The pn junction [Fonstad, Ghione] Band diagram On the vertical - PowerPoint PPT Presentation

The pn junction [Fonstad, Ghione]

Band diagram  On the vertical axis: potential energy of the electrons  On the horizontal axis: now there is nothing: later we’ll put the position  q F s : work function ( F s : extraction potential), depends on doping (which moves E F )  q c s : electron affinity (~ 4 eV for Si)  E 0 : vacuum level  Now we’ll consider the band diagrams in thermal equilibrium only

The same material, with different doping  If the two samples are isolated from each other, we have ->  note: the electron affinity does not change  We’ll first assume uniform doping in each sample  The work function is larger for the p side p side n side

The same material, with different doping  We have a junction !  How to build the band diagram:  the Fermi level is constant (if not, electrons would move from zones with higher f(E) to zones with lower f(E), due to the different probability of occupation of the states)  electron affinity and forbidden gap width are constant (depend on the material only)  far from the junction, the band structure is that of the isolated material  the vacuum level is continuous

Band diagram in contact  1 st approx. drawing... complete drawing E 0 is now the energy of the electron just outside the material

Depletion region  Where bands bend, there are two depletion layers, where r ~ N D or r ~ N A  These are space-charge regions => E => potential  The contact potential, or built-in potential, is  in order to compute the width of the depletion region, we’ll compute with ( global neutrality)

The two components of the built-in potential V i

Depletion region and potentials  Two integration steps: - [ div E = r/e 0 ] constant r => linear E (x) - [ E =-dv/dx ] linear E (x) -> parabolic j (x) j: electric potential – conventionally referred to E Fi U : potential energy of the electrons at E Fi (well, they do not exist...) U=-q j U is also (apart from an offset) the potential energy of the electrons e.g. at the bottom of CB

Depletion region  So we get  so that  And we get with N A x p =N D x n and N eq =(N A N D )/(N A +N D )  So, higher doping => thinner depletion region

Out of thermal equilibrium  Reverse bias....................................... forward bias

Out of thermal equilibrium

<-- minus! Electrons potential energy and concentrations  E and j in the depletion region ->  E => we have to spend energy to move electrons from right to left  i.e. their potential energy U increases when moving from right to left  U has the same behaviour of E Fi (and the same of j but with opposite sign)  So from [sl. 110/24] we can write

The current  Let us consider a 1D model (i.e., the section is constant along x )  Hypothesis:  low injection  n, p neglectable in the depletion layers  the sides of the junctions ( w p , w n ) are much longer than the diffusion lengths L n =(D n t n ) 1/2 and L p =(D p t p ) 1/2  In any section dr dr  low injection ->  We need to compute the concentrations of minority carriers

Junction law  The solutions of the diffusion equations are here  We need to know n’ p (-x p ) and p’ n (x n )  From we get

Junction law  From we get  At equilibrium ( V =0)  Out of equilibrium (but with low injection): junction law ->

Out of equilibrium  Carriers profile ->

The current  E.g. at x=x n  J tot (x n )=J p (x n )+J n (x n )=J p, diff (x n )+J n (x n )  Assuming that recombination is neglectable in the space charge region , J n (x n )=J n (-x p )=J n, diff (-x p )  so that J tot (x n )=J p, diff (x n )+J n, diff (-x p ) p n n p

The current  And we finally get with

Effect of temperature  Higher T implies higher current, due to changes in - increase in carrier concentration - V T - changes in D n/p and L n/p  At constant voltage, I doubles for an increase of 10 o C  At constant current, V decreases of about 2.5 mV/ o C

Depletion (or transition) capacitance Due to variations in the width of the depletion layer -> variations of the charge

Diffusion capacitance  Due to variations in the profile of the carriers in the proximity of the depletion layer  The “extra” charge in the profile of the carriers concentrations, close to the space charge region, is proportional to J :  The corresponding capacitive effect is also proportional to J :

Diodes and switching  [See spice Simulation]

Large signal model  So the large signal model of a diode is ->  All these 3 parameters (1 conductance and 2 capacitors) change with bias  In inverse bias C j prevails on C d  the opposite in direct bias

Small signal model  By linearising the model around the bias point we obtain the small signal model ->  g d0 =I/ h V T where I is the bias current  It is a linear model!

Zener diodes  All diodes are subject to breakdown for sufficiently high inverse bias  Zener diodes are designed to work in this condition, normally as voltage references  Two physical mechanisms: tunnel and avalanche  Higher doping => thinner depletion region => higher E => easier breakdown => lower V z  In order to have higher V z : lower doping (at least on one side)

Tunnel and avalanche breakdown  Tunnel: for V Z <6V, high doping; V Z decreases with T  Avalanche: for V Z >6V, lower doping; V Z increases with T

Photodiodes • If suitable photons reach the junction, they I may generate electron-hole couples V • minority (and also majority) concentrations will increase • In reverse bias, current will be larger due to an extra photo-generated current I S proportional to the # of photons • and we have I = -I S + I 0 (exp V/V T -1) • Photodiodes can be used as light detectors I’ • Considering the opposite current ( I’= -I ) V

Photodiodes as power generators I’ V • So we have an open circuit voltage I’ (photovoltaic effect) V • and a short circuit current I’ V • And we can also extract power! • Photovoltaic cells are diodes with very large area (to get many I’ V photons), optimized for power generation

LEDs

LEDs [Please note: the right drawing is deceitful, electron and holes are on the lines, not above or below the lines]

Heterojunctions  E G ~1.4 eV for GaAs  E G ~1.7 eV for AlGaAs Isolated semiconductors 1. Ideally, immediately after junction 2. (1) creation At thermal equilibrium 3. (2) (3)