We present our new ALMA observations targeting [OIII]88um, [CII]158um, [NII]122um, and dust continuum emission for three Lyman break galaxies at z=6.0293-6.2037 identified in the Subaru/Hyper Suprime-Cam survey. We clearly detect [OIII] and [CII] lines from all of the galaxies at 4.3-11.8sigma levels, and identify multi-band dust continuum emission in two of the three galaxies, allowing us to estimate infrared luminosities and dust temperatures simultaneously. In conjunction with previous ALMA observations for six galaxies at z>6, we confirm that all the nine z=6-9 galaxies have high [OIII]/[CII] ratios of L[OIII]/L[CII]~3-20, ~10 times higher than z~0 galaxies. We also find a positive correlation between the [OIII]/[CII] ratio and the Lya equivalent width (EW) at the ~90% confidence level. We carefully investigate physical origins of the high [OIII]/[CII] ratios at z=6-9 using Cloudy, and find that high density of the interstellar medium, low C/O abundance ratio, and the cosmic microwave background attenuation are responsible to only a part of the z=6-9 galaxies. Instead, the observed high [OIII]/[CII] ratios are explained by 10-100 times higher ionization parameters or low photodissociation region (PDR) covering fractions of 0-10%, both of which are consistent with our [NII] observations. The latter scenario can be reproduced with a density bounded nebula with PDR deficit, which would enhance the Lya, Lyman continuum, and C+ ionizing photons escape from galaxies, consistent with the [OIII]/[CII]-Lya EW correlation we find.
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