Time domain study of frequency-power correlation in spin-torque oscillators

Transcription

1 Time domain study o requency-power correlation in spin-torque oscillators G. Finocchio, 1 G. Siracusano, 1 V. Tiberkevich, 2 I. N. Krivorotov, 3 L. Torres, 4 B. Azzerboni 1 1 Dipartimento di Fisica della Materia e Ingegneria Elettronica, University o Messina, Salita Sperone 31, Messina, Italy. 2 Department o Physics, Oakland University, Rochester, Michigan 48309, USA 3 Department o Physics and Astronomy, University o Caliornia, Irvine, CA, USA. 4 Departamento de Fisica Aplicada, University o Salamanca, Plaza de la Merced s/n, Salamanca, Spain. Keywords: exchange biased spin valves, magnetization oscillation, micromagnetic model, spin-torque, nanooscillators, wavelet. Abstract This paper describes a numerical experiment, based on ull micromagnetic simulations o current-driven magnetization dynamics in nanoscale spin valves, to identiy the origins o spectral linewidth broadening in spin torque oscillators. Our numerical results show two qualitatively dierent regimes o magnetization dynamics at zero temperature: regular (single-mode precessional dynamics) and chaotic. In the regular regime, the dependence o the oscillator integrated power on requency is linear, and consequently the dynamics is well described by the analytical theory o current-driven magnetization dynamics or moderate amplitudes o oscillations. We observe that or higher oscillator amplitudes, the unctional dependence o the oscillator integrated power as a unction o requency is not a single-valued unction and can be described numerically via introduction o nonlinear oscillator power. For a range o currents in the regular regime, the oscillator spectral linewidth is a linear unction o temperature. In the chaotic regime ound at large current values, the linewidth is not described by the analytical theory. In this regime we observe the oscillator linewidth broadening, which originates rom sudden jumps o requency o the oscillator arising rom random domain wall nucleation and propagation through the sample. This intermittent behavior is revealed through a wavelet analysis that gives superior description o the requency jumps compared to several other techniques. 1

2 PACS : Ba, a, g * corresponding author. Electronic address: ginocchio@ingegneria.unime.it 2

3 I. INTRODUCTION Microwave emission driven by spin-polarized current 1 in metallic magnetic nanostructures holds great promise or the development o nanoscale microwave oscillators tunable by magnetic ield and current (spin torque oscillators, STO) 2,3,4. Because the STO requency is a strong unction o the STO output power (p), the STO spectral linewidth ( ω ) depends on the STO power noise via the non-linear requency shit, N: 5,6,7 2 ( N EFF ) ω = ω0 1 + ( / Γ ) (1) where ω0 and Γ EFF are the generation linewidth o a conventional auto-oscillator (that depends on output power and temperature) 6 and the eective non-linear damping, respectively (see Re. [6] or details). Eq. (1) shows that the stochastic dynamics o strongly non-linear STOs qualitatively diers rom the dynamics o classical linear auto-oscillators, in which the linewidth is determined primarily by the phase noise with a negligible contribution rom the power noise. Experimental data o the emitted microwave power as unction o the current or the spinvalves described in Re. [8] are well described by the analytical theory. 7 Furthermore, analysis o a complete set o low temperature experimentally determined parameters o STOs described in Re.[5] shows that the analytical single-mode theory 6 gives good description o the data or magnetization precession angles up to 70 degrees, where the unctional dependence between the output power and the oscillation requency is approximately linear. However, deviations rom the analytical theory are experimentally observed or larger precession angles. 5 In particular, the parametric plots o dimensionless integrated output power (P IOP ) vs. current (I) as well as requency () vs. P IOP (see Fig. 3 in Re. [5]) show that the relationship between these parameters is not a single-valued unction at large enough I and P IOP. Similar results are also observed or point contact geometries 9 and or low temperature measurements in nanoscale exchange biased spin valves with elliptical (130 nm x 60 nm) 4-nm-thick Py-ree layer. 10 In the regime o large-amplitude magnetization dynamics, the measured integrated output power P IOP is not a monotonic unction o the current (e.g. see Fig. 7b o Re. [10] or Fig. 6a o Re. [9]) while the STO requency is a monotonically decreasing unction o I. In the present work, we numerically study power-requency correlations or an STO operating in the regime o large precession angles. In our simulations, we identiy two qualitatively dierent regimes o magnetization dynamics at zero temperature: regular (single-mode precessional dynamics) and chaotic. In the regular regime, we ind that the dependence o the integrated output 3

4 power P IOP (the power measured experimentally as the integral o the power spectrum) on the oscillation requency can be quantitatively predicted rom the dependence o the STO non-linear power 11,12 (P NL, characterized by non-linear dependence o the power on amplitude) on the oscillation requency. Our results suggest that the deviations o the predictions o the analytical theory predictions rom the experimental data 5 or large precession angles can be explained i P NL is used or data analysis instead o the linear power P L. For a range o current densities in the regular regime, our simulations predict a linear dependence o the linewidth on temperature or a ixed current density as expected rom Eq. (1) in the low temperature regime (linewidth suiciently small). 13 In the chaotic regime observed at large current values, we ind that well-deined powerrequency correlation observed in the regular regime o oscillations is lost. Our wavelet-based analysis shows that the chaotic magnetization dynamics in this regime is characterized by sudden jumps o requency versus time corresponding to micromagnetic events such as domain wall nucleation and propagation through the ree layer o the spin valve. Similar requency jumps have been observed recently in MgO-based magnetic tunnel junction STOs. 14 We show that the large phase noise in this regime o chaotic dynamics can be estimated by integrating the waveletinstantaneous requency o the dominant excited STO mode. II. NUMERICAL SIMULATIONS We perormed numerical experiments by means o micromagnetic simulations or an STO system similar to that experimentally studied in Re. [10]. This system is exchange biased spin valve IrMn(8)/Py(4)/Cu(8)/Py(4) (the thicknesses are in nm; Py=Ni 80 Fe 20 ) o elliptical shape (130nm x 60nm). The exchange biased Py layer acts as the pinned layer while the other Py layer is the ree layer. We introduce a Cartesian coordinate system where the x-axis and y-axis are the long and the short axes o the ellipse. We simulate, by solving the Landau-Lishitz-Gilbert-Slonczweski equation, the entire spin-valve with the account o eedback eect o the spin-torque rom the ree layer on the pinned layer. 15 The eective ield consists o the exchange, the magnetostatic and the Oersted ields, and exchange bias ield acting on the pinned layer. We use Slonczweski orm o spin torque 16 T( mp, m ) (where m p and respectively) or both the pinned and ree layers: m are the normalized pinned and ree layer magnetizations, T( m, m ) = = p j ε( m, m ) m ( m m ) p p Tree g µ B LF 2 Tpinned e γ j 0 M s ε mp m mp mp m LP (, ) ( ) (2) 4

5 where g is the gyromagnetic actor, µ B is the Bohr magneton, e is the electron charge, j is the current density (positive or current lowing lows rom the pinned to the ree layer), L F and L P are the thicknesses o the ree and pinned layer respectively, and ε ( mp, m ) is the polarization unction given by: ( ) p P p ε ( m, m ) = 0.5 Λ / 1 + Λ + (1 Λ ) m m (3) where P is the current polarization and 2 Λ is related to the asymmetric giant-magneto-resistance 2 (GMR) parameter χ as Λ = 1+ χ. 16 The magneto-resistance signal is computed as 1 r(, ) = r (, ) m m m m, where ri ( mi, p, m i, ) is the magneto-resistance signal o the i th p i i, p i, N i = 1... N computational cell o the ree layer computed with respect to the i th computational cell o the pinned 2 2 layer r = θ + χ θ ( m, m ) [1 cos ( / 2)]/[1 cos ( / 2)] ( cos( θ i ) = i, p i, i i, p i, i i computational cell o the ree layer). 16,17 m m and N is the number o We make simulations or a ixed external ield (68 mt applied at -45 degrees with respect to the long axis o the ellipse) or a range o bias currents. In these simulations we use saturation magnetization o Py A/m, Py exchange constant J/m, Gilbert damping parameter α F = or the ree layer and α P = 0.2 or the pinned layer, 17 the exchange bias ield acting on the pinned layer is 70 mt (applied along +45 degrees with respect to the easy axis o the ellipse), P and χ are 0.38 and 1.5, respectively. The computational cell is 5x5x4 nm 3 and the integration time step is 0.2 ps. For the inite temperature simulations, the thermal luctuations have been taken into account via adding a stochastic ield to the eective ield o each computational cell o the pinned and ree layers. 18,19,20 III. RESULTS A) Regular regime The magnetization dynamics or this system is excited in a broad range o currents with a critical current density or the onset o auto-oscillations J C = A/cm 2. For zero temperature simulations and or the bias current densities rom J= A/cm 2 to J= A/cm 2, a regular regime where the magnetization dynamics exhibits a single-mode character is observed. Fig.1(a) shows an example o the power spectrum in the regular regime computed or J= A/cm 2 by means o the micromagnetic spectral mapping technique (MSMT). 21,22 In this regime, the inite 5

6 generation linewidth at zero temperature is caused solely by the inite computational time. The regular dynamics is characterized by a spatially uniorm coherent precession o the magnetization as illustrated in the inset o Fig. 1(a). Figure 1(b) displays the micromagnetic and the experimental (rom Re. [10]) data o requency versus current density. The oscillation requency red shit as a unction o current is in qualitative agreement with the analytical non-linear theory. 11,23 Figure 1(b) also shows normalized powers (P IOP, P L, and P NL ) as unctions o the current density. The integrated output power P IOP corresponds to the power measured in the experiments and it is computed by integrating the power spectrum: S / PIOP = Ri ( ) d (4) N 0 i= 1... N where Ri ( ) is the Fourier transorm o ri ( mi, p, m i, ) and S is the sampling requency. For a single excited mode, the linear power P L is deined as the square o the amplitude o the resistance oscillations o the giant-magneto-resistance signal r( mp, m ). 11 From the numerical point o view, we use the ollowing approximation r mp m = a0 + a1 π At + ϕ1 + a2 π At + ϕ2 ( A is the requency o the excited mode) to compute (, ) sin(2 ) sin(4 ) 2 2 L 1 2 P = a + a. The nonlinear power P NL o the excited spin wave mode is computed as ollows: in the regular regime, the trajectory γ o the magnetization vector is closed or every computational cell (see Fig. 2(a) or an example) and, thereore, the curve γ splits the surace o the unit sphere into two parts (the smaller S 1 and the larger S 2 ). The power P NL is deined as the sum o the S 1 or all the computational cells o the ree (N ) and the pinned (N p ) layers o the STO: 1 1 PNL = ds + ds 4π N 4π N (5) i= 1... N S p i= 1... N 1i p S1i Deined in such a way, the power P NL is proportional to the canonical action corresponding to the closed loop γ and represents proper generalization o the oscillation power or the case o spatially-nonuniorm and non-circular magnetization precession. The power P IOP is computed in the requency domain, while P L, and the P NL are computed in the time domain. The red shit unctional dependence o the oscillation requency on current density (see Fig. 1(a)) or the regular region given by our simulations is in agreement with the experimental data. 10 The calculated requencies o magnetization sel-oscillations are smaller than 6

7 the erromagnetic resonance requency, FMR, estimated to be 6.7GHz by means o spin torque erromagnetic resonance calculations. 24,25 Simulations o requency as a unction o power give negative non-linear requency shit N /(2 π ) -1.45GHz computed near the critical current density as: 7 N = 2 π ( ) / P ( J ) (6) JC FMR L C The parametric dependence o P NL and P L on the oscillation requency is shown in Fig. 1(c). This dependence is non-monotonic or P NL with the maximum o P NL achieved around the current density J max = A/cm 2. Fig. 2(b) shows the trajectories (x-z plane) o the magnetization o one computational cell (the trend is independent o the computational cell) or J=1.5, 1.8, A/cm 2. As the current density increases, the magnetization trajectories irst expand up to A/cm 2, and then become deormed and intersections with curves at lower current appear. In this current density range the P NL decreases. In contrast, P L is monotonic in requency. It shows behaviour qualitatively similar to that o P NL or small current densities J < J max, but approaches saturation or J > J max. Fig. 1(b) clearly demonstrates that or large values o current, P NL has very similar unctional dependence on current as the experimentally measurable quantity, P IOP. In contrast, the dependence o P L on current is qualitatively dierent rom that o P IOP. This means that one can not use P L or calculation o such parameters as nonlinear requency shit or current densities ar above the supercritical current density. 5 Our numerical results show the unctional dependence o P NL and P IOP on oscillation requency can be well approximated by an empirical unction third-order polynomial: 3 2 IOP P = a + a + a + a (7) (see or example the solid line in Fig. 1(c) or P NL () itting). By considering the current density range where the P IOP and P N are proportional to each other (J = A/cm 2 ), the polynomial coeicients computed or P IOP are proportional to the polynomial coeicients o P NL (i.e. a / a = a / a = a / a = a / a ). 3, IOP 3, NL 2, IOP 2, NL 1, IOP 1, NL 0, IOP 0, NL B) Chaotic regime For current densities larger than A/cm 2 we observe zero-temperature broadening o the linewidth (chaotic regime), as evident rom the magnetoresistance power spectrum computed 7

8 via the MSMT in Fig. 3(a) or the current density o A/cm 2. This spectral broadening is related to the loss o temporal coherence due to the presence o non-uniormities in the spatiotemporal distribution o the magnetization. 26,27 In contrast to the regular regime, our numerical results show a lack o correlation between power and requency in the chaotic regime. A time-requency characterization o the magnetorestance signal or large currents gives the possibility to identiy the origin o the linewidth broadening at zero temperature. We systematically study the micromagnetic wavelet scalogram (MWS) 28 o the magnetoresistance signal. We use the complex Morlet unction as the wavelet mother: ψ u, s t u t u j π c / B s s = e e (8) sπ B which has the best time( σ t )-requency( σ ) resolution because o its Gaussian envelope ( σ ( 2 π s ) 1 B =, σ = 0.5s t B ). 29 The variables u and s are the translation and scale parameters, B and C are the characteristic parameters. For the complex Morlet wavelet unction with C =1, it is possible to relate the Fourier requency directly to the scale requency. 30 S =, where S is the sampling Fig. 3(b) shows the MWS (white/black color corresponds to the largest/smallest wavelet amplitude) computed or J= A/cm 2 ( B =300, C =1). This igure shows that there are two sources o the linewidth broadening: (i) continuous modulation o the requency o the dominant excited mode and (ii) discontinuous requency jumps (see points A and B in Fig. 3(b)). By analyzing the time evolution o the spatial distribution o the magnetization close the points A and B, we observe nucleation o a domain wall at one side o the device and its subsequent propagation to the other side (see supplementary movie in Re.31). Another advantage o the wavelet analysis is the possibility to estimate the phase noise φ ( t) directly rom the time domain data through integration o the instantaneous wavelet requency i (t) o the dominant exited mode (the requency that has the largest value o the wavelet transorm at a t given moment o time) φ( t) = 2 π ( τ ) dτ + φ(0). The instantaneous phase and requency can also 0 i be estimated in a simple way rom the zero crossing o the voltage time traces as described in Re. [32]: i 0 is the oscillation requency the instantaneous phase, or the time ( t i ) where the zero crossing occurs, can be computed as φ( ti ) = nπ 2π 0ti (n is even (odd) or crossing with a positive s 8

9 (negative) slope). Fig. 3(c) shows a comparison o the instantaneous requency i computed with the wavelet analysis (blue circles) and the voltage crossing (solid line). The wavelet-based analysis gives the range o instantaneous requency luctuations that is in a better agreement with the linewidth o the power spectrum in Fig. 3(a) (the linewidth computed with a Lorentzian it is 278MHz). C) Eect o the thermal luctuations In the presence o thermal luctuations, we calculate the P IOP power as a unction o temperature. We ound that the unctional dependence o the P IOP on the oscillation requency is qualitatively independent o temperature as displayed in Fig. 4(a) (temperatures rom 50K to 300K at step o 50K, an oset is used or clarity) or the current values in the regular regime. We also calculated the temperature dependence o the linewidth (the simulation time was 1 µ s which corresponds to a requency resolution o 1MHz); the results are summarized in Fig. 4(b) or three current densities J=1.2, 1.8, and A/cm 2. In agreement with the analytical theory 33 and experimental data (e.g. see Fig. 4(d) in Re.[5]), we ind broad spectra near the threshold current with non-linear dependence o the linewidth on temperature (e.g. J= A/cm 2 ) corresponding to a GMR-signal with high noise level. For higher current densities, the linewidth decreases (Fig.4(b), J=1.8 and A/cm 2 ) by approximately one order o magnitude and the dependence o the linewidth on temperature is approximately linear at a ixed current density value. In this current region, the signal-to-noise ratio is signiicantly improved. Our results are in qualitative agreement with analytical calculations as predicted by Eq. (1) in the low temperature regime 13 and experimental measurements. 8 IV. CONCLUSIONS In conclusion, we perormed a numerical experiment to identiy the origin o the discrepancy o analytical theory and experimental data observed in some experimental studies o large-amplitude STO dynamics. We ound two qualitatively dierent regimes o magnetization dynamics at zero temperature: regular (single-mode precessional dynamics) and chaotic. In the regular regime or moderate amplitudes o oscillations, the unctional dependence o the integrated output power (the experimentally measured quantity) on the oscillation requency is linear, and it is well described by the analytical theory 11. Our studies reveal that micromagnetically calculated nonlinear power P NL gives a better description o the integrated output power or large amplitudes o magnetization oscillations o the regular regime than the linear power P L used in the current analytical theories. We also ound the linewidth as a unction o temperature is a linear unction in a 9

10 range o currents larger than the threshold current. These indings are important or modeling o STOs working in a wide range o current densities. In contrast, the chaotic regime is not described by the single-mode analytical theory. We ind that in the chaotic regime, the oscillator linewidth broadening originates rom sudden jumps o the oscillator requency arising rom random domain wall nucleation and propagation across the STO ree layer. This intermittent behaviour is revealed through a wavelet analysis that gives superior description o the requency jumps compared to other techniques. ACKNOWLEDGMENTS This work was supported by Spanish Project under Contracts No. MAT /NAN and No. SA025A08. I. N. K. grateully acknowledges the support o DARPA, NSF (grants DMR and ECCS ) and the Nanoelectronics Research Initiative through the Western Institute o Nanoelectronics. REFERENCES 1 J. Slonczewski. J. Magn. Magn. Mater. 159, L1-L7, (1996). 2 S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkop, R. A. Buhrman, D. C. Ralph. Nature, 425, 380, (2003). 3 W. H. Rippard, M. R. Puall, and S. E. Russek. Phys. Rev. B 74, , (2006). 4 K. V. Thadani, G. Finocchio, Z.-P. Li, O. Ozatay, J. C. Sankey, I. N. Krivorotov, Y.-T. Cui, R. A. Buhrman, D. C. Ralph. Phys. Rev. B 78, (2008). 5 C. Boone, J. A. Katine, J. R. Childress, J. Zhu, X. Cheng, I. N. Krivorotov. Phys. Rev. B 79, (R), (2009). 6 J.-V. Kim, V. Tiberkevich, A. Slavin. Phys. Rev. Lett. 100, , (2008). 7 V. Tiberkevich, A. Slavin, J.-V. Kim. Appl. Phys. Lett. 91, , (2007). 8 Q. Mistral, J.-V. Kim, T. Devolder, P. Crozat, C. Chappert, J. A. Katine, M. J. Carey, K. Ito. Appl. Phys. Lett. 88, , (2006). 9 M. L. Schneider, W. H. Rippard, M. R. Puall, T. Cecil, T. J. Silva, S. E. Russek. Phys. Rev. B 80, , (2009). 10 I. N. Krivorotov, D. V. Berkov and N. L. Gorn, N. C. Emley, J. C. Sankey, D. C. Ralph, R. A. Buhrman. Phys. Rev. B 76, (2007). 11 A. Slavin, V. Tiberkevich, IEEE Trans. on Magnetics, 45, , (2009). 12 The non-linear power describes the power o a non-circular magnetization precession (non-uniorm spin wave mode). 13 V. S. Tiberkevich, A. N. Slavin, J.-V. Kim. Phys. Rev. B 78, , (2008). 14 D. Houssameddine, U. Ebels, B. Dieny, K. Garello, J.-P. Michel, B. Delaet, B. Viala, M.-C. Cyrille, J. A. Katine, and D. Mauri, Phys. Rev. Lett. 102, (2009). 15 G. Siracusano, G. Finocchio, I. N. Krivorotov, L. Torres, G. Consolo, B. Azzerboni. J. Appl. Phys. 105, 07D107, (2009). 16 J. Slonczewski. J. Magn. Magn. Mater. 247, 324, (2002). 17 G. Finocchio, I. N. Krivorotov, L. Torres, R. A. Buhrman, D. C. Ralph, B. Azzerboni. Phys. Rev. B 76, , (2007). 18 W. F. Brown, Jr. Phys. Rev. 130, 1677, (1963). 19 D. M. Apalkov, P. B. Visscher. Phys. Rev. B 72, (R), (2005). 20 G. Finocchio, M. Carpentieri, B. Azzerboni, L. Torres, E. Martinez, L. Lopez-Diaz. J. Appl. Phys. 99, 08G522, (2006). 21 R. D. McMichael, M. D. Stiles. J. Appl. Phys. 97, 10J901, (2005). 22 L. Torres, L. Lopez-Diaz, E. Martinez, G. Finocchio, M. Carpentieri, B. Azzerboni. J. Appl. Phys. 101, , (2007). 10

11 23 S. M. Rezende, F. M. de Aguiar, A. Azevedo, Phys. Rev. Lett. 94, (2005). S. M. Rezende, F. M. de Aguiar, A. Azevedo, Physical Review B, 73, , (2006). 24 A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, S. Yuasa. Nature, 438, 339, (2005). 25 J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, D. C. Ralph. Phys. Rev. Lett. 96, , (2006). 26 Z. Yang, S. Zhang, Y. Charles Li. Phys. Rev. Lett. 99, , (2007). 27 D. Berkov, N. Gorn. Rev. B 71, , (2005). 28 G. Siracusano, G. Finocchio, A. La Corte, G. Consolo, L. Torres, B. Azzerboni. Phys. Rev. B 79, , (2009). 29 L. Hudgins, C.A. Friehe, M.E. Mayer. Phys. Rev. Lett. 71, , (1993). S. Mallat. A Wavelet Tour o Signal Processing. San Diego: Academic Press (1998). J. C. Goswami, A. K. Chan. Fundamentals o Wavelets Theory, Algorithms and Applications, John Wiley & Sons (2000). 30 C. Torrence, G.P. Compo. Bull. o the American Metereological Soc. 79, 61-78, (1998). 31 EPAPS Document No. [] or movie o the magnetization dynamics near the point A. For more inormation on EPAPS, see []. 32 M. W. Keller, A. B. Kos, T. J. Silva, W. H. Rippard, M. R. Puall. Appl. Phys. Lett. 94, , (2009). 33 J-V. Kim, Q. Mistral, C. Chapper, V. S. Tiberkevich, A. N. Slavin. Phys. Rev. Lett. 100, , (2008). 11

12 Figure 1. (color online) (a) Example o Fourier spectrum computed by means o the micromagnetic spectral mapping technique in the regular dynamics regime or a current density A/cm 2. Inset: spatial distribution o the dominant excited mode. (b) Micromagnetic (solid line) and experimental rom reerence [10] (circle o ) requency o the dominant excited mode, linear power (P L ), integrated output power (P IOP ), and non-linear power (P NL ) as unction o the current density. (c) Parametric dependence o the non-linear power and linear power on the generated requency and a cubic it o the non-linear power. 12

13 Figure 2 (color online) (a) Example o the calculated magnetization trajectory and illustration o the nonlinear power deinition. (b) Example o the trajectories o the magnetization projected in the x-z plane) o one computational cell (the trend is independent o the computational cell) or J=1.5, 1.8, A/cm 2. Figure 3: (color online) (a) and (b) Fourier spectrum computed by means o the micromagnetic spectral mapping technique and micromagnetic wavelet scalogram (white/black color corresponds to the largest/smallest wavelet amplitude) or a current density A/cm 2 respectively. (c) Comparison between instantaneous requency computed via the wavelet transorm (blue circle) and the technique developed in Re [30] (black solid line). 13

14 Figure 4: (color online) (a) Temperature dependence o integrated output power on oscillation requency o the dominant excited mode (an oset is applied or each curve). (b) Temperature dependence o the linewidth or three current density values J=1.2, 1.8, and A/cm 2 ). 14