Normal metal-superconductor action (revealing Andreev reflection)

Question

Short Example

A conventional Josephson junction, which consists of a superconductor-insulator-superconductor (SIS) sandwich and is biased by a constant current $I$, has the following action: $$S_{SIS}\left[\phi(t)\right]=\int_{t_i}^{t_f} dt\left[\frac{(\hbar \dot\phi)^2}{4E_C}+E_J\left(\cos\phi+\frac{I}{I_c}\phi\right)\right]$$ where $\phi(t)$ is a generalized coordinate, whose meaning is the difference between an order parameter phase $\phi_1$ of the first superconductor and a phase $\phi_2$ of the second one (i.e. $\phi = \phi_1-\phi_2$).

If one needs to include quasiparticle dissipation, a second-quantized hamiltonian can be used to determine the effective action:

$$S_{SIS}^{\text{eff}}\left[\phi(t),\chi(t)\right]=\int_{t_i}^{t_f} dt\left[-\frac{\hbar ^2\ddot\phi}{2E_C}\cdot\chi+E_J\left(\cos(\phi+\chi/2)-\cos(\phi-\chi/2)+\frac{I}{I_c}\chi\right)\right]+S_{SIS}^{\text{dissipation}}\left[\phi(t),\chi(t)\right]$$

where $\phi(t)$ is now an average phase difference and $\chi(t)$ characterizes some fluctuations near the average. I don't write a large expression for $S_{SIS}^{\text{dissipation}}$ explicitly, but it's known quantity (for details see Gerd Schön, A.D. Zaikin, the formulas (3.47), (3.48), (3.70)).

My Question

I wonder how to write down an (effective?) action $S_{NS}$ describing normal metal-superconductor (NS or NIS) connection biased by a voltage/current. I don't understand Green functions yet, so I'd like to see the answer in a form similar to the equations above.

Possible but Incomplete Answer

So far, I've found something describing Andreev conductance (Keldysh action for disordered superconductors, the formula (6.8)):

$$S_{A}\left[\phi(\omega),\chi(\omega)\right]=\frac{iG_A}{2}\int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\hbar\omega\sin\chi(\omega)\cdot\\\cdot\left[\sin\left\{\phi(\omega)-\phi(-\omega)\right\}\cos\chi(-\omega)+\coth\left(\frac{\hbar\omega}{2T}\right)e^{i\phi(-\omega)-i\phi(\omega)}\sin\chi(-\omega)\right]\tag{6.8}$$

where $G_A$ is Andreev conductance, $T$ is temperature of the superconductor, $\phi$ (or $\theta_1$) is an average superconducting phase and $\chi$ (or $\theta_2$) characterizes some fluctuations near the average.

As can be seen, the dimension of the action $S_A$ is not $[\hbar]$. If I understand right the formulas (6.11) and (6.13) from the same article, $G_A$ is dimensionless. Then the dimension of $S_A$ is $[\hbar/t^2]$.

Moreover, there is no integration over time in $S_A$, so a real action $S_{NS}\left[\phi(t),\chi(t)\right]$ has to be determined somehow from $S_A$. It seems a procedure I need was used to obtain (6.11):

$$S_A\left[\phi(t),\chi(t)\right]=-eG_A\int_{-\infty}^{+\infty}dtV(t)\chi(t)+o[\chi]\tag{6.11}$$

where $V(t) = \frac{\hbar}{2e}\cdot\frac{d\phi(t)}{dt}$ is the voltage bias. But the first order on $\chi$ is not sufficient for me, so I have to understand what steps to perform from (6.8) to (6.11). Are they some integral transformation $S_{NS}=\int dt\int dt'S_A f(t,t')$?

Answer 1

The following answer still can be improved (see its end).

According to the article, an action describing Andreev conductance is:

$$S_A\left[\phi(t),\chi(t)\right]=\frac{iG_A}{4}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\omega\left[i\left(e^{i\phi}\cos\chi\right)_{-\omega}\left(e^{-i\phi}\sin\chi\right)_{\omega}-\\-i\left(e^{-i\phi}\cos\chi\right)_{-\omega}\left(e^{i\phi}\sin\chi\right)_{\omega}+2\coth\left(\frac{\omega}{2T}\right)\left(e^{i\phi}\sin\chi\right)_{-\omega}\left(e^{-i\phi}\sin\chi\right)_{\omega}\right]\tag{6.8}$$

where $G_A$ is dimensionless Andreev conductance, $T$ is dimensionless temperature of the superconductor, $\phi$ (or $\theta_1$) is an average superconducting phase and $\chi$ (or $\theta_2$) characterizes some fluctuations near the average.

As can be seen in my question above, my first attempt to understand this formula led to several mistakes:

• All frequencies $\omega$ are made dimensionless after dividing them by some characteristic frequency $\omega_c$, i.e. $\frac{\omega}{\omega_c}\rightarrow\omega$. The same is done with $T$, i.e. $\frac{T}{\hbar\omega_c}\rightarrow T$. So the original action is also dimensionless and measured in units of $\hbar$;
• $\left(e^{i\phi}\cos\chi\right)_{\pm\omega}$ and similar expressions are not products of frequency components $e^{i\phi(\pm\omega)}\cdot\cos\chi(\pm\omega)$, but rather Fourier transforms of $e^{i\phi(t)}\cos\chi(t)$: $$\left(e^{i\phi(t)}\cos\chi(t)\right)_{\pm\omega}=\int_{-\infty}^{+\infty}dt\ e^{i\phi(t)}\cos\chi(t)\cdot e^{\pm\omega t}$$ where $t$ is dimensionless, since $\omega$ is dimensionless ($\omega_c t\rightarrow t$).

Taking into account these corrections, the action can be rewritten as:

$$S_A\left[\phi(t),\chi(t)\right]=\frac{iG_A}{2}\int_{-\infty}^{+\infty}dt\int_{-\infty}^{+\infty}dt'\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\omega\ e^{-i\omega(t-t')}\cdot\\ \cdot\left[-\cos\chi(t)\sin\chi(t')\sin\left\{\phi(t)-\phi(t')\right\}+\coth\left(\frac{\omega}{2T}\right)e^{i\phi(t)-i\phi(t')}\sin\chi(t)\sin\chi(t')\right]$$

To get the formula (6.11) one implements smallness of $\chi$:

$$S_A\left[\phi(t),\chi(t)\right]\approx-\frac{iG_A}{2}\int_{-\infty}^{+\infty}dt\int_{-\infty}^{+\infty}dt'\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\omega\ e^{-i\omega(t-t')}\chi(t')\sin\left\{\phi(t)-\phi(t')\right\}$$

and uses the following relations:

$$\frac{d}{dt'}\delta(t-t')=i\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\omega\ e^{-i\omega(t-t')}$$

$$\int_{-\infty}^{+\infty}dt'f(t')\frac{d}{dt'}\delta(t-t') = -\int_{-\infty}^{+\infty}dt'\delta(t-t')\frac{d}{dt'}f(t')$$

Finally:

$$S_A\left[\phi(t),\chi(t)\right]\approx-\frac{G_A}{2}\int_{-\infty}^{+\infty}dt\chi(t)\frac{d}{dt}\phi(t)=-\frac{e G_A}{\hbar\omega_c}\int_{-\infty}^{+\infty}dt\ V(t)\chi(t)$$

To return dimensionality one needs to substitute $t\rightarrow\omega_c t$, $S_A\rightarrow\frac{S_A}{\hbar}$:

$$S_A\left[\phi(t),\chi(t)\right]\approx-e G_A\int_{-\infty}^{+\infty}dt\ V(t)\chi(t)\tag{6.11}$$

At $T\to 0$, but not small $\chi$:

$$S_A\left[\phi(t),\chi(t)\right]\approx-e G_A\int_{-\infty}^{+\infty}dt\ V(t)\sin\chi(t)\cos\chi(t) - \frac{i\hbar G_A}{2\pi}\int_{-\infty}^{+\infty}dt\int_{-\infty}^{+\infty}dt'\ \frac{\sin\chi(t)\sin\chi(t')}{(t-t')^2}e^{i\phi(t)-i\phi(t')}\tag{*}$$

However, there are still some gaps in my understanding:

• What is that characteristic $\omega_c$ used in the article?
• How to write $S_A$ for finite times $t_i$ and $t_f$? Is it sufficient just to change the integration limits: $\int_{-\infty}^{+\infty}dt\rightarrow\int_{t_i}^{t_f}dt\ $? It seems not so straightforward, because the time integral is obtained through the inverse Fourier transform.
• Is the action of Andreev conductance $S_A$ the whole story of NS-junction? There might be other terms corresponding to different phenomena: $S_{NS} = S_A + S_{\text{other effects}}$

But at the moment the equation ($^*$) is enough for my goals.