L8: Non-Uniform Doping Distribution

6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-1 Lecture 8 - Carrier Drift and Diffusion (cont.), Carrier Flow February 21, 2007 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 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 8-2 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 equilib­ rium? • In the presence of carrier flow in the bulk of a semiconductor, how does one formulate bookkeeping relationships for carriers? • How about at surfaces? 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 8-3 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 1. Quasi-Fermi levels � Interested in energy band diagram representations of complex sit­ uations in semiconductors outside thermal equilibrium. � In TE, Fermi level makes statement about energy distribution of carriers in bands ⇒ EF relates no with Nc and po with Nv : no = Nc F1/2( EF − Ec ) kT po = Nv F1/2 ( Ev − EF ) kT Outside TE, EF cannot be used. Define two ”quasi-Fermi levels” such that: n = NcF1/2 ( Efe − Ec ) kT p = Nv F1/2 ( Ev − Efh ) kT Under Maxwell-Boltzmann statistics (n  Nc, p  Nv ): n = Nc exp Efe − Ec kT p = Nv exp Ev − Efh kT What are quasi-Fermi levels good for? 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 8-4 � Take derivative of n = f (Efe ) with respect to x: n = Nc exp Efe − Ec kT n dEfe dEc n dEfe q dn = ( − )= − nE dx kT dx dx kT dx kT Plug into current equation: Je = qμenE + qDe dn dx To get: Je = μe n dEfe dx Jh = μh p dEfh dx Similarly for holes: Gradient of quasi-Fermi level: unifying driving force for carrier flow. 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 8-5 � Physical meaning of ∇Ef For electrons, Je = μe n dEfe = −qnve dx Then: dEfe q = − ve dx μe ∇Efe linearly proportional to electron velocity! Similarly for holes: dEfh q = vh dx μh 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 8-6 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 � Quasi-Fermi levels: effective way to visualize carrier phenomena outside equilibrium in energy band diagram 1. Visualize carrier concentrations and net recombination np = n2i exp Efe − Efh kT • If Efe > Efh ⇒ np > n2i ⇒ U > 0 • If Efe < Efh ⇒ np < n2i ⇒ U < 0 • If Efe = Efh ⇒ np = n2i ⇒ U = 0 (carrier conc’s in TE) Examples (same semiconductor): Ec EF Efe EF Efh Ev Efe Efh Efh Efe But can’t visualize Gext. 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 8-7 2. Visualize currents Je = μe n dEfe dx • ∇Efe = 0 ⇒ Je = 0  0 ⇒ Je = 0 • ∇Efe = • if n high, ∇Efe small to maintain a certain current level • if n low, ∇Efe large to maintain a certain current level Examples: thermal equilibrium n-type Ec EF Ev Ec p-type EF Ev under bias Ec Efe Ev Ec Efh Ev 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 8-8 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 � 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 and among themselves very quickly. In time scales of interest, carrier distribution is close to Maxwellian (i.e., well described by a Fermi level). Ec hν hν Ev (1) generation (2) thermalization (3) recombination 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 8-9 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 2. Continuity Equations Semiconductor physics so far:  · E = q (p − n + ND+ − NA− ) ∇  Gauss’ law:  Electron current equation: Je = −qn ve drif t + qDe ∇n Hole current equation:  Jh = qpvhdrif t − qDh∇p Total current equation: Jt = Je + Jh Carrier dynamics: 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) 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 8-10 Need ”book-keeping relationships” for particles: n Fe ΔV rate of increase of number of electrons in ΔV = rate of electron generation in ΔV - rate of electron recombination in ΔV - net flow of electrons leaving ΔV per unit time � ∂(nΔV )  = GΔV − RΔV − Fe .dS ∂t Dividing by ΔV and taking the limit of small ΔV : ∂n  Fe = G − R − ∇. ∂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]. 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-11 In terms of current density: ∂n 1  = G − R + ∇.J e ∂t q For holes: ∂p 1  = G − R − ∇.J h ∂t q 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 8-12 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 3. Surface continuity equations � ”Free” surface: cannot ”store” carriers: Surface generation - surface recombination= carrier flow out of surface |Gs − Rs| = |Fs| This equation is axis sensitive. Gs-Rs=Fs Gs-Rs=-Fs Fs>0 Fs<0 x Gs-Rs=-Fs Gs-Rs=Fs Fs<0 x x Fs>0 x Rewrite in terms of current densities normal to surface: 1 1 |Us| = |Jes| = |Jhs| q q Always, no net current into surface: Js = Jes + Jhs = 0 =⇒ Jes = −Jhs 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 8-13 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 � Ohmic contact: provides path for current to flow out of device. contact area: Ac For n-type: Jes I Jhs Kirchoff’s law demands current continuity at metal-semiconductor interface: |I| = Ac|Jes + Jhs| = qAc|Fes − Fhs | Equation is sign sensitive: • IEEE convention: I entering into device is positive • sign of Jes and Jhs depend of choice of axis in semiconductor Key result from metal-semiconductor junction: Metal in ohmic con­ tact only communicates with majority carriers in semiconductor ⇒ majority carrier current • in n-type material, current supported by electrons • in p-type material, current supported by holes I I Jes n Jhs p 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 8-14 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 � If there is net generation or recombination at surface right below ohmic contact ⇒ minority carrier current in addition to majority carrier current Three possible cases (examples for n-type): I Fes I Fes I Fes Fhs Fhs n n ohmic contact without recombination or generation ohmic contact with net recombination n ohmic contact with net generation • ohmic contact with equilibrium carrier concentrations: Us = 0 |I| = qAc|Fes| • ohmic contact with net recombination: Us = |Fhs | |I| = qAc|Fes − Fhs| > qAc|Fes| • ohmic contact with net generation: Us = −|Fhs| |I| = qAc|Fes − Fhs| < qAc|Fes| Important boundary condition (will understand soon): at metalsemiconductor interface: ns = ps = 0 or 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 8-15 Key conclusions • Quasi-Fermi levels describe carrier statistics outside equilib­ rium ⇒ powerful way to visualize carrier concentrations and currents in energy band diagrams. • Quasi-equilibrium: carrier distributions in energy not signif­ icantly 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 cur­ rent – excess carrier concentrations are zero 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 8-16 Self-study • Do examples 4.8, 4.9, 5.1, 5.2. • Study integral form of continuity equations and corollary (§5.1). 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].