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Unformatted text preview: 1 ECE 407 Spring 2009 Farhan Rana Cornell University Handout 3 Free Electron Gas in 2D and 1D In this lecture you will learn: Free electron gas in two dimensions and in one dimension ECE 407 Spring 2009 Farhan Rana Cornell University Electron Gases in 2D In several physical systems electron are confined to move in just 2 dimensions Examples, discussed in detail later in the course, are shown below: Semiconductor Quantum Wells: GaAs GaAs InGaAs quantum well (110 nm) Graphene: Semiconductor quantum wells can be composed of pretty much any semiconductor from the groups II, III, IV, V, and VI of the periodic table Graphene is a single atomic layer of carbon atoms arranged in a honeycomb lattice TEM micrograph STM micrograph 2 ECE 407 Spring 2009 Farhan Rana Cornell University Electron Gases in 1D In several physical systems electron are confined to move in just 1 dimension Examples, discussed in detail later in the course, are shown below: Semiconductor Quantum Wires (or Nanowires): GaAs InGaAs Nanowire GaAs Semiconductor Quantum Point Contacts (Electrostatic Gating): GaAs InGaAs Quantum well Carbon Nanotubes (Rolled Graphene Sheets): metal metal ECE 407 Spring 2009 Farhan Rana Cornell University Electrons in 2D Metals: The Free Electron Model The quantum state of an electron is described by the timeindependent Schrodinger equation: ( ) ( ) ( ) ( ) r E r r V r m r r r r h = + 2 2 2 Consider a large metal sheet of area A = L x L y : x L y L Use the Sommerfeld model: The electrons inside the sheet are confined in a twodimensional infinite potential well with zero potential inside the sheet and infinite potential outside the sheet The electron states inside the sheet are given by the Schrodinger equation ( ) ( ) sheet the outside for sheet the inside for r r V r r V r r r r = = free electrons (experience no potential when inside the sheet) y x L L A = 3 ECE 407 Spring 2009 Farhan Rana Cornell University Born Von Karman Periodic Boundary Conditions in 2D ( ) ( ) r E r m r r h = 2 2 2 Solve: Use periodic boundary conditions: ( ) ( ) ( ) ( ) z y x z L y x z y x z y L x y x , , , , , , , , = + = + These imply that each edge of the sheet is folded and joined to the opposite edge Solution is: ( ) ( ) y k x k i r k i y x e A e A r + = = 1 1 . r r r The boundary conditions dictate that the allowed values of k x , and k y are such that: ( ) ( ) y y L k i x x L k i L m k e L n k e y y x x 2 1 2 1 = = = = n = 0, 1, 2, 3,. m = 0, 1, 2, 3,. x L y L y x L L A = x y ECE 407 Spring 2009 Farhan Rana Cornell University Born Von Karman Periodic Boundary Conditions in 2D Labeling Scheme: All electron states and energies can be labeled by the corresponding kvector ( ) m k k E 2 2 2 h r = ( ) r k i k e A r r r r r ....
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 RANA

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