A group-III-nitride layer structure for a normally-off transistor, the layer structure comprising, - an electron-supply layer (704, 705) made of at least one first group-III-nitride material having a first band-gap energy; - a back-barrier layer made of at least one second group-III-nitride material having a second band-gap energy; - a channel layer, which is arranged between the electron-supply layer (704, 705) and the back-barrier layer and which is made of a third group-III-nitride material having a third band-gap energy that is lower than the first and second band-gap energies, wherein - the second group-III-nitride material of the back-barrier layer is of p-type conductivity, while the first group-III-nitride material of the electron-supply layer (704, 705) and the third group-III-nitride material of the channel layer are not of p-type conductivity, - the first band-gap energy of the first group-III-nitride material of the electron-supply layer (704,705) is smaller than the second band-gap energy of the second group-III-nitride material of the back-barrier layer; and wherein - in absence of an external voltage applied to the layer structure a lower conduction-band-edge of the third group-III-nitride material in the channel layer is higher in energy than a Fermi level of the third group-III-nitride material in the channel layer, characterized in that - the electron-supply layer comprises a first sublayer (310.1) and a second sublayer (310.2) and the first sublayer (310.1) has a larger band gap than the second sublayer (310.2) whereby the band gap of the electron-supply layer decreases with increasing distance from the channel layer, - the group-III-nitride layer structure further comprising a spacer layer between the channel layer and the electron-supply layer, the spacer layer having a thickness of not more than 3 nanometers and being made of AlN or AlGaN having a higher Al mole fraction than the electron-supply layer and - the group-III-nitride layer structure further comprising exactly two recesses extending vertically down to and terminating at an upper edge of the channel layer, each recess being suitable for accommodating either a source contact or a drain contact. - an electron-supply layer (704, 705) made of at least one first group-III-nitride material having a first band-gap energy; - a back-barrier layer made of at least one second group-III-nitride material having a second band-gap energy; - a channel layer, which is arranged between the electron-supply layer (704, 705) and the back-barrier layer and which is made of a third group-III-nitride material having a third band-gap energy that is lower than the first and second band-gap energies, wherein - the second group-III-nitride material of the back-barrier layer is of p-type conductivity, while the first group-III-nitride material of the electron-supply layer (704, 705) and the third group-III-nitride material of the channel layer are not of p-type conductivity, - the first band-gap energy of the first group-III-nitride material of the electron-supply layer (704,705) is smaller than the second band-gap energy of the second group-III-nitride material of the back-barrier layer; and wherein - in absence of an external voltage applied to the layer structure a lower conduction-band-edge of the third group-III-nitride material in the channel layer is higher in energy than a Fermi level of the third group-III-nitride material in the channel layer, - the electron-supply layer comprises a first sublayer (310.1) and a second sublayer (310.2) and the first sublayer (310.1) has a larger band gap than the second sublayer (310.2) whereby the band gap of the electron-supply layer decreases with increasing distance from the channel layer, - the group-III-nitride layer structure further comprising a spacer layer between the channel layer and the electron-supply layer, the spacer layer having a thickness of not more than 3 nanometers and being made of AlN or AlGaN having a higher Al mole fraction than the electron-supply layer and - the group-III-nitride layer structure further comprising exactly two recesses extending vertically down to and terminating at an upper edge of the channel layer, each recess being suitable for accommodating either a source contact or a drain contact.