Transistor as a switch for 3A collector current
From your schematic, the current flowing from PB0 to T1's base-emitter junction is \$I_B = (5V - 0.6V) / 10\Omega = 440mA\$, which is extremely high for a MCU-pin's drive capability. Thus PB0 may limit the output (I'm not sure, but I hope).

To saturate the transistor, a base current of \$I_{Bs} = \frac{I_C}{ (\beta_{min}/10)}\$ 
is sufficient (I cannot prove this, but this comes from my experiences). From your circuit, this base current is 3000mA / (400/10) = 75mA, which is still extremely high for MCU's port/pin.

The most efficient way is to use an N-Channel MOSFET (with logic-level gate) instead.