IEEE Circuits and Systems Magazine - Q2 2018 - 69

The interface between the titanium oxide layer and the top titanium
OE features a low energy barrier, and its resistance is included
in the resistor Rseries together with the electrode resistance.
activation energy, T symbolises the homogeneous
temperature within the filament, and, as stated earlier,
k B represents the Boltzmann constant.
In the reset (set) process, a negative (positive) voltage v applied at the OE pushes oxygen vacancies from the
disc (plug) into the plug (disc)17, increasing (decreasing)
its resistance according to
R disc =

D
l disc
exp c W ac m,
ez Vo N disc n n A
kB T

i Schottky = - AA * T 2 exp c -

(38)

where v schottky is the voltage drop across the diode, A *
is the effective Richardson constant, ez B n stands for
the effective Schottky energy barrier height, which,
due to the action of the applied electric field, is lower
than the nominal Schottky energy barrier height ez B n0
according to
ez B n = ez Bn0 - e 4

e 3 z Vo N disc ^z B n0 - z n + v Schottky h
, (39)
8r 2 f z3 B

in which ez n defines the energy difference between
Fermi level and conduction band edge for the oxide,
whereas f z B is the dielectric permittivity the oxide
features as a result of the process of image force barrier lowering.
During set, i.e. under v applied 2 0 V, the current through
the reverse-biased Schottky diode consists of electrons
which tunnel through the energy barrier (a thermallyactivated process known as thermionic field emission),
and is expressed by
17
In the high resistance state (HRS) the oxygen vacancy concentration
Ndisc within the disc attains the minimum value Ndisc,min. In the low resistance state (LRS) the oxygen vacancy concentration Ndisc within the
disc attains the maximum value Ndisc,max, taken equal to the fixed oxygen vacancy concentration Nplug within the conductive plug.

sEcOnd quartEr 2018

· exp c -

rW 00 e v Schottky +

f

z Bn

p
cosh 2 c W 00 m
kB T

ez B n
ev Schottky
m - 1 m,
m · c exp c
p
W0

(40)

where W 00, W 0, and p are respectively defined as
W 00 = eh
4r

(37)

and forward (reverse) biases the Schottky diode modelling the interface between the hafnium oxide layer
and the active electrode (AE). During reset, i.e. under
v applied 1 0 V, the current through the forward-biased
Schottky diode originates from the process of thermal
emission of electrons, and is expressed by
ez B n
m
kB T
e (- v Schottky)
m - 1 m,
· c exp c
kB T

i Schottky = AA * T
kB

z V0 N disc
,
m) f

W 0 = W 00 coth c W 00 m,
kB T
p=

W 00
,
W 00 - tanh W 00
c
m
KB T
kB T

(41)
(42)
(43)

where h is the Planck constant, and m ) denotes the effective electron mass, taken equal to the electron mass
m in this model. The interface between the titanium oxide layer and the top titanium OE features a low energy
barrier, and its resistance is included in the resistor
R series together with the electrode resistance. The time
evolution of the memory state is governed by the following nonlinear differential equation
dN disc =
i ion
,
ez Vo Al disc
dt

(44)

where i ion represents the ion current between plug and
disc due to ion hopping conduction over an energy barrier of height DW A, and is regulated by the Mott-Gurney
law according to
az eE
i ion = 2A z Vo e c Vo a o 0 exp c - DW A m sinh c Vo
m (45)
kB T
2k B T
where
c Vo =

N plug + N disc
2

(46)

is the average between the concentration of oxygen vacancies within the plug and the concentration of oxygen vacancies within the disc a defines the hopping
distance, v 0 stands for the attempt frequency, while E
represents the electric field, calculated as
E=

v R disc
for v applied 2 0 (set),
l disc

(47)

E=

v R plug + v R disc + v Schottky
for v applied 1 0 (reset),
l cell

(48)

IEEE cIrcuIts and systEms magazInE

69



Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2018

Contents
IEEE Circuits and Systems Magazine - Q2 2018 - Cover1
IEEE Circuits and Systems Magazine - Q2 2018 - Cover2
IEEE Circuits and Systems Magazine - Q2 2018 - Contents
IEEE Circuits and Systems Magazine - Q2 2018 - 2
IEEE Circuits and Systems Magazine - Q2 2018 - 3
IEEE Circuits and Systems Magazine - Q2 2018 - 4
IEEE Circuits and Systems Magazine - Q2 2018 - 5
IEEE Circuits and Systems Magazine - Q2 2018 - 6
IEEE Circuits and Systems Magazine - Q2 2018 - 7
IEEE Circuits and Systems Magazine - Q2 2018 - 8
IEEE Circuits and Systems Magazine - Q2 2018 - 9
IEEE Circuits and Systems Magazine - Q2 2018 - 10
IEEE Circuits and Systems Magazine - Q2 2018 - 11
IEEE Circuits and Systems Magazine - Q2 2018 - 12
IEEE Circuits and Systems Magazine - Q2 2018 - 13
IEEE Circuits and Systems Magazine - Q2 2018 - 14
IEEE Circuits and Systems Magazine - Q2 2018 - 15
IEEE Circuits and Systems Magazine - Q2 2018 - 16
IEEE Circuits and Systems Magazine - Q2 2018 - 17
IEEE Circuits and Systems Magazine - Q2 2018 - 18
IEEE Circuits and Systems Magazine - Q2 2018 - 19
IEEE Circuits and Systems Magazine - Q2 2018 - 20
IEEE Circuits and Systems Magazine - Q2 2018 - 21
IEEE Circuits and Systems Magazine - Q2 2018 - 22
IEEE Circuits and Systems Magazine - Q2 2018 - 23
IEEE Circuits and Systems Magazine - Q2 2018 - 24
IEEE Circuits and Systems Magazine - Q2 2018 - 25
IEEE Circuits and Systems Magazine - Q2 2018 - 26
IEEE Circuits and Systems Magazine - Q2 2018 - 27
IEEE Circuits and Systems Magazine - Q2 2018 - 28
IEEE Circuits and Systems Magazine - Q2 2018 - 29
IEEE Circuits and Systems Magazine - Q2 2018 - 30
IEEE Circuits and Systems Magazine - Q2 2018 - 31
IEEE Circuits and Systems Magazine - Q2 2018 - 32
IEEE Circuits and Systems Magazine - Q2 2018 - 33
IEEE Circuits and Systems Magazine - Q2 2018 - 34
IEEE Circuits and Systems Magazine - Q2 2018 - 35
IEEE Circuits and Systems Magazine - Q2 2018 - 36
IEEE Circuits and Systems Magazine - Q2 2018 - 37
IEEE Circuits and Systems Magazine - Q2 2018 - 38
IEEE Circuits and Systems Magazine - Q2 2018 - 39
IEEE Circuits and Systems Magazine - Q2 2018 - 40
IEEE Circuits and Systems Magazine - Q2 2018 - 41
IEEE Circuits and Systems Magazine - Q2 2018 - 42
IEEE Circuits and Systems Magazine - Q2 2018 - 43
IEEE Circuits and Systems Magazine - Q2 2018 - 44
IEEE Circuits and Systems Magazine - Q2 2018 - 45
IEEE Circuits and Systems Magazine - Q2 2018 - 46
IEEE Circuits and Systems Magazine - Q2 2018 - 47
IEEE Circuits and Systems Magazine - Q2 2018 - 48
IEEE Circuits and Systems Magazine - Q2 2018 - 49
IEEE Circuits and Systems Magazine - Q2 2018 - 50
IEEE Circuits and Systems Magazine - Q2 2018 - 51
IEEE Circuits and Systems Magazine - Q2 2018 - 52
IEEE Circuits and Systems Magazine - Q2 2018 - 53
IEEE Circuits and Systems Magazine - Q2 2018 - 54
IEEE Circuits and Systems Magazine - Q2 2018 - 55
IEEE Circuits and Systems Magazine - Q2 2018 - 56
IEEE Circuits and Systems Magazine - Q2 2018 - 57
IEEE Circuits and Systems Magazine - Q2 2018 - 58
IEEE Circuits and Systems Magazine - Q2 2018 - 59
IEEE Circuits and Systems Magazine - Q2 2018 - 60
IEEE Circuits and Systems Magazine - Q2 2018 - 61
IEEE Circuits and Systems Magazine - Q2 2018 - 62
IEEE Circuits and Systems Magazine - Q2 2018 - 63
IEEE Circuits and Systems Magazine - Q2 2018 - 64
IEEE Circuits and Systems Magazine - Q2 2018 - 65
IEEE Circuits and Systems Magazine - Q2 2018 - 66
IEEE Circuits and Systems Magazine - Q2 2018 - 67
IEEE Circuits and Systems Magazine - Q2 2018 - 68
IEEE Circuits and Systems Magazine - Q2 2018 - 69
IEEE Circuits and Systems Magazine - Q2 2018 - 70
IEEE Circuits and Systems Magazine - Q2 2018 - 71
IEEE Circuits and Systems Magazine - Q2 2018 - 72
IEEE Circuits and Systems Magazine - Q2 2018 - 73
IEEE Circuits and Systems Magazine - Q2 2018 - 74
IEEE Circuits and Systems Magazine - Q2 2018 - 75
IEEE Circuits and Systems Magazine - Q2 2018 - 76
IEEE Circuits and Systems Magazine - Q2 2018 - 77
IEEE Circuits and Systems Magazine - Q2 2018 - 78
IEEE Circuits and Systems Magazine - Q2 2018 - 79
IEEE Circuits and Systems Magazine - Q2 2018 - 80
IEEE Circuits and Systems Magazine - Q2 2018 - 81
IEEE Circuits and Systems Magazine - Q2 2018 - 82
IEEE Circuits and Systems Magazine - Q2 2018 - 83
IEEE Circuits and Systems Magazine - Q2 2018 - 84
IEEE Circuits and Systems Magazine - Q2 2018 - 85
IEEE Circuits and Systems Magazine - Q2 2018 - 86
IEEE Circuits and Systems Magazine - Q2 2018 - 87
IEEE Circuits and Systems Magazine - Q2 2018 - 88
IEEE Circuits and Systems Magazine - Q2 2018 - 89
IEEE Circuits and Systems Magazine - Q2 2018 - 90
IEEE Circuits and Systems Magazine - Q2 2018 - 91
IEEE Circuits and Systems Magazine - Q2 2018 - 92
IEEE Circuits and Systems Magazine - Q2 2018 - 93
IEEE Circuits and Systems Magazine - Q2 2018 - 94
IEEE Circuits and Systems Magazine - Q2 2018 - 95
IEEE Circuits and Systems Magazine - Q2 2018 - 96
IEEE Circuits and Systems Magazine - Q2 2018 - 97
IEEE Circuits and Systems Magazine - Q2 2018 - 98
IEEE Circuits and Systems Magazine - Q2 2018 - 99
IEEE Circuits and Systems Magazine - Q2 2018 - 100
IEEE Circuits and Systems Magazine - Q2 2018 - 101
IEEE Circuits and Systems Magazine - Q2 2018 - 102
IEEE Circuits and Systems Magazine - Q2 2018 - 103
IEEE Circuits and Systems Magazine - Q2 2018 - 104
IEEE Circuits and Systems Magazine - Q2 2018 - 105
IEEE Circuits and Systems Magazine - Q2 2018 - 106
IEEE Circuits and Systems Magazine - Q2 2018 - 107
IEEE Circuits and Systems Magazine - Q2 2018 - 108
IEEE Circuits and Systems Magazine - Q2 2018 - Cover3
IEEE Circuits and Systems Magazine - Q2 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1
https://www.nxtbookmedia.com