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4 Pages

### Phys560Notes-8

Course: PHYS 560, Spring 2012
School: University of Illinois,...
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Word Count: 606

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of Density States: g Definition: g d = number of states between and d (per unit volume) For a single band k , g d 1 V 1 8 d 3 k ,s d d 2 3 k 1 4 3 dS dk d dk dS = area element on constant energy surface k dk = change in k, along k k (perpendicular to dS), corresponding to d dS d k k dk k dk 1 k k dk k k dk v g dk 1 4 3 v g = group velocity...

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Coursehero >> Illinois >> University of Illinois, Urbana Champaign >> PHYS 560

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of Density States: g Definition: g d = number of states between and d (per unit volume) For a single band k , g d 1 V 1 8 d 3 k ,s d d 2 3 k 1 4 3 dS dk d dk dS = area element on constant energy surface k dk = change in k, along k k (perpendicular to dS), corresponding to d dS d k k dk k dk 1 k k dk k k dk v g dk 1 4 3 v g = group velocity With dk k k d , g 1 4 3 k k k k gn n dS For multiple bands: g n n k n k dS Another formula for single band: g 1 V k 4 1 3 k ,s BZ d 3 k k Pf: 2 1 1 1 k d V V ks 1 k 2 k ,s 1 QED van Hove singularities in g arise when k n k 0 (denominator = 0); these occur at critical points k c . Consider a single band in repeated zone near k c . 2 1 k smooth k c k k c k c k k c kc 2 k c quadratic terms in k k c Make a principal axis transformation (diagonalize a 3 3 symmetric matrix) 2 2 k k c qi q k kc qi referred to principal axes. i 2mi mi = effective mass Class (# of negative mi's) 0 1 2 3 Name m1,2,3 0 m1,2 0, m3 0, etc. m1 0, m2,3 0, etc. m1,23 0 M0 (minimum; parabolic) M1 (saddle point) M2 (saddle point) M3 (maximum; parabolic) van Hove's theorem: a periodic function of variables (e.g. k in repeated-zone) ! must have at least Cn critical points of type Mn in each -dimensional n ! n ! primitive cell. 2D 3 n0 c0 1 Mo 3 Max 1 c1 3 M1 Example: in 3D; 3 Min 3 2 c2 3 M2 3 3 c3 1 M3 In 3D, there are at least one minimum, one maximum, and three saddle points of each type. This follows from topological considerations. Consider the region near a M0 of a single band. 1 1 2 2 g d 3 k k d 3k c 4 qi 3 BZ 4 3 i 2mi qi q k k c , define pi 2mi g 1 4 3 3 o 8m1m2 m3 3 d 3 p c p2 8m 1 m2 m3 4 3 3 o 4 p2 c p2 dp 8m 1 m2 m3 4 3 2 p2 c p 2 dp2 2m1m2 m 3 g 2 3 0, c , c c g So, it is a weak singularity. However, g' = at c . c Many bands: the critical points for each band are generally at different energies. So, each singularity may sit atop a smooth "background" of density of states from other bands. Mo , g c1 c2 c c3 c1 c2 c c , c 0 , c , c M1 , g c1 c2 c c1 c2 c c3 c1 c2 c c1 c2 c c3 c , c , c M2 , g c , c M3 , g c1 c2 c c3 c1 c2 c c , c , c Example of 3d g (schematic): g Experimentally, these critical point features can be seen by high-resolution XPS (x-ray photoelectron spectroscopy). XPS: an incident monochromatic x-ray beam ejects electrons from the sample; the energy distribution of the photoelectrons shows the valence density of states (modulated by cross section variation). XPS spectrum of GaP [L. Ley, R.A. Pollack, F.R. McFeely, G. P. Kowalczyk, and D.A. Shirley, Phys. Rev. B 9, 600 (1974)] Top: XPS spectrum with background removed 2nd: DOS broadened by lifetime & instrumental resolution 3rd: DOS (note the singularities) 4th: band structure of GaP Density of states for a free-electron-like material 0D: a set of delta functions at n (atomic levels, or particle in a box) 2 n 2 k z 1D: a set of subbands n ; at each band edge g 2m n 2D: a set of subbands n 2 2 2 k x k y 3D: a free-electron-like band Wikipedia: 2m 2 2 2 2 k x k y k z ; at each band edge g n 2m ; at band bottom g 0D 2D 1D 3D
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