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Transport equations [Fonstad, Sze02, Ghione] Carriers - - PowerPoint PPT Presentation
Transport equations [Fonstad, Sze02, Ghione] Carriers - - PowerPoint PPT Presentation
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
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Concentrations with and without thermal equilibrium
We add a subscript to concentrations to specify doping:
in thermal equilibrium
n doping: nn0, pn0 p doping: np0, pp0
Out of thermal equilibrium
n doping: nn, pn p doping: np, pp
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’
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Injection
E.g. for n doping Low level injection: n~n0~ND
n does not change significantly p does change significantly, but in any case p<<n
High level injection: n>ND
n does change significantly p does change significantly p ~ n equations are much more complex in this case
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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 mn/mp electron/hole mobility; for silicon
- mn ~ 1000 cm2/ Vs
- mp ~ 400 cm2/ 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
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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 me*
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 mh* the big dots here are impurities, defects…
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Mobility and E
For large E,
the velocity saturates (velocity saturation)
- r even
decreases
electrons holes velocity field
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Mobility and doping
With high
doping levels, carriers collide more often => lower mobility
mobility doping electrons holes
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Mobility and T
At high T, lattice
vibration are stronger => more collisions with phonons => lower mobility
mobility electrons holes temperature
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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=sE
so that s=qnmn In general and the total drift current is Jdrift=sE
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r=1/s versus doping
increase in carriers concentration is larger than reduction in mobility
resistivity doping
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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 with diffusion coefficients Dn, Dp >0. Pay attention to the signs! concentration diffusion
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Diffusion coefficients
It normally holds (Einstein equation)
with “thermal voltage”, ~25 mV at 300 K
In silicon
Dn ~25 cm2/s Dp ~10 cm2/s
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The total current
is the sum of all the contributions (drift-diffusion model): i.e.
J= qDndn/dx + qnmnE – qDpdp/dx + qpmpE
dr dr
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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 Et), in the forbidden gap
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Generation and recombination (first approximation)
Generation: either thermal (a phonon is absorbed) or optical (a
photon is absorbed)
Similarly for recombination
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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: Dptot=(h/2p)Dk with h Planck constant, h ~ 6.6 . 10-34 Js
In any transition, k must be
preserved
VB is more complicated that CB, three branches do exist
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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
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Direct gap semiconductor: photon absorption
If the photon energy is higher
than EG:
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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
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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
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Absorption coefficients
In parentheses: the
cutoff wavelengths lc
a-Si is amorphous
silicon
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Neutral regions
At thermal equilibrium Out of thermal equilibrium, for the quasi-neutrality we have so that
1/q
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GR processes
Generation rate G: number of carriers generated per unit of
volume and time
Recombination rate R: number of carriers recombined per unit
- f volume and time
Net recombination rate: Of course, at thermal equilibrium Average lifetime approximation:
with tn, tp, electron/holes excess carriers average lifetimes (indeed, for minority carriers only it is the average lifetime)
Of course Un=Up because electrons and holes are generated in
pairs; for the quasi-neutrality n’=p’, then tn=tp
with light: Un~n’/tn-Glight Up~p’/tp-Glight
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Continuity equations
We’ll get the time evolution of n and p basing
- n 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
attention to signs! 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)
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Continuity equations
With and with dx -> 0, we get the continuity equation for
electrons
Similarly, for holes (note: a sign is different)
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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 In the 1D case (the 3D case will have gradients, not derivatives)
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Semiconductor mathematical model
Some approximations are needed to proceed analytically
constant carrier mobility full ionisation of the dopants
then we get
with
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In low injection regime
the drift current of minority carriers is neglectable
(because their concentration is << than that of the majority
- nes, and m are similar)
so that, for minority carriers only, we can write the diffusion
equations
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