Confined states and direction-dependent transmission in graphene quantum wells

We report the existence of confined massless fermion states in a graphene quantum well (QW) by means of analytical and numerical calculations. These states show an unusual quasilinear dependence on the momentum parallel to the QW: their number depends on the wave vector and is constrained by electron-hole conversion in the barrier regions. An essential difference with nonrelativistic electron states is a mixing between free and confined states at the edges of the free-particle continua, demonstrated by the direction-dependent resonant transmission across a potential well.

Figure 1

(Color online) A square quantum well on a graphene layer.

Figure 2

Spectrum of confined states in a graphene square QW vs $k_y$ for $U_0=50\mathrm{meV}$, $L=200\mathrm{nm}$, and $m{v}_F^2=0$. The lower inset shows (a) ${\varphi }_A$ and $i{\varphi }_B$ for the state shown by the solid triangle and (b) the related probability density. The upper inset shows the effect of a nonzero mass, for $m{v}_F^2=10\mathrm{meV}$.

Figure 3

As in Fig. 2 but for a parabolic QW.

Figure 4

(Color online) Contour plot of the transmission coefficient of electrons incident on a graphene square well, with energy $E>U_0$, as a function of $\alpha$, for $U_0=50\mathrm{meV}$, $L=200\mathrm{nm}$, and $m=0$.

Figure 5

(Color online) (a) Transmission $T$ vs angle of incidence $\theta$ for different energies as indicated. (b) $T$ vs energy E for $\theta =\pi ∕3$. The maxima are marked by the numbers of the confined states.