Low-temperature specific heat has been measured and extensively analyzed on a series of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ single crystals from underdoped to overdoped regime. From these data the quasiparticle density of state in the mixed state is derived and compared to the predicted scaling law $C_{\mathit{vol}}∕T\sqrt{H}=f(T∕\sqrt{H})$ of $d$-wave superconductivity. It is found that the scaling law can be nicely followed by the optimally doped sample $(x=0.15)$ in quite a wide region of $(T∕\sqrt{H}⩽8K∕\sqrt{T})$. However, the region for this scaling becomes smaller and smaller toward more underdoped region: a clear trend can be seen for samples from $x=0.15\text{to}0.069$. Therefore, generally speaking, the scaling quality becomes worse on the underdoped samples in terms of scalable region of $T∕\sqrt{H}$. This feature in the underdoped region is explained as due to the low-energy excitations from a second order (for example, antiferromagnetic correlation, $d$-density wave, spin-density wave, or charge-density wave order) that may coexist or compete with superconductivity. Surprisingly, deviations from the $d$-wave scaling law have also been found for the overdoped sample $(x=0.22)$, while the scaling law is reconciled for the overdoped sample, when the core size effect is taken into account. An important discovery of present work is that the zero-temperature data follow the Volovik’s relation $\Delta \gamma (T=0)=A\sqrt{H}$ quite well for all samples investigated here; although the applicability of the $d$-wave scaling law to the data at finite temperatures varies with doped-hole concentration. We also present the doping dependence of some parameters, such as the residual linear term ${\gamma }_0$, the $\alpha$ value, etc. It is suggested that the residual linear term $\left({\gamma }_{0}T\right)$ of the electronic specific heat observed in all cuprate superconductors is probably due to the inhomogeneity, either chemical or electronic in origin. The field-induced reduction of the specific heat in the mixed state is also reported. Finally, implications on the electronic phase diagram are suggested.

• Received 8 March 2004

DOI:https://doi.org/10.1103/PhysRevB.70.214505