Lecture+C5-3

Lecture+C5-3 - Free electron gas model Fermi Level Model:...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Free electron gas model Fermi Level Model: 1. Outer most valence electrons become detached and shared throughout the solid 2. Electrons in a 3-d box with infinite walk Box size: *Ignore electron-electron & electron-nucleus interaction. *V=0 when electron is inside the box, V= when outside. More ElectronsSolid lx ly lz ( 29 ,0 ,0 , , x y z x l y l z l V x y z otherwise < < < < < < = + , , x y z l l l Separation of variables Shrodinger`s equation Recall: Solution to infinite potential well ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 , , 2 2 2 x y z x y z X x Y y Z z d X d Y d Z E X E Y E Z m dx m dy m dz = - =- =- = h h h ( 29 2 2 2 2 2 2 2 2 x y z E E m m - = - + + = h h ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 sin cos sin cos sin cos x y z x x x x y y y y z z z z E E E E X x A k x B k x Y y A k y B k y Z z A k z B k z = + + = + = + = + Boundary conditions ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 x y z X X l Y Y l Z Z l = = = = = = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 y y y z cos = sin cos = sin y cos = sin y x x x x y y z z B k x A k x B k y A k B k z A k = = = 2 2 2 2 2 2 2 2 2 2 , , 1,2,3 , 1,2,3 , 1,2,3 8 sin sin sin 2 2 x y z x y z x y z x x x y y y z z z x y z y x z n n n x y z x y z y x z n n n x y z B B B k l n k l n k l n n n n n n n x y z l l l l l l n n n k E m l l l m = = = = = = = = = = = + + = L L L h h in k space Each lattice point represents a distinct state (different kx, ky, kz) Volume occupied by each state in k space from a 3-d reciprocal lattice in momentum space x y z x y z l l l k spaced apart by k spaced apart by k spaced apart by 2 2 2 2 x y z k k k k = + + 3 3 x y z x y z l l l l l l V = = g g x k y k 2 D k space x l y l If we have N atoms in solid, q free electron per atom, What is the ground state? Volume in k space Volume occupied by each state # of states ( each state) Octant of a sphere The electron will fill the lower energy (smaller |k|) first Fermi level: 3 3 1 4 8 3 2 F Nq k V = F k 2 2 2 2 2 2 2 2 2 2 2 2 x y z y x z n n n x y z n n n k E m l l l m = + + =...
View Full Document

Page1 / 24

Lecture+C5-3 - Free electron gas model Fermi Level Model:...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online