# Why are phonons and photons orthogonal in $k$-space?

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?

• No, your first is not accurate. Review what a band structure means compared with free particles (where you can also plot energy vs momentum). Second, changes in temperature do more than just increase the number of phonons. – Jon Custer Aug 14 '17 at 13:11
• @JonCuster Can you clarify? My first...what? I don't think I'm dealing with any free particles except for freed electrons. Are you talking about increased temperature spreading out the crystal structure or something else? – horta Aug 14 '17 at 13:21
• The fact that you can plot energy vs momentum of phonons does not imply that they are 'orthogonal' to photons. You can also plot energy vs momentum of, well, anything including photons. – Jon Custer Aug 14 '17 at 13:36
• @JonCuster That's actually a great point that is probably causing all of my hang-ups. This person's plot in the link below helps clear up that energy and momentum don't have to be separate at all. The graph shows both photons and phonons having both energy and momentum in different amounts: physics.stackexchange.com/questions/1725/… – horta Aug 14 '17 at 18:26