Free-standing III-N substrate, wherein III means at least one element of group III of the periodic table selected from Al, Ga and In, having a diameter greater than 40 mm and having the following properties: (i) when mapping on the surface by white light interferometry, the standard deviation of the rms values is 5% or less, and (ii) for a Rocking curve mapping on the surface parallel to the growth plane, the standard deviation of the measured half-widths is 5% or less; and/or (iii) for a micro-Raman mapping on the surface parallel to the growth plane, the standard deviation of the measured half-widths of the E 2 phonon is 5% or less; wherein said mappings are measurable with the following conditions: ad (i) - Mapping of the wafer surface with a white light interferometer: after dividing the wafer surface into a grid, each with a maximum spacing of 5 mm in two mutually orthogonal directions, a surface scan is carried out, considering an edge exclusion range of 5 mm in each grid, the scan range being at least 1% of the grid size, and the rms values are measured with a white light interferometer using white light; ad (ii) - Rocking curve mapping: X-ray diffraction curves corresponding to diffraction at specific mesh plane arrays are generated, where ω-scans are acquired at different sample locations of the surface in a plane parallel to the growth plane, after which the standard deviation of the mean values of the half-widths of the acquired ω-scans are determined; ad (iii) - micro-Raman measurements to determine the half-width of the E 2: - Laser excitation wavelength 532nm (frequency doubled Nd:YAG laser) - Excitation power 3 mW - Focusing the laser line to a beam diameter of ∼ 1μm on the sample with microscope optics, wherein the spectrometer is calibrated using Ne plasma lines, the measurement is performed in backscatter geometry, when scanning on the surface in the x and y directions the step size is ∼2.5 mm and the edge exclusion region is 2 mm from the wafer edge and when scanning on the slit face of the wafer perpendicular to the surface the step size in the z direction is ∼10μm, and wherein the half width of the E2 phonon is determined by Lorentz line shape analyses. (i) when mapping on the surface by white light interferometry, the standard deviation of the rms values is 5% or less, and (ii) for a Rocking curve mapping on the surface parallel to the growth plane, the standard deviation of the measured half-widths is 5% or less; and/or (iii) for a micro-Raman mapping on the surface parallel to the growth plane, the standard deviation of the measured half-widths of the E 2 phonon is 5% or less; ad (i) - Mapping of the wafer surface with a white light interferometer: after dividing the wafer surface into a grid, each with a maximum spacing of 5 mm in two mutually orthogonal directions, a surface scan is carried out, considering an edge exclusion range of 5 mm in each grid, the scan range being at least 1% of the grid size, and the rms values are measured with a white light interferometer using white light; ad (ii) - Rocking curve mapping: X-ray diffraction curves corresponding to diffraction at specific mesh plane arrays are generated, where ω-scans are acquired at different sample locations of the surface in a plane parallel to the growth plane, after which the standard deviation of the mean values of the half-widths of the acquired ω-scans are determined; ad (iii) - micro-Raman measurements to determine the half-width of the E 2: - Laser excitation wavelength 532nm (frequency doubled Nd:YAG laser) - Excitation power 3 mW - Focusing the laser line to a beam diameter of ∼ 1μm on the sample with microscope optics, - Laser excitation wavelength 532nm (frequency doubled Nd:YAG laser) - Excitation power 3 mW - Focusing the laser line to a beam diameter of ∼ 1μm on the sample with microscope optics,