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Ultrasound studies of single crystal thulium in an applied magnetic field
Journal of Magnetism and Magnetic Materials 234 (2001) 387–394
Ultrasound studies of single crystal thulium in an applied magnetic field C.M. Lima, C. Edwardsb,*, S. Dixonb, S.B. Palmerb a
Department of Physics, University Brunei Darussalam, Tungku Link Road, BE1410, Brunei b Department of Physics, University of Warwick, Coventry, CV4 7AL, UK Received 3 April 2001; received in revised form 1 June 2001
Abstract The paper reports the first measurements of the single crystal elastic constants of the heavy rare earth metal thulium as a function of temperature and magnetic field. The constants were obtained from ultrasonic velocity measurements over a temperature range of 4.2–296 K and in applied magnetic fields of up to 5 T. The elastic constants; C11, C33, C44 and C66=(C11–C12)/2 were determined from the ultrasonic velocities. Anomalies in the elastic constants were observed at 58 K from the c-axis propagated shear wave measurements and at 55 K from the c-axis propagated longitudinal wave measurements. Significant softening of the elastic constants C33 and C44 was observed close to TN : Application of a magnetic field (>2 T) along the c-axis direction induced further softening of the material. Electromagnetic acoustic transducers (EMATs) were also employed in addition to conventional piezoelectric quartz transducers. A marked increase in the EMATs acoustic coupling efficiency (generation and detection efficiency) occurred close to TN : r 2001 Elsevier Science B.V. All rights reserved. PACS: 62.20.Dc; 75.30.Kz Keywords: Rare earth magnetism; Elastic constants
1. Introduction In zero applied field, thulium (Tm) exhibits a c-axis sinusoidally modulated antiferromagnetic ordering below a N!eel temperature, TN ; of 56 K. The wave vector Q is slightly less than 2/7c* at TN in this incommensurate c-axis modulated (CAM) phase. The magnetic structure begins to square up with higher order odd harmonics of the Q appear*Corresponding author. E-mail addresses: [email protected] (C.M. Lim), [email protected] (C. Edwards).
ing at B42 K. Q gradually increases to 2/7c* and forms a commensurate antiphase ferrimagnetic structure at a lock-in temperature around 32 K [1–5]. The commensurate ferrimagnetic structure is aligned along the c-axis and has a periodicity of seven atomic layers, with four layers parallel and three layers antiparallel, the net magnetization corresponds to 1/7 of the ionic moment of Tm (B1 mB). In the early studies of Tm it was reported that the magnetic moment does not fully develop in the ferrimagnetic phase at temperatures above 25–20 K and this was referred to as the Curie temperature, TC [1–3]. In applied
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magnetic fields over 2 T, below TN ; Tm develops a higher net moment and it becomes fully ferromagnetic (B7 mB) in magnetic fields greater than a critical field of 2.8 T below the lock-in temperature [2,6]. Anomalous magnetoresistive behaviour has been observed in between the CAM and 3,4ferrimagnetic phases indicating at least one mixed region (referred to as the A phase) with some incommensurate features [7]. Two additional intermediate regions have also been observed between the ferri- and ferromagnetic phases with boundaries at 3.3 and 3.6 T from magnetostriction data [6]. It was suggested that they may be due to the demagnetizing field of the sample (B0.8 T); there was no evidence of these phases reported in Ref. [7] which used a sample with a much lower demagnetizing factor. We report on measurements of the ultrasound velocity as a function of temperature and applied magnetic field in single crystal Tm using both quartz and electromagnetic acoustic transducers (EMATs) allowing the elastic and magnetoelastic behaviour of the material to be studied. The quartz transducers have to be bonded to the sample with an acoustic couplant which functions over the entire temperature range, whereas the EMATs operate via a non-contacting transduction mechanism and do not require any acoustic couplant.
2. Experimental procedures A Tm single crystal was grown by the solid-state method and cut in the form of a cuboid (4.355 3.588 3.77270.001 mm) with a- and caxis perpendicular to the spark planed sides of the cuboid [8]. The maximum c-axis demagnetizing field is 0.7 T in the fully ferromagnetic phase and 0.1 T in the 3,4-ferrimagnetic phase. These fields should be subtracted from the applied field to give the internal field in the ferromagnetic (>3.5 T, the critical field plus the demagnetizing field) and the ferrimagnetic phases (o2.8 T). In the intermediate region (2.8–3.5 T) we might expect some phase coexistence giving additional regions similar to those reported in Ref. [6]. Ultrasound measurements were performed in a send/receive mode where the 10 MHz quartz
transducers were driven by an RF tone burst. The time of flight of the ultrasound pulse echoes was determined using an auto-correlation technique. Shear and longitudinal ultrasonic waves were propagated along both the a- and c-axis. The relationship between the various velocities of propagation and the elastic constants for a hexagonal crystal allowed the five independent elastic constants (C11, C33, C44 and C12) to be calculated [9,10]. A more detailed description of the experimental set-up is given in Ref. [11]. The velocity measurements were carried out by attaching the quartz transducers to the sample surface using GE varnish as an acoustic couplant. The GE varnish acoustic bond did not fracture even when the sample and transducer were repeatedly cycled from room temperature down to 4.2 K. The EMAT design was based on the Lorentz mechanism [12]. The EMAT coil had the spiral pan-cake geometry and used 0.08 mm diameter copper wire. A static magnetic field was applied perpendicular to the coil by a superconducting magnet, hence generating in-plane radially polarised shear waves. The sample was placed in light contact with the EMAT coil in a sample holder. For a non-magnetic conductor, the EMAT functions purely via the Lorentz mechanism where generation and detection efficiencies are linearly proportional to the static field. In magnetic materials other magnetoelastic mechanisms are possible, these mechanisms vary non-linearly with applied field and are easily revealed by plotting the square root of the EMAT signal amplitude against applied field [13].
3. Results 3.1. Longitudinal elastic constants, C11 and C33 Fig. 1 shows C11 and C33 as a function of temperature in zero applied magnetic field (the errors are represented by the size of the markers). A signal processing technique described in Ref. [14], which looks for local changes in gradient, was applied to the data to extract any anomalies. Anomalies in C33 were observed at 55 K and
C.M. Lim et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 387–394
Fig. 1. C11 and C33 as a function of temperature at zero applied magnetic field.
B108 K where C33 shows a small change in gradient. The anomaly at 55 K corresponds to the N!eel temperature of Tm [1–6]. On further cooling C33 increased from 83.6 to 88.2 GPa at 4.2 K. In zero magnetic field, C11 gradually increased as the sample was cooled with no anomalies until a small change in gradient at 91 K. The experiment was discontinued at 64 K due to the loss of signal. The high temperature (>90 K) gradient changes do no correspond to any known phase changes. Application of a magnetic field along the c-axis has a minimal effect on C33 below 2 T, see Fig. 2, where the general features of the C33 curve are similar to the zero field curve. Note that for presentation purposes the C33 data are displaced as indicated in the legend and only 1 in 2 data points are shown. The N!eel temperature minimum remained unchanged at 54 K for c-axis applied magnetic fields of 1 and 2 T but decreased to 50 K at 3 T. The 3 T C33 data, below 54 K is lower than the zero field values due to a modification of the magnetic structure of the material. Increasing the c-axis applied field above 3 T induces further softening of C33 below 50 K. The distinct change in gradient drops to B40 K at 4 T and may correspond to squaring up of the magnetic structure, suggesting that the CAM to ferrimagnetic transition, below 50 K, is not a single stage transition. This is in agreement with the magnetoresistance measurements reported in Ref. [7] where it was suggested that a mixed A phase exists between the CAM and ferrimagnetic phases.
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Fig. 2. C33 as a function temperature in a c-axis applied magnetic field of 1–5 T.
The differences between the 4 and 5 T C33 data are relatively small, thus implying that the material has a stable magnetic structure once the c-axis applied field is greater than 4 T. We do not see any evidence of the 32 K lock-in but there are small anomalies present between 25–27 K in magnetic fields greater than 3 T. Fig. 3 shows a 3D plot of the C33 data to enable the large drop in elastic constant between the 3,4ferrimagnetic and ferromagnetic phases to be seen more clearly. The low temperature C33 data up to 2 T is almost constant, the 3 T data is slightly lower whereas the data over 4 T is much lower. This is consistent with the demagnetizing considerations for the crystal, the demagnetizing field could reduce the internal field to below 2.8 T for an applied field of 3 T.
3.2. Shear elastic constants C44 and C66 Fig. 4 shows the zero field temperature dependence of C44 and C66=(C11–C11)/2. C44 increases almost linearly as the sample cools from room temperature to 142 K. Below 142 K it decreases gradually and steepens below 100 K before dropping to a minimum value of 27.4 GPa at 59 K. On further cooling below 59 K, C44 increases steadily. Although no clear indication of the zero field lockin (32 K) was observed, changes in gradient are observed at 40 and 25 K. The 40 K anomaly may again be associated with the CAM/A phase
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Fig. 5. C44 as a function of temperature with a c-axis applied magnetic field of 1–3 T.
Fig. 3. 3D C33 plot.
coupling between the ultrasonic waves and the magnetic structure. Fig. 5 shows C44 as a function of temperature with applied magnetic field parallel to the c-axis. The magnetic field causes a reduction in C44 from 27.5 GPa at zero field to 25.0 GPa at 3 T and a shift in the C44 anomalies to lower temperatures. The C44 minimum, which corresponds to T N, moves from 59 K in zero field to 47 K at 3 T, the 2 T curve shows a distinct gradient change at 23 K, again possibly corresponding to T C. At 3 T, the shear wave is highly attenuated as the sample temperature approached 31 K possibly due to the formation of the 3,4-ferrimagnetic structure. 3.3. EMAT signal amplitude measurements
Fig. 4. C44 and C66=(C11+C12)/2 as a function of temperature at zero applied magnetic field.
boundary and 25 K could correspond to the TC quoted in the early work. Within the temperature range, 91–296 K, C66 increases smoothly as the sample is cooled. However, it was not possible to obtain any C66 results below 90 K due to loss of signal. The propagation of both shear and longitudinal waves down the a-axis is difficult as a result of very high attenuation resulting from strong magnetoelastic
The EMATs used in this work generated inplane radially polarised shear waves and were used for both the generation and detection of ultrasound. The EMAT acoustic coupling efficiency is a combination of the acoustic generation and detection efficiencies, hence the raw data is proportional to the efficiency squared. The amplitude of the third shear wave echo corresponding to a shear wave polarised along the c-axis was measured, this will be discussed more fully in the next section. The temperature dependence of EMAT efficiency with an applied field of 0.6 T, for shear waves propagating down the c-axis, is shown in Fig. 6. The present system could only detect an EMAT signal when the applied field was at least 0.5 T. The
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Fig. 6. EMAT signal amplitude (3rd. shear echo) as a function of temperature with a c-axis applied magnetic field of 0.6 T.
largest change in the EMAT acoustic coupling efficiency was observed in the vicinity of the N!eel temperature. A series of EMAT isofield signal amplitude measurements was carried out as a function of temperature (cooling) with applied fields ranging from 0.5–2 T. The EMAT efficiency was calculated by taking the square root of the measured voltage. The efficiency was then divided by the applied field to remove the Lorentz contribution and reveal the magnetoelastic effects and is presented as a contour plot in Fig. 7. Lines were drawn over the image plot to indicate the boundaries of the magnetic phases of Tm as reported in Refs. [1–7]. The ultrasonic anomalies are presented with the previously reported phase boundaries over a wider magnetic field range in Fig. 8. Anomalies are found in the vicinity of the paramagnetic/CAM phase boundary, vertically above the CAM/A phase boundary and delineating a near vertical anomaly at 25 K, implying that there may be additional phases above the critical field. The EMAT experiment could not be extended beyond 2 T because the ultrasound waveforms became complicated with mode converted signals interfering with the main shear wave echoes. One would expect a shear wave to be attenuated as the magnetic field is increased due to eddy current loses. Since the sample is a good conductor, eddy currents will be induced as the shear vibration direction is orthogonal to the applied field, whereas the mode converted longitudinal waves vibrate parallel to the magnetic field and suffer no eddy current losses. Variations in
Fig. 7. Contour plot of EMAT signal amplitude of Tm. The markers show anomalies observed in the ultrasound velocity measurements. The lines are magnetic phase transition boundaries as reported in [4–6].
Fig. 8. Comparison of ultrasonic anomalies with previously reported phase boundaries [1–7].
EMAT acoustic coupling efficiency has been previously been reported for single crystals of Gd and Dy [15,16]. 3.4. Tm slowness surfaces The propagation of EMAT generated in-plane radially polarised shear waves down the a-axis shows the anisotropic nature of the material. Fig. 9
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Fig. 9. Birefringence effect down the a-axis at 60, 70 and 96 K. Applied magnetic field of 2 T directed along the a-axis. Sb and Sc are the b-axis and c-axis polarised shear echoes, respectively. Echoes are labelled according to the number of transits through.
shows ultrasonic waveforms taken at 60, 70 and 96 K in an a-axis magnetic field of 2 T. Birefringent behaviour is clearly evident where the ultrasound energy is steered into the two orthogonal polarised shear waves as a result of the crystal anisotropy. This results in two sets of shear waves propagating down the a-axis, indicated as Sb and Sc with shear wave polarisation directions along the b- and c-axis, respectively. At T ¼ 96 K the generation efficiency and attenuation of both Sb and Sc are very similar. As the sample was cooled (while remaining in the paramagnetic phase) the generation efficiency of Sc increased strongly while Sb showed only a marginal increase possibly indicating that the energy is being preferentially steered into the C44 mode. The other possible explanation for the high generation efficiency of Sc ; is that the magnetoelastic coupling is larger along the c-axis. It should be noted that marked increases in EMAT efficiency often occur over a range of temperature above or below magnetic phase transitions, see Fig. 6 where the increase starts at B80 K. The slowness (inverse velocity) surfaces have been calculated from the measured elastic constants using equations given in Ref. [10]. This is plotted to illustrate the effect of beam steering in single crystal Tm at 90 K, see Fig. 10. Slowness
Fig. 10. Calculated slowness surfaces for Tm at 90 K in the a–c plane in units of m1 s, assuming C13=25 GPa. L is the longitudinal wave surface and S1 and S2 are a2c plane polarised and the b-axis polarised shear wave surfaces.
surfaces are useful for evaluating beam steering effects as the direction of energy transport is defined by the normal to the slowness surface. Flat regions of the slowness surface indicate strong beam steering effects. It must be emphasised that in constructing the plot all the elastic moduli were known from the velocity data except for C13. The C13 measurement requires the propagation of ultrasonic waves at an angle of 451 to the c-axis [9,10]. The small dimensions of the sample prevented us from cutting the additional parallel faces required for the C13 measurement. The present data for Tm, together with all the previously known elastic constants at 300 K for the heavy rare earth metals with a hexagonal closepacked (hcp) crystallographic structure are listed in Table 1 [17–19]. An interpolated C13 value of 25 GPa was used to construct the slowness surfaces shown in Fig. 8. The polar plot of Fig. 10 shows the S1 and S2 shear wave and L longitudinal wave slowness surfaces as a function of the propagation direction with respect to the c-axis (01). The shear wave labelled as S2 (dotted line) has a b-axis
C.M. Lim et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 387–394
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Table 1 The elastic moduli of the heavy rare earth elements at room temperature and pressure Elements
Gd [15] Tb [15] Dy [16] Ho [16] Er [15] Tm Lu [17]
Elastic moduli (GPa) at 300 K and 0 T C11
C33
C44
C12
C13
C66
67.8 69.2 73.1 76.1 83.7 92.5 91.0
71.2 74.4 78.1 78.6 84.5 81.5 84.0
20.8 21.8 24.0 24.2 27.5 28.2 26.8
25.6 25.0 25.3 24.7 29.3 33.5 32.0
20.7 21.8 22.3 20.6 22.3
21.1 22.1 23.9 25.7 27.1 29.5 29.5
polarisation direction and the S1 shear wave (solid line) is polarised in the a–c plane. The two shear wave surfaces are coincident along the c-axis. When the shear waves are propagated down the a-axis, the two shear wave slowness surfaces are well separated. This is shown in Fig. 10; where the c-axis polarised shear wave, S1 ¼ Sc ; is the slower of the two shear waves. The slowness surface of the b-axis polarised shear wave, S2 ¼ Sb ; has flat regions on it, indicating the preferential propagation directions, i.e. the preferred wave propagation being perpendicular to the flat regions. Off axis b-axis polarised shear waves are steered in the 451 direction.
4. Conclusions The elastic and magnetoelastic properties of Tm have been investigated via ultrasound. We believe this is the first comprehensive study of the elastic constants of Tm. Anomalies in the elastic moduli were observed at the paramagnetic/CAM phase transition and also close to the CAM/A phase boundary. There was no evidence of the onset of the 3,4-ferrimagnetic phase but we did observe anomalies at B25 K. The data also indicates that there may be additional phase boundaries in the ferromagnetic phase. Shear waves were observed to couple more strongly to the magnetic structure than longitudinal waves, particularly in the vicinity of the magnetic phase transitions. Application of a magnetic field >2 T, along the c-axis, produces significant softening in the material. Further ultrasonic
28.0
investigations (and other techniques) are required with a c-axis applied field between 2.5 and 5 T in single crystal Tm to provide conclusive evidence of the behaviour within the ferromagnetic phase. A series of isothermal C33 measurements up to 5 T below 40 K would locate the 3,4-ferri/ ferromagnetic phase boundary more precisely. Further work is also required to investigate the reason for the inability to propagate both longitudinal and shear waves down the a-axis at low temperatures. EMATs are found to provide additional information. The large increases in EMAT signal close to the phase transition temperatures, TN and TC ; indicates that the magnetoelastic contribution to acoustic generation dominates over the Lorentz mechanism at these temperatures. The use of linearly polarised EMATs would enable the shear wave birefringence and the EMAT efficiencies parallel to the c- and a-axis to be studied independently. In addition the use of EMATs could avoid problems associated with the acoustic coupling required between a quartz transducer and the sample.
Acknowledgements The authors wish to thank Dr. M.R. Lees and J. Reed for technical advice. C.M. Lim wishes to thank the University Brunei Darussalam for funding his Ph.D. training at the University of Warwick. The Tm single crystal was grown by D. Fort, School of Metallurgy and Materials, University of Birmingham, UK.
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References [1] W.C. Koehler, J.W. Cable, E.O. Wollan, M.K. Wilkinson, Phys. Rev. 126 (1962) 1672. [2] S.J. Pickart, H.A. Alperin, Bull Am. Phys. Soc. 10 (1965) 32. [3] B.L. Rhodes, S. Legvold, F.H. Spedding, Phys. Rev. 109 (1958) 1547. [4] T.O. Brun, S.K. Sinha, N. Wakabayashi, G.H. Lander, L.R. Edwards, F.H. Spedding, Phys. Rev. B 1 (1970) 1251. [5] J. Bohr, D. Gibbs, K. Huang, Phys. Rev. B 42 (1990) 4322. [6] S.W. Zochowski, K.A. McEwen, J. Magn. Magn. Mater. 104–107 (1992) 1515. [7] M. Ellerby, K.A. McEwan, J. Jensen, Phys. Rev. B 57 (1998) 8416. [8] D. Fort, J. Alloys Compounds 177 (1991) 31. [9] R. Truell, C. Elbaum, B.B. Chick, Ultrasonic Methods in Solid State Physics, Academic Press, New York, 1969, p. 16. [10] M.J.P. Musgrave, Crystal AcousticsIntro. To Study of Elastic Waves and Vibrations in Crystals, Holden-Day, California, 1970.
[11] C.M. Lim, S. Dixon, C. Edwards, S.B. Palmer, J. Phys. D 31 (1998) 1362. [12] E.R. Dobbs, in: W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, Vol. 10, Academic Press, London, 1973. [13] P. Crowther, C. Edwards, S.B. Palmer, S.D. Kenney, Insight 39 (1997) 618. [14] R.D. Greenough, J. Magn. Magn. Mater. 8 (1978) 286. [15] A.V. Andrianov, V.D. Buchel’nikov, A.N. Vasil’ev, Yu.P. Gaidukov, R.S. Ll’yasov, V.G. Shavrov, Sov. Phys. JETP 67 (1988) 2331. [16] A.V. Andrianov, V.D. Buchel’nikov, A.N. Vasil’ev, Yu.P. Gaidukov, R.S. Ll’yasov, V.G. Shavrov, Sov. Phys. JETP 70 (1990) 944. [17] S.B. Palmer, E.W. Lee, M.N. Islam, Proc. Roy. Soc. (London) A328 (1974) 341. [18] S.B. Palmer, E.W. Lee, Proc. Roy. Soc. (London) A327 (1972) 519. [19] J.J. Tonnies, K.A. Gschneidner, F.H. Spedding, J. Appl. Phys. 42 (1971) 3275.
Ultrasound studies of single crystal thulium in an applied magnetic field C.M. Lima, C. Edwardsb,*, S. Dixonb, S.B. Palmerb a
Department of Physics, University Brunei Darussalam, Tungku Link Road, BE1410, Brunei b Department of Physics, University of Warwick, Coventry, CV4 7AL, UK Received 3 April 2001; received in revised form 1 June 2001
Abstract The paper reports the first measurements of the single crystal elastic constants of the heavy rare earth metal thulium as a function of temperature and magnetic field. The constants were obtained from ultrasonic velocity measurements over a temperature range of 4.2–296 K and in applied magnetic fields of up to 5 T. The elastic constants; C11, C33, C44 and C66=(C11–C12)/2 were determined from the ultrasonic velocities. Anomalies in the elastic constants were observed at 58 K from the c-axis propagated shear wave measurements and at 55 K from the c-axis propagated longitudinal wave measurements. Significant softening of the elastic constants C33 and C44 was observed close to TN : Application of a magnetic field (>2 T) along the c-axis direction induced further softening of the material. Electromagnetic acoustic transducers (EMATs) were also employed in addition to conventional piezoelectric quartz transducers. A marked increase in the EMATs acoustic coupling efficiency (generation and detection efficiency) occurred close to TN : r 2001 Elsevier Science B.V. All rights reserved. PACS: 62.20.Dc; 75.30.Kz Keywords: Rare earth magnetism; Elastic constants
1. Introduction In zero applied field, thulium (Tm) exhibits a c-axis sinusoidally modulated antiferromagnetic ordering below a N!eel temperature, TN ; of 56 K. The wave vector Q is slightly less than 2/7c* at TN in this incommensurate c-axis modulated (CAM) phase. The magnetic structure begins to square up with higher order odd harmonics of the Q appear*Corresponding author. E-mail addresses: [email protected] (C.M. Lim), [email protected] (C. Edwards).
ing at B42 K. Q gradually increases to 2/7c* and forms a commensurate antiphase ferrimagnetic structure at a lock-in temperature around 32 K [1–5]. The commensurate ferrimagnetic structure is aligned along the c-axis and has a periodicity of seven atomic layers, with four layers parallel and three layers antiparallel, the net magnetization corresponds to 1/7 of the ionic moment of Tm (B1 mB). In the early studies of Tm it was reported that the magnetic moment does not fully develop in the ferrimagnetic phase at temperatures above 25–20 K and this was referred to as the Curie temperature, TC [1–3]. In applied
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magnetic fields over 2 T, below TN ; Tm develops a higher net moment and it becomes fully ferromagnetic (B7 mB) in magnetic fields greater than a critical field of 2.8 T below the lock-in temperature [2,6]. Anomalous magnetoresistive behaviour has been observed in between the CAM and 3,4ferrimagnetic phases indicating at least one mixed region (referred to as the A phase) with some incommensurate features [7]. Two additional intermediate regions have also been observed between the ferri- and ferromagnetic phases with boundaries at 3.3 and 3.6 T from magnetostriction data [6]. It was suggested that they may be due to the demagnetizing field of the sample (B0.8 T); there was no evidence of these phases reported in Ref. [7] which used a sample with a much lower demagnetizing factor. We report on measurements of the ultrasound velocity as a function of temperature and applied magnetic field in single crystal Tm using both quartz and electromagnetic acoustic transducers (EMATs) allowing the elastic and magnetoelastic behaviour of the material to be studied. The quartz transducers have to be bonded to the sample with an acoustic couplant which functions over the entire temperature range, whereas the EMATs operate via a non-contacting transduction mechanism and do not require any acoustic couplant.
2. Experimental procedures A Tm single crystal was grown by the solid-state method and cut in the form of a cuboid (4.355 3.588 3.77270.001 mm) with a- and caxis perpendicular to the spark planed sides of the cuboid [8]. The maximum c-axis demagnetizing field is 0.7 T in the fully ferromagnetic phase and 0.1 T in the 3,4-ferrimagnetic phase. These fields should be subtracted from the applied field to give the internal field in the ferromagnetic (>3.5 T, the critical field plus the demagnetizing field) and the ferrimagnetic phases (o2.8 T). In the intermediate region (2.8–3.5 T) we might expect some phase coexistence giving additional regions similar to those reported in Ref. [6]. Ultrasound measurements were performed in a send/receive mode where the 10 MHz quartz
transducers were driven by an RF tone burst. The time of flight of the ultrasound pulse echoes was determined using an auto-correlation technique. Shear and longitudinal ultrasonic waves were propagated along both the a- and c-axis. The relationship between the various velocities of propagation and the elastic constants for a hexagonal crystal allowed the five independent elastic constants (C11, C33, C44 and C12) to be calculated [9,10]. A more detailed description of the experimental set-up is given in Ref. [11]. The velocity measurements were carried out by attaching the quartz transducers to the sample surface using GE varnish as an acoustic couplant. The GE varnish acoustic bond did not fracture even when the sample and transducer were repeatedly cycled from room temperature down to 4.2 K. The EMAT design was based on the Lorentz mechanism [12]. The EMAT coil had the spiral pan-cake geometry and used 0.08 mm diameter copper wire. A static magnetic field was applied perpendicular to the coil by a superconducting magnet, hence generating in-plane radially polarised shear waves. The sample was placed in light contact with the EMAT coil in a sample holder. For a non-magnetic conductor, the EMAT functions purely via the Lorentz mechanism where generation and detection efficiencies are linearly proportional to the static field. In magnetic materials other magnetoelastic mechanisms are possible, these mechanisms vary non-linearly with applied field and are easily revealed by plotting the square root of the EMAT signal amplitude against applied field [13].
3. Results 3.1. Longitudinal elastic constants, C11 and C33 Fig. 1 shows C11 and C33 as a function of temperature in zero applied magnetic field (the errors are represented by the size of the markers). A signal processing technique described in Ref. [14], which looks for local changes in gradient, was applied to the data to extract any anomalies. Anomalies in C33 were observed at 55 K and
C.M. Lim et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 387–394
Fig. 1. C11 and C33 as a function of temperature at zero applied magnetic field.
B108 K where C33 shows a small change in gradient. The anomaly at 55 K corresponds to the N!eel temperature of Tm [1–6]. On further cooling C33 increased from 83.6 to 88.2 GPa at 4.2 K. In zero magnetic field, C11 gradually increased as the sample was cooled with no anomalies until a small change in gradient at 91 K. The experiment was discontinued at 64 K due to the loss of signal. The high temperature (>90 K) gradient changes do no correspond to any known phase changes. Application of a magnetic field along the c-axis has a minimal effect on C33 below 2 T, see Fig. 2, where the general features of the C33 curve are similar to the zero field curve. Note that for presentation purposes the C33 data are displaced as indicated in the legend and only 1 in 2 data points are shown. The N!eel temperature minimum remained unchanged at 54 K for c-axis applied magnetic fields of 1 and 2 T but decreased to 50 K at 3 T. The 3 T C33 data, below 54 K is lower than the zero field values due to a modification of the magnetic structure of the material. Increasing the c-axis applied field above 3 T induces further softening of C33 below 50 K. The distinct change in gradient drops to B40 K at 4 T and may correspond to squaring up of the magnetic structure, suggesting that the CAM to ferrimagnetic transition, below 50 K, is not a single stage transition. This is in agreement with the magnetoresistance measurements reported in Ref. [7] where it was suggested that a mixed A phase exists between the CAM and ferrimagnetic phases.
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Fig. 2. C33 as a function temperature in a c-axis applied magnetic field of 1–5 T.
The differences between the 4 and 5 T C33 data are relatively small, thus implying that the material has a stable magnetic structure once the c-axis applied field is greater than 4 T. We do not see any evidence of the 32 K lock-in but there are small anomalies present between 25–27 K in magnetic fields greater than 3 T. Fig. 3 shows a 3D plot of the C33 data to enable the large drop in elastic constant between the 3,4ferrimagnetic and ferromagnetic phases to be seen more clearly. The low temperature C33 data up to 2 T is almost constant, the 3 T data is slightly lower whereas the data over 4 T is much lower. This is consistent with the demagnetizing considerations for the crystal, the demagnetizing field could reduce the internal field to below 2.8 T for an applied field of 3 T.
3.2. Shear elastic constants C44 and C66 Fig. 4 shows the zero field temperature dependence of C44 and C66=(C11–C11)/2. C44 increases almost linearly as the sample cools from room temperature to 142 K. Below 142 K it decreases gradually and steepens below 100 K before dropping to a minimum value of 27.4 GPa at 59 K. On further cooling below 59 K, C44 increases steadily. Although no clear indication of the zero field lockin (32 K) was observed, changes in gradient are observed at 40 and 25 K. The 40 K anomaly may again be associated with the CAM/A phase
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Fig. 5. C44 as a function of temperature with a c-axis applied magnetic field of 1–3 T.
Fig. 3. 3D C33 plot.
coupling between the ultrasonic waves and the magnetic structure. Fig. 5 shows C44 as a function of temperature with applied magnetic field parallel to the c-axis. The magnetic field causes a reduction in C44 from 27.5 GPa at zero field to 25.0 GPa at 3 T and a shift in the C44 anomalies to lower temperatures. The C44 minimum, which corresponds to T N, moves from 59 K in zero field to 47 K at 3 T, the 2 T curve shows a distinct gradient change at 23 K, again possibly corresponding to T C. At 3 T, the shear wave is highly attenuated as the sample temperature approached 31 K possibly due to the formation of the 3,4-ferrimagnetic structure. 3.3. EMAT signal amplitude measurements
Fig. 4. C44 and C66=(C11+C12)/2 as a function of temperature at zero applied magnetic field.
boundary and 25 K could correspond to the TC quoted in the early work. Within the temperature range, 91–296 K, C66 increases smoothly as the sample is cooled. However, it was not possible to obtain any C66 results below 90 K due to loss of signal. The propagation of both shear and longitudinal waves down the a-axis is difficult as a result of very high attenuation resulting from strong magnetoelastic
The EMATs used in this work generated inplane radially polarised shear waves and were used for both the generation and detection of ultrasound. The EMAT acoustic coupling efficiency is a combination of the acoustic generation and detection efficiencies, hence the raw data is proportional to the efficiency squared. The amplitude of the third shear wave echo corresponding to a shear wave polarised along the c-axis was measured, this will be discussed more fully in the next section. The temperature dependence of EMAT efficiency with an applied field of 0.6 T, for shear waves propagating down the c-axis, is shown in Fig. 6. The present system could only detect an EMAT signal when the applied field was at least 0.5 T. The
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Fig. 6. EMAT signal amplitude (3rd. shear echo) as a function of temperature with a c-axis applied magnetic field of 0.6 T.
largest change in the EMAT acoustic coupling efficiency was observed in the vicinity of the N!eel temperature. A series of EMAT isofield signal amplitude measurements was carried out as a function of temperature (cooling) with applied fields ranging from 0.5–2 T. The EMAT efficiency was calculated by taking the square root of the measured voltage. The efficiency was then divided by the applied field to remove the Lorentz contribution and reveal the magnetoelastic effects and is presented as a contour plot in Fig. 7. Lines were drawn over the image plot to indicate the boundaries of the magnetic phases of Tm as reported in Refs. [1–7]. The ultrasonic anomalies are presented with the previously reported phase boundaries over a wider magnetic field range in Fig. 8. Anomalies are found in the vicinity of the paramagnetic/CAM phase boundary, vertically above the CAM/A phase boundary and delineating a near vertical anomaly at 25 K, implying that there may be additional phases above the critical field. The EMAT experiment could not be extended beyond 2 T because the ultrasound waveforms became complicated with mode converted signals interfering with the main shear wave echoes. One would expect a shear wave to be attenuated as the magnetic field is increased due to eddy current loses. Since the sample is a good conductor, eddy currents will be induced as the shear vibration direction is orthogonal to the applied field, whereas the mode converted longitudinal waves vibrate parallel to the magnetic field and suffer no eddy current losses. Variations in
Fig. 7. Contour plot of EMAT signal amplitude of Tm. The markers show anomalies observed in the ultrasound velocity measurements. The lines are magnetic phase transition boundaries as reported in [4–6].
Fig. 8. Comparison of ultrasonic anomalies with previously reported phase boundaries [1–7].
EMAT acoustic coupling efficiency has been previously been reported for single crystals of Gd and Dy [15,16]. 3.4. Tm slowness surfaces The propagation of EMAT generated in-plane radially polarised shear waves down the a-axis shows the anisotropic nature of the material. Fig. 9
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Fig. 9. Birefringence effect down the a-axis at 60, 70 and 96 K. Applied magnetic field of 2 T directed along the a-axis. Sb and Sc are the b-axis and c-axis polarised shear echoes, respectively. Echoes are labelled according to the number of transits through.
shows ultrasonic waveforms taken at 60, 70 and 96 K in an a-axis magnetic field of 2 T. Birefringent behaviour is clearly evident where the ultrasound energy is steered into the two orthogonal polarised shear waves as a result of the crystal anisotropy. This results in two sets of shear waves propagating down the a-axis, indicated as Sb and Sc with shear wave polarisation directions along the b- and c-axis, respectively. At T ¼ 96 K the generation efficiency and attenuation of both Sb and Sc are very similar. As the sample was cooled (while remaining in the paramagnetic phase) the generation efficiency of Sc increased strongly while Sb showed only a marginal increase possibly indicating that the energy is being preferentially steered into the C44 mode. The other possible explanation for the high generation efficiency of Sc ; is that the magnetoelastic coupling is larger along the c-axis. It should be noted that marked increases in EMAT efficiency often occur over a range of temperature above or below magnetic phase transitions, see Fig. 6 where the increase starts at B80 K. The slowness (inverse velocity) surfaces have been calculated from the measured elastic constants using equations given in Ref. [10]. This is plotted to illustrate the effect of beam steering in single crystal Tm at 90 K, see Fig. 10. Slowness
Fig. 10. Calculated slowness surfaces for Tm at 90 K in the a–c plane in units of m1 s, assuming C13=25 GPa. L is the longitudinal wave surface and S1 and S2 are a2c plane polarised and the b-axis polarised shear wave surfaces.
surfaces are useful for evaluating beam steering effects as the direction of energy transport is defined by the normal to the slowness surface. Flat regions of the slowness surface indicate strong beam steering effects. It must be emphasised that in constructing the plot all the elastic moduli were known from the velocity data except for C13. The C13 measurement requires the propagation of ultrasonic waves at an angle of 451 to the c-axis [9,10]. The small dimensions of the sample prevented us from cutting the additional parallel faces required for the C13 measurement. The present data for Tm, together with all the previously known elastic constants at 300 K for the heavy rare earth metals with a hexagonal closepacked (hcp) crystallographic structure are listed in Table 1 [17–19]. An interpolated C13 value of 25 GPa was used to construct the slowness surfaces shown in Fig. 8. The polar plot of Fig. 10 shows the S1 and S2 shear wave and L longitudinal wave slowness surfaces as a function of the propagation direction with respect to the c-axis (01). The shear wave labelled as S2 (dotted line) has a b-axis
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Table 1 The elastic moduli of the heavy rare earth elements at room temperature and pressure Elements
Gd [15] Tb [15] Dy [16] Ho [16] Er [15] Tm Lu [17]
Elastic moduli (GPa) at 300 K and 0 T C11
C33
C44
C12
C13
C66
67.8 69.2 73.1 76.1 83.7 92.5 91.0
71.2 74.4 78.1 78.6 84.5 81.5 84.0
20.8 21.8 24.0 24.2 27.5 28.2 26.8
25.6 25.0 25.3 24.7 29.3 33.5 32.0
20.7 21.8 22.3 20.6 22.3
21.1 22.1 23.9 25.7 27.1 29.5 29.5
polarisation direction and the S1 shear wave (solid line) is polarised in the a–c plane. The two shear wave surfaces are coincident along the c-axis. When the shear waves are propagated down the a-axis, the two shear wave slowness surfaces are well separated. This is shown in Fig. 10; where the c-axis polarised shear wave, S1 ¼ Sc ; is the slower of the two shear waves. The slowness surface of the b-axis polarised shear wave, S2 ¼ Sb ; has flat regions on it, indicating the preferential propagation directions, i.e. the preferred wave propagation being perpendicular to the flat regions. Off axis b-axis polarised shear waves are steered in the 451 direction.
4. Conclusions The elastic and magnetoelastic properties of Tm have been investigated via ultrasound. We believe this is the first comprehensive study of the elastic constants of Tm. Anomalies in the elastic moduli were observed at the paramagnetic/CAM phase transition and also close to the CAM/A phase boundary. There was no evidence of the onset of the 3,4-ferrimagnetic phase but we did observe anomalies at B25 K. The data also indicates that there may be additional phase boundaries in the ferromagnetic phase. Shear waves were observed to couple more strongly to the magnetic structure than longitudinal waves, particularly in the vicinity of the magnetic phase transitions. Application of a magnetic field >2 T, along the c-axis, produces significant softening in the material. Further ultrasonic
28.0
investigations (and other techniques) are required with a c-axis applied field between 2.5 and 5 T in single crystal Tm to provide conclusive evidence of the behaviour within the ferromagnetic phase. A series of isothermal C33 measurements up to 5 T below 40 K would locate the 3,4-ferri/ ferromagnetic phase boundary more precisely. Further work is also required to investigate the reason for the inability to propagate both longitudinal and shear waves down the a-axis at low temperatures. EMATs are found to provide additional information. The large increases in EMAT signal close to the phase transition temperatures, TN and TC ; indicates that the magnetoelastic contribution to acoustic generation dominates over the Lorentz mechanism at these temperatures. The use of linearly polarised EMATs would enable the shear wave birefringence and the EMAT efficiencies parallel to the c- and a-axis to be studied independently. In addition the use of EMATs could avoid problems associated with the acoustic coupling required between a quartz transducer and the sample.
Acknowledgements The authors wish to thank Dr. M.R. Lees and J. Reed for technical advice. C.M. Lim wishes to thank the University Brunei Darussalam for funding his Ph.D. training at the University of Warwick. The Tm single crystal was grown by D. Fort, School of Metallurgy and Materials, University of Birmingham, UK.
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