Quantum Phenomena & Nanotechnology (4B5)

Transcription

1 Quantum Phenomena & Nanotechnology (4B5) The 2-dimensional electron gas (2DEG), Resonant Tunneling diodes, Hot electron transistors Lecture 11 In this lecture, we are going to look at 2-dimensional electron gasses (2-DEGs). We will see that when we bring together two different semiconductor materials to form a heterojunction, band bending results in the formation of a triangular potential well, which confines electrons into a thin sheet parallel to the interface. This band bending results from the fact that the two Fermi levels must match when the materials are in contact. The electrons in a 2-DEG are free to move in the plane parallel to the interface with very high mobility, as the 2-DEG can be formed in an undoped layer, which has very few impurities, as opposed to the case in for example a FET. This high mobility gives rise to very fast devices. These high-electron-mobility-transistors, or HEMTs use a 2-DEG as the conducting channel between the source and drain. The fact of having high mobility also means that these devices are much less noisy than FETs. The formation of this type of device is an application of Band Engineering.

2 Band Engineering: combining materials of different band-gaps to produce a specific potential profile, in order to achieve a required functionality. E.g. Resonant tunneling devices. What are the factors determining the speed of operation of integrated circuits? Mainly determined by the time required for a transistor to switch it s state, which is in turn determined by: Distance over which charge carriers have to travel to perform switching operation. i.e. gate length. Speed at which charge carriers travel: related to mobility. The mobility (m) is the relationship between electron speed and applied electric field strength (v=me) In order to make faster devices, we just make them smaller, and/or increase the mobility. What determines mobility? The more scattering, the lower the mobility. If we dope a semiconductor, we increase the number of charge carriers, but the dopant atoms themselves act as scatterers. This actually reduces the mobility relative to that in intrinsic semiconductors. If there was a way to increase the number of charge carriers in a semiconductor without having to dope it, we could attain very high mobilities. Here we are going to look at heterojunctions.

3 A heterojunction is what we have when we bring two different materials into contact. Consider what happens when we take a piece of GaAs and dope it on one side to make AlGaAs, which will be n-type. Due to the band offset (GaAs has a lower band-gap than AlGaAs), some electrons will flow from the n-type material into the GaAs. The AlGaAs and GaAs will then develop a slight positive and negative charge, respectively. This sets up a dipole layer of charge, and causes the bands to bend. i.e. Conduction bands before contact: AlGaAs GaAs In contact: Few nm Band offset

4 The electrons in the GaAs are in a triangular well, and are confined to a sheet parallel to the surface a few nm thick. This is known as a 2-Dimensional electron Gas, or 2-DEG for short. Why is this important? Well, remember that the GaAs layer is undoped (in practice it is often very lightly doped), so electrons will have very high mobility. Now that we know we have a triangular confining potential a few nm across, what can we learn? States in a triangular potential are not unlike those in a square well. We will approximate the actual triangular potential by an infinite square one. Suppose the direction normal and parallel to the interface are z, and x-y, respectively. The confinement in the z-direction gives rise to discrete energy eigenvalues E n = ħ 2 π 2 n 2 /2mL 2

5 However, the electrons are free to move in the x-y plane. Therefore, we can split the Hamiltonian into a part due to the confining z-potential and the free x-y motion. The wave-function can then be written as Ψ(x,y,z) = Ψ(z)ϕ(x,y) where the Ψ(z) are the square well eigenfunctions. The ϕ(x,y) are just plane waves, giving a total wave-function Ψ(x,y,z) = Ψ(z)e i(k x x + k y y) With energy eigenvalues of E n,kx,ky = E n + ħ 2 k x2 /2m + ħ 2 k y2 /2m Of course, this is all in the free-electron approximation. Using the nearlyfree electron approximation, we get extra terms, and Ψ is modulated by a Bloch function. Sub-bands in 2-DEG, according to the freeelectron model of electronic conduction

6 Applications of 2-DEGs: HEMT or High-Electron-Mobility-Transistor The carrier density in the 2-DEG can be controlled by a Gate, as in a standard FET.

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8 Now, we are going to look at phenomena and devices utilising hot electrons. Hot electrons are generated whenever electrons pass from regions of higher to lower potential energy. As total energy is always conserved, when potential energy is decreased, the kinetic energy will increase. Immediately after passing the potential step, the carrier velocity can be much larger than the drift velocity, speeding up the operation of the device. The word Hot simply means that they have a lot of kinetic energy. Scattering processes will eventually reduce the velocity back down to the drift velocity. Until this happens, we say the carriers are traveling ballistically (just a word meaning no scattering). Transistors can be made faster by using electrons of higher kinetic energy and hence, speed. One way to achieve this is to inject electrons into a transistor via a resonant tunneling device. The key is to ensure the active region is short enough (i.e. shorter than the meanfree path) to allow totally ballistic transport. In principle, such devices can reach up to THz operation, which is above the expected limit to bipolar devices at 400 GHz.

9 Resonant Tunneling Revisited Let us again consider the case where we have an electron encountering a doublepotential barrier. We saw earlier in the course that there are resonances in the Transmission probability, T at energies whereby the barrier separation is an integer number of half electron-wavelengths. Energy +L V 0 I II III IV V E x 0 a a+l 2a e.g. (Numerically calculated current density, J for 3 Å barriers separated by 6 Å, of height 3 ev). NDR means negative differential resistance. NDR

10 In practice Theory always predicts a more pronounced drop in current (i.e. a lower valley current ) than is experimentally observed. Also, features in I-V curves are sharper at lower temperatures (less phonons). Experiment Theory

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12 In order to make a resonant tunnelling device, how would we go about fabricating the potential barriers? Well, in a conductor or semiconductor, all of the electrons responsible for carrying current tend to be at approximately the same energy in a given conductor. That energy is called the Fermi Energy. This Fermi energy is directly related to the density of electrons in the conductor, so is different for different materials. It is very easy to regulate the number of electrons in semiconductors simply by doping. It is well known that GaAs has a lower Fermi energy than AlGaAs, so if we grow alternate layers of these materials, we can get a potential structure similar to that shown above, or even much more complex ones. The use of different materials in this way is called band-gap engineering. Consider a resonant tunneling device: What are these useful for? Microwave circuits.why?

13 If we operate a tunneling diode at an operating point in the NDR regime, then any small signals will see a decreasing resistance with increasing voltage, and will therefore be amplified. If the diode is part of a LRC circuit, amplification will be maximum at the resonance frequency. NDR causes the signal at the resonance frequency to grow with the time dependence e t/rc. If the diode is placed inside a resonant cavity, it will act as a microwave amplifier and generator. What happens when carriers impinge on a potential step? Well, E = K.E. + P.E. = Constant low K.E. high K.E. Top view

14 Energy step gives a boost to K.E. in the direction perpendicular to the step, so velocity lies in a narrow cone: carriers become collimated. Various ways of utilising hot electrons. To date, these sort of devices only work at 77K and not at room temperature, due to thermionic emission. Characteristics of hot electron transistors

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16 Lateral structures: possible alternative to vertical structures