PHYSICAL REVIEW B VOLUME 49, NUMBER 9 1 MARCH 1994-I

Temperature dependence of the nuclear spin-lattice relaxation time in copper metal to below 10pK

M. P. Enrico, S. N. Fisher, A. M. Guenault, I. E. Miller, and G. R. PickettSchool ofPhysics and Materials, Lancaster University, Lancaster LA1 4YB, United Kingdom

(Received 7 September 1993)

We have cooled a high-purity specimen of copper to a lattice temperature of 7+1 pK in a field of 6.8mT. We have followed the time evolution of the lattice temperature as the specimen subsequently

warms in order to investigate to lower temperatures the apparent anomaly in the spin-lattice relaxation

time recently observed in this metal. We find the behavior of the lattice temperature to be entirely con-

sistent with the conventional Korringa relation.

1

Te

we find

d 1

dt T„%e can express the time derivative of T„in terms of thenuclear heat capacity C„=AB/T„and the incomingheat leak Q to yield Q =(n AB /T„)T„Introducing th. equantity n AB /a which has dimensions of power, as the"scale" heat flow Q„wemay also write

nQ,(2)

In two recent papers' it has been suggested that thenuclear spin-lattice relaxation rate in copper metal be-comes anomalously slow at microkelvin temperatures.The apparent Korringa constant was observed to increaserapidly at the lowest temperatures. This behavior hasbeen ascribed to the appearance of a gap in the electronicdensity of states in the metal. In the present paper we re-port an investigation of the temperature dependence ofthe Korringa constant in copper to well below 10 pK. Inour sample of very pure copper we see no evidence of anydeviation from the Korringa relation.

A direct measurement of the spin-lattice relaxationtine ~& is not easy at the lowest temperatures since it im-plies a simultaneous knowledge of the nuclear and latticetemperatures and their time derivatives. At microkelvintemperatures, with present knowledge, it is difBcult tomake very sophisticated measurements of any parametersince the cooling of the sample is so marginal. However,we have chosen a straightforward test for the Korringarelation by simply observing the time evolution of thetemperature of the electronic system in a sample ofcopper after a series of adiabatic nuclear demagnetiza-tions.

In the following we make the convenient assumptionthat the electron temperature T, is greater than the split-ting of the nuclear spin levels, i.e., that the high-temperature Curie-law limit applies. Starting from theKorringa relation ~,T, =~ and the basic dynamic equa-tion for the nuclear temperature T„,

Since our thermometer measures T, rather than T„wecombine (1) and (2) to obtain

1

dt T~ tt(Q+ Q, )

On the assumption that the heat leak Q is constant, thenboth the nuclei and the lattice warm up linearly in in-verse temperature. Consequently a plot of 1/T, vs timeshould yield a straight line if tt is constant. Since the ob-servations of Refs. 1 and 2 suggested that at 6.9 mT and20 pK in copper ~& was already a factor of 10 greaterthan that suggested by the conventional relation, such adeviation would readily be seen in any warm-up curve.

The experiment is made in a nuclear demagnetizationstage hung from a low-temperature dilution refrigerator.The demagnetization apparatus is a four-stage devicemade in two nested parts, and is a more sophisticated de-velopment of an arrangement we have used earlier. Theouter cell, containing copper Sakes and liquid He, actsas a guard cell. This copper-flake refrigerant has a largequadrupole heat capacity and after demagnetizationremains at around 1 mK, thus shielding the inner cellfrom most sources of external heat leak. The inner cell ofepoxy contains three copper plates as shown in Fig. 1.The three plates are arranged in series thermally and areseparated from each other and from the outer cell bymassive single-crystal aluminum superconducting heatswitches. The warmest of the three plates is supportedfrom the epoxy container by a 1-mm-diameter Ag wireleading up to the main heat switch and by two bent Agwire springs at the lower end which allow for diFerentialcontraction between the copper and the epoxy container.The other two plates are supported only by the heatswitches. The lowest-temperature plate is of 99.999%pure copper annealed in oxygen with a residual resistanceratio of around 3000, and constitutes the specimen. Thisplate is approximately 5 cm X2 cm X 1 mm and consistsof 0.134 moles. The plate is furnished with a heater to al-low calibration against the known nuclear heat capacityand against the known high-temperature Korringa con-stant as explained below. We have used throughout thethermal average value of 1.18 s K. To avoid problemswith unknown heat capacities, the heater is made of thincopper wire and has a resistance of around 1.5 mQ. Theheater is connected by thin pure-tin leads (thermally an-

0163-1829/94/49(9)/6339(4)/$06. 00 49 6339 1994 The American Physical Society

6340 BRIEF REPORTS 49

Ag thermallink to heatswitch

Plate 3(5N Cu)

IIt ~II

II

lII

Single crystalsuperconductingAl heat switch

Pt NMRthermometer

II

II g ~

NMR coil in

I

I

Iip' "~ ', :

outer cellI

I

I

Cu heaterI

I'Iwire

Plate 2(4N5 Cu)

II

I

Plate 1

(4N5CU) ~ I

I

II

I

Sprung Ag8Upports

Inner cellepoxy wall

FIG. 1. The inner ce11. The warmest plate (plate 1) is sup-

ported from the epoxy ce11 wall by the upward-going silver wire

to the main heat switch and by two lower silver wires which are

sprung to take account of differential therma1 contraction.Plates 2 and 3 are suspended in turn from plate 1 only by thea1uminum heat switches. The Sna1 specimen plate 3 carries a

copper wire heater, connected by tin leads, and a platinum wire

thermometer soldered by silver to a silver wire spot-welded tothe copper plate. The coil driving the Pt thermometer is in theouter ce11. A further short silver wire is attached to plate 3 forfuture thermal connection of experiments but none was includ-

ed in this experiment.

chored to the warmest plate) to a NbTi four-lead systemat the wall of the inner cell. To minimize the amount oforganic material in the system, the heater is glued to thespecimen plate by a minimum quantity of Stycast 1266epoxy diluted in acetone and applied under a microscope.The aluminum heat switches are spark cut from a bulksingle crystal and attached to the copper via pure silverlinks. The Pt NMR thermometer consists of a bundle of1300 25-pm-diameter Pt wires melted into a 1-mm-

diameter Ag wire which is then spot-welded to thecopper. The final stage is thus almost entirely pure metalwith only a microscopic quantity of insulation material.

The specimen plate is designed to be very compactthermally. All the Cu nuclei are exposed to the samemagnetic field and are demagnetized together. The onlymajor component which is not in direct contact with therefrigerant is the Pt thermometer, which has negligibleheat capacity compared with the rest of the system. Thiscompactness means that during demagnetization irrever-sibilities are minimized and there is little entropy change.The thermometer configuration is designed with twothings in mind. First, we want the thermometer torespond quickly, and secondly we want to reduce heatleaks into the Pt itself. Therefore the thermal path be-tween Pt and copper refrigerant is made to be as good aspossible, with the ends of the Pt wires only a few millime-ters from contact via high-conductivity silver with thecopper refrigerant. The driving coil is physically separat-

ed from the Pt wires and is placed in the outer cell.There is no nonconducting material in contact with thethermometer. The thermometer time constant is a fewminutes at 10 pK in a 6.8-mT field. This design is verysuccessful in reducing heat leaks into the Pt but meansthat a calibration against some other temperature stan-dard is not possible, since a particular calibration isunique to the thermometer-coil arrangement and is prob-ably not stable after cycling to room temperature, owingto relative movement of the Pt wires and the coil.

The thermometer is calibrated by two methods, thefirst against the known high-temperature value of theKorringa constant and the second against the nuclearheat capacity of the copper. The copper sample is ofhigh purity so that the heat capacity can be safely takenas the calculated value. The Korringa relation allows avery simple calibration based on the maximum value ofthe gradient of inverse electron temperature, as follows.Assuming that the y value registered by the Pt thermom-eter corresponds to the lattice temperature of the copper(i.e., ignoring any thermometer lag for the time being),then the Pt NMR calibration has the Curie-law formT, = alp. Substituting this into (3) we find

a(Q+Q, )

This means that for a constant high heat leak into thesample there is a maximum rate of decrease in y given by—u/z. Thus the calibration constant a can be directlyderived from the temperature evolution of the thermome-ter on heating and a knowledge of the Korringa constant.

The platinum thermometer is addressed by a PLM-4pulsed NMR system manufactured by RV-ElektronikkaOY, Helsinki, and modified for low-frequency use. Thisinstrument is very much microprocessor driven, but,when isolated from the NMR coil except during the actu-al measurement, gives an impressively small heat leakinto the nuclear stage.

The experimental procedure is straightforward. Thenuclear stage is magnetized in a rms field of 6.8 T andprecooled over several days to a starting temperature be-tween 10 and 5 mK. The system is then demagnetizedover a period of some six hours to a final field of 6.8 or3.4 mT, corresponding to Pt NMR frequencies of 62.5and 31.25 kHz, respectively. We make no great effort tooptimize the demagnetization profile since in our systemirreversibilities are quite small. The main solenoid in theHe bath has a p-metal lining which effectively switches

off the remanent field, and therefore towards the end ofthe demagnetization the field is gradually transferredfrom the main solenoid to a subsidiary coil at the temper-ature of the still which is driven by a second current sup-

ply. When the final field configuration is achieved, bothcoils are persisted and the current supplies switched off.Working with the PLM-4 electronics where the frequen-cies are submultiples of 250 kHz, we can very quicklycheck that the NMR frequency is correct by simply ob-serving the aliasing of the signal on a Gould digital oscil-loscope which happens to have a commensurate digitiz-ing frequency. Therefore, while the measurements arebeing made, all electronic and electrical equipment inside

49 BRIEF REPORTS 6341

89

& 10

~ 15

E 20

I-50—

100—00

50 100 150Time after demagnetization ( h )

200

FIG. 2. Two warming curves after demagnetization to 6.8mT plotted as inverse temperature against time. Curve A wasmade from a starting temperature of 6 mK and curve 8 from 5mK. The large discontinuities in the curves represent the tran-sients from filling refrigerants. The data of curve A were takenafter the system had been below 4.2 K for 47 days and curve 8after 95 days. The solid and dotted lines represent the calculat-ed time evolution of the copper nuclear and lattice tempera-tures, respectively.

the shielded room is switched off, other than the low-

frequency still regulator, the room lighting, the PLM-4electronics, and occasionally the Gould oscilloscope.While we are doing a calibration run, a battery-drivenheater is also switched on. We have made a whole seriesof demagnetizations from different starting temperatures,and with a variety of subsequent heating periods applied,to determine the thermometer calibration.

There is a design fault in our system which makes thisnot quite straightforward. The Al heat switches were cutby spark erosion and have sharp rectangular edges. Inthe final fields of both 6.8 and 3.4 mT a very small frac-tion of the material along the edges remains normal andthe switches are very slightly leaky. This means that dur-

ing the calibration a small fraction of the heat delivered

by the heater is lost through the heat switch to the nextplate. Consequently, we have to make allowances bycomputer simulation for the heat flow between the plates.This is readily done by taking together a large selectionof warming and heating curves. Since the heat leak intothe final plate began at around 100 pW at the beginningof the run, falling to below 1 pW three months later, wehad plenty of time for recording these data while waitingfor the heat leak to subside. The final plate-temperaturecalibration at 6.8 mT we estimate to be accurate to 15%%uo,

that is to say, 61 pK at the lowest temperatures, withthat at 3.4 mT somewhat poorer.

The data for two warming processes without any heat-ing periods in a final field of 6.8 mT are shown in Fig. 2.Curve A represents data taken after a demagnetizationfrom 6 mK, when the system had been below 4 K for 47days and curve B represents a demagnetization from 5mK taken after 95 days. The discontinuities in thecurves represent the filling of refrigerants which has alarge impact on the specimen owing to the microphonicnature of the support. Making allowances for the refri-

gerant fills, we see that the curves show a gradually in-

creasing slope towards higher temperatures. This is afunction of small leakage of the heat switches betweenthe plates. The heat leak into the warmest platerepresents almost the entire heat input to the wholethree-plate system as this plate is in direct thermal con-tact with the epoxy ceil wa11 which continues to evolveheat over a period much longer than the total duration ofthe experiment. As the warmest plate warms the heatleak into the second plate increases which in turn beginsto warm more rapidly. For the last 50% of both wartn-

ing curves a large fraction of the total heat leak has beentransferred to the coldest plate as the other two havelargely exhausted their heat capacities.

Discounting the heat input generated by the refri-gerant fills, we note that for the run corresponding tocurve A, the final plate would have remained below 20pK for 80 hours while the corresponding figure for curveB is nearly 200 h. Without the heat leaks between theplates, the warming of the coldest plate would have beengoverned by the collapse of the temperature of the outercell, a much longer period.

The curves superimposed on the experimental data in

Fig. 2 represent computed simulations of the time evolu-tion of the copper nuclear temperature (solid curves), andof the lattice temperature (dashed curves). In these sitnu-lations we have assumed that the Korringa relationholds, with sc at its high-temperature value. As describedabove, allowance is made for a steady heat leak, with ad-ditional heat input on cryogenic transfers, and for imper-fect aluminum heat switches. In particular we note thatin the temperature range above about 12 pK, where theheat leak into the final plate is almost constant for bothcurves, the rate of decrease of inverse temperature isquite linear. There is no sign of the anomalous behaviorsuggested by Refs. 1 and 2, where at this field at 20 pKthe apparent Korringa constant was observed to have in-creased by an order of magnitude over its high-temperature value. We have taken similar data at a finalfield of 3.4 mT, although the quality is much lower, as forsome reason the heat leaks tend to be higher than thoseat 6.8 mT. However, the warming curves from 10 pKupward are again almost linear and show no sign ofanomalous ~& behavior.

The curves in Fig. 2 are calculated in the high-temperature approximation, both for the copper and forthe platinum nuclei. At 6.8 mT, the splitting for thecopper nuclear levels is 3.76 pK, this being a weightedaverage over the two isotopes Cu and Cu. This meansthat at the lowest temperatures the simple Korringa rela-tion is slightly modified, and the copper nuclear heatcapacity begins to deviate slightly from the 1/T rela-tion. Second, the two platinum nuclear levels lie at+1.49 pK and thus, assuming no cooperative effects inthe Pt thermometer, the temperature deviates at thelowest temperatures from that suggested by the high-temperature calibration. However, at 7 pK the error inthe temperature scale is only 1.S%, and the curves calcu-lated for the evolution of the nuclear and lattice tempera-tures are almost indistinguishable from those shown inthe figure.

6342 BRIEF REPORTS

Apart from monitoring the behavior of ~& in this exper-iment, we have made a number of technical observations.The first appeared because either we had a very smallleak from the refrigerator into the inner vacuum jacket orwe did not make a very good job of pumping out the ex-change gas on the original cool down. Thus, after twomonths' running, we had to warm the system to around 2K in order to pump the vacuum jacket to reduce thebackground pressure. We fully expected that the heatleak into the final plate would have been reset to its initialvalue after the first cool down, but in fact there was virtu-ally no efFect on the residual heat leak. This shows thatthe energy scale of whatever causes the time-dependentheat leak is much larger than 2 K. Secondly, during theearliest runs, with relatively large amounts of heat being

evolved by the plates, we saw a number of very sharppulses in which T, increases suddenly by a factor of 2—4and then recovered over the next few minutes. Initiallythese pulses occurred with a frequency of about one perhour and became less frequent as the run progressed.From this behavior we infer that a large part of our heatleak arises from the relaxation of mechanical strain in thelow-temperature parts. Thirdly, we find that the ther-mometer appears to be in excellent thermal contact with

the sample even at the lowest temperatures, confirmingthat the thermal design is indeed very compact, and thatirreversibilities are rather small. We are thus fairlyconfident that with some redesign of the cell considerablylower temperatures should be possible.

~K. Gloos, P. Smeibidl, and F. Pobell, in Proceedings of the 19thInternational Conference on Low-Temperature Physics [Physi-ca B 165&166, 789 (1990)].

K. Gloos, P. Smeibidl, and F. Pobell, Z. Phys. B 82, 227 (1991).A. M. Guenault, V. Keith, C. J. Kennedy, I. E. Miller, S. G.

Mussett, and G. R. Pickett, in Proceedings of the 17th Inter

national Conference on Low Temper-ature Physics, edited by

U. Eckern, A. Schmid, W. Weber, and H. Wuhl (North-

Holland, Amsterdam, 1984), p. 1157.4M. I. Aalto, P. M. Berglund, H. K. Collan, G. J. Ehnholm, R.

G. Gylling, M. Krusius, and G. R. Pickett, Cryogenics 12,184 (1972).