A piece sacrifice on e6 (e3) is a typical middle game attacking theme to destroy the pawn structure around the enemy monarch and hopefully get a decisive attack. Sometimes it has positional characteristic too in order to secure outpost on e5 (usually a knight). Here are some instructive examples:
Yuri Balashov against Rifat Sabjanov in 1994 – White to Move
Q: Is it worth considering e5-e6 here?
A: Yes, White can get a strong attack as follows:
1.e6 Bxe6 2.Rxe6!!
This creates strong hold on e5 for White’s knight which completely dominates the position.
2…fxe6 3.Ne5 Qb6
White would have a winning position against other moves too, for example 3…g6 4.Qf3 or 3…Qd6/b8 4.Bf4. These may have prolonged the fight but woudn’t change the outcome.
Actually Qf3 was even better.
And White went on win after few more moves.
Sacrifice to destroy pawn cover – Kramnik against Nigel Short in 1995 – White to Move
Q: Which piece wwould you sacrifice on e6?
A: The bishop of course because if 1.Nxe6 then 1…Qxh4 wins
1.Bxe6!! fxe6 2.Qxg6 Nxe5 3. Qh7 Kf8 4. Nf4
The position is totally lost so Black resigned.
Sacrifice to use lead in development –
Helgi Olafsson against Jonathan Levitt in 1990 – White to Move
Q: How would you proceed with the White pieces?
A: I would sacrifice on e6 as follows:
There is no way to decline the sacrifice. If the bishop moves then there is mate on e6 and if knight moves then the bishop is lost. Meanwhile f5 can be met by Ng5.
The bishop on g2 can’t be taken because of the spectacular Qxe6+! leading to either a back rank mate or a smothered mate.
Threatening 4.Nf6+. Note that 4.Bxe4 would be a mistake because of Nc6 when you still need a move to save the knight so you can’t win pawn on c5.
3…Nc6 4.Nxc5 Qc7 5.Nxd7! Rac8
5…Rxd7 is not possible because of 6.Qxe6+ Rf7 7.Bxc6 etc..
6.Qxe6+ Kh8 7.Be4 Ne7 8.Rd6
Stronger than Qxe7. Black resigned after 3 more moves.