MIS or Metal Insulator Semiconductors can lead to very high performance inventors. However, they have a fundamental limitation in the fact that they have no internal gain. And so this really limits the applications to which the MIS device can be applied. Gain is really important for detecting weak signals or for high frequency operation, where electronic amplifier noise can be a problem. So there are really two main ways to deliver gain in solid-state detectors, photoconductive gain and multiplicative gain. So let's start by talking about photoconductive gain. So in a photoconductive device you essentially have a length of semiconductor with electrodes on each side. And then you have light that hits the device. And so essentially this is based on space charged neutrality and upon photogeneration each carrier species drifts under the applied field to their respective context. And if we think about what produces gain, what produces gain is actually the difference in the electron and the hole transit times, for this particular device. So let's think a little bit more about how the photoconductor operates. When light hits the device, it generates an electron-hole pair. So the electron moves to the positively biased electrode, and the hole moves to the negatively biased electrode. And if the transit time of the electrons in the holes is vastly different, which is the case for many III-V compounds, then the electron will transfer, traverse the device in much less time than the hole and it will exit, leaving the device with that positive charge. And so in order to maintain charge neutrality, an additional electron is injected from the device from an external circuit and it starts to drift across the device. And if the second electron traverses the device before the original hole does, then another electron is inducted into the device. And so this process continues until the hole arrives at the electrode, and exits the device. And so the source of gain is really the additional electrons that are injected into the device during the transit of the hole. So now, let's take an extreme case. So let's take the case where holes can't move but electrons can. And so in this case when there's a photogeneration event, the electron will drift towards the positively biased contact or the hole is localized at the position where it was generated. And when the electron arrives at the contact, and exits the semiconductor, it leaves behind positive charge and an additional electron is injected to maintain charge neutrality. And then the injected carrier drifts towards the positively biased contact exit and results in another electron injected into the semi-conductor. And so this is repeated many, many times until the hole finally recombines. So if we take the case of a one micron device, the drift time is approximately 10 picoseconds. And the hole recombination time is 10 nanoseconds. There is a three orders of magnitude difference. And so what that means is that on average, there are 1000 electrons that traverse the device before the hole recombines. And so that results in a photoconductive gain of 1000. The photoconductive gain is really given by the ratio of tau to t between the electron transit time and the hole recombination time. So let's talk actually about a practical way to implement this. A practical way to implement this kind of behavior where only one carrier can move freely is an effective mass filter. So it uses the unique band structure of a multi-quantum-well structure. And when the quantum wells are brought close together in this type of structure, you get quantized energy levels in the individual wells. That start to interact and form an extended state, which is referred to as a mini band. The mini band allows the electrons to freely move from well to well via tunneling. And this formation of this mini band really depends on the hype and the thickness of the barriers and how deep the quantized energy levels is in the quantum well. So in general the wave function of the lower line state is more localized and the wave form of the higher line level penetrates more into the barrier. So for certain barrier heights and thicknesses we can create a situation where the lower lining levels are isolated and the higher lining levels are extended to form mini bands. So the electrons in the ground state are localized but when they are excited to higher energy levels they can tunnel through the barriers. And so this location of the quantized energy levels inversely proportional to the effect of mass. And so what we find is that the holes and the valence band in this type of structure have deeper quantized energies than the electrons because of a larger effect of mass. And so that leaves the holes highly localized in the quantum wells. And so what this means is we can now design and fabricate a device where holes are highly localized and electrons are free to move across the device. So the result here is that this affected mass filter has high photoconductive gain. So let's now talk about some of the limitations of these photoconductors. Basically, the principal limitation is that the frequency response decreases with increasing gain. So the actual device response time is inversely related to the hole lifetime. Until the longer the hole survives the longer it takes the signal to decay. But the photoconductive gain is directly proportional to the hole lifetime. And so that means that the device is gain bandwidth limited. And it means that the gain bandwidth has to be constant. We can take an example of a typical photoconductive device. If we take mercury germanium, mercury is an acceptor with a 0.09 electron volt ionization energy. Because it's got a lower ionization, it can detect really long wavelengths in this case up to 50 microns. But we need to cool in order to avoid thermally generated carriers. In photoconductors, we can think about what the principal noise mechanism is. It's the randomness inherent in the current flow, essentially almost short noise. So if we think about the contributions of the carriers to the charge flow, it's going to be e times tau over tau d. Were essentially, tau is the lifetime of the carrier, which is not constant. We can go ahead and write down an expression for the average current, if we write I tilde, and I'll say this is average current. This is going to be equal to the average number of electrons which I'll denote with N tilde, times tau over tau d. Okay, so we can Fourier transform this to get the density. So S of nu is equal to 4eI, and then here we have, tau over tau d, over 1 + 4 pi squared, nu squared, tau squared. And so from this, we can write down the noise that we would expect. It's just going to be, iN squared is equal to s of nu times delta new which is equal nothing to more than 4e charge on the electron i tilde tau over tau d times delta new over 1 plus 4 pi squared nu squared tau squared. So nu is the frequency, I is the average electric current. Delta nu is essentially the bandwidth and then tau d is the transit time due to drift. And tau is the actual carrier lifetime. So we can consider an example here. Let's consider the case of a mercury doped germanium detector. And we're going to assume here that we're at 20k. My length of my device is equal to 10 to the minus 1 centimeters. And tau is equal to 10 to the minus 9 seconds. The voltage which is across the length, Is equal to 10 volts. And so from this we can write down an electric field. So the electric field is going to be nothing more than 10 to the 2 volts per centimeter. We can also, we are also given the carrier mobility, it's 3 times 10 to the 4 centimeter squared per volt second. So the first thing to do is to calculate the drift velocity. The drift velocity is nothing more than mu times the electric field. And so that is equal to 3 times 10 to the 6 centimeters per second. We can also calculate the drift time. The drift time is going to be nothing more than the length the carrier has to traverse divided by mu. And so this is equal to 3.3 times 10 to the minus 8 seconds. So what this tells you is that the carrier actually traverses 3% of the length before recombining because we know that D is equal to 1 millimeter. So from this, we can go ahead and we can write down what we'd expect for our signal to know it. So if we have i squared of rg so that's the recombination over i squared shot, What we get here is we get 2 tau over tau d. And so this is equal to one over 15 when omega times tau is much, much less than 1. So we can write if we want that i squared of a shot, so essentially, the shot now is current is equal to 2 eId times mu. So I think the bottom line here is that, with a photo conductor, an increasing gain means that you decrease the bandwidth. And there is device response time will be proportional to the minority carrier lifetime. So while photoconductors are an attractive technology, they're fundamentally limited by gained bandwidth. And that means that it's really hard to apply them to high-bandwidth systems that require high gain, such as optical communication systems.