Filament Growth and Resistive Switching in Hafnium Oxide Memristive

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Functional Inorganic Materials and Devices

Filament growth and resistive switching in hafnium oxide memristive devices Sven Dirkmann, Jan Kaiser, Christian Wenger, and Thomas Mussenbrock ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b19836 • Publication Date (Web): 30 Mar 2018 Downloaded from http://pubs.acs.org on March 31, 2018

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ACS Applied Materials & Interfaces

Filament growth and resistive switching in hafnium oxide memristive devices Sven Dirkmann,∗,† Jan Kaiser,‡ Christian Wenger,¶,§ and Thomas Mussenbrock† Electrodynamics and Physical Electronics Group, Brandenburg University of Technology, 03046 Cottbus, Germany, Institute of Theoretical Electrical Engineering, Ruhr University Bochum, 44780 Bochum, Germany, Innovations for High Performance (IHP), 15236 Frankfurt (Oder), Germany, and Brandenburg Medical School Theodor Fontane, 16816 Neuruppin, Germany E-mail: [email protected]

KEYWORDS: Resistive Switching, HfO2 , Filament, RRAM, Kinetic Monte Carlo, Simulation, Memristor, Oxygen Vacancy Abstract We report on the resistive switching in TiN/Ti/HfO2 /TiN memristive devices. A resistive switching model for the device is proposed, taking into account important experimental and theoretical findings. The proposed switching model is validated using 2D and 3D kinetic Monte Carlo simulation models. The models are consistently coupled to the electric field and different current transport mechanisms as direct tunneling, trap assisted tunneling (TAT), ohmic transport, and transport through a quantum point contact (QPC) have been considered. We find that the numerical results are in excellent agreement with experimentally obtained ∗ To

whom correspondence should be addressed and Physical Electronics Group, Brandenburg University of Technology, 03046 Cottbus, Ger-

† Electrodynamics

many ‡ Institute of Theoretical Electrical Engineering, Ruhr University Bochum, 44780 Bochum, Germany ¶ Innovations for High Performance (IHP), 15236 Frankfurt (Oder), Germany § Brandenburg Medical School Theodor Fontane, 16816 Neuruppin, Germany

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data. Important device parameters, which are difficult or impossible to measure in experiments, are calculated. This includes the shape of the conductive filament, width of filament constriction, current density, and temperature distribution. To obtain insights in the operation of the device, consecutive cycles have been simulated. Furthermore, the switching kinetic for the forming and set process for different applied voltages is investigated. Finally, the influence of an annealing process on the filament growth, especially on the filament growth direction, is discussed.

Introduction With the increasing miniaturization in electronics, new scalable memory types have become subjects of current research. Memristive nanostructures are devices that change their resistance when voltage is applied to them. Specific kinds of memristive switches are based on redox reactions and ion movement. These types of memristors are called resistive random-access memory (RRAM). Non-volatile memories based on the storage of electrons need relatively high and thick energy barriers to prevent the loss of the memory state due to tunneling currents. Due to the significant higher mass of ions compared to electrons, memories based on the change of their atomic state do not require such barriers and may therefore be advantageous with regard to scalability. The resistive switching process in RRAM devices can either be unipolar or bipolar. The bipolar resistive switching mechanism needs different voltage polarities to program the high resistance state (HRS) and the low resistance state (LRS), whereas for the unipolar resistive switching mechanism a change in polarity is not required. An example of unipolar RRAMs are thermochemical memory systems (TCM). Here the switching is dominated by thermal diffusion and redox reactions. 1 Typical bipolar RRAMs are electrochemical metallication (ECM) and valence change memory (VCM) devices. The switching in ECM-devices is based on the growth and dissolution of a conductive metal filament of an active electrode through an ion-conducting layer. 2 VCM-devices typically consist of an ionic-electronic mixed conductor sandwiched between two metal electrodes. The resistive switching process relies on the movement of oxygen defects and can be either an interface 2


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type switching i.e. due to the change of a Schottky- or a tunnelbarrier 3,4 or due to the formation and dissolution of a high conductive oxygen deficient region. 5 The resistive switching in the regarded HfO2 RRAM device is known to be based on the latter switching mechanism. 6 HfO2 based RRAMs are under investigation due to their scalability (< 10 nm), simple fabrication, fast switching speeds and their compatibility with conventional complementary metal-oxidesemiconductor (CMOS) technology. 7 Hence, they are promising candidates for future applications as nonvolatile memories as well as artificial synapses in neuronal systems. 8–11 Lee et al. developed a TiN/Ti/HfO2 /TiN memristive device that showed excellent memristive behavior. 12,13 The titanium layer works as a getter material for the oxygen in the HfO2 layer. Different memristive devices containing hafnium oxide have been presented up to now. 14–16 However, major challenges for industrial use of RRAM devices still have to be addressed. These challenges include a profound physical understanding of the switching and current transport mechanisms. To simulate the switching behavior a model for the microscopic processes inside the memristive device is necessary. Different simulations on HfO2 based RRAMs have been reported, containing analytical models 17–19 or continuum models. 20,21 Furthermore, also kinetic Monte Carlo (KMC) models on resistive switching in RRAMs have been published. 22–25 Bersuker et al. and Yu et al. reported KMC models explaining filament growth in hafnium oxide. 26,27 In their models, Frenkel pairs are generated within the bulk of the HfO2 layer. Mobile oxygen ions move to interstitial positions, leaving immobile oxygen vacancies behind. This leads to a change in the local stoichiometry and to the formation of a conductive filament. Many other KMC simulations of resistive switching in memristive hafnium oxide have been published so far, using this filament growth model . 28–31 A different KMC model, similar to the model of this work, has been presented, assuming oxygen vacancies to be mobile and formed at the HfO2 /electrode interface. 32,33 However, the model is limited to the forming process up to now. Theoretical investigations showed a very high energy barrier for the formation of Frenkel defects within the bulk of HfO2 and that these defects are unstable. 34,35 Furthermore, density functional theory (DFT) calculations predict that oxygen vacancies should be mobile in hafnium oxide. 26,36 Yalon et al. 37 investigated the growth direction in memristive

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hafnium oxide devices including titanium as a getter material for oxygen. They reported that the growth direction of the filament is depending on the post fabrication thermal annealing. Without thermal annealing the filament growth direction was found to be from the cathode to the anode and with annealing from the anode to the cathode. 37 This cannot be explained by the presented simulation models for resistive switching in HfO2 memristive devices. In this paper, a different model for the resistive switching in TiN/Ti/HfO2 /TiN devices without thermal annealing is presented. Our model is compatible with theoretical findings, similar to switching models presented by other groups and able to explain experimentally obtained data. 6,32,33,38–40 Furthermore, the conditions for filament growth from the anode to the cathode are discussed. 2D and 3D models have different advantages. 2D models are faster whereas 3D models are more exact and, of course, physically correct. To validate the presented resistive switching model, a suitable 2D as well as a 3D model has been developed. The simulations are not limited to just one set and reset cycle and the results have been compared to experiments. The KMC method offers the possibility to calculate the inner atomic state of a resistive switching device on experimental length and time scales. Hence, the simulation model is able to visualize hardly measurable internal processes like ion motion and filament growth. All ionic and atomic processes have been calculated using this KMC approach. To investigate its influence on the resistive switching, the local temperature distribution in the device is calculated using the heat transfer equation. The calculated temperature is coupled to the KMC-model. The second important parameter for the ionic processes is the electric field. The electric field is calculated solving either Laplace’s equation or the current continuity equation, depending on the atomic state of the device. Finally different electronic transport mechanisms have been implemented including trap assisted tunneling, direct elastic tunneling, ohmic current and QPC transport. The main scientific questions which are addressed with this work are: i) How does resistive switching in TiN/Ti/HfO2 /TiN memristive devices without thermal annealing work? ii) What is the effective size of the conductive filament? iii) What is the influence of the temperature on resistive switching? iv) How does the device behave during consecutive set and reset cycles? v) What is the forming and set kinetic of the TiN/Ti/HfO2 /TiN

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memristive devices without thermal annealing? vi) What are the conditions for filament growth from the anode? At first the model for resistive switching and the simulation model is presented. Afterwards the simulation results are compared to experimental findings. The simulation results are in good agreement with experimental findings.

Resistive Switching Model The formation of Frenkel pairs within the bulk of the HfO2 has a very high activation energy. Due to the high oxygen affinity of titanium, the activation energy for Frenkel pair formation is significantly smaller at the Ti/HfO2 interface. 41 Furthermore, measurements that investigate the Ti/HfO2 interface before and after the forming operation, show an enhanced oxidation of the titanium at the interface and an enhanced number of oxygen vacancies within the HfO2 layer close to the interface. 42 Consequently, during the forming process generation of Frenkel pairs happens at the Ti/HfO2 interface. The oxygen moves from the HfO2 into the titanium and leaves an oxygen vacancy behind. The energy barrier of this process has been shown to be about 0.9 eV. 41 Oxygen vacancies within the HfO2 that are isolated and not captured within a filament, are favored to be in the +2 charge state 43,44 and are mobile with an activation energy for diffusion of 0.7 eV. 26,36 By applying a positive voltage to the TiN/Ti electrode, the mobile oxygen vacancies move through the HfO2 towards the opposite TiN electrode and accumulate there. Following Ref. 43, it is energetically favored for oxygen vacancies to build chains within the HfO2 . Within these chains, the oxygen vacancies prefer to be in the neutral or +1 charge state. In this charge state the activation energy for oxygen vacancy diffusion is about 3 eV. 44 Therefore, the oxygen vacancies can be treated as immobile when within the filament. Moreover, the formation of a nucleus of a conductive filament is initially a many particle process. According to Ref. 45 the ideal local stoichiometry for the formation of a filament nucleus is between HfO1.5 and HfO1.75 . For this stoichiometry the calculated activation energy for the formation of filament nuclei within the bulk of HfO2 is about

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6-8 eV. The favored charge state for isolated oxygen vacancies is +2. For oxygen vacancies within the filament the preferred charge state is neutral or +1. Therefore, a charge transfer for the formation of a filament nucleus is necessary. Due to the significantly higher electron density in the TiN compared to the HfO2 the charge transfer is favored close to the electrode. Also in 6 it is expected that the nucleation barrier is much lower at the electrodes. In our model, the nucleation process is considered to happen at the electrodes. Starting from a nucleation seed at the TiN electrode a filament can grow in the direction towards the Ti electrode. This leads to the growth of conductive filaments through the device, bridging the Ti and the TiN electrodes. The existence of a conductive bridge between the electrodes brings the device in the LRS. By switching the voltage polarity, the device can be reset to the HRS. Due to the high oxygen affinity of titanium there are nearly no oxygen interstitials within the HfO2 . Therefore, the reason for the reset process is a thinning of the conductive filament. Driven by an enhanced local temperature due to Joule heating and by the electric field, oxygen vacancies oxidize and move out of the filament towards the Ti electrode. There, the oxygen vacancies are able to recombine with oxygen of the titanium.

Simulation Approach A suitable simulation model is developed to validate the presented mechanisms responsible for resistive switching in TiN/Ti/HfO2 /TiN memristive devices. The simulation results are compared to experiments. A 2D model in addition to a 3D model is developed to take advantage of both models. Some tasks are addressed by the 2D model, especially when they are very time consuming. The other tasks are addressed by the 3D model. The simulation box is designed to visualize the active part of a real device which usually has a larger spatial dimension. In 2D, the size of the simulation box is 30 nm × 20 nm. The titanium layer as well as the HfO2 layer has a thickness of 10 nm. In 3D, the simulation box has a 20 nm × 20 nm large base, a 9 nm thick HfO2 layer, and a 10 nm thick titanium layer. Since DFT calculations showed that incorporation of oxygen into perfect

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Figure 1: Illustration of the processes included in the simulation: (I) Formation of Frenkel pairs, (II) surface diffusion, (III) diffusion of mobile oxygen vacancies within HfO2 , (IV) reduction at filament nucleus, (V) reduction at kink side, (VI) nucleation and (VII) diffusion of oxygen within titanium. structured TiN is energetically unfavorable 46 (even though the incorporation over grain boundaries cannot be completely disregarded), the TiN electrodes have been modeled as a stopping layer for oxygen and oxygen vacancies. At all other boundaries periodic boundary conditions for the motion of oxygen and oxygen vacancies have been applied. At room temperature HfO2 crystallizes in a ˚ b = 5.165 A˚ und c = 5.281 A. ˚ 47 monoclinic crystal structure with grid constants of a = 5.106 A, Furthermore, the crystal structure of the HfO2 within the compared device was supposed to be monoclinic. 48 Therefore, a cubic grid, using a grid constant of 5 A˚ has been used. The grid points represent the lattice sites in the HfO2 layer which can be occupied by oxygen vacancies/ions. Furthermore, the surface roughness of the electrodes is taken into account. 49 Figure 1 shows the included atomic processes within the simulation. At the Ti/HfO2 interface Frenkel pairs can be generated. The oxygen and the oxygen vacancies can move within the titanium and within the HfO2 , respectively. The rate for the generation of Frenkel pairs, and for the diffusion of oxygen

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within titanium and of oxygen vacancies within the HfO2 , are given by Ea − 0.5zdeE k = ν exp − . kB T

(1)

Here Ea stands for the activation energies EH f O2 , ETi and EFP which are the activation energies for diffusion of oxygen vacancies in HfO2 , the diffusion of oxygen within Ti and for the formation of Frenkel pairs, respectively. ν is the phonon frequency, z is the charge number, d is the hopping distance, E is the local electric field, T is the local temperature, and kB is the Boltzmann constant. Important parameter are collected within table Table 1. Table 1: Important simulation parameter of the KMC simulation. Physical quantity Meaning EFP Activation energy for Frenkel pair formation EH f O 2 Activation energy for diffusion of oxygen vacancies in HfO2 ETi Activation energy for diffusion of oxygen in Ti Ered Activation energy for reduction (surface/kink/hole)) Enuc Activation energy for nucleation Esur f Activation energy for surface diffusion of oxygen vacancies Eox Activation energy for oxidation Erec Activation energy for recombination ρH f O2 Mass density of HfO2 ρTi Mass density of Ti ρH f Mass density of Hf cH f O2 Heat capacity of HfO2 cTi Heat capacity of Ti cTi Heat capacity of Hf λTi Thermal conductivity of Ti λH f O2 Thermal conductivity of HfO2 λH f Thermal conductivity of Hf σFilament Electrical conductivity of the filament σTi Electrical conductivity of Ti σH f O 2 Electrical conductivity of HfO2 z Charge number of mobile oxygen vacancies ν Phonon frequency

Value 0.9 eV 41 0.7 eV 36 0.7 eV 0.67/0.66/0.66 eV 0.8 eV 0.67 eV 1.1 eV 40 0.8 eV 9680 kg/m3 50 4506 kg/m3 51 13300 kg/m3 52 120 J/(kgK) 50 524 J/(kgK) 51 144 J/(kgK) 52 21 W/(mK) 53 1.1 W/(mK) 50 23 W/(mK) 54 1 · 105 S/(m) 55,56 2 · 106 S/(m) 53 1 · 10−9 S/(m) 50 2 43,44 1012 Hz 57

When the oxygen vacancies reach the TiN electrode, surface diffusion can occur. Typically surface diffusion happens much faster than diffusion in the bulk. Therefore, the rate for surface

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diffusion is calculated using equation 1 with a reduced activation energy Esur f . Nucleation can happen at the TiN electrode. The nucleation process has been shown to be a many particle process. Therefore, the rate for the nucleation process depends on the number of neighboring oxygen vacancies N and the activation energy for nucleation Enuc . The rate for nucleation can be expressed as ENuk − (N + 0.5) zdeE . kNuc = ν exp − kB T

(2)

Due to the high conductivity of the filament, mobile oxygen vacancies can reduce at the starting filament. Furthermore, oxygen vacancies of the filament can oxidize. The rates for oxidation and reduction are deduced from the Butler-Volmer equation and are given by 6,25,58

kred

Ered − αze∆Φ = ν exp − kB T

(3)

and Eox + (1 − α)ze∆Φ kox = ν exp − kB T

(4)

α = 0.5 is the charge transfer coefficient and ∆Φ is the overpotential at the filament/electrolyte interfaces. The overpotential at the Filament/HfO2 interface is given as ∆Φ = ΦH f O2 -ΦFilament . During the reset process oxygen vacancies move towards the Ti/HfO2 interface and can recombine there with oxygen of the titanium. The rate for recombination is given by equation 1 where Ea is equal to the activation energy for recombination Erec . The activation energy for recombination is higher than the diffusion barrier of oxygen vacancies due to the oxygen affinity of titanium. A detailed description of the KMC procedure can be found in. 57,59 All ionic processes are influenced by the electric field. Due to the small electric conductivity of HfO2 , the dominant transport mechanism of electrons through the HfO2 layer in the HRS is trap

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assisted tunneling. 60–62 Therefore, the electric potential is calculated solving Laplace’s equation:

∇2 Φ(~r) = 0.

(5)

The conductivity of the titanium is assumed to be constant during the complete simulation. 63 Therefore, the electric field is assumed to vanish within the titanium and the Ti as well as the opposite TiN electrode are modeled as Dirichlet boundary conditions for the electric potential in HfO2 . When a conductive filament is growing through the HfO2 layer, it is modeled as virtual electrode and Dirichlet boundary condition are applied at the filament. At all other boundaries of the simulation box, periodic boundary conditions are applied. When the device is in the LRS, a linear relation between voltage and current has been measured. This has been interpreted either by metallic-like ohmic transport behavior or by conduction through a QPC. 64,65 Since the size of the QPC in most metals is in the order of 5 A˚ in this model the QPC model has been applied to single particle contacts. 6 Therefore, for single particle contacts the quantum conductance G0 =

2e2 h

has been applied. Furthermore, the power is not dissipated

in the QPC but in the surrounding contacts. This has been considered in the calculation of the temperature distribution. At the other regions with a larger filament size, ohmic current with a local conductivity is assumed to be the dominant transport mechanism for electrons. Within these regions the current is calculated using an ohmic current model. To calculate the potential in the ~ r) = 0 is solved. The current density J(~ ~ r) has been calculated LRS, the continuity equation ∇ · J(~ ~ r) = σ (~r) ~E (~r) depending on the local conductivity σ (~r) of by a generalized form of Ohm’s law J(~ the local stoichiometry, and on the local electric field ~E (~r). Since the dynamics of the system can be assumed to be quasi static, the electric potential can be expressed as ~E (~r) = −∇Φ(~r). Thus, the continuity equation can be rewritten with the electric potential:

∇ · (σ (~r) ∇Φ (~r)) = 0

The boundary conditions are chosen to be the same as for Laplace’s equation. 10


(6)

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Table 2: Parameter for the simulation of trap assisted tunneling in memristive TiN/Ti/HfO2 /TiN structures Physical quantity Meaning a0 Attenuation length R0 Vibrational frequency of electrons

Value 0.33 nm 29 1012 Hz 29

R0 N T,B En+,−

Coupling factor between electrodes and HfO2

1014 Hz 29

m∗e,H f O2 + En,0 − EF − En,0 − EF EC − EF

Effective mass of electrons in HfO2 Energy state of unoccupied traps Energy state of occupied traps Energy state of conduction band

0.1·me 66 0.07 eV 67 -0.07 eV 67 2 eV 29

During the forming process the current through the device is dominated by trap assisted tunneling. Important parameter concerning the trap assisted tunneling model are shown in table Table 2. In this model phonon assisted as well as elastic trap assisted tunneling has been considered. The used trap assisted tunneling model is based on Ref. 29. The current through one electrode can be written as N oT ITAT = −e ∑ (1 − fn ) RiT n − Rn f n ,

(7)

n=1

oB using the occupation probability fn ∈ [0, 1] of the nth trap and the hopping rates RoT n /Rn from the iB nth trap to the top/bottom electrode and the hopping rates RiT n /Rn from the top/bottom electrode

to the nth trap. Trap assisted tunneling happens here over all mobile oxygen vacancies. The occupation probability of the nth trap can be calculated solving the continuity equation for the current: N N d fn oB iB oT fn . = (1 − fn ) ∑ Rmn fm − fn ∑ Rnm (1 − fm ) + RiT n + Rn (1 − f n ) − Rn + Rn dt m=1 m=1

(8)

Here, Rmn is the hopping rate of electrons from the mth trap to the nth trap. The hopping rate can

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be calculated by the Mott hopping model,

Rnm =

    0    

when m = n,

dnm when V m −Vn < 0, R0 exp − a0       R0 exp − dnm + e(φn −φm ) when V m −Vn > 0, a0 kB T

(9)

including the distance dnm between two traps n and m and the attenuation length a0 of the wave function of the electrons. φn and φm are the potentials at the positions of the traps n and m and R0 is the vibrational frequency of the electrons. The hopping rates between the electrodes and the nth trap are given by RiT,B = R0 N T,B En+ FinT,B En+ TnT,B+ n

(10)

T,B − T,B− T,B − RoT,B = R N E , 0 n n Fout En Tn

(11)

and

with R0

N T,B

+,− +,− T,B En being the coupling factor between electrodes and HfO2 , consisting of N En ,

as the number of states within the electrodes at a given energy and R0 is a fitting parameter to relate +,− T,B the simulated TAT current to experimental findings. Here, R0 N En was adopted from Ref. 29. The energies En+ and En− of the filled and empty traps are given by +,− En+,− = En,0 − eφn .

(12)

FinT,B (En+ ) is the fermi integral representing the number of occupied states within the electrodes T,B above En+ . Those electrons are able to tunnel into the nth trap. Fout (En− ) is the fermi integral,

representing the empty states within the electrodes below En− which can get filled by the electrons from the nth trap. The tunnel probability TnT,B+,− from the electrodes to the nth trap and from the

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nth trap to the electrodes is given by the WKB approximation, " TnT,B+,− = exp −2

Z b r 1 a

h¯

# h i +,− 2m∗e,H f O2 EC − En,0 − eφ (x) + eφn dx ,

(13)

with a and b being the initial and final position of the tunneling process, m∗e,H f O2 is the effective mass of electrons within HfO2 and φ (x) and φn are the electrical potentials at the position x and at the position of the nth trap. Besides tunneling over trap states also direct elastic tunneling from electrode to electrode and from electrode to filament have been considered. In order to calculate the elastic tunneling current, Simmons equation has been applied: 68 e J≈ 2πhl 2

" ! !# r r eV eV eV eV exp −A φ − exp −A φ + φ− − φ+ 2 2 2 2

With the tunnelling length l and A is given as: 4πl

(14)

q

2m∗e,H f O /h. The barrier height φ of about 2

2.1 eV is obtained by calculating the band offset at the Ti/HfO2 interface. V is the voltage drop over the tunneling barrier. When the filament bridges the top and bottom electrode, the current is dominated either by QPC transport or by diffusive ohmic transport. This depends on the local thickness of the filament. The current density is calculated by solving

~ r) = σ (~r) ~E (~r) , J(~

(15)

and the current is given by a surface integral Z

I=

σ (~r) ~E (~r)~nda

(16)

sur f ace

At the instant of time when the current overcomes the compliance current, the voltage source is substituted by an ideal current source that provides the constant compliance current. The applied voltage to the device can then be calculated from the compliance current and the actual device resistance. Since all ionic processes depend on the temperature, the temperature is a crucial pa-

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rameter for the simulation. The local temperature has been calculated solving the heat transfer equation,

c p (~r)ρ(~r)

∂ T (~r,t) j2 (~r,t) − ∇ · [λ (~r)∇T (~r,t)] = , ∂t σ (~r)

(17)

where c p (~r) is the local heat capacity, ρ(~r) is the mass density, and λ (~r) the thermal conductivity. Those parameters depend on the local materials and stoichiometry. Finally, to account for the temperature dependent processes, the calculated temperature distribution is coupled to the calculation of ionic process rates.

Results and discussion At first, the simulation results for the device without post-fabrication annealing are compared to experimental results, reported by Traoré et al. 69 The fabrication process of the device is presented in 48 . One forming, reset and set operation is examined in detail using the 3D model. Then 5 consecutive cycles and the switching kinetic are investigated. Because annealed devices show better switching characteristics an annealed TiN/Ti/HfO2 /TiN structure has been fabricated. For the switching mechanisms especially the forming process is important and the involved mechanisms are not completely understood. Hence, the 3D simulation model is applied for the structure. The simulation results are compared to the measurements. Finally, the switching mechanisms necessary for a filament growth from the Ti layer (as proposed by Yalon et al. 37 ) are examined.

Results for the non annealed device To characterize the device, a voltage ramp is applied to the simulation box. The source voltage and applied voltage to the device are shown in Figure 2b). Here, 7 important points in time are marked. In order to address the first question: “How does resistive switching in TiN/Ti/HfO2 /TiN memristive devices without thermal annealing work?”, the forming process as well as the reset and

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Figure 2: a): Calculated (red) and measured (black) 69 IV-characteristic of the TiN/Ti/HfO2 /TiN memristive device. b): Source voltage (red) and applied voltage to the device (blue). 7 important c points in time are marked by the labels (1)-(7). 2013 IEEE. Reprinted, with permission, from Ref. 69. the set process have been simulated. In Figure 2a) the calculated (red) and measured (black) 69 IV-characteristic is shown. The calculated and measured IV-characteristic are in very good agreement. During the forming process it can be seen that the measured IV-characteristic is much more smooth than the calculated one. The reason for this is the small area of the simulation box compared to the experimental setup. For larger areas of the simulation box an averaging effect due to an increased number of ions within the simulation box leads to a smoothing effect of the IVcharacteristic. However, also during the forming process the trend of the calculated and measured IV-characteristic is in very good agreement. To simulate the forming process a voltage ramp of 1 V/s has been applied. Figure 3 shows the positions of the mobile oxygen vacancies within the HfO2 (red), the positions of the oxygen within the Ti layer (blue) and the positions of the oxygen vacancies within the filament (white) for 4 chosen points in time (1)-(4) of the forming process. Initially, 13 oxygen vacancies have been randomly distributed within the HfO2 layer, which corresponds to an oxygen vacancy density of 3.6 · 1018 cm−3 (time point (1)). The initial number of oxygen vacancies has been chosen so that the calculated current fits the measured current. Furthermore, the inhomogeneity of the electrode is depicted. In Figure 4a) the electric field at the beginning of the forming process for an applied voltage of 0.1 V is shown. As expected, the electric field is significantly enhanced at the positions of the inhomogeneous electrode. This indi-

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Figure 3: Positions of charged and mobile oxygen vacancies (red) within the HfO2 layer, of oxygen within Ti layer (blue) and of immobile oxygen vacancies within the filament (white) for the four different points in time (1)-(4). The white spheres that can be seen in time point 1 are not an artificially placed nucleus of a conductive filament but represent the inhomogeneity of the TiN electrode.

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cates the inhomogeneity of the electrodes to have a significant influence on the filament growth. When the voltage at the device increases, Frenkel pairs are generated at the Ti/HfO2 interface (time point (2)). As a result, the density of oxygen vacancies within the HfO2 increases. The effective distances between the charged oxygen vacancies decrease. This makes tunneling processes more likely. Thus, the current through the device increases. In Figure 4b), the electric field for an applied voltage of 1.37 V is shown. The inhomogeneity decreases the effective gap between both electrodes. The voltage drops over a smaller distance which results in an increase of the electric field. The high electric field enhances ionic processes especially at the tip of the filament. Therefore, at this position the filament growth is accelerated and the filament growth of possible other smaller filaments is suppressed. Time point (3) presents the device right before the forming pro-

Figure 4: a): Distribution of the electric field (V/m) at time point (1) for 0.1 V applied voltage, b): Electric field at time point (2) for 1.37 V applied to the device. cess. Due to the enhanced electric field close to the filament tip, a large number of Frenkel pairs are formed here and the titanium layer oxidizes at this position. The density of oxygen vacancies is mainly dominated by the relation between the activation energy for formation of Frenkel pairs and the activation energy for oxygen vacancy diffusion in HfO2 . Furthermore, these two activation energies, which are extracted from literature, dominate the growth kinetic of the conductive filament. The behavior of the calculated forming current as well as the calculated time point of

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the breakthrough fits well with experimental data. This indicates the correct order of magnitude of the activation energies for Frenkel pair formation and oxygen vacancy diffusion as well as the correctness of the presented forming model from cathode to anode in memristive TiN/Ti/HfO2 /TiN devices. When the conductive filament connects the TiN and the Ti layer (time point (4)), current is flowing through the filament leading to an abrupt increase in the current. In order to reach the measured current compliance, more than one filament has to form through the HfO2 layer. Finally, an effective contact area of about 20 nm × 10 nm has been calculated. Two critical parameters, which are hardly measurable in experiments, are the current density and the temperature distribution within the device during resistive switching. In order to answer the third question, Figure 5 shows the current density (right hand side) and the corresponding temperature distribution (left hand side) for the four different points in time (4)-(7). Right after breakthrough the filament is thin and therefore the applied voltage to the device is relatively high. This leads to a large current density and to a high temperature around 861 K (time point (4)). Due to the high temperature, all ionic processes are extremely accelerated and the filament thickens rapidly. Consequently, the device resistance drops and the temperature drops to around 481 K (time point (5)). The calculated temperature is comparable with calculated temperatures of other groups. 31 When the voltage polarity is switched, the reset process starts and a voltage ramp of -1 V/s is applied to the device. The probability for oxidation of immobile oxygen vacancies increases. Furthermore, when oxygen vacancies and oxygen of the titanium layer meet at the Ti/HfO2 interface, the oxygen vacancies can recombine with oxygen with increasing probability. As a result, the filament thins out and ruptures finally. Time point (6) of Figure 5 shows the current density and temperature distribution right before rupture of the conductive filament. The calculated temperature is about 543 K. This indicates a significant influence of the temperature on the reset process. Figure 6 shows the conductive filament (white) and the distribution of the oxygen within the titanium (blue) right before (a) and after (b) the rupture of the conductive filament. In the insets of Figure 6 the position of the rupture of the filament is shown. The position where the filament breaks is not random but depends strongly on the local temperature distribution as well as on the

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Figure 5: Left hand side: Calculated temperature distribution (K) and right hand side: Calculated current density distribution (A/m2 ) at the four different points in time (4)-(7).

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Figure 6: Shape of the conductive filament right before (a) and after (b) rupture of the filament at time point 6. The inset shows the exact position of filament rupture. local electric field. Both parameters are large at thinnest position of the conductive filament. Due to the growth mechanism, the thinnest part of the conductive filament is close to the titanium layer. Therefore, typically the conductive filament ruptures close to the titanium layer. After the rupture, elastic tunneling is the dominant electron transport mechanism. After 3.7 s the set process starts. The voltage is initially set to 0 V and afterwards a voltage ramp of 1 V/s is applied since 4.7 s are reached. Also at the beginning of the set process the elastic tunneling transport mechanism dominates the current transport. Due to the electric field new Frenkel pairs are generated at the Ti/HfO2 interface closing the fracture point of the filament. The corresponding current density and temperature distribution right after the set process is presented in time point (7) of Figure 5. To answer the third central question: “How does the device behave during consecutive set and reset cycles?”, the behavior of the device in operation is investigated. Five consecutive switching cycles of set and reset operation after the forming process have been simulated. Here, the 2D model has been used, due to the fast simulation time. Figure 7a) shows the IV-characteristic and Figure 7b) shows the evolution of the calculated device resistance vs. time. During the reset process, more than one jump in the current can be seen in the simulation during the reset process as well as in the measurements (Figure 2). The reason for the first abrupt jump is, as explained 20


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Figure 7: a): Calculated IV-characteristic for five consecutive set and reset cycles, b): Evolution of the calculated resistance over time during the consecutive cycles. above, the rupture of the conductive filament. After the rupture of the filament the electric field in the tunneling gap between filament and electrode is enhanced. More oxygen vacancies oxidize, leading to an increase of tunneling gap. Due to the discrete mesh the tunneling gap can only increase in multiples of the distance between the lattice sites. The stepwise enlargement of the tunneling gap leads to a stepwise decrease in the current. During the consecutive set and reset cycles the stochastic nature of the device becomes obvious. As also seen in experiments, the reset time as well as the set time varies from cycle to cycle. 70 Besides this, especially the effective resistance of the HRS lowers slightly from cycle to cycle. This can be explained by a thickening effect of the conductive filament. In Figure 8 the filament after the first (a) and after the fifth (b)

Figure 8: Shape of the conductive filament (black) and of the oxygen within the Ti layer (blue) after the first (a) and fifth (b) set process. set process is shown. During the set process Frenkel pairs are generated not only at the thinnest 21



gap between the filament and the electrodes but also around it. This leads to an effective increase of the filament width. The LRS is dominated mainly by the thickness of the filament constriction which stays nearly constant during the consecutive cycles. However, since the width of the filament body increases from cycle to cycle, the tunneling current increases due to an enhanced effective tunneling area. The observed resistance reduction agrees with measurements of the endurance behavior of TiN/Ti/HfO2 /TiN RRAMs. 12 In the application of non-volatile memory, the switching between HRS and LRS usually happens with constant voltage pulses applied to the device. Depending on the value of the constant voltage, the switching time differs. In this section the switching behavior with a constant applied voltage is examined. The question: “What is the forming and set kinetic of the TiN/Ti/HfO2 /TiN memristive devices without thermal annealing?” is answered. Both the forming and the set kinetics are investigated from the 2D model. To account for the stochastic nature of the filament forming 100 cycles for each voltage value are simulated and the time is averaged. Figure 9 shows

Figure 9: Calculated forming and set kinetic for different constant applied voltages. the calculated forming kinetic as well as the calculated set kinetic. The expected trend is observed. With higher constant voltages applied the switching time declines. The applied voltage defines the electric field between the gap in the insulating layer. The higher the electric field the higher is the probability of Frenkel pairs to be generated at the interface. For the FORMING the gap between 22


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both electrodes is large so that the electric field is lower than for the SET operation. For this reason the forming time is always higher than the set time at the same applied voltage However, since all atomic processes depend exponentially on the electric field, the switching kinetic is nearly a straight line.

Results for the annealed device In order to answer the last question, the growth direction of the device and the influence of the fabrication process on the growth direction of the conductive filament is discussed. Yalon et al. investigated HfO2 RRAM devices with a Ti electrode with and without post fabrication thermal annealing. 37 They detected a filament growth from cathode to anode if the device was not thermally annealed. A filament growth from anode to cathode was observed for an annealed device. A model for the filament growth from the cathode to the anode has been presented in the previous section. However, the conditions for filament growth from anode to cathode still have to be investigated. During thermal annealing, oxygen vacancies are formed at the Ti/HfO2 interface and are able to diffuse due to the high temperature within the HfO2 layer. This leads to a vacancy gradient with a high oxygen vacancy density close to the Ti layer. Therefore, a filament is pre-formed starting from the Ti layer towards the opposite TiN electrode. As a consequence, a reduced effective gap exists between the pre-formed filament and the TiN electrode. If the gap is small enough electrons can tunnel through this gap. A tunneling gap of about 2 nm has been calculated using the 3D model. The tunneling current is presented in Figure 10 and compared to the measured current during the forming of a thermally annealed TiN/Ti/HfO2 /TiN 1T-1R memory device. The 1T1R memory devices are constituted by a select nMOS transistor (1T) manufactured in a CMOS technology (width of 1.14 µm and length of 0.24 µm), which also sets the current compliance, and the drain of the NMOS is connected to the MIM stack (1R). The area of the MIM cell is 0.4 µm2 . The MIM stack was integrated on the metal line 2 of the CMOS process. The TiN based bottom metal electrode of the resistive MIM cell was prepared by physical vapor deposition (PVD). The HfO2 layer with thickness of 9 nm was deposited at 320 ◦ C by the reaction of O2 and 23



Figure 10: Calculated (red) and measured (black) IV-characteristic during the forming process of thermally annealed TiN/Ti/HfO2 /TiN memristive devices. tetrakis(ethylmethylamido)-hafnium [Hf(NMeEt)4] in an atomic vapor deposition (AVD) chamber. The HfO2 layer was capped by 7-nm PVD Ti and 150-nm PVD TiN layers. Finally, the 1T-1R memory cells are annealed at 400 ◦ C for 30 min to activate the scavenging properties of the Ti layer. Details of the fabricated 1T-1R memory device are shown in Figure 11. The simulated current is in excellent agreement with the measured current. During the forming operation additional defects have to be existing for the dielectric breakdown of the HfO2 layer. Since the generation of Frenkel pairs in the bulk is not favorable Frenkel pairs have to generate at the interface of the preformed filament and the HfO2 layer. Then, the newly generated oxygen vacancies can either diffuse through the HfO2 layer or directly reduce at the filament. Thus, important parameter for filament growth dynamic are the reduction barrier and the diffusion barrier of oxygen vacancies. Since the diffusion barrier for oxygen vacancies in HfO2 is known (0.7 eV, Ref. 36), the filament growth is investigated for different reduction rates for the oxygen vacancies at the filament. Figure 12a) shows the test structure of the simulation. The blue framed circles show the pre-formed filament, the black circles are neutral vacancies. Variable D represents the distance between pre-formed filament and opposite TiN electrode. Variable d is the distance between TiN electrode and of the position of the last reduction process before the conductive filament connects both electrodes. The calculated normalized distance d/D vs. the activation energy for reduction is depicted in Figure 12b) for an activation energy for Frenkel pair formation of 0.9 eV and for different distances

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Figure 11: a) Schematic diagram, b) cross-sectional Transmission Electron Microscopy (TEM) image of the 1T-1R cell, c) MIM stack insight, and d) high magnification TEM image of the memristive MIM device.

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Figure 12: a): Schematic of the simulation procedure for investigating the conditions for filament growth from anode to cathode. D is the distance between pre-formed filament and TiN electrod and d is the distance between TiN electrode and last reduction position before set process. b): Normalized distance d/D vs. the reduction barrier at the pre-formed filament for different distances D between 4 nm and 10 nm. D between 4 nm and 10 nm. The simulation results show that the last reduction position d is always close to the pre-formed filament position. This shows that even for extremely small reduction barriers the filament mainly grows from the cathode side. This can be explained by the electric field working against the reduction process at the pre-formed filament and favoring the diffusion of oxygen vacancies. A process not considered here but mentioned by Traoré

48,63

is that Frenkel pairs consisting of

neutral oxygen vacancies could be generated due to charge injection from the cathode. This process could be more likely if the distance between the Frenkel pair generation and the TiN electrode is very small. These neutral oxygen vacancies are not accelerated by the electric field and have a high diffusion barrier which makes them immobile. 36 If this type of Frenkel pair generation is more favorable than other possible processes, a filament growth mode from anode to cathode could be explained and is supported by the simulations made within this work.

Conclusion We have presented a model explaining the resistive switching in TiN/Ti/HfO2 /TiN memristive devices without thermal annealing. This model is able to explain experimental findings showing a

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filament growing from the cathode side. In order to profit from the exactness of 3D models and from the high simulation speeds that are provided by 2D models, both, a 2D and a 3D simulation model have been developed in this work. Ionic processes have been simulated using the kinetic Monte Carlo method, coupled to a field solver based on Laplace’s equation and on the current continuity equation. Considered current transport mechanisms are elastic tunneling, trap assisted tunneling, ohmic electron transport, and transport through a QPC. The temperature has been calculated solving the heat flow equation. The calculated temperature is coupled to the ionic processes. To validate the proposed resistive switching model, simulation results have been compared to experimental results. Simulation and experimental results are in very good agreement, supporting the presented resistive switching model. Furthermore, parameters that are very difficult to measure in experiments as the shape and the size of the filament, the electric field distribution, the current density and the temperature distribution have been calculated. As a result, a maximum temperature of about 860 K, a current density of about 5·1013 A/m2 and a size at the filament constriction of about 200 nm2 have been obtained. A significant influence of the temperature on the resistive switching could be found. Since in the application of non-volatile memories the switching behavior usually happens with constant voltage pulses, the forming as well as the set kinetic have been calculated for constant voltages. An exponential trend of the set and forming kinetic on the applied voltage has been found. The set kinetic is much faster than the forming kinetic. In addition, the simulation model is not limited to just one set and reset cycle but consecutive cycles have been simulated. Experimental findings of a decrease of the resistance of the HRS in operation could be explained by a thickening effect of the conductive filament and subsequently by an increase of the tunneling area. Finally, TiN/Ti/HfO2 /TiN devices with annealing have been investigated. The simulated and measured forming currents have been compared. The current could be explained by a pre-formed filament due to the thermal annealing, resulting in an effective tunneling gap of about 2 nm. Furthermore, since filament growth in annealed TiN/Ti/HfO2 /TiN devices has been experimentally observed to happen from anode to cathode, the conditions for this filament growth mode has been investigated. It has been found, that filament growth, which is dominated by charged

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oxygen vacancies, nearly always happens from cathode to anode due to the high local electric field. However, oxygen vacancies might be neutral after generation due to charge injection from the cathode. Thus, they are not affected by the applied electric field and stay at the anode. In this case a filament growth mode from the anode side could be explained. We showed that our model is a powerful simulation tool that has been used to gain deeper understanding of the relevant physical and chemical processes for resistive switching on both, atomistic length scales and experimental time scales.

Acknowledgement The authors gratefully acknowledge financial support by the German Research Foundation (DFG) in the frame of research group FOR2093.

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Filament Growth and Resistive Switching in Hafnium Oxide Memristive

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