The IceCube Neutrino Observatory has observed a diffuse flux of TeV-PeV astrophysical neutrinos at 5.7{sigma} significance from an all-flavor search. The direct detection of tau neutrinos in this flux has yet to occur. Tau neutrinos become distinguishable from other flavors in IceCube at energies above a few hundred TeV, when the cascade from the tau neutrino charged current interaction becomes resolvable from the cascade from the tau lepton decay. This paper presents results from a dedicated search for tau neutrinos with energies between 214 TeV and 72 PeV. The analysis searches for IceCube optical sensors that observe two separate pulses in a single event - one from the tau neutrino interaction, and a second from the tau decay. This is the first IceCube tau neutrino search to be more sensitive to tau neutrinos than to any other neutrino flavor. No candidate events were observed in three years of IceCube data. For the first time, a differential upper limit on astrophysical tau neutrinos is derived around the PeV energy region, which is nearly three orders of magnitude lower in energy than previous limits from dedicated tau neutrino searches.
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