We investigated AlAs0.56Sb0.44 epitaxial layers lattice-matched to InP grown by molecular beam epitaxy (MBE). Silicon (Si) and tellurium (Te) were studied as n-type dopants in AlAs0.56Sb0.44 material. Similar to most Sb-based materials, AlAs0.56Sb0.44 demonstrates a maximum active carrier concentration around low-1018 cm-3 when using Te as a dopant. We propose the use of a heavily Si-doped InAlAs layer embedded in the AlAsSb barrier as a modulation-doped layer. The In0.53Ga0.47As/AlAs0.56Sb0.44 double heterostructures with a 10 nm InGaAs well show an electron mobility of about 9400 cm2/V・s at 295 K and 32000 cm2/V・s at 46 K. A thinner 5 nm InGaAs well has an electron mobility of about 4300 cm2/V・s at 295 K. This study demonstrates that AlAs0.56Sb0.44 is a promising barrier material for highly scaled InGaAs MOSFETs and HEMTs.