Fulde-Ferrell states in inverse proximity-coupled magnetically doped topological heterostructures

Moon Jip Park, Junyoung Yang, Youngseok Kim, and Matthew J. Gilbert

Phys. Rev. B 96, 064518 – Published 17 August 2017

Abstract

We study the superconducting properties of the thin film BCS superconductor proximity coupled to a magnetically doped time-reversal invariant topological insulator (TI). Using mean-field theory, we show that Fulde-Ferrell (FF) pairing can be induced in the conventional superconductor through the inverse proximity effect (IPE). This occurs when the IPE of the TI to the superconductor is large enough that the normal bands of the superconductor possess a proximity induced spin-orbit coupling and magnetization. We find that the energetics of the different pairings are dependent on the coupling strength between the TI and the BCS superconductor and the thickness of the superconductor film. As the thickness of the superconductor film is increased, we find a crossover from the FF pairing to the BCS pairing phase. This is a consequence of the increased number of the superconducting bands, which favor the BCS pairing, implying that the FF phase can only be observed in the thin film limit. In addition, we also propose transport experiments that show distinct signatures of the FF phase.

Received 21 March 2017

DOI:https://doi.org/10.1103/PhysRevB.96.064518

Physics Subject Headings (PhySH)

FFLO

Topological materials Unconventional superconductors

Mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Moon Jip Park^1,2, Junyoung Yang¹, Youngseok Kim^2,3, and Matthew J. Gilbert^2,3

¹Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
²Micro and Nanotechnology Laboratory, University of Illinois, Urbana, Illinois 61801, USA
³Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA

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Vol. 96, Iss. 6 — 1 August 2017

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Figure 1
The schematic figure of the magnetically doped TI superconductor heterostructure. On the top of the TI surface, a thin film of the BCS superconductor is deposited. The magnetization points out the parallel direction to the surface of the TI to shift the location of the Dirac cone in momentum space.
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Figure 2
The local density of the state in momentum space at the TI and the superconductor with $m = 0.4$ eV and $t_{c} = 0.5$ eV. (a) and (b) The LDOS of the TI and the first layer of the superconductor when $t_{c} = 0$ . The magnetically ordered dopants shift the Dirac cone along the $x$ direction because the magnetism is aligned in the $x$ direction. As $t_{c}$ is turned on, the IPE starts to hybridize the Fermi surface. (c) and (d) The LDOS and the spin texture. We find the nonzero spin-orbit coupling and the effective Zeeman field in the superconductor. (e)–(h) The LDOS of the TI surface and the first, second, and third layer of the superconductor. We find that the Fermi surface starts to recover the isotropy and the IPE decays as we look further away from the surface of the TI.
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Figure 3
(a) The energy contour of the ground state energy as a function of $q$ and $m$ . The region with $q = 0$ corresponds to the BCS energy. As $m$ increases, we find the minimum of the energy occurs at nonzero $q$ which signals the FF ground state. The blue line shows the evolution of the location of the minimum as $m$ increases. We find that $q_{min}$ increases as $m$ increases. (b) The calculation of $q_{min}$ with various values of $t_{c} = 0.3, 0.6, 1$ eV. We find a clear linear dependence of $q_{min}$ with respect to $m$ . As $t_{c}$ increases the slope of the line increases due to the enhanced IPE. (c) By sweeping all possible value of $q_{min}$ , we determine the pairing of the ground state with different values of $t_{c}$ and $m$ . We find that the stronger $t_{c}$ and $m$ enhance the stability of the FF phase.
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Figure 4
The evolution of the ground state energy difference between the FF phase and the BCS phase. The negative value indicates the FF phase is more energetically favored. (a) The energy difference when $t_{c} > t_{m} = (0.1)$ eV. We find that the energy gain of having the FF phase linearly decreases as $N_{layer}$ increases. The different value of $t_{c}$ sets the initial energy gain when $N_{layer} = 1$ . The red, green, and blue indicates the value of $t_{c} = 0.3, 0.6, 1$ eV, respectively. (b) The same plot when $t_{c} < t_{m} = (1 eV)$ . The linear dependence disappears in the small $N_{layer}$ , however the same trend still holds in large $N_{layer}$ . (c) The calculation of the energy difference between the interface FF pairing and the BCS pairing. Unlike the homogeneous FF, we find that the energy difference saturates as $N_{layer}$ increases, indicating that the interface FF might still survive in large $N_{layer}$ limit. The red, green, and blue indicates the value of $t_{m} = 0.1, 0.3, 0.6$ eV, respectively.
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Figure 5
(a) The schematic figure of the Josephson junction setup. On the top of the magnetically doped TI-superconductor junction, the normal insulator barrier is deposited, and the another superconductor is placed on the top of the normal insulator. The four terminal current is placed on the top of the superconductors. The two are attached on the different superconductors to drive the Josephson current. The other two are attached on the top BCS superconductor to drive the current in a direction parallel to the FF momentum. (b) Numerically calculated Josephson current as a function of the transverse momentum $q_{per}$ . The blue, black, and red lines represent the Josephson current in the BCS pairing, the homogeneous FF pairing, and the interface FF pairing, respectively. We find that the blue (BCS) line has the maximum located at the $q_{per} = 0$ and the black (homogenous FF) line has the maximum located at the $q_{per} = q = 0.3$ . The red (interface FF) line which has the peak at the $q_{per} = 0$ are the interface FF phase with $N_{layer} = 2$ .
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Figure 6
(a) A schematic of the Andreev interferometer using Y-junction method to measure the unconventional superconductivity. (b) A plot of conductance as a function of finite momentum angle $θ$ of the FF phase. From the outermost to innermost, we plot the calculation results with different $q$ where $| q | = 0, π / 8, π / 6, and π / 4$ , respectively. We use a value of $v_{F} = 6.61 \times 10^{5}$ m/s for the Fermi velocity, which has been extracted from the metal Hamiltonian parameter, and plot the conductance at the incident electron the energy of $E = 0.5 Δ$ , where the $Δ$ is superconducting gap. We set the barrier height at the interface of the metal arms and superconductor to be transparent.
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