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QUANTUM HALL EFFECT Classical Hall effect B 1879 - E. Hall v e F 𝑒𝑬𝑻 = 𝑒𝒗 × 𝑩 Carrier type, 𝑉𝐻 = 𝑏 𝑬𝑻 = http://topocondmat.org/w3_pump_QHE/figures/hall_bar.svg 𝑏 JB, 𝑒𝑛 𝜎𝑥 = 𝑒𝑛𝜇𝐻 , 𝜇𝐻 = 𝑟𝜇𝑒𝑓𝑓 r – Hall scattering factor 1 Classical Hall effect The Drude model: ∗𝒗 𝑑𝒗 𝑚 𝑚∗ = −𝑒𝑬 − 𝑒𝒗 × 𝑩 − 𝑑𝑡 𝜏 Applied Lorentz Scattering electric field force 𝐉 = −𝑒𝑛v 𝑱 = 𝜎𝑬 𝑒𝐵 𝑤𝐵 = ∗ 𝑚 𝜎= 𝜎(𝐵=0) 1+𝑤𝐵 2 𝜏2 1 𝑤𝐵 𝜏 𝜎𝑥𝑥 −𝑤𝐵 𝜏 = −𝜎 1 𝑥𝑦 𝜎𝑥𝑦 𝜎𝑥𝑥 𝜎 𝐵=0 = 𝑒 2 𝑛𝜏 𝑚∗ 2 Classical Hall effect 𝜌= 𝜌𝑥𝑥 = 𝜌𝑥𝑦 𝜎 −1 1 𝜎(𝐵=0) = 1 𝜎(𝐵=0) 1 −𝑤𝐵 𝜏 𝑚∗ = 2 𝑒 𝑛𝜏 𝑒𝐵 𝑤𝐵 𝜏 𝐵 ∗𝜏 𝑚 = = 2 = 𝜎(𝐵=0) 𝑒 𝑛𝜏 𝑒𝑛 𝑚∗ 𝜌𝑥𝑥 𝑤𝐵 𝜏 = −𝜌 1 𝑥𝑦 𝜌𝑥𝑦 𝜌𝑥𝑥 𝜌𝑥𝑦 𝜌𝑥𝑥 𝐵 3 Change experiment conditions 𝜌= 1 𝜎(𝐵=0) 1 −𝑤𝐵 𝜏 𝑤𝐵 𝜏 1 High 𝑤𝐵 𝜏: High B >2T High 𝜇 material 2DEG Low T (< 4K) 4 Quantum Hall Effect Klitzing, 1980 (𝜌𝑥𝑦 ) (𝜌𝑥𝑥 ) M. Houssa, lecture series, KU Leuven. 5 Quantum Hall Effect Why only 2 odd plateaus? 𝟐𝝅ℏ 𝟏 𝝆𝒙𝒚 = 𝟐 𝒆 𝝂 Why 5 is not Plateaus pronounced? over a range of B 𝑷𝒆𝒂𝒌𝒔 𝑻𝒘𝒐 𝒑𝒆𝒂𝒌𝒔 𝝆𝒙𝒙 ≠ 0 𝝆𝒙𝒙 = 0 M. Houssa, lecture series, KU Leuven. 6 Semiclassical approach Semiclassical approach: v ℎ 2𝜋𝑟 = 𝑛λ = 𝑛 𝑝 r 2 B 𝑛ℏ2 𝐸𝑛 = 𝑤𝐵 2 A bit wrong, but not bad for semiclassical approach D.Tong 7 Quantum physics approach Quantum mechanics approach: 1 𝐸𝑛 = ℏ𝑤𝐵 (n + ) 2 Landau levels (LL) E n=0, 1, 2 DOS at each LL: 𝑁= 2𝑒𝐵 ℎ Peaks spaced by ℏ𝑤𝐵 . 8 M. Houssa, lecture series, KU Leuven. Quantum physics approach Increasing B 𝑁= 2𝑒𝐵 ℎ Energy B=0 𝐸𝐹 ℏ𝑤𝐵 Density of states Fermi level between LL – no available states crossing LL – states availabe 9 Quantum physics approach Landau levels (LL) E Quantum mechanics approach: 1 𝐸𝑛 = ℏ𝑤𝐵 (n + ) 2 n=0, 1, 2 Edge channel Edge channel Localised e M. Houssa, lecture series, KU Leuven. http://d29qn7q9z0j1p6.cloudfront.net/content/roypta/363/1834/2203/F8.large.jpg http://i.stack.imgur.com/LT9cB.jpg 10 𝝆𝒙𝒚 quantization 𝝆𝒙𝒚 = M. Houssa, lecture series, KU Leuven. 𝟐𝝅ℏ 𝟏 𝒆𝟐 𝝂 11 𝝆𝒙𝒚 quantization Edge states in equilibrium with mL mL mR 𝑉𝐻 𝜌𝑥𝑦 = 𝐼 𝑒(𝜇𝑅 − 𝜇𝐿 ) 𝑒𝑉𝐻 = 𝜇𝑅 − 𝜇𝐿 𝐼 = −𝑒𝑣𝑛 = 2𝜋ℏ 𝑣 - from a dispertion relation n – sum over all filled states Edge states in equilibrium with mR 𝜌𝑥𝑦 2𝜋ℏ 1 = 2 𝑒 ν ν - number of occupied LL 12 𝝆𝒙𝒚 quantization Increasing B 𝑁= Energy B=0 2𝑒𝐵 ℎ 𝐸𝐹 ℏ𝑤𝐵 𝜌𝑥𝑦 2𝜋ℏ 1 = 2 𝑒 ν Density of states 𝐵 ↑ → ν ↓→ 𝜌𝑥𝑦 ↑→ 𝑉𝐻 ↑ 12 Origin of the plateaus Plateaus over a range of B M. Houssa, lecture series, KU Leuven. 13 Origin of the plateaux 1 𝐸𝑛 = ℏ𝑤𝐵 (n + ) 2 DOS at each LL: 𝑁 = 2𝑒𝐵 ℎ Increasing B Energy B=0 𝐸𝐹 Density of states 14 Density of states Origin of the plateaus 2𝑒𝐵 𝑁= ℎ Density of states Energy Defects Disorder Energy increases (within reason) – plateaus are more pronounced no disorder – plateaus expected to disappear David Tong, https://inspirehep.net/record/1127711/files/landau_levels.png 15 𝝆𝒙𝒙 oscillations 𝑷𝒆𝒂𝒌𝒔 𝝆𝒙𝒙 = 0 M. Houssa, lecture series, KU Leuven. 16 𝝆𝒙𝒙 oscillations No backscattering on the edges Edge states in equilibrium with mL Edge channel Edge channel mL mR Edge states in equilibrium with mR Localised e Energy Increasing B B=0 𝑒𝑉𝑥𝑥 = 𝜇𝑅 − 𝜇𝑅 = 0 𝐸𝐹 Density of states Fermi level between LL – no available states – no scattering - 𝝆𝒙𝒙 = 0 crossing LL – availabe bulk states between the edges - 𝝆𝒙𝒙 ≠ 0 17 Odd plateaus 𝑻𝒘𝒐 𝒑𝒆𝒂𝒌𝒔 M. Houssa, lecture series, KU Leuven. 𝝆𝒙𝒙 ≠ 0 18 Zeeman splitting ℏ𝑤𝐵 = B=0 ℏ𝑒𝐵 𝑚∗ ≫ 𝐸𝑧 = 𝑔𝑠 𝜇𝐵 𝐵 B≠0 ν =2 Energy 𝑔𝑠 𝜇𝐵 𝐵 ν=3 𝐸𝐹 ℏ𝑤𝐵 ↑ ↑ ↑ ↑ ν =1 ↑ ↑ ↑ ↑ ν =2 𝐸𝐹 ν =1 Density of states 19 Zeeman splitting 𝑚∗ 𝐸𝑧 = 𝑔𝑠 𝜇𝐵 𝐵 ≫ ν =2 ↑ ↑ ↑ ↑ ν=3 Energy Energy ν=3 𝑔𝑠 𝜇𝐵 𝐵 ℏ𝑤𝐵 ↑ ↑ ↑ ↑ Density of states Easy to jump ν =2 ↑ ↑ ↑ ↑ ℏ𝑤𝐵 = ℏ𝑒𝐵 Density of states Thermal excitation Difficult to jump Even number of LL occupied – pronounced plateaus Odd number of LL occupied – not pronounced plateaus 20 Odd plateaus 𝑻𝒘𝒐 𝒑𝒆𝒂𝒌𝒔 M. Houssa, lecture series, KU Leuven. 𝝆𝒙𝒙 ≠ 0 21 Applications Standard of measuring resistivity: 𝜌𝑥𝑦 = Ratio of fundamental constants: 2𝜋ℏ1 𝑒2 ν 𝜋ℏ 𝑒2 Quantum computers: error-free topologically protected states 22 Hall effects Classical Integer QHE Hall effect Fractional QHE Spin QHE 23 Conclusions QHE – quantization of Hall voltage (transverse resistivity) as a function of applied magnetic field at special conditions (2DEG, high B and 𝜇, low T). Quantization of energy levels and modulation of the DOS by magnetic field stands behind QHE. Due to finite size of the sample edge conducting states are formed. On each edge current can propagate in only one direction. Thus backscattering is only possible through the bulk of the sample when there are available states. This leads to oscillations of the longitudinal resistivity. Disorder is important. 24 Thank you for your attention! References Klitzing, K. V., Gerhard Dorda, and Michael Pepper. "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance." Physical Review Letters 45.6 (1980): 494. The Quantum Hall Effect. TIFR Infosys Lectures. D.Tong. Cambridge. The Quantum Hall Effect. Lecture series on NanoHub. S. Data. Perdue University. The Quantum Hall Effect. Lectures series. M.Houssa. KU Leuven. 26 Zeeman spliting The gyromagnetic ratio of a particle or system is the ratio of its magnetic momentum in an atom to its angular momentum. Bohr magneton is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum. Classical Hall effect