The electronic transport efficiency of a graphene charge carrier guider and an Aharanov–Bohm interferometer

We start by discussing the characteristic of a single gate defined charge carrier guider, as shown in figure 1(a). In figure 1(b), transmission coefficient T₁₃ versus V_in is shown for $V_{\rm out}=0.005$ and $V_{\rm out}=-0.005$. The transmission coefficient of an ideal graphene nanoribbon (the same width as the channel) is also displayed for comparison. For an ideal graphene nanoribbon, T₁₃ is quantized with value 2n  +  1. For the charge carrier guider, T₁₃ shows plateaus of value 2n. As $|V_{\rm in}|$ increases, a new plateau indicates two valley extra degenerate modes are propagating inside the channel [25]. The positions of the plateau edges are highlighted by the vertical dotted lines. Compared with the curve for an ideal graphene nanoribbon, the transmission coefficient of a charge carrier guider has a difference of 1. It means that the transmission of the edge states through the charge carrier guider is eliminated: the edge states flow from terminal 1 into two lateral terminals (2 and 4). In previous studies [26, 29, 30], the conductance in a two-terminal graphene guider is (2n  +  1)e²/h. The edge states transmit along the boundary of the ribbon contribute e²/h to the total conductance. In our work, the present model is a four-terminal device, similar to [25], thus the conductance is only contributed by the states inside the channel. When comparing these two curves, it can be noticed that the conductance for $V_{\rm out}=0.005/V_{\rm out}=-0.005$ is large when V_in is positive/negative, respectively. For example, for positive V_in, T₁₃ for negative V_out is better quantized with flat plateaus. The density of states at the channel boundary is depleted, when the gate voltage across the boundaries changes its sign, and the modes are well restricted inside the channel.

Now we focus on the transmission coefficients from terminal 1 to terminal 2 and 4. As T₁₄ and T₁₂ are the same considering device symmetry, we only display the dependence of T₁₂ on V_in. In figure 1(c), when $|V_{\rm in}|>0.02$, T₁₂ remains around 0.4. Therefore, the total transmission coefficient from terminal 1 to terminal 2 and 4 is close to unity. Several small peaks (guided by the grey dotted line) can be noticed. The peak positions are the same as the new appeared conductance plateaus in figure 1(b). It is contributed by an arising mode leaking from the channel. If we assume that a single channel is confined in the y direction and the charge carriers flow in the x direction (hence k_x is a good quantum number), then, for a given Fermi level E, the relationship between the two momentum components can be expressed as $E=3at\sqrt{k_x^2+k_y^2}/2$, with a the nearest C–C bond length in graphene and t the nearest coupling energy in equation (1). When an emerging mode appears in the channel, a low momentum k_x results in a small injecting angle to the channel boundaries (see the red dotted line in figure 1(a)). So the mode is likely to leak out the channel through the Klein tunneling [62, 63]. As the Fermi energy increases, the k_x is increased and the incident angle is large. When the incident angle becomes larger than the critical angle, the mode propagates well in the channel (see the blue solid line in figure 1(a)). From the above discussion, it is not surprising to see that T₂₄ versus V_in show a few peaks. These peaks are contributed by the resonant tunneling of the charge carriers across the channel, in which a Fabry–Pérot interferometer is formed [64, 65]. The peaks become higher when V_in and V_out have opposite signs, because the confinement of the states in the channel becomes stronger. To study the transmitting efficiency of the charge carrier guider, we defined $\newcommand{\e}{{\rm e}} \eta=T_{13}/(T_{13}+T_{12}+T_{14})$ and the results are shown in figure 1(e). When $|V_{\rm in}|$ is small, η is small as well. As $|V_{\rm in}|$ increases, η increases and shows several plateaus. The results agree with the experiment work in [25].

To further demonstrate transport of confined states inside the channel, the spatial LDOS $\rho_{\bf i}$ of a two-terminal graphene nanoribbon are shown for different V_in in figure 2. Here $\rho_{\bf i}$ was calculated from the Green's function in the central region $\rho_{\bf i}(V_{\rm in})=-\mathrm{Im}[{\bf G}_{\bf i}^{\rm r}(V_{\rm in})]/\pi$ [66]. For small values of V_in (e.g. $V_{\rm in}=0.1$ in figure 2(a)), there are two valley degenerate states (corresponding to T₁₃  =  2 in figure 1(b)), and the LDOS inside the channel shows a single peak. For slightly large values as $V_{\rm in}=0.135$ and 0.15 in figures 2(b) and (c), the LDOS inside the channel has two peaks, which correspond to two states at each valley, and the four states make a transmission contribution of 4. As V_in further increases (figures 2(d)–(f)), more states are propagating inside the channel and the number of LDOS peaks are related to the transmission coefficient (divided by 2). The multi-peaks are evenly distributed in the transverse direction. It is worth noting that, for $V_{\rm in}=0.135$ and $V_{\rm in}=0.21$ (figures 2(b) and (d)), the LDOS expands beyond the channel boundaries, indicating current leaking. At these two V_in values, new T₁₃ plateaus only occurs with small k_x, and newly appeared states tend to leak out the channel. Interestingly, in all cases in figure 2, the LDOS shows periodical quantum scars along the channel. The quantum scars in figures 2(b), (d) and (f) are stronger than the others, which can be explained by the diffusive transport of confined relativistic Dirac fermions in graphene [67, 68].

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We now discuss the dependence of V_in and V_out on transmission coefficients. Figure 3 shows two-dimensional (2D) maps of T₁₃, T₁₂, T₂₄ and η as functions of V_in and V_out. In figure 3(a), T₁₃ has small values near $V_{\rm in}=0$, indicating an unformed charge carrier guider. As V_in increases, plateau appears for T₁₃. At small $|V_{\rm out}|$ values, the plateaus become clear when V_in and V_out have opposite signs (the bipolar regime). At large $|V_{\rm out}|$ values, the plateaus become indistinct. Larger T₁₃ values can be found in the region where V_in and V_out have the same sign (the OF regime), because the current flow outside the channel can contribute. T₁₂ and T₂₄ show similar features: the values are small when either V_in or V_out is close to zero, and the values are large when both V_in and V_out are large. In addition, oscillation pattern was found for T₂₄ when $V_{\rm out}>0$, $V_{\rm in}<0$ or $V_{\rm out}<0$, $V_{\rm in}>0$ (the bipolar regime), because of the resonant tunneling across the charge carrier channel. In figure 3(d), η has high guiding efficiency when $|V_{\rm in}|>|V_{\rm out}|$ (the OF regime). When $|V_{\rm in}|<|V_{\rm out}|$, the guiding efficiency η is small. Similar results are found in a recent experimental work [25].

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Next we investigate the electronic transport properties in a ring composed of two charge carrier guiders, as shown in figure 4(a). The relationships between T₁₃ and V_in are shown in figure 4(b) for both two-terminal and four-terminal devices. Both curves show vibration and step behaviors at low $|V_{\rm in}|$, indicating that there are only a few modes in the channel. When $|V_{\rm in}|$ is large, T₁₃ shows strong irregular vibrations, which can be attributed to the intravalley scattering. We will discuss it later in details. The V_in values between two neighbor plateaus are the same to figure 1(a). T₁₃ values for the four-terminal device are smaller than two-terminal device, which means that the edge states leak out of the channel to lateral terminals (2 and 4).

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The LDOS of two-terminal devices are shown in figure 5 for different V_in values, guided by arrows in figure 4(b). For $V_{\rm in}=0.04$, the AB ring carries no state and the transmission is mainly through the ribbon edges (figure 5(a)). In figures 5(b)–(h), the states are well restricted within the channel. Two main features can be found: (i) the quantum scars occurs inside the channel. Periodical LDOS pattern is more obvious (figures 5(b) and (e)) when the propagating states are just formed. In such a case, k_x is smaller and the scattering is strong (figure 4(b)). In experiment, the radial quantum scars were observed in the ring area [52, 53]. The appearance of quantum scars is associated to the chaotic nature of states in a cavity, in which the conductance fluctuates. (ii) When there are more than two states in each arm of the AB ring (e.g. in figures 5(e)–(g)/(h), there are four/six states), scattering happens between these states. Since the states are valley degenerate, the scattering is an intravalley one [56]. Under a weak magnetic field, the scattering changes the dynamic phase of each state, and then the interference varies accordingly.

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To demonstrate how the quantum scars are affected by the shape of the AB ring, four kinds of AB rings are shown in figures 5(i)–(l) with $V_{\rm in}=0.15$ and $V_{\rm out}=-0.005$. In figure 5(i), LDOS pattern of a narrow ring is similar to that in figure 5(e). For a diamond shaped AB ring shown in figure 5(j), LDOS has a zigzag pattern. For a half AB ring (i.e. a curve single channel) where the confinement effect is weak, the quantum scars disappear, and the transport of charge carriers is nearly ballistic. For an incomplete ring with a gap (figure 5(l)), the quantum scars are obvious, and the states are scattered strongly at the gap, and the incomplete ring serves as a cavity. In experiments, such gate voltage dependent chaotic transport and quantum scars are expected using scanning gate microscopy techniques [52, 53, 55].

In figure 6, 2D maps of T₁₃ (versus ϕ and V_in) are displayed for both two-terminal and a four-terminal devices. The magnetic field is rather weak, therefore quantum Hall edge states are not formed, and the AB effect is predominant. In both figures, when V_in is smaller than 0.13, clear AB interference can be observed as ϕ changes. There are two valley degenerate propagating modes inside the channel. The period of the oscillation is about h/e, indicating the well performance of the interferometer. T₁₃ shows a maximum at $\phi=0$. When $V_{\rm in}>0.13$, irregular oscillations appear in addition to the regular oscillation with period h/e. Figure 7 gives a more clear demonstration of the oscillations. For example, when $V_{\rm in}=0.22$ and 0.23, T₁₃ shows a maximum (figures 7(a) and (b)). While for $V_{\rm in}=0.25$ and 0.28, T₁₃ shows a minimum (figures 7(c) and (d)). There are more than two modes propagating inside the channel. The minimal value at zero magnetic field should be attributed to the intravalley scattering (figures 5(f) and (h)). In previous experimental work, the linear conductance of an AB ring at zero magnetic field either shows a maximum or a minimum [26, 45, 48, 50, 54]. Here we show that this behavior is due to scattering between discrete propagating states inside a single valley, i.e. intravalley scattering.

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Now we explain the above results with a toy model. Figures 6(c)–(e) display symmetric AB rings sandwiched by two leads in three conditions: (i) a single mode, (ii) two modes with no scattering, and (iii) two modes with scattering. The incoming wave from the left lead, which is split at the left cross section of the interferometer, after traveling along both arms of the interferometer, flows into the right lead. The interference is determined both by the magnetic flux and the dynamic phase acquired by propagating waves.

• (i)  
  When there is only a single mode in the channel (figure 6(c)), the total transmission coefficient contributed by two waves from the upper and lower arms is given by $\tau=|{\rm e}^{{\rm i}(kl+\Phi/2)}+{\rm e}^{{\rm i}(kl-\Phi/2)}|={2+2\cos \Phi}$ with k the momentum along the channel, l the length of each arm and Φ the magnetic phase enclosed by the interferometer. So τ always shows a maximum at $\Phi=0$.
• (ii)  
  When there are two independent modes in the channel (figure 6(d)). The transmission coefficient can be expressed as:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau&= |{\rm e}^{{\rm i}(k_{1}l+\Phi/2)}+{\rm e}^{{\rm i}(k_{2}l+\Phi/2)}+{\rm e}^{{\rm i}(k_{1}l-\Phi/2)}+{\rm e}^{{\rm i}(k_{2}l-\Phi/2)}|^2\nonumber \\ &= 4(1+\cos\Phi)(1+\cos\delta kl)\label{Eq3} \nonumber \end{align} \tag{ 3 }$$

  with $\delta k=k_1-k_2$. So for two independent modes, there is always a constructive interference and τ is a maximum when $\Phi=0$.
• (iii)  
  When there are two modes in the channel (figure 6(d)), under scattering, the two states may interchange (figure 5(f)). For the simplest case, we assume that the charge carrier of momentum k₁ in the upper arm, after traveling a distance l₁, is scattered to the momentum k₂ and then propagates a distance of l₂ ($l_2=l-l_1$), to the right lead. Meanwhile, the charge carriers of momentum k₂ are transferred to the momentum k₁ due to the same process. The transmission coefficient reads:

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau=& |{\rm e}^{{\rm i}(k_{1}l_1+k_{2}l_2+\Phi/2)}+{\rm e}^{{\rm i}(k_{2}l_1+k_{2}l_2+\Phi/2)}+{\rm e}^{{\rm i}(k_{1}l_1+k_{2}l_2-\Phi/2)}\nonumber \\ &+{\rm e}^{{\rm i}(k_{2}l_1+k_{2}l_2-\Phi/2)}|^2= 4+2\cos\delta k l\nonumber \\ &+2\cos[\delta k (l_1-l_2)]+8\cos\Phi\cos\frac{\delta kl}{2}\cos\frac{\delta k (l_1-l_2)}{2}.\label{Eq4} \nonumber \end{align} \tag{ 4 }$$

Easy to notice that only the last term in the equation (4) is Φ dependent. If $\delta k=0$, τ always shows a maximum when $\Phi=0$. If $\cos\frac{\delta kl}{2}\cos\frac{\delta k (l_1-l_2)}{2}<0$, τ shows a minimum when $\Phi=0$.

The above discussion explains our numerical results. When there are only two modes (valley degenerated) in the channel, T₁₃ are displayed for ϕ shows periodical oscillation and a constructive interference at zero magnetic field, because the intervalley scattering is rather small [29]. When there are more than two modes (e.g. four, six or more modes), T₁₃ are displayed for ϕ shows irregular oscillation pattern due to the scattering. Since the propagating modes for a single valley are quite nearby to each other in the momentum space, the scattering is caused by the intravalley scattering. In all cases, T₁₃ is either a maximum or a minimum at $\phi=0$.

In the experiments of AB interferometers, when the conductance is larger than 2e²/h, both constructive interference and destructive interference can occur due to the scattering. When the conductance is smaller than 2e²/h, there are two possibilities. First, there are more than two modes and the conductance is suppressed to less than 2e²/h by either disorder or scattering. Second, there are only two modes and the disorder or scattering is rather weak due to a clean graphene. Only for the latter case the uniform constructive interference can be realized. Thus the quantum interference and the role of intravalley scattering in graphene AB interferometer can only be verified in controllable atomically precise samples in which the scattering can be deliberated to be on or off. With the present state-of-the-art graphene techniques, high-quality AB interferometers can be fabricated (e.g. with bottom–up approach) in which either the width or the charge carrier population can be precisely controlled by doping or a gate voltage [13, 16, 18]. So our finding for quantum interference in graphene AB ring around the zero magnetic field can be confirmed by experiment with nowaday graphene techniques. The spatial resolved states and chaotic transport related quantum scars in AB interferometer can also be detected by using scanning gate microscopy techniques [52, 53, 55].