Problem set 2

Solid State theory, spring '23

Due March 30, Th. Before class.

1. Equations of motion

We have obtained the equations of motion for a Bloch electron

where $f_a$$f_a$ is the external force, and $F_{ab}={ϵ}_{abc}{\mathrm{\Omega }}^c$$F_{ab} = \epsilon_{abc}\Omega^c$, where ${\mathrm{\Omega }}^c$$\Omega^c$ is the Berry curvature of the Bloch band.

(1) Show that the real space trajectory (轨迹) satisfies the following modified Newton's law

where $\mathrm{\Lambda }$$\Lambda$ is the inverse effective mass tensor.

(2) In the simplest case, we take $F_{ab}=0$$F_{ab} =0$, and ${\mathrm{\Lambda }}_{ab}={\delta }_{ab}/m$$\Lambda_{ab}=\delta_{ab}/m$, where $m$$m$ is the effective mass. We will also introduce a damping (阻尼) term to the force, to account for the dissipation (耗散) by scattering (with impurities or phonons, for example), as $f_a\to f_a-\frac{m}{\tau }{\dot{r}}_a$$f_a \rightarrow f_a -\frac{m}{\tau} \dot r_a$, such that

where $\tau >0$$\tau>0$ is the relaxation time (弛豫时间: average time elapsed between successive scatterings). The force is purely electric and periodic in time $f_a=q{E}_a(\omega )e^{-\text{i}\omega t}$$f_a = q E_a(\omega) e^{-\text i\omega t}$ in which $q$$q$ is the charge. The current density is $j_a(\omega )e^{-\text{i}\omega t}=qn{\dot{r}}_a$$j_a(\omega) e^{-\text i \omega t} = q n \dot r_a$, where $n$$n$ is the number density of charged particles. Note that in this heuristic approach, ${\dot{r}}_a$$\dot r_a$ is thought of as the drift velocity (漂移速度), the velocity along the force direction superposed on the Fermi velocity. The filled Fermi sea does not have net contribution to the induced current.

Then show that

where the conductivity is

This is the Drude's law of a.c. (交流) electric conductivity.

(3) Now consider a situation with uniform d.c. (直流) electric and magnetic fields, where $\mathit{E}$$\boldsymbol E$ is in the $x$$x$-$y$$y$ plane and $\mathit{B}=B\hat{\mathit{z}}$$\boldsymbol B=B \hat {\boldsymbol z}$, so that

where $q$$q$ is the charge of the particle.

Show that, with same damping term as in (2) and $F_{ab}=0$$F_{ab}=0$, the d.c. conductivity tensor for current density in the $x$$x$-$y$$y$ plane is

where ${\omega }_c=qB/m$$\omega_c = qB/m$ is the cyclotron frequency (回旋频率) with the sign of the particle charge.

Suppose $\mathit{E}=E_x\hat{\mathit{x}}$$\boldsymbol E = E_x \hat{\boldsymbol x}$ and $\mathit{j}=j_x\hat{\mathit{x}}$$\boldsymbol j = j_x \hat{\boldsymbol x}$.

What is the magnetoresistance (磁阻),

in this case?

2. Semiclassical dynamics

Let us some of the results derived in class. For Bloch dynamics in a uniform magnetic field

where $I$$I$ is the identity matrix, and $B$$B$ and $\mathrm{\Omega }$$\Omega$ are rank-2 antisymmetric tensors, related to their pseudovector counterparts via

Show that

In the presence of a magnetic field and nonzero Berry curvature, it is important to take into consideration the modified phase space density when studying a Boltzmann transport theory.

3. Exercises in Girvin & Yang

8.5, 8.7, 8.8, 8.9.