6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 1 Lecture 6 Carrier drift and di usion February 16 2007 Contents 1 Thermal motion and scattering 2 Drift 3 Di usion Reading assignment del Alamo Ch 4 4 1 4 3 Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 2 Key questions Are carriers sitting still in thermal equilibrium How do carriers move in an electric eld What are the key dependencies of the drift velocity How do the energy band diagrams represent the pres ence of an electric eld How does a concentration gradient a ect carriers Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 3 1 Thermal motion and scattering We can think of carriers as particles in an ideal gas At nite T carriers have nite thermal energy All this energy resides in the kinetic energy of the particles Carriers move in random directions no net velocity but average carrier velocity is thermal velocity vth 8 kT m c Where m c conductivity e ective mass eV s2 cm2 m c accounts for all interactions between the carriers and the perfect periodic potential of the lattice For electrons in Si at 300 K m ce 0 28mo and vthe 2 107 cm s Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 4 But semiconductor crystal is not perfect the Si atoms themselves are vibrating around their equilibrium position in the lattice there are impurities and crystal imperfections As carriers move around they su er frequent collisions De ne Mean free path lc average distance travelled be tween collisions cm Scattering time c average time between collisions s Then lc vth c Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 5 Scattering mechanisms 1 lattice or phonon scattering carriers collide with vibrating lattice atoms phonon absorption and emis sion some energy exchanged tens of meV 2 ionized impurity scattering Coulombic interaction between charged impurities and carriers no energy exchanged 3 surface scattering in inversion layer 4 neutral impurity scattering with neutral dopants in terstitials vacancies etc 5 carrier carrier scattering No need for detailed models Order of magnitude of c 1 ps see how to estimate in notes Then order of magnitude of lc 50 nm Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Lecture 6 6 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 2 Drift In the presence of an electric eld electrons drift net velocity in direction of field ce average net velocity time Drift velocity electric eld E electrostatic force on electron qE E acceleration between collisions q mce velocity acquired during time ce vedrif t qE ce mce or vedrif t eE e electron mobility cm2 V s Mobility suggests ease of carrier motion in response to E Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Lecture 6 7 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 vedrif t eE vhdrif t hE Mobility depends on doping level and whether carrier is majority or minority type Si at 300 K 1600 1400 electrons Mobility cm2 V s 1200 1000 800 600 holes n Si 400 p Si 200 0 1E 14 p Si 1E 15 1E 16 1E 17 n Si 1E 18 1E 19 1E 20 1E 21 Doping cm 3 at low N limited by phonon scattering at high N limited by ionized impurity scattering Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 8 Velocity saturation Implicit assumption quasi equilibrium that is scatter ing rates not much a ected from equilibrium v drift E only if v drif t vth For high E carriers acquire substantial energy from E optical phonon emission strongly enhanced scattering time 1 E drift velocity saturates 8 Eopt vsat 3 m c For Si at 300 K vsat 107 cm s for electrons vsat 6 106 cm s for holes independent of T Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Lecture 6 9 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 vdrift vsat 0 0 sat Drift velocity vs electric eld fairly well described by v drif t E 1 v E sat Field required to saturate velocity vsat Esat Velocity saturation crucial in modern devices if 500 cm2 V s Esat 2 104 V cm 2 V across 1 m Since depends on doping Esat depends on doping too Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Lecture 6 10 Particle ux and current density particle ux particles crossing unity surface normal to ow per unit time cm 2 s 1 current density electrical charge crossing unity surface normal to ow per unit time C cm 2 s 1 Je qFe Fe ve dt nve dt Fe nve dt Then Je qnve Jh qpvh Cite as Jes s del Alamo course materials for 6 720J Integrated Microelectronic Devices Spring 2007 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Lecture 6 11 6 720J 3 43J Integrated Microelectronic Devices Spring 2007 Drift current low elds Je q enE Jh q hpE total J q e n hp E Electrical conductivity cm 1 q e n hp Electrical resistivity cm 1 q e n hp Check