1EE130 Lecture 5, Slide 1Spring 2003Lecture #5ANNOUNCEMENT•Discussion Section 102 (Th 10-11AM) moved to 105 LatimerOUTLINE–Mobility dependence on temperature– Diffusion current– Relationship between band diagrams & V, – Non-uniformly doped semiconductor– Einstein relationship– Quasi-neutrality approximationRead: Chapter 3.2EE130 Lecture 5, Slide 2Spring 2003Dominant scattering mechanisms:1. Phonon scattering (lattice scattering)2. Impurity (dopant) ion scattering2/32/11 velocityermalcarrier thdensityphonon 1−∝×∝×∝∝ TTTphononphononτµPhonon scattering mobility decreases when T increases:µ= qτ/ mMechanisms of Carrier ScatteringTvth∝2EE130 Lecture 5, Slide 3Spring 2003_+--ElectronBoron IonElectronArsenicIonDADAthimpurityNNTNNv+∝+∝2/33µThere is less change in the electron’s direction of travel if the electron zips by the ion at a higher speed.Impurity Ion ScatteringEE130 Lecture 5, Slide 4Spring 2003Temperature Effect on Mobilityimpurityphononimpurityphononµµµτττ111111+=+=3EE130 Lecture 5, Slide 5Spring 2003Consider a Si sample doped with 1017cm-3 As.How will its resistivity change when the temperature is increased from T=300K to T=400K?Solution:The temperature dependent factor in σ (and therefore ρ) is µn. From the mobility vs. temperature curve for 1017cm-3, we find that µn decreases from 770 at 300K to 400 at 400K. As a result, ρ increases byExample: Temperature Dependence of ρ93.1400770=EE130 Lecture 5, Slide 6Spring 2003Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion.Diffusion4EE130 Lecture 5, Slide 7Spring 2003dxdnqDJN=diffN,dxdpqDJP−=diffP,D is the diffusion constant, or diffusivity.x xDiffusion CurrentEE130 Lecture 5, Slide 8Spring 2003JN= JN,drift+ JN,diff= qnµn+ dxdnqDNJP= JP,drift+ JP,diff= qpµp–dxdpqDPJ= JN+ JPTotal Current5EE130 Lecture 5, Slide 9Spring 2003Band Diagram: Potential vs. Kinetic Energyelectron kinetic energyincreasing electron energyEcEvhole kinetic energyincreasing hole energyreferencecP.E. EE −=Ecrepresents the electron potential energy:EE130 Lecture 5, Slide 10Spring 2003N-+– 0.7VSiEV(x)0.7V x0Electrostatic Potential V• Potential energy of a –q charged particle is related to the electrostatic potential V(x):)(1creferenceEEqV −=qV−=P.E.6EE130 Lecture 5, Slide 11Spring 2003N-+– 0.7VSiEV(x)0.7V x0Electric Field dxdEqdxdVc1=−=• Variation of Ecwith position is called “band bending.”EE130 Lecture 5, Slide 12Spring 2003Non-Uniformly-Doped Semiconductor• The position of EFrelative to the band edges is determined by the carrier concentrations, which is determined by the dopant concentrations.• In equilibrium, EFis constant; therefore, the band energies vary with position:Ev(x)Ec(x)EF7EE130 Lecture 5, Slide 13Spring 2003• In equilibrium, there is no net flow of electrons or holesÎ The drift and diffusion current components must balance each other exactly. (A built-in electric field exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient.)0=+=dxdnqDqnJNnNµJN= 0 and JP= 0EE130 Lecture 5, Slide 14Spring 2003 n-type semiconductorDecreasing donor concentrationdxdEekTNdxdnckTEEcFc/)( −−−=dxdEkTnc−=kTEEcFceNn/)( −−=Consider a piece of a non-uniformly doped semiconductor:Ev(x)Ec(x)EFqkTn−=8EE130 Lecture 5, Slide 15Spring 20030=+=dxdnqDqnJNnNµUnder equilibrium conditions, JN= 0 and JP= 0kTqDqnqnNn−=µ0nqkTDµ=NSimilarly,Einstein Relationship between D and µpqkTDµ=PNote: The Einstein relationship is valid for a non-degenerate semiconductor, even under non-equilibrium conditionsEE130 Lecture 5, Slide 16Spring 2003What is the hole diffusion constant in a sample of silicon with µp = 410 cm2/ V s ?Solution:Remember: kT/q = 26 mV at room temperature./scm 11sVcm 410)mV 26(2112P=⋅==−−pqkTDµExample: Diffusion Constant9EE130 Lecture 5, Slide 17Spring 2003Potential Difference due to n(x), p(x)• The ratio of carrier densities (n, p) at two points depends exponentially on the potential difference between these points:()=−=−=−=−−=−==>=−12i2i11212i1i2i2i1i2Fi2i1Fi1i1i1Fln1lnlnln Therefore ln Similarly,ln lnnnqkTEEqVVnnkTnnnnkTEEnnkTEEnnkTEEnnkTEEEE130 Lecture 5, Slide 18Spring 2003Quasi-Neutrality Approximation• If the dopant concentration profile varies gradually with position, then the majority-carrier concentration distribution does not differ much from the dopant concentration distribution.10EE130 Lecture 5, Slide 19Spring 2003Summary• Carrier mobility varies with temperature– decreases w/ increasing T if lattice scattering dominant– decreases w/ decreasing T if impurity scattering dominant• Electron/hole concentration gradient Æ diffusion• Current flowing in a semiconductor is comprised of drift & diffusion components for electrons & holes– In equilibrium, JN= JN,drift+ JN,diff= 0dxdnqDJN=diffN,dxdpqDJP−=diffP,J= JN,drift+ JN,diff+ JP,drift+ JP,diffEE130 Lecture 5, Slide 20Spring 2003• The characteristic constants of drift and diffusion are related:• In thermal equilibrium, EFis constant• Ecrepresents the electron potential energyVariation in Ec(x) Æ variation in electric potential VElectric field • E - Ec= electron kinetic
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