CN9-transport - Charge Transport 1 Outline Drift current...

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1 1 Charge Transport 2 Outline ± “Drift” current and “mobility” … ± … and High-field effects ± … and “Non-local” field effects ± “Diffusion current” ± Spatial invariance of the Fermi level (in Equilibrium) ± The Einstein relation and the Drift-Diffusion equations ± Voltage and quasi-Fermi levels ± Current continuity, charge injection and diffusion lengths ± Summary
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2 3 Basic mechanisms of charge transport: drift and diffusion There are two basic mechanism of charge transport: o“ drift ” of the charge carriers due to applied forces including electric fields … … and magnetic fields. diffusion ” due to random motion of charge carriers combined with spatial variations in the carrier concentrations … … and/or spatial variations in the carrier temperature (average energy) and/or spatial variations in carriers scattering rate with material variation. 4 Drift and mobility Consider electrons in energy valleys subject to the effective mass approximation. In the absence of applied forces the average velocity < v > a vector with magnitude and direction of carriers is zero. (However, even in equilibrium, the average energy and speed |< v > | a scaler of carriers can be significant, as it turns out in each direction i .; However … ) T k v m E B i i i 2 1 2 1 2 = =
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3 If an electric field ε (a vector) is applied, electrons will gain velocity/accelerate a in the direction of the force, F n = – q ε , on average as (remember) where m n,cond is conductivity effective mass for electrons (… which, from “motion” notes, is for electrons in Si or in terms of the “longitudinal” and “transverse” masses m l and m t respectively.) (Singh, 5.15, modified) m l = 0.98 m e Δ -valleys + t l cond n m m m 2 1 3 1 1 , cond n cond n n drift n m q m t , , , ε F a v ε = = = e cond n m m 26 . 0 , m t = 0.19 m e 6 scatt n drift n scattering drift n t , , , τ v v = However, the electrons are also frequently (!) ~ 10 12 to 10 14 times/sec typically subject to velocity randomizing/reducing (and energy dissipating) scattering processes … … that is, dopants, lattice vibrations or anything else that makes the crystal potential not perfectly periodic (such that the Bloch functions ψ k ( r ) with associated expected velocities v k are no longer true energy eigenstates and, thus, must evolve in time/scatter). If τ n,scatt is the average time required for an electron to have its velocity direction randomized by scattering processes, then the rate of deceleration due to scattering can be written,
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4 7 For a constant field and in steady-state where there can be no net acceleration, balancing velocity gain from the field and velocity loss due to scattering gives or Defining mobility μ for electrons as gives 0 , , , , , = = + scatt n drift n cond n scattering drift n drift n m q t t τ v ε v v ε ε v cond n scatt n drift n m q , , , = ε v drift n n , μ cond n scatt n n n drift n m q , , , , = = ε v 8 The electron drift current j n,drift
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CN9-transport - Charge Transport 1 Outline Drift current...

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