Characterization of the size and position of electron–hole puddles at a graphene p–n junction

Interesting is the bottom right part of figure 1(c). This is the region of small gate density and large magnetic field which means that the cyclotron radius in both regions of the p–n junction are small and the results from equation (2) deviate most strongly from our numerical results. Furthermore, equation (2) with ${L}_{\mathrm{PN}}=2$ suggests that this region should have many more peaks than observed. r_BG and r_TG in this region are smaller than the radius of the puddle which implies that all electrons that reach the p–n interface will be scattered by the puddle. Thus, the puddle effectively shortens the p–n interface to ${L}_{\mathrm{PN}}^{*}=W-{y}_{{\rm{p}}}$, which is the length of the p–n interface from the puddle. In figure 2(c) we show the results of equation (2) with ${L}_{\mathrm{PN}}=W-{y}_{{\rm{p}}}=1.25$ by the blue curves which agree with the numerical data. In this way we are able to extract information about the position of the e–h puddle.

Figure 1(c) shows interesting features in the high magnetic field region—curves that connect neighbouring curves that could be described by equation (2). Their origin can be traced back to the presence of disorder in the system. We find that these features satisfy equation (4) for a specific value of r_TG. In other words, these curves connect points with the same cyclotron radius in the gated region. Using equation (4) we took r_TG as a fitting parameter which results in the red curves in figure 2(b) for the following values of r_TG: 0.1, 0.072, 0.061, 0.05, 0.042, 0.036, and 0.03.

In order to better understand the latter features we use current density plots. The plots in figures 3(a)–(e) show the flow of the current carriers in the system for the points 1–5 indicated in figure 1(c). Notice the difference between figures 3(a) and (b). In both figures the injected beam flows around the puddle and only a small amount of carriers is affected by it. However, in figure 3(a) we show the case when the transmission has a local maximum and in figure 3(b) when the reflection has a local maximum. This is clearly seen if we follow the flow of the injected beam. In figure 3(a) the beam injected at terminal i bends to the right and hits the p–n interface from the bottom. We can follow its transmissions through the p–n interface until it finally ends up at the gated side of the p–n junction and is collected by the collector c. In figure 3(b) we have a similar situation however now a larger part of the beam ends up at the non-gated side and is transported back to the injector. Figure 3(c) shows a similar situation as figure 3(a) with the difference that the electron beam is now strongly affected by the e–h puddle. Notice the shift of the beam after it encounters the puddle. Similar shift can be observed in figure 3(d) which is for point 4 in figure 1(c) on the curve with constant ${r}_{\mathrm{TG}}=0.061$ (see also figure 2(b)). This curve appears in the region where the first derivative is negative similar as for point 2 in figure 1(b). In the latter case the electron beam is not affected by the puddle and it ends up at the injector's side. However, in figure 3(d) we see that the part of the beam that is scattered by the puddle ends up at the collector side of the p–n junction. Therefore, this feature appears when the shift of the beam that interacts with the puddle is such that it changes the region in which the beam will end up. If we follow this curve of constant r_TG eventually we will reach one of the snake state curves. This is shown in figure 3(e) where we see that for this point a part of the injected beam is still affected by the puddle but the part that flows without interacting with it is much larger than in previous case.

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Next, we will look for a way to obtain information about the size of the e–h puddle. A rough estimate can be made using previous results. Firstly, one needs to locate a region of ${\rm{d}}G/{\rm{d}}{V}_{\mathrm{TG}}$ with the new features. Secondly, one draws a curve of constant r_TG such that the located region lays below it. Since we know that the conductance is most strongly affected by the puddle if ${r}_{\mathrm{TG}}\lt {r}_{{\rm{p}}}$, chosen value of r_TG can serve as a rough estimate of r_p. Inspecting figures 2(b) and (c) we find that this region lays below the ${r}_{\mathrm{TG}}=0.72$ curve and this can be a rough estimate for r_p. However, for a more precise estimate we proceed as follows. We decrease the back-gate voltage to a very low value, fix the magnetic field and change the top-gate voltage. The idea is to have a very small cyclotron radius in the non-gated region and a large r_TG. In this way the signal will be affected by the e–h puddle only when the part of the beam on the n-side goes around the puddle. This is shown in figure 3(f). However, we should be careful that $2{r}_{\mathrm{TG}}$ is never larger than the width of the collector otherwise carriers will not be collected and will propagate to the p–n interface. This can be controlled by the magnetic field.

In figure 4(a) we compare the signals when there is no puddle (red curve) in the system with the one when an e–h puddle is present (blue curve) with (${x}_{{\rm{p}}},{y}_{{\rm{p}}},{r}_{{\rm{p}}}$) = (0.48, 0.75, 0.05). The two signals differ in the region of densities between n₁ and n₂. Therefore, this is the region affected by the e–h puddle. The snake trajectory for density n₁ is given schematically in the inset of figure 4(a). In order to find the size of the puddle underneath the gated region we need to calculate the distance that every part of the electron beam on the n-side (small orbit) covers when we increase the density below the top-gate voltage from n₁ to n₂. This is given by

$$\begin{eqnarray}&&{d}_{{\rm{p}}}=\displaystyle \frac{{r}_{\mathrm{TG}}({n}_{2})-{r}_{\mathrm{TG}}({n}_{1})}{2{n}_{c}}-2{r}_{\mathrm{BG}},\end{eqnarray} \tag{ 5 }$$

where d_p stands for the diameter of the puddle and n_c is the number of orbits that the beam forms on the p-side below the puddle's position. In figure 4(a) n_c = 2 because there are only two orbits below the centre of the puddle. The second term in equation (5) is a correction because at point n₁ the beam touches the puddle however at point n₂ the whole orbit on the n-side needs to be outside the puddle. Using this approach we get for the system of figure 4(b) d_p = 0.108. Figure 4(b) shows the results for B = 10 and (${x}_{{\rm{p}}},{y}_{{\rm{p}}},{r}_{{\rm{p}}}$) = (0.49, 1.25, 0.035). In this case n_c = 3 because the beam executes three orbits below y_p = 1.25 and using equation (5) this results in d_p = 0.076.

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Alternatively, we can also estimate the puddle size by varying the magnetic field instead of the carrier density. For example, we decrease the cyclotron radius in the gated region by lowering the density in that region and by changing the magnetic field. The idea is to have a small cyclotron radius in the gated region and a large cyclotron radius in the non-gated region. The reason to do so is to obtain a better estimate of its size on the p-side of the junction. We proceed as in the previous case and search for the region where the two responses differ. However, in this case the cyclotron radii in both regions are changing with the magnetic field and therefore we need to change equation (5) so it will also include a change of r_BG. In this case we need to sum all cyclotron orbits below y_p for points B₁ and B₂ and subtract them to get the diameter of the puddle. This is shown in figure 4(c) from which we derive d_p = 0.115.

In figure 4(d) we show another approach. Here, we increase the cyclotron radius in both regions to values much larger than the expected size of the puddle. The idea is to get rid of the correction from equation (5). Calculated diameter is d_p = 0.09. Notice that the previous methods always overestimate puddle's diameter which is due to the small radius on one side of the junction, however using the last method one arrives at an estimate which is smaller than the actual size of the puddle and thus in the end we realize a lower and upper bound for the size of the puddle.

In the case where there is more than one puddle, disentangling the effects of each puddle can be very difficult. However, in certain cases it can be done. For example, in the case of two puddles we will still be able to determine the position of both puddles. Let us say we have two puddles at positions y₁ and y₂ and y₂ > y₁, where we used a coordinate system as in figure 1(a). Furthermore, let us assume that the direction of current is the same as in this figure. We showed that for determining the position effective length of the p–n interface is important, i.e. the length from the puddle to the end of the p–n interface, since the new features satisfy equation (1) with L_PN = L^∗_PN. In the case of two (or more) puddles this will be the length from the puddle that is situated closest to the opposite end of the p–n interface with respect to the current flow direction, in our case y₂ and we have L^∗_PN = W – y₂. However, if we flip the direction of the current by e.g. changing the direction of the magnetic field, L^∗_PN will also change because now the effective length of the p–n interface will be L^∗_PN = y₁. In this way, we are able to obtain y₁ and y₂.

In this paper we focused on the e–h puddles that are near the p–n interface. However, using this technique we are also able to determine the position of the e–h puddle formed further away from the p–n interface. The procedure is as follows. Let us assume that a puddle is formed at distance d_p from the p–n interface. We can set parameters such that the cyclotron radius in both regions of the p–n interface is small. Then we start increasing the cyclotron radius by e.g. increasing the Fermi energy. We know that the conductance will be affected by the puddle only for r_c ≥ d_p, i.e. when r_c is large enough for electrons to reach the e–h puddle. Hence, by determining d_p we obtain x_p as x_p = D − d_p. On the other hand, the procedure to obtain y_p is the same as in the case of an e–h puddle close to the p–n interface.