Absence of superconducting dome at the charge-density-wave quantum phase transition in 2H-NbSe2

Superconductivity is often found in a dome around quantum critical points, i.e. 2nd-order quantum phase transitions. Here, we show that an enhancement of superconductivity is avoided at the critical pressure of the charge-density-wave (CDW) state in NbSe$_2$. We present comprehensive high-pressure Hall effect and magnetic susceptibility measurements of the CDW and superconducting state in NbSe$_2$. Initially, the 2nd-order CDW transition is suppressed smoothly but it drops to zero abruptly at PCDW = 4.4 GPa thus indicating a change to 1st order whilstthe superconducting transition temperature Tc rises continuously up to PCDW but is constant above. The putative 1st-order nature of the CDW transition is suggested as the cause for the absence of a superconducting dome at PCDW. Indeed, we show that the suppression of the superconducting state at low pressures is due to the loss of density of states inside the CDW phase whilst the initial suppression of the CDW state is accounted for by the stiffening of the underlying bare phonon mode.