- Email: [email protected]
de Haas van Alphen effect in heavy fermion compounds—effective mass and non-Fermi-liquid behaviour
ARTICLE IN PRESS
Physica B 403 (2008) 717–720 www.elsevier.com/locate/physb
de Haas van Alphen effect in heavy fermion compounds—effective mass and non-Fermi-liquid behaviour A. McCollama, J.-S. Xiab, J. Flouquetc, D. Aokic, S.R. Juliana, a
Department of Physics, University of Toronto, Toronto, Ont., Canada M5S 1A7 b National High Magnetic Field Laboratory, Gainesville, FL 32611-8440, USA c DRFMC-SPSMS, CEA Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France
Abstract One reason the de Haas van Alphen (dHvA) effect plays a central role in heavy fermion physics is that the temperature dependence of quantum oscillations can be used to measure effective masses on a Fermi surface specific basis. We present a simple picture of the physics behind this temperature dependence, and discuss the observation of non-Fermi-liquid properties at low millikelvin temperatures via the dHvA effect. r 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Hf; 71.27.+a; 71.18.+y Keywords: Non-Fermi-liquid; de Haas van Alphen; Heavy fermions; Quantum oscillations
1. Introduction The de Haas van Alphen (dHvA) effect has contributed enormously to our present understanding of heavy fermion metals. From the early demonstration of the existence of heavy charged quasi-particles [1] to issues such as the fate of the Fermi surface at a quantum phase transition [2], the capabilities of this experimental technique are particularly well suited to this field of research. The use of the dHvA effect to map Fermi surfaces via the Onsager relation is well established [3], and the underlying physics is widely known. Moreover, the most important application of this capability in heavy fermion physics, to the question of ‘‘small’’ vs. ‘‘large’’ Fermi surfaces in cerium compounds near a quantum critical point, has recently been reviewed [4,5]. That the dHvA effect permits the measurement of quasiparticle masses on a Fermi-surface-specific basis is also well known, but the possibility of observing non-Fermi-liquid behaviour is not generally appreciated. In this paper, we therefore review this aspect of quantum oscillation studies Corresponding author. Tel.: +1 416 978 8188; fax: +1 416 978 2537.
E-mail address: [email protected] (S.R. Julian). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.021
of heavy fermion systems, giving a simple derivation of the central formula—which reduces trivially to the Fermiliquid form when non-Fermi-liquid effects are absent—and describing our recent efforts to observe these effects in CeCoIn5, using a standard dHvA setup, and an ultra-low temperature facility. Our observations illustrate two important advantages of dHvA measurements over more conventional probes of non-Fermi-liquid behaviour such as transport, specific heat or susceptibility: the dHvA effect measures specific quasi-particle orbits on specific Fermi surface sheets, and its sensitivity increases as T ! 0. 2. A simple picture of the temperature dependence The temperature dependence of the amplitude of quantum oscillations in a free Fermi gas follows the Lifshitz–Kosevich formula: X = sinhðX Þ, with X ¼ 2p2 kB T=_oc , where oc ¼ eB=me , T is the temperature, B is the applied magnetic field, and me is the electron mass. The corresponding temperature dependence for a Fermi liquid, or even a weak non-Fermi-liquid, can be obtained quite simply, but this is not apparent in the literature [6]. We therefore give a simplified, non-rigorous, treatment, considering Landau levels crossing a Fermi surface in a
ARTICLE IN PRESS A. McCollam et al. / Physica B 403 (2008) 717–720
718
smoothly varying magnetic field [7], and arrive at the central result without making use of many-body techniques. In a Fermi gas in an applied magnetic field, electrons condense onto Landau levels with energies ðm þ 12Þ_oc , where m is an integer. As the magnetic field increases, _oc increases and successive Landau levels pass out through the Fermi energy. Quantum oscillations appear in many properties but, as the temperature dependence is the same for all, we calculate the simplest, which is the electron number N at constant chemical potential: Z 1 P1 dð ðm þ 1=2Þ_oc Þ N ¼ DðBÞ d m¼0 ðmÞ=k T . (1) B e þ1 0 DðBÞ is the degeneracy per Landau level, which does not enter into the temperature dependence. Assume that the effect of interactions is to shift each Landau level by the real part of the self-energy, and broaden it by the imaginary part. The sum over delta functions becomes a sum over Lorentzians, Z DðBÞ 1 1 d ðmÞ=k T N¼ B p e þ1 0 1 X S00m ðÞ ð2Þ 2 2 0 00 m¼0 ð ðm þ 1=2Þ_oc Sm ðÞÞ þ Sm ðÞ (note that the imaginary part of the electron self-energy is negative), and the sum over Lorentzians is then reexpressed as a sum over their Fourier transforms: Z 1 X 1 0 00 dt eiððmþ1=2Þ_oc Sm ðÞÞtþSm ðÞt . (3) Re 0
m¼0
P Þ ¼ p dðt 2pp=_oc Þ, where the p’s are As m ðe integers [8], the integral over t can be carried out to obtain Z 1 P1 ið_oc =2S0 ðÞiS00 ðÞÞ2pp=_oc p¼0 e N / Re d . (4) eðmÞ=kB T þ 1 0 P
i_oc t m
Eqs. (1)–(4) are illustrated in Fig. 1. At energies far from the Fermi energy, broadening of the Landau levels by S00 ðÞ is large compared to their separation, so distinct Landau levels are only identifiable in the vicinity of F ¼ m. This means that the oscillatory terms in the sum (those with p40) are zero except near F , as shown, so the lower limit of integration can be extended to 1. Each integral in the sum (except p ¼ 0, which is not of interest) can then be evaluated by contour integration. Via Cauchy’s theorem, this corresponds to evaluating the terms in the sum at the poles of the Fermi function, ð mÞ ¼ ion ¼ ið2n þ 1ÞpkB T, which gives X 2ppi p ð1Þ exp fm þ ion Sðion Þg . (5) N / Re _oc p;on Thus, to within some constants, the oscillatory part of the pth harmonic can be expressed as X 2pp 00 ð1Þp eð2pp=_oc Þfon S ðion Þg cos ðm S0 ðion ÞÞ . (6) _oc on
Fig. 1. Top: Fermi gas temperature dependence is given by the integral of the product of a set of delta functions at the Landau level energies with the Fermi function. Middle: When interactions are turned on, each Landau level is shifted by the real part of the self-energy, and broadened by the imaginary part of the self-energy. We then analyse the set of Lorentzians into ‘‘harmonics’’. The bottom panel shows the p ¼ 1 term: integrating the product of this with the Fermi function gives the p ¼ 1 term in Eq. (7).
This corresponds to Eq. (58) of Ref. [6], which gives a more rigorous, but abstract, derivation that includes the effect of a three-dimensional band structure; the self-energy in this formula is assumed to be an average around the quasiparticle orbit. A further modification in the presence of the lattice is that the bare electron mass, me , is replaced by mb , the average of the band mass around the quasiparticle orbit. The cosine term in Eq. (6) gives the oscillatory dependence on field [9]. The self-energy is usually expanded around the Fermi energy, so that the real part of the selfenergy on the imaginary axis is zero, and the cosine term may be brought in front of the sum over on . The temperature dependence, X 2pp AðTÞ ¼ ð1Þp exp fon S00 ðion Þg (7) _oc on is thus controlled by the imaginary part of the self-energy on the imaginary axis. It is interesting to consider various forms of the selfenergy. Firstly, for the Fermi gas SðoÞ ¼ 0 and the series P1 sums to n¼0 expð2ppon =_oc Þ ¼ ðpX Þ= sinh ðpX Þ, the Lifshitz–Kosevich form. If impurity damping is included, all levels are broadened equally, so S00 ðion Þ ¼ g, giving the famous exponential ‘‘Dingle factor’’ damping e2ppg=_oc which multiplies ðpX Þ= sinhðpX Þ. In a Fermi liquid SðoÞ ¼ lo iGo2 , so that the imaginary part of the self-energy on the imaginary axis is S00 ðion Þ ¼ lon þ Go2n . To leading order this is the same dependence on on as in the Fermi gas, so the oscillations still follow an X = sinh X behaviour [10], but with a modified cyclotron frequency oc ¼ eB=ð1 þ lÞmb , equivalent to an enhanced mass m ¼ ð1 þ lÞmb . This temperature
ARTICLE IN PRESS A. McCollam et al. / Physica B 403 (2008) 717–720
Fig. 2. Amplitude vs. temperature. (a) The solid curve is X = sinh X for a Fermi liquid with m ¼ 44me ; (b) the dashed curve is for a marginal Fermi liquid with S00 ðion Þ ¼ 10on lnðon =pkB T X Þ with T X ¼ 10 K; (c) the dotted line is for a two-dimensional antiferromagnetic quantum point pffiffiffiffiffiffiffiffiffiffiffiffiffiffifficritical pffiffiffiffiffiffi (entire orbit assumed critical) with S00 ðion Þ ¼ 5:3 pkB T X on . Parameters are chosen so that all curves fit on the same plot. In the inset, the amplitudes are rescaled to show the similarity of the temperature dependence at high temperature.
dependence is illustrated in Fig. 2, for quasi-particles with a mass of 44me . Assumption of the X = sinh X temperature dependence has, to the best of our knowledge, underpinned all previous dHvA studies of heavy fermion systems, and has led to landmark results such as the original observation of quasiparticle masses exceeding 100 times the bare electron mass in UPt3 [1], and the observation of a peak—possibly a divergence—in the effective mass at a pressure-induced quantum critical point [4,5]. A third important observation is the coexistence in some materials of light and heavy quasi-particles, which may be found on separate Fermi surfaces that have different degrees of coupling to the f-electrons [11], or on the so-called ‘‘cold’’ and ‘‘hot’’ regions of the same Fermi surface [12]. X = sinh X behaviour is shown in curve (a) of Fig. 2, and can be understood with reference to the top panel of Fig. 1: at high temperature, when the Fermi function is broad compared to the Landau level splitting, the oscillations vanish; as T ! 0 and the Fermi function becomes narrow compared to the Landau level splitting, the oscillation amplitude saturates. With a non-Fermi-liquid form of self-energy the oscillations deviate from X = sinh X , but in a rather subtle way. For example, Wasserman et al. [13] discuss the possibility of observing quantum oscillations in a marginal Fermi liquid, for which S00 ðion Þ / on logðon =oX Þ, where oX is a constant. The resulting temperature dependence (curve (b) in Fig. 2) is very similar in shape to X = sinh X . For a quasi-two-dimensional antiferromagnetic quanpffiffiffiffiffiffi tum critical point, S00 ðion Þ / on on regions of the Fermi surface that shrink to ‘‘hot lines’’ as T ! 0 [14]. If a dHvA pffiffiffiffiffiffi orbit traces the hot line, its self-energy is on (giving curve (c) in Fig. 2). If it does not intersect the hot line it will, in contrast, have a Fermi liquid temperature dependence. Finally, intermediate temperature dependence could, in
719
principle, be observed if the orbit crosses the hot line at T ¼ 0: at T40 the self-energy averaged around the orbit would be a superposition of a Fermi liquid part that grows, plus a non-Fermi-liquid part that shrinks to zero, as T ! 0. It is important to note that the curves in Fig. 2 have very little structure and, as shown in the inset, could all be mistaken for X = sinh X unless measurements are extremely precise and carried out to very low temperature. This is a drawback of this approach to measuring non-Fermi-liquid properties. The key identifying feature, especially evident in curve (c), is the ‘‘failure to saturate’’ in the T ! 0 limit. Even this, however, can be mimicked by contributions from spin-up and down components with very different masses, if they happen to add in phase [18]. Note too that the non-Fermi-liquid curves are suppressed relative to the Fermi liquid as T ! 0 [19]. Thus if non-Fermi-liquid coupling is too strong the oscillations will be unobservably small, no matter how low the measurement temperature. This suppression has anomalous (nonDingle) field dependence [6]. Therefore, to observe nonFermi-liquid behaviour requires either extremely high magnetic fields or quasi-particle orbits that are comparatively weakly coupled to the f-electrons. The latter seems to be the case for the a orbits in CeCoIn5, discussed in the following section. The ability to examine specific quasi-particle orbits would be an enormous advantage in the study of nonFermi-liquid effects. Bulk transport and thermodynamic properties are averaged over the Fermi surface in a complicated way that could miss non-Fermi-liquid contributions from hot regions. In addition, the key nonFermi-liquid signature in the dHvA effect emerges in the T ! 0 limit, where other measurements tend to lose sensitivity.
3. Quantum oscillations in CeCoIn5 CeCoIn5 is a heavy fermion compound with a field-tuned quantum critical point close to 5 T. Specific heat and transport measurements suggest non-Fermi-liquid behaviour over a wide region of the temperature-field phase diagram around this point [15–17]. In the context of quantum oscillation experiments, this material has two main Fermi surfaces: the very strongly renormalised b-sheet, for which oscillations have so far been undetectable in the neighbourhood of the critical point; and the comparatively weakly renormalised a-sheet, which supports three quasi-particle orbits (see inset of Fig. 3), observable at all fields down to 5 T. In a recent dHvA study [18] we reported unconventional temperature dependence of the oscillation amplitudes of the a-orbits, between 5 and 7 T. The amplitudes did not saturate, but continued to increase with decreasing temperature down to 6 mK, which was the lowest temperature measured; deviation from X = sinh X was only
ARTICLE IN PRESS A. McCollam et al. / Physica B 403 (2008) 717–720
720
0.14 α3
dHvA amplitude (a.u.)
0.12
rules out an explanation, discussed in Ref. [18], in terms of spin-dependent masses. Rather, it indicates a clear departure from the X = sinh X form, and a breakdown of Fermi liquid theory in the T ! 0 limit. This work demonstrates the potential of the dHvA effect to measure non-Fermi-liquid behaviour, and is an example of such behaviour in a metal at significantly lower temperatures than has previously been reported.
α3 α1 α2
0.1 α1
0.08 0.06 0.04
Acknowledgements
α2
0.02 0 0
50
100 Temperature (mK)
150
200
Fig. 3. Amplitude vs temperature for CeCoIn5 a-orbits between 6 and 7 T: open points are data from the high B/T facility in Gainesville; filled points are from our earlier study. The dashed lines are least-squares fits to these data of X = sinh X for TX20 mK. It is not possible to obtain even moderately good w2 values for a2 and a1 if the To20 mK data are included in the fit. Inset: a Fermi surface and quasi-particle orbits.
observed below 20 mK. We believe this result to be a manifestation of non-Fermi liquid behaviour. Here we report further measurements, carried out at the NHMFL High B/T facility in Gainesville, Florida, which extend these results to even lower temperatures. Fig. 3 shows amplitude vs. temperature data for CeCoIn5 from the two sets of experiments. The field modulation technique was employed in both cases, but in Gainesville a small modulation coil, mounted directly on the dilution refrigerator, was used, rather than a large coil built permanently into the main magnet. This arrangement meant that only small amplitude modulation fields were possible: concerns about eddy current heating at high excitation currents limited the modulation field to p1 G, and detection to the fundamental. A field closer to 30 G would have been required for optimum detection of the more usual second harmonic. The signal-to-noise ratio was consequently much smaller than in the earlier measurements. The temperature, however, was measured with very high accuracy, using a ruthenium oxide resistance thermometer and, below 12 mK, a 3He melting curve thermometer. To ensure that eddy current heating was not distorting our results, possible heating was carefully modelled and the temperatures for the open points are given with error bars on the basis of these calculations. In order to avoid heating effects due to sweeping the main magnetic field, sweeps were carried out at 11 h/Tesla. Oscillations from the a2 orbit were not resolvable in the Gainesville experiment, but there is excellent agreement between the old and new data for a1 and a3 , confirming the accuracy of our earlier measurements. Moreover, new data points below 6 mK show a continuing increase of amplitude with decreasing temperature. We believe this
This work is supported in part by NSERC Canada and the Canadian Institute for Advanced Research. We thank A. Chubukov for helpful discussions. References [1] L. Taillefer, R. Newbury, G.G. Lonzarich, Z. Fisk, J.L. Smith, J. Magn. Magn. Mater. 63 & 64 (1987) 372. [2] S. Araki, R. Settai, T.C. Kobayashi, H. Harima, Y. Onuki, Phys. Rev. B 64 (2001) 224417. [3] e.g., D. Shoenberg in: Magnetic Oscillations in Metals, Cambridge University Press, Cambridge, 1984. [4] Y. Onuki, R. Settai, K. Sugiyama, T. Takeuchi, T.C. Kobayashi, Y. Haga, E. Yamamoto, J. Phys. Soc. Japan 73 (2004) 769. [5] Y. Onuki, R. Settai, H. Shishido, T. Kubo, Y. Yasuda, K. Betsuyaku, H. Harima, J. Alloys Comp. 408–412 (2006) 27. [6] A. Wasserman, M. Springford, Adv. Phys. 45 (1996) 471 and references therein. [7] We ignore the dependence of energy on wave-vector parallel to the applied field, equivalent to working in a two-dimensional electron system. This saves a considerable amount of complexity, but still leads to the central result, Eq. (7). [8] The justification for ignoring the apparent m -dependence of the selfenergy is that the Landau-levels are actually quantised in k-space (they enclose equal k-space areas); the use of _oc is really just a shorthand for this quantisation. So provided that the k-dependence of the self-energy can be ignored this is valid. [9] Note that m=_oc ¼ _Ak =2peB F =B, giving the usual Onsager relation between the Fermi surface area Ak and the dHvA frequency F. [10] Yu.A. Bychkov, L.P. Gor’kov, JETP 14 (1962) 1132. [11] G.G. Lonzarich, J. Magn. Magn. Mater. 76 & 77 (1988) 1. [12] T. Ebihara, N. Harrison, M. Jaime, S. Uji, J.C. Lashley, Phys. Rev. Lett. 93 (2004) 246401. [13] A. Wasserman, M. Springford, F. Han, J. Phys.: Condens. Matter 3 (1991) 5335. [14] A.J. Millis, Phys. Rev. B 48 (1993) 7183; A. Abanov, A.V. Chubukov, J. Schmalian, Adv. Phys. 52 (2003) 119. [15] J. Paglione, M.A. Tanatar, D.G. Hawthorn, E. Boaknin, R.W. Hill, F. Ronning, M. Sutherland, L. Taillefer, C. Petrovich, P.C. Canfield, Phys. Rev. Lett. 91 (2003) 246405. [16] A. Bianchi, R. Movshovich, I. Vekhter, P.G. Pagliuso, J.L. Sarrao, Phys. Rev. Lett. 91 (2003) 257001. [17] J.S. Kim, J. Alwood, G.R. Stewart, J.L. Sarrao, J.D. Thompson, Phys. Rev. B 64 (2001) 134524. [18] A. McCollam, S.R. Julian, P.M.C. Rourke, D. Aoki, J. Flouquet, Phys. Rev. Lett. 94 (2005) 186401. [19] In reality the Dingle factor and other pre-factors (see e.g. Ref. [3]) enter the amplitude, so absolute amplitudes cannot be directly compared on different orbits—the point is that strong non-Fermiliquid behaviour can suppress oscillations to below the measurement threshold.
Physica B 403 (2008) 717–720 www.elsevier.com/locate/physb
de Haas van Alphen effect in heavy fermion compounds—effective mass and non-Fermi-liquid behaviour A. McCollama, J.-S. Xiab, J. Flouquetc, D. Aokic, S.R. Juliana, a
Department of Physics, University of Toronto, Toronto, Ont., Canada M5S 1A7 b National High Magnetic Field Laboratory, Gainesville, FL 32611-8440, USA c DRFMC-SPSMS, CEA Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France
Abstract One reason the de Haas van Alphen (dHvA) effect plays a central role in heavy fermion physics is that the temperature dependence of quantum oscillations can be used to measure effective masses on a Fermi surface specific basis. We present a simple picture of the physics behind this temperature dependence, and discuss the observation of non-Fermi-liquid properties at low millikelvin temperatures via the dHvA effect. r 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Hf; 71.27.+a; 71.18.+y Keywords: Non-Fermi-liquid; de Haas van Alphen; Heavy fermions; Quantum oscillations
1. Introduction The de Haas van Alphen (dHvA) effect has contributed enormously to our present understanding of heavy fermion metals. From the early demonstration of the existence of heavy charged quasi-particles [1] to issues such as the fate of the Fermi surface at a quantum phase transition [2], the capabilities of this experimental technique are particularly well suited to this field of research. The use of the dHvA effect to map Fermi surfaces via the Onsager relation is well established [3], and the underlying physics is widely known. Moreover, the most important application of this capability in heavy fermion physics, to the question of ‘‘small’’ vs. ‘‘large’’ Fermi surfaces in cerium compounds near a quantum critical point, has recently been reviewed [4,5]. That the dHvA effect permits the measurement of quasiparticle masses on a Fermi-surface-specific basis is also well known, but the possibility of observing non-Fermi-liquid behaviour is not generally appreciated. In this paper, we therefore review this aspect of quantum oscillation studies Corresponding author. Tel.: +1 416 978 8188; fax: +1 416 978 2537.
E-mail address: [email protected] (S.R. Julian). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.10.021
of heavy fermion systems, giving a simple derivation of the central formula—which reduces trivially to the Fermiliquid form when non-Fermi-liquid effects are absent—and describing our recent efforts to observe these effects in CeCoIn5, using a standard dHvA setup, and an ultra-low temperature facility. Our observations illustrate two important advantages of dHvA measurements over more conventional probes of non-Fermi-liquid behaviour such as transport, specific heat or susceptibility: the dHvA effect measures specific quasi-particle orbits on specific Fermi surface sheets, and its sensitivity increases as T ! 0. 2. A simple picture of the temperature dependence The temperature dependence of the amplitude of quantum oscillations in a free Fermi gas follows the Lifshitz–Kosevich formula: X = sinhðX Þ, with X ¼ 2p2 kB T=_oc , where oc ¼ eB=me , T is the temperature, B is the applied magnetic field, and me is the electron mass. The corresponding temperature dependence for a Fermi liquid, or even a weak non-Fermi-liquid, can be obtained quite simply, but this is not apparent in the literature [6]. We therefore give a simplified, non-rigorous, treatment, considering Landau levels crossing a Fermi surface in a
ARTICLE IN PRESS A. McCollam et al. / Physica B 403 (2008) 717–720
718
smoothly varying magnetic field [7], and arrive at the central result without making use of many-body techniques. In a Fermi gas in an applied magnetic field, electrons condense onto Landau levels with energies ðm þ 12Þ_oc , where m is an integer. As the magnetic field increases, _oc increases and successive Landau levels pass out through the Fermi energy. Quantum oscillations appear in many properties but, as the temperature dependence is the same for all, we calculate the simplest, which is the electron number N at constant chemical potential: Z 1 P1 dð ðm þ 1=2Þ_oc Þ N ¼ DðBÞ d m¼0 ðmÞ=k T . (1) B e þ1 0 DðBÞ is the degeneracy per Landau level, which does not enter into the temperature dependence. Assume that the effect of interactions is to shift each Landau level by the real part of the self-energy, and broaden it by the imaginary part. The sum over delta functions becomes a sum over Lorentzians, Z DðBÞ 1 1 d ðmÞ=k T N¼ B p e þ1 0 1 X S00m ðÞ ð2Þ 2 2 0 00 m¼0 ð ðm þ 1=2Þ_oc Sm ðÞÞ þ Sm ðÞ (note that the imaginary part of the electron self-energy is negative), and the sum over Lorentzians is then reexpressed as a sum over their Fourier transforms: Z 1 X 1 0 00 dt eiððmþ1=2Þ_oc Sm ðÞÞtþSm ðÞt . (3) Re 0
m¼0
P Þ ¼ p dðt 2pp=_oc Þ, where the p’s are As m ðe integers [8], the integral over t can be carried out to obtain Z 1 P1 ið_oc =2S0 ðÞiS00 ðÞÞ2pp=_oc p¼0 e N / Re d . (4) eðmÞ=kB T þ 1 0 P
i_oc t m
Eqs. (1)–(4) are illustrated in Fig. 1. At energies far from the Fermi energy, broadening of the Landau levels by S00 ðÞ is large compared to their separation, so distinct Landau levels are only identifiable in the vicinity of F ¼ m. This means that the oscillatory terms in the sum (those with p40) are zero except near F , as shown, so the lower limit of integration can be extended to 1. Each integral in the sum (except p ¼ 0, which is not of interest) can then be evaluated by contour integration. Via Cauchy’s theorem, this corresponds to evaluating the terms in the sum at the poles of the Fermi function, ð mÞ ¼ ion ¼ ið2n þ 1ÞpkB T, which gives X 2ppi p ð1Þ exp fm þ ion Sðion Þg . (5) N / Re _oc p;on Thus, to within some constants, the oscillatory part of the pth harmonic can be expressed as X 2pp 00 ð1Þp eð2pp=_oc Þfon S ðion Þg cos ðm S0 ðion ÞÞ . (6) _oc on
Fig. 1. Top: Fermi gas temperature dependence is given by the integral of the product of a set of delta functions at the Landau level energies with the Fermi function. Middle: When interactions are turned on, each Landau level is shifted by the real part of the self-energy, and broadened by the imaginary part of the self-energy. We then analyse the set of Lorentzians into ‘‘harmonics’’. The bottom panel shows the p ¼ 1 term: integrating the product of this with the Fermi function gives the p ¼ 1 term in Eq. (7).
This corresponds to Eq. (58) of Ref. [6], which gives a more rigorous, but abstract, derivation that includes the effect of a three-dimensional band structure; the self-energy in this formula is assumed to be an average around the quasiparticle orbit. A further modification in the presence of the lattice is that the bare electron mass, me , is replaced by mb , the average of the band mass around the quasiparticle orbit. The cosine term in Eq. (6) gives the oscillatory dependence on field [9]. The self-energy is usually expanded around the Fermi energy, so that the real part of the selfenergy on the imaginary axis is zero, and the cosine term may be brought in front of the sum over on . The temperature dependence, X 2pp AðTÞ ¼ ð1Þp exp fon S00 ðion Þg (7) _oc on is thus controlled by the imaginary part of the self-energy on the imaginary axis. It is interesting to consider various forms of the selfenergy. Firstly, for the Fermi gas SðoÞ ¼ 0 and the series P1 sums to n¼0 expð2ppon =_oc Þ ¼ ðpX Þ= sinh ðpX Þ, the Lifshitz–Kosevich form. If impurity damping is included, all levels are broadened equally, so S00 ðion Þ ¼ g, giving the famous exponential ‘‘Dingle factor’’ damping e2ppg=_oc which multiplies ðpX Þ= sinhðpX Þ. In a Fermi liquid SðoÞ ¼ lo iGo2 , so that the imaginary part of the self-energy on the imaginary axis is S00 ðion Þ ¼ lon þ Go2n . To leading order this is the same dependence on on as in the Fermi gas, so the oscillations still follow an X = sinh X behaviour [10], but with a modified cyclotron frequency oc ¼ eB=ð1 þ lÞmb , equivalent to an enhanced mass m ¼ ð1 þ lÞmb . This temperature
ARTICLE IN PRESS A. McCollam et al. / Physica B 403 (2008) 717–720
Fig. 2. Amplitude vs. temperature. (a) The solid curve is X = sinh X for a Fermi liquid with m ¼ 44me ; (b) the dashed curve is for a marginal Fermi liquid with S00 ðion Þ ¼ 10on lnðon =pkB T X Þ with T X ¼ 10 K; (c) the dotted line is for a two-dimensional antiferromagnetic quantum point pffiffiffiffiffiffiffiffiffiffiffiffiffiffifficritical pffiffiffiffiffiffi (entire orbit assumed critical) with S00 ðion Þ ¼ 5:3 pkB T X on . Parameters are chosen so that all curves fit on the same plot. In the inset, the amplitudes are rescaled to show the similarity of the temperature dependence at high temperature.
dependence is illustrated in Fig. 2, for quasi-particles with a mass of 44me . Assumption of the X = sinh X temperature dependence has, to the best of our knowledge, underpinned all previous dHvA studies of heavy fermion systems, and has led to landmark results such as the original observation of quasiparticle masses exceeding 100 times the bare electron mass in UPt3 [1], and the observation of a peak—possibly a divergence—in the effective mass at a pressure-induced quantum critical point [4,5]. A third important observation is the coexistence in some materials of light and heavy quasi-particles, which may be found on separate Fermi surfaces that have different degrees of coupling to the f-electrons [11], or on the so-called ‘‘cold’’ and ‘‘hot’’ regions of the same Fermi surface [12]. X = sinh X behaviour is shown in curve (a) of Fig. 2, and can be understood with reference to the top panel of Fig. 1: at high temperature, when the Fermi function is broad compared to the Landau level splitting, the oscillations vanish; as T ! 0 and the Fermi function becomes narrow compared to the Landau level splitting, the oscillation amplitude saturates. With a non-Fermi-liquid form of self-energy the oscillations deviate from X = sinh X , but in a rather subtle way. For example, Wasserman et al. [13] discuss the possibility of observing quantum oscillations in a marginal Fermi liquid, for which S00 ðion Þ / on logðon =oX Þ, where oX is a constant. The resulting temperature dependence (curve (b) in Fig. 2) is very similar in shape to X = sinh X . For a quasi-two-dimensional antiferromagnetic quanpffiffiffiffiffiffi tum critical point, S00 ðion Þ / on on regions of the Fermi surface that shrink to ‘‘hot lines’’ as T ! 0 [14]. If a dHvA pffiffiffiffiffiffi orbit traces the hot line, its self-energy is on (giving curve (c) in Fig. 2). If it does not intersect the hot line it will, in contrast, have a Fermi liquid temperature dependence. Finally, intermediate temperature dependence could, in
719
principle, be observed if the orbit crosses the hot line at T ¼ 0: at T40 the self-energy averaged around the orbit would be a superposition of a Fermi liquid part that grows, plus a non-Fermi-liquid part that shrinks to zero, as T ! 0. It is important to note that the curves in Fig. 2 have very little structure and, as shown in the inset, could all be mistaken for X = sinh X unless measurements are extremely precise and carried out to very low temperature. This is a drawback of this approach to measuring non-Fermi-liquid properties. The key identifying feature, especially evident in curve (c), is the ‘‘failure to saturate’’ in the T ! 0 limit. Even this, however, can be mimicked by contributions from spin-up and down components with very different masses, if they happen to add in phase [18]. Note too that the non-Fermi-liquid curves are suppressed relative to the Fermi liquid as T ! 0 [19]. Thus if non-Fermi-liquid coupling is too strong the oscillations will be unobservably small, no matter how low the measurement temperature. This suppression has anomalous (nonDingle) field dependence [6]. Therefore, to observe nonFermi-liquid behaviour requires either extremely high magnetic fields or quasi-particle orbits that are comparatively weakly coupled to the f-electrons. The latter seems to be the case for the a orbits in CeCoIn5, discussed in the following section. The ability to examine specific quasi-particle orbits would be an enormous advantage in the study of nonFermi-liquid effects. Bulk transport and thermodynamic properties are averaged over the Fermi surface in a complicated way that could miss non-Fermi-liquid contributions from hot regions. In addition, the key nonFermi-liquid signature in the dHvA effect emerges in the T ! 0 limit, where other measurements tend to lose sensitivity.
3. Quantum oscillations in CeCoIn5 CeCoIn5 is a heavy fermion compound with a field-tuned quantum critical point close to 5 T. Specific heat and transport measurements suggest non-Fermi-liquid behaviour over a wide region of the temperature-field phase diagram around this point [15–17]. In the context of quantum oscillation experiments, this material has two main Fermi surfaces: the very strongly renormalised b-sheet, for which oscillations have so far been undetectable in the neighbourhood of the critical point; and the comparatively weakly renormalised a-sheet, which supports three quasi-particle orbits (see inset of Fig. 3), observable at all fields down to 5 T. In a recent dHvA study [18] we reported unconventional temperature dependence of the oscillation amplitudes of the a-orbits, between 5 and 7 T. The amplitudes did not saturate, but continued to increase with decreasing temperature down to 6 mK, which was the lowest temperature measured; deviation from X = sinh X was only
ARTICLE IN PRESS A. McCollam et al. / Physica B 403 (2008) 717–720
720
0.14 α3
dHvA amplitude (a.u.)
0.12
rules out an explanation, discussed in Ref. [18], in terms of spin-dependent masses. Rather, it indicates a clear departure from the X = sinh X form, and a breakdown of Fermi liquid theory in the T ! 0 limit. This work demonstrates the potential of the dHvA effect to measure non-Fermi-liquid behaviour, and is an example of such behaviour in a metal at significantly lower temperatures than has previously been reported.
α3 α1 α2
0.1 α1
0.08 0.06 0.04
Acknowledgements
α2
0.02 0 0
50
100 Temperature (mK)
150
200
Fig. 3. Amplitude vs temperature for CeCoIn5 a-orbits between 6 and 7 T: open points are data from the high B/T facility in Gainesville; filled points are from our earlier study. The dashed lines are least-squares fits to these data of X = sinh X for TX20 mK. It is not possible to obtain even moderately good w2 values for a2 and a1 if the To20 mK data are included in the fit. Inset: a Fermi surface and quasi-particle orbits.
observed below 20 mK. We believe this result to be a manifestation of non-Fermi liquid behaviour. Here we report further measurements, carried out at the NHMFL High B/T facility in Gainesville, Florida, which extend these results to even lower temperatures. Fig. 3 shows amplitude vs. temperature data for CeCoIn5 from the two sets of experiments. The field modulation technique was employed in both cases, but in Gainesville a small modulation coil, mounted directly on the dilution refrigerator, was used, rather than a large coil built permanently into the main magnet. This arrangement meant that only small amplitude modulation fields were possible: concerns about eddy current heating at high excitation currents limited the modulation field to p1 G, and detection to the fundamental. A field closer to 30 G would have been required for optimum detection of the more usual second harmonic. The signal-to-noise ratio was consequently much smaller than in the earlier measurements. The temperature, however, was measured with very high accuracy, using a ruthenium oxide resistance thermometer and, below 12 mK, a 3He melting curve thermometer. To ensure that eddy current heating was not distorting our results, possible heating was carefully modelled and the temperatures for the open points are given with error bars on the basis of these calculations. In order to avoid heating effects due to sweeping the main magnetic field, sweeps were carried out at 11 h/Tesla. Oscillations from the a2 orbit were not resolvable in the Gainesville experiment, but there is excellent agreement between the old and new data for a1 and a3 , confirming the accuracy of our earlier measurements. Moreover, new data points below 6 mK show a continuing increase of amplitude with decreasing temperature. We believe this
This work is supported in part by NSERC Canada and the Canadian Institute for Advanced Research. We thank A. Chubukov for helpful discussions. References [1] L. Taillefer, R. Newbury, G.G. Lonzarich, Z. Fisk, J.L. Smith, J. Magn. Magn. Mater. 63 & 64 (1987) 372. [2] S. Araki, R. Settai, T.C. Kobayashi, H. Harima, Y. Onuki, Phys. Rev. B 64 (2001) 224417. [3] e.g., D. Shoenberg in: Magnetic Oscillations in Metals, Cambridge University Press, Cambridge, 1984. [4] Y. Onuki, R. Settai, K. Sugiyama, T. Takeuchi, T.C. Kobayashi, Y. Haga, E. Yamamoto, J. Phys. Soc. Japan 73 (2004) 769. [5] Y. Onuki, R. Settai, H. Shishido, T. Kubo, Y. Yasuda, K. Betsuyaku, H. Harima, J. Alloys Comp. 408–412 (2006) 27. [6] A. Wasserman, M. Springford, Adv. Phys. 45 (1996) 471 and references therein. [7] We ignore the dependence of energy on wave-vector parallel to the applied field, equivalent to working in a two-dimensional electron system. This saves a considerable amount of complexity, but still leads to the central result, Eq. (7). [8] The justification for ignoring the apparent m -dependence of the selfenergy is that the Landau-levels are actually quantised in k-space (they enclose equal k-space areas); the use of _oc is really just a shorthand for this quantisation. So provided that the k-dependence of the self-energy can be ignored this is valid. [9] Note that m=_oc ¼ _Ak =2peB F =B, giving the usual Onsager relation between the Fermi surface area Ak and the dHvA frequency F. [10] Yu.A. Bychkov, L.P. Gor’kov, JETP 14 (1962) 1132. [11] G.G. Lonzarich, J. Magn. Magn. Mater. 76 & 77 (1988) 1. [12] T. Ebihara, N. Harrison, M. Jaime, S. Uji, J.C. Lashley, Phys. Rev. Lett. 93 (2004) 246401. [13] A. Wasserman, M. Springford, F. Han, J. Phys.: Condens. Matter 3 (1991) 5335. [14] A.J. Millis, Phys. Rev. B 48 (1993) 7183; A. Abanov, A.V. Chubukov, J. Schmalian, Adv. Phys. 52 (2003) 119. [15] J. Paglione, M.A. Tanatar, D.G. Hawthorn, E. Boaknin, R.W. Hill, F. Ronning, M. Sutherland, L. Taillefer, C. Petrovich, P.C. Canfield, Phys. Rev. Lett. 91 (2003) 246405. [16] A. Bianchi, R. Movshovich, I. Vekhter, P.G. Pagliuso, J.L. Sarrao, Phys. Rev. Lett. 91 (2003) 257001. [17] J.S. Kim, J. Alwood, G.R. Stewart, J.L. Sarrao, J.D. Thompson, Phys. Rev. B 64 (2001) 134524. [18] A. McCollam, S.R. Julian, P.M.C. Rourke, D. Aoki, J. Flouquet, Phys. Rev. Lett. 94 (2005) 186401. [19] In reality the Dingle factor and other pre-factors (see e.g. Ref. [3]) enter the amplitude, so absolute amplitudes cannot be directly compared on different orbits—the point is that strong non-Fermiliquid behaviour can suppress oscillations to below the measurement threshold.