How can I motivate an employee to do more than just the bare required minimum of work?

Make them an owner!

Build your teams culture. Once the culture is to the point where the majority of the members are motivated to improve, those who have checked out will identify themselves. Try to mentor those few on getting to where they want to be career-wise, terminate those unable or unwilling to change.

Make him realise about the extra benefit for his smart and hard work.

Are there any rewards to work more? Any career opportunities? Promotions on the horizon? A bit of hope can go a long way. If you don't have anything for them don't expect more. You can also raise the minimum requirements for work until you are satisfied with them or tell them to leave.

Why is the employee not motivated by himself/herself in the first place?

You could make it a challenge amongst employees, or a weekly or monthly goal for employees to reach regularly. Bribery is a great concept I believe because it can promote great habits of routine responsibility's in employees.

People are motivated by incentives, whether it's recognition, compensation, opportunity for promotion. If there are no incentives in place, it wound actually be silly for someone to work harder than the bare minimum. That employee is better off doing the bare minimum and meanwhile use his or her extra time doing side gigs whether it's for monetary or personal reasons such as hobbies or feeling of self worth

If that employee can see the value in the skills you’re willing to offer and how they will benefit them profitably, a dedicated worker will perform more

Train, mentor, reward, and don't micro manage. Also offer alternative work schedules other than the basic 8 to 5. Maybe offer an extra day off a week or 9/80 schedule.

In my experience praising employees for accomplishing more than the minimum has yielded good results with those employees but also encouraging those who do less and giving them the training and tools they need and making them feel like a valued member of the team will increase performance

Engage in finding out what they need to take care of the personal life better

Working with the employee, set 3-5 SMART goals with one of them a stretch goal. Tie incentives ($, PTO, etc.) to said goals. If they don't measure up, loop them into HR policies that address poor performance which may lead to termination.

Incentives get everyone working to a common goal.

Summer Fridays, ask for opinions rather than give directives, find out how they prefer to work (a lot of supervision or a lot of free range), commend them on a good job

Learn about your employee's motivations of where she wants to be in her career or personal growth; align her goals and tasks with that future self, which will make her motivated because that's where she wants to be.

Set the bare minimum higher. They work within the confines of your rules. If you expect more, you get more. If you don't you know you have to make a change.