Screening of an Impurity Charge: Friedel Oscillations

We want to study the response of an electron gas to a static impurity with charge $Ze$. At the level of random phase approximation, the change in the electron distribution induced by the impurity charge is given by

$$\delta \rho(r) = - \frac{Ze}{2 \pi^2} \frac{1}{r} \int_0^{\i... ...) \frac{q_{TF}^2 \pi_0 (q/k_F)}{q^2 + q_{TF}^2 \pi_0 (q/k_F)},$$ (1)

where $q_{TF}$ is the Thomas-Fermi screening length, and $\pi_0(x)$ is the Lindhard function for density-density correlation and is given by

$$\pi_0 (x) = \frac{1}{2} - \frac{1}{2x} \left( 1- \frac{x^2}{4} \right) \log \left\vert \frac{1- x/2}{1 + x/2} \right\vert.$$ (2)

We note that the above function is non-analytic at $x=2$. This is due to low energy electron-hole excitations in a degenerate electron gas at wave-vector $2 k_F$. These processes are important in screening the impurity charge, and they give rise to the Friedel oscillations. The regular part of the integral in equation (1) gives the usual Thomas-Fermi screening, which has the asymptotic ( $r \rightarrow \infty$) form $\delta \rho (r) \sim - Ze q_{TF}^2 (4 \pi r)^{-1} e^{- q_{TF}r}$. In this exercise we will calculate only the irregular part of the integral in equation (1) which shows the Friedel oscillations. After two partial integrals we get

$$\delta \rho (r) = \frac{Ze}{2 \pi^2} \frac{1}{r^3} \int_0^{\infty } dq \sin(qr) \frac{d^2}{d q^2} \left[ \frac{q q_{TF}^2 \pi_0}{q^2 + q_{TF}^2 \pi_0} \right]$$

$$= - \frac{Ze}{2 \pi^2} \frac{1}{r^3} \int_0^{\infty } dq \sin(qr) \frac{d^2}{d q^2} \left[ \frac{q^3}{q^2 + q_{TF}^2 \pi_0 (q/k_F)} \right].$$ (3)

We will keep track of the most singular term which is

$$\frac{d^2}{d q^2} \left[ \frac{q^3}{q^2 + q_{TF}^2 \pi_0 (q/... ...)^2}{(q^2 + q_{TF}^2 \pi_0)^2} \pi_0^{\prime \prime } (q/k_F),$$

with

$$\pi_0^{\prime \prime } (x) \approx - \frac{1}{8} P \frac{1}{1- x/2}$$

around $x=2$. Close to $q = 2 k_F$ we get,

$$\delta \rho (r) \approx - \left( \frac{Ze}{2 \pi^2} \right) \left( \frac{q_{TF}^2/k_F^3}{... ... \int_{2 k_F- \Lambda }^{2 k_F + \Lambda } dq \sin (qr) P \frac{1}{1 - q/2 k_F}$$

$$= \left( \frac{Ze}{\pi^2} \right) \left( \frac{q_{TF}^2/k_F^2} {(4 ... ... \frac{1}{r^3} \int_{- \Lambda }^{\Lambda } dy \sin (2 k_Fr + yr) P \frac{1}{y}$$ (4)

where $\Lambda$ is a finite cut-off. The integrand $\sin (2 k_Fr) \cos (yr) /y$ drops out because it is odd over the range of integration. We also have

$$\lim_{r \rightarrow \infty } P \int_{- \Lambda }^{\Lambda } dy \sin (yr) \frac{1}{y} = \pi$$ (5)

for any finite $\Lambda$. Using these results we get

$$\lim_{r \rightarrow \infty } \delta \rho (r) \approx \frac{Z... ...2/k_F^2}{(4 + q_{TF}^2/2k_F^2)^2} \frac{1}{r^3} \cos (2 k_Fr).$$ (6)