When dealing with the classical Hall effect and the quantum Hall effect, the magnetic field B,...

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When dealing with the classical Hall effect and the quantum Hall effect, the magnetic field B, is often pointed along the z direction perpendicular to the sample plane. Now consider that the magnetic field points towards the x direction, parallel to the electron flow in the 2DEG (i.e. B). The 2DEG is confined with a quantum well potential of V (2) = m* wz = (a) Write the Schrdinger equation for this system using the vector potential A (r) eikteikyy (2) and we = (* remember that H = 2 (p q) + V(z)) - (b) Show that the Schrdinger equation can be expressed in the following form [k k w + 2m* qB m*. kywe m*w (B) where = + + 1 m w 2m* w(B) 2 (0, -zB, 0), plane wave ansatz *w (B)(z + c) V = EV and w (B) = w + w. (c) Sketch the confining potential along the z direction (i.e. E vs z(ky)) for B||| = 0 and B|| #0 (d) What can be said about the effective masses for x and y at B 0 ? (e) What can be said about the energy levels of the new system? (2) When dealing with the classical Hall effect and the quantum Hall effect, the magnetic field B, is often pointed along the z direction perpendicular to the sample plane. Now consider that the magnetic field points towards the x direction, parallel to the electron flow in the 2DEG (i.e. B). The 2DEG is confined with a quantum well potential of V (2) = m* wz = (a) Write the Schrdinger equation for this system using the vector potential A (r) eikteikyy (2) and we = (* remember that H = 2 (p q) + V(z)) - (b) Show that the Schrdinger equation can be expressed in the following form [k k w + 2m* qB m*. kywe m*w (B) where = + + 1 m w 2m* w(B) 2 (0, -zB, 0), plane wave ansatz *w (B)(z + c) V = EV and w (B) = w + w. (c) Sketch the confining potential along the z direction (i.e. E vs z(ky)) for B||| = 0 and B|| #0 (d) What can be said about the effective masses for x and y at B 0 ? (e) What can be said about the energy levels of the new system? (2)

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The image contains a set of problems related to the quantum Hall effect and a twodimensional electron gas 2DEG in the context of a magnetic field applied in a direction parallel to electron flow Ill g... View the full answer