A process for the manufacture of semiconductor devices comprising the chemical mechanical polishing of elemental germanium and/or Si 1-x Ge x material with 0.1 ≤ x < 1 in the presence of a chemical mechanical polishing composition comprising: inorganic particles, organic particles, or a mixture or composite thereof, at least one type of an oxidizing agent, at least one type of an organic compounds which comprises at least {k} moieties, but excluding salts whose anions are inorganic and whose only organic cation is [NR 11 R 12 R 13 R 14] +, wherein {k} is 1, 2 or 3, is a hydroxyl (-OH), alkoxy (-OR 1), heterocyclic alkoxy (-OR 1 as part of a heterocyclic structure), carboxylic acid (-COOH), amino (-NR 3 R 4), heterocyclic amino (-NR 3 R 4 as part of a heterocyclic structure), imino (=N-R 5 or -N=R 6), heterocyclic imino (=N-R 5 or -N=R 6 as part of a heterocyclic structure), or their protonated or deprotonated forms, R 1 is alkyl, aryl, alkylaryl, or arylalkyl, R 3, R 4, R 5 is - independently from each other - H, alkyl, aryl, alkylaryl, or arylalkyl, R 6 is alkylene, or arylalkylene, R 11, R 12, R 13 is - independently from each other - H, alkyl, aryl, alkylaryl, or arylalkyl, and R 11, R 12, R 13 does not comprise any moiety, R 14 is alkyl, aryl, alkylaryl, or arylalkyl, and R does not comprise any moiety, and said organic compound is selected from the group consisting of alpha-amino acid or a salt thereof, an organic compound comprising from two to five carboxylic acid (-COOH) groups or a salt thereof, a mono-, di-, trialkanolamine or a salt thereof, an amino ether comprising a further amino, hydroxyl, alkoxy, carboxyl moiety or a salt thereof, an organic compound comprising two to four hydroxyl groups or a salt thereof, a heterocyclic compound comprising a 5-or 6-membered ring having one to three nitrogen atoms as ring members atom or a salt thereof, N,N,N',N'-tetrakis(2-hydroxypropyl)ethylenediamine, 4-(2-hydroxyethyl)morpholine, pentamethyldiethylenetriamine, a salt or adduct of triethanolamine (=2,2',2"-nitrilotris(ethanol)) and 4-[(2-ethylhexyl)amino]-4-oxoisocrotonic acid and 2,2'-dimorpholinodiethylether, and an aqueous medium, wherein the pH value of the CMP composition is in the range of from 2.5 to 5.5. inorganic particles, organic particles, or a mixture or composite thereof, at least one type of an oxidizing agent, at least one type of an organic compounds which comprises at least {k} moieties, but excluding salts whose anions are inorganic and whose only organic cation is [NR 11 R 12 R 13 R 14] +, wherein {k} is 1, 2 or 3, is a hydroxyl (-OH), alkoxy (-OR 1), heterocyclic alkoxy (-OR 1 as part of a heterocyclic structure), carboxylic acid (-COOH), amino (-NR 3 R 4), heterocyclic amino (-NR 3 R 4 as part of a heterocyclic structure), imino (=N-R 5 or -N=R 6), heterocyclic imino (=N-R 5 or -N=R 6 as part of a heterocyclic structure), or their protonated or deprotonated forms, R 1 is alkyl, aryl, alkylaryl, or arylalkyl, R 3, R 4, R 5 is - independently from each other - H, alkyl, aryl, alkylaryl, or arylalkyl, R 6 is alkylene, or arylalkylene, R 11, R 12, R 13 is - independently from each other - H, alkyl, aryl, alkylaryl, or arylalkyl, and R 11, R 12, R 13 does not comprise any moiety, R 14 is alkyl, aryl, alkylaryl, or arylalkyl, and R does not comprise any moiety, and said organic compound is selected from the group consisting of alpha-amino acid or a salt thereof, an organic compound comprising from two to five carboxylic acid (-COOH) groups or a salt thereof, a mono-, di-, trialkanolamine or a salt thereof, an amino ether comprising a further amino, hydroxyl, alkoxy, carboxyl moiety or a salt thereof, an organic compound comprising two to four hydroxyl groups or a salt thereof, a heterocyclic compound comprising a 5-or 6-membered ring having one to three nitrogen atoms as ring members atom or a salt thereof, N,N,N',N'-tetrakis(2-hydroxypropyl)ethylenediamine, 4-(2-hydroxyethyl)morpholine, pentamethyldiethylenetriamine, a salt or adduct of triethanolamine (=2,2',2"-nitrilotris(ethanol)) and 4-[(2-ethylhexyl)amino]-4-oxoisocrotonic acid and 2,2'-dimorpholinodiethylether, and an aqueous medium, wherein the pH value of the CMP composition is in the range of from 2.5 to 5.5.