F Second Order Co Tunneling Rate at Zero Temperature

  The second order co-tunneling rate can be calculated analytically for zero temperature. Starting from (2.44) for N=2, the co-tunneling rate may be written as
$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \iiii... ...}^4(1-f(\omega_i)) \,d\omega_1\,d\omega_2\,d\omega_3\,d\omega_4. \end{multline}$$
At zero temperature the Fermi functions are either zero or one and may therefore be moved in the boundaries of the integrals.
$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \iiii... ...\omega_3+\omega_4) \,d\omega_1\,d\omega_2\,d\omega_3\,d\omega_4. \end{multline}$$
The integration over $\omega_4$ is eliminated by the $\delta$-function. The condition $\omega_4=eV_b-\omega_1-\omega_2-\omega_3$ demanded by the $\delta$-function and the integration regions of $\omega_x>0$ give the inequalities
$$\begin{gather}\omega_3<eV_b-\omega_1-\omega_2, \qquad \omega_2<eV_b-\omega_1, \qquad \omega_1<eV_b, \end{gather}$$
for the new upper boundaries of integration.
$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \int_... ...+\omega_2}+ \frac{1}{\Delta F_2+eV_b-\omega_1-\omega_2}\right)^2 \end{multline}$$
Integrating over $\omega_3$ results in
$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \int_... ...a F_2+eV_b-\omega_1-\omega_2}\right)^2 (eV_b-\omega_1-\omega_2). \end{multline}$$
Substituting $u=\omega_1+\omega_2$ gives
$$\begin{gather}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \int_0^... ...ac{1}{\Delta F_1+u}+\frac{1}{\Delta F_2+eV_b-u}\right)^2 (eV_b-u). \end{gather}$$
The integrand can be changed to a partial sum
$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \int_... ...(\frac{1}{\Delta F_1+u}+\frac{1}{\Delta F_2+eV_b-u}\right)\Biggr] \end{multline}$$

$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \int_... ...{\Delta F_2+eV_b-\omega_1}{\Delta F_2}\right\vert\right) \Biggr] \end{multline}$$

$$\begin{multline}\Gamma^{(2)}=\frac{\hbar}{2\pi}\frac{1}{e^4 R_{T1}R_{T2}} \Bigg... ...2\ln\left\vert 1+\frac{eV_b}{\Delta F_2}\right\vert\right)\Biggr] \end{multline}$$
And finally collecting the logarithms yields
$$\begin{gather}\Gamma^{(2)}\arrowvert_{T=0}=\frac{\hbar V_b}{2\pi e^3R_{T1}R_{T2}... ...n \left\vert 1+\frac{eV_b}{\Delta F_i}\right\vert\right)-2\right]. \end{gather}$$