Information-Centric Approach for Smart Grid Communications
The C-DAX Consortium, led by Chris Develder from Ghent University iMinds, focuses on an information-centric approach for enhancing smart grid communications. Through funding from the European Union's Seventh Framework Programme, the project aims to revolutionize the delivery and management of energy data services in a smart grid context. By adopting Information-Centric Networking (ICN), the network transforms its behavior to prioritize content delivery over traditional host-based routing. This shift enables efficient data dissemination, decoupling communicating entities and reducing configuration complexity.
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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 7: Reading: Chapter 7 in MPM; Electronic Structure 1. Bloch s Theorem 2. Eigenstates of a simple model potential 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 1
1/30/2015 PHY 752 Spring 2015 -- Lecture 7 2
Consider an electron moving in a one-dimensional model potential (Kronig and Penney, Proc. Roy. Soc. (London) 130, 499 (1931) a V0 Schroedinger equation for electron: + 2 2 = ( ) ( ) x ( ) x V x E 2 m 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 3
a V0 = + Effects of periodicity: ( ) + + + ( ) V x V x na 2 2 = ( ) ( ) x ( ) x V x E 2 m 2 2 + + = + ( ) ( ) ( ) V x na x na E x na 2 m 2 2 + = + ( ) ( ) ( ) V x x na E x na 2 m 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 4
a V0 + Since ( eigenvalue problem: ( Assume K = ) and ( ) are solutions of the same ) x na + = = x na x ( ) x K i ikna e e + = ikna Bloch theorem: ( ) ( ) x x na = e k k ikx where ( ) x na + ( ) ( ) e u x u x = kk ( ) u x k k 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 5
a V0 Eigenstates: 2 2 + = ( ) ( ) x ( ) ( ) V x E k x k k 2 m Band gap E(k) 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 6
What causes band gaps in the electronic structure? Consider a single potential well Spectrum E 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 7
( ) 2 m V E 2 mE a = = 0 Define: 2 2 V0 b = = + + x x For 0 ( ) ( ) x x a b Ae Be 1 i x i x For ( ) ( ) x a b x a Ce De 2 (0) (0) d d = = Continuity conditions: (0) (0) 1 2 x 1 2 dx d ( dx ) ( dx ) d a b d a b = = ( ) ( ) 1 2 a b a b 1 2 + = ika Also note: ( ) ( ) x x a e 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 8
a V0 b Matching conditions redu cos( ) ( )cos ka F E = ce to: ( ) ( ) E b 1/2 2 V ( ) + 2 ( ) E 1 sinh ( 0 F a b ( ) 4 E E V 0 2 V E V E ( ) = tan ( ) E tanh ( ) 0 a b ( ) 2 E 0 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 9
a V0 b ( ) b 2 2 49 2 16 17 = = Details for ; V b a 0 2 2 16 m E V Let f 0 ( ) 1/2 2 sinh 0.687 1 f ( ) = + cos( ) 1 cos 11.7 ( ) f k a f ( ) 4 1 f f ( ) 1 2 1 f = t an ( ) tanh 0.687 1 f f ( ) 2 f f 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 10
( ) 1/2 2 sinh 0.687 1 f ( ) = + cos( ) 1 co s 11.7 ( ) f k a f ( ) 4 1 f f ( ) 1 2 1 f = t an ( ) f tanh 0.687 1 f ( ) 2 f f Forbidden states v f Forbidden states v ( ) 1/2 2 sinh 0.687 1 f ( ) = + ( ) 1 cos 11.7 ( ) f X f f ( ) 4 1 f f 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 11
f Band gap Band gap ka 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 12
Electrons in the presence of a weak periodic potential V0 x x = + 0 1 H H H 2 2 2 d = = 0 1 H H cos V 0 2 2 mdx a 2 + i k n x = a ( ) x e C k n a a n 2 2 + 2 k n a = 0 nk E 2 m 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 13
2 2 2 d x = = 0 1 H H cos V 0 2 2 md x a 2 + i k n x = a ( ) x e C k n a a n 2 2 + 2 k n a = 0 nk E 2 m 2 2 = = 0 0 Note that E E 2 2 m a 0 1 a a Degenerate perturbation theory = 2 2 V C C C C 0 2 2 ma 0 0 = 1 E 2 2 1 1 V 0 2 2 ma 1 E V 0 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 14
ka 1/30/2015 PHY 752 Spring 2015 -- Lecture 7 15
