homework7 - Department of Electrical and Computer...

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1 Department of Electrical and Computer Engineering, Cornell University ECE 407: Physics of Semiconductor and Nanostructures Spring 2009 Homework 7 ` Due on March 24, 2009 at 5:00 PM Suggested Readings: a) Lecture notes b) Start homework early. Problem 7.1 (Constant energy surfaces) The concept of constant energy surfaces (or contours in 2D) arises in many problems. These surfaces are in the reciprocal space and are such that the energy of every point on the surface is the same. For example, suppose if the energy band dispersion for conduction band is given by: Then all points in k-space that are equidistant from the origin ( Γ -point) have the same energy. Therefore the constant energy surfaces in 3D are spherical shells, and in 2D are circles, with the origin as their center. The equation in reciprocal space for a constant energy surface in with energy o E can be found by putting the energy dispersion relation equal to o E .For example, ( ) o e c E m k E k E = + = 2 2 2 h r The equation obtained, ( ) c o e z y x E E m k k k = + + 2 2 2 2 2 h is clearly the equation of a sphere in k-space of radius: ( ) h c o e E E m 2 . Things become more complicated with anisotropic energy band dispersion relations. For example, in silicon the energy band dispersion for the conduction band minima near the ( ) 0 , 0 , 2 X a π -point is: ( ) ( ) ( ) ( ) t oz z t oy y ox x c c m k k m k k m k k E k E 2 2 2 2 2 2 2 2 2 + + + = h h h r l To find the equation of a constant energy surface with energy o E we put the energy dispersion relation equal to o E , ( ) ( ) ( ) ( ) o t oz z t oy y ox x c c E m k k m k k m k k E k E = + + + = 2 2 2 2 2 2 2 2 2 h h h r l The equation obtained: ( ) ( ) ( ) ( ) c o t oz z t oy y ox x E E m k k m k k m k k = + + 2 2 2 2 2 h l ( ) e c c m k E k E 2 2 2 h r + =
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2 is that of an ellipsoid. Finally, note that the Fermi surface is also a constant energy surface with energy equal to F E . a) Show that for an electron with wavevector k r the velocity in real space given by ( ) k v c r r is always perpendicular to the constant energy surface that passes through k r .
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