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Materials Science and Engineering - Electronic and Mechanical Properties of Materials Part 4

Tham khảo tài liệu 'materials science and engineering - electronic and mechanical properties of materials part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Diffraction Picture of the Origin of Band Gaps Probability Density probability volume of finding electron M2 ka 12 4 sin2 nx a k s I2 4 cos2 n x a a a Only two solutions for a diffracted wave Electron density on atoms Electron density off atoms No other solutions possible at this wavelength no free traveling wave 3.225 E. Fitzgerald-1999 9 Nearly-Free Electron Model Assume electrons with wave vectors k s far from diffraction condition are still free and look like traveling waves and see ion potential U as a weak background potential Electrons near diffraction condition have only two possible solutions electron densities between ions E Efree-U electron densities on ions E Efree U Exact solution using HT ET shows that E near diffraction conditions is also parabolic in k E k2 ---------------------------------------------------------------------------------------- 3.225 E. Fitzgerald-1999 10 5 Electron Wave Functions in Periodic Lattice Often called Bloch Electrons or Bloch Wavefunctions n a k Away from Bragg condition free electron H -h-V2 Uo - -h V2 V e 2m 2m J h 2k2 E 2m Near Bragg condition standing wave electron ft V2 Uo Uo x V cosGx or sinGx u x E Uo x 2m __ Since both are solutions to the S.E. general wave is Ỹ V freeV lattice x termed Bloch functions E. Fitzgerald-1999 12 6 Block Theorem If the potential on the lattice is U r and therefore U r R U r then the wave solutions to the S.E. are a plane wave with a periodic part u r that has the periodicity of the lattice T r eikru r u r u r R Note the probability density spatial info is in u r T T tJ2 r u r An equivalent way of writing the Bloch theorem in terms of T r R eik r R Ru r R eik r R T r R eikkR T r - . . z 3.225 E. Fitzgerald-1999 . 13