Thomas Haining shook Mikayla in the early hours of the morning resulting in the tot's head striking a door at the family home.
Haining, then aged 19, later woke up the child's mother Shannon Davies shouting that the baby had stopped breathing and urging her to call an ambulance while he made resuscitation efforts.
But a subsequent examination of his mobile phone revealed he carried out a series of internet searches beginning almost an hour before he alerted her to the baby's plight .
The first Google search used the term: "baby took a panic attack and now she's unresponsive".
It was followed minutes later by "what happens when a newborn is shaked hard", with a final search for "newborn in a coma".
Despite this Haining told paramedics that his daughter had been unwell for two or three days and was crying as if in pain when fed and that her breathing went "funny" then stopped.
Mikayla was found to have suffered a skull fracture and seven broken ribs. The cause of death was given as head trauma.
A judge told the killer: "Causing the death of a child by a violent assault is an extremely serious offence which must attract a lengthy period of imprisonment."
Lord Pentland told Haining: "No sentence I can impose can restore the damage you have caused."
The judge said that while the attack was "the work of a few moments" and the Crown accepted there was a sudden loss of control by Haining it was unquestionably a violent assault on a vulnerable baby.
He pointed out that Haining had the presence of mind to carry out three internet searches before he woke the child's mother.
The judge said: "Your immediate reaction was to protect yourself rather than to seek help for Mikayla."
He told Haining that he would have jailed him for nine years for the crime but for his guilty plea.
Lord Pentland said: "Mikayla was 23 days old. She was in your sole care at the time."
Advocate depute Michael Meehan earlier told the High Court in Edinburgh: "The rib injuries are consistent with significant pressure being applied to the baby's torso during shaking."
Haining, now 21, was originally charged with murdering his daughter on June 8 in 2017 by repeatedly inflicting blunt force trauma to the baby's head and body by means unknown to the prosecutor.
But the Crown earlier accepted his guilty plea to a reduced charge of culpable homicide. He admitted killing his daughter by shaking her causing her head to hit a door at the family home in Mackay Road, in Inverness.
Haining, formerly of a bail address at High Street, Grantown on Spey, was remanded in custody following his plea last month to await sentence today.
The court heard that Haining, a former shop assistant, began a relationship with Ms Davies who moved in with him to the house in Inverness.
Mr Meehan said: "At this time Shannon was pregnant with Mikayla. The accused is the father of Mikayla and it was a planned pregnancy."
Mikayla was born on May 17 in 2017 at Raigmore Hospital, in Inverness, and was a healthy baby. Her death was confirmed at the same hospital just weeks later following Haining's fatal attack after her ventilator was switched off and she was placed in her mother's arms.
Medical experts who reviewed CT images of the child concluded that she suffered brain injuries which were "catastrophic and un-survivable", the court heard.
The scans showed widespread bleeding to the brain and damage. Mr Meehan said: "The consultant radiologist who viewed the scan expressed the opinion that the severe brain injury was caused by trauma."
The court heard that in the two days before she died Mikayla was unsettled and crying more than usual.
Ms Davies fed her daughter and put her in a Moses basket in the living room before going to bed. Haining said he would stay downstairs with the child and try to settle her.
The mother went downstairs after 2 am and saw her daughter appeared to be sleeping. She told Haining to feed her and then bring her upstairs while she went back to bed.
Haining later woke her seeking a thermometer and claiming the child seemed cold. Her temperature was checked and the mother told him to keep an eye on her as she appeared to be sleeping.
But about 5.20 am Haining woke her again shouting that Mikayla had stopped breathing.
The first offender later told a doctor at the hospital that he fed his daughter some milk, but afterwards she was screaming before she suddenly went quiet and floppy.
He said he later noticed that she was making abnormal slow movements with her arms and that she had stopped breathing.
Defence counsel Shelagh McCall QC said Haining has shown remorse and now considered that he should have "manned up" at the time and told the truth.
She said it was reported that he felt disgusted with himself following the fatal attack.
"This is a violent incident, but it occurs in a moment and it occurs in a context of an inexperienced parent, exhausted and feeling himself under pressure, without the capacity or facility to cope with the situation," she said.
"He recognises a significant custodial sentence is appropriate and warranted because of the severity of his actions," she told the court.