Andreev Reflection at NS Interfaces

Hello! Let's continue our journey into the fascinating world of superconductivity.

In our last lesson, we saw how the formation of the BCS ground state leads to a gapped density of states (DOS), $N_S(E)$. We established that for a Normal-Insulator-Superconductor (NIS) junction, the differential conductance $dI/dV$ is proportional to $N_S(eV)$. This means that for energies within the gap ($|eV|<\Delta$), single-particle tunneling is forbidden, and the conductance drops to zero.

This raises a natural question: what happens if the interface is not an insulating tunnel barrier but a clean, direct contact between a normal metal (N) and a superconductor (S)? Can current flow below the gap? The answer is yes, through a remarkable process known as Andreev reflection. Today, we will explore the mechanism of this process, its theoretical underpinnings via the Bogoliubov-de Gennes equations, and its striking experimental signature.

1. The Mechanism: Electron-to-Hole Conversion

Imagine an electron in a normal metal with energy $ϵ$ (measured from the Fermi level, $E_F$) approaching a clean N-S interface. If its energy is below the superconducting gap ($ϵ<\Delta$), it cannot enter the superconductor as a single quasiparticle because there are no available states.

Instead of being simply reflected, a novel process occurs:

1. The incident electron (charge $-e$, momentum $\mathbf{k}$, spin $\uparrow$) pairs up with another electron from the normal metal's Fermi sea (charge $-e$, momentum $-\mathbf{k}$, spin $\downarrow$).
2. This pair enters the superconductor as a single Cooper pair, which joins the collective condensate.
3. To conserve charge, momentum, and spin, the act of removing the second electron from the Fermi sea creates a hole in the normal metal. This hole has charge $+e$, momentum $\approx -\mathbf{k}$, and energy $-ϵ$.
4. This hole propagates back into the normal metal along the path of the incident electron. This is called retro-reflection, a key feature distinguishing it from specular reflection where the angle of incidence equals the angle of reflection.

The net result is that a charge of $2e$ has been transferred into the superconductor. In the normal metal, the incident electron current (moving towards the interface) is replaced by a reflected hole current (also moving towards the interface, since a hole with momentum $-\mathbf{k}$ has a group velocity opposite to an electron with momentum $-\mathbf{k}$). This leads to a doubling of the current for the same applied voltage.

The following video provides an excellent conceptual visualization of this process.

2. Theoretical Framework: The BTK Model

To describe this process quantum mechanically, we need a formalism that can handle both electrons and holes simultaneously. This is achieved through the Bogoliubov-de Gennes (BdG) equations, which are a pair of coupled Schrödinger-like equations for the electron-like ($u$) and hole-like ($v$) components of a quasiparticle wavefunction.

A seminal model that applies this formalism to the N-S interface was developed by Blonder, Tinkham, and Klapwijk (BTK). The BTK model treats the problem as a 1D scattering problem at the interface, which can have a barrier of variable strength, modeled by a delta-function potential $H\delta (x)$. The strength of this barrier is encapsulated in a dimensionless parameter $Z=H/\hslash v_F$.

Given your background, you may appreciate a more formal development of the theory. The first resource below provides an elegant derivation of the Andreev equations from the more general Bogoliubov-de Gennes framework. The second resource applies this to the specific BTK model to calculate the scattering probabilities.

By solving the scattering problem, the BTK model yields the probabilities for:

• Andreev Reflection (A): The incident electron is converted into a reflected hole.
• Normal Reflection (B): The incident electron is reflected as an electron.
• Transmission (C): A quasiparticle is transmitted into the superconductor (only possible for $ϵ>\Delta$).

A crucial result from this analysis (which is also derived in the "Andreev bound states" paper, section (i)) is that for a perfect interface ($Z=0$) and an incident electron with energy $ϵ<\Delta$, the probability of Andreev reflection is exactly one: $A(ϵ)=1$. The electron has no other channel to go through, so it must be Andreev reflected.

3. Experimental Signature: Sub-gap Conductance Enhancement

The most direct experimental consequence of Andreev reflection is its effect on the electrical conductance of the N-S junction.

The total current flowing from the normal metal into the junction is proportional to the number of electrons that effectively cross the boundary.

• A normally transmitted electron contributes 1 unit of charge.
• An Andreev-reflected electron results in a charge of $2e$ entering the superconductor, so it contributes 2 units.
• A normally reflected electron contributes 0.

The total charge transmission coefficient can be written as $1+A(E)-B(E)$. At zero temperature, the differential conductance $G_{NS}=dI/dV$ is directly proportional to this coefficient evaluated at energy $E=eV$.

Let's consider the two limiting cases for an ideal interface ($Z=0$):

1. Sub-gap voltage ($|eV|<\Delta$): As we established, Andreev reflection is perfect. $A=1$ and $B=0$.
   $G_{NS}=2G_N$
   The conductance is doubled compared to the normal state conductance $G_N$ (the conductance the junction would have if the superconductor were in its normal state).

2. Above-gap voltage ($|eV|>\Delta$): Normal transmission of single quasiparticles becomes the dominant process. The probability of Andreev reflection $A$ drops to zero.
   $G_{NS}\approx G_N$
   The conductance returns to its normal value.

This sharp doubling of the sub-gap conductance is the classic signature of Andreev reflection. In real experiments, the interface is never perfect ($Z>0$), and thermal effects cause broadening. The BTK model beautifully captures this transition from perfect Andreev reflection ($Z=0$) to standard NIS tunneling (large $Z$).

This video segment shows experimental data and explains how the BTK model accounts for real-world measurements.

Conclusion

Today we've uncovered the fascinating process that governs charge transport across a clean normal-superconductor interface. By moving beyond the single-particle picture, we can understand how current flows even when no single-particle states are available.

Key Takeaways:

• Andreev Reflection is a scattering process at an N-S interface where an incident electron ($ϵ<\Delta$) is retro-reflected as a hole, transferring a charge of $2e$ (a Cooper pair) into the superconductor.
• This process is described theoretically by the Bogoliubov-de Gennes (BdG) equations, and the Blonder-Tinkham-Klapwijk (BTK) model provides a quantitative description of the reflection and transmission probabilities as a function of energy and interface barrier strength ($Z$).
• The primary experimental signature of Andreev reflection is the doubling of the differential conductance for sub-gap bias voltages ($|eV|<\Delta$) in a clean N-S junction, a direct consequence of the two-particle nature of the current.

Preview of the Next Lesson

So far, we have assumed that the superconducting gap $\Delta$ is isotropic (the same in all directions), which is characteristic of conventional s-wave superconductors. However, many of the most interesting and technologically relevant superconductors, such as the high-temperature cuprates, are unconventional, with anisotropic gaps that have nodes (points or lines on the Fermi surface where the gap is zero). In our next lesson, we will contrast conventional s-wave pairing with unconventional d-wave pairing and discuss how experimental techniques like ARPES can directly visualize these different gap structures.