The thermal conductivity of amorphous TaOx memristive films having variable oxygen content is measured using time domain thermoreflectance. Thermal transport is described by a two-part model where the electrical contribution is quantified via the Wiedemann-Franz relation and the vibrational contribution by the minimum thermal conductivity limit for amorphous solids. The vibrational contribution remains constant near 0.9 W/mK regardless of oxygen concentration, while the electrical contribution varies from 0 to 3.3 W/mK. Thus, the dominant thermal carrier in TaOx switches between vibrations and charge carriers and is controllable either by oxygen content during deposition, or dynamically by field-induced charge state migration.

While transition metal oxides such as HfO2, TiO2, and Ta2O5 are strong dielectrics, depletion of the oxygen content creates charged states that can dramatically lower the electrical resistance. The mobility of the charged states in response to electrical field and the concomitant heating results in nonvolatile changes of resistivity.1,2 This behavior is a unique property of the fourth circuit element, namely, the memristor (memory resistor),3 and has important implications for analog memory and neuromorphic computing architectures.4–9 Despite this general understanding, description of the physical processes belying memristor operation remains incomplete. For example, while it is accepted that the thermal environment during switching contributes to the concentration and distribution of oxygen,10 quantification or prediction of the temperature field has proven elusive. Characterization of this field is difficult owing to the small time- and length-scales—on the order of 1 ns and 10 nm, respectively11,12—involved in switching. Computational prediction of the temperature field, in turn, has relied on approximations13–16 as the thermal properties of the switching materials have not been investigated.

In response, the thermal conductivity of tantalum oxide films deposited with varying oxygen content is examined. Using electrical resistivity and X-ray photoelectron spectroscopy (XPS) measurements, the homogeneous films are shown to span the memristive transition from electrically insulating to conducting.17,18 Employing time domain thermoreflectance (TDTR), the thermal conductivity is found to have a lower bound of ∼0.9 W/mK for large oxygen content that grows to ∼4 W/mK at the lowest concentration of oxygen used in this study. While more metallic films should have even higher thermal conductivities, they are not included because the nature of conduction changes as continuous conductive pathways are formed.17 The variation in thermal conductivity during the transition from insulating toward metallic is well described by the framework of an amorphous oxide vibrational contribution plus an electrical contribution quantified using the Wiedemann-Franz relation. Given the simplicity of this framework, it can be readily extended to thermal property prediction in other memristive material systems.

Seven tantalum oxide films were deposited simultaneously on both silicon and silica substrates utilizing a Kurt J. Lesker Lab 18 system with 30° off-axis sputter geometry, tantalum metal target (100 mm in diameter), and target to substrate distance of 18.4 cm. In order to vary the oxygen content of the film, argon and O2 gas were supplied at the flow rates detailed in Table I, while the sputtering pressure was maintained at 4 mTorr, power at 400 W, and power density at 4.90 W/cm2. All samples are labeled by the ratio of O2 to Ar in the deposition environment. Prior to all depositions, the tantalum target was “seasoned” through the creation of a separate—dummy—film using an O2:Ar ratio of 9:91. Seasoning ensures that the surface of the target is saturated with oxygen at the outset of the deposition sequence—leading to better reproducibility and greater control of the oxygen content of the final films. The TaOx depositions lasted about 6400 s yielding the thicknesses shown in Table I as measured with a Veeco Dektak 6 M stylus profilometer. A thin platinum film ∼70 nm was then sputtered onto the the films synthesized atop silicon to facilitate the thermal conductivity measurements.

TABLE I.

Gas flow rates for sputtering of TaOx films and resulting film thickness and electrical resistivity.

O2:ArArgon flowO2 flowThicknessResistivity
Ratio(SCCM)(SCCM)(nm)Ω⋅ m × 10−6
3:97 57.7 1.9 634 ± 18 2.46 ± 0.07 
4:96 57.8 2.3 604 ± 27 3.44 ± 0.2 
5:95 58.3 2.9 492 ± 20 7.95 ± 0.3 
6:94 58.4 3.5 540 ± 11 228 ± 0.5 
7:93 58.4 4.1 571 ± 19 3.31 × 105 ± 1000 
8:92 58.4 4.6 615 ± 16 3.68 × 106 ± 9000 
9:91 58.2 5.2 620 ± 16  > 1 × 107a 
O2:ArArgon flowO2 flowThicknessResistivity
Ratio(SCCM)(SCCM)(nm)Ω⋅ m × 10−6
3:97 57.7 1.9 634 ± 18 2.46 ± 0.07 
4:96 57.8 2.3 604 ± 27 3.44 ± 0.2 
5:95 58.3 2.9 492 ± 20 7.95 ± 0.3 
6:94 58.4 3.5 540 ± 11 228 ± 0.5 
7:93 58.4 4.1 571 ± 19 3.31 × 105 ± 1000 
8:92 58.4 4.6 615 ± 16 3.68 × 106 ± 9000 
9:91 58.2 5.2 620 ± 16  > 1 × 107a 
a

Currents as small as 1 nA took the source to compliance at 50 V and precluded reliable measurements.

Due to the complexities of reactive sputtering and non-linearities in the oxygen control of the TaOx system,19 post-deposition characterization is required to identify the composition of the films. While a variety of techniques can be used to measure the films' oxygen content, electrical resistivity is known to be correlated with oxygen content17 and is more simply measured. To this end, van Der Pauw resitivity measurements were performed by depositing 100 nm Pt electrodes in the corners of the TaOx films deposited atop 1 cm2 square silica substrates. Current was sourced using a Keithley 237 source meter while the voltage was monitored with a HP3478 multimeter. Measurements were performed over several orders of magnitude of current levels to ensure the films were ohmic. The results are shown in Table I, where, as expected, decreasing oxygen content trends with lower resistivity.

Thermal conductivity of the films was measured using TDTR. For this measurement, the thin metal transducer is heated by an ultrafast laser (Ti-Sapphire oscillator at 82 MHz doubled via a 2 mm thick BiB3O5 crystal to 400 nm and modulated with a Conoptics 350–160 EOM at 11.3 MHz). The temperature of the transducer in response to the pulsed heating is monitored using the temperature dependent reflectivity (thermoreflectance) probed with a collinear 800 nm beam. The probe beam travels along a variable delay stage to monitor the change in the reflected intensity at various delay times after the pump pulse arrival. An SR844 lock-in amplifier is used to pick out the signal at the modulation frequency. By fitting the ratio of the in-phase signal versus the out-of-phase signal to a model explained elsewhere,20–23 the thermal transport properties can be obtained. For each of the three layers (Pt transducer / TaOx layer / silicon substrate), the thickness, volumetric specific heat, thermal conductivity, and thermal boundary conductance (TBC) must either be known or fit to the data. In this work, only the TBC between the Pt transducer and the TaOx layer along with the thermal conductivity of the latter are fitting variables—all other relevant material properties and their temperature dependence are taken from supporting experiments or literature values (see supplementary material24).

Seven TDTR measurements were taken at four distinct locations—which were optically determined to be free of surface defects—on each sample. Data is collected from −20 ps to 100 ps in steps of 1 ps and then from 100 to 6000 ps in steps of 100 ps, with t = 0 being the midpoint of the signal rise due to the arrival of the pump pulse. The phase is corrected by averaging the signal for 20 ps before and after t = 0 and then rotating the signal so that the averaged out-of-phase component is constant before and after t = 0.25 Fitting is performed for the window from 200 to 6000 ps.

Examples of the TDTR data and model fit are shown in Figure 1 and a summary of all measured thermal conductivity values are shown in Figure 2 and tabulated along with the thermal boundary conductance in Table II. To estimate uncertainty, fitting was performed first using the specified value for all “known” input parameters and then repeated using high and low values associated with the uncertainty of each input parameter. Uncertainty of the output parameters due to an input parameter is taken as half the difference between the low and high values resulting from these calculations. The spatial/experimental variation is also considered by taking the standard deviation of the seven scans for each sample. The reported uncertainty is the square root of the sum of all the squared uncertainties.

FIG. 1.

Examples of TDTR signal from lock-in amplifier (–Vin/Vout) and fit for three of the samples. Black solid lines show the model at the best fit values, and the gray areas show the uncertainty.

FIG. 1.

Examples of TDTR signal from lock-in amplifier (–Vin/Vout) and fit for three of the samples. Black solid lines show the model at the best fit values, and the gray areas show the uncertainty.

Close modal
FIG. 2.

Reduction of thermal conductivity with increasing oxygen content for TaOx films as measured by TDTR. Thermal conductivities of insulating films agree with the minimum thermal conductivity estimate of Eq. (2).

FIG. 2.

Reduction of thermal conductivity with increasing oxygen content for TaOx films as measured by TDTR. Thermal conductivities of insulating films agree with the minimum thermal conductivity estimate of Eq. (2).

Close modal
TABLE II.

Complete TDTR results.

O2:Ar ratioκtotal (W/mK)unc.G (MW/m2K)unc.
3:97 4.2 0.4 138 16 
4:96 3.1 0.4 142 10 
5:95 2.0 0.2 100 19 
6:94 1.0 0.1 121 15 
7:93 0.93 0.1 131 25 
8:92 0.92 0.1 134 19 
9:91 0.94 0.1 135 21 
O2:Ar ratioκtotal (W/mK)unc.G (MW/m2K)unc.
3:97 4.2 0.4 138 16 
4:96 3.1 0.4 142 10 
5:95 2.0 0.2 100 19 
6:94 1.0 0.1 121 15 
7:93 0.93 0.1 131 25 
8:92 0.92 0.1 134 19 
9:91 0.94 0.1 135 21 

Only modest changes were observed in the measured TBC exhibiting values ranging from 100–140 MW/m2K (See Table II). The TBC's magnitude implies that vibrations are the dominant energy carrier across the interface despite the changing electrical resistivity of the underlying TaOx. The consistency and magnitude in TBC are attributed to the formation of a surface oxide layer of Ta2O5, which is a consequence of exposing the films to atmosphere between the TaOx and Pt depositions as necessitated for the application of a shadow mask. XPS confirmed the presence of this Ta2O5 surface layer in a separate set of films prepared in a nominally similar fashion. Unlike TBC, thermal conductivity of the TaOx dramatically increased for the low oxygen content (more metallic) films. The cause for this increase is discussed hereafter.

In most electrical conductors, the vibrational contribution to thermal conductivity is negligible compared to the electronic contribution. However, for these samples, the vibrational contribution—as indicated by the three insulating samples—is a significant fraction of the total thermal conductivity. While the two components of thermal conductivity (electrical and vibrational) cannot be separately measured, the measured electrical conductivity provides additional insight. Specifically, the Wiedemann-Franz law26 describes the relationship between electrical conductivity and thermal conductivity as

κelec=LTσ,
(1)

where σ = 1/ρ is the electrical conductivity and L is the Lorenz number which has a theoretical value of 2.45 × 10−8 WΩ/K2. A linear regression fits the thermal versus electrical conductivity data well (R2 = 0.996), as shown in Figure 3. The calculated Lorentz number from the regression is L = 2.6 × 10−8 ± 0.2 × 10−8 WΩ/K2 (95% confidence bounds) and is consistent with the theoretical Lorentz number. Furthermore, the regression yields an intercept of κmin = 0.95 ± 0.1 W/mK (95% confidence bounds), which agrees well with the thermal conductivity of the three insulating samples (See Fig. 2 and Table II). Thus by utilizing the Wiedemann-Franz law and resistivity measurements, the electrical and vibrational contributions can be separated and quantified.

FIG. 3.

The measured thermal conductivity vs. electrical conductivity is well fit by a linear model consistent with the Wiedemann-Franz law, Eq. (1), plus a vibrational contribution (the y-intercept).

FIG. 3.

The measured thermal conductivity vs. electrical conductivity is well fit by a linear model consistent with the Wiedemann-Franz law, Eq. (1), plus a vibrational contribution (the y-intercept).

Close modal

While the paragraph above focuses on interpreting the experimental data, this two part model can also be used in a predictive manner. Specifically, the minimum thermal conductivity model27 of an amorphous material gives the vibrational contribution as

κmin=3(π6)1/3vkBn2/3(Tθ)20(θ/T)x3ex(ex1)dx,
(2)

where the cutoff frequency (Debye temperature) is given by θ=v/(kB6π2n),kB is the Boltzmann constant and the velocity (v = 5000 m/s) corresponds to the speed of sound in amorphous tantalum oxide.28,29 Due to the uncertainty surrounding thermal transport in amorphous solids,30 the number density n is taken as both the atomic density and the molecular density. This strategy was found to bound the thermal conductivity in hafnium oxide films31 and also bounds the measured vibrational contribution in this study (see Figure 2). There is strong agreement between the minimum thermal conductivity estimate and the vibrational contribution calculated from the experimental data. Thus, with only a knowledge of the speed of sound in the amorphous oxide and the electrical resistivity, the thermal conductivity of a memristive device can be predicted. This model is expected to extend to other metal oxide memristors (e.g., TiOx and HfOx).

These results also demonstrate that the dominant thermal carrier in TaOx switches from vibrational to electronic as the amount of oxygen is decreased during deposition. As shown in Fig. 2, the crossover is observed near 6% oxygen content in the growth chamber. This crossover is also expected to occur when resistivity is controlled in real time during memristive switching. Consequently, in addition to exhibiting real-time tunable electrical conductivity as is leveraged for memristive purposes, TaOx should be expected to also exhibit real-time tunable thermal conductivity.

In summary, thermal transport has been characterized in samples fabricated to mimic ionic states present in TaOx memristive devices. The thermal conductivity is shown to increase with decreasing oxygen content due to changing electrical resistivity. It is found that the predominant thermal energy carrier changes from vibrational to electronic. Representing the thermal conductivity as an amorphous vibrational contribution via the minimum thermal conductivity model plus an electrical contribution via the Wiedemann-Franz law appears to be a simple and effective strategy to interpret thermal conductivity measurements and to predict the same in the ongoing modeling efforts such as Refs. 13–16, 32, and 33. Furthermore, these results point to the possibility of using metal oxides for actively controlled thermal conductivity both in quantity and character (vibrational vs. electronic).

The authors gratefully acknowledge financial support from Sandia National Laboratories Laboratory Directed Research and Development Program. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

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Supplementary Material