Magnetization study of the energy scales in YbRh${}_2$Si${}_2$ under chemical pressure

We present a systematic study of the magnetization in YbRh${}_2$Si${}_2$ under slightly negative (6?% Ir substitution) and positive (7% Co substitution) chemical pressure. We show how the critical field $H_0$, associated with the high-field Lifshitz transitions, is shifted to lower (higher) values with Co (Ir) substitution. The critical field $H_{\mathrm{N}}$, which identifies the boundary line of the antiferromagnetic (AFM) phase $T_{\mathrm{N}}(H)$ increases with positive pressure and it approaches zero with 6% Ir substitution. On the other side, the crossover field $H^{\ast }$, associated with the energy scale $T^{\ast }(H)$ where a reconstruction of the Fermi surface has been observed, is not much influenced by the chemical substitution.}{Following the analysis proposed in Refs.,\cite{Paschen2004,Gegenwart2007,Friedemann2009,Tokiwa2009a} we have fitted the quantity $\tilde{M}(H)=M+(dM\mathrm{/}dH)H$ with a crossover function to indentify $H^{\ast }$. The $T^{\ast }(H)$ line follows an almost linear $H$-dependence at sufficiently high fields outside the AFM phase, but it deviates from linearity at $T\le T_{\mathrm{N}}(0)$ and in Yb(Rh${}_{0.93}$Co${}_{0.07}$)${}_2$Si${}_2$ it changes slope clearly inside the AFM phase. Moreover, the FWHM of the fit function depends linearly on temperature outside the phase, but remains constant inside, suggesting either that such an analysis is valid only for $T\ge T_{\mathrm{N}}(0)$ or that the Fermi surface changes continuously at $T=0$ inside the AFM phase.}}