Advanced Solid State Physics

9 Magnetism and Response to Electric and Magnetic Fields 9.1 Introduction: Electric and Magnetic Fields Physical quantities in s ...

Figure 58: Particle moving on a circle in cartesian coordinates and in k-space. Something moving on a circle at a located positi ...

So with eqn. (67) the momentum can be calculated as px= ∂L ∂vx =mvx+qAx (69) In this case, beside the kinetic momentum, there is ...

The second part of the Hamiltonian can be written as H 1 = −i~eBz 4 m ( −y ∂ ∂x +x ∂ ∂y ) + e^2 B^2 z 8 m ( x^2 +y^2 ) +gμBSzBz ...

Figure 59: Magnetic behavior of molecules and ions with closed electron shell configuration. The blue line is the expected behav ...

9.3.2 Paramagnetic Response Whenever there is a non-zero angular momentum in the z-direction (spin), there are more than one sta ...

Figure 60: Magnetization for different values of the total angular momentum at variable values of magnetic field and temperature ...

N+= 1 2 ∫EF −μB D(E+μ 0 μBB)dE≈ 1 2 ∫EF 0 D(E)dE+ 1 2 μ 0 μBBD(EF) N−= 1 2 ∫EF μB D(E−μ 0 μBB)dE≈ 1 2 ∫EF 0 D(E)dE− 1 2 μ 0 μBBD ...

Figure 62: Left: Fermi sphere for zero applied magnetic field. Right: Split of Fermi sphere into different Landau levels when ma ...

There are two limits where one can solve this problem: High temperatures / Landau diamagnetism (weak magnetic fields) For high t ...

Figure 63: De Haas-van Alphen effect for gold as a good approximation for a free electron gas. The magnetic moment is plotted as ...

classically seen due to electric interaction as we shall see in the following. Imagine a two electron system described by the st ...

This additional magnetic field may now be used for statistical considerations to calculate the mag- netization as a function of ...

Ferromagnetic materials show spontaneous magnetization without an external field below the critical temperature. The emerging o ...

Figure 64: Conductivity (green line) and Hall voltage (red) line over magnetic field. Figure 65: Possible states for a two elect ...

Figure 67: Curie-Weiss law. [from Kittel] ...

10 Transport Regimes Before we start with Transport Regimes we will discuss the Linear Response Theory which we need to understa ...

For the damped spring-mass system from above (eq. (95)) this gives us u(t) = ∫ − 1 b exp ( −b(t−t′) 2 m ) sin (√ 4 mk−b^2 2 m (t ...

Figure 69: Responseg(t)of a system to aδ(t)driving force splitted into an even and an odd component Since the even component of ...

This is a differential equation for the velocityv. The velocity can be determined to be v = −eEt m +v 0 and is proportional to t ...