I want to switch to my external display via shortcut and not by moving my cursor to the other screen.
The best thing to do for this case is to create a Keyboard Shortcut to switch desktops. To do this, you can do the following:
System Preferences -> Keyboard -> Shortcuts
From this menu, select Mission Control on the left hand side, and at the bottom of this list, you should see an option for "Switch to Desktop #." This is the screen where you can create a keyboard shortcut in order to change focus between your desktops.