V OLUME 93, N UMBER 3
3.2 Interlayer Capacitance
3.2.3 Compressibility and Interactions
Up until now we have ignored electron–electron interactions. Using the Hartree-Fock approximation, interactions can be treated by including both a Hartree term and exchange term for the bilayer energy Ebilayer [93]. While the total Hartree term is positive and proportional to (∆N)2, the exchange term for either layer is negative and proportional to −(Ni)3/2. This negative exchange contribution to the compress- ibility of the 2DES has been measured by Eisenstein et al. [28, 29] by detecting the electric field penetrating a single 2DES. Such a method bypasses the geometric capacitance and directly reveals the single-layer compressibility ∂N∂E
i. The negative exchange energy is expected to influence interlayer capacitance by lowering the total energy cost of transferring charge from one layer to another.
At high magnetic fields, the large degeneracy of the Landau levels will quench the kinetic energy term in Ebilayer(NT,∆N). Interaction effects will become more important. As we will see in a later chapter, an instability similar in nature to the one proposed by Ruden and Wu [93] can occur within highly imbalanced bilayer systems at large magnetic fields. In that particular case, the exchange-driven instability causes an unexpectedly large number of electrons to transfer from one layer to another.
Thus, capacitance measurements can reveal interesting physics atνT = 1. Due to excitonic effects, one expects that charge can transfer more easily from one layer to an- other. Consequently, one might anticipate anomalies in the temperature dependence of interlayer capacitance at νT = 1 (for example, see reference [8]). Once again, the geometric contribution complicates the interpretation of interlayer capacitance mea- surements. The next subsection will consider these issues and provide suggestions for future measurements of interlayer capacitance at νT = 1.
without a magnetic field. First, we will briefly describe our measurement technique.
Then we will present measurements of interlayer capacitance versus electron density atB = 0. Finally, we will consider the influence of a magnetic field on the interlayer capacitance.
Measurements of interlayer capacitance essentially use the same circuit as in tun- neling. An AC voltage (usually 20 µV and 13 Hz) is applied to one layer and the resulting current from the other layer is measured with a current preamp and lock-in amplifier. In the limit of zero sheet resistance, one then expects the total conduc- tance to be Gtotal = Gtunneling +iωC, where Gtunneling is the tunneling conductance and ωC is the capacitive admittance. Thus, the out-of-phase current is proportional to the interlayer capacitance. So long as sheet resistance is not too large compared with 1/|Gtotal|, the presence of tunneling is not expected to significantly affect the capacitance measurement because the tunneling currents and displacement currents effectively flow in parallel with one another.
A major concern in capacitance measurements is the presence of background ca- pacitance. For example, there might be stray capacitance between the measurement wires. The use of independently shielded coax wires helps to strongly reduce this stray capacitance, which can often be of order ∼1 nF for the meters-long pairs of twisted wires commonly employed in cryostats.
Another source of background capacitance is within the bilayer system itself. Our samples generally have both gated and ungated regions, with the gated regions being of central interest during measurements. For tunneling measurements, the ungated regions usually provide only a small amount of background tunneling because they are either imbalanced (which suppresses tunneling near zero bias at B = 0) or not at νT = 1 (and thus do not tunnel strongly at high magnetic fields). However, such ungated regions can still provide interlayer capacitive coupling despite their imbalanced state. Furthermore, the capacitance from the ungated regions is generally
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not suppressed at high magnetic fields. Most annoyingly, this background signal can vary significantly with magnetic field as the ungated regions enter and leave incompressible QH states.
Fortunately, one can measure the background signal from the ungated regions with a special geometry. In figure 3.3a we show the topside of sample 11-1-04.1M.
The geometry for this sample was created through improvised use of a variety of photolithography masks intended for other types of samples, hence its unusual ap- pearance. Here, we have defined a central mesa with ohmic contacts, top and bottom arm gates for selective depletion, and a main gated region in the center of the photo- graph. Normally the mesa pattern used has four arms leading to the central region.
However, we performed an additional etch to completely remove the bilayer 2DES in the two right arms. Thus, the bilayer 2DES only consists of two arms (with one ohmic contact each) leading to the main gated region.
We show a simplified drawing of the sample in figure 3.3b. The black dots denote ohmic contacts. The grey rectangles are the top and bottom arm gates that implement the selective depletion scheme. The hatched square shows the approximately 275 × 200 µm2 region of the sample that is covered by the main top gate. The clear section symbolizes the ungated regions, which are not covered by the main top gate but might partially be depleted by the main bottom gate.
The geometry depicted in 3.3b allows one to directly measure the background capacitance signal from the ungated regions. To do this, one first performs a capac- itance measurement while the main top and bottom gates are tuned to the desired biases. This will result in a capacitance signal containing the interlayer capacitance from both the gated and ungated regions. One then applies a large negative bias to the main top gate, depleting the gated region. But the ungated regions are essen- tially left undisturbed. By repeating the same capacitance measurement, one then directly measures the interlayer capacitance in the ungated regions alone. Subtract- ing the second measurement from the first results in the desired capacitance of only the gated region.
There is a small amount of error in this subtraction process due to the influence