* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Semiclassical motion in a perpendicular uniform electric
Introduction to gauge theory wikipedia , lookup
Electron mobility wikipedia , lookup
Weightlessness wikipedia , lookup
Aristotelian physics wikipedia , lookup
Maxwell's equations wikipedia , lookup
Electromagnet wikipedia , lookup
Electromagnetism wikipedia , lookup
History of fluid mechanics wikipedia , lookup
Work (physics) wikipedia , lookup
Superconductivity wikipedia , lookup
History of quantum field theory wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Centripetal force wikipedia , lookup
Electrostatics wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Lorentz force wikipedia , lookup
Speed of gravity wikipedia , lookup
Field (physics) wikipedia , lookup
Semiclassical motion in a perpendicular uniform electric and magentic field Sankalpa Ghosh, Physics Department, I I T Delhi September 4, 2008 1 Semiclassical Motion in a crossed electric and magnetic field Let us recall the semiclassical motion again. It describes then the Bloch wave packets localized over several lattice spacing moving in a slowly varying external field. Understanding this is the route to understanding electronic or any other transport in an otherwise weakly interacting electronic system. The motion of the electrons in the periodic crystal potential is treated quantum mechanically. That is how we got wave packets of Bloch waves. But then ) the rest of the motions was treated almost classically (only ṙ 6= ~k m We consider semiclassical motion in mutually perpendicular electric and magentic field. We already know that in the presence of a uniform field the k-space orbits are in the plane perpendicular to the magentic field H. The set of equations are again the same ones ṙ = 1 ∂E ~ ∂k 1 ~k̇ = (−e)[E + v(k) × H] c with E · H = 0 ( since they are mutually perpendicular) We note that E = E Ê, where Ê is a unit vector along the direction of the electric field. Again we multiply both side of the second equation with 1 Ĥ. This gives us e Ĥ × ~k̇ = −eĤ × E − Ĥ × ṙ × H) c eH (ṙ − Ĥ(Ĥ · ṙ) = −eE[Ĥ × Ê] − c eH = −eE[Ĥ × Ê] − r˙⊥ c Now integrating over time we obtain (1) Z Z dk d r˙⊥ = −eE[Ĥ × Ê] dt − Ĥ × ~ dt dt dt ~c r⊥ (t) − r⊥ (0) = − Ĥ × [k(t) − k(0)] + wt eH eH c Z Here E (Ê × Ĥ) (2) H As compared to the motion in presence of the uniform magentic field alone, namely w=c ~c Ĥ × [k(t) − k(0)] eH this motion has an additional drift velocity w. This velocity is the velocity of the reference frame in which the electric field actually vanishes. Home assignment; Show explicitly the above result. Hint: Recall that the Lorenz transformation of the electric field is given by r⊥ (t) − r⊥ (0) = − Ek0 = Ek 1 E⊥0 = q 1− v2 c2 [E⊥ + (v × H)⊥ ] (3) Here v is the relative velocity between two frames. The direction of this velocity is perpendicular to the electric as well as the magentic field . Thus the motion in the real space perpendicular to H is the superposition of 2 • the k-space orbit rotated and scaled just as it would be if only the magnetic field were present. • a uniform drift with velocity w To determine the k-space orbit we note that when E and H are perpendicular then |Ê × Ĥ| = sin π =1 2 Therefore we can write 1 ~k̇ = (−e)[E + v(k) × H] c e cE e 1 ∂E(k) ~k̇ = − H Ê − ×H cH c ~ ∂k e 1 ∂E(k) e1 ×H ~k̇ = − ~|w|H Ê − c~ c ~ ∂k e1 ∂ ~k̇ = − (E − ~k · w) × H c ~ ∂k (4) The last line is possible since w × H = wH By definition w is perpendicular to H. Thus the equation of motion can be written in a fashion such that it is the equation of motion an electron would have if only the magentic field H is present and if the band structure is given by Ē(k) = E(k) − ~k · w For a free electron Ē is simply the electrons energy in the frame moving with velocity w. We can therefore conclude from this analysis that k-space orbits are given by the intersections of surfaces of constant Ē with planes perpendicular to the magnetic field. 3