Figure 1

(Color online) Considering the Fermi-Dirac distribution $f_{\text{FD}}={\left(1+e^z\right)}^{-1}$ (where $z=\left[\epsilon -{\epsilon }_{\text{F}}\right]/k_{\text{B}}T$), the $T$-dependent step in occupation number (lines, dotted green for higher temperature) causes the oscillatory density of states $\tilde{g}=g_0{e}^{i2\pi \epsilon /\hslash {\omega }_{\text{c}}}$ (assuming that $g_0$ is approximately constant on the scale of the cyclotron energy $\hslash {\omega }_{\text{c}}=\hslash eB/m^{\ast }$) shown by the blue (sinusoidal) line to be thermally smeared by the probability distribution $\left|f_{\text{FD}}^{\prime }\left(z\right)\right|=1/\left(2+2\text{cosh}z\right)$ (shaded regions). The consequent reduction in amplitude is equivalent to a Fourier transform of $\left|f_{\text{FD}}^{\prime }\left(z\right)\right|$, yielding oscillations $\propto e^{i\left(2\pi F/B\right)}$ periodic in $1/B$ [$F$ is the conventional quantum oscillation frequency (Ref. 3)] modulated by a $T$-dependent prefactor $a\left(T\right)=a_0\pi \eta /\text{sinh}\pi \eta$ (where $\eta =2\pi k_{\text{B}}T{m}^{\ast }/e\hslash B$ and $a_0$ is a constant). The quantum oscillatory magnetization and resistivity can be expressed in terms of the above thermally averaged density of states, hence the same thermal amplitude factor $a\left(T\right)$.Reuse & Permissions

Figure 2

(Color online) Quantum oscillations measured by the PDO contactless conductivity technique (Ref. 15) in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x\approx 0.56$ (after background polynomial subtraction). This restricted interval in $B=\left|\mathbf{B}\right|$ furnishes a dynamic range of $\sim 50\text{dB}$ between $T=1$ and 18 K. The actual $T$ values are provided in Fig. 3.Reuse & Permissions

Figure 3

(Color online) Quantum oscillation amplitude $a$ versus $T$ extracted by Fourier analysis of the data in Fig. 2 for $40\le B\le 54\text{T}$, corresponding to an average $B=45.96\text{T}$. The upper inset shows the same data (filled circles) plotted versus $T/B$ together with similar points extracted from the field interval $30\le B\le 45\text{T}$ in which the amplitude has been normalized (open circles), matching the field range of the dc field torque measurements (lower inset). The close correspondence between field ranges implies that the dependence on $T/B$ is independent of the interval in $B$ within the resolution of the experiment, suggesting a magnetic field-independent $m^{\ast }$.Reuse & Permissions

Figure 4

(Color online) (a) Data (circles) from Fig. 3 plotted versus $\eta =2\pi k_{\text{B}}T{m}^{\ast }/e\hslash B$, where $m^{\ast }=1.676m_{\text{e}}$. Data from the lower inset to Fig. 3 is included after amplitude normalization relative to the contactless conductivity data at $\eta \approx 0.15$. The line shows a linear interpolation. b Numerical Fourier transform of $a\left(\eta \right)$ in (a) (solid red line) obtained using Eq. (1), plotted over a restricted range of positive $z=\epsilon /k_{\text{B}}T$, compared with the Fermi-Dirac (dotted black line), Gaussian (dashed green line) and Lorentzian (dot-dashed blue line) distributions.Reuse & Permissions

Figure 5

(Color online) Comparison of the fitted probability distributions shown in Fig. 4 with that obtained from experiment. The Fermi-Dirac (solid red line), Gaussian (dashed green line) and Lorentzian (dot-dashed blue line) probability distributions are shown on the $y$ axis as a function of the experimentally obtained probability distribution $\left|f^{\prime }\left(z\right)\right|$ on the $x$ axis. The + symbols locate integer values of $z$ (the black dotted line is a guide to the eye), with the error bars indicating the extent to which experimental uncertainties (chiefly associated with the sample temperature) can cause $\left|f^{\prime }\left(z\right)\right|$ to depart from $\left|f_{\text{FD}}^{\prime }\left(z\right)\right|$.Reuse & Permissions