  The LOFAR Two-metre Sky Survey (LoTSS) will cover the full northern sky and,
additionally, aims to observe the LoTSS deep fields to a noise level of ~10
microJy/bm over several tens of square degrees in areas that have the most
extensive ancillary data. This paper presents the ELAIS-N1 deep field, the
deepest of the LoTSS deep fields to date. With an effective observing time of
163.7 hours, it reaches a root mean square (RMS) noise level below 20
microJy/bm in the central region (and below 30 microJy/bm over 10 square
degrees). The resolution is 6 arcsecs and 84862 radio sources were detected in
the full area (68 sq. deg.) with 74127 sources in the highest quality area at
less than 3 degrees from the pointing centre. The observation reaches a sky
density of more than 5000 sources per sq. deg. in the central ~5 sq. deg.
region. We present the calibration procedure, which addresses the special
configuration of some observations and the extended bandwidth covered (115 to
177 MHz; central frequency 146.2 MHz) compared to standard LoTSS. We also
describe the methods used to calibrate the flux density scale using
cross-matching with sources detected by other radio surveys in the literature.
We find the flux density uncertainty related to the flux density scale to be
~6.5%. By studying the variations of the flux density measurements between
different epochs, we show that relative flux density calibration is reliable
out to about a 3 degree radius, but that additional flux density uncertainty is
present for all sources at about the 3 per cent level; this is likely to be
associated with residual calibration errors, and is shown to be more
significant in datasets with poorer ionosphere conditions. We also provide
intra-band spectral indices, which can be useful to detect sources with unusual
spectral properties. The final uncertainty in the flux densities is estimated
to be ~10% for ELAIS-N1.
