Band structure and effective mass

(available at http:tiny.cc/solidstatephys/Effective_mass.ipynb)

Last week

a chain of atoms with one orbital per atom has a tight-binding Hamiltonian $$H = h_0 + h_1 e^{ika} + h_1 e^{-ika}$$ Test question: how does this Hamiltonian look if we choose a unit cell with two atoms in it?

This week

• Motion of electrons in electric field
• Effective mass

Reminder: I use $\hbar = 1$, so that $p = k$.

Problem formulation

• Periodic atomic potential + electric field $V = V_0(x) + eEx$
• Electron energies belong to only a single band, kinetic energy $H = -2 h_1 \cos(ka)$ (or any other $E(k)$)
• Works when $E$ is small, $eEa \ll t$

Question: how do electrons move?

Begin with velocity

Electron velocity is $$ v = \frac{d E(k)}{d k} $$ several ways to understand:

• Electrons are waves, this is wave group velocity
• Hamiltonian mechanics $$ \frac{dx}{dt} = \frac{\partial H}{\partial p} $$

For the simplest example $$ v = 2h_1a\sin(ka) $$

Effect of the force

Conservation of energy after moving by $\delta x$ after $\delta t$: $$ \delta E(k) = -e E \delta x $$

$$ \frac{d E(k)}{d k}\delta k = -e E \delta x $$$$ v\frac{\delta k}{\delta t} = -e E \frac{\delta x}{\delta t} $$$$ \frac{d k}{d t} = -eE = F$$

Once again, this is just Hamiltonian mechanics!

Bloch oscillations

Let's solve equations of motion: $$k = F t$$ $$v = 2 h_1 a \sin(Fta)$$ $$x = 2 \frac{h_1}{F}\cos(Fta)$$

$\Rightarrow$ motion of electrons is periodic!

Very hard to observe, requires the relaxation time $\tau \gg (Fa)^{-1}$.

Acceleration

$$ a = \frac{dv}{dt} $$

using $v = d E(k)/d k$ we get

$$ a = \frac{d^2 E}{dk^2} \frac{dk}{dt} = \frac{d^2 E}{dk^2} F $$

Now compare with the Newton's law $a = F/m$.

$\Rightarrow$ we introduce effective mass $$m_{\text{eff}} = \left(\frac{d^2 E}{dk^2}\right)^{-1}$$ Electrons behave like their mass is momentum-dependent.

Effective mass properties

For the single band model: $$ m_{\text{eff}} = [2 h_1 a^2 \cos(ka)]^{-1}$$

• When $k\ll a^{-1}$: $m_{\text{eff}} = 1/2h_1a^2$, almost constant
• When $|k| > \pi/2$: $m_{\text{eff}} < 0$, these are holes (topic of next lecture), they accelerate opposite to the force applied.
• In real materials $m_{\text{eff}}$ varies from $\sim 1000 m_e$ to $\sim 0.01 m_e$.

Conclusions

• Electric field increases momentum at a constant rate
• Acceleration is controlled by effective mass
• Half of a band is holes that have negative mass.