Field-dependent mass enhancement in ${\text{Pr}}_{1\ensuremath{-}x}{\text{La}}_{x}{\text{Os}}_{4}{\text{Sb}}_{12}$ from aspherical Coulomb scattering

Field-dependent mass enhancement in ${Pr}_{1 - x} {La}_{x} {Os}_{4} {Sb}_{12}$ from aspherical Coulomb scattering

Gertrud Zwicknagl, Peter Thalmeier, and Peter Fulde

Phys. Rev. B 79, 115132 – Published 27 March 2009

Abstract

The scattering of conduction electrons by crystalline electric field (CEF) excitations may enhance their effective quasiparticle mass similar to scattering from phonons. A well-known example is Pr metal where the isotropic exchange scattering from inelastic singlet-singlet excitations causes the mass enhancement. An analogous mechanism may be at work in the skutterudite compounds ${Pr}_{1 - x} {La}_{x} {Os}_{4} {Sb}_{12}$ where close to $x = 1$ the compound develops heavy quasiparticles with a large specific heat $γ$ . There the low-lying CEF states are a singlet ground state and a triplet at $Δ = 8 K$ . Due to the tetrahedral CEF the main scattering mechanism must be the aspherical Coulomb scattering. We derive the expression for mass enhancement in this model including also the case of dispersive excitations. We show that for small to moderate dispersion there is a strongly field-dependent mass enhancement due to the field-induced triplet splitting. It is suggested that this effect may be seen in ${Pr}_{1 - x} {La}_{x} {Os}_{4} {Sb}_{12}$ with suitably large $x$ when the dispersion is small.

Received 16 January 2009

DOI:https://doi.org/10.1103/PhysRevB.79.115132

Authors & Affiliations

Gertrud Zwicknagl¹, Peter Thalmeier², and Peter Fulde^3,4

¹Technische Universität Braunschweig, 38106 Braunschweig, Germany
²Max-Planck-Institut für Chemische Physik fester Stoffe, 01187 Dresden, Germany
³Max-Planck-Institut für Physik Komplexer Systeme, 01187 Dresden, Germany
⁴Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790–784, Korea

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Vol. 79, Iss. 11 — 15 March 2009

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Figure 1
Left (a): Fermi surface of NNN tight-binding model in hole representation in the bcc Brillouin zone (BZ). Right (b): schematic Fermi surface in electron representation in the two-dimensional (2D) projected Brillouin zone. It consists of spheroids around the equivalent $H$ points $(\frac{2 π}{a}, 0, 0)$ . The polar angle $θ$ of $q$ is given by $θ = \frac{1}{2} (π - θ^{'})$ . Furthermore we have $q = 2 p_{F} sin \frac{θ^{'}}{2} = 2 p_{F} cos θ$ where $p_{F}$ is the Fermi momentum. The geometric restrictions require $0 \leq q \leq 2 p_{F}$ and $0 \leq θ \leq \frac{π}{2}$ .Reuse & Permissions
Figure 2
Static (normalized) quadrupolar susceptibility $Δ χ_{Q}$ as a function of magnetic field with ( $d^{2} = 0.067$ or $| d | = 0.26$ ) and without $(d^{2} = 0)$ tetrahedral CEF. Full lines correspond to evaluation with Eq. (24) while the dashed line corresponds to the low-field expansion Eq. (26). Approaching the $Γ_{s} - Γ_{t}^{+}$ level crossing leads to an increasing $χ_{Q}$ and mass enhancement which becomes singular at the crossing $h_{c} / Δ = {(1 - δ^{2})}^{- 1}$ . The latter is pushed to higher field for increasing $d^{2}$ which reduces the increase in $χ_{Q}$ . For $d^{2} \geq 0.42$ $χ_{Q}$ decreases with field strength because the tetrahedral CEF leads to $Γ_{s} - Γ_{t}^{0}$ repulsion. The mass enhancement is proportional to the quadrupolar susceptibility with $δ m^{*} / m = g_{eff} (Δ χ_{Q})$ and $g_{eff} = (\tilde{g} / Δ) \bar{f}$ . For $d^{2} = 0.067$ and $g_{eff} = 0.077$ (Sec. 5) one has ${(δ m^{*} / m)}_{h = 0} = 16$ .Reuse & Permissions
Figure 3
Left (a): variation with magnetic field of $δ m^{*} / m$ calculated from the self-consistent calculation for finite bandwidth $ϵ_{c} = 20 δ_{n} (0)$ (full line) and $ϵ_{c} \to \infty$ (broken line). Right (b): variation with magnetic field of the specific-heat coefficient $γ$ calculated self-consistently for $ϵ_{c} = 20 δ_{n} (0)$ .Reuse & Permissions
Figure 4
Left (a): dispersion of lowest triplet quadrupolar exciton $ω_{+} (q)$ at the critical field $h_{c} (g_{Q})$ where $ω_{+} (Q)$ becomes soft ( $Q$ in units of $2 π / a$ ). As $g_{Q}$ is reduced the dispersion becomes flat increasing the phase space for low-energy conduction-electron scattering. Right (b): field dependence of $Δ χ_{Q} \sim δ m^{*} / m$ for various strengths of intersite quadrupolar coupling $g_{Q}$ (left). For small $g_{Q}$ mass renormalization close to $h_{c}$ is large due to flat $ω_{+} (q)$ dispersion. For larger $g_{Q}$ the dispersion becomes stronger and $ω_{+} (q)$ softens only in the vicinity of $Q = (0, 0, 1)$ , leading to a much smaller mass enhancement at $h_{c}$ . The curves correspond to $g_{Q}$ given in the legend in decreasing order. For the value $g_{Q} = 0.3$ corresponding to ${PrOs}_{4} {Sb}_{12}$ little field dependence of $δ m^{*} / m$ remains.Reuse & Permissions
Figure 5
Quadrupolar susceptibility $Δ χ_{Q} \sim δ m^{*} / m$ as a function of effective intersite coupling $g_{Q}$ . For zero field the intersite coupling or dispersive width of $ω (q)$ has little effect on the mass enhancement. It actually decreases slightly when $g_{Q}^{c} = 0.5$ is approached despite the appearance of the soft mode $ω (Q) \to 0$ . At the critical field $h_{c}$ the dispersion has much stronger influence; when the latter becomes small (decreasing $g_{Q}$ ) the mass enhancement at $h_{c}$ strongly increases. For ${PrOs}_{4} {Sb}_{12}$ $(g_{Q} = 0.3)$ one may expect little field dependence of $δ m^{*} / m$ between $h = 0$ and $h = h_{c}$ . The inset shows the dependence of the AFQ critical field $h_{c}$ on $g_{Q}$ with the approximation $d^{2} ≃ 0$ .Reuse & Permissions

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Field-dependent mass enhancement in ${Pr}_{1 - x} {La}_{x} {Os}_{4} {Sb}_{12}$ from aspherical Coulomb scattering

Gertrud Zwicknagl, Peter Thalmeier, and Peter Fulde

Phys. Rev. B 79, 115132 – Published 27 March 2009

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Field-dependent mass enhancement in Pr1−xLaxOs4Sb12 from aspherical Coulomb scattering

Gertrud Zwicknagl, Peter Thalmeier, and Peter Fulde

Phys. Rev. B 79, 115132 – Published 27 March 2009

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Field-dependent mass enhancement in ${Pr}_{1 - x} {La}_{x} {Os}_{4} {Sb}_{12}$ from aspherical Coulomb scattering