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  • Letter
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Theory for tunnel magnetoresistance oscillation

Keisuke Masuda1,*, Thomas Scheike1, Hiroaki Sukegawa1, Yusuke Kozuka2, Seiji Mitani1,3, and Yoshio Miura1,4,5

  • *Contact author: MASUDA.Keisuke@nims.go.jp
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Phys. Rev. B 111, L220406 – Published 9 June, 2025

DOI: https://doi.org/10.1103/PhysRevB.111.L220406

Abstract

The universal oscillation of the tunnel magnetoresistance (TMR) ratio as a function of the insulating barrier thickness in crystalline magnetic tunnel junctions (MTJs) is a long-standing unsolved problem in condensed matter physics. To explain this, we here introduce a superposition of wave functions with opposite spins and different Fermi momenta, based on the fact that spin-flip scattering near the interface provides a hybridization between majority- and minority-spin states. In a typical Fe/MgO/Fe MTJ, we solve the tunneling problem and show that the TMR ratio oscillates with a period of ∼3Å by varying the MgO thickness, consistent with previous and present experimental observations.

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Supplemental Material

References (48)

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  37. We also considered a similar superposition of wave functions in the incident and reflection waves. This additional analysis clarified that these do not affect the oscillatory behavior of the transmittance given by the superposition in the transmitted wave. Thus, we omit these effects for simplicity.
  38. We used the same continuation conditions as mentioned in the Supplemental Material.
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  42. In our results, differences in both the shape and phase between the TP,↑−1 and TAP,↑−1 oscillations provide the TMR oscillation. Although the oscillation shapes of TP,↑−1 and TAP,↑−1 look quite similar, these have a slight difference owing to the difference in the effective mass and the replacement of the coefficients (u and v) in the expression of the transmittance.
  43. TMR ratios at low temperature are more than twice as high as those at room temperature and can be reproduced by using a smaller value of α.
  44. The amplitude of the RA oscillation hardly changes as the barrier thickness d increases, consistent with the results in Fig. 2. In our theory, the amplitude of the transmittance oscillation is determined by the values of u and v in Eq. (7). Since u and v are assumed to be constant, the amplitude of the oscillation hardly changes with increasing d. This assumption is reasonable, since the interfacial exchange interaction determining u and v is independent on d.
  45. Note that the phase difference in RA is smaller than that in the inverse of the transmittance in Fig. 2; however, this does not mean that our theory is inconsistent with experimental observations [5, 6, 8, 9, 10, 11]. It is experimentally known that the phase difference highly depends on the quality of the sample and is not a universal feature for the TMR oscillation.
  46. We introduced nonoscillatory terms βe−2κd decreasing exponentially as the barrier thickness increases.
  47. For a better comparison with experimental results, we applied a shift in d by 0.8 Å for Eq. (12).
  48. Experimentally, the shape of the TMR oscillation is different for different samples. Actually, Mg4AlOx-based MTJs have saw-tooth-like shapes [10] similarly to our present results, while MgO-based MTJs have sinelike shapes [9, 11]. These different shapes can be reproduced by tuning the parameters in our model.

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Phys. Rev. B 111, L220406– Published 9 June, 2025
  • Received 17 June 2024
  • Revised 25 March 2025
  • Accepted 23 May 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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