Tunneling versus θ

Bottom arm gate Top arm gate

6.5 Transport in a Tilted Magnetic Field

6.5.1 Tunneling versus θ

By tilting the sample at an angleθ with respect to the magnetic field, we introduce a in-plane magnetic fieldB|| that is parallel to the bilayer system. Assume for now that this in-plane field is in the y direction and we choose to express its vector potential in the Landau gauge, such that A~|| = zB||x. If we also include the perpendicularˆ magnetic field with a Landau gauge, the canonical momentum in the x direction is thus P_X = ¯hk_x +ezB||/c−eyB⊥/c. Thus, there is a shift in canonical momentum between the two layers equal to edB_||/c. This shift will not affect purely in-plane motion, but it can influence electrons moving from one layer to another.

In the limit of small B_||, most of the transport properties at ν_T = 1 such as Hall drag [67, 38] are not qualitatively altered.³ However, the coherent interlayer tunneling that is linked to phase-coherent excitons can be strongly influenced by a relatively small B||. With respect to interlayer charge transport, the parallel field provides a wave vector q = eB_||d/¯hc, where d is the interlayer separation. In a semiclassical picture, this wave vector corresponds to the displacement in canonical momentum between the two quantum wells due to the vector potential of the in- plane magnetic field [53]. Loosely speaking, electrons tunneling from one layer to

3More precisely, when both`_|| ≡p

¯

h/eB_|| is much larger than the quantum well width and the Zeeman energy is not significantly increased.

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another will then access the collective Goldstone mode at finite q vector. In the absence of a parallel field, tunneling electrons will probe the linearly dispersing mode at q = 0. Because the mode is gapless in the limit of zero bare tunneling energy, the tunneling resonance will occur at zero energy and thus zero interlayer bias. But as B_|| becomes nonzero, the tunneling peak should split into two peaks located at finite bias eV = ±¯hω(q), where ¯hω(q) is the energy associated with the Goldstone mode [6, 107, 40]. In the low wave-length limit, ¯hω(q) = ¯h¯cq ∝ B_||, where ¯c is the velocity of the linearly dispersive mode. This prediction was confirmed by Spielman et al. [104], who observed the appearance of side resonances in the tunneling spectra when an in-plane field was applied. Meanwhile, in the absence of disorder the central peak should become suppressed as the Goldstone mode is no longer accessible at q= 0. Spielman et al. did see the central peak at zero bias decrease in height as they introduced an in-plane field, but the decline was much slower than expected, most likely due to disorder [102].

We now comment on the coherent ν_T = 1 interlayer tunneling that we observe in our sample in the presence of a parallel magnetic field. The two-terminal differential tunneling conductance for various tilt angles are plotted in figure 6.10. The zero bias conductance becomes suppressed as the tilt angle is increased, as expected. The incoherent tunneling persists at this moderate in-plane field, as explained later on in this section. One can also see the side resonances in tunneling dI/dV at finite bias that were first observed by Spielman et al. [104]. As expected, the side resonances move out to higher bias as the parallel field is increased.

Figure 6.10 also indicates that the zero bias tunneling resonance appears to be- come narrower in our two-terminal measurement as the tilt angle is increased. This can be understood in terms of the finite series resistance R_series. In the limit of van- ishing tunneling resistance, when a bias is applied nearly all of the voltage drops are occurring along the series resistance. Thus, the two-terminal voltage width of the zero bias tunneling resonance is given by ∆V = 2I_maxR_series, where I_max is the maxi- mum tunneling current that can flow at zero interlayer voltage. As both the intrinsic ν_T = 1 tunneling conductance and I_max become reduced by the parallel field, the

m

m

q q q q

m

m

q q ! q q

"

Figure 6.10: Interlayer tunneling conductance at d/` = 1.49 and T = 25 mK for various tilt angles.

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two-terminal voltage width of the resonance decreases. The role of series resistance in determining the apparent width diminishes and other factors such as noise, thermal fluctuations, disorder, and the magnitude of the AC excitations (20 µV in this case) become relevant. This is why the width of the resonance saturates to a small value at high tilt angle in figure 6.10b.

As a side note, we point out that a large B|| can also reduce the incoherent interlayer tunneling, which is not associated with the interlayer correlations present atν_T = 1. This can be illustrated by measuring tunneling at high temperature, where the coherent tunneling is negligible and only incoherent tunneling remains. We plot the tunneling current as a function of applied interlayer bias for four different tilt angles in figure 6.11. For each measurement, T = 600 mK and d/` = 1.49. Under these conditions, we observe that the zero biasν_T = 1 tunneling feature is essentially gone. Each tunnelingI−V has a maximum currentI₊ at positive interlayer bias and a minimum current I− at negative bias. To characterize the strength of tunneling in the presence of B_||, we compute the average peak tunneling I_avg = ¹₂(I₊+|I−|) and plot the results versus tanθ. We focus on I_avg to reduce systematic errors stemming from preamp offsets and any interlayer asymmetries within the bilayer sample.

The average peak tunneling should be proportional to the square of the symmetric- antisymmetric tunneling splitting ∆_SAS. The tunneling splitting is reduced because the wave function overlap between sets of states in different layers and equal in- plane wave vector is reduced by the parallel magnetic field. Recall that in a large perpendicular magnetic field one can choose a gauge for the vector potential in which the wave functions are extended plane waves in thexdirection and localized Gaussians in theydirection. Without an in-plane field each Gaussian is localized at the guiding centery₀ =`²k_x, wherek_x is the wave vector in thexdirection. We next consider the effect of an in-plane magnetic field, neglecting the finite thickness of the 2DESs. If an in-plane field is applied in the ydirection, one must replacek_x in the Hamiltonian with the expressionk_x+ezB||/¯hc. Thus, states in two different layers that correspond to the same guiding center y₀ will differ in k_x by an amount equal to q =edB_||/¯hc.

Because k_x still commutes with the Hamiltonian and must represent a conserved

states’ wave functions to decrease asB||grows. The tunneling splitting is proportional to this wave function overlap and, as shown by Hu and MacDonald [53], this will lead to the equation

Iavg(θ) = I0exp

"

1 2

d

` tanθ 2#

, (6.2)

where we defineI0 ≡Iavg atθ = 0. We see that there is good agreement between ex- periment and equation (6.2). While only a small parallel field is sufficient to suppress coherent νT = 1 tunneling (B|| ≈0.1 T is required to reduce the coherent tunneling by half [102]), a much larger parallel field (∼1.4 T ford/` = 1.49) is needed to shrink the incoherent tunneling by the same fraction. However, because the incoherent tun- neling is orders of magnitude smaller than the coherent tunneling at B|| = 0, we can ignore its effects on counterflow measurements.