11 1 Anomalous Velocity Consider The Effect Of A Weak Uniform Electric Field E Perturbation On A Crystal In The Absenc 1 (51.04 KiB) Viewed 31 times
11.1 Anomalous velocity: Consider the effect of a weak uniform electric field E perturbation on a crystal. In the absence of the perturbation, the Bloch Hamiltonian is given by 1 H(k) = (p+ ħk)2 + V(x). 1 2m Notice that k behaves like a vector potential. What are the eigenfunctions of H(k)? (a) What are the drawbacks of using an electrostatic potential 0 (x) that produces the uniform E? (b) We can avoid these obstacles by introducing a time-varying uniform vector potential Ao(t). Describe how such a potential is manifest in the crystal's Bloch Hamiltonian.
(c) You may define a modified time-dependent crystal momentum, q = k - eEt, with its corresponding Bloch Hamiltonian H(q(t)). What is the condition for adiabatic change? (d) Write down an expression for the velocity operator v(q) in terms of H(q). (e) Use the expression you obtained for the first-order corrected adiabatic eigenket in Exercise 10.2 to show that the velocity vn = (un(q,t) v(9) \un(q,t)) is given by den (k) Vn(k) = ExFn (k). hak The second term involving the Berry curvature is the so-called anomalous veloc- ity - note that its direction is transverse to the electric field and thus will give rise to a Hall current. (f) Comment on the effect of time-reversal symmetry on the anomalous velocity and its consequences. =
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