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November 4, 2010 Class 17 Hall Effect If a magnetic field is applied perpendicular to a current, the electrons will suffer a force perpendicular to both, the drift velocity and B, of a magnitude F=qvxB This force produces an accumulation of charge and an electric field that generates an electrical force that opposes to the magnetic force. In equilibrium -eEH=evB or EH=-vB By defining the Hall coefficient RH as EH=-RH JB vB= RH JB or RH=v/J (notice that RH can be measured by measuring the electric field perpendicular to the current and it is a property of the material). Using J=nqv RH=1/nq (where q is negative for electrons) For Li, monovalent, the measured Hall coefficient at room temperature is -1.7x10-10 m3/C and the calculated value -1.4x10-10 m3/C. For Al (trivalent) both values are ~ -0.3x10-10 m3/C. But for Zinc, the calculated value is -0.5x10-10 m3/C while the measured one is +0.3x10-10 m3/C. The reason is that the carriers are positive. Although electrons are always the ones moving, some material, especially divalent, behave as if the carriers were positive. We’ll talk about that later. Nearly Free-electron Theory Free electron (FE) theory produces very good results for many things but there are fundamental issues that are not accounted for within that simple theory. Band structure will not come out from FE, thus there is no way to understand or define conductors, insulators, semiconductors, etc. Also FE, as indicated above, does not account for a positive Hall coefficient, in other words, from the FE point of view only free electrons can conduct. Real life shows us that the above is a way too simplified picture, electrons states do NOT form a continuous from 0 to infinity but the energy breaks in bands with gaps of forbidden energy. Also experiments show that some materials behave as if the particles being dragged by the electric field were actually positive particles. In addition, it is observed that the mass of the electrons being conducted (or that of the positive particles) is NOT necessarily equal to the free 1 electron mass. All of this can be explained with a “simply correction” to the FE, we know this as Nearly-FE (NFE) theory. Empty Lattice Approximation – used for calculating Energy Band Gaps Figures taken http://people.deas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u7/ls2_unit_7.html from One way of approximating the band structure is to use the free electron approximation within the brillouin zones, 2k 2 . Using that k k' G , where k’ is within the first brillouin zone we 2m can bring all the bands into the first zone for drawing. If G is a reciprocal translation vector-- the 2 ( k' G )2 solution within the free electron theory then becomes . Supporting info. 2m Bands (nearly free electrons) For >> a (small k), the waves do not see structures and electron behave as free electrons, then E=ħ2k2/2m (this is the low energy regime, electrons near the bottom of the band). Let’s suppose that instead /2=a. Now there is diffraction (the Bragg condition is met) since k=/a. This condition means standing waves and electrons are not traveling back and forth. The solutions to the Schrödinger equation cannot be a single complex exponential, since a complex exponential is never zero and standing waves must have fixed nodes. However, a linear combination of complex exponentials is also a solution, and the more appropriate solutions are sine’s and cosines (which can be obtained as a linear combination of two plane waves, one moving to the right and the other moving to the left). (+)=2A cos(x/a)Aeiπ/a+A e-iπ/a (-)=2Bi sin(x/a)= Beiπ/a-Be-iπ/a A and B do not need to be real numbers since what is physically observable is the density which is the square of the wave functions, thus 2 (+)=|(+)|2cos2(x/a) (-)=|(-)|2sin2(x/a) For x=0,a,2a,..,na (+) is maximum (cos=1) and it is zero in between (a/2, 3a/2,…) etc thus the electron density is maximum in the position of the nucleus and the nodes are in between. The opposite happens for (-) thus the first wavefunction represent electrons piled up at the position of the nucleus while the second wave function represents electrons piled up in between atoms. If the wave’s maximum is on the nuclei position, there is maximum (negative) interaction with the lattice and electrons have lower energy than free electrons. If instead the wave’s maximum is in between nuclei positions, there is maximum (positive) interaction with the lattice and electrons have higher energy than free electrons, this is the origin of the band gap. When NFE is considered, near the band edge, the E(k) differs from a parabola determining two values at the border separated by a gap. By taking these values to within the first Brillouin zone (for visual purposes) we get a clear picture of the gaps. You will find energy band plots of the form shown in the figure. http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm Bloch Functions “When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal… By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation.” F. Bloch Bloch proved that the solution of the Schrödinger equation in a periodic potential must be of the form k ( r ) uk r e ik .r Where u(r) has the periodicity of the lattice, thus u(r+T)= u(r) where T is a translation vector of the lattice. A proof restricted to the case where the states are non-degenerated is as follows: Consider a ring with N identical lattice points at a distance a from each other. By symmetry, let’s look at solutions of the form 3 x a C x where C is a complex number, thus x Na C N x Since x+Na=x in the circle, x Na x , then CN=1 and C is one of the N roots of the unity Ce i 2 s N Thus x ux e i 2sx Na where u(x) must satisfy u(x+a)= u(x) such that x a C x This is a simply proof to the block theorem for the case where non-degenerate solutions exist. Block functions are the electron wave functions within the NFE 4