Disorder Suppression in Topological Semiconductor-Superconductor Junctions

arXiv:2207.05109 · cond-mat.mes-hall · Submitted 2022-07-11 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Disorder Suppression in Topological Semiconductor-Superconductor Junctions".

Mira: The gist Disorder in a proximitizing bulk superconductor can scatter quasiparticles in a putative topological superconductor and eventually destroy the topological superconducting state<ref:2207.05109#pg2>.

Kai: First, who's behind it and why it matters.

Paper summary: Mira: Thinking about the title, "Disorder Suppression in Topological Semiconductor-Superconductor Junctions," it really tells us that the main challenge here is managing disorder when trying to get topological superconductivity via proximity coupling. The authors Stuart N. Thomas, Sankar Das Sarma, and Jay D Sau are using a scattering approach and random-matrix calculations for this work one <ref:2207.05109#pg1>.

Kai: And what this means in practice is that they show how to suppress the disorder scattering from the bulk superconductor by tuning the Fermi momenta between the semiconductor and superconductor one <ref:2207.05109#pg1,Fermi momenta between the semiconductor and superconductor>. They also point out that near perfect Andreev reflection at these interfaces signals that we've hit a condition where the proximity effect is very clean one <ref:2207.05109#pg1,near perfect Andreev reflection at>.

Lev: From an engineering standpoint, this suggests that if we can engineer a junction where those Fermi momentum ratios are right, we might be able to operate in a regime where the topological state is more robust against the inevitable disorder in our fabrication process.

Mira: Exactly. They also showed that classical estimates, which view electrons as random walks due to collisions with impurities, turn out to be qualitatively consistent with their results one <ref:2207.05109#pg1>. That's good because it gives us a simpler intuition for what's happening at the fundamental level.

Kai: So, the paper shows that the disorder isn't just noise; it can actually be managed or even suppressed in this specific configuration. This work helps us understand how to design junctions where we get the clean proximity effect we need for topological physics one <ref:2207.05109#pg1>.

Conclusion: Kai: So, we're wrapping up on this paper about disorder suppression in topological semiconductor-superconductor junctions one. Basically, they're looking at how getting electrons scattered by impurities inside the bulk superconductor messes up that delicate topological state they're trying to create one.

Mira: Right. They use random-matrix theory and look at how different materials interact—like the density of states and the tunnel coupling—to figure out what kind of scattering rate you get one.

Lev: From where I sit with error correction, I wonder if this suppression holds up when you try to build something real with finite temperature or noise one. Does this clean interface effect actually survive in a noisy lab setting?

Kai: It seems the authors found that by tuning things, specifically the Fermi wave-length mismatch between the semiconductor and superconductor, they can really suppress that bulk disorder scattering one. They show that near perfect Andreev reflection at these interfaces is a good sign that you've hit those conditions without needing an external magnetic field one.

Mira: It’s interesting because it points to a specific requirement—that Fermi momentum mismatch—to get that clean proximity effect where the topological state can survive even with some disorder one. It tells us which assumptions in the model are actually necessary for this clean result.

Lev: If we can engineer those material properties to hit that mismatch condition, then maybe it makes running low-error systems much easier because the interface itself becomes less of a source of decoherence one. But what about when you add a real magnetic field?

Kai: That's actually what they looked at too. They showed that applying a small magnetic field breaks time-reversal symmetry and can suppress some of that diagonal Andreev reflection, which is another part of the scattering one.

Mira: So it’s not just about the zero-field case; there are other ways to manipulate the scattering process using external fields one. It shows that disorder suppression isn't a one-size-fits-all thing; it depends on how you tune the system.

Lev: That means for experimentalists, we have a clear target—find those specific material parameters that give you the best disorder protection for your junction design one. We need to know if building this actual structure with those specific conditions is feasible.

Stuart N. Thomas, *Sankar Das Sarma, Jay D. Sau

Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland

cond-mat.mes-hall

Submitted: 2022-07-11

Updated: 2022-07-11

Comments: 13 pages, 2 figures

Journal ref: Phys. Rev. B 106, 174501 (2022)

DOI: 10.1103/PhysRevB.106.174501

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist Disorder in a proximitizing bulk superconductor can scatter quasiparticles in a putative topological superconductor and eventually destroy the topological superconducting

Key concepts

Random-Matrix Theory Approach to Scattering Rate
This mathematical method is used to calculate how often electrons scatter off impurities in the junction. It involves composing scattering matrices from different regions—the interface, a diffusive region, and the superconductor—to find the normal reflection probability and thus determine the effective scattering rate.
Disordered Superconducting Coherence Length ($\xi_d$)
This length describes how far an electron can travel before its superconducting properties are significantly affected by disorder. It is defined as $\sqrt{\xi_0\lambda}$, where $\xi_0$ is the clean coherence length and $\lambda$ is the mean free path. This concept helps define the characteristic scale for disorder effects in the superconducting material.
Magnetic Field Enhancement of Disorder Scattering
Applying a small magnetic field breaks time-reversal symmetry, which introduces a momentum shift ($\delta e_k$) to electron trajectories. This shift allows for an estimate of the mean free path induced by the magnetic field ($\lambda_{mag}$). This effect can substantially influence topological superconductivity if the ratio of coherence length to this magnetic scattering length is comparable to unity.

Terminology

Summary

The gist

Disorder in a proximitizing bulk superconductor can scatter quasiparticles in a putative topological superconductor and eventually destroy the topological superconducting state<ref:2207.05109#pg2>.

Model of Josephson junction

The paper uses a random-matrix theory approach to understand the density and tunnel coupling dependence of disorder scattering from the proximitizing superconductor<ref:2207.05109#pg2>. It models a planar Josephson junction with a short semiconducting region, where strong coupling allows one to cross into a topological phase by tuning the phase difference between the superconductors to π<ref:2207.05109#pg2>. The effective model for the right interface of the JJ is shown in Fig. 1b<ref:2207.05109#pg2>.

Random-Matrix Theory Approach to Scattering Rate

The calculation involves composing scattering matrices of the SMN interface, the diffusive region, and the superconductor to solve for the normal reflection probability ree2 and thus the effective scattering rate according to Eq. 1<ref:2207.05109#pg3>. The zero-energy conductance resulting from Andreev reflection is bounded by GNS > 4e2h/Nsm - 4Tr h rLr†L<ref:2207.05109#pg4>. The total disorder scattering from the bulk superconductor is a combination of the above normal reflection as well as the off-diagonal part of the Andreev reflection matrix<ref:2207.05109#pg5>.

Scattering Rates at Real Material Interfaces

The disordered superconducting coherence length is defined as ξd = √ξ0λ where ξ0 = vF,sc/∆ is the clean superconducting coherence length and λ is the mean free path<ref:2207.05109#pg5>. Setting the normal region length to the disordered superconducting coherence length ξd yields Z2 ≈ Nsm/N rξ0λ<ref:2207.05109#pg5>. The resulting minimum bound for the mean free path is λ ≥ 242 m∗sm/m∗scµsm∆k−1F,sc<ref:2207.05109#pg5>.

Classical Estimate of Scattering Rates

The results for both the planar Josephson junction as well as the thin superconducting shell turn out to be qualitatively consistent with a semi-classical estimate of Andreev scattering<ref:2207.05109#pg7>. This classical model views electrons as classical particles executing a random walk as a result of collisions with impurities<ref:2207.05109#pg7>. The total probability of a scattered electron reflecting into the semiconductor is P (e → e) = 2/π kF,sm/kF,sc rξ0λ<ref:2207.05109#pg7>.

Magnetic Field Enhancement of Disorder Scattering

The application of a small magnetic field breaks time-reversal symmetry and adds a momentum δek = qe(δr ×B) over the electron part of the trajectory r(t)<ref:2207.05109#pg8>. The average magnitude of the momentum shift provides an estimate of the mean-free path from this magnetic-field induced scattering, i.e. λmag = δk2/2<ref:2207.05109#pg8>. This field can have a significant effect on topological superconductivity if the ratio ξ0/λmag ∼ 2qeBξ0q/hδr2⊥i is comparable to unity<ref:2207.05109#pg8>.

Numerical Evaluation of Diagonal Andreev Reflection

A direct numerical simulation of Andreev reflection using Kwant shows that the diagonal part of the Andreev reflection dominates the scattering process in the time-reversal invariant case with large momentum mismatch<ref:2207.05109#pg9>. The numerical results for the no-field case shown in Fig. 3 agree quite well with the random matrix result in Eqs. 14 and 13<ref:2207.05109#pg9>. Application of a magnetic field (green dots in Fig. 3) is enough to suppress diagonal Andreev reflection substantially<ref:2207.05109#pg9>.

Conclusion

Using a combination of scattering theory and random-matrix calculations, the effects of disorder scattering in the bulk superconductor are suppressed as the ratio of the Fermi wave-length in the semiconductor to that in the superconductor<ref:2207.05109#pg8>. The demonstration of near perfect Andreev reflection at a semiconductorsuperconductor interface is a good indicator that such a condition has been met in the absence of a magnetic field<ref:2207.05109#pg9>.

Acknowledgments

We thank Anton Akhmerov for valuable discussions that alerted us to the importance of Fermi momentum mismatch in suppressing disorder scattering<ref:2207.05109#pg9>. We also thank David Goldhaber Gordon for discussions of controlling disorder scattering at the semiconductor/superconductor interface that motivated this work<ref:2207.05109#pg9>. Stuart Thomas thanks the Joint Quantum Institute at the University of Maryland for support through a JQI fellowship<ref:2207.05109#pg9>. JS acknowledges NSF DMR – 1555135 for support<ref:2207.05109#pg9>. This work is also supported by the Laboratory for Physical Sciences through its continuous support of the Condensed Matter Theory Center at the University of Maryland<ref:2207.05109#pg9>.

Appendix A

Using the Haar measure, we can calculate the integrals of unitary matrix moments as Weingarten functions [19]<ref:2207.05109#pg10>. For leading order in N, the Weingarten function of a permutation σ can be expressed as a product over cycle lengths Ci<ref:2207.05109#pg10>.

Appendix B

The Andreev reflection matrix for time-reversal invariant systems can be written in terms of rLr†L as reh = −i(1 − rLr†L1 + rLr†L!)<ref:2207.05109#pg11>. This gives the derivation of Eq. 12.

Appendix C

Following Beenakker [25], the Andreev reflection matrix reh for zero-energy Andreev reflection of electrons coming from the left in Fig. 1b is given by reh = −itLR (1 + r∗RrR)−1 t∗RL. This describes the total Andreev reflection of the SMN-SC system.

Appendix D

The average transmission eigenvalue of the combined metal-barrier region is TNB = ΓTr 1 − V†q(1 − T0N) (1 − Γ)V∗−1 × V†T0N V. The full geometric sum then becomes TNB = (T0N−1 + Γ−1 − 1)−1<ref:2207.05109#pg4>.

References

[48] T. D. Stanescu and S. D. Sarma, Proximity-induced superconductivity generated by thin films: Effects of fermi surface mismatch and disorder in the superconductor (2022).

[49] M. Aghaee, A. Akkala, Z. Alam, R. Ali, A.

Improvements for AI systems

  1. A new AI system can accurately predict topological superconducting states in semiconductor-superconductor (SM-SC) heterostructures by using a random-matrix theory approach to estimate disorder scattering time, as described in We use a scattering approach and a random-matrix calculation to estimate the disorder scattering time in a topological Josephson junction.

  2. The improved system can distinguish between regimes where topological superconductivity is destroyed and where it is suppressed, specifically by analyzing the ratio of Fermi momenta between the semiconductor and superconductor to predict if the disorder scattering rate from the bulk of the superconductor, even in the strong coupling limit, is suppressed in the ratio of Fermi momenta.

  3. The AI can serve as a tool for material design by calculating a minimum mean free path required for a topological gap to exist, using Combining Eqs. 1, 11 and 15 gives a resulting minimum bound for the mean free path: λ ≥ (242 m∗sm/m∗sc µsm/kF,sc − 1).

  4. The system can perform high-fidelity simulations of Andreev reflection probability by implementing a numerical evaluation using Kwant [44], enabling it to verify if the diagonal part of the Andreev reflection dominates the scattering process in the time-reversal invariant case with large momentum mismatch.

  5. The AI can predict how external fields influence disorder suppression, specifically by estimating the magnetic field required to eliminate protection, noting that the ratio ξ0/λmag ∼ 2qeBξ0/q hδr 2⊥i (20) is comparable to unity for certain system parameters.

  6. The AI can provide a unified view of previous theoretical results by applying both random-matrix theory and semiclassical estimates, as the paper demonstrates that "the combination of random-matrix theory and semiclassical approaches can overcome the computational challenge associated with the large length-scale difference between the Fermi wavelength of the superconductor and the coherence length in the semiconductor."

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