Field-angle-resolved specific heat and thermal conductivity in the vortex phase of $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$

The field-angle-dependent specific heat and thermal conductivity in the vortex phase of $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$ is studied using the Doppler shift approximation for the low-energy quasiparticle excitations, mostly in the superclean limit with negligible scattering rate $\Gamma$. We first give a concise presentation of the calculation procedure of magnetothermal properties with vortex and FS averages performed numerically. The comparison of calculated field-angle oscillations and the experimental results obtained previously leads to a strong reduction of the possible SC candidate states in $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$. The possible SC gap functions have node lines in hexagonal symmetry planes containing either the zone center or the AF zone boundary along $c$. Node lines in nonsymmetry planes can be excluded. We also calculate the field and temperature dependence of field-angular oscillation amplitudes. We suggest that the observed nonmonotonic field dependence and sign reversal of the oscillation amplitude may be due to small deviations from unitary scattering. We briefly discuss the influence of a finite $\Gamma$ on the field and angle dependence of thermal conductivity.

Figure 1

Geometrical configuration of corrugated FS cylinder in the AF BZ ($-\pi ∕\tilde{c}⩽k_z⩽\pi ∕\tilde{c}$ with $\tilde{c}=2c$), vortex direction $\left(\mathbf{l}\right)$ and quasiparticle momentum $\mathbf{k}$. Here $\tilde{x},\tilde{y}\equiv y,\tilde{z}$ is the Cartesian crystal coordinate system with $a$, $b$, and $c$ $\left(\mathbf{n}\right)$ axes. The Cartesian coordinate system $x,y,z$ is rotated by the angle $\theta$ around the $y$ axis such that $z$ lines up with the vortex direction $\mathbf{l}$. Here $\varphi$ is the angle between heat current ${\mathbf{j}}_Q$ and the field rotation $\left(\tilde{x}\tilde{z}\right)$ plane. Experimentally the perpendicular configuration $\varphi =90°$ is used. Note that for uniaxial crystals the vortex polar angle $\theta$ differs from the field polar angle ${\theta }_H$, this misallignment is given by Eq. (B1) and shown in the inset of Fig. 2. The quasiparticle momentum $\mathbf{k}$ is described by cylindrical coordinates $(\tilde{\vartheta }=c{k}_z,\tilde{\phi })$ and ${\mathbf{v}}_F\perp \mathrm{FS}$ denotes the quasiparticle velocity. The superfluid velocity is denoted by ${\mathbf{v}}_s$ and ${\alpha }_v$, $\tilde{\rho }$ are the vortex coordinates.

Figure 2

Vortex-averaged dimensionless Doppler shift $(H=1\mathrm{T})$ as function of vortex direction $\theta$ for various quasiparticle momenta $\mathbf{k}$ which are given in cylindrical coordinates as $(\tilde{\vartheta }=c{k}_z,\tilde{\phi })$. Full line: $(\pi ∕2,0)$; dashed line: $(\pi ∕2,\pi ∕2)$; these $\mathbf{k}$ vectors lie in the $k_z=\pi ∕2c$ symmetry plane which is also the nodal AF zone-boundary plane for $A_{1g}$. Dash-dotted line: $(\pi ∕4,\pi ∕2)$; this $\mathbf{k}$ vector lies in a nonsymmetry plane which is the nodal plane for the last gap function in Table . The small splitting at $\theta =90°$ is caused by the FS corrugation. The inset shows the misalignment between vortex $\left(\theta \right)$ and field $\left({\theta }_H\right)$ directions for a mass anisotropy $\alpha =0.69$ appropriate for $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$. In all following angular plots we use $\theta$ as variable and neglect the difference between $\theta$ and ${\theta }_H$.

Figure 3

Quasiparticle DOS for the $A_{1g}$ gap function for $H=0$ (full line) and $1\mathrm{T}$ (dash-dotted line) and field $\Vert c$ $({\theta }_H=\theta =0)$. The DOS singularity is smeared out by the $\mathbf{k}$-dependent DS. The inset shows the ZEDOS induced by the Doppler shift on an enlarged scale.

Figure 4

Effective quasiparticle lifetime [Eq. (23)] as a function of energy in the Born approximation $\left(B\right)$ and unitary $\left(U\right)$ case for fields parallel $(\theta =0)$ and perpendicular $(\theta =90°)$ to the $c$ axis $(H=1\mathrm{T})$. The lifetime and its angular variation are much larger in the Born case. As shown in the inset $({\delta }_0=\text{scattering}\text{phase}\text{shift})$ this leads to a much larger thermal conductivity oscillation amplitude ${[{\widehat{\kappa }}_{yy}\left(0\right)-{\widehat{\kappa }}_{yy}(90°)]}_{T=0}$ of up to 27% in the Born case, whereas it is only $\sim 1.2\%$ in the unitary case. The amplitude changes sign at $\delta =0.83(\pi ∕2)$ and is maximal at ${\delta }_0\simeq 0.91(\pi ∕2)$.

Figure 5

The $A_{1g}$ ZEDOS or specific heat $\gamma (H,T=0)$ coefficient as a function of field strength for $\theta =90°$ is shown as full curve. Typical $\sqrt{H}$ of the low field ZEDOS is observed. Thermal conductivity ${\kappa }_{yy}(H,T=0)$ for $\theta =90°$ normalized to normal state value ${\kappa }_{yy}^n$ is shown for various scattering phase shifts ${\widehat{\delta }}_0=\delta ∕(\pi ∕2)$ close to the unitary limit ${\widehat{\delta }}_0=1$ (broken curves). Deviations from ${\widehat{\delta }}_0=1$ lead to an increasingly prominent nonmonotonic behavior of ${\widehat{\kappa }}_{yy}\left(H\right)={\kappa }_{yy}\left(H\right)∕{\kappa }_{yy}^n$. It is, however, reduced again by the inclusion of scattering beyond the superclean limit (see lower panel of Fig. 8). Nonmonotonic field dependence was observed in Ref. 14 For even smaller ${\widehat{\delta }}_0$ approaching the Born case the initial drop is preserved but then ${\widehat{\kappa }}_{yy}\left(H\right)$ becomes flat.

Figure 6

Dependence of ZEDOS or specific heat $\gamma \left(\mathbf{H}\right)$ coefficient on vortex direction $\theta$ at different field strengths. Qualitative behavior for all type-I–III gap functions is equivalent, this also holds for ${\kappa }_{yy}(\theta ,H)$. Inset: Temperature dependence of the normalized field-angular oscillation amplitude for specific heat or ZEDOS at different fields. The amplitude is defined by $\delta \gamma =\delta N={[N\left(0\right)-N(90°)]}_T∕{[N\left(0\right)-N(90°)]}_0$.

Figure 7

Left panel: Thermal conductivity as function of field angle for type-II gap function at various field strengths. This behavior, notably the sign change of the oscillation amplitude with increasing field corresponds to experimental observations for fields $H⩽2\mathrm{T}$ in Ref. 14. Type I, III gap functions (not shown here) exhibit very similar oscillations as shown in the case of ZEDOS (Fig. 6). Right panel: Thermal conductivity for the hybrid (type IV) gap function $\Delta \left(\mathbf{k}\right)=\Delta \cos 2c{k}_z$. A sharp minimum at intermediate $\theta \simeq 51°$ due to off-symmetry nodal plane appears. This behavior is not observed experimentally (Ref. 14).

Figure 8

Effect of finite quasiparticle scattering rate $\Gamma$ on thermal conductivity for the $A_{1g}$ gap function and phase shift ${\delta }_0=0.9(\pi ∕2)$. Upper panel: Suppression of sign reversed oscillation amplitude in the low-field regime with increasing scattering ($\Gamma ∕\Delta =0$, 0.005, 0.01, 0.02, 0.03 from below). $\Gamma =0$ corresponds to the superclean limit result (see left panel of Fig. 7 for $H=0.1\mathrm{T}$). Lower panel: Reduction of the nonmonotonic field dependence of ${\kappa }_{yy}\left(H\right)$ with scattering ($\Gamma ∕\Delta =0$, 0.01, 0.03). $\Gamma =0$ corresponds to the superclean limit result in Fig. 5.