Where did the decisive battle take place that ended with the fall of Persia?

The decisive battle of Alexander the Great with the army of the Persian king Darius III took place near the city of Arbela (Erbil) after the Macedonian army crossed the Tigris River. Darius strove to give battle in an open area, where it was possible to give space for his serpentine chariots and numerous cavalry with elephants. However, the result of this battle in 331 BC. e. was the defeat of the army of the Persians and the surrender of Babylon. The key moment of the battle was the assassination of the charioteer Darius in the center of his army. Because of what he left the battle.