We study the field-induced quantum critical point (QCP) in YbRh$_2$Si$_2$ by low-temperature magnetization, $M(T)$, and magnetic Gr\"uneisen ratio, $\Gamma_{ \rm mag}$, measurements and compare the results with previous thermal expansion, $\beta(T)$, and critical Gr\"uneisen ratio, $\Gamma^{ cr}(T)$, data on YbRh$_2$(Si$_{ 0.95}$Ge$_{ 0.05}$)$_2$. In the latter case, a slightly negative chemical pressure has been used to tune the system towards its zero-field QCP. The magnetization derivative $-dM/dT$ is far more singular than thermal expansion, reflecting a strongly temperature dependent pressure derivative of the field at constant entropy, $(dH/dP)_S=V_m\beta/(dM/dT)$ ($V_m$: molar volume), which saturates at $(0.15\pm 0.04)$ T/GPa for $T\to 0$. The line $T^\star(H)$, previously observed in Hall- and thermodynamic measurements, separates regimes in $T$-$H$ phase space of stronger $(\epsilon>1$) and weaker $(\epsilon<1$) divergent $\Gamma_{ \rm mag}(T)\propto T^{ -\epsilon}$.
Categories: cond-mat.str-el