Spontaneous Time-Reversal Symmetry Breaking at Individual Grain Boundaries in Graphene

Graphene grain boundaries (GBs) have attracted interest for their ability to host nearly dispersionless electronic bands and magnetic instabilities. Here, we employ quantum transport and universal conductance fluctuation measurements to experimentally demonstrate a spontaneous breaking of time-reversal symmetry across individual GBs of chemical vapor deposited graphene. While quantum transport across the GBs indicate spin-scattering-induced dephasing and hence formation of local magnetic moments, below $T\lesssim 4\mathrm{K}$ we observe complete lifting of time-reversal symmetry at high carrier densities ($n\gtrsim 5\times {10}^{12}{\mathrm{cm}}^{-2}$) and low temperature ($T\lesssim 2\mathrm{K}$). An unprecedented thirtyfold reduction in the universal conductance fluctuation magnitude with increasing doping density further supports the possibility of an emergent frozen magnetic state at the GBs. Our experimental results suggest that realistic GBs of graphene can be a promising resource for new electronic phases and spin-based applications.

Figure 1

(a) Scanning electron micrograph of a typical pair of graphene grains with grain size $\approx 15\mu \mathrm{m}$ forming a GB in between. Scale bar, $10\mu \mathrm{m}$ (b) Bright field TEM image of the GB formed between two grains. The selected area electron diffraction pattern in the inset shows the misorientation angle between the grains $\approx 23°$. Scale bar, 5 nm. (c) HRTEM image of the GB region where line and point defects are outlined. Scale bar, 1 nm.

Figure 2

(a) Schematic showing (a1), a pair of crossings representing diffusons; (a2), a pair of crossings representing cooperons; and (a3), the expected behavior of $\nu (B)$ as a function of $B$ for TRS-invariant and TRS-broken systems. (b) Sheet resistance ($R_{\square }$) as a function of gate voltage ($V_{\mathrm{BG}}$) for intragrain (SG) and intergrain (GB) regions of D1 at $T=0.3\mathrm{K}$. Inset: Optical micrograph of a typical device. The morphology of the original pair of coalesced grains is shown (black line) along with the approximate GB location (red line). Scale bar, $10\mu \mathrm{m}$. (c) Magnetoconductance measurements are shown for $n=-6\times {10}^{12}{\mathrm{cm}}^{-2}$ at $T=0.3\mathrm{K}$ for the SG region and $T=0.3$, 0.8, and 4 K for the GB region, clearly exhibiting WL. Dashed lines correspond to HLN fits. (d) Quantum correction to conductivity $\mathrm{\Delta }{\sigma }_c$ in units of $e^2/\pi h$ plotted for both SG and GB regions as a function of $T$ for $n=-1\times {10}^{12}{\mathrm{cm}}^{-2}$ and $-6\times {10}^{12}{\mathrm{cm}}^{-2}$. (e) Scattering rate $\gamma$ normalized to its value ${\gamma }_{40\mathrm{K}}$ at $T=40\mathrm{K}$, plotted versus $T$ for the SG and GB regions at $n=-6\times {10}^{12}{\mathrm{cm}}^{-2}$, is shown, where the black dotted line indicates the temperature regime where Nyquist scattering dominates.

Figure 3

(a) Conductivity fluctuations for GB region of D1 at $n=-0.8\times {10}^{12}{\mathrm{cm}}^{-2}$ for $B=0\mathrm{mT}$ (pink) and $B=47\mathrm{mT}$ (purple), clearly indicating a reduction in the fluctuation magnitude at $B\gg B_{\varphi }$. (b) $\nu (B)$ plotted for three different $n$ at $T=0.3\mathrm{K}$ for D1 showing spontaneous TRS breaking at zero field as $n$ is increased. Solid lines are fits to Eq. (1). (c) $\nu (B)$ for the SG and GB regions plotted at $T=0.3\mathrm{K}$ for D1 at $n=-7\times {10}^{12}{\mathrm{cm}}^{-2}$ indicating that spontaneous TRS breaking occurs only in the presence of a GB. (d) Noise reduction factor $\nu (B=0)$ for the SG and GB regions of D1 (circles) measured at $T=0.3\mathrm{K}$ (darker) and D2 (diamonds) at $T=0.3\mathrm{K}$ (darker) and $T=4.5\mathrm{K}$ (lighter) as a function of $n$.

Figure 4

(a) Schematic describing the contribution of diffuson (blue) and cooperon (red) singlet ($|S⟩$) and triplet ($|T_1⟩$, $|T_2⟩$ and $|T_3⟩$) states to the UCF magnitude $⟨\delta G_{\varphi }^2⟩$ in different symmetry classes. Valley hybridization leads to a factor of 4 reduction in $⟨\delta G_{\varphi }^2⟩$, while magnetic impurities reduce $⟨\delta G_{\varphi }^2⟩$ by a further factor of 8 due to gapping of spin diffuson triplets and all cooperons. (b) The variance in conductance $⟨\delta G_{\varphi }^2⟩$ within a phase-coherent box of $l_{\varphi }^2$ normalized to its value at the Dirac point $\delta ⟨G_0^2⟩$ as a function of $n$ at $T=0.3\mathrm{K}$, showing a factor of $\approx 4$ reduction in the SG region and a factor of $\approx 30$ reduction in the GB region.

Figure 5

(a) $\nu (B)$ plotted for D2 at $n=6.1\times {10}^{12}{\mathrm{cm}}^{-2}$ showing spontaneous TRS breaking at zero field only at temperatures $T\lesssim 2\mathrm{K}$. (b) $T$ dependence of the reduced sheet resistance ${\rho }_{\square }$ (in units of $h/e^2$) averaged from resistance fluctuations measurements at $B=0\mathrm{T}$ for SG (blue) and GB (red) regions at $n=-6\times {10}^{12}{\mathrm{cm}}^{-2}$ for D1. (c) Normalized variance $N=S_{\sigma }/{\sigma }^2$ at $B=0$ as a function of temperature is plotted for GB region of D1 at $n=-6.9\times {10}^{12}{\mathrm{cm}}^{-2}$, clearly indicating a sharp increase in $N(B=0)$ at lower temperatures.