A process for the manufacture of semiconductor devices comprises the chemical mechanical polishing of elemental germanium and/or Si1-xGex material with 0.1 = x < 1 in the presence of a chemical mechanical polishing (CMP) composition comprising: (A) inorganic particles, organic particles, or a mixture or composite thereof, (B) at least one type of an oxidizing agent, (C) at least one type of an organic compound which comprises at least {k} moieties (Z), but excluding salts whose anions are inorganic and whose only organic cation is [NR11R12R13R14]+, wherein {k} is 1, 2 or 3, (Z) is a hydroxyl (-OH), alkoxy (-OR1), heterocyclic alkoxy (-OR1 as part of a heterocyclic structure), carboxylic acid (-COOH), carboxylate (-COOR2), amino (-NR3R4), heterocyclic amino (-NR3R4 as part of a heterocyclic structure), imino (=N-R5 or -N=R6),; heterocyclic imino (=N-R5 or -N=R6 as part of a heterocyclic structure), phosphonate (-P(=0)(OR7)(OR8) ), phosphate (-0-P(=0)(OR9)(OR10) ), phosphonic acid (-P(=0)(OH)2), phosphoric acid (-0-P(=0)(OH)2) moiety, or their protonated or deprotonated forms, R1, R2, R7, R9 is - independently from each other - alkyl, aryl, alkylaryl, or arylalkyl, R3, R4, R5, R8, R10 is - independently from each other - H, alkyl, aryl, alkylaryl, or arylalkyl, R6 is alkylene, or arylalkylene, R11, R12, R13 is - independently from each other - H, alkyl, aryl, alkylaryl, or arylalkyl, and R11, R12, R13 does not comprise any moiety (Z), R14 is alkyl, aryl, alkylaryl, or arylalkyl, and R14 does not comprise any moiety (Z), and (D) an aqueous medium.