Electron paths and double-slit interference in the scanning gate microscopy

We begin the discussion by setting equal widths of both input and output channels within QPCs, ${{w}_{{\rm qpc}1}}={{w}_{{\rm qpc}2}}=40$ nm. In both the input and the output channels we have a single sub-band transport for the applied Fermi energy of 8 meV. According to our previous paper [14], the single sub-band transport on the input part allows us to use the SGM technique to resolve the Young's interference pattern.

The amplitude of the scattering wave function in the absence of the tip is shown in figure 1(b). In addition to the well-resolved Young interference pattern with five essentially straight lines of constructive and destructive interference for the waves passing through both the slits (see figure 2(c)), one also notices additional vertical interference fringes. This feature is caused by the presence of the output QPC2, which reflects the incoming wave function back to the double-slit device.

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The role of QPC2 in the scattering is more clearly visible in the real part of the wave function, displayed in figure 1(c). We found that a very similar wave function can be reproduced by a superposition of three point sources positioned at the center of each slit of the system, following the Huygens–Fresnel principle for the circular waves propagating from the input slits and QPC2; the latter is due to backscattering of the incoming wave [10]. In the limit of the thin slit, the Fermi level wave function at position ${\bf r}$ traveling from the slit located at position ${{{\bf r}}_{{\rm source}}}$ can be well described by the angle-modulated Henkel function

$$\begin{eqnarray}&&{{\psi }_{{\rm Henkel}}}(r,{{r}_{{\rm source}}},{{\alpha }_{{\rm source}}})={{\alpha }_{{\rm source}}}{\rm cos} (\theta )\frac{{{{\rm e}}^{{\rm i}{{k}_{{\rm F}}}|{\bf r}-{{{\bf r}}_{{\rm source}}}|}}}{\sqrt{{{k}_{{\rm F}}}|{\bf r}-{{{\bf r}}_{{\rm source}}}|}},\end{eqnarray} \tag{ 8 }$$

where θ is the angle between the ${\bf r}-{{{\bf r}}_{{\rm source}}}$ vector and the x-axis (i.e., ${\rm cos} (\theta )=|x-{{x}_{{\rm source}|}}/r$ and ${{\alpha }_{{\rm source}}}$ is the scattering amplitude). For the superposition of the three point sources, one has

$$\begin{eqnarray}&&\psi (r)\approx {{\psi }_{{\rm Henkel}}}(r,{{r}_{1}},1)+{{\psi }_{{\rm Henkel}}}(r,{{r}_{2}},1)+{{\psi }_{{\rm Henkel}}}(r,{{r}_{3}},{{\alpha }_{{\rm reflection}}}),\end{eqnarray} \tag{ 9 }$$

where ${{r}_{1}}=(110\;{\rm nm},-80\;{\rm nm})$, ${{r}_{2}}=(110\;{\rm nm},+80\;{\rm nm}),$ and ${{r}_{3}}=(670\;{\rm nm},0\;{\rm nm})$ are the positions of slits in the system in figure 1(b). We set ${{\alpha }_{{\rm reflection}}}={{\alpha }_{{\rm source}}}/2$, for which we get the best agreement with the exact solution obtained from equation (2). The real part of the wave function obtained from equation (9) is plotted in figure 1(d). Note that the fitted value of ${{\alpha }_{{\rm reflection}}}$ is quite large, which suggests that the backscattering from QPC2 will have a substantial influence on the SGM images discussed below.

The Young interference (figure 2(d)) and the interference due to backscattering by the QPC2 detector (figures 1(b) and (c)) are present already without the tip. The Young's interference involves the superposition of waves passing through each of the slits with the modulation of the scattering density dependent on the relative phase

$$\begin{eqnarray}&&\rho ={\rm cos} ({{k}_{{\rm F}}}({{r}_{1}}-{{r}_{2}})),\end{eqnarray} \tag{ 10 }$$

where r₁, r₂, stand for the distance from the lower and upper slit of the input QPC1. The density modulation in the Young's interference pattern is plotted in figure 2(d), with the constant ρ values (${{r}_{1}}-{{r}_{2}}={\rm const}$) forming hyperbolae with the slits of QPC1 as the focal points.

The backscattering by the tip introduces additional interference effects with the electrons reflected to the source QPC (figure 2(a)), the tip-induced double slit interference (figure 2(b)), and the interference of the direct wave with the scattered wave (figure 2(c)). These interference effects will be discussed later in the paper. The conductance map fringes due to the scattering of the type given by figure 2(a) are well known and have been discussed in a number of papers [5–12]. The backscattering (R) is enhanced when the phase shift of the wave going to the tip and back (the red arrows in figure 2(a)) produces a constructive interference with the incident wave (the blue arrow in figure 2(a)) at the input slit. The R signal is then proportional to

$$\begin{eqnarray}&&I={\rm cos} ({{k}_{{\rm F}}}({{r}_{1}}+{{r}_{1}})),\end{eqnarray} \tag{ 11 }$$

where in the argument, ${{k}_{{\rm F}}}\times 2{{r}_{1}}$ is the phase acquired by the electron wave function on its way from a slit of QPC1 to the tip and back from the tip to the same slit. The interference of figure 2(a) leads to the angle-independent modulation of the R maps, which oscillate as a function of distance from the slit with the period of half the Fermi wavelength, ${{\lambda }_{{\rm F}}}/2$ [9, 10].

The tip induces a double-slit interference of the wave passing through one of the slits of QPC1, which is then scattered by the tip with the wave incident from the other QPC1 slit (figure 2(b)). An enhanced backscattering can be expected when the interference of the incident and the returning path is positive, or the R signal will be proportional to

$$\begin{eqnarray}&&I={\rm cos} \left( {{k}_{{\rm F}}}\left( {{r}_{1}}+{{r}_{2}} \right) \right).\end{eqnarray} \tag{ 12 }$$

The result of formula (12) is plotted in figure 2(b). The fringes corresponding to a constant argument of the cosine function in equation (12) are elliptic, with both the slits of QPC1 as the focal points. The double-slit interference according to the mechanism of figure 2(b) and the resulting SGM effects were never discussed before. Equations (11) and (12) produce the same periodicity of ${{\lambda }_{{\rm F}}}/2$ at a large distance from QPC1.

The tip induces interference of the waves incident directly from QPC1 to QPC2, with the waves scattered by the tip depicted in figure 2(c). The electron current passing to QPC2, and thus the conductance G, is then proportional to

$$\begin{eqnarray}&&I={\rm cos} \left( {{k}_{{\rm F}}}({{r}_{1}}+{{r}_{3}}) \right)\end{eqnarray} \tag{ 13 }$$

where r₃ is the distance from the tip to QPC2. The results of equation (13) are displayed in the lower panel of figure 2(d). The interference fringes resulting from this mechanism are again elliptic, but with one slit of QPC1 and the QPC2 detector as the focal points. This type of interference leads to lateral fringe patterns in SGM-G images, which will be discussed below.

In figures 3(a)–(f) we show the SGM maps of resistance (SGM-R) as functions of the position of the AFM tip. The area of the scan is shown by the blue rectangle in figures 1(a) and (b). We consider a single or both QPC1 slits to be open, as illustrated schematically in the insets in the top right corner. For each system in figures 3(h)–(m), we plotted the corresponding probability current distribution in the second row. In figures 3(a) and (b) we show the results of SGM-R images obtained for a single open (lower) input slit. In the absence of QPC2 (figure 3(b)), we observe circular fringes in SGM map due to interference of the type given in figure 2(a) between the wave incoming from the slit and the wave backscattered by the AFM tip (see the arrow in figures 3(d) and 2(a), and equation (11)).

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Figure 3(d) presents the sum of the SGM-R maps of figures 3(a) and (c) obtained for a single open slit. The sum is quite different from the image obtained for the system where both QPC1 slits are open (see figure 3(e)), which is a signature of the double-slit interference effects and involves both the Young's interference (figure 2(d)) and the tip-induced interference (figure 2(b)). The Young's interference is more clearly resolved in the absence of QPC2 (figure 3(f)), although it is also present when the QPC2 detector is a part of the setup (figure 3(e)). To demonstrate this more clearly, we extracted the enlarged fragments of figures 3(e) and (f) in figures 4(b) and (c). For a simple sum of SGM images for separate slits (figure 3(d)), instead of the elliptic fringes and Young's pattern, we observe a checkerboard pattern (figures 3(d) and 4(a)). This checkerboard pattern should be observed in the experiments for a large number of incident sub-bands (i.e., at a higher Fermi energy or a wider input channel, as discussed in [14]).

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In figures 3(h)–(m) we plotted the current density within the system in the absence of the tip. Note that for the double-slit systems (figures 3(l) and (m), the current distribution is very similar with or without QPC2. The deformation of the Young's interference for the system with QPC2 is due to the lateral interference pattern involving the paths marked in figure 2(b), which are introduced by QPC2.

The Young's interference pattern given by equation (10) and calculated for the Fermi wave vector is displayed in figure 3(n), and it has a good agreement with the probability current distribution of figures 3(l) and (m) and the features of the SGM map of figure 3(f). A resemblance to figure 3(e) with QPC2 present can also be spotted, although the lines of flat R in figure 3(e) are curved. The curvature and the presence of the lateral fringes can be reduced by placing the QPC2 further from QPC1, thus reducing the backscattering by QPC2, but at the expense of the reduced G map contrast (see below).

The SGM maps of source-drain conductance (SGM-G) in figure 5 exhibit a valley of minimal values along the shortest path from the source channel to the drain. The pattern of the image obtained for both slits open (figure 5(d)) is exactly the same as the one given by the sum of images in figures 5(a) and (b) (see figure 5(c)). This shows that in SGM-G images, the double-slit interference is absent, which may be quite surprising, given that the SGM-R image clearly resolved the interference.

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In figure 5(a), a lateral fringe pattern is observed along the classical trajectory. This valley of minimal G values coincides with the line of maximal backscattering R observed in figure 3(a). Figure 6(a) shows that the tip separates the electron wave into beams, which arise from the interference of waves incoming from QPC1 and scattered by the tip. The variation of the electron density depends on the relative phase of the wave reaching point ${\bf r}$ from QPC1 directly, and the one that first gets to the tip and then is scattered to r,

$$\begin{eqnarray}&&\rho ({\bf r})={\rm cos} \left( {{k}_{{\rm F}}}(r-({{r}_{1}}+{{r}_{4}})) \right),\end{eqnarray} \tag{ 14 }$$

where r₁ is displayed in figure 2 and ${{r}_{4}}=|{\bf r}-{{{\bf r}}_{1}}|$. The results of formula (14) are plotted in figure 6(b) with a very good agreement with the numerical result of figure 6(a), which also contains the fringes due to backscattering by QPC2. Figure 6(a) corresponds to the position of the tip above the point A and explains the valley of low conductance in the SGM-G image (e.g., see figure 5(a)), since QPC2 is in the shadow cast by the tip. Figures 6(c) and (d) correspond to tip above points B and C for a single slit and explain the increase in the conductance around the classical path, which is easily visible in figures 5(a) and (b). If we move the tip to position B or C (see figures 6(b) and (c)) the electron wave is still separated into two main beams, with one of these beams reaching QPC2, which produces the characteristic envelope of high conductance around the 'classical path.' The minima of the lateral pattern in figure 5 appear when a nodal line of the interference passes to the QPC2 detector.

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The points $A,B,C$ in figure 3(a) are the same as in figure 5(a), and they correspond to plots in figures 6(a)–(d). Note that point B corresponds to a minimum of R and a maximum of G, while point C corresponds to a maximum of both quantities. The lateral pattern in the R map appears only in presence of the QPC2 detector (cf figures 3(a) and (b)), and is a result of scattering first by the tip, and next by QPC2. The effect of the interference of the type in figure 2(c) depends on both the position of the tip with respect to the QPC2-QCP1 return path for the backscattered electrons and the incidence angle of the waves deflected by the tip on the edges of the QPC2 electron.

Let us now consider both slits open with the AFM tip at a position where the electron flow from the upper slit is totally blocked (see figure 6(e)). The current flux from the lower slit will be the only one reaching QPC2. In this manner, the tip turns off one of the slits of the source channel. Using the symmetry arguments for the considered device, we will get the same conclusion with second slit blocked and the first transmitting current. Such an argument can explain why there is no visible interference at certain points in the obtained SGM-G maps. Nevertheless, there are points where the potential of the tip does not totally block the current from any slit, so interference could be expected. Such a case is presented in figure 6(f), with the tip located at a position where the current distribution is almost the same as in the unperturbed system (for comparison, see figure 3(l)). We conclude that the double-slit interference is clearly visible in the current distribution, but not in the SGM-G images.

The lack of Young's interference in the source-drain conductance images can be explained using a reversed bias and the current flowing in the reverse direction (i.e., from QPC2 to QPC1). The upper row of figure 7 shows the SGM-G, SGM-R, and probability density for the current flowing from the double slit to QPC2, while the lower row presents the quantities for the opposite current direction. We can see that the results for SGM-G are identical for both current directions (figures 7(a) and (d)) in spite of the fact that the electron scattering is very different in both setups, with a clear Young's interference in figure 7(c) and no similar counterpart in figure 7(f).

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The SGM-G images for both the cases (figures 7(a) and (d)) are bound to be identical due to the Onsager microreversibility relation for the single sub-band transport, ${{T}_{{\rm qpc}2,{\rm qpc}1}}={{T}_{{\rm qpc}1,{\rm qpc}2}}$, which in terms of the Landauer approach implies no current flow for zero bias. For the electron incident from the right, there is no reason to expect the presence of Young's interference, which is indeed missing in figure 7(f). The absence of the double-slit interference in SGM-G follows from this observation and the microreversibility relation. Note that the SGM-R image for the reversed current orientation (figure 7(e)) clearly indicates the classical paths that the electron can follow from QPC2 to QPC1. Also, note that QPC2 is 100 nm further to the right of the end of the figure, and that two bright beams are emitted from QPC2, which is only 40 nm wide.

There is a way to restore the interference in the SGM-G images. Each of the ${{M}_{{\rm in}}}$ sub-bands is with a certain probability backscattered, transferred to QPC2, or exits the computational box through the transparent boundary conditions to the rest of the system (see equation (1)); M_in is independent of the tip position. Now if we consider that ${{w}_{{\rm qpc}2}}$ becomes wider, the ratio ${{G}_{{\rm rest}}}/G$ decreases, since QPC2 will be able to transfer more current, and the probability of the electron transfer form QPC1 to QPC2 increases with the width of the latter. Thus, for a large width, ${{w}_{{\rm qpc}2}},$ the value of ${{G}_{{\rm rest}}}$ in equation (1) will be small, and hence $\frac{2{{{\rm e}}^{2}}}{h}{{M}_{in}}\approx R+G$. We can express the conductance of the system in $G\approx \frac{2{{{\rm e}}^{2}}}{h}{{M}_{in}}-R\propto -R$. We can see that for large values of ${{w}_{{\rm qpc}2}}$, the SGM-G should start to exhibit the interference pattern as the SGM-R images do. In figures 8(a)–(e), we show the results obtained for a large value of QPC2 width, ${{w}_{{\rm qpc}2}}=800$ nm. Figure 8(a) shows the SGM-G image for the having the bottom slits open, and figure 8(b) is a sum of the images obtained for bottom and upper slit open separately. Now, the SGM-G image for both slits open (figure 8(c)) is clearly different from figure 8(b), with distinct stripes due to the Young's interference. We can compare this result with the SGM-R image of figure 8(d), which is now quite similar to the SGM-G image, particularly in the center and close to QPC2. In this region, the approximated relation, $G\propto -R,$ is easily visible and the SGM-G image is indeed a negative of the SGM-R image. To illustrate this further, in figure 9 we plotted a cross section of these images along the axis of the device. In figure 9 we find that the G and R change almost in antiphase for ${{x}_{{\rm tip}}}\gt 450$ nm, which is not exactly the case for lower values of x_tip. In figure 9 we also plotted the leakage conductance, G_rest, which corresponds to electrons going out of the device by the region marked by the blue dashed line in figure 1. Note that $R+{{G}_{rest}}+G={{G}_{0}}$, and G_rest was multiplied by 3 for the presentation in figure 9. For the tip close to QPC1, ${{x}_{{\rm tip}}}\lt 250$ nm, G_rest, and R are almost in phase. On the other hand, for a large x_tip, G_rest gets in phase with G. The change of the correlation can be explained when we consider that the tip is a source of a circular scattered wave, which produces a larger electron flux to the closer QPC. Scattering by the tip increases G_rest independent of x_tip and one of two quantities, R or G, depending whether QPC1 or QPC2 is closer to the tip. Varying correlation G_rest and the other conductance probabilities seems responsible for the variation of the phase between G and R in figure 9.

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Note that the increased width of QPC2 allows us to restore the interference in the SGM-G images, but at the expense of the lost information on the electron trajectories from QPC2 to QPC1. The double-slit interference is present as long as one does not interfere with the measurement by trying to determine through which slit the particle passes [15]. Here, if we set the detector (QCP2) for mapping the electron trajectories with a small value of ${{w}_{{\rm qpc}2}},$ we gain the information about electron trajectories, but we lose the interference pattern. Increasing the width, ${{w}_{{\rm qpc}2}},$ leads to a reduction of the spatial resolution of the detector, so we lose the paths in the images, but restore the interference pattern.

The classical trajectories as extracted from the SGM-G images should be stable against the geometrical parameters the system, the width of the QPC1 channel, ${{w}_{{\rm qpc}1}}$ (see figure 1 (a)), or the distance between the QPC1 slits, ${{d}_{{\rm qpc}1}}$ in particular. For each value of the width of the QPC1 channel, ${{w}_{{\rm qpc}1}},$ we calculated the SGM-G(${{w}_{{\rm qpc}1}}$) and SGM-R(w_qpc1) images. The calculations were performed for ${{d}_{{\rm qpc}1}}=400$ nm. Both slits had the same width equal 40 nm and unchanged position as the width of the feeding channel, w_qcp1, is changed. In figure 10(a), we plotted the Pearson correlation coefficient [22] between SGM-G (${{w}_{{\rm qpc}1}}$) and the last image of SGM-G (400 nm), which is the r(G) curve, and the correlation between SGM-R (${{w}_{{\rm qpc}1}}$) and SGM-R (400 nm), which is the r(R) curve. We chose the SGM image for ${{w}_{{\rm qpc}1}}=400\;{\rm nm}$ as a reference. One can see in figure 10(a) that lines r(G) and r(R) quickly stabilize around 1 after ${{w}_{{\rm qpc}1}}=150$ nm, which means that through all the values from around ${{w}_{{\rm qpc}1}}=150$ nm to 400 nm, both images stay almost unchanged.

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We found that the SGM-G images are generally stable in the function of ${{d}_{{\rm qpc}1}}$, which means that paths are always visible, except when the distance between slits is small enough that both trajectories (from each QPC) overlap, which makes the calculated SGM-G maps difficult to interpret. Sample images of SGM-G and SGM-R for small values of ${{d}_{{\rm qpc}1}}=160\;{\rm nm}$ are displayed in figures 10(b) and (e), respectively. For larger values of ${{d}_{{\rm qpc}1}},$ the classical trajectories are restored (see figures 10(c) and (d)). Note that for figures 10(c) and (d), the difference between the values of ${{d}_{{\rm qpc}1}}$ in each case is equal to 16 nm. The results for SGM-G are nearly identical. However, the SGM-R images, which are sensitive to the interference effects, very strongly depend on a specific value of ${{d}_{{\rm qpc}1}}$ (see figures 10(f) and (g)). The wave function passing through both the input slits interferes with the AFM tip, as well as with QPC2. The result of the interference in terms of the backscattering depends on the variation of the distance between the slits of the order of the period of the waves formed by interference at the Fermi level, which is equal to ${{\lambda }_{{\rm F}}}/2$.

Let us now consider the wider tip potential, ${{d}_{{\rm tip}}}=50$ nm. The results for conductance G for a single slit in figure 11(a) and both slits in figure 11(b) still indicate the classical current paths, although naturally the widths of the G minima are significantly increased. The G map pattern is still very similar to the one obtained by a sum of maps for separate slits (figure 11), indicating a lack of double-slit interference features in the source-drain conductance maps for ${{M}_{{\rm out}}}=1$, as discussed above.

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The 'resistance' map (figure 11(e)) for both QPC1 slits open contains the circular fringes near the input slits (single-slit interference of figure 2(a)), elliptical fringes (double-slit interference of figure 2(d)), and the lateral fringes (figure 2(b)). All these effects are tip related and in this case dominate the Young's interference, which is weaker but still detectable. The Young's interference is restored when the QPC2 detector is placed further from the QPC1 (see figure 11(h)), but naturally at the expense of the amplitude of the G signal (figure 11(g)). The reduction of the lateral fringe pattern is observed for R since it involves backscattering by QPC2 (figure 11(h) and (i)).