6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-1 Lecture 9 - Carrier Drift and Diffusion (cont.), Carrier Flow September 24, 2001 Contents: 1. Quasi-Fermi levels 2. Continuity equations 3. Surface continuity equations Reading assignment: del Alamo, Ch. 4, 4.6; Ch. 5, 5.1, 5.2 Announcement: The final exam of 6.720J/3.43J has been scheduled for December 17, 9AM-noon, at du Pont.
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-2 Key questions Is there something equivalent to the Fermi level that can be used outside equilibrium? How do carrier distributions in energy look like outside equilibrium? In the presence of carrier flow in the bulk of a semiconductor, how does one formulate bookkeeping relationships for carriers? How about at surfaces?
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-3 1. Quasi-Fermi levels In TE, Fermi level makes statement about energy distribution of carriers in bands E F relates n o with N c and p o with N v. Outside TE, E F cannot be used. Define two quasi-fermi levels such that n = N c F 1/2 ( E fe E c ) kt p = N v F 1/2 ( E v E fh ) kt Under Maxwell-Boltzmann statistics (n N c, p N v ): n = N c exp E fe E c kt p = N v exp E v E fh kt What are quasi-fermi levels good for?
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-4 Take derivative of n = f(e fe ) with respect to x: dn dx = n kt Plug into current equation: To get: de fe dx q kt ne J e = qµ e ne + qd e dn dx Similarly for holes: J e = µ e n de fe dx J h = µ h p de fh dx Gradient of quasi-fermi level: unifying driving force for carrier flow.
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-5 Physical meaning of E f For electrons, J e = µ e n de fe dx = qnv e Then: de fe dx = q µ e v e E fe linearly proportional to electron velocity! Similarly for holes: de fh dx = q µ h v h
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-6 Quasi-Fermi levels: effective way to visualize carrier phenomena outside equilibrium in energy band diagram 1. Visualize currents E fe =0 J e =0 E fe 0 J e 0 J e = µ e n de fe dx if n high, E fe small to maintain a certain current level if n low, E fe large to maintain a certain current level Examples: thermal equilibrium under bias n-type E c E F E c E fe E v E v p-type E c E F E v E c E fh E v
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-7 2. Visualize carrier concentrations np = n 2 i exp E fe E fh kt If E fe >E fh np > n 2 i U>0 If E fe <E fh np < n 2 i U<0 If E fe = E fh np = n 2 i U = 0 (carrier conc s in TE) Examples (same semiconductor): E c E F E fe EF E fe Efh E v E fh E fh E fe
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-8 The concept of Quasi-Fermi level hinges on notion of: Quasi-equilibrium: carrier distributions in energy never depart too far from TE in times scales of practical interest. Quasi-equilibrium appropriate if: scattering time dominant device time constant carriers undergo many collisions and attain thermal quasi-equilibrium with the lattice very quickly. In time scales of interest, carrier distribution is close to Maxwellian (i.e., well described by a Fermi level). E c hν hν E v (1) generation (2) thermalization (3) recombination
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-9 Semiconductor physics so far: 2. Continuity Equations Gauss law: Electron current equation: Hole current equation: Total current equation: Carrier dynamics: E = q ɛ (p n + N + D N A ) Je = qn v e drift + qd e n Jh = qp v h drift qd h p Jt = J e + J h dn dt = dp dt = G R Still, can t solve problems like this: hυ S S n -L/2 0 L/2 x Equation system does not capture: impact of carrier movement on carrier concentration (i.e. when carriers move away from a point, their concentration drops!) boundary conditions (surfaces are not infinitely far away)
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-10 Need book-keeping relationships for particles: n F e V rate of increase of number of electrons in V = rate of electron generation in V - rate of electron recombination in V - net flow of electrons leaving V per unit time (n V ) t = G V R V Fe. ds Dividing by V and taking the limit of small V : n t = G R. F e
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-11 In terms of current density: For holes: n t = G R + 1 q. J e p t = G R 1 q. J h
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-12 3. Surface continuity equations Free surface: cannot store carriers: Surface generation - surface recombination= carrier flow out of surface G s R s = F s This equation is axis sensitive. Gs-Rs=Fs Gs-Rs=-Fs Gs-Rs=-Fs Gs-Rs=Fs Fs Fs Fs Fs x x x x Rewrite in terms of current densities normal to surface: U s = 1 q J es = 1 q J hs Always no net current into surface: J s = J es + J hs =0= J es = J hs
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-13 Ohmic contact: provides path for current to flow out of device. For n-type: contact area: A c I J hs J es n Several considerations (for n-type semiconductor): Kirchoff s law: current continuity I = A c J es + J hs = qa c F es F hs Metal in ohmic contact only communicates with majority carriers in semiconductor majority carrier current If there is net generation or recombination at surface minority carrier current (plus additional majority carrier current too) All equations are sign sensitive: U s = 1 q J hs typically, I positive if entering into device (convention) sign of J es and J hs depend of choice of axis in semiconductor At metal-semiconductor interface: n s = p s =0 or S =
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-14 Three possible cases: I I F es I F es F hs n n Fhs F es n ohmic contact without recombination or generation ohmic contact with net recombination ohmic contact with net generation ohmic contact with equilibrium carrier concentrations: U s =0 I = qa c F es ohmic contact with net recombination: U s = F hs I = qa c F es F hs >qa c F es ohmic contact with net generation: U s = F hs I = qa c F es F hs <qa c F es
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-15 Key conclusions Quasi-Fermi levels describe carrier statistics outside equilibrium powerful way to visualize carrier concentrations and currents in energy band diagrams. Quasi-equilibrium: carrier distributions in energy not significantly different from TE in time scales of interest for device operation. Continuity equations: book-keeping relations for carriers. Surfaces cannot store carriers: at all times must have current balance at surface. At free surface: electron and hole currents result from carrier generation or recombination at surface (but net current is zero). At ohmic contact: additional majority carrier current to support terminal current excess carrier concentrations are zero
6.720J/3.43J - Integrated Microelectronic Devices - Fall 2001 Lecture 9-16 Self-study Do examples 4.8, 4.9, 5.1, 5.2. Study integral form of continuity equations and corollary ( 5.1).