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ECSE-6230 Semiconductor Devices and Models I Lecture 7

ECSE-6230 Semiconductor Devices and Models I Lecture 7. Prof. Shayla Sawyer Bldg. CII, Rooms 8225 Rensselaer Polytechnic Institute Troy, NY 12180-3590 Tel. (518)276-2164 Fax. (518)276-2990 e-mail: sawyes@rpi.edu. June 2, 2014. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html . 1.

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ECSE-6230 Semiconductor Devices and Models I Lecture 7

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  1. ECSE-6230Semiconductor Devices and Models ILecture 7 Prof. Shayla Sawyer Bldg. CII, Rooms 8225 Rensselaer Polytechnic Institute Troy, NY 12180-3590 Tel. (518)276-2164 Fax. (518)276-2990 e-mail: sawyes@rpi.edu June 2, 2014 sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html 1

  2. Carrier Motion Drift and Mobility Impact Ionization Diffusion Basic Equations Continuity Equations sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  3. Drift and Mobility μ [cm2/Vsec] is the “mobility” of the semiconductor and measures the ease with which carriers can move through the crystal μn ~ 1360 cm2/Vsec for Silicon @ 300K μp ~ 460 cm2/Vsec for Silicon @ 300K μn ~ 8000 cm2/Vsec for GaAs @ 300K μp ~ 400 cm2/Vsec for GaAs @ 300K Question: How does the donor concentration affect the temperature dependence of mobility? sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  4. Drift and Mobility • Electron and hole mobility on doping concentration in silicon sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  5. Drift: High Field Transport At low electric field, <vd >   The proportionality constant  that is independent of the electric field f ( ) At sufficiently high fields, <vd > is no longer prop. to  (nonlinearities in mobility) <vd >  vsat Larger fields, impact ionization occurs sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  6. Impact Ionization At still higher field, impact ionization occurs e-  e- + e- + h+ Auger recombination Carriers gain enough energy to excite electron hole pairs by this process http://www.iue.tuwien.ac.at/phd/entner/img263.png sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  7. Impact Ionization Energetic carriers, when colliding with the lattice, creates e- + h+ pairs. Generation Rate G = n n vn + p n vp n, p - ionization rates Si - n > p Ge - p > n = f () Ionization rate increases or decreases with increasing bandgap? What important device parameter does this affect? sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  8. Carrier Transport - Diffusion Diffusion Do particles have to be charged to diffuse? What is the mechanism behind carrier motion? Carrier transport due to a concentration gradient. Ex. Perfume in air, ink in water pairs. Particles or scent or color spread until the concentration becomes uniform. In metals, the carrier mobility is so high that diffusion can be ignored. In semiconductors, diffusion, the slower of the two, can be the major carrier transport mechanism. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  9. Diffusion - One Dimensional Analysis The flow or flux of carriers for electron (for example) is governed by Fick’s law Change in concentration over time is determined in part by a proportionality constant Dn is the proportionality constant called the Diffusion coefficient or diffusivity Flux of carriers constitutes a diffusion current given by To find the relationship between mobility and the diffusion current consider an n-type semiconductor non-uniformly doped What is the net current inside the semiconductor? Why? x sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  10. Diffusion - One Dimensional Analysis E-field is created by the non-uniform doping Take carrier concentration equation and substitute to find Einstein Relation (diffusion and mobility) Diffusion length And EF is constant in equil. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  11. Other methods of conduction Thermionic Emission Majority carrier current that is always associated with a potential barrier Critical parameter is the barrier height, not the shape Common devices? Tunneling Wavefunction does not terminate abruptly on a wall of finite potential height barrier Ec Metal Ev n-Semiconductor sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  12. Electron and Hole Current Densities Hole Flux p (x) = - Dp dp/dx (1-dim.) (or - Dp p for 3-dim.) Diffusion Current Densities Jn (diffusion) = (-q) n (x) = - (-q) Dn dn/dx (or - (-q) Dn n for 3-dim.) Jp (diffusion) = (+q) p (x) = - (+q) Dp dp/dx (or - (+q) Dp p for 3-dim.) Drift Current Densities Jn (drift) = +q n n(x) (x) Jp (drift) = +q p p(x) (x) Conductivity and Resistivity n = q n n and n = 1 / n p = q p p and p = 1 / p  = n + p = q n n0 + q p p0 For n-type semi.,   n = q n n0 since n0 >> p0 sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  13. Basic Equations Total Current Densities (using the drift + diffusion formalism) Jn (total) = Jn (drift) + Jn (diffusion) Jp (total) = Jp (drift) + Jp (diffusion) J (total) = Jn (total) + Jp (total) In three dimensions, Jn = q n n  + q Dn  n Jp = q p p  - q Dp  p In one dimension, Jn = q n n  + q Dn dn/dx Jp = q p p  - q Dp dp/dx sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  14. Continuity Equations Rate Equation for holes Rate of hole conc. = Increase of hole conc. - recomb. + generation built-up in Area · x per unit time rate rate • Previous equations are used for steady state conditions • Continuity equations deal with time dependent phenomena such as low level injection, generation and recombination • The net change of carrier concentration is the difference between generation and recombination plus the net current flowing in and out o the region of interest

  15. Continuity Equations Rate Equation for electrons Rate of elec. conc. = Increase of elec. conc. - recomb. + generation built-up in Area · x per unit time rate rate Minority Carrier Diffusion Equations (low level injection) • Assumptions: • One dimensional • Restricted to minority carriers • E=0 in region of semiconductor • Minority concentration not f(x) • Low level injection prevails • No other processes like photogeneration pn is the hole (minority carrier) concentration in n-type material np is the electron (minority carrier) concentration in p-type material

  16. Derivation: Minority Carrier Diffusion Deriving the minority carrier diffusion equations from general continuity equations Simplifying JN for electrons 0 Continuity Equation for electrons:

  17. Simplifications (Special Cases) • Steady state: • No diffusion current: • No R-G: • No light:

  18. Continuity Equations (Sample 1) A uniformly donor-doped silicon wafer maintained at room temperature is suddenly illuminated with light at time t=0. Assuming ND=1015, τp =10-6 sec, and a light creation of 1017 electrons and holes per cm3-sec throughout the semiconductor, determine Δpn(t) for t > 0. Step 1: Given information Semiconductor: Silicon T: Room Temperature Donor doping: ND=1015 /cm3same everywhere GL= 1017 cm3-sec Step 2: Characterize System under Equilibrium Silicon at room temperature ni=1010 /cm3 ND>>ni, no = ND = 1015 /cm3 Law of mass action po=ni2/ND = 105 /cm3

  19. Continuity Equations (Sample 1) A uniformly donor-doped silicon wafer maintained at room temperature is suddenly illuminated with light at time t=0. Assuming ND=1015, τp =10-6 sec, and a light creation of 1017 electrons and holes per cm3-sec throughout the semiconductor, determine Δpn(t) for t > 0. Step 3: Analyze Qualitatively Prior to t=0, equilibrium conditions prevail and Δpn=0 Light creates added electron and holes and Δpn will increase The thermal recombination rate will enhance due to excess carriers Photogenerated holes will be eliminated per second by thermal recombination Eventually carriers annihilated will balance the carrier created by the light (steady state) Step 4: Perform quantitative analysis Use minority carrier continuity equation

  20. Continuity Equations (Sample 1) A uniformly donor-doped silicon wafer maintained at room temperature is suddenly illuminated with light at time t=0. Assuming ND=1015, τp =10-6 sec, and a light creation of 1017 electrons and holes per cm3-sec throughout the semiconductor, determine Δpn(t) for t > 0. Step 4: Perform quantitative analysis Use minority carrier continuity equation Simplifies to General solution from Diff. Eq. Boundary condition yields solution

  21. Assume low-level injection and uniformly doped semiconductor ( = 0). Semi-infinite long bar: pn(x=0) = pn(0) = constant << nn0, pn(x) = pn0. where (Minority carrier diffusion length) Continuity Equations (Sample 2)One-Sided Injection sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  22. Assume low-level injection and uniformly doped semiconductor ( = 0). Semi-infinite long bar: General solution Continuity Equations (Sample 2)One-Sided Injection sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

  23. (b) Short bar Boundary Conditions: pn(x=0) = pn(0) = constant << nn0, pn(x=W) = pn0. Continuity Equations (Sample 2)One-Sided Injection Current density at x=W is given by: Related to current gain for BJTs sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

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