Semiclassical Theory of Electrons

Abstract

In the presence of an electric field ${\mathbf {E}}$ and a dc magnetic field ${\mathbf {B}}$, the equation of motion of a Bloch electron in k-space takes the form

$$\begin{aligned} \hbar \dot{\mathbf{k}} = -e{\mathbf {E}} -\frac{e}{c}{{\mathbf {v}}}\times {\mathbf {B}}. \end{aligned}$$

Appendices

Problems

13.1

The energy of an electron in a particular band of a solid is given by

$$ \varepsilon (k_x,k_y, k_z) = \frac{\hbar ^2k_x^2}{2m_x} + \frac{\hbar ^2k_y^2}{2m_y}, $$

where $-\frac{\pi }{a}<k_i <\frac{\pi }{a}$ is the first Brillouin zone of a simple cubic lattice.

• (a) Determine $v_i({\mathbf {k}})$ for $i=x, y$, and z.

• (b) Show that $\hbar \left( k_i(t), k_j(t)\right) = \left( \sqrt{2m_i\varepsilon }\cos \omega _\mathrm{c} t,\sqrt{2m_j\varepsilon }\sin \omega _\mathrm{c} t\right)$ where $(i, j)=x \text{ or } y$ for a d.c. magnetic field ${\mathbf {B}}_0$ in the z-direction.

• (c) Determine $\omega _\mathrm{c}$ in terms of $m_i$, $B_0$, etc.

13.2

Consider an electron in a two-dimensional system subject to a dc magnetic field B perpendicular to the system. The constant energy surface of the particle is shown in in Fig. 13.8.

• (a) Sketch the orbit of the particle in real space.

• (b) Sketch the velocity $v_y(t)$ as a function of t.

Fig. 13.8

A constant energy surface $\varepsilon ({\mathbf {k}})$ in a two-dimensional system

13.3

Take direction of current flow to make an angle $\theta$ with x axis as is shown in Fig. 13.9. First, transform to $x^\prime -y^\prime$ frame. Then, put $j_{y^\prime }=0$, and check for what angles $\theta$ the magnetoresistance fails to saturate.

Fig. 13.9

A simple geometry of current flow

13.4

Consider a band for a simple cubic structure with energy $\varepsilon (k)$ given by $\varepsilon ({\mathbf {k}}) = \varepsilon _0[\cos (k_x a)+\cos (k_y a)+\cos (k_z a)]$, where a is the lattice constant. Let an electron at rest ($k = 0$) at $t = 0$ feel a uniform external electric field ${\mathbf {E}}$ which is constant in time.

• (a) Find the real space trajectory [x(t), y(t), z(t)].

• (b) Sketch the trajectory in the k-space for the electric field ${\mathbf {E}}$ in a [120] direction.

13.5

Consider an electron in a state with a linear energy dispersion given by $\varepsilon ({\mathbf {k}}) = \pm \hbar v_\mathrm{F} |{\mathbf {k}}|$, where ${\mathbf {k}}$ is a two-dimensional wave vector. (It occurs in the low energy states in a graphene–a single layer of graphite.)

• (a) When a dc magnetic field ${\mathbf {B}}$ is applied perpendicular to the graphene layer, write down the area $\mathcal {S}(\varepsilon )$ and sketch $\mathcal {S}(\varepsilon _n)$ for various values of n.

• (b) Solve for the quantized energies $\varepsilon _n$ and plot the resulting $\varepsilon _n$ for $-5\hbar \omega _\mathrm{c} \le \varepsilon _n \le 5\hbar \omega _\mathrm{c}$.

• (b) What can you say about the effective mass of the particle in a graphene subject to the magnetic field ${\mathbf {B}}$?

13.6

Consider two-dimensional electrons with a linear dispersion given by $\varepsilon ({{\mathbf {k}}}) = \hbar v_\mathrm{F} |{\mathbf {k}}|$, where ${\mathbf {k}}$ is a two-dimensional wave vector. Now apply a dc magnetic field B perpendicular to the system. We shall assume that $\tau$ is constant.

• (a) Write down the ${\mathbf {v}}(\varepsilon , s)$ and the periodic part of the position vector ${\mathbf {R}}_\mathrm{p}(\varepsilon , s)$.

• (b) Evaluate the Fourier coefficients ${\mathbf {v}}_n(\varepsilon )$, and discuss the conductivity tensor $\underline{\sigma }$ defined by ${\mathbf {j}}=\underline{\sigma }\cdot {\mathbf {E}}$.

Summary

In this chapter we study behaviors of Bloch electrons in the presence of a dc magnetic field. Energy levels and possible trajectories of electrons are discussed, and simple two band model of magnetoresistance is illustrated including the effect of collisions. General expression of semiclassical magnetoconductivity tensor is derived by solving the Boltzmann equation of the distribution function, and the results are applied to the case of free electrons. The relationship between the local and nonlocal descriptions are discussed. Finally quantum mechanical theory of magnetoconductivity tensor is described and quantum oscillatory behavior in magnetoconductivity of Bloch electrons is compared with its semiclassical counterpart.

In the presence of an electric field ${\mathbf {E}}$ and a dc magnetic field ${\mathbf {B}} (= {\mathbf {\nabla }}\times \mathbf A)$, an effective Hamiltonian is given by $\mathcal {H} = \varepsilon \left( \frac{{\mathbf {p}}}{\hbar } +\frac{e}{\hbar c}\mathbf A\right) -e\phi ,$ where $\varepsilon ({\mathbf {k}})$ is the energy as a function of ${\mathbf {k}}$ in the absence of ${\mathbf {B}}$. The equation of motion of a Bloch electron in k-space takes the form

$$ \hbar \dot{\mathbf{k}} = -e{\mathbf {E}} -\frac{e}{c}{{\mathbf {v}}}\times {\mathbf {B}}. $$

Here ${{\mathbf {v}}} = \frac{1}{\hbar }{{\mathbf {\nabla }}}_{{\mathbf {k}}}\varepsilon ({\mathbf {k}})$ is the velocity of the Bloch electron whose energy $\varepsilon ({\mathbf {k}})$ is an arbitrary function of wave vector ${\mathbf {k}}$. The orbit in real space will be exactly the same shape as the orbit in k-space except that it is rotated by 90$^\circ$ and scaled by a factor $\frac{eB}{\hbar c}$: $\mathbf{k}_\perp = \frac{e{\mathbf {B}}}{\hbar c}\times {\mathbf {r}}_\perp .$ The factor $\frac{eB}{\hbar c}$ is $l_0^{-2}$, where $l_0$ is the magnetic length. The orbit of a particle in ${\varvec{k}}$-space is the intersection of a constant energy surface $\varepsilon ({\varvec{k}})=\varepsilon$ and a plane of constant $k_z$:

$$ \frac{d\varepsilon }{dt}={{\mathbf {\nabla }}}_k\varepsilon \cdot \frac{d{\mathbf {k}}}{dt} =\hbar {{{\mathbf {v}}}}\cdot \left( -\frac{e}{\hbar c}{{\mathbf {v}}}\times {\mathbf {B}}\right) =0. $$

The area of the orbit $\mathcal {A}(\varepsilon , k_z)$ in real space is proportional to the area $\mathcal {S}(\varepsilon , k_z)$ of the orbit in ${\varvec{k}}$-space: $\mathcal {S}(\varepsilon , k_z)=\left( \frac{eB}{\hbar c}\right) ^2 \mathcal {A}(\varepsilon , k_z)$. The area $\mathcal {S}(\varepsilon , k_z)$ is quantized by $\mathcal {S}(\varepsilon , k_z)=\frac{2\pi eB}{\hbar c}(n+\gamma )$ and the cyclotron effective mass is given by $m^*(\varepsilon , k_z) = \frac{\hbar ^2}{2\pi } \frac{\partial S(\varepsilon , k_z)}{\partial \varepsilon }.$ The Bloch electron velocity parallel to the magnetic field becomes

$$ v_z(\varepsilon , k_z)=-\frac{\hbar }{2\pi m^*(\varepsilon ,k_z)} \frac{\partial S(\varepsilon , k_z)}{\partial k_z}. $$

The transverse magnetoresistance is defined by $\frac{R(B_z)-R(0)}{R(0)} = \varDelta R(B_z).$ The simple free electron model gives $\varDelta R(B_z)=0$, which is different from the experimental results.

The current density is given by ${\mathbf {j}}({\mathbf {r}}, t)=\frac{2}{(2\pi )^3}\int (-e){\mathbf {v}} f_1 \, d^3k.$ In the presence of a uniform dc magnetic field ${\mathbf {B}}_0$, the semiclassical magnetoconductivity of an electron gas is written as

$$ \underline{\sigma }=\frac{e^2}{2\pi ^2\hbar ^2}\tau (\varepsilon _\mathrm{F}) \int _\mathrm{F.S.} dk_z \,m^*(k_z) \sum _{n=-\infty }^{\infty }\frac{{\mathbf {v}}_n(\varepsilon _\mathrm{F},k_z) {\mathbf {v}}_n^*(\varepsilon _\mathrm{F}, k_z)}{1+i\tau (\varepsilon _\mathrm{F})[\omega -{\mathbf {q}}\cdot {\mathbf {v}}_\mathrm{s}-n\omega _\mathrm{c}(\varepsilon _\mathrm{F}, k_z)]}, $$

where ${\mathbf {v}}_n(\varepsilon , k_z)$ is defined by

$$ {\mathbf {v}}_n(\varepsilon ,k_z) =\frac{\omega _\mathrm{c}(\varepsilon , k_z)}{2\pi } \int _0^{2\pi /\omega _\mathrm{c}} ds\, {\mathbf {v}}(\varepsilon ,k_z,s)\mathrm{{e}}^{i{\mathbf {q}}\cdot {\mathbf {R}}_\mathrm{p}(\varepsilon ,k_z, s)-in\omega _\mathrm{c}s}. $$

Here $R_\mathrm{p}(\varepsilon ,k_z, s)$ denotes the periodic part of the position vector in real space.

For the free electron model $m^*(k_z)=m$ is a constant independent of $k_z$ and the periodic part of the position vector is given by ${\mathbf {R}}_\mathrm{p}(\varepsilon ,k_z,s)=\frac{v_\perp }{\omega _\mathrm{c}}\left( \sin \omega _\mathrm{c}s,\right. \left. -\cos \omega _\mathrm{c}s, 0\right) .$ For the propagation ${\mathbf {q}} \perp {\mathbf {B}}_0$, i.e. ${\mathbf {q}}=(0,q, 0)$, the nonvanishing components of semiclassical conductivity $\underline{\sigma }$ are

$$ \sigma _{xx} =3\sigma _0\sum _{n=-\infty }^{\infty }\frac{s_n(w)}{1- i\tau (n\omega _\mathrm{c}-\omega )}; \sigma _{yy} =\frac{3\sigma _0}{w^2}\sum _{n=-\infty }^{\infty }\frac{n^2 g_n(w)}{1- i\tau (n\omega _\mathrm{c}-\omega )}; $$

$$ \sigma _{xy} =-\sigma _{yx} =\frac{3\sigma _0 i}{2w}\sum _{n=-\infty }^{\infty }\frac{n g_n^\prime (w)}{1- i\tau (n\omega _\mathrm{c}-\omega )}; \sigma _{zz} =3\sigma _0\sum _{n=-\infty }^{\infty }\frac{c_n(w)}{1- i\tau (n\omega _\mathrm{c}-\omega )}. $$

Here $c_n(w)=\frac{1}{2}\int _{-1}^1 d(\cos \theta )\,\cos ^2\theta J_n^2(w\sin \theta )$, $s_n(w)=\frac{1}{2}\int _{-1}^1 d(\cos \theta )\,\sin ^2\theta [J_n^\prime (w\sin \theta )]^2$, and $g_n(w)=\frac{1}{2}\int _{-1}^1 d(\cos \theta )\, J_n^2(w\sin \theta )$.

In the presence of a vector potential ${\mathbf {A}}_0 = (0,xB_0,0)$, the electronic states are described by $H_0 = \frac{1}{2m}\left[ p_x^2+\left( p_y+\frac{e}{c}B_0 x\right) ^2+p_z^2\right]$. The eigenfunctions and eigenvalues of $H_0$ can be written as

$$ \begin{array}{ll} |\nu>&{}=|n k_y k_z>=\frac{1}{L}\mathrm{{e}}^{ik_yy+ik_zz}u_n\left( x+\frac{\hbar k_y}{m\omega _\mathrm{c}}\right) ,\\ \varepsilon _\nu &{}=\varepsilon _{n,k_y, k_z}=\frac{\hbar ^2 k_z^2}{2m}+\hbar \omega _\mathrm{c}\left( n+\frac{1}{2}\right) . \end{array} $$

The quantum mechanical version of the nonvanishing components of $\underline{\sigma }(q,\omega )$ are given, for the case of ${\mathbf {q}}=(0,q, 0)$, by

$$ \begin{array}{ll} \sigma _{xx}(q,\omega )&{}= \frac{\omega _\mathrm{p}^2}{4\pi i \omega }\left[ 1-\frac{2m\omega _\mathrm{c}}{\hbar }\frac{1}{N}\sum _{n k_y k_z\alpha }^\prime f_0(\varepsilon _{nk_z})\left( \frac{\partial f_{n+\alpha , n}}{\partial q}\right) ^2\frac{\alpha }{\alpha ^2-(\omega /\omega _\mathrm{c})^2}\right] ,\\ \sigma _{yy}(q,\omega )&{}= \frac{i m\omega _\mathrm{p}^2\omega }{2\pi \hbar \omega _\mathrm{c}}\frac{1}{N}\sum _{n k_y k_z\alpha }^\prime f_0(\varepsilon _{nk_z}) f_{n+\alpha , n}^2\frac{\alpha }{\alpha ^2-(\omega /\omega _\mathrm{c})^2},\\ \sigma _{xy}(q,\omega )&{}= -\sigma _{yx}=\frac{i \omega _\mathrm{c}}{2\omega q}\frac{\partial (q^2\sigma _{yy})}{\partial q},\\ \sigma _{zz}(q,\omega )&{}= \frac{\omega _\mathrm{p}^2}{4\pi i \omega }\left[ 1-\frac{2\hbar }{m\omega _\mathrm{c}}\frac{1}{N}\sum _{n k_y k_z\alpha }^\prime f_0(\varepsilon _{nk_z})k_z^2 f_{n+\alpha , n}^2 \frac{\alpha }{\alpha ^2-(\omega /\omega _\mathrm{c})^2}\right] . \end{array} $$

where $f_{n^\prime n}(q_y)$ is the two-center harmonic oscillator integral:

$$ f_{n^\prime n}(q_y) =\int _{-\infty }^\infty u_{n^\prime }\left( x+\frac{\hbar q_y}{m\omega _\mathrm{c}}\right) u_n(x)dx $$

and

$$ X_{n^\prime n}^{(\pm )}(q_y) =(n+1)^{1/2} f_{n^\prime , n+1}(q_y)\pm n^{1/2} f_{n^\prime , n-1}(q_y). $$

The quantum mechanical conductivity tensor is the sum of a semiclassical term and a quantum oscillatory part:

$$ \underline{\sigma }(q,\omega ) = \underline{\sigma }^\mathrm{SC}(q,\omega ) + \underline{\sigma }^\mathrm{QO}(q,\omega ). $$