Thermopower evolution in Yb(Rh${}_{1-x}$Co${}_x$)${}_2$Si${}_2$ upon 4f localization

YbRh${}_2$Si${}_2$ is a heavy-fermion system, which exhibits a large negative thermopower S(T) with a single minimum around 80 K due to Kondo scattering from the full 4f multiplet. Substitution of Rh by Co leads to a strong reduction of the exchange coupling between the 4f and the conduction electron states and to a lowering of the Kondo scale T${}_K$. This drives the system to a stable trivalent state in YbCo${}_2$Si${}_2$. We performed thermopower measurements on Yb(Rh${}_{1-x}$Co${}_x$)${}_2$Si${}_2$ to study the evolution of S(T) upon 4f localization. As T${}_K$ decreases with increasing x we observe the appearance of a second minimum that shifts subsequently to lower T. Simultaneously, the absolute thermopower values are strongly reduced due to the weaker exchange coupling between the 4f and the conduction electron states. Pure YbCo${}_2$Si${}_2$ still exhibits two minima in S(T) indicative of weak residual Kondo scattering. The value at the high-T minimum is found to be proportional to the Sommerfeld coefficient for the whole series. We discuss this unexpected finding in relation to recent measurements of the valence and Fermi surface evolution with temperature.