Transport equations [Fonstad, Sze02, Ghione]
Carriers concentrations normally are functions of position and time; in a 1D model Their variations depend on drift currents diffusion currents generatio n and recombination other effects (only at very very high frequencies; we will not consider these)
Concentrations with and without thermal equilibrium We add a subscript to concentrations to specify doping: in thermal equilibrium n doping: n n0 , p n0 p doping: n p0 , p p0 O ut of thermal equilibrium n doping: n n , p n p doping: n p , p p Out of thermal equilibrium, law of mass action does not hold any more!! Excess concentrations If n’ or p’ >0: injection If n’ or p’ <0: depletion Neutrality hypothesis - > n’~p’
Injection E.g. for n doping Low level injection : n~n 0 ~N D n does not change significantly p does change significantly, but in any case p<<n High level injection : n>N D n does change significantly p does change significantly p ~ n equations are much more complex in this case
Conduction/drift current Free carriers are accelerated by the electric field E => electric current For low E , the (average) drift speed is proportional to E with m n / m p electron/hole mobility; for silicon - m n ~ 1000 cm 2 / Vs - m p ~ 400 cm 2 / Vs During their motion, carriers collide with - lattice imperfections - dopants and other (i.e., not Si) atoms (these may be seen as lattice imperfections) - lattice vibration (described by phonons ): atoms are not exactly in their place, so that they may be seen as (moving) lattice imperfections
Effective mass The motion of an electron in a lattice is very complex But the lattice is very regular. If the crystal is perfect, infinite, with fixed atoms, it may be shown that CB electrons - can be modeled as particles which move through the lattice without any collision or deflection ( as they were in free space! ) - but which respond to external forces as if they had a different mass: electron effective mass m e * This is the effective mass model These particles can collide with lattice defects, dopants and impurities, phonons (lattice vibrations) Similarly, for holes in VB: hole effective mass m h * the big dots here are impurities, defects…
Mobility and E For large E , the velocity saturates ( velocity saturation ) velocity or even decreases electrons holes field
Mobility and doping With high doping levels, carriers collide more often => electrons lower mobility holes mobility doping
Mobility and T At high T , lattice vibration are stronger => more collisions with phonons => lower mobility mobility electrons holes temperature
Ohm law at a microscopic level Considering for the moment electrons only: with uniform E this is the drift , or conduction , current Current density is J= s E so that s =qn m n In general and the total drift current is J drift = s E
r=1/s versus doping increase in carriers concentration is larger than reduction in mobility resistivity doping
Diffusion current Normally, gas particles tend to diffuse in all the available space (e.g. a gas in a room): they tend to a uniform concentration The same is done by carriers in semiconductors The intensity of motion is proportional to the gradient of the concentration It holds concentration diffusion with diffusion coefficients D n , D p >0. Pay attention to the signs!
Diffusion coefficients It normally holds (Einstein equation) with “thermal voltage”, ~25 mV at 300 K In silicon D n ~25 cm 2 /s D p ~10 cm 2 /s
The total current is the sum of all the contributions ( drift-diffusion model ): dr dr i.e. J= qD n dn/dx + qn m n E – qD p dp/dx + qp m p E
Generation and recombination (GR processes) Generation: an electron jumps from VB to CB, so that we also have a hole in VB Recombination: an electron moves from CB to VB, where it recombines with a hole (they both disappear) Direct processes: direct jump from VB to CB or vice-versa Indirect processes: jump from VB to CB or vice- versa “assisted” by recombination centers, or traps , (at energy level E t ), in the forbidden gap
Generation and recombination (first approximation) Generation: either thermal (a phonon is absorbed) or optical (a photon is absorbed) Similarly for recombination
Direct and indirect gap Up to now we had considered the energy of the electrons, but this is only a part of the picture: they also have momentum The energy band structure plots energy versus some directions of the wave vector k k is proportional to total momentum : D p tot =(h/2 p ) D k with h Planck constant, VB is more complicated that CB, h ~ 6.6 . 10 -34 Js three branches do exist In any transition, k must be preserved
Direct gap semiconductor A direct energy gap semiconductor, such as GaAs, has the minimum of CB at the same abscissa as the maximum of VB So, during the transition, there is no change in k , which is preserved This is a band-to-band process
Direct gap semiconductor: photon absorption If the photon energy is higher than E G :
Indirect gap semiconductor Photons have neglectable momentum; phonons normally have neglectable energy For an indirect energy gap material both a photon and a phonon are needed or generated Much lower probability of photon emission/absorption wrt direct gap
Band-to-bound events In an indirect gap material, normally G-R processes occur via more (e.g. 2) steps, relying on recombination centers (traps): Shockley-Read- Hall (SRH) recombination processes These are non radiative transitions
Absorption coefficients In parentheses: the cutoff wavelengths l c a -Si is amorphous silicon
Neutral regions At thermal equilibrium Out of thermal equilibrium, for the quasi-neutrality we have 1/ q so that
GR processes Generation rate G : number of carriers generated per unit of volume and time Recombination rate R : number of carriers recombined per unit of volume and time Net recombination rate : Of course, at thermal equilibrium Average lifetime approximation : with light: U n ~n ’/ t n -G light U p ~p ’/ t p -G light with t n , t p , electron/holes excess carriers average lifetimes (indeed, for minority carriers only it is the average lifetime ) Of course U n =U p because electrons and holes are generated in pairs; for the quasi-neutrality n’=p’, then t n = t p
Continuity equations We’ll get the time evolution of n and p basing on the principle of preservation of charge Let us consider the electrons which travel through the volume dV=Adx The variation of the number of electrons in dV will be There are 4 contributions: electrons entering or exiting at x (a) electrons entering or exiting at x+dx (b) electrons generated (c) electrons recombined (d) attention to signs!
Continuity equations With and with dx -> 0, we get the continuity equation for electrons Similarly, for holes (note: a sign is different)
Semiconductor mathematical model In the 1D case (the 3D case will have gradients, not derivatives ) In the average lifetime approximation: Remember that for the currents we had We need another equation: the Poisson equation with r net charge density
Semiconductor mathematical model Some approximations are needed to proceed analytically constant carrier mobility full ionisation of the dopants then we get with
In low injection regime the drift current of minority carriers is neglectable (because their concentration is << than that of the majority ones, and m are similar) so that, for minority carriers only , we can write the diffusion equations
In low injection and stationary regime When we have For minority carriers only and uniform doping n’=Ae x/Ln +Be -x/Ln or p’=Ae x/Lp +Be -x/Lp with L n =(D n t n ) 1/2 and L p =(D p t p ) 1/2 diffusion lengths , and A and B suitable integration constants
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