I. Carrier Transport: Drift B.A. DiffusionDrift Velocity Current Density- Holes • Current density = (charge) x (# carriers per second per area): 1 1 (a) Thermal Equilibrium, E =- -0 px()–λAλ–--(b)px Electric()+ FieldλA E λ> 0 diff 2 2 J = q --------------------------------------------------------------- Electron # 1 p Electronτ # 1 Ac E • If we assume the mean free path is much smaller than the dimensions x x x x of our device,i f,then1 we can consider λ = dx xandi xf, 1use Electrona Taylor# 2 expansion on p(x - λ) andElectron p(x #+ 2 λ): E diff dp x λ2x τ x • J xf,2 =xi –qD , where Dp = f, 2/ c is xthei diffusion coefficent p pdx Electron # 3 Electron # 3 • Holes diffuse down the concentration gradient and carryE positive charge. x xf,3 xi x xf,3 xi * xi = initial position * xf, n = final position of electron n after 7 collisions • Electrons drifting in an Electric Field move (on average) with a drift velocity which is proportional to the Electric Field, E. EECS 6.012 Spring 1998 Lecture 3 Drift Velocity (Cont.) Velocity in Direction of E Field t qE vat==± ------- t m qτ c vave =– ---------- E 2mn • mn is an effective mass to take into account quantum mechanics µ 2 • Lump it into a quantity we call mobility n (units:cm /V-s) µ vd = - nE EECS 6.012 Spring 1998 Lecture 3 B. Electron and Hole Mobility • mobilities vary with doping concentration-- plot is for 300K 1400 1200 electrons 1000 Vs) / 2 800 600 mobility (cm holes 400 200 0 1013 1014 1015 1016 1017 1018 1019 1020 + −3 Nd Na total dopant concentration (cm ) • “typical values” for bulk silicon -- 300 K µ 2 n = 1000 cm /(Vs) µ 2 p = 400 cm /(Vs) EECS 6.012 Spring 1998 Lecture 3 C. Velocity Saturation • At electric fields greater >~ 104 V/cm • Drift velocities saturate --> max. out at around 107 V/cm. • Velocity saturation is very common in VLSI devices, due to sub- micron dimensions − vdn, vdp (cm/s) 108 107 electrons 106 holes 105 104 103 10 102 103 104 105 E (V/cm) EECS 6.012 Spring 1998 Lecture 3 D. Drift Current Density • Electrons drift against electric field • Electrons carry negative charge dr -2 -1 -2 • Jn = (-q) n vd units: Ccm s = Acm dr µ µ Jn = (-q) n (- n E) = q n n E • Holes drift with electric field µ • vd = p E dr • Jp = (+q) p vd dr µ Jp = q p p E Caution:: The linear relationship between field and drift velocity breaks down for high electric fields. EECS 6.012 Spring 1998 Lecture 3 II. Carrier Transport: Diffusion A. Diffusion • Diffusion is a transport process driven by gradients in the concentration of particles in random motion undergoing frequent collisions. 7 • Average carrier velocity = vth = 10 cm/s τ -13 • Average interval between collisions = c = 10 s = 0.1 picoseconds λ τ -6 • mean free path = = vth c = 10 cm = 10 nm reference plane (area = A) p(x) hole diffusion diff Jp (positive) − λ p(xr ) volume Aλ: volume Aλ: holes moving p(x + λ) r holes moving in + x direction cross in − x direction cross reference plane within ∆ = τ reference plane within t c. ∆ = τ t c. − λ + λ x xr xr xr • Since their motion is random, half of the carriers in each volume will pass through the plane before their next collision. EECS 6.012 Spring 1998 Lecture 3 B. Diffusion Current Density- Holes • Current density = (charge) x (# carriers per second per area): 1 1 -- []px()–λ Aλ–--[]px()+λ Aλ diff 2 2 J = q ---------------------------------------------------------------------------- p τ Ac : • If we assume the mean free path is much smaller than the dimensions of our device, then we can consider λ = dx and use a Taylor expansion on p(x - λ) and p(x + λ): diff dp λ2 τ • J = –qD where Dp = / c is the diffusion coefficent p pdx • Holes diffuse down the concentration gradient and carry positive charge. p(x) diff Jp ( > 0) diff Jp ( < 0) x EECS 6.012 Spring 1998 Lecture 3 C. Diffusion Current Density - Electrons • Electrons diffuse down the concentration gradient, yet carry negative charge --> electron diffusion current density points in the direction of the gradient n(x) diff Jn ( < 0) diff Jn ( > 0) x D. Einstein Relation • Both mobility and the diffusion constant are related to the mean time τ between collisions c. There is a relation between these important quantities called the Einstein Relationship. D D p kT n kT -------- =------ and ------- = ------ µ q µ q p n EECS 6.012 Spring 1998 Lecture 3 III. Total Current Density dr diff dn J ==J + J qnµ EqD+ n n n n ndx dr diff dp J==J+ J qpµ EqD– ppp p pdx • Fortunately, we will be able to eliminate one or the other component in finding the internal currents in microelectronic devices. EECS 6.012 Spring 1998 Lecture 3.