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Mean field theory of electron spin resonance g-factors in -YbAlB4
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 382201 (http://iopscience.iop.org/0953-8984/26/38/382201) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 08/07/2017 at 14:29 Please note that terms and conditions apply.
You may also be interested in: Electron spin resonance in YbRh2Si2: local-moment, unlike-spin and quasiparticle descriptions D L Huber Conduction electron spin resonance in the -Yb1xFexAlB4 (0 x 0.50) and -LuAlB4 compounds L M Holanda, G G Lesseux, E T Magnavita et al. Low-energy spin fluctuations in filled skutterudites YbFe4Sb12 and LaFe4Sb12investigated through 121Sb nuclear quadrupole and 139La nuclear magnetic resonancemeasurements A Yamamoto, S Iemura, S Wada et al. Magnetic susceptibility of YbRh2Si2 and YbIr2Si2 on the basis of a localized 4f electronapproach A S Kutuzov, A M Skvortsova, S I Belov et al. Electron spin resonance of YbIr2Si2 below the Kondo temperature J Sichelschmidt, J Wykhoff, H-A Krug von Nidda et al. Spin dynamics of YbRh2Si2 observed by electron spin resonance J Sichelschmidt, J Wykhoff, H-A Krug von Nidda et al. Crossover between Fermi liquid and non-Fermi liquid behavior in the non-centrosymmetric compound Yb2Ni12P7 S Jang, B D White, P-C Ho et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 382201 (2pp)
doi:10.1088/0953-8984/26/38/382201
Fast Track Communication
Mean field theory of electron spin resonance g-factors in β-YbAlB4 D L Huber Department of Physics, University of Wisconsin-Madison, Madison, WI 53706 USA E-mail: [email protected] Received 29 May 2014, revised 5 August 2014 Accepted for publication Published 28 August 2014 Abstract
We present a mean-field theory of the conduction electron spin resonance g-factors in β-YbAlB4. The temperature-dependent shift in the g-factors is attributed to an isotropic exchange interaction between the spin of the conduction electrons and the spin of the Yb3+ ion. In the mean field approximation, the difference between the parallel and perpendicular g-factors is proportional to the difference between the parallel and perpendicular susceptibilities. Using experimental values for the susceptibilities, and fitting the data at 4.2 K, we predict the temperature dependence of the g-shifts at higher temperatures. Keywords: ESR g-factors, RE heavy fermion compounds, mean field theory (Some figures may appear in colour only in the online journal)
where SYb is the total spin of the Yb ion and Λ is an exchange parameter, we separate the interaction into two components
Electron spin resonance (ESR) provides information about metallic compounds with local magnetic moments that can’t be obtained with other techniques. This is especially important in the case of heavy fermion systems with rare earth sublattices that display unusual properties at low temperatures such as superconducting transitions and quantum critical points. In 2011 Holanda et al [1] reported results for the conduction electron (CE) g-factors of the uniaxial heavy fermion superconductor β-YbAlB4. They found that the g-factors were isotropic at room temperature; at 4.2 K, g⊥ was approximately the same as at 290 K, 2.4, whereas gǁ increased by approximately 0.4 to the value 2.8. Recently, Ramires and Coleman [2] presented a microscopic theory of ESR in β-YbAlB4 that accounted for the qualitative change in the anisotropy of the g-factor. In this note we present a theory of temperature dependence of the g-factors that relates the g-factor behavior to the temperature dependence of the magnetic susceptibilities. Our analysis is based on a mean field approximation (MFA) for the contribution of the Yb ions to the g-factors of the CE. The contribution originates in the isotropic exchange interaction between the electron spins and the spins of the Yb3+ ions. In detail, assuming an isotropic interaction between the CE and the trivalent Yb ions of the form −ΛSCE·SYb 0953-8984/14/382201+2$33.00
ΛSCE·SYb = ΛSCE ·+ΛSCE· [SYb− ]
(1)
where the brackets denote a thermal average. Since is proportional to the applied field, the first term gives rise to the shift in the g-factor, while the second term, which characterizes the fluctuations in the effective field, contributes to the ESR line width. The large difference in the temperature dependence of the g-factors of β-YbAlB4 and β-LuAlB4 reported in [1] is attributed to the presence of a filled shell of 4f electrons in the Lu3+ compound with the consequence that SLu = 0 so there is no exchange interaction between the Lu3+ ions and the conduction electrons. Within the ground state (J = 7/2) manifold of the Yb3+ ion we have
= − (gL − 1) (μBgL )−1
(2)
where gL (= 8/7) is the Landé g-factor,μB denotes the Bohr magneton and is the average magnetic moment of the Yb ion. At low fields, is proportional to the applied magnetic field, with the constant of proportionality being the Yb ion susceptibility. Allowing for the anisotropy of the Yb 1
© 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 382201
susceptibility, the mean field equations for the g-factors take the form
g∥ = gCE + A∥χ∥Yb (T )
g⊥ = gCE + A⊥ χ⊥Yb (T )
(3) (4)
where gCE is the conduction electron g-factor, and χ⊥Yb are the susceptibilities of the uniaxial Yb sublattice, and Aǁ and A⊥ are constants. Since the susceptibility of β-YbAlB4 is dominated by the contribution from the Yb ions, we can replace χ∥Yb and χ⊥Yb with the experimental values for β-YbAlB4, χ∥ and χ⊥. As shown in [3], χ⊥ is nearly independent of temperature, while χ∥ approaches χ⊥ for T > 300 K, similar to the longitudinal and transverse g-factors. As a result, at high temperatures we have χ∥Yb
Figure 1. Parallel and perpendicular g-factors for β-YbAlB4 versus
g∥ − g⊥ ≈(A∥ − A⊥ )χ → 0
temperature. The curves are calculated using susceptibility data from [3].
Taking Aǁ and A⊥ to be temperature-independent, we conclude that Aǁ = A⊥ , consistent with an isotropic exchange interaction. As a consequence, (3) and (4) reduce to
g∥ = gCE + Aχ∥
(5)
g⊥ = gCE + Aχ⊥
(6)
upturn in perpendicular susceptibility for T < 10K [3]. Within the framework of mean field theory, such behavior is consistent with a much weaker coupling between the conduction electrons and the Yb ions than is the case in β-YbAlB4, which has a different crystal structure and ground state. Finally, we point out that the appearance of the Yb susceptibility in the g-factor is consistent with a general approach to g-factors in anisotropic exchange-coupled systems where the g-factors are related to the magnetization of the resonating spins along the direction of the applied field and the static susceptibilities in directions perpendicular to the field [5–7]. In the case of the CE resonance in RE-heavy fermion compounds, where the relevant susceptibilities are those of the CE, the CE magnetization contains a term reflecting the induced magnetization arising from the interactions with the RE ions. In the mean field approximation, this term is proportional to the RE susceptibility.
With values of the susceptibility taken from [3], we can use the experimental results for gǁ (4.2 K), 2.8 and g⊥ (4.2 K), 2.4 [1], to infer gCE and A, obtaining gCE ≈ 2.3 and A ≈ 25 (mole-Yb/emu). The difference between gCE and g⊥ reflects the presence of a temperature-independent Van Vleck term in the Yb susceptibility. Note that the positive value obtained for A indicates a positive value for Λ in the exchange interaction. In figure 1, we show the predicted temperature dependence of gǁ along with the temperature-independent value of g⊥. The results obtained for gǁ, displayed in figure 1, are qualitatively similar to the results for g-factor of the powder sample shown in [1] in that gǁ increases slowly with decreasing temperature until a cross-over temperature of ≈ 50K at which point it begins to rise comparatively rapidly. This behavior is significantly different from what was found in [2] where gǁ has a steep rise at 150 K followed by a slow increase to a zero-temperature limit ≈5.8. The striking difference in the predicted temperature variation between the high and low temperature limits points to the need for detailed measurements of the g-factor in the intermediate temperature range. It should be noted that CESR results similar to what are observed in β-LuAlB4 have been reported for α-YbAlB4 [4] even though the temperature dependence of the static susceptibility resembles that of β-YbAlB4 apart from a Curie-like
Acknowledgment We would like to thank Carlos Rettori for helpful comments. References [1] Holanda L M et al 2011 Phys. Rev. Lett. 107 026402 [2] Ramires A and Coleman P 2014 Phys. Rev. Lett. 112 116405 [3] Macaluso R T et al 2007 Chem. Mater. 19 1918 [4] Holanda L M et al 2013 J. Phys. Condens. Matter 25 216001 [5] Huber D L 2009 J. Phys. Condens. Matter 21 322203 [6] Huber D L 2012 J. Phys. Condens. Matter 24 226001 [7] Huber D L 2012 Mod. Phys. Lett. B 29 1230021
2
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My IOPscience
Mean field theory of electron spin resonance g-factors in -YbAlB4
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 382201 (http://iopscience.iop.org/0953-8984/26/38/382201) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 08/07/2017 at 14:29 Please note that terms and conditions apply.
You may also be interested in: Electron spin resonance in YbRh2Si2: local-moment, unlike-spin and quasiparticle descriptions D L Huber Conduction electron spin resonance in the -Yb1xFexAlB4 (0 x 0.50) and -LuAlB4 compounds L M Holanda, G G Lesseux, E T Magnavita et al. Low-energy spin fluctuations in filled skutterudites YbFe4Sb12 and LaFe4Sb12investigated through 121Sb nuclear quadrupole and 139La nuclear magnetic resonancemeasurements A Yamamoto, S Iemura, S Wada et al. Magnetic susceptibility of YbRh2Si2 and YbIr2Si2 on the basis of a localized 4f electronapproach A S Kutuzov, A M Skvortsova, S I Belov et al. Electron spin resonance of YbIr2Si2 below the Kondo temperature J Sichelschmidt, J Wykhoff, H-A Krug von Nidda et al. Spin dynamics of YbRh2Si2 observed by electron spin resonance J Sichelschmidt, J Wykhoff, H-A Krug von Nidda et al. Crossover between Fermi liquid and non-Fermi liquid behavior in the non-centrosymmetric compound Yb2Ni12P7 S Jang, B D White, P-C Ho et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 382201 (2pp)
doi:10.1088/0953-8984/26/38/382201
Fast Track Communication
Mean field theory of electron spin resonance g-factors in β-YbAlB4 D L Huber Department of Physics, University of Wisconsin-Madison, Madison, WI 53706 USA E-mail: [email protected] Received 29 May 2014, revised 5 August 2014 Accepted for publication Published 28 August 2014 Abstract
We present a mean-field theory of the conduction electron spin resonance g-factors in β-YbAlB4. The temperature-dependent shift in the g-factors is attributed to an isotropic exchange interaction between the spin of the conduction electrons and the spin of the Yb3+ ion. In the mean field approximation, the difference between the parallel and perpendicular g-factors is proportional to the difference between the parallel and perpendicular susceptibilities. Using experimental values for the susceptibilities, and fitting the data at 4.2 K, we predict the temperature dependence of the g-shifts at higher temperatures. Keywords: ESR g-factors, RE heavy fermion compounds, mean field theory (Some figures may appear in colour only in the online journal)
where SYb is the total spin of the Yb ion and Λ is an exchange parameter, we separate the interaction into two components
Electron spin resonance (ESR) provides information about metallic compounds with local magnetic moments that can’t be obtained with other techniques. This is especially important in the case of heavy fermion systems with rare earth sublattices that display unusual properties at low temperatures such as superconducting transitions and quantum critical points. In 2011 Holanda et al [1] reported results for the conduction electron (CE) g-factors of the uniaxial heavy fermion superconductor β-YbAlB4. They found that the g-factors were isotropic at room temperature; at 4.2 K, g⊥ was approximately the same as at 290 K, 2.4, whereas gǁ increased by approximately 0.4 to the value 2.8. Recently, Ramires and Coleman [2] presented a microscopic theory of ESR in β-YbAlB4 that accounted for the qualitative change in the anisotropy of the g-factor. In this note we present a theory of temperature dependence of the g-factors that relates the g-factor behavior to the temperature dependence of the magnetic susceptibilities. Our analysis is based on a mean field approximation (MFA) for the contribution of the Yb ions to the g-factors of the CE. The contribution originates in the isotropic exchange interaction between the electron spins and the spins of the Yb3+ ions. In detail, assuming an isotropic interaction between the CE and the trivalent Yb ions of the form −ΛSCE·SYb 0953-8984/14/382201+2$33.00
ΛSCE·SYb = ΛSCE ·+ΛSCE· [SYb− ]
(1)
where the brackets denote a thermal average. Since is proportional to the applied field, the first term gives rise to the shift in the g-factor, while the second term, which characterizes the fluctuations in the effective field, contributes to the ESR line width. The large difference in the temperature dependence of the g-factors of β-YbAlB4 and β-LuAlB4 reported in [1] is attributed to the presence of a filled shell of 4f electrons in the Lu3+ compound with the consequence that SLu = 0 so there is no exchange interaction between the Lu3+ ions and the conduction electrons. Within the ground state (J = 7/2) manifold of the Yb3+ ion we have
= − (gL − 1) (μBgL )−1
(2)
where gL (= 8/7) is the Landé g-factor,μB denotes the Bohr magneton and is the average magnetic moment of the Yb ion. At low fields, is proportional to the applied magnetic field, with the constant of proportionality being the Yb ion susceptibility. Allowing for the anisotropy of the Yb 1
© 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 382201
susceptibility, the mean field equations for the g-factors take the form
g∥ = gCE + A∥χ∥Yb (T )
g⊥ = gCE + A⊥ χ⊥Yb (T )
(3) (4)
where gCE is the conduction electron g-factor, and χ⊥Yb are the susceptibilities of the uniaxial Yb sublattice, and Aǁ and A⊥ are constants. Since the susceptibility of β-YbAlB4 is dominated by the contribution from the Yb ions, we can replace χ∥Yb and χ⊥Yb with the experimental values for β-YbAlB4, χ∥ and χ⊥. As shown in [3], χ⊥ is nearly independent of temperature, while χ∥ approaches χ⊥ for T > 300 K, similar to the longitudinal and transverse g-factors. As a result, at high temperatures we have χ∥Yb
Figure 1. Parallel and perpendicular g-factors for β-YbAlB4 versus
g∥ − g⊥ ≈(A∥ − A⊥ )χ → 0
temperature. The curves are calculated using susceptibility data from [3].
Taking Aǁ and A⊥ to be temperature-independent, we conclude that Aǁ = A⊥ , consistent with an isotropic exchange interaction. As a consequence, (3) and (4) reduce to
g∥ = gCE + Aχ∥
(5)
g⊥ = gCE + Aχ⊥
(6)
upturn in perpendicular susceptibility for T < 10K [3]. Within the framework of mean field theory, such behavior is consistent with a much weaker coupling between the conduction electrons and the Yb ions than is the case in β-YbAlB4, which has a different crystal structure and ground state. Finally, we point out that the appearance of the Yb susceptibility in the g-factor is consistent with a general approach to g-factors in anisotropic exchange-coupled systems where the g-factors are related to the magnetization of the resonating spins along the direction of the applied field and the static susceptibilities in directions perpendicular to the field [5–7]. In the case of the CE resonance in RE-heavy fermion compounds, where the relevant susceptibilities are those of the CE, the CE magnetization contains a term reflecting the induced magnetization arising from the interactions with the RE ions. In the mean field approximation, this term is proportional to the RE susceptibility.
With values of the susceptibility taken from [3], we can use the experimental results for gǁ (4.2 K), 2.8 and g⊥ (4.2 K), 2.4 [1], to infer gCE and A, obtaining gCE ≈ 2.3 and A ≈ 25 (mole-Yb/emu). The difference between gCE and g⊥ reflects the presence of a temperature-independent Van Vleck term in the Yb susceptibility. Note that the positive value obtained for A indicates a positive value for Λ in the exchange interaction. In figure 1, we show the predicted temperature dependence of gǁ along with the temperature-independent value of g⊥. The results obtained for gǁ, displayed in figure 1, are qualitatively similar to the results for g-factor of the powder sample shown in [1] in that gǁ increases slowly with decreasing temperature until a cross-over temperature of ≈ 50K at which point it begins to rise comparatively rapidly. This behavior is significantly different from what was found in [2] where gǁ has a steep rise at 150 K followed by a slow increase to a zero-temperature limit ≈5.8. The striking difference in the predicted temperature variation between the high and low temperature limits points to the need for detailed measurements of the g-factor in the intermediate temperature range. It should be noted that CESR results similar to what are observed in β-LuAlB4 have been reported for α-YbAlB4 [4] even though the temperature dependence of the static susceptibility resembles that of β-YbAlB4 apart from a Curie-like
Acknowledgment We would like to thank Carlos Rettori for helpful comments. References [1] Holanda L M et al 2011 Phys. Rev. Lett. 107 026402 [2] Ramires A and Coleman P 2014 Phys. Rev. Lett. 112 116405 [3] Macaluso R T et al 2007 Chem. Mater. 19 1918 [4] Holanda L M et al 2013 J. Phys. Condens. Matter 25 216001 [5] Huber D L 2009 J. Phys. Condens. Matter 21 322203 [6] Huber D L 2012 J. Phys. Condens. Matter 24 226001 [7] Huber D L 2012 Mod. Phys. Lett. B 29 1230021
2
