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1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 3Free Electron Gas in 2D and 1DIn this lecture you will learn:• Free electron gas in two dimensions and in one dimension• Density of States in k-space and in energy in lower dimensionsECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectron 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:GaAsGaAsInGaAs quantum well (1-10 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 tableGraphene is a single atomic layer of carbon atoms arranged in a honeycomb lattice TEM micrographSTM micrograph2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectron 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):GaAsInGaAsNanowireGaAsSemiconductor Quantum Point Contacts (Electrostatic Gating):GaAsInGaAsQuantum wellCarbon Nanotubes (Rolled Graphene Sheets):metalmetalECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectrons in 2D Metals: The Free Electron ModelThe quantum state of an electron is described by the time-independent Schrodinger equation: rErrVrm222Consider a large metal sheet of area A= LxLy: xLyLUse the Sommerfeld model:• The electrons inside the sheet are confined in a two-dimensional 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 equationsheet the outside forsheet the inside for0rrVrrVfree electrons (experience no potential when inside the sheet)yxLLA3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBorn Von Karman Periodic Boundary Conditions in 2D rErm222Solve:Use periodic boundary conditions:zyxzLyxzyxzyLxyx,,,,,,,,These imply that each edge of the sheet is folded and joined to the opposite edgeSolution is:ykxkirkiyxeAeAr11.The boundary conditions dictate that the allowed values of kx , and ky are such that:yyLkixxLkiLmkeLnkeyyxx2121n = 0, ±1, ±2, ±3,…….m = 0, ±1, ±2, ±3,…….xLyLyxLLAxyECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBorn Von Karman Periodic Boundary Conditions in 2DLabeling Scheme:All electron states and energies can be labeled by the corresponding k-vectormkkE222rkikeAr.1Momentum Eigenstates:Another advantage of using the plane-wave energy eigenstates (as opposed to the “sine” energy eigenstates) is that the plane-wave states are also momentum eigenstatesMomentum operator:ipˆ rkrirpkkkˆNormalization: The wavefunction is properly normalized:122rrdkOrthogonality: Wavefunctions of two different states are orthogonal: kkrkkikkAerdrrrd,'.'2*'2Velocity:Velocity of eigenstates is: kEmkkvk14ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityStates in 2D k-SpacexL2yL2k-space Visualization:The allowed quantum states states can be visualized as a 2D grid of points in the entire “k-space”yyxxLmkLnk22Density of Grid Points in k-space:Looking at the figure, in k-space there is only one grid point in every small area of size:ALLyx222222A There are grid points per unit area of k-spaceVery important resultn, m = 0, ±1, ±2, ±3, …….xkykECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityThe Electron Gas in 2D at Zero Temperature - I• Suppose we have N electrons in the sheet. • Then how do we start filling the allowed quantum states? • Suppose T~0K and we are interested in a filling scheme that gives the lowest total energy.xkykNThe energy of a quantum state is:mkmkkkEyx2222222Strategy:• Each grid-point can be occupied by two electrons (spin up and spin down)• Start filling up the grid-points (with two electrons each) in circular regions of increasing radii until you have a total of N electrons• When we are done, all filled (i.e. occupied) quantum states correspond to grid-points that are inside a circular region of radius kFFkxLyLyxLLAxy5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityxkykFkFermi circle• Each grid-point can be occupied by two electrons (spin up and spin down)• All filled quantum states correspond to grid-points that are inside a circular region of radius kFArea of the circular region = Number of grid-points in the circular region = 2Fk222FkANumber of quantum states (including spin) in the circular region = 222222FFkAkABut the above must equal the total number N of electrons inside the box:22FkAN2density electron2FkANn212 nkFThe Electron Gas in 2D at Zero Temperature - IIUnits of the electron density n are #/cm2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityxkykFkFermi circle• All quantum states inside the Fermi circle are filled (i.e. occupied by electrons) • All quantum states outside the Fermi circle are emptyFermi Momentum:The largest momentum of the electrons is:This is called the Fermi momentumFermi momentum can be found if one knows the electron density:Fk212 nkFFermi Energy:The largest energy of the electrons is:This is called the Fermi energy EF:mkF222mkEFF222Fermi Velocity:The largest velocity of the electrons is called the Fermi velocity vF:mkvFFThe Electron Gas in 2D at Zero Temperature - IIImnEF2orFEmn2Also:6ECE 407 – Spring 2009 – Farhan Rana – Cornell University22ARecall that there are grid points per unit area of k-space So in area of k-space the number of grid points is: yxdkdk

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