Transcript Lecture 13
These PowerPoint color diagrams can only be used by instructors if the 3rd Edition has been adopted for his/her course. Permission is given to individuals who have purchased a copy of the third edition with CD-ROM Electronic Materials and Devices to use these slides in seminar, symposium and conference presentations provided that the book title, author and © McGraw-Hill are displayed under each diagram. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Minority Carrier Lifetime Fig 5.24 Low-level photoinjection into an n-type semiconductor in which nn > n0 • Consider an n-type semiconductor doped 5E16 cm-3 that is uniformly illuminated with an appropriate l of light to photogenerate EHPs • In n-type semiconductors, electrons are the majority carriers and holes are the minority carriers • When we illuminate the semiconductor, EHPs are created • Then, at an instant during illumination, the semiconductor will be comprised of EHP generated e- and h+, plus what was already there at thermal equilibrium From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 5.25 • nno: majority carrier concentration (econcentration in an n-type semiconductor) in thermal equilibrium in the dark. These e- are thermally ionized from the donors • pno: minority carrier concentration (h+ concentration in an n-type semiconductor) in thermal equilibrium in the dark. These h+ are thermally generated across the bandgap • In both cases, the subscript no refers to an n-type semiconductor and thermal equilibrium conditions, respectively. • Thermal equilibrium means that mass action law is obeyed: Low-level injection in an n-type semiconductor does not significantly affect nn but drastically affects the minority carrier concentration pn. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • The electron and hole concentrations at any instant during illumination are denoted by nn and pn. These are defined as the intstaneous majority (e-) and minority (h+) concentrations, respectively. • At any instant and at any location in the semiconductor, we define the departure from equilibrium by excess concentrations as follows: Dnn is excess e- (majority) concentration: Fig 5.25 Low-level injection in an n-type semiconductor does not significantly affect nn but drastically affects the minority carrier concentration pn. Dpn is excess h+ (majority) concentration: From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • So, under illumination, at any instant: • Photoexcitation creates EHPs or an equal number of e- and h+ such that and mass action law is obeyed • Recall that: Fig 5.25 Low-level injection in an n-type semiconductor does not significantly affect nn but drastically affects the minority carrier concentration pn. since nno and pno depend only on temp From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • So, under illumination, at any instant: • Photoexcitation creates EHPs or an equal number of e- and h+ such that and mass action law is obeyed • Recall that: Fig 5.25 Low-level injection in an n-type semiconductor does not significantly affect nn but drastically affects the minority carrier concentration pn. Since nno and pno depend only on temp From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Light is switched on (B), then off again (C) Fig 5.26 Illumination of an n-type semiconductor results in excess electron and hole concentrations. After the illumination, the recombination process restores equilibrium; the excess electrons and holes simply recombine. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Minority Carrier Lifetime Recombination time depends on the semiconductor material, impurities, crystal defects, temp, etc. • Right after the light is turned off the condition pn = Dpn must eventually go back to the dark case, pn = pno, thus the excess minority carriers, Dpn, and excess majority carriers, Dnn, must be removed • This removal occurs by recombination • Excess h+ recombine with available e• However, this process takes time because the electrons and holes need to find each other • Thus, minority carrier lifetime, th, describes the rate of recombination • This is defined as the mean time the h+ is free in the VB before recombining with an e• Alternatively, an equivalent definition is that 1/th is the average probability per unit time that a hole will recombine with an electron. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Excess Minority Carrier Concentration • Ex: Let’s say minority carrier lifetime is 10 s, and there are ~1000 holes. Then, these excess holes will be disappearing at a rate of 100 per second • The rate of recombination of excess minority carriers is simply: Dpn/tn • Thus, at any instant: Rate of increase in excess h+ = Rate of photogeneration – Rate of recombination of excess h+ dDpn Dpn Gph dt th This assumes weak injection: Dpn < nno Dpn = excess hole (minority carrier) concentration in n-type t = time Gph = rate of photogeneration th = minority carrier lifetime (mean recombination time) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 5.27 Illumination is switched on at time t = 0 and then off at t= toff. The excess minority carrier concentration Dpn(t) rises exponentially to its steady-state value with a time constant th. From toff, the excess minority carrier concentration decays exponentially to its equilibrium value. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Particle Flux Example typically used: diffusion If concentration gradient, then diffusion occurs from high to low concentration DN ADt = particle flux, DN = number of particles crossing A in a time interval Dt, A = area, Dt = time interval Definition of Current Density J Q J = electric current density, Q = charge of the particle, = particle flux From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 5.29 (a) Arbitrary electron concentration n (x, t) profile in a semiconductor. There is a net diffusion (flux) of electrons from higher to lower concentrations. (b) Expanded view of two adjacent sections at x0. There are more electrons crossing x0 coming from the left (x0-) than coming from the right (x0+) • Suppose e- concentration in a semiconductor, at some point t, decreases in the x direction and has a profile n(x,t) • Electron motion is random when there is no E-field, caused by lattice vibration and impurity scattering • Recall l is the mean free path and t is mean free time between collisions/scattering events • So, an e- moves a mean distance l in the +x or –x direction and then scatters • Mean speed along x is vx = l/t From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 5.29 (a) Arbitrary electron concentration n (x, t) profile in a semiconductor. There is a net diffusion (flux) of electrons from higher to lower concentrations. (b) Expanded view of two adjacent sections at x0. There are more electrons crossing x0 coming from the left (x0-) than coming from the right (x0+) • Let’s evaluate the flow of electrons in the –x and +x directions through plane at xo to thereby find the net flow in the +x direction • Divide the x axis into segments of length l corresponding to a mean free path • During one mean free time, half of the e- in xo -l would move toward xo, and the other half away from xo • In time t, half will then reach xo and cross • If n1 is the concentration of e- at xo-½l, then the # of e- moving toward the right to cross xo is ½ n1Al • Similarly, half of the e- in xo+½l would be moving toward the left and in time t would reach xo • Their number is ½ n2Al where is the concentration at xo+½l From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • The net # of e- crossing xo per unit time per unit area in the +x direction is the electron flux e, Fig 5.29 (a) Arbitrary electron concentration n (x, t) profile in a semiconductor. There is a net diffusion (flux) of electrons from higher to lower concentrations. (b) Expanded view of two adjacent sections at x0. There are more electrons crossing x0 coming from the left (x0-) than coming from the right (x0+) • Because the mean free path l is small, calculus allows us to calculate n2-n1 from the concentration gradient using: From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fick’s First Law dn e De dx Relationship between the net particle flux and driving force (i.e., the gradient) e = electron flux, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient Electron Diffusion Current Density J D,e dn ee eDe dx JD, e = electric current density due to electron diffusion, e = electron flux, e = electronic charge, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • Similar situation if it were holes involved: J D ,h Fig 5.30 Arbitrary hole concentration p (x, t) profile in a semiconductor. There is a net diffusion (flux) of holes from higher to lower concentrations. There are more holes crossing x0 coming from the left (x0-) than coming from the right (x0+). dp eh eDh dx JD, h = electric current density due to hole diffusion, e = electronic charge, h = hole flux, Dh = diffusion coefficient of holes, dp/dx = hole concentration gradient From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 5.31 When there is an electric field and also a concentration gradient, charge carriers move both by drift and diffusion. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Total Electron Current Due to Drift and Diffusion dn J e en eE x eDe dx Je = electron current due to drift and diffusion, n = electron concentration, e = electron drift mobility, Ex = electric field in the x direction, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Total Hole Current Due to Drift and Diffusion dp J h ep hE x eDh dx Jh = hole current due to drift and diffusion, p = hole concentration, h = hole drift mobility, Ex = electric field in the x direction, Dh = diffusion coefficient of holes, dp/dx = hole concentration gradient Einstein Relation De kT e e Dh kT h e • Diffusion coefficients are a measure of the ease with which the diffusing charge carriers move in the medium. • So is drift mobility • Both quantities are related through this expression De = diffusion coefficient of electrons, e = electron drift, Dh = diffusion coefficient of the holes, h = hole drift mobility From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Time-Dependent Continuity Equation • Many semiconductor devices operate on the principle of excess carriers being injected • This injection of carriers by external means upsets the equilibrium concentration • To determine the carrier concentration at any instant we need to solve the continuity equation, which accounts for the total charge at that location in the semiconductor pn(x,t) n-type Fig 5.33 Consider an elemental volume Adx in which the hole concentration is pn(x, t) • Consider infinitesimally thin elemental volume Adx in which h+ is pn(x,t) • The current density at x due to h+ flowing into volume is Jh and that due to h+ flowing out at x+d x is Jh +d Jh • There is a change in hole current density Jh where Jh(x,t) is not uniform along x. (assume it does not change in y and z directions) • If d Jh is negative, then the current leaving the volume is less thanthat entering the volume, which leads to an increase in the h+ concentration in Adx From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Time-Dependent Continuity Equation • Therefore, Rate of increase in h+ concentration due to the change in Jh pn(x,t) n-type Fig 5.33 Consider an elemental volume Adx in which the hole concentration is pn(x, t) • The negative sign ensures that d Jh leads to an increase in pn • Recombination taken place in Adx removes h+ from this volume • In addition, photogeneration at x at time t could be happening • Therefore, the net rate increase in the h+ concentration pn in Adx = Rate of increase due to decrease in Jh – Rate of recombination + Rate of photogeneration From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Continuity Equation for Holes Rate of increase due to decrease in Jh Net rate of increase Rate of Recomb. Rate of Photogen. pn 1 Jh pn pno Gph t e x th pn = hole concentration in an n-type semiconductor, pno = equilibrium minority carrier (hole concentration in an n-type semiconductor) concentration, Jh = hole current due to drift and diffusion, th = hole recombination time (lifetime), Gph = photogeneration rate at x at time t, x = position, t = time Continuity Equation with Uniform Photogeneration Dpn Dpn Gph t th Dpn = pn pno is the excess hole concentration • Photogeneration and current density do not vary with distance along the sample length, so first term above = 0 • If Dpn is the excess concentration, then the time derviative of pn is the same as Dpn From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Steady-State Continuity Equation for Holes 1 J h pn pno e x th Jh = hole current due to drift and diffusion, pn = hole concentration in an n-type semiconductor, pno = equilibrium minority carrier (hole concentration in an n-type semiconductor) concentration, th = hole recombination time (lifetime) Steady-State Continuity Equation with E = 0 2 d Dpn Dpn 2 2 dx Lh Dpn = pn pno is the excess hole concentration, Lh = diffusion length of the holes From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) Steady state excess carrier concentration profiles in an n-type semiconductor that is continuously illuminated at one end. (b) Majority and minority carrier current components in open circuit. Total current is zero. Fig 5.34 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Minority Carrier Concentration, Long Bar x Dpn (x) Dpn (0)exp Lh Dpn = pn pno is the excess hole concentration, Lh = diffusion length of the holes Steady State Hole Diffusion Current Ih ID,h x dpn (x) AeDh AeDh Dpn (0)exp dx Lh Lh Ih = hole current, ID, h = hole diffusion current, A = cross-sectional area, Dh = diffusion coefficient of holes, pn(x) = hole concentration in an n-type semiconductor as a function of position x, Lh = diffusion length of holes, Dpn = pn pno is the excess hole concentration From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Majority Carrier Concentration, Long Bar x Dnn (x) Dnn (0)exp Le Dnn(x) = the excess electron concentration, x = position, Le = diffusion length of the electrons Electron Diffusion Current ID, e x dnn (x) AeDe AeDe Dnn (0)exp dx Le Le ID, e = electron diffusion current, De = diffusion coefficient of electrons, nn(x) = electron concentration in an n-type semiconductor as a function of position x, Le = diffusion length of the electrons, Dnn = the excess electron concentration From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electron Drift Current: Use the Open Circuit Condition I drift ,e I D ,e I D ,h 0 Idrift, e = electron drift current, ID, e = electron diffusion current, ID, h = hole diffusion current, Electric Field E I drift ,e Aenno e E = electric field, Idrift, e = electron drift current, nno = equilibrium majority carrier (electron concentration in an n-type semiconductor) concentration, e = electron drift mobility From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Optical absorption generates electron-hole pairs. Energetic electrons must lose their excess energy to lattice vibrations until their average energy is (3/2)kT in the CB. Fig 5.35 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Absorption of photons within a small elemental volume of width Fig 5.36 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) x Definition of Optical Absorption Coefficient dI Idx = absorption coefficient, I = light intensity, dI = change in the light intensity in a small elemental volume of thickness dx at x Beer-Lambert Law I ( x) I o exp( x) I(x) = light intensity at x, Io = initial light intensity, = absorption coefficient, x = distance from the surface (location) where I = Io. Note: Light propagates along x. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) The absorption coefficient depends on the photon energy h and hence on the wavelength. Density of states increases from band edges and usually exhibits peaks and troughs. Generally increases with the photon energy greater than Eg because more energetic photons can excite electrons from populated regions of the VB to numerous available states deep in the CB. Fig 5.37 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Piezoresistivity and its applications. (a) Stress m along the current (longitudinal) direction changes the resistivity by . (b) Stresses L and T cause a resistivity change. (c) A force applied to a cantilever bends it. A piezoresistor at the support end (where the stress is large) measures the stress, which is proportional to the force. Fig 5.38 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (d) A pressure sensor has four piezoresistor R1, R2, R3, R4 embedded in a diaphragm. The pressures bends the diaphragm, which generates stresses that are sensed by the four piezoresistors. Fig 5.38 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)