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

Superconductivity is often found in a dome around quantum critical points, i.e., second-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 2H-NbSe_{2}. We present comprehensive high-pressure Hall effect and magnetic susceptibility measurements of the CDW and superconducting state in 2H-NbSe_{2}. Initially, the second-order CDW transition is suppressed smoothly but it drops to zero abruptly at P_{CDW}=4.4GPa thus indicating a change to first order, while the superconducting transition temperature T_{c} rises continuously up to P_{CDW} but is constant above. The putative first-order nature of the CDW transition is suggested as the cause for the absence of a superconducting dome at P_{CDW}. 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, while the initial suppression of the CDW state is accounted for by the stiffening of the underlying bare phonon mode.