Within solid-state physics and semi-conductor theory, the band-gaps of various semiconductors are often shown using $k$-space diagrams. Within the diagram, momentum is on the $x$-axis while on the $y$-axis there is energy. This implies to me that phonons/momentum cannot give any energy to electrons/holes since energy/momentum is orthogonal within the diagram. Is this accurate?
Further, my confusion is compounded when an increase of temperature (phonon activity) results in a decrease in the band-gap usually. This would imply that phonons can and are giving energy to electron hole pairs by raising the entire valence band, lowering the entire conduction band, or by simply pushing more electrons into the conduction band. If this is the case, why aren't phonons ever shown as giving a vertical bump from the valence band to the conduction band and instead are relegated to only horizontal movements for indirect band-gap semiconductors?
