![Fig. 6: a) Experimental RHRS as a function of stop voltage during Reset operation and b) switching time for Set operation as a function of voltage presented in [23] and corresponding simulation results.](/figures/fig-6-a-experimental-rhrs-as-a-function-of-stop-voltage-252ll2uu.webp)
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Robust Compact Model for Bipolar Oxide-Based
Resistive Switching Memories
Marc Bocquet, Damien Deleruyelle, Hassen Aziza, Christophe Muller,
Jean-Michel Portal, Thomas Cabout, Eric Jalaguier
To cite this version:
Marc Bocquet, Damien Deleruyelle, Hassen Aziza, Christophe Muller, Jean-Michel Portal, et al..
Robust Compact Model for Bipolar Oxide-Based Resistive Switching Memories. IEEE Transactions
on Electron Devices, Institute of Electrical and Electronics Engineers, 2014, 61 (3), pp.674 - 681.
�10.1109/TED.2013.2296793�. �hal-01737291�
1
Robust compact model for bipolar oxide-based
resistive switching memories
Marc Bocquet
∗
, Damien Deleruyelle
∗
, Hassen Aziza
∗
, Christophe Muller
∗
,
Jean-Michel Portal
∗
, Thomas Cabout
†
and Eric Jalaguier
†
∗
IM2NP, UMR CNRS 7334, Aix-Marseille Universit
´
e, 38 rue Joliot Curie, F-13451 Marseille Cedex 20, France
Email: marc.bocquet@im2np.fr
†
CEA-L
´
eti, Campus MINATEC, 17 avenue des Martyrs, F-38054 Grenoble Cedex 9, France
Email: thomas.cabout@cea.fr
Abstract—Emerging non-volatile memories based on resistive
switching mechanisms pull intense R&D efforts from both
academia and industry. Oxide-based Resistive Random Access
Memories (namely OxRAM) gather noteworthy performances,
such as fast write/read speed, low power, high endurance and
large integration density that outperform conventional Flash
memories. To fully explore new design concepts such as dis-
tributed memory in logic or biomimetic architectures, robust
OxRAM compact models must be developed and implemented
into electrical simulators to assess performances at a circuit
level. In this paper, we propose a physics-based compact model
used in electrical simulator for bipolar OxRAM memories. After
uncovering the theoretical background and the set of relevant
physical parameters, this model is confronted to experimental
electrical data. The excellent agreement with these data suggests
that this model can be confidently implemented into circuit
simulators for design purpose.
INTRODUCTION
Memory devices based on resistive switching materials
are currently pointed out as promising candidates to replace
conventional non-volatile memory (NVM) devices based on
charge-storage beyond 2x nm technological nodes [1]. Indeed,
as compared to conventional floating gate technologies, Re-
sistive RAMs (so-called RRAM) gather fast write/read oper-
ations, low power consumption, CMOS voltage compatibility
and high endurance [2]. Moreover, the resistive memory ele-
ment generally consists in simple Metal/Insulator/Metal (MIM)
structure. Hence, whatever the underlying physics, the resistive
switching memory elements may be advantageously integrated
into back-end-of-line (BEOL) enabling NVM solutions to be
distributed over CMOS logic. Depending on fundamental phys-
ical mechanisms responsible for resistance switching, various
RRAM technologies are now categorized by the ITRS. The
Redox Memory category, covered in this study, includes Con-
ductive Bridge RAM (CBRAM) [3] and Oxide Resistive RAM
(OxRAM) [4] that exhibit voltage polarity-dependent bipolar
switching. Besides, RRAM technologies referred as Thermo-
Chemical Memories (TCM), or fuse-antifuse memories, are
mostly based on materials such as nickel oxide (NiO) that
exhibit unipolar switching.
In the OxRAM memory elements addressed in this paper, a
MIM structure is generally composed of two passive metallic
electrodes sandwiching an active layer, usually an oxygen-
deficient oxide. A large number of resistive switching oxides,
like HfO
2
or Ta
2
O
5
, are reported in the literature [5], [6]. Even
if OxRAM technology is still in its infancy, it is commonly
accepted that the Valency Change Mechanism (VCM) occurs
in specific transition metal oxides and the field-assisted motion
of anions, such as oxygen ions O
2−
, governs the bipolar
resistance switching [7].
After an initial Electroforming step (cf FIG. 1b), the memory
element may be reversibly switched between a High Resis-
tance State (FIG. 1d-HRS) , and a Low Resistance State
(FIG. 1c-LRS) . The Electroforming stage corresponds to
a voltage-induced resistance switching from an initial very
high resistance state (pristine state) to a conductive state.
Resistive switching in OxRAM elements corresponds to an
abrupt change between HRS (R
HRS
) and LRS (R
LRS
) re-
sistances. This resistance change is achieved by sweeping a
voltage across the MIM structure: set operation corresponds
to a HRS-to-LRS transition at V
Set
while reset operation
enables turning back the structure into HRS state by applying
V
Reset
of opposite polarity. It has to be mentioned that the
Electroforming voltage V
F orming
is generally larger than V
Set
even if several groups have recently demonstrated forming-free
structures by adjusting the oxygen stoichiometry of the active
layer [8].
Thanks to their low operating voltage (typically 1 V ), their
fast read/write access times (tens of nanosecond) [2] and their
advantageous integration into BEOL, OxRAM memories pave
the way to new design solutions such as distributed memory
in logic or biomimetic architectures. Targeting efficient design
solutions, the compatibility between memory elements and
logic blocks must be beforehand evaluated. Hence, a robust
OxRAM compact model is required to assess and validate
new concepts before fabrication. This compact model must
(i) rely on realistic physical mechanisms described by a set of
relevant parameters; (ii) match actual experimental data (quasi-
static measurements, temperature dependence, timing...); (iii)
guarantee a good predictability up to the system level; (iv)
fulfill a computational-efficient implementation.
In the literature, many groups have proposed physical mod-
els for Set/Reset mechanisms [8]–[14], but their intrinsic com-
plexity excludes any implementation into electrical simulators.
In contrast, other models are fully compatible with simula-
2
Fig. 1: Schematic representation of a) the different regions con-
sidered within the MIM structure and the different mechanism
between the states of the cell: b) Pristine state, c) LRS and d)
HRS.
tors (e.g. SPICE model) but do not include typical OxRAM
voltage or time dependencies [15]–[17]. In this context, we
propose a robust physics-based compact model for bipolar
OxRAM memories that supports efficient implementation into
electrical simulators. After uncovering the theoretical back-
ground and the set of relevant physical parameters, this model
is confronted to experimental electrical data (DC behavior,
temperature dependence, dynamical characteristics...) based on
transient simulations.
I. COMPACT MODEL FOR OXRAM MEMORY ELEMENTS
In the literature, many works modeled the resistance switch-
ing effect by drift/diffusion of oxygen vacancies [9]–[12].
Nevertheless, recent results showed that Set/Reset processes
are triggered by voltage amplitude and that V
Set
/ V
Reset
are
weakly dependent on temperature [18]. Given these recent
insights, we propose an OxRAM model relying on local
redox processes controlled by voltage polarity. In this model,
the creation/destruction of a Conductive Filament (CF) is
described within a switching layer by a local modification of
the material (redox process). The model is based on a single
master equation in which both Set and Reset operations are
accounted simultaneously and controlled by the radius of the
conduction pathway also called conducting filament.
To ease the implementation into electrical simulators, the
model assumes an uniform CF radius and electric field within
the oxide layer in which the temperature increase (triggered by
Joule effect) may control the switching mechanisms. FIG. 1a
depicts the proposed model for the switchable MIM structure.
This latter is divided into three parts: a pristine oxide, switch-
able sub-oxide region originating from soft-breakdown events
and a conductive area. In this system, the two state variables
are the radius of the conductive filament r
CF
and the radius of
the switchable oxide r
CF max
. The oxide thickness is named
L
x
and x = 0 corresponds to the middle of the structure.
Set and Reset operations are described by electrochemical
redox reactions [13] relying on the Butler-Volmer equation
[19]. In LRS, in which conduction is controlled by CF, charge
transport is assumed to be ohmic accordingly to previous
works reported in literature [20], [21]. On the contrary, HRS
is dominated by a leakage current within the sub-oxide region.
The main mechanism reported in literature is the trap-assisted
conduction (Poole-Frenkel, Schottky emission, space charge
limited current...), but an ohmic behavior is considered for the
sake of simplicity.
A. Set/Reset operation
Set operation relies on an electrochemical reaction whose
charge transfer rate can be described by the Butler-Volmer
equation [19]. From this equation the electrochemical reduc-
tion rate τ
Red
(EQ. 1) and oxidation rate τ
Ox
(EQ. 2) can
be derived, here k
b
denotes the Boltzmann constant, E
a
an
activation energy, α the charge transfer coefficient (ranging
between 0 and 1) and τ
RedOx
the nominal redox rate. The
growth/destruction of the filament then results from the inter-
play between both redox reaction velocities through the master
EQ. 3, with the CF radius r
CF
ranging from 0 to r
CF max
.
τ
Red
= τ
RedOx
· e
E
a
− q · α · V
Cell
k
b
· T
(1)
τ
Ox
= τ
RedOx
· e
E
a
+ q · (1 − α) · V
Cell
k
b
· T
(2)
dr
CF
dt
=
r
CF
max
− r
CF
τ
Red
−
r
CF
τ
Ox
(3)
where V
Cell
is the voltage applied between the top and the
bottom electrodes, q is the elementary charge of electron, k
b
is
the Boltzmann constant, T is the temperature in the structure.
B. Electroforming stage
In addition to the Set operation, Electroforming converts
a highly resistive pristine oxide into a switchable sub-oxide
region. After this step, standard Set/Reset operation may then
occur. Due to the higher voltage bias required during Forming,
with respect to Set operation, a CF is generally formed
concomittantly to the sub-oxide region after Forming (FIG. 1c).
The Electroforming rate τ
F orm
is given in EQ. 4. The growth
of the sub-oxide region is controlled by EQ. 4, where E
a
F orm
is the activation energy for Electroforming and τ
F orm0
the
nominal forming rate.
τ
F orm
= τ
F orm0
· e
E
a
F orm
− q · α · V
Cell
k
b
· T
(4)
dr
CF
max
dt
=
r
work
− r
CF
max
τ
F orm
(5)
3
C. Temperature dependence
As shown by Govoreanu et al. [22], temperature plays
crucial role on the reaction rates. In our model, the local
temperature of the filament is computed from the heat equation
given in EQ. 6. Considering a cylinder-shaped filament, the
temperature is given by EQ. 7. For instance, EQ. 8 gives
maximum temperature reached into the CF at x = 0, the
middle of filament.
σ(x) ·F(x)
2
= −K
th
∂
2
T (x)
∂x
2
(6)
T (x) = T
amb
+
V
2
Cell
2 · L
2
x
· K
th
·
L
2
x
4
− x
2
σ
eq
(7)
T = T
amb
+
V
2
Cell
8 · K
th
· σ
eq
(8)
σ
eq
= σ
CF
·
r
2
CF
r
2
work
− σ
OX
·
r
2
CF max
− r
2
CF
r
2
work
(9)
where σ(x) is the local electrical conductivity, F (x) is the
local electric field, T
amb
is the ambient temperature and
σ
CF
(resp. σ
OX
) the electrical conductivity of the conductive
filament (resp. switchable sub-oxide) and K
th
is the thermal
conductivity. σ
eq
is the equivalent electrical conductivity in the
work area.
Let’s us mention that during Set operation, the temperature
increases due to the increase of CF radius: a positive feed-
back loop thus occurs leading to a self-accelerated reaction.
Conversely, during Reset operation, both the radius and the
temperature decrease in the CF, this feature lead to a self-
limited reaction also referred as a soft Reset [23].
D. Current through the MIM structure
The total current flowing through the OxRAM memory
element is the sum of three different contributions (EQ. 10):
the first one is related to the conductive area (I
CF
); the second
one that describes the conduction through the switchable sub-
oxide (I
Sub−oxide
); the last contribution arises from conduc-
tion through the unswitched pristine oxide (I
P ristine
). I
CF
and I
Sub−oxide
(EQs. 11 & 12 respectively) are described as
ohmic contributions; this assumption as already been applied
efficiently for TCM [24] and has proven to be accurate without
sacrificing the ease of numerical implementation. Conduction
in the pristine oxide region is described by means of a
tunneling current given in EQ. 13 [25].
I
Cell
= I
Sub−oxide
+ I
CF
+ I
P ristine
(10)
I
CF
= F · π · σ
CF
· r
2
CF
(11)
I
Sub−oxide
= F · π · σ
OX
·
r
2
CF max
− r
2
CF
(12)
I
P ristine
= S
Cell
· A · F
2
· exp
−B
F
(13)
A =
m
e
· q
3
8π · h · m
ox
e
· φ
b
!
"
#
$%
&''
Fig. 2: Program flowchart employed for numerical simulation
of OxRAM memory elements.
if φ
b
≥ qL
x
F : B =
8π
√
2m
ox
e
3 · h · q
h
φ
3/2
b
− (φ
b
− qL
x
F )
3/2
i
otherwise: B =
8π
√
2m
ox
e
3 · h · q
· φ
3/2
b
where F =
V
Cell
L
x
being the electric field across the active layer;
m
e
and m
ox
e
the effective electron masses into the cathode and
oxide respectively; h is the Planck constant; φ
b
the metal-oxide
barrier height; S
Cell
the section of the device.
E. Numerical implementation
The implementation of a compact model into electrical
simulation tools requires a discrete resolution of a set of
differential equations. If the time step is sufficiently small,
τ
Red
, τ
Ox
and τ
F orm
be assumed constant and the discrete
forms of EQs. 3&5 are given in EQs. 14&15, respectively.
Solving these differential equations step by step ensures a
better convergence of the simulation.
r
CF max
i+1
= (r
CF max
i
− r
work
) · e
−∆t
τ
F orm
+ r
work
(14)
r
CF
i+1
=
r
CF
i
− r
CF max
i
·
τ
eq
τ
Red
·e
−∆t
τ
eq
+r
CF max
i
·
τ
eq
τ
Red
(15)
where τ
eq
=
τ
Red
· τ
Ox
τ
Red
+ τ
Ox
These equations were then implemented within an ELDO
compact model following the flowchart given in FIG. 2. At
each call of the OxRAM instance during a transient simulation,
the previous state of the filament as well as the applied voltage
are provided to the model in order to take into account for the
memory effect. The new filament state and the current are then
computed as function of these inputs and the given time step.
4
V
BE
V
TE
OxRAM
V
Gate
(a)
-1
0
1
2
-600
-400
-200
0
200
0.05 0.10 0.15 0.20 0.25
0
1
2
3
4
5
Set
Reset
Voltage (V)
V
TE
V
BE
V
Gate
Forming
(b)
(c)
ICell (µA)
(d)
radius (nm)
Time (s)
r
CF
r
CFMAX
Fig. 3: Electrical simulation (ELDO) of 1T/1R OxRAM mem-
ory cell. a) Schematic of 1T1R structure. Chronogram of b)
voltages applied (V
T E
& V
Gate
) and obtained (V
BE
) to the
structure, c) current though the cell and d) radius of conductive
filament.
II. MODEL VALIDATION
To validate the proposed theoretical approach, the model
was confronted to quasi-static and dynamic experimental data
extracted. First, the compact model was calibrated on recent
electrical data measured on HfO
2
-based OxRAM devices [18].
In this study, the memory elements consisted in a Ti/HfO
2
/TiN
stack with a 5 nm tick hafnium oxide. The set of physical
parameters used for simulations are summarized in Table I.
A. DC behavior
To fully validate the compact model and its integration into
the electrical simulator, FIG. 3 gives an example of 1T/1R
bipolar OxRAM memory cell simulated at a circuit level. The
model used for the MOS transistor comes from Design Kit
65nm STM. The simulation was performed by Eldo. This first
simulation enables checking the stability of the model in a
system environment, the current flowing the OxRAM being
controlled by the gate voltage of the transistor. The model
showed a very good stability and we have simulated 2048-bits
memory array to verify the robustness and effectiveness of the
model through a complex simulation.
TABLE I: Physical parameters used in bipolar OxRAM com-
pact model
r
work
= 5 nm L
x
= 5 nm
Scell = 1µm × 1µm T
amb
= 300 K
τ
RedOx
= 1 × 10
−5
s E
a
= 0.7 eV
τ
F orm
= 1 × 10
−21
s E
a
F orm
= 2.7 eV
α = 0.7 K
th
= 2 W/(K · m)
φ
b
= 2 eV m
ox
e
= 0.1 · m
e
σ
Ox
= 50 m · S σ
CF
= 5 × 10
6
m · S
- 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
1 0
- 1 3
1 0
- 1 2
1 0
- 1 1
1 0
- 1 0
1 0
- 9
1 0
- 8
1 0
- 7
1 0
- 6
1 0
- 5
1 0
- 4
1 0
- 3
F o r m i n g
S e t
R e s e t
S y m b o l s : D a t a
L i n e s : S i m u .
I C e l l ( A )
V C e l l ( V )
Fig. 4: Experimental I(V ) characteristics measured on a HfO
2
-
based memory structures presented in [18] and corresponding
simulation results from FIG. 3. The parameters of the Table I
were used for simulations.
Based on this waveform, I(V ) characteristics were ex-
tracted. FIG. 4 shows quasi-static Electroforming, Set and
Reset current-voltage I(V ) characteristics measured on actual
HfO
2
-based memory elements [18]. Using the set of physical
parameters given in Table I, together with the actual hafnium
oxide film thickness, the present model shows an excellent
agreement with the experimental data for both Set and Reset
operations.
A peculiar attention must be paid to the dependence of
the resistance in LRS state against the maximum current
allowed during Set operations. As reported in previous works,
the resistance in LRS state (noted R
LRS
) and Reset current
strongly depend on the maximum current reached during the
preceding Set operation [4], [26]–[30] (referred as I
CompSet
).
This feature can be understood in terms of reduction of CF
radius that concomitantly increases the resistance of the MIM
structure [26]. FIG. 5 shows the experimental evolutions of
reset current I
reset
as a function of maximum set current
I
CompSet
[26]. The proposed model matches well the experi-
mental data obtained by various authors and confirms the scal-
ability trend of Reset current I
reset
in resistive memories. The
reset current may be scaled down by limiting the maximum
Set current through an integrated select device (e.g. transistor
or load resistor) in series with memory element.
Another marker of OxRAM is the soft-Reset that induces a
dependency between resistance in HRS (R
HRS
) and the stop
voltage (V
stop
) during the preceding reset operation [23]. This
feature can be understood as an incomplete destruction of the
CF as shown in FIG. 1d. FIG. 6a shows that the dependence of
R
HRS
versus V
stop
is well captured by the model thanks to the
thermal activation of Set and Reset mechanisms. Indeed, since
the filament has a uniform section, no thermal confinement
occurs. There is an interplay between temperature, current