# The Quantum Hall Effect Biology Essay

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The Quantum Hall Effect is the quantum mechanical version of the Hall Effect, and is observed under low temperature condition, when the object of interest is placed under a strong magnetic field.

When a voltage is applied across a material then charges get accelerated in the line of electric field generated as an effect. Now if we apply a magnetic field across the material in such a way that the magnetic field is perpendicular to the direction of voltage, then the accelerated charges experience a force in a direction that is perpendicular to both, the magnetic field and the voltage. This is because, the electron themselves generate a magnetic field, when they are accelerated. The polarity of this magnetic field depends on the spin of the electrons. This magnetic field of the electrons interact with the external magnetic field. As a result the electrons start to accumulate in the direction of force, creating a gradient of charge across the width.

Figure 1: The Hall Effect

But, if the electrons are confined to a two dimensional electron system, then things are observed differently. A two dimensional electron system is when, a system of electrons are confined only to a plane. A common example of the 2D electron system is the channel formed in an inversion mode. In such conditions, the electrons occupy well defined energy levels. If the size of the material is also reduced and temperature lowered under a high magnetic field, then the energy bands in the material start to split into definite energy levels. Under such conditions and due to such split energy level, the conductivity of the material would not increase in linear manner. Instead it increases in steps. This is because earlier all the energy levels were packed so closely that the electron could move along these energy levels in a seemingly linear manner, but with the splitting of these energy levels, the transition of the electron from one level to another means an increase of conductivity 'by a step'.

This increase in the conductivity is given by:

\sigma = \nu \; \frac{e^2}{h},

where, e is the charge of electron, h is Planck's constant and v is replaced by integral values 1,2,3,4…. Or fractional value like1/2, 2/5, 3/7 etc. Thus we observe that the conductivity is not continuous, but discrete in nature.

Figure 2: The Quantum Hall Effect

The figure above shows the energy levels with the conductivity graph together. The bands shown in the left side are the orbitals at different energy levels. These orbitals are also referred to as the Landau Level, after the physicist who predicted this effect long before it was actually observed.

As the magnetic field increase, the distance of separation (DOS) between the landau levels also increases. As these levels cross the Fermi level, the conductivity increases by a step.

## Can Schottky Diode be used as photovoltaic cell?

The Schottky diode is formed due the metal- semiconductor interface, instead of semiconductor- semiconductor interface. When a metal is joined with a semiconductor, then depending upon the work function of the metal and the electron affinity of the semiconductor different type of Schottky contacts are formed.

For the sake of convenience, we will consider an n-type semiconductor and a metal whose work function is greater than the electronic affinity of the semiconductor. This is depicted in the figure given below.

Figure 3: Energy Band Diagram before the contact of metal and semiconductor

As is given in the figure, this is the band diagram before the metal and the semiconductor come in contact. Since the semiconductor is n- type, the Fermi level is near the conduction band. Now, since the work function of metal is greater, the electrons will mostly try to flow from the semiconductor to the metal. As more and more electrons flow into the metal, a state of equilibrium will be finally achieved. At this point, the Fermi Level of the semiconductor and the metal will coincide. This gives us an energy band diagram as given below.

Figure 4: The bending of the energy band after the contact of metal and semiconductor.

Here, most of the flow of electrons happens due to thermionic emission. Meaning that all the electron, having the energy greater the barrier, climb over the barrier, onto the other side.

Figure 5: The thermionic emission of electron. The electron having greater energy than the barrier would jump above the barrier and pass into the metal.

So if the electrons can be given imparted with energy, enough to climb the interface barrier, then electrons will start flowing through the semiconductor into the metal. Thus, varying the frequency of light incident, we can vary the energy to be imparted to the electrons. This means we can vary the voltage of the photoelectricity produced by this method. Also the amount of light falling will determine the number of electrons 'crossing over', thereby adjusting the amount of current.

Figure 6: An electron gaining energy enough to cross the barrier by a photon of light. Light can be made to incident on the semiconductor such that an electron flow would take place when light falls on it.

## Temperature dependence of resistivity of semiconductor

The conductance of any material depends upon the availability of carriers for conduction in the material. For semiconductors, this availability depends upon the number of electron in the conduction band. As the temperature increases, the number of electrons transiting from the valence band to the conduction band increases. This means an increase in the number of effective carriers with increase in the temperature, leading to a change in the conductance of semiconductors with changing temperature.

This means that the resistance (which is the reciprocal of conductance) will also vary with varying temperature. Resistance is basically the opposition offered by a material to the flow of charges. This opposition offered by a material may be related to the geometry and structure, but more specifically depends upon the resistivity of the material. Resistivity is different for different materials. The resistivity of a material depends upon several factors. For semiconductors, factors like carrier concentration, lattice structure and carrier mobility affects the resistivity.

As the charges flow through the semiconductor, it experiences an opposition due to the scattering of charges. This scattering obstructs the flow of electrons, and consequently the resistivity. There are typically two types of scattering, the lattice scattering and the impurities scattering. The lattice scattering occurs when the motion of the charges are obstructed by the vibrations of the crystal lattice. These vibrations increase with the increase in temperature, leading to a decrease in the electron mobility. This means increase in resistivity with increase in temperature. But, on the other hand the impurities scattering occur due to the interaction of the charge carrier with the charged impurities in the lattice. At lower temperature, due to less mobility of the charges, they interact more with the impurities. As a result, the carriers get lost and the conductivity reduces (increasing resistivity) with decrease in temperature. But, this is more prominent at a lower scale of temperature.

The relation of resistivity for a semiconductor is as given below,

In the above equation, the resistivity is exponentially related to the temperature. But since the temperature is in the denominator, the value of resistivity would decrease with the increase in the temperature. Thus, while for the conductors the overall resistance increases, in semiconductors the resistance, more specifically, the resistivity decreases with increases in temperature.

## Moore's law and 3D Transistor

Moore's law states that the computing capability and number of transistors that can be cramped in a single wafer will double every two years. Though the trend with the computing capability, specially the speed of computing, has not been in agreement with the Moore's Law, the later part has been curiously been accurate. The size of transistors has reduced rapidly, thereby making it possible to fit more and more transistors per unit area. Analysing the history of semiconductors, we would observe that the number of transistors in a CPU increased from a 37.5 million, in 2000, to a staggering 904 million, in 2009. That's why the Moore's Law is more compatible with the number of transistors per unit area than with any other parameter.

But another report shows that this trend has also shown a decrease in the growth. The growth would decrease to doubling the number of transistors, in every three years from so forth. This is because the reduction in the size of transistors is nearing the point of saturation. The transistors cannot be reduced more. This is because, as the size of transistor reduces, the area over which the gate has control over the channel reduces.

Figure 7: The figure shows the leakage current path. As the gate is reduced, it loses control over the channel and regions far from the channel, due to which the amount of leakage current increases.

The gate starts losing the control over the channel, due to which the leakage current, the off current and the switching time increases, leading to the degradation of the performance of the transistor.

Figure 8: Graph showing the degradation of the transistor performance, with decrease of gate length below 32 nm. There is an increase in the off current and the threshold voltage.

Also, the length of the channel must also be equal to an integral value of the wavelength of electrons in the channel. So for the operating voltage of the processor, the wavelength of the electron is greater than 22 nm. Due to this the conventional structure, also known as the planar structure, cannot accommodate a channel less than 32 nm.

But a new type of MOSFET architecture, known as the 3D transistor, or the Trigate transistor, has led to the development of MOSFETs with gate length of 22 nm. In 3D transistor, the gate has been extended beyond the plane of the plane of the source and the drain. This extension of the gate region above is known as the fin. This fin help increase the channel length even when the gate length has been reduced.

Figure 9: The Intel image showing the planar, 32 nm structure on the left and the 3D, 22 nm structure on the right.

Due to this even though the gate the gate has greater control over the channel. The gate can control the channel over both the sides. This helps increase the control over the channel. The leakage current, particularly the 'ideal current', which is the off current reduces. Also the switching time increases.

Figure 10: The 3D transistor

The 3D transistor has thus helped in shaping smaller and faster transistors. It has also helped reduce the power consumption, as the leakage current has also reduced. Along with this, the 3D transistor can be further reduced to a 10 nm gate length. Thus even when the gate length is reduced, the channel length doesn't. In fact with the increase in the height of the fins, the channel length can be comfortably adjusted.

## List of Figures

The Hall Effect

Quantum Hall Effect

Energy Band Diagram before the contact of metal and semiconductor

The bending of the energy band after the contact of metal and semiconductor.

The thermionic emission of electron. The electron having greater energy than the barrier would jump above the barrier and pass into the metal.

An electron gaining energy enough to cross the barrier by a photon of light. Light can be made to incident on the semiconductor such that an electron flow would take place when light falls on it.

The figure shows the leakage current path. As the gate is reduced, it loses control over the channel and regions far from the channel, due to which the amount of leakage current increases.

Graph showing the degradation of the transistor performance, with decrease of gate length below 32 nm. There is an increase in the off current and the threshold voltage.

The Intel image showing the planar, 32 nm structure on the left and the 3D, 22 nm structure on the right.

The 3D transistor