Page 2 Reminder: The Lorentz Force F = q[E + (v B)]
Page 3 Lorentz Force: Review Velocity Filter Undeflected trajectories in crossed E & B fields: v = (E/B) Cyclotron Motion F B = ma r qvB = (mv 2 /r) Orbit Radius r = [(mv)/(|q|B)] = [p/(q|B|]) Orbit Frequency ω = 2πf = (|q|B)/m Orbit Energy K = (½)mv 2 = (q 2 B 2 R 2 )/2m A Momentum Filter!! A Mass Measurement Method!
Page 4 Standard Hall Effect Experiment Setup Current from the applied E-field Lorentz force from the magnetic field on a moving electron or hole e- v Top view: Electrons drift from back to front e+ v E field e - leaves + & – charge on the back & front surfaces Hall Voltage The sign is reversed for holes
Page 5 Electrons Flowing With No Magnetic Field t d Semiconductor Slice + _ II
Page 6 When the Magnetic Field is Turned On.. B-Field I qBv
Page 7 As time goes by... I qBv = qE Low Potential High Potential qE
Page 9 Hall Resistance R xy (Ohms) The Classical Hall Effect The Lorentz Force tends to deflect j x. However, this sets up an E-field which balances that Lorentz Force. Balance occurs when E y = v x B z = V y /l y. But j x = nev x (or i x = nev x A x ). So R xy = V y / i x = R H B z × (l y /A x ) where R H = 1/ne l y is the transverse width of the sample A x is the transverse cross sectional area 02 4 6810 200 0 1200 1000 800 600 400 1400 Slope Related to R H & Sample Dimensions Magnetic Field (Tesla) AxAx lyly
Page 10 Why is the Hall Effect useful? It can determine the carrier type (electron vs. hole) & the carrier density n for a semiconductor. How? Place the semiconductor into external B field, push current along one axis, & measure the induced Hall voltage V H along the perpendicular axis. The following can be derived: Derived from the Lorentz force F E = qE = F B = (qvB). n = [(IB)/(qwV H )] Semiconductors: Charge Carrier Density via Hall Effect Hole Electron + charge – charge
Page 11 The surface current density is s x = v x σ q, where σ is the surface charge density. Again, R H = 1/σe. However, now: R xy = V y / i x = R H B z. since s x = i x /l y & E y = V y /l y. So, R xy does NOT depend on the shape of the sample. This is a very important aspect of the Quantum Hall Effect The 2D Hall Effect
Page 12 The Integer Quantum Hall Effect Hall Conductance is quantized in units of e 2 /h, or Hall Resistance R xy = h/ie 2, where i is an integer. The quantum of conductance h/e 2 is now known as the "Klitzing“!! Has been measured to 1 part in 10 8 Very important: For a 2D electron system only First observed in 1980 by Klaus von Klitzing Awarded 1985 Nobel Prize.
Page 13 The Fractional Quantum Hall Effect The Royal Swedish Academy of Sciences awarded The 1998 Nobel Prize in Physics jointly to Robert B. Laughlin (Stanford), Horst L. Störmer (Columbia & Bell Labs) & Daniel C. Tsui, (Princeton) The researchers were awarded the Nobel Prize for discovering that electrons acting together in strong magnetic fields can form new types of "particles", with charges that are fractions of electron charges. Citation: “For their discovery of a new form of quantum fluid with fractionally charged excitations” Störmer & Tsui made the discovery in 1982 in an experiment using extremely powerful magnetic fields & low temperatures. Within a year of the discovery Laughlin had succeeded in explaining their result. His theory showed that electrons in a powerful magnetic field can condense to form a kind of quantum fluid related to the quantum fluids that occur in superconductivity & liquid helium. What makes these fluids particularly important is that events in a drop of quantum fluid can afford more profound insights into the general inner structure dynamics of matter. The contributions of the three laureates have thus led to yet another breakthrough in our understanding of quantum physics & to the development of new theoretical concepts of significance in many branches of modern physics.