Graphene Electron States in a Quantum Well

The paper reports on theoretical estimation for electron spectra of free-standing graphene monolayer in presence of a rectangular quantum well. We have shown that 1D quasi-localized states localized mainly in the quantum well may exist, forming a peculiar quantum “rod” in a grapheme monolayer. Conventional hole tunneling through the quantum well is also described in addition to Klein tunneling usually considered in such structure.


Introduction
Since the first report of successful fabrication by pulling out of atomic monolayer out of bulk graphite and its description [1,2] graphene has been attracting unfading interest as a fascinating system for fundamental studies and a promising candidate material for various future applications [3] (see also the review article [4]). Using technological tailoring or electric field a potential relief can be created in a graphene layer which gives new opportunities for its application.
Graphene is a mono-atomic layer of carbon atoms arranged on a 2D honeycomb lattice. Near each corner of the hexagonal first Brillouin zone (also called Dirac points or K-points) the quasiparticle excitations obey a 2D linear dispersion relation and behave like massless "relativistic" particles. That leads to a number of unusual electronic properties; one of them is so called Klein paradox [5] well known in quantum electrodynamics. It predicts that the electron can perfectly pass through the potential barrier independently of its height in contrast to the conventional nonrelativistic tunneling where the transmission probability exponentially decays with the barrier height increasing. As was shown in [6,7] the similar phenomenon takes place in graphene (see also [8]). So some electron states in graphene with a quantum well (which can be considered as a barrier for holes) are those corresponding just to that Klein (or "chiral") tunneling in both directions. Klein chiral tunneling in graphene has been further studied by various authors (see, e. g., [9][10][11]). However, chiral tunneling states, though quite essential, are far from exhausting the whole spectrum of graphene electron-hole excitations in presence of a quantum well (or a quantum barrier). In that paper we make an attempt to study all variety of allowed states in such a system, using a rectangular quantum well (QW) as a model potential relief. The detailed theoretical study of electron spectrum even in such a simple structure has not been performed yet. Underscore that we consider the QW inside formally unbounded graphene layer, not touching upon tunneling out from graphene through its edge [12].

Basic Premises
Electron excitations in graphene near Dirac points are described by two-component wave functions ( . ) [ , ] A B x y y j j = which are the envelope amplitude on sublattices A and B of graphene honeycomb lattice ( Fig. 1). They obey 2D Dirac-Weyl-like equation [8,[13][14][15]: where v 0 ≈10 8 cm/c is the characteristic velocity (called Fermi velocity by some authors [7,8]), σ =(σ x , σ y ) are the Pauli matrices, I is the 2x2 unit matrix, , ) x y ∂ ∂ ∂ ∂ ∇ = ( / / , V is the potential energy. We let the potential V be a square quantum well (QW) of finite width a and depth U, and limit our consideration to the most interesting electron energies 0>E>U.
Introduce the notations: κ>0, k >0, u>0 and μ>0 by the relations: is sometimes called the "power" of a potential well). It is also convenient to use normalized values by , etc., with normalized k and κ in the interval 0 ( , ) 1 k k £ £ . The dimensionless (κ, k)-domain is naturally fragmented into 4 sectors as shown in Fig. 2 as the electron energy spectrum in them is utterly different.

Figure 2. Specific sectors of dimensionless (κ, k)-domain
Below we consider the boundary problem for Eq.1 arising for elementary excitation stationary states in each of those sectors.

Excitations Spectrum
In sector I of Fig.2: -the abovementioned chiral tunneling (Klein tunneling) takes place. We remind briefly its main points referring to [6] for details.
Eq. (1) is satisfied by the set of wave functions: The coefficients r, b, c and t are found by matching boundary conditions at x = 0, a for any k and κ in that sector.
Since the electron velocity is directed opposite to the wave vector for E<0, the incident and transmitted waves may be interpreted as incident and transmitted hole flews respectively, while inside the well the resulting electron flew is formed when E>U, as shown in Fig. 3.
As was shown in [6][7][8] the transmission T=|t| 2 in that case does not decay exponentially with a increasing, and may reach resonant value T=1 for some incident angles and when the condition qa = πn is realized with an integer n, which is just the peculiar feature of Klein tunneling. We however draw one's attention to the essential fact that such chiral tunneling is possible only if the y-component of an electron wave vector 0 0 -that is for electron energy E=U/2, but the tunneling probability in that case is equal to zero as the electron wave vector is parallel to the barrier border. When 0 / 2 E U > > chiral tunneling takes place for any incident angle of the electron wave to the barrier border 0 / 2 q p < < , but when / 2 U E U > > chiral tunneling is possible only if the incident angle: The same is obviously valid for opposite k or/and p.
In Sector II of Fig. 2: (1) is now satisfied by the following set of the wave functions: where: Pay attention that in contrast to (2) The current density j is defined by the formula (see, e.g., [16]): [ , ] A B y j j = being the two component amplitude. Whence the incident hole flew is For the transmission T and reflection R we find using (5): where we turn to dimensional variables. It is seen from those expressions that in domain II the convenient (not chiral!) quantum tunneling takes place. One can also easily see that the above consideration may be extended to the enlarged domain, ( 1) κ κ − = − − , and thus finds that hole scattering on the barrier is described with the same scenario in that case. Fig. 2: (1) is satisfied by the following set of the wave functions:

In sector III of
, gives: which, for non-zero B and C solution, leads to the equation for κ, i.e. for the energy of electron-like states localized in the quantum well at given k (in dimensionless form): First of all we draw one's attention that such states may exist for normalized κ ≤1/2 only (see Fig. 2) -that is for energies no lower than a half of QW depth: |E|<|U|/2, and with y-component of the momentum Note also that the obvious solution q = 0 does not imply any electron state as in that case B = -C in (9) due to the boundary conditions and consequently φ A = φ B = 0.
With that in mind one concludes that Eq. (11) i.e. in the domain between to hyperbolas inside sector III of (1 ) As follows from above, a solution of (10) may exist provided 2 q π µ > . As q<1, there is no solution of (10) when 2 π µ < .
Let's now consider the states κ=0. In that case we have for normalized values: Graphical interpretation of those equations is presented below in Fig. 6 (alb). One can see that there is no other solution for 0≤μ <π/2 but κ=0, k=1 (q = z/μ = 0). However for μ ≥π/2 new solutions appear: particularly, there are n other solutions k=k i for a given μ if (2n -1)π/2≤ μ <(2n +1)π/2, (n=1, 2, ...). They are: where z i ≤ μ (i = 1, 2,…, n) are the solutions of Eq. (13.1) or (13.2). We draw one's attention to the fact that the electron in such a state with zero energy has non-zero momentum: Evidently there is no solution 0 ( ) 1/ 2, 0 1 k k κ ≤ ≤ < ≤ for π/2≤μ <π but the above found (14). That is because for κ>0 the left side of Eq. (10.1): 180 Graphene Electron States in a Quantum Well (we omit m=0 as q=0 in that case -see comment above). appear only when μ >π, as was discussed above. Actually the continuous brunches 0 ( ) k k κ κ < = ≤ appear in that case: the first one (and the only one as yet) -when π<μ<3π/2. They start at κ=0 from k(0)=k i from (14) and end at κ=k m from (16) with i=m; evidently k(0)>k m , so one can suggest that the energy spectrum is of the kind shown schematically in Fig. 7 for some values of μ.
When μ exceeds 3π/2 new spectral branch appear similar to described above, while the first one goes up closely to the line k=1-κ. So on, each time μ exceeds (2n+1)π/2, one more spectral branch of that kind emerges.