Bottom arm gate Top arm gate
6.2 Description of Corbino Sample
In order to probe the bulk conductance of counterflow currents, we studied a bilayer sample with a Corbino geometry. A photograph of this device (sample 7-12-99.1JJ) is shown in figure 6.2a. This sample was fabricated from a wafer with the usual double quantum well structure: two 18 nm wide GaAs quantum wells separated by a 10 nm Al0.9Ga0.1As barrier. Each quantum well is populated with a 2DES with nominal density n ≈ 5.5×1010 cm−2 per layer and low temperature mobility µ ≈ 1×106 cm2/Vs.
The sample is patterned into an annulus with inner diameter 1 mm and outer diameter 1.4 mm. Due to the relatively large size of the device, great care was taken during its fabrication in order to avoid any significant defects in the original GaAs wafer that could short the two layers together. Six 100 µm wide arms extend from the annulus to diffused NiAuGe contacts. There are four arms on the outer edge of the annulus and two on the inner edge. Each arm is crossed by front and/or back aluminum gates to implement the selective depletion technique. While each of the outer arms has both a top and bottom depletion gate associated with it, the two inner arms have either just a top depletion gate or just a bottom depletion gate. The annulus itself is covered by a large top gate and has a bottom gate directly underneath it. These two gates allow us to independently tune the 2DES densities in the upper and lower layers within the annulus. We will confine ourselves to the case where the two layers have equal densities.
In figure 6.2b we show a simplified picture of the sample, with the gates omitted.
Each ohmic contact is numbered. Contacts 1 through 4 are connected to the outer edge of the annulus while contacts 5 and 6 are along the inner edge. We will refer to this numbering scheme in circuit diagrams throughout this chapter.
In figure 6.3, we show measurements of interlayer tunneling conductance dI/dV versus interlayer bias at zero magnetic field andT = 14 mK. Note we have subtracted off from the recorded bias a small offset (20 µV) induced by the input of the current preamp. Here, we show traces for two different densities: NT = 1.11×1010 cm−2
(b)
4
6 5
2 1
3
Top arm gate
Main top gate
Top arm gate
Top ar Top ar
Top arm gate
Ohmic contacts Ohmic contacts
Ohmic contact Ohmic
contact
Figure 6.2: (a) Photograph of top side of sample 7-12-99.1JJ. (b) Simplified diagram of sample with labeled ohmic contacts. This labeling will be used throughout this chapter.
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(nominal density) and NT = 0.45 × 1010 cm−2 (corresponding to d/` = 1.49 at νT = 1). Note that the peak tunneling conductance G(0) ≈ 1.5 µS is much larger than seen in other samples made from the same GaAs/AlGaAs wafer. We attribute this primarily to the large size of the device, which has 12 times the area of the usual 250 µm square samples. The sample is wired to a rotating sample mount, allowing us to tilt the sample with respect to the magnetic field and introduce a field parallel to either 2DES. This will permit us to suppress the νT = 1 tunneling current. We will reveal below why this is vital for our counterflow measurements. It is important to note that each wire is thermally sunk to the cold-finger of the dilution fridge using an RC filters, with R = 10 kΩ and C = 500 pF. The resistors in the RC filters will contribute to the series resistance in each measurement.
m
m
´ ´
Figure 6.3: Interlayer tunneling at zero magnetic field and T = 14 mK for total densityNT = 1.11×1011cm−2 (solid black trace) andNT = 0.45×1011 cm−2 (dotted red trace).
Corbino conductance σxx is a measure of the ability for charged excitations to travel through the bulk of the bilayer system. In a single-layer system, σxx can be found by inverting the resistivity matrix:
σxx = ρxx
ρ2xx+ρ2xy. (6.1)
In a classical 2DES, σxx can be severely reduced by a sizable ρxy because current that is injected into an interior contact must circulate within the bulk multiple times before it reaches the outer edge. Within the bulk of a standard quantum Hall state, the Fermi energy lies within the energy gap separating two different bands of extended states. Thus, there are no states near the Fermi energy that can transport charge from one edge of the sample to the other. This implies that bothρxx = 0 andσxx = 0.
In a bilayer sample, one can consider both parallel Corbino conductance σ||xx and counterflow Corbino conductance σxxCF. We first focus on parallel Corbino conduc- tance, in which one drives currents within the same direction in the two layers from one edge of the annulus to another. As depicted in 6.4a, we realize this current flow pattern in our device by applying a small AC excitation voltage (20 µV at 13 Hz) to an ohmic contact on the outer rim (for example, contact 1) and detecting the current flowing to ground via a contact along the inner rim (contact 5). These two ohmic contacts are connected to both layers at the same time while all other ohmic contacts are fully disconnected from the annulus. The white triangle in the circuit di- agram symbolizes a low impedance current preamp whose output is read by a lock-in amplifier.
In figure 6.4b, we plot parallel Corbino conductance versus magnetic field while the sample is near nominal density and T = 50 mK. Deep minima can be seen each time the individual layers enter a quantum Hall state. Both integer and fractional
2We will use the symbolσxxto denote both conductance and conductivity. In reality, the Corbino conductance is equal to the Corbino conductivity times the geometric factor ln (R2π
2/R1), where R1
andR2 are respectively the inner and outer diameters of the annulus. In our device, this geometric factor is approximately 18.7.
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QH states are visible. Nominal density corresponds to d/` = 2.34 at νT = 1, which is well above the critical interlayer separation (d/`)c ≈ 1.8. Thus, no quantum Hall state is observed atνT = 1 in figure 6.4b.
In figure 6.4c, the density has been lowered to NT = 0.45× 1011 cm−2, which corresponds to d/` = 1.49 at νT = 1. At this low d/` and low temperature (T = 25 mK), the νT = 1 quantum Hall state is well formed and is centered on B⊥ = 1.88 T.
As has been reported before by Tiemann et al. [117], parallel Corbino conductance vanishes at νT = 1 because charged excitations are gapped out in the bulk. Note that the minimum inσxx atνT = 1 is not as well developed as the minima associated with integer QH states in the individual layers. This is consistent with the disparity between the charge gaps for the relatively fragile νT = 1 QH state and the robust integer states.