Physica E 6 (2000) 234–237www.elsevier.nl/locate/physe

Imaging electron and conduction-band-hole trajectories through oneand two series constrictions

R. Crook ∗, C.G. Smith, C.H.W. Barnes, M.Y. Simmons, D.A. Ritchie

Semiconductor Physics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK

Abstract

A charged scanning probe has been used to investigate electron transport in a two-dimensional electron gas (2DEG)patterned with 1D constrictions in a GaAs=AlGaAs heterojunction at 1.5 K. The probe creates a local perturbation in the2DEG electrostatic potential capable of scattering transport electrons. A highly collimated hot-electron beam emanating froma single constriction is imaged by backscattering from the probe perturbation. Images of electron and conduction-band-holecyclotron orbits and small angle scattering sites between two series constrictions are also presented. Conduction-band-holesare positively charged quasi-particles existing below the Fermi energy, which are equivalent to unoccupied electron states.? 2000 Elsevier Science B.V. All rights reserved.

PACS: 73.23.Ad; 73.20.Hb; 07.79.−v; 73.20.DxKeywords: Scanning probe; Collimation; Conduction-band-hole; Small angle scattering

Low-temperature scanning probe technology hasrecently been developed to explore quantum phe-nomena of subsurface semiconductor nanostructures.For example, images of electron compressibility inthe quantum Hall regime [1] and images of staticelectronic charge [2] have been produced. In thisletter experiments are reported that use an invasiveprobe which perturbs the two-dimensional electrongas (2DEG) electrostatic potential to scatter trans-port electrons [3–5] and conduction-band-holes.Conduction-band-holes or Cb-holes are the positively

∗ Corresponding author. Tel.: +44-1223-337482; fax: +44-1223-337271.E-mail address: rc230@cam.ac.uk (R. Crook)

charged quasi-particles existing below the Fermi en-ergy, which are equivalent to unoccupied electronstates.The device has a 2DEG at 98 nm beneath its surface

created at a GaAs=AlGaAs heterojunction. After illu-mination the high-mobility modulation-doped 2DEGwas found to have a carrier concentration of ns = 2:4×1011 cm−2, a mobility of 2:5× 106 cm2 V−1 s−1,and a mean free path of 20 �m. Negatively biasedsurface electrodes locally deplete electrons from theunderlying 2DEG, and are arranged in split-gate con-�gurations to de�ne 1D constrictions. The lithographicconstriction width is 700 nm and the electrodes ex-tend 30 nm above the GaAs surface. The scanningprobe is a modi�ed atomic force microscope with

1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(99)00117 -4

R. Crook et al. = Physica E 6 (2000) 234–237 235

Fig. 1. (a) Image made by scanning the tip 60 nm o� the device surface with a 0:5 Vrms AC tip bias. The scanned region is on the drainside of a single constriction where the surface electrodes are outlined, with a source bias of −2 mV to inject electrons, at B = 0 T andT = 1:5 K. Backscattered ratio �b = Ibackscattered=Itransmitted. Dashed lines enclose the ±34◦ semiclassical collimation cone, while the dottedlines enclose the ±3◦ quantum mechanical di�raction limited cone; (b) Pro�le corresponding the single y-direction sweep indicated bythe white arrow in (a); (c) Schematic band diagram of occupied electron states on both sides of the constriction in the absence of the tipperturbation. Where U is the electrostatic potential, � is the electron chemical potential, EF is the Fermi energy, Esd=e is the source-drainbias, and Ee and Ecbh are the energies of injected hot-electrons and Cb-holes, respectively.

a conductive boron-doped silicon tip fabricated ona piezoresistive cantilever, [6,7] and operates to 1.8K in magnetic �elds of up to 3 T. The resolution ofthis technique is intrinsically limited by the 2DEGdepth, [3] but in these experiments a resolution ofapproximately 300 nm is achieved due to the 100 nmtip radius of curvature and the separation between thesurface and tip.When the tip scans over the device surface on the

drain side of a single constriction, the resulting pertur-bation scatters electrons back through the constrictionto modify the constriction’s conductance [4]. Cb-holesare also scattered by the tip perturbation, as althoughthey possess positive charge they also have negativee�ective mass. Fig. 1(a) shows an image where thecontrast is proportional to the constriction AC draincurrent. A −2 mV DC bias was applied to the 2DEGon the source side of the constriction. A low-frequency0.5 Vrms AC signal was applied to the tip and becausethe contact potential from the tip to surface is +0:8 V,the tip perturbation will always repel electrons andCb-holes. It is estimated that the maximum perturba-tion to the carrier concentration is �ns=ns ≈ 0:6 fromprevious experiments [4] and straightforward models[3,5]. The surface electrodes, which are outside the im-age, were biased to de�ne a constriction with 5 trans-missive subbands. In Fig. 1(a) a wide cone divergingat ±34◦ is observed enclosed within the dashed lines

and seen more clearly in the single-sweep pro�leof Fig. 1(b). This wide cone is in agreement withthe established semiclassical collimation theory [8].A highly collimated beam which diverges at ±3◦

is also observed emanating from the constriction,in Fig. 1(a). This beam is interpreted as a beamof hot-electrons, being restricted to those electronswhich retain adiabatic transport from the constrictionto the backscattering event and back to the con-striction. The minimum beam divergence is limitedby quantum mechanical di�raction derived from theuncertainty principle |�y�ky|¿1 applied in the con-striction. For this device �y =W ≈ 200 nm giving�ky¿5× 106 m−1. The Fermi energy for this deviceEF = 8:5 meV is approximately equal to the elec-tron energy at the constriction. Using the dispersionrelation k2x + k

2y = 2m

∗EF=˜2, quantum mechanicaldi�raction therefore limits the collimation to an ap-proximate minimum of ±2:3◦, in good agreementwith the ±3◦ divergence of Fig. 1(a). When a +2mV bias is applied to the source, the wide cone andthe highly collimated beam are still observed to anequal intensity, and are attributed to Cb-holes beinginjected from the constriction.Fig. 1(c) shows schematically the generation of a

Cb-hole with a hot-electron at a 1D constriction, andinjected in opposite directions into the neighbouring2DEG regions. The hot-electron energy Ee is larger

236 R. Crook et al. = Physica E 6 (2000) 234–237

Fig. 2. (a) Schematic layout of the surface electrodes to de�ne two series constrictions; (b) Plot of transfer ratio � = Icentre;drain=Isource;centreagainst perpendicular magnetic �eld B, at T = 4:2 K; (c)–(f) Images made by scanning the tip 60 nm o� the surface with a 0:5 Vrms ACtip bias. In (c) and (d) the source bias is −1 mV DC to inject electrons while in (e) and (f) the bias is +1 mV DC to inject Cb-holes. In(c) and (e) B =−15 mT while in (d) and (f) B = +15 mT. Dotted and dashed lines indicate the electron and Cb-hole orbits, respectively.

Fig. 3. (a) Experimental image made by vibrating the tip at 10 nm o� the surface with a −0:5 V DC tip bias, at T = 4:2 K; (b) Imageminima spatially corresponding to (a).

than the Cb-hole energy Ecbh, and charge is trans-ported equally between the electrons and Cb-holes.Additional structure in Fig. 1(a), such as the reducedbackscattering above the highly collimated beam, isbelieved to be due to either electrode asymmetry orelectrostatic disorder originating from donors posi-tioned randomly in the locality of the constriction.Fig. 2(a) illustrates the gate layout of a di�erent de-

vice fabricated from the same wafer. Two series con-strictions are de�ned to separate three 2DEG regions,referred to as the source, the drain, and the central2DEG region. A −1 mV DC + 0:5 mVrms AC biaswas applied to the source side of the device, and the

central 2DEG region held at 0 V. In the absence ofthe tip, the magnetic �eld was swept to steer the in-jected electron beam over the collection constriction[9]. The AC drain current was recorded to producethe plot of Fig. 2(b) where the central peak is due tothose electrons transmitted directly between the twoconstrictions. The large side peaks are later interpretedas the result of small angle scattering processes, andare not due to quantum mechanical mode di�raction.In a di�erent experiment using the same device,

a 0:5 Vrms AC signal was applied to the scanningtip with the central 2DEG region again held at 0 V.Figs. 2(c)–(f) present images where the AC drain

R. Crook et al. = Physica E 6 (2000) 234–237 237

current was recorded to determine the image contrast.These images were made in magnetic �elds of either−15 or +15 mT with a source bias of either −1 mVto inject electrons or +1 mV to inject Cb-holes. Inlarger magnetic �elds the images become weaker andat larger source biases the structure becomes broaderand less distinct. The tip perturbation reduces the elec-tron transfer from the injection to the collection con-strictions when it scatters those electrons which wouldhave been transmitted through both constrictions in theabsence of the tip. Such electrons are ballistic as theconstrictions are separated by 4 �m and the transportmean free path is 20 �m. The images therefore revealthe possible trajectories that travel through both con-strictions. In a weak magnetic �eld the images show anarc of the cyclotron orbit, originating from electronsor Cb-holes at the edge of the semiclassical injectioncone. A least-squares �t to the peak transfer of theimages in Fig. 2 give cyclotron radii of 6:1± 0:1 and5:9± 0:1 �m for electrons and Cb-holes, respectively.The calculated energies are Ee = 11:1± 0:3 meV andEcbh = 10:3± 0:3 meV which are positioned 0.4 meVto the expected sides of the measured Fermi energy forthis device EF = 10:7 meV. The Cb-holes are magnet-ically steered in the same direction as the hot-electronsbecause the Cb-holes have a negative e�ective massas well as positive charge. Enhanced transfer is ob-served on the inside of the cyclotron orbits where thetip scatters additional electrons from the injection con-striction through the collection constriction, giving thee�ect of mirror image orbits. The enhanced transferseen at both ends of the paths is due to the tip directlymodifying the constriction’s conductance.Upon closer inspection, �ner structure is seen in

the images of Fig. 2. The paths bend slightly and thecontrast varies along their length. The image shownin Fig. 3(a) was generated from the same device us-ing a di�erent imaging technique. The tip vibrated at22 Hz with an amplitude of 50 nm positioned 60 nmo� the surface, with a +0:5 V DC tip bias. With asource bias of +1 mV and the central 2DEG held at0 V, the AC drain current was recorded. Structure insuch images was found to be enhanced, probably dueto the reduced background potential uctuations. InFig. 3(b) the minima of the image in Fig. 3(a) are plot-ted to the same spatial scale. The path is seen to bendand sometimes split, and such behaviour is believed tobe due to disorder originating in the donor layer. Pre-

vious experiments [10] have suggested that the trans-port mean free path is typically ten times longer thanthe quantum mean free path which is determined bysmall angle scattering events. This 2DEG was mea-sured to have a transport mean free path of 20 �m sothere are predicted to be on average two small anglescattering events between the constrictions, in goodagreement with Fig. 3. The large side peaks observedin Fig. 1(b) are also interpreted as the result of relatedsmall angle scattering events.In conclusion, images of electron and conduction-

band-hole trajectories backscattered from a singleconstriction, and scattered between two series con-strictions have been presented. A scanning probetechnique has been demonstrated that is capable ofimaging transport phenomena in subsurface devices.A backscattered highly collimated beam has beenobserved emanating from a single constriction, witha divergence limited by quantum mechanical di�rac-tion. Between two series constrictions the cyclotrontrajectories of electrons and conduction-band-holeshave been imaged, enabling a measure of their en-ergies. Small angle scattering sites originating fromdonor disorder have been directly imaged.

Acknowledgements

We thank C.J.B. Ford and J.T. Nicholls for help-ful discussions. We acknowledge �nancial supportfrom the EPSRC and from the RW Paul InstrumentFund, and DAR also acknowledges Toshiba ResearchEurope Limited.

References

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