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Electron-phonon interaction in heavy fermion compounds
Physica B 161 (1989) 333-336 North-Holland, Amsterdam
ELECTRON-PHONON ELECTROMAGNETIC
G. HAMPEL,
INTERACTION
GENERATION
G. BRULS,
IN HEAVY FERMION
OF ULTRASONIC
D. WEBER,
Physikalisches Institut, Universitiit Frankfurt,
COMPOUNDS
WAVES
I. KOUROUDIS
D-6000 Frankfurt,
and B. LUTHI
FRG
Electromagnetic generation of sound in the heavy fermion systems CeRu,S& and UPt, is investigated. Two coupling mechanisms are identified: Griineisen parameter coupling and Lorentz force coupling. The latter mechanism can be used to determine experimentally the skin effect penetration depth. Some special effects in UPt, are mentioned.
1. Introduction
In heavy fermion compounds the so-called Griineisen parameter coupling between electrons and phonons [l] is responsible for anomalies of the elastic constants as a function of temperature as well as of magnetic field, for the thermal expansion anomaly and the strong magnetostriction effects at low temperatures. Therefore we have investigated the possibility of generating sound waves electromagnetically
0
2
f,
6
8
10
B ITI Fig. 1. Electromagnetically generated longitudinal strain (arbitrary units) vs. applied field (B//c-axis//q) in CeRu,Si, at 4.2 K (circles and triangles) and 1.3 K (triangles). Solid lines: fit according to eq. (2). Inset: sample with one coil and a transducer in the applied field B.
via the Griineisen parameter effect. The inset in fig. 1 shows schematically the arrangement of the sample, with one coil and one transducer in the external magnetic field B. Actually we had, a coil and a transducer on both sides. In this way we could measure and compare transducer-transducer, coil-coil, coil-transducer and transducercoil signals. We describe experiments on CeRu,Si, and UPt, at low temperatures and in magnetic fields B up to 10T in CeRu,Si, and 22 T in UPt,, respectively, applied along an easy axis of magnetization. In the case of UPt, we used a “He system, allowing temperatures down to 0.37 K in 22T. It turns out that we have to take into account at least two different mechanisms of conversion between electromagnetic and acoustic energy. One mechanism work? via the Griineisen parameter coupling and predicts a longitudinal strain E, (see ref. [2]):
where c is an elastic constant (which can be magnetic field-dependent), 0 = - 1 lB, 6B,/& is the uniaxial Griineisen parameter, B, the critical field for the metamagnetic transition, B the magnetic field strength and M the magnetization. Our modulation technique yields the derivative of the static magnetostriction, 6~~ /6B b(w), with b(w) the alternating field of the small coil
0921-4526/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
G. Hampel et al. i Electron-phonon
131
(of order g
interactiort in healsy fermion
The transverse modes in CeRu,Si, show strong Lorentz-force effects. In addition a sharp spike was observed at B = B, which is reminiscent of helicon-phonon coupling. A spike was also visible in transverse transducer-transducer measurements.
100 G):
b(w) =
(
; R(xB
+ M) - $
$j
RBM
1
b(o)
(2)
where x = 6Mi6B is the magnetic susceptibility. In addition. one always has a Lorentz-force effect on the eddy currents within the skin depth (see, e.g. ref. [3]) which gives a strain E,,: Bh(o) EL=Z$
1 (3)
v-
where u, is the sound velocity, d the density of the sample material and /3 = q’6’/2 with 6 the skin depth and q the wave vector. The polarization of the generated sound wave is in principle perpendicular to the applied magnetic field B, but sometimes anomalous polarization along B occurs (see below). If we measure the response with a transducer, we get a signal directly proportional to eq. (3). However, if we detect with a second coil, the conversion efficiency enters once more and we measure a signal which is proportional to B’il + p’ (see ref. [3]).
2. Experiments
compounds
3. Experiments
on UPt,
In fig. 2 we show data for UPt,. Again the magnetic field B was parallel to q (and the u-axis), but we looked for the transverse modes. as could be seen from the sound velocity. In this geometry one expects only Lorentz force contributions, which is shown to be the case by the following analysis of the skin depth 6 with the help of eq. (3). Since the transverse modes change their velocities by less than 1% in the region below 10 K even in fields up to 20T and since their attenuation is also negligible (see ref. [6]), the temperature dependence of eL of eq. (3) must come from changes in the skin depth 6 in the factor li 1 + /3’ (without square root because fig. 2 are coil-coil measurements). To get absolute values for 6 we
id 6~ 1
B=20T
on CeRu,Si, ul
In fig. 1 we show a comparison of the predictions of the Gruneisen parameter theory (eq. (2)) with our data for CeRu,Si,. Values for R (0 = 120 for c33 in CeRu,Si,) we took from ref. [4], for x from ref. [5]. The delay time of the pulse generated by the coil coincided with the delay of the longitudinal mode generated and detected by two longitudinal transducers. In this geometry (B parallel to the q-vector of the sound wave) we do not expect a longitudinal contribution from the Lorentz-force effect and indeed the agreement between our data and eq. (2) is very good. The difference in peak height in the 4.2 K curve and the disappearance of the signal inside the peak in the 1.3 K curve both are due to strong attenuation (see ref. [6]) for which we did not correct.
-3
\
5
d
k
0.5
C ._ tl L z
0
0
2
4
6
8
1
Fig. 2. Electromagnetic excitation data for shear strain VS. temperature in UPt, for B = 12 T and B = 20 T (B//a-axis/i q), measured with 2 coils, c,, mode. 12MHz. Inset. upper curve: enlargement of the low-temperature end of the 2UT electromagnetic peak. Inset, lower curve: attenuation peak in zero field, measured with 2 longitudinal transducers at 270 MHz.
G. Hampel et al. I Electron-phonon
assumed that the flattening of the 12 T curve towards 0.4 K corresponds to the saturation of l/ 1 + p2 toward 1 for small p. Values for S in UPt, (for B = 12 T) obtained in this way agree very well with values calculated by entering zerofield resistivity values of a similar UPt, sample (from ref. [7]) into the classical equation for the skin depth, S = cV=, with c the light velocity, p the electrical resistivity, p the magnetic permeability and w the frequency. See fig. 3 for skin depth values from the measurements with w = 12 MHz and B = 12 T, and values calculated from the zero field resistivities of ref. [7]. We have taken p = 1, which is allowed for paramagnetic substances. The linear dependence of S on temperature T between 0.4 K and 7 K reflects nicely the low-temperature T2-behaviour of the resistivity. Similar agreement was found for measurements at a second frequency (5 MHz) and at fields of 5 and 15 T, where the magnetoresistance is still small. We took this as sufficient evidence for the applicability of eq. (3) in these cases. The 20 T curve in fig. 2 shows some additional interesting features. In the first place, it rises much more steeply towards lower temperatures than in the 12 T case. This can be explained by assuming that the skin depth changes as m, the magnetoresistance p(B) having a strong peak around 20 T (see ref. [8]). Secondly, as shown in
loo1
interaction in heavy fermion
2
4
6
T
8
10
[Kl
Fig. 3. Skin depth 6 as a function of temperature from our measurements at 12 MHz and 12 T (solid line) and from the classical formula 6 = cdwith resistivity data from ref. [7] (circles).
335
the inset of fig. 2, the 20T curve shows a clear decrease below 0.48 K, which is already absent at 19 T (not shown). This decrease cannot be explained from magnetoresistance, since the resistance at 20T decreases with decreasing temperature, as can be seen from the data of ref. [8]. Curiously enough, 0.48 K is exactly the superconducting transition temperature of this sample at B = 0, as determined from a peak in the longitudinal sound attenuation in zero field. At this moment we have no idea whether this coincidence is accidental or not. We also looked for electromagnetic generation of longitudinal sound in UPt,. Here the situation is more complicated. For the geometry where one does not expect longitudinal Lorentz waves (B parallel q), we nevertheless saw a signal with the delay time of a longitudinal mode and which was perfectly linear in B up to 18 T, like the Lorentz force effect. It was not due to a misalignment of field or sample, because in contrast to the true longitudinal and transverse Lorentz force waves this signal decreased in amplitude with decreasing temperature. Currently, we have no explanation for this anomalous generation or mode conversion. Dobbs [3] mentions anomalous generation in magnetic materials, also without explanation. At 20 T there is a peak which we associate with the Griineisen parameter effect. The peak is neither present in the transverse modes, nor in the true (with B perpendicular to q) longitudinal Lorentz effect mode for upt,.
4. Additional
00
compounds
experiments
In the presently used UPt,-crystal the oscillations in the transverse sound velocities which we reported on earlier [4] were much smaller in amplitude. In the new CeRu,Si, crystal we still observe the sharp spike at B, (see above), but the oscillations in the neighbourhood also practically disappeared. The samples of ref. [4] being much smaller and irregularly shaped, we must conclude that mode conversions due to multiple and indirect reflections, and consecutive interference, were responsible for the oscillations.
336
G. Hampel et al.
I Electron-phonon
We started to investigate nonlinear elastic effects in UPt, and CeRu,Si?. If we write for the additional stress Au = Ace + d ~‘12, the higher order term d can be related to the elastic constant using scaling arguments as dlhc = R. With a typical strain of lo-’ and R = 10’ this gives d&/c = 10mh. We tried to observe this effect by studying amplitude dependent phase changes and frequency doubling, but the effects are extremely small. Nonlinear effects have been recently reported for polycrystalline multiphase CeAl, [9].
Acknowledgements This research was supported by the DFG through SFB 252. We thank J. Flouquet and P. Lejay for the CeRu,Si,, E. Bucher for the UPt, crystal and L. Taillefer and A. de Visser for unpublished results and stimulating discussions. We thank the Max Planck Institut fur Festkorperforschung and P. Wyder for the use of the
interaction in heavy fermion
high-field noble.
facilities
compounrls
of the
MPI-SNCI
in Gre-
References B. Luthi. M. Yoshizawa. J. Magn. Magn. Mat. 63 ci: 64 (1087) 274. P. Thalmcier, J. Magn. Magn. Mat. 76 & 77 (19X8) 2YY. See, c.g. E.R. Dobbs. Physical Acoustics X (Academic Press. New York. lY73) 127. I. Kouroudis. D. Weher. M. Yoshizawa. B. Liithi. L. Puech. P. Haen. J. Flouquet. G. Bruls, U. Welp, J.J.M. Franse. A. Menovsky, E. Bucher and J. Hufnagel. Phys. Rev. Lett. 58 (1987) 820. P. Haen, J. Floquet. F. Lapierrc. P. Lejay and G Remenyi. J. Low Temp. Phys. 67 (1987) 3Y 1. D. Weher. I. Kouroudis. B. Liithi. G. Bruls. M. Yoshizawa. P. Haen. J. Flouquct. E. Buchcr and J. Hufnagel. J. Magn. Ma&n. Mat. 76 Rr 77 (1988) 315. A. de Visser, Ph.D. Thesis. University of Amsterdam (lY86). L. Taillefer, J. Flouquct, W. Joss, J. Magn. Magn. Mat. 76 Kr 77 (1988) 21X. and private communication. B.B. Voronov. A.I. Korohov. V.V. Moschalkov. J. Mapn Magn. Mat. 76 Kr 77 (198X) 317.
ELECTRON-PHONON ELECTROMAGNETIC
G. HAMPEL,
INTERACTION
GENERATION
G. BRULS,
IN HEAVY FERMION
OF ULTRASONIC
D. WEBER,
Physikalisches Institut, Universitiit Frankfurt,
COMPOUNDS
WAVES
I. KOUROUDIS
D-6000 Frankfurt,
and B. LUTHI
FRG
Electromagnetic generation of sound in the heavy fermion systems CeRu,S& and UPt, is investigated. Two coupling mechanisms are identified: Griineisen parameter coupling and Lorentz force coupling. The latter mechanism can be used to determine experimentally the skin effect penetration depth. Some special effects in UPt, are mentioned.
1. Introduction
In heavy fermion compounds the so-called Griineisen parameter coupling between electrons and phonons [l] is responsible for anomalies of the elastic constants as a function of temperature as well as of magnetic field, for the thermal expansion anomaly and the strong magnetostriction effects at low temperatures. Therefore we have investigated the possibility of generating sound waves electromagnetically
0
2
f,
6
8
10
B ITI Fig. 1. Electromagnetically generated longitudinal strain (arbitrary units) vs. applied field (B//c-axis//q) in CeRu,Si, at 4.2 K (circles and triangles) and 1.3 K (triangles). Solid lines: fit according to eq. (2). Inset: sample with one coil and a transducer in the applied field B.
via the Griineisen parameter effect. The inset in fig. 1 shows schematically the arrangement of the sample, with one coil and one transducer in the external magnetic field B. Actually we had, a coil and a transducer on both sides. In this way we could measure and compare transducer-transducer, coil-coil, coil-transducer and transducercoil signals. We describe experiments on CeRu,Si, and UPt, at low temperatures and in magnetic fields B up to 10T in CeRu,Si, and 22 T in UPt,, respectively, applied along an easy axis of magnetization. In the case of UPt, we used a “He system, allowing temperatures down to 0.37 K in 22T. It turns out that we have to take into account at least two different mechanisms of conversion between electromagnetic and acoustic energy. One mechanism work? via the Griineisen parameter coupling and predicts a longitudinal strain E, (see ref. [2]):
where c is an elastic constant (which can be magnetic field-dependent), 0 = - 1 lB, 6B,/& is the uniaxial Griineisen parameter, B, the critical field for the metamagnetic transition, B the magnetic field strength and M the magnetization. Our modulation technique yields the derivative of the static magnetostriction, 6~~ /6B b(w), with b(w) the alternating field of the small coil
0921-4526/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
G. Hampel et al. i Electron-phonon
131
(of order g
interactiort in healsy fermion
The transverse modes in CeRu,Si, show strong Lorentz-force effects. In addition a sharp spike was observed at B = B, which is reminiscent of helicon-phonon coupling. A spike was also visible in transverse transducer-transducer measurements.
100 G):
b(w) =
(
; R(xB
+ M) - $
$j
RBM
1
b(o)
(2)
where x = 6Mi6B is the magnetic susceptibility. In addition. one always has a Lorentz-force effect on the eddy currents within the skin depth (see, e.g. ref. [3]) which gives a strain E,,: Bh(o) EL=Z$
1 (3)
v-
where u, is the sound velocity, d the density of the sample material and /3 = q’6’/2 with 6 the skin depth and q the wave vector. The polarization of the generated sound wave is in principle perpendicular to the applied magnetic field B, but sometimes anomalous polarization along B occurs (see below). If we measure the response with a transducer, we get a signal directly proportional to eq. (3). However, if we detect with a second coil, the conversion efficiency enters once more and we measure a signal which is proportional to B’il + p’ (see ref. [3]).
2. Experiments
compounds
3. Experiments
on UPt,
In fig. 2 we show data for UPt,. Again the magnetic field B was parallel to q (and the u-axis), but we looked for the transverse modes. as could be seen from the sound velocity. In this geometry one expects only Lorentz force contributions, which is shown to be the case by the following analysis of the skin depth 6 with the help of eq. (3). Since the transverse modes change their velocities by less than 1% in the region below 10 K even in fields up to 20T and since their attenuation is also negligible (see ref. [6]), the temperature dependence of eL of eq. (3) must come from changes in the skin depth 6 in the factor li 1 + /3’ (without square root because fig. 2 are coil-coil measurements). To get absolute values for 6 we
id 6~ 1
B=20T
on CeRu,Si, ul
In fig. 1 we show a comparison of the predictions of the Gruneisen parameter theory (eq. (2)) with our data for CeRu,Si,. Values for R (0 = 120 for c33 in CeRu,Si,) we took from ref. [4], for x from ref. [5]. The delay time of the pulse generated by the coil coincided with the delay of the longitudinal mode generated and detected by two longitudinal transducers. In this geometry (B parallel to the q-vector of the sound wave) we do not expect a longitudinal contribution from the Lorentz-force effect and indeed the agreement between our data and eq. (2) is very good. The difference in peak height in the 4.2 K curve and the disappearance of the signal inside the peak in the 1.3 K curve both are due to strong attenuation (see ref. [6]) for which we did not correct.
-3
\
5
d
k
0.5
C ._ tl L z
0
0
2
4
6
8
1
Fig. 2. Electromagnetic excitation data for shear strain VS. temperature in UPt, for B = 12 T and B = 20 T (B//a-axis/i q), measured with 2 coils, c,, mode. 12MHz. Inset. upper curve: enlargement of the low-temperature end of the 2UT electromagnetic peak. Inset, lower curve: attenuation peak in zero field, measured with 2 longitudinal transducers at 270 MHz.
G. Hampel et al. I Electron-phonon
assumed that the flattening of the 12 T curve towards 0.4 K corresponds to the saturation of l/ 1 + p2 toward 1 for small p. Values for S in UPt, (for B = 12 T) obtained in this way agree very well with values calculated by entering zerofield resistivity values of a similar UPt, sample (from ref. [7]) into the classical equation for the skin depth, S = cV=, with c the light velocity, p the electrical resistivity, p the magnetic permeability and w the frequency. See fig. 3 for skin depth values from the measurements with w = 12 MHz and B = 12 T, and values calculated from the zero field resistivities of ref. [7]. We have taken p = 1, which is allowed for paramagnetic substances. The linear dependence of S on temperature T between 0.4 K and 7 K reflects nicely the low-temperature T2-behaviour of the resistivity. Similar agreement was found for measurements at a second frequency (5 MHz) and at fields of 5 and 15 T, where the magnetoresistance is still small. We took this as sufficient evidence for the applicability of eq. (3) in these cases. The 20 T curve in fig. 2 shows some additional interesting features. In the first place, it rises much more steeply towards lower temperatures than in the 12 T case. This can be explained by assuming that the skin depth changes as m, the magnetoresistance p(B) having a strong peak around 20 T (see ref. [8]). Secondly, as shown in
loo1
interaction in heavy fermion
2
4
6
T
8
10
[Kl
Fig. 3. Skin depth 6 as a function of temperature from our measurements at 12 MHz and 12 T (solid line) and from the classical formula 6 = cdwith resistivity data from ref. [7] (circles).
335
the inset of fig. 2, the 20T curve shows a clear decrease below 0.48 K, which is already absent at 19 T (not shown). This decrease cannot be explained from magnetoresistance, since the resistance at 20T decreases with decreasing temperature, as can be seen from the data of ref. [8]. Curiously enough, 0.48 K is exactly the superconducting transition temperature of this sample at B = 0, as determined from a peak in the longitudinal sound attenuation in zero field. At this moment we have no idea whether this coincidence is accidental or not. We also looked for electromagnetic generation of longitudinal sound in UPt,. Here the situation is more complicated. For the geometry where one does not expect longitudinal Lorentz waves (B parallel q), we nevertheless saw a signal with the delay time of a longitudinal mode and which was perfectly linear in B up to 18 T, like the Lorentz force effect. It was not due to a misalignment of field or sample, because in contrast to the true longitudinal and transverse Lorentz force waves this signal decreased in amplitude with decreasing temperature. Currently, we have no explanation for this anomalous generation or mode conversion. Dobbs [3] mentions anomalous generation in magnetic materials, also without explanation. At 20 T there is a peak which we associate with the Griineisen parameter effect. The peak is neither present in the transverse modes, nor in the true (with B perpendicular to q) longitudinal Lorentz effect mode for upt,.
4. Additional
00
compounds
experiments
In the presently used UPt,-crystal the oscillations in the transverse sound velocities which we reported on earlier [4] were much smaller in amplitude. In the new CeRu,Si, crystal we still observe the sharp spike at B, (see above), but the oscillations in the neighbourhood also practically disappeared. The samples of ref. [4] being much smaller and irregularly shaped, we must conclude that mode conversions due to multiple and indirect reflections, and consecutive interference, were responsible for the oscillations.
336
G. Hampel et al.
I Electron-phonon
We started to investigate nonlinear elastic effects in UPt, and CeRu,Si?. If we write for the additional stress Au = Ace + d ~‘12, the higher order term d can be related to the elastic constant using scaling arguments as dlhc = R. With a typical strain of lo-’ and R = 10’ this gives d&/c = 10mh. We tried to observe this effect by studying amplitude dependent phase changes and frequency doubling, but the effects are extremely small. Nonlinear effects have been recently reported for polycrystalline multiphase CeAl, [9].
Acknowledgements This research was supported by the DFG through SFB 252. We thank J. Flouquet and P. Lejay for the CeRu,Si,, E. Bucher for the UPt, crystal and L. Taillefer and A. de Visser for unpublished results and stimulating discussions. We thank the Max Planck Institut fur Festkorperforschung and P. Wyder for the use of the
interaction in heavy fermion
high-field noble.
facilities
compounrls
of the
MPI-SNCI
in Gre-
References B. Luthi. M. Yoshizawa. J. Magn. Magn. Mat. 63 ci: 64 (1087) 274. P. Thalmcier, J. Magn. Magn. Mat. 76 & 77 (19X8) 2YY. See, c.g. E.R. Dobbs. Physical Acoustics X (Academic Press. New York. lY73) 127. I. Kouroudis. D. Weher. M. Yoshizawa. B. Liithi. L. Puech. P. Haen. J. Flouquet. G. Bruls, U. Welp, J.J.M. Franse. A. Menovsky, E. Bucher and J. Hufnagel. Phys. Rev. Lett. 58 (1987) 820. P. Haen, J. Floquet. F. Lapierrc. P. Lejay and G Remenyi. J. Low Temp. Phys. 67 (1987) 3Y 1. D. Weher. I. Kouroudis. B. Liithi. G. Bruls. M. Yoshizawa. P. Haen. J. Flouquct. E. Buchcr and J. Hufnagel. J. Magn. Ma&n. Mat. 76 Rr 77 (1988) 315. A. de Visser, Ph.D. Thesis. University of Amsterdam (lY86). L. Taillefer, J. Flouquct, W. Joss, J. Magn. Magn. Mat. 76 Kr 77 (1988) 21X. and private communication. B.B. Voronov. A.I. Korohov. V.V. Moschalkov. J. Mapn Magn. Mat. 76 Kr 77 (198X) 317.