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The importance of Joule heating on the voltage-triggered insulator-to-metal transition in VO₂ Spitzig, Alyson 2017

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The importance of Joule heating on thevoltage-triggered insulator-to-metaltransition in VO2byAlyson SpitzigB.Sc., Dalhousie University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Alyson Spitzig 2017AbstractThe large change in resistivity in the material VO2 has attracted consider-able attention since it was first discovered in 1959. Recently, the ability totrigger the insulator-to-metal transition (IMT) with a strong electric fieldhas been observed, but there has been debate about whether the transitionis due to field-effects. We apply a voltage bias across a VO2 thin film viaa conductive atomic force microscope (CAFM) tip and measure the resul-tant current. We observe the IMT as a jump in the measured current inthe IV curves. We fit the IV curves to the Poole-Frenkel (PF) conductionmechanism in the insulating state, immediately preceding the IMT. The PFconduction mechanism describes the thermal excitation of electrons into theconduction band in insulators, facilitated by strong electric fields. The PFmechanism is temperature dependent, and we use the temperature depen-dence to calculate the local temperature of the film before the transition.We directly compare the local electric field and local temperature of the filmimmediately preceding the IMT. We determine that the transition is notsolely due to the applied electric field, but rather that the tip has locallywarmed the film close to its IMT temperature through Joule heating.iiLay SummaryThe material VO2 is an insulator at room temperature, but above 340 K (67◦C) it transitions to a metal. We ramp the voltage bias across a thin film, andmeasure the induced current. The Poole-Frenkel (PF) mechanism describesof how electrons conduct electricity in an insulator through strong electricfield and temperature. This description of the induced current is temperaturedependent, so by measuring the current and voltage was can fit to the PFmechanism to calculate the temperature. We calculate a temperature of 334K (61 ◦C), which is well above room temperature. By application of a strongelectric field and inducing a current in the film we have warmed the film closeto its transition temperature. This indicates that the transition was causedby thermal effects.iiiPrefaceThe results of this thesis are to be published later in 2017. The film studiedwas grown and characterized by Changhyun Ko and Shiriram Ramanthan.IV measurements, and preliminary analysis were performed by Hoffmangroup members Adam E. Pivonka, Harry Mickalide, Alex Frenzel, JeehoonKim, Kevin O’Connor, and Eric Hudson. Dielectric constant measurementsof VO2 were taken by You Zhou and Changhyun Ko. The author, AlysonSpitzig completed all calculations and conclusions therein, with mentorshipby Jennifer E. Hoffman and Jason D. Hoffman.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 52 Data Collection and Correction . . . . . . . . . . . . . . . . . 92.1 Conductive Atomic Force Microscopy . . . . . . . . . . . . . 92.2 VO2 Film Growth and Characterization . . . . . . . . . . . . 92.2.1 Physical Film Properties . . . . . . . . . . . . . . . . 92.2.2 Dielectric constant of VO2 . . . . . . . . . . . . . . . 112.3 IV Collection and Correction . . . . . . . . . . . . . . . . . . 142.3.1 External Resistance . . . . . . . . . . . . . . . . . . . 162.3.2 Stray Capacitance . . . . . . . . . . . . . . . . . . . . 172.4 Conduction Mechanisms in Insulators . . . . . . . . . . . . . 242.5 A Second Data Set . . . . . . . . . . . . . . . . . . . . . . . . 293 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Poole-Frenkel Fitting . . . . . . . . . . . . . . . . . . . . . . 33vTable of Contents3.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Number of Jumps . . . . . . . . . . . . . . . . . . . . 364 Nanoscale Thermometry . . . . . . . . . . . . . . . . . . . . . 394.1 Our Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Other Nanoscale Thermometry Techniques . . . . . . . . . . 404.3 Extension of Thermometry Technique . . . . . . . . . . . . . 405 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43AppendicesA Power Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.1 Input energy from tip . . . . . . . . . . . . . . . . . . . . . . 55A.2 Heat required to cause IMT in VO2 . . . . . . . . . . . . . . 56A.3 Thermal Conductivity of SiO2 . . . . . . . . . . . . . . . . . 56B Uncertainty in the calculated Temperature . . . . . . . . . 61B.1 Dielectric contant, εV(T ) . . . . . . . . . . . . . . . . . . . . 62B.2 Thickness, d . . . . . . . . . . . . . . . . . . . . . . . . . . . 63B.3 PF slope, P . . . . . . . . . . . . . . . . . . . . . . . . . . . 63B.4 Temperature, T . . . . . . . . . . . . . . . . . . . . . . . . . 66viList of Tables2.1 Summary of previous measurements of εV. . . . . . . . . . . . 122.2 Decay constant and resultant capacitance values in the in-sulating and metallic state for the different effective circuitdiagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Summary of conduction mechanisms in insulators. . . . . . . 252.4 List of variables in conduction mechanisms. . . . . . . . . . . 26viiList of Figures1.1 Morin’s first measurements of the metal-to-insulator transi-tion of Ti- and V- oxides . . . . . . . . . . . . . . . . . . . . . 21.2 Structure of VO2 above (left) and below (right) the structuralphase transition, around 340 K. . . . . . . . . . . . . . . . . . 31.3 Observation of voltage triggered IMT. . . . . . . . . . . . . . 41.4 Band representation above (left) and below (right) the elec-tronic transition temperature. . . . . . . . . . . . . . . . . . . 51.5 Hall carrier density across the IMT. . . . . . . . . . . . . . . . 61.6 Plane representation of the V atoms’ distortion from the rutilestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Diagram of CAFM . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Bulk film characterization. . . . . . . . . . . . . . . . . . . . . 102.3 Nanoscale film characterization. . . . . . . . . . . . . . . . . . 122.4 All measured dielectric constant data for VO2 on various sub-strates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 The raw data with the current spike and exponential decay,in both the ramping up (blue) and down (red) data. . . . . . 152.6 Possible effective circuit diagrams. . . . . . . . . . . . . . . . 152.7 Exponential decay in the IV data when ramping voltage up. . 172.8 Exponential decay in the IV data when ramping voltage down. 182.9 Possible effective circuit diagram and resultant IV curve for astray capacitance in parallel with both the film and externalresistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.10 Possible effective circuit diagram and resultant IV curve fora stray capacitance in parallel with only the film. . . . . . . . 212.11 Possible effective circuit diagram and resultant IV curve fora stray capacitance in parallel only the external resistance. . . 212.12 Possible effective circuit diagram and resultant IV curve fora stray capacitance in parallel with only part of the externalresistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22viiiList of Figures2.13 Final IV simulation, decomposed into signal from each com-ponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.14 Final IV simulation, for voltage ramping down. . . . . . . . . 232.15 A typical IV curve before the transition from the data an-alyzed in this paper, linearized for each of the conductionmechanisms in insulators. . . . . . . . . . . . . . . . . . . . . 272.16 Topography from the first data set . . . . . . . . . . . . . . . 292.17 A typical IV curve from the first data set. . . . . . . . . . . . 302.18 A typical IV curve before the transition from data set A,linearized for each of the conduction mechanisms in insulators. 313.1 Electric field at the transition. . . . . . . . . . . . . . . . . . . 333.2 A representative IV curve with the minimized PF fit. . . . . 343.3 The PF slope at each point on the image. . . . . . . . . . . . 343.4 Relation between the dielectric constant, PF slope and tem-perature of the film. . . . . . . . . . . . . . . . . . . . . . . . 353.5 Temperature of the film immediately preceding the transition. 373.6 A trimodal map of the number of jumps in each IV curve. . . 38A.1 Energy input into the film via the tip. . . . . . . . . . . . . . 56A.2 Approximation of the temperature using a sweeping PF fit. . 58A.3 SiO2 thermal conductivity κ(T ). . . . . . . . . . . . . . . . . 59A.4 Energy leaked through the SiO2 interface as a function of SiO2thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59B.1 The uncertainty associated with fitting a straight line to anIV curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63B.2 A representative curve, with the maximum domain PF slope. 64B.3 A map of the slope found by considering the largest voltagerange at each IV curve. . . . . . . . . . . . . . . . . . . . . . 64B.4 Comparison of fitting the slope with different conditions onthe fitting region. . . . . . . . . . . . . . . . . . . . . . . . . . 65B.5 The uncertainty in the PF slope due to choosing a fitting region. 65B.6 The total uncertainty in the PF slope at each point in theimage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66B.7 The uncertainty in the temperature at each point in the image. 67ixAcknowledgementsI would like to take this opportunity to thank my supervisor Professor Jen-nifer Hoffman, for first taking me on as a Master’s student, and then a PhDcandidate. From the start of my Master’s degree at The University of BritishColumbia, and continuing with my studies at Harvard University, she haschallenged me and helped me improve as a researcher and a scientist.I would also like to thank Dr. Jason Hoffman, for his continued patience,feedback, and improvements on my work.Thank you to Professor Mona Berciu for fitting my thesis into her busyschedule.xDedicationTo all my friends, old and new, who helped me get to this point.And of course to my family, without which I probably would have gottenhere sooner.And of course.. coffee ¬xiChapter 1IntroductionTransition metal oxides host a wide range of properties. Due to their par-tially filled d bands and many valence states, transition metals can formmany different compounds when combined with oxygen. Transition metaloxides can be paramagnetic (Ti2O3, VO2), antiferromagnetic (FeO, NiO),ferrimagnetic (Mn3O4), ferromagnetic (CrO2), or diamagnetic (TiO2, CrO3).They also display various electronic properties such as charge density waves(K0.3MoO3), defect ordering (Ca2Mn2O5, Ca2Fe2O5), insulator-to-metal tran-sition (IMT) (VO2, V2O3, La1-xSrxVO3), and superconductivity (YBa2Cu3O7)[1]. Of particular interest, and focus of this paper, is the IMT observed invanadium dioxide, VO2.In 1959, Morin first reported the 4 order of magnitude change in theconductivity of VO2 at 340 K (Fig. 1.1) [2]. Morin also reports the IMTin the lower oxide states; V2O3 has an IMT at 160 K and VO at 120 K.The IMT temperature of VO2 is the closest to room temperature of theV-O phases, and of the 2 vanadium oxides that show an IMT above roomtemperature it has the largest change in conductivity [3]. The large andsudden change in conductivity close to and above room temperature makesVO2 particularly applicable for a range of sensors and switches, such asbroad-band all-optical switching [4], nano-mechanical resonators [5], windowcoatings [6–8], and electro-optic modulators [9]. The phase transition in VO2is very close to room temperature, thermodynamically stable, and can betriggered by an applied voltage [3, 10], making it well suited for use in nextgeneration Mott transition field effect transistors [11], or a phase changememristor, the missing circuit element [12–14].In 1961, shortly after the discovery of the IMT, Westman observed thatVO2 undergoes a structural phase transition (SPT) at the same temperature,340 K [15]. VO2 transitions from monoclinic (M1) at room temperature (asassumed by Morin), to a rutile (R) structure when heated above 340 K(nominally, in the metallic state). A schematic of the two lattice structuresis shown in Fig. 1.2, with the rutile phase on the left and monoclinic on theright.These two transitions initiated a long debate of whether VO2 is a Mott1Chapter 1. IntroductionFigure 1.1: Morin’s first measurements of the metal-to-insulator transitionof Ti- and V- oxides. Conductivity as a function of reciprocal temperaturefor the lower oxides of titanium and vanadium. Measurements were madealong the [100] direction in VO, and along the c axis in V2O3 and VO2 [2].or a Peierls insulator. A Mott insulator is a material that, by conventionalband theory, would conduct electricity but instead, due to strong on-siteelectron-electron repulsion, it is energetically unfavorable for two electronsto occupy the same site. In these materials, half-filling locks each electronto its site, and resists the flow of electricity, resulting in an insulator. In theinsulating state of a Mott insulator the electrons tend to anti-align, formingan antiferromagnetic phase [17], however VO2 does not have a magneticallyordered ground state. The Peierls distortion is based on a one-dimensionalchain of atoms minimizing energy by forming dimers. In a Peierls insulator,when the dimers from the lattice constant doubles and halves the Brillouinzone, which opens a gap at the Fermi vector. In this case, once again, half-filling results in a full band, and thus an insulator. Due to how the V-V2Chapter 1. IntroductionFigure 1.2: Structure of VO2 above (left) and below (right) the SPT, around340 K. Open circles are O2− ions, dark circles are V4+ ions. Notice thatapproximately 2 rutile unit cells (left) make up one monoclinic unit cell(right). Reproduced from Goodenough [16].atoms pair and the electronic structure of the orbital polarization, VO2 iseffectively a one-dimensional system [18].In Sect. 1.2 we give a more detailed overview of the developments inthe literature with regard to VO2. Many theories and experiments findarguments for the material to be Mott [19], Peierls [20], or some combinationof both [21].Regardless of the driving force behind the IMT, multiple groups havereported a voltage triggered transition to the metallic state, observed as asudden jump in the measured current, shown in Fig. 1.3 [10, 21–26]. How-ever, this voltage induced transition has raised several important questionsthat we address in this work: By which mechanism is the applied voltagetriggering the transition? When high electric fields are applied across a VO2film, what is the local temperature of the film?To answer these question we image the temperature of a VO2 thin filmon the nanoscale under an applied voltage bias. Film growth and characteri-zation details are given in Sect. 2.2. We analyze current-voltage (IV ) curvesmeasured with a conductive atomic force microscope (CAFM) (Sect. 2.1).First, the raw IV curves must be corrected to ensure that we are analyzingthe current and voltage through the film, not any external resistances orcomponents (Sect. 2.3). Using the IV through the film, we conclude thatthe Poole-Frenkel (PF) conduction mechanism is the dominant mechanismin our film (Sect. 2.4).3Chapter 1. IntroductionFigure 1.3: Abrupt IMT induced by a DC voltage in a VO2 thin film. Theinset displays the circuit used for experiments. The left y-axis, IDS is thedrain-source current and the right y-axis, JDS is the corresponding currentdensity. Reproduced from Chae et al. [22].Next, we find the applied bias voltage at the transition and, along withthe local film thickness, calculate the electric field at the transition (Sect.3.1). We fit the IV curves to the PF conduction mechanism (Sect. 3.2),which then enables us to extract the temperature of the film immediatelypreceding the transition (Sect. 3.3).In Chapter 4 we briefly consider nanoscale thermometry. We first reflecton our novel technique (Sect 4.1), then discuss our thermometry techniquein relation to existing methods (Sect. 4.2), and consider the possibility ofextending our technique to other materials (Sect. 4.3).In Chapter 5, we compare the electric field and temperature at the tran-sition. The electric field at the transition is found to be consistent withprevious reports of an electric-field assisted transition [21, 24, 27]. We cal-culate that the local temperature immediately preceding the transition isaround 334 K, and conclude that Joule heating plays an important role intriggering the IMT.41.1. Literature Review1.1 Literature ReviewWith the discovery of the SPT in VO2, many theories arose with the assump-tion that the structural transition caused the electronic transition [28, 29].In 1971, Goodenough described the electronic transition in terms of the V3d band splitting into t2g and eg levels, with the t2g bands arising from the dorbitals parallel with the rutile c axis, d||. Through the structural transition,the pairing of the V-V bonds splits the d|| orbitals, and the antiferroelectriczig-zag-type displacement raises the pi∗ band, opening a gap at the Fermi en-ergy, as shown in Fig. 1.4. Nevertheless, Goodenough claims the electronictransition is predominantly due to the antiferroelectric distortion in the VO6octahedra as opposed to the V-V pairing [16]. Numerical calculations alsosuggested that the insulating state arises from the band splitting causedby the monoclinic distortion [30]. Eventually DFT and LDA calculationsvalidated Goodenough’s predictions [31].Figure 1.4: Band representation above (left) and below (right) the electronictransition temperature [16].In these early studies, defects such as V-vacancies and oxygen non-stoichiometry strongly influenced properties such as conductivity, thermalproperties, optical properties, and activation energy, which led to large vari-ations between samples [28]. Nevertheless, it was concluded that the carriersin VO2 are electrons and have relatively low mobility and density. Mobilitiesof 0.1 - 1 cm2/(V·s) in the insulating state and 1 - 10 cm2/(V·s) in the metal-lic state were found. This relatively small change in mobility cannot accountfor the large change in conductivity, suggesting that metallic state is due toan increase in carrier density [28, 29]. Zylbersztejn and Mott studied thenature of the carriers in VO2 through Cr and Nb doping. They concluded51.1. Literature Reviewthat Goodenough’s band picture is correct for the metallic state, but thatthe insulating state arises from electron correlations, and a local Hubbardmodel [32].Figure 1.5: Carrier density of a VO2 film measured using 1.4 T fixed fieldapparatus. The resistivity of the film is displayed in the inset. Reproducedfrom Ruzmetov et al. [33].Doping and strain have been used to synthesize and investigate newphases of VO2. Marezio stabilized an intermediate lattice distortion, la-belled M2, with Cr-doping. In this intermediate phase, half the V chainspair and move closer, while the other half rotate out of the rutile c axis [34].The two distorted phases, which occur on alternating V chains, are shown inFig. 1.6, with respect to a rutile lattice plane. The same intermediate phasewas also found to be stabilized with Al doping [35], and through uniaxialstress along the [110]R direction [36]. Pouget and Launois also stabilized atransitional phase (T) between M1 and M2, once again through Cr-dopingand uniaxial stress. In the T phase the alternating chains of V-V pairs twistslightly off-axis, and the off-axis V atoms form pairs, although neither chaincompletely enters the M1 phase [37]. The M2 phase showed that the zig-zagdistortion is coupled to the V-V pairing distortion on alternating chains. TheM2 phase is a metastable modification of the M1 phase, whose free energy isonly slightly higher than the M1 phase at room temperature and pressure.Interestingly, the M2 phase transition to the metallic state is consistent witha Mott-Hubbard transition [37, 38]. The discovery of the M2 phase separatedV into two sub-lattices, which was difficult to account for by band theoriesat the time. Up to this point, theories had described the insulating state asa consequence of the monoclinic crystal distortion, but now electron correla-61.1. Literature Reviewtions were deemed essential to the transition [31]. Although the importanceof electron correlations cannot be denied, the question remains whether theCoulomb interaction opens the Peierls gap [18, 39, 40], or if the pairing Vatoms induce a Mott instability [41].Figure 1.6: Plane representation of the V atoms distortion from the rutilestructure [34]. In the M1 phase (open circles) all the V atoms both pair andtwist from the rutile positions. In the M2 phase (filled circles) half of the Vatoms pair but do not twist and the other half form un-paired zig-zag chains.(The distortions from the rutile structure are exaggerated by a factor of 2for clarity.)Using a metal-oxide-metal sandwich structure, Stefanovich triggers thetransition with voltage pulses, and finds that the delay time of the tran-sition is dependent on the magnitude of the voltage pulse [42]. He findsthat a simple thermal model does not accurately describe the delay time,and he concludes that in strong electric fields the transition cannot be de-scribed by an electrothermal model alone. Shortly after, Boriskov surmisedthat the conductivity must be electric field dependent, and concluded thatVO2 must follow the PF conduction mechanism for increasing charge carrierdensity [43, 44]. There are many mechanisms by which electrons can con-duct through an insulator, influenced either by high fields or temperature,71.1. Literature Reviewincluding Schottky emission, Fowler-Nordheim Tunneling, thermionic-fieldemission, direct tunneling, PF conduction, hopping conduction, Ohmic con-duction, space charge-limited conduction and ionic conduction [45]. In Sect.2.4 we will consider all mechanisms and independently conclude that thedominant mechanism is in fact PF conduction.While the microscopic origin of the IMT remains unclear, in recent yearsthe focus has turned to fabricating high quality films, and directly controllingthe transition for potential applications. It has been found that IMT canbe induced by ultra fast laser pulses [5, 46–49] or the application of strongelectric fields or voltage pulses [10, 21–26]. The electronic transition can beaffected by changes in strain, either by varying the substrate [19, 50–54], ordoping the film [35, 55, 56]. Films have been grown on Si(001), c-, r-, andm- plane Al2O3 [57, 58], TiO2(001) and TiO2(110) [59], and Ge(100) [60].It is found that by stretching the c axis the IMT temperature is increased,and by compressing the c axis it is reduced [59].In the last couple of years, studies on VO2 have shifted toward for morespecialized potential applications. The reversibility of the phase transitionsare being investigated for applications in H storage [61], memristors [62],and hybrid metamaterial switching [63]. Films grown by oxide molecularbeam epitaxy can finely control V-O stoichiometry, and is it found thatthe IMT temperature can be lowered through the formation of O-vacancies,but in doing so the resistivity change is also reduced [64]. O-vacancies canalso induced through strong applied electric fields, reducing and eventuallysuppressing the IMT [53]. O-vacancy formation, diffusion, and recovery arestudied for the potential application of ionic-liquid gating on VO2 [65].A delay between the IMT and SPT has been realized, where a metallicmonoclinic phase has been observed [66–69]. Strain has also been used tocompletely suppress the SPT, while preserving the IMT. [19]Many groups have observed the coexistence of the insulating and metallicphases, dating back to 1966 [70], but the phenomenon has been brought tothe forefront more recently [10, 71, 72]. Phase coexistence suggests that theIMT originates in one region, and slowly spreads to the rest of the crystal orfilm. The percolation behavior was first observed indirectly through smallavalanches the IMT in bulk IV measurements [73], followed by the obser-vation of persistent metallic domains below the IMT [74], and finally wasprobed on the nanoscale in local IV maps [10, 72]. The SPT is also foundto percolate through the film [75]. The percolation of the transitions hassparked interest in nanoscale measurements to shed some light on the originof the IMT and SPT.8Chapter 2Data Collection and Correction2.1 Conductive Atomic Force MicroscopyA conductive atomic force microscope (CAFM) uses a probe consisting of acantilever with a tip that comes to a point with a diameter on the order of a30 nm. A diagram of the basic geometry is shown in Fig. 2.1. A laser lightis split, and 90% of the signal is fed into a variable optical attenuator as areference signal. This signal is also used to subtract off any noise from laserfluctuations. The remaining 10% intensity enters a custom single mode fiber,exits the flat end of the fiber and hits the end of the cantilever where it inter-feres with the incoming signal. The interference is measured by a balancedphotodiode detector. The interference measures tip deflection, which is usedto image the three-dimensional shape of the surface and provide z deflectionfeedback to keep the force between the tip and the surface constant. Thetip can also be used to manipulate the surface through strong applied volt-ages. In contact mode, the the current through the film is simultaneouslymeasured by the conductive cantilever. CAFM can measure currents in therange of 2 pA to 1 µA.We use a home-built AFM with a conductive cantilever in contact mode(cantilever spring constant kc = 40 N/m 1). An interferometer of wavelength1550 nm is used. We maintain a constant force of ∼380 nN between the probeand sample, which was calculated by taking the deflection from the tip setpoint, and the cantilever spring constant.2.2 VO2 Film Growth and Characterization2.2.1 Physical Film PropertiesThe VO2 film we study was grown by rf sputtering from a VO2 target (99.5%AJA International Inc.) [76]. A VO2 target is used in place of typical reactiveV sputtering because using a VO2 target preserves the stoichiometry, so there1part number: HQ:NSC16/Cr-Au, purchased from µmasch: VO2 Film Growth and CharacterizationVO2 thin filmConductive CantileverVreference signalphotodiode detectorcurrentapplied voltagelaservariable optical attenuatorinterferenceoptical fiberFigure 2.1: Diagram of a CAFM. The tip rasters the surface of the film andthe z deflection is recorded by laser interferometry measured by a photodiode.When collecting electronic measurements the voltage bias between the filmand tip is a wider range of stable growth parameters [57]. The chamber base pressurewas maintained at 3 ×10−8 Torr. The rf gun power is set to 270 W.A heavily As-doped Si(001) substrate (ρ = 0.002 - 0.005 Ω cm) washeated at 550 ◦C during growth [76]. X-ray diffraction (XRD) data, usinga ScintagXDS2000, with Cu Kα radiation (λ ∼ 1.5418 Å) indicates a poly-crystalline monoclinic VO2 film, with primary surface the (011) orientation(Fig. 2.2 (a)).300 320 340 360 380102103104105  Heating Cooling25 30 35 40 45 500102030 T (Kelvin)2θ (Degrees)CPSR (Ω)(011)(200)(020)(012)Monoclinic VO2(room temperature)(a) (b)VO2n-type SiPdFigure 2.2: (a) XRD profile of the VO2 film studied here. Multiple peaks areobserved, corresponding to a polycrystalline film, but the (011) orientationis most prominent. (b) The resistance-temperature measurement of the VO2film studied. Reproduced from Kim et al. [10].Cross sectional transmission electron microscopy (TEM) measured theaverage thickness of the film to be 187 nm (Fig. 2.3 (a)). Also observed in102.2. VO2 Film Growth and Characterizationthe TEM image is a thin SiOx interfacial layer, measured to be ∼ 2 nm thick.Topography from z deflection of CAFM gives the spatial height variation ofa 500 × 500 nm2 region of the film surface. The topography is shown in Fig.2.3 (c), where individual grains can be seen. The local thickness of the filmis calculated by adding the z deflection measured and the average thickness,giving local film thicknesses in the range of 167 nm to 207 nm. A typicalline cut trace (shown in Fig. 2.3 (b)) demonstrates the surface roughnessand approximate grain size. The root mean square (rms) roughness RRMSis calculated byRRMS =√√√√ 1nn∑1(∆y)2.RRMS of the image is 6.3 nm.We estimate the average grain diameter in two ways. First, using thecross sectional TEM image, we measure the total length of the image tobe 545 nm, and divide by the number of grains (7). This gives a roughestimate of the grain diameter to be 78 nm. The second method uses thefull topography, shown in Fig. 2.3 (c), where the total grains counted, andthe length and width are measured. The average of the length and width istaken to be the diameter of the grain. This gives an average diameter of 67nm, with average deviation of 13 nm. For simplicity we estimate the averagegrain diameter to be ∼ 72 nm.Resistance measurements of the film as a function of temperature givesan IMT temperature of about 339 K upon heating, and around 334 K whilecooling (Fig. 2.2 (b)). A voltage bias was applied between two neighboring500 × 500 µm2 Pd contact pads, 1 mm apart. The resistance through theVO2 film was calculated from the measured current between the pads. Vary-ing the distance between the pads did not change the resistance, verifyingthat the current flowed vertically through the film (as shown in the inset inFig. 2.2 (b)), and that there was negligible resistance from the Si substrateor SiOx interface layer.2.2.2 Dielectric constant of VO2There have been many measurements of the dielectric constant of VO2, εV,with some of these measurements presented in Table 2.2.2 and Fig. 2.4.In general, εV is frequency dependent, and since our IV curves weretaken using a DC measurement we do not consider the high frequency mea-112.2. VO2 Film Growth and Characterization 20 nm(a)(b)20 nm-20 nm100 nmfast slow(c)100 nm4002000Distance across map (nm)-20 nmVO2SiFigure 2.3: (a) Cross sectional TEM image of the VO2 film studied here. Anaverage thickness of 187 nm is measured. (b) A line cut from the topographymeasured by AFM showing the typical height variation across the map. (c)An AFM image of the surface of the VO2 film, where individual grains canbe seen.Table 2.1: Summary of previous measurements of εV. Also plotted Fig. 2.4.Author Film Substrate Temp. (K) εVZylbersztejn [32] single crystal - 298 39Barker [70] single crystal - 299 40.6Yang [25] 100 nm HfO2/n-Si 293 - 373 35 - 66000Hood [77] 160 nm Al2O3(0001) 348 - 371 1000 - 16700Ruzmetov [24] 100 nm Si(001) 293 240Yang [60] 100 nm Ge(100) 293, 333 13.5, 7.9Ko, unpublished 370 nm Al(100) 293 - 328 4.5 - 920Ko, unpublished 370 nm Si(100) 293 - 333 4.5 - 16.2surements reported by Yang [25] and Hood [77]. Ruzmetov’s [24], Yang’s[60], and Ko’s measurements of εV were performed by using the PF slope.For Ko’s measurement the temperature of the film is fixed by depositinglarge Pd pads in thermal equilibrium with the sample. By using the knowntemperature and measuring the PF slope, the dielectric constant can becalculated.Measurements of εV grown on Si are the most similar to our experimentand minimize the difference in the strain of the film. Ruzmetov’s measure-ment on Si was performed by an AFM tip on a 200 nm Au contact. Becauseof the size of the contact, thermal effects could play a role, since smaller de-vices are more susceptible to heat dissipation due to surface-to-volume ratio.Since the PF slope is inversely proportional to T and εV, if the temperature122.2. VO2 Film Growth and CharacterizationεVO2Temperature (K)Figure 2.4: All measured dielectric constant data for VO2 on various sub-strates. Notice the log scale of the y-axis. All citations in Table 2.2.2.of the film was higher than assumed, this would increase the calculated valueof εV. Therefore, it may be possible that during this measurement the filmwarmed, which artificially increased the measurement of εV.We will use Ko’s measurement of εV grown on Si, since it is the mostsimilar to our film, and less susceptible to thermal effects than Ruzmetov’smeasurement.We fit a functional form to the temperature dependent εV data to in-terpolate and extrapolate over the 293 K - 340 K range. An exponentialdependence of the formεV(T ) = a+ b · e(T−273.15)/c (2.1)was found to fit the desired data range the closest, with fit parameters andstandard error ofa = 4.44± 0.12b = 0.0176± 0.0047c = 9.22± 0.37 K. (2.2)Input temperature for Eq. 2.1 should be in Kelvin. Fitting was done incelsius to avoid numerical rounding errors. Fitting in Kelvin resulted inthe coefficient b being very small, with a large relative error in b. When132.3. IV Collection and Correctioncalculating the film temperature the input temperature must be in Kelvinfor consistency with the fundamental constants in the PF slope. Therefore,the fit was performed in celsius, then shifted to Kelvin.This exponential form was found to fit all temperature dependent sets ofεV data before the transition temperature from Fig. 2.4. We are primarilyusing the fit to interpolate and extrapolate data over the range 293 K to 340K, assuming the temperature in the insulating state can’t be outside of thisrange. One might expect a diverging dependence, for example of the form∼ 1/(TC − T ) where TC is the IMT temperature. We use the exponentialform since the rms error is 0.347, where as the rms error for a 3-parameterfit of the form a1 +b1/(TC−T )c1 has a rms error of 2.798, almost an order ofmagnitude larger. Even using a 4-parameter fit of this form, and treating TCas a parameter, results in a rms error of 1.919, and a TC = 334.3 K, whichis lower than the transition temperature of the film.2.3 IV Collection and CorrectionWith the AFM in contact mode, the voltage bias is ramped from 0 to 15 Vand back, measuring the current at 0.05 V intervals at each point on a 255 ×255 grid over the 500 × 500 nm2 image area. A typical IV curve is shown inFig. 2.5, with the voltage ramp up shown in blue, and ramp down shown inred. At each point four consecutive sweeps are performed, and are found tobe consistent with each other, ruling out sudden changes in film quality orcontact resistance. Data from the second sweep is analyzed throughout thiswork. We use the data from ramping to voltage up to calculate the electricfield and temperature.At low voltages the film is in the insulating state, and at high voltagesthe film is in the metallic state. The IMT can be seen as the sudden increasein the measured current (decrease for voltage ramping down).In each raw IV curve there are two artifacts that must be addressed. Thefirst is the presence of a current spike and subsequent exponential decay, bothwhen ramping up and down, immediately following the IMT (Fig. 2.5). Thelarge spike and decay is due to a stray capacitance (C) in the system, whichwe will show does not affect the current through the film in the insulatingstate. The second is the presence of a large resistance in the metallic state,which occurs at voltages greater than the transition. We expect a negligiblefilm resistance, but observe a non-negligible relationship between I and V .This non-negligible resistance does not arise from the VO2 film, but it doesaffect the voltage through the film.142.3. IV Collection and CorrectionVoltage (V)0 5 10 15Current (A)×10-400.511.522.5Figure 2.5: The raw data with the current spike, in both the ramping up(blue) and down (red) data. The linear region at higher voltages is due tothe film being in the metallic state, and is fit to calculate Rext.We model the film as a simple resistor (Rfilm), and the metallic stateresistance as a second external resistance (Rext) in series with the film. Wemodel the data collection by the CAFM as an effective circuit diagram usingthese components to reproduce the observed current spike and decay in theIV data. Fig. 2.6 shows the three representative configurations that are firstconsidered.RfilmRextRfilmRextVinVout = 0 VIout = ImeasRfilmRextVout = 0 VIout = ImeasVout = 0 VIout = ImeasVin VinCCC(a) (b) (c)Figure 2.6: Possible effective circuit diagrams. (a) The stray capacitance is inparallel with both the film and external resistance. (b) The stray capacitanceis in parallel with only the film resistance. (c) The stray capacitance inparallel with only the external resistance.152.3. IV Collection and Correction2.3.1 External ResistanceIn each case shown in Fig. 2.6, because the tip-sample geometry is effectivelya 2-probe IV measurement, we must correct for the external series resistancein the contacts and leads.First, we estimate the metallic state resistance of VO2 to confirm it ismuch less than the measured resistance. The metallic state resistance ofVO2 is calculated usingRmetallic =dσmetal ·A, (2.3)where d is the thickness of the film, σmetal is the metallic state conductivity,and A is the average area of a grain. We use the average area of a grain,assuming the conductivity and thermal conductivity is sufficiently large atthe grain boundary such that the induced current or temperature does notdiffuse into neighboring grains, thus only the grain in contact with the tip isin the metallic state. We can make this assumption based on the observationof percolation of the IMT through separate grains [10]. We use the metallicstate conductivity previously measured to be σmetal = 105 Ω−1 m−1 [29, 33].Assuming a film thickness of 187 nm and grain diameter of 72 nm, thesevalues give an upper bound estimate of the metallic state resistance asRmetallic =dσmetal · pir2 ≈1.87× 10−7 m105 Ω−1 m−1 · 4× 10−15 m2 ≈ 470 Ω. (2.4)This is an upper bound, since we do not consider and current leak intoneighboring grains.We fit the linear region of the IV plot (red line in Fig. 2.7 (a)) post-transition to find the resistance, and calculate an average value of 66 kΩ forthe image. The metallic state resistance of VO2 is therefore much less thanthe external state resistance, and we attribute the measured resistance inthe metallic state entirely to the external resistance Rext. Approximatingthe SiOx layer as a 2 nm thick combination of amorphous SiO2 and Si wecan estimate the resistance due to the interface. However, using d ' 2×10−9m, A ' 4× 10−15 m2 and the limiting behaviors of the resistivity of that ofSiO2 (ρSiO2 = 1015 Ω m) and Si (ρSi = 3.2×103 Ω m) results in an estimatedresistance of the interface in the range 1.6 ×109Ω and 4.9 ×1020 Ω, ordersof magnitude larger resistance than 6.6 ×104 Ω.Since we are interested in the voltage across the film, we must subtractthe voltage drop due to the external resistance. It is calculated byVfilm = Vin − Imeas ·Rext. (2.5)162.3. IV Collection and CorrectionWe use this corrected voltage in our calculation of the electric field andtemperature of the film preceding the transition.2.3.2 Stray CapacitanceIn the voltage ramping up case we observe a large current spike at the transi-tion, which decays exponentially into the metallic state. And in the voltageramping down case we observe a smaller spike that decays into the insu-lating state. This spike indicates a stray capacitance in our system, whichdischarges as the film undergoes the IMT. In both cases we have fit to theexponential decay and calculated a decay constant, τ . Figs. 2.7 and 2.8focus on the exponential decay in the IV curves in both the ramping up anddown data sets. To fit the decay the linear offset is subtracted off in bothcases, and is indicated by the red line in panel (a) in both figures. In thevoltage ramped up case this resistance is identified as the external resistance.In the voltage ramped down case this resistance is identified as an effectivetotal resistance. An effective insulating state film resistance, Rinsulating, iscalculated by subtracting the external resistance from the total effective re-sistance, and has an average value of 100 kΩ. Panel (b) shows the shiftedexponential decay and corresponding fit, where the ramp rate of 500 V/s hasbeen used to convert the x-axis from volts to seconds.Voltage (V)9 10 11 12 13 14 15Current (A)×10-40.511.52Current (A)Current (A)(a) (c) Steps of correcting: a) cor I, b) cPF linearized axes(a) (b)(a)time (s) ×10-30 2 4 6 8I (A)10-810-710-610-510-4fitted curve(b)Figure 2.7: Exponential decay in the IV data when ramping applied voltageup. (a) The capacitance spike in the data set when the applied voltage isramped up. The red line indicates the line that was subtracted from thedata to leave the exponential decay. (b) The exponential decay and fit.We have so far defined three effective resistances: the resistance of thefilm in the insulating state, Rinsulating ≈ 100 kΩ, the resistance of the film in172.3. IV Collection and Correctiontime (s) ×10-30 2 4 6 8I (A)10-910-810-710-610-5fitted curveVoltage (V)0 2 4 6 8Current (A)×10-5-202468(a) (b)Figure 2.8: Exponential decay in the IV data when ramping voltage down.(a) The capacitance spike in the data set when the voltage is ramped down.The red line indicates the line that was subtracted from the data to leavethe exponential decay. (b) The exponential decay and fit. Recall that thevoltage is ramped down, so when converting to time the spike happens first,then the decay, as expected of the usual exponential decay.the metallic state, Rmetallic ≈ 470 Ω, and the external resistance, Rext ≈ 66kΩ. We find a different decay time in the insulating and metallic state. Inthe insulating state the decay time is 6.3× 10−4 s, and in the metallic statethe decay time is 3.0×10−4 s. This is not surprising since the film resistanceis changing between the two states, and τ = RC.The Thevenin equivalent circuit will be used to determine the voltageand resistance that should be used when calculating the decay constant τ .To calculate the Thevenin equivalent circuit, one first identifies the load, inthis case the capacitor, and calculates the open circuit voltage at the loadterminals. The Thevenin resistance is the resistance seen across the termi-nals when the voltage source is shorted.The case where the capacitance is in parallel with both resistances (Fig. 2.6(a)) is the simplest. The Thevenin voltage is the input voltage, and theThevenin resistance is the sum of the resistances in series, Rtot = Rfilm+Rext.Once the load capacitor is reinstated into the circuit, the time constant be-comes τ = RtotC.The cases with the capacitance in parallel with one of the resistors(Fig. 2.6 (b) and (c)) have very similar Thevenin equivalent circuit calcula-tions. The Thevenin voltage when the capacitor is in parallel with the film(external resistance) can be seen as the voltage through the film (external182.3. IV Collection and Correctionresistance). The Thevenin resistance in both cases is the film and externalresistance added in parallel,RTH =Rfilm ·RextRfilm +Rext.The time constant for both of these cases is τ = RTHC.Using the Thevenin equivalent resistances in each case, and the mea-sured decay constant and resistances we can calculate the capacitance inthe insulating and metallic state, shown in Table 2.2. We can estimate thecapacitance due to considering the tip and substrate as parallel plates, witha dielectric film between using C = ε0εiA/d. The dielectric constant of VO2has been measured and that of SiO2 and Si are well studied, with knowndielectric constants of 3.9 and 11.7 respectively. The interface will have adielectric constant somewhere between the values of SiO2 and Si, and we cancalculate the capacitance of the two limiting cases assuming the interface isentirely SiO2 or Si.Table 2.2: Decay constant and resultant capacitance values in the insulatingand metallic state for the different effective circuit diagrams.R = Rtot R = RTHInsulating Metallic Insulating Metallicτ 6.3× 10−4 s 3.0× 10−4 s 6.3× 10−4 s 3.0× 10−4 sR 166 kΩ 66 kΩ 40 kΩ 470 ΩC 3.8 nF 4.5 nF 16 nF 640 nFWe calculate the temperature dependent capacitance from the insulatingVO2 layer in series with the SiOx. In a capacitor with different plate areasthe smaller plate area will give a reasonable estimate of the capacitance ofthe device. In this case, we use the area of the tip, A ' 7 × 10−16 m2,dVO2 ' 1.87 × 10−7 m and dSiOx ' 2 × 10−9 m. The insulating statecapacitance ranges from 1.7 ×10−19 F at room temperature to 1.4 ×10−18F above the transition at 345 K using the temperature-dependent dielectricconstant of VO2 from Sect. 2.2.2 and assuming the interface is entirelySiO2. In the other limiting behavior case, assuming that the interface isentirely Si, an upper bound of 1.5 ×10−18 F is calculated. The estimatedcapacitances arising from the film are orders of magnitude too small to causethe capacitance spike and decay we observe.We use LT Spice2 to simulate each circuit, ramping the voltage at thesame rate as the collected data (500 V/s in 0.05 V intervals), and instan-taneously change the film resistance from its insulating state resistance (∼ IV Collection and Correction100 kΩ) to its metallic state resistance (∼ 470 Ω) when the input voltage is10 V. The measured output current for each scenario is shown in Figs. 2.9 -2.11, with the corresponding effective circuit redrawn in panel (a).RfilmRextVinVout = 0 VIout = ImeasC(a) (b)CurrentFigure 2.9: Possible effective circuit diagram and resultant IV curve for astray capacitance in parallel with both the film and external resistance. (a)Possible effective circuit diagram. In the simulation Rfilm is set to 100 kΩuntil the input voltage is 10 V, then it is instantaneously set to 470 Ω. Rextis set to 66 kΩ. (b) The simulated result of the circuit from (a), with valuesfrom the metallic state from the left side of Table 2.2. The measured currentis the sum of the displayed measurements.Of the three cases, only the configuration with the capacitance in parallelwith the external resistance results in a capacitance spike and subsequentdecay at the IMT. In this situation however, the magnitude of the simu-lated spike is orders of magnitude larger than the measured spike. A fourthconfiguration was then considered, where the external resistance is split be-tween two resistors in series, and the stray capacitance is in parallel with one(Fig. 2.12 (a)). Calling the part of the resistance that is in parallel with thecapacitor Rext1, and the remaining external resistance Rext2, the Theveninequivalent resistance is found by first adding the film resistance and Rext2 inseries, then adding this with Rext1 in parallel. The result is thatRTH =(Rfilm +Rext2) ·Rext1Rfilm +Rext2 +Rext1=(Rfilm +Rext2) ·Rext1RtotPhysically, the resistance Rext1 could describe the contact resistance betweenthe tip and the film surface, or across a thin oxide layer either on the film ortip, which creates a capacitor. This oxide would be very thin, which could202.3. IV Collection and CorrectionRfilmRextVout = 0 VIout = ImeasVinC(a) (b)CurrentFigure 2.10: (a) Possible effective circuit diagram. In the simulation Rfilm isset to 100 kΩ until the input voltage is 10 V, then it is instantaneously setto 470 Ω. Rext is set to 66 kΩ. (b) The simulated result of the circuit from(a), with values from the metallic state from the right side of Table 2.2. Themeasured current is the current through the external resistance.RfilmRextVout = 0 VIout = ImeasVinC(a) (b)CurrentFigure 2.11: (a) Possible effective circuit diagram. In the simulation Rfilm isset to 100 kΩ until the input voltage is 10 V, then it is instantaneously setto 470 Ω. Rext is set to 66 kΩ. (b) The simulated result of the circuit from(a), with values from the metallic state from the right side of Table 2.2. Themeasured current is the current through the film.account for the relatively high values of the stray capacitance calculatedabove. We note that the resistance Rext2 can physically be anywhere in thecircuit, provided it is in series with the film resistance.212.3. IV Collection and CorrectionUsing the measured values of the time constant in the insulating andmetallic state and the film resistance in the insulating and metallic state, wewrite two equations for τ = RC, one for each the insulating and the metallicstate;6.3× 10−4 s = (100 kΩ +Rext2) ·Rext1166 kΩC, (2.6)3.0× 10−4 s = (0.47 kΩ +Rext2) ·Rext166.5 kΩC. (2.7)Assuming the stray capacitance is constant between the two states, alongwith the condition that Rext1 +Rext2 = 66 kΩ, we can uniquely solve for thestray capacitance value and find a value of 19.8 nF. This gives Rext1 = 43kΩ, and Rext2 = 23 kΩ. We simulate this situation and find that this outputmatches our measured data on the correct order of magnitude.a)RfilmRext1Vout = 0 VIout = ImeasVinRext2CCurrent(b)Figure 2.12: (a) Final effective circuit diagram where the capacitance is inparallel with only part of the external resistance. (b)The simulated result ofthe circuit from panel (a). The magnitude of both the spike and the decaytime qualitatively match our measured data.In Fig. 2.13, the current contributions from various elements have beenexplicitly plotted. The current through Rfilm and Rext2 are equal due toKirchhoff’s voltage law, and are equal to the current measured by the AFM.We see that the current contribution from the capacitor before the IMT spikeis negligible compared to the simulated current through the film. We alsosee that the simulated current contribution from the capacitor is an order ofmagnitude smaller (∼ 5µA) than the measured current before the transition(∼ 50µA).222.3. IV Collection and CorrectionCurrentFigure 2.13: Final IV simulation, decomposed into signal from each compo-nent.One final check that is performed to justify this configuration is thesimulation for when the voltage is ramped down. Using the same circuit asshown in Fig. 2.12 (a), but starting Vin at 15 V and ramping to 0 V resultsin a similar order of magnitude spike and decay as we observe in the downcase. The simulation is shown in Fig. 2.14.CurrentFigure 2.14: Final IV simulation, for voltage ramping down.To better describe the current in the insulating state and the IMT, anon-linear approximation of the film resistance could be considered. How-ever, even with this over-simplified approximation of the film resistance, we232.4. Conduction Mechanisms in Insulatorshave shown qualitatively that the equivalent circuit is of the form shown inFig. 2.12. Most importantly, we have shown that this configuration has anegligible effect on the current through the film in the insulating state, sowe do not need to correct for the current spike in our analysis.2.4 Conduction Mechanisms in InsulatorsTo determine which conduction mechanism dominates our VO2 film, we re-view all conduction mechanisms in insulators [45]. Table 2.4 summarizesthese mechanisms, with the current I and voltage V dependence, and typ-ical electric field (E) and temperature ranges where each mechanism is ap-plicable. The table also includes the film properties that must be known ormeasured in order to accurately calculate the temperature.242.4.ConductionMechanismsinInsulatorsTable 2.3: Summary of conduction mechanisms in insulators [45]. Additional sources citied individually.Mechanism Limited I and V linearized formRequiredproperties Typical E Typical TSchottky Emission [78] Electrode ln(I) =e3/2(4piε0εid)1/2kBTV 1/2 + C εi, d E & 100 MV/m T & 400 KFowler-NordheimTunneling [78, 79] Electrodeln(IV 2)=−8pi(2em∗)1/2φ3/2B d3h1V+ C m∗, φB, d E & 100 MV/m,d . 10 nmT . 100 KThermionic-FieldEmission [78] Electrodeln(IV)=~2e224m∗d2(kBT )3V 2 + C m∗, d E & 100 MV/m 100 K - 400 KDirect Tunneling [79, 80] Electrode I = −ae22pihd2V exp(−4pi(2em∗φB)1/2dh)m∗, φB, d E . 100 MV/m,d . 10 nmT . 100 KPoole-FrenkelConduction [25, 81] Bulkln(IV)=e3/2(piε0εid)1/2kBTV 1/2 + C εi, d E & 15 MV/m 100 K - 400 KHoppingConduction [82] Bulk ln(I) =erdkBTV + C r, d E . 50 MV/m . 300 KOhmic Conduction Bulk I =aeµndV exp( −Eg2kBT)Eg E . 10 MV/m 200 K - 500 KSpace Charge-Limited Conduction BulkI =9aµεsθ8d3V 2 µ, d E & 10 MV/m 200 K - 500 KIonic Conduction [83] Bulk ln(I) =eri2dkBTV + C ri, d T dependent* T & 500 K252.4. Conduction Mechanisms in InsulatorsTable 2.4: List of variables in conduction mechanisms.Variable Meaninge electron chargeε0 free space dielectric constantεi dielectric constant of the insulator (dynamic, ie frequency dependent)d film thicknessm∗ effective electron massφB barrier heighta tip contact arear carrier hopping distance, distance between trapsµ electron mobilityn density of states (in the conduction band)Eg (band) energy gapεs static dielectric constant of the film (i.e. low frequency limit)θ ratio of free-carrier density to the total (free plus trapped) carrier densityri ion hopping distanceWe linearize one representative voltage-corrected IV curve before thetransition for each mechanism, and plot each in Fig. 2.15 to determine thedominant mechanism. Before considering these plots, we can eliminate somemechanisms by looking at the electric field or temperature ranges. We knowwe fit our IV data in the insulating regime, so the temperature must bebetween room temperature and the IMT temperature, giving a range of 293K - 340 K. We calculate the electric field at the IMT by finding the voltageat the transition and dividing by film thickness, giving a maximum electricfield magnitude around 30 MV/m (Sect. 3.1).Given our relatively low applied electric field and the average thickness ofour film (187 nm), we can immediately rule out Fowler-Nordheim tunneling,thermionic-field emission and direct tunneling. However, these mechanismsmay describe the motion of the carrier through a surface oxide. In the casewhere the carrier tunnels through the oxide, the expected signal would bemuch less then the measured signal.Next, we look to temperature range to eliminate some possible mecha-nisms. Schottky emission is characterized as the field-assisted thermal ex-citation of electrons from a metal into a dielectric, and as a consequencestrongly temperature dependent. At lower temperatures, stronger fields arerequired to observe the effect. For temperatures . 400 K, very strong fieldsare required, & 100 MV/m, which we do not reach. Since we are interestedin the insulating phase of VO2, we are confident we are not reaching thiscombination of temperature and field we can rule out Schottky emission.Additionally, ionic conduction is in general only observed at high temper-ature, plus the mechanism would destroy the crystal lattice by moving theions, which we do not observe between four sweeps. This leaves PF, hopping,262.4. Conduction Mechanisms in Insulatorsohmic, and space charge-limited conduction as having acceptable tempera-ture and electric field ranges.Ohmic behavior is the usual metallic conduction, which we observe inthe metallic state, so we don’t expect VO2 to follow the ohmic behaviorin the insulating state. Space charge-limited conduction occurs when theinjected carrier density is greater than the thermally excited carrier densityin an Ohmic material, which follows an I ∝ V 2 dependence. Since we donot observe Ohmic conduction in the insulating state, we do not expect toobserve the space charge-limited conduction [84]. Furthermore, from bothof these plots, we see that neither exhibits a linear dependence.V1/21 1.5 2ln(I)-16-14-12-10 (a) Schottky Emission1/V0.2 0.4 0.6 0.8ln(I/V2)-16-15-14(b) Fowler-NordheimV20 10 20 30ln(I/V)-16-14-12(c) Thermionic-FieldV0 2 4 6I(µA)0204060(d) Direct TunnelingV1/21 1.5 2ln(I/V)-16-14-12(e) Poole-FrenkelV0 2 4 6ln(I)-16-14-12-10 (f) Hopping ConductionV0 2 4 6I(µA)0204060(g) Ohmic ConductionV20 10 20 30I(µA)0204060(h) Space-Charge LimitedV0 2 4 6ln(I)-16-14-12-10 (i) Ionic ConductionV1 2 3 4 5 6 i itV1 2 3 4 5 6V1 2 3 4 5 6V65 4 3 2 1V1 2 3 4 5 6i iFigure 2.15: A typical IV curve from the data analyzed in this paper, lin-earized for each of the conduction mechanisms in insulators. Where the scalehas been altered, the true voltage scale is shown on the top axis. Currentis displayed in A, and voltage in V unless otherwise indicated. (a) SchottkyEmission. (b) Fowler-Nordheim Tunneling. (c) Thermionic-Field Emission.(d) Direct Tunneling. (e) Poole-Frenkel Emission. (f) Hopping Conduction.(g) Ohmic Conduction. (h) Space Charge-Limited Conduction. (i) IonicConduction.272.4. Conduction Mechanisms in InsulatorsPF conduction and hopping conduction are the only remaining mecha-nisms that are applicable in our electric field and temperature range. Lookingat the linearized plots (Fig. 2.15), both PF (e) and hopping (f) appear tohave a linear region at the higher end of the applied voltage just before thetransition. The current density due to the hopping conduction mechanismis described byJ = ernCν exp(er|E| − EakBT)(2.8)where e is the electric charge, r is the hopping distance, nC is the concentra-tion of carriers, ν is frequency of thermal vibrations, and Ea is the activationenergy. We fit the hopping mechanism by plotting ln I as a function of V .In a hopping-dominated material the slope of the linear fit would be equalto er/(dkBT ). VO2 has not been demonstrated to follow the hopping mech-anism, and the hopping distance, r, has not been reported. Since both r andT are unknown, it is challenging to accurately calculate the temperature ofthe film. However, we can check for consistency by fitting the IV curve andassuming reasonable values for the hopping distance or temperature. Whenthe hopping distance is fixed at a reasonable length, in the 0.1 - 3 nm range[81], the temperature is well below room temperature which we know is notaccurate. Fixing the temperature in the range 293 K - 340 K results ina hopping distance on the order of tens of nanometers. We conclude thatour film is not dominated by the hopping mechanism because it predictsunreasonable film properties.Taking into consideration the electric field and temperature ranges, andcomparing the linearized plots, we conclude that PF emission is the dominantconduction mechanism in our VO2 film, in agreement with literature [21, 25].PF conduction is when high fields and temperatures excite carriers out oftraps into the conduction band. Poole first described the effects of thermalexcitations on trapped carriers, and in 1938 Frenkel incorporated the effectof high electric fields, describing how high fields lower the effective trapheight, and therefore lower the thermal energy required for trapped carriersto become excited into the conduction band [44].The IV linearized form of the PF mechanism isln(IV)=e3/2(piε0εVd)1/2kBTV 1/2 + C (2.9)We plot ln(I/V ) as a function of V 1/2, and fit to the linear region imme-diately preceding the transition. We define the linear relationship between282.5. A Second Data Setln(I/V ) and V 1/2 as P ,P ≡ e3/2(piε0εVd)1/2kBT. (2.10)2.5 A Second Data SetTwo separate maps of IV curves and topography were collected on the sameVO2 film. The first data set, hereafter referred to as data set A, was overa 1 × 1 µm2 region, splitting the image into a 63 × 63 grid, and collectingIV curves at each point. The second data set, referred to as data set B,was presented earlier in this thesis and is the data used for electric field andtemperature analysis. The IV curves were collected on a 255 × 255 gridover a 500 × 500 nm2 region.Data set A was problematic for a number of reasons. The first was thetopography showed signs of drift during measurement. The tip had not com-pletely settled, and continued to move during the start of the measurement.Although this would not affect the IV measurements, the topography ap-pears stretched along the left and top of the image. The second issue wasthat voltage bias was only ramped up to 12 V, and approximately a third ofthe pixels in the center of the image did not transition, leaving our analysisdifficult. In our analysis we must calculate the voltage across the film. Todo this we use the post-transition resistance to subtract off the external re-sistance. With little to no data points in the metallic state we could not getaccurate fits to correct the data.200 nm25 nm-25 nmFigure 2.16: Topography from the first data set292.5. A Second Data SetA typical IV curve, with sufficient data in the metallic state from dataset A is shown in Fig. 2.17. IV curves from this data set do not showthe capacitance spike at the transition. We also note the much more narrowhysteresis in this IV curve. If we consider the transition to be due to thermaleffects, then because these IV curves were collected at the slower ramp ratethe film has more time to cool while ramping down, and therefore wouldtransition to the insulating state at a higher voltage than observed in dataset B (Fig. 2.5).0 2 4 6 8 10 12×10-400. (V)Current (A)Figure 2.17: A typical IV curve from the first data set.We attempted to perform the same PF analysis on data set A. However,after attempting to linearize data set A in the aforementioned conductionmechanisms it was clear it behaved fundamentally different than data set B(Fig. 2.18).Similar to data set B, all mechanisms except for PF and hopping con-duction can be eliminated based on electric field and temperature ranges.Also similar to data set B we considered the hopping mechanism, but thisfitting produced either an unreasonable hopping distance or an unreasonabletemperature.Unlike data set B however, the PF dependence does not appear linear.All mechanisms with explicit T dependence are inversely proportional to Twhen the IV curves are linearized. So all curves must, at the very least,flatten out indicating a constant or increasing T, which we do not observefor the PF conduction mechanism (Fig. 2.18 (e)). The slightly concavequadratic behavior seen in set A means that as the bias is increased, the302.5. A Second Data SetPF slope increases, which would indicate a decreasing temperature, which iscounter-intuitive.V1/20 1 2ln(I)-16-14-12-10 (a) Schottky Emission1/V0 2 4 6ln(I/ V2)-15-14-13-12-11(b) Fowler-NordheimV20 20 40ln(I/V)-13.5-13-12.5-12(c) Thermionic-FieldV0 2 4 6I (µ A)0 102030 (d) Direct TunnelingV1/20 1 2ln(I/V)-13-12.8-12.6-12.4-12.2 (e) Poole-FrenkelV0 2 4 6ln(I)-16-14-12-10 (f) Hopping ConductionV0 2 4 6I (µ A)0 102030 (g) Ohmic ConductionV20 10 20 30I (µ A)0 102030 (h) Space-Charge LimitedV0 2 4 6ln(I)-16-14-12-10 (i) Ionic ConductionFigure 2.18: A typical IV curve from the first data set, linearized for eachof the conduction mechanisms in insulators. Current is displayed in A, andvoltage in V unless otherwise indicated. (a) Schottky Emission. (b) Fowler-Nordheim Tunneling. (c) Thermionic-Field Emission. (d) Direct Tunneling.(e) Poole-Frenkel Emission. (f) Hopping Conduction. (g) Ohmic Conduc-tion. (h) Space Charge-Limited Conduction. (i) Ionic Conduction.Because of these reasons a second data set was collected, with a smallerimage size, and smaller step size between collecting IV curves. This seconddata set did not have the drift issue, and the voltage was ramped to 15 V,resulting in all pixels transitioning. There was, however, a stray capacitanceduring measurement that produced a large current spike and subsequentdecay at the transition. However, we were able to account for the straycapacitance (Sect. 2.3), and proceed with the temperature calculation.31Chapter 3Analysis3.1 Electric fieldFor each IV curve on the map, we find the voltage at which the currentjumps. The voltage through the film at the transition, and the local filmthickness are used to calculate the applied field across the film at the transi-tion. The local field at the transition for each point on the image is shown inFig. 3.1 (a) with histogram and color bar for the map in (b). The histogramshows a narrow range of electric fields at the transition, which indicates thatby applying a voltage bias we are causing the transition. We find an aver-age applied electric field at the transition of 31 ± 3 MV/m, consistent withprevious reports of an electric-field driven transition [21, 24, 27].We notice that the electric field values for the top two thirds of the imageare smaller than those required at the bottom third of the image. Thisbehavior is also observed in the voltage and the current at the transition,so we are confident it is not due to a changing VO2 film thickness. InAppendix A we consider the energy input by the tip to cause the transition,and conclude that an interfacial layer thickness increase on the order of anm could account for an increase in required energy to warm the film, andan increase in voltage and current at the transition.Comparing with the topography of the film, we can distinguish individualgrains in the electric field map. The electric field at the transition shows littlevariation within each grain, but grain edges require a higher applied electricfield to transition. This could be due to variations within the film, eithera change in V-O stoichiometry or strain, or is the consequence of the tipbeing in contact with multiple grains at once. We can rule out changesin tip-sample resistance since we correct for the external resistance whencorrecting for the voltage through the film.323.2. Poole-Frenkel Fitting2.5 3 3.5 4100 nm(a) (b)25       30       35       40Electric Field (MV/m)NumberFigure 3.1: (a) Map of the electric field at the IMT. (b) Histogram of theelectric field values at the transition. The color corresponds to the color barfor the map in (a).3.2 Poole-Frenkel FittingWhen fitting the PF slope, first the inflection point is found by taking twonumerical derivatives and finding the minimum value. The inflection pointis marked on a PF linearized IV curve with a red asterisk in Fig. 3.2.The inflection point finds the most flat region, so we use the inflection as amiddle point in the fit. We allow the fit start point and endpoint to moveup and down curve, but force the fit region to include at least 10 points oneach side of the inflection point. We vary both endpoints and minimize thefinal PF slope, while keeping the overall linear fit rms error below a pre-defined value. The pre-defined value was found by trial and error. A coupleof representative IV curves were were fit in this manner, using a range ofallowed maximum rms error values. From these trials a best maximum rmserror was chosen.Since the temperature is inversely proportional to the slope, the proce-dure of minimizing the slope will maximize the temperature. In Fig. 3.2 weshow the fit PF slope in yellow, found through allowing a variable numberof data points, and choosing the fit region with the smallest slope.We notice two features about the fit. First, that the particular pixelshown in Fig. 3.2, more than 21 points were included in the fit. In this case,it looks that by including data from higher up the curve a slightly flatter333.2. Poole-Frenkel FittingV1/21.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6ln(I/V)-12.6-12.4-12.2-12-11.8-11.6-11.4PS slope = 1.185928 V-1/2FFigure 3.2: A representative IV curve, with the minimized PF fit. Thevoltage range shown is the largest region considered that could possibly bedominated by the PF mechanism11.522.53Used PF slope (V-1/2)Number1 1.5 2 2.5 30200040006000800010000Used PF slope (V-1/2)(a)(b)Figure 3.3: (a) The PF slope found by allowing both endpoints to vary andminimizing the slope. (b) The corresponding histogram.curve would be fit. Second, the fit does not fit all of the data up to theIMT, which is the end of the data shown. If we were to fit a slope to thedata closest to the IMT, the slope would be larger that the slope indicatedin yellow, and a lower temperature would be calculated. In both cases thisccould be attributed to the fact that the IV curve is slightly ‘S’-shaped,and that the upturn near the transition may be due to a the film not beingdominated by the PF mechanism at the onset of the transition.The method of minimizing the slope is done for each IV curve in the2-D the map, and Fig. 3.3 shows the resultant PF slope at each point, alongwith a histogram of the values.Looking back at the dielectric constant of VO2 we can check that thesePF slopes are reasonable. Ko’s measured dielectric data and error and our343.3. Temperature290 300 310 320 330 3405 101520251.522.533.5Temperature (K)VO2Poole-Frenkel Slope (V-1/2) 2211  5290    300    310    320    330    3403.532.521.5Figure 3.4: Relation between the dielectric constant, PF slope and temper-ature of the film. The raw dielectric constant data for a similar film of VO2grown on n-type Si(100) substrate (black points) with reported errors and aphenomenological exponential fit to the data (black line) are overlaid on thecorresponding PF slope P for the average film thickness d = 187 nm.temperature fit, εV(T ) are shown in Fig. 3.4, along with the correspondingPF slope as a color-scale for reasonable values of T and εV and a constantfilm thickness of 187 nm. From this plot, we expect the PF slope to bebetween about 1 and 3 V −1/2, which we do find across the image (Fig. 3.3).3.3 TemperatureWe fit the PF linearized IV curves, and use the defined relation P =e3/2/[(piε0εVd)1/2 kBT ] (Eq. 2.10). Most of the factors in P are fundamentalconstants, leaving only the film thickness d and relative dielectric constantεV to be measured, before determining T , both of which have already beenmeasured.The interpolated temperature-dependent dielectric constant, εV(T ) isused to self-consistently determine the temperature of each point on themap in the IV curves immediately preceding the IMT. The temperature iscalculated at each point on the map, and the resulting temperatures areshown in Fig. 3.5 (a), along with the corresponding histogram (b). We find353.3. Temperaturean average temperature of 334 K, with a standard deviation of 5 K. We notethe sharp cut off of calculated temperatures at 340 K, the usual bulk IMTtemperature, and very close to the specific IMT temperature of or film.We once again notice that the temperature varies little within grains,but that the calculated temperature seems to abruptly change across somegrain boundaries. Some of the sharp changes in temperatures across grainboundaries correspond to grains that exhibit more than one current jumpin the IV curves. The grains with multiple jumps can be correlated withsome of the cooler grains in the temperature map. In the following sectionwe discuss the number of jumps, and how considering curves with multiplejumps does not affect the overall average temperature of the image.The uncertainty in the temperature extracted from each IV curve isaround 4 K, which is determined by combining the uncertainty from filmthickness d, the dielectric constant εV, and the PF slope, P . The details ofwhich are outlined in Appendix B.The VO2 film was not heated by an external heater, all IV curves werecollected at room temperature, and yet we calculate an average temperatureof the image to be 334 K. This is evidence that Joule heating has occurredvia the voltage-induced current. We note that due to the necessity of fitting aslope, the calculated temperature is a temporal average immediately preced-ing the transition and the real temperature at the transition is presumablylarger, and even closer to the IMT temperature.3.3.1 Number of JumpsWe observe about 20% of IV curves with more than one transition. Thenumber of transitions is generally consistent within a grain. A map of thenumber of transitions is shown in Fig. 3.6 (a). For over half of IV curveswith multiple jumps, the region preceding the largest and final jump is fit.Consequently, in less than 10% of all curves there are not enough data pointsor the data after the first jump is too noisy to be accurately fit, so in thesecases the curve is fit before the first and smaller jump as an approximation.We suggest that in these grains the curve was fit before the first jump,and the calculated temperature was lower than neighboring grains whereeither there were not multiple jumps, or the second and final jump could befit. This might be due to a slightly O-deficient V-O stoichiometry in partor all of the grain. O-vacancies lowers the transition temperature in VO2,causing some sections of the grain to transition before the stoichiometricVO2 regions of the grain, and the rest of the film. In Fig. 3.6 (b) we showthe temperature histogram in dark green and overlay the temperature values363.3. Temperature310 320 330 340310     320      330     340Temperature (K)Number100 nm(a) (b)Figure 3.5: (a) Map of the temperature of the film immediately preceding theIMT as calculated through the PF conduction mechanism. (b) Histogram ofthe temperature map. The color corresponds to the color bar for the map in(a).from curves with two or more transitions light green to show that fitting thecurves with multiple transitions does not alter the overall behavior of thetemperature of the map.373.3. TemperatureNumber of Transitions(a) (b)NumberTemperature (K)310     320     330     340 100 nmNumberFigure 3.6: (a) A trimodal map of the number of jumps in each IV curve.The regions of dark green have one transition, the light green regions havetwo transitions in the IV curves, and the yellow points have more than twotransitions in their IV data. (b) The temperature histogram, with the valuesdue to pixels with more than one jump indicated by the light green color.38Chapter 4Nanoscale Thermometry4.1 Our TechniqueBoth the electric field and temperature maps show little variation withineach grain, but larger values at the grain boundaries. While the grains areevident in the topographic maps, the behavior observed in the electric fieldand temperature maps are not an artifact from using the topography in ourcalculations. Differences between the interior and boundary of the grains areobserved in the raw IV data and in the PF slope map (Fig. 3.3), where thefilm thickness has not yet been taken into account. This is evidence that weare in fact measuring local properties of the film.The temperature of the film must lie in a narrow window between roomtemperature and the measured IMT temperature (i.e., 293 K - 340 K). If thetemperature calculation produced values outside of this range, we would beforced to concede that either the PF mechanism is not applicable or someother aspect of the calculation is incorrect. However, using only the indepen-dently measured dielectric constant and fundamental constants, 98.8 % (64,236 of 65,025 pixels) of the measured temperatures fall within this range.This magnitude of self consistency gives us confidence in our calculations.The pixels that don’t fall into the reasonable temperature range fail todo so because of errors when fitting. In these pixels the IV curves are noisy,so we relax the fit rms error condition, which allows the fitting programto include many more points. In some cases performing a fit over a largerrange still captures the general PF trend of the curve, and so a reasonabletemperature approximation is still extracted. However, in 41 cases the noiseis too large so we disregard the pixel, giving it a ‘not a number’ value. In748 pixels, the fit condition is relaxed, the data is not disregarded, but thenoise is so great that it does affect the fit. If the data is sufficiently noisyapproaching the IMT, as is the case for 734 pixels, the fit curve is flatterthan the general trend of the curve, which results in a larger calculatedtemperature. In 14 curves the fit slope is too steep, either due to fittingtoo close to the jump, or fitting across a small jump in the curve, and theresultant temperature calculated is below 293 K.394.2. Other Nanoscale Thermometry Techniques4.2 Other Nanoscale Thermometry TechniquesHere we discuss other thermometry techniques. Most nanoscale thermometrytechniques require some calibration. In some methods this involves a prelim-inary measurement to calibrate how the material will respond to a change intemperature. For example, a couple groups have reported using the a Pt-Au[85] or Au-Cr [86] thermocouple integrated into a tip, which requires themeasurement of the Seebeck coefficient of the tip. Additionally, fluorescentparticles made from Er/Yb are doped onto a film, and their excited statesare temperature dependent [26, 87]. Another method uses a temperature de-pendent shift in the Raman peaks, but the shift must first be calibrated [88].Alternatively, if the thermal expansion of a material is known, a scanningtransmission electron microscope can be used measure the thermal expan-sion of Al wires through plasmon electron energy loss measurements [89]. Afew of methods use a CCD camera to capture near-IR radiation based onPlanck’s law [90] or near-IR reflectivity [91, 92]. Once again, the measuredsignal must be correlated to the temperature of the film before the measure-ment. These techniques that use near-IR or a CCD camera are also spatiallylimited by the detection wavelength, and can only probe length scales onthe order of 400 nm. Even though these techniques have their advantages,to measure an accurate temperature a calibration must be performed at oneknown temperature. In our case, if we suspect our film to be susceptibleto nanoscale Joule heating, then we cannot be sure the initial calibrationaccurate.Other techniques that don’t require a calibration measure the temper-ature variation with respect to some reference temperature. By combiningthe thermal resistive scanning probe measurements, and tip-sample heat fluxmeasurements one can calculate the temperature field [93]. Fluorescent NVcenters have temperature dependent crystal field splitting [94, 95]. The cur-rent measured by a superconducting junction has strong a temperature de-pendence but only in the vicinity of the critical temperature of the junction[96]. Measuring temperature fluctuations on the nanoscale is useful, how-ever the range of temperatures probed is limited and the initial referencetemperature must be known.4.3 Extension of Thermometry TechniqueWe have demonstrated the use of IV curves fit to the temperature-dependentPF conduction mechanism to calculate the local temperature of a thin VO2404.3. Extension of Thermometry Techniquefilm immediately preceding the IMT. The technique described here is the firstof its kind, and is unique in that it can measure the temperature of a film onthe nanoscale without the use of a reference temperature or calibration. Thethickness and temperature-dependent dielectric constant of the film must beknown or measurable by another method, but this leaves no free parametersto vary and results in the calculation of the temperature of the film.This novel thermometry technique has addressed the long debate sur-rounding the field-induced IMT in VO2. The technique can readily be ex-tended to other materials dominated by the PF mechanism. For exampleZnO has been considered for application in resistance random access mem-ory devices [97], for thin film transistors in displays, and for electron transferlayers in solar cells [98, 99]. Other materials that follow PF conduction, andalso have potential applications in resistive switching devices include SnOx,AlOx, CeOx, and WOx [100].Furthermore, the more general technique of using IV data to calculatethe local temperature of the film can be extended to other temperaturedependent conduction mechanisms. Section 2.4 discusses other conductionmechanisms in detail, but here we note that Schottky field emission, Fowler-Nordheim tunneling, thermionic-field emission, hopping conduction, Ohmicconduction, and ionic conduction can all be linearized in terms of current andvoltage which will produce a slope that is dependent on the film temperature.Depending on the mechanism, various film properties must either be knownor measured in order to accurately calculate the temperature of the film.As an example, which is discussed in Sect. 2.4, to reliably use the hoppingmechanism to calculate the temperature of the film the hopping distancemust either be known, measured, or assumed.41Chapter 5ConclusionWe use a CAFM to ramp a voltage bias across a VO2 thin film, and observethe IMT in the current. Using the voltage at which the film transitionsand the local film thickness, we calculate an average electric field of 31 ± 3MV/m. 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These pixels also display a larger current at the transition.Approximately double the energy is input into the film in the bottomthird of the image than at the top two thirds, but the calculated tempera-ture did not vary on the same order of magnitude. We do notice that thecalculated temperature follows the same pattern, but the difference betweenthe top and bottom of the image is only on the order of 5 K, or less thana 2 % change over the image. We hypothesize that a slightly varying SiO2interface would increase the amount of energy leaked from the VO2 film tothe substrate and would explain how a higher voltage and current is requiredto cause the transition in the lower portion of the image. Here, we calculatethe input energy from the tip, the energy required to cause a transition inVO2, and the energy leaked into a thin SiO2 interface to determine if smallchange of 1 nm in the SiO2 interface could result in the energy input changeobserved.A.1 Input energy from tipThe first step is to calculate the energy put into the film to cause the tran-sition. We are interested in the energy input into the film, so the correctedvoltage through the film is used for the calculation, not the applied voltage.The data points are sufficiently close to calculate the input power as the sumof rectangles of width ∆V , and height I. The power input isP =∑iI(i) ·∆Vfilm(i) =∑iI(i) · [Vfilm(i+ 1)− Vfilm(i)] ≈ 7.66× 10−5 W(A.1)for a representative pixel in the top half of the film. To calculate the energywe then multiply by the time taken for the film to transition. The total timeis t ≈ 0.0175 s, calculated by multiplying the number of steps (∼ 175) andthe step time, ∆t = 10−4 s.55A.2. Heat required to cause IMT in VO2Row down the map0 50 100 150 200 250Average energy input×10-60.511.522.5Energy input (J)(a) (b)Figure A.1: (a) Map showing the energy input to the film through the tip.(b) The average energy per row input to the film through the tip.Using this method, the average energy input in of the first two thirds ofthe map is 1.05×10−6 J, and the average for the bottom third is 1.99×10−6J. There’s an average of about 1×10−6 J of extra energy needed to cause thetransition at the bottom of the image. A map of the calculated input energyis shown in figure A.1, with the average input per row shown in panel (b).In panel (b) you can see the distinct change from ∼ 1×10−6 J to ∼ 2×10−6J about two thirds of the way down the map.A.2 Heat required to cause IMT in VO2The specific heat capacity of VO2 near room temperature is C = 690 J/(kgK)[20]. Assuming the heat capacity is constant up to the transition, thetotal energy is∆E = mgrainC∆T = ρV O2VgrainC∆T.Using a grain diameter of 72 nm, a film thickness of 187 nm, the densityof VO2 in the insulating state of 4571 kg/m2, and a ∆T of 45 K, gives∆E ∼ 1× 10−13 J. This value seems unreasonably low for amount of energyrequired to cause the transition.A.3 Thermal Conductivity of SiO2The thermal conductivity, κ, is defined as the constant of proportionalitybetween the power dQ/dt, per contact area A, and the thermal gradient∆T/∆x,dQdt1A= κ∆T∆x. (A.2)56A.3. Thermal Conductivity of SiO2To calculate the energy lost through the SiO2 interface, we must approx-imate the instantaneous temperature of the film, and the time spent at eachtemperature. Using a 5-point approximation of the PF slope, then usingthe slope to estimate the temperature, we calculate the temperature at eachvoltage. Figure A.2 shows the estimated film temperature baed on the esti-mated PF slope and film thickness 187 nm for each voltage through the film.We assume that the film starts at room temperature, and any result below293 K is a consequence of applying the PF fit to a region where it was notthe dominant mechanism. We notice that the film warms quickly at first,and spends most of the ramp time at 335 K. This allows us to approximatethe heat lost through the SiO2 interface as∆Q =κ ·Adi∆T∆t, (A.3)where di is the interface thickness, ∆T is the change in temperature, and ∆tis the time the thermal gradient is maintained. The contact area is estimatedassuming an average grain diameter of 72 nm. From the cross sectional TEMimage we estimate the interface to be about 2 nm ± 1 nm thick. The averagetemperature different is about 40 K, since the film temperature increasesquickly from 293 K to 335 K. To estimate the time the thermal gradient ismaintained, we notice the transition occurs roughly at 10 V, which occursat 0.02 s, and from figure A.2, we estimate the film only starts to warmaround 2 V, which occurs at 4 ms. The ramp rate is 0.05 V / 0.1 ms, whichgives a total ∆t = 0.016 s. We remark that a PF starting voltage of ∼ 2V is also consistent with the approximate electric field range where the PFmechanism is applicable for a 187 nm film.Lee has measured the thermal conductivity of SiO2 films ranging from32 nm to 190 nm thick, in a temperature range of 80 K - 300 K [101], witha comparison to a 15 nm SiO2 film from Swartz [102]. Costescu reports thethermal conductivity of thinner SiO2 films, ranging from 6.5 nm to 25 nmin the same temperature range from 80 K at 300 K (Fig. A.3) [103]. Swartzlooked at a range of thickness as well, but the general trend as a functionof thickness was opposite that of Lee and Costescu, so we do not considerSwartz’s data. Using the general trend in the bulk values, and the overlapfor thickness of the SiO2 of the two measurements we estimate a curve foran SiO2 film about 2 nm thick, marking in both plots with a dashed line infigure A.3. We read off the thermal conductivity of a 2 nm thick SiO2 filmat 335 K to estimate the heat leaked through the silicon interface.Using both extrapolated values as a minimum and maximum estimatewe plot the energy leaked as a function of SiO2 thickness for the two values57A.3. Thermal Conductivity of SiO2V1/20.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4ln(I/A)-17-16-15-14-13-12-11Temperature (K)50100150200250300350V1 2 3 4 5Figure A.2: Approximation of the temperature (red) using a sweeping PFslope on the IV data (blue). The true voltage scale is shown on the top axis.of thermal conductivity at 335 K. The thermal energy leak as a function ofSiO2 thickness is shown in Fig. A.4, where blue corresponds to a minimumthermal conductivity taken from Lee’s data of about 0.65 W/m K, and redcorresponds to a maximum value from Costescu’s data of about 1.0 W/ mK.58A.3. Thermal Conductivity of SiO2Figure A.3: SiO2 thermal conductivity κ(T ). (a) Lee [101]; (b) Costescu[103]. In both cases the dashed line is the extrapolation of the thermalconductivity as a function of temperature down to a 2 nm film. the dottedline are guides to reading off the thermal conductivity at 300 K and 340 K.SiO2interface thickness (nm)1 1.5 2 2.5 3 3.5EnergyleakthroughSiO2(J) ×10-60.511.522.53Figure A.4: Energy leaked through the SiO2 interface as a function of SiO2thickness for two separate thermal conductivity values (blue: 0.65 W/m·K,and red 1.0 W/m·K).Approximately 1×10−6 J are input at the top of the film, but 2×10−6J in the bottom third. Assuming a 2 nm thick SiO2 interfacial layer, theenergy loss could range anywhere from 1× 10−6 to 1.5× 10−6 J.59A.3. Thermal Conductivity of SiO2Looking instead at possible interface thicknesses resulting in a particularenergy loss, we notice that an energy around 1×10−6 J could be expected foran interface between 2 nm and 3 nm. These thicknesses are consistent withwhat we see in the TEM cross sectional image. For an energy loss around2× 10−6 J we expect in interface thickness between 1 nm and 1.5 nm, whichis also consistent with our observed thickness, and very plausible change ininterface thickness. We see that a small change in interface thickness (. 1nm) can change the thermal energy leak on the order of 10−6 J, for eithervalue of thermal conductivity. We conclude that if the SiO2 layer is 1 nmthinner on the bottom third of the map than the top, then it will leak morethermal energy, which would in tern require more energy input to the filmbefore the VO2 transitions.60Appendix BUncertainty in the calculatedTemperatureTo calculate the uncertainty of the temperature T , we use uncertainties as-sociated with the film thickness, the PF fit, and the dielectric constant, andpropagate uncertainty. Since the dielectric constant is a function of tem-perature we must propagate the uncertainty for an implicit expression. Wecollect the varying parameters and set it equal to the fundamental constants.We can express εV in terms of its parameters, a, b, and c, and writeT · P · d1/2 · (a+ b · e(T−273.15)/c)1/2 = e3/2(piε0)1/2kB. (B.1)For simplicity, combine the fundamental constants into one constant,K ≡ e3/2(piε0)1/2kB. (B.2)Write the parameters as a function, f ,f(T, P, d, a, b, c) = T · P · d1/2 · (a+ b · e(T−273.15)/c)1/2 = K (B.3)where five of the variables (a, b, c, d, P ) have known uncertainties, and wewant to calculate the uncertainty of the sixth, T .To confirm implicit uncertainty propagation, consider the simpler exam-ple of z(x, y) = k/xy, where x and y are variables and k is a constant. Fromthe usual propagation of uncertainty, we know thatδz2 =(∂z∂x)2δx2 +(∂z∂y)2δy2. (B.4)δz2 =(−kx2y)2δx2 +(−kxy2)2δy2 = z2(δx2x2+δy2y2). (B.5)61B.1. Dielectric contant, εV(T )If instead we represent this as zxy = k, we can still use Eq. B.4 even if thereis no explicit function for z, by defining the function g = g(x, y, z) = zxy,and using the fact that ∣∣∣∣∂z∂x∣∣∣∣ = ∣∣∣∣∂g∂x∣∣∣∣ / ∣∣∣∣∂g∂z∣∣∣∣ . (B.6)Equation B.4 then becomesδz2 =(∂g∂x/∂g∂z)2δx2 +(∂g∂y/∂g∂z)2δy2, (B.7)and taking the partial derivatives gives,δz2 =(zyxy)2δx2 +(zxxy)2δy2 = z2(δx2x2+δy2y2). (B.8)We produce the same expression for the uncertainty of z. We now apply thismethod to our temperature expression given by Eq. B.3 to calculate δT ,δT 2 =∑α(∂f∂α/∂f∂T)2δα2, α = a, b, c, d, P. (B.9)We can now calculate the temperature uncertainty at each pixel, but we firstquantify δa, δb, δc, δd, and δP .B.1 Dielectric contant, εV(T )The uncertainty of the dielectric constant is calculated using the standarderror values associated with each fit parameter. To check these are reason-able uncertainties we self-consistently use the temperature uncertainty andcalculate the associated uncertainty of εV. The uncertainty of εV is given byδε2V = δa2 + (e(T−273.15)/c)2δb2+(bc2(T − 273.15)e(T−273.15)/c)2δc2 +(bc· e(T−273.15)/c)2δT 2. (B.10)If we assume the the standard error stated above (Eq. 2.2), and the finaltemperature uncertainty (around 4 K), the average uncertainty of the di-electric constant is calculated to be about 6.5. Ko has reported errors ofapproximately 1 for each of the data points. The uncertainty of εV is calcu-lated this way is higher than the reported error, and on the same order ofmagnitude as some of the lower values.62B.2. Thickness, d00. in linear fit (V-1/2) 0 0.1 0.2 0.3 0.4 0.5x 10400.511.522.53NumberUncertainty in linear fit (V-1/2)(a)(b)Figure B.1: (a) The uncertainty at each point in the map from performinga linear fit on the IV data. (b) The corresponding histogram.B.2 Thickness, dThe largest source of uncertainty in the thickness stems from measuring theaverage overall thickness. The uncertainty in the average thickness comesfrom the possibility that the TEM image and IV curves may be taken ondifferent regions of the sample with different average thickness. We assumea constant uncertainty of δd = 5 nm.B.3 PF slope, PWhen calculating the PF slopes we concurrently calculate the error associ-ated with fitting a straight line to the data. A map of the uncertainty dueto the linear fit is shown in Fig. B.1, with corresponding histogram.To estimate the uncertainty from varying the PF fitting region, we find arepresentative slope that gives a reasonable estimate of the PF slope had nominimizing been done. Since the data is slightly ‘S’-shaped, we know the PFmechanism does not dominate for all applied voltages. During the fitting wedefine a minimum voltage the start point can be, defined by the approximateminimum electric field for PF dominated conduction. From Sect. 2.4, PFeffect has been observed down to ∼ 15 MV/m. For an average film thicknessof 187 nm, the minimum applied voltage we expect the PF conduction todominate is ∼ 2.8 V. The maximum value of the endpoint of the fit regionis constrained by the position of the jump. The slope obtained by fixing theendpoints to their maximum values is defined as the maximum domain PF63B.3. PF slope, PV1/21.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6ln(I/V)-12.6-12.4-12.2-12-11.8-11.6-11.4PS slope = 1.345213 V-1/2Figure B.2: A representative curve, with the maximum domain PF slope.The voltage range shown is the largest region considered that could possiblybe dominated by the PF mechanism1 1.5 2 2.5 3020004000600080001000011.522.53 Maximum domain PF slope (V-1/2)Number(a)(b)Maximum domain PF slope (V-1/2)Figure B.3: (a) A map of the PF slope found by taking the maximum regionconsidered to be PF (b) The corresponding histogram.slope, and a representative IV curve is shown in Fig. B.2. Fig. B.3 shows themaximum domain PF slope at each point on the map along with histogram inpanel (b). We compare the minimized PF slopes and the maximum domainPF slopes by comparing histograms (Fig. B.4). As expected the histogramof minimized PF slope is slightly narrower and translated to lower values inrelation to the maximum domain PF slope.The uncertainty due to allowing both endpoints to vary is the differencebetween the minimized PF slope and the maximum domain PF slope. Thisuncertainty is shown in Fig. B.5, with histogram in panel (b).The total uncertainty in the used PF slope at each point on the map iscalculated by combining in quadrature the uncertainty from fitting a linearline and the uncertainty from allowing the endpoints to vary. Fig. B.6 shows64B.3. PF slope, PComparison of PF slopes (V-1/2)1 1.5 2 2.5 30200040006000800010000Maximum domain PF slope (V-1/2)1 1.5 2 2.5 30200040006000800010000Used PF slope (V-1/2)1 1.5 2 2.5 30200040006000800010000NumberNumberNumber(a) (b) (c)Figure B.4: (a) The histogram of the PF slope values used to calculatethe temperature of the film. (b) The maximum domain PF slope values.(c) Superimposed histograms of minimizing the PF slope (blue) and themaximum domain slope (yellow outline).0 0.5 1 1.50100020003000400050006000700000. in varying PF region (V-1/2)NumberUncertainty in varying PF region (V-1/2)(a)(b)Figure B.5: (a) Map showing uncertainty due to a varying fit region. Thisuncertainty is the difference between the minimized PF slope at each point,and the PF slope by fitting the maximum domain. (b) Histogram of thedifference between the minimized PF slope and the maximum domain PFslope.65B.4. Temperature, T0 0.5 1 1.50100020003000400050006000700000. PF Uncertainty (V-1/2)NumberTotal PF Uncertainty (V-1/2)(a) (b)Figure B.6: (a) Map showing the total PF uncertainty due to the uncer-tainty in the linear fit added to the uncertainty due to the endpoints of thefit. (b) Histogram of the map of the total PF slope uncertainty.the total uncertainty at each point and the corresponding histogram.B.4 Temperature, TWe present a map of the uncertainty of the measured temperature for eachpixel (Fig. B.7) using the equation for the temperature uncertainty (Eq. B.9),along with the uncertainty in the dielectric constant fit (Eq. B.10), the un-certainty in the film thickness (5 nm), and the uncertainty in fitting thePF slope (Fig. B.6). As shown in the histogram, most of the temperatureuncertainty values are between 3 - 5 K, with a long tail for values greaterthan 5 K. The average uncertainty is 5.12 K, with a median of 4.14 K. Bycomparing to the PF uncertainty map (Fig. B.6) we note that the regionswith the largest PF uncertainty also have the largest temperature uncer-tainty. For the majority of the pixels in the map, the uncertainty of eachindividual temperature measurement is less than ∼ 4.2 K. This temperatureuncertainty is less than the overall standard deviation of the temperaturemap (5 K), and corroborates our calculated temperature values.66B.4. Temperature, TNumber345678910Total Temperature Uncertainty (K) Total Temperature Uncertainty (K)4 6 8 100100020003000400050006000(a)(b)Figure B.7: (a) Map showing the uncertainty of T at each pixel. (b)Histogram of the δT map.67


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