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The nature of elementary excitations below and above the metamagnetic transition in CeRu2Si2
PHYSICA ELSEVIER
Physica B 206 & 207 (1995) 29-32
The nature of elementary excitations below and above the metamagnetic transition in CeRu2Si 2 F.S. Tautz*, S.R. Julian, G.J. McMullan, G.G. Lonzarich Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK
Abstract
We discuss, with reference to our de Haas van Alphen (dHvA) measurements and supporting band calculations, the two low temperature states of the heavy fermion paramagnet CeRu2Si2 which are separated by a steep metamagnetic cross-over. While we are able to propose a consistent Fermi surface model for the paramagnetic state, the high field state continues to evade comprehensive understanding.
Heavy fermion systems take their name from the attribution of their unusual thermodynamic and transport properties at low temperatures to heavy quasiparticles at the Fermi surface. The standard model of these quasi-particles is based on a subtle and intricate coupling of the conduction electrons with a lattice of essentially localised moments (for a review see Ref. [1]). A key question, not readily settled by means of conventional macroscopic probes, is whether the coupling between these two subsystems proves strong enough to amalgamate them both, so that only one type of elementary excitation survives into the coherent regime, as postulated in the standard model. To gain deeper understanding of this issue, we have carried out a series of studies [2,3] of the de Haas van Alphen (dHvA) effect in pure samples of the comparatively simple Ce-based compound CeRu2Si 2 for magnetic fields below and above the metamagnetic transition. Here we focus mainly on our recent findings.
* Corresponding author; present address: Van der WaalsZeeman Laboratory, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands.
Below the metamagnetic transition, the dHvA branches detected include that labelled /z, reported in Refs. [3] and [4], and two other heavy components, v and ~¢, with frequencies 4.9-+0.1 and 8.1---0.1 kT, respectively (not reported in Ref. [4]). The temperature dependence of the amplitude down to nuclear demagnetisation temperatures is consistent with that expected in a model of Fermi liquid excitations with masses of magnitude m*(~:) = 140m e, m*(/x) = ll0rne, m*(v) = 105m e. Comparisons of the experimental results with energy band models based on Density Functional Theory (DFT) in the Local Density Approximation (LDA) as well as the Renormalised Band method (RB) [1], lead to a natural interpretation of these important branches in terms of a single large fluted hole sheet centred at the Z-point in the Brillouin zone. As indicated in Fig. 1, along 100 we observe a central (v) and an off-central (/z) extremal orbit. Together with a complicated electron sheet centred at F and two smaller ellipsoids at Z studied previously, this surface completes a picture in which the quasiparticles account for almost 80% of the observed specific heat for B = 0 [5] (below the metamagnetic transition). Table 1 contains details of the estimate of the specific heat coefficient. The calculated density of states at the Fermi surface of various sheets was
0921-4526/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD! 0921-4526(94)00359-9
30
F.S. Tautz / Physica B 206 & 207 (1995) 29-32
80i~\1,40(15) ~,g,v
60
'~-110(~ ~. (a)
1Ls. _ _L
_ _t _-sl
°°°ogooo l
0 11o
angle
100
Fig. 1. Overview of our results for B.l_c. Numbers in the diagram on the left give renormalised masses, with typical uncertainty in parentheses. Dotted lines are results of an f-band calculation. For each branch, the Fermi level has been adjusted to fit the data along I00. The required shifts are: v, 4.8; e, -2.8; y, -3.9; /3, -4.2 mRy.
Table 1 Specific heat contribution of the various branches. In the second column the unrenormalised contribution from the band calculation is listed. Units of C / T and C ' / T : mJ m o l i K-2. mb is the LDA-calculated mass. Branch
1 2 3 4 5 Sum
C/T
m* /m b
0.31
1.7
0.71 0.88 4.67 6.86 13.42
1.6 41.5 -+ 2.5 9.3 - 4
C*/T
0.5 1.1 194 -+ 12 64 _+27 260 -+ 40
renormalised by individual mass e n h a n c e m e n t factors based on the measured quasi-particle masses. Considering that this estimate is necessarily based entirely on data in the basal plane and that one expects the specific heat to decrease as the magnetic field is increased from 0 T, the above result has to be taken as very good agreement indeed. Thus we conclude that neutral excitations [6] are unlikely to play a m a j o r role in the low field state of CeRu2Si z. We stress that the correspondence between the observed Fermi surface and that predicted in a band model in which the f-orbitals are incorporated among the band states does not imply that the f-electrons are in the conventional sense itinerant. Local moments, dynamically screened by conduction electrons and neighbouring m o m e n t s , may be viewed as contributing to the Fermi volume. The R B description yields a Fermi surface topology similar to that of the L D A , except that it predicts correctly, in agreement with experiment, only two rather than three small ellipsoids about Z. M o r e o v e r , these calculations provide a natural explanation for the variation of the mass renormalisation from band to band, and thus beautifully reproduce the existence of essentially three groups of quasi-particles in this material [7], with cyclotron masses around 2, 20, and 200m e, respectively, i.e. for quasi-particles which are virtually bare conduction electrons and those that involve the intricacies discussed above, in varying degrees. We now turn to a discussion of our results for applied magnetic fields parallel to the c-axis close to the metamagnetic transition (B c ~ 7.8 T) and above. The transition region narrows continuously with decreasing temperature, tending to a value of only 0.025 T as T---~0, an order of magnitude lower than in earlier samples investigated (Fig. 2). A transition to long-range order appears to be suppressed, a result which may be consistent with the prediction of a model in which the coupling of strong longitudinal magnetic fluctuations is treated in a self-consistent field approximation. The behaviour of the resistivity, which is markedly different from that of the susceptibility near B e should provide a searching test of the applicability of this magnetic fluctuation description. On approaching Be, all d H v A frequencies show some effect; the most dramatic change occurs for the heavy branch K below B c which evolves into the light branch/~ above B c. We find a mass of m* = 12 m e for K (instead of the 2 0 m e given in Ref. [4]). The behaviour of the measured frequency, defined as the derivative of the d H v A phase with respect to the
F.S. Tautz / Physica B 206 & 207 (1995) 29-32
I
I
I
I
I
I
1
400 m K 600 m K 00 m K
J I
I
7.4
7.6
000 m K
I
[
I
7.8
I
I
8
8.2
B (T) Fig. 2. Susceptibility peak at the metamagnetic transition. reciprocal field l / B , is expected to be radically different from that of the true frequency or extremal area of the Fermi surface near B~ where the area is strongly field dependent. The behaviour of the extremal area may be inferred by a procedure outlined in Ref. [8]. This analysis leads us to suggest that the K branch is associated with the legs of the electron surface (rather than the torus as proposed in Ref. [4]). The unadjusted LDA calculation predicts a frequency of 0.9 kT for the legs along the c-axis; if the Fermi level
2.5
I
I
I
m*=12
2
31
is adjusted to account for the observed disappearance of the signal at about 35°, the frequency expands to a value close to that observed for 6 (1.2 kT). The sharpness of the transition enables us to follow some of the oscillatory components to within 0.005 T of B c. This has allowed us to observe that the K oscillations behave in a dramatically different way from the /3 and 3' components arising from the Zcentred ellipsoids. The latter decay in amplitude as B c is approached, while the former appear to continue without attenuation up to the transition point itself. This difference in behaviour would appear to provide an important clue as to the microscopic nature of the metamagnetic transition. In the high field state above Be, it has been possible to detect a high frequency branch to with F = 28.2-+ 0.3 kT and m * = 8-+ l m e [3,4]. Its amplitude is extremely sensitive to alignment in the magnetic field and at optimum alignment it may be followed over a wide field range above Be. In spite of the observation of this crucially important branch, the state above Be continues to evade comprehensive understanding. If, as the size of the to frequency would seem to support, a band model in which the f-electrons are treated as core states rather than band states is appropriate for B > B~, then one expects to observe additional frequencies for field along c, which are not seen. One extra branch F('0) = 3.2 kT does seem to exist, but it is very weak and its origin is difficult to reconcile with the above core model. Moreover, if the specific heat is estimated in a similar fashion as for the low field state, one finds that the observed sheets at 17 T contribute 15 mJ mo1-1 K -2, while 9 0 m J mol -~ K -2 are observed [5]. This leaves 80% of 3' unaccounted for. Finally, unambiguous evidence for exchange splittings of all the sheets of the Fermi surface in the polarised state (B >B~) of the form observed in the related polarised material CeRu2Ge 2, well described by the core model [9], has not yet been resolved.
FAcknowledgement
1.5
m,=4 1
I
5
t
The work reported in this article was supported by EPSRC, UK.
.,v I
7.5
10 12.5 15 B (T) Fig. 3. Frequency of orbits K and 6 vs. magnetic field, inset: the 'leg' of branch 5 on which these orbits are situated, rn* is in units of m e.
References [1] P. Fulde, J. Keller, G. Zwicknagl, Solid State Phys. 41 (1988) 1.
32
F.S. Tautz / Physica B 206 & 207 (1995) 29-32
[2] G.G. Lonzarich, J. Magn. Magn. Mater. 76 & 77 (1988) 1. [3] S.R. Julian, F.S. Tautz, G.J. McMullan, G.G. Lonzarich, Physica B 199 & 200 (1994) 63. [4] H. Aoki, S. Uji, A.K. Albessard, Y. Onuki, Phys. Rev. Lett. 71 (1993) 2110. [5] H.P. van der Meulen, A de Visser, J.J.M. Franse, T.T.J.M. Berendschot, J.A.A.J. Perenboom, H. van Kempen, A. Lacerda, P. Lejay, J. Flouquet, Phys. Rev. B 44 (1991) 814.
[6] Y. Kagan, K.A. Kikoin, N.V. Prokof'ev, Physica B 182 (1992) 201. [7] E.K.R. Runge, Ph.D. Thesis, Technische Hochschule, Darmstadt (1990). [8] S.R. Julian, P.A.A. Teunissen, S.A.J. Wiegers, Phys. Rev. B 46 (1992) 9821. [9] C.A. King, G.G. Lonzarich, Physica B 171 (1991) 161.
Physica B 206 & 207 (1995) 29-32
The nature of elementary excitations below and above the metamagnetic transition in CeRu2Si 2 F.S. Tautz*, S.R. Julian, G.J. McMullan, G.G. Lonzarich Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK
Abstract
We discuss, with reference to our de Haas van Alphen (dHvA) measurements and supporting band calculations, the two low temperature states of the heavy fermion paramagnet CeRu2Si2 which are separated by a steep metamagnetic cross-over. While we are able to propose a consistent Fermi surface model for the paramagnetic state, the high field state continues to evade comprehensive understanding.
Heavy fermion systems take their name from the attribution of their unusual thermodynamic and transport properties at low temperatures to heavy quasiparticles at the Fermi surface. The standard model of these quasi-particles is based on a subtle and intricate coupling of the conduction electrons with a lattice of essentially localised moments (for a review see Ref. [1]). A key question, not readily settled by means of conventional macroscopic probes, is whether the coupling between these two subsystems proves strong enough to amalgamate them both, so that only one type of elementary excitation survives into the coherent regime, as postulated in the standard model. To gain deeper understanding of this issue, we have carried out a series of studies [2,3] of the de Haas van Alphen (dHvA) effect in pure samples of the comparatively simple Ce-based compound CeRu2Si 2 for magnetic fields below and above the metamagnetic transition. Here we focus mainly on our recent findings.
* Corresponding author; present address: Van der WaalsZeeman Laboratory, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands.
Below the metamagnetic transition, the dHvA branches detected include that labelled /z, reported in Refs. [3] and [4], and two other heavy components, v and ~¢, with frequencies 4.9-+0.1 and 8.1---0.1 kT, respectively (not reported in Ref. [4]). The temperature dependence of the amplitude down to nuclear demagnetisation temperatures is consistent with that expected in a model of Fermi liquid excitations with masses of magnitude m*(~:) = 140m e, m*(/x) = ll0rne, m*(v) = 105m e. Comparisons of the experimental results with energy band models based on Density Functional Theory (DFT) in the Local Density Approximation (LDA) as well as the Renormalised Band method (RB) [1], lead to a natural interpretation of these important branches in terms of a single large fluted hole sheet centred at the Z-point in the Brillouin zone. As indicated in Fig. 1, along 100 we observe a central (v) and an off-central (/z) extremal orbit. Together with a complicated electron sheet centred at F and two smaller ellipsoids at Z studied previously, this surface completes a picture in which the quasiparticles account for almost 80% of the observed specific heat for B = 0 [5] (below the metamagnetic transition). Table 1 contains details of the estimate of the specific heat coefficient. The calculated density of states at the Fermi surface of various sheets was
0921-4526/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD! 0921-4526(94)00359-9
30
F.S. Tautz / Physica B 206 & 207 (1995) 29-32
80i~\1,40(15) ~,g,v
60
'~-110(~ ~. (a)
1Ls. _ _L
_ _t _-sl
°°°ogooo l
0 11o
angle
100
Fig. 1. Overview of our results for B.l_c. Numbers in the diagram on the left give renormalised masses, with typical uncertainty in parentheses. Dotted lines are results of an f-band calculation. For each branch, the Fermi level has been adjusted to fit the data along I00. The required shifts are: v, 4.8; e, -2.8; y, -3.9; /3, -4.2 mRy.
Table 1 Specific heat contribution of the various branches. In the second column the unrenormalised contribution from the band calculation is listed. Units of C / T and C ' / T : mJ m o l i K-2. mb is the LDA-calculated mass. Branch
1 2 3 4 5 Sum
C/T
m* /m b
0.31
1.7
0.71 0.88 4.67 6.86 13.42
1.6 41.5 -+ 2.5 9.3 - 4
C*/T
0.5 1.1 194 -+ 12 64 _+27 260 -+ 40
renormalised by individual mass e n h a n c e m e n t factors based on the measured quasi-particle masses. Considering that this estimate is necessarily based entirely on data in the basal plane and that one expects the specific heat to decrease as the magnetic field is increased from 0 T, the above result has to be taken as very good agreement indeed. Thus we conclude that neutral excitations [6] are unlikely to play a m a j o r role in the low field state of CeRu2Si z. We stress that the correspondence between the observed Fermi surface and that predicted in a band model in which the f-orbitals are incorporated among the band states does not imply that the f-electrons are in the conventional sense itinerant. Local moments, dynamically screened by conduction electrons and neighbouring m o m e n t s , may be viewed as contributing to the Fermi volume. The R B description yields a Fermi surface topology similar to that of the L D A , except that it predicts correctly, in agreement with experiment, only two rather than three small ellipsoids about Z. M o r e o v e r , these calculations provide a natural explanation for the variation of the mass renormalisation from band to band, and thus beautifully reproduce the existence of essentially three groups of quasi-particles in this material [7], with cyclotron masses around 2, 20, and 200m e, respectively, i.e. for quasi-particles which are virtually bare conduction electrons and those that involve the intricacies discussed above, in varying degrees. We now turn to a discussion of our results for applied magnetic fields parallel to the c-axis close to the metamagnetic transition (B c ~ 7.8 T) and above. The transition region narrows continuously with decreasing temperature, tending to a value of only 0.025 T as T---~0, an order of magnitude lower than in earlier samples investigated (Fig. 2). A transition to long-range order appears to be suppressed, a result which may be consistent with the prediction of a model in which the coupling of strong longitudinal magnetic fluctuations is treated in a self-consistent field approximation. The behaviour of the resistivity, which is markedly different from that of the susceptibility near B e should provide a searching test of the applicability of this magnetic fluctuation description. On approaching Be, all d H v A frequencies show some effect; the most dramatic change occurs for the heavy branch K below B c which evolves into the light branch/~ above B c. We find a mass of m* = 12 m e for K (instead of the 2 0 m e given in Ref. [4]). The behaviour of the measured frequency, defined as the derivative of the d H v A phase with respect to the
F.S. Tautz / Physica B 206 & 207 (1995) 29-32
I
I
I
I
I
I
1
400 m K 600 m K 00 m K
J I
I
7.4
7.6
000 m K
I
[
I
7.8
I
I
8
8.2
B (T) Fig. 2. Susceptibility peak at the metamagnetic transition. reciprocal field l / B , is expected to be radically different from that of the true frequency or extremal area of the Fermi surface near B~ where the area is strongly field dependent. The behaviour of the extremal area may be inferred by a procedure outlined in Ref. [8]. This analysis leads us to suggest that the K branch is associated with the legs of the electron surface (rather than the torus as proposed in Ref. [4]). The unadjusted LDA calculation predicts a frequency of 0.9 kT for the legs along the c-axis; if the Fermi level
2.5
I
I
I
m*=12
2
31
is adjusted to account for the observed disappearance of the signal at about 35°, the frequency expands to a value close to that observed for 6 (1.2 kT). The sharpness of the transition enables us to follow some of the oscillatory components to within 0.005 T of B c. This has allowed us to observe that the K oscillations behave in a dramatically different way from the /3 and 3' components arising from the Zcentred ellipsoids. The latter decay in amplitude as B c is approached, while the former appear to continue without attenuation up to the transition point itself. This difference in behaviour would appear to provide an important clue as to the microscopic nature of the metamagnetic transition. In the high field state above Be, it has been possible to detect a high frequency branch to with F = 28.2-+ 0.3 kT and m * = 8-+ l m e [3,4]. Its amplitude is extremely sensitive to alignment in the magnetic field and at optimum alignment it may be followed over a wide field range above Be. In spite of the observation of this crucially important branch, the state above Be continues to evade comprehensive understanding. If, as the size of the to frequency would seem to support, a band model in which the f-electrons are treated as core states rather than band states is appropriate for B > B~, then one expects to observe additional frequencies for field along c, which are not seen. One extra branch F('0) = 3.2 kT does seem to exist, but it is very weak and its origin is difficult to reconcile with the above core model. Moreover, if the specific heat is estimated in a similar fashion as for the low field state, one finds that the observed sheets at 17 T contribute 15 mJ mo1-1 K -2, while 9 0 m J mol -~ K -2 are observed [5]. This leaves 80% of 3' unaccounted for. Finally, unambiguous evidence for exchange splittings of all the sheets of the Fermi surface in the polarised state (B >B~) of the form observed in the related polarised material CeRu2Ge 2, well described by the core model [9], has not yet been resolved.
FAcknowledgement
1.5
m,=4 1
I
5
t
The work reported in this article was supported by EPSRC, UK.
.,v I
7.5
10 12.5 15 B (T) Fig. 3. Frequency of orbits K and 6 vs. magnetic field, inset: the 'leg' of branch 5 on which these orbits are situated, rn* is in units of m e.
References [1] P. Fulde, J. Keller, G. Zwicknagl, Solid State Phys. 41 (1988) 1.
32
F.S. Tautz / Physica B 206 & 207 (1995) 29-32
[2] G.G. Lonzarich, J. Magn. Magn. Mater. 76 & 77 (1988) 1. [3] S.R. Julian, F.S. Tautz, G.J. McMullan, G.G. Lonzarich, Physica B 199 & 200 (1994) 63. [4] H. Aoki, S. Uji, A.K. Albessard, Y. Onuki, Phys. Rev. Lett. 71 (1993) 2110. [5] H.P. van der Meulen, A de Visser, J.J.M. Franse, T.T.J.M. Berendschot, J.A.A.J. Perenboom, H. van Kempen, A. Lacerda, P. Lejay, J. Flouquet, Phys. Rev. B 44 (1991) 814.
[6] Y. Kagan, K.A. Kikoin, N.V. Prokof'ev, Physica B 182 (1992) 201. [7] E.K.R. Runge, Ph.D. Thesis, Technische Hochschule, Darmstadt (1990). [8] S.R. Julian, P.A.A. Teunissen, S.A.J. Wiegers, Phys. Rev. B 46 (1992) 9821. [9] C.A. King, G.G. Lonzarich, Physica B 171 (1991) 161.