SQL Between clause with strings columns

I want to make a search using "between" clause over a string column. Doing some test I got this:

Let's assume that there is a country table with a "name" column of type varchar. If I execute this query:

Select * from country where name between 'a' and 'b'

I got this result:

Argentina
.
.
.
Argelia.

It excludes those countries that starts with B which I found a little bit weird.

Is there a way to do this search in a more accurate way? Any other ideas for make this search?

Thanks in advance