IEEE Circuits and Systems Magazine - Q2 2018 - 71

1028

.
Ndisc/(m-3s-1)

10

Vapplied = 1 V
Vapplied = 0.9 V
Vapplied = 0.8 V

1028
1026

1027
Ndisc/m-3

30

1025

Vapplied = 0.7 V

1024

Ndisc,0 = 1.20 . 10+25 m-3
Ndisc,0 = 6.00 . 10+24 m-3
1024
10-10 10-8 10-6 10-4 10-2 100 102 104
t /s

Vapplied = 0.6 V

1022
1025

1026
Ndisc /m-3

1027

(a)

(a)

1028

1022

1027
Vapplied = -0.6 V

1024

Vapplied = -0.7 V

1026

Ndisc/m-3

.
Ndisc/(m-3s-1)

1026

1026
1025

Vapplied = -0.9 V

1028

Ndisc,0 = 6.00 . 10+25 m-3
Ndisc,0 = 3.00 . 10+25 m-3

Vapplied = -0.8 V

1024

Ndisc,0 = 2.50 . 10+27 m-3
Ndisc,0 = 5.00 . 10+26 m-3
Ndisc,0 = 2.50 . 10+26 m-3
Ndisc,0 = 5.00 . 10+25 m-3
Ndisc,0 = 2.50 . 10+25 m-3
Ndisc,0 = 6.00 . 10+24 m-3

10-10 10-8 10-6
1025

1026
Ndisc /m-3

1027

10-4 10-2
t /s

100

102

104

(b)

Figure 30. drm of the aachen memristor model under a
set of positive (a) and negative (b) dc values for the voltage
v applied at the top OE. the bottom aE is grounded, as shown in
Fig. 29. under a given positive (negative) input-refer to plot
(a) ((b))-the state rate No disc at N = N disc, max ^N = N disc, min h
drops suddenly from the respective value indicated by a
white-filled circle to 0 m -3 s -1, since the upper (lower) state
bound represents an equilibrium for the state equation (45)
under v applied 2 (1) 0 V. From table IV N disc, min = 6 · 10 24 m -3,
while N disc, max = 2.5 · 10 27 m -3 .

Figure 31. Evidence for dc fading memory in the hafnium
oxide/titanium oxide bilayer memristor from aachen. (a)
memory state over time during set, in response to the
application of a positive dc voltage Vapplied = 0.6 V to the
top OE with the bottom aE grounded for initial states
N disc,0 ! " 6.00 · 10 24, 1.20 · 10 25, 3.00 · 10 25, 6.00 · 10 25 , m -3
(note that the voltage applied across the memristor nano-device of Fig. 29 is here simply computed as Vm = Vapplied h . (b)
memory state over time during reset, in response to the application of a negative dc voltage Vm = - 0.6 V to the top OE with
the bottom aE grounded for initial states N disc,0 ! " 6.00·10 24,
2.50·10 25, 5.00·10 25, 2.50·10 26, 5.00·10 26, 2.50·10 27 , m -3 .

value for Vapplied, the time evolution of the memory state
from any initial condition leads inevitably to its convergence to the lower bound N disc,min .
The Aachen mathematical description predicts DC
and AC fading memory for all the nano-devices it is
able to model, i.e. for memristors based upon tantalum oxide [64], strontium titanate [20], and hafnium
oxide/titanium oxide [21] grown at RWTH Aachen.
With reference to the parameter set of Table IV, fitted
to the bilayer nano-device, Fig. 31(a) gives evidence
for the history erase phenomenon emerging in the hafnium oxide/titanium oxide memristor under application of a DC voltage of value 0.6 V to the top OE with
T
the bottom AE grounded for initial conditions N disc,0 =
24
25
25
25
-3
N disc (0) ! {6.00 · 10 , 1.20 · 10 , 3.00 · 10 , 6.00 · 10 } m .
All state solutions asymptotically saturate at the upper
bound N disc,max = 2.5·10 27 m -3 . Fig. 31(b) illustrates the progressive memory loss the same nano-device experien -

ces as a DC voltage of value -0.6 V is applied to the OE with
grounded AE and initial states within the set expres s e d by N disc,0 ! {6.00·10 24, 2.50·10 25, 5.00·10 25, 2.50·10 26,
5.00·10 26, 2.50·10 27} m -3 . Irrespective of the initial condition the memory state saturates at the lower bound
N disc,min = 6.00·10 24 m -3 before the end of the numerical simulation.
The Aachen mathematical description predicts also
the emergence of AC fading memory in the bilayer
nano-device. To show but one example, Fig. 32 shows the
memory state response to the application of a sine wave
voltage of the form v applied = vt applied sin (2r ft) with amplitude vt applied = 1.2 V and frequency f = 1 Hz to the top
OE with the bottom AE grounded for initial conditions
within the set N disc,0 ! {0.06·10 26, 12.0·10 26, 25.0·10 26} m -3 .
All state solutions progressively lose the information
embedded in the initial condition, featuring a unique asymptotic oscillatory behaviour.

(b)

sEcOnd quartEr 2018

IEEE cIrcuIts and systEms magazInE

71



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