Problem: State the Poisson-Boltzmann mean-field equation for the electrostatic potential \(\phi(\mathbf x)\) at temperature \(T\) in a dielectric of isotropic permittivity \(\varepsilon\) containing mobile and immobile charge carriers. Explain what it means to apply Debye-Huckel linearization to the Poisson-Boltzmann mean-field equation and state the conditions under which such an approximation is valid.
Solution: It is just Poisson’s electrostatic equation \(\biggr|\frac{\partial}{\partial\mathbf x}\biggr|^2\phi=-\rho_f/\varepsilon\) with the free charge density \(\rho_f=\rho_{\text{mobile}}+\rho_{\text{immobile}}\) and the mobile charge carrier density \(\rho_{\text{mobile}}(\mathbf x)=\sum_i\rho^{\infty}_ie^{-\beta q_i\phi (\mathbf x)}\) given by a mean-field Boltzmann distribution where \(\rho^{\infty}_i=q_in^{\infty}_i\) is the bulk \(|\mathbf x|,1/\phi(\mathbf x)\to\infty\) density of the \(i^{\text{th}}\) species of mobile charge carriers.
To apply Debye-Huckel linearization means Taylor expanding the Boltzmann factors \(e^{-\beta q_i\phi (\mathbf x)}\approx 1-\beta q_i\phi(\mathbf x)\) for \(\beta q_i\phi(\mathbf x)\ll 1\). This results in the Debye-Huckel equation:
\[\left(\biggr|\frac{\partial}{\partial\mathbf x}\biggr|^2-\frac{1}{r^2_D}\right)\phi(\mathbf x)=-\frac{\rho_{\text{immobile}}(\mathbf x)+\sum_i\rho_i^{\infty}}{\varepsilon}\]
with Debye screening length \(r_D:=\sqrt{\varepsilon k_BT/\sum_iq_i^2n_i^{\infty}}\) (note: the Debye screening length is sometimes denoted \(\lambda_D\) which is misleading because it is purely the length scale of an exponential decay, corresponding at best to an imaginary wavelength. Also, it’s a bit strange that the screening potential \(\phi\) is named after both Debye and Huckel but its screening length \(r_D\) is named after Debye only…).
Problem: Show that for a uniform jellium background of charge density \(-\rho^{\infty}\) with an impurity charge \(Q\) at the origin \(r=0\), the classical Debye-Huckel screening potential \(\phi\) around this impurity takes the isotropic Yukawa form:
\[\phi(r)=\frac{Q}{4\pi\varepsilon_0 r}e^{-r/r_D}\]
where the Debye screening length may be written \(r_D:=v_T/\omega_p\) with thermal velocity \(v_T:=\sqrt{k_BT/m}\) and plasma frequency \(\omega_p:=\sqrt{n^{\infty}q^2/\varepsilon m}\).
Solution: In this case, for a single charge carrier species \(q\) (e.g. electrons in a plasma or metal):
\[\rho_{\text{immobile}}(\mathbf x)=Q\delta^3(\mathbf x)-\rho^{\infty}\]
So it boils down to finding the (physical) Green’s function of the Helmholtz operator:
\[\left(\biggr|\frac{\partial}{\partial\mathbf x}\biggr|^2-\frac{1}{r^2_D}\right)\phi(\mathbf x)=-\frac{Q\delta^3(\mathbf x)}{\varepsilon}\]
which yields the answer. Incidentally, this shows the general solution of the Debye-Huckel equation may be written as a convolution with this Yukawa kernel:
\[\phi(\mathbf x)=\frac{1}{4\pi\varepsilon}\int d^3\mathbf x’\frac{\rho_{\text{immobile}}(\mathbf x)+\sum_i\rho_i^{\infty}}{|\mathbf x-\mathbf x’|}e^{-|\mathbf x-\mathbf x’|/r_D}\]
Problem: Consider a \(T=0\) fully degenerate ideal Fermi gas. Often, one is taught that the Fermi energy \(E_F:=\mu(T=0)\) is just the zero-temperature chemical potential. Now, suppose one applies a uniform external potential \(V\) across the entire ideal Fermi gas, so that the Hamiltonian now reads \(H=\sum_i\frac{|\mathbf p_i|^2}{2m}+V\). How do the new Fermi energy \(E’_F\) and zero-temperature chemical potential \(\mu'(T=0)\) of this gauge-shifted Hamiltonian compare to the original \(V=0\) case.
Solution: The Fermi energy is gauge invariant, so in particular \(E’_F=E_F\) (think “Fermi kinetic energy”). By contrast, the chemical potential is gauge covariant, so \(\mu'(T=0)=\mu(T=0)+V\).
Problem: A key property of the Debye screening length is that \(r_D\propto (n^{\infty})^{-1/2}\). However, this was a classical treatment of screening. By performing an analogous quantum treatment of screening, show that the corresponding Thomas-Fermi screening length \(r_{TF}\propto (n^{\infty})^{-1/6}\).
Solution: One has to make the following assumptions:
- (Fully degenerate Fermi gas) \(T=0\) and the charge carriers have to actually be fermions (usually electrons).
- (Local Density Approximation) The potential \(\phi(\mathbf x)\) varies on length scales much longer than the de Broglie wavelength \(\lambda_F\) of fermions on the Fermi surface, i.e. \(|\partial\phi/\partial\mathbf x|\ll\phi(\mathbf x)/\lambda_F\). Note \(\lambda_F=\sqrt{\pi}\lambda_{T_F}\) is almost but not quite the same thing as the thermal de Broglie wavelength \(\lambda_{T_F}\) at the Fermi temperature \(T_F\)). Intuitively, this means one can approximate the potential \(\phi(\mathbf x)\) by a suitable “staircase” where the size of each stair is \(\gg\lambda_F(\mathbf x)\propto n^{-1/3}(\mathbf x)\) but also \(\ll\phi(\mathbf x)/|\partial\phi/\partial\mathbf x|\).
- (Linear Response) \(|q\phi(\mathbf x)|\ll E_F^{\infty}\)
- (Mean-Field) Same as Poisson-Boltzmann (namely, neglecting electron-electron many-body exchange correlations, just working with the mean, self-consistent field \(\phi(\mathbf x)\)).
- (Jellium Background) Same as the above Debye-Huckel model.
Then, the idea is that the electron number density will be non-uniform \(n=n(\mathbf x)\). By the local density approximation, one can coarse grain the domain into boxes of suitable size (which will be modelled as point-like \(\mathbf x\) but not thought of like that), such that each box \(\mathbf x\in\mathbf R^3\) is a locally uniform ideal Fermi gas in the presence of an external mean-field potential \(\phi(\mathbf x)\) (this should be read as saying “each box \(\mathbf x\) receives its own tailored uniform potential \(\phi(\mathbf x)\)”). However, because these boxes are all connected to each other and can thus exchange particles, at equilibrium there will be some global uniform value of chemical potential \(\mu=E_F^{\infty}\) across the entire system. The local density \(n(\mathbf x)\) competes with the local potential \(\phi(\mathbf x)\) in such a way that their sum is conserved:
\[E_F^{\infty}=E_F(\mathbf x)+q\phi(\mathbf x)\]
Whereas for Poisson-Boltzmann one had \(n(\phi)=n^{\infty}e^{-\beta q\phi}\), here the analogous relation is thus \(n(\phi)=\frac{1}{3\pi^2}\left(\frac{2m(E^{\infty}_F-q\phi)}{\hbar^2}\right)^{3/2}\), and whereas applying Debye-Huckel linearization to Poisson-Boltzmann meant linearizing the exponential, here it means linearization via the binomial expansion \(n(\phi)\approx n^{\infty}\left(1-\frac{3q}{2E^{\infty}_F}\phi\right)\) which is valid thanks to the linear response assumption above. Note the \(3\)-dimensional density of states per unit volume at the Fermi surface is \(g_V(E_F^{\infty})=3n^{\infty}/2E_F^{\infty}\) s0 one can equivalently write \(n(\phi)\approx n^{\infty}-qg_V(E_F^{\infty})\phi\), and from here the rest of the analysis is identical to the above (e.g. \(\rho_{\text{immobile}}(\mathbf x)=Q\delta^3(\mathbf x)-\rho^{\infty}\) is the exact same thing), and one ends up with the same Yukawa form for the Thomas-Fermi screened potential:
\[\phi(r)=\frac{Q}{4\pi\varepsilon_0 r}e^{-r/r_{TF}}\]
only now the Thomas-Fermi screening length \(r_{TF}:=\sqrt{\varepsilon/q^2g_V(E_F^{\infty})}=\left(\frac{\pi^4\hbar^6\varepsilon^3}{3q^6m^3n^{\infty}}\right)^{1/6}\). This is related to the Debye screening length by:
\[\frac{r_D^2}{r_{TF}^2}=\frac{3}{2}\frac{T}{T^{\infty}_F}\]
Hence for typical metals at room temperature \(T\ll T^{\infty}_F\sim 10^3\text{ K}\), quantum mechanics (specifically the Pauli exclusion principle) weakens the effect of screening \(r_{TF}\gg r_D\) compared to classical predictions.
Problem: