The magnetocaloric effect or "magnetic Gr\"uneisen ratio"
$\Gamma_H=T^{-1}(dT/dH)_S$ quantifies the cooling or heating of a material when
an applied magnetic field is changed under adiabatic conditions. Recently this
property has attracted considerable interest in the field of quantum
criticality. Here we report the development of a low-frequency alternating
field technique which allows to perform continuous temperature scans of
$\Gamma_H(T)$ on small single crystals with very high precision and down to
very low temperatures. Measurements on doped YbRh$_2$Si$_2$ show that
$\Gamma_H(T)$ can be determined with this technique in a faster and much more
accurate way than by calculation from magnetization and specific heat
measurements

Gegenwart, Philipp

Tokiwa, Yoshi

English

arXiv

The magnetocaloric effect or "magnetic Gruneisen ratio" Gamma(H) = T(-1)(dT Id H)s quantifies the cooling or heating of a material when an applied magnetic field is changed under adiabatic conditions. Recently, this property has attracted considerable interest in the field of quantum criticality. Here, we report the development of a low-frequency alternating-field technique for measurements of the magnetocaloric effect down to very low temperatures, which is an important property for the study of quantum critical points. We focus, in particular, on highly conducting metallic samples and discuss the influence of eddy current heating. By comparison with magnetization and specific heat measurements, we demonstrate that our fast and accurate technique gives quantitatively correct values for the magnetocaloric effect under truly adiabatic conditions. (C) 2011 American Institute of Physics. [doi :10.1063/1.3529433]German Science Foundation [SFB 602 TPA19, FOR 960

GRO.publications (Univ. Göttingen)

High-resolution alternating-field technique to determine the magnetocaloric effect of metals down to very low temperatures

GRO.publications

Tokiwa, Y.

OPUS Augsburg

REVIEW OF SCIENTIFIC INSTRUMENTS 82, 013905 (2011)High-resolution alternating-field technique to determine the magnetocaloriceffect of metals down to very low temperaturesY. Tokiwaa) and P. GegenwartI. Physik. Institut, Georg-August-Universität Göttingen, D-37077 Göttingen, Germany(Received 9 September 2010; accepted 26 November 2010; published online 27 January 2011)The magnetocaloric effect or “magnetic Grüneisen ratio” H = T −1(dT/d H )S quantifies the cool-ing or heating of a material when an applied magnetic field is changed under adiabatic conditions.Recently, this property has attracted considerable interest in the field of quantum criticality. Here,we report the development of a low-frequency alternating-field technique for measurements of themagnetocaloric effect down to very low temperatures, which is an important property for the study ofquantum critical points. We focus, in particular, on highly conducting metallic samples and discussthe influence of eddy current heating. By comparison with magnetization and specific heat measure-ments, we demonstrate that our fast and accurate technique gives quantitatively correct values forthe magnetocaloric effect under truly adiabatic conditions. © 2011 American Institute of Physics.[doi:10.1063/1.3529433]I. INTRODUCTIONUnderstanding the phenomena close to quantum criti-cal points (QCPs) is one of the major challenges in con-densed matter physics. A QCP emerges when the character-istic temperature of a system, e.g., the Néel temperature, issuppressed to zero by variation of a nonthermal parameter rlike pressure, magnetic field or doping. Upon pressure-tuninga system toward a QCP, the Grüneisen ratio  is expected todisplay singular behavior, namely a divergence toward zerotemperature, while by tuning a system by magnetic field to-ward quantum criticality H, as defined below, diverges.1, 2Both Grüneisen parameters consist of ratios of the control-parameter and temperature derivatives of the entropy, whichis accumulated close to the QCP (S, α, C , and M denote theentropy, volume thermal expansion, specific heat, and magne-tization, respectively): = − 1Vm T(∂S/∂p)T(∂S/∂T )p= αC p, (1)H = − 1T(∂S/∂ H )T(∂S/∂T )H= − (∂ M/∂T )CH= 1TdTd H∣∣∣∣S.Indeed, both  and H have been experimentally shownto diverge in materials near QCPs.3–6 Since the magnetic field,in contrast to pressure or doping, can easily be tuned continu-ously in situ, the magnetic Grüneisen ratio is ideally suited toinvestigate the nature of QCPs. So far, the magnetic Grüneisenratio close to QCPs has been determined only indirectly bycalculation from magnetization and specific heat data.6 Note,that H can be determined by a single measurement of themagnetocaloric effect (MCE), (dT/d H )S , as shown above.Very recently, the field dependence of the entropy close toa QCP has been calculated from highly nonadiabatic (quasiisothermal) T (H ) traces, separating the effect of heat flowfrom/to the bath and the magnetocaloric effect of the sam-a)Electronic mail: ytokiwa@gwdg.de.ple, using independent measurements of the thermal con-ductance between sample and bath.7 However, a more directmeasurement of the MCE under truly adiabatic conditions ishighly desired for comparison with theoretical scenarios. Be-low, we show that the alternating-field technique is suitablefor this purpose. Previously, the spin–flip transition in insulat-ing GdVO4 has been investigated using such technique downto 0.5 K.8 In their study, not much care has been taken on theabsolute values of the MCE and eddy current heating playsno role, since the material is an electrical insulator. Below, wereport the application of the low-frequency alternating-fieldtechnique to clean metals and down to mK temperatures. Wecarefully investigate the effect of eddy current heating. Fur-thermore, we prove, by comparison with magnetization andspecific heat measurements, that accurate absolute values ofthe adiabatic MCE are obtained.II. EXPERIMENTAL SETUPThe definition of the MCE implies that a change of thesample temperature T caused by a small change of the mag-netic field H under adiabatic conditions needs to be de-tected. Below, we demonstrate how the MCE could be de-tected directly by applying a low-frequency alternating field,H0 sin(2π f t), and detecting the temperature oscillation of thesample with the same frequency, T0 sin[2π f (t + δt)] (δt is asmall delay due to the thermal resistance between the sampleand thermometer).Our setup for generating the alternating field isshown in Fig. 1. A SR830 Lock-In amplifier (StanfordResearch Systems) is used to produce a current with low fre-quency (typically 0.01–0.1 Hz, the choice of the frequencywill be discussed later). The current is amplified by a Kepcobipolar power amplifier up to 8 A and sent through a modula-tion coil (8 A current corresponds to 25 mT) which is insertedto the main superconducting magnet and cooled within theliquid 4-He. The current is determined by measuring the volt-age across a shunt resistance (4 m ) with a Keithley 195A0034-6748/2011/82(1)/013905/4/$30.00 © 2011 American Institute of Physics82, 013905-1013905-2 Y. Tokiwa and P. Gegenwart Rev. Sci. Instrum. 82, 013905 (2011)main magnetmodulation coilSR830Lock-In amplifierKepco Bipolar power amplifierKeithley 195AShunt resistanceFIG. 1. Generation and control of the alternating magnetic field superposedto the large magnetic field of the main superconducting magnet.multimeter. The modulation field is superimposed to the de-sired static magnetic field of interest, which is generated by alarge superconducting magnet. Note that a finite static mag-netic field is required since the MCE depends on the magne-tization of the system [cf. Eq. (1)].Figure 2 shows the setup to detect the sample temperatureunder quasiadiabatic conditions. The sample is glued on a thinsapphire plate and is cooled or heated through the weak ther-mal link (thin CuNi wire). If τ denotes the thermal relaxationthrough the link, the frequency of the field modulation mustbe adjusted such that 1/ f  τ in order to obtain effective adi-abatic conditions. Such adjustment can easily be performedat several different temperatures by taking temperature traceswith differing frequency and calculation of the MCE (see be-low). If the MCE is frequency independent, the measurementsare performed under adiabatic conditions. The RuO2 resistivethermometer is glued on the sample and its leads are madefrom superconducting filaments to avoid an additional heatleak. The sample temperature is measured using a LS370 re-sistance bridge (Lake Shore). The heat sink is weakly coupledto the mixing chamber and its temperature is proportional in-tegral differential (PID) controlled with a heater within thefield-compensated region of the large superconducting mag-net. In a typical measurement of the temperature dependenceof the MCE, we slowly sweep the temperature of the heatsink leading to a slow sweep of the average sample temper-ature. Due to the alternating magnetic field, the sample tem-perature oscillates through its average value. The amplitude ofFIG. 2. Schematic view of the platform used for alternating field magne-tocaloric effect measurements under quasiadiabatic conditions.this oscillation gives the MCE, while the average value deter-mines T .III. APPLICATION TO STRONGLY CORRELATEDMATERIALSFigure 3 shows a typical example of a temperature sweepduring an alternating field MCE measurement. We have cho-sen YbRh2Si2 as this is a system close to a QCP, at which apronounced divergence of the magnetic Grüneisen ratio hasbeen observed.6 The data discussed here are obtained on aslightly Fe-doped single crystal of 20.1 mg mass in a shapeof plate with a thickness of 0.25 mm (residual resistivity ra-tio ∼12).9 A relatively large piece of single crystal was cho-sen to determine precisely its magnetization and specific heat,which we will discuss later. The field was applied within theeasy magnetic plane of the system (parallel to the plane of thesample plate), which is perpendicular to the tetragonal c-axis.The static field is set to 50 mT which is the critical field forundoped YbRh2Si2 and the alternating field has an amplitudeof 4.5 mT with frequency of 0.1 Hz. A clear oscillation of thesample temperature due to the MCE is resolved. The inset ofFig. 3(a) displays the temperature variation on a larger timescale such that individual oscillations could not be resolved.Here the thickness of the line is a measure of the oscillationamplitude. Indeed the thickness of the T (t) trace increases astemperature is decreased, indicating a larger temperature os-cillation dT/d H with decreasing temperature. This is a clearevidence of quantum criticality in this material, since dT/d His expected to decrease linearly with decreasing temperaturefor a noncritical material with constant magnetic Grüneisenratio H = const.0.1350.1300.125T (K)3200316031203080Time (s)0.30.20.1T (K)400030002000Time (s)Yb(Rh 0.95Fe0.05)2Si 2H     c(a)(b)4648505254H (mT)TavFIG. 3. (Color online) Sample temperature (a) and magnetic field (b) vs timefor a measurement on doped YbRh2Si2. The red solid and dotted lines show afit to T (t) = a0 + a1t + a2t2 + T0 sin[2π f (t − δt)] and its background term,a0 + a1t + a2t2, respectively. The inset displays the temperature variation ofthe sample over a longer time scale.013905-3 Y. Tokiwa and P. Gegenwart Rev. Sci. Instrum. 82, 013905 (2011)0.010.11Γ mag (1/T)6 7 8 90.12 3 4 5 6 7 8 912T (K)Yb(Rh 0.95Fe0.05)2Si 2H = 0.05T     cAlternating field method-(dM/dT)/CFIG. 4. Magnetic Grüneisen ratio for Yb(Rh0.95 Fe0.05)2Si2 as a function oftemperature. The open circles are determined by the alternating field tech-nique while the solid squares are calculated from magnetization and specificheat data using −(d M/dT )/C .In order to quantify the MCE at a given averagetemperature Tav, we chose time intervals of five periods(5×1/ f s in time) and determine the amplitude of thetemperature oscillation by fitting the temperature trace toT (t) = a0 + a1t + a2t2 + T0 sin[2π f (t − δt)] [red solid linein Fig. 3(a)], where f is the frequency of the field modula-tion H (t) = Hav + H0 sin(2π f t). The second order polyno-mial in time [a0 + a1t + a2t2, cf. red dotted line in Fig. 3(a)]is added to the fitting function to take into account the back-ground temperature drift due to the slow sweep of the heatsink temperature. The amplitude of the temperature oscilla-tion T0 divided by the field modulation amplitude H0 givesthe MCE. This value is associated to the average temper-ature Tav given by the background temperature at the cen-ter of the time interval, Tav = a0 + a1(t0 + 2.5/ f ) + a2(t0+ 2.5/ f )2, where t0 is the time of the first data point forfitting. The magnetic Grüneisen ratio is then calculated byH (Tav) = 1/Tav × T0/H0. By shifting the fitting intervalacross the temperature trace, a dense and continuous curve ofH is obtained, as shown in Fig. 4. In this measurement from3 K down to 0.3 K, the temperature of the heat sink was slowlydecreased, using PID control, and below 0.3 K the heater forthe heat sink was turned off, since the thermal conductance ofthe thermal link is very small at low temperatures, and thus,the decrease of the average sample temperature is very slow.We have also performed a measurement with half value ofthe frequency ( f = 0.05 Hz) and the same H0 in the entiretemperature range from 3 to 60 mK, using the same coolingrate of the heat sink, and confirmed that the result is exactlythe same. This proves that the measurement is taken under ef-fective adiabatic conditions (1/ f  τ ; i.e., heat flow betweensample and heat sink within one period is negligible.). Thedata are compared with H (T ) calculated using Eq. (1) frommagnetization and specific heat data9 taken at the same fieldof 0.05 T and measured using the same piece of single crystal.The excellent agreement between the two independent waysto obtain the magnetic Grüneisen ratio prove that this tech-nique is best suited to quantitatively obtain the true adiabaticMCE. Furthermore, the comparison shows the extraordinarilyhigh resolution of the alternating field technique.At last we discuss the problem of Joule heating causedby the field modulation. If the sample is very clean and themean free path of the electrons is rather high, the frequencyand the amplitude of alternating field have to be low to avoidJoule heating due to eddy currents. Figure 5 shows data ob-tained in the normal state of a high-quality single crystal ofthe clean heavy-fermion superconductor CeCoIn5 in magneticfield applied parallel to c-axis. Note here that the material hasa very large electronic mean free path of 810 Å (Ref. 10)and alternating field is applied perpendicular to the plane of aplatelet sample (1.3×1.1 ×0.33 mm3) with a twice larger fre-quency of 0.2 Hz than the one used for Fe-doped YbRh2Si2.This condition causes a severe problem with eddy currentheating. As clearly seen in Fig. 5, the temperature of the sam-ple oscillates with twice the frequency of the field modula-tion which is due to Joule heating. Eddy currents induced bythe variation of a magnetic field are proportional to the timederivative of the field, d H/dt = 2π f H0 cos(2π f t). Sinceeddy current heating is proportional to the square of the cur-rent, (d H/dt)2 ∼ f 2 H 20 [1 + cos(2π2 f t)], it results in an os-cillation with 2 f frequency. Note that the heating effect hastwo components; constant and 2 f -oscillating ones. Figure 5also shows that as soon as the alternating field is turned on,the average sample temperature raises immediately due to theconstant heating component, followed by the 2 f oscillation.We have also verified that the amplitude of 2 f componentincreases rapidly with amplitude H0 and frequency of thealternating field, whereas the 1 f component due to the in-trinsic MCE remains constant. The eddy current heating canvery effectively be suppressed by a reduction of the cross sec-tion of the sample perpendicular to the field axis and as wellas by reducing the frequency and/or amplitude of the fieldmodulation.At an elevated temperature, where the electrical con-ductivity is smaller, and therefore, the eddy current heatingis less severe, a mixture of intrinsic 1 f and extrinsic 2 fT (K)5.0155.0004.985H (T)(a)(b)0.1000.0950.0900.08560504030Time (s)CeCoIn 5H//[001]FIG. 5. Sample temperature (a) and magnetic field (b) vs time for a high-quality single crystal of CeCoIn5 for H ‖ [001]. Note that the temperatureoscillation has twice the frequency of the field modulation.013905-4 Y. Tokiwa and P. Gegenwart Rev. Sci. Instrum. 82, 013905 (2011)0.2880.2860.284T (K)344034303420341034003390Time (s)CeCoIn 5H//[001]FIG. 6. (Color online) Mixture of 1 f - and 2 f -oscillations of sample temper-ature of CeCoIn5 under the same condition as Fig. 5. The red line is a fit toT (t) = a0 + a1t + a2t2 + T0 sin[2π f (t − δt)] + T 2 f0 cos[2π2 f (t − δt2 f )].oscillations can be found, as shown in Fig. 6. The temperatureas a function of time is well fitted by a function with two os-cillating components, T (t) = a0 + a1t + a2t2 + T0 sin[2π f (t− δt)] + T 2 f0 cos[2π2 f (t − δt2 f )] (red line in Fig. 6). We de-duced T0, using several different frequencies but the same H0and found a constant T0. This indicates that the obtained T0 isstill valid under the influence of eddy current heating.IV. SUMMARYWe developed a high-resolution low-temperaturealternating-field technique for quasiadiabatic measurementsof the MCE and magnetic Grüneisen ratio for clean metallicsamples down to mK temperatures and in large magneticfields. The performance check has revealed excellentagreement with the magnetic Grüneisen ratio calculated bymagnetization and specific heat measurements. The mainadvantages of the new technique compared to the latter arethat it is much faster and easier and that its resolution is muchhigher. The capability of continuous temperature sweepsmakes possible thorough investigations of the magneticGrüneisen ratio in the entire H -T phase space, which is ofparticular interest for systems close to magnetic-field inducedQCPs.ACKNOWLEDGMENTSWe are grateful to M. Brando, M. Lang, R. Movshovic,A. Steppke, K. Winzer, and B. Wolf for valuable conversa-tions. We thank H. S. Jeevan and E. D. Bauer for providingthe Yb(Rh0.95 Fe0.05)2Si2 and CeCoIn5 single crystals, respec-tively. Work supported by the German Science Foundationthrough SFB 602 TPA19 and FOR 960 (quantum phase tran-sitions).1L. Zhu, M. Garst, A. Rosch, and Q Si, Phys. Rev. Lett. 91, 066404 (2003).2M. Garst and A. Rosch, Phys. Rev. B: Condens. Matter 72, 205129(2005).3R. Kuchler, N. Oeschler, P. Gegenwart, T. Cichorek, K. Neumaier, O.Tegus, C. Geibel, J. A. Mydosh, F. Steglich, L. Zhu, and Q. Si, Phys. Rev.Lett. 91, 066405 (2003).4R. Kuchler, P. Gegenwart, K. Heuser, E. W. Scheidt, G. R. Stewart, and F.Steglich, Phys. Rev. Lett. 93, 096402 (2004).5T. Lorenz, S. Stark, O. Heyer, N. Hollmann, A. Vasiliev, A. Oosawa, andH. Tanaka, J. Magn. Magn. Mater. 316, 291 (2007).6Y. Tokiwa, T. Radu, C. Geibel, F. Steglich, and P. Gegenwart, Phys. Rev.Lett. 10, 066401 (2009).7A. W. Rost, R. S. Perry, J.-F. Mercure, A. P. Mackenzie, and S. A. Grigera,Science 325, 1360 (2009).8B. Fischer, J. Hoffmann, H. G. Kahle, and W. Paul, J. Magn. Magn. Mater.94, 79 (1991).9Y. Tokiwa, M. Schubert, P. Gegenwart, J. S. Hirale, and R. Movshovich,(unpublished).10R. Movshovich, M. Jaime, J. D. Thompson, C. Petrovic, Z. Fisk, P. G.Pagliuso, and J. L. Sarrao, Rev. Lett. 86, 5152 (2001).

Y. Tokiwa

P. Gegenwart

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Review of Scientific Instruments

High-resolution alternating-field technique to determine the
  magnetocaloric effect of metals down to very low temperatures

High-resolution alternating-field technique to determine the magnetocaloric effect of metals down to very low temperatures

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