  We report the observation of second-harmonic generation in stoichiometric
silicon nitride waveguides grown via low-pressure chemical vapour deposition.
Quasi-rectangular waveguides with a large cross section were used, with a
height of 1 {\mu}m and various different widths, from 0.6 to 1.2 {\mu}m, and
with various lengths from 22 to 74 mm. Using a mode-locked laser delivering
6-ps pulses at 1064 nm wavelength with a repetition rate of 20 MHz, 15% of the
incoming power was coupled through the waveguide, making maximum average powers
of up to 15 mW available in the waveguide. Second-harmonic output was observed
with a delay of minutes to several hours after the initial turn-on of pump
radiation, showing a fast growth rate between 10$^{-4}$ to 10$^{-2}$ s$^{-1}$,
with the shortest delay and highest growth rate at the highest input power.
After this first, initial build-up, the second-harmonic became generated
instantly with each new turn-on of the pump laser power. Phase matching was
found to be present independent of the used waveguide width, although the
latter changes the fundamental and second-harmonic phase velocities. We address
the presence of a second-order nonlinearity and phase matching, involving an
initial, power-dependent build-up, to the coherent photogalvanic effect. The
effect, via the third-order nonlinearity and multiphoton absorption leads to a
spatially patterned charge separation, which generates a spatially periodic,
semi-permanent, DC-field-induced second-order susceptibility with a period that
is appropriate for quasi-phase matching. The maximum measured second-harmonic
conversion efficiency amounts to 0.4% in a waveguide with 0.9 x 1 {\mu}m$^2$
cross section and 36 mm length, corresponding to 53 {\mu}W at 532 nm with 13 mW
of IR input coupled into the waveguide. The according $\chi^{(2)}$ amounts to
3.7 pm/V, as retrieved from the measured conversion efficiency.
