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 • Density of States in k-space and in energy in lower dimensions 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 STM micrograph • Examples, discussed in detail later in the course, are shown below: Semiconductor Quantum Wells: Graphene: InGaAs GaAs quantum well (1-10 nm) GaAs Semiconductor quantum wells can be composed of Graphene is a single atomic layer pretty much any of carbon atoms arranged in a semiconductor from the TEM honeycomb lattice groups II, III, IV, V, and VI of micrograph the periodic table ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 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 Semiconductor Quantum Carbon Nanotubes Wires (or Nanowires): Point Contacts (Rolled Graphene (Electrostatic Gating): Sheets): GaAs metal metal InGaAs InGaAs Nanowire Quantum well GaAs GaAs 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 time-independent Schrodinger equation: 2 2 r V r r E r 2m Consider a large metal sheet of area A= Lx Ly : A LxLy Use the Sommerfeld model: Ly • The electrons inside the sheet are confined in a Lx two-dimensional infinite potential well with zero potential inside the sheet and infinite potential outside the sheet V r 0 for r inside the sheet V r for r outside the sheet free electrons • The electron states inside the sheet are given (experience no by the Schrodinger equation potential when inside the sheet) ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 Born Von Karman Periodic Boundary Conditions in 2D 2 Solve: 2 r E r 2m Use periodic boundary conditions: y A LxLy These imply that each x L ,y,z x, y,z x L x edge of the sheet is y L x, y Ly ,z x, y,z folded and joined to x the opposite edge 1 i k . r 1 i k x x ky y Solution is: r e e A A The boundary conditions dictate that the allowed values of kx , and ky are such that: i kxLx 2 e 1 kx n n = 0, ±1, ±2, ±3,……. Lx i ky Ly 2 e 1 ky m m = 0, ±1, ±2, ±3,……. Ly 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 k-vector 2 2 1 i k . r k r e Ek k A 2 m 2 2 Normalization: The wavefunction is properly normalized: d r k r 1 Orthogonality: Wavefunctions of two different states are orthogonal: i k k'. r 2 * 2 e d r r r d r k' k A k' , k 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 eigenstates Momentum operator: pˆ pˆ r r k r i k i k k Velocity: k 1 Velocity of eigenstates is: vk k Ek m ECE 407 – Spring 2009 – Farhan Rana – Cornell University 3 States in 2D k-Space ky 2 k-space Visualization: Lx The allowed quantum states states can be visualized as a 2D grid of points in the entire “k-space” 2 2 kx n ky m Lx Ly kx 2 n, m = 0, ±1, ±2, ±3, ……. Ly 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: 2 2 2 2 Lx Ly A A There are grid points per unit area of k-space Very important 2 2 result ECE 407 – Spring 2009 – Farhan Rana – Cornell University The 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? N y A LxLy • Suppose T~0K and we are interested in a filling scheme x Ly that gives the lowest total energy. Lx The energy of a quantum state is: ky 2 2 2 2 2 kx ky k Ek 2 m 2 m Strategy: • Each grid-point can be occupied by two electrons (spin up and spin down) • Start filling up the grid-points (with two electrons kx each) in circular regions of increasing radii until k you have a total of N electrons F • When we are done, all filled (i.e. occupied) quantum states correspond to grid-points that are inside a circular region of radius kF ECE 407 – Spring 2009 – Farhan Rana – Cornell University 4 The Electron Gas in 2D at Zero Temperature - II ky • Each grid-point can be occupied by two electrons (spin up and spin down) kF • All filled quantum states correspond to grid-points that are inside a circular region of radius kF kx 2 Area of the circular region = kF Fermi circle A 2 Number of grid-points in the circular region = kF 2 2 Number of quantum states (including A 2 A 2 spin) in the circular region = 2 kF kF 2 2 2 But the above must equal the total number N of electrons inside the box: A N k 2 2 F 2 N kF Units of the electron n electron density 2 A 2 density n are #/cm 1 2 kF 2 n ECE 407 – Spring 2009 – Farhan Rana – Cornell University The Electron Gas in 2D at Zero Temperature - III k • All quantum states inside the Fermi circle are filled (i.e. y occupied by electrons) • All quantum states outside the Fermi circle are empty kF Fermi Momentum: kx The largest momentum of the electrons is: kF This is called the Fermi momentum Fermi momentum can be found if one knows the electron density: 1 Fermi circle 2 kF 2 n Fermi Energy: 2 2 The largest energy of the electrons is: kF 2m 2k2 This is called the Fermi energy E : E F F F 2m 2 n m Also: E or n E F m 2 F Fermi Velocity: k The largest velocity of the electrons is called the Fermi velocity v : v F F F m ECE 407 – Spring 2009 – Farhan Rana – Cornell University 5 The Electron Gas in 2D at Non-Zero Temperature - I ky dkx A Recall that there are grid points per unit area of k- dk 2 y space 2 kx So in area d k x d k y of k-space the number of grid points is: A A 2 dkx dky d k 2 2 2 2 The summation over all grid points in k-space can be replaced by an area integral d 2k A 2 all k 2 Therefore: d 2k N 2 f k 2 A f k 2 all k 2 f k is the occupation probability of a quantum state ECE 407 – Spring 2009 – Farhan Rana – Cornell University The Electron Gas in 2D at Non-Zero Temperature - II The probability f k that the quantum state of wavevector k is occupied by an electron is given by the Fermi-Dirac distribution function: 1 2 2 2 2 2 kx ky k f k Where: Ek 1 eEk Ef KT 2 m 2 m Therefore: d 2k d 2k 1 N 2 A f k 2 A 2 2 2 2 1 eEk Ef KT Density of States: The k-space integral is cumbersome. We need to convert into a simpler form – an energy space integral – using the following steps: 2 2k2 2k d k 2 k dk and E dE dk 2 m m Therefore: d 2k k dk m 2 A 2 A A 2 dE 2 0 0 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 6 The Electron Gas in 2D at Non-Zero Temperature - III d 2k 1 1 N 2 A A dE g E 2 E k E KT 2D E E KT 2 1 e f 0 1 e f m Where: g2D E 2 Density of states function is constant (independent of energy) in 2D 2 g2D(E) has units: # / Joule-cm ky The product g(E) dE represents the number of quantum states available in the energy interval between E and (E+dE) per cm2 of the metal Suppose E corresponds to the inner circle from the relation: k 2k 2 x E 2 m And suppose (E+dE) corresponds to the outer circle, then g2D(E) dE corresponds to twice the number of the grid points between the two circles ECE 407 – Spring 2009 – Farhan Rana – Cornell University The Electron Gas in 2D at Non-Zero Temperature - IV 1 N A dE g E A dE g E f E E 2D E E KT 2D f 0 1 e f 0 g2D E m Where: g2D E 2 f E E The expression for N can be visualized as the f integration over the product of the two functions: Ef E Check: Suppose T=0K: Ef N A dE g2D E f E Ef A dE g2D E f E Ef 0 0 1 m T = 0K A 2 Ef 0 Ef E m n 2 Ef Compare with the previous result at T=0K: m n E At T=0K (and only at T=0K) the Fermi level 2 F E is the same as the Fermi energy E f F ECE 407 – Spring 2009 – Farhan Rana – Cornell University 7 The Electron Gas in 2D at Non-Zero Temperature - V For T ≠ 0K: Since the carrier density is known, and does not change with temperature, the Fermi level at temperature T is found from the expression Ef 1 m n dE g E K T log1 eK T 2D E E KT 2 0 1 e f In general, the Fermi level Ef is a function of temperature and decreases from EF as the temperature increases.