Why the crystal momentum k in 2D band structure is defined on a torus?

[lyylynn]

Hi,

Could someone help to explain that why for the 2 D system, like 2 dimensional electronic gas, the crystal momentum k in 2D band structure is defined on a torus?

Thanks a lot.:)

 

[read]

Probably to satisfy Born–von Karman boundary condition...

 

[kanato]

Because it's periodic in x and y with period 2*pi/a.