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Very low temperature calorimetry Unai Urd´ıroz Urricelqui Trabajo fin de M´aster en F´ısica y Tecnolog´ıas F´ısicas Departamento de F´ısica de la Materia Condensada Facultad de Ciencias - Universida de Zaragoza Curso 2013/2014 Director: Fernando Bartolom´e Instituto de Ciencia de Materiales de Arag´on - CSIC 1 Contents 1 Brief introduction and motivation 3 1.1 Low temperature heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Physical problem: geometrical frustration . . . . . . . . . . . . . . . . . . . . 3 2 Cooling method 5 2.1 Historicalintroduction............................... 5 2.2 3He-4Hemixtureproperties ............................ 5 2.3 Dilutionrefrigerator ................................ 6 3 Low temperature thermometry 7 3.1 Thermometers calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Integrationtime .................................. 8 3.3 Readingexcitation ................................. 10 3.4 Magnetoresistance ................................. 11 4 Calorimetric methods 12 4.1 Quasiadiabaticmethod............................... 13 4.1.1 Theory ................................... 13 4.1.2 Experimentalsetup............................. 13 4.2 Relaxationmethod................................. 14 4.2.1 Theory ................................... 14 4.2.2 Datatreatment............................... 16 4.2.3 Experimentalsetup............................. 17 5 Measurements 18 5.1 Experimental conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Conclusions 22 2 1 Brief introduction and motivation 1.1 Low temperature heat capacity The thermal response of a body to applied heat, measured by the specific heat is one of the most important basic parameter of solid state research, providing information about the energy contributions playing a role in the studied system. The specific heat studies have been closely related to low temperature. As temperature is lowered, the thermal noise that avoids the observation of weak quantum phenomena disappears allowing its study. So that is a key technique to observe quantum phenomena by classical measuremets. The proposition of the 3rd law of Thermodynamics by Nerst, established that the entropy and thus the specific heat should tend to zero when the absolute zero temperature limit is approached. Einstein and Debye models for the specific heat established the existence of quantum effects at very low temperature that can be related to the energy states of the studied systems. Statistical mechanics allow us to obtain thermodynamic macroscopic parameters of a system such as the heat capacity or entropy, so that these measurements are useful to check theoretical models. Low temperature calorimetry involve a large number of technical difficulties to deal with, so that, several non-comercial experimental methods have been developed to obtain accurate measurements [1]. All of them have something in common, the use of a heating source that applies a known amount of heat to the sample while its temperature is monitorized. Both phase transitions, and Schottky anomalies, have a clear fingerprint in heat capacity. Cp(T) is useful in magnetism to obtain energy level configuration [2], in quantum relaxation studies, etc. Calorimeters are also used as radiation and particle detector as an alternative to ionization chamber ones because they do not require efficient charge transport and operate in equilibrium [3]. Their low temperature operation made them the best option in particular applications such as astrophysics, because of their versatility and measurement sensitivity [4]. The aim of this work is the development of a low temperature calorimeter with high magnetic fields using a 9 Tesla coil for the study of magnetic frustrated systems. Besides, the set up may be used to study any kind of temperature and field dependent Cp of any solid system of interest. 1.2 Physical problem: geometrical frustration In magnets, geometrical frustration prevents the formation of a single-ordered magnetic ground state creating a highly degenerated one [5]. This phenomenon appears when the magnetic ions occupying an ordered lattice are not able to find an absolute energy minimum. This is due to the imposibility to satisfy all contradictory interactions between them. The simplest example is a triangle in which three magnetic moments occupy the corners suffering antiferromagnetic interactions. When the physical conditions acting on the system change (such as: T, H, P or anisotropy), the frustration may be broken and a long-range order (LRO) state could be established. 3 Frustrated magnets thus lie at the crossroads of two fundamental work fields in condensed matter physics. On the applied side, the instabilities exhibited by frustrated magnets open a window to the richness of nature realized in different materials. On the fundamental side, it is a complex and interesting physical problem to be descrived using different theories [6]. Magnetic materials where lattice symmetries are based on triangles and tetrahedra, such as the pyrochlores or the garnets, may exhibit this phenomenon. In this case, the target of the study will be the Gadollinium Gallium Garnet (Gd3Ga5O12, GGG) that is very sensitive to the presence of an external magnetic field and has a rich phase diagram [7–11]. Fig.1: GGG phase diagram depending on temperature and magnetic field based on specific heat and susceptibility measurements. In this system, the magnetic Gd ions are located at the nodes of two interpenetrating corner-sharing triangle sublattices within the garnet, and the exchanges are almost purely AFM with ΘWeiss ∼-2 K [12,13]. The Gd spins are isotropic, although there is a small single ion anisotropy in the GGG of less than 0.04 K. This isotropy leads to the high degree of frustration in GGG which prevents ordering at low fields as observed in isomorphic magnetic garnets such as Dy3Ga5O12 or other rare earth gallium garnets like Sm3Ga5O12, Er3Ga5O12 and Nd3G5O12 with N´eel temperatures below 1 K. The long term scientific objective is to study the possibility to induce magnetic order in GGG by dilution of a non-isotropic rare earth ion in the Gd site. In particular, we have chosen Eu3+. It differs from other trivalent rare earth ions in having a particular ground state in which there is J=0 while |L|=|S| 6=0 and L is aligned antiparallel with S. The introduction of an easy magnetization direction may help to establish the LRO state. It is interesting to check, in case the transition to LRO state appears, if the Eu3+ concentration in GGG is higher or lower to the percolation limit. In case the percolation limit is exceeded the studied system changes from GGG with Eu impurities to Gd diluted EuGG. In that case, the conclusions of our study would be totally different. The stablishment of LRO will be verified by the observation of a clear lambda peak in the heat capacity as a function of external magnetic field applied and temperature. With this data, the phase diagram evolution with Eu impurities would be built. 4 2 Cooling method 2.1 Historical introduction Temperature is one of the most important thermodynamic variable that can be modified to study the properties of matter and phase transitions. Nowadays, several experimental techniques have been developed to achieve temperatures as low as tens of µK or even effective spin temperatures of nK. These techniques are based on very different experimental effects such as: cryogenic liquids, quanta phenomena or even laser traps. Low temperature physics have grown since the last decades of the XIX century due to work performed on different laboratories in their race of gas liquefaction: air (N2+O2), H2, and finally He. This last one, was a big temperature jump made by Kamerlingh-Ones in 1908 [14] reaching temperatures below one Kelvin. In 1927 Giauque propose the paramagnetic adiabatic demagnetization [15] allowing to get temperatures of 0.1 K. Finally, in 1962 London, Clarke and Mendoza published a new refrigerator proposal based on the He isotopes mixture properties [16] : the dilution refrigerator. The main advantages of this refrigeration method is the continuous cooling nature and the availability of using magnetic fields in the experiments. The lowest temperature achieved in these refrigerators has been 2 mK and usually are used as precooling stages for nuclear adiabatic desimanation experiments. 2.2 3He-4He mixture properties The cooling power of a dilution refrigerator is based on the physical properties of the 3He -4He mixture [17]. These isotopes have completely different behavior because of their different nuclear spin value. In one hand, 4He has zero nuclear spin obeying the Bose-Einstein statistics, so that, all the atoms can occupy the same energy level. Indeed, it undergoes a phase transition to superfluid state at T=2.18 K where all the atoms are in the ground state. In the other hand, 3He has a half-integer spin value obeying Fermi statistics and Pauli exclusion principle avoiding the presence of two identical atoms in the same energy level. A key fact is that the 3He and 4He are miscible with a rich phase diagram. 5 Fig.2: He isotopes mixture phase diagram depending on temperature and the 3He molar fraction x=n3 n3 + n4 In the phase diagram, the temperature of the superfluid phase transition of 4He (lambda line) decreases with the concentration of 3He in the mixture disappearing in the triple point for x=67.5 and 0.876 K. Cooling the mixture below this point, a separation in two different phases appears: one rich in 4He and the other in 3He. Due to its lower density, the 3He rich phase floats on top of the other becoming pure 3He phase when zero temperature is approaching. However, in the 4He rich phase, the 3He does not disappear but rather reaches a constant concentration of 6.6%. The reason for x not been zero at T=0 is, obviously that the binding energy of an 3He atom is bigger in 4He liquid than in 3He liquid. In the same way that a liquid 3He needs heat contribution in order to evaporate, the 3He atom absorbs heat from the surroundings, the so called mixture enthalpy, when it changes from the concentred to the diluted phase. So that, the cooling down process is obtained forcing this transitions pumping 3He atoms from the diluted phase. The cooling power, is proportional to the 3He flux trogugh the main circuit: ˙ Q= ˙n3[Hd(T)−Hc(T)] 2.3 Dilution refrigerator An Oxford Instruments MX40 dilution refrigerator has been used with: base temperature 27 mK, 80 µW cooling power at 100mK, 66 liters of total volume dewar and the possibility of applying magnetic fields up to 9T. The mixture volume is 58 l at 1atm and room temperature with 3He partial volume of 7 l [18]. The different parts involved in the mixture handling procedure are: insert, 4He rotary pump,3He pump, gas cabinet, N2traps and He trap. 6 Fig.3: Schematic description of dilution refrigerator [19]. Inner vacuum chamber delimits the inset space inmersed in liquid 4He bath. Arrows indicate 3He flow through the system. The cooling process could be described as a down-stairs road to the base temperature. In the first step, the system is cooled down to 90 K with liquid Nitrogen. After that, is filled up with liquid Helium taking it down to 4.2 K. Helium is pumped to obtain the working temperature of the 1 K Pot. After this, the 3He -4He mixture is condensed and circulated. Then the system achieves its base temperature. In the main system circuit, the 3He is pumped from the poor phase in the mixing chamber. It rises exchanging heat with the incoming one in the different stages. Before going back to the mixing chamber, it must pass through the N2and He traps and cool down in the heat exchangers. The still must have a vapour pressure fixed in order to maintain an efficient 3He flux [17,18]. There is a secondary circuit in which the 4He pump takes it from the 1 K Pot to have a fixed cold point. The whole procedure must be performed maintaining a high vacuum enclosure, and a very clean and leak-free 3He circuit through valvues, pumps and meter-long tubes. All this make the cooling down procedure a slow and rather expensive one. 3 Low temperature thermometry A very important point in low temperature physics and particularly in calorimetry, is thermometry. In order to measure accurately temperature, we need to have a good thermal contact of the sample with a thermometer: a device whose dependence of a given physical magnitude with temperature is well known. In our case, the resistance thermometry is used since is the best choice at temperatures from 50 mK to 1 K [1,17]. From the two main options, 7 Germanium and RuO2, the second one is selected due to its low changes with magnetic fields. To obtain the best possible measurements, the 4-wire configuration is used in order to eliminate the error caused by resistance of the wires. To measure resistances accurately at very low temperatures, a bridge resistance must be used to apply very low currents, to avoid the self-heating. In this case, a Picowat AVS 47 bridge is used with its analogic exit connected to the input of a HP 3458A multimeter with 8.5 digits to improve the reading resolution. Temperature measurement is the main source of error in calorimetry measurements. Three main problems have to be considered: the multimeter integation time for each reading, thermometer self-heating and changes in R due to magnetic field(magnetoresistance) [20–23]. Several studies of stability with different excitations and integration times help to choose the best measurement conditions in order to minimize the noise and improve accuracy. In the other hand, magnetic field sweeps at different temperatures will give information about the magnetoresistance of the thermometer and its thermal dependence. 3.1 Thermometers calibration A Lakeshore full-precision calibrated 2.21 KΩ RuO2thermometer is available to be used by the calorimetric set up. It is covered by a brass case holding the RuO2chip within a low 3He gas pressure for thermalization but makes it very heavy (3g). This leads to a big thermometer contribution to the measured specific heat. A calibration experiment has been performed with three thermometers: the calibrated one used as reference and other two RuO2chips (of 1.75 KΩ and 2.21 KΩ at RT) with masses of 11 mg. The objective is to have three full calibrated thermometers that could be used on the experiments. The chips have a heat capacity contribution 2 orders of magnitude lower, improving the accuracy in future measurements. Due to unclear reasons, the thermometer holder did not cooled down below 150 mK in this experiments. RuO2resistors are metal-ceramic composites, consisting of a mixture of conductive RuO2 and Bi2RuO2embedded in a lead silicate glass matrix, deposited on an alumina substrate. For thermal contact, they must be glued to clean surfaces with epoxy or GE Varnish. Their resistance as a function of temperature R(T), can be well fitted by the empirical equation [17]: ln(T) = 1 ΣiAiln (R−R0)iwith 0 ≤i≤3 As it is shown in Fig.5 there is a good quality fitting with errors of 3%-7% between the calculated and the measured temperature for both thermometers. 3.2 Integration time Integration time is the period of time that the HP multimeter measures the input signal in the multimeter. This value determines the measurement frequency, accuracy, maximum digits of resolution and the noise automatic rejection. With longer integration times, the measurement resolution and accuracy increase, but measurement speed decreases. This parameter will be controlled by the Number of Power Line Cycles (NPLC) selected. The multimeter multiplies 8 4000 6000 8000 10000 12000 14000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2.21KΩ 1.75KΩ T(K) R(Ohm) a0=-0.66086 a1= 0.76656 a2= 0.36587 a3=-0.09497 Polynom values for 1.75 KΩ ΩΩ Ω a0=-0.85142 a1= 1.10619 a2= 0.06629 a3=-0.02149 Polynom values for 2.21 KΩ ΩΩ Ω Fig.4: T(R) curves of the 2.21 KΩ and 1.75 KΩ thermometers taking as reference the calibrated one. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -10 -8 -6 -4 -2 0 2 4 6 8 10 Tfitt-Tref(%) 2.21KΩ 1.75KΩ T(K) Fig.5: Differences between fitting obtained temperature and measured in percentage. the specified number of PLCs by its reading period to determine the integration time [24]. In order to determinate the best integration time to minimize noise at different temperature ranges, the resistance has been measured with the same excitation but changing the NPLC of the multimeter. Noise windows will be compared in order to get the ideal NPLC. 0 100 200 300 400 500 10200 10300 10400 10500 10600 10700 10800 10900 11000 Ref 2.21KΩ 1.75KΩ R(Ohm) time(s) Resistance noise window T=100mK NPLC=80 Fig.6: Resistances measured at 150 mK with an excitation lecture of 3 µV and an integration time of NPLC=80(Number of Power Line Cycles). 9 Cp determination starts. This procedure can be repeated at will to reduce the uncertainty of the measurement. Fig.13: Experimental points obtained using the normal relaxation method(a) and the alternative fast one (b) extracted from [36]. Fig.14: Isoperibol calorimeter brought from Kamerlingh Onnes Laboratorium of the University of Leiden, used as example in the development of our own. Some parts have been used in our model. 4.2.2 Data treatment The procedure starts with the linear fitting of the exponential relaxation decay in a semi-log plot (Fig.15) obtainig the time constant τ. As τis known, a simple variable change is done, t→δ=[1 −exp(−t/τ)], turning the heating curve into ∆T(δ) = (P/KH)δ. Thus, a simple least-square linear fit provides the missing parameter KH, and heat capacity is obtained again from Cp =KH·τ(Fig.16). However, in real systems the thermal resistance between the sample and the holder do not satisfy τsτth. So that relaxation curves show two different contributions: the first and the fast one, related to the relaxation of the sample with the calorimeter cell; and the second one, the relaxation of the whole system with the surrondings. In fact, the Quantum Design Physical Properties Measurement System (PPMS) uses its own theoretical “two τ” model [37]. In this model, two main pre-measurement studies must be done in order to obtain an accurated specific heat value: empty calorimeter and thermal conductance of the link as a function of temperature and external field is necesary. Also is important to consider that this first contribution usually is almost negligible in the PPMS because the cell and mass amount are selected to optimize the measurement. In our case, relaxation measures have been made with large sample mass that avoid the optimization. Dr. Ana Arauzo from the SMF (Servicio de Medidas F´ısicas) confirmed that contribution in our case was much larger than in PPMS cases due probably to the large sample mass and surface-volume ratio in the sample pill. Empty calorimeter and thermal link response have not been measured. Copper and Apiezon N heat capacity are known [17] so that despite not having an empty calorimeter 16 0 100 200 300 400 500 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 ln(∆ ∆ ∆ ∆T) t relax(s) Relaxation sample-calorimeter τ ττ τ1=42s Relaxation system-enviroment τ τ τ τ2=444.4s Fig.15: Experimental points on a semi-logarithmic representation of the system relaxation. Linear fitting have been made to obtain time constant. 0.00 0.05 0.10 0.15 0.20 0.25 -0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 ∆ ∆ ∆ ∆T(K) δ δ δ δ heat= == =( (( (1− 1−1− 1−exp(-t/τ2 τ2τ2 τ2)) slope=(P/Kh)=0.0356 with P=1.2866x10 -7 W Sample heating Cp=5.576 (J/KmolGd) Fig.16: Linear fitting of heating points gave thermal conductance with which the specific heat is obtained. measurement, a rough Cp estimation has been done using: Cp =mco(γco ·T+Aco ·T3) + mApiez(AApiez ·T3) with 12 g of copper and 8.6 mg of Apiezon N. In the other hand, the thermal link response is unknown. The obtained values from data treatment are supported on an approach neglecting the first fast relaxation. 4.2.3 Experimental setup Several calorimeteres are reported in literature [38–41]. In our case (Fig.17), the calorimetric holder is a 12mm sapphire platelet mechanically attached to the thermal reservoir by kapton strips. Thermal reservoir is made of OFHC copper. The sapphire is used due to its low Cp (Cp≈0.1 nJ/gK at 0.1 K), good thermal conductivity (κth ≈20µW/cmK at 0.1 K) and mechanical strength. Sample holder is connected to the mixing chamber stage with the same stick-isolator-thermal link configuration as in the previous design. To control the reservoir temperature via PID, Evanohm wire heater and full calibrated thermometer are disposed. In the sapphire holder a chip thermometer is placed and a thin film is deposited to be the heater with R=259.95Ω at RT. 17 Current source Voltmeter Resistance bridge ELECTRONICSELECTRONICS PID temperature controller Sapphire holder Sample Thermometer Thermometer Thermal reservoir Reservoir heater Reservoir heater CALORIMETERCALORIMETER Sapphire holder Fig.17: Schematic description of the isoperibol calorimeter control with electronic equipment. Dashed lines, correspond to RT electrical wiring. The electronic components used in this case are the same the PID temperature controller introduction to monitorize the thermal reservoir. 5 Measurements First measurements have been performed in a GGG pure sample of 97 obtaining in total 20 Cp points. As the most interesting contribution in the GGG heat capacity is the magnetic one related to the Gd3+ ions, the obtained data are exposed in units of J/K per Gd mol. Zero field measurements at 270 mK, 300 mK and 320 mK are shown in the Fig.18 where are compared with Schieffer data [7] and PPMS obtained data. Then temperature has been stabilized at 130 mK to measure Cp as a function of externeal the field from 0.2 T to 2 T with a higher point density near the phase transition. The same was done at 320 mK with fields from 0 T to 1.8 T. This points are shown as Cp(H) curves at a fixed T in Fig.20 in comparison again with Scieffer data [7]. Finally experimental points for 0.65 T, 0.7 T and 0.75 T at temperatures from 87 mK to 345 mK are compared with Schieffer [7] and Tsui [11] data in Fig.19. 18 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 PPMS calorim Schieffer Cp(J/KmolGd) T(K) GGG Cp without ext. field 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3 4 5 6 Cp GGG 1T PPMS calori Cp(J/KmolGd) T(K) Fig.18: Specific heat measurements at zero field with handmade calorimeter compared with the exposed by Schieffer on [7] and with other values obtained using PPMS. Inset: Experimental point for H=1T in comparison with PPMS data. In Fig.18, the differences with Schieffer data go from a 10% to a 15% within the error bar that in this case reach a 30% in the three points due to the noisy data curves. In the inset, the point measured for 1 T field applied has an error of a 25% respect to the PPMS data curve out of its 9% error bar. 0.1 1 0 1 2 3 4 5 6 7 Schieffer 0.70T 0.65T 0.75T Tsui -- Spin-wave GGG Cp meas. with fields around 0.7T Cp(J/KmolGd) T(K) exp.points Fig.19: Experimental values at fields of 0.65T. 0.7T and 0.75T compared with the exposed on [11]. 19 Figure 19 is a compilation of the measurements performed near the phase transitio at fields around 0.7 T. A fit of the Schieffer and Tsui [7] [11] low temperature specific heat to a curve C∝T3/2corresponding to the contribution of the antiferromagnetic spin waves on the ordered phase is considered as reference to compare with experimental points. All the points agree with literature with differences respect to the T3/2fit within the error bars. 0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 5 6 exp. points at 130 mK PM AFM PM Cp(J/KmolGd) T(K) exp. points at 320 mK Fig.20: GGG specific heat as a function of the applied field at several low temperatures. From the bottom to the top in temperature increase order: 65 mK (blue), 100 mK(Schieffer)(red), 130 mK(exp.) (green), 150 mK(Schieffer)(purlple) and 320 mK(exp.)(yellow). Finally in Fig.20 Cp(H) is shown at different temperatures to observe magnetic phase transitions. Observing the GGG phase diagram (Fig.1) [7] is clear that two different phase transitions should be observed at the exposed temperatures. According to Tsui data, it seems that the phase transition that occurs at fields around 0.7 T is less abrupt than the one at field around 1.4 T. The experimental data for 130 mK has a qualitative agreement with Tsui Cp(H) curves [11] with error bars of a 15%. Furthermore, the phase transition at 0.7 T is observed. However at 320 mK, there are not enough measured points to observe neither of the phase transitions. Also, the error achieves values of a 30% because of the noisy data curves again. 5.1 Experimental conclusions A normal calorimeter has been designed, developed and proved. The sample mass and the calorimeter used in the experiment were optimized to measure using the quasiadiabatic method. However, relaxation measurements were performed due to the setup response. Although the experimental setup was not appropriate to use the relaxation method, 20 experimentnal points were obtained in agreement with literature [?, 7] and errors up to a 30% in 20 the worse fittings. To optimize the experimental setup for relaxation method an isoperibol calorimeter is being developed. Besides the cell contribution is minimum, the sample mass decrease critically as it does the first relaxation improving measurement accuracy. Nowadays the isoperibol calorimeter is in mounting process. 21 6 Conclusions This work is the starting point of a very low temperature calorimetric measurement system development with applied magnetic fields. An important effort has been done to get into low temperature physics. Also, the complete experimental setup building has been done, from the design to the mechanical assembly. a) Dilution refrigerator operation processes have been studied. Now a complete cooling down procedure can be completed without any help. b) Three thermometer complete calibration has been done to obtain future accurate temperature measurements. Self-heating, integration time noise and magnetoresistance have been studied and characterized. c) A normal calorimetric cell and measurement software have been built and used on a first attempt of quasiadabatic measure. An isoperibol calorimeter and relaxation software are being developed. d) Some experimental points are obtained using an approach. 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