Lecture%2014

Lecture%2014 - EEE 352: Lecture 14 More on Energy Bands *...

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1 EEE 352: Lecture 14 More on Energy Bands * Crystal diffraction in many directions Brillouin zone * Density of states within a band * Holes Alloys and heterostructures X The random atom in the lattice X Alloy band structures X Heterojunctions X Superlattices X When is a superlattice an alloy? The tetrahedral bond in Cu 2 O, as measured at ASU in 1999. This material is BCC—the atoms are indicated by the small red circles. The bond charge residing between the atoms is colored and shows that this material is covalent, and not ionic, as previously thought. Crystal Diffraction in Many Directions Previously we saw how electron diffraction gives rise to FORBIDDEN GAPS in the energy spectrum of electrons in a crystal * Diffraction by a PARTICULAR set of crystal planes of spacing d is found if the wavenumber PERPENDICULAR to the plane equals n π / d * As a result of this diffraction, an ENERGY GAP develops FORMATION OF GAPS IN THE ENERGY SPECTRUM OF ELECTRONS IN A CRYSTAL d d 2 d 3 d 4 d d 2 d 3 d 4 First Brillouin Zone Second Brillouin Zone Crystal Diffraction in Many Directions ± In real crystals electron diffraction can occur in MANY different directions ± This leads us to define the FIRST BRILLOUIN ZONE as the k -space region which electrons may occupy WITHOUT being diffracted ± The concept of Brillouin zones is easily demonstrated in TWO dimensions by focusing on the SQUARE lattice THE SHADED REGION IS THE FIRST BRILLOUIN ZONE AND RANGES FROM - π / d < k x < π / d AND - π / d < k y < π / d FOR A SQUARE LATTICE OF SPACING d WHEN THE ELECTRON WAVEVECTOR LIES WITHIN THE FIRST BRILLOUIN ZONE THE ELECTRON WILL NOT BE DIFFRACTED BY THE CRYSTAL IF THE WAVEVECTOR TOUCHES THE PERIMETER OF THE FIRST BRILLOUIN ZONE HOWEVER THEN THE ELECTRON WILL BE DIFFRACTED BY THE CRYSTAL k x k y 2 nd ZONE π / a π / a −π / a −π / a
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2 Crystal Diffraction in Many Directions EXAMPLE * For a two-dimensional rectangular lattice with translation vectors a and 2 a in the x and y directions draw the first Brillouin zone. Also estimate the highest energy electrons may have without being diffracted in the case where a = 5 Å eV 85 . 1 10 1 . 9 2 10 9 . 4 10 1 . 1 2 m 10 7 . 0 4 5 4 5 2 , 31 19 68 2 2 1 10 2 2 2 2 = × × × × × = × = = = + = = = m k E a a k k k a k a k has state energy highest The y x y x h π k x k y k a k a a k a y x 2 2 , < < < < Electron Motion in a Periodic Potential φ 2 N n = If we write φ = kd , then L n Nd n k 2 2 = = n =1, 2, …, N There are N independent values of k , spaced by k = 2 π / L.
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Lecture%2014 - EEE 352: Lecture 14 More on Energy Bands *...

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