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Unformatted text preview: 1 EEE 352: Lecture 20 Electrical Properties of the Electrons and Holes * Electrical conductivity and Ohm’s Law ⇒ Scattering (Relaxation) Time ⇒ Drift and average velocities ⇒ Mobility and Conductivity * Hall Effect * Quantum Hall Effect • We now consider how the electron moves under the influence of an external field * We want to know what happens when an ELECTRIC FIELD is applied to generate an electrical CURRENT . • Newton’s second law of motion for the electrons may be written as • In the ABSENCE of any collisions, this equation predicts the electron is CONSTANTLY accelerated by the ELECTRIC FIELD Electrical Conductivity and Ohm’s Law ( ) E k v F e dt d dt m d − = = = h * t e t h E k k − = ) ( ) ( Electrical Conductivity and Ohm’s Law Hence, the average momentum is E E k h h τ τ τ e dt e t e t − = − = − ∞ ∫ / 1 And the average velocity is E k v v * * e e d m e m τ − = = ≡ h Defines the MOBILITY * e e e m e τ µ = DRIFT VELOCITY Effective mass of the electron • We can also calculate the CURRENT that flows as a result of the application of the ELECTRIC FIELD * The AVERAGE velocity of each electron is given as * This allows us to compute the corresponding CURRENT DENSITY Electrical Conductivity and Ohm’s Law * 2 * 2 , e e e e e e d m ne where m ne ne τ σ σ τ = = = − = E E v j E v E d µ − = THIS IS JUST OHM’S LAW We define the RESISTIVITY as σ ρ 1 = Conductivity A L A L R σ ρ = = Resistance: 2 d l T T τ τ τ 1 ) ( 1 ) ( 1 + = THE TOTAL RELAXATION RATE IS COMPOSED OF TEMPERATURE DEPENDENT AND INDEPENDENT COMPONENTS τ l : RELAXATION TIME DUE TO SCATTERING FROM LATTICE VIBRATIONS τ d : RELAXATION TIME DUE TO SCATTERING FROM FIXED DEFECTS such as IMPURITIES Relaxation Time Note that RATES add, not TIMES. This is known as Mattheisen’s Rule. 1 1 1 2 / 3 2 / 3 ) ( ) ( ) ( : ) ( : ) ( : − − − − + = ∝ ∝ T T T RATE Relaxation Total T T Dopants Ionized From Scattering T T Vibrations Thermal From Scattering i l i l τ τ τ τ τ • An important quantity related to the scattering time is the electron MOBILITY * This provides a measure of how MOBILE charge is * At low temperatures µ is dominated by IONIZED IMPURITY scattering * But at high temperatures µ is dominated by scattering from the LATTICE Relaxation Time * / m e τ µ = APPROXIMATE TEMPERATURE DEPENDENCE OF MOBILITY WITH LATTICE AND IONIZED IMPURITY SCATTERING T ( K ) ( LOG SCALE ) MOBILITY ( LOG SCALE ) T 3/2 T- 3/2 IMPURITY SCATTERING LATTICE SCATTERING T ~ 75-100 K 300 K Relaxation Time LATTICE scattering dominates the mobility The large value of the exponent arises from scattering between the conduction band (or valence band) valleys via high energy (optical) modes of the lattice vibrations....
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- Spring '08