COMSOL 6.4 - Memristor

A memristor, short for “memory resistor,” is a two-terminal electronic component that controls the flow of electric current while retaining a memory of the charge that has previously passed through it. As a nonvolatile memory device, it preserves stored information even after power is removed. This distinctive feature makes memristors promising candidates for next-generation memory storage systems and emerging technologies such as neuromorphic computing, resistive random-access memory (ReRAM), and artificial synapses in neural networks.

Structurally, a memristor comprises a thin semiconductor layer — typically a metal oxide — sandwiched between two metal electrodes. When a voltage is applied, charge carriers such as oxygen vacancies migrate within the oxide layer, redistributing in a way that alters the material’s conductivity and thus its resistance. This change in resistance is what gives the memristor its memory capabilities. Once the voltage is removed, the distribution of charge carriers remains stable, allowing the device to retain its resistance state without power.

The model presented here simulates the dynamic resistive switching behavior of an oxide-based memristor.

Model Definition

This model adopts the approach outlined in Ref. 1 to simulate the dynamic resistive switching behavior of an oxide-based memristor. It uses a 2D-axisymmetric geometry, which enables the simulation of a 3D system through a 2D cross-sectional representation. This approach significantly reduces computational costs while effectively capturing the 3D behavior of the device. The structure consists of two oxide layers — a highly resistive layer stacked atop a low-resistivity layer — with thicknesses of 30 nm and 5 nm, respectively. These oxide layers are sandwiched between two metal electrodes, each 30 nm thick, positioned at the top and bottom. Figure 1 illustrates the configuration and geometry of the modeled memristor. The domain with a high concentration of charge carriers is identified as the conductive filament (CF), where local electrical conduction exhibits metallic behavior. The concept of CF formation and rupture has been employed to explain resistive switching behavior. The material properties, such as electric conductivity and thermal conductivity for each domain, are defined according to the approach described in Ref. 1. The dynamic resistive switching behavior of the modeled memristor is governed by oxygen vacancy drift–diffusion, current continuity, and heat transfer, along with their fully coupled multiphysics interactions. The model utilizes the Joule Heating interface to account for resistive heating, while the Transport of Charge Carriers interface captures the transport equations for oxygen vacancies as charge carriers.

Figure 1: Configuration of the modeled memristor in a 2D-axisymmetric geometry, including metal electrodes and oxide layers.

The outermost boundaries of the two electrodes are defined with specific boundary conditions to ensure an accurate simulation of Joule heating effects. The bottom electrode is grounded for the electric potential, while a time-dependent voltage is applied to the top electrode. Additionally, a constant temperature is maintained at both the top and bottom contacts. For charge transport, a No flux boundary condition is imposed on all outer boundaries of the oxide domains, including the interfaces between the oxides and metal electrodes.

The detailed procedure of the implementation is described in the Modeling Instructions section.

Results and Discussion

Figure 2 presents the I–V characteristics of the modeled memristor, with the color table illustrating the evolution of time with respect to the applied voltage. At the beginning of the simulation, the device is initialized in a high-conductivity state. When a positive time-dependent voltage is applied, the device undergoes a “reset” transition, shifting from high to low conductivity. In contrast, applying a negative time-dependent voltage induces a “set” transition, restoring the device to its high-conductivity state. These transitions demonstrate the resistive switching behavior characteristic of oxide-based memristors.

Figure 2: I–V characteristics of the oxide-based memristor.

An analysis of the concentration profiles provides insight into the physical mechanisms governing the set and reset processes. Figure 3 shows the concentration ratio (defined as the charge carrier concentration divided by the initial concentration, n/n0) along the CF at r = 0, at the initial state, reset, and set processes. During the reset process, a 1 nm depletion region forms close to the CF–metal electrode interface, resulting in a decrease in electric conductivity. In contrast, during the set process, charge accumulation occurs at the interface, restoring the high-conductivity state of the memristor. The low- and high-conductivity states, corresponding to the reset and set processes, are illustrated in Figure 4 and Figure 5, respectively.

Figure 3: Concentration ratio of the charge carriers along the CF at r = 0 at the initial state, reset, and set states.

Figure 4: Electric conductivity profile illustrating the region of low electric conductivity near the CF–metal electrode interface following the reset state.

Figure 5: Electric conductivity profile illustrating the region of high electric conductivity near the CF–metal electrode interface following the set state.

Reference

1. S. Kim and others, “Comprehensive physical model of dynamic resistive switching in an oxide memristor,” ACS nano., vol. 8, no. 3, p. 2369–2376, 2014.

Semiconductor_Module/Device_Building_Blocks/memristor

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
d_BE	30[nm]	3E-8 m	Bottom electrode thickness
d_TE	30[nm]	3E-8 m	Top electrode thickness
d_ox1	30[nm]	3E-8 m	Oxide1 thickness
d_ox2	5[nm]	5E-9 m	Oxide2 thickness
w	20[nm]	2E-8 m	Oxide radius
r_CF	5[nm]	5E-9 m	Conductive filament radius
T0	300[K]	300 K	Reference temperature
n0	1e21[1/cm^3]	1E27 1/m³	Initial concentration
a	0.1[nm]	1E-10 m	Hopping distance
f	1e12[Hz]	1E12 Hz	Escape-attempt frequency
Ea	0.85[eV]	1.3619E-19 J	Diffusion barrier

Definitions

Interpolation 1 (int1)

In the toolbar, click

In the window for , locate the section.

In the text field, type sigma0.

In the table, enter the following settings:


t	f(t)
0	10
1	940

Locate the section. In the table, enter the following settings:


Function	Unit
sigma0	S/cm

In the table, enter the following settings:


Argument	Unit
t	1

Interpolation 2 (int2)

In the toolbar, click

In the window for , locate the section.

In the text field, type E_AC.

In the table, enter the following settings:


t	f(t)
0	0.05
0.5	-0.006
1	-0.006

Locate the section. In the table, enter the following settings:


Function	Unit
E_AC	eV

In the table, enter the following settings:


Argument	Unit
t	1

Interpolation 3 (int3)

In the toolbar, click

In the window for , locate the section.

In the text field, type k_th0.

In the table, enter the following settings:


t	f(t)
0	0.12
1	57.5

Locate the section. In the table, enter the following settings:


Function	Unit
k_th0	W/(m*K)

In the table, enter the following settings:


Argument	Unit
t	1

Applied Voltage

In the toolbar, click

In the window for , type Applied Voltage in the text field.

Locate the section. In the text field, type V0.

In the table, enter the following settings:


t	f(t)
0	0
0.6	1.2
1.2	0
1.7	-1
2.2	0

Locate the section. In the table, enter the following settings:


Function	Unit
V0	V

In the table, enter the following settings:


Argument	Unit
t	s

Click

A time-dependent voltage is applied at a rate of 2 V/s.

Variables 1

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
D_n	(0.5a^2fexp(-Ea/(k_B_constT)))	m²/s	Diffusion coefficient
S	-Ea/(k_B_const*T^2)	1/K	Soret diffusion coefficient
v_Er	afexp(-Ea/(k_B_constT))sinh(e_constaec.ER/(k_B_const*T))	m/s	Velocity due to Electric field, r component
v_Ez	afexp(-Ea/(k_B_constT))sinh(e_constaec.EZ/(k_B_const*T))	m/s	Velocity due to Electric field, z component
v_Sr	D_nTrS	m/s	Velocity due to Soret effect, r component
v_Sz	D_nTzS	m/s	Velocity due to Soret effect, z component
vr	v_Er-v_Sr	m/s	Total velocity, r component
vz	v_Ez-v_Sz	m/s	Total velocity, z component
k	n/n0		Concentration ratio
sigma	sigma0(k)exp(-E_AC(k)/(k_B_constT))	S/m	Electric conductivity
k_th	k_th0(k)(1+0.1[1/K](T-T0))	W/(m·K)	Thermal conductivity

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type w.

In the text field, type d_BE.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type w.

In the text field, type d_ox1.

Locate the section. In the text field, type d_BE.

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type w.

In the text field, type d_ox2.

Locate the section. In the text field, type d_BE+d_ox1.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (nm)
Layer 1	r_CF

Select the checkbox.

Clear the checkbox.

Rectangle 4 (r4)

In the toolbar, click

In the window for , locate the section.

In the text field, type w.

In the text field, type d_TE.

Locate the section. In the text field, type d_BE+d_ox1+d_ox2.

Click

Definitions

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click the

button in the toolbar.

Select Boundary 9 only.

Variables 1

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
I	-intop1(ec.JZ)	A	Current

Materials

Palladium

In the window, under right-click and choose .

In the window for , type Palladium in the text field.

Select Domains 1 and 4 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electric conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	9.5E4[S/cm]	S/m	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	71.8[W/(m*K)]	W/(m·K)	Basic

TaxOy

Right-click and choose .

In the window for , type TaxOy in the text field.

Select Domains 2, 3, and 5 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electric conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	sigma	S/m	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	k_th	W/(m·K)	Basic