This Experiment! - Quantum Design Education
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1. Heat Capacity Measurement of Vanadium Oxide Powder Heat Capacity Option Prof Richard Averitt UC San Diego Description The objective of this educational module is to measure the heat capacity of vanadium dioxide VO2 a material that exhibits a structural phase transition at 340K that coincides with a transition from a low temperature monoclinic insulating phase to a high temperature rutile metallic phase 1 VO2 is a well known correlated electron material that is being actively investigated to elucidate the relative importance of structural changes versus electronic correlations as the fundamental cause of the transition The insulator to metal transition in VO2 is first order resulting in a large latent heat which can be measured using the heat capacity option of the VersaLab In the following module we will use VO2 powder a precursor material for growing thin films that is composed of micro crystallites for the heat capacity measurements highlighting the sensitivity of this measurement technique The procedures described in this module are generally applicable to other samples Before proceeding to the instructions for performing this measurement we discuss some the general background of heat capacity measurements from theoretical and practical points of view The heat capacity J K characterizes the increase in the internal energy of a system for a given temperature increase Considering the first law of thermodynamics we have2. Chapter 4 of the heat capacity user Heater Power 0 Time Temperature Time 0 to Figure 2 heat capacity measurement approach manual is used the fit the time dependence of the temperature change which can be used to determine the heat capacity For accurate measurements the heat capacity hardware must be designed to have a low thermal mass and appropriate thermal conductance and thermal isolation Fig 3a gives a schematic depiction of the hardware while Fig 3b is a picture of the heat capacity puck used for the VersaLab Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 4 Grease a Thermal Bath T t b ee er a ran E Pa Fi Pi P ff 2 Pi P F aT EEFE a ff ff rd F x VU 5 Fi a fi ra ra ft ra Heater Thermometer Figure 3 a Schematic of heat capacity hardware bo Heat capacity puck As shown in Fig 3 a platform heater and platform thermometer are attached to the bottom side of the sample platform Small wires provide the electrical connection to the platform heater and platform thermometer and also provide the thermal connection and structural support for the platform The sample is mounted to the platform by using a thin layer of grease which provides the required thermal contact to the platform The integrated vacuum system in the cryostat provides a sufficient vacuum so that the thermal conductance between the sample platform and the thermal ba
3. This provides reasonable agreement with the room temperature measurement of many solids e g Aluminum 24 2 Fe 25 1 However strong deviations of materials such as diamond 6 1 and more importantly the temperature dependence of the heat capacity necessitated the use of quantum mechanics to obtain a more complete understanding In Einstein s theory the solid is treated as a harmonic oscillator with a single characteristic vibrational frequency w 4 This approach was able fo provide a basic understanding of the decrease in the heat capacity with temperature The data for diamond with Einstein s fit is shown in Fig 1 In this plot the units are cal mol K which can be converted to J mol K by multiplying by 4 184 J cal For diamond the Einstein temperature is given by Te ha kp 1320K indicating the approximate temperature at which the heat capacity reaches the classical Dulong Petit value The deviations at low temperature between experiment and theory in Fig 1 is real and a better fit is obtained with the Debye model which essentially quantizes sound waves Amongst other things the Debye model correctly predicts the TS dependence of the heat capacity 3 ae bbe 4 EERE 7 af 0 QF O 0 2 kpT hus Figure 1 Comparison of Einstein model dashed line to experiment circles for the molar specific heat of diamond from 4 Importantly other degrees of freedom in solids also have a heat capacity For example in metals
4. mass 5 6 Heavy Fermions are but one fairly exotic example More generally heat capacity measurements are sensitive to phase transitions This includes magnetic ordering structural transitions ferroelectric polarization and superconductivity This applies to both first order and second order phase transitions In the case of d first order transition a discontinuity appears in the entropy which in turn leads to a divergence in the specific heat since cy T dS dT This singularity is the latent heat L J kg and is the increase in the internal energy needed to drive the phase transition 7 In this module the goal is fo measure L for VO2 powder In the case of a second order or continuous phase transition a kink appears in the entropy S leading to a discontinuity in the specific heat The Importance of this can be understood from considering thermodynamic potentials For example for the Helmholtz free energy we have F U TS From this we can see that the entropy is the driving force for a phase transition At low temperature below the phase transition temperature Te the entropy is not too important and F can be minimized by having U minimized This leads to ordering e g of spins in a magnet However with increasing temperature the entropy becomes increasingly important and minimizing F benefits from increased S corresponding to increasing disorder In fact the total entropy associated with the ordering can be determin
5. the specific heat exhibits a linear dependence on temperature c M arising from the free electrons yis the Sommerfeld coefficient This was first correctly obtained in free electron or Sommerfeld Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 2 theory using Fermi Dirac statistics for the electrons 3 Of Course c 7 is just for the electrons and the lattice must also be included At low temperatures c x Wf PE where the first term is for the electrons and the second term the lattice At high temperatures the lattice will dominate but the electron contribution becomes important in metals at low temperatures In contemporary condensed matter physics the fact that heat capacity measurements reveal interactions between various degrees of freedom is extremely important As one example we consider heavy Fermion HF materials As the name implies in these materials below a cross over temperature the electrons really quasiparticles i e dressed electrons become extremely heavy In some HF materials the quasiparticles exhibit an effective mass approaching 1000 times the mass of a free electron This arises from interactions of the conduction electrons with localized f moments in these materials Importantly the onset of HF phenomena appears in heat capacity measurements This is because the heat capacity is proportional to the density of states at the Fermi level which in turn is related to the effective
6. 3 34 1959 see also C N Berglund H J Guggenheim Phys Rev 185 1022 1969 N Mott Metal Insulator Transitions Taylor and Francis London 1977 D V Schroeder An Introduction to thermal physics Addison Wesley New York 2000 Steven H Simon The Oxford Solid State Basics Oxford University Press Oxford 2013 A Einstein Ann Phys 22 180 1907 see also any solid state physics book such as Simon s book in reference three htto en wikipedia org wiki Heavy_fermion P Coleman Heavy Fermions Electrons at the edge of magnetism http arxiv org abs cond mat 0612006v3 David L Sidebottom Fundamentals of Condensed Matter and Crystalline Physics Cambridge University Press Cambridge 2012 Chapter 15 D Knhomskii Basics Aspects of the Quantum Theory of Solids Order and Elementary Excitations Cambridge University Press Cambridge 2010 P M Chaikin and T C Lubensky Principles of Condensed Matter Physics Cambridge University Press Cambridge 1995 Instructions In this section we provide guidance on preparing and measuring the heat capacity of VO2 powder Our focus will be on observing the first order phase transition occurring at 340K As such we will utilize the slope analysis of relaxation curves to obtain the data Several items are needed for this experiment which includes VO powder available from Alfa Aesar at http www alfa com en catalog 22957 Apiezon H grease Importantly the specif
7. ast squares to obtain a best fit for the heat capacity Performing these thermal time constant measurements at a series of temperatures allows for the determination of the heat capacity as a function of temperature It is important to note that Crota is the total heat capacity of the sample platform the grease and the sample of interest Thus several measurements are actually required to obtain Cp of the sample First the puck must be calibrated That is a measurement must be performed without the grease or the sample This procedure needs to be performed for each new puck to determine the heat capacity of the sample platform and Kw The data for this calibration is saved in a cal file for reference with the subsequent measurements For each new sample to be measured an addenda must first be obtained This is essentially a measurement of the heat capacity of the grease and the sample platform without the sample This is also saved in the calibration file Finally the sample grease sample platform heat capacity is measured From this series of three measurements it is possible to obtain Cp of the sample of interest While this approach for measuring the heat capacity enables measurements over a wide temperature range it could easily miss features in the specific heat associated with first or second order phase transitions if for example the selected number of temperatures is too sparse This is because the heat capacity associated with p
8. ation with f an offset calculated from heating cooling overlap C astatic offset of conductance at base fo W Use heating curve W Use cooling curve Use only overlap View Checked Traces Analyze Checked Traces Save to File Save to File Figure 14 Screenshot of control panel and post processing window Select the single slope curve analysis method appropriate for a first order phase transition You can then view the checked traces of which there are three as performed from our sequence which will bring up a window of the raw data as in Figure 15 From this data you can see the kinks that are expected at a first order transition If desired this data can be saved to a Tile Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 14 __ Multi uRawPreview dat fe xj Time minutes relative Figure 15 Raw data of first order phase transition in VO2 t Alternatively one can select to analyze the checked traces The data will be analyzed and plotted as in Figure 16 The plot shows the experimentally obtained heat capacity in uJ K The latent heat peaks are clearly evident as is the expected hysteresis This data can be saved to a file for subsequent analysis and processing Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 15 _ Multi uHC2Preview dat 5 x Figure 16 Heat capacity data for VO2 powder Questions 1 Anal
9. dU dQ dW where dU is the change in energy when either work dW is performed or heat dQ is added to the system This expression can also be written as QU TdS PdV where the notation follows standard conventions The heat capacity is defined as C dQ dT which for constant volume is Cy dU dT T dS dT 2 3 The heat capacity is extensive i e it depends on the quantity of material It is more useful to express it as an intensive quantity Two common intensive descriptions are the mass heat capacity often called the mass specific heat with units J kg K and the molar heat capacity often called the molar specific heat with units of J mol K The same symbol cy is typically used for either specific heat quantity so care must be taken to specify Quantum Design Educational Module Heat Capacity Measurement of VOz v 1 1 the units For solid state measurements it is Usually the specific heat at constant pressure Cp that is measured The connection between cy and Cp IS Cp Cv 9aeVT K Where a 1 K is the linear coefficient of expansion x ms2 kg is the compressibility and V m kg is the specific volume We will not worry about this distinction in what follows Heat capacity measurements provide fundamental insight into the properties of a material As you may recall the classical result of Dulong Petit for the molar heat capacity of a solid is c 3R 24 9 J mol K where R 8 31J mol K is the ideal gas constant
10. ed trom S f cy T AT Eqn 1 In determining the entropy associated with ordering it is important to exclude other contributions such as the lattice specific heat An insightful description of second order and also first order phase transitions is Landau s mean field theory which provides a description in terms of an order parameter 8 However while providing considerable insight and a general framework for phase transitions this Quantum Design Educational Module Heat Capacity Measurement of VOQOz v 1 3 theory does not include fluctuations which affect the thermodynamic response and lead to interesting Phenomena such as critical behavior in phase transitions 8 9 A detailed description of critical phenomena ordering and broken symmetry etc can be found in the references Suffice to say heat capacity measurements are a primary means to study these fundamentally important effects In solids The next question we must address is how are heat capacity measurements performed The basic idea is fo heat the sample in a precise manner to add a precise amount of energy and measure the corresponding temperature change In the VersaLab heat capacity option this is accomplished by applying a known amount of heat at constant power for a fixed amount of time followed by a cooling period while measuring the temperature as a function of time This heating cooling process is depicted in Figure 2 An appropriate model discussed below and in
11. hase transitions can be quite narrow this is particularly true for first order transitions Thus an alternative approach to measure or search for the phase transition must be utilized In our study of VOz2 we will utilize the slope analysis method of relaxation curves If both sides of Eqn 2 are divided by dI t dt one obtains P t Ky T t Tp Ceotal arjap Eqn 4 This provides an operational approach to obtain the heat capacity as a function of temperature from a single curve such as that shown in Fig 2 At each time the slope is calculated providing a means to obtain Ciota at each temperature on the curve In the case of a first order phase transition there should be a distinct decrease in the slope at the transition temperature This intuitively makes sense since the latent heat requires the addition of energy to the sample without a temperature increase Further since first order transitions exhibit hysteresis the warming and cooling curves will have different kinks in the slopes Section 4 3 of the heat capacity user manual presents additional details while section 4 6 Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 6 provides examples of single slope analysis of a first order phase transition in Figures 4 6 and 4 7 It is strongly advised that chapter 1 4 of the heat capacity users manual is read prior to performing these measurements Notes E F J Morin Phys Rev Lett
12. heat capacity It will further allow for a determination of the latent heat of the first order phase transition Return the remaining VO2 powder to the vial from which it was obtained Remove the heat capacity puck from the vacuum chuck seal it with the thermal shield i e as in Figure 9 and place it in the VersaLab to prepare for the heat capacity measurement After the sample is loaded you are ready to perform the heat capacity measurement In the heat capacity control center it is possible to initiate a measurement However we will write a simple sequence that will focus on measuring the heat capacity over the temperature range from 325 to 350K appropriate for measuring the first order phase transition in the VO2 powder Here is a sample sequence to accomplish this Set Temperature 325K at 12K min Fast Settle Wait For Temperature Delay 0 secs No Action Sample HC at current temperature 25 K rise 3 times 3 tau Meas Time Set Temperature 300K at 12K min Fast Settle Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 13 S This sequence sets the initial temperature to 325K then performs a measurement three times ramping the temperature up to 350K Once the sequence is initiated the measurement status viewer will appear on the screen to enable tracking of the measurement You may obtain a warning about inaccurate Cp values arising from this sequence This warning is given since a large temperature ra
13. ic heat of N grease is strongly temperature dependent above 200K so H grease Is recommended to minimize errors Grease applicator which can be the wooden end of a cotton swab A microscope fo facilitate the application of the grease and sample A precision scale with 0 1 mg or better resolution Weigh boats or wax paper for handling the VOz2 powder a Prior fo preparing for the sample measurement the heat capacity module should be installed and activated In addition make certain that Quantum Design Educational Module Heat Capacity Measurement of VOz v 1 7 the vacuum chuck and pump that are part of the heat capacity option are in an easy to access location b Locate the heat capacity puck and thermal radiation shield K g E f Ai 4 Figure 4 Heat capacity puck and shield C Verify that the serial number of the puck is consistent with the calibration file that in MultiVu This can be checked in the heat capacity control center see section 4 4 of the user manual Click on the files tab will enable identification of the calibration file In the following we will assume that the puck in use has been properly calibrated If not the procedure in Chapter 5 of the heat capacity user s manual must be followed d The next step is to prepare the puck for the addenda measurement This requires placing H grease on the puck For this the vacuum chuck and pump are needed This is to stabilize the puck while wor
14. in Figure 12 Quantum Design Educational Module Heat Capacity Measurement of VOz2 v 1 11 kre Jap Figure 12 There are 45 1 mgs of VO2 powder in the weigh boat k The next step is fo place a small amount of this powder on the wooden portion of a cotton swap like a spoon cut at an angle to have a tip The powder Is very sensitive to electrostatic change and we found that it was not possible to scoop up a small portion with a metallic spatula The heat capacity puck should be placed back in the vacuum chuck for the addition of the powder You may find it useful to have the puck under the microscope to assist in depositing the powder m The powder should be gently dropped onto the platform in the area where the H grease is located It is important to not touch the grease as this will invalidate the previously obtained addenda Note that this can be challenging but as Figure 13 shows it can be done Quantum Design Educational Module Heat Capacity Measurement of VOQOz v 1 12 n q Figure 13 Heat capacity puck with VO2 powder deposited If there is any left over VO2 powder on your stick add it back to the weigh boat By subtracting off the remaining mass from the initial mass that was measured you can obtain an estimate of the mass added to the puck While this approach may have fairly large error bars it will allow for the calculation of the molar or mass specific heat from the
15. king with It since the sample platform is very fragile and it is easy to break the wires Figure 5 shows the chuck with the heat capacity puck inserted With the silver arm in the open position as shown the vacuum Is not on Upon closing see Fig 6 the vacuum line will be activated make sure the pump is on which will gently pull the sample platform into place thereby stabilizing it for grease application Figure 5 Heat capacity puck on vacuum Chuck Quantum Design Educational Module Heat Capacity Measurement of VOz v 1 8 Figure 6 With the silver lever closed the platform is stabilized e The next step is to place the H grease on the platform being careful to not touch the wires There are two reasons for this First the wires Could break Secondly any grease that gets on the wires will change the thermal conductance which could invalidate the calibration Figure 7 shows the grease being placed on the sample stage and Figure 8 shows a close up of the sample stage after the grease application i Figure 7 H grease application to sample platform Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 9 Figure 8 Sample platform with grease applied f The next step is to perform the addenda measurement First ensure the thermal radiation shield is firmly fastened to the puck as shown in Figure 9 Figure 9 Heat capacity puck ready for addenda measurement g The
16. nge is being covered which would lead to errors in a conventional fitting with a 1 tau or 2 tau model However for the slope analysis method a sufficient temperature rise covering the first order phase transition is what we want The data can be analyzed using MultiVu From the heat capacity control panel Figure 14 left side under tab files select Raw Data File Viewing and Post Processing A screen will appear as in Figure 14 on the right Post Processing of RAW data Ioj x CejHeat Capacity NO DATA FILE O x G x Installation Wizards Measurement Files Diagnostics Measurements in Data File WH 224435 K 723 79 W K 00e 2 324 64K 837 8 pJ K OOe Output Data Fil BE ee 3 324 721 K 830 89 wl K 00e Open New Appendto Calibration File Switch to New Addenda Tables File Manipulation Raw Data File Viewing and Post Processing Raw Data File AGdVersaLab Data Heat Capacitywoe powder FO raw View Preconditioning of Relaxation Curves Moving average Exclude Initial width 0 to 100 i E 0 to 100 a E View Meas Status Sample Info Calibration LPucki627 cal Addenda 14 measured on 3 20 2015 from 202 13169 to 390 26766 K Idle Thmz_vos_ Cal Res _1 9 9194378E 8 Thermeyvos_SROM 1 5406672E Curve Analysis C Dualslope method Combines heating and cooling curves f Single slope method Best for 1st order peaks Use wire heat loss from pass 2 calibr
17. puck can now be loaded into the VersaLab Figure 10 and the VersaLab chamber sealed using the cap with the vacuum bellows Figure 11 Quantum Design Educational Module Heat Capacity Measurement of VOz v 1 10 Figure 10 Loading the puck into the VersaLab el rs he 7 5s 5 5 A p Figure 11 Sealing the VersaLab for the addenda measurement h In the heat capacity control center again see section 4 4 of the users manual the addenda measurement can easily performed Under the measurement tab there is the option to create a new addenda table The temperature range can be selected For these measurements we are interested in the temperature range from approximately 325K 350K which extends well below and above the transition temperature in VOz For the addenda measurement choose 310K 370K to be certain that the range of interest is covered The addenda measurement will take some time and will create an addition hence the term addenda to the calibration file that will oe used for the subsequent sample measurements After the addenda measurement is completed and the sample puck is at room temperature it can be removed from the VersaLab in preparation for placing the sample onto the heat capacity platform The first step is to weigh out a portion of the powder As only 1 2 mg are required for the measurement the best approach is to measure out an amount in excess of this using an electronic balance as shown
18. th puck is dominated by the conductance of the wires This gives a reproducible heat link to the bath with a corresponding time constant large enough to allow both the platform and sample to achieve sufficient thermal equilibrium during the measurement The VersaLab measures heat capacity curves like that shown in Fig 2 i e the change in temperature versus time and the data is fitted in MultiVu using one of several models To give a feel for this we describe the 1 tau model that fits the data using a single time constant The details of the 2 tau model which accurately takes info account the thermal conductance between the sample and the platform are described in section 4 3 of the heat capacity user manual The 1 tau model describes the flow of power into and out of the sample Crotat gt P t Kw T t Ts Eqn 2 where Crotal Is the total heat capacity P t is the applied power Kw is the thermal conductance of the wires T t is the time dependent temperature and Tp is the bath temperature For P t see Fig 2 we have P t Po 0 lt t lt fo and P t 0 t gt fo With the initial conditions Ton 0 Tp and Ton to Tort To Eqn 2 can be solved yielding Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 5 Pot 1 7 ay are Ty 0 lt t lt tp T t total Eqn 3 Pot 1 675 ee t0 Ty 0 gt to Cto tal where t Ctotai Kw is the thermal time constant MultiVu uses le
19. yze your data to estimate the mass specific heat below the phase temperature Provide an estimate of the error bars and compare your result to experimentally published data for crystals 2 Now analyze your data to obtain an estimate of the latent heat Again provide error bars compare fo literature values Note for 1 and 2 you could use the data from C N Berglund H J Guggenheim Phys Rev 185 1022 1969 for comparison Or you can find other data in the literature if you so choose 3 Find a review paper on VO and write a paragraph summarizing the interesting properties of this material Discuss how the heat capacity plays an important role in understanding the physics of this material Quantum Design Educational Module Heat Capacity Measurement of VO2 v 1 16
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