We present semiconductor saturable absorber mirrors (SESAMs) that can potentially support femtosecond pulses from ultrafast thin disk lasers (TDLs) with high average power approaching the kW-power level and high pulse energy in the range of 100 µJ to 1 mJ at megahertz pulse repetition rates. For high-power operation, the SESAM parameters will ultimately limit the shortest pulse duration from a soliton mode-locked laser before mode locking instabilities such as multiple pulsing instabilities and continuous wave (cw) breakthrough start to occur. Currently shorter pulses are prevented due to the inverse saturable absorption that becomes stronger with shorter pulses and results in a shift of the "rollover" of the nonlinear SESAM reflectivity towards lower fluences. Here we discuss a novel SESAM design that addresses these issues and can be grown by metal-organic vapor phase epitaxy (MOVPE), an attractive epitaxial growth technology for manufacturing.