THE INTERACTION OF HEAT PULSES WITH PARAMAGNETIC SPINS

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THE INTERACTION OF HEAT PULSES WITH PARAMAGNETIC SPINS

J. Wigmore

To cite this version:

J. Wigmore. THE INTERACTION OF HEAT PULSES WITH PARAMAGNETIC SPINS. Journal

de Physique Colloques, 1972, 33 (C4), pp.C4-107-C4-110. �10.1051/jphyscol:1972423�. �jpa-00215100�

JOURNAL DE PHYSIQUE

Colloque C4, suppliment au no 10, Octobre 1972, page C4-107

TEE INTERACTION OF HEAT PULSES WITH PARAMAGNETIC SPINS

J. K. WIGMORE

Department of Physics, University of Lancaster, Lancaster, England

RksumB.

On prksente un expos6 bref des objectifs, de la thkorie et des expkriences. La dis- persion par les spins des pulses thermiques balistiques est decrite en terme d'un coefficient d'att6 nuation, cornme pour la resonance acoustique paramagnetique, tandis que la dispersion spin- phonon dans la limite diffusive est analysQ en utilisant un temps de relaxation des phonons.

Quelques rksultats sont r6sumks sur la dktermination de la distribution en frequence d'un pulse thermique, sur la dispersion klectron-phonon dans une couche mince, sur la diffusion par effet de diffkrence de masse dans MgO, et sur les separations en knergie dans A1203 : Fe.

Abstract.

A brief review of objectives, theory and experiments is presented. The scattering by spins of ballistic heat pulses is described in terms of an attenuation coefficient, analogous to acoustic paramagnetic resonance, while spin-phonon scattering in the diffusive limit is analysed via a phonon relaxation time. Some results are summarised on the determination of the frequency spectrum of a ballistic heat pulse, the electron-phonon scattering in a thin film, mass difference scattering in MgO, and energy splittings in A1203 : Fe.

I. Introduction. - The study of heat pulse propa- gation in paramagnetic materials was begun with two primaxy~ohjectiues.~

Firstly, as a means of determining the phonon frequency distribution of ballistic heat pulses, it was suggested that a heat pulse phonon spectrometer could be devised using a paramagnetic spin system as the frequency sensitive element [I]. Phonons of energies equal to the spin splittings would be absorbed from the heat pulse, and tuning could be obtained by varying the splittings with a magnetic field, analogously to acoustic paramagnetic resonance (APR) 121.

Secondly, the very high frequency phonons of the heat pulse could be used to study some less well-known impurity systems having excited states within about 30 cm-' of the ground state [3]. Few spectroscopic techniques work in this energy regime, and the use of heat pulses for this purpose is similar to the technique of magnetothermal conductivity [4], but with the additional advantage of temporal and spatial resolu- tion.

I t was necessary to find a new bolometer for these investigations since superconducting types do not function in a large magnetic field. The semiconducting avalanche bolometer was developed specially for these experiments. Complete details of this device have been published elsewhere (51, but the use of GaAs : Zn instead of Si or Ge has led to an increase in sensitivity of about 30 dB since this publication. At best, the bolometer will detect approximately 1 milliwatt 100 nanosecond pulses into the heater, and its sensiti- vity decreases by about 10 % in a magnetic field of 7 T.

A calibration against field is made with heat pulses propagating through a second, non-magnetic, crystal

bonded on to the back of the bolometer. The response time of the device is about 500 nanoseconds, which is slower than thin-film types but still fast enough to observe ballistic heat pulses and ultrasonic pulses in solids.

11. Theory. Ballistic flow.

The theory of heat pulse propagation in solids [6] is conveniently sepa- rated into the two regimes, ballistic and diffusive, defined by i(v) 2 d, and I(v) < d, respectively, where A(v) is the mean free path of phonons of frequency v, and d is an average dimension of the specimen.

In the ballistic situation, the magnitude of a mode is monitored experimentally as a function of magnetic field. The effect of the spins is to absorb phonons from the heat pulse, and to re-emit them again in random directions at an average time TI later, where T I is the spin lattice relaxation time, typically 0.1-10 milli- seconds at helium temperatures. The time interval between successive heat pulses should clearly be longer than TI. An attenuation coefficient P(v, H ) is used to represent the spin-phonon absorption so that the total heat pulse power in a transmitted mode, E(H), a function of magnetic field, H, can be written

00

E(H) = A J hvp(v) eap j - a(v) - ~ ( v , H) 1 ^{dv (1)}

0

where A is a normalisation constant, depending mainly on the geometry of the specimen, p(v) is the frequency distribution of phonons leaving the heater, and a(v) is a second attenuation coefficient representing other scattering processes that may be removing phonons from the ballistic envelope. Note, however, that normal phonon interactions cannot be represented by such a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972423

C4-108 J. K. WIGMORE

term, but these can be neglected at helium tempera- bolometer are effectively both points, the power m tures. The crystal is assumed to be dispersionless so takes the value 3 ; this is closer to the experimental that the heat pulse frequency spectrum is neither time situation than the one-dimensional case m

1. Some nor space dependent. error will result from assuming the crystal to be semi- An expression for P(v, H ) can be taken over directly infinite, although agreement should be improved by from APR [2] for a specimen length, L sand-blasting the sides of the specimen, since Ander-

2 n2 vL son [9] has shown that a roughened surface in contact

P(v, H)

-y- C M ; pi(v, H) gi(v, H ) (2) with liquid helium has a transmission coefficient

pv h

of -- 100 % for thermal phonons.

where p is the density of the material, u the appropriate The effect of the spins is to make an additional phonon velocity, h Planck's constant and Mi, pi(v, H ) contribution to il(v), and thence to z(v), the phonon and g,(v, H ) are respectively the matrix element of the relaxation time. Orbach [lo] has given an expression spin-lattice Hamiltonian per unit strain, the population for zs(v), the spin-phonon contribution to z(v)

difference per unit volume, and the lineshape, for the 2 n 2 v -

ith transition. The value of the ballistic heat pulse

( ⁷ _pV ^I:

sf ~ i ( v , HI gi(v7 H) ) ⁽⁴⁾

technique lies in the fact that different transitions may

be isolated by selecting different combinations of Here %i is a spin-phonon matrix element which must phonon polarisation, heat pulse direction, and magne- be averaged over phonons of all wavevectors and tic field orientation. polarisations since these are reaching the bolometer However, there are some significant differences simultaneously. For the same reason, all possible tran- between the present use of (2) and the acoustic situa- sitions must be included. The similarity to magneto- tion. Firstly, because of the high magnetic fields thermal conductivity is clear [4]. Finally the term for involved in the heat pulse experiments, both quadru- z(v) is substituted into an expression for the thermal polar and dipolar terms in the spin-lattice Hamiltonian conductivity [l 11, whence (again neglecting normal must be included in both Mi and g,(v, H) [7]. In APR processes)

experiments, involving fields below 1 T, the dipolar a = - C T ~ ex x4

contribution, proportional to H ~ , is neglected. The z(x) dx

ps So ^{(ex - 1)' (5)}

second feature of (2) as applied to heat pulses is that

population differences pi(v, H) must be calculated as a where x = hv/kT and C i s equal to 4 zk4/h3 v. This function not only of magnetic field but also of fre- approach is valid, however, only in the extreme diffu- quency [4]. The reason is that, unlike acoustic probes, sive limit since (5) cannot be used if, anywhere in the the heat pulse is not monochromatic and there is a specimen, the phonon distribution is very far different large absorption due to spins out on the wings of the from the local one.

resonances. Finally, (2) applies only if the spins are not Undoubtedly, the analysis for the diffusive situation saturated. In the heat pulse experiment, it is impossible is complex and as with magnetothermal conductivity, to apply the conventional APR check of varying the only qualitative agreement can be looked for. For an power level since the frequency spectrum also changes. order-of-magnitude estimate, K/ps can be written However, a simple calculation shows that 10'' spins

1 u2 Z(V) and z(v) calculated by assuming that all the in a field of 1 T require a minimum energy of 100 ergs P

for saturation, whilst a typical (1 watt 100 nanosecond) phonons are transverse polarisation with energies heat pulse contains only 1 erg spread over a wide fre- ^" kT.

quency band.

IV. Summary of experiments.

To date, relatively 111. Theory. Diffusive flow.

In the diffusive limit, few investigations have been carried out in this field.

the spin-phonon absorption is small compared with Most detailed have been ballistic experiments on other scattering in the crystal. The solution of the heat MgO : Fez+ aimed at determining the frequency distri- diffusion equation for a semi-infinite medium excited bution of a heat pulse [12]. MgO : Fez+ is an iso- by a &function of thermal flux at t = 0 and r = 0, tropic spin system with S = 1, but transverse ballistic

is [8] modes along a [I001 direction parallel to the magnetic

field interact only with the two AM

1 transitions, AT(r, t ) = B exp (2) which are identical in energy. As a magnetic field was (2 a Jnt>" ^{4 a 2 t (3)} applied, at constant heater power, the magnitude ofthe

transverse modes initially decreased, reached a mini- where AT(r, t ) is the increase of temperature above mum, and thenincreased again. The minimum at Hmi, ambient at a distance r from the source, and time t after in the transmitted heat pulse intensity corresponded to excitation, Bis a constant determined by the power and a maximum in the number of phonons absorbed by the duration of the applied heat pulse, and a is the quantity spins, and therefore to the maximum at v,,, of the fre- Kips, K being the thermal conductivity p the density, quency spectrum. Thus, approximately,g~Hmi,

hv,,,.

and s the specific heat of the material. If heater and A quantitative analysis by the methods of section I1 was

THE INTERACTION OF HEAT PULSES WITH PARAMAGNETIC SPINS

used to determine the characteristic temperature, 6, of the heat pulses for different heater power levels (Fig. 1 ; data for 680 A heater) and a comparison was made with theoretical curves based on the acoustic impedance and the perfect thermal contact models [13].

The results show clearly that for a constantan-MgO boundary, the interfacial thermal conductance can be described accurately by a Little-type calculation with phonon transmission coefficients determined by the relative acoustic impedances.

I I ' l c ' l ' l generator thickness 680

8.

0 "

cox

ambient temperature

thermal contact

Preliminary results have also been obtained with a specimen of Alz03 containing Fez+ and Fe3+. The interest in this material lies in determining the level scheme of the Fez+ impurity, which again has S = 1 but with a finite zero-field splitting of which neither the magnitude nor the sign has been unambiguously determined [16], [17]. The variation with H of the transverse modes along the c-axis parallel to H i s shown in figure 2. Such traces have proved difficult to analyse,

,

..=. . ".--

(arb~trary units)

1.05

-.

FIG.

Values of heat pulse temperature, 8, plotted against

generator power,

for two different thicknesses of the cons-

FIG.

Variation of transverse ballistic modes in

Fe tantan generator film. Also shown are theoretical curves calcu-

propagating along the c-axis parallel to the magnetic field, lated using

acoustic impedance and perfect thermal contact

models. ^H, at

Also presented in figure 1 are preliminary data taken using a heater film only 80 A thick. In this limit, the mean free path of the phonons emitted by the electrons is greater than the thickness of the film, so that they emerge directly into the MgO. From more refined measurements, it will be possible to determine the electron-phonon relaxation time of the electrons even though the resistance of the film is dominated by electron-impurity collisions. For the thin film, v,,, appears proportional to p1I2, rather than the

found in the thick film experiments.

Besides the spin-phonon scattering due to the Fe2+, the specimen also exhibited significant mass-defect scattering [14]. A term a(v)

D V ~ was included in (1) to take account of this, and the parameter D determined by a computer fit to data taken for 6 - 15 OK. In this regime, the shape of the phonon frequency = spectrum was much more sensitive to a(v) than to 6. The value of D obtained was in good agreement with calculations from the known concentrations of impurities in the MgO [15].

partly because of the additional presence of Fe3 +, but mainly because in this geometry phonons can be scattered by both AM

1 and AM

2 transitions.

The relatively fine structure is caused by the accidental coincidence in energy of two transitions. Since the absorptions due to the two transitions do not add linearly, the result is a change in the total absorption and therefore in the transmitted pulse. The magnetic fields at which these coincidences occur do not depend on the heat pulse temperature, and provide a valuable means of fixing positions of energy levels [4]. The technique can also be used on the diffusive component of the heat pulse.

Variation of transmitted heat pulse intensity with magnetic field has also been observed with the spin systems Ni3+ and V3+ in A1,03 and NiZ+ in MgO and TiO,. Data on these systems will be published later.

The original suggestion for this project was due to Dr. N. S. Shiven, and the work was begun while the author was at the IBM Research Center, Yorktown Heights.

References

El] WIGMORE (J. K.), Appl. Phys. Lett., 1968, 13, 73. [6] KWOK (P. C.), Phys. Rev., 1968, 175, 1208.

[2] For example, ROSENBERG (H. M.) and WIGMORE [7] MCMAHON (D. H.), Phys. Rev., 1964, 1344, 128.

( J . K . ) , Proc. R. Soc., 1967, 302A, 69. [8] VON GUTFELD (R. J.), Physical Acoustics (Academic [3] SHIREN (N. S.), private communication. Press : Editor Masson), 1969, 5, 133.

141 example, MCCLINT~CK (P. ^V. E.1, MORTON _[9] ANDERSON (C. H.), Physical Acoustics (Academic (I. P.), ORBACH (R.) and ROSENBERG (H. M.), Press : Editor Masson), 1972, 8, 1.

Proc. R. Soc., 1967,298A, 359 ;

CHALLIS (L. J.), MCCONACHIE (M. A.) and ^[lo] ORBACH (R.), Phys. Rev. Lett., 1962, 8, 393.

WILLIAMS(D. J.), Proc. R. sot., 1968, 3 0 8 ~ , 355. 1111 KLEMENS (P. G.), Solid State Phys., 1958, 7, 1.

[5] WIGMORE (J. K.), J. Appl. Phys., 1970, 41, 1996. [12] WIGMORE (J. K.), Phys. Rev., 1972, 5B, 700.

8

*

C4-110 J.

K . WIGMORE

[13] LITTLE (W. A.), Can. J. Phys., 1959, 37, 334. [16] LEWINER (J.), MEIJER (P. H. E.) and WIGMORE (J. K.), [14] WIGMORE (J. K.), J. Physique, 1971, SuppI. C 1, Phys. Rev., 1969, 185, 546.

766. [17] GANAPOL'SKII

M.), SOY. Phys. Sol. St., 1971,

1151 KLEMENS (P. G.), PI'OC. Phys.

1955, 68A, 1113. 12, 2095.

DISCUSSION

H. F. BUDD. - I don't understand the insensitivity K. WIGMORE. - The semiconducting avalanche of your detector up to 70 kilogauss. The breakdown bolometer depends on a thick, badly dislocated layer field is a sensitive function of the magnetic field in at the surface of the semiconductor, and it is not both longitudinal and transverse configurations, parti- unreasonable that the properties of this layer should cularly in the latter. Since your detector is biased just be very different from those of the bulk material.

in this avalanche range, how do you explain this ?