Topic 11 - Phonons and vibrations

Topic 11 - Phonons and vibrations - Phonons in crystals...

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Unformatted text preview: Phonons in crystals Phonons MS&E 803: Structure and Properties of Advanced Electronic Materials Fall 2009, Prof. Mike Arnold Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Acoustic vs. Optical phonons at low‐k Acoustic http://physics‐animations.com/Physics/English/phon_tmp.htm Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 1D chain with 6 degrees of freedom 1D Hǁ=1 H┴=0.8 Gǁ=0.5 H ┴ =0.3 d=1/3 a=1 m=1 ω 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -4 optical acoustic -3 -2 -1 0 1 2 3 4 k Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Nearest Nearest neighbors in graphene Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Force Force constants Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Graphene Graphene (1st nearest neigh.) Phonons in graphene 1500 omega (1/cm) 1000 500 0 0 Г wavevector (1/A) (1/A) M K 1 2 3 4 Г Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Graphene Graphene (2nd nearest neigh.) Phonons in graphene 1500 omega (1/cm) 1000 500 0 0 Г wavevector (1/A) (1/A) M K 1 2 3 4 Г Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Graphene (3rd nearest neigh.) Graphene Phonons in graphene 1500 omega (1/cm) 1000 500 0 0 Г wavevector (1/A) (1/A) M K 1 2 3 4 Г Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Graphene (4th nearest neigh.) Graphene Phonons in graphene 1500 omega (1/cm) 1000 500 0 0 Г wavevector (1/A) (1/A) M K 1 2 3 4 Г Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Physical and electronic structure Physical (4, 2) nanotube Primitive lattice Quantum-confined reciprocal vectors Saito and Dresselhaus2 Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Collapse Collapse segments into 1st BZ 2 y-wavevector (1/A) y-wavevector (1/A) 0 5 (n=4, m=2) 1 0 0 K -1 -5 -5 x-wavevector (1/A) 5 10 15 -2 -2 x-wavevector (1/A) (1/A) -1 0 1 2 Translate each point in each segment by linear combination li of integer multiples of b1 and b2 Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Evaluate Evaluate E(k) 2 y-wavevector (1/A) 1 E K 0 -1 k -2 -2 x-wavevector (1/A) -1 0 1 2 Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Band structure of (4, 2) CNT Band 1st BZ of (4, 2) CNT (4, 2) nanotube 3 2 28 Pi* Pi subbands Eg 28 Pi subbands Energy / t K 1 0 -1 -2 -3 -0.2 0 0.2 E k x-wavevector (1/A) (1/A) Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Same B.C. for phonons Same (4, 2) nanotube Primitive lattice Allowed reciprocal vectors Saito and Dresselhaus2 Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Phonon dispersion relationship Phonon (4, 2) nanotube 1st BZ of (4, 2) CNT (n=4, m=2) 1600 1400 1200 omega (1/cm) 1000 800 600 400 200 0 0 0.05 0.1 wavevector (1/A) 0.15 0.2 0.25 Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Phonon dispersion relationship Phonon (10, 10) nanotube 1st BZ of (10, 10) CNT (n=10, m=10) 1600 1400 1200 omega (1/cm) 1000 800 600 400 200 0 0 0.2 0.4 wavevector (1/A) 0.6 0.8 1 1.2 Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Measuring phonon dispersions with neutrons Measuring Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 A&M Example: Phonons in Al Example: Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 A&M Raman Raman in graphene Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 Vibrational spectra Vibrational Phonons in graphene 1500 omega (1/cm) 1000 500 0 0 wavevector (1/A) (1/A) 1 2 3 4 Г M K Г Prof. Michael Arnold, MS&E 803 – Structure and Properties of Advanced Electronic Materials, Fall 2009 ...
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This note was uploaded on 03/23/2010 for the course MATERIAL S 803 taught by Professor Samuel during the Spring '10 term at Wisconsin.

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