Modeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors
MM ODELING E LECTRICAL R ESISTANCE D RIFT WITH U LTRAFAST S ATURATION OF
OTS S
ELECTORS
A P
REPRINT
Yi˘git Demira˘g
Department of Electrical and Electronics EngineeringÉcole Polytechnique Fédérale de Lausanne (EPFL)Lausanne, Switzerland [email protected]
Ekmel Özbay
Department of Electrical and Electronics EngineeringDepartment of PhysicsBilkent UniversityAnkara, Turkey [email protected]
Yusuf Leblebici
Department of Electrical and Electronics EngineeringÉcole Polytechnique Fédérale de Lausanne (EPFL)Lausanne, Switzerland [email protected]
February 16, 2021 A BSTRACT
Crossbar array architecture is an essential design element for densely connected Non-Volatile Memory(NVM) applications. To overcome intrinsic sneak current problem of crossbar arrays, each memoryunit is serially attached to a selector unit with highly nonlinear current-voltage (I-V) characteris-tics. Recently, Ovonic Threshold Switching (OTS) materials are preferred as selectors due to theirfabrication compatibility with PRAM, MRAM or ReRAM technologies; however, OTS selectorssuffer from the temporal drift of its threshold voltage. First, based on Poole-Frenkel conduction, wepresent time and temperature dependent model that predicts temporally evolving I-V characteristics,including threshold voltage of OTS selectors. Second, we report an ultrafast saturation (at ∼ seconds) of the drift and extend the model to predict the time of drift saturation. Our model showsexcellent agreement with OTS devices fabricated with 8 nm technology node at 25 °C and 85 °Cambient temperatures. The proposed model plays a significant role in understanding OTS deviceinternals and the development of reliable threshold voltage jump table. K eywords OTS selectors · Drift problem · Device modeling · Neuromorphic hardware
An OTS material is a thin-film two terminal amorphous chalcogenide, whom electrical conductivity can rapidly changefrom the high resistive state (HRS) to low resistive state (LRS) by applying large potential exceeding a specific thresholdvoltage ( V th ). The conductivity difference between HRS and LRS can be as high as 4 orders of magnitude; nevertheless,the device immediately switches back when the applied potential is cut [1]. Its high ON/OFF current ratio and fastswitching make OTS a promising candidate material for selector applications.OTS is a chalcogenide material; hence it can crystallize. However, once crystallization starts, it is not feasible withindevice operation range to initiate melting and recover amorphous state. Therefore, crystallized OTS selector units arealways assumed non-operational. As a solution, OTS materials can be carefully selected to have lower ionicity andhigher hybridization, which lead to more directed covalent bonds to significantly slow crystallization process [2]. a r X i v : . [ phy s i c s . a pp - ph ] F e b odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT
OTS selectors also perfectly match to physical and electrical scaling properties of NVM technologies. Owing to thinfilm compatibility with mature metallization techniques and CMOS stack support, densely connected 3D crossbararrays with OTS selectors have been demonstrated [3].
The major problem of OTS technology is that the electrical conductivity of the selector decreases over time (Fig. 1(a)),called the drift problem. We observe that this conductivity decrease is not consistent, but saturates in time. Whether theconductance is drifting or already saturated, application of any potential higher than the threshold voltage (
V > V th ),resets the drift and revert to selector’s initial HRS level. Saturation of the drift is a rarely reported physical phenomenonin the literature, but critically important for understanding and developing OTS technology. In our OTS devices, weobserve an ultrafast drift saturation (Fig. 1(b)) which takes places at least 2-3 orders of magnitude faster than reporteddrift measurements [4, 5].The main problem is due to the increase of V th as conductance drifts. READ and WRITE operations require a known V th level of OTS. If V th increases and applied READ/WRITE pulses could not pass threshold value, then the selectordevice stays in HRS, hence READ/WRITE operations fail. One practical solution is to determine new V th withapplying various prior READ voltages and detecting the threshold voltage value, which certain current level is reached[6]. However, this solution requires additional support circuitry and increases the power consumption of the device;therefore, it is unfavorable. On the other hand, physical modeling of time-dependent resistance drift may lead to efficientsolutions.Figure 1: (a) Experimental (dot) and simulated (line) resistivity measurements of the same OTS device with 800 nm contact area at different ambient temperatures. The READ voltage values are 2.48 V and 2.64 V respectively for 25 °Cand 85 °C experiments. The resistivity difference is due to thermally activated Poole-Frenkel behavior. (b) Resistancedrift saturation measurements. The saturation of drift is faster at 85 °C due to faster annihilation of the defects.Modeling the drift behavior of OTS selectors is of vital importance for two reasons. First, a validated model canbe useful for developing reliable time and temperature dependent jump-table of V th . Second, a physically groundedmodel can provide a comprehensive understanding of temporally evolving non-measurable material properties such asactivation energy ( E a ) and inter-trap distance (∆ z ) to investigate the fabricated material in more detail. Although thereexist models capturing the drift behavior, these methods are either validated only for phase change memory (PCM)on a short range of time and ambient temperature or unable to predict the saturation of drift due to employing simplepower-law-like models [7, 8] . 2odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT
T = 85 °CT = 25 °C E A ( e V ) Δ z ( n m ) T = 85 °CT = 25 °C
Figure 2: Temporally evolving activation energy ( E a ) and inter-trap distance (∆ z ) calculated by the drift model. Upperand lower plots are generated for OTS selectors whom time and temperature dependent resistivity measurements areshown at Fig. 1 The physical phenomena governing the resistance drift on amorphous chalcogenide materials is yet to be fullyunderstood. Raty et al. [9], Gabardi et al. [10] and Zipoli et al. [11] have recently provided a significant insight into themicroscopic picture of the the drift mechanism. Using ab-initio simulations, it was found that there exist energeticallyunstable homopolar bonds and defects in melt-quenched amorphous. As these unstable defects naturally transforminto lower-energy structures with time, the distance between intrinsic traps increase. The structure evolves into a morecrystalline-like state, however without the necessary long-range order (Fig. 3).
Unstable configurationsafter V>V th SR(t) E A Δz PF-model(E A , Δz, T, E) RV th P e i e r l s d i s t o r ti on L ong r a ng e R e m ov a l o f ho m opo l a r bond s S ho r t r a ng e (Why SR seems like crystallization)(Why σ seems like amorphization)• Ge - Ge• Ge T • Ge H Figure 3: From left to right: Homopolar bonds in amorphous OTS are energetically unstable hence naturally disappearwith time. This mechanism is called structural relaxation (SR) and affects activation energy ( E a ) and inter-trap distance( ∆ z ). Change of mainly these two causes the drift of device resistivity and threshold voltage as the electrical transportmodel of OTS materials follow the Poole-Frenkel conduction.To model the kinetics of the structural relaxation, we started with a drift model developed by Le Gallo et al., whichhas been previously validated on GeTe and GST [5]. In this model, the bond network state of chalcogenide is denotedwith an order parameter Σ . Σ is a normalized parameter between 0 (low-ordered highly stressed state) and 1 (ideal,energetically favorable relaxed state). Whenever V > V th is applied to OTS selector, amorphous network state resetsand stressed with initial distance Σ(0) = Σ from the equilibrium. As network collectively relaxed through more3odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT favorable states with time, the energy barrier required to overcome, E b , monotonically increases and it is assumed to belinearly dependent to Σ : E b ( t ) = E s (1 − Σ( t )) , (1)where E s is the final energy barrier to reach the most relaxed state at Σ = 0 . With an Arrhenius type temperature-dependence, this relaxation occurs at the rate of r ( t ) = v exp ( − E b ( t ) /k B T ) , where v is an attempt-to-relaxfrequency, k B is the Boltzmann constant. After that the evolution of E b ( t ) can be calculated by: d Σ( t ) dt = − v ∆ Σ exp (cid:18) − E b k B T ( t ) (cid:19) . (2)At a constant temperature, Eq. 2 can be solved analytically to track the progress of structural relaxation, such that Σ( t ) = − k B TE s log (cid:18) t + τ τ (cid:19) , (3)where τ = ( k B T /ν ∆ Σ E s ) exp ( E s /k B T ) and τ = τ exp ( − Σ E s /k B T ) . Once Σ is calculated by Eq. 3, anempirical linear relationship between structural relaxation, activation energy and inter-trap distance can be written as: E a ( t ) = E ∗ − α Σ( t ) , ∆ z ( t ) = s / Σ( t ) , (4)where E ∗ is the activation energy at the equilibrium, α and s are the constants linking change in Σ to material properties.Finally, the temperature dependence of activation energy is assumed to follow the Varshni effect (cid:0) E a = E a − ξT (cid:1) ,as the optical bandgap of the material depends on the temperature [12]. Amorphous chalcogenide materials are known to follow Poole-Frenkel subthreshold conductivity behavior [13]. ThePoole-Frenkel effect suggests that thermal excitation and strong electric field release trapped carriers from ionizabledefect centers, which are believed to create Coulomb potential. In this work, we used a previously developed field andtemperature dependent 3D Poole-Frenkel emission model with field independent mobility [14]. We first calculatedthe potential profile between defect centers located at r = 0 and r = ∆ z in all directions using Eq. 5, where β is thePoole-Frenkel constant, e the electronic charge, θ the direction of escape relative to applied E-field F . V ( r, θ, F ) = − eF r cos( θ ) − β e (cid:18) r + 1∆ z − r (cid:19) + β e ∆ z (5)The potential profile between Coulombic defect centers separated by ∆ z is shown for OTS material in Fig. 4. Theenergy barrier lowering due to the Poole-Frenkel effect then can be calculated by E PF ( F, θ ) = − max r V ( r, θ, F ) . (6)Finally, assuming Boltzmann statistics, we calculated the subthreshold electrical conductivity of the selector with: σ = eµ K π (cid:90) π exp (cid:18) − E a − E P F ( F, θ ) k B T (cid:19) π sin( θ ) dθ. (7)4odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT θ = 0θ = π P o t e n ti a l ( e V ) r (nm)0 1 2-1-21.000.750.500.250.00-0.25-0.50-0.75-1.00 0 Figure 4: The potential profile between OTS defect centers are calculated via Eq. 5. The amount of energy barrier mayincrease (left) or decrease (right) significantly due to the angle ( θ ) between strong electric field × V/m and thedirection of escape.
In our experiments, we observed an unusually ultrafast saturation of the electrical resistance drift. Moreover, the driftsaturation point changes as a function of the ambient temperature.In the strong form of the drift model proposed by Le Gallo et al. [15], the evolution of subthreshold electrical resistancecan be predicted; however, it falls short predicting a drift saturation point. To extend the previous model to predict thesaturation time, we hypothesize that identical devices at different ambient temperatures that saturate at different times,eventually converge to the same E a and ∆ z at the time of the saturation. This hypothesis requires Σ( t ) to be the sameand constant for identical devices at different temperatures after the saturation time, t SAT : d Σ( t ) dt = − r ( t )∆ Σ = 0 for t > t SAT . (8)During the training of the model, structural relaxation parameters are tuned according to this constraint given at Eq. 8. t SAT values used in the training are experimentally gathered as the saturation times of V th (see Fig. 1 (b)). Figure 5shows the evolution of Σ( t ) for a trained model. T = 85 °CT = 25 °C S R C on s t a n t Time (AU) t
SAT = 4950 st
SAT = 11000 s0.30.40.50.60.70.80.91.0
Figure 5: Calculated structural relaxation constant Σ given for two ambient temperatures 25 °C and 85 °C.5odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT
As the electrical resistance of the selector drifts with time the threshold voltage also drifts. To predict the time evolutionof V th , a mere time and temperature dependent subthreshold electrical model would not suffice. The model requires anextension to explain the moment of threshold switching for OTS.To account the sudden increase of conductivity during threshold switching, we combined Poole-Frenkel subthresholdtransportation with Okuto-Crowell impact ionization. Okuto-Crowell impact ionization is an empirical model which isbased on electron-avalanche multiplication effect due to the high electric field ( ∼ × V/m) inside the OTS material[16]. With this extension illustrated in Fig. 6 (a), we demonstrated a successful prediction of time-evolution of V th ofOTS selectors at different ambient temperatures (Fig. 6 (b)).Figure 6: (a) Illustration of Poole-Frenkel (blue) conduction model and Okuto-Crowell impact ionization model (red).Together two models are sufficient for describing threshold switching of OTS. (b) Prediction of V th for T=25 °C, 85 °Cand t = 300 s. To adjust model parameters according to experimental measurements, the model I/O is matched with experimentalconditions. The implemented model takes the same control inputs with the fabricated device (voltage, ambienttemperature) and returns the same measurable quantity (resistivity). Figure 7 shows that fabricated OTS device can bemodeled as a black box whom physical characteristics are represented by a set of parameters θ . VoltageAmbient Temperature R(t)Ovonic Threshold Switching Device
Figure 7: For OTS device, the only control parameters are ambient temperature and applied voltage.Figure 8 shows the proposed drift saturation model with 17 parameters. To successfully optimize these model parametersto match the fabricated device, we consider two requirements. First, in an ideal situation, the proposed model andthe fabricated device must output the same resistivity level when applied the same voltage and ambient temperature.Therefore, the aim is to minimize the difference between measured resistivity of the fabricated device, R ( θ ) , and themodeled resistivity, R (ˆ θ ) , by tuning model parameters, θ .Second, the model parameters with physical correspondence must stay within their physically-realistic ranges. To limitevery parameter with different upper (UB) and lower bounds (LB) as in Eq. 9, several optimization methods could beused e.g., simulated annealing, evolutionary or gradient-based search algorithms. We utilized simulated annealing forits easy implementation and despite its computationally-heavy search, it successfully minimized the loss function insideof Eq. 9 with physically-realistic parameters [17]. x = arg min θ (cid:88) (cid:16) R ( t ) − ˆ R θ ( t ) (cid:17) , subject to LB i < θ i < U B i . (9)6odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT
Subthreshold Conduction (Poole-Frenkel)Impact Ionization (Okuto-Crowell)Structural Relaxation Δz(t)E A (t) R θ (t) R θ (t)Parameter ParameterParameterValue ValueValue0.91×10 s-12.3 eV0.415 eV0.275 eV1.39×10 -9 m0.5×10 -6 eV K -2 m -3 -4 cm -2 V -1 s -1 -1 Vm -1 -4 K -1 -4 K -1 -6 sVoltageAmbientTemperature Figure 8: Model parameters used in this work. The temporal nature of electrical conductance is due to structuralrelaxation model, which affects ∆ z and E A of OTS. Poole-Frenkel model and impact ionization combined, define theelectrical behavior of the OTS and enable tracking of time-dependent V th . We reported an ultrafast saturation phenomenon (at ∼ seconds) of resistance drift on OTS materials, which arepromising selector candidates in the next generation NVM (PRAM, MRAM and ReRAM) crossbar technologies. Anelectrical transport model is proposed to describe time and temperature dependent OTS I-V characteristics. The modelbased on structural relaxation, Poole-Frenkel conduction, and impact ionization, is shown to be in close agreementwith our devices fabricated with 8 nm node technology and tested at 25 °C and 85 °C ambient temperatures for ∼ seconds. The models and physical parameters (including E a and ∆ z ) provide valuable insight into non-measurablematerial properties. With the support of drift saturation and V th prediction, our model may play a significant role in thedevelopment of reliable V th jump tables. 7odeling Electrical Resistance Drift with Ultrafast Saturation of OTS Selectors PREPRINT