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We present band structure calculations and quantum oscillation measurements on LuRh _2 Si _2 , which is an ideal reference to the intensively studied quantum critical heavy-fermion system YbRh _2 Si _2 . Our band structure calculations show a strong sensitivity of the Fermi surface on the position of the silicon atoms z _Si within the unit cell. Single crystal structure refinement and comparison of predicted and observed quantum oscillation frequencies and masses yield z _Si  = 0.379  c in good agreement with numerical lattice relaxation. This value of z _Si is suggested for future band structure calculations on LuRh _2 Si _2  and YbRh _2 Si _2 . LuRh _2 Si _2  with a full f electron shell represents the ‘small’ Fermi surface configuration of YbRh _2 Si _2 . Our experimentally and ab intio derived quantum oscillation frequencies of LuRh _2 Si _2 differ significantly from the results of earlier measurements on YbRh _2 Si _2 . Consequently, our results confirm the contribution of the f electrons to the Fermi surface of YbRh _2 Si _2  at high magnetic fields. Yet, the limited agreement with refined fully itinerant local density approximation calculations highlights the need for more elaborate models to describe the Fermi surface of YbRh _2 Si _2 .