Magnetic-Field-Tunable Valley-Contrasting Pseudomagnetic Confinement in Graphene

Introducing quantum confinement has uncovered a rich set of interesting quantum phenomena and allows one to directly probe the physics of confined (quasi-)particles. In most experiments, however, an electrostatic potential is the only available method to generate quantum dots in a continuous system to confine (quasi-)particles. Here we demonstrate experimentally that inhomogeneous pseudomagnetic fields in strained graphene can introduce exotic quantum confinement of massless Dirac fermions. The pseudomagnetic fields have opposite directions in the two distinct valleys of graphene. By adding and tuning real magnetic fields, the total effective magnetic fields in the two valleys are imbalanced. By that we realized valley-contrasting spatial confinement, which lifts the valley degeneracy and results in field-tunable valley-polarized confined states in graphene. Our results provide a new avenue to manipulate the valley degree of freedom.

Figure 1

(Pseudo-)magnetic confinement in graphene. Schematic of electron transmission with inhomogeneous magnetic fields $B$ (a), inhomogeneous pseudomagnetic fields $B_S$ (b), and superposition of inhomogeneous pseudomagnetic fields and homogeneous external magnetic fields (c). The pseudomagnetic field has opposite signs in the two valleys. $l_B$ ($l_B^{\prime }$) is the magnetic length of (quasi-) particles in the $K$ ($K^{\prime }$) valley under (pseudo-)magnetic fields.

Figure 2

(a) A three-dimensional STM image of rippled regions in graphene monolayer ($V_{\text{sample}}=30\mathrm{mV}$ and $I=300\mathrm{nA}$). (b) Enlarged STM image showing clearly ripple structures ($V_{\text{sample}}=-60\mathrm{mV}$ and $I=40\mathrm{pA}$). Inset: Atomic resolution STM image of the ripple. Bottom: Schematic of the cross section of three adjacent ripples. (c) A typical $dI/dV$ spectrum recorded at blue dot in panel (b), showing ${\mathrm{pLL}}_0$. (d) A typical $dI/dV$ spectrum recorded at red dot in panel (b), showing a series of confined states. (e) and (f) The experimental and theoretical negative second derivative of differential conductance map versus the spatial position obtained along the black arrow in panel (b), respectively. The dark blue lines denote the boundaries between the wide and narrow ripples. Red arrows denote the peaks for pseudomagnetic confined states. Green and dark blue arrows denote the ${\mathrm{pLL}}_0$.

Figure 3

Imaging pseudomagnetic confined states. (a) and (b) Schematic plot of the LDOS for the electric potential and magnetic confinement with a fixed $k_x$. The black line in (a) and dark blue line in (b) denote the configuration of the electric potential field and magnetic field along the $y$ direction, respectively. The red lines in (a),(b) represent squares of corresponding wave functions. (c) STS maps recorded at different energies of the confined states in Fig. 2. The white lines denote the boundaries between the wide and narrow ripples. The scale bar is 10 nm. Note that the fine oscillatory signal extending over the total map area is due to experimental noise (See Fig. S5 [39]).

Figure 4

Field-tuned valley-contrasting pseudomagnetic confinement. (a) Experimental (left panel) and calculated (right panel) differential conductance maps of the confined states as a function of magnetic fields. The black (red) dashed lines guide the trend of bound states for the valley $K$ ($K^{\prime }$). The green dashed lines denote emergent ${\mathrm{LL}}_0$. The dark blue lines denote the two lowest bound states near the Dirac point. (b) and (c) The calculated spatial distribution for the $K$ and $K^{\prime }$ valleys at $B=10\mathrm{T}$. The dark blue lines denote the boundaries between the wide and narrow ripples.