
)(
qEF
4
Introduction to Drude Model (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
vdx = drift velocity of electrons due to applied field Ex, and it is also the average velocity for all electrons along x.
For i=1 to N electrons,
(3)
Considering is the mean time between collisions,
(4)
Nvvvv Nxxx
dx
...21
e
ixdx
mtteEv )(
itt
e
xdx m
Eev
5
Introduction to Drude Model (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
Where drift mobility is
(5)
From equations (4) and (5), (6)
xddx Ev
6
ed m
e
Premises
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
The following assumptions were made:1. In metals, electrons undergo random motion
but for a TiO2 memristor, the oxygen vacancies remain stationary.
2. The Drude Model may model the behavior of positive oxygen vacancies in the doped titanium dioxide material.
7
Physical Model Electrical Model
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
Comparison of TiO2 models
Doped Undoped
wD
Platinum Electrodes
Figure 2: Physical model of a Titanium Dioxide memristor
Doped
Undoped
A
Doped Undoped
Ron
Roff
Ron w/D Roff (1-w/D)
w
D
AC
Figure 3: Electrical model of a Titanium Dioxide memristor
8
(TiO2)(TiO2-x)
Deriving Memristor Equations
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
For a uniform electric field, (7)
(8)
From (6) and (8),
(9)
xiR
xV
dxdV
=E
DtiRE on )(
DtiRv ond
dx)(
()() d ondx
R i tdw tvdx D
)()()(
)()(
0twD
tqRtw
DdttiR
dttdw
ond
ond
9
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
According to the electrical model, (10)
(11)
From (10) and (11),
))(1()()(DtwR
DtwRwM offon
() () ( ())() ()() (1 ) ()on off
v t i t M w tw t w tv t R R i tD D
)(*)(1)()(
)(*)(
1)(
)(
222 ti
DtqRR
DtqRtv
tiDD
tqRR
DD
tqRRtv
ondoff
don
ond
off
ond
on
10
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
So,
Dividing (9) by D,
(12)
In uniform electric field,
)(1))((
)(1))(()()(
2
2
tqRRtqM
DD
tqRRtqMtitv
onoff
d
ondoff
DtwtqR
Dtw
Dtw
DtqR
Dtw
on
ond
)()()(
)()()(
0
02
tvD dx
11
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
12
is the amount of charge required to move the boundary from w(t0) to form a pure conductive channel.
(13)
Using (12) and (13), (14)
From (11) and (14) (15)
itQD
onD R
Q
DQtq
Dtw
Dtw )()()( 0
DQtqtxtx )()()( 0
)(*)(1)()( titxRtxRtv offon
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
13
Initially,
Memristance at time t is, (16)
(17)
Therefore ,
)(1)()()(
0000
0 txRtxRMtitv
offon
)(1)(
)(1)(
000
000
txrtxRM
txRR
txRM
on
on
offon
DQtqRMtqM )())(( 0
() ( ())* ()v t M q t i t
0()() ()D
q tv t M R i tQ
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
14
(18)
By integrating (18), (19)
(20)
Solving the above quadratic equation,
(21)
0
0
() ()() *
() ()() () *
D
D
q t dq tv t M RQ dt
q t dq tt v t M R dtQ dt
2
0()() () 2 D
Rq tt M q tQ
20() () () 02 D D
R q t M Q q t Q t
22 ()() 1 1 d
Dtq t Q
rD
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
15
From (13) and (14),
(22)
Restating (11),
0
2
()() ( )
2 ()() 1 1
D
d
q tx t x tQ
tx trD
2
() ()() 1 ()
()() 1 ()
() 1 () ()
2 ()() 1 1 1 ()
on off
off
off
doff
w t w tv t R R i tD D
w tv t R i tD
v t R x t i t
tv t R i trD
Deriving Memristor Equations (contd.)
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
16
Therefore, (23)
22 ()() 1 ()d
offtv t R i t
rD
2
()()2 ()1 d
off
v ti ttR
rD
Significance of Equations
(16)
(21)
(22)
(23)
17
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
0
2
2
2
()( ())
2 ()() 1 1
2 ()() 1 1
()()2 ()1
D
dD
d
doff
q tM q t M RQ
tq t QrD
tx trD
v ti ttR
rD
All four of these equations play a vital role in the MATLAB simulation of a TiO2 memristor.
18
Figure 4: Nonlinear charge-flux relationship
The only circuit element to have a nonlinear relationship between its quantities.
More positive voltage, more flux linkage, so the width of doped material increases.
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
Flux (W b)
Charg
e (mC
)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
19
Figure 5: Current-Voltage characteristics curve
Hysteresis loop between current and voltage.
This curve indicates that memristors have memory.
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
Voltage (V)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Current (mA)
-2.5-2
-1.5-1
-0.50
0.5
1
1.5
22.5
System Parameters
7th IEEE Conference on Industrial Electronics and Applications – ICIEA 2012
20
The frequency ω was set to 1 rad s-1.
An input voltage of 1 V.
Ron was 1 Ω and Roff was 600 Ω.The total width D of the memristor was 10 nm and the initial width of the doped region was 0 nm.The drift velocity of the dopants was 10-14 m2s-1V-1.
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