Recitation 4 Electrostatic Potential & Carrier Concentration 6.012 Spring 2009 Recitation 4: Electrostatic Potential & Carrier Concentration Yesterday in lecture, we learned that under T.E. (thermal equilibrium), there is a funda-mental relationship between the electrostatic potential φ(x), at one location (x) within thesemiconductor and the carrier concentration at that location.φ(x)=kTlnno(x)= −kTlnpo(x)qniqniBy the Boltzman relationship (or the 60 mV rule),φ(x) = (60 mV) lnno(x)= −(60 mV) lnpo(x)niniHow did this relation come about?Revisit Thermal Equilibrium1. Under T.E. can we have electrostatic field (or voltage) within the semiconductor?Yes. (we do not have “external” energy source). There can be static electrostatic fieldinside the semiconductor generated by “space” charges.2. Under T.E. can we have an overall current? No. That will give rise to charge piling up. Some Fundamentals about ElectrostaticsRelationship between electrostatic potential φ(x), electric field E(x), and (space) chargedensity ρ(x):1. First, the charge density we are talking about here is the “Net” charge density, wecall space charge density.in n-type material: ρ(x)=q(Nd(x) − n(x))p-type material: ρ(x)=q(p(x) − Na(x))(Nd,Naare space charges. ρ(x) is the extra which can not be compensated by e−orhole free charges) space charge density.→dE ρ2. = ( = electric permitivity Farad/cm). In other words,dx x1E(x) − E(0) =0e(x) dx1 Recitation 4 Electrostatic Potential & Carrier Concentration 6.012 Spring 2009 dφAlso, = −E(x). In other words:dxxφ(x) − φ(0) = − E(x) dx0The two equations above can be combined to give the following relation:d2φ(x) dE(x) ρ(x)dx2= −dx= −Boltzman RelationBasically the Boltzman relationship exists due to thermal equilibrium. Under T.E., forn-type:Jn=0dnoBut Jn= q · no· E + q · Dn·dx=0μn·dφ dno=−q · no· μn·dx−q · Dn·dxμndφ=1 dnoDndx nodxqdφ=d(ln(no))k T dx dx· Integrate:q(φ − φref) = ln(no) − ln(no,ref)=lnnokT no,ref·Similarly, for p-type:Jp=0dpoBut Jp= q · po· μp· E − q · Dp·dx=0dφ dpo−q · po· μp·dx= q · Dp·dxμpdφ=1 dpo−Dpdx podxqdφ d(ln(po))−kTdx=dx ·Integrate: −q(φ − φref) = ln(po) − ln(po,ref)=lnpok · Tpo,refSet φref=0atno, ref= po, ref= ni. Then:−kqTφ =lnpo· ni kT poφ = −q·lnniqφor po= nie−kT·2 Recitation 4 Electrostatic Potential & Carrier Concentration 6.012 Spring 2009 ExampleNow let us look at a particular example. We have a doping profile Nd(x)=Ndo+ΔNd(1 −e−fracxLc). Ndo=1016cm−3, ΔNd=9× 1016cm−3, Lc=10μm. We would like to know:1. What is the electrostatic profile φ(x)?2. How about electric field E(x)?3. Space charge density ρ(x)?First, we have φ(x) vs. no(x),po(x) from the Boltzman relationships. If we know no(x), or po, we can find φ(x). But does no(x)=Nd(x)? In reality, it should not, if no(x)=Nd(x), we will have a net current due to diffusion Not T.E. anymore. To obtain an accurate solution, we have: Jn= qno· μn· E + qDn·dno=0(Nddoping, electron majority carrier, only consider Jnhere.)··dxdE q=(Nd− no)dx SiWith these two, we get:Dn1 dnokTd2(ln no)1E = −μnnodx=⇒q·dx2=Si(no(x) − Nd)But very hard to solve no(x). In most cases, an analytical solution is impossible. Can wedo something simpler?We make approximations! The first type of scenario is no(x) Nd(“quasi-neutrality”). If−xwe assume no(x) Nd= Ndo+ΔNd(1 − eLc):−x Define a Ndo+ΔNd(1 − eLc) φ(x)=kq· Tlnnon(ix) kq· Tlnnai xE(x)=−dφdx(x) −kq· Ta1ΔNdL1ce−Lc⎛ ⎞ x−2x11ρ(x) = SidE(x) dx Si11 −kT ⎜⎝ ⎟⎠ ΔNd2LcLc· ΔNd+ e e 2L2cL2cqa a 1 ΔNd(Ndo+ΔNd)kT xe−· SiLcqa2Lc23 Recitation 4 Electrostatic Potential & Carrier Concentration 6.012 Spring 2009 To satisfy quasi-neutrality, we need: no(x) − Nd(x)Nd(x)ρ(x) qx 1, we know (no(x) − Nd(x)) = −= q · ρ(x)Nd(x) Sino(x) − Nd(x)Nd(x)kTΔNd(Ndo+ΔNd)1·−Lce q 2a3L2cxΔNd(Ndo+ΔNd)ΔNd(Ndo+ΔNd)e−Lc 1 (for x>0), anda<(Ndo)33no(x) − Nd(x)Nd(x)<SikTΔNd(Ndo+ΔNd)1·q2(Ndo)3L2c=1.46 × 10−4 1 Therefore, our quasi-neutrality is valid. This quasi-neutrality satisfies when doping profileis gradual. If we have time, we can verify Jndiff= qDndno(x)= Jndrift= qμnno(x)E(x)·dxusing the above equations.4MIT OpenCourseWarehttp://ocw.mit.edu 6.012 Microelectronic Devices and CircuitsSpring 2009 For information about citing these materials or our Terms of Use, visit:
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