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by: Miss Damien Crooks


Miss Damien Crooks
GPA 3.52

E. Schubert

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E. Schubert
Class Notes
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This 25 page Class Notes was uploaded by Miss Damien Crooks on Monday October 19, 2015. The Class Notes belongs to ECSE 2210 at Rensselaer Polytechnic Institute taught by E. Schubert in Fall. Since its upload, it has received 45 views. For similar materials see /class/224773/ecse-2210-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.





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Date Created: 10/19/15
Chapter 33 Relating diffusion coefficients and mobilities In this class we will learn about Constancy of Fermilevel Indicates equilibrium conditions Current ow under thermal equilibrium Net hole current diffusion drift should be zero Net electron current diffusion drift should be zero Einstein relationship relates diffusion coefficient to mobility Introduction to generationrecombination Constancv of Fermi level Under equilibrium conditions dEF dx O the Fermi level inside a material or a group of materials in intimate contact is invariant as a function of position EF appears as a horizontal line on equilibrium energy band diagram If the Fermi level is not constant with position charge transfer will take place resulting in a net current ow in contrast to the assumption of equilibrium conditions Constancy of Fermi level ND Doping concentration varies With position This results in a gradient in carrier concentration EC EF represents the change in carrier concentration With position 0 LI The graph as drawn indicates that there Will be an electric eld This eld causes drift current The concentration gradient gives diffusion if current These two currents exactly cancel each E other so that the net current is zero A nonhorizontal Fermi level means there Will 6 L39 be a continuous movement of carriers from one 13 s1de to the other 1nd1cat1ng current ow Figure 314 against the assumptiOn Current ow under equilibrium conditions The total current under equilibrium conditions is equal to zero Total electron current JH and total hole current Jp must also be zero Why J diff Jnldrift and Jpldiff Jpldrift 11 Under equilibrium conditions both drift and diffusion components will vanish only if E O and dn dx dp dx 0 Even under thermal equilibrium conditions nonuniform doping will give rise to carrier concentration gradient a builtin E fz ela and nonzero current components Einstein relationship Consider a nonuniformly doped semiconductor under equilibrium I l qs eld J Id Net current O Jnldiff dn JndriftJndiff qunnf ana 0 and n nieEFEikT Einstein relationship Manipulation of the above equations leads to Einstein relationship for electrons Einstein relationship for holes If an 1350 cmzVs then Then D11 kTqu11 00256 V x 1350 cmzVs 35 cmZs Always be careful about units Example 31 Exercise 32 Plot electrostatic potential V and E eld versus x for the case shown below O p x 39 H o 51 511 cm gt E electrons o E holes Example 32 Consider the diagram shown in example 31 How can you say that the semiconductor is in equilibrium What is the electron current density Jn and hole current density JP at x i L2 Roughly sketch n and p versus x inside the sample Is there an electron diffusion current at x i L2 If so What is its direction Is there an electron drift current at x i L2 If so What is its direction EXample 3 2 Log scale E Jnldiff Jpldiff Recombination and generation processes Recombination a process whereby electrons and holes are destroyed or annihilated Generation a process whereby electrons and holes are created Under steady state the generation rate and recombination rate exactly cancel each other such that some net steady state carrier concentration results Recombinationgeneration processes RG processes play a major role in shaping the characteristics exhibited by a device Various RG processes Bandtoband recombination generally results in light emission Bandtoband generation direct thermal generation generation by light absorption photogeneration RG center recombination generation Involves an RG center indirect generation recombination Auger recombination Generation Via impact ionization Various R G processes i EC W Photon Light gtltlt o Ev a Band to band recombination EC 39I39herrnal ET gt energy 0 W Ev b R G center recombination 8 Ev c Auger recombination E Thermal c gt energy or Light W o Ev d Band to band generation EC Thermal gt ET energy 0 Ev e R G center generation Ev f Carrier generation via impact ionization Figure 315 Midgap energy levels due to atomic impurities Ec Figure 316 Recombination in Si is mainly Via RG centers introduced by various unwanted impurities 13 Momentum considerations T EV a Direct semiconductor b Indirect semiconductor Figure 317 E vs k diagrams for direct and indirect semiconductors 14 Ek Visualizations of recombination in direct and indirect semiconductors W Photon a Direct semiconductor Figure 318 E N Phonons W m L k b Indirect semiconductor 15 Room temperature properties 0fSi Ge and GaAs Quantity Symbol Si Ge GaAs Unit Crystal structure D D Z Gap Direct D Indirect I I I D Lattice constant a0 543095 564613 56533 A Bandgap energy Eg 112 066 142 eV Intrinsic carrier concentration ni 10 X 1010 20 X 1013 20 X 106 cm 3 Effective DOS at CB edge Nc 7 28 X 1019 10 X 1019 44 X 1017 em 3 Effective DOS at VB edge Nv 7 10 X 1019 60 X 1018 77 X 1018 em 3 Electron mobility u 1500 3900 8500 cmzVs Hole mobility up 450 1900 400 cm2 Vs Electron diffusion constant Dn 39 101 220 cmzs Hole diffusion constant DP 12 49 10 cmzs Electron affinity x 405 40 407 V Minority carrier lifetime 139 10 6 10 6 10 8 s Electron effective mass me 098 me 164 me 0067 me 7 Heavy hole effective mass mhh 049 me 028 me 045 me 7 Relative dielectric constant 8r 119 160 131 7 Refractive index near Eg E 33 40 34 7 Absorption coefficient near Eg 0c 103 103 104 cm 1 D Diamond Z Zincblende W Wurtzite DOS Density of states VB Valence band CB Conduction band The Einstein relation relates the diffusion constant and mobility in a nondegenerately doped semiconductor D ukT e Minority carrier diffusion lengths are given by Ln Dn39cn12 and LP DP tp12 The mobilities and diffusion constants apply to low doping concentrations m 1015 cm 3 As the doping concentration increases mobilities and diffusion constants decrease The minority carrier lifetime 139 applies to doping concentrations of 1018 cm 3 For other doping concentrations the lifetime is given by 139 B 1n p 1 where Bsi m 5 X 10 14 cm3s BGe m 5 X 10 13 cm3s and BGaAs 10 10 cm3s Chapter 35 Continuity equation The continuity equation satis es the condition that particles should be conserved Electrons and holes cannot mysteriously appear or disappear at a given point but must be transported to or created at the given point via some type of carrier action Inside a given volume of a semiconductor 919 thermal at R G 9p ap ap diffusion at at at 11 drift 8 1 others light etc There is a corresponding equation for electrons Continuity equation consider 1D case Jpx J x Ax q Flux Volume A Ax x of holes Area4 a pAAx Jpx JpxAx Am 8 19 at q q at thermalR G llght etc A A a X a x Jx p AxAAxp 61 p qi p 9X at 3123 G J 2 BJ AAx 819 E q 8x a thermal R G light etc ap laJ8p at qax at thermal R G light etc Continuity eqn for holes Continuity eqn for electrons These are general equations for one dimension indicating that particles are conserved Minority carrier diffusion equations Apply the continuity equations to minority carriers with the following assumptions Electric eld E O at the region of analysis Equilibrium minority carrier concentrations are not functions of position ie no i n0x p0 7t p0x Lowlevel injection The dominant RG mechanism is thermal RG process The only external generation process is photo generation Minority carrier diffusion equations Consider electrons for p type and make the following simpli cations a a Jn qunnf an n 3 q n n 8x 8x 8 i 720 I BE 8x 8x 8x E and 2 GL at thermal R G In at light etc 872 a BAn A at at no n at Minority carrier diffusion equations The subscripts refer to type of materials either n type or ptype Why are these called diffusion equations Why are these called minority carrier diffusion equations Example 1 Consider an ntype Si uniformly illuminated such that the excess carrier generation rate is GL eh pairs s cm3 Use MCDE to predict how excess carriers decay after the light is turnedoff tlt O uniform 9 ddx is zero steady state 9 ddt 0 So applying to holes Apt lt O GLTP tgt 0 GL O uniform 9 ddx 0 83A 2 so Apn Apn0 exp 1 2 1p 13 Apt gt O GLTP exp since Ap0 GLTp P Example 2 Consider a uniformly doped Si With ND1015 cm 3 is illuminated such that Apno 1010 cm 3 at x O No light penetrates inside Si Determine Apnx see page 129 in text ApnO Apnx a b Light absorbed at 39 x Oedgeofbar Si bar Solution is r WAN n Where 7 ilk Minority carrier diffusion length In the previous example the exponential falloff in the excess carrier concentration is characterized by a decay length Lp which appears often in semiconductor analysis Lp Dp 1p 2 associated with minority carrier holes in ntype materials Ln D11 Tn12 associated with minority carrier electrons in p type materials Physically L11 and Lp represent the average distance minority carriers can diffuse into a sea of majority carriers before being annihilated What are typical values for Lp and Ln


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