Lifting of the Landau level degeneracy in graphene devices in a tilted magnetic field

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Lifting of the Landau level degeneracy in graphene devices in a tilted magnetic field

F. Chiappini, S. Wiedmann, K. Novoselov, A. Mishchenko, A. K. Geim, J. C. Maan, and U. Zeitler

Phys. Rev. B 92, 201412(R) – Published 24 November 2015

Abstract

We report on transport and capacitance measurements of graphene devices in magnetic fields up to 30 T. In both techniques, we observe the full splitting of Landau levels and we employ tilted field experiments to address the origin of the observed broken symmetry states. In the lowest energy level, the spin degeneracy is removed at filling factors $ν = \pm 1$ and we observe an enhanced energy gap. In the higher levels, the valley degeneracy is removed at odd filling factors while spin polarized states are formed at even $ν$ . Although the observation of odd filling factors in the higher levels points towards the spontaneous origin of the splitting, we find that the main contribution to the gap at $ν = - 4, - 8$ , and $- 12$ is due to the Zeeman energy.

Received 15 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.201412

Authors & Affiliations

F. Chiappini^1,*, S. Wiedmann¹, K. Novoselov², A. Mishchenko², A. K. Geim², J. C. Maan¹, and U. Zeitler^1,†

¹High Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands
²Department of Physics, University of Manchester, M13 9PL Manchester, United Kingdom

^*f.chiappini@science.ru.nl
^†u.zeitler@science.ru.nl

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Vol. 92, Iss. 20 — 15 November 2015

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Images

Figure 1
Transport and capacitance measurements at 1.4 K. (a) $R_{x x}$ (black line) and $R_{x y}$ (gray line) as a function of the back gate voltage $V_{BG}$ at 25 T. Inset: $R_{x x}$ as a function of $V_{BG}$ at 0 T. (b) $C$ as a function of the top gate voltage $V_{TG}$ at 15 T. The curve inside the red box is expanded five times and shifted in order to match the constant background of the original curve. The numbers close to the minima of $R_{x x}$ and $C$ indicate the filling factors. Inset: $C$ as a function of $V_{TG}$ at 0 T.
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Figure 2
Activation gaps $Δ_{ν}$ as a function of $B$ for $ν = - 1, - 3, - 4, - 7$ , and $- 8$ . The gray dashed line indicates $E_{Z}$ calculated with $g = 2$ , and the solid lines are the linear (green) and square root (blue) fits to the data. Inset: $R_{x x}$ minima of $ν = - 1$ as a function of temperature at 25 T (black triangles) and 30 T (orange circles), and the solid lines are a fit to the experimental data according to the Fermi-Dirac distribution. The relatively large error bars for gaps exceeding 20 K are due to the fact that the temperature range used (1.4–18 K) was not large enough to access them more accurately.
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Figure 3
Splitting of the higher Landau levels in a tilted magnetic field at 1.4 K. (a) $R_{x x}$ as a function of the back gate voltage at $B_{⊥} = 8.4$ T and $θ = 0^{\circ}$ (black line) and at $θ = 73 . 7^{\circ}$ (red line). Inset: Sketch of the tilting configuration. (b) $R_{x x}$ for $N = - 1$ and $- 2$ at $B_{⊥} = 14.1$ T and $θ = 0^{\circ}$ (black line) and $θ = 59 . 1^{\circ}$ (red line). (c) Splitting of $N = 1$ observed in the capacitance signal at $B_{⊥} = 13.5$ T and $θ = 0^{\circ}$ (black line), $θ = 43 . 1^{\circ}$ (red line), and $θ = 63 . 2^{\circ}$ (blue line). (d) Activation gaps for $ν = - 4$ (black squares), $ν = - 8$ (orange circles), and $ν = - 12$ (blue triangles) as a function of $B_{T}$ at $B_{⊥} = 10.4$ T. The gray dashed line represents $E_{Z}$ for $g = 2$ . (e) $g$ as a function of $ν$ at $B_{⊥} = 10.4$ T (top panel) and 11.3 T (bottom panel). (f) $Δ_{- 3}$ as a function of $B_{T}$ at $B_{⊥} = 20$ T (blue triangles), 22.5 T (green squares), and 25 T (orange circles).
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Figure 4
Splitting of $N = 0$ in a tilted magnetic field. (a) $R_{x x}$ at $ν = - 1$ in $B_{⊥} = 14.1$ T and $θ = 0^{\circ}$ (black line) and $θ = 59 . 1^{\circ}$ (red line). (b) Activation gap for $ν = - 1$ as a function of $B_{T}$ at different $B_{⊥}$ ; the gray dotted line is $E_{Z}$ calculated with $g = 2$ . (c) Value of the resistance maximum at $ν = 0$ in a tilted magnetic field as a function of $B_{T}$ at different $9.4 \leq B_{⊥} \leq 22.5$ T and 4.2 K. (d) $C$ at $B_{⊥} = 10.1$ T and $θ = 0^{\circ}$ (black line) and $θ = 59 . 7^{\circ}$ (red line).
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