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Supplemental material Current-induced electrical self-oscillations across out-of-plane threshold switches based on VO2 layers integrated in crossbars geometry A. Beaumont1, J. Leroy1, J.- C. Orlianges2 and A. Crunteanu1,a) 1 XLIM Research Institute UMR 7252, CNRS/ University of Limoges, 123 avenue Albert Thomas, 87060 Limoges, FRANCE 2 SPCTS UMR 7513, CNRS/ University of Limoges, 12 rue Atlantis, 87068 Limoges, FRANCE 1. STUDY OF THE DISPERSION OF ELECTRICAL PARAMETERS The study of the variability of the activation bias is described in the paper but the dispersion of other parameters was also assessed and a brief description is presented hereafter. Low bias resistivity was determined for every device of the batch by performing a linear regression on the experimental data measured for biases inferior to 0.1 V (Figure S1). It may be observed that both the median low bias resistivity and the dispersion of low bias resistivity are decreased when the size of capacitors is diminished. The high bias resistivity was also determined for every device of the batch with a linear regression on the experimental data measured for biases superior to the transition bias (Figure S2). Similarly as in the case of low bias resistivity, both the median high bias a) Author to whom correspondence should be addressed. Electronic mail: [email protected] 1 resistivity and the dispersion of high bias resistivity are decreased when the size of capacitors is diminished. FIG. S1. : Histograms representing the dispersion of the resistivity measured before MIT at low bias (0.1V) of (a) large (9x9 µm2), (b) medium (5x5 µm2) and (c) small (3x3 µm2) VO2 capacitors. FIG. S2.: Histograms representing the dispersion of the resistivity measured after MIT at high bias (the bias used depended on the activation bias but was chosen to ensure that the 2 VO2 was in conductive state) of (a) large (9x9 µm2), (b) medium (5x5 µm2) and (c) small (3x3 µm2) VO2 capacitors. 2. MODELLING A. Choice of the model type In this work, an analytical modelling of the self-oscillations in the circuit presented in the Figure 3a of the paper. Several physical models have been published and we attempted to use several of them to simulate our devices. However they all failed to reproduce the experimental data presented in this work, certainly because all the pertinent physical phenomena are not taken into account. An example of how these models fail to reproduce our data is given in Figure S3 where the dotted blue line shows the results obtained with the model of Pickett et al.S1, a recent and advanced model based on the thermal triggering of MIT in oxide materials. FIG. S3.: DC experimental I-V characteristics of the device L11 described in the paper (black circles) and bias simulated with the Pickett et al.’s model [S1] (blue dashed line) as a function of the current injected in the MIM device. 3 This model assumes that when small currents are injected in a metal-oxide-metal structure, a conductive filament inside the oxide is created. When the current is increased, the surrounding oxide material is transformed to a metallic phase, thus gradually increasing the diameter of the filament. Since the part of metallic oxide is growing, the out-of-plane resistivity of the layer between the metallic electrodes becomes lower, as observed in the experimental data. For the simulation of our VO2-based devices, the following parameters were used: a resistivity in the metallic phase of 10-2 Ω.m, a resistivity in the insulating phase of 0.255 Ω.m, a channel diameter of 6.06 µm, a VO2 thickness of 130 nm, a thermal conductivity of 7.2 W.m-1.K-1, an external temperature of 298 K, a transition temperature of 341 K and a total series resistance of 130 Ω. The thermodynamic parameters were found not relevant given the static measurements that were to simulate. As shown on Figure S3, although both the low bias insulating state and the high bias metallic state are well simulated by this thermalmodel, the part of the insulating state I-V curve for V>0.5 V and the negative differential region are not fitted and this situation persists independent of the parameters used to carry out the simulations. B. Derivation of the oscillation frequency As described in the paper, a simple model defining three points (named as A, B and C as shown in Figure 2a of the paper) is used to model the static I-V characteristic when the MOM device is biased by a current sweep. The main assumption of the model is that the I-V characteristic is purely linear between each of the three points. The complete derivation leading to the final equations is presented hereafter. When considering the equivalent circuit presented in the Figure 3a of the paper, one may write the value of the total current I0 as a function of the other currents: I 0 iC iDUT , (S1) 4 where iC is the current through the external capacitor and iDUT is the current flowing through the MIM device. Outside of the NDR part of the I-V characteristic, the current iDUT may be expressed as a function of the bias vDUT across the MIM device: iDUT v DUT , (S2) where σ is the conductance of VO2 and is worth σi or σm depending on whether the VO2 is in insulating or metallic phase and β is the value of the current when the voltage is null (βi=0 in insulating phase but βm≠0 in metallic phase). Similarly, the current ic depends on the capacitance of the external capacitor (CEXT) and on the bias vC across it: iC C EXT dv C . dt (S3) In order to take the parasitic inductance of cables into account, the bias vL which develops across the equivalent inductor L is calculated. It depends on the current iL flowing through it according to the following relation: vL L di L . dt (S4) Then one may derive a differential equation by noting that vC = vDUT + vL and iL=iDUT: d 2v C C EXT dv C I . LC EXT vC 0 2 dt dt (S5) Equation S5 is a second order differential equation with a constant right hand member therefore its characteristic equation has to be written as follows: LC EXT 2 C EXT 1 0, (S6) where λ is an arbitrary variable. The discriminant of this equation relation is 2 C EXT 4LC EXT (S7) 5 and its sign has to be studied. It is positive when L, CEXT and σ fulfil the following conditions: C EXT 4L . 2 (S8) This condition is expected to correspond to the charging of CEXT because usually, σi is low when the VO2 is in insulating phase and the solution of Equation S5 is then: 1 I0 1 1 1 V exp i 0 t 2 L 2 2 i L i i i 1 I0 1 1 1 V exp t , 0 i 2 L 2 2 i L i i i I 0 v C (t ) (S9) i where V0 is the initial voltage for t=0 and δi is a constant homogeneous to a frequency defined by 2 i 2LC EXT C EXT 4LC EXT i . 2LC EXT (S10) To get an analytical expression for the duration of a charge or a discharge of the external capacitor, one may approximate Equation S9. Since the two terms multiplied by the time in the exponential are commonly of the same order of magnitude, it is possible to write that 1 exp i t 2 L i 1 exp i t . 2 L i (S11) This enables one to rewrite Equation S9 as: 6 I 1 1 1 V 0 0 2 2 i L i v C (t ) 1 i t exp 2 i L I0 . i (S12) Then it is straightforward to derive the time needed to charge the external capacitor so that the voltage across it varies from V1 to V2: tV0V 1 2 iV 1 1 I 1 1 1 . log 1 0 2 2 i L i 1 iV 2 1 i 2 i L I0 (S13) The sign of the discriminant in Equation S7 may also be negative corresponding to an opposite condition to Equation S8 on L, CEXT and σ. This usually corresponds to the discharge of the capacitor CEXT because for the VO2 in metallic state, σm is high. Then the solution of Equation S5 is: v C (t ) V 0 t I 0 I 0 1 , cos t sin t exp m m m 2 L 2 L m m m (S14) where V0 is the initial voltage for t=0 and δm is defined by m 2LC EXT 4LC EXT C EXT m 2LC EXT 2 . (S15) To get an analytical expression for the duration of a charge or a discharge of the external capacitor, one may approximate Equation S14 by pointing out that the term containing the sine is negligible beside the one containing the cosine and that the exponential term may be approximated to 1 in practical circuits. Then Equation S14 may be approximated as follows: 7 v C (t ) V 0 I0 I0 . cos mt m m (S16) Here again it is possible to derive the time needed to charge the external capacitor so that the voltage across it varies from V1 to V2: tV0V 1 2 mV 2 1 I 1 . arccos 0 mV1 m 1 I0 (S17) It is finally possible to derive the duration of a full oscillation period consisting of a discharge from point A to point B and a charge from point B to point A. The expression of the frequency of the oscillations is: f t 0 VB V A 1 tV0V A . (S18) B iV B mV B 1 1 I 0 i I 1 1 1 1 m log 1 arccos 0 mV A 2 2 i L i iV A 1 m 1 1 i 2 i L I I m 0 i 0 1 It is worth noting that this relation has to be adapted as a function of the parameters of the experimental set-up and the devices, concerning the sign of the discriminant given in Equation S7 in particular. 8 FIG. S4. Simulation of the voltage across the MIM device as a function of time for the device L11 with a current bias of 0.8 mA. The simulations made with the numerical simulator (black line) are compared with the approximate analytical expression for the charge (Equation S12, red line) and for the discharge (Equation S16, blue line). C. Comparison with a numerical simulator In order to confirm the values of frequency oscillations obtained with Equation S18, a numerical simulator resolving the basic equation of circuits (Kirchhoff laws) as a function of the time has been scripted with Python programming languageS2 and the numpy libraryS3. For each time step, the simulator uses a dichotomy algorithm to solve the system of equations defined by Equations S1, S2, S3 and S4 if an inductor is inserted in series with the MOM device. The capability of the expressions derived here above was assessed by comparing them to the results of simulations performed with the numerical simulator. The Figure S4 shows an example of the good agreement between the simulations and the approximate expressions of the bias as a function of the time. REFERENCES S1 M.D. Pickett and R.S. Williams, Nanotechnology, 23 215202 (2012). 9 S2 G. Van Rossum, The python language reference, http://docs.python.org/release/2.7.6/reference/. S3 P.F. Dubois, K. Hinsen and J. Hugunin, Computers in Physics, 10, 262 (1996). 10