5.2.1 Setup and measurement procedure
The experimental setup used in Refs. [14, 15] is depicted in Fig. 5.1(a). A two-dimensional electron gas (2DEG), formed at the interface of a GaAs/AlGaAs het-erostructure, is electrically connected to two Ohmic contacts which can be used to pas a current through the electron gas. Two metallic split gates are attached on top of the structure. Applying a negative voltage on these gates results in a depletion of the electron gas underneath them, due to Coulomb repulsion. As such, a narrow constriction (quantum point contact) can be formed in the 2DEG for the electrons to flow through, and the width of this constriction can be tuned by varying the voltage on the split gates.
Now suppose the tip of a scanning tunneling microscope (STM) is put at a cer-tain position above the electron gas. The electrostatic potential resulting from a
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Figure 5.1: Measurement setup for imaging electron flow through a quantum point contact (a). The conductance of a point contact quantized (b). Image from Ref. [14].
negative voltage on the tip can deplete a small spot of the 2DEG at the tip position, which can act as a local scatterer for electrons. As a result, the conductance of the sample will decrease compared to the situation with no tip present. This conduc-tance decrease will be large when the tip is positioned over a region where a lot of electrons are flowing since then a lot of electrons can be backscattered, while it will be small if there are less electrons flowing underneath the tip. As such, by moving the tip across the whole sample and by measuring the conductance decrease for every single tip position, one can obtain an image of where electrons are flowing through the sample.
It is obvious that this imaging method is sufficiently general so that it can be used for a wide range of two-dimensional geometries. Nevertheless, at the moment of this writing, it has only been applied to the quantum point contact (QPC) geom-etry [14, 15, 53]: the experimental results for this system will be presented in the next section.
5.2.2 Experimental results
When measuring the conductance of the quantum point contact (in the absence of the STM tip), it is found to be quantized in steps of 2e2/h as a function of the voltage on the split gates, as depicted in Fig. 5.1(b). Conductance quantization in point contacts is not new: it was already observed almost two decades ago [4, 5]
and can be fully understood within the Landauer-B¨uttiker formalism, as will be explained next.
The conductance of the sample can be expressed in terms of the transmission coefficientT between the sample edges (see Chap. 2):
G= 2e2
h T = 2e2 h
X
n
Tn. (5.1)
The point contact creates a quasi-1D channel in the 2DEG, so that different discrete transverse modes n with energies En can be defined. In Eq. (5.1), T has been
subdivided in transmission probabilitiesTnfor these individual modes. When the motion of the electrons is ballistic (i.e., there is no impurity scattering) and the width of the constriction varies smoothly along the propagating direction, there will be no scattering between different modes and the coefficientsTncan take only values of either0 or1[56, 57]: Tn = 1if the Fermi energyEF > En, and zero otherwise1. In this case, the conductance of the point contact is thus proportional to the number of modes transmitting through it. A larger number of modes can transmit when the QPC is made wider (since all En will shift to lower values), which is done by adjusting the voltage on the split gates to less negative values.
Every additional transmitting mode increases the conductance by2e2/haccording to Eq. (5.1) and gives rise to a discrete step in the conductance. Such steps are clearly visible in Fig. 5.1(b), although they are not as sharp as expected from the theory above. This is because the motion of the electrons is never fully ballistic in an experimental situation and because the width of a quantum point contact cannot change in a perfectly adiabatic manner.
Electron flow images in a QPC obtained with the scanning probe technique described in the preceding section are shown in Fig. 5.2. In the first picture, the voltage on the split gates is tuned so that the conductance of the point contact lies on the first plateau (G = 2e2/h). In this case the region of large conductance decrease, and thus large electron flow, is concentrated in one lobe, corresponding to electrons in the first mode of the QPC flowing through the constriction. With two channels transmitting through the QPC [Fig. 5.2(b)], one can see two lobes of electron flow. In general, modenwill contributenlobes to the spatial pattern of electron flow. As such, electron flow shows a modal pattern reflecting the different channels transmitting through the QPC.
Figure 5.2: Electron flow maps obtained with the scanning probe technique. The amount of channels transmitting through the QPC can be varied by changing the split gate voltage. Results are shown for a single channel (a), two channels (b), and three channels (c) open for transmission. The QPC contour is depicted in gray. The black strip on both sides of the QPC corresponds to a region where no data is available:
placing the tip in such a region would pinch off the QPC, making a conductance measurement useless. Image from Ref. [53].
1The quantum mechanical possibility of tunneling will be neglected here.
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Another striking feature of the flow maps are fringes spaced at half the Fermi wavelength. These are an interference effect resulting from back-and-forth scat-tering between the scanning probe tip and the quantum point contact. As such, they are remnants of the experimental technique, but nevertheless they are inter-esting features to study: for instance, the fringe spacing is directly related to the Fermi wavelength and thus to the electron density, so that spatial variations of this spacing can be used to measure the local electron density [58].
At larger distances from the point contact, the flow maps show a quite surpris-ing branchsurpris-ing behavior, as shown in Fig. 5.3. The branches in this picture are not just continuations of the different lobes seen in Fig. 5.2. For example, in Fig. 5.3 there are multiple branches although it is taken on the first conductance plateau of the QPC with only a single channel open [and thus a single lobe in Fig. 5.2(a)].
Furthermore, existent branches fork into new branches in an irregular way so that the number of branches increases as one moves further from the point contact [59].
The source of this branching behavior is disorder in the system: the Coulomb
po-Figure 5.3: Scanning probe map showing branching electron flow at larger distances from the QPC. Interference fringes are present throughout the sample. Only the part on the left of the QPC is shown in (a), while in (b), the branching behavior is shown on both sides of a different QPC. Picture from Ref. [15].
tential of the donor atoms used to inject carriers in the 2DEG creates a potential landscape that consists of small dips and bumps. The branches are not resulting from electrons flowing in the valleys of this impurity potential, since the Fermi en-ergy of the electrons in the experiment was large compared to the height of these structures. Rather it was proven that they result from multiple small-angle scatter-ing events off the bumps and dips in the potential [59–61]. Every bump or dip can be understood to act as a small lens for the electron flow, and a large number of such lenses can then contribute to the “collimation” of electron flow in branches.
Please note also that all branches remain decorated with the interference fringes spaced at half the Fermi wavelength.