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Quantum critical point in CePd1-xRhx ferromagnet
ARTICLE IN PRESS
Physica B 403 (2008) 755–757 www.elsevier.com/locate/physb
Quantum critical point in CePd1xRhx ferromagnet V.R. Shaginyana,, E.V. Kirichenkob, V.A. Stephanovichb a
Petersburg Nuclear Physics Institute, RAS, Gatchina 188300, Russia Institute of Mathematics and Informatics, Opole University, 45-052 Opole, Poland
b
Abstract The heavy-fermion metal CePd1x Rhx can be tuned from ferromagnetism at x ¼ 0 to non-magnetic state at the critical concentration xc . The non-Fermi liquid behavior at x ’ xc is recognized by power law dependence of the specific heat CðTÞ given by p the ffiffiffiffi electronic contribution, susceptibility wðTÞ and volume expansion coefficient aðTÞ at low temperatures: C=T / wðTÞ / aðTÞ=T / 1= T . We show that this alloy exhibits a universal thermodynamic non-Fermi liquid behavior independent of magnetic ground state. This can be well understood utilizing the quasiparticle picture and the concept of fermion condensation quantum phase transition at the density r ¼ p3F =3p2 ¼ rFC , pF is Fermi momentum. r 2007 Published by Elsevier B.V. PACS: 71.27.þa; 74.20.Fg; 74.25.Jb Keywords: Heavy-fermion metal; Non-Fermi liquid behavior; Fermion condensation quantum phase transition
The nature of the non-Fermi liquid (NFL) behavior of heavy-fermion (HF) metals is still hotly debated. It is widely believed that the observed behavior is determined by quantum phase transitions taking place at quantum critical points (QCP), while the proximity of a system to above points generates its NFL behavior due to corresponding thermal and quantum fluctuations suppressing quasiparticle excitations [1]. The NFL behavior around QCP manifests itself in various anomalies. One of them is power in T variations of the specific heat CðTÞ, thermal expansion aðTÞ, magnetic susceptibility wðTÞ, etc. [2]. The phase diagram of HF metal CePd1x Rhx demonstrates two peculiar features—the percolation (due to effects of atomic disorder in the above diluted alloy) ‘‘tail’’ in the dependence of the phase transition temperature T c ðxÞ as well as the NFL behavior at x ’ xc . This means that around concentration x ¼ xc ’ 0:9 the suppression of the ferromagnetism occurs so that above alloy is tuned to a non-magneticpstate ffiffiffiffi at x ¼ xc . At x ¼ xc , the specific heat CðTÞ=T / 1= T , while around that concentration C=T and wðTÞ coincide in their temperature dependence, Corresponding author.
E-mail address: [email protected] (V.R. Shaginyan). 0921-4526/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.physb.2007.10.223
pffiffiffiffi CðTÞ=T / wðTÞ / 1= T [3]. pffiffiffiffiMoreover, as we shall see, it proved to be aðTÞ=T / 1= T . The above power laws can be hardly accounted for within scenarios based on the realization of the QCP with quantum and thermal fluctuations [3,4]. These facts suggest that the fluctuations might not be responsible for the observed behavior, and if they are not, what kind of physics determines the NFL behavior? Fortunately, the direct observations of quasiparticles in CeCoIn5 have been reported recently [5]. They permit to conclude safely that quasiparticles are responsible for the above NFL behavior [4,6]. In this paper, we show that the NFL behavior of the ferromagnet CePd1x Rhx related to the uniform temperapffiffiffiffi ture dependence of CðTÞ=T / wðTÞ / 1= T can be well understood in terms of quasiparticles and fermion condensation quantum phase transition (FCQPT) picture. This behavior is independent of specifics of the given alloy such as its lattice structure, magnetic state, etc. For the description of above percolational ‘‘tail’’ in T c ðxÞ we use so-called random local field method (RFM), developed earlier, see Ref. [7] and references therein. This method is dealing with diluted systems of Heisenberg or Ising (or pseudospins like electric dipoles) spins. It is based on the introduction of a random molecular field (rather
ARTICLE IN PRESS V.R. Shaginyan et al. / Physica B 403 (2008) 755–757
756
than mean field) acting between the spins. The distribution function of the above field is derived self-consistently. This method permits to derive the equation for T c ðxÞ with respect to the thermal and spatial fluctuations in the spin ensemble. The theoretical dependence T c ðxÞ=T c ðx ¼ 0Þ is reported in Fig. 1 along with experimental points from Ref. [3]. It is seen the pretty good coincidence between theory and experiment especially in the region of the ‘‘tail’’. Now we turn to a description of the temperature dependence of the effective mass when the system at x4xc approaches FCQPT r ! rFC , r p3F =ð3p2 Þ, pF is Fermi momentum. For that we apply the Landau equation relating the effective mass M ðTÞ with bare mass M and Landau interaction amplitude [8]. At T ¼ 0 the quasiparticle occupation number n ¼ yðpF pÞ and we obtain [8–10] M ðrÞ ¼
M B . ’Aþ 1 r rFC 1 N 0 F ðpF ; pF Þ=3
(1)
Here N 0 is the density of states of a free electron gas, pF is Fermi momentum, F 1 ðpF ; pF Þ F 1 ðrÞ is the p-wave component of Landau interaction amplitude, A and B are constants. When r ! rFC , F 1 ðrÞ achieves certain value making the denominator to be zero and M to diverge, see Eq. (1). At elevated temperatures we introduce small deviation dnðp; TÞ ¼ nðp; TÞ yðpF pÞ and consider finite temperature corrections to zero temperature result (1). It can be shown that in the FCQPT point, where 1=M ðrÞ ! 0, the evaluation of corresponding integrals yields 1 M ðTÞ / pffiffiffiffi . T
(2)
Eq. (2) shows the universal power-law behavior of the effective mass which does not depend on the details of interparticle interaction. Now we consider the thermal expansion coefficient aðTÞ given in Ref. [11]: 1 qðlog V Þ 1 qðS=xÞ aðTÞ ¼ ¼ . (3) 3 qT 3V qP P T
Fig. 1. The comparison between theory (line) and experiment [3] (points) of normalized dependence T c ðxÞ=T c ðx ¼ 0Þ for CePd1x Rhx .
Here, P is the pressure and V is the volume. In the FCQPT point, the compressibility K is approximately constant pffiffiffiffi [12]. Inserting the expression for the entropy SðTÞp/ffiffiffiffi T into Eq. (3), we find that aðTÞ ’ ðM TÞ=ðp2F KÞ / T . Onpthe ffiffiffiffi other hand, the specific heat CðTÞ ¼ TðqSðTÞ=qTÞ / T . As a result, at T ! 0 the Gru¨neisen ratio GðTÞ ¼ aðTÞ=CðTÞ tends to some constant rather than diverges as in the case when the electronic system is on the ordered side of FCQPT [13]. Since the magnetic susceptibility wðTÞ / M ðTÞ, we conclude that CðTÞ aðTÞ / wðTÞ / / M ðTÞ. T T
(4)
At this point, we consider how Eq. (4) and the behavior of the effective mass given by Eq. (2) coincides with experimental observations on CePd1x Rhx [3]. Measurements of wðTÞ and CðTÞ show that our results are in accord with facts. Measurements of aðTÞ on CePd1x Rhx with x ¼ 0:87 and x ¼ 0:90 [3] are shown pffiffiffiffi in Fig. 2. It is seen that the approximation aðTÞ ¼ c1 T for x ¼ 0:90 is in good agreement with observed behavior over two temperature decades from 6 K down to 100 mK. Note that measurements on CeNi2 Ge2 [14] and CeIn3x Snx [15] demonstrate the same behavior despite the different magnetic ground states. Namely, while CePd1x Rhx is a three dimensional ferromagnet [3], CeNi2 Ge2 has a paramagnetic ground state [14] and CeIn3x Snx is antiferromagnetic cubic metal [15]. This permits to conclude that the observed uniform behavior of the thermal expansion coefficient of these metals is determined by quasiparticles and FCQPT rather than by different magnetic quantum critical points and corresponding fluctuations. This work was supported by RFBR, Grant #05-0216085.
Fig. 2. The thermal expansion coefficient aðTÞ for 0:1pTp6 K. Full lines are fits for x ¼ 0:87 and x ¼ 0:90 data [3] based on Eq. (2) and represented pffiffiffiffi by function aðTÞ ¼ c1 T with c1 being a fitting parameter.
ARTICLE IN PRESS V.R. Shaginyan et al. / Physica B 403 (2008) 755–757
References [1] M. Vojta, Rep. Progr. Phys. 66 (2003) 2069. [2] G.R. Stewart, Rev. Modern Phys. 73 (2001) 797. [3] J.S. Sereni, et al., cond-mat/0602588; A.P. Pikul, et al., J. Phys.: Condens. Matter 18 (2006) L535. [4] V.R. Shaginyan, JETP Lett. 79 (2004) 286. [5] J. Paglione, et al., Phys. Rev. Lett. 97 (2006) 106606. [6] V.R. Shaginyan, A.Z. Msezane, V.A. Stephanovich, E.V. Kirichenko, Europhys. Lett. 76 (2006) 898. [7] Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 67 (2003) 195203.
757
[8] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 2, Butterworth-Heinemann, Oxford, 1999. [9] M. Pfitzner, P. Wo¨lfle, Phys. Rev. B 33 (1986) 2003. [10] V.R. Shaginyan, JETP Lett. 77 (2003) 99. [11] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 1, Butterworth-Heinemann, 2000, pp. 168. [12] P. Nozie`res, J. Phys. I (France) 2 (1992) 443. [13] M.Ya. Amusia, A.Z. Msezane, V.R. Shaginyan, Phys. Lett. A 320 (2004) 459. [14] R. Ku¨chler, et al., Phys. Rev. Lett. 91 (2003) 066405. [15] R. Ku¨chler, et al., Phys. Rev. Lett. 96 (2006) 256403.
Physica B 403 (2008) 755–757 www.elsevier.com/locate/physb
Quantum critical point in CePd1xRhx ferromagnet V.R. Shaginyana,, E.V. Kirichenkob, V.A. Stephanovichb a
Petersburg Nuclear Physics Institute, RAS, Gatchina 188300, Russia Institute of Mathematics and Informatics, Opole University, 45-052 Opole, Poland
b
Abstract The heavy-fermion metal CePd1x Rhx can be tuned from ferromagnetism at x ¼ 0 to non-magnetic state at the critical concentration xc . The non-Fermi liquid behavior at x ’ xc is recognized by power law dependence of the specific heat CðTÞ given by p the ffiffiffiffi electronic contribution, susceptibility wðTÞ and volume expansion coefficient aðTÞ at low temperatures: C=T / wðTÞ / aðTÞ=T / 1= T . We show that this alloy exhibits a universal thermodynamic non-Fermi liquid behavior independent of magnetic ground state. This can be well understood utilizing the quasiparticle picture and the concept of fermion condensation quantum phase transition at the density r ¼ p3F =3p2 ¼ rFC , pF is Fermi momentum. r 2007 Published by Elsevier B.V. PACS: 71.27.þa; 74.20.Fg; 74.25.Jb Keywords: Heavy-fermion metal; Non-Fermi liquid behavior; Fermion condensation quantum phase transition
The nature of the non-Fermi liquid (NFL) behavior of heavy-fermion (HF) metals is still hotly debated. It is widely believed that the observed behavior is determined by quantum phase transitions taking place at quantum critical points (QCP), while the proximity of a system to above points generates its NFL behavior due to corresponding thermal and quantum fluctuations suppressing quasiparticle excitations [1]. The NFL behavior around QCP manifests itself in various anomalies. One of them is power in T variations of the specific heat CðTÞ, thermal expansion aðTÞ, magnetic susceptibility wðTÞ, etc. [2]. The phase diagram of HF metal CePd1x Rhx demonstrates two peculiar features—the percolation (due to effects of atomic disorder in the above diluted alloy) ‘‘tail’’ in the dependence of the phase transition temperature T c ðxÞ as well as the NFL behavior at x ’ xc . This means that around concentration x ¼ xc ’ 0:9 the suppression of the ferromagnetism occurs so that above alloy is tuned to a non-magneticpstate ffiffiffiffi at x ¼ xc . At x ¼ xc , the specific heat CðTÞ=T / 1= T , while around that concentration C=T and wðTÞ coincide in their temperature dependence, Corresponding author.
E-mail address: [email protected] (V.R. Shaginyan). 0921-4526/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.physb.2007.10.223
pffiffiffiffi CðTÞ=T / wðTÞ / 1= T [3]. pffiffiffiffiMoreover, as we shall see, it proved to be aðTÞ=T / 1= T . The above power laws can be hardly accounted for within scenarios based on the realization of the QCP with quantum and thermal fluctuations [3,4]. These facts suggest that the fluctuations might not be responsible for the observed behavior, and if they are not, what kind of physics determines the NFL behavior? Fortunately, the direct observations of quasiparticles in CeCoIn5 have been reported recently [5]. They permit to conclude safely that quasiparticles are responsible for the above NFL behavior [4,6]. In this paper, we show that the NFL behavior of the ferromagnet CePd1x Rhx related to the uniform temperapffiffiffiffi ture dependence of CðTÞ=T / wðTÞ / 1= T can be well understood in terms of quasiparticles and fermion condensation quantum phase transition (FCQPT) picture. This behavior is independent of specifics of the given alloy such as its lattice structure, magnetic state, etc. For the description of above percolational ‘‘tail’’ in T c ðxÞ we use so-called random local field method (RFM), developed earlier, see Ref. [7] and references therein. This method is dealing with diluted systems of Heisenberg or Ising (or pseudospins like electric dipoles) spins. It is based on the introduction of a random molecular field (rather
ARTICLE IN PRESS V.R. Shaginyan et al. / Physica B 403 (2008) 755–757
756
than mean field) acting between the spins. The distribution function of the above field is derived self-consistently. This method permits to derive the equation for T c ðxÞ with respect to the thermal and spatial fluctuations in the spin ensemble. The theoretical dependence T c ðxÞ=T c ðx ¼ 0Þ is reported in Fig. 1 along with experimental points from Ref. [3]. It is seen the pretty good coincidence between theory and experiment especially in the region of the ‘‘tail’’. Now we turn to a description of the temperature dependence of the effective mass when the system at x4xc approaches FCQPT r ! rFC , r p3F =ð3p2 Þ, pF is Fermi momentum. For that we apply the Landau equation relating the effective mass M ðTÞ with bare mass M and Landau interaction amplitude [8]. At T ¼ 0 the quasiparticle occupation number n ¼ yðpF pÞ and we obtain [8–10] M ðrÞ ¼
M B . ’Aþ 1 r rFC 1 N 0 F ðpF ; pF Þ=3
(1)
Here N 0 is the density of states of a free electron gas, pF is Fermi momentum, F 1 ðpF ; pF Þ F 1 ðrÞ is the p-wave component of Landau interaction amplitude, A and B are constants. When r ! rFC , F 1 ðrÞ achieves certain value making the denominator to be zero and M to diverge, see Eq. (1). At elevated temperatures we introduce small deviation dnðp; TÞ ¼ nðp; TÞ yðpF pÞ and consider finite temperature corrections to zero temperature result (1). It can be shown that in the FCQPT point, where 1=M ðrÞ ! 0, the evaluation of corresponding integrals yields 1 M ðTÞ / pffiffiffiffi . T
(2)
Eq. (2) shows the universal power-law behavior of the effective mass which does not depend on the details of interparticle interaction. Now we consider the thermal expansion coefficient aðTÞ given in Ref. [11]: 1 qðlog V Þ 1 qðS=xÞ aðTÞ ¼ ¼ . (3) 3 qT 3V qP P T
Fig. 1. The comparison between theory (line) and experiment [3] (points) of normalized dependence T c ðxÞ=T c ðx ¼ 0Þ for CePd1x Rhx .
Here, P is the pressure and V is the volume. In the FCQPT point, the compressibility K is approximately constant pffiffiffiffi [12]. Inserting the expression for the entropy SðTÞp/ffiffiffiffi T into Eq. (3), we find that aðTÞ ’ ðM TÞ=ðp2F KÞ / T . Onpthe ffiffiffiffi other hand, the specific heat CðTÞ ¼ TðqSðTÞ=qTÞ / T . As a result, at T ! 0 the Gru¨neisen ratio GðTÞ ¼ aðTÞ=CðTÞ tends to some constant rather than diverges as in the case when the electronic system is on the ordered side of FCQPT [13]. Since the magnetic susceptibility wðTÞ / M ðTÞ, we conclude that CðTÞ aðTÞ / wðTÞ / / M ðTÞ. T T
(4)
At this point, we consider how Eq. (4) and the behavior of the effective mass given by Eq. (2) coincides with experimental observations on CePd1x Rhx [3]. Measurements of wðTÞ and CðTÞ show that our results are in accord with facts. Measurements of aðTÞ on CePd1x Rhx with x ¼ 0:87 and x ¼ 0:90 [3] are shown pffiffiffiffi in Fig. 2. It is seen that the approximation aðTÞ ¼ c1 T for x ¼ 0:90 is in good agreement with observed behavior over two temperature decades from 6 K down to 100 mK. Note that measurements on CeNi2 Ge2 [14] and CeIn3x Snx [15] demonstrate the same behavior despite the different magnetic ground states. Namely, while CePd1x Rhx is a three dimensional ferromagnet [3], CeNi2 Ge2 has a paramagnetic ground state [14] and CeIn3x Snx is antiferromagnetic cubic metal [15]. This permits to conclude that the observed uniform behavior of the thermal expansion coefficient of these metals is determined by quasiparticles and FCQPT rather than by different magnetic quantum critical points and corresponding fluctuations. This work was supported by RFBR, Grant #05-0216085.
Fig. 2. The thermal expansion coefficient aðTÞ for 0:1pTp6 K. Full lines are fits for x ¼ 0:87 and x ¼ 0:90 data [3] based on Eq. (2) and represented pffiffiffiffi by function aðTÞ ¼ c1 T with c1 being a fitting parameter.
ARTICLE IN PRESS V.R. Shaginyan et al. / Physica B 403 (2008) 755–757
References [1] M. Vojta, Rep. Progr. Phys. 66 (2003) 2069. [2] G.R. Stewart, Rev. Modern Phys. 73 (2001) 797. [3] J.S. Sereni, et al., cond-mat/0602588; A.P. Pikul, et al., J. Phys.: Condens. Matter 18 (2006) L535. [4] V.R. Shaginyan, JETP Lett. 79 (2004) 286. [5] J. Paglione, et al., Phys. Rev. Lett. 97 (2006) 106606. [6] V.R. Shaginyan, A.Z. Msezane, V.A. Stephanovich, E.V. Kirichenko, Europhys. Lett. 76 (2006) 898. [7] Yu.G. Semenov, V.A. Stephanovich, Phys. Rev. B 67 (2003) 195203.
757
[8] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 2, Butterworth-Heinemann, Oxford, 1999. [9] M. Pfitzner, P. Wo¨lfle, Phys. Rev. B 33 (1986) 2003. [10] V.R. Shaginyan, JETP Lett. 77 (2003) 99. [11] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part 1, Butterworth-Heinemann, 2000, pp. 168. [12] P. Nozie`res, J. Phys. I (France) 2 (1992) 443. [13] M.Ya. Amusia, A.Z. Msezane, V.R. Shaginyan, Phys. Lett. A 320 (2004) 459. [14] R. Ku¨chler, et al., Phys. Rev. Lett. 91 (2003) 066405. [15] R. Ku¨chler, et al., Phys. Rev. Lett. 96 (2006) 256403.