IEEE Circuits and Systems Magazine - Q2 2018 - 70

As expected of real-world memristors, irrespective of the positive value
for Vapplied, from any initial condition the route of the memory state
inevitably leads to its saturation at the upper bound Ndisc,max.
where v R plug = R plug i Schottky and v R disc = R disc i Schottky denote
the voltage drops across plug and disc resistances, respectively. The temperature T within the filament is
evaluated through
T = T0 + R th,eff ^v R plug + v R disc + v Schottky h i schottky,

(49)

where T0 stands for the ambient temperature, while
R th, eff represents the effective thermal resistance of the
nanodevice.
With the voltage across R series calculated through
v R series = R series i Schottky, a simple application of the Kirchhoff's Voltage Law (KVL) around the mesh formed by
the ground, the Schottky diode, the disc, plug, and
series resistances, and the voltage source v applied leads
to the following equation
v applied - v R plug - v R disc - v R series - v Schottky = 0,

(50)

which is solved for v Schottky at each time step using Newton's method. Using the new value for the voltage across
the Schottky diode and the values for temperature T
and memory state N disc from the previous time step,
we calculate the new values for R plug, R disc, ez B n, i Schottky
^W 00, W 0, and p need to be determined before the Schottky

current for the set process may be computed), and E.
Then, the new value for the temperature T is obtained
via equation (49). Finally, after determining the new values for c V0 and i ion, an Euler method is employed to solve
the state equation (44) for the new value for N disc .
The values for the model parameters of the Aachen
mathematical description fitted to the hafnium oxide/
titanium oxide bilayer memristor [21] are reported in
Table IV.
Fig. 30(a) shows the DRM of the Aachen hafnium oxide/titanium oxide bilayer memristor under a number of
positive DC values for the device voltage v applied, specifically Vapplied ! {0.6, 0.7, 0.8, 0.9, 1.0} V, over the memory
state existence domain, i.e. for N disc ! {N disc,min, N disc,max}.
As expected of real-world memristors, irrespective
of the positive value for Vapplied, from any initial condition the route of the memory state inevitably leads to
its saturation at the upper bound N disc,max . The illustration of the DRM of the Aachen nano-device under
the set of negative memristor DC voltages defined as
Vapplied ! {- 0.6, - 0.7, - 0 - 8, - 0.9} V for all values of
the oxygen vacancy concentration within the allowable state variation range is reported in Fig. 30(b).
Inspecting the arrows drawn on all curves of the
T
No disc = ^dN disc /dt h - N disc loci, it is clear that under any

Table IV.
Parameter values of the Aachen model fitted to the hafnium oxide/titanium oxide bilayer memristor [21]. f 0 and f
denote the static vacuum and oxide dielectric permittivities, respectively.
l cell /nm

l disc /nm

rfil /nm

z Vo

6

1.5

30

2

a /nm

o0 /THz

DW A /eV

N disc,min /m -3

0.4

4

0.95

6 · 10 24

N disc,max = N plug /m -3

A ) / (A $ m -2 $ K -2)

e

f zB

2.5 · 10 27

6.01 · 10 5

17f 0

5.5f 0

f 0 / (F $ m )

ez B n0 /eV

ez n /eV

n n / (m 2 V -1 s -1)

8.85 · 10 -12

0.3

0.1

3.0 · 10 -5

DW ac /eV

R series /kX

R th, eff / (K $ W -1)

-1

0.03

T0 /K

1.16

5.3 · 10

k B / (m kgs K )

e /C

m /Kg

h/ (m 2 $ kg $ s -1)

1.38 · 10 -23

1.60 · 10 -19

9.11 · 10 -31

6.63 · 10 -34

2

70

5

-2

-1

IEEE cIrcuIts and systEms magazInE

293

sEcOnd quartEr 2018



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