A Physics-Based Compact Model of Metal-Oxide-Based RRAM DC and AC Operations

4090 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 12, DECEMBER 2013

A Physics-Based Compact Model ofMetal-Oxide-Based RRAM DC and

AC OperationsPeng Huang, Student Member, IEEE, Xiao Yan Liu, Bing Chen, Student Member, IEEE, Hai Tong Li,

Yi Jiao Wang, Ye Xin Deng, Kang Liang Wei, Lang Zeng, Bin Gao, Student Member, IEEE,Gang Du, Xing Zhang, and Jin Feng Kang

Abstract— A physics-based compact model of metal-oxide-based resistive-switching random access memory (RRAM) cellunder dc and ac operation modes is presented. In this model,the conductive filament evolution corresponding to the resis-tive switching process is modeled by considering the transportbehaviors of oxygen vacancies and oxygen ions together with thetemperature effect. Both the metallic-like and electron hoppingconduction transports are considered to model the conductionof RRAM. The model can reproduce both the typical I–Vcharacteristics of RRAM in high-/low-resistance state (LRS) andthe nonlinear characteristics in LRS. Moreover, to accuratelymodel ac operation mode, the effects of parasitic capacitanceand resistance are included in our model. The developed compactmodel is verified and calibrated by measured data in differentHfOx-based RRAM devices under dc and ac operation modes.The excellent agreement between the model predictions andexperimental results shows a promising prospect of the futureimplementation of this compact model in large-scale circuitsimulation to optimize the design of RRAM.

Index Terms— Circuit simulation, compact model, conductionof resistive-switching random access memory (RRAM), conduc-tive filament’s evolution, parasitic effect, pulse mode, resistiveswitching.

I. INTRODUCTION

METAL-OXIDE-BASED resistive-switching randomaccess memory (RRAM) has attracted considerable

interests as next generation of memory technology due toits excellent memory performance such as simple structure,compatibility with current CMOS technology, and greatscalability potential [1]–[4]. Extensive researches have beenconducted to understand the resistive switching mechanismsand improve the performance of RRAM [5]–[8]. The resistiveswitching behavior is associated with the formation andrupture of conductivity filament (CF) [7]–[9]. Although greatprogress has been made on researches of RRAM, there are

Manuscript received May 22, 2013; revised October 7, 2013; acceptedOctober 21, 2013. Date of publication November 5, 2013; date of currentversion November 20, 2013. This work was supported by the 973 Programunder Grant 2011CBA00604. The review of this paper was arranged by EditorG. Jeong.

The authors are with the Key Laboratory of Microelectronic Devices andCircuits, Institute of Microelectronics, Peking University, Beijing 100871,China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected];[email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2013.2287755

still some critical technological issues needed to be solvedfor the future memory application [9]. The compact model ofan RRAM cell for circuit simulation is required to investigatethe sneak current path and voltage degradation caused by theinterconnect resistance in the RRAM array [10]. So far, muchworks have been done for modeling the switching behavior ofmetal–oxide-based RRAM [11]–[14] and some SPICE modelsbased on the subcircuits have been presented to describe thefeatures of a single device [15]–[17]. However, there are afew works addressing the physics-based compact model ofRRAM operation available for circuit simulation of RRAMarray [18], [19].

In this paper, a physics-based compact model of RRAMcell under dc and ac operation modes is developed based onthe insight of switching behavior simulation. This paper isorganized as follows. In Section II, the physics-based compactmodel is described, including physical mechanism of resistiveswitching, CF evolution process model, electrical transportmodel, heat conduction model, and parasitic effect model.Under an appropriate approximation, concise mathematicalformulas are deduced to describe the electrical characteristicsof RRAM device. In Section III, the developed compact modelis verified by the experimental results of different HfOx -based RRAM devices under both static and transient operationmodes. The excellent agreement between the model predic-tions and measured data shows that the developed physics-based compact model can be implemented in circuit simulationfor the design of RRAM array and give guidance to optimizeRRAM cell performance.

II. COMPACT MODEL OF RRAM OPERATION

A. Physical Mechanism

To develop an accurate model of RRAM for circuit simu-lation, the resistive switching and conduction behaviors arefirst modeled. For metal–oxide-based RRAM, it has beencommonly accepted that the resistive switching is due to theformation and rupture of CF. The switching characteristics arestrongly correlated with the geometry of CF as a direct resultof generation and recombination of oxygen vacancies (Vo) inthe switching oxide layer [12]–[14]. The physical process ofthe switching operation in an RRAM device is schematicallyshown in Fig. 1. In the SET process, the generation of Vo andthe drift of dissociated oxygen ion (O2−) to the top electrodecause the formation of CF connecting anode and cathode,

0018-9383 © 2013 IEEE

HUANG et al.: PHYSICS-BASED COMPACT MODEL 4091

Fig. 1. Schematic physical process of resistive switching used in the model,including the generation of oxygen vacancies, oxygen ions hopping, electrodeabsorbing and release oxygen ions, recombination between oxygen vacanciesand oxygen ions, and local temperature increase due to local current andelectron transport.

which results in the cell switching from the high-resistancestate (HRS) to the low-resistance state (LRS). The generationprobability (Pg) of Vo under electric field during dt is governedby the following equation [12]:

Pg(E, T, dt) = f dt exp

(− Ea − αa ZeE

kB T

)(1)

where E is the electric field, T is the local temperature, f isthe vibration frequency of oxygen atom, Ea is the averageactive energy of Vo, αa is the enhancement factor of theelectric field for the lowering of Ea , Z = 2 is the chargenumber of oxygen ion, e is the unit charge, and kB is theBoltzmann constant. The hopping probability (Ph) of thedissociated O2− in the oxide layer under the electric fieldduring dt can be described as [14]

Ph(E, T, dt) = f dt exp

(− Eh − αh ZeE

kB T

)(2)

where Eh is the hopping barrier of O2− and αh is theenhancement factor of the electric field for the lowering of Eh .During the RESET process, the dissociated O2− is released bythe top electrode and then hop to the nearby of Vo to recombinewith them, which causes the rupture of CF and the RRAMswitching from the LRS to the HRS. The recombinationbetween O2− and electron-depleted Vo is modeled as an energyrelaxation process and governed by the following equation[12]:

Pr (T, dt) = f dt exp

(−�Er

kB T

)(3)

where Pr (T , dt) is the probability of the recombinationbetween Vo and O2− during dt and �Er is the relaxation

Fig. 2. Simulated I–V curves in the (a) RESET and (b) SET processesunder the dc sweep mode.

energy during the recombination process. The top electrodeis the active electrode and acts as an O2− reservoir thatcan release O2− with the probability (Pm) governed by thefollowing equation [14]:

Pm(V , T, dt) = f dt exp

(− Ei − γ ZeV

kB T

)(4)

where Pm(V , T , dt) is the probability of electrode release O2−during dt, V is the external voltage, Ei is the energy barrierbetween the electrode and oxide, and γ is the enhancementfactor of the external voltage during the O2− release process.

The electron transport is a metallic-like transport along CFand the conductivity decreases with increasing T , accordingto Arrhenius law [20]

σ = σ0 exp (EAC/kB T ) (5)

where EAC is the activation energy for conduction and σ0is the Arrhenius preexponential factor for conductivity. Theelectron transport among the dispersive Vo is hopping andthe hopping rate between two vacancies can be calculated bythe Mott hopping model [21]

Wm→n = f ph exp(−2αRmn − Emn/kB T ) (6)

where the fph is the vibration frequency of electron, Rmn isthe distance between two Vo, α is the attenuation length ofthe electron wave function, and Emn is the change of barrierheight induced by the applied external field.

The local temperature plays an important role during theswitching processes. In general, the temperature distributioncan be obtained by solving the Fourier heat-flow equation [11]

C∂T

∂ t= ∇(k∇T ) + Q (7)

where C is the specific heat per unit volume of oxide, k is thethermal conductivity of oxide, and Q is the sum of multipleheat sources due to the inelastic electron–phonon.

B. Modeling of CF Evolution

The model of CF evolution process is critical for thecompact model for circuit simulation. With the mechanismof resistive switching, we developed a stochastic simulatorto investigate the characteristics of CF evolution during theresistive switching process [14]. The simulated I–V curvesand corresponding CF geometry evolution processes duringthe switching process are shown in Figs. 2 and 3, respectively.


Fig. 3. CF geometry evolution process during the (a) RESET and (b) SETprocesses, which is corresponding to Fig. 2.

Fig. 4. Schematic diagram of the CF evolution model. The reduction ofx in RESET process is determined by the slowest process (I–III) among thedominant processes for the RESET. The SET process is divided into two steps.First step: CF growth from the rupture CF tip to the electrode. Second step:CF expanding along the radius direction of the formed CF.

The results show that the rupture process of CF during theRESET operation is corresponding to the whole filamentdisconnecting first at the top electrode and then extendingtoward the interior step by step as the voltage increases.Whereas, the growing process of CF during the SET operationis corresponding to the formation of a fine filament in therupture region first connecting the tip of the residual CF withthe top electrode and then gradually enlarging along the radiusdirection as the current increases.

With the insight of the stochastic simulation, we modelthe 3-D CF evolution process as shown in Fig. 4. First, acylindrical CF with a diameter of w0 has emerged between thetwo electrodes and RESET process is modeled as the variationof gap distance x between the top electrode and the tip ofCF. In this case, the gap distance x between the rupture CFtip and the top electrode determines the resistance of HRSduring the RESET process [22], [23]. Hence, x and dx/dt arecritical factors for RESET process. With the mechanism of

resistive switching, we can conclude that the value of dx/dtis determined by three physical processes: 1) electrode releaseO2−; 2) O2− hopping in the oxide layer; and 3) recombinationbetween O2− and Vo. The reduction of x is determined by theslowest process among those three processes for the RESET.When O2− hopping is dominant, the flow rate of O2− (J 2−

O ),which is the amount of O2− through unit area in unit time,can be written as

JO2− = 1/2(Ph(E, T, dt) − Ph(−E, T, dt))/(a2dt) (8)

where a is the distance between two Vo and the coefficient 1/2results from two different directions for O2− hopping. Thus,the amount of O2− hopping to Vo during dt is

NO2− = JO2−π (w0/2)2 dt (9)

and meanwhile the amount (NV o) of Vo recombined with O2−is

NVo = π (w0/2)2 dx/a3. (10)

Combining (9) with (10), dx/dt can be deduced as

dx

dt= a f exp

(− Eh

kB T

)sinh

(αh ZeE

kB T

). (11)

Likewise, when the process of electrode release O2− is dom-inant, dx/dt can be written as

dx

dt= a f exp

(− Ei − γ ZeV

kB T

)(12)

and when the process of recombination between O2− and Vo

is the slowest, dx/dt is written as

dx/dt = a f exp (−�Er/kB T ) . (13)

After RESET operation, x is fixed to x0. Then, SET processis performed based on the residual CF with x0. With the sim-ulated results, the CF evolution process during SET operationis modeled by two steps: 1) a thin CF grows from the residualCF tip to the top electrode and 2) the newly grown CF expandsalong the radius direction of the previously formed CF. Thetransient resistance during SET operation is determined by thex0 and w of the newly formed CF. Hence, dx/dt and dw/dt arethe key factors to model the SET operation. With the resistiveswitching mechanism, the key physical process during the SETprocess is the generation of Vo. The amount of Vo generatedin the rupture region during dt in the first step of SET processis given by the following equation:

NVo g1 = Nsum Pg(E, T, dt) = Pg(E, T, dt)aπr2/

a3 (14)

where r is the radius of newly grown thin CF. The absolutevalue of dx is equal to the variation length of the newly grownCF (l)

dx = − NV Og1a3

πr2 = −a f exp

(− Ea − αa ZeE

kB T

)dt . (15)

In the second step, the generation of Vo mainly occurs inthe extended region along the radial direction. The volume ofthis region is

Vregion = x0π((w + �w)2 − w2)/4 (16)


Fig. 5. (a) Equivalent circuit of conduction in LRS and HRS. (b) and(c) LRS/HRS I–V curves reproduced by the conduction model. The nonlinearcharacteristic in LRS is reproduced by the developed conduction model withsmall LRS (w).

where �w is the effective expanding width of CF. Thus, theamount of Vo generation during dt is

NVog2 = Vregion Pg(E, T, dt)/a3. (17)

Hence, the changing rate of the width of CF is given as

dw

dt=

(�w + �w2

2w

)f exp

(− Ea − αa ZeE

kB T

). (18)

Until now, we can obtain the CF geometry evolution basedon (11)–(13), (15), (18) with the given initial conditions.

C. Modeling of Conduction

According to the aforementioned discussion of physicalmechanism, the metallic-like conduction and hopping con-duction are considered in this paper. The conduction of CFregion is modeled as metallic-like conduction with conductiv-ity described by (5), while the conduction of gap region ishopping conduction. The current in the CF can be calculatedbased on (5) and Ohm’s law. In general, the hopping currentcan be calculated by (6) with distribution of Vo in real andenergy spaces [23]. In this paper, we have adopted a compactpicture, in which the hopping current is correlated with thevoltage and gap distance by [24]

I = I0 exp(−x/xT ) sinh(V/VT ) (19)

where I0 is 10 μA/nm2 and xT and VT are the characteristiclength and voltage, respectively. Hence, the LRS is modeledby the conduction of CF with a width of w paralleled with anonlinear hopping current, as shown in Fig. 5(a). Under theultralow current condition, w is small and the nonlinear I–Vcurves at LRS can be reproduced, as shown in Fig. 5(b). TheI–V characteristics at HRS are shown in Fig. 5(c), which canbe modeled as two parallel nonlinear hopping currents withdifferent x , as shown in Fig. 5(a).

Fig. 6. Fitting of temperature of RRAM in HRS. The calculated temperatureis the average value of temperature in the CF growth region.

Fig. 7. Equivalent circuit of an RRAM cell with parasitic elements. Theparasitic resistance and capacitance are considered in this model.

D. Modeling of Temperature Effect

During switching process, the formation and rupture of CFare sensitive to the temperature. Therefore, the temperatureeffect is important to the resistive switching behavior andshould be considered in the model.

For the purpose of simplification, uniform temperature isused in our model. The description of temperature at LRSemploys the previously proposed analytic model [11]

T = T0 + I V Rth (20)

where T0 is the ambient temperature and Rth is the effectivethermal resistance. It should be noted that the transient effectof thermal capacitance in (20) is neglected to simplify themodel where the temperature variation is assumed to be muchfaster than the voltage variations imposed by the setup; inother words, the model is based on a quasi-static approach,where the thermal equilibrium is assumed to be reached oncethe operation voltage is applied to the cell. The temperaturein the resistive switching region at HRS was calculated basedon (7) and fits an equation, as shown in Fig. 6. The equationis the same as previous analytic model at LRS.

E. Modeling of Parasitic Effect

To accurately model the transient operation of RRAM, theparasitic effects should be considered. The equivalent circuitincluding the parasitic elements is shown in Fig. 7. It consistsof a parallel capacitance (Cp), a large parallel resistance (Rp),


TABLE I

PARAMETERS USED IN THIS PAPER

Fig. 8. Modeled dc I–V curves during the RESET and SET operation,together with the measured data.

contact resistance (Rc), and the resistive switching element.Cp is the result of the intrinsic MIM structure of metal–oxide-based RRAM. Rp includes the conduction associated with theleakage current between two electrodes. Rc represents the totalresistance of the electrodes itself and the contact resistancebetween the CF and electrodes. All the parameters used inthis model are listed in Table I.

III. MODEL VERIFICATION

Both dc and ac electrical characteristics of metal–oxide-based RRAM are modeled and compared with the measureddata of the fabricated TiN/TiOx /HfOx /TiOx /HfOx /Pt RRAMdevices [25]. The electrical measurements were performedwith Keithley 4200 and Agilent 81150A.

Fig. 8 shows the measured and calculated dc I–V curvesduring RESET and SET processes, respectively. The compactmodel can reproduce the gradual RESET and the abruptSET processes, which are consistent with the measured data.According to the physical mechanism of resistive switchingdiscussed in Section II, three physical processes determinethe RESET process. Under dc sweep mode, the voltage issmall in the prophase of the RESET process; hence, therecombination between O2− and Vo is faster than the othertwo physical processes. Meanwhile, short gap distance induceshigh electric field in the gap region and fast O2− hopping.

Fig. 9. Transient response during (a) RESET and (b) SET processes.Excellent agreement between the measured and modeled data is presented.

Therefore, the process of electrode release O2− is dominantin the prophase of the RESET process and it will determinethe RESET voltage. In the anaphase of RESET process, theelectric field in the gap region becomes weak with the increaseof gap distance and O2−hopping becomes slow. Therefore,O2− hopping is the dominant physical process in the anaphaseof the RESET process and it will determine the gap distance xand the resistance of HRS after the RESET process. During theRESET process, high stop voltage results in large resistanceof HRS as the model predicted in Fig. 8, since high RESETvoltage causes large gap distance x of the rupture region [22].Hence, multilevel HRS can be achieved with different stopvoltages during the RESET process [27]. The evolution rateof CF during the SET process depends on the electric fieldin the gap region, as described by (15) and (18). Hence, theSET voltage will increase with the increase in the initial gapdistance x0 as the model predicted in Fig. 8.

Fig. 9 shows the calculated and measured transientresponses during the RESET/SET process. Under ac modewith low pulse height or long rising edge, the dominantphysical process during the RESET process is the same asthe condition under dcsweep mode. However, the dominantphysical process is different under ac mode with large pulseheight and short rising edge. In the prophase of the RESETprocess, the dominant physical process is still the electroderelease O2−. As the voltage increases quickly whereas the gapdistance x is still small, the recombination between O2− andVo becomes the dominant physical process in the metaphaseof RESET process. As the voltage becomes steady and the gapdistance x starts to increase, the electric field in the gap regionbecomes weaken and O2− hopping becomes slow. Therefore,O2− hopping turns to be the dominant physical process inthe anaphase of RESET process. As the CF evolution rateincreases with the pulse height as described by (11)–(13), (15),and (18), the switching speed with large pulse height is fast, asshown in Fig. 9. The detailed relations of the pulse height withthe pulsewidth during the SET/RESET process are shown inFig. 10. The voltage–time dilemma phenomenon is reproducedby the compact model since the evolution rate of CF showsthe approximately exponential increase with the pulse height.Therefore, we can obtain high switching speed by a properincrease in the operation voltage. It can be found that theswitching speed is fast with low HRS resistance, as shownin Fig. 10. Low HRS resistance means small gap distance x .Hence, short time is required for the CF rupture during RESETprocess and the CF formation during SET process.


Fig. 10. Relation between (a) RESET and (b) SET voltages with differentpulsewidths, which coincides with the model prediction. Three lines ofdifferent colors in (a) represent the relation between the RESET voltage andpulsewidth with different target resistance of HRS. Three lines of differentcolors in (b) represent the relation between the SET voltage and pulsewidthwith different initial resistances of HRS.

Fig. 11. I–V characteristics of the TiN/HfOx /Pt/AlOx bilayer RRAM devicesmeasured by dc double sweep together with the modeled data.

To further verify the compact model and confirm the univer-sality, the model predictions are compared with the measureddata of the TiN/HfOx /AlOx /Pt bilayer RRAM devices [28].Accompanied with the change of resistive switching material,parts of the parameters have to be tuned as follows [29]:Ea ∼ 1.0 eV, Eh ∼ 0.9 eV, and Ei ∼ 1.05 eV. Fig. 11shows the measured and calculated I–V curves under the dcsweep mode. It can be found that the SET voltage decreasescompared with TiN/TiOx /HfOx /TiOx /HfOx /Pt RRAM devicesresulting from the decrease in the active energy of Vo (Ea ∼1.2 eV). It can be also found that the HRS resistance is highwith large stop voltage during the RESET process and willcause high SET voltage in the following SET process. This isdue to the increased gap distance x with RESET voltage duringthe RESET process, which is consistent with the previousanalysis. Fig. 12 shows that to achieve a target ∼50 k� froman initial resistance ∼10 k�, two pulse schemes are used: oneis to apply 2.3 V/50 ns and the other is to apply 2 V/500 ns.It demonstrates that the switching speed is fast under highoperation voltage, as shown in Fig. 10.

Excellent agreement between the modeling and measureddata in different RRAM devices shows the validity and uni-versality of the developed compact model to describe the mainfeatures of the RRAM cell operations under both static andtransient conditions. With the equations, which describe themain features of the RRAM cell operations, a module ofRRAM electrical characteristics in HSPICE circuit simulationcan be obtained.

Fig. 12. Calculated and measured transient response current waveformsfor the two RESET programming schemes in the TiN/HfOx/AlOx/Pt bilayerRRAM devices.

IV. CONCLUSION

A physics-based compact model of metal–oxide-basedRRAM cell operation is developed. In the model, the gen-eration of Vo, drift of O2−, and recombination between O2−and Vo are considered. The developed compact model consistsof CF evolution, conduction, temperature, and parasitic effectmodel. The typical I–V characteristics in HRS/LRS can bereproduced by the model. The model is verified by measureddata in the different HfOx -based RRAM devices under dcand ac operation modes. The excellent agreement between themodel predictions and the measured data shows the promisingprospects of the future implementation of this compact modelin circuit simulation and the design optimization of RRAM.

ACKNOWLEDGMENT

The authors would like to thank P. Wong at StanfordUniversity for the valuable discussion.

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Peng Huang (S’10) received the B.S. degree fromXi’dian University, Xi’an, China, in 2010. He is cur-rently pursuing the Ph.D. degree with the Institute ofMicroelectronics, Peking University, Beijing, China.

Xiao Yan Liu received the B.S., M.S., and Ph.D.degrees in microelectronics from Peking University,Beijing, China, in 1988, 1991, and 2001, respec-tively.

She is currently a Professor with the Institute ofMicroelectronics, Peking University.

Bing Chen (S’11) received the B.S. degree fromSichuan University, Chengdu, China, in 2008. He iscurrently pursuing the Ph.D. degree with the Insti-tute of Microelectronics, Peking University, Beijing,China.

Hai Tong Li is currently pursuing the B.S. degreewith the Institute of Microelectronics, Peking Uni-versity, Beijing, China.

He has been an Undergraduate Research Assistantwith the Institute of Microelectronics, Peking Uni-versity, since 2012.

Yi Jiao Wang received the B.S. degree in micro-electronics from Xi’dian University, Xi’an, China,in 2011. She is currently pursuing the Ph.D. degreewith the Institute of Microelectronics, Peking Uni-versity, Beijing, China.

Ye Xin Deng received the B.S. degree from PekingUniversity, Beijing, China, in 2013. He is currentlypursuing the Ph.D. degree with the School of Elec-trical and Computer Engineering, Purdue University,West Lafayette, IN, USA.


Kang Liang Wei received the B.S. degree fromPeking University, Beijing, China, in 2009. He iscurrently pursuing the Ph.D. degree with the Instituteof Microelectronics, Peking University.

Lang Zeng received the B.S. and Ph.D. degreesin microelectronics from Peking University, Beijing,China, in 2007 and 2012, respectively.

He is currently a Post-Doctoral Associate with theInstitute of Microelectronics, Peking University.

Bin Gao (S’08) received the B.S. degree in physicsfrom Peking University, Beijing, China, in 2008,where he is currently pursuing the Ph.D. degree inmicroelectronics.

Gang Du received the B.S. and Ph.D. degrees inmicroelectronics from Peking University, Beijing,China, in 1998 and 2002, respectively.

His current research interests include Monte Carlosimulation method for nanoscale devices, carrierquasi-ballistic transport effect, and MOSFET com-pact model parameter extraction.

Xing Zhang received M.S. and Ph.D. degree inmicroelectronics from the Shaanxi MicroelectronicsInstitute, Shaanxi, China, in 1989 and 1993, respec-tively.

He is currently a Professor with the Institute ofMicroelectronics, Peking University, Beijing, China.

Jin Feng Kang received the Ph.D. degree in solidstate electronics from Peking University, Beijing,China, in 1995.

He is currently a Professor with the Institute ofMicroelectronics, Peking University.

Documents

A Physics-Based Compact Model of Metal-Oxide-Based RRAM DC and AC Operations