In-Plane Magnetic Field Effect on the Transport Properties in a Quasi-3D Quantum We11 Structure

The transport properties of a quasi-three-dimensional, 200 layer quantum well structure are investigated at integer filling in the quantum Hall state. We find that the transverse ma,onetoresistance R,, the Hall resistance R, , and the vertical resistance R, all follow a similar behavior with both temperature and in-plane masetic field. A general feature of the influence of increasing in-plane field B, is that the Hall conductance quantization first improves, but above a characteristic value B,: , the quantization is systematically removed. We consider the interplay of the chiral ed, m e state transport and the bulk (quantum Hall) transport properties. This mechanism may arise from the competition of the cyclotron energy with the superlattice band structure energies. A comparison of the results with existing theories of the chiral edge state transport with in-plane field is also discussed.


INTRODUCTION
. The integer quantum Hall effect has been observed in many quasi-three dimensional ( Q30') structures1-5, where the interlayer tunneling band-width is much smaller than the two dimensional ( 2 0 ) quantum H a l l gap Eg = Au, . In such materials, in a quantum H A (QH) state, the edze of the sample is enveloped by a sheath of currentcarrying Chiral edge states which is the 2 0 extension of the 10 states at the edge of a single layer QH fluid. These chiral edge states have been predicted theoretically69 7, and confmed experimentally2. A curious property of these Q3D systems at integer filling is the observation of a very similar temperature dependence of the in-plane resistance at the center of the QH state (I?:") and the inverse of the vertical transport resistance (G, L -l/R= ). In particular, with magnetic field perpendicular to the layers, both follow an activated behavior (with a gap much smaller than the Landau gap h o c ) at higher temperatures, typically above 0.3 K, and a Coulomb gap-like behavior at lower temperatures where both R F and GE approach some asymptotic (residual) values..This is the case for the system discussed here, and also appears to be the case for R F and G-L in an independent study2. The motivation for the present work has been to try to understand the origin of this universal temperature dependence, and to consider the influence of f~t e in-plane field on the properties of the transport tensor. This second point is particularly important since there are theoretical predictions9 for the in-plane field dependence of the chiral surface transport.
BFUEF REVIEW OF OUR RESULTS.
The main results of the present work are the observation that in tilted magnetic fields, the quantization first improves at integer filling (as seen by the reduction i n dissipation in R,, the enhancement of R, , and the broadening of the R, Hall plateau ), followed by the gradual disappearance of the quantum Hall state above a characteristic in-plane magnetic .field BH. We further note that the temperature dependence of all measured components of the transport tensor also follow the same behavior. The results suggest an interplay between the bulk quantum Hall state and the chiral edge state systems, and provide a test of the theoretical models for the transport properties of the A conventional Hall bar configuration was used in measuring the in-plane magnetoresistance R, and R,, as shown in Figs. 2 and 3. To measure the vertical R,transport, sections from the same sample were processed by a vertical etching process'o to provide a mesa-like structure as shown in Fig. 4. We note that R-was a four-terminal measurement, and that independent two-terminal measurements11. 12 did not show any mixing of R, and R, into the R, signal. Measurements were carried out with standard ac lock-in methods with a current of 50 nA./layer. No evidence for heating or hot-electron gas effects were observed. For all measurements shown here, a rotating platform immersed in the mixture of a dilution refrigerator, associated with a superconducing magnet was employed.
The evidence that the Q3D sample exhibited complete quantization is shown in Fig. 3. At 8.7 tesla, concomitant with R, minima ( R E ) in Fig, 2, and the R,maximum (R,"") in Fig. 4 there is a well developed Hall plateau in Rq, which saturates at a below, the crossover may be a result of the modification of the mini-band structure with increasing in-plane magnetic field. We note that in our four terminal measurements, we find no systematic evidence for universal conductance fluctuations, although extensive efforts were made to measure these effects, as reported in Ref. 2.

DISCUSSION
The results discussed above indicate an unusual coupling of the bulk quantum well transport (R, and R,?.) at integer filling with the corresponding behavior of the chiral edge state transport ( l/R=). This is true both in terms of the temperature dependence, and in the angular (in-plane field ) dependence. At zero angle, both R$" and l/R,"remainfinite in the low temperature limit, indicating the presence of dissipation in both the bulk quantum Hall state and in the chiral edge state. And, in finite in-plane field, the angular dependence of these two tensor components consistently track each other, first as the quantization improves, and then as it is removed. Indeed, the-two states appear to be interconnected. The crossover from activated-to-power law behavior above the optimum angle also appears in both parameters. The finite dissipation at low temperatures in 1/ R, " " c , which is also observed in the measurements of Druist et al.2, is at odds with theoretical expectations for a dissipationless 2D chiral metal as predicted by Balents and Fisher7. Although the theoretically predicted geometrical relationship6 of G"" iL = C L (where C and L are the. height and circumference of a mesa-type structure respectively) has been demonstrated in experiment2, the anomalous low temperature dissipation is observed in both reported measurments of GI". Given the apparent coupling of the bulk and edge state transport properties, it is not clezi that the surface transport is truly decoupled from the bulk.
We next turn to a discussion of the effects of the h-plane field. Chalker and Sondhi have treated the chiral edge state conductivity o(B,) with in-plane field9 Bin . Their results show that R, should exhibit positive magnetoresistance, following a Fig. 5 a more detailed study of these trends is presented in the form R,"" and 1/ R , " " . For the case of no in-plane field, there is a clear relationship between the measured transverse resistance RE and the conductivity, G, -1/ R, . In this paper, we take the view point that a variable titled field at integer filling is simply the case where the transverse field is constant, and the in-plane field varies. Hence we retain the definition that G, -l/Rz (at v=2 filling in this case) with'ihcreasing in-plane field. Since the total field B must be increased to maintain the v=2 filling with increasing angle, the optimum angle (more accurateiy 27' corresponds to a perpendicular field of B, = 8.7 T and an in-plane field B, = 4.4 T. We will return to the influence of the in-plane magnetic field in the discussion.
Another striking feature of the transport properties is their temperature dependence, as is shown in Fig. 6. Here we show R F and 1 / R Y ( - temperature for O=Oo, and also for 0=36", which is well above the optimum angle where the quantization is reduced. There are two temperature ranges of interest. First, at high temperatures (above 0.3 K) and 0=Oo7 R E and l / R Y both show actjvated behavior ( R = and G = Goe-A'2n respectively), and at low temperatures (below 0.3 K) -&F both exhibit Coulomb gap-like behavior ( R = R, + R,e--and G = G, + G,e respectively). In contrast, in tilted magnetic field, and above the optimum angle, both exhibit power law (non activated) behavior at high temperatures ( R = R, + R,T" and G = Go + GIT" respectively). However, the low temperature behavior remains Coulomb gap-like. (The parameter values are defined in the caption of Fig-6.) The nearly identical temperature dependence of R,"" and l/R,""is further shown in Fig. 7a where they are plotted against each other for the different temperature ranges and angles. The crossover from activated to power law dependence with angle is more clearly demonstrated in Fig. 7b7 where, for each tensor component, the zero angle and tilted values are plotted against each other. This is to emphasize the different functional dependence for the two orientations. The crossover from activated to power law behavior is an additional indication of the degradation of the quantum Hall state. As discussed . sequentially deplete individual layers with a top gate structure'. Such a procedure also reduces or eliminates parallel conductance channels. However, in the presence of a gated sample with several integer layers depleted, the asymptotic behavior of R z w a s still evident, as was the full quantization in R, . Hence it is unlikely that the finite, residual dissipation observed in R F in the low temperature limit is due to non-integer depletion or a parallel conductance channel.

RESULTS
In Figs v=2 there is a &mum in the dissipation near 30°, Le. R Z approaches the lowest value at this angle, and at higher angles the dissipation increases. In Fig. 3 the width of the v=2 Hall plateau is largest at 30' then rapidly decreases with increasing angle (note the inset is plotted vs. B, , not the total field B). And finally, in Fig. 4 the maxium of the vertical resistance Ram has a slight extremum at 30°, followed by a reduction at higher angles. In all three cases the effect of increasing angle corresponds to an initial improvement of the quantization, followed by a rapid removal of quantization above an optimum angle. In Drude formula, with a field scaling field Bo = O, / Q !~~, where Oo is the flux quanta, Q is the lattice spacing and .le, is the in-plane elastic 1en-d We may further consider the behavior of the transport tensor components above the optimum angle, where the quantization is removed with increasing in-plane field. There are several treatments of multiple well structures in titled magnetic fields.
Marlow and co-workers have studied cyclotron resonance in a coupled two layer quantum well in titled magnetic fields'3. They find wavefunction hybridization and subband energy splitting which result from the in-plane magnetic field. Although a detailed description of the removal of the quantization must be worked out theoretically, a general argument for the mechanism may be made. In reference to Fig-1, we note that the condition for quantization, with the Fermi level in a gap, will change with increasing in-plane field due to fact that the eigen values of the Hamiltonian will change when the in-plane field is added. It would appear then, from our experiments, that the band structure as given in Fig. 1 starts to change si,Onificantly above an in-plane field of 4 T, and it is no longer possible to maintain the quantization condition with the Fermi level in a gap. Hence the results may be viewed a crossover from a quasi-two-dimensional to a quasi-three-dimensional electronic structure (Le. bands closing with respect to the Fermi level ) with increasing in-plane field.

SUMMARY '
In summary, we have studied the transport properties associated with the integer quantum Hall effect in a 200 layer quantum well superlattice in the Hall-bar configuration, and also the vertical transport associated with the chiral ed, ue state in the mesa configuration, both on the same material. Furthermore, the transport properties have also been studied in tilted magnetic fields while maintaining the position of the v = 2 filling, for increasing in-plane field. We find that there is a direct correlation between the in-plane magnetoresistance R,, and the vertical conductivity lR=, both in terms of the temperature dependence, and in the angular dependence. For in-plane fields up to 4T, the quantization improves, but for larger in-plane fields, the quantization is removed. These results suggest a strong correlation between the bulk quantum Hall states and the chiral edge states. This correlation persists even as the system crosses over from a quasi-twodimensional to a quasi-three-dimensional electronic structure in tilted field, an effect which is most likely due to the modification of the subband structure at high in-plane magnetic fields. If we assume that the chiral edge state is present, then its properties are substantially different from theoretical expectations, since there appears to be considerable dissipation in the low temperature limit, and, since the chiral behavior is correlated with the bulk behavior. Interaction effects may play an important role in the modification of the mini-band structure with in-plane field, and further theoretical work is needed to fully understand the behavior in tilted magnetic fields.
.   The mesa configuration is also indicated. Figure 5a. In-plane magnetic field dependence of l/R,"' "x and Rxxmin at 30mK There is an optimal value of the in-plane field BII. Here, 1/R,"' " and Rumin are plotted on different scales. With the in-plane field increasing from zero to 8 tesla, Ramin changes by almost an order of magnitude, but the change in l/Rn"'" is very small by comparison. Figure 5b. In-plane magnetic field dependence of 1/RZm" and Rnmin at 548mK.    RXxmi"(~=36') and 1R="""(0=O0) vs. 1/R~mx(0=360) are plotted to show that the functional dependence with temperature has changed for each between the two field .

Sandia is
orientations. (0=36' is above the optimal angle for quantization.) Tfie non-linear slopes indicate the functional difference between activated and power-law dependence between O=Oo and 0=36'.