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:   rErrVrm222Consider 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 equationsheet the outside forsheet the inside for0rrVrrVfree electrons (experience no potential when inside the sheet)yxLLA3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBorn Von Karman Periodic Boundary Conditions in 2D rErm222Solve:Use periodic boundary conditions:zyxzLyxzyxzyLxyx,,,,,,,,These imply that each edge of the sheet is folded and joined to the opposite edgeSolution is:ykxkirkiyxeAeAr11.The boundary conditions dictate that the allowed values of kx , and ky are such that:yyLkixxLkiLmkeLnkeyyxx2121n = 0, ±1, ±2, ±3,…….m = 0, ±1, ±2, ±3,…….xLyLyxLLAxyECE 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-vectormkkE222rkikeAr.1Momentum 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:ipˆ  rkrirpkkkˆNormalization: The wavefunction is properly normalized:122rrdkOrthogonality: Wavefunctions of two different states are orthogonal: kkrkkikkAerdrrrd,'.'2*'2Velocity:Velocity of eigenstates is:  kEmkkvk14ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityStates in 2D k-SpacexL2yL2k-space Visualization:The allowed quantum states states can be visualized as a 2D grid of points in the entire “k-space”yyxxLmkLnk22Density 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:ALLyx222222A 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:mkmkkkEyx2222222Strategy:• 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 kFFkxLyLyxLLAxy5ECE 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 = 2Fk222FkANumber of quantum states (including spin) in the circular region = 222222FFkAkABut the above must equal the total number N of electrons inside the box:22FkAN2density electron2FkANn212 nkFThe 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:Fk212 nkFFermi Energy:The largest energy of the electrons is:This is called the Fermi energy EF:mkF222mkEFF222Fermi Velocity:The largest velocity of the electrons is called the Fermi velocity vF:mkvFFThe Electron Gas in 2D at Zero Temperature - IIImnEF2orFEmn2Also:6ECE 407 – Spring 2009 – Farhan Rana – Cornell University22ARecall that there are grid points per unit area of k-space So in area of k-space the number of grid points is: yxdkdk