L6: Carrier Dynamics, Thermal Motion

6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 6-1 Lecture 6 - Carrier drift and diffusion February 16, 2007 Contents: 1. Thermal motion and scattering 2. Drift 3. Diffusion Reading assignment: del Alamo, Ch. 4, §§4.1-4.3. Cite as: Jesús 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 field? What are the key dependencies of the drift velocity? • How do the energy band diagrams represent the pres­ ence of an electric field? • How does a concentration gradient affect carriers? Cite as: Jesús 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 finite T , carriers have finite 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 π m∗c Where: m∗c ≡ conductivity effective 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: Jesús 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 suffer frequent collisions: Define: • Mean free path, lc : average distance travelled be­ tween collisions [cm]. • Scattering time, τc: average time between collisions [s]. Then: lc = vthτc Cite as: Jesús 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 τc < 1 ps (see how to estimate in notes). Then, order of magnitude of lc < 50 nm. Cite as: Jesús 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 field, electrons drift: net velocity in direction of field τce average net velocity time � Drift velocity -electric field: E -electrostatic force on electron: −qE E -acceleration between collisions: −q ∗ mce -velocity acquired during time τce: vedrif t qEτce =− ∗ mce or vedrif t = −μeE μe ≡ electron mobility [cm2 /V · s] Mobility suggests ease of carrier motion in response to E. Cite as: Jesús 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 = −μeE vhdrif t = μhE 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: Jesús 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 affected 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  3π 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: Jesús 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 μ 0 0 ε εsat Drift velocity vs. electric field fairly well described by: v drif t = ∓ μE 1 + | vμE | sat Field required to saturate velocity: vsat Esat = μ Velocity saturation crucial in modern devices: if μ = 500 cm2/V.s, Esat = 2 × 104 V /cm (2 V across 1 μm) Since μ depends on doping, Esat depends on doping too. Cite as: Jesús 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 flux and current density particle flux ≡ # particles crossing unity surface (normal to flow) per unit time [cm−2 · s−1] current density ≡ electrical charge crossing unity surface (normal to flow) 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: Jesús 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 fields): Je = qμenE Jh = qμhpE total: J = q(μe n + μhp)E Electrical conductivity [(Ω · cm)−1 ]: σ = q(μe n + μhp) Electrical resistivity [Ω · cm]: ρ= 1 q(μe n + μhp) Check signs: vdn vdp - + Jpdrift Jndrift x x Cite as: Jesús 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 ρ= Lecture 6-12 1 q(μe n + μhp) ρ strong function of doping ⇒ frequently used by wafer vendors to specify doping level of substrates -for n-type: ρn  1 qμe ND -for p-type: ρp  1 qμh NA Si at 300K: 1E+4 1E+3 Resistivity (ohm.cm) 1E+2 1E+1 p-Si 1E+0 n-Si 1E-1 1E-2 1E-3 1E-4 1E+12 1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 Doping (cm-3) Cite as: Jesús 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 6-13 • Drift current (high fields): Jesat = qnvesat Jhsat = qpvhsat The only way to get more current is to increase carrier concentration. Cite as: Jesús 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 6-14 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 � Energy band diagram under electric field Energy band diagram needs to account for potential en­ ergy of electric field • Vacuum: + φ ε - - x Ep=-qφ - x Electron trades potential energy by kinetic energy as it moves to the left → total electron energy unchanged Cite as: Jesús 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 6-15 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 • Energy band diagram is picture of electron energy ⇒ must add Ep to semiconductor energy band diagram ⇒ bands tilt ε + - Ec Ev Meauring from an arbitrary energy reference, Eref : Ec + Eref = Ep = −qφ Then: dφ 1 dEc 1 dEv E =− = = dx q dx q dx Shape of energy bands = shape of φ with a minus sign. Can easily compute E from energy band diagram. Cite as: Jesús 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 6-16 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 3. Diffusion Movement of particles from regions of high concentration to regions of low concentration. Diffusion produced by collisions with background medium (i.e., vibrating Si lattice). n x • Diffusion flux ∝ concentration gradient [Fick’s first law] dn Fe = −De dx dp Fh = −Dh dx D ≡ diffusion coefficient [cm2 /s] Cite as: Jesús 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 6-17 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 dn dx dp Fh = −Dh dx Fe = −De • Diffusion current: dn dx dp Jh = −qDh dx Je = qDe Check signs: n p Fe Fh Jhdiff Jediff x x Cite as: Jesús 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 6-18 Key conclusions • At finite temperatures, carriers move around in a ran­ dom way suffering many collisions (thermal motion). • Dominant scattering mechanisms in bulk Si at 300K: phonon scattering and ionized impurity scattering. • Two processes for carrier flow in semiconductors: drift and diffusion. • General relationship between carrier net velocity (by drift or diffusion) and current density: Je = −qnve Jh = qpvh • For low fields, v drif t ∼ E. • For high fields, v drift ∼ vsat. • Driving force for diffusion: concentration gradient. • Order of magnitude of key parameters for Si at 300K: – vth ∼ 2 × 107 cm/s – τc < 1 ps – lc < 50 nm – electron mobility: μe ∼ 100 − 1400 cm2 /V · s – hole mobility: μh ∼ 50 − 500 cm2/V · s – saturation velocity: vsat ∼ 107 cm/s Cite as: Jesús 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 6-19 Self study • Study estimation of τc and lc. • Study doping dependence of Esat. • Study phenomenological diffusion model in §4.3. Cite as: Jesús 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].