Layer interdependence of transport in an undoped electron-hole bilayer

Christian P. Morath,* John A. Seamons, John L. Reno, and Mike P. LillySandia National Laboratories, Albuquerque, New Mexico 87185, USA

�Received 7 March 2008; revised manuscript received 8 August 2008; published 23 September 2008�

The layer interdependence of transport in an undoped electron-hole bilayer �UEHBL� device was studied asa function of carrier density, interlayer electric field, and temperature. The UEHBL device consisted of adensity tunable, independently contacted two-dimensional electron gas �2DEG� and two-dimensional hole gas�2DHG� in distinct �100� GaAs quantum wells separated by a 30 nm Al0.9Ga0.1As barrier. The 2DEG and2DHG are induced via field effect by top and bottom gates, respectively. Transport measurements were madesimultaneously on each layer using the van der Pauw method. An increase in 2DHG mobility with increasing2DEG density was observed, while the 2DEG mobility showed minimal dependence on the 2DHG density.Changes in both surface-gate voltages for the mobility-density measurements were observed. Decreasing theinterlayer electric field and thereby increasing interlayer separation also increased the 2DHG mobility withnegligible effects on the 2DEG mobility. The change in interlayer separation as interlayer electric field changedwas estimated using 2DHG Coulomb drag measurements. A general discussion of the results is given in whichthe possible sources for the apparent interlayer dependence of the hole transport are examined.

DOI: 10.1103/PhysRevB.78.115318 PACS number�s�: 73.63.Hs, 73.40.Rw, 73.40.Ty

I. INTRODUCTION

Interest in electron-hole bilayers necessarily arose fromthe prospect of observing Bose-Einstein condensation of ex-citons in semiconductor double quantum well systems1,2 andsignificant progress toward this goal has been made.3,4 Thistrend in bilayer research centered on the behavior of theelectron-hole pair. The work presented in this paper, how-ever, focuses on the individual transport in each layer and thelayer interdependence. The latter is the primary question weseek to answer here and for which a bilayer device is singu-larly, exceptionally suited: to what extent will the mere pres-ence of a nearby two-dimensional electron gas �2DEG� affectthe transport in a two-dimensional hole gas �2DHG� and viceversa?

The general transport properties of the 2DEG system inmodulation-doped heterostructures were well established de-cades ago.5,6 Evidence for how the carrier mobility in thesesystems can be varied, via changes to scattering time or ef-fective mass, will be emerged later. Exploiting Coulombscattering’s dependence on the shape of the wave function,Hirakawa et al.7 demonstrated that a 2DEG’s mobility couldbe altered by deforming the wave function using externalfields from gates.8,9 Calculations by Kurobe10 also showedthat the 2DEG wave function can be squeezed by changingsurface-gate voltages; when the 2DEG’s confining potentialis tilted by an electric field, the wave function is squeezedagainst an interface. Furthermore, the calculations showedthat this squeezing reduces remote and space impurity scat-tering times but enhances the channel impurity scatteringtime. From other studies on similar devices it was deter-mined that background channel impurities dominate scatter-ing at low densities, while interface roughness dominates athigher densities.11,12 More recently, Das Sarma et al.13 andManfra et al.14 showed that background impurity scatteringin GaAs heterostructures is rudimentary to the two-dimensional �2D� metal-insulator transition, which occurs asdensity is reduced and screening of the random potential

landscape, due to these impurities, becomes progressivelyweaker. Changes to the effective mass, a second possiblemechanism to vary mobility, have also been recently inves-tigated. Zu et al.15 showed how a 2DHG effective mass var-ies with well parameters and density due to the highly non-parabolic valence subband structure. Lastly, several studiesof spin-orbit coupling induced Rashba effect have shown thatthe spin splitting of the hole subband could be controlled viasurface-gate voltages and how this changes the densities ofthe spin-split subbands.16,17 These subbands have differenteffective mass, and thus, a change in the relative populationsof each may influence mobility. These studies, however,were all on unipolar devices and an experimental study ofwhether general transport, specifically carrier mobility, is af-fected by a 2D system of opposite charge in close proximityhas not been reported.

In this paper, the results of an investigation into the layerinterdependence of transport in an undoped electron-hole bi-layer �UEHBL� are presented. The UEHBL device understudy consists of a density tunable, independently contacted2DEG and 2DHG induced via field effect in distinct GaAsquantum wells separated by a 30 nm Al0.9Ga0.1As barrier.18

To populate the undoped wells an interlayer electric field EILis necessarily established to account for the energy differencebetween the conduction and valence bands. This design af-fords the following advantages: �1� independent contacts al-low for simultaneous transport measurements of each layerand Coulomb drag measurements; �2� a tunable density2DEG and 2DHG allows for these measurements to be madeas functions of the density in each layer, n and p; �3� anundoped structure reduces scattering by remote ionized im-purities; and �4� for equal densities, the interlayer separationd between the 2DEG and 2DHG can be varied by changingEIL and both gate voltages. The investigation included mo-bility and resistivity measurements measured in each layer asfunctions of n and p, EIL, and temperature T. Coulomb dragmeasurements were used to estimate the change in d as EILwas varied.

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The paper is organized as follows. An explanation of thedevice material and fabrication is given in Sec. II. Detailsregarding the device operation and experiment are presentedin Sec. III. The results are presented in Sec. IV. They indicatethat 2DHG transport changed by varying n or EIL, while thetransport observed in the 2DEG was largely immune to simi-lar magnitude changes in p or EIL. The apparent layer inter-dependence demonstrated by the 2DHG transport may onlyhave been indirect; however, since increasing n also neces-sitated changes to the surface-gate voltage predominantlycontrolling p, which also would have affected the shape ofthe 2DHG confinement potential. Changes in confinementpotential are known to affect 2DHG transport and whetherany mechanisms are appropriate to the changes in hole mo-bility observed in the UEHBL is discussed in Sec. VI. Thisdiscussion is augmented with Coulomb drag measurementsthat were used to estimate the change in hole wave-functionposition with VIL. Finally, in Sec. VI the conclusions aresummarized.

II. MATERIAL AND FABRICATION

A description of the design and fabrication of the deviceused in this study was given by Seamons et al.18 The UE-HBL device was formed from molecular-beam epitaxygrown GaAs/AlGaAs double quantum well material �waferEA1286� grown on the �100� surface of a GaAs substrate. Aside profile diagram of the device after full processing isshown in Fig. 1�a�. The top and bottom 18 nm GaAs quan-tum wells were separated by a 30 nm Al0.9Ga0.1As barrier.Above the top quantum well is a 200 nm Al0.3Ga0.7As clad-ding layer and a 60 nm Si-doped n+ GaAs cap layer. Beneaththe bottom quantum well is a 125 nm Al0.3Ga0.7As claddinglayer and a 310 nm growth superlattice. Beneath that is a 15nm GaAs layer, which acts as the second etch stop duringback-side processing and effectively becomes the cap layer.The first stop etch �not visible in diagram� is a 500 nmAl0.55Ga0.45As layer and that is completely removed duringprocessing.

Because our approach to electron-hole bilayers is to in-duce carriers with gate-generated electric fields, the process-ing for these devices is quite involved. The key steps are todefine a Hall bar with an integrated top gate, define then-type and p-type contacts, thin the structure for back-sideprocessing, and apply a final metallization layer for both aback gate. An image of the final device is shown in Fig. 1�b�,and the key steps are described below.

To process the UEHBL device an �6�8 mm2 piece wascleaved from the wafer and mesa etched into the shape of a200 �m wide Hall bar with five arms extending from it oneach side. The only intentional dopants present in the mate-rial were in the n+ cap layer that forms the top gate. Thesedopants do not provide the carriers to populate either of thequantum wells. Instead, electrons are pulled into the topquantum well from n-type NiGeAu self-aligned Ohmic con-tacts with the top gate.19 A shallow annealing of PdGeAumetal was used to make Ohmic contact to the top gate �theright end of Fig. 1�b��. In Fig. 1�b�, the n-type contact andgate arms are outlined by the black dashed lines. The arms

are very difficult to see in the image because following back-side processing they end up beneath the epilayer, as will bediscussed further below. The actual contact pads are not vis-ible in the micrograph. These contacts served a dual role asreservoirs supplying electrons to the top quantum well and asthe n-type Ohmic contacts for transport measurements on the2DEG. At the ends of the remaining arms, AuBe p-typeOhmic contacts were formed.

The carriers in each quantum well were induced via ex-ternal fields generated by gates on the top and bottom of thedevice, as depicted in Fig. 1�a�. Each gate covers the centralHall bar region and extended along the entire length of halfthe arms in a geometry which allows for both longitudinalresistance Rxx and Hall resistance Rxy measurements in bothwells independently. The gate contact pads were positionedat opposite ends of the long axis of the central Hall barregion �arms visible at right and left ends of Fig. 1�b��.

The patterning of the bottom gate of the device requires aprocedure for thinning GaAs heterostructures called epoxybond and stop etch �EBASE�.20 This technique entails epoxy-ing the sample to a GaAs host substrate with the completely

(a)

(c)

125 nm Al Ga As0.3 0.7

18 nm GaAs Lower Well 2DHG

30 nm Al Ga As Barrier.9 .1

18 nm GaAs Upper Well 2DEG

200 nm Al Ga As0.3 0.7

60 nm n+ GaAs

PdGeAu Top Gate

100 nm SiN

TiAu TiAuTiAu Back Gate

Self-AlignedContact

NiGeAun-type

(SAC)

OverlapContactBeAup-type

Vias

Vias

310 nm Superlattice

15 nm GaAs

200 m�

bottom-gate top-gate

overlap region

p - n

(b)

CB

VB

VIL

2DEG

2DHGEF,p

EF,n

VTG

VBG

FIG. 1. �Color online� �a� Schematic of the UEHBL device. �b�Micrograph of the device. The top gate and n-type contacts arebarely visible through the epimaterial and are outlined by blackdashed lines. The back gate covering the Hall bar and p-type con-tact arms is the gold region. Top and bottom gate contact arms arelocated at the ends of the Hall bar; the actual contacts are notvisible. �c� Schematic of the energy-band diagram depicting VTG,VIL, and VBG.

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processed top-side face down on the host substrate. Theoriginal GaAs substrate is removed down to the first stop-etch layer �Al0.55Ga0.45As, not shown in Fig. 1�a�� using acombination of lapping and selective etching of GaAs usingcitric acid. The first stop-etch layer is removed with an HFetch, which stops on the second stop-etch layer of GaAs.Once completed SiN was deposited over the entire mesa sur-face by plasma enhanced chemical vapor deposition and viaswere etched through the heterostructure to contact the topelectrical layers �n-type Ohmics, p-type Ohmics, and topgate�. The back-gate metal, TiAu, was then deposited overthe top of the SiN. The five p-type contact and the back-gatearms are clearly visible as gold regions in Fig. 1�b�.

The back gate covers the central Hall bar region, the fivep-type Ohmic contact arms, and a small portion of the p-typecontacts. In this so-called overlap configuration, holes arepulled by the back gate into the bottom well from the p-typeOhmic contacts;21 this is analogous to the aforementionedn-type Ohmic contacts’ function. While both quantum wellsare physically in contact with the n-type and p-type metalcontacts, the gates and mesa configuration are such that onlyone type carrier is induced in each well. The positions of thecontacts along the Hall bar were chosen to allow Rxx and Rxymeasurements of both quantum wells independently; how-ever, due to processing problems, only four contacts to eachlayer worked on this device and this led to measurementsbeing made using the van der Pauw method.22

III. EXPERIMENT

A schematic of the energy-band diagram of the UEHBLduring typical operation is given in Fig. 1�c�. To simulta-neously establish 2DEG and 2DHG in the UEHBL devicesthree different bias voltages, top-gate bias VTG, bottom-gatebias VBG, and interlayer bias VIL, are necessarily used; allthese voltages are referenced to ground. As depicted in Fig.1�c�, VTG, VIL, and VBG predominantly adjust the electricfields across the 2DEG, barrier region, and 2DHG, respec-tively. During operation at least one p-type contact alwaysremains grounded. The 2DEG is held at VIL, which accountsfor the difference in the electron and hole Fermi levels andends up being slightly less ��1.43–1.45 meV� than theGaAs band-gap energy ��1.51 eV�, due to the presence ofother field sources, VTG, VBG, and the carriers in each well.Ideally, it is expected that n and p are controlled only bytheir nearest gate, the top and bottom gates, respectively, dueto screening. However, as will be demonstrated below,changing the density in one well causes a small change in thedensity of the other well, requiring simultaneous adjustmentof VTG and VBG for the mobility versus density measure-ments at constant VIL. Additionally, the system is overdeter-mined �two densities and three voltage settings� so the samedensities can be achieved at different gate voltage settings.Finally, with the 2DEG held at VIL with respect to ground, allthe circuitry connected to it must also be held at VIL, neces-sitating the use of an isolation transformer to break theground of the signal source. This also means that n is mainlyproportional to �VTG−VIL�, while p� �VBG�.

The n and p in the UEHBL were set by adjusting VTG,VBG, and VIL and measured using low-magnetic-field Rxy

measurements. To characterize transport, the resistivity �was measured in each layer as a function of n and p, EIL, andtemperature T. The mobility in each layer was calculatedfrom the resistivity and density according to �p=1 / pe�p and�n=1 /ne�n. Rxy and � measurements were made by standardvan der Pauw methods using low-frequency lock-in tech-nique with separate 20 nA excitation currents in both layers.Coulomb drag measurements were used to estimate thechange in d as VIL changed. For these measurements a 10 nAcurrent was driven in the 2DEG, while the induced voltage inthe 2DHG was measured with a high-impedance detectioncircuit. The constant-temperature measurements were alltaken at T=0.3 K in a He3 refrigerator.

IV. RESULTS

To establish some basis for comparison, the mobility ofeach layer is plotted in Fig. 2 as a function of its density, n orp, at different densities in the adjacent layer with VIL=

(a)

(b)

FIG. 2. �Color online� Mobilities �a� �p and �b� �n as a functionof p and n, respectively, and the adjacent well density at T=0.3 K and VIL=−1.44 V.

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−1.44 V. The resulting interlayer electric field EIL is ex-pected to be on the order of 10 kV/cm, far below the�500 kV /cm breakdown field of Al0.9Ga0.1As in this tem-perature range.23 In the traces of Fig. 2�a� an increase in holemobility �p with increasing n is visible with a weakeningdependence as p increases. In direct contrast, the traces inFig. 2�b� of the electron mobility �n show minimal depen-dence on p, the density in its adjacent well. This mobilitylayer interdependence is more closely investigated in Fig. 3.First, these data show that the mobility in both layers in-creases with its density, as expected from Fig. 2. Second, theplots in Fig. 3�a� also show that there is a monotonicallyincreasing relationship between the hole mobility �p andelectron density n, the density in the adjacent well. At thelowest hole density, p=5.0�1010 cm−2, a�23% change in�p was observed. As p was increased to 10.0�1010 cm−2

the percent change apparently becomes weaker, which is il-lustrated by the visible decrease in slope between the datasets. This decrease in slope corresponds to the data in Fig.

2�a�, where the spread between plots was much larger forsmaller p. The traces in Fig. 3�b� show that �n was roughlyindependent of p, confirming what was apparent in Fig. 2�b�.

The inset plot of Fig. 3�a� shows the change in bottom-gate voltage �VBG required to maintain a constant p while nwas increased from 3.0�1010 to 13.0�1010 cm−2 using VTGfor each trace in the main plot. As previously mentioned,changing the density in one well often leads to a smallchange in the density of the other well, and thus, both gatevoltages VTG and VBG must be simultaneously adjusted to setthe densities. For all the measurements taken for this work,making VTG less negative increased n, while making VBGmore negative increased p. According to the inset plot, as nwas increased for each trace VBG had to be made slightlymore negative for p to remain constant. The increases in nfor each trace thus result in small decreases in p since in-creasing VTG �making VTG less negative� to increase n re-quired making VBG more negative to keep p constant. Fur-thermore, the steepness of the slopes of each trace in Fig.3�a� correlates with the magnitude of the change in VBG inthe inset plot; as �VBG� increases, the slope becomes steeper.

In Fig. 4 the same �p data at p=5�1010 cm−2 and VIL=−1.44 V from Fig. 3�a� are plotted alongside similar mea-surements at VIL=−1.45 and −1.43 V. The data in Fig. 4were taken to examine the role of VIL in determining �p, aschanging VIL was also expected to alter the 2DHG and2DEG wave functions. With the 2DHG held at ground, mak-ing VIL, the voltage dropped across the barrier and less nega-tive pulled the 2DEG energy level down toward the 2DHGenergy and, thereby, increased EIL. Measurements of �n un-der similar conditions �not shown� were also taken butshowed no discernable dependence on VIL. The inset plot ofFig. 4 shows the �VBG required to maintain constant p whileincreasing n from 3.0�1010 to 12.0�1010 cm−2 at each VIL.

Based on the previous results in Fig. 3�a�, the slopes ofeach data sets in Fig. 4 suggest, and the inset data confirm,that the largest �VBG occurred at VIL=−1.43 V since it hasthe steepest slope and that �VBG increases with decreasing

(a)

(b)

FIG. 3. �Color online� Mobilities �a� �p and �b� �n as a functionof adjacent well carrier density, n and p, respectively, at T=0.3 Kand VIL=−1.44 V. The inset plot shows �VBG as a function of p forthe data in �a�.

FIG. 4. �Color online� Mobility �p as a function of n forVIL=−1.43 to −1.45 V and �inset� �VBG as a function of VIL at T=0.3 K.

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VIL. For fixed n, the traces also show an increase in �p asVIL decreases. In this case, however, the increase in �p wasaccompanied by an increase in VBG �making it less negative�as VIL decreased. For example, at n=3�1010 cm−2 aVBG=−1.647 V resulted for VIL=−1.43 V, which was morenegative than VBG=−1.5735 V at VIL=−1.45 V. This im-plies that changing VIL also has a large impact on �p since acomparison of the slopes of the traces in Figs. 3�a� and 4with the inset data shows that making VBG less negativewould typically be associated with a reduction in �p, not anenhancement.

V. DISCUSSION

A discussion of the mechanisms that might qualitativelydescribe the results in Sec. IV, the apparent increase in �p asn increased, the lack of similar magnitude changes in �n, andthe changes in hole transport and Coulomb drag with VIL, isgiven in the following. The analysis of the UEHBL is com-plicated by the bipolar nature of the device and the relateduse of three different voltages to control the carrier densitiesand the two different field-effect transistor �FET� structuresused to generate them. These make it difficult to determinethe exact shape of the confining potentials in the device;however, the action of the gates on the confining potentialsand the wave-function shape can still be qualitatively de-scribed sufficient for one of these mechanisms to seem like-lier than the other two.

Mobility is directly proportional to �p and inversely pro-portional to mh

�; changes in either could lead to the behaviorof �p above. One previously mentioned method for varying�p is via squeezing of the wave function. If scattering islimited by background impurity scattering in the well regionthen increased squeezing of the hole wave function wouldlead to an increase in �p. Squeezing is known to reducebackground impurity scattering in undoped heterostructuresin this density range ��1011 cm−2�.10,12 In the UEHBL, thehole wave-function squeezing would increase with increas-ing n because VBG, which tilts the 2DHG confinement poten-tial, was simultaneously decreased �made more negative� tomaintain a constant p, as shown in the inset of Fig. 3�a�. ABoltzmann transport calculation for the mobility-density dataof a similar structure showed that transport in each layer wasqualitatively consistent with scattering being dominated by auniform background impurity density.24 Finally, the decreasein steepness of the slopes of each data set in Fig. 3�a� as pwas increased is also consistent with squeezing since thechange in �p is proportional to −�VBG, which correspond-ingly drops in magnitude as p increased �see inset of Fig.3�a��.

While qualitatively �VBG would move the hole wavefunction in the right direction for squeezing to occur thequestion that arises is whether the effect was large enough tocause a change in �p. An analysis of calculations on a 2DEGin an undoped 100 nm wide quantum well showed that a�E�18 kV /cm led to an increase in background impurityscattering time of ���3�1012 s at n=1�1011 cm−2.10 Ifbackground impurity scattering was completely limiting �nthen this effect would change it by �5.0�103 cm2 /V s. In

comparison, the �p results in Fig. 3�a� show a �VBG=3 mV at p=1�1011 cm−2 which led to a ��p�4�104 cm2 /V s. It is not clear how these results wouldchange for the case of holes or for a smaller well; however,squeezing effect on mobility increases as density isreduced.10

A second question which arises is why similar magnitudechanges were not visible in the �n results since VTG was alsovaried as p increased. If background impurity scattering alsolimited the 2DEG mobility then the results in Fig. 3�b�, that�n is roughly independent of p, imply that squeezing of theelectron wave function was much weaker than squeezing ofthe hole wave function as density in the respective adjacentwell increased. A direct comparison of the changes in the topand bottom gate voltages is invalidated by the device’s asym-metry with regard to the cladding layer widths and relativelydifferent gate leakages. The latter was a function of the typeof gate and contact combinations for either 2D system �seeFig. 1�a��. Thus, it is possible that changes to VTG causedrelatively smaller changes in the electron wave functioncompared with VBG’s effect on the hole wave function.

The second means by which to alter mobility is by vary-ing m�. In GaAs/AlGaAs heterostructures, the conduction-band structure is parabolic and can be described by a con-stant mn

�, while the valence-band structure is highlynonparabolic due to admixing of the heavy-hole and light-hole subbands and that leads to mh

� being a function of sev-eral factors, such as p, the orientation of the grown surface,and the confining potential’s height, width, and symmetry.Changes in mh

� with p and well width W in �100� GaAsquantum wells were recently measured by Zu et al.15 usingcyclotron resonance. They showed that mh

� increased as pincreases for fixed W and as W increases for fixed p. In theresults of Sec. IV above, however, �p increases as the adja-cent well density n increased, while p and W were both heldconstant. Furthermore, from the traces in both Figs. 2�a� and3�a�, �p always increases as p increased, which means thatany increase in mh

� due to increasing p, as implied by theresults of Zu et al.,15 must have been weaker than the in-creases in scattering time from increased p. Finally, thechange in mh

� required for a ��p�2�104 cm2 /V s, which,for example, was roughly the change observed at p=5.0�1010 cm−2 in Fig. 3�a�, is roughly 0.06me. Based on ananalysis of the results of Zu et al.,15 this would equate toeither a �W�5 nm or a �p�2.0�1110 cm−2, neither ofwhich is expected.

Variation in m� is also effectively possible via the Rashbaeffect, a spin-orbit interaction found in systems lacking in-version symmetry that leads to spin splitting of subbands inthe absence of a magnetic field.25,26 In such systems the mov-ing carriers feel an effective magnetic field proportional tothe vector product of the carrier’s in-plane velocity and anelectric field which is present because of the inversionasymmetry.27,28 In a quantum well the application of asurface-gate bias provides a source of inversion asymmetryand the spin splitting can be varied using the surfacegate.29,30 In GaAs/AlGaAs heterostructures the smaller bandmass and weaker spin-orbit couplings of the electrons leadto, for our purposes, negligible Rashba effect in the conduc-tion band; in the valence band, however, measurable effects

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are expected.16,17,31 In the uppermost hole subband pair 2Dheavy-hole and light-hole spin-split subbands result from theheavy-hole band.32 By changing the spin splitting, the rela-tive populations of the spin-split subbands can be varied andsince these subbands have different m� it is expected that theRashba effect may thereby alter the effective mass of thesystem and correspondingly the mobility.

The confining potential for the 2DHG in the UEHBL wasexpected to be asymmetric due to VBG, which generate theholes. Thus, making VBG more negative was expected to in-crease the spin splitting of the 2DHG subbands. This would,in turn, increase the difference in relative populations of eachspin-split hole subband, with a corresponding increase�decrease� in the population of the spin-split heavy �light�hole subband for constant p.16 This change in relative popu-lations should lead to an increase in mh

� and, thereby, reduce�p. This type of behavior was not observed, however, asevidenced by the results in Fig. 3�a�, where �p increases as nincreased and simultaneously VBG was made more negativeso p remained constant. Thus, it appears that at least quali-tatively the Rashba effect cannot account for the apparentlayer interdependence of �p in the UEHBL data from Figs. 2and 3. Furthermore, the observed changes in hole mobilitywere strongest at low density, while the Rashba effect isexpected to occur mainly at larger k and vanish as k→0.Based on this analysis, it seems likely that squeezing of thehole wave function was the source of the apparent interlayerdependence manifest between �p and n.

To further elucidate the nature of the squeezing of thehole wave function in the UEHBL similar transport measure-ments, shown in Fig. 4, and Coulomb drag measurements,shown in Fig. 5, were then made at various VIL. Making VILless �more� negative presumably increased �decreased� thefield across the barrier since the 2DEG energy level wasbeing pulled down toward �up away from� the 2DHG energylevel, which was held at ground. Inspecting data from Fig. 4it is apparent that squeezing also occurs at different VIL and

that VIL significantly affects the relationship between �p andn and, thus, the amount of squeezing that occurs. Combinedwith the inset data of Fig. 4, these results are basically con-sistent with the previous discussion above; for each VIL traceas n increased VBG was made more negative and, presum-ably, this increased squeezing of the 2DHG wave functionleads to an increase in �p. Considering these results moreclosely, however, another question arose, which is discussedin the following.

From the data in Fig. 4 and the related discussion in Sec.IV it appears that for constant p the presence of the weakerbarrier field nearby caused �p to increase while having anegligible effect on �n. Furthermore, the process of makingVIL more negative was accompanied by an increase in VBG tomaintain a constant p. Based on the discussion above, theincrease in VBG would have reduced the squeezing effect anddecreased �p. This crucial difference made the role playedby VIL less obvious and worthy of more investigation.

To further illustrate how the 2DHG was affected bychanging the interlayer electric field and to provide an esti-mate of the e-h scattering contribution some Coulomb dragmeasurements, shown in Fig. 5, were taken as function of Tat three different VIL for matched density n= p�5.0�1010 cm−2. At T=0.3 K a �drag�0.1 � /sq was equiva-lently measured for each VIL. Using �drag=mh

� /e2p�h→e thetime it takes for a hole to transfer its momentum to an elec-tron is �h→e�313 ns.33 This was much longer than the holescattering time �p, which varies from �12.7 to 33.2 ps as VILdecreases at n= p=5�1010 cm−2 in Fig. 4, which eliminatese-h scattering as the dominant scattering mechanism.

The 2DHG Coulomb drag �drag measurements in Fig. 5also show that at a fixed T1.5 K, �drag decreased as VILwas made more negative, which was expected, based onBoltzmann theory, to occur if d were to have increased.33 Toanalyze the drag results more closely the ratio��drag�A� /�drag�B��1/4 was determined, where A and B werethe various VIL and AB. For a constant density this ratio isproportional to the ratio of interlayer separation d�B� /d�A� ateach VIL. From the ratio calculation, the decrease of�VIL=−10 mV led to an increase �d�5%. For the nomi-nally expected separation d=38 nm, where the 2DEG and2DHG wave functions would be located in the center of thewells, this change equates to roughly 1.9 nm. Intuitively, theincrease in d makes sense; the strength of the actual electricfield between the 2DEG and 2DHG is primarily due to VILand when the field is weaker than the charges it would beexpected to be further apart from each other. Most impor-tantly, however, this increase in d as VIL decreased suggeststhe hole wave function moved away from the 2DEG andcloser to the edge of the confinement potential where squeez-ing presumably occurs. Thus, making VIL more negative alsoqualitatively suggests an increase in squeezing and, there-fore, a commensurate increase in �p.

VI. CONCLUSION

The layer interdependence of low-temperature transport ina 30 nm barrier UEHBL device was investigated. An in-crease in �p with increasing n was observed at various p,

FIG. 5. �Color online� The hole drag resistivity �drag as a func-tion of T at matched density n= p�5.0�1010 cm2 for various VIL

and �inset� schematic of the drag measurement.

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while minimal change in �n with increasing p was observedat any n. The former was accompanied by a simultaneousdecrease in VBG �made more negative� to maintain constant pwhile n was increased, which was expected to have changedthe 2DHG confinement potential. Similar �p versus n resultswere seen at three different VIL. Making VIL more negativewas also observed to increase both �p and d. A �d�5%increase with �VIL=−10 mV was determined by measure-ments of �drag�T�. The �p results, which manifested an ap-parent layer interdependence on adjacent layer density n,were then discussed with regard to the following mecha-nisms related to varying mobility: wave-function squeezing,anisotropic band structure, and spin splitting of the subbandsdue to the Rashba effect. Based on the analysis it appearsthat hole wave-function squeezing was, at least qualitatively,the best candidate as the source of the apparent layer inter-

dependence. Bolstering this argument, the increase in d, sug-gested by the 2DHG drag measurements, as VIL was mademore negative was also consistent with squeezing of the holewave function modulating �p.

ACKNOWLEDGMENTS

The authors acknowledge Denise Tibbets for her technicalassistance. The authors also wish to thank S. Das Sarma andE. H. Hwang for their helpful discussions of the relatedtheory. This work has been supported by the Division ofMaterials Sciences and Engineering, Office of Basic EnergySciences, U.S. Department of Energy. Sandia is a multipro-gram laboratory operated by Sandia Corporation, a LockheedMartin Co., for the United States Department of Energy un-der Contract No. DE-AC04-94AL85000.

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