Integral Quantum Hall Effect in Mbe Grown Thin Films

We aim to study Integral Quantum Hall Effect in MBE-grown thin films like Graphene,Bismuth,GaAs and FeSiO2.We aim to look at how the quantum size effects dominate the electrical transport properties like the resistivity, hall coefficient, magnetoresistance coefficient and low-frequency electrical noise. We are also interested to look at the measurements on the MBE-grown AlGaAs/GaAs heterojunctions which indicate a difference of nearly an order of magnitude in the transport properties. INTRODUCTION AND BACKGROUND The Quantum Hall Effect (QHE) is one of the most remarkable phenomena in condensed matter discovered in the second half of the 20 century. At low temperatures and in strong magnetic fields, it is found that the Hall resistance of a two dimensional electron system has plateaus as a function of the number of electrons. In 1985 Klaus von Klitzing won the Nobel Prize for discovery of the quantized Hall effect. The previous Nobel Prize awarded in the area of semiconductor physics was to Bardeen, Shockley and Brattain for invention of the transistor. Everyone knows how important transistors are in all walks of life, but why is a quantized Hall effect significant? It was over 100 years ago E.H. Hall discovered that when a magnetic field is applied perpendicular to the direction of a current flowing through a metal a voltage is developed in the third perpendicular direction. This is well understood and is due to the charge carriers within the current being deflected towards the edge of the sample by the magentic field. Equilibrium is achieved when the magnetic force is balanced by the electrostatic force from the build up of charge at the edge. This happens when Ey = vxBz .The Hall coefficient is defined as RH = Ey /Bzjx and since the current density is jx = vxNq , RH =1/Nq in the case of a single species of charge carrier. RH can thus be measured to find N the density of carriers in the material. Often this transverse voltage is measured at fixed current and the Hall resistance recorded. It can easily be seen that this Hall resistance increases linearly with magnetic field. The FIGURE 1 above shows the normal Hall Effect experimental set-up. In a two-dimensional metal or semiconductor the Hall effect is also observed, but at low temperatures a series of steps appear in the Hall resistance as a function of magnetic field instead of the monotonic increase. What is more, these steps occur at incredibly precise values of resistance which are the same no matter what sample is investigated. The resistance is quantized in units of h/e divided by an integer. This is the QUANTUM HALL EFFECT. The FIGURE 2 shows the integer quantum Hall effect in a GaAs-GaAlAs heterojunction, recorded at 30mK. The basic experimental fact characterizing QHE is that the diagonal electrical conductivity of a two-dimensional electron system in a strong magnetic field is vanishingly small 0 xx σ → ,while the non-diagonal conductivity is quantized in multiples of 2 / e h : 2 / xy e h σ ν = − , where ν is an interger [The Integer Quantum Hall Effect (IQHE)] or a fractional number [The fractional Quantum Hall Effect]. THEORETICAL EXPLANATION OF QUANTUM HALL EFFECT Derivation of the Quantum Hall Coefficient In the Drude theory of the electrical conductivity of a metal, an electron is accelerated by the electric field for an average timeτ before being scattered by impurities, lattice imperfections and phonons to a state which has average velocity zero. The average drift velocity of the electron is / d v eE m τ = − where E is the electric field and m is the electron mass. The current density is thus 0 d j nev E σ = − = where 2 0 / ne m σ τ = where n is the electron density. In the presence of a steady magnetic field, the conductivity(σ ) and resistivity( ρ ) become tensors: . j E σ = and . E j ρ = always hold. From the Lorentz Force Law, we find ( ) d d v B v e E c m τ × = − + We assume that the magnetic field is in the z direction, then in the x-y plane 0 x c y x E j j σ ω τ = + and 0 y c x y E j j σ ω τ = − + where 0 σ is defined above and c eB mc ω = . Finally, we get the relationship between conductivity and resistivity as follows 2 2 xx xx xx xy ρ σ ρ ρ = + and 2 2 xy xy xx xy ρ σ ρ ρ = − + Here we see that if 0 xy ρ ≠ , then conductivity xx σ vanishes when xx ρ vanishes. On the other hand, we have xx xy c nec B σ σ ω τ = − + . So, when xx σ = 0, x xy y j E σ = where xy σ which is the Hall Conductivity is given by H xy nec B σ σ = = − . Now, in Quantum Mechanics, the Hamiltonian is 2 1 ( ) 2 eA H p eEx m c = + + ( E is along the x direction) For this problem it is convenient to choose the Landau gauge, in which the vector potential is independent of y coordinate: (0, ,0) x A B = . Solving the Schrodinger Equation which can be transformed to the familiar harmonic oscillator equation, the eigen values are as follows: 2 2 ( ) ( 1/ 2) ( / 2 ) i c c y c E i e I k e m ω ω Ε = + + Ε − Ε h where 1 2 ( ) c c I eB = h ( classical cyclotron orbit radius) and i=0,1,2,… The different oscillator levels are also called Landau Levels. The electric field simply shifts the eigen values by a value without changing the structure of the energy spectrum. From the wave functions, we can calculate the mean value of the velocities as / y v c B < >= −Ε and 0 x v < >= Thus, / y j ne c B = − Ε which is the same as classical results derived above.The current along the direction of electric field (x) is zero at Landau levels. Explanation of the plateaus and zeroes (FIG 2) The zeros and plateaus in the two components of the resistivity tensor are intimately connected and both can be understood in terms of the Landau levels (LLs) formed in a magnetic field. In the absence of magnetic field the density of states in 2D is constant as a function of energy, but in field the available states clump into Landau levels separated by the cyclotron energy, with regions of energy between the LLs where there are no allowed states. As the magnetic field is swept the LLs move relative to the Fermi energy. When the Fermi energy lies in a gap between LLs electrons can not move to new states and so there is no scattering. Thus the transport is dissipationless and the resistance falls to zero. The classical Hall resistance was just given by B/Ne. However, the number of current carrying states in each LL is eB/h, so when there are i LLs at energies below the Fermi energy completely filled with ieB/h electrons, the Hall resistance is h/ie. At integer filling factor this is exactly the same as the classical case. The difference in the QHE is that the Hall resistance can not change from the quantised value for the whole time the Fermi energy is in a gap, i.e between the fields (a) and (b) in the diagram, and so a plateau results. Only when case (c) is reached, with the Fermi energy in the Landau level, can the Hall voltage change and a finite value of resistance appear. EXPERIMENTAL SETUPS FOR QHE Quantum Hall Effect has been observed in two types of 2-D electron systems achieved in experiment. They are as follows:• MOSFET-Metal Oxide Semiconductor Field Effect Transistor FIG 4: Schematic side view of a silicon MOSFET In MOSFET, inversion layers are formed at the interface between a semiconductor and an insulator or between two semiconductors, with one of them acting as an insulator. The system in which the Quantum Hall Effect (QHE) was discovered has Si for the semiconductor, 2 SiO for the insulator. Figure 2 is a schematic side view of a silicon MOSFET showing the aluminum gate, the 2 SiO insulator and the p-type Si crystal substrate. The principle of the inversion layer is quite simple. It is arranged that an electric field perpendicular to the interface attracts electrons from the semiconductor to it. These electrons sit in a quantum well created by this field and the interface. The motion perpendicular to the interface is quantized and thus has a fundamental rigidity which freezes out motional degrees of freedom in this direction. The result is a two-dimensional system of electrons.