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Nearly lattice-matched InAs/AlSb/GaSb-based heterostructure backward diodes for zero-bias millimeter wave detection were fabricated and measured. A record-high curvature, γ=39.1 V-1, at zero bias was measured. On-wafer sensitivity measurements from 1 to 110 GHz gave a record-high average sensitivity of 3687 V/W for zero-bias operation. Further enhancement of detector sensitivity was observed with applied dc bias, with a sensitivity of 7996 V/W obtained for a 0.9 μA bias. Extrapolating the conjugately-matched measured sensitivity suggests that 1000 V/W should be achievable at a record-high 541 GHz. The temperature dependence of detector sensitivity was evaluated from measured dc current-voltage characteristics and gave expected sensitivities ranging from 3910 V/W at 293 K to 7740 V/W at 4.2 K.