42. Diodes under Bias
Qualitatively the function of the diode under bias can be described best in terms of energy band levels.
In the intrinsic case, the diffusion current is matched by a drift current in the opposite direction, which is driven by the contact potential.
In biased conditions, the driving voltage is combined with the contact potential. In the forward biased condition it acts to minimise the contact potential or even reverse it. In the reverse bias situation it increases the effective contact potential.
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43. Diodes under Bias
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44. Diodes under Bias
Diffusion Current
The effect of the reduced energy barrier in the forward bias condition is that the diffusion current increases as more majority holes and majority electrons have the energy to jump the now reduced energy barrier. In forward bias conditions, the diffusion current can be quite large.
In the reverse bias situation, there is a higher energy wall and there are almost no particles with sufficient energy to jump the barrier and hence the diffusion current is almost zero.
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45. Diodes under Bias Drift Current
The drift current however is relatively insensitive to the height of the energy barrier as it is more dependent on the quantity of available electrons and holes to acts as carriers.
Drift current across the junction is dependent on electrons falling down the energy bands from the P side to the N, and similarly for holes. The problem is the quantity of minority carriers available for the drift current to use, for example electrons on the P side, which depends primarily on thermal generation.
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46. Diodes under Bias Diffusion Drift
At zero voltage, the drift and diffusion currents equal zero, the unbiased case.
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47. Carrier Injection
The equilibrium level for minority carriers on each side of a pn junction varies with the applied bias because of variations in diffusion current across the junction while drift remains about the same.
The hole concentration on either side is given by the following equation in unbiased cases
Under bias conditions, this changes to
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48. Carrier Injection
For low bias levels we can ignore changes in the majority carrier concentrations.
The increase in majority carriers is the same as the minority carriers, in order to maintain charge neutrality.
However the relative increase of the minor carriers with respect to their equilibrium values is significantly greater.
This is demonstrated by the following expression where  is same order of magnitude as a
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49. Carrier Injection
The ratio of increase in minority hole concentration is given by
Thus there is a significant increase in minority hole concentration with increasing forward bias voltage.
This is also true for minority electron concentrations on the P side, near the junction.
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50. Carrier Injection The increase in minority carriers is given by
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51. Diode Diffusion Current
Now as we know the diffusion current is going to be a dominant current mechanism in the diode under bias conditions, we’ll proceed to ascertain the diffusion current.
The first stage is to calculate the distribution of excess carriers on each side and hence work out the current using the equations from earlier.
From our previous work we expect that as the excess carriers move away from the junction they’ll recombine with the opposing type, for example holes will recombine with elections.
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52. Diode Diffusion Current
The diffusion equation can give the distribution but if the length of the material on each side of the junction is long compared to the diffusion length, then the decay rate is exponential.
We’ll use new reference co-ordinates, measuring from the edge of the depletion bands in each regard, where xn is now the distance from the depletion band in the n-doped material
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53. Schockley Ideal Diode Approximation
In the following analysis, we will assume that there is no recombination of the carriers in the depletion region. This is the Schockley Ideal Diode Approximation. It’s premise is that there are no un-recombined carriers in the depletion region left.
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54. Diode Diffusion Current
From before, we have the equations
So combining with the expression for the increase in carriers, we can get their distribution as they recombine away from the junction.
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55. Diode Diffusion Current
The diffusion current at any point xn in the n-doped material can then be calculated from the diffusion equation current. so
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56. Diode Diffusion Current
A similar equation can be found for the electron current in the P type material
(minus sign is due to difference in carrier charge)
Assuming a junction of cross-sectional area A the junction diffusion current is given by
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57. Diode Equation
So quickly replacing the expression for p and n, and evaluating at the edge of the depletion region.
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58. Diode Equation
So subtracting In from Ip (due to charge direction) will give the total diffusion current across the junction.
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59. Diode Equation This is the familiar diode equation.
Some important points worth noting
The equation makes no statement on the bias voltage, so this equation is applicable for diffusion current for forward and reverse bias. It will be very small in reverse bias.
If the bias voltage is more than a few kT negative, then Io must equal to the total reverse saturation current.
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60. A Quick Way to the Diode Equation
A quicker instructive way of developing the hole current is to think of it as the current is the current required to maintain the distribution of excess carriers that was predicted by the Poisson’s equation as they recombine.
i.e. recombination rate must be matched by injection of new carriers
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61. A Quick Way to the Diode Equation
The total positive charge of excess carriers at any instant of time is
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62. A Quick Way to the Diode Equation
The average lifetime for a hole is  p, thus the current must replenish this quantity of holes every p seconds, giving
This is back to the equation that we had earlier. From here we can proceed as before.
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63. Diode Diffusion Current
Outside of the depletion region, the diffusion current tends to zero as the number of minority carriers is very low.
The current is then carried by majority carrier drift current.
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64. Assumptions and Drawbacks
It is important to note a couple of inaccuracies in the above approaches.
First it assumes that all the voltage is dropped across the junction.
However if this were the case then there would be no voltage in the neutral areas to cause the majority carriers to drift, but there is a current, so there must be some voltage dropped across the neutral area.
In reality there is a resistance in the neutral area which means that voltage is dropped in these areas.
This means that the applied junction voltage is less than assumed in the derivation so the currents will be less.
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65. Assumptions and Drawbacks
The next problem is that normally the diodes are not long with respect to the diffusion length and so the excess carrier distribution is no longer exponential.
The behaviour is more complicated than our simple models and equations indicate. However they do provide a good approximation to the behaviour, sufficient for a feeling of how diodes work.
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66. Assumptions and Drawbacks
One common practical difference between what we’ve been talking about and real diodes is that normally one side of the junction is much more heavily doped than the other, normally.
What this means is that on one side the depletion layer is very thin (the heavily doped side) and the other side contains most of the depletion layer.
This is common in manufacturing of these devices as you would use a lightly doped substrate and then heavily dope it to turn it to the other type of semiconductor, then cut it and start again.
In terms of our derivation, with this knowledge halve of the equations, the lightly doped side, can be discarded as being insignificant in terms of current contribution.
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67. Assumptions and Drawbacks
One common practical difference between what we’ve been talking about and real diodes is that normally one side of the junction is much more heavily doped than the other, normally.
What this means is that on one side the depletion layer is very thin (the heavily doped side) and the other side contains most of the depletion layer.
This is common in manufacturing of these devices as you would use a lightly doped substrate and then heavily dope it to turn it to the other type of semiconductor, then cut it and start again.
In terms of our derivation, with this knowledge halve of the equations, the lightly doped side, can be discarded as being insignificant in terms of current contribution.
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68. Reverse Bias
In reverse bias the electric field is such that minority carriers in each depletion region is swept out down the energy barrier across the junction.
However due to the increased energy barrier at the junction, diffusion back across the junction is not possible and so the density of excess carriers at the junction tends to zero very quickly.
The minority carriers are only restored by a diffusion current from the neutral regions into the depletion region.
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69. Reverse Bias
Electrons “fall down” the electric field gradient, from left to right
Holes “fall up” the electric field curve, from right to left
Diffusion
electrons
Drift
Drift
Diffusion
holes
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70. Reverse Bias
Free carrier concentration near the depletion region is effectively zero
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71. Reverse Bias
The drift current rate could be much greater but it doesn’t have the free carriers near the junction
The free carriers are provided by diffusion from the bulk of the diode.
The junction is too great for diffusion across the junction.
Bulk diffusion rate limited by thermal generation of electron-hole pairs.
It is very small.
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72. Reverse Bias
The rate of thermal generation of holes within a volume (Area A * one hole diffusion length) is given by
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73. Reverse Bias This simplifies down to a reverse-bias hole current of
This is one half of the current across the reverse bias junction. It is easy to see that this is the same as Io from the diffusion current equation from before.
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74. Reverse Bias Breakdown - Zener
When there is sufficient reverse bias and heavily doped material, the energy levels are so shifted that the conduction band on the N side dips below the energy requirement of the valence band on the P side. When this happens current still cannot flow because there is a forbidden energy gap.
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75. Reverse Bias Breakdown - Zener
However if the gap between the bands, the depletion region, is narrow it is possible for electrons to tunnel across the gap from P to N and provide a conduction path.
This requires very heavy doping and a very abrupt change in doping across the junction to keep the width of the depletion region small. The width of the depletion region is very important.
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76. Reverse Bias Breakdown - Zener
An easy way to understand the mechanism is that the heavily doped P side of the junction has a large voltage across it which ionises the atoms and strips out electrons which are accelerated towards the N side. These high energy electrons have sufficient energy to jump the gap.
A more quantum physics explanation is to say that the higher energy electrons have a more energetic wave which can extend across the energy gap and where it may find a lower energy state. In this case the electron will transfer to the lower energy state.
The sort of electric field required for this behavior is typically of the order of 106 V/cm
With controlled doping it is possible to set the required voltage to cause tunneling to some desired value, normally about 6V typically. This is not a precise value as it will begin to turn on before and after and is temperature and process dependent.
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77. Reverse Bias Breakdown - Avalanche
Basically, electrons and holes are so accelerated near junction that when they hit another atom they ionise it and bounce off and repeat the process on the next collision.
This process rapidly increases the number of carriers at the junction, allowing the drift current to climb significantly.
This process is not harmful to the diode unless it melts.
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78. Reverse Bias Breakdown - Avalanche
In lightly doped junctions, the depletion width is too great to allow tunnelling so the breakdown mechanism is impact ionisation.
In any lattice scattering event where an energetic carrier hits an atom it is possible, if there is sufficient energy, to create an electron-hole pair. If the new electron-hole pair has sufficient energy each half of the pair may cause another ionising impact and an exponential effect can occur
Under these circumstances the diode places no limit on the maximum current that can be passed.
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79. Diode Models
The model for the diode depends on how you want to imagine the diode. Many people consider the diode to be ideal when it says no reverse current, infinite forward current. (Figure a).
Another model is to assume that significant current flows after a certain voltage, this is where unlimited current begins to flow, zero resistance. (figure b).
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80. Diode Models
A more realistic one is to add in a resistance to indicate that the neutral regions do have a resistance which takes away some voltage (with increasing current) from the junction.
There is no right or wrong model, all usable models are inaccurate. It depends on your application.
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81. PN Junction Capacitance
In reverse bias, the dominant capacitance source is the charge distribution in the depletion region.
The junction capacitor of a reverse biased PN junction is quite commonly used in varactor diodes.
In many circuits diodes are placed in reverse bias to prevent voltage spikes from reaching sensitive components. In these cases the reverse bias diode capacitance is visible to the circuit and needs to be considered.
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82. PN Junction Capacitance
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83. PN Junction Capacitance
However the charge in the depletion region changes with applied voltage. From earlier we have the equation for the depletion layer width
and with an applied voltage, this becomes
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84. PN Junction Capacitance
The charge in the depletion region on each side is given from before as
where the width of each region is defined by the doping and the depletion width.
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85. PN Junction Capacitance
The charge on each side of the junction can be given by
Now we have Q in terms of V, and using the original definition we can proceed to get the junction capacitance.
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86. PN Junction Capacitance
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87. PN Junction Capacitance
But the term in the brackets is the formula for the depletion layer width. Swapping this for 1/W brings us neatly back to the structure of the parallel plate capacitor formula, where W is the width, A the area and e corresponds to the charge. Note, NOT the SAME, just similar form.
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88. PN Junction Capacitance
We’ll encounter this phenomenon again and again in semiconductor devices, the capacitance of device junctions is highly voltage dependent.
The depletion layer width is voltage dependent, hence the capacitance is voltage dependent.
This can be used to our advantage in some applications, but in many cases, especially with parasitic capacitances, it can lead to increased complexity in our circuit models. This is particularly critical for high frequency applications where even fento-farads can have a significant effect..
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