Past exam of the quantum condensed matter course of the University of Cambridge / 2015 / Paper 1 / 1

In the isotropic parabolic-band approximation implied by the single supplied curvature, the hole mass from a valence-band maximum isThe hole has charge $+e$, so its cyclotron angular frequency isIf “frequency” means cycles per second, $f_c={\omega }_c/(2\pi )=5.59\times 10^{10}\mathrm{Hz}$. The curvature unit is $\mathrm{J}{\mathrm{m}}^2$, so ${\hslash }^2/E^{\prime \prime }$ has mass units. The external constants used are $\hslash =1.054571800\times 10^{-34}\mathrm{J}\mathrm{s}$, $e=1.6021766208\times 10^{-19}\mathrm{C}$ and $m_e=9.10938356\times 10^{-31}\mathrm{kg}$.^[ref] A genuinely anisotropic band would require the cyclotron mass for the orbit plane, not a single arbitrary directional curvature.

Let the atomic spacing be $a$ and define the nearest-neighbour matrix elements $t_{\alpha }=⟨j,p_{\alpha }|H|j+1,p_{\alpha }⟩$, using the same translated orbital convention on every site. ThenFor the opposite sigma and pi hopping signs in a p-orbital chain, consider $p_z$ first: neighbouring lobes above the chain have the same sign, so the in-phase Bloch sum is bonding. Thus $t_{\pi }<0$ andFor $p_x$, the lobes facing one another have opposite signs for equal site coefficients. Bonding therefore needs alternating coefficients, giving $t_{\sigma }>0$ andEnd-on $\sigma$ overlap is normally stronger than side-on $\pi$ overlap, so the $p_x$ band is wider, with bandwidths $4|t_{\sigma }|$ and $4|t_{\pi }|$. The curve orientations require a stated hopping-sign convention; a Hamiltonian written with an explicit minus sign in front of the hopping uses correspondingly opposite parameter signs.^[ref]

A hydrogenic donor has the effective Bohr radius of a hydrogenic donorusing the external $a_0=0.52917721067\times 10^{-10}\mathrm{m}$.^[ref] Increasing donor density causes donor states to overlap and screens their binding, allowing a metallic impurity system rather than isolated frozen-out donors.

Using the standard approximate Mott criterion, $n_D^{1/3}{a}_B^{\ast }\simeq 0.25$, givesThe factor 0.25 is an externally documented empirical criterion, not supplied exam data or an exact universal threshold.^[ref] An order-of-magnitude overlap argument without that prefactor would give $(a_B^{\ast }{)}^{-3}\simeq 1.3\times 10^{25}{\mathrm{m}}^{-3}$. Compensation, disorder and multivalley anisotropy affect a real material's transition; this estimate uses the scalar mass and permittivity actually supplied, without substituting other germanium parameters.