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Compact Two-State-Variable Second-Order Memristor Model

Sungho Kim , Hee-Dong Kim , * and Sung-Jin Choi *

and temperatures can emulate the resistive switching

behavior with a reasonable accuracy; [ 6–11 ] however, because

such an approach requires too much computational power,

the reported models are not suitable for large-scale circuit

simulations.

Alternatively, several compact memristor models have

been developed [ 12–21 ] in which the current–voltage ( I – V ) rela-

tion depends on a state-variable that can evolve in time when

a current is fl owing through the device. Unfortunately, most

of these models have utilized only a one-state-variable, [ 12–19 ]

and they are typically derived by assuming a particular phys-

ical mechanism for the resistive switching and by fi tting the

experimental data to the equations corresponding to this

mechanism. For example, most models are fi tted by assuming

the modulation of the depletion gap length. This assumption

is certainly a simplifi cation of the actual physical mechanism,

and as a result, these models are still not suffi ciently accu-

rate because other multiple mechanisms are involved in the

RS. [ 22 ] Moreover, these models are either nondynamic and

can only predict the steady-state properties under a DC input

signal, [ 12–18 ] or they are phenomenological without including

a physical mechanism. [ 19 ] Recently, the advanced memristor

model based on a two-state-variable has been proposed,

which relies on the evolution of the conductive fi lament (CF)

in terms of both its diameter and depletion gap length. [ 20,21 ]

While these two-state-variable models reproduced the

A key requirement for using memristors in functional circuits is a predictive physical model to capture the resistive switching behavior, which shall be compact enough to be implemented using a circuit simulator. Although a number of memristor models have been developed, most of these models (i.e., fi rst-order memristor models) have utilized only a one-state-variable. However, such simplifi cation is not adequate for accurate modeling because multiple mechanisms are involved in resistive switching. Here, a two-state-variable based second-order memristor model is presented, which considers the axial drift of the charged vacancies in an applied electric fi eld and the radial vacancy motion caused by the thermophoresis and diffusion. In particular, this model emulates the details of the intrinsic short-term dynamics, such as decay and temporal heat summation, and therefore, it accurately predicts the resistive switching characteristics for both DC and AC input signals.

Memristors

DOI: 10.1002/smll.201600088

Prof. S. Kim, Prof. H. -D. Kim Department of Electrical Engineering Sejong University Seoul 05006 , South Korea E-mail: [email protected]

Prof. S.-J. Choi School of Electrical Engineering Kookmin University Seoul 02707 , South Korea E-mail: [email protected]

1. Introduction

The recent progress in memristive [ 1,2 ] devices is promising

for various applications. [ 3,4 ] The development of such appli-

cations and the utilization of the analog switching properties

of memristive devices [ 5 ] will rely on the availability of accu-

rate predictive device models. In oxide-based memristors,

especially relying on the valence change memory effect, the

principle of resistive switching (RS) is believed to be caused

by ion transports (i.e., oxygen vacancies (V O s)) in the oxide

layer, where regions with high concentrations of accumulated

V O s control the conductivity of the device according to the

history of the applied voltage and current. In principle, the

solving of the coupled continuity equations for ions, currents,

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experimental data more reliably over the

input voltage range, they still cannot pre-

dict the fast transient of the RS under an

AC input, [ 20 ] and consequently, they need

to use a different set of parameters for the

simulation of an AC transient. [ 21 ]

The reason why previous compact

models could not fully emulate the RS is

that the previous ones have overlooked

the second-order memristor effect. [ 23,24 ]

In previous models, the RS of oxide-based

memristors was only described by one-

state-variable, which represents the shape

of CF in the oxide-based memristors that

directly affects the resistance of the device.

However, it has been experimentally dem-

onstrated that the second-order memristor

effects can also be signifi cant in oxide-

based memristors. In other words, the

short-term temperature dynamics play an

important role in governing the long-term

RS of the device. [ 24 ] This secondary and

temporal effects affects the resistance of

the device indirectly, i.e., the second-order

memristor effects. Because the previous

compact memristor models have missed

the temporal heat summation and decay

effect, an accurate and fast RS transient

has not been reproduced by a model.

Therefore, in this study, we present a two-

state-variable second-order memristor

model for the memristor based on the

valence change memory effect. The proposed model combines

the axial drift of the oxygen vacancies in an applied electric

fi eld and the radial vacancy movement caused by the thermo-

phoresis and diffusion. In particular, this model illustrates the

details of the short-term dynamics of the temperature under

a fast AC input, which allows for the accurate estimation of a

thermally activated RS in an oxide-based memristor.

2. Second-Order Memristor Model

As mentioned earlier, the RS behavior in oxide-based mem-

ristors is associated with the CF growth and rupture due to

the migration of the oxygen vacancies. [ 25 ] To construct an

analytical model by following the memristor’s theoretical

frame, we simplify the model to a single dominant fi lament.

Figure 1 a schematically illustrates the CF evolution during

the reset and set processes. Here, the RS is caused by the cre-

ation of a depletion gap for the V O s near the electrode during

the reset and by the refi lling of the gap with V O s during the

set. [ 9,10,24 ] The CF is defi ned to be that region in the otherwise

insulating fi lm that contains enough V O s to have a metallic

conductivity. [ 25 ] The CF is approximated here as a cylinder

with a radius r and a length l , and the distance between the

tip of the cylinder and the top electrode (TE) is defi ned as a

depleted gap length g . As a result, the entire device can be

regarded as three serial subdivided parts: the depleted gap,

the CF, and the parasitic series resistance ( R S = 350 Ω), as

shown in Figure 1 b. The currents fl owing through the depleted

gap, CF, and R S are expressed as

sinh{( )/ }gap 0/

b 0I I e V V Vg gTEm= −− ( 1a)

( )/{ /( )}CF b a2I V V l rρ π= − ( 1b)

( )/S a BE SI V V R= − ( 1c)

where ρ is the resistivity of the CF, V TE and V BE are the volt-

ages applied to the top and bottom electrodes (TE and BE),

respectively, and V a and V b are the voltages between the subdi-

vided parts. The fi lament resistivity ρ is calculated based on the

Fuchs–Sondheimer approximation for the nanowire resistivity, [ 25 ]

(1 (3 /4 )(1 ))0 r pρ ρ λ= + − , where ρ 0 is the resistivity of the bulk

tantalum, λ is the electron mean free path, and p is the specularity

factor (i.e., the probability for elastic scattering at the CF surface).

In addition, the current through the depleted gap region can be

modeled as a tunneling current [ 13 ] and exponentially depends on

the gap length g . The fi tting parameters are I 0 , V 0 , and g m . Addi-

tionally, the current continuity conditions require that

gap CF S outI I I I= = = ( 1d)

Equations ( 1c – d) determine the device output current–

input voltage ( I out – V appl ) relations. Specifi cally, the device

small 2016, 12, No. 24, 3320–3326

Figure 1. a) Schematic illustration of the model for the reset and set transitions. In the reset transition, the V O migration toward the top electrode leads to the opening of a depleted gap with an increasing g . In the set transition, the V O injection from the tip of the CF into the depleted gap results in the growth of the CF length (step-1) with an increasing diameter (step-2). b) Equivalent model of the CF with the CF considered to be three subdivided parts: the depleted gap, the fi lament, and the series resistance parts. Metallic conduction dominates in the fi lament and series resistance parts, and the tunneling mechanism governs the conduction in the depleted gap region. c) Schematics of V O concentration ( n ) and its gradient in the vertical (d n /d y ) and lateral (d n /d x ) directions.



resistance R (i.e., the fi rst state-variable) is directly controlled

by g and r. The time-dependent output current ( I out ) for a

given input voltage ( V appl = V TE − V BE ) can be calculated if

the variables g and r are known.

The migration of V O s, determined by the V O drift,

Fick’s diffusion, and Soret diffusion processes (charac-

terized by J drift , J Fick , and J Soret ), causes the evolution

of g and r . The V O drift and Fick’s diffusion fl uxes are

driftJ vn= − and FickJ D n= ∇ , respectively. Here, n is the

V O concentration, D is the diffusion coeffi cient given by

(1/2) exp( / )2a CFD a f E kT= − , and v is the drift velocity given

by exp( / )sinh{( )/(2 )}CF CFv af E kT qaE kTa= − , where f is

the escape-attempt frequency, a is the effective hopping dis-

tance, and E a is the activation energy for V O migration. When

a positive reset voltage is applied to the TE, a fi eld-driven V O

movement (i.e., the drift fl ux of the V O s, J drift ) will migrate in

the direction from the TE to the BE as the V O s are positively

charged, while the V O diffusion (i.e., the Fick’s diffusion fl ux of

V O , J Fick ) produces a net fl ux in the opposite direction from the

BE to the TE because the segments close to the BE have higher

V O concentrations, as shown in Figure 1 a. These two fl uxes par-

tially cancel each other out. In contrast, when a negative set

voltage is applied to the TE, J drift and J Fick have the same direc-

tion, which results in the fast fi lling of the depletion gap, fol-

lowed by the gradual enlargement of the CF along the lateral

direction, as shown in Figure 1 a. For the dynamic rate equations

for variable g (d g /d t ), we fi rst need to obtain expressions for

the V O concentration n along the vertical direction (d n /d y ). For

simplicity, the V O concentration gradient is assumed to be such

that d n /d y = ( α 1 n max )/( y − l 0 ) at y ≥ l 0 , as shown in Figure 1 c. This

V O concentration profi le has the following expected boundary

conditions: having a small value near y = L (the TE) and

increasing values as y → l 0 , n max is the maximum V O concen-

tration of the CF, and α 1 is a fi tting parameter. Evaluating this

gradient at the tip of the CF, y = l = L − g , we consequently

obtain J drift = − vn max and J Fick = ( Dα 1 n max )/( L − g − l 0 ). The

total number of V O s ( N ) that have been transported to the CF

region can then be estimated by integrating the fl uxes over the

area and time as follows: N = Δ t ∫( J drift + J Fick )d A , with the area

A = π r 2 . This change in N leads to a modulation of the

CF length. Using the defi nitions for the concentration in the CF

( n max N /d V ) and the corresponding CF volume change

(d V = π r 2 d l ), and by noting that d l = −d g , d g /d t can be written as

12

2 sinh2

,

( ) / ,(set)

( ) / ( ),(reset)

12

0 CF

TE b

b a

CFdgdt e

a fL g l af

qEakT

EV V g

V V L g

EkT

z α= − − − − ⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

=−

− −

⎧⎨⎪

⎩⎪

−

( 2)

Here, the fi rst term represents the diffusion fl ux, and the

second term represents the drift fl ux. While these two effects

cancel each other with opposite signs when V appl > 0 (during

reset), they add to each other with the same sign when V appl < 0

(during set), which accounts for the relatively slower reset

process in comparison with the fast refi lling of the depletion

gap during the set process.

Similarly, the lateral expansion/shrinkage of the CF is

dominated by the Fick’s and Soret diffusion processes. By

Soret diffusion, the V O s will move toward the hotter region

of a temperature gradient. [ 26 ] When a positive reset voltage

is applied to the TE, the thermal-driven V O movement

(i.e., the Soret diffusion fl ux of V O , J Soret ) will migrate in an

inward direction toward the center of the CF (i.e., the lateral

shrinkage of the CF) because the current fl ow through the

formed CF activates the Joule heating. On the other hand,

the V O diffusion (i.e., the Fick’s diffusion fl ux of V O , J Fick )

produces a net fl ux in the opposite outward direction, as

shown in Figure 1 a, because the segments close to the center

of the CF have a higher V O concentration. These two fl uxes

compete against each other. When a negative set voltage is

applied to the TE, J Soret is negligible, and J Fick leads to the

lateral expansion of r , as shown in Figure 1 a, because the

ruptured CF suppresses the current fl ow and Joule heating.

For the dynamic rate equations for the variable r (d r /d t ),

we fi rst need to obtain expressions for J Soret and J Fick along

the lateral direction. First, the Soret diffusion fl ux is given

as ( / )Soret max CFJ DS n dT dxV= , where S V (= − E a /( kT CF 2 )) is

Soret coeffi cient. [ 26 ] Unfortunately, the temperature gradient

inside the CF cannot be described by a simple analytical

expression because many physical parameters (e.g., thermal

conductivity, electrical conductivity, joule heating genera-

tion/dissipation, and oxygen vacancy density profi le) are

coupled to each other. However, it can be assumed that the

temperature gradient inside the CF will increase with

the temperature of CF. Therefore, for simplicity, the tempera-

ture gradient is assumed to be such that β= β/CF 1 CF

2dT dx T ,

where β 1 and β 2 are the fi tting parameters. Next, to describe

J Fick , the V O concentration gradient along the lateral direction

(d n /d x ) needs to be developed. Similar to the case for d n /d y ,

a simple concentration gradient of d n /d x = ( α 2 n max )/( x − r 0 ) is

assumed for x ≥ r 0 , which has the expected boundary condi-

tions of starting with a small value near x = r and increasing

as x → r 0 , as shown in Figure 1 c. Here, r 0 and α 2 are the fi tting

parameters. Evaluating this gradient at the surface of the CF

( x = r ), J Fick can be obtained as J diff = ( Dα 2 n max )/( r − r 0 ). The

number of V O s ( N ) that have migrated to the sub-CF is then

estimated by integrating J diff over area and time as follows:

N = Δ t ·∫( J Fick + J Soret )d A . In this case, because J Fick and J Soret

are directed through the radial surface of the CF, A = 2π rl . Using the defi nitions of the concentration in the CF

( n max N /d V ) and the sub-CF volume change (d V = 2π rl d r ),

d r /d t can then be written as

α β= − −⎛⎝⎜

⎞⎠⎟

β−2 2

0 2 1 2drdt

a fer r

EkT

TTE

kT a

CFCF

a

CF

( 3)

Here, the fi rst term represents the Fick’s diffusion fl ux,

and the second term represents the Soret diffusion fl ux.

These two effects compete against each other, which deter-

mine the lateral expansion/shrinkage of the CF during the

reset and set processes.

Equations ( 2) and ( 3) correspond to the dynamic rate

equations of the variables g and r , respectively. These

dynamic rate equations for d g /d t and d r /d t in turn determine

the shape of the CF and the resultant device resistance R

(i.e., the fi rst state-variable). However, a close examination

of these equations show that the dynamics of both g and r

small 2016, 12, No. 24, 3320–3326



are strongly (i.e., through the exponential terms) affected

by the internal temperature. Consequently, the evolution of

the fi rst state-variable is governed by the temperature (i.e.,

the second state-variable). Here, for simplicity, we approxi-

mate the device as two regions: an inner region around the

fi lament, the temperature T CF of which directly affects the

V O migration through Equations ( 2) and ( 3) , and an outer

region, which surrounds the inner region with an effective

temperature of T bulk that affects the T CF dynamics. The out-

ermost boundary of the outer region is assumed to be in

thermal contact with the room temperature ( T 0 = 300 K).

The coupling between the inner and outer regions is modeled

by an effective thermal conductance k th . With these approxi-

mations, the dynamic equations for T CF and T bulk can be

written as

( )1

CF1 bulkC dT

dt VI k T Tp th CF= − −

( 4a)

( )2

bulk2 bulk 0C dT

dt VI k T Tp th= − −

( 4b)

where VI corresponds to the Joule heating through the fi la-

ment, C p1 and C p2 are the effective heat capacitances from

the inner region to the outer region and from the outer

region to the outermost boundary, respectively. The Joule

heating also elevates the temperature in the outer region to a

smaller effect, with k th1 and k th2 as the effective thermal con-

ductances. Both C p and k th are treated as fi tting parameters

to account for the unknown details of the thermal profi le

and fi lament structure in this simplifi ed model. Consequently,

Equations ( 4a) and ( 4b) correspond to the dynamic rate

equations of the second state-variable (i.e., the temperatures

determined by T CF and T bulk ).

By solving the set of I – V equations (Equations ( 1a – 1d) ),

the dynamic rate equations for the fi rst-order state variables

(i.e., the device resistance, Equations ( 2) and ( 3) ) and the

dynamic rate equations for the second-order state variables

(i.e., the temperature, Equations ( 4a) and ( 4b) ), the dynamics

of the memristor can be predicted. The physical and fi tting

parameters used in our model are summarized in Figure 2 a.

In particular, the memristor framework allows the device

dynamics with the internal state variables to be fully simu-

lated in a circuit simulator (LTSpice, in this case) as shown in

Figure 2 b. First, using the present values of the current ( I out )

and voltage ( V appl ), the temperatures ( T CF and T bulk ) and

their rates (d T /d t ) from Joule heating is determined. Then,

g , r , d g /d t , and d r /d t are calculated based on the value of T CF

and T bulk . Finally, I out is calculated after a time d t has passed

using the updated values of g and r .

3. Results and Discussions

To calibrate our model, we performed measurements and

compared our simulation results with our experimental data.

A tantalum-oxide-based bilayer memristor that consists of

a highly resistive Ta 2 O 5 layer on top of a less resistive TaO x base layer sandwiched by top and bottom Pt electrodes (TE

and BE) was fabricated, and the detailed fabrication pro-

cess has been reported elsewhere. [ 9,27 ] Figure 3 a shows the

calculated DC I – V characteristics during the set and reset

processes using this simple analytical second-order mem-

ristor model, and these results show good agreement with the

measured data. The physical nature of both these processes

can be studied by examining the change in the parameters

(i.e., g , r , T C F , and T bulk ), as shown in Figure 3 b. Specifi cally,

a depleted gap of ≈1.5 nm is formed during the reset, and

this increases the device resistance. The radius of the CF

also changes accordingly, and r is an important factor of the

device resistance during RS. In addition, T CF increases due

to Joule heating, while T bulk remains close to the room tem-

perature during the DC sweeps because the relatively slow

DC sweeps allow for suffi cient heat dissipation to the envi-

ronment. Therefore, the developed second-order memristor

model reveals the coupling of the different state variables

g , r , T C F , and T bulk during the RS and captures the nature of

the short-term temperature dynamics.

The applications of the proposed model are not limited to

reproducing the DC characteristics. The time-dependent AC

characteristics are now further explored. Figure 4 a shows the

simplifi ed schematics of the actual applied pulse trains used

for the measurement of analog switching behavior. Each

pulse train consists of 20 set or reset pulses (−0.9 and 1.1 V,

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Figure 2. a) Parameters used in the model during the simulation. b) Calculation procedure of the parameters during the simulation.



respectively) followed by small, nonperturbative read voltage

pulses (0.2 V, 1 ms) in intervals. Figure 4 b shows the meas-

ured resistance changes according to the applied pulse trains.

A gradual transition in both the set and reset responses is

clearly observed as the number of applied pulses increases.

The simulated results for this switching behavior match well

with the measured data for both the set and reset pulse trains.

Therefore, this simulation can describe the AC resistive

switching response reliably in addition to

the DC characteristics.

To further evaluate the model,

Figure 4 c shows the calculated g and r

during consecutive pulse trains. During

the reset process, J drift is increased due

to the high electric fi eld at the narrow

depleted gap, and the high temperature

caused by the high reset current leads to

simultaneous increases in both J Fick and

J Soret . An increase in g and a decrease in

r happen simultaneously because J Soret

and J drift , respectively, are more domi-

nant than J Fick . However, in the set pro-

cesses, the temperature inside the CF is

lower than that reached during the reset

pulse because the Joule heating does not

proceed effectively while the strong elec-

tric fi eld at the depleted gap increases

J drift , and this leads to a faster connection

between the tip of the CF and the TE.

Consequently, the depleted gap is pre-

dominantly refi lled by the V O migration

due to the J drift (here, both J drift and J Fick

have the same direction). Interestingly,

after the CF reconnects, the reconnected

CF allows the current fl ow to generate

Joule heating, which leads to an increase in J Fick . While the

temperature during the set process is lower than that of the

reset process as shown in Figure 3 b, J Soret is also increased

but temperature during the set process is lower than that of

the reset process as shown in Figure 3 b. As a result, J Fick is

more dominant than J Soret , and r is increased by J Fick during

the subsequent set pulses. Accordingly, the CF change is a

two-step process during the set train: (1) fast vertical gap

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Figure 3. a) Measured and calculated DC I – V characteristics of the Pt/Ta 2 O 5 /TaO x /Pt device. The measured device size is 5 µm × 5 µm, and the voltage sweep speed is 2 V s −1 . b) Calculated parameter changes as a function of time during the DC sweep.

Figure 4. a) Schematics of the applied pulse trains used for the measurement of the AC switching behavior. Each pulse train consists of 20 set or reset pulses (−0.9 and 1.1 V, respectively; 10 µs) followed by small, nonperturbative read voltage pulses (0.2 V, 1 ms) in intervals. b) Measured and calculated switching behavior. c) Calculated g and r at different points in the consecutive pulse trains.



fi lling by the fi eld-driven migration of the internal V O s and

( 2) the following lateral expansion of the CF by diffusion.

Therefore, a relatively abrupt transition in the device resist-

ance occurs during the set because of the faster connection

of the tip of the CF and the TE. From these results, it can be

inferred that the balance among the fl uxes and the internal

temperature dynamics are a crucial factor for governing the

resistive switching, and the proposed model accurately rep-

resents this switching process.

As mentioned above, the short-term temperature

dynamics (i.e., the second-order memristor effect) plays an

important role in governing the RS of the device. To sys-

tematically study how the RS behavior is affected by the

internal temperature dynamics, the transient response of the

RS was measured at different t SET and t interval confi gurations

(the measurement procedure is the same as the previously

reported one [ 24 ] ), as shown in Figure 5 a. In this measurement,

a train of set pulses were applied with the pulse amplitude

fi xed at V SET and t SET while t interval ranged from 100 ns to 1 µs.

Figure 5 b shows that two qualitatively different RS behaviors

can be observed depending on t interval : (1) when t interval is short

(100 ns), abrupt RS behavior is observed because the heat

generated by the set pulses can be temporarily accumulated

due to the short t interval ; ( 2) under a long t interval (1 µs), each

set pulse does not create a high enough temperature rise

(whereas the heat generated during the previous set pulses

will be effectively dissipated during the long interval), and

therefore, heat accumulation is not achieved, resulting in a

slower V O migration and a more gradual RS. Our second-

order memristor model captures the nature of the short-

term temperature dynamics accurately. Figure 5 c shows the

calculated transient response of I out , which is consistent with

the measured data. The model can emulate the different RS

behavior according to t i nterval , where the internal tempera-

ture dynamics are also accurately predicted by the model, as

shown in Figure 5 d. Figure 5 d shows the calculated transient

response of T CF and T bulk . Due to the temporal heat accumu-

lated with a short t i nterval , T CF exceeds 650 K, and T bulk is also

increased to near 400 K. In contrast, T CF is maintained ≈500 K

under a long t i nterval , with an almost constant T bulk at 300 K.

small 2016, 12, No. 24, 3320–3326

Figure 5. a) Schematic of pulse trains used during the measurements. A pulse train consists of a single reset pulse ( V RESET = 1.4 V, pulse duration t RESET = 40 µs) and 100 subsequent set pulses ( V SET = −0.9 V, pulse duration t SET = 100 ns), where the intervals between the set pulses ( t interval ) are either 1 µs or 100 ns. b) Measured transient responses with different t interval s, 1 µs and 100 ns. An abrupt RS with a t interval = 100 ns (left) and a gradual RS with a t interval = 1 µs (right) was observed. c) Simulated transient responses with different t interval s. d) Transient response of T CF and T bulk obtained from the analytical second-order memristor model.


3326 www.small-journal.com © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim small 2016, 12, No. 24, 3320–3326

The short-term dynamics of temperature and the temporal

summation effect allow for a different memristor resistance

change. As a result, this two-state-variable second-order

memristor model, with an emphasis on the internal dynamics,

enables the accurate prediction of the dynamic behavior of

the device, where the fi rst state-variable is the device resist-

ance determined by g and r , and the second state-variable is

given by T CF and T bulk .

4. Conclusion

In summary, we developed a two-state-variable second-

order memristor model and fi tted it using our experimental

measurement data in Pt/Ta 2 O 5 /TaO x /Pt memristor devices.

The model reproduced the experimental data accurately,

especially the transient response under the applying pulses.

The two-state variables determine the device resistance. The

depleted gap length g and the radius of the conducting fi la-

ment r were responsible directly for the change of the device

resistance (i.e., the fi rst state-variable). In addition, the unre-

vealed internal temperature dynamics described by T CF and

T bulk were also responsible for the shorter timescale system-

atic changes in the device resistance. This second-order effect

is important in understanding fast switching, and this com-

pact model is suitable for fast circuit simulations because of

the simple form of its equations. This study provided a useful

methodology to exploit second-order memristors, and it will

help to fi nd other applications of second-order memristors in

the future.

5. Experimental Section

Transient Response Measurements : For the transient meas-urements, the device was serially connected to a series resistor (50 Ω), which was used to read out the transient current through the device. A designed pulse train was applied to the device’s TE. A digital oscilloscope (with both input resistances of CH1 and CH2 set to 1 MΩ) was used to record the voltage transients at the TE and BE locations in the circuit. The net applied voltage and current in the device were calculated as V appl = V CH1 − V CH2 and I out = V CH2 /50 Ω, respectively.

Acknowledgements

The work was supported by the National Research Foun-dation of Korea through the Ministry of Education,

Science and Technology, Korean Government, under Grant 2013R1A1A1057870.

[1] L. Chua , IEEE Trans. Circuit Theory 1971 , 18 , 507 . [2] L. Chua , S. M. Kang , Proc. IEEE 1976 , 64 , 209 . [3] J. J. Yang , M. D. Pickett , X. Li , D. A. A. Ohlberg , D. R. Stewart ,

R. S. Williams , Nat. Nanotechnol. 2008 , 3 , 429 . [4] D. B. Strukov , G. S. Snider , D. R. Stewart , R. S. Williams , Nature

2008 , 453 , 80 . [5] F. Alibart , E. Zamanidoost , D. B. Strukov , Nat. Commun. 2013 , 4 ,

2072 . [6] X. Guan , S. Yu , H.-S. P. Wong , IEEE Trans. Electron Devices 2012 ,

59 , 1172 . [7] D. S. Jeong , H. Schroeder , R. Waser , Phys. Rev. B 2009 , 79 ,

195317 . [8] S. Kim , S.-J. Kim , K. M. Kim , S. R. Lee , M. Chang , E. Cho , Y.-B. Kim ,

C. J. Kim , U. -In Chung , I.-K. Yoo , Sci. Rep. 2013 , 3 , 1680 . [9] S. Kim , S. H. Choi , W. Lu , ACS Nano 2014 , 8 , 2369 .

[10] S. Larentis , F. Nardi , S. Balatti , D. C. Gilmer , D. Ielmini , IEEE Trans. Electron Devices 2012 , 59 , 2468 .

[11] D. B. Strukov , J. L. Borghetti , R. S. Williams , Small 2009 , 5 , 1058 .

[12] H. Abdalla , M. D. Pickett , in Proc. of ISCAS , IEEE , Rio de Janeiro, Brazil 2011 , p. 1832 .

[13] X. Guan , S. Yu , H. -S. P. Wong , IEEE Electron Device Lett. 2012 , 33 , 1405 .

[14] M. Pickett , D. B. Strukov , J. Borghetti , J. Yang , G. Snider , D. Stewart , R. S. Williams , J. Appl. Phys. 2009 , 106 , 074508 .

[15] P. Sheridan , K. H. Kim , S. Gaba , T. Chang , L. Chen , W. Lu , Nanoscale 2011 , 3 , 3833 .

[16] J. P. Strachan , A. C. Torrezan , F. Miao , M. D. Pickett , J. J. Yang , W. Yi , G. Medeiros-Ribeiro , R. S. Williams , IEEE Trans. Electron Devices 2013 , 60 , 2194 .

[17] J. Noh , M. Jo , C. Y. Kang , D. Gilmer , P. Kirsch , J. C. Lee , B. H. Lee , IEEE Electron Device Lett. 2013 , 34 , 1133 .

[18] Z. Biolek , D. Biolek , V. Biolkova , Radioengineering 2009 , 18 , 210 . [19] F. M. Bayat , B. Hoskins , D. B. Strukov , Appl. Phys. A 2015 , 118 ,

779 . [20] S. Ambrogio , S. Balatti , D. C. Gilmer , D. Ielmini , IEEE

Trans. Electron Devices 2014 , 61 , 2378 . [21] L. Zhang , N. Ge , J. J. Yang , Z. Li , R. S. Williams , Y. Chen ,

Appl. Phys. A 2015 , 119 , 1 . [22] J. J. Yang , D. B. Strukov , D. R. Stewart , Nat. Nanotechnol. 2013 ,

8 , 13 . [23] Y. V. Pershin , M. Di Ventra , Proc. IEEE 2012 , 100 , 2071 . [24] S. Kim , C. Du , P. Sheridan , W. Ma , S. H. Choi , W. D. Lu , Nano

Lett. 2015 , 15 , 2203 . [25] D. Ielmini , IEEE Trans. Electron Devices 2011 , 58 , 4309 . [26] D. B. Strukov , F. Alibart , R. S. Williams , Appl. Phys. A 2012 , 107 ,

509 . [27] S. Kim , S. Choi , J. Lee , W. D. Lu , ACS Nano 2014 , 8 , 10262 .

Received: January 11, 2016 Revised: March 31, 2016 Published online: May 6, 2016

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