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## Hall effect

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**VH**H J z y x Hall effect x**Low Field Magnetoresistance, Hall effect in Drude Theory**VH H J z y x x 2**One measures the voltage drop along y and the current along**x, not a resistance z y x Vx VH H J Hall resistanceisproportional to magneticfieldsince the phenomenonis due to the Lorentz force and the conductoris linear and inverselyproportional to n since the Lorentz force goes with the velocity, not with J, and J=neV. 4**Hall resistanceisproportional to magneticfieldin Drude**theory, however….. Quantized Hall effect of the two-dimensional electron gas in GaAs- AlxGa1-xAs heterojunctionsatlowtemperatures to 50 mK. In the small-current and low-temperature limitsharpstepsconnecting the quantized Hall resistanceplateaus. The diagonalresistivityρxxdecreases with decreasingTat the Shubnikov—de Haaspeaks, aswellasat the dips, and isvanishingly small atmagneticfieldsabove 40 kG 5**2d electron gas in (x,y) plane in magneticfield H**parallel to z axis z y x H We start with the Theory for spinless electrons Main alternative Choices of the gauge**y**kx**The energylevels do notdepend on kx, so theirdegeneracyis**just the number of kxpoints: itisfound by imposingthat y0cannotexceed sample size. y kx r(w) with H r(w) without H E E**m**Ht H Groundstate energy – 2DEG**Note!**All electrons contribute , not only the open LL**K. v. Klitzing**Used the depletionlayer of a GaAs MOSFET as a 2d electron gas**K. v. Klitzing**discovery: 1980 Nobel prize: 1985 Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0. Plateaux in Hall resistivity r=h/(ne2) with integer n correspond to the minima (From Datta page 25)