Parabolic E-k and quantized linear metal

E–k diagram is parabolic

In the free electron theory, the valence electrons of a metal move freely inside the metal; the potential inside is taken as constant ($V = 0$). All the energy is kinetic:

$$E = \frac{1}{2}mv^2 = \frac{p^2}{2m}$$

By de Broglie, $p = h/\lambda = \hbar k$ where $k = 2\pi/\lambda$ is the wave number:

$$E = \frac{\hbar^2k^2}{2m}$$

The same follows from the Schrödinger equation with $V = 0$: $\psi = Ae^{jkx}$ gives $-\frac{\hbar^2}{2m}(-k^2)\psi = E\psi$.

Since $\hbar^2/2m$ is constant, $E \propto k^2$: the E–k curve is a parabola, symmetric about $k = 0$.

   E
   ^
   |  \                 /
   |   \               /     E = (hbar k)^2 / 2m
   |    \             /
   |     `.         .'
   |       `-.___.-'
   +------------|-------------> k
                0

The slope gives velocity $v = \frac{1}{\hbar}\frac{dE}{dk} = \frac{\hbar k}{m}$, and the curvature gives $m^* = m$.

Energy in a linear metal is quantized

Consider a one-dimensional (linear) metal of length $L$, e.g. a thin wire. Electrons move freely inside ($V = 0$) but cannot leave it, because of the large surface barrier (work function). So it is a potential well with $V = \infty$ outside.

Inside, the Schrödinger equation is

$$\frac{d^2\psi}{dx^2} + k^2\psi = 0, \qquad k^2 = \frac{2mE}{\hbar^2}$$

with solution $\psi = A\sin kx + B\cos kx$. Boundary conditions:

• $\psi(0) = 0 \Rightarrow B = 0$
• $\psi(L) = A\sin kL = 0 \Rightarrow kL = n\pi$, so $k_n = \dfrac{n\pi}{L}$, $n = 1, 2, 3, \ldots$

Therefore

$$E_n = \frac{\hbar^2k_n^2}{2m} = \frac{n^2h^2}{8mL^2}$$

Only certain values of $k$ (spaced $\pi/L$ apart) and energy are allowed: the energy is quantized. The parabola above is really a set of closely spaced points.

Spacing of levels

For $L = 1$ cm, $E_1 = \dfrac{(6.626\times10^{-34})^2}{8\times9.1\times10^{-31}\times(10^{-2})^2} = 6.03\times10^{-34}$ J $= 3.8\times10^{-15}$ eV. The levels are so close that in a macroscopic metal the energy appears continuous, but strictly it is quantized; quantization becomes important only when $L$ is in the nanometre range (quantum wires).