A2.14

Quantum Physics
Part II, 2004

(i) A simple model of a crystal consists of an infinite linear array of sites equally spaced with separation bb. The probability amplitude for an electron to be at the nn-th site is cn,n=0,±1,±2,c_{n}, n=0, \pm 1, \pm 2, \ldots. The Schrödinger equation for the {cn}\left\{c_{n}\right\} is

Ecn=E0cnA(cn1+cn+1)E c_{n}=E_{0} c_{n}-A\left(c_{n-1}+c_{n+1}\right)

where AA is real and positive. Show that the allowed energies EE of the electron must lie in a band EE02A\left|E-E_{0}\right| \leq 2 A, and that the dispersion relation for EE written in terms of a certain parameter kk is given by

E=E02AcoskbE=E_{0}-2 A \cos k b

What is the physical interpretation of E0,AE_{0}, A and kk ?

(ii) Explain briefly the idea of group velocity and show that it is given by

v=1dE(k)dk,v=\frac{1}{\hbar} \frac{d E(k)}{d k},

for an electron of momentum k\hbar k and energy E(k)E(k).

An electron of charge qq confined to one dimension moves in a periodic potential under the influence of an electric field E\mathcal{E}. Show that the equation of motion for the electron is

v˙=qE2d2Edk2,\dot{v}=\frac{q \mathcal{E}}{\hbar^{2}} \frac{d^{2} E}{d k^{2}},

where v(t)v(t) is the group velocity of the electron at time tt. Explain why

m=2(d2Edk2)1m^{*}=\hbar^{2}\left(\frac{d^{2} E}{d k^{2}}\right)^{-1}

can be interpreted as an effective mass.

Show briefly how the absence from a band of an electron of charge qq and effective mass m<0m^{*}<0 can be interpreted as the presence of a 'hole' carrier of charge q-q and effective mass m-m^{*}.

In the model of Part (i) show that

(a) for k212/b2k^{2} \ll 12 / b^{2} an electron behaves like a free particle of mass 2/(2Ab2)\hbar^{2} /\left(2 A b^{2}\right);

(b) for (π/bk)212/b2(\pi / b-k)^{2} \ll 12 / b^{2} a hole behaves like a free particle of mass 2/(2Ab2)\hbar^{2} /\left(2 A b^{2}\right).