How do I fit a Half- T Distribution to data?

I have data which can be seen in the following image which is a histogram of my data

I want to fit a probability density function to this data. Since All my values are positive I first tried fitting a half normal distribution. However this does not give a good fit as there exists many values above a value of 7-8 and which increases my standard deviation too much to give a proper fit.

I thought about fitting a t-distribution to the data as it is more appropriate for data with heavier tails. However, while matlab has an option of fitting a 'halfnormal' distribution using fitdist, it does not have the option to create a half-T-distribution fit.

My question is this: How do I fit a half-t-distribution fit to this data?